5.12/2.20 YES 5.12/2.23 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 5.12/2.23 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.12/2.23 5.12/2.23 5.12/2.23 Quasi decreasingness of the given CTRS could be proven: 5.12/2.23 5.12/2.23 (0) CTRS 5.12/2.23 (1) CTRSToQTRSProof [SOUND, 0 ms] 5.12/2.23 (2) QTRS 5.12/2.23 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 5.12/2.23 (4) QDP 5.12/2.23 (5) DependencyGraphProof [EQUIVALENT, 4 ms] 5.12/2.23 (6) QDP 5.12/2.23 (7) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] 5.12/2.23 (8) QDP 5.12/2.23 (9) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (10) QDP 5.12/2.23 (11) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (12) QDP 5.12/2.23 (13) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (14) QDP 5.12/2.23 (15) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (16) QDP 5.12/2.23 (17) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (18) QDP 5.12/2.23 (19) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (20) QDP 5.12/2.23 (21) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (22) QDP 5.12/2.23 (23) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (24) QDP 5.12/2.23 (25) TransformationProof [EQUIVALENT, 2 ms] 5.12/2.23 (26) QDP 5.12/2.23 (27) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (28) QDP 5.12/2.23 (29) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (30) QDP 5.12/2.23 (31) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (32) QDP 5.12/2.23 (33) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (34) QDP 5.12/2.23 (35) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (36) QDP 5.12/2.23 (37) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (38) QDP 5.12/2.23 (39) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (40) QDP 5.12/2.23 (41) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (42) QDP 5.12/2.23 (43) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (44) QDP 5.12/2.23 (45) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (46) QDP 5.12/2.23 (47) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (48) QDP 5.12/2.23 (49) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (50) QDP 5.12/2.23 (51) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (52) QDP 5.12/2.23 (53) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (54) QDP 5.12/2.23 (55) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (56) QDP 5.12/2.23 (57) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (58) QDP 5.12/2.23 (59) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (60) QDP 5.12/2.23 (61) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (62) QDP 5.12/2.23 (63) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (64) QDP 5.12/2.23 (65) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (66) QDP 5.12/2.23 (67) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (68) QDP 5.12/2.23 (69) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (70) QDP 5.12/2.23 (71) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (72) QDP 5.12/2.23 (73) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (74) QDP 5.12/2.23 (75) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (76) QDP 5.12/2.23 (77) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (78) QDP 5.12/2.23 (79) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (80) QDP 5.12/2.23 (81) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (82) QDP 5.12/2.23 (83) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (84) QDP 5.12/2.23 (85) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (86) QDP 5.12/2.23 (87) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (88) QDP 5.12/2.23 (89) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (90) QDP 5.12/2.23 (91) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (92) QDP 5.12/2.23 (93) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (94) QDP 5.12/2.23 (95) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (96) QDP 5.12/2.23 (97) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (98) QDP 5.12/2.23 (99) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (100) QDP 5.12/2.23 (101) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (102) QDP 5.12/2.23 (103) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (104) QDP 5.12/2.23 (105) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (106) QDP 5.12/2.23 (107) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (108) QDP 5.12/2.23 (109) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (110) QDP 5.12/2.23 (111) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (112) QDP 5.12/2.23 (113) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (114) QDP 5.12/2.23 (115) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (116) QDP 5.12/2.23 (117) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (118) QDP 5.12/2.23 (119) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (120) QDP 5.12/2.23 (121) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (122) QDP 5.12/2.23 (123) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (124) QDP 5.12/2.23 (125) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (126) QDP 5.12/2.23 (127) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (128) QDP 5.12/2.23 (129) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (130) QDP 5.12/2.23 (131) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (132) QDP 5.12/2.23 (133) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (134) QDP 5.12/2.23 (135) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (136) QDP 5.12/2.23 (137) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (138) QDP 5.12/2.23 (139) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (140) QDP 5.12/2.23 (141) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (142) QDP 5.12/2.23 (143) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (144) QDP 5.12/2.23 (145) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (146) QDP 5.12/2.23 (147) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (148) QDP 5.12/2.23 (149) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (150) QDP 5.12/2.23 (151) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (152) QDP 5.12/2.23 (153) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (154) QDP 5.12/2.23 (155) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (156) QDP 5.12/2.23 (157) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (158) QDP 5.12/2.23 (159) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (160) QDP 5.12/2.23 (161) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (162) QDP 5.12/2.23 (163) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (164) QDP 5.12/2.23 (165) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (166) QDP 5.12/2.23 (167) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (168) QDP 5.12/2.23 (169) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (170) QDP 5.12/2.23 (171) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (172) QDP 5.12/2.23 (173) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (174) QDP 5.12/2.23 (175) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (176) QDP 5.12/2.23 (177) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (178) QDP 5.12/2.23 (179) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (180) QDP 5.12/2.23 (181) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (182) QDP 5.12/2.23 (183) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (184) QDP 5.12/2.23 (185) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (186) QDP 5.12/2.23 (187) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (188) QDP 5.12/2.23 (189) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (190) QDP 5.12/2.23 (191) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (192) QDP 5.12/2.23 (193) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (194) QDP 5.12/2.23 (195) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (196) QDP 5.12/2.23 (197) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (198) QDP 5.12/2.23 (199) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (200) QDP 5.12/2.23 (201) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (202) QDP 5.12/2.23 (203) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (204) QDP 5.12/2.23 (205) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (206) QDP 5.12/2.23 (207) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (208) QDP 5.12/2.23 (209) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (210) QDP 5.12/2.23 (211) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (212) QDP 5.12/2.23 (213) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (214) QDP 5.12/2.23 (215) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (216) QDP 5.12/2.23 (217) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (218) QDP 5.12/2.23 (219) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (220) QDP 5.12/2.23 (221) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (222) QDP 5.12/2.23 (223) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (224) QDP 5.12/2.23 (225) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (226) QDP 5.12/2.23 (227) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (228) QDP 5.12/2.23 (229) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (230) QDP 5.12/2.23 (231) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (232) QDP 5.12/2.23 (233) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (234) QDP 5.12/2.23 (235) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (236) QDP 5.12/2.23 (237) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (238) QDP 5.12/2.23 (239) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (240) QDP 5.12/2.23 (241) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (242) QDP 5.12/2.23 (243) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (244) QDP 5.12/2.23 (245) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (246) QDP 5.12/2.23 (247) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (248) QDP 5.12/2.23 (249) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (250) QDP 5.12/2.23 (251) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (252) QDP 5.12/2.23 (253) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (254) QDP 5.12/2.23 (255) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (256) QDP 5.12/2.23 (257) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (258) QDP 5.12/2.23 (259) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (260) QDP 5.12/2.23 (261) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (262) QDP 5.12/2.23 (263) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (264) QDP 5.12/2.23 (265) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (266) QDP 5.12/2.23 (267) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (268) QDP 5.12/2.23 (269) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (270) QDP 5.12/2.23 (271) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (272) QDP 5.12/2.23 (273) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (274) QDP 5.12/2.23 (275) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (276) QDP 5.12/2.23 (277) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (278) QDP 5.12/2.23 (279) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (280) QDP 5.12/2.23 (281) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (282) QDP 5.12/2.23 (283) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (284) QDP 5.12/2.23 (285) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (286) QDP 5.12/2.23 (287) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (288) QDP 5.12/2.23 (289) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (290) QDP 5.12/2.23 (291) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (292) QDP 5.12/2.23 (293) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (294) QDP 5.12/2.23 (295) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (296) QDP 5.12/2.23 (297) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (298) QDP 5.12/2.23 (299) DependencyGraphProof [EQUIVALENT, 1 ms] 5.12/2.23 (300) QDP 5.12/2.23 (301) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (302) QDP 5.12/2.23 (303) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (304) QDP 5.12/2.23 (305) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (306) QDP 5.12/2.23 (307) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (308) QDP 5.12/2.23 (309) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (310) QDP 5.12/2.23 (311) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (312) QDP 5.12/2.23 (313) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (314) QDP 5.12/2.23 (315) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (316) QDP 5.12/2.23 (317) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (318) QDP 5.12/2.23 (319) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (320) QDP 5.12/2.23 (321) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (322) QDP 5.12/2.23 (323) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (324) QDP 5.12/2.23 (325) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (326) QDP 5.12/2.23 (327) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (328) QDP 5.12/2.23 (329) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (330) QDP 5.12/2.23 (331) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (332) QDP 5.12/2.23 (333) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (334) QDP 5.12/2.23 (335) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (336) QDP 5.12/2.23 (337) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (338) QDP 5.12/2.23 (339) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (340) QDP 5.12/2.23 (341) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (342) QDP 5.12/2.23 (343) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (344) QDP 5.12/2.23 (345) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (346) QDP 5.12/2.23 (347) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (348) QDP 5.12/2.23 (349) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (350) QDP 5.12/2.23 (351) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (352) QDP 5.12/2.23 (353) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (354) QDP 5.12/2.23 (355) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (356) QDP 5.12/2.23 (357) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (358) QDP 5.12/2.23 (359) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (360) QDP 5.12/2.23 (361) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (362) QDP 5.12/2.23 (363) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (364) QDP 5.12/2.23 (365) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (366) QDP 5.12/2.23 (367) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (368) QDP 5.12/2.23 (369) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (370) QDP 5.12/2.23 (371) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (372) QDP 5.12/2.23 (373) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (374) QDP 5.12/2.23 (375) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (376) QDP 5.12/2.23 (377) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (378) QDP 5.12/2.23 (379) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (380) QDP 5.12/2.23 (381) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (382) QDP 5.12/2.23 (383) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (384) QDP 5.12/2.23 (385) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (386) QDP 5.12/2.23 (387) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (388) QDP 5.12/2.23 (389) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (390) QDP 5.12/2.23 (391) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (392) QDP 5.12/2.23 (393) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (394) QDP 5.12/2.23 (395) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (396) QDP 5.12/2.23 (397) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (398) QDP 5.12/2.23 (399) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (400) QDP 5.12/2.23 (401) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (402) QDP 5.12/2.23 (403) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (404) QDP 5.12/2.23 (405) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (406) QDP 5.12/2.23 (407) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (408) QDP 5.12/2.23 (409) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (410) QDP 5.12/2.23 (411) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (412) QDP 5.12/2.23 (413) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (414) QDP 5.12/2.23 (415) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (416) QDP 5.12/2.23 (417) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (418) QDP 5.12/2.23 (419) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (420) QDP 5.12/2.23 (421) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (422) QDP 5.12/2.23 (423) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (424) QDP 5.12/2.23 (425) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (426) QDP 5.12/2.23 (427) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (428) QDP 5.12/2.23 (429) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (430) QDP 5.12/2.23 (431) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (432) QDP 5.12/2.23 (433) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (434) QDP 5.12/2.23 (435) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (436) QDP 5.12/2.23 (437) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (438) QDP 5.12/2.23 (439) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (440) QDP 5.12/2.23 (441) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (442) QDP 5.12/2.23 (443) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (444) QDP 5.12/2.23 (445) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (446) QDP 5.12/2.23 (447) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (448) QDP 5.12/2.23 (449) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (450) QDP 5.12/2.23 (451) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (452) QDP 5.12/2.23 (453) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (454) QDP 5.12/2.23 (455) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (456) QDP 5.12/2.23 (457) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (458) QDP 5.12/2.23 (459) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (460) QDP 5.12/2.23 (461) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (462) QDP 5.12/2.23 (463) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (464) QDP 5.12/2.23 (465) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (466) QDP 5.12/2.23 (467) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (468) QDP 5.12/2.23 (469) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (470) QDP 5.12/2.23 (471) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (472) QDP 5.12/2.23 (473) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (474) QDP 5.12/2.23 (475) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (476) QDP 5.12/2.23 (477) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (478) QDP 5.12/2.23 (479) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (480) QDP 5.12/2.23 (481) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (482) QDP 5.12/2.23 (483) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (484) QDP 5.12/2.23 (485) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (486) QDP 5.12/2.23 (487) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (488) QDP 5.12/2.23 (489) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (490) QDP 5.12/2.23 (491) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (492) QDP 5.12/2.23 (493) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (494) QDP 5.12/2.23 (495) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (496) QDP 5.12/2.23 (497) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (498) QDP 5.12/2.23 (499) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (500) QDP 5.12/2.23 (501) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (502) QDP 5.12/2.23 (503) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (504) QDP 5.12/2.23 (505) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (506) QDP 5.12/2.23 (507) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (508) QDP 5.12/2.23 (509) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (510) QDP 5.12/2.23 (511) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (512) QDP 5.12/2.23 (513) TransformationProof [EQUIVALENT, 0 ms] 5.12/2.23 (514) QDP 5.12/2.23 (515) UsableRulesReductionPairsProof [EQUIVALENT, 4 ms] 5.12/2.23 (516) QDP 5.12/2.23 (517) MNOCProof [EQUIVALENT, 0 ms] 5.12/2.23 (518) QDP 5.12/2.23 (519) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.23 (520) TRUE 5.12/2.23 5.12/2.23 5.12/2.23 ---------------------------------------- 5.12/2.23 5.12/2.23 (0) 5.12/2.23 Obligation: 5.12/2.23 Conditional term rewrite system: 5.12/2.23 The TRS R consists of the following rules: 5.12/2.23 5.12/2.23 a -> d 5.12/2.23 b -> d 5.12/2.23 a -> e 5.12/2.23 b -> e 5.12/2.23 A -> h(f(a), f(b)) 5.12/2.23 h(x, x) -> g(x, x) 5.12/2.23 g(d, e) -> A 5.12/2.23 5.12/2.23 The conditional TRS C consists of the following conditional rules: 5.12/2.23 5.12/2.23 f(x) -> x <= x -> d 5.12/2.23 5.12/2.23 5.12/2.23 ---------------------------------------- 5.12/2.23 5.12/2.23 (1) CTRSToQTRSProof (SOUND) 5.12/2.23 The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC]. 5.12/2.23 ---------------------------------------- 5.12/2.23 5.12/2.23 (2) 5.12/2.23 Obligation: 5.12/2.23 Q restricted rewrite system: 5.12/2.23 The TRS R consists of the following rules: 5.12/2.23 5.12/2.23 f(x) -> U1(x, x) 5.12/2.23 U1(d, x) -> x 5.12/2.23 a -> d 5.12/2.23 b -> d 5.12/2.23 a -> e 5.12/2.23 b -> e 5.12/2.23 A -> h(f(a), f(b)) 5.12/2.23 h(x, x) -> g(x, x) 5.12/2.23 g(d, e) -> A 5.12/2.23 5.12/2.23 Q is empty. 5.12/2.23 5.12/2.23 ---------------------------------------- 5.12/2.23 5.12/2.23 (3) DependencyPairsProof (EQUIVALENT) 5.12/2.23 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 5.12/2.23 ---------------------------------------- 5.12/2.23 5.12/2.23 (4) 5.12/2.23 Obligation: 5.12/2.23 Q DP problem: 5.12/2.23 The TRS P consists of the following rules: 5.12/2.23 5.12/2.23 F(x) -> U1^1(x, x) 5.12/2.23 A^1 -> H(f(a), f(b)) 5.12/2.23 A^1 -> F(a) 5.12/2.23 A^1 -> A^2 5.12/2.23 A^1 -> F(b) 5.12/2.23 A^1 -> B 5.12/2.23 H(x, x) -> G(x, x) 5.12/2.23 G(d, e) -> A^1 5.12/2.23 5.12/2.23 The TRS R consists of the following rules: 5.12/2.23 5.12/2.23 f(x) -> U1(x, x) 5.12/2.23 U1(d, x) -> x 5.12/2.23 a -> d 5.12/2.23 b -> d 5.12/2.23 a -> e 5.12/2.23 b -> e 5.12/2.23 A -> h(f(a), f(b)) 5.12/2.23 h(x, x) -> g(x, x) 5.12/2.23 g(d, e) -> A 5.12/2.23 5.12/2.23 Q is empty. 5.12/2.23 We have to consider all minimal (P,Q,R)-chains. 5.12/2.23 ---------------------------------------- 5.12/2.23 5.12/2.23 (5) DependencyGraphProof (EQUIVALENT) 5.12/2.23 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes. 5.12/2.23 ---------------------------------------- 5.12/2.23 5.12/2.23 (6) 5.12/2.23 Obligation: 5.12/2.23 Q DP problem: 5.12/2.23 The TRS P consists of the following rules: 5.12/2.23 5.12/2.23 A^1 -> H(f(a), f(b)) 5.12/2.23 H(x, x) -> G(x, x) 5.12/2.23 G(d, e) -> A^1 5.12/2.23 5.12/2.23 The TRS R consists of the following rules: 5.12/2.23 5.12/2.23 f(x) -> U1(x, x) 5.12/2.23 U1(d, x) -> x 5.12/2.23 a -> d 5.12/2.23 b -> d 5.12/2.23 a -> e 5.12/2.23 b -> e 5.12/2.23 A -> h(f(a), f(b)) 5.12/2.23 h(x, x) -> g(x, x) 5.12/2.23 g(d, e) -> A 5.12/2.23 5.12/2.23 Q is empty. 5.12/2.23 We have to consider all minimal (P,Q,R)-chains. 5.12/2.23 ---------------------------------------- 5.12/2.23 5.12/2.23 (7) UsableRulesReductionPairsProof (EQUIVALENT) 5.12/2.23 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. 5.12/2.23 5.12/2.23 No dependency pairs are removed. 5.12/2.23 5.12/2.23 The following rules are removed from R: 5.12/2.23 5.12/2.23 A -> h(f(a), f(b)) 5.12/2.23 h(x, x) -> g(x, x) 5.12/2.23 g(d, e) -> A 5.12/2.23 Used ordering: POLO with Polynomial interpretation [POLO]: 5.12/2.23 5.12/2.23 POL(A^1) = 0 5.12/2.23 POL(G(x_1, x_2)) = x_1 + x_2 5.12/2.23 POL(H(x_1, x_2)) = 2*x_1 + 2*x_2 5.12/2.23 POL(U1(x_1, x_2)) = x_1 + x_2 5.12/2.23 POL(a) = 0 5.12/2.23 POL(b) = 0 5.12/2.23 POL(d) = 0 5.12/2.23 POL(e) = 0 5.12/2.23 POL(f(x_1)) = 2*x_1 5.12/2.23 5.12/2.23 5.12/2.23 ---------------------------------------- 5.12/2.23 5.12/2.23 (8) 5.12/2.23 Obligation: 5.12/2.23 Q DP problem: 5.12/2.23 The TRS P consists of the following rules: 5.12/2.23 5.12/2.23 A^1 -> H(f(a), f(b)) 5.12/2.23 H(x, x) -> G(x, x) 5.12/2.23 G(d, e) -> A^1 5.12/2.23 5.12/2.23 The TRS R consists of the following rules: 5.12/2.23 5.12/2.23 a -> d 5.12/2.23 a -> e 5.12/2.23 f(x) -> U1(x, x) 5.12/2.23 b -> d 5.12/2.23 b -> e 5.12/2.23 U1(d, x) -> x 5.12/2.23 5.12/2.23 Q is empty. 5.12/2.23 We have to consider all minimal (P,Q,R)-chains. 5.12/2.23 ---------------------------------------- 5.12/2.23 5.12/2.23 (9) TransformationProof (EQUIVALENT) 5.12/2.23 By narrowing [LPAR04] the rule A^1 -> H(f(a), f(b)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.23 5.12/2.23 (A^1 -> H(U1(a, a), f(b)),A^1 -> H(U1(a, a), f(b))) 5.12/2.23 (A^1 -> H(f(d), f(b)),A^1 -> H(f(d), f(b))) 5.12/2.23 (A^1 -> H(f(e), f(b)),A^1 -> H(f(e), f(b))) 5.12/2.23 (A^1 -> H(f(a), U1(b, b)),A^1 -> H(f(a), U1(b, b))) 5.12/2.23 (A^1 -> H(f(a), f(d)),A^1 -> H(f(a), f(d))) 5.12/2.23 (A^1 -> H(f(a), f(e)),A^1 -> H(f(a), f(e))) 5.12/2.23 5.12/2.23 5.12/2.23 ---------------------------------------- 5.12/2.23 5.12/2.23 (10) 5.12/2.23 Obligation: 5.12/2.23 Q DP problem: 5.12/2.23 The TRS P consists of the following rules: 5.12/2.23 5.12/2.23 H(x, x) -> G(x, x) 5.12/2.23 G(d, e) -> A^1 5.12/2.23 A^1 -> H(U1(a, a), f(b)) 5.12/2.23 A^1 -> H(f(d), f(b)) 5.12/2.23 A^1 -> H(f(e), f(b)) 5.12/2.23 A^1 -> H(f(a), U1(b, b)) 5.12/2.23 A^1 -> H(f(a), f(d)) 5.12/2.23 A^1 -> H(f(a), f(e)) 5.12/2.23 5.12/2.23 The TRS R consists of the following rules: 5.12/2.23 5.12/2.23 a -> d 5.12/2.23 a -> e 5.12/2.23 f(x) -> U1(x, x) 5.12/2.23 b -> d 5.12/2.23 b -> e 5.12/2.23 U1(d, x) -> x 5.12/2.23 5.12/2.23 Q is empty. 5.12/2.23 We have to consider all minimal (P,Q,R)-chains. 5.12/2.23 ---------------------------------------- 5.12/2.23 5.12/2.23 (11) TransformationProof (EQUIVALENT) 5.12/2.23 By narrowing [LPAR04] the rule A^1 -> H(U1(a, a), f(b)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.23 5.12/2.23 (A^1 -> H(U1(d, a), f(b)),A^1 -> H(U1(d, a), f(b))) 5.12/2.23 (A^1 -> H(U1(e, a), f(b)),A^1 -> H(U1(e, a), f(b))) 5.12/2.23 (A^1 -> H(U1(a, d), f(b)),A^1 -> H(U1(a, d), f(b))) 5.12/2.23 (A^1 -> H(U1(a, e), f(b)),A^1 -> H(U1(a, e), f(b))) 5.12/2.23 (A^1 -> H(U1(a, a), U1(b, b)),A^1 -> H(U1(a, a), U1(b, b))) 5.12/2.23 (A^1 -> H(U1(a, a), f(d)),A^1 -> H(U1(a, a), f(d))) 5.12/2.23 (A^1 -> H(U1(a, a), f(e)),A^1 -> H(U1(a, a), f(e))) 5.12/2.23 5.12/2.23 5.12/2.23 ---------------------------------------- 5.12/2.23 5.12/2.23 (12) 5.12/2.23 Obligation: 5.12/2.23 Q DP problem: 5.12/2.23 The TRS P consists of the following rules: 5.12/2.23 5.12/2.23 H(x, x) -> G(x, x) 5.12/2.23 G(d, e) -> A^1 5.12/2.23 A^1 -> H(f(d), f(b)) 5.12/2.23 A^1 -> H(f(e), f(b)) 5.12/2.23 A^1 -> H(f(a), U1(b, b)) 5.12/2.23 A^1 -> H(f(a), f(d)) 5.12/2.23 A^1 -> H(f(a), f(e)) 5.12/2.23 A^1 -> H(U1(d, a), f(b)) 5.12/2.23 A^1 -> H(U1(e, a), f(b)) 5.12/2.23 A^1 -> H(U1(a, d), f(b)) 5.12/2.23 A^1 -> H(U1(a, e), f(b)) 5.12/2.23 A^1 -> H(U1(a, a), U1(b, b)) 5.12/2.23 A^1 -> H(U1(a, a), f(d)) 5.12/2.23 A^1 -> H(U1(a, a), f(e)) 5.12/2.23 5.12/2.23 The TRS R consists of the following rules: 5.12/2.23 5.12/2.23 a -> d 5.12/2.23 a -> e 5.12/2.23 f(x) -> U1(x, x) 5.12/2.23 b -> d 5.12/2.23 b -> e 5.12/2.23 U1(d, x) -> x 5.12/2.23 5.12/2.23 Q is empty. 5.12/2.23 We have to consider all minimal (P,Q,R)-chains. 5.12/2.23 ---------------------------------------- 5.12/2.23 5.12/2.23 (13) TransformationProof (EQUIVALENT) 5.12/2.23 By narrowing [LPAR04] the rule A^1 -> H(f(d), f(b)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.23 5.12/2.23 (A^1 -> H(U1(d, d), f(b)),A^1 -> H(U1(d, d), f(b))) 5.12/2.23 (A^1 -> H(f(d), U1(b, b)),A^1 -> H(f(d), U1(b, b))) 5.12/2.23 (A^1 -> H(f(d), f(d)),A^1 -> H(f(d), f(d))) 5.12/2.23 (A^1 -> H(f(d), f(e)),A^1 -> H(f(d), f(e))) 5.12/2.23 5.12/2.23 5.12/2.23 ---------------------------------------- 5.12/2.23 5.12/2.23 (14) 5.12/2.23 Obligation: 5.12/2.23 Q DP problem: 5.12/2.23 The TRS P consists of the following rules: 5.12/2.23 5.12/2.23 H(x, x) -> G(x, x) 5.12/2.23 G(d, e) -> A^1 5.12/2.23 A^1 -> H(f(e), f(b)) 5.12/2.23 A^1 -> H(f(a), U1(b, b)) 5.12/2.23 A^1 -> H(f(a), f(d)) 5.12/2.23 A^1 -> H(f(a), f(e)) 5.12/2.23 A^1 -> H(U1(d, a), f(b)) 5.12/2.23 A^1 -> H(U1(e, a), f(b)) 5.12/2.23 A^1 -> H(U1(a, d), f(b)) 5.12/2.23 A^1 -> H(U1(a, e), f(b)) 5.12/2.23 A^1 -> H(U1(a, a), U1(b, b)) 5.12/2.23 A^1 -> H(U1(a, a), f(d)) 5.12/2.23 A^1 -> H(U1(a, a), f(e)) 5.12/2.23 A^1 -> H(U1(d, d), f(b)) 5.12/2.23 A^1 -> H(f(d), U1(b, b)) 5.12/2.23 A^1 -> H(f(d), f(d)) 5.12/2.23 A^1 -> H(f(d), f(e)) 5.12/2.23 5.12/2.23 The TRS R consists of the following rules: 5.12/2.23 5.12/2.23 a -> d 5.12/2.23 a -> e 5.12/2.23 f(x) -> U1(x, x) 5.12/2.23 b -> d 5.12/2.23 b -> e 5.12/2.23 U1(d, x) -> x 5.12/2.23 5.12/2.23 Q is empty. 5.12/2.23 We have to consider all minimal (P,Q,R)-chains. 5.12/2.23 ---------------------------------------- 5.12/2.23 5.12/2.23 (15) TransformationProof (EQUIVALENT) 5.12/2.23 By narrowing [LPAR04] the rule A^1 -> H(f(e), f(b)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.23 5.12/2.23 (A^1 -> H(U1(e, e), f(b)),A^1 -> H(U1(e, e), f(b))) 5.12/2.23 (A^1 -> H(f(e), U1(b, b)),A^1 -> H(f(e), U1(b, b))) 5.12/2.23 (A^1 -> H(f(e), f(d)),A^1 -> H(f(e), f(d))) 5.12/2.23 (A^1 -> H(f(e), f(e)),A^1 -> H(f(e), f(e))) 5.12/2.23 5.12/2.23 5.12/2.23 ---------------------------------------- 5.12/2.23 5.12/2.23 (16) 5.12/2.23 Obligation: 5.12/2.23 Q DP problem: 5.12/2.23 The TRS P consists of the following rules: 5.12/2.23 5.12/2.23 H(x, x) -> G(x, x) 5.12/2.23 G(d, e) -> A^1 5.12/2.23 A^1 -> H(f(a), U1(b, b)) 5.12/2.23 A^1 -> H(f(a), f(d)) 5.12/2.23 A^1 -> H(f(a), f(e)) 5.12/2.23 A^1 -> H(U1(d, a), f(b)) 5.12/2.23 A^1 -> H(U1(e, a), f(b)) 5.12/2.23 A^1 -> H(U1(a, d), f(b)) 5.12/2.23 A^1 -> H(U1(a, e), f(b)) 5.12/2.23 A^1 -> H(U1(a, a), U1(b, b)) 5.12/2.23 A^1 -> H(U1(a, a), f(d)) 5.12/2.23 A^1 -> H(U1(a, a), f(e)) 5.12/2.23 A^1 -> H(U1(d, d), f(b)) 5.12/2.23 A^1 -> H(f(d), U1(b, b)) 5.12/2.23 A^1 -> H(f(d), f(d)) 5.12/2.23 A^1 -> H(f(d), f(e)) 5.12/2.23 A^1 -> H(U1(e, e), f(b)) 5.12/2.23 A^1 -> H(f(e), U1(b, b)) 5.12/2.23 A^1 -> H(f(e), f(d)) 5.12/2.23 A^1 -> H(f(e), f(e)) 5.12/2.23 5.12/2.23 The TRS R consists of the following rules: 5.12/2.23 5.12/2.23 a -> d 5.12/2.23 a -> e 5.12/2.23 f(x) -> U1(x, x) 5.12/2.23 b -> d 5.12/2.23 b -> e 5.12/2.23 U1(d, x) -> x 5.12/2.23 5.12/2.23 Q is empty. 5.12/2.23 We have to consider all minimal (P,Q,R)-chains. 5.12/2.23 ---------------------------------------- 5.12/2.23 5.12/2.23 (17) TransformationProof (EQUIVALENT) 5.12/2.23 By narrowing [LPAR04] the rule A^1 -> H(f(a), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.23 5.12/2.23 (A^1 -> H(U1(a, a), U1(b, b)),A^1 -> H(U1(a, a), U1(b, b))) 5.12/2.23 (A^1 -> H(f(d), U1(b, b)),A^1 -> H(f(d), U1(b, b))) 5.12/2.23 (A^1 -> H(f(e), U1(b, b)),A^1 -> H(f(e), U1(b, b))) 5.12/2.23 (A^1 -> H(f(a), U1(d, b)),A^1 -> H(f(a), U1(d, b))) 5.12/2.23 (A^1 -> H(f(a), U1(e, b)),A^1 -> H(f(a), U1(e, b))) 5.12/2.23 (A^1 -> H(f(a), U1(b, d)),A^1 -> H(f(a), U1(b, d))) 5.12/2.23 (A^1 -> H(f(a), U1(b, e)),A^1 -> H(f(a), U1(b, e))) 5.12/2.23 5.12/2.23 5.12/2.23 ---------------------------------------- 5.12/2.23 5.12/2.23 (18) 5.12/2.23 Obligation: 5.12/2.23 Q DP problem: 5.12/2.23 The TRS P consists of the following rules: 5.12/2.23 5.12/2.23 H(x, x) -> G(x, x) 5.12/2.23 G(d, e) -> A^1 5.12/2.23 A^1 -> H(f(a), f(d)) 5.12/2.23 A^1 -> H(f(a), f(e)) 5.12/2.23 A^1 -> H(U1(d, a), f(b)) 5.12/2.23 A^1 -> H(U1(e, a), f(b)) 5.12/2.23 A^1 -> H(U1(a, d), f(b)) 5.12/2.23 A^1 -> H(U1(a, e), f(b)) 5.12/2.23 A^1 -> H(U1(a, a), U1(b, b)) 5.12/2.23 A^1 -> H(U1(a, a), f(d)) 5.12/2.23 A^1 -> H(U1(a, a), f(e)) 5.12/2.23 A^1 -> H(U1(d, d), f(b)) 5.12/2.23 A^1 -> H(f(d), U1(b, b)) 5.12/2.23 A^1 -> H(f(d), f(d)) 5.12/2.23 A^1 -> H(f(d), f(e)) 5.12/2.23 A^1 -> H(U1(e, e), f(b)) 5.12/2.23 A^1 -> H(f(e), U1(b, b)) 5.12/2.23 A^1 -> H(f(e), f(d)) 5.12/2.23 A^1 -> H(f(e), f(e)) 5.12/2.23 A^1 -> H(f(a), U1(d, b)) 5.12/2.23 A^1 -> H(f(a), U1(e, b)) 5.12/2.23 A^1 -> H(f(a), U1(b, d)) 5.12/2.23 A^1 -> H(f(a), U1(b, e)) 5.12/2.23 5.12/2.23 The TRS R consists of the following rules: 5.12/2.23 5.12/2.23 a -> d 5.12/2.23 a -> e 5.12/2.23 f(x) -> U1(x, x) 5.12/2.23 b -> d 5.12/2.23 b -> e 5.12/2.23 U1(d, x) -> x 5.12/2.23 5.12/2.23 Q is empty. 5.12/2.23 We have to consider all minimal (P,Q,R)-chains. 5.12/2.23 ---------------------------------------- 5.12/2.23 5.12/2.23 (19) TransformationProof (EQUIVALENT) 5.12/2.23 By narrowing [LPAR04] the rule A^1 -> H(f(a), f(d)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.23 5.12/2.23 (A^1 -> H(U1(a, a), f(d)),A^1 -> H(U1(a, a), f(d))) 5.12/2.23 (A^1 -> H(f(d), f(d)),A^1 -> H(f(d), f(d))) 5.12/2.23 (A^1 -> H(f(e), f(d)),A^1 -> H(f(e), f(d))) 5.12/2.23 (A^1 -> H(f(a), U1(d, d)),A^1 -> H(f(a), U1(d, d))) 5.12/2.23 5.12/2.23 5.12/2.23 ---------------------------------------- 5.12/2.23 5.12/2.23 (20) 5.12/2.23 Obligation: 5.12/2.23 Q DP problem: 5.12/2.23 The TRS P consists of the following rules: 5.12/2.23 5.12/2.23 H(x, x) -> G(x, x) 5.12/2.23 G(d, e) -> A^1 5.12/2.23 A^1 -> H(f(a), f(e)) 5.12/2.23 A^1 -> H(U1(d, a), f(b)) 5.12/2.23 A^1 -> H(U1(e, a), f(b)) 5.12/2.23 A^1 -> H(U1(a, d), f(b)) 5.12/2.23 A^1 -> H(U1(a, e), f(b)) 5.12/2.23 A^1 -> H(U1(a, a), U1(b, b)) 5.12/2.23 A^1 -> H(U1(a, a), f(d)) 5.12/2.23 A^1 -> H(U1(a, a), f(e)) 5.12/2.23 A^1 -> H(U1(d, d), f(b)) 5.12/2.23 A^1 -> H(f(d), U1(b, b)) 5.12/2.23 A^1 -> H(f(d), f(d)) 5.12/2.23 A^1 -> H(f(d), f(e)) 5.12/2.23 A^1 -> H(U1(e, e), f(b)) 5.12/2.23 A^1 -> H(f(e), U1(b, b)) 5.12/2.23 A^1 -> H(f(e), f(d)) 5.12/2.23 A^1 -> H(f(e), f(e)) 5.12/2.23 A^1 -> H(f(a), U1(d, b)) 5.12/2.23 A^1 -> H(f(a), U1(e, b)) 5.12/2.23 A^1 -> H(f(a), U1(b, d)) 5.12/2.23 A^1 -> H(f(a), U1(b, e)) 5.12/2.23 A^1 -> H(f(a), U1(d, d)) 5.12/2.23 5.12/2.23 The TRS R consists of the following rules: 5.12/2.23 5.12/2.23 a -> d 5.12/2.23 a -> e 5.12/2.23 f(x) -> U1(x, x) 5.12/2.23 b -> d 5.12/2.23 b -> e 5.12/2.23 U1(d, x) -> x 5.12/2.23 5.12/2.23 Q is empty. 5.12/2.23 We have to consider all minimal (P,Q,R)-chains. 5.12/2.23 ---------------------------------------- 5.12/2.23 5.12/2.23 (21) TransformationProof (EQUIVALENT) 5.12/2.23 By narrowing [LPAR04] the rule A^1 -> H(f(a), f(e)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.23 5.12/2.23 (A^1 -> H(U1(a, a), f(e)),A^1 -> H(U1(a, a), f(e))) 5.12/2.23 (A^1 -> H(f(d), f(e)),A^1 -> H(f(d), f(e))) 5.12/2.23 (A^1 -> H(f(e), f(e)),A^1 -> H(f(e), f(e))) 5.12/2.23 (A^1 -> H(f(a), U1(e, e)),A^1 -> H(f(a), U1(e, e))) 5.12/2.23 5.12/2.23 5.12/2.23 ---------------------------------------- 5.12/2.23 5.12/2.23 (22) 5.12/2.23 Obligation: 5.12/2.23 Q DP problem: 5.12/2.23 The TRS P consists of the following rules: 5.12/2.23 5.12/2.23 H(x, x) -> G(x, x) 5.12/2.23 G(d, e) -> A^1 5.12/2.23 A^1 -> H(U1(d, a), f(b)) 5.12/2.23 A^1 -> H(U1(e, a), f(b)) 5.12/2.23 A^1 -> H(U1(a, d), f(b)) 5.12/2.23 A^1 -> H(U1(a, e), f(b)) 5.12/2.23 A^1 -> H(U1(a, a), U1(b, b)) 5.12/2.23 A^1 -> H(U1(a, a), f(d)) 5.12/2.23 A^1 -> H(U1(a, a), f(e)) 5.12/2.23 A^1 -> H(U1(d, d), f(b)) 5.12/2.23 A^1 -> H(f(d), U1(b, b)) 5.12/2.23 A^1 -> H(f(d), f(d)) 5.12/2.23 A^1 -> H(f(d), f(e)) 5.12/2.23 A^1 -> H(U1(e, e), f(b)) 5.12/2.23 A^1 -> H(f(e), U1(b, b)) 5.12/2.23 A^1 -> H(f(e), f(d)) 5.12/2.23 A^1 -> H(f(e), f(e)) 5.12/2.23 A^1 -> H(f(a), U1(d, b)) 5.12/2.23 A^1 -> H(f(a), U1(e, b)) 5.12/2.23 A^1 -> H(f(a), U1(b, d)) 5.12/2.23 A^1 -> H(f(a), U1(b, e)) 5.12/2.23 A^1 -> H(f(a), U1(d, d)) 5.12/2.23 A^1 -> H(f(a), U1(e, e)) 5.12/2.23 5.12/2.23 The TRS R consists of the following rules: 5.12/2.23 5.12/2.23 a -> d 5.12/2.23 a -> e 5.12/2.23 f(x) -> U1(x, x) 5.12/2.23 b -> d 5.12/2.23 b -> e 5.12/2.23 U1(d, x) -> x 5.12/2.23 5.12/2.23 Q is empty. 5.12/2.23 We have to consider all minimal (P,Q,R)-chains. 5.12/2.23 ---------------------------------------- 5.12/2.23 5.12/2.23 (23) TransformationProof (EQUIVALENT) 5.12/2.23 By narrowing [LPAR04] the rule A^1 -> H(U1(d, a), f(b)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.23 5.12/2.23 (A^1 -> H(a, f(b)),A^1 -> H(a, f(b))) 5.12/2.23 (A^1 -> H(U1(d, d), f(b)),A^1 -> H(U1(d, d), f(b))) 5.12/2.23 (A^1 -> H(U1(d, e), f(b)),A^1 -> H(U1(d, e), f(b))) 5.12/2.23 (A^1 -> H(U1(d, a), U1(b, b)),A^1 -> H(U1(d, a), U1(b, b))) 5.12/2.23 (A^1 -> H(U1(d, a), f(d)),A^1 -> H(U1(d, a), f(d))) 5.12/2.23 (A^1 -> H(U1(d, a), f(e)),A^1 -> H(U1(d, a), f(e))) 5.12/2.23 5.12/2.23 5.12/2.23 ---------------------------------------- 5.12/2.23 5.12/2.23 (24) 5.12/2.23 Obligation: 5.12/2.23 Q DP problem: 5.12/2.23 The TRS P consists of the following rules: 5.12/2.23 5.12/2.23 H(x, x) -> G(x, x) 5.12/2.23 G(d, e) -> A^1 5.12/2.23 A^1 -> H(U1(e, a), f(b)) 5.12/2.23 A^1 -> H(U1(a, d), f(b)) 5.12/2.23 A^1 -> H(U1(a, e), f(b)) 5.12/2.23 A^1 -> H(U1(a, a), U1(b, b)) 5.12/2.23 A^1 -> H(U1(a, a), f(d)) 5.12/2.23 A^1 -> H(U1(a, a), f(e)) 5.12/2.23 A^1 -> H(U1(d, d), f(b)) 5.12/2.23 A^1 -> H(f(d), U1(b, b)) 5.12/2.23 A^1 -> H(f(d), f(d)) 5.12/2.23 A^1 -> H(f(d), f(e)) 5.12/2.23 A^1 -> H(U1(e, e), f(b)) 5.12/2.23 A^1 -> H(f(e), U1(b, b)) 5.12/2.23 A^1 -> H(f(e), f(d)) 5.12/2.23 A^1 -> H(f(e), f(e)) 5.12/2.23 A^1 -> H(f(a), U1(d, b)) 5.12/2.23 A^1 -> H(f(a), U1(e, b)) 5.12/2.23 A^1 -> H(f(a), U1(b, d)) 5.12/2.23 A^1 -> H(f(a), U1(b, e)) 5.12/2.23 A^1 -> H(f(a), U1(d, d)) 5.12/2.23 A^1 -> H(f(a), U1(e, e)) 5.12/2.23 A^1 -> H(a, f(b)) 5.12/2.23 A^1 -> H(U1(d, e), f(b)) 5.12/2.23 A^1 -> H(U1(d, a), U1(b, b)) 5.12/2.23 A^1 -> H(U1(d, a), f(d)) 5.12/2.23 A^1 -> H(U1(d, a), f(e)) 5.12/2.23 5.12/2.23 The TRS R consists of the following rules: 5.12/2.23 5.12/2.23 a -> d 5.12/2.23 a -> e 5.12/2.23 f(x) -> U1(x, x) 5.12/2.23 b -> d 5.12/2.23 b -> e 5.12/2.23 U1(d, x) -> x 5.12/2.23 5.12/2.23 Q is empty. 5.12/2.23 We have to consider all minimal (P,Q,R)-chains. 5.12/2.23 ---------------------------------------- 5.12/2.23 5.12/2.23 (25) TransformationProof (EQUIVALENT) 5.12/2.23 By narrowing [LPAR04] the rule A^1 -> H(U1(e, a), f(b)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.23 5.12/2.23 (A^1 -> H(U1(e, d), f(b)),A^1 -> H(U1(e, d), f(b))) 5.12/2.23 (A^1 -> H(U1(e, e), f(b)),A^1 -> H(U1(e, e), f(b))) 5.12/2.23 (A^1 -> H(U1(e, a), U1(b, b)),A^1 -> H(U1(e, a), U1(b, b))) 5.12/2.23 (A^1 -> H(U1(e, a), f(d)),A^1 -> H(U1(e, a), f(d))) 5.12/2.23 (A^1 -> H(U1(e, a), f(e)),A^1 -> H(U1(e, a), f(e))) 5.12/2.23 5.12/2.23 5.12/2.23 ---------------------------------------- 5.12/2.23 5.12/2.23 (26) 5.12/2.23 Obligation: 5.12/2.23 Q DP problem: 5.12/2.23 The TRS P consists of the following rules: 5.12/2.23 5.12/2.23 H(x, x) -> G(x, x) 5.12/2.23 G(d, e) -> A^1 5.12/2.23 A^1 -> H(U1(a, d), f(b)) 5.12/2.23 A^1 -> H(U1(a, e), f(b)) 5.12/2.23 A^1 -> H(U1(a, a), U1(b, b)) 5.12/2.23 A^1 -> H(U1(a, a), f(d)) 5.12/2.23 A^1 -> H(U1(a, a), f(e)) 5.12/2.23 A^1 -> H(U1(d, d), f(b)) 5.12/2.23 A^1 -> H(f(d), U1(b, b)) 5.12/2.23 A^1 -> H(f(d), f(d)) 5.12/2.23 A^1 -> H(f(d), f(e)) 5.12/2.23 A^1 -> H(U1(e, e), f(b)) 5.12/2.23 A^1 -> H(f(e), U1(b, b)) 5.12/2.23 A^1 -> H(f(e), f(d)) 5.12/2.23 A^1 -> H(f(e), f(e)) 5.12/2.23 A^1 -> H(f(a), U1(d, b)) 5.12/2.23 A^1 -> H(f(a), U1(e, b)) 5.12/2.23 A^1 -> H(f(a), U1(b, d)) 5.12/2.23 A^1 -> H(f(a), U1(b, e)) 5.12/2.23 A^1 -> H(f(a), U1(d, d)) 5.12/2.23 A^1 -> H(f(a), U1(e, e)) 5.12/2.23 A^1 -> H(a, f(b)) 5.12/2.23 A^1 -> H(U1(d, e), f(b)) 5.12/2.23 A^1 -> H(U1(d, a), U1(b, b)) 5.12/2.23 A^1 -> H(U1(d, a), f(d)) 5.12/2.23 A^1 -> H(U1(d, a), f(e)) 5.12/2.23 A^1 -> H(U1(e, d), f(b)) 5.12/2.23 A^1 -> H(U1(e, a), U1(b, b)) 5.12/2.23 A^1 -> H(U1(e, a), f(d)) 5.12/2.23 A^1 -> H(U1(e, a), f(e)) 5.12/2.23 5.12/2.23 The TRS R consists of the following rules: 5.12/2.23 5.12/2.23 a -> d 5.12/2.23 a -> e 5.12/2.23 f(x) -> U1(x, x) 5.12/2.23 b -> d 5.12/2.23 b -> e 5.12/2.23 U1(d, x) -> x 5.12/2.23 5.12/2.23 Q is empty. 5.12/2.23 We have to consider all minimal (P,Q,R)-chains. 5.12/2.23 ---------------------------------------- 5.12/2.23 5.12/2.23 (27) TransformationProof (EQUIVALENT) 5.12/2.23 By narrowing [LPAR04] the rule A^1 -> H(U1(a, d), f(b)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.23 5.12/2.23 (A^1 -> H(U1(d, d), f(b)),A^1 -> H(U1(d, d), f(b))) 5.12/2.23 (A^1 -> H(U1(e, d), f(b)),A^1 -> H(U1(e, d), f(b))) 5.12/2.23 (A^1 -> H(U1(a, d), U1(b, b)),A^1 -> H(U1(a, d), U1(b, b))) 5.12/2.23 (A^1 -> H(U1(a, d), f(d)),A^1 -> H(U1(a, d), f(d))) 5.12/2.23 (A^1 -> H(U1(a, d), f(e)),A^1 -> H(U1(a, d), f(e))) 5.12/2.23 5.12/2.23 5.12/2.23 ---------------------------------------- 5.12/2.23 5.12/2.23 (28) 5.12/2.23 Obligation: 5.12/2.23 Q DP problem: 5.12/2.23 The TRS P consists of the following rules: 5.12/2.23 5.12/2.23 H(x, x) -> G(x, x) 5.12/2.23 G(d, e) -> A^1 5.12/2.23 A^1 -> H(U1(a, e), f(b)) 5.12/2.23 A^1 -> H(U1(a, a), U1(b, b)) 5.12/2.23 A^1 -> H(U1(a, a), f(d)) 5.12/2.23 A^1 -> H(U1(a, a), f(e)) 5.12/2.23 A^1 -> H(U1(d, d), f(b)) 5.12/2.23 A^1 -> H(f(d), U1(b, b)) 5.12/2.23 A^1 -> H(f(d), f(d)) 5.12/2.23 A^1 -> H(f(d), f(e)) 5.12/2.23 A^1 -> H(U1(e, e), f(b)) 5.12/2.23 A^1 -> H(f(e), U1(b, b)) 5.12/2.23 A^1 -> H(f(e), f(d)) 5.12/2.23 A^1 -> H(f(e), f(e)) 5.12/2.23 A^1 -> H(f(a), U1(d, b)) 5.12/2.23 A^1 -> H(f(a), U1(e, b)) 5.12/2.23 A^1 -> H(f(a), U1(b, d)) 5.12/2.23 A^1 -> H(f(a), U1(b, e)) 5.12/2.23 A^1 -> H(f(a), U1(d, d)) 5.12/2.23 A^1 -> H(f(a), U1(e, e)) 5.12/2.23 A^1 -> H(a, f(b)) 5.12/2.23 A^1 -> H(U1(d, e), f(b)) 5.12/2.23 A^1 -> H(U1(d, a), U1(b, b)) 5.12/2.23 A^1 -> H(U1(d, a), f(d)) 5.12/2.23 A^1 -> H(U1(d, a), f(e)) 5.12/2.23 A^1 -> H(U1(e, d), f(b)) 5.12/2.23 A^1 -> H(U1(e, a), U1(b, b)) 5.12/2.23 A^1 -> H(U1(e, a), f(d)) 5.12/2.23 A^1 -> H(U1(e, a), f(e)) 5.12/2.23 A^1 -> H(U1(a, d), U1(b, b)) 5.12/2.23 A^1 -> H(U1(a, d), f(d)) 5.12/2.23 A^1 -> H(U1(a, d), f(e)) 5.12/2.23 5.12/2.23 The TRS R consists of the following rules: 5.12/2.23 5.12/2.23 a -> d 5.12/2.23 a -> e 5.12/2.23 f(x) -> U1(x, x) 5.12/2.23 b -> d 5.12/2.23 b -> e 5.12/2.23 U1(d, x) -> x 5.12/2.23 5.12/2.23 Q is empty. 5.12/2.23 We have to consider all minimal (P,Q,R)-chains. 5.12/2.23 ---------------------------------------- 5.12/2.23 5.12/2.23 (29) TransformationProof (EQUIVALENT) 5.12/2.23 By narrowing [LPAR04] the rule A^1 -> H(U1(a, e), f(b)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.23 5.12/2.23 (A^1 -> H(U1(d, e), f(b)),A^1 -> H(U1(d, e), f(b))) 5.12/2.23 (A^1 -> H(U1(e, e), f(b)),A^1 -> H(U1(e, e), f(b))) 5.12/2.23 (A^1 -> H(U1(a, e), U1(b, b)),A^1 -> H(U1(a, e), U1(b, b))) 5.12/2.23 (A^1 -> H(U1(a, e), f(d)),A^1 -> H(U1(a, e), f(d))) 5.12/2.23 (A^1 -> H(U1(a, e), f(e)),A^1 -> H(U1(a, e), f(e))) 5.12/2.23 5.12/2.23 5.12/2.23 ---------------------------------------- 5.12/2.23 5.12/2.23 (30) 5.12/2.23 Obligation: 5.12/2.23 Q DP problem: 5.12/2.23 The TRS P consists of the following rules: 5.12/2.23 5.12/2.23 H(x, x) -> G(x, x) 5.12/2.23 G(d, e) -> A^1 5.12/2.23 A^1 -> H(U1(a, a), U1(b, b)) 5.12/2.23 A^1 -> H(U1(a, a), f(d)) 5.12/2.23 A^1 -> H(U1(a, a), f(e)) 5.12/2.23 A^1 -> H(U1(d, d), f(b)) 5.12/2.23 A^1 -> H(f(d), U1(b, b)) 5.12/2.23 A^1 -> H(f(d), f(d)) 5.12/2.23 A^1 -> H(f(d), f(e)) 5.12/2.23 A^1 -> H(U1(e, e), f(b)) 5.12/2.23 A^1 -> H(f(e), U1(b, b)) 5.12/2.23 A^1 -> H(f(e), f(d)) 5.12/2.23 A^1 -> H(f(e), f(e)) 5.12/2.23 A^1 -> H(f(a), U1(d, b)) 5.12/2.23 A^1 -> H(f(a), U1(e, b)) 5.12/2.23 A^1 -> H(f(a), U1(b, d)) 5.12/2.23 A^1 -> H(f(a), U1(b, e)) 5.12/2.23 A^1 -> H(f(a), U1(d, d)) 5.12/2.23 A^1 -> H(f(a), U1(e, e)) 5.12/2.23 A^1 -> H(a, f(b)) 5.12/2.23 A^1 -> H(U1(d, e), f(b)) 5.12/2.23 A^1 -> H(U1(d, a), U1(b, b)) 5.12/2.23 A^1 -> H(U1(d, a), f(d)) 5.12/2.23 A^1 -> H(U1(d, a), f(e)) 5.12/2.23 A^1 -> H(U1(e, d), f(b)) 5.12/2.23 A^1 -> H(U1(e, a), U1(b, b)) 5.12/2.23 A^1 -> H(U1(e, a), f(d)) 5.12/2.23 A^1 -> H(U1(e, a), f(e)) 5.12/2.23 A^1 -> H(U1(a, d), U1(b, b)) 5.12/2.23 A^1 -> H(U1(a, d), f(d)) 5.12/2.23 A^1 -> H(U1(a, d), f(e)) 5.12/2.23 A^1 -> H(U1(a, e), U1(b, b)) 5.12/2.24 A^1 -> H(U1(a, e), f(d)) 5.12/2.24 A^1 -> H(U1(a, e), f(e)) 5.12/2.24 5.12/2.24 The TRS R consists of the following rules: 5.12/2.24 5.12/2.24 a -> d 5.12/2.24 a -> e 5.12/2.24 f(x) -> U1(x, x) 5.12/2.24 b -> d 5.12/2.24 b -> e 5.12/2.24 U1(d, x) -> x 5.12/2.24 5.12/2.24 Q is empty. 5.12/2.24 We have to consider all minimal (P,Q,R)-chains. 5.12/2.24 ---------------------------------------- 5.12/2.24 5.12/2.24 (31) TransformationProof (EQUIVALENT) 5.12/2.24 By narrowing [LPAR04] the rule A^1 -> H(U1(a, a), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.24 5.12/2.24 (A^1 -> H(U1(d, a), U1(b, b)),A^1 -> H(U1(d, a), U1(b, b))) 5.12/2.24 (A^1 -> H(U1(e, a), U1(b, b)),A^1 -> H(U1(e, a), U1(b, b))) 5.12/2.24 (A^1 -> H(U1(a, d), U1(b, b)),A^1 -> H(U1(a, d), U1(b, b))) 5.12/2.24 (A^1 -> H(U1(a, e), U1(b, b)),A^1 -> H(U1(a, e), U1(b, b))) 5.12/2.24 (A^1 -> H(U1(a, a), U1(d, b)),A^1 -> H(U1(a, a), U1(d, b))) 5.12/2.24 (A^1 -> H(U1(a, a), U1(e, b)),A^1 -> H(U1(a, a), U1(e, b))) 5.12/2.24 (A^1 -> H(U1(a, a), U1(b, d)),A^1 -> H(U1(a, a), U1(b, d))) 5.12/2.24 (A^1 -> H(U1(a, a), U1(b, e)),A^1 -> H(U1(a, a), U1(b, e))) 5.12/2.24 5.12/2.24 5.12/2.24 ---------------------------------------- 5.12/2.24 5.12/2.24 (32) 5.12/2.24 Obligation: 5.12/2.24 Q DP problem: 5.12/2.24 The TRS P consists of the following rules: 5.12/2.24 5.12/2.24 H(x, x) -> G(x, x) 5.12/2.24 G(d, e) -> A^1 5.12/2.24 A^1 -> H(U1(a, a), f(d)) 5.12/2.24 A^1 -> H(U1(a, a), f(e)) 5.12/2.24 A^1 -> H(U1(d, d), f(b)) 5.12/2.24 A^1 -> H(f(d), U1(b, b)) 5.12/2.24 A^1 -> H(f(d), f(d)) 5.12/2.24 A^1 -> H(f(d), f(e)) 5.12/2.24 A^1 -> H(U1(e, e), f(b)) 5.12/2.24 A^1 -> H(f(e), U1(b, b)) 5.12/2.24 A^1 -> H(f(e), f(d)) 5.12/2.24 A^1 -> H(f(e), f(e)) 5.12/2.24 A^1 -> H(f(a), U1(d, b)) 5.12/2.24 A^1 -> H(f(a), U1(e, b)) 5.12/2.24 A^1 -> H(f(a), U1(b, d)) 5.12/2.24 A^1 -> H(f(a), U1(b, e)) 5.12/2.24 A^1 -> H(f(a), U1(d, d)) 5.12/2.24 A^1 -> H(f(a), U1(e, e)) 5.12/2.24 A^1 -> H(a, f(b)) 5.12/2.24 A^1 -> H(U1(d, e), f(b)) 5.12/2.24 A^1 -> H(U1(d, a), U1(b, b)) 5.12/2.24 A^1 -> H(U1(d, a), f(d)) 5.12/2.24 A^1 -> H(U1(d, a), f(e)) 5.12/2.24 A^1 -> H(U1(e, d), f(b)) 5.12/2.24 A^1 -> H(U1(e, a), U1(b, b)) 5.12/2.24 A^1 -> H(U1(e, a), f(d)) 5.12/2.24 A^1 -> H(U1(e, a), f(e)) 5.12/2.24 A^1 -> H(U1(a, d), U1(b, b)) 5.12/2.24 A^1 -> H(U1(a, d), f(d)) 5.12/2.24 A^1 -> H(U1(a, d), f(e)) 5.12/2.24 A^1 -> H(U1(a, e), U1(b, b)) 5.12/2.24 A^1 -> H(U1(a, e), f(d)) 5.12/2.24 A^1 -> H(U1(a, e), f(e)) 5.12/2.24 A^1 -> H(U1(a, a), U1(d, b)) 5.12/2.24 A^1 -> H(U1(a, a), U1(e, b)) 5.12/2.24 A^1 -> H(U1(a, a), U1(b, d)) 5.12/2.24 A^1 -> H(U1(a, a), U1(b, e)) 5.12/2.24 5.12/2.24 The TRS R consists of the following rules: 5.12/2.24 5.12/2.24 a -> d 5.12/2.24 a -> e 5.12/2.24 f(x) -> U1(x, x) 5.12/2.24 b -> d 5.12/2.24 b -> e 5.12/2.24 U1(d, x) -> x 5.12/2.24 5.12/2.24 Q is empty. 5.12/2.24 We have to consider all minimal (P,Q,R)-chains. 5.12/2.24 ---------------------------------------- 5.12/2.24 5.12/2.24 (33) TransformationProof (EQUIVALENT) 5.12/2.24 By narrowing [LPAR04] the rule A^1 -> H(U1(a, a), f(d)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.24 5.12/2.24 (A^1 -> H(U1(d, a), f(d)),A^1 -> H(U1(d, a), f(d))) 5.12/2.24 (A^1 -> H(U1(e, a), f(d)),A^1 -> H(U1(e, a), f(d))) 5.12/2.24 (A^1 -> H(U1(a, d), f(d)),A^1 -> H(U1(a, d), f(d))) 5.12/2.24 (A^1 -> H(U1(a, e), f(d)),A^1 -> H(U1(a, e), f(d))) 5.12/2.24 (A^1 -> H(U1(a, a), U1(d, d)),A^1 -> H(U1(a, a), U1(d, d))) 5.12/2.24 5.12/2.24 5.12/2.24 ---------------------------------------- 5.12/2.24 5.12/2.24 (34) 5.12/2.24 Obligation: 5.12/2.24 Q DP problem: 5.12/2.24 The TRS P consists of the following rules: 5.12/2.24 5.12/2.24 H(x, x) -> G(x, x) 5.12/2.24 G(d, e) -> A^1 5.12/2.24 A^1 -> H(U1(a, a), f(e)) 5.12/2.24 A^1 -> H(U1(d, d), f(b)) 5.12/2.24 A^1 -> H(f(d), U1(b, b)) 5.12/2.24 A^1 -> H(f(d), f(d)) 5.12/2.24 A^1 -> H(f(d), f(e)) 5.12/2.24 A^1 -> H(U1(e, e), f(b)) 5.12/2.24 A^1 -> H(f(e), U1(b, b)) 5.12/2.24 A^1 -> H(f(e), f(d)) 5.12/2.24 A^1 -> H(f(e), f(e)) 5.12/2.24 A^1 -> H(f(a), U1(d, b)) 5.12/2.24 A^1 -> H(f(a), U1(e, b)) 5.12/2.24 A^1 -> H(f(a), U1(b, d)) 5.12/2.24 A^1 -> H(f(a), U1(b, e)) 5.12/2.24 A^1 -> H(f(a), U1(d, d)) 5.12/2.24 A^1 -> H(f(a), U1(e, e)) 5.12/2.24 A^1 -> H(a, f(b)) 5.12/2.24 A^1 -> H(U1(d, e), f(b)) 5.12/2.24 A^1 -> H(U1(d, a), U1(b, b)) 5.12/2.24 A^1 -> H(U1(d, a), f(d)) 5.12/2.24 A^1 -> H(U1(d, a), f(e)) 5.12/2.24 A^1 -> H(U1(e, d), f(b)) 5.12/2.24 A^1 -> H(U1(e, a), U1(b, b)) 5.12/2.24 A^1 -> H(U1(e, a), f(d)) 5.12/2.24 A^1 -> H(U1(e, a), f(e)) 5.12/2.24 A^1 -> H(U1(a, d), U1(b, b)) 5.12/2.24 A^1 -> H(U1(a, d), f(d)) 5.12/2.24 A^1 -> H(U1(a, d), f(e)) 5.12/2.24 A^1 -> H(U1(a, e), U1(b, b)) 5.12/2.24 A^1 -> H(U1(a, e), f(d)) 5.12/2.24 A^1 -> H(U1(a, e), f(e)) 5.12/2.24 A^1 -> H(U1(a, a), U1(d, b)) 5.12/2.24 A^1 -> H(U1(a, a), U1(e, b)) 5.12/2.24 A^1 -> H(U1(a, a), U1(b, d)) 5.12/2.24 A^1 -> H(U1(a, a), U1(b, e)) 5.12/2.24 A^1 -> H(U1(a, a), U1(d, d)) 5.12/2.24 5.12/2.24 The TRS R consists of the following rules: 5.12/2.24 5.12/2.24 a -> d 5.12/2.24 a -> e 5.12/2.24 f(x) -> U1(x, x) 5.12/2.24 b -> d 5.12/2.24 b -> e 5.12/2.24 U1(d, x) -> x 5.12/2.24 5.12/2.24 Q is empty. 5.12/2.24 We have to consider all minimal (P,Q,R)-chains. 5.12/2.24 ---------------------------------------- 5.12/2.24 5.12/2.24 (35) TransformationProof (EQUIVALENT) 5.12/2.24 By narrowing [LPAR04] the rule A^1 -> H(U1(a, a), f(e)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.24 5.12/2.24 (A^1 -> H(U1(d, a), f(e)),A^1 -> H(U1(d, a), f(e))) 5.12/2.24 (A^1 -> H(U1(e, a), f(e)),A^1 -> H(U1(e, a), f(e))) 5.12/2.24 (A^1 -> H(U1(a, d), f(e)),A^1 -> H(U1(a, d), f(e))) 5.12/2.24 (A^1 -> H(U1(a, e), f(e)),A^1 -> H(U1(a, e), f(e))) 5.12/2.24 (A^1 -> H(U1(a, a), U1(e, e)),A^1 -> H(U1(a, a), U1(e, e))) 5.12/2.24 5.12/2.24 5.12/2.24 ---------------------------------------- 5.12/2.24 5.12/2.24 (36) 5.12/2.24 Obligation: 5.12/2.24 Q DP problem: 5.12/2.24 The TRS P consists of the following rules: 5.12/2.24 5.12/2.24 H(x, x) -> G(x, x) 5.12/2.24 G(d, e) -> A^1 5.12/2.24 A^1 -> H(U1(d, d), f(b)) 5.12/2.24 A^1 -> H(f(d), U1(b, b)) 5.12/2.24 A^1 -> H(f(d), f(d)) 5.12/2.24 A^1 -> H(f(d), f(e)) 5.12/2.24 A^1 -> H(U1(e, e), f(b)) 5.12/2.24 A^1 -> H(f(e), U1(b, b)) 5.12/2.24 A^1 -> H(f(e), f(d)) 5.12/2.24 A^1 -> H(f(e), f(e)) 5.12/2.24 A^1 -> H(f(a), U1(d, b)) 5.12/2.24 A^1 -> H(f(a), U1(e, b)) 5.12/2.24 A^1 -> H(f(a), U1(b, d)) 5.12/2.24 A^1 -> H(f(a), U1(b, e)) 5.12/2.24 A^1 -> H(f(a), U1(d, d)) 5.12/2.24 A^1 -> H(f(a), U1(e, e)) 5.12/2.24 A^1 -> H(a, f(b)) 5.12/2.24 A^1 -> H(U1(d, e), f(b)) 5.12/2.24 A^1 -> H(U1(d, a), U1(b, b)) 5.12/2.24 A^1 -> H(U1(d, a), f(d)) 5.12/2.24 A^1 -> H(U1(d, a), f(e)) 5.12/2.24 A^1 -> H(U1(e, d), f(b)) 5.12/2.24 A^1 -> H(U1(e, a), U1(b, b)) 5.12/2.24 A^1 -> H(U1(e, a), f(d)) 5.12/2.24 A^1 -> H(U1(e, a), f(e)) 5.12/2.24 A^1 -> H(U1(a, d), U1(b, b)) 5.12/2.24 A^1 -> H(U1(a, d), f(d)) 5.12/2.24 A^1 -> H(U1(a, d), f(e)) 5.12/2.24 A^1 -> H(U1(a, e), U1(b, b)) 5.12/2.24 A^1 -> H(U1(a, e), f(d)) 5.12/2.24 A^1 -> H(U1(a, e), f(e)) 5.12/2.24 A^1 -> H(U1(a, a), U1(d, b)) 5.12/2.24 A^1 -> H(U1(a, a), U1(e, b)) 5.12/2.24 A^1 -> H(U1(a, a), U1(b, d)) 5.12/2.24 A^1 -> H(U1(a, a), U1(b, e)) 5.12/2.24 A^1 -> H(U1(a, a), U1(d, d)) 5.12/2.24 A^1 -> H(U1(a, a), U1(e, e)) 5.12/2.24 5.12/2.24 The TRS R consists of the following rules: 5.12/2.24 5.12/2.24 a -> d 5.12/2.24 a -> e 5.12/2.24 f(x) -> U1(x, x) 5.12/2.24 b -> d 5.12/2.24 b -> e 5.12/2.24 U1(d, x) -> x 5.12/2.24 5.12/2.24 Q is empty. 5.12/2.24 We have to consider all minimal (P,Q,R)-chains. 5.12/2.24 ---------------------------------------- 5.12/2.24 5.12/2.24 (37) TransformationProof (EQUIVALENT) 5.12/2.24 By narrowing [LPAR04] the rule A^1 -> H(U1(d, d), f(b)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.24 5.12/2.24 (A^1 -> H(d, f(b)),A^1 -> H(d, f(b))) 5.12/2.24 (A^1 -> H(U1(d, d), U1(b, b)),A^1 -> H(U1(d, d), U1(b, b))) 5.12/2.24 (A^1 -> H(U1(d, d), f(d)),A^1 -> H(U1(d, d), f(d))) 5.12/2.24 (A^1 -> H(U1(d, d), f(e)),A^1 -> H(U1(d, d), f(e))) 5.12/2.24 5.12/2.24 5.12/2.24 ---------------------------------------- 5.12/2.24 5.12/2.24 (38) 5.12/2.24 Obligation: 5.12/2.24 Q DP problem: 5.12/2.24 The TRS P consists of the following rules: 5.12/2.24 5.12/2.24 H(x, x) -> G(x, x) 5.12/2.24 G(d, e) -> A^1 5.12/2.24 A^1 -> H(f(d), U1(b, b)) 5.12/2.24 A^1 -> H(f(d), f(d)) 5.12/2.24 A^1 -> H(f(d), f(e)) 5.12/2.24 A^1 -> H(U1(e, e), f(b)) 5.12/2.24 A^1 -> H(f(e), U1(b, b)) 5.12/2.24 A^1 -> H(f(e), f(d)) 5.12/2.24 A^1 -> H(f(e), f(e)) 5.12/2.24 A^1 -> H(f(a), U1(d, b)) 5.12/2.24 A^1 -> H(f(a), U1(e, b)) 5.12/2.24 A^1 -> H(f(a), U1(b, d)) 5.12/2.24 A^1 -> H(f(a), U1(b, e)) 5.12/2.24 A^1 -> H(f(a), U1(d, d)) 5.12/2.24 A^1 -> H(f(a), U1(e, e)) 5.12/2.24 A^1 -> H(a, f(b)) 5.12/2.24 A^1 -> H(U1(d, e), f(b)) 5.12/2.24 A^1 -> H(U1(d, a), U1(b, b)) 5.12/2.24 A^1 -> H(U1(d, a), f(d)) 5.12/2.24 A^1 -> H(U1(d, a), f(e)) 5.12/2.24 A^1 -> H(U1(e, d), f(b)) 5.12/2.24 A^1 -> H(U1(e, a), U1(b, b)) 5.12/2.24 A^1 -> H(U1(e, a), f(d)) 5.12/2.24 A^1 -> H(U1(e, a), f(e)) 5.12/2.24 A^1 -> H(U1(a, d), U1(b, b)) 5.12/2.24 A^1 -> H(U1(a, d), f(d)) 5.12/2.24 A^1 -> H(U1(a, d), f(e)) 5.12/2.24 A^1 -> H(U1(a, e), U1(b, b)) 5.12/2.24 A^1 -> H(U1(a, e), f(d)) 5.12/2.24 A^1 -> H(U1(a, e), f(e)) 5.12/2.24 A^1 -> H(U1(a, a), U1(d, b)) 5.12/2.24 A^1 -> H(U1(a, a), U1(e, b)) 5.12/2.24 A^1 -> H(U1(a, a), U1(b, d)) 5.12/2.24 A^1 -> H(U1(a, a), U1(b, e)) 5.12/2.24 A^1 -> H(U1(a, a), U1(d, d)) 5.12/2.24 A^1 -> H(U1(a, a), U1(e, e)) 5.12/2.24 A^1 -> H(d, f(b)) 5.12/2.24 A^1 -> H(U1(d, d), U1(b, b)) 5.12/2.24 A^1 -> H(U1(d, d), f(d)) 5.12/2.24 A^1 -> H(U1(d, d), f(e)) 5.12/2.24 5.12/2.24 The TRS R consists of the following rules: 5.12/2.24 5.12/2.24 a -> d 5.12/2.24 a -> e 5.12/2.24 f(x) -> U1(x, x) 5.12/2.24 b -> d 5.12/2.24 b -> e 5.12/2.24 U1(d, x) -> x 5.12/2.24 5.12/2.24 Q is empty. 5.12/2.24 We have to consider all minimal (P,Q,R)-chains. 5.12/2.24 ---------------------------------------- 5.12/2.24 5.12/2.24 (39) TransformationProof (EQUIVALENT) 5.12/2.24 By narrowing [LPAR04] the rule A^1 -> H(f(d), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.24 5.12/2.24 (A^1 -> H(U1(d, d), U1(b, b)),A^1 -> H(U1(d, d), U1(b, b))) 5.12/2.24 (A^1 -> H(f(d), U1(d, b)),A^1 -> H(f(d), U1(d, b))) 5.12/2.24 (A^1 -> H(f(d), U1(e, b)),A^1 -> H(f(d), U1(e, b))) 5.12/2.24 (A^1 -> H(f(d), U1(b, d)),A^1 -> H(f(d), U1(b, d))) 5.12/2.24 (A^1 -> H(f(d), U1(b, e)),A^1 -> H(f(d), U1(b, e))) 5.12/2.25 5.12/2.25 5.12/2.25 ---------------------------------------- 5.12/2.25 5.12/2.25 (40) 5.12/2.25 Obligation: 5.12/2.25 Q DP problem: 5.12/2.25 The TRS P consists of the following rules: 5.12/2.25 5.12/2.25 H(x, x) -> G(x, x) 5.12/2.25 G(d, e) -> A^1 5.12/2.25 A^1 -> H(f(d), f(d)) 5.12/2.25 A^1 -> H(f(d), f(e)) 5.12/2.25 A^1 -> H(U1(e, e), f(b)) 5.12/2.25 A^1 -> H(f(e), U1(b, b)) 5.12/2.25 A^1 -> H(f(e), f(d)) 5.12/2.25 A^1 -> H(f(e), f(e)) 5.12/2.25 A^1 -> H(f(a), U1(d, b)) 5.12/2.25 A^1 -> H(f(a), U1(e, b)) 5.12/2.25 A^1 -> H(f(a), U1(b, d)) 5.12/2.25 A^1 -> H(f(a), U1(b, e)) 5.12/2.25 A^1 -> H(f(a), U1(d, d)) 5.12/2.25 A^1 -> H(f(a), U1(e, e)) 5.12/2.25 A^1 -> H(a, f(b)) 5.12/2.25 A^1 -> H(U1(d, e), f(b)) 5.12/2.25 A^1 -> H(U1(d, a), U1(b, b)) 5.12/2.25 A^1 -> H(U1(d, a), f(d)) 5.12/2.25 A^1 -> H(U1(d, a), f(e)) 5.12/2.25 A^1 -> H(U1(e, d), f(b)) 5.12/2.25 A^1 -> H(U1(e, a), U1(b, b)) 5.12/2.25 A^1 -> H(U1(e, a), f(d)) 5.12/2.25 A^1 -> H(U1(e, a), f(e)) 5.12/2.25 A^1 -> H(U1(a, d), U1(b, b)) 5.12/2.25 A^1 -> H(U1(a, d), f(d)) 5.12/2.25 A^1 -> H(U1(a, d), f(e)) 5.12/2.25 A^1 -> H(U1(a, e), U1(b, b)) 5.12/2.25 A^1 -> H(U1(a, e), f(d)) 5.12/2.25 A^1 -> H(U1(a, e), f(e)) 5.12/2.25 A^1 -> H(U1(a, a), U1(d, b)) 5.12/2.25 A^1 -> H(U1(a, a), U1(e, b)) 5.12/2.25 A^1 -> H(U1(a, a), U1(b, d)) 5.12/2.25 A^1 -> H(U1(a, a), U1(b, e)) 5.12/2.25 A^1 -> H(U1(a, a), U1(d, d)) 5.12/2.25 A^1 -> H(U1(a, a), U1(e, e)) 5.12/2.25 A^1 -> H(d, f(b)) 5.12/2.25 A^1 -> H(U1(d, d), U1(b, b)) 5.12/2.25 A^1 -> H(U1(d, d), f(d)) 5.12/2.25 A^1 -> H(U1(d, d), f(e)) 5.12/2.25 A^1 -> H(f(d), U1(d, b)) 5.12/2.25 A^1 -> H(f(d), U1(e, b)) 5.12/2.25 A^1 -> H(f(d), U1(b, d)) 5.12/2.25 A^1 -> H(f(d), U1(b, e)) 5.12/2.25 5.12/2.25 The TRS R consists of the following rules: 5.12/2.25 5.12/2.25 a -> d 5.12/2.25 a -> e 5.12/2.25 f(x) -> U1(x, x) 5.12/2.25 b -> d 5.12/2.25 b -> e 5.12/2.25 U1(d, x) -> x 5.12/2.25 5.12/2.25 Q is empty. 5.12/2.25 We have to consider all minimal (P,Q,R)-chains. 5.12/2.25 ---------------------------------------- 5.12/2.25 5.12/2.25 (41) TransformationProof (EQUIVALENT) 5.12/2.25 By narrowing [LPAR04] the rule A^1 -> H(f(d), f(e)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.25 5.12/2.25 (A^1 -> H(U1(d, d), f(e)),A^1 -> H(U1(d, d), f(e))) 5.12/2.25 (A^1 -> H(f(d), U1(e, e)),A^1 -> H(f(d), U1(e, e))) 5.12/2.25 5.12/2.25 5.12/2.25 ---------------------------------------- 5.12/2.25 5.12/2.25 (42) 5.12/2.25 Obligation: 5.12/2.25 Q DP problem: 5.12/2.25 The TRS P consists of the following rules: 5.12/2.25 5.12/2.25 H(x, x) -> G(x, x) 5.12/2.25 G(d, e) -> A^1 5.12/2.25 A^1 -> H(f(d), f(d)) 5.12/2.25 A^1 -> H(U1(e, e), f(b)) 5.12/2.25 A^1 -> H(f(e), U1(b, b)) 5.12/2.25 A^1 -> H(f(e), f(d)) 5.12/2.25 A^1 -> H(f(e), f(e)) 5.12/2.25 A^1 -> H(f(a), U1(d, b)) 5.12/2.25 A^1 -> H(f(a), U1(e, b)) 5.12/2.25 A^1 -> H(f(a), U1(b, d)) 5.12/2.25 A^1 -> H(f(a), U1(b, e)) 5.12/2.25 A^1 -> H(f(a), U1(d, d)) 5.12/2.25 A^1 -> H(f(a), U1(e, e)) 5.12/2.25 A^1 -> H(a, f(b)) 5.12/2.25 A^1 -> H(U1(d, e), f(b)) 5.12/2.25 A^1 -> H(U1(d, a), U1(b, b)) 5.12/2.25 A^1 -> H(U1(d, a), f(d)) 5.12/2.25 A^1 -> H(U1(d, a), f(e)) 5.12/2.25 A^1 -> H(U1(e, d), f(b)) 5.12/2.25 A^1 -> H(U1(e, a), U1(b, b)) 5.12/2.25 A^1 -> H(U1(e, a), f(d)) 5.12/2.25 A^1 -> H(U1(e, a), f(e)) 5.12/2.25 A^1 -> H(U1(a, d), U1(b, b)) 5.12/2.25 A^1 -> H(U1(a, d), f(d)) 5.12/2.25 A^1 -> H(U1(a, d), f(e)) 5.12/2.25 A^1 -> H(U1(a, e), U1(b, b)) 5.12/2.25 A^1 -> H(U1(a, e), f(d)) 5.12/2.25 A^1 -> H(U1(a, e), f(e)) 5.12/2.25 A^1 -> H(U1(a, a), U1(d, b)) 5.12/2.25 A^1 -> H(U1(a, a), U1(e, b)) 5.12/2.25 A^1 -> H(U1(a, a), U1(b, d)) 5.12/2.25 A^1 -> H(U1(a, a), U1(b, e)) 5.12/2.25 A^1 -> H(U1(a, a), U1(d, d)) 5.12/2.25 A^1 -> H(U1(a, a), U1(e, e)) 5.12/2.25 A^1 -> H(d, f(b)) 5.12/2.25 A^1 -> H(U1(d, d), U1(b, b)) 5.12/2.25 A^1 -> H(U1(d, d), f(d)) 5.12/2.25 A^1 -> H(U1(d, d), f(e)) 5.12/2.25 A^1 -> H(f(d), U1(d, b)) 5.12/2.25 A^1 -> H(f(d), U1(e, b)) 5.12/2.25 A^1 -> H(f(d), U1(b, d)) 5.12/2.25 A^1 -> H(f(d), U1(b, e)) 5.12/2.25 A^1 -> H(f(d), U1(e, e)) 5.12/2.25 5.12/2.25 The TRS R consists of the following rules: 5.12/2.25 5.12/2.25 a -> d 5.12/2.25 a -> e 5.12/2.25 f(x) -> U1(x, x) 5.12/2.25 b -> d 5.12/2.25 b -> e 5.12/2.25 U1(d, x) -> x 5.12/2.25 5.12/2.25 Q is empty. 5.12/2.25 We have to consider all minimal (P,Q,R)-chains. 5.12/2.25 ---------------------------------------- 5.12/2.25 5.12/2.25 (43) TransformationProof (EQUIVALENT) 5.12/2.25 By narrowing [LPAR04] the rule A^1 -> H(U1(e, e), f(b)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.25 5.12/2.25 (A^1 -> H(U1(e, e), U1(b, b)),A^1 -> H(U1(e, e), U1(b, b))) 5.12/2.25 (A^1 -> H(U1(e, e), f(d)),A^1 -> H(U1(e, e), f(d))) 5.12/2.25 (A^1 -> H(U1(e, e), f(e)),A^1 -> H(U1(e, e), f(e))) 5.12/2.25 5.12/2.25 5.12/2.25 ---------------------------------------- 5.12/2.25 5.12/2.25 (44) 5.12/2.25 Obligation: 5.12/2.25 Q DP problem: 5.12/2.25 The TRS P consists of the following rules: 5.12/2.25 5.12/2.25 H(x, x) -> G(x, x) 5.12/2.25 G(d, e) -> A^1 5.12/2.25 A^1 -> H(f(d), f(d)) 5.12/2.25 A^1 -> H(f(e), U1(b, b)) 5.12/2.25 A^1 -> H(f(e), f(d)) 5.12/2.25 A^1 -> H(f(e), f(e)) 5.12/2.25 A^1 -> H(f(a), U1(d, b)) 5.12/2.25 A^1 -> H(f(a), U1(e, b)) 5.12/2.25 A^1 -> H(f(a), U1(b, d)) 5.12/2.25 A^1 -> H(f(a), U1(b, e)) 5.12/2.25 A^1 -> H(f(a), U1(d, d)) 5.12/2.25 A^1 -> H(f(a), U1(e, e)) 5.12/2.25 A^1 -> H(a, f(b)) 5.12/2.25 A^1 -> H(U1(d, e), f(b)) 5.12/2.25 A^1 -> H(U1(d, a), U1(b, b)) 5.12/2.25 A^1 -> H(U1(d, a), f(d)) 5.12/2.25 A^1 -> H(U1(d, a), f(e)) 5.12/2.25 A^1 -> H(U1(e, d), f(b)) 5.12/2.25 A^1 -> H(U1(e, a), U1(b, b)) 5.12/2.25 A^1 -> H(U1(e, a), f(d)) 5.12/2.25 A^1 -> H(U1(e, a), f(e)) 5.12/2.25 A^1 -> H(U1(a, d), U1(b, b)) 5.12/2.25 A^1 -> H(U1(a, d), f(d)) 5.12/2.25 A^1 -> H(U1(a, d), f(e)) 5.12/2.25 A^1 -> H(U1(a, e), U1(b, b)) 5.12/2.25 A^1 -> H(U1(a, e), f(d)) 5.12/2.25 A^1 -> H(U1(a, e), f(e)) 5.12/2.25 A^1 -> H(U1(a, a), U1(d, b)) 5.12/2.25 A^1 -> H(U1(a, a), U1(e, b)) 5.12/2.25 A^1 -> H(U1(a, a), U1(b, d)) 5.12/2.25 A^1 -> H(U1(a, a), U1(b, e)) 5.12/2.25 A^1 -> H(U1(a, a), U1(d, d)) 5.12/2.25 A^1 -> H(U1(a, a), U1(e, e)) 5.12/2.25 A^1 -> H(d, f(b)) 5.12/2.25 A^1 -> H(U1(d, d), U1(b, b)) 5.12/2.25 A^1 -> H(U1(d, d), f(d)) 5.12/2.25 A^1 -> H(U1(d, d), f(e)) 5.12/2.25 A^1 -> H(f(d), U1(d, b)) 5.12/2.25 A^1 -> H(f(d), U1(e, b)) 5.12/2.25 A^1 -> H(f(d), U1(b, d)) 5.12/2.25 A^1 -> H(f(d), U1(b, e)) 5.12/2.25 A^1 -> H(f(d), U1(e, e)) 5.12/2.25 A^1 -> H(U1(e, e), U1(b, b)) 5.12/2.25 A^1 -> H(U1(e, e), f(d)) 5.12/2.25 A^1 -> H(U1(e, e), f(e)) 5.12/2.25 5.12/2.25 The TRS R consists of the following rules: 5.12/2.25 5.12/2.25 a -> d 5.12/2.25 a -> e 5.12/2.25 f(x) -> U1(x, x) 5.12/2.25 b -> d 5.12/2.25 b -> e 5.12/2.25 U1(d, x) -> x 5.12/2.25 5.12/2.25 Q is empty. 5.12/2.25 We have to consider all minimal (P,Q,R)-chains. 5.12/2.25 ---------------------------------------- 5.12/2.25 5.12/2.25 (45) TransformationProof (EQUIVALENT) 5.12/2.25 By narrowing [LPAR04] the rule A^1 -> H(f(e), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.25 5.12/2.25 (A^1 -> H(U1(e, e), U1(b, b)),A^1 -> H(U1(e, e), U1(b, b))) 5.12/2.25 (A^1 -> H(f(e), U1(d, b)),A^1 -> H(f(e), U1(d, b))) 5.12/2.25 (A^1 -> H(f(e), U1(e, b)),A^1 -> H(f(e), U1(e, b))) 5.12/2.25 (A^1 -> H(f(e), U1(b, d)),A^1 -> H(f(e), U1(b, d))) 5.12/2.25 (A^1 -> H(f(e), U1(b, e)),A^1 -> H(f(e), U1(b, e))) 5.12/2.25 5.12/2.25 5.12/2.25 ---------------------------------------- 5.12/2.25 5.12/2.25 (46) 5.12/2.25 Obligation: 5.12/2.25 Q DP problem: 5.12/2.25 The TRS P consists of the following rules: 5.12/2.25 5.12/2.25 H(x, x) -> G(x, x) 5.12/2.25 G(d, e) -> A^1 5.12/2.25 A^1 -> H(f(d), f(d)) 5.12/2.25 A^1 -> H(f(e), f(d)) 5.12/2.25 A^1 -> H(f(e), f(e)) 5.12/2.25 A^1 -> H(f(a), U1(d, b)) 5.12/2.25 A^1 -> H(f(a), U1(e, b)) 5.12/2.25 A^1 -> H(f(a), U1(b, d)) 5.12/2.25 A^1 -> H(f(a), U1(b, e)) 5.12/2.25 A^1 -> H(f(a), U1(d, d)) 5.12/2.25 A^1 -> H(f(a), U1(e, e)) 5.12/2.25 A^1 -> H(a, f(b)) 5.12/2.25 A^1 -> H(U1(d, e), f(b)) 5.12/2.25 A^1 -> H(U1(d, a), U1(b, b)) 5.12/2.25 A^1 -> H(U1(d, a), f(d)) 5.12/2.25 A^1 -> H(U1(d, a), f(e)) 5.12/2.25 A^1 -> H(U1(e, d), f(b)) 5.12/2.25 A^1 -> H(U1(e, a), U1(b, b)) 5.12/2.25 A^1 -> H(U1(e, a), f(d)) 5.12/2.25 A^1 -> H(U1(e, a), f(e)) 5.12/2.25 A^1 -> H(U1(a, d), U1(b, b)) 5.12/2.25 A^1 -> H(U1(a, d), f(d)) 5.12/2.25 A^1 -> H(U1(a, d), f(e)) 5.12/2.25 A^1 -> H(U1(a, e), U1(b, b)) 5.12/2.25 A^1 -> H(U1(a, e), f(d)) 5.12/2.25 A^1 -> H(U1(a, e), f(e)) 5.12/2.25 A^1 -> H(U1(a, a), U1(d, b)) 5.12/2.25 A^1 -> H(U1(a, a), U1(e, b)) 5.12/2.25 A^1 -> H(U1(a, a), U1(b, d)) 5.12/2.25 A^1 -> H(U1(a, a), U1(b, e)) 5.12/2.25 A^1 -> H(U1(a, a), U1(d, d)) 5.12/2.25 A^1 -> H(U1(a, a), U1(e, e)) 5.12/2.25 A^1 -> H(d, f(b)) 5.12/2.25 A^1 -> H(U1(d, d), U1(b, b)) 5.12/2.25 A^1 -> H(U1(d, d), f(d)) 5.12/2.25 A^1 -> H(U1(d, d), f(e)) 5.12/2.25 A^1 -> H(f(d), U1(d, b)) 5.12/2.25 A^1 -> H(f(d), U1(e, b)) 5.12/2.25 A^1 -> H(f(d), U1(b, d)) 5.12/2.25 A^1 -> H(f(d), U1(b, e)) 5.12/2.25 A^1 -> H(f(d), U1(e, e)) 5.12/2.25 A^1 -> H(U1(e, e), U1(b, b)) 5.12/2.25 A^1 -> H(U1(e, e), f(d)) 5.12/2.25 A^1 -> H(U1(e, e), f(e)) 5.12/2.25 A^1 -> H(f(e), U1(d, b)) 5.12/2.25 A^1 -> H(f(e), U1(e, b)) 5.12/2.25 A^1 -> H(f(e), U1(b, d)) 5.12/2.25 A^1 -> H(f(e), U1(b, e)) 5.12/2.25 5.12/2.25 The TRS R consists of the following rules: 5.12/2.25 5.12/2.25 a -> d 5.12/2.25 a -> e 5.12/2.25 f(x) -> U1(x, x) 5.12/2.25 b -> d 5.12/2.25 b -> e 5.12/2.25 U1(d, x) -> x 5.12/2.25 5.12/2.25 Q is empty. 5.12/2.25 We have to consider all minimal (P,Q,R)-chains. 5.12/2.25 ---------------------------------------- 5.12/2.25 5.12/2.25 (47) TransformationProof (EQUIVALENT) 5.12/2.25 By narrowing [LPAR04] the rule A^1 -> H(f(e), f(d)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.25 5.12/2.25 (A^1 -> H(U1(e, e), f(d)),A^1 -> H(U1(e, e), f(d))) 5.12/2.25 (A^1 -> H(f(e), U1(d, d)),A^1 -> H(f(e), U1(d, d))) 5.12/2.25 5.12/2.25 5.12/2.25 ---------------------------------------- 5.12/2.25 5.12/2.25 (48) 5.12/2.25 Obligation: 5.12/2.25 Q DP problem: 5.12/2.25 The TRS P consists of the following rules: 5.12/2.25 5.12/2.25 H(x, x) -> G(x, x) 5.12/2.25 G(d, e) -> A^1 5.12/2.25 A^1 -> H(f(d), f(d)) 5.12/2.25 A^1 -> H(f(e), f(e)) 5.12/2.25 A^1 -> H(f(a), U1(d, b)) 5.12/2.25 A^1 -> H(f(a), U1(e, b)) 5.12/2.25 A^1 -> H(f(a), U1(b, d)) 5.12/2.25 A^1 -> H(f(a), U1(b, e)) 5.12/2.25 A^1 -> H(f(a), U1(d, d)) 5.12/2.25 A^1 -> H(f(a), U1(e, e)) 5.12/2.25 A^1 -> H(a, f(b)) 5.12/2.25 A^1 -> H(U1(d, e), f(b)) 5.12/2.25 A^1 -> H(U1(d, a), U1(b, b)) 5.12/2.25 A^1 -> H(U1(d, a), f(d)) 5.12/2.25 A^1 -> H(U1(d, a), f(e)) 5.12/2.25 A^1 -> H(U1(e, d), f(b)) 5.12/2.25 A^1 -> H(U1(e, a), U1(b, b)) 5.12/2.25 A^1 -> H(U1(e, a), f(d)) 5.12/2.25 A^1 -> H(U1(e, a), f(e)) 5.12/2.25 A^1 -> H(U1(a, d), U1(b, b)) 5.12/2.25 A^1 -> H(U1(a, d), f(d)) 5.12/2.25 A^1 -> H(U1(a, d), f(e)) 5.12/2.25 A^1 -> H(U1(a, e), U1(b, b)) 5.12/2.25 A^1 -> H(U1(a, e), f(d)) 5.12/2.25 A^1 -> H(U1(a, e), f(e)) 5.12/2.25 A^1 -> H(U1(a, a), U1(d, b)) 5.12/2.25 A^1 -> H(U1(a, a), U1(e, b)) 5.12/2.25 A^1 -> H(U1(a, a), U1(b, d)) 5.12/2.25 A^1 -> H(U1(a, a), U1(b, e)) 5.12/2.25 A^1 -> H(U1(a, a), U1(d, d)) 5.12/2.25 A^1 -> H(U1(a, a), U1(e, e)) 5.12/2.25 A^1 -> H(d, f(b)) 5.12/2.25 A^1 -> H(U1(d, d), U1(b, b)) 5.12/2.25 A^1 -> H(U1(d, d), f(d)) 5.12/2.25 A^1 -> H(U1(d, d), f(e)) 5.12/2.25 A^1 -> H(f(d), U1(d, b)) 5.12/2.25 A^1 -> H(f(d), U1(e, b)) 5.12/2.25 A^1 -> H(f(d), U1(b, d)) 5.12/2.25 A^1 -> H(f(d), U1(b, e)) 5.12/2.25 A^1 -> H(f(d), U1(e, e)) 5.12/2.25 A^1 -> H(U1(e, e), U1(b, b)) 5.12/2.25 A^1 -> H(U1(e, e), f(d)) 5.12/2.25 A^1 -> H(U1(e, e), f(e)) 5.12/2.25 A^1 -> H(f(e), U1(d, b)) 5.12/2.25 A^1 -> H(f(e), U1(e, b)) 5.12/2.25 A^1 -> H(f(e), U1(b, d)) 5.12/2.25 A^1 -> H(f(e), U1(b, e)) 5.12/2.25 A^1 -> H(f(e), U1(d, d)) 5.12/2.25 5.12/2.25 The TRS R consists of the following rules: 5.12/2.25 5.12/2.25 a -> d 5.12/2.25 a -> e 5.12/2.25 f(x) -> U1(x, x) 5.12/2.25 b -> d 5.12/2.25 b -> e 5.12/2.25 U1(d, x) -> x 5.12/2.25 5.12/2.25 Q is empty. 5.12/2.25 We have to consider all minimal (P,Q,R)-chains. 5.12/2.25 ---------------------------------------- 5.12/2.25 5.12/2.25 (49) TransformationProof (EQUIVALENT) 5.12/2.25 By narrowing [LPAR04] the rule A^1 -> H(f(a), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.25 5.12/2.25 (A^1 -> H(U1(a, a), U1(d, b)),A^1 -> H(U1(a, a), U1(d, b))) 5.12/2.25 (A^1 -> H(f(d), U1(d, b)),A^1 -> H(f(d), U1(d, b))) 5.12/2.25 (A^1 -> H(f(e), U1(d, b)),A^1 -> H(f(e), U1(d, b))) 5.12/2.25 (A^1 -> H(f(a), b),A^1 -> H(f(a), b)) 5.12/2.25 (A^1 -> H(f(a), U1(d, d)),A^1 -> H(f(a), U1(d, d))) 5.12/2.25 (A^1 -> H(f(a), U1(d, e)),A^1 -> H(f(a), U1(d, e))) 5.12/2.25 5.12/2.25 5.12/2.25 ---------------------------------------- 5.12/2.25 5.12/2.25 (50) 5.12/2.25 Obligation: 5.12/2.25 Q DP problem: 5.12/2.25 The TRS P consists of the following rules: 5.12/2.25 5.12/2.25 H(x, x) -> G(x, x) 5.12/2.25 G(d, e) -> A^1 5.12/2.25 A^1 -> H(f(d), f(d)) 5.12/2.25 A^1 -> H(f(e), f(e)) 5.12/2.25 A^1 -> H(f(a), U1(e, b)) 5.12/2.25 A^1 -> H(f(a), U1(b, d)) 5.12/2.25 A^1 -> H(f(a), U1(b, e)) 5.12/2.25 A^1 -> H(f(a), U1(d, d)) 5.12/2.25 A^1 -> H(f(a), U1(e, e)) 5.12/2.25 A^1 -> H(a, f(b)) 5.12/2.25 A^1 -> H(U1(d, e), f(b)) 5.12/2.25 A^1 -> H(U1(d, a), U1(b, b)) 5.12/2.25 A^1 -> H(U1(d, a), f(d)) 5.12/2.25 A^1 -> H(U1(d, a), f(e)) 5.12/2.25 A^1 -> H(U1(e, d), f(b)) 5.12/2.25 A^1 -> H(U1(e, a), U1(b, b)) 5.12/2.25 A^1 -> H(U1(e, a), f(d)) 5.12/2.25 A^1 -> H(U1(e, a), f(e)) 5.12/2.25 A^1 -> H(U1(a, d), U1(b, b)) 5.12/2.25 A^1 -> H(U1(a, d), f(d)) 5.12/2.25 A^1 -> H(U1(a, d), f(e)) 5.12/2.25 A^1 -> H(U1(a, e), U1(b, b)) 5.12/2.25 A^1 -> H(U1(a, e), f(d)) 5.12/2.25 A^1 -> H(U1(a, e), f(e)) 5.12/2.25 A^1 -> H(U1(a, a), U1(d, b)) 5.12/2.25 A^1 -> H(U1(a, a), U1(e, b)) 5.12/2.25 A^1 -> H(U1(a, a), U1(b, d)) 5.12/2.25 A^1 -> H(U1(a, a), U1(b, e)) 5.12/2.25 A^1 -> H(U1(a, a), U1(d, d)) 5.12/2.25 A^1 -> H(U1(a, a), U1(e, e)) 5.12/2.25 A^1 -> H(d, f(b)) 5.12/2.25 A^1 -> H(U1(d, d), U1(b, b)) 5.12/2.25 A^1 -> H(U1(d, d), f(d)) 5.12/2.25 A^1 -> H(U1(d, d), f(e)) 5.12/2.25 A^1 -> H(f(d), U1(d, b)) 5.12/2.25 A^1 -> H(f(d), U1(e, b)) 5.12/2.25 A^1 -> H(f(d), U1(b, d)) 5.12/2.25 A^1 -> H(f(d), U1(b, e)) 5.12/2.25 A^1 -> H(f(d), U1(e, e)) 5.12/2.25 A^1 -> H(U1(e, e), U1(b, b)) 5.12/2.25 A^1 -> H(U1(e, e), f(d)) 5.12/2.25 A^1 -> H(U1(e, e), f(e)) 5.12/2.25 A^1 -> H(f(e), U1(d, b)) 5.12/2.25 A^1 -> H(f(e), U1(e, b)) 5.12/2.25 A^1 -> H(f(e), U1(b, d)) 5.12/2.25 A^1 -> H(f(e), U1(b, e)) 5.12/2.25 A^1 -> H(f(e), U1(d, d)) 5.12/2.25 A^1 -> H(f(a), b) 5.12/2.25 A^1 -> H(f(a), U1(d, e)) 5.12/2.25 5.12/2.25 The TRS R consists of the following rules: 5.12/2.25 5.12/2.25 a -> d 5.12/2.25 a -> e 5.12/2.25 f(x) -> U1(x, x) 5.12/2.25 b -> d 5.12/2.25 b -> e 5.12/2.25 U1(d, x) -> x 5.12/2.25 5.12/2.25 Q is empty. 5.12/2.25 We have to consider all minimal (P,Q,R)-chains. 5.12/2.25 ---------------------------------------- 5.12/2.25 5.12/2.25 (51) TransformationProof (EQUIVALENT) 5.12/2.25 By narrowing [LPAR04] the rule A^1 -> H(f(a), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.25 5.12/2.25 (A^1 -> H(U1(a, a), U1(e, b)),A^1 -> H(U1(a, a), U1(e, b))) 5.12/2.25 (A^1 -> H(f(d), U1(e, b)),A^1 -> H(f(d), U1(e, b))) 5.12/2.25 (A^1 -> H(f(e), U1(e, b)),A^1 -> H(f(e), U1(e, b))) 5.12/2.25 (A^1 -> H(f(a), U1(e, d)),A^1 -> H(f(a), U1(e, d))) 5.12/2.25 (A^1 -> H(f(a), U1(e, e)),A^1 -> H(f(a), U1(e, e))) 5.12/2.25 5.12/2.25 5.12/2.25 ---------------------------------------- 5.12/2.25 5.12/2.25 (52) 5.12/2.25 Obligation: 5.12/2.25 Q DP problem: 5.12/2.25 The TRS P consists of the following rules: 5.12/2.25 5.12/2.25 H(x, x) -> G(x, x) 5.12/2.25 G(d, e) -> A^1 5.12/2.25 A^1 -> H(f(d), f(d)) 5.12/2.25 A^1 -> H(f(e), f(e)) 5.12/2.25 A^1 -> H(f(a), U1(b, d)) 5.12/2.25 A^1 -> H(f(a), U1(b, e)) 5.12/2.25 A^1 -> H(f(a), U1(d, d)) 5.12/2.25 A^1 -> H(f(a), U1(e, e)) 5.12/2.25 A^1 -> H(a, f(b)) 5.12/2.25 A^1 -> H(U1(d, e), f(b)) 5.12/2.25 A^1 -> H(U1(d, a), U1(b, b)) 5.12/2.25 A^1 -> H(U1(d, a), f(d)) 5.12/2.25 A^1 -> H(U1(d, a), f(e)) 5.12/2.25 A^1 -> H(U1(e, d), f(b)) 5.12/2.25 A^1 -> H(U1(e, a), U1(b, b)) 5.12/2.25 A^1 -> H(U1(e, a), f(d)) 5.12/2.25 A^1 -> H(U1(e, a), f(e)) 5.12/2.25 A^1 -> H(U1(a, d), U1(b, b)) 5.12/2.25 A^1 -> H(U1(a, d), f(d)) 5.12/2.25 A^1 -> H(U1(a, d), f(e)) 5.12/2.25 A^1 -> H(U1(a, e), U1(b, b)) 5.12/2.25 A^1 -> H(U1(a, e), f(d)) 5.12/2.25 A^1 -> H(U1(a, e), f(e)) 5.12/2.25 A^1 -> H(U1(a, a), U1(d, b)) 5.12/2.25 A^1 -> H(U1(a, a), U1(e, b)) 5.12/2.25 A^1 -> H(U1(a, a), U1(b, d)) 5.12/2.25 A^1 -> H(U1(a, a), U1(b, e)) 5.12/2.25 A^1 -> H(U1(a, a), U1(d, d)) 5.12/2.25 A^1 -> H(U1(a, a), U1(e, e)) 5.12/2.25 A^1 -> H(d, f(b)) 5.12/2.25 A^1 -> H(U1(d, d), U1(b, b)) 5.12/2.25 A^1 -> H(U1(d, d), f(d)) 5.12/2.25 A^1 -> H(U1(d, d), f(e)) 5.12/2.25 A^1 -> H(f(d), U1(d, b)) 5.12/2.25 A^1 -> H(f(d), U1(e, b)) 5.12/2.25 A^1 -> H(f(d), U1(b, d)) 5.12/2.25 A^1 -> H(f(d), U1(b, e)) 5.12/2.25 A^1 -> H(f(d), U1(e, e)) 5.12/2.25 A^1 -> H(U1(e, e), U1(b, b)) 5.12/2.25 A^1 -> H(U1(e, e), f(d)) 5.12/2.25 A^1 -> H(U1(e, e), f(e)) 5.12/2.25 A^1 -> H(f(e), U1(d, b)) 5.12/2.25 A^1 -> H(f(e), U1(e, b)) 5.12/2.25 A^1 -> H(f(e), U1(b, d)) 5.12/2.25 A^1 -> H(f(e), U1(b, e)) 5.12/2.25 A^1 -> H(f(e), U1(d, d)) 5.12/2.25 A^1 -> H(f(a), b) 5.12/2.25 A^1 -> H(f(a), U1(d, e)) 5.12/2.25 A^1 -> H(f(a), U1(e, d)) 5.12/2.25 5.12/2.25 The TRS R consists of the following rules: 5.12/2.25 5.12/2.25 a -> d 5.12/2.25 a -> e 5.12/2.25 f(x) -> U1(x, x) 5.12/2.25 b -> d 5.12/2.25 b -> e 5.12/2.25 U1(d, x) -> x 5.12/2.25 5.12/2.25 Q is empty. 5.12/2.25 We have to consider all minimal (P,Q,R)-chains. 5.12/2.25 ---------------------------------------- 5.12/2.25 5.12/2.25 (53) TransformationProof (EQUIVALENT) 5.12/2.25 By narrowing [LPAR04] the rule A^1 -> H(f(a), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.25 5.12/2.25 (A^1 -> H(U1(a, a), U1(b, d)),A^1 -> H(U1(a, a), U1(b, d))) 5.12/2.25 (A^1 -> H(f(d), U1(b, d)),A^1 -> H(f(d), U1(b, d))) 5.12/2.25 (A^1 -> H(f(e), U1(b, d)),A^1 -> H(f(e), U1(b, d))) 5.12/2.25 (A^1 -> H(f(a), U1(d, d)),A^1 -> H(f(a), U1(d, d))) 5.12/2.25 (A^1 -> H(f(a), U1(e, d)),A^1 -> H(f(a), U1(e, d))) 5.12/2.25 5.12/2.25 5.12/2.25 ---------------------------------------- 5.12/2.25 5.12/2.25 (54) 5.12/2.25 Obligation: 5.12/2.25 Q DP problem: 5.12/2.25 The TRS P consists of the following rules: 5.12/2.25 5.12/2.25 H(x, x) -> G(x, x) 5.12/2.25 G(d, e) -> A^1 5.12/2.25 A^1 -> H(f(d), f(d)) 5.12/2.25 A^1 -> H(f(e), f(e)) 5.12/2.25 A^1 -> H(f(a), U1(b, e)) 5.12/2.25 A^1 -> H(f(a), U1(d, d)) 5.12/2.25 A^1 -> H(f(a), U1(e, e)) 5.12/2.25 A^1 -> H(a, f(b)) 5.12/2.25 A^1 -> H(U1(d, e), f(b)) 5.12/2.25 A^1 -> H(U1(d, a), U1(b, b)) 5.12/2.25 A^1 -> H(U1(d, a), f(d)) 5.12/2.25 A^1 -> H(U1(d, a), f(e)) 5.12/2.25 A^1 -> H(U1(e, d), f(b)) 5.12/2.25 A^1 -> H(U1(e, a), U1(b, b)) 5.12/2.25 A^1 -> H(U1(e, a), f(d)) 5.12/2.25 A^1 -> H(U1(e, a), f(e)) 5.12/2.25 A^1 -> H(U1(a, d), U1(b, b)) 5.12/2.25 A^1 -> H(U1(a, d), f(d)) 5.12/2.25 A^1 -> H(U1(a, d), f(e)) 5.12/2.25 A^1 -> H(U1(a, e), U1(b, b)) 5.12/2.25 A^1 -> H(U1(a, e), f(d)) 5.12/2.25 A^1 -> H(U1(a, e), f(e)) 5.12/2.25 A^1 -> H(U1(a, a), U1(d, b)) 5.12/2.25 A^1 -> H(U1(a, a), U1(e, b)) 5.12/2.25 A^1 -> H(U1(a, a), U1(b, d)) 5.12/2.25 A^1 -> H(U1(a, a), U1(b, e)) 5.12/2.25 A^1 -> H(U1(a, a), U1(d, d)) 5.12/2.25 A^1 -> H(U1(a, a), U1(e, e)) 5.12/2.25 A^1 -> H(d, f(b)) 5.12/2.25 A^1 -> H(U1(d, d), U1(b, b)) 5.12/2.25 A^1 -> H(U1(d, d), f(d)) 5.12/2.25 A^1 -> H(U1(d, d), f(e)) 5.12/2.25 A^1 -> H(f(d), U1(d, b)) 5.12/2.25 A^1 -> H(f(d), U1(e, b)) 5.12/2.25 A^1 -> H(f(d), U1(b, d)) 5.12/2.25 A^1 -> H(f(d), U1(b, e)) 5.12/2.25 A^1 -> H(f(d), U1(e, e)) 5.12/2.25 A^1 -> H(U1(e, e), U1(b, b)) 5.12/2.25 A^1 -> H(U1(e, e), f(d)) 5.12/2.25 A^1 -> H(U1(e, e), f(e)) 5.12/2.25 A^1 -> H(f(e), U1(d, b)) 5.12/2.25 A^1 -> H(f(e), U1(e, b)) 5.12/2.25 A^1 -> H(f(e), U1(b, d)) 5.12/2.25 A^1 -> H(f(e), U1(b, e)) 5.12/2.25 A^1 -> H(f(e), U1(d, d)) 5.12/2.25 A^1 -> H(f(a), b) 5.12/2.25 A^1 -> H(f(a), U1(d, e)) 5.12/2.25 A^1 -> H(f(a), U1(e, d)) 5.12/2.25 5.12/2.25 The TRS R consists of the following rules: 5.12/2.25 5.12/2.25 a -> d 5.12/2.25 a -> e 5.12/2.25 f(x) -> U1(x, x) 5.12/2.25 b -> d 5.12/2.25 b -> e 5.12/2.25 U1(d, x) -> x 5.12/2.25 5.12/2.25 Q is empty. 5.12/2.25 We have to consider all minimal (P,Q,R)-chains. 5.12/2.25 ---------------------------------------- 5.12/2.25 5.12/2.25 (55) TransformationProof (EQUIVALENT) 5.12/2.25 By narrowing [LPAR04] the rule A^1 -> H(f(a), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.25 5.12/2.25 (A^1 -> H(U1(a, a), U1(b, e)),A^1 -> H(U1(a, a), U1(b, e))) 5.12/2.25 (A^1 -> H(f(d), U1(b, e)),A^1 -> H(f(d), U1(b, e))) 5.12/2.25 (A^1 -> H(f(e), U1(b, e)),A^1 -> H(f(e), U1(b, e))) 5.12/2.25 (A^1 -> H(f(a), U1(d, e)),A^1 -> H(f(a), U1(d, e))) 5.12/2.25 (A^1 -> H(f(a), U1(e, e)),A^1 -> H(f(a), U1(e, e))) 5.12/2.25 5.12/2.25 5.12/2.25 ---------------------------------------- 5.12/2.25 5.12/2.25 (56) 5.12/2.25 Obligation: 5.12/2.25 Q DP problem: 5.12/2.25 The TRS P consists of the following rules: 5.12/2.25 5.12/2.25 H(x, x) -> G(x, x) 5.12/2.25 G(d, e) -> A^1 5.12/2.25 A^1 -> H(f(d), f(d)) 5.12/2.25 A^1 -> H(f(e), f(e)) 5.12/2.25 A^1 -> H(f(a), U1(d, d)) 5.12/2.25 A^1 -> H(f(a), U1(e, e)) 5.12/2.25 A^1 -> H(a, f(b)) 5.12/2.25 A^1 -> H(U1(d, e), f(b)) 5.12/2.25 A^1 -> H(U1(d, a), U1(b, b)) 5.12/2.25 A^1 -> H(U1(d, a), f(d)) 5.12/2.25 A^1 -> H(U1(d, a), f(e)) 5.12/2.25 A^1 -> H(U1(e, d), f(b)) 5.12/2.25 A^1 -> H(U1(e, a), U1(b, b)) 5.12/2.25 A^1 -> H(U1(e, a), f(d)) 5.12/2.25 A^1 -> H(U1(e, a), f(e)) 5.12/2.25 A^1 -> H(U1(a, d), U1(b, b)) 5.12/2.25 A^1 -> H(U1(a, d), f(d)) 5.12/2.25 A^1 -> H(U1(a, d), f(e)) 5.12/2.25 A^1 -> H(U1(a, e), U1(b, b)) 5.12/2.25 A^1 -> H(U1(a, e), f(d)) 5.12/2.25 A^1 -> H(U1(a, e), f(e)) 5.12/2.25 A^1 -> H(U1(a, a), U1(d, b)) 5.12/2.25 A^1 -> H(U1(a, a), U1(e, b)) 5.12/2.25 A^1 -> H(U1(a, a), U1(b, d)) 5.12/2.25 A^1 -> H(U1(a, a), U1(b, e)) 5.12/2.25 A^1 -> H(U1(a, a), U1(d, d)) 5.12/2.25 A^1 -> H(U1(a, a), U1(e, e)) 5.12/2.25 A^1 -> H(d, f(b)) 5.12/2.25 A^1 -> H(U1(d, d), U1(b, b)) 5.12/2.25 A^1 -> H(U1(d, d), f(d)) 5.12/2.25 A^1 -> H(U1(d, d), f(e)) 5.12/2.25 A^1 -> H(f(d), U1(d, b)) 5.12/2.25 A^1 -> H(f(d), U1(e, b)) 5.12/2.25 A^1 -> H(f(d), U1(b, d)) 5.12/2.25 A^1 -> H(f(d), U1(b, e)) 5.12/2.25 A^1 -> H(f(d), U1(e, e)) 5.12/2.25 A^1 -> H(U1(e, e), U1(b, b)) 5.12/2.25 A^1 -> H(U1(e, e), f(d)) 5.12/2.25 A^1 -> H(U1(e, e), f(e)) 5.12/2.25 A^1 -> H(f(e), U1(d, b)) 5.12/2.25 A^1 -> H(f(e), U1(e, b)) 5.12/2.25 A^1 -> H(f(e), U1(b, d)) 5.12/2.25 A^1 -> H(f(e), U1(b, e)) 5.12/2.25 A^1 -> H(f(e), U1(d, d)) 5.12/2.25 A^1 -> H(f(a), b) 5.12/2.25 A^1 -> H(f(a), U1(d, e)) 5.12/2.25 A^1 -> H(f(a), U1(e, d)) 5.12/2.25 5.12/2.25 The TRS R consists of the following rules: 5.12/2.25 5.12/2.25 a -> d 5.12/2.25 a -> e 5.12/2.25 f(x) -> U1(x, x) 5.12/2.25 b -> d 5.12/2.25 b -> e 5.12/2.25 U1(d, x) -> x 5.12/2.25 5.12/2.25 Q is empty. 5.12/2.25 We have to consider all minimal (P,Q,R)-chains. 5.12/2.25 ---------------------------------------- 5.12/2.25 5.12/2.25 (57) TransformationProof (EQUIVALENT) 5.12/2.25 By narrowing [LPAR04] the rule A^1 -> H(f(a), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.25 5.12/2.25 (A^1 -> H(U1(a, a), U1(d, d)),A^1 -> H(U1(a, a), U1(d, d))) 5.12/2.25 (A^1 -> H(f(d), U1(d, d)),A^1 -> H(f(d), U1(d, d))) 5.12/2.25 (A^1 -> H(f(e), U1(d, d)),A^1 -> H(f(e), U1(d, d))) 5.12/2.25 (A^1 -> H(f(a), d),A^1 -> H(f(a), d)) 5.12/2.25 5.12/2.25 5.12/2.25 ---------------------------------------- 5.12/2.25 5.12/2.26 (58) 5.12/2.26 Obligation: 5.12/2.26 Q DP problem: 5.12/2.26 The TRS P consists of the following rules: 5.12/2.26 5.12/2.26 H(x, x) -> G(x, x) 5.12/2.26 G(d, e) -> A^1 5.12/2.26 A^1 -> H(f(d), f(d)) 5.12/2.26 A^1 -> H(f(e), f(e)) 5.12/2.26 A^1 -> H(f(a), U1(e, e)) 5.12/2.26 A^1 -> H(a, f(b)) 5.12/2.26 A^1 -> H(U1(d, e), f(b)) 5.12/2.26 A^1 -> H(U1(d, a), U1(b, b)) 5.12/2.26 A^1 -> H(U1(d, a), f(d)) 5.12/2.26 A^1 -> H(U1(d, a), f(e)) 5.12/2.26 A^1 -> H(U1(e, d), f(b)) 5.12/2.26 A^1 -> H(U1(e, a), U1(b, b)) 5.12/2.26 A^1 -> H(U1(e, a), f(d)) 5.12/2.26 A^1 -> H(U1(e, a), f(e)) 5.12/2.26 A^1 -> H(U1(a, d), U1(b, b)) 5.12/2.26 A^1 -> H(U1(a, d), f(d)) 5.12/2.26 A^1 -> H(U1(a, d), f(e)) 5.12/2.26 A^1 -> H(U1(a, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(a, e), f(d)) 5.12/2.26 A^1 -> H(U1(a, e), f(e)) 5.12/2.26 A^1 -> H(U1(a, a), U1(d, b)) 5.12/2.26 A^1 -> H(U1(a, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(a, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(a, a), U1(b, e)) 5.12/2.26 A^1 -> H(U1(a, a), U1(d, d)) 5.12/2.26 A^1 -> H(U1(a, a), U1(e, e)) 5.12/2.26 A^1 -> H(d, f(b)) 5.12/2.26 A^1 -> H(U1(d, d), U1(b, b)) 5.12/2.26 A^1 -> H(U1(d, d), f(d)) 5.12/2.26 A^1 -> H(U1(d, d), f(e)) 5.12/2.26 A^1 -> H(f(d), U1(d, b)) 5.12/2.26 A^1 -> H(f(d), U1(e, b)) 5.12/2.26 A^1 -> H(f(d), U1(b, d)) 5.12/2.26 A^1 -> H(f(d), U1(b, e)) 5.12/2.26 A^1 -> H(f(d), U1(e, e)) 5.12/2.26 A^1 -> H(U1(e, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(e, e), f(d)) 5.12/2.26 A^1 -> H(U1(e, e), f(e)) 5.12/2.26 A^1 -> H(f(e), U1(d, b)) 5.12/2.26 A^1 -> H(f(e), U1(e, b)) 5.12/2.26 A^1 -> H(f(e), U1(b, d)) 5.12/2.26 A^1 -> H(f(e), U1(b, e)) 5.12/2.26 A^1 -> H(f(e), U1(d, d)) 5.12/2.26 A^1 -> H(f(a), b) 5.12/2.26 A^1 -> H(f(a), U1(d, e)) 5.12/2.26 A^1 -> H(f(a), U1(e, d)) 5.12/2.26 A^1 -> H(f(d), U1(d, d)) 5.12/2.26 A^1 -> H(f(a), d) 5.12/2.26 5.12/2.26 The TRS R consists of the following rules: 5.12/2.26 5.12/2.26 a -> d 5.12/2.26 a -> e 5.12/2.26 f(x) -> U1(x, x) 5.12/2.26 b -> d 5.12/2.26 b -> e 5.12/2.26 U1(d, x) -> x 5.12/2.26 5.12/2.26 Q is empty. 