171.66/44.50 MAYBE 172.08/44.54 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 172.08/44.54 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 172.08/44.54 172.08/44.54 172.08/44.54 Left Termination of the query pattern 172.08/44.54 172.08/44.54 reach(g,g,g,g) 172.08/44.54 172.08/44.54 w.r.t. the given Prolog program could not be shown: 172.08/44.54 172.08/44.54 (0) Prolog 172.08/44.54 (1) PrologToPiTRSProof [SOUND, 0 ms] 172.08/44.54 (2) PiTRS 172.08/44.54 (3) DependencyPairsProof [EQUIVALENT, 4 ms] 172.08/44.54 (4) PiDP 172.08/44.54 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 172.08/44.54 (6) AND 172.08/44.54 (7) PiDP 172.08/44.54 (8) UsableRulesProof [EQUIVALENT, 0 ms] 172.08/44.54 (9) PiDP 172.08/44.54 (10) PiDPToQDPProof [SOUND, 13 ms] 172.08/44.54 (11) QDP 172.08/44.54 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 172.08/44.54 (13) YES 172.08/44.54 (14) PiDP 172.08/44.54 (15) UsableRulesProof [EQUIVALENT, 0 ms] 172.08/44.54 (16) PiDP 172.08/44.54 (17) PiDPToQDPProof [EQUIVALENT, 0 ms] 172.08/44.54 (18) QDP 172.08/44.54 (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] 172.08/44.54 (20) YES 172.08/44.54 (21) PiDP 172.08/44.54 (22) UsableRulesProof [EQUIVALENT, 0 ms] 172.08/44.54 (23) PiDP 172.08/44.54 (24) PiDPToQDPProof [SOUND, 0 ms] 172.08/44.54 (25) QDP 172.08/44.54 (26) TransformationProof [SOUND, 0 ms] 172.08/44.54 (27) QDP 172.08/44.54 (28) TransformationProof [SOUND, 0 ms] 172.08/44.54 (29) QDP 172.08/44.54 (30) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (31) QDP 172.08/44.54 (32) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (33) QDP 172.08/44.54 (34) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (35) QDP 172.08/44.54 (36) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (37) QDP 172.08/44.54 (38) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (39) QDP 172.08/44.54 (40) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (41) QDP 172.08/44.54 (42) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (43) QDP 172.08/44.54 (44) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (45) QDP 172.08/44.54 (46) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (47) QDP 172.08/44.54 (48) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (49) QDP 172.08/44.54 (50) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (51) QDP 172.08/44.54 (52) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (53) QDP 172.08/44.54 (54) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (55) QDP 172.08/44.54 (56) TransformationProof [EQUIVALENT, 3 ms] 172.08/44.54 (57) QDP 172.08/44.54 (58) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (59) QDP 172.08/44.54 (60) TransformationProof [EQUIVALENT, 8 ms] 172.08/44.54 (61) QDP 172.08/44.54 (62) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (63) QDP 172.08/44.54 (64) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (65) QDP 172.08/44.54 (66) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (67) QDP 172.08/44.54 (68) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (69) QDP 172.08/44.54 (70) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (71) QDP 172.08/44.54 (72) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (73) QDP 172.08/44.54 (74) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (75) QDP 172.08/44.54 (76) PrologToPiTRSProof [SOUND, 0 ms] 172.08/44.54 (77) PiTRS 172.08/44.54 (78) DependencyPairsProof [EQUIVALENT, 3 ms] 172.08/44.54 (79) PiDP 172.08/44.54 (80) DependencyGraphProof [EQUIVALENT, 0 ms] 172.08/44.54 (81) AND 172.08/44.54 (82) PiDP 172.08/44.54 (83) UsableRulesProof [EQUIVALENT, 0 ms] 172.08/44.54 (84) PiDP 172.08/44.54 (85) PiDPToQDPProof [SOUND, 2 ms] 172.08/44.54 (86) QDP 172.08/44.54 (87) QDPSizeChangeProof [EQUIVALENT, 0 ms] 172.08/44.54 (88) YES 172.08/44.54 (89) PiDP 172.08/44.54 (90) UsableRulesProof [EQUIVALENT, 0 ms] 172.08/44.54 (91) PiDP 172.08/44.54 (92) PiDPToQDPProof [EQUIVALENT, 0 ms] 172.08/44.54 (93) QDP 172.08/44.54 (94) QDPSizeChangeProof [EQUIVALENT, 0 ms] 172.08/44.54 (95) YES 172.08/44.54 (96) PiDP 172.08/44.54 (97) UsableRulesProof [EQUIVALENT, 0 ms] 172.08/44.54 (98) PiDP 172.08/44.54 (99) PiDPToQDPProof [SOUND, 2 ms] 172.08/44.54 (100) QDP 172.08/44.54 (101) TransformationProof [SOUND, 0 ms] 172.08/44.54 (102) QDP 172.08/44.54 (103) TransformationProof [SOUND, 0 ms] 172.08/44.54 (104) QDP 172.08/44.54 (105) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (106) QDP 172.08/44.54 (107) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (108) QDP 172.08/44.54 (109) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (110) QDP 172.08/44.54 (111) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (112) QDP 172.08/44.54 (113) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (114) QDP 172.08/44.54 (115) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (116) QDP 172.08/44.54 (117) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (118) QDP 172.08/44.54 (119) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (120) QDP 172.08/44.54 (121) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (122) QDP 172.08/44.54 (123) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (124) QDP 172.08/44.54 (125) TransformationProof [EQUIVALENT, 1 ms] 172.08/44.54 (126) QDP 172.08/44.54 (127) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (128) QDP 172.08/44.54 (129) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (130) QDP 172.08/44.54 (131) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (132) QDP 172.08/44.54 (133) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (134) QDP 172.08/44.54 (135) TransformationProof [EQUIVALENT, 10 ms] 172.08/44.54 (136) QDP 172.08/44.54 (137) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (138) QDP 172.08/44.54 (139) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (140) QDP 172.08/44.54 (141) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (142) QDP 172.08/44.54 (143) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (144) QDP 172.08/44.54 (145) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (146) QDP 172.08/44.54 (147) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (148) QDP 172.08/44.54 (149) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (150) QDP 172.08/44.54 (151) PrologToDTProblemTransformerProof [SOUND, 65 ms] 172.08/44.54 (152) TRIPLES 172.08/44.54 (153) TriplesToPiDPProof [SOUND, 0 ms] 172.08/44.54 (154) PiDP 172.08/44.54 (155) DependencyGraphProof [EQUIVALENT, 0 ms] 172.08/44.54 (156) AND 172.08/44.54 (157) PiDP 172.08/44.54 (158) UsableRulesProof [EQUIVALENT, 0 ms] 172.08/44.54 (159) PiDP 172.08/44.54 (160) PiDPToQDPProof [EQUIVALENT, 1 ms] 172.08/44.54 (161) QDP 172.08/44.54 (162) QDPSizeChangeProof [EQUIVALENT, 0 ms] 172.08/44.54 (163) YES 172.08/44.54 (164) PiDP 172.08/44.54 (165) UsableRulesProof [EQUIVALENT, 0 ms] 172.08/44.54 (166) PiDP 172.08/44.54 (167) PiDPToQDPProof [SOUND, 0 ms] 172.08/44.54 (168) QDP 172.08/44.54 (169) QDPSizeChangeProof [EQUIVALENT, 0 ms] 172.08/44.54 (170) YES 172.08/44.54 (171) PiDP 172.08/44.54 (172) UsableRulesProof [EQUIVALENT, 0 ms] 172.08/44.54 (173) PiDP 172.08/44.54 (174) PiDPToQDPProof [EQUIVALENT, 0 ms] 172.08/44.54 (175) QDP 172.08/44.54 (176) QDPSizeChangeProof [EQUIVALENT, 0 ms] 172.08/44.54 (177) YES 172.08/44.54 (178) PiDP 172.08/44.54 (179) PiDPToQDPProof [SOUND, 0 ms] 172.08/44.54 (180) QDP 172.08/44.54 (181) TransformationProof [SOUND, 0 ms] 172.08/44.54 (182) QDP 172.08/44.54 (183) TransformationProof [SOUND, 0 ms] 172.08/44.54 (184) QDP 172.08/44.54 (185) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (186) QDP 172.08/44.54 (187) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (188) QDP 172.08/44.54 (189) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (190) QDP 172.08/44.54 (191) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (192) QDP 172.08/44.54 (193) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (194) QDP 172.08/44.54 (195) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (196) QDP 172.08/44.54 (197) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (198) QDP 172.08/44.54 (199) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (200) QDP 172.08/44.54 (201) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (202) QDP 172.08/44.54 (203) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (204) QDP 172.08/44.54 (205) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (206) QDP 172.08/44.54 (207) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (208) QDP 172.08/44.54 (209) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (210) QDP 172.08/44.54 (211) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (212) QDP 172.08/44.54 (213) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (214) QDP 172.08/44.54 (215) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (216) QDP 172.08/44.54 (217) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (218) QDP 172.08/44.54 (219) TransformationProof [EQUIVALENT, 8 ms] 172.08/44.54 (220) QDP 172.08/44.54 (221) TransformationProof [EQUIVALENT, 0 ms] 172.08/44.54 (222) QDP 172.08/44.54 (223) PrologToTRSTransformerProof [SOUND, 39 ms] 172.08/44.54 (224) QTRS 172.08/44.54 (225) DependencyPairsProof [EQUIVALENT, 0 ms] 172.08/44.54 (226) QDP 172.08/44.54 (227) DependencyGraphProof [EQUIVALENT, 0 ms] 172.08/44.54 (228) AND 172.08/44.54 (229) QDP 172.08/44.54 (230) UsableRulesProof [EQUIVALENT, 0 ms] 172.08/44.54 (231) QDP 172.08/44.54 (232) QDPSizeChangeProof [EQUIVALENT, 0 ms] 172.08/44.54 (233) YES 172.08/44.54 (234) QDP 172.08/44.54 (235) UsableRulesProof [EQUIVALENT, 0 ms] 172.08/44.54 (236) QDP 172.08/44.54 (237) QDPSizeChangeProof [EQUIVALENT, 0 ms] 172.08/44.54 (238) YES 172.08/44.54 (239) QDP 172.08/44.54 (240) UsableRulesProof [EQUIVALENT, 0 ms] 172.08/44.54 (241) QDP 172.08/44.54 (242) QDPSizeChangeProof [EQUIVALENT, 0 ms] 172.08/44.54 (243) YES 172.08/44.54 (244) QDP 172.08/44.54 (245) NonLoopProof [COMPLETE, 8572 ms] 172.08/44.54 (246) NO 172.08/44.54 (247) PrologToIRSwTTransformerProof [SOUND, 38 ms] 172.08/44.54 (248) AND 172.08/44.54 (249) IRSwT 172.08/44.54 (250) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] 172.08/44.54 (251) TRUE 172.08/44.54 (252) IRSwT 172.08/44.54 (253) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] 172.08/44.54 (254) IRSwT 172.08/44.54 (255) IntTRSCompressionProof [EQUIVALENT, 28 ms] 172.08/44.54 (256) IRSwT 172.08/44.54 (257) IRSFormatTransformerProof [EQUIVALENT, 1 ms] 172.08/44.54 (258) IRSwT 172.08/44.54 (259) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms] 172.08/44.54 (260) IRSwT 172.08/44.54 (261) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] 172.08/44.54 (262) IRSwT 172.08/44.54 (263) TempFilterProof [SOUND, 3 ms] 172.08/44.54 (264) IRSwT 172.08/44.54 (265) IRSwTToQDPProof [SOUND, 0 ms] 172.08/44.54 (266) QDP 172.08/44.54 (267) QDPSizeChangeProof [EQUIVALENT, 0 ms] 172.08/44.54 (268) YES 172.08/44.54 (269) IRSwT 172.08/44.54 (270) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] 172.08/44.54 (271) IRSwT 172.08/44.54 (272) IntTRSCompressionProof [EQUIVALENT, 4 ms] 172.08/44.54 (273) IRSwT 172.08/44.54 (274) IRSFormatTransformerProof [EQUIVALENT, 0 ms] 172.08/44.54 (275) IRSwT 172.08/44.54 (276) IRSwTTerminationDigraphProof [EQUIVALENT, 2 ms] 172.08/44.54 (277) IRSwT 172.08/44.54 (278) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] 172.08/44.54 (279) IRSwT 172.08/44.54 (280) TempFilterProof [SOUND, 2 ms] 172.08/44.54 (281) IRSwT 172.08/44.54 (282) IRSwTToQDPProof [SOUND, 0 ms] 172.08/44.54 (283) QDP 172.08/44.54 (284) QDPSizeChangeProof [EQUIVALENT, 0 ms] 172.08/44.54 (285) YES 172.08/44.54 (286) IRSwT 172.08/44.54 (287) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] 172.08/44.54 (288) IRSwT 172.08/44.54 (289) IntTRSCompressionProof [EQUIVALENT, 22 ms] 172.08/44.54 (290) IRSwT 172.08/44.54 (291) IRSFormatTransformerProof [EQUIVALENT, 0 ms] 172.08/44.54 (292) IRSwT 172.08/44.54 (293) IRSwTTerminationDigraphProof [EQUIVALENT, 40 ms] 172.08/44.54 (294) IRSwT 172.08/44.54 (295) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] 172.08/44.54 (296) IRSwT 172.08/44.54 (297) IRSwTToIntTRSProof [SOUND, 21 ms] 172.08/44.54 (298) IRSwT 172.08/44.54 (299) IntTRSCompressionProof [EQUIVALENT, 9 ms] 172.08/44.54 (300) IRSwT 172.08/44.54 172.08/44.54 172.08/44.54 ---------------------------------------- 172.08/44.54 172.08/44.54 (0) 172.08/44.54 Obligation: 172.08/44.54 Clauses: 172.08/44.54 172.08/44.54 reach(X, Y, Edges, Visited) :- member(.(X, .(Y, [])), Edges). 172.08/44.54 reach(X, Z, Edges, Visited) :- ','(member1(.(X, .(Y, [])), Edges), ','(member(Y, Visited), reach(Y, Z, Edges, .(Y, Visited)))). 172.08/44.54 member(H, .(H, L)). 172.08/44.54 member(X, .(H, L)) :- member(X, L). 172.08/44.54 member1(H, .(H, L)). 172.08/44.54 member1(X, .(H, L)) :- member1(X, L). 172.08/44.54 172.08/44.54 172.08/44.54 Query: reach(g,g,g,g) 172.08/44.54 ---------------------------------------- 172.08/44.54 172.08/44.54 (1) PrologToPiTRSProof (SOUND) 172.08/44.54 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 172.08/44.54 172.08/44.54 reach_in_4: (b,b,b,b) 172.08/44.54 172.08/44.54 member_in_2: (b,b) 172.08/44.54 172.08/44.54 member1_in_2: (f,b) 172.08/44.54 172.08/44.54 Transforming Prolog into the following Term Rewriting System: 172.08/44.54 172.08/44.54 Pi-finite rewrite system: 172.08/44.54 The TRS R consists of the following rules: 172.08/44.54 172.08/44.54 reach_in_gggg(X, Y, Edges, Visited) -> U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges)) 172.08/44.54 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.54 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.54 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.54 U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_gggg(X, Y, Edges, Visited) 172.08/44.54 reach_in_gggg(X, Z, Edges, Visited) -> U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) 172.08/44.54 member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.54 member1_in_ag(X, .(H, L)) -> U6_ag(X, H, L, member1_in_ag(X, L)) 172.08/44.54 U6_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.54 U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 172.08/44.54 U3_gggg(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> U4_gggg(X, Z, Edges, Visited, reach_in_gggg(Y, Z, Edges, .(Y, Visited))) 172.08/44.54 U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) -> reach_out_gggg(X, Z, Edges, Visited) 172.08/44.54 172.08/44.54 The argument filtering Pi contains the following mapping: 172.08/44.54 reach_in_gggg(x1, x2, x3, x4) = reach_in_gggg(x1, x2, x3, x4) 172.08/44.54 172.08/44.54 U1_gggg(x1, x2, x3, x4, x5) = U1_gggg(x1, x2, x3, x4, x5) 172.08/44.54 172.08/44.54 member_in_gg(x1, x2) = member_in_gg(x1, x2) 172.08/44.54 172.08/44.54 .(x1, x2) = .(x1, x2) 172.08/44.54 172.08/44.54 member_out_gg(x1, x2) = member_out_gg(x1, x2) 172.08/44.54 172.08/44.54 U5_gg(x1, x2, x3, x4) = U5_gg(x1, x2, x3, x4) 172.08/44.54 172.08/44.54 [] = [] 172.08/44.54 172.08/44.54 reach_out_gggg(x1, x2, x3, x4) = reach_out_gggg(x1, x2, x3, x4) 172.08/44.54 172.08/44.54 U2_gggg(x1, x2, x3, x4, x5) = U2_gggg(x1, x2, x3, x4, x5) 172.08/44.54 172.08/44.54 member1_in_ag(x1, x2) = member1_in_ag(x2) 172.08/44.54 172.08/44.54 member1_out_ag(x1, x2) = member1_out_ag(x1, x2) 172.08/44.54 172.08/44.54 U6_ag(x1, x2, x3, x4) = U6_ag(x2, x3, x4) 172.08/44.54 172.08/44.54 U3_gggg(x1, x2, x3, x4, x5, x6) = U3_gggg(x1, x2, x3, x4, x5, x6) 172.08/44.54 172.08/44.54 U4_gggg(x1, x2, x3, x4, x5) = U4_gggg(x1, x2, x3, x4, x5) 172.08/44.54 172.08/44.54 172.08/44.54 172.08/44.54 172.08/44.54 172.08/44.54 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 172.08/44.54 172.08/44.54 172.08/44.54 172.08/44.54 ---------------------------------------- 172.08/44.54 172.08/44.54 (2) 172.08/44.54 Obligation: 172.08/44.54 Pi-finite rewrite system: 172.08/44.54 The TRS R consists of the following rules: 172.08/44.54 172.08/44.54 reach_in_gggg(X, Y, Edges, Visited) -> U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges)) 172.08/44.54 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.54 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.54 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.54 U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_gggg(X, Y, Edges, Visited) 172.08/44.54 reach_in_gggg(X, Z, Edges, Visited) -> U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) 172.08/44.54 member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.54 member1_in_ag(X, .(H, L)) -> U6_ag(X, H, L, member1_in_ag(X, L)) 172.08/44.54 U6_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.54 U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 172.08/44.54 U3_gggg(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> U4_gggg(X, Z, Edges, Visited, reach_in_gggg(Y, Z, Edges, .(Y, Visited))) 172.08/44.54 U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) -> reach_out_gggg(X, Z, Edges, Visited) 172.08/44.54 172.08/44.54 The argument filtering Pi contains the following mapping: 172.08/44.54 reach_in_gggg(x1, x2, x3, x4) = reach_in_gggg(x1, x2, x3, x4) 172.08/44.54 172.08/44.54 U1_gggg(x1, x2, x3, x4, x5) = U1_gggg(x1, x2, x3, x4, x5) 172.08/44.54 172.08/44.54 member_in_gg(x1, x2) = member_in_gg(x1, x2) 172.08/44.54 172.08/44.54 .(x1, x2) = .(x1, x2) 172.08/44.54 172.08/44.54 member_out_gg(x1, x2) = member_out_gg(x1, x2) 172.08/44.54 172.08/44.54 U5_gg(x1, x2, x3, x4) = U5_gg(x1, x2, x3, x4) 172.08/44.54 172.08/44.54 [] = [] 172.08/44.54 172.08/44.54 reach_out_gggg(x1, x2, x3, x4) = reach_out_gggg(x1, x2, x3, x4) 172.08/44.54 172.08/44.54 U2_gggg(x1, x2, x3, x4, x5) = U2_gggg(x1, x2, x3, x4, x5) 172.08/44.54 172.08/44.54 member1_in_ag(x1, x2) = member1_in_ag(x2) 172.08/44.54 172.08/44.54 member1_out_ag(x1, x2) = member1_out_ag(x1, x2) 172.08/44.54 172.08/44.54 U6_ag(x1, x2, x3, x4) = U6_ag(x2, x3, x4) 172.08/44.54 172.08/44.54 U3_gggg(x1, x2, x3, x4, x5, x6) = U3_gggg(x1, x2, x3, x4, x5, x6) 172.08/44.54 172.08/44.54 U4_gggg(x1, x2, x3, x4, x5) = U4_gggg(x1, x2, x3, x4, x5) 172.08/44.54 172.08/44.54 172.08/44.54 172.08/44.54 ---------------------------------------- 172.08/44.54 172.08/44.54 (3) DependencyPairsProof (EQUIVALENT) 172.08/44.54 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 172.08/44.54 Pi DP problem: 172.08/44.54 The TRS P consists of the following rules: 172.08/44.54 172.08/44.54 REACH_IN_GGGG(X, Y, Edges, Visited) -> U1_GGGG(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges)) 172.08/44.54 REACH_IN_GGGG(X, Y, Edges, Visited) -> MEMBER_IN_GG(.(X, .(Y, [])), Edges) 172.08/44.54 MEMBER_IN_GG(X, .(H, L)) -> U5_GG(X, H, L, member_in_gg(X, L)) 172.08/44.54 MEMBER_IN_GG(X, .(H, L)) -> MEMBER_IN_GG(X, L) 172.08/44.54 REACH_IN_GGGG(X, Z, Edges, Visited) -> U2_GGGG(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) 172.08/44.54 REACH_IN_GGGG(X, Z, Edges, Visited) -> MEMBER1_IN_AG(.(X, .(Y, [])), Edges) 172.08/44.54 MEMBER1_IN_AG(X, .(H, L)) -> U6_AG(X, H, L, member1_in_ag(X, L)) 172.08/44.54 MEMBER1_IN_AG(X, .(H, L)) -> MEMBER1_IN_AG(X, L) 172.08/44.54 U2_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_GGGG(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 172.08/44.54 U2_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> MEMBER_IN_GG(Y, Visited) 172.08/44.54 U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> U4_GGGG(X, Z, Edges, Visited, reach_in_gggg(Y, Z, Edges, .(Y, Visited))) 172.08/44.54 U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited)) 172.08/44.54 172.08/44.54 The TRS R consists of the following rules: 172.08/44.54 172.08/44.54 reach_in_gggg(X, Y, Edges, Visited) -> U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges)) 172.08/44.54 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.54 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.54 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.54 U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_gggg(X, Y, Edges, Visited) 172.08/44.54 reach_in_gggg(X, Z, Edges, Visited) -> U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) 172.08/44.54 member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.54 member1_in_ag(X, .(H, L)) -> U6_ag(X, H, L, member1_in_ag(X, L)) 172.08/44.54 U6_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.54 U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 172.08/44.54 U3_gggg(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> U4_gggg(X, Z, Edges, Visited, reach_in_gggg(Y, Z, Edges, .(Y, Visited))) 172.08/44.54 U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) -> reach_out_gggg(X, Z, Edges, Visited) 172.08/44.54 172.08/44.54 The argument filtering Pi contains the following mapping: 172.08/44.54 reach_in_gggg(x1, x2, x3, x4) = reach_in_gggg(x1, x2, x3, x4) 172.08/44.54 172.08/44.54 U1_gggg(x1, x2, x3, x4, x5) = U1_gggg(x1, x2, x3, x4, x5) 172.08/44.54 172.08/44.54 member_in_gg(x1, x2) = member_in_gg(x1, x2) 172.08/44.54 172.08/44.54 .(x1, x2) = .(x1, x2) 172.08/44.54 172.08/44.54 member_out_gg(x1, x2) = member_out_gg(x1, x2) 172.08/44.54 172.08/44.54 U5_gg(x1, x2, x3, x4) = U5_gg(x1, x2, x3, x4) 172.08/44.54 172.08/44.54 [] = [] 172.08/44.54 172.08/44.54 reach_out_gggg(x1, x2, x3, x4) = reach_out_gggg(x1, x2, x3, x4) 172.08/44.54 172.08/44.54 U2_gggg(x1, x2, x3, x4, x5) = U2_gggg(x1, x2, x3, x4, x5) 172.08/44.54 172.08/44.54 member1_in_ag(x1, x2) = member1_in_ag(x2) 172.08/44.54 172.08/44.54 member1_out_ag(x1, x2) = member1_out_ag(x1, x2) 172.08/44.54 172.08/44.54 U6_ag(x1, x2, x3, x4) = U6_ag(x2, x3, x4) 172.08/44.54 172.08/44.54 U3_gggg(x1, x2, x3, x4, x5, x6) = U3_gggg(x1, x2, x3, x4, x5, x6) 172.08/44.54 172.08/44.54 U4_gggg(x1, x2, x3, x4, x5) = U4_gggg(x1, x2, x3, x4, x5) 172.08/44.54 172.08/44.54 REACH_IN_GGGG(x1, x2, x3, x4) = REACH_IN_GGGG(x1, x2, x3, x4) 172.08/44.54 172.08/44.54 U1_GGGG(x1, x2, x3, x4, x5) = U1_GGGG(x1, x2, x3, x4, x5) 172.08/44.54 172.08/44.54 MEMBER_IN_GG(x1, x2) = MEMBER_IN_GG(x1, x2) 172.08/44.54 172.08/44.54 U5_GG(x1, x2, x3, x4) = U5_GG(x1, x2, x3, x4) 172.08/44.54 172.08/44.54 U2_GGGG(x1, x2, x3, x4, x5) = U2_GGGG(x1, x2, x3, x4, x5) 172.08/44.54 172.08/44.54 MEMBER1_IN_AG(x1, x2) = MEMBER1_IN_AG(x2) 172.08/44.54 172.08/44.54 U6_AG(x1, x2, x3, x4) = U6_AG(x2, x3, x4) 172.08/44.54 172.08/44.54 U3_GGGG(x1, x2, x3, x4, x5, x6) = U3_GGGG(x1, x2, x3, x4, x5, x6) 172.08/44.54 172.08/44.54 U4_GGGG(x1, x2, x3, x4, x5) = U4_GGGG(x1, x2, x3, x4, x5) 172.08/44.54 172.08/44.54 172.08/44.54 We have to consider all (P,R,Pi)-chains 172.08/44.54 ---------------------------------------- 172.08/44.54 172.08/44.54 (4) 172.08/44.54 Obligation: 172.08/44.54 Pi DP problem: 172.08/44.54 The TRS P consists of the following rules: 172.08/44.54 172.08/44.54 REACH_IN_GGGG(X, Y, Edges, Visited) -> U1_GGGG(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges)) 172.08/44.54 REACH_IN_GGGG(X, Y, Edges, Visited) -> MEMBER_IN_GG(.(X, .(Y, [])), Edges) 172.08/44.54 MEMBER_IN_GG(X, .(H, L)) -> U5_GG(X, H, L, member_in_gg(X, L)) 172.08/44.54 MEMBER_IN_GG(X, .(H, L)) -> MEMBER_IN_GG(X, L) 172.08/44.54 REACH_IN_GGGG(X, Z, Edges, Visited) -> U2_GGGG(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) 172.08/44.54 REACH_IN_GGGG(X, Z, Edges, Visited) -> MEMBER1_IN_AG(.(X, .(Y, [])), Edges) 172.08/44.54 MEMBER1_IN_AG(X, .(H, L)) -> U6_AG(X, H, L, member1_in_ag(X, L)) 172.08/44.54 MEMBER1_IN_AG(X, .(H, L)) -> MEMBER1_IN_AG(X, L) 172.08/44.54 U2_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_GGGG(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 172.08/44.54 U2_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> MEMBER_IN_GG(Y, Visited) 172.08/44.54 U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> U4_GGGG(X, Z, Edges, Visited, reach_in_gggg(Y, Z, Edges, .(Y, Visited))) 172.08/44.54 U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited)) 172.08/44.54 172.08/44.54 The TRS R consists of the following rules: 172.08/44.54 172.08/44.54 reach_in_gggg(X, Y, Edges, Visited) -> U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges)) 172.08/44.54 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.54 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.54 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.54 U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_gggg(X, Y, Edges, Visited) 172.08/44.54 reach_in_gggg(X, Z, Edges, Visited) -> U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) 172.08/44.54 member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.54 member1_in_ag(X, .(H, L)) -> U6_ag(X, H, L, member1_in_ag(X, L)) 172.08/44.54 U6_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.54 U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 172.08/44.54 U3_gggg(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> U4_gggg(X, Z, Edges, Visited, reach_in_gggg(Y, Z, Edges, .(Y, Visited))) 172.08/44.54 U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) -> reach_out_gggg(X, Z, Edges, Visited) 172.08/44.54 172.08/44.54 The argument filtering Pi contains the following mapping: 172.08/44.54 reach_in_gggg(x1, x2, x3, x4) = reach_in_gggg(x1, x2, x3, x4) 172.08/44.54 172.08/44.54 U1_gggg(x1, x2, x3, x4, x5) = U1_gggg(x1, x2, x3, x4, x5) 172.08/44.54 172.08/44.54 member_in_gg(x1, x2) = member_in_gg(x1, x2) 172.08/44.54 172.08/44.54 .(x1, x2) = .(x1, x2) 172.08/44.54 172.08/44.54 member_out_gg(x1, x2) = member_out_gg(x1, x2) 172.08/44.54 172.08/44.54 U5_gg(x1, x2, x3, x4) = U5_gg(x1, x2, x3, x4) 172.08/44.54 172.08/44.54 [] = [] 172.08/44.54 172.08/44.54 reach_out_gggg(x1, x2, x3, x4) = reach_out_gggg(x1, x2, x3, x4) 172.08/44.54 172.08/44.54 U2_gggg(x1, x2, x3, x4, x5) = U2_gggg(x1, x2, x3, x4, x5) 172.08/44.54 172.08/44.54 member1_in_ag(x1, x2) = member1_in_ag(x2) 172.08/44.54 172.08/44.54 member1_out_ag(x1, x2) = member1_out_ag(x1, x2) 172.08/44.54 172.08/44.54 U6_ag(x1, x2, x3, x4) = U6_ag(x2, x3, x4) 172.08/44.54 172.08/44.54 U3_gggg(x1, x2, x3, x4, x5, x6) = U3_gggg(x1, x2, x3, x4, x5, x6) 172.08/44.54 172.08/44.54 U4_gggg(x1, x2, x3, x4, x5) = U4_gggg(x1, x2, x3, x4, x5) 172.08/44.54 172.08/44.54 REACH_IN_GGGG(x1, x2, x3, x4) = REACH_IN_GGGG(x1, x2, x3, x4) 172.08/44.54 172.08/44.54 U1_GGGG(x1, x2, x3, x4, x5) = U1_GGGG(x1, x2, x3, x4, x5) 172.08/44.54 172.08/44.54 MEMBER_IN_GG(x1, x2) = MEMBER_IN_GG(x1, x2) 172.08/44.54 172.08/44.54 U5_GG(x1, x2, x3, x4) = U5_GG(x1, x2, x3, x4) 172.08/44.54 172.08/44.54 U2_GGGG(x1, x2, x3, x4, x5) = U2_GGGG(x1, x2, x3, x4, x5) 172.08/44.54 172.08/44.54 MEMBER1_IN_AG(x1, x2) = MEMBER1_IN_AG(x2) 172.08/44.54 172.08/44.54 U6_AG(x1, x2, x3, x4) = U6_AG(x2, x3, x4) 172.08/44.54 172.08/44.54 U3_GGGG(x1, x2, x3, x4, x5, x6) = U3_GGGG(x1, x2, x3, x4, x5, x6) 172.08/44.54 172.08/44.54 U4_GGGG(x1, x2, x3, x4, x5) = U4_GGGG(x1, x2, x3, x4, x5) 172.08/44.54 172.08/44.54 172.08/44.54 We have to consider all (P,R,Pi)-chains 172.08/44.54 ---------------------------------------- 172.08/44.54 172.08/44.54 (5) DependencyGraphProof (EQUIVALENT) 172.08/44.54 The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 7 less nodes. 172.08/44.54 ---------------------------------------- 172.08/44.54 172.08/44.54 (6) 172.08/44.54 Complex Obligation (AND) 172.08/44.54 172.08/44.54 ---------------------------------------- 172.08/44.54 172.08/44.54 (7) 172.08/44.54 Obligation: 172.08/44.54 Pi DP problem: 172.08/44.54 The TRS P consists of the following rules: 172.08/44.54 172.08/44.54 MEMBER1_IN_AG(X, .(H, L)) -> MEMBER1_IN_AG(X, L) 172.08/44.54 172.08/44.54 The TRS R consists of the following rules: 172.08/44.54 172.08/44.54 reach_in_gggg(X, Y, Edges, Visited) -> U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges)) 172.08/44.54 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.54 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.54 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.54 U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_gggg(X, Y, Edges, Visited) 172.08/44.54 reach_in_gggg(X, Z, Edges, Visited) -> U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) 172.08/44.54 member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.54 member1_in_ag(X, .(H, L)) -> U6_ag(X, H, L, member1_in_ag(X, L)) 172.08/44.54 U6_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.54 U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 172.08/44.54 U3_gggg(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> U4_gggg(X, Z, Edges, Visited, reach_in_gggg(Y, Z, Edges, .(Y, Visited))) 172.08/44.54 U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) -> reach_out_gggg(X, Z, Edges, Visited) 172.08/44.54 172.08/44.54 The argument filtering Pi contains the following mapping: 172.08/44.54 reach_in_gggg(x1, x2, x3, x4) = reach_in_gggg(x1, x2, x3, x4) 172.08/44.54 172.08/44.54 U1_gggg(x1, x2, x3, x4, x5) = U1_gggg(x1, x2, x3, x4, x5) 172.08/44.54 172.08/44.54 member_in_gg(x1, x2) = member_in_gg(x1, x2) 172.08/44.54 172.08/44.54 .(x1, x2) = .(x1, x2) 172.08/44.54 172.08/44.54 member_out_gg(x1, x2) = member_out_gg(x1, x2) 172.08/44.54 172.08/44.54 U5_gg(x1, x2, x3, x4) = U5_gg(x1, x2, x3, x4) 172.08/44.54 172.08/44.54 [] = [] 172.08/44.54 172.08/44.54 reach_out_gggg(x1, x2, x3, x4) = reach_out_gggg(x1, x2, x3, x4) 172.08/44.54 172.08/44.54 U2_gggg(x1, x2, x3, x4, x5) = U2_gggg(x1, x2, x3, x4, x5) 172.08/44.54 172.08/44.54 member1_in_ag(x1, x2) = member1_in_ag(x2) 172.08/44.54 172.08/44.54 member1_out_ag(x1, x2) = member1_out_ag(x1, x2) 172.08/44.54 172.08/44.54 U6_ag(x1, x2, x3, x4) = U6_ag(x2, x3, x4) 172.08/44.54 172.08/44.54 U3_gggg(x1, x2, x3, x4, x5, x6) = U3_gggg(x1, x2, x3, x4, x5, x6) 172.08/44.54 172.08/44.54 U4_gggg(x1, x2, x3, x4, x5) = U4_gggg(x1, x2, x3, x4, x5) 172.08/44.54 172.08/44.54 MEMBER1_IN_AG(x1, x2) = MEMBER1_IN_AG(x2) 172.08/44.54 172.08/44.54 172.08/44.54 We have to consider all (P,R,Pi)-chains 172.08/44.54 ---------------------------------------- 172.08/44.54 172.08/44.54 (8) UsableRulesProof (EQUIVALENT) 172.08/44.54 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 172.08/44.54 ---------------------------------------- 172.08/44.54 172.08/44.54 (9) 172.08/44.54 Obligation: 172.08/44.54 Pi DP problem: 172.08/44.54 The TRS P consists of the following rules: 172.08/44.54 172.08/44.54 MEMBER1_IN_AG(X, .(H, L)) -> MEMBER1_IN_AG(X, L) 172.08/44.54 172.08/44.54 R is empty. 172.08/44.54 The argument filtering Pi contains the following mapping: 172.08/44.54 .(x1, x2) = .(x1, x2) 172.08/44.54 172.08/44.54 MEMBER1_IN_AG(x1, x2) = MEMBER1_IN_AG(x2) 172.08/44.54 172.08/44.54 172.08/44.54 We have to consider all (P,R,Pi)-chains 172.08/44.54 ---------------------------------------- 172.08/44.54 172.08/44.54 (10) PiDPToQDPProof (SOUND) 172.08/44.54 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 172.08/44.54 ---------------------------------------- 172.08/44.54 172.08/44.54 (11) 172.08/44.54 Obligation: 172.08/44.54 Q DP problem: 172.08/44.54 The TRS P consists of the following rules: 172.08/44.54 172.08/44.54 MEMBER1_IN_AG(.(H, L)) -> MEMBER1_IN_AG(L) 172.08/44.54 172.08/44.54 R is empty. 172.08/44.54 Q is empty. 172.08/44.54 We have to consider all (P,Q,R)-chains. 172.08/44.54 ---------------------------------------- 172.08/44.54 172.08/44.54 (12) QDPSizeChangeProof (EQUIVALENT) 172.08/44.54 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 172.08/44.54 172.08/44.54 From the DPs we obtained the following set of size-change graphs: 172.08/44.54 *MEMBER1_IN_AG(.(H, L)) -> MEMBER1_IN_AG(L) 172.08/44.54 The graph contains the following edges 1 > 1 172.08/44.54 172.08/44.54 172.08/44.54 ---------------------------------------- 172.08/44.54 172.08/44.54 (13) 172.08/44.54 YES 172.08/44.54 172.08/44.54 ---------------------------------------- 172.08/44.54 172.08/44.54 (14) 172.08/44.54 Obligation: 172.08/44.54 Pi DP problem: 172.08/44.54 The TRS P consists of the following rules: 172.08/44.54 172.08/44.54 MEMBER_IN_GG(X, .(H, L)) -> MEMBER_IN_GG(X, L) 172.08/44.54 172.08/44.54 The TRS R consists of the following rules: 172.08/44.54 172.08/44.54 reach_in_gggg(X, Y, Edges, Visited) -> U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges)) 172.08/44.54 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.54 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.54 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.54 U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_gggg(X, Y, Edges, Visited) 172.08/44.54 reach_in_gggg(X, Z, Edges, Visited) -> U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) 172.08/44.54 member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.54 member1_in_ag(X, .(H, L)) -> U6_ag(X, H, L, member1_in_ag(X, L)) 172.08/44.54 U6_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.54 U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 172.08/44.54 U3_gggg(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> U4_gggg(X, Z, Edges, Visited, reach_in_gggg(Y, Z, Edges, .(Y, Visited))) 172.08/44.54 U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) -> reach_out_gggg(X, Z, Edges, Visited) 172.08/44.54 172.08/44.54 The argument filtering Pi contains the following mapping: 172.08/44.54 reach_in_gggg(x1, x2, x3, x4) = reach_in_gggg(x1, x2, x3, x4) 172.08/44.54 172.08/44.54 U1_gggg(x1, x2, x3, x4, x5) = U1_gggg(x1, x2, x3, x4, x5) 172.08/44.54 172.08/44.54 member_in_gg(x1, x2) = member_in_gg(x1, x2) 172.08/44.54 172.08/44.54 .(x1, x2) = .(x1, x2) 172.08/44.54 172.08/44.54 member_out_gg(x1, x2) = member_out_gg(x1, x2) 172.08/44.54 172.08/44.54 U5_gg(x1, x2, x3, x4) = U5_gg(x1, x2, x3, x4) 172.08/44.54 172.08/44.54 [] = [] 172.08/44.54 172.08/44.54 reach_out_gggg(x1, x2, x3, x4) = reach_out_gggg(x1, x2, x3, x4) 172.08/44.54 172.08/44.54 U2_gggg(x1, x2, x3, x4, x5) = U2_gggg(x1, x2, x3, x4, x5) 172.08/44.54 172.08/44.54 member1_in_ag(x1, x2) = member1_in_ag(x2) 172.08/44.54 172.08/44.54 member1_out_ag(x1, x2) = member1_out_ag(x1, x2) 172.08/44.54 172.08/44.54 U6_ag(x1, x2, x3, x4) = U6_ag(x2, x3, x4) 172.08/44.54 172.08/44.54 U3_gggg(x1, x2, x3, x4, x5, x6) = U3_gggg(x1, x2, x3, x4, x5, x6) 172.08/44.54 172.08/44.54 U4_gggg(x1, x2, x3, x4, x5) = U4_gggg(x1, x2, x3, x4, x5) 172.08/44.54 172.08/44.54 MEMBER_IN_GG(x1, x2) = MEMBER_IN_GG(x1, x2) 172.08/44.54 172.08/44.54 172.08/44.54 We have to consider all (P,R,Pi)-chains 172.08/44.54 ---------------------------------------- 172.08/44.54 172.08/44.54 (15) UsableRulesProof (EQUIVALENT) 172.08/44.54 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 172.08/44.54 ---------------------------------------- 172.08/44.54 172.08/44.54 (16) 172.08/44.54 Obligation: 172.08/44.54 Pi DP problem: 172.08/44.54 The TRS P consists of the following rules: 172.08/44.54 172.08/44.54 MEMBER_IN_GG(X, .(H, L)) -> MEMBER_IN_GG(X, L) 172.08/44.54 172.08/44.54 R is empty. 172.08/44.54 Pi is empty. 172.08/44.54 We have to consider all (P,R,Pi)-chains 172.08/44.54 ---------------------------------------- 172.08/44.54 172.08/44.54 (17) PiDPToQDPProof (EQUIVALENT) 172.08/44.54 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 172.08/44.54 ---------------------------------------- 172.08/44.54 172.08/44.54 (18) 172.08/44.54 Obligation: 172.08/44.54 Q DP problem: 172.08/44.54 The TRS P consists of the following rules: 172.08/44.54 172.08/44.54 MEMBER_IN_GG(X, .(H, L)) -> MEMBER_IN_GG(X, L) 172.08/44.54 172.08/44.54 R is empty. 172.08/44.54 Q is empty. 172.08/44.54 We have to consider all (P,Q,R)-chains. 172.08/44.54 ---------------------------------------- 172.08/44.54 172.08/44.54 (19) QDPSizeChangeProof (EQUIVALENT) 172.08/44.54 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 172.08/44.54 172.08/44.54 From the DPs we obtained the following set of size-change graphs: 172.08/44.54 *MEMBER_IN_GG(X, .(H, L)) -> MEMBER_IN_GG(X, L) 172.08/44.54 The graph contains the following edges 1 >= 1, 2 > 2 172.08/44.54 172.08/44.54 172.08/44.54 ---------------------------------------- 172.08/44.54 172.08/44.54 (20) 172.08/44.54 YES 172.08/44.54 172.08/44.54 ---------------------------------------- 172.08/44.54 172.08/44.54 (21) 172.08/44.54 Obligation: 172.08/44.54 Pi DP problem: 172.08/44.54 The TRS P consists of the following rules: 172.08/44.54 172.08/44.54 REACH_IN_GGGG(X, Z, Edges, Visited) -> U2_GGGG(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) 172.08/44.54 U2_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_GGGG(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 172.08/44.54 U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited)) 172.08/44.54 172.08/44.54 The TRS R consists of the following rules: 172.08/44.54 172.08/44.54 reach_in_gggg(X, Y, Edges, Visited) -> U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges)) 172.08/44.54 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.54 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.54 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.54 U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_gggg(X, Y, Edges, Visited) 172.08/44.54 reach_in_gggg(X, Z, Edges, Visited) -> U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) 172.08/44.54 member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.54 member1_in_ag(X, .(H, L)) -> U6_ag(X, H, L, member1_in_ag(X, L)) 172.08/44.54 U6_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.54 U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 172.08/44.54 U3_gggg(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> U4_gggg(X, Z, Edges, Visited, reach_in_gggg(Y, Z, Edges, .(Y, Visited))) 172.08/44.54 U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) -> reach_out_gggg(X, Z, Edges, Visited) 172.08/44.54 172.08/44.54 The argument filtering Pi contains the following mapping: 172.08/44.54 reach_in_gggg(x1, x2, x3, x4) = reach_in_gggg(x1, x2, x3, x4) 172.08/44.54 172.08/44.54 U1_gggg(x1, x2, x3, x4, x5) = U1_gggg(x1, x2, x3, x4, x5) 172.08/44.54 172.08/44.54 member_in_gg(x1, x2) = member_in_gg(x1, x2) 172.08/44.54 172.08/44.54 .(x1, x2) = .(x1, x2) 172.08/44.54 172.08/44.54 member_out_gg(x1, x2) = member_out_gg(x1, x2) 172.08/44.54 172.08/44.54 U5_gg(x1, x2, x3, x4) = U5_gg(x1, x2, x3, x4) 172.08/44.54 172.08/44.54 [] = [] 172.08/44.54 172.08/44.54 reach_out_gggg(x1, x2, x3, x4) = reach_out_gggg(x1, x2, x3, x4) 172.08/44.55 172.08/44.55 U2_gggg(x1, x2, x3, x4, x5) = U2_gggg(x1, x2, x3, x4, x5) 172.08/44.55 172.08/44.55 member1_in_ag(x1, x2) = member1_in_ag(x2) 172.08/44.55 172.08/44.55 member1_out_ag(x1, x2) = member1_out_ag(x1, x2) 172.08/44.55 172.08/44.55 U6_ag(x1, x2, x3, x4) = U6_ag(x2, x3, x4) 172.08/44.55 172.08/44.55 U3_gggg(x1, x2, x3, x4, x5, x6) = U3_gggg(x1, x2, x3, x4, x5, x6) 172.08/44.55 172.08/44.55 U4_gggg(x1, x2, x3, x4, x5) = U4_gggg(x1, x2, x3, x4, x5) 172.08/44.55 172.08/44.55 REACH_IN_GGGG(x1, x2, x3, x4) = REACH_IN_GGGG(x1, x2, x3, x4) 172.08/44.55 172.08/44.55 U2_GGGG(x1, x2, x3, x4, x5) = U2_GGGG(x1, x2, x3, x4, x5) 172.08/44.55 172.08/44.55 U3_GGGG(x1, x2, x3, x4, x5, x6) = U3_GGGG(x1, x2, x3, x4, x5, x6) 172.08/44.55 172.08/44.55 172.08/44.55 We have to consider all (P,R,Pi)-chains 172.08/44.55 ---------------------------------------- 172.08/44.55 172.08/44.55 (22) UsableRulesProof (EQUIVALENT) 172.08/44.55 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 172.08/44.55 ---------------------------------------- 172.08/44.55 172.08/44.55 (23) 172.08/44.55 Obligation: 172.08/44.55 Pi DP problem: 172.08/44.55 The TRS P consists of the following rules: 172.08/44.55 172.08/44.55 REACH_IN_GGGG(X, Z, Edges, Visited) -> U2_GGGG(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) 172.08/44.55 U2_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_GGGG(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 172.08/44.55 U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited)) 172.08/44.55 172.08/44.55 The TRS R consists of the following rules: 172.08/44.55 172.08/44.55 member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.55 member1_in_ag(X, .(H, L)) -> U6_ag(X, H, L, member1_in_ag(X, L)) 172.08/44.55 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.55 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.55 U6_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.55 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.55 172.08/44.55 The argument filtering Pi contains the following mapping: 172.08/44.55 member_in_gg(x1, x2) = member_in_gg(x1, x2) 172.08/44.55 172.08/44.55 .(x1, x2) = .(x1, x2) 172.08/44.55 172.08/44.55 member_out_gg(x1, x2) = member_out_gg(x1, x2) 172.08/44.55 172.08/44.55 U5_gg(x1, x2, x3, x4) = U5_gg(x1, x2, x3, x4) 172.08/44.55 172.08/44.55 [] = [] 172.08/44.55 172.08/44.55 member1_in_ag(x1, x2) = member1_in_ag(x2) 172.08/44.55 172.08/44.55 member1_out_ag(x1, x2) = member1_out_ag(x1, x2) 172.08/44.55 172.08/44.55 U6_ag(x1, x2, x3, x4) = U6_ag(x2, x3, x4) 172.08/44.55 172.08/44.55 REACH_IN_GGGG(x1, x2, x3, x4) = REACH_IN_GGGG(x1, x2, x3, x4) 172.08/44.55 172.08/44.55 U2_GGGG(x1, x2, x3, x4, x5) = U2_GGGG(x1, x2, x3, x4, x5) 172.08/44.55 172.08/44.55 U3_GGGG(x1, x2, x3, x4, x5, x6) = U3_GGGG(x1, x2, x3, x4, x5, x6) 172.08/44.55 172.08/44.55 172.08/44.55 We have to consider all (P,R,Pi)-chains 172.08/44.55 ---------------------------------------- 172.08/44.55 172.08/44.55 (24) PiDPToQDPProof (SOUND) 172.08/44.55 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 172.08/44.55 ---------------------------------------- 172.08/44.55 172.08/44.55 (25) 172.08/44.55 Obligation: 172.08/44.55 Q DP problem: 172.08/44.55 The TRS P consists of the following rules: 172.08/44.55 172.08/44.55 REACH_IN_GGGG(X, Z, Edges, Visited) -> U2_GGGG(X, Z, Edges, Visited, member1_in_ag(Edges)) 172.08/44.55 U2_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_GGGG(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 172.08/44.55 U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited)) 172.08/44.55 172.08/44.55 The TRS R consists of the following rules: 172.08/44.55 172.08/44.55 member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.55 member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) 172.08/44.55 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.55 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.55 U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.55 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.55 172.08/44.55 The set Q consists of the following terms: 172.08/44.55 172.08/44.55 member1_in_ag(x0) 172.08/44.55 member_in_gg(x0, x1) 172.08/44.55 U6_ag(x0, x1, x2) 172.08/44.55 U5_gg(x0, x1, x2, x3) 172.08/44.55 172.08/44.55 We have to consider all (P,Q,R)-chains. 172.08/44.55 ---------------------------------------- 172.08/44.55 172.08/44.55 (26) TransformationProof (SOUND) 172.08/44.55 By narrowing [LPAR04] the rule REACH_IN_GGGG(X, Z, Edges, Visited) -> U2_GGGG(X, Z, Edges, Visited, member1_in_ag(Edges)) at position [4] we obtained the following new rules [LPAR04]: 172.08/44.55 172.08/44.55 (REACH_IN_GGGG(y0, y1, .(x0, x1), y3) -> U2_GGGG(y0, y1, .(x0, x1), y3, member1_out_ag(x0, .(x0, x1))),REACH_IN_GGGG(y0, y1, .(x0, x1), y3) -> U2_GGGG(y0, y1, .(x0, x1), y3, member1_out_ag(x0, .(x0, x1)))) 172.08/44.55 (REACH_IN_GGGG(y0, y1, .(x0, x1), y3) -> U2_GGGG(y0, y1, .(x0, x1), y3, U6_ag(x0, x1, member1_in_ag(x1))),REACH_IN_GGGG(y0, y1, .(x0, x1), y3) -> U2_GGGG(y0, y1, .(x0, x1), y3, U6_ag(x0, x1, member1_in_ag(x1)))) 172.08/44.55 172.08/44.55 172.08/44.55 ---------------------------------------- 172.08/44.55 172.08/44.55 (27) 172.08/44.55 Obligation: 172.08/44.55 Q DP problem: 172.08/44.55 The TRS P consists of the following rules: 172.08/44.55 172.08/44.55 U2_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_GGGG(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 172.08/44.55 U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited)) 172.08/44.55 REACH_IN_GGGG(y0, y1, .(x0, x1), y3) -> U2_GGGG(y0, y1, .(x0, x1), y3, member1_out_ag(x0, .(x0, x1))) 172.08/44.55 REACH_IN_GGGG(y0, y1, .(x0, x1), y3) -> U2_GGGG(y0, y1, .(x0, x1), y3, U6_ag(x0, x1, member1_in_ag(x1))) 172.08/44.55 172.08/44.55 The TRS R consists of the following rules: 172.08/44.55 172.08/44.55 member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.55 member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) 172.08/44.55 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.55 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.55 U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.55 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.55 172.08/44.55 The set Q consists of the following terms: 172.08/44.55 172.08/44.55 member1_in_ag(x0) 172.08/44.55 member_in_gg(x0, x1) 172.08/44.55 U6_ag(x0, x1, x2) 172.08/44.55 U5_gg(x0, x1, x2, x3) 172.08/44.55 172.08/44.55 We have to consider all (P,Q,R)-chains. 172.08/44.55 ---------------------------------------- 172.08/44.55 172.08/44.55 (28) TransformationProof (SOUND) 172.08/44.55 By narrowing [LPAR04] the rule U2_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_GGGG(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) at position [5] we obtained the following new rules [LPAR04]: 172.08/44.55 172.08/44.55 (U2_GGGG(y0, y1, y2, .(x0, x1), member1_out_ag(.(y0, .(x0, [])), y2)) -> U3_GGGG(y0, y1, y2, .(x0, x1), x0, member_out_gg(x0, .(x0, x1))),U2_GGGG(y0, y1, y2, .(x0, x1), member1_out_ag(.(y0, .(x0, [])), y2)) -> U3_GGGG(y0, y1, y2, .(x0, x1), x0, member_out_gg(x0, .(x0, x1)))) 172.08/44.55 (U2_GGGG(y0, y1, y2, .(x1, x2), member1_out_ag(.(y0, .(x0, [])), y2)) -> U3_GGGG(y0, y1, y2, .(x1, x2), x0, U5_gg(x0, x1, x2, member_in_gg(x0, x2))),U2_GGGG(y0, y1, y2, .(x1, x2), member1_out_ag(.(y0, .(x0, [])), y2)) -> U3_GGGG(y0, y1, y2, .(x1, x2), x0, U5_gg(x0, x1, x2, member_in_gg(x0, x2)))) 172.08/44.55 172.08/44.55 172.08/44.55 ---------------------------------------- 172.08/44.55 172.08/44.55 (29) 172.08/44.55 Obligation: 172.08/44.55 Q DP problem: 172.08/44.55 The TRS P consists of the following rules: 172.08/44.55 172.08/44.55 U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited)) 172.08/44.55 REACH_IN_GGGG(y0, y1, .(x0, x1), y3) -> U2_GGGG(y0, y1, .(x0, x1), y3, member1_out_ag(x0, .(x0, x1))) 172.08/44.55 REACH_IN_GGGG(y0, y1, .(x0, x1), y3) -> U2_GGGG(y0, y1, .(x0, x1), y3, U6_ag(x0, x1, member1_in_ag(x1))) 172.08/44.55 U2_GGGG(y0, y1, y2, .(x0, x1), member1_out_ag(.(y0, .(x0, [])), y2)) -> U3_GGGG(y0, y1, y2, .(x0, x1), x0, member_out_gg(x0, .(x0, x1))) 172.08/44.55 U2_GGGG(y0, y1, y2, .(x1, x2), member1_out_ag(.(y0, .(x0, [])), y2)) -> U3_GGGG(y0, y1, y2, .(x1, x2), x0, U5_gg(x0, x1, x2, member_in_gg(x0, x2))) 172.08/44.55 172.08/44.55 The TRS R consists of the following rules: 172.08/44.55 172.08/44.55 member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.55 member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) 172.08/44.55 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.55 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.55 U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.55 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.55 172.08/44.55 The set Q consists of the following terms: 172.08/44.55 172.08/44.55 member1_in_ag(x0) 172.08/44.55 member_in_gg(x0, x1) 172.08/44.55 U6_ag(x0, x1, x2) 172.08/44.55 U5_gg(x0, x1, x2, x3) 172.08/44.55 172.08/44.55 We have to consider all (P,Q,R)-chains. 172.08/44.55 ---------------------------------------- 172.08/44.55 172.08/44.55 (30) TransformationProof (EQUIVALENT) 172.08/44.55 By instantiating [LPAR04] the rule U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited)) we obtained the following new rules [LPAR04]: 172.08/44.55 172.08/44.55 (U3_GGGG(z0, z1, z2, .(z3, z4), z3, member_out_gg(z3, .(z3, z4))) -> REACH_IN_GGGG(z3, z1, z2, .(z3, .(z3, z4))),U3_GGGG(z0, z1, z2, .(z3, z4), z3, member_out_gg(z3, .(z3, z4))) -> REACH_IN_GGGG(z3, z1, z2, .(z3, .(z3, z4)))) 172.08/44.55 (U3_GGGG(z0, z1, z2, .(z3, z4), z5, member_out_gg(z5, .(z3, z4))) -> REACH_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4))),U3_GGGG(z0, z1, z2, .(z3, z4), z5, member_out_gg(z5, .(z3, z4))) -> REACH_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4)))) 172.08/44.55 172.08/44.55 172.08/44.55 ---------------------------------------- 172.08/44.55 172.08/44.55 (31) 172.08/44.55 Obligation: 172.08/44.55 Q DP problem: 172.08/44.55 The TRS P consists of the following rules: 172.08/44.55 172.08/44.55 REACH_IN_GGGG(y0, y1, .(x0, x1), y3) -> U2_GGGG(y0, y1, .(x0, x1), y3, member1_out_ag(x0, .(x0, x1))) 172.08/44.55 REACH_IN_GGGG(y0, y1, .(x0, x1), y3) -> U2_GGGG(y0, y1, .(x0, x1), y3, U6_ag(x0, x1, member1_in_ag(x1))) 172.08/44.55 U2_GGGG(y0, y1, y2, .(x0, x1), member1_out_ag(.(y0, .(x0, [])), y2)) -> U3_GGGG(y0, y1, y2, .(x0, x1), x0, member_out_gg(x0, .(x0, x1))) 172.08/44.55 U2_GGGG(y0, y1, y2, .(x1, x2), member1_out_ag(.(y0, .(x0, [])), y2)) -> U3_GGGG(y0, y1, y2, .(x1, x2), x0, U5_gg(x0, x1, x2, member_in_gg(x0, x2))) 172.08/44.55 U3_GGGG(z0, z1, z2, .(z3, z4), z3, member_out_gg(z3, .(z3, z4))) -> REACH_IN_GGGG(z3, z1, z2, .(z3, .(z3, z4))) 172.08/44.55 U3_GGGG(z0, z1, z2, .(z3, z4), z5, member_out_gg(z5, .(z3, z4))) -> REACH_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4))) 172.08/44.55 172.08/44.55 The TRS R consists of the following rules: 172.08/44.55 172.08/44.55 member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.55 member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) 172.08/44.55 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.55 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.55 U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.55 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.55 172.08/44.55 The set Q consists of the following terms: 172.08/44.55 172.08/44.55 member1_in_ag(x0) 172.08/44.55 member_in_gg(x0, x1) 172.08/44.55 U6_ag(x0, x1, x2) 172.08/44.55 U5_gg(x0, x1, x2, x3) 172.08/44.55 172.08/44.55 We have to consider all (P,Q,R)-chains. 172.08/44.55 ---------------------------------------- 172.08/44.55 172.08/44.55 (32) TransformationProof (EQUIVALENT) 172.08/44.55 By instantiating [LPAR04] the rule REACH_IN_GGGG(y0, y1, .(x0, x1), y3) -> U2_GGGG(y0, y1, .(x0, x1), y3, member1_out_ag(x0, .(x0, x1))) we obtained the following new rules [LPAR04]: 172.08/44.55 172.08/44.55 (REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) -> U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), member1_out_ag(x2, .(x2, x3))),REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) -> U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), member1_out_ag(x2, .(x2, x3)))) 172.08/44.55 (REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), member1_out_ag(x2, .(x2, x3))),REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), member1_out_ag(x2, .(x2, x3)))) 172.08/44.55 172.08/44.55 172.08/44.55 ---------------------------------------- 172.08/44.55 172.08/44.55 (33) 172.08/44.55 Obligation: 172.08/44.55 Q DP problem: 172.08/44.55 The TRS P consists of the following rules: 172.08/44.55 172.08/44.55 REACH_IN_GGGG(y0, y1, .(x0, x1), y3) -> U2_GGGG(y0, y1, .(x0, x1), y3, U6_ag(x0, x1, member1_in_ag(x1))) 172.08/44.55 U2_GGGG(y0, y1, y2, .(x0, x1), member1_out_ag(.(y0, .(x0, [])), y2)) -> U3_GGGG(y0, y1, y2, .(x0, x1), x0, member_out_gg(x0, .(x0, x1))) 172.08/44.55 U2_GGGG(y0, y1, y2, .(x1, x2), member1_out_ag(.(y0, .(x0, [])), y2)) -> U3_GGGG(y0, y1, y2, .(x1, x2), x0, U5_gg(x0, x1, x2, member_in_gg(x0, x2))) 172.08/44.55 U3_GGGG(z0, z1, z2, .(z3, z4), z3, member_out_gg(z3, .(z3, z4))) -> REACH_IN_GGGG(z3, z1, z2, .(z3, .(z3, z4))) 172.08/44.55 U3_GGGG(z0, z1, z2, .(z3, z4), z5, member_out_gg(z5, .(z3, z4))) -> REACH_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4))) 172.08/44.55 REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) -> U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), member1_out_ag(x2, .(x2, x3))) 172.08/44.55 REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), member1_out_ag(x2, .(x2, x3))) 172.08/44.55 172.08/44.55 The TRS R consists of the following rules: 172.08/44.55 172.08/44.55 member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.55 member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) 172.08/44.55 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.55 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.55 U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.55 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.55 172.08/44.55 The set Q consists of the following terms: 172.08/44.55 172.08/44.55 member1_in_ag(x0) 172.08/44.55 member_in_gg(x0, x1) 172.08/44.55 U6_ag(x0, x1, x2) 172.08/44.55 U5_gg(x0, x1, x2, x3) 172.08/44.55 172.08/44.55 We have to consider all (P,Q,R)-chains. 172.08/44.55 ---------------------------------------- 172.08/44.55 172.08/44.55 (34) TransformationProof (EQUIVALENT) 172.08/44.55 By instantiating [LPAR04] the rule REACH_IN_GGGG(y0, y1, .(x0, x1), y3) -> U2_GGGG(y0, y1, .(x0, x1), y3, U6_ag(x0, x1, member1_in_ag(x1))) we obtained the following new rules [LPAR04]: 172.08/44.55 172.08/44.55 (REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) -> U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3))),REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) -> U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3)))) 172.08/44.55 (REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3))),REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3)))) 172.08/44.55 172.08/44.55 172.08/44.55 ---------------------------------------- 172.08/44.55 172.08/44.55 (35) 172.08/44.55 Obligation: 172.08/44.55 Q DP problem: 172.08/44.55 The TRS P consists of the following rules: 172.08/44.55 172.08/44.55 U2_GGGG(y0, y1, y2, .(x0, x1), member1_out_ag(.(y0, .(x0, [])), y2)) -> U3_GGGG(y0, y1, y2, .(x0, x1), x0, member_out_gg(x0, .(x0, x1))) 172.08/44.55 U2_GGGG(y0, y1, y2, .(x1, x2), member1_out_ag(.(y0, .(x0, [])), y2)) -> U3_GGGG(y0, y1, y2, .(x1, x2), x0, U5_gg(x0, x1, x2, member_in_gg(x0, x2))) 172.08/44.55 U3_GGGG(z0, z1, z2, .(z3, z4), z3, member_out_gg(z3, .(z3, z4))) -> REACH_IN_GGGG(z3, z1, z2, .(z3, .(z3, z4))) 172.08/44.55 U3_GGGG(z0, z1, z2, .(z3, z4), z5, member_out_gg(z5, .(z3, z4))) -> REACH_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4))) 172.08/44.55 REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) -> U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), member1_out_ag(x2, .(x2, x3))) 172.08/44.55 REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), member1_out_ag(x2, .(x2, x3))) 172.08/44.55 REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) -> U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3))) 172.08/44.55 REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3))) 172.08/44.55 172.08/44.55 The TRS R consists of the following rules: 172.08/44.55 172.08/44.55 member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.55 member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) 172.08/44.55 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.55 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.55 U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.55 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.55 172.08/44.55 The set Q consists of the following terms: 172.08/44.55 172.08/44.55 member1_in_ag(x0) 172.08/44.55 member_in_gg(x0, x1) 172.08/44.55 U6_ag(x0, x1, x2) 172.08/44.55 U5_gg(x0, x1, x2, x3) 172.08/44.55 172.08/44.55 We have to consider all (P,Q,R)-chains. 172.08/44.55 ---------------------------------------- 172.08/44.55 172.08/44.55 (36) TransformationProof (EQUIVALENT) 172.08/44.55 By instantiating [LPAR04] the rule U2_GGGG(y0, y1, y2, .(x0, x1), member1_out_ag(.(y0, .(x0, [])), y2)) -> U3_GGGG(y0, y1, y2, .(x0, x1), x0, member_out_gg(x0, .(x0, x1))) we obtained the following new rules [LPAR04]: 172.08/44.55 172.08/44.55 (U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))),U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4))))) 172.08/44.55 (U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))),U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5))))) 172.08/44.55 (U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))),U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4))))) 172.08/44.55 (U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))),U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5))))) 172.08/44.55 172.08/44.55 172.08/44.55 ---------------------------------------- 172.08/44.55 172.08/44.55 (37) 172.08/44.55 Obligation: 172.08/44.55 Q DP problem: 172.08/44.55 The TRS P consists of the following rules: 172.08/44.55 172.08/44.55 U2_GGGG(y0, y1, y2, .(x1, x2), member1_out_ag(.(y0, .(x0, [])), y2)) -> U3_GGGG(y0, y1, y2, .(x1, x2), x0, U5_gg(x0, x1, x2, member_in_gg(x0, x2))) 172.08/44.55 U3_GGGG(z0, z1, z2, .(z3, z4), z3, member_out_gg(z3, .(z3, z4))) -> REACH_IN_GGGG(z3, z1, z2, .(z3, .(z3, z4))) 172.08/44.55 U3_GGGG(z0, z1, z2, .(z3, z4), z5, member_out_gg(z5, .(z3, z4))) -> REACH_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4))) 172.08/44.55 REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) -> U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), member1_out_ag(x2, .(x2, x3))) 172.08/44.55 REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), member1_out_ag(x2, .(x2, x3))) 172.08/44.55 REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) -> U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3))) 172.08/44.55 REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3))) 172.08/44.55 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) 172.08/44.55 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) 172.08/44.55 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) 172.08/44.55 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) 172.08/44.55 172.08/44.55 The TRS R consists of the following rules: 172.08/44.55 172.08/44.55 member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.55 member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) 172.08/44.55 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.55 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.55 U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.55 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.55 172.08/44.55 The set Q consists of the following terms: 172.08/44.55 172.08/44.55 member1_in_ag(x0) 172.08/44.55 member_in_gg(x0, x1) 172.08/44.55 U6_ag(x0, x1, x2) 172.08/44.55 U5_gg(x0, x1, x2, x3) 172.08/44.55 172.08/44.55 We have to consider all (P,Q,R)-chains. 172.08/44.55 ---------------------------------------- 172.08/44.55 172.08/44.55 (38) TransformationProof (EQUIVALENT) 172.08/44.55 By instantiating [LPAR04] the rule U2_GGGG(y0, y1, y2, .(x1, x2), member1_out_ag(.(y0, .(x0, [])), y2)) -> U3_GGGG(y0, y1, y2, .(x1, x2), x0, U5_gg(x0, x1, x2, member_in_gg(x0, x2))) we obtained the following new rules [LPAR04]: 172.08/44.55 172.08/44.55 (U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(x5, z0, .(z0, z4), member_in_gg(x5, .(z0, z4)))),U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(x5, z0, .(z0, z4), member_in_gg(x5, .(z0, z4))))) 172.08/44.55 (U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(x5, z0, .(z4, z5), member_in_gg(x5, .(z4, z5)))),U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(x5, z0, .(z4, z5), member_in_gg(x5, .(z4, z5))))) 172.08/44.55 (U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(x5, z0, .(z0, z4), member_in_gg(x5, .(z0, z4)))),U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(x5, z0, .(z0, z4), member_in_gg(x5, .(z0, z4))))) 172.08/44.55 (U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(x5, z0, .(z4, z5), member_in_gg(x5, .(z4, z5)))),U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(x5, z0, .(z4, z5), member_in_gg(x5, .(z4, z5))))) 172.08/44.55 172.08/44.55 172.08/44.55 ---------------------------------------- 172.08/44.55 172.08/44.55 (39) 172.08/44.55 Obligation: 172.08/44.55 Q DP problem: 172.08/44.55 The TRS P consists of the following rules: 172.08/44.55 172.08/44.55 U3_GGGG(z0, z1, z2, .(z3, z4), z3, member_out_gg(z3, .(z3, z4))) -> REACH_IN_GGGG(z3, z1, z2, .(z3, .(z3, z4))) 172.08/44.55 U3_GGGG(z0, z1, z2, .(z3, z4), z5, member_out_gg(z5, .(z3, z4))) -> REACH_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4))) 172.08/44.55 REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) -> U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), member1_out_ag(x2, .(x2, x3))) 172.08/44.55 REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), member1_out_ag(x2, .(x2, x3))) 172.08/44.55 REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) -> U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3))) 172.08/44.55 REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3))) 172.08/44.55 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) 172.08/44.55 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) 172.08/44.55 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) 172.08/44.55 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) 172.08/44.55 U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(x5, z0, .(z0, z4), member_in_gg(x5, .(z0, z4)))) 172.08/44.55 U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(x5, z0, .(z4, z5), member_in_gg(x5, .(z4, z5)))) 172.08/44.55 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(x5, z0, .(z0, z4), member_in_gg(x5, .(z0, z4)))) 172.08/44.55 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(x5, z0, .(z4, z5), member_in_gg(x5, .(z4, z5)))) 172.08/44.55 172.08/44.55 The TRS R consists of the following rules: 172.08/44.55 172.08/44.55 member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.55 member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) 172.08/44.55 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.55 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.55 U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.55 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.55 172.08/44.55 The set Q consists of the following terms: 172.08/44.55 172.08/44.55 member1_in_ag(x0) 172.08/44.55 member_in_gg(x0, x1) 172.08/44.55 U6_ag(x0, x1, x2) 172.08/44.55 U5_gg(x0, x1, x2, x3) 172.08/44.55 172.08/44.55 We have to consider all (P,Q,R)-chains. 172.08/44.55 ---------------------------------------- 172.08/44.55 172.08/44.55 (40) TransformationProof (EQUIVALENT) 172.08/44.55 By instantiating [LPAR04] the rule U3_GGGG(z0, z1, z2, .(z3, z4), z3, member_out_gg(z3, .(z3, z4))) -> REACH_IN_GGGG(z3, z1, z2, .(z3, .(z3, z4))) we obtained the following new rules [LPAR04]: 172.08/44.57 172.08/44.57 (U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member_out_gg(z0, .(z0, .(z0, z3)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))),U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member_out_gg(z0, .(z0, .(z0, z3)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))))) 172.08/44.57 (U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, member_out_gg(z0, .(z0, .(z3, z4)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))),U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, member_out_gg(z0, .(z0, .(z3, z4)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))))) 172.08/44.57 (U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))),U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))))) 172.08/44.57 (U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))),U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))))) 172.08/44.57 172.08/44.57 172.08/44.57 ---------------------------------------- 172.08/44.57 172.08/44.57 (41) 172.08/44.57 Obligation: 172.08/44.57 Q DP problem: 172.08/44.57 The TRS P consists of the following rules: 172.08/44.57 172.08/44.57 U3_GGGG(z0, z1, z2, .(z3, z4), z5, member_out_gg(z5, .(z3, z4))) -> REACH_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4))) 172.08/44.57 REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) -> U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), member1_out_ag(x2, .(x2, x3))) 172.08/44.57 REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), member1_out_ag(x2, .(x2, x3))) 172.08/44.57 REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) -> U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3))) 172.08/44.57 REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(x5, z0, .(z0, z4), member_in_gg(x5, .(z0, z4)))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(x5, z0, .(z4, z5), member_in_gg(x5, .(z4, z5)))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(x5, z0, .(z0, z4), member_in_gg(x5, .(z0, z4)))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(x5, z0, .(z4, z5), member_in_gg(x5, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member_out_gg(z0, .(z0, .(z0, z3)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, member_out_gg(z0, .(z0, .(z3, z4)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) 172.08/44.57 172.08/44.57 The TRS R consists of the following rules: 172.08/44.57 172.08/44.57 member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.57 member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) 172.08/44.57 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.57 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.57 U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.57 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.57 172.08/44.57 The set Q consists of the following terms: 172.08/44.57 172.08/44.57 member1_in_ag(x0) 172.08/44.57 member_in_gg(x0, x1) 172.08/44.57 U6_ag(x0, x1, x2) 172.08/44.57 U5_gg(x0, x1, x2, x3) 172.08/44.57 172.08/44.57 We have to consider all (P,Q,R)-chains. 172.08/44.57 ---------------------------------------- 172.08/44.57 172.08/44.57 (42) TransformationProof (EQUIVALENT) 172.08/44.57 By instantiating [LPAR04] the rule U3_GGGG(z0, z1, z2, .(z3, z4), z5, member_out_gg(z5, .(z3, z4))) -> REACH_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4))) we obtained the following new rules [LPAR04]: 172.08/44.57 172.08/44.57 (U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member_out_gg(z0, .(z0, .(z0, z3)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))),U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member_out_gg(z0, .(z0, .(z0, z3)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))))) 172.08/44.57 (U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, member_out_gg(z0, .(z0, .(z3, z4)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))),U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, member_out_gg(z0, .(z0, .