/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.pl /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- MAYBE proof of /export/starexec/sandbox/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern subset(a,g) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) CutEliminatorProof [SOUND, 0 ms] (2) Prolog (3) PrologToPiTRSProof [SOUND, 0 ms] (4) PiTRS (5) DependencyPairsProof [EQUIVALENT, 10 ms] (6) PiDP (7) DependencyGraphProof [EQUIVALENT, 0 ms] (8) AND (9) PiDP (10) UsableRulesProof [EQUIVALENT, 0 ms] (11) PiDP (12) PiDPToQDPProof [EQUIVALENT, 0 ms] (13) QDP (14) QDPSizeChangeProof [EQUIVALENT, 0 ms] (15) YES (16) PiDP (17) UsableRulesProof [EQUIVALENT, 0 ms] (18) PiDP (19) PiDPToQDPProof [SOUND, 0 ms] (20) QDP (21) QDPSizeChangeProof [EQUIVALENT, 0 ms] (22) YES (23) PiDP (24) UsableRulesProof [EQUIVALENT, 0 ms] (25) PiDP (26) PiDPToQDPProof [SOUND, 0 ms] (27) QDP (28) QDPQMonotonicMRRProof [EQUIVALENT, 37 ms] (29) QDP (30) UsableRulesProof [EQUIVALENT, 0 ms] (31) QDP (32) QReductionProof [EQUIVALENT, 0 ms] (33) QDP (34) TransformationProof [EQUIVALENT, 1 ms] (35) QDP (36) UsableRulesProof [EQUIVALENT, 0 ms] (37) QDP (38) QReductionProof [EQUIVALENT, 0 ms] (39) QDP (40) TransformationProof [SOUND, 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, 0 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) NonTerminationLoopProof [COMPLETE, 0 ms] (67) NO (68) PrologToPiTRSProof [SOUND, 0 ms] (69) PiTRS (70) DependencyPairsProof [EQUIVALENT, 1 ms] (71) PiDP (72) DependencyGraphProof [EQUIVALENT, 0 ms] (73) AND (74) PiDP (75) UsableRulesProof [EQUIVALENT, 0 ms] (76) PiDP (77) PiDPToQDPProof [EQUIVALENT, 0 ms] (78) QDP (79) QDPSizeChangeProof [EQUIVALENT, 0 ms] (80) YES (81) PiDP (82) UsableRulesProof [EQUIVALENT, 0 ms] (83) PiDP (84) PiDPToQDPProof [SOUND, 0 ms] (85) QDP (86) QDPSizeChangeProof [EQUIVALENT, 0 ms] (87) YES (88) PiDP (89) UsableRulesProof [EQUIVALENT, 0 ms] (90) PiDP (91) PiDPToQDPProof [SOUND, 1 ms] (92) QDP (93) QDPQMonotonicMRRProof [EQUIVALENT, 27 ms] (94) QDP (95) UsableRulesProof [EQUIVALENT, 0 ms] (96) QDP (97) QReductionProof [EQUIVALENT, 0 ms] (98) QDP (99) TransformationProof [EQUIVALENT, 0 ms] (100) QDP (101) UsableRulesProof [EQUIVALENT, 0 ms] (102) QDP (103) QReductionProof [EQUIVALENT, 0 ms] (104) QDP (105) TransformationProof [SOUND, 0 ms] (106) QDP (107) TransformationProof [EQUIVALENT, 0 ms] (108) QDP (109) TransformationProof [EQUIVALENT, 0 ms] (110) QDP (111) TransformationProof [EQUIVALENT, 0 ms] (112) QDP (113) TransformationProof [EQUIVALENT, 0 ms] (114) QDP (115) TransformationProof [EQUIVALENT, 0 ms] (116) QDP (117) TransformationProof [EQUIVALENT, 0 ms] (118) QDP (119) TransformationProof [EQUIVALENT, 0 ms] (120) QDP (121) TransformationProof [EQUIVALENT, 0 ms] (122) QDP (123) TransformationProof [EQUIVALENT, 0 ms] (124) QDP (125) TransformationProof [EQUIVALENT, 0 ms] (126) QDP (127) TransformationProof [EQUIVALENT, 0 ms] (128) QDP (129) TransformationProof [EQUIVALENT, 0 ms] (130) QDP (131) NonTerminationLoopProof [COMPLETE, 0 ms] (132) NO (133) PrologToDTProblemTransformerProof [SOUND, 284 ms] (134) TRIPLES (135) TriplesToPiDPProof [SOUND, 94 ms] (136) PiDP (137) DependencyGraphProof [EQUIVALENT, 0 ms] (138) AND (139) PiDP (140) UsableRulesProof [EQUIVALENT, 0 ms] (141) PiDP (142) PiDPToQDPProof [EQUIVALENT, 0 ms] (143) QDP (144) QDPSizeChangeProof [EQUIVALENT, 0 ms] (145) YES (146) PiDP (147) UsableRulesProof [EQUIVALENT, 0 ms] (148) PiDP (149) PiDPToQDPProof [SOUND, 0 ms] (150) QDP (151) QDPSizeChangeProof [EQUIVALENT, 0 ms] (152) YES (153) PiDP (154) UsableRulesProof [EQUIVALENT, 0 ms] (155) PiDP (156) PiDPToQDPProof [SOUND, 0 ms] (157) QDP (158) TransformationProof [EQUIVALENT, 0 ms] (159) QDP (160) UsableRulesProof [EQUIVALENT, 0 ms] (161) QDP (162) QReductionProof [EQUIVALENT, 0 ms] (163) QDP (164) TransformationProof [SOUND, 0 ms] (165) QDP (166) TransformationProof [EQUIVALENT, 0 ms] (167) QDP (168) TransformationProof [EQUIVALENT, 0 ms] (169) QDP (170) TransformationProof [EQUIVALENT, 0 ms] (171) QDP (172) TransformationProof [EQUIVALENT, 0 ms] (173) QDP (174) TransformationProof [EQUIVALENT, 0 ms] (175) QDP (176) TransformationProof [EQUIVALENT, 0 ms] (177) QDP (178) TransformationProof [EQUIVALENT, 1 ms] (179) QDP (180) TransformationProof [EQUIVALENT, 0 ms] (181) QDP (182) TransformationProof [EQUIVALENT, 0 ms] (183) QDP (184) TransformationProof [EQUIVALENT, 0 ms] (185) QDP (186) TransformationProof [EQUIVALENT, 0 ms] (187) QDP (188) TransformationProof [EQUIVALENT, 0 ms] (189) QDP (190) NonTerminationLoopProof [COMPLETE, 9 ms] (191) NO (192) PrologToTRSTransformerProof [SOUND, 339 ms] (193) QTRS (194) DependencyPairsProof [EQUIVALENT, 0 ms] (195) QDP (196) DependencyGraphProof [EQUIVALENT, 0 ms] (197) AND (198) QDP (199) UsableRulesProof [EQUIVALENT, 0 ms] (200) QDP (201) QDPSizeChangeProof [EQUIVALENT, 0 ms] (202) YES (203) QDP (204) UsableRulesProof [EQUIVALENT, 0 ms] (205) QDP (206) QDPSizeChangeProof [EQUIVALENT, 0 ms] (207) YES (208) QDP (209) NonLoopProof [COMPLETE, 44.5 s] (210) NO (211) PrologToIRSwTTransformerProof [SOUND, 485 ms] (212) AND (213) IRSwT (214) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (215) TRUE (216) IRSwT (217) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (218) IRSwT (219) IntTRSCompressionProof [EQUIVALENT, 24 ms] (220) IRSwT (221) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (222) IRSwT (223) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms] (224) IRSwT (225) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 15 ms] (226) IRSwT (227) TempFilterProof [SOUND, 3 ms] (228) IRSwT (229) IRSwTToQDPProof [SOUND, 0 ms] (230) QDP (231) QDPSizeChangeProof [EQUIVALENT, 0 ms] (232) YES (233) IRSwT (234) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (235) AND (236) IRSwT (237) IntTRSCompressionProof [EQUIVALENT, 3 ms] (238) IRSwT (239) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (240) IRSwT (241) IRSwTTerminationDigraphProof [EQUIVALENT, 1 ms] (242) IRSwT (243) IRSwTToIntTRSProof [SOUND, 9 ms] (244) IRSwT (245) IRSwT (246) IRSwT ---------------------------------------- (0) Obligation: Clauses: subset(X, Y) :- subsetchecked(X, [], Y). subsetchecked([], X1, X2). subsetchecked(.(X, Xs), Ys, Zs) :- ','(member(X, Zs), ','(not_member(X, Ys), subsetchecked(Xs, .(X, Ys), Zs))). member(X, .(X, X3)). member(X, .(X4, Xs)) :- member(X, Xs). not_member(X, Y) :- ','(member(X, Y), ','(!, failure(a))). not_member(X5, X6). failure(b). Query: subset(a,g) ---------------------------------------- (1) CutEliminatorProof (SOUND) Eliminated all cuts by simply ignoring them[PROLOG]. ---------------------------------------- (2) Obligation: Clauses: subset(X, Y) :- subsetchecked(X, [], Y). subsetchecked([], X1, X2). subsetchecked(.(X, Xs), Ys, Zs) :- ','(member(X, Zs), ','(not_member(X, Ys), subsetchecked(Xs, .(X, Ys), Zs))). member(X, .(X, X3)). member(X, .(X4, Xs)) :- member(X, Xs). not_member(X, Y) :- ','(member(X, Y), failure(a)). not_member(X5, X6). failure(b). Query: subset(a,g) ---------------------------------------- (3) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: subset_in_2: (f,b) subsetchecked_in_3: (f,b,b) member_in_2: (f,b) (b,b) not_member_in_2: (b,b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: subset_in_ag(X, Y) -> U1_ag(X, Y, subsetchecked_in_agg(X, [], Y)) subsetchecked_in_agg([], X1, X2) -> subsetchecked_out_agg([], X1, X2) subsetchecked_in_agg(.(X, Xs), Ys, Zs) -> U2_agg(X, Xs, Ys, Zs, member_in_ag(X, Zs)) member_in_ag(X, .(X, X3)) -> member_out_ag(X, .(X, X3)) member_in_ag(X, .(X4, Xs)) -> U5_ag(X, X4, Xs, member_in_ag(X, Xs)) U5_ag(X, X4, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X4, Xs)) U2_agg(X, Xs, Ys, Zs, member_out_ag(X, Zs)) -> U3_agg(X, Xs, Ys, Zs, not_member_in_gg(X, Ys)) not_member_in_gg(X, Y) -> U6_gg(X, Y, member_in_gg(X, Y)) member_in_gg(X, .(X, X3)) -> member_out_gg(X, .(X, X3)) member_in_gg(X, .(X4, Xs)) -> U5_gg(X, X4, Xs, member_in_gg(X, Xs)) U5_gg(X, X4, Xs, member_out_gg(X, Xs)) -> member_out_gg(X, .(X4, Xs)) U6_gg(X, Y, member_out_gg(X, Y)) -> U7_gg(X, Y, failure_in_g(a)) failure_in_g(b) -> failure_out_g(b) U7_gg(X, Y, failure_out_g(a)) -> not_member_out_gg(X, Y) not_member_in_gg(X5, X6) -> not_member_out_gg(X5, X6) U3_agg(X, Xs, Ys, Zs, not_member_out_gg(X, Ys)) -> U4_agg(X, Xs, Ys, Zs, subsetchecked_in_agg(Xs, .(X, Ys), Zs)) U4_agg(X, Xs, Ys, Zs, subsetchecked_out_agg(Xs, .(X, Ys), Zs)) -> subsetchecked_out_agg(.(X, Xs), Ys, Zs) U1_ag(X, Y, subsetchecked_out_agg(X, [], Y)) -> subset_out_ag(X, Y) The argument filtering Pi contains the following mapping: subset_in_ag(x1, x2) = subset_in_ag(x2) U1_ag(x1, x2, x3) = U1_ag(x2, x3) subsetchecked_in_agg(x1, x2, x3) = subsetchecked_in_agg(x2, x3) subsetchecked_out_agg(x1, x2, x3) = subsetchecked_out_agg(x1, x2, x3) U2_agg(x1, x2, x3, x4, x5) = U2_agg(x3, x4, x5) member_in_ag(x1, x2) = member_in_ag(x2) .(x1, x2) = .(x1, x2) member_out_ag(x1, x2) = member_out_ag(x1, x2) U5_ag(x1, x2, x3, x4) = U5_ag(x2, x3, x4) U3_agg(x1, x2, x3, x4, x5) = U3_agg(x1, x3, x4, x5) not_member_in_gg(x1, x2) = not_member_in_gg(x1, x2) U6_gg(x1, x2, x3) = U6_gg(x1, x2, x3) member_in_gg(x1, x2) = member_in_gg(x1, x2) member_out_gg(x1, x2) = member_out_gg(x1, x2) U5_gg(x1, x2, x3, x4) = U5_gg(x1, x2, x3, x4) U7_gg(x1, x2, x3) = U7_gg(x1, x2, x3) failure_in_g(x1) = failure_in_g(x1) b = b failure_out_g(x1) = failure_out_g(x1) a = a not_member_out_gg(x1, x2) = not_member_out_gg(x1, x2) U4_agg(x1, x2, x3, x4, x5) = U4_agg(x1, x3, x4, x5) [] = [] subset_out_ag(x1, x2) = subset_out_ag(x1, x2) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (4) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: subset_in_ag(X, Y) -> U1_ag(X, Y, subsetchecked_in_agg(X, [], Y)) subsetchecked_in_agg([], X1, X2) -> subsetchecked_out_agg([], X1, X2) subsetchecked_in_agg(.(X, Xs), Ys, Zs) -> U2_agg(X, Xs, Ys, Zs, member_in_ag(X, Zs)) member_in_ag(X, .(X, X3)) -> member_out_ag(X, .(X, X3)) member_in_ag(X, .(X4, Xs)) -> U5_ag(X, X4, Xs, member_in_ag(X, Xs)) U5_ag(X, X4, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X4, Xs)) U2_agg(X, Xs, Ys, Zs, member_out_ag(X, Zs)) -> U3_agg(X, Xs, Ys, Zs, not_member_in_gg(X, Ys)) not_member_in_gg(X, Y) -> U6_gg(X, Y, member_in_gg(X, Y)) member_in_gg(X, .(X, X3)) -> member_out_gg(X, .(X, X3)) member_in_gg(X, .(X4, Xs)) -> U5_gg(X, X4, Xs, member_in_gg(X, Xs)) U5_gg(X, X4, Xs, member_out_gg(X, Xs)) -> member_out_gg(X, .(X4, Xs)) U6_gg(X, Y, member_out_gg(X, Y)) -> U7_gg(X, Y, failure_in_g(a)) failure_in_g(b) -> failure_out_g(b) U7_gg(X, Y, failure_out_g(a)) -> not_member_out_gg(X, Y) not_member_in_gg(X5, X6) -> not_member_out_gg(X5, X6) U3_agg(X, Xs, Ys, Zs, not_member_out_gg(X, Ys)) -> U4_agg(X, Xs, Ys, Zs, subsetchecked_in_agg(Xs, .(X, Ys), Zs)) U4_agg(X, Xs, Ys, Zs, subsetchecked_out_agg(Xs, .(X, Ys), Zs)) -> subsetchecked_out_agg(.(X, Xs), Ys, Zs) U1_ag(X, Y, subsetchecked_out_agg(X, [], Y)) -> subset_out_ag(X, Y) The argument filtering Pi contains the following mapping: subset_in_ag(x1, x2) = subset_in_ag(x2) U1_ag(x1, x2, x3) = U1_ag(x2, x3) subsetchecked_in_agg(x1, x2, x3) = subsetchecked_in_agg(x2, x3) subsetchecked_out_agg(x1, x2, x3) = subsetchecked_out_agg(x1, x2, x3) U2_agg(x1, x2, x3, x4, x5) = U2_agg(x3, x4, x5) member_in_ag(x1, x2) = member_in_ag(x2) .(x1, x2) = .(x1, x2) member_out_ag(x1, x2) = member_out_ag(x1, x2) U5_ag(x1, x2, x3, x4) = U5_ag(x2, x3, x4) U3_agg(x1, x2, x3, x4, x5) = U3_agg(x1, x3, x4, x5) not_member_in_gg(x1, x2) = not_member_in_gg(x1, x2) U6_gg(x1, x2, x3) = U6_gg(x1, x2, x3) member_in_gg(x1, x2) = member_in_gg(x1, x2) member_out_gg(x1, x2) = member_out_gg(x1, x2) U5_gg(x1, x2, x3, x4) = U5_gg(x1, x2, x3, x4) U7_gg(x1, x2, x3) = U7_gg(x1, x2, x3) failure_in_g(x1) = failure_in_g(x1) b = b failure_out_g(x1) = failure_out_g(x1) a = a not_member_out_gg(x1, x2) = not_member_out_gg(x1, x2) U4_agg(x1, x2, x3, x4, x5) = U4_agg(x1, x3, x4, x5) [] = [] subset_out_ag(x1, x2) = subset_out_ag(x1, x2) ---------------------------------------- (5) 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: SUBSET_IN_AG(X, Y) -> U1_AG(X, Y, subsetchecked_in_agg(X, [], Y)) SUBSET_IN_AG(X, Y) -> SUBSETCHECKED_IN_AGG(X, [], Y) SUBSETCHECKED_IN_AGG(.(X, Xs), Ys, Zs) -> U2_AGG(X, Xs, Ys, Zs, member_in_ag(X, Zs)) SUBSETCHECKED_IN_AGG(.(X, Xs), Ys, Zs) -> MEMBER_IN_AG(X, Zs) MEMBER_IN_AG(X, .(X4, Xs)) -> U5_AG(X, X4, Xs, member_in_ag(X, Xs)) MEMBER_IN_AG(X, .(X4, Xs)) -> MEMBER_IN_AG(X, Xs) U2_AGG(X, Xs, Ys, Zs, member_out_ag(X, Zs)) -> U3_AGG(X, Xs, Ys, Zs, not_member_in_gg(X, Ys)) U2_AGG(X, Xs, Ys, Zs, member_out_ag(X, Zs)) -> NOT_MEMBER_IN_GG(X, Ys) NOT_MEMBER_IN_GG(X, Y) -> U6_GG(X, Y, member_in_gg(X, Y)) NOT_MEMBER_IN_GG(X, Y) -> MEMBER_IN_GG(X, Y) MEMBER_IN_GG(X, .(X4, Xs)) -> U5_GG(X, X4, Xs, member_in_gg(X, Xs)) MEMBER_IN_GG(X, .(X4, Xs)) -> MEMBER_IN_GG(X, Xs) U6_GG(X, Y, member_out_gg(X, Y)) -> U7_GG(X, Y, failure_in_g(a)) U6_GG(X, Y, member_out_gg(X, Y)) -> FAILURE_IN_G(a) U3_AGG(X, Xs, Ys, Zs, not_member_out_gg(X, Ys)) -> U4_AGG(X, Xs, Ys, Zs, subsetchecked_in_agg(Xs, .(X, Ys), Zs)) U3_AGG(X, Xs, Ys, Zs, not_member_out_gg(X, Ys)) -> SUBSETCHECKED_IN_AGG(Xs, .(X, Ys), Zs) The TRS R consists of the following rules: subset_in_ag(X, Y) -> U1_ag(X, Y, subsetchecked_in_agg(X, [], Y)) subsetchecked_in_agg([], X1, X2) -> subsetchecked_out_agg([], X1, X2) subsetchecked_in_agg(.(X, Xs), Ys, Zs) -> U2_agg(X, Xs, Ys, Zs, member_in_ag(X, Zs)) member_in_ag(X, .(X, X3)) -> member_out_ag(X, .(X, X3)) member_in_ag(X, .(X4, Xs)) -> U5_ag(X, X4, Xs, member_in_ag(X, Xs)) U5_ag(X, X4, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X4, Xs)) U2_agg(X, Xs, Ys, Zs, member_out_ag(X, Zs)) -> U3_agg(X, Xs, Ys, Zs, not_member_in_gg(X, Ys)) not_member_in_gg(X, Y) -> U6_gg(X, Y, member_in_gg(X, Y)) member_in_gg(X, .(X, X3)) -> member_out_gg(X, .(X, X3)) member_in_gg(X, .(X4, Xs)) -> U5_gg(X, X4, Xs, member_in_gg(X, Xs)) U5_gg(X, X4, Xs, member_out_gg(X, Xs)) -> member_out_gg(X, .(X4, Xs)) U6_gg(X, Y, member_out_gg(X, Y)) -> U7_gg(X, Y, failure_in_g(a)) failure_in_g(b) -> failure_out_g(b) U7_gg(X, Y, failure_out_g(a)) -> not_member_out_gg(X, Y) not_member_in_gg(X5, X6) -> not_member_out_gg(X5, X6) U3_agg(X, Xs, Ys, Zs, not_member_out_gg(X, Ys)) -> U4_agg(X, Xs, Ys, Zs, subsetchecked_in_agg(Xs, .(X, Ys), Zs)) U4_agg(X, Xs, Ys, Zs, subsetchecked_out_agg(Xs, .(X, Ys), Zs)) -> subsetchecked_out_agg(.(X, Xs), Ys, Zs) U1_ag(X, Y, subsetchecked_out_agg(X, [], Y)) -> subset_out_ag(X, Y) The argument filtering Pi contains the following mapping: subset_in_ag(x1, x2) = subset_in_ag(x2) U1_ag(x1, x2, x3) = U1_ag(x2, x3) subsetchecked_in_agg(x1, x2, x3) = subsetchecked_in_agg(x2, x3) subsetchecked_out_agg(x1, x2, x3) = subsetchecked_out_agg(x1, x2, x3) U2_agg(x1, x2, x3, x4, x5) = U2_agg(x3, x4, x5) member_in_ag(x1, x2) = member_in_ag(x2) .(x1, x2) = .(x1, x2) member_out_ag(x1, x2) = member_out_ag(x1, x2) U5_ag(x1, x2, x3, x4) = U5_ag(x2, x3, x4) U3_agg(x1, x2, x3, x4, x5) = U3_agg(x1, x3, x4, x5) not_member_in_gg(x1, x2) = not_member_in_gg(x1, x2) U6_gg(x1, x2, x3) = U6_gg(x1, x2, x3) member_in_gg(x1, x2) = member_in_gg(x1, x2) member_out_gg(x1, x2) = member_out_gg(x1, x2) U5_gg(x1, x2, x3, x4) = U5_gg(x1, x2, x3, x4) U7_gg(x1, x2, x3) = U7_gg(x1, x2, x3) failure_in_g(x1) = failure_in_g(x1) b = b failure_out_g(x1) = failure_out_g(x1) a = a not_member_out_gg(x1, x2) = not_member_out_gg(x1, x2) U4_agg(x1, x2, x3, x4, x5) = U4_agg(x1, x3, x4, x5) [] = [] subset_out_ag(x1, x2) = subset_out_ag(x1, x2) SUBSET_IN_AG(x1, x2) = SUBSET_IN_AG(x2) U1_AG(x1, x2, x3) = U1_AG(x2, x3) SUBSETCHECKED_IN_AGG(x1, x2, x3) = SUBSETCHECKED_IN_AGG(x2, x3) U2_AGG(x1, x2, x3, x4, x5) = U2_AGG(x3, x4, x5) MEMBER_IN_AG(x1, x2) = MEMBER_IN_AG(x2) U5_AG(x1, x2, x3, x4) = U5_AG(x2, x3, x4) U3_AGG(x1, x2, x3, x4, x5) = U3_AGG(x1, x3, x4, x5) NOT_MEMBER_IN_GG(x1, x2) = NOT_MEMBER_IN_GG(x1, x2) U6_GG(x1, x2, x3) = U6_GG(x1, x2, x3) MEMBER_IN_GG(x1, x2) = MEMBER_IN_GG(x1, x2) U5_GG(x1, x2, x3, x4) = U5_GG(x1, x2, x3, x4) U7_GG(x1, x2, x3) = U7_GG(x1, x2, x3) FAILURE_IN_G(x1) = FAILURE_IN_G(x1) U4_AGG(x1, x2, x3, x4, x5) = U4_AGG(x1, x3, x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (6) Obligation: Pi DP problem: The TRS P consists of the following rules: SUBSET_IN_AG(X, Y) -> U1_AG(X, Y, subsetchecked_in_agg(X, [], Y)) SUBSET_IN_AG(X, Y) -> SUBSETCHECKED_IN_AGG(X, [], Y) SUBSETCHECKED_IN_AGG(.(X, Xs), Ys, Zs) -> U2_AGG(X, Xs, Ys, Zs, member_in_ag(X, Zs)) SUBSETCHECKED_IN_AGG(.(X, Xs), Ys, Zs) -> MEMBER_IN_AG(X, Zs) MEMBER_IN_AG(X, .(X4, Xs)) -> U5_AG(X, X4, Xs, member_in_ag(X, Xs)) MEMBER_IN_AG(X, .(X4, Xs)) -> MEMBER_IN_AG(X, Xs) U2_AGG(X, Xs, Ys, Zs, member_out_ag(X, Zs)) -> U3_AGG(X, Xs, Ys, Zs, not_member_in_gg(X, Ys)) U2_AGG(X, Xs, Ys, Zs, member_out_ag(X, Zs)) -> NOT_MEMBER_IN_GG(X, Ys) NOT_MEMBER_IN_GG(X, Y) -> U6_GG(X, Y, member_in_gg(X, Y)) NOT_MEMBER_IN_GG(X, Y) -> MEMBER_IN_GG(X, Y) MEMBER_IN_GG(X, .(X4, Xs)) -> U5_GG(X, X4, Xs, member_in_gg(X, Xs)) MEMBER_IN_GG(X, .(X4, Xs)) -> MEMBER_IN_GG(X, Xs) U6_GG(X, Y, member_out_gg(X, Y)) -> U7_GG(X, Y, failure_in_g(a)) U6_GG(X, Y, member_out_gg(X, Y)) -> FAILURE_IN_G(a) U3_AGG(X, Xs, Ys, Zs, not_member_out_gg(X, Ys)) -> U4_AGG(X, Xs, Ys, Zs, subsetchecked_in_agg(Xs, .(X, Ys), Zs)) U3_AGG(X, Xs, Ys, Zs, not_member_out_gg(X, Ys)) -> SUBSETCHECKED_IN_AGG(Xs, .(X, Ys), Zs) The TRS R consists of the following rules: subset_in_ag(X, Y) -> U1_ag(X, Y, subsetchecked_in_agg(X, [], Y)) subsetchecked_in_agg([], X1, X2) -> subsetchecked_out_agg([], X1, X2) subsetchecked_in_agg(.(X, Xs), Ys, Zs) -> U2_agg(X, Xs, Ys, Zs, member_in_ag(X, Zs)) member_in_ag(X, .(X, X3)) -> member_out_ag(X, .(X, X3)) member_in_ag(X, .(X4, Xs)) -> U5_ag(X, X4, Xs, member_in_ag(X, Xs)) U5_ag(X, X4, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X4, Xs)) U2_agg(X, Xs, Ys, Zs, member_out_ag(X, Zs)) -> U3_agg(X, Xs, Ys, Zs, not_member_in_gg(X, Ys)) not_member_in_gg(X, Y) -> U6_gg(X, Y, member_in_gg(X, Y)) member_in_gg(X, .(X, X3)) -> member_out_gg(X, .(X, X3)) member_in_gg(X, .(X4, Xs)) -> U5_gg(X, X4, Xs, member_in_gg(X, Xs)) U5_gg(X, X4, Xs, member_out_gg(X, Xs)) -> member_out_gg(X, .(X4, Xs)) U6_gg(X, Y, member_out_gg(X, Y)) -> U7_gg(X, Y, failure_in_g(a)) failure_in_g(b) -> failure_out_g(b) U7_gg(X, Y, failure_out_g(a)) -> not_member_out_gg(X, Y) not_member_in_gg(X5, X6) -> not_member_out_gg(X5, X6) U3_agg(X, Xs, Ys, Zs, not_member_out_gg(X, Ys)) -> U4_agg(X, Xs, Ys, Zs, subsetchecked_in_agg(Xs, .(X, Ys), Zs)) U4_agg(X, Xs, Ys, Zs, subsetchecked_out_agg(Xs, .(X, Ys), Zs)) -> subsetchecked_out_agg(.(X, Xs), Ys, Zs) U1_ag(X, Y, subsetchecked_out_agg(X, [], Y)) -> subset_out_ag(X, Y) The argument filtering Pi contains the following mapping: subset_in_ag(x1, x2) = subset_in_ag(x2) U1_ag(x1, x2, x3) = U1_ag(x2, x3) subsetchecked_in_agg(x1, x2, x3) = subsetchecked_in_agg(x2, x3) subsetchecked_out_agg(x1, x2, x3) = subsetchecked_out_agg(x1, x2, x3) U2_agg(x1, x2, x3, x4, x5) = U2_agg(x3, x4, x5) member_in_ag(x1, x2) = member_in_ag(x2) .(x1, x2) = .(x1, x2) member_out_ag(x1, x2) = member_out_ag(x1, x2) U5_ag(x1, x2, x3, x4) = U5_ag(x2, x3, x4) U3_agg(x1, x2, x3, x4, x5) = U3_agg(x1, x3, x4, x5) not_member_in_gg(x1, x2) = not_member_in_gg(x1, x2) U6_gg(x1, x2, x3) = U6_gg(x1, x2, x3) member_in_gg(x1, x2) = member_in_gg(x1, x2) member_out_gg(x1, x2) = member_out_gg(x1, x2) U5_gg(x1, x2, x3, x4) = U5_gg(x1, x2, x3, x4) U7_gg(x1, x2, x3) = U7_gg(x1, x2, x3) failure_in_g(x1) = failure_in_g(x1) b = b failure_out_g(x1) = failure_out_g(x1) a = a not_member_out_gg(x1, x2) = not_member_out_gg(x1, x2) U4_agg(x1, x2, x3, x4, x5) = U4_agg(x1, x3, x4, x5) [] = [] subset_out_ag(x1, x2) = subset_out_ag(x1, x2) SUBSET_IN_AG(x1, x2) = SUBSET_IN_AG(x2) U1_AG(x1, x2, x3) = U1_AG(x2, x3) SUBSETCHECKED_IN_AGG(x1, x2, x3) = SUBSETCHECKED_IN_AGG(x2, x3) U2_AGG(x1, x2, x3, x4, x5) = U2_AGG(x3, x4, x5) MEMBER_IN_AG(x1, x2) = MEMBER_IN_AG(x2) U5_AG(x1, x2, x3, x4) = U5_AG(x2, x3, x4) U3_AGG(x1, x2, x3, x4, x5) = U3_AGG(x1, x3, x4, x5) NOT_MEMBER_IN_GG(x1, x2) = NOT_MEMBER_IN_GG(x1, x2) U6_GG(x1, x2, x3) = U6_GG(x1, x2, x3) MEMBER_IN_GG(x1, x2) = MEMBER_IN_GG(x1, x2) U5_GG(x1, x2, x3, x4) = U5_GG(x1, x2, x3, x4) U7_GG(x1, x2, x3) = U7_GG(x1, x2, x3) FAILURE_IN_G(x1) = FAILURE_IN_G(x1) U4_AGG(x1, x2, x3, x4, x5) = U4_AGG(x1, x3, x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (7) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 11 less nodes. ---------------------------------------- (8) Complex Obligation (AND) ---------------------------------------- (9) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER_IN_GG(X, .(X4, Xs)) -> MEMBER_IN_GG(X, Xs) The TRS R consists of the following rules: subset_in_ag(X, Y) -> U1_ag(X, Y, subsetchecked_in_agg(X, [], Y)) subsetchecked_in_agg([], X1, X2) -> subsetchecked_out_agg([], X1, X2) subsetchecked_in_agg(.(X, Xs), Ys, Zs) -> U2_agg(X, Xs, Ys, Zs, member_in_ag(X, Zs)) member_in_ag(X, .(X, X3)) -> member_out_ag(X, .(X, X3)) member_in_ag(X, .(X4, Xs)) -> U5_ag(X, X4, Xs, member_in_ag(X, Xs)) U5_ag(X, X4, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X4, Xs)) U2_agg(X, Xs, Ys, Zs, member_out_ag(X, Zs)) -> U3_agg(X, Xs, Ys, Zs, not_member_in_gg(X, Ys)) not_member_in_gg(X, Y) -> U6_gg(X, Y, member_in_gg(X, Y)) member_in_gg(X, .(X, X3)) -> member_out_gg(X, .(X, X3)) member_in_gg(X, .(X4, Xs)) -> U5_gg(X, X4, Xs, member_in_gg(X, Xs)) U5_gg(X, X4, Xs, member_out_gg(X, Xs)) -> member_out_gg(X, .(X4, Xs)) U6_gg(X, Y, member_out_gg(X, Y)) -> U7_gg(X, Y, failure_in_g(a)) failure_in_g(b) -> failure_out_g(b) U7_gg(X, Y, failure_out_g(a)) -> not_member_out_gg(X, Y) not_member_in_gg(X5, X6) -> not_member_out_gg(X5, X6) U3_agg(X, Xs, Ys, Zs, not_member_out_gg(X, Ys)) -> U4_agg(X, Xs, Ys, Zs, subsetchecked_in_agg(Xs, .(X, Ys), Zs)) U4_agg(X, Xs, Ys, Zs, subsetchecked_out_agg(Xs, .(X, Ys), Zs)) -> subsetchecked_out_agg(.(X, Xs), Ys, Zs) U1_ag(X, Y, subsetchecked_out_agg(X, [], Y)) -> subset_out_ag(X, Y) The argument filtering Pi contains the following mapping: subset_in_ag(x1, x2) = subset_in_ag(x2) U1_ag(x1, x2, x3) = U1_ag(x2, x3) subsetchecked_in_agg(x1, x2, x3) = subsetchecked_in_agg(x2, x3) subsetchecked_out_agg(x1, x2, x3) = subsetchecked_out_agg(x1, x2, x3) U2_agg(x1, x2, x3, x4, x5) = U2_agg(x3, x4, x5) member_in_ag(x1, x2) = member_in_ag(x2) .(x1, x2) = .(x1, x2) member_out_ag(x1, x2) = member_out_ag(x1, x2) U5_ag(x1, x2, x3, x4) = U5_ag(x2, x3, x4) U3_agg(x1, x2, x3, x4, x5) = U3_agg(x1, x3, x4, x5) not_member_in_gg(x1, x2) = not_member_in_gg(x1, x2) U6_gg(x1, x2, x3) = U6_gg(x1, x2, x3) member_in_gg(x1, x2) = member_in_gg(x1, x2) member_out_gg(x1, x2) = member_out_gg(x1, x2) U5_gg(x1, x2, x3, x4) = U5_gg(x1, x2, x3, x4) U7_gg(x1, x2, x3) = U7_gg(x1, x2, x3) failure_in_g(x1) = failure_in_g(x1) b = b failure_out_g(x1) = failure_out_g(x1) a = a not_member_out_gg(x1, x2) = not_member_out_gg(x1, x2) U4_agg(x1, x2, x3, x4, x5) = U4_agg(x1, x3, x4, x5) [] = [] subset_out_ag(x1, x2) = subset_out_ag(x1, x2) MEMBER_IN_GG(x1, x2) = MEMBER_IN_GG(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (10) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (11) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER_IN_GG(X, .(X4, Xs)) -> MEMBER_IN_GG(X, Xs) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (12) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (13) Obligation: Q DP problem: The TRS P consists of the following rules: MEMBER_IN_GG(X, .(X4, Xs)) -> MEMBER_IN_GG(X, Xs) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (14) 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, .(X4, Xs)) -> MEMBER_IN_GG(X, Xs) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (15) YES ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER_IN_AG(X, .(X4, Xs)) -> MEMBER_IN_AG(X, Xs) The TRS R consists of the following rules: subset_in_ag(X, Y) -> U1_ag(X, Y, subsetchecked_in_agg(X, [], Y)) subsetchecked_in_agg([], X1, X2) -> subsetchecked_out_agg([], X1, X2) subsetchecked_in_agg(.(X, Xs), Ys, Zs) -> U2_agg(X, Xs, Ys, Zs, member_in_ag(X, Zs)) member_in_ag(X, .(X, X3)) -> member_out_ag(X, .(X, X3)) member_in_ag(X, .(X4, Xs)) -> U5_ag(X, X4, Xs, member_in_ag(X, Xs)) U5_ag(X, X4, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X4, Xs)) U2_agg(X, Xs, Ys, Zs, member_out_ag(X, Zs)) -> U3_agg(X, Xs, Ys, Zs, not_member_in_gg(X, Ys)) not_member_in_gg(X, Y) -> U6_gg(X, Y, member_in_gg(X, Y)) member_in_gg(X, .(X, X3)) -> member_out_gg(X, .(X, X3)) member_in_gg(X, .(X4, Xs)) -> U5_gg(X, X4, Xs, member_in_gg(X, Xs)) U5_gg(X, X4, Xs, member_out_gg(X, Xs)) -> member_out_gg(X, .(X4, Xs)) U6_gg(X, Y, member_out_gg(X, Y)) -> U7_gg(X, Y, failure_in_g(a)) failure_in_g(b) -> failure_out_g(b) U7_gg(X, Y, failure_out_g(a)) -> not_member_out_gg(X, Y) not_member_in_gg(X5, X6) -> not_member_out_gg(X5, X6) U3_agg(X, Xs, Ys, Zs, not_member_out_gg(X, Ys)) -> U4_agg(X, Xs, Ys, Zs, subsetchecked_in_agg(Xs, .(X, Ys), Zs)) U4_agg(X, Xs, Ys, Zs, subsetchecked_out_agg(Xs, .(X, Ys), Zs)) -> subsetchecked_out_agg(.(X, Xs), Ys, Zs) U1_ag(X, Y, subsetchecked_out_agg(X, [], Y)) -> subset_out_ag(X, Y) The argument filtering Pi contains the following mapping: subset_in_ag(x1, x2) = subset_in_ag(x2) U1_ag(x1, x2, x3) = U1_ag(x2, x3) subsetchecked_in_agg(x1, x2, x3) = subsetchecked_in_agg(x2, x3) subsetchecked_out_agg(x1, x2, x3) = subsetchecked_out_agg(x1, x2, x3) U2_agg(x1, x2, x3, x4, x5) = U2_agg(x3, x4, x5) member_in_ag(x1, x2) = member_in_ag(x2) .(x1, x2) = .(x1, x2) member_out_ag(x1, x2) = member_out_ag(x1, x2) U5_ag(x1, x2, x3, x4) = U5_ag(x2, x3, x4) U3_agg(x1, x2, x3, x4, x5) = U3_agg(x1, x3, x4, x5) not_member_in_gg(x1, x2) = not_member_in_gg(x1, x2) U6_gg(x1, x2, x3) = U6_gg(x1, x2, x3) member_in_gg(x1, x2) = member_in_gg(x1, x2) member_out_gg(x1, x2) = member_out_gg(x1, x2) U5_gg(x1, x2, x3, x4) = U5_gg(x1, x2, x3, x4) U7_gg(x1, x2, x3) = U7_gg(x1, x2, x3) failure_in_g(x1) = failure_in_g(x1) b = b failure_out_g(x1) = failure_out_g(x1) a = a not_member_out_gg(x1, x2) = not_member_out_gg(x1, x2) U4_agg(x1, x2, x3, x4, x5) = U4_agg(x1, x3, x4, x5) [] = [] subset_out_ag(x1, x2) = subset_out_ag(x1, x2) MEMBER_IN_AG(x1, x2) = MEMBER_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (18) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER_IN_AG(X, .(X4, Xs)) -> MEMBER_IN_AG(X, Xs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) MEMBER_IN_AG(x1, x2) = MEMBER_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (19) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: MEMBER_IN_AG(.(X4, Xs)) -> MEMBER_IN_AG(Xs) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (21) 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_AG(.(X4, Xs)) -> MEMBER_IN_AG(Xs) The graph contains the following edges 1 > 1 ---------------------------------------- (22) YES ---------------------------------------- (23) Obligation: Pi DP problem: The TRS P consists of the following rules: U2_AGG(X, Xs, Ys, Zs, member_out_ag(X, Zs)) -> U3_AGG(X, Xs, Ys, Zs, not_member_in_gg(X, Ys)) U3_AGG(X, Xs, Ys, Zs, not_member_out_gg(X, Ys)) -> SUBSETCHECKED_IN_AGG(Xs, .(X, Ys), Zs) SUBSETCHECKED_IN_AGG(.(X, Xs), Ys, Zs) -> U2_AGG(X, Xs, Ys, Zs, member_in_ag(X, Zs)) The TRS R consists of the following rules: subset_in_ag(X, Y) -> U1_ag(X, Y, subsetchecked_in_agg(X, [], Y)) subsetchecked_in_agg([], X1, X2) -> subsetchecked_out_agg([], X1, X2) subsetchecked_in_agg(.(X, Xs), Ys, Zs) -> U2_agg(X, Xs, Ys, Zs, member_in_ag(X, Zs)) member_in_ag(X, .(X, X3)) -> member_out_ag(X, .(X, X3)) member_in_ag(X, .(X4, Xs)) -> U5_ag(X, X4, Xs, member_in_ag(X, Xs)) U5_ag(X, X4, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X4, Xs)) U2_agg(X, Xs, Ys, Zs, member_out_ag(X, Zs)) -> U3_agg(X, Xs, Ys, Zs, not_member_in_gg(X, Ys)) not_member_in_gg(X, Y) -> U6_gg(X, Y, member_in_gg(X, Y)) member_in_gg(X, .(X, X3)) -> member_out_gg(X, .(X, X3)) member_in_gg(X, .(X4, Xs)) -> U5_gg(X, X4, Xs, member_in_gg(X, Xs)) U5_gg(X, X4, Xs, member_out_gg(X, Xs)) -> member_out_gg(X, .(X4, Xs)) U6_gg(X, Y, member_out_gg(X, Y)) -> U7_gg(X, Y, failure_in_g(a)) failure_in_g(b) -> failure_out_g(b) U7_gg(X, Y, failure_out_g(a)) -> not_member_out_gg(X, Y) not_member_in_gg(X5, X6) -> not_member_out_gg(X5, X6) U3_agg(X, Xs, Ys, Zs, not_member_out_gg(X, Ys)) -> U4_agg(X, Xs, Ys, Zs, subsetchecked_in_agg(Xs, .(X, Ys), Zs)) U4_agg(X, Xs, Ys, Zs, subsetchecked_out_agg(Xs, .(X, Ys), Zs)) -> subsetchecked_out_agg(.(X, Xs), Ys, Zs) U1_ag(X, Y, subsetchecked_out_agg(X, [], Y)) -> subset_out_ag(X, Y) The argument filtering Pi contains the following mapping: subset_in_ag(x1, x2) = subset_in_ag(x2) U1_ag(x1, x2, x3) = U1_ag(x2, x3) subsetchecked_in_agg(x1, x2, x3) = subsetchecked_in_agg(x2, x3) subsetchecked_out_agg(x1, x2, x3) = subsetchecked_out_agg(x1, x2, x3) U2_agg(x1, x2, x3, x4, x5) = U2_agg(x3, x4, x5) member_in_ag(x1, x2) = member_in_ag(x2) .(x1, x2) = .(x1, x2) member_out_ag(x1, x2) = member_out_ag(x1, x2) U5_ag(x1, x2, x3, x4) = U5_ag(x2, x3, x4) U3_agg(x1, x2, x3, x4, x5) = U3_agg(x1, x3, x4, x5) not_member_in_gg(x1, x2) = not_member_in_gg(x1, x2) U6_gg(x1, x2, x3) = U6_gg(x1, x2, x3) member_in_gg(x1, x2) = member_in_gg(x1, x2) member_out_gg(x1, x2) = member_out_gg(x1, x2) U5_gg(x1, x2, x3, x4) = U5_gg(x1, x2, x3, x4) U7_gg(x1, x2, x3) = U7_gg(x1, x2, x3) failure_in_g(x1) = failure_in_g(x1) b = b failure_out_g(x1) = failure_out_g(x1) a = a not_member_out_gg(x1, x2) = not_member_out_gg(x1, x2) U4_agg(x1, x2, x3, x4, x5) = U4_agg(x1, x3, x4, x5) [] = [] subset_out_ag(x1, x2) = subset_out_ag(x1, x2) SUBSETCHECKED_IN_AGG(x1, x2, x3) = SUBSETCHECKED_IN_AGG(x2, x3) U2_AGG(x1, x2, x3, x4, x5) = U2_AGG(x3, x4, x5) U3_AGG(x1, x2, x3, x4, x5) = U3_AGG(x1, x3, x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (24) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (25) Obligation: Pi DP problem: The TRS P consists of the following rules: U2_AGG(X, Xs, Ys, Zs, member_out_ag(X, Zs)) -> U3_AGG(X, Xs, Ys, Zs, not_member_in_gg(X, Ys)) U3_AGG(X, Xs, Ys, Zs, not_member_out_gg(X, Ys)) -> SUBSETCHECKED_IN_AGG(Xs, .(X, Ys), Zs) SUBSETCHECKED_IN_AGG(.(X, Xs), Ys, Zs) -> U2_AGG(X, Xs, Ys, Zs, member_in_ag(X, Zs)) The TRS R consists of the following rules: not_member_in_gg(X, Y) -> U6_gg(X, Y, member_in_gg(X, Y)) not_member_in_gg(X5, X6) -> not_member_out_gg(X5, X6) member_in_ag(X, .(X, X3)) -> member_out_ag(X, .(X, X3)) member_in_ag(X, .(X4, Xs)) -> U5_ag(X, X4, Xs, member_in_ag(X, Xs)) U6_gg(X, Y, member_out_gg(X, Y)) -> U7_gg(X, Y, failure_in_g(a)) U5_ag(X, X4, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X4, Xs)) member_in_gg(X, .(X, X3)) -> member_out_gg(X, .(X, X3)) member_in_gg(X, .(X4, Xs)) -> U5_gg(X, X4, Xs, member_in_gg(X, Xs)) U5_gg(X, X4, Xs, member_out_gg(X, Xs)) -> member_out_gg(X, .(X4, Xs)) The argument filtering Pi contains the following mapping: member_in_ag(x1, x2) = member_in_ag(x2) .(x1, x2) = .(x1, x2) member_out_ag(x1, x2) = member_out_ag(x1, x2) U5_ag(x1, x2, x3, x4) = U5_ag(x2, x3, x4) not_member_in_gg(x1, x2) = not_member_in_gg(x1, x2) U6_gg(x1, x2, x3) = U6_gg(x1, x2, x3) member_in_gg(x1, x2) = member_in_gg(x1, x2) member_out_gg(x1, x2) = member_out_gg(x1, x2) U5_gg(x1, x2, x3, x4) = U5_gg(x1, x2, x3, x4) U7_gg(x1, x2, x3) = U7_gg(x1, x2, x3) failure_in_g(x1) = failure_in_g(x1) a = a not_member_out_gg(x1, x2) = not_member_out_gg(x1, x2) SUBSETCHECKED_IN_AGG(x1, x2, x3) = SUBSETCHECKED_IN_AGG(x2, x3) U2_AGG(x1, x2, x3, x4, x5) = U2_AGG(x3, x4, x5) U3_AGG(x1, x2, x3, x4, x5) = U3_AGG(x1, x3, x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (26) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (27) Obligation: Q DP problem: The TRS P consists of the following rules: U2_AGG(Ys, Zs, member_out_ag(X, Zs)) -> U3_AGG(X, Ys, Zs, not_member_in_gg(X, Ys)) U3_AGG(X, Ys, Zs, not_member_out_gg(X, Ys)) -> SUBSETCHECKED_IN_AGG(.(X, Ys), Zs) SUBSETCHECKED_IN_AGG(Ys, Zs) -> U2_AGG(Ys, Zs, member_in_ag(Zs)) The TRS R consists of the following rules: not_member_in_gg(X, Y) -> U6_gg(X, Y, member_in_gg(X, Y)) not_member_in_gg(X5, X6) -> not_member_out_gg(X5, X6) member_in_ag(.(X, X3)) -> member_out_ag(X, .(X, X3)) member_in_ag(.(X4, Xs)) -> U5_ag(X4, Xs, member_in_ag(Xs)) U6_gg(X, Y, member_out_gg(X, Y)) -> U7_gg(X, Y, failure_in_g(a)) U5_ag(X4, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X4, Xs)) member_in_gg(X, .(X, X3)) -> member_out_gg(X, .(X, X3)) member_in_gg(X, .(X4, Xs)) -> U5_gg(X, X4, Xs, member_in_gg(X, Xs)) U5_gg(X, X4, Xs, member_out_gg(X, Xs)) -> member_out_gg(X, .(X4, Xs)) The set Q consists of the following terms: not_member_in_gg(x0, x1) member_in_ag(x0) U6_gg(x0, x1, x2) U5_ag(x0, x1, x2) member_in_gg(x0, x1) U5_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (28) QDPQMonotonicMRRProof (EQUIVALENT) By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain. Strictly oriented rules of the TRS R: not_member_in_gg(X, Y) -> U6_gg(X, Y, member_in_gg(X, Y)) Used ordering: Polynomial interpretation [POLO]: POL(.(x_1, x_2)) = 2*x_2 POL(SUBSETCHECKED_IN_AGG(x_1, x_2)) = 2 POL(U2_AGG(x_1, x_2, x_3)) = 2 POL(U3_AGG(x_1, x_2, x_3, x_4)) = 2*x_4 POL(U5_ag(x_1, x_2, x_3)) = 0 POL(U5_gg(x_1, x_2, x_3, x_4)) = 0 POL(U6_gg(x_1, x_2, x_3)) = 0 POL(U7_gg(x_1, x_2, x_3)) = 0 POL(a) = 0 POL(failure_in_g(x_1)) = 1 + 2*x_1 POL(member_in_ag(x_1)) = 0 POL(member_in_gg(x_1, x_2)) = x_2 POL(member_out_ag(x_1, x_2)) = 0 POL(member_out_gg(x_1, x_2)) = 0 POL(not_member_in_gg(x_1, x_2)) = 1 POL(not_member_out_gg(x_1, x_2)) = 1 ---------------------------------------- (29) Obligation: Q DP problem: The TRS P consists of the following rules: U2_AGG(Ys, Zs, member_out_ag(X, Zs)) -> U3_AGG(X, Ys, Zs, not_member_in_gg(X, Ys)) U3_AGG(X, Ys, Zs, not_member_out_gg(X, Ys)) -> SUBSETCHECKED_IN_AGG(.(X, Ys), Zs) SUBSETCHECKED_IN_AGG(Ys, Zs) -> U2_AGG(Ys, Zs, member_in_ag(Zs)) The TRS R consists of the following rules: not_member_in_gg(X5, X6) -> not_member_out_gg(X5, X6) member_in_ag(.(X, X3)) -> member_out_ag(X, .(X, X3)) member_in_ag(.(X4, Xs)) -> U5_ag(X4, Xs, member_in_ag(Xs)) U6_gg(X, Y, member_out_gg(X, Y)) -> U7_gg(X, Y, failure_in_g(a)) U5_ag(X4, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X4, Xs)) member_in_gg(X, .(X, X3)) -> member_out_gg(X, .(X, X3)) member_in_gg(X, .(X4, Xs)) -> U5_gg(X, X4, Xs, member_in_gg(X, Xs)) U5_gg(X, X4, Xs, member_out_gg(X, Xs)) -> member_out_gg(X, .(X4, Xs)) The set Q consists of the following terms: not_member_in_gg(x0, x1) member_in_ag(x0) U6_gg(x0, x1, x2) U5_ag(x0, x1, x2) member_in_gg(x0, x1) U5_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (30) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (31) Obligation: Q DP problem: The TRS P consists of the following rules: U2_AGG(Ys, Zs, member_out_ag(X, Zs)) -> U3_AGG(X, Ys, Zs, not_member_in_gg(X, Ys)) U3_AGG(X, Ys, Zs, not_member_out_gg(X, Ys)) -> SUBSETCHECKED_IN_AGG(.(X, Ys), Zs) SUBSETCHECKED_IN_AGG(Ys, Zs) -> U2_AGG(Ys, Zs, member_in_ag(Zs)) The TRS R consists of the following rules: member_in_ag(.(X, X3)) -> member_out_ag(X, .(X, X3)) member_in_ag(.(X4, Xs)) -> U5_ag(X4, Xs, member_in_ag(Xs)) U5_ag(X4, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X4, Xs)) not_member_in_gg(X5, X6) -> not_member_out_gg(X5, X6) The set Q consists of the following terms: not_member_in_gg(x0, x1) member_in_ag(x0) U6_gg(x0, x1, x2) U5_ag(x0, x1, x2) member_in_gg(x0, x1) U5_gg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (32) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. U6_gg(x0, x1, x2) member_in_gg(x0, x1) U5_gg(x0, x1, x2, x3) ---------------------------------------- (33) Obligation: Q DP problem: The TRS P consists of the following rules: U2_AGG(Ys, Zs, member_out_ag(X, Zs)) -> U3_AGG(X, Ys, Zs, not_member_in_gg(X, Ys)) U3_AGG(X, Ys, Zs, not_member_out_gg(X, Ys)) -> SUBSETCHECKED_IN_AGG(.(X, Ys), Zs) SUBSETCHECKED_IN_AGG(Ys, Zs) -> U2_AGG(Ys, Zs, member_in_ag(Zs)) The TRS R consists of the following rules: member_in_ag(.(X, X3)) -> member_out_ag(X, .(X, X3)) member_in_ag(.(X4, Xs)) -> U5_ag(X4, Xs, member_in_ag(Xs)) U5_ag(X4, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X4, Xs)) not_member_in_gg(X5, X6) -> not_member_out_gg(X5, X6) The set Q consists of the following terms: not_member_in_gg(x0, x1) member_in_ag(x0) U5_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (34) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule U2_AGG(Ys, Zs, member_out_ag(X, Zs)) -> U3_AGG(X, Ys, Zs, not_member_in_gg(X, Ys)) at position [3] we obtained the following new rules [LPAR04]: (U2_AGG(Ys, Zs, member_out_ag(X, Zs)) -> U3_AGG(X, Ys, Zs, not_member_out_gg(X, Ys)),U2_AGG(Ys, Zs, member_out_ag(X, Zs)) -> U3_AGG(X, Ys, Zs, not_member_out_gg(X, Ys))) ---------------------------------------- (35) Obligation: Q DP problem: The TRS P consists of the following rules: U3_AGG(X, Ys, Zs, not_member_out_gg(X, Ys)) -> SUBSETCHECKED_IN_AGG(.(X, Ys), Zs) SUBSETCHECKED_IN_AGG(Ys, Zs) -> U2_AGG(Ys, Zs, member_in_ag(Zs)) U2_AGG(Ys, Zs, member_out_ag(X, Zs)) -> U3_AGG(X, Ys, Zs, not_member_out_gg(X, Ys)) The TRS R consists of the following rules: member_in_ag(.(X, X3)) -> member_out_ag(X, .