5.12/2.26 We have to consider all minimal (P,Q,R)-chains. 5.12/2.26 ---------------------------------------- 5.12/2.26 5.12/2.26 (59) TransformationProof (EQUIVALENT) 5.12/2.26 By narrowing [LPAR04] the rule A^1 -> H(f(a), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.26 5.12/2.26 (A^1 -> H(U1(a, a), U1(e, e)),A^1 -> H(U1(a, a), U1(e, e))) 5.12/2.26 (A^1 -> H(f(d), U1(e, e)),A^1 -> H(f(d), U1(e, e))) 5.12/2.26 (A^1 -> H(f(e), U1(e, e)),A^1 -> H(f(e), U1(e, e))) 5.12/2.26 5.12/2.26 5.12/2.26 ---------------------------------------- 5.12/2.26 5.12/2.26 (60) 5.12/2.26 Obligation: 5.12/2.26 Q DP problem: 5.12/2.26 The TRS P consists of the following rules: 5.12/2.26 5.12/2.26 H(x, x) -> G(x, x) 5.12/2.26 G(d, e) -> A^1 5.12/2.26 A^1 -> H(f(d), f(d)) 5.12/2.26 A^1 -> H(f(e), f(e)) 5.12/2.26 A^1 -> H(a, f(b)) 5.12/2.26 A^1 -> H(U1(d, e), f(b)) 5.12/2.26 A^1 -> H(U1(d, a), U1(b, b)) 5.12/2.26 A^1 -> H(U1(d, a), f(d)) 5.12/2.26 A^1 -> H(U1(d, a), f(e)) 5.12/2.26 A^1 -> H(U1(e, d), f(b)) 5.12/2.26 A^1 -> H(U1(e, a), U1(b, b)) 5.12/2.26 A^1 -> H(U1(e, a), f(d)) 5.12/2.26 A^1 -> H(U1(e, a), f(e)) 5.12/2.26 A^1 -> H(U1(a, d), U1(b, b)) 5.12/2.26 A^1 -> H(U1(a, d), f(d)) 5.12/2.26 A^1 -> H(U1(a, d), f(e)) 5.12/2.26 A^1 -> H(U1(a, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(a, e), f(d)) 5.12/2.26 A^1 -> H(U1(a, e), f(e)) 5.12/2.26 A^1 -> H(U1(a, a), U1(d, b)) 5.12/2.26 A^1 -> H(U1(a, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(a, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(a, a), U1(b, e)) 5.12/2.26 A^1 -> H(U1(a, a), U1(d, d)) 5.12/2.26 A^1 -> H(U1(a, a), U1(e, e)) 5.12/2.26 A^1 -> H(d, f(b)) 5.12/2.26 A^1 -> H(U1(d, d), U1(b, b)) 5.12/2.26 A^1 -> H(U1(d, d), f(d)) 5.12/2.26 A^1 -> H(U1(d, d), f(e)) 5.12/2.26 A^1 -> H(f(d), U1(d, b)) 5.12/2.26 A^1 -> H(f(d), U1(e, b)) 5.12/2.26 A^1 -> H(f(d), U1(b, d)) 5.12/2.26 A^1 -> H(f(d), U1(b, e)) 5.12/2.26 A^1 -> H(f(d), U1(e, e)) 5.12/2.26 A^1 -> H(U1(e, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(e, e), f(d)) 5.12/2.26 A^1 -> H(U1(e, e), f(e)) 5.12/2.26 A^1 -> H(f(e), U1(d, b)) 5.12/2.26 A^1 -> H(f(e), U1(e, b)) 5.12/2.26 A^1 -> H(f(e), U1(b, d)) 5.12/2.26 A^1 -> H(f(e), U1(b, e)) 5.12/2.26 A^1 -> H(f(e), U1(d, d)) 5.12/2.26 A^1 -> H(f(a), b) 5.12/2.26 A^1 -> H(f(a), U1(d, e)) 5.12/2.26 A^1 -> H(f(a), U1(e, d)) 5.12/2.26 A^1 -> H(f(d), U1(d, d)) 5.12/2.26 A^1 -> H(f(a), d) 5.12/2.26 A^1 -> H(f(e), U1(e, e)) 5.12/2.26 5.12/2.26 The TRS R consists of the following rules: 5.12/2.26 5.12/2.26 a -> d 5.12/2.26 a -> e 5.12/2.26 f(x) -> U1(x, x) 5.12/2.26 b -> d 5.12/2.26 b -> e 5.12/2.26 U1(d, x) -> x 5.12/2.26 5.12/2.26 Q is empty. 5.12/2.26 We have to consider all minimal (P,Q,R)-chains. 5.12/2.26 ---------------------------------------- 5.12/2.26 5.12/2.26 (61) TransformationProof (EQUIVALENT) 5.12/2.26 By narrowing [LPAR04] the rule A^1 -> H(a, f(b)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.26 5.12/2.26 (A^1 -> H(d, f(b)),A^1 -> H(d, f(b))) 5.12/2.26 (A^1 -> H(e, f(b)),A^1 -> H(e, f(b))) 5.12/2.26 (A^1 -> H(a, U1(b, b)),A^1 -> H(a, U1(b, b))) 5.12/2.26 (A^1 -> H(a, f(d)),A^1 -> H(a, f(d))) 5.12/2.26 (A^1 -> H(a, f(e)),A^1 -> H(a, f(e))) 5.12/2.26 5.12/2.26 5.12/2.26 ---------------------------------------- 5.12/2.26 5.12/2.26 (62) 5.12/2.26 Obligation: 5.12/2.26 Q DP problem: 5.12/2.26 The TRS P consists of the following rules: 5.12/2.26 5.12/2.26 H(x, x) -> G(x, x) 5.12/2.26 G(d, e) -> A^1 5.12/2.26 A^1 -> H(f(d), f(d)) 5.12/2.26 A^1 -> H(f(e), f(e)) 5.12/2.26 A^1 -> H(U1(d, e), f(b)) 5.12/2.26 A^1 -> H(U1(d, a), U1(b, b)) 5.12/2.26 A^1 -> H(U1(d, a), f(d)) 5.12/2.26 A^1 -> H(U1(d, a), f(e)) 5.12/2.26 A^1 -> H(U1(e, d), f(b)) 5.12/2.26 A^1 -> H(U1(e, a), U1(b, b)) 5.12/2.26 A^1 -> H(U1(e, a), f(d)) 5.12/2.26 A^1 -> H(U1(e, a), f(e)) 5.12/2.26 A^1 -> H(U1(a, d), U1(b, b)) 5.12/2.26 A^1 -> H(U1(a, d), f(d)) 5.12/2.26 A^1 -> H(U1(a, d), f(e)) 5.12/2.26 A^1 -> H(U1(a, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(a, e), f(d)) 5.12/2.26 A^1 -> H(U1(a, e), f(e)) 5.12/2.26 A^1 -> H(U1(a, a), U1(d, b)) 5.12/2.26 A^1 -> H(U1(a, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(a, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(a, a), U1(b, e)) 5.12/2.26 A^1 -> H(U1(a, a), U1(d, d)) 5.12/2.26 A^1 -> H(U1(a, a), U1(e, e)) 5.12/2.26 A^1 -> H(d, f(b)) 5.12/2.26 A^1 -> H(U1(d, d), U1(b, b)) 5.12/2.26 A^1 -> H(U1(d, d), f(d)) 5.12/2.26 A^1 -> H(U1(d, d), f(e)) 5.12/2.26 A^1 -> H(f(d), U1(d, b)) 5.12/2.26 A^1 -> H(f(d), U1(e, b)) 5.12/2.26 A^1 -> H(f(d), U1(b, d)) 5.12/2.26 A^1 -> H(f(d), U1(b, e)) 5.12/2.26 A^1 -> H(f(d), U1(e, e)) 5.12/2.26 A^1 -> H(U1(e, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(e, e), f(d)) 5.12/2.26 A^1 -> H(U1(e, e), f(e)) 5.12/2.26 A^1 -> H(f(e), U1(d, b)) 5.12/2.26 A^1 -> H(f(e), U1(e, b)) 5.12/2.26 A^1 -> H(f(e), U1(b, d)) 5.12/2.26 A^1 -> H(f(e), U1(b, e)) 5.12/2.26 A^1 -> H(f(e), U1(d, d)) 5.12/2.26 A^1 -> H(f(a), b) 5.12/2.26 A^1 -> H(f(a), U1(d, e)) 5.12/2.26 A^1 -> H(f(a), U1(e, d)) 5.12/2.26 A^1 -> H(f(d), U1(d, d)) 5.12/2.26 A^1 -> H(f(a), d) 5.12/2.26 A^1 -> H(f(e), U1(e, e)) 5.12/2.26 A^1 -> H(e, f(b)) 5.12/2.26 A^1 -> H(a, U1(b, b)) 5.12/2.26 A^1 -> H(a, f(d)) 5.12/2.26 A^1 -> H(a, f(e)) 5.12/2.26 5.12/2.26 The TRS R consists of the following rules: 5.12/2.26 5.12/2.26 a -> d 5.12/2.26 a -> e 5.12/2.26 f(x) -> U1(x, x) 5.12/2.26 b -> d 5.12/2.26 b -> e 5.12/2.26 U1(d, x) -> x 5.12/2.26 5.12/2.26 Q is empty. 5.12/2.26 We have to consider all minimal (P,Q,R)-chains. 5.12/2.26 ---------------------------------------- 5.12/2.26 5.12/2.26 (63) TransformationProof (EQUIVALENT) 5.12/2.26 By narrowing [LPAR04] the rule A^1 -> H(U1(d, e), f(b)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.26 5.12/2.26 (A^1 -> H(e, f(b)),A^1 -> H(e, f(b))) 5.12/2.26 (A^1 -> H(U1(d, e), U1(b, b)),A^1 -> H(U1(d, e), U1(b, b))) 5.12/2.26 (A^1 -> H(U1(d, e), f(d)),A^1 -> H(U1(d, e), f(d))) 5.12/2.26 (A^1 -> H(U1(d, e), f(e)),A^1 -> H(U1(d, e), f(e))) 5.12/2.26 5.12/2.26 5.12/2.26 ---------------------------------------- 5.12/2.26 5.12/2.26 (64) 5.12/2.26 Obligation: 5.12/2.26 Q DP problem: 5.12/2.26 The TRS P consists of the following rules: 5.12/2.26 5.12/2.26 H(x, x) -> G(x, x) 5.12/2.26 G(d, e) -> A^1 5.12/2.26 A^1 -> H(f(d), f(d)) 5.12/2.26 A^1 -> H(f(e), f(e)) 5.12/2.26 A^1 -> H(U1(d, a), U1(b, b)) 5.12/2.26 A^1 -> H(U1(d, a), f(d)) 5.12/2.26 A^1 -> H(U1(d, a), f(e)) 5.12/2.26 A^1 -> H(U1(e, d), f(b)) 5.12/2.26 A^1 -> H(U1(e, a), U1(b, b)) 5.12/2.26 A^1 -> H(U1(e, a), f(d)) 5.12/2.26 A^1 -> H(U1(e, a), f(e)) 5.12/2.26 A^1 -> H(U1(a, d), U1(b, b)) 5.12/2.26 A^1 -> H(U1(a, d), f(d)) 5.12/2.26 A^1 -> H(U1(a, d), f(e)) 5.12/2.26 A^1 -> H(U1(a, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(a, e), f(d)) 5.12/2.26 A^1 -> H(U1(a, e), f(e)) 5.12/2.26 A^1 -> H(U1(a, a), U1(d, b)) 5.12/2.26 A^1 -> H(U1(a, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(a, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(a, a), U1(b, e)) 5.12/2.26 A^1 -> H(U1(a, a), U1(d, d)) 5.12/2.26 A^1 -> H(U1(a, a), U1(e, e)) 5.12/2.26 A^1 -> H(d, f(b)) 5.12/2.26 A^1 -> H(U1(d, d), U1(b, b)) 5.12/2.26 A^1 -> H(U1(d, d), f(d)) 5.12/2.26 A^1 -> H(U1(d, d), f(e)) 5.12/2.26 A^1 -> H(f(d), U1(d, b)) 5.12/2.26 A^1 -> H(f(d), U1(e, b)) 5.12/2.26 A^1 -> H(f(d), U1(b, d)) 5.12/2.26 A^1 -> H(f(d), U1(b, e)) 5.12/2.26 A^1 -> H(f(d), U1(e, e)) 5.12/2.26 A^1 -> H(U1(e, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(e, e), f(d)) 5.12/2.26 A^1 -> H(U1(e, e), f(e)) 5.12/2.26 A^1 -> H(f(e), U1(d, b)) 5.12/2.26 A^1 -> H(f(e), U1(e, b)) 5.12/2.26 A^1 -> H(f(e), U1(b, d)) 5.12/2.26 A^1 -> H(f(e), U1(b, e)) 5.12/2.26 A^1 -> H(f(e), U1(d, d)) 5.12/2.26 A^1 -> H(f(a), b) 5.12/2.26 A^1 -> H(f(a), U1(d, e)) 5.12/2.26 A^1 -> H(f(a), U1(e, d)) 5.12/2.26 A^1 -> H(f(d), U1(d, d)) 5.12/2.26 A^1 -> H(f(a), d) 5.12/2.26 A^1 -> H(f(e), U1(e, e)) 5.12/2.26 A^1 -> H(e, f(b)) 5.12/2.26 A^1 -> H(a, U1(b, b)) 5.12/2.26 A^1 -> H(a, f(d)) 5.12/2.26 A^1 -> H(a, f(e)) 5.12/2.26 A^1 -> H(U1(d, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(d, e), f(d)) 5.12/2.26 A^1 -> H(U1(d, e), f(e)) 5.12/2.26 5.12/2.26 The TRS R consists of the following rules: 5.12/2.26 5.12/2.26 a -> d 5.12/2.26 a -> e 5.12/2.26 f(x) -> U1(x, x) 5.12/2.26 b -> d 5.12/2.26 b -> e 5.12/2.26 U1(d, x) -> x 5.12/2.26 5.12/2.26 Q is empty. 5.12/2.26 We have to consider all minimal (P,Q,R)-chains. 5.12/2.26 ---------------------------------------- 5.12/2.26 5.12/2.26 (65) TransformationProof (EQUIVALENT) 5.12/2.26 By narrowing [LPAR04] the rule A^1 -> H(U1(d, a), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.26 5.12/2.26 (A^1 -> H(a, U1(b, b)),A^1 -> H(a, U1(b, b))) 5.12/2.26 (A^1 -> H(U1(d, d), U1(b, b)),A^1 -> H(U1(d, d), U1(b, b))) 5.12/2.26 (A^1 -> H(U1(d, e), U1(b, b)),A^1 -> H(U1(d, e), U1(b, b))) 5.12/2.26 (A^1 -> H(U1(d, a), U1(d, b)),A^1 -> H(U1(d, a), U1(d, b))) 5.12/2.26 (A^1 -> H(U1(d, a), U1(e, b)),A^1 -> H(U1(d, a), U1(e, b))) 5.12/2.26 (A^1 -> H(U1(d, a), U1(b, d)),A^1 -> H(U1(d, a), U1(b, d))) 5.12/2.26 (A^1 -> H(U1(d, a), U1(b, e)),A^1 -> H(U1(d, a), U1(b, e))) 5.12/2.26 5.12/2.26 5.12/2.26 ---------------------------------------- 5.12/2.26 5.12/2.26 (66) 5.12/2.26 Obligation: 5.12/2.26 Q DP problem: 5.12/2.26 The TRS P consists of the following rules: 5.12/2.26 5.12/2.26 H(x, x) -> G(x, x) 5.12/2.26 G(d, e) -> A^1 5.12/2.26 A^1 -> H(f(d), f(d)) 5.12/2.26 A^1 -> H(f(e), f(e)) 5.12/2.26 A^1 -> H(U1(d, a), f(d)) 5.12/2.26 A^1 -> H(U1(d, a), f(e)) 5.12/2.26 A^1 -> H(U1(e, d), f(b)) 5.12/2.26 A^1 -> H(U1(e, a), U1(b, b)) 5.12/2.26 A^1 -> H(U1(e, a), f(d)) 5.12/2.26 A^1 -> H(U1(e, a), f(e)) 5.12/2.26 A^1 -> H(U1(a, d), U1(b, b)) 5.12/2.26 A^1 -> H(U1(a, d), f(d)) 5.12/2.26 A^1 -> H(U1(a, d), f(e)) 5.12/2.26 A^1 -> H(U1(a, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(a, e), f(d)) 5.12/2.26 A^1 -> H(U1(a, e), f(e)) 5.12/2.26 A^1 -> H(U1(a, a), U1(d, b)) 5.12/2.26 A^1 -> H(U1(a, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(a, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(a, a), U1(b, e)) 5.12/2.26 A^1 -> H(U1(a, a), U1(d, d)) 5.12/2.26 A^1 -> H(U1(a, a), U1(e, e)) 5.12/2.26 A^1 -> H(d, f(b)) 5.12/2.26 A^1 -> H(U1(d, d), U1(b, b)) 5.12/2.26 A^1 -> H(U1(d, d), f(d)) 5.12/2.26 A^1 -> H(U1(d, d), f(e)) 5.12/2.26 A^1 -> H(f(d), U1(d, b)) 5.12/2.26 A^1 -> H(f(d), U1(e, b)) 5.12/2.26 A^1 -> H(f(d), U1(b, d)) 5.12/2.26 A^1 -> H(f(d), U1(b, e)) 5.12/2.26 A^1 -> H(f(d), U1(e, e)) 5.12/2.26 A^1 -> H(U1(e, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(e, e), f(d)) 5.12/2.26 A^1 -> H(U1(e, e), f(e)) 5.12/2.26 A^1 -> H(f(e), U1(d, b)) 5.12/2.26 A^1 -> H(f(e), U1(e, b)) 5.12/2.26 A^1 -> H(f(e), U1(b, d)) 5.12/2.26 A^1 -> H(f(e), U1(b, e)) 5.12/2.26 A^1 -> H(f(e), U1(d, d)) 5.12/2.26 A^1 -> H(f(a), b) 5.12/2.26 A^1 -> H(f(a), U1(d, e)) 5.12/2.26 A^1 -> H(f(a), U1(e, d)) 5.12/2.26 A^1 -> H(f(d), U1(d, d)) 5.12/2.26 A^1 -> H(f(a), d) 5.12/2.26 A^1 -> H(f(e), U1(e, e)) 5.12/2.26 A^1 -> H(e, f(b)) 5.12/2.26 A^1 -> H(a, U1(b, b)) 5.12/2.26 A^1 -> H(a, f(d)) 5.12/2.26 A^1 -> H(a, f(e)) 5.12/2.26 A^1 -> H(U1(d, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(d, e), f(d)) 5.12/2.26 A^1 -> H(U1(d, e), f(e)) 5.12/2.26 A^1 -> H(U1(d, a), U1(d, b)) 5.12/2.26 A^1 -> H(U1(d, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(d, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(d, a), U1(b, e)) 5.12/2.26 5.12/2.26 The TRS R consists of the following rules: 5.12/2.26 5.12/2.26 a -> d 5.12/2.26 a -> e 5.12/2.26 f(x) -> U1(x, x) 5.12/2.26 b -> d 5.12/2.26 b -> e 5.12/2.26 U1(d, x) -> x 5.12/2.26 5.12/2.26 Q is empty. 5.12/2.26 We have to consider all minimal (P,Q,R)-chains. 5.12/2.26 ---------------------------------------- 5.12/2.26 5.12/2.26 (67) TransformationProof (EQUIVALENT) 5.12/2.26 By narrowing [LPAR04] the rule A^1 -> H(U1(d, a), f(d)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.26 5.12/2.26 (A^1 -> H(a, f(d)),A^1 -> H(a, f(d))) 5.12/2.26 (A^1 -> H(U1(d, d), f(d)),A^1 -> H(U1(d, d), f(d))) 5.12/2.26 (A^1 -> H(U1(d, e), f(d)),A^1 -> H(U1(d, e), f(d))) 5.12/2.26 (A^1 -> H(U1(d, a), U1(d, d)),A^1 -> H(U1(d, a), U1(d, d))) 5.12/2.26 5.12/2.26 5.12/2.26 ---------------------------------------- 5.12/2.26 5.12/2.26 (68) 5.12/2.26 Obligation: 5.12/2.26 Q DP problem: 5.12/2.26 The TRS P consists of the following rules: 5.12/2.26 5.12/2.26 H(x, x) -> G(x, x) 5.12/2.26 G(d, e) -> A^1 5.12/2.26 A^1 -> H(f(d), f(d)) 5.12/2.26 A^1 -> H(f(e), f(e)) 5.12/2.26 A^1 -> H(U1(d, a), f(e)) 5.12/2.26 A^1 -> H(U1(e, d), f(b)) 5.12/2.26 A^1 -> H(U1(e, a), U1(b, b)) 5.12/2.26 A^1 -> H(U1(e, a), f(d)) 5.12/2.26 A^1 -> H(U1(e, a), f(e)) 5.12/2.26 A^1 -> H(U1(a, d), U1(b, b)) 5.12/2.26 A^1 -> H(U1(a, d), f(d)) 5.12/2.26 A^1 -> H(U1(a, d), f(e)) 5.12/2.26 A^1 -> H(U1(a, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(a, e), f(d)) 5.12/2.26 A^1 -> H(U1(a, e), f(e)) 5.12/2.26 A^1 -> H(U1(a, a), U1(d, b)) 5.12/2.26 A^1 -> H(U1(a, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(a, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(a, a), U1(b, e)) 5.12/2.26 A^1 -> H(U1(a, a), U1(d, d)) 5.12/2.26 A^1 -> H(U1(a, a), U1(e, e)) 5.12/2.26 A^1 -> H(d, f(b)) 5.12/2.26 A^1 -> H(U1(d, d), U1(b, b)) 5.12/2.26 A^1 -> H(U1(d, d), f(d)) 5.12/2.26 A^1 -> H(U1(d, d), f(e)) 5.12/2.26 A^1 -> H(f(d), U1(d, b)) 5.12/2.26 A^1 -> H(f(d), U1(e, b)) 5.12/2.26 A^1 -> H(f(d), U1(b, d)) 5.12/2.26 A^1 -> H(f(d), U1(b, e)) 5.12/2.26 A^1 -> H(f(d), U1(e, e)) 5.12/2.26 A^1 -> H(U1(e, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(e, e), f(d)) 5.12/2.26 A^1 -> H(U1(e, e), f(e)) 5.12/2.26 A^1 -> H(f(e), U1(d, b)) 5.12/2.26 A^1 -> H(f(e), U1(e, b)) 5.12/2.26 A^1 -> H(f(e), U1(b, d)) 5.12/2.26 A^1 -> H(f(e), U1(b, e)) 5.12/2.26 A^1 -> H(f(e), U1(d, d)) 5.12/2.26 A^1 -> H(f(a), b) 5.12/2.26 A^1 -> H(f(a), U1(d, e)) 5.12/2.26 A^1 -> H(f(a), U1(e, d)) 5.12/2.26 A^1 -> H(f(d), U1(d, d)) 5.12/2.26 A^1 -> H(f(a), d) 5.12/2.26 A^1 -> H(f(e), U1(e, e)) 5.12/2.26 A^1 -> H(e, f(b)) 5.12/2.26 A^1 -> H(a, U1(b, b)) 5.12/2.26 A^1 -> H(a, f(d)) 5.12/2.26 A^1 -> H(a, f(e)) 5.12/2.26 A^1 -> H(U1(d, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(d, e), f(d)) 5.12/2.26 A^1 -> H(U1(d, e), f(e)) 5.12/2.26 A^1 -> H(U1(d, a), U1(d, b)) 5.12/2.26 A^1 -> H(U1(d, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(d, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(d, a), U1(b, e)) 5.12/2.26 A^1 -> H(U1(d, a), U1(d, d)) 5.12/2.26 5.12/2.26 The TRS R consists of the following rules: 5.12/2.26 5.12/2.26 a -> d 5.12/2.26 a -> e 5.12/2.26 f(x) -> U1(x, x) 5.12/2.26 b -> d 5.12/2.26 b -> e 5.12/2.26 U1(d, x) -> x 5.12/2.26 5.12/2.26 Q is empty. 5.12/2.26 We have to consider all minimal (P,Q,R)-chains. 5.12/2.26 ---------------------------------------- 5.12/2.26 5.12/2.26 (69) TransformationProof (EQUIVALENT) 5.12/2.26 By narrowing [LPAR04] the rule A^1 -> H(U1(d, a), f(e)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.26 5.12/2.26 (A^1 -> H(a, f(e)),A^1 -> H(a, f(e))) 5.12/2.26 (A^1 -> H(U1(d, d), f(e)),A^1 -> H(U1(d, d), f(e))) 5.12/2.26 (A^1 -> H(U1(d, e), f(e)),A^1 -> H(U1(d, e), f(e))) 5.12/2.26 (A^1 -> H(U1(d, a), U1(e, e)),A^1 -> H(U1(d, a), U1(e, e))) 5.12/2.26 5.12/2.26 5.12/2.26 ---------------------------------------- 5.12/2.26 5.12/2.26 (70) 5.12/2.26 Obligation: 5.12/2.26 Q DP problem: 5.12/2.26 The TRS P consists of the following rules: 5.12/2.26 5.12/2.26 H(x, x) -> G(x, x) 5.12/2.26 G(d, e) -> A^1 5.12/2.26 A^1 -> H(f(d), f(d)) 5.12/2.26 A^1 -> H(f(e), f(e)) 5.12/2.26 A^1 -> H(U1(e, d), f(b)) 5.12/2.26 A^1 -> H(U1(e, a), U1(b, b)) 5.12/2.26 A^1 -> H(U1(e, a), f(d)) 5.12/2.26 A^1 -> H(U1(e, a), f(e)) 5.12/2.26 A^1 -> H(U1(a, d), U1(b, b)) 5.12/2.26 A^1 -> H(U1(a, d), f(d)) 5.12/2.26 A^1 -> H(U1(a, d), f(e)) 5.12/2.26 A^1 -> H(U1(a, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(a, e), f(d)) 5.12/2.26 A^1 -> H(U1(a, e), f(e)) 5.12/2.26 A^1 -> H(U1(a, a), U1(d, b)) 5.12/2.26 A^1 -> H(U1(a, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(a, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(a, a), U1(b, e)) 5.12/2.26 A^1 -> H(U1(a, a), U1(d, d)) 5.12/2.26 A^1 -> H(U1(a, a), U1(e, e)) 5.12/2.26 A^1 -> H(d, f(b)) 5.12/2.26 A^1 -> H(U1(d, d), U1(b, b)) 5.12/2.26 A^1 -> H(U1(d, d), f(d)) 5.12/2.26 A^1 -> H(U1(d, d), f(e)) 5.12/2.26 A^1 -> H(f(d), U1(d, b)) 5.12/2.26 A^1 -> H(f(d), U1(e, b)) 5.12/2.26 A^1 -> H(f(d), U1(b, d)) 5.12/2.26 A^1 -> H(f(d), U1(b, e)) 5.12/2.26 A^1 -> H(f(d), U1(e, e)) 5.12/2.26 A^1 -> H(U1(e, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(e, e), f(d)) 5.12/2.26 A^1 -> H(U1(e, e), f(e)) 5.12/2.26 A^1 -> H(f(e), U1(d, b)) 5.12/2.26 A^1 -> H(f(e), U1(e, b)) 5.12/2.26 A^1 -> H(f(e), U1(b, d)) 5.12/2.26 A^1 -> H(f(e), U1(b, e)) 5.12/2.26 A^1 -> H(f(e), U1(d, d)) 5.12/2.26 A^1 -> H(f(a), b) 5.12/2.26 A^1 -> H(f(a), U1(d, e)) 5.12/2.26 A^1 -> H(f(a), U1(e, d)) 5.12/2.26 A^1 -> H(f(d), U1(d, d)) 5.12/2.26 A^1 -> H(f(a), d) 5.12/2.26 A^1 -> H(f(e), U1(e, e)) 5.12/2.26 A^1 -> H(e, f(b)) 5.12/2.26 A^1 -> H(a, U1(b, b)) 5.12/2.26 A^1 -> H(a, f(d)) 5.12/2.26 A^1 -> H(a, f(e)) 5.12/2.26 A^1 -> H(U1(d, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(d, e), f(d)) 5.12/2.26 A^1 -> H(U1(d, e), f(e)) 5.12/2.26 A^1 -> H(U1(d, a), U1(d, b)) 5.12/2.26 A^1 -> H(U1(d, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(d, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(d, a), U1(b, e)) 5.12/2.26 A^1 -> H(U1(d, a), U1(d, d)) 5.12/2.26 A^1 -> H(U1(d, a), U1(e, e)) 5.12/2.26 5.12/2.26 The TRS R consists of the following rules: 5.12/2.26 5.12/2.26 a -> d 5.12/2.26 a -> e 5.12/2.26 f(x) -> U1(x, x) 5.12/2.26 b -> d 5.12/2.26 b -> e 5.12/2.26 U1(d, x) -> x 5.12/2.26 5.12/2.26 Q is empty. 5.12/2.26 We have to consider all minimal (P,Q,R)-chains. 5.12/2.26 ---------------------------------------- 5.12/2.26 5.12/2.26 (71) TransformationProof (EQUIVALENT) 5.12/2.26 By narrowing [LPAR04] the rule A^1 -> H(U1(e, d), f(b)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.26 5.12/2.26 (A^1 -> H(U1(e, d), U1(b, b)),A^1 -> H(U1(e, d), U1(b, b))) 5.12/2.26 (A^1 -> H(U1(e, d), f(d)),A^1 -> H(U1(e, d), f(d))) 5.12/2.26 (A^1 -> H(U1(e, d), f(e)),A^1 -> H(U1(e, d), f(e))) 5.12/2.26 5.12/2.26 5.12/2.26 ---------------------------------------- 5.12/2.26 5.12/2.26 (72) 5.12/2.26 Obligation: 5.12/2.26 Q DP problem: 5.12/2.26 The TRS P consists of the following rules: 5.12/2.26 5.12/2.26 H(x, x) -> G(x, x) 5.12/2.26 G(d, e) -> A^1 5.12/2.26 A^1 -> H(f(d), f(d)) 5.12/2.26 A^1 -> H(f(e), f(e)) 5.12/2.26 A^1 -> H(U1(e, a), U1(b, b)) 5.12/2.26 A^1 -> H(U1(e, a), f(d)) 5.12/2.26 A^1 -> H(U1(e, a), f(e)) 5.12/2.26 A^1 -> H(U1(a, d), U1(b, b)) 5.12/2.26 A^1 -> H(U1(a, d), f(d)) 5.12/2.26 A^1 -> H(U1(a, d), f(e)) 5.12/2.26 A^1 -> H(U1(a, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(a, e), f(d)) 5.12/2.26 A^1 -> H(U1(a, e), f(e)) 5.12/2.26 A^1 -> H(U1(a, a), U1(d, b)) 5.12/2.26 A^1 -> H(U1(a, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(a, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(a, a), U1(b, e)) 5.12/2.26 A^1 -> H(U1(a, a), U1(d, d)) 5.12/2.26 A^1 -> H(U1(a, a), U1(e, e)) 5.12/2.26 A^1 -> H(d, f(b)) 5.12/2.26 A^1 -> H(U1(d, d), U1(b, b)) 5.12/2.26 A^1 -> H(U1(d, d), f(d)) 5.12/2.26 A^1 -> H(U1(d, d), f(e)) 5.12/2.26 A^1 -> H(f(d), U1(d, b)) 5.12/2.26 A^1 -> H(f(d), U1(e, b)) 5.12/2.26 A^1 -> H(f(d), U1(b, d)) 5.12/2.26 A^1 -> H(f(d), U1(b, e)) 5.12/2.26 A^1 -> H(f(d), U1(e, e)) 5.12/2.26 A^1 -> H(U1(e, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(e, e), f(d)) 5.12/2.26 A^1 -> H(U1(e, e), f(e)) 5.12/2.26 A^1 -> H(f(e), U1(d, b)) 5.12/2.26 A^1 -> H(f(e), U1(e, b)) 5.12/2.26 A^1 -> H(f(e), U1(b, d)) 5.12/2.26 A^1 -> H(f(e), U1(b, e)) 5.12/2.26 A^1 -> H(f(e), U1(d, d)) 5.12/2.26 A^1 -> H(f(a), b) 5.12/2.26 A^1 -> H(f(a), U1(d, e)) 5.12/2.26 A^1 -> H(f(a), U1(e, d)) 5.12/2.26 A^1 -> H(f(d), U1(d, d)) 5.12/2.26 A^1 -> H(f(a), d) 5.12/2.26 A^1 -> H(f(e), U1(e, e)) 5.12/2.26 A^1 -> H(e, f(b)) 5.12/2.26 A^1 -> H(a, U1(b, b)) 5.12/2.26 A^1 -> H(a, f(d)) 5.12/2.26 A^1 -> H(a, f(e)) 5.12/2.26 A^1 -> H(U1(d, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(d, e), f(d)) 5.12/2.26 A^1 -> H(U1(d, e), f(e)) 5.12/2.26 A^1 -> H(U1(d, a), U1(d, b)) 5.12/2.26 A^1 -> H(U1(d, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(d, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(d, a), U1(b, e)) 5.12/2.26 A^1 -> H(U1(d, a), U1(d, d)) 5.12/2.26 A^1 -> H(U1(d, a), U1(e, e)) 5.12/2.26 A^1 -> H(U1(e, d), U1(b, b)) 5.12/2.26 A^1 -> H(U1(e, d), f(d)) 5.12/2.26 A^1 -> H(U1(e, d), f(e)) 5.12/2.26 5.12/2.26 The TRS R consists of the following rules: 5.12/2.26 5.12/2.26 a -> d 5.12/2.26 a -> e 5.12/2.26 f(x) -> U1(x, x) 5.12/2.26 b -> d 5.12/2.26 b -> e 5.12/2.26 U1(d, x) -> x 5.12/2.26 5.12/2.26 Q is empty. 5.12/2.26 We have to consider all minimal (P,Q,R)-chains. 5.12/2.26 ---------------------------------------- 5.12/2.26 5.12/2.26 (73) TransformationProof (EQUIVALENT) 5.12/2.26 By narrowing [LPAR04] the rule A^1 -> H(U1(e, a), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.26 5.12/2.26 (A^1 -> H(U1(e, d), U1(b, b)),A^1 -> H(U1(e, d), U1(b, b))) 5.12/2.26 (A^1 -> H(U1(e, e), U1(b, b)),A^1 -> H(U1(e, e), U1(b, b))) 5.12/2.26 (A^1 -> H(U1(e, a), U1(d, b)),A^1 -> H(U1(e, a), U1(d, b))) 5.12/2.26 (A^1 -> H(U1(e, a), U1(e, b)),A^1 -> H(U1(e, a), U1(e, b))) 5.12/2.26 (A^1 -> H(U1(e, a), U1(b, d)),A^1 -> H(U1(e, a), U1(b, d))) 5.12/2.26 (A^1 -> H(U1(e, a), U1(b, e)),A^1 -> H(U1(e, a), U1(b, e))) 5.12/2.26 5.12/2.26 5.12/2.26 ---------------------------------------- 5.12/2.26 5.12/2.26 (74) 5.12/2.26 Obligation: 5.12/2.26 Q DP problem: 5.12/2.26 The TRS P consists of the following rules: 5.12/2.26 5.12/2.26 H(x, x) -> G(x, x) 5.12/2.26 G(d, e) -> A^1 5.12/2.26 A^1 -> H(f(d), f(d)) 5.12/2.26 A^1 -> H(f(e), f(e)) 5.12/2.26 A^1 -> H(U1(e, a), f(d)) 5.12/2.26 A^1 -> H(U1(e, a), f(e)) 5.12/2.26 A^1 -> H(U1(a, d), U1(b, b)) 5.12/2.26 A^1 -> H(U1(a, d), f(d)) 5.12/2.26 A^1 -> H(U1(a, d), f(e)) 5.12/2.26 A^1 -> H(U1(a, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(a, e), f(d)) 5.12/2.26 A^1 -> H(U1(a, e), f(e)) 5.12/2.26 A^1 -> H(U1(a, a), U1(d, b)) 5.12/2.26 A^1 -> H(U1(a, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(a, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(a, a), U1(b, e)) 5.12/2.26 A^1 -> H(U1(a, a), U1(d, d)) 5.12/2.26 A^1 -> H(U1(a, a), U1(e, e)) 5.12/2.26 A^1 -> H(d, f(b)) 5.12/2.26 A^1 -> H(U1(d, d), U1(b, b)) 5.12/2.26 A^1 -> H(U1(d, d), f(d)) 5.12/2.26 A^1 -> H(U1(d, d), f(e)) 5.12/2.26 A^1 -> H(f(d), U1(d, b)) 5.12/2.26 A^1 -> H(f(d), U1(e, b)) 5.12/2.26 A^1 -> H(f(d), U1(b, d)) 5.12/2.26 A^1 -> H(f(d), U1(b, e)) 5.12/2.26 A^1 -> H(f(d), U1(e, e)) 5.12/2.26 A^1 -> H(U1(e, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(e, e), f(d)) 5.12/2.26 A^1 -> H(U1(e, e), f(e)) 5.12/2.26 A^1 -> H(f(e), U1(d, b)) 5.12/2.26 A^1 -> H(f(e), U1(e, b)) 5.12/2.26 A^1 -> H(f(e), U1(b, d)) 5.12/2.26 A^1 -> H(f(e), U1(b, e)) 5.12/2.26 A^1 -> H(f(e), U1(d, d)) 5.12/2.26 A^1 -> H(f(a), b) 5.12/2.26 A^1 -> H(f(a), U1(d, e)) 5.12/2.26 A^1 -> H(f(a), U1(e, d)) 5.12/2.26 A^1 -> H(f(d), U1(d, d)) 5.12/2.26 A^1 -> H(f(a), d) 5.12/2.26 A^1 -> H(f(e), U1(e, e)) 5.12/2.26 A^1 -> H(e, f(b)) 5.12/2.26 A^1 -> H(a, U1(b, b)) 5.12/2.26 A^1 -> H(a, f(d)) 5.12/2.26 A^1 -> H(a, f(e)) 5.12/2.26 A^1 -> H(U1(d, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(d, e), f(d)) 5.12/2.26 A^1 -> H(U1(d, e), f(e)) 5.12/2.26 A^1 -> H(U1(d, a), U1(d, b)) 5.12/2.26 A^1 -> H(U1(d, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(d, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(d, a), U1(b, e)) 5.12/2.26 A^1 -> H(U1(d, a), U1(d, d)) 5.12/2.26 A^1 -> H(U1(d, a), U1(e, e)) 5.12/2.26 A^1 -> H(U1(e, d), U1(b, b)) 5.12/2.26 A^1 -> H(U1(e, d), f(d)) 5.12/2.26 A^1 -> H(U1(e, d), f(e)) 5.12/2.26 A^1 -> H(U1(e, a), U1(d, b)) 5.12/2.26 A^1 -> H(U1(e, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(e, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(e, a), U1(b, e)) 5.12/2.26 5.12/2.26 The TRS R consists of the following rules: 5.12/2.26 5.12/2.26 a -> d 5.12/2.26 a -> e 5.12/2.26 f(x) -> U1(x, x) 5.12/2.26 b -> d 5.12/2.26 b -> e 5.12/2.26 U1(d, x) -> x 5.12/2.26 5.12/2.26 Q is empty. 5.12/2.26 We have to consider all minimal (P,Q,R)-chains. 5.12/2.26 ---------------------------------------- 5.12/2.26 5.12/2.26 (75) TransformationProof (EQUIVALENT) 5.12/2.26 By narrowing [LPAR04] the rule A^1 -> H(U1(e, a), f(d)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.26 5.12/2.26 (A^1 -> H(U1(e, d), f(d)),A^1 -> H(U1(e, d), f(d))) 5.12/2.26 (A^1 -> H(U1(e, e), f(d)),A^1 -> H(U1(e, e), f(d))) 5.12/2.26 (A^1 -> H(U1(e, a), U1(d, d)),A^1 -> H(U1(e, a), U1(d, d))) 5.12/2.26 5.12/2.26 5.12/2.26 ---------------------------------------- 5.12/2.26 5.12/2.26 (76) 5.12/2.26 Obligation: 5.12/2.26 Q DP problem: 5.12/2.26 The TRS P consists of the following rules: 5.12/2.26 5.12/2.26 H(x, x) -> G(x, x) 5.12/2.26 G(d, e) -> A^1 5.12/2.26 A^1 -> H(f(d), f(d)) 5.12/2.26 A^1 -> H(f(e), f(e)) 5.12/2.26 A^1 -> H(U1(e, a), f(e)) 5.12/2.26 A^1 -> H(U1(a, d), U1(b, b)) 5.12/2.26 A^1 -> H(U1(a, d), f(d)) 5.12/2.26 A^1 -> H(U1(a, d), f(e)) 5.12/2.26 A^1 -> H(U1(a, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(a, e), f(d)) 5.12/2.26 A^1 -> H(U1(a, e), f(e)) 5.12/2.26 A^1 -> H(U1(a, a), U1(d, b)) 5.12/2.26 A^1 -> H(U1(a, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(a, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(a, a), U1(b, e)) 5.12/2.26 A^1 -> H(U1(a, a), U1(d, d)) 5.12/2.26 A^1 -> H(U1(a, a), U1(e, e)) 5.12/2.26 A^1 -> H(d, f(b)) 5.12/2.26 A^1 -> H(U1(d, d), U1(b, b)) 5.12/2.26 A^1 -> H(U1(d, d), f(d)) 5.12/2.26 A^1 -> H(U1(d, d), f(e)) 5.12/2.26 A^1 -> H(f(d), U1(d, b)) 5.12/2.26 A^1 -> H(f(d), U1(e, b)) 5.12/2.26 A^1 -> H(f(d), U1(b, d)) 5.12/2.26 A^1 -> H(f(d), U1(b, e)) 5.12/2.26 A^1 -> H(f(d), U1(e, e)) 5.12/2.26 A^1 -> H(U1(e, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(e, e), f(d)) 5.12/2.26 A^1 -> H(U1(e, e), f(e)) 5.12/2.26 A^1 -> H(f(e), U1(d, b)) 5.12/2.26 A^1 -> H(f(e), U1(e, b)) 5.12/2.26 A^1 -> H(f(e), U1(b, d)) 5.12/2.26 A^1 -> H(f(e), U1(b, e)) 5.12/2.26 A^1 -> H(f(e), U1(d, d)) 5.12/2.26 A^1 -> H(f(a), b) 5.12/2.26 A^1 -> H(f(a), U1(d, e)) 5.12/2.26 A^1 -> H(f(a), U1(e, d)) 5.12/2.26 A^1 -> H(f(d), U1(d, d)) 5.12/2.26 A^1 -> H(f(a), d) 5.12/2.26 A^1 -> H(f(e), U1(e, e)) 5.12/2.26 A^1 -> H(e, f(b)) 5.12/2.26 A^1 -> H(a, U1(b, b)) 5.12/2.26 A^1 -> H(a, f(d)) 5.12/2.26 A^1 -> H(a, f(e)) 5.12/2.26 A^1 -> H(U1(d, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(d, e), f(d)) 5.12/2.26 A^1 -> H(U1(d, e), f(e)) 5.12/2.26 A^1 -> H(U1(d, a), U1(d, b)) 5.12/2.26 A^1 -> H(U1(d, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(d, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(d, a), U1(b, e)) 5.12/2.26 A^1 -> H(U1(d, a), U1(d, d)) 5.12/2.26 A^1 -> H(U1(d, a), U1(e, e)) 5.12/2.26 A^1 -> H(U1(e, d), U1(b, b)) 5.12/2.26 A^1 -> H(U1(e, d), f(d)) 5.12/2.26 A^1 -> H(U1(e, d), f(e)) 5.12/2.26 A^1 -> H(U1(e, a), U1(d, b)) 5.12/2.26 A^1 -> H(U1(e, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(e, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(e, a), U1(b, e)) 5.12/2.26 A^1 -> H(U1(e, a), U1(d, d)) 5.12/2.26 5.12/2.26 The TRS R consists of the following rules: 5.12/2.26 5.12/2.26 a -> d 5.12/2.26 a -> e 5.12/2.26 f(x) -> U1(x, x) 5.12/2.26 b -> d 5.12/2.26 b -> e 5.12/2.26 U1(d, x) -> x 5.12/2.26 5.12/2.26 Q is empty. 5.12/2.26 We have to consider all minimal (P,Q,R)-chains. 5.12/2.26 ---------------------------------------- 5.12/2.26 5.12/2.26 (77) TransformationProof (EQUIVALENT) 5.12/2.26 By narrowing [LPAR04] the rule A^1 -> H(U1(e, a), f(e)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.26 5.12/2.26 (A^1 -> H(U1(e, d), f(e)),A^1 -> H(U1(e, d), f(e))) 5.12/2.26 (A^1 -> H(U1(e, e), f(e)),A^1 -> H(U1(e, e), f(e))) 5.12/2.26 (A^1 -> H(U1(e, a), U1(e, e)),A^1 -> H(U1(e, a), U1(e, e))) 5.12/2.26 5.12/2.26 5.12/2.26 ---------------------------------------- 5.12/2.26 5.12/2.26 (78) 5.12/2.26 Obligation: 5.12/2.26 Q DP problem: 5.12/2.26 The TRS P consists of the following rules: 5.12/2.26 5.12/2.26 H(x, x) -> G(x, x) 5.12/2.26 G(d, e) -> A^1 5.12/2.26 A^1 -> H(f(d), f(d)) 5.12/2.26 A^1 -> H(f(e), f(e)) 5.12/2.26 A^1 -> H(U1(a, d), U1(b, b)) 5.12/2.26 A^1 -> H(U1(a, d), f(d)) 5.12/2.26 A^1 -> H(U1(a, d), f(e)) 5.12/2.26 A^1 -> H(U1(a, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(a, e), f(d)) 5.12/2.26 A^1 -> H(U1(a, e), f(e)) 5.12/2.26 A^1 -> H(U1(a, a), U1(d, b)) 5.12/2.26 A^1 -> H(U1(a, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(a, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(a, a), U1(b, e)) 5.12/2.26 A^1 -> H(U1(a, a), U1(d, d)) 5.12/2.26 A^1 -> H(U1(a, a), U1(e, e)) 5.12/2.26 A^1 -> H(d, f(b)) 5.12/2.26 A^1 -> H(U1(d, d), U1(b, b)) 5.12/2.26 A^1 -> H(U1(d, d), f(d)) 5.12/2.26 A^1 -> H(U1(d, d), f(e)) 5.12/2.26 A^1 -> H(f(d), U1(d, b)) 5.12/2.26 A^1 -> H(f(d), U1(e, b)) 5.12/2.26 A^1 -> H(f(d), U1(b, d)) 5.12/2.26 A^1 -> H(f(d), U1(b, e)) 5.12/2.26 A^1 -> H(f(d), U1(e, e)) 5.12/2.26 A^1 -> H(U1(e, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(e, e), f(d)) 5.12/2.26 A^1 -> H(U1(e, e), f(e)) 5.12/2.26 A^1 -> H(f(e), U1(d, b)) 5.12/2.26 A^1 -> H(f(e), U1(e, b)) 5.12/2.26 A^1 -> H(f(e), U1(b, d)) 5.12/2.26 A^1 -> H(f(e), U1(b, e)) 5.12/2.26 A^1 -> H(f(e), U1(d, d)) 5.12/2.26 A^1 -> H(f(a), b) 5.12/2.26 A^1 -> H(f(a), U1(d, e)) 5.12/2.26 A^1 -> H(f(a), U1(e, d)) 5.12/2.26 A^1 -> H(f(d), U1(d, d)) 5.12/2.26 A^1 -> H(f(a), d) 5.12/2.26 A^1 -> H(f(e), U1(e, e)) 5.12/2.26 A^1 -> H(e, f(b)) 5.12/2.26 A^1 -> H(a, U1(b, b)) 5.12/2.26 A^1 -> H(a, f(d)) 5.12/2.26 A^1 -> H(a, f(e)) 5.12/2.26 A^1 -> H(U1(d, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(d, e), f(d)) 5.12/2.26 A^1 -> H(U1(d, e), f(e)) 5.12/2.26 A^1 -> H(U1(d, a), U1(d, b)) 5.12/2.26 A^1 -> H(U1(d, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(d, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(d, a), U1(b, e)) 5.12/2.26 A^1 -> H(U1(d, a), U1(d, d)) 5.12/2.26 A^1 -> H(U1(d, a), U1(e, e)) 5.12/2.26 A^1 -> H(U1(e, d), U1(b, b)) 5.12/2.26 A^1 -> H(U1(e, d), f(d)) 5.12/2.26 A^1 -> H(U1(e, d), f(e)) 5.12/2.26 A^1 -> H(U1(e, a), U1(d, b)) 5.12/2.26 A^1 -> H(U1(e, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(e, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(e, a), U1(b, e)) 5.12/2.26 A^1 -> H(U1(e, a), U1(d, d)) 5.12/2.26 A^1 -> H(U1(e, a), U1(e, e)) 5.12/2.26 5.12/2.26 The TRS R consists of the following rules: 5.12/2.26 5.12/2.26 a -> d 5.12/2.26 a -> e 5.12/2.26 f(x) -> U1(x, x) 5.12/2.26 b -> d 5.12/2.26 b -> e 5.12/2.26 U1(d, x) -> x 5.12/2.26 5.12/2.26 Q is empty. 5.12/2.26 We have to consider all minimal (P,Q,R)-chains. 5.12/2.26 ---------------------------------------- 5.12/2.26 5.12/2.26 (79) TransformationProof (EQUIVALENT) 5.12/2.26 By narrowing [LPAR04] the rule A^1 -> H(U1(a, d), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.26 5.12/2.26 (A^1 -> H(U1(d, d), U1(b, b)),A^1 -> H(U1(d, d), U1(b, b))) 5.12/2.26 (A^1 -> H(U1(e, d), U1(b, b)),A^1 -> H(U1(e, d), U1(b, b))) 5.12/2.26 (A^1 -> H(U1(a, d), U1(d, b)),A^1 -> H(U1(a, d), U1(d, b))) 5.12/2.26 (A^1 -> H(U1(a, d), U1(e, b)),A^1 -> H(U1(a, d), U1(e, b))) 5.12/2.26 (A^1 -> H(U1(a, d), U1(b, d)),A^1 -> H(U1(a, d), U1(b, d))) 5.12/2.26 (A^1 -> H(U1(a, d), U1(b, e)),A^1 -> H(U1(a, d), U1(b, e))) 5.12/2.26 5.12/2.26 5.12/2.26 ---------------------------------------- 5.12/2.26 5.12/2.26 (80) 5.12/2.26 Obligation: 5.12/2.26 Q DP problem: 5.12/2.26 The TRS P consists of the following rules: 5.12/2.26 5.12/2.26 H(x, x) -> G(x, x) 5.12/2.26 G(d, e) -> A^1 5.12/2.26 A^1 -> H(f(d), f(d)) 5.12/2.26 A^1 -> H(f(e), f(e)) 5.12/2.26 A^1 -> H(U1(a, d), f(d)) 5.12/2.26 A^1 -> H(U1(a, d), f(e)) 5.12/2.26 A^1 -> H(U1(a, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(a, e), f(d)) 5.12/2.26 A^1 -> H(U1(a, e), f(e)) 5.12/2.26 A^1 -> H(U1(a, a), U1(d, b)) 5.12/2.26 A^1 -> H(U1(a, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(a, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(a, a), U1(b, e)) 5.12/2.26 A^1 -> H(U1(a, a), U1(d, d)) 5.12/2.26 A^1 -> H(U1(a, a), U1(e, e)) 5.12/2.26 A^1 -> H(d, f(b)) 5.12/2.26 A^1 -> H(U1(d, d), U1(b, b)) 5.12/2.26 A^1 -> H(U1(d, d), f(d)) 5.12/2.26 A^1 -> H(U1(d, d), f(e)) 5.12/2.26 A^1 -> H(f(d), U1(d, b)) 5.12/2.26 A^1 -> H(f(d), U1(e, b)) 5.12/2.26 A^1 -> H(f(d), U1(b, d)) 5.12/2.26 A^1 -> H(f(d), U1(b, e)) 5.12/2.26 A^1 -> H(f(d), U1(e, e)) 5.12/2.26 A^1 -> H(U1(e, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(e, e), f(d)) 5.12/2.26 A^1 -> H(U1(e, e), f(e)) 5.12/2.26 A^1 -> H(f(e), U1(d, b)) 5.12/2.26 A^1 -> H(f(e), U1(e, b)) 5.12/2.26 A^1 -> H(f(e), U1(b, d)) 5.12/2.26 A^1 -> H(f(e), U1(b, e)) 5.12/2.26 A^1 -> H(f(e), U1(d, d)) 5.12/2.26 A^1 -> H(f(a), b) 5.12/2.26 A^1 -> H(f(a), U1(d, e)) 5.12/2.26 A^1 -> H(f(a), U1(e, d)) 5.12/2.26 A^1 -> H(f(d), U1(d, d)) 5.12/2.26 A^1 -> H(f(a), d) 5.12/2.26 A^1 -> H(f(e), U1(e, e)) 5.12/2.26 A^1 -> H(e, f(b)) 5.12/2.26 A^1 -> H(a, U1(b, b)) 5.12/2.26 A^1 -> H(a, f(d)) 5.12/2.26 A^1 -> H(a, f(e)) 5.12/2.26 A^1 -> H(U1(d, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(d, e), f(d)) 5.12/2.26 A^1 -> H(U1(d, e), f(e)) 5.12/2.26 A^1 -> H(U1(d, a), U1(d, b)) 5.12/2.26 A^1 -> H(U1(d, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(d, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(d, a), U1(b, e)) 5.12/2.26 A^1 -> H(U1(d, a), U1(d, d)) 5.12/2.26 A^1 -> H(U1(d, a), U1(e, e)) 5.12/2.26 A^1 -> H(U1(e, d), U1(b, b)) 5.12/2.26 A^1 -> H(U1(e, d), f(d)) 5.12/2.26 A^1 -> H(U1(e, d), f(e)) 5.12/2.26 A^1 -> H(U1(e, a), U1(d, b)) 5.12/2.26 A^1 -> H(U1(e, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(e, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(e, a), U1(b, e)) 5.12/2.26 A^1 -> H(U1(e, a), U1(d, d)) 5.12/2.26 A^1 -> H(U1(e, a), U1(e, e)) 5.12/2.26 A^1 -> H(U1(a, d), U1(d, b)) 5.12/2.26 A^1 -> H(U1(a, d), U1(e, b)) 5.12/2.26 A^1 -> H(U1(a, d), U1(b, d)) 5.12/2.26 A^1 -> H(U1(a, d), U1(b, e)) 5.12/2.26 5.12/2.26 The TRS R consists of the following rules: 5.12/2.26 5.12/2.26 a -> d 5.12/2.26 a -> e 5.12/2.26 f(x) -> U1(x, x) 5.12/2.26 b -> d 5.12/2.26 b -> e 5.12/2.26 U1(d, x) -> x 5.12/2.26 5.12/2.26 Q is empty. 5.12/2.26 We have to consider all minimal (P,Q,R)-chains. 5.12/2.26 ---------------------------------------- 5.12/2.26 5.12/2.26 (81) TransformationProof (EQUIVALENT) 5.12/2.26 By narrowing [LPAR04] the rule A^1 -> H(U1(a, d), f(d)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.26 5.12/2.26 (A^1 -> H(U1(d, d), f(d)),A^1 -> H(U1(d, d), f(d))) 5.12/2.26 (A^1 -> H(U1(e, d), f(d)),A^1 -> H(U1(e, d), f(d))) 5.12/2.26 (A^1 -> H(U1(a, d), U1(d, d)),A^1 -> H(U1(a, d), U1(d, d))) 5.12/2.26 5.12/2.26 5.12/2.26 ---------------------------------------- 5.12/2.26 5.12/2.26 (82) 5.12/2.26 Obligation: 5.12/2.26 Q DP problem: 5.12/2.26 The TRS P consists of the following rules: 5.12/2.26 5.12/2.26 H(x, x) -> G(x, x) 5.12/2.26 G(d, e) -> A^1 5.12/2.26 A^1 -> H(f(d), f(d)) 5.12/2.26 A^1 -> H(f(e), f(e)) 5.12/2.26 A^1 -> H(U1(a, d), f(e)) 5.12/2.26 A^1 -> H(U1(a, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(a, e), f(d)) 5.12/2.26 A^1 -> H(U1(a, e), f(e)) 5.12/2.26 A^1 -> H(U1(a, a), U1(d, b)) 5.12/2.26 A^1 -> H(U1(a, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(a, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(a, a), U1(b, e)) 5.12/2.26 A^1 -> H(U1(a, a), U1(d, d)) 5.12/2.26 A^1 -> H(U1(a, a), U1(e, e)) 5.12/2.26 A^1 -> H(d, f(b)) 5.12/2.26 A^1 -> H(U1(d, d), U1(b, b)) 5.12/2.26 A^1 -> H(U1(d, d), f(d)) 5.12/2.26 A^1 -> H(U1(d, d), f(e)) 5.12/2.26 A^1 -> H(f(d), U1(d, b)) 5.12/2.26 A^1 -> H(f(d), U1(e, b)) 5.12/2.26 A^1 -> H(f(d), U1(b, d)) 5.12/2.26 A^1 -> H(f(d), U1(b, e)) 5.12/2.26 A^1 -> H(f(d), U1(e, e)) 5.12/2.26 A^1 -> H(U1(e, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(e, e), f(d)) 5.12/2.26 A^1 -> H(U1(e, e), f(e)) 5.12/2.26 A^1 -> H(f(e), U1(d, b)) 5.12/2.26 A^1 -> H(f(e), U1(e, b)) 5.12/2.26 A^1 -> H(f(e), U1(b, d)) 5.12/2.26 A^1 -> H(f(e), U1(b, e)) 5.12/2.26 A^1 -> H(f(e), U1(d, d)) 5.12/2.26 A^1 -> H(f(a), b) 5.12/2.26 A^1 -> H(f(a), U1(d, e)) 5.12/2.26 A^1 -> H(f(a), U1(e, d)) 5.12/2.26 A^1 -> H(f(d), U1(d, d)) 5.12/2.26 A^1 -> H(f(a), d) 5.12/2.26 A^1 -> H(f(e), U1(e, e)) 5.12/2.26 A^1 -> H(e, f(b)) 5.12/2.26 A^1 -> H(a, U1(b, b)) 5.12/2.26 A^1 -> H(a, f(d)) 5.12/2.26 A^1 -> H(a, f(e)) 5.12/2.26 A^1 -> H(U1(d, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(d, e), f(d)) 5.12/2.26 A^1 -> H(U1(d, e), f(e)) 5.12/2.26 A^1 -> H(U1(d, a), U1(d, b)) 5.12/2.26 A^1 -> H(U1(d, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(d, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(d, a), U1(b, e)) 5.12/2.26 A^1 -> H(U1(d, a), U1(d, d)) 5.12/2.26 A^1 -> H(U1(d, a), U1(e, e)) 5.12/2.26 A^1 -> H(U1(e, d), U1(b, b)) 5.12/2.26 A^1 -> H(U1(e, d), f(d)) 5.12/2.26 A^1 -> H(U1(e, d), f(e)) 5.12/2.26 A^1 -> H(U1(e, a), U1(d, b)) 5.12/2.26 A^1 -> H(U1(e, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(e, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(e, a), U1(b, e)) 5.12/2.26 A^1 -> H(U1(e, a), U1(d, d)) 5.12/2.26 A^1 -> H(U1(e, a), U1(e, e)) 5.12/2.26 A^1 -> H(U1(a, d), U1(d, b)) 5.12/2.26 A^1 -> H(U1(a, d), U1(e, b)) 5.12/2.26 A^1 -> H(U1(a, d), U1(b, d)) 5.12/2.26 A^1 -> H(U1(a, d), U1(b, e)) 5.12/2.26 A^1 -> H(U1(a, d), U1(d, d)) 5.12/2.26 5.12/2.26 The TRS R consists of the following rules: 5.12/2.26 5.12/2.26 a -> d 5.12/2.26 a -> e 5.12/2.26 f(x) -> U1(x, x) 5.12/2.26 b -> d 5.12/2.26 b -> e 5.12/2.26 U1(d, x) -> x 5.12/2.26 5.12/2.26 Q is empty. 5.12/2.26 We have to consider all minimal (P,Q,R)-chains. 5.12/2.26 ---------------------------------------- 5.12/2.26 5.12/2.26 (83) TransformationProof (EQUIVALENT) 5.12/2.26 By narrowing [LPAR04] the rule A^1 -> H(U1(a, d), f(e)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.26 5.12/2.26 (A^1 -> H(U1(d, d), f(e)),A^1 -> H(U1(d, d), f(e))) 5.12/2.26 (A^1 -> H(U1(e, d), f(e)),A^1 -> H(U1(e, d), f(e))) 5.12/2.26 (A^1 -> H(U1(a, d), U1(e, e)),A^1 -> H(U1(a, d), U1(e, e))) 5.12/2.26 5.12/2.26 5.12/2.26 ---------------------------------------- 5.12/2.26 5.12/2.26 (84) 5.12/2.26 Obligation: 5.12/2.26 Q DP problem: 5.12/2.26 The TRS P consists of the following rules: 5.12/2.26 5.12/2.26 H(x, x) -> G(x, x) 5.12/2.26 G(d, e) -> A^1 5.12/2.26 A^1 -> H(f(d), f(d)) 5.12/2.26 A^1 -> H(f(e), f(e)) 5.12/2.26 A^1 -> H(U1(a, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(a, e), f(d)) 5.12/2.26 A^1 -> H(U1(a, e), f(e)) 5.12/2.26 A^1 -> H(U1(a, a), U1(d, b)) 5.12/2.26 A^1 -> H(U1(a, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(a, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(a, a), U1(b, e)) 5.12/2.26 A^1 -> H(U1(a, a), U1(d, d)) 5.12/2.26 A^1 -> H(U1(a, a), U1(e, e)) 5.12/2.26 A^1 -> H(d, f(b)) 5.12/2.26 A^1 -> H(U1(d, d), U1(b, b)) 5.12/2.26 A^1 -> H(U1(d, d), f(d)) 5.12/2.26 A^1 -> H(U1(d, d), f(e)) 5.12/2.26 A^1 -> H(f(d), U1(d, b)) 5.12/2.26 A^1 -> H(f(d), U1(e, b)) 5.12/2.26 A^1 -> H(f(d), U1(b, d)) 5.12/2.26 A^1 -> H(f(d), U1(b, e)) 5.12/2.26 A^1 -> H(f(d), U1(e, e)) 5.12/2.26 A^1 -> H(U1(e, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(e, e), f(d)) 5.12/2.26 A^1 -> H(U1(e, e), f(e)) 5.12/2.26 A^1 -> H(f(e), U1(d, b)) 5.12/2.26 A^1 -> H(f(e), U1(e, b)) 5.12/2.26 A^1 -> H(f(e), U1(b, d)) 5.12/2.26 A^1 -> H(f(e), U1(b, e)) 5.12/2.26 A^1 -> H(f(e), U1(d, d)) 5.12/2.26 A^1 -> H(f(a), b) 5.12/2.26 A^1 -> H(f(a), U1(d, e)) 5.12/2.26 A^1 -> H(f(a), U1(e, d)) 5.12/2.26 A^1 -> H(f(d), U1(d, d)) 5.12/2.26 A^1 -> H(f(a), d) 5.12/2.26 A^1 -> H(f(e), U1(e, e)) 5.12/2.26 A^1 -> H(e, f(b)) 5.12/2.26 A^1 -> H(a, U1(b, b)) 5.12/2.26 A^1 -> H(a, f(d)) 5.12/2.26 A^1 -> H(a, f(e)) 5.12/2.26 A^1 -> H(U1(d, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(d, e), f(d)) 5.12/2.26 A^1 -> H(U1(d, e), f(e)) 5.12/2.26 A^1 -> H(U1(d, a), U1(d, b)) 5.12/2.26 A^1 -> H(U1(d, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(d, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(d, a), U1(b, e)) 5.12/2.26 A^1 -> H(U1(d, a), U1(d, d)) 5.12/2.26 A^1 -> H(U1(d, a), U1(e, e)) 5.12/2.26 A^1 -> H(U1(e, d), U1(b, b)) 5.12/2.26 A^1 -> H(U1(e, d), f(d)) 5.12/2.26 A^1 -> H(U1(e, d), f(e)) 5.12/2.26 A^1 -> H(U1(e, a), U1(d, b)) 5.12/2.26 A^1 -> H(U1(e, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(e, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(e, a), U1(b, e)) 5.12/2.26 A^1 -> H(U1(e, a), U1(d, d)) 5.12/2.26 A^1 -> H(U1(e, a), U1(e, e)) 5.12/2.26 A^1 -> H(U1(a, d), U1(d, b)) 5.12/2.26 A^1 -> H(U1(a, d), U1(e, b)) 5.12/2.26 A^1 -> H(U1(a, d), U1(b, d)) 5.12/2.26 A^1 -> H(U1(a, d), U1(b, e)) 5.12/2.26 A^1 -> H(U1(a, d), U1(d, d)) 5.12/2.26 A^1 -> H(U1(a, d), U1(e, e)) 5.12/2.26 5.12/2.26 The TRS R consists of the following rules: 5.12/2.26 5.12/2.26 a -> d 5.12/2.26 a -> e 5.12/2.26 f(x) -> U1(x, x) 5.12/2.26 b -> d 5.12/2.26 b -> e 5.12/2.26 U1(d, x) -> x 5.12/2.26 5.12/2.26 Q is empty. 5.12/2.26 We have to consider all minimal (P,Q,R)-chains. 5.12/2.26 ---------------------------------------- 5.12/2.26 5.12/2.26 (85) TransformationProof (EQUIVALENT) 5.12/2.26 By narrowing [LPAR04] the rule A^1 -> H(U1(a, e), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.26 5.12/2.26 (A^1 -> H(U1(d, e), U1(b, b)),A^1 -> H(U1(d, e), U1(b, b))) 5.12/2.26 (A^1 -> H(U1(e, e), U1(b, b)),A^1 -> H(U1(e, e), U1(b, b))) 5.12/2.26 (A^1 -> H(U1(a, e), U1(d, b)),A^1 -> H(U1(a, e), U1(d, b))) 5.12/2.26 (A^1 -> H(U1(a, e), U1(e, b)),A^1 -> H(U1(a, e), U1(e, b))) 5.12/2.26 (A^1 -> H(U1(a, e), U1(b, d)),A^1 -> H(U1(a, e), U1(b, d))) 5.12/2.26 (A^1 -> H(U1(a, e), U1(b, e)),A^1 -> H(U1(a, e), U1(b, e))) 5.12/2.26 5.12/2.26 5.12/2.26 ---------------------------------------- 5.12/2.26 5.12/2.26 (86) 5.12/2.26 Obligation: 5.12/2.26 Q DP problem: 5.12/2.26 The TRS P consists of the following rules: 5.12/2.26 5.12/2.26 H(x, x) -> G(x, x) 5.12/2.26 G(d, e) -> A^1 5.12/2.26 A^1 -> H(f(d), f(d)) 5.12/2.26 A^1 -> H(f(e), f(e)) 5.12/2.26 A^1 -> H(U1(a, e), f(d)) 5.12/2.26 A^1 -> H(U1(a, e), f(e)) 5.12/2.26 A^1 -> H(U1(a, a), U1(d, b)) 5.12/2.26 A^1 -> H(U1(a, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(a, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(a, a), U1(b, e)) 5.12/2.26 A^1 -> H(U1(a, a), U1(d, d)) 5.12/2.26 A^1 -> H(U1(a, a), U1(e, e)) 5.12/2.26 A^1 -> H(d, f(b)) 5.12/2.26 A^1 -> H(U1(d, d), U1(b, b)) 5.12/2.26 A^1 -> H(U1(d, d), f(d)) 5.12/2.26 A^1 -> H(U1(d, d), f(e)) 5.12/2.26 A^1 -> H(f(d), U1(d, b)) 5.12/2.26 A^1 -> H(f(d), U1(e, b)) 5.12/2.26 A^1 -> H(f(d), U1(b, d)) 5.12/2.26 A^1 -> H(f(d), U1(b, e)) 5.12/2.26 A^1 -> H(f(d), U1(e, e)) 5.12/2.26 A^1 -> H(U1(e, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(e, e), f(d)) 5.12/2.26 A^1 -> H(U1(e, e), f(e)) 5.12/2.26 A^1 -> H(f(e), U1(d, b)) 5.12/2.26 A^1 -> H(f(e), U1(e, b)) 5.12/2.26 A^1 -> H(f(e), U1(b, d)) 5.12/2.26 A^1 -> H(f(e), U1(b, e)) 5.12/2.26 A^1 -> H(f(e), U1(d, d)) 5.12/2.26 A^1 -> H(f(a), b) 5.12/2.26 A^1 -> H(f(a), U1(d, e)) 5.12/2.26 A^1 -> H(f(a), U1(e, d)) 5.12/2.26 A^1 -> H(f(d), U1(d, d)) 5.12/2.26 A^1 -> H(f(a), d) 5.12/2.26 A^1 -> H(f(e), U1(e, e)) 5.12/2.26 A^1 -> H(e, f(b)) 5.12/2.26 A^1 -> H(a, U1(b, b)) 5.12/2.26 A^1 -> H(a, f(d)) 5.12/2.26 A^1 -> H(a, f(e)) 5.12/2.26 A^1 -> H(U1(d, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(d, e), f(d)) 5.12/2.26 A^1 -> H(U1(d, e), f(e)) 5.12/2.26 A^1 -> H(U1(d, a), U1(d, b)) 5.12/2.26 A^1 -> H(U1(d, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(d, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(d, a), U1(b, e)) 5.12/2.26 A^1 -> H(U1(d, a), U1(d, d)) 5.12/2.26 A^1 -> H(U1(d, a), U1(e, e)) 5.12/2.26 A^1 -> H(U1(e, d), U1(b, b)) 5.12/2.26 A^1 -> H(U1(e, d), f(d)) 5.12/2.26 A^1 -> H(U1(e, d), f(e)) 5.12/2.26 A^1 -> H(U1(e, a), U1(d, b)) 5.12/2.26 A^1 -> H(U1(e, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(e, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(e, a), U1(b, e)) 5.12/2.26 A^1 -> H(U1(e, a), U1(d, d)) 5.12/2.26 A^1 -> H(U1(e, a), U1(e, e)) 5.12/2.26 A^1 -> H(U1(a, d), U1(d, b)) 5.12/2.26 A^1 -> H(U1(a, d), U1(e, b)) 5.12/2.26 A^1 -> H(U1(a, d), U1(b, d)) 5.12/2.26 A^1 -> H(U1(a, d), U1(b, e)) 5.12/2.26 A^1 -> H(U1(a, d), U1(d, d)) 5.12/2.26 A^1 -> H(U1(a, d), U1(e, e)) 5.12/2.26 A^1 -> H(U1(a, e), U1(d, b)) 5.12/2.26 A^1 -> H(U1(a, e), U1(e, b)) 5.12/2.26 A^1 -> H(U1(a, e), U1(b, d)) 5.12/2.26 A^1 -> H(U1(a, e), U1(b, e)) 5.12/2.26 5.12/2.26 The TRS R consists of the following rules: 5.12/2.26 5.12/2.26 a -> d 5.12/2.26 a -> e 5.12/2.26 f(x) -> U1(x, x) 5.12/2.26 b -> d 5.12/2.26 b -> e 5.12/2.26 U1(d, x) -> x 5.12/2.26 5.12/2.26 Q is empty. 5.12/2.26 We have to consider all minimal (P,Q,R)-chains. 5.12/2.26 ---------------------------------------- 5.12/2.26 5.12/2.26 (87) TransformationProof (EQUIVALENT) 5.12/2.26 By narrowing [LPAR04] the rule A^1 -> H(U1(a, e), f(d)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.26 5.12/2.26 (A^1 -> H(U1(d, e), f(d)),A^1 -> H(U1(d, e), f(d))) 5.12/2.26 (A^1 -> H(U1(e, e), f(d)),A^1 -> H(U1(e, e), f(d))) 5.12/2.26 (A^1 -> H(U1(a, e), U1(d, d)),A^1 -> H(U1(a, e), U1(d, d))) 5.12/2.26 5.12/2.26 5.12/2.26 ---------------------------------------- 5.12/2.26 5.12/2.26 (88) 5.12/2.26 Obligation: 5.12/2.26 Q DP problem: 5.12/2.26 The TRS P consists of the following rules: 5.12/2.26 5.12/2.26 H(x, x) -> G(x, x) 5.12/2.26 G(d, e) -> A^1 5.12/2.26 A^1 -> H(f(d), f(d)) 5.12/2.26 A^1 -> H(f(e), f(e)) 5.12/2.26 A^1 -> H(U1(a, e), f(e)) 5.12/2.26 A^1 -> H(U1(a, a), U1(d, b)) 5.12/2.26 A^1 -> H(U1(a, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(a, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(a, a), U1(b, e)) 5.12/2.26 A^1 -> H(U1(a, a), U1(d, d)) 5.12/2.26 A^1 -> H(U1(a, a), U1(e, e)) 5.12/2.26 A^1 -> H(d, f(b)) 5.12/2.26 A^1 -> H(U1(d, d), U1(b, b)) 5.12/2.26 A^1 -> H(U1(d, d), f(d)) 5.12/2.26 A^1 -> H(U1(d, d), f(e)) 5.12/2.26 A^1 -> H(f(d), U1(d, b)) 5.12/2.26 A^1 -> H(f(d), U1(e, b)) 5.12/2.26 A^1 -> H(f(d), U1(b, d)) 5.12/2.26 A^1 -> H(f(d), U1(b, e)) 5.12/2.26 A^1 -> H(f(d), U1(e, e)) 5.12/2.26 A^1 -> H(U1(e, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(e, e), f(d)) 5.12/2.26 A^1 -> H(U1(e, e), f(e)) 5.12/2.26 A^1 -> H(f(e), U1(d, b)) 5.12/2.26 A^1 -> H(f(e), U1(e, b)) 5.12/2.26 A^1 -> H(f(e), U1(b, d)) 5.12/2.26 A^1 -> H(f(e), U1(b, e)) 5.12/2.26 A^1 -> H(f(e), U1(d, d)) 5.12/2.26 A^1 -> H(f(a), b) 5.12/2.26 A^1 -> H(f(a), U1(d, e)) 5.12/2.26 A^1 -> H(f(a), U1(e, d)) 5.12/2.26 A^1 -> H(f(d), U1(d, d)) 5.12/2.26 A^1 -> H(f(a), d) 5.12/2.26 A^1 -> H(f(e), U1(e, e)) 5.12/2.26 A^1 -> H(e, f(b)) 5.12/2.26 A^1 -> H(a, U1(b, b)) 5.12/2.26 A^1 -> H(a, f(d)) 5.12/2.26 A^1 -> H(a, f(e)) 5.12/2.26 A^1 -> H(U1(d, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(d, e), f(d)) 5.12/2.26 A^1 -> H(U1(d, e), f(e)) 5.12/2.26 A^1 -> H(U1(d, a), U1(d, b)) 5.12/2.26 A^1 -> H(U1(d, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(d, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(d, a), U1(b, e)) 5.12/2.26 A^1 -> H(U1(d, a), U1(d, d)) 5.12/2.26 A^1 -> H(U1(d, a), U1(e, e)) 5.12/2.26 A^1 -> H(U1(e, d), U1(b, b)) 5.12/2.26 A^1 -> H(U1(e, d), f(d)) 5.12/2.26 A^1 -> H(U1(e, d), f(e)) 5.12/2.26 A^1 -> H(U1(e, a), U1(d, b)) 5.12/2.26 A^1 -> H(U1(e, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(e, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(e, a), U1(b, e)) 5.12/2.26 A^1 -> H(U1(e, a), U1(d, d)) 5.12/2.26 A^1 -> H(U1(e, a), U1(e, e)) 5.12/2.26 A^1 -> H(U1(a, d), U1(d, b)) 5.12/2.26 A^1 -> H(U1(a, d), U1(e, b)) 5.12/2.26 A^1 -> H(U1(a, d), U1(b, d)) 5.12/2.26 A^1 -> H(U1(a, d), U1(b, e)) 5.12/2.26 A^1 -> H(U1(a, d), U1(d, d)) 5.12/2.26 A^1 -> H(U1(a, d), U1(e, e)) 5.12/2.26 A^1 -> H(U1(a, e), U1(d, b)) 5.12/2.26 A^1 -> H(U1(a, e), U1(e, b)) 5.12/2.26 A^1 -> H(U1(a, e), U1(b, d)) 5.12/2.26 A^1 -> H(U1(a, e), U1(b, e)) 5.12/2.26 A^1 -> H(U1(a, e), U1(d, d)) 5.12/2.26 5.12/2.26 The TRS R consists of the following rules: 5.12/2.26 5.12/2.26 a -> d 5.12/2.26 a -> e 5.12/2.26 f(x) -> U1(x, x) 5.12/2.26 b -> d 5.12/2.26 b -> e 5.12/2.26 U1(d, x) -> x 5.12/2.26 5.12/2.26 Q is empty. 5.12/2.26 We have to consider all minimal (P,Q,R)-chains. 5.12/2.26 ---------------------------------------- 5.12/2.26 5.12/2.26 (89) TransformationProof (EQUIVALENT) 5.12/2.26 By narrowing [LPAR04] the rule A^1 -> H(U1(a, e), f(e)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.26 5.12/2.26 (A^1 -> H(U1(d, e), f(e)),A^1 -> H(U1(d, e), f(e))) 5.12/2.26 (A^1 -> H(U1(e, e), f(e)),A^1 -> H(U1(e, e), f(e))) 5.12/2.26 (A^1 -> H(U1(a, e), U1(e, e)),A^1 -> H(U1(a, e), U1(e, e))) 5.12/2.26 5.12/2.26 5.12/2.26 ---------------------------------------- 5.12/2.26 5.12/2.26 (90) 5.12/2.26 Obligation: 5.12/2.26 Q DP problem: 5.12/2.26 The TRS P consists of the following rules: 5.12/2.26 5.12/2.26 H(x, x) -> G(x, x) 5.12/2.26 G(d, e) -> A^1 5.12/2.26 A^1 -> H(f(d), f(d)) 5.12/2.26 A^1 -> H(f(e), f(e)) 5.12/2.26 A^1 -> H(U1(a, a), U1(d, b)) 5.12/2.26 A^1 -> H(U1(a, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(a, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(a, a), U1(b, e)) 5.12/2.26 A^1 -> H(U1(a, a), U1(d, d)) 5.12/2.26 A^1 -> H(U1(a, a), U1(e, e)) 5.12/2.26 A^1 -> H(d, f(b)) 5.12/2.26 A^1 -> H(U1(d, d), U1(b, b)) 5.12/2.26 A^1 -> H(U1(d, d), f(d)) 5.12/2.26 A^1 -> H(U1(d, d), f(e)) 5.12/2.26 A^1 -> H(f(d), U1(d, b)) 5.12/2.26 A^1 -> H(f(d), U1(e, b)) 5.12/2.26 A^1 -> H(f(d), U1(b, d)) 5.12/2.26 A^1 -> H(f(d), U1(b, e)) 5.12/2.26 A^1 -> H(f(d), U1(e, e)) 5.12/2.26 A^1 -> H(U1(e, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(e, e), f(d)) 5.12/2.26 A^1 -> H(U1(e, e), f(e)) 5.12/2.26 A^1 -> H(f(e), U1(d, b)) 5.12/2.26 A^1 -> H(f(e), U1(e, b)) 5.12/2.26 A^1 -> H(f(e), U1(b, d)) 5.12/2.26 A^1 -> H(f(e), U1(b, e)) 5.12/2.26 A^1 -> H(f(e), U1(d, d)) 5.12/2.26 A^1 -> H(f(a), b) 5.12/2.26 A^1 -> H(f(a), U1(d, e)) 5.12/2.26 A^1 -> H(f(a), U1(e, d)) 5.12/2.26 A^1 -> H(f(d), U1(d, d)) 5.12/2.26 A^1 -> H(f(a), d) 5.12/2.26 A^1 -> H(f(e), U1(e, e)) 5.12/2.26 A^1 -> H(e, f(b)) 5.12/2.26 A^1 -> H(a, U1(b, b)) 5.12/2.26 A^1 -> H(a, f(d)) 5.12/2.26 A^1 -> H(a, f(e)) 5.12/2.26 A^1 -> H(U1(d, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(d, e), f(d)) 5.12/2.26 A^1 -> H(U1(d, e), f(e)) 5.12/2.26 A^1 -> H(U1(d, a), U1(d, b)) 5.12/2.26 A^1 -> H(U1(d, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(d, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(d, a), U1(b, e)) 5.12/2.26 A^1 -> H(U1(d, a), U1(d, d)) 5.12/2.26 A^1 -> H(U1(d, a), U1(e, e)) 5.12/2.26 A^1 -> H(U1(e, d), U1(b, b)) 5.12/2.26 A^1 -> H(U1(e, d), f(d)) 5.12/2.26 A^1 -> H(U1(e, d), f(e)) 5.12/2.26 A^1 -> H(U1(e, a), U1(d, b)) 5.12/2.26 A^1 -> H(U1(e, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(e, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(e, a), U1(b, e)) 5.12/2.26 A^1 -> H(U1(e, a), U1(d, d)) 5.12/2.26 A^1 -> H(U1(e, a), U1(e, e)) 5.12/2.26 A^1 -> H(U1(a, d), U1(d, b)) 5.12/2.26 A^1 -> H(U1(a, d), U1(e, b)) 5.12/2.26 A^1 -> H(U1(a, d), U1(b, d)) 5.12/2.26 A^1 -> H(U1(a, d), U1(b, e)) 5.12/2.26 A^1 -> H(U1(a, d), U1(d, d)) 5.12/2.26 A^1 -> H(U1(a, d), U1(e, e)) 5.12/2.26 A^1 -> H(U1(a, e), U1(d, b)) 5.12/2.26 A^1 -> H(U1(a, e), U1(e, b)) 5.12/2.26 A^1 -> H(U1(a, e), U1(b, d)) 5.12/2.26 A^1 -> H(U1(a, e), U1(b, e)) 5.12/2.26 A^1 -> H(U1(a, e), U1(d, d)) 5.12/2.26 A^1 -> H(U1(a, e), U1(e, e)) 5.12/2.26 5.12/2.26 The TRS R consists of the following rules: 5.12/2.26 5.12/2.26 a -> d 5.12/2.26 a -> e 5.12/2.26 f(x) -> U1(x, x) 5.12/2.26 b -> d 5.12/2.26 b -> e 5.12/2.26 U1(d, x) -> x 5.12/2.26 5.12/2.26 Q is empty. 5.12/2.26 We have to consider all minimal (P,Q,R)-chains. 5.12/2.26 ---------------------------------------- 5.12/2.26 5.12/2.26 (91) TransformationProof (EQUIVALENT) 5.12/2.26 By narrowing [LPAR04] the rule A^1 -> H(U1(a, a), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.26 5.12/2.26 (A^1 -> H(U1(d, a), U1(d, b)),A^1 -> H(U1(d, a), U1(d, b))) 5.12/2.26 (A^1 -> H(U1(e, a), U1(d, b)),A^1 -> H(U1(e, a), U1(d, b))) 5.12/2.26 (A^1 -> H(U1(a, d), U1(d, b)),A^1 -> H(U1(a, d), U1(d, b))) 5.12/2.26 (A^1 -> H(U1(a, e), U1(d, b)),A^1 -> H(U1(a, e), U1(d, b))) 5.12/2.26 (A^1 -> H(U1(a, a), b),A^1 -> H(U1(a, a), b)) 5.12/2.26 (A^1 -> H(U1(a, a), U1(d, d)),A^1 -> H(U1(a, a), U1(d, d))) 5.12/2.26 (A^1 -> H(U1(a, a), U1(d, e)),A^1 -> H(U1(a, a), U1(d, e))) 5.12/2.26 5.12/2.26 5.12/2.26 ---------------------------------------- 5.12/2.26 5.12/2.26 (92) 5.12/2.26 Obligation: 5.12/2.26 Q DP problem: 5.12/2.26 The TRS P consists of the following rules: 5.12/2.26 5.12/2.26 H(x, x) -> G(x, x) 5.12/2.26 G(d, e) -> A^1 5.12/2.26 A^1 -> H(f(d), f(d)) 5.12/2.26 A^1 -> H(f(e), f(e)) 5.12/2.26 A^1 -> H(U1(a, a), U1(e, b)) 5.12/2.26 A^1 -> H(U1(a, a), U1(b, d)) 5.12/2.26 A^1 -> H(U1(a, a), U1(b, e)) 5.12/2.26 A^1 -> H(U1(a, a), U1(d, d)) 5.12/2.26 A^1 -> H(U1(a, a), U1(e, e)) 5.12/2.26 A^1 -> H(d, f(b)) 5.12/2.26 A^1 -> H(U1(d, d), U1(b, b)) 5.12/2.26 A^1 -> H(U1(d, d), f(d)) 5.12/2.26 A^1 -> H(U1(d, d), f(e)) 5.12/2.26 A^1 -> H(f(d), U1(d, b)) 5.12/2.26 A^1 -> H(f(d), U1(e, b)) 5.12/2.26 A^1 -> H(f(d), U1(b, d)) 5.12/2.26 A^1 -> H(f(d), U1(b, e)) 5.12/2.26 A^1 -> H(f(d), U1(e, e)) 5.12/2.26 A^1 -> H(U1(e, e), U1(b, b)) 5.12/2.26 A^1 -> H(U1(e, e), f(d)) 5.12/2.26 A^1 -> H(U1(e, e), f(e)) 5.12/2.26 A^1 -> H(f(e), U1(d, b)) 5.12/2.26 A^1 -> H(f(e), U1(e, b)) 5.12/2.26 A^1 -> H(f(e), U1(b, d)) 5.12/2.26 A^1 -> H(f(e), U1(b, e)) 5.12/2.26 A^1 -> H(f(e), U1(d, d)) 5.12/2.26 A^1 -> H(f(a), b) 5.12/2.26 A^1 -> H(f(a), U1(d, e)) 5.12/2.27 A^1 -> H(f(a), U1(e, d)) 5.12/2.27 A^1 -> H(f(d), U1(d, d)) 5.12/2.27 A^1 -> H(f(a), d) 5.12/2.27 A^1 -> H(f(e), U1(e, e)) 5.12/2.27 A^1 -> H(e, f(b)) 5.12/2.27 A^1 -> H(a, U1(b, b)) 5.12/2.27 A^1 -> H(a, f(d)) 5.12/2.27 A^1 -> H(a, f(e)) 5.12/2.27 A^1 -> H(U1(d, e), U1(b, b)) 5.12/2.27 A^1 -> H(U1(d, e), f(d)) 5.12/2.27 A^1 -> H(U1(d, e), f(e)) 5.12/2.27 A^1 -> H(U1(d, a), U1(d, b)) 5.12/2.27 A^1 -> H(U1(d, a), U1(e, b)) 5.12/2.27 A^1 -> H(U1(d, a), U1(b, d)) 5.12/2.27 A^1 -> H(U1(d, a), U1(b, e)) 5.12/2.27 A^1 -> H(U1(d, a), U1(d, d)) 5.12/2.27 A^1 -> H(U1(d, a), U1(e, e)) 5.12/2.27 A^1 -> H(U1(e, d), U1(b, b)) 5.12/2.27 A^1 -> H(U1(e, d), f(d)) 5.12/2.27 A^1 -> H(U1(e, d), f(e)) 5.12/2.27 A^1 -> H(U1(e, a), U1(d, b)) 5.12/2.27 A^1 -> H(U1(e, a), U1(e, b)) 5.12/2.27 A^1 -> H(U1(e, a), U1(b, d)) 5.12/2.27 A^1 -> H(U1(e, a), U1(b, e)) 5.12/2.27 A^1 -> H(U1(e, a), U1(d, d)) 5.12/2.27 A^1 -> H(U1(e, a), U1(e, e)) 5.12/2.27 A^1 -> H(U1(a, d), U1(d, b)) 5.12/2.27 A^1 -> H(U1(a, d), U1(e, b)) 5.12/2.27 A^1 -> H(U1(a, d), U1(b, d)) 5.12/2.27 A^1 -> H(U1(a, d), U1(b, e)) 5.12/2.27 A^1 -> H(U1(a, d), U1(d, d)) 5.12/2.27 A^1 -> H(U1(a, d), U1(e, e)) 5.12/2.27 A^1 -> H(U1(a, e), U1(d, b)) 5.12/2.27 A^1 -> H(U1(a, e), U1(e, b)) 5.12/2.27 A^1 -> H(U1(a, e), U1(b, d)) 5.12/2.27 A^1 -> H(U1(a, e), U1(b, e)) 5.12/2.27 A^1 -> H(U1(a, e), U1(d, d)) 5.12/2.27 A^1 -> H(U1(a, e), U1(e, e)) 5.12/2.27 A^1 -> H(U1(a, a), b) 5.12/2.27 A^1 -> H(U1(a, a), U1(d, e)) 5.12/2.27 5.12/2.27 The TRS R consists of the following rules: 5.12/2.27 5.12/2.27 a -> d 5.12/2.27 a -> e 5.12/2.27 f(x) -> U1(x, x) 5.12/2.27 b -> d 5.12/2.27 b -> e 5.12/2.27 U1(d, x) -> x 5.12/2.27 5.12/2.27 Q is empty. 5.12/2.27 We have to consider all minimal (P,Q,R)-chains. 5.12/2.27 ---------------------------------------- 5.12/2.27 5.12/2.27 (93) TransformationProof (EQUIVALENT) 5.12/2.27 By narrowing [LPAR04] the rule A^1 -> H(U1(a, a), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.27 5.12/2.27 (A^1 -> H(U1(d, a), U1(e, b)),A^1 -> H(U1(d, a), U1(e, b))) 5.12/2.27 (A^1 -> H(U1(e, a), U1(e, b)),A^1 -> H(U1(e, a), U1(e, b))) 5.12/2.27 (A^1 -> H(U1(a, d), U1(e, b)),A^1 -> H(U1(a, d), U1(e, b))) 5.12/2.27 (A^1 -> H(U1(a, e), U1(e, b)),A^1 -> H(U1(a, e), U1(e, b))) 5.12/2.27 (A^1 -> H(U1(a, a), U1(e, d)),A^1 -> H(U1(a, a), U1(e, d))) 5.12/2.27 (A^1 -> H(U1(a, a), U1(e, e)),A^1 -> H(U1(a, a), U1(e, e))) 5.12/2.27 5.12/2.27 5.12/2.27 ---------------------------------------- 5.12/2.27 5.12/2.27 (94) 5.12/2.27 Obligation: 5.12/2.27 Q DP problem: 5.12/2.27 The TRS P consists of the following rules: 5.12/2.27 5.12/2.27 H(x, x) -> G(x, x) 5.12/2.27 G(d, e) -> A^1 5.12/2.27 A^1 -> H(f(d), f(d)) 5.12/2.27 A^1 -> H(f(e), f(e)) 5.12/2.27 A^1 -> H(U1(a, a), U1(b, d)) 5.12/2.27 A^1 -> H(U1(a, a), U1(b, e)) 5.12/2.27 A^1 -> H(U1(a, a), U1(d, d)) 5.12/2.27 A^1 -> H(U1(a, a), U1(e, e)) 5.12/2.27 A^1 -> H(d, f(b)) 5.12/2.27 A^1 -> H(U1(d, d), U1(b, b)) 5.12/2.27 A^1 -> H(U1(d, d), f(d)) 5.12/2.27 A^1 -> H(U1(d, d), f(e)) 5.12/2.27 A^1 -> H(f(d), U1(d, b)) 5.12/2.27 A^1 -> H(f(d), U1(e, b)) 5.12/2.27 A^1 -> H(f(d), U1(b, d)) 5.12/2.27 A^1 -> H(f(d), U1(b, e)) 5.12/2.27 A^1 -> H(f(d), U1(e, e)) 5.12/2.27 A^1 -> H(U1(e, e), U1(b, b)) 5.12/2.27 A^1 -> H(U1(e, e), f(d)) 5.12/2.27 A^1 -> H(U1(e, e), f(e)) 5.12/2.27 A^1 -> H(f(e), U1(d, b)) 5.12/2.27 A^1 -> H(f(e), U1(e, b)) 5.12/2.27 A^1 -> H(f(e), U1(b, d)) 5.12/2.27 A^1 -> H(f(e), U1(b, e)) 5.12/2.27 A^1 -> H(f(e), U1(d, d)) 5.12/2.27 A^1 -> H(f(a), b) 5.12/2.27 A^1 -> H(f(a), U1(d, e)) 5.12/2.27 A^1 -> H(f(a), U1(e, d)) 5.12/2.27 A^1 -> H(f(d), U1(d, d)) 5.12/2.27 A^1 -> H(f(a), d) 5.12/2.27 A^1 -> H(f(e), U1(e, e)) 5.12/2.27 A^1 -> H(e, f(b)) 5.12/2.27 A^1 -> H(a, U1(b, b)) 5.12/2.27 A^1 -> H(a, f(d)) 5.12/2.27 A^1 -> H(a, f(e)) 5.12/2.27 A^1 -> H(U1(d, e), U1(b, b)) 5.12/2.27 A^1 -> H(U1(d, e), f(d)) 5.12/2.27 A^1 -> H(U1(d, e), f(e)) 5.12/2.27 A^1 -> H(U1(d, a), U1(d, b)) 5.12/2.27 A^1 -> H(U1(d, a), U1(e, b)) 5.12/2.27 A^1 -> H(U1(d, a), U1(b, d)) 5.12/2.27 A^1 -> H(U1(d, a), U1(b, e)) 5.12/2.27 A^1 -> H(U1(d, a), U1(d, d)) 5.12/2.27 A^1 -> H(U1(d, a), U1(e, e)) 5.12/2.27 A^1 -> H(U1(e, d), U1(b, b)) 5.12/2.27 A^1 -> H(U1(e, d), f(d)) 5.12/2.27 A^1 -> H(U1(e, d), f(e)) 5.12/2.27 A^1 -> H(U1(e, a), U1(d, b)) 5.12/2.27 A^1 -> H(U1(e, a), U1(e, b)) 5.12/2.27 A^1 -> H(U1(e, a), U1(b, d)) 5.12/2.27 A^1 -> H(U1(e, a), U1(b, e)) 5.12/2.27 A^1 -> H(U1(e, a), U1(d, d)) 5.12/2.27 A^1 -> H(U1(e, a), U1(e, e)) 5.12/2.27 A^1 -> H(U1(a, d), U1(d, b)) 5.12/2.27 A^1 -> H(U1(a, d), U1(e, b)) 5.12/2.27 A^1 -> H(U1(a, d), U1(b, d)) 5.12/2.27 A^1 -> H(U1(a, d), U1(b, e)) 5.12/2.27 A^1 -> H(U1(a, d), U1(d, d)) 5.12/2.27 A^1 -> H(U1(a, d), U1(e, e)) 5.12/2.27 A^1 -> H(U1(a, e), U1(d, b)) 5.12/2.27 A^1 -> H(U1(a, e), U1(e, b)) 5.12/2.27 A^1 -> H(U1(a, e), U1(b, d)) 5.12/2.27 A^1 -> H(U1(a, e), U1(b, e)) 5.12/2.27 A^1 -> H(U1(a, e), U1(d, d)) 5.12/2.27 A^1 -> H(U1(a, e), U1(e, e)) 5.12/2.27 A^1 -> H(U1(a, a), b) 5.12/2.27 A^1 -> H(U1(a, a), U1(d, e)) 5.12/2.27 A^1 -> H(U1(a, a), U1(e, d)) 5.12/2.27 5.12/2.27 The TRS R consists of the following rules: 5.12/2.27 5.12/2.27 a -> d 5.12/2.27 a -> e 5.12/2.27 f(x) -> U1(x, x) 5.12/2.27 b -> d 5.12/2.27 b -> e 5.12/2.27 U1(d, x) -> x 5.12/2.27 5.12/2.27 Q is empty. 5.12/2.27 We have to consider all minimal (P,Q,R)-chains. 5.12/2.27 ---------------------------------------- 5.12/2.27 5.12/2.27 (95) TransformationProof (EQUIVALENT) 5.12/2.27 By narrowing [LPAR04] the rule A^1 -> H(U1(a, a), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: 5.12/2.27 5.12/2.27 (A^1 -> H(U1(d, a), U1(b, d)),A^1 -> H(U1(d, a), U1(b, d))) 5.12/2.27 (A^1 -> H(U1(e, a), U1(b, d)),A^1 -> H(U1(e, a), U1(b, d))) 5.12/2.27 (A^1 -> H(U1(a, d), U1(b, d)),A^1 -> H(U1(a, d), U1(b, d))) 5.12/2.27 (A^1 -> H(U1(a, e), U1(b, d)),A^1 -> H(U1(a, e), U1(b, d))) 5.12/2.27 (A^1 -> H(U1(a, a), U1(d, d)),A^1 -> H(U1(a, a), U1(d, d))) 5.12/2.27 (A^1 -> H(U1(a, a), U1(e, d)),A^1 -> H(U1(a, a), U1(e, d))) 5.12/2.27 5.12/2.27 5.12/2.27 ---------------------------------------- 5.12/2.27 5.12/2.27 (96) 5.12/2.27 Obligation: 5.12/2.27 Q DP problem: 5.12/2.27 The TRS P consists of the following rules: 5.12/2.27 5.12/2.27 H(x, x) -> G(x, x) 5.12/2.27 G(d, e) -> A^1 5.12/2.27 A^1 -> H(f(d), f(d)) 5.12/2.27 A^1 -> H(f(e), f(e)) 5.12/2.27 A^1 -> H(U1(a, a), U1(b, e)) 5.12/2.27 A^1 -> H(U1(a, a), U1(d, d)) 5.12/2.27 A^1 -> H(U1(a, a), U1(e, e)) 5.12/2.27 A^1 -> H(d, f(b)) 5.44/2.27 A^1 -> H(U1(d, d), U1(b, b)) 5.44/2.27 A^1 -> H(U1(d, d), f(d)) 5.44/2.27 A^1 -> H(U1(d, d), f(e)) 5.44/2.27 A^1 -> H(f(d), U1(d, b)) 5.44/2.27 A^1 -> H(f(d), U1(e, b)) 5.44/2.27 A^1 -> H(f(d), U1(b, d)) 5.44/2.27 A^1 -> H(f(d), U1(b, e)) 5.44/2.27 A^1 -> H(f(d), U1(e, e)) 5.44/2.27 A^1 -> H(U1(e, e), U1(b, b)) 5.44/2.27 A^1 -> H(U1(e, e), f(d)) 5.44/2.27 A^1 -> H(U1(e, e), f(e)) 5.44/2.27 A^1 -> H(f(e), U1(d, b)) 5.44/2.27 A^1 -> H(f(e), U1(e, b)) 5.44/2.27 A^1 -> H(f(e), U1(b, d)) 5.44/2.27 A^1 -> H(f(e), U1(b, e)) 5.44/2.27 A^1 -> H(f(e), U1(d, d)) 5.44/2.27 A^1 -> H(f(a), b) 5.44/2.27 A^1 -> H(f(a), U1(d, e)) 5.44/2.27 A^1 -> H(f(a), U1(e, d)) 5.44/2.27 A^1 -> H(f(d), U1(d, d)) 5.44/2.27 A^1 -> H(f(a), d) 5.44/2.27 A^1 -> H(f(e), U1(e, e)) 5.44/2.27 A^1 -> H(e, f(b)) 5.44/2.27 A^1 -> H(a, U1(b, b)) 5.44/2.27 A^1 -> H(a, f(d)) 5.44/2.27 A^1 -> H(a, f(e)) 5.44/2.27 A^1 -> H(U1(d, e), U1(b, b)) 5.44/2.27 A^1 -> H(U1(d, e), f(d)) 5.44/2.27 A^1 -> H(U1(d, e), f(e)) 5.44/2.27 A^1 -> H(U1(d, a), U1(d, b)) 5.44/2.27 A^1 -> H(U1(d, a), U1(e, b)) 5.44/2.27 A^1 -> H(U1(d, a), U1(b, d)) 5.44/2.27 A^1 -> H(U1(d, a), U1(b, e)) 5.44/2.27 A^1 -> H(U1(d, a), U1(d, d)) 5.44/2.27 A^1 -> H(U1(d, a), U1(e, e)) 5.44/2.27 A^1 -> H(U1(e, d), U1(b, b)) 5.44/2.27 A^1 -> H(U1(e, d), f(d)) 5.44/2.27 A^1 -> H(U1(e, d), f(e)) 5.44/2.27 A^1 -> H(U1(e, a), U1(d, b)) 5.44/2.27 A^1 -> H(U1(e, a), U1(e, b)) 5.44/2.27 A^1 -> H(U1(e, a), U1(b, d)) 5.44/2.27 A^1 -> H(U1(e, a), U1(b, e)) 5.44/2.27 A^1 -> H(U1(e, a), U1(d, d)) 5.44/2.27 A^1 -> H(U1(e, a), U1(e, e)) 5.44/2.27 A^1 -> H(U1(a, d), U1(d, b)) 5.44/2.27 A^1 -> H(U1(a, d), U1(e, b)) 5.44/2.27 A^1 -> H(U1(a, d), U1(b, d)) 5.44/2.27 A^1 -> H(U1(a, d), U1(b, e)) 5.44/2.27 A^1 -> H(U1(a, d), U1(d, d)) 5.44/2.27 A^1 -> H(U1(a, d), U1(e, e)) 5.44/2.27 A^1 -> H(U1(a, e), U1(d, b)) 5.44/2.27 A^1 -> H(U1(a, e), U1(e, b)) 5.44/2.27 A^1 -> H(U1(a, e), U1(b, d)) 5.44/2.27 A^1 -> H(U1(a, e), U1(b, e)) 5.44/2.27 A^1 -> H(U1(a, e), U1(d, d)) 5.44/2.27 A^1 -> H(U1(a, e), U1(e, e)) 5.44/2.27 A^1 -> H(U1(a, a), b) 5.44/2.27 A^1 -> H(U1(a, a), U1(d, e)) 5.44/2.27 A^1 -> H(U1(a, a), U1(e, d)) 5.44/2.27 5.44/2.27 The TRS R consists of the following rules: 5.44/2.27 5.44/2.27 a -> d 5.44/2.27 a -> e 5.44/2.27 f(x) -> U1(x, x) 5.44/2.27 b -> d 5.44/2.27 b -> e 5.44/2.27 U1(d, x) -> x 5.44/2.27 5.44/2.27 Q is empty. 5.44/2.27 We have to consider all minimal (P,Q,R)-chains. 5.44/2.27 ---------------------------------------- 5.44/2.27 5.44/2.27 (97) TransformationProof (EQUIVALENT) 5.44/2.27 By narrowing [LPAR04] the rule A^1 -> H(U1(a, a), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: 5.44/2.27 5.44/2.27 (A^1 -> H(U1(d, a), U1(b, e)),A^1 -> H(U1(d, a), U1(b, e))) 5.44/2.27 (A^1 -> H(U1(e, a), U1(b, e)),A^1 -> H(U1(e, a), U1(b, e))) 5.44/2.27 (A^1 -> H(U1(a, d), U1(b, e)),A^1 -> H(U1(a, d), U1(b, e))) 5.44/2.27 (A^1 -> H(U1(a, e), U1(b, e)),A^1 -> H(U1(a, e), U1(b, e))) 5.44/2.27 (A^1 -> H(U1(a, a), U1(d, e)),A^1 -> H(U1(a, a), U1(d, e))) 5.44/2.27 (A^1 -> H(U1(a, a), U1(e, e)),A^1 -> H(U1(a, a), U1(e, e))) 5.44/2.27 5.44/2.27 5.44/2.27 ---------------------------------------- 5.44/2.27 5.44/2.27 (98) 5.44/2.27 Obligation: 5.44/2.27 Q DP problem: 5.44/2.27 The TRS P consists of the following rules: 5.44/2.27 5.44/2.27 H(x, x) -> G(x, x) 5.44/2.27 G(d, e) -> A^1 5.44/2.27 A^1 -> H(f(d), f(d)) 5.44/2.27 A^1 -> H(f(e), f(e)) 5.44/2.27 A^1 -> H(U1(a, a), U1(d, d)) 5.44/2.27 A^1 -> H(U1(a, a), U1(e, e)) 5.44/2.27 A^1 -> H(d, f(b)) 5.44/2.27 A^1 -> H(U1(d, d), U1(b, b)) 5.44/2.27 A^1 -> H(U1(d, d), f(d)) 5.44/2.27 A^1 -> H(U1(d, d), f(e)) 5.44/2.27 A^1 -> H(f(d), U1(d, b)) 5.44/2.27 A^1 -> H(f(d), U1(e, b)) 5.44/2.27 A^1 -> H(f(d), U1(b, d)) 5.44/2.27 A^1 -> H(f(d), U1(b, e)) 5.44/2.27 A^1 -> H(f(d), U1(e, e)) 5.44/2.27 A^1 -> H(U1(e, e), U1(b, b)) 5.44/2.27 A^1 -> H(U1(e, e), f(d)) 5.44/2.27 A^1 -> H(U1(e, e), f(e)) 5.44/2.27 A^1 -> H(f(e), U1(d, b)) 5.44/2.27 A^1 -> H(f(e), U1(e, b)) 5.44/2.27 A^1 -> H(f(e), U1(b, d)) 5.44/2.27 A^1 -> H(f(e), U1(b, e)) 5.44/2.27 A^1 -> H(f(e), U1(d, d)) 5.44/2.27 A^1 -> H(f(a), b) 5.44/2.27 A^1 -> H(f(a), U1(d, e)) 5.44/2.27 A^1 -> H(f(a), U1(e, d)) 5.44/2.27 A^1 -> H(f(d), U1(d, d)) 5.44/2.27 A^1 -> H(f(a), d) 5.44/2.27 A^1 -> H(f(e), U1(e, e)) 5.44/2.27 A^1 -> H(e, f(b)) 5.44/2.27 A^1 -> H(a, U1(b, b)) 5.44/2.27 A^1 -> H(a, f(d)) 5.44/2.27 A^1 -> H(a, f(e)) 5.44/2.27 A^1 -> H(U1(d, e), U1(b, b)) 5.44/2.27 A^1 -> H(U1(d, e), f(d)) 5.44/2.27 A^1 -> H(U1(d, e), f(e)) 5.44/2.27 A^1 -> H(U1(d, a), U1(d, b)) 5.44/2.27 A^1 -> H(U1(d, a), U1(e, b)) 5.44/2.27 A^1 -> H(U1(d, a), U1(b, d)) 5.44/2.27 A^1 -> H(U1(d, a), U1(b, e)) 5.44/2.27 A^1 -> H(U1(d, a), U1(d, d)) 5.44/2.27 A^1 -> H(U1(d, a), U1(e, e)) 5.44/2.27 A^1 -> H(U1(e, d), U1(b, b)) 5.44/2.27 A^1 -> H(U1(e, d), f(d)) 5.44/2.27 A^1 -> H(U1(e, d), f(e)) 5.44/2.27 A^1 -> H(U1(e, a), U1(d, b)) 5.44/2.27 A^1 -> H(U1(e, a), U1(e, b)) 5.44/2.27 A^1 -> H(U1(e, a), U1(b, d)) 5.44/2.27 A^1 -> H(U1(e, a), U1(b, e)) 5.44/2.27 A^1 -> H(U1(e, a), U1(d, d)) 5.44/2.27 A^1 -> H(U1(e, a), U1(e, e)) 5.44/2.27 A^1 -> H(U1(a, d), U1(d, b)) 5.44/2.27 A^1 -> H(U1(a, d), U1(e, b)) 5.44/2.27 A^1 -> H(U1(a, d), U1(b, d)) 5.44/2.27 A^1 -> H(U1(a, d), U1(b, e)) 5.44/2.27 A^1 -> H(U1(a, d), U1(d, d)) 5.44/2.27 A^1 -> H(U1(a, d), U1(e, e)) 5.44/2.27 A^1 -> H(U1(a, e), U1(d, b)) 5.44/2.27 A^1 -> H(U1(a, e), U1(e, b)) 5.44/2.27 A^1 -> H(U1(a, e), U1(b, d)) 5.44/2.27 A^1 -> H(U1(a, e), U1(b, e)) 5.44/2.27 A^1 -> H(U1(a, e), U1(d, d)) 5.44/2.27 A^1 -> H(U1(a, e), U1(e, e)) 5.44/2.27 A^1 -> H(U1(a, a), b) 5.44/2.27 A^1 -> H(U1(a, a), U1(d, e)) 5.44/2.27 A^1 -> H(U1(a, a), U1(e, d)) 5.44/2.27 5.44/2.27 The TRS R consists of the following rules: 5.44/2.27 5.44/2.27 a -> d 5.44/2.27 a -> e 5.44/2.27 f(x) -> U1(x, x) 5.44/2.27 b -> d 5.44/2.27 b -> e 5.44/2.27 U1(d, x) -> x 5.44/2.27 5.44/2.27 Q is empty. 5.44/2.27 We have to consider all minimal (P,Q,R)-chains. 5.44/2.27 ---------------------------------------- 5.44/2.27 5.44/2.27 (99) TransformationProof (EQUIVALENT) 5.44/2.27 By narrowing [LPAR04] the rule A^1 -> H(U1(a, a), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: 5.44/2.27 5.44/2.27 (A^1 -> H(U1(d, a), U1(d, d)),A^1 -> H(U1(d, a), U1(d, d))) 5.44/2.27 (A^1 -> H(U1(e, a), U1(d, d)),A^1 -> H(U1(e, a), U1(d, d))) 5.44/2.27 (A^1 -> H(U1(a, d), U1(d, d)),A^1 -> H(U1(a, d), U1(d, d))) 5.44/2.27 (A^1 -> H(U1(a, e), U1(d, d)),A^1 -> H(U1(a, e), U1(d, d))) 5.44/2.27 (A^1 -> H(U1(a, a), d),A^1 -> H(U1(a, a), d)) 5.44/2.27 5.44/2.27 5.44/2.27 ---------------------------------------- 5.44/2.27 5.44/2.27 (100) 5.44/2.27 Obligation: 5.44/2.27 Q DP problem: 5.44/2.27 The TRS P consists of the following rules: 5.44/2.27 5.44/2.27 H(x, x) -> G(x, x) 5.44/2.27 G(d, e) -> A^1 5.44/2.27 A^1 -> H(f(d), f(d)) 5.44/2.27 A^1 -> H(f(e), f(e)) 5.44/2.27 A^1 -> H(U1(a, a), U1(e, e)) 5.44/2.27 A^1 -> H(d, f(b)) 5.44/2.27 A^1 -> H(U1(d, d), U1(b, b)) 5.44/2.27 A^1 -> H(U1(d, d), f(d)) 5.44/2.27 A^1 -> H(U1(d, d), f(e)) 5.44/2.27 A^1 -> H(f(d), U1(d, b)) 5.44/2.27 A^1 -> H(f(d), U1(e, b)) 5.44/2.27 A^1 -> H(f(d), U1(b, d)) 5.44/2.27 A^1 -> H(f(d), U1(b, e)) 5.44/2.27 A^1 -> H(f(d), U1(e, e)) 5.44/2.27 A^1 -> H(U1(e, e), U1(b, b)) 5.44/2.27 A^1 -> H(U1(e, e), f(d)) 5.44/2.27 A^1 -> H(U1(e, e), f(e)) 5.44/2.27 A^1 -> H(f(e), U1(d, b)) 5.44/2.27 A^1 -> H(f(e), U1(e, b)) 5.44/2.27 A^1 -> H(f(e), U1(b, d)) 5.44/2.27 A^1 -> H(f(e), U1(b, e)) 5.44/2.27 A^1 -> H(f(e), U1(d, d)) 5.44/2.27 A^1 -> H(f(a), b) 5.44/2.27 A^1 -> H(f(a), U1(d, e)) 5.44/2.27 A^1 -> H(f(a), U1(e, d)) 5.44/2.27 A^1 -> H(f(d), U1(d, d)) 5.44/2.27 A^1 -> H(f(a), d) 5.44/2.27 A^1 -> H(f(e), U1(e, e)) 5.44/2.27 A^1 -> H(e, f(b)) 5.44/2.27 A^1 -> H(a, U1(b, b)) 5.44/2.27 A^1 -> H(a, f(d)) 5.44/2.27 A^1 -> H(a, f(e)) 5.44/2.27 A^1 -> H(U1(d, e), U1(b, b)) 5.44/2.27 A^1 -> H(U1(d, e), f(d)) 5.44/2.27 A^1 -> H(U1(d, e), f(e)) 5.44/2.27 A^1 -> H(U1(d, a), U1(d, b)) 5.44/2.27 A^1 -> H(U1(d, a), U1(e, b)) 5.44/2.27 A^1 -> H(U1(d, a), U1(b, d)) 5.44/2.27 A^1 -> H(U1(d, a), U1(b, e)) 5.44/2.27 A^1 -> H(U1(d, a), U1(d, d)) 5.44/2.27 A^1 -> H(U1(d, a), U1(e, e)) 5.44/2.27 A^1 -> H(U1(e, d), U1(b, b)) 5.44/2.27 A^1 -> H(U1(e, d), f(d)) 5.44/2.27 A^1 -> H(U1(e, d), f(e)) 5.44/2.27 A^1 -> H(U1(e, a), U1(d, b)) 5.44/2.27 A^1 -> H(U1(e, a), U1(e, b)) 5.44/2.27 A^1 -> H(U1(e, a), U1(b, d)) 5.44/2.27 A^1 -> H(U1(e, a), U1(b, e)) 5.44/2.27 A^1 -> H(U1(e, a), U1(d, d)) 5.44/2.27 A^1 -> H(U1(e, a), U1(e, e)) 5.44/2.27 A^1 -> H(U1(a, d), U1(d, b)) 5.44/2.27 A^1 -> H(U1(a, d), U1(e, b)) 5.44/2.27 A^1 -> H(U1(a, d), U1(b, d)) 5.44/2.27 A^1 -> H(U1(a, d), U1(b, e)) 5.44/2.27 A^1 -> H(U1(a, d), U1(d, d)) 5.44/2.27 A^1 -> H(U1(a, d), U1(e, e)) 5.44/2.27 A^1 -> H(U1(a, e), U1(d, b)) 5.44/2.27 A^1 -> H(U1(a, e), U1(e, b)) 5.44/2.27 A^1 -> H(U1(a, e), U1(b, d)) 5.44/2.27 A^1 -> H(U1(a, e), U1(b, e)) 5.44/2.27 A^1 -> H(U1(a, e), U1(d, d)) 5.44/2.27 A^1 -> H(U1(a, e), U1(e, e)) 5.44/2.27 A^1 -> H(U1(a, a), b) 5.44/2.27 A^1 -> H(U1(a, a), U1(d, e)) 5.44/2.27 A^1 -> H(U1(a, a), U1(e, d)) 5.44/2.27 A^1 -> H(U1(a, a), d) 5.44/2.27 5.44/2.27 The TRS R consists of the following rules: 5.44/2.27 5.44/2.27 a -> d 5.44/2.27 a -> e 5.44/2.27 f(x) -> U1(x, x) 5.44/2.27 b -> d 5.44/2.27 b -> e 5.44/2.27 U1(d, x) -> x 5.44/2.27 5.44/2.27 Q is empty. 5.44/2.27 We have to consider all minimal (P,Q,R)-chains. 5.44/2.27 ---------------------------------------- 5.44/2.27 5.44/2.27 (101) TransformationProof (EQUIVALENT) 5.44/2.27 By narrowing [LPAR04] the rule A^1 -> H(U1(a, a), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: 5.44/2.27 5.44/2.27 (A^1 -> H(U1(d, a), U1(e, e)),A^1 -> H(U1(d, a), U1(e, e))) 5.44/2.27 (A^1 -> H(U1(e, a), U1(e, e)),A^1 -> H(U1(e, a), U1(e, e))) 5.44/2.27 (A^1 -> H(U1(a, d), U1(e, e)),A^1 -> H(U1(a, d), U1(e, e))) 5.44/2.27 (A^1 -> H(U1(a, e), U1(e, e)),A^1 -> H(U1(a, e), U1(e, e))) 5.44/2.27 5.44/2.27 5.44/2.27 ---------------------------------------- 5.44/2.27 5.44/2.27 (102) 5.44/2.27 Obligation: 5.44/2.27 Q DP problem: 5.44/2.27 The TRS P consists of the following rules: 5.44/2.27 5.44/2.27 H(x, x) -> G(x, x) 5.44/2.27 G(d, e) -> A^1 5.44/2.27 A^1 -> H(f(d), f(d)) 5.44/2.27 A^1 -> H(f(e), f(e)) 5.44/2.27 A^1 -> H(d, f(b)) 5.44/2.27 A^1 -> H(U1(d, d), U1(b, b)) 5.44/2.27 A^1 -> H(U1(d, d), f(d)) 5.44/2.27 A^1 -> H(U1(d, d), f(e)) 5.44/2.27 A^1 -> H(f(d), U1(d, b)) 5.44/2.27 A^1 -> H(f(d), U1(e, b)) 5.44/2.27 A^1 -> H(f(d), U1(b, d)) 5.44/2.27 A^1 -> H(f(d), U1(b, e)) 5.44/2.27 A^1 -> H(f(d), U1(e, e)) 5.44/2.27 A^1 -> H(U1(e, e), U1(b, b)) 5.44/2.27 A^1 -> H(U1(e, e), f(d)) 5.44/2.27 A^1 -> H(U1(e, e), f(e)) 5.44/2.27 A^1 -> H(f(e), U1(d, b)) 5.44/2.27 A^1 -> H(f(e), U1(e, b)) 5.44/2.27 A^1 -> H(f(e), U1(b, d)) 5.44/2.27 A^1 -> H(f(e), U1(b, e)) 5.44/2.27 A^1 -> H(f(e), U1(d, d)) 5.44/2.27 A^1 -> H(f(a), b) 5.44/2.27 A^1 -> H(f(a), U1(d, e)) 5.44/2.27 A^1 -> H(f(a), U1(e, d)) 5.44/2.27 A^1 -> H(f(d), U1(d, d)) 5.44/2.27 A^1 -> H(f(a), d) 5.44/2.27 A^1 -> H(f(e), U1(e, e)) 5.44/2.27 A^1 -> H(e, f(b)) 5.44/2.27 A^1 -> H(a, U1(b, b)) 5.44/2.27 A^1 -> H(a, f(d)) 5.44/2.27 A^1 -> H(a, f(e)) 5.44/2.27 A^1 -> H(U1(d, e), U1(b, b)) 5.44/2.27 A^1 -> H(U1(d, e), f(d)) 5.44/2.27 A^1 -> H(U1(d, e), f(e)) 5.44/2.27 A^1 -> H(U1(d, a), U1(d, b)) 5.44/2.27 A^1 -> H(U1(d, a), U1(e, b)) 5.44/2.27 A^1 -> H(U1(d, a), U1(b, d)) 5.44/2.27 A^1 -> H(U1(d, a), U1(b, e)) 5.44/2.27 A^1 -> H(U1(d, a), U1(d, d)) 5.44/2.27 A^1 -> H(U1(d, a), U1(e, e)) 5.44/2.27 A^1 -> H(U1(e, d), U1(b, b)) 5.44/2.27 A^1 -> H(U1(e, d), f(d)) 5.44/2.27 A^1 -> H(U1(e, d), f(e)) 5.44/2.27 A^1 -> H(U1(e, a), U1(d, b)) 5.44/2.27 A^1 -> H(U1(e, a), U1(e, b)) 5.44/2.27 A^1 -> H(U1(e, a), U1(b, d)) 5.44/2.27 A^1 -> H(U1(e, a), U1(b, e)) 5.44/2.27 A^1 -> H(U1(e, a), U1(d, d)) 5.44/2.27 A^1 -> H(U1(e, a), U1(e, e)) 5.44/2.27 A^1 -> H(U1(a, d), U1(d, b)) 5.44/2.27 A^1 -> H(U1(a, d), U1(e, b)) 5.44/2.27 A^1 -> H(U1(a, d), U1(b, d)) 5.44/2.27 A^1 -> H(U1(a, d), U1(b, e)) 5.44/2.27 A^1 -> H(U1(a, d), U1(d, d)) 5.44/2.27 A^1 -> H(U1(a, d), U1(e, e)) 5.44/2.27 A^1 -> H(U1(a, e), U1(d, b)) 5.44/2.27 A^1 -> H(U1(a, e), U1(e, b)) 5.44/2.27 A^1 -> H(U1(a, e), U1(b, d)) 5.44/2.27 A^1 -> H(U1(a, e), U1(b, e)) 5.44/2.27 A^1 -> H(U1(a, e), U1(d, d)) 5.44/2.27 A^1 -> H(U1(a, e), U1(e, e)) 5.44/2.27 A^1 -> H(U1(a, a), b) 5.44/2.27 A^1 -> H(U1(a, a), U1(d, e)) 5.44/2.27 A^1 -> H(U1(a, a), U1(e, d)) 5.44/2.27 A^1 -> H(U1(a, a), d) 5.44/2.27 5.44/2.27 The TRS R consists of the following rules: 5.44/2.27 5.44/2.27 a -> d 5.44/2.27 a -> e 5.44/2.27 f(x) -> U1(x, x) 5.44/2.27 b -> d 5.44/2.27 b -> e 5.44/2.27 U1(d, x) -> x 5.44/2.27 5.44/2.27 Q is empty. 5.44/2.27 We have to consider all minimal (P,Q,R)-chains. 5.44/2.27 ---------------------------------------- 5.44/2.27 5.44/2.27 (103) TransformationProof (EQUIVALENT) 5.44/2.27 By narrowing [LPAR04] the rule A^1 -> H(d, f(b)) at position [] we obtained the following new rules [LPAR04]: 5.44/2.27 5.44/2.27 (A^1 -> H(d, U1(b, b)),A^1 -> H(d, U1(b, b))) 5.44/2.27 (A^1 -> H(d, f(d)),A^1 -> H(d, f(d))) 5.44/2.27 (A^1 -> H(d, f(e)),A^1 -> H(d, f(e))) 5.44/2.27 5.44/2.27 5.44/2.27 ---------------------------------------- 5.44/2.27 5.44/2.27 (104) 5.44/2.27 Obligation: 5.44/2.27 Q DP problem: 5.44/2.27 The TRS P consists of the following rules: 5.44/2.27 5.44/2.27 H(x, x) -> G(x, x) 5.44/2.27 G(d, e) -> A^1 5.44/2.27 A^1 -> H(f(d), f(d)) 5.44/2.27 A^1 -> H(f(e), f(e)) 5.44/2.27 A^1 -> H(U1(d, d), U1(b, b)) 5.44/2.27 A^1 -> H(U1(d, d), f(d)) 5.44/2.27 A^1 -> H(U1(d, d), f(e)) 5.44/2.27 A^1 -> H(f(d), U1(d, b)) 5.44/2.27 A^1 -> H(f(d), U1(e, b)) 5.44/2.27 A^1 -> H(f(d), U1(b, d)) 5.44/2.27 A^1 -> H(f(d), U1(b, e)) 5.44/2.27 A^1 -> H(f(d), U1(e, e)) 5.44/2.27 A^1 -> H(U1(e, e), U1(b, b)) 5.44/2.27 A^1 -> H(U1(e, e), f(d)) 5.44/2.27 A^1 -> H(U1(e, e), f(e)) 5.44/2.27 A^1 -> H(f(e), U1(d, b)) 5.44/2.27 A^1 -> H(f(e), U1(e, b)) 5.44/2.27 A^1 -> H(f(e), U1(b, d)) 5.44/2.27 A^1 -> H(f(e), U1(b, e)) 5.44/2.27 A^1 -> H(f(e), U1(d, d)) 5.44/2.27 A^1 -> H(f(a), b) 5.44/2.27 A^1 -> H(f(a), U1(d, e)) 5.44/2.27 A^1 -> H(f(a), U1(e, d)) 5.44/2.27 A^1 -> H(f(d), U1(d, d)) 5.44/2.27 A^1 -> H(f(a), d) 5.44/2.27 A^1 -> H(f(e), U1(e, e)) 5.44/2.27 A^1 -> H(e, f(b)) 5.44/2.27 A^1 -> H(a, U1(b, b)) 5.44/2.27 A^1 -> H(a, f(d)) 5.44/2.27 A^1 -> H(a, f(e)) 5.44/2.27 A^1 -> H(U1(d, e), U1(b, b)) 5.44/2.27 A^1 -> H(U1(d, e), f(d)) 5.44/2.27 A^1 -> H(U1(d, e), f(e)) 5.44/2.27 A^1 -> H(U1(d, a), U1(d, b)) 5.44/2.27 A^1 -> H(U1(d, a), U1(e, b)) 5.44/2.27 A^1 -> H(U1(d, a), U1(b, d)) 5.44/2.27 A^1 -> H(U1(d, a), U1(b, e)) 5.44/2.27 A^1 -> H(U1(d, a), U1(d, d)) 5.44/2.27 A^1 -> H(U1(d, a), U1(e, e)) 5.44/2.27 A^1 -> H(U1(e, d), U1(b, b)) 5.44/2.27 A^1 -> H(U1(e, d), f(d)) 5.44/2.27 A^1 -> H(U1(e, d), f(e)) 5.44/2.27 A^1 -> H(U1(e, a), U1(d, b)) 5.44/2.27 A^1 -> H(U1(e, a), U1(e, b)) 5.44/2.27 A^1 -> H(U1(e, a), U1(b, d)) 5.44/2.27 A^1 -> H(U1(e, a), U1(b, e)) 5.44/2.27 A^1 -> H(U1(e, a), U1(d, d)) 5.44/2.27 A^1 -> H(U1(e, a), U1(e, e)) 5.44/2.27 A^1 -> H(U1(a, d), U1(d, b)) 5.44/2.27 A^1 -> H(U1(a, d), U1(e, b)) 5.44/2.27 A^1 -> H(U1(a, d), U1(b, d)) 5.44/2.27 A^1 -> H(U1(a, d), U1(b, e)) 5.44/2.27 A^1 -> H(U1(a, d), U1(d, d)) 5.44/2.27 A^1 -> H(U1(a, d), U1(e, e)) 5.44/2.27 A^1 -> H(U1(a, e), U1(d, b)) 5.44/2.27 A^1 -> H(U1(a, e), U1(e, b)) 5.44/2.27 A^1 -> H(U1(a, e), U1(b, d)) 5.44/2.27 A^1 -> H(U1(a, e), U1(b, e)) 5.44/2.27 A^1 -> H(U1(a, e), U1(d, d)) 5.44/2.27 A^1 -> H(U1(a, e), U1(e, e)) 5.44/2.27 A^1 -> H(U1(a, a), b) 5.44/2.27 A^1 -> H(U1(a, a), U1(d, e)) 5.44/2.27 A^1 -> H(U1(a, a), U1(e, d)) 5.44/2.29 A^1 -> H(U1(a, a), d) 5.44/2.29 A^1 -> H(d, U1(b, b)) 5.44/2.29 A^1 -> H(d, f(d)) 5.44/2.29 A^1 -> H(d, f(e)) 5.44/2.29 5.44/2.29 The TRS R consists of the following rules: 5.44/2.29 5.44/2.29 a -> d 5.44/2.29 a -> e 5.44/2.29 f(x) -> U1(x, x) 5.44/2.29 b -> d 5.44/2.29 b -> e 5.44/2.29 U1(d, x) -> x 5.44/2.29 5.44/2.29 Q is empty. 5.44/2.29 We have to consider all minimal (P,Q,R)-chains. 5.44/2.29 ---------------------------------------- 5.44/2.29 5.44/2.29 (105) TransformationProof (EQUIVALENT) 5.44/2.29 By narrowing [LPAR04] the rule A^1 -> H(U1(d, d), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: 5.44/2.29 5.44/2.29 (A^1 -> H(d, U1(b, b)),A^1 -> H(d, U1(b, b))) 5.44/2.29 (A^1 -> H(U1(d, d), U1(d, b)),A^1 -> H(U1(d, d), U1(d, b))) 5.44/2.29 (A^1 -> H(U1(d, d), U1(e, b)),A^1 -> H(U1(d, d), U1(e, b))) 5.44/2.29 (A^1 -> H(U1(d, d), U1(b, d)),A^1 -> H(U1(d, d), U1(b, d))) 5.44/2.29 (A^1 -> H(U1(d, d), U1(b, e)),A^1 -> H(U1(d, d), U1(b, e))) 5.44/2.29 5.44/2.29 5.44/2.29 ---------------------------------------- 5.44/2.29 5.44/2.29 (106) 5.44/2.29 Obligation: 5.44/2.29 Q DP problem: 5.44/2.29 The TRS P consists of the following rules: 5.44/2.29 5.44/2.29 H(x, x) -> G(x, x) 5.44/2.29 G(d, e) -> A^1 5.44/2.29 A^1 -> H(f(d), f(d)) 5.44/2.29 A^1 -> H(f(e), f(e)) 5.44/2.29 A^1 -> H(U1(d, d), f(d)) 5.44/2.29 A^1 -> H(U1(d, d), f(e)) 5.44/2.29 A^1 -> H(f(d), U1(d, b)) 5.44/2.29 A^1 -> H(f(d), U1(e, b)) 5.44/2.29 A^1 -> H(f(d), U1(b, d)) 5.44/2.29 A^1 -> H(f(d), U1(b, e)) 5.44/2.29 A^1 -> H(f(d), U1(e, e)) 5.44/2.29 A^1 -> H(U1(e, e), U1(b, b)) 5.44/2.29 A^1 -> H(U1(e, e), f(d)) 5.44/2.29 A^1 -> H(U1(e, e), f(e)) 5.44/2.29 A^1 -> H(f(e), U1(d, b)) 5.44/2.29 A^1 -> H(f(e), U1(e, b)) 5.44/2.29 A^1 -> H(f(e), U1(b, d)) 5.44/2.29 A^1 -> H(f(e), U1(b, e)) 5.44/2.29 A^1 -> H(f(e), U1(d, d)) 5.44/2.29 A^1 -> H(f(a), b) 5.44/2.29 A^1 -> H(f(a), U1(d, e)) 5.44/2.29 A^1 -> H(f(a), U1(e, d)) 5.44/2.29 A^1 -> H(f(d), U1(d, d)) 5.44/2.29 A^1 -> H(f(a), d) 5.44/2.29 A^1 -> H(f(e), U1(e, e)) 5.44/2.29 A^1 -> H(e, f(b)) 5.44/2.29 A^1 -> H(a, U1(b, b)) 5.44/2.29 A^1 -> H(a, f(d)) 5.44/2.29 A^1 -> H(a, f(e)) 5.44/2.29 A^1 -> H(U1(d, e), U1(b, b)) 5.44/2.29 A^1 -> H(U1(d, e), f(d)) 5.44/2.29 A^1 -> H(U1(d, e), f(e)) 5.44/2.29 A^1 -> H(U1(d, a), U1(d, b)) 5.44/2.29 A^1 -> H(U1(d, a), U1(e, b)) 5.44/2.29 A^1 -> H(U1(d, a), U1(b, d)) 5.44/2.29 A^1 -> H(U1(d, a), U1(b, e)) 5.44/2.29 A^1 -> H(U1(d, a), U1(d, d)) 5.44/2.29 A^1 -> H(U1(d, a), U1(e, e)) 5.44/2.29 A^1 -> H(U1(e, d), U1(b, b)) 5.44/2.29 A^1 -> H(U1(e, d), f(d)) 5.44/2.29 A^1 -> H(U1(e, d), f(e)) 5.44/2.29 A^1 -> H(U1(e, a), U1(d, b)) 5.44/2.29 A^1 -> H(U1(e, a), U1(e, b)) 5.44/2.29 A^1 -> H(U1(e, a), U1(b, d)) 5.44/2.29 A^1 -> H(U1(e, a), U1(b, e)) 5.44/2.29 A^1 -> H(U1(e, a), U1(d, d)) 5.44/2.29 A^1 -> H(U1(e, a), U1(e, e)) 5.44/2.29 A^1 -> H(U1(a, d), U1(d, b)) 5.44/2.29 A^1 -> H(U1(a, d), U1(e, b)) 5.44/2.29 A^1 -> H(U1(a, d), U1(b, d)) 5.44/2.29 A^1 -> H(U1(a, d), U1(b, e)) 5.44/2.29 A^1 -> H(U1(a, d), U1(d, d)) 5.44/2.29 A^1 -> H(U1(a, d), U1(e, e)) 5.44/2.29 A^1 -> H(U1(a, e), U1(d, b)) 5.44/2.29 A^1 -> H(U1(a, e), U1(e, b)) 5.44/2.29 A^1 -> H(U1(a, e), U1(b, d)) 5.44/2.29 A^1 -> H(U1(a, e), U1(b, e)) 5.44/2.29 A^1 -> H(U1(a, e), U1(d, d)) 5.44/2.29 A^1 -> H(U1(a, e), U1(e, e)) 5.44/2.29 A^1 -> H(U1(a, a), b) 5.44/2.29 A^1 -> H(U1(a, a), U1(d, e)) 5.44/2.29 A^1 -> H(U1(a, a), U1(e, d)) 5.44/2.29 A^1 -> H(U1(a, a), d) 5.44/2.29 A^1 -> H(d, U1(b, b)) 5.44/2.29 A^1 -> H(d, f(d)) 5.44/2.29 A^1 -> H(d, f(e)) 5.44/2.29 A^1 -> H(U1(d, d), U1(d, b)) 5.44/2.29 A^1 -> H(U1(d, d), U1(e, b)) 5.44/2.29 A^1 -> H(U1(d, d), U1(b, d)) 5.44/2.29 A^1 -> H(U1(d, d), U1(b, e)) 5.44/2.29 5.44/2.29 The TRS R consists of the following rules: 5.44/2.29 5.44/2.29 a -> d 5.44/2.29 a -> e 5.44/2.29 f(x) -> U1(x, x) 5.44/2.29 b -> d 5.44/2.29 b -> e 5.44/2.29 U1(d, x) -> x 5.44/2.29 5.44/2.29 Q is empty. 