(z3, z4)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))))) 172.08/44.57 (U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))),U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))))) 172.08/44.57 (U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))),U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))))) 172.08/44.57 (U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, member_out_gg(z2, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))),U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, member_out_gg(z2, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))))) 172.08/44.57 (U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, member_out_gg(z2, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))),U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, member_out_gg(z2, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))))) 172.08/44.57 (U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, member_out_gg(z5, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))),U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, member_out_gg(z5, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))))) 172.08/44.57 (U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, member_out_gg(z6, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))),U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, member_out_gg(z6, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))))) 172.08/44.57 172.08/44.57 172.08/44.57 ---------------------------------------- 172.08/44.57 172.08/44.57 (43) 172.08/44.57 Obligation: 172.08/44.57 Q DP problem: 172.08/44.57 The TRS P consists of the following rules: 172.08/44.57 172.08/44.57 REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) -> U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), member1_out_ag(x2, .(x2, x3))) 172.08/44.57 REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), member1_out_ag(x2, .(x2, x3))) 172.08/44.57 REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) -> U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3))) 172.08/44.57 REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(x5, z0, .(z0, z4), member_in_gg(x5, .(z0, z4)))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(x5, z0, .(z4, z5), member_in_gg(x5, .(z4, z5)))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(x5, z0, .(z0, z4), member_in_gg(x5, .(z0, z4)))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(x5, z0, .(z4, z5), member_in_gg(x5, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member_out_gg(z0, .(z0, .(z0, z3)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, member_out_gg(z0, .(z0, .(z3, z4)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, member_out_gg(z2, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, member_out_gg(z2, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, member_out_gg(z5, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, member_out_gg(z6, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 172.08/44.57 172.08/44.57 The TRS R consists of the following rules: 172.08/44.57 172.08/44.57 member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.57 member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) 172.08/44.57 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.57 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.57 U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.57 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.57 172.08/44.57 The set Q consists of the following terms: 172.08/44.57 172.08/44.57 member1_in_ag(x0) 172.08/44.57 member_in_gg(x0, x1) 172.08/44.57 U6_ag(x0, x1, x2) 172.08/44.57 U5_gg(x0, x1, x2, x3) 172.08/44.57 172.08/44.57 We have to consider all (P,Q,R)-chains. 172.08/44.57 ---------------------------------------- 172.08/44.57 172.08/44.57 (44) TransformationProof (EQUIVALENT) 172.08/44.57 By instantiating [LPAR04] the rule REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) -> U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), member1_out_ag(x2, .(x2, x3))) we obtained the following new rules [LPAR04]: 172.08/44.57 172.08/44.57 (REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))),REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2)))) 172.08/44.57 (REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))),REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2)))) 172.08/44.57 (REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(z2, .(z2, z3))),REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(z2, .(z2, z3)))) 172.08/44.57 (REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3))),REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3)))) 172.08/44.57 172.08/44.57 172.08/44.57 ---------------------------------------- 172.08/44.57 172.08/44.57 (45) 172.08/44.57 Obligation: 172.08/44.57 Q DP problem: 172.08/44.57 The TRS P consists of the following rules: 172.08/44.57 172.08/44.57 REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), member1_out_ag(x2, .(x2, x3))) 172.08/44.57 REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) -> U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3))) 172.08/44.57 REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(x5, z0, .(z0, z4), member_in_gg(x5, .(z0, z4)))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(x5, z0, .(z4, z5), member_in_gg(x5, .(z4, z5)))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(x5, z0, .(z0, z4), member_in_gg(x5, .(z0, z4)))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(x5, z0, .(z4, z5), member_in_gg(x5, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member_out_gg(z0, .(z0, .(z0, z3)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, member_out_gg(z0, .(z0, .(z3, z4)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, member_out_gg(z2, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, member_out_gg(z2, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, member_out_gg(z5, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, member_out_gg(z6, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 172.08/44.57 The TRS R consists of the following rules: 172.08/44.57 172.08/44.57 member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.57 member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) 172.08/44.57 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.57 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.57 U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.57 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.57 172.08/44.57 The set Q consists of the following terms: 172.08/44.57 172.08/44.57 member1_in_ag(x0) 172.08/44.57 member_in_gg(x0, x1) 172.08/44.57 U6_ag(x0, x1, x2) 172.08/44.57 U5_gg(x0, x1, x2, x3) 172.08/44.57 172.08/44.57 We have to consider all (P,Q,R)-chains. 172.08/44.57 ---------------------------------------- 172.08/44.57 172.08/44.57 (46) TransformationProof (EQUIVALENT) 172.08/44.57 By instantiating [LPAR04] the rule REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), member1_out_ag(x2, .(x2, x3))) we obtained the following new rules [LPAR04]: 172.08/44.57 172.08/44.57 (REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))),REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2)))) 172.08/44.57 (REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))),REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2)))) 172.08/44.57 (REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(z2, .(z2, z3))),REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(z2, .(z2, z3)))) 172.08/44.57 (REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3))),REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3)))) 172.08/44.57 (REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(z2, [])), z3))),REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(z2, [])), z3)))) 172.08/44.57 (REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(z2, [])), z3))),REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(z2, [])), z3)))) 172.08/44.57 (REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U2_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), member1_out_ag(z2, .(z2, z3))),REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U2_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), member1_out_ag(z2, .(z2, z3)))) 172.08/44.57 (REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U2_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3))),REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U2_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3)))) 172.08/44.57 172.08/44.57 172.08/44.57 ---------------------------------------- 172.08/44.57 172.08/44.57 (47) 172.08/44.57 Obligation: 172.08/44.57 Q DP problem: 172.08/44.57 The TRS P consists of the following rules: 172.08/44.57 172.08/44.57 REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) -> U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3))) 172.08/44.57 REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(x5, z0, .(z0, z4), member_in_gg(x5, .(z0, z4)))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(x5, z0, .(z4, z5), member_in_gg(x5, .(z4, z5)))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(x5, z0, .(z0, z4), member_in_gg(x5, .(z0, z4)))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(x5, z0, .(z4, z5), member_in_gg(x5, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member_out_gg(z0, .(z0, .(z0, z3)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, member_out_gg(z0, .(z0, .(z3, z4)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, member_out_gg(z2, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, member_out_gg(z2, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, member_out_gg(z5, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, member_out_gg(z6, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(z2, [])), z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(z2, [])), z3))) 172.08/44.57 REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U2_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U2_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 172.08/44.57 The TRS R consists of the following rules: 172.08/44.57 172.08/44.57 member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.57 member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) 172.08/44.57 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.57 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.57 U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.57 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.57 172.08/44.57 The set Q consists of the following terms: 172.08/44.57 172.08/44.57 member1_in_ag(x0) 172.08/44.57 member_in_gg(x0, x1) 172.08/44.57 U6_ag(x0, x1, x2) 172.08/44.57 U5_gg(x0, x1, x2, x3) 172.08/44.57 172.08/44.57 We have to consider all (P,Q,R)-chains. 172.08/44.57 ---------------------------------------- 172.08/44.57 172.08/44.57 (48) TransformationProof (EQUIVALENT) 172.08/44.57 By instantiating [LPAR04] the rule REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) -> U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3))) we obtained the following new rules [LPAR04]: 172.08/44.57 172.08/44.57 (REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2))),REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2)))) 172.08/44.57 (REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2))),REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2)))) 172.08/44.57 (REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U6_ag(z2, z3, member1_in_ag(z3))),REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U6_ag(z2, z3, member1_in_ag(z3)))) 172.08/44.57 (REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U6_ag(z2, z3, member1_in_ag(z3))),REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U6_ag(z2, z3, member1_in_ag(z3)))) 172.08/44.57 172.08/44.57 172.08/44.57 ---------------------------------------- 172.08/44.57 172.08/44.57 (49) 172.08/44.57 Obligation: 172.08/44.57 Q DP problem: 172.08/44.57 The TRS P consists of the following rules: 172.08/44.57 172.08/44.57 REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(x5, z0, .(z0, z4), member_in_gg(x5, .(z0, z4)))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(x5, z0, .(z4, z5), member_in_gg(x5, .(z4, z5)))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(x5, z0, .(z0, z4), member_in_gg(x5, .(z0, z4)))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(x5, z0, .(z4, z5), member_in_gg(x5, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member_out_gg(z0, .(z0, .(z0, z3)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, member_out_gg(z0, .(z0, .(z3, z4)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, member_out_gg(z2, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, member_out_gg(z2, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, member_out_gg(z5, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, member_out_gg(z6, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(z2, [])), z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(z2, [])), z3))) 172.08/44.57 REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U2_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U2_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 172.08/44.57 The TRS R consists of the following rules: 172.08/44.57 172.08/44.57 member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.57 member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) 172.08/44.57 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.57 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.57 U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.57 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.57 172.08/44.57 The set Q consists of the following terms: 172.08/44.57 172.08/44.57 member1_in_ag(x0) 172.08/44.57 member_in_gg(x0, x1) 172.08/44.57 U6_ag(x0, x1, x2) 172.08/44.57 U5_gg(x0, x1, x2, x3) 172.08/44.57 172.08/44.57 We have to consider all (P,Q,R)-chains. 172.08/44.57 ---------------------------------------- 172.08/44.57 172.08/44.57 (50) TransformationProof (EQUIVALENT) 172.08/44.57 By instantiating [LPAR04] the rule REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3))) we obtained the following new rules [LPAR04]: 172.08/44.57 172.08/44.57 (REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2))),REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2)))) 172.08/44.57 (REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2))),REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2)))) 172.08/44.57 (REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U6_ag(z2, z3, member1_in_ag(z3))),REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U6_ag(z2, z3, member1_in_ag(z3)))) 172.08/44.57 (REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U6_ag(z2, z3, member1_in_ag(z3))),REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U6_ag(z2, z3, member1_in_ag(z3)))) 172.08/44.57 (REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), U6_ag(.(z0, .(z2, [])), z3, member1_in_ag(z3))),REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), U6_ag(.(z0, .(z2, [])), z3, member1_in_ag(z3)))) 172.08/44.57 (REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), U6_ag(.(z0, .(z2, [])), z3, member1_in_ag(z3))),REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), U6_ag(.(z0, .(z2, [])), z3, member1_in_ag(z3)))) 172.08/44.57 (REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U2_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), U6_ag(z2, z3, member1_in_ag(z3))),REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U2_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), U6_ag(z2, z3, member1_in_ag(z3)))) 172.08/44.57 (REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U2_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), U6_ag(z2, z3, member1_in_ag(z3))),REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U2_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), U6_ag(z2, z3, member1_in_ag(z3)))) 172.08/44.57 172.08/44.57 172.08/44.57 ---------------------------------------- 172.08/44.57 172.08/44.57 (51) 172.08/44.57 Obligation: 172.08/44.57 Q DP problem: 172.08/44.57 The TRS P consists of the following rules: 172.08/44.57 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(x5, z0, .(z0, z4), member_in_gg(x5, .(z0, z4)))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(x5, z0, .(z4, z5), member_in_gg(x5, .(z4, z5)))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(x5, z0, .(z0, z4), member_in_gg(x5, .(z0, z4)))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(x5, z0, .(z4, z5), member_in_gg(x5, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member_out_gg(z0, .(z0, .(z0, z3)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, member_out_gg(z0, .(z0, .(z3, z4)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, member_out_gg(z2, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, member_out_gg(z2, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, member_out_gg(z5, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, member_out_gg(z6, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(z2, [])), z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(z2, [])), z3))) 172.08/44.57 REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U2_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U2_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), U6_ag(.(z0, .(z2, [])), z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), U6_ag(.(z0, .(z2, [])), z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U2_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U2_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 172.08/44.57 The TRS R consists of the following rules: 172.08/44.57 172.08/44.57 member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.57 member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) 172.08/44.57 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.57 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.57 U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.57 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.57 172.08/44.57 The set Q consists of the following terms: 172.08/44.57 172.08/44.57 member1_in_ag(x0) 172.08/44.57 member_in_gg(x0, x1) 172.08/44.57 U6_ag(x0, x1, x2) 172.08/44.57 U5_gg(x0, x1, x2, x3) 172.08/44.57 172.08/44.57 We have to consider all (P,Q,R)-chains. 172.08/44.57 ---------------------------------------- 172.08/44.57 172.08/44.57 (52) TransformationProof (EQUIVALENT) 172.08/44.57 By instantiating [LPAR04] the rule U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) we obtained the following new rules [LPAR04]: 172.08/44.57 172.08/44.57 (U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z3))))),U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z3)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member_out_gg(z0, .(z0, .(z0, .(z3, z4))))),U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member_out_gg(z0, .(z0, .(z0, .(z3, z4)))))) 172.08/44.57 172.08/44.57 172.08/44.57 ---------------------------------------- 172.08/44.57 172.08/44.57 (53) 172.08/44.57 Obligation: 172.08/44.57 Q DP problem: 172.08/44.57 The TRS P consists of the following rules: 172.08/44.57 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(x5, z0, .(z0, z4), member_in_gg(x5, .(z0, z4)))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(x5, z0, .(z4, z5), member_in_gg(x5, .(z4, z5)))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(x5, z0, .(z0, z4), member_in_gg(x5, .(z0, z4)))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(x5, z0, .(z4, z5), member_in_gg(x5, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member_out_gg(z0, .(z0, .(z0, z3)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, member_out_gg(z0, .(z0, .(z3, z4)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, member_out_gg(z2, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, member_out_gg(z2, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, member_out_gg(z5, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, member_out_gg(z6, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(z2, [])), z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(z2, [])), z3))) 172.08/44.57 REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U2_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U2_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), U6_ag(.(z0, .(z2, [])), z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), U6_ag(.(z0, .(z2, [])), z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U2_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U2_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z3))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member_out_gg(z0, .(z0, .(z0, .(z3, z4))))) 172.08/44.57 172.08/44.57 The TRS R consists of the following rules: 172.08/44.57 172.08/44.57 member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.57 member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) 172.08/44.57 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.57 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.57 U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.57 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.57 172.08/44.57 The set Q consists of the following terms: 172.08/44.57 172.08/44.57 member1_in_ag(x0) 172.08/44.57 member_in_gg(x0, x1) 172.08/44.57 U6_ag(x0, x1, x2) 172.08/44.57 U5_gg(x0, x1, x2, x3) 172.08/44.57 172.08/44.57 We have to consider all (P,Q,R)-chains. 172.08/44.57 ---------------------------------------- 172.08/44.57 172.08/44.57 (54) TransformationProof (EQUIVALENT) 172.08/44.57 By instantiating [LPAR04] the rule U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) we obtained the following new rules [LPAR04]: 172.08/44.57 172.08/44.57 (U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z3))))),U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z3)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member_out_gg(z0, .(z0, .(z0, .(z3, z4))))),U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member_out_gg(z0, .(z0, .(z0, .(z3, z4)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z4, .(z4, z5))))),U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z4, .(z4, z5)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg(z0, .(z0, .(z4, .(z5, z6))))),U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg(z0, .(z0, .(z4, .(z5, z6)))))) 172.08/44.57 172.08/44.57 172.08/44.57 ---------------------------------------- 172.08/44.57 172.08/44.57 (55) 172.08/44.57 Obligation: 172.08/44.57 Q DP problem: 172.08/44.57 The TRS P consists of the following rules: 172.08/44.57 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(x5, z0, .(z0, z4), member_in_gg(x5, .(z0, z4)))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(x5, z0, .(z4, z5), member_in_gg(x5, .(z4, z5)))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(x5, z0, .(z0, z4), member_in_gg(x5, .(z0, z4)))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(x5, z0, .(z4, z5), member_in_gg(x5, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member_out_gg(z0, .(z0, .(z0, z3)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, member_out_gg(z0, .(z0, .(z3, z4)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, member_out_gg(z2, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, member_out_gg(z2, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, member_out_gg(z5, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, member_out_gg(z6, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(z2, [])), z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(z2, [])), z3))) 172.08/44.57 REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U2_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U2_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), U6_ag(.(z0, .(z2, [])), z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), U6_ag(.(z0, .(z2, [])), z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U2_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U2_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z3))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member_out_gg(z0, .(z0, .(z0, .(z3, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z4, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg(z0, .(z0, .(z4, .(z5, z6))))) 172.08/44.57 172.08/44.57 The TRS R consists of the following rules: 172.08/44.57 172.08/44.57 member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.57 member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) 172.08/44.57 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.57 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.57 U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.57 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.57 172.08/44.57 The set Q consists of the following terms: 172.08/44.57 172.08/44.57 member1_in_ag(x0) 172.08/44.57 member_in_gg(x0, x1) 172.08/44.57 U6_ag(x0, x1, x2) 172.08/44.57 U5_gg(x0, x1, x2, x3) 172.08/44.57 172.08/44.57 We have to consider all (P,Q,R)-chains. 172.08/44.57 ---------------------------------------- 172.08/44.57 172.08/44.57 (56) TransformationProof (EQUIVALENT) 172.08/44.57 By instantiating [LPAR04] the rule U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) we obtained the following new rules [LPAR04]: 172.08/44.57 172.08/44.57 (U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z3))))),U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z3)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member_out_gg(z0, .(z0, .(z0, .(z3, z4))))),U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member_out_gg(z0, .(z0, .(z0, .(z3, z4)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z4))))),U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z4)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z0, .(z4, z5))))),U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z0, .(z4, z5)))))) 172.08/44.57 172.08/44.57 172.08/44.57 ---------------------------------------- 172.08/44.57 172.08/44.57 (57) 172.08/44.57 Obligation: 172.08/44.57 Q DP problem: 172.08/44.57 The TRS P consists of the following rules: 172.08/44.57 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(x5, z0, .(z0, z4), member_in_gg(x5, .(z0, z4)))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(x5, z0, .(z4, z5), member_in_gg(x5, .(z4, z5)))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(x5, z0, .(z0, z4), member_in_gg(x5, .(z0, z4)))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(x5, z0, .(z4, z5), member_in_gg(x5, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member_out_gg(z0, .(z0, .(z0, z3)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, member_out_gg(z0, .(z0, .(z3, z4)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, member_out_gg(z2, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, member_out_gg(z2, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, member_out_gg(z5, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, member_out_gg(z6, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(z2, [])), z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(z2, [])), z3))) 172.08/44.57 REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U2_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U2_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), U6_ag(.(z0, .(z2, [])), z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), U6_ag(.(z0, .(z2, [])), z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U2_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U2_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z3))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member_out_gg(z0, .(z0, .(z0, .(z3, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z4, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg(z0, .(z0, .(z4, .(z5, z6))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z0, .(z4, z5))))) 172.08/44.57 172.08/44.57 The TRS R consists of the following rules: 172.08/44.57 172.08/44.57 member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.57 member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) 172.08/44.57 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.57 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.57 U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.57 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.57 172.08/44.57 The set Q consists of the following terms: 172.08/44.57 172.08/44.57 member1_in_ag(x0) 172.08/44.57 member_in_gg(x0, x1) 172.08/44.57 U6_ag(x0, x1, x2) 172.08/44.57 U5_gg(x0, x1, x2, x3) 172.08/44.57 172.08/44.57 We have to consider all (P,Q,R)-chains. 172.08/44.57 ---------------------------------------- 172.08/44.57 172.08/44.57 (58) TransformationProof (EQUIVALENT) 172.08/44.57 By instantiating [LPAR04] the rule U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) we obtained the following new rules [LPAR04]: 172.08/44.57 172.08/44.57 (U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z3))))),U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z3)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member_out_gg(z0, .(z0, .(z0, .(z3, z4))))),U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member_out_gg(z0, .(z0, .(z0, .(z3, z4)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z4, .(z4, z5))))),U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z4, .(z4, z5)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg(z0, .(z0, .(z4, .(z5, z6))))),U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg(z0, .(z0, .(z4, .(z5, z6)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z4))))),U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z4)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z0, .(z4, z5))))),U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z0, .(z4, z5)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z2, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), z0, member_out_gg(z0, .(z0, .(z2, .(z2, z4))))),U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z2, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), z0, member_out_gg(z0, .(z0, .(z2, .(z2, z4)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(.(z2, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z2, .(z4, z5))))),U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(.(z2, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z2, .(z4, z5)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z4, .(z4, z5))))),U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z4, .(z4, z5)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg(z0, .(z0, .(z4, .(z5, z6))))),U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg(z0, .(z0, .(z4, .(z5, z6)))))) 172.08/44.57 172.08/44.57 172.08/44.57 ---------------------------------------- 172.08/44.57 172.08/44.57 (59) 172.08/44.57 Obligation: 172.08/44.57 Q DP problem: 172.08/44.57 The TRS P consists of the following rules: 172.08/44.57 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(x5, z0, .(z0, z4), member_in_gg(x5, .(z0, z4)))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(x5, z0, .(z4, z5), member_in_gg(x5, .(z4, z5)))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(x5, z0, .(z0, z4), member_in_gg(x5, .(z0, z4)))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(x5, z0, .(z4, z5), member_in_gg(x5, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member_out_gg(z0, .(z0, .(z0, z3)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, member_out_gg(z0, .(z0, .(z3, z4)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, member_out_gg(z2, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, member_out_gg(z2, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, member_out_gg(z5, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, member_out_gg(z6, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(z2, [])), z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(z2, [])), z3))) 172.08/44.57 REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U2_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U2_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), U6_ag(.(z0, .(z2, [])), z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), U6_ag(.(z0, .(z2, [])), z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U2_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U2_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z3))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member_out_gg(z0, .(z0, .(z0, .(z3, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z4, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg(z0, .(z0, .(z4, .(z5, z6))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z0, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z2, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), z0, member_out_gg(z0, .(z0, .(z2, .(z2, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(.(z2, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z2, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z4, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg(z0, .(z0, .(z4, .(z5, z6))))) 172.08/44.57 172.08/44.57 The TRS R consists of the following rules: 172.08/44.57 172.08/44.57 member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.57 member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) 172.08/44.57 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.57 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.57 U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.57 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.57 172.08/44.57 The set Q consists of the following terms: 172.08/44.57 172.08/44.57 member1_in_ag(x0) 172.08/44.57 member_in_gg(x0, x1) 172.08/44.57 U6_ag(x0, x1, x2) 172.08/44.57 U5_gg(x0, x1, x2, x3) 172.08/44.57 172.08/44.57 We have to consider all (P,Q,R)-chains. 172.08/44.57 ---------------------------------------- 172.08/44.57 172.08/44.57 (60) TransformationProof (EQUIVALENT) 172.08/44.57 By instantiating [LPAR04] the rule U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(x5, z0, .(z0, z4), member_in_gg(x5, .(z0, z4)))) we obtained the following new rules [LPAR04]: 172.08/44.57 172.08/44.57 (U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U5_gg(z0, z0, .(z0, .(z0, z3)), member_in_gg(z0, .(z0, .(z0, z3))))),U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U5_gg(z0, z0, .(z0, .(z0, z3)), member_in_gg(z0, .(z0, .(z0, z3)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U5_gg(z0, z0, .(z0, .(z3, z4)), member_in_gg(z0, .(z0, .(z3, z4))))),U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U5_gg(z0, z0, .(z0, .(z3, z4)), member_in_gg(z0, .(z0, .(z3, z4)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(x2, z0, .(z0, .(z0, z4)), member_in_gg(x2, .(z0, .(z0, z4))))),U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(x2, z0, .(z0, .(z0, z4)), member_in_gg(x2, .(z0, .(z0, z4)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(x2, z0, .(z0, .(z4, z5)), member_in_gg(x2, .(z0, .(z4, z5))))),U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(x2, z0, .(z0, .(z4, z5)), member_in_gg(x2, .(z0, .(z4, z5)))))) 172.08/44.57 172.08/44.57 172.08/44.57 ---------------------------------------- 172.08/44.57 172.08/44.57 (61) 172.08/44.57 Obligation: 172.08/44.57 Q DP problem: 172.08/44.57 The TRS P consists of the following rules: 172.08/44.57 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(x5, z0, .(z4, z5), member_in_gg(x5, .(z4, z5)))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(x5, z0, .(z0, z4), member_in_gg(x5, .(z0, z4)))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(x5, z0, .(z4, z5), member_in_gg(x5, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member_out_gg(z0, .(z0, .(z0, z3)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, member_out_gg(z0, .(z0, .(z3, z4)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, member_out_gg(z2, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, member_out_gg(z2, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, member_out_gg(z5, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, member_out_gg(z6, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(z2, [])), z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(z2, [])), z3))) 172.08/44.57 REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U2_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U2_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), U6_ag(.(z0, .(z2, [])), z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), U6_ag(.(z0, .(z2, [])), z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U2_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U2_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z3))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member_out_gg(z0, .(z0, .(z0, .(z3, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z4, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg(z0, .(z0, .(z4, .(z5, z6))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z0, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z2, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), z0, member_out_gg(z0, .(z0, .(z2, .(z2, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(.(z2, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z2, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z4, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg(z0, .(z0, .(z4, .(z5, z6))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U5_gg(z0, z0, .(z0, .(z0, z3)), member_in_gg(z0, .(z0, .(z0, z3))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U5_gg(z0, z0, .(z0, .(z3, z4)), member_in_gg(z0, .(z0, .(z3, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(x2, z0, .(z0, .(z0, z4)), member_in_gg(x2, .(z0, .(z0, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(x2, z0, .(z0, .(z4, z5)), member_in_gg(x2, .(z0, .(z4, z5))))) 172.08/44.57 172.08/44.57 The TRS R consists of the following rules: 172.08/44.57 172.08/44.57 member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.57 member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) 172.08/44.57 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.57 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.57 U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.57 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.57 172.08/44.57 The set Q consists of the following terms: 172.08/44.57 172.08/44.57 member1_in_ag(x0) 172.08/44.57 member_in_gg(x0, x1) 172.08/44.57 U6_ag(x0, x1, x2) 172.08/44.57 U5_gg(x0, x1, x2, x3) 172.08/44.57 172.08/44.57 We have to consider all (P,Q,R)-chains. 172.08/44.57 ---------------------------------------- 172.08/44.57 172.08/44.57 (62) TransformationProof (EQUIVALENT) 172.08/44.57 By instantiating [LPAR04] the rule U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(x5, z0, .(z4, z5), member_in_gg(x5, .(z4, z5)))) we obtained the following new rules [LPAR04]: 172.08/44.57 172.08/44.57 (U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U5_gg(z0, z0, .(z0, .(z0, z3)), member_in_gg(z0, .(z0, .(z0, z3))))),U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U5_gg(z0, z0, .(z0, .(z0, z3)), member_in_gg(z0, .(z0, .(z0, z3)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U5_gg(z0, z0, .(z0, .(z3, z4)), member_in_gg(z0, .(z0, .(z3, z4))))),U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U5_gg(z0, z0, .(z0, .(z3, z4)), member_in_gg(z0, .(z0, .(z3, z4)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(x2, z0, .(z0, .(z0, z4)), member_in_gg(x2, .(z0, .(z0, z4))))),U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(x2, z0, .(z0, .(z0, z4)), member_in_gg(x2, .(z0, .(z0, z4)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(x2, z0, .(z0, .(z4, z5)), member_in_gg(x2, .(z0, .(z4, z5))))),U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(x2, z0, .(z0, .(z4, z5)), member_in_gg(x2, .(z0, .(z4, z5)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), x2, U5_gg(x2, z0, .(z4, .(z4, z5)), member_in_gg(x2, .(z4, .(z4, z5))))),U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), x2, U5_gg(x2, z0, .(z4, .(z4, z5)), member_in_gg(x2, .(z4, .(z4, z5)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), x2, U5_gg(x2, z0, .(z4, .(z5, z6)), member_in_gg(x2, .(z4, .(z5, z6))))),U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), x2, U5_gg(x2, z0, .(z4, .(z5, z6)), member_in_gg(x2, .(z4, .(z5, z6)))))) 172.08/44.57 172.08/44.57 172.08/44.57 ---------------------------------------- 172.08/44.57 172.08/44.57 (63) 172.08/44.57 Obligation: 172.08/44.57 Q DP problem: 172.08/44.57 The TRS P consists of the following rules: 172.08/44.57 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(x5, z0, .(z0, z4), member_in_gg(x5, .(z0, z4)))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(x5, z0, .(z4, z5), member_in_gg(x5, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member_out_gg(z0, .(z0, .(z0, z3)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, member_out_gg(z0, .(z0, .(z3, z4)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, member_out_gg(z2, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, member_out_gg(z2, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, member_out_gg(z5, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, member_out_gg(z6, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(z2, [])), z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(z2, [])), z3))) 172.08/44.57 REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U2_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U2_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), U6_ag(.(z0, .(z2, [])), z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), U6_ag(.(z0, .(z2, [])), z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U2_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U2_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z3))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member_out_gg(z0, .(z0, .(z0, .(z3, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z4, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg(z0, .(z0, .(z4, .(z5, z6))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z0, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z2, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), z0, member_out_gg(z0, .(z0, .(z2, .(z2, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(.(z2, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z2, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z4, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg(z0, .(z0, .(z4, .(z5, z6))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U5_gg(z0, z0, .(z0, .(z0, z3)), member_in_gg(z0, .(z0, .(z0, z3))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U5_gg(z0, z0, .(z0, .(z3, z4)), member_in_gg(z0, .(z0, .(z3, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(x2, z0, .(z0, .(z0, z4)), member_in_gg(x2, .(z0, .(z0, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(x2, z0, .(z0, .(z4, z5)), member_in_gg(x2, .(z0, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), x2, U5_gg(x2, z0, .(z4, .(z4, z5)), member_in_gg(x2, .(z4, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), x2, U5_gg(x2, z0, .(z4, .(z5, z6)), member_in_gg(x2, .(z4, .(z5, z6))))) 172.08/44.57 172.08/44.57 The TRS R consists of the following rules: 172.08/44.57 172.08/44.57 member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.57 member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) 172.08/44.57 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.57 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.57 U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.57 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.57 172.08/44.57 The set Q consists of the following terms: 172.08/44.57 172.08/44.57 member1_in_ag(x0) 172.08/44.57 member_in_gg(x0, x1) 172.08/44.57 U6_ag(x0, x1, x2) 172.08/44.57 U5_gg(x0, x1, x2, x3) 172.08/44.57 172.08/44.57 We have to consider all (P,Q,R)-chains. 172.08/44.57 ---------------------------------------- 172.08/44.57 172.08/44.57 (64) TransformationProof (EQUIVALENT) 172.08/44.57 By instantiating [LPAR04] the rule U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(x5, z0, .(z0, z4), member_in_gg(x5, .(z0, z4)))) we obtained the following new rules [LPAR04]: 172.08/44.57 172.08/44.57 (U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U5_gg(z0, z0, .(z0, .(z0, z3)), member_in_gg(z0, .(z0, .(z0, z3))))),U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U5_gg(z0, z0, .(z0, .(z0, z3)), member_in_gg(z0, .(z0, .(z0, z3)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U5_gg(z0, z0, .(z0, .(z3, z4)), member_in_gg(z0, .(z0, .(z3, z4))))),U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U5_gg(z0, z0, .(z0, .(z3, z4)), member_in_gg(z0, .(z0, .(z3, z4)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z0, .(z0, z4))), x5, U5_gg(x5, z0, .(z0, .(z0, z4)), member_in_gg(x5, .(z0, .(z0, z4))))),U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(x2, z0, .(z0, .(z0, z4)), member_in_gg(x2, .(z0, .(z0, z4)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z0, .(z4, z5))), x5, U5_gg(x5, z0, .(z0, .(z4, z5)), member_in_gg(x5, .(z0, .(z4, z5))))),U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(x2, z0, .(z0, .(z4, z5)), member_in_gg(x2, .(z0, .(z4, z5)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), x5, U5_gg(x5, z0, .(z0, .(z0, z3)), member_in_gg(x5, .(z0, .(z0, z3))))),U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), x5, U5_gg(x5, z0, .(z0, .(z0, z3)), member_in_gg(x5, .(z0, .(z0, z3)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), x5, U5_gg(x5, z0, .(z0, .(z3, z4)), member_in_gg(x5, .(z0, .(z3, z4))))),U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), x5, U5_gg(x5, z0, .(z0, .(z3, z4)), member_in_gg(x5, .(z0, .(z3, z4)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x5, U5_gg(x5, z0, .(z0, .(z0, z4)), member_in_gg(x5, .(z0, .(z0, z4))))),U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x5, U5_gg(x5, z0, .(z0, .(z0, z4)), member_in_gg(x5, .(z0, .(z0, z4)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x5, U5_gg(x5, z0, .(z0, .(z4, z5)), member_in_gg(x5, .(z0, .(z4, z5))))),U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x5, U5_gg(x5, z0, .(z0, .(z4, z5)), member_in_gg(x5, .(z0, .(z4, z5)))))) 172.08/44.57 172.08/44.57 172.08/44.57 ---------------------------------------- 172.08/44.57 172.08/44.57 (65) 172.08/44.57 Obligation: 172.08/44.57 Q DP problem: 172.08/44.57 The TRS P consists of the following rules: 172.08/44.57 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(x5, z0, .(z4, z5), member_in_gg(x5, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member_out_gg(z0, .(z0, .(z0, z3)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, member_out_gg(z0, .(z0, .(z3, z4)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, member_out_gg(z2, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, member_out_gg(z2, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, member_out_gg(z5, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, member_out_gg(z6, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(z2, [])), z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(z2, [])), z3))) 172.08/44.57 REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U2_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U2_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), U6_ag(.(z0, .(z2, [])), z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), U6_ag(.(z0, .(z2, [])), z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U2_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U2_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z3))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member_out_gg(z0, .(z0, .(z0, .(z3, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z4, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg(z0, .(z0, .(z4, .(z5, z6))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z0, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z2, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), z0, member_out_gg(z0, .(z0, .(z2, .(z2, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(.(z2, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z2, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z4, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg(z0, .(z0, .(z4, .(z5, z6))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U5_gg(z0, z0, .(z0, .(z0, z3)), member_in_gg(z0, .(z0, .(z0, z3))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U5_gg(z0, z0, .(z0, .(z3, z4)), member_in_gg(z0, .(z0, .(z3, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(x2, z0, .(z0, .(z0, z4)), member_in_gg(x2, .(z0, .(z0, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(x2, z0, .(z0, .(z4, z5)), member_in_gg(x2, .(z0, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), x2, U5_gg(x2, z0, .(z4, .(z4, z5)), member_in_gg(x2, .(z4, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), x2, U5_gg(x2, z0, .(z4, .(z5, z6)), member_in_gg(x2, .(z4, .(z5, z6))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), x5, U5_gg(x5, z0, .(z0, .(z0, z3)), member_in_gg(x5, .(z0, .(z0, z3))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), x5, U5_gg(x5, z0, .(z0, .(z3, z4)), member_in_gg(x5, .(z0, .(z3, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x5, U5_gg(x5, z0, .(z0, .(z0, z4)), member_in_gg(x5, .(z0, .(z0, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x5, U5_gg(x5, z0, .(z0, .(z4, z5)), member_in_gg(x5, .(z0, .(z4, z5))))) 172.08/44.57 172.08/44.57 The TRS R consists of the following rules: 172.08/44.57 172.08/44.57 member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.57 member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) 172.08/44.57 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.57 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.57 U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.57 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.57 172.08/44.57 The set Q consists of the following terms: 172.08/44.57 172.08/44.57 member1_in_ag(x0) 172.08/44.57 member_in_gg(x0, x1) 172.08/44.57 U6_ag(x0, x1, x2) 172.08/44.57 U5_gg(x0, x1, x2, x3) 172.08/44.57 172.08/44.57 We have to consider all (P,Q,R)-chains. 172.08/44.57 ---------------------------------------- 172.08/44.57 172.08/44.57 (66) TransformationProof (EQUIVALENT) 172.08/44.57 By instantiating [LPAR04] the rule U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(x5, z0, .(z4, z5), member_in_gg(x5, .(z4, z5)))) we obtained the following new rules [LPAR04]: 172.08/44.57 172.08/44.57 (U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U5_gg(z0, z0, .(z0, .(z0, z3)), member_in_gg(z0, .(z0, .(z0, z3))))),U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U5_gg(z0, z0, .(z0, .(z0, z3)), member_in_gg(z0, .(z0, .(z0, z3)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U5_gg(z0, z0, .(z0, .(z3, z4)), member_in_gg(z0, .(z0, .(z3, z4))))),U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U5_gg(z0, z0, .(z0, .(z3, z4)), member_in_gg(z0, .(z0, .(z3, z4)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(x6, [])), .(.(z0, .(x6, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z0, .(z0, z4))), x6, U5_gg(x6, z0, .(z0, .(z0, z4)), member_in_gg(x6, .(z0, .(z0, z4))))),U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(x2, z0, .(z0, .(z0, z4)), member_in_gg(x2, .(z0, .(z0, z4)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(x6, [])), .(.(z0, .(x6, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z0, .(z4, z5))), x6, U5_gg(x6, z0, .(z0, .(z4, z5)), member_in_gg(x6, .(z0, .(z4, z5))))),U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(x2, z0, .(z0, .(z4, z5)), member_in_gg(x2, .(z0, .(z4, z5)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(x6, [])), .(.(z0, .(x6, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z4, .(z4, z5))), x6, U5_gg(x6, z0, .(z4, .(z4, z5)), member_in_gg(x6, .(z4, .(z4, z5))))),U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), x2, U5_gg(x2, z0, .(z4, .(z4, z5)), member_in_gg(x2, .(z4, .(z4, z5)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(x6, [])), .(.(z0, .(x6, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z4, .(z5, z6))), x6, U5_gg(x6, z0, .(z4, .(z5, z6)), member_in_gg(x6, .(z4, .(z5, z6))))),U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), x2, U5_gg(x2, z0, .(z4, .(z5, z6)), member_in_gg(x2, .(z4, .(z5, z6)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(x6, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), x6, U5_gg(x6, z0, .(z0, .(z0, z3)), member_in_gg(x6, .(z0, .(z0, z3))))),U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), x5, U5_gg(x5, z0, .(z0, .(z0, z3)), member_in_gg(x5, .(z0, .(z0, z3)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(x6, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), x6, U5_gg(x6, z0, .(z0, .(z3, z4)), member_in_gg(x6, .(z0, .(z3, z4))))),U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), x5, U5_gg(x5, z0, .(z0, .(z3, z4)), member_in_gg(x5, .(z0, .(z3, z4)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(x6, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x6, U5_gg(x6, z0, .(z0, .(z0, z4)), member_in_gg(x6, .(z0, .(z0, z4))))),U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x5, U5_gg(x5, z0, .(z0, .(z0, z4)), member_in_gg(x5, .(z0, .(z0, z4)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(x6, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x6, U5_gg(x6, z0, .(z0, .(z4, z5)), member_in_gg(x6, .(z0, .(z4, z5))))),U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x5, U5_gg(x5, z0, .(z0, .(z4, z5)), member_in_gg(x5, .(z0, .(z4, z5)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member1_out_ag(.(z0, .(x6, [])), .(.(z2, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), x6, U5_gg(x6, z0, .(z2, .(z2, z4)), member_in_gg(x6, .(z2, .(z2, z4))))),U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member1_out_ag(.(z0, .(x6, [])), .(.(z2, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), x6, U5_gg(x6, z0, .(z2, .(z2, z4)), member_in_gg(x6, .(z2, .(z2, z4)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member1_out_ag(.(z0, .(x6, [])), .(.(z2, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), x6, U5_gg(x6, z0, .(z2, .(z4, z5)), member_in_gg(x6, .(z2, .(z4, z5))))),U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member1_out_ag(.(z0, .(x6, [])), .(.(z2, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), x6, U5_gg(x6, z0, .(z2, .(z4, z5)), member_in_gg(x6, .(z2, .(z4, z5)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(x6, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), x6, U5_gg(x6, z0, .(z4, .(z4, z5)), member_in_gg(x6, .(z4, .(z4, z5))))),U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(x6, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), x6, U5_gg(x6, z0, .(z4, .(z4, z5)), member_in_gg(x6, .(z4, .(z4, z5)))))) 172.08/44.57 (U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(x6, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), x6, U5_gg(x6, z0, .(z4, .(z5, z6)), member_in_gg(x6, .(z4, .(z5, z6))))),U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(x6, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), x6, U5_gg(x6, z0, .(z4, .(z5, z6)), member_in_gg(x6, .(z4, .(z5, z6)))))) 172.08/44.57 172.08/44.57 172.08/44.57 ---------------------------------------- 172.08/44.57 172.08/44.57 (67) 172.08/44.57 Obligation: 172.08/44.57 Q DP problem: 172.08/44.57 The TRS P consists of the following rules: 172.08/44.57 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member_out_gg(z0, .(z0, .(z0, z3)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, member_out_gg(z0, .(z0, .(z3, z4)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, member_out_gg(z2, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, member_out_gg(z2, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, member_out_gg(z5, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, member_out_gg(z6, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(z2, [])), z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(z2, [])), z3))) 172.08/44.57 REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U2_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U2_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), U6_ag(.(z0, .(z2, [])), z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), U6_ag(.(z0, .(z2, [])), z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U2_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U2_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z3))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member_out_gg(z0, .(z0, .(z0, .(z3, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z4, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg(z0, .(z0, .(z4, .(z5, z6))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z0, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z2, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), z0, member_out_gg(z0, .(z0, .(z2, .(z2, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(.(z2, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z2, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z4, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg(z0, .(z0, .(z4, .(z5, z6))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U5_gg(z0, z0, .(z0, .(z0, z3)), member_in_gg(z0, .(z0, .(z0, z3))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U5_gg(z0, z0, .(z0, .(z3, z4)), member_in_gg(z0, .(z0, .(z3, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(x2, z0, .(z0, .(z0, z4)), member_in_gg(x2, .(z0, .(z0, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(x2, z0, .(z0, .(z4, z5)), member_in_gg(x2, .(z0, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), x2, U5_gg(x2, z0, .(z4, .(z4, z5)), member_in_gg(x2, .(z4, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), x2, U5_gg(x2, z0, .(z4, .(z5, z6)), member_in_gg(x2, .(z4, .(z5, z6))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), x5, U5_gg(x5, z0, .(z0, .(z0, z3)), member_in_gg(x5, .(z0, .(z0, z3))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), x5, U5_gg(x5, z0, .(z0, .(z3, z4)), member_in_gg(x5, .(z0, .(z3, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x5, U5_gg(x5, z0, .(z0, .(z0, z4)), member_in_gg(x5, .(z0, .(z0, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x5, U5_gg(x5, z0, .(z0, .(z4, z5)), member_in_gg(x5, .(z0, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member1_out_ag(.(z0, .(x6, [])), .(.(z2, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), x6, U5_gg(x6, z0, .(z2, .(z2, z4)), member_in_gg(x6, .(z2, .(z2, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member1_out_ag(.(z0, .(x6, [])), .(.(z2, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), x6, U5_gg(x6, z0, .(z2, .(z4, z5)), member_in_gg(x6, .(z2, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(x6, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), x6, U5_gg(x6, z0, .(z4, .(z4, z5)), member_in_gg(x6, .(z4, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(x6, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), x6, U5_gg(x6, z0, .(z4, .(z5, z6)), member_in_gg(x6, .(z4, .(z5, z6))))) 172.08/44.57 172.08/44.57 The TRS R consists of the following rules: 172.08/44.57 172.08/44.57 member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.57 member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) 172.08/44.57 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.57 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.57 U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.57 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.57 172.08/44.57 The set Q consists of the following terms: 172.08/44.57 172.08/44.57 member1_in_ag(x0) 172.08/44.57 member_in_gg(x0, x1) 172.08/44.57 U6_ag(x0, x1, x2) 172.08/44.57 U5_gg(x0, x1, x2, x3) 172.08/44.57 172.08/44.57 We have to consider all (P,Q,R)-chains. 172.08/44.57 ---------------------------------------- 172.08/44.57 172.08/44.57 (68) TransformationProof (EQUIVALENT) 172.08/44.57 By forward instantiating [JAR06] the rule REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(z2, .(z2, z3))) we obtained the following new rules [LPAR04]: 172.08/44.57 172.08/44.57 (REACH_IN_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x0, .(x0, x4)))) -> U2_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x0, .(x0, x4))), member1_out_ag(.(y_2, .(y_3, [])), .(.(y_2, .(y_3, [])), x3))),REACH_IN_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x0, .(x0, x4)))) -> U2_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x0, .(x0, x4))), member1_out_ag(.(y_2, .(y_3, [])), .(.(y_2, .(y_3, [])), x3)))) 172.08/44.57 172.08/44.57 172.08/44.57 ---------------------------------------- 172.08/44.57 172.08/44.57 (69) 172.08/44.57 Obligation: 172.08/44.57 Q DP problem: 172.08/44.57 The TRS P consists of the following rules: 172.08/44.57 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member_out_gg(z0, .(z0, .(z0, z3)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, member_out_gg(z0, .(z0, .(z3, z4)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, member_out_gg(z2, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, member_out_gg(z2, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, member_out_gg(z5, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, member_out_gg(z6, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(z2, [])), z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(z2, [])), z3))) 172.08/44.57 REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U2_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U2_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), U6_ag(.(z0, .(z2, [])), z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), U6_ag(.(z0, .(z2, [])), z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U2_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U2_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z3))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member_out_gg(z0, .(z0, .(z0, .(z3, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z4, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg(z0, .(z0, .(z4, .(z5, z6))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z0, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z2, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), z0, member_out_gg(z0, .(z0, .(z2, .(z2, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(.(z2, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z2, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z4, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg(z0, .(z0, .(z4, .(z5, z6))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U5_gg(z0, z0, .(z0, .(z0, z3)), member_in_gg(z0, .(z0, .(z0, z3))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U5_gg(z0, z0, .(z0, .(z3, z4)), member_in_gg(z0, .(z0, .(z3, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(x2, z0, .(z0, .(z0, z4)), member_in_gg(x2, .(z0, .(z0, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(x2, z0, .(z0, .(z4, z5)), member_in_gg(x2, .(z0, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), x2, U5_gg(x2, z0, .(z4, .(z4, z5)), member_in_gg(x2, .(z4, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), x2, U5_gg(x2, z0, .(z4, .(z5, z6)), member_in_gg(x2, .(z4, .(z5, z6))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), x5, U5_gg(x5, z0, .(z0, .(z0, z3)), member_in_gg(x5, .(z0, .(z0, z3))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), x5, U5_gg(x5, z0, .(z0, .(z3, z4)), member_in_gg(x5, .(z0, .(z3, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x5, U5_gg(x5, z0, .(z0, .(z0, z4)), member_in_gg(x5, .(z0, .(z0, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x5, U5_gg(x5, z0, .(z0, .(z4, z5)), member_in_gg(x5, .(z0, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member1_out_ag(.(z0, .(x6, [])), .(.(z2, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), x6, U5_gg(x6, z0, .(z2, .(z2, z4)), member_in_gg(x6, .(z2, .(z2, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member1_out_ag(.(z0, .(x6, [])), .(.(z2, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), x6, U5_gg(x6, z0, .(z2, .(z4, z5)), member_in_gg(x6, .(z2, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(x6, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), x6, U5_gg(x6, z0, .(z4, .(z4, z5)), member_in_gg(x6, .(z4, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(x6, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), x6, U5_gg(x6, z0, .(z4, .(z5, z6)), member_in_gg(x6, .(z4, .(z5, z6))))) 172.08/44.57 REACH_IN_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x0, .(x0, x4)))) -> U2_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x0, .(x0, x4))), member1_out_ag(.(y_2, .(y_3, [])), .(.(y_2, .(y_3, [])), x3))) 172.08/44.57 172.08/44.57 The TRS R consists of the following rules: 172.08/44.57 172.08/44.57 member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.57 member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) 172.08/44.57 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.57 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.57 U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.57 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.57 172.08/44.57 The set Q consists of the following terms: 172.08/44.57 172.08/44.57 member1_in_ag(x0) 172.08/44.57 member_in_gg(x0, x1) 172.08/44.57 U6_ag(x0, x1, x2) 172.08/44.57 U5_gg(x0, x1, x2, x3) 172.08/44.57 172.08/44.57 We have to consider all (P,Q,R)-chains. 172.08/44.57 ---------------------------------------- 172.08/44.57 172.08/44.57 (70) TransformationProof (EQUIVALENT) 172.08/44.57 By forward instantiating [JAR06] the rule REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3))) we obtained the following new rules [LPAR04]: 172.08/44.57 172.08/44.57 (REACH_IN_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x0, .(x4, x5)))) -> U2_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x0, .(x4, x5))), member1_out_ag(.(y_2, .(y_3, [])), .(.(y_2, .(y_3, [])), x3))),REACH_IN_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x0, .(x4, x5)))) -> U2_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x0, .(x4, x5))), member1_out_ag(.(y_2, .(y_3, [])), .(.(y_2, .(y_3, [])), x3)))) 172.08/44.57 172.08/44.57 172.08/44.57 ---------------------------------------- 172.08/44.57 172.08/44.57 (71) 172.08/44.57 Obligation: 172.08/44.57 Q DP problem: 172.08/44.57 The TRS P consists of the following rules: 172.08/44.57 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member_out_gg(z0, .(z0, .(z0, z3)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, member_out_gg(z0, .(z0, .(z3, z4)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, member_out_gg(z2, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, member_out_gg(z2, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, member_out_gg(z5, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, member_out_gg(z6, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(z2, [])), z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(z2, [])), z3))) 172.08/44.57 REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U2_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U2_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), U6_ag(.