/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.pl /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- MAYBE proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern reach(g,g,g,g) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2) PiTRS (3) DependencyPairsProof [EQUIVALENT, 7 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [SOUND, 0 ms] (11) QDP (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] (13) YES (14) PiDP (15) UsableRulesProof [EQUIVALENT, 0 ms] (16) PiDP (17) PiDPToQDPProof [EQUIVALENT, 0 ms] (18) QDP (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] (20) YES (21) PiDP (22) UsableRulesProof [EQUIVALENT, 0 ms] (23) PiDP (24) PiDPToQDPProof [SOUND, 5 ms] (25) QDP (26) TransformationProof [SOUND, 0 ms] (27) QDP (28) TransformationProof [SOUND, 0 ms] (29) QDP (30) TransformationProof [EQUIVALENT, 0 ms] (31) QDP (32) TransformationProof [EQUIVALENT, 0 ms] (33) QDP (34) TransformationProof [EQUIVALENT, 0 ms] (35) QDP (36) TransformationProof [EQUIVALENT, 0 ms] (37) QDP (38) TransformationProof [EQUIVALENT, 0 ms] (39) QDP (40) TransformationProof [EQUIVALENT, 0 ms] (41) QDP (42) TransformationProof [EQUIVALENT, 0 ms] (43) QDP (44) TransformationProof [EQUIVALENT, 0 ms] (45) QDP (46) TransformationProof [EQUIVALENT, 0 ms] (47) QDP (48) TransformationProof [EQUIVALENT, 3 ms] (49) QDP (50) TransformationProof [EQUIVALENT, 0 ms] (51) QDP (52) TransformationProof [EQUIVALENT, 0 ms] (53) QDP (54) TransformationProof [EQUIVALENT, 0 ms] (55) QDP (56) TransformationProof [EQUIVALENT, 0 ms] (57) QDP (58) TransformationProof [EQUIVALENT, 0 ms] (59) QDP (60) TransformationProof [EQUIVALENT, 0 ms] (61) QDP (62) TransformationProof [EQUIVALENT, 0 ms] (63) QDP (64) TransformationProof [EQUIVALENT, 0 ms] (65) QDP (66) TransformationProof [EQUIVALENT, 9 ms] (67) QDP (68) TransformationProof [EQUIVALENT, 0 ms] (69) QDP (70) TransformationProof [EQUIVALENT, 0 ms] (71) QDP (72) TransformationProof [EQUIVALENT, 0 ms] (73) QDP (74) TransformationProof [EQUIVALENT, 0 ms] (75) QDP (76) PrologToPiTRSProof [SOUND, 0 ms] (77) PiTRS (78) DependencyPairsProof [EQUIVALENT, 6 ms] (79) PiDP (80) DependencyGraphProof [EQUIVALENT, 0 ms] (81) AND (82) PiDP (83) UsableRulesProof [EQUIVALENT, 0 ms] (84) PiDP (85) PiDPToQDPProof [SOUND, 0 ms] (86) QDP (87) QDPSizeChangeProof [EQUIVALENT, 0 ms] (88) YES (89) PiDP (90) UsableRulesProof [EQUIVALENT, 0 ms] (91) PiDP (92) PiDPToQDPProof [EQUIVALENT, 0 ms] (93) QDP (94) QDPSizeChangeProof [EQUIVALENT, 0 ms] (95) YES (96) PiDP (97) UsableRulesProof [EQUIVALENT, 0 ms] (98) PiDP (99) PiDPToQDPProof [SOUND, 0 ms] (100) QDP (101) TransformationProof [SOUND, 0 ms] (102) QDP (103) TransformationProof [SOUND, 0 ms] (104) QDP (105) TransformationProof [EQUIVALENT, 0 ms] (106) QDP (107) TransformationProof [EQUIVALENT, 0 ms] (108) QDP (109) TransformationProof [EQUIVALENT, 0 ms] (110) QDP (111) TransformationProof [EQUIVALENT, 0 ms] (112) QDP (113) TransformationProof [EQUIVALENT, 0 ms] (114) QDP (115) TransformationProof [EQUIVALENT, 0 ms] (116) QDP (117) TransformationProof [EQUIVALENT, 9 ms] (118) QDP (119) TransformationProof [EQUIVALENT, 0 ms] (120) QDP (121) TransformationProof [EQUIVALENT, 10 ms] (122) QDP (123) TransformationProof [EQUIVALENT, 0 ms] (124) QDP (125) TransformationProof [EQUIVALENT, 0 ms] (126) QDP (127) TransformationProof [EQUIVALENT, 0 ms] (128) QDP (129) TransformationProof [EQUIVALENT, 0 ms] (130) QDP (131) TransformationProof [EQUIVALENT, 12 ms] (132) QDP (133) TransformationProof [EQUIVALENT, 0 ms] (134) QDP (135) TransformationProof [EQUIVALENT, 0 ms] (136) QDP (137) TransformationProof [EQUIVALENT, 4 ms] (138) QDP (139) TransformationProof [EQUIVALENT, 0 ms] (140) QDP (141) TransformationProof [EQUIVALENT, 0 ms] (142) QDP (143) TransformationProof [EQUIVALENT, 0 ms] (144) QDP (145) TransformationProof [EQUIVALENT, 0 ms] (146) QDP (147) TransformationProof [EQUIVALENT, 0 ms] (148) QDP (149) TransformationProof [EQUIVALENT, 0 ms] (150) QDP (151) PrologToDTProblemTransformerProof [SOUND, 46 ms] (152) TRIPLES (153) TriplesToPiDPProof [SOUND, 0 ms] (154) PiDP (155) DependencyGraphProof [EQUIVALENT, 0 ms] (156) AND (157) PiDP (158) UsableRulesProof [EQUIVALENT, 0 ms] (159) PiDP (160) PiDPToQDPProof [EQUIVALENT, 0 ms] (161) QDP (162) QDPSizeChangeProof [EQUIVALENT, 0 ms] (163) YES (164) PiDP (165) UsableRulesProof [EQUIVALENT, 0 ms] (166) PiDP (167) PiDPToQDPProof [SOUND, 0 ms] (168) QDP (169) QDPSizeChangeProof [EQUIVALENT, 0 ms] (170) YES (171) PiDP (172) UsableRulesProof [EQUIVALENT, 0 ms] (173) PiDP (174) PiDPToQDPProof [EQUIVALENT, 0 ms] (175) QDP (176) QDPSizeChangeProof [EQUIVALENT, 0 ms] (177) YES (178) PiDP (179) PiDPToQDPProof [SOUND, 0 ms] (180) QDP (181) TransformationProof [SOUND, 0 ms] (182) QDP (183) TransformationProof [SOUND, 0 ms] (184) QDP (185) TransformationProof [EQUIVALENT, 0 ms] (186) QDP (187) TransformationProof [EQUIVALENT, 0 ms] (188) QDP (189) TransformationProof [EQUIVALENT, 0 ms] (190) QDP (191) TransformationProof [EQUIVALENT, 0 ms] (192) QDP (193) TransformationProof [EQUIVALENT, 0 ms] (194) QDP (195) TransformationProof [EQUIVALENT, 0 ms] (196) QDP (197) TransformationProof [EQUIVALENT, 0 ms] (198) QDP (199) TransformationProof [EQUIVALENT, 0 ms] (200) QDP (201) TransformationProof [EQUIVALENT, 0 ms] (202) QDP (203) TransformationProof [EQUIVALENT, 0 ms] (204) QDP (205) TransformationProof [EQUIVALENT, 0 ms] (206) QDP (207) TransformationProof [EQUIVALENT, 0 ms] (208) QDP (209) TransformationProof [EQUIVALENT, 0 ms] (210) QDP (211) TransformationProof [EQUIVALENT, 0 ms] (212) QDP (213) TransformationProof [EQUIVALENT, 0 ms] (214) QDP (215) TransformationProof [EQUIVALENT, 0 ms] (216) QDP (217) TransformationProof [EQUIVALENT, 8 ms] (218) QDP (219) TransformationProof [EQUIVALENT, 0 ms] (220) QDP (221) TransformationProof [EQUIVALENT, 0 ms] (222) QDP (223) PrologToTRSTransformerProof [SOUND, 35 ms] (224) QTRS (225) DependencyPairsProof [EQUIVALENT, 8 ms] (226) QDP (227) DependencyGraphProof [EQUIVALENT, 1 ms] (228) AND (229) QDP (230) UsableRulesProof [EQUIVALENT, 0 ms] (231) QDP (232) QDPSizeChangeProof [EQUIVALENT, 0 ms] (233) YES (234) QDP (235) UsableRulesProof [EQUIVALENT, 0 ms] (236) QDP (237) QDPSizeChangeProof [EQUIVALENT, 0 ms] (238) YES (239) QDP (240) UsableRulesProof [EQUIVALENT, 0 ms] (241) QDP (242) QDPSizeChangeProof [EQUIVALENT, 0 ms] (243) YES (244) QDP (245) NonLoopProof [COMPLETE, 8377 ms] (246) NO (247) PrologToIRSwTTransformerProof [SOUND, 32 ms] (248) AND (249) IRSwT (250) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (251) TRUE (252) IRSwT (253) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (254) IRSwT (255) IntTRSCompressionProof [EQUIVALENT, 26 ms] (256) IRSwT (257) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (258) IRSwT (259) IRSwTTerminationDigraphProof [EQUIVALENT, 6 ms] (260) IRSwT (261) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 16 ms] (262) IRSwT (263) TempFilterProof [SOUND, 2 ms] (264) IRSwT (265) IRSwTToQDPProof [SOUND, 0 ms] (266) QDP (267) QDPSizeChangeProof [EQUIVALENT, 0 ms] (268) YES (269) IRSwT (270) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (271) IRSwT (272) IntTRSCompressionProof [EQUIVALENT, 2 ms] (273) IRSwT (274) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (275) IRSwT (276) IRSwTTerminationDigraphProof [EQUIVALENT, 1 ms] (277) IRSwT (278) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (279) IRSwT (280) TempFilterProof [SOUND, 2 ms] (281) IRSwT (282) IRSwTToQDPProof [SOUND, 0 ms] (283) QDP (284) QDPSizeChangeProof [EQUIVALENT, 0 ms] (285) YES (286) IRSwT (287) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (288) IRSwT (289) IntTRSCompressionProof [EQUIVALENT, 19 ms] (290) IRSwT (291) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (292) IRSwT (293) IRSwTTerminationDigraphProof [EQUIVALENT, 34 ms] (294) IRSwT (295) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (296) IRSwT (297) IRSwTToIntTRSProof [SOUND, 20 ms] (298) IRSwT (299) IntTRSCompressionProof [EQUIVALENT, 10 ms] (300) IRSwT ---------------------------------------- (0) Obligation: Clauses: reach(X, Y, Edges, Visited) :- member(.(X, .(Y, [])), Edges). reach(X, Z, Edges, Visited) :- ','(member1(.(X, .(Y, [])), Edges), ','(member(Y, Visited), reach(Y, Z, Edges, .(Y, Visited)))). member(H, .(H, L)). member(X, .(H, L)) :- member(X, L). member1(H, .(H, L)). member1(X, .(H, L)) :- member1(X, L). Query: reach(g,g,g,g) ---------------------------------------- (1) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: reach_in_4: (b,b,b,b) member_in_2: (b,b) member1_in_2: (f,b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: reach_in_gggg(X, Y, Edges, Visited) -> U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_gggg(X, Y, Edges, Visited) reach_in_gggg(X, Z, Edges, Visited) -> U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(X, .(H, L)) -> U6_ag(X, H, L, member1_in_ag(X, L)) U6_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 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))) U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) -> reach_out_gggg(X, Z, Edges, Visited) The argument filtering Pi contains the following mapping: reach_in_gggg(x1, x2, x3, x4) = reach_in_gggg(x1, x2, x3, x4) U1_gggg(x1, x2, x3, x4, x5) = U1_gggg(x1, x2, x3, x4, x5) member_in_gg(x1, x2) = member_in_gg(x1, x2) .(x1, x2) = .(x1, x2) member_out_gg(x1, x2) = member_out_gg(x1, x2) U5_gg(x1, x2, x3, x4) = U5_gg(x1, x2, x3, x4) [] = [] reach_out_gggg(x1, x2, x3, x4) = reach_out_gggg(x1, x2, x3, x4) U2_gggg(x1, x2, x3, x4, x5) = U2_gggg(x1, x2, x3, x4, x5) member1_in_ag(x1, x2) = member1_in_ag(x2) member1_out_ag(x1, x2) = member1_out_ag(x1, x2) U6_ag(x1, x2, x3, x4) = U6_ag(x2, x3, x4) U3_gggg(x1, x2, x3, x4, x5, x6) = U3_gggg(x1, x2, x3, x4, x5, x6) U4_gggg(x1, x2, x3, x4, x5) = U4_gggg(x1, x2, x3, x4, x5) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: reach_in_gggg(X, Y, Edges, Visited) -> U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_gggg(X, Y, Edges, Visited) reach_in_gggg(X, Z, Edges, Visited) -> U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(X, .(H, L)) -> U6_ag(X, H, L, member1_in_ag(X, L)) U6_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 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))) U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) -> reach_out_gggg(X, Z, Edges, Visited) The argument filtering Pi contains the following mapping: reach_in_gggg(x1, x2, x3, x4) = reach_in_gggg(x1, x2, x3, x4) U1_gggg(x1, x2, x3, x4, x5) = U1_gggg(x1, x2, x3, x4, x5) member_in_gg(x1, x2) = member_in_gg(x1, x2) .(x1, x2) = .(x1, x2) member_out_gg(x1, x2) = member_out_gg(x1, x2) U5_gg(x1, x2, x3, x4) = U5_gg(x1, x2, x3, x4) [] = [] reach_out_gggg(x1, x2, x3, x4) = reach_out_gggg(x1, x2, x3, x4) U2_gggg(x1, x2, x3, x4, x5) = U2_gggg(x1, x2, x3, x4, x5) member1_in_ag(x1, x2) = member1_in_ag(x2) member1_out_ag(x1, x2) = member1_out_ag(x1, x2) U6_ag(x1, x2, x3, x4) = U6_ag(x2, x3, x4) U3_gggg(x1, x2, x3, x4, x5, x6) = U3_gggg(x1, x2, x3, x4, x5, x6) U4_gggg(x1, x2, x3, x4, x5) = U4_gggg(x1, x2, x3, x4, x5) ---------------------------------------- (3) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: Pi DP problem: The TRS P consists of the following rules: REACH_IN_GGGG(X, Y, Edges, Visited) -> U1_GGGG(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges)) REACH_IN_GGGG(X, Y, Edges, Visited) -> MEMBER_IN_GG(.(X, .(Y, [])), Edges) MEMBER_IN_GG(X, .(H, L)) -> U5_GG(X, H, L, member_in_gg(X, L)) MEMBER_IN_GG(X, .(H, L)) -> MEMBER_IN_GG(X, L) REACH_IN_GGGG(X, Z, Edges, Visited) -> U2_GGGG(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) REACH_IN_GGGG(X, Z, Edges, Visited) -> MEMBER1_IN_AG(.(X, .(Y, [])), Edges) MEMBER1_IN_AG(X, .(H, L)) -> U6_AG(X, H, L, member1_in_ag(X, L)) MEMBER1_IN_AG(X, .(H, L)) -> MEMBER1_IN_AG(X, L) U2_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_GGGG(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) U2_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> MEMBER_IN_GG(Y, Visited) 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))) U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited)) The TRS R consists of the following rules: reach_in_gggg(X, Y, Edges, Visited) -> U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_gggg(X, Y, Edges, Visited) reach_in_gggg(X, Z, Edges, Visited) -> U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(X, .(H, L)) -> U6_ag(X, H, L, member1_in_ag(X, L)) U6_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 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))) U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) -> reach_out_gggg(X, Z, Edges, Visited) The argument filtering Pi contains the following mapping: reach_in_gggg(x1, x2, x3, x4) = reach_in_gggg(x1, x2, x3, x4) U1_gggg(x1, x2, x3, x4, x5) = U1_gggg(x1, x2, x3, x4, x5) member_in_gg(x1, x2) = member_in_gg(x1, x2) .(x1, x2) = .(x1, x2) member_out_gg(x1, x2) = member_out_gg(x1, x2) U5_gg(x1, x2, x3, x4) = U5_gg(x1, x2, x3, x4) [] = [] reach_out_gggg(x1, x2, x3, x4) = reach_out_gggg(x1, x2, x3, x4) U2_gggg(x1, x2, x3, x4, x5) = U2_gggg(x1, x2, x3, x4, x5) member1_in_ag(x1, x2) = member1_in_ag(x2) member1_out_ag(x1, x2) = member1_out_ag(x1, x2) U6_ag(x1, x2, x3, x4) = U6_ag(x2, x3, x4) U3_gggg(x1, x2, x3, x4, x5, x6) = U3_gggg(x1, x2, x3, x4, x5, x6) U4_gggg(x1, x2, x3, x4, x5) = U4_gggg(x1, x2, x3, x4, x5) REACH_IN_GGGG(x1, x2, x3, x4) = REACH_IN_GGGG(x1, x2, x3, x4) U1_GGGG(x1, x2, x3, x4, x5) = U1_GGGG(x1, x2, x3, x4, x5) MEMBER_IN_GG(x1, x2) = MEMBER_IN_GG(x1, x2) U5_GG(x1, x2, x3, x4) = U5_GG(x1, x2, x3, x4) U2_GGGG(x1, x2, x3, x4, x5) = U2_GGGG(x1, x2, x3, x4, x5) MEMBER1_IN_AG(x1, x2) = MEMBER1_IN_AG(x2) U6_AG(x1, x2, x3, x4) = U6_AG(x2, x3, x4) U3_GGGG(x1, x2, x3, x4, x5, x6) = U3_GGGG(x1, x2, x3, x4, x5, x6) U4_GGGG(x1, x2, x3, x4, x5) = U4_GGGG(x1, x2, x3, x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: REACH_IN_GGGG(X, Y, Edges, Visited) -> U1_GGGG(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges)) REACH_IN_GGGG(X, Y, Edges, Visited) -> MEMBER_IN_GG(.(X, .(Y, [])), Edges) MEMBER_IN_GG(X, .(H, L)) -> U5_GG(X, H, L, member_in_gg(X, L)) MEMBER_IN_GG(X, .(H, L)) -> MEMBER_IN_GG(X, L) REACH_IN_GGGG(X, Z, Edges, Visited) -> U2_GGGG(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) REACH_IN_GGGG(X, Z, Edges, Visited) -> MEMBER1_IN_AG(.(X, .(Y, [])), Edges) MEMBER1_IN_AG(X, .(H, L)) -> U6_AG(X, H, L, member1_in_ag(X, L)) MEMBER1_IN_AG(X, .(H, L)) -> MEMBER1_IN_AG(X, L) U2_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_GGGG(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) U2_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> MEMBER_IN_GG(Y, Visited) 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))) U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited)) The TRS R consists of the following rules: reach_in_gggg(X, Y, Edges, Visited) -> U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_gggg(X, Y, Edges, Visited) reach_in_gggg(X, Z, Edges, Visited) -> U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(X, .(H, L)) -> U6_ag(X, H, L, member1_in_ag(X, L)) U6_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 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))) U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) -> reach_out_gggg(X, Z, Edges, Visited) The argument filtering Pi contains the following mapping: reach_in_gggg(x1, x2, x3, x4) = reach_in_gggg(x1, x2, x3, x4) U1_gggg(x1, x2, x3, x4, x5) = U1_gggg(x1, x2, x3, x4, x5) member_in_gg(x1, x2) = member_in_gg(x1, x2) .(x1, x2) = .(x1, x2) member_out_gg(x1, x2) = member_out_gg(x1, x2) U5_gg(x1, x2, x3, x4) = U5_gg(x1, x2, x3, x4) [] = [] reach_out_gggg(x1, x2, x3, x4) = reach_out_gggg(x1, x2, x3, x4) U2_gggg(x1, x2, x3, x4, x5) = U2_gggg(x1, x2, x3, x4, x5) member1_in_ag(x1, x2) = member1_in_ag(x2) member1_out_ag(x1, x2) = member1_out_ag(x1, x2) U6_ag(x1, x2, x3, x4) = U6_ag(x2, x3, x4) U3_gggg(x1, x2, x3, x4, x5, x6) = U3_gggg(x1, x2, x3, x4, x5, x6) U4_gggg(x1, x2, x3, x4, x5) = U4_gggg(x1, x2, x3, x4, x5) REACH_IN_GGGG(x1, x2, x3, x4) = REACH_IN_GGGG(x1, x2, x3, x4) U1_GGGG(x1, x2, x3, x4, x5) = U1_GGGG(x1, x2, x3, x4, x5) MEMBER_IN_GG(x1, x2) = MEMBER_IN_GG(x1, x2) U5_GG(x1, x2, x3, x4) = U5_GG(x1, x2, x3, x4) U2_GGGG(x1, x2, x3, x4, x5) = U2_GGGG(x1, x2, x3, x4, x5) MEMBER1_IN_AG(x1, x2) = MEMBER1_IN_AG(x2) U6_AG(x1, x2, x3, x4) = U6_AG(x2, x3, x4) U3_GGGG(x1, x2, x3, x4, x5, x6) = U3_GGGG(x1, x2, x3, x4, x5, x6) U4_GGGG(x1, x2, x3, x4, x5) = U4_GGGG(x1, x2, x3, x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 7 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER1_IN_AG(X, .(H, L)) -> MEMBER1_IN_AG(X, L) The TRS R consists of the following rules: reach_in_gggg(X, Y, Edges, Visited) -> U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_gggg(X, Y, Edges, Visited) reach_in_gggg(X, Z, Edges, Visited) -> U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(X, .(H, L)) -> U6_ag(X, H, L, member1_in_ag(X, L)) U6_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 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))) U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) -> reach_out_gggg(X, Z, Edges, Visited) The argument filtering Pi contains the following mapping: reach_in_gggg(x1, x2, x3, x4) = reach_in_gggg(x1, x2, x3, x4) U1_gggg(x1, x2, x3, x4, x5) = U1_gggg(x1, x2, x3, x4, x5) member_in_gg(x1, x2) = member_in_gg(x1, x2) .(x1, x2) = .(x1, x2) member_out_gg(x1, x2) = member_out_gg(x1, x2) U5_gg(x1, x2, x3, x4) = U5_gg(x1, x2, x3, x4) [] = [] reach_out_gggg(x1, x2, x3, x4) = reach_out_gggg(x1, x2, x3, x4) U2_gggg(x1, x2, x3, x4, x5) = U2_gggg(x1, x2, x3, x4, x5) member1_in_ag(x1, x2) = member1_in_ag(x2) member1_out_ag(x1, x2) = member1_out_ag(x1, x2) U6_ag(x1, x2, x3, x4) = U6_ag(x2, x3, x4) U3_gggg(x1, x2, x3, x4, x5, x6) = U3_gggg(x1, x2, x3, x4, x5, x6) U4_gggg(x1, x2, x3, x4, x5) = U4_gggg(x1, x2, x3, x4, x5) MEMBER1_IN_AG(x1, x2) = MEMBER1_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (8) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (9) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER1_IN_AG(X, .(H, L)) -> MEMBER1_IN_AG(X, L) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) MEMBER1_IN_AG(x1, x2) = MEMBER1_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (10) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: MEMBER1_IN_AG(.(H, L)) -> MEMBER1_IN_AG(L) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (12) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *MEMBER1_IN_AG(.(H, L)) -> MEMBER1_IN_AG(L) The graph contains the following edges 1 > 1 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER_IN_GG(X, .(H, L)) -> MEMBER_IN_GG(X, L) The TRS R consists of the following rules: reach_in_gggg(X, Y, Edges, Visited) -> U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_gggg(X, Y, Edges, Visited) reach_in_gggg(X, Z, Edges, Visited) -> U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(X, .(H, L)) -> U6_ag(X, H, L, member1_in_ag(X, L)) U6_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 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))) U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) -> reach_out_gggg(X, Z, Edges, Visited) The argument filtering Pi contains the following mapping: reach_in_gggg(x1, x2, x3, x4) = reach_in_gggg(x1, x2, x3, x4) U1_gggg(x1, x2, x3, x4, x5) = U1_gggg(x1, x2, x3, x4, x5) member_in_gg(x1, x2) = member_in_gg(x1, x2) .(x1, x2) = .(x1, x2) member_out_gg(x1, x2) = member_out_gg(x1, x2) U5_gg(x1, x2, x3, x4) = U5_gg(x1, x2, x3, x4) [] = [] reach_out_gggg(x1, x2, x3, x4) = reach_out_gggg(x1, x2, x3, x4) U2_gggg(x1, x2, x3, x4, x5) = U2_gggg(x1, x2, x3, x4, x5) member1_in_ag(x1, x2) = member1_in_ag(x2) member1_out_ag(x1, x2) = member1_out_ag(x1, x2) U6_ag(x1, x2, x3, x4) = U6_ag(x2, x3, x4) U3_gggg(x1, x2, x3, x4, x5, x6) = U3_gggg(x1, x2, x3, x4, x5, x6) U4_gggg(x1, x2, x3, x4, x5) = U4_gggg(x1, x2, x3, x4, x5) MEMBER_IN_GG(x1, x2) = MEMBER_IN_GG(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (15) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER_IN_GG(X, .(H, L)) -> MEMBER_IN_GG(X, L) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: MEMBER_IN_GG(X, .(H, L)) -> MEMBER_IN_GG(X, L) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (19) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *MEMBER_IN_GG(X, .(H, L)) -> MEMBER_IN_GG(X, L) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (20) YES ---------------------------------------- (21) Obligation: Pi DP problem: The TRS P consists of the following rules: REACH_IN_GGGG(X, Z, Edges, Visited) -> U2_GGGG(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) U2_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_GGGG(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited)) The TRS R consists of the following rules: reach_in_gggg(X, Y, Edges, Visited) -> U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_gggg(X, Y, Edges, Visited) reach_in_gggg(X, Z, Edges, Visited) -> U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(X, .(H, L)) -> U6_ag(X, H, L, member1_in_ag(X, L)) U6_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 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))) U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) -> reach_out_gggg(X, Z, Edges, Visited) The argument filtering Pi contains the following mapping: reach_in_gggg(x1, x2, x3, x4) = reach_in_gggg(x1, x2, x3, x4) U1_gggg(x1, x2, x3, x4, x5) = U1_gggg(x1, x2, x3, x4, x5) member_in_gg(x1, x2) = member_in_gg(x1, x2) .(x1, x2) = .(x1, x2) member_out_gg(x1, x2) = member_out_gg(x1, x2) U5_gg(x1, x2, x3, x4) = U5_gg(x1, x2, x3, x4) [] = [] reach_out_gggg(x1, x2, x3, x4) = reach_out_gggg(x1, x2, x3, x4) U2_gggg(x1, x2, x3, x4, x5) = U2_gggg(x1, x2, x3, x4, x5) member1_in_ag(x1, x2) = member1_in_ag(x2) member1_out_ag(x1, x2) = member1_out_ag(x1, x2) U6_ag(x1, x2, x3, x4) = U6_ag(x2, x3, x4) U3_gggg(x1, x2, x3, x4, x5, x6) = U3_gggg(x1, x2, x3, x4, x5, x6) U4_gggg(x1, x2, x3, x4, x5) = U4_gggg(x1, x2, x3, x4, x5) REACH_IN_GGGG(x1, x2, x3, x4) = REACH_IN_GGGG(x1, x2, x3, x4) U2_GGGG(x1, x2, x3, x4, x5) = U2_GGGG(x1, x2, x3, x4, x5) U3_GGGG(x1, x2, x3, x4, x5, x6) = U3_GGGG(x1, x2, x3, x4, x5, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (22) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (23) Obligation: Pi DP problem: The TRS P consists of the following rules: REACH_IN_GGGG(X, Z, Edges, Visited) -> U2_GGGG(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) U2_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_GGGG(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited)) The TRS R consists of the following rules: member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(X, .(H, L)) -> U6_ag(X, H, L, member1_in_ag(X, L)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U6_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) The argument filtering Pi contains the following mapping: member_in_gg(x1, x2) = member_in_gg(x1, x2) .(x1, x2) = .(x1, x2) member_out_gg(x1, x2) = member_out_gg(x1, x2) U5_gg(x1, x2, x3, x4) = U5_gg(x1, x2, x3, x4) [] = [] member1_in_ag(x1, x2) = member1_in_ag(x2) member1_out_ag(x1, x2) = member1_out_ag(x1, x2) U6_ag(x1, x2, x3, x4) = U6_ag(x2, x3, x4) REACH_IN_GGGG(x1, x2, x3, x4) = REACH_IN_GGGG(x1, x2, x3, x4) U2_GGGG(x1, x2, x3, x4, x5) = U2_GGGG(x1, x2, x3, x4, x5) U3_GGGG(x1, x2, x3, x4, x5, x6) = U3_GGGG(x1, x2, x3, x4, x5, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (24) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (25) Obligation: Q DP problem: The TRS P consists of the following rules: REACH_IN_GGGG(X, Z, Edges, Visited) -> U2_GGGG(X, Z, Edges, Visited, member1_in_ag(Edges)) U2_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_GGGG(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited)) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0, x1, x2) U5_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (26) TransformationProof (SOUND) 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]: (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)))) (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)))) ---------------------------------------- (27) Obligation: Q DP problem: The TRS P consists of the following rules: U2_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_GGGG(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited)) 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, U6_ag(x0, x1, member1_in_ag(x1))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0, x1, x2) U5_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (28) TransformationProof (SOUND) 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]: (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)))) (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)))) ---------------------------------------- (29) Obligation: Q DP problem: The TRS P consists of the following rules: U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited)) 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, U6_ag(x0, x1, member1_in_ag(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))) 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))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0, x1, x2) U5_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (30) TransformationProof (EQUIVALENT) 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]: (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)))) (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)))) ---------------------------------------- (31) Obligation: Q DP problem: The TRS P consists of the following rules: 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, U6_ag(x0, x1, member1_in_ag(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))) 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))) 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), z5, member_out_gg(z5, .(z3, z4))) -> REACH_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0, x1, x2) U5_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (32) TransformationProof (EQUIVALENT) 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]: (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)))) (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)))) ---------------------------------------- (33) Obligation: Q DP problem: The TRS P consists of the following rules: REACH_IN_GGGG(y0, y1, .(x0, x1), y3) -> U2_GGGG(y0, y1, .(x0, x1), y3, U6_ag(x0, x1, member1_in_ag(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))) 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))) 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), z5, member_out_gg(z5, .(z3, z4))) -> REACH_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4))) 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(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), member1_out_ag(x2, .(x2, x3))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0, x1, x2) U5_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (34) TransformationProof (EQUIVALENT) 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]: (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)))) (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)))) ---------------------------------------- (35) Obligation: Q DP problem: The TRS P consists of the following rules: 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, .(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))) 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), z5, member_out_gg(z5, .(z3, z4))) -> REACH_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4))) 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(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(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(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0, x1, x2) U5_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (36) TransformationProof (EQUIVALENT) 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]: (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))))) (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))))) (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))))) (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))))) ---------------------------------------- (37) Obligation: Q DP problem: The TRS P consists of the following rules: 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))) 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), z5, member_out_gg(z5, .(z3, z4))) -> REACH_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4))) 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(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(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(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3))) 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, .(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, .(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, .(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)))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0, x1, x2) U5_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (38) TransformationProof (EQUIVALENT) 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]: (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))))) (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))))) (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))))) (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))))) ---------------------------------------- (39) Obligation: Q DP problem: The TRS P consists of the following rules: 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), z5, member_out_gg(z5, .(z3, z4))) -> REACH_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4))) 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(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(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(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3))) 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(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)))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0, x1, x2) U5_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (40) TransformationProof (EQUIVALENT) 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]: (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))))) (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))))) (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))))) (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))))) ---------------------------------------- (41) Obligation: Q DP problem: The TRS P consists of the following rules: U3_GGGG(z0, z1, z2, .(z3, z4), z5, member_out_gg(z5, .(z3, z4))) -> REACH_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4))) 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(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(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(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3))) 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(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)))) 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, .(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, .(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, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0, x1, x2) U5_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (42) TransformationProof (EQUIVALENT) 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]: (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))))) (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))))) (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))))) (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))))) (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))))) (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))))) (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))))) (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))))) ---------------------------------------- (43) Obligation: Q DP problem: The TRS P consists of the following rules: 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(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(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(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3))) 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(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)))) 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(z4, z5)), z6, member_out_gg(z6, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0, x1, x2) U5_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (44) TransformationProof (EQUIVALENT) 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]: (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)))) (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)))) (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)))) (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)))) ---------------------------------------- (45) Obligation: Q DP problem: The TRS P consists of the following rules: 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(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(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3))) 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(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)))) 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(z4, z5)), z6, member_out_gg(z6, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 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, .