5.44/2.29 We have to consider all minimal (P,Q,R)-chains. 5.44/2.29 ---------------------------------------- 5.44/2.29 5.44/2.29 (107) TransformationProof (EQUIVALENT) 5.44/2.29 By narrowing [LPAR04] the rule A^1 -> H(U1(d, d), f(d)) at position [] we obtained the following new rules [LPAR04]: 5.44/2.29 5.44/2.29 (A^1 -> H(d, f(d)),A^1 -> H(d, f(d))) 5.44/2.29 (A^1 -> H(U1(d, d), U1(d, d)),A^1 -> H(U1(d, d), U1(d, d))) 5.44/2.29 5.44/2.29 5.44/2.29 ---------------------------------------- 5.44/2.29 5.44/2.29 (108) 5.44/2.29 Obligation: 5.44/2.29 Q DP problem: 5.44/2.29 The TRS P consists of the following rules: 5.44/2.29 5.44/2.29 H(x, x) -> G(x, x) 5.44/2.29 G(d, e) -> A^1 5.44/2.29 A^1 -> H(f(d), f(d)) 5.44/2.29 A^1 -> H(f(e), f(e)) 5.44/2.29 A^1 -> H(U1(d, d), f(e)) 5.44/2.29 A^1 -> H(f(d), U1(d, b)) 5.44/2.29 A^1 -> H(f(d), U1(e, b)) 5.44/2.29 A^1 -> H(f(d), U1(b, d)) 5.44/2.29 A^1 -> H(f(d), U1(b, e)) 5.44/2.29 A^1 -> H(f(d), U1(e, e)) 5.44/2.29 A^1 -> H(U1(e, e), U1(b, b)) 5.44/2.29 A^1 -> H(U1(e, e), f(d)) 5.44/2.29 A^1 -> H(U1(e, e), f(e)) 5.44/2.29 A^1 -> H(f(e), U1(d, b)) 5.44/2.29 A^1 -> H(f(e), U1(e, b)) 5.44/2.29 A^1 -> H(f(e), U1(b, d)) 5.44/2.29 A^1 -> H(f(e), U1(b, e)) 5.44/2.29 A^1 -> H(f(e), U1(d, d)) 5.44/2.29 A^1 -> H(f(a), b) 5.44/2.29 A^1 -> H(f(a), U1(d, e)) 5.44/2.29 A^1 -> H(f(a), U1(e, d)) 5.44/2.29 A^1 -> H(f(d), U1(d, d)) 5.44/2.29 A^1 -> H(f(a), d) 5.44/2.29 A^1 -> H(f(e), U1(e, e)) 5.44/2.29 A^1 -> H(e, f(b)) 5.44/2.29 A^1 -> H(a, U1(b, b)) 5.44/2.29 A^1 -> H(a, f(d)) 5.44/2.29 A^1 -> H(a, f(e)) 5.44/2.29 A^1 -> H(U1(d, e), U1(b, b)) 5.44/2.29 A^1 -> H(U1(d, e), f(d)) 5.44/2.29 A^1 -> H(U1(d, e), f(e)) 5.44/2.29 A^1 -> H(U1(d, a), U1(d, b)) 5.44/2.29 A^1 -> H(U1(d, a), U1(e, b)) 5.44/2.29 A^1 -> H(U1(d, a), U1(b, d)) 5.44/2.29 A^1 -> H(U1(d, a), U1(b, e)) 5.44/2.29 A^1 -> H(U1(d, a), U1(d, d)) 5.44/2.29 A^1 -> H(U1(d, a), U1(e, e)) 5.44/2.29 A^1 -> H(U1(e, d), U1(b, b)) 5.44/2.29 A^1 -> H(U1(e, d), f(d)) 5.44/2.29 A^1 -> H(U1(e, d), f(e)) 5.44/2.29 A^1 -> H(U1(e, a), U1(d, b)) 5.44/2.29 A^1 -> H(U1(e, a), U1(e, b)) 5.44/2.29 A^1 -> H(U1(e, a), U1(b, d)) 5.44/2.29 A^1 -> H(U1(e, a), U1(b, e)) 5.44/2.29 A^1 -> H(U1(e, a), U1(d, d)) 5.44/2.29 A^1 -> H(U1(e, a), U1(e, e)) 5.44/2.29 A^1 -> H(U1(a, d), U1(d, b)) 5.44/2.29 A^1 -> H(U1(a, d), U1(e, b)) 5.44/2.29 A^1 -> H(U1(a, d), U1(b, d)) 5.44/2.29 A^1 -> H(U1(a, d), U1(b, e)) 5.44/2.29 A^1 -> H(U1(a, d), U1(d, d)) 5.44/2.29 A^1 -> H(U1(a, d), U1(e, e)) 5.44/2.29 A^1 -> H(U1(a, e), U1(d, b)) 5.44/2.29 A^1 -> H(U1(a, e), U1(e, b)) 5.44/2.29 A^1 -> H(U1(a, e), U1(b, d)) 5.44/2.29 A^1 -> H(U1(a, e), U1(b, e)) 5.44/2.29 A^1 -> H(U1(a, e), U1(d, d)) 5.44/2.29 A^1 -> H(U1(a, e), U1(e, e)) 5.44/2.29 A^1 -> H(U1(a, a), b) 5.44/2.29 A^1 -> H(U1(a, a), U1(d, e)) 5.44/2.29 A^1 -> H(U1(a, a), U1(e, d)) 5.44/2.29 A^1 -> H(U1(a, a), d) 5.44/2.29 A^1 -> H(d, U1(b, b)) 5.44/2.29 A^1 -> H(d, f(d)) 5.44/2.29 A^1 -> H(d, f(e)) 5.44/2.29 A^1 -> H(U1(d, d), U1(d, b)) 5.44/2.29 A^1 -> H(U1(d, d), U1(e, b)) 5.44/2.29 A^1 -> H(U1(d, d), U1(b, d)) 5.44/2.29 A^1 -> H(U1(d, d), U1(b, e)) 5.44/2.29 A^1 -> H(U1(d, d), U1(d, d)) 5.44/2.29 5.44/2.29 The TRS R consists of the following rules: 5.44/2.29 5.44/2.29 a -> d 5.44/2.29 a -> e 5.44/2.29 f(x) -> U1(x, x) 5.44/2.29 b -> d 5.44/2.29 b -> e 5.44/2.29 U1(d, x) -> x 5.44/2.29 5.44/2.29 Q is empty. 5.44/2.29 We have to consider all minimal (P,Q,R)-chains. 5.44/2.29 ---------------------------------------- 5.44/2.29 5.44/2.29 (109) TransformationProof (EQUIVALENT) 5.44/2.29 By narrowing [LPAR04] the rule A^1 -> H(U1(d, d), f(e)) at position [] we obtained the following new rules [LPAR04]: 5.44/2.29 5.44/2.29 (A^1 -> H(d, f(e)),A^1 -> H(d, f(e))) 5.44/2.29 (A^1 -> H(U1(d, d), U1(e, e)),A^1 -> H(U1(d, d), U1(e, e))) 5.44/2.29 5.44/2.29 5.44/2.29 ---------------------------------------- 5.44/2.29 5.44/2.29 (110) 5.44/2.29 Obligation: 5.44/2.29 Q DP problem: 5.44/2.29 The TRS P consists of the following rules: 5.44/2.29 5.44/2.29 H(x, x) -> G(x, x) 5.44/2.29 G(d, e) -> A^1 5.44/2.29 A^1 -> H(f(d), f(d)) 5.44/2.29 A^1 -> H(f(e), f(e)) 5.44/2.29 A^1 -> H(f(d), U1(d, b)) 5.44/2.29 A^1 -> H(f(d), U1(e, b)) 5.44/2.29 A^1 -> H(f(d), U1(b, d)) 5.44/2.29 A^1 -> H(f(d), U1(b, e)) 5.44/2.29 A^1 -> H(f(d), U1(e, e)) 5.44/2.29 A^1 -> H(U1(e, e), U1(b, b)) 5.44/2.29 A^1 -> H(U1(e, e), f(d)) 5.44/2.29 A^1 -> H(U1(e, e), f(e)) 5.44/2.29 A^1 -> H(f(e), U1(d, b)) 5.44/2.29 A^1 -> H(f(e), U1(e, b)) 5.44/2.29 A^1 -> H(f(e), U1(b, d)) 5.44/2.29 A^1 -> H(f(e), U1(b, e)) 5.44/2.29 A^1 -> H(f(e), U1(d, d)) 5.44/2.29 A^1 -> H(f(a), b) 5.44/2.29 A^1 -> H(f(a), U1(d, e)) 5.44/2.29 A^1 -> H(f(a), U1(e, d)) 5.44/2.29 A^1 -> H(f(d), U1(d, d)) 5.44/2.29 A^1 -> H(f(a), d) 5.44/2.29 A^1 -> H(f(e), U1(e, e)) 5.44/2.29 A^1 -> H(e, f(b)) 5.44/2.29 A^1 -> H(a, U1(b, b)) 5.44/2.29 A^1 -> H(a, f(d)) 5.44/2.29 A^1 -> H(a, f(e)) 5.44/2.29 A^1 -> H(U1(d, e), U1(b, b)) 5.44/2.29 A^1 -> H(U1(d, e), f(d)) 5.44/2.29 A^1 -> H(U1(d, e), f(e)) 5.44/2.29 A^1 -> H(U1(d, a), U1(d, b)) 5.44/2.29 A^1 -> H(U1(d, a), U1(e, b)) 5.44/2.29 A^1 -> H(U1(d, a), U1(b, d)) 5.44/2.29 A^1 -> H(U1(d, a), U1(b, e)) 5.44/2.29 A^1 -> H(U1(d, a), U1(d, d)) 5.44/2.29 A^1 -> H(U1(d, a), U1(e, e)) 5.44/2.29 A^1 -> H(U1(e, d), U1(b, b)) 5.44/2.29 A^1 -> H(U1(e, d), f(d)) 5.44/2.29 A^1 -> H(U1(e, d), f(e)) 5.44/2.29 A^1 -> H(U1(e, a), U1(d, b)) 5.44/2.29 A^1 -> H(U1(e, a), U1(e, b)) 5.44/2.29 A^1 -> H(U1(e, a), U1(b, d)) 5.44/2.29 A^1 -> H(U1(e, a), U1(b, e)) 5.44/2.29 A^1 -> H(U1(e, a), U1(d, d)) 5.44/2.29 A^1 -> H(U1(e, a), U1(e, e)) 5.44/2.29 A^1 -> H(U1(a, d), U1(d, b)) 5.44/2.29 A^1 -> H(U1(a, d), U1(e, b)) 5.44/2.29 A^1 -> H(U1(a, d), U1(b, d)) 5.44/2.29 A^1 -> H(U1(a, d), U1(b, e)) 5.44/2.29 A^1 -> H(U1(a, d), U1(d, d)) 5.44/2.29 A^1 -> H(U1(a, d), U1(e, e)) 5.44/2.29 A^1 -> H(U1(a, e), U1(d, b)) 5.44/2.29 A^1 -> H(U1(a, e), U1(e, b)) 5.44/2.29 A^1 -> H(U1(a, e), U1(b, d)) 5.44/2.29 A^1 -> H(U1(a, e), U1(b, e)) 5.44/2.29 A^1 -> H(U1(a, e), U1(d, d)) 5.44/2.29 A^1 -> H(U1(a, e), U1(e, e)) 5.44/2.29 A^1 -> H(U1(a, a), b) 5.44/2.29 A^1 -> H(U1(a, a), U1(d, e)) 5.44/2.29 A^1 -> H(U1(a, a), U1(e, d)) 5.44/2.29 A^1 -> H(U1(a, a), d) 5.44/2.29 A^1 -> H(d, U1(b, b)) 5.44/2.29 A^1 -> H(d, f(d)) 5.44/2.29 A^1 -> H(d, f(e)) 5.44/2.29 A^1 -> H(U1(d, d), U1(d, b)) 5.44/2.29 A^1 -> H(U1(d, d), U1(e, b)) 5.44/2.29 A^1 -> H(U1(d, d), U1(b, d)) 5.44/2.29 A^1 -> H(U1(d, d), U1(b, e)) 5.44/2.29 A^1 -> H(U1(d, d), U1(d, d)) 5.44/2.29 A^1 -> H(U1(d, d), U1(e, e)) 5.44/2.29 5.44/2.29 The TRS R consists of the following rules: 5.44/2.29 5.44/2.29 a -> d 5.44/2.29 a -> e 5.44/2.29 f(x) -> U1(x, x) 5.44/2.29 b -> d 5.44/2.29 b -> e 5.44/2.29 U1(d, x) -> x 5.44/2.29 5.44/2.29 Q is empty. 5.44/2.29 We have to consider all minimal (P,Q,R)-chains. 5.44/2.29 ---------------------------------------- 5.44/2.29 5.44/2.29 (111) TransformationProof (EQUIVALENT) 5.44/2.29 By narrowing [LPAR04] the rule A^1 -> H(f(d), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: 5.44/2.29 5.44/2.29 (A^1 -> H(U1(d, d), U1(d, b)),A^1 -> H(U1(d, d), U1(d, b))) 5.44/2.29 (A^1 -> H(f(d), b),A^1 -> H(f(d), b)) 5.44/2.29 (A^1 -> H(f(d), U1(d, d)),A^1 -> H(f(d), U1(d, d))) 5.44/2.29 (A^1 -> H(f(d), U1(d, e)),A^1 -> H(f(d), U1(d, e))) 5.44/2.29 5.44/2.29 5.44/2.29 ---------------------------------------- 5.44/2.29 5.44/2.29 (112) 5.44/2.29 Obligation: 5.44/2.29 Q DP problem: 5.44/2.29 The TRS P consists of the following rules: 5.44/2.29 5.44/2.29 H(x, x) -> G(x, x) 5.44/2.29 G(d, e) -> A^1 5.44/2.29 A^1 -> H(f(d), f(d)) 5.44/2.29 A^1 -> H(f(e), f(e)) 5.44/2.29 A^1 -> H(f(d), U1(e, b)) 5.44/2.29 A^1 -> H(f(d), U1(b, d)) 5.44/2.29 A^1 -> H(f(d), U1(b, e)) 5.44/2.29 A^1 -> H(f(d), U1(e, e)) 5.44/2.29 A^1 -> H(U1(e, e), U1(b, b)) 5.44/2.29 A^1 -> H(U1(e, e), f(d)) 5.44/2.29 A^1 -> H(U1(e, e), f(e)) 5.44/2.29 A^1 -> H(f(e), U1(d, b)) 5.44/2.29 A^1 -> H(f(e), U1(e, b)) 5.44/2.29 A^1 -> H(f(e), U1(b, d)) 5.44/2.29 A^1 -> H(f(e), U1(b, e)) 5.44/2.29 A^1 -> H(f(e), U1(d, d)) 5.44/2.29 A^1 -> H(f(a), b) 5.44/2.29 A^1 -> H(f(a), U1(d, e)) 5.44/2.29 A^1 -> H(f(a), U1(e, d)) 5.44/2.29 A^1 -> H(f(d), U1(d, d)) 5.44/2.29 A^1 -> H(f(a), d) 5.44/2.29 A^1 -> H(f(e), U1(e, e)) 5.44/2.29 A^1 -> H(e, f(b)) 5.44/2.29 A^1 -> H(a, U1(b, b)) 5.44/2.29 A^1 -> H(a, f(d)) 5.44/2.29 A^1 -> H(a, f(e)) 5.44/2.29 A^1 -> H(U1(d, e), U1(b, b)) 5.44/2.29 A^1 -> H(U1(d, e), f(d)) 5.44/2.29 A^1 -> H(U1(d, e), f(e)) 5.44/2.29 A^1 -> H(U1(d, a), U1(d, b)) 5.44/2.29 A^1 -> H(U1(d, a), U1(e, b)) 5.44/2.29 A^1 -> H(U1(d, a), U1(b, d)) 5.44/2.29 A^1 -> H(U1(d, a), U1(b, e)) 5.44/2.29 A^1 -> H(U1(d, a), U1(d, d)) 5.44/2.29 A^1 -> H(U1(d, a), U1(e, e)) 5.44/2.29 A^1 -> H(U1(e, d), U1(b, b)) 5.44/2.29 A^1 -> H(U1(e, d), f(d)) 5.44/2.29 A^1 -> H(U1(e, d), f(e)) 5.44/2.29 A^1 -> H(U1(e, a), U1(d, b)) 5.44/2.29 A^1 -> H(U1(e, a), U1(e, b)) 5.44/2.29 A^1 -> H(U1(e, a), U1(b, d)) 5.44/2.29 A^1 -> H(U1(e, a), U1(b, e)) 5.44/2.29 A^1 -> H(U1(e, a), U1(d, d)) 5.44/2.29 A^1 -> H(U1(e, a), U1(e, e)) 5.44/2.29 A^1 -> H(U1(a, d), U1(d, b)) 5.44/2.29 A^1 -> H(U1(a, d), U1(e, b)) 5.44/2.29 A^1 -> H(U1(a, d), U1(b, d)) 5.44/2.29 A^1 -> H(U1(a, d), U1(b, e)) 5.44/2.29 A^1 -> H(U1(a, d), U1(d, d)) 5.44/2.29 A^1 -> H(U1(a, d), U1(e, e)) 5.44/2.29 A^1 -> H(U1(a, e), U1(d, b)) 5.44/2.29 A^1 -> H(U1(a, e), U1(e, b)) 5.44/2.29 A^1 -> H(U1(a, e), U1(b, d)) 5.44/2.29 A^1 -> H(U1(a, e), U1(b, e)) 5.44/2.29 A^1 -> H(U1(a, e), U1(d, d)) 5.44/2.29 A^1 -> H(U1(a, e), U1(e, e)) 5.44/2.29 A^1 -> H(U1(a, a), b) 5.44/2.29 A^1 -> H(U1(a, a), U1(d, e)) 5.44/2.29 A^1 -> H(U1(a, a), U1(e, d)) 5.44/2.29 A^1 -> H(U1(a, a), d) 5.44/2.29 A^1 -> H(d, U1(b, b)) 5.44/2.29 A^1 -> H(d, f(d)) 5.44/2.29 A^1 -> H(d, f(e)) 5.44/2.29 A^1 -> H(U1(d, d), U1(d, b)) 5.44/2.29 A^1 -> H(U1(d, d), U1(e, b)) 5.44/2.29 A^1 -> H(U1(d, d), U1(b, d)) 5.44/2.29 A^1 -> H(U1(d, d), U1(b, e)) 5.44/2.29 A^1 -> H(U1(d, d), U1(d, d)) 5.44/2.29 A^1 -> H(U1(d, d), U1(e, e)) 5.44/2.29 A^1 -> H(f(d), b) 5.44/2.29 A^1 -> H(f(d), U1(d, e)) 5.44/2.29 5.44/2.29 The TRS R consists of the following rules: 5.44/2.29 5.44/2.29 a -> d 5.44/2.29 a -> e 5.44/2.29 f(x) -> U1(x, x) 5.44/2.29 b -> d 5.44/2.29 b -> e 5.44/2.29 U1(d, x) -> x 5.44/2.29 5.44/2.29 Q is empty. 5.44/2.29 We have to consider all minimal (P,Q,R)-chains. 5.44/2.29 ---------------------------------------- 5.44/2.29 5.44/2.29 (113) TransformationProof (EQUIVALENT) 5.44/2.29 By narrowing [LPAR04] the rule A^1 -> H(f(d), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: 5.44/2.29 5.44/2.29 (A^1 -> H(U1(d, d), U1(e, b)),A^1 -> H(U1(d, d), U1(e, b))) 5.44/2.29 (A^1 -> H(f(d), U1(e, d)),A^1 -> H(f(d), U1(e, d))) 5.44/2.29 (A^1 -> H(f(d), U1(e, e)),A^1 -> H(f(d), U1(e, e))) 5.44/2.29 5.44/2.29 5.44/2.29 ---------------------------------------- 5.44/2.29 5.44/2.29 (114) 5.44/2.29 Obligation: 5.44/2.29 Q DP problem: 5.44/2.29 The TRS P consists of the following rules: 5.44/2.29 5.44/2.29 H(x, x) -> G(x, x) 5.44/2.29 G(d, e) -> A^1 5.44/2.29 A^1 -> H(f(d), f(d)) 5.44/2.29 A^1 -> H(f(e), f(e)) 5.44/2.29 A^1 -> H(f(d), U1(b, d)) 5.44/2.29 A^1 -> H(f(d), U1(b, e)) 5.44/2.29 A^1 -> H(f(d), U1(e, e)) 5.44/2.29 A^1 -> H(U1(e, e), U1(b, b)) 5.44/2.29 A^1 -> H(U1(e, e), f(d)) 5.44/2.29 A^1 -> H(U1(e, e), f(e)) 5.44/2.29 A^1 -> H(f(e), U1(d, b)) 5.44/2.29 A^1 -> H(f(e), U1(e, b)) 5.44/2.29 A^1 -> H(f(e), U1(b, d)) 5.44/2.29 A^1 -> H(f(e), U1(b, e)) 5.44/2.29 A^1 -> H(f(e), U1(d, d)) 5.44/2.29 A^1 -> H(f(a), b) 5.44/2.29 A^1 -> H(f(a), U1(d, e)) 5.44/2.29 A^1 -> H(f(a), U1(e, d)) 5.44/2.29 A^1 -> H(f(d), U1(d, d)) 5.44/2.29 A^1 -> H(f(a), d) 5.44/2.29 A^1 -> H(f(e), U1(e, e)) 5.44/2.29 A^1 -> H(e, f(b)) 5.44/2.29 A^1 -> H(a, U1(b, b)) 5.44/2.29 A^1 -> H(a, f(d)) 5.44/2.29 A^1 -> H(a, f(e)) 5.44/2.29 A^1 -> H(U1(d, e), U1(b, b)) 5.44/2.29 A^1 -> H(U1(d, e), f(d)) 5.44/2.29 A^1 -> H(U1(d, e), f(e)) 5.44/2.29 A^1 -> H(U1(d, a), U1(d, b)) 5.44/2.29 A^1 -> H(U1(d, a), U1(e, b)) 5.44/2.29 A^1 -> H(U1(d, a), U1(b, d)) 5.44/2.29 A^1 -> H(U1(d, a), U1(b, e)) 5.44/2.29 A^1 -> H(U1(d, a), U1(d, d)) 5.44/2.29 A^1 -> H(U1(d, a), U1(e, e)) 5.44/2.29 A^1 -> H(U1(e, d), U1(b, b)) 5.44/2.29 A^1 -> H(U1(e, d), f(d)) 5.44/2.29 A^1 -> H(U1(e, d), f(e)) 5.44/2.29 A^1 -> H(U1(e, a), U1(d, b)) 5.44/2.29 A^1 -> H(U1(e, a), U1(e, b)) 5.44/2.29 A^1 -> H(U1(e, a), U1(b, d)) 5.44/2.29 A^1 -> H(U1(e, a), U1(b, e)) 5.44/2.29 A^1 -> H(U1(e, a), U1(d, d)) 5.44/2.29 A^1 -> H(U1(e, a), U1(e, e)) 5.44/2.29 A^1 -> H(U1(a, d), U1(d, b)) 5.44/2.29 A^1 -> H(U1(a, d), U1(e, b)) 5.44/2.29 A^1 -> H(U1(a, d), U1(b, d)) 5.44/2.29 A^1 -> H(U1(a, d), U1(b, e)) 5.44/2.29 A^1 -> H(U1(a, d), U1(d, d)) 5.44/2.29 A^1 -> H(U1(a, d), U1(e, e)) 5.44/2.29 A^1 -> H(U1(a, e), U1(d, b)) 5.44/2.29 A^1 -> H(U1(a, e), U1(e, b)) 5.44/2.29 A^1 -> H(U1(a, e), U1(b, d)) 5.44/2.29 A^1 -> H(U1(a, e), U1(b, e)) 5.44/2.29 A^1 -> H(U1(a, e), U1(d, d)) 5.44/2.29 A^1 -> H(U1(a, e), U1(e, e)) 5.44/2.29 A^1 -> H(U1(a, a), b) 5.44/2.29 A^1 -> H(U1(a, a), U1(d, e)) 5.44/2.29 A^1 -> H(U1(a, a), U1(e, d)) 5.44/2.29 A^1 -> H(U1(a, a), d) 5.44/2.29 A^1 -> H(d, U1(b, b)) 5.44/2.29 A^1 -> H(d, f(d)) 5.44/2.29 A^1 -> H(d, f(e)) 5.44/2.29 A^1 -> H(U1(d, d), U1(d, b)) 5.44/2.29 A^1 -> H(U1(d, d), U1(e, b)) 5.44/2.29 A^1 -> H(U1(d, d), U1(b, d)) 5.44/2.29 A^1 -> H(U1(d, d), U1(b, e)) 5.44/2.29 A^1 -> H(U1(d, d), U1(d, d)) 5.44/2.29 A^1 -> H(U1(d, d), U1(e, e)) 5.44/2.29 A^1 -> H(f(d), b) 5.44/2.29 A^1 -> H(f(d), U1(d, e)) 5.44/2.29 A^1 -> H(f(d), U1(e, d)) 5.44/2.29 5.44/2.29 The TRS R consists of the following rules: 5.44/2.29 5.44/2.29 a -> d 5.44/2.29 a -> e 5.44/2.29 f(x) -> U1(x, x) 5.44/2.29 b -> d 5.44/2.29 b -> e 5.44/2.29 U1(d, x) -> x 5.44/2.29 5.44/2.29 Q is empty. 5.44/2.29 We have to consider all minimal (P,Q,R)-chains. 5.44/2.29 ---------------------------------------- 5.44/2.29 5.44/2.29 (115) TransformationProof (EQUIVALENT) 5.44/2.29 By narrowing [LPAR04] the rule A^1 -> H(f(d), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: 5.44/2.29 5.44/2.29 (A^1 -> H(U1(d, d), U1(b, d)),A^1 -> H(U1(d, d), U1(b, d))) 5.44/2.29 (A^1 -> H(f(d), U1(d, d)),A^1 -> H(f(d), U1(d, d))) 5.44/2.29 (A^1 -> H(f(d), U1(e, d)),A^1 -> H(f(d), U1(e, d))) 5.44/2.29 5.44/2.29 5.44/2.29 ---------------------------------------- 5.44/2.29 5.44/2.29 (116) 5.44/2.29 Obligation: 5.44/2.29 Q DP problem: 5.44/2.29 The TRS P consists of the following rules: 5.44/2.29 5.44/2.29 H(x, x) -> G(x, x) 5.44/2.29 G(d, e) -> A^1 5.44/2.29 A^1 -> H(f(d), f(d)) 5.44/2.29 A^1 -> H(f(e), f(e)) 5.44/2.29 A^1 -> H(f(d), U1(b, e)) 5.44/2.29 A^1 -> H(f(d), U1(e, e)) 5.44/2.29 A^1 -> H(U1(e, e), U1(b, b)) 5.44/2.29 A^1 -> H(U1(e, e), f(d)) 5.44/2.29 A^1 -> H(U1(e, e), f(e)) 5.44/2.29 A^1 -> H(f(e), U1(d, b)) 5.44/2.29 A^1 -> H(f(e), U1(e, b)) 5.44/2.29 A^1 -> H(f(e), U1(b, d)) 5.44/2.29 A^1 -> H(f(e), U1(b, e)) 5.44/2.29 A^1 -> H(f(e), U1(d, d)) 5.44/2.29 A^1 -> H(f(a), b) 5.44/2.29 A^1 -> H(f(a), U1(d, e)) 5.44/2.29 A^1 -> H(f(a), U1(e, d)) 5.44/2.29 A^1 -> H(f(d), U1(d, d)) 5.44/2.29 A^1 -> H(f(a), d) 5.44/2.29 A^1 -> H(f(e), U1(e, e)) 5.44/2.29 A^1 -> H(e, f(b)) 5.44/2.29 A^1 -> H(a, U1(b, b)) 5.44/2.29 A^1 -> H(a, f(d)) 5.44/2.29 A^1 -> H(a, f(e)) 5.44/2.29 A^1 -> H(U1(d, e), U1(b, b)) 5.44/2.29 A^1 -> H(U1(d, e), f(d)) 5.44/2.29 A^1 -> H(U1(d, e), f(e)) 5.44/2.29 A^1 -> H(U1(d, a), U1(d, b)) 5.44/2.29 A^1 -> H(U1(d, a), U1(e, b)) 5.44/2.29 A^1 -> H(U1(d, a), U1(b, d)) 5.44/2.29 A^1 -> H(U1(d, a), U1(b, e)) 5.44/2.29 A^1 -> H(U1(d, a), U1(d, d)) 5.44/2.29 A^1 -> H(U1(d, a), U1(e, e)) 5.44/2.29 A^1 -> H(U1(e, d), U1(b, b)) 5.44/2.29 A^1 -> H(U1(e, d), f(d)) 5.44/2.29 A^1 -> H(U1(e, d), f(e)) 5.44/2.29 A^1 -> H(U1(e, a), U1(d, b)) 5.44/2.29 A^1 -> H(U1(e, a), U1(e, b)) 5.44/2.29 A^1 -> H(U1(e, a), U1(b, d)) 5.44/2.29 A^1 -> H(U1(e, a), U1(b, e)) 5.44/2.29 A^1 -> H(U1(e, a), U1(d, d)) 5.44/2.29 A^1 -> H(U1(e, a), U1(e, e)) 5.44/2.29 A^1 -> H(U1(a, d), U1(d, b)) 5.44/2.29 A^1 -> H(U1(a, d), U1(e, b)) 5.44/2.29 A^1 -> H(U1(a, d), U1(b, d)) 5.44/2.29 A^1 -> H(U1(a, d), U1(b, e)) 5.44/2.29 A^1 -> H(U1(a, d), U1(d, d)) 5.44/2.29 A^1 -> H(U1(a, d), U1(e, e)) 5.44/2.29 A^1 -> H(U1(a, e), U1(d, b)) 5.44/2.29 A^1 -> H(U1(a, e), U1(e, b)) 5.44/2.29 A^1 -> H(U1(a, e), U1(b, d)) 5.44/2.29 A^1 -> H(U1(a, e), U1(b, e)) 5.44/2.29 A^1 -> H(U1(a, e), U1(d, d)) 5.44/2.29 A^1 -> H(U1(a, e), U1(e, e)) 5.44/2.29 A^1 -> H(U1(a, a), b) 5.44/2.29 A^1 -> H(U1(a, a), U1(d, e)) 5.44/2.29 A^1 -> H(U1(a, a), U1(e, d)) 5.44/2.29 A^1 -> H(U1(a, a), d) 5.44/2.29 A^1 -> H(d, U1(b, b)) 5.44/2.29 A^1 -> H(d, f(d)) 5.44/2.29 A^1 -> H(d, f(e)) 5.44/2.29 A^1 -> H(U1(d, d), U1(d, b)) 5.44/2.29 A^1 -> H(U1(d, d), U1(e, b)) 5.44/2.29 A^1 -> H(U1(d, d), U1(b, d)) 5.44/2.29 A^1 -> H(U1(d, d), U1(b, e)) 5.44/2.29 A^1 -> H(U1(d, d), U1(d, d)) 5.44/2.29 A^1 -> H(U1(d, d), U1(e, e)) 5.44/2.29 A^1 -> H(f(d), b) 5.44/2.29 A^1 -> H(f(d), U1(d, e)) 5.44/2.29 A^1 -> H(f(d), U1(e, d)) 5.44/2.29 5.44/2.29 The TRS R consists of the following rules: 5.44/2.29 5.44/2.29 a -> d 5.44/2.29 a -> e 5.44/2.29 f(x) -> U1(x, x) 5.44/2.29 b -> d 5.44/2.29 b -> e 5.44/2.29 U1(d, x) -> x 5.44/2.29 5.44/2.29 Q is empty. 5.44/2.29 We have to consider all minimal (P,Q,R)-chains. 5.44/2.29 ---------------------------------------- 5.44/2.29 5.44/2.29 (117) TransformationProof (EQUIVALENT) 5.44/2.29 By narrowing [LPAR04] the rule A^1 -> H(f(d), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: 5.44/2.29 5.44/2.29 (A^1 -> H(U1(d, d), U1(b, e)),A^1 -> H(U1(d, d), U1(b, e))) 5.44/2.29 (A^1 -> H(f(d), U1(d, e)),A^1 -> H(f(d), U1(d, e))) 5.44/2.29 (A^1 -> H(f(d), U1(e, e)),A^1 -> H(f(d), U1(e, e))) 5.44/2.29 5.44/2.29 5.44/2.29 ---------------------------------------- 5.44/2.29 5.44/2.29 (118) 5.44/2.29 Obligation: 5.44/2.29 Q DP problem: 5.44/2.29 The TRS P consists of the following rules: 5.44/2.29 5.44/2.29 H(x, x) -> G(x, x) 5.44/2.29 G(d, e) -> A^1 5.44/2.29 A^1 -> H(f(d), f(d)) 5.44/2.29 A^1 -> H(f(e), f(e)) 5.44/2.29 A^1 -> H(f(d), U1(e, e)) 5.44/2.29 A^1 -> H(U1(e, e), U1(b, b)) 5.44/2.29 A^1 -> H(U1(e, e), f(d)) 5.44/2.29 A^1 -> H(U1(e, e), f(e)) 5.44/2.29 A^1 -> H(f(e), U1(d, b)) 5.44/2.29 A^1 -> H(f(e), U1(e, b)) 5.44/2.29 A^1 -> H(f(e), U1(b, d)) 5.44/2.29 A^1 -> H(f(e), U1(b, e)) 5.44/2.29 A^1 -> H(f(e), U1(d, d)) 5.44/2.29 A^1 -> H(f(a), b) 5.44/2.29 A^1 -> H(f(a), U1(d, e)) 5.44/2.29 A^1 -> H(f(a), U1(e, d)) 5.44/2.29 A^1 -> H(f(d), U1(d, d)) 5.44/2.29 A^1 -> H(f(a), d) 5.44/2.29 A^1 -> H(f(e), U1(e, e)) 5.44/2.29 A^1 -> H(e, f(b)) 5.44/2.29 A^1 -> H(a, U1(b, b)) 5.44/2.29 A^1 -> H(a, f(d)) 5.44/2.29 A^1 -> H(a, f(e)) 5.44/2.29 A^1 -> H(U1(d, e), U1(b, b)) 5.44/2.29 A^1 -> H(U1(d, e), f(d)) 5.44/2.29 A^1 -> H(U1(d, e), f(e)) 5.44/2.29 A^1 -> H(U1(d, a), U1(d, b)) 5.44/2.29 A^1 -> H(U1(d, a), U1(e, b)) 5.44/2.29 A^1 -> H(U1(d, a), U1(b, d)) 5.44/2.29 A^1 -> H(U1(d, a), U1(b, e)) 5.44/2.29 A^1 -> H(U1(d, a), U1(d, d)) 5.44/2.29 A^1 -> H(U1(d, a), U1(e, e)) 5.44/2.29 A^1 -> H(U1(e, d), U1(b, b)) 5.44/2.29 A^1 -> H(U1(e, d), f(d)) 5.44/2.29 A^1 -> H(U1(e, d), f(e)) 5.44/2.29 A^1 -> H(U1(e, a), U1(d, b)) 5.44/2.29 A^1 -> H(U1(e, a), U1(e, b)) 5.44/2.29 A^1 -> H(U1(e, a), U1(b, d)) 5.44/2.29 A^1 -> H(U1(e, a), U1(b, e)) 5.44/2.29 A^1 -> H(U1(e, a), U1(d, d)) 5.44/2.29 A^1 -> H(U1(e, a), U1(e, e)) 5.44/2.29 A^1 -> H(U1(a, d), U1(d, b)) 5.44/2.29 A^1 -> H(U1(a, d), U1(e, b)) 5.44/2.29 A^1 -> H(U1(a, d), U1(b, d)) 5.44/2.29 A^1 -> H(U1(a, d), U1(b, e)) 5.44/2.29 A^1 -> H(U1(a, d), U1(d, d)) 5.44/2.29 A^1 -> H(U1(a, d), U1(e, e)) 5.44/2.29 A^1 -> H(U1(a, e), U1(d, b)) 5.44/2.29 A^1 -> H(U1(a, e), U1(e, b)) 5.44/2.29 A^1 -> H(U1(a, e), U1(b, d)) 5.44/2.29 A^1 -> H(U1(a, e), U1(b, e)) 5.44/2.29 A^1 -> H(U1(a, e), U1(d, d)) 5.44/2.29 A^1 -> H(U1(a, e), U1(e, e)) 5.44/2.29 A^1 -> H(U1(a, a), b) 5.44/2.29 A^1 -> H(U1(a, a), U1(d, e)) 5.44/2.29 A^1 -> H(U1(a, a), U1(e, d)) 5.44/2.29 A^1 -> H(U1(a, a), d) 5.44/2.29 A^1 -> H(d, U1(b, b)) 5.44/2.29 A^1 -> H(d, f(d)) 5.44/2.29 A^1 -> H(d, f(e)) 5.44/2.29 A^1 -> H(U1(d, d), U1(d, b)) 5.44/2.29 A^1 -> H(U1(d, d), U1(e, b)) 5.44/2.29 A^1 -> H(U1(d, d), U1(b, d)) 5.44/2.29 A^1 -> H(U1(d, d), U1(b, e)) 5.44/2.29 A^1 -> H(U1(d, d), U1(d, d)) 5.44/2.29 A^1 -> H(U1(d, d), U1(e, e)) 5.44/2.29 A^1 -> H(f(d), b) 5.44/2.29 A^1 -> H(f(d), U1(d, e)) 5.44/2.29 A^1 -> H(f(d), U1(e, d)) 5.44/2.29 5.44/2.29 The TRS R consists of the following rules: 5.44/2.29 5.44/2.29 a -> d 5.44/2.29 a -> e 5.44/2.29 f(x) -> U1(x, x) 5.44/2.29 b -> d 5.44/2.29 b -> e 5.44/2.29 U1(d, x) -> x 5.44/2.29 5.44/2.29 Q is empty. 5.44/2.29 We have to consider all minimal (P,Q,R)-chains. 5.44/2.29 ---------------------------------------- 5.44/2.29 5.44/2.29 (119) TransformationProof (EQUIVALENT) 5.44/2.29 By narrowing [LPAR04] the rule A^1 -> H(f(d), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: 5.44/2.29 5.44/2.29 (A^1 -> H(U1(d, d), U1(e, e)),A^1 -> H(U1(d, d), U1(e, e))) 5.44/2.29 5.44/2.29 5.44/2.29 ---------------------------------------- 5.44/2.29 5.44/2.29 (120) 5.44/2.29 Obligation: 5.44/2.29 Q DP problem: 5.44/2.29 The TRS P consists of the following rules: 5.44/2.29 5.44/2.29 H(x, x) -> G(x, x) 5.44/2.29 G(d, e) -> A^1 5.44/2.29 A^1 -> H(f(d), f(d)) 5.44/2.29 A^1 -> H(f(e), f(e)) 5.44/2.29 A^1 -> H(U1(e, e), U1(b, b)) 5.44/2.29 A^1 -> H(U1(e, e), f(d)) 5.44/2.29 A^1 -> H(U1(e, e), f(e)) 5.44/2.29 A^1 -> H(f(e), U1(d, b)) 5.44/2.29 A^1 -> H(f(e), U1(e, b)) 5.44/2.29 A^1 -> H(f(e), U1(b, d)) 5.44/2.29 A^1 -> H(f(e), U1(b, e)) 5.44/2.29 A^1 -> H(f(e), U1(d, d)) 5.44/2.29 A^1 -> H(f(a), b) 5.44/2.29 A^1 -> H(f(a), U1(d, e)) 5.44/2.29 A^1 -> H(f(a), U1(e, d)) 5.44/2.29 A^1 -> H(f(d), U1(d, d)) 5.44/2.29 A^1 -> H(f(a), d) 5.44/2.29 A^1 -> H(f(e), U1(e, e)) 5.44/2.29 A^1 -> H(e, f(b)) 5.44/2.29 A^1 -> H(a, U1(b, b)) 5.44/2.29 A^1 -> H(a, f(d)) 5.44/2.29 A^1 -> H(a, f(e)) 5.44/2.29 A^1 -> H(U1(d, e), U1(b, b)) 5.44/2.29 A^1 -> H(U1(d, e), f(d)) 5.44/2.29 A^1 -> H(U1(d, e), f(e)) 5.44/2.29 A^1 -> H(U1(d, a), U1(d, b)) 5.44/2.29 A^1 -> H(U1(d, a), U1(e, b)) 5.44/2.29 A^1 -> H(U1(d, a), U1(b, d)) 5.44/2.29 A^1 -> H(U1(d, a), U1(b, e)) 5.44/2.29 A^1 -> H(U1(d, a), U1(d, d)) 5.44/2.29 A^1 -> H(U1(d, a), U1(e, e)) 5.44/2.29 A^1 -> H(U1(e, d), U1(b, b)) 5.44/2.29 A^1 -> H(U1(e, d), f(d)) 5.44/2.29 A^1 -> H(U1(e, d), f(e)) 5.44/2.29 A^1 -> H(U1(e, a), U1(d, b)) 5.44/2.30 A^1 -> H(U1(e, a), U1(e, b)) 5.44/2.30 A^1 -> H(U1(e, a), U1(b, d)) 5.44/2.30 A^1 -> H(U1(e, a), U1(b, e)) 5.44/2.30 A^1 -> H(U1(e, a), U1(d, d)) 5.44/2.30 A^1 -> H(U1(e, a), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, d), U1(d, b)) 5.44/2.30 A^1 -> H(U1(a, d), U1(e, b)) 5.44/2.30 A^1 -> H(U1(a, d), U1(b, d)) 5.44/2.30 A^1 -> H(U1(a, d), U1(b, e)) 5.44/2.30 A^1 -> H(U1(a, d), U1(d, d)) 5.44/2.30 A^1 -> H(U1(a, d), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, e), U1(d, b)) 5.44/2.30 A^1 -> H(U1(a, e), U1(e, b)) 5.44/2.30 A^1 -> H(U1(a, e), U1(b, d)) 5.44/2.30 A^1 -> H(U1(a, e), U1(b, e)) 5.44/2.30 A^1 -> H(U1(a, e), U1(d, d)) 5.44/2.30 A^1 -> H(U1(a, e), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, a), b) 5.44/2.30 A^1 -> H(U1(a, a), U1(d, e)) 5.44/2.30 A^1 -> H(U1(a, a), U1(e, d)) 5.44/2.30 A^1 -> H(U1(a, a), d) 5.44/2.30 A^1 -> H(d, U1(b, b)) 5.44/2.30 A^1 -> H(d, f(d)) 5.44/2.30 A^1 -> H(d, f(e)) 5.44/2.30 A^1 -> H(U1(d, d), U1(d, b)) 5.44/2.30 A^1 -> H(U1(d, d), U1(e, b)) 5.44/2.30 A^1 -> H(U1(d, d), U1(b, d)) 5.44/2.30 A^1 -> H(U1(d, d), U1(b, e)) 5.44/2.30 A^1 -> H(U1(d, d), U1(d, d)) 5.44/2.30 A^1 -> H(U1(d, d), U1(e, e)) 5.44/2.30 A^1 -> H(f(d), b) 5.44/2.30 A^1 -> H(f(d), U1(d, e)) 5.44/2.30 A^1 -> H(f(d), U1(e, d)) 5.44/2.30 5.44/2.30 The TRS R consists of the following rules: 5.44/2.30 5.44/2.30 a -> d 5.44/2.30 a -> e 5.44/2.30 f(x) -> U1(x, x) 5.44/2.30 b -> d 5.44/2.30 b -> e 5.44/2.30 U1(d, x) -> x 5.44/2.30 5.44/2.30 Q is empty. 5.44/2.30 We have to consider all minimal (P,Q,R)-chains. 5.44/2.30 ---------------------------------------- 5.44/2.30 5.44/2.30 (121) TransformationProof (EQUIVALENT) 5.44/2.30 By narrowing [LPAR04] the rule A^1 -> H(U1(e, e), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: 5.44/2.30 5.44/2.30 (A^1 -> H(U1(e, e), U1(d, b)),A^1 -> H(U1(e, e), U1(d, b))) 5.44/2.30 (A^1 -> H(U1(e, e), U1(e, b)),A^1 -> H(U1(e, e), U1(e, b))) 5.44/2.30 (A^1 -> H(U1(e, e), U1(b, d)),A^1 -> H(U1(e, e), U1(b, d))) 5.44/2.30 (A^1 -> H(U1(e, e), U1(b, e)),A^1 -> H(U1(e, e), U1(b, e))) 5.44/2.30 5.44/2.30 5.44/2.30 ---------------------------------------- 5.44/2.30 5.44/2.30 (122) 5.44/2.30 Obligation: 5.44/2.30 Q DP problem: 5.44/2.30 The TRS P consists of the following rules: 5.44/2.30 5.44/2.30 H(x, x) -> G(x, x) 5.44/2.30 G(d, e) -> A^1 5.44/2.30 A^1 -> H(f(d), f(d)) 5.44/2.30 A^1 -> H(f(e), f(e)) 5.44/2.30 A^1 -> H(U1(e, e), f(d)) 5.44/2.30 A^1 -> H(U1(e, e), f(e)) 5.44/2.30 A^1 -> H(f(e), U1(d, b)) 5.44/2.30 A^1 -> H(f(e), U1(e, b)) 5.44/2.30 A^1 -> H(f(e), U1(b, d)) 5.44/2.30 A^1 -> H(f(e), U1(b, e)) 5.44/2.30 A^1 -> H(f(e), U1(d, d)) 5.44/2.30 A^1 -> H(f(a), b) 5.44/2.30 A^1 -> H(f(a), U1(d, e)) 5.44/2.30 A^1 -> H(f(a), U1(e, d)) 5.44/2.30 A^1 -> H(f(d), U1(d, d)) 5.44/2.30 A^1 -> H(f(a), d) 5.44/2.30 A^1 -> H(f(e), U1(e, e)) 5.44/2.30 A^1 -> H(e, f(b)) 5.44/2.30 A^1 -> H(a, U1(b, b)) 5.44/2.30 A^1 -> H(a, f(d)) 5.44/2.30 A^1 -> H(a, f(e)) 5.44/2.30 A^1 -> H(U1(d, e), U1(b, b)) 5.44/2.30 A^1 -> H(U1(d, e), f(d)) 5.44/2.30 A^1 -> H(U1(d, e), f(e)) 5.44/2.30 A^1 -> H(U1(d, a), U1(d, b)) 5.44/2.30 A^1 -> H(U1(d, a), U1(e, b)) 5.44/2.30 A^1 -> H(U1(d, a), U1(b, d)) 5.44/2.30 A^1 -> H(U1(d, a), U1(b, e)) 5.44/2.30 A^1 -> H(U1(d, a), U1(d, d)) 5.44/2.30 A^1 -> H(U1(d, a), U1(e, e)) 5.44/2.30 A^1 -> H(U1(e, d), U1(b, b)) 5.44/2.30 A^1 -> H(U1(e, d), f(d)) 5.44/2.30 A^1 -> H(U1(e, d), f(e)) 5.44/2.30 A^1 -> H(U1(e, a), U1(d, b)) 5.44/2.30 A^1 -> H(U1(e, a), U1(e, b)) 5.44/2.30 A^1 -> H(U1(e, a), U1(b, d)) 5.44/2.30 A^1 -> H(U1(e, a), U1(b, e)) 5.44/2.30 A^1 -> H(U1(e, a), U1(d, d)) 5.44/2.30 A^1 -> H(U1(e, a), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, d), U1(d, b)) 5.44/2.30 A^1 -> H(U1(a, d), U1(e, b)) 5.44/2.30 A^1 -> H(U1(a, d), U1(b, d)) 5.44/2.30 A^1 -> H(U1(a, d), U1(b, e)) 5.44/2.30 A^1 -> H(U1(a, d), U1(d, d)) 5.44/2.30 A^1 -> H(U1(a, d), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, e), U1(d, b)) 5.44/2.30 A^1 -> H(U1(a, e), U1(e, b)) 5.44/2.30 A^1 -> H(U1(a, e), U1(b, d)) 5.44/2.30 A^1 -> H(U1(a, e), U1(b, e)) 5.44/2.30 A^1 -> H(U1(a, e), U1(d, d)) 5.44/2.30 A^1 -> H(U1(a, e), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, a), b) 5.44/2.30 A^1 -> H(U1(a, a), U1(d, e)) 5.44/2.30 A^1 -> H(U1(a, a), U1(e, d)) 5.44/2.30 A^1 -> H(U1(a, a), d) 5.44/2.30 A^1 -> H(d, U1(b, b)) 5.44/2.30 A^1 -> H(d, f(d)) 5.44/2.30 A^1 -> H(d, f(e)) 5.44/2.30 A^1 -> H(U1(d, d), U1(d, b)) 5.44/2.30 A^1 -> H(U1(d, d), U1(e, b)) 5.44/2.30 A^1 -> H(U1(d, d), U1(b, d)) 5.44/2.30 A^1 -> H(U1(d, d), U1(b, e)) 5.44/2.30 A^1 -> H(U1(d, d), U1(d, d)) 5.44/2.30 A^1 -> H(U1(d, d), U1(e, e)) 5.44/2.30 A^1 -> H(f(d), b) 5.44/2.30 A^1 -> H(f(d), U1(d, e)) 5.44/2.30 A^1 -> H(f(d), U1(e, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(d, b)) 5.44/2.30 A^1 -> H(U1(e, e), U1(e, b)) 5.44/2.30 A^1 -> H(U1(e, e), U1(b, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(b, e)) 5.44/2.30 5.44/2.30 The TRS R consists of the following rules: 5.44/2.30 5.44/2.30 a -> d 5.44/2.30 a -> e 5.44/2.30 f(x) -> U1(x, x) 5.44/2.30 b -> d 5.44/2.30 b -> e 5.44/2.30 U1(d, x) -> x 5.44/2.30 5.44/2.30 Q is empty. 5.44/2.30 We have to consider all minimal (P,Q,R)-chains. 5.44/2.30 ---------------------------------------- 5.44/2.30 5.44/2.30 (123) TransformationProof (EQUIVALENT) 5.44/2.30 By narrowing [LPAR04] the rule A^1 -> H(U1(e, e), f(d)) at position [] we obtained the following new rules [LPAR04]: 5.44/2.30 5.44/2.30 (A^1 -> H(U1(e, e), U1(d, d)),A^1 -> H(U1(e, e), U1(d, d))) 5.44/2.30 5.44/2.30 5.44/2.30 ---------------------------------------- 5.44/2.30 5.44/2.30 (124) 5.44/2.30 Obligation: 5.44/2.30 Q DP problem: 5.44/2.30 The TRS P consists of the following rules: 5.44/2.30 5.44/2.30 H(x, x) -> G(x, x) 5.44/2.30 G(d, e) -> A^1 5.44/2.30 A^1 -> H(f(d), f(d)) 5.44/2.30 A^1 -> H(f(e), f(e)) 5.44/2.30 A^1 -> H(U1(e, e), f(e)) 5.44/2.30 A^1 -> H(f(e), U1(d, b)) 5.44/2.30 A^1 -> H(f(e), U1(e, b)) 5.44/2.30 A^1 -> H(f(e), U1(b, d)) 5.44/2.30 A^1 -> H(f(e), U1(b, e)) 5.44/2.30 A^1 -> H(f(e), U1(d, d)) 5.44/2.30 A^1 -> H(f(a), b) 5.44/2.30 A^1 -> H(f(a), U1(d, e)) 5.44/2.30 A^1 -> H(f(a), U1(e, d)) 5.44/2.30 A^1 -> H(f(d), U1(d, d)) 5.44/2.30 A^1 -> H(f(a), d) 5.44/2.30 A^1 -> H(f(e), U1(e, e)) 5.44/2.30 A^1 -> H(e, f(b)) 5.44/2.30 A^1 -> H(a, U1(b, b)) 5.44/2.30 A^1 -> H(a, f(d)) 5.44/2.30 A^1 -> H(a, f(e)) 5.44/2.30 A^1 -> H(U1(d, e), U1(b, b)) 5.44/2.30 A^1 -> H(U1(d, e), f(d)) 5.44/2.30 A^1 -> H(U1(d, e), f(e)) 5.44/2.30 A^1 -> H(U1(d, a), U1(d, b)) 5.44/2.30 A^1 -> H(U1(d, a), U1(e, b)) 5.44/2.30 A^1 -> H(U1(d, a), U1(b, d)) 5.44/2.30 A^1 -> H(U1(d, a), U1(b, e)) 5.44/2.30 A^1 -> H(U1(d, a), U1(d, d)) 5.44/2.30 A^1 -> H(U1(d, a), U1(e, e)) 5.44/2.30 A^1 -> H(U1(e, d), U1(b, b)) 5.44/2.30 A^1 -> H(U1(e, d), f(d)) 5.44/2.30 A^1 -> H(U1(e, d), f(e)) 5.44/2.30 A^1 -> H(U1(e, a), U1(d, b)) 5.44/2.30 A^1 -> H(U1(e, a), U1(e, b)) 5.44/2.30 A^1 -> H(U1(e, a), U1(b, d)) 5.44/2.30 A^1 -> H(U1(e, a), U1(b, e)) 5.44/2.30 A^1 -> H(U1(e, a), U1(d, d)) 5.44/2.30 A^1 -> H(U1(e, a), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, d), U1(d, b)) 5.44/2.30 A^1 -> H(U1(a, d), U1(e, b)) 5.44/2.30 A^1 -> H(U1(a, d), U1(b, d)) 5.44/2.30 A^1 -> H(U1(a, d), U1(b, e)) 5.44/2.30 A^1 -> H(U1(a, d), U1(d, d)) 5.44/2.30 A^1 -> H(U1(a, d), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, e), U1(d, b)) 5.44/2.30 A^1 -> H(U1(a, e), U1(e, b)) 5.44/2.30 A^1 -> H(U1(a, e), U1(b, d)) 5.44/2.30 A^1 -> H(U1(a, e), U1(b, e)) 5.44/2.30 A^1 -> H(U1(a, e), U1(d, d)) 5.44/2.30 A^1 -> H(U1(a, e), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, a), b) 5.44/2.30 A^1 -> H(U1(a, a), U1(d, e)) 5.44/2.30 A^1 -> H(U1(a, a), U1(e, d)) 5.44/2.30 A^1 -> H(U1(a, a), d) 5.44/2.30 A^1 -> H(d, U1(b, b)) 5.44/2.30 A^1 -> H(d, f(d)) 5.44/2.30 A^1 -> H(d, f(e)) 5.44/2.30 A^1 -> H(U1(d, d), U1(d, b)) 5.44/2.30 A^1 -> H(U1(d, d), U1(e, b)) 5.44/2.30 A^1 -> H(U1(d, d), U1(b, d)) 5.44/2.30 A^1 -> H(U1(d, d), U1(b, e)) 5.44/2.30 A^1 -> H(U1(d, d), U1(d, d)) 5.44/2.30 A^1 -> H(U1(d, d), U1(e, e)) 5.44/2.30 A^1 -> H(f(d), b) 5.44/2.30 A^1 -> H(f(d), U1(d, e)) 5.44/2.30 A^1 -> H(f(d), U1(e, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(d, b)) 5.44/2.30 A^1 -> H(U1(e, e), U1(e, b)) 5.44/2.30 A^1 -> H(U1(e, e), U1(b, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(b, e)) 5.44/2.30 A^1 -> H(U1(e, e), U1(d, d)) 5.44/2.30 5.44/2.30 The TRS R consists of the following rules: 5.44/2.30 5.44/2.30 a -> d 5.44/2.30 a -> e 5.44/2.30 f(x) -> U1(x, x) 5.44/2.30 b -> d 5.44/2.30 b -> e 5.44/2.30 U1(d, x) -> x 5.44/2.30 5.44/2.30 Q is empty. 5.44/2.30 We have to consider all minimal (P,Q,R)-chains. 5.44/2.30 ---------------------------------------- 5.44/2.30 5.44/2.30 (125) TransformationProof (EQUIVALENT) 5.44/2.30 By narrowing [LPAR04] the rule A^1 -> H(U1(e, e), f(e)) at position [] we obtained the following new rules [LPAR04]: 5.44/2.30 5.44/2.30 (A^1 -> H(U1(e, e), U1(e, e)),A^1 -> H(U1(e, e), U1(e, e))) 5.44/2.30 5.44/2.30 5.44/2.30 ---------------------------------------- 5.44/2.30 5.44/2.30 (126) 5.44/2.30 Obligation: 5.44/2.30 Q DP problem: 5.44/2.30 The TRS P consists of the following rules: 5.44/2.30 5.44/2.30 H(x, x) -> G(x, x) 5.44/2.30 G(d, e) -> A^1 5.44/2.30 A^1 -> H(f(d), f(d)) 5.44/2.30 A^1 -> H(f(e), f(e)) 5.44/2.30 A^1 -> H(f(e), U1(d, b)) 5.44/2.30 A^1 -> H(f(e), U1(e, b)) 5.44/2.30 A^1 -> H(f(e), U1(b, d)) 5.44/2.30 A^1 -> H(f(e), U1(b, e)) 5.44/2.30 A^1 -> H(f(e), U1(d, d)) 5.44/2.30 A^1 -> H(f(a), b) 5.44/2.30 A^1 -> H(f(a), U1(d, e)) 5.44/2.30 A^1 -> H(f(a), U1(e, d)) 5.44/2.30 A^1 -> H(f(d), U1(d, d)) 5.44/2.30 A^1 -> H(f(a), d) 5.44/2.30 A^1 -> H(f(e), U1(e, e)) 5.44/2.30 A^1 -> H(e, f(b)) 5.44/2.30 A^1 -> H(a, U1(b, b)) 5.44/2.30 A^1 -> H(a, f(d)) 5.44/2.30 A^1 -> H(a, f(e)) 5.44/2.30 A^1 -> H(U1(d, e), U1(b, b)) 5.44/2.30 A^1 -> H(U1(d, e), f(d)) 5.44/2.30 A^1 -> H(U1(d, e), f(e)) 5.44/2.30 A^1 -> H(U1(d, a), U1(d, b)) 5.44/2.30 A^1 -> H(U1(d, a), U1(e, b)) 5.44/2.30 A^1 -> H(U1(d, a), U1(b, d)) 5.44/2.30 A^1 -> H(U1(d, a), U1(b, e)) 5.44/2.30 A^1 -> H(U1(d, a), U1(d, d)) 5.44/2.30 A^1 -> H(U1(d, a), U1(e, e)) 5.44/2.30 A^1 -> H(U1(e, d), U1(b, b)) 5.44/2.30 A^1 -> H(U1(e, d), f(d)) 5.44/2.30 A^1 -> H(U1(e, d), f(e)) 5.44/2.30 A^1 -> H(U1(e, a), U1(d, b)) 5.44/2.30 A^1 -> H(U1(e, a), U1(e, b)) 5.44/2.30 A^1 -> H(U1(e, a), U1(b, d)) 5.44/2.30 A^1 -> H(U1(e, a), U1(b, e)) 5.44/2.30 A^1 -> H(U1(e, a), U1(d, d)) 5.44/2.30 A^1 -> H(U1(e, a), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, d), U1(d, b)) 5.44/2.30 A^1 -> H(U1(a, d), U1(e, b)) 5.44/2.30 A^1 -> H(U1(a, d), U1(b, d)) 5.44/2.30 A^1 -> H(U1(a, d), U1(b, e)) 5.44/2.30 A^1 -> H(U1(a, d), U1(d, d)) 5.44/2.30 A^1 -> H(U1(a, d), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, e), U1(d, b)) 5.44/2.30 A^1 -> H(U1(a, e), U1(e, b)) 5.44/2.30 A^1 -> H(U1(a, e), U1(b, d)) 5.44/2.30 A^1 -> H(U1(a, e), U1(b, e)) 5.44/2.30 A^1 -> H(U1(a, e), U1(d, d)) 5.44/2.30 A^1 -> H(U1(a, e), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, a), b) 5.44/2.30 A^1 -> H(U1(a, a), U1(d, e)) 5.44/2.30 A^1 -> H(U1(a, a), U1(e, d)) 5.44/2.30 A^1 -> H(U1(a, a), d) 5.44/2.30 A^1 -> H(d, U1(b, b)) 5.44/2.30 A^1 -> H(d, f(d)) 5.44/2.30 A^1 -> H(d, f(e)) 5.44/2.30 A^1 -> H(U1(d, d), U1(d, b)) 5.44/2.30 A^1 -> H(U1(d, d), U1(e, b)) 5.44/2.30 A^1 -> H(U1(d, d), U1(b, d)) 5.44/2.30 A^1 -> H(U1(d, d), U1(b, e)) 5.44/2.30 A^1 -> H(U1(d, d), U1(d, d)) 5.44/2.30 A^1 -> H(U1(d, d), U1(e, e)) 5.44/2.30 A^1 -> H(f(d), b) 5.44/2.30 A^1 -> H(f(d), U1(d, e)) 5.44/2.30 A^1 -> H(f(d), U1(e, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(d, b)) 5.44/2.30 A^1 -> H(U1(e, e), U1(e, b)) 5.44/2.30 A^1 -> H(U1(e, e), U1(b, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(b, e)) 5.44/2.30 A^1 -> H(U1(e, e), U1(d, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(e, e)) 5.44/2.30 5.44/2.30 The TRS R consists of the following rules: 5.44/2.30 5.44/2.30 a -> d 5.44/2.30 a -> e 5.44/2.30 f(x) -> U1(x, x) 5.44/2.30 b -> d 5.44/2.30 b -> e 5.44/2.30 U1(d, x) -> x 5.44/2.30 5.44/2.30 Q is empty. 5.44/2.30 We have to consider all minimal (P,Q,R)-chains. 5.44/2.30 ---------------------------------------- 5.44/2.30 5.44/2.30 (127) TransformationProof (EQUIVALENT) 5.44/2.30 By narrowing [LPAR04] the rule A^1 -> H(f(e), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: 5.44/2.30 5.44/2.30 (A^1 -> H(U1(e, e), U1(d, b)),A^1 -> H(U1(e, e), U1(d, b))) 5.44/2.30 (A^1 -> H(f(e), b),A^1 -> H(f(e), b)) 5.44/2.30 (A^1 -> H(f(e), U1(d, d)),A^1 -> H(f(e), U1(d, d))) 5.44/2.30 (A^1 -> H(f(e), U1(d, e)),A^1 -> H(f(e), U1(d, e))) 5.44/2.30 5.44/2.30 5.44/2.30 ---------------------------------------- 5.44/2.30 5.44/2.30 (128) 5.44/2.30 Obligation: 5.44/2.30 Q DP problem: 5.44/2.30 The TRS P consists of the following rules: 5.44/2.30 5.44/2.30 H(x, x) -> G(x, x) 5.44/2.30 G(d, e) -> A^1 5.44/2.30 A^1 -> H(f(d), f(d)) 5.44/2.30 A^1 -> H(f(e), f(e)) 5.44/2.30 A^1 -> H(f(e), U1(e, b)) 5.44/2.30 A^1 -> H(f(e), U1(b, d)) 5.44/2.30 A^1 -> H(f(e), U1(b, e)) 5.44/2.30 A^1 -> H(f(e), U1(d, d)) 5.44/2.30 A^1 -> H(f(a), b) 5.44/2.30 A^1 -> H(f(a), U1(d, e)) 5.44/2.30 A^1 -> H(f(a), U1(e, d)) 5.44/2.30 A^1 -> H(f(d), U1(d, d)) 5.44/2.30 A^1 -> H(f(a), d) 5.44/2.30 A^1 -> H(f(e), U1(e, e)) 5.44/2.30 A^1 -> H(e, f(b)) 5.44/2.30 A^1 -> H(a, U1(b, b)) 5.44/2.30 A^1 -> H(a, f(d)) 5.44/2.30 A^1 -> H(a, f(e)) 5.44/2.30 A^1 -> H(U1(d, e), U1(b, b)) 5.44/2.30 A^1 -> H(U1(d, e), f(d)) 5.44/2.30 A^1 -> H(U1(d, e), f(e)) 5.44/2.30 A^1 -> H(U1(d, a), U1(d, b)) 5.44/2.30 A^1 -> H(U1(d, a), U1(e, b)) 5.44/2.30 A^1 -> H(U1(d, a), U1(b, d)) 5.44/2.30 A^1 -> H(U1(d, a), U1(b, e)) 5.44/2.30 A^1 -> H(U1(d, a), U1(d, d)) 5.44/2.30 A^1 -> H(U1(d, a), U1(e, e)) 5.44/2.30 A^1 -> H(U1(e, d), U1(b, b)) 5.44/2.30 A^1 -> H(U1(e, d), f(d)) 5.44/2.30 A^1 -> H(U1(e, d), f(e)) 5.44/2.30 A^1 -> H(U1(e, a), U1(d, b)) 5.44/2.30 A^1 -> H(U1(e, a), U1(e, b)) 5.44/2.30 A^1 -> H(U1(e, a), U1(b, d)) 5.44/2.30 A^1 -> H(U1(e, a), U1(b, e)) 5.44/2.30 A^1 -> H(U1(e, a), U1(d, d)) 5.44/2.30 A^1 -> H(U1(e, a), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, d), U1(d, b)) 5.44/2.30 A^1 -> H(U1(a, d), U1(e, b)) 5.44/2.30 A^1 -> H(U1(a, d), U1(b, d)) 5.44/2.30 A^1 -> H(U1(a, d), U1(b, e)) 5.44/2.30 A^1 -> H(U1(a, d), U1(d, d)) 5.44/2.30 A^1 -> H(U1(a, d), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, e), U1(d, b)) 5.44/2.30 A^1 -> H(U1(a, e), U1(e, b)) 5.44/2.30 A^1 -> H(U1(a, e), U1(b, d)) 5.44/2.30 A^1 -> H(U1(a, e), U1(b, e)) 5.44/2.30 A^1 -> H(U1(a, e), U1(d, d)) 5.44/2.30 A^1 -> H(U1(a, e), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, a), b) 5.44/2.30 A^1 -> H(U1(a, a), U1(d, e)) 5.44/2.30 A^1 -> H(U1(a, a), U1(e, d)) 5.44/2.30 A^1 -> H(U1(a, a), d) 5.44/2.30 A^1 -> H(d, U1(b, b)) 5.44/2.30 A^1 -> H(d, f(d)) 5.44/2.30 A^1 -> H(d, f(e)) 5.44/2.30 A^1 -> H(U1(d, d), U1(d, b)) 5.44/2.30 A^1 -> H(U1(d, d), U1(e, b)) 5.44/2.30 A^1 -> H(U1(d, d), U1(b, d)) 5.44/2.30 A^1 -> H(U1(d, d), U1(b, e)) 5.44/2.30 A^1 -> H(U1(d, d), U1(d, d)) 5.44/2.30 A^1 -> H(U1(d, d), U1(e, e)) 5.44/2.30 A^1 -> H(f(d), b) 5.44/2.30 A^1 -> H(f(d), U1(d, e)) 5.44/2.30 A^1 -> H(f(d), U1(e, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(d, b)) 5.44/2.30 A^1 -> H(U1(e, e), U1(e, b)) 5.44/2.30 A^1 -> H(U1(e, e), U1(b, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(b, e)) 5.44/2.30 A^1 -> H(U1(e, e), U1(d, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(e, e)) 5.44/2.30 A^1 -> H(f(e), b) 5.44/2.30 A^1 -> H(f(e), U1(d, e)) 5.44/2.30 5.44/2.30 The TRS R consists of the following rules: 5.44/2.30 5.44/2.30 a -> d 5.44/2.30 a -> e 5.44/2.30 f(x) -> U1(x, x) 5.44/2.30 b -> d 5.44/2.30 b -> e 5.44/2.30 U1(d, x) -> x 5.44/2.30 5.44/2.30 Q is empty. 5.44/2.30 We have to consider all minimal (P,Q,R)-chains. 5.44/2.30 ---------------------------------------- 5.44/2.30 5.44/2.30 (129) TransformationProof (EQUIVALENT) 5.44/2.30 By narrowing [LPAR04] the rule A^1 -> H(f(e), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: 5.44/2.30 5.44/2.30 (A^1 -> H(U1(e, e), U1(e, b)),A^1 -> H(U1(e, e), U1(e, b))) 5.44/2.30 (A^1 -> H(f(e), U1(e, d)),A^1 -> H(f(e), U1(e, d))) 5.44/2.30 (A^1 -> H(f(e), U1(e, e)),A^1 -> H(f(e), U1(e, e))) 5.44/2.30 5.44/2.30 5.44/2.30 ---------------------------------------- 5.44/2.30 5.44/2.30 (130) 5.44/2.30 Obligation: 5.44/2.30 Q DP problem: 5.44/2.30 The TRS P consists of the following rules: 5.44/2.30 5.44/2.30 H(x, x) -> G(x, x) 5.44/2.30 G(d, e) -> A^1 5.44/2.30 A^1 -> H(f(d), f(d)) 5.44/2.30 A^1 -> H(f(e), f(e)) 5.44/2.30 A^1 -> H(f(e), U1(b, d)) 5.44/2.30 A^1 -> H(f(e), U1(b, e)) 5.44/2.30 A^1 -> H(f(e), U1(d, d)) 5.44/2.30 A^1 -> H(f(a), b) 5.44/2.30 A^1 -> H(f(a), U1(d, e)) 5.44/2.30 A^1 -> H(f(a), U1(e, d)) 5.44/2.30 A^1 -> H(f(d), U1(d, d)) 5.44/2.30 A^1 -> H(f(a), d) 5.44/2.30 A^1 -> H(f(e), U1(e, e)) 5.44/2.30 A^1 -> H(e, f(b)) 5.44/2.30 A^1 -> H(a, U1(b, b)) 5.44/2.30 A^1 -> H(a, f(d)) 5.44/2.30 A^1 -> H(a, f(e)) 5.44/2.30 A^1 -> H(U1(d, e), U1(b, b)) 5.44/2.30 A^1 -> H(U1(d, e), f(d)) 5.44/2.30 A^1 -> H(U1(d, e), f(e)) 5.44/2.30 A^1 -> H(U1(d, a), U1(d, b)) 5.44/2.30 A^1 -> H(U1(d, a), U1(e, b)) 5.44/2.30 A^1 -> H(U1(d, a), U1(b, d)) 5.44/2.30 A^1 -> H(U1(d, a), U1(b, e)) 5.44/2.30 A^1 -> H(U1(d, a), U1(d, d)) 5.44/2.30 A^1 -> H(U1(d, a), U1(e, e)) 5.44/2.30 A^1 -> H(U1(e, d), U1(b, b)) 5.44/2.30 A^1 -> H(U1(e, d), f(d)) 5.44/2.30 A^1 -> H(U1(e, d), f(e)) 5.44/2.30 A^1 -> H(U1(e, a), U1(d, b)) 5.44/2.30 A^1 -> H(U1(e, a), U1(e, b)) 5.44/2.30 A^1 -> H(U1(e, a), U1(b, d)) 5.44/2.30 A^1 -> H(U1(e, a), U1(b, e)) 5.44/2.30 A^1 -> H(U1(e, a), U1(d, d)) 5.44/2.30 A^1 -> H(U1(e, a), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, d), U1(d, b)) 5.44/2.30 A^1 -> H(U1(a, d), U1(e, b)) 5.44/2.30 A^1 -> H(U1(a, d), U1(b, d)) 5.44/2.30 A^1 -> H(U1(a, d), U1(b, e)) 5.44/2.30 A^1 -> H(U1(a, d), U1(d, d)) 5.44/2.30 A^1 -> H(U1(a, d), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, e), U1(d, b)) 5.44/2.30 A^1 -> H(U1(a, e), U1(e, b)) 5.44/2.30 A^1 -> H(U1(a, e), U1(b, d)) 5.44/2.30 A^1 -> H(U1(a, e), U1(b, e)) 5.44/2.30 A^1 -> H(U1(a, e), U1(d, d)) 5.44/2.30 A^1 -> H(U1(a, e), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, a), b) 5.44/2.30 A^1 -> H(U1(a, a), U1(d, e)) 5.44/2.30 A^1 -> H(U1(a, a), U1(e, d)) 5.44/2.30 A^1 -> H(U1(a, a), d) 5.44/2.30 A^1 -> H(d, U1(b, b)) 5.44/2.30 A^1 -> H(d, f(d)) 5.44/2.30 A^1 -> H(d, f(e)) 5.44/2.30 A^1 -> H(U1(d, d), U1(d, b)) 5.44/2.30 A^1 -> H(U1(d, d), U1(e, b)) 5.44/2.30 A^1 -> H(U1(d, d), U1(b, d)) 5.44/2.30 A^1 -> H(U1(d, d), U1(b, e)) 5.44/2.30 A^1 -> H(U1(d, d), U1(d, d)) 5.44/2.30 A^1 -> H(U1(d, d), U1(e, e)) 5.44/2.30 A^1 -> H(f(d), b) 5.44/2.30 A^1 -> H(f(d), U1(d, e)) 5.44/2.30 A^1 -> H(f(d), U1(e, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(d, b)) 5.44/2.30 A^1 -> H(U1(e, e), U1(e, b)) 5.44/2.30 A^1 -> H(U1(e, e), U1(b, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(b, e)) 5.44/2.30 A^1 -> H(U1(e, e), U1(d, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(e, e)) 5.44/2.30 A^1 -> H(f(e), b) 5.44/2.30 A^1 -> H(f(e), U1(d, e)) 5.44/2.30 A^1 -> H(f(e), U1(e, d)) 5.44/2.30 5.44/2.30 The TRS R consists of the following rules: 5.44/2.30 5.44/2.30 a -> d 5.44/2.30 a -> e 5.44/2.30 f(x) -> U1(x, x) 5.44/2.30 b -> d 5.44/2.30 b -> e 5.44/2.30 U1(d, x) -> x 5.44/2.30 5.44/2.30 Q is empty. 5.44/2.30 We have to consider all minimal (P,Q,R)-chains. 5.44/2.30 ---------------------------------------- 5.44/2.30 5.44/2.30 (131) TransformationProof (EQUIVALENT) 5.44/2.30 By narrowing [LPAR04] the rule A^1 -> H(f(e), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: 5.44/2.30 5.44/2.30 (A^1 -> H(U1(e, e), U1(b, d)),A^1 -> H(U1(e, e), U1(b, d))) 5.44/2.30 (A^1 -> H(f(e), U1(d, d)),A^1 -> H(f(e), U1(d, d))) 5.44/2.30 (A^1 -> H(f(e), U1(e, d)),A^1 -> H(f(e), U1(e, d))) 5.44/2.30 5.44/2.30 5.44/2.30 ---------------------------------------- 5.44/2.30 5.44/2.30 (132) 5.44/2.30 Obligation: 5.44/2.30 Q DP problem: 5.44/2.30 The TRS P consists of the following rules: 5.44/2.30 5.44/2.30 H(x, x) -> G(x, x) 5.44/2.30 G(d, e) -> A^1 5.44/2.30 A^1 -> H(f(d), f(d)) 5.44/2.30 A^1 -> H(f(e), f(e)) 5.44/2.30 A^1 -> H(f(e), U1(b, e)) 5.44/2.30 A^1 -> H(f(e), U1(d, d)) 5.44/2.30 A^1 -> H(f(a), b) 5.44/2.30 A^1 -> H(f(a), U1(d, e)) 5.44/2.30 A^1 -> H(f(a), U1(e, d)) 5.44/2.30 A^1 -> H(f(d), U1(d, d)) 5.44/2.30 A^1 -> H(f(a), d) 5.44/2.30 A^1 -> H(f(e), U1(e, e)) 5.44/2.30 A^1 -> H(e, f(b)) 5.44/2.30 A^1 -> H(a, U1(b, b)) 5.44/2.30 A^1 -> H(a, f(d)) 5.44/2.30 A^1 -> H(a, f(e)) 5.44/2.30 A^1 -> H(U1(d, e), U1(b, b)) 5.44/2.30 A^1 -> H(U1(d, e), f(d)) 5.44/2.30 A^1 -> H(U1(d, e), f(e)) 5.44/2.30 A^1 -> H(U1(d, a), U1(d, b)) 5.44/2.30 A^1 -> H(U1(d, a), U1(e, b)) 5.44/2.30 A^1 -> H(U1(d, a), U1(b, d)) 5.44/2.30 A^1 -> H(U1(d, a), U1(b, e)) 5.44/2.30 A^1 -> H(U1(d, a), U1(d, d)) 5.44/2.30 A^1 -> H(U1(d, a), U1(e, e)) 5.44/2.30 A^1 -> H(U1(e, d), U1(b, b)) 5.44/2.30 A^1 -> H(U1(e, d), f(d)) 5.44/2.30 A^1 -> H(U1(e, d), f(e)) 5.44/2.30 A^1 -> H(U1(e, a), U1(d, b)) 5.44/2.30 A^1 -> H(U1(e, a), U1(e, b)) 5.44/2.30 A^1 -> H(U1(e, a), U1(b, d)) 5.44/2.30 A^1 -> H(U1(e, a), U1(b, e)) 5.44/2.30 A^1 -> H(U1(e, a), U1(d, d)) 5.44/2.30 A^1 -> H(U1(e, a), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, d), U1(d, b)) 5.44/2.30 A^1 -> H(U1(a, d), U1(e, b)) 5.44/2.30 A^1 -> H(U1(a, d), U1(b, d)) 5.44/2.30 A^1 -> H(U1(a, d), U1(b, e)) 5.44/2.30 A^1 -> H(U1(a, d), U1(d, d)) 5.44/2.30 A^1 -> H(U1(a, d), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, e), U1(d, b)) 5.44/2.30 A^1 -> H(U1(a, e), U1(e, b)) 5.44/2.30 A^1 -> H(U1(a, e), U1(b, d)) 5.44/2.30 A^1 -> H(U1(a, e), U1(b, e)) 5.44/2.30 A^1 -> H(U1(a, e), U1(d, d)) 5.44/2.30 A^1 -> H(U1(a, e), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, a), b) 5.44/2.30 A^1 -> H(U1(a, a), U1(d, e)) 5.44/2.30 A^1 -> H(U1(a, a), U1(e, d)) 5.44/2.30 A^1 -> H(U1(a, a), d) 5.44/2.30 A^1 -> H(d, U1(b, b)) 5.44/2.30 A^1 -> H(d, f(d)) 5.44/2.30 A^1 -> H(d, f(e)) 5.44/2.30 A^1 -> H(U1(d, d), U1(d, b)) 5.44/2.30 A^1 -> H(U1(d, d), U1(e, b)) 5.44/2.30 A^1 -> H(U1(d, d), U1(b, d)) 5.44/2.30 A^1 -> H(U1(d, d), U1(b, e)) 5.44/2.30 A^1 -> H(U1(d, d), U1(d, d)) 5.44/2.30 A^1 -> H(U1(d, d), U1(e, e)) 5.44/2.30 A^1 -> H(f(d), b) 5.44/2.30 A^1 -> H(f(d), U1(d, e)) 5.44/2.30 A^1 -> H(f(d), U1(e, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(d, b)) 5.44/2.30 A^1 -> H(U1(e, e), U1(e, b)) 5.44/2.30 A^1 -> H(U1(e, e), U1(b, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(b, e)) 5.44/2.30 A^1 -> H(U1(e, e), U1(d, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(e, e)) 5.44/2.30 A^1 -> H(f(e), b) 5.44/2.30 A^1 -> H(f(e), U1(d, e)) 5.44/2.30 A^1 -> H(f(e), U1(e, d)) 5.44/2.30 5.44/2.30 The TRS R consists of the following rules: 5.44/2.30 5.44/2.30 a -> d 5.44/2.30 a -> e 5.44/2.30 f(x) -> U1(x, x) 5.44/2.30 b -> d 5.44/2.30 b -> e 5.44/2.30 U1(d, x) -> x 5.44/2.30 5.44/2.30 Q is empty. 5.44/2.30 We have to consider all minimal (P,Q,R)-chains. 5.44/2.30 ---------------------------------------- 5.44/2.30 5.44/2.30 (133) TransformationProof (EQUIVALENT) 5.44/2.30 By narrowing [LPAR04] the rule A^1 -> H(f(e), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: 5.44/2.30 5.44/2.30 (A^1 -> H(U1(e, e), U1(b, e)),A^1 -> H(U1(e, e), U1(b, e))) 5.44/2.30 (A^1 -> H(f(e), U1(d, e)),A^1 -> H(f(e), U1(d, e))) 5.44/2.30 (A^1 -> H(f(e), U1(e, e)),A^1 -> H(f(e), U1(e, e))) 5.44/2.30 5.44/2.30 5.44/2.30 ---------------------------------------- 5.44/2.30 5.44/2.30 (134) 5.44/2.30 Obligation: 5.44/2.30 Q DP problem: 5.44/2.30 The TRS P consists of the following rules: 5.44/2.30 5.44/2.30 H(x, x) -> G(x, x) 5.44/2.30 G(d, e) -> A^1 5.44/2.30 A^1 -> H(f(d), f(d)) 5.44/2.30 A^1 -> H(f(e), f(e)) 5.44/2.30 A^1 -> H(f(e), U1(d, d)) 5.44/2.30 A^1 -> H(f(a), b) 5.44/2.30 A^1 -> H(f(a), U1(d, e)) 5.44/2.30 A^1 -> H(f(a), U1(e, d)) 5.44/2.30 A^1 -> H(f(d), U1(d, d)) 5.44/2.30 A^1 -> H(f(a), d) 5.44/2.30 A^1 -> H(f(e), U1(e, e)) 5.44/2.30 A^1 -> H(e, f(b)) 5.44/2.30 A^1 -> H(a, U1(b, b)) 5.44/2.30 A^1 -> H(a, f(d)) 5.44/2.30 A^1 -> H(a, f(e)) 5.44/2.30 A^1 -> H(U1(d, e), U1(b, b)) 5.44/2.30 A^1 -> H(U1(d, e), f(d)) 5.44/2.30 A^1 -> H(U1(d, e), f(e)) 5.44/2.30 A^1 -> H(U1(d, a), U1(d, b)) 5.44/2.30 A^1 -> H(U1(d, a), U1(e, b)) 5.44/2.30 A^1 -> H(U1(d, a), U1(b, d)) 5.44/2.30 A^1 -> H(U1(d, a), U1(b, e)) 5.44/2.30 A^1 -> H(U1(d, a), U1(d, d)) 5.44/2.30 A^1 -> H(U1(d, a), U1(e, e)) 5.44/2.30 A^1 -> H(U1(e, d), U1(b, b)) 5.44/2.30 A^1 -> H(U1(e, d), f(d)) 5.44/2.30 A^1 -> H(U1(e, d), f(e)) 5.44/2.30 A^1 -> H(U1(e, a), U1(d, b)) 5.44/2.30 A^1 -> H(U1(e, a), U1(e, b)) 5.44/2.30 A^1 -> H(U1(e, a), U1(b, d)) 5.44/2.30 A^1 -> H(U1(e, a), U1(b, e)) 5.44/2.30 A^1 -> H(U1(e, a), U1(d, d)) 5.44/2.30 A^1 -> H(U1(e, a), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, d), U1(d, b)) 5.44/2.30 A^1 -> H(U1(a, d), U1(e, b)) 5.44/2.30 A^1 -> H(U1(a, d), U1(b, d)) 5.44/2.30 A^1 -> H(U1(a, d), U1(b, e)) 5.44/2.30 A^1 -> H(U1(a, d), U1(d, d)) 5.44/2.30 A^1 -> H(U1(a, d), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, e), U1(d, b)) 5.44/2.30 A^1 -> H(U1(a, e), U1(e, b)) 5.44/2.30 A^1 -> H(U1(a, e), U1(b, d)) 5.44/2.30 A^1 -> H(U1(a, e), U1(b, e)) 5.44/2.30 A^1 -> H(U1(a, e), U1(d, d)) 5.44/2.30 A^1 -> H(U1(a, e), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, a), b) 5.44/2.30 A^1 -> H(U1(a, a), U1(d, e)) 5.44/2.30 A^1 -> H(U1(a, a), U1(e, d)) 5.44/2.30 A^1 -> H(U1(a, a), d) 5.44/2.30 A^1 -> H(d, U1(b, b)) 5.44/2.30 A^1 -> H(d, f(d)) 5.44/2.30 A^1 -> H(d, f(e)) 5.44/2.30 A^1 -> H(U1(d, d), U1(d, b)) 5.44/2.30 A^1 -> H(U1(d, d), U1(e, b)) 5.44/2.30 A^1 -> H(U1(d, d), U1(b, d)) 5.44/2.30 A^1 -> H(U1(d, d), U1(b, e)) 5.44/2.30 A^1 -> H(U1(d, d), U1(d, d)) 5.44/2.30 A^1 -> H(U1(d, d), U1(e, e)) 5.44/2.30 A^1 -> H(f(d), b) 5.44/2.30 A^1 -> H(f(d), U1(d, e)) 5.44/2.30 A^1 -> H(f(d), U1(e, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(d, b)) 5.44/2.30 A^1 -> H(U1(e, e), U1(e, b)) 5.44/2.30 A^1 -> H(U1(e, e), U1(b, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(b, e)) 5.44/2.30 A^1 -> H(U1(e, e), U1(d, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(e, e)) 5.44/2.30 A^1 -> H(f(e), b) 5.44/2.30 A^1 -> H(f(e), U1(d, e)) 5.44/2.30 A^1 -> H(f(e), U1(e, d)) 5.44/2.30 5.44/2.30 The TRS R consists of the following rules: 5.44/2.30 5.44/2.30 a -> d 5.44/2.30 a -> e 5.44/2.30 f(x) -> U1(x, x) 5.44/2.30 b -> d 5.44/2.30 b -> e 5.44/2.30 U1(d, x) -> x 5.44/2.30 5.44/2.30 Q is empty. 5.44/2.30 We have to consider all minimal (P,Q,R)-chains. 5.44/2.30 ---------------------------------------- 5.44/2.30 5.44/2.30 (135) TransformationProof (EQUIVALENT) 5.44/2.30 By narrowing [LPAR04] the rule A^1 -> H(f(e), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: 5.44/2.30 5.44/2.30 (A^1 -> H(U1(e, e), U1(d, d)),A^1 -> H(U1(e, e), U1(d, d))) 5.44/2.30 (A^1 -> H(f(e), d),A^1 -> H(f(e), d)) 5.44/2.30 5.44/2.30 5.44/2.30 ---------------------------------------- 5.44/2.30 5.44/2.30 (136) 5.44/2.30 Obligation: 5.44/2.30 Q DP problem: 5.44/2.30 The TRS P consists of the following rules: 5.44/2.30 5.44/2.30 H(x, x) -> G(x, x) 5.44/2.30 G(d, e) -> A^1 5.44/2.30 A^1 -> H(f(d), f(d)) 5.44/2.30 A^1 -> H(f(e), f(e)) 5.44/2.30 A^1 -> H(f(a), b) 5.44/2.30 A^1 -> H(f(a), U1(d, e)) 5.44/2.30 A^1 -> H(f(a), U1(e, d)) 5.44/2.30 A^1 -> H(f(d), U1(d, d)) 5.44/2.30 A^1 -> H(f(a), d) 5.44/2.30 A^1 -> H(f(e), U1(e, e)) 5.44/2.30 A^1 -> H(e, f(b)) 5.44/2.30 A^1 -> H(a, U1(b, b)) 5.44/2.30 A^1 -> H(a, f(d)) 5.44/2.30 A^1 -> H(a, f(e)) 5.44/2.30 A^1 -> H(U1(d, e), U1(b, b)) 5.44/2.30 A^1 -> H(U1(d, e), f(d)) 5.44/2.30 A^1 -> H(U1(d, e), f(e)) 5.44/2.30 A^1 -> H(U1(d, a), U1(d, b)) 5.44/2.30 A^1 -> H(U1(d, a), U1(e, b)) 5.44/2.30 A^1 -> H(U1(d, a), U1(b, d)) 5.44/2.30 A^1 -> H(U1(d, a), U1(b, e)) 5.44/2.30 A^1 -> H(U1(d, a), U1(d, d)) 5.44/2.30 A^1 -> H(U1(d, a), U1(e, e)) 5.44/2.30 A^1 -> H(U1(e, d), U1(b, b)) 5.44/2.30 A^1 -> H(U1(e, d), f(d)) 5.44/2.30 A^1 -> H(U1(e, d), f(e)) 5.44/2.30 A^1 -> H(U1(e, a), U1(d, b)) 5.44/2.30 A^1 -> H(U1(e, a), U1(e, b)) 5.44/2.30 A^1 -> H(U1(e, a), U1(b, d)) 5.44/2.30 A^1 -> H(U1(e, a), U1(b, e)) 5.44/2.30 A^1 -> H(U1(e, a), U1(d, d)) 5.44/2.30 A^1 -> H(U1(e, a), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, d), U1(d, b)) 5.44/2.30 A^1 -> H(U1(a, d), U1(e, b)) 5.44/2.30 A^1 -> H(U1(a, d), U1(b, d)) 5.44/2.30 A^1 -> H(U1(a, d), U1(b, e)) 5.44/2.30 A^1 -> H(U1(a, d), U1(d, d)) 5.44/2.30 A^1 -> H(U1(a, d), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, e), U1(d, b)) 5.44/2.30 A^1 -> H(U1(a, e), U1(e, b)) 5.44/2.30 A^1 -> H(U1(a, e), U1(b, d)) 5.44/2.30 A^1 -> H(U1(a, e), U1(b, e)) 5.44/2.30 A^1 -> H(U1(a, e), U1(d, d)) 5.44/2.30 A^1 -> H(U1(a, e), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, a), b) 5.44/2.30 A^1 -> H(U1(a, a), U1(d, e)) 5.44/2.30 A^1 -> H(U1(a, a), U1(e, d)) 5.44/2.30 A^1 -> H(U1(a, a), d) 5.44/2.30 A^1 -> H(d, U1(b, b)) 5.44/2.30 A^1 -> H(d, f(d)) 5.44/2.30 A^1 -> H(d, f(e)) 5.44/2.30 A^1 -> H(U1(d, d), U1(d, b)) 5.44/2.30 A^1 -> H(U1(d, d), U1(e, b)) 5.44/2.30 A^1 -> H(U1(d, d), U1(b, d)) 5.44/2.30 A^1 -> H(U1(d, d), U1(b, e)) 5.44/2.30 A^1 -> H(U1(d, d), U1(d, d)) 5.44/2.30 A^1 -> H(U1(d, d), U1(e, e)) 5.44/2.30 A^1 -> H(f(d), b) 5.44/2.30 A^1 -> H(f(d), U1(d, e)) 5.44/2.30 A^1 -> H(f(d), U1(e, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(d, b)) 5.44/2.30 A^1 -> H(U1(e, e), U1(e, b)) 5.44/2.30 A^1 -> H(U1(e, e), U1(b, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(b, e)) 5.44/2.30 A^1 -> H(U1(e, e), U1(d, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(e, e)) 5.44/2.30 A^1 -> H(f(e), b) 5.44/2.30 A^1 -> H(f(e), U1(d, e)) 5.44/2.30 A^1 -> H(f(e), U1(e, d)) 5.44/2.30 A^1 -> H(f(e), d) 5.44/2.30 5.44/2.30 The TRS R consists of the following rules: 5.44/2.30 5.44/2.30 a -> d 5.44/2.30 a -> e 5.44/2.30 f(x) -> U1(x, x) 5.44/2.30 b -> d 5.44/2.30 b -> e 5.44/2.30 U1(d, x) -> x 5.44/2.30 5.44/2.30 Q is empty. 5.44/2.30 We have to consider all minimal (P,Q,R)-chains. 5.44/2.30 ---------------------------------------- 5.44/2.30 5.44/2.30 (137) TransformationProof (EQUIVALENT) 5.44/2.30 By narrowing [LPAR04] the rule A^1 -> H(f(a), b) at position [] we obtained the following new rules [LPAR04]: 5.44/2.30 5.44/2.30 (A^1 -> H(U1(a, a), b),A^1 -> H(U1(a, a), b)) 5.44/2.30 (A^1 -> H(f(d), b),A^1 -> H(f(d), b)) 5.44/2.30 (A^1 -> H(f(e), b),A^1 -> H(f(e), b)) 5.44/2.30 (A^1 -> H(f(a), d),A^1 -> H(f(a), d)) 5.44/2.30 (A^1 -> H(f(a), e),A^1 -> H(f(a), e)) 5.44/2.30 5.44/2.30 5.44/2.30 ---------------------------------------- 5.44/2.30 5.44/2.30 (138) 5.44/2.30 Obligation: 5.44/2.30 Q DP problem: 5.44/2.30 The TRS P consists of the following rules: 5.44/2.30 5.44/2.30 H(x, x) -> G(x, x) 5.44/2.30 G(d, e) -> A^1 5.44/2.30 A^1 -> H(f(d), f(d)) 5.44/2.30 A^1 -> H(f(e), f(e)) 5.44/2.30 A^1 -> H(f(a), U1(d, e)) 5.44/2.30 A^1 -> H(f(a), U1(e, d)) 5.44/2.30 A^1 -> H(f(d), U1(d, d)) 5.44/2.30 A^1 -> H(f(a), d) 5.44/2.30 A^1 -> H(f(e), U1(e, e)) 5.44/2.30 A^1 -> H(e, f(b)) 5.44/2.30 A^1 -> H(a, U1(b, b)) 5.44/2.30 A^1 -> H(a, f(d)) 5.44/2.30 A^1 -> H(a, f(e)) 5.44/2.30 A^1 -> H(U1(d, e), U1(b, b)) 5.44/2.30 A^1 -> H(U1(d, e), f(d)) 5.44/2.30 A^1 -> H(U1(d, e), f(e)) 5.44/2.30 A^1 -> H(U1(d, a), U1(d, b)) 5.44/2.30 A^1 -> H(U1(d, a), U1(e, b)) 5.44/2.30 A^1 -> H(U1(d, a), U1(b, d)) 5.44/2.30 A^1 -> H(U1(d, a), U1(b, e)) 5.44/2.30 A^1 -> H(U1(d, a), U1(d, d)) 5.44/2.30 A^1 -> H(U1(d, a), U1(e, e)) 5.44/2.30 A^1 -> H(U1(e, d), U1(b, b)) 5.44/2.30 A^1 -> H(U1(e, d), f(d)) 5.44/2.30 A^1 -> H(U1(e, d), f(e)) 5.44/2.30 A^1 -> H(U1(e, a), U1(d, b)) 5.44/2.30 A^1 -> H(U1(e, a), U1(e, b)) 5.44/2.30 A^1 -> H(U1(e, a), U1(b, d)) 5.44/2.30 A^1 -> H(U1(e, a), U1(b, e)) 5.44/2.30 A^1 -> H(U1(e, a), U1(d, d)) 5.44/2.30 A^1 -> H(U1(e, a), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, d), U1(d, b)) 5.44/2.30 A^1 -> H(U1(a, d), U1(e, b)) 5.44/2.30 A^1 -> H(U1(a, d), U1(b, d)) 5.44/2.30 A^1 -> H(U1(a, d), U1(b, e)) 5.44/2.30 A^1 -> H(U1(a, d), U1(d, d)) 5.44/2.30 A^1 -> H(U1(a, d), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, e), U1(d, b)) 5.44/2.30 A^1 -> H(U1(a, e), U1(e, b)) 5.44/2.30 A^1 -> H(U1(a, e), U1(b, d)) 5.44/2.30 A^1 -> H(U1(a, e), U1(b, e)) 5.44/2.30 A^1 -> H(U1(a, e), U1(d, d)) 5.44/2.30 A^1 -> H(U1(a, e), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, a), b) 5.44/2.30 A^1 -> H(U1(a, a), U1(d, e)) 5.44/2.30 A^1 -> H(U1(a, a), U1(e, d)) 5.44/2.30 A^1 -> H(U1(a, a), d) 5.44/2.30 A^1 -> H(d, U1(b, b)) 5.44/2.30 A^1 -> H(d, f(d)) 5.44/2.30 A^1 -> H(d, f(e)) 5.44/2.30 A^1 -> H(U1(d, d), U1(d, b)) 5.44/2.30 A^1 -> H(U1(d, d), U1(e, b)) 5.44/2.30 A^1 -> H(U1(d, d), U1(b, d)) 5.44/2.30 A^1 -> H(U1(d, d), U1(b, e)) 5.44/2.30 A^1 -> H(U1(d, d), U1(d, d)) 5.44/2.30 A^1 -> H(U1(d, d), U1(e, e)) 5.44/2.30 A^1 -> H(f(d), b) 5.44/2.30 A^1 -> H(f(d), U1(d, e)) 5.44/2.30 A^1 -> H(f(d), U1(e, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(d, b)) 5.44/2.30 A^1 -> H(U1(e, e), U1(e, b)) 5.44/2.30 A^1 -> H(U1(e, e), U1(b, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(b, e)) 5.44/2.30 A^1 -> H(U1(e, e), U1(d, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(e, e)) 5.44/2.30 A^1 -> H(f(e), b) 5.44/2.30 A^1 -> H(f(e), U1(d, e)) 5.44/2.30 A^1 -> H(f(e), U1(e, d)) 5.44/2.30 A^1 -> H(f(e), d) 5.44/2.30 A^1 -> H(f(a), e) 5.44/2.30 5.44/2.30 The TRS R consists of the following rules: 5.44/2.30 5.44/2.30 a -> d 5.44/2.30 a -> e 5.44/2.30 f(x) -> U1(x, x) 5.44/2.30 b -> d 5.44/2.30 b -> e 5.44/2.30 U1(d, x) -> x 5.44/2.30 5.44/2.30 Q is empty. 5.44/2.30 We have to consider all minimal (P,Q,R)-chains. 5.44/2.30 ---------------------------------------- 5.44/2.30 5.44/2.30 (139) TransformationProof (EQUIVALENT) 5.44/2.30 By narrowing [LPAR04] the rule A^1 -> H(f(a), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: 5.44/2.30 5.44/2.30 (A^1 -> H(U1(a, a), U1(d, e)),A^1 -> H(U1(a, a), U1(d, e))) 5.44/2.30 (A^1 -> H(f(d), U1(d, e)),A^1 -> H(f(d), U1(d, e))) 5.44/2.30 (A^1 -> H(f(e), U1(d, e)),A^1 -> H(f(e), U1(d, e))) 5.44/2.30 (A^1 -> H(f(a), e),A^1 -> H(f(a), e)) 5.44/2.30 5.44/2.30 5.44/2.30 ---------------------------------------- 5.44/2.30 5.44/2.30 (140) 5.44/2.30 Obligation: 5.44/2.30 Q DP problem: 5.44/2.30 The TRS P consists of the following rules: 5.44/2.30 5.44/2.30 H(x, x) -> G(x, x) 5.44/2.30 G(d, e) -> A^1 5.44/2.30 A^1 -> H(f(d), f(d)) 5.44/2.30 A^1 -> H(f(e), f(e)) 5.44/2.30 A^1 -> H(f(a), U1(e, d)) 5.44/2.30 A^1 -> H(f(d), U1(d, d)) 5.44/2.30 A^1 -> H(f(a), d) 5.44/2.30 A^1 -> H(f(e), U1(e, e)) 5.44/2.30 A^1 -> H(e, f(b)) 5.44/2.30 A^1 -> H(a, U1(b, b)) 5.44/2.30 A^1 -> H(a, f(d)) 5.44/2.30 A^1 -> H(a, f(e)) 5.44/2.30 A^1 -> H(U1(d, e), U1(b, b)) 5.44/2.30 A^1 -> H(U1(d, e), f(d)) 5.44/2.30 A^1 -> H(U1(d, e), f(e)) 5.44/2.30 A^1 -> H(U1(d, a), U1(d, b)) 5.44/2.30 A^1 -> H(U1(d, a), U1(e, b)) 5.44/2.30 A^1 -> H(U1(d, a), U1(b, d)) 5.44/2.30 A^1 -> H(U1(d, a), U1(b, e)) 5.44/2.30 A^1 -> H(U1(d, a), U1(d, d)) 5.44/2.30 A^1 -> H(U1(d, a), U1(e, e)) 5.44/2.30 A^1 -> H(U1(e, d), U1(b, b)) 5.44/2.30 A^1 -> H(U1(e, d), f(d)) 5.44/2.30 A^1 -> H(U1(e, d), f(e)) 5.44/2.30 A^1 -> H(U1(e, a), U1(d, b)) 5.44/2.30 A^1 -> H(U1(e, a), U1(e, b)) 5.44/2.30 A^1 -> H(U1(e, a), U1(b, d)) 5.44/2.30 A^1 -> H(U1(e, a), U1(b, e)) 5.44/2.30 A^1 -> H(U1(e, a), U1(d, d)) 5.44/2.30 A^1 -> H(U1(e, a), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, d), U1(d, b)) 5.44/2.30 A^1 -> H(U1(a, d), U1(e, b)) 5.44/2.30 A^1 -> H(U1(a, d), U1(b, d)) 5.44/2.30 A^1 -> H(U1(a, d), U1(b, e)) 5.44/2.30 A^1 -> H(U1(a, d), U1(d, d)) 5.44/2.30 A^1 -> H(U1(a, d), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, e), U1(d, b)) 5.44/2.30 A^1 -> H(U1(a, e), U1(e, b)) 5.44/2.30 A^1 -> H(U1(a, e), U1(b, d)) 5.44/2.30 A^1 -> H(U1(a, e), U1(b, e)) 5.44/2.30 A^1 -> H(U1(a, e), U1(d, d)) 5.44/2.30 A^1 -> H(U1(a, e), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, a), b) 5.44/2.30 A^1 -> H(U1(a, a), U1(d, e)) 5.44/2.30 A^1 -> H(U1(a, a), U1(e, d)) 5.44/2.30 A^1 -> H(U1(a, a), d) 5.44/2.30 A^1 -> H(d, U1(b, b)) 5.44/2.30 A^1 -> H(d, f(d)) 5.44/2.30 A^1 -> H(d, f(e)) 5.44/2.30 A^1 -> H(U1(d, d), U1(d, b)) 5.44/2.30 A^1 -> H(U1(d, d), U1(e, b)) 5.44/2.30 A^1 -> H(U1(d, d), U1(b, d)) 5.44/2.30 A^1 -> H(U1(d, d), U1(b, e)) 5.44/2.30 A^1 -> H(U1(d, d), U1(d, d)) 5.44/2.30 A^1 -> H(U1(d, d), U1(e, e)) 5.44/2.30 A^1 -> H(f(d), b) 5.44/2.30 A^1 -> H(f(d), U1(d, e)) 5.44/2.30 A^1 -> H(f(d), U1(e, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(d, b)) 5.44/2.30 A^1 -> H(U1(e, e), U1(e, b)) 5.44/2.30 A^1 -> H(U1(e, e), U1(b, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(b, e)) 5.44/2.30 A^1 -> H(U1(e, e), U1(d, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(e, e)) 5.44/2.30 A^1 -> H(f(e), b) 5.44/2.30 A^1 -> H(f(e), U1(d, e)) 5.44/2.30 A^1 -> H(f(e), U1(e, d)) 5.44/2.30 A^1 -> H(f(e), d) 5.44/2.30 A^1 -> H(f(a), e) 5.44/2.30 5.44/2.30 The TRS R consists of the following rules: 5.44/2.30 5.44/2.30 a -> d 5.44/2.30 a -> e 5.44/2.30 f(x) -> U1(x, x) 5.44/2.30 b -> d 5.44/2.30 b -> e 5.44/2.30 U1(d, x) -> x 5.44/2.30 5.44/2.30 Q is empty. 5.44/2.30 We have to consider all minimal (P,Q,R)-chains. 5.44/2.30 ---------------------------------------- 5.44/2.30 5.44/2.30 (141) TransformationProof (EQUIVALENT) 5.44/2.30 By narrowing [LPAR04] the rule A^1 -> H(f(a), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: 5.44/2.30 5.44/2.30 (A^1 -> H(U1(a, a), U1(e, d)),A^1 -> H(U1(a, a), U1(e, d))) 5.44/2.30 (A^1 -> H(f(d), U1(e, d)),A^1 -> H(f(d), U1(e, d))) 5.44/2.30 (A^1 -> H(f(e), U1(e, d)),A^1 -> H(f(e), U1(e, d))) 5.44/2.30 5.44/2.30 5.44/2.30 ---------------------------------------- 5.44/2.30 5.44/2.30 (142) 5.44/2.30 Obligation: 5.44/2.30 Q DP problem: 5.44/2.30 The TRS P consists of the following rules: 5.44/2.30 5.44/2.30 H(x, x) -> G(x, x) 5.44/2.30 G(d, e) -> A^1 5.44/2.30 A^1 -> H(f(d), f(d)) 5.44/2.30 A^1 -> H(f(e), f(e)) 5.44/2.30 A^1 -> H(f(d), U1(d, d)) 5.44/2.30 A^1 -> H(f(a), d) 5.44/2.30 A^1 -> H(f(e), U1(e, e)) 5.44/2.30 A^1 -> H(e, f(b)) 5.44/2.30 A^1 -> H(a, U1(b, b)) 5.44/2.30 A^1 -> H(a, f(d)) 5.44/2.30 A^1 -> H(a, f(e)) 5.44/2.30 A^1 -> H(U1(d, e), U1(b, b)) 5.44/2.30 A^1 -> H(U1(d, e), f(d)) 5.44/2.30 A^1 -> H(U1(d, e), f(e)) 5.44/2.30 A^1 -> H(U1(d, a), U1(d, b)) 5.44/2.30 A^1 -> H(U1(d, a), U1(e, b)) 5.44/2.30 A^1 -> H(U1(d, a), U1(b, d)) 5.44/2.30 A^1 -> H(U1(d, a), U1(b, e)) 5.44/2.30 A^1 -> H(U1(d, a), U1(d, d)) 5.44/2.30 A^1 -> H(U1(d, a), U1(e, e)) 5.44/2.30 A^1 -> H(U1(e, d), U1(b, b)) 5.44/2.30 A^1 -> H(U1(e, d), f(d)) 5.44/2.30 A^1 -> H(U1(e, d), f(e)) 5.44/2.30 A^1 -> H(U1(e, a), U1(d, b)) 5.44/2.30 A^1 -> H(U1(e, a), U1(e, b)) 5.44/2.30 A^1 -> H(U1(e, a), U1(b, d)) 5.44/2.30 A^1 -> H(U1(e, a), U1(b, e)) 5.44/2.30 A^1 -> H(U1(e, a), U1(d, d)) 5.44/2.30 A^1 -> H(U1(e, a), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, d), U1(d, b)) 5.44/2.30 A^1 -> H(U1(a, d), U1(e, b)) 5.44/2.30 A^1 -> H(U1(a, d), U1(b, d)) 5.44/2.30 A^1 -> H(U1(a, d), U1(b, e)) 5.44/2.30 A^1 -> H(U1(a, d), U1(d, d)) 5.44/2.30 A^1 -> H(U1(a, d), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, e), U1(d, b)) 5.44/2.30 A^1 -> H(U1(a, e), U1(e, b)) 5.44/2.30 A^1 -> H(U1(a, e), U1(b, d)) 5.44/2.30 A^1 -> H(U1(a, e), U1(b, e)) 5.44/2.30 A^1 -> H(U1(a, e), U1(d, d)) 5.44/2.30 A^1 -> H(U1(a, e), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, a), b) 5.44/2.30 A^1 -> H(U1(a, a), U1(d, e)) 5.44/2.30 A^1 -> H(U1(a, a), U1(e, d)) 5.44/2.30 A^1 -> H(U1(a, a), d) 5.44/2.30 A^1 -> H(d, U1(b, b)) 5.44/2.30 A^1 -> H(d, f(d)) 5.44/2.30 A^1 -> H(d, f(e)) 5.44/2.30 A^1 -> H(U1(d, d), U1(d, b)) 5.44/2.30 A^1 -> H(U1(d, d), U1(e, b)) 5.44/2.30 A^1 -> H(U1(d, d), U1(b, d)) 5.44/2.30 A^1 -> H(U1(d, d), U1(b, e)) 5.44/2.30 A^1 -> H(U1(d, d), U1(d, d)) 5.44/2.30 A^1 -> H(U1(d, d), U1(e, e)) 5.44/2.30 A^1 -> H(f(d), b) 5.44/2.30 A^1 -> H(f(d), U1(d, e)) 5.44/2.30 A^1 -> H(f(d), U1(e, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(d, b)) 5.44/2.30 A^1 -> H(U1(e, e), U1(e, b)) 5.44/2.30 A^1 -> H(U1(e, e), U1(b, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(b, e)) 5.44/2.30 A^1 -> H(U1(e, e), U1(d, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(e, e)) 5.44/2.30 A^1 -> H(f(e), b) 5.44/2.30 A^1 -> H(f(e), U1(d, e)) 5.44/2.30 A^1 -> H(f(e), U1(e, d)) 5.44/2.30 A^1 -> H(f(e), d) 5.44/2.30 A^1 -> H(f(a), e) 5.44/2.30 5.44/2.30 The TRS R consists of the following rules: 5.44/2.30 5.44/2.30 a -> d 5.44/2.30 a -> e 5.44/2.30 f(x) -> U1(x, x) 5.44/2.30 b -> d 5.44/2.30 b -> e 5.44/2.30 U1(d, x) -> x 5.44/2.30 5.44/2.30 Q is empty. 5.44/2.30 We have to consider all minimal (P,Q,R)-chains. 5.44/2.30 ---------------------------------------- 5.44/2.30 5.44/2.30 (143) TransformationProof (EQUIVALENT) 5.44/2.30 By narrowing [LPAR04] the rule A^1 -> H(f(d), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: 5.44/2.30 5.44/2.30 (A^1 -> H(U1(d, d), U1(d, d)),A^1 -> H(U1(d, d), U1(d, d))) 5.44/2.30 (A^1 -> H(f(d), d),A^1 -> H(f(d), d)) 5.44/2.30 5.44/2.30 5.44/2.30 ---------------------------------------- 5.44/2.30 5.44/2.30 (144) 5.44/2.30 Obligation: 5.44/2.30 Q DP problem: 5.44/2.30 The TRS P consists of the following rules: 5.44/2.30 5.44/2.30 H(x, x) -> G(x, x) 5.44/2.30 G(d, e) -> A^1 5.44/2.30 A^1 -> H(f(d), f(d)) 5.44/2.30 A^1 -> H(f(e), f(e)) 5.44/2.30 A^1 -> H(f(a), d) 5.44/2.30 A^1 -> H(f(e), U1(e, e)) 5.44/2.30 A^1 -> H(e, f(b)) 5.44/2.30 A^1 -> H(a, U1(b, b)) 5.44/2.30 A^1 -> H(a, f(d)) 5.44/2.30 A^1 -> H(a, f(e)) 5.44/2.30 A^1 -> H(U1(d, e), U1(b, b)) 5.44/2.30 A^1 -> H(U1(d, e), f(d)) 5.44/2.30 A^1 -> H(U1(d, e), f(e)) 5.44/2.30 A^1 -> H(U1(d, a), U1(d, b)) 5.44/2.30 A^1 -> H(U1(d, a), U1(e, b)) 5.44/2.30 A^1 -> H(U1(d, a), U1(b, d)) 5.44/2.30 A^1 -> H(U1(d, a), U1(b, e)) 5.44/2.30 A^1 -> H(U1(d, a), U1(d, d)) 5.44/2.30 A^1 -> H(U1(d, a), U1(e, e)) 5.44/2.30 A^1 -> H(U1(e, d), U1(b, b)) 5.44/2.30 A^1 -> H(U1(e, d), f(d)) 5.44/2.30 A^1 -> H(U1(e, d), f(e)) 5.44/2.30 A^1 -> H(U1(e, a), U1(d, b)) 5.44/2.30 A^1 -> H(U1(e, a), U1(e, b)) 5.44/2.30 A^1 -> H(U1(e, a), U1(b, d)) 5.44/2.30 A^1 -> H(U1(e, a), U1(b, e)) 5.44/2.30 A^1 -> H(U1(e, a), U1(d, d)) 5.44/2.30 A^1 -> H(U1(e, a), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, d), U1(d, b)) 5.44/2.30 A^1 -> H(U1(a, d), U1(e, b)) 5.44/2.30 A^1 -> H(U1(a, d), U1(b, d)) 5.44/2.30 A^1 -> H(U1(a, d), U1(b, e)) 5.44/2.30 A^1 -> H(U1(a, d), U1(d, d)) 5.44/2.30 A^1 -> H(U1(a, d), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, e), U1(d, b)) 5.44/2.30 A^1 -> H(U1(a, e), U1(e, b)) 5.44/2.30 A^1 -> H(U1(a, e), U1(b, d)) 5.44/2.30 A^1 -> H(U1(a, e), U1(b, e)) 5.44/2.30 A^1 -> H(U1(a, e), U1(d, d)) 5.44/2.30 A^1 -> H(U1(a, e), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, a), b) 5.44/2.30 A^1 -> H(U1(a, a), U1(d, e)) 5.44/2.30 A^1 -> H(U1(a, a), U1(e, d)) 5.44/2.30 A^1 -> H(U1(a, a), d) 5.44/2.30 A^1 -> H(d, U1(b, b)) 5.44/2.30 A^1 -> H(d, f(d)) 5.44/2.30 A^1 -> H(d, f(e)) 5.44/2.30 A^1 -> H(U1(d, d), U1(d, b)) 5.44/2.30 A^1 -> H(U1(d, d), U1(e, b)) 5.44/2.30 A^1 -> H(U1(d, d), U1(b, d)) 5.44/2.30 A^1 -> H(U1(d, d), U1(b, e)) 5.44/2.30 A^1 -> H(U1(d, d), U1(d, d)) 5.44/2.30 A^1 -> H(U1(d, d), U1(e, e)) 5.44/2.30 A^1 -> H(f(d), b) 5.44/2.30 A^1 -> H(f(d), U1(d, e)) 5.44/2.30 A^1 -> H(f(d), U1(e, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(d, b)) 5.44/2.30 A^1 -> H(U1(e, e), U1(e, b)) 5.44/2.30 A^1 -> H(U1(e, e), U1(b, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(b, e)) 5.44/2.30 A^1 -> H(U1(e, e), U1(d, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(e, e)) 5.44/2.30 A^1 -> H(f(e), b) 5.44/2.30 A^1 -> H(f(e), U1(d, e)) 5.44/2.30 A^1 -> H(f(e), U1(e, d)) 5.44/2.30 A^1 -> H(f(e), d) 5.44/2.30 A^1 -> H(f(a), e) 5.44/2.30 A^1 -> H(f(d), d) 5.44/2.30 5.44/2.30 The TRS R consists of the following rules: 5.44/2.30 5.44/2.30 a -> d 5.44/2.30 a -> e 5.44/2.30 f(x) -> U1(x, x) 5.44/2.30 b -> d 5.44/2.30 b -> e 5.44/2.30 U1(d, x) -> x 5.44/2.30 5.44/2.30 Q is empty. 5.44/2.30 We have to consider all minimal (P,Q,R)-chains. 5.44/2.30 ---------------------------------------- 5.44/2.30 5.44/2.30 (145) TransformationProof (EQUIVALENT) 5.44/2.30 By narrowing [LPAR04] the rule A^1 -> H(f(a), d) at position [] we obtained the following new rules [LPAR04]: 5.44/2.30 5.44/2.30 (A^1 -> H(U1(a, a), d),A^1 -> H(U1(a, a), d)) 5.44/2.30 (A^1 -> H(f(d), d),A^1 -> H(f(d), d)) 5.44/2.30 (A^1 -> H(f(e), d),A^1 -> H(f(e), d)) 5.44/2.30 5.44/2.30 5.44/2.30 ---------------------------------------- 5.44/2.30 5.44/2.30 (146) 5.44/2.30 Obligation: 5.44/2.30 Q DP problem: 5.44/2.30 The TRS P consists of the following rules: 5.44/2.30 5.44/2.30 H(x, x) -> G(x, x) 5.44/2.30 G(d, e) -> A^1 5.44/2.30 A^1 -> H(f(d), f(d)) 5.44/2.30 A^1 -> H(f(e), f(e)) 5.44/2.30 A^1 -> H(f(e), U1(e, e)) 5.44/2.30 A^1 -> H(e, f(b)) 5.44/2.30 A^1 -> H(a, U1(b, b)) 5.44/2.30 A^1 -> H(a, f(d)) 5.44/2.30 A^1 -> H(a, f(e)) 5.44/2.30 A^1 -> H(U1(d, e), U1(b, b)) 5.44/2.30 A^1 -> H(U1(d, e), f(d)) 5.44/2.30 A^1 -> H(U1(d, e), f(e)) 5.44/2.30 A^1 -> H(U1(d, a), U1(d, b)) 5.44/2.30 A^1 -> H(U1(d, a), U1(e, b)) 5.44/2.30 A^1 -> H(U1(d, a), U1(b, d)) 5.44/2.30 A^1 -> H(U1(d, a), U1(b, e)) 5.44/2.30 A^1 -> H(U1(d, a), U1(d, d)) 5.44/2.30 A^1 -> H(U1(d, a), U1(e, e)) 5.44/2.30 A^1 -> H(U1(e, d), U1(b, b)) 5.44/2.30 A^1 -> H(U1(e, d), f(d)) 5.44/2.30 A^1 -> H(U1(e, d), f(e)) 5.44/2.30 A^1 -> H(U1(e, a), U1(d, b)) 5.44/2.30 A^1 -> H(U1(e, a), U1(e, b)) 5.44/2.30 A^1 -> H(U1(e, a), U1(b, d)) 5.44/2.30 A^1 -> H(U1(e, a), U1(b, e)) 5.44/2.30 A^1 -> H(U1(e, a), U1(d, d)) 5.44/2.30 A^1 -> H(U1(e, a), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, d), U1(d, b)) 5.44/2.30 A^1 -> H(U1(a, d), U1(e, b)) 5.44/2.30 A^1 -> H(U1(a, d), U1(b, d)) 5.44/2.30 A^1 -> H(U1(a, d), U1(b, e)) 5.44/2.30 A^1 -> H(U1(a, d), U1(d, d)) 5.44/2.30 A^1 -> H(U1(a, d), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, e), U1(d, b)) 5.44/2.30 A^1 -> H(U1(a, e), U1(e, b)) 5.44/2.30 A^1 -> H(U1(a, e), U1(b, d)) 5.44/2.30 A^1 -> H(U1(a, e), U1(b, e)) 5.44/2.30 A^1 -> H(U1(a, e), U1(d, d)) 5.44/2.30 A^1 -> H(U1(a, e), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, a), b) 5.44/2.30 A^1 -> H(U1(a, a), U1(d, e)) 5.44/2.30 A^1 -> H(U1(a, a), U1(e, d)) 5.44/2.30 A^1 -> H(U1(a, a), d) 5.44/2.30 A^1 -> H(d, U1(b, b)) 5.44/2.30 A^1 -> H(d, f(d)) 5.44/2.30 A^1 -> H(d, f(e)) 5.44/2.30 A^1 -> H(U1(d, d), U1(d, b)) 5.44/2.30 A^1 -> H(U1(d, d), U1(e, b)) 5.44/2.30 A^1 -> H(U1(d, d), U1(b, d)) 5.44/2.30 A^1 -> H(U1(d, d), U1(b, e)) 5.44/2.30 A^1 -> H(U1(d, d), U1(d, d)) 5.44/2.30 A^1 -> H(U1(d, d), U1(e, e)) 5.44/2.30 A^1 -> H(f(d), b) 5.44/2.30 A^1 -> H(f(d), U1(d, e)) 5.44/2.30 A^1 -> H(f(d), U1(e, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(d, b)) 5.44/2.30 A^1 -> H(U1(e, e), U1(e, b)) 5.44/2.30 A^1 -> H(U1(e, e), U1(b, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(b, e)) 5.44/2.30 A^1 -> H(U1(e, e), U1(d, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(e, e)) 5.44/2.30 A^1 -> H(f(e), b) 5.44/2.30 A^1 -> H(f(e), U1(d, e)) 5.44/2.30 A^1 -> H(f(e), U1(e, d)) 5.44/2.30 A^1 -> H(f(e), d) 5.44/2.30 A^1 -> H(f(a), e) 5.44/2.30 A^1 -> H(f(d), d) 5.44/2.30 5.44/2.30 The TRS R consists of the following rules: 5.44/2.30 5.44/2.30 a -> d 5.44/2.30 a -> e 5.44/2.30 f(x) -> U1(x, x) 5.44/2.30 b -> d 5.44/2.30 b -> e 5.44/2.30 U1(d, x) -> x 5.44/2.30 5.44/2.30 Q is empty. 5.44/2.30 We have to consider all minimal (P,Q,R)-chains. 5.44/2.30 ---------------------------------------- 5.44/2.30 5.44/2.30 (147) TransformationProof (EQUIVALENT) 5.44/2.30 By narrowing [LPAR04] the rule A^1 -> H(f(e), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: 5.44/2.30 5.44/2.30 (A^1 -> H(U1(e, e), U1(e, e)),A^1 -> H(U1(e, e), U1(e, e))) 5.44/2.30 5.44/2.30 5.44/2.30 ---------------------------------------- 5.44/2.30 5.44/2.30 (148) 5.44/2.30 Obligation: 5.44/2.30 Q DP problem: 5.44/2.30 The TRS P consists of the following rules: 5.44/2.30 5.44/2.30 H(x, x) -> G(x, x) 5.44/2.30 G(d, e) -> A^1 5.44/2.30 A^1 -> H(f(d), f(d)) 5.44/2.30 A^1 -> H(f(e), f(e)) 5.44/2.30 A^1 -> H(e, f(b)) 5.44/2.30 A^1 -> H(a, U1(b, b)) 5.44/2.30 A^1 -> H(a, f(d)) 5.44/2.30 A^1 -> H(a, f(e)) 5.44/2.30 A^1 -> H(U1(d, e), U1(b, b)) 5.44/2.30 A^1 -> H(U1(d, e), f(d)) 5.44/2.30 A^1 -> H(U1(d, e), f(e)) 5.44/2.30 A^1 -> H(U1(d, a), U1(d, b)) 5.44/2.30 A^1 -> H(U1(d, a), U1(e, b)) 5.44/2.30 A^1 -> H(U1(d, a), U1(b, d)) 5.44/2.30 A^1 -> H(U1(d, a), U1(b, e)) 5.44/2.30 A^1 -> H(U1(d, a), U1(d, d)) 5.44/2.30 A^1 -> H(U1(d, a), U1(e, e)) 5.44/2.30 A^1 -> H(U1(e, d), U1(b, b)) 5.44/2.30 A^1 -> H(U1(e, d), f(d)) 5.44/2.30 A^1 -> H(U1(e, d), f(e)) 5.44/2.30 A^1 -> H(U1(e, a), U1(d, b)) 5.44/2.30 A^1 -> H(U1(e, a), U1(e, b)) 5.44/2.30 A^1 -> H(U1(e, a), U1(b, d)) 5.44/2.30 A^1 -> H(U1(e, a), U1(b, e)) 5.44/2.30 A^1 -> H(U1(e, a), U1(d, d)) 5.44/2.30 A^1 -> H(U1(e, a), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, d), U1(d, b)) 5.44/2.30 A^1 -> H(U1(a, d), U1(e, b)) 5.44/2.30 A^1 -> H(U1(a, d), U1(b, d)) 5.44/2.30 A^1 -> H(U1(a, d), U1(b, e)) 5.44/2.30 A^1 -> H(U1(a, d), U1(d, d)) 5.44/2.30 A^1 -> H(U1(a, d), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, e), U1(d, b)) 5.44/2.30 A^1 -> H(U1(a, e), U1(e, b)) 5.44/2.30 A^1 -> H(U1(a, e), U1(b, d)) 5.44/2.30 A^1 -> H(U1(a, e), U1(b, e)) 5.44/2.30 A^1 -> H(U1(a, e), U1(d, d)) 5.44/2.30 A^1 -> H(U1(a, e), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, a), b) 5.44/2.30 A^1 -> H(U1(a, a), U1(d, e)) 5.44/2.30 A^1 -> H(U1(a, a), U1(e, d)) 5.44/2.30 A^1 -> H(U1(a, a), d) 5.44/2.30 A^1 -> H(d, U1(b, b)) 5.44/2.30 A^1 -> H(d, f(d)) 5.44/2.30 A^1 -> H(d, f(e)) 5.44/2.30 A^1 -> H(U1(d, d), U1(d, b)) 5.44/2.30 A^1 -> H(U1(d, d), U1(e, b)) 5.44/2.30 A^1 -> H(U1(d, d), U1(b, d)) 5.44/2.30 A^1 -> H(U1(d, d), U1(b, e)) 5.44/2.30 A^1 -> H(U1(d, d), U1(d, d)) 5.44/2.30 A^1 -> H(U1(d, d), U1(e, e)) 5.44/2.30 A^1 -> H(f(d), b) 5.44/2.30 A^1 -> H(f(d), U1(d, e)) 5.44/2.30 A^1 -> H(f(d), U1(e, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(d, b)) 5.44/2.30 A^1 -> H(U1(e, e), U1(e, b)) 5.44/2.30 A^1 -> H(U1(e, e), U1(b, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(b, e)) 5.44/2.30 A^1 -> H(U1(e, e), U1(d, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(e, e)) 5.44/2.30 A^1 -> H(f(e), b) 5.44/2.30 A^1 -> H(f(e), U1(d, e)) 5.44/2.30 A^1 -> H(f(e), U1(e, d)) 5.44/2.30 A^1 -> H(f(e), d) 5.44/2.30 A^1 -> H(f(a), e) 5.44/2.30 A^1 -> H(f(d), d) 5.44/2.30 5.44/2.30 The TRS R consists of the following rules: 5.44/2.30 5.44/2.30 a -> d 5.44/2.30 a -> e 5.44/2.30 f(x) -> U1(x, x) 5.44/2.30 b -> d 5.44/2.30 b -> e 5.44/2.30 U1(d, x) -> x 5.44/2.30 5.44/2.30 Q is empty. 5.44/2.30 We have to consider all minimal (P,Q,R)-chains. 5.44/2.30 ---------------------------------------- 5.44/2.30 5.44/2.30 (149) TransformationProof (EQUIVALENT) 5.44/2.30 By narrowing [LPAR04] the rule A^1 -> H(e, f(b)) at position [] we obtained the following new rules [LPAR04]: 5.44/2.30 5.44/2.30 (A^1 -> H(e, U1(b, b)),A^1 -> H(e, U1(b, b))) 5.44/2.30 (A^1 -> H(e, f(d)),A^1 -> H(e, f(d))) 5.44/2.30 (A^1 -> H(e, f(e)),A^1 -> H(e, f(e))) 5.44/2.30 5.44/2.30 5.44/2.30 ---------------------------------------- 5.44/2.30 5.44/2.30 (150) 5.44/2.30 Obligation: 5.44/2.30 Q DP problem: 5.44/2.30 The TRS P consists of the following rules: 5.44/2.30 5.44/2.30 H(x, x) -> G(x, x) 5.44/2.30 G(d, e) -> A^1 5.44/2.30 A^1 -> H(f(d), f(d)) 5.44/2.30 A^1 -> H(f(e), f(e)) 5.44/2.30 A^1 -> H(a, U1(b, b)) 5.44/2.30 A^1 -> H(a, f(d)) 5.44/2.30 A^1 -> H(a, f(e)) 5.44/2.30 A^1 -> H(U1(d, e), U1(b, b)) 5.44/2.30 A^1 -> H(U1(d, e), f(d)) 5.44/2.30 A^1 -> H(U1(d, e), f(e)) 5.44/2.30 A^1 -> H(U1(d, a), U1(d, b)) 5.44/2.30 A^1 -> H(U1(d, a), U1(e, b)) 5.44/2.30 A^1 -> H(U1(d, a), U1(b, d)) 5.44/2.30 A^1 -> H(U1(d, a), U1(b, e)) 5.44/2.30 A^1 -> H(U1(d, a), U1(d, d)) 5.44/2.30 A^1 -> H(U1(d, a), U1(e, e)) 5.44/2.30 A^1 -> H(U1(e, d), U1(b, b)) 5.44/2.30 A^1 -> H(U1(e, d), f(d)) 5.44/2.30 A^1 -> H(U1(e, d), f(e)) 5.44/2.30 A^1 -> H(U1(e, a), U1(d, b)) 5.44/2.30 A^1 -> H(U1(e, a), U1(e, b)) 5.44/2.30 A^1 -> H(U1(e, a), U1(b, d)) 5.44/2.30 A^1 -> H(U1(e, a), U1(b, e)) 5.44/2.30 A^1 -> H(U1(e, a), U1(d, d)) 5.44/2.30 A^1 -> H(U1(e, a), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, d), U1(d, b)) 5.44/2.30 A^1 -> H(U1(a, d), U1(e, b)) 5.44/2.30 A^1 -> H(U1(a, d), U1(b, d)) 5.44/2.30 A^1 -> H(U1(a, d), U1(b, e)) 5.44/2.30 A^1 -> H(U1(a, d), U1(d, d)) 5.44/2.30 A^1 -> H(U1(a, d), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, e), U1(d, b)) 5.44/2.30 A^1 -> H(U1(a, e), U1(e, b)) 5.44/2.30 A^1 -> H(U1(a, e), U1(b, d)) 5.44/2.30 A^1 -> H(U1(a, e), U1(b, e)) 5.44/2.30 A^1 -> H(U1(a, e), U1(d, d)) 5.44/2.30 A^1 -> H(U1(a, e), U1(e, e)) 5.44/2.30 A^1 -> H(U1(a, a), b) 5.44/2.30 A^1 -> H(U1(a, a), U1(d, e)) 5.44/2.30 A^1 -> H(U1(a, a), U1(e, d)) 5.44/2.30 A^1 -> H(U1(a, a), d) 5.44/2.30 A^1 -> H(d, U1(b, b)) 5.44/2.30 A^1 -> H(d, f(d)) 5.44/2.30 A^1 -> H(d, f(e)) 5.44/2.30 A^1 -> H(U1(d, d), U1(d, b)) 5.44/2.30 A^1 -> H(U1(d, d), U1(e, b)) 5.44/2.30 A^1 -> H(U1(d, d), U1(b, d)) 5.44/2.30 A^1 -> H(U1(d, d), U1(b, e)) 5.44/2.30 A^1 -> H(U1(d, d), U1(d, d)) 5.44/2.30 A^1 -> H(U1(d, d), U1(e, e)) 5.44/2.30 A^1 -> H(f(d), b) 5.44/2.30 A^1 -> H(f(d), U1(d, e)) 5.44/2.30 A^1 -> H(f(d), U1(e, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(d, b)) 5.44/2.30 A^1 -> H(U1(e, e), U1(e, b)) 5.44/2.30 A^1 -> H(U1(e, e), U1(b, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(b, e)) 5.44/2.30 A^1 -> H(U1(e, e), U1(d, d)) 5.44/2.30 A^1 -> H(U1(e, e), U1(e, e)) 5.44/2.30 A^1 -> H(f(e), b) 5.44/2.30 A^1 -> H(f(e), U1(d, e)) 5.44/2.30 A^1 -> H(f(e), U1(e, d)) 5.44/2.30 A^1 -> H(f(e), d) 5.44/2.30 A^1 -> H(f(a), e) 5.44/2.30 A^1 -> H(f(d), d) 5.44/2.30 A^1 -> H(e, U1(b, b)) 5.44/2.30 A^1 -> H(e, f(d)) 5.44/2.30 A^1 -> H(e, f(e)) 5.44/2.30 5.44/2.30 The TRS R consists of the following rules: 5.44/2.30 5.44/2.30 a -> d 5.44/2.30 a -> e 5.44/2.30 f(x) -> U1(x, x) 5.44/2.30 b -> d 5.44/2.30 b -> e 5.44/2.30 U1(d, x) -> x 5.54/2.30 5.54/2.30 Q is empty. 5.54/2.30 We have to consider all minimal (P,Q,R)-chains. 5.54/2.30 ---------------------------------------- 5.54/2.30 5.54/2.30 (151) TransformationProof (EQUIVALENT) 5.54/2.30 By narrowing [LPAR04] the rule A^1 -> H(a, U1(b, b)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.30 5.54/2.30 (A^1 -> H(d, U1(b, b)),A^1 -> H(d, U1(b, b))) 5.54/2.30 (A^1 -> H(e, U1(b, b)),A^1 -> H(e, U1(b, b))) 5.54/2.30 (A^1 -> H(a, U1(d, b)),A^1 -> H(a, U1(d, b))) 5.54/2.30 (A^1 -> H(a, U1(e, b)),A^1 -> H(a, U1(e, b))) 5.54/2.30 (A^1 -> H(a, U1(b, d)),A^1 -> H(a, U1(b, d))) 5.54/2.30 (A^1 -> H(a, U1(b, e)),A^1 -> H(a, U1(b, e))) 5.54/2.30 5.54/2.30 5.54/2.30 ---------------------------------------- 5.54/2.30 5.54/2.30 (152) 5.54/2.30 Obligation: 5.54/2.30 Q DP problem: 5.54/2.30 The TRS P consists of the following rules: 5.54/2.30 5.54/2.30 H(x, x) -> G(x, x) 5.54/2.30 G(d, e) -> A^1 5.54/2.30 A^1 -> H(f(d), f(d)) 5.54/2.30 A^1 -> H(f(e), f(e)) 5.54/2.30 A^1 -> H(a, f(d)) 5.54/2.30 A^1 -> H(a, f(e)) 5.54/2.30 A^1 -> H(U1(d, e), U1(b, b)) 5.54/2.30 A^1 -> H(U1(d, e), f(d)) 5.54/2.30 A^1 -> H(U1(d, e), f(e)) 5.54/2.30 A^1 -> H(U1(d, a), U1(d, b)) 5.54/2.30 A^1 -> H(U1(d, a), U1(e, b)) 5.54/2.30 A^1 -> H(U1(d, a), U1(b, d)) 5.54/2.30 A^1 -> H(U1(d, a), U1(b, e)) 5.54/2.30 A^1 -> H(U1(d, a), U1(d, d)) 5.54/2.30 A^1 -> H(U1(d, a), U1(e, e)) 5.54/2.30 A^1 -> H(U1(e, d), U1(b, b)) 5.54/2.30 A^1 -> H(U1(e, d), f(d)) 5.54/2.30 A^1 -> H(U1(e, d), f(e)) 5.54/2.30 A^1 -> H(U1(e, a), U1(d, b)) 5.54/2.30 A^1 -> H(U1(e, a), U1(e, b)) 5.54/2.30 A^1 -> H(U1(e, a), U1(b, d)) 5.54/2.30 A^1 -> H(U1(e, a), U1(b, e)) 5.54/2.30 A^1 -> H(U1(e, a), U1(d, d)) 5.54/2.30 A^1 -> H(U1(e, a), U1(e, e)) 5.54/2.30 A^1 -> H(U1(a, d), U1(d, b)) 5.54/2.30 A^1 -> H(U1(a, d), U1(e, b)) 5.54/2.30 A^1 -> H(U1(a, d), U1(b, d)) 5.54/2.30 A^1 -> H(U1(a, d), U1(b, e)) 5.54/2.30 A^1 -> H(U1(a, d), U1(d, d)) 5.54/2.30 A^1 -> H(U1(a, d), U1(e, e)) 5.54/2.30 A^1 -> H(U1(a, e), U1(d, b)) 5.54/2.30 A^1 -> H(U1(a, e), U1(e, b)) 5.54/2.30 A^1 -> H(U1(a, e), U1(b, d)) 5.54/2.30 A^1 -> H(U1(a, e), U1(b, e)) 5.54/2.30 A^1 -> H(U1(a, e), U1(d, d)) 5.54/2.30 A^1 -> H(U1(a, e), U1(e, e)) 5.54/2.30 A^1 -> H(U1(a, a), b) 5.54/2.30 A^1 -> H(U1(a, a), U1(d, e)) 5.54/2.30 A^1 -> H(U1(a, a), U1(e, d)) 5.54/2.30 A^1 -> H(U1(a, a), d) 5.54/2.30 A^1 -> H(d, U1(b, b)) 5.54/2.30 A^1 -> H(d, f(d)) 5.54/2.30 A^1 -> H(d, f(e)) 5.54/2.30 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.30 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.30 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.30 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.30 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.30 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.30 A^1 -> H(f(d), b) 5.54/2.30 A^1 -> H(f(d), U1(d, e)) 5.54/2.30 A^1 -> H(f(d), U1(e, d)) 5.54/2.30 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.30 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.30 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.30 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.30 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.30 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.30 A^1 -> H(f(e), b) 5.54/2.30 A^1 -> H(f(e), U1(d, e)) 5.54/2.30 A^1 -> H(f(e), U1(e, d)) 5.54/2.30 A^1 -> H(f(e), d) 5.54/2.30 A^1 -> H(f(a), e) 5.54/2.30 A^1 -> H(f(d), d) 5.54/2.30 A^1 -> H(e, U1(b, b)) 5.54/2.30 A^1 -> H(e, f(d)) 5.54/2.30 A^1 -> H(e, f(e)) 5.54/2.30 A^1 -> H(a, U1(d, b)) 5.54/2.30 A^1 -> H(a, U1(e, b)) 5.54/2.30 A^1 -> H(a, U1(b, d)) 5.54/2.30 A^1 -> H(a, U1(b, e)) 5.54/2.30 5.54/2.30 The TRS R consists of the following rules: 5.54/2.30 5.54/2.30 a -> d 5.54/2.30 a -> e 5.54/2.30 f(x) -> U1(x, x) 5.54/2.30 b -> d 5.54/2.30 b -> e 5.54/2.30 U1(d, x) -> x 5.54/2.30 5.54/2.30 Q is empty. 