(X, X3)) member_in_ag(.(X4, Xs)) -> U5_ag(X4, Xs, member_in_ag(Xs)) U5_ag(X4, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X4, Xs)) not_member_in_gg(X5, X6) -> not_member_out_gg(X5, X6) The set Q consists of the following terms: not_member_in_gg(x0, x1) member_in_ag(x0) U5_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (36) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (37) Obligation: Q DP problem: The TRS P consists of the following rules: U3_AGG(X, Ys, Zs, not_member_out_gg(X, Ys)) -> SUBSETCHECKED_IN_AGG(.(X, Ys), Zs) SUBSETCHECKED_IN_AGG(Ys, Zs) -> U2_AGG(Ys, Zs, member_in_ag(Zs)) U2_AGG(Ys, Zs, member_out_ag(X, Zs)) -> U3_AGG(X, Ys, Zs, not_member_out_gg(X, Ys)) The TRS R consists of the following rules: member_in_ag(.(X, X3)) -> member_out_ag(X, .(X, X3)) member_in_ag(.(X4, Xs)) -> U5_ag(X4, Xs, member_in_ag(Xs)) U5_ag(X4, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X4, Xs)) The set Q consists of the following terms: not_member_in_gg(x0, x1) member_in_ag(x0) U5_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (38) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. not_member_in_gg(x0, x1) ---------------------------------------- (39) Obligation: Q DP problem: The TRS P consists of the following rules: U3_AGG(X, Ys, Zs, not_member_out_gg(X, Ys)) -> SUBSETCHECKED_IN_AGG(.(X, Ys), Zs) SUBSETCHECKED_IN_AGG(Ys, Zs) -> U2_AGG(Ys, Zs, member_in_ag(Zs)) U2_AGG(Ys, Zs, member_out_ag(X, Zs)) -> U3_AGG(X, Ys, Zs, not_member_out_gg(X, Ys)) The TRS R consists of the following rules: member_in_ag(.(X, X3)) -> member_out_ag(X, .(X, X3)) member_in_ag(.(X4, Xs)) -> U5_ag(X4, Xs, member_in_ag(Xs)) U5_ag(X4, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X4, Xs)) The set Q consists of the following terms: member_in_ag(x0) U5_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (40) TransformationProof (SOUND) By narrowing [LPAR04] the rule SUBSETCHECKED_IN_AGG(Ys, Zs) -> U2_AGG(Ys, Zs, member_in_ag(Zs)) at position [2] we obtained the following new rules [LPAR04]: (SUBSETCHECKED_IN_AGG(y0, .(x0, x1)) -> U2_AGG(y0, .(x0, x1), member_out_ag(x0, .(x0, x1))),SUBSETCHECKED_IN_AGG(y0, .(x0, x1)) -> U2_AGG(y0, .(x0, x1), member_out_ag(x0, .(x0, x1)))) (SUBSETCHECKED_IN_AGG(y0, .(x0, x1)) -> U2_AGG(y0, .(x0, x1), U5_ag(x0, x1, member_in_ag(x1))),SUBSETCHECKED_IN_AGG(y0, .(x0, x1)) -> U2_AGG(y0, .(x0, x1), U5_ag(x0, x1, member_in_ag(x1)))) ---------------------------------------- (41) Obligation: Q DP problem: The TRS P consists of the following rules: U3_AGG(X, Ys, Zs, not_member_out_gg(X, Ys)) -> SUBSETCHECKED_IN_AGG(.(X, Ys), Zs) U2_AGG(Ys, Zs, member_out_ag(X, Zs)) -> U3_AGG(X, Ys, Zs, not_member_out_gg(X, Ys)) SUBSETCHECKED_IN_AGG(y0, .(x0, x1)) -> U2_AGG(y0, .(x0, x1), member_out_ag(x0, .(x0, x1))) SUBSETCHECKED_IN_AGG(y0, .(x0, x1)) -> U2_AGG(y0, .(x0, x1), U5_ag(x0, x1, member_in_ag(x1))) The TRS R consists of the following rules: member_in_ag(.(X, X3)) -> member_out_ag(X, .(X, X3)) member_in_ag(.(X4, Xs)) -> U5_ag(X4, Xs, member_in_ag(Xs)) U5_ag(X4, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X4, Xs)) The set Q consists of the following terms: member_in_ag(x0) U5_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (42) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U2_AGG(Ys, Zs, member_out_ag(X, Zs)) -> U3_AGG(X, Ys, Zs, not_member_out_gg(X, Ys)) we obtained the following new rules [LPAR04]: (U2_AGG(z0, .(z1, z2), member_out_ag(z1, .(z1, z2))) -> U3_AGG(z1, z0, .(z1, z2), not_member_out_gg(z1, z0)),U2_AGG(z0, .(z1, z2), member_out_ag(z1, .(z1, z2))) -> U3_AGG(z1, z0, .(z1, z2), not_member_out_gg(z1, z0))) (U2_AGG(z0, .(z1, z2), member_out_ag(x2, .(z1, z2))) -> U3_AGG(x2, z0, .(z1, z2), not_member_out_gg(x2, z0)),U2_AGG(z0, .(z1, z2), member_out_ag(x2, .(z1, z2))) -> U3_AGG(x2, z0, .(z1, z2), not_member_out_gg(x2, z0))) ---------------------------------------- (43) Obligation: Q DP problem: The TRS P consists of the following rules: U3_AGG(X, Ys, Zs, not_member_out_gg(X, Ys)) -> SUBSETCHECKED_IN_AGG(.(X, Ys), Zs) SUBSETCHECKED_IN_AGG(y0, .(x0, x1)) -> U2_AGG(y0, .(x0, x1), member_out_ag(x0, .(x0, x1))) SUBSETCHECKED_IN_AGG(y0, .(x0, x1)) -> U2_AGG(y0, .(x0, x1), U5_ag(x0, x1, member_in_ag(x1))) U2_AGG(z0, .(z1, z2), member_out_ag(z1, .(z1, z2))) -> U3_AGG(z1, z0, .(z1, z2), not_member_out_gg(z1, z0)) U2_AGG(z0, .(z1, z2), member_out_ag(x2, .(z1, z2))) -> U3_AGG(x2, z0, .(z1, z2), not_member_out_gg(x2, z0)) The TRS R consists of the following rules: member_in_ag(.(X, X3)) -> member_out_ag(X, .(X, X3)) member_in_ag(.(X4, Xs)) -> U5_ag(X4, Xs, member_in_ag(Xs)) U5_ag(X4, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X4, Xs)) The set Q consists of the following terms: member_in_ag(x0) U5_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (44) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U3_AGG(X, Ys, Zs, not_member_out_gg(X, Ys)) -> SUBSETCHECKED_IN_AGG(.(X, Ys), Zs) we obtained the following new rules [LPAR04]: (U3_AGG(z1, z0, .(z1, z2), not_member_out_gg(z1, z0)) -> SUBSETCHECKED_IN_AGG(.(z1, z0), .(z1, z2)),U3_AGG(z1, z0, .(z1, z2), not_member_out_gg(z1, z0)) -> SUBSETCHECKED_IN_AGG(.(z1, z0), .(z1, z2))) (U3_AGG(z3, z0, .(z1, z2), not_member_out_gg(z3, z0)) -> SUBSETCHECKED_IN_AGG(.(z3, z0), .(z1, z2)),U3_AGG(z3, z0, .(z1, z2), not_member_out_gg(z3, z0)) -> SUBSETCHECKED_IN_AGG(.(z3, z0), .(z1, z2))) ---------------------------------------- (45) Obligation: Q DP problem: The TRS P consists of the following rules: SUBSETCHECKED_IN_AGG(y0, .(x0, x1)) -> U2_AGG(y0, .(x0, x1), member_out_ag(x0, .(x0, x1))) SUBSETCHECKED_IN_AGG(y0, .(x0, x1)) -> U2_AGG(y0, .(x0, x1), U5_ag(x0, x1, member_in_ag(x1))) U2_AGG(z0, .(z1, z2), member_out_ag(z1, .(z1, z2))) -> U3_AGG(z1, z0, .(z1, z2), not_member_out_gg(z1, z0)) U2_AGG(z0, .(z1, z2), member_out_ag(x2, .(z1, z2))) -> U3_AGG(x2, z0, .(z1, z2), not_member_out_gg(x2, z0)) U3_AGG(z1, z0, .(z1, z2), not_member_out_gg(z1, z0)) -> SUBSETCHECKED_IN_AGG(.(z1, z0), .(z1, z2)) U3_AGG(z3, z0, .(z1, z2), not_member_out_gg(z3, z0)) -> SUBSETCHECKED_IN_AGG(.(z3, z0), .(z1, z2)) The TRS R consists of the following rules: member_in_ag(.(X, X3)) -> member_out_ag(X, .(X, X3)) member_in_ag(.(X4, Xs)) -> U5_ag(X4, Xs, member_in_ag(Xs)) U5_ag(X4, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X4, Xs)) The set Q consists of the following terms: member_in_ag(x0) U5_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (46) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule SUBSETCHECKED_IN_AGG(y0, .(x0, x1)) -> U2_AGG(y0, .(x0, x1), member_out_ag(x0, .(x0, x1))) we obtained the following new rules [LPAR04]: (SUBSETCHECKED_IN_AGG(.(z0, z1), .(z0, z2)) -> U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0, .(z0, z2))),SUBSETCHECKED_IN_AGG(.(z0, z1), .(z0, z2)) -> U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0, .(z0, z2)))) (SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2, .(z2, z3))),SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2, .(z2, z3)))) ---------------------------------------- (47) Obligation: Q DP problem: The TRS P consists of the following rules: SUBSETCHECKED_IN_AGG(y0, .(x0, x1)) -> U2_AGG(y0, .(x0, x1), U5_ag(x0, x1, member_in_ag(x1))) U2_AGG(z0, .(z1, z2), member_out_ag(z1, .(z1, z2))) -> U3_AGG(z1, z0, .(z1, z2), not_member_out_gg(z1, z0)) U2_AGG(z0, .(z1, z2), member_out_ag(x2, .(z1, z2))) -> U3_AGG(x2, z0, .(z1, z2), not_member_out_gg(x2, z0)) U3_AGG(z1, z0, .(z1, z2), not_member_out_gg(z1, z0)) -> SUBSETCHECKED_IN_AGG(.(z1, z0), .(z1, z2)) U3_AGG(z3, z0, .(z1, z2), not_member_out_gg(z3, z0)) -> SUBSETCHECKED_IN_AGG(.(z3, z0), .(z1, z2)) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z0, z2)) -> U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0, .(z0, z2))) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2, .(z2, z3))) The TRS R consists of the following rules: member_in_ag(.(X, X3)) -> member_out_ag(X, .(X, X3)) member_in_ag(.(X4, Xs)) -> U5_ag(X4, Xs, member_in_ag(Xs)) U5_ag(X4, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X4, Xs)) The set Q consists of the following terms: member_in_ag(x0) U5_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (48) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule SUBSETCHECKED_IN_AGG(y0, .(x0, x1)) -> U2_AGG(y0, .(x0, x1), U5_ag(x0, x1, member_in_ag(x1))) we obtained the following new rules [LPAR04]: (SUBSETCHECKED_IN_AGG(.(z0, z1), .(z0, z2)) -> U2_AGG(.(z0, z1), .(z0, z2), U5_ag(z0, z2, member_in_ag(z2))),SUBSETCHECKED_IN_AGG(.(z0, z1), .(z0, z2)) -> U2_AGG(.(z0, z1), .(z0, z2), U5_ag(z0, z2, member_in_ag(z2)))) (SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), U5_ag(z2, z3, member_in_ag(z3))),SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), U5_ag(z2, z3, member_in_ag(z3)))) ---------------------------------------- (49) Obligation: Q DP problem: The TRS P consists of the following rules: U2_AGG(z0, .(z1, z2), member_out_ag(z1, .(z1, z2))) -> U3_AGG(z1, z0, .(z1, z2), not_member_out_gg(z1, z0)) U2_AGG(z0, .(z1, z2), member_out_ag(x2, .(z1, z2))) -> U3_AGG(x2, z0, .(z1, z2), not_member_out_gg(x2, z0)) U3_AGG(z1, z0, .(z1, z2), not_member_out_gg(z1, z0)) -> SUBSETCHECKED_IN_AGG(.(z1, z0), .(z1, z2)) U3_AGG(z3, z0, .(z1, z2), not_member_out_gg(z3, z0)) -> SUBSETCHECKED_IN_AGG(.(z3, z0), .(z1, z2)) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z0, z2)) -> U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0, .(z0, z2))) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2, .(z2, z3))) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z0, z2)) -> U2_AGG(.(z0, z1), .(z0, z2), U5_ag(z0, z2, member_in_ag(z2))) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), U5_ag(z2, z3, member_in_ag(z3))) The TRS R consists of the following rules: member_in_ag(.(X, X3)) -> member_out_ag(X, .(X, X3)) member_in_ag(.(X4, Xs)) -> U5_ag(X4, Xs, member_in_ag(Xs)) U5_ag(X4, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X4, Xs)) The set Q consists of the following terms: member_in_ag(x0) U5_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (50) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U2_AGG(z0, .(z1, z2), member_out_ag(z1, .(z1, z2))) -> U3_AGG(z1, z0, .(z1, z2), not_member_out_gg(z1, z0)) we obtained the following new rules [LPAR04]: (U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0, .(z0, z2))) -> U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg(z0, .(z0, z1))),U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0, .(z0, z2))) -> U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg(z0, .(z0, z1)))) (U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2, .(z2, z3))) -> U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg(z2, .(z0, z1))),U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2, .(z2, z3))) -> U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg(z2, .(z0, z1)))) ---------------------------------------- (51) Obligation: Q DP problem: The TRS P consists of the following rules: U2_AGG(z0, .(z1, z2), member_out_ag(x2, .(z1, z2))) -> U3_AGG(x2, z0, .(z1, z2), not_member_out_gg(x2, z0)) U3_AGG(z1, z0, .(z1, z2), not_member_out_gg(z1, z0)) -> SUBSETCHECKED_IN_AGG(.(z1, z0), .(z1, z2)) U3_AGG(z3, z0, .(z1, z2), not_member_out_gg(z3, z0)) -> SUBSETCHECKED_IN_AGG(.(z3, z0), .(z1, z2)) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z0, z2)) -> U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0, .(z0, z2))) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2, .(z2, z3))) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z0, z2)) -> U2_AGG(.(z0, z1), .(z0, z2), U5_ag(z0, z2, member_in_ag(z2))) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), U5_ag(z2, z3, member_in_ag(z3))) U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0, .(z0, z2))) -> U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg(z0, .(z0, z1))) U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2, .(z2, z3))) -> U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg(z2, .(z0, z1))) The TRS R consists of the following rules: member_in_ag(.(X, X3)) -> member_out_ag(X, .(X, X3)) member_in_ag(.(X4, Xs)) -> U5_ag(X4, Xs, member_in_ag(Xs)) U5_ag(X4, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X4, Xs)) The set Q consists of the following terms: member_in_ag(x0) U5_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (52) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U2_AGG(z0, .(z1, z2), member_out_ag(x2, .(z1, z2))) -> U3_AGG(x2, z0, .(z1, z2), not_member_out_gg(x2, z0)) we obtained the following new rules [LPAR04]: (U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0, .(z0, z2))) -> U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg(z0, .(z0, z1))),U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0, .(z0, z2))) -> U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg(z0, .(z0, z1)))) (U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2, .(z2, z3))) -> U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg(z2, .(z0, z1))),U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2, .(z2, z3))) -> U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg(z2, .(z0, z1)))) (U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(x3, .(z0, z2))) -> U3_AGG(x3, .(z0, z1), .(z0, z2), not_member_out_gg(x3, .(z0, z1))),U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(x3, .(z0, z2))) -> U3_AGG(x3, .(z0, z1), .(z0, z2), not_member_out_gg(x3, .(z0, z1)))) (U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(x3, .(z2, z3))) -> U3_AGG(x3, .(z0, z1), .(z2, z3), not_member_out_gg(x3, .(z0, z1))),U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(x3, .(z2, z3))) -> U3_AGG(x3, .(z0, z1), .(z2, z3), not_member_out_gg(x3, .(z0, z1)))) ---------------------------------------- (53) Obligation: Q DP problem: The TRS P consists of the following rules: U3_AGG(z1, z0, .(z1, z2), not_member_out_gg(z1, z0)) -> SUBSETCHECKED_IN_AGG(.(z1, z0), .(z1, z2)) U3_AGG(z3, z0, .(z1, z2), not_member_out_gg(z3, z0)) -> SUBSETCHECKED_IN_AGG(.(z3, z0), .(z1, z2)) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z0, z2)) -> U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0, .(z0, z2))) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2, .(z2, z3))) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z0, z2)) -> U2_AGG(.(z0, z1), .(z0, z2), U5_ag(z0, z2, member_in_ag(z2))) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), U5_ag(z2, z3, member_in_ag(z3))) U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0, .(z0, z2))) -> U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg(z0, .(z0, z1))) U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2, .(z2, z3))) -> U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg(z2, .(z0, z1))) U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(x3, .(z0, z2))) -> U3_AGG(x3, .(z0, z1), .(z0, z2), not_member_out_gg(x3, .(z0, z1))) U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(x3, .(z2, z3))) -> U3_AGG(x3, .(z0, z1), .(z2, z3), not_member_out_gg(x3, .(z0, z1))) The TRS R consists of the following rules: member_in_ag(.(X, X3)) -> member_out_ag(X, .(X, X3)) member_in_ag(.(X4, Xs)) -> U5_ag(X4, Xs, member_in_ag(Xs)) U5_ag(X4, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X4, Xs)) The set Q consists of the following terms: member_in_ag(x0) U5_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (54) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U3_AGG(z1, z0, .(z1, z2), not_member_out_gg(z1, z0)) -> SUBSETCHECKED_IN_AGG(.(z1, z0), .(z1, z2)) we obtained the following new rules [LPAR04]: (U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg(z0, .(z0, z1))) -> SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)),U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg(z0, .(z0, z1))) -> SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2))) (U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg(z2, .(z0, z1))) -> SUBSETCHECKED_IN_AGG(.(z2, .(z0, z1)), .(z2, z3)),U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg(z2, .(z0, z1))) -> SUBSETCHECKED_IN_AGG(.(z2, .(z0, z1)), .(z2, z3))) ---------------------------------------- (55) Obligation: Q DP problem: The TRS P consists of the following rules: U3_AGG(z3, z0, .(z1, z2), not_member_out_gg(z3, z0)) -> SUBSETCHECKED_IN_AGG(.(z3, z0), .(z1, z2)) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z0, z2)) -> U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0, .(z0, z2))) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2, .(z2, z3))) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z0, z2)) -> U2_AGG(.(z0, z1), .(z0, z2), U5_ag(z0, z2, member_in_ag(z2))) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), U5_ag(z2, z3, member_in_ag(z3))) U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0, .(z0, z2))) -> U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg(z0, .(z0, z1))) U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2, .(z2, z3))) -> U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg(z2, .(z0, z1))) U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(x3, .(z0, z2))) -> U3_AGG(x3, .(z0, z1), .(z0, z2), not_member_out_gg(x3, .(z0, z1))) U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(x3, .(z2, z3))) -> U3_AGG(x3, .(z0, z1), .(z2, z3), not_member_out_gg(x3, .(z0, z1))) U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg(z0, .(z0, z1))) -> SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg(z2, .(z0, z1))) -> SUBSETCHECKED_IN_AGG(.(z2, .(z0, z1)), .(z2, z3)) The TRS R consists of the following rules: member_in_ag(.(X, X3)) -> member_out_ag(X, .(X, X3)) member_in_ag(.(X4, Xs)) -> U5_ag(X4, Xs, member_in_ag(Xs)) U5_ag(X4, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X4, Xs)) The set Q consists of the following terms: member_in_ag(x0) U5_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (56) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U3_AGG(z3, z0, .(z1, z2), not_member_out_gg(z3, z0)) -> SUBSETCHECKED_IN_AGG(.(z3, z0), .(z1, z2)) we obtained the following new rules [LPAR04]: (U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg(z0, .(z0, z1))) -> SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)),U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg(z0, .(z0, z1))) -> SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2))) (U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg(z2, .(z0, z1))) -> SUBSETCHECKED_IN_AGG(.(z2, .(z0, z1)), .(z2, z3)),U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg(z2, .(z0, z1))) -> SUBSETCHECKED_IN_AGG(.(z2, .(z0, z1)), .(z2, z3))) (U3_AGG(z3, .(z0, z1), .(z0, z2), not_member_out_gg(z3, .(z0, z1))) -> SUBSETCHECKED_IN_AGG(.(z3, .(z0, z1)), .(z0, z2)),U3_AGG(z3, .(z0, z1), .(z0, z2), not_member_out_gg(z3, .(z0, z1))) -> SUBSETCHECKED_IN_AGG(.(z3, .(z0, z1)), .(z0, z2))) (U3_AGG(z4, .(z0, z1), .(z2, z3), not_member_out_gg(z4, .(z0, z1))) -> SUBSETCHECKED_IN_AGG(.(z4, .(z0, z1)), .(z2, z3)),U3_AGG(z4, .(z0, z1), .(z2, z3), not_member_out_gg(z4, .(z0, z1))) -> SUBSETCHECKED_IN_AGG(.(z4, .(z0, z1)), .(z2, z3))) ---------------------------------------- (57) Obligation: Q DP problem: The TRS P consists of the following rules: SUBSETCHECKED_IN_AGG(.(z0, z1), .(z0, z2)) -> U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0, .(z0, z2))) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2, .(z2, z3))) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z0, z2)) -> U2_AGG(.(z0, z1), .(z0, z2), U5_ag(z0, z2, member_in_ag(z2))) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), U5_ag(z2, z3, member_in_ag(z3))) U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0, .(z0, z2))) -> U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg(z0, .(z0, z1))) U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2, .(z2, z3))) -> U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg(z2, .(z0, z1))) U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(x3, .(z0, z2))) -> U3_AGG(x3, .(z0, z1), .(z0, z2), not_member_out_gg(x3, .(z0, z1))) U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(x3, .(z2, z3))) -> U3_AGG(x3, .(z0, z1), .(z2, z3), not_member_out_gg(x3, .(z0, z1))) U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg(z0, .(z0, z1))) -> SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg(z2, .(z0, z1))) -> SUBSETCHECKED_IN_AGG(.(z2, .(z0, z1)), .(z2, z3)) U3_AGG(z3, .(z0, z1), .(z0, z2), not_member_out_gg(z3, .(z0, z1))) -> SUBSETCHECKED_IN_AGG(.(z3, .(z0, z1)), .(z0, z2)) U3_AGG(z4, .(z0, z1), .(z2, z3), not_member_out_gg(z4, .(z0, z1))) -> SUBSETCHECKED_IN_AGG(.(z4, .(z0, z1)), .(z2, z3)) The TRS R consists of the following rules: member_in_ag(.(X, X3)) -> member_out_ag(X, .(X, X3)) member_in_ag(.(X4, Xs)) -> U5_ag(X4, Xs, member_in_ag(Xs)) U5_ag(X4, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X4, Xs)) The set Q consists of the following terms: member_in_ag(x0) U5_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (58) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule SUBSETCHECKED_IN_AGG(.(z0, z1), .(z0, z2)) -> U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0, .(z0, z2))) we obtained the following new rules [LPAR04]: (SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) -> U2_AGG(.(z0, .(z0, z1)), .(z0, z2), member_out_ag(z0, .(z0, z2))),SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) -> U2_AGG(.(z0, .(z0, z1)), .(z0, z2), member_out_ag(z0, .(z0, z2)))) (SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z0, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z0, z3), member_out_ag(z0, .(z0, z3))),SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z0, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z0, z3), member_out_ag(z0, .(z0, z3)))) ---------------------------------------- (59) Obligation: Q DP problem: The TRS P consists of the following rules: SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2, .(z2, z3))) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z0, z2)) -> U2_AGG(.(z0, z1), .(z0, z2), U5_ag(z0, z2, member_in_ag(z2))) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), U5_ag(z2, z3, member_in_ag(z3))) U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0, .(z0, z2))) -> U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg(z0, .(z0, z1))) U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2, .(z2, z3))) -> U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg(z2, .(z0, z1))) U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(x3, .(z0, z2))) -> U3_AGG(x3, .(z0, z1), .(z0, z2), not_member_out_gg(x3, .(z0, z1))) U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(x3, .(z2, z3))) -> U3_AGG(x3, .(z0, z1), .(z2, z3), not_member_out_gg(x3, .(z0, z1))) U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg(z0, .(z0, z1))) -> SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg(z2, .(z0, z1))) -> SUBSETCHECKED_IN_AGG(.(z2, .(z0, z1)), .(z2, z3)) U3_AGG(z3, .(z0, z1), .(z0, z2), not_member_out_gg(z3, .(z0, z1))) -> SUBSETCHECKED_IN_AGG(.(z3, .(z0, z1)), .(z0, z2)) U3_AGG(z4, .(z0, z1), .(z2, z3), not_member_out_gg(z4, .(z0, z1))) -> SUBSETCHECKED_IN_AGG(.(z4, .(z0, z1)), .(z2, z3)) SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) -> U2_AGG(.(z0, .(z0, z1)), .(z0, z2), member_out_ag(z0, .(z0, z2))) SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z0, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z0, z3), member_out_ag(z0, .(z0, z3))) The TRS R consists of the following rules: member_in_ag(.(X, X3)) -> member_out_ag(X, .(X, X3)) member_in_ag(.(X4, Xs)) -> U5_ag(X4, Xs, member_in_ag(Xs)) U5_ag(X4, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X4, Xs)) The set Q consists of the following terms: member_in_ag(x0) U5_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (60) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2, .(z2, z3))) we obtained the following new rules [LPAR04]: (SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) -> U2_AGG(.(z0, .(z0, z1)), .(z0, z2), member_out_ag(z0, .(z0, z2))),SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) -> U2_AGG(.(z0, .(z0, z1)), .(z0, z2), member_out_ag(z0, .(z0, z2)))) (SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z0, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z0, z3), member_out_ag(z0, .(z0, z3))),SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z0, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z0, z3), member_out_ag(z0, .(z0, z3)))) (SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z1, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z1, z3), member_out_ag(z1, .(z1, z3))),SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z1, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z1, z3), member_out_ag(z1, .(z1, z3)))) (SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z3, z4)) -> U2_AGG(.(z0, .(z1, z2)), .(z3, z4), member_out_ag(z3, .(z3, z4))),SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z3, z4)) -> U2_AGG(.(z0, .(z1, z2)), .(z3, z4), member_out_ag(z3, .(z3, z4)))) ---------------------------------------- (61) Obligation: Q DP problem: The TRS P consists of the following rules: SUBSETCHECKED_IN_AGG(.(z0, z1), .(z0, z2)) -> U2_AGG(.(z0, z1), .(z0, z2), U5_ag(z0, z2, member_in_ag(z2))) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), U5_ag(z2, z3, member_in_ag(z3))) U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0, .(z0, z2))) -> U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg(z0, .(z0, z1))) U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2, .(z2, z3))) -> U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg(z2, .(z0, z1))) U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(x3, .(z0, z2))) -> U3_AGG(x3, .(z0, z1), .(z0, z2), not_member_out_gg(x3, .(z0, z1))) U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(x3, .(z2, z3))) -> U3_AGG(x3, .(z0, z1), .(z2, z3), not_member_out_gg(x3, .(z0, z1))) U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg(z0, .(z0, z1))) -> SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg(z2, .(z0, z1))) -> SUBSETCHECKED_IN_AGG(.(z2, .(z0, z1)), .(z2, z3)) U3_AGG(z3, .(z0, z1), .(z0, z2), not_member_out_gg(z3, .(z0, z1))) -> SUBSETCHECKED_IN_AGG(.(z3, .(z0, z1)), .(z0, z2)) U3_AGG(z4, .(z0, z1), .(z2, z3), not_member_out_gg(z4, .(z0, z1))) -> SUBSETCHECKED_IN_AGG(.(z4, .(z0, z1)), .(z2, z3)) SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) -> U2_AGG(.(z0, .(z0, z1)), .(z0, z2), member_out_ag(z0, .(z0, z2))) SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z0, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z0, z3), member_out_ag(z0, .(z0, z3))) SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z1, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z1, z3), member_out_ag(z1, .(z1, z3))) SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z3, z4)) -> U2_AGG(.(z0, .(z1, z2)), .(z3, z4), member_out_ag(z3, .(z3, z4))) The TRS R consists of the following rules: member_in_ag(.(X, X3)) -> member_out_ag(X, .(X, X3)) member_in_ag(.(X4, Xs)) -> U5_ag(X4, Xs, member_in_ag(Xs)) U5_ag(X4, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X4, Xs)) The set Q consists of the following terms: member_in_ag(x0) U5_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (62) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule SUBSETCHECKED_IN_AGG(.(z0, z1), .(z0, z2)) -> U2_AGG(.(z0, z1), .(z0, z2), U5_ag(z0, z2, member_in_ag(z2))) we obtained the following new rules [LPAR04]: (SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) -> U2_AGG(.(z0, .(z0, z1)), .(z0, z2), U5_ag(z0, z2, member_in_ag(z2))),SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) -> U2_AGG(.(z0, .(z0, z1)), .(z0, z2), U5_ag(z0, z2, member_in_ag(z2)))) (SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z0, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z0, z3), U5_ag(z0, z3, member_in_ag(z3))),SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z0, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z0, z3), U5_ag(z0, z3, member_in_ag(z3)))) ---------------------------------------- (63) Obligation: Q DP problem: The TRS P consists of the following rules: SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), U5_ag(z2, z3, member_in_ag(z3))) U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0, .(z0, z2))) -> U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg(z0, .(z0, z1))) U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2, .(z2, z3))) -> U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg(z2, .(z0, z1))) U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(x3, .(z0, z2))) -> U3_AGG(x3, .(z0, z1), .(z0, z2), not_member_out_gg(x3, .(z0, z1))) U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(x3, .(z2, z3))) -> U3_AGG(x3, .(z0, z1), .(z2, z3), not_member_out_gg(x3, .(z0, z1))) U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg(z0, .(z0, z1))) -> SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg(z2, .(z0, z1))) -> SUBSETCHECKED_IN_AGG(.(z2, .(z0, z1)), .(z2, z3)) U3_AGG(z3, .(z0, z1), .(z0, z2), not_member_out_gg(z3, .(z0, z1))) -> SUBSETCHECKED_IN_AGG(.(z3, .(z0, z1)), .(z0, z2)) U3_AGG(z4, .(z0, z1), .(z2, z3), not_member_out_gg(z4, .(z0, z1))) -> SUBSETCHECKED_IN_AGG(.(z4, .(z0, z1)), .(z2, z3)) SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) -> U2_AGG(.(z0, .(z0, z1)), .(z0, z2), member_out_ag(z0, .(z0, z2))) SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z0, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z0, z3), member_out_ag(z0, .(z0, z3))) SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z1, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z1, z3), member_out_ag(z1, .(z1, z3))) SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z3, z4)) -> U2_AGG(.(z0, .(z1, z2)), .(z3, z4), member_out_ag(z3, .(z3, z4))) SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) -> U2_AGG(.(z0, .(z0, z1)), .(z0, z2), U5_ag(z0, z2, member_in_ag(z2))) SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z0, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z0, z3), U5_ag(z0, z3, member_in_ag(z3))) The TRS R consists of the following rules: member_in_ag(.(X, X3)) -> member_out_ag(X, .(X, X3)) member_in_ag(.(X4, Xs)) -> U5_ag(X4, Xs, member_in_ag(Xs)) U5_ag(X4, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X4, Xs)) The set Q consists of the following terms: member_in_ag(x0) U5_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (64) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), U5_ag(z2, z3, member_in_ag(z3))) we obtained the following new rules [LPAR04]: (SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) -> U2_AGG(.(z0, .(z0, z1)), .(z0, z2), U5_ag(z0, z2, member_in_ag(z2))),SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) -> U2_AGG(.(z0, .(z0, z1)), .(z0, z2), U5_ag(z0, z2, member_in_ag(z2)))) (SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z0, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z0, z3), U5_ag(z0, z3, member_in_ag(z3))),SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z0, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z0, z3), U5_ag(z0, z3, member_in_ag(z3)))) (SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z1, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z1, z3), U5_ag(z1, z3, member_in_ag(z3))),SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z1, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z1, z3), U5_ag(z1, z3, member_in_ag(z3)))) (SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z3, z4)) -> U2_AGG(.(z0, .(z1, z2)), .(z3, z4), U5_ag(z3, z4, member_in_ag(z4))),SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z3, z4)) -> U2_AGG(.(z0, .(z1, z2)), .(z3, z4), U5_ag(z3, z4, member_in_ag(z4)))) ---------------------------------------- (65) Obligation: Q DP problem: The TRS P consists of the following rules: U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0, .(z0, z2))) -> U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg(z0, .(z0, z1))) U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2, .(z2, z3))) -> U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg(z2, .(z0, z1))) U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(x3, .(z0, z2))) -> U3_AGG(x3, .(z0, z1), .(z0, z2), not_member_out_gg(x3, .(z0, z1))) U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(x3, .(z2, z3))) -> U3_AGG(x3, .(z0, z1), .(z2, z3), not_member_out_gg(x3, .(z0, z1))) U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg(z0, .(z0, z1))) -> SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg(z2, .(z0, z1))) -> SUBSETCHECKED_IN_AGG(.(z2, .(z0, z1)), .(z2, z3)) U3_AGG(z3, .(z0, z1), .(z0, z2), not_member_out_gg(z3, .(z0, z1))) -> SUBSETCHECKED_IN_AGG(.(z3, .(z0, z1)), .(z0, z2)) U3_AGG(z4, .(z0, z1), .(z2, z3), not_member_out_gg(z4, .(z0, z1))) -> SUBSETCHECKED_IN_AGG(.(z4, .(z0, z1)), .(z2, z3)) SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) -> U2_AGG(.(z0, .(z0, z1)), .(z0, z2), member_out_ag(z0, .(z0, z2))) SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z0, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z0, z3), member_out_ag(z0, .(z0, z3))) SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z1, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z1, z3), member_out_ag(z1, .(z1, z3))) SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z3, z4)) -> U2_AGG(.(z0, .(z1, z2)), .(z3, z4), member_out_ag(z3, .(z3, z4))) SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) -> U2_AGG(.(z0, .(z0, z1)), .(z0, z2), U5_ag(z0, z2, member_in_ag(z2))) SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z0, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z0, z3), U5_ag(z0, z3, member_in_ag(z3))) SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z1, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z1, z3), U5_ag(z1, z3, member_in_ag(z3))) SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z3, z4)) -> U2_AGG(.(z0, .(z1, z2)), .(z3, z4), U5_ag(z3, z4, member_in_ag(z4))) The TRS R consists of the following rules: member_in_ag(.(X, X3)) -> member_out_ag(X, .(X, X3)) member_in_ag(.(X4, Xs)) -> U5_ag(X4, Xs, member_in_ag(Xs)) U5_ag(X4, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X4, Xs)) The set Q consists of the following terms: member_in_ag(x0) U5_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (66) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by narrowing to the left: s = U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg(z0, .(z0, z1))) evaluates to t =U3_AGG(z0, .(z0, .(z0, z1)), .(z0, z2), not_member_out_gg(z0, .(z0, .(z0, z1)))) Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [z1 / .(z0, z1)] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg(z0, .(z0, z1))) -> SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) with rule U3_AGG(z0', .(z0', z1'), .(z0', z2'), not_member_out_gg(z0', .(z0', z1'))) -> SUBSETCHECKED_IN_AGG(.(z0', .(z0', z1')), .(z0', z2')) at position [] and matcher [z0' / z0, z1' / z1, z2' / z2] SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) -> U2_AGG(.(z0, .(z0, z1)), .(z0, z2), member_out_ag(z0, .(z0, z2))) with rule SUBSETCHECKED_IN_AGG(.(z0', .(z0', z1')), .(z0', z2')) -> U2_AGG(.(z0', .(z0', z1')), .(z0', z2'), member_out_ag(z0', .(z0', z2'))) at position [] and matcher [z0' / z0, z1' / z1, z2' / z2] U2_AGG(.(z0, .(z0, z1)), .(z0, z2), member_out_ag(z0, .(z0, z2))) -> U3_AGG(z0, .(z0, .(z0, z1)), .(z0, z2), not_member_out_gg(z0, .(z0, .(z0, z1)))) with rule U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0, .(z0, z2))) -> U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg(z0, .(z0, z1))) Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence All these steps are and every following step will be a correct step w.r.t to Q. ---------------------------------------- (67) NO ---------------------------------------- (68) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: subset_in_2: (f,b) subsetchecked_in_3: (f,b,b) member_in_2: (f,b) (b,b) not_member_in_2: (b,b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: subset_in_ag(X, Y) -> U1_ag(X, Y, subsetchecked_in_agg(X, [], Y)) subsetchecked_in_agg([], X1, X2) -> subsetchecked_out_agg([], X1, X2) subsetchecked_in_agg(.(X, Xs), Ys, Zs) -> U2_agg(X, Xs, Ys, Zs, member_in_ag(X, Zs)) member_in_ag(X, .(X, X3)) -> member_out_ag(X, .(X, X3)) member_in_ag(X, .(X4, Xs)) -> U5_ag(X, X4, Xs, member_in_ag(X, Xs)) U5_ag(X, X4, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X4, Xs)) U2_agg(X, Xs, Ys, Zs, member_out_ag(X, Zs)) -> U3_agg(X, Xs, Ys, Zs, not_member_in_gg(X, Ys)) not_member_in_gg(X, Y) -> U6_gg(X, Y, member_in_gg(X, Y)) member_in_gg(X, .(X, X3)) -> member_out_gg(X, .(X, X3)) member_in_gg(X, .(X4, Xs)) -> U5_gg(X, X4, Xs, member_in_gg(X, Xs)) U5_gg(X, X4, Xs, member_out_gg(X, Xs)) -> member_out_gg(X, .(X4, Xs)) U6_gg(X, Y, member_out_gg(X, Y)) -> U7_gg(X, Y, failure_in_g(a)) failure_in_g(b) -> failure_out_g(b) U7_gg(X, Y, failure_out_g(a)) -> not_member_out_gg(X, Y) not_member_in_gg(X5, X6) -> not_member_out_gg(X5, X6) U3_agg(X, Xs, Ys, Zs, not_member_out_gg(X, Ys)) -> U4_agg(X, Xs, Ys, Zs, subsetchecked_in_agg(Xs, .(X, Ys), Zs)) U4_agg(X, Xs, Ys, Zs, subsetchecked_out_agg(Xs, .