/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Quasi decreasingness of the given CTRS could be proven: (0) CTRS (1) CTRSToQTRSProof [SOUND, 0 ms] (2) QTRS (3) DependencyPairsProof [EQUIVALENT, 0 ms] (4) QDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) QDP (7) UsableRulesReductionPairsProof [EQUIVALENT, 14 ms] (8) QDP (9) TransformationProof [EQUIVALENT, 0 ms] (10) QDP (11) TransformationProof [EQUIVALENT, 0 ms] (12) QDP (13) TransformationProof [EQUIVALENT, 0 ms] (14) QDP (15) TransformationProof [EQUIVALENT, 0 ms] (16) QDP (17) TransformationProof [EQUIVALENT, 0 ms] (18) QDP (19) TransformationProof [EQUIVALENT, 0 ms] (20) QDP (21) TransformationProof [EQUIVALENT, 0 ms] (22) QDP (23) TransformationProof [EQUIVALENT, 0 ms] (24) QDP (25) TransformationProof [EQUIVALENT, 0 ms] (26) QDP (27) TransformationProof [EQUIVALENT, 0 ms] (28) QDP (29) TransformationProof [EQUIVALENT, 0 ms] (30) QDP (31) TransformationProof [EQUIVALENT, 0 ms] (32) QDP (33) TransformationProof [EQUIVALENT, 0 ms] (34) QDP (35) TransformationProof [EQUIVALENT, 0 ms] (36) QDP (37) TransformationProof [EQUIVALENT, 0 ms] (38) QDP (39) TransformationProof [EQUIVALENT, 0 ms] (40) QDP (41) TransformationProof [EQUIVALENT, 0 ms] (42) QDP (43) TransformationProof [EQUIVALENT, 0 ms] (44) QDP (45) TransformationProof [EQUIVALENT, 3 ms] (46) QDP (47) TransformationProof [EQUIVALENT, 0 ms] (48) QDP (49) TransformationProof [EQUIVALENT, 0 ms] (50) QDP (51) TransformationProof [EQUIVALENT, 0 ms] (52) QDP (53) TransformationProof [EQUIVALENT, 0 ms] (54) QDP (55) TransformationProof [EQUIVALENT, 0 ms] (56) QDP (57) TransformationProof [EQUIVALENT, 0 ms] (58) QDP (59) TransformationProof [EQUIVALENT, 0 ms] (60) QDP (61) TransformationProof [EQUIVALENT, 0 ms] (62) QDP (63) TransformationProof [EQUIVALENT, 0 ms] (64) QDP (65) TransformationProof [EQUIVALENT, 0 ms] (66) QDP (67) TransformationProof [EQUIVALENT, 0 ms] (68) QDP (69) TransformationProof [EQUIVALENT, 0 ms] (70) QDP (71) TransformationProof [EQUIVALENT, 0 ms] (72) QDP (73) TransformationProof [EQUIVALENT, 0 ms] (74) QDP (75) TransformationProof [EQUIVALENT, 0 ms] (76) QDP (77) TransformationProof [EQUIVALENT, 0 ms] (78) QDP (79) TransformationProof [EQUIVALENT, 0 ms] (80) QDP (81) TransformationProof [EQUIVALENT, 0 ms] (82) QDP (83) TransformationProof [EQUIVALENT, 0 ms] (84) QDP (85) TransformationProof [EQUIVALENT, 0 ms] (86) QDP (87) TransformationProof [EQUIVALENT, 0 ms] (88) QDP (89) TransformationProof [EQUIVALENT, 0 ms] (90) QDP (91) TransformationProof [EQUIVALENT, 0 ms] (92) QDP (93) TransformationProof [EQUIVALENT, 0 ms] (94) QDP (95) TransformationProof [EQUIVALENT, 0 ms] (96) QDP (97) TransformationProof [EQUIVALENT, 0 ms] (98) QDP (99) TransformationProof [EQUIVALENT, 0 ms] (100) QDP (101) TransformationProof [EQUIVALENT, 0 ms] (102) QDP (103) TransformationProof [EQUIVALENT, 0 ms] (104) QDP (105) TransformationProof [EQUIVALENT, 0 ms] (106) QDP (107) TransformationProof [EQUIVALENT, 0 ms] (108) QDP (109) TransformationProof [EQUIVALENT, 0 ms] (110) QDP (111) TransformationProof [EQUIVALENT, 0 ms] (112) QDP (113) TransformationProof [EQUIVALENT, 0 ms] (114) QDP (115) TransformationProof [EQUIVALENT, 0 ms] (116) QDP (117) TransformationProof [EQUIVALENT, 0 ms] (118) QDP (119) TransformationProof [EQUIVALENT, 0 ms] (120) QDP (121) TransformationProof [EQUIVALENT, 0 ms] (122) QDP (123) TransformationProof [EQUIVALENT, 0 ms] (124) QDP (125) TransformationProof [EQUIVALENT, 0 ms] (126) QDP (127) TransformationProof [EQUIVALENT, 0 ms] (128) QDP (129) TransformationProof [EQUIVALENT, 0 ms] (130) QDP (131) TransformationProof [EQUIVALENT, 0 ms] (132) QDP (133) TransformationProof [EQUIVALENT, 0 ms] (134) QDP (135) TransformationProof [EQUIVALENT, 0 ms] (136) QDP (137) TransformationProof [EQUIVALENT, 0 ms] (138) QDP (139) TransformationProof [EQUIVALENT, 0 ms] (140) QDP (141) TransformationProof [EQUIVALENT, 0 ms] (142) QDP (143) TransformationProof [EQUIVALENT, 0 ms] (144) QDP (145) TransformationProof [EQUIVALENT, 0 ms] (146) QDP (147) TransformationProof [EQUIVALENT, 0 ms] (148) QDP (149) TransformationProof [EQUIVALENT, 0 ms] (150) QDP (151) TransformationProof [EQUIVALENT, 0 ms] (152) QDP (153) TransformationProof [EQUIVALENT, 0 ms] (154) QDP (155) TransformationProof [EQUIVALENT, 0 ms] (156) QDP (157) TransformationProof [EQUIVALENT, 0 ms] (158) QDP (159) TransformationProof [EQUIVALENT, 0 ms] (160) QDP (161) TransformationProof [EQUIVALENT, 0 ms] (162) QDP (163) TransformationProof [EQUIVALENT, 0 ms] (164) QDP (165) TransformationProof [EQUIVALENT, 0 ms] (166) QDP (167) TransformationProof [EQUIVALENT, 0 ms] (168) QDP (169) TransformationProof [EQUIVALENT, 0 ms] (170) QDP (171) TransformationProof [EQUIVALENT, 0 ms] (172) QDP (173) TransformationProof [EQUIVALENT, 0 ms] (174) QDP (175) TransformationProof [EQUIVALENT, 0 ms] (176) QDP (177) TransformationProof [EQUIVALENT, 0 ms] (178) QDP (179) TransformationProof [EQUIVALENT, 0 ms] (180) QDP (181) DependencyGraphProof [EQUIVALENT, 0 ms] (182) QDP (183) TransformationProof [EQUIVALENT, 0 ms] (184) QDP (185) TransformationProof [EQUIVALENT, 0 ms] (186) QDP (187) TransformationProof [EQUIVALENT, 0 ms] (188) QDP (189) TransformationProof [EQUIVALENT, 0 ms] (190) QDP (191) TransformationProof [EQUIVALENT, 0 ms] (192) QDP (193) DependencyGraphProof [EQUIVALENT, 0 ms] (194) QDP (195) TransformationProof [EQUIVALENT, 0 ms] (196) QDP (197) DependencyGraphProof [EQUIVALENT, 0 ms] (198) QDP (199) TransformationProof [EQUIVALENT, 0 ms] (200) QDP (201) TransformationProof [EQUIVALENT, 0 ms] (202) QDP (203) TransformationProof [EQUIVALENT, 0 ms] (204) QDP (205) TransformationProof [EQUIVALENT, 0 ms] (206) QDP (207) TransformationProof [EQUIVALENT, 0 ms] (208) QDP (209) TransformationProof [EQUIVALENT, 0 ms] (210) QDP (211) DependencyGraphProof [EQUIVALENT, 0 ms] (212) QDP (213) TransformationProof [EQUIVALENT, 0 ms] (214) QDP (215) TransformationProof [EQUIVALENT, 0 ms] (216) QDP (217) TransformationProof [EQUIVALENT, 0 ms] (218) QDP (219) TransformationProof [EQUIVALENT, 0 ms] (220) QDP (221) TransformationProof [EQUIVALENT, 0 ms] (222) QDP (223) TransformationProof [EQUIVALENT, 0 ms] (224) QDP (225) TransformationProof [EQUIVALENT, 0 ms] (226) QDP (227) TransformationProof [EQUIVALENT, 0 ms] (228) QDP (229) TransformationProof [EQUIVALENT, 0 ms] (230) QDP (231) TransformationProof [EQUIVALENT, 0 ms] (232) QDP (233) DependencyGraphProof [EQUIVALENT, 0 ms] (234) QDP (235) TransformationProof [EQUIVALENT, 0 ms] (236) QDP (237) DependencyGraphProof [EQUIVALENT, 0 ms] (238) QDP (239) TransformationProof [EQUIVALENT, 0 ms] (240) QDP (241) TransformationProof [EQUIVALENT, 0 ms] (242) QDP (243) DependencyGraphProof [EQUIVALENT, 0 ms] (244) QDP (245) TransformationProof [EQUIVALENT, 0 ms] (246) QDP (247) TransformationProof [EQUIVALENT, 0 ms] (248) QDP (249) DependencyGraphProof [EQUIVALENT, 0 ms] (250) QDP (251) TransformationProof [EQUIVALENT, 0 ms] (252) QDP (253) TransformationProof [EQUIVALENT, 0 ms] (254) QDP (255) TransformationProof [EQUIVALENT, 0 ms] (256) QDP (257) DependencyGraphProof [EQUIVALENT, 0 ms] (258) QDP (259) TransformationProof [EQUIVALENT, 0 ms] (260) QDP (261) TransformationProof [EQUIVALENT, 0 ms] (262) QDP (263) TransformationProof [EQUIVALENT, 0 ms] (264) QDP (265) TransformationProof [EQUIVALENT, 0 ms] (266) QDP (267) TransformationProof [EQUIVALENT, 0 ms] (268) QDP (269) DependencyGraphProof [EQUIVALENT, 0 ms] (270) QDP (271) TransformationProof [EQUIVALENT, 0 ms] (272) QDP (273) DependencyGraphProof [EQUIVALENT, 0 ms] (274) QDP (275) TransformationProof [EQUIVALENT, 0 ms] (276) QDP (277) TransformationProof [EQUIVALENT, 0 ms] (278) QDP (279) DependencyGraphProof [EQUIVALENT, 0 ms] (280) QDP (281) TransformationProof [EQUIVALENT, 0 ms] (282) QDP (283) TransformationProof [EQUIVALENT, 0 ms] (284) QDP (285) TransformationProof [EQUIVALENT, 0 ms] (286) QDP (287) DependencyGraphProof [EQUIVALENT, 0 ms] (288) QDP (289) TransformationProof [EQUIVALENT, 0 ms] (290) QDP (291) DependencyGraphProof [EQUIVALENT, 0 ms] (292) QDP (293) TransformationProof [EQUIVALENT, 0 ms] (294) QDP (295) TransformationProof [EQUIVALENT, 0 ms] (296) QDP (297) TransformationProof [EQUIVALENT, 0 ms] (298) QDP (299) DependencyGraphProof [EQUIVALENT, 0 ms] (300) QDP (301) TransformationProof [EQUIVALENT, 0 ms] (302) QDP (303) TransformationProof [EQUIVALENT, 0 ms] (304) QDP (305) DependencyGraphProof [EQUIVALENT, 0 ms] (306) QDP (307) TransformationProof [EQUIVALENT, 0 ms] (308) QDP (309) TransformationProof [EQUIVALENT, 0 ms] (310) QDP (311) DependencyGraphProof [EQUIVALENT, 0 ms] (312) QDP (313) TransformationProof [EQUIVALENT, 0 ms] (314) QDP (315) TransformationProof [EQUIVALENT, 0 ms] (316) QDP (317) TransformationProof [EQUIVALENT, 0 ms] (318) QDP (319) TransformationProof [EQUIVALENT, 0 ms] (320) QDP (321) DependencyGraphProof [EQUIVALENT, 0 ms] (322) QDP (323) TransformationProof [EQUIVALENT, 0 ms] (324) QDP (325) TransformationProof [EQUIVALENT, 0 ms] (326) QDP (327) DependencyGraphProof [EQUIVALENT, 0 ms] (328) QDP (329) TransformationProof [EQUIVALENT, 0 ms] (330) QDP (331) TransformationProof [EQUIVALENT, 0 ms] (332) QDP (333) TransformationProof [EQUIVALENT, 0 ms] (334) QDP (335) TransformationProof [EQUIVALENT, 0 ms] (336) QDP (337) DependencyGraphProof [EQUIVALENT, 0 ms] (338) QDP (339) TransformationProof [EQUIVALENT, 0 ms] (340) QDP (341) TransformationProof [EQUIVALENT, 0 ms] (342) QDP (343) TransformationProof [EQUIVALENT, 0 ms] (344) QDP (345) TransformationProof [EQUIVALENT, 0 ms] (346) QDP (347) TransformationProof [EQUIVALENT, 0 ms] (348) QDP (349) TransformationProof [EQUIVALENT, 0 ms] (350) QDP (351) DependencyGraphProof [EQUIVALENT, 0 ms] (352) QDP (353) TransformationProof [EQUIVALENT, 0 ms] (354) QDP (355) TransformationProof [EQUIVALENT, 0 ms] (356) QDP (357) DependencyGraphProof [EQUIVALENT, 0 ms] (358) QDP (359) TransformationProof [EQUIVALENT, 0 ms] (360) QDP (361) DependencyGraphProof [EQUIVALENT, 0 ms] (362) QDP (363) TransformationProof [EQUIVALENT, 0 ms] (364) QDP (365) DependencyGraphProof [EQUIVALENT, 0 ms] (366) QDP (367) TransformationProof [EQUIVALENT, 0 ms] (368) QDP (369) DependencyGraphProof [EQUIVALENT, 0 ms] (370) QDP (371) TransformationProof [EQUIVALENT, 0 ms] (372) QDP (373) DependencyGraphProof [EQUIVALENT, 0 ms] (374) QDP (375) TransformationProof [EQUIVALENT, 0 ms] (376) QDP (377) TransformationProof [EQUIVALENT, 0 ms] (378) QDP (379) TransformationProof [EQUIVALENT, 0 ms] (380) QDP (381) TransformationProof [EQUIVALENT, 0 ms] (382) QDP (383) DependencyGraphProof [EQUIVALENT, 0 ms] (384) QDP (385) TransformationProof [EQUIVALENT, 0 ms] (386) QDP (387) TransformationProof [EQUIVALENT, 0 ms] (388) QDP (389) TransformationProof [EQUIVALENT, 0 ms] (390) QDP (391) DependencyGraphProof [EQUIVALENT, 0 ms] (392) QDP (393) TransformationProof [EQUIVALENT, 0 ms] (394) QDP (395) DependencyGraphProof [EQUIVALENT, 0 ms] (396) QDP (397) TransformationProof [EQUIVALENT, 0 ms] (398) QDP (399) DependencyGraphProof [EQUIVALENT, 0 ms] (400) QDP (401) TransformationProof [EQUIVALENT, 0 ms] (402) QDP (403) TransformationProof [EQUIVALENT, 0 ms] (404) QDP (405) DependencyGraphProof [EQUIVALENT, 0 ms] (406) QDP (407) TransformationProof [EQUIVALENT, 0 ms] (408) QDP (409) DependencyGraphProof [EQUIVALENT, 0 ms] (410) QDP (411) TransformationProof [EQUIVALENT, 0 ms] (412) QDP (413) TransformationProof [EQUIVALENT, 0 ms] (414) QDP (415) TransformationProof [EQUIVALENT, 0 ms] (416) QDP (417) TransformationProof [EQUIVALENT, 0 ms] (418) QDP (419) DependencyGraphProof [EQUIVALENT, 0 ms] (420) QDP (421) TransformationProof [EQUIVALENT, 0 ms] (422) QDP (423) TransformationProof [EQUIVALENT, 0 ms] (424) QDP (425) DependencyGraphProof [EQUIVALENT, 0 ms] (426) QDP (427) TransformationProof [EQUIVALENT, 0 ms] (428) QDP (429) DependencyGraphProof [EQUIVALENT, 0 ms] (430) QDP (431) TransformationProof [EQUIVALENT, 0 ms] (432) QDP (433) DependencyGraphProof [EQUIVALENT, 0 ms] (434) QDP (435) TransformationProof [EQUIVALENT, 0 ms] (436) QDP (437) TransformationProof [EQUIVALENT, 0 ms] (438) QDP (439) TransformationProof [EQUIVALENT, 0 ms] (440) QDP (441) DependencyGraphProof [EQUIVALENT, 0 ms] (442) QDP (443) TransformationProof [EQUIVALENT, 0 ms] (444) QDP (445) DependencyGraphProof [EQUIVALENT, 0 ms] (446) QDP (447) TransformationProof [EQUIVALENT, 0 ms] (448) QDP (449) DependencyGraphProof [EQUIVALENT, 0 ms] (450) QDP (451) TransformationProof [EQUIVALENT, 0 ms] (452) QDP (453) TransformationProof [EQUIVALENT, 0 ms] (454) QDP (455) TransformationProof [EQUIVALENT, 0 ms] (456) QDP (457) DependencyGraphProof [EQUIVALENT, 0 ms] (458) QDP (459) TransformationProof [EQUIVALENT, 0 ms] (460) QDP (461) DependencyGraphProof [EQUIVALENT, 0 ms] (462) QDP (463) TransformationProof [EQUIVALENT, 0 ms] (464) QDP (465) TransformationProof [EQUIVALENT, 0 ms] (466) QDP (467) DependencyGraphProof [EQUIVALENT, 0 ms] (468) QDP (469) TransformationProof [EQUIVALENT, 0 ms] (470) QDP (471) DependencyGraphProof [EQUIVALENT, 0 ms] (472) QDP (473) TransformationProof [EQUIVALENT, 0 ms] (474) QDP (475) TransformationProof [EQUIVALENT, 0 ms] (476) QDP (477) DependencyGraphProof [EQUIVALENT, 0 ms] (478) QDP (479) TransformationProof [EQUIVALENT, 0 ms] (480) QDP (481) DependencyGraphProof [EQUIVALENT, 0 ms] (482) QDP (483) TransformationProof [EQUIVALENT, 0 ms] (484) QDP (485) DependencyGraphProof [EQUIVALENT, 0 ms] (486) QDP (487) TransformationProof [EQUIVALENT, 0 ms] (488) QDP (489) DependencyGraphProof [EQUIVALENT, 0 ms] (490) QDP (491) TransformationProof [EQUIVALENT, 0 ms] (492) QDP (493) DependencyGraphProof [EQUIVALENT, 0 ms] (494) QDP (495) TransformationProof [EQUIVALENT, 0 ms] (496) QDP (497) DependencyGraphProof [EQUIVALENT, 0 ms] (498) QDP (499) TransformationProof [EQUIVALENT, 0 ms] (500) QDP (501) DependencyGraphProof [EQUIVALENT, 0 ms] (502) QDP (503) TransformationProof [EQUIVALENT, 0 ms] (504) QDP (505) DependencyGraphProof [EQUIVALENT, 0 ms] (506) QDP (507) TransformationProof [EQUIVALENT, 0 ms] (508) QDP (509) TransformationProof [EQUIVALENT, 0 ms] (510) QDP (511) DependencyGraphProof [EQUIVALENT, 0 ms] (512) QDP (513) TransformationProof [EQUIVALENT, 0 ms] (514) QDP (515) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] (516) QDP (517) MNOCProof [EQUIVALENT, 0 ms] (518) QDP (519) DependencyGraphProof [EQUIVALENT, 0 ms] (520) TRUE ---------------------------------------- (0) Obligation: Conditional term rewrite system: The TRS R consists of the following rules: a -> d b -> d a -> e b -> e A -> h(f(a), f(b)) h(x, x) -> g(x, x) g(d, e) -> A The conditional TRS C consists of the following conditional rules: f(x) -> x <= x -> d ---------------------------------------- (1) CTRSToQTRSProof (SOUND) The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC]. ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f(x) -> U1(x, x) U1(d, x) -> x a -> d b -> d a -> e b -> e A -> h(f(a), f(b)) h(x, x) -> g(x, x) g(d, e) -> A Q is empty. ---------------------------------------- (3) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (4) Obligation: Q DP problem: The TRS P consists of the following rules: F(x) -> U1^1(x, x) A^1 -> H(f(a), f(b)) A^1 -> F(a) A^1 -> A^2 A^1 -> F(b) A^1 -> B H(x, x) -> G(x, x) G(d, e) -> A^1 The TRS R consists of the following rules: f(x) -> U1(x, x) U1(d, x) -> x a -> d b -> d a -> e b -> e A -> h(f(a), f(b)) h(x, x) -> g(x, x) g(d, e) -> A Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes. ---------------------------------------- (6) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(a), f(b)) H(x, x) -> G(x, x) G(d, e) -> A^1 The TRS R consists of the following rules: f(x) -> U1(x, x) U1(d, x) -> x a -> d b -> d a -> e b -> e A -> h(f(a), f(b)) h(x, x) -> g(x, x) g(d, e) -> A Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (7) UsableRulesReductionPairsProof (EQUIVALENT) By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well. No dependency pairs are removed. The following rules are removed from R: A -> h(f(a), f(b)) h(x, x) -> g(x, x) g(d, e) -> A Used ordering: POLO with Polynomial interpretation [POLO]: POL(A^1) = 0 POL(G(x_1, x_2)) = x_1 + x_2 POL(H(x_1, x_2)) = 2*x_1 + 2*x_2 POL(U1(x_1, x_2)) = x_1 + x_2 POL(a) = 0 POL(b) = 0 POL(d) = 0 POL(e) = 0 POL(f(x_1)) = 2*x_1 ---------------------------------------- (8) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(a), f(b)) H(x, x) -> G(x, x) G(d, e) -> A^1 The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (9) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(a), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(a, a), f(b)),A^1 -> H(U1(a, a), f(b))) (A^1 -> H(f(d), f(b)),A^1 -> H(f(d), f(b))) (A^1 -> H(f(e), f(b)),A^1 -> H(f(e), f(b))) (A^1 -> H(f(a), U1(b, b)),A^1 -> H(f(a), U1(b, b))) (A^1 -> H(f(a), f(d)),A^1 -> H(f(a), f(d))) (A^1 -> H(f(a), f(e)),A^1 -> H(f(a), f(e))) ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(U1(a, a), f(b)) A^1 -> H(f(d), f(b)) A^1 -> H(f(e), f(b)) A^1 -> H(f(a), U1(b, b)) A^1 -> H(f(a), f(d)) A^1 -> H(f(a), f(e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (11) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, a), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, a), f(b)),A^1 -> H(U1(d, a), f(b))) (A^1 -> H(U1(e, a), f(b)),A^1 -> H(U1(e, a), f(b))) (A^1 -> H(U1(a, d), f(b)),A^1 -> H(U1(a, d), f(b))) (A^1 -> H(U1(a, e), f(b)),A^1 -> H(U1(a, e), f(b))) (A^1 -> H(U1(a, a), U1(b, b)),A^1 -> H(U1(a, a), U1(b, b))) (A^1 -> H(U1(a, a), f(d)),A^1 -> H(U1(a, a), f(d))) (A^1 -> H(U1(a, a), f(e)),A^1 -> H(U1(a, a), f(e))) ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(b)) A^1 -> H(f(e), f(b)) A^1 -> H(f(a), U1(b, b)) A^1 -> H(f(a), f(d)) A^1 -> H(f(a), f(e)) A^1 -> H(U1(d, a), f(b)) A^1 -> H(U1(e, a), f(b)) A^1 -> H(U1(a, d), f(b)) A^1 -> H(U1(a, e), f(b)) A^1 -> H(U1(a, a), U1(b, b)) A^1 -> H(U1(a, a), f(d)) A^1 -> H(U1(a, a), f(e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (13) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(d), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, d), f(b)),A^1 -> H(U1(d, d), f(b))) (A^1 -> H(f(d), U1(b, b)),A^1 -> H(f(d), U1(b, b))) (A^1 -> H(f(d), f(d)),A^1 -> H(f(d), f(d))) (A^1 -> H(f(d), f(e)),A^1 -> H(f(d), f(e))) ---------------------------------------- (14) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(b)) A^1 -> H(f(a), U1(b, b)) A^1 -> H(f(a), f(d)) A^1 -> H(f(a), f(e)) A^1 -> H(U1(d, a), f(b)) A^1 -> H(U1(e, a), f(b)) A^1 -> H(U1(a, d), f(b)) A^1 -> H(U1(a, e), f(b)) A^1 -> H(U1(a, a), U1(b, b)) A^1 -> H(U1(a, a), f(d)) A^1 -> H(U1(a, a), f(e)) A^1 -> H(U1(d, d), f(b)) A^1 -> H(f(d), U1(b, b)) A^1 -> H(f(d), f(d)) A^1 -> H(f(d), f(e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (15) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(e), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, e), f(b)),A^1 -> H(U1(e, e), f(b))) (A^1 -> H(f(e), U1(b, b)),A^1 -> H(f(e), U1(b, b))) (A^1 -> H(f(e), f(d)),A^1 -> H(f(e), f(d))) (A^1 -> H(f(e), f(e)),A^1 -> H(f(e), f(e))) ---------------------------------------- (16) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(a), U1(b, b)) A^1 -> H(f(a), f(d)) A^1 -> H(f(a), f(e)) A^1 -> H(U1(d, a), f(b)) A^1 -> H(U1(e, a), f(b)) A^1 -> H(U1(a, d), f(b)) A^1 -> H(U1(a, e), f(b)) A^1 -> H(U1(a, a), U1(b, b)) A^1 -> H(U1(a, a), f(d)) A^1 -> H(U1(a, a), f(e)) A^1 -> H(U1(d, d), f(b)) A^1 -> H(f(d), U1(b, b)) A^1 -> H(f(d), f(d)) A^1 -> H(f(d), f(e)) A^1 -> H(U1(e, e), f(b)) A^1 -> H(f(e), U1(b, b)) A^1 -> H(f(e), f(d)) A^1 -> H(f(e), f(e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (17) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(a), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(a, a), U1(b, b)),A^1 -> H(U1(a, a), U1(b, b))) (A^1 -> H(f(d), U1(b, b)),A^1 -> H(f(d), U1(b, b))) (A^1 -> H(f(e), U1(b, b)),A^1 -> H(f(e), U1(b, b))) (A^1 -> H(f(a), U1(d, b)),A^1 -> H(f(a), U1(d, b))) (A^1 -> H(f(a), U1(e, b)),A^1 -> H(f(a), U1(e, b))) (A^1 -> H(f(a), U1(b, d)),A^1 -> H(f(a), U1(b, d))) (A^1 -> H(f(a), U1(b, e)),A^1 -> H(f(a), U1(b, e))) ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(a), f(d)) A^1 -> H(f(a), f(e)) A^1 -> H(U1(d, a), f(b)) A^1 -> H(U1(e, a), f(b)) A^1 -> H(U1(a, d), f(b)) A^1 -> H(U1(a, e), f(b)) A^1 -> H(U1(a, a), U1(b, b)) A^1 -> H(U1(a, a), f(d)) A^1 -> H(U1(a, a), f(e)) A^1 -> H(U1(d, d), f(b)) A^1 -> H(f(d), U1(b, b)) A^1 -> H(f(d), f(d)) A^1 -> H(f(d), f(e)) A^1 -> H(U1(e, e), f(b)) A^1 -> H(f(e), U1(b, b)) A^1 -> H(f(e), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(f(a), U1(d, b)) A^1 -> H(f(a), U1(e, b)) A^1 -> H(f(a), U1(b, d)) A^1 -> H(f(a), U1(b, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (19) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(a), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(a, a), f(d)),A^1 -> H(U1(a, a), f(d))) (A^1 -> H(f(d), f(d)),A^1 -> H(f(d), f(d))) (A^1 -> H(f(e), f(d)),A^1 -> H(f(e), f(d))) (A^1 -> H(f(a), U1(d, d)),A^1 -> H(f(a), U1(d, d))) ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(a), f(e)) A^1 -> H(U1(d, a), f(b)) A^1 -> H(U1(e, a), f(b)) A^1 -> H(U1(a, d), f(b)) A^1 -> H(U1(a, e), f(b)) A^1 -> H(U1(a, a), U1(b, b)) A^1 -> H(U1(a, a), f(d)) A^1 -> H(U1(a, a), f(e)) A^1 -> H(U1(d, d), f(b)) A^1 -> H(f(d), U1(b, b)) A^1 -> H(f(d), f(d)) A^1 -> H(f(d), f(e)) A^1 -> H(U1(e, e), f(b)) A^1 -> H(f(e), U1(b, b)) A^1 -> H(f(e), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(f(a), U1(d, b)) A^1 -> H(f(a), U1(e, b)) A^1 -> H(f(a), U1(b, d)) A^1 -> H(f(a), U1(b, e)) A^1 -> H(f(a), U1(d, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (21) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(a), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(a, a), f(e)),A^1 -> H(U1(a, a), f(e))) (A^1 -> H(f(d), f(e)),A^1 -> H(f(d), f(e))) (A^1 -> H(f(e), f(e)),A^1 -> H(f(e), f(e))) (A^1 -> H(f(a), U1(e, e)),A^1 -> H(f(a), U1(e, e))) ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(U1(d, a), f(b)) A^1 -> H(U1(e, a), f(b)) A^1 -> H(U1(a, d), f(b)) A^1 -> H(U1(a, e), f(b)) A^1 -> H(U1(a, a), U1(b, b)) A^1 -> H(U1(a, a), f(d)) A^1 -> H(U1(a, a), f(e)) A^1 -> H(U1(d, d), f(b)) A^1 -> H(f(d), U1(b, b)) A^1 -> H(f(d), f(d)) A^1 -> H(f(d), f(e)) A^1 -> H(U1(e, e), f(b)) A^1 -> H(f(e), U1(b, b)) A^1 -> H(f(e), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(f(a), U1(d, b)) A^1 -> H(f(a), U1(e, b)) A^1 -> H(f(a), U1(b, d)) A^1 -> H(f(a), U1(b, e)) A^1 -> H(f(a), U1(d, d)) A^1 -> H(f(a), U1(e, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (23) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, a), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(a, f(b)),A^1 -> H(a, f(b))) (A^1 -> H(U1(d, d), f(b)),A^1 -> H(U1(d, d), f(b))) (A^1 -> H(U1(d, e), f(b)),A^1 -> H(U1(d, e), f(b))) (A^1 -> H(U1(d, a), U1(b, b)),A^1 -> H(U1(d, a), U1(b, b))) (A^1 -> H(U1(d, a), f(d)),A^1 -> H(U1(d, a), f(d))) (A^1 -> H(U1(d, a), f(e)),A^1 -> H(U1(d, a), f(e))) ---------------------------------------- (24) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(U1(e, a), f(b)) A^1 -> H(U1(a, d), f(b)) A^1 -> H(U1(a, e), f(b)) A^1 -> H(U1(a, a), U1(b, b)) A^1 -> H(U1(a, a), f(d)) A^1 -> H(U1(a, a), f(e)) A^1 -> H(U1(d, d), f(b)) A^1 -> H(f(d), U1(b, b)) A^1 -> H(f(d), f(d)) A^1 -> H(f(d), f(e)) A^1 -> H(U1(e, e), f(b)) A^1 -> H(f(e), U1(b, b)) A^1 -> H(f(e), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(f(a), U1(d, b)) A^1 -> H(f(a), U1(e, b)) A^1 -> H(f(a), U1(b, d)) A^1 -> H(f(a), U1(b, e)) A^1 -> H(f(a), U1(d, d)) A^1 -> H(f(a), U1(e, e)) A^1 -> H(a, f(b)) A^1 -> H(U1(d, e), f(b)) A^1 -> H(U1(d, a), U1(b, b)) A^1 -> H(U1(d, a), f(d)) A^1 -> H(U1(d, a), f(e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (25) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(e, a), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, d), f(b)),A^1 -> H(U1(e, d), f(b))) (A^1 -> H(U1(e, e), f(b)),A^1 -> H(U1(e, e), f(b))) (A^1 -> H(U1(e, a), U1(b, b)),A^1 -> H(U1(e, a), U1(b, b))) (A^1 -> H(U1(e, a), f(d)),A^1 -> H(U1(e, a), f(d))) (A^1 -> H(U1(e, a), f(e)),A^1 -> H(U1(e, a), f(e))) ---------------------------------------- (26) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(U1(a, d), f(b)) A^1 -> H(U1(a, e), f(b)) A^1 -> H(U1(a, a), U1(b, b)) A^1 -> H(U1(a, a), f(d)) A^1 -> H(U1(a, a), f(e)) A^1 -> H(U1(d, d), f(b)) A^1 -> H(f(d), U1(b, b)) A^1 -> H(f(d), f(d)) A^1 -> H(f(d), f(e)) A^1 -> H(U1(e, e), f(b)) A^1 -> H(f(e), U1(b, b)) A^1 -> H(f(e), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(f(a), U1(d, b)) A^1 -> H(f(a), U1(e, b)) A^1 -> H(f(a), U1(b, d)) A^1 -> H(f(a), U1(b, e)) A^1 -> H(f(a), U1(d, d)) A^1 -> H(f(a), U1(e, e)) A^1 -> H(a, f(b)) A^1 -> H(U1(d, e), f(b)) A^1 -> H(U1(d, a), U1(b, b)) A^1 -> H(U1(d, a), f(d)) A^1 -> H(U1(d, a), f(e)) A^1 -> H(U1(e, d), f(b)) A^1 -> H(U1(e, a), U1(b, b)) A^1 -> H(U1(e, a), f(d)) A^1 -> H(U1(e, a), f(e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (27) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, d), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, d), f(b)),A^1 -> H(U1(d, d), f(b))) (A^1 -> H(U1(e, d), f(b)),A^1 -> H(U1(e, d), f(b))) (A^1 -> H(U1(a, d), U1(b, b)),A^1 -> H(U1(a, d), U1(b, b))) (A^1 -> H(U1(a, d), f(d)),A^1 -> H(U1(a, d), f(d))) (A^1 -> H(U1(a, d), f(e)),A^1 -> H(U1(a, d), f(e))) ---------------------------------------- (28) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(U1(a, e), f(b)) A^1 -> H(U1(a, a), U1(b, b)) A^1 -> H(U1(a, a), f(d)) A^1 -> H(U1(a, a), f(e)) A^1 -> H(U1(d, d), f(b)) A^1 -> H(f(d), U1(b, b)) A^1 -> H(f(d), f(d)) A^1 -> H(f(d), f(e)) A^1 -> H(U1(e, e), f(b)) A^1 -> H(f(e), U1(b, b)) A^1 -> H(f(e), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(f(a), U1(d, b)) A^1 -> H(f(a), U1(e, b)) A^1 -> H(f(a), U1(b, d)) A^1 -> H(f(a), U1(b, e)) A^1 -> H(f(a), U1(d, d)) A^1 -> H(f(a), U1(e, e)) A^1 -> H(a, f(b)) A^1 -> H(U1(d, e), f(b)) A^1 -> H(U1(d, a), U1(b, b)) A^1 -> H(U1(d, a), f(d)) A^1 -> H(U1(d, a), f(e)) A^1 -> H(U1(e, d), f(b)) A^1 -> H(U1(e, a), U1(b, b)) A^1 -> H(U1(e, a), f(d)) A^1 -> H(U1(e, a), f(e)) A^1 -> H(U1(a, d), U1(b, b)) A^1 -> H(U1(a, d), f(d)) A^1 -> H(U1(a, d), f(e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (29) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, e), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, e), f(b)),A^1 -> H(U1(d, e), f(b))) (A^1 -> H(U1(e, e), f(b)),A^1 -> H(U1(e, e), f(b))) (A^1 -> H(U1(a, e), U1(b, b)),A^1 -> H(U1(a, e), U1(b, b))) (A^1 -> H(U1(a, e), f(d)),A^1 -> H(U1(a, e), f(d))) (A^1 -> H(U1(a, e), f(e)),A^1 -> H(U1(a, e), f(e))) ---------------------------------------- (30) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(U1(a, a), U1(b, b)) A^1 -> H(U1(a, a), f(d)) A^1 -> H(U1(a, a), f(e)) A^1 -> H(U1(d, d), f(b)) A^1 -> H(f(d), U1(b, b)) A^1 -> H(f(d), f(d)) A^1 -> H(f(d), f(e)) A^1 -> H(U1(e, e), f(b)) A^1 -> H(f(e), U1(b, b)) A^1 -> H(f(e), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(f(a), U1(d, b)) A^1 -> H(f(a), U1(e, b)) A^1 -> H(f(a), U1(b, d)) A^1 -> H(f(a), U1(b, e)) A^1 -> H(f(a), U1(d, d)) A^1 -> H(f(a), U1(e, e)) A^1 -> H(a, f(b)) A^1 -> H(U1(d, e), f(b)) A^1 -> H(U1(d, a), U1(b, b)) A^1 -> H(U1(d, a), f(d)) A^1 -> H(U1(d, a), f(e)) A^1 -> H(U1(e, d), f(b)) A^1 -> H(U1(e, a), U1(b, b)) A^1 -> H(U1(e, a), f(d)) A^1 -> H(U1(e, a), f(e)) A^1 -> H(U1(a, d), U1(b, b)) A^1 -> H(U1(a, d), f(d)) A^1 -> H(U1(a, d), f(e)) A^1 -> H(U1(a, e), U1(b, b)) A^1 -> H(U1(a, e), f(d)) A^1 -> H(U1(a, e), f(e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (31) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, a), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, a), U1(b, b)),A^1 -> H(U1(d, a), U1(b, b))) (A^1 -> H(U1(e, a), U1(b, b)),A^1 -> H(U1(e, a), U1(b, b))) (A^1 -> H(U1(a, d), U1(b, b)),A^1 -> H(U1(a, d), U1(b, b))) (A^1 -> H(U1(a, e), U1(b, b)),A^1 -> H(U1(a, e), U1(b, b))) (A^1 -> H(U1(a, a), U1(d, b)),A^1 -> H(U1(a, a), U1(d, b))) (A^1 -> H(U1(a, a), U1(e, b)),A^1 -> H(U1(a, a), U1(e, b))) (A^1 -> H(U1(a, a), U1(b, d)),A^1 -> H(U1(a, a), U1(b, d))) (A^1 -> H(U1(a, a), U1(b, e)),A^1 -> H(U1(a, a), U1(b, e))) ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(U1(a, a), f(d)) A^1 -> H(U1(a, a), f(e)) A^1 -> H(U1(d, d), f(b)) A^1 -> H(f(d), U1(b, b)) A^1 -> H(f(d), f(d)) A^1 -> H(f(d), f(e)) A^1 -> H(U1(e, e), f(b)) A^1 -> H(f(e), U1(b, b)) A^1 -> H(f(e), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(f(a), U1(d, b)) A^1 -> H(f(a), U1(e, b)) A^1 -> H(f(a), U1(b, d)) A^1 -> H(f(a), U1(b, e)) A^1 -> H(f(a), U1(d, d)) A^1 -> H(f(a), U1(e, e)) A^1 -> H(a, f(b)) A^1 -> H(U1(d, e), f(b)) A^1 -> H(U1(d, a), U1(b, b)) A^1 -> H(U1(d, a), f(d)) A^1 -> H(U1(d, a), f(e)) A^1 -> H(U1(e, d), f(b)) A^1 -> H(U1(e, a), U1(b, b)) A^1 -> H(U1(e, a), f(d)) A^1 -> H(U1(e, a), f(e)) A^1 -> H(U1(a, d), U1(b, b)) A^1 -> H(U1(a, d), f(d)) A^1 -> H(U1(a, d), f(e)) A^1 -> H(U1(a, e), U1(b, b)) A^1 -> H(U1(a, e), f(d)) A^1 -> H(U1(a, e), f(e)) A^1 -> H(U1(a, a), U1(d, b)) A^1 -> H(U1(a, a), U1(e, b)) A^1 -> H(U1(a, a), U1(b, d)) A^1 -> H(U1(a, a), U1(b, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (33) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, a), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, a), f(d)),A^1 -> H(U1(d, a), f(d))) (A^1 -> H(U1(e, a), f(d)),A^1 -> H(U1(e, a), f(d))) (A^1 -> H(U1(a, d), f(d)),A^1 -> H(U1(a, d), f(d))) (A^1 -> H(U1(a, e), f(d)),A^1 -> H(U1(a, e), f(d))) (A^1 -> H(U1(a, a), U1(d, d)),A^1 -> H(U1(a, a), U1(d, d))) ---------------------------------------- (34) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(U1(a, a), f(e)) A^1 -> H(U1(d, d), f(b)) A^1 -> H(f(d), U1(b, b)) A^1 -> H(f(d), f(d)) A^1 -> H(f(d), f(e)) A^1 -> H(U1(e, e), f(b)) A^1 -> H(f(e), U1(b, b)) A^1 -> H(f(e), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(f(a), U1(d, b)) A^1 -> H(f(a), U1(e, b)) A^1 -> H(f(a), U1(b, d)) A^1 -> H(f(a), U1(b, e)) A^1 -> H(f(a), U1(d, d)) A^1 -> H(f(a), U1(e, e)) A^1 -> H(a, f(b)) A^1 -> H(U1(d, e), f(b)) A^1 -> H(U1(d, a), U1(b, b)) A^1 -> H(U1(d, a), f(d)) A^1 -> H(U1(d, a), f(e)) A^1 -> H(U1(e, d), f(b)) A^1 -> H(U1(e, a), U1(b, b)) A^1 -> H(U1(e, a), f(d)) A^1 -> H(U1(e, a), f(e)) A^1 -> H(U1(a, d), U1(b, b)) A^1 -> H(U1(a, d), f(d)) A^1 -> H(U1(a, d), f(e)) A^1 -> H(U1(a, e), U1(b, b)) A^1 -> H(U1(a, e), f(d)) A^1 -> H(U1(a, e), f(e)) A^1 -> H(U1(a, a), U1(d, b)) A^1 -> H(U1(a, a), U1(e, b)) A^1 -> H(U1(a, a), U1(b, d)) A^1 -> H(U1(a, a), U1(b, e)) A^1 -> H(U1(a, a), U1(d, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (35) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, a), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, a), f(e)),A^1 -> H(U1(d, a), f(e))) (A^1 -> H(U1(e, a), f(e)),A^1 -> H(U1(e, a), f(e))) (A^1 -> H(U1(a, d), f(e)),A^1 -> H(U1(a, d), f(e))) (A^1 -> H(U1(a, e), f(e)),A^1 -> H(U1(a, e), f(e))) (A^1 -> H(U1(a, a), U1(e, e)),A^1 -> H(U1(a, a), U1(e, e))) ---------------------------------------- (36) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(U1(d, d), f(b)) A^1 -> H(f(d), U1(b, b)) A^1 -> H(f(d), f(d)) A^1 -> H(f(d), f(e)) A^1 -> H(U1(e, e), f(b)) A^1 -> H(f(e), U1(b, b)) A^1 -> H(f(e), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(f(a), U1(d, b)) A^1 -> H(f(a), U1(e, b)) A^1 -> H(f(a), U1(b, d)) A^1 -> H(f(a), U1(b, e)) A^1 -> H(f(a), U1(d, d)) A^1 -> H(f(a), U1(e, e)) A^1 -> H(a, f(b)) A^1 -> H(U1(d, e), f(b)) A^1 -> H(U1(d, a), U1(b, b)) A^1 -> H(U1(d, a), f(d)) A^1 -> H(U1(d, a), f(e)) A^1 -> H(U1(e, d), f(b)) A^1 -> H(U1(e, a), U1(b, b)) A^1 -> H(U1(e, a), f(d)) A^1 -> H(U1(e, a), f(e)) A^1 -> H(U1(a, d), U1(b, b)) A^1 -> H(U1(a, d), f(d)) A^1 -> H(U1(a, d), f(e)) A^1 -> H(U1(a, e), U1(b, b)) A^1 -> H(U1(a, e), f(d)) A^1 -> H(U1(a, e), f(e)) A^1 -> H(U1(a, a), U1(d, b)) A^1 -> H(U1(a, a), U1(e, b)) A^1 -> H(U1(a, a), U1(b, d)) A^1 -> H(U1(a, a), U1(b, e)) A^1 -> H(U1(a, a), U1(d, d)) A^1 -> H(U1(a, a), U1(e, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (37) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, d), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(d, f(b)),A^1 -> H(d, f(b))) (A^1 -> H(U1(d, d), U1(b, b)),A^1 -> H(U1(d, d), U1(b, b))) (A^1 -> H(U1(d, d), f(d)),A^1 -> H(U1(d, d), f(d))) (A^1 -> H(U1(d, d), f(e)),A^1 -> H(U1(d, d), f(e))) ---------------------------------------- (38) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), U1(b, b)) A^1 -> H(f(d), f(d)) A^1 -> H(f(d), f(e)) A^1 -> H(U1(e, e), f(b)) A^1 -> H(f(e), U1(b, b)) A^1 -> H(f(e), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(f(a), U1(d, b)) A^1 -> H(f(a), U1(e, b)) A^1 -> H(f(a), U1(b, d)) A^1 -> H(f(a), U1(b, e)) A^1 -> H(f(a), U1(d, d)) A^1 -> H(f(a), U1(e, e)) A^1 -> H(a, f(b)) A^1 -> H(U1(d, e), f(b)) A^1 -> H(U1(d, a), U1(b, b)) A^1 -> H(U1(d, a), f(d)) A^1 -> H(U1(d, a), f(e)) A^1 -> H(U1(e, d), f(b)) A^1 -> H(U1(e, a), U1(b, b)) A^1 -> H(U1(e, a), f(d)) A^1 -> H(U1(e, a), f(e)) A^1 -> H(U1(a, d), U1(b, b)) A^1 -> H(U1(a, d), f(d)) A^1 -> H(U1(a, d), f(e)) A^1 -> H(U1(a, e), U1(b, b)) A^1 -> H(U1(a, e), f(d)) A^1 -> H(U1(a, e), f(e)) A^1 -> H(U1(a, a), U1(d, b)) A^1 -> H(U1(a, a), U1(e, b)) A^1 -> H(U1(a, a), U1(b, d)) A^1 -> H(U1(a, a), U1(b, e)) A^1 -> H(U1(a, a), U1(d, d)) A^1 -> H(U1(a, a), U1(e, e)) A^1 -> H(d, f(b)) A^1 -> H(U1(d, d), U1(b, b)) A^1 -> H(U1(d, d), f(d)) A^1 -> H(U1(d, d), f(e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (39) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(d), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, d), U1(b, b)),A^1 -> H(U1(d, d), U1(b, b))) (A^1 -> H(f(d), U1(d, b)),A^1 -> H(f(d), U1(d, b))) (A^1 -> H(f(d), U1(e, b)),A^1 -> H(f(d), U1(e, b))) (A^1 -> H(f(d), U1(b, d)),A^1 -> H(f(d), U1(b, d))) (A^1 -> H(f(d), U1(b, e)),A^1 -> H(f(d), U1(b, e))) ---------------------------------------- (40) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(d), f(e)) A^1 -> H(U1(e, e), f(b)) A^1 -> H(f(e), U1(b, b)) A^1 -> H(f(e), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(f(a), U1(d, b)) A^1 -> H(f(a), U1(e, b)) A^1 -> H(f(a), U1(b, d)) A^1 -> H(f(a), U1(b, e)) A^1 -> H(f(a), U1(d, d)) A^1 -> H(f(a), U1(e, e)) A^1 -> H(a, f(b)) A^1 -> H(U1(d, e), f(b)) A^1 -> H(U1(d, a), U1(b, b)) A^1 -> H(U1(d, a), f(d)) A^1 -> H(U1(d, a), f(e)) A^1 -> H(U1(e, d), f(b)) A^1 -> H(U1(e, a), U1(b, b)) A^1 -> H(U1(e, a), f(d)) A^1 -> H(U1(e, a), f(e)) A^1 -> H(U1(a, d), U1(b, b)) A^1 -> H(U1(a, d), f(d)) A^1 -> H(U1(a, d), f(e)) A^1 -> H(U1(a, e), U1(b, b)) A^1 -> H(U1(a, e), f(d)) A^1 -> H(U1(a, e), f(e)) A^1 -> H(U1(a, a), U1(d, b)) A^1 -> H(U1(a, a), U1(e, b)) A^1 -> H(U1(a, a), U1(b, d)) A^1 -> H(U1(a, a), U1(b, e)) A^1 -> H(U1(a, a), U1(d, d)) A^1 -> H(U1(a, a), U1(e, e)) A^1 -> H(d, f(b)) A^1 -> H(U1(d, d), U1(b, b)) A^1 -> H(U1(d, d), f(d)) A^1 -> H(U1(d, d), f(e)) A^1 -> H(f(d), U1(d, b)) A^1 -> H(f(d), U1(e, b)) A^1 -> H(f(d), U1(b, d)) A^1 -> H(f(d), U1(b, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (41) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(d), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, d), f(e)),A^1 -> H(U1(d, d), f(e))) (A^1 -> H(f(d), U1(e, e)),A^1 -> H(f(d), U1(e, e))) ---------------------------------------- (42) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(U1(e, e), f(b)) A^1 -> H(f(e), U1(b, b)) A^1 -> H(f(e), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(f(a), U1(d, b)) A^1 -> H(f(a), U1(e, b)) A^1 -> H(f(a), U1(b, d)) A^1 -> H(f(a), U1(b, e)) A^1 -> H(f(a), U1(d, d)) A^1 -> H(f(a), U1(e, e)) A^1 -> H(a, f(b)) A^1 -> H(U1(d, e), f(b)) A^1 -> H(U1(d, a), U1(b, b)) A^1 -> H(U1(d, a), f(d)) A^1 -> H(U1(d, a), f(e)) A^1 -> H(U1(e, d), f(b)) A^1 -> H(U1(e, a), U1(b, b)) A^1 -> H(U1(e, a), f(d)) A^1 -> H(U1(e, a), f(e)) A^1 -> H(U1(a, d), U1(b, b)) A^1 -> H(U1(a, d), f(d)) A^1 -> H(U1(a, d), f(e)) A^1 -> H(U1(a, e), U1(b, b)) A^1 -> H(U1(a, e), f(d)) A^1 -> H(U1(a, e), f(e)) A^1 -> H(U1(a, a), U1(d, b)) A^1 -> H(U1(a, a), U1(e, b)) A^1 -> H(U1(a, a), U1(b, d)) A^1 -> H(U1(a, a), U1(b, e)) A^1 -> H(U1(a, a), U1(d, d)) A^1 -> H(U1(a, a), U1(e, e)) A^1 -> H(d, f(b)) A^1 -> H(U1(d, d), U1(b, b)) A^1 -> H(U1(d, d), f(d)) A^1 -> H(U1(d, d), f(e)) A^1 -> H(f(d), U1(d, b)) A^1 -> H(f(d), U1(e, b)) A^1 -> H(f(d), U1(b, d)) A^1 -> H(f(d), U1(b, e)) A^1 -> H(f(d), U1(e, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (43) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(e, e), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, e), U1(b, b)),A^1 -> H(U1(e, e), U1(b, b))) (A^1 -> H(U1(e, e), f(d)),A^1 -> H(U1(e, e), f(d))) (A^1 -> H(U1(e, e), f(e)),A^1 -> H(U1(e, e), f(e))) ---------------------------------------- (44) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), U1(b, b)) A^1 -> H(f(e), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(f(a), U1(d, b)) A^1 -> H(f(a), U1(e, b)) A^1 -> H(f(a), U1(b, d)) A^1 -> H(f(a), U1(b, e)) A^1 -> H(f(a), U1(d, d)) A^1 -> H(f(a), U1(e, e)) A^1 -> H(a, f(b)) A^1 -> H(U1(d, e), f(b)) A^1 -> H(U1(d, a), U1(b, b)) A^1 -> H(U1(d, a), f(d)) A^1 -> H(U1(d, a), f(e)) A^1 -> H(U1(e, d), f(b)) A^1 -> H(U1(e, a), U1(b, b)) A^1 -> H(U1(e, a), f(d)) A^1 -> H(U1(e, a), f(e)) A^1 -> H(U1(a, d), U1(b, b)) A^1 -> H(U1(a, d), f(d)) A^1 -> H(U1(a, d), f(e)) A^1 -> H(U1(a, e), U1(b, b)) A^1 -> H(U1(a, e), f(d)) A^1 -> H(U1(a, e), f(e)) A^1 -> H(U1(a, a), U1(d, b)) A^1 -> H(U1(a, a), U1(e, b)) A^1 -> H(U1(a, a), U1(b, d)) A^1 -> H(U1(a, a), U1(b, e)) A^1 -> H(U1(a, a), U1(d, d)) A^1 -> H(U1(a, a), U1(e, e)) A^1 -> H(d, f(b)) A^1 -> H(U1(d, d), U1(b, b)) A^1 -> H(U1(d, d), f(d)) A^1 -> H(U1(d, d), f(e)) A^1 -> H(f(d), U1(d, b)) A^1 -> H(f(d), U1(e, b)) A^1 -> H(f(d), U1(b, d)) A^1 -> H(f(d), U1(b, e)) A^1 -> H(f(d), U1(e, e)) A^1 -> H(U1(e, e), U1(b, b)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (45) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(e), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, e), U1(b, b)),A^1 -> H(U1(e, e), U1(b, b))) (A^1 -> H(f(e), U1(d, b)),A^1 -> H(f(e), U1(d, b))) (A^1 -> H(f(e), U1(e, b)),A^1 -> H(f(e), U1(e, b))) (A^1 -> H(f(e), U1(b, d)),A^1 -> H(f(e), U1(b, d))) (A^1 -> H(f(e), U1(b, e)),A^1 -> H(f(e), U1(b, e))) ---------------------------------------- (46) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(f(a), U1(d, b)) A^1 -> H(f(a), U1(e, b)) A^1 -> H(f(a), U1(b, d)) A^1 -> H(f(a), U1(b, e)) A^1 -> H(f(a), U1(d, d)) A^1 -> H(f(a), U1(e, e)) A^1 -> H(a, f(b)) A^1 -> H(U1(d, e), f(b)) A^1 -> H(U1(d, a), U1(b, b)) A^1 -> H(U1(d, a), f(d)) A^1 -> H(U1(d, a), f(e)) A^1 -> H(U1(e, d), f(b)) A^1 -> H(U1(e, a), U1(b, b)) A^1 -> H(U1(e, a), f(d)) A^1 -> H(U1(e, a), f(e)) A^1 -> H(U1(a, d), U1(b, b)) A^1 -> H(U1(a, d), f(d)) A^1 -> H(U1(a, d), f(e)) A^1 -> H(U1(a, e), U1(b, b)) A^1 -> H(U1(a, e), f(d)) A^1 -> H(U1(a, e), f(e)) A^1 -> H(U1(a, a), U1(d, b)) A^1 -> H(U1(a, a), U1(e, b)) A^1 -> H(U1(a, a), U1(b, d)) A^1 -> H(U1(a, a), U1(b, e)) A^1 -> H(U1(a, a), U1(d, d)) A^1 -> H(U1(a, a), U1(e, e)) A^1 -> H(d, f(b)) A^1 -> H(U1(d, d), U1(b, b)) A^1 -> H(U1(d, d), f(d)) A^1 -> H(U1(d, d), f(e)) A^1 -> H(f(d), U1(d, b)) A^1 -> H(f(d), U1(e, b)) A^1 -> H(f(d), U1(b, d)) A^1 -> H(f(d), U1(b, e)) A^1 -> H(f(d), U1(e, e)) A^1 -> H(U1(e, e), U1(b, b)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (47) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(e), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, e), f(d)),A^1 -> H(U1(e, e), f(d))) (A^1 -> H(f(e), U1(d, d)),A^1 -> H(f(e), U1(d, d))) ---------------------------------------- (48) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(f(a), U1(d, b)) A^1 -> H(f(a), U1(e, b)) A^1 -> H(f(a), U1(b, d)) A^1 -> H(f(a), U1(b, e)) A^1 -> H(f(a), U1(d, d)) A^1 -> H(f(a), U1(e, e)) A^1 -> H(a, f(b)) A^1 -> H(U1(d, e), f(b)) A^1 -> H(U1(d, a), U1(b, b)) A^1 -> H(U1(d, a), f(d)) A^1 -> H(U1(d, a), f(e)) A^1 -> H(U1(e, d), f(b)) A^1 -> H(U1(e, a), U1(b, b)) A^1 -> H(U1(e, a), f(d)) A^1 -> H(U1(e, a), f(e)) A^1 -> H(U1(a, d), U1(b, b)) A^1 -> H(U1(a, d), f(d)) A^1 -> H(U1(a, d), f(e)) A^1 -> H(U1(a, e), U1(b, b)) A^1 -> H(U1(a, e), f(d)) A^1 -> H(U1(a, e), f(e)) A^1 -> H(U1(a, a), U1(d, b)) A^1 -> H(U1(a, a), U1(e, b)) A^1 -> H(U1(a, a), U1(b, d)) A^1 -> H(U1(a, a), U1(b, e)) A^1 -> H(U1(a, a), U1(d, d)) A^1 -> H(U1(a, a), U1(e, e)) A^1 -> H(d, f(b)) A^1 -> H(U1(d, d), U1(b, b)) A^1 -> H(U1(d, d), f(d)) A^1 -> H(U1(d, d), f(e)) A^1 -> H(f(d), U1(d, b)) A^1 -> H(f(d), U1(e, b)) A^1 -> H(f(d), U1(b, d)) A^1 -> H(f(d), U1(b, e)) A^1 -> H(f(d), U1(e, e)) A^1 -> H(U1(e, e), U1(b, b)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (49) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(a), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(a, a), U1(d, b)),A^1 -> H(U1(a, a), U1(d, b))) (A^1 -> H(f(d), U1(d, b)),A^1 -> H(f(d), U1(d, b))) (A^1 -> H(f(e), U1(d, b)),A^1 -> H(f(e), U1(d, b))) (A^1 -> H(f(a), b),A^1 -> H(f(a), b)) (A^1 -> H(f(a), U1(d, d)),A^1 -> H(f(a), U1(d, d))) (A^1 -> H(f(a), U1(d, e)),A^1 -> H(f(a), U1(d, e))) ---------------------------------------- (50) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(f(a), U1(e, b)) A^1 -> H(f(a), U1(b, d)) A^1 -> H(f(a), U1(b, e)) A^1 -> H(f(a), U1(d, d)) A^1 -> H(f(a), U1(e, e)) A^1 -> H(a, f(b)) A^1 -> H(U1(d, e), f(b)) A^1 -> H(U1(d, a), U1(b, b)) A^1 -> H(U1(d, a), f(d)) A^1 -> H(U1(d, a), f(e)) A^1 -> H(U1(e, d), f(b)) A^1 -> H(U1(e, a), U1(b, b)) A^1 -> H(U1(e, a), f(d)) A^1 -> H(U1(e, a), f(e)) A^1 -> H(U1(a, d), U1(b, b)) A^1 -> H(U1(a, d), f(d)) A^1 -> H(U1(a, d), f(e)) A^1 -> H(U1(a, e), U1(b, b)) A^1 -> H(U1(a, e), f(d)) A^1 -> H(U1(a, e), f(e)) A^1 -> H(U1(a, a), U1(d, b)) A^1 -> H(U1(a, a), U1(e, b)) A^1 -> H(U1(a, a), U1(b, d)) A^1 -> H(U1(a, a), U1(b, e)) A^1 -> H(U1(a, a), U1(d, d)) A^1 -> H(U1(a, a), U1(e, e)) A^1 -> H(d, f(b)) A^1 -> H(U1(d, d), U1(b, b)) A^1 -> H(U1(d, d), f(d)) A^1 -> H(U1(d, d), f(e)) A^1 -> H(f(d), U1(d, b)) A^1 -> H(f(d), U1(e, b)) A^1 -> H(f(d), U1(b, d)) A^1 -> H(f(d), U1(b, e)) A^1 -> H(f(d), U1(e, e)) A^1 -> H(U1(e, e), U1(b, b)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (51) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(a), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(a, a), U1(e, b)),A^1 -> H(U1(a, a), U1(e, b))) (A^1 -> H(f(d), U1(e, b)),A^1 -> H(f(d), U1(e, b))) (A^1 -> H(f(e), U1(e, b)),A^1 -> H(f(e), U1(e, b))) (A^1 -> H(f(a), U1(e, d)),A^1 -> H(f(a), U1(e, d))) (A^1 -> H(f(a), U1(e, e)),A^1 -> H(f(a), U1(e, e))) ---------------------------------------- (52) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(f(a), U1(b, d)) A^1 -> H(f(a), U1(b, e)) A^1 -> H(f(a), U1(d, d)) A^1 -> H(f(a), U1(e, e)) A^1 -> H(a, f(b)) A^1 -> H(U1(d, e), f(b)) A^1 -> H(U1(d, a), U1(b, b)) A^1 -> H(U1(d, a), f(d)) A^1 -> H(U1(d, a), f(e)) A^1 -> H(U1(e, d), f(b)) A^1 -> H(U1(e, a), U1(b, b)) A^1 -> H(U1(e, a), f(d)) A^1 -> H(U1(e, a), f(e)) A^1 -> H(U1(a, d), U1(b, b)) A^1 -> H(U1(a, d), f(d)) A^1 -> H(U1(a, d), f(e)) A^1 -> H(U1(a, e), U1(b, b)) A^1 -> H(U1(a, e), f(d)) A^1 -> H(U1(a, e), f(e)) A^1 -> H(U1(a, a), U1(d, b)) A^1 -> H(U1(a, a), U1(e, b)) A^1 -> H(U1(a, a), U1(b, d)) A^1 -> H(U1(a, a), U1(b, e)) A^1 -> H(U1(a, a), U1(d, d)) A^1 -> H(U1(a, a), U1(e, e)) A^1 -> H(d, f(b)) A^1 -> H(U1(d, d), U1(b, b)) A^1 -> H(U1(d, d), f(d)) A^1 -> H(U1(d, d), f(e)) A^1 -> H(f(d), U1(d, b)) A^1 -> H(f(d), U1(e, b)) A^1 -> H(f(d), U1(b, d)) A^1 -> H(f(d), U1(b, e)) A^1 -> H(f(d), U1(e, e)) A^1 -> H(U1(e, e), U1(b, b)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (53) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(a), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(a, a), U1(b, d)),A^1 -> H(U1(a, a), U1(b, d))) (A^1 -> H(f(d), U1(b, d)),A^1 -> H(f(d), U1(b, d))) (A^1 -> H(f(e), U1(b, d)),A^1 -> H(f(e), U1(b, d))) (A^1 -> H(f(a), U1(d, d)),A^1 -> H(f(a), U1(d, d))) (A^1 -> H(f(a), U1(e, d)),A^1 -> H(f(a), U1(e, d))) ---------------------------------------- (54) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(f(a), U1(b, e)) A^1 -> H(f(a), U1(d, d)) A^1 -> H(f(a), U1(e, e)) A^1 -> H(a, f(b)) A^1 -> H(U1(d, e), f(b)) A^1 -> H(U1(d, a), U1(b, b)) A^1 -> H(U1(d, a), f(d)) A^1 -> H(U1(d, a), f(e)) A^1 -> H(U1(e, d), f(b)) A^1 -> H(U1(e, a), U1(b, b)) A^1 -> H(U1(e, a), f(d)) A^1 -> H(U1(e, a), f(e)) A^1 -> H(U1(a, d), U1(b, b)) A^1 -> H(U1(a, d), f(d)) A^1 -> H(U1(a, d), f(e)) A^1 -> H(U1(a, e), U1(b, b)) A^1 -> H(U1(a, e), f(d)) A^1 -> H(U1(a, e), f(e)) A^1 -> H(U1(a, a), U1(d, b)) A^1 -> H(U1(a, a), U1(e, b)) A^1 -> H(U1(a, a), U1(b, d)) A^1 -> H(U1(a, a), U1(b, e)) A^1 -> H(U1(a, a), U1(d, d)) A^1 -> H(U1(a, a), U1(e, e)) A^1 -> H(d, f(b)) A^1 -> H(U1(d, d), U1(b, b)) A^1 -> H(U1(d, d), f(d)) A^1 -> H(U1(d, d), f(e)) A^1 -> H(f(d), U1(d, b)) A^1 -> H(f(d), U1(e, b)) A^1 -> H(f(d), U1(b, d)) A^1 -> H(f(d), U1(b, e)) A^1 -> H(f(d), U1(e, e)) A^1 -> H(U1(e, e), U1(b, b)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (55) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(a), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(a, a), U1(b, e)),A^1 -> H(U1(a, a), U1(b, e))) (A^1 -> H(f(d), U1(b, e)),A^1 -> H(f(d), U1(b, e))) (A^1 -> H(f(e), U1(b, e)),A^1 -> H(f(e), U1(b, e))) (A^1 -> H(f(a), U1(d, e)),A^1 -> H(f(a), U1(d, e))) (A^1 -> H(f(a), U1(e, e)),A^1 -> H(f(a), U1(e, e))) ---------------------------------------- (56) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(f(a), U1(d, d)) A^1 -> H(f(a), U1(e, e)) A^1 -> H(a, f(b)) A^1 -> H(U1(d, e), f(b)) A^1 -> H(U1(d, a), U1(b, b)) A^1 -> H(U1(d, a), f(d)) A^1 -> H(U1(d, a), f(e)) A^1 -> H(U1(e, d), f(b)) A^1 -> H(U1(e, a), U1(b, b)) A^1 -> H(U1(e, a), f(d)) A^1 -> H(U1(e, a), f(e)) A^1 -> H(U1(a, d), U1(b, b)) A^1 -> H(U1(a, d), f(d)) A^1 -> H(U1(a, d), f(e)) A^1 -> H(U1(a, e), U1(b, b)) A^1 -> H(U1(a, e), f(d)) A^1 -> H(U1(a, e), f(e)) A^1 -> H(U1(a, a), U1(d, b)) A^1 -> H(U1(a, a), U1(e, b)) A^1 -> H(U1(a, a), U1(b, d)) A^1 -> H(U1(a, a), U1(b, e)) A^1 -> H(U1(a, a), U1(d, d)) A^1 -> H(U1(a, a), U1(e, e)) A^1 -> H(d, f(b)) A^1 -> H(U1(d, d), U1(b, b)) A^1 -> H(U1(d, d), f(d)) A^1 -> H(U1(d, d), f(e)) A^1 -> H(f(d), U1(d, b)) A^1 -> H(f(d), U1(e, b)) A^1 -> H(f(d), U1(b, d)) A^1 -> H(f(d), U1(b, e)) A^1 -> H(f(d), U1(e, e)) A^1 -> H(U1(e, e), U1(b, b)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (57) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(a), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(a, a), U1(d, d)),A^1 -> H(U1(a, a), U1(d, d))) (A^1 -> H(f(d), U1(d, d)),A^1 -> H(f(d), U1(d, d))) (A^1 -> H(f(e), U1(d, d)),A^1 -> H(f(e), U1(d, d))) (A^1 -> H(f(a), d),A^1 -> H(f(a), d)) ---------------------------------------- (58) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(f(a), U1(e, e)) A^1 -> H(a, f(b)) A^1 -> H(U1(d, e), f(b)) A^1 -> H(U1(d, a), U1(b, b)) A^1 -> H(U1(d, a), f(d)) A^1 -> H(U1(d, a), f(e)) A^1 -> H(U1(e, d), f(b)) A^1 -> H(U1(e, a), U1(b, b)) A^1 -> H(U1(e, a), f(d)) A^1 -> H(U1(e, a), f(e)) A^1 -> H(U1(a, d), U1(b, b)) A^1 -> H(U1(a, d), f(d)) A^1 -> H(U1(a, d), f(e)) A^1 -> H(U1(a, e), U1(b, b)) A^1 -> H(U1(a, e), f(d)) A^1 -> H(U1(a, e), f(e)) A^1 -> H(U1(a, a), U1(d, b)) A^1 -> H(U1(a, a), U1(e, b)) A^1 -> H(U1(a, a), U1(b, d)) A^1 -> H(U1(a, a), U1(b, e)) A^1 -> H(U1(a, a), U1(d, d)) A^1 -> H(U1(a, a), U1(e, e)) A^1 -> H(d, f(b)) A^1 -> H(U1(d, d), U1(b, b)) A^1 -> H(U1(d, d), f(d)) A^1 -> H(U1(d, d), f(e)) A^1 -> H(f(d), U1(d, b)) A^1 -> H(f(d), U1(e, b)) A^1 -> H(f(d), U1(b, d)) A^1 -> H(f(d), U1(b, e)) A^1 -> H(f(d), U1(e, e)) A^1 -> H(U1(e, e), U1(b, b)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (59) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(a), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(a, a), U1(e, e)),A^1 -> H(U1(a, a), U1(e, e))) (A^1 -> H(f(d), U1(e, e)),A^1 -> H(f(d), U1(e, e))) (A^1 -> H(f(e), U1(e, e)),A^1 -> H(f(e), U1(e, e))) ---------------------------------------- (60) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(a, f(b)) A^1 -> H(U1(d, e), f(b)) A^1 -> H(U1(d, a), U1(b, b)) A^1 -> H(U1(d, a), f(d)) A^1 -> H(U1(d, a), f(e)) A^1 -> H(U1(e, d), f(b)) A^1 -> H(U1(e, a), U1(b, b)) A^1 -> H(U1(e, a), f(d)) A^1 -> H(U1(e, a), f(e)) A^1 -> H(U1(a, d), U1(b, b)) A^1 -> H(U1(a, d), f(d)) A^1 -> H(U1(a, d), f(e)) A^1 -> H(U1(a, e), U1(b, b)) A^1 -> H(U1(a, e), f(d)) A^1 -> H(U1(a, e), f(e)) A^1 -> H(U1(a, a), U1(d, b)) A^1 -> H(U1(a, a), U1(e, b)) A^1 -> H(U1(a, a), U1(b, d)) A^1 -> H(U1(a, a), U1(b, e)) A^1 -> H(U1(a, a), U1(d, d)) A^1 -> H(U1(a, a), U1(e, e)) A^1 -> H(d, f(b)) A^1 -> H(U1(d, d), U1(b, b)) A^1 -> H(U1(d, d), f(d)) A^1 -> H(U1(d, d), f(e)) A^1 -> H(f(d), U1(d, b)) A^1 -> H(f(d), U1(e, b)) A^1 -> H(f(d), U1(b, d)) A^1 -> H(f(d), U1(b, e)) A^1 -> H(f(d), U1(e, e)) A^1 -> H(U1(e, e), U1(b, b)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (61) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(a, f(b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(d, f(b)),A^1 -> H(d, f(b))) (A^1 -> H(e, f(b)),A^1 -> H(e, f(b))) (A^1 -> H(a, U1(b, b)),A^1 -> H(a, U1(b, b))) (A^1 -> H(a, f(d)),A^1 -> H(a, f(d))) (A^1 -> H(a, f(e)),A^1 -> H(a, f(e))) ---------------------------------------- (62) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, e), f(b)) A^1 -> H(U1(d, a), U1(b, b)) A^1 -> H(U1(d, a), f(d)) A^1 -> H(U1(d, a), f(e)) A^1 -> H(U1(e, d), f(b)) A^1 -> H(U1(e, a), U1(b, b)) A^1 -> H(U1(e, a), f(d)) A^1 -> H(U1(e, a), f(e)) A^1 -> H(U1(a, d), U1(b, b)) A^1 -> H(U1(a, d), f(d)) A^1 -> H(U1(a, d), f(e)) A^1 -> H(U1(a, e), U1(b, b)) A^1 -> H(U1(a, e), f(d)) A^1 -> H(U1(a, e), f(e)) A^1 -> H(U1(a, a), U1(d, b)) A^1 -> H(U1(a, a), U1(e, b)) A^1 -> H(U1(a, a), U1(b, d)) A^1 -> H(U1(a, a), U1(b, e)) A^1 -> H(U1(a, a), U1(d, d)) A^1 -> H(U1(a, a), U1(e, e)) A^1 -> H(d, f(b)) A^1 -> H(U1(d, d), U1(b, b)) A^1 -> H(U1(d, d), f(d)) A^1 -> H(U1(d, d), f(e)) A^1 -> H(f(d), U1(d, b)) A^1 -> H(f(d), U1(e, b)) A^1 -> H(f(d), U1(b, d)) A^1 -> H(f(d), U1(b, e)) A^1 -> H(f(d), U1(e, e)) A^1 -> H(U1(e, e), U1(b, b)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (63) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, e), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(e, f(b)),A^1 -> H(e, f(b))) (A^1 -> H(U1(d, e), U1(b, b)),A^1 -> H(U1(d, e), U1(b, b))) (A^1 -> H(U1(d, e), f(d)),A^1 -> H(U1(d, e), f(d))) (A^1 -> H(U1(d, e), f(e)),A^1 -> H(U1(d, e), f(e))) ---------------------------------------- (64) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, a), U1(b, b)) A^1 -> H(U1(d, a), f(d)) A^1 -> H(U1(d, a), f(e)) A^1 -> H(U1(e, d), f(b)) A^1 -> H(U1(e, a), U1(b, b)) A^1 -> H(U1(e, a), f(d)) A^1 -> H(U1(e, a), f(e)) A^1 -> H(U1(a, d), U1(b, b)) A^1 -> H(U1(a, d), f(d)) A^1 -> H(U1(a, d), f(e)) A^1 -> H(U1(a, e), U1(b, b)) A^1 -> H(U1(a, e), f(d)) A^1 -> H(U1(a, e), f(e)) A^1 -> H(U1(a, a), U1(d, b)) A^1 -> H(U1(a, a), U1(e, b)) A^1 -> H(U1(a, a), U1(b, d)) A^1 -> H(U1(a, a), U1(b, e)) A^1 -> H(U1(a, a), U1(d, d)) A^1 -> H(U1(a, a), U1(e, e)) A^1 -> H(d, f(b)) A^1 -> H(U1(d, d), U1(b, b)) A^1 -> H(U1(d, d), f(d)) A^1 -> H(U1(d, d), f(e)) A^1 -> H(f(d), U1(d, b)) A^1 -> H(f(d), U1(e, b)) A^1 -> H(f(d), U1(b, d)) A^1 -> H(f(d), U1(b, e)) A^1 -> H(f(d), U1(e, e)) A^1 -> H(U1(e, e), U1(b, b)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (65) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, a), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(a, U1(b, b)),A^1 -> H(a, U1(b, b))) (A^1 -> H(U1(d, d), U1(b, b)),A^1 -> H(U1(d, d), U1(b, b))) (A^1 -> H(U1(d, e), U1(b, b)),A^1 -> H(U1(d, e), U1(b, b))) (A^1 -> H(U1(d, a), U1(d, b)),A^1 -> H(U1(d, a), U1(d, b))) (A^1 -> H(U1(d, a), U1(e, b)),A^1 -> H(U1(d, a), U1(e, b))) (A^1 -> H(U1(d, a), U1(b, d)),A^1 -> H(U1(d, a), U1(b, d))) (A^1 -> H(U1(d, a), U1(b, e)),A^1 -> H(U1(d, a), U1(b, e))) ---------------------------------------- (66) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, a), f(d)) A^1 -> H(U1(d, a), f(e)) A^1 -> H(U1(e, d), f(b)) A^1 -> H(U1(e, a), U1(b, b)) A^1 -> H(U1(e, a), f(d)) A^1 -> H(U1(e, a), f(e)) A^1 -> H(U1(a, d), U1(b, b)) A^1 -> H(U1(a, d), f(d)) A^1 -> H(U1(a, d), f(e)) A^1 -> H(U1(a, e), U1(b, b)) A^1 -> H(U1(a, e), f(d)) A^1 -> H(U1(a, e), f(e)) A^1 -> H(U1(a, a), U1(d, b)) A^1 -> H(U1(a, a), U1(e, b)) A^1 -> H(U1(a, a), U1(b, d)) A^1 -> H(U1(a, a), U1(b, e)) A^1 -> H(U1(a, a), U1(d, d)) A^1 -> H(U1(a, a), U1(e, e)) A^1 -> H(d, f(b)) A^1 -> H(U1(d, d), U1(b, b)) A^1 -> H(U1(d, d), f(d)) A^1 -> H(U1(d, d), f(e)) A^1 -> H(f(d), U1(d, b)) A^1 -> H(f(d), U1(e, b)) A^1 -> H(f(d), U1(b, d)) A^1 -> H(f(d), U1(b, e)) A^1 -> H(f(d), U1(e, e)) A^1 -> H(U1(e, e), U1(b, b)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (67) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, a), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(a, f(d)),A^1 -> H(a, f(d))) (A^1 -> H(U1(d, d), f(d)),A^1 -> H(U1(d, d), f(d))) (A^1 -> H(U1(d, e), f(d)),A^1 -> H(U1(d, e), f(d))) (A^1 -> H(U1(d, a), U1(d, d)),A^1 -> H(U1(d, a), U1(d, d))) ---------------------------------------- (68) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, a), f(e)) A^1 -> H(U1(e, d), f(b)) A^1 -> H(U1(e, a), U1(b, b)) A^1 -> H(U1(e, a), f(d)) A^1 -> H(U1(e, a), f(e)) A^1 -> H(U1(a, d), U1(b, b)) A^1 -> H(U1(a, d), f(d)) A^1 -> H(U1(a, d), f(e)) A^1 -> H(U1(a, e), U1(b, b)) A^1 -> H(U1(a, e), f(d)) A^1 -> H(U1(a, e), f(e)) A^1 -> H(U1(a, a), U1(d, b)) A^1 -> H(U1(a, a), U1(e, b)) A^1 -> H(U1(a, a), U1(b, d)) A^1 -> H(U1(a, a), U1(b, e)) A^1 -> H(U1(a, a), U1(d, d)) A^1 -> H(U1(a, a), U1(e, e)) A^1 -> H(d, f(b)) A^1 -> H(U1(d, d), U1(b, b)) A^1 -> H(U1(d, d), f(d)) A^1 -> H(U1(d, d), f(e)) A^1 -> H(f(d), U1(d, b)) A^1 -> H(f(d), U1(e, b)) A^1 -> H(f(d), U1(b, d)) A^1 -> H(f(d), U1(b, e)) A^1 -> H(f(d), U1(e, e)) A^1 -> H(U1(e, e), U1(b, b)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (69) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, a), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(a, f(e)),A^1 -> H(a, f(e))) (A^1 -> H(U1(d, d), f(e)),A^1 -> H(U1(d, d), f(e))) (A^1 -> H(U1(d, e), f(e)),A^1 -> H(U1(d, e), f(e))) (A^1 -> H(U1(d, a), U1(e, e)),A^1 -> H(U1(d, a), U1(e, e))) ---------------------------------------- (70) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(e, d), f(b)) A^1 -> H(U1(e, a), U1(b, b)) A^1 -> H(U1(e, a), f(d)) A^1 -> H(U1(e, a), f(e)) A^1 -> H(U1(a, d), U1(b, b)) A^1 -> H(U1(a, d), f(d)) A^1 -> H(U1(a, d), f(e)) A^1 -> H(U1(a, e), U1(b, b)) A^1 -> H(U1(a, e), f(d)) A^1 -> H(U1(a, e), f(e)) A^1 -> H(U1(a, a), U1(d, b)) A^1 -> H(U1(a, a), U1(e, b)) A^1 -> H(U1(a, a), U1(b, d)) A^1 -> H(U1(a, a), U1(b, e)) A^1 -> H(U1(a, a), U1(d, d)) A^1 -> H(U1(a, a), U1(e, e)) A^1 -> H(d, f(b)) A^1 -> H(U1(d, d), U1(b, b)) A^1 -> H(U1(d, d), f(d)) A^1 -> H(U1(d, d), f(e)) A^1 -> H(f(d), U1(d, b)) A^1 -> H(f(d), U1(e, b)) A^1 -> H(f(d), U1(b, d)) A^1 -> H(f(d), U1(b, e)) A^1 -> H(f(d), U1(e, e)) A^1 -> H(U1(e, e), U1(b, b)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (71) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(e, d), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, d), U1(b, b)),A^1 -> H(U1(e, d), U1(b, b))) (A^1 -> H(U1(e, d), f(d)),A^1 -> H(U1(e, d), f(d))) (A^1 -> H(U1(e, d), f(e)),A^1 -> H(U1(e, d), f(e))) ---------------------------------------- (72) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(e, a), U1(b, b)) A^1 -> H(U1(e, a), f(d)) A^1 -> H(U1(e, a), f(e)) A^1 -> H(U1(a, d), U1(b, b)) A^1 -> H(U1(a, d), f(d)) A^1 -> H(U1(a, d), f(e)) A^1 -> H(U1(a, e), U1(b, b)) A^1 -> H(U1(a, e), f(d)) A^1 -> H(U1(a, e), f(e)) A^1 -> H(U1(a, a), U1(d, b)) A^1 -> H(U1(a, a), U1(e, b)) A^1 -> H(U1(a, a), U1(b, d)) A^1 -> H(U1(a, a), U1(b, e)) A^1 -> H(U1(a, a), U1(d, d)) A^1 -> H(U1(a, a), U1(e, e)) A^1 -> H(d, f(b)) A^1 -> H(U1(d, d), U1(b, b)) A^1 -> H(U1(d, d), f(d)) A^1 -> H(U1(d, d), f(e)) A^1 -> H(f(d), U1(d, b)) A^1 -> H(f(d), U1(e, b)) A^1 -> H(f(d), U1(b, d)) A^1 -> H(f(d), U1(b, e)) A^1 -> H(f(d), U1(e, e)) A^1 -> H(U1(e, e), U1(b, b)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (73) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(e, a), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, d), U1(b, b)),A^1 -> H(U1(e, d), U1(b, b))) (A^1 -> H(U1(e, e), U1(b, b)),A^1 -> H(U1(e, e), U1(b, b))) (A^1 -> H(U1(e, a), U1(d, b)),A^1 -> H(U1(e, a), U1(d, b))) (A^1 -> H(U1(e, a), U1(e, b)),A^1 -> H(U1(e, a), U1(e, b))) (A^1 -> H(U1(e, a), U1(b, d)),A^1 -> H(U1(e, a), U1(b, d))) (A^1 -> H(U1(e, a), U1(b, e)),A^1 -> H(U1(e, a), U1(b, e))) ---------------------------------------- (74) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(e, a), f(d)) A^1 -> H(U1(e, a), f(e)) A^1 -> H(U1(a, d), U1(b, b)) A^1 -> H(U1(a, d), f(d)) A^1 -> H(U1(a, d), f(e)) A^1 -> H(U1(a, e), U1(b, b)) A^1 -> H(U1(a, e), f(d)) A^1 -> H(U1(a, e), f(e)) A^1 -> H(U1(a, a), U1(d, b)) A^1 -> H(U1(a, a), U1(e, b)) A^1 -> H(U1(a, a), U1(b, d)) A^1 -> H(U1(a, a), U1(b, e)) A^1 -> H(U1(a, a), U1(d, d)) A^1 -> H(U1(a, a), U1(e, e)) A^1 -> H(d, f(b)) A^1 -> H(U1(d, d), U1(b, b)) A^1 -> H(U1(d, d), f(d)) A^1 -> H(U1(d, d), f(e)) A^1 -> H(f(d), U1(d, b)) A^1 -> H(f(d), U1(e, b)) A^1 -> H(f(d), U1(b, d)) A^1 -> H(f(d), U1(b, e)) A^1 -> H(f(d), U1(e, e)) A^1 -> H(U1(e, e), U1(b, b)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (75) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(e, a), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, d), f(d)),A^1 -> H(U1(e, d), f(d))) (A^1 -> H(U1(e, e), f(d)),A^1 -> H(U1(e, e), f(d))) (A^1 -> H(U1(e, a), U1(d, d)),A^1 -> H(U1(e, a), U1(d, d))) ---------------------------------------- (76) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(e, a), f(e)) A^1 -> H(U1(a, d), U1(b, b)) A^1 -> H(U1(a, d), f(d)) A^1 -> H(U1(a, d), f(e)) A^1 -> H(U1(a, e), U1(b, b)) A^1 -> H(U1(a, e), f(d)) A^1 -> H(U1(a, e), f(e)) A^1 -> H(U1(a, a), U1(d, b)) A^1 -> H(U1(a, a), U1(e, b)) A^1 -> H(U1(a, a), U1(b, d)) A^1 -> H(U1(a, a), U1(b, e)) A^1 -> H(U1(a, a), U1(d, d)) A^1 -> H(U1(a, a), U1(e, e)) A^1 -> H(d, f(b)) A^1 -> H(U1(d, d), U1(b, b)) A^1 -> H(U1(d, d), f(d)) A^1 -> H(U1(d, d), f(e)) A^1 -> H(f(d), U1(d, b)) A^1 -> H(f(d), U1(e, b)) A^1 -> H(f(d), U1(b, d)) A^1 -> H(f(d), U1(b, e)) A^1 -> H(f(d), U1(e, e)) A^1 -> H(U1(e, e), U1(b, b)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (77) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(e, a), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, d), f(e)),A^1 -> H(U1(e, d), f(e))) (A^1 -> H(U1(e, e), f(e)),A^1 -> H(U1(e, e), f(e))) (A^1 -> H(U1(e, a), U1(e, e)),A^1 -> H(U1(e, a), U1(e, e))) ---------------------------------------- (78) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(a, d), U1(b, b)) A^1 -> H(U1(a, d), f(d)) A^1 -> H(U1(a, d), f(e)) A^1 -> H(U1(a, e), U1(b, b)) A^1 -> H(U1(a, e), f(d)) A^1 -> H(U1(a, e), f(e)) A^1 -> H(U1(a, a), U1(d, b)) A^1 -> H(U1(a, a), U1(e, b)) A^1 -> H(U1(a, a), U1(b, d)) A^1 -> H(U1(a, a), U1(b, e)) A^1 -> H(U1(a, a), U1(d, d)) A^1 -> H(U1(a, a), U1(e, e)) A^1 -> H(d, f(b)) A^1 -> H(U1(d, d), U1(b, b)) A^1 -> H(U1(d, d), f(d)) A^1 -> H(U1(d, d), f(e)) A^1 -> H(f(d), U1(d, b)) A^1 -> H(f(d), U1(e, b)) A^1 -> H(f(d), U1(b, d)) A^1 -> H(f(d), U1(b, e)) A^1 -> H(f(d), U1(e, e)) A^1 -> H(U1(e, e), U1(b, b)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (79) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, d), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, d), U1(b, b)),A^1 -> H(U1(d, d), U1(b, b))) (A^1 -> H(U1(e, d), U1(b, b)),A^1 -> H(U1(e, d), U1(b, b))) (A^1 -> H(U1(a, d), U1(d, b)),A^1 -> H(U1(a, d), U1(d, b))) (A^1 -> H(U1(a, d), U1(e, b)),A^1 -> H(U1(a, d), U1(e, b))) (A^1 -> H(U1(a, d), U1(b, d)),A^1 -> H(U1(a, d), U1(b, d))) (A^1 -> H(U1(a, d), U1(b, e)),A^1 -> H(U1(a, d), U1(b, e))) ---------------------------------------- (80) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(a, d), f(d)) A^1 -> H(U1(a, d), f(e)) A^1 -> H(U1(a, e), U1(b, b)) A^1 -> H(U1(a, e), f(d)) A^1 -> H(U1(a, e), f(e)) A^1 -> H(U1(a, a), U1(d, b)) A^1 -> H(U1(a, a), U1(e, b)) A^1 -> H(U1(a, a), U1(b, d)) A^1 -> H(U1(a, a), U1(b, e)) A^1 -> H(U1(a, a), U1(d, d)) A^1 -> H(U1(a, a), U1(e, e)) A^1 -> H(d, f(b)) A^1 -> H(U1(d, d), U1(b, b)) A^1 -> H(U1(d, d), f(d)) A^1 -> H(U1(d, d), f(e)) A^1 -> H(f(d), U1(d, b)) A^1 -> H(f(d), U1(e, b)) A^1 -> H(f(d), U1(b, d)) A^1 -> H(f(d), U1(b, e)) A^1 -> H(f(d), U1(e, e)) A^1 -> H(U1(e, e), U1(b, b)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (81) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, d), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, d), f(d)),A^1 -> H(U1(d, d), f(d))) (A^1 -> H(U1(e, d), f(d)),A^1 -> H(U1(e, d), f(d))) (A^1 -> H(U1(a, d), U1(d, d)),A^1 -> H(U1(a, d), U1(d, d))) ---------------------------------------- (82) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(a, d), f(e)) A^1 -> H(U1(a, e), U1(b, b)) A^1 -> H(U1(a, e), f(d)) A^1 -> H(U1(a, e), f(e)) A^1 -> H(U1(a, a), U1(d, b)) A^1 -> H(U1(a, a), U1(e, b)) A^1 -> H(U1(a, a), U1(b, d)) A^1 -> H(U1(a, a), U1(b, e)) A^1 -> H(U1(a, a), U1(d, d)) A^1 -> H(U1(a, a), U1(e, e)) A^1 -> H(d, f(b)) A^1 -> H(U1(d, d), U1(b, b)) A^1 -> H(U1(d, d), f(d)) A^1 -> H(U1(d, d), f(e)) A^1 -> H(f(d), U1(d, b)) A^1 -> H(f(d), U1(e, b)) A^1 -> H(f(d), U1(b, d)) A^1 -> H(f(d), U1(b, e)) A^1 -> H(f(d), U1(e, e)) A^1 -> H(U1(e, e), U1(b, b)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (83) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, d), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, d), f(e)),A^1 -> H(U1(d, d), f(e))) (A^1 -> H(U1(e, d), f(e)),A^1 -> H(U1(e, d), f(e))) (A^1 -> H(U1(a, d), U1(e, e)),A^1 -> H(U1(a, d), U1(e, e))) ---------------------------------------- (84) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(a, e), U1(b, b)) A^1 -> H(U1(a, e), f(d)) A^1 -> H(U1(a, e), f(e)) A^1 -> H(U1(a, a), U1(d, b)) A^1 -> H(U1(a, a), U1(e, b)) A^1 -> H(U1(a, a), U1(b, d)) A^1 -> H(U1(a, a), U1(b, e)) A^1 -> H(U1(a, a), U1(d, d)) A^1 -> H(U1(a, a), U1(e, e)) A^1 -> H(d, f(b)) A^1 -> H(U1(d, d), U1(b, b)) A^1 -> H(U1(d, d), f(d)) A^1 -> H(U1(d, d), f(e)) A^1 -> H(f(d), U1(d, b)) A^1 -> H(f(d), U1(e, b)) A^1 -> H(f(d), U1(b, d)) A^1 -> H(f(d), U1(b, e)) A^1 -> H(f(d), U1(e, e)) A^1 -> H(U1(e, e), U1(b, b)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (85) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, e), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, e), U1(b, b)),A^1 -> H(U1(d, e), U1(b, b))) (A^1 -> H(U1(e, e), U1(b, b)),A^1 -> H(U1(e, e), U1(b, b))) (A^1 -> H(U1(a, e), U1(d, b)),A^1 -> H(U1(a, e), U1(d, b))) (A^1 -> H(U1(a, e), U1(e, b)),A^1 -> H(U1(a, e), U1(e, b))) (A^1 -> H(U1(a, e), U1(b, d)),A^1 -> H(U1(a, e), U1(b, d))) (A^1 -> H(U1(a, e), U1(b, e)),A^1 -> H(U1(a, e), U1(b, e))) ---------------------------------------- (86) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(a, e), f(d)) A^1 -> H(U1(a, e), f(e)) A^1 -> H(U1(a, a), U1(d, b)) A^1 -> H(U1(a, a), U1(e, b)) A^1 -> H(U1(a, a), U1(b, d)) A^1 -> H(U1(a, a), U1(b, e)) A^1 -> H(U1(a, a), U1(d, d)) A^1 -> H(U1(a, a), U1(e, e)) A^1 -> H(d, f(b)) A^1 -> H(U1(d, d), U1(b, b)) A^1 -> H(U1(d, d), f(d)) A^1 -> H(U1(d, d), f(e)) A^1 -> H(f(d), U1(d, b)) A^1 -> H(f(d), U1(e, b)) A^1 -> H(f(d), U1(b, d)) A^1 -> H(f(d), U1(b, e)) A^1 -> H(f(d), U1(e, e)) A^1 -> H(U1(e, e), U1(b, b)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (87) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, e), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, e), f(d)),A^1 -> H(U1(d, e), f(d))) (A^1 -> H(U1(e, e), f(d)),A^1 -> H(U1(e, e), f(d))) (A^1 -> H(U1(a, e), U1(d, d)),A^1 -> H(U1(a, e), U1(d, d))) ---------------------------------------- (88) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(a, e), f(e)) A^1 -> H(U1(a, a), U1(d, b)) A^1 -> H(U1(a, a), U1(e, b)) A^1 -> H(U1(a, a), U1(b, d)) A^1 -> H(U1(a, a), U1(b, e)) A^1 -> H(U1(a, a), U1(d, d)) A^1 -> H(U1(a, a), U1(e, e)) A^1 -> H(d, f(b)) A^1 -> H(U1(d, d), U1(b, b)) A^1 -> H(U1(d, d), f(d)) A^1 -> H(U1(d, d), f(e)) A^1 -> H(f(d), U1(d, b)) A^1 -> H(f(d), U1(e, b)) A^1 -> H(f(d), U1(b, d)) A^1 -> H(f(d), U1(b, e)) A^1 -> H(f(d), U1(e, e)) A^1 -> H(U1(e, e), U1(b, b)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (89) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, e), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, e), f(e)),A^1 -> H(U1(d, e), f(e))) (A^1 -> H(U1(e, e), f(e)),A^1 -> H(U1(e, e), f(e))) (A^1 -> H(U1(a, e), U1(e, e)),A^1 -> H(U1(a, e), U1(e, e))) ---------------------------------------- (90) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(a, a), U1(d, b)) A^1 -> H(U1(a, a), U1(e, b)) A^1 -> H(U1(a, a), U1(b, d)) A^1 -> H(U1(a, a), U1(b, e)) A^1 -> H(U1(a, a), U1(d, d)) A^1 -> H(U1(a, a), U1(e, e)) A^1 -> H(d, f(b)) A^1 -> H(U1(d, d), U1(b, b)) A^1 -> H(U1(d, d), f(d)) A^1 -> H(U1(d, d), f(e)) A^1 -> H(f(d), U1(d, b)) A^1 -> H(f(d), U1(e, b)) A^1 -> H(f(d), U1(b, d)) A^1 -> H(f(d), U1(b, e)) A^1 -> H(f(d), U1(e, e)) A^1 -> H(U1(e, e), U1(b, b)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (91) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, a), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, a), U1(d, b)),A^1 -> H(U1(d, a), U1(d, b))) (A^1 -> H(U1(e, a), U1(d, b)),A^1 -> H(U1(e, a), U1(d, b))) (A^1 -> H(U1(a, d), U1(d, b)),A^1 -> H(U1(a, d), U1(d, b))) (A^1 -> H(U1(a, e), U1(d, b)),A^1 -> H(U1(a, e), U1(d, b))) (A^1 -> H(U1(a, a), b),A^1 -> H(U1(a, a), b)) (A^1 -> H(U1(a, a), U1(d, d)),A^1 -> H(U1(a, a), U1(d, d))) (A^1 -> H(U1(a, a), U1(d, e)),A^1 -> H(U1(a, a), U1(d, e))) ---------------------------------------- (92) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(a, a), U1(e, b)) A^1 -> H(U1(a, a), U1(b, d)) A^1 -> H(U1(a, a), U1(b, e)) A^1 -> H(U1(a, a), U1(d, d)) A^1 -> H(U1(a, a), U1(e, e)) A^1 -> H(d, f(b)) A^1 -> H(U1(d, d), U1(b, b)) A^1 -> H(U1(d, d), f(d)) A^1 -> H(U1(d, d), f(e)) A^1 -> H(f(d), U1(d, b)) A^1 -> H(f(d), U1(e, b)) A^1 -> H(f(d), U1(b, d)) A^1 -> H(f(d), U1(b, e)) A^1 -> H(f(d), U1(e, e)) A^1 -> H(U1(e, e), U1(b, b)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (93) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, a), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, a), U1(e, b)),A^1 -> H(U1(d, a), U1(e, b))) (A^1 -> H(U1(e, a), U1(e, b)),A^1 -> H(U1(e, a), U1(e, b))) (A^1 -> H(U1(a, d), U1(e, b)),A^1 -> H(U1(a, d), U1(e, b))) (A^1 -> H(U1(a, e), U1(e, b)),A^1 -> H(U1(a, e), U1(e, b))) (A^1 -> H(U1(a, a), U1(e, d)),A^1 -> H(U1(a, a), U1(e, d))) (A^1 -> H(U1(a, a), U1(e, e)),A^1 -> H(U1(a, a), U1(e, e))) ---------------------------------------- (94) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(a, a), U1(b, d)) A^1 -> H(U1(a, a), U1(b, e)) A^1 -> H(U1(a, a), U1(d, d)) A^1 -> H(U1(a, a), U1(e, e)) A^1 -> H(d, f(b)) A^1 -> H(U1(d, d), U1(b, b)) A^1 -> H(U1(d, d), f(d)) A^1 -> H(U1(d, d), f(e)) A^1 -> H(f(d), U1(d, b)) A^1 -> H(f(d), U1(e, b)) A^1 -> H(f(d), U1(b, d)) A^1 -> H(f(d), U1(b, e)) A^1 -> H(f(d), U1(e, e)) A^1 -> H(U1(e, e), U1(b, b)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (95) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, a), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, a), U1(b, d)),A^1 -> H(U1(d, a), U1(b, d))) (A^1 -> H(U1(e, a), U1(b, d)),A^1 -> H(U1(e, a), U1(b, d))) (A^1 -> H(U1(a, d), U1(b, d)),A^1 -> H(U1(a, d), U1(b, d))) (A^1 -> H(U1(a, e), U1(b, d)),A^1 -> H(U1(a, e), U1(b, d))) (A^1 -> H(U1(a, a), U1(d, d)),A^1 -> H(U1(a, a), U1(d, d))) (A^1 -> H(U1(a, a), U1(e, d)),A^1 -> H(U1(a, a), U1(e, d))) ---------------------------------------- (96) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(a, a), U1(b, e)) A^1 -> H(U1(a, a), U1(d, d)) A^1 -> H(U1(a, a), U1(e, e)) A^1 -> H(d, f(b)) A^1 -> H(U1(d, d), U1(b, b)) A^1 -> H(U1(d, d), f(d)) A^1 -> H(U1(d, d), f(e)) A^1 -> H(f(d), U1(d, b)) A^1 -> H(f(d), U1(e, b)) A^1 -> H(f(d), U1(b, d)) A^1 -> H(f(d), U1(b, e)) A^1 -> H(f(d), U1(e, e)) A^1 -> H(U1(e, e), U1(b, b)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (97) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, a), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, a), U1(b, e)),A^1 -> H(U1(d, a), U1(b, e))) (A^1 -> H(U1(e, a), U1(b, e)),A^1 -> H(U1(e, a), U1(b, e))) (A^1 -> H(U1(a, d), U1(b, e)),A^1 -> H(U1(a, d), U1(b, e))) (A^1 -> H(U1(a, e), U1(b, e)),A^1 -> H(U1(a, e), U1(b, e))) (A^1 -> H(U1(a, a), U1(d, e)),A^1 -> H(U1(a, a), U1(d, e))) (A^1 -> H(U1(a, a), U1(e, e)),A^1 -> H(U1(a, a), U1(e, e))) ---------------------------------------- (98) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(a, a), U1(d, d)) A^1 -> H(U1(a, a), U1(e, e)) A^1 -> H(d, f(b)) A^1 -> H(U1(d, d), U1(b, b)) A^1 -> H(U1(d, d), f(d)) A^1 -> H(U1(d, d), f(e)) A^1 -> H(f(d), U1(d, b)) A^1 -> H(f(d), U1(e, b)) A^1 -> H(f(d), U1(b, d)) A^1 -> H(f(d), U1(b, e)) A^1 -> H(f(d), U1(e, e)) A^1 -> H(U1(e, e), U1(b, b)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (99) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, a), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, a), U1(d, d)),A^1 -> H(U1(d, a), U1(d, d))) (A^1 -> H(U1(e, a), U1(d, d)),A^1 -> H(U1(e, a), U1(d, d))) (A^1 -> H(U1(a, d), U1(d, d)),A^1 -> H(U1(a, d), U1(d, d))) (A^1 -> H(U1(a, e), U1(d, d)),A^1 -> H(U1(a, e), U1(d, d))) (A^1 -> H(U1(a, a), d),A^1 -> H(U1(a, a), d)) ---------------------------------------- (100) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(a, a), U1(e, e)) A^1 -> H(d, f(b)) A^1 -> H(U1(d, d), U1(b, b)) A^1 -> H(U1(d, d), f(d)) A^1 -> H(U1(d, d), f(e)) A^1 -> H(f(d), U1(d, b)) A^1 -> H(f(d), U1(e, b)) A^1 -> H(f(d), U1(b, d)) A^1 -> H(f(d), U1(b, e)) A^1 -> H(f(d), U1(e, e)) A^1 -> H(U1(e, e), U1(b, b)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (101) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, a), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, a), U1(e, e)),A^1 -> H(U1(d, a), U1(e, e))) (A^1 -> H(U1(e, a), U1(e, e)),A^1 -> H(U1(e, a), U1(e, e))) (A^1 -> H(U1(a, d), U1(e, e)),A^1 -> H(U1(a, d), U1(e, e))) (A^1 -> H(U1(a, e), U1(e, e)),A^1 -> H(U1(a, e), U1(e, e))) ---------------------------------------- (102) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(d, f(b)) A^1 -> H(U1(d, d), U1(b, b)) A^1 -> H(U1(d, d), f(d)) A^1 -> H(U1(d, d), f(e)) A^1 -> H(f(d), U1(d, b)) A^1 -> H(f(d), U1(e, b)) A^1 -> H(f(d), U1(b, d)) A^1 -> H(f(d), U1(b, e)) A^1 -> H(f(d), U1(e, e)) A^1 -> H(U1(e, e), U1(b, b)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (103) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(d, f(b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(d, U1(b, b)),A^1 -> H(d, U1(b, b))) (A^1 -> H(d, f(d)),A^1 -> H(d, f(d))) (A^1 -> H(d, f(e)),A^1 -> H(d, f(e))) ---------------------------------------- (104) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(b, b)) A^1 -> H(U1(d, d), f(d)) A^1 -> H(U1(d, d), f(e)) A^1 -> H(f(d), U1(d, b)) A^1 -> H(f(d), U1(e, b)) A^1 -> H(f(d), U1(b, d)) A^1 -> H(f(d), U1(b, e)) A^1 -> H(f(d), U1(e, e)) A^1 -> H(U1(e, e), U1(b, b)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (105) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, d), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(d, U1(b, b)),A^1 -> H(d, U1(b, b))) (A^1 -> H(U1(d, d), U1(d, b)),A^1 -> H(U1(d, d), U1(d, b))) (A^1 -> H(U1(d, d), U1(e, b)),A^1 -> H(U1(d, d), U1(e, b))) (A^1 -> H(U1(d, d), U1(b, d)),A^1 -> H(U1(d, d), U1(b, d))) (A^1 -> H(U1(d, d), U1(b, e)),A^1 -> H(U1(d, d), U1(b, e))) ---------------------------------------- (106) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), f(d)) A^1 -> H(U1(d, d), f(e)) A^1 -> H(f(d), U1(d, b)) A^1 -> H(f(d), U1(e, b)) A^1 -> H(f(d), U1(b, d)) A^1 -> H(f(d), U1(b, e)) A^1 -> H(f(d), U1(e, e)) A^1 -> H(U1(e, e), U1(b, b)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (107) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, d), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(d, f(d)),A^1 -> H(d, f(d))) (A^1 -> H(U1(d, d), U1(d, d)),A^1 -> H(U1(d, d), U1(d, d))) ---------------------------------------- (108) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), f(e)) A^1 -> H(f(d), U1(d, b)) A^1 -> H(f(d), U1(e, b)) A^1 -> H(f(d), U1(b, d)) A^1 -> H(f(d), U1(b, e)) A^1 -> H(f(d), U1(e, e)) A^1 -> H(U1(e, e), U1(b, b)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (109) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, d), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(d, f(e)),A^1 -> H(d, f(e))) (A^1 -> H(U1(d, d), U1(e, e)),A^1 -> H(U1(d, d), U1(e, e))) ---------------------------------------- (110) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(f(d), U1(d, b)) A^1 -> H(f(d), U1(e, b)) A^1 -> H(f(d), U1(b, d)) A^1 -> H(f(d), U1(b, e)) A^1 -> H(f(d), U1(e, e)) A^1 -> H(U1(e, e), U1(b, b)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (111) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(d), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, d), U1(d, b)),A^1 -> H(U1(d, d), U1(d, b))) (A^1 -> H(f(d), b),A^1 -> H(f(d), b)) (A^1 -> H(f(d), U1(d, d)),A^1 -> H(f(d), U1(d, d))) (A^1 -> H(f(d), U1(d, e)),A^1 -> H(f(d), U1(d, e))) ---------------------------------------- (112) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(f(d), U1(e, b)) A^1 -> H(f(d), U1(b, d)) A^1 -> H(f(d), U1(b, e)) A^1 -> H(f(d), U1(e, e)) A^1 -> H(U1(e, e), U1(b, b)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (113) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(d), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, d), U1(e, b)),A^1 -> H(U1(d, d), U1(e, b))) (A^1 -> H(f(d), U1(e, d)),A^1 -> H(f(d), U1(e, d))) (A^1 -> H(f(d), U1(e, e)),A^1 -> H(f(d), U1(e, e))) ---------------------------------------- (114) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(f(d), U1(b, d)) A^1 -> H(f(d), U1(b, e)) A^1 -> H(f(d), U1(e, e)) A^1 -> H(U1(e, e), U1(b, b)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (115) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(d), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, d), U1(b, d)),A^1 -> H(U1(d, d), U1(b, d))) (A^1 -> H(f(d), U1(d, d)),A^1 -> H(f(d), U1(d, d))) (A^1 -> H(f(d), U1(e, d)),A^1 -> H(f(d), U1(e, d))) ---------------------------------------- (116) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(f(d), U1(b, e)) A^1 -> H(f(d), U1(e, e)) A^1 -> H(U1(e, e), U1(b, b)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (117) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(d), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, d), U1(b, e)),A^1 -> H(U1(d, d), U1(b, e))) (A^1 -> H(f(d), U1(d, e)),A^1 -> H(f(d), U1(d, e))) (A^1 -> H(f(d), U1(e, e)),A^1 -> H(f(d), U1(e, e))) ---------------------------------------- (118) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(f(d), U1(e, e)) A^1 -> H(U1(e, e), U1(b, b)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (119) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(d), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, d), U1(e, e)),A^1 -> H(U1(d, d), U1(e, e))) ---------------------------------------- (120) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(e, e), U1(b, b)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (121) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(e, e), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, e), U1(d, b)),A^1 -> H(U1(e, e), U1(d, b))) (A^1 -> H(U1(e, e), U1(e, b)),A^1 -> H(U1(e, e), U1(e, b))) (A^1 -> H(U1(e, e), U1(b, d)),A^1 -> H(U1(e, e), U1(b, d))) (A^1 -> H(U1(e, e), U1(b, e)),A^1 -> H(U1(e, e), U1(b, e))) ---------------------------------------- (122) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(e, e), f(d)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (123) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(e, e), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, e), U1(d, d)),A^1 -> H(U1(e, e), U1(d, d))) ---------------------------------------- (124) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(e, e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (125) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(e, e), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, e), U1(e, e)),A^1 -> H(U1(e, e), U1(e, e))) ---------------------------------------- (126) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(f(e), U1(d, b)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (127) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(e), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, e), U1(d, b)),A^1 -> H(U1(e, e), U1(d, b))) (A^1 -> H(f(e), b),A^1 -> H(f(e), b)) (A^1 -> H(f(e), U1(d, d)),A^1 -> H(f(e), U1(d, d))) (A^1 -> H(f(e), U1(d, e)),A^1 -> H(f(e), U1(d, e))) ---------------------------------------- (128) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(f(e), U1(e, b)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (129) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(e), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, e), U1(e, b)),A^1 -> H(U1(e, e), U1(e, b))) (A^1 -> H(f(e), U1(e, d)),A^1 -> H(f(e), U1(e, d))) (A^1 -> H(f(e), U1(e, e)),A^1 -> H(f(e), U1(e, e))) ---------------------------------------- (130) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(f(e), U1(b, d)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (131) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(e), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, e), U1(b, d)),A^1 -> H(U1(e, e), U1(b, d))) (A^1 -> H(f(e), U1(d, d)),A^1 -> H(f(e), U1(d, d))) (A^1 -> H(f(e), U1(e, d)),A^1 -> H(f(e), U1(e, d))) ---------------------------------------- (132) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(f(e), U1(b, e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (133) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(e), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, e), U1(b, e)),A^1 -> H(U1(e, e), U1(b, e))) (A^1 -> H(f(e), U1(d, e)),A^1 -> H(f(e), U1(d, e))) (A^1 -> H(f(e), U1(e, e)),A^1 -> H(f(e), U1(e, e))) ---------------------------------------- (134) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(f(e), U1(d, d)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (135) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(e), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, e), U1(d, d)),A^1 -> H(U1(e, e), U1(d, d))) (A^1 -> H(f(e), d),A^1 -> H(f(e), d)) ---------------------------------------- (136) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(f(a), b) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (137) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(a), b) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(a, a), b),A^1 -> H(U1(a, a), b)) (A^1 -> H(f(d), b),A^1 -> H(f(d), b)) (A^1 -> H(f(e), b),A^1 -> H(f(e), b)) (A^1 -> H(f(a), d),A^1 -> H(f(a), d)) (A^1 -> H(f(a), e),A^1 -> H(f(a), e)) ---------------------------------------- (138) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(f(a), U1(d, e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (139) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(a), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(a, a), U1(d, e)),A^1 -> H(U1(a, a), U1(d, e))) (A^1 -> H(f(d), U1(d, e)),A^1 -> H(f(d), U1(d, e))) (A^1 -> H(f(e), U1(d, e)),A^1 -> H(f(e), U1(d, e))) (A^1 -> H(f(a), e),A^1 -> H(f(a), e)) ---------------------------------------- (140) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(f(a), U1(e, d)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (141) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(a), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(a, a), U1(e, d)),A^1 -> H(U1(a, a), U1(e, d))) (A^1 -> H(f(d), U1(e, d)),A^1 -> H(f(d), U1(e, d))) (A^1 -> H(f(e), U1(e, d)),A^1 -> H(f(e), U1(e, d))) ---------------------------------------- (142) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(f(d), U1(d, d)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (143) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(d), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, d), U1(d, d)),A^1 -> H(U1(d, d), U1(d, d))) (A^1 -> H(f(d), d),A^1 -> H(f(d), d)) ---------------------------------------- (144) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(f(a), d) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (145) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(a), d) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(a, a), d),A^1 -> H(U1(a, a), d)) (A^1 -> H(f(d), d),A^1 -> H(f(d), d)) (A^1 -> H(f(e), d),A^1 -> H(f(e), d)) ---------------------------------------- (146) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(f(e), U1(e, e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (147) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(e), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, e), U1(e, e)),A^1 -> H(U1(e, e), U1(e, e))) ---------------------------------------- (148) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(e, f(b)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (149) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(e, f(b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(e, U1(b, b)),A^1 -> H(e, U1(b, b))) (A^1 -> H(e, f(d)),A^1 -> H(e, f(d))) (A^1 -> H(e, f(e)),A^1 -> H(e, f(e))) ---------------------------------------- (150) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(a, U1(b, b)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (151) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(a, U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(d, U1(b, b)),A^1 -> H(d, U1(b, b))) (A^1 -> H(e, U1(b, b)),A^1 -> H(e, U1(b, b))) (A^1 -> H(a, U1(d, b)),A^1 -> H(a, U1(d, b))) (A^1 -> H(a, U1(e, b)),A^1 -> H(a, U1(e, b))) (A^1 -> H(a, U1(b, d)),A^1 -> H(a, U1(b, d))) (A^1 -> H(a, U1(b, e)),A^1 -> H(a, U1(b, e))) ---------------------------------------- (152) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(a, f(d)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (153) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(a, f(d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(d, f(d)),A^1 -> H(d, f(d))) (A^1 -> H(e, f(d)),A^1 -> H(e, f(d))) (A^1 -> H(a, U1(d, d)),A^1 -> H(a, U1(d, d))) ---------------------------------------- (154) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(a, f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (155) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(a, f(e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(d, f(e)),A^1 -> H(d, f(e))) (A^1 -> H(e, f(e)),A^1 -> H(e, f(e))) (A^1 -> H(a, U1(e, e)),A^1 -> H(a, U1(e, e))) ---------------------------------------- (156) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, e), U1(b, b)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (157) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, e), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(e, U1(b, b)),A^1 -> H(e, U1(b, b))) (A^1 -> H(U1(d, e), U1(d, b)),A^1 -> H(U1(d, e), U1(d, b))) (A^1 -> H(U1(d, e), U1(e, b)),A^1 -> H(U1(d, e), U1(e, b))) (A^1 -> H(U1(d, e), U1(b, d)),A^1 -> H(U1(d, e), U1(b, d))) (A^1 -> H(U1(d, e), U1(b, e)),A^1 -> H(U1(d, e), U1(b, e))) ---------------------------------------- (158) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, e), f(d)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (159) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, e), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(e, f(d)),A^1 -> H(e, f(d))) (A^1 -> H(U1(d, e), U1(d, d)),A^1 -> H(U1(d, e), U1(d, d))) ---------------------------------------- (160) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (161) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, e), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(e, f(e)),A^1 -> H(e, f(e))) (A^1 -> H(U1(d, e), U1(e, e)),A^1 -> H(U1(d, e), U1(e, e))) ---------------------------------------- (162) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, a), U1(d, b)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (163) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, a), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(a, U1(d, b)),A^1 -> H(a, U1(d, b))) (A^1 -> H(U1(d, d), U1(d, b)),A^1 -> H(U1(d, d), U1(d, b))) (A^1 -> H(U1(d, e), U1(d, b)),A^1 -> H(U1(d, e), U1(d, b))) (A^1 -> H(U1(d, a), b),A^1 -> H(U1(d, a), b)) (A^1 -> H(U1(d, a), U1(d, d)),A^1 -> H(U1(d, a), U1(d, d))) (A^1 -> H(U1(d, a), U1(d, e)),A^1 -> H(U1(d, a), U1(d, e))) ---------------------------------------- (164) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, a), U1(e, b)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (165) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, a), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(a, U1(e, b)),A^1 -> H(a, U1(e, b))) (A^1 -> H(U1(d, d), U1(e, b)),A^1 -> H(U1(d, d), U1(e, b))) (A^1 -> H(U1(d, e), U1(e, b)),A^1 -> H(U1(d, e), U1(e, b))) (A^1 -> H(U1(d, a), U1(e, d)),A^1 -> H(U1(d, a), U1(e, d))) (A^1 -> H(U1(d, a), U1(e, e)),A^1 -> H(U1(d, a), U1(e, e))) ---------------------------------------- (166) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, a), U1(b, d)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (167) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, a), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(a, U1(b, d)),A^1 -> H(a, U1(b, d))) (A^1 -> H(U1(d, d), U1(b, d)),A^1 -> H(U1(d, d), U1(b, d))) (A^1 -> H(U1(d, e), U1(b, d)),A^1 -> H(U1(d, e), U1(b, d))) (A^1 -> H(U1(d, a), U1(d, d)),A^1 -> H(U1(d, a), U1(d, d))) (A^1 -> H(U1(d, a), U1(e, d)),A^1 -> H(U1(d, a), U1(e, d))) ---------------------------------------- (168) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, a), U1(b, e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (169) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, a), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(a, U1(b, e)),A^1 -> H(a, U1(b, e))) (A^1 -> H(U1(d, d), U1(b, e)),A^1 -> H(U1(d, d), U1(b, e))) (A^1 -> H(U1(d, e), U1(b, e)),A^1 -> H(U1(d, e), U1(b, e))) (A^1 -> H(U1(d, a), U1(d, e)),A^1 -> H(U1(d, a), U1(d, e))) (A^1 -> H(U1(d, a), U1(e, e)),A^1 -> H(U1(d, a), U1(e, e))) ---------------------------------------- (170) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, a), U1(d, d)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (171) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, a), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(a, U1(d, d)),A^1 -> H(a, U1(d, d))) (A^1 -> H(U1(d, d), U1(d, d)),A^1 -> H(U1(d, d), U1(d, d))) (A^1 -> H(U1(d, e), U1(d, d)),A^1 -> H(U1(d, e), U1(d, d))) (A^1 -> H(U1(d, a), d),A^1 -> H(U1(d, a), d)) ---------------------------------------- (172) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, a), U1(e, e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (173) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, a), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(a, U1(e, e)),A^1 -> H(a, U1(e, e))) (A^1 -> H(U1(d, d), U1(e, e)),A^1 -> H(U1(d, d), U1(e, e))) (A^1 -> H(U1(d, e), U1(e, e)),A^1 -> H(U1(d, e), U1(e, e))) ---------------------------------------- (174) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(e, d), U1(b, b)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (175) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(e, d), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, d), U1(d, b)),A^1 -> H(U1(e, d), U1(d, b))) (A^1 -> H(U1(e, d), U1(e, b)),A^1 -> H(U1(e, d), U1(e, b))) (A^1 -> H(U1(e, d), U1(b, d)),A^1 -> H(U1(e, d), U1(b, d))) (A^1 -> H(U1(e, d), U1(b, e)),A^1 -> H(U1(e, d), U1(b, e))) ---------------------------------------- (176) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(e, d), f(d)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (177) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(e, d), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, d), U1(d, d)),A^1 -> H(U1(e, d), U1(d, d))) ---------------------------------------- (178) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(e, d), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (179) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(e, d), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, d), U1(e, e)),A^1 -> H(U1(e, d), U1(e, e))) ---------------------------------------- (180) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, d), U1(e, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (181) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (182) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(e, a), U1(d, b)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (183) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(e, a), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, d), U1(d, b)),A^1 -> H(U1(e, d), U1(d, b))) (A^1 -> H(U1(e, e), U1(d, b)),A^1 -> H(U1(e, e), U1(d, b))) (A^1 -> H(U1(e, a), b),A^1 -> H(U1(e, a), b)) (A^1 -> H(U1(e, a), U1(d, d)),A^1 -> H(U1(e, a), U1(d, d))) (A^1 -> H(U1(e, a), U1(d, e)),A^1 -> H(U1(e, a), U1(d, e))) ---------------------------------------- (184) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(e, a), U1(e, b)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (185) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(e, a), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, d), U1(e, b)),A^1 -> H(U1(e, d), U1(e, b))) (A^1 -> H(U1(e, e), U1(e, b)),A^1 -> H(U1(e, e), U1(e, b))) (A^1 -> H(U1(e, a), U1(e, d)),A^1 -> H(U1(e, a), U1(e, d))) (A^1 -> H(U1(e, a), U1(e, e)),A^1 -> H(U1(e, a), U1(e, e))) ---------------------------------------- (186) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(e, a), U1(b, d)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (187) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(e, a), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, d), U1(b, d)),A^1 -> H(U1(e, d), U1(b, d))) (A^1 -> H(U1(e, e), U1(b, d)),A^1 -> H(U1(e, e), U1(b, d))) (A^1 -> H(U1(e, a), U1(d, d)),A^1 -> H(U1(e, a), U1(d, d))) (A^1 -> H(U1(e, a), U1(e, d)),A^1 -> H(U1(e, a), U1(e, d))) ---------------------------------------- (188) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(e, a), U1(b, e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (189) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(e, a), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, d), U1(b, e)),A^1 -> H(U1(e, d), U1(b, e))) (A^1 -> H(U1(e, e), U1(b, e)),A^1 -> H(U1(e, e), U1(b, e))) (A^1 -> H(U1(e, a), U1(d, e)),A^1 -> H(U1(e, a), U1(d, e))) (A^1 -> H(U1(e, a), U1(e, e)),A^1 -> H(U1(e, a), U1(e, e))) ---------------------------------------- (190) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(e, a), U1(d, d)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (191) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(e, a), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, d), U1(d, d)),A^1 -> H(U1(e, d), U1(d, d))) (A^1 -> H(U1(e, e), U1(d, d)),A^1 -> H(U1(e, e), U1(d, d))) (A^1 -> H(U1(e, a), d),A^1 -> H(U1(e, a), d)) ---------------------------------------- (192) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(e, a), d) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (193) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (194) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(e, a), U1(e, e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (195) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(e, a), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, d), U1(e, e)),A^1 -> H(U1(e, d), U1(e, e))) (A^1 -> H(U1(e, e), U1(e, e)),A^1 -> H(U1(e, e), U1(e, e))) ---------------------------------------- (196) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(e, d), U1(e, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (197) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (198) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(a, d), U1(d, b)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (199) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, d), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, d), U1(d, b)),A^1 -> H(U1(d, d), U1(d, b))) (A^1 -> H(U1(e, d), U1(d, b)),A^1 -> H(U1(e, d), U1(d, b))) (A^1 -> H(U1(a, d), b),A^1 -> H(U1(a, d), b)) (A^1 -> H(U1(a, d), U1(d, d)),A^1 -> H(U1(a, d), U1(d, d))) (A^1 -> H(U1(a, d), U1(d, e)),A^1 -> H(U1(a, d), U1(d, e))) ---------------------------------------- (200) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(a, d), U1(e, b)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (201) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, d), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, d), U1(e, b)),A^1 -> H(U1(d, d), U1(e, b))) (A^1 -> H(U1(e, d), U1(e, b)),A^1 -> H(U1(e, d), U1(e, b))) (A^1 -> H(U1(a, d), U1(e, d)),A^1 -> H(U1(a, d), U1(e, d))) (A^1 -> H(U1(a, d), U1(e, e)),A^1 -> H(U1(a, d), U1(e, e))) ---------------------------------------- (202) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(a, d), U1(b, d)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (203) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, d), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, d), U1(b, d)),A^1 -> H(U1(d, d), U1(b, d))) (A^1 -> H(U1(e, d), U1(b, d)),A^1 -> H(U1(e, d), U1(b, d))) (A^1 -> H(U1(a, d), U1(d, d)),A^1 -> H(U1(a, d), U1(d, d))) (A^1 -> H(U1(a, d), U1(e, d)),A^1 -> H(U1(a, d), U1(e, d))) ---------------------------------------- (204) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(a, d), U1(b, e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (205) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, d), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, d), U1(b, e)),A^1 -> H(U1(d, d), U1(b, e))) (A^1 -> H(U1(e, d), U1(b, e)),A^1 -> H(U1(e, d), U1(b, e))) (A^1 -> H(U1(a, d), U1(d, e)),A^1 -> H(U1(a, d), U1(d, e))) (A^1 -> H(U1(a, d), U1(e, e)),A^1 -> H(U1(a, d), U1(e, e))) ---------------------------------------- (206) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(a, d), U1(d, d)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (207) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, d), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, d), U1(d, d)),A^1 -> H(U1(d, d), U1(d, d))) (A^1 -> H(U1(e, d), U1(d, d)),A^1 -> H(U1(e, d), U1(d, d))) (A^1 -> H(U1(a, d), d),A^1 -> H(U1(a, d), d)) ---------------------------------------- (208) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(a, d), U1(e, e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (209) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, d), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, d), U1(e, e)),A^1 -> H(U1(d, d), U1(e, e))) (A^1 -> H(U1(e, d), U1(e, e)),A^1 -> H(U1(e, d), U1(e, e))) ---------------------------------------- (210) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(e, d), U1(e, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (211) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (212) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(a, e), U1(d, b)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (213) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, e), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, e), U1(d, b)),A^1 -> H(U1(d, e), U1(d, b))) (A^1 -> H(U1(e, e), U1(d, b)),A^1 -> H(U1(e, e), U1(d, b))) (A^1 -> H(U1(a, e), b),A^1 -> H(U1(a, e), b)) (A^1 -> H(U1(a, e), U1(d, d)),A^1 -> H(U1(a, e), U1(d, d))) (A^1 -> H(U1(a, e), U1(d, e)),A^1 -> H(U1(a, e), U1(d, e))) ---------------------------------------- (214) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(a, e), U1(e, b)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (215) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, e), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, e), U1(e, b)),A^1 -> H(U1(d, e), U1(e, b))) (A^1 -> H(U1(e, e), U1(e, b)),A^1 -> H(U1(e, e), U1(e, b))) (A^1 -> H(U1(a, e), U1(e, d)),A^1 -> H(U1(a, e), U1(e, d))) (A^1 -> H(U1(a, e), U1(e, e)),A^1 -> H(U1(a, e), U1(e, e))) ---------------------------------------- (216) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(a, e), U1(b, d)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (217) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, e), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, e), U1(b, d)),A^1 -> H(U1(d, e), U1(b, d))) (A^1 -> H(U1(e, e), U1(b, d)),A^1 -> H(U1(e, e), U1(b, d))) (A^1 -> H(U1(a, e), U1(d, d)),A^1 -> H(U1(a, e), U1(d, d))) (A^1 -> H(U1(a, e), U1(e, d)),A^1 -> H(U1(a, e), U1(e, d))) ---------------------------------------- (218) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(a, e), U1(b, e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (219) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, e), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, e), U1(b, e)),A^1 -> H(U1(d, e), U1(b, e))) (A^1 -> H(U1(e, e), U1(b, e)),A^1 -> H(U1(e, e), U1(b, e))) (A^1 -> H(U1(a, e), U1(d, e)),A^1 -> H(U1(a, e), U1(d, e))) (A^1 -> H(U1(a, e), U1(e, e)),A^1 -> H(U1(a, e), U1(e, e))) ---------------------------------------- (220) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(a, e), U1(d, d)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (221) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, e), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, e), U1(d, d)),A^1 -> H(U1(d, e), U1(d, d))) (A^1 -> H(U1(e, e), U1(d, d)),A^1 -> H(U1(e, e), U1(d, d))) (A^1 -> H(U1(a, e), d),A^1 -> H(U1(a, e), d)) ---------------------------------------- (222) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(a, e), U1(e, e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (223) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, e), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, e), U1(e, e)),A^1 -> H(U1(d, e), U1(e, e))) (A^1 -> H(U1(e, e), U1(e, e)),A^1 -> H(U1(e, e), U1(e, e))) ---------------------------------------- (224) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(a, a), b) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (225) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, a), b) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, a), b),A^1 -> H(U1(d, a), b)) (A^1 -> H(U1(e, a), b),A^1 -> H(U1(e, a), b)) (A^1 -> H(U1(a, d), b),A^1 -> H(U1(a, d), b)) (A^1 -> H(U1(a, e), b),A^1 -> H(U1(a, e), b)) (A^1 -> H(U1(a, a), d),A^1 -> H(U1(a, a), d)) (A^1 -> H(U1(a, a), e),A^1 -> H(U1(a, a), e)) ---------------------------------------- (226) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(a, a), U1(d, e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (227) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, a), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, a), U1(d, e)),A^1 -> H(U1(d, a), U1(d, e))) (A^1 -> H(U1(e, a), U1(d, e)),A^1 -> H(U1(e, a), U1(d, e))) (A^1 -> H(U1(a, d), U1(d, e)),A^1 -> H(U1(a, d), U1(d, e))) (A^1 -> H(U1(a, e), U1(d, e)),A^1 -> H(U1(a, e), U1(d, e))) (A^1 -> H(U1(a, a), e),A^1 -> H(U1(a, a), e)) ---------------------------------------- (228) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(a, a), U1(e, d)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (229) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, a), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, a), U1(e, d)),A^1 -> H(U1(d, a), U1(e, d))) (A^1 -> H(U1(e, a), U1(e, d)),A^1 -> H(U1(e, a), U1(e, d))) (A^1 -> H(U1(a, d), U1(e, d)),A^1 -> H(U1(a, d), U1(e, d))) (A^1 -> H(U1(a, e), U1(e, d)),A^1 -> H(U1(a, e), U1(e, d))) ---------------------------------------- (230) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(a, a), d) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (231) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, a), d) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, a), d),A^1 -> H(U1(d, a), d)) (A^1 -> H(U1(e, a), d),A^1 -> H(U1(e, a), d)) (A^1 -> H(U1(a, d), d),A^1 -> H(U1(a, d), d)) (A^1 -> H(U1(a, e), d),A^1 -> H(U1(a, e), d)) ---------------------------------------- (232) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(U1(e, a), d) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (233) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (234) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(d, U1(b, b)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (235) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(d, U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(d, U1(d, b)),A^1 -> H(d, U1(d, b))) (A^1 -> H(d, U1(e, b)),A^1 -> H(d, U1(e, b))) (A^1 -> H(d, U1(b, d)),A^1 -> H(d, U1(b, d))) (A^1 -> H(d, U1(b, e)),A^1 -> H(d, U1(b, e))) ---------------------------------------- (236) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(e, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (237) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (238) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(d, f(d)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (239) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(d, f(d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(d, U1(d, d)),A^1 -> H(d, U1(d, d))) ---------------------------------------- (240) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(d, f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (241) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(d, f(e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(d, U1(e, e)),A^1 -> H(d, U1(e, e))) ---------------------------------------- (242) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(d, U1(e, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (243) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (244) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, b)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (245) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, d), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(d, U1(d, b)),A^1 -> H(d, U1(d, b))) (A^1 -> H(U1(d, d), b),A^1 -> H(U1(d, d), b)) (A^1 -> H(U1(d, d), U1(d, d)),A^1 -> H(U1(d, d), U1(d, d))) (A^1 -> H(U1(d, d), U1(d, e)),A^1 -> H(U1(d, d), U1(d, e))) ---------------------------------------- (246) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(e, b)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (247) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, d), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(d, U1(e, b)),A^1 -> H(d, U1(e, b))) (A^1 -> H(U1(d, d), U1(e, d)),A^1 -> H(U1(d, d), U1(e, d))) (A^1 -> H(U1(d, d), U1(e, e)),A^1 -> H(U1(d, d), U1(e, e))) ---------------------------------------- (248) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(d, U1(e, b)) A^1 -> H(U1(d, d), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (249) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (250) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(b, d)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (251) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, d), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(d, U1(b, d)),A^1 -> H(d, U1(b, d))) (A^1 -> H(U1(d, d), U1(d, d)),A^1 -> H(U1(d, d), U1(d, d))) (A^1 -> H(U1(d, d), U1(e, d)),A^1 -> H(U1(d, d), U1(e, d))) ---------------------------------------- (252) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(b, e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (253) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, d), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(d, U1(b, e)),A^1 -> H(d, U1(b, e))) (A^1 -> H(U1(d, d), U1(d, e)),A^1 -> H(U1(d, d), U1(d, e))) (A^1 -> H(U1(d, d), U1(e, e)),A^1 -> H(U1(d, d), U1(e, e))) ---------------------------------------- (254) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(d, d), U1(e, e)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (255) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, d), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(d, U1(e, e)),A^1 -> H(d, U1(e, e))) ---------------------------------------- (256) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(d, U1(e, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (257) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (258) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(f(d), b) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (259) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(d), b) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, d), b),A^1 -> H(U1(d, d), b)) (A^1 -> H(f(d), d),A^1 -> H(f(d), d)) (A^1 -> H(f(d), e),A^1 -> H(f(d), e)) ---------------------------------------- (260) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(f(d), U1(d, e)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (261) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(d), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, d), U1(d, e)),A^1 -> H(U1(d, d), U1(d, e))) (A^1 -> H(f(d), e),A^1 -> H(f(d), e)) ---------------------------------------- (262) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(f(d), U1(e, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (263) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(d), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, d), U1(e, d)),A^1 -> H(U1(d, d), U1(e, d))) ---------------------------------------- (264) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(d, b)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (265) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(e, e), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, e), b),A^1 -> H(U1(e, e), b)) (A^1 -> H(U1(e, e), U1(d, d)),A^1 -> H(U1(e, e), U1(d, d))) (A^1 -> H(U1(e, e), U1(d, e)),A^1 -> H(U1(e, e), U1(d, e))) ---------------------------------------- (266) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, b)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (267) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(e, e), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, e), U1(e, d)),A^1 -> H(U1(e, e), U1(e, d))) (A^1 -> H(U1(e, e), U1(e, e)),A^1 -> H(U1(e, e), U1(e, e))) ---------------------------------------- (268) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(U1(e, e), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (269) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (270) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(b, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (271) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(e, e), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, e), U1(d, d)),A^1 -> H(U1(e, e), U1(d, d))) (A^1 -> H(U1(e, e), U1(e, d)),A^1 -> H(U1(e, e), U1(e, d))) ---------------------------------------- (272) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(U1(e, e), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (273) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (274) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(b, e)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (275) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(e, e), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, e), U1(d, e)),A^1 -> H(U1(e, e), U1(d, e))) (A^1 -> H(U1(e, e), U1(e, e)),A^1 -> H(U1(e, e), U1(e, e))) ---------------------------------------- (276) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (277) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(e, e), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, e), d),A^1 -> H(U1(e, e), d)) ---------------------------------------- (278) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(U1(e, e), d) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (279) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (280) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), b) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (281) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(e), b) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, e), b),A^1 -> H(U1(e, e), b)) (A^1 -> H(f(e), d),A^1 -> H(f(e), d)) (A^1 -> H(f(e), e),A^1 -> H(f(e), e)) ---------------------------------------- (282) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), U1(d, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (283) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(e), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, e), U1(d, e)),A^1 -> H(U1(e, e), U1(d, e))) (A^1 -> H(f(e), e),A^1 -> H(f(e), e)) ---------------------------------------- (284) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), U1(e, d)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (285) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(e), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, e), U1(e, d)),A^1 -> H(U1(e, e), U1(e, d))) ---------------------------------------- (286) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(e, e), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (287) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (288) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), d) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (289) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(e), d) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, e), d),A^1 -> H(U1(e, e), d)) ---------------------------------------- (290) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(e, e), d) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (291) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (292) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(a), e) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (293) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(a), e) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(a, a), e),A^1 -> H(U1(a, a), e)) (A^1 -> H(f(d), e),A^1 -> H(f(d), e)) (A^1 -> H(f(e), e),A^1 -> H(f(e), e)) ---------------------------------------- (294) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(d), d) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (295) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(d), d) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, d), d),A^1 -> H(U1(d, d), d)) ---------------------------------------- (296) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(e, U1(b, b)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (297) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(e, U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(e, U1(d, b)),A^1 -> H(e, U1(d, b))) (A^1 -> H(e, U1(e, b)),A^1 -> H(e, U1(e, b))) (A^1 -> H(e, U1(b, d)),A^1 -> H(e, U1(b, d))) (A^1 -> H(e, U1(b, e)),A^1 -> H(e, U1(b, e))) ---------------------------------------- (298) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(e, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (299) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (300) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(e, f(d)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (301) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(e, f(d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(e, U1(d, d)),A^1 -> H(e, U1(d, d))) ---------------------------------------- (302) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(e, f(e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (303) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(e, f(e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(e, U1(e, e)),A^1 -> H(e, U1(e, e))) ---------------------------------------- (304) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(e, U1(e, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (305) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (306) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(a, U1(d, b)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (307) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(a, U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(d, U1(d, b)),A^1 -> H(d, U1(d, b))) (A^1 -> H(e, U1(d, b)),A^1 -> H(e, U1(d, b))) (A^1 -> H(a, b),A^1 -> H(a, b)) (A^1 -> H(a, U1(d, d)),A^1 -> H(a, U1(d, d))) (A^1 -> H(a, U1(d, e)),A^1 -> H(a, U1(d, e))) ---------------------------------------- (308) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(a, U1(e, b)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (309) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(a, U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(d, U1(e, b)),A^1 -> H(d, U1(e, b))) (A^1 -> H(e, U1(e, b)),A^1 -> H(e, U1(e, b))) (A^1 -> H(a, U1(e, d)),A^1 -> H(a, U1(e, d))) (A^1 -> H(a, U1(e, e)),A^1 -> H(a, U1(e, e))) ---------------------------------------- (310) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(d, U1(e, b)) A^1 -> H(e, U1(e, b)) A^1 -> H(a, U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (311) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. ---------------------------------------- (312) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(a, U1(b, d)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (313) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(a, U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(d, U1(b, d)),A^1 -> H(d, U1(b, d))) (A^1 -> H(e, U1(b, d)),A^1 -> H(e, U1(b, d))) (A^1 -> H(a, U1(d, d)),A^1 -> H(a, U1(d, d))) (A^1 -> H(a, U1(e, d)),A^1 -> H(a, U1(e, d))) ---------------------------------------- (314) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(a, U1(b, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (315) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(a, U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(d, U1(b, e)),A^1 -> H(d, U1(b, e))) (A^1 -> H(e, U1(b, e)),A^1 -> H(e, U1(b, e))) (A^1 -> H(a, U1(d, e)),A^1 -> H(a, U1(d, e))) (A^1 -> H(a, U1(e, e)),A^1 -> H(a, U1(e, e))) ---------------------------------------- (316) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(a, U1(d, d)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (317) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(a, U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(d, U1(d, d)),A^1 -> H(d, U1(d, d))) (A^1 -> H(e, U1(d, d)),A^1 -> H(e, U1(d, d))) (A^1 -> H(a, d),A^1 -> H(a, d)) ---------------------------------------- (318) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(a, U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (319) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(a, U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(d, U1(e, e)),A^1 -> H(d, U1(e, e))) (A^1 -> H(e, U1(e, e)),A^1 -> H(e, U1(e, e))) ---------------------------------------- (320) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(d, U1(e, e)) A^1 -> H(e, U1(e, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (321) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. ---------------------------------------- (322) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, e), U1(d, b)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (323) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, e), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(e, U1(d, b)),A^1 -> H(e, U1(d, b))) (A^1 -> H(U1(d, e), b),A^1 -> H(U1(d, e), b)) (A^1 -> H(U1(d, e), U1(d, d)),A^1 -> H(U1(d, e), U1(d, d))) (A^1 -> H(U1(d, e), U1(d, e)),A^1 -> H(U1(d, e), U1(d, e))) ---------------------------------------- (324) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, e), U1(e, b)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (325) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, e), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(e, U1(e, b)),A^1 -> H(e, U1(e, b))) (A^1 -> H(U1(d, e), U1(e, d)),A^1 -> H(U1(d, e), U1(e, d))) (A^1 -> H(U1(d, e), U1(e, e)),A^1 -> H(U1(d, e), U1(e, e))) ---------------------------------------- (326) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(e, U1(e, b)) A^1 -> H(U1(d, e), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (327) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (328) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, e), U1(b, d)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (329) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, e), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(e, U1(b, d)),A^1 -> H(e, U1(b, d))) (A^1 -> H(U1(d, e), U1(d, d)),A^1 -> H(U1(d, e), U1(d, d))) (A^1 -> H(U1(d, e), U1(e, d)),A^1 -> H(U1(d, e), U1(e, d))) ---------------------------------------- (330) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, e), U1(b, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (331) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, e), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(e, U1(b, e)),A^1 -> H(e, U1(b, e))) (A^1 -> H(U1(d, e), U1(d, e)),A^1 -> H(U1(d, e), U1(d, e))) (A^1 -> H(U1(d, e), U1(e, e)),A^1 -> H(U1(d, e), U1(e, e))) ---------------------------------------- (332) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, e), U1(d, d)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (333) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, e), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(e, U1(d, d)),A^1 -> H(e, U1(d, d))) (A^1 -> H(U1(d, e), d),A^1 -> H(U1(d, e), d)) ---------------------------------------- (334) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (335) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, e), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(e, U1(e, e)),A^1 -> H(e, U1(e, e))) ---------------------------------------- (336) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(e, U1(e, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (337) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (338) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, a), b) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (339) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, a), b) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(a, b),A^1 -> H(a, b)) (A^1 -> H(U1(d, d), b),A^1 -> H(U1(d, d), b)) (A^1 -> H(U1(d, e), b),A^1 -> H(U1(d, e), b)) (A^1 -> H(U1(d, a), d),A^1 -> H(U1(d, a), d)) (A^1 -> H(U1(d, a), e),A^1 -> H(U1(d, a), e)) ---------------------------------------- (340) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, a), U1(d, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (341) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, a), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(a, U1(d, e)),A^1 -> H(a, U1(d, e))) (A^1 -> H(U1(d, d), U1(d, e)),A^1 -> H(U1(d, d), U1(d, e))) (A^1 -> H(U1(d, e), U1(d, e)),A^1 -> H(U1(d, e), U1(d, e))) (A^1 -> H(U1(d, a), e),A^1 -> H(U1(d, a), e)) ---------------------------------------- (342) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, a), U1(e, d)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (343) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, a), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(a, U1(e, d)),A^1 -> H(a, U1(e, d))) (A^1 -> H(U1(d, d), U1(e, d)),A^1 -> H(U1(d, d), U1(e, d))) (A^1 -> H(U1(d, e), U1(e, d)),A^1 -> H(U1(d, e), U1(e, d))) ---------------------------------------- (344) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, a), d) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (345) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, a), d) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(a, d),A^1 -> H(a, d)) (A^1 -> H(U1(d, d), d),A^1 -> H(U1(d, d), d)) (A^1 -> H(U1(d, e), d),A^1 -> H(U1(d, e), d)) ---------------------------------------- (346) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(e, d), U1(d, b)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (347) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(e, d), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, d), b),A^1 -> H(U1(e, d), b)) (A^1 -> H(U1(e, d), U1(d, d)),A^1 -> H(U1(e, d), U1(d, d))) (A^1 -> H(U1(e, d), U1(d, e)),A^1 -> H(U1(e, d), U1(d, e))) ---------------------------------------- (348) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(e, d), U1(e, b)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (349) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(e, d), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, d), U1(e, d)),A^1 -> H(U1(e, d), U1(e, d))) (A^1 -> H(U1(e, d), U1(e, e)),A^1 -> H(U1(e, d), U1(e, e))) ---------------------------------------- (350) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(e, d), U1(e, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (351) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (352) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(e, d), U1(b, d)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (353) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(e, d), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, d), U1(d, d)),A^1 -> H(U1(e, d), U1(d, d))) (A^1 -> H(U1(e, d), U1(e, d)),A^1 -> H(U1(e, d), U1(e, d))) ---------------------------------------- (354) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(e, d), U1(b, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (355) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(e, d), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, d), U1(d, e)),A^1 -> H(U1(e, d), U1(d, e))) (A^1 -> H(U1(e, d), U1(e, e)),A^1 -> H(U1(e, d), U1(e, e))) ---------------------------------------- (356) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(e, d), U1(e, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (357) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (358) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(e, d), U1(d, d)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (359) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(e, d), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, d), d),A^1 -> H(U1(e, d), d)) ---------------------------------------- (360) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(e, d), d) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (361) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (362) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(e, a), b) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (363) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(e, a), b) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, d), b),A^1 -> H(U1(e, d), b)) (A^1 -> H(U1(e, e), b),A^1 -> H(U1(e, e), b)) (A^1 -> H(U1(e, a), d),A^1 -> H(U1(e, a), d)) (A^1 -> H(U1(e, a), e),A^1 -> H(U1(e, a), e)) ---------------------------------------- (364) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(e, a), d) A^1 -> H(U1(e, a), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (365) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. ---------------------------------------- (366) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(e, a), U1(d, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (367) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(e, a), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, d), U1(d, e)),A^1 -> H(U1(e, d), U1(d, e))) (A^1 -> H(U1(e, e), U1(d, e)),A^1 -> H(U1(e, e), U1(d, e))) (A^1 -> H(U1(e, a), e),A^1 -> H(U1(e, a), e)) ---------------------------------------- (368) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(e, a), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (369) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (370) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(e, a), U1(e, d)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (371) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(e, a), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, d), U1(e, d)),A^1 -> H(U1(e, d), U1(e, d))) (A^1 -> H(U1(e, e), U1(e, d)),A^1 -> H(U1(e, e), U1(e, d))) ---------------------------------------- (372) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(e, e), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (373) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (374) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(a, d), b) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (375) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, d), b) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, d), b),A^1 -> H(U1(d, d), b)) (A^1 -> H(U1(e, d), b),A^1 -> H(U1(e, d), b)) (A^1 -> H(U1(a, d), d),A^1 -> H(U1(a, d), d)) (A^1 -> H(U1(a, d), e),A^1 -> H(U1(a, d), e)) ---------------------------------------- (376) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(a, d), U1(d, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (377) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, d), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, d), U1(d, e)),A^1 -> H(U1(d, d), U1(d, e))) (A^1 -> H(U1(e, d), U1(d, e)),A^1 -> H(U1(e, d), U1(d, e))) (A^1 -> H(U1(a, d), e),A^1 -> H(U1(a, d), e)) ---------------------------------------- (378) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(a, d), U1(e, d)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (379) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, d), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, d), U1(e, d)),A^1 -> H(U1(d, d), U1(e, d))) (A^1 -> H(U1(e, d), U1(e, d)),A^1 -> H(U1(e, d), U1(e, d))) ---------------------------------------- (380) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(a, d), d) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (381) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, d), d) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, d), d),A^1 -> H(U1(d, d), d)) (A^1 -> H(U1(e, d), d),A^1 -> H(U1(e, d), d)) ---------------------------------------- (382) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(e, d), d) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (383) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (384) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(a, e), b) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (385) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, e), b) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, e), b),A^1 -> H(U1(d, e), b)) (A^1 -> H(U1(e, e), b),A^1 -> H(U1(e, e), b)) (A^1 -> H(U1(a, e), d),A^1 -> H(U1(a, e), d)) (A^1 -> H(U1(a, e), e),A^1 -> H(U1(a, e), e)) ---------------------------------------- (386) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(a, e), U1(d, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (387) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, e), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, e), U1(d, e)),A^1 -> H(U1(d, e), U1(d, e))) (A^1 -> H(U1(e, e), U1(d, e)),A^1 -> H(U1(e, e), U1(d, e))) (A^1 -> H(U1(a, e), e),A^1 -> H(U1(a, e), e)) ---------------------------------------- (388) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(a, e), U1(e, d)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (389) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, e), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, e), U1(e, d)),A^1 -> H(U1(d, e), U1(e, d))) (A^1 -> H(U1(e, e), U1(e, d)),A^1 -> H(U1(e, e), U1(e, d))) ---------------------------------------- (390) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(U1(e, e), U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (391) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (392) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(a, e), d) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (393) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, e), d) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, e), d),A^1 -> H(U1(d, e), d)) (A^1 -> H(U1(e, e), d),A^1 -> H(U1(e, e), d)) ---------------------------------------- (394) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(U1(e, e), d) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (395) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (396) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(a, a), e) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (397) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, a), e) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, a), e),A^1 -> H(U1(d, a), e)) (A^1 -> H(U1(e, a), e),A^1 -> H(U1(e, a), e)) (A^1 -> H(U1(a, d), e),A^1 -> H(U1(a, d), e)) (A^1 -> H(U1(a, e), e),A^1 -> H(U1(a, e), e)) ---------------------------------------- (398) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(U1(e, a), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (399) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (400) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(d, U1(d, b)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (401) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(d, U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(d, b),A^1 -> H(d, b)) (A^1 -> H(d, U1(d, d)),A^1 -> H(d, U1(d, d))) (A^1 -> H(d, U1(d, e)),A^1 -> H(d, U1(d, e))) ---------------------------------------- (402) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(d, U1(b, d)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (403) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(d, U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(d, U1(d, d)),A^1 -> H(d, U1(d, d))) (A^1 -> H(d, U1(e, d)),A^1 -> H(d, U1(e, d))) ---------------------------------------- (404) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (405) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (406) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(d, U1(b, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (407) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(d, U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(d, U1(d, e)),A^1 -> H(d, U1(d, e))) (A^1 -> H(d, U1(e, e)),A^1 -> H(d, U1(e, e))) ---------------------------------------- (408) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, U1(e, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (409) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (410) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(d, U1(d, d)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (411) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(d, U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(d, d),A^1 -> H(d, d)) ---------------------------------------- (412) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, d), b) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (413) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, d), b) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(d, b),A^1 -> H(d, b)) (A^1 -> H(U1(d, d), d),A^1 -> H(U1(d, d), d)) (A^1 -> H(U1(d, d), e),A^1 -> H(U1(d, d), e)) ---------------------------------------- (414) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, d), U1(d, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (415) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, d), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(d, U1(d, e)),A^1 -> H(d, U1(d, e))) (A^1 -> H(U1(d, d), e),A^1 -> H(U1(d, d), e)) ---------------------------------------- (416) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, d), U1(e, d)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (417) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, d), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(d, U1(e, d)),A^1 -> H(d, U1(e, d))) ---------------------------------------- (418) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) A^1 -> H(d, U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (419) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (420) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(d), e) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (421) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(d), e) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, d), e),A^1 -> H(U1(d, d), e)) ---------------------------------------- (422) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(e, e), b) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (423) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(e, e), b) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, e), d),A^1 -> H(U1(e, e), d)) (A^1 -> H(U1(e, e), e),A^1 -> H(U1(e, e), e)) ---------------------------------------- (424) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) A^1 -> H(U1(e, e), d) A^1 -> H(U1(e, e), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (425) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. ---------------------------------------- (426) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(e, e), U1(d, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (427) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(e, e), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, e), e),A^1 -> H(U1(e, e), e)) ---------------------------------------- (428) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) A^1 -> H(U1(e, e), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (429) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (430) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(f(e), e) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (431) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(f(e), e) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, e), e),A^1 -> H(U1(e, e), e)) ---------------------------------------- (432) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) A^1 -> H(U1(e, e), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (433) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (434) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, d), d) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (435) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, d), d) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(d, d),A^1 -> H(d, d)) ---------------------------------------- (436) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(e, U1(d, b)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (437) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(e, U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(e, b),A^1 -> H(e, b)) (A^1 -> H(e, U1(d, d)),A^1 -> H(e, U1(d, d))) (A^1 -> H(e, U1(d, e)),A^1 -> H(e, U1(d, e))) ---------------------------------------- (438) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(e, U1(b, d)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) A^1 -> H(e, b) A^1 -> H(e, U1(d, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (439) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(e, U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(e, U1(d, d)),A^1 -> H(e, U1(d, d))) (A^1 -> H(e, U1(e, d)),A^1 -> H(e, U1(e, d))) ---------------------------------------- (440) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) A^1 -> H(e, b) A^1 -> H(e, U1(d, e)) A^1 -> H(e, U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (441) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (442) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(e, U1(b, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) A^1 -> H(e, b) A^1 -> H(e, U1(d, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (443) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(e, U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(e, U1(d, e)),A^1 -> H(e, U1(d, e))) (A^1 -> H(e, U1(e, e)),A^1 -> H(e, U1(e, e))) ---------------------------------------- (444) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) A^1 -> H(e, b) A^1 -> H(e, U1(d, e)) A^1 -> H(e, U1(e, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (445) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (446) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(e, U1(d, d)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) A^1 -> H(e, b) A^1 -> H(e, U1(d, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (447) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(e, U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(e, d),A^1 -> H(e, d)) ---------------------------------------- (448) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) A^1 -> H(e, b) A^1 -> H(e, U1(d, e)) A^1 -> H(e, d) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (449) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (450) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(a, b) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) A^1 -> H(e, b) A^1 -> H(e, U1(d, e)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (451) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(a, b) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(d, b),A^1 -> H(d, b)) (A^1 -> H(e, b),A^1 -> H(e, b)) (A^1 -> H(a, d),A^1 -> H(a, d)) (A^1 -> H(a, e),A^1 -> H(a, e)) ---------------------------------------- (452) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(a, U1(d, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) A^1 -> H(e, b) A^1 -> H(e, U1(d, e)) A^1 -> H(a, e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (453) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(a, U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(d, U1(d, e)),A^1 -> H(d, U1(d, e))) (A^1 -> H(e, U1(d, e)),A^1 -> H(e, U1(d, e))) (A^1 -> H(a, e),A^1 -> H(a, e)) ---------------------------------------- (454) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(a, U1(e, d)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) A^1 -> H(e, b) A^1 -> H(e, U1(d, e)) A^1 -> H(a, e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (455) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(a, U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(d, U1(e, d)),A^1 -> H(d, U1(e, d))) (A^1 -> H(e, U1(e, d)),A^1 -> H(e, U1(e, d))) ---------------------------------------- (456) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) A^1 -> H(e, b) A^1 -> H(e, U1(d, e)) A^1 -> H(a, e) A^1 -> H(d, U1(e, d)) A^1 -> H(e, U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (457) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. ---------------------------------------- (458) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(a, d) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) A^1 -> H(e, b) A^1 -> H(e, U1(d, e)) A^1 -> H(a, e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (459) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(a, d) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(d, d),A^1 -> H(d, d)) (A^1 -> H(e, d),A^1 -> H(e, d)) ---------------------------------------- (460) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) A^1 -> H(e, b) A^1 -> H(e, U1(d, e)) A^1 -> H(a, e) A^1 -> H(e, d) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (461) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (462) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, e), b) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) A^1 -> H(e, b) A^1 -> H(e, U1(d, e)) A^1 -> H(a, e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (463) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, e), b) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(e, b),A^1 -> H(e, b)) (A^1 -> H(U1(d, e), d),A^1 -> H(U1(d, e), d)) (A^1 -> H(U1(d, e), e),A^1 -> H(U1(d, e), e)) ---------------------------------------- (464) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), U1(e, d)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) A^1 -> H(e, b) A^1 -> H(e, U1(d, e)) A^1 -> H(a, e) A^1 -> H(U1(d, e), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (465) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, e), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(e, U1(e, d)),A^1 -> H(e, U1(e, d))) ---------------------------------------- (466) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) A^1 -> H(e, b) A^1 -> H(e, U1(d, e)) A^1 -> H(a, e) A^1 -> H(U1(d, e), e) A^1 -> H(e, U1(e, d)) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (467) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (468) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, e), d) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) A^1 -> H(e, b) A^1 -> H(e, U1(d, e)) A^1 -> H(a, e) A^1 -> H(U1(d, e), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (469) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, e), d) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(e, d),A^1 -> H(e, d)) ---------------------------------------- (470) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) A^1 -> H(e, b) A^1 -> H(e, U1(d, e)) A^1 -> H(a, e) A^1 -> H(U1(d, e), e) A^1 -> H(e, d) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (471) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (472) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(d, a), e) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) A^1 -> H(e, b) A^1 -> H(e, U1(d, e)) A^1 -> H(a, e) A^1 -> H(U1(d, e), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (473) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, a), e) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(a, e),A^1 -> H(a, e)) (A^1 -> H(U1(d, d), e),A^1 -> H(U1(d, d), e)) (A^1 -> H(U1(d, e), e),A^1 -> H(U1(d, e), e)) ---------------------------------------- (474) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(e, d), b) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) A^1 -> H(e, b) A^1 -> H(e, U1(d, e)) A^1 -> H(a, e) A^1 -> H(U1(d, e), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (475) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(e, d), b) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, d), d),A^1 -> H(U1(e, d), d)) (A^1 -> H(U1(e, d), e),A^1 -> H(U1(e, d), e)) ---------------------------------------- (476) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) A^1 -> H(e, b) A^1 -> H(e, U1(d, e)) A^1 -> H(a, e) A^1 -> H(U1(d, e), e) A^1 -> H(U1(e, d), d) A^1 -> H(U1(e, d), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (477) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. ---------------------------------------- (478) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(e, d), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) A^1 -> H(e, b) A^1 -> H(e, U1(d, e)) A^1 -> H(a, e) A^1 -> H(U1(d, e), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (479) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(e, d), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(e, d), e),A^1 -> H(U1(e, d), e)) ---------------------------------------- (480) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) A^1 -> H(e, b) A^1 -> H(e, U1(d, e)) A^1 -> H(a, e) A^1 -> H(U1(d, e), e) A^1 -> H(U1(e, d), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (481) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (482) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, d), e) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) A^1 -> H(e, b) A^1 -> H(e, U1(d, e)) A^1 -> H(a, e) A^1 -> H(U1(d, e), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (483) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, d), e) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, d), e),A^1 -> H(U1(d, d), e)) (A^1 -> H(U1(e, d), e),A^1 -> H(U1(e, d), e)) ---------------------------------------- (484) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) A^1 -> H(e, b) A^1 -> H(e, U1(d, e)) A^1 -> H(a, e) A^1 -> H(U1(d, e), e) A^1 -> H(U1(e, d), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (485) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (486) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(U1(a, e), e) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) A^1 -> H(e, b) A^1 -> H(e, U1(d, e)) A^1 -> H(a, e) A^1 -> H(U1(d, e), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (487) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(a, e), e) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(U1(d, e), e),A^1 -> H(U1(d, e), e)) (A^1 -> H(U1(e, e), e),A^1 -> H(U1(e, e), e)) ---------------------------------------- (488) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) A^1 -> H(e, b) A^1 -> H(e, U1(d, e)) A^1 -> H(a, e) A^1 -> H(U1(d, e), e) A^1 -> H(U1(e, e), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (489) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (490) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(d, b) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) A^1 -> H(e, b) A^1 -> H(e, U1(d, e)) A^1 -> H(a, e) A^1 -> H(U1(d, e), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (491) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(d, b) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(d, d),A^1 -> H(d, d)) (A^1 -> H(d, e),A^1 -> H(d, e)) ---------------------------------------- (492) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) A^1 -> H(e, b) A^1 -> H(e, U1(d, e)) A^1 -> H(a, e) A^1 -> H(U1(d, e), e) A^1 -> H(d, e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (493) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (494) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(d, U1(d, e)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) A^1 -> H(e, b) A^1 -> H(e, U1(d, e)) A^1 -> H(a, e) A^1 -> H(U1(d, e), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (495) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(d, U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(d, e),A^1 -> H(d, e)) ---------------------------------------- (496) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) A^1 -> H(e, b) A^1 -> H(e, U1(d, e)) A^1 -> H(a, e) A^1 -> H(U1(d, e), e) A^1 -> H(d, e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (497) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (498) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(d, d) A^1 -> H(U1(d, d), e) A^1 -> H(e, b) A^1 -> H(e, U1(d, e)) A^1 -> H(a, e) A^1 -> H(U1(d, e), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (499) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, d), e) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(d, e),A^1 -> H(d, e)) ---------------------------------------- (500) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(d, d) A^1 -> H(e, b) A^1 -> H(e, U1(d, e)) A^1 -> H(a, e) A^1 -> H(U1(d, e), e) A^1 -> H(d, e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (501) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (502) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(d, d) A^1 -> H(e, b) A^1 -> H(e, U1(d, e)) A^1 -> H(a, e) A^1 -> H(U1(d, e), e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (503) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(e, b) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(e, d),A^1 -> H(e, d)) (A^1 -> H(e, e),A^1 -> H(e, e)) ---------------------------------------- (504) Obligation: Q DP problem: The TRS P consists of the following rules: A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(d, d) A^1 -> H(e, U1(d, e)) A^1 -> H(a, e) A^1 -> H(U1(d, e), e) A^1 -> H(e, d) A^1 -> H(e, e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (505) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (506) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(d, d) A^1 -> H(e, U1(d, e)) A^1 -> H(a, e) A^1 -> H(U1(d, e), e) A^1 -> H(e, e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (507) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(e, U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(e, e),A^1 -> H(e, e)) ---------------------------------------- (508) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(d, d) A^1 -> H(a, e) A^1 -> H(U1(d, e), e) A^1 -> H(e, e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (509) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(a, e) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(d, e),A^1 -> H(d, e)) (A^1 -> H(e, e),A^1 -> H(e, e)) ---------------------------------------- (510) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x) G(d, e) -> A^1 A^1 -> H(f(d), f(d)) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(d, d) A^1 -> H(U1(d, e), e) A^1 -> H(e, e) A^1 -> H(d, e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (511) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (512) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(d, d) A^1 -> H(U1(d, e), e) A^1 -> H(e, e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (513) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^1 -> H(U1(d, e), e) at position [] we obtained the following new rules [LPAR04]: (A^1 -> H(e, e),A^1 -> H(e, e)) ---------------------------------------- (514) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(d, d) A^1 -> H(e, e) The TRS R consists of the following rules: a -> d a -> e f(x) -> U1(x, x) b -> d b -> e U1(d, x) -> x Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (515) UsableRulesReductionPairsProof (EQUIVALENT) By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well. No dependency pairs are removed. The following rules are removed from R: a -> d a -> e b -> d b -> e Used ordering: POLO with Polynomial interpretation [POLO]: POL(A^1) = 0 POL(G(x_1, x_2)) = x_1 + x_2 POL(H(x_1, x_2)) = x_1 + x_2 POL(U1(x_1, x_2)) = x_1 + x_2 POL(d) = 0 POL(e) = 0 POL(f(x_1)) = 2*x_1 ---------------------------------------- (516) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(d, d) A^1 -> H(e, e) The TRS R consists of the following rules: U1(d, x) -> x f(x) -> U1(x, x) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (517) MNOCProof (EQUIVALENT) We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R. ---------------------------------------- (518) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, e) -> A^1 A^1 -> H(f(d), f(d)) H(x, x) -> G(x, x) A^1 -> H(f(e), f(e)) A^1 -> H(U1(d, d), U1(d, d)) A^1 -> H(U1(e, e), U1(e, e)) A^1 -> H(U1(d, e), U1(d, e)) A^1 -> H(U1(e, d), U1(e, d)) A^1 -> H(d, d) A^1 -> H(e, e) The TRS R consists of the following rules: U1(d, x) -> x f(x) -> U1(x, x) The set Q consists of the following terms: U1(d, x0) f(x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (519) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 10 less nodes. ---------------------------------------- (520) TRUE