5.54/2.30 We have to consider all minimal (P,Q,R)-chains. 5.54/2.30 ---------------------------------------- 5.54/2.30 5.54/2.30 (153) TransformationProof (EQUIVALENT) 5.54/2.30 By narrowing [LPAR04] the rule A^1 -> H(a, f(d)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.30 5.54/2.30 (A^1 -> H(d, f(d)),A^1 -> H(d, f(d))) 5.54/2.30 (A^1 -> H(e, f(d)),A^1 -> H(e, f(d))) 5.54/2.30 (A^1 -> H(a, U1(d, d)),A^1 -> H(a, U1(d, d))) 5.54/2.30 5.54/2.30 5.54/2.30 ---------------------------------------- 5.54/2.30 5.54/2.30 (154) 5.54/2.30 Obligation: 5.54/2.30 Q DP problem: 5.54/2.30 The TRS P consists of the following rules: 5.54/2.30 5.54/2.30 H(x, x) -> G(x, x) 5.54/2.30 G(d, e) -> A^1 5.54/2.30 A^1 -> H(f(d), f(d)) 5.54/2.30 A^1 -> H(f(e), f(e)) 5.54/2.30 A^1 -> H(a, f(e)) 5.54/2.30 A^1 -> H(U1(d, e), U1(b, b)) 5.54/2.30 A^1 -> H(U1(d, e), f(d)) 5.54/2.30 A^1 -> H(U1(d, e), f(e)) 5.54/2.30 A^1 -> H(U1(d, a), U1(d, b)) 5.54/2.30 A^1 -> H(U1(d, a), U1(e, b)) 5.54/2.30 A^1 -> H(U1(d, a), U1(b, d)) 5.54/2.30 A^1 -> H(U1(d, a), U1(b, e)) 5.54/2.30 A^1 -> H(U1(d, a), U1(d, d)) 5.54/2.30 A^1 -> H(U1(d, a), U1(e, e)) 5.54/2.30 A^1 -> H(U1(e, d), U1(b, b)) 5.54/2.30 A^1 -> H(U1(e, d), f(d)) 5.54/2.30 A^1 -> H(U1(e, d), f(e)) 5.54/2.30 A^1 -> H(U1(e, a), U1(d, b)) 5.54/2.30 A^1 -> H(U1(e, a), U1(e, b)) 5.54/2.30 A^1 -> H(U1(e, a), U1(b, d)) 5.54/2.30 A^1 -> H(U1(e, a), U1(b, e)) 5.54/2.30 A^1 -> H(U1(e, a), U1(d, d)) 5.54/2.30 A^1 -> H(U1(e, a), U1(e, e)) 5.54/2.30 A^1 -> H(U1(a, d), U1(d, b)) 5.54/2.30 A^1 -> H(U1(a, d), U1(e, b)) 5.54/2.30 A^1 -> H(U1(a, d), U1(b, d)) 5.54/2.30 A^1 -> H(U1(a, d), U1(b, e)) 5.54/2.30 A^1 -> H(U1(a, d), U1(d, d)) 5.54/2.30 A^1 -> H(U1(a, d), U1(e, e)) 5.54/2.30 A^1 -> H(U1(a, e), U1(d, b)) 5.54/2.30 A^1 -> H(U1(a, e), U1(e, b)) 5.54/2.30 A^1 -> H(U1(a, e), U1(b, d)) 5.54/2.30 A^1 -> H(U1(a, e), U1(b, e)) 5.54/2.30 A^1 -> H(U1(a, e), U1(d, d)) 5.54/2.30 A^1 -> H(U1(a, e), U1(e, e)) 5.54/2.30 A^1 -> H(U1(a, a), b) 5.54/2.30 A^1 -> H(U1(a, a), U1(d, e)) 5.54/2.30 A^1 -> H(U1(a, a), U1(e, d)) 5.54/2.30 A^1 -> H(U1(a, a), d) 5.54/2.30 A^1 -> H(d, U1(b, b)) 5.54/2.30 A^1 -> H(d, f(d)) 5.54/2.30 A^1 -> H(d, f(e)) 5.54/2.30 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.30 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.30 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.30 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.30 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.30 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.30 A^1 -> H(f(d), b) 5.54/2.30 A^1 -> H(f(d), U1(d, e)) 5.54/2.30 A^1 -> H(f(d), U1(e, d)) 5.54/2.30 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.30 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.30 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.30 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.30 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.30 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.30 A^1 -> H(f(e), b) 5.54/2.30 A^1 -> H(f(e), U1(d, e)) 5.54/2.30 A^1 -> H(f(e), U1(e, d)) 5.54/2.30 A^1 -> H(f(e), d) 5.54/2.30 A^1 -> H(f(a), e) 5.54/2.30 A^1 -> H(f(d), d) 5.54/2.30 A^1 -> H(e, U1(b, b)) 5.54/2.30 A^1 -> H(e, f(d)) 5.54/2.30 A^1 -> H(e, f(e)) 5.54/2.30 A^1 -> H(a, U1(d, b)) 5.54/2.30 A^1 -> H(a, U1(e, b)) 5.54/2.30 A^1 -> H(a, U1(b, d)) 5.54/2.30 A^1 -> H(a, U1(b, e)) 5.54/2.30 A^1 -> H(a, U1(d, d)) 5.54/2.30 5.54/2.30 The TRS R consists of the following rules: 5.54/2.30 5.54/2.30 a -> d 5.54/2.30 a -> e 5.54/2.30 f(x) -> U1(x, x) 5.54/2.30 b -> d 5.54/2.30 b -> e 5.54/2.30 U1(d, x) -> x 5.54/2.30 5.54/2.30 Q is empty. 5.54/2.30 We have to consider all minimal (P,Q,R)-chains. 5.54/2.30 ---------------------------------------- 5.54/2.30 5.54/2.30 (155) TransformationProof (EQUIVALENT) 5.54/2.30 By narrowing [LPAR04] the rule A^1 -> H(a, f(e)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.30 5.54/2.30 (A^1 -> H(d, f(e)),A^1 -> H(d, f(e))) 5.54/2.30 (A^1 -> H(e, f(e)),A^1 -> H(e, f(e))) 5.54/2.30 (A^1 -> H(a, U1(e, e)),A^1 -> H(a, U1(e, e))) 5.54/2.30 5.54/2.30 5.54/2.30 ---------------------------------------- 5.54/2.30 5.54/2.30 (156) 5.54/2.30 Obligation: 5.54/2.30 Q DP problem: 5.54/2.30 The TRS P consists of the following rules: 5.54/2.30 5.54/2.30 H(x, x) -> G(x, x) 5.54/2.30 G(d, e) -> A^1 5.54/2.30 A^1 -> H(f(d), f(d)) 5.54/2.30 A^1 -> H(f(e), f(e)) 5.54/2.30 A^1 -> H(U1(d, e), U1(b, b)) 5.54/2.30 A^1 -> H(U1(d, e), f(d)) 5.54/2.30 A^1 -> H(U1(d, e), f(e)) 5.54/2.30 A^1 -> H(U1(d, a), U1(d, b)) 5.54/2.30 A^1 -> H(U1(d, a), U1(e, b)) 5.54/2.30 A^1 -> H(U1(d, a), U1(b, d)) 5.54/2.30 A^1 -> H(U1(d, a), U1(b, e)) 5.54/2.30 A^1 -> H(U1(d, a), U1(d, d)) 5.54/2.30 A^1 -> H(U1(d, a), U1(e, e)) 5.54/2.30 A^1 -> H(U1(e, d), U1(b, b)) 5.54/2.30 A^1 -> H(U1(e, d), f(d)) 5.54/2.30 A^1 -> H(U1(e, d), f(e)) 5.54/2.30 A^1 -> H(U1(e, a), U1(d, b)) 5.54/2.30 A^1 -> H(U1(e, a), U1(e, b)) 5.54/2.30 A^1 -> H(U1(e, a), U1(b, d)) 5.54/2.30 A^1 -> H(U1(e, a), U1(b, e)) 5.54/2.30 A^1 -> H(U1(e, a), U1(d, d)) 5.54/2.30 A^1 -> H(U1(e, a), U1(e, e)) 5.54/2.30 A^1 -> H(U1(a, d), U1(d, b)) 5.54/2.30 A^1 -> H(U1(a, d), U1(e, b)) 5.54/2.30 A^1 -> H(U1(a, d), U1(b, d)) 5.54/2.30 A^1 -> H(U1(a, d), U1(b, e)) 5.54/2.30 A^1 -> H(U1(a, d), U1(d, d)) 5.54/2.30 A^1 -> H(U1(a, d), U1(e, e)) 5.54/2.30 A^1 -> H(U1(a, e), U1(d, b)) 5.54/2.30 A^1 -> H(U1(a, e), U1(e, b)) 5.54/2.30 A^1 -> H(U1(a, e), U1(b, d)) 5.54/2.30 A^1 -> H(U1(a, e), U1(b, e)) 5.54/2.30 A^1 -> H(U1(a, e), U1(d, d)) 5.54/2.30 A^1 -> H(U1(a, e), U1(e, e)) 5.54/2.30 A^1 -> H(U1(a, a), b) 5.54/2.30 A^1 -> H(U1(a, a), U1(d, e)) 5.54/2.30 A^1 -> H(U1(a, a), U1(e, d)) 5.54/2.30 A^1 -> H(U1(a, a), d) 5.54/2.30 A^1 -> H(d, U1(b, b)) 5.54/2.30 A^1 -> H(d, f(d)) 5.54/2.30 A^1 -> H(d, f(e)) 5.54/2.30 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.30 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.30 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.30 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.30 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.30 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.30 A^1 -> H(f(d), b) 5.54/2.30 A^1 -> H(f(d), U1(d, e)) 5.54/2.30 A^1 -> H(f(d), U1(e, d)) 5.54/2.30 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.30 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.30 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.30 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.30 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.30 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.30 A^1 -> H(f(e), b) 5.54/2.30 A^1 -> H(f(e), U1(d, e)) 5.54/2.30 A^1 -> H(f(e), U1(e, d)) 5.54/2.30 A^1 -> H(f(e), d) 5.54/2.30 A^1 -> H(f(a), e) 5.54/2.30 A^1 -> H(f(d), d) 5.54/2.30 A^1 -> H(e, U1(b, b)) 5.54/2.30 A^1 -> H(e, f(d)) 5.54/2.30 A^1 -> H(e, f(e)) 5.54/2.30 A^1 -> H(a, U1(d, b)) 5.54/2.30 A^1 -> H(a, U1(e, b)) 5.54/2.30 A^1 -> H(a, U1(b, d)) 5.54/2.30 A^1 -> H(a, U1(b, e)) 5.54/2.30 A^1 -> H(a, U1(d, d)) 5.54/2.30 A^1 -> H(a, U1(e, e)) 5.54/2.30 5.54/2.30 The TRS R consists of the following rules: 5.54/2.30 5.54/2.30 a -> d 5.54/2.30 a -> e 5.54/2.30 f(x) -> U1(x, x) 5.54/2.30 b -> d 5.54/2.30 b -> e 5.54/2.30 U1(d, x) -> x 5.54/2.30 5.54/2.30 Q is empty. 5.54/2.30 We have to consider all minimal (P,Q,R)-chains. 5.54/2.30 ---------------------------------------- 5.54/2.30 5.54/2.30 (157) TransformationProof (EQUIVALENT) 5.54/2.30 By narrowing [LPAR04] the rule A^1 -> H(U1(d, e), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.30 5.54/2.30 (A^1 -> H(e, U1(b, b)),A^1 -> H(e, U1(b, b))) 5.54/2.30 (A^1 -> H(U1(d, e), U1(d, b)),A^1 -> H(U1(d, e), U1(d, b))) 5.54/2.30 (A^1 -> H(U1(d, e), U1(e, b)),A^1 -> H(U1(d, e), U1(e, b))) 5.54/2.30 (A^1 -> H(U1(d, e), U1(b, d)),A^1 -> H(U1(d, e), U1(b, d))) 5.54/2.30 (A^1 -> H(U1(d, e), U1(b, e)),A^1 -> H(U1(d, e), U1(b, e))) 5.54/2.30 5.54/2.30 5.54/2.30 ---------------------------------------- 5.54/2.30 5.54/2.30 (158) 5.54/2.30 Obligation: 5.54/2.30 Q DP problem: 5.54/2.30 The TRS P consists of the following rules: 5.54/2.30 5.54/2.30 H(x, x) -> G(x, x) 5.54/2.30 G(d, e) -> A^1 5.54/2.30 A^1 -> H(f(d), f(d)) 5.54/2.30 A^1 -> H(f(e), f(e)) 5.54/2.30 A^1 -> H(U1(d, e), f(d)) 5.54/2.30 A^1 -> H(U1(d, e), f(e)) 5.54/2.30 A^1 -> H(U1(d, a), U1(d, b)) 5.54/2.30 A^1 -> H(U1(d, a), U1(e, b)) 5.54/2.30 A^1 -> H(U1(d, a), U1(b, d)) 5.54/2.30 A^1 -> H(U1(d, a), U1(b, e)) 5.54/2.30 A^1 -> H(U1(d, a), U1(d, d)) 5.54/2.30 A^1 -> H(U1(d, a), U1(e, e)) 5.54/2.30 A^1 -> H(U1(e, d), U1(b, b)) 5.54/2.30 A^1 -> H(U1(e, d), f(d)) 5.54/2.30 A^1 -> H(U1(e, d), f(e)) 5.54/2.30 A^1 -> H(U1(e, a), U1(d, b)) 5.54/2.30 A^1 -> H(U1(e, a), U1(e, b)) 5.54/2.30 A^1 -> H(U1(e, a), U1(b, d)) 5.54/2.30 A^1 -> H(U1(e, a), U1(b, e)) 5.54/2.30 A^1 -> H(U1(e, a), U1(d, d)) 5.54/2.30 A^1 -> H(U1(e, a), U1(e, e)) 5.54/2.30 A^1 -> H(U1(a, d), U1(d, b)) 5.54/2.30 A^1 -> H(U1(a, d), U1(e, b)) 5.54/2.30 A^1 -> H(U1(a, d), U1(b, d)) 5.54/2.30 A^1 -> H(U1(a, d), U1(b, e)) 5.54/2.30 A^1 -> H(U1(a, d), U1(d, d)) 5.54/2.30 A^1 -> H(U1(a, d), U1(e, e)) 5.54/2.30 A^1 -> H(U1(a, e), U1(d, b)) 5.54/2.30 A^1 -> H(U1(a, e), U1(e, b)) 5.54/2.30 A^1 -> H(U1(a, e), U1(b, d)) 5.54/2.30 A^1 -> H(U1(a, e), U1(b, e)) 5.54/2.30 A^1 -> H(U1(a, e), U1(d, d)) 5.54/2.30 A^1 -> H(U1(a, e), U1(e, e)) 5.54/2.30 A^1 -> H(U1(a, a), b) 5.54/2.30 A^1 -> H(U1(a, a), U1(d, e)) 5.54/2.30 A^1 -> H(U1(a, a), U1(e, d)) 5.54/2.30 A^1 -> H(U1(a, a), d) 5.54/2.30 A^1 -> H(d, U1(b, b)) 5.54/2.30 A^1 -> H(d, f(d)) 5.54/2.30 A^1 -> H(d, f(e)) 5.54/2.30 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.30 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.30 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.30 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.30 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.30 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.30 A^1 -> H(f(d), b) 5.54/2.30 A^1 -> H(f(d), U1(d, e)) 5.54/2.30 A^1 -> H(f(d), U1(e, d)) 5.54/2.30 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.30 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.30 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.30 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.30 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.30 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.30 A^1 -> H(f(e), b) 5.54/2.30 A^1 -> H(f(e), U1(d, e)) 5.54/2.30 A^1 -> H(f(e), U1(e, d)) 5.54/2.30 A^1 -> H(f(e), d) 5.54/2.30 A^1 -> H(f(a), e) 5.54/2.30 A^1 -> H(f(d), d) 5.54/2.30 A^1 -> H(e, U1(b, b)) 5.54/2.30 A^1 -> H(e, f(d)) 5.54/2.30 A^1 -> H(e, f(e)) 5.54/2.30 A^1 -> H(a, U1(d, b)) 5.54/2.30 A^1 -> H(a, U1(e, b)) 5.54/2.30 A^1 -> H(a, U1(b, d)) 5.54/2.30 A^1 -> H(a, U1(b, e)) 5.54/2.30 A^1 -> H(a, U1(d, d)) 5.54/2.30 A^1 -> H(a, U1(e, e)) 5.54/2.30 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.30 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.30 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.30 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.30 5.54/2.30 The TRS R consists of the following rules: 5.54/2.30 5.54/2.30 a -> d 5.54/2.30 a -> e 5.54/2.30 f(x) -> U1(x, x) 5.54/2.30 b -> d 5.54/2.30 b -> e 5.54/2.30 U1(d, x) -> x 5.54/2.30 5.54/2.30 Q is empty. 5.54/2.30 We have to consider all minimal (P,Q,R)-chains. 5.54/2.30 ---------------------------------------- 5.54/2.30 5.54/2.30 (159) TransformationProof (EQUIVALENT) 5.54/2.30 By narrowing [LPAR04] the rule A^1 -> H(U1(d, e), f(d)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.31 5.54/2.31 (A^1 -> H(e, f(d)),A^1 -> H(e, f(d))) 5.54/2.31 (A^1 -> H(U1(d, e), U1(d, d)),A^1 -> H(U1(d, e), U1(d, d))) 5.54/2.31 5.54/2.31 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (160) 5.54/2.31 Obligation: 5.54/2.31 Q DP problem: 5.54/2.31 The TRS P consists of the following rules: 5.54/2.31 5.54/2.31 H(x, x) -> G(x, x) 5.54/2.31 G(d, e) -> A^1 5.54/2.31 A^1 -> H(f(d), f(d)) 5.54/2.31 A^1 -> H(f(e), f(e)) 5.54/2.31 A^1 -> H(U1(d, e), f(e)) 5.54/2.31 A^1 -> H(U1(d, a), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, a), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, a), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, a), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, a), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, a), U1(e, e)) 5.54/2.31 A^1 -> H(U1(e, d), U1(b, b)) 5.54/2.31 A^1 -> H(U1(e, d), f(d)) 5.54/2.31 A^1 -> H(U1(e, d), f(e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, a), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, a), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, a), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, a), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, a), b) 5.54/2.31 A^1 -> H(U1(a, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(a, a), U1(e, d)) 5.54/2.31 A^1 -> H(U1(a, a), d) 5.54/2.31 A^1 -> H(d, U1(b, b)) 5.54/2.31 A^1 -> H(d, f(d)) 5.54/2.31 A^1 -> H(d, f(e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.31 A^1 -> H(f(d), b) 5.54/2.31 A^1 -> H(f(d), U1(d, e)) 5.54/2.31 A^1 -> H(f(d), U1(e, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.31 A^1 -> H(f(e), b) 5.54/2.31 A^1 -> H(f(e), U1(d, e)) 5.54/2.31 A^1 -> H(f(e), U1(e, d)) 5.54/2.31 A^1 -> H(f(e), d) 5.54/2.31 A^1 -> H(f(a), e) 5.54/2.31 A^1 -> H(f(d), d) 5.54/2.31 A^1 -> H(e, U1(b, b)) 5.54/2.31 A^1 -> H(e, f(d)) 5.54/2.31 A^1 -> H(e, f(e)) 5.54/2.31 A^1 -> H(a, U1(d, b)) 5.54/2.31 A^1 -> H(a, U1(e, b)) 5.54/2.31 A^1 -> H(a, U1(b, d)) 5.54/2.31 A^1 -> H(a, U1(b, e)) 5.54/2.31 A^1 -> H(a, U1(d, d)) 5.54/2.31 A^1 -> H(a, U1(e, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.31 5.54/2.31 The TRS R consists of the following rules: 5.54/2.31 5.54/2.31 a -> d 5.54/2.31 a -> e 5.54/2.31 f(x) -> U1(x, x) 5.54/2.31 b -> d 5.54/2.31 b -> e 5.54/2.31 U1(d, x) -> x 5.54/2.31 5.54/2.31 Q is empty. 5.54/2.31 We have to consider all minimal (P,Q,R)-chains. 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (161) TransformationProof (EQUIVALENT) 5.54/2.31 By narrowing [LPAR04] the rule A^1 -> H(U1(d, e), f(e)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.31 5.54/2.31 (A^1 -> H(e, f(e)),A^1 -> H(e, f(e))) 5.54/2.31 (A^1 -> H(U1(d, e), U1(e, e)),A^1 -> H(U1(d, e), U1(e, e))) 5.54/2.31 5.54/2.31 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (162) 5.54/2.31 Obligation: 5.54/2.31 Q DP problem: 5.54/2.31 The TRS P consists of the following rules: 5.54/2.31 5.54/2.31 H(x, x) -> G(x, x) 5.54/2.31 G(d, e) -> A^1 5.54/2.31 A^1 -> H(f(d), f(d)) 5.54/2.31 A^1 -> H(f(e), f(e)) 5.54/2.31 A^1 -> H(U1(d, a), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, a), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, a), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, a), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, a), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, a), U1(e, e)) 5.54/2.31 A^1 -> H(U1(e, d), U1(b, b)) 5.54/2.31 A^1 -> H(U1(e, d), f(d)) 5.54/2.31 A^1 -> H(U1(e, d), f(e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, a), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, a), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, a), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, a), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, a), b) 5.54/2.31 A^1 -> H(U1(a, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(a, a), U1(e, d)) 5.54/2.31 A^1 -> H(U1(a, a), d) 5.54/2.31 A^1 -> H(d, U1(b, b)) 5.54/2.31 A^1 -> H(d, f(d)) 5.54/2.31 A^1 -> H(d, f(e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.31 A^1 -> H(f(d), b) 5.54/2.31 A^1 -> H(f(d), U1(d, e)) 5.54/2.31 A^1 -> H(f(d), U1(e, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.31 A^1 -> H(f(e), b) 5.54/2.31 A^1 -> H(f(e), U1(d, e)) 5.54/2.31 A^1 -> H(f(e), U1(e, d)) 5.54/2.31 A^1 -> H(f(e), d) 5.54/2.31 A^1 -> H(f(a), e) 5.54/2.31 A^1 -> H(f(d), d) 5.54/2.31 A^1 -> H(e, U1(b, b)) 5.54/2.31 A^1 -> H(e, f(d)) 5.54/2.31 A^1 -> H(e, f(e)) 5.54/2.31 A^1 -> H(a, U1(d, b)) 5.54/2.31 A^1 -> H(a, U1(e, b)) 5.54/2.31 A^1 -> H(a, U1(b, d)) 5.54/2.31 A^1 -> H(a, U1(b, e)) 5.54/2.31 A^1 -> H(a, U1(d, d)) 5.54/2.31 A^1 -> H(a, U1(e, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.31 5.54/2.31 The TRS R consists of the following rules: 5.54/2.31 5.54/2.31 a -> d 5.54/2.31 a -> e 5.54/2.31 f(x) -> U1(x, x) 5.54/2.31 b -> d 5.54/2.31 b -> e 5.54/2.31 U1(d, x) -> x 5.54/2.31 5.54/2.31 Q is empty. 5.54/2.31 We have to consider all minimal (P,Q,R)-chains. 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (163) TransformationProof (EQUIVALENT) 5.54/2.31 By narrowing [LPAR04] the rule A^1 -> H(U1(d, a), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.31 5.54/2.31 (A^1 -> H(a, U1(d, b)),A^1 -> H(a, U1(d, b))) 5.54/2.31 (A^1 -> H(U1(d, d), U1(d, b)),A^1 -> H(U1(d, d), U1(d, b))) 5.54/2.31 (A^1 -> H(U1(d, e), U1(d, b)),A^1 -> H(U1(d, e), U1(d, b))) 5.54/2.31 (A^1 -> H(U1(d, a), b),A^1 -> H(U1(d, a), b)) 5.54/2.31 (A^1 -> H(U1(d, a), U1(d, d)),A^1 -> H(U1(d, a), U1(d, d))) 5.54/2.31 (A^1 -> H(U1(d, a), U1(d, e)),A^1 -> H(U1(d, a), U1(d, e))) 5.54/2.31 5.54/2.31 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (164) 5.54/2.31 Obligation: 5.54/2.31 Q DP problem: 5.54/2.31 The TRS P consists of the following rules: 5.54/2.31 5.54/2.31 H(x, x) -> G(x, x) 5.54/2.31 G(d, e) -> A^1 5.54/2.31 A^1 -> H(f(d), f(d)) 5.54/2.31 A^1 -> H(f(e), f(e)) 5.54/2.31 A^1 -> H(U1(d, a), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, a), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, a), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, a), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, a), U1(e, e)) 5.54/2.31 A^1 -> H(U1(e, d), U1(b, b)) 5.54/2.31 A^1 -> H(U1(e, d), f(d)) 5.54/2.31 A^1 -> H(U1(e, d), f(e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, a), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, a), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, a), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, a), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, a), b) 5.54/2.31 A^1 -> H(U1(a, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(a, a), U1(e, d)) 5.54/2.31 A^1 -> H(U1(a, a), d) 5.54/2.31 A^1 -> H(d, U1(b, b)) 5.54/2.31 A^1 -> H(d, f(d)) 5.54/2.31 A^1 -> H(d, f(e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.31 A^1 -> H(f(d), b) 5.54/2.31 A^1 -> H(f(d), U1(d, e)) 5.54/2.31 A^1 -> H(f(d), U1(e, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.31 A^1 -> H(f(e), b) 5.54/2.31 A^1 -> H(f(e), U1(d, e)) 5.54/2.31 A^1 -> H(f(e), U1(e, d)) 5.54/2.31 A^1 -> H(f(e), d) 5.54/2.31 A^1 -> H(f(a), e) 5.54/2.31 A^1 -> H(f(d), d) 5.54/2.31 A^1 -> H(e, U1(b, b)) 5.54/2.31 A^1 -> H(e, f(d)) 5.54/2.31 A^1 -> H(e, f(e)) 5.54/2.31 A^1 -> H(a, U1(d, b)) 5.54/2.31 A^1 -> H(a, U1(e, b)) 5.54/2.31 A^1 -> H(a, U1(b, d)) 5.54/2.31 A^1 -> H(a, U1(b, e)) 5.54/2.31 A^1 -> H(a, U1(d, d)) 5.54/2.31 A^1 -> H(a, U1(e, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.31 A^1 -> H(U1(d, a), b) 5.54/2.31 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.31 5.54/2.31 The TRS R consists of the following rules: 5.54/2.31 5.54/2.31 a -> d 5.54/2.31 a -> e 5.54/2.31 f(x) -> U1(x, x) 5.54/2.31 b -> d 5.54/2.31 b -> e 5.54/2.31 U1(d, x) -> x 5.54/2.31 5.54/2.31 Q is empty. 5.54/2.31 We have to consider all minimal (P,Q,R)-chains. 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (165) TransformationProof (EQUIVALENT) 5.54/2.31 By narrowing [LPAR04] the rule A^1 -> H(U1(d, a), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.31 5.54/2.31 (A^1 -> H(a, U1(e, b)),A^1 -> H(a, U1(e, b))) 5.54/2.31 (A^1 -> H(U1(d, d), U1(e, b)),A^1 -> H(U1(d, d), U1(e, b))) 5.54/2.31 (A^1 -> H(U1(d, e), U1(e, b)),A^1 -> H(U1(d, e), U1(e, b))) 5.54/2.31 (A^1 -> H(U1(d, a), U1(e, d)),A^1 -> H(U1(d, a), U1(e, d))) 5.54/2.31 (A^1 -> H(U1(d, a), U1(e, e)),A^1 -> H(U1(d, a), U1(e, e))) 5.54/2.31 5.54/2.31 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (166) 5.54/2.31 Obligation: 5.54/2.31 Q DP problem: 5.54/2.31 The TRS P consists of the following rules: 5.54/2.31 5.54/2.31 H(x, x) -> G(x, x) 5.54/2.31 G(d, e) -> A^1 5.54/2.31 A^1 -> H(f(d), f(d)) 5.54/2.31 A^1 -> H(f(e), f(e)) 5.54/2.31 A^1 -> H(U1(d, a), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, a), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, a), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, a), U1(e, e)) 5.54/2.31 A^1 -> H(U1(e, d), U1(b, b)) 5.54/2.31 A^1 -> H(U1(e, d), f(d)) 5.54/2.31 A^1 -> H(U1(e, d), f(e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, a), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, a), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, a), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, a), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, a), b) 5.54/2.31 A^1 -> H(U1(a, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(a, a), U1(e, d)) 5.54/2.31 A^1 -> H(U1(a, a), d) 5.54/2.31 A^1 -> H(d, U1(b, b)) 5.54/2.31 A^1 -> H(d, f(d)) 5.54/2.31 A^1 -> H(d, f(e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.31 A^1 -> H(f(d), b) 5.54/2.31 A^1 -> H(f(d), U1(d, e)) 5.54/2.31 A^1 -> H(f(d), U1(e, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.31 A^1 -> H(f(e), b) 5.54/2.31 A^1 -> H(f(e), U1(d, e)) 5.54/2.31 A^1 -> H(f(e), U1(e, d)) 5.54/2.31 A^1 -> H(f(e), d) 5.54/2.31 A^1 -> H(f(a), e) 5.54/2.31 A^1 -> H(f(d), d) 5.54/2.31 A^1 -> H(e, U1(b, b)) 5.54/2.31 A^1 -> H(e, f(d)) 5.54/2.31 A^1 -> H(e, f(e)) 5.54/2.31 A^1 -> H(a, U1(d, b)) 5.54/2.31 A^1 -> H(a, U1(e, b)) 5.54/2.31 A^1 -> H(a, U1(b, d)) 5.54/2.31 A^1 -> H(a, U1(b, e)) 5.54/2.31 A^1 -> H(a, U1(d, d)) 5.54/2.31 A^1 -> H(a, U1(e, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.31 A^1 -> H(U1(d, a), b) 5.54/2.31 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.31 5.54/2.31 The TRS R consists of the following rules: 5.54/2.31 5.54/2.31 a -> d 5.54/2.31 a -> e 5.54/2.31 f(x) -> U1(x, x) 5.54/2.31 b -> d 5.54/2.31 b -> e 5.54/2.31 U1(d, x) -> x 5.54/2.31 5.54/2.31 Q is empty. 5.54/2.31 We have to consider all minimal (P,Q,R)-chains. 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (167) TransformationProof (EQUIVALENT) 5.54/2.31 By narrowing [LPAR04] the rule A^1 -> H(U1(d, a), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.31 5.54/2.31 (A^1 -> H(a, U1(b, d)),A^1 -> H(a, U1(b, d))) 5.54/2.31 (A^1 -> H(U1(d, d), U1(b, d)),A^1 -> H(U1(d, d), U1(b, d))) 5.54/2.31 (A^1 -> H(U1(d, e), U1(b, d)),A^1 -> H(U1(d, e), U1(b, d))) 5.54/2.31 (A^1 -> H(U1(d, a), U1(d, d)),A^1 -> H(U1(d, a), U1(d, d))) 5.54/2.31 (A^1 -> H(U1(d, a), U1(e, d)),A^1 -> H(U1(d, a), U1(e, d))) 5.54/2.31 5.54/2.31 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (168) 5.54/2.31 Obligation: 5.54/2.31 Q DP problem: 5.54/2.31 The TRS P consists of the following rules: 5.54/2.31 5.54/2.31 H(x, x) -> G(x, x) 5.54/2.31 G(d, e) -> A^1 5.54/2.31 A^1 -> H(f(d), f(d)) 5.54/2.31 A^1 -> H(f(e), f(e)) 5.54/2.31 A^1 -> H(U1(d, a), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, a), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, a), U1(e, e)) 5.54/2.31 A^1 -> H(U1(e, d), U1(b, b)) 5.54/2.31 A^1 -> H(U1(e, d), f(d)) 5.54/2.31 A^1 -> H(U1(e, d), f(e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, a), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, a), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, a), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, a), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, a), b) 5.54/2.31 A^1 -> H(U1(a, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(a, a), U1(e, d)) 5.54/2.31 A^1 -> H(U1(a, a), d) 5.54/2.31 A^1 -> H(d, U1(b, b)) 5.54/2.31 A^1 -> H(d, f(d)) 5.54/2.31 A^1 -> H(d, f(e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.31 A^1 -> H(f(d), b) 5.54/2.31 A^1 -> H(f(d), U1(d, e)) 5.54/2.31 A^1 -> H(f(d), U1(e, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.31 A^1 -> H(f(e), b) 5.54/2.31 A^1 -> H(f(e), U1(d, e)) 5.54/2.31 A^1 -> H(f(e), U1(e, d)) 5.54/2.31 A^1 -> H(f(e), d) 5.54/2.31 A^1 -> H(f(a), e) 5.54/2.31 A^1 -> H(f(d), d) 5.54/2.31 A^1 -> H(e, U1(b, b)) 5.54/2.31 A^1 -> H(e, f(d)) 5.54/2.31 A^1 -> H(e, f(e)) 5.54/2.31 A^1 -> H(a, U1(d, b)) 5.54/2.31 A^1 -> H(a, U1(e, b)) 5.54/2.31 A^1 -> H(a, U1(b, d)) 5.54/2.31 A^1 -> H(a, U1(b, e)) 5.54/2.31 A^1 -> H(a, U1(d, d)) 5.54/2.31 A^1 -> H(a, U1(e, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.31 A^1 -> H(U1(d, a), b) 5.54/2.31 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.31 5.54/2.31 The TRS R consists of the following rules: 5.54/2.31 5.54/2.31 a -> d 5.54/2.31 a -> e 5.54/2.31 f(x) -> U1(x, x) 5.54/2.31 b -> d 5.54/2.31 b -> e 5.54/2.31 U1(d, x) -> x 5.54/2.31 5.54/2.31 Q is empty. 5.54/2.31 We have to consider all minimal (P,Q,R)-chains. 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (169) TransformationProof (EQUIVALENT) 5.54/2.31 By narrowing [LPAR04] the rule A^1 -> H(U1(d, a), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.31 5.54/2.31 (A^1 -> H(a, U1(b, e)),A^1 -> H(a, U1(b, e))) 5.54/2.31 (A^1 -> H(U1(d, d), U1(b, e)),A^1 -> H(U1(d, d), U1(b, e))) 5.54/2.31 (A^1 -> H(U1(d, e), U1(b, e)),A^1 -> H(U1(d, e), U1(b, e))) 5.54/2.31 (A^1 -> H(U1(d, a), U1(d, e)),A^1 -> H(U1(d, a), U1(d, e))) 5.54/2.31 (A^1 -> H(U1(d, a), U1(e, e)),A^1 -> H(U1(d, a), U1(e, e))) 5.54/2.31 5.54/2.31 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (170) 5.54/2.31 Obligation: 5.54/2.31 Q DP problem: 5.54/2.31 The TRS P consists of the following rules: 5.54/2.31 5.54/2.31 H(x, x) -> G(x, x) 5.54/2.31 G(d, e) -> A^1 5.54/2.31 A^1 -> H(f(d), f(d)) 5.54/2.31 A^1 -> H(f(e), f(e)) 5.54/2.31 A^1 -> H(U1(d, a), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, a), U1(e, e)) 5.54/2.31 A^1 -> H(U1(e, d), U1(b, b)) 5.54/2.31 A^1 -> H(U1(e, d), f(d)) 5.54/2.31 A^1 -> H(U1(e, d), f(e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, a), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, a), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, a), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, a), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, a), b) 5.54/2.31 A^1 -> H(U1(a, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(a, a), U1(e, d)) 5.54/2.31 A^1 -> H(U1(a, a), d) 5.54/2.31 A^1 -> H(d, U1(b, b)) 5.54/2.31 A^1 -> H(d, f(d)) 5.54/2.31 A^1 -> H(d, f(e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.31 A^1 -> H(f(d), b) 5.54/2.31 A^1 -> H(f(d), U1(d, e)) 5.54/2.31 A^1 -> H(f(d), U1(e, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.31 A^1 -> H(f(e), b) 5.54/2.31 A^1 -> H(f(e), U1(d, e)) 5.54/2.31 A^1 -> H(f(e), U1(e, d)) 5.54/2.31 A^1 -> H(f(e), d) 5.54/2.31 A^1 -> H(f(a), e) 5.54/2.31 A^1 -> H(f(d), d) 5.54/2.31 A^1 -> H(e, U1(b, b)) 5.54/2.31 A^1 -> H(e, f(d)) 5.54/2.31 A^1 -> H(e, f(e)) 5.54/2.31 A^1 -> H(a, U1(d, b)) 5.54/2.31 A^1 -> H(a, U1(e, b)) 5.54/2.31 A^1 -> H(a, U1(b, d)) 5.54/2.31 A^1 -> H(a, U1(b, e)) 5.54/2.31 A^1 -> H(a, U1(d, d)) 5.54/2.31 A^1 -> H(a, U1(e, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.31 A^1 -> H(U1(d, a), b) 5.54/2.31 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.31 5.54/2.31 The TRS R consists of the following rules: 5.54/2.31 5.54/2.31 a -> d 5.54/2.31 a -> e 5.54/2.31 f(x) -> U1(x, x) 5.54/2.31 b -> d 5.54/2.31 b -> e 5.54/2.31 U1(d, x) -> x 5.54/2.31 5.54/2.31 Q is empty. 5.54/2.31 We have to consider all minimal (P,Q,R)-chains. 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (171) TransformationProof (EQUIVALENT) 5.54/2.31 By narrowing [LPAR04] the rule A^1 -> H(U1(d, a), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.31 5.54/2.31 (A^1 -> H(a, U1(d, d)),A^1 -> H(a, U1(d, d))) 5.54/2.31 (A^1 -> H(U1(d, d), U1(d, d)),A^1 -> H(U1(d, d), U1(d, d))) 5.54/2.31 (A^1 -> H(U1(d, e), U1(d, d)),A^1 -> H(U1(d, e), U1(d, d))) 5.54/2.31 (A^1 -> H(U1(d, a), d),A^1 -> H(U1(d, a), d)) 5.54/2.31 5.54/2.31 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (172) 5.54/2.31 Obligation: 5.54/2.31 Q DP problem: 5.54/2.31 The TRS P consists of the following rules: 5.54/2.31 5.54/2.31 H(x, x) -> G(x, x) 5.54/2.31 G(d, e) -> A^1 5.54/2.31 A^1 -> H(f(d), f(d)) 5.54/2.31 A^1 -> H(f(e), f(e)) 5.54/2.31 A^1 -> H(U1(d, a), U1(e, e)) 5.54/2.31 A^1 -> H(U1(e, d), U1(b, b)) 5.54/2.31 A^1 -> H(U1(e, d), f(d)) 5.54/2.31 A^1 -> H(U1(e, d), f(e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, a), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, a), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, a), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, a), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, a), b) 5.54/2.31 A^1 -> H(U1(a, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(a, a), U1(e, d)) 5.54/2.31 A^1 -> H(U1(a, a), d) 5.54/2.31 A^1 -> H(d, U1(b, b)) 5.54/2.31 A^1 -> H(d, f(d)) 5.54/2.31 A^1 -> H(d, f(e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.31 A^1 -> H(f(d), b) 5.54/2.31 A^1 -> H(f(d), U1(d, e)) 5.54/2.31 A^1 -> H(f(d), U1(e, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.31 A^1 -> H(f(e), b) 5.54/2.31 A^1 -> H(f(e), U1(d, e)) 5.54/2.31 A^1 -> H(f(e), U1(e, d)) 5.54/2.31 A^1 -> H(f(e), d) 5.54/2.31 A^1 -> H(f(a), e) 5.54/2.31 A^1 -> H(f(d), d) 5.54/2.31 A^1 -> H(e, U1(b, b)) 5.54/2.31 A^1 -> H(e, f(d)) 5.54/2.31 A^1 -> H(e, f(e)) 5.54/2.31 A^1 -> H(a, U1(d, b)) 5.54/2.31 A^1 -> H(a, U1(e, b)) 5.54/2.31 A^1 -> H(a, U1(b, d)) 5.54/2.31 A^1 -> H(a, U1(b, e)) 5.54/2.31 A^1 -> H(a, U1(d, d)) 5.54/2.31 A^1 -> H(a, U1(e, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.31 A^1 -> H(U1(d, a), b) 5.54/2.31 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.31 A^1 -> H(U1(d, a), d) 5.54/2.31 5.54/2.31 The TRS R consists of the following rules: 5.54/2.31 5.54/2.31 a -> d 5.54/2.31 a -> e 5.54/2.31 f(x) -> U1(x, x) 5.54/2.31 b -> d 5.54/2.31 b -> e 5.54/2.31 U1(d, x) -> x 5.54/2.31 5.54/2.31 Q is empty. 5.54/2.31 We have to consider all minimal (P,Q,R)-chains. 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (173) TransformationProof (EQUIVALENT) 5.54/2.31 By narrowing [LPAR04] the rule A^1 -> H(U1(d, a), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.31 5.54/2.31 (A^1 -> H(a, U1(e, e)),A^1 -> H(a, U1(e, e))) 5.54/2.31 (A^1 -> H(U1(d, d), U1(e, e)),A^1 -> H(U1(d, d), U1(e, e))) 5.54/2.31 (A^1 -> H(U1(d, e), U1(e, e)),A^1 -> H(U1(d, e), U1(e, e))) 5.54/2.31 5.54/2.31 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (174) 5.54/2.31 Obligation: 5.54/2.31 Q DP problem: 5.54/2.31 The TRS P consists of the following rules: 5.54/2.31 5.54/2.31 H(x, x) -> G(x, x) 5.54/2.31 G(d, e) -> A^1 5.54/2.31 A^1 -> H(f(d), f(d)) 5.54/2.31 A^1 -> H(f(e), f(e)) 5.54/2.31 A^1 -> H(U1(e, d), U1(b, b)) 5.54/2.31 A^1 -> H(U1(e, d), f(d)) 5.54/2.31 A^1 -> H(U1(e, d), f(e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, a), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, a), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, a), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, a), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, a), b) 5.54/2.31 A^1 -> H(U1(a, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(a, a), U1(e, d)) 5.54/2.31 A^1 -> H(U1(a, a), d) 5.54/2.31 A^1 -> H(d, U1(b, b)) 5.54/2.31 A^1 -> H(d, f(d)) 5.54/2.31 A^1 -> H(d, f(e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.31 A^1 -> H(f(d), b) 5.54/2.31 A^1 -> H(f(d), U1(d, e)) 5.54/2.31 A^1 -> H(f(d), U1(e, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.31 A^1 -> H(f(e), b) 5.54/2.31 A^1 -> H(f(e), U1(d, e)) 5.54/2.31 A^1 -> H(f(e), U1(e, d)) 5.54/2.31 A^1 -> H(f(e), d) 5.54/2.31 A^1 -> H(f(a), e) 5.54/2.31 A^1 -> H(f(d), d) 5.54/2.31 A^1 -> H(e, U1(b, b)) 5.54/2.31 A^1 -> H(e, f(d)) 5.54/2.31 A^1 -> H(e, f(e)) 5.54/2.31 A^1 -> H(a, U1(d, b)) 5.54/2.31 A^1 -> H(a, U1(e, b)) 5.54/2.31 A^1 -> H(a, U1(b, d)) 5.54/2.31 A^1 -> H(a, U1(b, e)) 5.54/2.31 A^1 -> H(a, U1(d, d)) 5.54/2.31 A^1 -> H(a, U1(e, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.31 A^1 -> H(U1(d, a), b) 5.54/2.31 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.31 A^1 -> H(U1(d, a), d) 5.54/2.31 5.54/2.31 The TRS R consists of the following rules: 5.54/2.31 5.54/2.31 a -> d 5.54/2.31 a -> e 5.54/2.31 f(x) -> U1(x, x) 5.54/2.31 b -> d 5.54/2.31 b -> e 5.54/2.31 U1(d, x) -> x 5.54/2.31 5.54/2.31 Q is empty. 5.54/2.31 We have to consider all minimal (P,Q,R)-chains. 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (175) TransformationProof (EQUIVALENT) 5.54/2.31 By narrowing [LPAR04] the rule A^1 -> H(U1(e, d), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.31 5.54/2.31 (A^1 -> H(U1(e, d), U1(d, b)),A^1 -> H(U1(e, d), U1(d, b))) 5.54/2.31 (A^1 -> H(U1(e, d), U1(e, b)),A^1 -> H(U1(e, d), U1(e, b))) 5.54/2.31 (A^1 -> H(U1(e, d), U1(b, d)),A^1 -> H(U1(e, d), U1(b, d))) 5.54/2.31 (A^1 -> H(U1(e, d), U1(b, e)),A^1 -> H(U1(e, d), U1(b, e))) 5.54/2.31 5.54/2.31 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (176) 5.54/2.31 Obligation: 5.54/2.31 Q DP problem: 5.54/2.31 The TRS P consists of the following rules: 5.54/2.31 5.54/2.31 H(x, x) -> G(x, x) 5.54/2.31 G(d, e) -> A^1 5.54/2.31 A^1 -> H(f(d), f(d)) 5.54/2.31 A^1 -> H(f(e), f(e)) 5.54/2.31 A^1 -> H(U1(e, d), f(d)) 5.54/2.31 A^1 -> H(U1(e, d), f(e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, a), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, a), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, a), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, a), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, a), b) 5.54/2.31 A^1 -> H(U1(a, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(a, a), U1(e, d)) 5.54/2.31 A^1 -> H(U1(a, a), d) 5.54/2.31 A^1 -> H(d, U1(b, b)) 5.54/2.31 A^1 -> H(d, f(d)) 5.54/2.31 A^1 -> H(d, f(e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.31 A^1 -> H(f(d), b) 5.54/2.31 A^1 -> H(f(d), U1(d, e)) 5.54/2.31 A^1 -> H(f(d), U1(e, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.31 A^1 -> H(f(e), b) 5.54/2.31 A^1 -> H(f(e), U1(d, e)) 5.54/2.31 A^1 -> H(f(e), U1(e, d)) 5.54/2.31 A^1 -> H(f(e), d) 5.54/2.31 A^1 -> H(f(a), e) 5.54/2.31 A^1 -> H(f(d), d) 5.54/2.31 A^1 -> H(e, U1(b, b)) 5.54/2.31 A^1 -> H(e, f(d)) 5.54/2.31 A^1 -> H(e, f(e)) 5.54/2.31 A^1 -> H(a, U1(d, b)) 5.54/2.31 A^1 -> H(a, U1(e, b)) 5.54/2.31 A^1 -> H(a, U1(b, d)) 5.54/2.31 A^1 -> H(a, U1(b, e)) 5.54/2.31 A^1 -> H(a, U1(d, d)) 5.54/2.31 A^1 -> H(a, U1(e, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.31 A^1 -> H(U1(d, a), b) 5.54/2.31 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.31 A^1 -> H(U1(d, a), d) 5.54/2.31 A^1 -> H(U1(e, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, d), U1(b, e)) 5.54/2.31 5.54/2.31 The TRS R consists of the following rules: 5.54/2.31 5.54/2.31 a -> d 5.54/2.31 a -> e 5.54/2.31 f(x) -> U1(x, x) 5.54/2.31 b -> d 5.54/2.31 b -> e 5.54/2.31 U1(d, x) -> x 5.54/2.31 5.54/2.31 Q is empty. 5.54/2.31 We have to consider all minimal (P,Q,R)-chains. 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (177) TransformationProof (EQUIVALENT) 5.54/2.31 By narrowing [LPAR04] the rule A^1 -> H(U1(e, d), f(d)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.31 5.54/2.31 (A^1 -> H(U1(e, d), U1(d, d)),A^1 -> H(U1(e, d), U1(d, d))) 5.54/2.31 5.54/2.31 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (178) 5.54/2.31 Obligation: 5.54/2.31 Q DP problem: 5.54/2.31 The TRS P consists of the following rules: 5.54/2.31 5.54/2.31 H(x, x) -> G(x, x) 5.54/2.31 G(d, e) -> A^1 5.54/2.31 A^1 -> H(f(d), f(d)) 5.54/2.31 A^1 -> H(f(e), f(e)) 5.54/2.31 A^1 -> H(U1(e, d), f(e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, a), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, a), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, a), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, a), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, a), b) 5.54/2.31 A^1 -> H(U1(a, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(a, a), U1(e, d)) 5.54/2.31 A^1 -> H(U1(a, a), d) 5.54/2.31 A^1 -> H(d, U1(b, b)) 5.54/2.31 A^1 -> H(d, f(d)) 5.54/2.31 A^1 -> H(d, f(e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.31 A^1 -> H(f(d), b) 5.54/2.31 A^1 -> H(f(d), U1(d, e)) 5.54/2.31 A^1 -> H(f(d), U1(e, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.31 A^1 -> H(f(e), b) 5.54/2.31 A^1 -> H(f(e), U1(d, e)) 5.54/2.31 A^1 -> H(f(e), U1(e, d)) 5.54/2.31 A^1 -> H(f(e), d) 5.54/2.31 A^1 -> H(f(a), e) 5.54/2.31 A^1 -> H(f(d), d) 5.54/2.31 A^1 -> H(e, U1(b, b)) 5.54/2.31 A^1 -> H(e, f(d)) 5.54/2.31 A^1 -> H(e, f(e)) 5.54/2.31 A^1 -> H(a, U1(d, b)) 5.54/2.31 A^1 -> H(a, U1(e, b)) 5.54/2.31 A^1 -> H(a, U1(b, d)) 5.54/2.31 A^1 -> H(a, U1(b, e)) 5.54/2.31 A^1 -> H(a, U1(d, d)) 5.54/2.31 A^1 -> H(a, U1(e, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.31 A^1 -> H(U1(d, a), b) 5.54/2.31 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.31 A^1 -> H(U1(d, a), d) 5.54/2.31 A^1 -> H(U1(e, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, d), U1(d, d)) 5.54/2.31 5.54/2.31 The TRS R consists of the following rules: 5.54/2.31 5.54/2.31 a -> d 5.54/2.31 a -> e 5.54/2.31 f(x) -> U1(x, x) 5.54/2.31 b -> d 5.54/2.31 b -> e 5.54/2.31 U1(d, x) -> x 5.54/2.31 5.54/2.31 Q is empty. 5.54/2.31 We have to consider all minimal (P,Q,R)-chains. 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (179) TransformationProof (EQUIVALENT) 5.54/2.31 By narrowing [LPAR04] the rule A^1 -> H(U1(e, d), f(e)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.31 5.54/2.31 (A^1 -> H(U1(e, d), U1(e, e)),A^1 -> H(U1(e, d), U1(e, e))) 5.54/2.31 5.54/2.31 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (180) 5.54/2.31 Obligation: 5.54/2.31 Q DP problem: 5.54/2.31 The TRS P consists of the following rules: 5.54/2.31 5.54/2.31 H(x, x) -> G(x, x) 5.54/2.31 G(d, e) -> A^1 5.54/2.31 A^1 -> H(f(d), f(d)) 5.54/2.31 A^1 -> H(f(e), f(e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, a), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, a), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, a), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, a), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, a), b) 5.54/2.31 A^1 -> H(U1(a, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(a, a), U1(e, d)) 5.54/2.31 A^1 -> H(U1(a, a), d) 5.54/2.31 A^1 -> H(d, U1(b, b)) 5.54/2.31 A^1 -> H(d, f(d)) 5.54/2.31 A^1 -> H(d, f(e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.31 A^1 -> H(f(d), b) 5.54/2.31 A^1 -> H(f(d), U1(d, e)) 5.54/2.31 A^1 -> H(f(d), U1(e, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.31 A^1 -> H(f(e), b) 5.54/2.31 A^1 -> H(f(e), U1(d, e)) 5.54/2.31 A^1 -> H(f(e), U1(e, d)) 5.54/2.31 A^1 -> H(f(e), d) 5.54/2.31 A^1 -> H(f(a), e) 5.54/2.31 A^1 -> H(f(d), d) 5.54/2.31 A^1 -> H(e, U1(b, b)) 5.54/2.31 A^1 -> H(e, f(d)) 5.54/2.31 A^1 -> H(e, f(e)) 5.54/2.31 A^1 -> H(a, U1(d, b)) 5.54/2.31 A^1 -> H(a, U1(e, b)) 5.54/2.31 A^1 -> H(a, U1(b, d)) 5.54/2.31 A^1 -> H(a, U1(b, e)) 5.54/2.31 A^1 -> H(a, U1(d, d)) 5.54/2.31 A^1 -> H(a, U1(e, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.31 A^1 -> H(U1(d, a), b) 5.54/2.31 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.31 A^1 -> H(U1(d, a), d) 5.54/2.31 A^1 -> H(U1(e, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, d), U1(e, e)) 5.54/2.31 5.54/2.31 The TRS R consists of the following rules: 5.54/2.31 5.54/2.31 a -> d 5.54/2.31 a -> e 5.54/2.31 f(x) -> U1(x, x) 5.54/2.31 b -> d 5.54/2.31 b -> e 5.54/2.31 U1(d, x) -> x 5.54/2.31 5.54/2.31 Q is empty. 5.54/2.31 We have to consider all minimal (P,Q,R)-chains. 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (181) DependencyGraphProof (EQUIVALENT) 5.54/2.31 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (182) 5.54/2.31 Obligation: 5.54/2.31 Q DP problem: 5.54/2.31 The TRS P consists of the following rules: 5.54/2.31 5.54/2.31 G(d, e) -> A^1 5.54/2.31 A^1 -> H(f(d), f(d)) 5.54/2.31 H(x, x) -> G(x, x) 5.54/2.31 A^1 -> H(f(e), f(e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, a), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, a), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, a), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, a), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, a), b) 5.54/2.31 A^1 -> H(U1(a, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(a, a), U1(e, d)) 5.54/2.31 A^1 -> H(U1(a, a), d) 5.54/2.31 A^1 -> H(d, U1(b, b)) 5.54/2.31 A^1 -> H(d, f(d)) 5.54/2.31 A^1 -> H(d, f(e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.31 A^1 -> H(f(d), b) 5.54/2.31 A^1 -> H(f(d), U1(d, e)) 5.54/2.31 A^1 -> H(f(d), U1(e, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.31 A^1 -> H(f(e), b) 5.54/2.31 A^1 -> H(f(e), U1(d, e)) 5.54/2.31 A^1 -> H(f(e), U1(e, d)) 5.54/2.31 A^1 -> H(f(e), d) 5.54/2.31 A^1 -> H(f(a), e) 5.54/2.31 A^1 -> H(f(d), d) 5.54/2.31 A^1 -> H(e, U1(b, b)) 5.54/2.31 A^1 -> H(e, f(d)) 5.54/2.31 A^1 -> H(e, f(e)) 5.54/2.31 A^1 -> H(a, U1(d, b)) 5.54/2.31 A^1 -> H(a, U1(e, b)) 5.54/2.31 A^1 -> H(a, U1(b, d)) 5.54/2.31 A^1 -> H(a, U1(b, e)) 5.54/2.31 A^1 -> H(a, U1(d, d)) 5.54/2.31 A^1 -> H(a, U1(e, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.31 A^1 -> H(U1(d, a), b) 5.54/2.31 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.31 A^1 -> H(U1(d, a), d) 5.54/2.31 A^1 -> H(U1(e, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, d), U1(d, d)) 5.54/2.31 5.54/2.31 The TRS R consists of the following rules: 5.54/2.31 5.54/2.31 a -> d 5.54/2.31 a -> e 5.54/2.31 f(x) -> U1(x, x) 5.54/2.31 b -> d 5.54/2.31 b -> e 5.54/2.31 U1(d, x) -> x 5.54/2.31 5.54/2.31 Q is empty. 5.54/2.31 We have to consider all minimal (P,Q,R)-chains. 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (183) TransformationProof (EQUIVALENT) 5.54/2.31 By narrowing [LPAR04] the rule A^1 -> H(U1(e, a), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.31 5.54/2.31 (A^1 -> H(U1(e, d), U1(d, b)),A^1 -> H(U1(e, d), U1(d, b))) 5.54/2.31 (A^1 -> H(U1(e, e), U1(d, b)),A^1 -> H(U1(e, e), U1(d, b))) 5.54/2.31 (A^1 -> H(U1(e, a), b),A^1 -> H(U1(e, a), b)) 5.54/2.31 (A^1 -> H(U1(e, a), U1(d, d)),A^1 -> H(U1(e, a), U1(d, d))) 5.54/2.31 (A^1 -> H(U1(e, a), U1(d, e)),A^1 -> H(U1(e, a), U1(d, e))) 5.54/2.31 5.54/2.31 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (184) 5.54/2.31 Obligation: 5.54/2.31 Q DP problem: 5.54/2.31 The TRS P consists of the following rules: 5.54/2.31 5.54/2.31 G(d, e) -> A^1 5.54/2.31 A^1 -> H(f(d), f(d)) 5.54/2.31 H(x, x) -> G(x, x) 5.54/2.31 A^1 -> H(f(e), f(e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, a), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, a), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, a), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, a), b) 5.54/2.31 A^1 -> H(U1(a, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(a, a), U1(e, d)) 5.54/2.31 A^1 -> H(U1(a, a), d) 5.54/2.31 A^1 -> H(d, U1(b, b)) 5.54/2.31 A^1 -> H(d, f(d)) 5.54/2.31 A^1 -> H(d, f(e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.31 A^1 -> H(f(d), b) 5.54/2.31 A^1 -> H(f(d), U1(d, e)) 5.54/2.31 A^1 -> H(f(d), U1(e, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.31 A^1 -> H(f(e), b) 5.54/2.31 A^1 -> H(f(e), U1(d, e)) 5.54/2.31 A^1 -> H(f(e), U1(e, d)) 5.54/2.31 A^1 -> H(f(e), d) 5.54/2.31 A^1 -> H(f(a), e) 5.54/2.31 A^1 -> H(f(d), d) 5.54/2.31 A^1 -> H(e, U1(b, b)) 5.54/2.31 A^1 -> H(e, f(d)) 5.54/2.31 A^1 -> H(e, f(e)) 5.54/2.31 A^1 -> H(a, U1(d, b)) 5.54/2.31 A^1 -> H(a, U1(e, b)) 5.54/2.31 A^1 -> H(a, U1(b, d)) 5.54/2.31 A^1 -> H(a, U1(b, e)) 5.54/2.31 A^1 -> H(a, U1(d, d)) 5.54/2.31 A^1 -> H(a, U1(e, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.31 A^1 -> H(U1(d, a), b) 5.54/2.31 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.31 A^1 -> H(U1(d, a), d) 5.54/2.31 A^1 -> H(U1(e, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, a), b) 5.54/2.31 A^1 -> H(U1(e, a), U1(d, e)) 5.54/2.31 5.54/2.31 The TRS R consists of the following rules: 5.54/2.31 5.54/2.31 a -> d 5.54/2.31 a -> e 5.54/2.31 f(x) -> U1(x, x) 5.54/2.31 b -> d 5.54/2.31 b -> e 5.54/2.31 U1(d, x) -> x 5.54/2.31 5.54/2.31 Q is empty. 5.54/2.31 We have to consider all minimal (P,Q,R)-chains. 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (185) TransformationProof (EQUIVALENT) 5.54/2.31 By narrowing [LPAR04] the rule A^1 -> H(U1(e, a), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.31 5.54/2.31 (A^1 -> H(U1(e, d), U1(e, b)),A^1 -> H(U1(e, d), U1(e, b))) 5.54/2.31 (A^1 -> H(U1(e, e), U1(e, b)),A^1 -> H(U1(e, e), U1(e, b))) 5.54/2.31 (A^1 -> H(U1(e, a), U1(e, d)),A^1 -> H(U1(e, a), U1(e, d))) 5.54/2.31 (A^1 -> H(U1(e, a), U1(e, e)),A^1 -> H(U1(e, a), U1(e, e))) 5.54/2.31 5.54/2.31 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (186) 5.54/2.31 Obligation: 5.54/2.31 Q DP problem: 5.54/2.31 The TRS P consists of the following rules: 5.54/2.31 5.54/2.31 G(d, e) -> A^1 5.54/2.31 A^1 -> H(f(d), f(d)) 5.54/2.31 H(x, x) -> G(x, x) 5.54/2.31 A^1 -> H(f(e), f(e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, a), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, a), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, a), b) 5.54/2.31 A^1 -> H(U1(a, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(a, a), U1(e, d)) 5.54/2.31 A^1 -> H(U1(a, a), d) 5.54/2.31 A^1 -> H(d, U1(b, b)) 5.54/2.31 A^1 -> H(d, f(d)) 5.54/2.31 A^1 -> H(d, f(e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.31 A^1 -> H(f(d), b) 5.54/2.31 A^1 -> H(f(d), U1(d, e)) 5.54/2.31 A^1 -> H(f(d), U1(e, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.31 A^1 -> H(f(e), b) 5.54/2.31 A^1 -> H(f(e), U1(d, e)) 5.54/2.31 A^1 -> H(f(e), U1(e, d)) 5.54/2.31 A^1 -> H(f(e), d) 5.54/2.31 A^1 -> H(f(a), e) 5.54/2.31 A^1 -> H(f(d), d) 5.54/2.31 A^1 -> H(e, U1(b, b)) 5.54/2.31 A^1 -> H(e, f(d)) 5.54/2.31 A^1 -> H(e, f(e)) 5.54/2.31 A^1 -> H(a, U1(d, b)) 5.54/2.31 A^1 -> H(a, U1(e, b)) 5.54/2.31 A^1 -> H(a, U1(b, d)) 5.54/2.31 A^1 -> H(a, U1(b, e)) 5.54/2.31 A^1 -> H(a, U1(d, d)) 5.54/2.31 A^1 -> H(a, U1(e, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.31 A^1 -> H(U1(d, a), b) 5.54/2.31 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.31 A^1 -> H(U1(d, a), d) 5.54/2.31 A^1 -> H(U1(e, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, a), b) 5.54/2.31 A^1 -> H(U1(e, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(e, d)) 5.54/2.31 5.54/2.31 The TRS R consists of the following rules: 5.54/2.31 5.54/2.31 a -> d 5.54/2.31 a -> e 5.54/2.31 f(x) -> U1(x, x) 5.54/2.31 b -> d 5.54/2.31 b -> e 5.54/2.31 U1(d, x) -> x 5.54/2.31 5.54/2.31 Q is empty. 5.54/2.31 We have to consider all minimal (P,Q,R)-chains. 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (187) TransformationProof (EQUIVALENT) 5.54/2.31 By narrowing [LPAR04] the rule A^1 -> H(U1(e, a), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.31 5.54/2.31 (A^1 -> H(U1(e, d), U1(b, d)),A^1 -> H(U1(e, d), U1(b, d))) 5.54/2.31 (A^1 -> H(U1(e, e), U1(b, d)),A^1 -> H(U1(e, e), U1(b, d))) 5.54/2.31 (A^1 -> H(U1(e, a), U1(d, d)),A^1 -> H(U1(e, a), U1(d, d))) 5.54/2.31 (A^1 -> H(U1(e, a), U1(e, d)),A^1 -> H(U1(e, a), U1(e, d))) 5.54/2.31 5.54/2.31 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (188) 5.54/2.31 Obligation: 5.54/2.31 Q DP problem: 5.54/2.31 The TRS P consists of the following rules: 5.54/2.31 5.54/2.31 G(d, e) -> A^1 5.54/2.31 A^1 -> H(f(d), f(d)) 5.54/2.31 H(x, x) -> G(x, x) 5.54/2.31 A^1 -> H(f(e), f(e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, a), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, a), b) 5.54/2.31 A^1 -> H(U1(a, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(a, a), U1(e, d)) 5.54/2.31 A^1 -> H(U1(a, a), d) 5.54/2.31 A^1 -> H(d, U1(b, b)) 5.54/2.31 A^1 -> H(d, f(d)) 5.54/2.31 A^1 -> H(d, f(e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.31 A^1 -> H(f(d), b) 5.54/2.31 A^1 -> H(f(d), U1(d, e)) 5.54/2.31 A^1 -> H(f(d), U1(e, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.31 A^1 -> H(f(e), b) 5.54/2.31 A^1 -> H(f(e), U1(d, e)) 5.54/2.31 A^1 -> H(f(e), U1(e, d)) 5.54/2.31 A^1 -> H(f(e), d) 5.54/2.31 A^1 -> H(f(a), e) 5.54/2.31 A^1 -> H(f(d), d) 5.54/2.31 A^1 -> H(e, U1(b, b)) 5.54/2.31 A^1 -> H(e, f(d)) 5.54/2.31 A^1 -> H(e, f(e)) 5.54/2.31 A^1 -> H(a, U1(d, b)) 5.54/2.31 A^1 -> H(a, U1(e, b)) 5.54/2.31 A^1 -> H(a, U1(b, d)) 5.54/2.31 A^1 -> H(a, U1(b, e)) 5.54/2.31 A^1 -> H(a, U1(d, d)) 5.54/2.31 A^1 -> H(a, U1(e, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.31 A^1 -> H(U1(d, a), b) 5.54/2.31 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.31 A^1 -> H(U1(d, a), d) 5.54/2.31 A^1 -> H(U1(e, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, a), b) 5.54/2.31 A^1 -> H(U1(e, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(e, d)) 5.54/2.31 5.54/2.31 The TRS R consists of the following rules: 5.54/2.31 5.54/2.31 a -> d 5.54/2.31 a -> e 5.54/2.31 f(x) -> U1(x, x) 5.54/2.31 b -> d 5.54/2.31 b -> e 5.54/2.31 U1(d, x) -> x 5.54/2.31 5.54/2.31 Q is empty. 5.54/2.31 We have to consider all minimal (P,Q,R)-chains. 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (189) TransformationProof (EQUIVALENT) 5.54/2.31 By narrowing [LPAR04] the rule A^1 -> H(U1(e, a), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.31 5.54/2.31 (A^1 -> H(U1(e, d), U1(b, e)),A^1 -> H(U1(e, d), U1(b, e))) 5.54/2.31 (A^1 -> H(U1(e, e), U1(b, e)),A^1 -> H(U1(e, e), U1(b, e))) 5.54/2.31 (A^1 -> H(U1(e, a), U1(d, e)),A^1 -> H(U1(e, a), U1(d, e))) 5.54/2.31 (A^1 -> H(U1(e, a), U1(e, e)),A^1 -> H(U1(e, a), U1(e, e))) 5.54/2.31 5.54/2.31 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (190) 5.54/2.31 Obligation: 5.54/2.31 Q DP problem: 5.54/2.31 The TRS P consists of the following rules: 5.54/2.31 5.54/2.31 G(d, e) -> A^1 5.54/2.31 A^1 -> H(f(d), f(d)) 5.54/2.31 H(x, x) -> G(x, x) 5.54/2.31 A^1 -> H(f(e), f(e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, a), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, a), b) 5.54/2.31 A^1 -> H(U1(a, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(a, a), U1(e, d)) 5.54/2.31 A^1 -> H(U1(a, a), d) 5.54/2.31 A^1 -> H(d, U1(b, b)) 5.54/2.31 A^1 -> H(d, f(d)) 5.54/2.31 A^1 -> H(d, f(e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.31 A^1 -> H(f(d), b) 5.54/2.31 A^1 -> H(f(d), U1(d, e)) 5.54/2.31 A^1 -> H(f(d), U1(e, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.31 A^1 -> H(f(e), b) 5.54/2.31 A^1 -> H(f(e), U1(d, e)) 5.54/2.31 A^1 -> H(f(e), U1(e, d)) 5.54/2.31 A^1 -> H(f(e), d) 5.54/2.31 A^1 -> H(f(a), e) 5.54/2.31 A^1 -> H(f(d), d) 5.54/2.31 A^1 -> H(e, U1(b, b)) 5.54/2.31 A^1 -> H(e, f(d)) 5.54/2.31 A^1 -> H(e, f(e)) 5.54/2.31 A^1 -> H(a, U1(d, b)) 5.54/2.31 A^1 -> H(a, U1(e, b)) 5.54/2.31 A^1 -> H(a, U1(b, d)) 5.54/2.31 A^1 -> H(a, U1(b, e)) 5.54/2.31 A^1 -> H(a, U1(d, d)) 5.54/2.31 A^1 -> H(a, U1(e, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.31 A^1 -> H(U1(d, a), b) 5.54/2.31 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.31 A^1 -> H(U1(d, a), d) 5.54/2.31 A^1 -> H(U1(e, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, a), b) 5.54/2.31 A^1 -> H(U1(e, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(e, d)) 5.54/2.31 5.54/2.31 The TRS R consists of the following rules: 5.54/2.31 5.54/2.31 a -> d 5.54/2.31 a -> e 5.54/2.31 f(x) -> U1(x, x) 5.54/2.31 b -> d 5.54/2.31 b -> e 5.54/2.31 U1(d, x) -> x 5.54/2.31 5.54/2.31 Q is empty. 5.54/2.31 We have to consider all minimal (P,Q,R)-chains. 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (191) TransformationProof (EQUIVALENT) 5.54/2.31 By narrowing [LPAR04] the rule A^1 -> H(U1(e, a), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.31 5.54/2.31 (A^1 -> H(U1(e, d), U1(d, d)),A^1 -> H(U1(e, d), U1(d, d))) 5.54/2.31 (A^1 -> H(U1(e, e), U1(d, d)),A^1 -> H(U1(e, e), U1(d, d))) 5.54/2.31 (A^1 -> H(U1(e, a), d),A^1 -> H(U1(e, a), d)) 5.54/2.31 5.54/2.31 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (192) 5.54/2.31 Obligation: 5.54/2.31 Q DP problem: 5.54/2.31 The TRS P consists of the following rules: 5.54/2.31 5.54/2.31 G(d, e) -> A^1 5.54/2.31 A^1 -> H(f(d), f(d)) 5.54/2.31 H(x, x) -> G(x, x) 5.54/2.31 A^1 -> H(f(e), f(e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, a), b) 5.54/2.31 A^1 -> H(U1(a, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(a, a), U1(e, d)) 5.54/2.31 A^1 -> H(U1(a, a), d) 5.54/2.31 A^1 -> H(d, U1(b, b)) 5.54/2.31 A^1 -> H(d, f(d)) 5.54/2.31 A^1 -> H(d, f(e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.31 A^1 -> H(f(d), b) 5.54/2.31 A^1 -> H(f(d), U1(d, e)) 5.54/2.31 A^1 -> H(f(d), U1(e, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.31 A^1 -> H(f(e), b) 5.54/2.31 A^1 -> H(f(e), U1(d, e)) 5.54/2.31 A^1 -> H(f(e), U1(e, d)) 5.54/2.31 A^1 -> H(f(e), d) 5.54/2.31 A^1 -> H(f(a), e) 5.54/2.31 A^1 -> H(f(d), d) 5.54/2.31 A^1 -> H(e, U1(b, b)) 5.54/2.31 A^1 -> H(e, f(d)) 5.54/2.31 A^1 -> H(e, f(e)) 5.54/2.31 A^1 -> H(a, U1(d, b)) 5.54/2.31 A^1 -> H(a, U1(e, b)) 5.54/2.31 A^1 -> H(a, U1(b, d)) 5.54/2.31 A^1 -> H(a, U1(b, e)) 5.54/2.31 A^1 -> H(a, U1(d, d)) 5.54/2.31 A^1 -> H(a, U1(e, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.31 A^1 -> H(U1(d, a), b) 5.54/2.31 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.31 A^1 -> H(U1(d, a), d) 5.54/2.31 A^1 -> H(U1(e, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, a), b) 5.54/2.31 A^1 -> H(U1(e, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(e, d)) 5.54/2.31 A^1 -> H(U1(e, a), d) 5.54/2.31 5.54/2.31 The TRS R consists of the following rules: 5.54/2.31 5.54/2.31 a -> d 5.54/2.31 a -> e 5.54/2.31 f(x) -> U1(x, x) 5.54/2.31 b -> d 5.54/2.31 b -> e 5.54/2.31 U1(d, x) -> x 5.54/2.31 5.54/2.31 Q is empty. 5.54/2.31 We have to consider all minimal (P,Q,R)-chains. 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (193) DependencyGraphProof (EQUIVALENT) 5.54/2.31 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (194) 5.54/2.31 Obligation: 5.54/2.31 Q DP problem: 5.54/2.31 The TRS P consists of the following rules: 5.54/2.31 5.54/2.31 A^1 -> H(f(d), f(d)) 5.54/2.31 H(x, x) -> G(x, x) 5.54/2.31 G(d, e) -> A^1 5.54/2.31 A^1 -> H(f(e), f(e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, a), b) 5.54/2.31 A^1 -> H(U1(a, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(a, a), U1(e, d)) 5.54/2.31 A^1 -> H(U1(a, a), d) 5.54/2.31 A^1 -> H(d, U1(b, b)) 5.54/2.31 A^1 -> H(d, f(d)) 5.54/2.31 A^1 -> H(d, f(e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.31 A^1 -> H(f(d), b) 5.54/2.31 A^1 -> H(f(d), U1(d, e)) 5.54/2.31 A^1 -> H(f(d), U1(e, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.31 A^1 -> H(f(e), b) 5.54/2.31 A^1 -> H(f(e), U1(d, e)) 5.54/2.31 A^1 -> H(f(e), U1(e, d)) 5.54/2.31 A^1 -> H(f(e), d) 5.54/2.31 A^1 -> H(f(a), e) 5.54/2.31 A^1 -> H(f(d), d) 5.54/2.31 A^1 -> H(e, U1(b, b)) 5.54/2.31 A^1 -> H(e, f(d)) 5.54/2.31 A^1 -> H(e, f(e)) 5.54/2.31 A^1 -> H(a, U1(d, b)) 5.54/2.31 A^1 -> H(a, U1(e, b)) 5.54/2.31 A^1 -> H(a, U1(b, d)) 5.54/2.31 A^1 -> H(a, U1(b, e)) 5.54/2.31 A^1 -> H(a, U1(d, d)) 5.54/2.31 A^1 -> H(a, U1(e, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.31 A^1 -> H(U1(d, a), b) 5.54/2.31 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.31 A^1 -> H(U1(d, a), d) 5.54/2.31 A^1 -> H(U1(e, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, a), b) 5.54/2.31 A^1 -> H(U1(e, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(e, d)) 5.54/2.31 5.54/2.31 The TRS R consists of the following rules: 5.54/2.31 5.54/2.31 a -> d 5.54/2.31 a -> e 5.54/2.31 f(x) -> U1(x, x) 5.54/2.31 b -> d 5.54/2.31 b -> e 5.54/2.31 U1(d, x) -> x 5.54/2.31 5.54/2.31 Q is empty. 5.54/2.31 We have to consider all minimal (P,Q,R)-chains. 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (195) TransformationProof (EQUIVALENT) 5.54/2.31 By narrowing [LPAR04] the rule A^1 -> H(U1(e, a), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.31 5.54/2.31 (A^1 -> H(U1(e, d), U1(e, e)),A^1 -> H(U1(e, d), U1(e, e))) 5.54/2.31 (A^1 -> H(U1(e, e), U1(e, e)),A^1 -> H(U1(e, e), U1(e, e))) 5.54/2.31 5.54/2.31 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (196) 5.54/2.31 Obligation: 5.54/2.31 Q DP problem: 5.54/2.31 The TRS P consists of the following rules: 5.54/2.31 5.54/2.31 A^1 -> H(f(d), f(d)) 5.54/2.31 H(x, x) -> G(x, x) 5.54/2.31 G(d, e) -> A^1 5.54/2.31 A^1 -> H(f(e), f(e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, a), b) 5.54/2.31 A^1 -> H(U1(a, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(a, a), U1(e, d)) 5.54/2.31 A^1 -> H(U1(a, a), d) 5.54/2.31 A^1 -> H(d, U1(b, b)) 5.54/2.31 A^1 -> H(d, f(d)) 5.54/2.31 A^1 -> H(d, f(e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.31 A^1 -> H(f(d), b) 5.54/2.31 A^1 -> H(f(d), U1(d, e)) 5.54/2.31 A^1 -> H(f(d), U1(e, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.31 A^1 -> H(f(e), b) 5.54/2.31 A^1 -> H(f(e), U1(d, e)) 5.54/2.31 A^1 -> H(f(e), U1(e, d)) 5.54/2.31 A^1 -> H(f(e), d) 5.54/2.31 A^1 -> H(f(a), e) 5.54/2.31 A^1 -> H(f(d), d) 5.54/2.31 A^1 -> H(e, U1(b, b)) 5.54/2.31 A^1 -> H(e, f(d)) 5.54/2.31 A^1 -> H(e, f(e)) 5.54/2.31 A^1 -> H(a, U1(d, b)) 5.54/2.31 A^1 -> H(a, U1(e, b)) 5.54/2.31 A^1 -> H(a, U1(b, d)) 5.54/2.31 A^1 -> H(a, U1(b, e)) 5.54/2.31 A^1 -> H(a, U1(d, d)) 5.54/2.31 A^1 -> H(a, U1(e, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.31 A^1 -> H(U1(d, a), b) 5.54/2.31 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.31 A^1 -> H(U1(d, a), d) 5.54/2.31 A^1 -> H(U1(e, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, a), b) 5.54/2.31 A^1 -> H(U1(e, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(e, d)) 5.54/2.31 A^1 -> H(U1(e, d), U1(e, e)) 5.54/2.31 5.54/2.31 The TRS R consists of the following rules: 5.54/2.31 5.54/2.31 a -> d 5.54/2.31 a -> e 5.54/2.31 f(x) -> U1(x, x) 5.54/2.31 b -> d 5.54/2.31 b -> e 5.54/2.31 U1(d, x) -> x 5.54/2.31 5.54/2.31 Q is empty. 5.54/2.31 We have to consider all minimal (P,Q,R)-chains. 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (197) DependencyGraphProof (EQUIVALENT) 5.54/2.31 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (198) 5.54/2.31 Obligation: 5.54/2.31 Q DP problem: 5.54/2.31 The TRS P consists of the following rules: 5.54/2.31 5.54/2.31 H(x, x) -> G(x, x) 5.54/2.31 G(d, e) -> A^1 5.54/2.31 A^1 -> H(f(d), f(d)) 5.54/2.31 A^1 -> H(f(e), f(e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, d), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(a, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(a, e), U1(e, e)) 5.54/2.31 A^1 -> H(U1(a, a), b) 5.54/2.31 A^1 -> H(U1(a, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(a, a), U1(e, d)) 5.54/2.31 A^1 -> H(U1(a, a), d) 5.54/2.31 A^1 -> H(d, U1(b, b)) 5.54/2.31 A^1 -> H(d, f(d)) 5.54/2.31 A^1 -> H(d, f(e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.31 A^1 -> H(f(d), b) 5.54/2.31 A^1 -> H(f(d), U1(d, e)) 5.54/2.31 A^1 -> H(f(d), U1(e, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.31 A^1 -> H(f(e), b) 5.54/2.31 A^1 -> H(f(e), U1(d, e)) 5.54/2.31 A^1 -> H(f(e), U1(e, d)) 5.54/2.31 A^1 -> H(f(e), d) 5.54/2.31 A^1 -> H(f(a), e) 5.54/2.31 A^1 -> H(f(d), d) 5.54/2.31 A^1 -> H(e, U1(b, b)) 5.54/2.31 A^1 -> H(e, f(d)) 5.54/2.31 A^1 -> H(e, f(e)) 5.54/2.31 A^1 -> H(a, U1(d, b)) 5.54/2.31 A^1 -> H(a, U1(e, b)) 5.54/2.31 A^1 -> H(a, U1(b, d)) 5.54/2.31 A^1 -> H(a, U1(b, e)) 5.54/2.31 A^1 -> H(a, U1(d, d)) 5.54/2.31 A^1 -> H(a, U1(e, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.31 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.31 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.31 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.31 A^1 -> H(U1(d, a), b) 5.54/2.31 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.31 A^1 -> H(U1(d, a), d) 5.54/2.31 A^1 -> H(U1(e, d), U1(d, b)) 5.54/2.31 A^1 -> H(U1(e, d), U1(e, b)) 5.54/2.31 A^1 -> H(U1(e, d), U1(b, d)) 5.54/2.31 A^1 -> H(U1(e, d), U1(b, e)) 5.54/2.31 A^1 -> H(U1(e, d), U1(d, d)) 5.54/2.31 A^1 -> H(U1(e, a), b) 5.54/2.31 A^1 -> H(U1(e, a), U1(d, e)) 5.54/2.31 A^1 -> H(U1(e, a), U1(e, d)) 5.54/2.31 5.54/2.31 The TRS R consists of the following rules: 5.54/2.31 5.54/2.31 a -> d 5.54/2.31 a -> e 5.54/2.31 f(x) -> U1(x, x) 5.54/2.31 b -> d 5.54/2.31 b -> e 5.54/2.31 U1(d, x) -> x 5.54/2.31 5.54/2.31 Q is empty. 5.54/2.31 We have to consider all minimal (P,Q,R)-chains. 5.54/2.31 ---------------------------------------- 5.54/2.31 5.54/2.31 (199) TransformationProof (EQUIVALENT) 5.54/2.31 By narrowing [LPAR04] the rule A^1 -> H(U1(a, d), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.31 5.54/2.31 (A^1 -> H(U1(d, d), U1(d, b)),A^1 -> H(U1(d, d), U1(d, b))) 5.54/2.31 (A^1 -> H(U1(e, d), U1(d, b)),A^1 -> H(U1(e, d), U1(d, b))) 5.54/2.31 (A^1 -> H(U1(a, d), b),A^1 -> H(U1(a, d), b)) 5.54/2.31 (A^1 -> H(U1(a, d), U1(d, d)),A^1 -> H(U1(a, d), U1(d, d))) 5.54/2.31 (A^1 -> H(U1(a, d), U1(d, e)),A^1 -> H(U1(a, d), U1(d, e))) 5.54/2.31 5.54/2.31 5.54/2.31 ---------------------------------------- 5.54/2.32 5.54/2.32 (200) 5.54/2.32 Obligation: 5.54/2.32 Q DP problem: 5.54/2.32 The TRS P consists of the following rules: 5.54/2.32 5.54/2.32 H(x, x) -> G(x, x) 5.54/2.32 G(d, e) -> A^1 5.54/2.32 A^1 -> H(f(d), f(d)) 5.54/2.32 A^1 -> H(f(e), f(e)) 5.54/2.32 A^1 -> H(U1(a, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(a, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(a, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(a, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(a, d), U1(e, e)) 5.54/2.32 A^1 -> H(U1(a, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(a, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(a, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(a, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(a, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(a, e), U1(e, e)) 5.54/2.32 A^1 -> H(U1(a, a), b) 5.54/2.32 A^1 -> H(U1(a, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, a), d) 5.54/2.32 A^1 -> H(d, U1(b, b)) 5.54/2.32 A^1 -> H(d, f(d)) 5.54/2.32 A^1 -> H(d, f(e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.32 A^1 -> H(f(d), b) 5.54/2.32 A^1 -> H(f(d), U1(d, e)) 5.54/2.32 A^1 -> H(f(d), U1(e, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.32 A^1 -> H(f(e), b) 5.54/2.32 A^1 -> H(f(e), U1(d, e)) 5.54/2.32 A^1 -> H(f(e), U1(e, d)) 5.54/2.32 A^1 -> H(f(e), d) 5.54/2.32 A^1 -> H(f(a), e) 5.54/2.32 A^1 -> H(f(d), d) 5.54/2.32 A^1 -> H(e, U1(b, b)) 5.54/2.32 A^1 -> H(e, f(d)) 5.54/2.32 A^1 -> H(e, f(e)) 5.54/2.32 A^1 -> H(a, U1(d, b)) 5.54/2.32 A^1 -> H(a, U1(e, b)) 5.54/2.32 A^1 -> H(a, U1(b, d)) 5.54/2.32 A^1 -> H(a, U1(b, e)) 5.54/2.32 A^1 -> H(a, U1(d, d)) 5.54/2.32 A^1 -> H(a, U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, a), b) 5.54/2.32 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(d, a), d) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, a), b) 5.54/2.32 A^1 -> H(U1(e, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(e, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, d), b) 5.54/2.32 A^1 -> H(U1(a, d), U1(d, e)) 5.54/2.32 5.54/2.32 The TRS R consists of the following rules: 5.54/2.32 5.54/2.32 a -> d 5.54/2.32 a -> e 5.54/2.32 f(x) -> U1(x, x) 5.54/2.32 b -> d 5.54/2.32 b -> e 5.54/2.32 U1(d, x) -> x 5.54/2.32 5.54/2.32 Q is empty. 5.54/2.32 We have to consider all minimal (P,Q,R)-chains. 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (201) TransformationProof (EQUIVALENT) 5.54/2.32 By narrowing [LPAR04] the rule A^1 -> H(U1(a, d), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.32 5.54/2.32 (A^1 -> H(U1(d, d), U1(e, b)),A^1 -> H(U1(d, d), U1(e, b))) 5.54/2.32 (A^1 -> H(U1(e, d), U1(e, b)),A^1 -> H(U1(e, d), U1(e, b))) 5.54/2.32 (A^1 -> H(U1(a, d), U1(e, d)),A^1 -> H(U1(a, d), U1(e, d))) 5.54/2.32 (A^1 -> H(U1(a, d), U1(e, e)),A^1 -> H(U1(a, d), U1(e, e))) 5.54/2.32 5.54/2.32 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (202) 5.54/2.32 Obligation: 5.54/2.32 Q DP problem: 5.54/2.32 The TRS P consists of the following rules: 5.54/2.32 5.54/2.32 H(x, x) -> G(x, x) 5.54/2.32 G(d, e) -> A^1 5.54/2.32 A^1 -> H(f(d), f(d)) 5.54/2.32 A^1 -> H(f(e), f(e)) 5.54/2.32 A^1 -> H(U1(a, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(a, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(a, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(a, d), U1(e, e)) 5.54/2.32 A^1 -> H(U1(a, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(a, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(a, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(a, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(a, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(a, e), U1(e, e)) 5.54/2.32 A^1 -> H(U1(a, a), b) 5.54/2.32 A^1 -> H(U1(a, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, a), d) 5.54/2.32 A^1 -> H(d, U1(b, b)) 5.54/2.32 A^1 -> H(d, f(d)) 5.54/2.32 A^1 -> H(d, f(e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.32 A^1 -> H(f(d), b) 5.54/2.32 A^1 -> H(f(d), U1(d, e)) 5.54/2.32 A^1 -> H(f(d), U1(e, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.32 A^1 -> H(f(e), b) 5.54/2.32 A^1 -> H(f(e), U1(d, e)) 5.54/2.32 A^1 -> H(f(e), U1(e, d)) 5.54/2.32 A^1 -> H(f(e), d) 5.54/2.32 A^1 -> H(f(a), e) 5.54/2.32 A^1 -> H(f(d), d) 5.54/2.32 A^1 -> H(e, U1(b, b)) 5.54/2.32 A^1 -> H(e, f(d)) 5.54/2.32 A^1 -> H(e, f(e)) 5.54/2.32 A^1 -> H(a, U1(d, b)) 5.54/2.32 A^1 -> H(a, U1(e, b)) 5.54/2.32 A^1 -> H(a, U1(b, d)) 5.54/2.32 A^1 -> H(a, U1(b, e)) 5.54/2.32 A^1 -> H(a, U1(d, d)) 5.54/2.32 A^1 -> H(a, U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, a), b) 5.54/2.32 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(d, a), d) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, a), b) 5.54/2.32 A^1 -> H(U1(e, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(e, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, d), b) 5.54/2.32 A^1 -> H(U1(a, d), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, d), U1(e, d)) 5.54/2.32 5.54/2.32 The TRS R consists of the following rules: 5.54/2.32 5.54/2.32 a -> d 5.54/2.32 a -> e 5.54/2.32 f(x) -> U1(x, x) 5.54/2.32 b -> d 5.54/2.32 b -> e 5.54/2.32 U1(d, x) -> x 5.54/2.32 5.54/2.32 Q is empty. 5.54/2.32 We have to consider all minimal (P,Q,R)-chains. 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (203) TransformationProof (EQUIVALENT) 5.54/2.32 By narrowing [LPAR04] the rule A^1 -> H(U1(a, d), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.32 5.54/2.32 (A^1 -> H(U1(d, d), U1(b, d)),A^1 -> H(U1(d, d), U1(b, d))) 5.54/2.32 (A^1 -> H(U1(e, d), U1(b, d)),A^1 -> H(U1(e, d), U1(b, d))) 5.54/2.32 (A^1 -> H(U1(a, d), U1(d, d)),A^1 -> H(U1(a, d), U1(d, d))) 5.54/2.32 (A^1 -> H(U1(a, d), U1(e, d)),A^1 -> H(U1(a, d), U1(e, d))) 5.54/2.32 5.54/2.32 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (204) 5.54/2.32 Obligation: 5.54/2.32 Q DP problem: 5.54/2.32 The TRS P consists of the following rules: 5.54/2.32 5.54/2.32 H(x, x) -> G(x, x) 5.54/2.32 G(d, e) -> A^1 5.54/2.32 A^1 -> H(f(d), f(d)) 5.54/2.32 A^1 -> H(f(e), f(e)) 5.54/2.32 A^1 -> H(U1(a, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(a, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(a, d), U1(e, e)) 5.54/2.32 A^1 -> H(U1(a, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(a, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(a, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(a, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(a, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(a, e), U1(e, e)) 5.54/2.32 A^1 -> H(U1(a, a), b) 5.54/2.32 A^1 -> H(U1(a, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, a), d) 5.54/2.32 A^1 -> H(d, U1(b, b)) 5.54/2.32 A^1 -> H(d, f(d)) 5.54/2.32 A^1 -> H(d, f(e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.32 A^1 -> H(f(d), b) 5.54/2.32 A^1 -> H(f(d), U1(d, e)) 5.54/2.32 A^1 -> H(f(d), U1(e, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.32 A^1 -> H(f(e), b) 5.54/2.32 A^1 -> H(f(e), U1(d, e)) 5.54/2.32 A^1 -> H(f(e), U1(e, d)) 5.54/2.32 A^1 -> H(f(e), d) 5.54/2.32 A^1 -> H(f(a), e) 5.54/2.32 A^1 -> H(f(d), d) 5.54/2.32 A^1 -> H(e, U1(b, b)) 5.54/2.32 A^1 -> H(e, f(d)) 5.54/2.32 A^1 -> H(e, f(e)) 5.54/2.32 A^1 -> H(a, U1(d, b)) 5.54/2.32 A^1 -> H(a, U1(e, b)) 5.54/2.32 A^1 -> H(a, U1(b, d)) 5.54/2.32 A^1 -> H(a, U1(b, e)) 5.54/2.32 A^1 -> H(a, U1(d, d)) 5.54/2.32 A^1 -> H(a, U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, a), b) 5.54/2.32 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(d, a), d) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, a), b) 5.54/2.32 A^1 -> H(U1(e, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(e, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, d), b) 5.54/2.32 A^1 -> H(U1(a, d), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, d), U1(e, d)) 5.54/2.32 5.54/2.32 The TRS R consists of the following rules: 5.54/2.32 5.54/2.32 a -> d 5.54/2.32 a -> e 5.54/2.32 f(x) -> U1(x, x) 5.54/2.32 b -> d 5.54/2.32 b -> e 5.54/2.32 U1(d, x) -> x 5.54/2.32 5.54/2.32 Q is empty. 5.54/2.32 We have to consider all minimal (P,Q,R)-chains. 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (205) TransformationProof (EQUIVALENT) 5.54/2.32 By narrowing [LPAR04] the rule A^1 -> H(U1(a, d), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.32 5.54/2.32 (A^1 -> H(U1(d, d), U1(b, e)),A^1 -> H(U1(d, d), U1(b, e))) 5.54/2.32 (A^1 -> H(U1(e, d), U1(b, e)),A^1 -> H(U1(e, d), U1(b, e))) 5.54/2.32 (A^1 -> H(U1(a, d), U1(d, e)),A^1 -> H(U1(a, d), U1(d, e))) 5.54/2.32 (A^1 -> H(U1(a, d), U1(e, e)),A^1 -> H(U1(a, d), U1(e, e))) 5.54/2.32 5.54/2.32 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (206) 5.54/2.32 Obligation: 5.54/2.32 Q DP problem: 5.54/2.32 The TRS P consists of the following rules: 5.54/2.32 5.54/2.32 H(x, x) -> G(x, x) 5.54/2.32 G(d, e) -> A^1 5.54/2.32 A^1 -> H(f(d), f(d)) 5.54/2.32 A^1 -> H(f(e), f(e)) 5.54/2.32 A^1 -> H(U1(a, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(a, d), U1(e, e)) 5.54/2.32 A^1 -> H(U1(a, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(a, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(a, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(a, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(a, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(a, e), U1(e, e)) 5.54/2.32 A^1 -> H(U1(a, a), b) 5.54/2.32 A^1 -> H(U1(a, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, a), d) 5.54/2.32 A^1 -> H(d, U1(b, b)) 5.54/2.32 A^1 -> H(d, f(d)) 5.54/2.32 A^1 -> H(d, f(e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.32 A^1 -> H(f(d), b) 5.54/2.32 A^1 -> H(f(d), U1(d, e)) 5.54/2.32 A^1 -> H(f(d), U1(e, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.32 A^1 -> H(f(e), b) 5.54/2.32 A^1 -> H(f(e), U1(d, e)) 5.54/2.32 A^1 -> H(f(e), U1(e, d)) 5.54/2.32 A^1 -> H(f(e), d) 5.54/2.32 A^1 -> H(f(a), e) 5.54/2.32 A^1 -> H(f(d), d) 5.54/2.32 A^1 -> H(e, U1(b, b)) 5.54/2.32 A^1 -> H(e, f(d)) 5.54/2.32 A^1 -> H(e, f(e)) 5.54/2.32 A^1 -> H(a, U1(d, b)) 5.54/2.32 A^1 -> H(a, U1(e, b)) 5.54/2.32 A^1 -> H(a, U1(b, d)) 5.54/2.32 A^1 -> H(a, U1(b, e)) 5.54/2.32 A^1 -> H(a, U1(d, d)) 5.54/2.32 A^1 -> H(a, U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, a), b) 5.54/2.32 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(d, a), d) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, a), b) 5.54/2.32 A^1 -> H(U1(e, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(e, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, d), b) 5.54/2.32 A^1 -> H(U1(a, d), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, d), U1(e, d)) 5.54/2.32 5.54/2.32 The TRS R consists of the following rules: 5.54/2.32 5.54/2.32 a -> d 5.54/2.32 a -> e 5.54/2.32 f(x) -> U1(x, x) 5.54/2.32 b -> d 5.54/2.32 b -> e 5.54/2.32 U1(d, x) -> x 5.54/2.32 5.54/2.32 Q is empty. 5.54/2.32 We have to consider all minimal (P,Q,R)-chains. 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (207) TransformationProof (EQUIVALENT) 5.54/2.32 By narrowing [LPAR04] the rule A^1 -> H(U1(a, d), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.32 5.54/2.32 (A^1 -> H(U1(d, d), U1(d, d)),A^1 -> H(U1(d, d), U1(d, d))) 5.54/2.32 (A^1 -> H(U1(e, d), U1(d, d)),A^1 -> H(U1(e, d), U1(d, d))) 5.54/2.32 (A^1 -> H(U1(a, d), d),A^1 -> H(U1(a, d), d)) 5.54/2.32 5.54/2.32 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (208) 5.54/2.32 Obligation: 5.54/2.32 Q DP problem: 5.54/2.32 The TRS P consists of the following rules: 5.54/2.32 5.54/2.32 H(x, x) -> G(x, x) 5.54/2.32 G(d, e) -> A^1 5.54/2.32 A^1 -> H(f(d), f(d)) 5.54/2.32 A^1 -> H(f(e), f(e)) 5.54/2.32 A^1 -> H(U1(a, d), U1(e, e)) 5.54/2.32 A^1 -> H(U1(a, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(a, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(a, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(a, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(a, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(a, e), U1(e, e)) 5.54/2.32 A^1 -> H(U1(a, a), b) 5.54/2.32 A^1 -> H(U1(a, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, a), d) 5.54/2.32 A^1 -> H(d, U1(b, b)) 5.54/2.32 A^1 -> H(d, f(d)) 5.54/2.32 A^1 -> H(d, f(e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.32 A^1 -> H(f(d), b) 5.54/2.32 A^1 -> H(f(d), U1(d, e)) 5.54/2.32 A^1 -> H(f(d), U1(e, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.32 A^1 -> H(f(e), b) 5.54/2.32 A^1 -> H(f(e), U1(d, e)) 5.54/2.32 A^1 -> H(f(e), U1(e, d)) 5.54/2.32 A^1 -> H(f(e), d) 5.54/2.32 A^1 -> H(f(a), e) 5.54/2.32 A^1 -> H(f(d), d) 5.54/2.32 A^1 -> H(e, U1(b, b)) 5.54/2.32 A^1 -> H(e, f(d)) 5.54/2.32 A^1 -> H(e, f(e)) 5.54/2.32 A^1 -> H(a, U1(d, b)) 5.54/2.32 A^1 -> H(a, U1(e, b)) 5.54/2.32 A^1 -> H(a, U1(b, d)) 5.54/2.32 A^1 -> H(a, U1(b, e)) 5.54/2.32 A^1 -> H(a, U1(d, d)) 5.54/2.32 A^1 -> H(a, U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, a), b) 5.54/2.32 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(d, a), d) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, a), b) 5.54/2.32 A^1 -> H(U1(e, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(e, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, d), b) 5.54/2.32 A^1 -> H(U1(a, d), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, d), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, d), d) 5.54/2.32 5.54/2.32 The TRS R consists of the following rules: 5.54/2.32 5.54/2.32 a -> d 5.54/2.32 a -> e 5.54/2.32 f(x) -> U1(x, x) 5.54/2.32 b -> d 5.54/2.32 b -> e 5.54/2.32 U1(d, x) -> x 5.54/2.32 5.54/2.32 Q is empty. 5.54/2.32 We have to consider all minimal (P,Q,R)-chains. 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (209) TransformationProof (EQUIVALENT) 5.54/2.32 By narrowing [LPAR04] the rule A^1 -> H(U1(a, d), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.32 5.54/2.32 (A^1 -> H(U1(d, d), U1(e, e)),A^1 -> H(U1(d, d), U1(e, e))) 5.54/2.32 (A^1 -> H(U1(e, d), U1(e, e)),A^1 -> H(U1(e, d), U1(e, e))) 5.54/2.32 5.54/2.32 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (210) 5.54/2.32 Obligation: 5.54/2.32 Q DP problem: 5.54/2.32 The TRS P consists of the following rules: 5.54/2.32 5.54/2.32 H(x, x) -> G(x, x) 5.54/2.32 G(d, e) -> A^1 5.54/2.32 A^1 -> H(f(d), f(d)) 5.54/2.32 A^1 -> H(f(e), f(e)) 5.54/2.32 A^1 -> H(U1(a, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(a, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(a, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(a, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(a, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(a, e), U1(e, e)) 5.54/2.32 A^1 -> H(U1(a, a), b) 5.54/2.32 A^1 -> H(U1(a, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, a), d) 5.54/2.32 A^1 -> H(d, U1(b, b)) 5.54/2.32 A^1 -> H(d, f(d)) 5.54/2.32 A^1 -> H(d, f(e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.32 A^1 -> H(f(d), b) 5.54/2.32 A^1 -> H(f(d), U1(d, e)) 5.54/2.32 A^1 -> H(f(d), U1(e, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.32 A^1 -> H(f(e), b) 5.54/2.32 A^1 -> H(f(e), U1(d, e)) 5.54/2.32 A^1 -> H(f(e), U1(e, d)) 5.54/2.32 A^1 -> H(f(e), d) 5.54/2.32 A^1 -> H(f(a), e) 5.54/2.32 A^1 -> H(f(d), d) 5.54/2.32 A^1 -> H(e, U1(b, b)) 5.54/2.32 A^1 -> H(e, f(d)) 5.54/2.32 A^1 -> H(e, f(e)) 5.54/2.32 A^1 -> H(a, U1(d, b)) 5.54/2.32 A^1 -> H(a, U1(e, b)) 5.54/2.32 A^1 -> H(a, U1(b, d)) 5.54/2.32 A^1 -> H(a, U1(b, e)) 5.54/2.32 A^1 -> H(a, U1(d, d)) 5.54/2.32 A^1 -> H(a, U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, a), b) 5.54/2.32 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(d, a), d) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, a), b) 5.54/2.32 A^1 -> H(U1(e, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(e, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, d), b) 5.54/2.32 A^1 -> H(U1(a, d), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, d), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, d), d) 5.54/2.32 A^1 -> H(U1(e, d), U1(e, e)) 5.54/2.32 5.54/2.32 The TRS R consists of the following rules: 5.54/2.32 5.54/2.32 a -> d 5.54/2.32 a -> e 5.54/2.32 f(x) -> U1(x, x) 5.54/2.32 b -> d 5.54/2.32 b -> e 5.54/2.32 U1(d, x) -> x 5.54/2.32 5.54/2.32 Q is empty. 5.54/2.32 We have to consider all minimal (P,Q,R)-chains. 