(X, Ys), Zs)) -> subsetchecked_out_agg(.(X, Xs), Ys, Zs) U1_ag(X, Y, subsetchecked_out_agg(X, [], Y)) -> subset_out_ag(X, Y) The argument filtering Pi contains the following mapping: subset_in_ag(x1, x2) = subset_in_ag(x2) U1_ag(x1, x2, x3) = U1_ag(x3) subsetchecked_in_agg(x1, x2, x3) = subsetchecked_in_agg(x2, x3) subsetchecked_out_agg(x1, x2, x3) = subsetchecked_out_agg(x1) U2_agg(x1, x2, x3, x4, x5) = U2_agg(x3, x4, x5) member_in_ag(x1, x2) = member_in_ag(x2) .(x1, x2) = .(x1, x2) member_out_ag(x1, x2) = member_out_ag(x1) U5_ag(x1, x2, x3, x4) = U5_ag(x4) U3_agg(x1, x2, x3, x4, x5) = U3_agg(x1, x3, x4, x5) not_member_in_gg(x1, x2) = not_member_in_gg(x1, x2) U6_gg(x1, x2, x3) = U6_gg(x3) member_in_gg(x1, x2) = member_in_gg(x1, x2) member_out_gg(x1, x2) = member_out_gg U5_gg(x1, x2, x3, x4) = U5_gg(x4) U7_gg(x1, x2, x3) = U7_gg(x3) failure_in_g(x1) = failure_in_g(x1) b = b failure_out_g(x1) = failure_out_g a = a not_member_out_gg(x1, x2) = not_member_out_gg U4_agg(x1, x2, x3, x4, x5) = U4_agg(x1, x5) [] = [] subset_out_ag(x1, x2) = subset_out_ag(x1) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (69) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: subset_in_ag(X, Y) -> U1_ag(X, Y, subsetchecked_in_agg(X, [], Y)) subsetchecked_in_agg([], X1, X2) -> subsetchecked_out_agg([], X1, X2) subsetchecked_in_agg(.(X, Xs), Ys, Zs) -> U2_agg(X, Xs, Ys, Zs, member_in_ag(X, Zs)) member_in_ag(X, .(X, X3)) -> member_out_ag(X, .(X, X3)) member_in_ag(X, .(X4, Xs)) -> U5_ag(X, X4, Xs, member_in_ag(X, Xs)) U5_ag(X, X4, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X4, Xs)) U2_agg(X, Xs, Ys, Zs, member_out_ag(X, Zs)) -> U3_agg(X, Xs, Ys, Zs, not_member_in_gg(X, Ys)) not_member_in_gg(X, Y) -> U6_gg(X, Y, member_in_gg(X, Y)) member_in_gg(X, .(X, X3)) -> member_out_gg(X, .(X, X3)) member_in_gg(X, .(X4, Xs)) -> U5_gg(X, X4, Xs, member_in_gg(X, Xs)) U5_gg(X, X4, Xs, member_out_gg(X, Xs)) -> member_out_gg(X, .(X4, Xs)) U6_gg(X, Y, member_out_gg(X, Y)) -> U7_gg(X, Y, failure_in_g(a)) failure_in_g(b) -> failure_out_g(b) U7_gg(X, Y, failure_out_g(a)) -> not_member_out_gg(X, Y) not_member_in_gg(X5, X6) -> not_member_out_gg(X5, X6) U3_agg(X, Xs, Ys, Zs, not_member_out_gg(X, Ys)) -> U4_agg(X, Xs, Ys, Zs, subsetchecked_in_agg(Xs, .(X, Ys), Zs)) U4_agg(X, Xs, Ys, Zs, subsetchecked_out_agg(Xs, .(X, Ys), Zs)) -> subsetchecked_out_agg(.(X, Xs), Ys, Zs) U1_ag(X, Y, subsetchecked_out_agg(X, [], Y)) -> subset_out_ag(X, Y) The argument filtering Pi contains the following mapping: subset_in_ag(x1, x2) = subset_in_ag(x2) U1_ag(x1, x2, x3) = U1_ag(x3) subsetchecked_in_agg(x1, x2, x3) = subsetchecked_in_agg(x2, x3) subsetchecked_out_agg(x1, x2, x3) = subsetchecked_out_agg(x1) U2_agg(x1, x2, x3, x4, x5) = U2_agg(x3, x4, x5) member_in_ag(x1, x2) = member_in_ag(x2) .(x1, x2) = .(x1, x2) member_out_ag(x1, x2) = member_out_ag(x1) U5_ag(x1, x2, x3, x4) = U5_ag(x4) U3_agg(x1, x2, x3, x4, x5) = U3_agg(x1, x3, x4, x5) not_member_in_gg(x1, x2) = not_member_in_gg(x1, x2) U6_gg(x1, x2, x3) = U6_gg(x3) member_in_gg(x1, x2) = member_in_gg(x1, x2) member_out_gg(x1, x2) = member_out_gg U5_gg(x1, x2, x3, x4) = U5_gg(x4) U7_gg(x1, x2, x3) = U7_gg(x3) failure_in_g(x1) = failure_in_g(x1) b = b failure_out_g(x1) = failure_out_g a = a not_member_out_gg(x1, x2) = not_member_out_gg U4_agg(x1, x2, x3, x4, x5) = U4_agg(x1, x5) [] = [] subset_out_ag(x1, x2) = subset_out_ag(x1) ---------------------------------------- (70) 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: SUBSET_IN_AG(X, Y) -> U1_AG(X, Y, subsetchecked_in_agg(X, [], Y)) SUBSET_IN_AG(X, Y) -> SUBSETCHECKED_IN_AGG(X, [], Y) SUBSETCHECKED_IN_AGG(.(X, Xs), Ys, Zs) -> U2_AGG(X, Xs, Ys, Zs, member_in_ag(X, Zs)) SUBSETCHECKED_IN_AGG(.(X, Xs), Ys, Zs) -> MEMBER_IN_AG(X, Zs) MEMBER_IN_AG(X, .(X4, Xs)) -> U5_AG(X, X4, Xs, member_in_ag(X, Xs)) MEMBER_IN_AG(X, .(X4, Xs)) -> MEMBER_IN_AG(X, Xs) U2_AGG(X, Xs, Ys, Zs, member_out_ag(X, Zs)) -> U3_AGG(X, Xs, Ys, Zs, not_member_in_gg(X, Ys)) U2_AGG(X, Xs, Ys, Zs, member_out_ag(X, Zs)) -> NOT_MEMBER_IN_GG(X, Ys) NOT_MEMBER_IN_GG(X, Y) -> U6_GG(X, Y, member_in_gg(X, Y)) NOT_MEMBER_IN_GG(X, Y) -> MEMBER_IN_GG(X, Y) MEMBER_IN_GG(X, .(X4, Xs)) -> U5_GG(X, X4, Xs, member_in_gg(X, Xs)) MEMBER_IN_GG(X, .(X4, Xs)) -> MEMBER_IN_GG(X, Xs) U6_GG(X, Y, member_out_gg(X, Y)) -> U7_GG(X, Y, failure_in_g(a)) U6_GG(X, Y, member_out_gg(X, Y)) -> FAILURE_IN_G(a) U3_AGG(X, Xs, Ys, Zs, not_member_out_gg(X, Ys)) -> U4_AGG(X, Xs, Ys, Zs, subsetchecked_in_agg(Xs, .(X, Ys), Zs)) U3_AGG(X, Xs, Ys, Zs, not_member_out_gg(X, Ys)) -> SUBSETCHECKED_IN_AGG(Xs, .(X, Ys), Zs) The TRS R consists of the following rules: subset_in_ag(X, Y) -> U1_ag(X, Y, subsetchecked_in_agg(X, [], Y)) subsetchecked_in_agg([], X1, X2) -> subsetchecked_out_agg([], X1, X2) subsetchecked_in_agg(.(X, Xs), Ys, Zs) -> U2_agg(X, Xs, Ys, Zs, member_in_ag(X, Zs)) member_in_ag(X, .(X, X3)) -> member_out_ag(X, .(X, X3)) member_in_ag(X, .(X4, Xs)) -> U5_ag(X, X4, Xs, member_in_ag(X, Xs)) U5_ag(X, X4, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X4, Xs)) U2_agg(X, Xs, Ys, Zs, member_out_ag(X, Zs)) -> U3_agg(X, Xs, Ys, Zs, not_member_in_gg(X, Ys)) not_member_in_gg(X, Y) -> U6_gg(X, Y, member_in_gg(X, Y)) member_in_gg(X, .(X, X3)) -> member_out_gg(X, .(X, X3)) member_in_gg(X, .(X4, Xs)) -> U5_gg(X, X4, Xs, member_in_gg(X, Xs)) U5_gg(X, X4, Xs, member_out_gg(X, Xs)) -> member_out_gg(X, .(X4, Xs)) U6_gg(X, Y, member_out_gg(X, Y)) -> U7_gg(X, Y, failure_in_g(a)) failure_in_g(b) -> failure_out_g(b) U7_gg(X, Y, failure_out_g(a)) -> not_member_out_gg(X, Y) not_member_in_gg(X5, X6) -> not_member_out_gg(X5, X6) U3_agg(X, Xs, Ys, Zs, not_member_out_gg(X, Ys)) -> U4_agg(X, Xs, Ys, Zs, subsetchecked_in_agg(Xs, .(X, Ys), Zs)) U4_agg(X, Xs, Ys, Zs, subsetchecked_out_agg(Xs, .(X, Ys), Zs)) -> subsetchecked_out_agg(.(X, Xs), Ys, Zs) U1_ag(X, Y, subsetchecked_out_agg(X, [], Y)) -> subset_out_ag(X, Y) The argument filtering Pi contains the following mapping: subset_in_ag(x1, x2) = subset_in_ag(x2) U1_ag(x1, x2, x3) = U1_ag(x3) subsetchecked_in_agg(x1, x2, x3) = subsetchecked_in_agg(x2, x3) subsetchecked_out_agg(x1, x2, x3) = subsetchecked_out_agg(x1) U2_agg(x1, x2, x3, x4, x5) = U2_agg(x3, x4, x5) member_in_ag(x1, x2) = member_in_ag(x2) .(x1, x2) = .(x1, x2) member_out_ag(x1, x2) = member_out_ag(x1) U5_ag(x1, x2, x3, x4) = U5_ag(x4) U3_agg(x1, x2, x3, x4, x5) = U3_agg(x1, x3, x4, x5) not_member_in_gg(x1, x2) = not_member_in_gg(x1, x2) U6_gg(x1, x2, x3) = U6_gg(x3) member_in_gg(x1, x2) = member_in_gg(x1, x2) member_out_gg(x1, x2) = member_out_gg U5_gg(x1, x2, x3, x4) = U5_gg(x4) U7_gg(x1, x2, x3) = U7_gg(x3) failure_in_g(x1) = failure_in_g(x1) b = b failure_out_g(x1) = failure_out_g a = a not_member_out_gg(x1, x2) = not_member_out_gg U4_agg(x1, x2, x3, x4, x5) = U4_agg(x1, x5) [] = [] subset_out_ag(x1, x2) = subset_out_ag(x1) SUBSET_IN_AG(x1, x2) = SUBSET_IN_AG(x2) U1_AG(x1, x2, x3) = U1_AG(x3) SUBSETCHECKED_IN_AGG(x1, x2, x3) = SUBSETCHECKED_IN_AGG(x2, x3) U2_AGG(x1, x2, x3, x4, x5) = U2_AGG(x3, x4, x5) MEMBER_IN_AG(x1, x2) = MEMBER_IN_AG(x2) U5_AG(x1, x2, x3, x4) = U5_AG(x4) U3_AGG(x1, x2, x3, x4, x5) = U3_AGG(x1, x3, x4, x5) NOT_MEMBER_IN_GG(x1, x2) = NOT_MEMBER_IN_GG(x1, x2) U6_GG(x1, x2, x3) = U6_GG(x3) MEMBER_IN_GG(x1, x2) = MEMBER_IN_GG(x1, x2) U5_GG(x1, x2, x3, x4) = U5_GG(x4) U7_GG(x1, x2, x3) = U7_GG(x3) FAILURE_IN_G(x1) = FAILURE_IN_G(x1) U4_AGG(x1, x2, x3, x4, x5) = U4_AGG(x1, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (71) Obligation: Pi DP problem: The TRS P consists of the following rules: SUBSET_IN_AG(X, Y) -> U1_AG(X, Y, subsetchecked_in_agg(X, [], Y)) SUBSET_IN_AG(X, Y) -> SUBSETCHECKED_IN_AGG(X, [], Y) SUBSETCHECKED_IN_AGG(.(X, Xs), Ys, Zs) -> U2_AGG(X, Xs, Ys, Zs, member_in_ag(X, Zs)) SUBSETCHECKED_IN_AGG(.(X, Xs), Ys, Zs) -> MEMBER_IN_AG(X, Zs) MEMBER_IN_AG(X, .(X4, Xs)) -> U5_AG(X, X4, Xs, member_in_ag(X, Xs)) MEMBER_IN_AG(X, .(X4, Xs)) -> MEMBER_IN_AG(X, Xs) U2_AGG(X, Xs, Ys, Zs, member_out_ag(X, Zs)) -> U3_AGG(X, Xs, Ys, Zs, not_member_in_gg(X, Ys)) U2_AGG(X, Xs, Ys, Zs, member_out_ag(X, Zs)) -> NOT_MEMBER_IN_GG(X, Ys) NOT_MEMBER_IN_GG(X, Y) -> U6_GG(X, Y, member_in_gg(X, Y)) NOT_MEMBER_IN_GG(X, Y) -> MEMBER_IN_GG(X, Y) MEMBER_IN_GG(X, .(X4, Xs)) -> U5_GG(X, X4, Xs, member_in_gg(X, Xs)) MEMBER_IN_GG(X, .(X4, Xs)) -> MEMBER_IN_GG(X, Xs) U6_GG(X, Y, member_out_gg(X, Y)) -> U7_GG(X, Y, failure_in_g(a)) U6_GG(X, Y, member_out_gg(X, Y)) -> FAILURE_IN_G(a) U3_AGG(X, Xs, Ys, Zs, not_member_out_gg(X, Ys)) -> U4_AGG(X, Xs, Ys, Zs, subsetchecked_in_agg(Xs, .(X, Ys), Zs)) U3_AGG(X, Xs, Ys, Zs, not_member_out_gg(X, Ys)) -> SUBSETCHECKED_IN_AGG(Xs, .(X, Ys), Zs) The TRS R consists of the following rules: subset_in_ag(X, Y) -> U1_ag(X, Y, subsetchecked_in_agg(X, [], Y)) subsetchecked_in_agg([], X1, X2) -> subsetchecked_out_agg([], X1, X2) subsetchecked_in_agg(.(X, Xs), Ys, Zs) -> U2_agg(X, Xs, Ys, Zs, member_in_ag(X, Zs)) member_in_ag(X, .(X, X3)) -> member_out_ag(X, .(X, X3)) member_in_ag(X, .(X4, Xs)) -> U5_ag(X, X4, Xs, member_in_ag(X, Xs)) U5_ag(X, X4, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X4, Xs)) U2_agg(X, Xs, Ys, Zs, member_out_ag(X, Zs)) -> U3_agg(X, Xs, Ys, Zs, not_member_in_gg(X, Ys)) not_member_in_gg(X, Y) -> U6_gg(X, Y, member_in_gg(X, Y)) member_in_gg(X, .(X, X3)) -> member_out_gg(X, .(X, X3)) member_in_gg(X, .(X4, Xs)) -> U5_gg(X, X4, Xs, member_in_gg(X, Xs)) U5_gg(X, X4, Xs, member_out_gg(X, Xs)) -> member_out_gg(X, .(X4, Xs)) U6_gg(X, Y, member_out_gg(X, Y)) -> U7_gg(X, Y, failure_in_g(a)) failure_in_g(b) -> failure_out_g(b) U7_gg(X, Y, failure_out_g(a)) -> not_member_out_gg(X, Y) not_member_in_gg(X5, X6) -> not_member_out_gg(X5, X6) U3_agg(X, Xs, Ys, Zs, not_member_out_gg(X, Ys)) -> U4_agg(X, Xs, Ys, Zs, subsetchecked_in_agg(Xs, .(X, Ys), Zs)) U4_agg(X, Xs, Ys, Zs, subsetchecked_out_agg(Xs, .(X, Ys), Zs)) -> subsetchecked_out_agg(.(X, Xs), Ys, Zs) U1_ag(X, Y, subsetchecked_out_agg(X, [], Y)) -> subset_out_ag(X, Y) The argument filtering Pi contains the following mapping: subset_in_ag(x1, x2) = subset_in_ag(x2) U1_ag(x1, x2, x3) = U1_ag(x3) subsetchecked_in_agg(x1, x2, x3) = subsetchecked_in_agg(x2, x3) subsetchecked_out_agg(x1, x2, x3) = subsetchecked_out_agg(x1) U2_agg(x1, x2, x3, x4, x5) = U2_agg(x3, x4, x5) member_in_ag(x1, x2) = member_in_ag(x2) .(x1, x2) = .(x1, x2) member_out_ag(x1, x2) = member_out_ag(x1) U5_ag(x1, x2, x3, x4) = U5_ag(x4) U3_agg(x1, x2, x3, x4, x5) = U3_agg(x1, x3, x4, x5) not_member_in_gg(x1, x2) = not_member_in_gg(x1, x2) U6_gg(x1, x2, x3) = U6_gg(x3) member_in_gg(x1, x2) = member_in_gg(x1, x2) member_out_gg(x1, x2) = member_out_gg U5_gg(x1, x2, x3, x4) = U5_gg(x4) U7_gg(x1, x2, x3) = U7_gg(x3) failure_in_g(x1) = failure_in_g(x1) b = b failure_out_g(x1) = failure_out_g a = a not_member_out_gg(x1, x2) = not_member_out_gg U4_agg(x1, x2, x3, x4, x5) = U4_agg(x1, x5) [] = [] subset_out_ag(x1, x2) = subset_out_ag(x1) SUBSET_IN_AG(x1, x2) = SUBSET_IN_AG(x2) U1_AG(x1, x2, x3) = U1_AG(x3) SUBSETCHECKED_IN_AGG(x1, x2, x3) = SUBSETCHECKED_IN_AGG(x2, x3) U2_AGG(x1, x2, x3, x4, x5) = U2_AGG(x3, x4, x5) MEMBER_IN_AG(x1, x2) = MEMBER_IN_AG(x2) U5_AG(x1, x2, x3, x4) = U5_AG(x4) U3_AGG(x1, x2, x3, x4, x5) = U3_AGG(x1, x3, x4, x5) NOT_MEMBER_IN_GG(x1, x2) = NOT_MEMBER_IN_GG(x1, x2) U6_GG(x1, x2, x3) = U6_GG(x3) MEMBER_IN_GG(x1, x2) = MEMBER_IN_GG(x1, x2) U5_GG(x1, x2, x3, x4) = U5_GG(x4) U7_GG(x1, x2, x3) = U7_GG(x3) FAILURE_IN_G(x1) = FAILURE_IN_G(x1) U4_AGG(x1, x2, x3, x4, x5) = U4_AGG(x1, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (72) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 11 less nodes. ---------------------------------------- (73) Complex Obligation (AND) ---------------------------------------- (74) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER_IN_GG(X, .(X4, Xs)) -> MEMBER_IN_GG(X, Xs) The TRS R consists of the following rules: subset_in_ag(X, Y) -> U1_ag(X, Y, subsetchecked_in_agg(X, [], Y)) subsetchecked_in_agg([], X1, X2) -> subsetchecked_out_agg([], X1, X2) subsetchecked_in_agg(.(X, Xs), Ys, Zs) -> U2_agg(X, Xs, Ys, Zs, member_in_ag(X, Zs)) member_in_ag(X, .(X, X3)) -> member_out_ag(X, .(X, X3)) member_in_ag(X, .(X4, Xs)) -> U5_ag(X, X4, Xs, member_in_ag(X, Xs)) U5_ag(X, X4, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X4, Xs)) U2_agg(X, Xs, Ys, Zs, member_out_ag(X, Zs)) -> U3_agg(X, Xs, Ys, Zs, not_member_in_gg(X, Ys)) not_member_in_gg(X, Y) -> U6_gg(X, Y, member_in_gg(X, Y)) member_in_gg(X, .(X, X3)) -> member_out_gg(X, .(X, X3)) member_in_gg(X, .(X4, Xs)) -> U5_gg(X, X4, Xs, member_in_gg(X, Xs)) U5_gg(X, X4, Xs, member_out_gg(X, Xs)) -> member_out_gg(X, .(X4, Xs)) U6_gg(X, Y, member_out_gg(X, Y)) -> U7_gg(X, Y, failure_in_g(a)) failure_in_g(b) -> failure_out_g(b) U7_gg(X, Y, failure_out_g(a)) -> not_member_out_gg(X, Y) not_member_in_gg(X5, X6) -> not_member_out_gg(X5, X6) U3_agg(X, Xs, Ys, Zs, not_member_out_gg(X, Ys)) -> U4_agg(X, Xs, Ys, Zs, subsetchecked_in_agg(Xs, .(X, Ys), Zs)) U4_agg(X, Xs, Ys, Zs, subsetchecked_out_agg(Xs, .(X, Ys), Zs)) -> subsetchecked_out_agg(.(X, Xs), Ys, Zs) U1_ag(X, Y, subsetchecked_out_agg(X, [], Y)) -> subset_out_ag(X, Y) The argument filtering Pi contains the following mapping: subset_in_ag(x1, x2) = subset_in_ag(x2) U1_ag(x1, x2, x3) = U1_ag(x3) subsetchecked_in_agg(x1, x2, x3) = subsetchecked_in_agg(x2, x3) subsetchecked_out_agg(x1, x2, x3) = subsetchecked_out_agg(x1) U2_agg(x1, x2, x3, x4, x5) = U2_agg(x3, x4, x5) member_in_ag(x1, x2) = member_in_ag(x2) .(x1, x2) = .(x1, x2) member_out_ag(x1, x2) = member_out_ag(x1) U5_ag(x1, x2, x3, x4) = U5_ag(x4) U3_agg(x1, x2, x3, x4, x5) = U3_agg(x1, x3, x4, x5) not_member_in_gg(x1, x2) = not_member_in_gg(x1, x2) U6_gg(x1, x2, x3) = U6_gg(x3) member_in_gg(x1, x2) = member_in_gg(x1, x2) member_out_gg(x1, x2) = member_out_gg U5_gg(x1, x2, x3, x4) = U5_gg(x4) U7_gg(x1, x2, x3) = U7_gg(x3) failure_in_g(x1) = failure_in_g(x1) b = b failure_out_g(x1) = failure_out_g a = a not_member_out_gg(x1, x2) = not_member_out_gg U4_agg(x1, x2, x3, x4, x5) = U4_agg(x1, x5) [] = [] subset_out_ag(x1, x2) = subset_out_ag(x1) MEMBER_IN_GG(x1, x2) = MEMBER_IN_GG(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (75) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (76) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER_IN_GG(X, .(X4, Xs)) -> MEMBER_IN_GG(X, Xs) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (77) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (78) Obligation: Q DP problem: The TRS P consists of the following rules: MEMBER_IN_GG(X, .(X4, Xs)) -> MEMBER_IN_GG(X, Xs) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (79) 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, .(X4, Xs)) -> MEMBER_IN_GG(X, Xs) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (80) YES ---------------------------------------- (81) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER_IN_AG(X, .(X4, Xs)) -> MEMBER_IN_AG(X, Xs) The TRS R consists of the following rules: subset_in_ag(X, Y) -> U1_ag(X, Y, subsetchecked_in_agg(X, [], Y)) subsetchecked_in_agg([], X1, X2) -> subsetchecked_out_agg([], X1, X2) subsetchecked_in_agg(.(X, Xs), Ys, Zs) -> U2_agg(X, Xs, Ys, Zs, member_in_ag(X, Zs)) member_in_ag(X, .(X, X3)) -> member_out_ag(X, .(X, X3)) member_in_ag(X, .(X4, Xs)) -> U5_ag(X, X4, Xs, member_in_ag(X, Xs)) U5_ag(X, X4, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X4, Xs)) U2_agg(X, Xs, Ys, Zs, member_out_ag(X, Zs)) -> U3_agg(X, Xs, Ys, Zs, not_member_in_gg(X, Ys)) not_member_in_gg(X, Y) -> U6_gg(X, Y, member_in_gg(X, Y)) member_in_gg(X, .(X, X3)) -> member_out_gg(X, .(X, X3)) member_in_gg(X, .(X4, Xs)) -> U5_gg(X, X4, Xs, member_in_gg(X, Xs)) U5_gg(X, X4, Xs, member_out_gg(X, Xs)) -> member_out_gg(X, .(X4, Xs)) U6_gg(X, Y, member_out_gg(X, Y)) -> U7_gg(X, Y, failure_in_g(a)) failure_in_g(b) -> failure_out_g(b) U7_gg(X, Y, failure_out_g(a)) -> not_member_out_gg(X, Y) not_member_in_gg(X5, X6) -> not_member_out_gg(X5, X6) U3_agg(X, Xs, Ys, Zs, not_member_out_gg(X, Ys)) -> U4_agg(X, Xs, Ys, Zs, subsetchecked_in_agg(Xs, .(X, Ys), Zs)) U4_agg(X, Xs, Ys, Zs, subsetchecked_out_agg(Xs, .(X, Ys), Zs)) -> subsetchecked_out_agg(.(X, Xs), Ys, Zs) U1_ag(X, Y, subsetchecked_out_agg(X, [], Y)) -> subset_out_ag(X, Y) The argument filtering Pi contains the following mapping: subset_in_ag(x1, x2) = subset_in_ag(x2) U1_ag(x1, x2, x3) = U1_ag(x3) subsetchecked_in_agg(x1, x2, x3) = subsetchecked_in_agg(x2, x3) subsetchecked_out_agg(x1, x2, x3) = subsetchecked_out_agg(x1) U2_agg(x1, x2, x3, x4, x5) = U2_agg(x3, x4, x5) member_in_ag(x1, x2) = member_in_ag(x2) .(x1, x2) = .(x1, x2) member_out_ag(x1, x2) = member_out_ag(x1) U5_ag(x1, x2, x3, x4) = U5_ag(x4) U3_agg(x1, x2, x3, x4, x5) = U3_agg(x1, x3, x4, x5) not_member_in_gg(x1, x2) = not_member_in_gg(x1, x2) U6_gg(x1, x2, x3) = U6_gg(x3) member_in_gg(x1, x2) = member_in_gg(x1, x2) member_out_gg(x1, x2) = member_out_gg U5_gg(x1, x2, x3, x4) = U5_gg(x4) U7_gg(x1, x2, x3) = U7_gg(x3) failure_in_g(x1) = failure_in_g(x1) b = b failure_out_g(x1) = failure_out_g a = a not_member_out_gg(x1, x2) = not_member_out_gg U4_agg(x1, x2, x3, x4, x5) = U4_agg(x1, x5) [] = [] subset_out_ag(x1, x2) = subset_out_ag(x1) MEMBER_IN_AG(x1, x2) = MEMBER_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (82) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (83) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER_IN_AG(X, .(X4, Xs)) -> MEMBER_IN_AG(X, Xs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) MEMBER_IN_AG(x1, x2) = MEMBER_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (84) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (85) Obligation: Q DP problem: The TRS P consists of the following rules: MEMBER_IN_AG(.(X4, Xs)) -> MEMBER_IN_AG(Xs) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (86) 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_AG(.(X4, Xs)) -> MEMBER_IN_AG(Xs) The graph contains the following edges 1 > 1 ---------------------------------------- (87) YES ---------------------------------------- (88) Obligation: Pi DP problem: The TRS P consists of the following rules: U2_AGG(X, Xs, Ys, Zs, member_out_ag(X, Zs)) -> U3_AGG(X, Xs, Ys, Zs, not_member_in_gg(X, Ys)) U3_AGG(X, Xs, Ys, Zs, not_member_out_gg(X, Ys)) -> SUBSETCHECKED_IN_AGG(Xs, .(X, Ys), Zs) SUBSETCHECKED_IN_AGG(.(X, Xs), Ys, Zs) -> U2_AGG(X, Xs, Ys, Zs, member_in_ag(X, Zs)) The TRS R consists of the following rules: subset_in_ag(X, Y) -> U1_ag(X, Y, subsetchecked_in_agg(X, [], Y)) subsetchecked_in_agg([], X1, X2) -> subsetchecked_out_agg([], X1, X2) subsetchecked_in_agg(.(X, Xs), Ys, Zs) -> U2_agg(X, Xs, Ys, Zs, member_in_ag(X, Zs)) member_in_ag(X, .(X, X3)) -> member_out_ag(X, .(X, X3)) member_in_ag(X, .(X4, Xs)) -> U5_ag(X, X4, Xs, member_in_ag(X, Xs)) U5_ag(X, X4, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X4, Xs)) U2_agg(X, Xs, Ys, Zs, member_out_ag(X, Zs)) -> U3_agg(X, Xs, Ys, Zs, not_member_in_gg(X, Ys)) not_member_in_gg(X, Y) -> U6_gg(X, Y, member_in_gg(X, Y)) member_in_gg(X, .(X, X3)) -> member_out_gg(X, .(X, X3)) member_in_gg(X, .(X4, Xs)) -> U5_gg(X, X4, Xs, member_in_gg(X, Xs)) U5_gg(X, X4, Xs, member_out_gg(X, Xs)) -> member_out_gg(X, .(X4, Xs)) U6_gg(X, Y, member_out_gg(X, Y)) -> U7_gg(X, Y, failure_in_g(a)) failure_in_g(b) -> failure_out_g(b) U7_gg(X, Y, failure_out_g(a)) -> not_member_out_gg(X, Y) not_member_in_gg(X5, X6) -> not_member_out_gg(X5, X6) U3_agg(X, Xs, Ys, Zs, not_member_out_gg(X, Ys)) -> U4_agg(X, Xs, Ys, Zs, subsetchecked_in_agg(Xs, .(X, Ys), Zs)) U4_agg(X, Xs, Ys, Zs, subsetchecked_out_agg(Xs, .(X, Ys), Zs)) -> subsetchecked_out_agg(.(X, Xs), Ys, Zs) U1_ag(X, Y, subsetchecked_out_agg(X, [], Y)) -> subset_out_ag(X, Y) The argument filtering Pi contains the following mapping: subset_in_ag(x1, x2) = subset_in_ag(x2) U1_ag(x1, x2, x3) = U1_ag(x3) subsetchecked_in_agg(x1, x2, x3) = subsetchecked_in_agg(x2, x3) subsetchecked_out_agg(x1, x2, x3) = subsetchecked_out_agg(x1) U2_agg(x1, x2, x3, x4, x5) = U2_agg(x3, x4, x5) member_in_ag(x1, x2) = member_in_ag(x2) .(x1, x2) = .(x1, x2) member_out_ag(x1, x2) = member_out_ag(x1) U5_ag(x1, x2, x3, x4) = U5_ag(x4) U3_agg(x1, x2, x3, x4, x5) = U3_agg(x1, x3, x4, x5) not_member_in_gg(x1, x2) = not_member_in_gg(x1, x2) U6_gg(x1, x2, x3) = U6_gg(x3) member_in_gg(x1, x2) = member_in_gg(x1, x2) member_out_gg(x1, x2) = member_out_gg U5_gg(x1, x2, x3, x4) = U5_gg(x4) U7_gg(x1, x2, x3) = U7_gg(x3) failure_in_g(x1) = failure_in_g(x1) b = b failure_out_g(x1) = failure_out_g a = a not_member_out_gg(x1, x2) = not_member_out_gg U4_agg(x1, x2, x3, x4, x5) = U4_agg(x1, x5) [] = [] subset_out_ag(x1, x2) = subset_out_ag(x1) SUBSETCHECKED_IN_AGG(x1, x2, x3) = SUBSETCHECKED_IN_AGG(x2, x3) U2_AGG(x1, x2, x3, x4, x5) = U2_AGG(x3, x4, x5) U3_AGG(x1, x2, x3, x4, x5) = U3_AGG(x1, x3, x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (89) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (90) Obligation: Pi DP problem: The TRS P consists of the following rules: U2_AGG(X, Xs, Ys, Zs, member_out_ag(X, Zs)) -> U3_AGG(X, Xs, Ys, Zs, not_member_in_gg(X, Ys)) U3_AGG(X, Xs, Ys, Zs, not_member_out_gg(X, Ys)) -> SUBSETCHECKED_IN_AGG(Xs, .(X, Ys), Zs) SUBSETCHECKED_IN_AGG(.(X, Xs), Ys, Zs) -> U2_AGG(X, Xs, Ys, Zs, member_in_ag(X, Zs)) The TRS R consists of the following rules: not_member_in_gg(X, Y) -> U6_gg(X, Y, member_in_gg(X, Y)) not_member_in_gg(X5, X6) -> not_member_out_gg(X5, X6) member_in_ag(X, .(X, X3)) -> member_out_ag(X, .(X, X3)) member_in_ag(X, .(X4, Xs)) -> U5_ag(X, X4, Xs, member_in_ag(X, Xs)) U6_gg(X, Y, member_out_gg(X, Y)) -> U7_gg(X, Y, failure_in_g(a)) U5_ag(X, X4, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X4, Xs)) member_in_gg(X, .(X, X3)) -> member_out_gg(X, .(X, X3)) member_in_gg(X, .(X4, Xs)) -> U5_gg(X, X4, Xs, member_in_gg(X, Xs)) U5_gg(X, X4, Xs, member_out_gg(X, Xs)) -> member_out_gg(X, .(X4, Xs)) The argument filtering Pi contains the following mapping: member_in_ag(x1, x2) = member_in_ag(x2) .(x1, x2) = .(x1, x2) member_out_ag(x1, x2) = member_out_ag(x1) U5_ag(x1, x2, x3, x4) = U5_ag(x4) not_member_in_gg(x1, x2) = not_member_in_gg(x1, x2) U6_gg(x1, x2, x3) = U6_gg(x3) member_in_gg(x1, x2) = member_in_gg(x1, x2) member_out_gg(x1, x2) = member_out_gg U5_gg(x1, x2, x3, x4) = U5_gg(x4) U7_gg(x1, x2, x3) = U7_gg(x3) failure_in_g(x1) = failure_in_g(x1) a = a not_member_out_gg(x1, x2) = not_member_out_gg SUBSETCHECKED_IN_AGG(x1, x2, x3) = SUBSETCHECKED_IN_AGG(x2, x3) U2_AGG(x1, x2, x3, x4, x5) = U2_AGG(x3, x4, x5) U3_AGG(x1, x2, x3, x4, x5) = U3_AGG(x1, x3, x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (91) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (92) Obligation: Q DP problem: The TRS P consists of the following rules: U2_AGG(Ys, Zs, member_out_ag(X)) -> U3_AGG(X, Ys, Zs, not_member_in_gg(X, Ys)) U3_AGG(X, Ys, Zs, not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(X, Ys), Zs) SUBSETCHECKED_IN_AGG(Ys, Zs) -> U2_AGG(Ys, Zs, member_in_ag(Zs)) The TRS R consists of the following rules: not_member_in_gg(X, Y) -> U6_gg(member_in_gg(X, Y)) not_member_in_gg(X5, X6) -> not_member_out_gg member_in_ag(.(X, X3)) -> member_out_ag(X) member_in_ag(.(X4, Xs)) -> U5_ag(member_in_ag(Xs)) U6_gg(member_out_gg) -> U7_gg(failure_in_g(a)) U5_ag(member_out_ag(X)) -> member_out_ag(X) member_in_gg(X, .(X, X3)) -> member_out_gg member_in_gg(X, .(X4, Xs)) -> U5_gg(member_in_gg(X, Xs)) U5_gg(member_out_gg) -> member_out_gg The set Q consists of the following terms: not_member_in_gg(x0, x1) member_in_ag(x0) U6_gg(x0) U5_ag(x0) member_in_gg(x0, x1) U5_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (93) QDPQMonotonicMRRProof (EQUIVALENT) By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain. Strictly oriented rules of the TRS R: not_member_in_gg(X, Y) -> U6_gg(member_in_gg(X, Y)) Used ordering: Polynomial interpretation [POLO]: POL(.(x_1, x_2)) = 0 POL(SUBSETCHECKED_IN_AGG(x_1, x_2)) = 1 + 2*x_2 POL(U2_AGG(x_1, x_2, x_3)) = 1 + 2*x_2 POL(U3_AGG(x_1, x_2, x_3, x_4)) = 2*x_3 + x_4 POL(U5_ag(x_1)) = 1 POL(U5_gg(x_1)) = 0 POL(U6_gg(x_1)) = 0 POL(U7_gg(x_1)) = 0 POL(a) = 0 POL(failure_in_g(x_1)) = 2 + 2*x_1 POL(member_in_ag(x_1)) = 2 POL(member_in_gg(x_1, x_2)) = 2*x_1 POL(member_out_ag(x_1)) = 0 POL(member_out_gg) = 0 POL(not_member_in_gg(x_1, x_2)) = 1 POL(not_member_out_gg) = 1 ---------------------------------------- (94) Obligation: Q DP problem: The TRS P consists of the following rules: U2_AGG(Ys, Zs, member_out_ag(X)) -> U3_AGG(X, Ys, Zs, not_member_in_gg(X, Ys)) U3_AGG(X, Ys, Zs, not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(X, Ys), Zs) SUBSETCHECKED_IN_AGG(Ys, Zs) -> U2_AGG(Ys, Zs, member_in_ag(Zs)) The TRS R consists of the following rules: not_member_in_gg(X5, X6) -> not_member_out_gg member_in_ag(.(X, X3)) -> member_out_ag(X) member_in_ag(.(X4, Xs)) -> U5_ag(member_in_ag(Xs)) U6_gg(member_out_gg) -> U7_gg(failure_in_g(a)) U5_ag(member_out_ag(X)) -> member_out_ag(X) member_in_gg(X, .(X, X3)) -> member_out_gg member_in_gg(X, .(X4, Xs)) -> U5_gg(member_in_gg(X, Xs)) U5_gg(member_out_gg) -> member_out_gg The set Q consists of the following terms: not_member_in_gg(x0, x1) member_in_ag(x0) U6_gg(x0) U5_ag(x0) member_in_gg(x0, x1) U5_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (95) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (96) Obligation: Q DP problem: The TRS P consists of the following rules: U2_AGG(Ys, Zs, member_out_ag(X)) -> U3_AGG(X, Ys, Zs, not_member_in_gg(X, Ys)) U3_AGG(X, Ys, Zs, not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(X, Ys), Zs) SUBSETCHECKED_IN_AGG(Ys, Zs) -> U2_AGG(Ys, Zs, member_in_ag(Zs)) The TRS R consists of the following rules: member_in_ag(.(X, X3)) -> member_out_ag(X) member_in_ag(.(X4, Xs)) -> U5_ag(member_in_ag(Xs)) U5_ag(member_out_ag(X)) -> member_out_ag(X) not_member_in_gg(X5, X6) -> not_member_out_gg The set Q consists of the following terms: not_member_in_gg(x0, x1) member_in_ag(x0) U6_gg(x0) U5_ag(x0) member_in_gg(x0, x1) U5_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (97) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. U6_gg(x0) member_in_gg(x0, x1) U5_gg(x0) ---------------------------------------- (98) Obligation: Q DP problem: The TRS P consists of the following rules: U2_AGG(Ys, Zs, member_out_ag(X)) -> U3_AGG(X, Ys, Zs, not_member_in_gg(X, Ys)) U3_AGG(X, Ys, Zs, not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(X, Ys), Zs) SUBSETCHECKED_IN_AGG(Ys, Zs) -> U2_AGG(Ys, Zs, member_in_ag(Zs)) The TRS R consists of the following rules: member_in_ag(.(X, X3)) -> member_out_ag(X) member_in_ag(.(X4, Xs)) -> U5_ag(member_in_ag(Xs)) U5_ag(member_out_ag(X)) -> member_out_ag(X) not_member_in_gg(X5, X6) -> not_member_out_gg The set Q consists of the following terms: not_member_in_gg(x0, x1) member_in_ag(x0) U5_ag(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (99) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule U2_AGG(Ys, Zs, member_out_ag(X)) -> U3_AGG(X, Ys, Zs, not_member_in_gg(X, Ys)) at position [3] we obtained the following new rules [LPAR04]: (U2_AGG(Ys, Zs, member_out_ag(X)) -> U3_AGG(X, Ys, Zs, not_member_out_gg),U2_AGG(Ys, Zs, member_out_ag(X)) -> U3_AGG(X, Ys, Zs, not_member_out_gg)) ---------------------------------------- (100) Obligation: Q DP problem: The TRS P consists of the following rules: U3_AGG(X, Ys, Zs, not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(X, Ys), Zs) SUBSETCHECKED_IN_AGG(Ys, Zs) -> U2_AGG(Ys, Zs, member_in_ag(Zs)) U2_AGG(Ys, Zs, member_out_ag(X)) -> U3_AGG(X, Ys, Zs, not_member_out_gg) The TRS R consists of the following rules: member_in_ag(.(X, X3)) -> member_out_ag(X) member_in_ag(.(X4, Xs)) -> U5_ag(member_in_ag(Xs)) U5_ag(member_out_ag(X)) -> member_out_ag(X) not_member_in_gg(X5, X6) -> not_member_out_gg The set Q consists of the following terms: not_member_in_gg(x0, x1) member_in_ag(x0) U5_ag(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (101) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (102) Obligation: Q DP problem: The TRS P consists of the following rules: U3_AGG(X, Ys, Zs, not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(X, Ys), Zs) SUBSETCHECKED_IN_AGG(Ys, Zs) -> U2_AGG(Ys, Zs, member_in_ag(Zs)) U2_AGG(Ys, Zs, member_out_ag(X)) -> U3_AGG(X, Ys, Zs, not_member_out_gg) The TRS R consists of the following rules: member_in_ag(.(X, X3)) -> member_out_ag(X) member_in_ag(.(X4, Xs)) -> U5_ag(member_in_ag(Xs)) U5_ag(member_out_ag(X)) -> member_out_ag(X) The set Q consists of the following terms: not_member_in_gg(x0, x1) member_in_ag(x0) U5_ag(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (103) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. not_member_in_gg(x0, x1) ---------------------------------------- (104) Obligation: Q DP problem: The TRS P consists of the following rules: U3_AGG(X, Ys, Zs, not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(X, Ys), Zs) SUBSETCHECKED_IN_AGG(Ys, Zs) -> U2_AGG(Ys, Zs, member_in_ag(Zs)) U2_AGG(Ys, Zs, member_out_ag(X)) -> U3_AGG(X, Ys, Zs, not_member_out_gg) The TRS R consists of the following rules: member_in_ag(.(X, X3)) -> member_out_ag(X) member_in_ag(.(X4, Xs)) -> U5_ag(member_in_ag(Xs)) U5_ag(member_out_ag(X)) -> member_out_ag(X) The set Q consists of the following terms: member_in_ag(x0) U5_ag(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (105) TransformationProof (SOUND) By narrowing [LPAR04] the rule SUBSETCHECKED_IN_AGG(Ys, Zs) -> U2_AGG(Ys, Zs, member_in_ag(Zs)) at position [2] we obtained the following new rules [LPAR04]: (SUBSETCHECKED_IN_AGG(y0, .(x0, x1)) -> U2_AGG(y0, .(x0, x1), member_out_ag(x0)),SUBSETCHECKED_IN_AGG(y0, .(x0, x1)) -> U2_AGG(y0, .(x0, x1), member_out_ag(x0))) (SUBSETCHECKED_IN_AGG(y0, .(x0, x1)) -> U2_AGG(y0, .(x0, x1), U5_ag(member_in_ag(x1))),SUBSETCHECKED_IN_AGG(y0, .(x0, x1)) -> U2_AGG(y0, .(x0, x1), U5_ag(member_in_ag(x1)))) ---------------------------------------- (106) Obligation: Q DP problem: The TRS P consists of the following rules: U3_AGG(X, Ys, Zs, not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(X, Ys), Zs) U2_AGG(Ys, Zs, member_out_ag(X)) -> U3_AGG(X, Ys, Zs, not_member_out_gg) SUBSETCHECKED_IN_AGG(y0, .(x0, x1)) -> U2_AGG(y0, .(x0, x1), member_out_ag(x0)) SUBSETCHECKED_IN_AGG(y0, .(x0, x1)) -> U2_AGG(y0, .(x0, x1), U5_ag(member_in_ag(x1))) The TRS R consists of the following rules: member_in_ag(.(X, X3)) -> member_out_ag(X) member_in_ag(.(X4, Xs)) -> U5_ag(member_in_ag(Xs)) U5_ag(member_out_ag(X)) -> member_out_ag(X) The set Q consists of the following terms: member_in_ag(x0) U5_ag(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (107) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U2_AGG(Ys, Zs, member_out_ag(X)) -> U3_AGG(X, Ys, Zs, not_member_out_gg) we obtained the following new rules [LPAR04]: (U2_AGG(z0, .(z1, z2), member_out_ag(z1)) -> U3_AGG(z1, z0, .(z1, z2), not_member_out_gg),U2_AGG(z0, .(z1, z2), member_out_ag(z1)) -> U3_AGG(z1, z0, .(z1, z2), not_member_out_gg)) (U2_AGG(z0, .(z1, z2), member_out_ag(x2)) -> U3_AGG(x2, z0, .(z1, z2), not_member_out_gg),U2_AGG(z0, .(z1, z2), member_out_ag(x2)) -> U3_AGG(x2, z0, .(z1, z2), not_member_out_gg)) ---------------------------------------- (108) Obligation: Q DP problem: The TRS P consists of the following rules: U3_AGG(X, Ys, Zs, not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(X, Ys), Zs) SUBSETCHECKED_IN_AGG(y0, .(x0, x1)) -> U2_AGG(y0, .(x0, x1), member_out_ag(x0)) SUBSETCHECKED_IN_AGG(y0, .(x0, x1)) -> U2_AGG(y0, .(x0, x1), U5_ag(member_in_ag(x1))) U2_AGG(z0, .(z1, z2), member_out_ag(z1)) -> U3_AGG(z1, z0, .(z1, z2), not_member_out_gg) U2_AGG(z0, .(z1, z2), member_out_ag(x2)) -> U3_AGG(x2, z0, .(z1, z2), not_member_out_gg) The TRS R consists of the following rules: member_in_ag(.(X, X3)) -> member_out_ag(X) member_in_ag(.(X4, Xs)) -> U5_ag(member_in_ag(Xs)) U5_ag(member_out_ag(X)) -> member_out_ag(X) The set Q consists of the following terms: member_in_ag(x0) U5_ag(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (109) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U3_AGG(X, Ys, Zs, not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(X, Ys), Zs) we obtained the following new rules [LPAR04]: (U3_AGG(z1, z0, .(z1, z2), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z1, z0), .(z1, z2)),U3_AGG(z1, z0, .(z1, z2), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z1, z0), .(z1, z2))) (U3_AGG(z3, z0, .(z1, z2), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z3, z0), .(z1, z2)),U3_AGG(z3, z0, .(z1, z2), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z3, z0), .(z1, z2))) ---------------------------------------- (110) Obligation: Q DP problem: The TRS P consists of the following rules: SUBSETCHECKED_IN_AGG(y0, .(x0, x1)) -> U2_AGG(y0, .(x0, x1), member_out_ag(x0)) SUBSETCHECKED_IN_AGG(y0, .(x0, x1)) -> U2_AGG(y0, .(x0, x1), U5_ag(member_in_ag(x1))) U2_AGG(z0, .(z1, z2), member_out_ag(z1)) -> U3_AGG(z1, z0, .(z1, z2), not_member_out_gg) U2_AGG(z0, .(z1, z2), member_out_ag(x2)) -> U3_AGG(x2, z0, .(z1, z2), not_member_out_gg) U3_AGG(z1, z0, .(z1, z2), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z1, z0), .(z1, z2)) U3_AGG(z3, z0, .(z1, z2), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z3, z0), .(z1, z2)) The TRS R consists of the following rules: member_in_ag(.(X, X3)) -> member_out_ag(X) member_in_ag(.(X4, Xs)) -> U5_ag(member_in_ag(Xs)) U5_ag(member_out_ag(X)) -> member_out_ag(X) The set Q consists of the following terms: member_in_ag(x0) U5_ag(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (111) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule SUBSETCHECKED_IN_AGG(y0, .(x0, x1)) -> U2_AGG(y0, .(x0, x1), member_out_ag(x0)) we obtained the following new rules [LPAR04]: (SUBSETCHECKED_IN_AGG(.(z0, z1), .(z0, z2)) -> U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0)),SUBSETCHECKED_IN_AGG(.(z0, z1), .(z0, z2)) -> U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0))) (SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2)),SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2))) ---------------------------------------- (112) Obligation: Q DP problem: The TRS P consists of the following rules: SUBSETCHECKED_IN_AGG(y0, .(x0, x1)) -> U2_AGG(y0, .(x0, x1), U5_ag(member_in_ag(x1))) U2_AGG(z0, .(z1, z2), member_out_ag(z1)) -> U3_AGG(z1, z0, .(z1, z2), not_member_out_gg) U2_AGG(z0, .(z1, z2), member_out_ag(x2)) -> U3_AGG(x2, z0, .(z1, z2), not_member_out_gg) U3_AGG(z1, z0, .(z1, z2), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z1, z0), .(z1, z2)) U3_AGG(z3, z0, .(z1, z2), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z3, z0), .(z1, z2)) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z0, z2)) -> U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0)) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2)) The TRS R consists of the following rules: member_in_ag(.(X, X3)) -> member_out_ag(X) member_in_ag(.(X4, Xs)) -> U5_ag(member_in_ag(Xs)) U5_ag(member_out_ag(X)) -> member_out_ag(X) The set Q consists of the following terms: member_in_ag(x0) U5_ag(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (113) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule SUBSETCHECKED_IN_AGG(y0, .(x0, x1)) -> U2_AGG(y0, .(x0, x1), U5_ag(member_in_ag(x1))) we obtained the following new rules [LPAR04]: (SUBSETCHECKED_IN_AGG(.(z0, z1), .(z0, z2)) -> U2_AGG(.(z0, z1), .(z0, z2), U5_ag(member_in_ag(z2))),SUBSETCHECKED_IN_AGG(.(z0, z1), .(z0, z2)) -> U2_AGG(.(z0, z1), .(z0, z2), U5_ag(member_in_ag(z2)))) (SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), U5_ag(member_in_ag(z3))),SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), U5_ag(member_in_ag(z3)))) ---------------------------------------- (114) Obligation: Q DP problem: The TRS P consists of the following rules: U2_AGG(z0, .(z1, z2), member_out_ag(z1)) -> U3_AGG(z1, z0, .(z1, z2), not_member_out_gg) U2_AGG(z0, .(z1, z2), member_out_ag(x2)) -> U3_AGG(x2, z0, .(z1, z2), not_member_out_gg) U3_AGG(z1, z0, .(z1, z2), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z1, z0), .(z1, z2)) U3_AGG(z3, z0, .(z1, z2), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z3, z0), .(z1, z2)) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z0, z2)) -> U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0)) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2)) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z0, z2)) -> U2_AGG(.(z0, z1), .(z0, z2), U5_ag(member_in_ag(z2))) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), U5_ag(member_in_ag(z3))) The TRS R consists of the following rules: member_in_ag(.(X, X3)) -> member_out_ag(X) member_in_ag(.(X4, Xs)) -> U5_ag(member_in_ag(Xs)) U5_ag(member_out_ag(X)) -> member_out_ag(X) The set Q consists of the following terms: member_in_ag(x0) U5_ag(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (115) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U2_AGG(z0, .(z1, z2), member_out_ag(z1)) -> U3_AGG(z1, z0, .