(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, .(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, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0, x1, x2) U5_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (46) TransformationProof (EQUIVALENT) 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]: (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)))) (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)))) (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)))) (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)))) (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)))) (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)))) (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)))) (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)))) ---------------------------------------- (47) Obligation: Q DP problem: The TRS P consists of the following rules: 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(z5, z1, .(x2, x3), .(z5, .(z3, z4))) -> U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3))) 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(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)))) 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(z4, z5)), z6, member_out_gg(z6, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 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, .(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, .(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, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(z2, .(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))) 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(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(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) -> U2_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0, x1, x2) U5_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (48) TransformationProof (EQUIVALENT) 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]: (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)))) (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)))) (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)))) (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)))) ---------------------------------------- (49) Obligation: Q DP problem: The TRS P consists of the following rules: 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))) 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(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)))) 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(z4, z5)), z6, member_out_gg(z6, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 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, .(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, .(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, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(z2, .(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))) 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(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(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(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, .(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, .(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, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U6_ag(z2, z3, member1_in_ag(z3))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0, x1, x2) U5_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (50) TransformationProof (EQUIVALENT) 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]: (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)))) (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)))) (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)))) (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)))) (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)))) (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)))) (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)))) (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)))) ---------------------------------------- (51) Obligation: Q DP problem: The TRS P consists of the following rules: 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(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)))) 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(z4, z5)), z6, member_out_gg(z6, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 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, .(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, .(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, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(z2, .(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))) 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(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(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(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, .(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, .(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, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U6_ag(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))) 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(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(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))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0, x1, x2) U5_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (52) TransformationProof (EQUIVALENT) 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]: (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)))))) (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)))))) ---------------------------------------- (53) Obligation: Q DP problem: The TRS P consists of the following rules: 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, .(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, .(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, .(.(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, .(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, .(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, .(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)))) 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(z4, z5)), z6, member_out_gg(z6, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 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, .(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, .(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, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(z2, .(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))) 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(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(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(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, .(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, .(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, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U6_ag(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))) 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(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(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))) 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, .(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))))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0, x1, x2) U5_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (54) TransformationProof (EQUIVALENT) 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]: (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)))))) (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)))))) (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)))))) (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)))))) ---------------------------------------- (55) Obligation: Q DP problem: The TRS P consists of the following rules: 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, .(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, .(.(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, .(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, .(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, .(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)))) 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(z4, z5)), z6, member_out_gg(z6, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 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, .(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, .(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, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(z2, .(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))) 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(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(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(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, .(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, .(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, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U6_ag(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))) 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(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(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))) 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, .(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, [])), 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, .(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))))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0, x1, x2) U5_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (56) TransformationProof (EQUIVALENT) 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]: (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)))))) (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)))))) (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)))))) (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)))))) ---------------------------------------- (57) Obligation: Q DP problem: The TRS P consists of the following rules: 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, .(.(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, .(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, .(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, .(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)))) 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(z4, z5)), z6, member_out_gg(z6, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 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, .(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, .(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, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(z2, .(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))) 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(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(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(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, .(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, .(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, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U6_ag(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))) 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(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(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))) 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, .(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, [])), 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, .(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, .(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, .(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))))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0, x1, x2) U5_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (58) TransformationProof (EQUIVALENT) 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]: (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) ---------------------------------------- (59) Obligation: Q DP problem: The TRS P consists of the following rules: 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, .(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, .(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, .(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)))) 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(z4, z5)), z6, member_out_gg(z6, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 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, .(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, .(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, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(z2, .(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))) 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(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(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(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, .(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, .(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, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U6_ag(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))) 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(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(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))) 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, .(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, [])), 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, .(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, .(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, .(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, .(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, .(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, 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, .(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))))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0, x1, x2) U5_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (60) TransformationProof (EQUIVALENT) 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]: (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)))))) (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)))))) (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)))))) (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)))))) ---------------------------------------- (61) Obligation: Q DP problem: The TRS P consists of the following rules: 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, .(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, .(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)))) 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(z4, z5)), z6, member_out_gg(z6, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 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, .(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, .(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, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(z2, .(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))) 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(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(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(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, .(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, .(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, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U6_ag(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))) 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(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(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))) 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, .(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, [])), 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, .(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, .(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, .(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, .(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, .(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, 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, .(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, .(.(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, .(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, .(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, .(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))))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0, x1, x2) U5_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (62) TransformationProof (EQUIVALENT) 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]: (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) ---------------------------------------- (63) Obligation: Q DP problem: The TRS P consists of the following rules: 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, .(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)))) 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(z4, z5)), z6, member_out_gg(z6, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 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, .(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, .(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, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(z2, .(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))) 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(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(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(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, .(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, .(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, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U6_ag(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))) 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(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(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))) 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, .(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, [])), 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, .(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, .(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, .(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, .(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, .(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, 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, .(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, .(.(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, .(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, .(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, .(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, .(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, .(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))))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0, x1, x2) U5_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (64) TransformationProof (EQUIVALENT) 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]: (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) ---------------------------------------- (65) Obligation: Q DP problem: The TRS P consists of the following rules: 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)))) 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(z4, z5)), z6, member_out_gg(z6, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 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, .(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, .(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, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(z2, .(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))) 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(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(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(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, .(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, .(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, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U6_ag(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))) 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(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(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))) 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, .(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, [])), 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, .(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, .(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, .(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, .(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, .(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, 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, .(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, .(.(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, .(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, .(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, .(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, .(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, .(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, .(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, .(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, .(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, .(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))))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0, x1, x2) U5_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (66) TransformationProof (EQUIVALENT) 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]: (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) ---------------------------------------- (67) Obligation: Q DP problem: The TRS P consists of the following rules: 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(z4, z5)), z6, member_out_gg(z6, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 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, .(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, .(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, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(z2, .(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))) 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(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(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(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, .(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, .(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, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U6_ag(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))) 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(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(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))) 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, .(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, [])), 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, .(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, .(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, .(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, .(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, .(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, 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, .(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, .(.(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, .(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, .(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, .(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, .(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, .(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, .(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, .(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, .(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, .(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, .(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, .(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, 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, .(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))))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0, x1, x2) U5_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (68) TransformationProof (EQUIVALENT) 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]: (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)))) ---------------------------------------- (69) Obligation: Q DP problem: The TRS P consists of the following rules: 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(z4, z5)), z6, member_out_gg(z6, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 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, .(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, .(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(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, .(z4, z5)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(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))) 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(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, .(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, .(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, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U6_ag(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))) 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(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(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))) 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, .(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, [])), 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, .(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, .(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, .(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, .(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, .(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, 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, .(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, .(.(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, .(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, .(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, .(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, .(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, .(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, .(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, .(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, .(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, .(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, .(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, .(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, 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, .(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))))) 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))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0, x1, x2) U5_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (70) TransformationProof (EQUIVALENT) 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]: (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)))) ---------------------------------------- (71) Obligation: Q DP problem: The TRS P consists of the following rules: 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(z4, z5)), z6, member_out_gg(z6, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 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, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) 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, .(z4, z5)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(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))) 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(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, .(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, .(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, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U6_ag(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))) 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(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(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))) 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, .(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, [])), 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, .(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, .(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, .(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, .(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, .(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, 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, .(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, .(.(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, .(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, .(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, .(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, .(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, .(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, .(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, .(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, .(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, .(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, .(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, .(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, 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, .(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))))) 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, .(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))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0, x1, x2) U5_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (72) TransformationProof (EQUIVALENT) 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]: (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)))) ---------------------------------------- (73) Obligation: Q DP problem: The TRS P consists of the following rules: 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(z4, z5)), z6, member_out_gg(z6, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 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, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) 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, .(z4, z5)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(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))) 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, .(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, .(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, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U6_ag(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))) 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(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(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))) 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, .(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, [])), 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, .(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, .(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, .(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, .(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, .(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, 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, .(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, .(.(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, .(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, .(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, .(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, .(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, .(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, .(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, .(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, .(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, .(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, .(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, .(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, 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, .(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))))) 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, .(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, .