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (211) DependencyGraphProof (EQUIVALENT) 5.54/2.32 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (212) 5.54/2.32 Obligation: 5.54/2.32 Q DP problem: 5.54/2.32 The TRS P consists of the following rules: 5.54/2.32 5.54/2.32 G(d, e) -> A^1 5.54/2.32 A^1 -> H(f(d), f(d)) 5.54/2.32 H(x, x) -> G(x, x) 5.54/2.32 A^1 -> H(f(e), f(e)) 5.54/2.32 A^1 -> H(U1(a, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(a, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(a, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(a, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(a, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(a, e), U1(e, e)) 5.54/2.32 A^1 -> H(U1(a, a), b) 5.54/2.32 A^1 -> H(U1(a, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, a), d) 5.54/2.32 A^1 -> H(d, U1(b, b)) 5.54/2.32 A^1 -> H(d, f(d)) 5.54/2.32 A^1 -> H(d, f(e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.32 A^1 -> H(f(d), b) 5.54/2.32 A^1 -> H(f(d), U1(d, e)) 5.54/2.32 A^1 -> H(f(d), U1(e, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.32 A^1 -> H(f(e), b) 5.54/2.32 A^1 -> H(f(e), U1(d, e)) 5.54/2.32 A^1 -> H(f(e), U1(e, d)) 5.54/2.32 A^1 -> H(f(e), d) 5.54/2.32 A^1 -> H(f(a), e) 5.54/2.32 A^1 -> H(f(d), d) 5.54/2.32 A^1 -> H(e, U1(b, b)) 5.54/2.32 A^1 -> H(e, f(d)) 5.54/2.32 A^1 -> H(e, f(e)) 5.54/2.32 A^1 -> H(a, U1(d, b)) 5.54/2.32 A^1 -> H(a, U1(e, b)) 5.54/2.32 A^1 -> H(a, U1(b, d)) 5.54/2.32 A^1 -> H(a, U1(b, e)) 5.54/2.32 A^1 -> H(a, U1(d, d)) 5.54/2.32 A^1 -> H(a, U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, a), b) 5.54/2.32 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(d, a), d) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, a), b) 5.54/2.32 A^1 -> H(U1(e, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(e, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, d), b) 5.54/2.32 A^1 -> H(U1(a, d), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, d), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, d), d) 5.54/2.32 5.54/2.32 The TRS R consists of the following rules: 5.54/2.32 5.54/2.32 a -> d 5.54/2.32 a -> e 5.54/2.32 f(x) -> U1(x, x) 5.54/2.32 b -> d 5.54/2.32 b -> e 5.54/2.32 U1(d, x) -> x 5.54/2.32 5.54/2.32 Q is empty. 5.54/2.32 We have to consider all minimal (P,Q,R)-chains. 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (213) TransformationProof (EQUIVALENT) 5.54/2.32 By narrowing [LPAR04] the rule A^1 -> H(U1(a, e), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.32 5.54/2.32 (A^1 -> H(U1(d, e), U1(d, b)),A^1 -> H(U1(d, e), U1(d, b))) 5.54/2.32 (A^1 -> H(U1(e, e), U1(d, b)),A^1 -> H(U1(e, e), U1(d, b))) 5.54/2.32 (A^1 -> H(U1(a, e), b),A^1 -> H(U1(a, e), b)) 5.54/2.32 (A^1 -> H(U1(a, e), U1(d, d)),A^1 -> H(U1(a, e), U1(d, d))) 5.54/2.32 (A^1 -> H(U1(a, e), U1(d, e)),A^1 -> H(U1(a, e), U1(d, e))) 5.54/2.32 5.54/2.32 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (214) 5.54/2.32 Obligation: 5.54/2.32 Q DP problem: 5.54/2.32 The TRS P consists of the following rules: 5.54/2.32 5.54/2.32 G(d, e) -> A^1 5.54/2.32 A^1 -> H(f(d), f(d)) 5.54/2.32 H(x, x) -> G(x, x) 5.54/2.32 A^1 -> H(f(e), f(e)) 5.54/2.32 A^1 -> H(U1(a, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(a, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(a, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(a, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(a, e), U1(e, e)) 5.54/2.32 A^1 -> H(U1(a, a), b) 5.54/2.32 A^1 -> H(U1(a, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, a), d) 5.54/2.32 A^1 -> H(d, U1(b, b)) 5.54/2.32 A^1 -> H(d, f(d)) 5.54/2.32 A^1 -> H(d, f(e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.32 A^1 -> H(f(d), b) 5.54/2.32 A^1 -> H(f(d), U1(d, e)) 5.54/2.32 A^1 -> H(f(d), U1(e, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.32 A^1 -> H(f(e), b) 5.54/2.32 A^1 -> H(f(e), U1(d, e)) 5.54/2.32 A^1 -> H(f(e), U1(e, d)) 5.54/2.32 A^1 -> H(f(e), d) 5.54/2.32 A^1 -> H(f(a), e) 5.54/2.32 A^1 -> H(f(d), d) 5.54/2.32 A^1 -> H(e, U1(b, b)) 5.54/2.32 A^1 -> H(e, f(d)) 5.54/2.32 A^1 -> H(e, f(e)) 5.54/2.32 A^1 -> H(a, U1(d, b)) 5.54/2.32 A^1 -> H(a, U1(e, b)) 5.54/2.32 A^1 -> H(a, U1(b, d)) 5.54/2.32 A^1 -> H(a, U1(b, e)) 5.54/2.32 A^1 -> H(a, U1(d, d)) 5.54/2.32 A^1 -> H(a, U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, a), b) 5.54/2.32 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(d, a), d) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, a), b) 5.54/2.32 A^1 -> H(U1(e, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(e, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, d), b) 5.54/2.32 A^1 -> H(U1(a, d), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, d), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, d), d) 5.54/2.32 A^1 -> H(U1(a, e), b) 5.54/2.32 A^1 -> H(U1(a, e), U1(d, e)) 5.54/2.32 5.54/2.32 The TRS R consists of the following rules: 5.54/2.32 5.54/2.32 a -> d 5.54/2.32 a -> e 5.54/2.32 f(x) -> U1(x, x) 5.54/2.32 b -> d 5.54/2.32 b -> e 5.54/2.32 U1(d, x) -> x 5.54/2.32 5.54/2.32 Q is empty. 5.54/2.32 We have to consider all minimal (P,Q,R)-chains. 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (215) TransformationProof (EQUIVALENT) 5.54/2.32 By narrowing [LPAR04] the rule A^1 -> H(U1(a, e), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.32 5.54/2.32 (A^1 -> H(U1(d, e), U1(e, b)),A^1 -> H(U1(d, e), U1(e, b))) 5.54/2.32 (A^1 -> H(U1(e, e), U1(e, b)),A^1 -> H(U1(e, e), U1(e, b))) 5.54/2.32 (A^1 -> H(U1(a, e), U1(e, d)),A^1 -> H(U1(a, e), U1(e, d))) 5.54/2.32 (A^1 -> H(U1(a, e), U1(e, e)),A^1 -> H(U1(a, e), U1(e, e))) 5.54/2.32 5.54/2.32 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (216) 5.54/2.32 Obligation: 5.54/2.32 Q DP problem: 5.54/2.32 The TRS P consists of the following rules: 5.54/2.32 5.54/2.32 G(d, e) -> A^1 5.54/2.32 A^1 -> H(f(d), f(d)) 5.54/2.32 H(x, x) -> G(x, x) 5.54/2.32 A^1 -> H(f(e), f(e)) 5.54/2.32 A^1 -> H(U1(a, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(a, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(a, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(a, e), U1(e, e)) 5.54/2.32 A^1 -> H(U1(a, a), b) 5.54/2.32 A^1 -> H(U1(a, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, a), d) 5.54/2.32 A^1 -> H(d, U1(b, b)) 5.54/2.32 A^1 -> H(d, f(d)) 5.54/2.32 A^1 -> H(d, f(e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.32 A^1 -> H(f(d), b) 5.54/2.32 A^1 -> H(f(d), U1(d, e)) 5.54/2.32 A^1 -> H(f(d), U1(e, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.32 A^1 -> H(f(e), b) 5.54/2.32 A^1 -> H(f(e), U1(d, e)) 5.54/2.32 A^1 -> H(f(e), U1(e, d)) 5.54/2.32 A^1 -> H(f(e), d) 5.54/2.32 A^1 -> H(f(a), e) 5.54/2.32 A^1 -> H(f(d), d) 5.54/2.32 A^1 -> H(e, U1(b, b)) 5.54/2.32 A^1 -> H(e, f(d)) 5.54/2.32 A^1 -> H(e, f(e)) 5.54/2.32 A^1 -> H(a, U1(d, b)) 5.54/2.32 A^1 -> H(a, U1(e, b)) 5.54/2.32 A^1 -> H(a, U1(b, d)) 5.54/2.32 A^1 -> H(a, U1(b, e)) 5.54/2.32 A^1 -> H(a, U1(d, d)) 5.54/2.32 A^1 -> H(a, U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, a), b) 5.54/2.32 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(d, a), d) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, a), b) 5.54/2.32 A^1 -> H(U1(e, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(e, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, d), b) 5.54/2.32 A^1 -> H(U1(a, d), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, d), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, d), d) 5.54/2.32 A^1 -> H(U1(a, e), b) 5.54/2.32 A^1 -> H(U1(a, e), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, e), U1(e, d)) 5.54/2.32 5.54/2.32 The TRS R consists of the following rules: 5.54/2.32 5.54/2.32 a -> d 5.54/2.32 a -> e 5.54/2.32 f(x) -> U1(x, x) 5.54/2.32 b -> d 5.54/2.32 b -> e 5.54/2.32 U1(d, x) -> x 5.54/2.32 5.54/2.32 Q is empty. 5.54/2.32 We have to consider all minimal (P,Q,R)-chains. 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (217) TransformationProof (EQUIVALENT) 5.54/2.32 By narrowing [LPAR04] the rule A^1 -> H(U1(a, e), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.32 5.54/2.32 (A^1 -> H(U1(d, e), U1(b, d)),A^1 -> H(U1(d, e), U1(b, d))) 5.54/2.32 (A^1 -> H(U1(e, e), U1(b, d)),A^1 -> H(U1(e, e), U1(b, d))) 5.54/2.32 (A^1 -> H(U1(a, e), U1(d, d)),A^1 -> H(U1(a, e), U1(d, d))) 5.54/2.32 (A^1 -> H(U1(a, e), U1(e, d)),A^1 -> H(U1(a, e), U1(e, d))) 5.54/2.32 5.54/2.32 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (218) 5.54/2.32 Obligation: 5.54/2.32 Q DP problem: 5.54/2.32 The TRS P consists of the following rules: 5.54/2.32 5.54/2.32 G(d, e) -> A^1 5.54/2.32 A^1 -> H(f(d), f(d)) 5.54/2.32 H(x, x) -> G(x, x) 5.54/2.32 A^1 -> H(f(e), f(e)) 5.54/2.32 A^1 -> H(U1(a, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(a, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(a, e), U1(e, e)) 5.54/2.32 A^1 -> H(U1(a, a), b) 5.54/2.32 A^1 -> H(U1(a, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, a), d) 5.54/2.32 A^1 -> H(d, U1(b, b)) 5.54/2.32 A^1 -> H(d, f(d)) 5.54/2.32 A^1 -> H(d, f(e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.32 A^1 -> H(f(d), b) 5.54/2.32 A^1 -> H(f(d), U1(d, e)) 5.54/2.32 A^1 -> H(f(d), U1(e, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.32 A^1 -> H(f(e), b) 5.54/2.32 A^1 -> H(f(e), U1(d, e)) 5.54/2.32 A^1 -> H(f(e), U1(e, d)) 5.54/2.32 A^1 -> H(f(e), d) 5.54/2.32 A^1 -> H(f(a), e) 5.54/2.32 A^1 -> H(f(d), d) 5.54/2.32 A^1 -> H(e, U1(b, b)) 5.54/2.32 A^1 -> H(e, f(d)) 5.54/2.32 A^1 -> H(e, f(e)) 5.54/2.32 A^1 -> H(a, U1(d, b)) 5.54/2.32 A^1 -> H(a, U1(e, b)) 5.54/2.32 A^1 -> H(a, U1(b, d)) 5.54/2.32 A^1 -> H(a, U1(b, e)) 5.54/2.32 A^1 -> H(a, U1(d, d)) 5.54/2.32 A^1 -> H(a, U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, a), b) 5.54/2.32 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(d, a), d) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, a), b) 5.54/2.32 A^1 -> H(U1(e, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(e, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, d), b) 5.54/2.32 A^1 -> H(U1(a, d), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, d), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, d), d) 5.54/2.32 A^1 -> H(U1(a, e), b) 5.54/2.32 A^1 -> H(U1(a, e), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, e), U1(e, d)) 5.54/2.32 5.54/2.32 The TRS R consists of the following rules: 5.54/2.32 5.54/2.32 a -> d 5.54/2.32 a -> e 5.54/2.32 f(x) -> U1(x, x) 5.54/2.32 b -> d 5.54/2.32 b -> e 5.54/2.32 U1(d, x) -> x 5.54/2.32 5.54/2.32 Q is empty. 5.54/2.32 We have to consider all minimal (P,Q,R)-chains. 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (219) TransformationProof (EQUIVALENT) 5.54/2.32 By narrowing [LPAR04] the rule A^1 -> H(U1(a, e), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.32 5.54/2.32 (A^1 -> H(U1(d, e), U1(b, e)),A^1 -> H(U1(d, e), U1(b, e))) 5.54/2.32 (A^1 -> H(U1(e, e), U1(b, e)),A^1 -> H(U1(e, e), U1(b, e))) 5.54/2.32 (A^1 -> H(U1(a, e), U1(d, e)),A^1 -> H(U1(a, e), U1(d, e))) 5.54/2.32 (A^1 -> H(U1(a, e), U1(e, e)),A^1 -> H(U1(a, e), U1(e, e))) 5.54/2.32 5.54/2.32 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (220) 5.54/2.32 Obligation: 5.54/2.32 Q DP problem: 5.54/2.32 The TRS P consists of the following rules: 5.54/2.32 5.54/2.32 G(d, e) -> A^1 5.54/2.32 A^1 -> H(f(d), f(d)) 5.54/2.32 H(x, x) -> G(x, x) 5.54/2.32 A^1 -> H(f(e), f(e)) 5.54/2.32 A^1 -> H(U1(a, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(a, e), U1(e, e)) 5.54/2.32 A^1 -> H(U1(a, a), b) 5.54/2.32 A^1 -> H(U1(a, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, a), d) 5.54/2.32 A^1 -> H(d, U1(b, b)) 5.54/2.32 A^1 -> H(d, f(d)) 5.54/2.32 A^1 -> H(d, f(e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.32 A^1 -> H(f(d), b) 5.54/2.32 A^1 -> H(f(d), U1(d, e)) 5.54/2.32 A^1 -> H(f(d), U1(e, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.32 A^1 -> H(f(e), b) 5.54/2.32 A^1 -> H(f(e), U1(d, e)) 5.54/2.32 A^1 -> H(f(e), U1(e, d)) 5.54/2.32 A^1 -> H(f(e), d) 5.54/2.32 A^1 -> H(f(a), e) 5.54/2.32 A^1 -> H(f(d), d) 5.54/2.32 A^1 -> H(e, U1(b, b)) 5.54/2.32 A^1 -> H(e, f(d)) 5.54/2.32 A^1 -> H(e, f(e)) 5.54/2.32 A^1 -> H(a, U1(d, b)) 5.54/2.32 A^1 -> H(a, U1(e, b)) 5.54/2.32 A^1 -> H(a, U1(b, d)) 5.54/2.32 A^1 -> H(a, U1(b, e)) 5.54/2.32 A^1 -> H(a, U1(d, d)) 5.54/2.32 A^1 -> H(a, U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, a), b) 5.54/2.32 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(d, a), d) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, a), b) 5.54/2.32 A^1 -> H(U1(e, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(e, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, d), b) 5.54/2.32 A^1 -> H(U1(a, d), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, d), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, d), d) 5.54/2.32 A^1 -> H(U1(a, e), b) 5.54/2.32 A^1 -> H(U1(a, e), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, e), U1(e, d)) 5.54/2.32 5.54/2.32 The TRS R consists of the following rules: 5.54/2.32 5.54/2.32 a -> d 5.54/2.32 a -> e 5.54/2.32 f(x) -> U1(x, x) 5.54/2.32 b -> d 5.54/2.32 b -> e 5.54/2.32 U1(d, x) -> x 5.54/2.32 5.54/2.32 Q is empty. 5.54/2.32 We have to consider all minimal (P,Q,R)-chains. 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (221) TransformationProof (EQUIVALENT) 5.54/2.32 By narrowing [LPAR04] the rule A^1 -> H(U1(a, e), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.32 5.54/2.32 (A^1 -> H(U1(d, e), U1(d, d)),A^1 -> H(U1(d, e), U1(d, d))) 5.54/2.32 (A^1 -> H(U1(e, e), U1(d, d)),A^1 -> H(U1(e, e), U1(d, d))) 5.54/2.32 (A^1 -> H(U1(a, e), d),A^1 -> H(U1(a, e), d)) 5.54/2.32 5.54/2.32 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (222) 5.54/2.32 Obligation: 5.54/2.32 Q DP problem: 5.54/2.32 The TRS P consists of the following rules: 5.54/2.32 5.54/2.32 G(d, e) -> A^1 5.54/2.32 A^1 -> H(f(d), f(d)) 5.54/2.32 H(x, x) -> G(x, x) 5.54/2.32 A^1 -> H(f(e), f(e)) 5.54/2.32 A^1 -> H(U1(a, e), U1(e, e)) 5.54/2.32 A^1 -> H(U1(a, a), b) 5.54/2.32 A^1 -> H(U1(a, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, a), d) 5.54/2.32 A^1 -> H(d, U1(b, b)) 5.54/2.32 A^1 -> H(d, f(d)) 5.54/2.32 A^1 -> H(d, f(e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.32 A^1 -> H(f(d), b) 5.54/2.32 A^1 -> H(f(d), U1(d, e)) 5.54/2.32 A^1 -> H(f(d), U1(e, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.32 A^1 -> H(f(e), b) 5.54/2.32 A^1 -> H(f(e), U1(d, e)) 5.54/2.32 A^1 -> H(f(e), U1(e, d)) 5.54/2.32 A^1 -> H(f(e), d) 5.54/2.32 A^1 -> H(f(a), e) 5.54/2.32 A^1 -> H(f(d), d) 5.54/2.32 A^1 -> H(e, U1(b, b)) 5.54/2.32 A^1 -> H(e, f(d)) 5.54/2.32 A^1 -> H(e, f(e)) 5.54/2.32 A^1 -> H(a, U1(d, b)) 5.54/2.32 A^1 -> H(a, U1(e, b)) 5.54/2.32 A^1 -> H(a, U1(b, d)) 5.54/2.32 A^1 -> H(a, U1(b, e)) 5.54/2.32 A^1 -> H(a, U1(d, d)) 5.54/2.32 A^1 -> H(a, U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, a), b) 5.54/2.32 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(d, a), d) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, a), b) 5.54/2.32 A^1 -> H(U1(e, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(e, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, d), b) 5.54/2.32 A^1 -> H(U1(a, d), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, d), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, d), d) 5.54/2.32 A^1 -> H(U1(a, e), b) 5.54/2.32 A^1 -> H(U1(a, e), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, e), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, e), d) 5.54/2.32 5.54/2.32 The TRS R consists of the following rules: 5.54/2.32 5.54/2.32 a -> d 5.54/2.32 a -> e 5.54/2.32 f(x) -> U1(x, x) 5.54/2.32 b -> d 5.54/2.32 b -> e 5.54/2.32 U1(d, x) -> x 5.54/2.32 5.54/2.32 Q is empty. 5.54/2.32 We have to consider all minimal (P,Q,R)-chains. 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (223) TransformationProof (EQUIVALENT) 5.54/2.32 By narrowing [LPAR04] the rule A^1 -> H(U1(a, e), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.32 5.54/2.32 (A^1 -> H(U1(d, e), U1(e, e)),A^1 -> H(U1(d, e), U1(e, e))) 5.54/2.32 (A^1 -> H(U1(e, e), U1(e, e)),A^1 -> H(U1(e, e), U1(e, e))) 5.54/2.32 5.54/2.32 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (224) 5.54/2.32 Obligation: 5.54/2.32 Q DP problem: 5.54/2.32 The TRS P consists of the following rules: 5.54/2.32 5.54/2.32 G(d, e) -> A^1 5.54/2.32 A^1 -> H(f(d), f(d)) 5.54/2.32 H(x, x) -> G(x, x) 5.54/2.32 A^1 -> H(f(e), f(e)) 5.54/2.32 A^1 -> H(U1(a, a), b) 5.54/2.32 A^1 -> H(U1(a, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, a), d) 5.54/2.32 A^1 -> H(d, U1(b, b)) 5.54/2.32 A^1 -> H(d, f(d)) 5.54/2.32 A^1 -> H(d, f(e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.32 A^1 -> H(f(d), b) 5.54/2.32 A^1 -> H(f(d), U1(d, e)) 5.54/2.32 A^1 -> H(f(d), U1(e, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.32 A^1 -> H(f(e), b) 5.54/2.32 A^1 -> H(f(e), U1(d, e)) 5.54/2.32 A^1 -> H(f(e), U1(e, d)) 5.54/2.32 A^1 -> H(f(e), d) 5.54/2.32 A^1 -> H(f(a), e) 5.54/2.32 A^1 -> H(f(d), d) 5.54/2.32 A^1 -> H(e, U1(b, b)) 5.54/2.32 A^1 -> H(e, f(d)) 5.54/2.32 A^1 -> H(e, f(e)) 5.54/2.32 A^1 -> H(a, U1(d, b)) 5.54/2.32 A^1 -> H(a, U1(e, b)) 5.54/2.32 A^1 -> H(a, U1(b, d)) 5.54/2.32 A^1 -> H(a, U1(b, e)) 5.54/2.32 A^1 -> H(a, U1(d, d)) 5.54/2.32 A^1 -> H(a, U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, a), b) 5.54/2.32 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(d, a), d) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, a), b) 5.54/2.32 A^1 -> H(U1(e, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(e, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, d), b) 5.54/2.32 A^1 -> H(U1(a, d), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, d), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, d), d) 5.54/2.32 A^1 -> H(U1(a, e), b) 5.54/2.32 A^1 -> H(U1(a, e), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, e), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, e), d) 5.54/2.32 5.54/2.32 The TRS R consists of the following rules: 5.54/2.32 5.54/2.32 a -> d 5.54/2.32 a -> e 5.54/2.32 f(x) -> U1(x, x) 5.54/2.32 b -> d 5.54/2.32 b -> e 5.54/2.32 U1(d, x) -> x 5.54/2.32 5.54/2.32 Q is empty. 5.54/2.32 We have to consider all minimal (P,Q,R)-chains. 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (225) TransformationProof (EQUIVALENT) 5.54/2.32 By narrowing [LPAR04] the rule A^1 -> H(U1(a, a), b) at position [] we obtained the following new rules [LPAR04]: 5.54/2.32 5.54/2.32 (A^1 -> H(U1(d, a), b),A^1 -> H(U1(d, a), b)) 5.54/2.32 (A^1 -> H(U1(e, a), b),A^1 -> H(U1(e, a), b)) 5.54/2.32 (A^1 -> H(U1(a, d), b),A^1 -> H(U1(a, d), b)) 5.54/2.32 (A^1 -> H(U1(a, e), b),A^1 -> H(U1(a, e), b)) 5.54/2.32 (A^1 -> H(U1(a, a), d),A^1 -> H(U1(a, a), d)) 5.54/2.32 (A^1 -> H(U1(a, a), e),A^1 -> H(U1(a, a), e)) 5.54/2.32 5.54/2.32 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (226) 5.54/2.32 Obligation: 5.54/2.32 Q DP problem: 5.54/2.32 The TRS P consists of the following rules: 5.54/2.32 5.54/2.32 G(d, e) -> A^1 5.54/2.32 A^1 -> H(f(d), f(d)) 5.54/2.32 H(x, x) -> G(x, x) 5.54/2.32 A^1 -> H(f(e), f(e)) 5.54/2.32 A^1 -> H(U1(a, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, a), d) 5.54/2.32 A^1 -> H(d, U1(b, b)) 5.54/2.32 A^1 -> H(d, f(d)) 5.54/2.32 A^1 -> H(d, f(e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.32 A^1 -> H(f(d), b) 5.54/2.32 A^1 -> H(f(d), U1(d, e)) 5.54/2.32 A^1 -> H(f(d), U1(e, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.32 A^1 -> H(f(e), b) 5.54/2.32 A^1 -> H(f(e), U1(d, e)) 5.54/2.32 A^1 -> H(f(e), U1(e, d)) 5.54/2.32 A^1 -> H(f(e), d) 5.54/2.32 A^1 -> H(f(a), e) 5.54/2.32 A^1 -> H(f(d), d) 5.54/2.32 A^1 -> H(e, U1(b, b)) 5.54/2.32 A^1 -> H(e, f(d)) 5.54/2.32 A^1 -> H(e, f(e)) 5.54/2.32 A^1 -> H(a, U1(d, b)) 5.54/2.32 A^1 -> H(a, U1(e, b)) 5.54/2.32 A^1 -> H(a, U1(b, d)) 5.54/2.32 A^1 -> H(a, U1(b, e)) 5.54/2.32 A^1 -> H(a, U1(d, d)) 5.54/2.32 A^1 -> H(a, U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, a), b) 5.54/2.32 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(d, a), d) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, a), b) 5.54/2.32 A^1 -> H(U1(e, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(e, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, d), b) 5.54/2.32 A^1 -> H(U1(a, d), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, d), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, d), d) 5.54/2.32 A^1 -> H(U1(a, e), b) 5.54/2.32 A^1 -> H(U1(a, e), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, e), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, e), d) 5.54/2.32 A^1 -> H(U1(a, a), e) 5.54/2.32 5.54/2.32 The TRS R consists of the following rules: 5.54/2.32 5.54/2.32 a -> d 5.54/2.32 a -> e 5.54/2.32 f(x) -> U1(x, x) 5.54/2.32 b -> d 5.54/2.32 b -> e 5.54/2.32 U1(d, x) -> x 5.54/2.32 5.54/2.32 Q is empty. 5.54/2.32 We have to consider all minimal (P,Q,R)-chains. 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (227) TransformationProof (EQUIVALENT) 5.54/2.32 By narrowing [LPAR04] the rule A^1 -> H(U1(a, a), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.32 5.54/2.32 (A^1 -> H(U1(d, a), U1(d, e)),A^1 -> H(U1(d, a), U1(d, e))) 5.54/2.32 (A^1 -> H(U1(e, a), U1(d, e)),A^1 -> H(U1(e, a), U1(d, e))) 5.54/2.32 (A^1 -> H(U1(a, d), U1(d, e)),A^1 -> H(U1(a, d), U1(d, e))) 5.54/2.32 (A^1 -> H(U1(a, e), U1(d, e)),A^1 -> H(U1(a, e), U1(d, e))) 5.54/2.32 (A^1 -> H(U1(a, a), e),A^1 -> H(U1(a, a), e)) 5.54/2.32 5.54/2.32 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (228) 5.54/2.32 Obligation: 5.54/2.32 Q DP problem: 5.54/2.32 The TRS P consists of the following rules: 5.54/2.32 5.54/2.32 G(d, e) -> A^1 5.54/2.32 A^1 -> H(f(d), f(d)) 5.54/2.32 H(x, x) -> G(x, x) 5.54/2.32 A^1 -> H(f(e), f(e)) 5.54/2.32 A^1 -> H(U1(a, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, a), d) 5.54/2.32 A^1 -> H(d, U1(b, b)) 5.54/2.32 A^1 -> H(d, f(d)) 5.54/2.32 A^1 -> H(d, f(e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.32 A^1 -> H(f(d), b) 5.54/2.32 A^1 -> H(f(d), U1(d, e)) 5.54/2.32 A^1 -> H(f(d), U1(e, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.32 A^1 -> H(f(e), b) 5.54/2.32 A^1 -> H(f(e), U1(d, e)) 5.54/2.32 A^1 -> H(f(e), U1(e, d)) 5.54/2.32 A^1 -> H(f(e), d) 5.54/2.32 A^1 -> H(f(a), e) 5.54/2.32 A^1 -> H(f(d), d) 5.54/2.32 A^1 -> H(e, U1(b, b)) 5.54/2.32 A^1 -> H(e, f(d)) 5.54/2.32 A^1 -> H(e, f(e)) 5.54/2.32 A^1 -> H(a, U1(d, b)) 5.54/2.32 A^1 -> H(a, U1(e, b)) 5.54/2.32 A^1 -> H(a, U1(b, d)) 5.54/2.32 A^1 -> H(a, U1(b, e)) 5.54/2.32 A^1 -> H(a, U1(d, d)) 5.54/2.32 A^1 -> H(a, U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, a), b) 5.54/2.32 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(d, a), d) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, a), b) 5.54/2.32 A^1 -> H(U1(e, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(e, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, d), b) 5.54/2.32 A^1 -> H(U1(a, d), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, d), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, d), d) 5.54/2.32 A^1 -> H(U1(a, e), b) 5.54/2.32 A^1 -> H(U1(a, e), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, e), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, e), d) 5.54/2.32 A^1 -> H(U1(a, a), e) 5.54/2.32 5.54/2.32 The TRS R consists of the following rules: 5.54/2.32 5.54/2.32 a -> d 5.54/2.32 a -> e 5.54/2.32 f(x) -> U1(x, x) 5.54/2.32 b -> d 5.54/2.32 b -> e 5.54/2.32 U1(d, x) -> x 5.54/2.32 5.54/2.32 Q is empty. 5.54/2.32 We have to consider all minimal (P,Q,R)-chains. 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (229) TransformationProof (EQUIVALENT) 5.54/2.32 By narrowing [LPAR04] the rule A^1 -> H(U1(a, a), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.32 5.54/2.32 (A^1 -> H(U1(d, a), U1(e, d)),A^1 -> H(U1(d, a), U1(e, d))) 5.54/2.32 (A^1 -> H(U1(e, a), U1(e, d)),A^1 -> H(U1(e, a), U1(e, d))) 5.54/2.32 (A^1 -> H(U1(a, d), U1(e, d)),A^1 -> H(U1(a, d), U1(e, d))) 5.54/2.32 (A^1 -> H(U1(a, e), U1(e, d)),A^1 -> H(U1(a, e), U1(e, d))) 5.54/2.32 5.54/2.32 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (230) 5.54/2.32 Obligation: 5.54/2.32 Q DP problem: 5.54/2.32 The TRS P consists of the following rules: 5.54/2.32 5.54/2.32 G(d, e) -> A^1 5.54/2.32 A^1 -> H(f(d), f(d)) 5.54/2.32 H(x, x) -> G(x, x) 5.54/2.32 A^1 -> H(f(e), f(e)) 5.54/2.32 A^1 -> H(U1(a, a), d) 5.54/2.32 A^1 -> H(d, U1(b, b)) 5.54/2.32 A^1 -> H(d, f(d)) 5.54/2.32 A^1 -> H(d, f(e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.32 A^1 -> H(f(d), b) 5.54/2.32 A^1 -> H(f(d), U1(d, e)) 5.54/2.32 A^1 -> H(f(d), U1(e, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.32 A^1 -> H(f(e), b) 5.54/2.32 A^1 -> H(f(e), U1(d, e)) 5.54/2.32 A^1 -> H(f(e), U1(e, d)) 5.54/2.32 A^1 -> H(f(e), d) 5.54/2.32 A^1 -> H(f(a), e) 5.54/2.32 A^1 -> H(f(d), d) 5.54/2.32 A^1 -> H(e, U1(b, b)) 5.54/2.32 A^1 -> H(e, f(d)) 5.54/2.32 A^1 -> H(e, f(e)) 5.54/2.32 A^1 -> H(a, U1(d, b)) 5.54/2.32 A^1 -> H(a, U1(e, b)) 5.54/2.32 A^1 -> H(a, U1(b, d)) 5.54/2.32 A^1 -> H(a, U1(b, e)) 5.54/2.32 A^1 -> H(a, U1(d, d)) 5.54/2.32 A^1 -> H(a, U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, a), b) 5.54/2.32 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(d, a), d) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, a), b) 5.54/2.32 A^1 -> H(U1(e, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(e, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, d), b) 5.54/2.32 A^1 -> H(U1(a, d), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, d), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, d), d) 5.54/2.32 A^1 -> H(U1(a, e), b) 5.54/2.32 A^1 -> H(U1(a, e), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, e), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, e), d) 5.54/2.32 A^1 -> H(U1(a, a), e) 5.54/2.32 5.54/2.32 The TRS R consists of the following rules: 5.54/2.32 5.54/2.32 a -> d 5.54/2.32 a -> e 5.54/2.32 f(x) -> U1(x, x) 5.54/2.32 b -> d 5.54/2.32 b -> e 5.54/2.32 U1(d, x) -> x 5.54/2.32 5.54/2.32 Q is empty. 5.54/2.32 We have to consider all minimal (P,Q,R)-chains. 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (231) TransformationProof (EQUIVALENT) 5.54/2.32 By narrowing [LPAR04] the rule A^1 -> H(U1(a, a), d) at position [] we obtained the following new rules [LPAR04]: 5.54/2.32 5.54/2.32 (A^1 -> H(U1(d, a), d),A^1 -> H(U1(d, a), d)) 5.54/2.32 (A^1 -> H(U1(e, a), d),A^1 -> H(U1(e, a), d)) 5.54/2.32 (A^1 -> H(U1(a, d), d),A^1 -> H(U1(a, d), d)) 5.54/2.32 (A^1 -> H(U1(a, e), d),A^1 -> H(U1(a, e), d)) 5.54/2.32 5.54/2.32 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (232) 5.54/2.32 Obligation: 5.54/2.32 Q DP problem: 5.54/2.32 The TRS P consists of the following rules: 5.54/2.32 5.54/2.32 G(d, e) -> A^1 5.54/2.32 A^1 -> H(f(d), f(d)) 5.54/2.32 H(x, x) -> G(x, x) 5.54/2.32 A^1 -> H(f(e), f(e)) 5.54/2.32 A^1 -> H(d, U1(b, b)) 5.54/2.32 A^1 -> H(d, f(d)) 5.54/2.32 A^1 -> H(d, f(e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.32 A^1 -> H(f(d), b) 5.54/2.32 A^1 -> H(f(d), U1(d, e)) 5.54/2.32 A^1 -> H(f(d), U1(e, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.32 A^1 -> H(f(e), b) 5.54/2.32 A^1 -> H(f(e), U1(d, e)) 5.54/2.32 A^1 -> H(f(e), U1(e, d)) 5.54/2.32 A^1 -> H(f(e), d) 5.54/2.32 A^1 -> H(f(a), e) 5.54/2.32 A^1 -> H(f(d), d) 5.54/2.32 A^1 -> H(e, U1(b, b)) 5.54/2.32 A^1 -> H(e, f(d)) 5.54/2.32 A^1 -> H(e, f(e)) 5.54/2.32 A^1 -> H(a, U1(d, b)) 5.54/2.32 A^1 -> H(a, U1(e, b)) 5.54/2.32 A^1 -> H(a, U1(b, d)) 5.54/2.32 A^1 -> H(a, U1(b, e)) 5.54/2.32 A^1 -> H(a, U1(d, d)) 5.54/2.32 A^1 -> H(a, U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, a), b) 5.54/2.32 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(d, a), d) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, a), b) 5.54/2.32 A^1 -> H(U1(e, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(e, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, d), b) 5.54/2.32 A^1 -> H(U1(a, d), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, d), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, d), d) 5.54/2.32 A^1 -> H(U1(a, e), b) 5.54/2.32 A^1 -> H(U1(a, e), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, e), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, e), d) 5.54/2.32 A^1 -> H(U1(a, a), e) 5.54/2.32 A^1 -> H(U1(e, a), d) 5.54/2.32 5.54/2.32 The TRS R consists of the following rules: 5.54/2.32 5.54/2.32 a -> d 5.54/2.32 a -> e 5.54/2.32 f(x) -> U1(x, x) 5.54/2.32 b -> d 5.54/2.32 b -> e 5.54/2.32 U1(d, x) -> x 5.54/2.32 5.54/2.32 Q is empty. 5.54/2.32 We have to consider all minimal (P,Q,R)-chains. 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (233) DependencyGraphProof (EQUIVALENT) 5.54/2.32 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (234) 5.54/2.32 Obligation: 5.54/2.32 Q DP problem: 5.54/2.32 The TRS P consists of the following rules: 5.54/2.32 5.54/2.32 A^1 -> H(f(d), f(d)) 5.54/2.32 H(x, x) -> G(x, x) 5.54/2.32 G(d, e) -> A^1 5.54/2.32 A^1 -> H(f(e), f(e)) 5.54/2.32 A^1 -> H(d, U1(b, b)) 5.54/2.32 A^1 -> H(d, f(d)) 5.54/2.32 A^1 -> H(d, f(e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.32 A^1 -> H(f(d), b) 5.54/2.32 A^1 -> H(f(d), U1(d, e)) 5.54/2.32 A^1 -> H(f(d), U1(e, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.32 A^1 -> H(f(e), b) 5.54/2.32 A^1 -> H(f(e), U1(d, e)) 5.54/2.32 A^1 -> H(f(e), U1(e, d)) 5.54/2.32 A^1 -> H(f(e), d) 5.54/2.32 A^1 -> H(f(a), e) 5.54/2.32 A^1 -> H(f(d), d) 5.54/2.32 A^1 -> H(e, U1(b, b)) 5.54/2.32 A^1 -> H(e, f(d)) 5.54/2.32 A^1 -> H(e, f(e)) 5.54/2.32 A^1 -> H(a, U1(d, b)) 5.54/2.32 A^1 -> H(a, U1(e, b)) 5.54/2.32 A^1 -> H(a, U1(b, d)) 5.54/2.32 A^1 -> H(a, U1(b, e)) 5.54/2.32 A^1 -> H(a, U1(d, d)) 5.54/2.32 A^1 -> H(a, U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, a), b) 5.54/2.32 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(d, a), d) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, a), b) 5.54/2.32 A^1 -> H(U1(e, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(e, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, d), b) 5.54/2.32 A^1 -> H(U1(a, d), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, d), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, d), d) 5.54/2.32 A^1 -> H(U1(a, e), b) 5.54/2.32 A^1 -> H(U1(a, e), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, e), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, e), d) 5.54/2.32 A^1 -> H(U1(a, a), e) 5.54/2.32 5.54/2.32 The TRS R consists of the following rules: 5.54/2.32 5.54/2.32 a -> d 5.54/2.32 a -> e 5.54/2.32 f(x) -> U1(x, x) 5.54/2.32 b -> d 5.54/2.32 b -> e 5.54/2.32 U1(d, x) -> x 5.54/2.32 5.54/2.32 Q is empty. 5.54/2.32 We have to consider all minimal (P,Q,R)-chains. 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (235) TransformationProof (EQUIVALENT) 5.54/2.32 By narrowing [LPAR04] the rule A^1 -> H(d, U1(b, b)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.32 5.54/2.32 (A^1 -> H(d, U1(d, b)),A^1 -> H(d, U1(d, b))) 5.54/2.32 (A^1 -> H(d, U1(e, b)),A^1 -> H(d, U1(e, b))) 5.54/2.32 (A^1 -> H(d, U1(b, d)),A^1 -> H(d, U1(b, d))) 5.54/2.32 (A^1 -> H(d, U1(b, e)),A^1 -> H(d, U1(b, e))) 5.54/2.32 5.54/2.32 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (236) 5.54/2.32 Obligation: 5.54/2.32 Q DP problem: 5.54/2.32 The TRS P consists of the following rules: 5.54/2.32 5.54/2.32 A^1 -> H(f(d), f(d)) 5.54/2.32 H(x, x) -> G(x, x) 5.54/2.32 G(d, e) -> A^1 5.54/2.32 A^1 -> H(f(e), f(e)) 5.54/2.32 A^1 -> H(d, f(d)) 5.54/2.32 A^1 -> H(d, f(e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.32 A^1 -> H(f(d), b) 5.54/2.32 A^1 -> H(f(d), U1(d, e)) 5.54/2.32 A^1 -> H(f(d), U1(e, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.32 A^1 -> H(f(e), b) 5.54/2.32 A^1 -> H(f(e), U1(d, e)) 5.54/2.32 A^1 -> H(f(e), U1(e, d)) 5.54/2.32 A^1 -> H(f(e), d) 5.54/2.32 A^1 -> H(f(a), e) 5.54/2.32 A^1 -> H(f(d), d) 5.54/2.32 A^1 -> H(e, U1(b, b)) 5.54/2.32 A^1 -> H(e, f(d)) 5.54/2.32 A^1 -> H(e, f(e)) 5.54/2.32 A^1 -> H(a, U1(d, b)) 5.54/2.32 A^1 -> H(a, U1(e, b)) 5.54/2.32 A^1 -> H(a, U1(b, d)) 5.54/2.32 A^1 -> H(a, U1(b, e)) 5.54/2.32 A^1 -> H(a, U1(d, d)) 5.54/2.32 A^1 -> H(a, U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, a), b) 5.54/2.32 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(d, a), d) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, a), b) 5.54/2.32 A^1 -> H(U1(e, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(e, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, d), b) 5.54/2.32 A^1 -> H(U1(a, d), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, d), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, d), d) 5.54/2.32 A^1 -> H(U1(a, e), b) 5.54/2.32 A^1 -> H(U1(a, e), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, e), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, e), d) 5.54/2.32 A^1 -> H(U1(a, a), e) 5.54/2.32 A^1 -> H(d, U1(d, b)) 5.54/2.32 A^1 -> H(d, U1(e, b)) 5.54/2.32 A^1 -> H(d, U1(b, d)) 5.54/2.32 A^1 -> H(d, U1(b, e)) 5.54/2.32 5.54/2.32 The TRS R consists of the following rules: 5.54/2.32 5.54/2.32 a -> d 5.54/2.32 a -> e 5.54/2.32 f(x) -> U1(x, x) 5.54/2.32 b -> d 5.54/2.32 b -> e 5.54/2.32 U1(d, x) -> x 5.54/2.32 5.54/2.32 Q is empty. 5.54/2.32 We have to consider all minimal (P,Q,R)-chains. 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (237) DependencyGraphProof (EQUIVALENT) 5.54/2.32 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (238) 5.54/2.32 Obligation: 5.54/2.32 Q DP problem: 5.54/2.32 The TRS P consists of the following rules: 5.54/2.32 5.54/2.32 H(x, x) -> G(x, x) 5.54/2.32 G(d, e) -> A^1 5.54/2.32 A^1 -> H(f(d), f(d)) 5.54/2.32 A^1 -> H(f(e), f(e)) 5.54/2.32 A^1 -> H(d, f(d)) 5.54/2.32 A^1 -> H(d, f(e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.32 A^1 -> H(f(d), b) 5.54/2.32 A^1 -> H(f(d), U1(d, e)) 5.54/2.32 A^1 -> H(f(d), U1(e, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.32 A^1 -> H(f(e), b) 5.54/2.32 A^1 -> H(f(e), U1(d, e)) 5.54/2.32 A^1 -> H(f(e), U1(e, d)) 5.54/2.32 A^1 -> H(f(e), d) 5.54/2.32 A^1 -> H(f(a), e) 5.54/2.32 A^1 -> H(f(d), d) 5.54/2.32 A^1 -> H(e, U1(b, b)) 5.54/2.32 A^1 -> H(e, f(d)) 5.54/2.32 A^1 -> H(e, f(e)) 5.54/2.32 A^1 -> H(a, U1(d, b)) 5.54/2.32 A^1 -> H(a, U1(e, b)) 5.54/2.32 A^1 -> H(a, U1(b, d)) 5.54/2.32 A^1 -> H(a, U1(b, e)) 5.54/2.32 A^1 -> H(a, U1(d, d)) 5.54/2.32 A^1 -> H(a, U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, a), b) 5.54/2.32 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(d, a), d) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, a), b) 5.54/2.32 A^1 -> H(U1(e, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(e, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, d), b) 5.54/2.32 A^1 -> H(U1(a, d), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, d), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, d), d) 5.54/2.32 A^1 -> H(U1(a, e), b) 5.54/2.32 A^1 -> H(U1(a, e), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, e), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, e), d) 5.54/2.32 A^1 -> H(U1(a, a), e) 5.54/2.32 A^1 -> H(d, U1(d, b)) 5.54/2.32 A^1 -> H(d, U1(b, d)) 5.54/2.32 A^1 -> H(d, U1(b, e)) 5.54/2.32 5.54/2.32 The TRS R consists of the following rules: 5.54/2.32 5.54/2.32 a -> d 5.54/2.32 a -> e 5.54/2.32 f(x) -> U1(x, x) 5.54/2.32 b -> d 5.54/2.32 b -> e 5.54/2.32 U1(d, x) -> x 5.54/2.32 5.54/2.32 Q is empty. 5.54/2.32 We have to consider all minimal (P,Q,R)-chains. 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (239) TransformationProof (EQUIVALENT) 5.54/2.32 By narrowing [LPAR04] the rule A^1 -> H(d, f(d)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.32 5.54/2.32 (A^1 -> H(d, U1(d, d)),A^1 -> H(d, U1(d, d))) 5.54/2.32 5.54/2.32 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (240) 5.54/2.32 Obligation: 5.54/2.32 Q DP problem: 5.54/2.32 The TRS P consists of the following rules: 5.54/2.32 5.54/2.32 H(x, x) -> G(x, x) 5.54/2.32 G(d, e) -> A^1 5.54/2.32 A^1 -> H(f(d), f(d)) 5.54/2.32 A^1 -> H(f(e), f(e)) 5.54/2.32 A^1 -> H(d, f(e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.32 A^1 -> H(f(d), b) 5.54/2.32 A^1 -> H(f(d), U1(d, e)) 5.54/2.32 A^1 -> H(f(d), U1(e, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.32 A^1 -> H(f(e), b) 5.54/2.32 A^1 -> H(f(e), U1(d, e)) 5.54/2.32 A^1 -> H(f(e), U1(e, d)) 5.54/2.32 A^1 -> H(f(e), d) 5.54/2.32 A^1 -> H(f(a), e) 5.54/2.32 A^1 -> H(f(d), d) 5.54/2.32 A^1 -> H(e, U1(b, b)) 5.54/2.32 A^1 -> H(e, f(d)) 5.54/2.32 A^1 -> H(e, f(e)) 5.54/2.32 A^1 -> H(a, U1(d, b)) 5.54/2.32 A^1 -> H(a, U1(e, b)) 5.54/2.32 A^1 -> H(a, U1(b, d)) 5.54/2.32 A^1 -> H(a, U1(b, e)) 5.54/2.32 A^1 -> H(a, U1(d, d)) 5.54/2.32 A^1 -> H(a, U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.32 A^1 -> H(U1(d, a), b) 5.54/2.32 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(d, a), d) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, a), b) 5.54/2.32 A^1 -> H(U1(e, a), U1(d, e)) 5.54/2.32 A^1 -> H(U1(e, a), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, d), b) 5.54/2.32 A^1 -> H(U1(a, d), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, d), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, d), d) 5.54/2.32 A^1 -> H(U1(a, e), b) 5.54/2.32 A^1 -> H(U1(a, e), U1(d, e)) 5.54/2.32 A^1 -> H(U1(a, e), U1(e, d)) 5.54/2.32 A^1 -> H(U1(a, e), d) 5.54/2.32 A^1 -> H(U1(a, a), e) 5.54/2.32 A^1 -> H(d, U1(d, b)) 5.54/2.32 A^1 -> H(d, U1(b, d)) 5.54/2.32 A^1 -> H(d, U1(b, e)) 5.54/2.32 A^1 -> H(d, U1(d, d)) 5.54/2.32 5.54/2.32 The TRS R consists of the following rules: 5.54/2.32 5.54/2.32 a -> d 5.54/2.32 a -> e 5.54/2.32 f(x) -> U1(x, x) 5.54/2.32 b -> d 5.54/2.32 b -> e 5.54/2.32 U1(d, x) -> x 5.54/2.32 5.54/2.32 Q is empty. 5.54/2.32 We have to consider all minimal (P,Q,R)-chains. 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (241) TransformationProof (EQUIVALENT) 5.54/2.32 By narrowing [LPAR04] the rule A^1 -> H(d, f(e)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.32 5.54/2.32 (A^1 -> H(d, U1(e, e)),A^1 -> H(d, U1(e, e))) 5.54/2.32 5.54/2.32 5.54/2.32 ---------------------------------------- 5.54/2.32 5.54/2.32 (242) 5.54/2.32 Obligation: 5.54/2.32 Q DP problem: 5.54/2.32 The TRS P consists of the following rules: 5.54/2.32 5.54/2.32 H(x, x) -> G(x, x) 5.54/2.32 G(d, e) -> A^1 5.54/2.32 A^1 -> H(f(d), f(d)) 5.54/2.32 A^1 -> H(f(e), f(e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.32 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.32 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.32 A^1 -> H(f(d), b) 5.54/2.32 A^1 -> H(f(d), U1(d, e)) 5.54/2.32 A^1 -> H(f(d), U1(e, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.32 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.32 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.32 A^1 -> H(f(e), b) 5.54/2.32 A^1 -> H(f(e), U1(d, e)) 5.54/2.32 A^1 -> H(f(e), U1(e, d)) 5.54/2.32 A^1 -> H(f(e), d) 5.54/2.32 A^1 -> H(f(a), e) 5.54/2.32 A^1 -> H(f(d), d) 5.54/2.32 A^1 -> H(e, U1(b, b)) 5.54/2.32 A^1 -> H(e, f(d)) 5.54/2.33 A^1 -> H(e, f(e)) 5.54/2.33 A^1 -> H(a, U1(d, b)) 5.54/2.33 A^1 -> H(a, U1(e, b)) 5.54/2.33 A^1 -> H(a, U1(b, d)) 5.54/2.33 A^1 -> H(a, U1(b, e)) 5.54/2.33 A^1 -> H(a, U1(d, d)) 5.54/2.33 A^1 -> H(a, U1(e, e)) 5.54/2.33 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.33 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.33 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.33 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.33 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.33 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.33 A^1 -> H(U1(d, a), b) 5.54/2.33 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.33 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.33 A^1 -> H(U1(d, a), d) 5.54/2.33 A^1 -> H(U1(e, d), U1(d, b)) 5.54/2.33 A^1 -> H(U1(e, d), U1(e, b)) 5.54/2.33 A^1 -> H(U1(e, d), U1(b, d)) 5.54/2.33 A^1 -> H(U1(e, d), U1(b, e)) 5.54/2.33 A^1 -> H(U1(e, d), U1(d, d)) 5.54/2.33 A^1 -> H(U1(e, a), b) 5.54/2.33 A^1 -> H(U1(e, a), U1(d, e)) 5.54/2.33 A^1 -> H(U1(e, a), U1(e, d)) 5.54/2.33 A^1 -> H(U1(a, d), b) 5.54/2.33 A^1 -> H(U1(a, d), U1(d, e)) 5.54/2.33 A^1 -> H(U1(a, d), U1(e, d)) 5.54/2.33 A^1 -> H(U1(a, d), d) 5.54/2.33 A^1 -> H(U1(a, e), b) 5.54/2.33 A^1 -> H(U1(a, e), U1(d, e)) 5.54/2.33 A^1 -> H(U1(a, e), U1(e, d)) 5.54/2.33 A^1 -> H(U1(a, e), d) 5.54/2.33 A^1 -> H(U1(a, a), e) 5.54/2.33 A^1 -> H(d, U1(d, b)) 5.54/2.33 A^1 -> H(d, U1(b, d)) 5.54/2.33 A^1 -> H(d, U1(b, e)) 5.54/2.33 A^1 -> H(d, U1(d, d)) 5.54/2.33 A^1 -> H(d, U1(e, e)) 5.54/2.33 5.54/2.33 The TRS R consists of the following rules: 5.54/2.33 5.54/2.33 a -> d 5.54/2.33 a -> e 5.54/2.33 f(x) -> U1(x, x) 5.54/2.33 b -> d 5.54/2.33 b -> e 5.54/2.33 U1(d, x) -> x 5.54/2.33 5.54/2.33 Q is empty. 5.54/2.33 We have to consider all minimal (P,Q,R)-chains. 5.54/2.33 ---------------------------------------- 5.54/2.33 5.54/2.33 (243) DependencyGraphProof (EQUIVALENT) 5.54/2.33 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.54/2.33 ---------------------------------------- 5.54/2.33 5.54/2.33 (244) 5.54/2.33 Obligation: 5.54/2.33 Q DP problem: 5.54/2.33 The TRS P consists of the following rules: 5.54/2.33 5.54/2.33 G(d, e) -> A^1 5.54/2.33 A^1 -> H(f(d), f(d)) 5.54/2.33 H(x, x) -> G(x, x) 5.54/2.33 A^1 -> H(f(e), f(e)) 5.54/2.33 A^1 -> H(U1(d, d), U1(d, b)) 5.54/2.33 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.33 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.33 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.33 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.33 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.33 A^1 -> H(f(d), b) 5.54/2.33 A^1 -> H(f(d), U1(d, e)) 5.54/2.33 A^1 -> H(f(d), U1(e, d)) 5.54/2.33 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.33 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.33 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.33 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.33 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.33 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.33 A^1 -> H(f(e), b) 5.54/2.33 A^1 -> H(f(e), U1(d, e)) 5.54/2.33 A^1 -> H(f(e), U1(e, d)) 5.54/2.33 A^1 -> H(f(e), d) 5.54/2.33 A^1 -> H(f(a), e) 5.54/2.33 A^1 -> H(f(d), d) 5.54/2.33 A^1 -> H(e, U1(b, b)) 5.54/2.33 A^1 -> H(e, f(d)) 5.54/2.33 A^1 -> H(e, f(e)) 5.54/2.33 A^1 -> H(a, U1(d, b)) 5.54/2.33 A^1 -> H(a, U1(e, b)) 5.54/2.33 A^1 -> H(a, U1(b, d)) 5.54/2.33 A^1 -> H(a, U1(b, e)) 5.54/2.33 A^1 -> H(a, U1(d, d)) 5.54/2.33 A^1 -> H(a, U1(e, e)) 5.54/2.33 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.33 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.33 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.33 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.33 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.33 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.33 A^1 -> H(U1(d, a), b) 5.54/2.33 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.33 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.33 A^1 -> H(U1(d, a), d) 5.54/2.33 A^1 -> H(U1(e, d), U1(d, b)) 5.54/2.33 A^1 -> H(U1(e, d), U1(e, b)) 5.54/2.33 A^1 -> H(U1(e, d), U1(b, d)) 5.54/2.33 A^1 -> H(U1(e, d), U1(b, e)) 5.54/2.33 A^1 -> H(U1(e, d), U1(d, d)) 5.54/2.33 A^1 -> H(U1(e, a), b) 5.54/2.33 A^1 -> H(U1(e, a), U1(d, e)) 5.54/2.33 A^1 -> H(U1(e, a), U1(e, d)) 5.54/2.33 A^1 -> H(U1(a, d), b) 5.54/2.33 A^1 -> H(U1(a, d), U1(d, e)) 5.54/2.33 A^1 -> H(U1(a, d), U1(e, d)) 5.54/2.33 A^1 -> H(U1(a, d), d) 5.54/2.33 A^1 -> H(U1(a, e), b) 5.54/2.33 A^1 -> H(U1(a, e), U1(d, e)) 5.54/2.33 A^1 -> H(U1(a, e), U1(e, d)) 5.54/2.33 A^1 -> H(U1(a, e), d) 5.54/2.33 A^1 -> H(U1(a, a), e) 5.54/2.33 A^1 -> H(d, U1(d, b)) 5.54/2.33 A^1 -> H(d, U1(b, d)) 5.54/2.33 A^1 -> H(d, U1(b, e)) 5.54/2.33 A^1 -> H(d, U1(d, d)) 5.54/2.33 5.54/2.33 The TRS R consists of the following rules: 5.54/2.33 5.54/2.33 a -> d 5.54/2.33 a -> e 5.54/2.33 f(x) -> U1(x, x) 5.54/2.33 b -> d 5.54/2.33 b -> e 5.54/2.33 U1(d, x) -> x 5.54/2.33 5.54/2.33 Q is empty. 5.54/2.33 We have to consider all minimal (P,Q,R)-chains. 5.54/2.33 ---------------------------------------- 5.54/2.33 5.54/2.33 (245) TransformationProof (EQUIVALENT) 5.54/2.33 By narrowing [LPAR04] the rule A^1 -> H(U1(d, d), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.33 5.54/2.33 (A^1 -> H(d, U1(d, b)),A^1 -> H(d, U1(d, b))) 5.54/2.33 (A^1 -> H(U1(d, d), b),A^1 -> H(U1(d, d), b)) 5.54/2.33 (A^1 -> H(U1(d, d), U1(d, d)),A^1 -> H(U1(d, d), U1(d, d))) 5.54/2.33 (A^1 -> H(U1(d, d), U1(d, e)),A^1 -> H(U1(d, d), U1(d, e))) 5.54/2.33 5.54/2.33 5.54/2.33 ---------------------------------------- 5.54/2.33 5.54/2.33 (246) 5.54/2.33 Obligation: 5.54/2.33 Q DP problem: 5.54/2.33 The TRS P consists of the following rules: 5.54/2.33 5.54/2.33 G(d, e) -> A^1 5.54/2.33 A^1 -> H(f(d), f(d)) 5.54/2.33 H(x, x) -> G(x, x) 5.54/2.33 A^1 -> H(f(e), f(e)) 5.54/2.33 A^1 -> H(U1(d, d), U1(e, b)) 5.54/2.33 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.33 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.33 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.33 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.33 A^1 -> H(f(d), b) 5.54/2.33 A^1 -> H(f(d), U1(d, e)) 5.54/2.33 A^1 -> H(f(d), U1(e, d)) 5.54/2.33 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.33 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.33 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.33 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.33 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.33 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.33 A^1 -> H(f(e), b) 5.54/2.33 A^1 -> H(f(e), U1(d, e)) 5.54/2.33 A^1 -> H(f(e), U1(e, d)) 5.54/2.33 A^1 -> H(f(e), d) 5.54/2.33 A^1 -> H(f(a), e) 5.54/2.33 A^1 -> H(f(d), d) 5.54/2.33 A^1 -> H(e, U1(b, b)) 5.54/2.33 A^1 -> H(e, f(d)) 5.54/2.33 A^1 -> H(e, f(e)) 5.54/2.33 A^1 -> H(a, U1(d, b)) 5.54/2.33 A^1 -> H(a, U1(e, b)) 5.54/2.33 A^1 -> H(a, U1(b, d)) 5.54/2.33 A^1 -> H(a, U1(b, e)) 5.54/2.33 A^1 -> H(a, U1(d, d)) 5.54/2.33 A^1 -> H(a, U1(e, e)) 5.54/2.33 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.33 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.33 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.33 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.33 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.33 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.33 A^1 -> H(U1(d, a), b) 5.54/2.33 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.33 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.33 A^1 -> H(U1(d, a), d) 5.54/2.33 A^1 -> H(U1(e, d), U1(d, b)) 5.54/2.33 A^1 -> H(U1(e, d), U1(e, b)) 5.54/2.33 A^1 -> H(U1(e, d), U1(b, d)) 5.54/2.33 A^1 -> H(U1(e, d), U1(b, e)) 5.54/2.33 A^1 -> H(U1(e, d), U1(d, d)) 5.54/2.33 A^1 -> H(U1(e, a), b) 5.54/2.33 A^1 -> H(U1(e, a), U1(d, e)) 5.54/2.33 A^1 -> H(U1(e, a), U1(e, d)) 5.54/2.33 A^1 -> H(U1(a, d), b) 5.54/2.33 A^1 -> H(U1(a, d), U1(d, e)) 5.54/2.33 A^1 -> H(U1(a, d), U1(e, d)) 5.54/2.33 A^1 -> H(U1(a, d), d) 5.54/2.33 A^1 -> H(U1(a, e), b) 5.54/2.33 A^1 -> H(U1(a, e), U1(d, e)) 5.54/2.33 A^1 -> H(U1(a, e), U1(e, d)) 5.54/2.33 A^1 -> H(U1(a, e), d) 5.54/2.33 A^1 -> H(U1(a, a), e) 5.54/2.33 A^1 -> H(d, U1(d, b)) 5.54/2.33 A^1 -> H(d, U1(b, d)) 5.54/2.33 A^1 -> H(d, U1(b, e)) 5.54/2.33 A^1 -> H(d, U1(d, d)) 5.54/2.33 A^1 -> H(U1(d, d), b) 5.54/2.33 A^1 -> H(U1(d, d), U1(d, e)) 5.54/2.33 5.54/2.33 The TRS R consists of the following rules: 5.54/2.33 5.54/2.33 a -> d 5.54/2.33 a -> e 5.54/2.33 f(x) -> U1(x, x) 5.54/2.33 b -> d 5.54/2.33 b -> e 5.54/2.33 U1(d, x) -> x 5.54/2.33 5.54/2.33 Q is empty. 5.54/2.33 We have to consider all minimal (P,Q,R)-chains. 5.54/2.33 ---------------------------------------- 5.54/2.33 5.54/2.33 (247) TransformationProof (EQUIVALENT) 5.54/2.33 By narrowing [LPAR04] the rule A^1 -> H(U1(d, d), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.33 5.54/2.33 (A^1 -> H(d, U1(e, b)),A^1 -> H(d, U1(e, b))) 5.54/2.33 (A^1 -> H(U1(d, d), U1(e, d)),A^1 -> H(U1(d, d), U1(e, d))) 5.54/2.33 (A^1 -> H(U1(d, d), U1(e, e)),A^1 -> H(U1(d, d), U1(e, e))) 5.54/2.33 5.54/2.33 5.54/2.33 ---------------------------------------- 5.54/2.33 5.54/2.33 (248) 5.54/2.33 Obligation: 5.54/2.33 Q DP problem: 5.54/2.33 The TRS P consists of the following rules: 5.54/2.33 5.54/2.33 G(d, e) -> A^1 5.54/2.33 A^1 -> H(f(d), f(d)) 5.54/2.33 H(x, x) -> G(x, x) 5.54/2.33 A^1 -> H(f(e), f(e)) 5.54/2.33 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.33 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.33 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.33 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.33 A^1 -> H(f(d), b) 5.54/2.33 A^1 -> H(f(d), U1(d, e)) 5.54/2.33 A^1 -> H(f(d), U1(e, d)) 5.54/2.33 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.33 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.33 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.33 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.33 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.33 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.33 A^1 -> H(f(e), b) 5.54/2.33 A^1 -> H(f(e), U1(d, e)) 5.54/2.33 A^1 -> H(f(e), U1(e, d)) 5.54/2.33 A^1 -> H(f(e), d) 5.54/2.33 A^1 -> H(f(a), e) 5.54/2.33 A^1 -> H(f(d), d) 5.54/2.33 A^1 -> H(e, U1(b, b)) 5.54/2.33 A^1 -> H(e, f(d)) 5.54/2.33 A^1 -> H(e, f(e)) 5.54/2.33 A^1 -> H(a, U1(d, b)) 5.54/2.33 A^1 -> H(a, U1(e, b)) 5.54/2.33 A^1 -> H(a, U1(b, d)) 5.54/2.33 A^1 -> H(a, U1(b, e)) 5.54/2.33 A^1 -> H(a, U1(d, d)) 5.54/2.33 A^1 -> H(a, U1(e, e)) 5.54/2.33 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.33 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.33 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.33 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.33 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.33 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.33 A^1 -> H(U1(d, a), b) 5.54/2.33 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.33 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.33 A^1 -> H(U1(d, a), d) 5.54/2.33 A^1 -> H(U1(e, d), U1(d, b)) 5.54/2.33 A^1 -> H(U1(e, d), U1(e, b)) 5.54/2.33 A^1 -> H(U1(e, d), U1(b, d)) 5.54/2.33 A^1 -> H(U1(e, d), U1(b, e)) 5.54/2.33 A^1 -> H(U1(e, d), U1(d, d)) 5.54/2.33 A^1 -> H(U1(e, a), b) 5.54/2.33 A^1 -> H(U1(e, a), U1(d, e)) 5.54/2.33 A^1 -> H(U1(e, a), U1(e, d)) 5.54/2.33 A^1 -> H(U1(a, d), b) 5.54/2.33 A^1 -> H(U1(a, d), U1(d, e)) 5.54/2.33 A^1 -> H(U1(a, d), U1(e, d)) 5.54/2.33 A^1 -> H(U1(a, d), d) 5.54/2.33 A^1 -> H(U1(a, e), b) 5.54/2.33 A^1 -> H(U1(a, e), U1(d, e)) 5.54/2.33 A^1 -> H(U1(a, e), U1(e, d)) 5.54/2.33 A^1 -> H(U1(a, e), d) 5.54/2.33 A^1 -> H(U1(a, a), e) 5.54/2.33 A^1 -> H(d, U1(d, b)) 5.54/2.33 A^1 -> H(d, U1(b, d)) 5.54/2.33 A^1 -> H(d, U1(b, e)) 5.54/2.33 A^1 -> H(d, U1(d, d)) 5.54/2.33 A^1 -> H(U1(d, d), b) 5.54/2.33 A^1 -> H(U1(d, d), U1(d, e)) 5.54/2.33 A^1 -> H(d, U1(e, b)) 5.54/2.33 A^1 -> H(U1(d, d), U1(e, d)) 5.54/2.33 5.54/2.33 The TRS R consists of the following rules: 5.54/2.33 5.54/2.33 a -> d 5.54/2.33 a -> e 5.54/2.33 f(x) -> U1(x, x) 5.54/2.33 b -> d 5.54/2.33 b -> e 5.54/2.33 U1(d, x) -> x 5.54/2.33 5.54/2.33 Q is empty. 5.54/2.33 We have to consider all minimal (P,Q,R)-chains. 5.54/2.33 ---------------------------------------- 5.54/2.33 5.54/2.33 (249) DependencyGraphProof (EQUIVALENT) 5.54/2.33 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.54/2.33 ---------------------------------------- 5.54/2.33 5.54/2.33 (250) 5.54/2.33 Obligation: 5.54/2.33 Q DP problem: 5.54/2.33 The TRS P consists of the following rules: 5.54/2.33 5.54/2.33 A^1 -> H(f(d), f(d)) 5.54/2.33 H(x, x) -> G(x, x) 5.54/2.33 G(d, e) -> A^1 5.54/2.33 A^1 -> H(f(e), f(e)) 5.54/2.33 A^1 -> H(U1(d, d), U1(b, d)) 5.54/2.33 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.33 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.33 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.33 A^1 -> H(f(d), b) 5.54/2.33 A^1 -> H(f(d), U1(d, e)) 5.54/2.33 A^1 -> H(f(d), U1(e, d)) 5.54/2.33 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.33 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.33 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.33 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.33 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.33 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.33 A^1 -> H(f(e), b) 5.54/2.33 A^1 -> H(f(e), U1(d, e)) 5.54/2.33 A^1 -> H(f(e), U1(e, d)) 5.54/2.33 A^1 -> H(f(e), d) 5.54/2.33 A^1 -> H(f(a), e) 5.54/2.33 A^1 -> H(f(d), d) 5.54/2.33 A^1 -> H(e, U1(b, b)) 5.54/2.33 A^1 -> H(e, f(d)) 5.54/2.33 A^1 -> H(e, f(e)) 5.54/2.33 A^1 -> H(a, U1(d, b)) 5.54/2.33 A^1 -> H(a, U1(e, b)) 5.54/2.33 A^1 -> H(a, U1(b, d)) 5.54/2.33 A^1 -> H(a, U1(b, e)) 5.54/2.33 A^1 -> H(a, U1(d, d)) 5.54/2.33 A^1 -> H(a, U1(e, e)) 5.54/2.33 A^1 -> H(U1(d, e), U1(d, b)) 5.54/2.33 A^1 -> H(U1(d, e), U1(e, b)) 5.54/2.33 A^1 -> H(U1(d, e), U1(b, d)) 5.54/2.33 A^1 -> H(U1(d, e), U1(b, e)) 5.54/2.33 A^1 -> H(U1(d, e), U1(d, d)) 5.54/2.33 A^1 -> H(U1(d, e), U1(e, e)) 5.54/2.33 A^1 -> H(U1(d, a), b) 5.54/2.33 A^1 -> H(U1(d, a), U1(d, e)) 5.54/2.33 A^1 -> H(U1(d, a), U1(e, d)) 5.54/2.33 A^1 -> H(U1(d, a), d) 5.54/2.33 A^1 -> H(U1(e, d), U1(d, b)) 5.54/2.33 A^1 -> H(U1(e, d), U1(e, b)) 5.54/2.33 A^1 -> H(U1(e, d), U1(b, d)) 5.54/2.33 A^1 -> H(U1(e, d), U1(b, e)) 5.54/2.33 A^1 -> H(U1(e, d), U1(d, d)) 5.54/2.33 A^1 -> H(U1(e, a), b) 5.54/2.33 A^1 -> H(U1(e, a), U1(d, e)) 5.54/2.33 A^1 -> H(U1(e, a), U1(e, d)) 5.54/2.33 A^1 -> H(U1(a, d), b) 5.54/2.33 A^1 -> H(U1(a, d), U1(d, e)) 5.54/2.33 A^1 -> H(U1(a, d), U1(e, d)) 5.54/2.33 A^1 -> H(U1(a, d), d) 5.54/2.33 A^1 -> H(U1(a, e), b) 5.54/2.33 A^1 -> H(U1(a, e), U1(d, e)) 5.54/2.33 A^1 -> H(U1(a, e), U1(e, d)) 5.54/2.33 A^1 -> H(U1(a, e), d) 5.54/2.33 A^1 -> H(U1(a, a), e) 5.54/2.33 A^1 -> H(d, U1(d, b)) 5.54/2.33 A^1 -> H(d, U1(b, d)) 5.54/2.33 A^1 -> H(d, U1(b, e)) 5.54/2.33 A^1 -> H(d, U1(d, d)) 5.54/2.33 A^1 -> H(U1(d, d), b) 5.54/2.33 A^1 -> H(U1(d, d), U1(d, e)) 5.54/2.33 A^1 -> H(U1(d, d), U1(e, d)) 5.54/2.33 5.54/2.33 The TRS R consists of the following rules: 5.54/2.33 5.54/2.33 a -> d 5.54/2.33 a -> e 5.54/2.33 f(x) -> U1(x, x) 5.54/2.33 b -> d 5.54/2.33 b -> e 5.54/2.33 U1(d, x) -> x 5.54/2.33 5.54/2.33 Q is empty. 5.54/2.33 We have to consider all minimal (P,Q,R)-chains. 5.54/2.33 ---------------------------------------- 5.54/2.33 5.54/2.33 (251) TransformationProof (EQUIVALENT) 5.54/2.33 By narrowing [LPAR04] the rule A^1 -> H(U1(d, d), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: 5.54/2.33 5.54/2.33 (A^1 -> H(d, U1(b, d)),A^1 -> H(d, U1(b, d))) 5.54/2.33 (A^1 -> H(U1(d, d), U1(d, d)),A^1 -> H(U1(d, d), U1(d, d))) 5.54/2.33 (A^1 -> H(U1(d, d), U1(e, d)),A^1 -> H(U1(d, d), U1(e, d))) 5.54/2.33 5.54/2.33 5.54/2.33 ---------------------------------------- 5.54/2.33 5.54/2.33 (252) 5.54/2.33 Obligation: 5.54/2.33 Q DP problem: 5.54/2.33 The TRS P consists of the following rules: 5.54/2.33 5.54/2.33 A^1 -> H(f(d), f(d)) 5.54/2.33 H(x, x) -> G(x, x) 5.54/2.33 G(d, e) -> A^1 5.54/2.33 A^1 -> H(f(e), f(e)) 5.54/2.33 A^1 -> H(U1(d, d), U1(b, e)) 5.54/2.33 A^1 -> H(U1(d, d), U1(d, d)) 5.54/2.33 A^1 -> H(U1(d, d), U1(e, e)) 5.54/2.33 A^1 -> H(f(d), b) 5.54/2.33 A^1 -> H(f(d), U1(d, e)) 5.54/2.33 A^1 -> H(f(d), U1(e, d)) 5.54/2.33 A^1 -> H(U1(e, e), U1(d, b)) 5.54/2.33 A^1 -> H(U1(e, e), U1(e, b)) 5.54/2.33 A^1 -> H(U1(e, e), U1(b, d)) 5.54/2.33 A^1 -> H(U1(e, e), U1(b, e)) 5.54/2.33 A^1 -> H(U1(e, e), U1(d, d)) 5.54/2.33 A^1 -> H(U1(e, e), U1(e, e)) 5.54/2.33 A^1 -> H(f(e), b) 5.54/2.33 A^1 -> H(f(e), U1(d, e)) 5.54/2.33 A^1 -> H(f(e), U1(e, d)) 5.54/2.33 A^1 -> H(f(e), d) 5.54/2.33 A^1 -> H(f(a), e) 5.54/2.33 A^1 -> H(f(d), d) 5.54/2.33 A^1 -> H(e, U1(b, b)) 5.54/2.33 A^1 -> H(e, f(d)) 5.54/2.33 A^1 -> H(e, f(e)) 5.54/2.33 A^1 -> H(a, U1(d, b)) 5.54/2.33 A^1 -> H(a, U1(e, b)) 5.54/2.33 A^1 -> H(a, U1(b, d)) 5.54/2.33 A^1 -> H(a, U1(b, e)) 5.60/2.33 A^1 -> H(a, U1(d, d)) 5.60/2.33 A^1 -> H(a, U1(e, e)) 5.60/2.33 A^1 -> H(U1(d, e), U1(d, b)) 5.60/2.33 A^1 -> H(U1(d, e), U1(e, b)) 5.60/2.33 A^1 -> H(U1(d, e), U1(b, d)) 5.60/2.33 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.33 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.33 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.33 A^1 -> H(U1(d, a), b) 5.60/2.33 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.33 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.33 A^1 -> H(U1(d, a), d) 5.60/2.33 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.33 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.33 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.33 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.33 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, a), b) 5.60/2.33 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.33 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, d), b) 5.60/2.33 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.33 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, d), d) 5.60/2.33 A^1 -> H(U1(a, e), b) 5.60/2.33 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.33 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, e), d) 5.60/2.33 A^1 -> H(U1(a, a), e) 5.60/2.33 A^1 -> H(d, U1(d, b)) 5.60/2.33 A^1 -> H(d, U1(b, d)) 5.60/2.33 A^1 -> H(d, U1(b, e)) 5.60/2.33 A^1 -> H(d, U1(d, d)) 5.60/2.33 A^1 -> H(U1(d, d), b) 5.60/2.33 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.33 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.33 5.60/2.33 The TRS R consists of the following rules: 5.60/2.33 5.60/2.33 a -> d 5.60/2.33 a -> e 5.60/2.33 f(x) -> U1(x, x) 5.60/2.33 b -> d 5.60/2.33 b -> e 5.60/2.33 U1(d, x) -> x 5.60/2.33 5.60/2.33 Q is empty. 5.60/2.33 We have to consider all minimal (P,Q,R)-chains. 5.60/2.33 ---------------------------------------- 5.60/2.33 5.60/2.33 (253) TransformationProof (EQUIVALENT) 5.60/2.33 By narrowing [LPAR04] the rule A^1 -> H(U1(d, d), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.33 5.60/2.33 (A^1 -> H(d, U1(b, e)),A^1 -> H(d, U1(b, e))) 5.60/2.33 (A^1 -> H(U1(d, d), U1(d, e)),A^1 -> H(U1(d, d), U1(d, e))) 5.60/2.33 (A^1 -> H(U1(d, d), U1(e, e)),A^1 -> H(U1(d, d), U1(e, e))) 5.60/2.33 5.60/2.33 5.60/2.33 ---------------------------------------- 5.60/2.33 5.60/2.33 (254) 5.60/2.33 Obligation: 5.60/2.33 Q DP problem: 5.60/2.33 The TRS P consists of the following rules: 5.60/2.33 5.60/2.33 A^1 -> H(f(d), f(d)) 5.60/2.33 H(x, x) -> G(x, x) 5.60/2.33 G(d, e) -> A^1 5.60/2.33 A^1 -> H(f(e), f(e)) 5.60/2.33 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.33 A^1 -> H(U1(d, d), U1(e, e)) 5.60/2.33 A^1 -> H(f(d), b) 5.60/2.33 A^1 -> H(f(d), U1(d, e)) 5.60/2.33 A^1 -> H(f(d), U1(e, d)) 5.60/2.33 A^1 -> H(U1(e, e), U1(d, b)) 5.60/2.33 A^1 -> H(U1(e, e), U1(e, b)) 5.60/2.33 A^1 -> H(U1(e, e), U1(b, d)) 5.60/2.33 A^1 -> H(U1(e, e), U1(b, e)) 5.60/2.33 A^1 -> H(U1(e, e), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.33 A^1 -> H(f(e), b) 5.60/2.33 A^1 -> H(f(e), U1(d, e)) 5.60/2.33 A^1 -> H(f(e), U1(e, d)) 5.60/2.33 A^1 -> H(f(e), d) 5.60/2.33 A^1 -> H(f(a), e) 5.60/2.33 A^1 -> H(f(d), d) 5.60/2.33 A^1 -> H(e, U1(b, b)) 5.60/2.33 A^1 -> H(e, f(d)) 5.60/2.33 A^1 -> H(e, f(e)) 5.60/2.33 A^1 -> H(a, U1(d, b)) 5.60/2.33 A^1 -> H(a, U1(e, b)) 5.60/2.33 A^1 -> H(a, U1(b, d)) 5.60/2.33 A^1 -> H(a, U1(b, e)) 5.60/2.33 A^1 -> H(a, U1(d, d)) 5.60/2.33 A^1 -> H(a, U1(e, e)) 5.60/2.33 A^1 -> H(U1(d, e), U1(d, b)) 5.60/2.33 A^1 -> H(U1(d, e), U1(e, b)) 5.60/2.33 A^1 -> H(U1(d, e), U1(b, d)) 5.60/2.33 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.33 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.33 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.33 A^1 -> H(U1(d, a), b) 5.60/2.33 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.33 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.33 A^1 -> H(U1(d, a), d) 5.60/2.33 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.33 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.33 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.33 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.33 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, a), b) 5.60/2.33 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.33 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, d), b) 5.60/2.33 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.33 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, d), d) 5.60/2.33 A^1 -> H(U1(a, e), b) 5.60/2.33 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.33 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, e), d) 5.60/2.33 A^1 -> H(U1(a, a), e) 5.60/2.33 A^1 -> H(d, U1(d, b)) 5.60/2.33 A^1 -> H(d, U1(b, d)) 5.60/2.33 A^1 -> H(d, U1(b, e)) 5.60/2.33 A^1 -> H(d, U1(d, d)) 5.60/2.33 A^1 -> H(U1(d, d), b) 5.60/2.33 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.33 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.33 5.60/2.33 The TRS R consists of the following rules: 5.60/2.33 5.60/2.33 a -> d 5.60/2.33 a -> e 5.60/2.33 f(x) -> U1(x, x) 5.60/2.33 b -> d 5.60/2.33 b -> e 5.60/2.33 U1(d, x) -> x 5.60/2.33 5.60/2.33 Q is empty. 5.60/2.33 We have to consider all minimal (P,Q,R)-chains. 5.60/2.33 ---------------------------------------- 5.60/2.33 5.60/2.33 (255) TransformationProof (EQUIVALENT) 5.60/2.33 By narrowing [LPAR04] the rule A^1 -> H(U1(d, d), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.33 5.60/2.33 (A^1 -> H(d, U1(e, e)),A^1 -> H(d, U1(e, e))) 5.60/2.33 5.60/2.33 5.60/2.33 ---------------------------------------- 5.60/2.33 5.60/2.33 (256) 5.60/2.33 Obligation: 5.60/2.33 Q DP problem: 5.60/2.33 The TRS P consists of the following rules: 5.60/2.33 5.60/2.33 A^1 -> H(f(d), f(d)) 5.60/2.33 H(x, x) -> G(x, x) 5.60/2.33 G(d, e) -> A^1 5.60/2.33 A^1 -> H(f(e), f(e)) 5.60/2.33 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.33 A^1 -> H(f(d), b) 5.60/2.33 A^1 -> H(f(d), U1(d, e)) 5.60/2.33 A^1 -> H(f(d), U1(e, d)) 5.60/2.33 A^1 -> H(U1(e, e), U1(d, b)) 5.60/2.33 A^1 -> H(U1(e, e), U1(e, b)) 5.60/2.33 A^1 -> H(U1(e, e), U1(b, d)) 5.60/2.33 A^1 -> H(U1(e, e), U1(b, e)) 5.60/2.33 A^1 -> H(U1(e, e), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.33 A^1 -> H(f(e), b) 5.60/2.33 A^1 -> H(f(e), U1(d, e)) 5.60/2.33 A^1 -> H(f(e), U1(e, d)) 5.60/2.33 A^1 -> H(f(e), d) 5.60/2.33 A^1 -> H(f(a), e) 5.60/2.33 A^1 -> H(f(d), d) 5.60/2.33 A^1 -> H(e, U1(b, b)) 5.60/2.33 A^1 -> H(e, f(d)) 5.60/2.33 A^1 -> H(e, f(e)) 5.60/2.33 A^1 -> H(a, U1(d, b)) 5.60/2.33 A^1 -> H(a, U1(e, b)) 5.60/2.33 A^1 -> H(a, U1(b, d)) 5.60/2.33 A^1 -> H(a, U1(b, e)) 5.60/2.33 A^1 -> H(a, U1(d, d)) 5.60/2.33 A^1 -> H(a, U1(e, e)) 5.60/2.33 A^1 -> H(U1(d, e), U1(d, b)) 5.60/2.33 A^1 -> H(U1(d, e), U1(e, b)) 5.60/2.33 A^1 -> H(U1(d, e), U1(b, d)) 5.60/2.33 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.33 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.33 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.33 A^1 -> H(U1(d, a), b) 5.60/2.33 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.33 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.33 A^1 -> H(U1(d, a), d) 5.60/2.33 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.33 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.33 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.33 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.33 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, a), b) 5.60/2.33 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.33 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, d), b) 5.60/2.33 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.33 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, d), d) 5.60/2.33 A^1 -> H(U1(a, e), b) 5.60/2.33 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.33 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, e), d) 5.60/2.33 A^1 -> H(U1(a, a), e) 5.60/2.33 A^1 -> H(d, U1(d, b)) 5.60/2.33 A^1 -> H(d, U1(b, d)) 5.60/2.33 A^1 -> H(d, U1(b, e)) 5.60/2.33 A^1 -> H(d, U1(d, d)) 5.60/2.33 A^1 -> H(U1(d, d), b) 5.60/2.33 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.33 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.33 A^1 -> H(d, U1(e, e)) 5.60/2.33 5.60/2.33 The TRS R consists of the following rules: 5.60/2.33 5.60/2.33 a -> d 5.60/2.33 a -> e 5.60/2.33 f(x) -> U1(x, x) 5.60/2.33 b -> d 5.60/2.33 b -> e 5.60/2.33 U1(d, x) -> x 5.60/2.33 5.60/2.33 Q is empty. 5.60/2.33 We have to consider all minimal (P,Q,R)-chains. 5.60/2.33 ---------------------------------------- 5.60/2.33 5.60/2.33 (257) DependencyGraphProof (EQUIVALENT) 5.60/2.33 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.60/2.33 ---------------------------------------- 5.60/2.33 5.60/2.33 (258) 5.60/2.33 Obligation: 5.60/2.33 Q DP problem: 5.60/2.33 The TRS P consists of the following rules: 5.60/2.33 5.60/2.33 H(x, x) -> G(x, x) 5.60/2.33 G(d, e) -> A^1 5.60/2.33 A^1 -> H(f(d), f(d)) 5.60/2.33 A^1 -> H(f(e), f(e)) 5.60/2.33 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.33 A^1 -> H(f(d), b) 5.60/2.33 A^1 -> H(f(d), U1(d, e)) 5.60/2.33 A^1 -> H(f(d), U1(e, d)) 5.60/2.33 A^1 -> H(U1(e, e), U1(d, b)) 5.60/2.33 A^1 -> H(U1(e, e), U1(e, b)) 5.60/2.33 A^1 -> H(U1(e, e), U1(b, d)) 5.60/2.33 A^1 -> H(U1(e, e), U1(b, e)) 5.60/2.33 A^1 -> H(U1(e, e), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.33 A^1 -> H(f(e), b) 5.60/2.33 A^1 -> H(f(e), U1(d, e)) 5.60/2.33 A^1 -> H(f(e), U1(e, d)) 5.60/2.33 A^1 -> H(f(e), d) 5.60/2.33 A^1 -> H(f(a), e) 5.60/2.33 A^1 -> H(f(d), d) 5.60/2.33 A^1 -> H(e, U1(b, b)) 5.60/2.33 A^1 -> H(e, f(d)) 5.60/2.33 A^1 -> H(e, f(e)) 5.60/2.33 A^1 -> H(a, U1(d, b)) 5.60/2.33 A^1 -> H(a, U1(e, b)) 5.60/2.33 A^1 -> H(a, U1(b, d)) 5.60/2.33 A^1 -> H(a, U1(b, e)) 5.60/2.33 A^1 -> H(a, U1(d, d)) 5.60/2.33 A^1 -> H(a, U1(e, e)) 5.60/2.33 A^1 -> H(U1(d, e), U1(d, b)) 5.60/2.33 A^1 -> H(U1(d, e), U1(e, b)) 5.60/2.33 A^1 -> H(U1(d, e), U1(b, d)) 5.60/2.33 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.33 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.33 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.33 A^1 -> H(U1(d, a), b) 5.60/2.33 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.33 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.33 A^1 -> H(U1(d, a), d) 5.60/2.33 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.33 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.33 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.33 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.33 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, a), b) 5.60/2.33 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.33 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, d), b) 5.60/2.33 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.33 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, d), d) 5.60/2.33 A^1 -> H(U1(a, e), b) 5.60/2.33 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.33 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, e), d) 5.60/2.33 A^1 -> H(U1(a, a), e) 5.60/2.33 A^1 -> H(d, U1(d, b)) 5.60/2.33 A^1 -> H(d, U1(b, d)) 5.60/2.33 A^1 -> H(d, U1(b, e)) 5.60/2.33 A^1 -> H(d, U1(d, d)) 5.60/2.33 A^1 -> H(U1(d, d), b) 5.60/2.33 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.33 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.33 5.60/2.33 The TRS R consists of the following rules: 5.60/2.33 5.60/2.33 a -> d 5.60/2.33 a -> e 5.60/2.33 f(x) -> U1(x, x) 5.60/2.33 b -> d 5.60/2.33 b -> e 5.60/2.33 U1(d, x) -> x 5.60/2.33 5.60/2.33 Q is empty. 5.60/2.33 We have to consider all minimal (P,Q,R)-chains. 5.60/2.33 ---------------------------------------- 5.60/2.33 5.60/2.33 (259) TransformationProof (EQUIVALENT) 5.60/2.33 By narrowing [LPAR04] the rule A^1 -> H(f(d), b) at position [] we obtained the following new rules [LPAR04]: 5.60/2.33 5.60/2.33 (A^1 -> H(U1(d, d), b),A^1 -> H(U1(d, d), b)) 5.60/2.33 (A^1 -> H(f(d), d),A^1 -> H(f(d), d)) 5.60/2.33 (A^1 -> H(f(d), e),A^1 -> H(f(d), e)) 5.60/2.33 5.60/2.33 5.60/2.33 ---------------------------------------- 5.60/2.33 5.60/2.33 (260) 5.60/2.33 Obligation: 5.60/2.33 Q DP problem: 5.60/2.33 The TRS P consists of the following rules: 5.60/2.33 5.60/2.33 H(x, x) -> G(x, x) 5.60/2.33 G(d, e) -> A^1 5.60/2.33 A^1 -> H(f(d), f(d)) 5.60/2.33 A^1 -> H(f(e), f(e)) 5.60/2.33 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.33 A^1 -> H(f(d), U1(d, e)) 5.60/2.33 A^1 -> H(f(d), U1(e, d)) 5.60/2.33 A^1 -> H(U1(e, e), U1(d, b)) 5.60/2.33 A^1 -> H(U1(e, e), U1(e, b)) 5.60/2.33 A^1 -> H(U1(e, e), U1(b, d)) 5.60/2.33 A^1 -> H(U1(e, e), U1(b, e)) 5.60/2.33 A^1 -> H(U1(e, e), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.33 A^1 -> H(f(e), b) 5.60/2.33 A^1 -> H(f(e), U1(d, e)) 5.60/2.33 A^1 -> H(f(e), U1(e, d)) 5.60/2.33 A^1 -> H(f(e), d) 5.60/2.33 A^1 -> H(f(a), e) 5.60/2.33 A^1 -> H(f(d), d) 5.60/2.33 A^1 -> H(e, U1(b, b)) 5.60/2.33 A^1 -> H(e, f(d)) 5.60/2.33 A^1 -> H(e, f(e)) 5.60/2.33 A^1 -> H(a, U1(d, b)) 5.60/2.33 A^1 -> H(a, U1(e, b)) 5.60/2.33 A^1 -> H(a, U1(b, d)) 5.60/2.33 A^1 -> H(a, U1(b, e)) 5.60/2.33 A^1 -> H(a, U1(d, d)) 5.60/2.33 A^1 -> H(a, U1(e, e)) 5.60/2.33 A^1 -> H(U1(d, e), U1(d, b)) 5.60/2.33 A^1 -> H(U1(d, e), U1(e, b)) 5.60/2.33 A^1 -> H(U1(d, e), U1(b, d)) 5.60/2.33 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.33 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.33 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.33 A^1 -> H(U1(d, a), b) 5.60/2.33 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.33 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.33 A^1 -> H(U1(d, a), d) 5.60/2.33 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.33 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.33 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.33 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.33 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, a), b) 5.60/2.33 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.33 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, d), b) 5.60/2.33 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.33 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, d), d) 5.60/2.33 A^1 -> H(U1(a, e), b) 5.60/2.33 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.33 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, e), d) 5.60/2.33 A^1 -> H(U1(a, a), e) 5.60/2.33 A^1 -> H(d, U1(d, b)) 5.60/2.33 A^1 -> H(d, U1(b, d)) 5.60/2.33 A^1 -> H(d, U1(b, e)) 5.60/2.33 A^1 -> H(d, U1(d, d)) 5.60/2.33 A^1 -> H(U1(d, d), b) 5.60/2.33 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.33 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.33 A^1 -> H(f(d), e) 5.60/2.33 5.60/2.33 The TRS R consists of the following rules: 5.60/2.33 5.60/2.33 a -> d 5.60/2.33 a -> e 5.60/2.33 f(x) -> U1(x, x) 5.60/2.33 b -> d 5.60/2.33 b -> e 5.60/2.33 U1(d, x) -> x 5.60/2.33 5.60/2.33 Q is empty. 5.60/2.33 We have to consider all minimal (P,Q,R)-chains. 5.60/2.33 ---------------------------------------- 5.60/2.33 5.60/2.33 (261) TransformationProof (EQUIVALENT) 5.60/2.33 By narrowing [LPAR04] the rule A^1 -> H(f(d), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.33 5.60/2.33 (A^1 -> H(U1(d, d), U1(d, e)),A^1 -> H(U1(d, d), U1(d, e))) 5.60/2.33 (A^1 -> H(f(d), e),A^1 -> H(f(d), e)) 5.60/2.33 5.60/2.33 5.60/2.33 ---------------------------------------- 5.60/2.33 5.60/2.33 (262) 5.60/2.33 Obligation: 5.60/2.33 Q DP problem: 5.60/2.33 The TRS P consists of the following rules: 5.60/2.33 5.60/2.33 H(x, x) -> G(x, x) 5.60/2.33 G(d, e) -> A^1 5.60/2.33 A^1 -> H(f(d), f(d)) 5.60/2.33 A^1 -> H(f(e), f(e)) 5.60/2.33 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.33 A^1 -> H(f(d), U1(e, d)) 5.60/2.33 A^1 -> H(U1(e, e), U1(d, b)) 5.60/2.33 A^1 -> H(U1(e, e), U1(e, b)) 5.60/2.33 A^1 -> H(U1(e, e), U1(b, d)) 5.60/2.33 A^1 -> H(U1(e, e), U1(b, e)) 5.60/2.33 A^1 -> H(U1(e, e), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.33 A^1 -> H(f(e), b) 5.60/2.33 A^1 -> H(f(e), U1(d, e)) 5.60/2.33 A^1 -> H(f(e), U1(e, d)) 5.60/2.33 A^1 -> H(f(e), d) 5.60/2.33 A^1 -> H(f(a), e) 5.60/2.33 A^1 -> H(f(d), d) 5.60/2.33 A^1 -> H(e, U1(b, b)) 5.60/2.33 A^1 -> H(e, f(d)) 5.60/2.33 A^1 -> H(e, f(e)) 5.60/2.33 A^1 -> H(a, U1(d, b)) 5.60/2.33 A^1 -> H(a, U1(e, b)) 5.60/2.33 A^1 -> H(a, U1(b, d)) 5.60/2.33 A^1 -> H(a, U1(b, e)) 5.60/2.33 A^1 -> H(a, U1(d, d)) 5.60/2.33 A^1 -> H(a, U1(e, e)) 5.60/2.33 A^1 -> H(U1(d, e), U1(d, b)) 5.60/2.33 A^1 -> H(U1(d, e), U1(e, b)) 5.60/2.33 A^1 -> H(U1(d, e), U1(b, d)) 5.60/2.33 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.33 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.33 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.33 A^1 -> H(U1(d, a), b) 5.60/2.33 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.33 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.33 A^1 -> H(U1(d, a), d) 5.60/2.33 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.33 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.33 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.33 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.33 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, a), b) 5.60/2.33 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.33 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, d), b) 5.60/2.33 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.33 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, d), d) 5.60/2.33 A^1 -> H(U1(a, e), b) 5.60/2.33 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.33 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, e), d) 5.60/2.33 A^1 -> H(U1(a, a), e) 5.60/2.33 A^1 -> H(d, U1(d, b)) 5.60/2.33 A^1 -> H(d, U1(b, d)) 5.60/2.33 A^1 -> H(d, U1(b, e)) 5.60/2.33 A^1 -> H(d, U1(d, d)) 5.60/2.33 A^1 -> H(U1(d, d), b) 5.60/2.33 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.33 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.33 A^1 -> H(f(d), e) 5.60/2.33 5.60/2.33 The TRS R consists of the following rules: 5.60/2.33 5.60/2.33 a -> d 5.60/2.33 a -> e 5.60/2.33 f(x) -> U1(x, x) 5.60/2.33 b -> d 5.60/2.33 b -> e 5.60/2.33 U1(d, x) -> x 5.60/2.33 5.60/2.33 Q is empty. 5.60/2.33 We have to consider all minimal (P,Q,R)-chains. 5.60/2.33 ---------------------------------------- 5.60/2.33 5.60/2.33 (263) TransformationProof (EQUIVALENT) 5.60/2.33 By narrowing [LPAR04] the rule A^1 -> H(f(d), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.33 5.60/2.33 (A^1 -> H(U1(d, d), U1(e, d)),A^1 -> H(U1(d, d), U1(e, d))) 5.60/2.33 5.60/2.33 5.60/2.33 ---------------------------------------- 5.60/2.33 5.60/2.33 (264) 5.60/2.33 Obligation: 5.60/2.33 Q DP problem: 5.60/2.33 The TRS P consists of the following rules: 5.60/2.33 5.60/2.33 H(x, x) -> G(x, x) 5.60/2.33 G(d, e) -> A^1 5.60/2.33 A^1 -> H(f(d), f(d)) 5.60/2.33 A^1 -> H(f(e), f(e)) 5.60/2.33 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, e), U1(d, b)) 5.60/2.33 A^1 -> H(U1(e, e), U1(e, b)) 5.60/2.33 A^1 -> H(U1(e, e), U1(b, d)) 5.60/2.33 A^1 -> H(U1(e, e), U1(b, e)) 5.60/2.33 A^1 -> H(U1(e, e), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.33 A^1 -> H(f(e), b) 5.60/2.33 A^1 -> H(f(e), U1(d, e)) 5.60/2.33 A^1 -> H(f(e), U1(e, d)) 5.60/2.33 A^1 -> H(f(e), d) 5.60/2.33 A^1 -> H(f(a), e) 5.60/2.33 A^1 -> H(f(d), d) 5.60/2.33 A^1 -> H(e, U1(b, b)) 5.60/2.33 A^1 -> H(e, f(d)) 5.60/2.33 A^1 -> H(e, f(e)) 5.60/2.33 A^1 -> H(a, U1(d, b)) 5.60/2.33 A^1 -> H(a, U1(e, b)) 5.60/2.33 A^1 -> H(a, U1(b, d)) 5.60/2.33 A^1 -> H(a, U1(b, e)) 5.60/2.33 A^1 -> H(a, U1(d, d)) 5.60/2.33 A^1 -> H(a, U1(e, e)) 5.60/2.33 A^1 -> H(U1(d, e), U1(d, b)) 5.60/2.33 A^1 -> H(U1(d, e), U1(e, b)) 5.60/2.33 A^1 -> H(U1(d, e), U1(b, d)) 5.60/2.33 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.33 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.33 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.33 A^1 -> H(U1(d, a), b) 5.60/2.33 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.33 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.33 A^1 -> H(U1(d, a), d) 5.60/2.33 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.33 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.33 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.33 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.33 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, a), b) 5.60/2.33 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.33 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, d), b) 5.60/2.33 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.33 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, d), d) 5.60/2.33 A^1 -> H(U1(a, e), b) 5.60/2.33 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.33 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, e), d) 5.60/2.33 A^1 -> H(U1(a, a), e) 5.60/2.33 A^1 -> H(d, U1(d, b)) 5.60/2.33 A^1 -> H(d, U1(b, d)) 5.60/2.33 A^1 -> H(d, U1(b, e)) 5.60/2.33 A^1 -> H(d, U1(d, d)) 5.60/2.33 A^1 -> H(U1(d, d), b) 5.60/2.33 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.33 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.33 A^1 -> H(f(d), e) 5.60/2.33 5.60/2.33 The TRS R consists of the following rules: 5.60/2.33 5.60/2.33 a -> d 5.60/2.33 a -> e 5.60/2.33 f(x) -> U1(x, x) 5.60/2.33 b -> d 5.60/2.33 b -> e 5.60/2.33 U1(d, x) -> x 5.60/2.33 5.60/2.33 Q is empty. 5.60/2.33 We have to consider all minimal (P,Q,R)-chains. 5.60/2.33 ---------------------------------------- 5.60/2.33 5.60/2.33 (265) TransformationProof (EQUIVALENT) 5.60/2.33 By narrowing [LPAR04] the rule A^1 -> H(U1(e, e), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.33 5.60/2.33 (A^1 -> H(U1(e, e), b),A^1 -> H(U1(e, e), b)) 5.60/2.33 (A^1 -> H(U1(e, e), U1(d, d)),A^1 -> H(U1(e, e), U1(d, d))) 5.60/2.33 (A^1 -> H(U1(e, e), U1(d, e)),A^1 -> H(U1(e, e), U1(d, e))) 5.60/2.33 5.60/2.33 5.60/2.33 ---------------------------------------- 5.60/2.33 5.60/2.33 (266) 5.60/2.33 Obligation: 5.60/2.33 Q DP problem: 5.60/2.33 The TRS P consists of the following rules: 5.60/2.33 5.60/2.33 H(x, x) -> G(x, x) 5.60/2.33 G(d, e) -> A^1 5.60/2.33 A^1 -> H(f(d), f(d)) 5.60/2.33 A^1 -> H(f(e), f(e)) 5.60/2.33 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, e), U1(e, b)) 5.60/2.33 A^1 -> H(U1(e, e), U1(b, d)) 5.60/2.33 A^1 -> H(U1(e, e), U1(b, e)) 5.60/2.33 A^1 -> H(U1(e, e), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.33 A^1 -> H(f(e), b) 5.60/2.33 A^1 -> H(f(e), U1(d, e)) 5.60/2.33 A^1 -> H(f(e), U1(e, d)) 5.60/2.33 A^1 -> H(f(e), d) 5.60/2.33 A^1 -> H(f(a), e) 5.60/2.33 A^1 -> H(f(d), d) 5.60/2.33 A^1 -> H(e, U1(b, b)) 5.60/2.33 A^1 -> H(e, f(d)) 5.60/2.33 A^1 -> H(e, f(e)) 5.60/2.33 A^1 -> H(a, U1(d, b)) 5.60/2.33 A^1 -> H(a, U1(e, b)) 5.60/2.33 A^1 -> H(a, U1(b, d)) 5.60/2.33 A^1 -> H(a, U1(b, e)) 5.60/2.33 A^1 -> H(a, U1(d, d)) 5.60/2.33 A^1 -> H(a, U1(e, e)) 5.60/2.33 A^1 -> H(U1(d, e), U1(d, b)) 5.60/2.33 A^1 -> H(U1(d, e), U1(e, b)) 5.60/2.33 A^1 -> H(U1(d, e), U1(b, d)) 5.60/2.33 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.33 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.33 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.33 A^1 -> H(U1(d, a), b) 5.60/2.33 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.33 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.33 A^1 -> H(U1(d, a), d) 5.60/2.33 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.33 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.33 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.33 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.33 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, a), b) 5.60/2.33 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.33 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, d), b) 5.60/2.33 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.33 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, d), d) 5.60/2.33 A^1 -> H(U1(a, e), b) 5.60/2.33 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.33 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, e), d) 5.60/2.33 A^1 -> H(U1(a, a), e) 5.60/2.33 A^1 -> H(d, U1(d, b)) 5.60/2.33 A^1 -> H(d, U1(b, d)) 5.60/2.33 A^1 -> H(d, U1(b, e)) 5.60/2.33 A^1 -> H(d, U1(d, d)) 5.60/2.33 A^1 -> H(U1(d, d), b) 5.60/2.33 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.33 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.33 A^1 -> H(f(d), e) 5.60/2.33 A^1 -> H(U1(e, e), b) 5.60/2.33 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.33 5.60/2.33 The TRS R consists of the following rules: 5.60/2.33 5.60/2.33 a -> d 5.60/2.33 a -> e 5.60/2.33 f(x) -> U1(x, x) 5.60/2.33 b -> d 5.60/2.33 b -> e 5.60/2.33 U1(d, x) -> x 5.60/2.33 5.60/2.33 Q is empty. 