(z1, z2), not_member_out_gg) we obtained the following new rules [LPAR04]: (U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0)) -> U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg),U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0)) -> U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg)) (U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2)) -> U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg),U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2)) -> U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg)) ---------------------------------------- (116) Obligation: Q DP problem: The TRS P consists of the following rules: U2_AGG(z0, .(z1, z2), member_out_ag(x2)) -> U3_AGG(x2, z0, .(z1, z2), not_member_out_gg) U3_AGG(z1, z0, .(z1, z2), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z1, z0), .(z1, z2)) U3_AGG(z3, z0, .(z1, z2), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z3, z0), .(z1, z2)) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z0, z2)) -> U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0)) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2)) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z0, z2)) -> U2_AGG(.(z0, z1), .(z0, z2), U5_ag(member_in_ag(z2))) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), U5_ag(member_in_ag(z3))) U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0)) -> U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg) U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2)) -> U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg) The TRS R consists of the following rules: member_in_ag(.(X, X3)) -> member_out_ag(X) member_in_ag(.(X4, Xs)) -> U5_ag(member_in_ag(Xs)) U5_ag(member_out_ag(X)) -> member_out_ag(X) The set Q consists of the following terms: member_in_ag(x0) U5_ag(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (117) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U2_AGG(z0, .(z1, z2), member_out_ag(x2)) -> U3_AGG(x2, z0, .(z1, z2), not_member_out_gg) we obtained the following new rules [LPAR04]: (U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0)) -> U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg),U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0)) -> U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg)) (U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2)) -> U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg),U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2)) -> U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg)) (U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(x3)) -> U3_AGG(x3, .(z0, z1), .(z0, z2), not_member_out_gg),U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(x3)) -> U3_AGG(x3, .(z0, z1), .(z0, z2), not_member_out_gg)) (U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(x3)) -> U3_AGG(x3, .(z0, z1), .(z2, z3), not_member_out_gg),U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(x3)) -> U3_AGG(x3, .(z0, z1), .(z2, z3), not_member_out_gg)) ---------------------------------------- (118) Obligation: Q DP problem: The TRS P consists of the following rules: U3_AGG(z1, z0, .(z1, z2), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z1, z0), .(z1, z2)) U3_AGG(z3, z0, .(z1, z2), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z3, z0), .(z1, z2)) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z0, z2)) -> U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0)) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2)) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z0, z2)) -> U2_AGG(.(z0, z1), .(z0, z2), U5_ag(member_in_ag(z2))) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), U5_ag(member_in_ag(z3))) U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0)) -> U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg) U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2)) -> U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg) U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(x3)) -> U3_AGG(x3, .(z0, z1), .(z0, z2), not_member_out_gg) U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(x3)) -> U3_AGG(x3, .(z0, z1), .(z2, z3), not_member_out_gg) The TRS R consists of the following rules: member_in_ag(.(X, X3)) -> member_out_ag(X) member_in_ag(.(X4, Xs)) -> U5_ag(member_in_ag(Xs)) U5_ag(member_out_ag(X)) -> member_out_ag(X) The set Q consists of the following terms: member_in_ag(x0) U5_ag(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (119) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U3_AGG(z1, z0, .(z1, z2), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z1, z0), .(z1, z2)) we obtained the following new rules [LPAR04]: (U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)),U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2))) (U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z2, .(z0, z1)), .(z2, z3)),U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z2, .(z0, z1)), .(z2, z3))) ---------------------------------------- (120) Obligation: Q DP problem: The TRS P consists of the following rules: U3_AGG(z3, z0, .(z1, z2), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z3, z0), .(z1, z2)) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z0, z2)) -> U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0)) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2)) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z0, z2)) -> U2_AGG(.(z0, z1), .(z0, z2), U5_ag(member_in_ag(z2))) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), U5_ag(member_in_ag(z3))) U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0)) -> U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg) U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2)) -> U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg) U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(x3)) -> U3_AGG(x3, .(z0, z1), .(z0, z2), not_member_out_gg) U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(x3)) -> U3_AGG(x3, .(z0, z1), .(z2, z3), not_member_out_gg) U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z2, .(z0, z1)), .(z2, z3)) The TRS R consists of the following rules: member_in_ag(.(X, X3)) -> member_out_ag(X) member_in_ag(.(X4, Xs)) -> U5_ag(member_in_ag(Xs)) U5_ag(member_out_ag(X)) -> member_out_ag(X) The set Q consists of the following terms: member_in_ag(x0) U5_ag(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (121) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U3_AGG(z3, z0, .(z1, z2), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z3, z0), .(z1, z2)) we obtained the following new rules [LPAR04]: (U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)),U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2))) (U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z2, .(z0, z1)), .(z2, z3)),U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z2, .(z0, z1)), .(z2, z3))) (U3_AGG(z3, .(z0, z1), .(z0, z2), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z3, .(z0, z1)), .(z0, z2)),U3_AGG(z3, .(z0, z1), .(z0, z2), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z3, .(z0, z1)), .(z0, z2))) (U3_AGG(z4, .(z0, z1), .(z2, z3), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z4, .(z0, z1)), .(z2, z3)),U3_AGG(z4, .(z0, z1), .(z2, z3), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z4, .(z0, z1)), .(z2, z3))) ---------------------------------------- (122) Obligation: Q DP problem: The TRS P consists of the following rules: SUBSETCHECKED_IN_AGG(.(z0, z1), .(z0, z2)) -> U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0)) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2)) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z0, z2)) -> U2_AGG(.(z0, z1), .(z0, z2), U5_ag(member_in_ag(z2))) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), U5_ag(member_in_ag(z3))) U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0)) -> U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg) U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2)) -> U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg) U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(x3)) -> U3_AGG(x3, .(z0, z1), .(z0, z2), not_member_out_gg) U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(x3)) -> U3_AGG(x3, .(z0, z1), .(z2, z3), not_member_out_gg) U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z2, .(z0, z1)), .(z2, z3)) U3_AGG(z3, .(z0, z1), .(z0, z2), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z3, .(z0, z1)), .(z0, z2)) U3_AGG(z4, .(z0, z1), .(z2, z3), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z4, .(z0, z1)), .(z2, z3)) The TRS R consists of the following rules: member_in_ag(.(X, X3)) -> member_out_ag(X) member_in_ag(.(X4, Xs)) -> U5_ag(member_in_ag(Xs)) U5_ag(member_out_ag(X)) -> member_out_ag(X) The set Q consists of the following terms: member_in_ag(x0) U5_ag(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (123) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule SUBSETCHECKED_IN_AGG(.(z0, z1), .(z0, z2)) -> U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0)) we obtained the following new rules [LPAR04]: (SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) -> U2_AGG(.(z0, .(z0, z1)), .(z0, z2), member_out_ag(z0)),SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) -> U2_AGG(.(z0, .(z0, z1)), .(z0, z2), member_out_ag(z0))) (SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z0, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z0, z3), member_out_ag(z0)),SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z0, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z0, z3), member_out_ag(z0))) ---------------------------------------- (124) Obligation: Q DP problem: The TRS P consists of the following rules: SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2)) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z0, z2)) -> U2_AGG(.(z0, z1), .(z0, z2), U5_ag(member_in_ag(z2))) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), U5_ag(member_in_ag(z3))) U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0)) -> U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg) U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2)) -> U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg) U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(x3)) -> U3_AGG(x3, .(z0, z1), .(z0, z2), not_member_out_gg) U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(x3)) -> U3_AGG(x3, .(z0, z1), .(z2, z3), not_member_out_gg) U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z2, .(z0, z1)), .(z2, z3)) U3_AGG(z3, .(z0, z1), .(z0, z2), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z3, .(z0, z1)), .(z0, z2)) U3_AGG(z4, .(z0, z1), .(z2, z3), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z4, .(z0, z1)), .(z2, z3)) SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) -> U2_AGG(.(z0, .(z0, z1)), .(z0, z2), member_out_ag(z0)) SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z0, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z0, z3), member_out_ag(z0)) The TRS R consists of the following rules: member_in_ag(.(X, X3)) -> member_out_ag(X) member_in_ag(.(X4, Xs)) -> U5_ag(member_in_ag(Xs)) U5_ag(member_out_ag(X)) -> member_out_ag(X) The set Q consists of the following terms: member_in_ag(x0) U5_ag(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (125) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2)) we obtained the following new rules [LPAR04]: (SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) -> U2_AGG(.(z0, .(z0, z1)), .(z0, z2), member_out_ag(z0)),SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) -> U2_AGG(.(z0, .(z0, z1)), .(z0, z2), member_out_ag(z0))) (SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z0, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z0, z3), member_out_ag(z0)),SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z0, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z0, z3), member_out_ag(z0))) (SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z1, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z1, z3), member_out_ag(z1)),SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z1, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z1, z3), member_out_ag(z1))) (SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z3, z4)) -> U2_AGG(.(z0, .(z1, z2)), .(z3, z4), member_out_ag(z3)),SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z3, z4)) -> U2_AGG(.(z0, .(z1, z2)), .(z3, z4), member_out_ag(z3))) ---------------------------------------- (126) Obligation: Q DP problem: The TRS P consists of the following rules: SUBSETCHECKED_IN_AGG(.(z0, z1), .(z0, z2)) -> U2_AGG(.(z0, z1), .(z0, z2), U5_ag(member_in_ag(z2))) SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), U5_ag(member_in_ag(z3))) U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0)) -> U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg) U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2)) -> U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg) U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(x3)) -> U3_AGG(x3, .(z0, z1), .(z0, z2), not_member_out_gg) U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(x3)) -> U3_AGG(x3, .(z0, z1), .(z2, z3), not_member_out_gg) U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z2, .(z0, z1)), .(z2, z3)) U3_AGG(z3, .(z0, z1), .(z0, z2), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z3, .(z0, z1)), .(z0, z2)) U3_AGG(z4, .(z0, z1), .(z2, z3), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z4, .(z0, z1)), .(z2, z3)) SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) -> U2_AGG(.(z0, .(z0, z1)), .(z0, z2), member_out_ag(z0)) SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z0, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z0, z3), member_out_ag(z0)) SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z1, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z1, z3), member_out_ag(z1)) SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z3, z4)) -> U2_AGG(.(z0, .(z1, z2)), .(z3, z4), member_out_ag(z3)) The TRS R consists of the following rules: member_in_ag(.(X, X3)) -> member_out_ag(X) member_in_ag(.(X4, Xs)) -> U5_ag(member_in_ag(Xs)) U5_ag(member_out_ag(X)) -> member_out_ag(X) The set Q consists of the following terms: member_in_ag(x0) U5_ag(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (127) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule SUBSETCHECKED_IN_AGG(.(z0, z1), .(z0, z2)) -> U2_AGG(.(z0, z1), .(z0, z2), U5_ag(member_in_ag(z2))) we obtained the following new rules [LPAR04]: (SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) -> U2_AGG(.(z0, .(z0, z1)), .(z0, z2), U5_ag(member_in_ag(z2))),SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) -> U2_AGG(.(z0, .(z0, z1)), .(z0, z2), U5_ag(member_in_ag(z2)))) (SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z0, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z0, z3), U5_ag(member_in_ag(z3))),SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z0, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z0, z3), U5_ag(member_in_ag(z3)))) ---------------------------------------- (128) Obligation: Q DP problem: The TRS P consists of the following rules: SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), U5_ag(member_in_ag(z3))) U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0)) -> U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg) U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2)) -> U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg) U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(x3)) -> U3_AGG(x3, .(z0, z1), .(z0, z2), not_member_out_gg) U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(x3)) -> U3_AGG(x3, .(z0, z1), .(z2, z3), not_member_out_gg) U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z2, .(z0, z1)), .(z2, z3)) U3_AGG(z3, .(z0, z1), .(z0, z2), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z3, .(z0, z1)), .(z0, z2)) U3_AGG(z4, .(z0, z1), .(z2, z3), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z4, .(z0, z1)), .(z2, z3)) SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) -> U2_AGG(.(z0, .(z0, z1)), .(z0, z2), member_out_ag(z0)) SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z0, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z0, z3), member_out_ag(z0)) SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z1, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z1, z3), member_out_ag(z1)) SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z3, z4)) -> U2_AGG(.(z0, .(z1, z2)), .(z3, z4), member_out_ag(z3)) SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) -> U2_AGG(.(z0, .(z0, z1)), .(z0, z2), U5_ag(member_in_ag(z2))) SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z0, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z0, z3), U5_ag(member_in_ag(z3))) The TRS R consists of the following rules: member_in_ag(.(X, X3)) -> member_out_ag(X) member_in_ag(.(X4, Xs)) -> U5_ag(member_in_ag(Xs)) U5_ag(member_out_ag(X)) -> member_out_ag(X) The set Q consists of the following terms: member_in_ag(x0) U5_ag(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (129) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule SUBSETCHECKED_IN_AGG(.(z0, z1), .(z2, z3)) -> U2_AGG(.(z0, z1), .(z2, z3), U5_ag(member_in_ag(z3))) we obtained the following new rules [LPAR04]: (SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) -> U2_AGG(.(z0, .(z0, z1)), .(z0, z2), U5_ag(member_in_ag(z2))),SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) -> U2_AGG(.(z0, .(z0, z1)), .(z0, z2), U5_ag(member_in_ag(z2)))) (SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z0, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z0, z3), U5_ag(member_in_ag(z3))),SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z0, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z0, z3), U5_ag(member_in_ag(z3)))) (SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z1, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z1, z3), U5_ag(member_in_ag(z3))),SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z1, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z1, z3), U5_ag(member_in_ag(z3)))) (SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z3, z4)) -> U2_AGG(.(z0, .(z1, z2)), .(z3, z4), U5_ag(member_in_ag(z4))),SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z3, z4)) -> U2_AGG(.(z0, .(z1, z2)), .(z3, z4), U5_ag(member_in_ag(z4)))) ---------------------------------------- (130) Obligation: Q DP problem: The TRS P consists of the following rules: U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0)) -> U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg) U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(z2)) -> U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg) U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(x3)) -> U3_AGG(x3, .(z0, z1), .(z0, z2), not_member_out_gg) U2_AGG(.(z0, z1), .(z2, z3), member_out_ag(x3)) -> U3_AGG(x3, .(z0, z1), .(z2, z3), not_member_out_gg) U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) U3_AGG(z2, .(z0, z1), .(z2, z3), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z2, .(z0, z1)), .(z2, z3)) U3_AGG(z3, .(z0, z1), .(z0, z2), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z3, .(z0, z1)), .(z0, z2)) U3_AGG(z4, .(z0, z1), .(z2, z3), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z4, .(z0, z1)), .(z2, z3)) SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) -> U2_AGG(.(z0, .(z0, z1)), .(z0, z2), member_out_ag(z0)) SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z0, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z0, z3), member_out_ag(z0)) SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z1, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z1, z3), member_out_ag(z1)) SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z3, z4)) -> U2_AGG(.(z0, .(z1, z2)), .(z3, z4), member_out_ag(z3)) SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) -> U2_AGG(.(z0, .(z0, z1)), .(z0, z2), U5_ag(member_in_ag(z2))) SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z0, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z0, z3), U5_ag(member_in_ag(z3))) SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z1, z3)) -> U2_AGG(.(z0, .(z1, z2)), .(z1, z3), U5_ag(member_in_ag(z3))) SUBSETCHECKED_IN_AGG(.(z0, .(z1, z2)), .(z3, z4)) -> U2_AGG(.(z0, .(z1, z2)), .(z3, z4), U5_ag(member_in_ag(z4))) The TRS R consists of the following rules: member_in_ag(.(X, X3)) -> member_out_ag(X) member_in_ag(.(X4, Xs)) -> U5_ag(member_in_ag(Xs)) U5_ag(member_out_ag(X)) -> member_out_ag(X) The set Q consists of the following terms: member_in_ag(x0) U5_ag(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (131) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by narrowing to the left: s = U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg) evaluates to t =U3_AGG(z0, .(z0, .(z0, z1)), .(z0, z2), not_member_out_gg) Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [z1 / .(z0, z1)] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) with rule U3_AGG(z0', .(z0', z1'), .(z0', z2'), not_member_out_gg) -> SUBSETCHECKED_IN_AGG(.(z0', .(z0', z1')), .(z0', z2')) at position [] and matcher [z0' / z0, z1' / z1, z2' / z2] SUBSETCHECKED_IN_AGG(.(z0, .(z0, z1)), .(z0, z2)) -> U2_AGG(.(z0, .(z0, z1)), .(z0, z2), member_out_ag(z0)) with rule SUBSETCHECKED_IN_AGG(.(z0', .(z0', z1')), .(z0', z2')) -> U2_AGG(.(z0', .(z0', z1')), .(z0', z2'), member_out_ag(z0')) at position [] and matcher [z0' / z0, z1' / z1, z2' / z2] U2_AGG(.(z0, .(z0, z1)), .(z0, z2), member_out_ag(z0)) -> U3_AGG(z0, .(z0, .(z0, z1)), .(z0, z2), not_member_out_gg) with rule U2_AGG(.(z0, z1), .(z0, z2), member_out_ag(z0)) -> U3_AGG(z0, .(z0, z1), .(z0, z2), not_member_out_gg) Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence All these steps are and every following step will be a correct step w.r.t to Q. ---------------------------------------- (132) NO ---------------------------------------- (133) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 1, "program": { "directives": [], "clauses": [ [ "(subset X Y)", "(subsetchecked X ([]) Y)" ], [ "(subsetchecked ([]) X1 X2)", null ], [ "(subsetchecked (. 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T100 ([]))) T101)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T100", "T101", "T108" ], "free": [], "exprvars": [] } }, "234": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "477": { "goal": [{ "clause": 1, "scope": 9, "term": "(subsetchecked T109 (. T108 (. T100 ([]))) T101)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T100", "T101", "T108" ], "free": [], "exprvars": [] } }, "479": { "goal": [{ "clause": 2, "scope": 9, "term": "(subsetchecked T109 (. T108 (. 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T100 ([])))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T100", "T108" ], "free": [], "exprvars": [] } }, "566": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "325": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "567": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "326": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "571": { "goal": [ { "clause": 1, "scope": 12, "term": "(subsetchecked T185 (. T184 (. T175 (. T176 ([])))) T177)" }, { "clause": 2, "scope": 12, "term": "(subsetchecked T185 (. T184 (. T175 (. T176 ([])))) T177)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T175", "T176", "T177", "T184" ], "free": [], "exprvars": [] } }, "451": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (member T130 (. T131 ([]))) (',' (!_7) (failure (a))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T130", "T131" ], "free": [], "exprvars": [] } }, "572": { "goal": [{ "clause": 1, "scope": 12, "term": "(subsetchecked T185 (. T184 (. T175 (. T176 ([])))) T177)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T175", "T176", "T177", "T184" ], "free": [], "exprvars": [] } }, "573": { "goal": [{ "clause": 2, "scope": 12, "term": "(subsetchecked T185 (. T184 (. T175 (. T176 ([])))) T177)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T175", "T176", "T177", "T184" ], "free": [], "exprvars": [] } }, "574": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "575": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "696": { "goal": [{ "clause": 5, "scope": 16, "term": "(not_member T420 (. T409 (. T410 (. T411 (. T412 ([]))))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T409", "T410", "T411", "T412", "T420" ], "free": [], "exprvars": [] } }, "576": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "697": { "goal": [{ "clause": 6, "scope": 16, "term": "(not_member T420 (. T409 (. T410 (. T411 (. T412 ([]))))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T409", "T410", "T411", "T412", "T420" ], "free": [], "exprvars": [] } }, "457": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T130 (. T131 ([])))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T130", "T131" ], "free": [], "exprvars": [] } }, "579": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (member T282 T281) (',' (not_member T282 (. T278 (. T279 (. T280 ([]))))) (subsetchecked T283 (. T282 (. T278 (. T279 (. T280 ([]))))) T281)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T278", "T279", "T280", "T281" ], "free": [], "exprvars": [] } }, "459": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (!_7) (failure (a)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "580": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "582": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T282 T281)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T281"], "free": [], "exprvars": [] } }, "583": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (not_member T288 (. T278 (. T279 (. T280 ([]))))) (subsetchecked T289 (. T288 (. T278 (. T279 (. T280 ([]))))) T281))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T278", "T279", "T280", "T281", "T288" ], "free": [], "exprvars": [] } }, "463": { "goal": [{ "clause": -1, "scope": -1, "term": "(failure (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "464": { "goal": [{ "clause": 7, "scope": 8, "term": "(failure (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "465": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "467": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "468": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "902": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (member T978 T977) (',' (not_member T978 (. T970 (. T971 (. T972 (. T973 (. T974 (. T975 (. T976 ([]))))))))) (subsetchecked T979 (. T978 T984) T977)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T970", "T971", "T972", "T973", "T974", "T975", "T976", "T977", "T984" ], "free": [], "exprvars": [] } }, "905": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T978 T977)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T977"], "free": [], "exprvars": [] } }, "906": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (not_member T989 (. T970 (. T971 (. T972 (. T973 (. T974 (. T975 (. T976 ([]))))))))) (subsetchecked T990 (. T989 T984) T977))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T970", "T971", "T972", "T973", "T974", "T975", "T976", "T977", "T984", "T989" ], "free": [], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 4, "label": "CASE" }, { "from": 4, "to": 21, "label": "ONLY EVAL with clause\nsubset(X9, X10) :- subsetchecked(X9, [], X10).\nand substitutionT1 -> T7,\nX9 -> T7,\nT2 -> T6,\nX10 -> T6,\nT5 -> T7" }, { "from": 21, "to": 22, "label": "CASE" }, { "from": 22, "to": 144, "label": "PARALLEL" }, { "from": 22, "to": 159, "label": "PARALLEL" }, { "from": 144, "to": 232, "label": "EVAL with clause\nsubsetchecked([], X19, X20).\nand substitutionT7 -> [],\nX19 -> [],\nT6 -> T12,\nX20 -> T12" }, { "from": 144, "to": 233, "label": "EVAL-BACKTRACK" }, { "from": 159, "to": 238, "label": "EVAL with clause\nsubsetchecked(.(X29, X30), X31, X32) :- ','(member(X29, X32), ','(not_member(X29, X31), subsetchecked(X30, .(X29, X31), X32))).\nand substitutionX29 -> T22,\nX30 -> T23,\nT7 -> .(T22, T23),\nX31 -> [],\nT6 -> T21,\nX32 -> T21,\nT19 -> T22,\nT20 -> T23" }, { "from": 159, "to": 239, "label": "EVAL-BACKTRACK" }, { "from": 232, "to": 234, "label": "SUCCESS" }, { "from": 238, "to": 242, "label": "SPLIT 1" }, { "from": 238, "to": 244, "label": "SPLIT 2\nnew knowledge:\nT28 is ground\nT21 is ground\nreplacements:T22 -> T28,\nT23 -> T29" }, { "from": 242, "to": 247, "label": "CASE" }, { "from": 244, "to": 281, "label": "SPLIT 1" }, { "from": 244, "to": 282, "label": "SPLIT 2\nnew knowledge:\nT28 is ground" }, { "from": 247, "to": 248, "label": "PARALLEL" }, { "from": 247, "to": 249, "label": "PARALLEL" }, { "from": 248, "to": 252, "label": "EVAL with clause\nmember(X49, .(X49, X50)).\nand substitutionT22 -> T42,\nX49 -> T42,\nX50 -> T43,\nT21 -> .(T42, T43)" }, { "from": 248, "to": 253, "label": "EVAL-BACKTRACK" }, { "from": 249, "to": 255, "label": "EVAL with clause\nmember(X57, .(X58, X59)) :- member(X57, X59).\nand substitutionT22 -> T53,\nX57 -> T53,\nX58 -> T51,\nX59 -> T52,\nT21 -> .(T51, T52),\nT50 -> T53" }, { "from": 249, "to": 256, "label": "EVAL-BACKTRACK" }, { "from": 252, "to": 254, "label": "SUCCESS" }, { "from": 255, "to": 242, "label": "INSTANCE with matching:\nT22 -> T53\nT21 -> T52" }, { "from": 281, "to": 283, "label": "CASE" }, { "from": 282, "to": 313, "label": "CASE" }, { "from": 283, "to": 291, "label": "PARALLEL" }, { "from": 283, "to": 292, "label": "PARALLEL" }, { "from": 291, "to": 294, "label": "ONLY EVAL with clause\nnot_member(X84, X85) :- ','(member(X84, X85), ','(!_4, failure(a))).\nand substitutionT28 -> T68,\nX84 -> T68,\nX85 -> []" }, { "from": 292, "to": 309, "label": "ONLY EVAL with clause\nnot_member(X98, X99).\nand substitutionT28 -> T75,\nX98 -> T75,\nX99 -> []" }, { "from": 294, "to": 297, "label": "SPLIT 1" }, { "from": 294, "to": 298, "label": "SPLIT 2\nnew knowledge:\nT68 is ground" }, { "from": 297, "to": 242, "label": "INSTANCE with matching:\nT22 -> T68\nT21 -> []" }, { "from": 298, "to": 301, "label": "CUT" }, { "from": 301, "to": 303, "label": "CASE" }, { "from": 303, "to": 304, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 309, "to": 310, "label": "SUCCESS" }, { "from": 313, "to": 318, "label": "PARALLEL" }, { "from": 313, "to": 319, "label": "PARALLEL" }, { "from": 318, "to": 323, "label": "EVAL with clause\nsubsetchecked([], X112, X113).\nand substitutionT29 -> [],\nT28 -> T88,\nX112 -> .(T88, []),\nT21 -> T89,\nX113 -> T89" }, { "from": 318, "to": 325, "label": "EVAL-BACKTRACK" }, { "from": 319, "to": 361, "label": "EVAL with clause\nsubsetchecked(.(X122, X123), X124, X125) :- ','(member(X122, X125), ','(not_member(X122, X124), subsetchecked(X123, .(X122, X124), X125))).\nand substitutionX122 -> T102,\nX123 -> T103,\nT29 -> .(T102, T103),\nT28 -> T100,\nX124 -> .(T100, []),\nT21 -> T101,\nX125 -> T101,\nT98 -> T102,\nT99 -> T103" }, { "from": 319, "to": 362, "label": "EVAL-BACKTRACK" }, { "from": 323, "to": 326, "label": "SUCCESS" }, { "from": 361, "to": 398, "label": "SPLIT 1" }, { "from": 361, "to": 399, "label": "SPLIT 2\nnew knowledge:\nT108 is ground\nT101 is ground\nreplacements:T102 -> T108,\nT103 -> T109" }, { "from": 398, "to": 242, "label": "INSTANCE with matching:\nT22 -> T102\nT21 -> T101" }, { "from": 399, "to": 400, "label": "SPLIT 1" }, { "from": 399, "to": 401, "label": "SPLIT 2\nnew knowledge:\nT108 is ground\nT100 is ground" }, { "from": 400, "to": 404, "label": "CASE" }, { "from": 401, "to": 475, "label": "CASE" }, { "from": 404, "to": 443, "label": "PARALLEL" }, { "from": 404, "to": 444, "label": "PARALLEL" }, { "from": 443, "to": 451, "label": "ONLY EVAL with clause\nnot_member(X154, X155) :- ','(member(X154, X155), ','(!_7, failure(a))).\nand substitutionT108 -> T130,\nX154 -> T130,\nT100 -> T131,\nX155 -> .(T131, [])" }, { "from": 444, "to": 467, "label": "ONLY EVAL with clause\nnot_member(X168, X169).\nand substitutionT108 -> T140,\nX168 -> T140,\nT100 -> T141,\nX169 -> .(T141, [])" }, { "from": 451, "to": 457, "label": "SPLIT 1" }, { "from": 451, "to": 459, "label": "SPLIT 2\nnew knowledge:\nT130 is ground\nT131 is ground" }, { "from": 457, "to": 242, "label": "INSTANCE with matching:\nT22 -> T130\nT21 -> .(T131, [])" }, { "from": 459, "to": 463, "label": "CUT" }, { "from": 463, "to": 464, "label": "CASE" }, { "from": 464, "to": 465, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 467, "to": 468, "label": "SUCCESS" }, { "from": 475, "to": 477, "label": "PARALLEL" }, { "from": 475, "to": 479, "label": "PARALLEL" }, { "from": 477, "to": 521, "label": "EVAL with clause\nsubsetchecked([], X182, X183).\nand substitutionT109 -> [],\nT108 -> T160,\nT100 -> T161,\nX182 -> .(T160, .(T161, [])),\nT101 -> T162,\nX183 -> T162" }, { "from": 477, "to": 522, "label": "EVAL-BACKTRACK" }, { "from": 479, "to": 531, "label": "EVAL with clause\nsubsetchecked(.(X192, X193), X194, X195) :- ','(member(X192, X195), ','(not_member(X192, X194), subsetchecked(X193, .(X192, X194), X195))).\nand substitutionX192 -> T178,\nX193 -> T179,\nT109 -> .(T178, T179),\nT108 -> T175,\nT100 -> T176,\nX194 -> .(T175, .(T176, [])),\nT101 -> T177,\nX195 -> T177,\nT173 -> T178,\nT174 -> T179" }, { "from": 479, "to": 532, "label": "EVAL-BACKTRACK" }, { "from": 521, "to": 523, "label": "SUCCESS" }, { "from": 531, "to": 534, "label": "SPLIT 1" }, { "from": 531, "to": 536, "label": "SPLIT 2\nnew knowledge:\nT184 is ground\nT177 is ground\nreplacements:T178 -> T184,\nT179 -> T185" }, { "from": 534, "to": 242, "label": "INSTANCE with matching:\nT22 -> T178\nT21 -> T177" }, { "from": 536, "to": 539, "label": "SPLIT 1" }, { "from": 536, "to": 540, "label": "SPLIT 2\nnew knowledge:\nT184 is ground\nT175 is ground\nT176 is ground" }, { "from": 539, "to": 541, "label": "CASE" }, { "from": 540, "to": 571, "label": "CASE" }, { "from": 541, "to": 545, "label": "PARALLEL" }, { "from": 541, "to": 546, "label": "PARALLEL" }, { "from": 545, "to": 555, "label": "ONLY EVAL with clause\nnot_member(X224, X225) :- ','(member(X224, X225), ','(!_10, failure(a))).\nand substitutionT184 -> T216,\nX224 -> T216,\nT175 -> T217,\nT176 -> T218,\nX225 -> .(T217, .(T218, []))" }, { "from": 546, "to": 566, "label": "ONLY EVAL with clause\nnot_member(X238, X239).\nand substitutionT184 -> T233,\nX238 -> T233,\nT175 -> T234,\nT176 -> T235,\nX239 -> .(T234, .(T235, []))" }, { "from": 555, "to": 558, "label": "SPLIT 1" }, { "from": 555, "to": 559, "label": "SPLIT 2\nnew knowledge:\nT216 is ground\nT217 is ground\nT218 is ground" }, { "from": 558, "to": 242, "label": "INSTANCE with matching:\nT22 -> T216\nT21 -> .(T217, .(T218, []))" }, { "from": 559, "to": 562, "label": "CUT" }, { "from": 562, "to": 563, "label": "CASE" }, { "from": 563, "to": 564, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 566, "to": 567, "label": "SUCCESS" }, { "from": 571, "to": 572, "label": "PARALLEL" }, { "from": 571, "to": 573, "label": "PARALLEL" }, { "from": 572, "to": 574, "label": "EVAL with clause\nsubsetchecked([], X252, X253).\nand substitutionT185 -> [],\nT184 -> T260,\nT175 -> T261,\nT176 -> T262,\nX252 -> .(T260, .(T261, .(T262, []))),\nT177 -> T263,\nX253 -> T263" }, { "from": 572, "to": 575, "label": "EVAL-BACKTRACK" }, { "from": 573, "to": 579, "label": "EVAL with clause\nsubsetchecked(.(X262, X263), X264, X265) :- ','(member(X262, X265), ','(not_member(X262, X264), subsetchecked(X263, .(X262, X264), X265))).\nand substitutionX262 -> T282,\nX263 -> T283,\nT185 -> .(T282, T283),\nT184 -> T278,\nT175 -> T279,\nT176 -> T280,\nX264 -> .(T278, .(T279, .(T280, []))),\nT177 -> T281,\nX265 -> T281,\nT276 -> T282,\nT277 -> T283" }, { "from": 573, "to": 580, "label": "EVAL-BACKTRACK" }, { "from": 574, "to": 576, "label": "SUCCESS" }, { "from": 579, "to": 582, "label": "SPLIT 1" }, { "from": 579, "to": 583, "label": "SPLIT 2\nnew knowledge:\nT288 is ground\nT281 is ground\nreplacements:T282 -> T288,\nT283 -> T289" }, { "from": 582, "to": 242, "label": "INSTANCE with matching:\nT22 -> T282\nT21 -> T281" }, { "from": 583, "to": 614, "label": "SPLIT 1" }, { "from": 583, "to": 615, "label": "SPLIT 2\nnew knowledge:\nT288 is ground\nT278 is ground\nT279 is ground\nT280 is ground" }, { "from": 614, "to": 618, "label": "CASE" }, { "from": 615, "to": 653, "label": "CASE" }, { "from": 618, "to": 622, "label": "PARALLEL" }, { "from": 618, "to": 623, "label": "PARALLEL" }, { "from": 622, "to": 629, "label": "ONLY EVAL with clause\nnot_member(X294, X295) :- ','(member(X294, X295), ','(!_13, failure(a))).\nand substitutionT288 -> T330,\nX294 -> T330,\nT278 -> T331,\nT279 -> T332,\nT280 -> T333,\nX295 -> .(T331, .(T332, .(T333, [])))" }, { "from": 623, "to": 643, "label": "ONLY EVAL with clause\nnot_member(X308, X309).\nand substitutionT288 -> T354,\nX308 -> T354,\nT278 -> T355,\nT279 -> T356,\nT280 -> T357,\nX309 -> .(T355, .(T356, .(T357, [])))" }, { "from": 629, "to": 633, "label": "SPLIT 1" }, { "from": 629, "to": 634, "label": "SPLIT 2\nnew knowledge:\nT330 is ground\nT331 is ground\nT332 is ground\nT333 is ground" }, { "from": 633, "to": 242, "label": "INSTANCE with matching:\nT22 -> T330\nT21 -> .(T331, .(T332, .