(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))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0, x1, x2) U5_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (74) TransformationProof (EQUIVALENT) 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]: (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)))) ---------------------------------------- (75) Obligation: Q DP problem: The TRS P consists of the following rules: 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(z4, z5)), z6, member_out_gg(z6, .(z0, .(z4, z5)))) -> REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 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, .(z3, z4)))) -> U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) 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, .(z4, z5)))) -> U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(z2, [])), z3))) 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, .(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, .(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, .(z4, z5)))) -> U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U6_ag(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))) 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(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(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))) 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, .(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, [])), 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, .(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, .(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, .(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, .(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, .(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, 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, .(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, .(.(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, .(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, .(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, .(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, .(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, .(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, .(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, .(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, .(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, .(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, .(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, .(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, 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, .(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))))) 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, .(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, .(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, .(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))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(.(H, L)) -> U6_ag(H, L, member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U6_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0, x1, x2) U5_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (76) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: reach_in_4: (b,b,b,b) member_in_2: (b,b) member1_in_2: (f,b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: reach_in_gggg(X, Y, Edges, Visited) -> U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_gggg(X, Y, Edges, Visited) reach_in_gggg(X, Z, Edges, Visited) -> U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(X, .(H, L)) -> U6_ag(X, H, L, member1_in_ag(X, L)) U6_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 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))) U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) -> reach_out_gggg(X, Z, Edges, Visited) The argument filtering Pi contains the following mapping: reach_in_gggg(x1, x2, x3, x4) = reach_in_gggg(x1, x2, x3, x4) U1_gggg(x1, x2, x3, x4, x5) = U1_gggg(x5) member_in_gg(x1, x2) = member_in_gg(x1, x2) .(x1, x2) = .(x1, x2) member_out_gg(x1, x2) = member_out_gg U5_gg(x1, x2, x3, x4) = U5_gg(x4) [] = [] reach_out_gggg(x1, x2, x3, x4) = reach_out_gggg U2_gggg(x1, x2, x3, x4, x5) = U2_gggg(x2, x3, x4, x5) member1_in_ag(x1, x2) = member1_in_ag(x2) member1_out_ag(x1, x2) = member1_out_ag(x1) U6_ag(x1, x2, x3, x4) = U6_ag(x4) U3_gggg(x1, x2, x3, x4, x5, x6) = U3_gggg(x2, x3, x4, x5, x6) U4_gggg(x1, x2, x3, x4, x5) = U4_gggg(x5) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (77) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: reach_in_gggg(X, Y, Edges, Visited) -> U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_gggg(X, Y, Edges, Visited) reach_in_gggg(X, Z, Edges, Visited) -> U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(X, .(H, L)) -> U6_ag(X, H, L, member1_in_ag(X, L)) U6_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 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))) U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) -> reach_out_gggg(X, Z, Edges, Visited) The argument filtering Pi contains the following mapping: reach_in_gggg(x1, x2, x3, x4) = reach_in_gggg(x1, x2, x3, x4) U1_gggg(x1, x2, x3, x4, x5) = U1_gggg(x5) member_in_gg(x1, x2) = member_in_gg(x1, x2) .(x1, x2) = .(x1, x2) member_out_gg(x1, x2) = member_out_gg U5_gg(x1, x2, x3, x4) = U5_gg(x4) [] = [] reach_out_gggg(x1, x2, x3, x4) = reach_out_gggg U2_gggg(x1, x2, x3, x4, x5) = U2_gggg(x2, x3, x4, x5) member1_in_ag(x1, x2) = member1_in_ag(x2) member1_out_ag(x1, x2) = member1_out_ag(x1) U6_ag(x1, x2, x3, x4) = U6_ag(x4) U3_gggg(x1, x2, x3, x4, x5, x6) = U3_gggg(x2, x3, x4, x5, x6) U4_gggg(x1, x2, x3, x4, x5) = U4_gggg(x5) ---------------------------------------- (78) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: Pi DP problem: The TRS P consists of the following rules: REACH_IN_GGGG(X, Y, Edges, Visited) -> U1_GGGG(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges)) REACH_IN_GGGG(X, Y, Edges, Visited) -> MEMBER_IN_GG(.(X, .(Y, [])), Edges) MEMBER_IN_GG(X, .(H, L)) -> U5_GG(X, H, L, member_in_gg(X, L)) MEMBER_IN_GG(X, .(H, L)) -> MEMBER_IN_GG(X, L) REACH_IN_GGGG(X, Z, Edges, Visited) -> U2_GGGG(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) REACH_IN_GGGG(X, Z, Edges, Visited) -> MEMBER1_IN_AG(.(X, .(Y, [])), Edges) MEMBER1_IN_AG(X, .(H, L)) -> U6_AG(X, H, L, member1_in_ag(X, L)) MEMBER1_IN_AG(X, .(H, L)) -> MEMBER1_IN_AG(X, L) U2_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_GGGG(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) U2_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> MEMBER_IN_GG(Y, Visited) 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))) U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited)) The TRS R consists of the following rules: reach_in_gggg(X, Y, Edges, Visited) -> U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_gggg(X, Y, Edges, Visited) reach_in_gggg(X, Z, Edges, Visited) -> U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(X, .(H, L)) -> U6_ag(X, H, L, member1_in_ag(X, L)) U6_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 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))) U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) -> reach_out_gggg(X, Z, Edges, Visited) The argument filtering Pi contains the following mapping: reach_in_gggg(x1, x2, x3, x4) = reach_in_gggg(x1, x2, x3, x4) U1_gggg(x1, x2, x3, x4, x5) = U1_gggg(x5) member_in_gg(x1, x2) = member_in_gg(x1, x2) .(x1, x2) = .(x1, x2) member_out_gg(x1, x2) = member_out_gg U5_gg(x1, x2, x3, x4) = U5_gg(x4) [] = [] reach_out_gggg(x1, x2, x3, x4) = reach_out_gggg U2_gggg(x1, x2, x3, x4, x5) = U2_gggg(x2, x3, x4, x5) member1_in_ag(x1, x2) = member1_in_ag(x2) member1_out_ag(x1, x2) = member1_out_ag(x1) U6_ag(x1, x2, x3, x4) = U6_ag(x4) U3_gggg(x1, x2, x3, x4, x5, x6) = U3_gggg(x2, x3, x4, x5, x6) U4_gggg(x1, x2, x3, x4, x5) = U4_gggg(x5) REACH_IN_GGGG(x1, x2, x3, x4) = REACH_IN_GGGG(x1, x2, x3, x4) U1_GGGG(x1, x2, x3, x4, x5) = U1_GGGG(x5) MEMBER_IN_GG(x1, x2) = MEMBER_IN_GG(x1, x2) U5_GG(x1, x2, x3, x4) = U5_GG(x4) U2_GGGG(x1, x2, x3, x4, x5) = U2_GGGG(x2, x3, x4, x5) MEMBER1_IN_AG(x1, x2) = MEMBER1_IN_AG(x2) U6_AG(x1, x2, x3, x4) = U6_AG(x4) U3_GGGG(x1, x2, x3, x4, x5, x6) = U3_GGGG(x2, x3, x4, x5, x6) U4_GGGG(x1, x2, x3, x4, x5) = U4_GGGG(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (79) Obligation: Pi DP problem: The TRS P consists of the following rules: REACH_IN_GGGG(X, Y, Edges, Visited) -> U1_GGGG(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges)) REACH_IN_GGGG(X, Y, Edges, Visited) -> MEMBER_IN_GG(.(X, .(Y, [])), Edges) MEMBER_IN_GG(X, .(H, L)) -> U5_GG(X, H, L, member_in_gg(X, L)) MEMBER_IN_GG(X, .(H, L)) -> MEMBER_IN_GG(X, L) REACH_IN_GGGG(X, Z, Edges, Visited) -> U2_GGGG(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) REACH_IN_GGGG(X, Z, Edges, Visited) -> MEMBER1_IN_AG(.(X, .(Y, [])), Edges) MEMBER1_IN_AG(X, .(H, L)) -> U6_AG(X, H, L, member1_in_ag(X, L)) MEMBER1_IN_AG(X, .(H, L)) -> MEMBER1_IN_AG(X, L) U2_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_GGGG(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) U2_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> MEMBER_IN_GG(Y, Visited) 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))) U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited)) The TRS R consists of the following rules: reach_in_gggg(X, Y, Edges, Visited) -> U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_gggg(X, Y, Edges, Visited) reach_in_gggg(X, Z, Edges, Visited) -> U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(X, .(H, L)) -> U6_ag(X, H, L, member1_in_ag(X, L)) U6_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 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))) U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) -> reach_out_gggg(X, Z, Edges, Visited) The argument filtering Pi contains the following mapping: reach_in_gggg(x1, x2, x3, x4) = reach_in_gggg(x1, x2, x3, x4) U1_gggg(x1, x2, x3, x4, x5) = U1_gggg(x5) member_in_gg(x1, x2) = member_in_gg(x1, x2) .(x1, x2) = .(x1, x2) member_out_gg(x1, x2) = member_out_gg U5_gg(x1, x2, x3, x4) = U5_gg(x4) [] = [] reach_out_gggg(x1, x2, x3, x4) = reach_out_gggg U2_gggg(x1, x2, x3, x4, x5) = U2_gggg(x2, x3, x4, x5) member1_in_ag(x1, x2) = member1_in_ag(x2) member1_out_ag(x1, x2) = member1_out_ag(x1) U6_ag(x1, x2, x3, x4) = U6_ag(x4) U3_gggg(x1, x2, x3, x4, x5, x6) = U3_gggg(x2, x3, x4, x5, x6) U4_gggg(x1, x2, x3, x4, x5) = U4_gggg(x5) REACH_IN_GGGG(x1, x2, x3, x4) = REACH_IN_GGGG(x1, x2, x3, x4) U1_GGGG(x1, x2, x3, x4, x5) = U1_GGGG(x5) MEMBER_IN_GG(x1, x2) = MEMBER_IN_GG(x1, x2) U5_GG(x1, x2, x3, x4) = U5_GG(x4) U2_GGGG(x1, x2, x3, x4, x5) = U2_GGGG(x2, x3, x4, x5) MEMBER1_IN_AG(x1, x2) = MEMBER1_IN_AG(x2) U6_AG(x1, x2, x3, x4) = U6_AG(x4) U3_GGGG(x1, x2, x3, x4, x5, x6) = U3_GGGG(x2, x3, x4, x5, x6) U4_GGGG(x1, x2, x3, x4, x5) = U4_GGGG(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (80) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 7 less nodes. ---------------------------------------- (81) Complex Obligation (AND) ---------------------------------------- (82) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER1_IN_AG(X, .(H, L)) -> MEMBER1_IN_AG(X, L) The TRS R consists of the following rules: reach_in_gggg(X, Y, Edges, Visited) -> U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_gggg(X, Y, Edges, Visited) reach_in_gggg(X, Z, Edges, Visited) -> U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(X, .(H, L)) -> U6_ag(X, H, L, member1_in_ag(X, L)) U6_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 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))) U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) -> reach_out_gggg(X, Z, Edges, Visited) The argument filtering Pi contains the following mapping: reach_in_gggg(x1, x2, x3, x4) = reach_in_gggg(x1, x2, x3, x4) U1_gggg(x1, x2, x3, x4, x5) = U1_gggg(x5) member_in_gg(x1, x2) = member_in_gg(x1, x2) .(x1, x2) = .(x1, x2) member_out_gg(x1, x2) = member_out_gg U5_gg(x1, x2, x3, x4) = U5_gg(x4) [] = [] reach_out_gggg(x1, x2, x3, x4) = reach_out_gggg U2_gggg(x1, x2, x3, x4, x5) = U2_gggg(x2, x3, x4, x5) member1_in_ag(x1, x2) = member1_in_ag(x2) member1_out_ag(x1, x2) = member1_out_ag(x1) U6_ag(x1, x2, x3, x4) = U6_ag(x4) U3_gggg(x1, x2, x3, x4, x5, x6) = U3_gggg(x2, x3, x4, x5, x6) U4_gggg(x1, x2, x3, x4, x5) = U4_gggg(x5) MEMBER1_IN_AG(x1, x2) = MEMBER1_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (83) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (84) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER1_IN_AG(X, .(H, L)) -> MEMBER1_IN_AG(X, L) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) MEMBER1_IN_AG(x1, x2) = MEMBER1_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (85) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (86) Obligation: Q DP problem: The TRS P consists of the following rules: MEMBER1_IN_AG(.(H, L)) -> MEMBER1_IN_AG(L) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (87) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *MEMBER1_IN_AG(.(H, L)) -> MEMBER1_IN_AG(L) The graph contains the following edges 1 > 1 ---------------------------------------- (88) YES ---------------------------------------- (89) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER_IN_GG(X, .(H, L)) -> MEMBER_IN_GG(X, L) The TRS R consists of the following rules: reach_in_gggg(X, Y, Edges, Visited) -> U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_gggg(X, Y, Edges, Visited) reach_in_gggg(X, Z, Edges, Visited) -> U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(X, .(H, L)) -> U6_ag(X, H, L, member1_in_ag(X, L)) U6_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 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))) U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) -> reach_out_gggg(X, Z, Edges, Visited) The argument filtering Pi contains the following mapping: reach_in_gggg(x1, x2, x3, x4) = reach_in_gggg(x1, x2, x3, x4) U1_gggg(x1, x2, x3, x4, x5) = U1_gggg(x5) member_in_gg(x1, x2) = member_in_gg(x1, x2) .(x1, x2) = .(x1, x2) member_out_gg(x1, x2) = member_out_gg U5_gg(x1, x2, x3, x4) = U5_gg(x4) [] = [] reach_out_gggg(x1, x2, x3, x4) = reach_out_gggg U2_gggg(x1, x2, x3, x4, x5) = U2_gggg(x2, x3, x4, x5) member1_in_ag(x1, x2) = member1_in_ag(x2) member1_out_ag(x1, x2) = member1_out_ag(x1) U6_ag(x1, x2, x3, x4) = U6_ag(x4) U3_gggg(x1, x2, x3, x4, x5, x6) = U3_gggg(x2, x3, x4, x5, x6) U4_gggg(x1, x2, x3, x4, x5) = U4_gggg(x5) MEMBER_IN_GG(x1, x2) = MEMBER_IN_GG(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (90) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (91) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER_IN_GG(X, .(H, L)) -> MEMBER_IN_GG(X, L) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (92) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (93) Obligation: Q DP problem: The TRS P consists of the following rules: MEMBER_IN_GG(X, .(H, L)) -> MEMBER_IN_GG(X, L) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (94) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *MEMBER_IN_GG(X, .(H, L)) -> MEMBER_IN_GG(X, L) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (95) YES ---------------------------------------- (96) Obligation: Pi DP problem: The TRS P consists of the following rules: REACH_IN_GGGG(X, Z, Edges, Visited) -> U2_GGGG(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) U2_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_GGGG(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited)) The TRS R consists of the following rules: reach_in_gggg(X, Y, Edges, Visited) -> U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_gggg(X, Y, Edges, Visited) reach_in_gggg(X, Z, Edges, Visited) -> U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(X, .(H, L)) -> U6_ag(X, H, L, member1_in_ag(X, L)) U6_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) 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))) U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) -> reach_out_gggg(X, Z, Edges, Visited) The argument filtering Pi contains the following mapping: reach_in_gggg(x1, x2, x3, x4) = reach_in_gggg(x1, x2, x3, x4) U1_gggg(x1, x2, x3, x4, x5) = U1_gggg(x5) member_in_gg(x1, x2) = member_in_gg(x1, x2) .(x1, x2) = .(x1, x2) member_out_gg(x1, x2) = member_out_gg U5_gg(x1, x2, x3, x4) = U5_gg(x4) [] = [] reach_out_gggg(x1, x2, x3, x4) = reach_out_gggg U2_gggg(x1, x2, x3, x4, x5) = U2_gggg(x2, x3, x4, x5) member1_in_ag(x1, x2) = member1_in_ag(x2) member1_out_ag(x1, x2) = member1_out_ag(x1) U6_ag(x1, x2, x3, x4) = U6_ag(x4) U3_gggg(x1, x2, x3, x4, x5, x6) = U3_gggg(x2, x3, x4, x5, x6) U4_gggg(x1, x2, x3, x4, x5) = U4_gggg(x5) REACH_IN_GGGG(x1, x2, x3, x4) = REACH_IN_GGGG(x1, x2, x3, x4) U2_GGGG(x1, x2, x3, x4, x5) = U2_GGGG(x2, x3, x4, x5) U3_GGGG(x1, x2, x3, x4, x5, x6) = U3_GGGG(x2, x3, x4, x5, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (97) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (98) Obligation: Pi DP problem: The TRS P consists of the following rules: REACH_IN_GGGG(X, Z, Edges, Visited) -> U2_GGGG(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges)) U2_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_GGGG(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) -> REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited)) The TRS R consists of the following rules: member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L)) member1_in_ag(X, .(H, L)) -> U6_ag(X, H, L, member1_in_ag(X, L)) member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L)) member_in_gg(X, .(H, L)) -> U5_gg(X, H, L, member_in_gg(X, L)) U6_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L)) U5_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L)) The argument filtering Pi contains the following mapping: member_in_gg(x1, x2) = member_in_gg(x1, x2) .(x1, x2) = .(x1, x2) member_out_gg(x1, x2) = member_out_gg U5_gg(x1, x2, x3, x4) = U5_gg(x4) [] = [] member1_in_ag(x1, x2) = member1_in_ag(x2) member1_out_ag(x1, x2) = member1_out_ag(x1) U6_ag(x1, x2, x3, x4) = U6_ag(x4) REACH_IN_GGGG(x1, x2, x3, x4) = REACH_IN_GGGG(x1, x2, x3, x4) U2_GGGG(x1, x2, x3, x4, x5) = U2_GGGG(x2, x3, x4, x5) U3_GGGG(x1, x2, x3, x4, x5, x6) = U3_GGGG(x2, x3, x4, x5, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (99) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (100) Obligation: Q DP problem: The TRS P consists of the following rules: REACH_IN_GGGG(X, Z, Edges, Visited) -> U2_GGGG(Z, Edges, Visited, member1_in_ag(Edges)) U2_GGGG(Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])))) -> U3_GGGG(Z, Edges, Visited, Y, member_in_gg(Y, Visited)) U3_GGGG(Z, Edges, Visited, Y, member_out_gg) -> REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited)) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H) member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) U6_ag(member1_out_ag(X)) -> member1_out_ag(X) U5_gg(member_out_gg) -> member_out_gg The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0) U5_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (101) TransformationProof (SOUND) 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]: (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))) (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)))) ---------------------------------------- (102) Obligation: Q DP problem: The TRS P consists of the following rules: U2_GGGG(Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])))) -> U3_GGGG(Z, Edges, Visited, Y, member_in_gg(Y, Visited)) U3_GGGG(Z, Edges, Visited, Y, member_out_gg) -> REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited)) 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, U6_ag(member1_in_ag(x1))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H) member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) U6_ag(member1_out_ag(X)) -> member1_out_ag(X) U5_gg(member_out_gg) -> member_out_gg The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0) U5_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (103) TransformationProof (SOUND) 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]: (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)) (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)))) ---------------------------------------- (104) Obligation: Q DP problem: The TRS P consists of the following rules: U3_GGGG(Z, Edges, Visited, Y, member_out_gg) -> REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited)) 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, U6_ag(member1_in_ag(x1))) 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, .(x1, x2), member1_out_ag(.(y3, .(x0, [])))) -> U3_GGGG(y0, y1, .(x1, x2), x0, U5_gg(member_in_gg(x0, x2))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H) member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) U6_ag(member1_out_ag(X)) -> member1_out_ag(X) U5_gg(member_out_gg) -> member_out_gg The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0) U5_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (105) TransformationProof (EQUIVALENT) 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]: (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)))) (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)))) ---------------------------------------- (106) Obligation: Q DP problem: The TRS P consists of the following rules: 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, U6_ag(member1_in_ag(x1))) 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, .(x1, x2), member1_out_ag(.(y3, .(x0, [])))) -> U3_GGGG(y0, y1, .(x1, x2), x0, U5_gg(member_in_gg(x0, x2))) 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), z5, member_out_gg) -> REACH_IN_GGGG(z5, z0, z1, .(z5, .(z2, z3))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H) member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) U6_ag(member1_out_ag(X)) -> member1_out_ag(X) U5_gg(member_out_gg) -> member_out_gg The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0) U5_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (107) TransformationProof (EQUIVALENT) 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]: (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))) (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))) ---------------------------------------- (108) Obligation: Q DP problem: The TRS P consists of the following rules: REACH_IN_GGGG(y0, y1, .(x0, x1), y3) -> U2_GGGG(y1, .(x0, x1), y3, U6_ag(member1_in_ag(x1))) 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, .(x1, x2), member1_out_ag(.(y3, .(x0, [])))) -> U3_GGGG(y0, y1, .(x1, x2), x0, U5_gg(member_in_gg(x0, x2))) 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), z5, member_out_gg) -> REACH_IN_GGGG(z5, z0, z1, .(z5, .(z2, z3))) 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(z4, z0, .(x2, x3), .(z4, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), member1_out_ag(x2)) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H) member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) U6_ag(member1_out_ag(X)) -> member1_out_ag(X) U5_gg(member_out_gg) -> member_out_gg The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0) U5_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (109) TransformationProof (EQUIVALENT) 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]: (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)))) (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)))) ---------------------------------------- (110) Obligation: Q DP problem: The TRS P consists of the following rules: 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, .(x1, x2), member1_out_ag(.(y3, .(x0, [])))) -> U3_GGGG(y0, y1, .(x1, x2), x0, U5_gg(member_in_gg(x0, x2))) 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), z5, member_out_gg) -> REACH_IN_GGGG(z5, z0, z1, .(z5, .(z2, z3))) 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(z4, z0, .(x2, x3), .(z4, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), member1_out_ag(x2)) 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(z4, z0, .(x2, x3), .(z4, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U6_ag(member1_in_ag(x3))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H) member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) U6_ag(member1_out_ag(X)) -> member1_out_ag(X) U5_gg(member_out_gg) -> member_out_gg The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0) U5_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (111) TransformationProof (EQUIVALENT) 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]: (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)) (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)) (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)) (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)) ---------------------------------------- (112) Obligation: Q DP problem: The TRS P consists of the following rules: 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))) 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), z5, member_out_gg) -> REACH_IN_GGGG(z5, z0, z1, .(z5, .(z2, z3))) 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(z4, z0, .(x2, x3), .(z4, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), member1_out_ag(x2)) 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(z4, z0, .(x2, x3), .(z4, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U6_ag(member1_in_ag(x3))) 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, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), 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) 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) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H) member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) U6_ag(member1_out_ag(X)) -> member1_out_ag(X) U5_gg(member_out_gg) -> member_out_gg The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0) U5_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (113) TransformationProof (EQUIVALENT) 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]: (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))))) (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))))) (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))))) (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))))) ---------------------------------------- (114) Obligation: Q DP problem: The TRS P consists of the following rules: 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), z5, member_out_gg) -> REACH_IN_GGGG(z5, z0, z1, .(z5, .(z2, z3))) 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(z4, z0, .(x2, x3), .(z4, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), member1_out_ag(x2)) 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(z4, z0, .(x2, x3), .(z4, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U6_ag(member1_in_ag(x3))) 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, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), 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) 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, .(.(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, .(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, .(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, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H) member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) U6_ag(member1_out_ag(X)) -> member1_out_ag(X) U5_gg(member_out_gg) -> member_out_gg The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0) U5_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (115) TransformationProof (EQUIVALENT) 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]: (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))))) (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))))) (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))))) (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))))) ---------------------------------------- (116) Obligation: Q DP problem: The TRS P consists of the following rules: U3_GGGG(z0, z1, .(z2, z3), z5, member_out_gg) -> REACH_IN_GGGG(z5, z0, z1, .(z5, .(z2, z3))) 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(z4, z0, .(x2, x3), .(z4, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), member1_out_ag(x2)) 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(z4, z0, .(x2, x3), .(z4, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U6_ag(member1_in_ag(x3))) 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, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), 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) 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, .(.(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, .(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, .(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, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) 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, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) 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, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H) member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) U6_ag(member1_out_ag(X)) -> member1_out_ag(X) U5_gg(member_out_gg) -> member_out_gg The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0) U5_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (117) TransformationProof (EQUIVALENT) 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]: (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))))) (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))))) (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))))) (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))))) (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))))) (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))))) (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))))) (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))))) ---------------------------------------- (118) Obligation: Q DP problem: The TRS P consists of the following rules: 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(z4, z0, .(x2, x3), .(z4, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), member1_out_ag(x2)) 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(z4, z0, .(x2, x3), .(z4, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U6_ag(member1_in_ag(x3))) 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, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), 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) 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, .(.(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, .(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, .(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, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) 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, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) 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, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(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)))) 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, .(z3, z4)), z6, member_out_gg) -> REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4)))) U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) -> REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5)))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H) member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) U6_ag(member1_out_ag(X)) -> member1_out_ag(X) U5_gg(member_out_gg) -> member_out_gg The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0) U5_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (119) TransformationProof (EQUIVALENT) 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]: (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, []))))) (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, []))))) (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))) (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))) ---------------------------------------- (120) Obligation: Q DP problem: The TRS P consists of the following rules: 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(z2, z0, .(x2, x3), .(z2, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z2, .(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))) 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, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), 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) 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, .(.(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, .(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, .(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, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) 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, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) 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, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(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)))) 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, .