5.60/2.33 We have to consider all minimal (P,Q,R)-chains. 5.60/2.33 ---------------------------------------- 5.60/2.33 5.60/2.33 (267) TransformationProof (EQUIVALENT) 5.60/2.33 By narrowing [LPAR04] the rule A^1 -> H(U1(e, e), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.33 5.60/2.33 (A^1 -> H(U1(e, e), U1(e, d)),A^1 -> H(U1(e, e), U1(e, d))) 5.60/2.33 (A^1 -> H(U1(e, e), U1(e, e)),A^1 -> H(U1(e, e), U1(e, e))) 5.60/2.33 5.60/2.33 5.60/2.33 ---------------------------------------- 5.60/2.33 5.60/2.33 (268) 5.60/2.33 Obligation: 5.60/2.33 Q DP problem: 5.60/2.33 The TRS P consists of the following rules: 5.60/2.33 5.60/2.33 H(x, x) -> G(x, x) 5.60/2.33 G(d, e) -> A^1 5.60/2.33 A^1 -> H(f(d), f(d)) 5.60/2.33 A^1 -> H(f(e), f(e)) 5.60/2.33 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, e), U1(b, d)) 5.60/2.33 A^1 -> H(U1(e, e), U1(b, e)) 5.60/2.33 A^1 -> H(U1(e, e), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.33 A^1 -> H(f(e), b) 5.60/2.33 A^1 -> H(f(e), U1(d, e)) 5.60/2.33 A^1 -> H(f(e), U1(e, d)) 5.60/2.33 A^1 -> H(f(e), d) 5.60/2.33 A^1 -> H(f(a), e) 5.60/2.33 A^1 -> H(f(d), d) 5.60/2.33 A^1 -> H(e, U1(b, b)) 5.60/2.33 A^1 -> H(e, f(d)) 5.60/2.33 A^1 -> H(e, f(e)) 5.60/2.33 A^1 -> H(a, U1(d, b)) 5.60/2.33 A^1 -> H(a, U1(e, b)) 5.60/2.33 A^1 -> H(a, U1(b, d)) 5.60/2.33 A^1 -> H(a, U1(b, e)) 5.60/2.33 A^1 -> H(a, U1(d, d)) 5.60/2.33 A^1 -> H(a, U1(e, e)) 5.60/2.33 A^1 -> H(U1(d, e), U1(d, b)) 5.60/2.33 A^1 -> H(U1(d, e), U1(e, b)) 5.60/2.33 A^1 -> H(U1(d, e), U1(b, d)) 5.60/2.33 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.33 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.33 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.33 A^1 -> H(U1(d, a), b) 5.60/2.33 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.33 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.33 A^1 -> H(U1(d, a), d) 5.60/2.33 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.33 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.33 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.33 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.33 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, a), b) 5.60/2.33 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.33 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, d), b) 5.60/2.33 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.33 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, d), d) 5.60/2.33 A^1 -> H(U1(a, e), b) 5.60/2.33 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.33 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, e), d) 5.60/2.33 A^1 -> H(U1(a, a), e) 5.60/2.33 A^1 -> H(d, U1(d, b)) 5.60/2.33 A^1 -> H(d, U1(b, d)) 5.60/2.33 A^1 -> H(d, U1(b, e)) 5.60/2.33 A^1 -> H(d, U1(d, d)) 5.60/2.33 A^1 -> H(U1(d, d), b) 5.60/2.33 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.33 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.33 A^1 -> H(f(d), e) 5.60/2.33 A^1 -> H(U1(e, e), b) 5.60/2.33 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.33 A^1 -> H(U1(e, e), U1(e, d)) 5.60/2.33 5.60/2.33 The TRS R consists of the following rules: 5.60/2.33 5.60/2.33 a -> d 5.60/2.33 a -> e 5.60/2.33 f(x) -> U1(x, x) 5.60/2.33 b -> d 5.60/2.33 b -> e 5.60/2.33 U1(d, x) -> x 5.60/2.33 5.60/2.33 Q is empty. 5.60/2.33 We have to consider all minimal (P,Q,R)-chains. 5.60/2.33 ---------------------------------------- 5.60/2.33 5.60/2.33 (269) DependencyGraphProof (EQUIVALENT) 5.60/2.33 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.60/2.33 ---------------------------------------- 5.60/2.33 5.60/2.33 (270) 5.60/2.33 Obligation: 5.60/2.33 Q DP problem: 5.60/2.33 The TRS P consists of the following rules: 5.60/2.33 5.60/2.33 G(d, e) -> A^1 5.60/2.33 A^1 -> H(f(d), f(d)) 5.60/2.33 H(x, x) -> G(x, x) 5.60/2.33 A^1 -> H(f(e), f(e)) 5.60/2.33 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, e), U1(b, d)) 5.60/2.33 A^1 -> H(U1(e, e), U1(b, e)) 5.60/2.33 A^1 -> H(U1(e, e), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.33 A^1 -> H(f(e), b) 5.60/2.33 A^1 -> H(f(e), U1(d, e)) 5.60/2.33 A^1 -> H(f(e), U1(e, d)) 5.60/2.33 A^1 -> H(f(e), d) 5.60/2.33 A^1 -> H(f(a), e) 5.60/2.33 A^1 -> H(f(d), d) 5.60/2.33 A^1 -> H(e, U1(b, b)) 5.60/2.33 A^1 -> H(e, f(d)) 5.60/2.33 A^1 -> H(e, f(e)) 5.60/2.33 A^1 -> H(a, U1(d, b)) 5.60/2.33 A^1 -> H(a, U1(e, b)) 5.60/2.33 A^1 -> H(a, U1(b, d)) 5.60/2.33 A^1 -> H(a, U1(b, e)) 5.60/2.33 A^1 -> H(a, U1(d, d)) 5.60/2.33 A^1 -> H(a, U1(e, e)) 5.60/2.33 A^1 -> H(U1(d, e), U1(d, b)) 5.60/2.33 A^1 -> H(U1(d, e), U1(e, b)) 5.60/2.33 A^1 -> H(U1(d, e), U1(b, d)) 5.60/2.33 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.33 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.33 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.33 A^1 -> H(U1(d, a), b) 5.60/2.33 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.33 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.33 A^1 -> H(U1(d, a), d) 5.60/2.33 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.33 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.33 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.33 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.33 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, a), b) 5.60/2.33 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.33 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, d), b) 5.60/2.33 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.33 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, d), d) 5.60/2.33 A^1 -> H(U1(a, e), b) 5.60/2.33 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.33 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, e), d) 5.60/2.33 A^1 -> H(U1(a, a), e) 5.60/2.33 A^1 -> H(d, U1(d, b)) 5.60/2.33 A^1 -> H(d, U1(b, d)) 5.60/2.33 A^1 -> H(d, U1(b, e)) 5.60/2.33 A^1 -> H(d, U1(d, d)) 5.60/2.33 A^1 -> H(U1(d, d), b) 5.60/2.33 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.33 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.33 A^1 -> H(f(d), e) 5.60/2.33 A^1 -> H(U1(e, e), b) 5.60/2.33 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.33 5.60/2.33 The TRS R consists of the following rules: 5.60/2.33 5.60/2.33 a -> d 5.60/2.33 a -> e 5.60/2.33 f(x) -> U1(x, x) 5.60/2.33 b -> d 5.60/2.33 b -> e 5.60/2.33 U1(d, x) -> x 5.60/2.33 5.60/2.33 Q is empty. 5.60/2.33 We have to consider all minimal (P,Q,R)-chains. 5.60/2.33 ---------------------------------------- 5.60/2.33 5.60/2.33 (271) TransformationProof (EQUIVALENT) 5.60/2.33 By narrowing [LPAR04] the rule A^1 -> H(U1(e, e), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.33 5.60/2.33 (A^1 -> H(U1(e, e), U1(d, d)),A^1 -> H(U1(e, e), U1(d, d))) 5.60/2.33 (A^1 -> H(U1(e, e), U1(e, d)),A^1 -> H(U1(e, e), U1(e, d))) 5.60/2.33 5.60/2.33 5.60/2.33 ---------------------------------------- 5.60/2.33 5.60/2.33 (272) 5.60/2.33 Obligation: 5.60/2.33 Q DP problem: 5.60/2.33 The TRS P consists of the following rules: 5.60/2.33 5.60/2.33 G(d, e) -> A^1 5.60/2.33 A^1 -> H(f(d), f(d)) 5.60/2.33 H(x, x) -> G(x, x) 5.60/2.33 A^1 -> H(f(e), f(e)) 5.60/2.33 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, e), U1(b, e)) 5.60/2.33 A^1 -> H(U1(e, e), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.33 A^1 -> H(f(e), b) 5.60/2.33 A^1 -> H(f(e), U1(d, e)) 5.60/2.33 A^1 -> H(f(e), U1(e, d)) 5.60/2.33 A^1 -> H(f(e), d) 5.60/2.33 A^1 -> H(f(a), e) 5.60/2.33 A^1 -> H(f(d), d) 5.60/2.33 A^1 -> H(e, U1(b, b)) 5.60/2.33 A^1 -> H(e, f(d)) 5.60/2.33 A^1 -> H(e, f(e)) 5.60/2.33 A^1 -> H(a, U1(d, b)) 5.60/2.33 A^1 -> H(a, U1(e, b)) 5.60/2.33 A^1 -> H(a, U1(b, d)) 5.60/2.33 A^1 -> H(a, U1(b, e)) 5.60/2.33 A^1 -> H(a, U1(d, d)) 5.60/2.33 A^1 -> H(a, U1(e, e)) 5.60/2.33 A^1 -> H(U1(d, e), U1(d, b)) 5.60/2.33 A^1 -> H(U1(d, e), U1(e, b)) 5.60/2.33 A^1 -> H(U1(d, e), U1(b, d)) 5.60/2.33 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.33 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.33 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.33 A^1 -> H(U1(d, a), b) 5.60/2.33 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.33 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.33 A^1 -> H(U1(d, a), d) 5.60/2.33 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.33 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.33 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.33 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.33 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, a), b) 5.60/2.33 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.33 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, d), b) 5.60/2.33 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.33 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, d), d) 5.60/2.33 A^1 -> H(U1(a, e), b) 5.60/2.33 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.33 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, e), d) 5.60/2.33 A^1 -> H(U1(a, a), e) 5.60/2.33 A^1 -> H(d, U1(d, b)) 5.60/2.33 A^1 -> H(d, U1(b, d)) 5.60/2.33 A^1 -> H(d, U1(b, e)) 5.60/2.33 A^1 -> H(d, U1(d, d)) 5.60/2.33 A^1 -> H(U1(d, d), b) 5.60/2.33 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.33 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.33 A^1 -> H(f(d), e) 5.60/2.33 A^1 -> H(U1(e, e), b) 5.60/2.33 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.33 A^1 -> H(U1(e, e), U1(e, d)) 5.60/2.33 5.60/2.33 The TRS R consists of the following rules: 5.60/2.33 5.60/2.33 a -> d 5.60/2.33 a -> e 5.60/2.33 f(x) -> U1(x, x) 5.60/2.33 b -> d 5.60/2.33 b -> e 5.60/2.33 U1(d, x) -> x 5.60/2.33 5.60/2.33 Q is empty. 5.60/2.33 We have to consider all minimal (P,Q,R)-chains. 5.60/2.33 ---------------------------------------- 5.60/2.33 5.60/2.33 (273) DependencyGraphProof (EQUIVALENT) 5.60/2.33 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.60/2.33 ---------------------------------------- 5.60/2.33 5.60/2.33 (274) 5.60/2.33 Obligation: 5.60/2.33 Q DP problem: 5.60/2.33 The TRS P consists of the following rules: 5.60/2.33 5.60/2.33 A^1 -> H(f(d), f(d)) 5.60/2.33 H(x, x) -> G(x, x) 5.60/2.33 G(d, e) -> A^1 5.60/2.33 A^1 -> H(f(e), f(e)) 5.60/2.33 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, e), U1(b, e)) 5.60/2.33 A^1 -> H(U1(e, e), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.33 A^1 -> H(f(e), b) 5.60/2.33 A^1 -> H(f(e), U1(d, e)) 5.60/2.33 A^1 -> H(f(e), U1(e, d)) 5.60/2.33 A^1 -> H(f(e), d) 5.60/2.33 A^1 -> H(f(a), e) 5.60/2.33 A^1 -> H(f(d), d) 5.60/2.33 A^1 -> H(e, U1(b, b)) 5.60/2.33 A^1 -> H(e, f(d)) 5.60/2.33 A^1 -> H(e, f(e)) 5.60/2.33 A^1 -> H(a, U1(d, b)) 5.60/2.33 A^1 -> H(a, U1(e, b)) 5.60/2.33 A^1 -> H(a, U1(b, d)) 5.60/2.33 A^1 -> H(a, U1(b, e)) 5.60/2.33 A^1 -> H(a, U1(d, d)) 5.60/2.33 A^1 -> H(a, U1(e, e)) 5.60/2.33 A^1 -> H(U1(d, e), U1(d, b)) 5.60/2.33 A^1 -> H(U1(d, e), U1(e, b)) 5.60/2.33 A^1 -> H(U1(d, e), U1(b, d)) 5.60/2.33 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.33 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.33 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.33 A^1 -> H(U1(d, a), b) 5.60/2.33 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.33 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.33 A^1 -> H(U1(d, a), d) 5.60/2.33 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.33 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.33 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.33 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.33 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, a), b) 5.60/2.33 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.33 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, d), b) 5.60/2.33 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.33 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, d), d) 5.60/2.33 A^1 -> H(U1(a, e), b) 5.60/2.33 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.33 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, e), d) 5.60/2.33 A^1 -> H(U1(a, a), e) 5.60/2.33 A^1 -> H(d, U1(d, b)) 5.60/2.33 A^1 -> H(d, U1(b, d)) 5.60/2.33 A^1 -> H(d, U1(b, e)) 5.60/2.33 A^1 -> H(d, U1(d, d)) 5.60/2.33 A^1 -> H(U1(d, d), b) 5.60/2.33 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.33 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.33 A^1 -> H(f(d), e) 5.60/2.33 A^1 -> H(U1(e, e), b) 5.60/2.33 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.33 5.60/2.33 The TRS R consists of the following rules: 5.60/2.33 5.60/2.33 a -> d 5.60/2.33 a -> e 5.60/2.33 f(x) -> U1(x, x) 5.60/2.33 b -> d 5.60/2.33 b -> e 5.60/2.33 U1(d, x) -> x 5.60/2.33 5.60/2.33 Q is empty. 5.60/2.33 We have to consider all minimal (P,Q,R)-chains. 5.60/2.33 ---------------------------------------- 5.60/2.33 5.60/2.33 (275) TransformationProof (EQUIVALENT) 5.60/2.33 By narrowing [LPAR04] the rule A^1 -> H(U1(e, e), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.33 5.60/2.33 (A^1 -> H(U1(e, e), U1(d, e)),A^1 -> H(U1(e, e), U1(d, e))) 5.60/2.33 (A^1 -> H(U1(e, e), U1(e, e)),A^1 -> H(U1(e, e), U1(e, e))) 5.60/2.33 5.60/2.33 5.60/2.33 ---------------------------------------- 5.60/2.33 5.60/2.33 (276) 5.60/2.33 Obligation: 5.60/2.33 Q DP problem: 5.60/2.33 The TRS P consists of the following rules: 5.60/2.33 5.60/2.33 A^1 -> H(f(d), f(d)) 5.60/2.33 H(x, x) -> G(x, x) 5.60/2.33 G(d, e) -> A^1 5.60/2.33 A^1 -> H(f(e), f(e)) 5.60/2.33 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, e), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.33 A^1 -> H(f(e), b) 5.60/2.33 A^1 -> H(f(e), U1(d, e)) 5.60/2.33 A^1 -> H(f(e), U1(e, d)) 5.60/2.33 A^1 -> H(f(e), d) 5.60/2.33 A^1 -> H(f(a), e) 5.60/2.33 A^1 -> H(f(d), d) 5.60/2.33 A^1 -> H(e, U1(b, b)) 5.60/2.33 A^1 -> H(e, f(d)) 5.60/2.33 A^1 -> H(e, f(e)) 5.60/2.33 A^1 -> H(a, U1(d, b)) 5.60/2.33 A^1 -> H(a, U1(e, b)) 5.60/2.33 A^1 -> H(a, U1(b, d)) 5.60/2.33 A^1 -> H(a, U1(b, e)) 5.60/2.33 A^1 -> H(a, U1(d, d)) 5.60/2.33 A^1 -> H(a, U1(e, e)) 5.60/2.33 A^1 -> H(U1(d, e), U1(d, b)) 5.60/2.33 A^1 -> H(U1(d, e), U1(e, b)) 5.60/2.33 A^1 -> H(U1(d, e), U1(b, d)) 5.60/2.33 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.33 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.33 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.33 A^1 -> H(U1(d, a), b) 5.60/2.33 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.33 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.33 A^1 -> H(U1(d, a), d) 5.60/2.33 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.33 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.33 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.33 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.33 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, a), b) 5.60/2.33 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.33 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, d), b) 5.60/2.33 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.33 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, d), d) 5.60/2.33 A^1 -> H(U1(a, e), b) 5.60/2.33 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.33 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, e), d) 5.60/2.33 A^1 -> H(U1(a, a), e) 5.60/2.33 A^1 -> H(d, U1(d, b)) 5.60/2.33 A^1 -> H(d, U1(b, d)) 5.60/2.33 A^1 -> H(d, U1(b, e)) 5.60/2.33 A^1 -> H(d, U1(d, d)) 5.60/2.33 A^1 -> H(U1(d, d), b) 5.60/2.33 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.33 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.33 A^1 -> H(f(d), e) 5.60/2.33 A^1 -> H(U1(e, e), b) 5.60/2.33 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.33 5.60/2.33 The TRS R consists of the following rules: 5.60/2.33 5.60/2.33 a -> d 5.60/2.33 a -> e 5.60/2.33 f(x) -> U1(x, x) 5.60/2.33 b -> d 5.60/2.33 b -> e 5.60/2.33 U1(d, x) -> x 5.60/2.33 5.60/2.33 Q is empty. 5.60/2.33 We have to consider all minimal (P,Q,R)-chains. 5.60/2.33 ---------------------------------------- 5.60/2.33 5.60/2.33 (277) TransformationProof (EQUIVALENT) 5.60/2.33 By narrowing [LPAR04] the rule A^1 -> H(U1(e, e), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.33 5.60/2.33 (A^1 -> H(U1(e, e), d),A^1 -> H(U1(e, e), d)) 5.60/2.33 5.60/2.33 5.60/2.33 ---------------------------------------- 5.60/2.33 5.60/2.33 (278) 5.60/2.33 Obligation: 5.60/2.33 Q DP problem: 5.60/2.33 The TRS P consists of the following rules: 5.60/2.33 5.60/2.33 A^1 -> H(f(d), f(d)) 5.60/2.33 H(x, x) -> G(x, x) 5.60/2.33 G(d, e) -> A^1 5.60/2.33 A^1 -> H(f(e), f(e)) 5.60/2.33 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.33 A^1 -> H(f(e), b) 5.60/2.33 A^1 -> H(f(e), U1(d, e)) 5.60/2.33 A^1 -> H(f(e), U1(e, d)) 5.60/2.33 A^1 -> H(f(e), d) 5.60/2.33 A^1 -> H(f(a), e) 5.60/2.33 A^1 -> H(f(d), d) 5.60/2.33 A^1 -> H(e, U1(b, b)) 5.60/2.33 A^1 -> H(e, f(d)) 5.60/2.33 A^1 -> H(e, f(e)) 5.60/2.33 A^1 -> H(a, U1(d, b)) 5.60/2.33 A^1 -> H(a, U1(e, b)) 5.60/2.33 A^1 -> H(a, U1(b, d)) 5.60/2.33 A^1 -> H(a, U1(b, e)) 5.60/2.33 A^1 -> H(a, U1(d, d)) 5.60/2.33 A^1 -> H(a, U1(e, e)) 5.60/2.33 A^1 -> H(U1(d, e), U1(d, b)) 5.60/2.33 A^1 -> H(U1(d, e), U1(e, b)) 5.60/2.33 A^1 -> H(U1(d, e), U1(b, d)) 5.60/2.33 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.33 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.33 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.33 A^1 -> H(U1(d, a), b) 5.60/2.33 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.33 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.33 A^1 -> H(U1(d, a), d) 5.60/2.33 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.33 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.33 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.33 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.33 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, a), b) 5.60/2.33 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.33 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, d), b) 5.60/2.33 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.33 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, d), d) 5.60/2.33 A^1 -> H(U1(a, e), b) 5.60/2.33 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.33 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, e), d) 5.60/2.33 A^1 -> H(U1(a, a), e) 5.60/2.33 A^1 -> H(d, U1(d, b)) 5.60/2.33 A^1 -> H(d, U1(b, d)) 5.60/2.33 A^1 -> H(d, U1(b, e)) 5.60/2.33 A^1 -> H(d, U1(d, d)) 5.60/2.33 A^1 -> H(U1(d, d), b) 5.60/2.33 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.33 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.33 A^1 -> H(f(d), e) 5.60/2.33 A^1 -> H(U1(e, e), b) 5.60/2.33 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.33 A^1 -> H(U1(e, e), d) 5.60/2.33 5.60/2.33 The TRS R consists of the following rules: 5.60/2.33 5.60/2.33 a -> d 5.60/2.33 a -> e 5.60/2.33 f(x) -> U1(x, x) 5.60/2.33 b -> d 5.60/2.33 b -> e 5.60/2.33 U1(d, x) -> x 5.60/2.33 5.60/2.33 Q is empty. 5.60/2.33 We have to consider all minimal (P,Q,R)-chains. 5.60/2.33 ---------------------------------------- 5.60/2.33 5.60/2.33 (279) DependencyGraphProof (EQUIVALENT) 5.60/2.33 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.60/2.33 ---------------------------------------- 5.60/2.33 5.60/2.33 (280) 5.60/2.33 Obligation: 5.60/2.33 Q DP problem: 5.60/2.33 The TRS P consists of the following rules: 5.60/2.33 5.60/2.33 H(x, x) -> G(x, x) 5.60/2.33 G(d, e) -> A^1 5.60/2.33 A^1 -> H(f(d), f(d)) 5.60/2.33 A^1 -> H(f(e), f(e)) 5.60/2.33 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.33 A^1 -> H(f(e), b) 5.60/2.33 A^1 -> H(f(e), U1(d, e)) 5.60/2.33 A^1 -> H(f(e), U1(e, d)) 5.60/2.33 A^1 -> H(f(e), d) 5.60/2.33 A^1 -> H(f(a), e) 5.60/2.33 A^1 -> H(f(d), d) 5.60/2.33 A^1 -> H(e, U1(b, b)) 5.60/2.33 A^1 -> H(e, f(d)) 5.60/2.33 A^1 -> H(e, f(e)) 5.60/2.33 A^1 -> H(a, U1(d, b)) 5.60/2.33 A^1 -> H(a, U1(e, b)) 5.60/2.33 A^1 -> H(a, U1(b, d)) 5.60/2.33 A^1 -> H(a, U1(b, e)) 5.60/2.33 A^1 -> H(a, U1(d, d)) 5.60/2.33 A^1 -> H(a, U1(e, e)) 5.60/2.33 A^1 -> H(U1(d, e), U1(d, b)) 5.60/2.33 A^1 -> H(U1(d, e), U1(e, b)) 5.60/2.33 A^1 -> H(U1(d, e), U1(b, d)) 5.60/2.33 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.33 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.33 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.33 A^1 -> H(U1(d, a), b) 5.60/2.33 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.33 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.33 A^1 -> H(U1(d, a), d) 5.60/2.33 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.33 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.33 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.33 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.33 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, a), b) 5.60/2.33 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.33 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, d), b) 5.60/2.33 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.33 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, d), d) 5.60/2.33 A^1 -> H(U1(a, e), b) 5.60/2.33 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.33 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, e), d) 5.60/2.33 A^1 -> H(U1(a, a), e) 5.60/2.33 A^1 -> H(d, U1(d, b)) 5.60/2.33 A^1 -> H(d, U1(b, d)) 5.60/2.33 A^1 -> H(d, U1(b, e)) 5.60/2.33 A^1 -> H(d, U1(d, d)) 5.60/2.33 A^1 -> H(U1(d, d), b) 5.60/2.33 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.33 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.33 A^1 -> H(f(d), e) 5.60/2.33 A^1 -> H(U1(e, e), b) 5.60/2.33 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.33 5.60/2.33 The TRS R consists of the following rules: 5.60/2.33 5.60/2.33 a -> d 5.60/2.33 a -> e 5.60/2.33 f(x) -> U1(x, x) 5.60/2.33 b -> d 5.60/2.33 b -> e 5.60/2.33 U1(d, x) -> x 5.60/2.33 5.60/2.33 Q is empty. 5.60/2.33 We have to consider all minimal (P,Q,R)-chains. 5.60/2.33 ---------------------------------------- 5.60/2.33 5.60/2.33 (281) TransformationProof (EQUIVALENT) 5.60/2.33 By narrowing [LPAR04] the rule A^1 -> H(f(e), b) at position [] we obtained the following new rules [LPAR04]: 5.60/2.33 5.60/2.33 (A^1 -> H(U1(e, e), b),A^1 -> H(U1(e, e), b)) 5.60/2.33 (A^1 -> H(f(e), d),A^1 -> H(f(e), d)) 5.60/2.33 (A^1 -> H(f(e), e),A^1 -> H(f(e), e)) 5.60/2.33 5.60/2.33 5.60/2.33 ---------------------------------------- 5.60/2.33 5.60/2.33 (282) 5.60/2.33 Obligation: 5.60/2.33 Q DP problem: 5.60/2.33 The TRS P consists of the following rules: 5.60/2.33 5.60/2.33 H(x, x) -> G(x, x) 5.60/2.33 G(d, e) -> A^1 5.60/2.33 A^1 -> H(f(d), f(d)) 5.60/2.33 A^1 -> H(f(e), f(e)) 5.60/2.33 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.33 A^1 -> H(f(e), U1(d, e)) 5.60/2.33 A^1 -> H(f(e), U1(e, d)) 5.60/2.33 A^1 -> H(f(e), d) 5.60/2.33 A^1 -> H(f(a), e) 5.60/2.33 A^1 -> H(f(d), d) 5.60/2.33 A^1 -> H(e, U1(b, b)) 5.60/2.33 A^1 -> H(e, f(d)) 5.60/2.33 A^1 -> H(e, f(e)) 5.60/2.33 A^1 -> H(a, U1(d, b)) 5.60/2.33 A^1 -> H(a, U1(e, b)) 5.60/2.33 A^1 -> H(a, U1(b, d)) 5.60/2.33 A^1 -> H(a, U1(b, e)) 5.60/2.33 A^1 -> H(a, U1(d, d)) 5.60/2.33 A^1 -> H(a, U1(e, e)) 5.60/2.33 A^1 -> H(U1(d, e), U1(d, b)) 5.60/2.33 A^1 -> H(U1(d, e), U1(e, b)) 5.60/2.33 A^1 -> H(U1(d, e), U1(b, d)) 5.60/2.33 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.33 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.33 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.33 A^1 -> H(U1(d, a), b) 5.60/2.33 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.33 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.33 A^1 -> H(U1(d, a), d) 5.60/2.33 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.33 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.33 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.33 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.33 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, a), b) 5.60/2.33 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.33 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, d), b) 5.60/2.33 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.33 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, d), d) 5.60/2.33 A^1 -> H(U1(a, e), b) 5.60/2.33 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.33 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, e), d) 5.60/2.33 A^1 -> H(U1(a, a), e) 5.60/2.33 A^1 -> H(d, U1(d, b)) 5.60/2.33 A^1 -> H(d, U1(b, d)) 5.60/2.33 A^1 -> H(d, U1(b, e)) 5.60/2.33 A^1 -> H(d, U1(d, d)) 5.60/2.33 A^1 -> H(U1(d, d), b) 5.60/2.33 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.33 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.33 A^1 -> H(f(d), e) 5.60/2.33 A^1 -> H(U1(e, e), b) 5.60/2.33 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.33 A^1 -> H(f(e), e) 5.60/2.33 5.60/2.33 The TRS R consists of the following rules: 5.60/2.33 5.60/2.33 a -> d 5.60/2.33 a -> e 5.60/2.33 f(x) -> U1(x, x) 5.60/2.33 b -> d 5.60/2.33 b -> e 5.60/2.33 U1(d, x) -> x 5.60/2.33 5.60/2.33 Q is empty. 5.60/2.33 We have to consider all minimal (P,Q,R)-chains. 5.60/2.33 ---------------------------------------- 5.60/2.33 5.60/2.33 (283) TransformationProof (EQUIVALENT) 5.60/2.33 By narrowing [LPAR04] the rule A^1 -> H(f(e), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.33 5.60/2.33 (A^1 -> H(U1(e, e), U1(d, e)),A^1 -> H(U1(e, e), U1(d, e))) 5.60/2.33 (A^1 -> H(f(e), e),A^1 -> H(f(e), e)) 5.60/2.33 5.60/2.33 5.60/2.33 ---------------------------------------- 5.60/2.33 5.60/2.33 (284) 5.60/2.33 Obligation: 5.60/2.33 Q DP problem: 5.60/2.33 The TRS P consists of the following rules: 5.60/2.33 5.60/2.33 H(x, x) -> G(x, x) 5.60/2.33 G(d, e) -> A^1 5.60/2.33 A^1 -> H(f(d), f(d)) 5.60/2.33 A^1 -> H(f(e), f(e)) 5.60/2.33 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.33 A^1 -> H(f(e), U1(e, d)) 5.60/2.33 A^1 -> H(f(e), d) 5.60/2.33 A^1 -> H(f(a), e) 5.60/2.33 A^1 -> H(f(d), d) 5.60/2.33 A^1 -> H(e, U1(b, b)) 5.60/2.33 A^1 -> H(e, f(d)) 5.60/2.33 A^1 -> H(e, f(e)) 5.60/2.33 A^1 -> H(a, U1(d, b)) 5.60/2.33 A^1 -> H(a, U1(e, b)) 5.60/2.33 A^1 -> H(a, U1(b, d)) 5.60/2.33 A^1 -> H(a, U1(b, e)) 5.60/2.33 A^1 -> H(a, U1(d, d)) 5.60/2.33 A^1 -> H(a, U1(e, e)) 5.60/2.33 A^1 -> H(U1(d, e), U1(d, b)) 5.60/2.33 A^1 -> H(U1(d, e), U1(e, b)) 5.60/2.33 A^1 -> H(U1(d, e), U1(b, d)) 5.60/2.33 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.33 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.33 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.33 A^1 -> H(U1(d, a), b) 5.60/2.33 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.33 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.33 A^1 -> H(U1(d, a), d) 5.60/2.33 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.33 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.33 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.33 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.33 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.33 A^1 -> H(U1(e, a), b) 5.60/2.33 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.33 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, d), b) 5.60/2.33 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.33 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, d), d) 5.60/2.33 A^1 -> H(U1(a, e), b) 5.60/2.33 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.33 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.33 A^1 -> H(U1(a, e), d) 5.60/2.34 A^1 -> H(U1(a, a), e) 5.60/2.34 A^1 -> H(d, U1(d, b)) 5.60/2.34 A^1 -> H(d, U1(b, d)) 5.60/2.34 A^1 -> H(d, U1(b, e)) 5.60/2.34 A^1 -> H(d, U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, d), b) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.34 A^1 -> H(f(d), e) 5.60/2.34 A^1 -> H(U1(e, e), b) 5.60/2.34 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.34 A^1 -> H(f(e), e) 5.60/2.34 5.60/2.34 The TRS R consists of the following rules: 5.60/2.34 5.60/2.34 a -> d 5.60/2.34 a -> e 5.60/2.34 f(x) -> U1(x, x) 5.60/2.34 b -> d 5.60/2.34 b -> e 5.60/2.34 U1(d, x) -> x 5.60/2.34 5.60/2.34 Q is empty. 5.60/2.34 We have to consider all minimal (P,Q,R)-chains. 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (285) TransformationProof (EQUIVALENT) 5.60/2.34 By narrowing [LPAR04] the rule A^1 -> H(f(e), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.34 5.60/2.34 (A^1 -> H(U1(e, e), U1(e, d)),A^1 -> H(U1(e, e), U1(e, d))) 5.60/2.34 5.60/2.34 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (286) 5.60/2.34 Obligation: 5.60/2.34 Q DP problem: 5.60/2.34 The TRS P consists of the following rules: 5.60/2.34 5.60/2.34 H(x, x) -> G(x, x) 5.60/2.34 G(d, e) -> A^1 5.60/2.34 A^1 -> H(f(d), f(d)) 5.60/2.34 A^1 -> H(f(e), f(e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.34 A^1 -> H(f(e), d) 5.60/2.34 A^1 -> H(f(a), e) 5.60/2.34 A^1 -> H(f(d), d) 5.60/2.34 A^1 -> H(e, U1(b, b)) 5.60/2.34 A^1 -> H(e, f(d)) 5.60/2.34 A^1 -> H(e, f(e)) 5.60/2.34 A^1 -> H(a, U1(d, b)) 5.60/2.34 A^1 -> H(a, U1(e, b)) 5.60/2.34 A^1 -> H(a, U1(b, d)) 5.60/2.34 A^1 -> H(a, U1(b, e)) 5.60/2.34 A^1 -> H(a, U1(d, d)) 5.60/2.34 A^1 -> H(a, U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, a), b) 5.60/2.34 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(d, a), d) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, a), b) 5.60/2.34 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), b) 5.60/2.34 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), d) 5.60/2.34 A^1 -> H(U1(a, e), b) 5.60/2.34 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, e), d) 5.60/2.34 A^1 -> H(U1(a, a), e) 5.60/2.34 A^1 -> H(d, U1(d, b)) 5.60/2.34 A^1 -> H(d, U1(b, d)) 5.60/2.34 A^1 -> H(d, U1(b, e)) 5.60/2.34 A^1 -> H(d, U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, d), b) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.34 A^1 -> H(f(d), e) 5.60/2.34 A^1 -> H(U1(e, e), b) 5.60/2.34 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.34 A^1 -> H(f(e), e) 5.60/2.34 A^1 -> H(U1(e, e), U1(e, d)) 5.60/2.34 5.60/2.34 The TRS R consists of the following rules: 5.60/2.34 5.60/2.34 a -> d 5.60/2.34 a -> e 5.60/2.34 f(x) -> U1(x, x) 5.60/2.34 b -> d 5.60/2.34 b -> e 5.60/2.34 U1(d, x) -> x 5.60/2.34 5.60/2.34 Q is empty. 5.60/2.34 We have to consider all minimal (P,Q,R)-chains. 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (287) DependencyGraphProof (EQUIVALENT) 5.60/2.34 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (288) 5.60/2.34 Obligation: 5.60/2.34 Q DP problem: 5.60/2.34 The TRS P consists of the following rules: 5.60/2.34 5.60/2.34 G(d, e) -> A^1 5.60/2.34 A^1 -> H(f(d), f(d)) 5.60/2.34 H(x, x) -> G(x, x) 5.60/2.34 A^1 -> H(f(e), f(e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.34 A^1 -> H(f(e), d) 5.60/2.34 A^1 -> H(f(a), e) 5.60/2.34 A^1 -> H(f(d), d) 5.60/2.34 A^1 -> H(e, U1(b, b)) 5.60/2.34 A^1 -> H(e, f(d)) 5.60/2.34 A^1 -> H(e, f(e)) 5.60/2.34 A^1 -> H(a, U1(d, b)) 5.60/2.34 A^1 -> H(a, U1(e, b)) 5.60/2.34 A^1 -> H(a, U1(b, d)) 5.60/2.34 A^1 -> H(a, U1(b, e)) 5.60/2.34 A^1 -> H(a, U1(d, d)) 5.60/2.34 A^1 -> H(a, U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, a), b) 5.60/2.34 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(d, a), d) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, a), b) 5.60/2.34 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), b) 5.60/2.34 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), d) 5.60/2.34 A^1 -> H(U1(a, e), b) 5.60/2.34 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, e), d) 5.60/2.34 A^1 -> H(U1(a, a), e) 5.60/2.34 A^1 -> H(d, U1(d, b)) 5.60/2.34 A^1 -> H(d, U1(b, d)) 5.60/2.34 A^1 -> H(d, U1(b, e)) 5.60/2.34 A^1 -> H(d, U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, d), b) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.34 A^1 -> H(f(d), e) 5.60/2.34 A^1 -> H(U1(e, e), b) 5.60/2.34 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.34 A^1 -> H(f(e), e) 5.60/2.34 5.60/2.34 The TRS R consists of the following rules: 5.60/2.34 5.60/2.34 a -> d 5.60/2.34 a -> e 5.60/2.34 f(x) -> U1(x, x) 5.60/2.34 b -> d 5.60/2.34 b -> e 5.60/2.34 U1(d, x) -> x 5.60/2.34 5.60/2.34 Q is empty. 5.60/2.34 We have to consider all minimal (P,Q,R)-chains. 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (289) TransformationProof (EQUIVALENT) 5.60/2.34 By narrowing [LPAR04] the rule A^1 -> H(f(e), d) at position [] we obtained the following new rules [LPAR04]: 5.60/2.34 5.60/2.34 (A^1 -> H(U1(e, e), d),A^1 -> H(U1(e, e), d)) 5.60/2.34 5.60/2.34 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (290) 5.60/2.34 Obligation: 5.60/2.34 Q DP problem: 5.60/2.34 The TRS P consists of the following rules: 5.60/2.34 5.60/2.34 G(d, e) -> A^1 5.60/2.34 A^1 -> H(f(d), f(d)) 5.60/2.34 H(x, x) -> G(x, x) 5.60/2.34 A^1 -> H(f(e), f(e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.34 A^1 -> H(f(a), e) 5.60/2.34 A^1 -> H(f(d), d) 5.60/2.34 A^1 -> H(e, U1(b, b)) 5.60/2.34 A^1 -> H(e, f(d)) 5.60/2.34 A^1 -> H(e, f(e)) 5.60/2.34 A^1 -> H(a, U1(d, b)) 5.60/2.34 A^1 -> H(a, U1(e, b)) 5.60/2.34 A^1 -> H(a, U1(b, d)) 5.60/2.34 A^1 -> H(a, U1(b, e)) 5.60/2.34 A^1 -> H(a, U1(d, d)) 5.60/2.34 A^1 -> H(a, U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, a), b) 5.60/2.34 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(d, a), d) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, a), b) 5.60/2.34 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), b) 5.60/2.34 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), d) 5.60/2.34 A^1 -> H(U1(a, e), b) 5.60/2.34 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, e), d) 5.60/2.34 A^1 -> H(U1(a, a), e) 5.60/2.34 A^1 -> H(d, U1(d, b)) 5.60/2.34 A^1 -> H(d, U1(b, d)) 5.60/2.34 A^1 -> H(d, U1(b, e)) 5.60/2.34 A^1 -> H(d, U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, d), b) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.34 A^1 -> H(f(d), e) 5.60/2.34 A^1 -> H(U1(e, e), b) 5.60/2.34 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.34 A^1 -> H(f(e), e) 5.60/2.34 A^1 -> H(U1(e, e), d) 5.60/2.34 5.60/2.34 The TRS R consists of the following rules: 5.60/2.34 5.60/2.34 a -> d 5.60/2.34 a -> e 5.60/2.34 f(x) -> U1(x, x) 5.60/2.34 b -> d 5.60/2.34 b -> e 5.60/2.34 U1(d, x) -> x 5.60/2.34 5.60/2.34 Q is empty. 5.60/2.34 We have to consider all minimal (P,Q,R)-chains. 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (291) DependencyGraphProof (EQUIVALENT) 5.60/2.34 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (292) 5.60/2.34 Obligation: 5.60/2.34 Q DP problem: 5.60/2.34 The TRS P consists of the following rules: 5.60/2.34 5.60/2.34 A^1 -> H(f(d), f(d)) 5.60/2.34 H(x, x) -> G(x, x) 5.60/2.34 G(d, e) -> A^1 5.60/2.34 A^1 -> H(f(e), f(e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.34 A^1 -> H(f(a), e) 5.60/2.34 A^1 -> H(f(d), d) 5.60/2.34 A^1 -> H(e, U1(b, b)) 5.60/2.34 A^1 -> H(e, f(d)) 5.60/2.34 A^1 -> H(e, f(e)) 5.60/2.34 A^1 -> H(a, U1(d, b)) 5.60/2.34 A^1 -> H(a, U1(e, b)) 5.60/2.34 A^1 -> H(a, U1(b, d)) 5.60/2.34 A^1 -> H(a, U1(b, e)) 5.60/2.34 A^1 -> H(a, U1(d, d)) 5.60/2.34 A^1 -> H(a, U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, a), b) 5.60/2.34 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(d, a), d) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, a), b) 5.60/2.34 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), b) 5.60/2.34 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), d) 5.60/2.34 A^1 -> H(U1(a, e), b) 5.60/2.34 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, e), d) 5.60/2.34 A^1 -> H(U1(a, a), e) 5.60/2.34 A^1 -> H(d, U1(d, b)) 5.60/2.34 A^1 -> H(d, U1(b, d)) 5.60/2.34 A^1 -> H(d, U1(b, e)) 5.60/2.34 A^1 -> H(d, U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, d), b) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.34 A^1 -> H(f(d), e) 5.60/2.34 A^1 -> H(U1(e, e), b) 5.60/2.34 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.34 A^1 -> H(f(e), e) 5.60/2.34 5.60/2.34 The TRS R consists of the following rules: 5.60/2.34 5.60/2.34 a -> d 5.60/2.34 a -> e 5.60/2.34 f(x) -> U1(x, x) 5.60/2.34 b -> d 5.60/2.34 b -> e 5.60/2.34 U1(d, x) -> x 5.60/2.34 5.60/2.34 Q is empty. 5.60/2.34 We have to consider all minimal (P,Q,R)-chains. 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (293) TransformationProof (EQUIVALENT) 5.60/2.34 By narrowing [LPAR04] the rule A^1 -> H(f(a), e) at position [] we obtained the following new rules [LPAR04]: 5.60/2.34 5.60/2.34 (A^1 -> H(U1(a, a), e),A^1 -> H(U1(a, a), e)) 5.60/2.34 (A^1 -> H(f(d), e),A^1 -> H(f(d), e)) 5.60/2.34 (A^1 -> H(f(e), e),A^1 -> H(f(e), e)) 5.60/2.34 5.60/2.34 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (294) 5.60/2.34 Obligation: 5.60/2.34 Q DP problem: 5.60/2.34 The TRS P consists of the following rules: 5.60/2.34 5.60/2.34 A^1 -> H(f(d), f(d)) 5.60/2.34 H(x, x) -> G(x, x) 5.60/2.34 G(d, e) -> A^1 5.60/2.34 A^1 -> H(f(e), f(e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.34 A^1 -> H(f(d), d) 5.60/2.34 A^1 -> H(e, U1(b, b)) 5.60/2.34 A^1 -> H(e, f(d)) 5.60/2.34 A^1 -> H(e, f(e)) 5.60/2.34 A^1 -> H(a, U1(d, b)) 5.60/2.34 A^1 -> H(a, U1(e, b)) 5.60/2.34 A^1 -> H(a, U1(b, d)) 5.60/2.34 A^1 -> H(a, U1(b, e)) 5.60/2.34 A^1 -> H(a, U1(d, d)) 5.60/2.34 A^1 -> H(a, U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, a), b) 5.60/2.34 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(d, a), d) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, a), b) 5.60/2.34 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), b) 5.60/2.34 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), d) 5.60/2.34 A^1 -> H(U1(a, e), b) 5.60/2.34 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, e), d) 5.60/2.34 A^1 -> H(U1(a, a), e) 5.60/2.34 A^1 -> H(d, U1(d, b)) 5.60/2.34 A^1 -> H(d, U1(b, d)) 5.60/2.34 A^1 -> H(d, U1(b, e)) 5.60/2.34 A^1 -> H(d, U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, d), b) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.34 A^1 -> H(f(d), e) 5.60/2.34 A^1 -> H(U1(e, e), b) 5.60/2.34 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.34 A^1 -> H(f(e), e) 5.60/2.34 5.60/2.34 The TRS R consists of the following rules: 5.60/2.34 5.60/2.34 a -> d 5.60/2.34 a -> e 5.60/2.34 f(x) -> U1(x, x) 5.60/2.34 b -> d 5.60/2.34 b -> e 5.60/2.34 U1(d, x) -> x 5.60/2.34 5.60/2.34 Q is empty. 5.60/2.34 We have to consider all minimal (P,Q,R)-chains. 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (295) TransformationProof (EQUIVALENT) 5.60/2.34 By narrowing [LPAR04] the rule A^1 -> H(f(d), d) at position [] we obtained the following new rules [LPAR04]: 5.60/2.34 5.60/2.34 (A^1 -> H(U1(d, d), d),A^1 -> H(U1(d, d), d)) 5.60/2.34 5.60/2.34 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (296) 5.60/2.34 Obligation: 5.60/2.34 Q DP problem: 5.60/2.34 The TRS P consists of the following rules: 5.60/2.34 5.60/2.34 A^1 -> H(f(d), f(d)) 5.60/2.34 H(x, x) -> G(x, x) 5.60/2.34 G(d, e) -> A^1 5.60/2.34 A^1 -> H(f(e), f(e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.34 A^1 -> H(e, U1(b, b)) 5.60/2.34 A^1 -> H(e, f(d)) 5.60/2.34 A^1 -> H(e, f(e)) 5.60/2.34 A^1 -> H(a, U1(d, b)) 5.60/2.34 A^1 -> H(a, U1(e, b)) 5.60/2.34 A^1 -> H(a, U1(b, d)) 5.60/2.34 A^1 -> H(a, U1(b, e)) 5.60/2.34 A^1 -> H(a, U1(d, d)) 5.60/2.34 A^1 -> H(a, U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, a), b) 5.60/2.34 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(d, a), d) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, a), b) 5.60/2.34 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), b) 5.60/2.34 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), d) 5.60/2.34 A^1 -> H(U1(a, e), b) 5.60/2.34 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, e), d) 5.60/2.34 A^1 -> H(U1(a, a), e) 5.60/2.34 A^1 -> H(d, U1(d, b)) 5.60/2.34 A^1 -> H(d, U1(b, d)) 5.60/2.34 A^1 -> H(d, U1(b, e)) 5.60/2.34 A^1 -> H(d, U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, d), b) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.34 A^1 -> H(f(d), e) 5.60/2.34 A^1 -> H(U1(e, e), b) 5.60/2.34 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.34 A^1 -> H(f(e), e) 5.60/2.34 A^1 -> H(U1(d, d), d) 5.60/2.34 5.60/2.34 The TRS R consists of the following rules: 5.60/2.34 5.60/2.34 a -> d 5.60/2.34 a -> e 5.60/2.34 f(x) -> U1(x, x) 5.60/2.34 b -> d 5.60/2.34 b -> e 5.60/2.34 U1(d, x) -> x 5.60/2.34 5.60/2.34 Q is empty. 5.60/2.34 We have to consider all minimal (P,Q,R)-chains. 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (297) TransformationProof (EQUIVALENT) 5.60/2.34 By narrowing [LPAR04] the rule A^1 -> H(e, U1(b, b)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.34 5.60/2.34 (A^1 -> H(e, U1(d, b)),A^1 -> H(e, U1(d, b))) 5.60/2.34 (A^1 -> H(e, U1(e, b)),A^1 -> H(e, U1(e, b))) 5.60/2.34 (A^1 -> H(e, U1(b, d)),A^1 -> H(e, U1(b, d))) 5.60/2.34 (A^1 -> H(e, U1(b, e)),A^1 -> H(e, U1(b, e))) 5.60/2.34 5.60/2.34 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (298) 5.60/2.34 Obligation: 5.60/2.34 Q DP problem: 5.60/2.34 The TRS P consists of the following rules: 5.60/2.34 5.60/2.34 A^1 -> H(f(d), f(d)) 5.60/2.34 H(x, x) -> G(x, x) 5.60/2.34 G(d, e) -> A^1 5.60/2.34 A^1 -> H(f(e), f(e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.34 A^1 -> H(e, f(d)) 5.60/2.34 A^1 -> H(e, f(e)) 5.60/2.34 A^1 -> H(a, U1(d, b)) 5.60/2.34 A^1 -> H(a, U1(e, b)) 5.60/2.34 A^1 -> H(a, U1(b, d)) 5.60/2.34 A^1 -> H(a, U1(b, e)) 5.60/2.34 A^1 -> H(a, U1(d, d)) 5.60/2.34 A^1 -> H(a, U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, a), b) 5.60/2.34 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(d, a), d) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, a), b) 5.60/2.34 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), b) 5.60/2.34 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), d) 5.60/2.34 A^1 -> H(U1(a, e), b) 5.60/2.34 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, e), d) 5.60/2.34 A^1 -> H(U1(a, a), e) 5.60/2.34 A^1 -> H(d, U1(d, b)) 5.60/2.34 A^1 -> H(d, U1(b, d)) 5.60/2.34 A^1 -> H(d, U1(b, e)) 5.60/2.34 A^1 -> H(d, U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, d), b) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.34 A^1 -> H(f(d), e) 5.60/2.34 A^1 -> H(U1(e, e), b) 5.60/2.34 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.34 A^1 -> H(f(e), e) 5.60/2.34 A^1 -> H(U1(d, d), d) 5.60/2.34 A^1 -> H(e, U1(d, b)) 5.60/2.34 A^1 -> H(e, U1(e, b)) 5.60/2.34 A^1 -> H(e, U1(b, d)) 5.60/2.34 A^1 -> H(e, U1(b, e)) 5.60/2.34 5.60/2.34 The TRS R consists of the following rules: 5.60/2.34 5.60/2.34 a -> d 5.60/2.34 a -> e 5.60/2.34 f(x) -> U1(x, x) 5.60/2.34 b -> d 5.60/2.34 b -> e 5.60/2.34 U1(d, x) -> x 5.60/2.34 5.60/2.34 Q is empty. 5.60/2.34 We have to consider all minimal (P,Q,R)-chains. 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (299) DependencyGraphProof (EQUIVALENT) 5.60/2.34 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (300) 5.60/2.34 Obligation: 5.60/2.34 Q DP problem: 5.60/2.34 The TRS P consists of the following rules: 5.60/2.34 5.60/2.34 H(x, x) -> G(x, x) 5.60/2.34 G(d, e) -> A^1 5.60/2.34 A^1 -> H(f(d), f(d)) 5.60/2.34 A^1 -> H(f(e), f(e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.34 A^1 -> H(e, f(d)) 5.60/2.34 A^1 -> H(e, f(e)) 5.60/2.34 A^1 -> H(a, U1(d, b)) 5.60/2.34 A^1 -> H(a, U1(e, b)) 5.60/2.34 A^1 -> H(a, U1(b, d)) 5.60/2.34 A^1 -> H(a, U1(b, e)) 5.60/2.34 A^1 -> H(a, U1(d, d)) 5.60/2.34 A^1 -> H(a, U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, a), b) 5.60/2.34 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(d, a), d) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, a), b) 5.60/2.34 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), b) 5.60/2.34 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), d) 5.60/2.34 A^1 -> H(U1(a, e), b) 5.60/2.34 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, e), d) 5.60/2.34 A^1 -> H(U1(a, a), e) 5.60/2.34 A^1 -> H(d, U1(d, b)) 5.60/2.34 A^1 -> H(d, U1(b, d)) 5.60/2.34 A^1 -> H(d, U1(b, e)) 5.60/2.34 A^1 -> H(d, U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, d), b) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.34 A^1 -> H(f(d), e) 5.60/2.34 A^1 -> H(U1(e, e), b) 5.60/2.34 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.34 A^1 -> H(f(e), e) 5.60/2.34 A^1 -> H(U1(d, d), d) 5.60/2.34 A^1 -> H(e, U1(d, b)) 5.60/2.34 A^1 -> H(e, U1(b, d)) 5.60/2.34 A^1 -> H(e, U1(b, e)) 5.60/2.34 5.60/2.34 The TRS R consists of the following rules: 5.60/2.34 5.60/2.34 a -> d 5.60/2.34 a -> e 5.60/2.34 f(x) -> U1(x, x) 5.60/2.34 b -> d 5.60/2.34 b -> e 5.60/2.34 U1(d, x) -> x 5.60/2.34 5.60/2.34 Q is empty. 5.60/2.34 We have to consider all minimal (P,Q,R)-chains. 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (301) TransformationProof (EQUIVALENT) 5.60/2.34 By narrowing [LPAR04] the rule A^1 -> H(e, f(d)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.34 5.60/2.34 (A^1 -> H(e, U1(d, d)),A^1 -> H(e, U1(d, d))) 5.60/2.34 5.60/2.34 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (302) 5.60/2.34 Obligation: 5.60/2.34 Q DP problem: 5.60/2.34 The TRS P consists of the following rules: 5.60/2.34 5.60/2.34 H(x, x) -> G(x, x) 5.60/2.34 G(d, e) -> A^1 5.60/2.34 A^1 -> H(f(d), f(d)) 5.60/2.34 A^1 -> H(f(e), f(e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.34 A^1 -> H(e, f(e)) 5.60/2.34 A^1 -> H(a, U1(d, b)) 5.60/2.34 A^1 -> H(a, U1(e, b)) 5.60/2.34 A^1 -> H(a, U1(b, d)) 5.60/2.34 A^1 -> H(a, U1(b, e)) 5.60/2.34 A^1 -> H(a, U1(d, d)) 5.60/2.34 A^1 -> H(a, U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, a), b) 5.60/2.34 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(d, a), d) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, a), b) 5.60/2.34 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), b) 5.60/2.34 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), d) 5.60/2.34 A^1 -> H(U1(a, e), b) 5.60/2.34 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, e), d) 5.60/2.34 A^1 -> H(U1(a, a), e) 5.60/2.34 A^1 -> H(d, U1(d, b)) 5.60/2.34 A^1 -> H(d, U1(b, d)) 5.60/2.34 A^1 -> H(d, U1(b, e)) 5.60/2.34 A^1 -> H(d, U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, d), b) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.34 A^1 -> H(f(d), e) 5.60/2.34 A^1 -> H(U1(e, e), b) 5.60/2.34 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.34 A^1 -> H(f(e), e) 5.60/2.34 A^1 -> H(U1(d, d), d) 5.60/2.34 A^1 -> H(e, U1(d, b)) 5.60/2.34 A^1 -> H(e, U1(b, d)) 5.60/2.34 A^1 -> H(e, U1(b, e)) 5.60/2.34 A^1 -> H(e, U1(d, d)) 5.60/2.34 5.60/2.34 The TRS R consists of the following rules: 5.60/2.34 5.60/2.34 a -> d 5.60/2.34 a -> e 5.60/2.34 f(x) -> U1(x, x) 5.60/2.34 b -> d 5.60/2.34 b -> e 5.60/2.34 U1(d, x) -> x 5.60/2.34 5.60/2.34 Q is empty. 5.60/2.34 We have to consider all minimal (P,Q,R)-chains. 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (303) TransformationProof (EQUIVALENT) 5.60/2.34 By narrowing [LPAR04] the rule A^1 -> H(e, f(e)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.34 5.60/2.34 (A^1 -> H(e, U1(e, e)),A^1 -> H(e, U1(e, e))) 5.60/2.34 5.60/2.34 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (304) 5.60/2.34 Obligation: 5.60/2.34 Q DP problem: 5.60/2.34 The TRS P consists of the following rules: 5.60/2.34 5.60/2.34 H(x, x) -> G(x, x) 5.60/2.34 G(d, e) -> A^1 5.60/2.34 A^1 -> H(f(d), f(d)) 5.60/2.34 A^1 -> H(f(e), f(e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.34 A^1 -> H(a, U1(d, b)) 5.60/2.34 A^1 -> H(a, U1(e, b)) 5.60/2.34 A^1 -> H(a, U1(b, d)) 5.60/2.34 A^1 -> H(a, U1(b, e)) 5.60/2.34 A^1 -> H(a, U1(d, d)) 5.60/2.34 A^1 -> H(a, U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, a), b) 5.60/2.34 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(d, a), d) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, a), b) 5.60/2.34 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), b) 5.60/2.34 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), d) 5.60/2.34 A^1 -> H(U1(a, e), b) 5.60/2.34 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, e), d) 5.60/2.34 A^1 -> H(U1(a, a), e) 5.60/2.34 A^1 -> H(d, U1(d, b)) 5.60/2.34 A^1 -> H(d, U1(b, d)) 5.60/2.34 A^1 -> H(d, U1(b, e)) 5.60/2.34 A^1 -> H(d, U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, d), b) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.34 A^1 -> H(f(d), e) 5.60/2.34 A^1 -> H(U1(e, e), b) 5.60/2.34 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.34 A^1 -> H(f(e), e) 5.60/2.34 A^1 -> H(U1(d, d), d) 5.60/2.34 A^1 -> H(e, U1(d, b)) 5.60/2.34 A^1 -> H(e, U1(b, d)) 5.60/2.34 A^1 -> H(e, U1(b, e)) 5.60/2.34 A^1 -> H(e, U1(d, d)) 5.60/2.34 A^1 -> H(e, U1(e, e)) 5.60/2.34 5.60/2.34 The TRS R consists of the following rules: 5.60/2.34 5.60/2.34 a -> d 5.60/2.34 a -> e 5.60/2.34 f(x) -> U1(x, x) 5.60/2.34 b -> d 5.60/2.34 b -> e 5.60/2.34 U1(d, x) -> x 5.60/2.34 5.60/2.34 Q is empty. 5.60/2.34 We have to consider all minimal (P,Q,R)-chains. 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (305) DependencyGraphProof (EQUIVALENT) 5.60/2.34 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (306) 5.60/2.34 Obligation: 5.60/2.34 Q DP problem: 5.60/2.34 The TRS P consists of the following rules: 5.60/2.34 5.60/2.34 G(d, e) -> A^1 5.60/2.34 A^1 -> H(f(d), f(d)) 5.60/2.34 H(x, x) -> G(x, x) 5.60/2.34 A^1 -> H(f(e), f(e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.34 A^1 -> H(a, U1(d, b)) 5.60/2.34 A^1 -> H(a, U1(e, b)) 5.60/2.34 A^1 -> H(a, U1(b, d)) 5.60/2.34 A^1 -> H(a, U1(b, e)) 5.60/2.34 A^1 -> H(a, U1(d, d)) 5.60/2.34 A^1 -> H(a, U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, a), b) 5.60/2.34 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(d, a), d) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, a), b) 5.60/2.34 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), b) 5.60/2.34 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), d) 5.60/2.34 A^1 -> H(U1(a, e), b) 5.60/2.34 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, e), d) 5.60/2.34 A^1 -> H(U1(a, a), e) 5.60/2.34 A^1 -> H(d, U1(d, b)) 5.60/2.34 A^1 -> H(d, U1(b, d)) 5.60/2.34 A^1 -> H(d, U1(b, e)) 5.60/2.34 A^1 -> H(d, U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, d), b) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.34 A^1 -> H(f(d), e) 5.60/2.34 A^1 -> H(U1(e, e), b) 5.60/2.34 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.34 A^1 -> H(f(e), e) 5.60/2.34 A^1 -> H(U1(d, d), d) 5.60/2.34 A^1 -> H(e, U1(d, b)) 5.60/2.34 A^1 -> H(e, U1(b, d)) 5.60/2.34 A^1 -> H(e, U1(b, e)) 5.60/2.34 A^1 -> H(e, U1(d, d)) 5.60/2.34 5.60/2.34 The TRS R consists of the following rules: 5.60/2.34 5.60/2.34 a -> d 5.60/2.34 a -> e 5.60/2.34 f(x) -> U1(x, x) 5.60/2.34 b -> d 5.60/2.34 b -> e 5.60/2.34 U1(d, x) -> x 5.60/2.34 5.60/2.34 Q is empty. 5.60/2.34 We have to consider all minimal (P,Q,R)-chains. 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (307) TransformationProof (EQUIVALENT) 5.60/2.34 By narrowing [LPAR04] the rule A^1 -> H(a, U1(d, b)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.34 5.60/2.34 (A^1 -> H(d, U1(d, b)),A^1 -> H(d, U1(d, b))) 5.60/2.34 (A^1 -> H(e, U1(d, b)),A^1 -> H(e, U1(d, b))) 5.60/2.34 (A^1 -> H(a, b),A^1 -> H(a, b)) 5.60/2.34 (A^1 -> H(a, U1(d, d)),A^1 -> H(a, U1(d, d))) 5.60/2.34 (A^1 -> H(a, U1(d, e)),A^1 -> H(a, U1(d, e))) 5.60/2.34 5.60/2.34 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (308) 5.60/2.34 Obligation: 5.60/2.34 Q DP problem: 5.60/2.34 The TRS P consists of the following rules: 5.60/2.34 5.60/2.34 G(d, e) -> A^1 5.60/2.34 A^1 -> H(f(d), f(d)) 5.60/2.34 H(x, x) -> G(x, x) 5.60/2.34 A^1 -> H(f(e), f(e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.34 A^1 -> H(a, U1(e, b)) 5.60/2.34 A^1 -> H(a, U1(b, d)) 5.60/2.34 A^1 -> H(a, U1(b, e)) 5.60/2.34 A^1 -> H(a, U1(d, d)) 5.60/2.34 A^1 -> H(a, U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, a), b) 5.60/2.34 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(d, a), d) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, a), b) 5.60/2.34 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), b) 5.60/2.34 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), d) 5.60/2.34 A^1 -> H(U1(a, e), b) 5.60/2.34 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, e), d) 5.60/2.34 A^1 -> H(U1(a, a), e) 5.60/2.34 A^1 -> H(d, U1(d, b)) 5.60/2.34 A^1 -> H(d, U1(b, d)) 5.60/2.34 A^1 -> H(d, U1(b, e)) 5.60/2.34 A^1 -> H(d, U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, d), b) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.34 A^1 -> H(f(d), e) 5.60/2.34 A^1 -> H(U1(e, e), b) 5.60/2.34 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.34 A^1 -> H(f(e), e) 5.60/2.34 A^1 -> H(U1(d, d), d) 5.60/2.34 A^1 -> H(e, U1(d, b)) 5.60/2.34 A^1 -> H(e, U1(b, d)) 5.60/2.34 A^1 -> H(e, U1(b, e)) 5.60/2.34 A^1 -> H(e, U1(d, d)) 5.60/2.34 A^1 -> H(a, b) 5.60/2.34 A^1 -> H(a, U1(d, e)) 5.60/2.34 5.60/2.34 The TRS R consists of the following rules: 5.60/2.34 5.60/2.34 a -> d 5.60/2.34 a -> e 5.60/2.34 f(x) -> U1(x, x) 5.60/2.34 b -> d 5.60/2.34 b -> e 5.60/2.34 U1(d, x) -> x 5.60/2.34 5.60/2.34 Q is empty. 5.60/2.34 We have to consider all minimal (P,Q,R)-chains. 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (309) TransformationProof (EQUIVALENT) 5.60/2.34 By narrowing [LPAR04] the rule A^1 -> H(a, U1(e, b)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.34 5.60/2.34 (A^1 -> H(d, U1(e, b)),A^1 -> H(d, U1(e, b))) 5.60/2.34 (A^1 -> H(e, U1(e, b)),A^1 -> H(e, U1(e, b))) 5.60/2.34 (A^1 -> H(a, U1(e, d)),A^1 -> H(a, U1(e, d))) 5.60/2.34 (A^1 -> H(a, U1(e, e)),A^1 -> H(a, U1(e, e))) 5.60/2.34 5.60/2.34 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (310) 5.60/2.34 Obligation: 5.60/2.34 Q DP problem: 5.60/2.34 The TRS P consists of the following rules: 5.60/2.34 5.60/2.34 G(d, e) -> A^1 5.60/2.34 A^1 -> H(f(d), f(d)) 5.60/2.34 H(x, x) -> G(x, x) 5.60/2.34 A^1 -> H(f(e), f(e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.34 A^1 -> H(a, U1(b, d)) 5.60/2.34 A^1 -> H(a, U1(b, e)) 5.60/2.34 A^1 -> H(a, U1(d, d)) 5.60/2.34 A^1 -> H(a, U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, a), b) 5.60/2.34 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(d, a), d) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, a), b) 5.60/2.34 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), b) 5.60/2.34 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), d) 5.60/2.34 A^1 -> H(U1(a, e), b) 5.60/2.34 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, e), d) 5.60/2.34 A^1 -> H(U1(a, a), e) 5.60/2.34 A^1 -> H(d, U1(d, b)) 5.60/2.34 A^1 -> H(d, U1(b, d)) 5.60/2.34 A^1 -> H(d, U1(b, e)) 5.60/2.34 A^1 -> H(d, U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, d), b) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.34 A^1 -> H(f(d), e) 5.60/2.34 A^1 -> H(U1(e, e), b) 5.60/2.34 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.34 A^1 -> H(f(e), e) 5.60/2.34 A^1 -> H(U1(d, d), d) 5.60/2.34 A^1 -> H(e, U1(d, b)) 5.60/2.34 A^1 -> H(e, U1(b, d)) 5.60/2.34 A^1 -> H(e, U1(b, e)) 5.60/2.34 A^1 -> H(e, U1(d, d)) 5.60/2.34 A^1 -> H(a, b) 5.60/2.34 A^1 -> H(a, U1(d, e)) 5.60/2.34 A^1 -> H(d, U1(e, b)) 5.60/2.34 A^1 -> H(e, U1(e, b)) 5.60/2.34 A^1 -> H(a, U1(e, d)) 5.60/2.34 5.60/2.34 The TRS R consists of the following rules: 5.60/2.34 5.60/2.34 a -> d 5.60/2.34 a -> e 5.60/2.34 f(x) -> U1(x, x) 5.60/2.34 b -> d 5.60/2.34 b -> e 5.60/2.34 U1(d, x) -> x 5.60/2.34 5.60/2.34 Q is empty. 5.60/2.34 We have to consider all minimal (P,Q,R)-chains. 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (311) DependencyGraphProof (EQUIVALENT) 5.60/2.34 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (312) 5.60/2.34 Obligation: 5.60/2.34 Q DP problem: 5.60/2.34 The TRS P consists of the following rules: 5.60/2.34 5.60/2.34 A^1 -> H(f(d), f(d)) 5.60/2.34 H(x, x) -> G(x, x) 5.60/2.34 G(d, e) -> A^1 5.60/2.34 A^1 -> H(f(e), f(e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.34 A^1 -> H(a, U1(b, d)) 5.60/2.34 A^1 -> H(a, U1(b, e)) 5.60/2.34 A^1 -> H(a, U1(d, d)) 5.60/2.34 A^1 -> H(a, U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, a), b) 5.60/2.34 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(d, a), d) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, a), b) 5.60/2.34 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), b) 5.60/2.34 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), d) 5.60/2.34 A^1 -> H(U1(a, e), b) 5.60/2.34 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, e), d) 5.60/2.34 A^1 -> H(U1(a, a), e) 5.60/2.34 A^1 -> H(d, U1(d, b)) 5.60/2.34 A^1 -> H(d, U1(b, d)) 5.60/2.34 A^1 -> H(d, U1(b, e)) 5.60/2.34 A^1 -> H(d, U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, d), b) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.34 A^1 -> H(f(d), e) 5.60/2.34 A^1 -> H(U1(e, e), b) 5.60/2.34 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.34 A^1 -> H(f(e), e) 5.60/2.34 A^1 -> H(U1(d, d), d) 5.60/2.34 A^1 -> H(e, U1(d, b)) 5.60/2.34 A^1 -> H(e, U1(b, d)) 5.60/2.34 A^1 -> H(e, U1(b, e)) 5.60/2.34 A^1 -> H(e, U1(d, d)) 5.60/2.34 A^1 -> H(a, b) 5.60/2.34 A^1 -> H(a, U1(d, e)) 5.60/2.34 A^1 -> H(a, U1(e, d)) 5.60/2.34 5.60/2.34 The TRS R consists of the following rules: 5.60/2.34 5.60/2.34 a -> d 5.60/2.34 a -> e 5.60/2.34 f(x) -> U1(x, x) 5.60/2.34 b -> d 5.60/2.34 b -> e 5.60/2.34 U1(d, x) -> x 5.60/2.34 5.60/2.34 Q is empty. 5.60/2.34 We have to consider all minimal (P,Q,R)-chains. 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (313) TransformationProof (EQUIVALENT) 5.60/2.34 By narrowing [LPAR04] the rule A^1 -> H(a, U1(b, d)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.34 5.60/2.34 (A^1 -> H(d, U1(b, d)),A^1 -> H(d, U1(b, d))) 5.60/2.34 (A^1 -> H(e, U1(b, d)),A^1 -> H(e, U1(b, d))) 5.60/2.34 (A^1 -> H(a, U1(d, d)),A^1 -> H(a, U1(d, d))) 5.60/2.34 (A^1 -> H(a, U1(e, d)),A^1 -> H(a, U1(e, d))) 5.60/2.34 5.60/2.34 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (314) 5.60/2.34 Obligation: 5.60/2.34 Q DP problem: 5.60/2.34 The TRS P consists of the following rules: 5.60/2.34 5.60/2.34 A^1 -> H(f(d), f(d)) 5.60/2.34 H(x, x) -> G(x, x) 5.60/2.34 G(d, e) -> A^1 5.60/2.34 A^1 -> H(f(e), f(e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.34 A^1 -> H(a, U1(b, e)) 5.60/2.34 A^1 -> H(a, U1(d, d)) 5.60/2.34 A^1 -> H(a, U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, a), b) 5.60/2.34 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(d, a), d) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, a), b) 5.60/2.34 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), b) 5.60/2.34 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), d) 5.60/2.34 A^1 -> H(U1(a, e), b) 5.60/2.34 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, e), d) 5.60/2.34 A^1 -> H(U1(a, a), e) 5.60/2.34 A^1 -> H(d, U1(d, b)) 5.60/2.34 A^1 -> H(d, U1(b, d)) 5.60/2.34 A^1 -> H(d, U1(b, e)) 5.60/2.34 A^1 -> H(d, U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, d), b) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.34 A^1 -> H(f(d), e) 5.60/2.34 A^1 -> H(U1(e, e), b) 5.60/2.34 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.34 A^1 -> H(f(e), e) 5.60/2.34 A^1 -> H(U1(d, d), d) 5.60/2.34 A^1 -> H(e, U1(d, b)) 5.60/2.34 A^1 -> H(e, U1(b, d)) 5.60/2.34 A^1 -> H(e, U1(b, e)) 5.60/2.34 A^1 -> H(e, U1(d, d)) 5.60/2.34 A^1 -> H(a, b) 5.60/2.34 A^1 -> H(a, U1(d, e)) 5.60/2.34 A^1 -> H(a, U1(e, d)) 5.60/2.34 5.60/2.34 The TRS R consists of the following rules: 5.60/2.34 5.60/2.34 a -> d 5.60/2.34 a -> e 5.60/2.34 f(x) -> U1(x, x) 5.60/2.34 b -> d 5.60/2.34 b -> e 5.60/2.34 U1(d, x) -> x 5.60/2.34 5.60/2.34 Q is empty. 5.60/2.34 We have to consider all minimal (P,Q,R)-chains. 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (315) TransformationProof (EQUIVALENT) 5.60/2.34 By narrowing [LPAR04] the rule A^1 -> H(a, U1(b, e)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.34 5.60/2.34 (A^1 -> H(d, U1(b, e)),A^1 -> H(d, U1(b, e))) 5.60/2.34 (A^1 -> H(e, U1(b, e)),A^1 -> H(e, U1(b, e))) 5.60/2.34 (A^1 -> H(a, U1(d, e)),A^1 -> H(a, U1(d, e))) 5.60/2.34 (A^1 -> H(a, U1(e, e)),A^1 -> H(a, U1(e, e))) 5.60/2.34 5.60/2.34 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (316) 5.60/2.34 Obligation: 5.60/2.34 Q DP problem: 5.60/2.34 The TRS P consists of the following rules: 5.60/2.34 5.60/2.34 A^1 -> H(f(d), f(d)) 5.60/2.34 H(x, x) -> G(x, x) 5.60/2.34 G(d, e) -> A^1 5.60/2.34 A^1 -> H(f(e), f(e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.34 A^1 -> H(a, U1(d, d)) 5.60/2.34 A^1 -> H(a, U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, a), b) 5.60/2.34 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(d, a), d) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, a), b) 5.60/2.34 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), b) 5.60/2.34 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), d) 5.60/2.34 A^1 -> H(U1(a, e), b) 5.60/2.34 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, e), d) 5.60/2.34 A^1 -> H(U1(a, a), e) 5.60/2.34 A^1 -> H(d, U1(d, b)) 5.60/2.34 A^1 -> H(d, U1(b, d)) 5.60/2.34 A^1 -> H(d, U1(b, e)) 5.60/2.34 A^1 -> H(d, U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, d), b) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.34 A^1 -> H(f(d), e) 5.60/2.34 A^1 -> H(U1(e, e), b) 5.60/2.34 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.34 A^1 -> H(f(e), e) 5.60/2.34 A^1 -> H(U1(d, d), d) 5.60/2.34 A^1 -> H(e, U1(d, b)) 5.60/2.34 A^1 -> H(e, U1(b, d)) 5.60/2.34 A^1 -> H(e, U1(b, e)) 5.60/2.34 A^1 -> H(e, U1(d, d)) 5.60/2.34 A^1 -> H(a, b) 5.60/2.34 A^1 -> H(a, U1(d, e)) 5.60/2.34 A^1 -> H(a, U1(e, d)) 5.60/2.34 5.60/2.34 The TRS R consists of the following rules: 5.60/2.34 5.60/2.34 a -> d 5.60/2.34 a -> e 5.60/2.34 f(x) -> U1(x, x) 5.60/2.34 b -> d 5.60/2.34 b -> e 5.60/2.34 U1(d, x) -> x 5.60/2.34 5.60/2.34 Q is empty. 5.60/2.34 We have to consider all minimal (P,Q,R)-chains. 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (317) TransformationProof (EQUIVALENT) 5.60/2.34 By narrowing [LPAR04] the rule A^1 -> H(a, U1(d, d)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.34 5.60/2.34 (A^1 -> H(d, U1(d, d)),A^1 -> H(d, U1(d, d))) 5.60/2.34 (A^1 -> H(e, U1(d, d)),A^1 -> H(e, U1(d, d))) 5.60/2.34 (A^1 -> H(a, d),A^1 -> H(a, d)) 5.60/2.34 5.60/2.34 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (318) 5.60/2.34 Obligation: 5.60/2.34 Q DP problem: 5.60/2.34 The TRS P consists of the following rules: 5.60/2.34 5.60/2.34 A^1 -> H(f(d), f(d)) 5.60/2.34 H(x, x) -> G(x, x) 5.60/2.34 G(d, e) -> A^1 5.60/2.34 A^1 -> H(f(e), f(e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.34 A^1 -> H(a, U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, a), b) 5.60/2.34 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(d, a), d) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, a), b) 5.60/2.34 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), b) 5.60/2.34 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), d) 5.60/2.34 A^1 -> H(U1(a, e), b) 5.60/2.34 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, e), d) 5.60/2.34 A^1 -> H(U1(a, a), e) 5.60/2.34 A^1 -> H(d, U1(d, b)) 5.60/2.34 A^1 -> H(d, U1(b, d)) 5.60/2.34 A^1 -> H(d, U1(b, e)) 5.60/2.34 A^1 -> H(d, U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, d), b) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.34 A^1 -> H(f(d), e) 5.60/2.34 A^1 -> H(U1(e, e), b) 5.60/2.34 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.34 A^1 -> H(f(e), e) 5.60/2.34 A^1 -> H(U1(d, d), d) 5.60/2.34 A^1 -> H(e, U1(d, b)) 5.60/2.34 A^1 -> H(e, U1(b, d)) 5.60/2.34 A^1 -> H(e, U1(b, e)) 5.60/2.34 A^1 -> H(e, U1(d, d)) 5.60/2.34 A^1 -> H(a, b) 5.60/2.34 A^1 -> H(a, U1(d, e)) 5.60/2.34 A^1 -> H(a, U1(e, d)) 5.60/2.34 A^1 -> H(a, d) 5.60/2.34 5.60/2.34 The TRS R consists of the following rules: 5.60/2.34 5.60/2.34 a -> d 5.60/2.34 a -> e 5.60/2.34 f(x) -> U1(x, x) 5.60/2.34 b -> d 5.60/2.34 b -> e 5.60/2.34 U1(d, x) -> x 5.60/2.34 5.60/2.34 Q is empty. 5.60/2.34 We have to consider all minimal (P,Q,R)-chains. 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (319) TransformationProof (EQUIVALENT) 5.60/2.34 By narrowing [LPAR04] the rule A^1 -> H(a, U1(e, e)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.34 5.60/2.34 (A^1 -> H(d, U1(e, e)),A^1 -> H(d, U1(e, e))) 5.60/2.34 (A^1 -> H(e, U1(e, e)),A^1 -> H(e, U1(e, e))) 5.60/2.34 5.60/2.34 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (320) 5.60/2.34 Obligation: 5.60/2.34 Q DP problem: 5.60/2.34 The TRS P consists of the following rules: 5.60/2.34 5.60/2.34 A^1 -> H(f(d), f(d)) 5.60/2.34 H(x, x) -> G(x, x) 5.60/2.34 G(d, e) -> A^1 5.60/2.34 A^1 -> H(f(e), f(e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, a), b) 5.60/2.34 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(d, a), d) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, a), b) 5.60/2.34 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), b) 5.60/2.34 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), d) 5.60/2.34 A^1 -> H(U1(a, e), b) 5.60/2.34 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, e), d) 5.60/2.34 A^1 -> H(U1(a, a), e) 5.60/2.34 A^1 -> H(d, U1(d, b)) 5.60/2.34 A^1 -> H(d, U1(b, d)) 5.60/2.34 A^1 -> H(d, U1(b, e)) 5.60/2.34 A^1 -> H(d, U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, d), b) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.34 A^1 -> H(f(d), e) 5.60/2.34 A^1 -> H(U1(e, e), b) 5.60/2.34 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.34 A^1 -> H(f(e), e) 5.60/2.34 A^1 -> H(U1(d, d), d) 5.60/2.34 A^1 -> H(e, U1(d, b)) 5.60/2.34 A^1 -> H(e, U1(b, d)) 5.60/2.34 A^1 -> H(e, U1(b, e)) 5.60/2.34 A^1 -> H(e, U1(d, d)) 5.60/2.34 A^1 -> H(a, b) 5.60/2.34 A^1 -> H(a, U1(d, e)) 5.60/2.34 A^1 -> H(a, U1(e, d)) 5.60/2.34 A^1 -> H(a, d) 5.60/2.34 A^1 -> H(d, U1(e, e)) 5.60/2.34 A^1 -> H(e, U1(e, e)) 5.60/2.34 5.60/2.34 The TRS R consists of the following rules: 5.60/2.34 5.60/2.34 a -> d 5.60/2.34 a -> e 5.60/2.34 f(x) -> U1(x, x) 5.60/2.34 b -> d 5.60/2.34 b -> e 5.60/2.34 U1(d, x) -> x 5.60/2.34 5.60/2.34 Q is empty. 5.60/2.34 We have to consider all minimal (P,Q,R)-chains. 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (321) DependencyGraphProof (EQUIVALENT) 5.60/2.34 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (322) 5.60/2.34 Obligation: 5.60/2.34 Q DP problem: 5.60/2.34 The TRS P consists of the following rules: 5.60/2.34 5.60/2.34 H(x, x) -> G(x, x) 5.60/2.34 G(d, e) -> A^1 5.60/2.34 A^1 -> H(f(d), f(d)) 5.60/2.34 A^1 -> H(f(e), f(e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, a), b) 5.60/2.34 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(d, a), d) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, a), b) 5.60/2.34 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), b) 5.60/2.34 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), d) 5.60/2.34 A^1 -> H(U1(a, e), b) 5.60/2.34 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, e), d) 5.60/2.34 A^1 -> H(U1(a, a), e) 5.60/2.34 A^1 -> H(d, U1(d, b)) 5.60/2.34 A^1 -> H(d, U1(b, d)) 5.60/2.34 A^1 -> H(d, U1(b, e)) 5.60/2.34 A^1 -> H(d, U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, d), b) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.34 A^1 -> H(f(d), e) 5.60/2.34 A^1 -> H(U1(e, e), b) 5.60/2.34 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.34 A^1 -> H(f(e), e) 5.60/2.34 A^1 -> H(U1(d, d), d) 5.60/2.34 A^1 -> H(e, U1(d, b)) 5.60/2.34 A^1 -> H(e, U1(b, d)) 5.60/2.34 A^1 -> H(e, U1(b, e)) 5.60/2.34 A^1 -> H(e, U1(d, d)) 5.60/2.34 A^1 -> H(a, b) 5.60/2.34 A^1 -> H(a, U1(d, e)) 5.60/2.34 A^1 -> H(a, U1(e, d)) 5.60/2.34 A^1 -> H(a, d) 5.60/2.34 5.60/2.34 The TRS R consists of the following rules: 5.60/2.34 5.60/2.34 a -> d 5.60/2.34 a -> e 5.60/2.34 f(x) -> U1(x, x) 5.60/2.34 b -> d 5.60/2.34 b -> e 5.60/2.34 U1(d, x) -> x 5.60/2.34 5.60/2.34 Q is empty. 5.60/2.34 We have to consider all minimal (P,Q,R)-chains. 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (323) TransformationProof (EQUIVALENT) 5.60/2.34 By narrowing [LPAR04] the rule A^1 -> H(U1(d, e), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.34 5.60/2.34 (A^1 -> H(e, U1(d, b)),A^1 -> H(e, U1(d, b))) 5.60/2.34 (A^1 -> H(U1(d, e), b),A^1 -> H(U1(d, e), b)) 5.60/2.34 (A^1 -> H(U1(d, e), U1(d, d)),A^1 -> H(U1(d, e), U1(d, d))) 5.60/2.34 (A^1 -> H(U1(d, e), U1(d, e)),A^1 -> H(U1(d, e), U1(d, e))) 5.60/2.34 5.60/2.34 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (324) 5.60/2.34 Obligation: 5.60/2.34 Q DP problem: 5.60/2.34 The TRS P consists of the following rules: 5.60/2.34 5.60/2.34 H(x, x) -> G(x, x) 5.60/2.34 G(d, e) -> A^1 5.60/2.34 A^1 -> H(f(d), f(d)) 5.60/2.34 A^1 -> H(f(e), f(e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, a), b) 5.60/2.34 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(d, a), d) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, a), b) 5.60/2.34 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), b) 5.60/2.34 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), d) 5.60/2.34 A^1 -> H(U1(a, e), b) 5.60/2.34 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, e), d) 5.60/2.34 A^1 -> H(U1(a, a), e) 5.60/2.34 A^1 -> H(d, U1(d, b)) 5.60/2.34 A^1 -> H(d, U1(b, d)) 5.60/2.34 A^1 -> H(d, U1(b, e)) 5.60/2.34 A^1 -> H(d, U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, d), b) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.34 A^1 -> H(f(d), e) 5.60/2.34 A^1 -> H(U1(e, e), b) 5.60/2.34 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.34 A^1 -> H(f(e), e) 5.60/2.34 A^1 -> H(U1(d, d), d) 5.60/2.34 A^1 -> H(e, U1(d, b)) 5.60/2.34 A^1 -> H(e, U1(b, d)) 5.60/2.34 A^1 -> H(e, U1(b, e)) 5.60/2.34 A^1 -> H(e, U1(d, d)) 5.60/2.34 A^1 -> H(a, b) 5.60/2.34 A^1 -> H(a, U1(d, e)) 5.60/2.34 A^1 -> H(a, U1(e, d)) 5.60/2.34 A^1 -> H(a, d) 5.60/2.34 A^1 -> H(U1(d, e), b) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.34 5.60/2.34 The TRS R consists of the following rules: 5.60/2.34 5.60/2.34 a -> d 5.60/2.34 a -> e 5.60/2.34 f(x) -> U1(x, x) 5.60/2.34 b -> d 5.60/2.34 b -> e 5.60/2.34 U1(d, x) -> x 5.60/2.34 5.60/2.34 Q is empty. 5.60/2.34 We have to consider all minimal (P,Q,R)-chains. 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (325) TransformationProof (EQUIVALENT) 5.60/2.34 By narrowing [LPAR04] the rule A^1 -> H(U1(d, e), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.34 5.60/2.34 (A^1 -> H(e, U1(e, b)),A^1 -> H(e, U1(e, b))) 5.60/2.34 (A^1 -> H(U1(d, e), U1(e, d)),A^1 -> H(U1(d, e), U1(e, d))) 5.60/2.34 (A^1 -> H(U1(d, e), U1(e, e)),A^1 -> H(U1(d, e), U1(e, e))) 5.60/2.34 5.60/2.34 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (326) 5.60/2.34 Obligation: 5.60/2.34 Q DP problem: 5.60/2.34 The TRS P consists of the following rules: 5.60/2.34 5.60/2.34 H(x, x) -> G(x, x) 5.60/2.34 G(d, e) -> A^1 5.60/2.34 A^1 -> H(f(d), f(d)) 5.60/2.34 A^1 -> H(f(e), f(e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, a), b) 5.60/2.34 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(d, a), d) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, a), b) 5.60/2.34 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), b) 5.60/2.34 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), d) 5.60/2.34 A^1 -> H(U1(a, e), b) 5.60/2.34 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, e), d) 5.60/2.34 A^1 -> H(U1(a, a), e) 5.60/2.34 A^1 -> H(d, U1(d, b)) 5.60/2.34 A^1 -> H(d, U1(b, d)) 5.60/2.34 A^1 -> H(d, U1(b, e)) 5.60/2.34 A^1 -> H(d, U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, d), b) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.34 A^1 -> H(f(d), e) 5.60/2.34 A^1 -> H(U1(e, e), b) 5.60/2.34 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.34 A^1 -> H(f(e), e) 5.60/2.34 A^1 -> H(U1(d, d), d) 5.60/2.34 A^1 -> H(e, U1(d, b)) 5.60/2.34 A^1 -> H(e, U1(b, d)) 5.60/2.34 A^1 -> H(e, U1(b, e)) 5.60/2.34 A^1 -> H(e, U1(d, d)) 5.60/2.34 A^1 -> H(a, b) 5.60/2.34 A^1 -> H(a, U1(d, e)) 5.60/2.34 A^1 -> H(a, U1(e, d)) 5.60/2.34 A^1 -> H(a, d) 5.60/2.34 A^1 -> H(U1(d, e), b) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.34 A^1 -> H(e, U1(e, b)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.34 5.60/2.34 The TRS R consists of the following rules: 5.60/2.34 5.60/2.34 a -> d 5.60/2.34 a -> e 5.60/2.34 f(x) -> U1(x, x) 5.60/2.34 b -> d 5.60/2.34 b -> e 5.60/2.34 U1(d, x) -> x 5.60/2.34 5.60/2.34 Q is empty. 5.60/2.34 We have to consider all minimal (P,Q,R)-chains. 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (327) DependencyGraphProof (EQUIVALENT) 5.60/2.34 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (328) 5.60/2.34 Obligation: 5.60/2.34 Q DP problem: 5.60/2.34 The TRS P consists of the following rules: 5.60/2.34 5.60/2.34 G(d, e) -> A^1 5.60/2.34 A^1 -> H(f(d), f(d)) 5.60/2.34 H(x, x) -> G(x, x) 5.60/2.34 A^1 -> H(f(e), f(e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, a), b) 5.60/2.34 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(d, a), d) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, a), b) 5.60/2.34 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), b) 5.60/2.34 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), d) 5.60/2.34 A^1 -> H(U1(a, e), b) 5.60/2.34 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, e), d) 5.60/2.34 A^1 -> H(U1(a, a), e) 5.60/2.34 A^1 -> H(d, U1(d, b)) 5.60/2.34 A^1 -> H(d, U1(b, d)) 5.60/2.34 A^1 -> H(d, U1(b, e)) 5.60/2.34 A^1 -> H(d, U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, d), b) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.34 A^1 -> H(f(d), e) 5.60/2.34 A^1 -> H(U1(e, e), b) 5.60/2.34 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.34 A^1 -> H(f(e), e) 5.60/2.34 A^1 -> H(U1(d, d), d) 5.60/2.34 A^1 -> H(e, U1(d, b)) 5.60/2.34 A^1 -> H(e, U1(b, d)) 5.60/2.34 A^1 -> H(e, U1(b, e)) 5.60/2.34 A^1 -> H(e, U1(d, d)) 5.60/2.34 A^1 -> H(a, b) 5.60/2.34 A^1 -> H(a, U1(d, e)) 5.60/2.34 A^1 -> H(a, U1(e, d)) 5.60/2.34 A^1 -> H(a, d) 5.60/2.34 A^1 -> H(U1(d, e), b) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.34 5.60/2.34 The TRS R consists of the following rules: 5.60/2.34 5.60/2.34 a -> d 5.60/2.34 a -> e 5.60/2.34 f(x) -> U1(x, x) 5.60/2.34 b -> d 5.60/2.34 b -> e 5.60/2.34 U1(d, x) -> x 5.60/2.34 5.60/2.34 Q is empty. 5.60/2.34 We have to consider all minimal (P,Q,R)-chains. 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (329) TransformationProof (EQUIVALENT) 5.60/2.34 By narrowing [LPAR04] the rule A^1 -> H(U1(d, e), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.34 5.60/2.34 (A^1 -> H(e, U1(b, d)),A^1 -> H(e, U1(b, d))) 5.60/2.34 (A^1 -> H(U1(d, e), U1(d, d)),A^1 -> H(U1(d, e), U1(d, d))) 5.60/2.34 (A^1 -> H(U1(d, e), U1(e, d)),A^1 -> H(U1(d, e), U1(e, d))) 5.60/2.34 5.60/2.34 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (330) 5.60/2.34 Obligation: 5.60/2.34 Q DP problem: 5.60/2.34 The TRS P consists of the following rules: 5.60/2.34 5.60/2.34 G(d, e) -> A^1 5.60/2.34 A^1 -> H(f(d), f(d)) 5.60/2.34 H(x, x) -> G(x, x) 5.60/2.34 A^1 -> H(f(e), f(e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(b, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, a), b) 5.60/2.34 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(d, a), d) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, a), b) 5.60/2.34 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), b) 5.60/2.34 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), d) 5.60/2.34 A^1 -> H(U1(a, e), b) 5.60/2.34 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, e), d) 5.60/2.34 A^1 -> H(U1(a, a), e) 5.60/2.34 A^1 -> H(d, U1(d, b)) 5.60/2.34 A^1 -> H(d, U1(b, d)) 5.60/2.34 A^1 -> H(d, U1(b, e)) 5.60/2.34 A^1 -> H(d, U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, d), b) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.34 A^1 -> H(f(d), e) 5.60/2.34 A^1 -> H(U1(e, e), b) 5.60/2.34 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.34 A^1 -> H(f(e), e) 5.60/2.34 A^1 -> H(U1(d, d), d) 5.60/2.34 A^1 -> H(e, U1(d, b)) 5.60/2.34 A^1 -> H(e, U1(b, d)) 5.60/2.34 A^1 -> H(e, U1(b, e)) 5.60/2.34 A^1 -> H(e, U1(d, d)) 5.60/2.34 A^1 -> H(a, b) 5.60/2.34 A^1 -> H(a, U1(d, e)) 5.60/2.34 A^1 -> H(a, U1(e, d)) 5.60/2.34 A^1 -> H(a, d) 5.60/2.34 A^1 -> H(U1(d, e), b) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.34 5.60/2.34 The TRS R consists of the following rules: 5.60/2.34 5.60/2.34 a -> d 5.60/2.34 a -> e 5.60/2.34 f(x) -> U1(x, x) 5.60/2.34 b -> d 5.60/2.34 b -> e 5.60/2.34 U1(d, x) -> x 5.60/2.34 5.60/2.34 Q is empty. 5.60/2.34 We have to consider all minimal (P,Q,R)-chains. 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (331) TransformationProof (EQUIVALENT) 5.60/2.34 By narrowing [LPAR04] the rule A^1 -> H(U1(d, e), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.34 5.60/2.34 (A^1 -> H(e, U1(b, e)),A^1 -> H(e, U1(b, e))) 5.60/2.34 (A^1 -> H(U1(d, e), U1(d, e)),A^1 -> H(U1(d, e), U1(d, e))) 5.60/2.34 (A^1 -> H(U1(d, e), U1(e, e)),A^1 -> H(U1(d, e), U1(e, e))) 5.60/2.34 5.60/2.34 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (332) 5.60/2.34 Obligation: 5.60/2.34 Q DP problem: 5.60/2.34 The TRS P consists of the following rules: 5.60/2.34 5.60/2.34 G(d, e) -> A^1 5.60/2.34 A^1 -> H(f(d), f(d)) 5.60/2.34 H(x, x) -> G(x, x) 5.60/2.34 A^1 -> H(f(e), f(e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, a), b) 5.60/2.34 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(d, a), d) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, a), b) 5.60/2.34 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), b) 5.60/2.34 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), d) 5.60/2.34 A^1 -> H(U1(a, e), b) 5.60/2.34 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, e), d) 5.60/2.34 A^1 -> H(U1(a, a), e) 5.60/2.34 A^1 -> H(d, U1(d, b)) 5.60/2.34 A^1 -> H(d, U1(b, d)) 5.60/2.34 A^1 -> H(d, U1(b, e)) 5.60/2.34 A^1 -> H(d, U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, d), b) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.34 A^1 -> H(f(d), e) 5.60/2.34 A^1 -> H(U1(e, e), b) 5.60/2.34 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.34 A^1 -> H(f(e), e) 5.60/2.34 A^1 -> H(U1(d, d), d) 5.60/2.34 A^1 -> H(e, U1(d, b)) 5.60/2.34 A^1 -> H(e, U1(b, d)) 5.60/2.34 A^1 -> H(e, U1(b, e)) 5.60/2.34 A^1 -> H(e, U1(d, d)) 5.60/2.34 A^1 -> H(a, b) 5.60/2.34 A^1 -> H(a, U1(d, e)) 5.60/2.34 A^1 -> H(a, U1(e, d)) 5.60/2.34 A^1 -> H(a, d) 5.60/2.34 A^1 -> H(U1(d, e), b) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.34 5.60/2.34 The TRS R consists of the following rules: 5.60/2.34 5.60/2.34 a -> d 5.60/2.34 a -> e 5.60/2.34 f(x) -> U1(x, x) 5.60/2.34 b -> d 5.60/2.34 b -> e 5.60/2.34 U1(d, x) -> x 5.60/2.34 5.60/2.34 Q is empty. 5.60/2.34 We have to consider all minimal (P,Q,R)-chains. 