(z0, .(z2, [])), z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), U6_ag(.(z0, .(z2, [])), z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U2_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U2_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z3))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member_out_gg(z0, .(z0, .(z0, .(z3, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z4, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg(z0, .(z0, .(z4, .(z5, z6))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z0, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z2, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), z0, member_out_gg(z0, .(z0, .(z2, .(z2, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(.(z2, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z2, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z4, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg(z0, .(z0, .(z4, .(z5, z6))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U5_gg(z0, z0, .(z0, .(z0, z3)), member_in_gg(z0, .(z0, .(z0, z3))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U5_gg(z0, z0, .(z0, .(z3, z4)), member_in_gg(z0, .(z0, .(z3, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(x2, z0, .(z0, .(z0, z4)), member_in_gg(x2, .(z0, .(z0, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(x2, z0, .(z0, .(z4, z5)), member_in_gg(x2, .(z0, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), x2, U5_gg(x2, z0, .(z4, .(z4, z5)), member_in_gg(x2, .(z4, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), x2, U5_gg(x2, z0, .(z4, .(z5, z6)), member_in_gg(x2, .(z4, .(z5, z6))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), x5, U5_gg(x5, z0, .(z0, .(z0, z3)), member_in_gg(x5, .(z0, .(z0, z3))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), x5, U5_gg(x5, z0, .(z0, .(z3, z4)), member_in_gg(x5, .(z0, .(z3, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x5, U5_gg(x5, z0, .(z0, .(z0, z4)), member_in_gg(x5, .(z0, .(z0, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x5, U5_gg(x5, z0, .(z0, .(z4, z5)), member_in_gg(x5, .(z0, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member1_out_ag(.(z0, .(x6, [])), .(.(z2, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), x6, U5_gg(x6, z0, .(z2, .(z2, z4)), member_in_gg(x6, .(z2, .(z2, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member1_out_ag(.(z0, .(x6, [])), .(.(z2, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), x6, U5_gg(x6, z0, .(z2, .(z4, z5)), member_in_gg(x6, .(z2, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(x6, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), x6, U5_gg(x6, z0, .(z4, .(z4, z5)), member_in_gg(x6, .(z4, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(x6, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), x6, U5_gg(x6, z0, .(z4, .(z5, z6)), member_in_gg(x6, .(z4, .(z5, z6))))) 172.08/44.57 REACH_IN_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x0, .(x0, x4)))) -> U2_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x0, .(x0, x4))), member1_out_ag(.(y_2, .(y_3, [])), .(.(y_2, .(y_3, [])), x3))) 172.08/44.57 REACH_IN_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x0, .(x4, x5)))) -> U2_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x0, .(x4, x5))), member1_out_ag(.(y_2, .(y_3, [])), .(.(y_2, .(y_3, [])), x3))) 172.08/44.57 172.08/44.57 The TRS R consists of the following rules: 172.08/44.57 172.08/44.57 member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.57 member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) 172.08/44.57 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.57 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.57 U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.57 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.57 172.08/44.57 The set Q consists of the following terms: 172.08/44.57 172.08/44.57 member1_in_ag(x0) 172.08/44.57 member_in_gg(x0, x1) 172.08/44.57 U6_ag(x0, x1, x2) 172.08/44.57 U5_gg(x0, x1, x2, x3) 172.08/44.57 172.08/44.57 We have to consider all (P,Q,R)-chains. 172.08/44.57 ---------------------------------------- 172.08/44.57 172.08/44.57 (72) TransformationProof (EQUIVALENT) 172.08/44.57 By forward instantiating [JAR06] the rule REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U2_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), member1_out_ag(z2, .(z2, z3))) we obtained the following new rules [LPAR04]: 172.08/44.57 172.08/44.57 (REACH_IN_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x4, .(x4, x5)))) -> U2_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x4, .(x4, x5))), member1_out_ag(.(y_2, .(y_3, [])), .(.(y_2, .(y_3, [])), x3))),REACH_IN_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x4, .(x4, x5)))) -> U2_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x4, .(x4, x5))), member1_out_ag(.(y_2, .(y_3, [])), .(.(y_2, .(y_3, [])), x3)))) 172.08/44.57 172.08/44.57 172.08/44.57 ---------------------------------------- 172.08/44.57 172.08/44.57 (73) 172.08/44.57 Obligation: 172.08/44.57 Q DP problem: 172.08/44.57 The TRS P consists of the following rules: 172.08/44.57 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member_out_gg(z0, .(z0, .(z0, z3)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, member_out_gg(z0, .(z0, .(z3, z4)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, member_out_gg(z2, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, member_out_gg(z2, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, member_out_gg(z5, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, member_out_gg(z6, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(z2, [])), z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(z2, [])), z3))) 172.08/44.57 REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U2_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), U6_ag(.(z0, .(z2, [])), z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), U6_ag(.(z0, .(z2, [])), z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U2_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U2_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z3))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member_out_gg(z0, .(z0, .(z0, .(z3, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z4, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg(z0, .(z0, .(z4, .(z5, z6))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z0, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z2, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), z0, member_out_gg(z0, .(z0, .(z2, .(z2, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(.(z2, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z2, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z4, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg(z0, .(z0, .(z4, .(z5, z6))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U5_gg(z0, z0, .(z0, .(z0, z3)), member_in_gg(z0, .(z0, .(z0, z3))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U5_gg(z0, z0, .(z0, .(z3, z4)), member_in_gg(z0, .(z0, .(z3, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(x2, z0, .(z0, .(z0, z4)), member_in_gg(x2, .(z0, .(z0, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(x2, z0, .(z0, .(z4, z5)), member_in_gg(x2, .(z0, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), x2, U5_gg(x2, z0, .(z4, .(z4, z5)), member_in_gg(x2, .(z4, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), x2, U5_gg(x2, z0, .(z4, .(z5, z6)), member_in_gg(x2, .(z4, .(z5, z6))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), x5, U5_gg(x5, z0, .(z0, .(z0, z3)), member_in_gg(x5, .(z0, .(z0, z3))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), x5, U5_gg(x5, z0, .(z0, .(z3, z4)), member_in_gg(x5, .(z0, .(z3, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x5, U5_gg(x5, z0, .(z0, .(z0, z4)), member_in_gg(x5, .(z0, .(z0, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x5, U5_gg(x5, z0, .(z0, .(z4, z5)), member_in_gg(x5, .(z0, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member1_out_ag(.(z0, .(x6, [])), .(.(z2, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), x6, U5_gg(x6, z0, .(z2, .(z2, z4)), member_in_gg(x6, .(z2, .(z2, z4))))) 172.08/44.57 U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member1_out_ag(.(z0, .(x6, [])), .(.(z2, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), x6, U5_gg(x6, z0, .(z2, .(z4, z5)), member_in_gg(x6, .(z2, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(x6, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), x6, U5_gg(x6, z0, .(z4, .(z4, z5)), member_in_gg(x6, .(z4, .(z4, z5))))) 172.08/44.57 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(x6, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), x6, U5_gg(x6, z0, .(z4, .(z5, z6)), member_in_gg(x6, .(z4, .(z5, z6))))) 172.08/44.57 REACH_IN_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x0, .(x0, x4)))) -> U2_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x0, .(x0, x4))), member1_out_ag(.(y_2, .(y_3, [])), .(.(y_2, .(y_3, [])), x3))) 172.08/44.57 REACH_IN_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x0, .(x4, x5)))) -> U2_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x0, .(x4, x5))), member1_out_ag(.(y_2, .(y_3, [])), .(.(y_2, .(y_3, [])), x3))) 172.08/44.57 REACH_IN_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x4, .(x4, x5)))) -> U2_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x4, .(x4, x5))), member1_out_ag(.(y_2, .(y_3, [])), .(.(y_2, .(y_3, [])), x3))) 172.08/44.57 172.08/44.57 The TRS R consists of the following rules: 172.08/44.57 172.08/44.57 member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.57 member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) 172.08/44.57 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.57 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.57 U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.57 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.57 172.08/44.57 The set Q consists of the following terms: 172.08/44.57 172.08/44.57 member1_in_ag(x0) 172.08/44.57 member_in_gg(x0, x1) 172.08/44.57 U6_ag(x0, x1, x2) 172.08/44.57 U5_gg(x0, x1, x2, x3) 172.08/44.57 172.08/44.57 We have to consider all (P,Q,R)-chains. 172.08/44.57 ---------------------------------------- 172.08/44.57 172.08/44.57 (74) TransformationProof (EQUIVALENT) 172.08/44.57 By forward instantiating [JAR06] the rule REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U2_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3))) we obtained the following new rules [LPAR04]: 172.08/44.57 172.08/44.57 (REACH_IN_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x4, .(x5, x6)))) -> U2_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x4, .(x5, x6))), member1_out_ag(.(y_2, .(y_3, [])), .(.(y_2, .(y_3, [])), x3))),REACH_IN_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x4, .(x5, x6)))) -> U2_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x4, .(x5, x6))), member1_out_ag(.(y_2, .(y_3, [])), .(.(y_2, .(y_3, [])), x3)))) 172.08/44.57 172.08/44.57 172.08/44.57 ---------------------------------------- 172.08/44.57 172.08/44.57 (75) 172.08/44.57 Obligation: 172.08/44.57 Q DP problem: 172.08/44.57 The TRS P consists of the following rules: 172.08/44.57 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member_out_gg(z0, .(z0, .(z0, z3)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, member_out_gg(z0, .(z0, .(z3, z4)))) -> REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, member_out_gg(z2, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, member_out_gg(z2, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, member_out_gg(z5, .(z0, .(z0, z4)))) -> REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) 172.08/44.57 U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, member_out_gg(z6, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(z2, [])), z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(z2, [])), z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.57 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), U6_ag(.(z0, .(z2, [])), z3, member1_in_ag(z3))) 172.08/44.58 REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), U6_ag(.(z0, .(z2, [])), z3, member1_in_ag(z3))) 172.08/44.58 REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U2_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.58 REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U2_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), U6_ag(z2, z3, member1_in_ag(z3))) 172.08/44.58 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z3))))) 172.08/44.58 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member_out_gg(z0, .(z0, .(z0, .(z3, z4))))) 172.08/44.58 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z4, .(z4, z5))))) 172.08/44.58 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg(z0, .(z0, .(z4, .(z5, z6))))) 172.08/44.58 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z4))))) 172.08/44.58 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z0, .(z4, z5))))) 172.08/44.58 U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z2, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), z0, member_out_gg(z0, .(z0, .(z2, .(z2, z4))))) 172.08/44.58 U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(.(z2, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z2, .(z4, z5))))) 172.08/44.58 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z4, .(z4, z5))))) 172.08/44.58 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg(z0, .(z0, .(z4, .(z5, z6))))) 172.08/44.58 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U5_gg(z0, z0, .(z0, .(z0, z3)), member_in_gg(z0, .(z0, .(z0, z3))))) 172.08/44.58 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U5_gg(z0, z0, .(z0, .(z3, z4)), member_in_gg(z0, .(z0, .(z3, z4))))) 172.08/44.58 U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(x2, z0, .(z0, .(z0, z4)), member_in_gg(x2, .(z0, .(z0, z4))))) 172.08/44.58 U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(x2, z0, .(z0, .(z4, z5)), member_in_gg(x2, .(z0, .(z4, z5))))) 172.08/44.58 U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), x2, U5_gg(x2, z0, .(z4, .(z4, z5)), member_in_gg(x2, .(z4, .(z4, z5))))) 172.08/44.58 U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) -> U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), x2, U5_gg(x2, z0, .(z4, .(z5, z6)), member_in_gg(x2, .(z4, .(z5, z6))))) 172.08/44.58 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), x5, U5_gg(x5, z0, .(z0, .(z0, z3)), member_in_gg(x5, .(z0, .(z0, z3))))) 172.08/44.58 U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(z0, [])), z2))) -> U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), x5, U5_gg(x5, z0, .(z0, .(z3, z4)), member_in_gg(x5, .(z0, .(z3, z4))))) 172.08/44.58 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x5, U5_gg(x5, z0, .(z0, .(z0, z4)), member_in_gg(x5, .(z0, .(z0, z4))))) 172.08/44.58 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x5, U5_gg(x5, z0, .(z0, .(z4, z5)), member_in_gg(x5, .(z0, .(z4, z5))))) 172.08/44.58 U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member1_out_ag(.(z0, .(x6, [])), .(.(z2, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), x6, U5_gg(x6, z0, .(z2, .(z2, z4)), member_in_gg(x6, .(z2, .(z2, z4))))) 172.08/44.58 U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member1_out_ag(.(z0, .(x6, [])), .(.(z2, .(z0, [])), z3))) -> U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), x6, U5_gg(x6, z0, .(z2, .(z4, z5)), member_in_gg(x6, .(z2, .(z4, z5))))) 172.08/44.58 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(x6, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), x6, U5_gg(x6, z0, .(z4, .(z4, z5)), member_in_gg(x6, .(z4, .(z4, z5))))) 172.08/44.58 U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(x6, [])), .(z2, z3))) -> U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), x6, U5_gg(x6, z0, .(z4, .(z5, z6)), member_in_gg(x6, .(z4, .(z5, z6))))) 172.08/44.58 REACH_IN_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x0, .(x0, x4)))) -> U2_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x0, .(x0, x4))), member1_out_ag(.(y_2, .(y_3, [])), .(.(y_2, .(y_3, [])), x3))) 172.08/44.58 REACH_IN_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x0, .(x4, x5)))) -> U2_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x0, .(x4, x5))), member1_out_ag(.(y_2, .(y_3, [])), .(.(y_2, .(y_3, [])), x3))) 172.08/44.58 REACH_IN_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x4, .(x4, x5)))) -> U2_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x4, .(x4, x5))), member1_out_ag(.(y_2, .(y_3, [])), .(.(y_2, .(y_3, [])), x3))) 172.08/44.58 REACH_IN_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x4, .(x5, x6)))) -> U2_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x4, .(x5, x6))), member1_out_ag(.(y_2, .(y_3, [])), .(.(y_2, .(y_3, [])), x3))) 172.08/44.58 172.08/44.58 The TRS R consists of the following rules: 172.08/44.58 172.08/44.58 member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.58 member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) 172.08/44.58 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.58 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.58 U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.58 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.58 172.08/44.58 The set Q consists of the following terms: 172.08/44.58 172.08/44.58 member1_in_ag(x0) 172.08/44.58 member_in_gg(x0, x1) 172.08/44.58 U6_ag(x0, x1, x2) 172.08/44.58 U5_gg(x0, x1, x2, x3) 172.08/44.58 172.08/44.58 We have to consider all (P,Q,R)-chains. 172.08/44.58 ---------------------------------------- 172.08/44.58 172.08/44.58 (76) PrologToPiTRSProof (SOUND) 172.08/44.58 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 172.08/44.58 172.08/44.58 reach_in_4: (b,b,b,b) 172.08/44.58 172.08/44.58 member_in_2: (b,b) 172.08/44.58 172.08/44.58 member1_in_2: (f,b) 172.08/44.58 172.08/44.58 Transforming Prolog into the following Term Rewriting System: 172.08/44.58 172.08/44.58 Pi-finite rewrite system: 172.08/44.58 The TRS R consists of the following rules: 172.08/44.58 172.08/44.58 reach_in_gggg(X, Y, Edges, Visited) -> U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges)) 172.08/44.58 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.58 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.58 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.58 U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_gggg(X, Y, Edges, Visited) 172.08/44.58 reach_in_gggg(X, Z, Edges, Visited) -> U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) 172.08/44.58 member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.58 member1_in_ag(X, .(H, L)) -> U6_ag(X, H, L, member1_in_ag(X, L)) 172.08/44.58 U6_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.58 U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 172.08/44.58 U3_gggg(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> U4_gggg(X, Z, Edges, Visited, reach_in_gggg(Y, Z, Edges, .(Y, Visited))) 172.08/44.59 U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) -> reach_out_gggg(X, Z, Edges, Visited) 172.08/44.59 172.08/44.59 The argument filtering Pi contains the following mapping: 172.08/44.59 reach_in_gggg(x1, x2, x3, x4) = reach_in_gggg(x1, x2, x3, x4) 172.08/44.59 172.08/44.59 U1_gggg(x1, x2, x3, x4, x5) = U1_gggg(x5) 172.08/44.59 172.08/44.59 member_in_gg(x1, x2) = member_in_gg(x1, x2) 172.08/44.59 172.08/44.59 .(x1, x2) = .(x1, x2) 172.08/44.59 172.08/44.59 member_out_gg(x1, x2) = member_out_gg 172.08/44.59 172.08/44.59 U5_gg(x1, x2, x3, x4) = U5_gg(x4) 172.08/44.59 172.08/44.59 [] = [] 172.08/44.59 172.08/44.59 reach_out_gggg(x1, x2, x3, x4) = reach_out_gggg 172.08/44.59 172.08/44.59 U2_gggg(x1, x2, x3, x4, x5) = U2_gggg(x2, x3, x4, x5) 172.08/44.59 172.08/44.59 member1_in_ag(x1, x2) = member1_in_ag(x2) 172.08/44.59 172.08/44.59 member1_out_ag(x1, x2) = member1_out_ag(x1) 172.08/44.59 172.08/44.59 U6_ag(x1, x2, x3, x4) = U6_ag(x4) 172.08/44.59 172.08/44.59 U3_gggg(x1, x2, x3, x4, x5, x6) = U3_gggg(x2, x3, x4, x5, x6) 172.08/44.59 172.08/44.59 U4_gggg(x1, x2, x3, x4, x5) = U4_gggg(x5) 172.08/44.59 172.08/44.59 172.08/44.59 172.08/44.59 172.08/44.59 172.08/44.59 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 172.08/44.59 172.08/44.59 172.08/44.59 172.08/44.59 ---------------------------------------- 172.08/44.59 172.08/44.59 (77) 172.08/44.59 Obligation: 172.08/44.59 Pi-finite rewrite system: 172.08/44.59 The TRS R consists of the following rules: 172.08/44.59 172.08/44.59 reach_in_gggg(X, Y, Edges, Visited) -> U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges)) 172.08/44.59 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.59 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.59 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.59 U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_gggg(X, Y, Edges, Visited) 172.08/44.59 reach_in_gggg(X, Z, Edges, Visited) -> U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) 172.08/44.59 member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.59 member1_in_ag(X, .(H, L)) -> U6_ag(X, H, L, member1_in_ag(X, L)) 172.08/44.59 U6_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.59 U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 172.08/44.59 U3_gggg(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> U4_gggg(X, Z, Edges, Visited, reach_in_gggg(Y, Z, Edges, .(Y, Visited))) 172.08/44.59 U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) -> reach_out_gggg(X, Z, Edges, Visited) 172.08/44.59 172.08/44.59 The argument filtering Pi contains the following mapping: 172.08/44.59 reach_in_gggg(x1, x2, x3, x4) = reach_in_gggg(x1, x2, x3, x4) 172.08/44.59 172.08/44.59 U1_gggg(x1, x2, x3, x4, x5) = U1_gggg(x5) 172.08/44.59 172.08/44.59 member_in_gg(x1, x2) = member_in_gg(x1, x2) 172.08/44.59 172.08/44.59 .(x1, x2) = .(x1, x2) 172.08/44.59 172.08/44.59 member_out_gg(x1, x2) = member_out_gg 172.08/44.59 172.08/44.59 U5_gg(x1, x2, x3, x4) = U5_gg(x4) 172.08/44.59 172.08/44.59 [] = [] 172.08/44.59 172.08/44.59 reach_out_gggg(x1, x2, x3, x4) = reach_out_gggg 172.08/44.59 172.08/44.59 U2_gggg(x1, x2, x3, x4, x5) = U2_gggg(x2, x3, x4, x5) 172.08/44.59 172.08/44.59 member1_in_ag(x1, x2) = member1_in_ag(x2) 172.08/44.59 172.08/44.59 member1_out_ag(x1, x2) = member1_out_ag(x1) 172.08/44.59 172.08/44.59 U6_ag(x1, x2, x3, x4) = U6_ag(x4) 172.08/44.59 172.08/44.59 U3_gggg(x1, x2, x3, x4, x5, x6) = U3_gggg(x2, x3, x4, x5, x6) 172.08/44.59 172.08/44.59 U4_gggg(x1, x2, x3, x4, x5) = U4_gggg(x5) 172.08/44.59 172.08/44.59 172.08/44.59 172.08/44.59 ---------------------------------------- 172.08/44.59 172.08/44.59 (78) DependencyPairsProof (EQUIVALENT) 172.08/44.59 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 172.08/44.59 Pi DP problem: 172.08/44.59 The TRS P consists of the following rules: 172.08/44.59 172.08/44.59 REACH_IN_GGGG(X, Y, Edges, Visited) -> U1_GGGG(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges)) 172.08/44.59 REACH_IN_GGGG(X, Y, Edges, Visited) -> MEMBER_IN_GG(.(X, .(Y, [])), Edges) 172.08/44.59 MEMBER_IN_GG(X, .(H, L)) -> U5_GG(X, H, L, member_in_gg(X, L)) 172.08/44.59 MEMBER_IN_GG(X, .(H, L)) -> MEMBER_IN_GG(X, L) 172.08/44.59 REACH_IN_GGGG(X, Z, Edges, Visited) -> U2_GGGG(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) 172.08/44.59 REACH_IN_GGGG(X, Z, Edges, Visited) -> MEMBER1_IN_AG(.(X, .(Y, [])), Edges) 172.08/44.59 MEMBER1_IN_AG(X, .(H, L)) -> U6_AG(X, H, L, member1_in_ag(X, L)) 172.08/44.59 MEMBER1_IN_AG(X, .(H, L)) -> MEMBER1_IN_AG(X, L) 172.08/44.59 U2_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_GGGG(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 172.08/44.59 U2_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> MEMBER_IN_GG(Y, Visited) 172.08/44.59 U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> U4_GGGG(X, Z, Edges, Visited, reach_in_gggg(Y, Z, Edges, .(Y, Visited))) 172.08/44.59 U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited)) 172.08/44.59 172.08/44.59 The TRS R consists of the following rules: 172.08/44.59 172.08/44.59 reach_in_gggg(X, Y, Edges, Visited) -> U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges)) 172.08/44.59 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.59 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.59 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.59 U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_gggg(X, Y, Edges, Visited) 172.08/44.59 reach_in_gggg(X, Z, Edges, Visited) -> U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) 172.08/44.59 member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.59 member1_in_ag(X, .(H, L)) -> U6_ag(X, H, L, member1_in_ag(X, L)) 172.08/44.59 U6_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.59 U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 172.08/44.59 U3_gggg(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> U4_gggg(X, Z, Edges, Visited, reach_in_gggg(Y, Z, Edges, .(Y, Visited))) 172.08/44.59 U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) -> reach_out_gggg(X, Z, Edges, Visited) 172.08/44.59 172.08/44.59 The argument filtering Pi contains the following mapping: 172.08/44.59 reach_in_gggg(x1, x2, x3, x4) = reach_in_gggg(x1, x2, x3, x4) 172.08/44.59 172.08/44.59 U1_gggg(x1, x2, x3, x4, x5) = U1_gggg(x5) 172.08/44.59 172.08/44.59 member_in_gg(x1, x2) = member_in_gg(x1, x2) 172.08/44.59 172.08/44.59 .(x1, x2) = .(x1, x2) 172.08/44.59 172.08/44.59 member_out_gg(x1, x2) = member_out_gg 172.08/44.59 172.08/44.59 U5_gg(x1, x2, x3, x4) = U5_gg(x4) 172.08/44.59 172.08/44.59 [] = [] 172.08/44.59 172.08/44.59 reach_out_gggg(x1, x2, x3, x4) = reach_out_gggg 172.08/44.59 172.08/44.59 U2_gggg(x1, x2, x3, x4, x5) = U2_gggg(x2, x3, x4, x5) 172.08/44.59 172.08/44.59 member1_in_ag(x1, x2) = member1_in_ag(x2) 172.08/44.59 172.08/44.59 member1_out_ag(x1, x2) = member1_out_ag(x1) 172.08/44.59 172.08/44.59 U6_ag(x1, x2, x3, x4) = U6_ag(x4) 172.08/44.59 172.08/44.59 U3_gggg(x1, x2, x3, x4, x5, x6) = U3_gggg(x2, x3, x4, x5, x6) 172.08/44.59 172.08/44.59 U4_gggg(x1, x2, x3, x4, x5) = U4_gggg(x5) 172.08/44.59 172.08/44.59 REACH_IN_GGGG(x1, x2, x3, x4) = REACH_IN_GGGG(x1, x2, x3, x4) 172.08/44.59 172.08/44.59 U1_GGGG(x1, x2, x3, x4, x5) = U1_GGGG(x5) 172.08/44.59 172.08/44.59 MEMBER_IN_GG(x1, x2) = MEMBER_IN_GG(x1, x2) 172.08/44.59 172.08/44.59 U5_GG(x1, x2, x3, x4) = U5_GG(x4) 172.08/44.59 172.08/44.59 U2_GGGG(x1, x2, x3, x4, x5) = U2_GGGG(x2, x3, x4, x5) 172.08/44.59 172.08/44.59 MEMBER1_IN_AG(x1, x2) = MEMBER1_IN_AG(x2) 172.08/44.59 172.08/44.59 U6_AG(x1, x2, x3, x4) = U6_AG(x4) 172.08/44.59 172.08/44.59 U3_GGGG(x1, x2, x3, x4, x5, x6) = U3_GGGG(x2, x3, x4, x5, x6) 172.08/44.59 172.08/44.59 U4_GGGG(x1, x2, x3, x4, x5) = U4_GGGG(x5) 172.08/44.59 172.08/44.59 172.08/44.59 We have to consider all (P,R,Pi)-chains 172.08/44.59 ---------------------------------------- 172.08/44.59 172.08/44.59 (79) 172.08/44.59 Obligation: 172.08/44.59 Pi DP problem: 172.08/44.59 The TRS P consists of the following rules: 172.08/44.59 172.08/44.59 REACH_IN_GGGG(X, Y, Edges, Visited) -> U1_GGGG(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges)) 172.08/44.59 REACH_IN_GGGG(X, Y, Edges, Visited) -> MEMBER_IN_GG(.(X, .(Y, [])), Edges) 172.08/44.59 MEMBER_IN_GG(X, .(H, L)) -> U5_GG(X, H, L, member_in_gg(X, L)) 172.08/44.59 MEMBER_IN_GG(X, .(H, L)) -> MEMBER_IN_GG(X, L) 172.08/44.59 REACH_IN_GGGG(X, Z, Edges, Visited) -> U2_GGGG(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) 172.08/44.59 REACH_IN_GGGG(X, Z, Edges, Visited) -> MEMBER1_IN_AG(.(X, .(Y, [])), Edges) 172.08/44.59 MEMBER1_IN_AG(X, .(H, L)) -> U6_AG(X, H, L, member1_in_ag(X, L)) 172.08/44.59 MEMBER1_IN_AG(X, .(H, L)) -> MEMBER1_IN_AG(X, L) 172.08/44.59 U2_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_GGGG(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 172.08/44.59 U2_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> MEMBER_IN_GG(Y, Visited) 172.08/44.59 U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> U4_GGGG(X, Z, Edges, Visited, reach_in_gggg(Y, Z, Edges, .(Y, Visited))) 172.08/44.59 U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited)) 172.08/44.59 172.08/44.59 The TRS R consists of the following rules: 172.08/44.59 172.08/44.59 reach_in_gggg(X, Y, Edges, Visited) -> U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges)) 172.08/44.59 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.59 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.59 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.59 U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_gggg(X, Y, Edges, Visited) 172.08/44.59 reach_in_gggg(X, Z, Edges, Visited) -> U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) 172.08/44.59 member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.59 member1_in_ag(X, .(H, L)) -> U6_ag(X, H, L, member1_in_ag(X, L)) 172.08/44.59 U6_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.59 U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 172.08/44.59 U3_gggg(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> U4_gggg(X, Z, Edges, Visited, reach_in_gggg(Y, Z, Edges, .(Y, Visited))) 172.08/44.59 U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) -> reach_out_gggg(X, Z, Edges, Visited) 172.08/44.59 172.08/44.59 The argument filtering Pi contains the following mapping: 172.08/44.59 reach_in_gggg(x1, x2, x3, x4) = reach_in_gggg(x1, x2, x3, x4) 172.08/44.59 172.08/44.59 U1_gggg(x1, x2, x3, x4, x5) = U1_gggg(x5) 172.08/44.59 172.08/44.59 member_in_gg(x1, x2) = member_in_gg(x1, x2) 172.08/44.59 172.08/44.59 .(x1, x2) = .(x1, x2) 172.08/44.59 172.08/44.59 member_out_gg(x1, x2) = member_out_gg 172.08/44.59 172.08/44.59 U5_gg(x1, x2, x3, x4) = U5_gg(x4) 172.08/44.59 172.08/44.59 [] = [] 172.08/44.59 172.08/44.59 reach_out_gggg(x1, x2, x3, x4) = reach_out_gggg 172.08/44.59 172.08/44.59 U2_gggg(x1, x2, x3, x4, x5) = U2_gggg(x2, x3, x4, x5) 172.08/44.59 172.08/44.59 member1_in_ag(x1, x2) = member1_in_ag(x2) 172.08/44.59 172.08/44.59 member1_out_ag(x1, x2) = member1_out_ag(x1) 172.08/44.59 172.08/44.59 U6_ag(x1, x2, x3, x4) = U6_ag(x4) 172.08/44.59 172.08/44.59 U3_gggg(x1, x2, x3, x4, x5, x6) = U3_gggg(x2, x3, x4, x5, x6) 172.08/44.59 172.08/44.59 U4_gggg(x1, x2, x3, x4, x5) = U4_gggg(x5) 172.08/44.59 172.08/44.59 REACH_IN_GGGG(x1, x2, x3, x4) = REACH_IN_GGGG(x1, x2, x3, x4) 172.08/44.59 172.08/44.59 U1_GGGG(x1, x2, x3, x4, x5) = U1_GGGG(x5) 172.08/44.59 172.08/44.59 MEMBER_IN_GG(x1, x2) = MEMBER_IN_GG(x1, x2) 172.08/44.59 172.08/44.59 U5_GG(x1, x2, x3, x4) = U5_GG(x4) 172.08/44.59 172.08/44.59 U2_GGGG(x1, x2, x3, x4, x5) = U2_GGGG(x2, x3, x4, x5) 172.08/44.59 172.08/44.59 MEMBER1_IN_AG(x1, x2) = MEMBER1_IN_AG(x2) 172.08/44.59 172.08/44.59 U6_AG(x1, x2, x3, x4) = U6_AG(x4) 172.08/44.59 172.08/44.59 U3_GGGG(x1, x2, x3, x4, x5, x6) = U3_GGGG(x2, x3, x4, x5, x6) 172.08/44.59 172.08/44.59 U4_GGGG(x1, x2, x3, x4, x5) = U4_GGGG(x5) 172.08/44.59 172.08/44.59 172.08/44.59 We have to consider all (P,R,Pi)-chains 172.08/44.59 ---------------------------------------- 172.08/44.59 172.08/44.59 (80) DependencyGraphProof (EQUIVALENT) 172.08/44.59 The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 7 less nodes. 172.08/44.59 ---------------------------------------- 172.08/44.59 172.08/44.59 (81) 172.08/44.59 Complex Obligation (AND) 172.08/44.59 172.08/44.59 ---------------------------------------- 172.08/44.59 172.08/44.59 (82) 172.08/44.59 Obligation: 172.08/44.59 Pi DP problem: 172.08/44.59 The TRS P consists of the following rules: 172.08/44.59 172.08/44.59 MEMBER1_IN_AG(X, .(H, L)) -> MEMBER1_IN_AG(X, L) 172.08/44.59 172.08/44.59 The TRS R consists of the following rules: 172.08/44.59 172.08/44.59 reach_in_gggg(X, Y, Edges, Visited) -> U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges)) 172.08/44.59 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.59 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.59 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.59 U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_gggg(X, Y, Edges, Visited) 172.08/44.59 reach_in_gggg(X, Z, Edges, Visited) -> U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) 172.08/44.59 member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.59 member1_in_ag(X, .(H, L)) -> U6_ag(X, H, L, member1_in_ag(X, L)) 172.08/44.59 U6_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.59 U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 172.08/44.59 U3_gggg(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> U4_gggg(X, Z, Edges, Visited, reach_in_gggg(Y, Z, Edges, .(Y, Visited))) 172.08/44.59 U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) -> reach_out_gggg(X, Z, Edges, Visited) 172.08/44.59 172.08/44.59 The argument filtering Pi contains the following mapping: 172.08/44.59 reach_in_gggg(x1, x2, x3, x4) = reach_in_gggg(x1, x2, x3, x4) 172.08/44.59 172.08/44.59 U1_gggg(x1, x2, x3, x4, x5) = U1_gggg(x5) 172.08/44.59 172.08/44.59 member_in_gg(x1, x2) = member_in_gg(x1, x2) 172.08/44.59 172.08/44.59 .(x1, x2) = .(x1, x2) 172.08/44.59 172.08/44.59 member_out_gg(x1, x2) = member_out_gg 172.08/44.59 172.08/44.59 U5_gg(x1, x2, x3, x4) = U5_gg(x4) 172.08/44.59 172.08/44.59 [] = [] 172.08/44.59 172.08/44.59 reach_out_gggg(x1, x2, x3, x4) = reach_out_gggg 172.08/44.59 172.08/44.59 U2_gggg(x1, x2, x3, x4, x5) = U2_gggg(x2, x3, x4, x5) 172.08/44.59 172.08/44.59 member1_in_ag(x1, x2) = member1_in_ag(x2) 172.08/44.59 172.08/44.59 member1_out_ag(x1, x2) = member1_out_ag(x1) 172.08/44.59 172.08/44.59 U6_ag(x1, x2, x3, x4) = U6_ag(x4) 172.08/44.59 172.08/44.59 U3_gggg(x1, x2, x3, x4, x5, x6) = U3_gggg(x2, x3, x4, x5, x6) 172.08/44.59 172.08/44.59 U4_gggg(x1, x2, x3, x4, x5) = U4_gggg(x5) 172.08/44.59 172.08/44.59 MEMBER1_IN_AG(x1, x2) = MEMBER1_IN_AG(x2) 172.08/44.59 172.08/44.59 172.08/44.59 We have to consider all (P,R,Pi)-chains 172.08/44.59 ---------------------------------------- 172.08/44.59 172.08/44.59 (83) UsableRulesProof (EQUIVALENT) 172.08/44.59 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 172.08/44.59 ---------------------------------------- 172.08/44.59 172.08/44.59 (84) 172.08/44.59 Obligation: 172.08/44.59 Pi DP problem: 172.08/44.59 The TRS P consists of the following rules: 172.08/44.59 172.08/44.59 MEMBER1_IN_AG(X, .(H, L)) -> MEMBER1_IN_AG(X, L) 172.08/44.59 172.08/44.59 R is empty. 172.08/44.59 The argument filtering Pi contains the following mapping: 172.08/44.59 .(x1, x2) = .(x1, x2) 172.08/44.59 172.08/44.59 MEMBER1_IN_AG(x1, x2) = MEMBER1_IN_AG(x2) 172.08/44.59 172.08/44.59 172.08/44.59 We have to consider all (P,R,Pi)-chains 172.08/44.59 ---------------------------------------- 172.08/44.59 172.08/44.59 (85) PiDPToQDPProof (SOUND) 172.08/44.59 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 172.08/44.59 ---------------------------------------- 172.08/44.59 172.08/44.59 (86) 172.08/44.59 Obligation: 172.08/44.59 Q DP problem: 172.08/44.59 The TRS P consists of the following rules: 172.08/44.59 172.08/44.59 MEMBER1_IN_AG(.(H, L)) -> MEMBER1_IN_AG(L) 172.08/44.59 172.08/44.59 R is empty. 172.08/44.59 Q is empty. 172.08/44.59 We have to consider all (P,Q,R)-chains. 172.08/44.59 ---------------------------------------- 172.08/44.59 172.08/44.59 (87) QDPSizeChangeProof (EQUIVALENT) 172.08/44.59 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 172.08/44.59 172.08/44.59 From the DPs we obtained the following set of size-change graphs: 172.08/44.59 *MEMBER1_IN_AG(.(H, L)) -> MEMBER1_IN_AG(L) 172.08/44.59 The graph contains the following edges 1 > 1 172.08/44.59 172.08/44.59 172.08/44.59 ---------------------------------------- 172.08/44.59 172.08/44.59 (88) 172.08/44.59 YES 172.08/44.59 172.08/44.59 ---------------------------------------- 172.08/44.59 172.08/44.59 (89) 172.08/44.59 Obligation: 172.08/44.59 Pi DP problem: 172.08/44.59 The TRS P consists of the following rules: 172.08/44.59 172.08/44.59 MEMBER_IN_GG(X, .(H, L)) -> MEMBER_IN_GG(X, L) 172.08/44.59 172.08/44.59 The TRS R consists of the following rules: 172.08/44.59 172.08/44.59 reach_in_gggg(X, Y, Edges, Visited) -> U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges)) 172.08/44.59 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.59 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.59 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.59 U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_gggg(X, Y, Edges, Visited) 172.08/44.59 reach_in_gggg(X, Z, Edges, Visited) -> U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) 172.08/44.59 member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.59 member1_in_ag(X, .(H, L)) -> U6_ag(X, H, L, member1_in_ag(X, L)) 172.08/44.59 U6_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.59 U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 172.08/44.59 U3_gggg(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> U4_gggg(X, Z, Edges, Visited, reach_in_gggg(Y, Z, Edges, .(Y, Visited))) 172.08/44.59 U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) -> reach_out_gggg(X, Z, Edges, Visited) 172.08/44.59 172.08/44.59 The argument filtering Pi contains the following mapping: 172.08/44.59 reach_in_gggg(x1, x2, x3, x4) = reach_in_gggg(x1, x2, x3, x4) 172.08/44.59 172.08/44.59 U1_gggg(x1, x2, x3, x4, x5) = U1_gggg(x5) 172.08/44.59 172.08/44.59 member_in_gg(x1, x2) = member_in_gg(x1, x2) 172.08/44.59 172.08/44.59 .(x1, x2) = .(x1, x2) 172.08/44.59 172.08/44.59 member_out_gg(x1, x2) = member_out_gg 172.08/44.59 172.08/44.59 U5_gg(x1, x2, x3, x4) = U5_gg(x4) 172.08/44.59 172.08/44.59 [] = [] 172.08/44.59 172.08/44.59 reach_out_gggg(x1, x2, x3, x4) = reach_out_gggg 172.08/44.59 172.08/44.59 U2_gggg(x1, x2, x3, x4, x5) = U2_gggg(x2, x3, x4, x5) 172.08/44.59 172.08/44.59 member1_in_ag(x1, x2) = member1_in_ag(x2) 172.08/44.59 172.08/44.59 member1_out_ag(x1, x2) = member1_out_ag(x1) 172.08/44.59 172.08/44.59 U6_ag(x1, x2, x3, x4) = U6_ag(x4) 172.08/44.59 172.08/44.59 U3_gggg(x1, x2, x3, x4, x5, x6) = U3_gggg(x2, x3, x4, x5, x6) 172.08/44.59 172.08/44.59 U4_gggg(x1, x2, x3, x4, x5) = U4_gggg(x5) 172.08/44.59 172.08/44.59 MEMBER_IN_GG(x1, x2) = MEMBER_IN_GG(x1, x2) 172.08/44.59 172.08/44.59 172.08/44.59 We have to consider all (P,R,Pi)-chains 172.08/44.59 ---------------------------------------- 172.08/44.59 172.08/44.59 (90) UsableRulesProof (EQUIVALENT) 172.08/44.59 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 172.08/44.59 ---------------------------------------- 172.08/44.59 172.08/44.59 (91) 172.08/44.59 Obligation: 172.08/44.59 Pi DP problem: 172.08/44.59 The TRS P consists of the following rules: 172.08/44.59 172.08/44.59 MEMBER_IN_GG(X, .(H, L)) -> MEMBER_IN_GG(X, L) 172.08/44.59 172.08/44.59 R is empty. 172.08/44.59 Pi is empty. 172.08/44.59 We have to consider all (P,R,Pi)-chains 172.08/44.59 ---------------------------------------- 172.08/44.59 172.08/44.59 (92) PiDPToQDPProof (EQUIVALENT) 172.08/44.59 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 172.08/44.59 ---------------------------------------- 172.08/44.59 172.08/44.59 (93) 172.08/44.59 Obligation: 172.08/44.59 Q DP problem: 172.08/44.59 The TRS P consists of the following rules: 172.08/44.59 172.08/44.59 MEMBER_IN_GG(X, .(H, L)) -> MEMBER_IN_GG(X, L) 172.08/44.59 172.08/44.59 R is empty. 172.08/44.59 Q is empty. 172.08/44.59 We have to consider all (P,Q,R)-chains. 172.08/44.59 ---------------------------------------- 172.08/44.59 172.08/44.59 (94) QDPSizeChangeProof (EQUIVALENT) 172.08/44.59 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 172.08/44.59 172.08/44.59 From the DPs we obtained the following set of size-change graphs: 172.08/44.59 *MEMBER_IN_GG(X, .(H, L)) -> MEMBER_IN_GG(X, L) 172.08/44.59 The graph contains the following edges 1 >= 1, 2 > 2 172.08/44.59 172.08/44.59 172.08/44.59 ---------------------------------------- 172.08/44.59 172.08/44.59 (95) 172.08/44.59 YES 172.08/44.59 172.08/44.59 ---------------------------------------- 172.08/44.59 172.08/44.59 (96) 172.08/44.59 Obligation: 172.08/44.59 Pi DP problem: 172.08/44.59 The TRS P consists of the following rules: 172.08/44.59 172.08/44.59 REACH_IN_GGGG(X, Z, Edges, Visited) -> U2_GGGG(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) 172.08/44.59 U2_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_GGGG(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 172.08/44.59 U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited)) 172.08/44.59 172.08/44.59 The TRS R consists of the following rules: 172.08/44.59 172.08/44.59 reach_in_gggg(X, Y, Edges, Visited) -> U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges)) 172.08/44.59 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.59 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.59 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.59 U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_gggg(X, Y, Edges, Visited) 172.08/44.59 reach_in_gggg(X, Z, Edges, Visited) -> U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) 172.08/44.59 member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.59 member1_in_ag(X, .(H, L)) -> U6_ag(X, H, L, member1_in_ag(X, L)) 172.08/44.59 U6_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.59 U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 172.08/44.59 U3_gggg(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> U4_gggg(X, Z, Edges, Visited, reach_in_gggg(Y, Z, Edges, .(Y, Visited))) 172.08/44.59 U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) -> reach_out_gggg(X, Z, Edges, Visited) 172.08/44.59 172.08/44.59 The argument filtering Pi contains the following mapping: 172.08/44.59 reach_in_gggg(x1, x2, x3, x4) = reach_in_gggg(x1, x2, x3, x4) 172.08/44.59 172.08/44.59 U1_gggg(x1, x2, x3, x4, x5) = U1_gggg(x5) 172.08/44.59 172.08/44.59 member_in_gg(x1, x2) = member_in_gg(x1, x2) 172.08/44.59 172.08/44.59 .(x1, x2) = .(x1, x2) 172.08/44.59 172.08/44.59 member_out_gg(x1, x2) = member_out_gg 172.08/44.59 172.08/44.59 U5_gg(x1, x2, x3, x4) = U5_gg(x4) 172.08/44.59 172.08/44.59 [] = [] 172.08/44.59 172.08/44.59 reach_out_gggg(x1, x2, x3, x4) = reach_out_gggg 172.08/44.59 172.08/44.59 U2_gggg(x1, x2, x3, x4, x5) = U2_gggg(x2, x3, x4, x5) 172.08/44.59 172.08/44.59 member1_in_ag(x1, x2) = member1_in_ag(x2) 172.08/44.59 172.08/44.59 member1_out_ag(x1, x2) = member1_out_ag(x1) 172.08/44.59 172.08/44.59 U6_ag(x1, x2, x3, x4) = U6_ag(x4) 172.08/44.59 172.08/44.59 U3_gggg(x1, x2, x3, x4, x5, x6) = U3_gggg(x2, x3, x4, x5, x6) 172.08/44.59 172.08/44.59 U4_gggg(x1, x2, x3, x4, x5) = U4_gggg(x5) 172.08/44.59 172.08/44.59 REACH_IN_GGGG(x1, x2, x3, x4) = REACH_IN_GGGG(x1, x2, x3, x4) 172.08/44.59 172.08/44.59 U2_GGGG(x1, x2, x3, x4, x5) = U2_GGGG(x2, x3, x4, x5) 172.08/44.59 172.08/44.59 U3_GGGG(x1, x2, x3, x4, x5, x6) = U3_GGGG(x2, x3, x4, x5, x6) 172.08/44.59 172.08/44.59 172.08/44.59 We have to consider all (P,R,Pi)-chains 172.08/44.59 ---------------------------------------- 172.08/44.59 172.08/44.59 (97) UsableRulesProof (EQUIVALENT) 172.08/44.59 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 172.08/44.59 ---------------------------------------- 172.08/44.59 172.08/44.59 (98) 172.08/44.59 Obligation: 172.08/44.59 Pi DP problem: 172.08/44.59 The TRS P consists of the following rules: 172.08/44.59 172.08/44.59 REACH_IN_GGGG(X, Z, Edges, Visited) -> U2_GGGG(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) 172.08/44.59 U2_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_GGGG(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 172.08/44.59 U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited)) 172.08/44.59 172.08/44.59 The TRS R consists of the following rules: 172.08/44.59 172.08/44.59 member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L)) 172.08/44.59 member1_in_ag(X, .(H, L)) -> U6_ag(X, H, L, member1_in_ag(X, L)) 172.08/44.59 member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) 172.08/44.59 member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) 172.08/44.59 U6_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) 172.08/44.59 U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) 172.08/44.59 172.08/44.59 The argument filtering Pi contains the following mapping: 172.08/44.59 member_in_gg(x1, x2) = member_in_gg(x1, x2) 172.08/44.59 172.08/44.59 .(x1, x2) = .(x1, x2) 172.08/44.59 172.08/44.59 member_out_gg(x1, x2) = member_out_gg 172.08/44.59 172.08/44.59 U5_gg(x1, x2, x3, x4) = U5_gg(x4) 172.08/44.59 172.08/44.59 [] = [] 172.08/44.59 172.08/44.59 member1_in_ag(x1, x2) = member1_in_ag(x2) 172.08/44.59 172.08/44.59 member1_out_ag(x1, x2) = member1_out_ag(x1) 172.08/44.59 172.08/44.59 U6_ag(x1, x2, x3, x4) = U6_ag(x4) 172.08/44.59 172.08/44.59 REACH_IN_GGGG(x1, x2, x3, x4) = REACH_IN_GGGG(x1, x2, x3, x4) 172.08/44.59 172.08/44.59 U2_GGGG(x1, x2, x3, x4, x5) = U2_GGGG(x2, x3, x4, x5) 172.08/44.59 172.08/44.59 U3_GGGG(x1, x2, x3, x4, x5, x6) = U3_GGGG(x2, x3, x4, x5, x6) 172.08/44.59 172.08/44.59 172.08/44.59 We have to consider all (P,R,Pi)-chains 172.08/44.59 ---------------------------------------- 172.08/44.59 172.08/44.59 (99) PiDPToQDPProof (SOUND) 172.08/44.59 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 172.08/44.59 ---------------------------------------- 172.08/44.59 172.08/44.59 (100) 172.08/44.59 Obligation: 172.08/44.59 Q DP problem: 172.08/44.59 The TRS P consists of the following rules: 172.08/44.59 172.08/44.59 REACH_IN_GGGG(X, Z, Edges, Visited) -> U2_GGGG(Z, Edges, Visited, member1_in_ag(Edges)) 172.08/44.59 U2_GGGG(Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])))) -> U3_GGGG(Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 172.08/44.59 U3_GGGG(Z, Edges, Visited, Y, member_out_gg) -> REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited)) 172.08/44.59 172.08/44.59 The TRS R consists of the following rules: 172.08/44.59 172.08/44.59 member1_in_ag(.(H, L)) -> member1_out_ag(H) 172.08/44.59 member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) 172.08/44.59 member_in_gg(H, .(H, L)) -> member_out_gg 172.08/44.59 member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) 172.08/44.59 U6_ag(member1_out_ag(X)) -> member1_out_ag(X) 172.08/44.59 U5_gg(member_out_gg) -> member_out_gg 172.08/44.59 172.08/44.59 The set Q consists of the following terms: 172.08/44.59 172.08/44.59 member1_in_ag(x0) 172.08/44.59 member_in_gg(x0, x1) 172.08/44.59 U6_ag(x0) 172.08/44.59 U5_gg(x0) 172.08/44.59 172.08/44.59 We have to consider all (P,Q,R)-chains. 172.08/44.59 ---------------------------------------- 172.08/44.59 172.08/44.59 (101) TransformationProof (SOUND) 172.08/44.59 By narrowing [LPAR04] the rule REACH_IN_GGGG(X, Z, Edges, Visited) -> U2_GGGG(Z, Edges, Visited, member1_in_ag(Edges)) at position [3] we obtained the following new rules [LPAR04]: 172.08/44.59 172.08/44.59 (REACH_IN_GGGG(y0, y1, .(x0, x1), y3) -> U2_GGGG(y1, .(x0, x1), y3, member1_out_ag(x0)),REACH_IN_GGGG(y0, y1, .(x0, x1), y3) -> U2_GGGG(y1, .(x0, x1), y3, member1_out_ag(x0))) 172.08/44.59 (REACH_IN_GGGG(y0, y1, .(x0, x1), y3) -> U2_GGGG(y1, .(x0, x1), y3, U6_ag(member1_in_ag(x1))),REACH_IN_GGGG(y0, y1, .(x0, x1), y3) -> U2_GGGG(y1, .(x0, x1), y3, U6_ag(member1_in_ag(x1)))) 172.08/44.59 172.08/44.59 172.08/44.59 ---------------------------------------- 172.08/44.59 172.08/44.59 (102) 172.08/44.59 Obligation: 172.08/44.59 Q DP problem: 172.08/44.59 The TRS P consists of the following rules: 172.08/44.59 172.08/44.59 U2_GGGG(Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])))) -> U3_GGGG(Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 172.08/44.59 U3_GGGG(Z, Edges, Visited, Y, member_out_gg) -> REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited)) 172.08/44.59 REACH_IN_GGGG(y0, y1, .(x0, x1), y3) -> U2_GGGG(y1, .(x0, x1), y3, member1_out_ag(x0)) 172.08/44.59 REACH_IN_GGGG(y0, y1, .(x0, x1), y3) -> U2_GGGG(y1, .(x0, x1), y3, U6_ag(member1_in_ag(x1))) 172.08/44.59 172.08/44.59 The TRS R consists of the following rules: 172.08/44.59 172.08/44.59 member1_in_ag(.(H, L)) -> member1_out_ag(H) 172.08/44.59 member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) 172.08/44.59 member_in_gg(H, .(H, L)) -> member_out_gg 172.08/44.59 member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) 172.08/44.59 U6_ag(member1_out_ag(X)) -> member1_out_ag(X) 172.08/44.59 U5_gg(member_out_gg) -> member_out_gg 172.08/44.59 172.08/44.59 The set Q consists of the following terms: 172.08/44.59 172.08/44.59 member1_in_ag(x0) 172.08/44.59 member_in_gg(x0, x1) 172.08/44.59 U6_ag(x0) 172.08/44.59 U5_gg(x0) 172.08/44.59 172.08/44.59 We have to consider all (P,Q,R)-chains. 172.08/44.59 ---------------------------------------- 172.08/44.59 172.08/44.59 (103) TransformationProof (SOUND) 172.08/44.59 By narrowing [LPAR04] the rule U2_GGGG(Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])))) -> U3_GGGG(Z, Edges, Visited, Y, member_in_gg(Y, Visited)) at position [4] we obtained the following new rules [LPAR04]: 172.08/44.59 172.08/44.59 (U2_GGGG(y0, y1, .(x0, x1), member1_out_ag(.(y3, .(x0, [])))) -> U3_GGGG(y0, y1, .(x0, x1), x0, member_out_gg),U2_GGGG(y0, y1, .(x0, x1), member1_out_ag(.(y3, .(x0, [])))) -> U3_GGGG(y0, y1, .(x0, x1), x0, member_out_gg)) 172.08/44.59 (U2_GGGG(y0, y1, .(x1, x2), member1_out_ag(.(y3, .(x0, [])))) -> U3_GGGG(y0, y1, .(x1, x2), x0, U5_gg(member_in_gg(x0, x2))),U2_GGGG(y0, y1, .(x1, x2), member1_out_ag(.(y3, .(x0, [])))) -> U3_GGGG(y0, y1, .(x1, x2), x0, U5_gg(member_in_gg(x0, x2)))) 172.08/44.59 172.08/44.59 172.08/44.59 ---------------------------------------- 172.08/44.59 172.08/44.59 (104) 172.08/44.59 Obligation: 172.08/44.59 Q DP problem: 172.08/44.59 The TRS P consists of the following rules: 172.08/44.59 172.08/44.59 U3_GGGG(Z, Edges, Visited, Y, member_out_gg) -> REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited)) 172.08/44.59 REACH_IN_GGGG(y0, y1, .(x0, x1), y3) -> U2_GGGG(y1, .(x0, x1), y3, member1_out_ag(x0)) 172.08/44.59 REACH_IN_GGGG(y0, y1, .(x0, x1), y3) -> U2_GGGG(y1, .(x0, x1), y3, U6_ag(member1_in_ag(x1))) 172.08/44.59 U2_GGGG(y0, y1, .(x0, x1), member1_out_ag(.(y3, .(x0, [])))) -> U3_GGGG(y0, y1, .(x0, x1), x0, member_out_gg) 172.08/44.59 U2_GGGG(y0, y1, .(x1, x2), member1_out_ag(.(y3, .(x0, [])))) -> U3_GGGG(y0, y1, .(x1, x2), x0, U5_gg(member_in_gg(x0, x2))) 172.08/44.59 172.08/44.59 The TRS R consists of the following rules: 172.08/44.59 172.08/44.59 member1_in_ag(.(H, L)) -> member1_out_ag(H) 172.08/44.59 member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) 172.08/44.59 member_in_gg(H, .(H, L)) -> member_out_gg 172.08/44.59 member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) 172.08/44.59 U6_ag(member1_out_ag(X)) -> member1_out_ag(X) 172.08/44.59 U5_gg(member_out_gg) -> member_out_gg 172.08/44.59 172.08/44.59 The set Q consists of the following terms: 172.08/44.59 172.08/44.59 member1_in_ag(x0) 172.08/44.59 member_in_gg(x0, x1) 172.08/44.59 U6_ag(x0) 172.08/44.59 U5_gg(x0) 172.08/44.59 172.08/44.59 We have to consider all (P,Q,R)-chains. 172.08/44.59 ---------------------------------------- 172.08/44.59 172.08/44.59 (105) TransformationProof (EQUIVALENT) 172.08/44.59 By instantiating [LPAR04] the rule U3_GGGG(Z, Edges, Visited, Y, member_out_gg) -> REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited)) we obtained the following new rules [LPAR04]: 172.08/44.59 172.08/44.59 (U3_GGGG(z0, z1, .(z2, z3), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, z1, .(z2, .(z2, z3))),U3_GGGG(z0, z1, .(z2, z3), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, z1, .(z2, .(z2, z3)))) 172.08/44.59 (U3_GGGG(z0, z1, .(z2, z3), z5, member_out_gg) -> REACH_IN_GGGG(z5, z0, z1, .(z5, .(z2, z3))),U3_GGGG(z0, z1, .(z2, z3), z5, member_out_gg) -> REACH_IN_GGGG(z5, z0, z1, .(z5, .(z2, z3)))) 172.08/44.59 172.08/44.59 172.08/44.59 ---------------------------------------- 172.08/44.59 172.08/44.59 (106) 172.08/44.59 Obligation: 172.08/44.59 Q DP problem: 172.08/44.59 The TRS P consists of the following rules: 172.08/44.59 172.08/44.59 REACH_IN_GGGG(y0, y1, .(x0, x1), y3) -> U2_GGGG(y1, .(x0, x1), y3, member1_out_ag(x0)) 172.08/44.59 REACH_IN_GGGG(y0, y1, .(x0, x1), y3) -> U2_GGGG(y1, .(x0, x1), y3, U6_ag(member1_in_ag(x1))) 172.08/44.59 U2_GGGG(y0, y1, .(x0, x1), member1_out_ag(.(y3, .(x0, [])))) -> U3_GGGG(y0, y1, .(x0, x1), x0, member_out_gg) 172.08/44.59 U2_GGGG(y0, y1, .(x1, x2), member1_out_ag(.(y3, .(x0, [])))) -> U3_GGGG(y0, y1, .(x1, x2), x0, U5_gg(member_in_gg(x0, x2))) 172.08/44.59 U3_GGGG(z0, z1, .(z2, z3), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, z1, .(z2, .(z2, z3))) 172.08/44.59 U3_GGGG(z0, z1, .(z2, z3), z5, member_out_gg) -> REACH_IN_GGGG(z5, z0, z1, .(z5, .(z2, z3))) 172.08/44.59 172.08/44.59 The TRS R consists of the following rules: 172.08/44.59 172.08/44.59 member1_in_ag(.(H, L)) -> member1_out_ag(H) 172.08/44.59 member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) 172.08/44.59 member_in_gg(H, .(H, L)) -> member_out_gg 172.08/44.59 member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) 172.08/44.59 U6_ag(member1_out_ag(X)) -> member1_out_ag(X) 172.08/44.59 U5_gg(member_out_gg) -> member_out_gg 172.08/44.59 172.08/44.59 The set Q consists of the following terms: 172.08/44.59 172.08/44.59 member1_in_ag(x0) 172.08/44.59 member_in_gg(x0, x1) 172.08/44.59 U6_ag(x0) 172.08/44.59 U5_gg(x0) 172.08/44.59 172.08/44.59 We have to consider all (P,Q,R)-chains. 172.08/44.59 ---------------------------------------- 172.08/44.59 172.08/44.59 (107) TransformationProof (EQUIVALENT) 172.08/44.59 By instantiating [LPAR04] the rule REACH_IN_GGGG(y0, y1, .(x0, x1), y3) -> U2_GGGG(y1, .(x0, x1), y3, member1_out_ag(x0)) we obtained the following new rules [LPAR04]: 172.08/44.59 172.08/44.59 (REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), member1_out_ag(x2)),REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), member1_out_ag(x2))) 172.08/44.59 (REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), member1_out_ag(x2)),REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), member1_out_ag(x2))) 172.08/44.59 172.08/44.59 172.08/44.59 ---------------------------------------- 172.08/44.59 172.08/44.59 (108) 172.08/44.59 Obligation: 172.08/44.59 Q DP problem: 172.08/44.59 The TRS P consists of the following rules: 172.08/44.59 172.08/44.59 REACH_IN_GGGG(y0, y1, .(x0, x1), y3) -> U2_GGGG(y1, .(x0, x1), y3, U6_ag(member1_in_ag(x1))) 172.08/44.59 U2_GGGG(y0, y1, .(x0, x1), member1_out_ag(.(y3, .(x0, [])))) -> U3_GGGG(y0, y1, .(x0, x1), x0, member_out_gg) 172.08/44.59 U2_GGGG(y0, y1, .(x1, x2), member1_out_ag(.(y3, .(x0, [])))) -> U3_GGGG(y0, y1, .(x1, x2), x0, U5_gg(member_in_gg(x0, x2))) 172.08/44.59 U3_GGGG(z0, z1, .(z2, z3), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, z1, .(z2, .(z2, z3))) 172.08/44.59 U3_GGGG(z0, z1, .(z2, z3), z5, member_out_gg) -> REACH_IN_GGGG(z5, z0, z1, .(z5, .(z2, z3))) 172.08/44.59 REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), member1_out_ag(x2)) 172.08/44.59 REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), member1_out_ag(x2)) 172.08/44.59 172.08/44.59 The TRS R consists of the following rules: 172.08/44.59 172.08/44.59 member1_in_ag(.(H, L)) -> member1_out_ag(H) 172.08/44.59 member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) 172.08/44.59 member_in_gg(H, .(H, L)) -> member_out_gg 172.08/44.59 member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) 172.08/44.59 U6_ag(member1_out_ag(X)) -> member1_out_ag(X) 172.08/44.59 U5_gg(member_out_gg) -> member_out_gg 172.08/44.59 172.08/44.59 The set Q consists of the following terms: 172.08/44.59 172.08/44.59 member1_in_ag(x0) 172.08/44.59 member_in_gg(x0, x1) 172.08/44.59 U6_ag(x0) 172.08/44.59 U5_gg(x0) 172.08/44.59 172.08/44.59 We have to consider all (P,Q,R)-chains. 172.08/44.59 ---------------------------------------- 172.08/44.59 172.08/44.59 (109) TransformationProof (EQUIVALENT) 172.08/44.59 By instantiating [LPAR04] the rule REACH_IN_GGGG(y0, y1, .(x0, x1), y3) -> U2_GGGG(y1, .(x0, x1), y3, U6_ag(member1_in_ag(x1))) we obtained the following new rules [LPAR04]: 172.08/44.59 172.08/44.59 (REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), U6_ag(member1_in_ag(x3))),REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), U6_ag(member1_in_ag(x3)))) 172.08/44.59 (REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U6_ag(member1_in_ag(x3))),REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U6_ag(member1_in_ag(x3)))) 172.08/44.59 172.08/44.59 172.08/44.59 ---------------------------------------- 172.08/44.59 172.08/44.59 (110) 172.08/44.59 Obligation: 172.08/44.59 Q DP problem: 172.08/44.59 The TRS P consists of the following rules: 172.08/44.59 172.08/44.59 U2_GGGG(y0, y1, .(x0, x1), member1_out_ag(.(y3, .(x0, [])))) -> U3_GGGG(y0, y1, .(x0, x1), x0, member_out_gg) 172.08/44.59 U2_GGGG(y0, y1, .(x1, x2), member1_out_ag(.(y3, .(x0, [])))) -> U3_GGGG(y0, y1, .(x1, x2), x0, U5_gg(member_in_gg(x0, x2))) 172.08/44.59 U3_GGGG(z0, z1, .(z2, z3), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, z1, .(z2, .(z2, z3))) 172.08/44.59 U3_GGGG(z0, z1, .(z2, z3), z5, member_out_gg) -> REACH_IN_GGGG(z5, z0, z1, .(z5, .(z2, z3))) 172.08/44.59 REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), member1_out_ag(x2)) 172.08/44.59 REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), member1_out_ag(x2)) 172.08/44.59 REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), U6_ag(member1_in_ag(x3))) 172.08/44.59 REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U6_ag(member1_in_ag(x3))) 172.08/44.59 172.08/44.59 The TRS R consists of the following rules: 172.08/44.59 172.08/44.59 member1_in_ag(.(H, L)) -> member1_out_ag(H) 172.08/44.59 member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) 172.08/44.59 member_in_gg(H, .(H, L)) -> member_out_gg 172.08/44.59 member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) 172.08/44.59 U6_ag(member1_out_ag(X)) -> member1_out_ag(X) 172.08/44.59 U5_gg(member_out_gg) -> member_out_gg 172.08/44.59 172.08/44.59 The set Q consists of the following terms: 172.08/44.59 172.08/44.59 member1_in_ag(x0) 172.08/44.59 member_in_gg(x0, x1) 172.08/44.59 U6_ag(x0) 172.08/44.59 U5_gg(x0) 172.08/44.59 172.08/44.59 We have to consider all (P,Q,R)-chains. 172.08/44.59 ---------------------------------------- 172.08/44.59 172.08/44.59 (111) TransformationProof (EQUIVALENT) 172.08/44.59 By instantiating [LPAR04] the rule U2_GGGG(y0, y1, .(x0, x1), member1_out_ag(.(y3, .(x0, [])))) -> U3_GGGG(y0, y1, .(x0, x1), x0, member_out_gg) we obtained the following new rules [LPAR04]: 172.08/44.59 172.08/44.59 (U2_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg),U2_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg)) 172.08/44.59 (U2_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg),U2_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg)) 172.08/44.59 (U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg),U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg)) 172.08/44.59 (U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg),U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg)) 172.08/44.59 172.08/44.59 172.08/44.59 ---------------------------------------- 172.08/44.59 172.08/44.59 (112) 172.08/44.59 Obligation: 172.08/44.59 Q DP problem: 172.08/44.59 The TRS P consists of the following rules: 172.08/44.59 172.08/44.59 U2_GGGG(y0, y1, .(x1, x2), member1_out_ag(.(y3, .(x0, [])))) -> U3_GGGG(y0, y1, .(x1, x2), x0, U5_gg(member_in_gg(x0, x2))) 172.08/44.59 U3_GGGG(z0, z1, .(z2, z3), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, z1, .(z2, .(z2, z3))) 172.08/44.59 U3_GGGG(z0, z1, .(z2, z3), z5, member_out_gg) -> REACH_IN_GGGG(z5, z0, z1, .(z5, .(z2, z3))) 172.08/44.59 REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), member1_out_ag(x2)) 172.08/44.59 REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), member1_out_ag(x2)) 172.08/44.59 REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), U6_ag(member1_in_ag(x3))) 172.08/44.59 REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U6_ag(member1_in_ag(x3))) 172.08/44.59 U2_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg) 172.08/44.59 U2_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg) 172.08/44.59 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg) 172.08/44.59 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg) 172.08/44.59 172.08/44.59 The TRS R consists of the following rules: 172.08/44.59 172.08/44.59 member1_in_ag(.(H, L)) -> member1_out_ag(H) 172.08/44.59 member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) 172.08/44.59 member_in_gg(H, .(H, L)) -> member_out_gg 172.08/44.59 member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) 172.08/44.59 U6_ag(member1_out_ag(X)) -> member1_out_ag(X) 172.08/44.59 U5_gg(member_out_gg) -> member_out_gg 172.08/44.59 172.08/44.59 The set Q consists of the following terms: 172.08/44.59 172.08/44.59 member1_in_ag(x0) 172.08/44.59 member_in_gg(x0, x1) 172.08/44.59 U6_ag(x0) 172.08/44.59 U5_gg(x0) 172.08/44.59 172.08/44.59 We have to consider all (P,Q,R)-chains. 172.08/44.59 ---------------------------------------- 172.08/44.59 172.08/44.59 (113) TransformationProof (EQUIVALENT) 172.08/44.59 By instantiating [LPAR04] the rule U2_GGGG(y0, y1, .(x1, x2), member1_out_ag(.(y3, .(x0, [])))) -> U3_GGGG(y0, y1, .(x1, x2), x0, U5_gg(member_in_gg(x0, x2))) we obtained the following new rules [LPAR04]: 172.08/44.59 172.08/44.59 (U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4)))),U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4))))) 172.08/44.59 (U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))),U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5))))) 172.08/44.59 (U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4)))),U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4))))) 172.08/44.59 (U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))),U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5))))) 172.08/44.59 172.08/44.59 172.08/44.59 ---------------------------------------- 172.08/44.59 172.08/44.59 (114) 172.08/44.59 Obligation: 172.08/44.59 Q DP problem: 172.08/44.59 The TRS P consists of the following rules: 172.08/44.59 172.08/44.59 U3_GGGG(z0, z1, .(z2, z3), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, z1, .(z2, .(z2, z3))) 172.08/44.59 U3_GGGG(z0, z1, .(z2, z3), z5, member_out_gg) -> REACH_IN_GGGG(z5, z0, z1, .(z5, .(z2, z3))) 172.08/44.59 REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), member1_out_ag(x2)) 172.08/44.60 REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), member1_out_ag(x2)) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), U6_ag(member1_in_ag(x3))) 172.08/44.60 REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U6_ag(member1_in_ag(x3))) 172.08/44.60 U2_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4)))) 172.08/44.60 U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4)))) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) 172.08/44.60 172.08/44.60 The TRS R consists of the following rules: 172.08/44.60 172.08/44.60 member1_in_ag(.(H, L)) -> member1_out_ag(H) 172.08/44.60 member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) 172.08/44.60 member_in_gg(H, .(H, L)) -> member_out_gg 172.08/44.60 member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) 172.08/44.60 U6_ag(member1_out_ag(X)) -> member1_out_ag(X) 172.08/44.60 U5_gg(member_out_gg) -> member_out_gg 172.08/44.60 172.08/44.60 The set Q consists of the following terms: 172.08/44.60 172.08/44.60 member1_in_ag(x0) 172.08/44.60 member_in_gg(x0, x1) 172.08/44.60 U6_ag(x0) 172.08/44.60 U5_gg(x0) 172.08/44.60 172.08/44.60 We have to consider all (P,Q,R)-chains. 172.08/44.60 ---------------------------------------- 172.08/44.60 172.08/44.60 (115) TransformationProof (EQUIVALENT) 172.08/44.60 By instantiating [LPAR04] the rule U3_GGGG(z0, z1, .(z2, z3), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, z1, .(z2, .(z2, z3))) we obtained the following new rules [LPAR04]: 172.08/44.60 172.08/44.60 (U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))),U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))))) 172.08/44.60 (U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))),U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))))) 172.08/44.60 (U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))),U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4))))) 172.08/44.60 (U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))),U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5))))) 172.08/44.60 172.08/44.60 172.08/44.60 ---------------------------------------- 172.08/44.60 172.08/44.60 (116) 172.08/44.60 Obligation: 172.08/44.60 Q DP problem: 172.08/44.60 The TRS P consists of the following rules: 172.08/44.60 172.08/44.60 U3_GGGG(z0, z1, .(z2, z3), z5, member_out_gg) -> REACH_IN_GGGG(z5, z0, z1, .(z5, .(z2, z3))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), member1_out_ag(x2)) 172.08/44.60 REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), member1_out_ag(x2)) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), U6_ag(member1_in_ag(x3))) 172.08/44.60 REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U6_ag(member1_in_ag(x3))) 172.08/44.60 U2_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4)))) 172.08/44.60 U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4)))) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) 172.08/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) 172.08/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) 172.08/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) 172.08/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) 172.08/44.60 172.08/44.60 The TRS R consists of the following rules: 172.08/44.60 172.08/44.60 member1_in_ag(.(H, L)) -> member1_out_ag(H) 172.08/44.60 member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) 172.08/44.60 member_in_gg(H, .(H, L)) -> member_out_gg 172.08/44.60 member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) 172.08/44.60 U6_ag(member1_out_ag(X)) -> member1_out_ag(X) 172.08/44.60 U5_gg(member_out_gg) -> member_out_gg 172.08/44.60 172.08/44.60 The set Q consists of the following terms: 172.08/44.60 172.08/44.60 member1_in_ag(x0) 172.08/44.60 member_in_gg(x0, x1) 172.08/44.60 U6_ag(x0) 172.08/44.60 U5_gg(x0) 172.08/44.60 172.08/44.60 We have to consider all (P,Q,R)-chains. 