(z3, z4)), z6, member_out_gg) -> REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4)))) U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) -> REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5)))) 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, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 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, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1)) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H) member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) U6_ag(member1_out_ag(X)) -> member1_out_ag(X) U5_gg(member_out_gg) -> member_out_gg The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0) U5_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (121) TransformationProof (EQUIVALENT) 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]: (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, []))))) (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, []))))) (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))) (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))) (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, []))))) (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, []))))) (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))) (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))) ---------------------------------------- (122) Obligation: Q DP problem: The TRS P consists of the following rules: 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(z4, z0, .(x2, x3), .(z4, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U6_ag(member1_in_ag(x3))) 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, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), 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) 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, .(.(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, .(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, .(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, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) 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, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) 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, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(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)))) 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, .(z3, z4)), z6, member_out_gg) -> REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4)))) U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) -> REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5)))) 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, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 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, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1)) 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, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, [])))) 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(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1)) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H) member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) U6_ag(member1_out_ag(X)) -> member1_out_ag(X) U5_gg(member_out_gg) -> member_out_gg The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0) U5_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (123) TransformationProof (EQUIVALENT) 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]: (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)))) (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)))) (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)))) (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)))) ---------------------------------------- (124) Obligation: Q DP problem: The TRS P consists of the following rules: REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) -> U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U6_ag(member1_in_ag(x3))) 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, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), 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) 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, .(.(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, .(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, .(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, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) 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, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) 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, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(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)))) 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, .(z3, z4)), z6, member_out_gg) -> REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4)))) U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) -> REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5)))) 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, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 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, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1)) 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, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, [])))) 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(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1)) 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, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3))) 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, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H) member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) U6_ag(member1_out_ag(X)) -> member1_out_ag(X) U5_gg(member_out_gg) -> member_out_gg The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0) U5_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (125) TransformationProof (EQUIVALENT) 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]: (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)))) (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)))) (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)))) (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)))) (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)))) (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)))) (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)))) (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)))) ---------------------------------------- (126) Obligation: Q DP problem: The TRS P consists of the following rules: 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, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), 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) 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, .(.(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, .(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, .(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, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) 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, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) 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, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(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)))) 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, .(z3, z4)), z6, member_out_gg) -> REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4)))) U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) -> REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5)))) 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, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 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, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1)) 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, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, [])))) 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(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1)) 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, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3))) 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, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 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, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3))) 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(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H) member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) U6_ag(member1_out_ag(X)) -> member1_out_ag(X) U5_gg(member_out_gg) -> member_out_gg The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0) U5_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (127) TransformationProof (EQUIVALENT) 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]: (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)) (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)) ---------------------------------------- (128) Obligation: Q DP problem: The TRS P consists of the following rules: 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, .(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, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg) 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, .(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, .(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, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) 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, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) 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, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(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)))) 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, .(z3, z4)), z6, member_out_gg) -> REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4)))) U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) -> REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5)))) 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, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 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, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1)) 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, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, [])))) 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(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1)) 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, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3))) 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, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 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, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3))) 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(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 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, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H) member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) U6_ag(member1_out_ag(X)) -> member1_out_ag(X) U5_gg(member_out_gg) -> member_out_gg The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0) U5_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (129) TransformationProof (EQUIVALENT) 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]: (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)) (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)) (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)) (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)) ---------------------------------------- (130) Obligation: Q DP problem: The TRS P consists of the following rules: 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, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg) 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, .(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, .(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, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) 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, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) 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, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(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)))) 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, .(z3, z4)), z6, member_out_gg) -> REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4)))) U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) -> REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5)))) 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, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 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, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1)) 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, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, [])))) 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(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1)) 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, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3))) 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, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 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, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3))) 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(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 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, .(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, .(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, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) -> U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H) member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) U6_ag(member1_out_ag(X)) -> member1_out_ag(X) U5_gg(member_out_gg) -> member_out_gg The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0) U5_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (131) TransformationProof (EQUIVALENT) 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]: (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)) (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)) (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)) (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)) (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)) (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)) ---------------------------------------- (132) Obligation: Q DP problem: The TRS P consists of the following rules: 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, .(.(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, .(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, .(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, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) 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, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) 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, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(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)))) 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, .(z3, z4)), z6, member_out_gg) -> REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4)))) U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) -> REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5)))) 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, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 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, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1)) 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, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, [])))) 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(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1)) 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, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3))) 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, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 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, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3))) 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(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 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, .(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, .(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, .(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, .(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, .(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, 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, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H) member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) U6_ag(member1_out_ag(X)) -> member1_out_ag(X) U5_gg(member_out_gg) -> member_out_gg The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0) U5_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (133) TransformationProof (EQUIVALENT) 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]: (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)) (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)) (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)) (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)) (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)) (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)) (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)) (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)) (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)) (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)) (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)) (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)) ---------------------------------------- (134) Obligation: Q DP problem: The TRS P consists of the following rules: 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, .(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, .(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, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) 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, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) 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, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(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)))) 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, .(z3, z4)), z6, member_out_gg) -> REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4)))) U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) -> REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5)))) 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, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 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, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1)) 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, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, [])))) 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(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1)) 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, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3))) 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, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 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, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3))) 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(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 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, .(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, .(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, .(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, .(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, .(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, 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, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(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) 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, 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, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H) member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) U6_ag(member1_out_ag(X)) -> member1_out_ag(X) U5_gg(member_out_gg) -> member_out_gg The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0) U5_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (135) TransformationProof (EQUIVALENT) 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]: (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)))))) (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)))))) (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)))))) (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)))))) ---------------------------------------- (136) Obligation: Q DP problem: The TRS P consists of the following rules: 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, .(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, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) 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, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) 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, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(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)))) 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, .(z3, z4)), z6, member_out_gg) -> REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4)))) U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) -> REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5)))) 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, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 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, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1)) 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, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, [])))) 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(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1)) 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, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3))) 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, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 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, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3))) 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(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 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, .(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, .(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, .(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, .(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, .(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, 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, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(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) 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, 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, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), 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, U5_gg(member_in_gg(z0, .(z0, .(z0, z4))))) 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, .(.(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, .(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))))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H) member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) U6_ag(member1_out_ag(X)) -> member1_out_ag(X) U5_gg(member_out_gg) -> member_out_gg The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0) U5_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (137) TransformationProof (EQUIVALENT) 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]: (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) ---------------------------------------- (138) Obligation: Q DP problem: The TRS P consists of the following rules: 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, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) 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, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) 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, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(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)))) 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, .(z3, z4)), z6, member_out_gg) -> REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4)))) U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) -> REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5)))) 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, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 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, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1)) 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, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, [])))) 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(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1)) 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, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3))) 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, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 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, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3))) 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(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 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, .(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, .(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, .(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, .(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, .(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, 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, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(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) 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, 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, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), 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, U5_gg(member_in_gg(z0, .(z0, .(z0, z4))))) 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, .(.(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, .(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, .(.(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, .(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, .(.(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, .(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))))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H) member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) U6_ag(member1_out_ag(X)) -> member1_out_ag(X) U5_gg(member_out_gg) -> member_out_gg The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0) U5_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (139) TransformationProof (EQUIVALENT) 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]: (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) ---------------------------------------- (140) Obligation: Q DP problem: The TRS P consists of the following rules: 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)))) 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, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) 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, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(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)))) 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, .(z3, z4)), z6, member_out_gg) -> REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4)))) U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) -> REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5)))) 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, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 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, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1)) 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, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, [])))) 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(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1)) 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, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3))) 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, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 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, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3))) 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(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 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, .(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, .(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, .(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, .(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, .(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, 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, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(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) 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, 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, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), 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, U5_gg(member_in_gg(z0, .(z0, .(z0, z4))))) 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, .(.(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, .(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, .(.(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, .(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, .(.(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, .(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, .(.(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, .(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, 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, .(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))))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H) member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) U6_ag(member1_out_ag(X)) -> member1_out_ag(X) U5_gg(member_out_gg) -> member_out_gg The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0) U5_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (141) TransformationProof (EQUIVALENT) 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]: (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) ---------------------------------------- (142) Obligation: Q DP problem: The TRS P consists of the following rules: 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, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) 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, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(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)))) 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, .(z3, z4)), z6, member_out_gg) -> REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4)))) U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) -> REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5)))) 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, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 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, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1)) 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, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, [])))) 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(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1)) 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, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3))) 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, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 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, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3))) 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(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 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, .(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, .(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, .(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, .(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, .(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, 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, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(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) 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, 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, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), 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, U5_gg(member_in_gg(z0, .(z0, .(z0, z4))))) 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, .(.(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, .(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, .(.(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, .(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, .(.(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, .(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, .(.(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, .