(T333, [])))" }, { "from": 634, "to": 639, "label": "CUT" }, { "from": 639, "to": 640, "label": "CASE" }, { "from": 640, "to": 641, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 643, "to": 646, "label": "SUCCESS" }, { "from": 653, "to": 654, "label": "PARALLEL" }, { "from": 653, "to": 655, "label": "PARALLEL" }, { "from": 654, "to": 663, "label": "EVAL with clause\nsubsetchecked([], X322, X323).\nand substitutionT289 -> [],\nT288 -> T388,\nT278 -> T389,\nT279 -> T390,\nT280 -> T391,\nX322 -> .(T388, .(T389, .(T390, .(T391, [])))),\nT281 -> T392,\nX323 -> T392" }, { "from": 654, "to": 664, "label": "EVAL-BACKTRACK" }, { "from": 655, "to": 668, "label": "EVAL with clause\nsubsetchecked(.(X332, X333), X334, X335) :- ','(member(X332, X335), ','(not_member(X332, X334), subsetchecked(X333, .(X332, X334), X335))).\nand substitutionX332 -> T414,\nX333 -> T415,\nT289 -> .(T414, T415),\nT288 -> T409,\nT278 -> T410,\nT279 -> T411,\nT280 -> T412,\nX334 -> .(T409, .(T410, .(T411, .(T412, [])))),\nT281 -> T413,\nX335 -> T413,\nT407 -> T414,\nT408 -> T415" }, { "from": 655, "to": 669, "label": "EVAL-BACKTRACK" }, { "from": 663, "to": 665, "label": "SUCCESS" }, { "from": 668, "to": 673, "label": "SPLIT 1" }, { "from": 668, "to": 674, "label": "SPLIT 2\nnew knowledge:\nT420 is ground\nT413 is ground\nreplacements:T414 -> T420,\nT415 -> T421" }, { "from": 673, "to": 242, "label": "INSTANCE with matching:\nT22 -> T414\nT21 -> T413" }, { "from": 674, "to": 678, "label": "SPLIT 1" }, { "from": 674, "to": 679, "label": "SPLIT 2\nnew knowledge:\nT420 is ground\nT409 is ground\nT410 is ground\nT411 is ground\nT412 is ground" }, { "from": 678, "to": 680, "label": "CASE" }, { "from": 679, "to": 723, "label": "CASE" }, { "from": 680, "to": 696, "label": "PARALLEL" }, { "from": 680, "to": 697, "label": "PARALLEL" }, { "from": 696, "to": 706, "label": "ONLY EVAL with clause\nnot_member(X364, X365) :- ','(member(X364, X365), ','(!_16, failure(a))).\nand substitutionT420 -> T472,\nX364 -> T472,\nT409 -> T473,\nT410 -> T474,\nT411 -> T475,\nT412 -> T476,\nX365 -> .(T473, .(T474, .(T475, .(T476, []))))" }, { "from": 697, "to": 719, "label": "ONLY EVAL with clause\nnot_member(X378, X379).\nand substitutionT420 -> T503,\nX378 -> T503,\nT409 -> T504,\nT410 -> T505,\nT411 -> T506,\nT412 -> T507,\nX379 -> .(T504, .(T505, .(T506, .(T507, []))))" }, { "from": 706, "to": 710, "label": "SPLIT 1" }, { "from": 706, "to": 711, "label": "SPLIT 2\nnew knowledge:\nT472 is ground\nT473 is ground\nT474 is ground\nT475 is ground\nT476 is ground" }, { "from": 710, "to": 242, "label": "INSTANCE with matching:\nT22 -> T472\nT21 -> .(T473, .(T474, .(T475, .(T476, []))))" }, { "from": 711, "to": 713, "label": "CUT" }, { "from": 713, "to": 714, "label": "CASE" }, { "from": 714, "to": 715, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 719, "to": 720, "label": "SUCCESS" }, { "from": 723, "to": 726, "label": "PARALLEL" }, { "from": 723, "to": 727, "label": "PARALLEL" }, { "from": 726, "to": 731, "label": "EVAL with clause\nsubsetchecked([], X392, X393).\nand substitutionT421 -> [],\nT420 -> T544,\nT409 -> T545,\nT410 -> T546,\nT411 -> T547,\nT412 -> T548,\nX392 -> .(T544, .(T545, .(T546, .(T547, .(T548, []))))),\nT413 -> T549,\nX393 -> T549" }, { "from": 726, "to": 732, "label": "EVAL-BACKTRACK" }, { "from": 727, "to": 739, "label": "EVAL with clause\nsubsetchecked(.(X402, X403), X404, X405) :- ','(member(X402, X405), ','(not_member(X402, X404), subsetchecked(X403, .(X402, X404), X405))).\nand substitutionX402 -> T574,\nX403 -> T575,\nT421 -> .(T574, T575),\nT420 -> T568,\nT409 -> T569,\nT410 -> T570,\nT411 -> T571,\nT412 -> T572,\nX404 -> .(T568, .(T569, .(T570, .(T571, .(T572, []))))),\nT413 -> T573,\nX405 -> T573,\nT566 -> T574,\nT567 -> T575" }, { "from": 727, "to": 740, "label": "EVAL-BACKTRACK" }, { "from": 731, "to": 733, "label": "SUCCESS" }, { "from": 739, "to": 741, "label": "SPLIT 1" }, { "from": 739, "to": 742, "label": "SPLIT 2\nnew knowledge:\nT580 is ground\nT573 is ground\nreplacements:T574 -> T580,\nT575 -> T581" }, { "from": 741, "to": 242, "label": "INSTANCE with matching:\nT22 -> T574\nT21 -> T573" }, { "from": 742, "to": 743, "label": "SPLIT 1" }, { "from": 742, "to": 744, "label": "SPLIT 2\nnew knowledge:\nT580 is ground\nT568 is ground\nT569 is ground\nT570 is ground\nT571 is ground\nT572 is ground" }, { "from": 743, "to": 746, "label": "CASE" }, { "from": 744, "to": 795, "label": "CASE" }, { "from": 746, "to": 776, "label": "PARALLEL" }, { "from": 746, "to": 777, "label": "PARALLEL" }, { "from": 776, "to": 780, "label": "ONLY EVAL with clause\nnot_member(X434, X435) :- ','(member(X434, X435), ','(!_19, failure(a))).\nand substitutionT580 -> T642,\nX434 -> T642,\nT568 -> T643,\nT569 -> T644,\nT570 -> T645,\nT571 -> T646,\nT572 -> T647,\nX435 -> .(T643, .(T644, .(T645, .(T646, .(T647, [])))))" }, { "from": 777, "to": 791, "label": "ONLY EVAL with clause\nnot_member(X448, X449).\nand substitutionT580 -> T680,\nX448 -> T680,\nT568 -> T681,\nT569 -> T682,\nT570 -> T683,\nT571 -> T684,\nT572 -> T685,\nX449 -> .(T681, .(T682, .(T683, .(T684, .(T685, [])))))" }, { "from": 780, "to": 783, "label": "SPLIT 1" }, { "from": 780, "to": 784, "label": "SPLIT 2\nnew knowledge:\nT642 is ground\nT643 is ground\nT644 is ground\nT645 is ground\nT646 is ground\nT647 is ground" }, { "from": 783, "to": 242, "label": "INSTANCE with matching:\nT22 -> T642\nT21 -> .(T643, .(T644, .(T645, .(T646, .(T647, [])))))" }, { "from": 784, "to": 787, "label": "CUT" }, { "from": 787, "to": 788, "label": "CASE" }, { "from": 788, "to": 789, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 791, "to": 792, "label": "SUCCESS" }, { "from": 795, "to": 796, "label": "PARALLEL" }, { "from": 795, "to": 797, "label": "PARALLEL" }, { "from": 796, "to": 804, "label": "EVAL with clause\nsubsetchecked([], X462, X463).\nand substitutionT581 -> [],\nT580 -> T728,\nT568 -> T729,\nT569 -> T730,\nT570 -> T731,\nT571 -> T732,\nT572 -> T733,\nX462 -> .(T728, .(T729, .(T730, .(T731, .(T732, .(T733, [])))))),\nT573 -> T734,\nX463 -> T734" }, { "from": 796, "to": 805, "label": "EVAL-BACKTRACK" }, { "from": 797, "to": 813, "label": "EVAL with clause\nsubsetchecked(.(X472, X473), X474, X475) :- ','(member(X472, X475), ','(not_member(X472, X474), subsetchecked(X473, .(X472, X474), X475))).\nand substitutionX472 -> T762,\nX473 -> T763,\nT581 -> .(T762, T763),\nT580 -> T755,\nT568 -> T756,\nT569 -> T757,\nT570 -> T758,\nT571 -> T759,\nT572 -> T760,\nX474 -> .(T755, .(T756, .(T757, .(T758, .(T759, .(T760, [])))))),\nT573 -> T761,\nX475 -> T761,\nT753 -> T762,\nT754 -> T763" }, { "from": 797, "to": 814, "label": "EVAL-BACKTRACK" }, { "from": 804, "to": 806, "label": "SUCCESS" }, { "from": 813, "to": 815, "label": "SPLIT 1" }, { "from": 813, "to": 816, "label": "SPLIT 2\nnew knowledge:\nT768 is ground\nT761 is ground\nreplacements:T762 -> T768,\nT763 -> T769" }, { "from": 815, "to": 242, "label": "INSTANCE with matching:\nT22 -> T762\nT21 -> T761" }, { "from": 816, "to": 818, "label": "SPLIT 1" }, { "from": 816, "to": 819, "label": "SPLIT 2\nnew knowledge:\nT768 is ground\nT755 is ground\nT756 is ground\nT757 is ground\nT758 is ground\nT759 is ground\nT760 is ground" }, { "from": 818, "to": 822, "label": "CASE" }, { "from": 819, "to": 863, "label": "CASE" }, { "from": 822, "to": 826, "label": "PARALLEL" }, { "from": 822, "to": 827, "label": "PARALLEL" }, { "from": 826, "to": 837, "label": "ONLY EVAL with clause\nnot_member(X504, X505) :- ','(member(X504, X505), ','(!_22, failure(a))).\nand substitutionT768 -> T840,\nX504 -> T840,\nT755 -> T841,\nT756 -> T842,\nT757 -> T843,\nT758 -> T844,\nT759 -> T845,\nT760 -> T846,\nX505 -> .(T841, .(T842, .(T843, .(T844, .(T845, .(T846, []))))))" }, { "from": 827, "to": 858, "label": "ONLY EVAL with clause\nnot_member(X518, X519).\nand substitutionT768 -> T885,\nX518 -> T885,\nT755 -> T886,\nT756 -> T887,\nT757 -> T888,\nT758 -> T889,\nT759 -> T890,\nT760 -> T891,\nX519 -> .(T886, .(T887, .(T888, .(T889, .(T890, .(T891, []))))))" }, { "from": 837, "to": 841, "label": "SPLIT 1" }, { "from": 837, "to": 842, "label": "SPLIT 2\nnew knowledge:\nT840 is ground\nT841 is ground\nT842 is ground\nT843 is ground\nT844 is ground\nT845 is ground\nT846 is ground" }, { "from": 841, "to": 242, "label": "INSTANCE with matching:\nT22 -> T840\nT21 -> .(T841, .(T842, .(T843, .(T844, .(T845, .(T846, []))))))" }, { "from": 842, "to": 843, "label": "CUT" }, { "from": 843, "to": 844, "label": "CASE" }, { "from": 844, "to": 846, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 858, "to": 859, "label": "SUCCESS" }, { "from": 863, "to": 865, "label": "PARALLEL" }, { "from": 863, "to": 866, "label": "PARALLEL" }, { "from": 865, "to": 874, "label": "EVAL with clause\nsubsetchecked([], X532, X533).\nand substitutionT769 -> [],\nT768 -> T940,\nT755 -> T941,\nT756 -> T942,\nT757 -> T943,\nT758 -> T944,\nT759 -> T945,\nT760 -> T946,\nX532 -> .(T940, .(T941, .(T942, .(T943, .(T944, .(T945, .(T946, []))))))),\nT761 -> T947,\nX533 -> T947" }, { "from": 865, "to": 875, "label": "EVAL-BACKTRACK" }, { "from": 866, "to": 896, "label": "EVAL with clause\nsubsetchecked(.(X542, X543), X544, X545) :- ','(member(X542, X545), ','(not_member(X542, X544), subsetchecked(X543, .(X542, X544), X545))).\nand substitutionX542 -> T978,\nX543 -> T979,\nT769 -> .(T978, T979),\nT768 -> T970,\nT755 -> T971,\nT756 -> T972,\nT757 -> T973,\nT758 -> T974,\nT759 -> T975,\nT760 -> T976,\nX544 -> .(T970, .(T971, .(T972, .(T973, .(T974, .(T975, .(T976, []))))))),\nT761 -> T977,\nX545 -> T977,\nT968 -> T978,\nT969 -> T979" }, { "from": 866, "to": 897, "label": "EVAL-BACKTRACK" }, { "from": 874, "to": 876, "label": "SUCCESS" }, { "from": 896, "to": 902, "label": "GENERALIZATION\nT984 <-- .(T970, .(T971, .(T972, .(T973, .(T974, .(T975, .(T976, [])))))))\n\nNew Knowledge:\nT984 is ground" }, { "from": 902, "to": 905, "label": "SPLIT 1" }, { "from": 902, "to": 906, "label": "SPLIT 2\nnew knowledge:\nT989 is ground\nT977 is ground\nreplacements:T978 -> T989,\nT979 -> T990" }, { "from": 905, "to": 242, "label": "INSTANCE with matching:\nT22 -> T978\nT21 -> T977" }, { "from": 906, "to": 915, "label": "SPLIT 1" }, { "from": 906, "to": 916, "label": "SPLIT 2\nnew knowledge:\nT989 is ground\nT970 is ground\nT971 is ground\nT972 is ground\nT973 is ground\nT974 is ground\nT975 is ground\nT976 is ground" }, { "from": 915, "to": 926, "label": "CASE" }, { "from": 916, "to": 961, "label": "CASE" }, { "from": 926, "to": 934, "label": "PARALLEL" }, { "from": 926, "to": 935, "label": "PARALLEL" }, { "from": 934, "to": 940, "label": "ONLY EVAL with clause\nnot_member(X578, X579) :- ','(member(X578, X579), ','(!_25, failure(a))).\nand substitutionT989 -> T1071,\nX578 -> T1071,\nT970 -> T1072,\nT971 -> T1073,\nT972 -> T1074,\nT973 -> T1075,\nT974 -> T1076,\nT975 -> T1077,\nT976 -> T1078,\nX579 -> .(T1072, .(T1073, .(T1074, .(T1075, .(T1076, .(T1077, .(T1078, [])))))))" }, { "from": 935, "to": 956, "label": "ONLY EVAL with clause\nnot_member(X592, X593).\nand substitutionT989 -> T1123,\nX592 -> T1123,\nT970 -> T1124,\nT971 -> T1125,\nT972 -> T1126,\nT973 -> T1127,\nT974 -> T1128,\nT975 -> T1129,\nT976 -> T1130,\nX593 -> .(T1124, .(T1125, .(T1126, .(T1127, .(T1128, .(T1129, .(T1130, [])))))))" }, { "from": 940, "to": 945, "label": "SPLIT 1" }, { "from": 940, "to": 946, "label": "SPLIT 2\nnew knowledge:\nT1071 is ground\nT1072 is ground\nT1073 is ground\nT1074 is ground\nT1075 is ground\nT1076 is ground\nT1077 is ground\nT1078 is ground" }, { "from": 945, "to": 242, "label": "INSTANCE with matching:\nT22 -> T1071\nT21 -> .(T1072, .(T1073, .(T1074, .(T1075, .(T1076, .(T1077, .(T1078, [])))))))" }, { "from": 946, "to": 949, "label": "CUT" }, { "from": 949, "to": 950, "label": "CASE" }, { "from": 950, "to": 951, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 956, "to": 957, "label": "SUCCESS" }, { "from": 961, "to": 965, "label": "PARALLEL" }, { "from": 961, "to": 966, "label": "PARALLEL" }, { "from": 965, "to": 967, "label": "EVAL with clause\nsubsetchecked([], X606, X607).\nand substitutionT990 -> [],\nT989 -> T1149,\nT984 -> T1150,\nX606 -> .(T1149, T1150),\nT977 -> T1151,\nX607 -> T1151" }, { "from": 965, "to": 968, "label": "EVAL-BACKTRACK" }, { "from": 966, "to": 973, "label": "EVAL with clause\nsubsetchecked(.(X616, X617), X618, X619) :- ','(member(X616, X619), ','(not_member(X616, X618), subsetchecked(X617, .(X616, X618), X619))).\nand substitutionX616 -> T1167,\nX617 -> T1168,\nT990 -> .(T1167, T1168),\nT989 -> T1164,\nT984 -> T1165,\nX618 -> .(T1164, T1165),\nT977 -> T1166,\nX619 -> T1166,\nT1162 -> T1167,\nT1163 -> T1168" }, { "from": 966, "to": 974, "label": "EVAL-BACKTRACK" }, { "from": 967, "to": 969, "label": "SUCCESS" }, { "from": 973, "to": 979, "label": "SPLIT 1" }, { "from": 973, "to": 980, "label": "SPLIT 2\nnew knowledge:\nT1173 is ground\nT1166 is ground\nreplacements:T1167 -> T1173,\nT1168 -> T1174" }, { "from": 979, "to": 242, "label": "INSTANCE with matching:\nT22 -> T1167\nT21 -> T1166" }, { "from": 980, "to": 992, "label": "SPLIT 1" }, { "from": 980, "to": 993, "label": "SPLIT 2\nnew knowledge:\nT1173 is ground\nT1164 is ground\nT1165 is ground" }, { "from": 992, "to": 996, "label": "CASE" }, { "from": 993, "to": 916, "label": "INSTANCE with matching:\nT990 -> T1174\nT989 -> T1173\nT984 -> .(T1164, T1165)\nT977 -> T1166" }, { "from": 996, "to": 997, "label": "PARALLEL" }, { "from": 996, "to": 998, "label": "PARALLEL" }, { "from": 997, "to": 1000, "label": "ONLY EVAL with clause\nnot_member(X648, X649) :- ','(member(X648, X649), ','(!_28, failure(a))).\nand substitutionT1173 -> T1205,\nX648 -> T1205,\nT1164 -> T1206,\nT1165 -> T1207,\nX649 -> .(T1206, T1207)" }, { "from": 998, "to": 1014, "label": "ONLY EVAL with clause\nnot_member(X662, X663).\nand substitutionT1173 -> T1222,\nX662 -> T1222,\nT1164 -> T1223,\nT1165 -> T1224,\nX663 -> .(T1223, T1224)" }, { "from": 1000, "to": 1004, "label": "SPLIT 1" }, { "from": 1000, "to": 1005, "label": "SPLIT 2\nnew knowledge:\nT1205 is ground\nT1206 is ground\nT1207 is ground" }, { "from": 1004, "to": 242, "label": "INSTANCE with matching:\nT22 -> T1205\nT21 -> .(T1206, T1207)" }, { "from": 1005, "to": 1006, "label": "CUT" }, { "from": 1006, "to": 1007, "label": "CASE" }, { "from": 1007, "to": 1008, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 1014, "to": 1015, "label": "SUCCESS" } ], "type": "Graph" } } ---------------------------------------- (134) Obligation: Triples: memberA(X1, .(X2, X3)) :- memberA(X1, X3). subsetcheckedB(.(X1, X2), X3, X4, X5) :- memberA(X1, X5). subsetcheckedB(.(X1, X2), X3, X4, X5) :- ','(membercA(X1, X5), memberA(X1, .(X3, X4))). subsetcheckedB(.(X1, X2), X3, X4, X5) :- ','(membercA(X1, X5), ','(not_membercC(X1, X3, X4), subsetcheckedB(X2, X1, .(X3, X4), X5))). pD(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11) :- memberA(X1, X2). pD(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11) :- ','(membercA(X1, X2), memberA(X1, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, .(X9, []))))))))). pD(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11) :- ','(membercA(X1, X2), ','(not_membercE(X1, X3, X4, X5, X6, X7, X8, X9), subsetcheckedB(X10, X1, X11, X2))). subsetM(.(X1, X2), X3) :- memberA(X1, X3). subsetM(.(X1, X2), X3) :- ','(membercA(X1, X3), memberA(X1, [])). subsetM(.(X1, .(X2, X3)), X4) :- ','(membercA(X1, X4), ','(not_membercF(X1), memberA(X2, X4))). subsetM(.(X1, .(X2, X3)), X4) :- ','(membercA(X1, X4), ','(not_membercF(X1), ','(membercA(X2, X4), memberA(X2, .(X1, []))))). subsetM(.(X1, .(X2, .(X3, X4))), X5) :- ','(membercA(X1, X5), ','(not_membercF(X1), ','(membercA(X2, X5), ','(not_membercG(X2, X1), memberA(X3, X5))))). subsetM(.(X1, .(X2, .(X3, X4))), X5) :- ','(membercA(X1, X5), ','(not_membercF(X1), ','(membercA(X2, X5), ','(not_membercG(X2, X1), ','(membercA(X3, X5), memberA(X3, .(X2, .(X1, [])))))))). subsetM(.(X1, .(X2, .(X3, .(X4, X5)))), X6) :- ','(membercA(X1, X6), ','(not_membercF(X1), ','(membercA(X2, X6), ','(not_membercG(X2, X1), ','(membercA(X3, X6), ','(not_membercH(X3, X2, X1), memberA(X4, X6))))))). subsetM(.(X1, .(X2, .(X3, .(X4, X5)))), X6) :- ','(membercA(X1, X6), ','(not_membercF(X1), ','(membercA(X2, X6), ','(not_membercG(X2, X1), ','(membercA(X3, X6), ','(not_membercH(X3, X2, X1), ','(membercA(X4, X6), memberA(X4, .(X3, .(X2, .(X1, []))))))))))). subsetM(.(X1, .(X2, .(X3, .(X4, .(X5, X6))))), X7) :- ','(membercA(X1, X7), ','(not_membercF(X1), ','(membercA(X2, X7), ','(not_membercG(X2, X1), ','(membercA(X3, X7), ','(not_membercH(X3, X2, X1), ','(membercA(X4, X7), ','(not_membercI(X4, X3, X2, X1), memberA(X5, X7))))))))). subsetM(.(X1, .(X2, .(X3, .(X4, .(X5, X6))))), X7) :- ','(membercA(X1, X7), ','(not_membercF(X1), ','(membercA(X2, X7), ','(not_membercG(X2, X1), ','(membercA(X3, X7), ','(not_membercH(X3, X2, X1), ','(membercA(X4, X7), ','(not_membercI(X4, X3, X2, X1), ','(membercA(X5, X7), memberA(X5, .(X4, .(X3, .(X2, .(X1, [])))))))))))))). subsetM(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, X7)))))), X8) :- ','(membercA(X1, X8), ','(not_membercF(X1), ','(membercA(X2, X8), ','(not_membercG(X2, X1), ','(membercA(X3, X8), ','(not_membercH(X3, X2, X1), ','(membercA(X4, X8), ','(not_membercI(X4, X3, X2, X1), ','(membercA(X5, X8), ','(not_membercJ(X5, X4, X3, X2, X1), memberA(X6, X8))))))))))). subsetM(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, X7)))))), X8) :- ','(membercA(X1, X8), ','(not_membercF(X1), ','(membercA(X2, X8), ','(not_membercG(X2, X1), ','(membercA(X3, X8), ','(not_membercH(X3, X2, X1), ','(membercA(X4, X8), ','(not_membercI(X4, X3, X2, X1), ','(membercA(X5, X8), ','(not_membercJ(X5, X4, X3, X2, X1), ','(membercA(X6, X8), memberA(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))))))))))))). subsetM(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, X8))))))), X9) :- ','(membercA(X1, X9), ','(not_membercF(X1), ','(membercA(X2, X9), ','(not_membercG(X2, X1), ','(membercA(X3, X9), ','(not_membercH(X3, X2, X1), ','(membercA(X4, X9), ','(not_membercI(X4, X3, X2, X1), ','(membercA(X5, X9), ','(not_membercJ(X5, X4, X3, X2, X1), ','(membercA(X6, X9), ','(not_membercK(X6, X5, X4, X3, X2, X1), memberA(X7, X9))))))))))))). subsetM(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, X8))))))), X9) :- ','(membercA(X1, X9), ','(not_membercF(X1), ','(membercA(X2, X9), ','(not_membercG(X2, X1), ','(membercA(X3, X9), ','(not_membercH(X3, X2, X1), ','(membercA(X4, X9), ','(not_membercI(X4, X3, X2, X1), ','(membercA(X5, X9), ','(not_membercJ(X5, X4, X3, X2, X1), ','(membercA(X6, X9), ','(not_membercK(X6, X5, X4, X3, X2, X1), ','(membercA(X7, X9), memberA(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, [])))))))))))))))))))). subsetM(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) :- ','(membercA(X1, X10), ','(not_membercF(X1), ','(membercA(X2, X10), ','(not_membercG(X2, X1), ','(membercA(X3, X10), ','(not_membercH(X3, X2, X1), ','(membercA(X4, X10), ','(not_membercI(X4, X3, X2, X1), ','(membercA(X5, X10), ','(not_membercJ(X5, X4, X3, X2, X1), ','(membercA(X6, X10), ','(not_membercK(X6, X5, X4, X3, X2, X1), ','(membercA(X7, X10), ','(not_membercL(X7, X6, X5, X4, X3, X2, X1), pD(X8, X10, X7, X6, X5, X4, X3, X2, X1, X9, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, [])))))))))))))))))))))). Clauses: membercA(X1, .(X1, X2)). membercA(X1, .(X2, X3)) :- membercA(X1, X3). subsetcheckedcB([], X1, X2, X3). subsetcheckedcB(.(X1, X2), X3, X4, X5) :- ','(membercA(X1, X5), ','(not_membercC(X1, X3, X4), subsetcheckedcB(X2, X1, .(X3, X4), X5))). qcD(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11) :- ','(membercA(X1, X2), ','(not_membercE(X1, X3, X4, X5, X6, X7, X8, X9), subsetcheckedcB(X10, X1, X11, X2))). not_membercF(X1). not_membercG(X1, X2). not_membercH(X1, X2, X3). not_membercI(X1, X2, X3, X4). not_membercJ(X1, X2, X3, X4, X5). not_membercK(X1, X2, X3, X4, X5, X6). not_membercL(X1, X2, X3, X4, X5, X6, X7). not_membercE(X1, X2, X3, X4, X5, X6, X7, X8). not_membercC(X1, X2, X3). Afs: subsetM(x1, x2) = subsetM(x2) ---------------------------------------- (135) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: subsetM_in_2: (f,b) memberA_in_2: (f,b) (b,b) membercA_in_2: (f,b) pD_in_11: (f,b,b,b,b,b,b,b,b,f,b) subsetcheckedB_in_4: (f,b,b,b) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: SUBSETM_IN_AG(.(X1, X2), X3) -> U12_AG(X1, X2, X3, memberA_in_ag(X1, X3)) SUBSETM_IN_AG(.(X1, X2), X3) -> MEMBERA_IN_AG(X1, X3) MEMBERA_IN_AG(X1, .(X2, X3)) -> U1_AG(X1, X2, X3, memberA_in_ag(X1, X3)) MEMBERA_IN_AG(X1, .(X2, X3)) -> MEMBERA_IN_AG(X1, X3) SUBSETM_IN_AG(.(X1, X2), X3) -> U13_AG(X1, X2, X3, membercA_in_ag(X1, X3)) U13_AG(X1, X2, X3, membercA_out_ag(X1, X3)) -> U14_AG(X1, X2, X3, memberA_in_gg(X1, [])) U13_AG(X1, X2, X3, membercA_out_ag(X1, X3)) -> MEMBERA_IN_GG(X1, []) MEMBERA_IN_GG(X1, .(X2, X3)) -> U1_GG(X1, X2, X3, memberA_in_gg(X1, X3)) MEMBERA_IN_GG(X1, .(X2, X3)) -> MEMBERA_IN_GG(X1, X3) SUBSETM_IN_AG(.(X1, .(X2, X3)), X4) -> U15_AG(X1, X2, X3, X4, membercA_in_ag(X1, X4)) U15_AG(X1, X2, X3, X4, membercA_out_ag(X1, X4)) -> U16_AG(X1, X2, X3, X4, not_membercF_in_g(X1)) U16_AG(X1, X2, X3, X4, not_membercF_out_g(X1)) -> U17_AG(X1, X2, X3, X4, memberA_in_ag(X2, X4)) U16_AG(X1, X2, X3, X4, not_membercF_out_g(X1)) -> MEMBERA_IN_AG(X2, X4) U16_AG(X1, X2, X3, X4, not_membercF_out_g(X1)) -> U18_AG(X1, X2, X3, X4, membercA_in_ag(X2, X4)) U18_AG(X1, X2, X3, X4, membercA_out_ag(X2, X4)) -> U19_AG(X1, X2, X3, X4, memberA_in_gg(X2, .(X1, []))) U18_AG(X1, X2, X3, X4, membercA_out_ag(X2, X4)) -> MEMBERA_IN_GG(X2, .(X1, [])) SUBSETM_IN_AG(.(X1, .(X2, .(X3, X4))), X5) -> U20_AG(X1, X2, X3, X4, X5, membercA_in_ag(X1, X5)) U20_AG(X1, X2, X3, X4, X5, membercA_out_ag(X1, X5)) -> U21_AG(X1, X2, X3, X4, X5, not_membercF_in_g(X1)) U21_AG(X1, X2, X3, X4, X5, not_membercF_out_g(X1)) -> U22_AG(X1, X2, X3, X4, X5, membercA_in_ag(X2, X5)) U22_AG(X1, X2, X3, X4, X5, membercA_out_ag(X2, X5)) -> U23_AG(X1, X2, X3, X4, X5, not_membercG_in_gg(X2, X1)) U23_AG(X1, X2, X3, X4, X5, not_membercG_out_gg(X2, X1)) -> U24_AG(X1, X2, X3, X4, X5, memberA_in_ag(X3, X5)) U23_AG(X1, X2, X3, X4, X5, not_membercG_out_gg(X2, X1)) -> MEMBERA_IN_AG(X3, X5) U23_AG(X1, X2, X3, X4, X5, not_membercG_out_gg(X2, X1)) -> U25_AG(X1, X2, X3, X4, X5, membercA_in_ag(X3, X5)) U25_AG(X1, X2, X3, X4, X5, membercA_out_ag(X3, X5)) -> U26_AG(X1, X2, X3, X4, X5, memberA_in_gg(X3, .(X2, .(X1, [])))) U25_AG(X1, X2, X3, X4, X5, membercA_out_ag(X3, X5)) -> MEMBERA_IN_GG(X3, .(X2, .(X1, []))) SUBSETM_IN_AG(.(X1, .(X2, .(X3, .(X4, X5)))), X6) -> U27_AG(X1, X2, X3, X4, X5, X6, membercA_in_ag(X1, X6)) U27_AG(X1, X2, X3, X4, X5, X6, membercA_out_ag(X1, X6)) -> U28_AG(X1, X2, X3, X4, X5, X6, not_membercF_in_g(X1)) U28_AG(X1, X2, X3, X4, X5, X6, not_membercF_out_g(X1)) -> U29_AG(X1, X2, X3, X4, X5, X6, membercA_in_ag(X2, X6)) U29_AG(X1, X2, X3, X4, X5, X6, membercA_out_ag(X2, X6)) -> U30_AG(X1, X2, X3, X4, X5, X6, not_membercG_in_gg(X2, X1)) U30_AG(X1, X2, X3, X4, X5, X6, not_membercG_out_gg(X2, X1)) -> U31_AG(X1, X2, X3, X4, X5, X6, membercA_in_ag(X3, X6)) U31_AG(X1, X2, X3, X4, X5, X6, membercA_out_ag(X3, X6)) -> U32_AG(X1, X2, X3, X4, X5, X6, not_membercH_in_ggg(X3, X2, X1)) U32_AG(X1, X2, X3, X4, X5, X6, not_membercH_out_ggg(X3, X2, X1)) -> U33_AG(X1, X2, X3, X4, X5, X6, memberA_in_ag(X4, X6)) U32_AG(X1, X2, X3, X4, X5, X6, not_membercH_out_ggg(X3, X2, X1)) -> MEMBERA_IN_AG(X4, X6) U32_AG(X1, X2, X3, X4, X5, X6, not_membercH_out_ggg(X3, X2, X1)) -> U34_AG(X1, X2, X3, X4, X5, X6, membercA_in_ag(X4, X6)) U34_AG(X1, X2, X3, X4, X5, X6, membercA_out_ag(X4, X6)) -> U35_AG(X1, X2, X3, X4, X5, X6, memberA_in_gg(X4, .(X3, .(X2, .(X1, []))))) U34_AG(X1, X2, X3, X4, X5, X6, membercA_out_ag(X4, X6)) -> MEMBERA_IN_GG(X4, .(X3, .(X2, .(X1, [])))) SUBSETM_IN_AG(.(X1, .(X2, .(X3, .(X4, .(X5, X6))))), X7) -> U36_AG(X1, X2, X3, X4, X5, X6, X7, membercA_in_ag(X1, X7)) U36_AG(X1, X2, X3, X4, X5, X6, X7, membercA_out_ag(X1, X7)) -> U37_AG(X1, X2, X3, X4, X5, X6, X7, not_membercF_in_g(X1)) U37_AG(X1, X2, X3, X4, X5, X6, X7, not_membercF_out_g(X1)) -> U38_AG(X1, X2, X3, X4, X5, X6, X7, membercA_in_ag(X2, X7)) U38_AG(X1, X2, X3, X4, X5, X6, X7, membercA_out_ag(X2, X7)) -> U39_AG(X1, X2, X3, X4, X5, X6, X7, not_membercG_in_gg(X2, X1)) U39_AG(X1, X2, X3, X4, X5, X6, X7, not_membercG_out_gg(X2, X1)) -> U40_AG(X1, X2, X3, X4, X5, X6, X7, membercA_in_ag(X3, X7)) U40_AG(X1, X2, X3, X4, X5, X6, X7, membercA_out_ag(X3, X7)) -> U41_AG(X1, X2, X3, X4, X5, X6, X7, not_membercH_in_ggg(X3, X2, X1)) U41_AG(X1, X2, X3, X4, X5, X6, X7, not_membercH_out_ggg(X3, X2, X1)) -> U42_AG(X1, X2, X3, X4, X5, X6, X7, membercA_in_ag(X4, X7)) U42_AG(X1, X2, X3, X4, X5, X6, X7, membercA_out_ag(X4, X7)) -> U43_AG(X1, X2, X3, X4, X5, X6, X7, not_membercI_in_gggg(X4, X3, X2, X1)) U43_AG(X1, X2, X3, X4, X5, X6, X7, not_membercI_out_gggg(X4, X3, X2, X1)) -> U44_AG(X1, X2, X3, X4, X5, X6, X7, memberA_in_ag(X5, X7)) U43_AG(X1, X2, X3, X4, X5, X6, X7, not_membercI_out_gggg(X4, X3, X2, X1)) -> MEMBERA_IN_AG(X5, X7) U43_AG(X1, X2, X3, X4, X5, X6, X7, not_membercI_out_gggg(X4, X3, X2, X1)) -> U45_AG(X1, X2, X3, X4, X5, X6, X7, membercA_in_ag(X5, X7)) U45_AG(X1, X2, X3, X4, X5, X6, X7, membercA_out_ag(X5, X7)) -> U46_AG(X1, X2, X3, X4, X5, X6, X7, memberA_in_gg(X5, .(X4, .(X3, .(X2, .(X1, [])))))) U45_AG(X1, X2, X3, X4, X5, X6, X7, membercA_out_ag(X5, X7)) -> MEMBERA_IN_GG(X5, .(X4, .(X3, .(X2, .(X1, []))))) SUBSETM_IN_AG(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, X7)))))), X8) -> U47_AG(X1, X2, X3, X4, X5, X6, X7, X8, membercA_in_ag(X1, X8)) U47_AG(X1, X2, X3, X4, X5, X6, X7, X8, membercA_out_ag(X1, X8)) -> U48_AG(X1, X2, X3, X4, X5, X6, X7, X8, not_membercF_in_g(X1)) U48_AG(X1, X2, X3, X4, X5, X6, X7, X8, not_membercF_out_g(X1)) -> U49_AG(X1, X2, X3, X4, X5, X6, X7, X8, membercA_in_ag(X2, X8)) U49_AG(X1, X2, X3, X4, X5, X6, X7, X8, membercA_out_ag(X2, X8)) -> U50_AG(X1, X2, X3, X4, X5, X6, X7, X8, not_membercG_in_gg(X2, X1)) U50_AG(X1, X2, X3, X4, X5, X6, X7, X8, not_membercG_out_gg(X2, X1)) -> U51_AG(X1, X2, X3, X4, X5, X6, X7, X8, membercA_in_ag(X3, X8)) U51_AG(X1, X2, X3, X4, X5, X6, X7, X8, membercA_out_ag(X3, X8)) -> U52_AG(X1, X2, X3, X4, X5, X6, X7, X8, not_membercH_in_ggg(X3, X2, X1)) U52_AG(X1, X2, X3, X4, X5, X6, X7, X8, not_membercH_out_ggg(X3, X2, X1)) -> U53_AG(X1, X2, X3, X4, X5, X6, X7, X8, membercA_in_ag(X4, X8)) U53_AG(X1, X2, X3, X4, X5, X6, X7, X8, membercA_out_ag(X4, X8)) -> U54_AG(X1, X2, X3, X4, X5, X6, X7, X8, not_membercI_in_gggg(X4, X3, X2, X1)) U54_AG(X1, X2, X3, X4, X5, X6, X7, X8, not_membercI_out_gggg(X4, X3, X2, X1)) -> U55_AG(X1, X2, X3, X4, X5, X6, X7, X8, membercA_in_ag(X5, X8)) U55_AG(X1, X2, X3, X4, X5, X6, X7, X8, membercA_out_ag(X5, X8)) -> U56_AG(X1, X2, X3, X4, X5, X6, X7, X8, not_membercJ_in_ggggg(X5, X4, X3, X2, X1)) U56_AG(X1, X2, X3, X4, X5, X6, X7, X8, not_membercJ_out_ggggg(X5, X4, X3, X2, X1)) -> U57_AG(X1, X2, X3, X4, X5, X6, X7, X8, memberA_in_ag(X6, X8)) U56_AG(X1, X2, X3, X4, X5, X6, X7, X8, not_membercJ_out_ggggg(X5, X4, X3, X2, X1)) -> MEMBERA_IN_AG(X6, X8) U56_AG(X1, X2, X3, X4, X5, X6, X7, X8, not_membercJ_out_ggggg(X5, X4, X3, X2, X1)) -> U58_AG(X1, X2, X3, X4, X5, X6, X7, X8, membercA_in_ag(X6, X8)) U58_AG(X1, X2, X3, X4, X5, X6, X7, X8, membercA_out_ag(X6, X8)) -> U59_AG(X1, X2, X3, X4, X5, X6, X7, X8, memberA_in_gg(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))) U58_AG(X1, X2, X3, X4, X5, X6, X7, X8, membercA_out_ag(X6, X8)) -> MEMBERA_IN_GG(X6, .(X5, .(X4, .(X3, .(X2, .(X1, [])))))) SUBSETM_IN_AG(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, X8))))))), X9) -> U60_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, membercA_in_ag(X1, X9)) U60_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, membercA_out_ag(X1, X9)) -> U61_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, not_membercF_in_g(X1)) U61_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, not_membercF_out_g(X1)) -> U62_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, membercA_in_ag(X2, X9)) U62_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, membercA_out_ag(X2, X9)) -> U63_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, not_membercG_in_gg(X2, X1)) U63_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, not_membercG_out_gg(X2, X1)) -> U64_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, membercA_in_ag(X3, X9)) U64_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, membercA_out_ag(X3, X9)) -> U65_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, not_membercH_in_ggg(X3, X2, X1)) U65_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, not_membercH_out_ggg(X3, X2, X1)) -> U66_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, membercA_in_ag(X4, X9)) U66_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, membercA_out_ag(X4, X9)) -> U67_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, not_membercI_in_gggg(X4, X3, X2, X1)) U67_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, not_membercI_out_gggg(X4, X3, X2, X1)) -> U68_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, membercA_in_ag(X5, X9)) U68_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, membercA_out_ag(X5, X9)) -> U69_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, not_membercJ_in_ggggg(X5, X4, X3, X2, X1)) U69_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, not_membercJ_out_ggggg(X5, X4, X3, X2, X1)) -> U70_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, membercA_in_ag(X6, X9)) U70_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, membercA_out_ag(X6, X9)) -> U71_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, not_membercK_in_gggggg(X6, X5, X4, X3, X2, X1)) U71_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, not_membercK_out_gggggg(X6, X5, X4, X3, X2, X1)) -> U72_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, memberA_in_ag(X7, X9)) U71_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, not_membercK_out_gggggg(X6, X5, X4, X3, X2, X1)) -> MEMBERA_IN_AG(X7, X9) U71_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, not_membercK_out_gggggg(X6, X5, X4, X3, X2, X1)) -> U73_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, membercA_in_ag(X7, X9)) U73_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, membercA_out_ag(X7, X9)) -> U74_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, memberA_in_gg(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, [])))))))) U73_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, membercA_out_ag(X7, X9)) -> MEMBERA_IN_GG(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))) SUBSETM_IN_AG(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) -> U75_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, membercA_in_ag(X1, X10)) U75_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, membercA_out_ag(X1, X10)) -> U76_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, not_membercF_in_g(X1)) U76_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, not_membercF_out_g(X1)) -> U77_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, membercA_in_ag(X2, X10)) U77_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, membercA_out_ag(X2, X10)) -> U78_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, not_membercG_in_gg(X2, X1)) U78_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, not_membercG_out_gg(X2, X1)) -> U79_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, membercA_in_ag(X3, X10)) U79_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, membercA_out_ag(X3, X10)) -> U80_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, not_membercH_in_ggg(X3, X2, X1)) U80_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, not_membercH_out_ggg(X3, X2, X1)) -> U81_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, membercA_in_ag(X4, X10)) U81_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, membercA_out_ag(X4, X10)) -> U82_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, not_membercI_in_gggg(X4, X3, X2, X1)) U82_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, not_membercI_out_gggg(X4, X3, X2, X1)) -> U83_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, membercA_in_ag(X5, X10)) U83_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, membercA_out_ag(X5, X10)) -> U84_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, not_membercJ_in_ggggg(X5, X4, X3, X2, X1)) U84_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, not_membercJ_out_ggggg(X5, X4, X3, X2, X1)) -> U85_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, membercA_in_ag(X6, X10)) U85_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, membercA_out_ag(X6, X10)) -> U86_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, not_membercK_in_gggggg(X6, X5, X4, X3, X2, X1)) U86_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, not_membercK_out_gggggg(X6, X5, X4, X3, X2, X1)) -> U87_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, membercA_in_ag(X7, X10)) U87_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, membercA_out_ag(X7, X10)) -> U88_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, not_membercL_in_ggggggg(X7, X6, X5, X4, X3, X2, X1)) U88_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, not_membercL_out_ggggggg(X7, X6, X5, X4, X3, X2, X1)) -> U89_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, pD_in_aggggggggag(X8, X10, X7, X6, X5, X4, X3, X2, X1, X9, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))))) U88_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, not_membercL_out_ggggggg(X7, X6, X5, X4, X3, X2, X1)) -> PD_IN_AGGGGGGGGAG(X8, X10, X7, X6, X5, X4, X3, X2, X1, X9, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, [])))))))) PD_IN_AGGGGGGGGAG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11) -> U7_AGGGGGGGGAG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, memberA_in_ag(X1, X2)) PD_IN_AGGGGGGGGAG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11) -> MEMBERA_IN_AG(X1, X2) PD_IN_AGGGGGGGGAG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11) -> U8_AGGGGGGGGAG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, membercA_in_ag(X1, X2)) U8_AGGGGGGGGAG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, membercA_out_ag(X1, X2)) -> U9_AGGGGGGGGAG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, memberA_in_gg(X1, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, .(X9, []))))))))) U8_AGGGGGGGGAG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, membercA_out_ag(X1, X2)) -> MEMBERA_IN_GG(X1, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, .(X9, [])))))))) U8_AGGGGGGGGAG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, membercA_out_ag(X1, X2)) -> U10_AGGGGGGGGAG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, not_membercE_in_gggggggg(X1, X3, X4, X5, X6, X7, X8, X9)) U10_AGGGGGGGGAG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, not_membercE_out_gggggggg(X1, X3, X4, X5, X6, X7, X8, X9)) -> U11_AGGGGGGGGAG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, subsetcheckedB_in_aggg(X10, X1, X11, X2)) U10_AGGGGGGGGAG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, not_membercE_out_gggggggg(X1, X3, X4, X5, X6, X7, X8, X9)) -> SUBSETCHECKEDB_IN_AGGG(X10, X1, X11, X2) SUBSETCHECKEDB_IN_AGGG(.(X1, X2), X3, X4, X5) -> U2_AGGG(X1, X2, X3, X4, X5, memberA_in_ag(X1, X5)) SUBSETCHECKEDB_IN_AGGG(.(X1, X2), X3, X4, X5) -> MEMBERA_IN_AG(X1, X5) SUBSETCHECKEDB_IN_AGGG(.(X1, X2), X3, X4, X5) -> U3_AGGG(X1, X2, X3, X4, X5, membercA_in_ag(X1, X5)) U3_AGGG(X1, X2, X3, X4, X5, membercA_out_ag(X1, X5)) -> U4_AGGG(X1, X2, X3, X4, X5, memberA_in_gg(X1, .(X3, X4))) U3_AGGG(X1, X2, X3, X4, X5, membercA_out_ag(X1, X5)) -> MEMBERA_IN_GG(X1, .(X3, X4)) U3_AGGG(X1, X2, X3, X4, X5, membercA_out_ag(X1, X5)) -> U5_AGGG(X1, X2, X3, X4, X5, not_membercC_in_ggg(X1, X3, X4)) U5_AGGG(X1, X2, X3, X4, X5, not_membercC_out_ggg(X1, X3, X4)) -> U6_AGGG(X1, X2, X3, X4, X5, subsetcheckedB_in_aggg(X2, X1, .(X3, X4), X5)) U5_AGGG(X1, X2, X3, X4, X5, not_membercC_out_ggg(X1, X3, X4)) -> SUBSETCHECKEDB_IN_AGGG(X2, X1, .(X3, X4), X5) The TRS R consists of the following rules: membercA_in_ag(X1, .(X1, X2)) -> membercA_out_ag(X1, .(X1, X2)) membercA_in_ag(X1, .(X2, X3)) -> U91_ag(X1, X2, X3, membercA_in_ag(X1, X3)) U91_ag(X1, X2, X3, membercA_out_ag(X1, X3)) -> membercA_out_ag(X1, .(X2, X3)) not_membercF_in_g(X1) -> not_membercF_out_g(X1) not_membercG_in_gg(X1, X2) -> not_membercG_out_gg(X1, X2) not_membercH_in_ggg(X1, X2, X3) -> not_membercH_out_ggg(X1, X2, X3) not_membercI_in_gggg(X1, X2, X3, X4) -> not_membercI_out_gggg(X1, X2, X3, X4) not_membercJ_in_ggggg(X1, X2, X3, X4, X5) -> not_membercJ_out_ggggg(X1, X2, X3, X4, X5) not_membercK_in_gggggg(X1, X2, X3, X4, X5, X6) -> not_membercK_out_gggggg(X1, X2, X3, X4, X5, X6) not_membercL_in_ggggggg(X1, X2, X3, X4, X5, X6, X7) -> not_membercL_out_ggggggg(X1, X2, X3, X4, X5, X6, X7) not_membercE_in_gggggggg(X1, X2, X3, X4, X5, X6, X7, X8) -> not_membercE_out_gggggggg(X1, X2, X3, X4, X5, X6, X7, X8) not_membercC_in_ggg(X1, X2, X3) -> not_membercC_out_ggg(X1, X2, X3) The argument filtering Pi contains the following mapping: memberA_in_ag(x1, x2) = memberA_in_ag(x2) .