172.08/44.60 ---------------------------------------- 172.08/44.60 172.08/44.60 (117) TransformationProof (EQUIVALENT) 172.08/44.60 By instantiating [LPAR04] the rule U3_GGGG(z0, z1, .(z2, z3), z5, member_out_gg) -> REACH_IN_GGGG(z5, z0, z1, .(z5, .(z2, z3))) we obtained the following new rules [LPAR04]: 172.08/44.60 172.08/44.60 (U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))),U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))))) 172.08/44.60 (U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))),U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))))) 172.08/44.60 (U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))),U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4))))) 172.08/44.60 (U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))),U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5))))) 172.08/44.60 (U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))),U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))))) 172.08/44.60 (U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z5, z6)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))),U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z5, z6)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))))) 172.08/44.60 (U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z6, member_out_gg) -> REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4)))),U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z6, member_out_gg) -> REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4))))) 172.08/44.60 (U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) -> REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5)))),U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) -> REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5))))) 172.08/44.60 172.08/44.60 172.08/44.60 ---------------------------------------- 172.08/44.60 172.08/44.60 (118) 172.08/44.60 Obligation: 172.08/44.60 Q DP problem: 172.08/44.60 The TRS P consists of the following rules: 172.08/44.60 172.08/44.60 REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), member1_out_ag(x2)) 172.08/44.60 REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), member1_out_ag(x2)) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), U6_ag(member1_in_ag(x3))) 172.08/44.60 REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U6_ag(member1_in_ag(x3))) 172.08/44.60 U2_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4)))) 172.08/44.60 U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4)))) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) 172.08/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) 172.08/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) 172.08/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) 172.08/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) 172.08/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) 172.08/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z5, z6)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) 172.08/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z6, member_out_gg) -> REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4)))) 172.08/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) -> REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5)))) 172.08/44.60 172.08/44.60 The TRS R consists of the following rules: 172.08/44.60 172.08/44.60 member1_in_ag(.(H, L)) -> member1_out_ag(H) 172.08/44.60 member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) 172.08/44.60 member_in_gg(H, .(H, L)) -> member_out_gg 172.08/44.60 member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) 172.08/44.60 U6_ag(member1_out_ag(X)) -> member1_out_ag(X) 172.08/44.60 U5_gg(member_out_gg) -> member_out_gg 172.08/44.60 172.08/44.60 The set Q consists of the following terms: 172.08/44.60 172.08/44.60 member1_in_ag(x0) 172.08/44.60 member_in_gg(x0, x1) 172.08/44.60 U6_ag(x0) 172.08/44.60 U5_gg(x0) 172.08/44.60 172.08/44.60 We have to consider all (P,Q,R)-chains. 172.08/44.60 ---------------------------------------- 172.08/44.60 172.08/44.60 (119) TransformationProof (EQUIVALENT) 172.08/44.60 By instantiating [LPAR04] the rule REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), member1_out_ag(x2)) we obtained the following new rules [LPAR04]: 172.08/44.60 172.08/44.60 (REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), member1_out_ag(.(z1, .(z2, [])))),REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), member1_out_ag(.(z1, .(z2, []))))) 172.08/44.60 (REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))),REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, []))))) 172.08/44.60 (REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), member1_out_ag(z1)),REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), member1_out_ag(z1))) 172.08/44.60 (REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1)),REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1))) 172.08/44.60 172.08/44.60 172.08/44.60 ---------------------------------------- 172.08/44.60 172.08/44.60 (120) 172.08/44.60 Obligation: 172.08/44.60 Q DP problem: 172.08/44.60 The TRS P consists of the following rules: 172.08/44.60 172.08/44.60 REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), member1_out_ag(x2)) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), U6_ag(member1_in_ag(x3))) 172.08/44.60 REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U6_ag(member1_in_ag(x3))) 172.08/44.60 U2_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4)))) 172.08/44.60 U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4)))) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) 172.08/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) 172.08/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) 172.08/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) 172.08/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) 172.08/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) 172.08/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z5, z6)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) 172.08/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z6, member_out_gg) -> REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4)))) 172.08/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) -> REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5)))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), member1_out_ag(.(z1, .(z2, [])))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 172.08/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), member1_out_ag(z1)) 172.08/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1)) 172.08/44.60 172.08/44.60 The TRS R consists of the following rules: 172.08/44.60 172.08/44.60 member1_in_ag(.(H, L)) -> member1_out_ag(H) 172.08/44.60 member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) 172.08/44.60 member_in_gg(H, .(H, L)) -> member_out_gg 172.08/44.60 member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) 172.08/44.60 U6_ag(member1_out_ag(X)) -> member1_out_ag(X) 172.08/44.60 U5_gg(member_out_gg) -> member_out_gg 172.08/44.60 172.08/44.60 The set Q consists of the following terms: 172.08/44.60 172.08/44.60 member1_in_ag(x0) 172.08/44.60 member_in_gg(x0, x1) 172.08/44.60 U6_ag(x0) 172.08/44.60 U5_gg(x0) 172.08/44.60 172.08/44.60 We have to consider all (P,Q,R)-chains. 172.08/44.60 ---------------------------------------- 172.08/44.60 172.08/44.60 (121) TransformationProof (EQUIVALENT) 172.08/44.60 By instantiating [LPAR04] the rule REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), member1_out_ag(x2)) we obtained the following new rules [LPAR04]: 172.08/44.60 172.08/44.60 (REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), member1_out_ag(.(z1, .(z2, [])))),REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), member1_out_ag(.(z1, .(z2, []))))) 172.08/44.60 (REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))),REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, []))))) 172.08/44.60 (REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), member1_out_ag(z1)),REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), member1_out_ag(z1))) 172.08/44.60 (REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1)),REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1))) 172.08/44.60 (REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))),REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), member1_out_ag(.(z1, .(z2, []))))) 172.08/44.60 (REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, [])))),REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, []))))) 172.08/44.60 (REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), member1_out_ag(z1)),REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), member1_out_ag(z1))) 172.08/44.60 (REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1)),REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1))) 172.08/44.60 172.08/44.60 172.08/44.60 ---------------------------------------- 172.08/44.60 172.08/44.60 (122) 172.08/44.60 Obligation: 172.08/44.60 Q DP problem: 172.08/44.60 The TRS P consists of the following rules: 172.08/44.60 172.08/44.60 REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), U6_ag(member1_in_ag(x3))) 172.08/44.60 REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U6_ag(member1_in_ag(x3))) 172.08/44.60 U2_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4)))) 172.08/44.60 U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4)))) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) 172.08/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) 172.08/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) 172.08/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) 172.08/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) 172.08/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) 172.08/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z5, z6)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) 172.08/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z6, member_out_gg) -> REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4)))) 172.08/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) -> REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5)))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), member1_out_ag(.(z1, .(z2, [])))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 172.08/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), member1_out_ag(z1)) 172.08/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1)) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, [])))) 172.08/44.60 REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), member1_out_ag(z1)) 172.08/44.60 REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1)) 172.08/44.60 172.08/44.60 The TRS R consists of the following rules: 172.08/44.60 172.08/44.60 member1_in_ag(.(H, L)) -> member1_out_ag(H) 172.08/44.60 member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) 172.08/44.60 member_in_gg(H, .(H, L)) -> member_out_gg 172.08/44.60 member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) 172.08/44.60 U6_ag(member1_out_ag(X)) -> member1_out_ag(X) 172.08/44.60 U5_gg(member_out_gg) -> member_out_gg 172.08/44.60 172.08/44.60 The set Q consists of the following terms: 172.08/44.60 172.08/44.60 member1_in_ag(x0) 172.08/44.60 member_in_gg(x0, x1) 172.08/44.60 U6_ag(x0) 172.08/44.60 U5_gg(x0) 172.08/44.60 172.08/44.60 We have to consider all (P,Q,R)-chains. 172.08/44.60 ---------------------------------------- 172.08/44.60 172.08/44.60 (123) TransformationProof (EQUIVALENT) 172.08/44.60 By instantiating [LPAR04] the rule REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), U6_ag(member1_in_ag(x3))) we obtained the following new rules [LPAR04]: 172.08/44.60 172.08/44.60 (REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), U6_ag(member1_in_ag(z3))),REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), U6_ag(member1_in_ag(z3)))) 172.08/44.60 (REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3))),REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3)))) 172.08/44.60 (REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2))),REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2)))) 172.08/44.60 (REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))),REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2)))) 172.08/44.60 172.08/44.60 172.08/44.60 ---------------------------------------- 172.08/44.60 172.08/44.60 (124) 172.08/44.60 Obligation: 172.08/44.60 Q DP problem: 172.08/44.60 The TRS P consists of the following rules: 172.08/44.60 172.08/44.60 REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U6_ag(member1_in_ag(x3))) 172.08/44.60 U2_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4)))) 172.08/44.60 U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4)))) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) 172.08/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) 172.08/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) 172.08/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) 172.08/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) 172.08/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) 172.08/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z5, z6)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) 172.08/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z6, member_out_gg) -> REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4)))) 172.08/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) -> REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5)))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), member1_out_ag(.(z1, .(z2, [])))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 172.08/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), member1_out_ag(z1)) 172.08/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1)) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, [])))) 172.08/44.60 REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), member1_out_ag(z1)) 172.08/44.60 REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1)) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), U6_ag(member1_in_ag(z3))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3))) 172.08/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2))) 172.08/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 172.08/44.60 172.08/44.60 The TRS R consists of the following rules: 172.08/44.60 172.08/44.60 member1_in_ag(.(H, L)) -> member1_out_ag(H) 172.08/44.60 member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) 172.08/44.60 member_in_gg(H, .(H, L)) -> member_out_gg 172.08/44.60 member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) 172.08/44.60 U6_ag(member1_out_ag(X)) -> member1_out_ag(X) 172.08/44.60 U5_gg(member_out_gg) -> member_out_gg 172.08/44.60 172.08/44.60 The set Q consists of the following terms: 172.08/44.60 172.08/44.60 member1_in_ag(x0) 172.08/44.60 member_in_gg(x0, x1) 172.08/44.60 U6_ag(x0) 172.08/44.60 U5_gg(x0) 172.08/44.60 172.08/44.60 We have to consider all (P,Q,R)-chains. 172.08/44.60 ---------------------------------------- 172.08/44.60 172.08/44.60 (125) TransformationProof (EQUIVALENT) 172.08/44.60 By instantiating [LPAR04] the rule REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U6_ag(member1_in_ag(x3))) we obtained the following new rules [LPAR04]: 172.08/44.60 172.08/44.60 (REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), U6_ag(member1_in_ag(z3))),REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), U6_ag(member1_in_ag(z3)))) 172.08/44.60 (REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3))),REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3)))) 172.08/44.60 (REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2))),REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2)))) 172.08/44.60 (REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))),REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2)))) 172.08/44.60 (REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), U6_ag(member1_in_ag(z3))),REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), U6_ag(member1_in_ag(z3)))) 172.08/44.60 (REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3))),REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3)))) 172.08/44.60 (REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2))),REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2)))) 172.08/44.60 (REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))),REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2)))) 172.08/44.60 172.08/44.60 172.08/44.60 ---------------------------------------- 172.08/44.60 172.08/44.60 (126) 172.08/44.60 Obligation: 172.08/44.60 Q DP problem: 172.08/44.60 The TRS P consists of the following rules: 172.08/44.60 172.08/44.60 U2_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4)))) 172.08/44.60 U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4)))) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) 172.08/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) 172.08/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) 172.08/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) 172.08/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) 172.08/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) 172.08/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z5, z6)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) 172.08/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z6, member_out_gg) -> REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4)))) 172.08/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) -> REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5)))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), member1_out_ag(.(z1, .(z2, [])))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 172.08/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), member1_out_ag(z1)) 172.08/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1)) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, [])))) 172.08/44.60 REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), member1_out_ag(z1)) 172.08/44.60 REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1)) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), U6_ag(member1_in_ag(z3))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3))) 172.08/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2))) 172.08/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), U6_ag(member1_in_ag(z3))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3))) 172.08/44.60 REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2))) 172.08/44.60 REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 172.08/44.60 172.08/44.60 The TRS R consists of the following rules: 172.08/44.60 172.08/44.60 member1_in_ag(.(H, L)) -> member1_out_ag(H) 172.08/44.60 member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) 172.08/44.60 member_in_gg(H, .(H, L)) -> member_out_gg 172.08/44.60 member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) 172.08/44.60 U6_ag(member1_out_ag(X)) -> member1_out_ag(X) 172.08/44.60 U5_gg(member_out_gg) -> member_out_gg 172.08/44.60 172.08/44.60 The set Q consists of the following terms: 172.08/44.60 172.08/44.60 member1_in_ag(x0) 172.08/44.60 member_in_gg(x0, x1) 172.08/44.60 U6_ag(x0) 172.08/44.60 U5_gg(x0) 172.08/44.60 172.08/44.60 We have to consider all (P,Q,R)-chains. 172.08/44.60 ---------------------------------------- 172.08/44.60 172.08/44.60 (127) TransformationProof (EQUIVALENT) 172.08/44.60 By instantiating [LPAR04] the rule U2_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg) we obtained the following new rules [LPAR04]: 172.08/44.60 172.08/44.60 (U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg),U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)) 172.08/44.60 (U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg),U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)) 172.08/44.60 172.08/44.60 172.08/44.60 ---------------------------------------- 172.08/44.60 172.08/44.60 (128) 172.08/44.60 Obligation: 172.08/44.60 Q DP problem: 172.08/44.60 The TRS P consists of the following rules: 172.08/44.60 172.08/44.60 U2_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4)))) 172.08/44.60 U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4)))) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) 172.08/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) 172.08/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) 172.08/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) 172.08/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) 172.08/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) 172.08/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z5, z6)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) 172.08/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z6, member_out_gg) -> REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4)))) 172.08/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) -> REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5)))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), member1_out_ag(.(z1, .(z2, [])))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 172.08/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), member1_out_ag(z1)) 172.08/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1)) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, [])))) 172.08/44.60 REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), member1_out_ag(z1)) 172.08/44.60 REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1)) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), U6_ag(member1_in_ag(z3))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3))) 172.08/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2))) 172.08/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), U6_ag(member1_in_ag(z3))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3))) 172.08/44.60 REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2))) 172.08/44.60 REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 172.08/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg) 172.08/44.60 172.08/44.60 The TRS R consists of the following rules: 172.08/44.60 172.08/44.60 member1_in_ag(.(H, L)) -> member1_out_ag(H) 172.08/44.60 member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) 172.08/44.60 member_in_gg(H, .(H, L)) -> member_out_gg 172.08/44.60 member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) 172.08/44.60 U6_ag(member1_out_ag(X)) -> member1_out_ag(X) 172.08/44.60 U5_gg(member_out_gg) -> member_out_gg 172.08/44.60 172.08/44.60 The set Q consists of the following terms: 172.08/44.60 172.08/44.60 member1_in_ag(x0) 172.08/44.60 member_in_gg(x0, x1) 172.08/44.60 U6_ag(x0) 172.08/44.60 U5_gg(x0) 172.08/44.60 172.08/44.60 We have to consider all (P,Q,R)-chains. 172.08/44.60 ---------------------------------------- 172.08/44.60 172.08/44.60 (129) TransformationProof (EQUIVALENT) 172.08/44.60 By instantiating [LPAR04] the rule U2_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg) we obtained the following new rules [LPAR04]: 172.08/44.60 172.08/44.60 (U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg),U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)) 172.08/44.60 (U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg),U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)) 172.08/44.60 (U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg),U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg)) 172.08/44.60 (U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg),U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg)) 172.08/44.60 172.08/44.60 172.08/44.60 ---------------------------------------- 172.08/44.60 172.08/44.60 (130) 172.08/44.60 Obligation: 172.08/44.60 Q DP problem: 172.08/44.60 The TRS P consists of the following rules: 172.08/44.60 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4)))) 172.08/44.60 U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4)))) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) 172.08/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) 172.08/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) 172.08/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) 172.08/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) 172.08/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) 172.08/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z5, z6)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) 172.08/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z6, member_out_gg) -> REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4)))) 172.08/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) -> REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5)))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), member1_out_ag(.(z1, .(z2, [])))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 172.08/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), member1_out_ag(z1)) 172.08/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1)) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, [])))) 172.08/44.60 REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), member1_out_ag(z1)) 172.08/44.60 REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1)) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), U6_ag(member1_in_ag(z3))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3))) 172.08/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2))) 172.08/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), U6_ag(member1_in_ag(z3))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3))) 172.08/44.60 REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2))) 172.08/44.60 REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 172.08/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg) 172.08/44.60 172.08/44.60 The TRS R consists of the following rules: 172.08/44.60 172.08/44.60 member1_in_ag(.(H, L)) -> member1_out_ag(H) 172.08/44.60 member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) 172.08/44.60 member_in_gg(H, .(H, L)) -> member_out_gg 172.08/44.60 member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) 172.08/44.60 U6_ag(member1_out_ag(X)) -> member1_out_ag(X) 172.08/44.60 U5_gg(member_out_gg) -> member_out_gg 172.08/44.60 172.08/44.60 The set Q consists of the following terms: 172.08/44.60 172.08/44.60 member1_in_ag(x0) 172.08/44.60 member_in_gg(x0, x1) 172.08/44.60 U6_ag(x0) 172.08/44.60 U5_gg(x0) 172.08/44.60 172.08/44.60 We have to consider all (P,Q,R)-chains. 172.08/44.60 ---------------------------------------- 172.08/44.60 172.08/44.60 (131) TransformationProof (EQUIVALENT) 172.08/44.60 By instantiating [LPAR04] the rule U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg) we obtained the following new rules [LPAR04]: 172.08/44.60 172.08/44.60 (U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg),U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)) 172.08/44.60 (U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg),U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)) 172.08/44.60 (U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg),U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)) 172.08/44.60 (U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg),U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)) 172.08/44.60 (U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg),U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)) 172.08/44.60 (U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg),U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)) 172.08/44.60 172.08/44.60 172.08/44.60 ---------------------------------------- 172.08/44.60 172.08/44.60 (132) 172.08/44.60 Obligation: 172.08/44.60 Q DP problem: 172.08/44.60 The TRS P consists of the following rules: 172.08/44.60 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4)))) 172.08/44.60 U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4)))) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) 172.08/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) 172.08/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) 172.08/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) 172.08/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) 172.08/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) 172.08/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z5, z6)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) 172.08/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z6, member_out_gg) -> REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4)))) 172.08/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) -> REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5)))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), member1_out_ag(.(z1, .(z2, [])))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 172.08/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), member1_out_ag(z1)) 172.08/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1)) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, [])))) 172.08/44.60 REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), member1_out_ag(z1)) 172.08/44.60 REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1)) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), U6_ag(member1_in_ag(z3))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3))) 172.08/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2))) 172.08/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), U6_ag(member1_in_ag(z3))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3))) 172.08/44.60 REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2))) 172.08/44.60 REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 172.08/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg) 172.08/44.60 172.08/44.60 The TRS R consists of the following rules: 172.08/44.60 172.08/44.60 member1_in_ag(.(H, L)) -> member1_out_ag(H) 172.08/44.60 member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) 172.08/44.60 member_in_gg(H, .(H, L)) -> member_out_gg 172.08/44.60 member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) 172.08/44.60 U6_ag(member1_out_ag(X)) -> member1_out_ag(X) 172.08/44.60 U5_gg(member_out_gg) -> member_out_gg 172.08/44.60 172.08/44.60 The set Q consists of the following terms: 172.08/44.60 172.08/44.60 member1_in_ag(x0) 172.08/44.60 member_in_gg(x0, x1) 172.08/44.60 U6_ag(x0) 172.08/44.60 U5_gg(x0) 172.08/44.60 172.08/44.60 We have to consider all (P,Q,R)-chains. 172.08/44.60 ---------------------------------------- 172.08/44.60 172.08/44.60 (133) TransformationProof (EQUIVALENT) 172.08/44.60 By instantiating [LPAR04] the rule U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg) we obtained the following new rules [LPAR04]: 172.08/44.60 172.08/44.60 (U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg),U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)) 172.08/44.60 (U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg),U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)) 172.08/44.60 (U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg),U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg)) 172.08/44.60 (U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg),U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg)) 172.08/44.60 (U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg),U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)) 172.08/44.60 (U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg),U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)) 172.08/44.60 (U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg),U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)) 172.08/44.60 (U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg),U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)) 172.08/44.60 (U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg),U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg)) 172.08/44.60 (U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg),U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg)) 172.08/44.60 (U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg),U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg)) 172.08/44.60 (U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg),U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg)) 172.08/44.60 172.08/44.60 172.08/44.60 ---------------------------------------- 172.08/44.60 172.08/44.60 (134) 172.08/44.60 Obligation: 172.08/44.60 Q DP problem: 172.08/44.60 The TRS P consists of the following rules: 172.08/44.60 172.08/44.60 U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4)))) 172.08/44.60 U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4)))) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) 172.08/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) 172.08/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) 172.08/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) 172.08/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) 172.08/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) 172.08/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z5, z6)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) 172.08/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z6, member_out_gg) -> REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4)))) 172.08/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) -> REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5)))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), member1_out_ag(.(z1, .(z2, [])))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 172.08/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), member1_out_ag(z1)) 172.08/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1)) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, [])))) 172.08/44.60 REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), member1_out_ag(z1)) 172.08/44.60 REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1)) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), U6_ag(member1_in_ag(z3))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3))) 172.08/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2))) 172.08/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), U6_ag(member1_in_ag(z3))) 172.08/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3))) 172.08/44.60 REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2))) 172.08/44.60 REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 172.08/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg) 172.08/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg) 172.08/44.60 172.08/44.60 The TRS R consists of the following rules: 172.08/44.60 172.08/44.60 member1_in_ag(.(H, L)) -> member1_out_ag(H) 172.08/44.60 member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) 172.08/44.60 member_in_gg(H, .(H, L)) -> member_out_gg 172.08/44.60 member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) 172.08/44.60 U6_ag(member1_out_ag(X)) -> member1_out_ag(X) 172.08/44.60 U5_gg(member_out_gg) -> member_out_gg 172.08/44.60 172.08/44.60 The set Q consists of the following terms: 172.08/44.60 172.08/44.60 member1_in_ag(x0) 172.08/44.60 member_in_gg(x0, x1) 172.08/44.60 U6_ag(x0) 172.08/44.60 U5_gg(x0) 172.08/44.60 172.08/44.60 We have to consider all (P,Q,R)-chains. 172.08/44.60 ---------------------------------------- 172.08/44.60 172.08/44.60 (135) TransformationProof (EQUIVALENT) 172.08/44.60 By instantiating [LPAR04] the rule U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4)))) we obtained the following new rules [LPAR04]: 172.08/44.60 172.08/44.60 (U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, U5_gg(member_in_gg(z0, .(z0, .(z0, z4))))),U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, U5_gg(member_in_gg(z0, .(z0, .(z0, z4)))))) 172.08/44.60 (U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z0, .(z4, z5))))),U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z0, .(z4, z5)))))) 172.38/44.60 (U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(member_in_gg(x2, .(z0, .(z0, z4))))),U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(member_in_gg(x2, .(z0, .(z0, z4)))))) 172.38/44.60 (U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z0, .(z4, z5))))),U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z0, .(z4, z5)))))) 172.38/44.60 172.38/44.60 172.38/44.60 ---------------------------------------- 172.38/44.60 172.38/44.60 (136) 172.38/44.60 Obligation: 172.38/44.60 Q DP problem: 172.38/44.60 The TRS P consists of the following rules: 172.38/44.60 172.38/44.60 U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4)))) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) 172.38/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) 172.38/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) 172.38/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) 172.38/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) 172.38/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) 172.38/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z5, z6)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) 172.38/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z6, member_out_gg) -> REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4)))) 172.38/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) -> REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5)))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), member1_out_ag(.(z1, .(z2, [])))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 172.38/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), member1_out_ag(z1)) 172.38/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1)) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, [])))) 172.38/44.60 REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), member1_out_ag(z1)) 172.38/44.60 REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1)) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), U6_ag(member1_in_ag(z3))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3))) 172.38/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2))) 172.38/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), U6_ag(member1_in_ag(z3))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3))) 172.38/44.60 REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2))) 172.38/44.60 REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, U5_gg(member_in_gg(z0, .(z0, .(z0, z4))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z0, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(member_in_gg(x2, .(z0, .(z0, z4))))) 172.38/44.60 U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z0, .(z4, z5))))) 172.38/44.60 172.38/44.60 The TRS R consists of the following rules: 172.38/44.60 172.38/44.60 member1_in_ag(.(H, L)) -> member1_out_ag(H) 172.38/44.60 member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) 172.38/44.60 member_in_gg(H, .(H, L)) -> member_out_gg 172.38/44.60 member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) 172.38/44.60 U6_ag(member1_out_ag(X)) -> member1_out_ag(X) 172.38/44.60 U5_gg(member_out_gg) -> member_out_gg 172.38/44.60 172.38/44.60 The set Q consists of the following terms: 172.38/44.60 172.38/44.60 member1_in_ag(x0) 172.38/44.60 member_in_gg(x0, x1) 172.38/44.60 U6_ag(x0) 172.38/44.60 U5_gg(x0) 172.38/44.60 172.38/44.60 We have to consider all (P,Q,R)-chains. 172.38/44.60 ---------------------------------------- 172.38/44.60 172.38/44.60 (137) TransformationProof (EQUIVALENT) 172.38/44.60 By instantiating [LPAR04] the rule U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) we obtained the following new rules [LPAR04]: 172.38/44.60 172.38/44.60 (U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, U5_gg(member_in_gg(z0, .(z0, .(z0, z4))))),U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, U5_gg(member_in_gg(z0, .(z0, .(z0, z4)))))) 172.38/44.60 (U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z0, .(z4, z5))))),U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z0, .(z4, z5)))))) 172.38/44.60 (U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(member_in_gg(x2, .(z0, .(z0, z4))))),U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(member_in_gg(x2, .(z0, .(z0, z4)))))) 172.38/44.60 (U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z0, .(z4, z5))))),U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z0, .(z4, z5)))))) 172.38/44.60 (U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z4, .(z4, z5))))),U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z4, .(z4, z5)))))) 172.38/44.60 (U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, U5_gg(member_in_gg(z0, .(z4, .(z5, z6))))),U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, U5_gg(member_in_gg(z0, .(z4, .(z5, z6)))))) 172.38/44.60 (U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z4, .(z4, z5))))),U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z4, .(z4, z5)))))) 172.38/44.60 (U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), x2, U5_gg(member_in_gg(x2, .(z4, .(z5, z6))))),U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), x2, U5_gg(member_in_gg(x2, .(z4, .(z5, z6)))))) 172.38/44.60 172.38/44.60 172.38/44.60 ---------------------------------------- 172.38/44.60 172.38/44.60 (138) 172.38/44.60 Obligation: 172.38/44.60 Q DP problem: 172.38/44.60 The TRS P consists of the following rules: 172.38/44.60 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4)))) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) 172.38/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) 172.38/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) 172.38/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) 172.38/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) 172.38/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) 172.38/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z5, z6)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) 172.38/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z6, member_out_gg) -> REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4)))) 172.38/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) -> REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5)))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), member1_out_ag(.(z1, .(z2, [])))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 172.38/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), member1_out_ag(z1)) 172.38/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1)) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, [])))) 172.38/44.60 REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), member1_out_ag(z1)) 172.38/44.60 REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1)) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), U6_ag(member1_in_ag(z3))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3))) 172.38/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2))) 172.38/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), U6_ag(member1_in_ag(z3))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3))) 172.38/44.60 REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2))) 172.38/44.60 REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, U5_gg(member_in_gg(z0, .(z0, .(z0, z4))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z0, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(member_in_gg(x2, .(z0, .(z0, z4))))) 172.38/44.60 U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z0, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z4, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, U5_gg(member_in_gg(z0, .(z4, .(z5, z6))))) 172.38/44.60 U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z4, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), x2, U5_gg(member_in_gg(x2, .(z4, .(z5, z6))))) 172.38/44.60 172.38/44.60 The TRS R consists of the following rules: 172.38/44.60 172.38/44.60 member1_in_ag(.(H, L)) -> member1_out_ag(H) 172.38/44.60 member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) 172.38/44.60 member_in_gg(H, .(H, L)) -> member_out_gg 172.38/44.60 member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) 172.38/44.60 U6_ag(member1_out_ag(X)) -> member1_out_ag(X) 172.38/44.60 U5_gg(member_out_gg) -> member_out_gg 172.38/44.60 172.38/44.60 The set Q consists of the following terms: 172.38/44.60 172.38/44.60 member1_in_ag(x0) 172.38/44.60 member_in_gg(x0, x1) 172.38/44.60 U6_ag(x0) 172.38/44.60 U5_gg(x0) 172.38/44.60 172.38/44.60 We have to consider all (P,Q,R)-chains. 172.38/44.60 ---------------------------------------- 172.38/44.60 172.38/44.60 (139) TransformationProof (EQUIVALENT) 172.38/44.60 By instantiating [LPAR04] the rule U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4)))) we obtained the following new rules [LPAR04]: 172.38/44.60 172.38/44.60 (U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, U5_gg(member_in_gg(z0, .(z0, .(z0, z4))))),U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, U5_gg(member_in_gg(z0, .(z0, .(z0, z4)))))) 172.38/44.60 (U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z0, .(z4, z5))))),U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z0, .(z4, z5)))))) 172.38/44.60 (U2_GGGG(z1, .(.(x5, .(x6, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(x6, [])))) -> U3_GGGG(z1, .(.(x5, .(x6, [])), z3), .(z0, .(z0, .(z0, z4))), x6, U5_gg(member_in_gg(x6, .(z0, .(z0, z4))))),U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(member_in_gg(x2, .(z0, .(z0, z4)))))) 172.38/44.60 (U2_GGGG(z1, .(.(x5, .(x6, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(x6, [])))) -> U3_GGGG(z1, .(.(x5, .(x6, [])), z3), .(z0, .(z0, .(z4, z5))), x6, U5_gg(member_in_gg(x6, .(z0, .(z4, z5))))),U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z0, .(z4, z5)))))) 172.38/44.60 (U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(x6, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), x6, U5_gg(member_in_gg(x6, .(z0, .(z0, z4))))),U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(x6, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), x6, U5_gg(member_in_gg(x6, .(z0, .(z0, z4)))))) 172.38/44.60 (U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(x6, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), x6, U5_gg(member_in_gg(x6, .(z0, .(z4, z5))))),U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(x6, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), x6, U5_gg(member_in_gg(x6, .(z0, .(z4, z5)))))) 172.38/44.60 (U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(x6, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x6, U5_gg(member_in_gg(x6, .(z0, .(z0, z4))))),U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(x6, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x6, U5_gg(member_in_gg(x6, .(z0, .(z0, z4)))))) 172.38/44.60 (U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(x6, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x6, U5_gg(member_in_gg(x6, .(z0, .(z4, z5))))),U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(x6, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x6, U5_gg(member_in_gg(x6, .(z0, .(z4, z5)))))) 172.38/44.60 172.38/44.60 172.38/44.60 ---------------------------------------- 172.38/44.60 172.38/44.60 (140) 172.38/44.60 Obligation: 172.38/44.60 Q DP problem: 172.38/44.60 The TRS P consists of the following rules: 172.38/44.60 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) 172.38/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) 172.38/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) 172.38/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) 172.38/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) 172.38/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) 172.38/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z5, z6)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) 172.38/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z6, member_out_gg) -> REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4)))) 172.38/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) -> REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5)))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), member1_out_ag(.(z1, .(z2, [])))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 172.38/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), member1_out_ag(z1)) 172.38/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1)) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, [])))) 172.38/44.60 REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), member1_out_ag(z1)) 172.38/44.60 REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1)) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), U6_ag(member1_in_ag(z3))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3))) 172.38/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2))) 172.38/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), U6_ag(member1_in_ag(z3))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3))) 172.38/44.60 REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2))) 172.38/44.60 REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, U5_gg(member_in_gg(z0, .(z0, .(z0, z4))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z0, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(member_in_gg(x2, .(z0, .(z0, z4))))) 172.38/44.60 U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z0, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z4, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, U5_gg(member_in_gg(z0, .(z4, .(z5, z6))))) 172.38/44.60 U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z4, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), x2, U5_gg(member_in_gg(x2, .(z4, .(z5, z6))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(x6, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), x6, U5_gg(member_in_gg(x6, .(z0, .(z0, z4))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(x6, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), x6, U5_gg(member_in_gg(x6, .(z0, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(x6, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x6, U5_gg(member_in_gg(x6, .(z0, .(z0, z4))))) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(x6, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x6, U5_gg(member_in_gg(x6, .(z0, .(z4, z5))))) 172.38/44.60 172.38/44.60 The TRS R consists of the following rules: 172.38/44.60 172.38/44.60 member1_in_ag(.(H, L)) -> member1_out_ag(H) 172.38/44.60 member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) 172.38/44.60 member_in_gg(H, .(H, L)) -> member_out_gg 172.38/44.60 member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) 172.38/44.60 U6_ag(member1_out_ag(X)) -> member1_out_ag(X) 172.38/44.60 U5_gg(member_out_gg) -> member_out_gg 172.38/44.60 172.38/44.60 The set Q consists of the following terms: 172.38/44.60 172.38/44.60 member1_in_ag(x0) 172.38/44.60 member_in_gg(x0, x1) 172.38/44.60 U6_ag(x0) 172.38/44.60 U5_gg(x0) 172.38/44.60 172.38/44.60 We have to consider all (P,Q,R)-chains. 172.38/44.60 ---------------------------------------- 172.38/44.60 172.38/44.60 (141) TransformationProof (EQUIVALENT) 172.38/44.60 By instantiating [LPAR04] the rule U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) we obtained the following new rules [LPAR04]: 172.38/44.60 172.38/44.60 (U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, U5_gg(member_in_gg(z0, .(z0, .(z0, z4))))),U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, U5_gg(member_in_gg(z0, .(z0, .(z0, z4)))))) 172.38/44.60 (U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z0, .(z4, z5))))),U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z0, .(z4, z5)))))) 172.38/44.60 (U2_GGGG(z1, .(.(x6, .(x7, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x6, .(x7, [])))) -> U3_GGGG(z1, .(.(x6, .(x7, [])), z3), .(z0, .(z0, .(z0, z4))), x7, U5_gg(member_in_gg(x7, .(z0, .(z0, z4))))),U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(member_in_gg(x2, .(z0, .(z0, z4)))))) 172.38/44.60 (U2_GGGG(z1, .(.(x6, .(x7, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x6, .(x7, [])))) -> U3_GGGG(z1, .(.(x6, .(x7, [])), z3), .(z0, .(z0, .(z4, z5))), x7, U5_gg(member_in_gg(x7, .(z0, .(z4, z5))))),U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z0, .(z4, z5)))))) 172.38/44.60 (U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z4, .(z4, z5))))),U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z4, .(z4, z5)))))) 172.38/44.60 (U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, U5_gg(member_in_gg(z0, .(z4, .(z5, z6))))),U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, U5_gg(member_in_gg(z0, .(z4, .(z5, z6)))))) 172.38/44.60 (U2_GGGG(z1, .(.(x6, .(x7, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(x7, [])))) -> U3_GGGG(z1, .(.(x6, .(x7, [])), z3), .(z0, .(z4, .(z4, z5))), x7, U5_gg(member_in_gg(x7, .(z4, .(z4, z5))))),U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z4, .(z4, z5)))))) 172.38/44.60 (U2_GGGG(z1, .(.(x6, .(x7, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(x7, [])))) -> U3_GGGG(z1, .(.(x6, .(x7, [])), z3), .(z0, .(z4, .(z5, z6))), x7, U5_gg(member_in_gg(x7, .(z4, .(z5, z6))))),U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), x2, U5_gg(member_in_gg(x2, .(z4, .(z5, z6)))))) 172.38/44.60 (U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x6, .(x7, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), x7, U5_gg(member_in_gg(x7, .(z0, .(z0, z4))))),U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(x6, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), x6, U5_gg(member_in_gg(x6, .(z0, .(z0, z4)))))) 172.38/44.60 (U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x6, .(x7, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), x7, U5_gg(member_in_gg(x7, .(z0, .(z4, z5))))),U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(x6, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), x6, U5_gg(member_in_gg(x6, .(z0, .(z4, z5)))))) 172.38/44.60 (U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x6, .(x7, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x7, U5_gg(member_in_gg(x7, .(z0, .(z0, z4))))),U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(x6, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x6, U5_gg(member_in_gg(x6, .(z0, .(z0, z4)))))) 172.38/44.60 (U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x6, .(x7, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x7, U5_gg(member_in_gg(x7, .(z0, .(z4, z5))))),U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(x6, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x6, U5_gg(member_in_gg(x6, .(z0, .(z4, z5)))))) 172.38/44.60 (U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(x7, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), x7, U5_gg(member_in_gg(x7, .(z4, .(z4, z5))))),U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(x7, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), x7, U5_gg(member_in_gg(x7, .(z4, .(z4, z5)))))) 172.38/44.60 (U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(x7, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), x7, U5_gg(member_in_gg(x7, .(z4, .(z5, z6))))),U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(x7, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), x7, U5_gg(member_in_gg(x7, .(z4, .(z5, z6)))))) 172.38/44.60 (U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(x7, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), x7, U5_gg(member_in_gg(x7, .(z4, .(z4, z5))))),U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(x7, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), x7, U5_gg(member_in_gg(x7, .(z4, .(z4, z5)))))) 172.38/44.60 (U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(x7, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), x7, U5_gg(member_in_gg(x7, .(z4, .(z5, z6))))),U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(x7, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), x7, U5_gg(member_in_gg(x7, .(z4, .(z5, z6)))))) 172.38/44.60 172.38/44.60 172.38/44.60 ---------------------------------------- 172.38/44.60 172.38/44.60 (142) 172.38/44.60 Obligation: 172.38/44.60 Q DP problem: 172.38/44.60 The TRS P consists of the following rules: 172.38/44.60 172.38/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) 172.38/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) 172.38/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) 172.38/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) 172.38/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) 172.38/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z5, z6)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) 172.38/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z6, member_out_gg) -> REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4)))) 172.38/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) -> REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5)))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), member1_out_ag(.(z1, .(z2, [])))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 172.38/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), member1_out_ag(z1)) 172.38/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1)) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, [])))) 172.38/44.60 REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), member1_out_ag(z1)) 172.38/44.60 REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1)) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), U6_ag(member1_in_ag(z3))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3))) 172.38/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2))) 172.38/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), U6_ag(member1_in_ag(z3))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3))) 172.38/44.60 REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2))) 172.38/44.60 REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, U5_gg(member_in_gg(z0, .(z0, .(z0, z4))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z0, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(member_in_gg(x2, .(z0, .(z0, z4))))) 172.38/44.60 U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z0, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z4, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, U5_gg(member_in_gg(z0, .(z4, .(z5, z6))))) 172.38/44.60 U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z4, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), x2, U5_gg(member_in_gg(x2, .(z4, .(z5, z6))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(x6, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), x6, U5_gg(member_in_gg(x6, .(z0, .(z0, z4))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(x6, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), x6, U5_gg(member_in_gg(x6, .(z0, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(x6, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x6, U5_gg(member_in_gg(x6, .(z0, .(z0, z4))))) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(x6, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x6, U5_gg(member_in_gg(x6, .(z0, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(x7, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), x7, U5_gg(member_in_gg(x7, .(z4, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(x7, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), x7, U5_gg(member_in_gg(x7, .(z4, .(z5, z6))))) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(x7, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), x7, U5_gg(member_in_gg(x7, .(z4, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(x7, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), x7, U5_gg(member_in_gg(x7, .(z4, .(z5, z6))))) 172.38/44.60 172.38/44.60 The TRS R consists of the following rules: 172.38/44.60 172.38/44.60 member1_in_ag(.(H, L)) -> member1_out_ag(H) 172.38/44.60 member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) 172.38/44.60 member_in_gg(H, .(H, L)) -> member_out_gg 172.38/44.60 member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) 172.38/44.60 U6_ag(member1_out_ag(X)) -> member1_out_ag(X) 172.38/44.60 U5_gg(member_out_gg) -> member_out_gg 172.38/44.60 172.38/44.60 The set Q consists of the following terms: 172.38/44.60 172.38/44.60 member1_in_ag(x0) 172.38/44.60 member_in_gg(x0, x1) 172.38/44.60 U6_ag(x0) 172.38/44.60 U5_gg(x0) 172.38/44.60 172.38/44.60 We have to consider all (P,Q,R)-chains. 172.38/44.60 ---------------------------------------- 172.38/44.60 172.38/44.60 (143) TransformationProof (EQUIVALENT) 172.38/44.60 By forward instantiating [JAR06] the rule REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), member1_out_ag(z1)) we obtained the following new rules [LPAR04]: 172.38/44.60 172.38/44.60 (REACH_IN_GGGG(x0, x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x0, x4)))) -> U2_GGGG(x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x0, x4))), member1_out_ag(.(y_1, .(y_2, [])))),REACH_IN_GGGG(x0, x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x0, x4)))) -> U2_GGGG(x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x0, x4))), member1_out_ag(.(y_1, .(y_2, []))))) 172.38/44.60 172.38/44.60 172.38/44.60 ---------------------------------------- 172.38/44.60 172.38/44.60 (144) 172.38/44.60 Obligation: 172.38/44.60 Q DP problem: 172.38/44.60 The TRS P consists of the following rules: 172.38/44.60 172.38/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) 172.38/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) 172.38/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) 172.38/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) 172.38/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) 172.38/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z5, z6)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) 172.38/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z6, member_out_gg) -> REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4)))) 172.38/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) -> REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5)))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), member1_out_ag(.(z1, .(z2, [])))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 172.38/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1)) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, [])))) 172.38/44.60 REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), member1_out_ag(z1)) 172.38/44.60 REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1)) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), U6_ag(member1_in_ag(z3))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3))) 172.38/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2))) 172.38/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), U6_ag(member1_in_ag(z3))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3))) 172.38/44.60 REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2))) 172.38/44.60 REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, U5_gg(member_in_gg(z0, .(z0, .(z0, z4))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z0, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(member_in_gg(x2, .(z0, .(z0, z4))))) 172.38/44.60 U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z0, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z4, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, U5_gg(member_in_gg(z0, .(z4, .(z5, z6))))) 172.38/44.60 U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z4, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), x2, U5_gg(member_in_gg(x2, .(z4, .(z5, z6))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(x6, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), x6, U5_gg(member_in_gg(x6, .(z0, .(z0, z4))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(x6, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), x6, U5_gg(member_in_gg(x6, .(z0, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(x6, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x6, U5_gg(member_in_gg(x6, .(z0, .(z0, z4))))) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(x6, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x6, U5_gg(member_in_gg(x6, .(z0, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(x7, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), x7, U5_gg(member_in_gg(x7, .(z4, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(x7, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), x7, U5_gg(member_in_gg(x7, .(z4, .(z5, z6))))) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(x7, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), x7, U5_gg(member_in_gg(x7, .(z4, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(x7, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), x7, U5_gg(member_in_gg(x7, .(z4, .(z5, z6))))) 172.38/44.60 REACH_IN_GGGG(x0, x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x0, x4)))) -> U2_GGGG(x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x0, x4))), member1_out_ag(.(y_1, .(y_2, [])))) 172.38/44.60 172.38/44.60 The TRS R consists of the following rules: 172.38/44.60 172.38/44.60 member1_in_ag(.(H, L)) -> member1_out_ag(H) 172.38/44.60 member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) 172.38/44.60 member_in_gg(H, .(H, L)) -> member_out_gg 172.38/44.60 member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) 172.38/44.60 U6_ag(member1_out_ag(X)) -> member1_out_ag(X) 172.38/44.60 U5_gg(member_out_gg) -> member_out_gg 172.38/44.60 172.38/44.60 The set Q consists of the following terms: 172.38/44.60 172.38/44.60 member1_in_ag(x0) 172.38/44.60 member_in_gg(x0, x1) 172.38/44.60 U6_ag(x0) 172.38/44.60 U5_gg(x0) 172.38/44.60 172.38/44.60 We have to consider all (P,Q,R)-chains. 172.38/44.60 ---------------------------------------- 172.38/44.60 172.38/44.60 (145) TransformationProof (EQUIVALENT) 172.38/44.60 By forward instantiating [JAR06] the rule REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1)) we obtained the following new rules [LPAR04]: 172.38/44.60 172.38/44.60 (REACH_IN_GGGG(x0, x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x4, x5)))) -> U2_GGGG(x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x4, x5))), member1_out_ag(.(y_1, .(y_2, [])))),REACH_IN_GGGG(x0, x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x4, x5)))) -> U2_GGGG(x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x4, x5))), member1_out_ag(.(y_1, .