(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, 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, .(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, .(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, .(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, 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, .(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))))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H) member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) U6_ag(member1_out_ag(X)) -> member1_out_ag(X) U5_gg(member_out_gg) -> member_out_gg The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0) U5_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (143) TransformationProof (EQUIVALENT) 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]: (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, []))))) ---------------------------------------- (144) Obligation: Q DP problem: The TRS P consists of the following rules: 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, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) 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, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(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)))) 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, .(z3, z4)), z6, member_out_gg) -> REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4)))) U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) -> REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5)))) 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, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, [])))) 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(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, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, [])))) 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(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1)) 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, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3))) 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, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 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, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3))) 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(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 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, .(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, .(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, .(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, .(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, .(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, 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, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(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) 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, 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, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), 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, U5_gg(member_in_gg(z0, .(z0, .(z0, z4))))) 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, .(.(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, .(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, .(.(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, .(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, .(.(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, .(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, .(.(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, .(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, 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, .(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, .(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, .(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, 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, .(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))))) 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, [])))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H) member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) U6_ag(member1_out_ag(X)) -> member1_out_ag(X) U5_gg(member_out_gg) -> member_out_gg The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0) U5_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (145) TransformationProof (EQUIVALENT) 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]: (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, []))))) ---------------------------------------- (146) Obligation: Q DP problem: The TRS P consists of the following rules: 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, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) 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, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(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)))) 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, .(z3, z4)), z6, member_out_gg) -> REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4)))) U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) -> REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5)))) 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, .(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, .(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, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, [])))) 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(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1)) 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, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3))) 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, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 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, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3))) 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(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 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, .(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, .(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, .(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, .(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, .(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, 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, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(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) 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, 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, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), 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, U5_gg(member_in_gg(z0, .(z0, .(z0, z4))))) 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, .(.(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, .(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, .(.(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, .(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, .(.(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, .(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, .(.(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, .(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, 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, .(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, .(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, .(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, 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, .(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))))) 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, .(x4, x5)))) -> U2_GGGG(x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x4, x5))), member1_out_ag(.(y_1, .(y_2, [])))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H) member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) U6_ag(member1_out_ag(X)) -> member1_out_ag(X) U5_gg(member_out_gg) -> member_out_gg The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0) U5_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (147) TransformationProof (EQUIVALENT) 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]: (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, []))))) ---------------------------------------- (148) Obligation: Q DP problem: The TRS P consists of the following rules: 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, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) 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, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(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)))) 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, .(z3, z4)), z6, member_out_gg) -> REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4)))) U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) -> REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5)))) 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, .(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, .(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, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, [])))) 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(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, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3))) 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, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 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, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3))) 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(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 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, .(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, .(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, .(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, .(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, .(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, 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, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(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) 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, 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, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), 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, U5_gg(member_in_gg(z0, .(z0, .(z0, z4))))) 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, .(.(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, .(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, .(.(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, .(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, .(.(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, .(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, .(.(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, .(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, 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, .(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, .(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, .(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, 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, .(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))))) 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, .(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, .(x4, .(x4, x5)))) -> U2_GGGG(x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x4, .(x4, x5))), member1_out_ag(.(y_1, .(y_2, [])))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H) member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) U6_ag(member1_out_ag(X)) -> member1_out_ag(X) U5_gg(member_out_gg) -> member_out_gg The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0) U5_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (149) TransformationProof (EQUIVALENT) 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]: (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, []))))) ---------------------------------------- (150) Obligation: Q DP problem: The TRS P consists of the following rules: 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, .(z4, z5)), z2, member_out_gg) -> REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) 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, .(z4, z5)), z3, member_out_gg) -> REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(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)))) 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, .(z3, z4)), z6, member_out_gg) -> REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4)))) U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) -> REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5)))) 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, .(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, .(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, .(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, .(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, .(z4, z5)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3))) 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, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 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, .(z5, z6)))) -> U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3))) 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(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) -> U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2))) 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, .(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, .(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, .(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, .(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, .(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, 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, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(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) 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, 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, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) -> U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), 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, U5_gg(member_in_gg(z0, .(z0, .(z0, z4))))) 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, .(.(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, .(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, .(.(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, .(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, .(.(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, .(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, .(.(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, .(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, 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, .(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, .(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, .(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, 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, .(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))))) 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, .(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, .(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, .(x5, x6)))) -> U2_GGGG(x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x4, .(x5, x6))), member1_out_ag(.(y_1, .(y_2, [])))) The TRS R consists of the following rules: member1_in_ag(.(H, L)) -> member1_out_ag(H) member1_in_ag(.(H, L)) -> U6_ag(member1_in_ag(L)) member_in_gg(H, .(H, L)) -> member_out_gg member_in_gg(X, .(H, L)) -> U5_gg(member_in_gg(X, L)) U6_ag(member1_out_ag(X)) -> member1_out_ag(X) U5_gg(member_out_gg) -> member_out_gg The set Q consists of the following terms: member1_in_ag(x0) member_in_gg(x0, x1) U6_ag(x0) U5_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (151) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 287, "program": { "directives": [], "clauses": [ [ "(reach X Y Edges Visited)", "(member (. X (. Y ([]))) Edges)" ], [ "(reach X Z Edges Visited)", "(',' (member1 (. X (. Y ([]))) Edges) (',' (member Y Visited) (reach Y Z Edges (. Y Visited))))" ], [ "(member H (. H L))", null ], [ "(member X (. H L))", "(member X L)" ], [ "(member1 H (. H L))", null ], [ "(member1 X (. H L))", "(member1 X L)" ] ] }, "graph": { "nodes": { "290": { "goal": [ { "clause": 2, "scope": 2, "term": "(member (. T9 (. T10 ([]))) T11)" }, { "clause": 3, "scope": 2, "term": "(member (. T9 (. T10 ([]))) T11)" }, { "clause": -1, "scope": 2, "term": null }, { "clause": 1, "scope": 1, "term": "(reach T9 T10 T11 T12)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T9", "T10", "T11", "T12" ], "free": [], "exprvars": [] } }, "291": { "goal": [{ "clause": 2, "scope": 2, "term": "(member (. T9 (. T10 ([]))) T11)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T9", "T10", "T11" ], "free": [], "exprvars": [] } }, "292": { "goal": [ { "clause": 3, "scope": 2, "term": "(member (. T9 (. T10 ([]))) T11)" }, { "clause": -1, "scope": 2, "term": null }, { "clause": 1, "scope": 1, "term": "(reach T9 T10 T11 T12)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T9", "T10", "T11", "T12" ], "free": [], "exprvars": [] } }, "type": "Nodes", "293": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "294": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "295": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "350": { "goal": [{ "clause": -1, "scope": -1, "term": "(member1 (. T139 (. X119 ([]))) T141)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T139", "T141" ], "free": ["X119"], "exprvars": [] } }, "296": { "goal": [{ "clause": 3, "scope": 2, "term": "(member (. T9 (. T10 ([]))) T11)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T9", "T10", "T11" ], "free": [], "exprvars": [] } }, "351": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "297": { "goal": [ { "clause": -1, "scope": 2, "term": null }, { "clause": 1, "scope": 1, "term": "(reach T9 T10 T11 T12)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T9", "T10", "T11", "T12" ], "free": [], "exprvars": [] } }, "352": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T111 T104)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T104", "T111" ], "free": [], "exprvars": [] } }, "298": { "goal": [{ "clause": -1, "scope": -1, "term": "(member (. T44 (. T45 ([]))) T47)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T44", "T45", "T47" ], "free": [], "exprvars": [] } }, "353": { "goal": [{ "clause": -1, "scope": -1, "term": "(reach T111 T102 T103 (. T111 T104))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T102", "T103", "T104", "T111" ], "free": [], "exprvars": [] } }, "299": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "354": { "goal": [ { "clause": 2, "scope": 5, "term": "(member T111 T104)" }, { "clause": 3, "scope": 5, "term": "(member T111 T104)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T104", "T111" ], "free": [], "exprvars": [] } }, "355": { "goal": [{ "clause": 2, "scope": 5, "term": "(member T111 T104)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T104", "T111" ], "free": [], "exprvars": [] } }, "312": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (member1 (. T101 (. X80 ([]))) T103) (',' (member X80 T104) (reach X80 T102 T103 (. X80 T104))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T101", "T102", "T103", "T104" ], "free": ["X80"], "exprvars": [] } }, "356": { "goal": [{ "clause": 3, "scope": 5, "term": "(member T111 T104)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T104", "T111" ], "free": [], "exprvars": [] } }, "357": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "314": { "goal": [{ "clause": -1, "scope": -1, "term": "(member1 (. T101 (. X80 ([]))) T103)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T101", "T103" ], "free": ["X80"], "exprvars": [] } }, "358": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "315": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (member T111 T104) (reach T111 T102 T103 (. T111 T104)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T102", "T103", "T104", "T111" ], "free": [], "exprvars": [] } }, "359": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "318": { "goal": [ { "clause": 4, "scope": 4, "term": "(member1 (. T101 (. X80 ([]))) T103)" }, { "clause": 5, "scope": 4, "term": "(member1 (. T101 (. X80 ([]))) T103)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T101", "T103" ], "free": ["X80"], "exprvars": [] } }, "360": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T172 T174)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T172", "T174" ], "free": [], "exprvars": [] } }, "361": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "287": { "goal": [{ "clause": -1, "scope": -1, "term": "(reach T1 T2 T3 T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T4", "T1", "T2", "T3" ], "free": [], "exprvars": [] } }, "288": { "goal": [ { "clause": 0, "scope": 1, "term": "(reach T1 T2 T3 T4)" }, { "clause": 1, "scope": 1, "term": "(reach T1 T2 T3 T4)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T4", "T1", "T2", "T3" ], "free": [], "exprvars": [] } }, "289": { "goal": [ { "clause": -1, "scope": -1, "term": "(member (. T9 (. T10 ([]))) T11)" }, { "clause": 1, "scope": 1, "term": "(reach T9 T10 T11 T12)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T9", "T10", "T11", "T12" ], "free": [], "exprvars": [] } }, "300": { "goal": [ { "clause": 2, "scope": 3, "term": "(member (. T44 (. T45 ([]))) T47)" }, { "clause": 3, "scope": 3, "term": "(member (. T44 (. T45 ([]))) T47)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T44", "T45", "T47" ], "free": [], "exprvars": [] } }, "322": { "goal": [{ "clause": 4, "scope": 4, "term": "(member1 (. T101 (. X80 ([]))) T103)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T101", "T103" ], "free": ["X80"], "exprvars": [] } }, "301": { "goal": [{ "clause": 2, "scope": 3, "term": "(member (. T44 (. T45 ([]))) T47)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T44", "T45", "T47" ], "free": [], "exprvars": [] } }, "323": { "goal": [{ "clause": 5, "scope": 4, "term": "(member1 (. T101 (. X80 ([]))) T103)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T101", "T103" ], "free": ["X80"], "exprvars": [] } }, "302": { "goal": [{ "clause": 3, "scope": 3, "term": "(member (. T44 (. T45 ([]))) T47)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T44", "T45", "T47" ], "free": [], "exprvars": [] } }, "303": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "347": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "304": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "326": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "305": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "327": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "306": { "goal": [{ "clause": -1, "scope": -1, "term": "(member (. T77 (. T78 ([]))) T80)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T77", "T78", "T80" ], "free": [], "exprvars": [] } }, "307": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "309": { "goal": [{ "clause": 1, "scope": 1, "term": "(reach T9 T10 T11 T12)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T9", "T10", "T11", "T12" ], "free": [], "exprvars": [] } } }, "edges": [ { "from": 287, "to": 288, "label": "CASE" }, { "from": 288, "to": 289, "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" }, { "from": 289, "to": 290, "label": "CASE" }, { "from": 290, "to": 291, "label": "PARALLEL" }, { "from": 290, "to": 292, "label": "PARALLEL" }, { "from": 291, "to": 293, "label": "EVAL with clause\nmember(X17, .(X17, X18)).\nand substitutionT9 -> T25,\nT10 -> T26,\nX17 -> .(T25, .(T26, [])),\nX18 -> T27,\nT11 -> .(.(T25, .(T26, [])), T27)" }, { "from": 291, "to": 294, "label": "EVAL-BACKTRACK" }, { "from": 292, "to": 296, "label": "PARALLEL" }, { "from": 292, "to": 297, "label": "PARALLEL" }, { "from": 293, "to": 295, "label": "SUCCESS" }, { "from": 296, "to": 298, "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)" }, { "from": 296, "to": 299, "label": "EVAL-BACKTRACK" }, { "from": 297, "to": 309, "label": "FAILURE" }, { "from": 298, "to": 300, "label": "CASE" }, { "from": 300, "to": 301, "label": "PARALLEL" }, { "from": 300, "to": 302, "label": "PARALLEL" }, { "from": 301, "to": 303, "label": "EVAL with clause\nmember(X46, .(X46, X47)).\nand substitutionT44 -> T66,\nT45 -> T67,\nX46 -> .(T66, .(T67, [])),\nX47 -> T68,\nT47 -> .(.(T66, .(T67, [])), T68)" }, { "from": 301, "to": 304, "label": "EVAL-BACKTRACK" }, { "from": 302, "to": 306, "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)" }, { "from": 302, "to": 307, "label": "EVAL-BACKTRACK" }, { "from": 303, "to": 305, "label": "SUCCESS" }, { "from": 306, "to": 298, "label": "INSTANCE with matching:\nT44 -> T77\nT45 -> T78\nT47 -> T80" }, { "from": 309, "to": 312, "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" }, { "from": 312, "to": 314, "label": "SPLIT 1" }, { "from": 312, "to": 315, "label": "SPLIT 2\nnew knowledge:\nT101 is ground\nT111 is ground\nT103 is ground\nreplacements:X80 -> T111" }, { "from": 314, "to": 318, "label": "CASE" }, { "from": 315, "to": 352, "label": "SPLIT 1" }, { "from": 315, "to": 353, "label": "SPLIT 2\nnew knowledge:\nT111 is ground\nT104 is ground" }, { "from": 318, "to": 322, "label": "PARALLEL" }, { "from": 318, "to": 323, "label": "PARALLEL" }, { "from": 322, "to": 326, "label": "EVAL with clause\nmember1(X105, .(X105, X106)).\nand substitutionT101 -> T130,\nX80 -> T131,\nX105 -> .(T130, .(T131, [])),\nX107 -> T131,\nX106 -> T132,\nT103 -> .(.(T130, .(T131, [])), T132)" }, { "from": 322, "to": 327, "label": "EVAL-BACKTRACK" }, { "from": 323, "to": 350, "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)" }, { "from": 323, "to": 351, "label": "EVAL-BACKTRACK" }, { "from": 326, "to": 347, "label": "SUCCESS" }, { "from": 350, "to": 314, "label": "INSTANCE with matching:\nT101 -> T139\nX80 -> X119\nT103 -> T141" }, { "from": 352, "to": 354, "label": "CASE" }, { "from": 353, "to": 287, "label": "INSTANCE with matching:\nT1 -> T111\nT2 -> T102\nT3 -> T103\nT4 -> .(T111, T104)" }, { "from": 354, "to": 355, "label": "PARALLEL" }, { "from": 354, "to": 356, "label": "PARALLEL" }, { "from": 355, "to": 357, "label": "EVAL with clause\nmember(X142, .(X142, X143)).\nand substitutionT111 -> T164,\nX142 -> T164,\nX143 -> T165,\nT104 -> .(T164, T165)" }, { "from": 355, "to": 358, "label": "EVAL-BACKTRACK" }, { "from": 356, "to": 360, "label": "EVAL with clause\nmember(X150, .(X151, X152)) :- member(X150, X152).\nand substitutionT111 -> T172,\nX150 -> T172,\nX151 -> T173,\nX152 -> T174,\nT104 -> .(T173, T174)" }, { "from": 356, "to": 361, "label": "EVAL-BACKTRACK" }, { "from": 357, "to": 359, "label": "SUCCESS" }, { "from": 360, "to": 352, "label": "INSTANCE with matching:\nT111 -> T172\nT104 -> T174" } ], "type": "Graph" } } ---------------------------------------- (152) Obligation: Triples: memberA(X1, X2, .(X3, X4)) :- memberA(X1, X2, X4). member1B(X1, X2, .(X3, X4)) :- member1B(X1, X2, X4). memberD(X1, .(X2, X3)) :- memberD(X1, X3). reachC(X1, X2, .(X3, X4), X5) :- memberA(X1, X2, X4). reachC(X1, X2, X3, X4) :- member1B(X1, X5, X3). reachC(X1, X2, X3, X4) :- ','(member1cB(X1, X5, X3), memberD(X5, X4)). reachC(X1, X2, X3, X4) :- ','(member1cB(X1, X5, X3), ','(membercD(X5, X4), reachC(X5, X2, X3, .(X5, X4)))). Clauses: membercA(X1, X2, .(.(X1, .(X2, [])), X3)). membercA(X1, X2, .(X3, X4)) :- membercA(X1, X2, X4). member1cB(X1, X2, .(.(X1, .(X2, [])), X3)). member1cB(X1, X2, .(X3, X4)) :- member1cB(X1, X2, X4). reachcC(X1, X2, .(.(X1, .(X2, [])), X3), X4). reachcC(X1, X2, .(X3, X4), X5) :- membercA(X1, X2, X4). reachcC(X1, X2, X3, X4) :- ','(member1cB(X1, X5, X3), ','(membercD(X5, X4), reachcC(X5, X2, X3, .(X5, X4)))). membercD(X1, .(X1, X2)). membercD(X1, .(X2, X3)) :- membercD(X1, X3). Afs: reachC(x1, x2, x3, x4) = reachC(x1, x2, x3, x4) ---------------------------------------- (153) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: reachC_in_4: (b,b,b,b) memberA_in_3: (b,b,b) member1B_in_3: (b,f,b) member1cB_in_3: (b,f,b) memberD_in_2: (b,b) membercD_in_2: (b,b) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: REACHC_IN_GGGG(X1, X2, .(X3, X4), X5) -> U4_GGGG(X1, X2, X3, X4, X5, memberA_in_ggg(X1, X2, X4)) REACHC_IN_GGGG(X1, X2, .(X3, X4), X5) -> MEMBERA_IN_GGG(X1, X2, X4) MEMBERA_IN_GGG(X1, X2, .(X3, X4)) -> U1_GGG(X1, X2, X3, X4, memberA_in_ggg(X1, X2, X4)) MEMBERA_IN_GGG(X1, X2, .(X3, X4)) -> MEMBERA_IN_GGG(X1, X2, X4) REACHC_IN_GGGG(X1, X2, X3, X4) -> U5_GGGG(X1, X2, X3, X4, member1B_in_gag(X1, X5, X3)) REACHC_IN_GGGG(X1, X2, X3, X4) -> MEMBER1B_IN_GAG(X1, X5, X3) MEMBER1B_IN_GAG(X1, X2, .(X3, X4)) -> U2_GAG(X1, X2, X3, X4, member1B_in_gag(X1, X2, X4)) MEMBER1B_IN_GAG(X1, X2, .(X3, X4)) -> MEMBER1B_IN_GAG(X1, X2, X4) REACHC_IN_GGGG(X1, X2, X3, X4) -> U6_GGGG(X1, X2, X3, X4, member1cB_in_gag(X1, X5, X3)) U6_GGGG(X1, X2, X3, X4, member1cB_out_gag(X1, X5, X3)) -> U7_GGGG(X1, X2, X3, X4, memberD_in_gg(X5, X4)) U6_GGGG(X1, X2, X3, X4, member1cB_out_gag(X1, X5, X3)) -> MEMBERD_IN_GG(X5, X4) MEMBERD_IN_GG(X1, .(X2, X3)) -> U3_GG(X1, X2, X3, memberD_in_gg(X1, X3)) MEMBERD_IN_GG(X1, .(X2, X3)) -> MEMBERD_IN_GG(X1, X3) U6_GGGG(X1, X2, X3, X4, member1cB_out_gag(X1, X5, X3)) -> U8_GGGG(X1, X2, X3, X4, X5, membercD_in_gg(X5, X4)) 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))) U8_GGGG(X1, X2, X3, X4, X5, membercD_out_gg(X5, X4)) -> REACHC_IN_GGGG(X5, X2, X3, .(X5, X4)) The TRS R consists of the following rules: member1cB_in_gag(X1, X2, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) member1cB_in_gag(X1, X2, .(X3, X4)) -> U12_gag(X1, X2, X3, X4, member1cB_in_gag(X1, X2, X4)) U12_gag(X1, X2, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) The argument filtering Pi contains the following mapping: reachC_in_gggg(x1, x2, x3, x4) = reachC_in_gggg(x1, x2, x3, x4) .(x1, x2) = .(x1, x2) memberA_in_ggg(x1, x2, x3) = memberA_in_ggg(x1, x2, x3) member1B_in_gag(x1, x2, x3) = member1B_in_gag(x1, x3) member1cB_in_gag(x1, x2, x3) = member1cB_in_gag(x1, x3) [] = [] member1cB_out_gag(x1, x2, x3) = member1cB_out_gag(x1, x2, x3) U12_gag(x1, x2, x3, x4, x5) = U12_gag(x1, x3, x4, x5) memberD_in_gg(x1, x2) = memberD_in_gg(x1, x2) membercD_in_gg(x1, x2) = membercD_in_gg(x1, x2) membercD_out_gg(x1, x2) = membercD_out_gg(x1, x2) U17_gg(x1, x2, x3, x4) = U17_gg(x1, x2, x3, x4) REACHC_IN_GGGG(x1, x2, x3, x4) = REACHC_IN_GGGG(x1, x2, x3, x4) U4_GGGG(x1, x2, x3, x4, x5, x6) = U4_GGGG(x1, x2, x3, x4, x5, x6) MEMBERA_IN_GGG(x1, x2, x3) = MEMBERA_IN_GGG(x1, x2, x3) U1_GGG(x1, x2, x3, x4, x5) = U1_GGG(x1, x2, x3, x4, x5) U5_GGGG(x1, x2, x3, x4, x5) = U5_GGGG(x1, x2, x3, x4, x5) MEMBER1B_IN_GAG(x1, x2, x3) = MEMBER1B_IN_GAG(x1, x3) U2_GAG(x1, x2, x3, x4, x5) = U2_GAG(x1, x3, x4, x5) U6_GGGG(x1, x2, x3, x4, x5) = U6_GGGG(x1, x2, x3, x4, x5) U7_GGGG(x1, x2, x3, x4, x5) = U7_GGGG(x1, x2, x3, x4, x5) MEMBERD_IN_GG(x1, x2) = MEMBERD_IN_GG(x1, x2) U3_GG(x1, x2, x3, x4) = U3_GG(x1, x2, x3, x4) U8_GGGG(x1, x2, x3, x4, x5, x6) = U8_GGGG(x1, x2, x3, x4, x5, x6) U9_GGGG(x1, x2, x3, x4, x5) = U9_GGGG(x1, x2, x3, x4, x5) We have to consider all (P,R,Pi)-chains Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES ---------------------------------------- (154) Obligation: Pi DP problem: The TRS P consists of the following rules: REACHC_IN_GGGG(X1, X2, .(X3, X4), X5) -> U4_GGGG(X1, X2, X3, X4, X5, memberA_in_ggg(X1, X2, X4)) REACHC_IN_GGGG(X1, X2, .(X3, X4), X5) -> MEMBERA_IN_GGG(X1, X2, X4) MEMBERA_IN_GGG(X1, X2, .(X3, X4)) -> U1_GGG(X1, X2, X3, X4, memberA_in_ggg(X1, X2, X4)) MEMBERA_IN_GGG(X1, X2, .(X3, X4)) -> MEMBERA_IN_GGG(X1, X2, X4) REACHC_IN_GGGG(X1, X2, X3, X4) -> U5_GGGG(X1, X2, X3, X4, member1B_in_gag(X1, X5, X3)) REACHC_IN_GGGG(X1, X2, X3, X4) -> MEMBER1B_IN_GAG(X1, X5, X3) MEMBER1B_IN_GAG(X1, X2, .(X3, X4)) -> U2_GAG(X1, X2, X3, X4, member1B_in_gag(X1, X2, X4)) MEMBER1B_IN_GAG(X1, X2, .(X3, X4)) -> MEMBER1B_IN_GAG(X1, X2, X4) REACHC_IN_GGGG(X1, X2, X3, X4) -> U6_GGGG(X1, X2, X3, X4, member1cB_in_gag(X1, X5, X3)) U6_GGGG(X1, X2, X3, X4, member1cB_out_gag(X1, X5, X3)) -> U7_GGGG(X1, X2, X3, X4, memberD_in_gg(X5, X4)) U6_GGGG(X1, X2, X3, X4, member1cB_out_gag(X1, X5, X3)) -> MEMBERD_IN_GG(X5, X4) MEMBERD_IN_GG(X1, .(X2, X3)) -> U3_GG(X1, X2, X3, memberD_in_gg(X1, X3)) MEMBERD_IN_GG(X1, .(X2, X3)) -> MEMBERD_IN_GG(X1, X3) U6_GGGG(X1, X2, X3, X4, member1cB_out_gag(X1, X5, X3)) -> U8_GGGG(X1, X2, X3, X4, X5, membercD_in_gg(X5, X4)) 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))) U8_GGGG(X1, X2, X3, X4, X5, membercD_out_gg(X5, X4)) -> REACHC_IN_GGGG(X5, X2, X3, .(X5, X4)) The TRS R consists of the following rules: member1cB_in_gag(X1, X2, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) member1cB_in_gag(X1, X2, .(X3, X4)) -> U12_gag(X1, X2, X3, X4, member1cB_in_gag(X1, X2, X4)) U12_gag(X1, X2, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) The argument filtering Pi contains the following mapping: reachC_in_gggg(x1, x2, x3, x4) = reachC_in_gggg(x1, x2, x3, x4) .(x1, x2) = .