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (333) TransformationProof (EQUIVALENT) 5.60/2.34 By narrowing [LPAR04] the rule A^1 -> H(U1(d, e), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.34 5.60/2.34 (A^1 -> H(e, U1(d, d)),A^1 -> H(e, U1(d, d))) 5.60/2.34 (A^1 -> H(U1(d, e), d),A^1 -> H(U1(d, e), d)) 5.60/2.34 5.60/2.34 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (334) 5.60/2.34 Obligation: 5.60/2.34 Q DP problem: 5.60/2.34 The TRS P consists of the following rules: 5.60/2.34 5.60/2.34 G(d, e) -> A^1 5.60/2.34 A^1 -> H(f(d), f(d)) 5.60/2.34 H(x, x) -> G(x, x) 5.60/2.34 A^1 -> H(f(e), f(e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, a), b) 5.60/2.34 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(d, a), d) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, a), b) 5.60/2.34 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), b) 5.60/2.34 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), d) 5.60/2.34 A^1 -> H(U1(a, e), b) 5.60/2.34 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, e), d) 5.60/2.34 A^1 -> H(U1(a, a), e) 5.60/2.34 A^1 -> H(d, U1(d, b)) 5.60/2.34 A^1 -> H(d, U1(b, d)) 5.60/2.34 A^1 -> H(d, U1(b, e)) 5.60/2.34 A^1 -> H(d, U1(d, d)) 5.60/2.34 A^1 -> H(U1(d, d), b) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.34 A^1 -> H(f(d), e) 5.60/2.34 A^1 -> H(U1(e, e), b) 5.60/2.34 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.34 A^1 -> H(f(e), e) 5.60/2.34 A^1 -> H(U1(d, d), d) 5.60/2.34 A^1 -> H(e, U1(d, b)) 5.60/2.34 A^1 -> H(e, U1(b, d)) 5.60/2.34 A^1 -> H(e, U1(b, e)) 5.60/2.34 A^1 -> H(e, U1(d, d)) 5.60/2.34 A^1 -> H(a, b) 5.60/2.34 A^1 -> H(a, U1(d, e)) 5.60/2.34 A^1 -> H(a, U1(e, d)) 5.60/2.34 A^1 -> H(a, d) 5.60/2.34 A^1 -> H(U1(d, e), b) 5.60/2.34 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.34 A^1 -> H(U1(d, e), d) 5.60/2.34 5.60/2.34 The TRS R consists of the following rules: 5.60/2.34 5.60/2.34 a -> d 5.60/2.34 a -> e 5.60/2.34 f(x) -> U1(x, x) 5.60/2.34 b -> d 5.60/2.34 b -> e 5.60/2.34 U1(d, x) -> x 5.60/2.34 5.60/2.34 Q is empty. 5.60/2.34 We have to consider all minimal (P,Q,R)-chains. 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (335) TransformationProof (EQUIVALENT) 5.60/2.34 By narrowing [LPAR04] the rule A^1 -> H(U1(d, e), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.34 5.60/2.34 (A^1 -> H(e, U1(e, e)),A^1 -> H(e, U1(e, e))) 5.60/2.34 5.60/2.34 5.60/2.34 ---------------------------------------- 5.60/2.34 5.60/2.34 (336) 5.60/2.34 Obligation: 5.60/2.34 Q DP problem: 5.60/2.34 The TRS P consists of the following rules: 5.60/2.34 5.60/2.34 G(d, e) -> A^1 5.60/2.34 A^1 -> H(f(d), f(d)) 5.60/2.34 H(x, x) -> G(x, x) 5.60/2.34 A^1 -> H(f(e), f(e)) 5.60/2.34 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.34 A^1 -> H(U1(d, a), b) 5.60/2.34 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(d, a), d) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.34 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.34 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.34 A^1 -> H(U1(e, a), b) 5.60/2.34 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.34 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), b) 5.60/2.34 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, d), d) 5.60/2.34 A^1 -> H(U1(a, e), b) 5.60/2.34 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.34 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.34 A^1 -> H(U1(a, e), d) 5.60/2.34 A^1 -> H(U1(a, a), e) 5.60/2.34 A^1 -> H(d, U1(d, b)) 5.60/2.34 A^1 -> H(d, U1(b, d)) 5.60/2.35 A^1 -> H(d, U1(b, e)) 5.60/2.35 A^1 -> H(d, U1(d, d)) 5.60/2.35 A^1 -> H(U1(d, d), b) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.35 A^1 -> H(f(d), e) 5.60/2.35 A^1 -> H(U1(e, e), b) 5.60/2.35 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.35 A^1 -> H(f(e), e) 5.60/2.35 A^1 -> H(U1(d, d), d) 5.60/2.35 A^1 -> H(e, U1(d, b)) 5.60/2.35 A^1 -> H(e, U1(b, d)) 5.60/2.35 A^1 -> H(e, U1(b, e)) 5.60/2.35 A^1 -> H(e, U1(d, d)) 5.60/2.35 A^1 -> H(a, b) 5.60/2.35 A^1 -> H(a, U1(d, e)) 5.60/2.35 A^1 -> H(a, U1(e, d)) 5.60/2.35 A^1 -> H(a, d) 5.60/2.35 A^1 -> H(U1(d, e), b) 5.60/2.35 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(d, e), d) 5.60/2.35 A^1 -> H(e, U1(e, e)) 5.60/2.35 5.60/2.35 The TRS R consists of the following rules: 5.60/2.35 5.60/2.35 a -> d 5.60/2.35 a -> e 5.60/2.35 f(x) -> U1(x, x) 5.60/2.35 b -> d 5.60/2.35 b -> e 5.60/2.35 U1(d, x) -> x 5.60/2.35 5.60/2.35 Q is empty. 5.60/2.35 We have to consider all minimal (P,Q,R)-chains. 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (337) DependencyGraphProof (EQUIVALENT) 5.60/2.35 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (338) 5.60/2.35 Obligation: 5.60/2.35 Q DP problem: 5.60/2.35 The TRS P consists of the following rules: 5.60/2.35 5.60/2.35 A^1 -> H(f(d), f(d)) 5.60/2.35 H(x, x) -> G(x, x) 5.60/2.35 G(d, e) -> A^1 5.60/2.35 A^1 -> H(f(e), f(e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.35 A^1 -> H(U1(d, a), b) 5.60/2.35 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.35 A^1 -> H(U1(d, a), d) 5.60/2.35 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.35 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.35 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.35 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.35 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, a), b) 5.60/2.35 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.35 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), b) 5.60/2.35 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), d) 5.60/2.35 A^1 -> H(U1(a, e), b) 5.60/2.35 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, e), d) 5.60/2.35 A^1 -> H(U1(a, a), e) 5.60/2.35 A^1 -> H(d, U1(d, b)) 5.60/2.35 A^1 -> H(d, U1(b, d)) 5.60/2.35 A^1 -> H(d, U1(b, e)) 5.60/2.35 A^1 -> H(d, U1(d, d)) 5.60/2.35 A^1 -> H(U1(d, d), b) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.35 A^1 -> H(f(d), e) 5.60/2.35 A^1 -> H(U1(e, e), b) 5.60/2.35 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.35 A^1 -> H(f(e), e) 5.60/2.35 A^1 -> H(U1(d, d), d) 5.60/2.35 A^1 -> H(e, U1(d, b)) 5.60/2.35 A^1 -> H(e, U1(b, d)) 5.60/2.35 A^1 -> H(e, U1(b, e)) 5.60/2.35 A^1 -> H(e, U1(d, d)) 5.60/2.35 A^1 -> H(a, b) 5.60/2.35 A^1 -> H(a, U1(d, e)) 5.60/2.35 A^1 -> H(a, U1(e, d)) 5.60/2.35 A^1 -> H(a, d) 5.60/2.35 A^1 -> H(U1(d, e), b) 5.60/2.35 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(d, e), d) 5.60/2.35 5.60/2.35 The TRS R consists of the following rules: 5.60/2.35 5.60/2.35 a -> d 5.60/2.35 a -> e 5.60/2.35 f(x) -> U1(x, x) 5.60/2.35 b -> d 5.60/2.35 b -> e 5.60/2.35 U1(d, x) -> x 5.60/2.35 5.60/2.35 Q is empty. 5.60/2.35 We have to consider all minimal (P,Q,R)-chains. 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (339) TransformationProof (EQUIVALENT) 5.60/2.35 By narrowing [LPAR04] the rule A^1 -> H(U1(d, a), b) at position [] we obtained the following new rules [LPAR04]: 5.60/2.35 5.60/2.35 (A^1 -> H(a, b),A^1 -> H(a, b)) 5.60/2.35 (A^1 -> H(U1(d, d), b),A^1 -> H(U1(d, d), b)) 5.60/2.35 (A^1 -> H(U1(d, e), b),A^1 -> H(U1(d, e), b)) 5.60/2.35 (A^1 -> H(U1(d, a), d),A^1 -> H(U1(d, a), d)) 5.60/2.35 (A^1 -> H(U1(d, a), e),A^1 -> H(U1(d, a), e)) 5.60/2.35 5.60/2.35 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (340) 5.60/2.35 Obligation: 5.60/2.35 Q DP problem: 5.60/2.35 The TRS P consists of the following rules: 5.60/2.35 5.60/2.35 A^1 -> H(f(d), f(d)) 5.60/2.35 H(x, x) -> G(x, x) 5.60/2.35 G(d, e) -> A^1 5.60/2.35 A^1 -> H(f(e), f(e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.35 A^1 -> H(U1(d, a), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.35 A^1 -> H(U1(d, a), d) 5.60/2.35 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.35 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.35 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.35 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.35 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, a), b) 5.60/2.35 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.35 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), b) 5.60/2.35 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), d) 5.60/2.35 A^1 -> H(U1(a, e), b) 5.60/2.35 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, e), d) 5.60/2.35 A^1 -> H(U1(a, a), e) 5.60/2.35 A^1 -> H(d, U1(d, b)) 5.60/2.35 A^1 -> H(d, U1(b, d)) 5.60/2.35 A^1 -> H(d, U1(b, e)) 5.60/2.35 A^1 -> H(d, U1(d, d)) 5.60/2.35 A^1 -> H(U1(d, d), b) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.35 A^1 -> H(f(d), e) 5.60/2.35 A^1 -> H(U1(e, e), b) 5.60/2.35 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.35 A^1 -> H(f(e), e) 5.60/2.35 A^1 -> H(U1(d, d), d) 5.60/2.35 A^1 -> H(e, U1(d, b)) 5.60/2.35 A^1 -> H(e, U1(b, d)) 5.60/2.35 A^1 -> H(e, U1(b, e)) 5.60/2.35 A^1 -> H(e, U1(d, d)) 5.60/2.35 A^1 -> H(a, b) 5.60/2.35 A^1 -> H(a, U1(d, e)) 5.60/2.35 A^1 -> H(a, U1(e, d)) 5.60/2.35 A^1 -> H(a, d) 5.60/2.35 A^1 -> H(U1(d, e), b) 5.60/2.35 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(d, e), d) 5.60/2.35 A^1 -> H(U1(d, a), e) 5.60/2.35 5.60/2.35 The TRS R consists of the following rules: 5.60/2.35 5.60/2.35 a -> d 5.60/2.35 a -> e 5.60/2.35 f(x) -> U1(x, x) 5.60/2.35 b -> d 5.60/2.35 b -> e 5.60/2.35 U1(d, x) -> x 5.60/2.35 5.60/2.35 Q is empty. 5.60/2.35 We have to consider all minimal (P,Q,R)-chains. 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (341) TransformationProof (EQUIVALENT) 5.60/2.35 By narrowing [LPAR04] the rule A^1 -> H(U1(d, a), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.35 5.60/2.35 (A^1 -> H(a, U1(d, e)),A^1 -> H(a, U1(d, e))) 5.60/2.35 (A^1 -> H(U1(d, d), U1(d, e)),A^1 -> H(U1(d, d), U1(d, e))) 5.60/2.35 (A^1 -> H(U1(d, e), U1(d, e)),A^1 -> H(U1(d, e), U1(d, e))) 5.60/2.35 (A^1 -> H(U1(d, a), e),A^1 -> H(U1(d, a), e)) 5.60/2.35 5.60/2.35 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (342) 5.60/2.35 Obligation: 5.60/2.35 Q DP problem: 5.60/2.35 The TRS P consists of the following rules: 5.60/2.35 5.60/2.35 A^1 -> H(f(d), f(d)) 5.60/2.35 H(x, x) -> G(x, x) 5.60/2.35 G(d, e) -> A^1 5.60/2.35 A^1 -> H(f(e), f(e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.35 A^1 -> H(U1(d, a), U1(e, d)) 5.60/2.35 A^1 -> H(U1(d, a), d) 5.60/2.35 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.35 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.35 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.35 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.35 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, a), b) 5.60/2.35 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.35 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), b) 5.60/2.35 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), d) 5.60/2.35 A^1 -> H(U1(a, e), b) 5.60/2.35 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, e), d) 5.60/2.35 A^1 -> H(U1(a, a), e) 5.60/2.35 A^1 -> H(d, U1(d, b)) 5.60/2.35 A^1 -> H(d, U1(b, d)) 5.60/2.35 A^1 -> H(d, U1(b, e)) 5.60/2.35 A^1 -> H(d, U1(d, d)) 5.60/2.35 A^1 -> H(U1(d, d), b) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.35 A^1 -> H(f(d), e) 5.60/2.35 A^1 -> H(U1(e, e), b) 5.60/2.35 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.35 A^1 -> H(f(e), e) 5.60/2.35 A^1 -> H(U1(d, d), d) 5.60/2.35 A^1 -> H(e, U1(d, b)) 5.60/2.35 A^1 -> H(e, U1(b, d)) 5.60/2.35 A^1 -> H(e, U1(b, e)) 5.60/2.35 A^1 -> H(e, U1(d, d)) 5.60/2.35 A^1 -> H(a, b) 5.60/2.35 A^1 -> H(a, U1(d, e)) 5.60/2.35 A^1 -> H(a, U1(e, d)) 5.60/2.35 A^1 -> H(a, d) 5.60/2.35 A^1 -> H(U1(d, e), b) 5.60/2.35 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(d, e), d) 5.60/2.35 A^1 -> H(U1(d, a), e) 5.60/2.35 5.60/2.35 The TRS R consists of the following rules: 5.60/2.35 5.60/2.35 a -> d 5.60/2.35 a -> e 5.60/2.35 f(x) -> U1(x, x) 5.60/2.35 b -> d 5.60/2.35 b -> e 5.60/2.35 U1(d, x) -> x 5.60/2.35 5.60/2.35 Q is empty. 5.60/2.35 We have to consider all minimal (P,Q,R)-chains. 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (343) TransformationProof (EQUIVALENT) 5.60/2.35 By narrowing [LPAR04] the rule A^1 -> H(U1(d, a), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.35 5.60/2.35 (A^1 -> H(a, U1(e, d)),A^1 -> H(a, U1(e, d))) 5.60/2.35 (A^1 -> H(U1(d, d), U1(e, d)),A^1 -> H(U1(d, d), U1(e, d))) 5.60/2.35 (A^1 -> H(U1(d, e), U1(e, d)),A^1 -> H(U1(d, e), U1(e, d))) 5.60/2.35 5.60/2.35 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (344) 5.60/2.35 Obligation: 5.60/2.35 Q DP problem: 5.60/2.35 The TRS P consists of the following rules: 5.60/2.35 5.60/2.35 A^1 -> H(f(d), f(d)) 5.60/2.35 H(x, x) -> G(x, x) 5.60/2.35 G(d, e) -> A^1 5.60/2.35 A^1 -> H(f(e), f(e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.35 A^1 -> H(U1(d, a), d) 5.60/2.35 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.35 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.35 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.35 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.35 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, a), b) 5.60/2.35 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.35 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), b) 5.60/2.35 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), d) 5.60/2.35 A^1 -> H(U1(a, e), b) 5.60/2.35 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, e), d) 5.60/2.35 A^1 -> H(U1(a, a), e) 5.60/2.35 A^1 -> H(d, U1(d, b)) 5.60/2.35 A^1 -> H(d, U1(b, d)) 5.60/2.35 A^1 -> H(d, U1(b, e)) 5.60/2.35 A^1 -> H(d, U1(d, d)) 5.60/2.35 A^1 -> H(U1(d, d), b) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.35 A^1 -> H(f(d), e) 5.60/2.35 A^1 -> H(U1(e, e), b) 5.60/2.35 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.35 A^1 -> H(f(e), e) 5.60/2.35 A^1 -> H(U1(d, d), d) 5.60/2.35 A^1 -> H(e, U1(d, b)) 5.60/2.35 A^1 -> H(e, U1(b, d)) 5.60/2.35 A^1 -> H(e, U1(b, e)) 5.60/2.35 A^1 -> H(e, U1(d, d)) 5.60/2.35 A^1 -> H(a, b) 5.60/2.35 A^1 -> H(a, U1(d, e)) 5.60/2.35 A^1 -> H(a, U1(e, d)) 5.60/2.35 A^1 -> H(a, d) 5.60/2.35 A^1 -> H(U1(d, e), b) 5.60/2.35 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(d, e), d) 5.60/2.35 A^1 -> H(U1(d, a), e) 5.60/2.35 5.60/2.35 The TRS R consists of the following rules: 5.60/2.35 5.60/2.35 a -> d 5.60/2.35 a -> e 5.60/2.35 f(x) -> U1(x, x) 5.60/2.35 b -> d 5.60/2.35 b -> e 5.60/2.35 U1(d, x) -> x 5.60/2.35 5.60/2.35 Q is empty. 5.60/2.35 We have to consider all minimal (P,Q,R)-chains. 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (345) TransformationProof (EQUIVALENT) 5.60/2.35 By narrowing [LPAR04] the rule A^1 -> H(U1(d, a), d) at position [] we obtained the following new rules [LPAR04]: 5.60/2.35 5.60/2.35 (A^1 -> H(a, d),A^1 -> H(a, d)) 5.60/2.35 (A^1 -> H(U1(d, d), d),A^1 -> H(U1(d, d), d)) 5.60/2.35 (A^1 -> H(U1(d, e), d),A^1 -> H(U1(d, e), d)) 5.60/2.35 5.60/2.35 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (346) 5.60/2.35 Obligation: 5.60/2.35 Q DP problem: 5.60/2.35 The TRS P consists of the following rules: 5.60/2.35 5.60/2.35 A^1 -> H(f(d), f(d)) 5.60/2.35 H(x, x) -> G(x, x) 5.60/2.35 G(d, e) -> A^1 5.60/2.35 A^1 -> H(f(e), f(e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.35 A^1 -> H(U1(e, d), U1(d, b)) 5.60/2.35 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.35 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.35 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.35 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, a), b) 5.60/2.35 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.35 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), b) 5.60/2.35 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), d) 5.60/2.35 A^1 -> H(U1(a, e), b) 5.60/2.35 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, e), d) 5.60/2.35 A^1 -> H(U1(a, a), e) 5.60/2.35 A^1 -> H(d, U1(d, b)) 5.60/2.35 A^1 -> H(d, U1(b, d)) 5.60/2.35 A^1 -> H(d, U1(b, e)) 5.60/2.35 A^1 -> H(d, U1(d, d)) 5.60/2.35 A^1 -> H(U1(d, d), b) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.35 A^1 -> H(f(d), e) 5.60/2.35 A^1 -> H(U1(e, e), b) 5.60/2.35 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.35 A^1 -> H(f(e), e) 5.60/2.35 A^1 -> H(U1(d, d), d) 5.60/2.35 A^1 -> H(e, U1(d, b)) 5.60/2.35 A^1 -> H(e, U1(b, d)) 5.60/2.35 A^1 -> H(e, U1(b, e)) 5.60/2.35 A^1 -> H(e, U1(d, d)) 5.60/2.35 A^1 -> H(a, b) 5.60/2.35 A^1 -> H(a, U1(d, e)) 5.60/2.35 A^1 -> H(a, U1(e, d)) 5.60/2.35 A^1 -> H(a, d) 5.60/2.35 A^1 -> H(U1(d, e), b) 5.60/2.35 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(d, e), d) 5.60/2.35 A^1 -> H(U1(d, a), e) 5.60/2.35 5.60/2.35 The TRS R consists of the following rules: 5.60/2.35 5.60/2.35 a -> d 5.60/2.35 a -> e 5.60/2.35 f(x) -> U1(x, x) 5.60/2.35 b -> d 5.60/2.35 b -> e 5.60/2.35 U1(d, x) -> x 5.60/2.35 5.60/2.35 Q is empty. 5.60/2.35 We have to consider all minimal (P,Q,R)-chains. 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (347) TransformationProof (EQUIVALENT) 5.60/2.35 By narrowing [LPAR04] the rule A^1 -> H(U1(e, d), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.35 5.60/2.35 (A^1 -> H(U1(e, d), b),A^1 -> H(U1(e, d), b)) 5.60/2.35 (A^1 -> H(U1(e, d), U1(d, d)),A^1 -> H(U1(e, d), U1(d, d))) 5.60/2.35 (A^1 -> H(U1(e, d), U1(d, e)),A^1 -> H(U1(e, d), U1(d, e))) 5.60/2.35 5.60/2.35 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (348) 5.60/2.35 Obligation: 5.60/2.35 Q DP problem: 5.60/2.35 The TRS P consists of the following rules: 5.60/2.35 5.60/2.35 A^1 -> H(f(d), f(d)) 5.60/2.35 H(x, x) -> G(x, x) 5.60/2.35 G(d, e) -> A^1 5.60/2.35 A^1 -> H(f(e), f(e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.35 A^1 -> H(U1(e, d), U1(e, b)) 5.60/2.35 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.35 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.35 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, a), b) 5.60/2.35 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.35 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), b) 5.60/2.35 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), d) 5.60/2.35 A^1 -> H(U1(a, e), b) 5.60/2.35 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, e), d) 5.60/2.35 A^1 -> H(U1(a, a), e) 5.60/2.35 A^1 -> H(d, U1(d, b)) 5.60/2.35 A^1 -> H(d, U1(b, d)) 5.60/2.35 A^1 -> H(d, U1(b, e)) 5.60/2.35 A^1 -> H(d, U1(d, d)) 5.60/2.35 A^1 -> H(U1(d, d), b) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.35 A^1 -> H(f(d), e) 5.60/2.35 A^1 -> H(U1(e, e), b) 5.60/2.35 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.35 A^1 -> H(f(e), e) 5.60/2.35 A^1 -> H(U1(d, d), d) 5.60/2.35 A^1 -> H(e, U1(d, b)) 5.60/2.35 A^1 -> H(e, U1(b, d)) 5.60/2.35 A^1 -> H(e, U1(b, e)) 5.60/2.35 A^1 -> H(e, U1(d, d)) 5.60/2.35 A^1 -> H(a, b) 5.60/2.35 A^1 -> H(a, U1(d, e)) 5.60/2.35 A^1 -> H(a, U1(e, d)) 5.60/2.35 A^1 -> H(a, d) 5.60/2.35 A^1 -> H(U1(d, e), b) 5.60/2.35 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(d, e), d) 5.60/2.35 A^1 -> H(U1(d, a), e) 5.60/2.35 A^1 -> H(U1(e, d), b) 5.60/2.35 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.35 5.60/2.35 The TRS R consists of the following rules: 5.60/2.35 5.60/2.35 a -> d 5.60/2.35 a -> e 5.60/2.35 f(x) -> U1(x, x) 5.60/2.35 b -> d 5.60/2.35 b -> e 5.60/2.35 U1(d, x) -> x 5.60/2.35 5.60/2.35 Q is empty. 5.60/2.35 We have to consider all minimal (P,Q,R)-chains. 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (349) TransformationProof (EQUIVALENT) 5.60/2.35 By narrowing [LPAR04] the rule A^1 -> H(U1(e, d), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.35 5.60/2.35 (A^1 -> H(U1(e, d), U1(e, d)),A^1 -> H(U1(e, d), U1(e, d))) 5.60/2.35 (A^1 -> H(U1(e, d), U1(e, e)),A^1 -> H(U1(e, d), U1(e, e))) 5.60/2.35 5.60/2.35 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (350) 5.60/2.35 Obligation: 5.60/2.35 Q DP problem: 5.60/2.35 The TRS P consists of the following rules: 5.60/2.35 5.60/2.35 A^1 -> H(f(d), f(d)) 5.60/2.35 H(x, x) -> G(x, x) 5.60/2.35 G(d, e) -> A^1 5.60/2.35 A^1 -> H(f(e), f(e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.35 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.35 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.35 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, a), b) 5.60/2.35 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.35 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), b) 5.60/2.35 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), d) 5.60/2.35 A^1 -> H(U1(a, e), b) 5.60/2.35 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, e), d) 5.60/2.35 A^1 -> H(U1(a, a), e) 5.60/2.35 A^1 -> H(d, U1(d, b)) 5.60/2.35 A^1 -> H(d, U1(b, d)) 5.60/2.35 A^1 -> H(d, U1(b, e)) 5.60/2.35 A^1 -> H(d, U1(d, d)) 5.60/2.35 A^1 -> H(U1(d, d), b) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.35 A^1 -> H(f(d), e) 5.60/2.35 A^1 -> H(U1(e, e), b) 5.60/2.35 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.35 A^1 -> H(f(e), e) 5.60/2.35 A^1 -> H(U1(d, d), d) 5.60/2.35 A^1 -> H(e, U1(d, b)) 5.60/2.35 A^1 -> H(e, U1(b, d)) 5.60/2.35 A^1 -> H(e, U1(b, e)) 5.60/2.35 A^1 -> H(e, U1(d, d)) 5.60/2.35 A^1 -> H(a, b) 5.60/2.35 A^1 -> H(a, U1(d, e)) 5.60/2.35 A^1 -> H(a, U1(e, d)) 5.60/2.35 A^1 -> H(a, d) 5.60/2.35 A^1 -> H(U1(d, e), b) 5.60/2.35 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(d, e), d) 5.60/2.35 A^1 -> H(U1(d, a), e) 5.60/2.35 A^1 -> H(U1(e, d), b) 5.60/2.35 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.35 A^1 -> H(U1(e, d), U1(e, e)) 5.60/2.35 5.60/2.35 The TRS R consists of the following rules: 5.60/2.35 5.60/2.35 a -> d 5.60/2.35 a -> e 5.60/2.35 f(x) -> U1(x, x) 5.60/2.35 b -> d 5.60/2.35 b -> e 5.60/2.35 U1(d, x) -> x 5.60/2.35 5.60/2.35 Q is empty. 5.60/2.35 We have to consider all minimal (P,Q,R)-chains. 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (351) DependencyGraphProof (EQUIVALENT) 5.60/2.35 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (352) 5.60/2.35 Obligation: 5.60/2.35 Q DP problem: 5.60/2.35 The TRS P consists of the following rules: 5.60/2.35 5.60/2.35 H(x, x) -> G(x, x) 5.60/2.35 G(d, e) -> A^1 5.60/2.35 A^1 -> H(f(d), f(d)) 5.60/2.35 A^1 -> H(f(e), f(e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.35 A^1 -> H(U1(e, d), U1(b, d)) 5.60/2.35 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.35 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, a), b) 5.60/2.35 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.35 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), b) 5.60/2.35 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), d) 5.60/2.35 A^1 -> H(U1(a, e), b) 5.60/2.35 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, e), d) 5.60/2.35 A^1 -> H(U1(a, a), e) 5.60/2.35 A^1 -> H(d, U1(d, b)) 5.60/2.35 A^1 -> H(d, U1(b, d)) 5.60/2.35 A^1 -> H(d, U1(b, e)) 5.60/2.35 A^1 -> H(d, U1(d, d)) 5.60/2.35 A^1 -> H(U1(d, d), b) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.35 A^1 -> H(f(d), e) 5.60/2.35 A^1 -> H(U1(e, e), b) 5.60/2.35 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.35 A^1 -> H(f(e), e) 5.60/2.35 A^1 -> H(U1(d, d), d) 5.60/2.35 A^1 -> H(e, U1(d, b)) 5.60/2.35 A^1 -> H(e, U1(b, d)) 5.60/2.35 A^1 -> H(e, U1(b, e)) 5.60/2.35 A^1 -> H(e, U1(d, d)) 5.60/2.35 A^1 -> H(a, b) 5.60/2.35 A^1 -> H(a, U1(d, e)) 5.60/2.35 A^1 -> H(a, U1(e, d)) 5.60/2.35 A^1 -> H(a, d) 5.60/2.35 A^1 -> H(U1(d, e), b) 5.60/2.35 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(d, e), d) 5.60/2.35 A^1 -> H(U1(d, a), e) 5.60/2.35 A^1 -> H(U1(e, d), b) 5.60/2.35 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.35 5.60/2.35 The TRS R consists of the following rules: 5.60/2.35 5.60/2.35 a -> d 5.60/2.35 a -> e 5.60/2.35 f(x) -> U1(x, x) 5.60/2.35 b -> d 5.60/2.35 b -> e 5.60/2.35 U1(d, x) -> x 5.60/2.35 5.60/2.35 Q is empty. 5.60/2.35 We have to consider all minimal (P,Q,R)-chains. 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (353) TransformationProof (EQUIVALENT) 5.60/2.35 By narrowing [LPAR04] the rule A^1 -> H(U1(e, d), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.35 5.60/2.35 (A^1 -> H(U1(e, d), U1(d, d)),A^1 -> H(U1(e, d), U1(d, d))) 5.60/2.35 (A^1 -> H(U1(e, d), U1(e, d)),A^1 -> H(U1(e, d), U1(e, d))) 5.60/2.35 5.60/2.35 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (354) 5.60/2.35 Obligation: 5.60/2.35 Q DP problem: 5.60/2.35 The TRS P consists of the following rules: 5.60/2.35 5.60/2.35 H(x, x) -> G(x, x) 5.60/2.35 G(d, e) -> A^1 5.60/2.35 A^1 -> H(f(d), f(d)) 5.60/2.35 A^1 -> H(f(e), f(e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.35 A^1 -> H(U1(e, d), U1(b, e)) 5.60/2.35 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, a), b) 5.60/2.35 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.35 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), b) 5.60/2.35 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), d) 5.60/2.35 A^1 -> H(U1(a, e), b) 5.60/2.35 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, e), d) 5.60/2.35 A^1 -> H(U1(a, a), e) 5.60/2.35 A^1 -> H(d, U1(d, b)) 5.60/2.35 A^1 -> H(d, U1(b, d)) 5.60/2.35 A^1 -> H(d, U1(b, e)) 5.60/2.35 A^1 -> H(d, U1(d, d)) 5.60/2.35 A^1 -> H(U1(d, d), b) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.35 A^1 -> H(f(d), e) 5.60/2.35 A^1 -> H(U1(e, e), b) 5.60/2.35 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.35 A^1 -> H(f(e), e) 5.60/2.35 A^1 -> H(U1(d, d), d) 5.60/2.35 A^1 -> H(e, U1(d, b)) 5.60/2.35 A^1 -> H(e, U1(b, d)) 5.60/2.35 A^1 -> H(e, U1(b, e)) 5.60/2.35 A^1 -> H(e, U1(d, d)) 5.60/2.35 A^1 -> H(a, b) 5.60/2.35 A^1 -> H(a, U1(d, e)) 5.60/2.35 A^1 -> H(a, U1(e, d)) 5.60/2.35 A^1 -> H(a, d) 5.60/2.35 A^1 -> H(U1(d, e), b) 5.60/2.35 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(d, e), d) 5.60/2.35 A^1 -> H(U1(d, a), e) 5.60/2.35 A^1 -> H(U1(e, d), b) 5.60/2.35 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.35 5.60/2.35 The TRS R consists of the following rules: 5.60/2.35 5.60/2.35 a -> d 5.60/2.35 a -> e 5.60/2.35 f(x) -> U1(x, x) 5.60/2.35 b -> d 5.60/2.35 b -> e 5.60/2.35 U1(d, x) -> x 5.60/2.35 5.60/2.35 Q is empty. 5.60/2.35 We have to consider all minimal (P,Q,R)-chains. 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (355) TransformationProof (EQUIVALENT) 5.60/2.35 By narrowing [LPAR04] the rule A^1 -> H(U1(e, d), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.35 5.60/2.35 (A^1 -> H(U1(e, d), U1(d, e)),A^1 -> H(U1(e, d), U1(d, e))) 5.60/2.35 (A^1 -> H(U1(e, d), U1(e, e)),A^1 -> H(U1(e, d), U1(e, e))) 5.60/2.35 5.60/2.35 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (356) 5.60/2.35 Obligation: 5.60/2.35 Q DP problem: 5.60/2.35 The TRS P consists of the following rules: 5.60/2.35 5.60/2.35 H(x, x) -> G(x, x) 5.60/2.35 G(d, e) -> A^1 5.60/2.35 A^1 -> H(f(d), f(d)) 5.60/2.35 A^1 -> H(f(e), f(e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.35 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, a), b) 5.60/2.35 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.35 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), b) 5.60/2.35 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), d) 5.60/2.35 A^1 -> H(U1(a, e), b) 5.60/2.35 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, e), d) 5.60/2.35 A^1 -> H(U1(a, a), e) 5.60/2.35 A^1 -> H(d, U1(d, b)) 5.60/2.35 A^1 -> H(d, U1(b, d)) 5.60/2.35 A^1 -> H(d, U1(b, e)) 5.60/2.35 A^1 -> H(d, U1(d, d)) 5.60/2.35 A^1 -> H(U1(d, d), b) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.35 A^1 -> H(f(d), e) 5.60/2.35 A^1 -> H(U1(e, e), b) 5.60/2.35 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.35 A^1 -> H(f(e), e) 5.60/2.35 A^1 -> H(U1(d, d), d) 5.60/2.35 A^1 -> H(e, U1(d, b)) 5.60/2.35 A^1 -> H(e, U1(b, d)) 5.60/2.35 A^1 -> H(e, U1(b, e)) 5.60/2.35 A^1 -> H(e, U1(d, d)) 5.60/2.35 A^1 -> H(a, b) 5.60/2.35 A^1 -> H(a, U1(d, e)) 5.60/2.35 A^1 -> H(a, U1(e, d)) 5.60/2.35 A^1 -> H(a, d) 5.60/2.35 A^1 -> H(U1(d, e), b) 5.60/2.35 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(d, e), d) 5.60/2.35 A^1 -> H(U1(d, a), e) 5.60/2.35 A^1 -> H(U1(e, d), b) 5.60/2.35 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.35 A^1 -> H(U1(e, d), U1(e, e)) 5.60/2.35 5.60/2.35 The TRS R consists of the following rules: 5.60/2.35 5.60/2.35 a -> d 5.60/2.35 a -> e 5.60/2.35 f(x) -> U1(x, x) 5.60/2.35 b -> d 5.60/2.35 b -> e 5.60/2.35 U1(d, x) -> x 5.60/2.35 5.60/2.35 Q is empty. 5.60/2.35 We have to consider all minimal (P,Q,R)-chains. 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (357) DependencyGraphProof (EQUIVALENT) 5.60/2.35 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (358) 5.60/2.35 Obligation: 5.60/2.35 Q DP problem: 5.60/2.35 The TRS P consists of the following rules: 5.60/2.35 5.60/2.35 G(d, e) -> A^1 5.60/2.35 A^1 -> H(f(d), f(d)) 5.60/2.35 H(x, x) -> G(x, x) 5.60/2.35 A^1 -> H(f(e), f(e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.35 A^1 -> H(U1(e, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, a), b) 5.60/2.35 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.35 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), b) 5.60/2.35 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), d) 5.60/2.35 A^1 -> H(U1(a, e), b) 5.60/2.35 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, e), d) 5.60/2.35 A^1 -> H(U1(a, a), e) 5.60/2.35 A^1 -> H(d, U1(d, b)) 5.60/2.35 A^1 -> H(d, U1(b, d)) 5.60/2.35 A^1 -> H(d, U1(b, e)) 5.60/2.35 A^1 -> H(d, U1(d, d)) 5.60/2.35 A^1 -> H(U1(d, d), b) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.35 A^1 -> H(f(d), e) 5.60/2.35 A^1 -> H(U1(e, e), b) 5.60/2.35 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.35 A^1 -> H(f(e), e) 5.60/2.35 A^1 -> H(U1(d, d), d) 5.60/2.35 A^1 -> H(e, U1(d, b)) 5.60/2.35 A^1 -> H(e, U1(b, d)) 5.60/2.35 A^1 -> H(e, U1(b, e)) 5.60/2.35 A^1 -> H(e, U1(d, d)) 5.60/2.35 A^1 -> H(a, b) 5.60/2.35 A^1 -> H(a, U1(d, e)) 5.60/2.35 A^1 -> H(a, U1(e, d)) 5.60/2.35 A^1 -> H(a, d) 5.60/2.35 A^1 -> H(U1(d, e), b) 5.60/2.35 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(d, e), d) 5.60/2.35 A^1 -> H(U1(d, a), e) 5.60/2.35 A^1 -> H(U1(e, d), b) 5.60/2.35 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.35 5.60/2.35 The TRS R consists of the following rules: 5.60/2.35 5.60/2.35 a -> d 5.60/2.35 a -> e 5.60/2.35 f(x) -> U1(x, x) 5.60/2.35 b -> d 5.60/2.35 b -> e 5.60/2.35 U1(d, x) -> x 5.60/2.35 5.60/2.35 Q is empty. 5.60/2.35 We have to consider all minimal (P,Q,R)-chains. 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (359) TransformationProof (EQUIVALENT) 5.60/2.35 By narrowing [LPAR04] the rule A^1 -> H(U1(e, d), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.35 5.60/2.35 (A^1 -> H(U1(e, d), d),A^1 -> H(U1(e, d), d)) 5.60/2.35 5.60/2.35 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (360) 5.60/2.35 Obligation: 5.60/2.35 Q DP problem: 5.60/2.35 The TRS P consists of the following rules: 5.60/2.35 5.60/2.35 G(d, e) -> A^1 5.60/2.35 A^1 -> H(f(d), f(d)) 5.60/2.35 H(x, x) -> G(x, x) 5.60/2.35 A^1 -> H(f(e), f(e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.35 A^1 -> H(U1(e, a), b) 5.60/2.35 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.35 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), b) 5.60/2.35 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), d) 5.60/2.35 A^1 -> H(U1(a, e), b) 5.60/2.35 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, e), d) 5.60/2.35 A^1 -> H(U1(a, a), e) 5.60/2.35 A^1 -> H(d, U1(d, b)) 5.60/2.35 A^1 -> H(d, U1(b, d)) 5.60/2.35 A^1 -> H(d, U1(b, e)) 5.60/2.35 A^1 -> H(d, U1(d, d)) 5.60/2.35 A^1 -> H(U1(d, d), b) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.35 A^1 -> H(f(d), e) 5.60/2.35 A^1 -> H(U1(e, e), b) 5.60/2.35 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.35 A^1 -> H(f(e), e) 5.60/2.35 A^1 -> H(U1(d, d), d) 5.60/2.35 A^1 -> H(e, U1(d, b)) 5.60/2.35 A^1 -> H(e, U1(b, d)) 5.60/2.35 A^1 -> H(e, U1(b, e)) 5.60/2.35 A^1 -> H(e, U1(d, d)) 5.60/2.35 A^1 -> H(a, b) 5.60/2.35 A^1 -> H(a, U1(d, e)) 5.60/2.35 A^1 -> H(a, U1(e, d)) 5.60/2.35 A^1 -> H(a, d) 5.60/2.35 A^1 -> H(U1(d, e), b) 5.60/2.35 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(d, e), d) 5.60/2.35 A^1 -> H(U1(d, a), e) 5.60/2.35 A^1 -> H(U1(e, d), b) 5.60/2.35 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.35 A^1 -> H(U1(e, d), d) 5.60/2.35 5.60/2.35 The TRS R consists of the following rules: 5.60/2.35 5.60/2.35 a -> d 5.60/2.35 a -> e 5.60/2.35 f(x) -> U1(x, x) 5.60/2.35 b -> d 5.60/2.35 b -> e 5.60/2.35 U1(d, x) -> x 5.60/2.35 5.60/2.35 Q is empty. 5.60/2.35 We have to consider all minimal (P,Q,R)-chains. 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (361) DependencyGraphProof (EQUIVALENT) 5.60/2.35 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (362) 5.60/2.35 Obligation: 5.60/2.35 Q DP problem: 5.60/2.35 The TRS P consists of the following rules: 5.60/2.35 5.60/2.35 A^1 -> H(f(d), f(d)) 5.60/2.35 H(x, x) -> G(x, x) 5.60/2.35 G(d, e) -> A^1 5.60/2.35 A^1 -> H(f(e), f(e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.35 A^1 -> H(U1(e, a), b) 5.60/2.35 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.35 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), b) 5.60/2.35 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), d) 5.60/2.35 A^1 -> H(U1(a, e), b) 5.60/2.35 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, e), d) 5.60/2.35 A^1 -> H(U1(a, a), e) 5.60/2.35 A^1 -> H(d, U1(d, b)) 5.60/2.35 A^1 -> H(d, U1(b, d)) 5.60/2.35 A^1 -> H(d, U1(b, e)) 5.60/2.35 A^1 -> H(d, U1(d, d)) 5.60/2.35 A^1 -> H(U1(d, d), b) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.35 A^1 -> H(f(d), e) 5.60/2.35 A^1 -> H(U1(e, e), b) 5.60/2.35 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.35 A^1 -> H(f(e), e) 5.60/2.35 A^1 -> H(U1(d, d), d) 5.60/2.35 A^1 -> H(e, U1(d, b)) 5.60/2.35 A^1 -> H(e, U1(b, d)) 5.60/2.35 A^1 -> H(e, U1(b, e)) 5.60/2.35 A^1 -> H(e, U1(d, d)) 5.60/2.35 A^1 -> H(a, b) 5.60/2.35 A^1 -> H(a, U1(d, e)) 5.60/2.35 A^1 -> H(a, U1(e, d)) 5.60/2.35 A^1 -> H(a, d) 5.60/2.35 A^1 -> H(U1(d, e), b) 5.60/2.35 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(d, e), d) 5.60/2.35 A^1 -> H(U1(d, a), e) 5.60/2.35 A^1 -> H(U1(e, d), b) 5.60/2.35 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.35 5.60/2.35 The TRS R consists of the following rules: 5.60/2.35 5.60/2.35 a -> d 5.60/2.35 a -> e 5.60/2.35 f(x) -> U1(x, x) 5.60/2.35 b -> d 5.60/2.35 b -> e 5.60/2.35 U1(d, x) -> x 5.60/2.35 5.60/2.35 Q is empty. 5.60/2.35 We have to consider all minimal (P,Q,R)-chains. 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (363) TransformationProof (EQUIVALENT) 5.60/2.35 By narrowing [LPAR04] the rule A^1 -> H(U1(e, a), b) at position [] we obtained the following new rules [LPAR04]: 5.60/2.35 5.60/2.35 (A^1 -> H(U1(e, d), b),A^1 -> H(U1(e, d), b)) 5.60/2.35 (A^1 -> H(U1(e, e), b),A^1 -> H(U1(e, e), b)) 5.60/2.35 (A^1 -> H(U1(e, a), d),A^1 -> H(U1(e, a), d)) 5.60/2.35 (A^1 -> H(U1(e, a), e),A^1 -> H(U1(e, a), e)) 5.60/2.35 5.60/2.35 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (364) 5.60/2.35 Obligation: 5.60/2.35 Q DP problem: 5.60/2.35 The TRS P consists of the following rules: 5.60/2.35 5.60/2.35 A^1 -> H(f(d), f(d)) 5.60/2.35 H(x, x) -> G(x, x) 5.60/2.35 G(d, e) -> A^1 5.60/2.35 A^1 -> H(f(e), f(e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.35 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.35 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), b) 5.60/2.35 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), d) 5.60/2.35 A^1 -> H(U1(a, e), b) 5.60/2.35 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, e), d) 5.60/2.35 A^1 -> H(U1(a, a), e) 5.60/2.35 A^1 -> H(d, U1(d, b)) 5.60/2.35 A^1 -> H(d, U1(b, d)) 5.60/2.35 A^1 -> H(d, U1(b, e)) 5.60/2.35 A^1 -> H(d, U1(d, d)) 5.60/2.35 A^1 -> H(U1(d, d), b) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.35 A^1 -> H(f(d), e) 5.60/2.35 A^1 -> H(U1(e, e), b) 5.60/2.35 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.35 A^1 -> H(f(e), e) 5.60/2.35 A^1 -> H(U1(d, d), d) 5.60/2.35 A^1 -> H(e, U1(d, b)) 5.60/2.35 A^1 -> H(e, U1(b, d)) 5.60/2.35 A^1 -> H(e, U1(b, e)) 5.60/2.35 A^1 -> H(e, U1(d, d)) 5.60/2.35 A^1 -> H(a, b) 5.60/2.35 A^1 -> H(a, U1(d, e)) 5.60/2.35 A^1 -> H(a, U1(e, d)) 5.60/2.35 A^1 -> H(a, d) 5.60/2.35 A^1 -> H(U1(d, e), b) 5.60/2.35 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(d, e), d) 5.60/2.35 A^1 -> H(U1(d, a), e) 5.60/2.35 A^1 -> H(U1(e, d), b) 5.60/2.35 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.35 A^1 -> H(U1(e, a), d) 5.60/2.35 A^1 -> H(U1(e, a), e) 5.60/2.35 5.60/2.35 The TRS R consists of the following rules: 5.60/2.35 5.60/2.35 a -> d 5.60/2.35 a -> e 5.60/2.35 f(x) -> U1(x, x) 5.60/2.35 b -> d 5.60/2.35 b -> e 5.60/2.35 U1(d, x) -> x 5.60/2.35 5.60/2.35 Q is empty. 5.60/2.35 We have to consider all minimal (P,Q,R)-chains. 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (365) DependencyGraphProof (EQUIVALENT) 5.60/2.35 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (366) 5.60/2.35 Obligation: 5.60/2.35 Q DP problem: 5.60/2.35 The TRS P consists of the following rules: 5.60/2.35 5.60/2.35 H(x, x) -> G(x, x) 5.60/2.35 G(d, e) -> A^1 5.60/2.35 A^1 -> H(f(d), f(d)) 5.60/2.35 A^1 -> H(f(e), f(e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.35 A^1 -> H(U1(e, a), U1(d, e)) 5.60/2.35 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), b) 5.60/2.35 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), d) 5.60/2.35 A^1 -> H(U1(a, e), b) 5.60/2.35 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, e), d) 5.60/2.35 A^1 -> H(U1(a, a), e) 5.60/2.35 A^1 -> H(d, U1(d, b)) 5.60/2.35 A^1 -> H(d, U1(b, d)) 5.60/2.35 A^1 -> H(d, U1(b, e)) 5.60/2.35 A^1 -> H(d, U1(d, d)) 5.60/2.35 A^1 -> H(U1(d, d), b) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.35 A^1 -> H(f(d), e) 5.60/2.35 A^1 -> H(U1(e, e), b) 5.60/2.35 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.35 A^1 -> H(f(e), e) 5.60/2.35 A^1 -> H(U1(d, d), d) 5.60/2.35 A^1 -> H(e, U1(d, b)) 5.60/2.35 A^1 -> H(e, U1(b, d)) 5.60/2.35 A^1 -> H(e, U1(b, e)) 5.60/2.35 A^1 -> H(e, U1(d, d)) 5.60/2.35 A^1 -> H(a, b) 5.60/2.35 A^1 -> H(a, U1(d, e)) 5.60/2.35 A^1 -> H(a, U1(e, d)) 5.60/2.35 A^1 -> H(a, d) 5.60/2.35 A^1 -> H(U1(d, e), b) 5.60/2.35 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(d, e), d) 5.60/2.35 A^1 -> H(U1(d, a), e) 5.60/2.35 A^1 -> H(U1(e, d), b) 5.60/2.35 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.35 5.60/2.35 The TRS R consists of the following rules: 5.60/2.35 5.60/2.35 a -> d 5.60/2.35 a -> e 5.60/2.35 f(x) -> U1(x, x) 5.60/2.35 b -> d 5.60/2.35 b -> e 5.60/2.35 U1(d, x) -> x 5.60/2.35 5.60/2.35 Q is empty. 5.60/2.35 We have to consider all minimal (P,Q,R)-chains. 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (367) TransformationProof (EQUIVALENT) 5.60/2.35 By narrowing [LPAR04] the rule A^1 -> H(U1(e, a), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.35 5.60/2.35 (A^1 -> H(U1(e, d), U1(d, e)),A^1 -> H(U1(e, d), U1(d, e))) 5.60/2.35 (A^1 -> H(U1(e, e), U1(d, e)),A^1 -> H(U1(e, e), U1(d, e))) 5.60/2.35 (A^1 -> H(U1(e, a), e),A^1 -> H(U1(e, a), e)) 5.60/2.35 5.60/2.35 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (368) 5.60/2.35 Obligation: 5.60/2.35 Q DP problem: 5.60/2.35 The TRS P consists of the following rules: 5.60/2.35 5.60/2.35 H(x, x) -> G(x, x) 5.60/2.35 G(d, e) -> A^1 5.60/2.35 A^1 -> H(f(d), f(d)) 5.60/2.35 A^1 -> H(f(e), f(e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.35 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), b) 5.60/2.35 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), d) 5.60/2.35 A^1 -> H(U1(a, e), b) 5.60/2.35 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, e), d) 5.60/2.35 A^1 -> H(U1(a, a), e) 5.60/2.35 A^1 -> H(d, U1(d, b)) 5.60/2.35 A^1 -> H(d, U1(b, d)) 5.60/2.35 A^1 -> H(d, U1(b, e)) 5.60/2.35 A^1 -> H(d, U1(d, d)) 5.60/2.35 A^1 -> H(U1(d, d), b) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.35 A^1 -> H(f(d), e) 5.60/2.35 A^1 -> H(U1(e, e), b) 5.60/2.35 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.35 A^1 -> H(f(e), e) 5.60/2.35 A^1 -> H(U1(d, d), d) 5.60/2.35 A^1 -> H(e, U1(d, b)) 5.60/2.35 A^1 -> H(e, U1(b, d)) 5.60/2.35 A^1 -> H(e, U1(b, e)) 5.60/2.35 A^1 -> H(e, U1(d, d)) 5.60/2.35 A^1 -> H(a, b) 5.60/2.35 A^1 -> H(a, U1(d, e)) 5.60/2.35 A^1 -> H(a, U1(e, d)) 5.60/2.35 A^1 -> H(a, d) 5.60/2.35 A^1 -> H(U1(d, e), b) 5.60/2.35 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(d, e), d) 5.60/2.35 A^1 -> H(U1(d, a), e) 5.60/2.35 A^1 -> H(U1(e, d), b) 5.60/2.35 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.35 A^1 -> H(U1(e, a), e) 5.60/2.35 5.60/2.35 The TRS R consists of the following rules: 5.60/2.35 5.60/2.35 a -> d 5.60/2.35 a -> e 5.60/2.35 f(x) -> U1(x, x) 5.60/2.35 b -> d 5.60/2.35 b -> e 5.60/2.35 U1(d, x) -> x 5.60/2.35 5.60/2.35 Q is empty. 5.60/2.35 We have to consider all minimal (P,Q,R)-chains. 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (369) DependencyGraphProof (EQUIVALENT) 5.60/2.35 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (370) 5.60/2.35 Obligation: 5.60/2.35 Q DP problem: 5.60/2.35 The TRS P consists of the following rules: 5.60/2.35 5.60/2.35 G(d, e) -> A^1 5.60/2.35 A^1 -> H(f(d), f(d)) 5.60/2.35 H(x, x) -> G(x, x) 5.60/2.35 A^1 -> H(f(e), f(e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.35 A^1 -> H(U1(e, a), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), b) 5.60/2.35 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), d) 5.60/2.35 A^1 -> H(U1(a, e), b) 5.60/2.35 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, e), d) 5.60/2.35 A^1 -> H(U1(a, a), e) 5.60/2.35 A^1 -> H(d, U1(d, b)) 5.60/2.35 A^1 -> H(d, U1(b, d)) 5.60/2.35 A^1 -> H(d, U1(b, e)) 5.60/2.35 A^1 -> H(d, U1(d, d)) 5.60/2.35 A^1 -> H(U1(d, d), b) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.35 A^1 -> H(f(d), e) 5.60/2.35 A^1 -> H(U1(e, e), b) 5.60/2.35 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.35 A^1 -> H(f(e), e) 5.60/2.35 A^1 -> H(U1(d, d), d) 5.60/2.35 A^1 -> H(e, U1(d, b)) 5.60/2.35 A^1 -> H(e, U1(b, d)) 5.60/2.35 A^1 -> H(e, U1(b, e)) 5.60/2.35 A^1 -> H(e, U1(d, d)) 5.60/2.35 A^1 -> H(a, b) 5.60/2.35 A^1 -> H(a, U1(d, e)) 5.60/2.35 A^1 -> H(a, U1(e, d)) 5.60/2.35 A^1 -> H(a, d) 5.60/2.35 A^1 -> H(U1(d, e), b) 5.60/2.35 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(d, e), d) 5.60/2.35 A^1 -> H(U1(d, a), e) 5.60/2.35 A^1 -> H(U1(e, d), b) 5.60/2.35 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.35 5.60/2.35 The TRS R consists of the following rules: 5.60/2.35 5.60/2.35 a -> d 5.60/2.35 a -> e 5.60/2.35 f(x) -> U1(x, x) 5.60/2.35 b -> d 5.60/2.35 b -> e 5.60/2.35 U1(d, x) -> x 5.60/2.35 5.60/2.35 Q is empty. 5.60/2.35 We have to consider all minimal (P,Q,R)-chains. 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (371) TransformationProof (EQUIVALENT) 5.60/2.35 By narrowing [LPAR04] the rule A^1 -> H(U1(e, a), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.35 5.60/2.35 (A^1 -> H(U1(e, d), U1(e, d)),A^1 -> H(U1(e, d), U1(e, d))) 5.60/2.35 (A^1 -> H(U1(e, e), U1(e, d)),A^1 -> H(U1(e, e), U1(e, d))) 5.60/2.35 5.60/2.35 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (372) 5.60/2.35 Obligation: 5.60/2.35 Q DP problem: 5.60/2.35 The TRS P consists of the following rules: 5.60/2.35 5.60/2.35 G(d, e) -> A^1 5.60/2.35 A^1 -> H(f(d), f(d)) 5.60/2.35 H(x, x) -> G(x, x) 5.60/2.35 A^1 -> H(f(e), f(e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.35 A^1 -> H(U1(a, d), b) 5.60/2.35 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), d) 5.60/2.35 A^1 -> H(U1(a, e), b) 5.60/2.35 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, e), d) 5.60/2.35 A^1 -> H(U1(a, a), e) 5.60/2.35 A^1 -> H(d, U1(d, b)) 5.60/2.35 A^1 -> H(d, U1(b, d)) 5.60/2.35 A^1 -> H(d, U1(b, e)) 5.60/2.35 A^1 -> H(d, U1(d, d)) 5.60/2.35 A^1 -> H(U1(d, d), b) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.35 A^1 -> H(f(d), e) 5.60/2.35 A^1 -> H(U1(e, e), b) 5.60/2.35 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.35 A^1 -> H(f(e), e) 5.60/2.35 A^1 -> H(U1(d, d), d) 5.60/2.35 A^1 -> H(e, U1(d, b)) 5.60/2.35 A^1 -> H(e, U1(b, d)) 5.60/2.35 A^1 -> H(e, U1(b, e)) 5.60/2.35 A^1 -> H(e, U1(d, d)) 5.60/2.35 A^1 -> H(a, b) 5.60/2.35 A^1 -> H(a, U1(d, e)) 5.60/2.35 A^1 -> H(a, U1(e, d)) 5.60/2.35 A^1 -> H(a, d) 5.60/2.35 A^1 -> H(U1(d, e), b) 5.60/2.35 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(d, e), d) 5.60/2.35 A^1 -> H(U1(d, a), e) 5.60/2.35 A^1 -> H(U1(e, d), b) 5.60/2.35 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.35 A^1 -> H(U1(e, e), U1(e, d)) 5.60/2.35 5.60/2.35 The TRS R consists of the following rules: 5.60/2.35 5.60/2.35 a -> d 5.60/2.35 a -> e 5.60/2.35 f(x) -> U1(x, x) 5.60/2.35 b -> d 5.60/2.35 b -> e 5.60/2.35 U1(d, x) -> x 5.60/2.35 5.60/2.35 Q is empty. 5.60/2.35 We have to consider all minimal (P,Q,R)-chains. 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (373) DependencyGraphProof (EQUIVALENT) 5.60/2.35 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (374) 5.60/2.35 Obligation: 5.60/2.35 Q DP problem: 5.60/2.35 The TRS P consists of the following rules: 5.60/2.35 5.60/2.35 A^1 -> H(f(d), f(d)) 5.60/2.35 H(x, x) -> G(x, x) 5.60/2.35 G(d, e) -> A^1 5.60/2.35 A^1 -> H(f(e), f(e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.35 A^1 -> H(U1(a, d), b) 5.60/2.35 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), d) 5.60/2.35 A^1 -> H(U1(a, e), b) 5.60/2.35 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, e), d) 5.60/2.35 A^1 -> H(U1(a, a), e) 5.60/2.35 A^1 -> H(d, U1(d, b)) 5.60/2.35 A^1 -> H(d, U1(b, d)) 5.60/2.35 A^1 -> H(d, U1(b, e)) 5.60/2.35 A^1 -> H(d, U1(d, d)) 5.60/2.35 A^1 -> H(U1(d, d), b) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.35 A^1 -> H(f(d), e) 5.60/2.35 A^1 -> H(U1(e, e), b) 5.60/2.35 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.35 A^1 -> H(f(e), e) 5.60/2.35 A^1 -> H(U1(d, d), d) 5.60/2.35 A^1 -> H(e, U1(d, b)) 5.60/2.35 A^1 -> H(e, U1(b, d)) 5.60/2.35 A^1 -> H(e, U1(b, e)) 5.60/2.35 A^1 -> H(e, U1(d, d)) 5.60/2.35 A^1 -> H(a, b) 5.60/2.35 A^1 -> H(a, U1(d, e)) 5.60/2.35 A^1 -> H(a, U1(e, d)) 5.60/2.35 A^1 -> H(a, d) 5.60/2.35 A^1 -> H(U1(d, e), b) 5.60/2.35 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(d, e), d) 5.60/2.35 A^1 -> H(U1(d, a), e) 5.60/2.35 A^1 -> H(U1(e, d), b) 5.60/2.35 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.35 5.60/2.35 The TRS R consists of the following rules: 5.60/2.35 5.60/2.35 a -> d 5.60/2.35 a -> e 5.60/2.35 f(x) -> U1(x, x) 5.60/2.35 b -> d 5.60/2.35 b -> e 5.60/2.35 U1(d, x) -> x 5.60/2.35 5.60/2.35 Q is empty. 5.60/2.35 We have to consider all minimal (P,Q,R)-chains. 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (375) TransformationProof (EQUIVALENT) 5.60/2.35 By narrowing [LPAR04] the rule A^1 -> H(U1(a, d), b) at position [] we obtained the following new rules [LPAR04]: 5.60/2.35 5.60/2.35 (A^1 -> H(U1(d, d), b),A^1 -> H(U1(d, d), b)) 5.60/2.35 (A^1 -> H(U1(e, d), b),A^1 -> H(U1(e, d), b)) 5.60/2.35 (A^1 -> H(U1(a, d), d),A^1 -> H(U1(a, d), d)) 5.60/2.35 (A^1 -> H(U1(a, d), e),A^1 -> H(U1(a, d), e)) 5.60/2.35 5.60/2.35 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (376) 5.60/2.35 Obligation: 5.60/2.35 Q DP problem: 5.60/2.35 The TRS P consists of the following rules: 5.60/2.35 5.60/2.35 A^1 -> H(f(d), f(d)) 5.60/2.35 H(x, x) -> G(x, x) 5.60/2.35 G(d, e) -> A^1 5.60/2.35 A^1 -> H(f(e), f(e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.35 A^1 -> H(U1(a, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), d) 5.60/2.35 A^1 -> H(U1(a, e), b) 5.60/2.35 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, e), d) 5.60/2.35 A^1 -> H(U1(a, a), e) 5.60/2.35 A^1 -> H(d, U1(d, b)) 5.60/2.35 A^1 -> H(d, U1(b, d)) 5.60/2.35 A^1 -> H(d, U1(b, e)) 5.60/2.35 A^1 -> H(d, U1(d, d)) 5.60/2.35 A^1 -> H(U1(d, d), b) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.35 A^1 -> H(f(d), e) 5.60/2.35 A^1 -> H(U1(e, e), b) 5.60/2.35 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.35 A^1 -> H(f(e), e) 5.60/2.35 A^1 -> H(U1(d, d), d) 5.60/2.35 A^1 -> H(e, U1(d, b)) 5.60/2.35 A^1 -> H(e, U1(b, d)) 5.60/2.35 A^1 -> H(e, U1(b, e)) 5.60/2.35 A^1 -> H(e, U1(d, d)) 5.60/2.35 A^1 -> H(a, b) 5.60/2.35 A^1 -> H(a, U1(d, e)) 5.60/2.35 A^1 -> H(a, U1(e, d)) 5.60/2.35 A^1 -> H(a, d) 5.60/2.35 A^1 -> H(U1(d, e), b) 5.60/2.35 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(d, e), d) 5.60/2.35 A^1 -> H(U1(d, a), e) 5.60/2.35 A^1 -> H(U1(e, d), b) 5.60/2.35 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), e) 5.60/2.35 5.60/2.35 The TRS R consists of the following rules: 5.60/2.35 5.60/2.35 a -> d 5.60/2.35 a -> e 5.60/2.35 f(x) -> U1(x, x) 5.60/2.35 b -> d 5.60/2.35 b -> e 5.60/2.35 U1(d, x) -> x 5.60/2.35 5.60/2.35 Q is empty. 5.60/2.35 We have to consider all minimal (P,Q,R)-chains. 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (377) TransformationProof (EQUIVALENT) 5.60/2.35 By narrowing [LPAR04] the rule A^1 -> H(U1(a, d), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.35 5.60/2.35 (A^1 -> H(U1(d, d), U1(d, e)),A^1 -> H(U1(d, d), U1(d, e))) 5.60/2.35 (A^1 -> H(U1(e, d), U1(d, e)),A^1 -> H(U1(e, d), U1(d, e))) 5.60/2.35 (A^1 -> H(U1(a, d), e),A^1 -> H(U1(a, d), e)) 5.60/2.35 5.60/2.35 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (378) 5.60/2.35 Obligation: 5.60/2.35 Q DP problem: 5.60/2.35 The TRS P consists of the following rules: 5.60/2.35 5.60/2.35 A^1 -> H(f(d), f(d)) 5.60/2.35 H(x, x) -> G(x, x) 5.60/2.35 G(d, e) -> A^1 5.60/2.35 A^1 -> H(f(e), f(e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.35 A^1 -> H(U1(a, d), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), d) 5.60/2.35 A^1 -> H(U1(a, e), b) 5.60/2.35 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, e), d) 5.60/2.35 A^1 -> H(U1(a, a), e) 5.60/2.35 A^1 -> H(d, U1(d, b)) 5.60/2.35 A^1 -> H(d, U1(b, d)) 5.60/2.35 A^1 -> H(d, U1(b, e)) 5.60/2.35 A^1 -> H(d, U1(d, d)) 5.60/2.35 A^1 -> H(U1(d, d), b) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.35 A^1 -> H(f(d), e) 5.60/2.35 A^1 -> H(U1(e, e), b) 5.60/2.35 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.35 A^1 -> H(f(e), e) 5.60/2.35 A^1 -> H(U1(d, d), d) 5.60/2.35 A^1 -> H(e, U1(d, b)) 5.60/2.35 A^1 -> H(e, U1(b, d)) 5.60/2.35 A^1 -> H(e, U1(b, e)) 5.60/2.35 A^1 -> H(e, U1(d, d)) 5.60/2.35 A^1 -> H(a, b) 5.60/2.35 A^1 -> H(a, U1(d, e)) 5.60/2.35 A^1 -> H(a, U1(e, d)) 5.60/2.35 A^1 -> H(a, d) 5.60/2.35 A^1 -> H(U1(d, e), b) 5.60/2.35 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(d, e), d) 5.60/2.35 A^1 -> H(U1(d, a), e) 5.60/2.35 A^1 -> H(U1(e, d), b) 5.60/2.35 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), e) 5.60/2.35 5.60/2.35 The TRS R consists of the following rules: 5.60/2.35 5.60/2.35 a -> d 5.60/2.35 a -> e 5.60/2.35 f(x) -> U1(x, x) 5.60/2.35 b -> d 5.60/2.35 b -> e 5.60/2.35 U1(d, x) -> x 5.60/2.35 5.60/2.35 Q is empty. 5.60/2.35 We have to consider all minimal (P,Q,R)-chains. 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (379) TransformationProof (EQUIVALENT) 5.60/2.35 By narrowing [LPAR04] the rule A^1 -> H(U1(a, d), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.35 5.60/2.35 (A^1 -> H(U1(d, d), U1(e, d)),A^1 -> H(U1(d, d), U1(e, d))) 5.60/2.35 (A^1 -> H(U1(e, d), U1(e, d)),A^1 -> H(U1(e, d), U1(e, d))) 5.60/2.35 5.60/2.35 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (380) 5.60/2.35 Obligation: 5.60/2.35 Q DP problem: 5.60/2.35 The TRS P consists of the following rules: 5.60/2.35 5.60/2.35 A^1 -> H(f(d), f(d)) 5.60/2.35 H(x, x) -> G(x, x) 5.60/2.35 G(d, e) -> A^1 5.60/2.35 A^1 -> H(f(e), f(e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.35 A^1 -> H(U1(a, d), d) 5.60/2.35 A^1 -> H(U1(a, e), b) 5.60/2.35 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, e), d) 5.60/2.35 A^1 -> H(U1(a, a), e) 5.60/2.35 A^1 -> H(d, U1(d, b)) 5.60/2.35 A^1 -> H(d, U1(b, d)) 5.60/2.35 A^1 -> H(d, U1(b, e)) 5.60/2.35 A^1 -> H(d, U1(d, d)) 5.60/2.35 A^1 -> H(U1(d, d), b) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.35 A^1 -> H(f(d), e) 5.60/2.35 A^1 -> H(U1(e, e), b) 5.60/2.35 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.35 A^1 -> H(f(e), e) 5.60/2.35 A^1 -> H(U1(d, d), d) 5.60/2.35 A^1 -> H(e, U1(d, b)) 5.60/2.35 A^1 -> H(e, U1(b, d)) 5.60/2.35 A^1 -> H(e, U1(b, e)) 5.60/2.35 A^1 -> H(e, U1(d, d)) 5.60/2.35 A^1 -> H(a, b) 5.60/2.35 A^1 -> H(a, U1(d, e)) 5.60/2.35 A^1 -> H(a, U1(e, d)) 5.60/2.35 A^1 -> H(a, d) 5.60/2.35 A^1 -> H(U1(d, e), b) 5.60/2.35 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(d, e), d) 5.60/2.35 A^1 -> H(U1(d, a), e) 5.60/2.35 A^1 -> H(U1(e, d), b) 5.60/2.35 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), e) 5.60/2.35 5.60/2.35 The TRS R consists of the following rules: 5.60/2.35 5.60/2.35 a -> d 5.60/2.35 a -> e 5.60/2.35 f(x) -> U1(x, x) 5.60/2.35 b -> d 5.60/2.35 b -> e 5.60/2.35 U1(d, x) -> x 5.60/2.35 5.60/2.35 Q is empty. 5.60/2.35 We have to consider all minimal (P,Q,R)-chains. 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (381) TransformationProof (EQUIVALENT) 5.60/2.35 By narrowing [LPAR04] the rule A^1 -> H(U1(a, d), d) at position [] we obtained the following new rules [LPAR04]: 5.60/2.35 5.60/2.35 (A^1 -> H(U1(d, d), d),A^1 -> H(U1(d, d), d)) 5.60/2.35 (A^1 -> H(U1(e, d), d),A^1 -> H(U1(e, d), d)) 5.60/2.35 5.60/2.35 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (382) 5.60/2.35 Obligation: 5.60/2.35 Q DP problem: 5.60/2.35 The TRS P consists of the following rules: 5.60/2.35 5.60/2.35 A^1 -> H(f(d), f(d)) 5.60/2.35 H(x, x) -> G(x, x) 5.60/2.35 G(d, e) -> A^1 5.60/2.35 A^1 -> H(f(e), f(e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.35 A^1 -> H(U1(a, e), b) 5.60/2.35 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, e), d) 5.60/2.35 A^1 -> H(U1(a, a), e) 5.60/2.35 A^1 -> H(d, U1(d, b)) 5.60/2.35 A^1 -> H(d, U1(b, d)) 5.60/2.35 A^1 -> H(d, U1(b, e)) 5.60/2.35 A^1 -> H(d, U1(d, d)) 5.60/2.35 A^1 -> H(U1(d, d), b) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.35 A^1 -> H(f(d), e) 5.60/2.35 A^1 -> H(U1(e, e), b) 5.60/2.35 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.35 A^1 -> H(f(e), e) 5.60/2.35 A^1 -> H(U1(d, d), d) 5.60/2.35 A^1 -> H(e, U1(d, b)) 5.60/2.35 A^1 -> H(e, U1(b, d)) 5.60/2.35 A^1 -> H(e, U1(b, e)) 5.60/2.35 A^1 -> H(e, U1(d, d)) 5.60/2.35 A^1 -> H(a, b) 5.60/2.35 A^1 -> H(a, U1(d, e)) 5.60/2.35 A^1 -> H(a, U1(e, d)) 5.60/2.35 A^1 -> H(a, d) 5.60/2.35 A^1 -> H(U1(d, e), b) 5.60/2.35 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(d, e), d) 5.60/2.35 A^1 -> H(U1(d, a), e) 5.60/2.35 A^1 -> H(U1(e, d), b) 5.60/2.35 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), e) 5.60/2.35 A^1 -> H(U1(e, d), d) 5.60/2.35 5.60/2.35 The TRS R consists of the following rules: 5.60/2.35 5.60/2.35 a -> d 5.60/2.35 a -> e 5.60/2.35 f(x) -> U1(x, x) 5.60/2.35 b -> d 5.60/2.35 b -> e 5.60/2.35 U1(d, x) -> x 5.60/2.35 5.60/2.35 Q is empty. 5.60/2.35 We have to consider all minimal (P,Q,R)-chains. 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (383) DependencyGraphProof (EQUIVALENT) 5.60/2.35 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (384) 5.60/2.35 Obligation: 5.60/2.35 Q DP problem: 5.60/2.35 The TRS P consists of the following rules: 5.60/2.35 5.60/2.35 H(x, x) -> G(x, x) 5.60/2.35 G(d, e) -> A^1 5.60/2.35 A^1 -> H(f(d), f(d)) 5.60/2.35 A^1 -> H(f(e), f(e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.35 A^1 -> H(U1(a, e), b) 5.60/2.35 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, e), d) 5.60/2.35 A^1 -> H(U1(a, a), e) 5.60/2.35 A^1 -> H(d, U1(d, b)) 5.60/2.35 A^1 -> H(d, U1(b, d)) 5.60/2.35 A^1 -> H(d, U1(b, e)) 5.60/2.35 A^1 -> H(d, U1(d, d)) 5.60/2.35 A^1 -> H(U1(d, d), b) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.35 A^1 -> H(f(d), e) 5.60/2.35 A^1 -> H(U1(e, e), b) 5.60/2.35 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.35 A^1 -> H(f(e), e) 5.60/2.35 A^1 -> H(U1(d, d), d) 5.60/2.35 A^1 -> H(e, U1(d, b)) 5.60/2.35 A^1 -> H(e, U1(b, d)) 5.60/2.35 A^1 -> H(e, U1(b, e)) 5.60/2.35 A^1 -> H(e, U1(d, d)) 5.60/2.35 A^1 -> H(a, b) 5.60/2.35 A^1 -> H(a, U1(d, e)) 5.60/2.35 A^1 -> H(a, U1(e, d)) 5.60/2.35 A^1 -> H(a, d) 5.60/2.35 A^1 -> H(U1(d, e), b) 5.60/2.35 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(d, e), d) 5.60/2.35 A^1 -> H(U1(d, a), e) 5.60/2.35 A^1 -> H(U1(e, d), b) 5.60/2.35 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), e) 5.60/2.35 5.60/2.35 The TRS R consists of the following rules: 5.60/2.35 5.60/2.35 a -> d 5.60/2.35 a -> e 5.60/2.35 f(x) -> U1(x, x) 5.60/2.35 b -> d 5.60/2.35 b -> e 5.60/2.35 U1(d, x) -> x 5.60/2.35 5.60/2.35 Q is empty. 5.60/2.35 We have to consider all minimal (P,Q,R)-chains. 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (385) TransformationProof (EQUIVALENT) 5.60/2.35 By narrowing [LPAR04] the rule A^1 -> H(U1(a, e), b) at position [] we obtained the following new rules [LPAR04]: 5.60/2.35 5.60/2.35 (A^1 -> H(U1(d, e), b),A^1 -> H(U1(d, e), b)) 5.60/2.35 (A^1 -> H(U1(e, e), b),A^1 -> H(U1(e, e), b)) 5.60/2.35 (A^1 -> H(U1(a, e), d),A^1 -> H(U1(a, e), d)) 5.60/2.35 (A^1 -> H(U1(a, e), e),A^1 -> H(U1(a, e), e)) 5.60/2.35 5.60/2.35 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (386) 5.60/2.35 Obligation: 5.60/2.35 Q DP problem: 5.60/2.35 The TRS P consists of the following rules: 5.60/2.35 5.60/2.35 H(x, x) -> G(x, x) 5.60/2.35 G(d, e) -> A^1 5.60/2.35 A^1 -> H(f(d), f(d)) 5.60/2.35 A^1 -> H(f(e), f(e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.35 A^1 -> H(U1(a, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, e), d) 5.60/2.35 A^1 -> H(U1(a, a), e) 5.60/2.35 A^1 -> H(d, U1(d, b)) 5.60/2.35 A^1 -> H(d, U1(b, d)) 5.60/2.35 A^1 -> H(d, U1(b, e)) 5.60/2.35 A^1 -> H(d, U1(d, d)) 5.60/2.35 A^1 -> H(U1(d, d), b) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.35 A^1 -> H(f(d), e) 5.60/2.35 A^1 -> H(U1(e, e), b) 5.60/2.35 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.35 A^1 -> H(f(e), e) 5.60/2.35 A^1 -> H(U1(d, d), d) 5.60/2.35 A^1 -> H(e, U1(d, b)) 5.60/2.35 A^1 -> H(e, U1(b, d)) 5.60/2.35 A^1 -> H(e, U1(b, e)) 5.60/2.35 A^1 -> H(e, U1(d, d)) 5.60/2.35 A^1 -> H(a, b) 5.60/2.35 A^1 -> H(a, U1(d, e)) 5.60/2.35 A^1 -> H(a, U1(e, d)) 5.60/2.35 A^1 -> H(a, d) 5.60/2.35 A^1 -> H(U1(d, e), b) 5.60/2.35 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(d, e), d) 5.60/2.35 A^1 -> H(U1(d, a), e) 5.60/2.35 A^1 -> H(U1(e, d), b) 5.60/2.35 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), e) 5.60/2.35 A^1 -> H(U1(a, e), e) 5.60/2.35 5.60/2.35 The TRS R consists of the following rules: 5.60/2.35 5.60/2.35 a -> d 5.60/2.35 a -> e 5.60/2.35 f(x) -> U1(x, x) 5.60/2.35 b -> d 5.60/2.35 b -> e 5.60/2.35 U1(d, x) -> x 5.60/2.35 5.60/2.35 Q is empty. 5.60/2.35 We have to consider all minimal (P,Q,R)-chains. 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (387) TransformationProof (EQUIVALENT) 5.60/2.35 By narrowing [LPAR04] the rule A^1 -> H(U1(a, e), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.35 5.60/2.35 (A^1 -> H(U1(d, e), U1(d, e)),A^1 -> H(U1(d, e), U1(d, e))) 5.60/2.35 (A^1 -> H(U1(e, e), U1(d, e)),A^1 -> H(U1(e, e), U1(d, e))) 5.60/2.35 (A^1 -> H(U1(a, e), e),A^1 -> H(U1(a, e), e)) 5.60/2.35 5.60/2.35 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (388) 5.60/2.35 Obligation: 5.60/2.35 Q DP problem: 5.60/2.35 The TRS P consists of the following rules: 5.60/2.35 5.60/2.35 H(x, x) -> G(x, x) 5.60/2.35 G(d, e) -> A^1 5.60/2.35 A^1 -> H(f(d), f(d)) 5.60/2.35 A^1 -> H(f(e), f(e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.35 A^1 -> H(U1(a, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, e), d) 5.60/2.35 A^1 -> H(U1(a, a), e) 5.60/2.35 A^1 -> H(d, U1(d, b)) 5.60/2.35 A^1 -> H(d, U1(b, d)) 5.60/2.35 A^1 -> H(d, U1(b, e)) 5.60/2.35 A^1 -> H(d, U1(d, d)) 5.60/2.35 A^1 -> H(U1(d, d), b) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.35 A^1 -> H(f(d), e) 5.60/2.35 A^1 -> H(U1(e, e), b) 5.60/2.35 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.35 A^1 -> H(f(e), e) 5.60/2.35 A^1 -> H(U1(d, d), d) 5.60/2.35 A^1 -> H(e, U1(d, b)) 5.60/2.35 A^1 -> H(e, U1(b, d)) 5.60/2.35 A^1 -> H(e, U1(b, e)) 5.60/2.35 A^1 -> H(e, U1(d, d)) 5.60/2.35 A^1 -> H(a, b) 5.60/2.35 A^1 -> H(a, U1(d, e)) 5.60/2.35 A^1 -> H(a, U1(e, d)) 5.60/2.35 A^1 -> H(a, d) 5.60/2.35 A^1 -> H(U1(d, e), b) 5.60/2.35 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(d, e), d) 5.60/2.35 A^1 -> H(U1(d, a), e) 5.60/2.35 A^1 -> H(U1(e, d), b) 5.60/2.35 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), e) 5.60/2.35 A^1 -> H(U1(a, e), e) 5.60/2.35 5.60/2.35 The TRS R consists of the following rules: 5.60/2.35 5.60/2.35 a -> d 5.60/2.35 a -> e 5.60/2.35 f(x) -> U1(x, x) 5.60/2.35 b -> d 5.60/2.35 b -> e 5.60/2.35 U1(d, x) -> x 5.60/2.35 5.60/2.35 Q is empty. 5.60/2.35 We have to consider all minimal (P,Q,R)-chains. 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (389) TransformationProof (EQUIVALENT) 5.60/2.35 By narrowing [LPAR04] the rule A^1 -> H(U1(a, e), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.35 5.60/2.35 (A^1 -> H(U1(d, e), U1(e, d)),A^1 -> H(U1(d, e), U1(e, d))) 5.60/2.35 (A^1 -> H(U1(e, e), U1(e, d)),A^1 -> H(U1(e, e), U1(e, d))) 5.60/2.35 5.60/2.35 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (390) 5.60/2.35 Obligation: 5.60/2.35 Q DP problem: 5.60/2.35 The TRS P consists of the following rules: 5.60/2.35 5.60/2.35 H(x, x) -> G(x, x) 5.60/2.35 G(d, e) -> A^1 5.60/2.35 A^1 -> H(f(d), f(d)) 5.60/2.35 A^1 -> H(f(e), f(e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.35 A^1 -> H(U1(a, e), d) 5.60/2.35 A^1 -> H(U1(a, a), e) 5.60/2.35 A^1 -> H(d, U1(d, b)) 5.60/2.35 A^1 -> H(d, U1(b, d)) 5.60/2.35 A^1 -> H(d, U1(b, e)) 5.60/2.35 A^1 -> H(d, U1(d, d)) 5.60/2.35 A^1 -> H(U1(d, d), b) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.35 A^1 -> H(f(d), e) 5.60/2.35 A^1 -> H(U1(e, e), b) 5.60/2.35 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.35 A^1 -> H(f(e), e) 5.60/2.35 A^1 -> H(U1(d, d), d) 5.60/2.35 A^1 -> H(e, U1(d, b)) 5.60/2.35 A^1 -> H(e, U1(b, d)) 5.60/2.35 A^1 -> H(e, U1(b, e)) 5.60/2.35 A^1 -> H(e, U1(d, d)) 5.60/2.35 A^1 -> H(a, b) 5.60/2.35 A^1 -> H(a, U1(d, e)) 5.60/2.35 A^1 -> H(a, U1(e, d)) 5.60/2.35 A^1 -> H(a, d) 5.60/2.35 A^1 -> H(U1(d, e), b) 5.60/2.35 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(d, e), d) 5.60/2.35 A^1 -> H(U1(d, a), e) 5.60/2.35 A^1 -> H(U1(e, d), b) 5.60/2.35 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), e) 5.60/2.35 A^1 -> H(U1(a, e), e) 5.60/2.35 A^1 -> H(U1(e, e), U1(e, d)) 5.60/2.35 5.60/2.35 The TRS R consists of the following rules: 5.60/2.35 5.60/2.35 a -> d 5.60/2.35 a -> e 5.60/2.35 f(x) -> U1(x, x) 5.60/2.35 b -> d 5.60/2.35 b -> e 5.60/2.35 U1(d, x) -> x 5.60/2.35 5.60/2.35 Q is empty. 5.60/2.35 We have to consider all minimal (P,Q,R)-chains. 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (391) DependencyGraphProof (EQUIVALENT) 5.60/2.35 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (392) 5.60/2.35 Obligation: 5.60/2.35 Q DP problem: 5.60/2.35 The TRS P consists of the following rules: 5.60/2.35 5.60/2.35 G(d, e) -> A^1 5.60/2.35 A^1 -> H(f(d), f(d)) 5.60/2.35 H(x, x) -> G(x, x) 5.60/2.35 A^1 -> H(f(e), f(e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.35 A^1 -> H(U1(a, e), d) 5.60/2.35 A^1 -> H(U1(a, a), e) 5.60/2.35 A^1 -> H(d, U1(d, b)) 5.60/2.35 A^1 -> H(d, U1(b, d)) 5.60/2.35 A^1 -> H(d, U1(b, e)) 5.60/2.35 A^1 -> H(d, U1(d, d)) 5.60/2.35 A^1 -> H(U1(d, d), b) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.35 A^1 -> H(f(d), e) 5.60/2.35 A^1 -> H(U1(e, e), b) 5.60/2.35 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.35 A^1 -> H(f(e), e) 5.60/2.35 A^1 -> H(U1(d, d), d) 5.60/2.35 A^1 -> H(e, U1(d, b)) 5.60/2.35 A^1 -> H(e, U1(b, d)) 5.60/2.35 A^1 -> H(e, U1(b, e)) 5.60/2.35 A^1 -> H(e, U1(d, d)) 5.60/2.35 A^1 -> H(a, b) 5.60/2.35 A^1 -> H(a, U1(d, e)) 5.60/2.35 A^1 -> H(a, U1(e, d)) 5.60/2.35 A^1 -> H(a, d) 5.60/2.35 A^1 -> H(U1(d, e), b) 5.60/2.35 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.35 A^1 -> H(U1(d, e), d) 5.60/2.35 A^1 -> H(U1(d, a), e) 5.60/2.35 A^1 -> H(U1(e, d), b) 5.60/2.35 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.35 A^1 -> H(U1(a, d), e) 5.60/2.35 A^1 -> H(U1(a, e), e) 5.60/2.35 5.60/2.35 The TRS R consists of the following rules: 5.60/2.35 5.60/2.35 a -> d 5.60/2.35 a -> e 5.60/2.35 f(x) -> U1(x, x) 5.60/2.35 b -> d 5.60/2.35 b -> e 5.60/2.35 U1(d, x) -> x 5.60/2.35 5.60/2.35 Q is empty. 5.60/2.35 We have to consider all minimal (P,Q,R)-chains. 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (393) TransformationProof (EQUIVALENT) 5.60/2.35 By narrowing [LPAR04] the rule A^1 -> H(U1(a, e), d) at position [] we obtained the following new rules [LPAR04]: 5.60/2.35 5.60/2.35 (A^1 -> H(U1(d, e), d),A^1 -> H(U1(d, e), d)) 5.60/2.35 (A^1 -> H(U1(e, e), d),A^1 -> H(U1(e, e), d)) 5.60/2.35 5.60/2.35 5.60/2.35 ---------------------------------------- 5.60/2.35 5.60/2.35 (394) 5.60/2.35 Obligation: 5.60/2.35 Q DP problem: 5.60/2.35 The TRS P consists of the following rules: 5.60/2.35 5.60/2.35 G(d, e) -> A^1 5.60/2.35 A^1 -> H(f(d), f(d)) 5.60/2.35 H(x, x) -> G(x, x) 5.60/2.35 A^1 -> H(f(e), f(e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.35 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.35 A^1 -> H(U1(a, a), e) 5.60/2.35 A^1 -> H(d, U1(d, b)) 5.60/2.35 A^1 -> H(d, U1(b, d)) 5.60/2.35 A^1 -> H(d, U1(b, e)) 5.60/2.35 A^1 -> H(d, U1(d, d)) 5.60/2.35 A^1 -> H(U1(d, d), b) 5.60/2.35 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.35 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.35 A^1 -> H(f(d), e) 5.60/2.35 A^1 -> H(U1(e, e), b) 5.60/2.35 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.35 A^1 -> H(f(e), e) 5.60/2.35 A^1 -> H(U1(d, d), d) 5.60/2.35 A^1 -> H(e, U1(d, b)) 5.60/2.35 A^1 -> H(e, U1(b, d)) 5.60/2.35 A^1 -> H(e, U1(b, e)) 5.60/2.35 A^1 -> H(e, U1(d, d)) 5.60/2.35 A^1 -> H(a, b) 5.60/2.36 A^1 -> H(a, U1(d, e)) 5.60/2.36 A^1 -> H(a, U1(e, d)) 5.60/2.36 A^1 -> H(a, d) 5.60/2.36 A^1 -> H(U1(d, e), b) 5.60/2.36 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.36 A^1 -> H(U1(d, e), d) 5.60/2.36 A^1 -> H(U1(d, a), e) 5.60/2.36 A^1 -> H(U1(e, d), b) 5.60/2.36 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.36 A^1 -> H(U1(a, d), e) 5.60/2.36 A^1 -> H(U1(a, e), e) 5.60/2.36 A^1 -> H(U1(e, e), d) 5.60/2.36 5.60/2.36 The TRS R consists of the following rules: 5.60/2.36 5.60/2.36 a -> d 5.60/2.36 a -> e 5.60/2.36 f(x) -> U1(x, x) 5.60/2.36 b -> d 5.60/2.36 b -> e 5.60/2.36 U1(d, x) -> x 5.60/2.36 5.60/2.36 Q is empty. 5.60/2.36 We have to consider all minimal (P,Q,R)-chains. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (395) DependencyGraphProof (EQUIVALENT) 5.60/2.36 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (396) 5.60/2.36 Obligation: 5.60/2.36 Q DP problem: 5.60/2.36 The TRS P consists of the following rules: 5.60/2.36 5.60/2.36 A^1 -> H(f(d), f(d)) 5.60/2.36 H(x, x) -> G(x, x) 5.60/2.36 G(d, e) -> A^1 5.60/2.36 A^1 -> H(f(e), f(e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.36 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.36 A^1 -> H(U1(a, a), e) 5.60/2.36 A^1 -> H(d, U1(d, b)) 5.60/2.36 A^1 -> H(d, U1(b, d)) 5.60/2.36 A^1 -> H(d, U1(b, e)) 5.60/2.36 A^1 -> H(d, U1(d, d)) 5.60/2.36 A^1 -> H(U1(d, d), b) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.36 A^1 -> H(f(d), e) 5.60/2.36 A^1 -> H(U1(e, e), b) 5.60/2.36 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.36 A^1 -> H(f(e), e) 5.60/2.36 A^1 -> H(U1(d, d), d) 5.60/2.36 A^1 -> H(e, U1(d, b)) 5.60/2.36 A^1 -> H(e, U1(b, d)) 5.60/2.36 A^1 -> H(e, U1(b, e)) 5.60/2.36 A^1 -> H(e, U1(d, d)) 5.60/2.36 A^1 -> H(a, b) 5.60/2.36 A^1 -> H(a, U1(d, e)) 5.60/2.36 A^1 -> H(a, U1(e, d)) 5.60/2.36 A^1 -> H(a, d) 5.60/2.36 A^1 -> H(U1(d, e), b) 5.60/2.36 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.36 A^1 -> H(U1(d, e), d) 5.60/2.36 A^1 -> H(U1(d, a), e) 5.60/2.36 A^1 -> H(U1(e, d), b) 5.60/2.36 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.36 A^1 -> H(U1(a, d), e) 5.60/2.36 A^1 -> H(U1(a, e), e) 5.60/2.36 5.60/2.36 The TRS R consists of the following rules: 5.60/2.36 5.60/2.36 a -> d 5.60/2.36 a -> e 5.60/2.36 f(x) -> U1(x, x) 5.60/2.36 b -> d 5.60/2.36 b -> e 5.60/2.36 U1(d, x) -> x 5.60/2.36 5.60/2.36 Q is empty. 5.60/2.36 We have to consider all minimal (P,Q,R)-chains. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (397) TransformationProof (EQUIVALENT) 5.60/2.36 By narrowing [LPAR04] the rule A^1 -> H(U1(a, a), e) at position [] we obtained the following new rules [LPAR04]: 5.60/2.36 5.60/2.36 (A^1 -> H(U1(d, a), e),A^1 -> H(U1(d, a), e)) 5.60/2.36 (A^1 -> H(U1(e, a), e),A^1 -> H(U1(e, a), e)) 5.60/2.36 (A^1 -> H(U1(a, d), e),A^1 -> H(U1(a, d), e)) 5.60/2.36 (A^1 -> H(U1(a, e), e),A^1 -> H(U1(a, e), e)) 5.60/2.36 5.60/2.36 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (398) 5.60/2.36 Obligation: 5.60/2.36 Q DP problem: 5.60/2.36 The TRS P consists of the following rules: 5.60/2.36 5.60/2.36 A^1 -> H(f(d), f(d)) 5.60/2.36 H(x, x) -> G(x, x) 5.60/2.36 G(d, e) -> A^1 5.60/2.36 A^1 -> H(f(e), f(e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.36 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.36 A^1 -> H(d, U1(d, b)) 5.60/2.36 A^1 -> H(d, U1(b, d)) 5.60/2.36 A^1 -> H(d, U1(b, e)) 5.60/2.36 A^1 -> H(d, U1(d, d)) 5.60/2.36 A^1 -> H(U1(d, d), b) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.36 A^1 -> H(f(d), e) 5.60/2.36 A^1 -> H(U1(e, e), b) 5.60/2.36 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.36 A^1 -> H(f(e), e) 5.60/2.36 A^1 -> H(U1(d, d), d) 5.60/2.36 A^1 -> H(e, U1(d, b)) 5.60/2.36 A^1 -> H(e, U1(b, d)) 5.60/2.36 A^1 -> H(e, U1(b, e)) 5.60/2.36 A^1 -> H(e, U1(d, d)) 5.60/2.36 A^1 -> H(a, b) 5.60/2.36 A^1 -> H(a, U1(d, e)) 5.60/2.36 A^1 -> H(a, U1(e, d)) 5.60/2.36 A^1 -> H(a, d) 5.60/2.36 A^1 -> H(U1(d, e), b) 5.60/2.36 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.36 A^1 -> H(U1(d, e), d) 5.60/2.36 A^1 -> H(U1(d, a), e) 5.60/2.36 A^1 -> H(U1(e, d), b) 5.60/2.36 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.36 A^1 -> H(U1(a, d), e) 5.60/2.36 A^1 -> H(U1(a, e), e) 5.60/2.36 A^1 -> H(U1(e, a), e) 5.60/2.36 5.60/2.36 The TRS R consists of the following rules: 5.60/2.36 5.60/2.36 a -> d 5.60/2.36 a -> e 5.60/2.36 f(x) -> U1(x, x) 5.60/2.36 b -> d 5.60/2.36 b -> e 5.60/2.36 U1(d, x) -> x 5.60/2.36 5.60/2.36 Q is empty. 5.60/2.36 We have to consider all minimal (P,Q,R)-chains. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (399) DependencyGraphProof (EQUIVALENT) 5.60/2.36 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (400) 5.60/2.36 Obligation: 5.60/2.36 Q DP problem: 5.60/2.36 The TRS P consists of the following rules: 5.60/2.36 5.60/2.36 H(x, x) -> G(x, x) 5.60/2.36 G(d, e) -> A^1 5.60/2.36 A^1 -> H(f(d), f(d)) 5.60/2.36 A^1 -> H(f(e), f(e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.36 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.36 A^1 -> H(d, U1(d, b)) 5.60/2.36 A^1 -> H(d, U1(b, d)) 5.60/2.36 A^1 -> H(d, U1(b, e)) 5.60/2.36 A^1 -> H(d, U1(d, d)) 5.60/2.36 A^1 -> H(U1(d, d), b) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.36 A^1 -> H(f(d), e) 5.60/2.36 A^1 -> H(U1(e, e), b) 5.60/2.36 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.36 A^1 -> H(f(e), e) 5.60/2.36 A^1 -> H(U1(d, d), d) 5.60/2.36 A^1 -> H(e, U1(d, b)) 5.60/2.36 A^1 -> H(e, U1(b, d)) 5.60/2.36 A^1 -> H(e, U1(b, e)) 5.60/2.36 A^1 -> H(e, U1(d, d)) 5.60/2.36 A^1 -> H(a, b) 5.60/2.36 A^1 -> H(a, U1(d, e)) 5.60/2.36 A^1 -> H(a, U1(e, d)) 5.60/2.36 A^1 -> H(a, d) 5.60/2.36 A^1 -> H(U1(d, e), b) 5.60/2.36 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.36 A^1 -> H(U1(d, e), d) 5.60/2.36 A^1 -> H(U1(d, a), e) 5.60/2.36 A^1 -> H(U1(e, d), b) 5.60/2.36 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.36 A^1 -> H(U1(a, d), e) 5.60/2.36 A^1 -> H(U1(a, e), e) 5.60/2.36 5.60/2.36 The TRS R consists of the following rules: 5.60/2.36 5.60/2.36 a -> d 5.60/2.36 a -> e 5.60/2.36 f(x) -> U1(x, x) 5.60/2.36 b -> d 5.60/2.36 b -> e 5.60/2.36 U1(d, x) -> x 5.60/2.36 5.60/2.36 Q is empty. 5.60/2.36 We have to consider all minimal (P,Q,R)-chains. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (401) TransformationProof (EQUIVALENT) 5.60/2.36 By narrowing [LPAR04] the rule A^1 -> H(d, U1(d, b)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.36 5.60/2.36 (A^1 -> H(d, b),A^1 -> H(d, b)) 5.60/2.36 (A^1 -> H(d, U1(d, d)),A^1 -> H(d, U1(d, d))) 5.60/2.36 (A^1 -> H(d, U1(d, e)),A^1 -> H(d, U1(d, e))) 5.60/2.36 5.60/2.36 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (402) 5.60/2.36 Obligation: 5.60/2.36 Q DP problem: 5.60/2.36 The TRS P consists of the following rules: 5.60/2.36 5.60/2.36 H(x, x) -> G(x, x) 5.60/2.36 G(d, e) -> A^1 5.60/2.36 A^1 -> H(f(d), f(d)) 5.60/2.36 A^1 -> H(f(e), f(e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.36 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.36 A^1 -> H(d, U1(b, d)) 5.60/2.36 A^1 -> H(d, U1(b, e)) 5.60/2.36 A^1 -> H(d, U1(d, d)) 5.60/2.36 A^1 -> H(U1(d, d), b) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.36 A^1 -> H(f(d), e) 5.60/2.36 A^1 -> H(U1(e, e), b) 5.60/2.36 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.36 A^1 -> H(f(e), e) 5.60/2.36 A^1 -> H(U1(d, d), d) 5.60/2.36 A^1 -> H(e, U1(d, b)) 5.60/2.36 A^1 -> H(e, U1(b, d)) 5.60/2.36 A^1 -> H(e, U1(b, e)) 5.60/2.36 A^1 -> H(e, U1(d, d)) 5.60/2.36 A^1 -> H(a, b) 5.60/2.36 A^1 -> H(a, U1(d, e)) 5.60/2.36 A^1 -> H(a, U1(e, d)) 5.60/2.36 A^1 -> H(a, d) 5.60/2.36 A^1 -> H(U1(d, e), b) 5.60/2.36 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.36 A^1 -> H(U1(d, e), d) 5.60/2.36 A^1 -> H(U1(d, a), e) 5.60/2.36 A^1 -> H(U1(e, d), b) 5.60/2.36 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.36 A^1 -> H(U1(a, d), e) 5.60/2.36 A^1 -> H(U1(a, e), e) 5.60/2.36 A^1 -> H(d, b) 5.60/2.36 A^1 -> H(d, U1(d, e)) 5.60/2.36 5.60/2.36 The TRS R consists of the following rules: 5.60/2.36 5.60/2.36 a -> d 5.60/2.36 a -> e 5.60/2.36 f(x) -> U1(x, x) 5.60/2.36 b -> d 5.60/2.36 b -> e 5.60/2.36 U1(d, x) -> x 5.60/2.36 5.60/2.36 Q is empty. 5.60/2.36 We have to consider all minimal (P,Q,R)-chains. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (403) TransformationProof (EQUIVALENT) 5.60/2.36 By narrowing [LPAR04] the rule A^1 -> H(d, U1(b, d)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.36 5.60/2.36 (A^1 -> H(d, U1(d, d)),A^1 -> H(d, U1(d, d))) 5.60/2.36 (A^1 -> H(d, U1(e, d)),A^1 -> H(d, U1(e, d))) 5.60/2.36 5.60/2.36 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (404) 5.60/2.36 Obligation: 5.60/2.36 Q DP problem: 5.60/2.36 The TRS P consists of the following rules: 5.60/2.36 5.60/2.36 H(x, x) -> G(x, x) 5.60/2.36 G(d, e) -> A^1 5.60/2.36 A^1 -> H(f(d), f(d)) 5.60/2.36 A^1 -> H(f(e), f(e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.36 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.36 A^1 -> H(d, U1(b, e)) 5.60/2.36 A^1 -> H(d, U1(d, d)) 5.60/2.36 A^1 -> H(U1(d, d), b) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.36 A^1 -> H(f(d), e) 5.60/2.36 A^1 -> H(U1(e, e), b) 5.60/2.36 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.36 A^1 -> H(f(e), e) 5.60/2.36 A^1 -> H(U1(d, d), d) 5.60/2.36 A^1 -> H(e, U1(d, b)) 5.60/2.36 A^1 -> H(e, U1(b, d)) 5.60/2.36 A^1 -> H(e, U1(b, e)) 5.60/2.36 A^1 -> H(e, U1(d, d)) 5.60/2.36 A^1 -> H(a, b) 5.60/2.36 A^1 -> H(a, U1(d, e)) 5.60/2.36 A^1 -> H(a, U1(e, d)) 5.60/2.36 A^1 -> H(a, d) 5.60/2.36 A^1 -> H(U1(d, e), b) 5.60/2.36 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.36 A^1 -> H(U1(d, e), d) 5.60/2.36 A^1 -> H(U1(d, a), e) 5.60/2.36 A^1 -> H(U1(e, d), b) 5.60/2.36 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.36 A^1 -> H(U1(a, d), e) 5.60/2.36 A^1 -> H(U1(a, e), e) 5.60/2.36 A^1 -> H(d, b) 5.60/2.36 A^1 -> H(d, U1(d, e)) 5.60/2.36 A^1 -> H(d, U1(e, d)) 5.60/2.36 5.60/2.36 The TRS R consists of the following rules: 5.60/2.36 5.60/2.36 a -> d 5.60/2.36 a -> e 5.60/2.36 f(x) -> U1(x, x) 5.60/2.36 b -> d 5.60/2.36 b -> e 5.60/2.36 U1(d, x) -> x 5.60/2.36 5.60/2.36 Q is empty. 5.60/2.36 We have to consider all minimal (P,Q,R)-chains. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (405) DependencyGraphProof (EQUIVALENT) 5.60/2.36 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (406) 5.60/2.36 Obligation: 5.60/2.36 Q DP problem: 5.60/2.36 The TRS P consists of the following rules: 5.60/2.36 5.60/2.36 G(d, e) -> A^1 5.60/2.36 A^1 -> H(f(d), f(d)) 5.60/2.36 H(x, x) -> G(x, x) 5.60/2.36 A^1 -> H(f(e), f(e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.36 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.36 A^1 -> H(d, U1(b, e)) 5.60/2.36 A^1 -> H(d, U1(d, d)) 5.60/2.36 A^1 -> H(U1(d, d), b) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.36 A^1 -> H(f(d), e) 5.60/2.36 A^1 -> H(U1(e, e), b) 5.60/2.36 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.36 A^1 -> H(f(e), e) 5.60/2.36 A^1 -> H(U1(d, d), d) 5.60/2.36 A^1 -> H(e, U1(d, b)) 5.60/2.36 A^1 -> H(e, U1(b, d)) 5.60/2.36 A^1 -> H(e, U1(b, e)) 5.60/2.36 A^1 -> H(e, U1(d, d)) 5.60/2.36 A^1 -> H(a, b) 5.60/2.36 A^1 -> H(a, U1(d, e)) 5.60/2.36 A^1 -> H(a, U1(e, d)) 5.60/2.36 A^1 -> H(a, d) 5.60/2.36 A^1 -> H(U1(d, e), b) 5.60/2.36 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.36 A^1 -> H(U1(d, e), d) 5.60/2.36 A^1 -> H(U1(d, a), e) 5.60/2.36 A^1 -> H(U1(e, d), b) 5.60/2.36 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.36 A^1 -> H(U1(a, d), e) 5.60/2.36 A^1 -> H(U1(a, e), e) 5.60/2.36 A^1 -> H(d, b) 5.60/2.36 A^1 -> H(d, U1(d, e)) 5.60/2.36 5.60/2.36 The TRS R consists of the following rules: 5.60/2.36 5.60/2.36 a -> d 5.60/2.36 a -> e 5.60/2.36 f(x) -> U1(x, x) 5.60/2.36 b -> d 5.60/2.36 b -> e 5.60/2.36 U1(d, x) -> x 5.60/2.36 5.60/2.36 Q is empty. 5.60/2.36 We have to consider all minimal (P,Q,R)-chains. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (407) TransformationProof (EQUIVALENT) 5.60/2.36 By narrowing [LPAR04] the rule A^1 -> H(d, U1(b, e)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.36 5.60/2.36 (A^1 -> H(d, U1(d, e)),A^1 -> H(d, U1(d, e))) 5.60/2.36 (A^1 -> H(d, U1(e, e)),A^1 -> H(d, U1(e, e))) 5.60/2.36 5.60/2.36 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (408) 5.60/2.36 Obligation: 5.60/2.36 Q DP problem: 5.60/2.36 The TRS P consists of the following rules: 5.60/2.36 5.60/2.36 G(d, e) -> A^1 5.60/2.36 A^1 -> H(f(d), f(d)) 5.60/2.36 H(x, x) -> G(x, x) 5.60/2.36 A^1 -> H(f(e), f(e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.36 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.36 A^1 -> H(d, U1(d, d)) 5.60/2.36 A^1 -> H(U1(d, d), b) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.36 A^1 -> H(f(d), e) 5.60/2.36 A^1 -> H(U1(e, e), b) 5.60/2.36 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.36 A^1 -> H(f(e), e) 5.60/2.36 A^1 -> H(U1(d, d), d) 5.60/2.36 A^1 -> H(e, U1(d, b)) 5.60/2.36 A^1 -> H(e, U1(b, d)) 5.60/2.36 A^1 -> H(e, U1(b, e)) 5.60/2.36 A^1 -> H(e, U1(d, d)) 5.60/2.36 A^1 -> H(a, b) 5.60/2.36 A^1 -> H(a, U1(d, e)) 5.60/2.36 A^1 -> H(a, U1(e, d)) 5.60/2.36 A^1 -> H(a, d) 5.60/2.36 A^1 -> H(U1(d, e), b) 5.60/2.36 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.36 A^1 -> H(U1(d, e), d) 5.60/2.36 A^1 -> H(U1(d, a), e) 5.60/2.36 A^1 -> H(U1(e, d), b) 5.60/2.36 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.36 A^1 -> H(U1(a, d), e) 5.60/2.36 A^1 -> H(U1(a, e), e) 5.60/2.36 A^1 -> H(d, b) 5.60/2.36 A^1 -> H(d, U1(d, e)) 5.60/2.36 A^1 -> H(d, U1(e, e)) 5.60/2.36 5.60/2.36 The TRS R consists of the following rules: 5.60/2.36 5.60/2.36 a -> d 5.60/2.36 a -> e 5.60/2.36 f(x) -> U1(x, x) 5.60/2.36 b -> d 5.60/2.36 b -> e 5.60/2.36 U1(d, x) -> x 5.60/2.36 5.60/2.36 Q is empty. 5.60/2.36 We have to consider all minimal (P,Q,R)-chains. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (409) DependencyGraphProof (EQUIVALENT) 5.60/2.36 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (410) 5.60/2.36 Obligation: 5.60/2.36 Q DP problem: 5.60/2.36 The TRS P consists of the following rules: 5.60/2.36 5.60/2.36 A^1 -> H(f(d), f(d)) 5.60/2.36 H(x, x) -> G(x, x) 5.60/2.36 G(d, e) -> A^1 5.60/2.36 A^1 -> H(f(e), f(e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.36 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.36 A^1 -> H(d, U1(d, d)) 5.60/2.36 A^1 -> H(U1(d, d), b) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.36 A^1 -> H(f(d), e) 5.60/2.36 A^1 -> H(U1(e, e), b) 5.60/2.36 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.36 A^1 -> H(f(e), e) 5.60/2.36 A^1 -> H(U1(d, d), d) 5.60/2.36 A^1 -> H(e, U1(d, b)) 5.60/2.36 A^1 -> H(e, U1(b, d)) 5.60/2.36 A^1 -> H(e, U1(b, e)) 5.60/2.36 A^1 -> H(e, U1(d, d)) 5.60/2.36 A^1 -> H(a, b) 5.60/2.36 A^1 -> H(a, U1(d, e)) 5.60/2.36 A^1 -> H(a, U1(e, d)) 5.60/2.36 A^1 -> H(a, d) 5.60/2.36 A^1 -> H(U1(d, e), b) 5.60/2.36 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.36 A^1 -> H(U1(d, e), d) 5.60/2.36 A^1 -> H(U1(d, a), e) 5.60/2.36 A^1 -> H(U1(e, d), b) 5.60/2.36 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.36 A^1 -> H(U1(a, d), e) 5.60/2.36 A^1 -> H(U1(a, e), e) 5.60/2.36 A^1 -> H(d, b) 5.60/2.36 A^1 -> H(d, U1(d, e)) 5.60/2.36 5.60/2.36 The TRS R consists of the following rules: 5.60/2.36 5.60/2.36 a -> d 5.60/2.36 a -> e 5.60/2.36 f(x) -> U1(x, x) 5.60/2.36 b -> d 5.60/2.36 b -> e 5.60/2.36 U1(d, x) -> x 5.60/2.36 5.60/2.36 Q is empty. 5.60/2.36 We have to consider all minimal (P,Q,R)-chains. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (411) TransformationProof (EQUIVALENT) 5.60/2.36 By narrowing [LPAR04] the rule A^1 -> H(d, U1(d, d)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.36 5.60/2.36 (A^1 -> H(d, d),A^1 -> H(d, d)) 5.60/2.36 5.60/2.36 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (412) 5.60/2.36 Obligation: 5.60/2.36 Q DP problem: 5.60/2.36 The TRS P consists of the following rules: 5.60/2.36 5.60/2.36 A^1 -> H(f(d), f(d)) 5.60/2.36 H(x, x) -> G(x, x) 5.60/2.36 G(d, e) -> A^1 5.60/2.36 A^1 -> H(f(e), f(e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.36 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.36 A^1 -> H(U1(d, d), b) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.36 A^1 -> H(f(d), e) 5.60/2.36 A^1 -> H(U1(e, e), b) 5.60/2.36 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.36 A^1 -> H(f(e), e) 5.60/2.36 A^1 -> H(U1(d, d), d) 5.60/2.36 A^1 -> H(e, U1(d, b)) 5.60/2.36 A^1 -> H(e, U1(b, d)) 5.60/2.36 A^1 -> H(e, U1(b, e)) 5.60/2.36 A^1 -> H(e, U1(d, d)) 5.60/2.36 A^1 -> H(a, b) 5.60/2.36 A^1 -> H(a, U1(d, e)) 5.60/2.36 A^1 -> H(a, U1(e, d)) 5.60/2.36 A^1 -> H(a, d) 5.60/2.36 A^1 -> H(U1(d, e), b) 5.60/2.36 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.36 A^1 -> H(U1(d, e), d) 5.60/2.36 A^1 -> H(U1(d, a), e) 5.60/2.36 A^1 -> H(U1(e, d), b) 5.60/2.36 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.36 A^1 -> H(U1(a, d), e) 5.60/2.36 A^1 -> H(U1(a, e), e) 5.60/2.36 A^1 -> H(d, b) 5.60/2.36 A^1 -> H(d, U1(d, e)) 5.60/2.36 A^1 -> H(d, d) 5.60/2.36 5.60/2.36 The TRS R consists of the following rules: 5.60/2.36 5.60/2.36 a -> d 5.60/2.36 a -> e 5.60/2.36 f(x) -> U1(x, x) 5.60/2.36 b -> d 5.60/2.36 b -> e 5.60/2.36 U1(d, x) -> x 5.60/2.36 5.60/2.36 Q is empty. 5.60/2.36 We have to consider all minimal (P,Q,R)-chains. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (413) TransformationProof (EQUIVALENT) 5.60/2.36 By narrowing [LPAR04] the rule A^1 -> H(U1(d, d), b) at position [] we obtained the following new rules [LPAR04]: 5.60/2.36 5.60/2.36 (A^1 -> H(d, b),A^1 -> H(d, b)) 5.60/2.36 (A^1 -> H(U1(d, d), d),A^1 -> H(U1(d, d), d)) 5.60/2.36 (A^1 -> H(U1(d, d), e),A^1 -> H(U1(d, d), e)) 5.60/2.36 5.60/2.36 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (414) 5.60/2.36 Obligation: 5.60/2.36 Q DP problem: 5.60/2.36 The TRS P consists of the following rules: 5.60/2.36 5.60/2.36 A^1 -> H(f(d), f(d)) 5.60/2.36 H(x, x) -> G(x, x) 5.60/2.36 G(d, e) -> A^1 5.60/2.36 A^1 -> H(f(e), f(e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.36 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.36 A^1 -> H(f(d), e) 5.60/2.36 A^1 -> H(U1(e, e), b) 5.60/2.36 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.36 A^1 -> H(f(e), e) 5.60/2.36 A^1 -> H(U1(d, d), d) 5.60/2.36 A^1 -> H(e, U1(d, b)) 5.60/2.36 A^1 -> H(e, U1(b, d)) 5.60/2.36 A^1 -> H(e, U1(b, e)) 5.60/2.36 A^1 -> H(e, U1(d, d)) 5.60/2.36 A^1 -> H(a, b) 5.60/2.36 A^1 -> H(a, U1(d, e)) 5.60/2.36 A^1 -> H(a, U1(e, d)) 5.60/2.36 A^1 -> H(a, d) 5.60/2.36 A^1 -> H(U1(d, e), b) 5.60/2.36 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.36 A^1 -> H(U1(d, e), d) 5.60/2.36 A^1 -> H(U1(d, a), e) 5.60/2.36 A^1 -> H(U1(e, d), b) 5.60/2.36 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.36 A^1 -> H(U1(a, d), e) 5.60/2.36 A^1 -> H(U1(a, e), e) 5.60/2.36 A^1 -> H(d, b) 5.60/2.36 A^1 -> H(d, U1(d, e)) 5.60/2.36 A^1 -> H(d, d) 5.60/2.36 A^1 -> H(U1(d, d), e) 5.60/2.36 5.60/2.36 The TRS R consists of the following rules: 5.60/2.36 5.60/2.36 a -> d 5.60/2.36 a -> e 5.60/2.36 f(x) -> U1(x, x) 5.60/2.36 b -> d 5.60/2.36 b -> e 5.60/2.36 U1(d, x) -> x 5.60/2.36 5.60/2.36 Q is empty. 5.60/2.36 We have to consider all minimal (P,Q,R)-chains. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (415) TransformationProof (EQUIVALENT) 5.60/2.36 By narrowing [LPAR04] the rule A^1 -> H(U1(d, d), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.36 5.60/2.36 (A^1 -> H(d, U1(d, e)),A^1 -> H(d, U1(d, e))) 5.60/2.36 (A^1 -> H(U1(d, d), e),A^1 -> H(U1(d, d), e)) 5.60/2.36 5.60/2.36 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (416) 5.60/2.36 Obligation: 5.60/2.36 Q DP problem: 5.60/2.36 The TRS P consists of the following rules: 5.60/2.36 5.60/2.36 A^1 -> H(f(d), f(d)) 5.60/2.36 H(x, x) -> G(x, x) 5.60/2.36 G(d, e) -> A^1 5.60/2.36 A^1 -> H(f(e), f(e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.36 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(e, d)) 5.60/2.36 A^1 -> H(f(d), e) 5.60/2.36 A^1 -> H(U1(e, e), b) 5.60/2.36 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.36 A^1 -> H(f(e), e) 5.60/2.36 A^1 -> H(U1(d, d), d) 5.60/2.36 A^1 -> H(e, U1(d, b)) 5.60/2.36 A^1 -> H(e, U1(b, d)) 5.60/2.36 A^1 -> H(e, U1(b, e)) 5.60/2.36 A^1 -> H(e, U1(d, d)) 5.60/2.36 A^1 -> H(a, b) 5.60/2.36 A^1 -> H(a, U1(d, e)) 5.60/2.36 A^1 -> H(a, U1(e, d)) 5.60/2.36 A^1 -> H(a, d) 5.60/2.36 A^1 -> H(U1(d, e), b) 5.60/2.36 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.36 A^1 -> H(U1(d, e), d) 5.60/2.36 A^1 -> H(U1(d, a), e) 5.60/2.36 A^1 -> H(U1(e, d), b) 5.60/2.36 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.36 A^1 -> H(U1(a, d), e) 5.60/2.36 A^1 -> H(U1(a, e), e) 5.60/2.36 A^1 -> H(d, b) 5.60/2.36 A^1 -> H(d, U1(d, e)) 5.60/2.36 A^1 -> H(d, d) 5.60/2.36 A^1 -> H(U1(d, d), e) 5.60/2.36 5.60/2.36 The TRS R consists of the following rules: 5.60/2.36 5.60/2.36 a -> d 5.60/2.36 a -> e 5.60/2.36 f(x) -> U1(x, x) 5.60/2.36 b -> d 5.60/2.36 b -> e 5.60/2.36 U1(d, x) -> x 5.60/2.36 5.60/2.36 Q is empty. 5.60/2.36 We have to consider all minimal (P,Q,R)-chains. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (417) TransformationProof (EQUIVALENT) 5.60/2.36 By narrowing [LPAR04] the rule A^1 -> H(U1(d, d), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.36 5.60/2.36 (A^1 -> H(d, U1(e, d)),A^1 -> H(d, U1(e, d))) 5.60/2.36 5.60/2.36 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (418) 5.60/2.36 Obligation: 5.60/2.36 Q DP problem: 5.60/2.36 The TRS P consists of the following rules: 5.60/2.36 5.60/2.36 A^1 -> H(f(d), f(d)) 5.60/2.36 H(x, x) -> G(x, x) 5.60/2.36 G(d, e) -> A^1 5.60/2.36 A^1 -> H(f(e), f(e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.36 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.36 A^1 -> H(f(d), e) 5.60/2.36 A^1 -> H(U1(e, e), b) 5.60/2.36 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.36 A^1 -> H(f(e), e) 5.60/2.36 A^1 -> H(U1(d, d), d) 5.60/2.36 A^1 -> H(e, U1(d, b)) 5.60/2.36 A^1 -> H(e, U1(b, d)) 5.60/2.36 A^1 -> H(e, U1(b, e)) 5.60/2.36 A^1 -> H(e, U1(d, d)) 5.60/2.36 A^1 -> H(a, b) 5.60/2.36 A^1 -> H(a, U1(d, e)) 5.60/2.36 A^1 -> H(a, U1(e, d)) 5.60/2.36 A^1 -> H(a, d) 5.60/2.36 A^1 -> H(U1(d, e), b) 5.60/2.36 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.36 A^1 -> H(U1(d, e), d) 5.60/2.36 A^1 -> H(U1(d, a), e) 5.60/2.36 A^1 -> H(U1(e, d), b) 5.60/2.36 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.36 A^1 -> H(U1(a, d), e) 5.60/2.36 A^1 -> H(U1(a, e), e) 5.60/2.36 A^1 -> H(d, b) 5.60/2.36 A^1 -> H(d, U1(d, e)) 5.60/2.36 A^1 -> H(d, d) 5.60/2.36 A^1 -> H(U1(d, d), e) 5.60/2.36 A^1 -> H(d, U1(e, d)) 5.60/2.36 5.60/2.36 The TRS R consists of the following rules: 5.60/2.36 5.60/2.36 a -> d 5.60/2.36 a -> e 5.60/2.36 f(x) -> U1(x, x) 5.60/2.36 b -> d 5.60/2.36 b -> e 5.60/2.36 U1(d, x) -> x 5.60/2.36 5.60/2.36 Q is empty. 5.60/2.36 We have to consider all minimal (P,Q,R)-chains. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (419) DependencyGraphProof (EQUIVALENT) 5.60/2.36 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (420) 5.60/2.36 Obligation: 5.60/2.36 Q DP problem: 5.60/2.36 The TRS P consists of the following rules: 5.60/2.36 5.60/2.36 H(x, x) -> G(x, x) 5.60/2.36 G(d, e) -> A^1 5.60/2.36 A^1 -> H(f(d), f(d)) 5.60/2.36 A^1 -> H(f(e), f(e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.36 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.36 A^1 -> H(f(d), e) 5.60/2.36 A^1 -> H(U1(e, e), b) 5.60/2.36 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.36 A^1 -> H(f(e), e) 5.60/2.36 A^1 -> H(U1(d, d), d) 5.60/2.36 A^1 -> H(e, U1(d, b)) 5.60/2.36 A^1 -> H(e, U1(b, d)) 5.60/2.36 A^1 -> H(e, U1(b, e)) 5.60/2.36 A^1 -> H(e, U1(d, d)) 5.60/2.36 A^1 -> H(a, b) 5.60/2.36 A^1 -> H(a, U1(d, e)) 5.60/2.36 A^1 -> H(a, U1(e, d)) 5.60/2.36 A^1 -> H(a, d) 5.60/2.36 A^1 -> H(U1(d, e), b) 5.60/2.36 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.36 A^1 -> H(U1(d, e), d) 5.60/2.36 A^1 -> H(U1(d, a), e) 5.60/2.36 A^1 -> H(U1(e, d), b) 5.60/2.36 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.36 A^1 -> H(U1(a, d), e) 5.60/2.36 A^1 -> H(U1(a, e), e) 5.60/2.36 A^1 -> H(d, b) 5.60/2.36 A^1 -> H(d, U1(d, e)) 5.60/2.36 A^1 -> H(d, d) 5.60/2.36 A^1 -> H(U1(d, d), e) 5.60/2.36 5.60/2.36 The TRS R consists of the following rules: 5.60/2.36 5.60/2.36 a -> d 5.60/2.36 a -> e 5.60/2.36 f(x) -> U1(x, x) 5.60/2.36 b -> d 5.60/2.36 b -> e 5.60/2.36 U1(d, x) -> x 5.60/2.36 5.60/2.36 Q is empty. 5.60/2.36 We have to consider all minimal (P,Q,R)-chains. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (421) TransformationProof (EQUIVALENT) 5.60/2.36 By narrowing [LPAR04] the rule A^1 -> H(f(d), e) at position [] we obtained the following new rules [LPAR04]: 5.60/2.36 5.60/2.36 (A^1 -> H(U1(d, d), e),A^1 -> H(U1(d, d), e)) 5.60/2.36 5.60/2.36 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (422) 5.60/2.36 Obligation: 5.60/2.36 Q DP problem: 5.60/2.36 The TRS P consists of the following rules: 5.60/2.36 5.60/2.36 H(x, x) -> G(x, x) 5.60/2.36 G(d, e) -> A^1 5.60/2.36 A^1 -> H(f(d), f(d)) 5.60/2.36 A^1 -> H(f(e), f(e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.36 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.36 A^1 -> H(U1(e, e), b) 5.60/2.36 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.36 A^1 -> H(f(e), e) 5.60/2.36 A^1 -> H(U1(d, d), d) 5.60/2.36 A^1 -> H(e, U1(d, b)) 5.60/2.36 A^1 -> H(e, U1(b, d)) 5.60/2.36 A^1 -> H(e, U1(b, e)) 5.60/2.36 A^1 -> H(e, U1(d, d)) 5.60/2.36 A^1 -> H(a, b) 5.60/2.36 A^1 -> H(a, U1(d, e)) 5.60/2.36 A^1 -> H(a, U1(e, d)) 5.60/2.36 A^1 -> H(a, d) 5.60/2.36 A^1 -> H(U1(d, e), b) 5.60/2.36 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.36 A^1 -> H(U1(d, e), d) 5.60/2.36 A^1 -> H(U1(d, a), e) 5.60/2.36 A^1 -> H(U1(e, d), b) 5.60/2.36 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.36 A^1 -> H(U1(a, d), e) 5.60/2.36 A^1 -> H(U1(a, e), e) 5.60/2.36 A^1 -> H(d, b) 5.60/2.36 A^1 -> H(d, U1(d, e)) 5.60/2.36 A^1 -> H(d, d) 5.60/2.36 A^1 -> H(U1(d, d), e) 5.60/2.36 5.60/2.36 The TRS R consists of the following rules: 5.60/2.36 5.60/2.36 a -> d 5.60/2.36 a -> e 5.60/2.36 f(x) -> U1(x, x) 5.60/2.36 b -> d 5.60/2.36 b -> e 5.60/2.36 U1(d, x) -> x 5.60/2.36 5.60/2.36 Q is empty. 5.60/2.36 We have to consider all minimal (P,Q,R)-chains. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (423) TransformationProof (EQUIVALENT) 5.60/2.36 By narrowing [LPAR04] the rule A^1 -> H(U1(e, e), b) at position [] we obtained the following new rules [LPAR04]: 5.60/2.36 5.60/2.36 (A^1 -> H(U1(e, e), d),A^1 -> H(U1(e, e), d)) 5.60/2.36 (A^1 -> H(U1(e, e), e),A^1 -> H(U1(e, e), e)) 5.60/2.36 5.60/2.36 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (424) 5.60/2.36 Obligation: 5.60/2.36 Q DP problem: 5.60/2.36 The TRS P consists of the following rules: 5.60/2.36 5.60/2.36 H(x, x) -> G(x, x) 5.60/2.36 G(d, e) -> A^1 5.60/2.36 A^1 -> H(f(d), f(d)) 5.60/2.36 A^1 -> H(f(e), f(e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.36 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.36 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.36 A^1 -> H(f(e), e) 5.60/2.36 A^1 -> H(U1(d, d), d) 5.60/2.36 A^1 -> H(e, U1(d, b)) 5.60/2.36 A^1 -> H(e, U1(b, d)) 5.60/2.36 A^1 -> H(e, U1(b, e)) 5.60/2.36 A^1 -> H(e, U1(d, d)) 5.60/2.36 A^1 -> H(a, b) 5.60/2.36 A^1 -> H(a, U1(d, e)) 5.60/2.36 A^1 -> H(a, U1(e, d)) 5.60/2.36 A^1 -> H(a, d) 5.60/2.36 A^1 -> H(U1(d, e), b) 5.60/2.36 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.36 A^1 -> H(U1(d, e), d) 5.60/2.36 A^1 -> H(U1(d, a), e) 5.60/2.36 A^1 -> H(U1(e, d), b) 5.60/2.36 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.36 A^1 -> H(U1(a, d), e) 5.60/2.36 A^1 -> H(U1(a, e), e) 5.60/2.36 A^1 -> H(d, b) 5.60/2.36 A^1 -> H(d, U1(d, e)) 5.60/2.36 A^1 -> H(d, d) 5.60/2.36 A^1 -> H(U1(d, d), e) 5.60/2.36 A^1 -> H(U1(e, e), d) 5.60/2.36 A^1 -> H(U1(e, e), e) 5.60/2.36 5.60/2.36 The TRS R consists of the following rules: 5.60/2.36 5.60/2.36 a -> d 5.60/2.36 a -> e 5.60/2.36 f(x) -> U1(x, x) 5.60/2.36 b -> d 5.60/2.36 b -> e 5.60/2.36 U1(d, x) -> x 5.60/2.36 5.60/2.36 Q is empty. 5.60/2.36 We have to consider all minimal (P,Q,R)-chains. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (425) DependencyGraphProof (EQUIVALENT) 5.60/2.36 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (426) 5.60/2.36 Obligation: 5.60/2.36 Q DP problem: 5.60/2.36 The TRS P consists of the following rules: 5.60/2.36 5.60/2.36 G(d, e) -> A^1 5.60/2.36 A^1 -> H(f(d), f(d)) 5.60/2.36 H(x, x) -> G(x, x) 5.60/2.36 A^1 -> H(f(e), f(e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.36 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.36 A^1 -> H(U1(e, e), U1(d, e)) 5.60/2.36 A^1 -> H(f(e), e) 5.60/2.36 A^1 -> H(U1(d, d), d) 5.60/2.36 A^1 -> H(e, U1(d, b)) 5.60/2.36 A^1 -> H(e, U1(b, d)) 5.60/2.36 A^1 -> H(e, U1(b, e)) 5.60/2.36 A^1 -> H(e, U1(d, d)) 5.60/2.36 A^1 -> H(a, b) 5.60/2.36 A^1 -> H(a, U1(d, e)) 5.60/2.36 A^1 -> H(a, U1(e, d)) 5.60/2.36 A^1 -> H(a, d) 5.60/2.36 A^1 -> H(U1(d, e), b) 5.60/2.36 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.36 A^1 -> H(U1(d, e), d) 5.60/2.36 A^1 -> H(U1(d, a), e) 5.60/2.36 A^1 -> H(U1(e, d), b) 5.60/2.36 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.36 A^1 -> H(U1(a, d), e) 5.60/2.36 A^1 -> H(U1(a, e), e) 5.60/2.36 A^1 -> H(d, b) 5.60/2.36 A^1 -> H(d, U1(d, e)) 5.60/2.36 A^1 -> H(d, d) 5.60/2.36 A^1 -> H(U1(d, d), e) 5.60/2.36 5.60/2.36 The TRS R consists of the following rules: 5.60/2.36 5.60/2.36 a -> d 5.60/2.36 a -> e 5.60/2.36 f(x) -> U1(x, x) 5.60/2.36 b -> d 5.60/2.36 b -> e 5.60/2.36 U1(d, x) -> x 5.60/2.36 5.60/2.36 Q is empty. 5.60/2.36 We have to consider all minimal (P,Q,R)-chains. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (427) TransformationProof (EQUIVALENT) 5.60/2.36 By narrowing [LPAR04] the rule A^1 -> H(U1(e, e), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.36 5.60/2.36 (A^1 -> H(U1(e, e), e),A^1 -> H(U1(e, e), e)) 5.60/2.36 5.60/2.36 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (428) 5.60/2.36 Obligation: 5.60/2.36 Q DP problem: 5.60/2.36 The TRS P consists of the following rules: 5.60/2.36 5.60/2.36 G(d, e) -> A^1 5.60/2.36 A^1 -> H(f(d), f(d)) 5.60/2.36 H(x, x) -> G(x, x) 5.60/2.36 A^1 -> H(f(e), f(e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.36 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.36 A^1 -> H(f(e), e) 5.60/2.36 A^1 -> H(U1(d, d), d) 5.60/2.36 A^1 -> H(e, U1(d, b)) 5.60/2.36 A^1 -> H(e, U1(b, d)) 5.60/2.36 A^1 -> H(e, U1(b, e)) 5.60/2.36 A^1 -> H(e, U1(d, d)) 5.60/2.36 A^1 -> H(a, b) 5.60/2.36 A^1 -> H(a, U1(d, e)) 5.60/2.36 A^1 -> H(a, U1(e, d)) 5.60/2.36 A^1 -> H(a, d) 5.60/2.36 A^1 -> H(U1(d, e), b) 5.60/2.36 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.36 A^1 -> H(U1(d, e), d) 5.60/2.36 A^1 -> H(U1(d, a), e) 5.60/2.36 A^1 -> H(U1(e, d), b) 5.60/2.36 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.36 A^1 -> H(U1(a, d), e) 5.60/2.36 A^1 -> H(U1(a, e), e) 5.60/2.36 A^1 -> H(d, b) 5.60/2.36 A^1 -> H(d, U1(d, e)) 5.60/2.36 A^1 -> H(d, d) 5.60/2.36 A^1 -> H(U1(d, d), e) 5.60/2.36 A^1 -> H(U1(e, e), e) 5.60/2.36 5.60/2.36 The TRS R consists of the following rules: 5.60/2.36 5.60/2.36 a -> d 5.60/2.36 a -> e 5.60/2.36 f(x) -> U1(x, x) 5.60/2.36 b -> d 5.60/2.36 b -> e 5.60/2.36 U1(d, x) -> x 5.60/2.36 5.60/2.36 Q is empty. 5.60/2.36 We have to consider all minimal (P,Q,R)-chains. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (429) DependencyGraphProof (EQUIVALENT) 5.60/2.36 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (430) 5.60/2.36 Obligation: 5.60/2.36 Q DP problem: 5.60/2.36 The TRS P consists of the following rules: 5.60/2.36 5.60/2.36 A^1 -> H(f(d), f(d)) 5.60/2.36 H(x, x) -> G(x, x) 5.60/2.36 G(d, e) -> A^1 5.60/2.36 A^1 -> H(f(e), f(e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.36 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.36 A^1 -> H(f(e), e) 5.60/2.36 A^1 -> H(U1(d, d), d) 5.60/2.36 A^1 -> H(e, U1(d, b)) 5.60/2.36 A^1 -> H(e, U1(b, d)) 5.60/2.36 A^1 -> H(e, U1(b, e)) 5.60/2.36 A^1 -> H(e, U1(d, d)) 5.60/2.36 A^1 -> H(a, b) 5.60/2.36 A^1 -> H(a, U1(d, e)) 5.60/2.36 A^1 -> H(a, U1(e, d)) 5.60/2.36 A^1 -> H(a, d) 5.60/2.36 A^1 -> H(U1(d, e), b) 5.60/2.36 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.36 A^1 -> H(U1(d, e), d) 5.60/2.36 A^1 -> H(U1(d, a), e) 5.60/2.36 A^1 -> H(U1(e, d), b) 5.60/2.36 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.36 A^1 -> H(U1(a, d), e) 5.60/2.36 A^1 -> H(U1(a, e), e) 5.60/2.36 A^1 -> H(d, b) 5.60/2.36 A^1 -> H(d, U1(d, e)) 5.60/2.36 A^1 -> H(d, d) 5.60/2.36 A^1 -> H(U1(d, d), e) 5.60/2.36 5.60/2.36 The TRS R consists of the following rules: 5.60/2.36 5.60/2.36 a -> d 5.60/2.36 a -> e 5.60/2.36 f(x) -> U1(x, x) 5.60/2.36 b -> d 5.60/2.36 b -> e 5.60/2.36 U1(d, x) -> x 5.60/2.36 5.60/2.36 Q is empty. 5.60/2.36 We have to consider all minimal (P,Q,R)-chains. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (431) TransformationProof (EQUIVALENT) 5.60/2.36 By narrowing [LPAR04] the rule A^1 -> H(f(e), e) at position [] we obtained the following new rules [LPAR04]: 5.60/2.36 5.60/2.36 (A^1 -> H(U1(e, e), e),A^1 -> H(U1(e, e), e)) 5.60/2.36 5.60/2.36 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (432) 5.60/2.36 Obligation: 5.60/2.36 Q DP problem: 5.60/2.36 The TRS P consists of the following rules: 5.60/2.36 5.60/2.36 A^1 -> H(f(d), f(d)) 5.60/2.36 H(x, x) -> G(x, x) 5.60/2.36 G(d, e) -> A^1 5.60/2.36 A^1 -> H(f(e), f(e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.36 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.36 A^1 -> H(U1(d, d), d) 5.60/2.36 A^1 -> H(e, U1(d, b)) 5.60/2.36 A^1 -> H(e, U1(b, d)) 5.60/2.36 A^1 -> H(e, U1(b, e)) 5.60/2.36 A^1 -> H(e, U1(d, d)) 5.60/2.36 A^1 -> H(a, b) 5.60/2.36 A^1 -> H(a, U1(d, e)) 5.60/2.36 A^1 -> H(a, U1(e, d)) 5.60/2.36 A^1 -> H(a, d) 5.60/2.36 A^1 -> H(U1(d, e), b) 5.60/2.36 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.36 A^1 -> H(U1(d, e), d) 5.60/2.36 A^1 -> H(U1(d, a), e) 5.60/2.36 A^1 -> H(U1(e, d), b) 5.60/2.36 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.36 A^1 -> H(U1(a, d), e) 5.60/2.36 A^1 -> H(U1(a, e), e) 5.60/2.36 A^1 -> H(d, b) 5.60/2.36 A^1 -> H(d, U1(d, e)) 5.60/2.36 A^1 -> H(d, d) 5.60/2.36 A^1 -> H(U1(d, d), e) 5.60/2.36 A^1 -> H(U1(e, e), e) 5.60/2.36 5.60/2.36 The TRS R consists of the following rules: 5.60/2.36 5.60/2.36 a -> d 5.60/2.36 a -> e 5.60/2.36 f(x) -> U1(x, x) 5.60/2.36 b -> d 5.60/2.36 b -> e 5.60/2.36 U1(d, x) -> x 5.60/2.36 5.60/2.36 Q is empty. 5.60/2.36 We have to consider all minimal (P,Q,R)-chains. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (433) DependencyGraphProof (EQUIVALENT) 5.60/2.36 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (434) 5.60/2.36 Obligation: 5.60/2.36 Q DP problem: 5.60/2.36 The TRS P consists of the following rules: 5.60/2.36 5.60/2.36 H(x, x) -> G(x, x) 5.60/2.36 G(d, e) -> A^1 5.60/2.36 A^1 -> H(f(d), f(d)) 5.60/2.36 A^1 -> H(f(e), f(e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.36 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.36 A^1 -> H(U1(d, d), d) 5.60/2.36 A^1 -> H(e, U1(d, b)) 5.60/2.36 A^1 -> H(e, U1(b, d)) 5.60/2.36 A^1 -> H(e, U1(b, e)) 5.60/2.36 A^1 -> H(e, U1(d, d)) 5.60/2.36 A^1 -> H(a, b) 5.60/2.36 A^1 -> H(a, U1(d, e)) 5.60/2.36 A^1 -> H(a, U1(e, d)) 5.60/2.36 A^1 -> H(a, d) 5.60/2.36 A^1 -> H(U1(d, e), b) 5.60/2.36 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.36 A^1 -> H(U1(d, e), d) 5.60/2.36 A^1 -> H(U1(d, a), e) 5.60/2.36 A^1 -> H(U1(e, d), b) 5.60/2.36 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.36 A^1 -> H(U1(a, d), e) 5.60/2.36 A^1 -> H(U1(a, e), e) 5.60/2.36 A^1 -> H(d, b) 5.60/2.36 A^1 -> H(d, U1(d, e)) 5.60/2.36 A^1 -> H(d, d) 5.60/2.36 A^1 -> H(U1(d, d), e) 5.60/2.36 5.60/2.36 The TRS R consists of the following rules: 5.60/2.36 5.60/2.36 a -> d 5.60/2.36 a -> e 5.60/2.36 f(x) -> U1(x, x) 5.60/2.36 b -> d 5.60/2.36 b -> e 5.60/2.36 U1(d, x) -> x 5.60/2.36 5.60/2.36 Q is empty. 5.60/2.36 We have to consider all minimal (P,Q,R)-chains. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (435) TransformationProof (EQUIVALENT) 5.60/2.36 By narrowing [LPAR04] the rule A^1 -> H(U1(d, d), d) at position [] we obtained the following new rules [LPAR04]: 5.60/2.36 5.60/2.36 (A^1 -> H(d, d),A^1 -> H(d, d)) 5.60/2.36 5.60/2.36 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (436) 5.60/2.36 Obligation: 5.60/2.36 Q DP problem: 5.60/2.36 The TRS P consists of the following rules: 5.60/2.36 5.60/2.36 H(x, x) -> G(x, x) 5.60/2.36 G(d, e) -> A^1 5.60/2.36 A^1 -> H(f(d), f(d)) 5.60/2.36 A^1 -> H(f(e), f(e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.36 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.36 A^1 -> H(e, U1(d, b)) 5.60/2.36 A^1 -> H(e, U1(b, d)) 5.60/2.36 A^1 -> H(e, U1(b, e)) 5.60/2.36 A^1 -> H(e, U1(d, d)) 5.60/2.36 A^1 -> H(a, b) 5.60/2.36 A^1 -> H(a, U1(d, e)) 5.60/2.36 A^1 -> H(a, U1(e, d)) 5.60/2.36 A^1 -> H(a, d) 5.60/2.36 A^1 -> H(U1(d, e), b) 5.60/2.36 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.36 A^1 -> H(U1(d, e), d) 5.60/2.36 A^1 -> H(U1(d, a), e) 5.60/2.36 A^1 -> H(U1(e, d), b) 5.60/2.36 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.36 A^1 -> H(U1(a, d), e) 5.60/2.36 A^1 -> H(U1(a, e), e) 5.60/2.36 A^1 -> H(d, b) 5.60/2.36 A^1 -> H(d, U1(d, e)) 5.60/2.36 A^1 -> H(d, d) 5.60/2.36 A^1 -> H(U1(d, d), e) 5.60/2.36 5.60/2.36 The TRS R consists of the following rules: 5.60/2.36 5.60/2.36 a -> d 5.60/2.36 a -> e 5.60/2.36 f(x) -> U1(x, x) 5.60/2.36 b -> d 5.60/2.36 b -> e 5.60/2.36 U1(d, x) -> x 5.60/2.36 5.60/2.36 Q is empty. 5.60/2.36 We have to consider all minimal (P,Q,R)-chains. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (437) TransformationProof (EQUIVALENT) 5.60/2.36 By narrowing [LPAR04] the rule A^1 -> H(e, U1(d, b)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.36 5.60/2.36 (A^1 -> H(e, b),A^1 -> H(e, b)) 5.60/2.36 (A^1 -> H(e, U1(d, d)),A^1 -> H(e, U1(d, d))) 5.60/2.36 (A^1 -> H(e, U1(d, e)),A^1 -> H(e, U1(d, e))) 5.60/2.36 5.60/2.36 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (438) 5.60/2.36 Obligation: 5.60/2.36 Q DP problem: 5.60/2.36 The TRS P consists of the following rules: 5.60/2.36 5.60/2.36 H(x, x) -> G(x, x) 5.60/2.36 G(d, e) -> A^1 5.60/2.36 A^1 -> H(f(d), f(d)) 5.60/2.36 A^1 -> H(f(e), f(e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.36 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.36 A^1 -> H(e, U1(b, d)) 5.60/2.36 A^1 -> H(e, U1(b, e)) 5.60/2.36 A^1 -> H(e, U1(d, d)) 5.60/2.36 A^1 -> H(a, b) 5.60/2.36 A^1 -> H(a, U1(d, e)) 5.60/2.36 A^1 -> H(a, U1(e, d)) 5.60/2.36 A^1 -> H(a, d) 5.60/2.36 A^1 -> H(U1(d, e), b) 5.60/2.36 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.36 A^1 -> H(U1(d, e), d) 5.60/2.36 A^1 -> H(U1(d, a), e) 5.60/2.36 A^1 -> H(U1(e, d), b) 5.60/2.36 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.36 A^1 -> H(U1(a, d), e) 5.60/2.36 A^1 -> H(U1(a, e), e) 5.60/2.36 A^1 -> H(d, b) 5.60/2.36 A^1 -> H(d, U1(d, e)) 5.60/2.36 A^1 -> H(d, d) 5.60/2.36 A^1 -> H(U1(d, d), e) 5.60/2.36 A^1 -> H(e, b) 5.60/2.36 A^1 -> H(e, U1(d, e)) 5.60/2.36 5.60/2.36 The TRS R consists of the following rules: 5.60/2.36 5.60/2.36 a -> d 5.60/2.36 a -> e 5.60/2.36 f(x) -> U1(x, x) 5.60/2.36 b -> d 5.60/2.36 b -> e 5.60/2.36 U1(d, x) -> x 5.60/2.36 5.60/2.36 Q is empty. 5.60/2.36 We have to consider all minimal (P,Q,R)-chains. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (439) TransformationProof (EQUIVALENT) 5.60/2.36 By narrowing [LPAR04] the rule A^1 -> H(e, U1(b, d)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.36 5.60/2.36 (A^1 -> H(e, U1(d, d)),A^1 -> H(e, U1(d, d))) 5.60/2.36 (A^1 -> H(e, U1(e, d)),A^1 -> H(e, U1(e, d))) 5.60/2.36 5.60/2.36 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (440) 5.60/2.36 Obligation: 5.60/2.36 Q DP problem: 5.60/2.36 The TRS P consists of the following rules: 5.60/2.36 5.60/2.36 H(x, x) -> G(x, x) 5.60/2.36 G(d, e) -> A^1 5.60/2.36 A^1 -> H(f(d), f(d)) 5.60/2.36 A^1 -> H(f(e), f(e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.36 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.36 A^1 -> H(e, U1(b, e)) 5.60/2.36 A^1 -> H(e, U1(d, d)) 5.60/2.36 A^1 -> H(a, b) 5.60/2.36 A^1 -> H(a, U1(d, e)) 5.60/2.36 A^1 -> H(a, U1(e, d)) 5.60/2.36 A^1 -> H(a, d) 5.60/2.36 A^1 -> H(U1(d, e), b) 5.60/2.36 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.36 A^1 -> H(U1(d, e), d) 5.60/2.36 A^1 -> H(U1(d, a), e) 5.60/2.36 A^1 -> H(U1(e, d), b) 5.60/2.36 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.36 A^1 -> H(U1(a, d), e) 5.60/2.36 A^1 -> H(U1(a, e), e) 5.60/2.36 A^1 -> H(d, b) 5.60/2.36 A^1 -> H(d, U1(d, e)) 5.60/2.36 A^1 -> H(d, d) 5.60/2.36 A^1 -> H(U1(d, d), e) 5.60/2.36 A^1 -> H(e, b) 5.60/2.36 A^1 -> H(e, U1(d, e)) 5.60/2.36 A^1 -> H(e, U1(e, d)) 5.60/2.36 5.60/2.36 The TRS R consists of the following rules: 5.60/2.36 5.60/2.36 a -> d 5.60/2.36 a -> e 5.60/2.36 f(x) -> U1(x, x) 5.60/2.36 b -> d 5.60/2.36 b -> e 5.60/2.36 U1(d, x) -> x 5.60/2.36 5.60/2.36 Q is empty. 5.60/2.36 We have to consider all minimal (P,Q,R)-chains. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (441) DependencyGraphProof (EQUIVALENT) 5.60/2.36 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (442) 5.60/2.36 Obligation: 5.60/2.36 Q DP problem: 5.60/2.36 The TRS P consists of the following rules: 5.60/2.36 5.60/2.36 G(d, e) -> A^1 5.60/2.36 A^1 -> H(f(d), f(d)) 5.60/2.36 H(x, x) -> G(x, x) 5.60/2.36 A^1 -> H(f(e), f(e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.36 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.36 A^1 -> H(e, U1(b, e)) 5.60/2.36 A^1 -> H(e, U1(d, d)) 5.60/2.36 A^1 -> H(a, b) 5.60/2.36 A^1 -> H(a, U1(d, e)) 5.60/2.36 A^1 -> H(a, U1(e, d)) 5.60/2.36 A^1 -> H(a, d) 5.60/2.36 A^1 -> H(U1(d, e), b) 5.60/2.36 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.36 A^1 -> H(U1(d, e), d) 5.60/2.36 A^1 -> H(U1(d, a), e) 5.60/2.36 A^1 -> H(U1(e, d), b) 5.60/2.36 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.36 A^1 -> H(U1(a, d), e) 5.60/2.36 A^1 -> H(U1(a, e), e) 5.60/2.36 A^1 -> H(d, b) 5.60/2.36 A^1 -> H(d, U1(d, e)) 5.60/2.36 A^1 -> H(d, d) 5.60/2.36 A^1 -> H(U1(d, d), e) 5.60/2.36 A^1 -> H(e, b) 5.60/2.36 A^1 -> H(e, U1(d, e)) 5.60/2.36 5.60/2.36 The TRS R consists of the following rules: 5.60/2.36 5.60/2.36 a -> d 5.60/2.36 a -> e 5.60/2.36 f(x) -> U1(x, x) 5.60/2.36 b -> d 5.60/2.36 b -> e 5.60/2.36 U1(d, x) -> x 5.60/2.36 5.60/2.36 Q is empty. 5.60/2.36 We have to consider all minimal (P,Q,R)-chains. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (443) TransformationProof (EQUIVALENT) 5.60/2.36 By narrowing [LPAR04] the rule A^1 -> H(e, U1(b, e)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.36 5.60/2.36 (A^1 -> H(e, U1(d, e)),A^1 -> H(e, U1(d, e))) 5.60/2.36 (A^1 -> H(e, U1(e, e)),A^1 -> H(e, U1(e, e))) 5.60/2.36 5.60/2.36 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (444) 5.60/2.36 Obligation: 5.60/2.36 Q DP problem: 5.60/2.36 The TRS P consists of the following rules: 5.60/2.36 5.60/2.36 G(d, e) -> A^1 5.60/2.36 A^1 -> H(f(d), f(d)) 5.60/2.36 H(x, x) -> G(x, x) 5.60/2.36 A^1 -> H(f(e), f(e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.36 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.36 A^1 -> H(e, U1(d, d)) 5.60/2.36 A^1 -> H(a, b) 5.60/2.36 A^1 -> H(a, U1(d, e)) 5.60/2.36 A^1 -> H(a, U1(e, d)) 5.60/2.36 A^1 -> H(a, d) 5.60/2.36 A^1 -> H(U1(d, e), b) 5.60/2.36 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.36 A^1 -> H(U1(d, e), d) 5.60/2.36 A^1 -> H(U1(d, a), e) 5.60/2.36 A^1 -> H(U1(e, d), b) 5.60/2.36 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.36 A^1 -> H(U1(a, d), e) 5.60/2.36 A^1 -> H(U1(a, e), e) 5.60/2.36 A^1 -> H(d, b) 5.60/2.36 A^1 -> H(d, U1(d, e)) 5.60/2.36 A^1 -> H(d, d) 5.60/2.36 A^1 -> H(U1(d, d), e) 5.60/2.36 A^1 -> H(e, b) 5.60/2.36 A^1 -> H(e, U1(d, e)) 5.60/2.36 A^1 -> H(e, U1(e, e)) 5.60/2.36 5.60/2.36 The TRS R consists of the following rules: 5.60/2.36 5.60/2.36 a -> d 5.60/2.36 a -> e 5.60/2.36 f(x) -> U1(x, x) 5.60/2.36 b -> d 5.60/2.36 b -> e 5.60/2.36 U1(d, x) -> x 5.60/2.36 5.60/2.36 Q is empty. 5.60/2.36 We have to consider all minimal (P,Q,R)-chains. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (445) DependencyGraphProof (EQUIVALENT) 5.60/2.36 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (446) 5.60/2.36 Obligation: 5.60/2.36 Q DP problem: 5.60/2.36 The TRS P consists of the following rules: 5.60/2.36 5.60/2.36 A^1 -> H(f(d), f(d)) 5.60/2.36 H(x, x) -> G(x, x) 5.60/2.36 G(d, e) -> A^1 5.60/2.36 A^1 -> H(f(e), f(e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.36 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.36 A^1 -> H(e, U1(d, d)) 5.60/2.36 A^1 -> H(a, b) 5.60/2.36 A^1 -> H(a, U1(d, e)) 5.60/2.36 A^1 -> H(a, U1(e, d)) 5.60/2.36 A^1 -> H(a, d) 5.60/2.36 A^1 -> H(U1(d, e), b) 5.60/2.36 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.36 A^1 -> H(U1(d, e), d) 5.60/2.36 A^1 -> H(U1(d, a), e) 5.60/2.36 A^1 -> H(U1(e, d), b) 5.60/2.36 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.36 A^1 -> H(U1(a, d), e) 5.60/2.36 A^1 -> H(U1(a, e), e) 5.60/2.36 A^1 -> H(d, b) 5.60/2.36 A^1 -> H(d, U1(d, e)) 5.60/2.36 A^1 -> H(d, d) 5.60/2.36 A^1 -> H(U1(d, d), e) 5.60/2.36 A^1 -> H(e, b) 5.60/2.36 A^1 -> H(e, U1(d, e)) 5.60/2.36 5.60/2.36 The TRS R consists of the following rules: 5.60/2.36 5.60/2.36 a -> d 5.60/2.36 a -> e 5.60/2.36 f(x) -> U1(x, x) 5.60/2.36 b -> d 5.60/2.36 b -> e 5.60/2.36 U1(d, x) -> x 5.60/2.36 5.60/2.36 Q is empty. 5.60/2.36 We have to consider all minimal (P,Q,R)-chains. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (447) TransformationProof (EQUIVALENT) 5.60/2.36 By narrowing [LPAR04] the rule A^1 -> H(e, U1(d, d)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.36 5.60/2.36 (A^1 -> H(e, d),A^1 -> H(e, d)) 5.60/2.36 5.60/2.36 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (448) 5.60/2.36 Obligation: 5.60/2.36 Q DP problem: 5.60/2.36 The TRS P consists of the following rules: 5.60/2.36 5.60/2.36 A^1 -> H(f(d), f(d)) 5.60/2.36 H(x, x) -> G(x, x) 5.60/2.36 G(d, e) -> A^1 5.60/2.36 A^1 -> H(f(e), f(e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.36 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.36 A^1 -> H(a, b) 5.60/2.36 A^1 -> H(a, U1(d, e)) 5.60/2.36 A^1 -> H(a, U1(e, d)) 5.60/2.36 A^1 -> H(a, d) 5.60/2.36 A^1 -> H(U1(d, e), b) 5.60/2.36 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.36 A^1 -> H(U1(d, e), d) 5.60/2.36 A^1 -> H(U1(d, a), e) 5.60/2.36 A^1 -> H(U1(e, d), b) 5.60/2.36 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.36 A^1 -> H(U1(a, d), e) 5.60/2.36 A^1 -> H(U1(a, e), e) 5.60/2.36 A^1 -> H(d, b) 5.60/2.36 A^1 -> H(d, U1(d, e)) 5.60/2.36 A^1 -> H(d, d) 5.60/2.36 A^1 -> H(U1(d, d), e) 5.60/2.36 A^1 -> H(e, b) 5.60/2.36 A^1 -> H(e, U1(d, e)) 5.60/2.36 A^1 -> H(e, d) 5.60/2.36 5.60/2.36 The TRS R consists of the following rules: 5.60/2.36 5.60/2.36 a -> d 5.60/2.36 a -> e 5.60/2.36 f(x) -> U1(x, x) 5.60/2.36 b -> d 5.60/2.36 b -> e 5.60/2.36 U1(d, x) -> x 5.60/2.36 5.60/2.36 Q is empty. 5.60/2.36 We have to consider all minimal (P,Q,R)-chains. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (449) DependencyGraphProof (EQUIVALENT) 5.60/2.36 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (450) 5.60/2.36 Obligation: 5.60/2.36 Q DP problem: 5.60/2.36 The TRS P consists of the following rules: 5.60/2.36 5.60/2.36 H(x, x) -> G(x, x) 5.60/2.36 G(d, e) -> A^1 5.60/2.36 A^1 -> H(f(d), f(d)) 5.60/2.36 A^1 -> H(f(e), f(e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.36 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.36 A^1 -> H(a, b) 5.60/2.36 A^1 -> H(a, U1(d, e)) 5.60/2.36 A^1 -> H(a, U1(e, d)) 5.60/2.36 A^1 -> H(a, d) 5.60/2.36 A^1 -> H(U1(d, e), b) 5.60/2.36 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.36 A^1 -> H(U1(d, e), d) 5.60/2.36 A^1 -> H(U1(d, a), e) 5.60/2.36 A^1 -> H(U1(e, d), b) 5.60/2.36 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.36 A^1 -> H(U1(a, d), e) 5.60/2.36 A^1 -> H(U1(a, e), e) 5.60/2.36 A^1 -> H(d, b) 5.60/2.36 A^1 -> H(d, U1(d, e)) 5.60/2.36 A^1 -> H(d, d) 5.60/2.36 A^1 -> H(U1(d, d), e) 5.60/2.36 A^1 -> H(e, b) 5.60/2.36 A^1 -> H(e, U1(d, e)) 5.60/2.36 5.60/2.36 The TRS R consists of the following rules: 5.60/2.36 5.60/2.36 a -> d 5.60/2.36 a -> e 5.60/2.36 f(x) -> U1(x, x) 5.60/2.36 b -> d 5.60/2.36 b -> e 5.60/2.36 U1(d, x) -> x 5.60/2.36 5.60/2.36 Q is empty. 5.60/2.36 We have to consider all minimal (P,Q,R)-chains. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (451) TransformationProof (EQUIVALENT) 5.60/2.36 By narrowing [LPAR04] the rule A^1 -> H(a, b) at position [] we obtained the following new rules [LPAR04]: 5.60/2.36 5.60/2.36 (A^1 -> H(d, b),A^1 -> H(d, b)) 5.60/2.36 (A^1 -> H(e, b),A^1 -> H(e, b)) 5.60/2.36 (A^1 -> H(a, d),A^1 -> H(a, d)) 5.60/2.36 (A^1 -> H(a, e),A^1 -> H(a, e)) 5.60/2.36 5.60/2.36 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (452) 5.60/2.36 Obligation: 5.60/2.36 Q DP problem: 5.60/2.36 The TRS P consists of the following rules: 5.60/2.36 5.60/2.36 H(x, x) -> G(x, x) 5.60/2.36 G(d, e) -> A^1 5.60/2.36 A^1 -> H(f(d), f(d)) 5.60/2.36 A^1 -> H(f(e), f(e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.36 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.36 A^1 -> H(a, U1(d, e)) 5.60/2.36 A^1 -> H(a, U1(e, d)) 5.60/2.36 A^1 -> H(a, d) 5.60/2.36 A^1 -> H(U1(d, e), b) 5.60/2.36 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.36 A^1 -> H(U1(d, e), d) 5.60/2.36 A^1 -> H(U1(d, a), e) 5.60/2.36 A^1 -> H(U1(e, d), b) 5.60/2.36 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.36 A^1 -> H(U1(a, d), e) 5.60/2.36 A^1 -> H(U1(a, e), e) 5.60/2.36 A^1 -> H(d, b) 5.60/2.36 A^1 -> H(d, U1(d, e)) 5.60/2.36 A^1 -> H(d, d) 5.60/2.36 A^1 -> H(U1(d, d), e) 5.60/2.36 A^1 -> H(e, b) 5.60/2.36 A^1 -> H(e, U1(d, e)) 5.60/2.36 A^1 -> H(a, e) 5.60/2.36 5.60/2.36 The TRS R consists of the following rules: 5.60/2.36 5.60/2.36 a -> d 5.60/2.36 a -> e 5.60/2.36 f(x) -> U1(x, x) 5.60/2.36 b -> d 5.60/2.36 b -> e 5.60/2.36 U1(d, x) -> x 5.60/2.36 5.60/2.36 Q is empty. 5.60/2.36 We have to consider all minimal (P,Q,R)-chains. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (453) TransformationProof (EQUIVALENT) 5.60/2.36 By narrowing [LPAR04] the rule A^1 -> H(a, U1(d, e)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.36 5.60/2.36 (A^1 -> H(d, U1(d, e)),A^1 -> H(d, U1(d, e))) 5.60/2.36 (A^1 -> H(e, U1(d, e)),A^1 -> H(e, U1(d, e))) 5.60/2.36 (A^1 -> H(a, e),A^1 -> H(a, e)) 5.60/2.36 5.60/2.36 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (454) 5.60/2.36 Obligation: 5.60/2.36 Q DP problem: 5.60/2.36 The TRS P consists of the following rules: 5.60/2.36 5.60/2.36 H(x, x) -> G(x, x) 5.60/2.36 G(d, e) -> A^1 5.60/2.36 A^1 -> H(f(d), f(d)) 5.60/2.36 A^1 -> H(f(e), f(e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.36 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.36 A^1 -> H(a, U1(e, d)) 5.60/2.36 A^1 -> H(a, d) 5.60/2.36 A^1 -> H(U1(d, e), b) 5.60/2.36 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.36 A^1 -> H(U1(d, e), d) 5.60/2.36 A^1 -> H(U1(d, a), e) 5.60/2.36 A^1 -> H(U1(e, d), b) 5.60/2.36 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.36 A^1 -> H(U1(a, d), e) 5.60/2.36 A^1 -> H(U1(a, e), e) 5.60/2.36 A^1 -> H(d, b) 5.60/2.36 A^1 -> H(d, U1(d, e)) 5.60/2.36 A^1 -> H(d, d) 5.60/2.36 A^1 -> H(U1(d, d), e) 5.60/2.36 A^1 -> H(e, b) 5.60/2.36 A^1 -> H(e, U1(d, e)) 5.60/2.36 A^1 -> H(a, e) 5.60/2.36 5.60/2.36 The TRS R consists of the following rules: 5.60/2.36 5.60/2.36 a -> d 5.60/2.36 a -> e 5.60/2.36 f(x) -> U1(x, x) 5.60/2.36 b -> d 5.60/2.36 b -> e 5.60/2.36 U1(d, x) -> x 5.60/2.36 5.60/2.36 Q is empty. 5.60/2.36 We have to consider all minimal (P,Q,R)-chains. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (455) TransformationProof (EQUIVALENT) 5.60/2.36 By narrowing [LPAR04] the rule A^1 -> H(a, U1(e, d)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.36 5.60/2.36 (A^1 -> H(d, U1(e, d)),A^1 -> H(d, U1(e, d))) 5.60/2.36 (A^1 -> H(e, U1(e, d)),A^1 -> H(e, U1(e, d))) 5.60/2.36 5.60/2.36 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (456) 5.60/2.36 Obligation: 5.60/2.36 Q DP problem: 5.60/2.36 The TRS P consists of the following rules: 5.60/2.36 5.60/2.36 H(x, x) -> G(x, x) 5.60/2.36 G(d, e) -> A^1 5.60/2.36 A^1 -> H(f(d), f(d)) 5.60/2.36 A^1 -> H(f(e), f(e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.36 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.36 A^1 -> H(a, d) 5.60/2.36 A^1 -> H(U1(d, e), b) 5.60/2.36 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.36 A^1 -> H(U1(d, e), d) 5.60/2.36 A^1 -> H(U1(d, a), e) 5.60/2.36 A^1 -> H(U1(e, d), b) 5.60/2.36 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.36 A^1 -> H(U1(a, d), e) 5.60/2.36 A^1 -> H(U1(a, e), e) 5.60/2.36 A^1 -> H(d, b) 5.60/2.36 A^1 -> H(d, U1(d, e)) 5.60/2.36 A^1 -> H(d, d) 5.60/2.36 A^1 -> H(U1(d, d), e) 5.60/2.36 A^1 -> H(e, b) 5.60/2.36 A^1 -> H(e, U1(d, e)) 5.60/2.36 A^1 -> H(a, e) 5.60/2.36 A^1 -> H(d, U1(e, d)) 5.60/2.36 A^1 -> H(e, U1(e, d)) 5.60/2.36 5.60/2.36 The TRS R consists of the following rules: 5.60/2.36 5.60/2.36 a -> d 5.60/2.36 a -> e 5.60/2.36 f(x) -> U1(x, x) 5.60/2.36 b -> d 5.60/2.36 b -> e 5.60/2.36 U1(d, x) -> x 5.60/2.36 5.60/2.36 Q is empty. 5.60/2.36 We have to consider all minimal (P,Q,R)-chains. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (457) DependencyGraphProof (EQUIVALENT) 5.60/2.36 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (458) 5.60/2.36 Obligation: 5.60/2.36 Q DP problem: 5.60/2.36 The TRS P consists of the following rules: 5.60/2.36 5.60/2.36 G(d, e) -> A^1 5.60/2.36 A^1 -> H(f(d), f(d)) 5.60/2.36 H(x, x) -> G(x, x) 5.60/2.36 A^1 -> H(f(e), f(e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.36 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.36 A^1 -> H(a, d) 5.60/2.36 A^1 -> H(U1(d, e), b) 5.60/2.36 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.36 A^1 -> H(U1(d, e), d) 5.60/2.36 A^1 -> H(U1(d, a), e) 5.60/2.36 A^1 -> H(U1(e, d), b) 5.60/2.36 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.36 A^1 -> H(U1(a, d), e) 5.60/2.36 A^1 -> H(U1(a, e), e) 5.60/2.36 A^1 -> H(d, b) 5.60/2.36 A^1 -> H(d, U1(d, e)) 5.60/2.36 A^1 -> H(d, d) 5.60/2.36 A^1 -> H(U1(d, d), e) 5.60/2.36 A^1 -> H(e, b) 5.60/2.36 A^1 -> H(e, U1(d, e)) 5.60/2.36 A^1 -> H(a, e) 5.60/2.36 5.60/2.36 The TRS R consists of the following rules: 5.60/2.36 5.60/2.36 a -> d 5.60/2.36 a -> e 5.60/2.36 f(x) -> U1(x, x) 5.60/2.36 b -> d 5.60/2.36 b -> e 5.60/2.36 U1(d, x) -> x 5.60/2.36 5.60/2.36 Q is empty. 5.60/2.36 We have to consider all minimal (P,Q,R)-chains. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (459) TransformationProof (EQUIVALENT) 5.60/2.36 By narrowing [LPAR04] the rule A^1 -> H(a, d) at position [] we obtained the following new rules [LPAR04]: 5.60/2.36 5.60/2.36 (A^1 -> H(d, d),A^1 -> H(d, d)) 5.60/2.36 (A^1 -> H(e, d),A^1 -> H(e, d)) 5.60/2.36 5.60/2.36 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (460) 5.60/2.36 Obligation: 5.60/2.36 Q DP problem: 5.60/2.36 The TRS P consists of the following rules: 5.60/2.36 5.60/2.36 G(d, e) -> A^1 5.60/2.36 A^1 -> H(f(d), f(d)) 5.60/2.36 H(x, x) -> G(x, x) 5.60/2.36 A^1 -> H(f(e), f(e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.36 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.36 A^1 -> H(U1(d, e), b) 5.60/2.36 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.36 A^1 -> H(U1(d, e), d) 5.60/2.36 A^1 -> H(U1(d, a), e) 5.60/2.36 A^1 -> H(U1(e, d), b) 5.60/2.36 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.36 A^1 -> H(U1(a, d), e) 5.60/2.36 A^1 -> H(U1(a, e), e) 5.60/2.36 A^1 -> H(d, b) 5.60/2.36 A^1 -> H(d, U1(d, e)) 5.60/2.36 A^1 -> H(d, d) 5.60/2.36 A^1 -> H(U1(d, d), e) 5.60/2.36 A^1 -> H(e, b) 5.60/2.36 A^1 -> H(e, U1(d, e)) 5.60/2.36 A^1 -> H(a, e) 5.60/2.36 A^1 -> H(e, d) 5.60/2.36 5.60/2.36 The TRS R consists of the following rules: 5.60/2.36 5.60/2.36 a -> d 5.60/2.36 a -> e 5.60/2.36 f(x) -> U1(x, x) 5.60/2.36 b -> d 5.60/2.36 b -> e 5.60/2.36 U1(d, x) -> x 5.60/2.36 5.60/2.36 Q is empty. 5.60/2.36 We have to consider all minimal (P,Q,R)-chains. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (461) DependencyGraphProof (EQUIVALENT) 5.60/2.36 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (462) 5.60/2.36 Obligation: 5.60/2.36 Q DP problem: 5.60/2.36 The TRS P consists of the following rules: 5.60/2.36 5.60/2.36 A^1 -> H(f(d), f(d)) 5.60/2.36 H(x, x) -> G(x, x) 5.60/2.36 G(d, e) -> A^1 5.60/2.36 A^1 -> H(f(e), f(e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.36 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.36 A^1 -> H(U1(d, e), b) 5.60/2.36 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.36 A^1 -> H(U1(d, e), d) 5.60/2.36 A^1 -> H(U1(d, a), e) 5.60/2.36 A^1 -> H(U1(e, d), b) 5.60/2.36 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.36 A^1 -> H(U1(a, d), e) 5.60/2.36 A^1 -> H(U1(a, e), e) 5.60/2.36 A^1 -> H(d, b) 5.60/2.36 A^1 -> H(d, U1(d, e)) 5.60/2.36 A^1 -> H(d, d) 5.60/2.36 A^1 -> H(U1(d, d), e) 5.60/2.36 A^1 -> H(e, b) 5.60/2.36 A^1 -> H(e, U1(d, e)) 5.60/2.36 A^1 -> H(a, e) 5.60/2.36 5.60/2.36 The TRS R consists of the following rules: 5.60/2.36 5.60/2.36 a -> d 5.60/2.36 a -> e 5.60/2.36 f(x) -> U1(x, x) 5.60/2.36 b -> d 5.60/2.36 b -> e 5.60/2.36 U1(d, x) -> x 5.60/2.36 5.60/2.36 Q is empty. 5.60/2.36 We have to consider all minimal (P,Q,R)-chains. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (463) TransformationProof (EQUIVALENT) 5.60/2.36 By narrowing [LPAR04] the rule A^1 -> H(U1(d, e), b) at position [] we obtained the following new rules [LPAR04]: 5.60/2.36 5.60/2.36 (A^1 -> H(e, b),A^1 -> H(e, b)) 5.60/2.36 (A^1 -> H(U1(d, e), d),A^1 -> H(U1(d, e), d)) 5.60/2.36 (A^1 -> H(U1(d, e), e),A^1 -> H(U1(d, e), e)) 5.60/2.36 5.60/2.36 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (464) 5.60/2.36 Obligation: 5.60/2.36 Q DP problem: 5.60/2.36 The TRS P consists of the following rules: 5.60/2.36 5.60/2.36 A^1 -> H(f(d), f(d)) 5.60/2.36 H(x, x) -> G(x, x) 5.60/2.36 G(d, e) -> A^1 5.60/2.36 A^1 -> H(f(e), f(e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.36 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.36 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, e), U1(e, d)) 5.60/2.36 A^1 -> H(U1(d, e), d) 5.60/2.36 A^1 -> H(U1(d, a), e) 5.60/2.36 A^1 -> H(U1(e, d), b) 5.60/2.36 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.36 A^1 -> H(U1(a, d), e) 5.60/2.36 A^1 -> H(U1(a, e), e) 5.60/2.36 A^1 -> H(d, b) 5.60/2.36 A^1 -> H(d, U1(d, e)) 5.60/2.36 A^1 -> H(d, d) 5.60/2.36 A^1 -> H(U1(d, d), e) 5.60/2.36 A^1 -> H(e, b) 5.60/2.36 A^1 -> H(e, U1(d, e)) 5.60/2.36 A^1 -> H(a, e) 5.60/2.36 A^1 -> H(U1(d, e), e) 5.60/2.36 5.60/2.36 The TRS R consists of the following rules: 5.60/2.36 5.60/2.36 a -> d 5.60/2.36 a -> e 5.60/2.36 f(x) -> U1(x, x) 5.60/2.36 b -> d 5.60/2.36 b -> e 5.60/2.36 U1(d, x) -> x 5.60/2.36 5.60/2.36 Q is empty. 5.60/2.36 We have to consider all minimal (P,Q,R)-chains. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (465) TransformationProof (EQUIVALENT) 5.60/2.36 By narrowing [LPAR04] the rule A^1 -> H(U1(d, e), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.36 5.60/2.36 (A^1 -> H(e, U1(e, d)),A^1 -> H(e, U1(e, d))) 5.60/2.36 5.60/2.36 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (466) 5.60/2.36 Obligation: 5.60/2.36 Q DP problem: 5.60/2.36 The TRS P consists of the following rules: 5.60/2.36 5.60/2.36 A^1 -> H(f(d), f(d)) 5.60/2.36 H(x, x) -> G(x, x) 5.60/2.36 G(d, e) -> A^1 5.60/2.36 A^1 -> H(f(e), f(e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.36 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.36 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, e), d) 5.60/2.36 A^1 -> H(U1(d, a), e) 5.60/2.36 A^1 -> H(U1(e, d), b) 5.60/2.36 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.36 A^1 -> H(U1(a, d), e) 5.60/2.36 A^1 -> H(U1(a, e), e) 5.60/2.36 A^1 -> H(d, b) 5.60/2.36 A^1 -> H(d, U1(d, e)) 5.60/2.36 A^1 -> H(d, d) 5.60/2.36 A^1 -> H(U1(d, d), e) 5.60/2.36 A^1 -> H(e, b) 5.60/2.36 A^1 -> H(e, U1(d, e)) 5.60/2.36 A^1 -> H(a, e) 5.60/2.36 A^1 -> H(U1(d, e), e) 5.60/2.36 A^1 -> H(e, U1(e, d)) 5.60/2.36 5.60/2.36 The TRS R consists of the following rules: 5.60/2.36 5.60/2.36 a -> d 5.60/2.36 a -> e 5.60/2.36 f(x) -> U1(x, x) 5.60/2.36 b -> d 5.60/2.36 b -> e 5.60/2.36 U1(d, x) -> x 5.60/2.36 5.60/2.36 Q is empty. 5.60/2.36 We have to consider all minimal (P,Q,R)-chains. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (467) DependencyGraphProof (EQUIVALENT) 5.60/2.36 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (468) 5.60/2.36 Obligation: 5.60/2.36 Q DP problem: 5.60/2.36 The TRS P consists of the following rules: 5.60/2.36 5.60/2.36 H(x, x) -> G(x, x) 5.60/2.36 G(d, e) -> A^1 5.60/2.36 A^1 -> H(f(d), f(d)) 5.60/2.36 A^1 -> H(f(e), f(e)) 5.60/2.36 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.36 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.36 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.36 A^1 -> H(U1(d, e), d) 5.60/2.36 A^1 -> H(U1(d, a), e) 5.60/2.36 A^1 -> H(U1(e, d), b) 5.60/2.36 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.36 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.36 A^1 -> H(U1(a, d), e) 5.60/2.36 A^1 -> H(U1(a, e), e) 5.60/2.36 A^1 -> H(d, b) 5.60/2.36 A^1 -> H(d, U1(d, e)) 5.60/2.36 A^1 -> H(d, d) 5.60/2.36 A^1 -> H(U1(d, d), e) 5.60/2.36 A^1 -> H(e, b) 5.60/2.36 A^1 -> H(e, U1(d, e)) 5.60/2.36 A^1 -> H(a, e) 5.60/2.36 A^1 -> H(U1(d, e), e) 5.60/2.36 5.60/2.36 The TRS R consists of the following rules: 5.60/2.36 5.60/2.36 a -> d 5.60/2.36 a -> e 5.60/2.36 f(x) -> U1(x, x) 5.60/2.36 b -> d 5.60/2.36 b -> e 5.60/2.36 U1(d, x) -> x 5.60/2.36 5.60/2.36 Q is empty. 5.60/2.36 We have to consider all minimal (P,Q,R)-chains. 5.60/2.36 ---------------------------------------- 5.60/2.36 5.60/2.36 (469) TransformationProof (EQUIVALENT) 5.60/2.36 By narrowing [LPAR04] the rule A^1 -> H(U1(d, e), d) at position [] we obtained the following new rules [LPAR04]: 5.60/2.37 5.60/2.37 (A^1 -> H(e, d),A^1 -> H(e, d)) 5.60/2.37 5.60/2.37 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (470) 5.60/2.37 Obligation: 5.60/2.37 Q DP problem: 5.60/2.37 The TRS P consists of the following rules: 5.60/2.37 5.60/2.37 H(x, x) -> G(x, x) 5.60/2.37 G(d, e) -> A^1 5.60/2.37 A^1 -> H(f(d), f(d)) 5.60/2.37 A^1 -> H(f(e), f(e)) 5.60/2.37 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.37 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.37 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.37 A^1 -> H(U1(d, a), e) 5.60/2.37 A^1 -> H(U1(e, d), b) 5.60/2.37 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.37 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.37 A^1 -> H(U1(a, d), e) 5.60/2.37 A^1 -> H(U1(a, e), e) 5.60/2.37 A^1 -> H(d, b) 5.60/2.37 A^1 -> H(d, U1(d, e)) 5.60/2.37 A^1 -> H(d, d) 5.60/2.37 A^1 -> H(U1(d, d), e) 5.60/2.37 A^1 -> H(e, b) 5.60/2.37 A^1 -> H(e, U1(d, e)) 5.60/2.37 A^1 -> H(a, e) 5.60/2.37 A^1 -> H(U1(d, e), e) 5.60/2.37 A^1 -> H(e, d) 5.60/2.37 5.60/2.37 The TRS R consists of the following rules: 5.60/2.37 5.60/2.37 a -> d 5.60/2.37 a -> e 5.60/2.37 f(x) -> U1(x, x) 5.60/2.37 b -> d 5.60/2.37 b -> e 5.60/2.37 U1(d, x) -> x 5.60/2.37 5.60/2.37 Q is empty. 5.60/2.37 We have to consider all minimal (P,Q,R)-chains. 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (471) DependencyGraphProof (EQUIVALENT) 5.60/2.37 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (472) 5.60/2.37 Obligation: 5.60/2.37 Q DP problem: 5.60/2.37 The TRS P consists of the following rules: 5.60/2.37 5.60/2.37 G(d, e) -> A^1 5.60/2.37 A^1 -> H(f(d), f(d)) 5.60/2.37 H(x, x) -> G(x, x) 5.60/2.37 A^1 -> H(f(e), f(e)) 5.60/2.37 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.37 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.37 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.37 A^1 -> H(U1(d, a), e) 5.60/2.37 A^1 -> H(U1(e, d), b) 5.60/2.37 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.37 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.37 A^1 -> H(U1(a, d), e) 5.60/2.37 A^1 -> H(U1(a, e), e) 5.60/2.37 A^1 -> H(d, b) 5.60/2.37 A^1 -> H(d, U1(d, e)) 5.60/2.37 A^1 -> H(d, d) 5.60/2.37 A^1 -> H(U1(d, d), e) 5.60/2.37 A^1 -> H(e, b) 5.60/2.37 A^1 -> H(e, U1(d, e)) 5.60/2.37 A^1 -> H(a, e) 5.60/2.37 A^1 -> H(U1(d, e), e) 5.60/2.37 5.60/2.37 The TRS R consists of the following rules: 5.60/2.37 5.60/2.37 a -> d 5.60/2.37 a -> e 5.60/2.37 f(x) -> U1(x, x) 5.60/2.37 b -> d 5.60/2.37 b -> e 5.60/2.37 U1(d, x) -> x 5.60/2.37 5.60/2.37 Q is empty. 5.60/2.37 We have to consider all minimal (P,Q,R)-chains. 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (473) TransformationProof (EQUIVALENT) 5.60/2.37 By narrowing [LPAR04] the rule A^1 -> H(U1(d, a), e) at position [] we obtained the following new rules [LPAR04]: 5.60/2.37 5.60/2.37 (A^1 -> H(a, e),A^1 -> H(a, e)) 5.60/2.37 (A^1 -> H(U1(d, d), e),A^1 -> H(U1(d, d), e)) 5.60/2.37 (A^1 -> H(U1(d, e), e),A^1 -> H(U1(d, e), e)) 5.60/2.37 5.60/2.37 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (474) 5.60/2.37 Obligation: 5.60/2.37 Q DP problem: 5.60/2.37 The TRS P consists of the following rules: 5.60/2.37 5.60/2.37 G(d, e) -> A^1 5.60/2.37 A^1 -> H(f(d), f(d)) 5.60/2.37 H(x, x) -> G(x, x) 5.60/2.37 A^1 -> H(f(e), f(e)) 5.60/2.37 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.37 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.37 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.37 A^1 -> H(U1(e, d), b) 5.60/2.37 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.37 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.37 A^1 -> H(U1(a, d), e) 5.60/2.37 A^1 -> H(U1(a, e), e) 5.60/2.37 A^1 -> H(d, b) 5.60/2.37 A^1 -> H(d, U1(d, e)) 5.60/2.37 A^1 -> H(d, d) 5.60/2.37 A^1 -> H(U1(d, d), e) 5.60/2.37 A^1 -> H(e, b) 5.60/2.37 A^1 -> H(e, U1(d, e)) 5.60/2.37 A^1 -> H(a, e) 5.60/2.37 A^1 -> H(U1(d, e), e) 5.60/2.37 5.60/2.37 The TRS R consists of the following rules: 5.60/2.37 5.60/2.37 a -> d 5.60/2.37 a -> e 5.60/2.37 f(x) -> U1(x, x) 5.60/2.37 b -> d 5.60/2.37 b -> e 5.60/2.37 U1(d, x) -> x 5.60/2.37 5.60/2.37 Q is empty. 5.60/2.37 We have to consider all minimal (P,Q,R)-chains. 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (475) TransformationProof (EQUIVALENT) 5.60/2.37 By narrowing [LPAR04] the rule A^1 -> H(U1(e, d), b) at position [] we obtained the following new rules [LPAR04]: 5.60/2.37 5.60/2.37 (A^1 -> H(U1(e, d), d),A^1 -> H(U1(e, d), d)) 5.60/2.37 (A^1 -> H(U1(e, d), e),A^1 -> H(U1(e, d), e)) 5.60/2.37 5.60/2.37 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (476) 5.60/2.37 Obligation: 5.60/2.37 Q DP problem: 5.60/2.37 The TRS P consists of the following rules: 5.60/2.37 5.60/2.37 G(d, e) -> A^1 5.60/2.37 A^1 -> H(f(d), f(d)) 5.60/2.37 H(x, x) -> G(x, x) 5.60/2.37 A^1 -> H(f(e), f(e)) 5.60/2.37 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.37 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.37 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.37 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.37 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.37 A^1 -> H(U1(a, d), e) 5.60/2.37 A^1 -> H(U1(a, e), e) 5.60/2.37 A^1 -> H(d, b) 5.60/2.37 A^1 -> H(d, U1(d, e)) 5.60/2.37 A^1 -> H(d, d) 5.60/2.37 A^1 -> H(U1(d, d), e) 5.60/2.37 A^1 -> H(e, b) 5.60/2.37 A^1 -> H(e, U1(d, e)) 5.60/2.37 A^1 -> H(a, e) 5.60/2.37 A^1 -> H(U1(d, e), e) 5.60/2.37 A^1 -> H(U1(e, d), d) 5.60/2.37 A^1 -> H(U1(e, d), e) 5.60/2.37 5.60/2.37 The TRS R consists of the following rules: 5.60/2.37 5.60/2.37 a -> d 5.60/2.37 a -> e 5.60/2.37 f(x) -> U1(x, x) 5.60/2.37 b -> d 5.60/2.37 b -> e 5.60/2.37 U1(d, x) -> x 5.60/2.37 5.60/2.37 Q is empty. 5.60/2.37 We have to consider all minimal (P,Q,R)-chains. 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (477) DependencyGraphProof (EQUIVALENT) 5.60/2.37 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (478) 5.60/2.37 Obligation: 5.60/2.37 Q DP problem: 5.60/2.37 The TRS P consists of the following rules: 5.60/2.37 5.60/2.37 A^1 -> H(f(d), f(d)) 5.60/2.37 H(x, x) -> G(x, x) 5.60/2.37 G(d, e) -> A^1 5.60/2.37 A^1 -> H(f(e), f(e)) 5.60/2.37 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.37 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.37 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.37 A^1 -> H(U1(e, d), U1(d, e)) 5.60/2.37 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.37 A^1 -> H(U1(a, d), e) 5.60/2.37 A^1 -> H(U1(a, e), e) 5.60/2.37 A^1 -> H(d, b) 5.60/2.37 A^1 -> H(d, U1(d, e)) 5.60/2.37 A^1 -> H(d, d) 5.60/2.37 A^1 -> H(U1(d, d), e) 5.60/2.37 A^1 -> H(e, b) 5.60/2.37 A^1 -> H(e, U1(d, e)) 5.60/2.37 A^1 -> H(a, e) 5.60/2.37 A^1 -> H(U1(d, e), e) 5.60/2.37 5.60/2.37 The TRS R consists of the following rules: 5.60/2.37 5.60/2.37 a -> d 5.60/2.37 a -> e 5.60/2.37 f(x) -> U1(x, x) 5.60/2.37 b -> d 5.60/2.37 b -> e 5.60/2.37 U1(d, x) -> x 5.60/2.37 5.60/2.37 Q is empty. 5.60/2.37 We have to consider all minimal (P,Q,R)-chains. 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (479) TransformationProof (EQUIVALENT) 5.60/2.37 By narrowing [LPAR04] the rule A^1 -> H(U1(e, d), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.37 5.60/2.37 (A^1 -> H(U1(e, d), e),A^1 -> H(U1(e, d), e)) 5.60/2.37 5.60/2.37 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (480) 5.60/2.37 Obligation: 5.60/2.37 Q DP problem: 5.60/2.37 The TRS P consists of the following rules: 5.60/2.37 5.60/2.37 A^1 -> H(f(d), f(d)) 5.60/2.37 H(x, x) -> G(x, x) 5.60/2.37 G(d, e) -> A^1 5.60/2.37 A^1 -> H(f(e), f(e)) 5.60/2.37 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.37 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.37 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.37 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.37 A^1 -> H(U1(a, d), e) 5.60/2.37 A^1 -> H(U1(a, e), e) 5.60/2.37 A^1 -> H(d, b) 5.60/2.37 A^1 -> H(d, U1(d, e)) 5.60/2.37 A^1 -> H(d, d) 5.60/2.37 A^1 -> H(U1(d, d), e) 5.60/2.37 A^1 -> H(e, b) 5.60/2.37 A^1 -> H(e, U1(d, e)) 5.60/2.37 A^1 -> H(a, e) 5.60/2.37 A^1 -> H(U1(d, e), e) 5.60/2.37 A^1 -> H(U1(e, d), e) 5.60/2.37 5.60/2.37 The TRS R consists of the following rules: 5.60/2.37 5.60/2.37 a -> d 5.60/2.37 a -> e 5.60/2.37 f(x) -> U1(x, x) 5.60/2.37 b -> d 5.60/2.37 b -> e 5.60/2.37 U1(d, x) -> x 5.60/2.37 5.60/2.37 Q is empty. 5.60/2.37 We have to consider all minimal (P,Q,R)-chains. 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (481) DependencyGraphProof (EQUIVALENT) 5.60/2.37 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (482) 5.60/2.37 Obligation: 5.60/2.37 Q DP problem: 5.60/2.37 The TRS P consists of the following rules: 5.60/2.37 5.60/2.37 H(x, x) -> G(x, x) 5.60/2.37 G(d, e) -> A^1 5.60/2.37 A^1 -> H(f(d), f(d)) 5.60/2.37 A^1 -> H(f(e), f(e)) 5.60/2.37 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.37 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.37 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.37 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.37 A^1 -> H(U1(a, d), e) 5.60/2.37 A^1 -> H(U1(a, e), e) 5.60/2.37 A^1 -> H(d, b) 5.60/2.37 A^1 -> H(d, U1(d, e)) 5.60/2.37 A^1 -> H(d, d) 5.60/2.37 A^1 -> H(U1(d, d), e) 5.60/2.37 A^1 -> H(e, b) 5.60/2.37 A^1 -> H(e, U1(d, e)) 5.60/2.37 A^1 -> H(a, e) 5.60/2.37 A^1 -> H(U1(d, e), e) 5.60/2.37 5.60/2.37 The TRS R consists of the following rules: 5.60/2.37 5.60/2.37 a -> d 5.60/2.37 a -> e 5.60/2.37 f(x) -> U1(x, x) 5.60/2.37 b -> d 5.60/2.37 b -> e 5.60/2.37 U1(d, x) -> x 5.60/2.37 5.60/2.37 Q is empty. 5.60/2.37 We have to consider all minimal (P,Q,R)-chains. 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (483) TransformationProof (EQUIVALENT) 5.60/2.37 By narrowing [LPAR04] the rule A^1 -> H(U1(a, d), e) at position [] we obtained the following new rules [LPAR04]: 5.60/2.37 5.60/2.37 (A^1 -> H(U1(d, d), e),A^1 -> H(U1(d, d), e)) 5.60/2.37 (A^1 -> H(U1(e, d), e),A^1 -> H(U1(e, d), e)) 5.60/2.37 5.60/2.37 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (484) 5.60/2.37 Obligation: 5.60/2.37 Q DP problem: 5.60/2.37 The TRS P consists of the following rules: 5.60/2.37 5.60/2.37 H(x, x) -> G(x, x) 5.60/2.37 G(d, e) -> A^1 5.60/2.37 A^1 -> H(f(d), f(d)) 5.60/2.37 A^1 -> H(f(e), f(e)) 5.60/2.37 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.37 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.37 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.37 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.37 A^1 -> H(U1(a, e), e) 5.60/2.37 A^1 -> H(d, b) 5.60/2.37 A^1 -> H(d, U1(d, e)) 5.60/2.37 A^1 -> H(d, d) 5.60/2.37 A^1 -> H(U1(d, d), e) 5.60/2.37 A^1 -> H(e, b) 5.60/2.37 A^1 -> H(e, U1(d, e)) 5.60/2.37 A^1 -> H(a, e) 5.60/2.37 A^1 -> H(U1(d, e), e) 5.60/2.37 A^1 -> H(U1(e, d), e) 5.60/2.37 5.60/2.37 The TRS R consists of the following rules: 5.60/2.37 5.60/2.37 a -> d 5.60/2.37 a -> e 5.60/2.37 f(x) -> U1(x, x) 5.60/2.37 b -> d 5.60/2.37 b -> e 5.60/2.37 U1(d, x) -> x 5.60/2.37 5.60/2.37 Q is empty. 5.60/2.37 We have to consider all minimal (P,Q,R)-chains. 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (485) DependencyGraphProof (EQUIVALENT) 5.60/2.37 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (486) 5.60/2.37 Obligation: 5.60/2.37 Q DP problem: 5.60/2.37 The TRS P consists of the following rules: 5.60/2.37 5.60/2.37 G(d, e) -> A^1 5.60/2.37 A^1 -> H(f(d), f(d)) 5.60/2.37 H(x, x) -> G(x, x) 5.60/2.37 A^1 -> H(f(e), f(e)) 5.60/2.37 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.37 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.37 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.37 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.37 A^1 -> H(U1(a, e), e) 5.60/2.37 A^1 -> H(d, b) 5.60/2.37 A^1 -> H(d, U1(d, e)) 5.60/2.37 A^1 -> H(d, d) 5.60/2.37 A^1 -> H(U1(d, d), e) 5.60/2.37 A^1 -> H(e, b) 5.60/2.37 A^1 -> H(e, U1(d, e)) 5.60/2.37 A^1 -> H(a, e) 5.60/2.37 A^1 -> H(U1(d, e), e) 5.60/2.37 5.60/2.37 The TRS R consists of the following rules: 5.60/2.37 5.60/2.37 a -> d 5.60/2.37 a -> e 5.60/2.37 f(x) -> U1(x, x) 5.60/2.37 b -> d 5.60/2.37 b -> e 5.60/2.37 U1(d, x) -> x 5.60/2.37 5.60/2.37 Q is empty. 5.60/2.37 We have to consider all minimal (P,Q,R)-chains. 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (487) TransformationProof (EQUIVALENT) 5.60/2.37 By narrowing [LPAR04] the rule A^1 -> H(U1(a, e), e) at position [] we obtained the following new rules [LPAR04]: 5.60/2.37 5.60/2.37 (A^1 -> H(U1(d, e), e),A^1 -> H(U1(d, e), e)) 5.60/2.37 (A^1 -> H(U1(e, e), e),A^1 -> H(U1(e, e), e)) 5.60/2.37 5.60/2.37 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (488) 5.60/2.37 Obligation: 5.60/2.37 Q DP problem: 5.60/2.37 The TRS P consists of the following rules: 5.60/2.37 5.60/2.37 G(d, e) -> A^1 5.60/2.37 A^1 -> H(f(d), f(d)) 5.60/2.37 H(x, x) -> G(x, x) 5.60/2.37 A^1 -> H(f(e), f(e)) 5.60/2.37 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.37 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.37 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.37 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.37 A^1 -> H(d, b) 5.60/2.37 A^1 -> H(d, U1(d, e)) 5.60/2.37 A^1 -> H(d, d) 5.60/2.37 A^1 -> H(U1(d, d), e) 5.60/2.37 A^1 -> H(e, b) 5.60/2.37 A^1 -> H(e, U1(d, e)) 5.60/2.37 A^1 -> H(a, e) 5.60/2.37 A^1 -> H(U1(d, e), e) 5.60/2.37 A^1 -> H(U1(e, e), e) 5.60/2.37 5.60/2.37 The TRS R consists of the following rules: 5.60/2.37 5.60/2.37 a -> d 5.60/2.37 a -> e 5.60/2.37 f(x) -> U1(x, x) 5.60/2.37 b -> d 5.60/2.37 b -> e 5.60/2.37 U1(d, x) -> x 5.60/2.37 5.60/2.37 Q is empty. 5.60/2.37 We have to consider all minimal (P,Q,R)-chains. 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (489) DependencyGraphProof (EQUIVALENT) 5.60/2.37 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (490) 5.60/2.37 Obligation: 5.60/2.37 Q DP problem: 5.60/2.37 The TRS P consists of the following rules: 5.60/2.37 5.60/2.37 A^1 -> H(f(d), f(d)) 5.60/2.37 H(x, x) -> G(x, x) 5.60/2.37 G(d, e) -> A^1 5.60/2.37 A^1 -> H(f(e), f(e)) 5.60/2.37 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.37 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.37 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.37 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.37 A^1 -> H(d, b) 5.60/2.37 A^1 -> H(d, U1(d, e)) 5.60/2.37 A^1 -> H(d, d) 5.60/2.37 A^1 -> H(U1(d, d), e) 5.60/2.37 A^1 -> H(e, b) 5.60/2.37 A^1 -> H(e, U1(d, e)) 5.60/2.37 A^1 -> H(a, e) 5.60/2.37 A^1 -> H(U1(d, e), e) 5.60/2.37 5.60/2.37 The TRS R consists of the following rules: 5.60/2.37 5.60/2.37 a -> d 5.60/2.37 a -> e 5.60/2.37 f(x) -> U1(x, x) 5.60/2.37 b -> d 5.60/2.37 b -> e 5.60/2.37 U1(d, x) -> x 5.60/2.37 5.60/2.37 Q is empty. 5.60/2.37 We have to consider all minimal (P,Q,R)-chains. 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (491) TransformationProof (EQUIVALENT) 5.60/2.37 By narrowing [LPAR04] the rule A^1 -> H(d, b) at position [] we obtained the following new rules [LPAR04]: 5.60/2.37 5.60/2.37 (A^1 -> H(d, d),A^1 -> H(d, d)) 5.60/2.37 (A^1 -> H(d, e),A^1 -> H(d, e)) 5.60/2.37 5.60/2.37 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (492) 5.60/2.37 Obligation: 5.60/2.37 Q DP problem: 5.60/2.37 The TRS P consists of the following rules: 5.60/2.37 5.60/2.37 A^1 -> H(f(d), f(d)) 5.60/2.37 H(x, x) -> G(x, x) 5.60/2.37 G(d, e) -> A^1 5.60/2.37 A^1 -> H(f(e), f(e)) 5.60/2.37 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.37 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.37 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.37 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.37 A^1 -> H(d, U1(d, e)) 5.60/2.37 A^1 -> H(d, d) 5.60/2.37 A^1 -> H(U1(d, d), e) 5.60/2.37 A^1 -> H(e, b) 5.60/2.37 A^1 -> H(e, U1(d, e)) 5.60/2.37 A^1 -> H(a, e) 5.60/2.37 A^1 -> H(U1(d, e), e) 5.60/2.37 A^1 -> H(d, e) 5.60/2.37 5.60/2.37 The TRS R consists of the following rules: 5.60/2.37 5.60/2.37 a -> d 5.60/2.37 a -> e 5.60/2.37 f(x) -> U1(x, x) 5.60/2.37 b -> d 5.60/2.37 b -> e 5.60/2.37 U1(d, x) -> x 5.60/2.37 5.60/2.37 Q is empty. 5.60/2.37 We have to consider all minimal (P,Q,R)-chains. 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (493) DependencyGraphProof (EQUIVALENT) 5.60/2.37 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (494) 5.60/2.37 Obligation: 5.60/2.37 Q DP problem: 5.60/2.37 The TRS P consists of the following rules: 5.60/2.37 5.60/2.37 H(x, x) -> G(x, x) 5.60/2.37 G(d, e) -> A^1 5.60/2.37 A^1 -> H(f(d), f(d)) 5.60/2.37 A^1 -> H(f(e), f(e)) 5.60/2.37 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.37 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.37 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.37 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.37 A^1 -> H(d, U1(d, e)) 5.60/2.37 A^1 -> H(d, d) 5.60/2.37 A^1 -> H(U1(d, d), e) 5.60/2.37 A^1 -> H(e, b) 5.60/2.37 A^1 -> H(e, U1(d, e)) 5.60/2.37 A^1 -> H(a, e) 5.60/2.37 A^1 -> H(U1(d, e), e) 5.60/2.37 5.60/2.37 The TRS R consists of the following rules: 5.60/2.37 5.60/2.37 a -> d 5.60/2.37 a -> e 5.60/2.37 f(x) -> U1(x, x) 5.60/2.37 b -> d 5.60/2.37 b -> e 5.60/2.37 U1(d, x) -> x 5.60/2.37 5.60/2.37 Q is empty. 5.60/2.37 We have to consider all minimal (P,Q,R)-chains. 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (495) TransformationProof (EQUIVALENT) 5.60/2.37 By narrowing [LPAR04] the rule A^1 -> H(d, U1(d, e)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.37 5.60/2.37 (A^1 -> H(d, e),A^1 -> H(d, e)) 5.60/2.37 5.60/2.37 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (496) 5.60/2.37 Obligation: 5.60/2.37 Q DP problem: 5.60/2.37 The TRS P consists of the following rules: 5.60/2.37 5.60/2.37 H(x, x) -> G(x, x) 5.60/2.37 G(d, e) -> A^1 5.60/2.37 A^1 -> H(f(d), f(d)) 5.60/2.37 A^1 -> H(f(e), f(e)) 5.60/2.37 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.37 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.37 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.37 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.37 A^1 -> H(d, d) 5.60/2.37 A^1 -> H(U1(d, d), e) 5.60/2.37 A^1 -> H(e, b) 5.60/2.37 A^1 -> H(e, U1(d, e)) 5.60/2.37 A^1 -> H(a, e) 5.60/2.37 A^1 -> H(U1(d, e), e) 5.60/2.37 A^1 -> H(d, e) 5.60/2.37 5.60/2.37 The TRS R consists of the following rules: 5.60/2.37 5.60/2.37 a -> d 5.60/2.37 a -> e 5.60/2.37 f(x) -> U1(x, x) 5.60/2.37 b -> d 5.60/2.37 b -> e 5.60/2.37 U1(d, x) -> x 5.60/2.37 5.60/2.37 Q is empty. 5.60/2.37 We have to consider all minimal (P,Q,R)-chains. 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (497) DependencyGraphProof (EQUIVALENT) 5.60/2.37 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (498) 5.60/2.37 Obligation: 5.60/2.37 Q DP problem: 5.60/2.37 The TRS P consists of the following rules: 5.60/2.37 5.60/2.37 G(d, e) -> A^1 5.60/2.37 A^1 -> H(f(d), f(d)) 5.60/2.37 H(x, x) -> G(x, x) 5.60/2.37 A^1 -> H(f(e), f(e)) 5.60/2.37 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.37 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.37 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.37 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.37 A^1 -> H(d, d) 5.60/2.37 A^1 -> H(U1(d, d), e) 5.60/2.37 A^1 -> H(e, b) 5.60/2.37 A^1 -> H(e, U1(d, e)) 5.60/2.37 A^1 -> H(a, e) 5.60/2.37 A^1 -> H(U1(d, e), e) 5.60/2.37 5.60/2.37 The TRS R consists of the following rules: 5.60/2.37 5.60/2.37 a -> d 5.60/2.37 a -> e 5.60/2.37 f(x) -> U1(x, x) 5.60/2.37 b -> d 5.60/2.37 b -> e 5.60/2.37 U1(d, x) -> x 5.60/2.37 5.60/2.37 Q is empty. 5.60/2.37 We have to consider all minimal (P,Q,R)-chains. 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (499) TransformationProof (EQUIVALENT) 5.60/2.37 By narrowing [LPAR04] the rule A^1 -> H(U1(d, d), e) at position [] we obtained the following new rules [LPAR04]: 5.60/2.37 5.60/2.37 (A^1 -> H(d, e),A^1 -> H(d, e)) 5.60/2.37 5.60/2.37 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (500) 5.60/2.37 Obligation: 5.60/2.37 Q DP problem: 5.60/2.37 The TRS P consists of the following rules: 5.60/2.37 5.60/2.37 G(d, e) -> A^1 5.60/2.37 A^1 -> H(f(d), f(d)) 5.60/2.37 H(x, x) -> G(x, x) 5.60/2.37 A^1 -> H(f(e), f(e)) 5.60/2.37 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.37 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.37 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.37 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.37 A^1 -> H(d, d) 5.60/2.37 A^1 -> H(e, b) 5.60/2.37 A^1 -> H(e, U1(d, e)) 5.60/2.37 A^1 -> H(a, e) 5.60/2.37 A^1 -> H(U1(d, e), e) 5.60/2.37 A^1 -> H(d, e) 5.60/2.37 5.60/2.37 The TRS R consists of the following rules: 5.60/2.37 5.60/2.37 a -> d 5.60/2.37 a -> e 5.60/2.37 f(x) -> U1(x, x) 5.60/2.37 b -> d 5.60/2.37 b -> e 5.60/2.37 U1(d, x) -> x 5.60/2.37 5.60/2.37 Q is empty. 5.60/2.37 We have to consider all minimal (P,Q,R)-chains. 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (501) DependencyGraphProof (EQUIVALENT) 5.60/2.37 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (502) 5.60/2.37 Obligation: 5.60/2.37 Q DP problem: 5.60/2.37 The TRS P consists of the following rules: 5.60/2.37 5.60/2.37 A^1 -> H(f(d), f(d)) 5.60/2.37 H(x, x) -> G(x, x) 5.60/2.37 G(d, e) -> A^1 5.60/2.37 A^1 -> H(f(e), f(e)) 5.60/2.37 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.37 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.37 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.37 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.37 A^1 -> H(d, d) 5.60/2.37 A^1 -> H(e, b) 5.60/2.37 A^1 -> H(e, U1(d, e)) 5.60/2.37 A^1 -> H(a, e) 5.60/2.37 A^1 -> H(U1(d, e), e) 5.60/2.37 5.60/2.37 The TRS R consists of the following rules: 5.60/2.37 5.60/2.37 a -> d 5.60/2.37 a -> e 5.60/2.37 f(x) -> U1(x, x) 5.60/2.37 b -> d 5.60/2.37 b -> e 5.60/2.37 U1(d, x) -> x 5.60/2.37 5.60/2.37 Q is empty. 5.60/2.37 We have to consider all minimal (P,Q,R)-chains. 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (503) TransformationProof (EQUIVALENT) 5.60/2.37 By narrowing [LPAR04] the rule A^1 -> H(e, b) at position [] we obtained the following new rules [LPAR04]: 5.60/2.37 5.60/2.37 (A^1 -> H(e, d),A^1 -> H(e, d)) 5.60/2.37 (A^1 -> H(e, e),A^1 -> H(e, e)) 5.60/2.37 5.60/2.37 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (504) 5.60/2.37 Obligation: 5.60/2.37 Q DP problem: 5.60/2.37 The TRS P consists of the following rules: 5.60/2.37 5.60/2.37 A^1 -> H(f(d), f(d)) 5.60/2.37 H(x, x) -> G(x, x) 5.60/2.37 G(d, e) -> A^1 5.60/2.37 A^1 -> H(f(e), f(e)) 5.60/2.37 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.37 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.37 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.37 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.37 A^1 -> H(d, d) 5.60/2.37 A^1 -> H(e, U1(d, e)) 5.60/2.37 A^1 -> H(a, e) 5.60/2.37 A^1 -> H(U1(d, e), e) 5.60/2.37 A^1 -> H(e, d) 5.60/2.37 A^1 -> H(e, e) 5.60/2.37 5.60/2.37 The TRS R consists of the following rules: 5.60/2.37 5.60/2.37 a -> d 5.60/2.37 a -> e 5.60/2.37 f(x) -> U1(x, x) 5.60/2.37 b -> d 5.60/2.37 b -> e 5.60/2.37 U1(d, x) -> x 5.60/2.37 5.60/2.37 Q is empty. 5.60/2.37 We have to consider all minimal (P,Q,R)-chains. 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (505) DependencyGraphProof (EQUIVALENT) 5.60/2.37 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (506) 5.60/2.37 Obligation: 5.60/2.37 Q DP problem: 5.60/2.37 The TRS P consists of the following rules: 5.60/2.37 5.60/2.37 H(x, x) -> G(x, x) 5.60/2.37 G(d, e) -> A^1 5.60/2.37 A^1 -> H(f(d), f(d)) 5.60/2.37 A^1 -> H(f(e), f(e)) 5.60/2.37 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.37 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.37 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.37 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.37 A^1 -> H(d, d) 5.60/2.37 A^1 -> H(e, U1(d, e)) 5.60/2.37 A^1 -> H(a, e) 5.60/2.37 A^1 -> H(U1(d, e), e) 5.60/2.37 A^1 -> H(e, e) 5.60/2.37 5.60/2.37 The TRS R consists of the following rules: 5.60/2.37 5.60/2.37 a -> d 5.60/2.37 a -> e 5.60/2.37 f(x) -> U1(x, x) 5.60/2.37 b -> d 5.60/2.37 b -> e 5.60/2.37 U1(d, x) -> x 5.60/2.37 5.60/2.37 Q is empty. 5.60/2.37 We have to consider all minimal (P,Q,R)-chains. 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (507) TransformationProof (EQUIVALENT) 5.60/2.37 By narrowing [LPAR04] the rule A^1 -> H(e, U1(d, e)) at position [] we obtained the following new rules [LPAR04]: 5.60/2.37 5.60/2.37 (A^1 -> H(e, e),A^1 -> H(e, e)) 5.60/2.37 5.60/2.37 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (508) 5.60/2.37 Obligation: 5.60/2.37 Q DP problem: 5.60/2.37 The TRS P consists of the following rules: 5.60/2.37 5.60/2.37 H(x, x) -> G(x, x) 5.60/2.37 G(d, e) -> A^1 5.60/2.37 A^1 -> H(f(d), f(d)) 5.60/2.37 A^1 -> H(f(e), f(e)) 5.60/2.37 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.37 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.37 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.37 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.37 A^1 -> H(d, d) 5.60/2.37 A^1 -> H(a, e) 5.60/2.37 A^1 -> H(U1(d, e), e) 5.60/2.37 A^1 -> H(e, e) 5.60/2.37 5.60/2.37 The TRS R consists of the following rules: 5.60/2.37 5.60/2.37 a -> d 5.60/2.37 a -> e 5.60/2.37 f(x) -> U1(x, x) 5.60/2.37 b -> d 5.60/2.37 b -> e 5.60/2.37 U1(d, x) -> x 5.60/2.37 5.60/2.37 Q is empty. 5.60/2.37 We have to consider all minimal (P,Q,R)-chains. 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (509) TransformationProof (EQUIVALENT) 5.60/2.37 By narrowing [LPAR04] the rule A^1 -> H(a, e) at position [] we obtained the following new rules [LPAR04]: 5.60/2.37 5.60/2.37 (A^1 -> H(d, e),A^1 -> H(d, e)) 5.60/2.37 (A^1 -> H(e, e),A^1 -> H(e, e)) 5.60/2.37 5.60/2.37 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (510) 5.60/2.37 Obligation: 5.60/2.37 Q DP problem: 5.60/2.37 The TRS P consists of the following rules: 5.60/2.37 5.60/2.37 H(x, x) -> G(x, x) 5.60/2.37 G(d, e) -> A^1 5.60/2.37 A^1 -> H(f(d), f(d)) 5.60/2.37 A^1 -> H(f(e), f(e)) 5.60/2.37 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.37 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.37 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.37 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.37 A^1 -> H(d, d) 5.60/2.37 A^1 -> H(U1(d, e), e) 5.60/2.37 A^1 -> H(e, e) 5.60/2.37 A^1 -> H(d, e) 5.60/2.37 5.60/2.37 The TRS R consists of the following rules: 5.60/2.37 5.60/2.37 a -> d 5.60/2.37 a -> e 5.60/2.37 f(x) -> U1(x, x) 5.60/2.37 b -> d 5.60/2.37 b -> e 5.60/2.37 U1(d, x) -> x 5.60/2.37 5.60/2.37 Q is empty. 5.60/2.37 We have to consider all minimal (P,Q,R)-chains. 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (511) DependencyGraphProof (EQUIVALENT) 5.60/2.37 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (512) 5.60/2.37 Obligation: 5.60/2.37 Q DP problem: 5.60/2.37 The TRS P consists of the following rules: 5.60/2.37 5.60/2.37 G(d, e) -> A^1 5.60/2.37 A^1 -> H(f(d), f(d)) 5.60/2.37 H(x, x) -> G(x, x) 5.60/2.37 A^1 -> H(f(e), f(e)) 5.60/2.37 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.37 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.37 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.37 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.37 A^1 -> H(d, d) 5.60/2.37 A^1 -> H(U1(d, e), e) 5.60/2.37 A^1 -> H(e, e) 5.60/2.37 5.60/2.37 The TRS R consists of the following rules: 5.60/2.37 5.60/2.37 a -> d 5.60/2.37 a -> e 5.60/2.37 f(x) -> U1(x, x) 5.60/2.37 b -> d 5.60/2.37 b -> e 5.60/2.37 U1(d, x) -> x 5.60/2.37 5.60/2.37 Q is empty. 5.60/2.37 We have to consider all minimal (P,Q,R)-chains. 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (513) TransformationProof (EQUIVALENT) 5.60/2.37 By narrowing [LPAR04] the rule A^1 -> H(U1(d, e), e) at position [] we obtained the following new rules [LPAR04]: 5.60/2.37 5.60/2.37 (A^1 -> H(e, e),A^1 -> H(e, e)) 5.60/2.37 5.60/2.37 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (514) 5.60/2.37 Obligation: 5.60/2.37 Q DP problem: 5.60/2.37 The TRS P consists of the following rules: 5.60/2.37 5.60/2.37 G(d, e) -> A^1 5.60/2.37 A^1 -> H(f(d), f(d)) 5.60/2.37 H(x, x) -> G(x, x) 5.60/2.37 A^1 -> H(f(e), f(e)) 5.60/2.37 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.37 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.37 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.37 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.37 A^1 -> H(d, d) 5.60/2.37 A^1 -> H(e, e) 5.60/2.37 5.60/2.37 The TRS R consists of the following rules: 5.60/2.37 5.60/2.37 a -> d 5.60/2.37 a -> e 5.60/2.37 f(x) -> U1(x, x) 5.60/2.37 b -> d 5.60/2.37 b -> e 5.60/2.37 U1(d, x) -> x 5.60/2.37 5.60/2.37 Q is empty. 5.60/2.37 We have to consider all minimal (P,Q,R)-chains. 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (515) UsableRulesReductionPairsProof (EQUIVALENT) 5.60/2.37 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. 5.60/2.37 5.60/2.37 No dependency pairs are removed. 5.60/2.37 5.60/2.37 The following rules are removed from R: 5.60/2.37 5.60/2.37 a -> d 5.60/2.37 a -> e 5.60/2.37 b -> d 5.60/2.37 b -> e 5.60/2.37 Used ordering: POLO with Polynomial interpretation [POLO]: 5.60/2.37 5.60/2.37 POL(A^1) = 0 5.60/2.37 POL(G(x_1, x_2)) = x_1 + x_2 5.60/2.37 POL(H(x_1, x_2)) = x_1 + x_2 5.60/2.37 POL(U1(x_1, x_2)) = x_1 + x_2 5.60/2.37 POL(d) = 0 5.60/2.37 POL(e) = 0 5.60/2.37 POL(f(x_1)) = 2*x_1 5.60/2.37 5.60/2.37 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (516) 5.60/2.37 Obligation: 5.60/2.37 Q DP problem: 5.60/2.37 The TRS P consists of the following rules: 5.60/2.37 5.60/2.37 G(d, e) -> A^1 5.60/2.37 A^1 -> H(f(d), f(d)) 5.60/2.37 H(x, x) -> G(x, x) 5.60/2.37 A^1 -> H(f(e), f(e)) 5.60/2.37 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.37 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.37 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.37 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.37 A^1 -> H(d, d) 5.60/2.37 A^1 -> H(e, e) 5.60/2.37 5.60/2.37 The TRS R consists of the following rules: 5.60/2.37 5.60/2.37 U1(d, x) -> x 5.60/2.37 f(x) -> U1(x, x) 5.60/2.37 5.60/2.37 Q is empty. 5.60/2.37 We have to consider all minimal (P,Q,R)-chains. 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (517) MNOCProof (EQUIVALENT) 5.60/2.37 We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R. 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (518) 5.60/2.37 Obligation: 5.60/2.37 Q DP problem: 5.60/2.37 The TRS P consists of the following rules: 5.60/2.37 5.60/2.37 G(d, e) -> A^1 5.60/2.37 A^1 -> H(f(d), f(d)) 5.60/2.37 H(x, x) -> G(x, x) 5.60/2.37 A^1 -> H(f(e), f(e)) 5.60/2.37 A^1 -> H(U1(d, d), U1(d, d)) 5.60/2.37 A^1 -> H(U1(e, e), U1(e, e)) 5.60/2.37 A^1 -> H(U1(d, e), U1(d, e)) 5.60/2.37 A^1 -> H(U1(e, d), U1(e, d)) 5.60/2.37 A^1 -> H(d, d) 5.60/2.37 A^1 -> H(e, e) 5.60/2.37 5.60/2.37 The TRS R consists of the following rules: 5.60/2.37 5.60/2.37 U1(d, x) -> x 5.60/2.37 f(x) -> U1(x, x) 5.60/2.37 5.60/2.37 The set Q consists of the following terms: 5.60/2.37 5.60/2.37 U1(d, x0) 5.60/2.37 f(x0) 5.60/2.37 5.60/2.37 We have to consider all minimal (P,Q,R)-chains. 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (519) DependencyGraphProof (EQUIVALENT) 5.60/2.37 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 10 less nodes. 5.60/2.37 ---------------------------------------- 5.60/2.37 5.60/2.37 (520) 5.60/2.37 TRUE 5.60/2.39 EOF