(y_2, []))))) 172.38/44.60 172.38/44.60 172.38/44.60 ---------------------------------------- 172.38/44.60 172.38/44.60 (146) 172.38/44.60 Obligation: 172.38/44.60 Q DP problem: 172.38/44.60 The TRS P consists of the following rules: 172.38/44.60 172.38/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) 172.38/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) 172.38/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) 172.38/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) 172.38/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) 172.38/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z5, z6)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) 172.38/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z6, member_out_gg) -> REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4)))) 172.38/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) -> REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5)))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), member1_out_ag(.(z1, .(z2, [])))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, [])))) 172.38/44.60 REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), member1_out_ag(z1)) 172.38/44.60 REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1)) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), U6_ag(member1_in_ag(z3))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3))) 172.38/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2))) 172.38/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), U6_ag(member1_in_ag(z3))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3))) 172.38/44.60 REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2))) 172.38/44.60 REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, U5_gg(member_in_gg(z0, .(z0, .(z0, z4))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z0, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(member_in_gg(x2, .(z0, .(z0, z4))))) 172.38/44.60 U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z0, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z4, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, U5_gg(member_in_gg(z0, .(z4, .(z5, z6))))) 172.38/44.60 U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z4, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), x2, U5_gg(member_in_gg(x2, .(z4, .(z5, z6))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(x6, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), x6, U5_gg(member_in_gg(x6, .(z0, .(z0, z4))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(x6, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), x6, U5_gg(member_in_gg(x6, .(z0, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(x6, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x6, U5_gg(member_in_gg(x6, .(z0, .(z0, z4))))) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(x6, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x6, U5_gg(member_in_gg(x6, .(z0, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(x7, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), x7, U5_gg(member_in_gg(x7, .(z4, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(x7, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), x7, U5_gg(member_in_gg(x7, .(z4, .(z5, z6))))) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(x7, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), x7, U5_gg(member_in_gg(x7, .(z4, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(x7, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), x7, U5_gg(member_in_gg(x7, .(z4, .(z5, z6))))) 172.38/44.60 REACH_IN_GGGG(x0, x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x0, x4)))) -> U2_GGGG(x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x0, x4))), member1_out_ag(.(y_1, .(y_2, [])))) 172.38/44.60 REACH_IN_GGGG(x0, x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x4, x5)))) -> U2_GGGG(x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x4, x5))), member1_out_ag(.(y_1, .(y_2, [])))) 172.38/44.60 172.38/44.60 The TRS R consists of the following rules: 172.38/44.60 172.38/44.60 member1_in_ag(.(H, L)) -> member1_out_ag(H) 172.38/44.60 member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) 172.38/44.60 member_in_gg(H, .(H, L)) -> member_out_gg 172.38/44.60 member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) 172.38/44.60 U6_ag(member1_out_ag(X)) -> member1_out_ag(X) 172.38/44.60 U5_gg(member_out_gg) -> member_out_gg 172.38/44.60 172.38/44.60 The set Q consists of the following terms: 172.38/44.60 172.38/44.60 member1_in_ag(x0) 172.38/44.60 member_in_gg(x0, x1) 172.38/44.60 U6_ag(x0) 172.38/44.60 U5_gg(x0) 172.38/44.60 172.38/44.60 We have to consider all (P,Q,R)-chains. 172.38/44.60 ---------------------------------------- 172.38/44.60 172.38/44.60 (147) TransformationProof (EQUIVALENT) 172.38/44.60 By forward instantiating [JAR06] the rule REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), member1_out_ag(z1)) we obtained the following new rules [LPAR04]: 172.38/44.60 172.38/44.60 (REACH_IN_GGGG(x0, x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x4, .(x4, x5)))) -> U2_GGGG(x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x4, .(x4, x5))), member1_out_ag(.(y_1, .(y_2, [])))),REACH_IN_GGGG(x0, x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x4, .(x4, x5)))) -> U2_GGGG(x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x4, .(x4, x5))), member1_out_ag(.(y_1, .(y_2, []))))) 172.38/44.60 172.38/44.60 172.38/44.60 ---------------------------------------- 172.38/44.60 172.38/44.60 (148) 172.38/44.60 Obligation: 172.38/44.60 Q DP problem: 172.38/44.60 The TRS P consists of the following rules: 172.38/44.60 172.38/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) 172.38/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) 172.38/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) 172.38/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) 172.38/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) 172.38/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z5, z6)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) 172.38/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z6, member_out_gg) -> REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4)))) 172.38/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) -> REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5)))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), member1_out_ag(.(z1, .(z2, [])))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, [])))) 172.38/44.60 REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1)) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), U6_ag(member1_in_ag(z3))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3))) 172.38/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2))) 172.38/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), U6_ag(member1_in_ag(z3))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3))) 172.38/44.60 REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2))) 172.38/44.60 REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, U5_gg(member_in_gg(z0, .(z0, .(z0, z4))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z0, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(member_in_gg(x2, .(z0, .(z0, z4))))) 172.38/44.60 U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z0, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z4, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, U5_gg(member_in_gg(z0, .(z4, .(z5, z6))))) 172.38/44.60 U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z4, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), x2, U5_gg(member_in_gg(x2, .(z4, .(z5, z6))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(x6, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), x6, U5_gg(member_in_gg(x6, .(z0, .(z0, z4))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(x6, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), x6, U5_gg(member_in_gg(x6, .(z0, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(x6, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x6, U5_gg(member_in_gg(x6, .(z0, .(z0, z4))))) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(x6, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x6, U5_gg(member_in_gg(x6, .(z0, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(x7, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), x7, U5_gg(member_in_gg(x7, .(z4, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(x7, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), x7, U5_gg(member_in_gg(x7, .(z4, .(z5, z6))))) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(x7, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), x7, U5_gg(member_in_gg(x7, .(z4, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(x7, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), x7, U5_gg(member_in_gg(x7, .(z4, .(z5, z6))))) 172.38/44.60 REACH_IN_GGGG(x0, x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x0, x4)))) -> U2_GGGG(x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x0, x4))), member1_out_ag(.(y_1, .(y_2, [])))) 172.38/44.60 REACH_IN_GGGG(x0, x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x4, x5)))) -> U2_GGGG(x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x4, x5))), member1_out_ag(.(y_1, .(y_2, [])))) 172.38/44.60 REACH_IN_GGGG(x0, x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x4, .(x4, x5)))) -> U2_GGGG(x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x4, .(x4, x5))), member1_out_ag(.(y_1, .(y_2, [])))) 172.38/44.60 172.38/44.60 The TRS R consists of the following rules: 172.38/44.60 172.38/44.60 member1_in_ag(.(H, L)) -> member1_out_ag(H) 172.38/44.60 member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) 172.38/44.60 member_in_gg(H, .(H, L)) -> member_out_gg 172.38/44.60 member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) 172.38/44.60 U6_ag(member1_out_ag(X)) -> member1_out_ag(X) 172.38/44.60 U5_gg(member_out_gg) -> member_out_gg 172.38/44.60 172.38/44.60 The set Q consists of the following terms: 172.38/44.60 172.38/44.60 member1_in_ag(x0) 172.38/44.60 member_in_gg(x0, x1) 172.38/44.60 U6_ag(x0) 172.38/44.60 U5_gg(x0) 172.38/44.60 172.38/44.60 We have to consider all (P,Q,R)-chains. 172.38/44.60 ---------------------------------------- 172.38/44.60 172.38/44.60 (149) TransformationProof (EQUIVALENT) 172.38/44.60 By forward instantiating [JAR06] the rule REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1)) we obtained the following new rules [LPAR04]: 172.38/44.60 172.38/44.60 (REACH_IN_GGGG(x0, x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x4, .(x5, x6)))) -> U2_GGGG(x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x4, .(x5, x6))), member1_out_ag(.(y_1, .(y_2, [])))),REACH_IN_GGGG(x0, x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x4, .(x5, x6)))) -> U2_GGGG(x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x4, .(x5, x6))), member1_out_ag(.(y_1, .(y_2, []))))) 172.38/44.60 172.38/44.60 172.38/44.60 ---------------------------------------- 172.38/44.60 172.38/44.60 (150) 172.38/44.60 Obligation: 172.38/44.60 Q DP problem: 172.38/44.60 The TRS P consists of the following rules: 172.38/44.60 172.38/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) 172.38/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) 172.38/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) 172.38/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) 172.38/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) 172.38/44.60 U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z5, z6)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) 172.38/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z6, member_out_gg) -> REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4)))) 172.38/44.60 U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) -> REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5)))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), member1_out_ag(.(z1, .(z2, [])))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, [])))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), U6_ag(member1_in_ag(z3))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3))) 172.38/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2))) 172.38/44.60 REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), U6_ag(member1_in_ag(z3))) 172.38/44.60 REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3))) 172.38/44.60 REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) -> U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2))) 172.38/44.60 REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, U5_gg(member_in_gg(z0, .(z0, .(z0, z4))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z0, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(member_in_gg(x2, .(z0, .(z0, z4))))) 172.38/44.60 U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z0, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z4, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, U5_gg(member_in_gg(z0, .(z4, .(z5, z6))))) 172.38/44.60 U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z4, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x1, .(x2, [])))) -> U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), x2, U5_gg(member_in_gg(x2, .(z4, .(z5, z6))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(x6, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), x6, U5_gg(member_in_gg(x6, .(z0, .(z0, z4))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(x6, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), x6, U5_gg(member_in_gg(x6, .(z0, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(x6, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x6, U5_gg(member_in_gg(x6, .(z0, .(z0, z4))))) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(x6, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x6, U5_gg(member_in_gg(x6, .(z0, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(x7, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), x7, U5_gg(member_in_gg(x7, .(z4, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(x7, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), x7, U5_gg(member_in_gg(x7, .(z4, .(z5, z6))))) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(x7, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), x7, U5_gg(member_in_gg(x7, .(z4, .(z4, z5))))) 172.38/44.60 U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(x7, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), x7, U5_gg(member_in_gg(x7, .(z4, .(z5, z6))))) 172.38/44.60 REACH_IN_GGGG(x0, x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x0, x4)))) -> U2_GGGG(x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x0, x4))), member1_out_ag(.(y_1, .(y_2, [])))) 172.38/44.60 REACH_IN_GGGG(x0, x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x4, x5)))) -> U2_GGGG(x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x4, x5))), member1_out_ag(.(y_1, .(y_2, [])))) 172.38/44.60 REACH_IN_GGGG(x0, x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x4, .(x4, x5)))) -> U2_GGGG(x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x4, .(x4, x5))), member1_out_ag(.(y_1, .(y_2, [])))) 172.38/44.60 REACH_IN_GGGG(x0, x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x4, .(x5, x6)))) -> U2_GGGG(x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x4, .(x5, x6))), member1_out_ag(.(y_1, .(y_2, [])))) 172.38/44.60 172.38/44.60 The TRS R consists of the following rules: 172.38/44.60 172.38/44.60 member1_in_ag(.(H, L)) -> member1_out_ag(H) 172.38/44.60 member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) 172.38/44.60 member_in_gg(H, .(H, L)) -> member_out_gg 172.38/44.60 member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) 172.38/44.60 U6_ag(member1_out_ag(X)) -> member1_out_ag(X) 172.38/44.60 U5_gg(member_out_gg) -> member_out_gg 172.38/44.60 172.38/44.60 The set Q consists of the following terms: 172.38/44.60 172.38/44.60 member1_in_ag(x0) 172.38/44.60 member_in_gg(x0, x1) 172.38/44.60 U6_ag(x0) 172.38/44.60 U5_gg(x0) 172.38/44.60 172.38/44.60 We have to consider all (P,Q,R)-chains. 172.38/44.60 ---------------------------------------- 172.38/44.60 172.38/44.60 (151) PrologToDTProblemTransformerProof (SOUND) 172.38/44.60 Built DT problem from termination graph DT10. 172.38/44.60 172.38/44.60 { 172.38/44.60 "root": 1, 172.38/44.60 "program": { 172.38/44.60 "directives": [], 172.38/44.60 "clauses": [ 172.38/44.60 [ 172.38/44.60 "(reach X Y Edges Visited)", 172.38/44.60 "(member (. X (. Y ([]))) Edges)" 172.38/44.60 ], 172.38/44.60 [ 172.38/44.60 "(reach X Z Edges Visited)", 172.38/44.60 "(',' (member1 (. X (. Y ([]))) Edges) (',' (member Y Visited) (reach Y Z Edges (. Y Visited))))" 172.38/44.60 ], 172.38/44.60 [ 172.38/44.60 "(member H (. H L))", 172.38/44.60 null 172.38/44.60 ], 172.38/44.60 [ 172.38/44.60 "(member X (. H L))", 172.38/44.60 "(member X L)" 172.38/44.60 ], 172.38/44.60 [ 172.38/44.60 "(member1 H (. H L))", 172.38/44.60 null 172.38/44.60 ], 172.38/44.60 [ 172.38/44.60 "(member1 X (. H L))", 172.38/44.60 "(member1 X L)" 172.38/44.60 ] 172.38/44.60 ] 172.38/44.60 }, 172.38/44.60 "graph": { 172.38/44.60 "nodes": { 172.38/44.60 "68": { 172.38/44.60 "goal": [{ 172.38/44.60 "clause": 2, 172.38/44.60 "scope": 2, 172.38/44.60 "term": "(member (. T9 (. T10 ([]))) T11)" 172.38/44.60 }], 172.38/44.60 "kb": { 172.38/44.60 "nonunifying": [], 172.38/44.60 "intvars": {}, 172.38/44.60 "arithmetic": { 172.38/44.60 "type": "PlainIntegerRelationState", 172.38/44.60 "relations": [] 172.38/44.60 }, 172.38/44.60 "ground": [ 172.38/44.60 "T9", 172.38/44.60 "T10", 172.38/44.60 "T11" 172.38/44.60 ], 172.38/44.60 "free": [], 172.38/44.60 "exprvars": [] 172.38/44.60 } 172.38/44.60 }, 172.38/44.60 "type": "Nodes", 172.38/44.60 "370": { 172.38/44.60 "goal": [], 172.38/44.60 "kb": { 172.38/44.60 "nonunifying": [], 172.38/44.60 "intvars": {}, 172.38/44.60 "arithmetic": { 172.38/44.60 "type": "PlainIntegerRelationState", 172.38/44.60 "relations": [] 172.38/44.60 }, 172.38/44.60 "ground": [], 172.38/44.60 "free": [], 172.38/44.60 "exprvars": [] 172.38/44.60 } 172.38/44.60 }, 172.38/44.60 "251": { 172.38/44.60 "goal": [], 172.38/44.60 "kb": { 172.38/44.60 "nonunifying": [], 172.38/44.60 "intvars": {}, 172.38/44.60 "arithmetic": { 172.38/44.60 "type": "PlainIntegerRelationState", 172.38/44.60 "relations": [] 172.38/44.60 }, 172.38/44.60 "ground": [], 172.38/44.60 "free": [], 172.38/44.60 "exprvars": [] 172.38/44.60 } 172.38/44.60 }, 172.38/44.60 "352": { 172.38/44.60 "goal": [{ 172.38/44.60 "clause": -1, 172.38/44.60 "scope": -1, 172.38/44.60 "term": "(member1 (. T139 (. X119 ([]))) T141)" 172.38/44.60 }], 172.38/44.60 "kb": { 172.38/44.60 "nonunifying": [], 172.38/44.60 "intvars": {}, 172.38/44.60 "arithmetic": { 172.38/44.60 "type": "PlainIntegerRelationState", 172.38/44.60 "relations": [] 172.38/44.60 }, 172.38/44.60 "ground": [ 172.38/44.60 "T139", 172.38/44.60 "T141" 172.38/44.60 ], 172.38/44.60 "free": ["X119"], 172.38/44.60 "exprvars": [] 172.38/44.60 } 172.38/44.60 }, 172.38/44.60 "111": { 172.38/44.60 "goal": [{ 172.38/44.60 "clause": -1, 172.38/44.60 "scope": -1, 172.38/44.60 "term": "(true)" 172.38/44.60 }], 172.38/44.60 "kb": { 172.38/44.60 "nonunifying": [], 172.38/44.60 "intvars": {}, 172.38/44.60 "arithmetic": { 172.38/44.60 "type": "PlainIntegerRelationState", 172.38/44.60 "relations": [] 172.38/44.60 }, 172.38/44.60 "ground": [], 172.38/44.60 "free": [], 172.38/44.60 "exprvars": [] 172.38/44.60 } 172.38/44.60 }, 172.38/44.60 "353": { 172.38/44.60 "goal": [], 172.38/44.60 "kb": { 172.38/44.60 "nonunifying": [], 172.38/44.60 "intvars": {}, 172.38/44.60 "arithmetic": { 172.38/44.60 "type": "PlainIntegerRelationState", 172.38/44.60 "relations": [] 172.38/44.60 }, 172.38/44.60 "ground": [], 172.38/44.60 "free": [], 172.38/44.60 "exprvars": [] 172.38/44.60 } 172.38/44.60 }, 172.38/44.60 "255": { 172.38/44.60 "goal": [{ 172.38/44.60 "clause": 3, 172.38/44.60 "scope": 2, 172.38/44.60 "term": "(member (. T9 (. T10 ([]))) T11)" 172.38/44.60 }], 172.38/44.60 "kb": { 172.38/44.60 "nonunifying": [], 172.38/44.60 "intvars": {}, 172.38/44.60 "arithmetic": { 172.38/44.60 "type": "PlainIntegerRelationState", 172.38/44.60 "relations": [] 172.38/44.60 }, 172.38/44.60 "ground": [ 172.38/44.60 "T9", 172.38/44.60 "T10", 172.38/44.60 "T11" 172.38/44.60 ], 172.38/44.60 "free": [], 172.38/44.60 "exprvars": [] 172.38/44.60 } 172.38/44.60 }, 172.38/44.60 "256": { 172.38/44.60 "goal": [ 172.38/44.60 { 172.38/44.60 "clause": -1, 172.38/44.60 "scope": 2, 172.38/44.60 "term": null 172.38/44.60 }, 172.38/44.60 { 172.38/44.60 "clause": 1, 172.38/44.60 "scope": 1, 172.38/44.60 "term": "(reach T9 T10 T11 T12)" 172.38/44.60 } 172.38/44.60 ], 172.38/44.60 "kb": { 172.38/44.60 "nonunifying": [], 172.38/44.60 "intvars": {}, 172.38/44.60 "arithmetic": { 172.38/44.60 "type": "PlainIntegerRelationState", 172.38/44.60 "relations": [] 172.38/44.60 }, 172.38/44.60 "ground": [ 172.38/44.60 "T9", 172.38/44.60 "T10", 172.38/44.60 "T11", 172.38/44.60 "T12" 172.38/44.60 ], 172.38/44.60 "free": [], 172.38/44.60 "exprvars": [] 172.38/44.60 } 172.38/44.60 }, 172.38/44.60 "259": { 172.38/44.60 "goal": [{ 172.38/44.60 "clause": -1, 172.38/44.60 "scope": -1, 172.38/44.60 "term": "(member (. T44 (. T45 ([]))) T47)" 172.38/44.60 }], 172.38/44.60 "kb": { 172.38/44.60 "nonunifying": [], 172.38/44.60 "intvars": {}, 172.38/44.60 "arithmetic": { 172.38/44.60 "type": "PlainIntegerRelationState", 172.38/44.60 "relations": [] 172.38/44.60 }, 172.38/44.60 "ground": [ 172.38/44.60 "T44", 172.38/44.60 "T45", 172.38/44.60 "T47" 172.38/44.60 ], 172.38/44.60 "free": [], 172.38/44.60 "exprvars": [] 172.38/44.60 } 172.38/44.60 }, 172.38/44.60 "336": { 172.38/44.60 "goal": [{ 172.38/44.60 "clause": -1, 172.38/44.60 "scope": -1, 172.38/44.60 "term": "(',' (member1 (. T101 (. X80 ([]))) T103) (',' (member X80 T104) (reach X80 T102 T103 (. X80 T104))))" 172.38/44.60 }], 172.38/44.60 "kb": { 172.38/44.60 "nonunifying": [], 172.38/44.60 "intvars": {}, 172.38/44.60 "arithmetic": { 172.38/44.60 "type": "PlainIntegerRelationState", 172.38/44.60 "relations": [] 172.38/44.60 }, 172.38/44.60 "ground": [ 172.38/44.60 "T101", 172.38/44.60 "T102", 172.38/44.60 "T103", 172.38/44.60 "T104" 172.38/44.60 ], 172.38/44.60 "free": ["X80"], 172.38/44.60 "exprvars": [] 172.38/44.60 } 172.38/44.60 }, 172.38/44.60 "339": { 172.38/44.60 "goal": [{ 172.38/44.60 "clause": -1, 172.38/44.60 "scope": -1, 172.38/44.60 "term": "(member1 (. T101 (. X80 ([]))) T103)" 172.38/44.60 }], 172.38/44.60 "kb": { 172.38/44.60 "nonunifying": [], 172.38/44.60 "intvars": {}, 172.38/44.60 "arithmetic": { 172.38/44.60 "type": "PlainIntegerRelationState", 172.38/44.60 "relations": [] 172.38/44.60 }, 172.38/44.60 "ground": [ 172.38/44.60 "T101", 172.38/44.60 "T103" 172.38/44.60 ], 172.38/44.60 "free": ["X80"], 172.38/44.60 "exprvars": [] 172.38/44.60 } 172.38/44.60 }, 172.38/44.60 "72": { 172.38/44.60 "goal": [ 172.38/44.60 { 172.38/44.60 "clause": 3, 172.38/44.60 "scope": 2, 172.38/44.60 "term": "(member (. T9 (. T10 ([]))) T11)" 172.38/44.60 }, 172.38/44.60 { 172.38/44.60 "clause": -1, 172.38/44.60 "scope": 2, 172.38/44.60 "term": null 172.38/44.60 }, 172.38/44.60 { 172.38/44.60 "clause": 1, 172.38/44.60 "scope": 1, 172.38/44.60 "term": "(reach T9 T10 T11 T12)" 172.38/44.60 } 172.38/44.60 ], 172.38/44.60 "kb": { 172.38/44.60 "nonunifying": [], 172.38/44.60 "intvars": {}, 172.38/44.60 "arithmetic": { 172.38/44.60 "type": "PlainIntegerRelationState", 172.38/44.60 "relations": [] 172.38/44.60 }, 172.38/44.60 "ground": [ 172.38/44.60 "T9", 172.38/44.60 "T10", 172.38/44.60 "T11", 172.38/44.60 "T12" 172.38/44.60 ], 172.38/44.60 "free": [], 172.38/44.60 "exprvars": [] 172.38/44.60 } 172.38/44.60 }, 172.38/44.60 "31": { 172.38/44.60 "goal": [ 172.38/44.60 { 172.38/44.60 "clause": 2, 172.38/44.60 "scope": 2, 172.38/44.60 "term": "(member (. T9 (. T10 ([]))) T11)" 172.38/44.60 }, 172.38/44.60 { 172.38/44.60 "clause": 3, 172.38/44.60 "scope": 2, 172.38/44.60 "term": "(member (. T9 (. T10 ([]))) T11)" 172.38/44.60 }, 172.38/44.60 { 172.38/44.60 "clause": -1, 172.38/44.60 "scope": 2, 172.38/44.60 "term": null 172.38/44.60 }, 172.38/44.60 { 172.38/44.60 "clause": 1, 172.38/44.60 "scope": 1, 172.38/44.60 "term": "(reach T9 T10 T11 T12)" 172.38/44.60 } 172.38/44.60 ], 172.38/44.61 "kb": { 172.38/44.61 "nonunifying": [], 172.38/44.61 "intvars": {}, 172.38/44.61 "arithmetic": { 172.38/44.61 "type": "PlainIntegerRelationState", 172.38/44.61 "relations": [] 172.38/44.61 }, 172.38/44.61 "ground": [ 172.38/44.61 "T9", 172.38/44.61 "T10", 172.38/44.61 "T11", 172.38/44.61 "T12" 172.38/44.61 ], 172.38/44.61 "free": [], 172.38/44.61 "exprvars": [] 172.38/44.61 } 172.38/44.61 }, 172.38/44.61 "18": { 172.38/44.61 "goal": [ 172.38/44.61 { 172.38/44.61 "clause": -1, 172.38/44.61 "scope": -1, 172.38/44.61 "term": "(member (. T9 (. T10 ([]))) T11)" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "clause": 1, 172.38/44.61 "scope": 1, 172.38/44.61 "term": "(reach T9 T10 T11 T12)" 172.38/44.61 } 172.38/44.61 ], 172.38/44.61 "kb": { 172.38/44.61 "nonunifying": [], 172.38/44.61 "intvars": {}, 172.38/44.61 "arithmetic": { 172.38/44.61 "type": "PlainIntegerRelationState", 172.38/44.61 "relations": [] 172.38/44.61 }, 172.38/44.61 "ground": [ 172.38/44.61 "T9", 172.38/44.61 "T10", 172.38/44.61 "T11", 172.38/44.61 "T12" 172.38/44.61 ], 172.38/44.61 "free": [], 172.38/44.61 "exprvars": [] 172.38/44.61 } 172.38/44.61 }, 172.38/44.61 "260": { 172.38/44.61 "goal": [], 172.38/44.61 "kb": { 172.38/44.61 "nonunifying": [], 172.38/44.61 "intvars": {}, 172.38/44.61 "arithmetic": { 172.38/44.61 "type": "PlainIntegerRelationState", 172.38/44.61 "relations": [] 172.38/44.61 }, 172.38/44.61 "ground": [], 172.38/44.61 "free": [], 172.38/44.61 "exprvars": [] 172.38/44.61 } 172.38/44.61 }, 172.38/44.61 "261": { 172.38/44.61 "goal": [ 172.38/44.61 { 172.38/44.61 "clause": 2, 172.38/44.61 "scope": 3, 172.38/44.61 "term": "(member (. T44 (. T45 ([]))) T47)" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "clause": 3, 172.38/44.61 "scope": 3, 172.38/44.61 "term": "(member (. T44 (. T45 ([]))) T47)" 172.38/44.61 } 172.38/44.61 ], 172.38/44.61 "kb": { 172.38/44.61 "nonunifying": [], 172.38/44.61 "intvars": {}, 172.38/44.61 "arithmetic": { 172.38/44.61 "type": "PlainIntegerRelationState", 172.38/44.61 "relations": [] 172.38/44.61 }, 172.38/44.61 "ground": [ 172.38/44.61 "T44", 172.38/44.61 "T45", 172.38/44.61 "T47" 172.38/44.61 ], 172.38/44.61 "free": [], 172.38/44.61 "exprvars": [] 172.38/44.61 } 172.38/44.61 }, 172.38/44.61 "360": { 172.38/44.61 "goal": [{ 172.38/44.61 "clause": -1, 172.38/44.61 "scope": -1, 172.38/44.61 "term": "(member T111 T104)" 172.38/44.61 }], 172.38/44.61 "kb": { 172.38/44.61 "nonunifying": [], 172.38/44.61 "intvars": {}, 172.38/44.61 "arithmetic": { 172.38/44.61 "type": "PlainIntegerRelationState", 172.38/44.61 "relations": [] 172.38/44.61 }, 172.38/44.61 "ground": [ 172.38/44.61 "T104", 172.38/44.61 "T111" 172.38/44.61 ], 172.38/44.61 "free": [], 172.38/44.61 "exprvars": [] 172.38/44.61 } 172.38/44.61 }, 172.38/44.61 "262": { 172.38/44.61 "goal": [{ 172.38/44.61 "clause": 2, 172.38/44.61 "scope": 3, 172.38/44.61 "term": "(member (. T44 (. T45 ([]))) T47)" 172.38/44.61 }], 172.38/44.61 "kb": { 172.38/44.61 "nonunifying": [], 172.38/44.61 "intvars": {}, 172.38/44.61 "arithmetic": { 172.38/44.61 "type": "PlainIntegerRelationState", 172.38/44.61 "relations": [] 172.38/44.61 }, 172.38/44.61 "ground": [ 172.38/44.61 "T44", 172.38/44.61 "T45", 172.38/44.61 "T47" 172.38/44.61 ], 172.38/44.61 "free": [], 172.38/44.61 "exprvars": [] 172.38/44.61 } 172.38/44.61 }, 172.38/44.61 "361": { 172.38/44.61 "goal": [{ 172.38/44.61 "clause": -1, 172.38/44.61 "scope": -1, 172.38/44.61 "term": "(reach T111 T102 T103 (. T111 T104))" 172.38/44.61 }], 172.38/44.61 "kb": { 172.38/44.61 "nonunifying": [], 172.38/44.61 "intvars": {}, 172.38/44.61 "arithmetic": { 172.38/44.61 "type": "PlainIntegerRelationState", 172.38/44.61 "relations": [] 172.38/44.61 }, 172.38/44.61 "ground": [ 172.38/44.61 "T102", 172.38/44.61 "T103", 172.38/44.61 "T104", 172.38/44.61 "T111" 172.38/44.61 ], 172.38/44.61 "free": [], 172.38/44.61 "exprvars": [] 172.38/44.61 } 172.38/44.61 }, 172.38/44.61 "120": { 172.38/44.61 "goal": [], 172.38/44.61 "kb": { 172.38/44.61 "nonunifying": [], 172.38/44.61 "intvars": {}, 172.38/44.61 "arithmetic": { 172.38/44.61 "type": "PlainIntegerRelationState", 172.38/44.61 "relations": [] 172.38/44.61 }, 172.38/44.61 "ground": [], 172.38/44.61 "free": [], 172.38/44.61 "exprvars": [] 172.38/44.61 } 172.38/44.61 }, 172.38/44.61 "263": { 172.38/44.61 "goal": [{ 172.38/44.61 "clause": 3, 172.38/44.61 "scope": 3, 172.38/44.61 "term": "(member (. T44 (. T45 ([]))) T47)" 172.38/44.61 }], 172.38/44.61 "kb": { 172.38/44.61 "nonunifying": [], 172.38/44.61 "intvars": {}, 172.38/44.61 "arithmetic": { 172.38/44.61 "type": "PlainIntegerRelationState", 172.38/44.61 "relations": [] 172.38/44.61 }, 172.38/44.61 "ground": [ 172.38/44.61 "T44", 172.38/44.61 "T45", 172.38/44.61 "T47" 172.38/44.61 ], 172.38/44.61 "free": [], 172.38/44.61 "exprvars": [] 172.38/44.61 } 172.38/44.61 }, 172.38/44.61 "340": { 172.38/44.61 "goal": [{ 172.38/44.61 "clause": -1, 172.38/44.61 "scope": -1, 172.38/44.61 "term": "(',' (member T111 T104) (reach T111 T102 T103 (. T111 T104)))" 172.38/44.61 }], 172.38/44.61 "kb": { 172.38/44.61 "nonunifying": [], 172.38/44.61 "intvars": {}, 172.38/44.61 "arithmetic": { 172.38/44.61 "type": "PlainIntegerRelationState", 172.38/44.61 "relations": [] 172.38/44.61 }, 172.38/44.61 "ground": [ 172.38/44.61 "T102", 172.38/44.61 "T103", 172.38/44.61 "T104", 172.38/44.61 "T111" 172.38/44.61 ], 172.38/44.61 "free": [], 172.38/44.61 "exprvars": [] 172.38/44.61 } 172.38/44.61 }, 172.38/44.61 "264": { 172.38/44.61 "goal": [{ 172.38/44.61 "clause": -1, 172.38/44.61 "scope": -1, 172.38/44.61 "term": "(true)" 172.38/44.61 }], 172.38/44.61 "kb": { 172.38/44.61 "nonunifying": [], 172.38/44.61 "intvars": {}, 172.38/44.61 "arithmetic": { 172.38/44.61 "type": "PlainIntegerRelationState", 172.38/44.61 "relations": [] 172.38/44.61 }, 172.38/44.61 "ground": [], 172.38/44.61 "free": [], 172.38/44.61 "exprvars": [] 172.38/44.61 } 172.38/44.61 }, 172.38/44.61 "363": { 172.38/44.61 "goal": [ 172.38/44.61 { 172.38/44.61 "clause": 2, 172.38/44.61 "scope": 5, 172.38/44.61 "term": "(member T111 T104)" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "clause": 3, 172.38/44.61 "scope": 5, 172.38/44.61 "term": "(member T111 T104)" 172.38/44.61 } 172.38/44.61 ], 172.38/44.61 "kb": { 172.38/44.61 "nonunifying": [], 172.38/44.61 "intvars": {}, 172.38/44.61 "arithmetic": { 172.38/44.61 "type": "PlainIntegerRelationState", 172.38/44.61 "relations": [] 172.38/44.61 }, 172.38/44.61 "ground": [ 172.38/44.61 "T104", 172.38/44.61 "T111" 172.38/44.61 ], 172.38/44.61 "free": [], 172.38/44.61 "exprvars": [] 172.38/44.61 } 172.38/44.61 }, 172.38/44.61 "1": { 172.38/44.61 "goal": [{ 172.38/44.61 "clause": -1, 172.38/44.61 "scope": -1, 172.38/44.61 "term": "(reach T1 T2 T3 T4)" 172.38/44.61 }], 172.38/44.61 "kb": { 172.38/44.61 "nonunifying": [], 172.38/44.61 "intvars": {}, 172.38/44.61 "arithmetic": { 172.38/44.61 "type": "PlainIntegerRelationState", 172.38/44.61 "relations": [] 172.38/44.61 }, 172.38/44.61 "ground": [ 172.38/44.61 "T4", 172.38/44.61 "T1", 172.38/44.61 "T2", 172.38/44.61 "T3" 172.38/44.61 ], 172.38/44.61 "free": [], 172.38/44.61 "exprvars": [] 172.38/44.61 } 172.38/44.61 }, 172.38/44.61 "265": { 172.38/44.61 "goal": [], 172.38/44.61 "kb": { 172.38/44.61 "nonunifying": [], 172.38/44.61 "intvars": {}, 172.38/44.61 "arithmetic": { 172.38/44.61 "type": "PlainIntegerRelationState", 172.38/44.61 "relations": [] 172.38/44.61 }, 172.38/44.61 "ground": [], 172.38/44.61 "free": [], 172.38/44.61 "exprvars": [] 172.38/44.61 } 172.38/44.61 }, 172.38/44.61 "342": { 172.38/44.61 "goal": [ 172.38/44.61 { 172.38/44.61 "clause": 4, 172.38/44.61 "scope": 4, 172.38/44.61 "term": "(member1 (. T101 (. X80 ([]))) T103)" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "clause": 5, 172.38/44.61 "scope": 4, 172.38/44.61 "term": "(member1 (. T101 (. X80 ([]))) T103)" 172.38/44.61 } 172.38/44.61 ], 172.38/44.61 "kb": { 172.38/44.61 "nonunifying": [], 172.38/44.61 "intvars": {}, 172.38/44.61 "arithmetic": { 172.38/44.61 "type": "PlainIntegerRelationState", 172.38/44.61 "relations": [] 172.38/44.61 }, 172.38/44.61 "ground": [ 172.38/44.61 "T101", 172.38/44.61 "T103" 172.38/44.61 ], 172.38/44.61 "free": ["X80"], 172.38/44.61 "exprvars": [] 172.38/44.61 } 172.38/44.61 }, 172.38/44.61 "364": { 172.38/44.61 "goal": [{ 172.38/44.61 "clause": 2, 172.38/44.61 "scope": 5, 172.38/44.61 "term": "(member T111 T104)" 172.38/44.61 }], 172.38/44.61 "kb": { 172.38/44.61 "nonunifying": [], 172.38/44.61 "intvars": {}, 172.38/44.61 "arithmetic": { 172.38/44.61 "type": "PlainIntegerRelationState", 172.38/44.61 "relations": [] 172.38/44.61 }, 172.38/44.61 "ground": [ 172.38/44.61 "T104", 172.38/44.61 "T111" 172.38/44.61 ], 172.38/44.61 "free": [], 172.38/44.61 "exprvars": [] 172.38/44.61 } 172.38/44.61 }, 172.38/44.61 "266": { 172.38/44.61 "goal": [], 172.38/44.61 "kb": { 172.38/44.61 "nonunifying": [], 172.38/44.61 "intvars": {}, 172.38/44.61 "arithmetic": { 172.38/44.61 "type": "PlainIntegerRelationState", 172.38/44.61 "relations": [] 172.38/44.61 }, 172.38/44.61 "ground": [], 172.38/44.61 "free": [], 172.38/44.61 "exprvars": [] 172.38/44.61 } 172.38/44.61 }, 172.38/44.61 "365": { 172.38/44.61 "goal": [{ 172.38/44.61 "clause": 3, 172.38/44.61 "scope": 5, 172.38/44.61 "term": "(member T111 T104)" 172.38/44.61 }], 172.38/44.61 "kb": { 172.38/44.61 "nonunifying": [], 172.38/44.61 "intvars": {}, 172.38/44.61 "arithmetic": { 172.38/44.61 "type": "PlainIntegerRelationState", 172.38/44.61 "relations": [] 172.38/44.61 }, 172.38/44.61 "ground": [ 172.38/44.61 "T104", 172.38/44.61 "T111" 172.38/44.61 ], 172.38/44.61 "free": [], 172.38/44.61 "exprvars": [] 172.38/44.61 } 172.38/44.61 }, 172.38/44.61 "267": { 172.38/44.61 "goal": [{ 172.38/44.61 "clause": -1, 172.38/44.61 "scope": -1, 172.38/44.61 "term": "(member (. T77 (. T78 ([]))) T80)" 172.38/44.61 }], 172.38/44.61 "kb": { 172.38/44.61 "nonunifying": [], 172.38/44.61 "intvars": {}, 172.38/44.61 "arithmetic": { 172.38/44.61 "type": "PlainIntegerRelationState", 172.38/44.61 "relations": [] 172.38/44.61 }, 172.38/44.61 "ground": [ 172.38/44.61 "T77", 172.38/44.61 "T78", 172.38/44.61 "T80" 172.38/44.61 ], 172.38/44.61 "free": [], 172.38/44.61 "exprvars": [] 172.38/44.61 } 172.38/44.61 }, 172.38/44.61 "322": { 172.38/44.61 "goal": [{ 172.38/44.61 "clause": 1, 172.38/44.61 "scope": 1, 172.38/44.61 "term": "(reach T9 T10 T11 T12)" 172.38/44.61 }], 172.38/44.61 "kb": { 172.38/44.61 "nonunifying": [], 172.38/44.61 "intvars": {}, 172.38/44.61 "arithmetic": { 172.38/44.61 "type": "PlainIntegerRelationState", 172.38/44.61 "relations": [] 172.38/44.61 }, 172.38/44.61 "ground": [ 172.38/44.61 "T9", 172.38/44.61 "T10", 172.38/44.61 "T11", 172.38/44.61 "T12" 172.38/44.61 ], 172.38/44.61 "free": [], 172.38/44.61 "exprvars": [] 172.38/44.61 } 172.38/44.61 }, 172.38/44.61 "366": { 172.38/44.61 "goal": [{ 172.38/44.61 "clause": -1, 172.38/44.61 "scope": -1, 172.38/44.61 "term": "(true)" 172.38/44.61 }], 172.38/44.61 "kb": { 172.38/44.61 "nonunifying": [], 172.38/44.61 "intvars": {}, 172.38/44.61 "arithmetic": { 172.38/44.61 "type": "PlainIntegerRelationState", 172.38/44.61 "relations": [] 172.38/44.61 }, 172.38/44.61 "ground": [], 172.38/44.61 "free": [], 172.38/44.61 "exprvars": [] 172.38/44.61 } 172.38/44.61 }, 172.38/44.61 "268": { 172.38/44.61 "goal": [], 172.38/44.61 "kb": { 172.38/44.61 "nonunifying": [], 172.38/44.61 "intvars": {}, 172.38/44.61 "arithmetic": { 172.38/44.61 "type": "PlainIntegerRelationState", 172.38/44.61 "relations": [] 172.38/44.61 }, 172.38/44.61 "ground": [], 172.38/44.61 "free": [], 172.38/44.61 "exprvars": [] 172.38/44.61 } 172.38/44.61 }, 172.38/44.61 "345": { 172.38/44.61 "goal": [{ 172.38/44.61 "clause": 4, 172.38/44.61 "scope": 4, 172.38/44.61 "term": "(member1 (. T101 (. X80 ([]))) T103)" 172.38/44.61 }], 172.38/44.61 "kb": { 172.38/44.61 "nonunifying": [], 172.38/44.61 "intvars": {}, 172.38/44.61 "arithmetic": { 172.38/44.61 "type": "PlainIntegerRelationState", 172.38/44.61 "relations": [] 172.38/44.61 }, 172.38/44.61 "ground": [ 172.38/44.61 "T101", 172.38/44.61 "T103" 172.38/44.61 ], 172.38/44.61 "free": ["X80"], 172.38/44.61 "exprvars": [] 172.38/44.61 } 172.38/44.61 }, 172.38/44.61 "367": { 172.38/44.61 "goal": [], 172.38/44.61 "kb": { 172.38/44.61 "nonunifying": [], 172.38/44.61 "intvars": {}, 172.38/44.61 "arithmetic": { 172.38/44.61 "type": "PlainIntegerRelationState", 172.38/44.61 "relations": [] 172.38/44.61 }, 172.38/44.61 "ground": [], 172.38/44.61 "free": [], 172.38/44.61 "exprvars": [] 172.38/44.61 } 172.38/44.61 }, 172.38/44.61 "346": { 172.38/44.61 "goal": [{ 172.38/44.61 "clause": 5, 172.38/44.61 "scope": 4, 172.38/44.61 "term": "(member1 (. T101 (. X80 ([]))) T103)" 172.38/44.61 }], 172.38/44.61 "kb": { 172.38/44.61 "nonunifying": [], 172.38/44.61 "intvars": {}, 172.38/44.61 "arithmetic": { 172.38/44.61 "type": "PlainIntegerRelationState", 172.38/44.61 "relations": [] 172.38/44.61 }, 172.38/44.61 "ground": [ 172.38/44.61 "T101", 172.38/44.61 "T103" 172.38/44.61 ], 172.38/44.61 "free": ["X80"], 172.38/44.61 "exprvars": [] 172.38/44.61 } 172.38/44.61 }, 172.38/44.61 "368": { 172.38/44.61 "goal": [], 172.38/44.61 "kb": { 172.38/44.61 "nonunifying": [], 172.38/44.61 "intvars": {}, 172.38/44.61 "arithmetic": { 172.38/44.61 "type": "PlainIntegerRelationState", 172.38/44.61 "relations": [] 172.38/44.61 }, 172.38/44.61 "ground": [], 172.38/44.61 "free": [], 172.38/44.61 "exprvars": [] 172.38/44.61 } 172.38/44.61 }, 172.38/44.61 "347": { 172.38/44.61 "goal": [{ 172.38/44.61 "clause": -1, 172.38/44.61 "scope": -1, 172.38/44.61 "term": "(true)" 172.38/44.61 }], 172.38/44.61 "kb": { 172.38/44.61 "nonunifying": [], 172.38/44.61 "intvars": {}, 172.38/44.61 "arithmetic": { 172.38/44.61 "type": "PlainIntegerRelationState", 172.38/44.61 "relations": [] 172.38/44.61 }, 172.38/44.61 "ground": [], 172.38/44.61 "free": [], 172.38/44.61 "exprvars": [] 172.38/44.61 } 172.38/44.61 }, 172.38/44.61 "369": { 172.38/44.61 "goal": [{ 172.38/44.61 "clause": -1, 172.38/44.61 "scope": -1, 172.38/44.61 "term": "(member T172 T174)" 172.38/44.61 }], 172.38/44.61 "kb": { 172.38/44.61 "nonunifying": [], 172.38/44.61 "intvars": {}, 172.38/44.61 "arithmetic": { 172.38/44.61 "type": "PlainIntegerRelationState", 172.38/44.61 "relations": [] 172.38/44.61 }, 172.38/44.61 "ground": [ 172.38/44.61 "T172", 172.38/44.61 "T174" 172.38/44.61 ], 172.38/44.61 "free": [], 172.38/44.61 "exprvars": [] 172.38/44.61 } 172.38/44.61 }, 172.38/44.61 "7": { 172.38/44.61 "goal": [ 172.38/44.61 { 172.38/44.61 "clause": 0, 172.38/44.61 "scope": 1, 172.38/44.61 "term": "(reach T1 T2 T3 T4)" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "clause": 1, 172.38/44.61 "scope": 1, 172.38/44.61 "term": "(reach T1 T2 T3 T4)" 172.38/44.61 } 172.38/44.61 ], 172.38/44.61 "kb": { 172.38/44.61 "nonunifying": [], 172.38/44.61 "intvars": {}, 172.38/44.61 "arithmetic": { 172.38/44.61 "type": "PlainIntegerRelationState", 172.38/44.61 "relations": [] 172.38/44.61 }, 172.38/44.61 "ground": [ 172.38/44.61 "T4", 172.38/44.61 "T1", 172.38/44.61 "T2", 172.38/44.61 "T3" 172.38/44.61 ], 172.38/44.61 "free": [], 172.38/44.61 "exprvars": [] 172.38/44.61 } 172.38/44.61 }, 172.38/44.61 "348": { 172.38/44.61 "goal": [], 172.38/44.61 "kb": { 172.38/44.61 "nonunifying": [], 172.38/44.61 "intvars": {}, 172.38/44.61 "arithmetic": { 172.38/44.61 "type": "PlainIntegerRelationState", 172.38/44.61 "relations": [] 172.38/44.61 }, 172.38/44.61 "ground": [], 172.38/44.61 "free": [], 172.38/44.61 "exprvars": [] 172.38/44.61 } 172.38/44.61 }, 172.38/44.61 "349": { 172.38/44.61 "goal": [], 172.38/44.61 "kb": { 172.38/44.61 "nonunifying": [], 172.38/44.61 "intvars": {}, 172.38/44.61 "arithmetic": { 172.38/44.61 "type": "PlainIntegerRelationState", 172.38/44.61 "relations": [] 172.38/44.61 }, 172.38/44.61 "ground": [], 172.38/44.61 "free": [], 172.38/44.61 "exprvars": [] 172.38/44.61 } 172.38/44.61 } 172.38/44.61 }, 172.38/44.61 "edges": [ 172.38/44.61 { 172.38/44.61 "from": 1, 172.38/44.61 "to": 7, 172.38/44.61 "label": "CASE" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 7, 172.38/44.61 "to": 18, 172.38/44.61 "label": "ONLY EVAL with clause\nreach(X5, X6, X7, X8) :- member(.(X5, .(X6, [])), X7).\nand substitutionT1 -> T9,\nX5 -> T9,\nT2 -> T10,\nX6 -> T10,\nT3 -> T11,\nX7 -> T11,\nT4 -> T12,\nX8 -> T12" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 18, 172.38/44.61 "to": 31, 172.38/44.61 "label": "CASE" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 31, 172.38/44.61 "to": 68, 172.38/44.61 "label": "PARALLEL" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 31, 172.38/44.61 "to": 72, 172.38/44.61 "label": "PARALLEL" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 68, 172.38/44.61 "to": 111, 172.38/44.61 "label": "EVAL with clause\nmember(X17, .(X17, X18)).\nand substitutionT9 -> T25,\nT10 -> T26,\nX17 -> .(T25, .(T26, [])),\nX18 -> T27,\nT11 -> .(.(T25, .(T26, [])), T27)" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 68, 172.38/44.61 "to": 120, 172.38/44.61 "label": "EVAL-BACKTRACK" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 72, 172.38/44.61 "to": 255, 172.38/44.61 "label": "PARALLEL" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 72, 172.38/44.61 "to": 256, 172.38/44.61 "label": "PARALLEL" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 111, 172.38/44.61 "to": 251, 172.38/44.61 "label": "SUCCESS" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 255, 172.38/44.61 "to": 259, 172.38/44.61 "label": "EVAL with clause\nmember(X31, .(X32, X33)) :- member(X31, X33).\nand substitutionT9 -> T44,\nT10 -> T45,\nX31 -> .(T44, .(T45, [])),\nX32 -> T46,\nX33 -> T47,\nT11 -> .(T46, T47)" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 255, 172.38/44.61 "to": 260, 172.38/44.61 "label": "EVAL-BACKTRACK" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 256, 172.38/44.61 "to": 322, 172.38/44.61 "label": "FAILURE" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 259, 172.38/44.61 "to": 261, 172.38/44.61 "label": "CASE" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 261, 172.38/44.61 "to": 262, 172.38/44.61 "label": "PARALLEL" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 261, 172.38/44.61 "to": 263, 172.38/44.61 "label": "PARALLEL" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 262, 172.38/44.61 "to": 264, 172.38/44.61 "label": "EVAL with clause\nmember(X46, .(X46, X47)).\nand substitutionT44 -> T66,\nT45 -> T67,\nX46 -> .(T66, .(T67, [])),\nX47 -> T68,\nT47 -> .(.(T66, .(T67, [])), T68)" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 262, 172.38/44.61 "to": 265, 172.38/44.61 "label": "EVAL-BACKTRACK" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 263, 172.38/44.61 "to": 267, 172.38/44.61 "label": "EVAL with clause\nmember(X54, .(X55, X56)) :- member(X54, X56).\nand substitutionT44 -> T77,\nT45 -> T78,\nX54 -> .(T77, .(T78, [])),\nX55 -> T79,\nX56 -> T80,\nT47 -> .(T79, T80)" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 263, 172.38/44.61 "to": 268, 172.38/44.61 "label": "EVAL-BACKTRACK" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 264, 172.38/44.61 "to": 266, 172.38/44.61 "label": "SUCCESS" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 267, 172.38/44.61 "to": 259, 172.38/44.61 "label": "INSTANCE with matching:\nT44 -> T77\nT45 -> T78\nT47 -> T80" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 322, 172.38/44.61 "to": 336, 172.38/44.61 "label": "ONLY EVAL with clause\nreach(X76, X77, X78, X79) :- ','(member1(.(X76, .(X80, [])), X78), ','(member(X80, X79), reach(X80, X77, X78, .(X80, X79)))).\nand substitutionT9 -> T101,\nX76 -> T101,\nT10 -> T102,\nX77 -> T102,\nT11 -> T103,\nX78 -> T103,\nT12 -> T104,\nX79 -> T104" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 336, 172.38/44.61 "to": 339, 172.38/44.61 "label": "SPLIT 1" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 336, 172.38/44.61 "to": 340, 172.38/44.61 "label": "SPLIT 2\nnew knowledge:\nT101 is ground\nT111 is ground\nT103 is ground\nreplacements:X80 -> T111" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 339, 172.38/44.61 "to": 342, 172.38/44.61 "label": "CASE" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 340, 172.38/44.61 "to": 360, 172.38/44.61 "label": "SPLIT 1" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 340, 172.38/44.61 "to": 361, 172.38/44.61 "label": "SPLIT 2\nnew knowledge:\nT111 is ground\nT104 is ground" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 342, 172.38/44.61 "to": 345, 172.38/44.61 "label": "PARALLEL" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 342, 172.38/44.61 "to": 346, 172.38/44.61 "label": "PARALLEL" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 345, 172.38/44.61 "to": 347, 172.38/44.61 "label": "EVAL with clause\nmember1(X105, .(X105, X106)).\nand substitutionT101 -> T130,\nX80 -> T131,\nX105 -> .(T130, .(T131, [])),\nX107 -> T131,\nX106 -> T132,\nT103 -> .(.(T130, .(T131, [])), T132)" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 345, 172.38/44.61 "to": 348, 172.38/44.61 "label": "EVAL-BACKTRACK" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 346, 172.38/44.61 "to": 352, 172.38/44.61 "label": "EVAL with clause\nmember1(X116, .(X117, X118)) :- member1(X116, X118).\nand substitutionT101 -> T139,\nX80 -> X119,\nX116 -> .(T139, .(X119, [])),\nX117 -> T140,\nX118 -> T141,\nT103 -> .(T140, T141)" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 346, 172.38/44.61 "to": 353, 172.38/44.61 "label": "EVAL-BACKTRACK" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 347, 172.38/44.61 "to": 349, 172.38/44.61 "label": "SUCCESS" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 352, 172.38/44.61 "to": 339, 172.38/44.61 "label": "INSTANCE with matching:\nT101 -> T139\nX80 -> X119\nT103 -> T141" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 360, 172.38/44.61 "to": 363, 172.38/44.61 "label": "CASE" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 361, 172.38/44.61 "to": 1, 172.38/44.61 "label": "INSTANCE with matching:\nT1 -> T111\nT2 -> T102\nT3 -> T103\nT4 -> .(T111, T104)" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 363, 172.38/44.61 "to": 364, 172.38/44.61 "label": "PARALLEL" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 363, 172.38/44.61 "to": 365, 172.38/44.61 "label": "PARALLEL" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 364, 172.38/44.61 "to": 366, 172.38/44.61 "label": "EVAL with clause\nmember(X142, .(X142, X143)).\nand substitutionT111 -> T164,\nX142 -> T164,\nX143 -> T165,\nT104 -> .(T164, T165)" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 364, 172.38/44.61 "to": 367, 172.38/44.61 "label": "EVAL-BACKTRACK" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 365, 172.38/44.61 "to": 369, 172.38/44.61 "label": "EVAL with clause\nmember(X150, .(X151, X152)) :- member(X150, X152).\nand substitutionT111 -> T172,\nX150 -> T172,\nX151 -> T173,\nX152 -> T174,\nT104 -> .(T173, T174)" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 365, 172.38/44.61 "to": 370, 172.38/44.61 "label": "EVAL-BACKTRACK" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 366, 172.38/44.61 "to": 368, 172.38/44.61 "label": "SUCCESS" 172.38/44.61 }, 172.38/44.61 { 172.38/44.61 "from": 369, 172.38/44.61 "to": 360, 172.38/44.61 "label": "INSTANCE with matching:\nT111 -> T172\nT104 -> T174" 172.38/44.61 } 172.38/44.61 ], 172.38/44.61 "type": "Graph" 172.38/44.61 } 172.38/44.61 } 172.38/44.61 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (152) 172.38/44.61 Obligation: 172.38/44.61 Triples: 172.38/44.61 172.38/44.61 memberA(X1, X2, .(X3, X4)) :- memberA(X1, X2, X4). 172.38/44.61 member1B(X1, X2, .(X3, X4)) :- member1B(X1, X2, X4). 172.38/44.61 memberD(X1, .(X2, X3)) :- memberD(X1, X3). 172.38/44.61 reachC(X1, X2, .(X3, X4), X5) :- memberA(X1, X2, X4). 172.38/44.61 reachC(X1, X2, X3, X4) :- member1B(X1, X5, X3). 172.38/44.61 reachC(X1, X2, X3, X4) :- ','(member1cB(X1, X5, X3), memberD(X5, X4)). 172.38/44.61 reachC(X1, X2, X3, X4) :- ','(member1cB(X1, X5, X3), ','(membercD(X5, X4), reachC(X5, X2, X3, .(X5, X4)))). 172.38/44.61 172.38/44.61 Clauses: 172.38/44.61 172.38/44.61 membercA(X1, X2, .(.(X1, .(X2, [])), X3)). 172.38/44.61 membercA(X1, X2, .(X3, X4)) :- membercA(X1, X2, X4). 172.38/44.61 member1cB(X1, X2, .(.(X1, .(X2, [])), X3)). 172.38/44.61 member1cB(X1, X2, .(X3, X4)) :- member1cB(X1, X2, X4). 172.38/44.61 reachcC(X1, X2, .(.(X1, .(X2, [])), X3), X4). 172.38/44.61 reachcC(X1, X2, .(X3, X4), X5) :- membercA(X1, X2, X4). 172.38/44.61 reachcC(X1, X2, X3, X4) :- ','(member1cB(X1, X5, X3), ','(membercD(X5, X4), reachcC(X5, X2, X3, .(X5, X4)))). 172.38/44.61 membercD(X1, .(X1, X2)). 172.38/44.61 membercD(X1, .(X2, X3)) :- membercD(X1, X3). 172.38/44.61 172.38/44.61 Afs: 172.38/44.61 172.38/44.61 reachC(x1, x2, x3, x4) = reachC(x1, x2, x3, x4) 172.38/44.61 172.38/44.61 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (153) TriplesToPiDPProof (SOUND) 172.38/44.61 We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: 172.38/44.61 172.38/44.61 reachC_in_4: (b,b,b,b) 172.38/44.61 172.38/44.61 memberA_in_3: (b,b,b) 172.38/44.61 172.38/44.61 member1B_in_3: (b,f,b) 172.38/44.61 172.38/44.61 member1cB_in_3: (b,f,b) 172.38/44.61 172.38/44.61 memberD_in_2: (b,b) 172.38/44.61 172.38/44.61 membercD_in_2: (b,b) 172.38/44.61 172.38/44.61 Transforming TRIPLES into the following Term Rewriting System: 172.38/44.61 172.38/44.61 Pi DP problem: 172.38/44.61 The TRS P consists of the following rules: 172.38/44.61 172.38/44.61 REACHC_IN_GGGG(X1, X2, .(X3, X4), X5) -> U4_GGGG(X1, X2, X3, X4, X5, memberA_in_ggg(X1, X2, X4)) 172.38/44.61 REACHC_IN_GGGG(X1, X2, .(X3, X4), X5) -> MEMBERA_IN_GGG(X1, X2, X4) 172.38/44.61 MEMBERA_IN_GGG(X1, X2, .(X3, X4)) -> U1_GGG(X1, X2, X3, X4, memberA_in_ggg(X1, X2, X4)) 172.38/44.61 MEMBERA_IN_GGG(X1, X2, .(X3, X4)) -> MEMBERA_IN_GGG(X1, X2, X4) 172.38/44.61 REACHC_IN_GGGG(X1, X2, X3, X4) -> U5_GGGG(X1, X2, X3, X4, member1B_in_gag(X1, X5, X3)) 172.38/44.61 REACHC_IN_GGGG(X1, X2, X3, X4) -> MEMBER1B_IN_GAG(X1, X5, X3) 172.38/44.61 MEMBER1B_IN_GAG(X1, X2, .(X3, X4)) -> U2_GAG(X1, X2, X3, X4, member1B_in_gag(X1, X2, X4)) 172.38/44.61 MEMBER1B_IN_GAG(X1, X2, .(X3, X4)) -> MEMBER1B_IN_GAG(X1, X2, X4) 172.38/44.61 REACHC_IN_GGGG(X1, X2, X3, X4) -> U6_GGGG(X1, X2, X3, X4, member1cB_in_gag(X1, X5, X3)) 172.38/44.61 U6_GGGG(X1, X2, X3, X4, member1cB_out_gag(X1, X5, X3)) -> U7_GGGG(X1, X2, X3, X4, memberD_in_gg(X5, X4)) 172.38/44.61 U6_GGGG(X1, X2, X3, X4, member1cB_out_gag(X1, X5, X3)) -> MEMBERD_IN_GG(X5, X4) 172.38/44.61 MEMBERD_IN_GG(X1, .(X2, X3)) -> U3_GG(X1, X2, X3, memberD_in_gg(X1, X3)) 172.38/44.61 MEMBERD_IN_GG(X1, .(X2, X3)) -> MEMBERD_IN_GG(X1, X3) 172.38/44.61 U6_GGGG(X1, X2, X3, X4, member1cB_out_gag(X1, X5, X3)) -> U8_GGGG(X1, X2, X3, X4, X5, membercD_in_gg(X5, X4)) 172.38/44.61 U8_GGGG(X1, X2, X3, X4, X5, membercD_out_gg(X5, X4)) -> U9_GGGG(X1, X2, X3, X4, reachC_in_gggg(X5, X2, X3, .(X5, X4))) 172.38/44.61 U8_GGGG(X1, X2, X3, X4, X5, membercD_out_gg(X5, X4)) -> REACHC_IN_GGGG(X5, X2, X3, .(X5, X4)) 172.38/44.61 172.38/44.61 The TRS R consists of the following rules: 172.38/44.61 172.38/44.61 member1cB_in_gag(X1, X2, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) 172.38/44.61 member1cB_in_gag(X1, X2, .(X3, X4)) -> U12_gag(X1, X2, X3, X4, member1cB_in_gag(X1, X2, X4)) 172.38/44.61 U12_gag(X1, X2, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) 172.38/44.61 membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) 172.38/44.61 membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) 172.38/44.61 U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) 172.38/44.61 172.38/44.61 The argument filtering Pi contains the following mapping: 172.38/44.61 reachC_in_gggg(x1, x2, x3, x4) = reachC_in_gggg(x1, x2, x3, x4) 172.38/44.61 172.38/44.61 .(x1, x2) = .(x1, x2) 172.38/44.61 172.38/44.61 memberA_in_ggg(x1, x2, x3) = memberA_in_ggg(x1, x2, x3) 172.38/44.61 172.38/44.61 member1B_in_gag(x1, x2, x3) = member1B_in_gag(x1, x3) 172.38/44.61 172.38/44.61 member1cB_in_gag(x1, x2, x3) = member1cB_in_gag(x1, x3) 172.38/44.61 172.38/44.61 [] = [] 172.38/44.61 172.38/44.61 member1cB_out_gag(x1, x2, x3) = member1cB_out_gag(x1, x2, x3) 172.38/44.61 172.38/44.61 U12_gag(x1, x2, x3, x4, x5) = U12_gag(x1, x3, x4, x5) 172.38/44.61 172.38/44.61 memberD_in_gg(x1, x2) = memberD_in_gg(x1, x2) 172.38/44.61 172.38/44.61 membercD_in_gg(x1, x2) = membercD_in_gg(x1, x2) 172.38/44.61 172.38/44.61 membercD_out_gg(x1, x2) = membercD_out_gg(x1, x2) 172.38/44.61 172.38/44.61 U17_gg(x1, x2, x3, x4) = U17_gg(x1, x2, x3, x4) 172.38/44.61 172.38/44.61 REACHC_IN_GGGG(x1, x2, x3, x4) = REACHC_IN_GGGG(x1, x2, x3, x4) 172.38/44.61 172.38/44.61 U4_GGGG(x1, x2, x3, x4, x5, x6) = U4_GGGG(x1, x2, x3, x4, x5, x6) 172.38/44.61 172.38/44.61 MEMBERA_IN_GGG(x1, x2, x3) = MEMBERA_IN_GGG(x1, x2, x3) 172.38/44.61 172.38/44.61 U1_GGG(x1, x2, x3, x4, x5) = U1_GGG(x1, x2, x3, x4, x5) 172.38/44.61 172.38/44.61 U5_GGGG(x1, x2, x3, x4, x5) = U5_GGGG(x1, x2, x3, x4, x5) 172.38/44.61 172.38/44.61 MEMBER1B_IN_GAG(x1, x2, x3) = MEMBER1B_IN_GAG(x1, x3) 172.38/44.61 172.38/44.61 U2_GAG(x1, x2, x3, x4, x5) = U2_GAG(x1, x3, x4, x5) 172.38/44.61 172.38/44.61 U6_GGGG(x1, x2, x3, x4, x5) = U6_GGGG(x1, x2, x3, x4, x5) 172.38/44.61 172.38/44.61 U7_GGGG(x1, x2, x3, x4, x5) = U7_GGGG(x1, x2, x3, x4, x5) 172.38/44.61 172.38/44.61 MEMBERD_IN_GG(x1, x2) = MEMBERD_IN_GG(x1, x2) 172.38/44.61 172.38/44.61 U3_GG(x1, x2, x3, x4) = U3_GG(x1, x2, x3, x4) 172.38/44.61 172.38/44.61 U8_GGGG(x1, x2, x3, x4, x5, x6) = U8_GGGG(x1, x2, x3, x4, x5, x6) 172.38/44.61 172.38/44.61 U9_GGGG(x1, x2, x3, x4, x5) = U9_GGGG(x1, x2, x3, x4, x5) 172.38/44.61 172.38/44.61 172.38/44.61 We have to consider all (P,R,Pi)-chains 172.38/44.61 172.38/44.61 172.38/44.61 Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES 172.38/44.61 172.38/44.61 172.38/44.61 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (154) 172.38/44.61 Obligation: 172.38/44.61 Pi DP problem: 172.38/44.61 The TRS P consists of the following rules: 172.38/44.61 172.38/44.61 REACHC_IN_GGGG(X1, X2, .(X3, X4), X5) -> U4_GGGG(X1, X2, X3, X4, X5, memberA_in_ggg(X1, X2, X4)) 172.38/44.61 REACHC_IN_GGGG(X1, X2, .(X3, X4), X5) -> MEMBERA_IN_GGG(X1, X2, X4) 172.38/44.61 MEMBERA_IN_GGG(X1, X2, .(X3, X4)) -> U1_GGG(X1, X2, X3, X4, memberA_in_ggg(X1, X2, X4)) 172.38/44.61 MEMBERA_IN_GGG(X1, X2, .(X3, X4)) -> MEMBERA_IN_GGG(X1, X2, X4) 172.38/44.61 REACHC_IN_GGGG(X1, X2, X3, X4) -> U5_GGGG(X1, X2, X3, X4, member1B_in_gag(X1, X5, X3)) 172.38/44.61 REACHC_IN_GGGG(X1, X2, X3, X4) -> MEMBER1B_IN_GAG(X1, X5, X3) 172.38/44.61 MEMBER1B_IN_GAG(X1, X2, .(X3, X4)) -> U2_GAG(X1, X2, X3, X4, member1B_in_gag(X1, X2, X4)) 172.38/44.61 MEMBER1B_IN_GAG(X1, X2, .(X3, X4)) -> MEMBER1B_IN_GAG(X1, X2, X4) 172.38/44.61 REACHC_IN_GGGG(X1, X2, X3, X4) -> U6_GGGG(X1, X2, X3, X4, member1cB_in_gag(X1, X5, X3)) 172.38/44.61 U6_GGGG(X1, X2, X3, X4, member1cB_out_gag(X1, X5, X3)) -> U7_GGGG(X1, X2, X3, X4, memberD_in_gg(X5, X4)) 172.38/44.61 U6_GGGG(X1, X2, X3, X4, member1cB_out_gag(X1, X5, X3)) -> MEMBERD_IN_GG(X5, X4) 172.38/44.61 MEMBERD_IN_GG(X1, .(X2, X3)) -> U3_GG(X1, X2, X3, memberD_in_gg(X1, X3)) 172.38/44.61 MEMBERD_IN_GG(X1, .(X2, X3)) -> MEMBERD_IN_GG(X1, X3) 172.38/44.61 U6_GGGG(X1, X2, X3, X4, member1cB_out_gag(X1, X5, X3)) -> U8_GGGG(X1, X2, X3, X4, X5, membercD_in_gg(X5, X4)) 172.38/44.61 U8_GGGG(X1, X2, X3, X4, X5, membercD_out_gg(X5, X4)) -> U9_GGGG(X1, X2, X3, X4, reachC_in_gggg(X5, X2, X3, .(X5, X4))) 172.38/44.61 U8_GGGG(X1, X2, X3, X4, X5, membercD_out_gg(X5, X4)) -> REACHC_IN_GGGG(X5, X2, X3, .(X5, X4)) 172.38/44.61 172.38/44.61 The TRS R consists of the following rules: 172.38/44.61 172.38/44.61 member1cB_in_gag(X1, X2, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) 172.38/44.61 member1cB_in_gag(X1, X2, .(X3, X4)) -> U12_gag(X1, X2, X3, X4, member1cB_in_gag(X1, X2, X4)) 172.38/44.61 U12_gag(X1, X2, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) 172.38/44.61 membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) 172.38/44.61 membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) 172.38/44.61 U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) 172.38/44.61 172.38/44.61 The argument filtering Pi contains the following mapping: 172.38/44.61 reachC_in_gggg(x1, x2, x3, x4) = reachC_in_gggg(x1, x2, x3, x4) 172.38/44.61 172.38/44.61 .(x1, x2) = .(x1, x2) 172.38/44.61 172.38/44.61 memberA_in_ggg(x1, x2, x3) = memberA_in_ggg(x1, x2, x3) 172.38/44.61 172.38/44.61 member1B_in_gag(x1, x2, x3) = member1B_in_gag(x1, x3) 172.38/44.61 172.38/44.61 member1cB_in_gag(x1, x2, x3) = member1cB_in_gag(x1, x3) 172.38/44.61 172.38/44.61 [] = [] 172.38/44.61 172.38/44.61 member1cB_out_gag(x1, x2, x3) = member1cB_out_gag(x1, x2, x3) 172.38/44.61 172.38/44.61 U12_gag(x1, x2, x3, x4, x5) = U12_gag(x1, x3, x4, x5) 172.38/44.61 172.38/44.61 memberD_in_gg(x1, x2) = memberD_in_gg(x1, x2) 172.38/44.61 172.38/44.61 membercD_in_gg(x1, x2) = membercD_in_gg(x1, x2) 172.38/44.61 172.38/44.61 membercD_out_gg(x1, x2) = membercD_out_gg(x1, x2) 172.38/44.61 172.38/44.61 U17_gg(x1, x2, x3, x4) = U17_gg(x1, x2, x3, x4) 172.38/44.61 172.38/44.61 REACHC_IN_GGGG(x1, x2, x3, x4) = REACHC_IN_GGGG(x1, x2, x3, x4) 172.38/44.61 172.38/44.61 U4_GGGG(x1, x2, x3, x4, x5, x6) = U4_GGGG(x1, x2, x3, x4, x5, x6) 172.38/44.61 172.38/44.61 MEMBERA_IN_GGG(x1, x2, x3) = MEMBERA_IN_GGG(x1, x2, x3) 172.38/44.61 172.38/44.61 U1_GGG(x1, x2, x3, x4, x5) = U1_GGG(x1, x2, x3, x4, x5) 172.38/44.61 172.38/44.61 U5_GGGG(x1, x2, x3, x4, x5) = U5_GGGG(x1, x2, x3, x4, x5) 172.38/44.61 172.38/44.61 MEMBER1B_IN_GAG(x1, x2, x3) = MEMBER1B_IN_GAG(x1, x3) 172.38/44.61 172.38/44.61 U2_GAG(x1, x2, x3, x4, x5) = U2_GAG(x1, x3, x4, x5) 172.38/44.61 172.38/44.61 U6_GGGG(x1, x2, x3, x4, x5) = U6_GGGG(x1, x2, x3, x4, x5) 172.38/44.61 172.38/44.61 U7_GGGG(x1, x2, x3, x4, x5) = U7_GGGG(x1, x2, x3, x4, x5) 172.38/44.61 172.38/44.61 MEMBERD_IN_GG(x1, x2) = MEMBERD_IN_GG(x1, x2) 172.38/44.61 172.38/44.61 U3_GG(x1, x2, x3, x4) = U3_GG(x1, x2, x3, x4) 172.38/44.61 172.38/44.61 U8_GGGG(x1, x2, x3, x4, x5, x6) = U8_GGGG(x1, x2, x3, x4, x5, x6) 172.38/44.61 172.38/44.61 U9_GGGG(x1, x2, x3, x4, x5) = U9_GGGG(x1, x2, x3, x4, x5) 172.38/44.61 172.38/44.61 172.38/44.61 We have to consider all (P,R,Pi)-chains 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (155) DependencyGraphProof (EQUIVALENT) 172.38/44.61 The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 10 less nodes. 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (156) 172.38/44.61 Complex Obligation (AND) 172.38/44.61 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (157) 172.38/44.61 Obligation: 172.38/44.61 Pi DP problem: 172.38/44.61 The TRS P consists of the following rules: 172.38/44.61 172.38/44.61 MEMBERD_IN_GG(X1, .(X2, X3)) -> MEMBERD_IN_GG(X1, X3) 172.38/44.61 172.38/44.61 The TRS R consists of the following rules: 172.38/44.61 172.38/44.61 member1cB_in_gag(X1, X2, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) 172.38/44.61 member1cB_in_gag(X1, X2, .(X3, X4)) -> U12_gag(X1, X2, X3, X4, member1cB_in_gag(X1, X2, X4)) 172.38/44.61 U12_gag(X1, X2, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) 172.38/44.61 membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) 172.38/44.61 membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) 172.38/44.61 U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) 172.38/44.61 172.38/44.61 The argument filtering Pi contains the following mapping: 172.38/44.61 .(x1, x2) = .(x1, x2) 172.38/44.61 172.38/44.61 member1cB_in_gag(x1, x2, x3) = member1cB_in_gag(x1, x3) 172.38/44.61 172.38/44.61 [] = [] 172.38/44.61 172.38/44.61 member1cB_out_gag(x1, x2, x3) = member1cB_out_gag(x1, x2, x3) 172.38/44.61 172.38/44.61 U12_gag(x1, x2, x3, x4, x5) = U12_gag(x1, x3, x4, x5) 172.38/44.61 172.38/44.61 membercD_in_gg(x1, x2) = membercD_in_gg(x1, x2) 172.38/44.61 172.38/44.61 membercD_out_gg(x1, x2) = membercD_out_gg(x1, x2) 172.38/44.61 172.38/44.61 U17_gg(x1, x2, x3, x4) = U17_gg(x1, x2, x3, x4) 172.38/44.61 172.38/44.61 MEMBERD_IN_GG(x1, x2) = MEMBERD_IN_GG(x1, x2) 172.38/44.61 172.38/44.61 172.38/44.61 We have to consider all (P,R,Pi)-chains 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (158) UsableRulesProof (EQUIVALENT) 172.38/44.61 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (159) 172.38/44.61 Obligation: 172.38/44.61 Pi DP problem: 172.38/44.61 The TRS P consists of the following rules: 172.38/44.61 172.38/44.61 MEMBERD_IN_GG(X1, .(X2, X3)) -> MEMBERD_IN_GG(X1, X3) 172.38/44.61 172.38/44.61 R is empty. 172.38/44.61 Pi is empty. 172.38/44.61 We have to consider all (P,R,Pi)-chains 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (160) PiDPToQDPProof (EQUIVALENT) 172.38/44.61 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (161) 172.38/44.61 Obligation: 172.38/44.61 Q DP problem: 172.38/44.61 The TRS P consists of the following rules: 172.38/44.61 172.38/44.61 MEMBERD_IN_GG(X1, .(X2, X3)) -> MEMBERD_IN_GG(X1, X3) 172.38/44.61 172.38/44.61 R is empty. 172.38/44.61 Q is empty. 172.38/44.61 We have to consider all (P,Q,R)-chains. 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (162) QDPSizeChangeProof (EQUIVALENT) 172.38/44.61 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 172.38/44.61 172.38/44.61 From the DPs we obtained the following set of size-change graphs: 172.38/44.61 *MEMBERD_IN_GG(X1, .(X2, X3)) -> MEMBERD_IN_GG(X1, X3) 172.38/44.61 The graph contains the following edges 1 >= 1, 2 > 2 172.38/44.61 172.38/44.61 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (163) 172.38/44.61 YES 172.38/44.61 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (164) 172.38/44.61 Obligation: 172.38/44.61 Pi DP problem: 172.38/44.61 The TRS P consists of the following rules: 172.38/44.61 172.38/44.61 MEMBER1B_IN_GAG(X1, X2, .(X3, X4)) -> MEMBER1B_IN_GAG(X1, X2, X4) 172.38/44.61 172.38/44.61 The TRS R consists of the following rules: 172.38/44.61 172.38/44.61 member1cB_in_gag(X1, X2, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) 172.38/44.61 member1cB_in_gag(X1, X2, .(X3, X4)) -> U12_gag(X1, X2, X3, X4, member1cB_in_gag(X1, X2, X4)) 172.38/44.61 U12_gag(X1, X2, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) 172.38/44.61 membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) 172.38/44.61 membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) 172.38/44.61 U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) 172.38/44.61 172.38/44.61 The argument filtering Pi contains the following mapping: 172.38/44.61 .(x1, x2) = .(x1, x2) 172.38/44.61 172.38/44.61 member1cB_in_gag(x1, x2, x3) = member1cB_in_gag(x1, x3) 172.38/44.61 172.38/44.61 [] = [] 172.38/44.61 172.38/44.61 member1cB_out_gag(x1, x2, x3) = member1cB_out_gag(x1, x2, x3) 172.38/44.61 172.38/44.61 U12_gag(x1, x2, x3, x4, x5) = U12_gag(x1, x3, x4, x5) 172.38/44.61 172.38/44.61 membercD_in_gg(x1, x2) = membercD_in_gg(x1, x2) 172.38/44.61 172.38/44.61 membercD_out_gg(x1, x2) = membercD_out_gg(x1, x2) 172.38/44.61 172.38/44.61 U17_gg(x1, x2, x3, x4) = U17_gg(x1, x2, x3, x4) 172.38/44.61 172.38/44.61 MEMBER1B_IN_GAG(x1, x2, x3) = MEMBER1B_IN_GAG(x1, x3) 172.38/44.61 172.38/44.61 172.38/44.61 We have to consider all (P,R,Pi)-chains 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (165) UsableRulesProof (EQUIVALENT) 172.38/44.61 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (166) 172.38/44.61 Obligation: 172.38/44.61 Pi DP problem: 172.38/44.61 The TRS P consists of the following rules: 172.38/44.61 172.38/44.61 MEMBER1B_IN_GAG(X1, X2, .(X3, X4)) -> MEMBER1B_IN_GAG(X1, X2, X4) 172.38/44.61 172.38/44.61 R is empty. 172.38/44.61 The argument filtering Pi contains the following mapping: 172.38/44.61 .(x1, x2) = .(x1, x2) 172.38/44.61 172.38/44.61 MEMBER1B_IN_GAG(x1, x2, x3) = MEMBER1B_IN_GAG(x1, x3) 172.38/44.61 172.38/44.61 172.38/44.61 We have to consider all (P,R,Pi)-chains 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (167) PiDPToQDPProof (SOUND) 172.38/44.61 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (168) 172.38/44.61 Obligation: 172.38/44.61 Q DP problem: 172.38/44.61 The TRS P consists of the following rules: 172.38/44.61 172.38/44.61 MEMBER1B_IN_GAG(X1, .(X3, X4)) -> MEMBER1B_IN_GAG(X1, X4) 172.38/44.61 172.38/44.61 R is empty. 172.38/44.61 Q is empty. 172.38/44.61 We have to consider all (P,Q,R)-chains. 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (169) QDPSizeChangeProof (EQUIVALENT) 172.38/44.61 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 172.38/44.61 172.38/44.61 From the DPs we obtained the following set of size-change graphs: 172.38/44.61 *MEMBER1B_IN_GAG(X1, .(X3, X4)) -> MEMBER1B_IN_GAG(X1, X4) 172.38/44.61 The graph contains the following edges 1 >= 1, 2 > 2 172.38/44.61 172.38/44.61 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (170) 172.38/44.61 YES 172.38/44.61 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (171) 172.38/44.61 Obligation: 172.38/44.61 Pi DP problem: 172.38/44.61 The TRS P consists of the following rules: 172.38/44.61 172.38/44.61 MEMBERA_IN_GGG(X1, X2, .(X3, X4)) -> MEMBERA_IN_GGG(X1, X2, X4) 172.38/44.61 172.38/44.61 The TRS R consists of the following rules: 172.38/44.61 172.38/44.61 member1cB_in_gag(X1, X2, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) 172.38/44.61 member1cB_in_gag(X1, X2, .(X3, X4)) -> U12_gag(X1, X2, X3, X4, member1cB_in_gag(X1, X2, X4)) 172.38/44.61 U12_gag(X1, X2, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) 172.38/44.61 membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) 172.38/44.61 membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) 172.38/44.61 U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) 172.38/44.61 172.38/44.61 The argument filtering Pi contains the following mapping: 172.38/44.61 .(x1, x2) = .(x1, x2) 172.38/44.61 172.38/44.61 member1cB_in_gag(x1, x2, x3) = member1cB_in_gag(x1, x3) 172.38/44.61 172.38/44.61 [] = [] 172.38/44.61 172.38/44.61 member1cB_out_gag(x1, x2, x3) = member1cB_out_gag(x1, x2, x3) 172.38/44.61 172.38/44.61 U12_gag(x1, x2, x3, x4, x5) = U12_gag(x1, x3, x4, x5) 172.38/44.61 172.38/44.61 membercD_in_gg(x1, x2) = membercD_in_gg(x1, x2) 172.38/44.61 172.38/44.61 membercD_out_gg(x1, x2) = membercD_out_gg(x1, x2) 172.38/44.61 172.38/44.61 U17_gg(x1, x2, x3, x4) = U17_gg(x1, x2, x3, x4) 172.38/44.61 172.38/44.61 MEMBERA_IN_GGG(x1, x2, x3) = MEMBERA_IN_GGG(x1, x2, x3) 172.38/44.61 172.38/44.61 172.38/44.61 We have to consider all (P,R,Pi)-chains 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (172) UsableRulesProof (EQUIVALENT) 172.38/44.61 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (173) 172.38/44.61 Obligation: 172.38/44.61 Pi DP problem: 172.38/44.61 The TRS P consists of the following rules: 172.38/44.61 172.38/44.61 MEMBERA_IN_GGG(X1, X2, .