(x1, x2) = .(x1, x2) membercA_in_ag(x1, x2) = membercA_in_ag(x2) membercA_out_ag(x1, x2) = membercA_out_ag(x1, x2) U91_ag(x1, x2, x3, x4) = U91_ag(x2, x3, x4) memberA_in_gg(x1, x2) = memberA_in_gg(x1, x2) [] = [] not_membercF_in_g(x1) = not_membercF_in_g(x1) not_membercF_out_g(x1) = not_membercF_out_g(x1) not_membercG_in_gg(x1, x2) = not_membercG_in_gg(x1, x2) not_membercG_out_gg(x1, x2) = not_membercG_out_gg(x1, x2) not_membercH_in_ggg(x1, x2, x3) = not_membercH_in_ggg(x1, x2, x3) not_membercH_out_ggg(x1, x2, x3) = not_membercH_out_ggg(x1, x2, x3) not_membercI_in_gggg(x1, x2, x3, x4) = not_membercI_in_gggg(x1, x2, x3, x4) not_membercI_out_gggg(x1, x2, x3, x4) = not_membercI_out_gggg(x1, x2, x3, x4) not_membercJ_in_ggggg(x1, x2, x3, x4, x5) = not_membercJ_in_ggggg(x1, x2, x3, x4, x5) not_membercJ_out_ggggg(x1, x2, x3, x4, x5) = not_membercJ_out_ggggg(x1, x2, x3, x4, x5) not_membercK_in_gggggg(x1, x2, x3, x4, x5, x6) = not_membercK_in_gggggg(x1, x2, x3, x4, x5, x6) not_membercK_out_gggggg(x1, x2, x3, x4, x5, x6) = not_membercK_out_gggggg(x1, x2, x3, x4, x5, x6) not_membercL_in_ggggggg(x1, x2, x3, x4, x5, x6, x7) = not_membercL_in_ggggggg(x1, x2, x3, x4, x5, x6, x7) not_membercL_out_ggggggg(x1, x2, x3, x4, x5, x6, x7) = not_membercL_out_ggggggg(x1, x2, x3, x4, x5, x6, x7) pD_in_aggggggggag(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = pD_in_aggggggggag(x2, x3, x4, x5, x6, x7, x8, x9, x11) not_membercE_in_gggggggg(x1, x2, x3, x4, x5, x6, x7, x8) = not_membercE_in_gggggggg(x1, x2, x3, x4, x5, x6, x7, x8) not_membercE_out_gggggggg(x1, x2, x3, x4, x5, x6, x7, x8) = not_membercE_out_gggggggg(x1, x2, x3, x4, x5, x6, x7, x8) subsetcheckedB_in_aggg(x1, x2, x3, x4) = subsetcheckedB_in_aggg(x2, x3, x4) not_membercC_in_ggg(x1, x2, x3) = not_membercC_in_ggg(x1, x2, x3) not_membercC_out_ggg(x1, x2, x3) = not_membercC_out_ggg(x1, x2, x3) SUBSETM_IN_AG(x1, x2) = SUBSETM_IN_AG(x2) U12_AG(x1, x2, x3, x4) = U12_AG(x3, x4) MEMBERA_IN_AG(x1, x2) = MEMBERA_IN_AG(x2) U1_AG(x1, x2, x3, x4) = U1_AG(x2, x3, x4) U13_AG(x1, x2, x3, x4) = U13_AG(x3, x4) U14_AG(x1, x2, x3, x4) = U14_AG(x3, x4) MEMBERA_IN_GG(x1, x2) = MEMBERA_IN_GG(x1, x2) U1_GG(x1, x2, x3, x4) = U1_GG(x1, x2, x3, x4) U15_AG(x1, x2, x3, x4, x5) = U15_AG(x4, x5) U16_AG(x1, x2, x3, x4, x5) = U16_AG(x4, x5) U17_AG(x1, x2, x3, x4, x5) = U17_AG(x4, x5) U18_AG(x1, x2, x3, x4, x5) = U18_AG(x1, x4, x5) U19_AG(x1, x2, x3, x4, x5) = U19_AG(x4, x5) U20_AG(x1, x2, x3, x4, x5, x6) = U20_AG(x5, x6) U21_AG(x1, x2, x3, x4, x5, x6) = U21_AG(x1, x5, x6) U22_AG(x1, x2, x3, x4, x5, x6) = U22_AG(x1, x5, x6) U23_AG(x1, x2, x3, x4, x5, x6) = U23_AG(x5, x6) U24_AG(x1, x2, x3, x4, x5, x6) = U24_AG(x5, x6) U25_AG(x1, x2, x3, x4, x5, x6) = U25_AG(x1, x2, x5, x6) U26_AG(x1, x2, x3, x4, x5, x6) = U26_AG(x5, x6) U27_AG(x1, x2, x3, x4, x5, x6, x7) = U27_AG(x6, x7) U28_AG(x1, x2, x3, x4, x5, x6, x7) = U28_AG(x1, x6, x7) U29_AG(x1, x2, x3, x4, x5, x6, x7) = U29_AG(x1, x6, x7) U30_AG(x1, x2, x3, x4, x5, x6, x7) = U30_AG(x1, x2, x6, x7) U31_AG(x1, x2, x3, x4, x5, x6, x7) = U31_AG(x1, x2, x6, x7) U32_AG(x1, x2, x3, x4, x5, x6, x7) = U32_AG(x6, x7) U33_AG(x1, x2, x3, x4, x5, x6, x7) = U33_AG(x6, x7) U34_AG(x1, x2, x3, x4, x5, x6, x7) = U34_AG(x1, x2, x3, x6, x7) U35_AG(x1, x2, x3, x4, x5, x6, x7) = U35_AG(x6, x7) U36_AG(x1, x2, x3, x4, x5, x6, x7, x8) = U36_AG(x7, x8) U37_AG(x1, x2, x3, x4, x5, x6, x7, x8) = U37_AG(x1, x7, x8) U38_AG(x1, x2, x3, x4, x5, x6, x7, x8) = U38_AG(x1, x7, x8) U39_AG(x1, x2, x3, x4, x5, x6, x7, x8) = U39_AG(x1, x2, x7, x8) U40_AG(x1, x2, x3, x4, x5, x6, x7, x8) = U40_AG(x1, x2, x7, x8) U41_AG(x1, x2, x3, x4, x5, x6, x7, x8) = U41_AG(x1, x2, x3, x7, x8) U42_AG(x1, x2, x3, x4, x5, x6, x7, x8) = U42_AG(x1, x2, x3, x7, x8) U43_AG(x1, x2, x3, x4, x5, x6, x7, x8) = U43_AG(x7, x8) U44_AG(x1, x2, x3, x4, x5, x6, x7, x8) = U44_AG(x7, x8) U45_AG(x1, x2, x3, x4, x5, x6, x7, x8) = U45_AG(x1, x2, x3, x4, x7, x8) U46_AG(x1, x2, x3, x4, x5, x6, x7, x8) = U46_AG(x7, x8) U47_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U47_AG(x8, x9) U48_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U48_AG(x1, x8, x9) U49_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U49_AG(x1, x8, x9) U50_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U50_AG(x1, x2, x8, x9) U51_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U51_AG(x1, x2, x8, x9) U52_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U52_AG(x1, x2, x3, x8, x9) U53_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U53_AG(x1, x2, x3, x8, x9) U54_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U54_AG(x1, x2, x3, x4, x8, x9) U55_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U55_AG(x1, x2, x3, x4, x8, x9) U56_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U56_AG(x8, x9) U57_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U57_AG(x8, x9) U58_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U58_AG(x1, x2, x3, x4, x5, x8, x9) U59_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U59_AG(x8, x9) U60_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U60_AG(x9, x10) U61_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U61_AG(x1, x9, x10) U62_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U62_AG(x1, x9, x10) U63_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U63_AG(x1, x2, x9, x10) U64_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U64_AG(x1, x2, x9, x10) U65_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U65_AG(x1, x2, x3, x9, x10) U66_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U66_AG(x1, x2, x3, x9, x10) U67_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U67_AG(x1, x2, x3, x4, x9, x10) U68_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U68_AG(x1, x2, x3, x4, x9, x10) U69_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U69_AG(x1, x2, x3, x4, x5, x9, x10) U70_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U70_AG(x1, x2, x3, x4, x5, x9, x10) U71_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U71_AG(x9, x10) U72_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U72_AG(x9, x10) U73_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U73_AG(x1, x2, x3, x4, x5, x6, x9, x10) U74_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U74_AG(x9, x10) U75_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U75_AG(x10, x11) U76_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U76_AG(x1, x10, x11) U77_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U77_AG(x1, x10, x11) U78_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U78_AG(x1, x2, x10, x11) U79_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U79_AG(x1, x2, x10, x11) U80_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U80_AG(x1, x2, x3, x10, x11) U81_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U81_AG(x1, x2, x3, x10, x11) U82_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U82_AG(x1, x2, x3, x4, x10, x11) U83_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U83_AG(x1, x2, x3, x4, x10, x11) U84_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U84_AG(x1, x2, x3, x4, x5, x10, x11) U85_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U85_AG(x1, x2, x3, x4, x5, x10, x11) U86_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U86_AG(x1, x2, x3, x4, x5, x6, x10, x11) U87_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U87_AG(x1, x2, x3, x4, x5, x6, x10, x11) U88_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U88_AG(x1, x2, x3, x4, x5, x6, x7, x10, x11) U89_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U89_AG(x10, x11) PD_IN_AGGGGGGGGAG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = PD_IN_AGGGGGGGGAG(x2, x3, x4, x5, x6, x7, x8, x9, x11) U7_AGGGGGGGGAG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U7_AGGGGGGGGAG(x2, x3, x4, x5, x6, x7, x8, x9, x11, x12) U8_AGGGGGGGGAG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U8_AGGGGGGGGAG(x2, x3, x4, x5, x6, x7, x8, x9, x11, x12) U9_AGGGGGGGGAG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U9_AGGGGGGGGAG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x11, x12) U10_AGGGGGGGGAG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U10_AGGGGGGGGAG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x11, x12) U11_AGGGGGGGGAG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U11_AGGGGGGGGAG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x11, x12) SUBSETCHECKEDB_IN_AGGG(x1, x2, x3, x4) = SUBSETCHECKEDB_IN_AGGG(x2, x3, x4) U2_AGGG(x1, x2, x3, x4, x5, x6) = U2_AGGG(x3, x4, x5, x6) U3_AGGG(x1, x2, x3, x4, x5, x6) = U3_AGGG(x3, x4, x5, x6) U4_AGGG(x1, x2, x3, x4, x5, x6) = U4_AGGG(x3, x4, x5, x6) U5_AGGG(x1, x2, x3, x4, x5, x6) = U5_AGGG(x1, x3, x4, x5, x6) U6_AGGG(x1, x2, x3, x4, x5, x6) = U6_AGGG(x3, x4, x5, x6) We have to consider all (P,R,Pi)-chains Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES ---------------------------------------- (136) Obligation: Pi DP problem: The TRS P consists of the following rules: SUBSETM_IN_AG(.(X1, X2), X3) -> U12_AG(X1, X2, X3, memberA_in_ag(X1, X3)) SUBSETM_IN_AG(.(X1, X2), X3) -> MEMBERA_IN_AG(X1, X3) MEMBERA_IN_AG(X1, .(X2, X3)) -> U1_AG(X1, X2, X3, memberA_in_ag(X1, X3)) MEMBERA_IN_AG(X1, .(X2, X3)) -> MEMBERA_IN_AG(X1, X3) SUBSETM_IN_AG(.(X1, X2), X3) -> U13_AG(X1, X2, X3, membercA_in_ag(X1, X3)) U13_AG(X1, X2, X3, membercA_out_ag(X1, X3)) -> U14_AG(X1, X2, X3, memberA_in_gg(X1, [])) U13_AG(X1, X2, X3, membercA_out_ag(X1, X3)) -> MEMBERA_IN_GG(X1, []) MEMBERA_IN_GG(X1, .(X2, X3)) -> U1_GG(X1, X2, X3, memberA_in_gg(X1, X3)) MEMBERA_IN_GG(X1, .(X2, X3)) -> MEMBERA_IN_GG(X1, X3) SUBSETM_IN_AG(.(X1, .(X2, X3)), X4) -> U15_AG(X1, X2, X3, X4, membercA_in_ag(X1, X4)) U15_AG(X1, X2, X3, X4, membercA_out_ag(X1, X4)) -> U16_AG(X1, X2, X3, X4, not_membercF_in_g(X1)) U16_AG(X1, X2, X3, X4, not_membercF_out_g(X1)) -> U17_AG(X1, X2, X3, X4, memberA_in_ag(X2, X4)) U16_AG(X1, X2, X3, X4, not_membercF_out_g(X1)) -> MEMBERA_IN_AG(X2, X4) U16_AG(X1, X2, X3, X4, not_membercF_out_g(X1)) -> U18_AG(X1, X2, X3, X4, membercA_in_ag(X2, X4)) U18_AG(X1, X2, X3, X4, membercA_out_ag(X2, X4)) -> U19_AG(X1, X2, X3, X4, memberA_in_gg(X2, .(X1, []))) U18_AG(X1, X2, X3, X4, membercA_out_ag(X2, X4)) -> MEMBERA_IN_GG(X2, .(X1, [])) SUBSETM_IN_AG(.(X1, .(X2, .(X3, X4))), X5) -> U20_AG(X1, X2, X3, X4, X5, membercA_in_ag(X1, X5)) U20_AG(X1, X2, X3, X4, X5, membercA_out_ag(X1, X5)) -> U21_AG(X1, X2, X3, X4, X5, not_membercF_in_g(X1)) U21_AG(X1, X2, X3, X4, X5, not_membercF_out_g(X1)) -> U22_AG(X1, X2, X3, X4, X5, membercA_in_ag(X2, X5)) U22_AG(X1, X2, X3, X4, X5, membercA_out_ag(X2, X5)) -> U23_AG(X1, X2, X3, X4, X5, not_membercG_in_gg(X2, X1)) U23_AG(X1, X2, X3, X4, X5, not_membercG_out_gg(X2, X1)) -> U24_AG(X1, X2, X3, X4, X5, memberA_in_ag(X3, X5)) U23_AG(X1, X2, X3, X4, X5, not_membercG_out_gg(X2, X1)) -> MEMBERA_IN_AG(X3, X5) U23_AG(X1, X2, X3, X4, X5, not_membercG_out_gg(X2, X1)) -> U25_AG(X1, X2, X3, X4, X5, membercA_in_ag(X3, X5)) U25_AG(X1, X2, X3, X4, X5, membercA_out_ag(X3, X5)) -> U26_AG(X1, X2, X3, X4, X5, memberA_in_gg(X3, .(X2, .(X1, [])))) U25_AG(X1, X2, X3, X4, X5, membercA_out_ag(X3, X5)) -> MEMBERA_IN_GG(X3, .(X2, .(X1, []))) SUBSETM_IN_AG(.(X1, .(X2, .(X3, .(X4, X5)))), X6) -> U27_AG(X1, X2, X3, X4, X5, X6, membercA_in_ag(X1, X6)) U27_AG(X1, X2, X3, X4, X5, X6, membercA_out_ag(X1, X6)) -> U28_AG(X1, X2, X3, X4, X5, X6, not_membercF_in_g(X1)) U28_AG(X1, X2, X3, X4, X5, X6, not_membercF_out_g(X1)) -> U29_AG(X1, X2, X3, X4, X5, X6, membercA_in_ag(X2, X6)) U29_AG(X1, X2, X3, X4, X5, X6, membercA_out_ag(X2, X6)) -> U30_AG(X1, X2, X3, X4, X5, X6, not_membercG_in_gg(X2, X1)) U30_AG(X1, X2, X3, X4, X5, X6, not_membercG_out_gg(X2, X1)) -> U31_AG(X1, X2, X3, X4, X5, X6, membercA_in_ag(X3, X6)) U31_AG(X1, X2, X3, X4, X5, X6, membercA_out_ag(X3, X6)) -> U32_AG(X1, X2, X3, X4, X5, X6, not_membercH_in_ggg(X3, X2, X1)) U32_AG(X1, X2, X3, X4, X5, X6, not_membercH_out_ggg(X3, X2, X1)) -> U33_AG(X1, X2, X3, X4, X5, X6, memberA_in_ag(X4, X6)) U32_AG(X1, X2, X3, X4, X5, X6, not_membercH_out_ggg(X3, X2, X1)) -> MEMBERA_IN_AG(X4, X6) U32_AG(X1, X2, X3, X4, X5, X6, not_membercH_out_ggg(X3, X2, X1)) -> U34_AG(X1, X2, X3, X4, X5, X6, membercA_in_ag(X4, X6)) U34_AG(X1, X2, X3, X4, X5, X6, membercA_out_ag(X4, X6)) -> U35_AG(X1, X2, X3, X4, X5, X6, memberA_in_gg(X4, .(X3, .(X2, .(X1, []))))) U34_AG(X1, X2, X3, X4, X5, X6, membercA_out_ag(X4, X6)) -> MEMBERA_IN_GG(X4, .(X3, .(X2, .(X1, [])))) SUBSETM_IN_AG(.(X1, .(X2, .(X3, .(X4, .(X5, X6))))), X7) -> U36_AG(X1, X2, X3, X4, X5, X6, X7, membercA_in_ag(X1, X7)) U36_AG(X1, X2, X3, X4, X5, X6, X7, membercA_out_ag(X1, X7)) -> U37_AG(X1, X2, X3, X4, X5, X6, X7, not_membercF_in_g(X1)) U37_AG(X1, X2, X3, X4, X5, X6, X7, not_membercF_out_g(X1)) -> U38_AG(X1, X2, X3, X4, X5, X6, X7, membercA_in_ag(X2, X7)) U38_AG(X1, X2, X3, X4, X5, X6, X7, membercA_out_ag(X2, X7)) -> U39_AG(X1, X2, X3, X4, X5, X6, X7, not_membercG_in_gg(X2, X1)) U39_AG(X1, X2, X3, X4, X5, X6, X7, not_membercG_out_gg(X2, X1)) -> U40_AG(X1, X2, X3, X4, X5, X6, X7, membercA_in_ag(X3, X7)) U40_AG(X1, X2, X3, X4, X5, X6, X7, membercA_out_ag(X3, X7)) -> U41_AG(X1, X2, X3, X4, X5, X6, X7, not_membercH_in_ggg(X3, X2, X1)) U41_AG(X1, X2, X3, X4, X5, X6, X7, not_membercH_out_ggg(X3, X2, X1)) -> U42_AG(X1, X2, X3, X4, X5, X6, X7, membercA_in_ag(X4, X7)) U42_AG(X1, X2, X3, X4, X5, X6, X7, membercA_out_ag(X4, X7)) -> U43_AG(X1, X2, X3, X4, X5, X6, X7, not_membercI_in_gggg(X4, X3, X2, X1)) U43_AG(X1, X2, X3, X4, X5, X6, X7, not_membercI_out_gggg(X4, X3, X2, X1)) -> U44_AG(X1, X2, X3, X4, X5, X6, X7, memberA_in_ag(X5, X7)) U43_AG(X1, X2, X3, X4, X5, X6, X7, not_membercI_out_gggg(X4, X3, X2, X1)) -> MEMBERA_IN_AG(X5, X7) U43_AG(X1, X2, X3, X4, X5, X6, X7, not_membercI_out_gggg(X4, X3, X2, X1)) -> U45_AG(X1, X2, X3, X4, X5, X6, X7, membercA_in_ag(X5, X7)) U45_AG(X1, X2, X3, X4, X5, X6, X7, membercA_out_ag(X5, X7)) -> U46_AG(X1, X2, X3, X4, X5, X6, X7, memberA_in_gg(X5, .(X4, .(X3, .(X2, .(X1, [])))))) U45_AG(X1, X2, X3, X4, X5, X6, X7, membercA_out_ag(X5, X7)) -> MEMBERA_IN_GG(X5, .(X4, .(X3, .(X2, .(X1, []))))) SUBSETM_IN_AG(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, X7)))))), X8) -> U47_AG(X1, X2, X3, X4, X5, X6, X7, X8, membercA_in_ag(X1, X8)) U47_AG(X1, X2, X3, X4, X5, X6, X7, X8, membercA_out_ag(X1, X8)) -> U48_AG(X1, X2, X3, X4, X5, X6, X7, X8, not_membercF_in_g(X1)) U48_AG(X1, X2, X3, X4, X5, X6, X7, X8, not_membercF_out_g(X1)) -> U49_AG(X1, X2, X3, X4, X5, X6, X7, X8, membercA_in_ag(X2, X8)) U49_AG(X1, X2, X3, X4, X5, X6, X7, X8, membercA_out_ag(X2, X8)) -> U50_AG(X1, X2, X3, X4, X5, X6, X7, X8, not_membercG_in_gg(X2, X1)) U50_AG(X1, X2, X3, X4, X5, X6, X7, X8, not_membercG_out_gg(X2, X1)) -> U51_AG(X1, X2, X3, X4, X5, X6, X7, X8, membercA_in_ag(X3, X8)) U51_AG(X1, X2, X3, X4, X5, X6, X7, X8, membercA_out_ag(X3, X8)) -> U52_AG(X1, X2, X3, X4, X5, X6, X7, X8, not_membercH_in_ggg(X3, X2, X1)) U52_AG(X1, X2, X3, X4, X5, X6, X7, X8, not_membercH_out_ggg(X3, X2, X1)) -> U53_AG(X1, X2, X3, X4, X5, X6, X7, X8, membercA_in_ag(X4, X8)) U53_AG(X1, X2, X3, X4, X5, X6, X7, X8, membercA_out_ag(X4, X8)) -> U54_AG(X1, X2, X3, X4, X5, X6, X7, X8, not_membercI_in_gggg(X4, X3, X2, X1)) U54_AG(X1, X2, X3, X4, X5, X6, X7, X8, not_membercI_out_gggg(X4, X3, X2, X1)) -> U55_AG(X1, X2, X3, X4, X5, X6, X7, X8, membercA_in_ag(X5, X8)) U55_AG(X1, X2, X3, X4, X5, X6, X7, X8, membercA_out_ag(X5, X8)) -> U56_AG(X1, X2, X3, X4, X5, X6, X7, X8, not_membercJ_in_ggggg(X5, X4, X3, X2, X1)) U56_AG(X1, X2, X3, X4, X5, X6, X7, X8, not_membercJ_out_ggggg(X5, X4, X3, X2, X1)) -> U57_AG(X1, X2, X3, X4, X5, X6, X7, X8, memberA_in_ag(X6, X8)) U56_AG(X1, X2, X3, X4, X5, X6, X7, X8, not_membercJ_out_ggggg(X5, X4, X3, X2, X1)) -> MEMBERA_IN_AG(X6, X8) U56_AG(X1, X2, X3, X4, X5, X6, X7, X8, not_membercJ_out_ggggg(X5, X4, X3, X2, X1)) -> U58_AG(X1, X2, X3, X4, X5, X6, X7, X8, membercA_in_ag(X6, X8)) U58_AG(X1, X2, X3, X4, X5, X6, X7, X8, membercA_out_ag(X6, X8)) -> U59_AG(X1, X2, X3, X4, X5, X6, X7, X8, memberA_in_gg(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))) U58_AG(X1, X2, X3, X4, X5, X6, X7, X8, membercA_out_ag(X6, X8)) -> MEMBERA_IN_GG(X6, .(X5, .(X4, .(X3, .(X2, .(X1, [])))))) SUBSETM_IN_AG(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, X8))))))), X9) -> U60_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, membercA_in_ag(X1, X9)) U60_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, membercA_out_ag(X1, X9)) -> U61_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, not_membercF_in_g(X1)) U61_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, not_membercF_out_g(X1)) -> U62_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, membercA_in_ag(X2, X9)) U62_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, membercA_out_ag(X2, X9)) -> U63_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, not_membercG_in_gg(X2, X1)) U63_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, not_membercG_out_gg(X2, X1)) -> U64_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, membercA_in_ag(X3, X9)) U64_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, membercA_out_ag(X3, X9)) -> U65_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, not_membercH_in_ggg(X3, X2, X1)) U65_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, not_membercH_out_ggg(X3, X2, X1)) -> U66_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, membercA_in_ag(X4, X9)) U66_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, membercA_out_ag(X4, X9)) -> U67_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, not_membercI_in_gggg(X4, X3, X2, X1)) U67_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, not_membercI_out_gggg(X4, X3, X2, X1)) -> U68_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, membercA_in_ag(X5, X9)) U68_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, membercA_out_ag(X5, X9)) -> U69_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, not_membercJ_in_ggggg(X5, X4, X3, X2, X1)) U69_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, not_membercJ_out_ggggg(X5, X4, X3, X2, X1)) -> U70_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, membercA_in_ag(X6, X9)) U70_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, membercA_out_ag(X6, X9)) -> U71_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, not_membercK_in_gggggg(X6, X5, X4, X3, X2, X1)) U71_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, not_membercK_out_gggggg(X6, X5, X4, X3, X2, X1)) -> U72_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, memberA_in_ag(X7, X9)) U71_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, not_membercK_out_gggggg(X6, X5, X4, X3, X2, X1)) -> MEMBERA_IN_AG(X7, X9) U71_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, not_membercK_out_gggggg(X6, X5, X4, X3, X2, X1)) -> U73_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, membercA_in_ag(X7, X9)) U73_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, membercA_out_ag(X7, X9)) -> U74_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, memberA_in_gg(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, [])))))))) U73_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, membercA_out_ag(X7, X9)) -> MEMBERA_IN_GG(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))) SUBSETM_IN_AG(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) -> U75_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, membercA_in_ag(X1, X10)) U75_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, membercA_out_ag(X1, X10)) -> U76_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, not_membercF_in_g(X1)) U76_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, not_membercF_out_g(X1)) -> U77_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, membercA_in_ag(X2, X10)) U77_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, membercA_out_ag(X2, X10)) -> U78_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, not_membercG_in_gg(X2, X1)) U78_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, not_membercG_out_gg(X2, X1)) -> U79_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, membercA_in_ag(X3, X10)) U79_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, membercA_out_ag(X3, X10)) -> U80_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, not_membercH_in_ggg(X3, X2, X1)) U80_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, not_membercH_out_ggg(X3, X2, X1)) -> U81_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, membercA_in_ag(X4, X10)) U81_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, membercA_out_ag(X4, X10)) -> U82_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, not_membercI_in_gggg(X4, X3, X2, X1)) U82_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, not_membercI_out_gggg(X4, X3, X2, X1)) -> U83_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, membercA_in_ag(X5, X10)) U83_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, membercA_out_ag(X5, X10)) -> U84_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, not_membercJ_in_ggggg(X5, X4, X3, X2, X1)) U84_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, not_membercJ_out_ggggg(X5, X4, X3, X2, X1)) -> U85_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, membercA_in_ag(X6, X10)) U85_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, membercA_out_ag(X6, X10)) -> U86_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, not_membercK_in_gggggg(X6, X5, X4, X3, X2, X1)) U86_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, not_membercK_out_gggggg(X6, X5, X4, X3, X2, X1)) -> U87_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, membercA_in_ag(X7, X10)) U87_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, membercA_out_ag(X7, X10)) -> U88_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, not_membercL_in_ggggggg(X7, X6, X5, X4, X3, X2, X1)) U88_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, not_membercL_out_ggggggg(X7, X6, X5, X4, X3, X2, X1)) -> U89_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, pD_in_aggggggggag(X8, X10, X7, X6, X5, X4, X3, X2, X1, X9, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))))) U88_AG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, not_membercL_out_ggggggg(X7, X6, X5, X4, X3, X2, X1)) -> PD_IN_AGGGGGGGGAG(X8, X10, X7, X6, X5, X4, X3, X2, X1, X9, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, [])))))))) PD_IN_AGGGGGGGGAG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11) -> U7_AGGGGGGGGAG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, memberA_in_ag(X1, X2)) PD_IN_AGGGGGGGGAG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11) -> MEMBERA_IN_AG(X1, X2) PD_IN_AGGGGGGGGAG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11) -> U8_AGGGGGGGGAG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, membercA_in_ag(X1, X2)) U8_AGGGGGGGGAG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, membercA_out_ag(X1, X2)) -> U9_AGGGGGGGGAG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, memberA_in_gg(X1, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, .(X9, []))))))))) U8_AGGGGGGGGAG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, membercA_out_ag(X1, X2)) -> MEMBERA_IN_GG(X1, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, .(X9, [])))))))) U8_AGGGGGGGGAG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, membercA_out_ag(X1, X2)) -> U10_AGGGGGGGGAG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, not_membercE_in_gggggggg(X1, X3, X4, X5, X6, X7, X8, X9)) U10_AGGGGGGGGAG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, not_membercE_out_gggggggg(X1, X3, X4, X5, X6, X7, X8, X9)) -> U11_AGGGGGGGGAG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, subsetcheckedB_in_aggg(X10, X1, X11, X2)) U10_AGGGGGGGGAG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, not_membercE_out_gggggggg(X1, X3, X4, X5, X6, X7, X8, X9)) -> SUBSETCHECKEDB_IN_AGGG(X10, X1, X11, X2) SUBSETCHECKEDB_IN_AGGG(.(X1, X2), X3, X4, X5) -> U2_AGGG(X1, X2, X3, X4, X5, memberA_in_ag(X1, X5)) SUBSETCHECKEDB_IN_AGGG(.(X1, X2), X3, X4, X5) -> MEMBERA_IN_AG(X1, X5) SUBSETCHECKEDB_IN_AGGG(.(X1, X2), X3, X4, X5) -> U3_AGGG(X1, X2, X3, X4, X5, membercA_in_ag(X1, X5)) U3_AGGG(X1, X2, X3, X4, X5, membercA_out_ag(X1, X5)) -> U4_AGGG(X1, X2, X3, X4, X5, memberA_in_gg(X1, .(X3, X4))) U3_AGGG(X1, X2, X3, X4, X5, membercA_out_ag(X1, X5)) -> MEMBERA_IN_GG(X1, .(X3, X4)) U3_AGGG(X1, X2, X3, X4, X5, membercA_out_ag(X1, X5)) -> U5_AGGG(X1, X2, X3, X4, X5, not_membercC_in_ggg(X1, X3, X4)) U5_AGGG(X1, X2, X3, X4, X5, not_membercC_out_ggg(X1, X3, X4)) -> U6_AGGG(X1, X2, X3, X4, X5, subsetcheckedB_in_aggg(X2, X1, .(X3, X4), X5)) U5_AGGG(X1, X2, X3, X4, X5, not_membercC_out_ggg(X1, X3, X4)) -> SUBSETCHECKEDB_IN_AGGG(X2, X1, .(X3, X4), X5) The TRS R consists of the following rules: membercA_in_ag(X1, .(X1, X2)) -> membercA_out_ag(X1, .(X1, X2)) membercA_in_ag(X1, .(X2, X3)) -> U91_ag(X1, X2, X3, membercA_in_ag(X1, X3)) U91_ag(X1, X2, X3, membercA_out_ag(X1, X3)) -> membercA_out_ag(X1, .(X2, X3)) not_membercF_in_g(X1) -> not_membercF_out_g(X1) not_membercG_in_gg(X1, X2) -> not_membercG_out_gg(X1, X2) not_membercH_in_ggg(X1, X2, X3) -> not_membercH_out_ggg(X1, X2, X3) not_membercI_in_gggg(X1, X2, X3, X4) -> not_membercI_out_gggg(X1, X2, X3, X4) not_membercJ_in_ggggg(X1, X2, X3, X4, X5) -> not_membercJ_out_ggggg(X1, X2, X3, X4, X5) not_membercK_in_gggggg(X1, X2, X3, X4, X5, X6) -> not_membercK_out_gggggg(X1, X2, X3, X4, X5, X6) not_membercL_in_ggggggg(X1, X2, X3, X4, X5, X6, X7) -> not_membercL_out_ggggggg(X1, X2, X3, X4, X5, X6, X7) not_membercE_in_gggggggg(X1, X2, X3, X4, X5, X6, X7, X8) -> not_membercE_out_gggggggg(X1, X2, X3, X4, X5, X6, X7, X8) not_membercC_in_ggg(X1, X2, X3) -> not_membercC_out_ggg(X1, X2, X3) The argument filtering Pi contains the following mapping: memberA_in_ag(x1, x2) = memberA_in_ag(x2) .(x1, x2) = .(x1, x2) membercA_in_ag(x1, x2) = membercA_in_ag(x2) membercA_out_ag(x1, x2) = membercA_out_ag(x1, x2) U91_ag(x1, x2, x3, x4) = U91_ag(x2, x3, x4) memberA_in_gg(x1, x2) = memberA_in_gg(x1, x2) [] = [] not_membercF_in_g(x1) = not_membercF_in_g(x1) not_membercF_out_g(x1) = not_membercF_out_g(x1) not_membercG_in_gg(x1, x2) = not_membercG_in_gg(x1, x2) not_membercG_out_gg(x1, x2) = not_membercG_out_gg(x1, x2) not_membercH_in_ggg(x1, x2, x3) = not_membercH_in_ggg(x1, x2, x3) not_membercH_out_ggg(x1, x2, x3) = not_membercH_out_ggg(x1, x2, x3) not_membercI_in_gggg(x1, x2, x3, x4) = not_membercI_in_gggg(x1, x2, x3, x4) not_membercI_out_gggg(x1, x2, x3, x4) = not_membercI_out_gggg(x1, x2, x3, x4) not_membercJ_in_ggggg(x1, x2, x3, x4, x5) = not_membercJ_in_ggggg(x1, x2, x3, x4, x5) not_membercJ_out_ggggg(x1, x2, x3, x4, x5) = not_membercJ_out_ggggg(x1, x2, x3, x4, x5) not_membercK_in_gggggg(x1, x2, x3, x4, x5, x6) = not_membercK_in_gggggg(x1, x2, x3, x4, x5, x6) not_membercK_out_gggggg(x1, x2, x3, x4, x5, x6) = not_membercK_out_gggggg(x1, x2, x3, x4, x5, x6) not_membercL_in_ggggggg(x1, x2, x3, x4, x5, x6, x7) = not_membercL_in_ggggggg(x1, x2, x3, x4, x5, x6, x7) not_membercL_out_ggggggg(x1, x2, x3, x4, x5, x6, x7) = not_membercL_out_ggggggg(x1, x2, x3, x4, x5, x6, x7) pD_in_aggggggggag(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = pD_in_aggggggggag(x2, x3, x4, x5, x6, x7, x8, x9, x11) not_membercE_in_gggggggg(x1, x2, x3, x4, x5, x6, x7, x8) = not_membercE_in_gggggggg(x1, x2, x3, x4, x5, x6, x7, x8) not_membercE_out_gggggggg(x1, x2, x3, x4, x5, x6, x7, x8) = not_membercE_out_gggggggg(x1, x2, x3, x4, x5, x6, x7, x8) subsetcheckedB_in_aggg(x1, x2, x3, x4) = subsetcheckedB_in_aggg(x2, x3, x4) not_membercC_in_ggg(x1, x2, x3) = not_membercC_in_ggg(x1, x2, x3) not_membercC_out_ggg(x1, x2, x3) = not_membercC_out_ggg(x1, x2, x3) SUBSETM_IN_AG(x1, x2) = SUBSETM_IN_AG(x2) U12_AG(x1, x2, x3, x4) = U12_AG(x3, x4) MEMBERA_IN_AG(x1, x2) = MEMBERA_IN_AG(x2) U1_AG(x1, x2, x3, x4) = U1_AG(x2, x3, x4) U13_AG(x1, x2, x3, x4) = U13_AG(x3, x4) U14_AG(x1, x2, x3, x4) = U14_AG(x3, x4) MEMBERA_IN_GG(x1, x2) = MEMBERA_IN_GG(x1, x2) U1_GG(x1, x2, x3, x4) = U1_GG(x1, x2, x3, x4) U15_AG(x1, x2, x3, x4, x5) = U15_AG(x4, x5) U16_AG(x1, x2, x3, x4, x5) = U16_AG(x4, x5) U17_AG(x1, x2, x3, x4, x5) = U17_AG(x4, x5) U18_AG(x1, x2, x3, x4, x5) = U18_AG(x1, x4, x5) U19_AG(x1, x2, x3, x4, x5) = U19_AG(x4, x5) U20_AG(x1, x2, x3, x4, x5, x6) = U20_AG(x5, x6) U21_AG(x1, x2, x3, x4, x5, x6) = U21_AG(x1, x5, x6) U22_AG(x1, x2, x3, x4, x5, x6) = U22_AG(x1, x5, x6) U23_AG(x1, x2, x3, x4, x5, x6) = U23_AG(x5, x6) U24_AG(x1, x2, x3, x4, x5, x6) = U24_AG(x5, x6) U25_AG(x1, x2, x3, x4, x5, x6) = U25_AG(x1, x2, x5, x6) U26_AG(x1, x2, x3, x4, x5, x6) = U26_AG(x5, x6) U27_AG(x1, x2, x3, x4, x5, x6, x7) = U27_AG(x6, x7) U28_AG(x1, x2, x3, x4, x5, x6, x7) = U28_AG(x1, x6, x7) U29_AG(x1, x2, x3, x4, x5, x6, x7) = U29_AG(x1, x6, x7) U30_AG(x1, x2, x3, x4, x5, x6, x7) = U30_AG(x1, x2, x6, x7) U31_AG(x1, x2, x3, x4, x5, x6, x7) = U31_AG(x1, x2, x6, x7) U32_AG(x1, x2, x3, x4, x5, x6, x7) = U32_AG(x6, x7) U33_AG(x1, x2, x3, x4, x5, x6, x7) = U33_AG(x6, x7) U34_AG(x1, x2, x3, x4, x5, x6, x7) = U34_AG(x1, x2, x3, x6, x7) U35_AG(x1, x2, x3, x4, x5, x6, x7) = U35_AG(x6, x7) U36_AG(x1, x2, x3, x4, x5, x6, x7, x8) = U36_AG(x7, x8) U37_AG(x1, x2, x3, x4, x5, x6, x7, x8) = U37_AG(x1, x7, x8) U38_AG(x1, x2, x3, x4, x5, x6, x7, x8) = U38_AG(x1, x7, x8) U39_AG(x1, x2, x3, x4, x5, x6, x7, x8) = U39_AG(x1, x2, x7, x8) U40_AG(x1, x2, x3, x4, x5, x6, x7, x8) = U40_AG(x1, x2, x7, x8) U41_AG(x1, x2, x3, x4, x5, x6, x7, x8) = U41_AG(x1, x2, x3, x7, x8) U42_AG(x1, x2, x3, x4, x5, x6, x7, x8) = U42_AG(x1, x2, x3, x7, x8) U43_AG(x1, x2, x3, x4, x5, x6, x7, x8) = U43_AG(x7, x8) U44_AG(x1, x2, x3, x4, x5, x6, x7, x8) = U44_AG(x7, x8) U45_AG(x1, x2, x3, x4, x5, x6, x7, x8) = U45_AG(x1, x2, x3, x4, x7, x8) U46_AG(x1, x2, x3, x4, x5, x6, x7, x8) = U46_AG(x7, x8) U47_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U47_AG(x8, x9) U48_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U48_AG(x1, x8, x9) U49_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U49_AG(x1, x8, x9) U50_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U50_AG(x1, x2, x8, x9) U51_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U51_AG(x1, x2, x8, x9) U52_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U52_AG(x1, x2, x3, x8, x9) U53_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U53_AG(x1, x2, x3, x8, x9) U54_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U54_AG(x1, x2, x3, x4, x8, x9) U55_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U55_AG(x1, x2, x3, x4, x8, x9) U56_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U56_AG(x8, x9) U57_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U57_AG(x8, x9) U58_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U58_AG(x1, x2, x3, x4, x5, x8, x9) U59_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U59_AG(x8, x9) U60_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U60_AG(x9, x10) U61_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U61_AG(x1, x9, x10) U62_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U62_AG(x1, x9, x10) U63_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U63_AG(x1, x2, x9, x10) U64_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U64_AG(x1, x2, x9, x10) U65_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U65_AG(x1, x2, x3, x9, x10) U66_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U66_AG(x1, x2, x3, x9, x10) U67_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U67_AG(x1, x2, x3, x4, x9, x10) U68_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U68_AG(x1, x2, x3, x4, x9, x10) U69_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U69_AG(x1, x2, x3, x4, x5, x9, x10) U70_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U70_AG(x1, x2, x3, x4, x5, x9, x10) U71_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U71_AG(x9, x10) U72_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U72_AG(x9, x10) U73_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U73_AG(x1, x2, x3, x4, x5, x6, x9, x10) U74_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U74_AG(x9, x10) U75_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U75_AG(x10, x11) U76_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U76_AG(x1, x10, x11) U77_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U77_AG(x1, x10, x11) U78_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U78_AG(x1, x2, x10, x11) U79_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U79_AG(x1, x2, x10, x11) U80_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U80_AG(x1, x2, x3, x10, x11) U81_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U81_AG(x1, x2, x3, x10, x11) U82_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U82_AG(x1, x2, x3, x4, x10, x11) U83_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U83_AG(x1, x2, x3, x4, x10, x11) U84_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U84_AG(x1, x2, x3, x4, x5, x10, x11) U85_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U85_AG(x1, x2, x3, x4, x5, x10, x11) U86_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U86_AG(x1, x2, x3, x4, x5, x6, x10, x11) U87_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U87_AG(x1, x2, x3, x4, x5, x6, x10, x11) U88_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U88_AG(x1, x2, x3, x4, x5, x6, x7, x10, x11) U89_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U89_AG(x10, x11) PD_IN_AGGGGGGGGAG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = PD_IN_AGGGGGGGGAG(x2, x3, x4, x5, x6, x7, x8, x9, x11) U7_AGGGGGGGGAG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U7_AGGGGGGGGAG(x2, x3, x4, x5, x6, x7, x8, x9, x11, x12) U8_AGGGGGGGGAG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U8_AGGGGGGGGAG(x2, x3, x4, x5, x6, x7, x8, x9, x11, x12) U9_AGGGGGGGGAG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U9_AGGGGGGGGAG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x11, x12) U10_AGGGGGGGGAG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U10_AGGGGGGGGAG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x11, x12) U11_AGGGGGGGGAG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U11_AGGGGGGGGAG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x11, x12) SUBSETCHECKEDB_IN_AGGG(x1, x2, x3, x4) = SUBSETCHECKEDB_IN_AGGG(x2, x3, x4) U2_AGGG(x1, x2, x3, x4, x5, x6) = U2_AGGG(x3, x4, x5, x6) U3_AGGG(x1, x2, x3, x4, x5, x6) = U3_AGGG(x3, x4, x5, x6) U4_AGGG(x1, x2, x3, x4, x5, x6) = U4_AGGG(x3, x4, x5, x6) U5_AGGG(x1, x2, x3, x4, x5, x6) = U5_AGGG(x1, x3, x4, x5, x6) U6_AGGG(x1, x2, x3, x4, x5, x6) = U6_AGGG(x3, x4, x5, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (137) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 108 less nodes. ---------------------------------------- (138) Complex Obligation (AND) ---------------------------------------- (139) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBERA_IN_GG(X1, .(X2, X3)) -> MEMBERA_IN_GG(X1, X3) The TRS R consists of the following rules: membercA_in_ag(X1, .(X1, X2)) -> membercA_out_ag(X1, .(X1, X2)) membercA_in_ag(X1, .(X2, X3)) -> U91_ag(X1, X2, X3, membercA_in_ag(X1, X3)) U91_ag(X1, X2, X3, membercA_out_ag(X1, X3)) -> membercA_out_ag(X1, .(X2, X3)) not_membercF_in_g(X1) -> not_membercF_out_g(X1) not_membercG_in_gg(X1, X2) -> not_membercG_out_gg(X1, X2) not_membercH_in_ggg(X1, X2, X3) -> not_membercH_out_ggg(X1, X2, X3) not_membercI_in_gggg(X1, X2, X3, X4) -> not_membercI_out_gggg(X1, X2, X3, X4) not_membercJ_in_ggggg(X1, X2, X3, X4, X5) -> not_membercJ_out_ggggg(X1, X2, X3, X4, X5) not_membercK_in_gggggg(X1, X2, X3, X4, X5, X6) -> not_membercK_out_gggggg(X1, X2, X3, X4, X5, X6) not_membercL_in_ggggggg(X1, X2, X3, X4, X5, X6, X7) -> not_membercL_out_ggggggg(X1, X2, X3, X4, X5, X6, X7) not_membercE_in_gggggggg(X1, X2, X3, X4, X5, X6, X7, X8) -> not_membercE_out_gggggggg(X1, X2, X3, X4, X5, X6, X7, X8) not_membercC_in_ggg(X1, X2, X3) -> not_membercC_out_ggg(X1, X2, X3) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) membercA_in_ag(x1, x2) = membercA_in_ag(x2) membercA_out_ag(x1, x2) = membercA_out_ag(x1, x2) U91_ag(x1, x2, x3, x4) = U91_ag(x2, x3, x4) not_membercF_in_g(x1) = not_membercF_in_g(x1) not_membercF_out_g(x1) = not_membercF_out_g(x1) not_membercG_in_gg(x1, x2) = not_membercG_in_gg(x1, x2) not_membercG_out_gg(x1, x2) = not_membercG_out_gg(x1, x2) not_membercH_in_ggg(x1, x2, x3) = not_membercH_in_ggg(x1, x2, x3) not_membercH_out_ggg(x1, x2, x3) = not_membercH_out_ggg(x1, x2, x3) not_membercI_in_gggg(x1, x2, x3, x4) = not_membercI_in_gggg(x1, x2, x3, x4) not_membercI_out_gggg(x1, x2, x3, x4) = not_membercI_out_gggg(x1, x2, x3, x4) not_membercJ_in_ggggg(x1, x2, x3, x4, x5) = not_membercJ_in_ggggg(x1, x2, x3, x4, x5) not_membercJ_out_ggggg(x1, x2, x3, x4, x5) = not_membercJ_out_ggggg(x1, x2, x3, x4, x5) not_membercK_in_gggggg(x1, x2, x3, x4, x5, x6) = not_membercK_in_gggggg(x1, x2, x3, x4, x5, x6) not_membercK_out_gggggg(x1, x2, x3, x4, x5, x6) = not_membercK_out_gggggg(x1, x2, x3, x4, x5, x6) not_membercL_in_ggggggg(x1, x2, x3, x4, x5, x6, x7) = not_membercL_in_ggggggg(x1, x2, x3, x4, x5, x6, x7) not_membercL_out_ggggggg(x1, x2, x3, x4, x5, x6, x7) = not_membercL_out_ggggggg(x1, x2, x3, x4, x5, x6, x7) not_membercE_in_gggggggg(x1, x2, x3, x4, x5, x6, x7, x8) = not_membercE_in_gggggggg(x1, x2, x3, x4, x5, x6, x7, x8) not_membercE_out_gggggggg(x1, x2, x3, x4, x5, x6, x7, x8) = not_membercE_out_gggggggg(x1, x2, x3, x4, x5, x6, x7, x8) not_membercC_in_ggg(x1, x2, x3) = not_membercC_in_ggg(x1, x2, x3) not_membercC_out_ggg(x1, x2, x3) = not_membercC_out_ggg(x1, x2, x3) MEMBERA_IN_GG(x1, x2) = MEMBERA_IN_GG(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (140) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (141) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBERA_IN_GG(X1, .