(x1, x2) memberA_in_ggg(x1, x2, x3) = memberA_in_ggg(x1, x2, x3) member1B_in_gag(x1, x2, x3) = member1B_in_gag(x1, x3) member1cB_in_gag(x1, x2, x3) = member1cB_in_gag(x1, x3) [] = [] member1cB_out_gag(x1, x2, x3) = member1cB_out_gag(x1, x2, x3) U12_gag(x1, x2, x3, x4, x5) = U12_gag(x1, x3, x4, x5) memberD_in_gg(x1, x2) = memberD_in_gg(x1, x2) membercD_in_gg(x1, x2) = membercD_in_gg(x1, x2) membercD_out_gg(x1, x2) = membercD_out_gg(x1, x2) U17_gg(x1, x2, x3, x4) = U17_gg(x1, x2, x3, x4) REACHC_IN_GGGG(x1, x2, x3, x4) = REACHC_IN_GGGG(x1, x2, x3, x4) U4_GGGG(x1, x2, x3, x4, x5, x6) = U4_GGGG(x1, x2, x3, x4, x5, x6) MEMBERA_IN_GGG(x1, x2, x3) = MEMBERA_IN_GGG(x1, x2, x3) U1_GGG(x1, x2, x3, x4, x5) = U1_GGG(x1, x2, x3, x4, x5) U5_GGGG(x1, x2, x3, x4, x5) = U5_GGGG(x1, x2, x3, x4, x5) MEMBER1B_IN_GAG(x1, x2, x3) = MEMBER1B_IN_GAG(x1, x3) U2_GAG(x1, x2, x3, x4, x5) = U2_GAG(x1, x3, x4, x5) U6_GGGG(x1, x2, x3, x4, x5) = U6_GGGG(x1, x2, x3, x4, x5) U7_GGGG(x1, x2, x3, x4, x5) = U7_GGGG(x1, x2, x3, x4, x5) MEMBERD_IN_GG(x1, x2) = MEMBERD_IN_GG(x1, x2) U3_GG(x1, x2, x3, x4) = U3_GG(x1, x2, x3, x4) U8_GGGG(x1, x2, x3, x4, x5, x6) = U8_GGGG(x1, x2, x3, x4, x5, x6) U9_GGGG(x1, x2, x3, x4, x5) = U9_GGGG(x1, x2, x3, x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (155) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 10 less nodes. ---------------------------------------- (156) Complex Obligation (AND) ---------------------------------------- (157) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBERD_IN_GG(X1, .(X2, X3)) -> MEMBERD_IN_GG(X1, X3) The TRS R consists of the following rules: member1cB_in_gag(X1, X2, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) member1cB_in_gag(X1, X2, .(X3, X4)) -> U12_gag(X1, X2, X3, X4, member1cB_in_gag(X1, X2, X4)) U12_gag(X1, X2, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) member1cB_in_gag(x1, x2, x3) = member1cB_in_gag(x1, x3) [] = [] member1cB_out_gag(x1, x2, x3) = member1cB_out_gag(x1, x2, x3) U12_gag(x1, x2, x3, x4, x5) = U12_gag(x1, x3, x4, x5) membercD_in_gg(x1, x2) = membercD_in_gg(x1, x2) membercD_out_gg(x1, x2) = membercD_out_gg(x1, x2) U17_gg(x1, x2, x3, x4) = U17_gg(x1, x2, x3, x4) MEMBERD_IN_GG(x1, x2) = MEMBERD_IN_GG(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (158) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (159) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBERD_IN_GG(X1, .(X2, X3)) -> MEMBERD_IN_GG(X1, X3) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (160) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (161) Obligation: Q DP problem: The TRS P consists of the following rules: MEMBERD_IN_GG(X1, .(X2, X3)) -> MEMBERD_IN_GG(X1, X3) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (162) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *MEMBERD_IN_GG(X1, .(X2, X3)) -> MEMBERD_IN_GG(X1, X3) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (163) YES ---------------------------------------- (164) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER1B_IN_GAG(X1, X2, .(X3, X4)) -> MEMBER1B_IN_GAG(X1, X2, X4) The TRS R consists of the following rules: member1cB_in_gag(X1, X2, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) member1cB_in_gag(X1, X2, .(X3, X4)) -> U12_gag(X1, X2, X3, X4, member1cB_in_gag(X1, X2, X4)) U12_gag(X1, X2, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) member1cB_in_gag(x1, x2, x3) = member1cB_in_gag(x1, x3) [] = [] member1cB_out_gag(x1, x2, x3) = member1cB_out_gag(x1, x2, x3) U12_gag(x1, x2, x3, x4, x5) = U12_gag(x1, x3, x4, x5) membercD_in_gg(x1, x2) = membercD_in_gg(x1, x2) membercD_out_gg(x1, x2) = membercD_out_gg(x1, x2) U17_gg(x1, x2, x3, x4) = U17_gg(x1, x2, x3, x4) MEMBER1B_IN_GAG(x1, x2, x3) = MEMBER1B_IN_GAG(x1, x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (165) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (166) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER1B_IN_GAG(X1, X2, .(X3, X4)) -> MEMBER1B_IN_GAG(X1, X2, X4) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) MEMBER1B_IN_GAG(x1, x2, x3) = MEMBER1B_IN_GAG(x1, x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (167) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (168) Obligation: Q DP problem: The TRS P consists of the following rules: MEMBER1B_IN_GAG(X1, .(X3, X4)) -> MEMBER1B_IN_GAG(X1, X4) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (169) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *MEMBER1B_IN_GAG(X1, .(X3, X4)) -> MEMBER1B_IN_GAG(X1, X4) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (170) YES ---------------------------------------- (171) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBERA_IN_GGG(X1, X2, .(X3, X4)) -> MEMBERA_IN_GGG(X1, X2, X4) The TRS R consists of the following rules: member1cB_in_gag(X1, X2, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) member1cB_in_gag(X1, X2, .(X3, X4)) -> U12_gag(X1, X2, X3, X4, member1cB_in_gag(X1, X2, X4)) U12_gag(X1, X2, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) member1cB_in_gag(x1, x2, x3) = member1cB_in_gag(x1, x3) [] = [] member1cB_out_gag(x1, x2, x3) = member1cB_out_gag(x1, x2, x3) U12_gag(x1, x2, x3, x4, x5) = U12_gag(x1, x3, x4, x5) membercD_in_gg(x1, x2) = membercD_in_gg(x1, x2) membercD_out_gg(x1, x2) = membercD_out_gg(x1, x2) U17_gg(x1, x2, x3, x4) = U17_gg(x1, x2, x3, x4) MEMBERA_IN_GGG(x1, x2, x3) = MEMBERA_IN_GGG(x1, x2, x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (172) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (173) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBERA_IN_GGG(X1, X2, .(X3, X4)) -> MEMBERA_IN_GGG(X1, X2, X4) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (174) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (175) Obligation: Q DP problem: The TRS P consists of the following rules: MEMBERA_IN_GGG(X1, X2, .(X3, X4)) -> MEMBERA_IN_GGG(X1, X2, X4) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (176) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *MEMBERA_IN_GGG(X1, X2, .(X3, X4)) -> MEMBERA_IN_GGG(X1, X2, X4) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 ---------------------------------------- (177) YES ---------------------------------------- (178) Obligation: Pi DP problem: The TRS P consists of the following rules: REACHC_IN_GGGG(X1, X2, X3, X4) -> U6_GGGG(X1, X2, X3, X4, member1cB_in_gag(X1, X5, X3)) U6_GGGG(X1, X2, X3, X4, member1cB_out_gag(X1, X5, X3)) -> U8_GGGG(X1, X2, X3, X4, X5, membercD_in_gg(X5, X4)) U8_GGGG(X1, X2, X3, X4, X5, membercD_out_gg(X5, X4)) -> REACHC_IN_GGGG(X5, X2, X3, .(X5, X4)) The TRS R consists of the following rules: member1cB_in_gag(X1, X2, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) member1cB_in_gag(X1, X2, .(X3, X4)) -> U12_gag(X1, X2, X3, X4, member1cB_in_gag(X1, X2, X4)) U12_gag(X1, X2, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) member1cB_in_gag(x1, x2, x3) = member1cB_in_gag(x1, x3) [] = [] member1cB_out_gag(x1, x2, x3) = member1cB_out_gag(x1, x2, x3) U12_gag(x1, x2, x3, x4, x5) = U12_gag(x1, x3, x4, x5) membercD_in_gg(x1, x2) = membercD_in_gg(x1, x2) membercD_out_gg(x1, x2) = membercD_out_gg(x1, x2) U17_gg(x1, x2, x3, x4) = U17_gg(x1, x2, x3, x4) REACHC_IN_GGGG(x1, x2, x3, x4) = REACHC_IN_GGGG(x1, x2, x3, x4) U6_GGGG(x1, x2, x3, x4, x5) = U6_GGGG(x1, x2, x3, x4, x5) U8_GGGG(x1, x2, x3, x4, x5, x6) = U8_GGGG(x1, x2, x3, x4, x5, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (179) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (180) Obligation: Q DP problem: The TRS P consists of the following rules: REACHC_IN_GGGG(X1, X2, X3, X4) -> U6_GGGG(X1, X2, X3, X4, member1cB_in_gag(X1, X3)) U6_GGGG(X1, X2, X3, X4, member1cB_out_gag(X1, X5, X3)) -> U8_GGGG(X1, X2, X3, X4, X5, membercD_in_gg(X5, X4)) U8_GGGG(X1, X2, X3, X4, X5, membercD_out_gg(X5, X4)) -> REACHC_IN_GGGG(X5, X2, X3, .(X5, X4)) The TRS R consists of the following rules: member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) The set Q consists of the following terms: member1cB_in_gag(x0, x1) U12_gag(x0, x1, x2, x3) membercD_in_gg(x0, x1) U17_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (181) TransformationProof (SOUND) 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]: (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)))) (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)))) ---------------------------------------- (182) Obligation: Q DP problem: The TRS P consists of the following rules: U6_GGGG(X1, X2, X3, X4, member1cB_out_gag(X1, X5, X3)) -> U8_GGGG(X1, X2, X3, X4, X5, membercD_in_gg(X5, X4)) U8_GGGG(X1, X2, X3, X4, X5, membercD_out_gg(X5, X4)) -> REACHC_IN_GGGG(X5, X2, X3, .(X5, X4)) 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, .(x1, x2), y3) -> U6_GGGG(x0, y1, .(x1, x2), y3, U12_gag(x0, x1, x2, member1cB_in_gag(x0, x2))) The TRS R consists of the following rules: member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) The set Q consists of the following terms: member1cB_in_gag(x0, x1) U12_gag(x0, x1, x2, x3) membercD_in_gg(x0, x1) U17_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (183) TransformationProof (SOUND) 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]: (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)))) (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)))) ---------------------------------------- (184) Obligation: Q DP problem: The TRS P consists of the following rules: U8_GGGG(X1, X2, X3, X4, X5, membercD_out_gg(X5, X4)) -> REACHC_IN_GGGG(X5, X2, X3, .(X5, X4)) 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, .(x1, x2), y3) -> U6_GGGG(x0, y1, .(x1, x2), y3, U12_gag(x0, x1, x2, member1cB_in_gag(x0, x2))) 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, .(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))) The TRS R consists of the following rules: member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) The set Q consists of the following terms: member1cB_in_gag(x0, x1) U12_gag(x0, x1, x2, x3) membercD_in_gg(x0, x1) U17_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (185) TransformationProof (EQUIVALENT) 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]: (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)))) (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)))) ---------------------------------------- (186) Obligation: Q DP problem: The TRS P consists of the following rules: 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, .(x1, x2), y3) -> U6_GGGG(x0, y1, .(x1, x2), y3, U12_gag(x0, x1, x2, member1cB_in_gag(x0, x2))) 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, .(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))) 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), z5, membercD_out_gg(z5, .(z3, z4))) -> REACHC_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4))) The TRS R consists of the following rules: member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) The set Q consists of the following terms: member1cB_in_gag(x0, x1) U12_gag(x0, x1, x2, x3) membercD_in_gg(x0, x1) U17_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (187) TransformationProof (EQUIVALENT) 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]: (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)))) (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)))) ---------------------------------------- (188) Obligation: Q DP problem: The TRS P consists of the following rules: 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))) 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, .(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))) 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), z5, membercD_out_gg(z5, .(z3, z4))) -> REACHC_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4))) 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(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))) The TRS R consists of the following rules: member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) The set Q consists of the following terms: member1cB_in_gag(x0, x1) U12_gag(x0, x1, x2, x3) membercD_in_gg(x0, x1) U17_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (189) TransformationProof (EQUIVALENT) 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]: (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)))) (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)))) ---------------------------------------- (190) Obligation: Q DP problem: The TRS P consists of the following rules: 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, .(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))) 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), z5, membercD_out_gg(z5, .(z3, z4))) -> REACHC_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4))) 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(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(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(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))) The TRS R consists of the following rules: member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) The set Q consists of the following terms: member1cB_in_gag(x0, x1) U12_gag(x0, x1, x2, x3) membercD_in_gg(x0, x1) U17_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (191) TransformationProof (EQUIVALENT) 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]: (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))))) (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))))) (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))))) (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))))) ---------------------------------------- (192) Obligation: Q DP problem: The TRS P consists of the following rules: 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))) 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), z5, membercD_out_gg(z5, .(z3, z4))) -> REACHC_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4))) 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(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(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(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))) 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, .(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, .(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, .(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)))) The TRS R consists of the following rules: member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) The set Q consists of the following terms: member1cB_in_gag(x0, x1) U12_gag(x0, x1, x2, x3) membercD_in_gg(x0, x1) U17_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (193) TransformationProof (EQUIVALENT) 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]: (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))))) (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))))) (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))))) (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))))) ---------------------------------------- (194) Obligation: Q DP problem: The TRS P consists of the following rules: 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), z5, membercD_out_gg(z5, .(z3, z4))) -> REACHC_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4))) 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(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(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(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))) 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(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)))) The TRS R consists of the following rules: member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) The set Q consists of the following terms: member1cB_in_gag(x0, x1) U12_gag(x0, x1, x2, x3) membercD_in_gg(x0, x1) U17_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (195) TransformationProof (EQUIVALENT) 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]: (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))))) (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))))) (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))))) (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))))) ---------------------------------------- (196) Obligation: Q DP problem: The TRS P consists of the following rules: U8_GGGG(z0, z1, z2, .(z3, z4), z5, membercD_out_gg(z5, .(z3, z4))) -> REACHC_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4))) 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(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(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(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))) 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(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)))) 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, .(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, .(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, .(z4, z5)), z0, membercD_out_gg(z0, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) The TRS R consists of the following rules: member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) The set Q consists of the following terms: member1cB_in_gag(x0, x1) U12_gag(x0, x1, x2, x3) membercD_in_gg(x0, x1) U17_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (197) TransformationProof (EQUIVALENT) 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]: (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))))) (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))))) (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))))) (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))))) (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))))) (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))))) (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))))) (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))))) ---------------------------------------- (198) Obligation: Q DP problem: The TRS P consists of the following rules: 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(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(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(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))) 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(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)))) 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(z4, z5)), z6, membercD_out_gg(z6, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) The TRS R consists of the following rules: member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) The set Q consists of the following terms: member1cB_in_gag(x0, x1) U12_gag(x0, x1, x2, x3) membercD_in_gg(x0, x1) U17_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (199) TransformationProof (EQUIVALENT) 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]: (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)))) (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)))) (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)))) (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)))) ---------------------------------------- (200) Obligation: Q DP problem: The TRS P consists of the following rules: 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(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(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))) 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(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)))) 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(z4, z5)), z6, membercD_out_gg(z6, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 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, .(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, .(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, .(z4, z5)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, x2, .(.(z0, .(x2, [])), z3))) The TRS R consists of the following rules: member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) The set Q consists of the following terms: member1cB_in_gag(x0, x1) U12_gag(x0, x1, x2, x3) membercD_in_gg(x0, x1) U17_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (201) TransformationProof (EQUIVALENT) 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]: (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)))) (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)))) (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)))) (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)))) (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)))) (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)))) ---------------------------------------- (202) Obligation: Q DP problem: The TRS P consists of the following rules: 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(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))) 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(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)))) 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(z4, z5)), z6, membercD_out_gg(z6, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 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, .(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, .(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, .(z4, z5)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, x2, .(.(z0, .(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))) 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))) The TRS R consists of the following rules: member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) The set Q consists of the following terms: member1cB_in_gag(x0, x1) U12_gag(x0, x1, x2, x3) membercD_in_gg(x0, x1) U17_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (203) TransformationProof (EQUIVALENT) 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]: (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)))) (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)))) (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)))) (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)))) ---------------------------------------- (204) Obligation: Q DP problem: The TRS P consists of the following rules: 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))) 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(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)))) 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(z4, z5)), z6, membercD_out_gg(z6, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 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, .(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, .(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, .(z4, z5)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, x2, .(.(z0, .(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))) 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(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, .(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, .(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, .(z4, z5)))) -> U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U12_gag(z0, z2, z3, member1cB_in_gag(z0, z3))) The TRS R consists of the following rules: member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) The set Q consists of the following terms: member1cB_in_gag(x0, x1) U12_gag(x0, x1, x2, x3) membercD_in_gg(x0, x1) U17_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (205) TransformationProof (EQUIVALENT) 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]: (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)))) (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)))) (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)))) (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)))) (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)))) (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)))) (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)))) (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)))) ---------------------------------------- (206) Obligation: Q DP problem: The TRS P consists of the following rules: 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(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)))) 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(z4, z5)), z6, membercD_out_gg(z6, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 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, .(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, .(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, .(z4, z5)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, x2, .(.(z0, .(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))) 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(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, .(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, .(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, .(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(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, .(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(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(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))) The TRS R consists of the following rules: member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) The set Q consists of the following terms: member1cB_in_gag(x0, x1) U12_gag(x0, x1, x2, x3) membercD_in_gg(x0, x1) U17_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (207) TransformationProof (EQUIVALENT) 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]: (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)))))) (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)))))) ---------------------------------------- (208) Obligation: Q DP problem: The TRS P consists of the following rules: 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, .(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, .(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, .(.(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, .(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, .(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, .(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)))) 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(z4, z5)), z6, membercD_out_gg(z6, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 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, .(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, .(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, .(z4, z5)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, x2, .(.(z0, .(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))) 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(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, .(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, .(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, .(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(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, .(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(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(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))) 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, .(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))))) The TRS R consists of the following rules: member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) The set Q consists of the following terms: member1cB_in_gag(x0, x1) U12_gag(x0, x1, x2, x3) membercD_in_gg(x0, x1) U17_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (209) TransformationProof (EQUIVALENT) 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]: (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)))))) (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)))))) (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)))))) (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)))))) ---------------------------------------- (210) Obligation: Q DP problem: The TRS P consists of the following rules: 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, .(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, .(.(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, .(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, .(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, .(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)))) 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(z4, z5)), z6, membercD_out_gg(z6, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 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, .(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, .(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, .(z4, z5)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, x2, .(.(z0, .(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))) 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(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, .(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, .(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, .(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(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, .(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(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(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))) 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, .(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, [])), 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, .(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))))) The TRS R consists of the following rules: member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) The set Q consists of the following terms: member1cB_in_gag(x0, x1) U12_gag(x0, x1, x2, x3) membercD_in_gg(x0, x1) U17_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (211) TransformationProof (EQUIVALENT) 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]: (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)))))) (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)))))) (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)))))) (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)))))) ---------------------------------------- (212) Obligation: Q DP problem: The TRS P consists of the following rules: 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, .(.(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, .(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, .(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, .(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)))) 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(z4, z5)), z6, membercD_out_gg(z6, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 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, .(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, .(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, .(z4, z5)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, x2, .(.(z0, .(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))) 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(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, .(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, .(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, .(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(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, .(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(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(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))) 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, .(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, [])), 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, .(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, .(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, .(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))))) The TRS R consists of the following rules: member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) The set Q consists of the following terms: member1cB_in_gag(x0, x1) U12_gag(x0, x1, x2, x3) membercD_in_gg(x0, x1) U17_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (213) TransformationProof (EQUIVALENT) 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]: (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) ---------------------------------------- (214) Obligation: Q DP problem: The TRS P consists of the following rules: 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, .(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, .(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, .(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)))) 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(z4, z5)), z6, membercD_out_gg(z6, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 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, .(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, .(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, .(z4, z5)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, x2, .(.(z0, .(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))) 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(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, .(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, .(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, .(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(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, .