(X3, X4)) -> MEMBERA_IN_GGG(X1, X2, X4) 172.38/44.61 172.38/44.61 R is empty. 172.38/44.61 Pi is empty. 172.38/44.61 We have to consider all (P,R,Pi)-chains 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (174) PiDPToQDPProof (EQUIVALENT) 172.38/44.61 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (175) 172.38/44.61 Obligation: 172.38/44.61 Q DP problem: 172.38/44.61 The TRS P consists of the following rules: 172.38/44.61 172.38/44.61 MEMBERA_IN_GGG(X1, X2, .(X3, X4)) -> MEMBERA_IN_GGG(X1, X2, X4) 172.38/44.61 172.38/44.61 R is empty. 172.38/44.61 Q is empty. 172.38/44.61 We have to consider all (P,Q,R)-chains. 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (176) QDPSizeChangeProof (EQUIVALENT) 172.38/44.61 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 172.38/44.61 172.38/44.61 From the DPs we obtained the following set of size-change graphs: 172.38/44.61 *MEMBERA_IN_GGG(X1, X2, .(X3, X4)) -> MEMBERA_IN_GGG(X1, X2, X4) 172.38/44.61 The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 172.38/44.61 172.38/44.61 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (177) 172.38/44.61 YES 172.38/44.61 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (178) 172.38/44.61 Obligation: 172.38/44.61 Pi DP problem: 172.38/44.61 The TRS P consists of the following rules: 172.38/44.61 172.38/44.61 REACHC_IN_GGGG(X1, X2, X3, X4) -> U6_GGGG(X1, X2, X3, X4, member1cB_in_gag(X1, X5, X3)) 172.38/44.61 U6_GGGG(X1, X2, X3, X4, member1cB_out_gag(X1, X5, X3)) -> U8_GGGG(X1, X2, X3, X4, X5, membercD_in_gg(X5, X4)) 172.38/44.61 U8_GGGG(X1, X2, X3, X4, X5, membercD_out_gg(X5, X4)) -> REACHC_IN_GGGG(X5, X2, X3, .(X5, X4)) 172.38/44.61 172.38/44.61 The TRS R consists of the following rules: 172.38/44.61 172.38/44.61 member1cB_in_gag(X1, X2, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) 172.38/44.61 member1cB_in_gag(X1, X2, .(X3, X4)) -> U12_gag(X1, X2, X3, X4, member1cB_in_gag(X1, X2, X4)) 172.38/44.61 U12_gag(X1, X2, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) 172.38/44.61 membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) 172.38/44.61 membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) 172.38/44.61 U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) 172.38/44.61 172.38/44.61 The argument filtering Pi contains the following mapping: 172.38/44.61 .(x1, x2) = .(x1, x2) 172.38/44.61 172.38/44.61 member1cB_in_gag(x1, x2, x3) = member1cB_in_gag(x1, x3) 172.38/44.61 172.38/44.61 [] = [] 172.38/44.61 172.38/44.61 member1cB_out_gag(x1, x2, x3) = member1cB_out_gag(x1, x2, x3) 172.38/44.61 172.38/44.61 U12_gag(x1, x2, x3, x4, x5) = U12_gag(x1, x3, x4, x5) 172.38/44.61 172.38/44.61 membercD_in_gg(x1, x2) = membercD_in_gg(x1, x2) 172.38/44.61 172.38/44.61 membercD_out_gg(x1, x2) = membercD_out_gg(x1, x2) 172.38/44.61 172.38/44.61 U17_gg(x1, x2, x3, x4) = U17_gg(x1, x2, x3, x4) 172.38/44.61 172.38/44.61 REACHC_IN_GGGG(x1, x2, x3, x4) = REACHC_IN_GGGG(x1, x2, x3, x4) 172.38/44.61 172.38/44.61 U6_GGGG(x1, x2, x3, x4, x5) = U6_GGGG(x1, x2, x3, x4, x5) 172.38/44.61 172.38/44.61 U8_GGGG(x1, x2, x3, x4, x5, x6) = U8_GGGG(x1, x2, x3, x4, x5, x6) 172.38/44.61 172.38/44.61 172.38/44.61 We have to consider all (P,R,Pi)-chains 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (179) PiDPToQDPProof (SOUND) 172.38/44.61 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (180) 172.38/44.61 Obligation: 172.38/44.61 Q DP problem: 172.38/44.61 The TRS P consists of the following rules: 172.38/44.61 172.38/44.61 REACHC_IN_GGGG(X1, X2, X3, X4) -> U6_GGGG(X1, X2, X3, X4, member1cB_in_gag(X1, X3)) 172.38/44.61 U6_GGGG(X1, X2, X3, X4, member1cB_out_gag(X1, X5, X3)) -> U8_GGGG(X1, X2, X3, X4, X5, membercD_in_gg(X5, X4)) 172.38/44.61 U8_GGGG(X1, X2, X3, X4, X5, membercD_out_gg(X5, X4)) -> REACHC_IN_GGGG(X5, X2, X3, .(X5, X4)) 172.38/44.61 172.38/44.61 The TRS R consists of the following rules: 172.38/44.61 172.38/44.61 member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) 172.38/44.61 member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) 172.38/44.61 U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) 172.38/44.61 membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) 172.38/44.61 membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) 172.38/44.61 U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) 172.38/44.61 172.38/44.61 The set Q consists of the following terms: 172.38/44.61 172.38/44.61 member1cB_in_gag(x0, x1) 172.38/44.61 U12_gag(x0, x1, x2, x3) 172.38/44.61 membercD_in_gg(x0, x1) 172.38/44.61 U17_gg(x0, x1, x2, x3) 172.38/44.61 172.38/44.61 We have to consider all (P,Q,R)-chains. 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (181) TransformationProof (SOUND) 172.38/44.61 By narrowing [LPAR04] the rule REACHC_IN_GGGG(X1, X2, X3, X4) -> U6_GGGG(X1, X2, X3, X4, member1cB_in_gag(X1, X3)) at position [4] we obtained the following new rules [LPAR04]: 172.38/44.61 172.38/44.61 (REACHC_IN_GGGG(x0, y1, .(.(x0, .(x1, [])), x2), y3) -> U6_GGGG(x0, y1, .(.(x0, .(x1, [])), x2), y3, member1cB_out_gag(x0, x1, .(.(x0, .(x1, [])), x2))),REACHC_IN_GGGG(x0, y1, .(.(x0, .(x1, [])), x2), y3) -> U6_GGGG(x0, y1, .(.(x0, .(x1, [])), x2), y3, member1cB_out_gag(x0, x1, .(.(x0, .(x1, [])), x2)))) 172.38/44.61 (REACHC_IN_GGGG(x0, y1, .(x1, x2), y3) -> U6_GGGG(x0, y1, .(x1, x2), y3, U12_gag(x0, x1, x2, member1cB_in_gag(x0, x2))),REACHC_IN_GGGG(x0, y1, .(x1, x2), y3) -> U6_GGGG(x0, y1, .(x1, x2), y3, U12_gag(x0, x1, x2, member1cB_in_gag(x0, x2)))) 172.38/44.61 172.38/44.61 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (182) 172.38/44.61 Obligation: 172.38/44.61 Q DP problem: 172.38/44.61 The TRS P consists of the following rules: 172.38/44.61 172.38/44.61 U6_GGGG(X1, X2, X3, X4, member1cB_out_gag(X1, X5, X3)) -> U8_GGGG(X1, X2, X3, X4, X5, membercD_in_gg(X5, X4)) 172.38/44.61 U8_GGGG(X1, X2, X3, X4, X5, membercD_out_gg(X5, X4)) -> REACHC_IN_GGGG(X5, X2, X3, .(X5, X4)) 172.38/44.61 REACHC_IN_GGGG(x0, y1, .(.(x0, .(x1, [])), x2), y3) -> U6_GGGG(x0, y1, .(.(x0, .(x1, [])), x2), y3, member1cB_out_gag(x0, x1, .(.(x0, .(x1, [])), x2))) 172.38/44.61 REACHC_IN_GGGG(x0, y1, .(x1, x2), y3) -> U6_GGGG(x0, y1, .(x1, x2), y3, U12_gag(x0, x1, x2, member1cB_in_gag(x0, x2))) 172.38/44.61 172.38/44.61 The TRS R consists of the following rules: 172.38/44.61 172.38/44.61 member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) 172.38/44.61 member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) 172.38/44.61 U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) 172.38/44.61 membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) 172.38/44.61 membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) 172.38/44.61 U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) 172.38/44.61 172.38/44.61 The set Q consists of the following terms: 172.38/44.61 172.38/44.61 member1cB_in_gag(x0, x1) 172.38/44.61 U12_gag(x0, x1, x2, x3) 172.38/44.61 membercD_in_gg(x0, x1) 172.38/44.61 U17_gg(x0, x1, x2, x3) 172.38/44.61 172.38/44.61 We have to consider all (P,Q,R)-chains. 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (183) TransformationProof (SOUND) 172.38/44.61 By narrowing [LPAR04] the rule U6_GGGG(X1, X2, X3, X4, member1cB_out_gag(X1, X5, X3)) -> U8_GGGG(X1, X2, X3, X4, X5, membercD_in_gg(X5, X4)) at position [5] we obtained the following new rules [LPAR04]: 172.38/44.61 172.38/44.61 (U6_GGGG(y0, y1, y2, .(x0, x1), member1cB_out_gag(y0, x0, y2)) -> U8_GGGG(y0, y1, y2, .(x0, x1), x0, membercD_out_gg(x0, .(x0, x1))),U6_GGGG(y0, y1, y2, .(x0, x1), member1cB_out_gag(y0, x0, y2)) -> U8_GGGG(y0, y1, y2, .(x0, x1), x0, membercD_out_gg(x0, .(x0, x1)))) 172.38/44.61 (U6_GGGG(y0, y1, y2, .(x1, x2), member1cB_out_gag(y0, x0, y2)) -> U8_GGGG(y0, y1, y2, .(x1, x2), x0, U17_gg(x0, x1, x2, membercD_in_gg(x0, x2))),U6_GGGG(y0, y1, y2, .(x1, x2), member1cB_out_gag(y0, x0, y2)) -> U8_GGGG(y0, y1, y2, .(x1, x2), x0, U17_gg(x0, x1, x2, membercD_in_gg(x0, x2)))) 172.38/44.61 172.38/44.61 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (184) 172.38/44.61 Obligation: 172.38/44.61 Q DP problem: 172.38/44.61 The TRS P consists of the following rules: 172.38/44.61 172.38/44.61 U8_GGGG(X1, X2, X3, X4, X5, membercD_out_gg(X5, X4)) -> REACHC_IN_GGGG(X5, X2, X3, .(X5, X4)) 172.38/44.61 REACHC_IN_GGGG(x0, y1, .(.(x0, .(x1, [])), x2), y3) -> U6_GGGG(x0, y1, .(.(x0, .(x1, [])), x2), y3, member1cB_out_gag(x0, x1, .(.(x0, .(x1, [])), x2))) 172.38/44.61 REACHC_IN_GGGG(x0, y1, .(x1, x2), y3) -> U6_GGGG(x0, y1, .(x1, x2), y3, U12_gag(x0, x1, x2, member1cB_in_gag(x0, x2))) 172.38/44.61 U6_GGGG(y0, y1, y2, .(x0, x1), member1cB_out_gag(y0, x0, y2)) -> U8_GGGG(y0, y1, y2, .(x0, x1), x0, membercD_out_gg(x0, .(x0, x1))) 172.38/44.61 U6_GGGG(y0, y1, y2, .(x1, x2), member1cB_out_gag(y0, x0, y2)) -> U8_GGGG(y0, y1, y2, .(x1, x2), x0, U17_gg(x0, x1, x2, membercD_in_gg(x0, x2))) 172.38/44.61 172.38/44.61 The TRS R consists of the following rules: 172.38/44.61 172.38/44.61 member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) 172.38/44.61 member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) 172.38/44.61 U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) 172.38/44.61 membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) 172.38/44.61 membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) 172.38/44.61 U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) 172.38/44.61 172.38/44.61 The set Q consists of the following terms: 172.38/44.61 172.38/44.61 member1cB_in_gag(x0, x1) 172.38/44.61 U12_gag(x0, x1, x2, x3) 172.38/44.61 membercD_in_gg(x0, x1) 172.38/44.61 U17_gg(x0, x1, x2, x3) 172.38/44.61 172.38/44.61 We have to consider all (P,Q,R)-chains. 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (185) TransformationProof (EQUIVALENT) 172.38/44.61 By instantiating [LPAR04] the rule U8_GGGG(X1, X2, X3, X4, X5, membercD_out_gg(X5, X4)) -> REACHC_IN_GGGG(X5, X2, X3, .(X5, X4)) we obtained the following new rules [LPAR04]: 172.38/44.61 172.38/44.61 (U8_GGGG(z0, z1, z2, .(z3, z4), z3, membercD_out_gg(z3, .(z3, z4))) -> REACHC_IN_GGGG(z3, z1, z2, .(z3, .(z3, z4))),U8_GGGG(z0, z1, z2, .(z3, z4), z3, membercD_out_gg(z3, .(z3, z4))) -> REACHC_IN_GGGG(z3, z1, z2, .(z3, .(z3, z4)))) 172.38/44.61 (U8_GGGG(z0, z1, z2, .(z3, z4), z5, membercD_out_gg(z5, .(z3, z4))) -> REACHC_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4))),U8_GGGG(z0, z1, z2, .(z3, z4), z5, membercD_out_gg(z5, .(z3, z4))) -> REACHC_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4)))) 172.38/44.61 172.38/44.61 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (186) 172.38/44.61 Obligation: 172.38/44.61 Q DP problem: 172.38/44.61 The TRS P consists of the following rules: 172.38/44.61 172.38/44.61 REACHC_IN_GGGG(x0, y1, .(.(x0, .(x1, [])), x2), y3) -> U6_GGGG(x0, y1, .(.(x0, .(x1, [])), x2), y3, member1cB_out_gag(x0, x1, .(.(x0, .(x1, [])), x2))) 172.38/44.61 REACHC_IN_GGGG(x0, y1, .(x1, x2), y3) -> U6_GGGG(x0, y1, .(x1, x2), y3, U12_gag(x0, x1, x2, member1cB_in_gag(x0, x2))) 172.38/44.61 U6_GGGG(y0, y1, y2, .(x0, x1), member1cB_out_gag(y0, x0, y2)) -> U8_GGGG(y0, y1, y2, .(x0, x1), x0, membercD_out_gg(x0, .(x0, x1))) 172.38/44.61 U6_GGGG(y0, y1, y2, .(x1, x2), member1cB_out_gag(y0, x0, y2)) -> U8_GGGG(y0, y1, y2, .(x1, x2), x0, U17_gg(x0, x1, x2, membercD_in_gg(x0, x2))) 172.38/44.61 U8_GGGG(z0, z1, z2, .(z3, z4), z3, membercD_out_gg(z3, .(z3, z4))) -> REACHC_IN_GGGG(z3, z1, z2, .(z3, .(z3, z4))) 172.38/44.61 U8_GGGG(z0, z1, z2, .(z3, z4), z5, membercD_out_gg(z5, .(z3, z4))) -> REACHC_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4))) 172.38/44.61 172.38/44.61 The TRS R consists of the following rules: 172.38/44.61 172.38/44.61 member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) 172.38/44.61 member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) 172.38/44.61 U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) 172.38/44.61 membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) 172.38/44.61 membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) 172.38/44.61 U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) 172.38/44.61 172.38/44.61 The set Q consists of the following terms: 172.38/44.61 172.38/44.61 member1cB_in_gag(x0, x1) 172.38/44.61 U12_gag(x0, x1, x2, x3) 172.38/44.61 membercD_in_gg(x0, x1) 172.38/44.61 U17_gg(x0, x1, x2, x3) 172.38/44.61 172.38/44.61 We have to consider all (P,Q,R)-chains. 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (187) TransformationProof (EQUIVALENT) 172.38/44.61 By instantiating [LPAR04] the rule REACHC_IN_GGGG(x0, y1, .(.(x0, .(x1, [])), x2), y3) -> U6_GGGG(x0, y1, .(.(x0, .(x1, [])), x2), y3, member1cB_out_gag(x0, x1, .(.(x0, .(x1, [])), x2))) we obtained the following new rules [LPAR04]: 172.38/44.61 172.38/44.61 (REACHC_IN_GGGG(z3, z1, .(.(z3, .(x2, [])), x3), .(z3, .(z3, z4))) -> U6_GGGG(z3, z1, .(.(z3, .(x2, [])), x3), .(z3, .(z3, z4)), member1cB_out_gag(z3, x2, .(.(z3, .(x2, [])), x3))),REACHC_IN_GGGG(z3, z1, .(.(z3, .(x2, [])), x3), .(z3, .(z3, z4))) -> U6_GGGG(z3, z1, .(.(z3, .(x2, [])), x3), .(z3, .(z3, z4)), member1cB_out_gag(z3, x2, .(.(z3, .(x2, [])), x3)))) 172.38/44.61 (REACHC_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), x3), .(z5, .(z3, z4))) -> U6_GGGG(z5, z1, .(.(z5, .(x2, [])), x3), .(z5, .(z3, z4)), member1cB_out_gag(z5, x2, .(.(z5, .(x2, [])), x3))),REACHC_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), x3), .(z5, .(z3, z4))) -> U6_GGGG(z5, z1, .(.(z5, .(x2, [])), x3), .(z5, .(z3, z4)), member1cB_out_gag(z5, x2, .(.(z5, .(x2, [])), x3)))) 172.38/44.61 172.38/44.61 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (188) 172.38/44.61 Obligation: 172.38/44.61 Q DP problem: 172.38/44.61 The TRS P consists of the following rules: 172.38/44.61 172.38/44.61 REACHC_IN_GGGG(x0, y1, .(x1, x2), y3) -> U6_GGGG(x0, y1, .(x1, x2), y3, U12_gag(x0, x1, x2, member1cB_in_gag(x0, x2))) 172.38/44.61 U6_GGGG(y0, y1, y2, .(x0, x1), member1cB_out_gag(y0, x0, y2)) -> U8_GGGG(y0, y1, y2, .(x0, x1), x0, membercD_out_gg(x0, .(x0, x1))) 172.38/44.61 U6_GGGG(y0, y1, y2, .(x1, x2), member1cB_out_gag(y0, x0, y2)) -> U8_GGGG(y0, y1, y2, .(x1, x2), x0, U17_gg(x0, x1, x2, membercD_in_gg(x0, x2))) 172.38/44.61 U8_GGGG(z0, z1, z2, .(z3, z4), z3, membercD_out_gg(z3, .(z3, z4))) -> REACHC_IN_GGGG(z3, z1, z2, .(z3, .(z3, z4))) 172.38/44.61 U8_GGGG(z0, z1, z2, .(z3, z4), z5, membercD_out_gg(z5, .(z3, z4))) -> REACHC_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4))) 172.38/44.61 REACHC_IN_GGGG(z3, z1, .(.(z3, .(x2, [])), x3), .(z3, .(z3, z4))) -> U6_GGGG(z3, z1, .(.(z3, .(x2, [])), x3), .(z3, .(z3, z4)), member1cB_out_gag(z3, x2, .(.(z3, .(x2, [])), x3))) 172.38/44.61 REACHC_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), x3), .(z5, .(z3, z4))) -> U6_GGGG(z5, z1, .(.(z5, .(x2, [])), x3), .(z5, .(z3, z4)), member1cB_out_gag(z5, x2, .(.(z5, .(x2, [])), x3))) 172.38/44.61 172.38/44.61 The TRS R consists of the following rules: 172.38/44.61 172.38/44.61 member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) 172.38/44.61 member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) 172.38/44.61 U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) 172.38/44.61 membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) 172.38/44.61 membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) 172.38/44.61 U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) 172.38/44.61 172.38/44.61 The set Q consists of the following terms: 172.38/44.61 172.38/44.61 member1cB_in_gag(x0, x1) 172.38/44.61 U12_gag(x0, x1, x2, x3) 172.38/44.61 membercD_in_gg(x0, x1) 172.38/44.61 U17_gg(x0, x1, x2, x3) 172.38/44.61 172.38/44.61 We have to consider all (P,Q,R)-chains. 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (189) TransformationProof (EQUIVALENT) 172.38/44.61 By instantiating [LPAR04] the rule REACHC_IN_GGGG(x0, y1, .(x1, x2), y3) -> U6_GGGG(x0, y1, .(x1, x2), y3, U12_gag(x0, x1, x2, member1cB_in_gag(x0, x2))) we obtained the following new rules [LPAR04]: 172.38/44.61 172.38/44.61 (REACHC_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) -> U6_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U12_gag(z3, x2, x3, member1cB_in_gag(z3, x3))),REACHC_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) -> U6_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U12_gag(z3, x2, x3, member1cB_in_gag(z3, x3)))) 172.38/44.61 (REACHC_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U6_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U12_gag(z5, x2, x3, member1cB_in_gag(z5, x3))),REACHC_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U6_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U12_gag(z5, x2, x3, member1cB_in_gag(z5, x3)))) 172.38/44.61 172.38/44.61 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (190) 172.38/44.61 Obligation: 172.38/44.61 Q DP problem: 172.38/44.61 The TRS P consists of the following rules: 172.38/44.61 172.38/44.61 U6_GGGG(y0, y1, y2, .(x0, x1), member1cB_out_gag(y0, x0, y2)) -> U8_GGGG(y0, y1, y2, .(x0, x1), x0, membercD_out_gg(x0, .(x0, x1))) 172.38/44.61 U6_GGGG(y0, y1, y2, .(x1, x2), member1cB_out_gag(y0, x0, y2)) -> U8_GGGG(y0, y1, y2, .(x1, x2), x0, U17_gg(x0, x1, x2, membercD_in_gg(x0, x2))) 172.38/44.61 U8_GGGG(z0, z1, z2, .(z3, z4), z3, membercD_out_gg(z3, .(z3, z4))) -> REACHC_IN_GGGG(z3, z1, z2, .(z3, .(z3, z4))) 172.38/44.61 U8_GGGG(z0, z1, z2, .(z3, z4), z5, membercD_out_gg(z5, .(z3, z4))) -> REACHC_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4))) 172.38/44.61 REACHC_IN_GGGG(z3, z1, .(.(z3, .(x2, [])), x3), .(z3, .(z3, z4))) -> U6_GGGG(z3, z1, .(.(z3, .(x2, [])), x3), .(z3, .(z3, z4)), member1cB_out_gag(z3, x2, .(.(z3, .(x2, [])), x3))) 172.38/44.61 REACHC_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), x3), .(z5, .(z3, z4))) -> U6_GGGG(z5, z1, .(.(z5, .(x2, [])), x3), .(z5, .(z3, z4)), member1cB_out_gag(z5, x2, .(.(z5, .(x2, [])), x3))) 172.38/44.61 REACHC_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) -> U6_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U12_gag(z3, x2, x3, member1cB_in_gag(z3, x3))) 172.38/44.61 REACHC_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U6_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U12_gag(z5, x2, x3, member1cB_in_gag(z5, x3))) 172.38/44.61 172.38/44.61 The TRS R consists of the following rules: 172.38/44.61 172.38/44.61 member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) 172.38/44.61 member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) 172.38/44.61 U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) 172.38/44.61 membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) 172.38/44.61 membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) 172.38/44.61 U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) 172.38/44.61 172.38/44.61 The set Q consists of the following terms: 172.38/44.61 172.38/44.61 member1cB_in_gag(x0, x1) 172.38/44.61 U12_gag(x0, x1, x2, x3) 172.38/44.61 membercD_in_gg(x0, x1) 172.38/44.61 U17_gg(x0, x1, x2, x3) 172.38/44.61 172.38/44.61 We have to consider all (P,Q,R)-chains. 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (191) TransformationProof (EQUIVALENT) 172.38/44.61 By instantiating [LPAR04] the rule U6_GGGG(y0, y1, y2, .(x0, x1), member1cB_out_gag(y0, x0, y2)) -> U8_GGGG(y0, y1, y2, .(x0, x1), x0, membercD_out_gg(x0, .(x0, x1))) we obtained the following new rules [LPAR04]: 172.38/44.61 172.38/44.61 (U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))),U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4))))) 172.38/44.61 (U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))),U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5))))) 172.38/44.61 (U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))),U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4))))) 172.38/44.61 (U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))),U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5))))) 172.38/44.61 172.38/44.61 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (192) 172.38/44.61 Obligation: 172.38/44.61 Q DP problem: 172.38/44.61 The TRS P consists of the following rules: 172.38/44.61 172.38/44.61 U6_GGGG(y0, y1, y2, .(x1, x2), member1cB_out_gag(y0, x0, y2)) -> U8_GGGG(y0, y1, y2, .(x1, x2), x0, U17_gg(x0, x1, x2, membercD_in_gg(x0, x2))) 172.38/44.61 U8_GGGG(z0, z1, z2, .(z3, z4), z3, membercD_out_gg(z3, .(z3, z4))) -> REACHC_IN_GGGG(z3, z1, z2, .(z3, .(z3, z4))) 172.38/44.61 U8_GGGG(z0, z1, z2, .(z3, z4), z5, membercD_out_gg(z5, .(z3, z4))) -> REACHC_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4))) 172.38/44.61 REACHC_IN_GGGG(z3, z1, .(.(z3, .(x2, [])), x3), .(z3, .(z3, z4))) -> U6_GGGG(z3, z1, .(.(z3, .(x2, [])), x3), .(z3, .(z3, z4)), member1cB_out_gag(z3, x2, .(.(z3, .(x2, [])), x3))) 172.38/44.61 REACHC_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), x3), .(z5, .(z3, z4))) -> U6_GGGG(z5, z1, .(.(z5, .(x2, [])), x3), .(z5, .(z3, z4)), member1cB_out_gag(z5, x2, .(.(z5, .(x2, [])), x3))) 172.38/44.61 REACHC_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) -> U6_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U12_gag(z3, x2, x3, member1cB_in_gag(z3, x3))) 172.38/44.61 REACHC_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U6_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U12_gag(z5, x2, x3, member1cB_in_gag(z5, x3))) 172.38/44.61 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))) 172.38/44.61 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) 172.38/44.61 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))) 172.38/44.61 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) 172.38/44.61 172.38/44.61 The TRS R consists of the following rules: 172.38/44.61 172.38/44.61 member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) 172.38/44.61 member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) 172.38/44.61 U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) 172.38/44.61 membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) 172.38/44.61 membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) 172.38/44.61 U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) 172.38/44.61 172.38/44.61 The set Q consists of the following terms: 172.38/44.61 172.38/44.61 member1cB_in_gag(x0, x1) 172.38/44.61 U12_gag(x0, x1, x2, x3) 172.38/44.61 membercD_in_gg(x0, x1) 172.38/44.61 U17_gg(x0, x1, x2, x3) 172.38/44.61 172.38/44.61 We have to consider all (P,Q,R)-chains. 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (193) TransformationProof (EQUIVALENT) 172.38/44.61 By instantiating [LPAR04] the rule U6_GGGG(y0, y1, y2, .(x1, x2), member1cB_out_gag(y0, x0, y2)) -> U8_GGGG(y0, y1, y2, .(x1, x2), x0, U17_gg(x0, x1, x2, membercD_in_gg(x0, x2))) we obtained the following new rules [LPAR04]: 172.38/44.61 172.38/44.61 (U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U17_gg(z2, z0, .(z0, z4), membercD_in_gg(z2, .(z0, z4)))),U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U17_gg(z2, z0, .(z0, z4), membercD_in_gg(z2, .(z0, z4))))) 172.38/44.61 (U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U17_gg(z2, z0, .(z4, z5), membercD_in_gg(z2, .(z4, z5)))),U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U17_gg(z2, z0, .(z4, z5), membercD_in_gg(z2, .(z4, z5))))) 172.38/44.61 (U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U17_gg(x5, z0, .(z0, z4), membercD_in_gg(x5, .(z0, z4)))),U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U17_gg(x5, z0, .(z0, z4), membercD_in_gg(x5, .(z0, z4))))) 172.38/44.61 (U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U17_gg(x5, z0, .(z4, z5), membercD_in_gg(x5, .(z4, z5)))),U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U17_gg(x5, z0, .(z4, z5), membercD_in_gg(x5, .(z4, z5))))) 172.38/44.61 172.38/44.61 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (194) 172.38/44.61 Obligation: 172.38/44.61 Q DP problem: 172.38/44.61 The TRS P consists of the following rules: 172.38/44.61 172.38/44.61 U8_GGGG(z0, z1, z2, .(z3, z4), z3, membercD_out_gg(z3, .(z3, z4))) -> REACHC_IN_GGGG(z3, z1, z2, .(z3, .(z3, z4))) 172.38/44.61 U8_GGGG(z0, z1, z2, .(z3, z4), z5, membercD_out_gg(z5, .(z3, z4))) -> REACHC_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4))) 172.38/44.61 REACHC_IN_GGGG(z3, z1, .(.(z3, .(x2, [])), x3), .(z3, .(z3, z4))) -> U6_GGGG(z3, z1, .(.(z3, .(x2, [])), x3), .(z3, .(z3, z4)), member1cB_out_gag(z3, x2, .(.(z3, .(x2, [])), x3))) 172.38/44.61 REACHC_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), x3), .(z5, .(z3, z4))) -> U6_GGGG(z5, z1, .(.(z5, .(x2, [])), x3), .(z5, .(z3, z4)), member1cB_out_gag(z5, x2, .(.(z5, .(x2, [])), x3))) 172.38/44.61 REACHC_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) -> U6_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U12_gag(z3, x2, x3, member1cB_in_gag(z3, x3))) 172.38/44.61 REACHC_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U6_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U12_gag(z5, x2, x3, member1cB_in_gag(z5, x3))) 172.38/44.61 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))) 172.38/44.61 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) 172.38/44.61 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))) 172.38/44.61 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) 172.38/44.61 U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U17_gg(z2, z0, .(z0, z4), membercD_in_gg(z2, .(z0, z4)))) 172.38/44.61 U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U17_gg(z2, z0, .(z4, z5), membercD_in_gg(z2, .(z4, z5)))) 172.38/44.61 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U17_gg(x5, z0, .(z0, z4), membercD_in_gg(x5, .(z0, z4)))) 172.38/44.61 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U17_gg(x5, z0, .(z4, z5), membercD_in_gg(x5, .(z4, z5)))) 172.38/44.61 172.38/44.61 The TRS R consists of the following rules: 172.38/44.61 172.38/44.61 member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) 172.38/44.61 member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) 172.38/44.61 U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) 172.38/44.61 membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) 172.38/44.61 membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) 172.38/44.61 U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) 172.38/44.61 172.38/44.61 The set Q consists of the following terms: 172.38/44.61 172.38/44.61 member1cB_in_gag(x0, x1) 172.38/44.61 U12_gag(x0, x1, x2, x3) 172.38/44.61 membercD_in_gg(x0, x1) 172.38/44.61 U17_gg(x0, x1, x2, x3) 172.38/44.61 172.38/44.61 We have to consider all (P,Q,R)-chains. 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (195) TransformationProof (EQUIVALENT) 172.38/44.61 By instantiating [LPAR04] the rule U8_GGGG(z0, z1, z2, .(z3, z4), z3, membercD_out_gg(z3, .(z3, z4))) -> REACHC_IN_GGGG(z3, z1, z2, .(z3, .(z3, z4))) we obtained the following new rules [LPAR04]: 172.38/44.61 172.38/44.61 (U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, membercD_out_gg(z0, .(z0, .(z0, z3)))) -> REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))),U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, membercD_out_gg(z0, .(z0, .(z0, z3)))) -> REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))))) 172.38/44.61 (U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, membercD_out_gg(z0, .(z0, .(z3, z4)))) -> REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))),U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, membercD_out_gg(z0, .(z0, .(z3, z4)))) -> REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))))) 172.38/44.61 (U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))),U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))))) 172.38/44.61 (U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))),U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))))) 172.38/44.61 172.38/44.61 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (196) 172.38/44.61 Obligation: 172.38/44.61 Q DP problem: 172.38/44.61 The TRS P consists of the following rules: 172.38/44.61 172.38/44.61 U8_GGGG(z0, z1, z2, .(z3, z4), z5, membercD_out_gg(z5, .(z3, z4))) -> REACHC_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4))) 172.38/44.61 REACHC_IN_GGGG(z3, z1, .(.(z3, .(x2, [])), x3), .(z3, .(z3, z4))) -> U6_GGGG(z3, z1, .(.(z3, .(x2, [])), x3), .(z3, .(z3, z4)), member1cB_out_gag(z3, x2, .(.(z3, .(x2, [])), x3))) 172.38/44.61 REACHC_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), x3), .(z5, .(z3, z4))) -> U6_GGGG(z5, z1, .(.(z5, .(x2, [])), x3), .(z5, .(z3, z4)), member1cB_out_gag(z5, x2, .(.(z5, .(x2, [])), x3))) 172.38/44.61 REACHC_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) -> U6_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U12_gag(z3, x2, x3, member1cB_in_gag(z3, x3))) 172.38/44.61 REACHC_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U6_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U12_gag(z5, x2, x3, member1cB_in_gag(z5, x3))) 172.38/44.61 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))) 172.38/44.61 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) 172.38/44.61 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))) 172.38/44.61 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) 172.38/44.61 U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U17_gg(z2, z0, .(z0, z4), membercD_in_gg(z2, .(z0, z4)))) 172.38/44.61 U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U17_gg(z2, z0, .(z4, z5), membercD_in_gg(z2, .(z4, z5)))) 172.38/44.61 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U17_gg(x5, z0, .(z0, z4), membercD_in_gg(x5, .(z0, z4)))) 172.38/44.61 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U17_gg(x5, z0, .(z4, z5), membercD_in_gg(x5, .(z4, z5)))) 172.38/44.61 U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, membercD_out_gg(z0, .(z0, .(z0, z3)))) -> REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) 172.38/44.61 U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, membercD_out_gg(z0, .(z0, .(z3, z4)))) -> REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) 172.38/44.61 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) 172.38/44.61 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) 172.38/44.61 172.38/44.61 The TRS R consists of the following rules: 172.38/44.61 172.38/44.61 member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) 172.38/44.61 member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) 172.38/44.61 U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) 172.38/44.61 membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) 172.38/44.61 membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) 172.38/44.61 U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) 172.38/44.61 172.38/44.61 The set Q consists of the following terms: 172.38/44.61 172.38/44.61 member1cB_in_gag(x0, x1) 172.38/44.61 U12_gag(x0, x1, x2, x3) 172.38/44.61 membercD_in_gg(x0, x1) 172.38/44.61 U17_gg(x0, x1, x2, x3) 172.38/44.61 172.38/44.61 We have to consider all (P,Q,R)-chains. 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (197) TransformationProof (EQUIVALENT) 172.38/44.61 By instantiating [LPAR04] the rule U8_GGGG(z0, z1, z2, .(z3, z4), z5, membercD_out_gg(z5, .(z3, z4))) -> REACHC_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4))) we obtained the following new rules [LPAR04]: 172.38/44.61 172.38/44.61 (U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, membercD_out_gg(z0, .(z0, .(z0, z3)))) -> REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))),U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, membercD_out_gg(z0, .(z0, .(z0, z3)))) -> REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))))) 172.38/44.61 (U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, membercD_out_gg(z0, .(z0, .(z3, z4)))) -> REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))),U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, membercD_out_gg(z0, .(z0, .(z3, z4)))) -> REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))))) 172.38/44.61 (U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))),U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))))) 172.38/44.61 (U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))),U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))))) 172.38/44.61 (U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, membercD_out_gg(z2, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))),U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, membercD_out_gg(z2, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))))) 172.38/44.61 (U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, membercD_out_gg(z2, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))),U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, membercD_out_gg(z2, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))))) 172.38/44.61 (U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, membercD_out_gg(z5, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))),U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, membercD_out_gg(z5, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))))) 172.38/44.61 (U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, membercD_out_gg(z6, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))),U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, membercD_out_gg(z6, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))))) 172.38/44.61 172.38/44.61 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (198) 172.38/44.61 Obligation: 172.38/44.61 Q DP problem: 172.38/44.61 The TRS P consists of the following rules: 172.38/44.61 172.38/44.61 REACHC_IN_GGGG(z3, z1, .(.(z3, .(x2, [])), x3), .(z3, .(z3, z4))) -> U6_GGGG(z3, z1, .(.(z3, .(x2, [])), x3), .(z3, .(z3, z4)), member1cB_out_gag(z3, x2, .(.(z3, .(x2, [])), x3))) 172.38/44.61 REACHC_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), x3), .(z5, .(z3, z4))) -> U6_GGGG(z5, z1, .(.(z5, .(x2, [])), x3), .(z5, .(z3, z4)), member1cB_out_gag(z5, x2, .(.(z5, .(x2, [])), x3))) 172.38/44.61 REACHC_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) -> U6_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U12_gag(z3, x2, x3, member1cB_in_gag(z3, x3))) 172.38/44.61 REACHC_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U6_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U12_gag(z5, x2, x3, member1cB_in_gag(z5, x3))) 172.38/44.61 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))) 172.38/44.61 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) 172.38/44.61 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))) 172.38/44.61 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) 172.38/44.61 U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U17_gg(z2, z0, .(z0, z4), membercD_in_gg(z2, .(z0, z4)))) 172.38/44.61 U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U17_gg(z2, z0, .(z4, z5), membercD_in_gg(z2, .(z4, z5)))) 172.38/44.61 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U17_gg(x5, z0, .(z0, z4), membercD_in_gg(x5, .(z0, z4)))) 172.38/44.61 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U17_gg(x5, z0, .(z4, z5), membercD_in_gg(x5, .(z4, z5)))) 172.38/44.61 U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, membercD_out_gg(z0, .(z0, .(z0, z3)))) -> REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) 172.38/44.61 U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, membercD_out_gg(z0, .(z0, .(z3, z4)))) -> REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) 172.38/44.61 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) 172.38/44.61 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) 172.38/44.61 U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, membercD_out_gg(z2, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) 172.38/44.61 U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, membercD_out_gg(z2, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) 172.38/44.61 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, membercD_out_gg(z5, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) 172.38/44.61 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, membercD_out_gg(z6, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 172.38/44.61 172.38/44.61 The TRS R consists of the following rules: 172.38/44.61 172.38/44.61 member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) 172.38/44.61 member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) 172.38/44.61 U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) 172.38/44.61 membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) 172.38/44.61 membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) 172.38/44.61 U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) 172.38/44.61 172.38/44.61 The set Q consists of the following terms: 172.38/44.61 172.38/44.61 member1cB_in_gag(x0, x1) 172.38/44.61 U12_gag(x0, x1, x2, x3) 172.38/44.61 membercD_in_gg(x0, x1) 172.38/44.61 U17_gg(x0, x1, x2, x3) 172.38/44.61 172.38/44.61 We have to consider all (P,Q,R)-chains. 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (199) TransformationProof (EQUIVALENT) 172.38/44.61 By instantiating [LPAR04] the rule REACHC_IN_GGGG(z3, z1, .(.(z3, .(x2, [])), x3), .(z3, .(z3, z4))) -> U6_GGGG(z3, z1, .(.(z3, .(x2, [])), x3), .(z3, .(z3, z4)), member1cB_out_gag(z3, x2, .(.(z3, .(x2, [])), x3))) we obtained the following new rules [LPAR04]: 172.38/44.61 172.38/44.61 (REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))),REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2)))) 172.38/44.61 (REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))),REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2)))) 172.38/44.61 (REACHC_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, x2, .(.(z0, .(x2, [])), z3))),REACHC_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, x2, .(.(z0, .(x2, [])), z3)))) 172.38/44.61 (REACHC_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, x2, .(.(z0, .(x2, [])), z3))),REACHC_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, x2, .(.(z0, .(x2, [])), z3)))) 172.38/44.61 172.38/44.61 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (200) 172.38/44.61 Obligation: 172.38/44.61 Q DP problem: 172.38/44.61 The TRS P consists of the following rules: 172.38/44.61 172.38/44.61 REACHC_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), x3), .(z5, .(z3, z4))) -> U6_GGGG(z5, z1, .(.(z5, .(x2, [])), x3), .(z5, .(z3, z4)), member1cB_out_gag(z5, x2, .(.(z5, .(x2, [])), x3))) 172.38/44.61 REACHC_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) -> U6_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U12_gag(z3, x2, x3, member1cB_in_gag(z3, x3))) 172.38/44.61 REACHC_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U6_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U12_gag(z5, x2, x3, member1cB_in_gag(z5, x3))) 172.38/44.61 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))) 172.38/44.61 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) 172.38/44.61 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))) 172.38/44.61 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) 172.38/44.61 U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U17_gg(z2, z0, .(z0, z4), membercD_in_gg(z2, .(z0, z4)))) 172.38/44.61 U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U17_gg(z2, z0, .(z4, z5), membercD_in_gg(z2, .(z4, z5)))) 172.38/44.61 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U17_gg(x5, z0, .(z0, z4), membercD_in_gg(x5, .(z0, z4)))) 172.38/44.61 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U17_gg(x5, z0, .(z4, z5), membercD_in_gg(x5, .(z4, z5)))) 172.38/44.61 U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, membercD_out_gg(z0, .(z0, .(z0, z3)))) -> REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) 172.38/44.61 U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, membercD_out_gg(z0, .(z0, .(z3, z4)))) -> REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) 172.38/44.61 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) 172.38/44.61 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) 172.38/44.61 U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, membercD_out_gg(z2, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) 172.38/44.61 U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, membercD_out_gg(z2, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) 172.38/44.61 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, membercD_out_gg(z5, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) 172.38/44.61 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, membercD_out_gg(z6, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 172.38/44.61 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) 172.38/44.61 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) 172.38/44.61 REACHC_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, x2, .(.(z0, .(x2, [])), z3))) 172.38/44.61 REACHC_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, x2, .(.(z0, .(x2, [])), z3))) 172.38/44.61 172.38/44.61 The TRS R consists of the following rules: 172.38/44.61 172.38/44.61 member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) 172.38/44.61 member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) 172.38/44.61 U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) 172.38/44.61 membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) 172.38/44.61 membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) 172.38/44.61 U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) 172.38/44.61 172.38/44.61 The set Q consists of the following terms: 172.38/44.61 172.38/44.61 member1cB_in_gag(x0, x1) 172.38/44.61 U12_gag(x0, x1, x2, x3) 172.38/44.61 membercD_in_gg(x0, x1) 172.38/44.61 U17_gg(x0, x1, x2, x3) 172.38/44.61 172.38/44.61 We have to consider all (P,Q,R)-chains. 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (201) TransformationProof (EQUIVALENT) 172.38/44.61 By instantiating [LPAR04] the rule REACHC_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), x3), .(z5, .(z3, z4))) -> U6_GGGG(z5, z1, .(.(z5, .(x2, [])), x3), .(z5, .(z3, z4)), member1cB_out_gag(z5, x2, .(.(z5, .(x2, [])), x3))) we obtained the following new rules [LPAR04]: 172.38/44.61 172.38/44.61 (REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))),REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2)))) 172.38/44.61 (REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))),REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2)))) 172.38/44.61 (REACHC_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, x2, .(.(z0, .(x2, [])), z3))),REACHC_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, x2, .(.(z0, .(x2, [])), z3)))) 172.38/44.61 (REACHC_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, x2, .(.(z0, .(x2, [])), z3))),REACHC_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, x2, .(.(z0, .(x2, [])), z3)))) 172.38/44.61 (REACHC_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4)))) -> U6_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4))), member1cB_out_gag(z5, x2, .(.(z5, .(x2, [])), z3))),REACHC_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4)))) -> U6_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4))), member1cB_out_gag(z5, x2, .(.(z5, .(x2, [])), z3)))) 172.38/44.61 (REACHC_IN_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5)))) -> U6_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5))), member1cB_out_gag(z6, x2, .(.(z6, .(x2, [])), z3))),REACHC_IN_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5)))) -> U6_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5))), member1cB_out_gag(z6, x2, .(.(z6, .(x2, [])), z3)))) 172.38/44.61 172.38/44.61 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (202) 172.38/44.61 Obligation: 172.38/44.61 Q DP problem: 172.38/44.61 The TRS P consists of the following rules: 172.38/44.61 172.38/44.61 REACHC_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) -> U6_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U12_gag(z3, x2, x3, member1cB_in_gag(z3, x3))) 172.38/44.61 REACHC_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U6_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U12_gag(z5, x2, x3, member1cB_in_gag(z5, x3))) 172.38/44.61 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))) 172.38/44.61 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) 172.38/44.61 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))) 172.38/44.61 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) 172.38/44.61 U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U17_gg(z2, z0, .(z0, z4), membercD_in_gg(z2, .(z0, z4)))) 172.38/44.61 U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U17_gg(z2, z0, .(z4, z5), membercD_in_gg(z2, .(z4, z5)))) 172.38/44.61 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U17_gg(x5, z0, .(z0, z4), membercD_in_gg(x5, .(z0, z4)))) 172.38/44.61 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U17_gg(x5, z0, .(z4, z5), membercD_in_gg(x5, .(z4, z5)))) 172.38/44.61 U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, membercD_out_gg(z0, .(z0, .(z0, z3)))) -> REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) 172.38/44.61 U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, membercD_out_gg(z0, .(z0, .(z3, z4)))) -> REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) 172.38/44.61 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) 172.38/44.61 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) 172.38/44.61 U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, membercD_out_gg(z2, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) 172.38/44.61 U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, membercD_out_gg(z2, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) 172.38/44.61 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, membercD_out_gg(z5, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) 172.38/44.61 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, membercD_out_gg(z6, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 172.38/44.61 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) 172.38/44.61 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) 172.38/44.61 REACHC_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, x2, .(.(z0, .(x2, [])), z3))) 172.38/44.61 REACHC_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, x2, .(.(z0, .(x2, [])), z3))) 172.38/44.61 REACHC_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4)))) -> U6_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4))), member1cB_out_gag(z5, x2, .(.(z5, .(x2, [])), z3))) 172.38/44.61 REACHC_IN_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5)))) -> U6_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5))), member1cB_out_gag(z6, x2, .(.(z6, .(x2, [])), z3))) 172.38/44.61 172.38/44.61 The TRS R consists of the following rules: 172.38/44.61 172.38/44.61 member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) 172.38/44.61 member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) 172.38/44.61 U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) 172.38/44.61 membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) 172.38/44.61 membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) 172.38/44.61 U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) 172.38/44.61 172.38/44.61 The set Q consists of the following terms: 172.38/44.61 172.38/44.61 member1cB_in_gag(x0, x1) 172.38/44.61 U12_gag(x0, x1, x2, x3) 172.38/44.61 membercD_in_gg(x0, x1) 172.38/44.61 U17_gg(x0, x1, x2, x3) 172.38/44.61 172.38/44.61 We have to consider all (P,Q,R)-chains. 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (203) TransformationProof (EQUIVALENT) 172.38/44.61 By instantiating [LPAR04] the rule REACHC_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) -> U6_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U12_gag(z3, x2, x3, member1cB_in_gag(z3, x3))) we obtained the following new rules [LPAR04]: 172.38/44.61 172.38/44.61 (REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U12_gag(z0, .(z0, .(z0, [])), z2, member1cB_in_gag(z0, z2))),REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U12_gag(z0, .(z0, .(z0, [])), z2, member1cB_in_gag(z0, z2)))) 172.38/44.61 (REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U12_gag(z0, .(z0, .(z0, [])), z2, member1cB_in_gag(z0, z2))),REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U12_gag(z0, .(z0, .(z0, [])), z2, member1cB_in_gag(z0, z2)))) 172.38/44.61 (REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U12_gag(z0, z2, z3, member1cB_in_gag(z0, z3))),REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U12_gag(z0, z2, z3, member1cB_in_gag(z0, z3)))) 172.38/44.61 (REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U12_gag(z0, z2, z3, member1cB_in_gag(z0, z3))),REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U12_gag(z0, z2, z3, member1cB_in_gag(z0, z3)))) 172.38/44.61 172.38/44.61 172.38/44.61 ---------------------------------------- 172.38/44.61 172.38/44.61 (204) 172.38/44.61 Obligation: 172.38/44.61 Q DP problem: 172.38/44.61 The TRS P consists of the following rules: 172.38/44.61 172.38/44.61 REACHC_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U6_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U12_gag(z5, x2, x3, member1cB_in_gag(z5, x3))) 172.38/44.61 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))) 172.38/44.61 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) 172.38/44.61 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))) 172.38/44.61 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U17_gg(z2, z0, .(z0, z4), membercD_in_gg(z2, .(z0, z4)))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U17_gg(z2, z0, .(z4, z5), membercD_in_gg(z2, .(z4, z5)))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U17_gg(x5, z0, .(z0, z4), membercD_in_gg(x5, .(z0, z4)))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U17_gg(x5, z0, .(z4, z5), membercD_in_gg(x5, .(z4, z5)))) 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, membercD_out_gg(z0, .(z0, .(z0, z3)))) -> REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, membercD_out_gg(z0, .(z0, .(z3, z4)))) -> REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, membercD_out_gg(z2, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, membercD_out_gg(z2, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, membercD_out_gg(z5, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, membercD_out_gg(z6, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, x2, .(.(z0, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, x2, .(.(z0, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4)))) -> U6_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4))), member1cB_out_gag(z5, x2, .(.(z5, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5)))) -> U6_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5))), member1cB_out_gag(z6, x2, .(.(z6, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U12_gag(z0, .(z0, .(z0, [])), z2, member1cB_in_gag(z0, z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U12_gag(z0, .(z0, .(z0, [])), z2, member1cB_in_gag(z0, z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U12_gag(z0, z2, z3, member1cB_in_gag(z0, z3))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U12_gag(z0, z2, z3, member1cB_in_gag(z0, z3))) 172.38/44.62 172.38/44.62 The TRS R consists of the following rules: 172.38/44.62 172.38/44.62 member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) 172.38/44.62 member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) 172.38/44.62 U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) 172.38/44.62 membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) 172.38/44.62 membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) 172.38/44.62 U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) 172.38/44.62 172.38/44.62 The set Q consists of the following terms: 172.38/44.62 172.38/44.62 member1cB_in_gag(x0, x1) 172.38/44.62 U12_gag(x0, x1, x2, x3) 172.38/44.62 membercD_in_gg(x0, x1) 172.38/44.62 U17_gg(x0, x1, x2, x3) 172.38/44.62 172.38/44.62 We have to consider all (P,Q,R)-chains. 172.38/44.62 ---------------------------------------- 172.38/44.62 172.38/44.62 (205) TransformationProof (EQUIVALENT) 172.38/44.62 By instantiating [LPAR04] the rule REACHC_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U6_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U12_gag(z5, x2, x3, member1cB_in_gag(z5, x3))) we obtained the following new rules [LPAR04]: 172.38/44.62 172.38/44.62 (REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U12_gag(z0, .(z0, .(z0, [])), z2, member1cB_in_gag(z0, z2))),REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U12_gag(z0, .(z0, .(z0, [])), z2, member1cB_in_gag(z0, z2)))) 172.38/44.62 (REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U12_gag(z0, .(z0, .(z0, [])), z2, member1cB_in_gag(z0, z2))),REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U12_gag(z0, .(z0, .(z0, [])), z2, member1cB_in_gag(z0, z2)))) 172.38/44.62 (REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U12_gag(z0, z2, z3, member1cB_in_gag(z0, z3))),REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U12_gag(z0, z2, z3, member1cB_in_gag(z0, z3)))) 172.38/44.62 (REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U12_gag(z0, z2, z3, member1cB_in_gag(z0, z3))),REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U12_gag(z0, z2, z3, member1cB_in_gag(z0, z3)))) 172.38/44.62 (REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), U12_gag(z2, .(z0, .(z2, [])), z3, member1cB_in_gag(z2, z3))),REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), U12_gag(z2, .(z0, .(z2, [])), z3, member1cB_in_gag(z2, z3)))) 172.38/44.62 (REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), U12_gag(z2, .(z0, .(z2, [])), z3, member1cB_in_gag(z2, z3))),REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), U12_gag(z2, .(z0, .(z2, [])), z3, member1cB_in_gag(z2, z3)))) 172.38/44.62 (REACHC_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U6_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), U12_gag(z5, z2, z3, member1cB_in_gag(z5, z3))),REACHC_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U6_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), U12_gag(z5, z2, z3, member1cB_in_gag(z5, z3)))) 172.38/44.62 (REACHC_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U6_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), U12_gag(z6, z2, z3, member1cB_in_gag(z6, z3))),REACHC_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U6_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), U12_gag(z6, z2, z3, member1cB_in_gag(z6, z3)))) 172.38/44.62 172.38/44.62 172.38/44.62 ---------------------------------------- 172.38/44.62 172.38/44.62 (206) 172.38/44.62 Obligation: 172.38/44.62 Q DP problem: 172.38/44.62 The TRS P consists of the following rules: 172.38/44.62 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U17_gg(z2, z0, .(z0, z4), membercD_in_gg(z2, .(z0, z4)))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U17_gg(z2, z0, .(z4, z5), membercD_in_gg(z2, .(z4, z5)))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U17_gg(x5, z0, .(z0, z4), membercD_in_gg(x5, .(z0, z4)))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U17_gg(x5, z0, .(z4, z5), membercD_in_gg(x5, .(z4, z5)))) 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, membercD_out_gg(z0, .(z0, .(z0, z3)))) -> REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, membercD_out_gg(z0, .(z0, .(z3, z4)))) -> REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, membercD_out_gg(z2, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, membercD_out_gg(z2, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, membercD_out_gg(z5, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, membercD_out_gg(z6, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, x2, .(.(z0, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, x2, .(.(z0, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4)))) -> U6_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4))), member1cB_out_gag(z5, x2, .(.(z5, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5)))) -> U6_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5))), member1cB_out_gag(z6, x2, .(.(z6, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U12_gag(z0, .(z0, .(z0, [])), z2, member1cB_in_gag(z0, z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U12_gag(z0, .(z0, .(z0, [])), z2, member1cB_in_gag(z0, z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U12_gag(z0, z2, z3, member1cB_in_gag(z0, z3))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U12_gag(z0, z2, z3, member1cB_in_gag(z0, z3))) 172.38/44.62 REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), U12_gag(z2, .(z0, .(z2, [])), z3, member1cB_in_gag(z2, z3))) 172.38/44.62 REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), U12_gag(z2, .(z0, .(z2, [])), z3, member1cB_in_gag(z2, z3))) 172.38/44.62 REACHC_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U6_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), U12_gag(z5, z2, z3, member1cB_in_gag(z5, z3))) 172.38/44.62 REACHC_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U6_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), U12_gag(z6, z2, z3, member1cB_in_gag(z6, z3))) 172.38/44.62 172.38/44.62 The TRS R consists of the following rules: 172.38/44.62 172.38/44.62 member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) 172.38/44.62 member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) 172.38/44.62 U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) 172.38/44.62 membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) 172.38/44.62 membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) 172.38/44.62 U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) 172.38/44.62 172.38/44.62 The set Q consists of the following terms: 172.38/44.62 172.38/44.62 member1cB_in_gag(x0, x1) 172.38/44.62 U12_gag(x0, x1, x2, x3) 172.38/44.62 membercD_in_gg(x0, x1) 172.38/44.62 U17_gg(x0, x1, x2, x3) 172.38/44.62 172.38/44.62 We have to consider all (P,Q,R)-chains. 172.38/44.62 ---------------------------------------- 172.38/44.62 172.38/44.62 (207) TransformationProof (EQUIVALENT) 172.38/44.62 By instantiating [LPAR04] the rule U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))) we obtained the following new rules [LPAR04]: 172.38/44.62 172.38/44.62 (U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z0, z3))))),U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z0, z3)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z3, z4))))),U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z3, z4)))))) 172.38/44.62 172.38/44.62 172.38/44.62 ---------------------------------------- 172.38/44.62 172.38/44.62 (208) 172.38/44.62 Obligation: 172.38/44.62 Q DP problem: 172.38/44.62 The TRS P consists of the following rules: 172.38/44.62 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U17_gg(z2, z0, .(z0, z4), membercD_in_gg(z2, .(z0, z4)))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U17_gg(z2, z0, .(z4, z5), membercD_in_gg(z2, .(z4, z5)))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U17_gg(x5, z0, .(z0, z4), membercD_in_gg(x5, .(z0, z4)))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U17_gg(x5, z0, .(z4, z5), membercD_in_gg(x5, .(z4, z5)))) 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, membercD_out_gg(z0, .(z0, .(z0, z3)))) -> REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, membercD_out_gg(z0, .(z0, .(z3, z4)))) -> REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, membercD_out_gg(z2, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, membercD_out_gg(z2, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, membercD_out_gg(z5, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, membercD_out_gg(z6, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, x2, .(.(z0, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, x2, .(.(z0, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4)))) -> U6_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4))), member1cB_out_gag(z5, x2, .(.(z5, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5)))) -> U6_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5))), member1cB_out_gag(z6, x2, .(.(z6, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U12_gag(z0, .(z0, .(z0, [])), z2, member1cB_in_gag(z0, z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U12_gag(z0, .(z0, .(z0, [])), z2, member1cB_in_gag(z0, z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U12_gag(z0, z2, z3, member1cB_in_gag(z0, z3))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U12_gag(z0, z2, z3, member1cB_in_gag(z0, z3))) 172.38/44.62 REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), U12_gag(z2, .(z0, .(z2, [])), z3, member1cB_in_gag(z2, z3))) 172.38/44.62 REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), U12_gag(z2, .(z0, .(z2, [])), z3, member1cB_in_gag(z2, z3))) 172.38/44.62 REACHC_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U6_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), U12_gag(z5, z2, z3, member1cB_in_gag(z5, z3))) 172.38/44.62 REACHC_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U6_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), U12_gag(z6, z2, z3, member1cB_in_gag(z6, z3))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z0, z3))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z3, z4))))) 172.38/44.62 172.38/44.62 The TRS R consists of the following rules: 172.38/44.62 172.38/44.62 member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) 172.38/44.62 member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) 172.38/44.62 U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) 172.38/44.62 membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) 172.38/44.62 membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) 172.38/44.62 U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) 172.38/44.62 172.38/44.62 The set Q consists of the following terms: 172.38/44.62 172.38/44.62 member1cB_in_gag(x0, x1) 172.38/44.62 U12_gag(x0, x1, x2, x3) 172.38/44.62 membercD_in_gg(x0, x1) 172.38/44.62 U17_gg(x0, x1, x2, x3) 172.38/44.62 172.38/44.62 We have to consider all (P,Q,R)-chains. 172.38/44.62 ---------------------------------------- 172.38/44.62 172.38/44.62 (209) TransformationProof (EQUIVALENT) 172.38/44.62 By instantiating [LPAR04] the rule U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) we obtained the following new rules [LPAR04]: 172.38/44.62 172.38/44.62 (U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z0, z3))))),U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z0, z3)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z3, z4))))),U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z3, z4)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, membercD_out_gg(z0, .(z0, .(z4, .(z4, z5))))),U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, membercD_out_gg(z0, .(z0, .(z4, .(z4, z5)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, membercD_out_gg(z0, .(z0, .(z4, .(z5, z6))))),U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, membercD_out_gg(z0, .(z0, .(z4, .(z5, z6)))))) 172.38/44.62 172.38/44.62 172.38/44.62 ---------------------------------------- 172.38/44.62 172.38/44.62 (210) 172.38/44.62 Obligation: 172.38/44.62 Q DP problem: 172.38/44.62 The TRS P consists of the following rules: 172.38/44.62 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U17_gg(z2, z0, .(z0, z4), membercD_in_gg(z2, .(z0, z4)))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U17_gg(z2, z0, .(z4, z5), membercD_in_gg(z2, .(z4, z5)))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U17_gg(x5, z0, .(z0, z4), membercD_in_gg(x5, .(z0, z4)))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U17_gg(x5, z0, .(z4, z5), membercD_in_gg(x5, .(z4, z5)))) 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, membercD_out_gg(z0, .(z0, .(z0, z3)))) -> REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, membercD_out_gg(z0, .(z0, .(z3, z4)))) -> REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, membercD_out_gg(z2, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, membercD_out_gg(z2, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, membercD_out_gg(z5, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, membercD_out_gg(z6, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, x2, .(.(z0, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, x2, .(.(z0, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4)))) -> U6_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4))), member1cB_out_gag(z5, x2, .(.(z5, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5)))) -> U6_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5))), member1cB_out_gag(z6, x2, .(.(z6, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U12_gag(z0, .(z0, .(z0, [])), z2, member1cB_in_gag(z0, z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U12_gag(z0, .(z0, .(z0, [])), z2, member1cB_in_gag(z0, z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U12_gag(z0, z2, z3, member1cB_in_gag(z0, z3))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U12_gag(z0, z2, z3, member1cB_in_gag(z0, z3))) 172.38/44.62 REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), U12_gag(z2, .(z0, .(z2, [])), z3, member1cB_in_gag(z2, z3))) 172.38/44.62 REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), U12_gag(z2, .(z0, .(z2, [])), z3, member1cB_in_gag(z2, z3))) 172.38/44.62 REACHC_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U6_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), U12_gag(z5, z2, z3, member1cB_in_gag(z5, z3))) 172.38/44.62 REACHC_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U6_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), U12_gag(z6, z2, z3, member1cB_in_gag(z6, z3))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z0, z3))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z3, z4))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, membercD_out_gg(z0, .(z0, .(z4, .(z4, z5))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, membercD_out_gg(z0, .(z0, .(z4, .