(X2, X3)) -> MEMBERA_IN_GG(X1, X3) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (142) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (143) Obligation: Q DP problem: The TRS P consists of the following rules: MEMBERA_IN_GG(X1, .(X2, X3)) -> MEMBERA_IN_GG(X1, X3) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (144) 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_GG(X1, .(X2, X3)) -> MEMBERA_IN_GG(X1, X3) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (145) YES ---------------------------------------- (146) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBERA_IN_AG(X1, .(X2, X3)) -> MEMBERA_IN_AG(X1, X3) The TRS R consists of the following rules: membercA_in_ag(X1, .(X1, X2)) -> membercA_out_ag(X1, .(X1, X2)) membercA_in_ag(X1, .(X2, X3)) -> U91_ag(X1, X2, X3, membercA_in_ag(X1, X3)) U91_ag(X1, X2, X3, membercA_out_ag(X1, X3)) -> membercA_out_ag(X1, .(X2, X3)) not_membercF_in_g(X1) -> not_membercF_out_g(X1) not_membercG_in_gg(X1, X2) -> not_membercG_out_gg(X1, X2) not_membercH_in_ggg(X1, X2, X3) -> not_membercH_out_ggg(X1, X2, X3) not_membercI_in_gggg(X1, X2, X3, X4) -> not_membercI_out_gggg(X1, X2, X3, X4) not_membercJ_in_ggggg(X1, X2, X3, X4, X5) -> not_membercJ_out_ggggg(X1, X2, X3, X4, X5) not_membercK_in_gggggg(X1, X2, X3, X4, X5, X6) -> not_membercK_out_gggggg(X1, X2, X3, X4, X5, X6) not_membercL_in_ggggggg(X1, X2, X3, X4, X5, X6, X7) -> not_membercL_out_ggggggg(X1, X2, X3, X4, X5, X6, X7) not_membercE_in_gggggggg(X1, X2, X3, X4, X5, X6, X7, X8) -> not_membercE_out_gggggggg(X1, X2, X3, X4, X5, X6, X7, X8) not_membercC_in_ggg(X1, X2, X3) -> not_membercC_out_ggg(X1, X2, X3) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) membercA_in_ag(x1, x2) = membercA_in_ag(x2) membercA_out_ag(x1, x2) = membercA_out_ag(x1, x2) U91_ag(x1, x2, x3, x4) = U91_ag(x2, x3, x4) not_membercF_in_g(x1) = not_membercF_in_g(x1) not_membercF_out_g(x1) = not_membercF_out_g(x1) not_membercG_in_gg(x1, x2) = not_membercG_in_gg(x1, x2) not_membercG_out_gg(x1, x2) = not_membercG_out_gg(x1, x2) not_membercH_in_ggg(x1, x2, x3) = not_membercH_in_ggg(x1, x2, x3) not_membercH_out_ggg(x1, x2, x3) = not_membercH_out_ggg(x1, x2, x3) not_membercI_in_gggg(x1, x2, x3, x4) = not_membercI_in_gggg(x1, x2, x3, x4) not_membercI_out_gggg(x1, x2, x3, x4) = not_membercI_out_gggg(x1, x2, x3, x4) not_membercJ_in_ggggg(x1, x2, x3, x4, x5) = not_membercJ_in_ggggg(x1, x2, x3, x4, x5) not_membercJ_out_ggggg(x1, x2, x3, x4, x5) = not_membercJ_out_ggggg(x1, x2, x3, x4, x5) not_membercK_in_gggggg(x1, x2, x3, x4, x5, x6) = not_membercK_in_gggggg(x1, x2, x3, x4, x5, x6) not_membercK_out_gggggg(x1, x2, x3, x4, x5, x6) = not_membercK_out_gggggg(x1, x2, x3, x4, x5, x6) not_membercL_in_ggggggg(x1, x2, x3, x4, x5, x6, x7) = not_membercL_in_ggggggg(x1, x2, x3, x4, x5, x6, x7) not_membercL_out_ggggggg(x1, x2, x3, x4, x5, x6, x7) = not_membercL_out_ggggggg(x1, x2, x3, x4, x5, x6, x7) not_membercE_in_gggggggg(x1, x2, x3, x4, x5, x6, x7, x8) = not_membercE_in_gggggggg(x1, x2, x3, x4, x5, x6, x7, x8) not_membercE_out_gggggggg(x1, x2, x3, x4, x5, x6, x7, x8) = not_membercE_out_gggggggg(x1, x2, x3, x4, x5, x6, x7, x8) not_membercC_in_ggg(x1, x2, x3) = not_membercC_in_ggg(x1, x2, x3) not_membercC_out_ggg(x1, x2, x3) = not_membercC_out_ggg(x1, x2, x3) MEMBERA_IN_AG(x1, x2) = MEMBERA_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (147) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (148) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBERA_IN_AG(X1, .(X2, X3)) -> MEMBERA_IN_AG(X1, X3) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) MEMBERA_IN_AG(x1, x2) = MEMBERA_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (149) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (150) Obligation: Q DP problem: The TRS P consists of the following rules: MEMBERA_IN_AG(.(X2, X3)) -> MEMBERA_IN_AG(X3) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (151) 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_AG(.(X2, X3)) -> MEMBERA_IN_AG(X3) The graph contains the following edges 1 > 1 ---------------------------------------- (152) YES ---------------------------------------- (153) Obligation: Pi DP problem: The TRS P consists of the following rules: SUBSETCHECKEDB_IN_AGGG(.(X1, X2), X3, X4, X5) -> U3_AGGG(X1, X2, X3, X4, X5, membercA_in_ag(X1, X5)) U3_AGGG(X1, X2, X3, X4, X5, membercA_out_ag(X1, X5)) -> U5_AGGG(X1, X2, X3, X4, X5, not_membercC_in_ggg(X1, X3, X4)) U5_AGGG(X1, X2, X3, X4, X5, not_membercC_out_ggg(X1, X3, X4)) -> SUBSETCHECKEDB_IN_AGGG(X2, X1, .(X3, X4), X5) The TRS R consists of the following rules: membercA_in_ag(X1, .(X1, X2)) -> membercA_out_ag(X1, .(X1, X2)) membercA_in_ag(X1, .(X2, X3)) -> U91_ag(X1, X2, X3, membercA_in_ag(X1, X3)) U91_ag(X1, X2, X3, membercA_out_ag(X1, X3)) -> membercA_out_ag(X1, .(X2, X3)) not_membercF_in_g(X1) -> not_membercF_out_g(X1) not_membercG_in_gg(X1, X2) -> not_membercG_out_gg(X1, X2) not_membercH_in_ggg(X1, X2, X3) -> not_membercH_out_ggg(X1, X2, X3) not_membercI_in_gggg(X1, X2, X3, X4) -> not_membercI_out_gggg(X1, X2, X3, X4) not_membercJ_in_ggggg(X1, X2, X3, X4, X5) -> not_membercJ_out_ggggg(X1, X2, X3, X4, X5) not_membercK_in_gggggg(X1, X2, X3, X4, X5, X6) -> not_membercK_out_gggggg(X1, X2, X3, X4, X5, X6) not_membercL_in_ggggggg(X1, X2, X3, X4, X5, X6, X7) -> not_membercL_out_ggggggg(X1, X2, X3, X4, X5, X6, X7) not_membercE_in_gggggggg(X1, X2, X3, X4, X5, X6, X7, X8) -> not_membercE_out_gggggggg(X1, X2, X3, X4, X5, X6, X7, X8) not_membercC_in_ggg(X1, X2, X3) -> not_membercC_out_ggg(X1, X2, X3) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) membercA_in_ag(x1, x2) = membercA_in_ag(x2) membercA_out_ag(x1, x2) = membercA_out_ag(x1, x2) U91_ag(x1, x2, x3, x4) = U91_ag(x2, x3, x4) not_membercF_in_g(x1) = not_membercF_in_g(x1) not_membercF_out_g(x1) = not_membercF_out_g(x1) not_membercG_in_gg(x1, x2) = not_membercG_in_gg(x1, x2) not_membercG_out_gg(x1, x2) = not_membercG_out_gg(x1, x2) not_membercH_in_ggg(x1, x2, x3) = not_membercH_in_ggg(x1, x2, x3) not_membercH_out_ggg(x1, x2, x3) = not_membercH_out_ggg(x1, x2, x3) not_membercI_in_gggg(x1, x2, x3, x4) = not_membercI_in_gggg(x1, x2, x3, x4) not_membercI_out_gggg(x1, x2, x3, x4) = not_membercI_out_gggg(x1, x2, x3, x4) not_membercJ_in_ggggg(x1, x2, x3, x4, x5) = not_membercJ_in_ggggg(x1, x2, x3, x4, x5) not_membercJ_out_ggggg(x1, x2, x3, x4, x5) = not_membercJ_out_ggggg(x1, x2, x3, x4, x5) not_membercK_in_gggggg(x1, x2, x3, x4, x5, x6) = not_membercK_in_gggggg(x1, x2, x3, x4, x5, x6) not_membercK_out_gggggg(x1, x2, x3, x4, x5, x6) = not_membercK_out_gggggg(x1, x2, x3, x4, x5, x6) not_membercL_in_ggggggg(x1, x2, x3, x4, x5, x6, x7) = not_membercL_in_ggggggg(x1, x2, x3, x4, x5, x6, x7) not_membercL_out_ggggggg(x1, x2, x3, x4, x5, x6, x7) = not_membercL_out_ggggggg(x1, x2, x3, x4, x5, x6, x7) not_membercE_in_gggggggg(x1, x2, x3, x4, x5, x6, x7, x8) = not_membercE_in_gggggggg(x1, x2, x3, x4, x5, x6, x7, x8) not_membercE_out_gggggggg(x1, x2, x3, x4, x5, x6, x7, x8) = not_membercE_out_gggggggg(x1, x2, x3, x4, x5, x6, x7, x8) not_membercC_in_ggg(x1, x2, x3) = not_membercC_in_ggg(x1, x2, x3) not_membercC_out_ggg(x1, x2, x3) = not_membercC_out_ggg(x1, x2, x3) SUBSETCHECKEDB_IN_AGGG(x1, x2, x3, x4) = SUBSETCHECKEDB_IN_AGGG(x2, x3, x4) U3_AGGG(x1, x2, x3, x4, x5, x6) = U3_AGGG(x3, x4, x5, x6) U5_AGGG(x1, x2, x3, x4, x5, x6) = U5_AGGG(x1, x3, x4, x5, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (154) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (155) Obligation: Pi DP problem: The TRS P consists of the following rules: SUBSETCHECKEDB_IN_AGGG(.(X1, X2), X3, X4, X5) -> U3_AGGG(X1, X2, X3, X4, X5, membercA_in_ag(X1, X5)) U3_AGGG(X1, X2, X3, X4, X5, membercA_out_ag(X1, X5)) -> U5_AGGG(X1, X2, X3, X4, X5, not_membercC_in_ggg(X1, X3, X4)) U5_AGGG(X1, X2, X3, X4, X5, not_membercC_out_ggg(X1, X3, X4)) -> SUBSETCHECKEDB_IN_AGGG(X2, X1, .(X3, X4), X5) The TRS R consists of the following rules: membercA_in_ag(X1, .(X1, X2)) -> membercA_out_ag(X1, .(X1, X2)) membercA_in_ag(X1, .(X2, X3)) -> U91_ag(X1, X2, X3, membercA_in_ag(X1, X3)) not_membercC_in_ggg(X1, X2, X3) -> not_membercC_out_ggg(X1, X2, X3) U91_ag(X1, X2, X3, membercA_out_ag(X1, X3)) -> membercA_out_ag(X1, .(X2, X3)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) membercA_in_ag(x1, x2) = membercA_in_ag(x2) membercA_out_ag(x1, x2) = membercA_out_ag(x1, x2) U91_ag(x1, x2, x3, x4) = U91_ag(x2, x3, x4) not_membercC_in_ggg(x1, x2, x3) = not_membercC_in_ggg(x1, x2, x3) not_membercC_out_ggg(x1, x2, x3) = not_membercC_out_ggg(x1, x2, x3) SUBSETCHECKEDB_IN_AGGG(x1, x2, x3, x4) = SUBSETCHECKEDB_IN_AGGG(x2, x3, x4) U3_AGGG(x1, x2, x3, x4, x5, x6) = U3_AGGG(x3, x4, x5, x6) U5_AGGG(x1, x2, x3, x4, x5, x6) = U5_AGGG(x1, x3, x4, x5, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (156) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (157) Obligation: Q DP problem: The TRS P consists of the following rules: SUBSETCHECKEDB_IN_AGGG(X3, X4, X5) -> U3_AGGG(X3, X4, X5, membercA_in_ag(X5)) U3_AGGG(X3, X4, X5, membercA_out_ag(X1, X5)) -> U5_AGGG(X1, X3, X4, X5, not_membercC_in_ggg(X1, X3, X4)) U5_AGGG(X1, X3, X4, X5, not_membercC_out_ggg(X1, X3, X4)) -> SUBSETCHECKEDB_IN_AGGG(X1, .(X3, X4), X5) The TRS R consists of the following rules: membercA_in_ag(.(X1, X2)) -> membercA_out_ag(X1, .(X1, X2)) membercA_in_ag(.(X2, X3)) -> U91_ag(X2, X3, membercA_in_ag(X3)) not_membercC_in_ggg(X1, X2, X3) -> not_membercC_out_ggg(X1, X2, X3) U91_ag(X2, X3, membercA_out_ag(X1, X3)) -> membercA_out_ag(X1, .(X2, X3)) The set Q consists of the following terms: membercA_in_ag(x0) not_membercC_in_ggg(x0, x1, x2) U91_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (158) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule U3_AGGG(X3, X4, X5, membercA_out_ag(X1, X5)) -> U5_AGGG(X1, X3, X4, X5, not_membercC_in_ggg(X1, X3, X4)) at position [4] we obtained the following new rules [LPAR04]: (U3_AGGG(X3, X4, X5, membercA_out_ag(X1, X5)) -> U5_AGGG(X1, X3, X4, X5, not_membercC_out_ggg(X1, X3, X4)),U3_AGGG(X3, X4, X5, membercA_out_ag(X1, X5)) -> U5_AGGG(X1, X3, X4, X5, not_membercC_out_ggg(X1, X3, X4))) ---------------------------------------- (159) Obligation: Q DP problem: The TRS P consists of the following rules: SUBSETCHECKEDB_IN_AGGG(X3, X4, X5) -> U3_AGGG(X3, X4, X5, membercA_in_ag(X5)) U5_AGGG(X1, X3, X4, X5, not_membercC_out_ggg(X1, X3, X4)) -> SUBSETCHECKEDB_IN_AGGG(X1, .(X3, X4), X5) U3_AGGG(X3, X4, X5, membercA_out_ag(X1, X5)) -> U5_AGGG(X1, X3, X4, X5, not_membercC_out_ggg(X1, X3, X4)) The TRS R consists of the following rules: membercA_in_ag(.(X1, X2)) -> membercA_out_ag(X1, .(X1, X2)) membercA_in_ag(.(X2, X3)) -> U91_ag(X2, X3, membercA_in_ag(X3)) not_membercC_in_ggg(X1, X2, X3) -> not_membercC_out_ggg(X1, X2, X3) U91_ag(X2, X3, membercA_out_ag(X1, X3)) -> membercA_out_ag(X1, .(X2, X3)) The set Q consists of the following terms: membercA_in_ag(x0) not_membercC_in_ggg(x0, x1, x2) U91_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (160) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (161) Obligation: Q DP problem: The TRS P consists of the following rules: SUBSETCHECKEDB_IN_AGGG(X3, X4, X5) -> U3_AGGG(X3, X4, X5, membercA_in_ag(X5)) U5_AGGG(X1, X3, X4, X5, not_membercC_out_ggg(X1, X3, X4)) -> SUBSETCHECKEDB_IN_AGGG(X1, .(X3, X4), X5) U3_AGGG(X3, X4, X5, membercA_out_ag(X1, X5)) -> U5_AGGG(X1, X3, X4, X5, not_membercC_out_ggg(X1, X3, X4)) The TRS R consists of the following rules: membercA_in_ag(.(X1, X2)) -> membercA_out_ag(X1, .(X1, X2)) membercA_in_ag(.(X2, X3)) -> U91_ag(X2, X3, membercA_in_ag(X3)) U91_ag(X2, X3, membercA_out_ag(X1, X3)) -> membercA_out_ag(X1, .(X2, X3)) The set Q consists of the following terms: membercA_in_ag(x0) not_membercC_in_ggg(x0, x1, x2) U91_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (162) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. not_membercC_in_ggg(x0, x1, x2) ---------------------------------------- (163) Obligation: Q DP problem: The TRS P consists of the following rules: SUBSETCHECKEDB_IN_AGGG(X3, X4, X5) -> U3_AGGG(X3, X4, X5, membercA_in_ag(X5)) U5_AGGG(X1, X3, X4, X5, not_membercC_out_ggg(X1, X3, X4)) -> SUBSETCHECKEDB_IN_AGGG(X1, .(X3, X4), X5) U3_AGGG(X3, X4, X5, membercA_out_ag(X1, X5)) -> U5_AGGG(X1, X3, X4, X5, not_membercC_out_ggg(X1, X3, X4)) The TRS R consists of the following rules: membercA_in_ag(.(X1, X2)) -> membercA_out_ag(X1, .(X1, X2)) membercA_in_ag(.(X2, X3)) -> U91_ag(X2, X3, membercA_in_ag(X3)) U91_ag(X2, X3, membercA_out_ag(X1, X3)) -> membercA_out_ag(X1, .(X2, X3)) The set Q consists of the following terms: membercA_in_ag(x0) U91_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (164) TransformationProof (SOUND) By narrowing [LPAR04] the rule SUBSETCHECKEDB_IN_AGGG(X3, X4, X5) -> U3_AGGG(X3, X4, X5, membercA_in_ag(X5)) at position [3] we obtained the following new rules [LPAR04]: (SUBSETCHECKEDB_IN_AGGG(y0, y1, .(x0, x1)) -> U3_AGGG(y0, y1, .(x0, x1), membercA_out_ag(x0, .(x0, x1))),SUBSETCHECKEDB_IN_AGGG(y0, y1, .(x0, x1)) -> U3_AGGG(y0, y1, .(x0, x1), membercA_out_ag(x0, .(x0, x1)))) (SUBSETCHECKEDB_IN_AGGG(y0, y1, .(x0, x1)) -> U3_AGGG(y0, y1, .(x0, x1), U91_ag(x0, x1, membercA_in_ag(x1))),SUBSETCHECKEDB_IN_AGGG(y0, y1, .(x0, x1)) -> U3_AGGG(y0, y1, .(x0, x1), U91_ag(x0, x1, membercA_in_ag(x1)))) ---------------------------------------- (165) Obligation: Q DP problem: The TRS P consists of the following rules: U5_AGGG(X1, X3, X4, X5, not_membercC_out_ggg(X1, X3, X4)) -> SUBSETCHECKEDB_IN_AGGG(X1, .(X3, X4), X5) U3_AGGG(X3, X4, X5, membercA_out_ag(X1, X5)) -> U5_AGGG(X1, X3, X4, X5, not_membercC_out_ggg(X1, X3, X4)) SUBSETCHECKEDB_IN_AGGG(y0, y1, .(x0, x1)) -> U3_AGGG(y0, y1, .(x0, x1), membercA_out_ag(x0, .(x0, x1))) SUBSETCHECKEDB_IN_AGGG(y0, y1, .(x0, x1)) -> U3_AGGG(y0, y1, .(x0, x1), U91_ag(x0, x1, membercA_in_ag(x1))) The TRS R consists of the following rules: membercA_in_ag(.(X1, X2)) -> membercA_out_ag(X1, .(X1, X2)) membercA_in_ag(.(X2, X3)) -> U91_ag(X2, X3, membercA_in_ag(X3)) U91_ag(X2, X3, membercA_out_ag(X1, X3)) -> membercA_out_ag(X1, .(X2, X3)) The set Q consists of the following terms: membercA_in_ag(x0) U91_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (166) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U3_AGGG(X3, X4, X5, membercA_out_ag(X1, X5)) -> U5_AGGG(X1, X3, X4, X5, not_membercC_out_ggg(X1, X3, X4)) we obtained the following new rules [LPAR04]: (U3_AGGG(z0, z1, .(z2, z3), membercA_out_ag(z2, .(z2, z3))) -> U5_AGGG(z2, z0, z1, .(z2, z3), not_membercC_out_ggg(z2, z0, z1)),U3_AGGG(z0, z1, .(z2, z3), membercA_out_ag(z2, .(z2, z3))) -> U5_AGGG(z2, z0, z1, .(z2, z3), not_membercC_out_ggg(z2, z0, z1))) (U3_AGGG(z0, z1, .(z2, z3), membercA_out_ag(x3, .(z2, z3))) -> U5_AGGG(x3, z0, z1, .(z2, z3), not_membercC_out_ggg(x3, z0, z1)),U3_AGGG(z0, z1, .(z2, z3), membercA_out_ag(x3, .(z2, z3))) -> U5_AGGG(x3, z0, z1, .(z2, z3), not_membercC_out_ggg(x3, z0, z1))) ---------------------------------------- (167) Obligation: Q DP problem: The TRS P consists of the following rules: U5_AGGG(X1, X3, X4, X5, not_membercC_out_ggg(X1, X3, X4)) -> SUBSETCHECKEDB_IN_AGGG(X1, .(X3, X4), X5) SUBSETCHECKEDB_IN_AGGG(y0, y1, .(x0, x1)) -> U3_AGGG(y0, y1, .(x0, x1), membercA_out_ag(x0, .(x0, x1))) SUBSETCHECKEDB_IN_AGGG(y0, y1, .(x0, x1)) -> U3_AGGG(y0, y1, .(x0, x1), U91_ag(x0, x1, membercA_in_ag(x1))) U3_AGGG(z0, z1, .(z2, z3), membercA_out_ag(z2, .(z2, z3))) -> U5_AGGG(z2, z0, z1, .(z2, z3), not_membercC_out_ggg(z2, z0, z1)) U3_AGGG(z0, z1, .(z2, z3), membercA_out_ag(x3, .(z2, z3))) -> U5_AGGG(x3, z0, z1, .(z2, z3), not_membercC_out_ggg(x3, z0, z1)) The TRS R consists of the following rules: membercA_in_ag(.(X1, X2)) -> membercA_out_ag(X1, .(X1, X2)) membercA_in_ag(.(X2, X3)) -> U91_ag(X2, X3, membercA_in_ag(X3)) U91_ag(X2, X3, membercA_out_ag(X1, X3)) -> membercA_out_ag(X1, .(X2, X3)) The set Q consists of the following terms: membercA_in_ag(x0) U91_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (168) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U5_AGGG(X1, X3, X4, X5, not_membercC_out_ggg(X1, X3, X4)) -> SUBSETCHECKEDB_IN_AGGG(X1, .(X3, X4), X5) we obtained the following new rules [LPAR04]: (U5_AGGG(z2, z0, z1, .(z2, z3), not_membercC_out_ggg(z2, z0, z1)) -> SUBSETCHECKEDB_IN_AGGG(z2, .(z0, z1), .(z2, z3)),U5_AGGG(z2, z0, z1, .(z2, z3), not_membercC_out_ggg(z2, z0, z1)) -> SUBSETCHECKEDB_IN_AGGG(z2, .(z0, z1), .(z2, z3))) (U5_AGGG(z4, z0, z1, .(z2, z3), not_membercC_out_ggg(z4, z0, z1)) -> SUBSETCHECKEDB_IN_AGGG(z4, .(z0, z1), .(z2, z3)),U5_AGGG(z4, z0, z1, .(z2, z3), not_membercC_out_ggg(z4, z0, z1)) -> SUBSETCHECKEDB_IN_AGGG(z4, .(z0, z1), .(z2, z3))) ---------------------------------------- (169) Obligation: Q DP problem: The TRS P consists of the following rules: SUBSETCHECKEDB_IN_AGGG(y0, y1, .(x0, x1)) -> U3_AGGG(y0, y1, .(x0, x1), membercA_out_ag(x0, .(x0, x1))) SUBSETCHECKEDB_IN_AGGG(y0, y1, .(x0, x1)) -> U3_AGGG(y0, y1, .(x0, x1), U91_ag(x0, x1, membercA_in_ag(x1))) U3_AGGG(z0, z1, .(z2, z3), membercA_out_ag(z2, .(z2, z3))) -> U5_AGGG(z2, z0, z1, .(z2, z3), not_membercC_out_ggg(z2, z0, z1)) U3_AGGG(z0, z1, .(z2, z3), membercA_out_ag(x3, .(z2, z3))) -> U5_AGGG(x3, z0, z1, .(z2, z3), not_membercC_out_ggg(x3, z0, z1)) U5_AGGG(z2, z0, z1, .(z2, z3), not_membercC_out_ggg(z2, z0, z1)) -> SUBSETCHECKEDB_IN_AGGG(z2, .(z0, z1), .(z2, z3)) U5_AGGG(z4, z0, z1, .(z2, z3), not_membercC_out_ggg(z4, z0, z1)) -> SUBSETCHECKEDB_IN_AGGG(z4, .(z0, z1), .(z2, z3)) The TRS R consists of the following rules: membercA_in_ag(.(X1, X2)) -> membercA_out_ag(X1, .(X1, X2)) membercA_in_ag(.(X2, X3)) -> U91_ag(X2, X3, membercA_in_ag(X3)) U91_ag(X2, X3, membercA_out_ag(X1, X3)) -> membercA_out_ag(X1, .(X2, X3)) The set Q consists of the following terms: membercA_in_ag(x0) U91_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (170) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule SUBSETCHECKEDB_IN_AGGG(y0, y1, .(x0, x1)) -> U3_AGGG(y0, y1, .(x0, x1), membercA_out_ag(x0, .(x0, x1))) we obtained the following new rules [LPAR04]: (SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z0, z3)) -> U3_AGGG(z0, .(z1, z2), .(z0, z3), membercA_out_ag(z0, .(z0, z3))),SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z0, z3)) -> U3_AGGG(z0, .(z1, z2), .(z0, z3), membercA_out_ag(z0, .(z0, z3)))) (SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z3, z4)) -> U3_AGGG(z0, .(z1, z2), .(z3, z4), membercA_out_ag(z3, .(z3, z4))),SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z3, z4)) -> U3_AGGG(z0, .(z1, z2), .(z3, z4), membercA_out_ag(z3, .(z3, z4)))) ---------------------------------------- (171) Obligation: Q DP problem: The TRS P consists of the following rules: SUBSETCHECKEDB_IN_AGGG(y0, y1, .(x0, x1)) -> U3_AGGG(y0, y1, .(x0, x1), U91_ag(x0, x1, membercA_in_ag(x1))) U3_AGGG(z0, z1, .(z2, z3), membercA_out_ag(z2, .(z2, z3))) -> U5_AGGG(z2, z0, z1, .(z2, z3), not_membercC_out_ggg(z2, z0, z1)) U3_AGGG(z0, z1, .(z2, z3), membercA_out_ag(x3, .(z2, z3))) -> U5_AGGG(x3, z0, z1, .(z2, z3), not_membercC_out_ggg(x3, z0, z1)) U5_AGGG(z2, z0, z1, .(z2, z3), not_membercC_out_ggg(z2, z0, z1)) -> SUBSETCHECKEDB_IN_AGGG(z2, .(z0, z1), .(z2, z3)) U5_AGGG(z4, z0, z1, .(z2, z3), not_membercC_out_ggg(z4, z0, z1)) -> SUBSETCHECKEDB_IN_AGGG(z4, .(z0, z1), .(z2, z3)) SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z0, z3)) -> U3_AGGG(z0, .(z1, z2), .(z0, z3), membercA_out_ag(z0, .(z0, z3))) SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z3, z4)) -> U3_AGGG(z0, .(z1, z2), .(z3, z4), membercA_out_ag(z3, .(z3, z4))) The TRS R consists of the following rules: membercA_in_ag(.(X1, X2)) -> membercA_out_ag(X1, .(X1, X2)) membercA_in_ag(.(X2, X3)) -> U91_ag(X2, X3, membercA_in_ag(X3)) U91_ag(X2, X3, membercA_out_ag(X1, X3)) -> membercA_out_ag(X1, .(X2, X3)) The set Q consists of the following terms: membercA_in_ag(x0) U91_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (172) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule SUBSETCHECKEDB_IN_AGGG(y0, y1, .(x0, x1)) -> U3_AGGG(y0, y1, .(x0, x1), U91_ag(x0, x1, membercA_in_ag(x1))) we obtained the following new rules [LPAR04]: (SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z0, z3)) -> U3_AGGG(z0, .(z1, z2), .(z0, z3), U91_ag(z0, z3, membercA_in_ag(z3))),SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z0, z3)) -> U3_AGGG(z0, .(z1, z2), .(z0, z3), U91_ag(z0, z3, membercA_in_ag(z3)))) (SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z3, z4)) -> U3_AGGG(z0, .(z1, z2), .(z3, z4), U91_ag(z3, z4, membercA_in_ag(z4))),SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z3, z4)) -> U3_AGGG(z0, .(z1, z2), .(z3, z4), U91_ag(z3, z4, membercA_in_ag(z4)))) ---------------------------------------- (173) Obligation: Q DP problem: The TRS P consists of the following rules: U3_AGGG(z0, z1, .(z2, z3), membercA_out_ag(z2, .(z2, z3))) -> U5_AGGG(z2, z0, z1, .(z2, z3), not_membercC_out_ggg(z2, z0, z1)) U3_AGGG(z0, z1, .(z2, z3), membercA_out_ag(x3, .(z2, z3))) -> U5_AGGG(x3, z0, z1, .(z2, z3), not_membercC_out_ggg(x3, z0, z1)) U5_AGGG(z2, z0, z1, .(z2, z3), not_membercC_out_ggg(z2, z0, z1)) -> SUBSETCHECKEDB_IN_AGGG(z2, .(z0, z1), .(z2, z3)) U5_AGGG(z4, z0, z1, .(z2, z3), not_membercC_out_ggg(z4, z0, z1)) -> SUBSETCHECKEDB_IN_AGGG(z4, .(z0, z1), .(z2, z3)) SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z0, z3)) -> U3_AGGG(z0, .(z1, z2), .(z0, z3), membercA_out_ag(z0, .(z0, z3))) SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z3, z4)) -> U3_AGGG(z0, .(z1, z2), .(z3, z4), membercA_out_ag(z3, .(z3, z4))) SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z0, z3)) -> U3_AGGG(z0, .(z1, z2), .(z0, z3), U91_ag(z0, z3, membercA_in_ag(z3))) SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z3, z4)) -> U3_AGGG(z0, .(z1, z2), .(z3, z4), U91_ag(z3, z4, membercA_in_ag(z4))) The TRS R consists of the following rules: membercA_in_ag(.(X1, X2)) -> membercA_out_ag(X1, .(X1, X2)) membercA_in_ag(.(X2, X3)) -> U91_ag(X2, X3, membercA_in_ag(X3)) U91_ag(X2, X3, membercA_out_ag(X1, X3)) -> membercA_out_ag(X1, .(X2, X3)) The set Q consists of the following terms: membercA_in_ag(x0) U91_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (174) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U3_AGGG(z0, z1, .(z2, z3), membercA_out_ag(z2, .(z2, z3))) -> U5_AGGG(z2, z0, z1, .(z2, z3), not_membercC_out_ggg(z2, z0, z1)) we obtained the following new rules [LPAR04]: (U3_AGGG(z0, .(z1, z2), .(z0, z3), membercA_out_ag(z0, .(z0, z3))) -> U5_AGGG(z0, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(z0, z0, .(z1, z2))),U3_AGGG(z0, .(z1, z2), .(z0, z3), membercA_out_ag(z0, .(z0, z3))) -> U5_AGGG(z0, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(z0, z0, .(z1, z2)))) (U3_AGGG(z0, .(z1, z2), .(z3, z4), membercA_out_ag(z3, .(z3, z4))) -> U5_AGGG(z3, z0, .(z1, z2), .(z3, z4), not_membercC_out_ggg(z3, z0, .(z1, z2))),U3_AGGG(z0, .(z1, z2), .(z3, z4), membercA_out_ag(z3, .(z3, z4))) -> U5_AGGG(z3, z0, .(z1, z2), .(z3, z4), not_membercC_out_ggg(z3, z0, .(z1, z2)))) ---------------------------------------- (175) Obligation: Q DP problem: The TRS P consists of the following rules: U3_AGGG(z0, z1, .(z2, z3), membercA_out_ag(x3, .(z2, z3))) -> U5_AGGG(x3, z0, z1, .(z2, z3), not_membercC_out_ggg(x3, z0, z1)) U5_AGGG(z2, z0, z1, .(z2, z3), not_membercC_out_ggg(z2, z0, z1)) -> SUBSETCHECKEDB_IN_AGGG(z2, .(z0, z1), .(z2, z3)) U5_AGGG(z4, z0, z1, .(z2, z3), not_membercC_out_ggg(z4, z0, z1)) -> SUBSETCHECKEDB_IN_AGGG(z4, .(z0, z1), .(z2, z3)) SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z0, z3)) -> U3_AGGG(z0, .(z1, z2), .(z0, z3), membercA_out_ag(z0, .(z0, z3))) SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z3, z4)) -> U3_AGGG(z0, .(z1, z2), .(z3, z4), membercA_out_ag(z3, .(z3, z4))) SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z0, z3)) -> U3_AGGG(z0, .(z1, z2), .(z0, z3), U91_ag(z0, z3, membercA_in_ag(z3))) SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z3, z4)) -> U3_AGGG(z0, .(z1, z2), .(z3, z4), U91_ag(z3, z4, membercA_in_ag(z4))) U3_AGGG(z0, .(z1, z2), .(z0, z3), membercA_out_ag(z0, .(z0, z3))) -> U5_AGGG(z0, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(z0, z0, .(z1, z2))) U3_AGGG(z0, .(z1, z2), .(z3, z4), membercA_out_ag(z3, .(z3, z4))) -> U5_AGGG(z3, z0, .(z1, z2), .(z3, z4), not_membercC_out_ggg(z3, z0, .(z1, z2))) The TRS R consists of the following rules: membercA_in_ag(.(X1, X2)) -> membercA_out_ag(X1, .(X1, X2)) membercA_in_ag(.(X2, X3)) -> U91_ag(X2, X3, membercA_in_ag(X3)) U91_ag(X2, X3, membercA_out_ag(X1, X3)) -> membercA_out_ag(X1, .(X2, X3)) The set Q consists of the following terms: membercA_in_ag(x0) U91_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (176) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U3_AGGG(z0, z1, .(z2, z3), membercA_out_ag(x3, .(z2, z3))) -> U5_AGGG(x3, z0, z1, .(z2, z3), not_membercC_out_ggg(x3, z0, z1)) we obtained the following new rules [LPAR04]: (U3_AGGG(z0, .(z1, z2), .(z0, z3), membercA_out_ag(z0, .(z0, z3))) -> U5_AGGG(z0, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(z0, z0, .(z1, z2))),U3_AGGG(z0, .(z1, z2), .(z0, z3), membercA_out_ag(z0, .(z0, z3))) -> U5_AGGG(z0, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(z0, z0, .(z1, z2)))) (U3_AGGG(z0, .(z1, z2), .(z3, z4), membercA_out_ag(z3, .(z3, z4))) -> U5_AGGG(z3, z0, .(z1, z2), .(z3, z4), not_membercC_out_ggg(z3, z0, .(z1, z2))),U3_AGGG(z0, .(z1, z2), .(z3, z4), membercA_out_ag(z3, .(z3, z4))) -> U5_AGGG(z3, z0, .(z1, z2), .(z3, z4), not_membercC_out_ggg(z3, z0, .(z1, z2)))) (U3_AGGG(z0, .(z1, z2), .(z0, z3), membercA_out_ag(x4, .(z0, z3))) -> U5_AGGG(x4, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(x4, z0, .(z1, z2))),U3_AGGG(z0, .(z1, z2), .(z0, z3), membercA_out_ag(x4, .(z0, z3))) -> U5_AGGG(x4, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(x4, z0, .(z1, z2)))) (U3_AGGG(z0, .(z1, z2), .(z3, z4), membercA_out_ag(x4, .(z3, z4))) -> U5_AGGG(x4, z0, .(z1, z2), .(z3, z4), not_membercC_out_ggg(x4, z0, .(z1, z2))),U3_AGGG(z0, .(z1, z2), .(z3, z4), membercA_out_ag(x4, .(z3, z4))) -> U5_AGGG(x4, z0, .(z1, z2), .(z3, z4), not_membercC_out_ggg(x4, z0, .(z1, z2)))) ---------------------------------------- (177) Obligation: Q DP problem: The TRS P consists of the following rules: U5_AGGG(z2, z0, z1, .(z2, z3), not_membercC_out_ggg(z2, z0, z1)) -> SUBSETCHECKEDB_IN_AGGG(z2, .(z0, z1), .(z2, z3)) U5_AGGG(z4, z0, z1, .(z2, z3), not_membercC_out_ggg(z4, z0, z1)) -> SUBSETCHECKEDB_IN_AGGG(z4, .(z0, z1), .(z2, z3)) SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z0, z3)) -> U3_AGGG(z0, .(z1, z2), .(z0, z3), membercA_out_ag(z0, .(z0, z3))) SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z3, z4)) -> U3_AGGG(z0, .(z1, z2), .(z3, z4), membercA_out_ag(z3, .(z3, z4))) SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z0, z3)) -> U3_AGGG(z0, .(z1, z2), .(z0, z3), U91_ag(z0, z3, membercA_in_ag(z3))) SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z3, z4)) -> U3_AGGG(z0, .(z1, z2), .(z3, z4), U91_ag(z3, z4, membercA_in_ag(z4))) U3_AGGG(z0, .(z1, z2), .(z0, z3), membercA_out_ag(z0, .(z0, z3))) -> U5_AGGG(z0, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(z0, z0, .(z1, z2))) U3_AGGG(z0, .(z1, z2), .(z3, z4), membercA_out_ag(z3, .(z3, z4))) -> U5_AGGG(z3, z0, .(z1, z2), .(z3, z4), not_membercC_out_ggg(z3, z0, .(z1, z2))) U3_AGGG(z0, .(z1, z2), .(z0, z3), membercA_out_ag(x4, .(z0, z3))) -> U5_AGGG(x4, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(x4, z0, .(z1, z2))) U3_AGGG(z0, .(z1, z2), .(z3, z4), membercA_out_ag(x4, .(z3, z4))) -> U5_AGGG(x4, z0, .(z1, z2), .(z3, z4), not_membercC_out_ggg(x4, z0, .(z1, z2))) The TRS R consists of the following rules: membercA_in_ag(.(X1, X2)) -> membercA_out_ag(X1, .(X1, X2)) membercA_in_ag(.(X2, X3)) -> U91_ag(X2, X3, membercA_in_ag(X3)) U91_ag(X2, X3, membercA_out_ag(X1, X3)) -> membercA_out_ag(X1, .(X2, X3)) The set Q consists of the following terms: membercA_in_ag(x0) U91_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (178) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U5_AGGG(z2, z0, z1, .(z2, z3), not_membercC_out_ggg(z2, z0, z1)) -> SUBSETCHECKEDB_IN_AGGG(z2, .(z0, z1), .(z2, z3)) we obtained the following new rules [LPAR04]: (U5_AGGG(z0, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(z0, z0, .(z1, z2))) -> SUBSETCHECKEDB_IN_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3)),U5_AGGG(z0, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(z0, z0, .(z1, z2))) -> SUBSETCHECKEDB_IN_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3))) (U5_AGGG(z3, z0, .(z1, z2), .(z3, z4), not_membercC_out_ggg(z3, z0, .(z1, z2))) -> SUBSETCHECKEDB_IN_AGGG(z3, .(z0, .(z1, z2)), .(z3, z4)),U5_AGGG(z3, z0, .(z1, z2), .(z3, z4), not_membercC_out_ggg(z3, z0, .(z1, z2))) -> SUBSETCHECKEDB_IN_AGGG(z3, .(z0, .(z1, z2)), .(z3, z4))) ---------------------------------------- (179) Obligation: Q DP problem: The TRS P consists of the following rules: U5_AGGG(z4, z0, z1, .(z2, z3), not_membercC_out_ggg(z4, z0, z1)) -> SUBSETCHECKEDB_IN_AGGG(z4, .(z0, z1), .(z2, z3)) SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z0, z3)) -> U3_AGGG(z0, .(z1, z2), .(z0, z3), membercA_out_ag(z0, .(z0, z3))) SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z3, z4)) -> U3_AGGG(z0, .(z1, z2), .(z3, z4), membercA_out_ag(z3, .(z3, z4))) SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z0, z3)) -> U3_AGGG(z0, .(z1, z2), .(z0, z3), U91_ag(z0, z3, membercA_in_ag(z3))) SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z3, z4)) -> U3_AGGG(z0, .(z1, z2), .(z3, z4), U91_ag(z3, z4, membercA_in_ag(z4))) U3_AGGG(z0, .(z1, z2), .(z0, z3), membercA_out_ag(z0, .(z0, z3))) -> U5_AGGG(z0, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(z0, z0, .(z1, z2))) U3_AGGG(z0, .(z1, z2), .(z3, z4), membercA_out_ag(z3, .(z3, z4))) -> U5_AGGG(z3, z0, .(z1, z2), .(z3, z4), not_membercC_out_ggg(z3, z0, .(z1, z2))) U3_AGGG(z0, .(z1, z2), .(z0, z3), membercA_out_ag(x4, .(z0, z3))) -> U5_AGGG(x4, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(x4, z0, .(z1, z2))) U3_AGGG(z0, .(z1, z2), .(z3, z4), membercA_out_ag(x4, .(z3, z4))) -> U5_AGGG(x4, z0, .(z1, z2), .(z3, z4), not_membercC_out_ggg(x4, z0, .(z1, z2))) U5_AGGG(z0, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(z0, z0, .(z1, z2))) -> SUBSETCHECKEDB_IN_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3)) U5_AGGG(z3, z0, .(z1, z2), .(z3, z4), not_membercC_out_ggg(z3, z0, .(z1, z2))) -> SUBSETCHECKEDB_IN_AGGG(z3, .(z0, .(z1, z2)), .(z3, z4)) The TRS R consists of the following rules: membercA_in_ag(.(X1, X2)) -> membercA_out_ag(X1, .(X1, X2)) membercA_in_ag(.(X2, X3)) -> U91_ag(X2, X3, membercA_in_ag(X3)) U91_ag(X2, X3, membercA_out_ag(X1, X3)) -> membercA_out_ag(X1, .(X2, X3)) The set Q consists of the following terms: membercA_in_ag(x0) U91_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (180) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U5_AGGG(z4, z0, z1, .(z2, z3), not_membercC_out_ggg(z4, z0, z1)) -> SUBSETCHECKEDB_IN_AGGG(z4, .(z0, z1), .(z2, z3)) we obtained the following new rules [LPAR04]: (U5_AGGG(z0, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(z0, z0, .(z1, z2))) -> SUBSETCHECKEDB_IN_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3)),U5_AGGG(z0, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(z0, z0, .(z1, z2))) -> SUBSETCHECKEDB_IN_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3))) (U5_AGGG(z3, z0, .(z1, z2), .(z3, z4), not_membercC_out_ggg(z3, z0, .(z1, z2))) -> SUBSETCHECKEDB_IN_AGGG(z3, .(z0, .(z1, z2)), .(z3, z4)),U5_AGGG(z3, z0, .(z1, z2), .(z3, z4), not_membercC_out_ggg(z3, z0, .(z1, z2))) -> SUBSETCHECKEDB_IN_AGGG(z3, .(z0, .(z1, z2)), .(z3, z4))) (U5_AGGG(z4, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(z4, z0, .(z1, z2))) -> SUBSETCHECKEDB_IN_AGGG(z4, .(z0, .(z1, z2)), .(z0, z3)),U5_AGGG(z4, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(z4, z0, .(z1, z2))) -> SUBSETCHECKEDB_IN_AGGG(z4, .(z0, .(z1, z2)), .(z0, z3))) (U5_AGGG(z5, z0, .(z1, z2), .(z3, z4), not_membercC_out_ggg(z5, z0, .(z1, z2))) -> SUBSETCHECKEDB_IN_AGGG(z5, .(z0, .(z1, z2)), .(z3, z4)),U5_AGGG(z5, z0, .(z1, z2), .(z3, z4), not_membercC_out_ggg(z5, z0, .(z1, z2))) -> SUBSETCHECKEDB_IN_AGGG(z5, .(z0, .(z1, z2)), .(z3, z4))) ---------------------------------------- (181) Obligation: Q DP problem: The TRS P consists of the following rules: SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z0, z3)) -> U3_AGGG(z0, .(z1, z2), .(z0, z3), membercA_out_ag(z0, .(z0, z3))) SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z3, z4)) -> U3_AGGG(z0, .(z1, z2), .(z3, z4), membercA_out_ag(z3, .(z3, z4))) SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z0, z3)) -> U3_AGGG(z0, .(z1, z2), .(z0, z3), U91_ag(z0, z3, membercA_in_ag(z3))) SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z3, z4)) -> U3_AGGG(z0, .(z1, z2), .(z3, z4), U91_ag(z3, z4, membercA_in_ag(z4))) U3_AGGG(z0, .(z1, z2), .(z0, z3), membercA_out_ag(z0, .(z0, z3))) -> U5_AGGG(z0, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(z0, z0, .(z1, z2))) U3_AGGG(z0, .(z1, z2), .(z3, z4), membercA_out_ag(z3, .(z3, z4))) -> U5_AGGG(z3, z0, .(z1, z2), .(z3, z4), not_membercC_out_ggg(z3, z0, .(z1, z2))) U3_AGGG(z0, .(z1, z2), .(z0, z3), membercA_out_ag(x4, .(z0, z3))) -> U5_AGGG(x4, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(x4, z0, .(z1, z2))) U3_AGGG(z0, .(z1, z2), .(z3, z4), membercA_out_ag(x4, .(z3, z4))) -> U5_AGGG(x4, z0, .(z1, z2), .(z3, z4), not_membercC_out_ggg(x4, z0, .(z1, z2))) U5_AGGG(z0, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(z0, z0, .(z1, z2))) -> SUBSETCHECKEDB_IN_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3)) U5_AGGG(z3, z0, .(z1, z2), .(z3, z4), not_membercC_out_ggg(z3, z0, .(z1, z2))) -> SUBSETCHECKEDB_IN_AGGG(z3, .(z0, .(z1, z2)), .(z3, z4)) U5_AGGG(z4, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(z4, z0, .(z1, z2))) -> SUBSETCHECKEDB_IN_AGGG(z4, .(z0, .(z1, z2)), .(z0, z3)) U5_AGGG(z5, z0, .(z1, z2), .(z3, z4), not_membercC_out_ggg(z5, z0, .(z1, z2))) -> SUBSETCHECKEDB_IN_AGGG(z5, .(z0, .(z1, z2)), .(z3, z4)) The TRS R consists of the following rules: membercA_in_ag(.(X1, X2)) -> membercA_out_ag(X1, .(X1, X2)) membercA_in_ag(.(X2, X3)) -> U91_ag(X2, X3, membercA_in_ag(X3)) U91_ag(X2, X3, membercA_out_ag(X1, X3)) -> membercA_out_ag(X1, .(X2, X3)) The set Q consists of the following terms: membercA_in_ag(x0) U91_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (182) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z0, z3)) -> U3_AGGG(z0, .(z1, z2), .(z0, z3), membercA_out_ag(z0, .(z0, z3))) we obtained the following new rules [LPAR04]: (SUBSETCHECKEDB_IN_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3)) -> U3_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3), membercA_out_ag(z0, .(z0, z3))),SUBSETCHECKEDB_IN_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3)) -> U3_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3), membercA_out_ag(z0, .(z0, z3)))) (SUBSETCHECKEDB_IN_AGGG(z0, .(z1, .(z2, z3)), .(z0, z4)) -> U3_AGGG(z0, .(z1, .(z2, z3)), .(z0, z4), membercA_out_ag(z0, .(z0, z4))),SUBSETCHECKEDB_IN_AGGG(z0, .(z1, .(z2, z3)), .(z0, z4)) -> U3_AGGG(z0, .(z1, .(z2, z3)), .(z0, z4), membercA_out_ag(z0, .(z0, z4)))) ---------------------------------------- (183) Obligation: Q DP problem: The TRS P consists of the following rules: SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z3, z4)) -> U3_AGGG(z0, .(z1, z2), .(z3, z4), membercA_out_ag(z3, .(z3, z4))) SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z0, z3)) -> U3_AGGG(z0, .(z1, z2), .(z0, z3), U91_ag(z0, z3, membercA_in_ag(z3))) SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z3, z4)) -> U3_AGGG(z0, .(z1, z2), .(z3, z4), U91_ag(z3, z4, membercA_in_ag(z4))) U3_AGGG(z0, .(z1, z2), .(z0, z3), membercA_out_ag(z0, .(z0, z3))) -> U5_AGGG(z0, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(z0, z0, .(z1, z2))) U3_AGGG(z0, .(z1, z2), .(z3, z4), membercA_out_ag(z3, .(z3, z4))) -> U5_AGGG(z3, z0, .(z1, z2), .(z3, z4), not_membercC_out_ggg(z3, z0, .(z1, z2))) U3_AGGG(z0, .(z1, z2), .(z0, z3), membercA_out_ag(x4, .(z0, z3))) -> U5_AGGG(x4, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(x4, z0, .(z1, z2))) U3_AGGG(z0, .(z1, z2), .(z3, z4), membercA_out_ag(x4, .(z3, z4))) -> U5_AGGG(x4, z0, .