(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(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(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))) 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, .(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, [])), 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, .(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, .(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, .(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, .(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, .(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, 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, .(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))))) The TRS R consists of the following rules: member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) The set Q consists of the following terms: member1cB_in_gag(x0, x1) U12_gag(x0, x1, x2, x3) membercD_in_gg(x0, x1) U17_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (215) TransformationProof (EQUIVALENT) 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]: (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)))))) (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)))))) (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)))))) (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)))))) ---------------------------------------- (216) Obligation: Q DP problem: The TRS P consists of the following rules: 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, .(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, .(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)))) 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(z4, z5)), z6, membercD_out_gg(z6, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 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, .(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, .(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, .(z4, z5)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, x2, .(.(z0, .(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))) 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(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, .(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, .(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, .(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(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, .(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(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(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))) 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, .(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, [])), 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, .(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, .(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, .(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, .(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, .(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, 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, .(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, .(.(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, .(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, .(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, .(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))))) The TRS R consists of the following rules: member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) The set Q consists of the following terms: member1cB_in_gag(x0, x1) U12_gag(x0, x1, x2, x3) membercD_in_gg(x0, x1) U17_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (217) TransformationProof (EQUIVALENT) 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]: (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) ---------------------------------------- (218) Obligation: Q DP problem: The TRS P consists of the following rules: 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, .(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)))) 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(z4, z5)), z6, membercD_out_gg(z6, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 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, .(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, .(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, .(z4, z5)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, x2, .(.(z0, .(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))) 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(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, .(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, .(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, .(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(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, .(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(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(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))) 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, .(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, [])), 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, .(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, .(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, .(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, .(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, .(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, 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, .(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, .(.(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, .(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, .(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, .(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, .(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, .(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))))) The TRS R consists of the following rules: member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) The set Q consists of the following terms: member1cB_in_gag(x0, x1) U12_gag(x0, x1, x2, x3) membercD_in_gg(x0, x1) U17_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (219) TransformationProof (EQUIVALENT) 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]: (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) ---------------------------------------- (220) Obligation: Q DP problem: The TRS P consists of the following rules: 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)))) 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(z4, z5)), z6, membercD_out_gg(z6, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 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, .(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, .(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, .(z4, z5)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, x2, .(.(z0, .(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))) 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(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, .(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, .(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, .(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(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, .(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(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(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))) 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, .(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, [])), 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, .(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, .(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, .(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, .(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, .(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, 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, .(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, .(.(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, .(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, .(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, .(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, .(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, .(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, .(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, .(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, .(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, .(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))))) The TRS R consists of the following rules: member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) The set Q consists of the following terms: member1cB_in_gag(x0, x1) U12_gag(x0, x1, x2, x3) membercD_in_gg(x0, x1) U17_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (221) TransformationProof (EQUIVALENT) 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]: (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) (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)))))) ---------------------------------------- (222) Obligation: Q DP problem: The TRS P consists of the following rules: 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, .(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, .(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, .(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, .(.(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, .(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, .(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, .(z4, z5)), z6, membercD_out_gg(z6, .(z0, .(z4, z5)))) -> REACHC_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) 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, .(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, .(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, .(z4, z5)))) -> U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1cB_out_gag(z0, x2, .(.(z0, .(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))) 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(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, .(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, .(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, .(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(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, .(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(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(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))) 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, .(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, [])), 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, .(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, .(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, .(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, .(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, .(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, 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, .(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, .(.(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, .(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, .(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, .(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, .(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, .(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, .(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, .(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, .(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, .(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, .(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, .(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, 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, .(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))))) The TRS R consists of the following rules: member1cB_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cB_out_gag(X1, X2, .(.(X1, .(X2, [])), X3)) member1cB_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cB_in_gag(X1, X4)) U12_gag(X1, X3, X4, member1cB_out_gag(X1, X2, X4)) -> member1cB_out_gag(X1, X2, .(X3, X4)) membercD_in_gg(X1, .(X1, X2)) -> membercD_out_gg(X1, .(X1, X2)) membercD_in_gg(X1, .(X2, X3)) -> U17_gg(X1, X2, X3, membercD_in_gg(X1, X3)) U17_gg(X1, X2, X3, membercD_out_gg(X1, X3)) -> membercD_out_gg(X1, .(X2, X3)) The set Q consists of the following terms: member1cB_in_gag(x0, x1) U12_gag(x0, x1, x2, x3) membercD_in_gg(x0, x1) U17_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (223) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 2, "program": { "directives": [], "clauses": [ [ "(reach X Y Edges Visited)", "(member (. X (. Y ([]))) Edges)" ], [ "(reach X Z Edges Visited)", "(',' (member1 (. X (. Y ([]))) Edges) (',' (member Y Visited) (reach Y Z Edges (. Y Visited))))" ], [ "(member H (. H L))", null ], [ "(member X (. H L))", "(member X L)" ], [ "(member1 H (. H L))", null ], [ "(member1 X (. H L))", "(member1 X L)" ] ] }, "graph": { "nodes": { "46": { "goal": [{ "clause": 0, "scope": 1, "term": "(reach T1 T2 T3 T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T4", "T1", "T2", "T3" ], "free": [], "exprvars": [] } }, "47": { "goal": [{ "clause": 1, "scope": 1, "term": "(reach T1 T2 T3 T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T4", "T1", "T2", "T3" ], "free": [], "exprvars": [] } }, "390": { "goal": [{ "clause": 3, "scope": 4, "term": "(member T100 T93)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T93", "T100" ], "free": [], "exprvars": [] } }, "270": { "goal": [{ "clause": -1, "scope": -1, "term": "(member (. T66 (. T67 ([]))) T69)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T66", "T67", "T69" ], "free": [], "exprvars": [] } }, "391": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "271": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "392": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "393": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "130": { "goal": [{ "clause": 3, "scope": 2, "term": "(member (. T33 (. T34 ([]))) T35)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T33", "T34", "T35" ], "free": [], "exprvars": [] } }, "394": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T161 T163)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T161", "T163" ], "free": [], "exprvars": [] } }, "395": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "330": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (member T100 T93) (reach T100 T91 T92 (. T100 T93)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T91", "T92", "T93", "T100" ], "free": [], "exprvars": [] } }, "333": { "goal": [ { "clause": 4, "scope": 3, "term": "(member1 (. T90 (. X75 ([]))) T92)" }, { "clause": 5, "scope": 3, "term": "(member1 (. T90 (. X75 ([]))) T92)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T90", "T92" ], "free": ["X75"], "exprvars": [] } }, "137": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "337": { "goal": [{ "clause": 4, "scope": 3, "term": "(member1 (. T90 (. X75 ([]))) T92)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T90", "T92" ], "free": ["X75"], "exprvars": [] } }, "338": { "goal": [{ "clause": 5, "scope": 3, "term": "(member1 (. T90 (. X75 ([]))) T92)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T90", "T92" ], "free": ["X75"], "exprvars": [] } }, "339": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "15": { "goal": [ { "clause": 0, "scope": 1, "term": "(reach T1 T2 T3 T4)" }, { "clause": 1, "scope": 1, "term": "(reach T1 T2 T3 T4)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T4", "T1", "T2", "T3" ], "free": [], "exprvars": [] } }, "140": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "121": { "goal": [{ "clause": -1, "scope": -1, "term": "(member (. T33 (. T34 ([]))) T35)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T33", "T34", "T35" ], "free": [], "exprvars": [] } }, "341": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "386": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T100 T93)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T93", "T100" ], "free": [], "exprvars": [] } }, "2": { "goal": [{ "clause": -1, "scope": -1, "term": "(reach T1 T2 T3 T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T4", "T1", "T2", "T3" ], "free": [], "exprvars": [] } }, "343": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "387": { "goal": [{ "clause": -1, "scope": -1, "term": "(reach T100 T91 T92 (. T100 T93))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T91", "T92", "T93", "T100" ], "free": [], "exprvars": [] } }, "388": { "goal": [ { "clause": 2, "scope": 4, "term": "(member T100 T93)" }, { "clause": 3, "scope": 4, "term": "(member T100 T93)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T93", "T100" ], "free": [], "exprvars": [] } }, "147": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "345": { "goal": [{ "clause": -1, "scope": -1, "term": "(member1 (. T128 (. X114 ([]))) T130)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T128", "T130" ], "free": ["X114"], "exprvars": [] } }, "389": { "goal": [{ "clause": 2, "scope": 4, "term": "(member T100 T93)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T93", "T100" ], "free": [], "exprvars": [] } }, "346": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "128": { "goal": [ { "clause": 2, "scope": 2, "term": "(member (. T33 (. T34 ([]))) T35)" }, { "clause": 3, "scope": 2, "term": "(member (. T33 (. T34 ([]))) T35)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T33", "T34", "T35" ], "free": [], "exprvars": [] } }, "129": { "goal": [{ "clause": 2, "scope": 2, "term": "(member (. T33 (. T34 ([]))) T35)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T33", "T34", "T35" ], "free": [], "exprvars": [] } }, "328": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (member1 (. T90 (. X75 ([]))) T92) (',' (member X75 T93) (reach X75 T91 T92 (. X75 T93))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T90", "T91", "T92", "T93" ], "free": ["X75"], "exprvars": [] } }, "329": { "goal": [{ "clause": -1, "scope": -1, "term": "(member1 (. T90 (. X75 ([]))) T92)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T90", "T92" ], "free": ["X75"], "exprvars": [] } } }, "edges": [ { "from": 2, "to": 15, "label": "CASE" }, { "from": 15, "to": 46, "label": "PARALLEL" }, { "from": 15, "to": 47, "label": "PARALLEL" }, { "from": 46, "to": 121, "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" }, { "from": 47, "to": 328, "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" }, { "from": 121, "to": 128, "label": "CASE" }, { "from": 128, "to": 129, "label": "PARALLEL" }, { "from": 128, "to": 130, "label": "PARALLEL" }, { "from": 129, "to": 137, "label": "EVAL with clause\nmember(X41, .(X41, X42)).\nand substitutionT33 -> T55,\nT34 -> T56,\nX41 -> .(T55, .(T56, [])),\nX42 -> T57,\nT35 -> .(.(T55, .(T56, [])), T57)" }, { "from": 129, "to": 140, "label": "EVAL-BACKTRACK" }, { "from": 130, "to": 270, "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)" }, { "from": 130, "to": 271, "label": "EVAL-BACKTRACK" }, { "from": 137, "to": 147, "label": "SUCCESS" }, { "from": 270, "to": 121, "label": "INSTANCE with matching:\nT33 -> T66\nT34 -> T67\nT35 -> T69" }, { "from": 328, "to": 329, "label": "SPLIT 1" }, { "from": 328, "to": 330, "label": "SPLIT 2\nnew knowledge:\nT90 is ground\nT100 is ground\nT92 is ground\nreplacements:X75 -> T100" }, { "from": 329, "to": 333, "label": "CASE" }, { "from": 330, "to": 386, "label": "SPLIT 1" }, { "from": 330, "to": 387, "label": "SPLIT 2\nnew knowledge:\nT100 is ground\nT93 is ground" }, { "from": 333, "to": 337, "label": "PARALLEL" }, { "from": 333, "to": 338, "label": "PARALLEL" }, { "from": 337, "to": 339, "label": "EVAL with clause\nmember1(X100, .(X100, X101)).\nand substitutionT90 -> T119,\nX75 -> T120,\nX100 -> .(T119, .(T120, [])),\nX102 -> T120,\nX101 -> T121,\nT92 -> .(.(T119, .(T120, [])), T121)" }, { "from": 337, "to": 341, "label": "EVAL-BACKTRACK" }, { "from": 338, "to": 345, "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)" }, { "from": 338, "to": 346, "label": "EVAL-BACKTRACK" }, { "from": 339, "to": 343, "label": "SUCCESS" }, { "from": 345, "to": 329, "label": "INSTANCE with matching:\nT90 -> T128\nX75 -> X114\nT92 -> T130" }, { "from": 386, "to": 388, "label": "CASE" }, { "from": 387, "to": 2, "label": "INSTANCE with matching:\nT1 -> T100\nT2 -> T91\nT3 -> T92\nT4 -> .(T100, T93)" }, { "from": 388, "to": 389, "label": "PARALLEL" }, { "from": 388, "to": 390, "label": "PARALLEL" }, { "from": 389, "to": 391, "label": "EVAL with clause\nmember(X137, .(X137, X138)).\nand substitutionT100 -> T153,\nX137 -> T153,\nX138 -> T154,\nT93 -> .(T153, T154)" }, { "from": 389, "to": 392, "label": "EVAL-BACKTRACK" }, { "from": 390, "to": 394, "label": "EVAL with clause\nmember(X145, .(X146, X147)) :- member(X145, X147).\nand substitutionT100 -> T161,\nX145 -> T161,\nX146 -> T162,\nX147 -> T163,\nT93 -> .(T162, T163)" }, { "from": 390, "to": 395, "label": "EVAL-BACKTRACK" }, { "from": 391, "to": 393, "label": "SUCCESS" }, { "from": 394, "to": 386, "label": "INSTANCE with matching:\nT100 -> T161\nT93 -> T163" } ], "type": "Graph" } } ---------------------------------------- (224) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f2_in(T33, T34, T35, T36) -> U1(f121_in(T33, T34, T35), T33, T34, T35, T36) U1(f121_out1, T33, T34, T35, T36) -> f2_out1 f2_in(T90, T91, T92, T93) -> U2(f328_in(T90, T92, T93, T91), T90, T91, T92, T93) U2(f328_out1(X75), T90, T91, T92, T93) -> f2_out1 f121_in(T55, T56, .(.(T55, .(T56, [])), T57)) -> f121_out1 f121_in(T66, T67, .(T68, T69)) -> U3(f121_in(T66, T67, T69), T66, T67, .(T68, T69)) U3(f121_out1, T66, T67, .(T68, T69)) -> f121_out1 f329_in(T119, .(.(T119, .(T120, [])), T121)) -> f329_out1(T120) f329_in(T128, .(T129, T130)) -> U4(f329_in(T128, T130), T128, .(T129, T130)) U4(f329_out1(X114), T128, .(T129, T130)) -> f329_out1(X114) f386_in(T153, .(T153, T154)) -> f386_out1 f386_in(T161, .(T162, T163)) -> U5(f386_in(T161, T163), T161, .(T162, T163)) U5(f386_out1, T161, .(T162, T163)) -> f386_out1 f328_in(T90, T92, T93, T91) -> U6(f329_in(T90, T92), T90, T92, T93, T91) U6(f329_out1(T100), T90, T92, T93, T91) -> U7(f330_in(T100, T93, T91, T92), T90, T92, T93, T91, T100) U7(f330_out1, T90, T92, T93, T91, T100) -> f328_out1(T100) f330_in(T100, T93, T91, T92) -> U8(f386_in(T100, T93), T100, T93, T91, T92) U8(f386_out1, T100, T93, T91, T92) -> U9(f2_in(T100, T91, T92, .(T100, T93)), T100, T93, T91, T92) U9(f2_out1, T100, T93, T91, T92) -> f330_out1 Q is empty. ---------------------------------------- (225) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (226) Obligation: Q DP problem: The TRS P consists of the following rules: F2_IN(T33, T34, T35, T36) -> U1^1(f121_in(T33, T34, T35), T33, T34, T35, T36) F2_IN(T33, T34, T35, T36) -> F121_IN(T33, T34, T35) F2_IN(T90, T91, T92, T93) -> U2^1(f328_in(T90, T92, T93, T91), T90, T91, T92, T93) F2_IN(T90, T91, T92, T93) -> F328_IN(T90, T92, T93, T91) F121_IN(T66, T67, .(T68, T69)) -> U3^1(f121_in(T66, T67, T69), T66, T67, .(T68, T69)) F121_IN(T66, T67, .(T68, T69)) -> F121_IN(T66, T67, T69) F329_IN(T128, .(T129, T130)) -> U4^1(f329_in(T128, T130), T128, .(T129, T130)) F329_IN(T128, .(T129, T130)) -> F329_IN(T128, T130) F386_IN(T161, .(T162, T163)) -> U5^1(f386_in(T161, T163), T161, .(T162, T163)) F386_IN(T161, .(T162, T163)) -> F386_IN(T161, T163) F328_IN(T90, T92, T93, T91) -> U6^1(f329_in(T90, T92), T90, T92, T93, T91) F328_IN(T90, T92, T93, T91) -> F329_IN(T90, T92) U6^1(f329_out1(T100), T90, T92, T93, T91) -> U7^1(f330_in(T100, T93, T91, T92), T90, T92, T93, T91, T100) U6^1(f329_out1(T100), T90, T92, T93, T91) -> F330_IN(T100, T93, T91, T92) F330_IN(T100, T93, T91, T92) -> U8^1(f386_in(T100, T93), T100, T93, T91, T92) F330_IN(T100, T93, T91, T92) -> F386_IN(T100, T93) U8^1(f386_out1, T100, T93, T91, T92) -> U9^1(f2_in(T100, T91, T92, .(T100, T93)), T100, T93, T91, T92) U8^1(f386_out1, T100, T93, T91, T92) -> F2_IN(T100, T91, T92, .(T100, T93)) The TRS R consists of the following rules: f2_in(T33, T34, T35, T36) -> U1(f121_in(T33, T34, T35), T33, T34, T35, T36) U1(f121_out1, T33, T34, T35, T36) -> f2_out1 f2_in(T90, T91, T92, T93) -> U2(f328_in(T90, T92, T93, T91), T90, T91, T92, T93) U2(f328_out1(X75), T90, T91, T92, T93) -> f2_out1 f121_in(T55, T56, .(.(T55, .(T56, [])), T57)) -> f121_out1 f121_in(T66, T67, .(T68, T69)) -> U3(f121_in(T66, T67, T69), T66, T67, .(T68, T69)) U3(f121_out1, T66, T67, .(T68, T69)) -> f121_out1 f329_in(T119, .(.(T119, .(T120, [])), T121)) -> f329_out1(T120) f329_in(T128, .(T129, T130)) -> U4(f329_in(T128, T130), T128, .(T129, T130)) U4(f329_out1(X114), T128, .(T129, T130)) -> f329_out1(X114) f386_in(T153, .(T153, T154)) -> f386_out1 f386_in(T161, .(T162, T163)) -> U5(f386_in(T161, T163), T161, .(T162, T163)) U5(f386_out1, T161, .(T162, T163)) -> f386_out1 f328_in(T90, T92, T93, T91) -> U6(f329_in(T90, T92), T90, T92, T93, T91) U6(f329_out1(T100), T90, T92, T93, T91) -> U7(f330_in(T100, T93, T91, T92), T90, T92, T93, T91, T100) U7(f330_out1, T90, T92, T93, T91, T100) -> f328_out1(T100) f330_in(T100, T93, T91, T92) -> U8(f386_in(T100, T93), T100, T93, T91, T92) U8(f386_out1, T100, T93, T91, T92) -> U9(f2_in(T100, T91, T92, .(T100, T93)), T100, T93, T91, T92) U9(f2_out1, T100, T93, T91, T92) -> f330_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (227) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 10 less nodes. ---------------------------------------- (228) Complex Obligation (AND) ---------------------------------------- (229) Obligation: Q DP problem: The TRS P consists of the following rules: F386_IN(T161, .(T162, T163)) -> F386_IN(T161, T163) The TRS R consists of the following rules: f2_in(T33, T34, T35, T36) -> U1(f121_in(T33, T34, T35), T33, T34, T35, T36) U1(f121_out1, T33, T34, T35, T36) -> f2_out1 f2_in(T90, T91, T92, T93) -> U2(f328_in(T90, T92, T93, T91), T90, T91, T92, T93) U2(f328_out1(X75), T90, T91, T92, T93) -> f2_out1 f121_in(T55, T56, .(.(T55, .(T56, [])), T57)) -> f121_out1 f121_in(T66, T67, .(T68, T69)) -> U3(f121_in(T66, T67, T69), T66, T67, .(T68, T69)) U3(f121_out1, T66, T67, .(T68, T69)) -> f121_out1 f329_in(T119, .(.(T119, .(T120, [])), T121)) -> f329_out1(T120) f329_in(T128, .(T129, T130)) -> U4(f329_in(T128, T130), T128, .(T129, T130)) U4(f329_out1(X114), T128, .(T129, T130)) -> f329_out1(X114) f386_in(T153, .(T153, T154)) -> f386_out1 f386_in(T161, .(T162, T163)) -> U5(f386_in(T161, T163), T161, .(T162, T163)) U5(f386_out1, T161, .(T162, T163)) -> f386_out1 f328_in(T90, T92, T93, T91) -> U6(f329_in(T90, T92), T90, T92, T93, T91) U6(f329_out1(T100), T90, T92, T93, T91) -> U7(f330_in(T100, T93, T91, T92), T90, T92, T93, T91, T100) U7(f330_out1, T90, T92, T93, T91, T100) -> f328_out1(T100) f330_in(T100, T93, T91, T92) -> U8(f386_in(T100, T93), T100, T93, T91, T92) U8(f386_out1, T100, T93, T91, T92) -> U9(f2_in(T100, T91, T92, .(T100, T93)), T100, T93, T91, T92) U9(f2_out1, T100, T93, T91, T92) -> f330_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (230) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (231) Obligation: Q DP problem: The TRS P consists of the following rules: F386_IN(T161, .(T162, T163)) -> F386_IN(T161, T163) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (232) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *F386_IN(T161, .(T162, T163)) -> F386_IN(T161, T163) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (233) YES ---------------------------------------- (234) Obligation: Q DP problem: The TRS P consists of the following rules: F329_IN(T128, .(T129, T130)) -> F329_IN(T128, T130) The TRS R consists of the following rules: f2_in(T33, T34, T35, T36) -> U1(f121_in(T33, T34, T35), T33, T34, T35, T36) U1(f121_out1, T33, T34, T35, T36) -> f2_out1 f2_in(T90, T91, T92, T93) -> U2(f328_in(T90, T92, T93, T91), T90, T91, T92, T93) U2(f328_out1(X75), T90, T91, T92, T93) -> f2_out1 f121_in(T55, T56, .(.(T55, .(T56, [])), T57)) -> f121_out1 f121_in(T66, T67, .(T68, T69)) -> U3(f121_in(T66, T67, T69), T66, T67, .(T68, T69)) U3(f121_out1, T66, T67, .(T68, T69)) -> f121_out1 f329_in(T119, .(.(T119, .(T120, [])), T121)) -> f329_out1(T120) f329_in(T128, .(T129, T130)) -> U4(f329_in(T128, T130), T128, .(T129, T130)) U4(f329_out1(X114), T128, .(T129, T130)) -> f329_out1(X114) f386_in(T153, .(T153, T154)) -> f386_out1 f386_in(T161, .(T162, T163)) -> U5(f386_in(T161, T163), T161, .(T162, T163)) U5(f386_out1, T161, .(T162, T163)) -> f386_out1 f328_in(T90, T92, T93, T91) -> U6(f329_in(T90, T92), T90, T92, T93, T91) U6(f329_out1(T100), T90, T92, T93, T91) -> U7(f330_in(T100, T93, T91, T92), T90, T92, T93, T91, T100) U7(f330_out1, T90, T92, T93, T91, T100) -> f328_out1(T100) f330_in(T100, T93, T91, T92) -> U8(f386_in(T100, T93), T100, T93, T91, T92) U8(f386_out1, T100, T93, T91, T92) -> U9(f2_in(T100, T91, T92, .(T100, T93)), T100, T93, T91, T92) U9(f2_out1, T100, T93, T91, T92) -> f330_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (235) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (236) Obligation: Q DP problem: The TRS P consists of the following rules: F329_IN(T128, .(T129, T130)) -> F329_IN(T128, T130) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (237) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *F329_IN(T128, .(T129, T130)) -> F329_IN(T128, T130) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (238) YES ---------------------------------------- (239) Obligation: Q DP problem: The TRS P consists of the following rules: F121_IN(T66, T67, .(T68, T69)) -> F121_IN(T66, T67, T69) The TRS R consists of the following rules: f2_in(T33, T34, T35, T36) -> U1(f121_in(T33, T34, T35), T33, T34, T35, T36) U1(f121_out1, T33, T34, T35, T36) -> f2_out1 f2_in(T90, T91, T92, T93) -> U2(f328_in(T90, T92, T93, T91), T90, T91, T92, T93) U2(f328_out1(X75), T90, T91, T92, T93) -> f2_out1 f121_in(T55, T56, .(.(T55, .(T56, [])), T57)) -> f121_out1 f121_in(T66, T67, .(T68, T69)) -> U3(f121_in(T66, T67, T69), T66, T67, .(T68, T69)) U3(f121_out1, T66, T67, .(T68, T69)) -> f121_out1 f329_in(T119, .(.(T119, .(T120, [])), T121)) -> f329_out1(T120) f329_in(T128, .(T129, T130)) -> U4(f329_in(T128, T130), T128, .(T129, T130)) U4(f329_out1(X114), T128, .(T129, T130)) -> f329_out1(X114) f386_in(T153, .(T153, T154)) -> f386_out1 f386_in(T161, .(T162, T163)) -> U5(f386_in(T161, T163), T161, .(T162, T163)) U5(f386_out1, T161, .(T162, T163)) -> f386_out1 f328_in(T90, T92, T93, T91) -> U6(f329_in(T90, T92), T90, T92, T93, T91) U6(f329_out1(T100), T90, T92, T93, T91) -> U7(f330_in(T100, T93, T91, T92), T90, T92, T93, T91, T100) U7(f330_out1, T90, T92, T93, T91, T100) -> f328_out1(T100) f330_in(T100, T93, T91, T92) -> U8(f386_in(T100, T93), T100, T93, T91, T92) U8(f386_out1, T100, T93, T91, T92) -> U9(f2_in(T100, T91, T92, .(T100, T93)), T100, T93, T91, T92) U9(f2_out1, T100, T93, T91, T92) -> f330_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (240) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (241) Obligation: Q DP problem: The TRS P consists of the following rules: F121_IN(T66, T67, .(T68, T69)) -> F121_IN(T66, T67, T69) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (242) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *F121_IN(T66, T67, .(T68, T69)) -> F121_IN(T66, T67, T69) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 ---------------------------------------- (243) YES ---------------------------------------- (244) Obligation: Q DP problem: The TRS P consists of the following rules: F2_IN(T90, T91, T92, T93) -> F328_IN(T90, T92, T93, T91) F328_IN(T90, T92, T93, T91) -> U6^1(f329_in(T90, T92), T90, T92, T93, T91) U6^1(f329_out1(T100), T90, T92, T93, T91) -> F330_IN(T100, T93, T91, T92) F330_IN(T100, T93, T91, T92) -> U8^1(f386_in(T100, T93), T100, T93, T91, T92) U8^1(f386_out1, T100, T93, T91, T92) -> F2_IN(T100, T91, T92, .(T100, T93)) The TRS R consists of the following rules: f2_in(T33, T34, T35, T36) -> U1(f121_in(T33, T34, T35), T33, T34, T35, T36) U1(f121_out1, T33, T34, T35, T36) -> f2_out1 f2_in(T90, T91, T92, T93) -> U2(f328_in(T90, T92, T93, T91), T90, T91, T92, T93) U2(f328_out1(X75), T90, T91, T92, T93) -> f2_out1 f121_in(T55, T56, .(.(T55, .(T56, [])), T57)) -> f121_out1 f121_in(T66, T67, .(T68, T69)) -> U3(f121_in(T66, T67, T69), T66, T67, .(T68, T69)) U3(f121_out1, T66, T67, .(T68, T69)) -> f121_out1 f329_in(T119, .(.(T119, .(T120, [])), T121)) -> f329_out1(T120) f329_in(T128, .(T129, T130)) -> U4(f329_in(T128, T130), T128, .(T129, T130)) U4(f329_out1(X114), T128, .(T129, T130)) -> f329_out1(X114) f386_in(T153, .(T153, T154)) -> f386_out1 f386_in(T161, .(T162, T163)) -> U5(f386_in(T161, T163), T161, .(T162, T163)) U5(f386_out1, T161, .(T162, T163)) -> f386_out1 f328_in(T90, T92, T93, T91) -> U6(f329_in(T90, T92), T90, T92, T93, T91) U6(f329_out1(T100), T90, T92, T93, T91) -> U7(f330_in(T100, T93, T91, T92), T90, T92, T93, T91, T100) U7(f330_out1, T90, T92, T93, T91, T100) -> f328_out1(T100) f330_in(T100, T93, T91, T92) -> U8(f386_in(T100, T93), T100, T93, T91, T92) U8(f386_out1, T100, T93, T91, T92) -> U9(f2_in(T100, T91, T92, .