(z5, z6))))) 172.38/44.62 172.38/44.62 The TRS R consists of the following rules: 172.38/44.62 172.38/44.62 member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) 172.38/44.62 member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) 172.38/44.62 U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) 172.38/44.62 membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) 172.38/44.62 membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) 172.38/44.62 U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) 172.38/44.62 172.38/44.62 The set Q consists of the following terms: 172.38/44.62 172.38/44.62 member1cB_in_gag(x0, x1) 172.38/44.62 U12_gag(x0, x1, x2, x3) 172.38/44.62 membercD_in_gg(x0, x1) 172.38/44.62 U17_gg(x0, x1, x2, x3) 172.38/44.62 172.38/44.62 We have to consider all (P,Q,R)-chains. 172.38/44.62 ---------------------------------------- 172.38/44.62 172.38/44.62 (211) TransformationProof (EQUIVALENT) 172.38/44.62 By instantiating [LPAR04] the rule U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))) we obtained the following new rules [LPAR04]: 172.38/44.62 172.38/44.62 (U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z0, z3))))),U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z0, z3)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z3, z4))))),U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z3, z4)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z0, z4))))),U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z0, z4)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z4, z5))))),U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z4, z5)))))) 172.38/44.62 172.38/44.62 172.38/44.62 ---------------------------------------- 172.38/44.62 172.38/44.62 (212) 172.38/44.62 Obligation: 172.38/44.62 Q DP problem: 172.38/44.62 The TRS P consists of the following rules: 172.38/44.62 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U17_gg(z2, z0, .(z0, z4), membercD_in_gg(z2, .(z0, z4)))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U17_gg(z2, z0, .(z4, z5), membercD_in_gg(z2, .(z4, z5)))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U17_gg(x5, z0, .(z0, z4), membercD_in_gg(x5, .(z0, z4)))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U17_gg(x5, z0, .(z4, z5), membercD_in_gg(x5, .(z4, z5)))) 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, membercD_out_gg(z0, .(z0, .(z0, z3)))) -> REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, membercD_out_gg(z0, .(z0, .(z3, z4)))) -> REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, membercD_out_gg(z2, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, membercD_out_gg(z2, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, membercD_out_gg(z5, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, membercD_out_gg(z6, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, x2, .(.(z0, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, x2, .(.(z0, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4)))) -> U6_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4))), member1cB_out_gag(z5, x2, .(.(z5, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5)))) -> U6_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5))), member1cB_out_gag(z6, x2, .(.(z6, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U12_gag(z0, .(z0, .(z0, [])), z2, member1cB_in_gag(z0, z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U12_gag(z0, .(z0, .(z0, [])), z2, member1cB_in_gag(z0, z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U12_gag(z0, z2, z3, member1cB_in_gag(z0, z3))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U12_gag(z0, z2, z3, member1cB_in_gag(z0, z3))) 172.38/44.62 REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), U12_gag(z2, .(z0, .(z2, [])), z3, member1cB_in_gag(z2, z3))) 172.38/44.62 REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), U12_gag(z2, .(z0, .(z2, [])), z3, member1cB_in_gag(z2, z3))) 172.38/44.62 REACHC_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U6_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), U12_gag(z5, z2, z3, member1cB_in_gag(z5, z3))) 172.38/44.62 REACHC_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U6_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), U12_gag(z6, z2, z3, member1cB_in_gag(z6, z3))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z0, z3))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z3, z4))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, membercD_out_gg(z0, .(z0, .(z4, .(z4, z5))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, membercD_out_gg(z0, .(z0, .(z4, .(z5, z6))))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z0, z4))))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z4, z5))))) 172.38/44.62 172.38/44.62 The TRS R consists of the following rules: 172.38/44.62 172.38/44.62 member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) 172.38/44.62 member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) 172.38/44.62 U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) 172.38/44.62 membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) 172.38/44.62 membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) 172.38/44.62 U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) 172.38/44.62 172.38/44.62 The set Q consists of the following terms: 172.38/44.62 172.38/44.62 member1cB_in_gag(x0, x1) 172.38/44.62 U12_gag(x0, x1, x2, x3) 172.38/44.62 membercD_in_gg(x0, x1) 172.38/44.62 U17_gg(x0, x1, x2, x3) 172.38/44.62 172.38/44.62 We have to consider all (P,Q,R)-chains. 172.38/44.62 ---------------------------------------- 172.38/44.62 172.38/44.62 (213) TransformationProof (EQUIVALENT) 172.38/44.62 By instantiating [LPAR04] the rule U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) we obtained the following new rules [LPAR04]: 172.38/44.62 172.38/44.62 (U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z0, z3))))),U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z0, z3)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z3, z4))))),U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z3, z4)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, membercD_out_gg(z0, .(z0, .(z4, .(z4, z5))))),U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, membercD_out_gg(z0, .(z0, .(z4, .(z4, z5)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, membercD_out_gg(z0, .(z0, .(z4, .(z5, z6))))),U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, membercD_out_gg(z0, .(z0, .(z4, .(z5, z6)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z0, z4))))),U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z0, z4)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z4, z5))))),U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z4, z5)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member1cB_out_gag(z0, z0, .(.(z2, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), z0, membercD_out_gg(z0, .(z0, .(z2, .(z2, z4))))),U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member1cB_out_gag(z0, z0, .(.(z2, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), z0, membercD_out_gg(z0, .(z0, .(z2, .(z2, z4)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member1cB_out_gag(z0, z0, .(.(z2, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), z0, membercD_out_gg(z0, .(z0, .(z2, .(z4, z5))))),U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member1cB_out_gag(z0, z0, .(.(z2, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), z0, membercD_out_gg(z0, .(z0, .(z2, .(z4, z5)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, membercD_out_gg(z0, .(z0, .(z4, .(z4, z5))))),U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, membercD_out_gg(z0, .(z0, .(z4, .(z4, z5)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, membercD_out_gg(z0, .(z0, .(z4, .(z5, z6))))),U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, membercD_out_gg(z0, .(z0, .(z4, .(z5, z6)))))) 172.38/44.62 172.38/44.62 172.38/44.62 ---------------------------------------- 172.38/44.62 172.38/44.62 (214) 172.38/44.62 Obligation: 172.38/44.62 Q DP problem: 172.38/44.62 The TRS P consists of the following rules: 172.38/44.62 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U17_gg(z2, z0, .(z0, z4), membercD_in_gg(z2, .(z0, z4)))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U17_gg(z2, z0, .(z4, z5), membercD_in_gg(z2, .(z4, z5)))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U17_gg(x5, z0, .(z0, z4), membercD_in_gg(x5, .(z0, z4)))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U17_gg(x5, z0, .(z4, z5), membercD_in_gg(x5, .(z4, z5)))) 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, membercD_out_gg(z0, .(z0, .(z0, z3)))) -> REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, membercD_out_gg(z0, .(z0, .(z3, z4)))) -> REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, membercD_out_gg(z2, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, membercD_out_gg(z2, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, membercD_out_gg(z5, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, membercD_out_gg(z6, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, x2, .(.(z0, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, x2, .(.(z0, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4)))) -> U6_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4))), member1cB_out_gag(z5, x2, .(.(z5, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5)))) -> U6_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5))), member1cB_out_gag(z6, x2, .(.(z6, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U12_gag(z0, .(z0, .(z0, [])), z2, member1cB_in_gag(z0, z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U12_gag(z0, .(z0, .(z0, [])), z2, member1cB_in_gag(z0, z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U12_gag(z0, z2, z3, member1cB_in_gag(z0, z3))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U12_gag(z0, z2, z3, member1cB_in_gag(z0, z3))) 172.38/44.62 REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), U12_gag(z2, .(z0, .(z2, [])), z3, member1cB_in_gag(z2, z3))) 172.38/44.62 REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), U12_gag(z2, .(z0, .(z2, [])), z3, member1cB_in_gag(z2, z3))) 172.38/44.62 REACHC_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U6_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), U12_gag(z5, z2, z3, member1cB_in_gag(z5, z3))) 172.38/44.62 REACHC_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U6_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), U12_gag(z6, z2, z3, member1cB_in_gag(z6, z3))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z0, z3))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z3, z4))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, membercD_out_gg(z0, .(z0, .(z4, .(z4, z5))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, membercD_out_gg(z0, .(z0, .(z4, .(z5, z6))))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z0, z4))))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z4, z5))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member1cB_out_gag(z0, z0, .(.(z2, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), z0, membercD_out_gg(z0, .(z0, .(z2, .(z2, z4))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member1cB_out_gag(z0, z0, .(.(z2, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), z0, membercD_out_gg(z0, .(z0, .(z2, .(z4, z5))))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, membercD_out_gg(z0, .(z0, .(z4, .(z4, z5))))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, membercD_out_gg(z0, .(z0, .(z4, .(z5, z6))))) 172.38/44.62 172.38/44.62 The TRS R consists of the following rules: 172.38/44.62 172.38/44.62 member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) 172.38/44.62 member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) 172.38/44.62 U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) 172.38/44.62 membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) 172.38/44.62 membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) 172.38/44.62 U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) 172.38/44.62 172.38/44.62 The set Q consists of the following terms: 172.38/44.62 172.38/44.62 member1cB_in_gag(x0, x1) 172.38/44.62 U12_gag(x0, x1, x2, x3) 172.38/44.62 membercD_in_gg(x0, x1) 172.38/44.62 U17_gg(x0, x1, x2, x3) 172.38/44.62 172.38/44.62 We have to consider all (P,Q,R)-chains. 172.38/44.62 ---------------------------------------- 172.38/44.62 172.38/44.62 (215) TransformationProof (EQUIVALENT) 172.38/44.62 By instantiating [LPAR04] the rule U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U17_gg(z2, z0, .(z0, z4), membercD_in_gg(z2, .(z0, z4)))) we obtained the following new rules [LPAR04]: 172.38/44.62 172.38/44.62 (U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U17_gg(z0, z0, .(z0, .(z0, z3)), membercD_in_gg(z0, .(z0, .(z0, z3))))),U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U17_gg(z0, z0, .(z0, .(z0, z3)), membercD_in_gg(z0, .(z0, .(z0, z3)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U17_gg(z0, z0, .(z0, .(z3, z4)), membercD_in_gg(z0, .(z0, .(z3, z4))))),U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U17_gg(z0, z0, .(z0, .(z3, z4)), membercD_in_gg(z0, .(z0, .(z3, z4)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), z2, U17_gg(z2, z0, .(z0, .(z0, z4)), membercD_in_gg(z2, .(z0, .(z0, z4))))),U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), z2, U17_gg(z2, z0, .(z0, .(z0, z4)), membercD_in_gg(z2, .(z0, .(z0, z4)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), z2, U17_gg(z2, z0, .(z0, .(z4, z5)), membercD_in_gg(z2, .(z0, .(z4, z5))))),U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), z2, U17_gg(z2, z0, .(z0, .(z4, z5)), membercD_in_gg(z2, .(z0, .(z4, z5)))))) 172.38/44.62 172.38/44.62 172.38/44.62 ---------------------------------------- 172.38/44.62 172.38/44.62 (216) 172.38/44.62 Obligation: 172.38/44.62 Q DP problem: 172.38/44.62 The TRS P consists of the following rules: 172.38/44.62 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U17_gg(z2, z0, .(z4, z5), membercD_in_gg(z2, .(z4, z5)))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U17_gg(x5, z0, .(z0, z4), membercD_in_gg(x5, .(z0, z4)))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U17_gg(x5, z0, .(z4, z5), membercD_in_gg(x5, .(z4, z5)))) 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, membercD_out_gg(z0, .(z0, .(z0, z3)))) -> REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, membercD_out_gg(z0, .(z0, .(z3, z4)))) -> REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, membercD_out_gg(z2, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, membercD_out_gg(z2, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, membercD_out_gg(z5, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, membercD_out_gg(z6, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, x2, .(.(z0, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, x2, .(.(z0, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4)))) -> U6_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4))), member1cB_out_gag(z5, x2, .(.(z5, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5)))) -> U6_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5))), member1cB_out_gag(z6, x2, .(.(z6, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U12_gag(z0, .(z0, .(z0, [])), z2, member1cB_in_gag(z0, z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U12_gag(z0, .(z0, .(z0, [])), z2, member1cB_in_gag(z0, z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U12_gag(z0, z2, z3, member1cB_in_gag(z0, z3))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U12_gag(z0, z2, z3, member1cB_in_gag(z0, z3))) 172.38/44.62 REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), U12_gag(z2, .(z0, .(z2, [])), z3, member1cB_in_gag(z2, z3))) 172.38/44.62 REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), U12_gag(z2, .(z0, .(z2, [])), z3, member1cB_in_gag(z2, z3))) 172.38/44.62 REACHC_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U6_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), U12_gag(z5, z2, z3, member1cB_in_gag(z5, z3))) 172.38/44.62 REACHC_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U6_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), U12_gag(z6, z2, z3, member1cB_in_gag(z6, z3))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z0, z3))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z3, z4))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, membercD_out_gg(z0, .(z0, .(z4, .(z4, z5))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, membercD_out_gg(z0, .(z0, .(z4, .(z5, z6))))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z0, z4))))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z4, z5))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member1cB_out_gag(z0, z0, .(.(z2, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), z0, membercD_out_gg(z0, .(z0, .(z2, .(z2, z4))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member1cB_out_gag(z0, z0, .(.(z2, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), z0, membercD_out_gg(z0, .(z0, .(z2, .(z4, z5))))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, membercD_out_gg(z0, .(z0, .(z4, .(z4, z5))))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, membercD_out_gg(z0, .(z0, .(z4, .(z5, z6))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U17_gg(z0, z0, .(z0, .(z0, z3)), membercD_in_gg(z0, .(z0, .(z0, z3))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U17_gg(z0, z0, .(z0, .(z3, z4)), membercD_in_gg(z0, .(z0, .(z3, z4))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), z2, U17_gg(z2, z0, .(z0, .(z0, z4)), membercD_in_gg(z2, .(z0, .(z0, z4))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), z2, U17_gg(z2, z0, .(z0, .(z4, z5)), membercD_in_gg(z2, .(z0, .(z4, z5))))) 172.38/44.62 172.38/44.62 The TRS R consists of the following rules: 172.38/44.62 172.38/44.62 member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) 172.38/44.62 member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) 172.38/44.62 U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) 172.38/44.62 membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) 172.38/44.62 membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) 172.38/44.62 U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) 172.38/44.62 172.38/44.62 The set Q consists of the following terms: 172.38/44.62 172.38/44.62 member1cB_in_gag(x0, x1) 172.38/44.62 U12_gag(x0, x1, x2, x3) 172.38/44.62 membercD_in_gg(x0, x1) 172.38/44.62 U17_gg(x0, x1, x2, x3) 172.38/44.62 172.38/44.62 We have to consider all (P,Q,R)-chains. 172.38/44.62 ---------------------------------------- 172.38/44.62 172.38/44.62 (217) TransformationProof (EQUIVALENT) 172.38/44.62 By instantiating [LPAR04] the rule U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U17_gg(z2, z0, .(z4, z5), membercD_in_gg(z2, .(z4, z5)))) we obtained the following new rules [LPAR04]: 172.38/44.62 172.38/44.62 (U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U17_gg(z0, z0, .(z0, .(z0, z3)), membercD_in_gg(z0, .(z0, .(z0, z3))))),U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U17_gg(z0, z0, .(z0, .(z0, z3)), membercD_in_gg(z0, .(z0, .(z0, z3)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U17_gg(z0, z0, .(z0, .(z3, z4)), membercD_in_gg(z0, .(z0, .(z3, z4))))),U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U17_gg(z0, z0, .(z0, .(z3, z4)), membercD_in_gg(z0, .(z0, .(z3, z4)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), z2, U17_gg(z2, z0, .(z0, .(z0, z4)), membercD_in_gg(z2, .(z0, .(z0, z4))))),U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), z2, U17_gg(z2, z0, .(z0, .(z0, z4)), membercD_in_gg(z2, .(z0, .(z0, z4)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), z2, U17_gg(z2, z0, .(z0, .(z4, z5)), membercD_in_gg(z2, .(z0, .(z4, z5))))),U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), z2, U17_gg(z2, z0, .(z0, .(z4, z5)), membercD_in_gg(z2, .(z0, .(z4, z5)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), z2, U17_gg(z2, z0, .(z4, .(z4, z5)), membercD_in_gg(z2, .(z4, .(z4, z5))))),U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), z2, U17_gg(z2, z0, .(z4, .(z4, z5)), membercD_in_gg(z2, .(z4, .(z4, z5)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), z2, U17_gg(z2, z0, .(z4, .(z5, z6)), membercD_in_gg(z2, .(z4, .(z5, z6))))),U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), z2, U17_gg(z2, z0, .(z4, .(z5, z6)), membercD_in_gg(z2, .(z4, .(z5, z6)))))) 172.38/44.62 172.38/44.62 172.38/44.62 ---------------------------------------- 172.38/44.62 172.38/44.62 (218) 172.38/44.62 Obligation: 172.38/44.62 Q DP problem: 172.38/44.62 The TRS P consists of the following rules: 172.38/44.62 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U17_gg(x5, z0, .(z0, z4), membercD_in_gg(x5, .(z0, z4)))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U17_gg(x5, z0, .(z4, z5), membercD_in_gg(x5, .(z4, z5)))) 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, membercD_out_gg(z0, .(z0, .(z0, z3)))) -> REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, membercD_out_gg(z0, .(z0, .(z3, z4)))) -> REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, membercD_out_gg(z2, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, membercD_out_gg(z2, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, membercD_out_gg(z5, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, membercD_out_gg(z6, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, x2, .(.(z0, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, x2, .(.(z0, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4)))) -> U6_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4))), member1cB_out_gag(z5, x2, .(.(z5, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5)))) -> U6_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5))), member1cB_out_gag(z6, x2, .(.(z6, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U12_gag(z0, .(z0, .(z0, [])), z2, member1cB_in_gag(z0, z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U12_gag(z0, .(z0, .(z0, [])), z2, member1cB_in_gag(z0, z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U12_gag(z0, z2, z3, member1cB_in_gag(z0, z3))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U12_gag(z0, z2, z3, member1cB_in_gag(z0, z3))) 172.38/44.62 REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), U12_gag(z2, .(z0, .(z2, [])), z3, member1cB_in_gag(z2, z3))) 172.38/44.62 REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), U12_gag(z2, .(z0, .(z2, [])), z3, member1cB_in_gag(z2, z3))) 172.38/44.62 REACHC_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U6_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), U12_gag(z5, z2, z3, member1cB_in_gag(z5, z3))) 172.38/44.62 REACHC_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U6_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), U12_gag(z6, z2, z3, member1cB_in_gag(z6, z3))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z0, z3))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z3, z4))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, membercD_out_gg(z0, .(z0, .(z4, .(z4, z5))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, membercD_out_gg(z0, .(z0, .(z4, .(z5, z6))))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z0, z4))))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z4, z5))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member1cB_out_gag(z0, z0, .(.(z2, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), z0, membercD_out_gg(z0, .(z0, .(z2, .(z2, z4))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member1cB_out_gag(z0, z0, .(.(z2, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), z0, membercD_out_gg(z0, .(z0, .(z2, .(z4, z5))))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, membercD_out_gg(z0, .(z0, .(z4, .(z4, z5))))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, membercD_out_gg(z0, .(z0, .(z4, .(z5, z6))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U17_gg(z0, z0, .(z0, .(z0, z3)), membercD_in_gg(z0, .(z0, .(z0, z3))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U17_gg(z0, z0, .(z0, .(z3, z4)), membercD_in_gg(z0, .(z0, .(z3, z4))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), z2, U17_gg(z2, z0, .(z0, .(z0, z4)), membercD_in_gg(z2, .(z0, .(z0, z4))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), z2, U17_gg(z2, z0, .(z0, .(z4, z5)), membercD_in_gg(z2, .(z0, .(z4, z5))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), z2, U17_gg(z2, z0, .(z4, .(z4, z5)), membercD_in_gg(z2, .(z4, .(z4, z5))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), z2, U17_gg(z2, z0, .(z4, .(z5, z6)), membercD_in_gg(z2, .(z4, .(z5, z6))))) 172.38/44.62 172.38/44.62 The TRS R consists of the following rules: 172.38/44.62 172.38/44.62 member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) 172.38/44.62 member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) 172.38/44.62 U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) 172.38/44.62 membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) 172.38/44.62 membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) 172.38/44.62 U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) 172.38/44.62 172.38/44.62 The set Q consists of the following terms: 172.38/44.62 172.38/44.62 member1cB_in_gag(x0, x1) 172.38/44.62 U12_gag(x0, x1, x2, x3) 172.38/44.62 membercD_in_gg(x0, x1) 172.38/44.62 U17_gg(x0, x1, x2, x3) 172.38/44.62 172.38/44.62 We have to consider all (P,Q,R)-chains. 172.38/44.62 ---------------------------------------- 172.38/44.62 172.38/44.62 (219) TransformationProof (EQUIVALENT) 172.38/44.62 By instantiating [LPAR04] the rule U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U17_gg(x5, z0, .(z0, z4), membercD_in_gg(x5, .(z0, z4)))) we obtained the following new rules [LPAR04]: 172.38/44.62 172.38/44.62 (U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U17_gg(z0, z0, .(z0, .(z0, z3)), membercD_in_gg(z0, .(z0, .(z0, z3))))),U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U17_gg(z0, z0, .(z0, .(z0, z3)), membercD_in_gg(z0, .(z0, .(z0, z3)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U17_gg(z0, z0, .(z0, .(z3, z4)), membercD_in_gg(z0, .(z0, .(z3, z4))))),U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U17_gg(z0, z0, .(z0, .(z3, z4)), membercD_in_gg(z0, .(z0, .(z3, z4)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), z2, U17_gg(z2, z0, .(z0, .(z0, z4)), membercD_in_gg(z2, .(z0, .(z0, z4))))),U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), z2, U17_gg(z2, z0, .(z0, .(z0, z4)), membercD_in_gg(z2, .(z0, .(z0, z4)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), z2, U17_gg(z2, z0, .(z0, .(z4, z5)), membercD_in_gg(z2, .(z0, .(z4, z5))))),U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), z2, U17_gg(z2, z0, .(z0, .(z4, z5)), membercD_in_gg(z2, .(z0, .(z4, z5)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, x5, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), x5, U17_gg(x5, z0, .(z0, .(z0, z3)), membercD_in_gg(x5, .(z0, .(z0, z3))))),U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, x5, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), x5, U17_gg(x5, z0, .(z0, .(z0, z3)), membercD_in_gg(x5, .(z0, .(z0, z3)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, x5, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), x5, U17_gg(x5, z0, .(z0, .(z3, z4)), membercD_in_gg(x5, .(z0, .(z3, z4))))),U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, x5, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), x5, U17_gg(x5, z0, .(z0, .(z3, z4)), membercD_in_gg(x5, .(z0, .(z3, z4)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x5, U17_gg(x5, z0, .(z0, .(z0, z4)), membercD_in_gg(x5, .(z0, .(z0, z4))))),U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x5, U17_gg(x5, z0, .(z0, .(z0, z4)), membercD_in_gg(x5, .(z0, .(z0, z4)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x5, U17_gg(x5, z0, .(z0, .(z4, z5)), membercD_in_gg(x5, .(z0, .(z4, z5))))),U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x5, U17_gg(x5, z0, .(z0, .(z4, z5)), membercD_in_gg(x5, .(z0, .(z4, z5)))))) 172.38/44.62 172.38/44.62 172.38/44.62 ---------------------------------------- 172.38/44.62 172.38/44.62 (220) 172.38/44.62 Obligation: 172.38/44.62 Q DP problem: 172.38/44.62 The TRS P consists of the following rules: 172.38/44.62 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U17_gg(x5, z0, .(z4, z5), membercD_in_gg(x5, .(z4, z5)))) 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, membercD_out_gg(z0, .(z0, .(z0, z3)))) -> REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, membercD_out_gg(z0, .(z0, .(z3, z4)))) -> REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, membercD_out_gg(z2, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, membercD_out_gg(z2, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, membercD_out_gg(z5, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, membercD_out_gg(z6, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, x2, .(.(z0, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, x2, .(.(z0, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4)))) -> U6_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4))), member1cB_out_gag(z5, x2, .(.(z5, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5)))) -> U6_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5))), member1cB_out_gag(z6, x2, .(.(z6, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U12_gag(z0, .(z0, .(z0, [])), z2, member1cB_in_gag(z0, z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U12_gag(z0, .(z0, .(z0, [])), z2, member1cB_in_gag(z0, z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U12_gag(z0, z2, z3, member1cB_in_gag(z0, z3))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U12_gag(z0, z2, z3, member1cB_in_gag(z0, z3))) 172.38/44.62 REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), U12_gag(z2, .(z0, .(z2, [])), z3, member1cB_in_gag(z2, z3))) 172.38/44.62 REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), U12_gag(z2, .(z0, .(z2, [])), z3, member1cB_in_gag(z2, z3))) 172.38/44.62 REACHC_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U6_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), U12_gag(z5, z2, z3, member1cB_in_gag(z5, z3))) 172.38/44.62 REACHC_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U6_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), U12_gag(z6, z2, z3, member1cB_in_gag(z6, z3))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z0, z3))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z3, z4))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, membercD_out_gg(z0, .(z0, .(z4, .(z4, z5))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, membercD_out_gg(z0, .(z0, .(z4, .(z5, z6))))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z0, z4))))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z4, z5))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member1cB_out_gag(z0, z0, .(.(z2, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), z0, membercD_out_gg(z0, .(z0, .(z2, .(z2, z4))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member1cB_out_gag(z0, z0, .(.(z2, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), z0, membercD_out_gg(z0, .(z0, .(z2, .(z4, z5))))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, membercD_out_gg(z0, .(z0, .(z4, .(z4, z5))))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, membercD_out_gg(z0, .(z0, .(z4, .(z5, z6))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U17_gg(z0, z0, .(z0, .(z0, z3)), membercD_in_gg(z0, .(z0, .(z0, z3))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U17_gg(z0, z0, .(z0, .(z3, z4)), membercD_in_gg(z0, .(z0, .(z3, z4))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), z2, U17_gg(z2, z0, .(z0, .(z0, z4)), membercD_in_gg(z2, .(z0, .(z0, z4))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), z2, U17_gg(z2, z0, .(z0, .(z4, z5)), membercD_in_gg(z2, .(z0, .(z4, z5))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), z2, U17_gg(z2, z0, .(z4, .(z4, z5)), membercD_in_gg(z2, .(z4, .(z4, z5))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), z2, U17_gg(z2, z0, .(z4, .(z5, z6)), membercD_in_gg(z2, .(z4, .(z5, z6))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, x5, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), x5, U17_gg(x5, z0, .(z0, .(z0, z3)), membercD_in_gg(x5, .(z0, .(z0, z3))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, x5, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), x5, U17_gg(x5, z0, .(z0, .(z3, z4)), membercD_in_gg(x5, .(z0, .(z3, z4))))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x5, U17_gg(x5, z0, .(z0, .(z0, z4)), membercD_in_gg(x5, .(z0, .(z0, z4))))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x5, U17_gg(x5, z0, .(z0, .(z4, z5)), membercD_in_gg(x5, .(z0, .(z4, z5))))) 172.38/44.62 172.38/44.62 The TRS R consists of the following rules: 172.38/44.62 172.38/44.62 member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) 172.38/44.62 member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) 172.38/44.62 U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) 172.38/44.62 membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) 172.38/44.62 membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) 172.38/44.62 U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) 172.38/44.62 172.38/44.62 The set Q consists of the following terms: 172.38/44.62 172.38/44.62 member1cB_in_gag(x0, x1) 172.38/44.62 U12_gag(x0, x1, x2, x3) 172.38/44.62 membercD_in_gg(x0, x1) 172.38/44.62 U17_gg(x0, x1, x2, x3) 172.38/44.62 172.38/44.62 We have to consider all (P,Q,R)-chains. 172.38/44.62 ---------------------------------------- 172.38/44.62 172.38/44.62 (221) TransformationProof (EQUIVALENT) 172.38/44.62 By instantiating [LPAR04] the rule U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U17_gg(x5, z0, .(z4, z5), membercD_in_gg(x5, .(z4, z5)))) we obtained the following new rules [LPAR04]: 172.38/44.62 172.38/44.62 (U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U17_gg(z0, z0, .(z0, .(z0, z3)), membercD_in_gg(z0, .(z0, .(z0, z3))))),U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U17_gg(z0, z0, .(z0, .(z0, z3)), membercD_in_gg(z0, .(z0, .(z0, z3)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U17_gg(z0, z0, .(z0, .(z3, z4)), membercD_in_gg(z0, .(z0, .(z3, z4))))),U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U17_gg(z0, z0, .(z0, .(z3, z4)), membercD_in_gg(z0, .(z0, .(z3, z4)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), z2, U17_gg(z2, z0, .(z0, .(z0, z4)), membercD_in_gg(z2, .(z0, .(z0, z4))))),U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), z2, U17_gg(z2, z0, .(z0, .(z0, z4)), membercD_in_gg(z2, .(z0, .(z0, z4)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), z2, U17_gg(z2, z0, .(z0, .(z4, z5)), membercD_in_gg(z2, .(z0, .(z4, z5))))),U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), z2, U17_gg(z2, z0, .(z0, .(z4, z5)), membercD_in_gg(z2, .(z0, .(z4, z5)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), z2, U17_gg(z2, z0, .(z4, .(z4, z5)), membercD_in_gg(z2, .(z4, .(z4, z5))))),U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), z2, U17_gg(z2, z0, .(z4, .(z4, z5)), membercD_in_gg(z2, .(z4, .(z4, z5)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), z2, U17_gg(z2, z0, .(z4, .(z5, z6)), membercD_in_gg(z2, .(z4, .(z5, z6))))),U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), z2, U17_gg(z2, z0, .(z4, .(z5, z6)), membercD_in_gg(z2, .(z4, .(z5, z6)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, x6, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), x6, U17_gg(x6, z0, .(z0, .(z0, z3)), membercD_in_gg(x6, .(z0, .(z0, z3))))),U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, x5, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), x5, U17_gg(x5, z0, .(z0, .(z0, z3)), membercD_in_gg(x5, .(z0, .(z0, z3)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, x6, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), x6, U17_gg(x6, z0, .(z0, .(z3, z4)), membercD_in_gg(x6, .(z0, .(z3, z4))))),U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, x5, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), x5, U17_gg(x5, z0, .(z0, .(z3, z4)), membercD_in_gg(x5, .(z0, .(z3, z4)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, x6, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x6, U17_gg(x6, z0, .(z0, .(z0, z4)), membercD_in_gg(x6, .(z0, .(z0, z4))))),U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x5, U17_gg(x5, z0, .(z0, .(z0, z4)), membercD_in_gg(x5, .(z0, .(z0, z4)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, x6, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x6, U17_gg(x6, z0, .(z0, .(z4, z5)), membercD_in_gg(x6, .(z0, .(z4, z5))))),U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x5, U17_gg(x5, z0, .(z0, .(z4, z5)), membercD_in_gg(x5, .(z0, .(z4, z5)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member1cB_out_gag(z0, x6, .(.(z2, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), x6, U17_gg(x6, z0, .(z2, .(z2, z4)), membercD_in_gg(x6, .(z2, .(z2, z4))))),U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member1cB_out_gag(z0, x6, .(.(z2, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), x6, U17_gg(x6, z0, .(z2, .(z2, z4)), membercD_in_gg(x6, .(z2, .(z2, z4)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member1cB_out_gag(z0, x6, .(.(z2, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), x6, U17_gg(x6, z0, .(z2, .(z4, z5)), membercD_in_gg(x6, .(z2, .(z4, z5))))),U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member1cB_out_gag(z0, x6, .(.(z2, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), x6, U17_gg(x6, z0, .(z2, .(z4, z5)), membercD_in_gg(x6, .(z2, .(z4, z5)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1cB_out_gag(z0, x6, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), x6, U17_gg(x6, z0, .(z4, .(z4, z5)), membercD_in_gg(x6, .(z4, .(z4, z5))))),U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1cB_out_gag(z0, x6, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), x6, U17_gg(x6, z0, .(z4, .(z4, z5)), membercD_in_gg(x6, .(z4, .(z4, z5)))))) 172.38/44.62 (U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1cB_out_gag(z0, x6, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), x6, U17_gg(x6, z0, .(z4, .(z5, z6)), membercD_in_gg(x6, .(z4, .(z5, z6))))),U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1cB_out_gag(z0, x6, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), x6, U17_gg(x6, z0, .(z4, .(z5, z6)), membercD_in_gg(x6, .(z4, .(z5, z6)))))) 172.38/44.62 172.38/44.62 172.38/44.62 ---------------------------------------- 172.38/44.62 172.38/44.62 (222) 172.38/44.62 Obligation: 172.38/44.62 Q DP problem: 172.38/44.62 The TRS P consists of the following rules: 172.38/44.62 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, membercD_out_gg(z0, .(z0, .(z0, z3)))) -> REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, membercD_out_gg(z0, .(z0, .(z3, z4)))) -> REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, membercD_out_gg(z0, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, membercD_out_gg(z2, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, membercD_out_gg(z2, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, membercD_out_gg(z5, .(z0, .(z0, z4)))) -> REACHC_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) 172.38/44.62 U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, membercD_out_gg(z6, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, x2, .(.(z0, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, x2, .(.(z0, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4)))) -> U6_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4))), member1cB_out_gag(z5, x2, .(.(z5, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5)))) -> U6_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5))), member1cB_out_gag(z6, x2, .(.(z6, .(x2, [])), z3))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U12_gag(z0, .(z0, .(z0, [])), z2, member1cB_in_gag(z0, z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) -> U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U12_gag(z0, .(z0, .(z0, [])), z2, member1cB_in_gag(z0, z2))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) -> U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U12_gag(z0, z2, z3, member1cB_in_gag(z0, z3))) 172.38/44.62 REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) -> U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U12_gag(z0, z2, z3, member1cB_in_gag(z0, z3))) 172.38/44.62 REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) -> U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), U12_gag(z2, .(z0, .(z2, [])), z3, member1cB_in_gag(z2, z3))) 172.38/44.62 REACHC_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) -> U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), U12_gag(z2, .(z0, .(z2, [])), z3, member1cB_in_gag(z2, z3))) 172.38/44.62 REACHC_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) -> U6_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), U12_gag(z5, z2, z3, member1cB_in_gag(z5, z3))) 172.38/44.62 REACHC_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U6_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), U12_gag(z6, z2, z3, member1cB_in_gag(z6, z3))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z0, z3))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z3, z4))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, membercD_out_gg(z0, .(z0, .(z4, .(z4, z5))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, membercD_out_gg(z0, .(z0, .(z4, .(z5, z6))))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z0, z4))))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, membercD_out_gg(z0, .(z0, .(z0, .(z4, z5))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member1cB_out_gag(z0, z0, .(.(z2, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), z0, membercD_out_gg(z0, .(z0, .(z2, .(z2, z4))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member1cB_out_gag(z0, z0, .(.(z2, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), z0, membercD_out_gg(z0, .(z0, .(z2, .(z4, z5))))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, membercD_out_gg(z0, .(z0, .(z4, .(z4, z5))))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1cB_out_gag(z0, z0, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, membercD_out_gg(z0, .(z0, .(z4, .(z5, z6))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U17_gg(z0, z0, .(z0, .(z0, z3)), membercD_in_gg(z0, .(z0, .(z0, z3))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U17_gg(z0, z0, .(z0, .(z3, z4)), membercD_in_gg(z0, .(z0, .(z3, z4))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), z2, U17_gg(z2, z0, .(z0, .(z0, z4)), membercD_in_gg(z2, .(z0, .(z0, z4))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), z2, U17_gg(z2, z0, .(z0, .(z4, z5)), membercD_in_gg(z2, .(z0, .(z4, z5))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), z2, U17_gg(z2, z0, .(z4, .(z4, z5)), membercD_in_gg(z2, .(z4, .(z4, z5))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), member1cB_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) -> U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), z2, U17_gg(z2, z0, .(z4, .(z5, z6)), membercD_in_gg(z2, .(z4, .(z5, z6))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1cB_out_gag(z0, x5, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), x5, U17_gg(x5, z0, .(z0, .(z0, z3)), membercD_in_gg(x5, .(z0, .(z0, z3))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1cB_out_gag(z0, x5, .(.(z0, .(z0, [])), z2))) -> U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), x5, U17_gg(x5, z0, .(z0, .(z3, z4)), membercD_in_gg(x5, .(z0, .(z3, z4))))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x5, U17_gg(x5, z0, .(z0, .(z0, z4)), membercD_in_gg(x5, .(z0, .(z0, z4))))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, x5, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x5, U17_gg(x5, z0, .(z0, .(z4, z5)), membercD_in_gg(x5, .(z0, .(z4, z5))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member1cB_out_gag(z0, x6, .(.(z2, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), x6, U17_gg(x6, z0, .(z2, .(z2, z4)), membercD_in_gg(x6, .(z2, .(z2, z4))))) 172.38/44.62 U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member1cB_out_gag(z0, x6, .(.(z2, .(z0, [])), z3))) -> U8_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), x6, U17_gg(x6, z0, .(z2, .(z4, z5)), membercD_in_gg(x6, .(z2, .(z4, z5))))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1cB_out_gag(z0, x6, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), x6, U17_gg(x6, z0, .(z4, .(z4, z5)), membercD_in_gg(x6, .(z4, .(z4, z5))))) 172.38/44.62 U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1cB_out_gag(z0, x6, .(z2, z3))) -> U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), x6, U17_gg(x6, z0, .(z4, .(z5, z6)), membercD_in_gg(x6, .(z4, .(z5, z6))))) 172.38/44.62 172.38/44.62 The TRS R consists of the following rules: 172.38/44.62 172.38/44.62 member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) 172.38/44.62 member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) 172.38/44.62 U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) 172.38/44.62 membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) 172.38/44.62 membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) 172.38/44.62 U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) 172.38/44.62 172.38/44.62 The set Q consists of the following terms: 172.38/44.62 172.38/44.62 member1cB_in_gag(x0, x1) 172.38/44.62 U12_gag(x0, x1, x2, x3) 172.38/44.62 membercD_in_gg(x0, x1) 172.38/44.62 U17_gg(x0, x1, x2, x3) 172.38/44.62 172.38/44.62 We have to consider all (P,Q,R)-chains. 172.38/44.62 ---------------------------------------- 172.38/44.62 172.38/44.62 (223) PrologToTRSTransformerProof (SOUND) 172.38/44.62 Transformed Prolog program to TRS. 172.38/44.62 172.38/44.62 { 172.38/44.62 "root": 3, 172.38/44.62 "program": { 172.38/44.62 "directives": [], 172.38/44.62 "clauses": [ 172.38/44.62 [ 172.38/44.62 "(reach X Y Edges Visited)", 172.38/44.62 "(member (. X (. Y ([]))) Edges)" 172.38/44.62 ], 172.38/44.62 [ 172.38/44.62 "(reach X Z Edges Visited)", 172.38/44.62 "(',' (member1 (. X (. Y ([]))) Edges) (',' (member Y Visited) (reach Y Z Edges (. Y Visited))))" 172.38/44.62 ], 172.38/44.62 [ 172.38/44.62 "(member H (. H L))", 172.38/44.62 null 172.38/44.62 ], 172.38/44.62 [ 172.38/44.62 "(member X (. H L))", 172.38/44.62 "(member X L)" 172.38/44.62 ], 172.38/44.62 [ 172.38/44.62 "(member1 H (. H L))", 172.38/44.62 null 172.38/44.62 ], 172.38/44.62 [ 172.38/44.62 "(member1 X (. H L))", 172.38/44.62 "(member1 X L)" 172.38/44.62 ] 172.38/44.62 ] 172.38/44.62 }, 172.38/44.62 "graph": { 172.38/44.62 "nodes": { 172.38/44.62 "type": "Nodes", 172.38/44.62 "371": { 172.38/44.62 "goal": [], 172.38/44.62 "kb": { 172.38/44.62 "nonunifying": [], 172.38/44.62 "intvars": {}, 172.38/44.62 "arithmetic": { 172.38/44.62 "type": "PlainIntegerRelationState", 172.38/44.62 "relations": [] 172.38/44.62 }, 172.38/44.62 "ground": [], 172.38/44.62 "free": [], 172.38/44.62 "exprvars": [] 172.38/44.62 } 172.38/44.62 }, 172.38/44.62 "350": { 172.38/44.62 "goal": [{ 172.38/44.62 "clause": -1, 172.38/44.62 "scope": -1, 172.38/44.62 "term": "(member T100 T93)" 172.38/44.62 }], 172.38/44.62 "kb": { 172.38/44.62 "nonunifying": [], 172.38/44.62 "intvars": {}, 172.38/44.62 "arithmetic": { 172.38/44.62 "type": "PlainIntegerRelationState", 172.38/44.62 "relations": [] 172.38/44.62 }, 172.38/44.62 "ground": [ 172.38/44.62 "T93", 172.38/44.62 "T100" 172.38/44.62 ], 172.38/44.62 "free": [], 172.38/44.62 "exprvars": [] 172.38/44.62 } 172.38/44.62 }, 172.38/44.62 "252": { 172.38/44.62 "goal": [{ 172.38/44.62 "clause": -1, 172.38/44.62 "scope": -1, 172.38/44.62 "term": "(true)" 172.38/44.62 }], 172.38/44.62 "kb": { 172.38/44.62 "nonunifying": [], 172.38/44.62 "intvars": {}, 172.38/44.62 "arithmetic": { 172.38/44.62 "type": "PlainIntegerRelationState", 172.38/44.62 "relations": [] 172.38/44.62 }, 172.38/44.62 "ground": [], 172.38/44.62 "free": [], 172.38/44.62 "exprvars": [] 172.38/44.62 } 172.38/44.62 }, 172.38/44.62 "351": { 172.38/44.62 "goal": [{ 172.38/44.62 "clause": -1, 172.38/44.62 "scope": -1, 172.38/44.62 "term": "(reach T100 T91 T92 (. T100 T93))" 172.38/44.62 }], 172.38/44.62 "kb": { 172.38/44.62 "nonunifying": [], 172.38/44.62 "intvars": {}, 172.38/44.62 "arithmetic": { 172.38/44.62 "type": "PlainIntegerRelationState", 172.38/44.62 "relations": [] 172.38/44.62 }, 172.38/44.62 "ground": [ 172.38/44.62 "T91", 172.38/44.62 "T92", 172.38/44.62 "T93", 172.38/44.62 "T100" 172.38/44.62 ], 172.38/44.62 "free": [], 172.38/44.62 "exprvars": [] 172.38/44.62 } 172.38/44.62 }, 172.38/44.62 "253": { 172.38/44.62 "goal": [], 172.38/44.62 "kb": { 172.38/44.62 "nonunifying": [], 172.38/44.62 "intvars": {}, 172.38/44.62 "arithmetic": { 172.38/44.62 "type": "PlainIntegerRelationState", 172.38/44.62 "relations": [] 172.38/44.62 }, 172.38/44.62 "ground": [], 172.38/44.62 "free": [], 172.38/44.62 "exprvars": [] 172.38/44.62 } 172.38/44.62 }, 172.38/44.62 "254": { 172.38/44.62 "goal": [], 172.38/44.62 "kb": { 172.38/44.62 "nonunifying": [], 172.38/44.62 "intvars": {}, 172.38/44.62 "arithmetic": { 172.38/44.62 "type": "PlainIntegerRelationState", 172.38/44.62 "relations": [] 172.38/44.62 }, 172.38/44.62 "ground": [], 172.38/44.62 "free": [], 172.38/44.62 "exprvars": [] 172.38/44.62 } 172.38/44.62 }, 172.38/44.62 "331": { 172.38/44.62 "goal": [ 172.38/44.62 { 172.38/44.62 "clause": 4, 172.38/44.62 "scope": 3, 172.38/44.62 "term": "(member1 (. T90 (. X75 ([]))) T92)" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "clause": 5, 172.38/44.62 "scope": 3, 172.38/44.62 "term": "(member1 (. T90 (. X75 ([]))) T92)" 172.38/44.62 } 172.38/44.62 ], 172.38/44.62 "kb": { 172.38/44.62 "nonunifying": [], 172.38/44.62 "intvars": {}, 172.38/44.62 "arithmetic": { 172.38/44.62 "type": "PlainIntegerRelationState", 172.38/44.62 "relations": [] 172.38/44.62 }, 172.38/44.62 "ground": [ 172.38/44.62 "T90", 172.38/44.62 "T92" 172.38/44.62 ], 172.38/44.62 "free": ["X75"], 172.38/44.62 "exprvars": [] 172.38/44.62 } 172.38/44.62 }, 172.38/44.62 "354": { 172.38/44.62 "goal": [ 172.38/44.62 { 172.38/44.62 "clause": 2, 172.38/44.62 "scope": 4, 172.38/44.62 "term": "(member T100 T93)" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "clause": 3, 172.38/44.62 "scope": 4, 172.38/44.62 "term": "(member T100 T93)" 172.38/44.62 } 172.38/44.62 ], 172.38/44.62 "kb": { 172.38/44.62 "nonunifying": [], 172.38/44.62 "intvars": {}, 172.38/44.62 "arithmetic": { 172.38/44.62 "type": "PlainIntegerRelationState", 172.38/44.62 "relations": [] 172.38/44.62 }, 172.38/44.62 "ground": [ 172.38/44.62 "T93", 172.38/44.62 "T100" 172.38/44.62 ], 172.38/44.62 "free": [], 172.38/44.62 "exprvars": [] 172.38/44.62 } 172.38/44.62 }, 172.38/44.62 "355": { 172.38/44.62 "goal": [{ 172.38/44.62 "clause": 2, 172.38/44.62 "scope": 4, 172.38/44.62 "term": "(member T100 T93)" 172.38/44.62 }], 172.38/44.62 "kb": { 172.38/44.62 "nonunifying": [], 172.38/44.62 "intvars": {}, 172.38/44.62 "arithmetic": { 172.38/44.62 "type": "PlainIntegerRelationState", 172.38/44.62 "relations": [] 172.38/44.62 }, 172.38/44.62 "ground": [ 172.38/44.62 "T93", 172.38/44.62 "T100" 172.38/44.62 ], 172.38/44.62 "free": [], 172.38/44.62 "exprvars": [] 172.38/44.62 } 172.38/44.62 }, 172.38/44.62 "257": { 172.38/44.62 "goal": [{ 172.38/44.62 "clause": -1, 172.38/44.62 "scope": -1, 172.38/44.62 "term": "(member (. T66 (. T67 ([]))) T69)" 172.38/44.62 }], 172.38/44.62 "kb": { 172.38/44.62 "nonunifying": [], 172.38/44.62 "intvars": {}, 172.38/44.62 "arithmetic": { 172.38/44.62 "type": "PlainIntegerRelationState", 172.38/44.62 "relations": [] 172.38/44.62 }, 172.38/44.62 "ground": [ 172.38/44.62 "T66", 172.38/44.62 "T67", 172.38/44.62 "T69" 172.38/44.62 ], 172.38/44.62 "free": [], 172.38/44.62 "exprvars": [] 172.38/44.62 } 172.38/44.62 }, 172.38/44.62 "334": { 172.38/44.62 "goal": [{ 172.38/44.62 "clause": 4, 172.38/44.62 "scope": 3, 172.38/44.62 "term": "(member1 (. T90 (. X75 ([]))) T92)" 172.38/44.62 }], 172.38/44.62 "kb": { 172.38/44.62 "nonunifying": [], 172.38/44.62 "intvars": {}, 172.38/44.62 "arithmetic": { 172.38/44.62 "type": "PlainIntegerRelationState", 172.38/44.62 "relations": [] 172.38/44.62 }, 172.38/44.62 "ground": [ 172.38/44.62 "T90", 172.38/44.62 "T92" 172.38/44.62 ], 172.38/44.62 "free": ["X75"], 172.38/44.62 "exprvars": [] 172.38/44.62 } 172.38/44.62 }, 172.38/44.62 "356": { 172.38/44.62 "goal": [{ 172.38/44.62 "clause": 3, 172.38/44.62 "scope": 4, 172.38/44.62 "term": "(member T100 T93)" 172.38/44.62 }], 172.38/44.62 "kb": { 172.38/44.62 "nonunifying": [], 172.38/44.62 "intvars": {}, 172.38/44.62 "arithmetic": { 172.38/44.62 "type": "PlainIntegerRelationState", 172.38/44.62 "relations": [] 172.38/44.62 }, 172.38/44.62 "ground": [ 172.38/44.62 "T93", 172.38/44.62 "T100" 172.38/44.62 ], 172.38/44.62 "free": [], 172.38/44.62 "exprvars": [] 172.38/44.62 } 172.38/44.62 }, 172.38/44.62 "258": { 172.38/44.62 "goal": [], 172.38/44.62 "kb": { 172.38/44.62 "nonunifying": [], 172.38/44.62 "intvars": {}, 172.38/44.62 "arithmetic": { 172.38/44.62 "type": "PlainIntegerRelationState", 172.38/44.62 "relations": [] 172.38/44.62 }, 172.38/44.62 "ground": [], 172.38/44.62 "free": [], 172.38/44.62 "exprvars": [] 172.38/44.62 } 172.38/44.62 }, 172.38/44.62 "335": { 172.38/44.62 "goal": [{ 172.38/44.62 "clause": 5, 172.38/44.62 "scope": 3, 172.38/44.62 "term": "(member1 (. T90 (. X75 ([]))) T92)" 172.38/44.62 }], 172.38/44.62 "kb": { 172.38/44.62 "nonunifying": [], 172.38/44.62 "intvars": {}, 172.38/44.62 "arithmetic": { 172.38/44.62 "type": "PlainIntegerRelationState", 172.38/44.62 "relations": [] 172.38/44.62 }, 172.38/44.62 "ground": [ 172.38/44.62 "T90", 172.38/44.62 "T92" 172.38/44.62 ], 172.38/44.62 "free": ["X75"], 172.38/44.62 "exprvars": [] 172.38/44.62 } 172.38/44.62 }, 172.38/44.62 "357": { 172.38/44.62 "goal": [{ 172.38/44.62 "clause": -1, 172.38/44.62 "scope": -1, 172.38/44.62 "term": "(true)" 172.38/44.62 }], 172.38/44.62 "kb": { 172.38/44.62 "nonunifying": [], 172.38/44.62 "intvars": {}, 172.38/44.62 "arithmetic": { 172.38/44.62 "type": "PlainIntegerRelationState", 172.38/44.62 "relations": [] 172.38/44.62 }, 172.38/44.62 "ground": [], 172.38/44.62 "free": [], 172.38/44.62 "exprvars": [] 172.38/44.62 } 172.38/44.62 }, 172.38/44.62 "358": { 172.38/44.62 "goal": [], 172.38/44.62 "kb": { 172.38/44.62 "nonunifying": [], 172.38/44.62 "intvars": {}, 172.38/44.62 "arithmetic": { 172.38/44.62 "type": "PlainIntegerRelationState", 172.38/44.62 "relations": [] 172.38/44.62 }, 172.38/44.62 "ground": [], 172.38/44.62 "free": [], 172.38/44.62 "exprvars": [] 172.38/44.62 } 172.38/44.62 }, 172.38/44.62 "337": { 172.38/44.62 "goal": [{ 172.38/44.62 "clause": -1, 172.38/44.62 "scope": -1, 172.38/44.62 "term": "(true)" 172.38/44.62 }], 172.38/44.62 "kb": { 172.38/44.62 "nonunifying": [], 172.38/44.62 "intvars": {}, 172.38/44.62 "arithmetic": { 172.38/44.62 "type": "PlainIntegerRelationState", 172.38/44.62 "relations": [] 172.38/44.62 }, 172.38/44.62 "ground": [], 172.38/44.62 "free": [], 172.38/44.62 "exprvars": [] 172.38/44.62 } 172.38/44.62 }, 172.38/44.62 "359": { 172.38/44.62 "goal": [], 172.38/44.62 "kb": { 172.38/44.62 "nonunifying": [], 172.38/44.62 "intvars": {}, 172.38/44.62 "arithmetic": { 172.38/44.62 "type": "PlainIntegerRelationState", 172.38/44.62 "relations": [] 172.38/44.62 }, 172.38/44.62 "ground": [], 172.38/44.62 "free": [], 172.38/44.62 "exprvars": [] 172.38/44.62 } 172.38/44.62 }, 172.38/44.62 "338": { 172.38/44.62 "goal": [], 172.38/44.62 "kb": { 172.38/44.62 "nonunifying": [], 172.38/44.62 "intvars": {}, 172.38/44.62 "arithmetic": { 172.38/44.62 "type": "PlainIntegerRelationState", 172.38/44.62 "relations": [] 172.38/44.62 }, 172.38/44.62 "ground": [], 172.38/44.62 "free": [], 172.38/44.62 "exprvars": [] 172.38/44.62 } 172.38/44.62 }, 172.38/44.62 "362": { 172.38/44.62 "goal": [{ 172.38/44.62 "clause": -1, 172.38/44.62 "scope": -1, 172.38/44.62 "term": "(member T161 T163)" 172.38/44.62 }], 172.38/44.62 "kb": { 172.38/44.62 "nonunifying": [], 172.38/44.62 "intvars": {}, 172.38/44.62 "arithmetic": { 172.38/44.62 "type": "PlainIntegerRelationState", 172.38/44.62 "relations": [] 172.38/44.62 }, 172.38/44.62 "ground": [ 172.38/44.62 "T161", 172.38/44.62 "T163" 172.38/44.62 ], 172.38/44.62 "free": [], 172.38/44.62 "exprvars": [] 172.38/44.62 } 172.38/44.62 }, 172.38/44.62 "341": { 172.38/44.62 "goal": [], 172.38/44.62 "kb": { 172.38/44.62 "nonunifying": [], 172.38/44.62 "intvars": {}, 172.38/44.62 "arithmetic": { 172.38/44.62 "type": "PlainIntegerRelationState", 172.38/44.62 "relations": [] 172.38/44.62 }, 172.38/44.62 "ground": [], 172.38/44.62 "free": [], 172.38/44.62 "exprvars": [] 172.38/44.62 } 172.38/44.62 }, 172.38/44.62 "243": { 172.38/44.62 "goal": [{ 172.38/44.62 "clause": -1, 172.38/44.62 "scope": -1, 172.38/44.62 "term": "(member (. T33 (. T34 ([]))) T35)" 172.38/44.62 }], 172.38/44.62 "kb": { 172.38/44.62 "nonunifying": [], 172.38/44.62 "intvars": {}, 172.38/44.62 "arithmetic": { 172.38/44.62 "type": "PlainIntegerRelationState", 172.38/44.62 "relations": [] 172.38/44.62 }, 172.38/44.62 "ground": [ 172.38/44.62 "T33", 172.38/44.62 "T34", 172.38/44.62 "T35" 172.38/44.62 ], 172.38/44.62 "free": [], 172.38/44.62 "exprvars": [] 172.38/44.62 } 172.38/44.62 }, 172.38/44.62 "343": { 172.38/44.62 "goal": [{ 172.38/44.62 "clause": -1, 172.38/44.62 "scope": -1, 172.38/44.62 "term": "(member1 (. T128 (. X114 ([]))) T130)" 172.38/44.62 }], 172.38/44.62 "kb": { 172.38/44.62 "nonunifying": [], 172.38/44.62 "intvars": {}, 172.38/44.62 "arithmetic": { 172.38/44.62 "type": "PlainIntegerRelationState", 172.38/44.62 "relations": [] 172.38/44.62 }, 172.38/44.62 "ground": [ 172.38/44.62 "T128", 172.38/44.62 "T130" 172.38/44.62 ], 172.38/44.62 "free": ["X114"], 172.38/44.62 "exprvars": [] 172.38/44.62 } 172.38/44.62 }, 172.38/44.62 "3": { 172.38/44.62 "goal": [{ 172.38/44.62 "clause": -1, 172.38/44.62 "scope": -1, 172.38/44.62 "term": "(reach T1 T2 T3 T4)" 172.38/44.62 }], 172.38/44.62 "kb": { 172.38/44.62 "nonunifying": [], 172.38/44.62 "intvars": {}, 172.38/44.62 "arithmetic": { 172.38/44.62 "type": "PlainIntegerRelationState", 172.38/44.62 "relations": [] 172.38/44.62 }, 172.38/44.62 "ground": [ 172.38/44.62 "T4", 172.38/44.62 "T1", 172.38/44.62 "T2", 172.38/44.62 "T3" 172.38/44.62 ], 172.38/44.62 "free": [], 172.38/44.62 "exprvars": [] 172.38/44.62 } 172.38/44.62 }, 172.38/44.62 "124": { 172.38/44.62 "goal": [ 172.38/44.62 { 172.38/44.62 "clause": 0, 172.38/44.62 "scope": 1, 172.38/44.62 "term": "(reach T1 T2 T3 T4)" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "clause": 1, 172.38/44.62 "scope": 1, 172.38/44.62 "term": "(reach T1 T2 T3 T4)" 172.38/44.62 } 172.38/44.62 ], 172.38/44.62 "kb": { 172.38/44.62 "nonunifying": [], 172.38/44.62 "intvars": {}, 172.38/44.62 "arithmetic": { 172.38/44.62 "type": "PlainIntegerRelationState", 172.38/44.62 "relations": [] 172.38/44.62 }, 172.38/44.62 "ground": [ 172.38/44.62 "T4", 172.38/44.62 "T1", 172.38/44.62 "T2", 172.38/44.62 "T3" 172.38/44.62 ], 172.38/44.62 "free": [], 172.38/44.62 "exprvars": [] 172.38/44.62 } 172.38/44.62 }, 172.38/44.62 "344": { 172.38/44.62 "goal": [], 172.38/44.62 "kb": { 172.38/44.62 "nonunifying": [], 172.38/44.62 "intvars": {}, 172.38/44.62 "arithmetic": { 172.38/44.62 "type": "PlainIntegerRelationState", 172.38/44.62 "relations": [] 172.38/44.62 }, 172.38/44.62 "ground": [], 172.38/44.62 "free": [], 172.38/44.62 "exprvars": [] 172.38/44.62 } 172.38/44.62 }, 172.38/44.62 "246": { 172.38/44.62 "goal": [ 172.38/44.62 { 172.38/44.62 "clause": 2, 172.38/44.62 "scope": 2, 172.38/44.62 "term": "(member (. T33 (. T34 ([]))) T35)" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "clause": 3, 172.38/44.62 "scope": 2, 172.38/44.62 "term": "(member (. T33 (. T34 ([]))) T35)" 172.38/44.62 } 172.38/44.62 ], 172.38/44.62 "kb": { 172.38/44.62 "nonunifying": [], 172.38/44.62 "intvars": {}, 172.38/44.62 "arithmetic": { 172.38/44.62 "type": "PlainIntegerRelationState", 172.38/44.62 "relations": [] 172.38/44.62 }, 172.38/44.62 "ground": [ 172.38/44.62 "T33", 172.38/44.62 "T34", 172.38/44.62 "T35" 172.38/44.62 ], 172.38/44.62 "free": [], 172.38/44.62 "exprvars": [] 172.38/44.62 } 172.38/44.62 }, 172.38/44.62 "323": { 172.38/44.62 "goal": [{ 172.38/44.62 "clause": -1, 172.38/44.62 "scope": -1, 172.38/44.62 "term": "(',' (member1 (. T90 (. X75 ([]))) T92) (',' (member X75 T93) (reach X75 T91 T92 (. X75 T93))))" 172.38/44.62 }], 172.38/44.62 "kb": { 172.38/44.62 "nonunifying": [], 172.38/44.62 "intvars": {}, 172.38/44.62 "arithmetic": { 172.38/44.62 "type": "PlainIntegerRelationState", 172.38/44.62 "relations": [] 172.38/44.62 }, 172.38/44.62 "ground": [ 172.38/44.62 "T90", 172.38/44.62 "T91", 172.38/44.62 "T92", 172.38/44.62 "T93" 172.38/44.62 ], 172.38/44.62 "free": ["X75"], 172.38/44.62 "exprvars": [] 172.38/44.62 } 172.38/44.62 }, 172.38/44.62 "247": { 172.38/44.62 "goal": [{ 172.38/44.62 "clause": 2, 172.38/44.62 "scope": 2, 172.38/44.62 "term": "(member (. T33 (. T34 ([]))) T35)" 172.38/44.62 }], 172.38/44.62 "kb": { 172.38/44.62 "nonunifying": [], 172.38/44.62 "intvars": {}, 172.38/44.62 "arithmetic": { 172.38/44.62 "type": "PlainIntegerRelationState", 172.38/44.62 "relations": [] 172.38/44.62 }, 172.38/44.62 "ground": [ 172.38/44.62 "T33", 172.38/44.62 "T34", 172.38/44.62 "T35" 172.38/44.62 ], 172.38/44.62 "free": [], 172.38/44.62 "exprvars": [] 172.38/44.62 } 172.38/44.62 }, 172.38/44.62 "127": { 172.38/44.62 "goal": [{ 172.38/44.62 "clause": 0, 172.38/44.62 "scope": 1, 172.38/44.62 "term": "(reach T1 T2 T3 T4)" 172.38/44.62 }], 172.38/44.62 "kb": { 172.38/44.62 "nonunifying": [], 172.38/44.62 "intvars": {}, 172.38/44.62 "arithmetic": { 172.38/44.62 "type": "PlainIntegerRelationState", 172.38/44.62 "relations": [] 172.38/44.62 }, 172.38/44.62 "ground": [ 172.38/44.62 "T4", 172.38/44.62 "T1", 172.38/44.62 "T2", 172.38/44.62 "T3" 172.38/44.62 ], 172.38/44.62 "free": [], 172.38/44.62 "exprvars": [] 172.38/44.62 } 172.38/44.62 }, 172.38/44.62 "248": { 172.38/44.62 "goal": [{ 172.38/44.62 "clause": 3, 172.38/44.62 "scope": 2, 172.38/44.62 "term": "(member (. T33 (. T34 ([]))) T35)" 172.38/44.62 }], 172.38/44.62 "kb": { 172.38/44.62 "nonunifying": [], 172.38/44.62 "intvars": {}, 172.38/44.62 "arithmetic": { 172.38/44.62 "type": "PlainIntegerRelationState", 172.38/44.62 "relations": [] 172.38/44.62 }, 172.38/44.62 "ground": [ 172.38/44.62 "T33", 172.38/44.62 "T34", 172.38/44.62 "T35" 172.38/44.62 ], 172.38/44.62 "free": [], 172.38/44.62 "exprvars": [] 172.38/44.62 } 172.38/44.62 }, 172.38/44.62 "128": { 172.38/44.62 "goal": [{ 172.38/44.62 "clause": 1, 172.38/44.62 "scope": 1, 172.38/44.62 "term": "(reach T1 T2 T3 T4)" 172.38/44.62 }], 172.38/44.62 "kb": { 172.38/44.62 "nonunifying": [], 172.38/44.62 "intvars": {}, 172.38/44.62 "arithmetic": { 172.38/44.62 "type": "PlainIntegerRelationState", 172.38/44.62 "relations": [] 172.38/44.62 }, 172.38/44.62 "ground": [ 172.38/44.62 "T4", 172.38/44.62 "T1", 172.38/44.62 "T2", 172.38/44.62 "T3" 172.38/44.62 ], 172.38/44.62 "free": [], 172.38/44.62 "exprvars": [] 172.38/44.62 } 172.38/44.62 }, 172.38/44.62 "327": { 172.38/44.62 "goal": [{ 172.38/44.62 "clause": -1, 172.38/44.62 "scope": -1, 172.38/44.62 "term": "(member1 (. T90 (. X75 ([]))) T92)" 172.38/44.62 }], 172.38/44.62 "kb": { 172.38/44.62 "nonunifying": [], 172.38/44.62 "intvars": {}, 172.38/44.62 "arithmetic": { 172.38/44.62 "type": "PlainIntegerRelationState", 172.38/44.62 "relations": [] 172.38/44.62 }, 172.38/44.62 "ground": [ 172.38/44.62 "T90", 172.38/44.62 "T92" 172.38/44.62 ], 172.38/44.62 "free": ["X75"], 172.38/44.62 "exprvars": [] 172.38/44.62 } 172.38/44.62 }, 172.38/44.62 "328": { 172.38/44.62 "goal": [{ 172.38/44.62 "clause": -1, 172.38/44.62 "scope": -1, 172.38/44.62 "term": "(',' (member T100 T93) (reach T100 T91 T92 (. T100 T93)))" 172.38/44.62 }], 172.38/44.62 "kb": { 172.38/44.62 "nonunifying": [], 172.38/44.62 "intvars": {}, 172.38/44.62 "arithmetic": { 172.38/44.62 "type": "PlainIntegerRelationState", 172.38/44.62 "relations": [] 172.38/44.62 }, 172.38/44.62 "ground": [ 172.38/44.62 "T91", 172.38/44.62 "T92", 172.38/44.62 "T93", 172.38/44.62 "T100" 172.38/44.62 ], 172.38/44.62 "free": [], 172.38/44.62 "exprvars": [] 172.38/44.62 } 172.38/44.62 } 172.38/44.62 }, 172.38/44.62 "edges": [ 172.38/44.62 { 172.38/44.62 "from": 3, 172.38/44.62 "to": 124, 172.38/44.62 "label": "CASE" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "from": 124, 172.38/44.62 "to": 127, 172.38/44.62 "label": "PARALLEL" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "from": 124, 172.38/44.62 "to": 128, 172.38/44.62 "label": "PARALLEL" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "from": 127, 172.38/44.62 "to": 243, 172.38/44.62 "label": "ONLY EVAL with clause\nreach(X25, X26, X27, X28) :- member(.(X25, .(X26, [])), X27).\nand substitutionT1 -> T33,\nX25 -> T33,\nT2 -> T34,\nX26 -> T34,\nT3 -> T35,\nX27 -> T35,\nT4 -> T36,\nX28 -> T36" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "from": 128, 172.38/44.62 "to": 323, 172.38/44.62 "label": "ONLY EVAL with clause\nreach(X71, X72, X73, X74) :- ','(member1(.(X71, .(X75, [])), X73), ','(member(X75, X74), reach(X75, X72, X73, .(X75, X74)))).\nand substitutionT1 -> T90,\nX71 -> T90,\nT2 -> T91,\nX72 -> T91,\nT3 -> T92,\nX73 -> T92,\nT4 -> T93,\nX74 -> T93" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "from": 243, 172.38/44.62 "to": 246, 172.38/44.62 "label": "CASE" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "from": 246, 172.38/44.62 "to": 247, 172.38/44.62 "label": "PARALLEL" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "from": 246, 172.38/44.62 "to": 248, 172.38/44.62 "label": "PARALLEL" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "from": 247, 172.38/44.62 "to": 252, 172.38/44.62 "label": "EVAL with clause\nmember(X41, .(X41, X42)).\nand substitutionT33 -> T55,\nT34 -> T56,\nX41 -> .(T55, .(T56, [])),\nX42 -> T57,\nT35 -> .(.(T55, .(T56, [])), T57)" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "from": 247, 172.38/44.62 "to": 253, 172.38/44.62 "label": "EVAL-BACKTRACK" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "from": 248, 172.38/44.62 "to": 257, 172.38/44.62 "label": "EVAL with clause\nmember(X49, .(X50, X51)) :- member(X49, X51).\nand substitutionT33 -> T66,\nT34 -> T67,\nX49 -> .(T66, .(T67, [])),\nX50 -> T68,\nX51 -> T69,\nT35 -> .(T68, T69)" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "from": 248, 172.38/44.62 "to": 258, 172.38/44.62 "label": "EVAL-BACKTRACK" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "from": 252, 172.38/44.62 "to": 254, 172.38/44.62 "label": "SUCCESS" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "from": 257, 172.38/44.62 "to": 243, 172.38/44.62 "label": "INSTANCE with matching:\nT33 -> T66\nT34 -> T67\nT35 -> T69" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "from": 323, 172.38/44.62 "to": 327, 172.38/44.62 "label": "SPLIT 1" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "from": 323, 172.38/44.62 "to": 328, 172.38/44.62 "label": "SPLIT 2\nnew knowledge:\nT90 is ground\nT100 is ground\nT92 is ground\nreplacements:X75 -> T100" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "from": 327, 172.38/44.62 "to": 331, 172.38/44.62 "label": "CASE" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "from": 328, 172.38/44.62 "to": 350, 172.38/44.62 "label": "SPLIT 1" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "from": 328, 172.38/44.62 "to": 351, 172.38/44.62 "label": "SPLIT 2\nnew knowledge:\nT100 is ground\nT93 is ground" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "from": 331, 172.38/44.62 "to": 334, 172.38/44.62 "label": "PARALLEL" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "from": 331, 172.38/44.62 "to": 335, 172.38/44.62 "label": "PARALLEL" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "from": 334, 172.38/44.62 "to": 337, 172.38/44.62 "label": "EVAL with clause\nmember1(X100, .(X100, X101)).\nand substitutionT90 -> T119,\nX75 -> T120,\nX100 -> .(T119, .(T120, [])),\nX102 -> T120,\nX101 -> T121,\nT92 -> .(.(T119, .(T120, [])), T121)" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "from": 334, 172.38/44.62 "to": 338, 172.38/44.62 "label": "EVAL-BACKTRACK" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "from": 335, 172.38/44.62 "to": 343, 172.38/44.62 "label": "EVAL with clause\nmember1(X111, .(X112, X113)) :- member1(X111, X113).\nand substitutionT90 -> T128,\nX75 -> X114,\nX111 -> .(T128, .(X114, [])),\nX112 -> T129,\nX113 -> T130,\nT92 -> .(T129, T130)" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "from": 335, 172.38/44.62 "to": 344, 172.38/44.62 "label": "EVAL-BACKTRACK" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "from": 337, 172.38/44.62 "to": 341, 172.38/44.62 "label": "SUCCESS" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "from": 343, 172.38/44.62 "to": 327, 172.38/44.62 "label": "INSTANCE with matching:\nT90 -> T128\nX75 -> X114\nT92 -> T130" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "from": 350, 172.38/44.62 "to": 354, 172.38/44.62 "label": "CASE" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "from": 351, 172.38/44.62 "to": 3, 172.38/44.62 "label": "INSTANCE with matching:\nT1 -> T100\nT2 -> T91\nT3 -> T92\nT4 -> .(T100, T93)" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "from": 354, 172.38/44.62 "to": 355, 172.38/44.62 "label": "PARALLEL" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "from": 354, 172.38/44.62 "to": 356, 172.38/44.62 "label": "PARALLEL" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "from": 355, 172.38/44.62 "to": 357, 172.38/44.62 "label": "EVAL with clause\nmember(X137, .(X137, X138)).\nand substitutionT100 -> T153,\nX137 -> T153,\nX138 -> T154,\nT93 -> .(T153, T154)" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "from": 355, 172.38/44.62 "to": 358, 172.38/44.62 "label": "EVAL-BACKTRACK" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "from": 356, 172.38/44.62 "to": 362, 172.38/44.62 "label": "EVAL with clause\nmember(X145, .(X146, X147)) :- member(X145, X147).\nand substitutionT100 -> T161,\nX145 -> T161,\nX146 -> T162,\nX147 -> T163,\nT93 -> .(T162, T163)" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "from": 356, 172.38/44.62 "to": 371, 172.38/44.62 "label": "EVAL-BACKTRACK" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "from": 357, 172.38/44.62 "to": 359, 172.38/44.62 "label": "SUCCESS" 172.38/44.62 }, 172.38/44.62 { 172.38/44.62 "from": 362, 172.38/44.62 "to": 350, 172.38/44.62 "label": "INSTANCE with matching:\nT100 -> T161\nT93 -> T163" 172.38/44.62 } 172.38/44.62 ], 172.38/44.62 "type": "Graph" 172.38/44.62 } 172.38/44.62 } 172.38/44.62 172.38/44.62 ---------------------------------------- 172.38/44.62 172.38/44.62 (224) 172.38/44.62 Obligation: 172.38/44.62 Q restricted rewrite system: 172.38/44.62 The TRS R consists of the following rules: 172.38/44.62 172.38/44.62 f3_in(T33, T34, T35, T36) -> U1(f243_in(T33, T34, T35), T33, T34, T35, T36) 172.38/44.62 U1(f243_out1, T33, T34, T35, T36) -> f3_out1 172.38/44.62 f3_in(T90, T91, T92, T93) -> U2(f323_in(T90, T92, T93, T91), T90, T91, T92, T93) 172.38/44.62 U2(f323_out1(X75), T90, T91, T92, T93) -> f3_out1 172.38/44.62 f243_in(T55, T56, .(.(T55, .(T56, [])), T57)) -> f243_out1 172.38/44.62 f243_in(T66, T67, .(T68, T69)) -> U3(f243_in(T66, T67, T69), T66, T67, .(T68, T69)) 172.38/44.62 U3(f243_out1, T66, T67, .(T68, T69)) -> f243_out1 172.38/44.62 f327_in(T119, .(.(T119, .(T120, [])), T121)) -> f327_out1(T120) 172.38/44.62 f327_in(T128, .(T129, T130)) -> U4(f327_in(T128, T130), T128, .(T129, T130)) 172.38/44.62 U4(f327_out1(X114), T128, .(T129, T130)) -> f327_out1(X114) 172.38/44.62 f350_in(T153, .(T153, T154)) -> f350_out1 172.38/44.62 f350_in(T161, .(T162, T163)) -> U5(f350_in(T161, T163), T161, .(T162, T163)) 172.38/44.62 U5(f350_out1, T161, .(T162, T163)) -> f350_out1 172.38/44.62 f323_in(T90, T92, T93, T91) -> U6(f327_in(T90, T92), T90, T92, T93, T91) 172.38/44.62 U6(f327_out1(T100), T90, T92, T93, T91) -> U7(f328_in(T100, T93, T91, T92), T90, T92, T93, T91, T100) 172.38/44.62 U7(f328_out1, T90, T92, T93, T91, T100) -> f323_out1(T100) 172.38/44.62 f328_in(T100, T93, T91, T92) -> U8(f350_in(T100, T93), T100, T93, T91, T92) 172.38/44.62 U8(f350_out1, T100, T93, T91, T92) -> U9(f3_in(T100, T91, T92, .(T100, T93)), T100, T93, T91, T92) 172.38/44.62 U9(f3_out1, T100, T93, T91, T92) -> f328_out1 172.38/44.62 172.38/44.62 Q is empty. 172.38/44.62 172.38/44.62 ---------------------------------------- 172.38/44.62 172.38/44.62 (225) DependencyPairsProof (EQUIVALENT) 172.38/44.62 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 172.38/44.62 ---------------------------------------- 172.38/44.62 172.38/44.62 (226) 172.38/44.62 Obligation: 172.38/44.62 Q DP problem: 172.38/44.62 The TRS P consists of the following rules: 172.38/44.62 172.38/44.62 F3_IN(T33, T34, T35, T36) -> U1^1(f243_in(T33, T34, T35), T33, T34, T35, T36) 172.38/44.62 F3_IN(T33, T34, T35, T36) -> F243_IN(T33, T34, T35) 172.38/44.62 F3_IN(T90, T91, T92, T93) -> U2^1(f323_in(T90, T92, T93, T91), T90, T91, T92, T93) 172.38/44.62 F3_IN(T90, T91, T92, T93) -> F323_IN(T90, T92, T93, T91) 172.38/44.62 F243_IN(T66, T67, .(T68, T69)) -> U3^1(f243_in(T66, T67, T69), T66, T67, .(T68, T69)) 172.38/44.62 F243_IN(T66, T67, .(T68, T69)) -> F243_IN(T66, T67, T69) 172.38/44.62 F327_IN(T128, .(T129, T130)) -> U4^1(f327_in(T128, T130), T128, .(T129, T130)) 172.38/44.62 F327_IN(T128, .(T129, T130)) -> F327_IN(T128, T130) 172.38/44.62 F350_IN(T161, .(T162, T163)) -> U5^1(f350_in(T161, T163), T161, .(T162, T163)) 172.38/44.62 F350_IN(T161, .(T162, T163)) -> F350_IN(T161, T163) 172.38/44.62 F323_IN(T90, T92, T93, T91) -> U6^1(f327_in(T90, T92), T90, T92, T93, T91) 172.38/44.62 F323_IN(T90, T92, T93, T91) -> F327_IN(T90, T92) 172.38/44.62 U6^1(f327_out1(T100), T90, T92, T93, T91) -> U7^1(f328_in(T100, T93, T91, T92), T90, T92, T93, T91, T100) 172.38/44.62 U6^1(f327_out1(T100), T90, T92, T93, T91) -> F328_IN(T100, T93, T91, T92) 172.38/44.62 F328_IN(T100, T93, T91, T92) -> U8^1(f350_in(T100, T93), T100, T93, T91, T92) 172.38/44.62 F328_IN(T100, T93, T91, T92) -> F350_IN(T100, T93) 172.38/44.62 U8^1(f350_out1, T100, T93, T91, T92) -> U9^1(f3_in(T100, T91, T92, .(T100, T93)), T100, T93, T91, T92) 172.38/44.62 U8^1(f350_out1, T100, T93, T91, T92) -> F3_IN(T100, T91, T92, .(T100, T93)) 172.38/44.62 172.38/44.62 The TRS R consists of the following rules: 172.38/44.62 172.38/44.62 f3_in(T33, T34, T35, T36) -> U1(f243_in(T33, T34, T35), T33, T34, T35, T36) 172.38/44.62 U1(f243_out1, T33, T34, T35, T36) -> f3_out1 172.38/44.62 f3_in(T90, T91, T92, T93) -> U2(f323_in(T90, T92, T93, T91), T90, T91, T92, T93) 172.38/44.62 U2(f323_out1(X75), T90, T91, T92, T93) -> f3_out1 172.38/44.62 f243_in(T55, T56, .(.(T55, .(T56, [])), T57)) -> f243_out1 172.38/44.62 f243_in(T66, T67, .(T68, T69)) -> U3(f243_in(T66, T67, T69), T66, T67, .(T68, T69)) 172.38/44.62 U3(f243_out1, T66, T67, .(T68, T69)) -> f243_out1 172.38/44.62 f327_in(T119, .(.(T119, .(T120, [])), T121)) -> f327_out1(T120) 172.38/44.62 f327_in(T128, .(T129, T130)) -> U4(f327_in(T128, T130), T128, .(T129, T130)) 172.38/44.62 U4(f327_out1(X114), T128, .(T129, T130)) -> f327_out1(X114) 172.38/44.62 f350_in(T153, .(T153, T154)) -> f350_out1 172.38/44.62 f350_in(T161, .(T162, T163)) -> U5(f350_in(T161, T163), T161, .(T162, T163)) 172.38/44.62 U5(f350_out1, T161, .(T162, T163)) -> f350_out1 172.38/44.62 f323_in(T90, T92, T93, T91) -> U6(f327_in(T90, T92), T90, T92, T93, T91) 172.38/44.62 U6(f327_out1(T100), T90, T92, T93, T91) -> U7(f328_in(T100, T93, T91, T92), T90, T92, T93, T91, T100) 172.38/44.62 U7(f328_out1, T90, T92, T93, T91, T100) -> f323_out1(T100) 172.38/44.62 f328_in(T100, T93, T91, T92) -> U8(f350_in(T100, T93), T100, T93, T91, T92) 172.38/44.62 U8(f350_out1, T100, T93, T91, T92) -> U9(f3_in(T100, T91, T92, .(T100, T93)), T100, T93, T91, T92) 172.38/44.62 U9(f3_out1, T100, T93, T91, T92) -> f328_out1 172.38/44.62 172.38/44.62 Q is empty. 172.38/44.62 We have to consider all minimal (P,Q,R)-chains. 172.38/44.62 ---------------------------------------- 172.38/44.62 172.38/44.62 (227) DependencyGraphProof (EQUIVALENT) 172.38/44.62 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 10 less nodes. 172.38/44.62 ---------------------------------------- 172.38/44.62 172.38/44.62 (228) 172.38/44.62 Complex Obligation (AND) 172.38/44.62 172.38/44.62 ---------------------------------------- 172.38/44.62 172.38/44.62 (229) 172.38/44.62 Obligation: 172.38/44.62 Q DP problem: 172.38/44.62 The TRS P consists of the following rules: 172.38/44.62 172.38/44.62 F350_IN(T161, .(T162, T163)) -> F350_IN(T161, T163) 172.38/44.62 172.38/44.62 The TRS R consists of the following rules: 172.38/44.62 172.38/44.62 f3_in(T33, T34, T35, T36) -> U1(f243_in(T33, T34, T35), T33, T34, T35, T36) 172.38/44.62 U1(f243_out1, T33, T34, T35, T36) -> f3_out1 172.38/44.62 f3_in(T90, T91, T92, T93) -> U2(f323_in(T90, T92, T93, T91), T90, T91, T92, T93) 172.38/44.62 U2(f323_out1(X75), T90, T91, T92, T93) -> f3_out1 172.38/44.62 f243_in(T55, T56, .(.(T55, .(T56, [])), T57)) -> f243_out1 172.38/44.62 f243_in(T66, T67, .(T68, T69)) -> U3(f243_in(T66, T67, T69), T66, T67, .(T68, T69)) 172.38/44.62 U3(f243_out1, T66, T67, .(T68, T69)) -> f243_out1 172.38/44.62 f327_in(T119, .(.(T119, .(T120, [])), T121)) -> f327_out1(T120) 172.38/44.62 f327_in(T128, .(T129, T130)) -> U4(f327_in(T128, T130), T128, .(T129, T130)) 172.38/44.62 U4(f327_out1(X114), T128, .(T129, T130)) -> f327_out1(X114) 172.38/44.62 f350_in(T153, .(T153, T154)) -> f350_out1 172.38/44.62 f350_in(T161, .(T162, T163)) -> U5(f350_in(T161, T163), T161, .(T162, T163)) 172.38/44.62 U5(f350_out1, T161, .(T162, T163)) -> f350_out1 172.38/44.62 f323_in(T90, T92, T93, T91) -> U6(f327_in(T90, T92), T90, T92, T93, T91) 172.38/44.62 U6(f327_out1(T100), T90, T92, T93, T91) -> U7(f328_in(T100, T93, T91, T92), T90, T92, T93, T91, T100) 172.38/44.62 U7(f328_out1, T90, T92, T93, T91, T100) -> f323_out1(T100) 172.38/44.62 f328_in(T100, T93, T91, T92) -> U8(f350_in(T100, T93), T100, T93, T91, T92) 172.38/44.62 U8(f350_out1, T100, T93, T91, T92) -> U9(f3_in(T100, T91, T92, .(T100, T93)), T100, T93, T91, T92) 172.38/44.62 U9(f3_out1, T100, T93, T91, T92) -> f328_out1 172.38/44.62 172.38/44.62 Q is empty. 172.38/44.62 We have to consider all minimal (P,Q,R)-chains. 172.38/44.62 ---------------------------------------- 172.38/44.62 172.38/44.62 (230) UsableRulesProof (EQUIVALENT) 172.38/44.62 We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. 172.38/44.62 ---------------------------------------- 172.38/44.62 172.38/44.62 (231) 172.38/44.62 Obligation: 172.38/44.62 Q DP problem: 172.38/44.62 The TRS P consists of the following rules: 172.38/44.62 172.38/44.62 F350_IN(T161, .(T162, T163)) -> F350_IN(T161, T163) 172.38/44.62 172.38/44.62 R is empty. 172.38/44.62 Q is empty. 172.38/44.62 We have to consider all minimal (P,Q,R)-chains. 172.38/44.62 ---------------------------------------- 172.38/44.62 172.38/44.62 (232) QDPSizeChangeProof (EQUIVALENT) 172.38/44.62 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 172.38/44.62 172.38/44.62 From the DPs we obtained the following set of size-change graphs: 172.38/44.62 *F350_IN(T161, .(T162, T163)) -> F350_IN(T161, T163) 172.38/44.62 The graph contains the following edges 1 >= 1, 2 > 2 172.38/44.62 172.38/44.62 172.38/44.62 ---------------------------------------- 172.38/44.62 172.38/44.62 (233) 172.38/44.62 YES 172.38/44.62 172.38/44.62 ---------------------------------------- 172.38/44.62 172.38/44.62 (234) 172.38/44.62 Obligation: 172.38/44.62 Q DP problem: 172.38/44.62 The TRS P consists of the following rules: 172.38/44.62 172.38/44.62 F327_IN(T128, .(T129, T130)) -> F327_IN(T128, T130) 172.38/44.62 172.38/44.62 The TRS R consists of the following rules: 172.38/44.62 172.38/44.62 f3_in(T33, T34, T35, T36) -> U1(f243_in(T33, T34, T35), T33, T34, T35, T36) 172.38/44.62 U1(f243_out1, T33, T34, T35, T36) -> f3_out1 172.38/44.62 f3_in(T90, T91, T92, T93) -> U2(f323_in(T90, T92, T93, T91), T90, T91, T92, T93) 172.38/44.62 U2(f323_out1(X75), T90, T91, T92, T93) -> f3_out1 172.38/44.62 f243_in(T55, T56, .(.(T55, .(T56, [])), T57)) -> f243_out1 172.38/44.62 f243_in(T66, T67, .(T68, T69)) -> U3(f243_in(T66, T67, T69), T66, T67, .(T68, T69)) 172.38/44.62 U3(f243_out1, T66, T67, .(T68, T69)) -> f243_out1 172.38/44.62 f327_in(T119, .(.(T119, .(T120, [])), T121)) -> f327_out1(T120) 172.38/44.62 f327_in(T128, .(T129, T130)) -> U4(f327_in(T128, T130), T128, .(T129, T130)) 172.38/44.62 U4(f327_out1(X114), T128, .(T129, T130)) -> f327_out1(X114) 172.38/44.62 f350_in(T153, .(T153, T154)) -> f350_out1 172.38/44.62 f350_in(T161, .(T162, T163)) -> U5(f350_in(T161, T163), T161, .(T162, T163)) 172.38/44.62 U5(f350_out1, T161, .(T162, T163)) -> f350_out1 172.38/44.62 f323_in(T90, T92, T93, T91) -> U6(f327_in(T90, T92), T90, T92, T93, T91) 172.38/44.62 U6(f327_out1(T100), T90, T92, T93, T91) -> U7(f328_in(T100, T93, T91, T92), T90, T92, T93, T91, T100) 172.38/44.62 U7(f328_out1, T90, T92, T93, T91, T100) -> f323_out1(T100) 172.38/44.62 f328_in(T100, T93, T91, T92) -> U8(f350_in(T100, T93), T100, T93, T91, T92) 172.38/44.62 U8(f350_out1, T100, T93, T91, T92) -> U9(f3_in(T100, T91, T92, .(T100, T93)), T100, T93, T91, T92) 172.38/44.62 U9(f3_out1, T100, T93, T91, T92) -> f328_out1 172.38/44.62 172.38/44.62 Q is empty. 172.38/44.62 We have to consider all minimal (P,Q,R)-chains. 172.38/44.62 ---------------------------------------- 172.38/44.62 172.38/44.62 (235) UsableRulesProof (EQUIVALENT) 172.38/44.62 We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. 172.38/44.62 ---------------------------------------- 172.38/44.62 172.38/44.62 (236) 172.38/44.62 Obligation: 172.38/44.62 Q DP problem: 172.38/44.62 The TRS P consists of the following rules: 172.38/44.62 172.38/44.62 F327_IN(T128, .(T129, T130)) -> F327_IN(T128, T130) 172.38/44.62 172.38/44.62 R is empty. 172.38/44.62 Q is empty. 172.38/44.62 We have to consider all minimal (P,Q,R)-chains. 172.38/44.62 ---------------------------------------- 172.38/44.62 172.38/44.62 (237) QDPSizeChangeProof (EQUIVALENT) 172.38/44.62 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 172.38/44.62 172.38/44.62 From the DPs we obtained the following set of size-change graphs: 172.38/44.62 *F327_IN(T128, .(T129, T130)) -> F327_IN(T128, T130) 172.38/44.62 The graph contains the following edges 1 >= 1, 2 > 2 172.38/44.62 172.38/44.62 172.38/44.62 ---------------------------------------- 172.38/44.62 172.38/44.62 (238) 172.38/44.62 YES 172.38/44.62 172.38/44.62 ---------------------------------------- 172.38/44.62 172.38/44.62 (239) 172.38/44.62 Obligation: 172.38/44.62 Q DP problem: 172.38/44.62 The TRS P consists of the following rules: 172.38/44.62 172.38/44.62 F243_IN(T66, T67, .(T68, T69)) -> F243_IN(T66, T67, T69) 172.38/44.62 172.38/44.62 The TRS R consists of the following rules: 172.38/44.62 172.38/44.62 f3_in(T33, T34, T35, T36) -> U1(f243_in(T33, T34, T35), T33, T34, T35, T36) 172.38/44.62 U1(f243_out1, T33, T34, T35, T36) -> f3_out1 172.38/44.62 f3_in(T90, T91, T92, T93) -> U2(f323_in(T90, T92, T93, T91), T90, T91, T92, T93) 172.38/44.62 U2(f323_out1(X75), T90, T91, T92, T93) -> f3_out1 172.38/44.62 f243_in(T55, T56, .(.(T55, .(T56, [])), T57)) -> f243_out1 172.38/44.62 f243_in(T66, T67, .(T68, T69)) -> U3(f243_in(T66, T67, T69), T66, T67, .(T68, T69)) 172.38/44.62 U3(f243_out1, T66, T67, .(T68, T69)) -> f243_out1 172.38/44.62 f327_in(T119, .(.(T119, .(T120, [])), T121)) -> f327_out1(T120) 172.38/44.62 f327_in(T128, .(T129, T130)) -> U4(f327_in(T128, T130), T128, .(T129, T130)) 172.38/44.62 U4(f327_out1(X114), T128, .(T129, T130)) -> f327_out1(X114) 172.38/44.62 f350_in(T153, .(T153, T154)) -> f350_out1 172.38/44.62 f350_in(T161, .(T162, T163)) -> U5(f350_in(T161, T163), T161, .(T162, T163)) 172.38/44.62 U5(f350_out1, T161, .(T162, T163)) -> f350_out1 172.38/44.62 f323_in(T90, T92, T93, T91) -> U6(f327_in(T90, T92), T90, T92, T93, T91) 172.38/44.62 U6(f327_out1(T100), T90, T92, T93, T91) -> U7(f328_in(T100, T93, T91, T92), T90, T92, T93, T91, T100) 172.38/44.62 U7(f328_out1, T90, T92, T93, T91, T100) -> f323_out1(T100) 172.38/44.62 f328_in(T100, T93, T91, T92) -> U8(f350_in(T100, T93), T100, T93, T91, T92) 172.38/44.62 U8(f350_out1, T100, T93, T91, T92) -> U9(f3_in(T100, T91, T92, .(T100, T93)), T100, T93, T91, T92) 172.38/44.62 U9(f3_out1, T100, T93, T91, T92) -> f328_out1 172.38/44.62 172.38/44.62 Q is empty. 172.38/44.62 We have to consider all minimal (P,Q,R)-chains. 172.38/44.62 ---------------------------------------- 172.38/44.62 172.38/44.62 (240) UsableRulesProof (EQUIVALENT) 172.38/44.62 We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. 172.38/44.62 ---------------------------------------- 172.38/44.62 172.38/44.62 (241) 172.38/44.62 Obligation: 172.38/44.62 Q DP problem: 172.38/44.62 The TRS P consists of the following rules: 172.38/44.62 172.38/44.62 F243_IN(T66, T67, .(T68, T69)) -> F243_IN(T66, T67, T69) 172.38/44.62 172.38/44.62 R is empty. 172.38/44.62 Q is empty. 172.38/44.62 We have to consider all minimal (P,Q,R)-chains. 172.38/44.62 ---------------------------------------- 172.38/44.62 172.38/44.62 (242) QDPSizeChangeProof (EQUIVALENT) 172.38/44.62 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 172.38/44.62 172.38/44.62 From the DPs we obtained the following set of size-change graphs: 172.38/44.62 *F243_IN(T66, T67, .(T68, T69)) -> F243_IN(T66, T67, T69) 172.38/44.62 The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 172.38/44.62 172.38/44.62 172.38/44.62 ---------------------------------------- 172.38/44.63 172.38/44.63 (243) 172.38/44.63 YES 172.38/44.63 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (244) 172.38/44.63 Obligation: 172.38/44.63 Q DP problem: 172.38/44.63 The TRS P consists of the following rules: 172.38/44.63 172.38/44.63 F3_IN(T90, T91, T92, T93) -> F323_IN(T90, T92, T93, T91) 172.38/44.63 F323_IN(T90, T92, T93, T91) -> U6^1(f327_in(T90, T92), T90, T92, T93, T91) 172.38/44.63 U6^1(f327_out1(T100), T90, T92, T93, T91) -> F328_IN(T100, T93, T91, T92) 172.38/44.63 F328_IN(T100, T93, T91, T92) -> U8^1(f350_in(T100, T93), T100, T93, T91, T92) 172.38/44.63 U8^1(f350_out1, T100, T93, T91, T92) -> F3_IN(T100, T91, T92, .(T100, T93)) 172.38/44.63 172.38/44.63 The TRS R consists of the following rules: 172.38/44.63 172.38/44.63 f3_in(T33, T34, T35, T36) -> U1(f243_in(T33, T34, T35), T33, T34, T35, T36) 172.38/44.63 U1(f243_out1, T33, T34, T35, T36) -> f3_out1 172.38/44.63 f3_in(T90, T91, T92, T93) -> U2(f323_in(T90, T92, T93, T91), T90, T91, T92, T93) 172.38/44.63 U2(f323_out1(X75), T90, T91, T92, T93) -> f3_out1 172.38/44.63 f243_in(T55, T56, .(.(T55, .(T56, [])), T57)) -> f243_out1 172.38/44.63 f243_in(T66, T67, .(T68, T69)) -> U3(f243_in(T66, T67, T69), T66, T67, .(T68, T69)) 172.38/44.63 U3(f243_out1, T66, T67, .(T68, T69)) -> f243_out1 172.38/44.63 f327_in(T119, .(.(T119, .(T120, [])), T121)) -> f327_out1(T120) 172.38/44.63 f327_in(T128, .(T129, T130)) -> U4(f327_in(T128, T130), T128, .(T129, T130)) 172.38/44.63 U4(f327_out1(X114), T128, .(T129, T130)) -> f327_out1(X114) 172.38/44.63 f350_in(T153, .(T153, T154)) -> f350_out1 172.38/44.63 f350_in(T161, .(T162, T163)) -> U5(f350_in(T161, T163), T161, .(T162, T163)) 172.38/44.63 U5(f350_out1, T161, .(T162, T163)) -> f350_out1 172.38/44.63 f323_in(T90, T92, T93, T91) -> U6(f327_in(T90, T92), T90, T92, T93, T91) 172.38/44.63 U6(f327_out1(T100), T90, T92, T93, T91) -> U7(f328_in(T100, T93, T91, T92), T90, T92, T93, T91, T100) 172.38/44.63 U7(f328_out1, T90, T92, T93, T91, T100) -> f323_out1(T100) 172.38/44.63 f328_in(T100, T93, T91, T92) -> U8(f350_in(T100, T93), T100, T93, T91, T92) 172.38/44.63 U8(f350_out1, T100, T93, T91, T92) -> U9(f3_in(T100, T91, T92, .(T100, T93)), T100, T93, T91, T92) 172.38/44.63 U9(f3_out1, T100, T93, T91, T92) -> f328_out1 172.38/44.63 172.38/44.63 Q is empty. 172.38/44.63 We have to consider all minimal (P,Q,R)-chains. 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (245) NonLoopProof (COMPLETE) 172.38/44.63 By Theorem 8 [NONLOOP] we deduce infiniteness of the QDP. 172.38/44.63 We apply the theorem with m = 1, b = 0, 172.38/44.63 σ' = [ ], and μ' = [x1 / .(x2, x1)] on the rule 172.38/44.63 U8^1(f350_out1, x2, .(x2, x1), x0, .(.(x2, .(x2, [])), x3))[ ]^n[ ] -> U8^1(f350_out1, x2, .(x2, x1), x0, .(.(x2, .(x2, [])), x3))[ ]^n[x1 / .(x2, x1)] 172.38/44.63 This rule is correct for the QDP as the following derivation shows: 172.38/44.63 172.38/44.63 U8^1(f350_out1, x2, .(x2, x1), x0, .(.(x2, .(x2, [])), x3))[ ]^n[ ] -> U8^1(f350_out1, x2, .(x2, x1), x0, .(.(x2, .(x2, [])), x3))[ ]^n[x1 / .(x2, x1)] 172.38/44.63 by Equivalency by Simplifying Mu with mu1: [x1 / .(x2, x1)] mu2: [ ] 172.38/44.63 intermediate steps: Instantiate mu - Instantiation 172.38/44.63 U8^1(f350_out1, x1, x0, x3, .(.(x1, .(x1, [])), y2))[ ]^n[ ] -> U8^1(f350_out1, x1, .(x1, x0), x3, .(.(x1, .(x1, [])), y2))[ ]^n[ ] 172.38/44.63 by Narrowing at position: [] 172.38/44.63 intermediate steps: Instantiation - Instantiation - Instantiation 172.38/44.63 U8^1(f350_out1, T100, T93, T91, T92)[ ]^n[ ] -> F3_IN(T100, T91, T92, .(T100, T93))[ ]^n[ ] 172.38/44.63 by Rule from TRS P 172.38/44.63 172.38/44.63 intermediate steps: Instantiation - Instantiation - Instantiation - Instantiation - Instantiation - Instantiation 172.38/44.63 F3_IN(x1, x0, .(.(x1, .(x4, [])), x2), .(x4, y0))[ ]^n[ ] -> U8^1(f350_out1, x4, .(x4, y0), x0, .(.(x1, .(x4, [])), x2))[ ]^n[ ] 172.38/44.63 by Narrowing at position: [0] 172.38/44.63 intermediate steps: Instantiation - Instantiation - Instantiation 172.38/44.63 F3_IN(x3, x1, .(.(x3, .(x2, [])), x0), x4)[ ]^n[ ] -> U8^1(f350_in(x2, x4), x2, x4, x1, .(.(x3, .(x2, [])), x0))[ ]^n[ ] 172.38/44.63 by Narrowing at position: [] 172.38/44.63 intermediate steps: Instantiation - Instantiation 172.38/44.63 F3_IN(x4, x3, .(.(x4, .(x0, [])), x1), x2)[ ]^n[ ] -> F328_IN(x0, x2, x3, .(.(x4, .(x0, [])), x1))[ ]^n[ ] 172.38/44.63 by Narrowing at position: [] 172.38/44.63 intermediate steps: Instantiation - Instantiation 172.38/44.63 F3_IN(x0, x3, .(.(x0, .(y0, [])), y1), x1)[ ]^n[ ] -> U6^1(f327_out1(y0), x0, .(.(x0, .(y0, [])), y1), x1, x3)[ ]^n[ ] 172.38/44.63 by Narrowing at position: [0] 172.38/44.63 intermediate steps: Instantiation - Instantiation - Instantiation 172.38/44.63 F3_IN(x0, x3, x2, x1)[ ]^n[ ] -> U6^1(f327_in(x0, x2), x0, x2, x1, x3)[ ]^n[ ] 172.38/44.63 by Narrowing at position: [] 172.38/44.63 intermediate steps: Instantiation - Instantiation 172.38/44.63 F3_IN(T90, T91, T92, T93)[ ]^n[ ] -> F323_IN(T90, T92, T93, T91)[ ]^n[ ] 172.38/44.63 by Rule from TRS P 172.38/44.63 172.38/44.63 intermediate steps: Instantiation - Instantiation - Instantiation - Instantiation - Instantiation - Instantiation 172.38/44.63 F323_IN(T90, T92, T93, T91)[ ]^n[ ] -> U6^1(f327_in(T90, T92), T90, T92, T93, T91)[ ]^n[ ] 172.38/44.63 by Rule from TRS P 172.38/44.63 172.38/44.63 intermediate steps: Instantiation - Instantiation - Instantiation 172.38/44.63 f327_in(T119, .(.(T119, .(T120, [])), T121))[ ]^n[ ] -> f327_out1(T120)[ ]^n[ ] 172.38/44.63 by Rule from TRS R 172.38/44.63 172.38/44.63 intermediate steps: Instantiation - Instantiation - Instantiation - Instantiation - Instantiation - Instantiation - Instantiation 172.38/44.63 U6^1(f327_out1(T100), T90, T92, T93, T91)[ ]^n[ ] -> F328_IN(T100, T93, T91, T92)[ ]^n[ ] 172.38/44.63 by Rule from TRS P 172.38/44.63 172.38/44.63 intermediate steps: Instantiation - Instantiation - Instantiation - Instantiation - Instantiation - Instantiation 172.38/44.63 F328_IN(T100, T93, T91, T92)[ ]^n[ ] -> U8^1(f350_in(T100, T93), T100, T93, T91, T92)[ ]^n[ ] 172.38/44.63 by Rule from TRS P 172.38/44.63 172.38/44.63 intermediate steps: Instantiation - Instantiation - Instantiation 172.38/44.63 f350_in(T153, .(T153, T154))[ ]^n[ ] -> f350_out1[ ]^n[ ] 172.38/44.63 by Rule from TRS R 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (246) 172.38/44.63 NO 172.38/44.63 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (247) PrologToIRSwTTransformerProof (SOUND) 172.38/44.63 Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert 172.38/44.63 172.38/44.63 { 172.38/44.63 "root": 2, 172.38/44.63 "program": { 172.38/44.63 "directives": [], 172.38/44.63 "clauses": [ 172.38/44.63 [ 172.38/44.63 "(reach X Y Edges Visited)", 172.38/44.63 "(member (. X (. Y ([]))) Edges)" 172.38/44.63 ], 172.38/44.63 [ 172.38/44.63 "(reach X Z Edges Visited)", 172.38/44.63 "(',' (member1 (. X (. Y ([]))) Edges) (',' (member Y Visited) (reach Y Z Edges (. Y Visited))))" 172.38/44.63 ], 172.38/44.63 [ 172.38/44.63 "(member H (. H L))", 172.38/44.63 null 172.38/44.63 ], 172.38/44.63 [ 172.38/44.63 "(member X (. H L))", 172.38/44.63 "(member X L)" 172.38/44.63 ], 172.38/44.63 [ 172.38/44.63 "(member1 H (. H L))", 172.38/44.63 null 172.38/44.63 ], 172.38/44.63 [ 172.38/44.63 "(member1 X (. 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T34 ([]))) T35)" 172.38/44.63 }], 172.38/44.63 "kb": { 172.38/44.63 "nonunifying": [], 172.38/44.63 "intvars": {}, 172.38/44.63 "arithmetic": { 172.38/44.63 "type": "PlainIntegerRelationState", 172.38/44.63 "relations": [] 172.38/44.63 }, 172.38/44.63 "ground": [ 172.38/44.63 "T33", 172.38/44.63 "T34", 172.38/44.63 "T35" 172.38/44.63 ], 172.38/44.63 "free": [], 172.38/44.63 "exprvars": [] 172.38/44.63 } 172.38/44.63 }, 172.38/44.63 "125": { 172.38/44.63 "goal": [{ 172.38/44.63 "clause": 0, 172.38/44.63 "scope": 1, 172.38/44.63 "term": "(reach T1 T2 T3 T4)" 172.38/44.63 }], 172.38/44.63 "kb": { 172.38/44.63 "nonunifying": [], 172.38/44.63 "intvars": {}, 172.38/44.63 "arithmetic": { 172.38/44.63 "type": "PlainIntegerRelationState", 172.38/44.63 "relations": [] 172.38/44.63 }, 172.38/44.63 "ground": [ 172.38/44.63 "T4", 172.38/44.63 "T1", 172.38/44.63 "T2", 172.38/44.63 "T3" 172.38/44.63 ], 172.38/44.63 "free": [], 172.38/44.63 "exprvars": [] 172.38/44.63 } 172.38/44.63 }, 172.38/44.63 "126": { 172.38/44.63 "goal": [{ 172.38/44.63 "clause": 1, 172.38/44.63 "scope": 1, 172.38/44.63 "term": "(reach T1 T2 T3 T4)" 172.38/44.63 }], 172.38/44.63 "kb": { 172.38/44.63 "nonunifying": [], 172.38/44.63 "intvars": {}, 172.38/44.63 "arithmetic": { 172.38/44.63 "type": "PlainIntegerRelationState", 172.38/44.63 "relations": [] 172.38/44.63 }, 172.38/44.63 "ground": [ 172.38/44.63 "T4", 172.38/44.63 "T1", 172.38/44.63 "T2", 172.38/44.63 "T3" 172.38/44.63 ], 172.38/44.63 "free": [], 172.38/44.63 "exprvars": [] 172.38/44.63 } 172.38/44.63 }, 172.38/44.63 "324": { 172.38/44.63 "goal": [{ 172.38/44.63 "clause": 2, 172.38/44.63 "scope": 4, 172.38/44.63 "term": "(member T100 T93)" 172.38/44.63 }], 172.38/44.63 "kb": { 172.38/44.63 "nonunifying": [], 172.38/44.63 "intvars": {}, 172.38/44.63 "arithmetic": { 172.38/44.63 "type": "PlainIntegerRelationState", 172.38/44.63 "relations": [] 172.38/44.63 }, 172.38/44.63 "ground": [ 172.38/44.63 "T93", 172.38/44.63 "T100" 172.38/44.63 ], 172.38/44.63 "free": [], 172.38/44.63 "exprvars": [] 172.38/44.63 } 172.38/44.63 }, 172.38/44.63 "325": { 172.38/44.63 "goal": [{ 172.38/44.63 "clause": 3, 172.38/44.63 "scope": 4, 172.38/44.63 "term": "(member T100 T93)" 172.38/44.63 }], 172.38/44.63 "kb": { 172.38/44.63 "nonunifying": [], 172.38/44.63 "intvars": {}, 172.38/44.63 "arithmetic": { 172.38/44.63 "type": "PlainIntegerRelationState", 172.38/44.63 "relations": [] 172.38/44.63 }, 172.38/44.63 "ground": [ 172.38/44.63 "T93", 172.38/44.63 "T100" 172.38/44.63 ], 172.38/44.63 "free": [], 172.38/44.63 "exprvars": [] 172.38/44.63 } 172.38/44.63 }, 172.38/44.63 "249": { 172.38/44.63 "goal": [{ 172.38/44.63 "clause": -1, 172.38/44.63 "scope": -1, 172.38/44.63 "term": "(true)" 172.38/44.63 }], 172.38/44.63 "kb": { 172.38/44.63 "nonunifying": [], 172.38/44.63 "intvars": {}, 172.38/44.63 "arithmetic": { 172.38/44.63 "type": "PlainIntegerRelationState", 172.38/44.63 "relations": [] 172.38/44.63 }, 172.38/44.63 "ground": [], 172.38/44.63 "free": [], 172.38/44.63 "exprvars": [] 172.38/44.63 } 172.38/44.63 }, 172.38/44.63 "326": { 172.38/44.63 "goal": [{ 172.38/44.63 "clause": -1, 172.38/44.63 "scope": -1, 172.38/44.63 "term": "(true)" 172.38/44.63 }], 172.38/44.63 "kb": { 172.38/44.63 "nonunifying": [], 172.38/44.63 "intvars": {}, 172.38/44.63 "arithmetic": { 172.38/44.63 "type": "PlainIntegerRelationState", 172.38/44.63 "relations": [] 172.38/44.63 }, 172.38/44.63 "ground": [], 172.38/44.63 "free": [], 172.38/44.63 "exprvars": [] 172.38/44.63 } 172.38/44.63 }, 172.38/44.63 "307": { 172.38/44.63 "goal": [ 172.38/44.63 { 172.38/44.63 "clause": 4, 172.38/44.63 "scope": 3, 172.38/44.63 "term": "(member1 (. T90 (. X75 ([]))) T92)" 172.38/44.63 }, 172.38/44.63 { 172.38/44.63 "clause": 5, 172.38/44.63 "scope": 3, 172.38/44.63 "term": "(member1 (. T90 (. X75 ([]))) T92)" 172.38/44.63 } 172.38/44.63 ], 172.38/44.63 "kb": { 172.38/44.63 "nonunifying": [], 172.38/44.63 "intvars": {}, 172.38/44.63 "arithmetic": { 172.38/44.63 "type": "PlainIntegerRelationState", 172.38/44.63 "relations": [] 172.38/44.63 }, 172.38/44.63 "ground": [ 172.38/44.63 "T90", 172.38/44.63 "T92" 172.38/44.63 ], 172.38/44.63 "free": ["X75"], 172.38/44.63 "exprvars": [] 172.38/44.63 } 172.38/44.63 }, 172.38/44.63 "329": { 172.38/44.63 "goal": [], 172.38/44.63 "kb": { 172.38/44.63 "nonunifying": [], 172.38/44.63 "intvars": {}, 172.38/44.63 "arithmetic": { 172.38/44.63 "type": "PlainIntegerRelationState", 172.38/44.63 "relations": [] 172.38/44.63 }, 172.38/44.63 "ground": [], 172.38/44.63 "free": [], 172.38/44.63 "exprvars": [] 172.38/44.63 } 172.38/44.63 } 172.38/44.63 }, 172.38/44.63 "edges": [ 172.38/44.63 { 172.38/44.63 "from": 2, 172.38/44.63 "to": 123, 172.38/44.63 "label": "CASE" 172.38/44.63 }, 172.38/44.63 { 172.38/44.63 "from": 123, 172.38/44.63 "to": 125, 172.38/44.63 "label": "PARALLEL" 172.38/44.63 }, 172.38/44.63 { 172.38/44.63 "from": 123, 172.38/44.63 "to": 126, 172.38/44.63 "label": "PARALLEL" 172.38/44.63 }, 172.38/44.63 { 172.38/44.63 "from": 125, 172.38/44.63 "to": 146, 172.38/44.63 "label": "ONLY EVAL with clause\nreach(X25, X26, X27, X28) :- member(.(X25, .(X26, [])), X27).\nand substitutionT1 -> T33,\nX25 -> T33,\nT2 -> T34,\nX26 -> T34,\nT3 -> T35,\nX27 -> T35,\nT4 -> T36,\nX28 -> T36" 172.38/44.63 }, 172.38/44.63 { 172.38/44.63 "from": 126, 172.38/44.63 "to": 286, 172.38/44.63 "label": "ONLY EVAL with clause\nreach(X71, X72, X73, X74) :- ','(member1(.(X71, .(X75, [])), X73), ','(member(X75, X74), reach(X75, X72, X73, .(X75, X74)))).\nand substitutionT1 -> T90,\nX71 -> T90,\nT2 -> T91,\nX72 -> T91,\nT3 -> T92,\nX73 -> T92,\nT4 -> T93,\nX74 -> T93" 172.38/44.63 }, 172.38/44.63 { 172.38/44.63 "from": 146, 172.38/44.63 "to": 242, 172.38/44.63 "label": "CASE" 172.38/44.63 }, 172.38/44.63 { 172.38/44.63 "from": 242, 172.38/44.63 "to": 244, 172.38/44.63 "label": "PARALLEL" 172.38/44.63 }, 172.38/44.63 { 172.38/44.63 "from": 242, 172.38/44.63 "to": 245, 172.38/44.63 "label": "PARALLEL" 172.38/44.63 }, 172.38/44.63 { 172.38/44.63 "from": 244, 172.38/44.63 "to": 249, 172.38/44.63 "label": "EVAL with clause\nmember(X41, .(X41, X42)).\nand substitutionT33 -> T55,\nT34 -> T56,\nX41 -> .(T55, .(T56, [])),\nX42 -> T57,\nT35 -> .(.(T55, .(T56, [])), T57)" 172.38/44.63 }, 172.38/44.63 { 172.38/44.63 "from": 244, 172.38/44.63 "to": 250, 172.38/44.63 "label": "EVAL-BACKTRACK" 172.38/44.63 }, 172.38/44.63 { 172.38/44.63 "from": 245, 172.38/44.63 "to": 278, 172.38/44.63 "label": "EVAL with clause\nmember(X49, .(X50, X51)) :- member(X49, X51).\nand substitutionT33 -> T66,\nT34 -> T67,\nX49 -> .(T66, .(T67, [])),\nX50 -> T68,\nX51 -> T69,\nT35 -> .(T68, T69)" 172.38/44.63 }, 172.38/44.63 { 172.38/44.63 "from": 245, 172.38/44.63 "to": 279, 172.38/44.63 "label": "EVAL-BACKTRACK" 172.38/44.63 }, 172.38/44.63 { 172.38/44.63 "from": 249, 172.38/44.63 "to": 270, 172.38/44.63 "label": "SUCCESS" 172.38/44.63 }, 172.38/44.63 { 172.38/44.63 "from": 278, 172.38/44.63 "to": 146, 172.38/44.63 "label": "INSTANCE with matching:\nT33 -> T66\nT34 -> T67\nT35 -> T69" 172.38/44.63 }, 172.38/44.63 { 172.38/44.63 "from": 286, 172.38/44.63 "to": 291, 172.38/44.63 "label": "SPLIT 1" 172.38/44.63 }, 172.38/44.63 { 172.38/44.63 "from": 286, 172.38/44.63 "to": 292, 172.38/44.63 "label": "SPLIT 2\nnew knowledge:\nT90 is ground\nT100 is ground\nT92 is ground\nreplacements:X75 -> T100" 172.38/44.63 }, 172.38/44.63 { 172.38/44.63 "from": 291, 172.38/44.63 "to": 307, 172.38/44.63 "label": "CASE" 172.38/44.63 }, 172.38/44.63 { 172.38/44.63 "from": 292, 172.38/44.63 "to": 319, 172.38/44.63 "label": "SPLIT 1" 172.38/44.63 }, 172.38/44.63 { 172.38/44.63 "from": 292, 172.38/44.63 "to": 320, 172.38/44.63 "label": "SPLIT 2\nnew knowledge:\nT100 is ground\nT93 is ground" 172.38/44.63 }, 172.38/44.63 { 172.38/44.63 "from": 307, 172.38/44.63 "to": 312, 172.38/44.63 "label": "PARALLEL" 172.38/44.63 }, 172.38/44.63 { 172.38/44.63 "from": 307, 172.38/44.63 "to": 313, 172.38/44.63 "label": "PARALLEL" 172.38/44.63 }, 172.38/44.63 { 172.38/44.63 "from": 312, 172.38/44.63 "to": 314, 172.38/44.63 "label": "EVAL with clause\nmember1(X100, .(X100, X101)).\nand substitutionT90 -> T119,\nX75 -> T120,\nX100 -> .(T119, .(T120, [])),\nX102 -> T120,\nX101 -> T121,\nT92 -> .(.(T119, .(T120, [])), T121)" 172.38/44.63 }, 172.38/44.63 { 172.38/44.63 "from": 312, 172.38/44.63 "to": 315, 172.38/44.63 "label": "EVAL-BACKTRACK" 172.38/44.63 }, 172.38/44.63 { 172.38/44.63 "from": 313, 172.38/44.63 "to": 317, 172.38/44.63 "label": "EVAL with clause\nmember1(X111, .(X112, X113)) :- member1(X111, X113).\nand substitutionT90 -> T128,\nX75 -> X114,\nX111 -> .(T128, .(X114, [])),\nX112 -> T129,\nX113 -> T130,\nT92 -> .(T129, T130)" 172.38/44.63 }, 172.38/44.63 { 172.38/44.63 "from": 313, 172.38/44.63 "to": 318, 172.38/44.63 "label": "EVAL-BACKTRACK" 172.38/44.63 }, 172.38/44.63 { 172.38/44.63 "from": 314, 172.38/44.63 "to": 316, 172.38/44.63 "label": "SUCCESS" 172.38/44.63 }, 172.38/44.63 { 172.38/44.63 "from": 317, 172.38/44.63 "to": 291, 172.38/44.63 "label": "INSTANCE with matching:\nT90 -> T128\nX75 -> X114\nT92 -> T130" 172.38/44.63 }, 172.38/44.63 { 172.38/44.63 "from": 319, 172.38/44.63 "to": 321, 172.38/44.63 "label": "CASE" 172.38/44.63 }, 172.38/44.63 { 172.38/44.63 "from": 320, 172.38/44.63 "to": 2, 172.38/44.63 "label": "INSTANCE with matching:\nT1 -> T100\nT2 -> T91\nT3 -> T92\nT4 -> .(T100, T93)" 172.38/44.63 }, 172.38/44.63 { 172.38/44.63 "from": 321, 172.38/44.63 "to": 324, 172.38/44.63 "label": "PARALLEL" 172.38/44.63 }, 172.38/44.63 { 172.38/44.63 "from": 321, 172.38/44.63 "to": 325, 172.38/44.63 "label": "PARALLEL" 172.38/44.63 }, 172.38/44.63 { 172.38/44.63 "from": 324, 172.38/44.63 "to": 326, 172.38/44.63 "label": "EVAL with clause\nmember(X137, .(X137, X138)).\nand substitutionT100 -> T153,\nX137 -> T153,\nX138 -> T154,\nT93 -> .(T153, T154)" 172.38/44.63 }, 172.38/44.63 { 172.38/44.63 "from": 324, 172.38/44.63 "to": 329, 172.38/44.63 "label": "EVAL-BACKTRACK" 172.38/44.63 }, 172.38/44.63 { 172.38/44.63 "from": 325, 172.38/44.63 "to": 332, 172.38/44.63 "label": "EVAL with clause\nmember(X145, .(X146, X147)) :- member(X145, X147).\nand substitutionT100 -> T161,\nX145 -> T161,\nX146 -> T162,\nX147 -> T163,\nT93 -> .(T162, T163)" 172.38/44.63 }, 172.38/44.63 { 172.38/44.63 "from": 325, 172.38/44.63 "to": 333, 172.38/44.63 "label": "EVAL-BACKTRACK" 172.38/44.63 }, 172.38/44.63 { 172.38/44.63 "from": 326, 172.38/44.63 "to": 330, 172.38/44.63 "label": "SUCCESS" 172.38/44.63 }, 172.38/44.63 { 172.38/44.63 "from": 332, 172.38/44.63 "to": 319, 172.38/44.63 "label": "INSTANCE with matching:\nT100 -> T161\nT93 -> T163" 172.38/44.63 } 172.38/44.63 ], 172.38/44.63 "type": "Graph" 172.38/44.63 } 172.38/44.63 } 172.38/44.63 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (248) 172.38/44.63 Complex Obligation (AND) 172.38/44.63 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (249) 172.38/44.63 Obligation: 172.38/44.63 Rules: 172.38/44.63 f333_out -> f325_out(T100, T93) :|: TRUE 172.38/44.63 f332_out(T161, T163) -> f325_out(T161, .(T162, T163)) :|: TRUE 172.38/44.63 f325_in(x, .(x1, x2)) -> f332_in(x, x2) :|: TRUE 172.38/44.63 f325_in(x3, x4) -> f333_in :|: TRUE 172.38/44.63 f319_out(x5, x6) -> f332_out(x5, x6) :|: TRUE 172.38/44.63 f332_in(x7, x8) -> f319_in(x7, x8) :|: TRUE 172.38/44.63 f319_in(x9, x10) -> f321_in(x9, x10) :|: TRUE 172.38/44.63 f321_out(x11, x12) -> f319_out(x11, x12) :|: TRUE 172.38/44.63 f321_in(x13, x14) -> f325_in(x13, x14) :|: TRUE 172.38/44.63 f325_out(x15, x16) -> f321_out(x15, x16) :|: TRUE 172.38/44.63 f321_in(x17, x18) -> f324_in(x17, x18) :|: TRUE 172.38/44.63 f324_out(x19, x20) -> f321_out(x19, x20) :|: TRUE 172.38/44.63 f2_in(T1, T2, T3, T4) -> f123_in(T1, T2, T3, T4) :|: TRUE 172.38/44.63 f123_out(x21, x22, x23, x24) -> f2_out(x21, x22, x23, x24) :|: TRUE 172.38/44.63 f125_out(x25, x26, x27, x28) -> f123_out(x25, x26, x27, x28) :|: TRUE 172.38/44.63 f126_out(x29, x30, x31, x32) -> f123_out(x29, x30, x31, x32) :|: TRUE 172.38/44.63 f123_in(x33, x34, x35, x36) -> f126_in(x33, x34, x35, x36) :|: TRUE 172.38/44.63 f123_in(x37, x38, x39, x40) -> f125_in(x37, x38, x39, x40) :|: TRUE 172.38/44.63 f286_out(x41, x42, x43, x44) -> f126_out(x41, x44, x42, x43) :|: TRUE 172.38/44.63 f126_in(x45, x46, x47, x48) -> f286_in(x45, x47, x48, x46) :|: TRUE 172.38/44.63 f291_out(x49, x50) -> f292_in(x51, x52, x53, x50) :|: TRUE 172.38/44.63 f286_in(x54, x55, x56, x57) -> f291_in(x54, x55) :|: TRUE 172.38/44.63 f292_out(x58, x59, x60, x61) -> f286_out(x62, x61, x59, x60) :|: TRUE 172.38/44.63 f320_out(x63, x64, x65, x66) -> f292_out(x63, x66, x64, x65) :|: TRUE 172.38/44.63 f292_in(x67, x68, x69, x70) -> f319_in(x67, x68) :|: TRUE 172.38/44.63 f319_out(x71, x72) -> f320_in(x71, x73, x74, x72) :|: TRUE 172.38/44.63 Start term: f2_in(T1, T2, T3, T4) 172.38/44.63 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (250) IRSwTSimpleDependencyGraphProof (EQUIVALENT) 172.38/44.63 Constructed simple dependency graph. 172.38/44.63 172.38/44.63 Simplified to the following IRSwTs: 172.38/44.63 172.38/44.63 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (251) 172.38/44.63 TRUE 172.38/44.63 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (252) 172.38/44.63 Obligation: 172.38/44.63 Rules: 172.38/44.63 f291_out(T128, T130) -> f317_out(T128, T130) :|: TRUE 172.38/44.63 f317_in(x, x1) -> f291_in(x, x1) :|: TRUE 172.38/44.63 f312_out(T90, T92) -> f307_out(T90, T92) :|: TRUE 172.38/44.63 f307_in(x2, x3) -> f313_in(x2, x3) :|: TRUE 172.38/44.63 f307_in(x4, x5) -> f312_in(x4, x5) :|: TRUE 172.38/44.63 f313_out(x6, x7) -> f307_out(x6, x7) :|: TRUE 172.38/44.63 f317_out(x8, x9) -> f313_out(x8, .(x10, x9)) :|: TRUE 172.38/44.63 f318_out -> f313_out(x11, x12) :|: TRUE 172.38/44.63 f313_in(x13, x14) -> f318_in :|: TRUE 172.38/44.63 f313_in(x15, .(x16, x17)) -> f317_in(x15, x17) :|: TRUE 172.38/44.63 f307_out(x18, x19) -> f291_out(x18, x19) :|: TRUE 172.38/44.63 f291_in(x20, x21) -> f307_in(x20, x21) :|: TRUE 172.38/44.63 f2_in(T1, T2, T3, T4) -> f123_in(T1, T2, T3, T4) :|: TRUE 172.38/44.63 f123_out(x22, x23, x24, x25) -> f2_out(x22, x23, x24, x25) :|: TRUE 172.38/44.63 f125_out(x26, x27, x28, x29) -> f123_out(x26, x27, x28, x29) :|: TRUE 172.38/44.63 f126_out(x30, x31, x32, x33) -> f123_out(x30, x31, x32, x33) :|: TRUE 172.38/44.63 f123_in(x34, x35, x36, x37) -> f126_in(x34, x35, x36, x37) :|: TRUE 172.38/44.63 f123_in(x38, x39, x40, x41) -> f125_in(x38, x39, x40, x41) :|: TRUE 172.38/44.63 f286_out(x42, x43, x44, x45) -> f126_out(x42, x45, x43, x44) :|: TRUE 172.38/44.63 f126_in(x46, x47, x48, x49) -> f286_in(x46, x48, x49, x47) :|: TRUE 172.38/44.63 f291_out(x50, x51) -> f292_in(x52, x53, x54, x51) :|: TRUE 172.38/44.63 f286_in(x55, x56, x57, x58) -> f291_in(x55, x56) :|: TRUE 172.38/44.63 f292_out(x59, x60, x61, x62) -> f286_out(x63, x62, x60, x61) :|: TRUE 172.38/44.63 Start term: f2_in(T1, T2, T3, T4) 172.38/44.63 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (253) IRSwTSimpleDependencyGraphProof (EQUIVALENT) 172.38/44.63 Constructed simple dependency graph. 172.38/44.63 172.38/44.63 Simplified to the following IRSwTs: 172.38/44.63 172.38/44.63 intTRSProblem: 172.38/44.63 f317_in(x, x1) -> f291_in(x, x1) :|: TRUE 172.38/44.63 f307_in(x2, x3) -> f313_in(x2, x3) :|: TRUE 172.38/44.63 f313_in(x15, .(x16, x17)) -> f317_in(x15, x17) :|: TRUE 172.38/44.63 f291_in(x20, x21) -> f307_in(x20, x21) :|: TRUE 172.38/44.63 172.38/44.63 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (254) 172.38/44.63 Obligation: 172.38/44.63 Rules: 172.38/44.63 f317_in(x, x1) -> f291_in(x, x1) :|: TRUE 172.38/44.63 f307_in(x2, x3) -> f313_in(x2, x3) :|: TRUE 172.38/44.63 f313_in(x15, .(x16, x17)) -> f317_in(x15, x17) :|: TRUE 172.38/44.63 f291_in(x20, x21) -> f307_in(x20, x21) :|: TRUE 172.38/44.63 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (255) IntTRSCompressionProof (EQUIVALENT) 172.38/44.63 Compressed rules. 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (256) 172.38/44.63 Obligation: 172.38/44.63 Rules: 172.38/44.63 f307_in(x2:0, .(x16:0, x17:0)) -> f307_in(x2:0, x17:0) :|: TRUE 172.38/44.63 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (257) IRSFormatTransformerProof (EQUIVALENT) 172.38/44.63 Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (258) 172.38/44.63 Obligation: 172.38/44.63 Rules: 172.38/44.63 f307_in(x2:0, .(x16:0, x17:0)) -> f307_in(x2:0, x17:0) :|: TRUE 172.38/44.63 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (259) IRSwTTerminationDigraphProof (EQUIVALENT) 172.38/44.63 Constructed termination digraph! 172.38/44.63 Nodes: 172.38/44.63 (1) f307_in(x2:0, .(x16:0, x17:0)) -> f307_in(x2:0, x17:0) :|: TRUE 172.38/44.63 172.38/44.63 Arcs: 172.38/44.63 (1) -> (1) 172.38/44.63 172.38/44.63 This digraph is fully evaluated! 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (260) 172.38/44.63 Obligation: 172.38/44.63 172.38/44.63 Termination digraph: 172.38/44.63 Nodes: 172.38/44.63 (1) f307_in(x2:0, .(x16:0, x17:0)) -> f307_in(x2:0, x17:0) :|: TRUE 172.38/44.63 172.38/44.63 Arcs: 172.38/44.63 (1) -> (1) 172.38/44.63 172.38/44.63 This digraph is fully evaluated! 172.38/44.63 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (261) IntTRSUnneededArgumentFilterProof (EQUIVALENT) 172.38/44.63 Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements: 172.38/44.63 172.38/44.63 f307_in(x1, x2) -> f307_in(x2) 172.38/44.63 .(x1, x2) -> .(x2) 172.38/44.63 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (262) 172.38/44.63 Obligation: 172.38/44.63 Rules: 172.38/44.63 f307_in(.(x17:0)) -> f307_in(x17:0) :|: TRUE 172.38/44.63 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (263) TempFilterProof (SOUND) 172.38/44.63 Used the following sort dictionary for filtering: 172.38/44.63 f307_in(VARIABLE) 172.38/44.63 .(VARIABLE) 172.38/44.63 Removed predefined arithmetic. 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (264) 172.38/44.63 Obligation: 172.38/44.63 Rules: 172.38/44.63 f307_in(.(x17:0)) -> f307_in(x17:0) 172.38/44.63 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (265) IRSwTToQDPProof (SOUND) 172.38/44.63 Removed the integers and created a QDP-Problem. 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (266) 172.38/44.63 Obligation: 172.38/44.63 Q DP problem: 172.38/44.63 The TRS P consists of the following rules: 172.38/44.63 172.38/44.63 f307_in(.(x17:0)) -> f307_in(x17:0) 172.38/44.63 172.38/44.63 R is empty. 172.38/44.63 Q is empty. 172.38/44.63 We have to consider all (P,Q,R)-chains. 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (267) QDPSizeChangeProof (EQUIVALENT) 172.38/44.63 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 172.38/44.63 172.38/44.63 From the DPs we obtained the following set of size-change graphs: 172.38/44.63 *f307_in(.(x17:0)) -> f307_in(x17:0) 172.38/44.63 The graph contains the following edges 1 > 1 172.38/44.63 172.38/44.63 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (268) 172.38/44.63 YES 172.38/44.63 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (269) 172.38/44.63 Obligation: 172.38/44.63 Rules: 172.38/44.63 f242_out(T33, T34, T35) -> f146_out(T33, T34, T35) :|: TRUE 172.38/44.63 f146_in(x, x1, x2) -> f242_in(x, x1, x2) :|: TRUE 172.38/44.63 f244_out(x3, x4, x5) -> f242_out(x3, x4, x5) :|: TRUE 172.38/44.63 f245_out(x6, x7, x8) -> f242_out(x6, x7, x8) :|: TRUE 172.38/44.63 f242_in(x9, x10, x11) -> f245_in(x9, x10, x11) :|: TRUE 172.38/44.63 f242_in(x12, x13, x14) -> f244_in(x12, x13, x14) :|: TRUE 172.38/44.63 f245_in(T66, T67, .(T68, T69)) -> f278_in(T66, T67, T69) :|: TRUE 172.38/44.63 f278_out(x15, x16, x17) -> f245_out(x15, x16, .(x18, x17)) :|: TRUE 172.38/44.63 f279_out -> f245_out(x19, x20, x21) :|: TRUE 172.38/44.63 f245_in(x22, x23, x24) -> f279_in :|: TRUE 172.38/44.63 f146_out(x25, x26, x27) -> f278_out(x25, x26, x27) :|: TRUE 172.38/44.63 f278_in(x28, x29, x30) -> f146_in(x28, x29, x30) :|: TRUE 172.38/44.63 f2_in(T1, T2, T3, T4) -> f123_in(T1, T2, T3, T4) :|: TRUE 172.38/44.63 f123_out(x31, x32, x33, x34) -> f2_out(x31, x32, x33, x34) :|: TRUE 172.38/44.63 f125_out(x35, x36, x37, x38) -> f123_out(x35, x36, x37, x38) :|: TRUE 172.38/44.63 f126_out(x39, x40, x41, x42) -> f123_out(x39, x40, x41, x42) :|: TRUE 172.38/44.63 f123_in(x43, x44, x45, x46) -> f126_in(x43, x44, x45, x46) :|: TRUE 172.38/44.63 f123_in(x47, x48, x49, x50) -> f125_in(x47, x48, x49, x50) :|: TRUE 172.38/44.63 f146_out(x51, x52, x53) -> f125_out(x51, x52, x53, x54) :|: TRUE 172.38/44.63 f125_in(x55, x56, x57, x58) -> f146_in(x55, x56, x57) :|: TRUE 172.38/44.63 Start term: f2_in(T1, T2, T3, T4) 172.38/44.63 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (270) IRSwTSimpleDependencyGraphProof (EQUIVALENT) 172.38/44.63 Constructed simple dependency graph. 172.38/44.63 172.38/44.63 Simplified to the following IRSwTs: 172.38/44.63 172.38/44.63 intTRSProblem: 172.38/44.63 f146_in(x, x1, x2) -> f242_in(x, x1, x2) :|: TRUE 172.38/44.63 f242_in(x9, x10, x11) -> f245_in(x9, x10, x11) :|: TRUE 172.38/44.63 f245_in(T66, T67, .(T68, T69)) -> f278_in(T66, T67, T69) :|: TRUE 172.38/44.63 f278_in(x28, x29, x30) -> f146_in(x28, x29, x30) :|: TRUE 172.38/44.63 172.38/44.63 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (271) 172.38/44.63 Obligation: 172.38/44.63 Rules: 172.38/44.63 f146_in(x, x1, x2) -> f242_in(x, x1, x2) :|: TRUE 172.38/44.63 f242_in(x9, x10, x11) -> f245_in(x9, x10, x11) :|: TRUE 172.38/44.63 f245_in(T66, T67, .(T68, T69)) -> f278_in(T66, T67, T69) :|: TRUE 172.38/44.63 f278_in(x28, x29, x30) -> f146_in(x28, x29, x30) :|: TRUE 172.38/44.63 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (272) IntTRSCompressionProof (EQUIVALENT) 172.38/44.63 Compressed rules. 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (273) 172.38/44.63 Obligation: 172.38/44.63 Rules: 172.38/44.63 f146_in(x:0, x1:0, .(T68:0, T69:0)) -> f146_in(x:0, x1:0, T69:0) :|: TRUE 172.38/44.63 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (274) IRSFormatTransformerProof (EQUIVALENT) 172.38/44.63 Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (275) 172.38/44.63 Obligation: 172.38/44.63 Rules: 172.38/44.63 f146_in(x:0, x1:0, .(T68:0, T69:0)) -> f146_in(x:0, x1:0, T69:0) :|: TRUE 172.38/44.63 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (276) IRSwTTerminationDigraphProof (EQUIVALENT) 172.38/44.63 Constructed termination digraph! 172.38/44.63 Nodes: 172.38/44.63 (1) f146_in(x:0, x1:0, .(T68:0, T69:0)) -> f146_in(x:0, x1:0, T69:0) :|: TRUE 172.38/44.63 172.38/44.63 Arcs: 172.38/44.63 (1) -> (1) 172.38/44.63 172.38/44.63 This digraph is fully evaluated! 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (277) 172.38/44.63 Obligation: 172.38/44.63 172.38/44.63 Termination digraph: 172.38/44.63 Nodes: 172.38/44.63 (1) f146_in(x:0, x1:0, .(T68:0, T69:0)) -> f146_in(x:0, x1:0, T69:0) :|: TRUE 172.38/44.63 172.38/44.63 Arcs: 172.38/44.63 (1) -> (1) 172.38/44.63 172.38/44.63 This digraph is fully evaluated! 172.38/44.63 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (278) IntTRSUnneededArgumentFilterProof (EQUIVALENT) 172.38/44.63 Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements: 172.38/44.63 172.38/44.63 f146_in(x1, x2, x3) -> f146_in(x3) 172.38/44.63 .(x1, x2) -> .(x2) 172.38/44.63 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (279) 172.38/44.63 Obligation: 172.38/44.63 Rules: 172.38/44.63 f146_in(.(T69:0)) -> f146_in(T69:0) :|: TRUE 172.38/44.63 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (280) TempFilterProof (SOUND) 172.38/44.63 Used the following sort dictionary for filtering: 172.38/44.63 f146_in(VARIABLE) 172.38/44.63 .(VARIABLE) 172.38/44.63 Removed predefined arithmetic. 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (281) 172.38/44.63 Obligation: 172.38/44.63 Rules: 172.38/44.63 f146_in(.(T69:0)) -> f146_in(T69:0) 172.38/44.63 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (282) IRSwTToQDPProof (SOUND) 172.38/44.63 Removed the integers and created a QDP-Problem. 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (283) 172.38/44.63 Obligation: 172.38/44.63 Q DP problem: 172.38/44.63 The TRS P consists of the following rules: 172.38/44.63 172.38/44.63 f146_in(.(T69:0)) -> f146_in(T69:0) 172.38/44.63 172.38/44.63 R is empty. 172.38/44.63 Q is empty. 172.38/44.63 We have to consider all (P,Q,R)-chains. 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (284) QDPSizeChangeProof (EQUIVALENT) 172.38/44.63 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 172.38/44.63 172.38/44.63 From the DPs we obtained the following set of size-change graphs: 172.38/44.63 *f146_in(.(T69:0)) -> f146_in(T69:0) 172.38/44.63 The graph contains the following edges 1 > 1 172.38/44.63 172.38/44.63 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (285) 172.38/44.63 YES 172.38/44.63 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (286) 172.38/44.63 Obligation: 172.38/44.63 Rules: 172.38/44.63 f319_in(T100, T93) -> f321_in(T100, T93) :|: TRUE 172.38/44.63 f321_out(x, x1) -> f319_out(x, x1) :|: TRUE 172.38/44.63 f291_out(x2, x3) -> f292_in(x4, x5, x6, x3) :|: TRUE 172.38/44.63 f286_in(x7, x8, x9, x10) -> f291_in(x7, x8) :|: TRUE 172.38/44.63 f292_out(x11, x12, x13, x14) -> f286_out(x15, x14, x12, x13) :|: TRUE 172.38/44.63 f307_out(T90, T92) -> f291_out(T90, T92) :|: TRUE 172.38/44.63 f291_in(x16, x17) -> f307_in(x16, x17) :|: TRUE 172.38/44.63 f2_in(T1, T2, T3, T4) -> f123_in(T1, T2, T3, T4) :|: TRUE 172.38/44.63 f123_out(x18, x19, x20, x21) -> f2_out(x18, x19, x20, x21) :|: TRUE 172.38/44.63 f321_in(x22, x23) -> f325_in(x22, x23) :|: TRUE 172.38/44.63 f325_out(x24, x25) -> f321_out(x24, x25) :|: TRUE 172.38/44.63 f321_in(x26, x27) -> f324_in(x26, x27) :|: TRUE 172.38/44.63 f324_out(x28, x29) -> f321_out(x28, x29) :|: TRUE 172.38/44.63 f320_out(x30, x31, x32, x33) -> f292_out(x30, x33, x31, x32) :|: TRUE 172.38/44.63 f292_in(x34, x35, x36, x37) -> f319_in(x34, x35) :|: TRUE 172.38/44.63 f319_out(x38, x39) -> f320_in(x38, x40, x41, x39) :|: TRUE 172.38/44.63 f291_out(T128, T130) -> f317_out(T128, T130) :|: TRUE 172.38/44.63 f317_in(x42, x43) -> f291_in(x42, x43) :|: TRUE 172.38/44.63 f333_out -> f325_out(x44, x45) :|: TRUE 172.38/44.63 f332_out(T161, T163) -> f325_out(T161, .(T162, T163)) :|: TRUE 172.38/44.63 f325_in(x46, .(x47, x48)) -> f332_in(x46, x48) :|: TRUE 172.38/44.63 f325_in(x49, x50) -> f333_in :|: TRUE 172.38/44.63 f286_out(x51, x52, x53, x54) -> f126_out(x51, x54, x52, x53) :|: TRUE 172.38/44.63 f126_in(x55, x56, x57, x58) -> f286_in(x55, x57, x58, x56) :|: TRUE 172.38/44.63 f320_in(x59, x60, x61, x62) -> f2_in(x59, x60, x61, .(x59, x62)) :|: TRUE 172.38/44.63 f2_out(x63, x64, x65, .(x63, x66)) -> f320_out(x63, x64, x65, x66) :|: TRUE 172.38/44.63 f326_out -> f324_out(T153, .(T153, T154)) :|: TRUE 172.38/44.63 f324_in(x67, .(x67, x68)) -> f326_in :|: TRUE 172.38/44.63 f329_out -> f324_out(x69, x70) :|: TRUE 172.38/44.63 f324_in(x71, x72) -> f329_in :|: TRUE 172.38/44.63 f125_out(x73, x74, x75, x76) -> f123_out(x73, x74, x75, x76) :|: TRUE 172.38/44.63 f126_out(x77, x78, x79, x80) -> f123_out(x77, x78, x79, x80) :|: TRUE 172.38/44.63 f123_in(x81, x82, x83, x84) -> f126_in(x81, x82, x83, x84) :|: TRUE 172.38/44.63 f123_in(x85, x86, x87, x88) -> f125_in(x85, x86, x87, x88) :|: TRUE 172.38/44.63 f312_in(x89, x90) -> f315_in :|: TRUE 172.38/44.63 f312_in(T119, .(.(T119, .(T120, [])), T121)) -> f314_in :|: TRUE 172.38/44.63 f315_out -> f312_out(x91, x92) :|: TRUE 172.38/44.63 f314_out -> f312_out(x93, .(.(x93, .(x94, [])), x95)) :|: TRUE 172.38/44.63 f312_out(x96, x97) -> f307_out(x96, x97) :|: TRUE 172.38/44.63 f307_in(x98, x99) -> f313_in(x98, x99) :|: TRUE 172.38/44.63 f307_in(x100, x101) -> f312_in(x100, x101) :|: TRUE 172.38/44.63 f313_out(x102, x103) -> f307_out(x102, x103) :|: TRUE 172.38/44.63 f319_out(x104, x105) -> f332_out(x104, x105) :|: TRUE 172.38/44.63 f332_in(x106, x107) -> f319_in(x106, x107) :|: TRUE 172.38/44.63 f317_out(x108, x109) -> f313_out(x108, .(x110, x109)) :|: TRUE 172.38/44.63 f318_out -> f313_out(x111, x112) :|: TRUE 172.38/44.63 f313_in(x113, x114) -> f318_in :|: TRUE 172.38/44.63 f313_in(x115, .(x116, x117)) -> f317_in(x115, x117) :|: TRUE 172.38/44.63 f314_in -> f314_out :|: TRUE 172.38/44.63 f326_in -> f326_out :|: TRUE 172.38/44.63 Start term: f2_in(T1, T2, T3, T4) 172.38/44.63 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (287) IRSwTSimpleDependencyGraphProof (EQUIVALENT) 172.38/44.63 Constructed simple dependency graph. 172.38/44.63 172.38/44.63 Simplified to the following IRSwTs: 172.38/44.63 172.38/44.63 intTRSProblem: 172.38/44.63 f319_in(T100, T93) -> f321_in(T100, T93) :|: TRUE 172.38/44.63 f321_out(x, x1) -> f319_out(x, x1) :|: TRUE 172.38/44.63 f291_out(x2, x3) -> f292_in(x4, x5, x6, x3) :|: TRUE 172.38/44.63 f286_in(x7, x8, x9, x10) -> f291_in(x7, x8) :|: TRUE 172.38/44.63 f307_out(T90, T92) -> f291_out(T90, T92) :|: TRUE 172.38/44.63 f291_in(x16, x17) -> f307_in(x16, x17) :|: TRUE 172.38/44.63 f2_in(T1, T2, T3, T4) -> f123_in(T1, T2, T3, T4) :|: TRUE 172.38/44.63 f321_in(x22, x23) -> f325_in(x22, x23) :|: TRUE 172.38/44.63 f325_out(x24, x25) -> f321_out(x24, x25) :|: TRUE 172.38/44.63 f321_in(x26, x27) -> f324_in(x26, x27) :|: TRUE 172.38/44.63 f324_out(x28, x29) -> f321_out(x28, x29) :|: TRUE 172.38/44.63 f292_in(x34, x35, x36, x37) -> f319_in(x34, x35) :|: TRUE 172.38/44.63 f319_out(x38, x39) -> f320_in(x38, x40, x41, x39) :|: TRUE 172.38/44.63 f291_out(T128, T130) -> f317_out(T128, T130) :|: TRUE 172.38/44.63 f317_in(x42, x43) -> f291_in(x42, x43) :|: TRUE 172.38/44.63 f332_out(T161, T163) -> f325_out(T161, .(T162, T163)) :|: TRUE 172.38/44.63 f325_in(x46, .(x47, x48)) -> f332_in(x46, x48) :|: TRUE 172.38/44.63 f126_in(x55, x56, x57, x58) -> f286_in(x55, x57, x58, x56) :|: TRUE 172.38/44.63 f320_in(x59, x60, x61, x62) -> f2_in(x59, x60, x61, .(x59, x62)) :|: TRUE 172.38/44.63 f326_out -> f324_out(T153, .(T153, T154)) :|: TRUE 172.38/44.63 f324_in(x67, .(x67, x68)) -> f326_in :|: TRUE 172.38/44.63 f123_in(x81, x82, x83, x84) -> f126_in(x81, x82, x83, x84) :|: TRUE 172.38/44.63 f312_in(T119, .(.(T119, .(T120, [])), T121)) -> f314_in :|: TRUE 172.38/44.63 f314_out -> f312_out(x93, .(.(x93, .(x94, [])), x95)) :|: TRUE 172.38/44.63 f312_out(x96, x97) -> f307_out(x96, x97) :|: TRUE 172.38/44.63 f307_in(x98, x99) -> f313_in(x98, x99) :|: TRUE 172.38/44.63 f307_in(x100, x101) -> f312_in(x100, x101) :|: TRUE 172.38/44.63 f313_out(x102, x103) -> f307_out(x102, x103) :|: TRUE 172.38/44.63 f319_out(x104, x105) -> f332_out(x104, x105) :|: TRUE 172.38/44.63 f332_in(x106, x107) -> f319_in(x106, x107) :|: TRUE 172.38/44.63 f317_out(x108, x109) -> f313_out(x108, .(x110, x109)) :|: TRUE 172.38/44.63 f313_in(x115, .(x116, x117)) -> f317_in(x115, x117) :|: TRUE 172.38/44.63 f314_in -> f314_out :|: TRUE 172.38/44.63 f326_in -> f326_out :|: TRUE 172.38/44.63 172.38/44.63 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (288) 172.38/44.63 Obligation: 172.38/44.63 Rules: 172.38/44.63 f319_in(T100, T93) -> f321_in(T100, T93) :|: TRUE 172.38/44.63 f321_out(x, x1) -> f319_out(x, x1) :|: TRUE 172.38/44.63 f291_out(x2, x3) -> f292_in(x4, x5, x6, x3) :|: TRUE 172.38/44.63 f286_in(x7, x8, x9, x10) -> f291_in(x7, x8) :|: TRUE 172.38/44.63 f307_out(T90, T92) -> f291_out(T90, T92) :|: TRUE 172.38/44.63 f291_in(x16, x17) -> f307_in(x16, x17) :|: TRUE 172.38/44.63 f2_in(T1, T2, T3, T4) -> f123_in(T1, T2, T3, T4) :|: TRUE 172.38/44.63 f321_in(x22, x23) -> f325_in(x22, x23) :|: TRUE 172.38/44.63 f325_out(x24, x25) -> f321_out(x24, x25) :|: TRUE 172.38/44.63 f321_in(x26, x27) -> f324_in(x26, x27) :|: TRUE 172.38/44.63 f324_out(x28, x29) -> f321_out(x28, x29) :|: TRUE 172.38/44.63 f292_in(x34, x35, x36, x37) -> f319_in(x34, x35) :|: TRUE 172.38/44.63 f319_out(x38, x39) -> f320_in(x38, x40, x41, x39) :|: TRUE 172.38/44.63 f291_out(T128, T130) -> f317_out(T128, T130) :|: TRUE 172.38/44.63 f317_in(x42, x43) -> f291_in(x42, x43) :|: TRUE 172.38/44.63 f332_out(T161, T163) -> f325_out(T161, .(T162, T163)) :|: TRUE 172.38/44.63 f325_in(x46, .(x47, x48)) -> f332_in(x46, x48) :|: TRUE 172.38/44.63 f126_in(x55, x56, x57, x58) -> f286_in(x55, x57, x58, x56) :|: TRUE 172.38/44.63 f320_in(x59, x60, x61, x62) -> f2_in(x59, x60, x61, .(x59, x62)) :|: TRUE 172.38/44.63 f326_out -> f324_out(T153, .(T153, T154)) :|: TRUE 172.38/44.63 f324_in(x67, .(x67, x68)) -> f326_in :|: TRUE 172.38/44.63 f123_in(x81, x82, x83, x84) -> f126_in(x81, x82, x83, x84) :|: TRUE 172.38/44.63 f312_in(T119, .(.(T119, .(T120, [])), T121)) -> f314_in :|: TRUE 172.38/44.63 f314_out -> f312_out(x93, .(.(x93, .(x94, [])), x95)) :|: TRUE 172.38/44.63 f312_out(x96, x97) -> f307_out(x96, x97) :|: TRUE 172.38/44.63 f307_in(x98, x99) -> f313_in(x98, x99) :|: TRUE 172.38/44.63 f307_in(x100, x101) -> f312_in(x100, x101) :|: TRUE 172.38/44.63 f313_out(x102, x103) -> f307_out(x102, x103) :|: TRUE 172.38/44.63 f319_out(x104, x105) -> f332_out(x104, x105) :|: TRUE 172.38/44.63 f332_in(x106, x107) -> f319_in(x106, x107) :|: TRUE 172.38/44.63 f317_out(x108, x109) -> f313_out(x108, .(x110, x109)) :|: TRUE 172.38/44.63 f313_in(x115, .(x116, x117)) -> f317_in(x115, x117) :|: TRUE 172.38/44.63 f314_in -> f314_out :|: TRUE 172.38/44.63 f326_in -> f326_out :|: TRUE 172.38/44.63 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (289) IntTRSCompressionProof (EQUIVALENT) 172.38/44.63 Compressed rules. 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (290) 172.38/44.63 Obligation: 172.38/44.63 Rules: 172.38/44.63 f319_in(T100:0, .(x47:0, x48:0)) -> f319_in(T100:0, x48:0) :|: TRUE 172.38/44.63 f307_in(x98:0, .(x116:0, x117:0)) -> f307_in(x98:0, x117:0) :|: TRUE 172.38/44.63 f321_out(x:0, x1:0) -> f307_in(x:0, x41:0) :|: TRUE 172.38/44.63 f319_in(x, .(x, x1)) -> f321_out(x2, .(x2, x3)) :|: TRUE 172.38/44.63 f307_out(T90:0, T92:0) -> f319_in(x4:0, x5:0) :|: TRUE 172.38/44.63 f307_out(x4, x5) -> f307_out(x4, .(x6, x5)) :|: TRUE 172.38/44.63 f307_in(x100:0, .(.(x100:0, .(T120:0, [])), T121:0)) -> f307_out(x93:0, .(.(x93:0, .(x94:0, [])), x95:0)) :|: TRUE 172.38/44.63 f321_out(x7, x8) -> f321_out(x7, .(x9, x8)) :|: TRUE 172.38/44.63 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (291) IRSFormatTransformerProof (EQUIVALENT) 172.38/44.63 Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (292) 172.38/44.63 Obligation: 172.38/44.63 Rules: 172.38/44.63 f319_in(T100:0, .(x47:0, x48:0)) -> f319_in(T100:0, x48:0) :|: TRUE 172.38/44.63 f307_in(x98:0, .(x116:0, x117:0)) -> f307_in(x98:0, x117:0) :|: TRUE 172.38/44.63 f321_out(x:0, x1:0) -> f307_in(x:0, x41:0) :|: TRUE 172.38/44.63 f319_in(x, .(x, x1)) -> f321_out(x2, .(x2, x3)) :|: TRUE 172.38/44.63 f307_out(T90:0, T92:0) -> f319_in(x4:0, x5:0) :|: TRUE 172.38/44.63 f307_out(x4, x5) -> f307_out(x4, .(x6, x5)) :|: TRUE 172.38/44.63 f307_in(x100:0, .(.(x100:0, .(T120:0, [])), T121:0)) -> f307_out(x93:0, .(.(x93:0, .(x94:0, [])), x95:0)) :|: TRUE 172.38/44.63 f321_out(x7, x8) -> f321_out(x7, .(x9, x8)) :|: TRUE 172.38/44.63 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (293) IRSwTTerminationDigraphProof (EQUIVALENT) 172.38/44.63 Constructed termination digraph! 172.38/44.63 Nodes: 172.38/44.63 (1) f319_in(T100:0, .(x47:0, x48:0)) -> f319_in(T100:0, x48:0) :|: TRUE 172.38/44.63 (2) f307_in(x98:0, .(x116:0, x117:0)) -> f307_in(x98:0, x117:0) :|: TRUE 172.38/44.63 (3) f321_out(x:0, x1:0) -> f307_in(x:0, x41:0) :|: TRUE 172.38/44.63 (4) f319_in(x, .(x, x1)) -> f321_out(x2, .(x2, x3)) :|: TRUE 172.38/44.63 (5) f307_out(T90:0, T92:0) -> f319_in(x4:0, x5:0) :|: TRUE 172.38/44.63 (6) f307_out(x4, x5) -> f307_out(x4, .(x6, x5)) :|: TRUE 172.38/44.63 (7) f307_in(x100:0, .(.(x100:0, .(T120:0, [])), T121:0)) -> f307_out(x93:0, .(.(x93:0, .(x94:0, [])), x95:0)) :|: TRUE 172.38/44.63 (8) f321_out(x7, x8) -> f321_out(x7, .(x9, x8)) :|: TRUE 172.38/44.63 172.38/44.63 Arcs: 172.38/44.63 (1) -> (1), (4) 172.38/44.63 (2) -> (2), (7) 172.38/44.63 (3) -> (2), (7) 172.38/44.63 (4) -> (3), (8) 172.38/44.63 (5) -> (1), (4) 172.38/44.63 (6) -> (5), (6) 172.38/44.63 (7) -> (5), (6) 172.38/44.63 (8) -> (3), (8) 172.38/44.63 172.38/44.63 This digraph is fully evaluated! 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (294) 172.38/44.63 Obligation: 172.38/44.63 172.38/44.63 Termination digraph: 172.38/44.63 Nodes: 172.38/44.63 (1) f319_in(T100:0, .(x47:0, x48:0)) -> f319_in(T100:0, x48:0) :|: TRUE 172.38/44.63 (2) f307_out(T90:0, T92:0) -> f319_in(x4:0, x5:0) :|: TRUE 172.38/44.63 (3) f307_out(x4, x5) -> f307_out(x4, .(x6, x5)) :|: TRUE 172.38/44.63 (4) f307_in(x100:0, .(.(x100:0, .(T120:0, [])), T121:0)) -> f307_out(x93:0, .(.(x93:0, .(x94:0, [])), x95:0)) :|: TRUE 172.38/44.63 (5) f307_in(x98:0, .(x116:0, x117:0)) -> f307_in(x98:0, x117:0) :|: TRUE 172.38/44.63 (6) f321_out(x:0, x1:0) -> f307_in(x:0, x41:0) :|: TRUE 172.38/44.63 (7) f321_out(x7, x8) -> f321_out(x7, .(x9, x8)) :|: TRUE 172.38/44.63 (8) f319_in(x, .(x, x1)) -> f321_out(x2, .(x2, x3)) :|: TRUE 172.38/44.63 172.38/44.63 Arcs: 172.38/44.63 (1) -> (1), (8) 172.38/44.63 (2) -> (1), (8) 172.38/44.63 (3) -> (2), (3) 172.38/44.63 (4) -> (2), (3) 172.38/44.63 (5) -> (4), (5) 172.38/44.63 (6) -> (4), (5) 172.38/44.63 (7) -> (6), (7) 172.38/44.63 (8) -> (6), (7) 172.38/44.63 172.38/44.63 This digraph is fully evaluated! 172.38/44.63 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (295) IntTRSUnneededArgumentFilterProof (EQUIVALENT) 172.38/44.63 Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements: 172.38/44.63 172.38/44.63 f321_out(x1, x2) -> f321_out(x1) 172.38/44.63 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (296) 172.38/44.63 Obligation: 172.38/44.63 Rules: 172.38/44.63 f319_in(T100:0, .(x47:0, x48:0)) -> f319_in(T100:0, x48:0) :|: TRUE 172.38/44.63 f307_out(T90:0, T92:0) -> f319_in(x4:0, x5:0) :|: TRUE 172.38/44.63 f307_out(x4, x5) -> f307_out(x4, .(x6, x5)) :|: TRUE 172.38/44.63 f307_in(x100:0, .(.(x100:0, .(T120:0, [])), T121:0)) -> f307_out(x93:0, .(.(x93:0, .(x94:0, [])), x95:0)) :|: TRUE 172.38/44.63 f307_in(x98:0, .(x116:0, x117:0)) -> f307_in(x98:0, x117:0) :|: TRUE 172.38/44.63 f321_out(x:0) -> f307_in(x:0, x41:0) :|: TRUE 172.38/44.63 f321_out(x7) -> f321_out(x7) :|: TRUE 172.38/44.63 f319_in(x, .(x, x1)) -> f321_out(x2) :|: TRUE 172.38/44.63 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (297) IRSwTToIntTRSProof (SOUND) 172.38/44.63 Applied path-length measure to transform intTRS with terms to intTRS. 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (298) 172.38/44.63 Obligation: 172.38/44.63 Rules: 172.38/44.63 f319_in(x, .(x1, x2)) -> f319_in(x, x2) :|: TRUE 172.38/44.63 f307_out(x3, x4) -> f319_in(x5, x6) :|: TRUE 172.38/44.63 f307_out(x7, x8) -> f307_out(x7, .(x9, x8)) :|: TRUE 172.38/44.63 f307_in(x101, .(.(x10, .(x11, [])), x12)) -> f307_out(x13, .(.(x13, .(x14, [])), x15)) :|: TRUE && x10 = x101 172.38/44.63 f307_in(x16, .(x17, x18)) -> f307_in(x16, x18) :|: TRUE 172.38/44.63 f321_out(x19) -> f307_in(x19, x20) :|: TRUE 172.38/44.63 f321_out(x21) -> f321_out(x21) :|: TRUE 172.38/44.63 f319_in(x221, .(x22, x23)) -> f321_out(x24) :|: TRUE && x22 = x221 172.38/44.63 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (299) IntTRSCompressionProof (EQUIVALENT) 172.38/44.63 Compressed rules. 172.38/44.63 ---------------------------------------- 172.38/44.63 172.38/44.63 (300) 172.38/44.63 Obligation: 172.38/44.63 Rules: 172.38/44.63 f319_in(x:0, .(x1:0, x2:0)) -> f319_in(x:0, x2:0) :|: TRUE 172.38/44.63 f307_in(x16:0, .(x17:0, x18:0)) -> f307_in(x16:0, x18:0) :|: TRUE 172.38/44.63 f307_out(x3:0, x4:0) -> f319_in(x5:0, x6:0) :|: TRUE 172.38/44.63 f307_in(x101:0, .(.(x101:0, .(x11:0, [])), x12:0)) -> f307_out(x13:0, .(.(x13:0, .(x14:0, [])), x15:0)) :|: TRUE 172.38/44.63 f307_out(x7:0, x8:0) -> f307_out(x7:0, .(x9:0, x8:0)) :|: TRUE 172.38/44.63 f321_out(x19:0) -> f307_in(x19:0, x20:0) :|: TRUE 172.38/44.63 f321_out(x21:0) -> f321_out(x21:0) :|: TRUE 172.38/44.63 f319_in(x221:0, .(x221:0, x23:0)) -> f321_out(x24:0) :|: TRUE 172.38/44.67 EOF