(T100, T93)), T100, T93, T91, T92) U9(f2_out1, T100, T93, T91, T92) -> f330_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (245) NonLoopProof (COMPLETE) By Theorem 8 [NONLOOP] we deduce infiniteness of the QDP. We apply the theorem with m = 1, b = 0, σ' = [ ], and μ' = [x1 / .(x2, x1)] on the rule U8^1(f386_out1, x2, .(x2, x1), x0, .(.(x2, .(x2, [])), x3))[ ]^n[ ] -> U8^1(f386_out1, x2, .(x2, x1), x0, .(.(x2, .(x2, [])), x3))[ ]^n[x1 / .(x2, x1)] This rule is correct for the QDP as the following derivation shows: U8^1(f386_out1, x2, .(x2, x1), x0, .(.(x2, .(x2, [])), x3))[ ]^n[ ] -> U8^1(f386_out1, x2, .(x2, x1), x0, .(.(x2, .(x2, [])), x3))[ ]^n[x1 / .(x2, x1)] by Equivalency by Simplifying Mu with mu1: [x1 / .(x2, x1)] mu2: [ ] intermediate steps: Instantiate mu - Instantiation U8^1(f386_out1, x1, x0, x3, .(.(x1, .(x1, [])), y2))[ ]^n[ ] -> U8^1(f386_out1, x1, .(x1, x0), x3, .(.(x1, .(x1, [])), y2))[ ]^n[ ] by Narrowing at position: [] intermediate steps: Instantiation - Instantiation - Instantiation U8^1(f386_out1, T100, T93, T91, T92)[ ]^n[ ] -> F2_IN(T100, T91, T92, .(T100, T93))[ ]^n[ ] by Rule from TRS P intermediate steps: Instantiation - Instantiation - Instantiation - Instantiation - Instantiation - Instantiation F2_IN(x1, x0, .(.(x1, .(x4, [])), x2), .(x4, y0))[ ]^n[ ] -> U8^1(f386_out1, x4, .(x4, y0), x0, .(.(x1, .(x4, [])), x2))[ ]^n[ ] by Narrowing at position: [0] intermediate steps: Instantiation - Instantiation - Instantiation F2_IN(x3, x1, .(.(x3, .(x2, [])), x0), x4)[ ]^n[ ] -> U8^1(f386_in(x2, x4), x2, x4, x1, .(.(x3, .(x2, [])), x0))[ ]^n[ ] by Narrowing at position: [] intermediate steps: Instantiation - Instantiation F2_IN(x4, x3, .(.(x4, .(x0, [])), x1), x2)[ ]^n[ ] -> F330_IN(x0, x2, x3, .(.(x4, .(x0, [])), x1))[ ]^n[ ] by Narrowing at position: [] intermediate steps: Instantiation - Instantiation F2_IN(x0, x3, .(.(x0, .(y0, [])), y1), x1)[ ]^n[ ] -> U6^1(f329_out1(y0), x0, .(.(x0, .(y0, [])), y1), x1, x3)[ ]^n[ ] by Narrowing at position: [0] intermediate steps: Instantiation - Instantiation - Instantiation F2_IN(x0, x3, x2, x1)[ ]^n[ ] -> U6^1(f329_in(x0, x2), x0, x2, x1, x3)[ ]^n[ ] by Narrowing at position: [] intermediate steps: Instantiation - Instantiation F2_IN(T90, T91, T92, T93)[ ]^n[ ] -> F328_IN(T90, T92, T93, T91)[ ]^n[ ] by Rule from TRS P intermediate steps: Instantiation - Instantiation - Instantiation - Instantiation - Instantiation - Instantiation F328_IN(T90, T92, T93, T91)[ ]^n[ ] -> U6^1(f329_in(T90, T92), T90, T92, T93, T91)[ ]^n[ ] by Rule from TRS P intermediate steps: Instantiation - Instantiation - Instantiation f329_in(T119, .(.(T119, .(T120, [])), T121))[ ]^n[ ] -> f329_out1(T120)[ ]^n[ ] by Rule from TRS R intermediate steps: Instantiation - Instantiation - Instantiation - Instantiation - Instantiation - Instantiation - Instantiation U6^1(f329_out1(T100), T90, T92, T93, T91)[ ]^n[ ] -> F330_IN(T100, T93, T91, T92)[ ]^n[ ] by Rule from TRS P intermediate steps: Instantiation - Instantiation - Instantiation - Instantiation - Instantiation - Instantiation F330_IN(T100, T93, T91, T92)[ ]^n[ ] -> U8^1(f386_in(T100, T93), T100, T93, T91, T92)[ ]^n[ ] by Rule from TRS P intermediate steps: Instantiation - Instantiation - Instantiation f386_in(T153, .(T153, T154))[ ]^n[ ] -> f386_out1[ ]^n[ ] by Rule from TRS R ---------------------------------------- (246) NO ---------------------------------------- (247) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 16, "program": { "directives": [], "clauses": [ [ "(reach X Y Edges Visited)", "(member (. X (. Y ([]))) Edges)" ], [ "(reach X Z Edges Visited)", "(',' (member1 (. X (. Y ([]))) Edges) (',' (member Y Visited) (reach Y Z Edges (. Y Visited))))" ], [ "(member H (. H L))", null ], [ "(member X (. H L))", "(member X L)" ], [ "(member1 H (. H L))", null ], [ "(member1 X (. H L))", "(member1 X L)" ] ] }, "graph": { "nodes": { "type": "Nodes", "272": { "goal": [{ "clause": -1, "scope": -1, "term": "(member (. T33 (. T34 ([]))) T35)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T33", "T34", "T35" ], "free": [], "exprvars": [] } }, "273": { "goal": [ { "clause": 2, "scope": 2, "term": "(member (. T33 (. T34 ([]))) T35)" }, { "clause": 3, "scope": 2, "term": "(member (. T33 (. T34 ([]))) T35)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T33", "T34", "T35" ], "free": [], "exprvars": [] } }, "274": { "goal": [{ "clause": 2, "scope": 2, "term": "(member (. T33 (. T34 ([]))) T35)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T33", "T34", "T35" ], "free": [], "exprvars": [] } }, "275": { "goal": [{ "clause": 3, "scope": 2, "term": "(member (. T33 (. T34 ([]))) T35)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T33", "T34", "T35" ], "free": [], "exprvars": [] } }, "276": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "331": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T100 T93)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T93", "T100" ], "free": [], "exprvars": [] } }, "277": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "310": { "goal": [{ "clause": -1, "scope": -1, "term": "(member1 (. T90 (. X75 ([]))) T92)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T90", "T92" ], "free": ["X75"], "exprvars": [] } }, "332": { "goal": [{ "clause": -1, "scope": -1, "term": "(reach T100 T91 T92 (. T100 T93))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T91", "T92", "T93", "T100" ], "free": [], "exprvars": [] } }, "278": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "311": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (member T100 T93) (reach T100 T91 T92 (. T100 T93)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T91", "T92", "T93", "T100" ], "free": [], "exprvars": [] } }, "279": { "goal": [{ "clause": -1, "scope": -1, "term": "(member (. T66 (. T67 ([]))) T69)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T66", "T67", "T69" ], "free": [], "exprvars": [] } }, "334": { "goal": [ { "clause": 2, "scope": 4, "term": "(member T100 T93)" }, { "clause": 3, "scope": 4, "term": "(member T100 T93)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T93", "T100" ], "free": [], "exprvars": [] } }, "313": { "goal": [ { "clause": 4, "scope": 3, "term": "(member1 (. T90 (. X75 ([]))) T92)" }, { "clause": 5, "scope": 3, "term": "(member1 (. T90 (. X75 ([]))) T92)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T90", "T92" ], "free": ["X75"], "exprvars": [] } }, "335": { "goal": [{ "clause": 2, "scope": 4, "term": "(member T100 T93)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T93", "T100" ], "free": [], "exprvars": [] } }, "336": { "goal": [{ "clause": 3, "scope": 4, "term": "(member T100 T93)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T93", "T100" ], "free": [], "exprvars": [] } }, "316": { "goal": [{ "clause": 4, "scope": 3, "term": "(member1 (. T90 (. X75 ([]))) T92)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T90", "T92" ], "free": ["X75"], "exprvars": [] } }, "317": { "goal": [{ "clause": 5, "scope": 3, "term": "(member1 (. T90 (. X75 ([]))) T92)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T90", "T92" ], "free": ["X75"], "exprvars": [] } }, "319": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "16": { "goal": [{ "clause": -1, "scope": -1, "term": "(reach T1 T2 T3 T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T4", "T1", "T2", "T3" ], "free": [], "exprvars": [] } }, "17": { "goal": [ { "clause": 0, "scope": 1, "term": "(reach T1 T2 T3 T4)" }, { "clause": 1, "scope": 1, "term": "(reach T1 T2 T3 T4)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T4", "T1", "T2", "T3" ], "free": [], "exprvars": [] } }, "18": { "goal": [{ "clause": 0, "scope": 1, "term": "(reach T1 T2 T3 T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T4", "T1", "T2", "T3" ], "free": [], "exprvars": [] } }, "19": { "goal": [{ "clause": 1, "scope": 1, "term": "(reach T1 T2 T3 T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T4", "T1", "T2", "T3" ], "free": [], "exprvars": [] } }, "280": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "340": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "320": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "342": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "321": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "344": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "324": { "goal": [{ "clause": -1, "scope": -1, "term": "(member1 (. T128 (. X114 ([]))) T130)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T128", "T130" ], "free": ["X114"], "exprvars": [] } }, "325": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "348": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T161 T163)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T161", "T163" ], "free": [], "exprvars": [] } }, "349": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "308": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (member1 (. T90 (. X75 ([]))) T92) (',' (member X75 T93) (reach X75 T91 T92 (. X75 T93))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T90", "T91", "T92", "T93" ], "free": ["X75"], "exprvars": [] } } }, "edges": [ { "from": 16, "to": 17, "label": "CASE" }, { "from": 17, "to": 18, "label": "PARALLEL" }, { "from": 17, "to": 19, "label": "PARALLEL" }, { "from": 18, "to": 272, "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" }, { "from": 19, "to": 308, "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" }, { "from": 272, "to": 273, "label": "CASE" }, { "from": 273, "to": 274, "label": "PARALLEL" }, { "from": 273, "to": 275, "label": "PARALLEL" }, { "from": 274, "to": 276, "label": "EVAL with clause\nmember(X41, .(X41, X42)).\nand substitutionT33 -> T55,\nT34 -> T56,\nX41 -> .(T55, .(T56, [])),\nX42 -> T57,\nT35 -> .(.(T55, .(T56, [])), T57)" }, { "from": 274, "to": 277, "label": "EVAL-BACKTRACK" }, { "from": 275, "to": 279, "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)" }, { "from": 275, "to": 280, "label": "EVAL-BACKTRACK" }, { "from": 276, "to": 278, "label": "SUCCESS" }, { "from": 279, "to": 272, "label": "INSTANCE with matching:\nT33 -> T66\nT34 -> T67\nT35 -> T69" }, { "from": 308, "to": 310, "label": "SPLIT 1" }, { "from": 308, "to": 311, "label": "SPLIT 2\nnew knowledge:\nT90 is ground\nT100 is ground\nT92 is ground\nreplacements:X75 -> T100" }, { "from": 310, "to": 313, "label": "CASE" }, { "from": 311, "to": 331, "label": "SPLIT 1" }, { "from": 311, "to": 332, "label": "SPLIT 2\nnew knowledge:\nT100 is ground\nT93 is ground" }, { "from": 313, "to": 316, "label": "PARALLEL" }, { "from": 313, "to": 317, "label": "PARALLEL" }, { "from": 316, "to": 319, "label": "EVAL with clause\nmember1(X100, .(X100, X101)).\nand substitutionT90 -> T119,\nX75 -> T120,\nX100 -> .(T119, .(T120, [])),\nX102 -> T120,\nX101 -> T121,\nT92 -> .(.(T119, .(T120, [])), T121)" }, { "from": 316, "to": 320, "label": "EVAL-BACKTRACK" }, { "from": 317, "to": 324, "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)" }, { "from": 317, "to": 325, "label": "EVAL-BACKTRACK" }, { "from": 319, "to": 321, "label": "SUCCESS" }, { "from": 324, "to": 310, "label": "INSTANCE with matching:\nT90 -> T128\nX75 -> X114\nT92 -> T130" }, { "from": 331, "to": 334, "label": "CASE" }, { "from": 332, "to": 16, "label": "INSTANCE with matching:\nT1 -> T100\nT2 -> T91\nT3 -> T92\nT4 -> .(T100, T93)" }, { "from": 334, "to": 335, "label": "PARALLEL" }, { "from": 334, "to": 336, "label": "PARALLEL" }, { "from": 335, "to": 340, "label": "EVAL with clause\nmember(X137, .(X137, X138)).\nand substitutionT100 -> T153,\nX137 -> T153,\nX138 -> T154,\nT93 -> .(T153, T154)" }, { "from": 335, "to": 342, "label": "EVAL-BACKTRACK" }, { "from": 336, "to": 348, "label": "EVAL with clause\nmember(X145, .(X146, X147)) :- member(X145, X147).\nand substitutionT100 -> T161,\nX145 -> T161,\nX146 -> T162,\nX147 -> T163,\nT93 -> .(T162, T163)" }, { "from": 336, "to": 349, "label": "EVAL-BACKTRACK" }, { "from": 340, "to": 344, "label": "SUCCESS" }, { "from": 348, "to": 331, "label": "INSTANCE with matching:\nT100 -> T161\nT93 -> T163" } ], "type": "Graph" } } ---------------------------------------- (248) Complex Obligation (AND) ---------------------------------------- (249) Obligation: Rules: f335_out(T100, T93) -> f334_out(T100, T93) :|: TRUE f334_in(x, x1) -> f335_in(x, x1) :|: TRUE f334_in(x2, x3) -> f336_in(x2, x3) :|: TRUE f336_out(x4, x5) -> f334_out(x4, x5) :|: TRUE f331_out(T161, T163) -> f348_out(T161, T163) :|: TRUE f348_in(x6, x7) -> f331_in(x6, x7) :|: TRUE f334_out(x8, x9) -> f331_out(x8, x9) :|: TRUE f331_in(x10, x11) -> f334_in(x10, x11) :|: TRUE f336_in(x12, .(x13, x14)) -> f348_in(x12, x14) :|: TRUE f349_out -> f336_out(x15, x16) :|: TRUE f348_out(x17, x18) -> f336_out(x17, .(x19, x18)) :|: TRUE f336_in(x20, x21) -> f349_in :|: TRUE f16_in(T1, T2, T3, T4) -> f17_in(T1, T2, T3, T4) :|: TRUE f17_out(x22, x23, x24, x25) -> f16_out(x22, x23, x24, x25) :|: TRUE f18_out(x26, x27, x28, x29) -> f17_out(x26, x27, x28, x29) :|: TRUE f19_out(x30, x31, x32, x33) -> f17_out(x30, x31, x32, x33) :|: TRUE f17_in(x34, x35, x36, x37) -> f19_in(x34, x35, x36, x37) :|: TRUE f17_in(x38, x39, x40, x41) -> f18_in(x38, x39, x40, x41) :|: TRUE f308_out(x42, x43, x44, x45) -> f19_out(x42, x45, x43, x44) :|: TRUE f19_in(x46, x47, x48, x49) -> f308_in(x46, x48, x49, x47) :|: TRUE f311_out(x50, x51, x52, x53) -> f308_out(x54, x53, x51, x52) :|: TRUE f308_in(x55, x56, x57, x58) -> f310_in(x55, x56) :|: TRUE f310_out(x59, x60) -> f311_in(x61, x62, x63, x60) :|: TRUE f311_in(x64, x65, x66, x67) -> f331_in(x64, x65) :|: TRUE f331_out(x68, x69) -> f332_in(x68, x70, x71, x69) :|: TRUE f332_out(x72, x73, x74, x75) -> f311_out(x72, x75, x73, x74) :|: TRUE Start term: f16_in(T1, T2, T3, T4) ---------------------------------------- (250) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (251) TRUE ---------------------------------------- (252) Obligation: Rules: f313_in(T90, T92) -> f317_in(T90, T92) :|: TRUE f316_out(x, x1) -> f313_out(x, x1) :|: TRUE f313_in(x2, x3) -> f316_in(x2, x3) :|: TRUE f317_out(x4, x5) -> f313_out(x4, x5) :|: TRUE f310_out(T128, T130) -> f324_out(T128, T130) :|: TRUE f324_in(x6, x7) -> f310_in(x6, x7) :|: TRUE f324_out(x8, x9) -> f317_out(x8, .(x10, x9)) :|: TRUE f317_in(x11, .(x12, x13)) -> f324_in(x11, x13) :|: TRUE f325_out -> f317_out(x14, x15) :|: TRUE f317_in(x16, x17) -> f325_in :|: TRUE f310_in(x18, x19) -> f313_in(x18, x19) :|: TRUE f313_out(x20, x21) -> f310_out(x20, x21) :|: TRUE f16_in(T1, T2, T3, T4) -> f17_in(T1, T2, T3, T4) :|: TRUE f17_out(x22, x23, x24, x25) -> f16_out(x22, x23, x24, x25) :|: TRUE f18_out(x26, x27, x28, x29) -> f17_out(x26, x27, x28, x29) :|: TRUE f19_out(x30, x31, x32, x33) -> f17_out(x30, x31, x32, x33) :|: TRUE f17_in(x34, x35, x36, x37) -> f19_in(x34, x35, x36, x37) :|: TRUE f17_in(x38, x39, x40, x41) -> f18_in(x38, x39, x40, x41) :|: TRUE f308_out(x42, x43, x44, x45) -> f19_out(x42, x45, x43, x44) :|: TRUE f19_in(x46, x47, x48, x49) -> f308_in(x46, x48, x49, x47) :|: TRUE f311_out(x50, x51, x52, x53) -> f308_out(x54, x53, x51, x52) :|: TRUE f308_in(x55, x56, x57, x58) -> f310_in(x55, x56) :|: TRUE f310_out(x59, x60) -> f311_in(x61, x62, x63, x60) :|: TRUE Start term: f16_in(T1, T2, T3, T4) ---------------------------------------- (253) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f313_in(T90, T92) -> f317_in(T90, T92) :|: TRUE f324_in(x6, x7) -> f310_in(x6, x7) :|: TRUE f317_in(x11, .(x12, x13)) -> f324_in(x11, x13) :|: TRUE f310_in(x18, x19) -> f313_in(x18, x19) :|: TRUE ---------------------------------------- (254) Obligation: Rules: f313_in(T90, T92) -> f317_in(T90, T92) :|: TRUE f324_in(x6, x7) -> f310_in(x6, x7) :|: TRUE f317_in(x11, .(x12, x13)) -> f324_in(x11, x13) :|: TRUE f310_in(x18, x19) -> f313_in(x18, x19) :|: TRUE ---------------------------------------- (255) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (256) Obligation: Rules: f313_in(T90:0, .(x12:0, x13:0)) -> f313_in(T90:0, x13:0) :|: TRUE ---------------------------------------- (257) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (258) Obligation: Rules: f313_in(T90:0, .(x12:0, x13:0)) -> f313_in(T90:0, x13:0) :|: TRUE ---------------------------------------- (259) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f313_in(T90:0, .(x12:0, x13:0)) -> f313_in(T90:0, x13:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (260) Obligation: Termination digraph: Nodes: (1) f313_in(T90:0, .(x12:0, x13:0)) -> f313_in(T90:0, x13:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (261) IntTRSUnneededArgumentFilterProof (EQUIVALENT) Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements: f313_in(x1, x2) -> f313_in(x2) .(x1, x2) -> .(x2) ---------------------------------------- (262) Obligation: Rules: f313_in(.(x13:0)) -> f313_in(x13:0) :|: TRUE ---------------------------------------- (263) TempFilterProof (SOUND) Used the following sort dictionary for filtering: f313_in(VARIABLE) .(VARIABLE) Removed predefined arithmetic. ---------------------------------------- (264) Obligation: Rules: f313_in(.(x13:0)) -> f313_in(x13:0) ---------------------------------------- (265) IRSwTToQDPProof (SOUND) Removed the integers and created a QDP-Problem. ---------------------------------------- (266) Obligation: Q DP problem: The TRS P consists of the following rules: f313_in(.(x13:0)) -> f313_in(x13:0) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (267) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *f313_in(.(x13:0)) -> f313_in(x13:0) The graph contains the following edges 1 > 1 ---------------------------------------- (268) YES ---------------------------------------- (269) Obligation: Rules: f272_out(T66, T67, T69) -> f279_out(T66, T67, T69) :|: TRUE f279_in(x, x1, x2) -> f272_in(x, x1, x2) :|: TRUE f275_in(T33, T34, T35) -> f280_in :|: TRUE f275_in(x3, x4, .(x5, x6)) -> f279_in(x3, x4, x6) :|: TRUE f280_out -> f275_out(x7, x8, x9) :|: TRUE f279_out(x10, x11, x12) -> f275_out(x10, x11, .(x13, x12)) :|: TRUE f273_out(x14, x15, x16) -> f272_out(x14, x15, x16) :|: TRUE f272_in(x17, x18, x19) -> f273_in(x17, x18, x19) :|: TRUE f273_in(x20, x21, x22) -> f275_in(x20, x21, x22) :|: TRUE f273_in(x23, x24, x25) -> f274_in(x23, x24, x25) :|: TRUE f275_out(x26, x27, x28) -> f273_out(x26, x27, x28) :|: TRUE f274_out(x29, x30, x31) -> f273_out(x29, x30, x31) :|: TRUE f16_in(T1, T2, T3, T4) -> f17_in(T1, T2, T3, T4) :|: TRUE f17_out(x32, x33, x34, x35) -> f16_out(x32, x33, x34, x35) :|: TRUE f18_out(x36, x37, x38, x39) -> f17_out(x36, x37, x38, x39) :|: TRUE f19_out(x40, x41, x42, x43) -> f17_out(x40, x41, x42, x43) :|: TRUE f17_in(x44, x45, x46, x47) -> f19_in(x44, x45, x46, x47) :|: TRUE f17_in(x48, x49, x50, x51) -> f18_in(x48, x49, x50, x51) :|: TRUE f18_in(x52, x53, x54, x55) -> f272_in(x52, x53, x54) :|: TRUE f272_out(x56, x57, x58) -> f18_out(x56, x57, x58, x59) :|: TRUE Start term: f16_in(T1, T2, T3, T4) ---------------------------------------- (270) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f279_in(x, x1, x2) -> f272_in(x, x1, x2) :|: TRUE f275_in(x3, x4, .(x5, x6)) -> f279_in(x3, x4, x6) :|: TRUE f272_in(x17, x18, x19) -> f273_in(x17, x18, x19) :|: TRUE f273_in(x20, x21, x22) -> f275_in(x20, x21, x22) :|: TRUE ---------------------------------------- (271) Obligation: Rules: f279_in(x, x1, x2) -> f272_in(x, x1, x2) :|: TRUE f275_in(x3, x4, .(x5, x6)) -> f279_in(x3, x4, x6) :|: TRUE f272_in(x17, x18, x19) -> f273_in(x17, x18, x19) :|: TRUE f273_in(x20, x21, x22) -> f275_in(x20, x21, x22) :|: TRUE ---------------------------------------- (272) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (273) Obligation: Rules: f275_in(x3:0, x4:0, .(x5:0, x6:0)) -> f275_in(x3:0, x4:0, x6:0) :|: TRUE ---------------------------------------- (274) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (275) Obligation: Rules: f275_in(x3:0, x4:0, .(x5:0, x6:0)) -> f275_in(x3:0, x4:0, x6:0) :|: TRUE ---------------------------------------- (276) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f275_in(x3:0, x4:0, .(x5:0, x6:0)) -> f275_in(x3:0, x4:0, x6:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (277) Obligation: Termination digraph: Nodes: (1) f275_in(x3:0, x4:0, .(x5:0, x6:0)) -> f275_in(x3:0, x4:0, x6:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (278) IntTRSUnneededArgumentFilterProof (EQUIVALENT) Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements: f275_in(x1, x2, x3) -> f275_in(x3) .(x1, x2) -> .(x2) ---------------------------------------- (279) Obligation: Rules: f275_in(.(x6:0)) -> f275_in(x6:0) :|: TRUE ---------------------------------------- (280) TempFilterProof (SOUND) Used the following sort dictionary for filtering: f275_in(VARIABLE) .(VARIABLE) Removed predefined arithmetic. ---------------------------------------- (281) Obligation: Rules: f275_in(.(x6:0)) -> f275_in(x6:0) ---------------------------------------- (282) IRSwTToQDPProof (SOUND) Removed the integers and created a QDP-Problem. ---------------------------------------- (283) Obligation: Q DP problem: The TRS P consists of the following rules: f275_in(.(x6:0)) -> f275_in(x6:0) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (284) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *f275_in(.(x6:0)) -> f275_in(x6:0) The graph contains the following edges 1 > 1 ---------------------------------------- (285) YES ---------------------------------------- (286) Obligation: Rules: f308_out(T90, T92, T93, T91) -> f19_out(T90, T91, T92, T93) :|: TRUE f19_in(x, x1, x2, x3) -> f308_in(x, x2, x3, x1) :|: TRUE f313_in(x4, x5) -> f317_in(x4, x5) :|: TRUE f316_out(x6, x7) -> f313_out(x6, x7) :|: TRUE f313_in(x8, x9) -> f316_in(x8, x9) :|: TRUE f317_out(x10, x11) -> f313_out(x10, x11) :|: TRUE f335_out(x12, x13) -> f334_out(x12, x13) :|: TRUE f334_in(x14, x15) -> f335_in(x14, x15) :|: TRUE f334_in(x16, x17) -> f336_in(x16, x17) :|: TRUE f336_out(x18, x19) -> f334_out(x18, x19) :|: TRUE f310_out(T128, T130) -> f324_out(T128, T130) :|: TRUE f324_in(x20, x21) -> f310_in(x20, x21) :|: TRUE f311_in(x22, x23, x24, x25) -> f331_in(x22, x23) :|: TRUE f331_out(x26, x27) -> f332_in(x26, x28, x29, x27) :|: TRUE f332_out(x30, x31, x32, x33) -> f311_out(x30, x33, x31, x32) :|: TRUE f18_out(T1, T2, T3, T4) -> f17_out(T1, T2, T3, T4) :|: TRUE f19_out(x34, x35, x36, x37) -> f17_out(x34, x35, x36, x37) :|: TRUE f17_in(x38, x39, x40, x41) -> f19_in(x38, x39, x40, x41) :|: TRUE f17_in(x42, x43, x44, x45) -> f18_in(x42, x43, x44, x45) :|: TRUE f16_in(x46, x47, x48, x49) -> f17_in(x46, x47, x48, x49) :|: TRUE f17_out(x50, x51, x52, x53) -> f16_out(x50, x51, x52, x53) :|: TRUE f334_out(x54, x55) -> f331_out(x54, x55) :|: TRUE f331_in(x56, x57) -> f334_in(x56, x57) :|: TRUE f336_in(T161, .(T162, T163)) -> f348_in(T161, T163) :|: TRUE f349_out -> f336_out(x58, x59) :|: TRUE f348_out(x60, x61) -> f336_out(x60, .(x62, x61)) :|: TRUE f336_in(x63, x64) -> f349_in :|: TRUE f324_out(x65, x66) -> f317_out(x65, .(x67, x66)) :|: TRUE f317_in(x68, .(x69, x70)) -> f324_in(x68, x70) :|: TRUE f325_out -> f317_out(x71, x72) :|: TRUE f317_in(x73, x74) -> f325_in :|: TRUE f340_in -> f340_out :|: TRUE f342_out -> f335_out(x75, x76) :|: TRUE f340_out -> f335_out(T153, .(T153, T154)) :|: TRUE f335_in(x77, x78) -> f342_in :|: TRUE f335_in(x79, .(x79, x80)) -> f340_in :|: TRUE f319_in -> f319_out :|: TRUE f311_out(x81, x82, x83, x84) -> f308_out(x85, x84, x82, x83) :|: TRUE f308_in(x86, x87, x88, x89) -> f310_in(x86, x87) :|: TRUE f310_out(x90, x91) -> f311_in(x92, x93, x94, x91) :|: TRUE f331_out(x95, x96) -> f348_out(x95, x96) :|: TRUE f348_in(x97, x98) -> f331_in(x97, x98) :|: TRUE f332_in(x99, x100, x101, x102) -> f16_in(x99, x100, x101, .(x99, x102)) :|: TRUE f16_out(x103, x104, x105, .(x103, x106)) -> f332_out(x103, x104, x105, x106) :|: TRUE f310_in(x107, x108) -> f313_in(x107, x108) :|: TRUE f313_out(x109, x110) -> f310_out(x109, x110) :|: TRUE f320_out -> f316_out(x111, x112) :|: TRUE f316_in(x113, x114) -> f320_in :|: TRUE f319_out -> f316_out(T119, .(.(T119, .(T120, [])), T121)) :|: TRUE f316_in(x115, .(.(x115, .(x116, [])), x117)) -> f319_in :|: TRUE Start term: f16_in(T1, T2, T3, T4) ---------------------------------------- (287) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f19_in(x, x1, x2, x3) -> f308_in(x, x2, x3, x1) :|: TRUE f313_in(x4, x5) -> f317_in(x4, x5) :|: TRUE f316_out(x6, x7) -> f313_out(x6, x7) :|: TRUE f313_in(x8, x9) -> f316_in(x8, x9) :|: TRUE f317_out(x10, x11) -> f313_out(x10, x11) :|: TRUE f335_out(x12, x13) -> f334_out(x12, x13) :|: TRUE f334_in(x14, x15) -> f335_in(x14, x15) :|: TRUE f334_in(x16, x17) -> f336_in(x16, x17) :|: TRUE f336_out(x18, x19) -> f334_out(x18, x19) :|: TRUE f310_out(T128, T130) -> f324_out(T128, T130) :|: TRUE f324_in(x20, x21) -> f310_in(x20, x21) :|: TRUE f311_in(x22, x23, x24, x25) -> f331_in(x22, x23) :|: TRUE f331_out(x26, x27) -> f332_in(x26, x28, x29, x27) :|: TRUE f17_in(x38, x39, x40, x41) -> f19_in(x38, x39, x40, x41) :|: TRUE f16_in(x46, x47, x48, x49) -> f17_in(x46, x47, x48, x49) :|: TRUE f334_out(x54, x55) -> f331_out(x54, x55) :|: TRUE f331_in(x56, x57) -> f334_in(x56, x57) :|: TRUE f336_in(T161, .(T162, T163)) -> f348_in(T161, T163) :|: TRUE f348_out(x60, x61) -> f336_out(x60, .(x62, x61)) :|: TRUE f324_out(x65, x66) -> f317_out(x65, .(x67, x66)) :|: TRUE f317_in(x68, .(x69, x70)) -> f324_in(x68, x70) :|: TRUE f340_in -> f340_out :|: TRUE f340_out -> f335_out(T153, .(T153, T154)) :|: TRUE f335_in(x79, .(x79, x80)) -> f340_in :|: TRUE f319_in -> f319_out :|: TRUE f308_in(x86, x87, x88, x89) -> f310_in(x86, x87) :|: TRUE f310_out(x90, x91) -> f311_in(x92, x93, x94, x91) :|: TRUE f331_out(x95, x96) -> f348_out(x95, x96) :|: TRUE f348_in(x97, x98) -> f331_in(x97, x98) :|: TRUE f332_in(x99, x100, x101, x102) -> f16_in(x99, x100, x101, .(x99, x102)) :|: TRUE f310_in(x107, x108) -> f313_in(x107, x108) :|: TRUE f313_out(x109, x110) -> f310_out(x109, x110) :|: TRUE f319_out -> f316_out(T119, .(.(T119, .(T120, [])), T121)) :|: TRUE f316_in(x115, .(.(x115, .(x116, [])), x117)) -> f319_in :|: TRUE ---------------------------------------- (288) Obligation: Rules: f19_in(x, x1, x2, x3) -> f308_in(x, x2, x3, x1) :|: TRUE f313_in(x4, x5) -> f317_in(x4, x5) :|: TRUE f316_out(x6, x7) -> f313_out(x6, x7) :|: TRUE f313_in(x8, x9) -> f316_in(x8, x9) :|: TRUE f317_out(x10, x11) -> f313_out(x10, x11) :|: TRUE f335_out(x12, x13) -> f334_out(x12, x13) :|: TRUE f334_in(x14, x15) -> f335_in(x14, x15) :|: TRUE f334_in(x16, x17) -> f336_in(x16, x17) :|: TRUE f336_out(x18, x19) -> f334_out(x18, x19) :|: TRUE f310_out(T128, T130) -> f324_out(T128, T130) :|: TRUE f324_in(x20, x21) -> f310_in(x20, x21) :|: TRUE f311_in(x22, x23, x24, x25) -> f331_in(x22, x23) :|: TRUE f331_out(x26, x27) -> f332_in(x26, x28, x29, x27) :|: TRUE f17_in(x38, x39, x40, x41) -> f19_in(x38, x39, x40, x41) :|: TRUE f16_in(x46, x47, x48, x49) -> f17_in(x46, x47, x48, x49) :|: TRUE f334_out(x54, x55) -> f331_out(x54, x55) :|: TRUE f331_in(x56, x57) -> f334_in(x56, x57) :|: TRUE f336_in(T161, .(T162, T163)) -> f348_in(T161, T163) :|: TRUE f348_out(x60, x61) -> f336_out(x60, .(x62, x61)) :|: TRUE f324_out(x65, x66) -> f317_out(x65, .(x67, x66)) :|: TRUE f317_in(x68, .(x69, x70)) -> f324_in(x68, x70) :|: TRUE f340_in -> f340_out :|: TRUE f340_out -> f335_out(T153, .(T153, T154)) :|: TRUE f335_in(x79, .(x79, x80)) -> f340_in :|: TRUE f319_in -> f319_out :|: TRUE f308_in(x86, x87, x88, x89) -> f310_in(x86, x87) :|: TRUE f310_out(x90, x91) -> f311_in(x92, x93, x94, x91) :|: TRUE f331_out(x95, x96) -> f348_out(x95, x96) :|: TRUE f348_in(x97, x98) -> f331_in(x97, x98) :|: TRUE f332_in(x99, x100, x101, x102) -> f16_in(x99, x100, x101, .(x99, x102)) :|: TRUE f310_in(x107, x108) -> f313_in(x107, x108) :|: TRUE f313_out(x109, x110) -> f310_out(x109, x110) :|: TRUE f319_out -> f316_out(T119, .(.(T119, .(T120, [])), T121)) :|: TRUE f316_in(x115, .(.(x115, .(x116, [])), x117)) -> f319_in :|: TRUE ---------------------------------------- (289) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (290) Obligation: Rules: f313_in(x4:0, .(x69:0, x70:0)) -> f313_in(x4:0, x70:0) :|: TRUE f310_out(x90:0, x91:0) -> f334_in(x92:0, x93:0) :|: TRUE f310_out(T128:0, T130:0) -> f310_out(T128:0, .(x67:0, T130:0)) :|: TRUE f331_out(x26:0, x27:0) -> f313_in(x26:0, x29:0) :|: TRUE f334_in(x14:0, .(x14:0, x80:0)) -> f331_out(T153:0, .(T153:0, T154:0)) :|: TRUE f334_in(x16:0, .(T162:0, T163:0)) -> f334_in(x16:0, T163:0) :|: TRUE f331_out(x95:0, x96:0) -> f331_out(x95:0, .(x62:0, x96:0)) :|: TRUE f313_in(x8:0, .(.(x8:0, .(x116:0, [])), x117:0)) -> f310_out(T119:0, .(.(T119:0, .(T120:0, [])), T121:0)) :|: TRUE ---------------------------------------- (291) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (292) Obligation: Rules: f313_in(x4:0, .(x69:0, x70:0)) -> f313_in(x4:0, x70:0) :|: TRUE f310_out(x90:0, x91:0) -> f334_in(x92:0, x93:0) :|: TRUE f310_out(T128:0, T130:0) -> f310_out(T128:0, .(x67:0, T130:0)) :|: TRUE f331_out(x26:0, x27:0) -> f313_in(x26:0, x29:0) :|: TRUE f334_in(x14:0, .(x14:0, x80:0)) -> f331_out(T153:0, .(T153:0, T154:0)) :|: TRUE f334_in(x16:0, .(T162:0, T163:0)) -> f334_in(x16:0, T163:0) :|: TRUE f331_out(x95:0, x96:0) -> f331_out(x95:0, .(x62:0, x96:0)) :|: TRUE f313_in(x8:0, .(.(x8:0, .(x116:0, [])), x117:0)) -> f310_out(T119:0, .(.(T119:0, .(T120:0, [])), T121:0)) :|: TRUE ---------------------------------------- (293) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f313_in(x4:0, .(x69:0, x70:0)) -> f313_in(x4:0, x70:0) :|: TRUE (2) f310_out(x90:0, x91:0) -> f334_in(x92:0, x93:0) :|: TRUE (3) f310_out(T128:0, T130:0) -> f310_out(T128:0, .(x67:0, T130:0)) :|: TRUE (4) f331_out(x26:0, x27:0) -> f313_in(x26:0, x29:0) :|: TRUE (5) f334_in(x14:0, .(x14:0, x80:0)) -> f331_out(T153:0, .(T153:0, T154:0)) :|: TRUE (6) f334_in(x16:0, .(T162:0, T163:0)) -> f334_in(x16:0, T163:0) :|: TRUE (7) f331_out(x95:0, x96:0) -> f331_out(x95:0, .(x62:0, x96:0)) :|: TRUE (8) f313_in(x8:0, .(.(x8:0, .(x116:0, [])), x117:0)) -> f310_out(T119:0, .(.(T119:0, .(T120:0, [])), T121:0)) :|: TRUE Arcs: (1) -> (1), (8) (2) -> (5), (6) (3) -> (2), (3) (4) -> (1), (8) (5) -> (4), (7) (6) -> (5), (6) (7) -> (4), (7) (8) -> (2), (3) This digraph is fully evaluated! ---------------------------------------- (294) Obligation: Termination digraph: Nodes: (1) f313_in(x4:0, .(x69:0, x70:0)) -> f313_in(x4:0, x70:0) :|: TRUE (2) f331_out(x26:0, x27:0) -> f313_in(x26:0, x29:0) :|: TRUE (3) f331_out(x95:0, x96:0) -> f331_out(x95:0, .(x62:0, x96:0)) :|: TRUE (4) f334_in(x14:0, .(x14:0, x80:0)) -> f331_out(T153:0, .(T153:0, T154:0)) :|: TRUE (5) f334_in(x16:0, .(T162:0, T163:0)) -> f334_in(x16:0, T163:0) :|: TRUE (6) f310_out(x90:0, x91:0) -> f334_in(x92:0, x93:0) :|: TRUE (7) f310_out(T128:0, T130:0) -> f310_out(T128:0, .(x67:0, T130:0)) :|: TRUE (8) f313_in(x8:0, .(.(x8:0, .(x116:0, [])), x117:0)) -> f310_out(T119:0, .(.(T119:0, .(T120:0, [])), T121:0)) :|: TRUE Arcs: (1) -> (1), (8) (2) -> (1), (8) (3) -> (2), (3) (4) -> (2), (3) (5) -> (4), (5) (6) -> (4), (5) (7) -> (6), (7) (8) -> (6), (7) This digraph is fully evaluated! ---------------------------------------- (295) IntTRSUnneededArgumentFilterProof (EQUIVALENT) Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements: f331_out(x1, x2) -> f331_out(x1) ---------------------------------------- (296) Obligation: Rules: f313_in(x4:0, .(x69:0, x70:0)) -> f313_in(x4:0, x70:0) :|: TRUE f331_out(x26:0) -> f313_in(x26:0, x29:0) :|: TRUE f331_out(x95:0) -> f331_out(x95:0) :|: TRUE f334_in(x14:0, .(x14:0, x80:0)) -> f331_out(T153:0) :|: TRUE f334_in(x16:0, .(T162:0, T163:0)) -> f334_in(x16:0, T163:0) :|: TRUE f310_out(x90:0, x91:0) -> f334_in(x92:0, x93:0) :|: TRUE f310_out(T128:0, T130:0) -> f310_out(T128:0, .(x67:0, T130:0)) :|: TRUE f313_in(x8:0, .(.(x8:0, .(x116:0, [])), x117:0)) -> f310_out(T119:0, .(.(T119:0, .(T120:0, [])), T121:0)) :|: TRUE ---------------------------------------- (297) IRSwTToIntTRSProof (SOUND) Applied path-length measure to transform intTRS with terms to intTRS. ---------------------------------------- (298) Obligation: Rules: f313_in(x, .(x1, x2)) -> f313_in(x, x2) :|: TRUE f331_out(x3) -> f313_in(x3, x4) :|: TRUE f331_out(x5) -> f331_out(x5) :|: TRUE f334_in(x61, .(x6, x7)) -> f331_out(x8) :|: TRUE && x6 = x61 f334_in(x9, .(x10, x11)) -> f334_in(x9, x11) :|: TRUE f310_out(x12, x13) -> f334_in(x14, x15) :|: TRUE f310_out(x16, x17) -> f310_out(x16, .(x18, x17)) :|: TRUE f313_in(x191, .(.(x19, .(x20, [])), x21)) -> f310_out(x22, .(.(x22, .(x23, [])), x24)) :|: TRUE && x19 = x191 ---------------------------------------- (299) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (300) Obligation: Rules: f313_in(x:0, .(x1:0, x2:0)) -> f313_in(x:0, x2:0) :|: TRUE f310_out(x12:0, x13:0) -> f334_in(x14:0, x15:0) :|: TRUE f310_out(x16:0, x17:0) -> f310_out(x16:0, .(x18:0, x17:0)) :|: TRUE f313_in(x191:0, .(.(x191:0, .(x20:0, [])), x21:0)) -> f310_out(x22:0, .(.(x22:0, .(x23:0, [])), x24:0)) :|: TRUE f334_in(x9:0, .(x10:0, x11:0)) -> f334_in(x9:0, x11:0) :|: TRUE f334_in(x61:0, .(x61:0, x7:0)) -> f331_out(x8:0) :|: TRUE f331_out(x5:0) -> f331_out(x5:0) :|: TRUE f331_out(x3:0) -> f313_in(x3:0, x4:0) :|: TRUE