(z1, z2), .(z3, z4), not_membercC_out_ggg(x4, z0, .(z1, z2))) U5_AGGG(z0, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(z0, z0, .(z1, z2))) -> SUBSETCHECKEDB_IN_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3)) U5_AGGG(z3, z0, .(z1, z2), .(z3, z4), not_membercC_out_ggg(z3, z0, .(z1, z2))) -> SUBSETCHECKEDB_IN_AGGG(z3, .(z0, .(z1, z2)), .(z3, z4)) U5_AGGG(z4, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(z4, z0, .(z1, z2))) -> SUBSETCHECKEDB_IN_AGGG(z4, .(z0, .(z1, z2)), .(z0, z3)) U5_AGGG(z5, z0, .(z1, z2), .(z3, z4), not_membercC_out_ggg(z5, z0, .(z1, z2))) -> SUBSETCHECKEDB_IN_AGGG(z5, .(z0, .(z1, z2)), .(z3, z4)) SUBSETCHECKEDB_IN_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3)) -> U3_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3), membercA_out_ag(z0, .(z0, z3))) SUBSETCHECKEDB_IN_AGGG(z0, .(z1, .(z2, z3)), .(z0, z4)) -> U3_AGGG(z0, .(z1, .(z2, z3)), .(z0, z4), membercA_out_ag(z0, .(z0, z4))) The TRS R consists of the following rules: membercA_in_ag(.(X1, X2)) -> membercA_out_ag(X1, .(X1, X2)) membercA_in_ag(.(X2, X3)) -> U91_ag(X2, X3, membercA_in_ag(X3)) U91_ag(X2, X3, membercA_out_ag(X1, X3)) -> membercA_out_ag(X1, .(X2, X3)) The set Q consists of the following terms: membercA_in_ag(x0) U91_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (184) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z3, z4)) -> U3_AGGG(z0, .(z1, z2), .(z3, z4), membercA_out_ag(z3, .(z3, z4))) we obtained the following new rules [LPAR04]: (SUBSETCHECKEDB_IN_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3)) -> U3_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3), membercA_out_ag(z0, .(z0, z3))),SUBSETCHECKEDB_IN_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3)) -> U3_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3), membercA_out_ag(z0, .(z0, z3)))) (SUBSETCHECKEDB_IN_AGGG(z0, .(z1, .(z2, z3)), .(z0, z4)) -> U3_AGGG(z0, .(z1, .(z2, z3)), .(z0, z4), membercA_out_ag(z0, .(z0, z4))),SUBSETCHECKEDB_IN_AGGG(z0, .(z1, .(z2, z3)), .(z0, z4)) -> U3_AGGG(z0, .(z1, .(z2, z3)), .(z0, z4), membercA_out_ag(z0, .(z0, z4)))) (SUBSETCHECKEDB_IN_AGGG(z0, .(z1, .(z2, z3)), .(z1, z4)) -> U3_AGGG(z0, .(z1, .(z2, z3)), .(z1, z4), membercA_out_ag(z1, .(z1, z4))),SUBSETCHECKEDB_IN_AGGG(z0, .(z1, .(z2, z3)), .(z1, z4)) -> U3_AGGG(z0, .(z1, .(z2, z3)), .(z1, z4), membercA_out_ag(z1, .(z1, z4)))) (SUBSETCHECKEDB_IN_AGGG(z0, .(z1, .(z2, z3)), .(z4, z5)) -> U3_AGGG(z0, .(z1, .(z2, z3)), .(z4, z5), membercA_out_ag(z4, .(z4, z5))),SUBSETCHECKEDB_IN_AGGG(z0, .(z1, .(z2, z3)), .(z4, z5)) -> U3_AGGG(z0, .(z1, .(z2, z3)), .(z4, z5), membercA_out_ag(z4, .(z4, z5)))) ---------------------------------------- (185) Obligation: Q DP problem: The TRS P consists of the following rules: SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z0, z3)) -> U3_AGGG(z0, .(z1, z2), .(z0, z3), U91_ag(z0, z3, membercA_in_ag(z3))) SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z3, z4)) -> U3_AGGG(z0, .(z1, z2), .(z3, z4), U91_ag(z3, z4, membercA_in_ag(z4))) U3_AGGG(z0, .(z1, z2), .(z0, z3), membercA_out_ag(z0, .(z0, z3))) -> U5_AGGG(z0, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(z0, z0, .(z1, z2))) U3_AGGG(z0, .(z1, z2), .(z3, z4), membercA_out_ag(z3, .(z3, z4))) -> U5_AGGG(z3, z0, .(z1, z2), .(z3, z4), not_membercC_out_ggg(z3, z0, .(z1, z2))) U3_AGGG(z0, .(z1, z2), .(z0, z3), membercA_out_ag(x4, .(z0, z3))) -> U5_AGGG(x4, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(x4, z0, .(z1, z2))) U3_AGGG(z0, .(z1, z2), .(z3, z4), membercA_out_ag(x4, .(z3, z4))) -> U5_AGGG(x4, z0, .(z1, z2), .(z3, z4), not_membercC_out_ggg(x4, z0, .(z1, z2))) U5_AGGG(z0, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(z0, z0, .(z1, z2))) -> SUBSETCHECKEDB_IN_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3)) U5_AGGG(z3, z0, .(z1, z2), .(z3, z4), not_membercC_out_ggg(z3, z0, .(z1, z2))) -> SUBSETCHECKEDB_IN_AGGG(z3, .(z0, .(z1, z2)), .(z3, z4)) U5_AGGG(z4, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(z4, z0, .(z1, z2))) -> SUBSETCHECKEDB_IN_AGGG(z4, .(z0, .(z1, z2)), .(z0, z3)) U5_AGGG(z5, z0, .(z1, z2), .(z3, z4), not_membercC_out_ggg(z5, z0, .(z1, z2))) -> SUBSETCHECKEDB_IN_AGGG(z5, .(z0, .(z1, z2)), .(z3, z4)) SUBSETCHECKEDB_IN_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3)) -> U3_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3), membercA_out_ag(z0, .(z0, z3))) SUBSETCHECKEDB_IN_AGGG(z0, .(z1, .(z2, z3)), .(z0, z4)) -> U3_AGGG(z0, .(z1, .(z2, z3)), .(z0, z4), membercA_out_ag(z0, .(z0, z4))) SUBSETCHECKEDB_IN_AGGG(z0, .(z1, .(z2, z3)), .(z1, z4)) -> U3_AGGG(z0, .(z1, .(z2, z3)), .(z1, z4), membercA_out_ag(z1, .(z1, z4))) SUBSETCHECKEDB_IN_AGGG(z0, .(z1, .(z2, z3)), .(z4, z5)) -> U3_AGGG(z0, .(z1, .(z2, z3)), .(z4, z5), membercA_out_ag(z4, .(z4, z5))) The TRS R consists of the following rules: membercA_in_ag(.(X1, X2)) -> membercA_out_ag(X1, .(X1, X2)) membercA_in_ag(.(X2, X3)) -> U91_ag(X2, X3, membercA_in_ag(X3)) U91_ag(X2, X3, membercA_out_ag(X1, X3)) -> membercA_out_ag(X1, .(X2, X3)) The set Q consists of the following terms: membercA_in_ag(x0) U91_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (186) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z0, z3)) -> U3_AGGG(z0, .(z1, z2), .(z0, z3), U91_ag(z0, z3, membercA_in_ag(z3))) we obtained the following new rules [LPAR04]: (SUBSETCHECKEDB_IN_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3)) -> U3_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3), U91_ag(z0, z3, membercA_in_ag(z3))),SUBSETCHECKEDB_IN_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3)) -> U3_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3), U91_ag(z0, z3, membercA_in_ag(z3)))) (SUBSETCHECKEDB_IN_AGGG(z0, .(z1, .(z2, z3)), .(z0, z4)) -> U3_AGGG(z0, .(z1, .(z2, z3)), .(z0, z4), U91_ag(z0, z4, membercA_in_ag(z4))),SUBSETCHECKEDB_IN_AGGG(z0, .(z1, .(z2, z3)), .(z0, z4)) -> U3_AGGG(z0, .(z1, .(z2, z3)), .(z0, z4), U91_ag(z0, z4, membercA_in_ag(z4)))) ---------------------------------------- (187) Obligation: Q DP problem: The TRS P consists of the following rules: SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z3, z4)) -> U3_AGGG(z0, .(z1, z2), .(z3, z4), U91_ag(z3, z4, membercA_in_ag(z4))) U3_AGGG(z0, .(z1, z2), .(z0, z3), membercA_out_ag(z0, .(z0, z3))) -> U5_AGGG(z0, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(z0, z0, .(z1, z2))) U3_AGGG(z0, .(z1, z2), .(z3, z4), membercA_out_ag(z3, .(z3, z4))) -> U5_AGGG(z3, z0, .(z1, z2), .(z3, z4), not_membercC_out_ggg(z3, z0, .(z1, z2))) U3_AGGG(z0, .(z1, z2), .(z0, z3), membercA_out_ag(x4, .(z0, z3))) -> U5_AGGG(x4, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(x4, z0, .(z1, z2))) U3_AGGG(z0, .(z1, z2), .(z3, z4), membercA_out_ag(x4, .(z3, z4))) -> U5_AGGG(x4, z0, .(z1, z2), .(z3, z4), not_membercC_out_ggg(x4, z0, .(z1, z2))) U5_AGGG(z0, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(z0, z0, .(z1, z2))) -> SUBSETCHECKEDB_IN_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3)) U5_AGGG(z3, z0, .(z1, z2), .(z3, z4), not_membercC_out_ggg(z3, z0, .(z1, z2))) -> SUBSETCHECKEDB_IN_AGGG(z3, .(z0, .(z1, z2)), .(z3, z4)) U5_AGGG(z4, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(z4, z0, .(z1, z2))) -> SUBSETCHECKEDB_IN_AGGG(z4, .(z0, .(z1, z2)), .(z0, z3)) U5_AGGG(z5, z0, .(z1, z2), .(z3, z4), not_membercC_out_ggg(z5, z0, .(z1, z2))) -> SUBSETCHECKEDB_IN_AGGG(z5, .(z0, .(z1, z2)), .(z3, z4)) SUBSETCHECKEDB_IN_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3)) -> U3_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3), membercA_out_ag(z0, .(z0, z3))) SUBSETCHECKEDB_IN_AGGG(z0, .(z1, .(z2, z3)), .(z0, z4)) -> U3_AGGG(z0, .(z1, .(z2, z3)), .(z0, z4), membercA_out_ag(z0, .(z0, z4))) SUBSETCHECKEDB_IN_AGGG(z0, .(z1, .(z2, z3)), .(z1, z4)) -> U3_AGGG(z0, .(z1, .(z2, z3)), .(z1, z4), membercA_out_ag(z1, .(z1, z4))) SUBSETCHECKEDB_IN_AGGG(z0, .(z1, .(z2, z3)), .(z4, z5)) -> U3_AGGG(z0, .(z1, .(z2, z3)), .(z4, z5), membercA_out_ag(z4, .(z4, z5))) SUBSETCHECKEDB_IN_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3)) -> U3_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3), U91_ag(z0, z3, membercA_in_ag(z3))) SUBSETCHECKEDB_IN_AGGG(z0, .(z1, .(z2, z3)), .(z0, z4)) -> U3_AGGG(z0, .(z1, .(z2, z3)), .(z0, z4), U91_ag(z0, z4, membercA_in_ag(z4))) The TRS R consists of the following rules: membercA_in_ag(.(X1, X2)) -> membercA_out_ag(X1, .(X1, X2)) membercA_in_ag(.(X2, X3)) -> U91_ag(X2, X3, membercA_in_ag(X3)) U91_ag(X2, X3, membercA_out_ag(X1, X3)) -> membercA_out_ag(X1, .(X2, X3)) The set Q consists of the following terms: membercA_in_ag(x0) U91_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (188) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule SUBSETCHECKEDB_IN_AGGG(z0, .(z1, z2), .(z3, z4)) -> U3_AGGG(z0, .(z1, z2), .(z3, z4), U91_ag(z3, z4, membercA_in_ag(z4))) we obtained the following new rules [LPAR04]: (SUBSETCHECKEDB_IN_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3)) -> U3_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3), U91_ag(z0, z3, membercA_in_ag(z3))),SUBSETCHECKEDB_IN_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3)) -> U3_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3), U91_ag(z0, z3, membercA_in_ag(z3)))) (SUBSETCHECKEDB_IN_AGGG(z0, .(z1, .(z2, z3)), .(z0, z4)) -> U3_AGGG(z0, .(z1, .(z2, z3)), .(z0, z4), U91_ag(z0, z4, membercA_in_ag(z4))),SUBSETCHECKEDB_IN_AGGG(z0, .(z1, .(z2, z3)), .(z0, z4)) -> U3_AGGG(z0, .(z1, .(z2, z3)), .(z0, z4), U91_ag(z0, z4, membercA_in_ag(z4)))) (SUBSETCHECKEDB_IN_AGGG(z0, .(z1, .(z2, z3)), .(z1, z4)) -> U3_AGGG(z0, .(z1, .(z2, z3)), .(z1, z4), U91_ag(z1, z4, membercA_in_ag(z4))),SUBSETCHECKEDB_IN_AGGG(z0, .(z1, .(z2, z3)), .(z1, z4)) -> U3_AGGG(z0, .(z1, .(z2, z3)), .(z1, z4), U91_ag(z1, z4, membercA_in_ag(z4)))) (SUBSETCHECKEDB_IN_AGGG(z0, .(z1, .(z2, z3)), .(z4, z5)) -> U3_AGGG(z0, .(z1, .(z2, z3)), .(z4, z5), U91_ag(z4, z5, membercA_in_ag(z5))),SUBSETCHECKEDB_IN_AGGG(z0, .(z1, .(z2, z3)), .(z4, z5)) -> U3_AGGG(z0, .(z1, .(z2, z3)), .(z4, z5), U91_ag(z4, z5, membercA_in_ag(z5)))) ---------------------------------------- (189) Obligation: Q DP problem: The TRS P consists of the following rules: U3_AGGG(z0, .(z1, z2), .(z0, z3), membercA_out_ag(z0, .(z0, z3))) -> U5_AGGG(z0, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(z0, z0, .(z1, z2))) U3_AGGG(z0, .(z1, z2), .(z3, z4), membercA_out_ag(z3, .(z3, z4))) -> U5_AGGG(z3, z0, .(z1, z2), .(z3, z4), not_membercC_out_ggg(z3, z0, .(z1, z2))) U3_AGGG(z0, .(z1, z2), .(z0, z3), membercA_out_ag(x4, .(z0, z3))) -> U5_AGGG(x4, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(x4, z0, .(z1, z2))) U3_AGGG(z0, .(z1, z2), .(z3, z4), membercA_out_ag(x4, .(z3, z4))) -> U5_AGGG(x4, z0, .(z1, z2), .(z3, z4), not_membercC_out_ggg(x4, z0, .(z1, z2))) U5_AGGG(z0, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(z0, z0, .(z1, z2))) -> SUBSETCHECKEDB_IN_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3)) U5_AGGG(z3, z0, .(z1, z2), .(z3, z4), not_membercC_out_ggg(z3, z0, .(z1, z2))) -> SUBSETCHECKEDB_IN_AGGG(z3, .(z0, .(z1, z2)), .(z3, z4)) U5_AGGG(z4, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(z4, z0, .(z1, z2))) -> SUBSETCHECKEDB_IN_AGGG(z4, .(z0, .(z1, z2)), .(z0, z3)) U5_AGGG(z5, z0, .(z1, z2), .(z3, z4), not_membercC_out_ggg(z5, z0, .(z1, z2))) -> SUBSETCHECKEDB_IN_AGGG(z5, .(z0, .(z1, z2)), .(z3, z4)) SUBSETCHECKEDB_IN_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3)) -> U3_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3), membercA_out_ag(z0, .(z0, z3))) SUBSETCHECKEDB_IN_AGGG(z0, .(z1, .(z2, z3)), .(z0, z4)) -> U3_AGGG(z0, .(z1, .(z2, z3)), .(z0, z4), membercA_out_ag(z0, .(z0, z4))) SUBSETCHECKEDB_IN_AGGG(z0, .(z1, .(z2, z3)), .(z1, z4)) -> U3_AGGG(z0, .(z1, .(z2, z3)), .(z1, z4), membercA_out_ag(z1, .(z1, z4))) SUBSETCHECKEDB_IN_AGGG(z0, .(z1, .(z2, z3)), .(z4, z5)) -> U3_AGGG(z0, .(z1, .(z2, z3)), .(z4, z5), membercA_out_ag(z4, .(z4, z5))) SUBSETCHECKEDB_IN_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3)) -> U3_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3), U91_ag(z0, z3, membercA_in_ag(z3))) SUBSETCHECKEDB_IN_AGGG(z0, .(z1, .(z2, z3)), .(z0, z4)) -> U3_AGGG(z0, .(z1, .(z2, z3)), .(z0, z4), U91_ag(z0, z4, membercA_in_ag(z4))) SUBSETCHECKEDB_IN_AGGG(z0, .(z1, .(z2, z3)), .(z1, z4)) -> U3_AGGG(z0, .(z1, .(z2, z3)), .(z1, z4), U91_ag(z1, z4, membercA_in_ag(z4))) SUBSETCHECKEDB_IN_AGGG(z0, .(z1, .(z2, z3)), .(z4, z5)) -> U3_AGGG(z0, .(z1, .(z2, z3)), .(z4, z5), U91_ag(z4, z5, membercA_in_ag(z5))) The TRS R consists of the following rules: membercA_in_ag(.(X1, X2)) -> membercA_out_ag(X1, .(X1, X2)) membercA_in_ag(.(X2, X3)) -> U91_ag(X2, X3, membercA_in_ag(X3)) U91_ag(X2, X3, membercA_out_ag(X1, X3)) -> membercA_out_ag(X1, .(X2, X3)) The set Q consists of the following terms: membercA_in_ag(x0) U91_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (190) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by narrowing to the left: s = U5_AGGG(z0, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(z0, z0, .(z1, z2))) evaluates to t =U5_AGGG(z0, z0, .(z0, .(z1, z2)), .(z0, z3), not_membercC_out_ggg(z0, z0, .(z0, .(z1, z2)))) Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [z1 / z0, z2 / .(z1, z2)] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence U5_AGGG(z0, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(z0, z0, .(z1, z2))) -> SUBSETCHECKEDB_IN_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3)) with rule U5_AGGG(z0', z0', .(z1', z2'), .(z0', z3'), not_membercC_out_ggg(z0', z0', .(z1', z2'))) -> SUBSETCHECKEDB_IN_AGGG(z0', .(z0', .(z1', z2')), .(z0', z3')) at position [] and matcher [z0' / z0, z1' / z1, z2' / z2, z3' / z3] SUBSETCHECKEDB_IN_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3)) -> U3_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3), membercA_out_ag(z0, .(z0, z3))) with rule SUBSETCHECKEDB_IN_AGGG(z0', .(z0', .(z1', z2')), .(z0', z3')) -> U3_AGGG(z0', .(z0', .(z1', z2')), .(z0', z3'), membercA_out_ag(z0', .(z0', z3'))) at position [] and matcher [z0' / z0, z1' / z1, z2' / z2, z3' / z3] U3_AGGG(z0, .(z0, .(z1, z2)), .(z0, z3), membercA_out_ag(z0, .(z0, z3))) -> U5_AGGG(z0, z0, .(z0, .(z1, z2)), .(z0, z3), not_membercC_out_ggg(z0, z0, .(z0, .(z1, z2)))) with rule U3_AGGG(z0, .(z1, z2), .(z0, z3), membercA_out_ag(z0, .(z0, z3))) -> U5_AGGG(z0, z0, .(z1, z2), .(z0, z3), not_membercC_out_ggg(z0, z0, .(z1, z2))) Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence All these steps are and every following step will be a correct step w.r.t to Q. ---------------------------------------- (191) NO ---------------------------------------- (192) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 3, "program": { "directives": [], "clauses": [ [ "(subset X Y)", "(subsetchecked X ([]) Y)" ], [ "(subsetchecked ([]) X1 X2)", null ], [ "(subsetchecked (. X Xs) Ys Zs)", "(',' (member X Zs) (',' (not_member X Ys) (subsetchecked Xs (. X Ys) Zs)))" ], [ "(member X (. X X3))", null ], [ "(member X (. X4 Xs))", "(member X Xs)" ], [ "(not_member X Y)", "(',' (member X Y) (',' (!) (failure (a))))" ], [ "(not_member X5 X6)", null ], [ "(failure (b))", null ] ] }, "graph": { "nodes": { "909": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (member T1028 (. T1029 (. T1030 (. T1031 (. T1032 (. T1033 (. T1034 ([])))))))) (',' (!_28) (failure (a))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1028", "T1029", "T1030", "T1031", "T1032", "T1033", "T1034" ], "free": [], "exprvars": [] } }, "590": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "591": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (member T329 T328) (',' (not_member T329 (. T325 (. T326 (. T327 ([]))))) (subsetchecked T330 (. T329 (. T325 (. T326 (. T327 ([]))))) T328)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T325", "T326", "T327", "T328" ], "free": [], "exprvars": [] } }, "350": { "goal": [{ "clause": 7, "scope": 6, "term": "(failure (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "471": { "goal": [{ "clause": -1, "scope": -1, "term": "(failure (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "592": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "351": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "472": { "goal": [{ "clause": 7, "scope": 10, "term": "(failure (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "593": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T329 T328)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T328"], "free": [], "exprvars": [] } }, "352": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "473": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "594": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (not_member T335 (. 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"goal": [{ "clause": 4, "scope": 5, "term": "(member T75 ([]))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T75"], "free": [], "exprvars": [] } }, "589": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "348": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "349": { "goal": [{ "clause": -1, "scope": -1, "term": "(failure (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 3, "to": 23, "label": "CASE" }, { "from": 23, "to": 24, "label": "ONLY EVAL with clause\nsubset(X15, X16) :- subsetchecked(X15, [], X16).\nand substitutionT1 -> T12,\nX15 -> T12,\nT2 -> T11,\nX16 -> T11,\nT10 -> T12" }, { "from": 24, "to": 41, "label": "CASE" }, { "from": 41, "to": 42, "label": "PARALLEL" }, { "from": 41, "to": 43, "label": "PARALLEL" }, { "from": 42, "to": 44, "label": "EVAL with clause\nsubsetchecked([], X29, X30).\nand substitutionT12 -> [],\nX29 -> [],\nT11 -> T19,\nX30 -> T19" }, { "from": 42, "to": 45, "label": "EVAL-BACKTRACK" }, { "from": 43, "to": 150, "label": "EVAL with clause\nsubsetchecked(.(X39, X40), X41, X42) :- ','(member(X39, X42), ','(not_member(X39, X41), subsetchecked(X40, .(X39, X41), X42))).\nand substitutionX39 -> T29,\nX40 -> T30,\nT12 -> .(T29, T30),\nX41 -> [],\nT11 -> T28,\nX42 -> T28,\nT26 -> T29,\nT27 -> T30" }, { "from": 43, "to": 151, "label": "EVAL-BACKTRACK" }, { "from": 44, "to": 52, "label": "SUCCESS" }, { "from": 150, "to": 235, "label": "SPLIT 1" }, { "from": 150, "to": 236, "label": "SPLIT 2\nnew knowledge:\nT35 is ground\nT28 is ground\nreplacements:T29 -> T35,\nT30 -> T36" }, { "from": 235, "to": 237, "label": "CASE" }, { "from": 236, "to": 289, "label": "SPLIT 1" }, { "from": 236, "to": 290, "label": "SPLIT 2\nnew knowledge:\nT35 is ground" }, { "from": 237, "to": 240, "label": "PARALLEL" }, { "from": 237, "to": 241, "label": "PARALLEL" }, { "from": 240, "to": 243, "label": "EVAL with clause\nmember(X59, .(X59, X60)).\nand substitutionT29 -> T49,\nX59 -> T49,\nX60 -> T50,\nT28 -> .(T49, T50)" }, { "from": 240, "to": 245, "label": "EVAL-BACKTRACK" }, { "from": 241, "to": 250, "label": "EVAL with clause\nmember(X67, .(X68, X69)) :- member(X67, X69).\nand substitutionT29 -> T60,\nX67 -> T60,\nX68 -> T58,\nX69 -> T59,\nT28 -> .(T58, T59),\nT57 -> T60" }, { "from": 241, "to": 251, "label": "EVAL-BACKTRACK" }, { "from": 243, "to": 246, "label": "SUCCESS" }, { "from": 250, "to": 235, "label": "INSTANCE with matching:\nT29 -> T60\nT28 -> T59" }, { "from": 289, "to": 293, "label": "CASE" }, { "from": 290, "to": 354, "label": "CASE" }, { "from": 293, "to": 295, "label": "PARALLEL" }, { "from": 293, "to": 296, "label": "PARALLEL" }, { "from": 295, "to": 299, "label": "ONLY EVAL with clause\nnot_member(X94, X95) :- ','(member(X94, X95), ','(!_4, failure(a))).\nand substitutionT35 -> T75,\nX94 -> T75,\nX95 -> []" }, { "from": 296, "to": 352, "label": "ONLY EVAL with clause\nnot_member(X113, X114).\nand substitutionT35 -> T84,\nX113 -> T84,\nX114 -> []" }, { "from": 299, "to": 316, "label": "SPLIT 1" }, { "from": 299, "to": 317, "label": "SPLIT 2\nnew knowledge:\nT75 is ground" }, { "from": 316, "to": 346, "label": "CASE" }, { "from": 317, "to": 349, "label": "CUT" }, { "from": 346, "to": 347, "label": "BACKTRACK\nfor clause: member(X, .(X, X3))because of non-unification" }, { "from": 347, "to": 348, "label": "BACKTRACK\nfor clause: member(X, .(X4, Xs)) :- member(X, Xs)because of non-unification" }, { "from": 349, "to": 350, "label": "CASE" }, { "from": 350, "to": 351, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 352, "to": 353, "label": "SUCCESS" }, { "from": 354, "to": 358, "label": "PARALLEL" }, { "from": 354, "to": 364, "label": "PARALLEL" }, { "from": 358, "to": 402, "label": "EVAL with clause\nsubsetchecked([], X127, X128).\nand substitutionT36 -> [],\nT35 -> T97,\nX127 -> .(T97, []),\nT28 -> T98,\nX128 -> T98" }, { "from": 358, "to": 403, "label": "EVAL-BACKTRACK" }, { "from": 364, "to": 408, "label": "EVAL with clause\nsubsetchecked(.(X137, X138), X139, X140) :- ','(member(X137, X140), ','(not_member(X137, X139), subsetchecked(X138, .(X137, X139), X140))).\nand substitutionX137 -> T111,\nX138 -> T112,\nT36 -> .(T111, T112),\nT35 -> T109,\nX139 -> .(T109, []),\nT28 -> T110,\nX140 -> T110,\nT107 -> T111,\nT108 -> T112" }, { "from": 364, "to": 409, "label": "EVAL-BACKTRACK" }, { "from": 402, "to": 405, "label": "SUCCESS" }, { "from": 408, "to": 410, "label": "SPLIT 1" }, { "from": 408, "to": 411, "label": "SPLIT 2\nnew knowledge:\nT117 is ground\nT110 is ground\nreplacements:T111 -> T117,\nT112 -> T118" }, { "from": 410, "to": 235, "label": "INSTANCE with matching:\nT29 -> T111\nT28 -> T110" }, { "from": 411, "to": 412, "label": "SPLIT 1" }, { "from": 411, "to": 413, "label": "SPLIT 2\nnew knowledge:\nT117 is ground\nT109 is ground" }, { "from": 412, "to": 414, "label": "CASE" }, { "from": 413, "to": 482, "label": "CASE" }, { "from": 414, "to": 434, "label": "PARALLEL" }, { "from": 414, "to": 436, "label": "PARALLEL" }, { "from": 434, "to": 439, "label": "ONLY EVAL with clause\nnot_member(X169, X170) :- ','(member(X169, X170), ','(!_8, failure(a))).\nand substitutionT117 -> T139,\nX169 -> T139,\nT109 -> T140,\nX170 -> .(T140, [])" }, { "from": 436, "to": 476, "label": "ONLY EVAL with clause\nnot_member(X210, X211).\nand substitutionT117 -> T164,\nX210 -> T164,\nT109 -> T165,\nX211 -> .(T165, [])" }, { "from": 439, "to": 445, "label": "SPLIT 1" }, { "from": 439, "to": 446, "label": "SPLIT 2\nnew knowledge:\nT139 is ground\nT140 is ground" }, { "from": 445, "to": 450, "label": "CASE" }, { "from": 446, "to": 471, "label": "CUT" }, { "from": 450, "to": 452, "label": "PARALLEL" }, { "from": 450, "to": 453, "label": "PARALLEL" }, { "from": 452, "to": 456, "label": "EVAL with clause\nmember(X187, .(X187, X188)).\nand substitutionT139 -> T149,\nX187 -> T149,\nT140 -> T149,\nX188 -> []" }, { "from": 452, "to": 458, "label": "EVAL-BACKTRACK" }, { "from": 453, "to": 466, "label": "ONLY EVAL with clause\nmember(X199, .(X200, X201)) :- member(X199, X201).\nand substitutionT139 -> T156,\nX199 -> T156,\nT140 -> T157,\nX200 -> T157,\nX201 -> []" }, { "from": 456, "to": 460, "label": "SUCCESS" }, { "from": 466, "to": 316, "label": "INSTANCE with matching:\nT75 -> T156" }, { "from": 471, "to": 472, "label": "CASE" }, { "from": 472, "to": 473, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 476, "to": 478, "label": "SUCCESS" }, { "from": 482, "to": 484, "label": "PARALLEL" }, { "from": 482, "to": 485, "label": "PARALLEL" }, { "from": 484, "to": 486, "label": "EVAL with clause\nsubsetchecked([], X224, X225).\nand substitutionT118 -> [],\nT117 -> T184,\nT109 -> T185,\nX224 -> .(T184, .(T185, [])),\nT110 -> T186,\nX225 -> T186" }, { "from": 484, "to": 487, "label": "EVAL-BACKTRACK" }, { "from": 485, "to": 505, "label": "EVAL with clause\nsubsetchecked(.(X234, X235), X236, X237) :- ','(member(X234, X237), ','(not_member(X234, X236), subsetchecked(X235, .(X234, X236), X237))).\nand substitutionX234 -> T202,\nX235 -> T203,\nT118 -> .(T202, T203),\nT117 -> T199,\nT109 -> T200,\nX236 -> .(T199, .(T200, [])),\nT110 -> T201,\nX237 -> T201,\nT197 -> T202,\nT198 -> T203" }, { "from": 485, "to": 506, "label": "EVAL-BACKTRACK" }, { "from": 486, "to": 488, "label": "SUCCESS" }, { "from": 505, "to": 509, "label": "SPLIT 1" }, { "from": 505, "to": 510, "label": "SPLIT 2\nnew knowledge:\nT208 is ground\nT201 is ground\nreplacements:T202 -> T208,\nT203 -> T209" }, { "from": 509, "to": 235, "label": "INSTANCE with matching:\nT29 -> T202\nT28 -> T201" }, { "from": 510, "to": 517, "label": "SPLIT 1" }, { "from": 510, "to": 518, "label": "SPLIT 2\nnew knowledge:\nT208 is ground\nT199 is ground\nT200 is ground" }, { "from": 517, "to": 519, "label": "CASE" }, { "from": 518, "to": 581, "label": "CASE" }, { "from": 519, "to": 527, "label": "PARALLEL" }, { "from": 519, "to": 528, "label": "PARALLEL" }, { "from": 527, "to": 535, "label": "ONLY EVAL with clause\nnot_member(X266, X267) :- ','(member(X266, X267), ','(!_12, failure(a))).\nand substitutionT208 -> T240,\nX266 -> T240,\nT199 -> T241,\nT200 -> T242,\nX267 -> .(T241, .(T242, []))" }, { "from": 528, "to": 577, "label": "ONLY EVAL with clause\nnot_member(X307, X308).\nand substitutionT208 -> T280,\nX307 -> T280,\nT199 -> T281,\nT200 -> T282,\nX308 -> .(T281, .(T282, []))" }, { "from": 535, "to": 537, "label": "SPLIT 1" }, { "from": 535, "to": 538, "label": "SPLIT 2\nnew knowledge:\nT240 is ground\nT241 is ground\nT242 is ground" }, { "from": 537, "to": 542, "label": "CASE" }, { "from": 538, "to": 568, "label": "CUT" }, { "from": 542, "to": 543, "label": "PARALLEL" }, { "from": 542, "to": 544, "label": "PARALLEL" }, { "from": 543, "to": 547, "label": "EVAL with clause\nmember(X284, .(X284, X285)).\nand substitutionT240 -> T259,\nX284 -> T259,\nT241 -> T259,\nT242 -> T260,\nX285 -> .(T260, [])" }, { "from": 543, "to": 548, "label": "EVAL-BACKTRACK" }, { "from": 544, "to": 565, "label": "ONLY EVAL with clause\nmember(X296, .(X297, X298)) :- member(X296, X298).\nand substitutionT240 -> T269,\nX296 -> T269,\nT241 -> T270,\nX297 -> T270,\nT242 -> T271,\nX298 -> .(T271, [])" }, { "from": 547, "to": 549, "label": "SUCCESS" }, { "from": 565, "to": 445, "label": "INSTANCE with matching:\nT139 -> T269\nT140 -> T271" }, { "from": 568, "to": 569, "label": "CASE" }, { "from": 569, "to": 570, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 577, "to": 578, "label": "SUCCESS" }, { "from": 581, "to": 584, "label": "PARALLEL" }, { "from": 581, "to": 585, "label": "PARALLEL" }, { "from": 584, "to": 588, "label": "EVAL with clause\nsubsetchecked([], X321, X322).\nand substitutionT209 -> [],\nT208 -> T307,\nT199 -> T308,\nT200 -> T309,\nX321 -> .(T307, .(T308, .(T309, []))),\nT201 -> T310,\nX322 -> T310" }, { "from": 584, "to": 589, "label": "EVAL-BACKTRACK" }, { "from": 585, "to": 591, "label": "EVAL with clause\nsubsetchecked(.(X331, X332), X333, X334) :- ','(member(X331, X334), ','(not_member(X331, X333), subsetchecked(X332, .(X331, X333), X334))).\nand substitutionX331 -> T329,\nX332 -> T330,\nT209 -> .(T329, T330),\nT208 -> T325,\nT199 -> T326,\nT200 -> T327,\nX333 -> .(T325, .(T326, .(T327, []))),\nT201 -> T328,\nX334 -> T328,\nT323 -> T329,\nT324 -> T330" }, { "from": 585, "to": 592, "label": "EVAL-BACKTRACK" }, { "from": 588, "to": 590, "label": "SUCCESS" }, { "from": 591, "to": 593, "label": "SPLIT 1" }, { "from": 591, "to": 594, "label": "SPLIT 2\nnew knowledge:\nT335 is ground\nT328 is ground\nreplacements:T329 -> T335,\nT330 -> T336" }, { "from": 593, "to": 235, "label": "INSTANCE with matching:\nT29 -> T329\nT28 -> T328" }, { "from": 594, "to": 596, "label": "SPLIT 1" }, { "from": 594, "to": 597, "label": "SPLIT 2\nnew knowledge:\nT335 is ground\nT325 is ground\nT326 is ground\nT327 is ground" }, { "from": 596, "to": 599, "label": "CASE" }, { "from": 597, "to": 670, "label": "CASE" }, { "from": 599, "to": 624, "label": "PARALLEL" }, { "from": 599, "to": 625, "label": "PARALLEL" }, { "from": 624, "to": 630, "label": "ONLY EVAL with clause\nnot_member(X363, X364) :- ','(member(X363, X364), ','(!_16, failure(a))).\nand substitutionT335 -> T377,\nX363 -> T377,\nT325 -> T378,\nT326 -> T379,\nT327 -> T380,\nX364 -> .(T378, .(T379, .(T380, [])))" }, { "from": 625, "to": 666, "label": "ONLY EVAL with clause\nnot_member(X404, X405).\nand substitutionT335 -> T436,\nX404 -> T436,\nT325 -> T437,\nT326 -> T438,\nT327 -> T439,\nX405 -> .(T437, .(T438, .(T439, [])))" }, { "from": 630, "to": 635, "label": "SPLIT 1" }, { "from": 630, "to": 636, "label": "SPLIT 2\nnew knowledge:\nT377 is ground\nT378 is ground\nT379 is ground\nT380 is ground" }, { "from": 635, "to": 642, "label": "CASE" }, { "from": 636, "to": 660, "label": "CUT" }, { "from": 642, "to": 644, "label": "PARALLEL" }, { "from": 642, "to": 645, "label": "PARALLEL" }, { "from": 644, "to": 649, "label": "EVAL with clause\nmember(X381, .(X381, X382)).\nand substitutionT377 -> T405,\nX381 -> T405,\nT378 -> T405,\nT379 -> T406,\nT380 -> T407,\nX382 -> .(T406, .(T407, []))" }, { "from": 644, "to": 650, "label": "EVAL-BACKTRACK" }, { "from": 645, "to": 658, "label": "ONLY EVAL with clause\nmember(X393, .(X394, X395)) :- member(X393, X395).\nand substitutionT377 -> T420,\nX393 -> T420,\nT378 -> T421,\nX394 -> T421,\nT379 -> T422,\nT380 -> T423,\nX395 -> .(T422, .(T423, []))" }, { "from": 649, "to": 651, "label": "SUCCESS" }, { "from": 658, "to": 537, "label": "INSTANCE with matching:\nT240 -> T420\nT241 -> T422\nT242 -> T423" }, { "from": 660, "to": 661, "label": "CASE" }, { "from": 661, "to": 662, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 666, "to": 667, "label": "SUCCESS" }, { "from": 670, "to": 671, "label": "PARALLEL" }, { "from": 670, "to": 672, "label": "PARALLEL" }, { "from": 671, "to": 675, "label": "EVAL with clause\nsubsetchecked([], X418, X419).\nand substitutionT336 -> [],\nT335 -> T470,\nT325 -> T471,\nT326 -> T472,\nT327 -> T473,\nX418 -> .(T470, .(T471, .(T472, .(T473, [])))),\nT328 -> T474,\nX419 -> T474" }, { "from": 671, "to": 676, "label": "EVAL-BACKTRACK" }, { "from": 672, "to": 681, "label": "EVAL with clause\nsubsetchecked(.(X428, X429), X430, X431) :- ','(member(X428, X431), ','(not_member(X428, X430), subsetchecked(X429, .(X428, X430), X431))).\nand substitutionX428 -> T496,\nX429 -> T497,\nT336 -> .(T496, T497),\nT335 -> T491,\nT325 -> T492,\nT326 -> T493,\nT327 -> T494,\nX430 -> .(T491, .(T492, .(T493, .(T494, [])))),\nT328 -> T495,\nX431 -> T495,\nT489 -> T496,\nT490 -> T497" }, { "from": 672, "to": 682, "label": "EVAL-BACKTRACK" }, { "from": 675, "to": 677, "label": "SUCCESS" }, { "from": 681, "to": 683, "label": "SPLIT 1" }, { "from": 681, "to": 684, "label": "SPLIT 2\nnew knowledge:\nT502 is ground\nT495 is ground\nreplacements:T496 -> T502,\nT497 -> T503" }, { "from": 683, "to": 235, "label": "INSTANCE with matching:\nT29 -> T496\nT28 -> T495" }, { "from": 684, "to": 694, "label": "SPLIT 1" }, { "from": 684, "to": 695, "label": "SPLIT 2\nnew knowledge:\nT502 is ground\nT491 is ground\nT492 is ground\nT493 is ground\nT494 is ground" }, { "from": 694, "to": 699, "label": "CASE" }, { "from": 695, "to": 763, "label": "CASE" }, { "from": 699, "to": 704, "label": "PARALLEL" }, { "from": 699, "to": 705, "label": "PARALLEL" }, { "from": 704, "to": 712, "label": "ONLY EVAL with clause\nnot_member(X460, X461) :- ','(member(X460, X461), ','(!_20, failure(a))).\nand substitutionT502 -> T554,\nX460 -> T554,\nT491 -> T555,\nT492 -> T556,\nT493 -> T557,\nT494 -> T558,\nX461 -> .(T555, .(T556, .(T557, .(T558, []))))" }, { "from": 705, "to": 759, "label": "ONLY EVAL with clause\nnot_member(X501, X502).\nand substitutionT502 -> T632,\nX501 -> T632,\nT491 -> T633,\nT492 -> T634,\nT493 -> T635,\nT494 -> T636,\nX502 -> .(T633, .(T634, .(T635, .(T636, []))))" }, { "from": 712, "to": 724, "label": "SPLIT 1" }, { "from": 712, "to": 725, "label": "SPLIT 2\nnew knowledge:\nT554 is ground\nT555 is ground\nT556 is ground\nT557 is ground\nT558 is ground" }, { "from": 724, "to": 730, "label": "CASE" }, { "from": 725, "to": 756, "label": "CUT" }, { "from": 730, "to": 734, "label": "PARALLEL" }, { "from": 730, "to": 735, "label": "PARALLEL" }, { "from": 734, "to": 736, "label": "EVAL with clause\nmember(X478, .(X478, X479)).\nand substitutionT554 -> T591,\nX478 -> T591,\nT555 -> T591,\nT556 -> T592,\nT557 -> T593,\nT558 -> T594,\nX479 -> .(T592, .(T593, .(T594, [])))" }, { "from": 734, "to": 737, "label": "EVAL-BACKTRACK" }, { "from": 735, "to": 753, "label": "ONLY EVAL with clause\nmember(X490, .(X491, X492)) :- member(X490, X492).\nand substitutionT554 -> T611,\nX490 -> T611,\nT555 -> T612,\nX491 -> T612,\nT556 -> T613,\nT557 -> T614,\nT558 -> T615,\nX492 -> .(T613, .(T614, .(T615, [])))" }, { "from": 736, "to": 738, "label": "SUCCESS" }, { "from": 753, "to": 635, "label": "INSTANCE with matching:\nT377 -> T611\nT378 -> T613\nT379 -> T614\nT380 -> T615" }, { "from": 756, "to": 757, "label": "CASE" }, { "from": 757, "to": 758, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 759, "to": 760, "label": "SUCCESS" }, { "from": 763, "to": 765, "label": "PARALLEL" }, { "from": 763, "to": 767, "label": "PARALLEL" }, { "from": 765, "to": 770, "label": "EVAL with clause\nsubsetchecked([], X515, X516).\nand substitutionT503 -> [],\nT502 -> T673,\nT491 -> T674,\nT492 -> T675,\nT493 -> T676,\nT494 -> T677,\nX515 -> .(T673, .(T674, .(T675, .(T676, .(T677, []))))),\nT495 -> T678,\nX516 -> T678" }, { "from": 765, "to": 771, "label": "EVAL-BACKTRACK" }, { "from": 767, "to": 778, "label": "EVAL with clause\nsubsetchecked(.(X525, X526), X527, X528) :- ','(member(X525, X528), ','(not_member(X525, X527), subsetchecked(X526, .(X525, X527), X528))).\nand substitutionX525 -> T703,\nX526 -> T704,\nT503 -> .(T703, T704),\nT502 -> T697,\nT491 -> T698,\nT492 -> T699,\nT493 -> T700,\nT494 -> T701,\nX527 -> .(T697, .(T698, .(T699, .(T700, .(T701, []))))),\nT495 -> T702,\nX528 -> T702,\nT695 -> T703,\nT696 -> T704" }, { "from": 767, "to": 779, "label": "EVAL-BACKTRACK" }, { "from": 770, "to": 772, "label": "SUCCESS" }, { "from": 778, "to": 781, "label": "SPLIT 1" }, { "from": 778, "to": 782, "label": "SPLIT 2\nnew knowledge:\nT709 is ground\nT702 is ground\nreplacements:T703 -> T709,\nT704 -> T710" }, { "from": 781, "to": 235, "label": "INSTANCE with matching:\nT29 -> T703\nT28 -> T702" }, { "from": 782, "to": 785, "label": "SPLIT 1" }, { "from": 782, "to": 786, "label": "SPLIT 2\nnew knowledge:\nT709 is ground\nT697 is ground\nT698 is ground\nT699 is ground\nT700 is ground\nT701 is ground" }, { "from": 785, "to": 790, "label": "CASE" }, { "from": 786, "to": 850, "label": "CASE" }, { "from": 790, "to": 793, "label": "PARALLEL" }, { "from": 790, "to": 794, "label": "PARALLEL" }, { "from": 793, "to": 817, "label": "ONLY EVAL with clause\nnot_member(X557, X558) :- ','(member(X557, X558), ','(!_24, failure(a))).\nand substitutionT709 -> T771,\nX557 -> T771,\nT697 -> T772,\nT698 -> T773,\nT699 -> T774,\nT700 -> T775,\nT701 -> T776,\nX558 -> .(T772, .(T773, .(T774, .(T775, .(T776, [])))))" }, { "from": 794, "to": 845, "label": "ONLY EVAL with clause\nnot_member(X598, X599).\nand substitutionT709 -> T868,\nX598 -> T868,\nT697 -> T869,\nT698 -> T870,\nT699 -> T871,\nT700 -> T872,\nT701 -> T873,\nX599 -> .(T869, .(T870, .(T871, .(T872, .(T873, [])))))" }, { "from": 817, "to": 820, "label": "SPLIT 1" }, { "from": 817, "to": 821, "label": "SPLIT 2\nnew knowledge:\nT771 is ground\nT772 is ground\nT773 is ground\nT774 is ground\nT775 is ground\nT776 is ground" }, { "from": 820, "to": 823, "label": "CASE" }, { "from": 821, "to": 838, "label": "CUT" }, { "from": 823, "to": 824, "label": "PARALLEL" }, { "from": 823, "to": 825, "label": "PARALLEL" }, { "from": 824, "to": 833, "label": "EVAL with clause\nmember(X575, .(X575, X576)).\nand substitutionT771 -> T817,\nX575 -> T817,\nT772 -> T817,\nT773 -> T818,\nT774 -> T819,\nT775 -> T820,\nT776 -> T821,\nX576 -> .(T818, .(T819, .(T820, .(T821, []))))" }, { "from": 824, "to": 834, "label": "EVAL-BACKTRACK" }, { "from": 825, "to": 836, "label": "ONLY EVAL with clause\nmember(X587, .(X588, X589)) :- member(X587, X589).\nand substitutionT771 -> T842,\nX587 -> T842,\nT772 -> T843,\nX588 -> T843,\nT773 -> T844,\nT774 -> T845,\nT775 -> T846,\nT776 -> T847,\nX589 -> .(T844, .(T845, .(T846, .(T847, []))))" }, { "from": 833, "to": 835, "label": "SUCCESS" }, { "from": 836, "to": 724, "label": "INSTANCE with matching:\nT554 -> T842\nT555 -> T844\nT556 -> T845\nT557 -> T846\nT558 -> T847" }, { "from": 838, "to": 839, "label": "CASE" }, { "from": 839, "to": 840, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 845, "to": 847, "label": "SUCCESS" }, { "from": 850, "to": 851, "label": "PARALLEL" }, { "from": 850, "to": 852, "label": "PARALLEL" }, { "from": 851, "to": 855, "label": "EVAL with clause\nsubsetchecked([], X612, X613).\nand substitutionT710 -> [],\nT709 -> T916,\nT697 -> T917,\nT698 -> T918,\nT699 -> T919,\nT700 -> T920,\nT701 -> T921,\nX612 -> .(T916, .(T917, .(T918, .(T919, .(T920, .(T921, [])))))),\nT702 -> T922,\nX613 -> T922" }, { "from": 851, "to": 856, "label": "EVAL-BACKTRACK" }, { "from": 852, "to": 872, "label": "EVAL with clause\nsubsetchecked(.(X622, X623), X624, X625) :- ','(member(X622, X625), ','(not_member(X622, X624), subsetchecked(X623, .(X622, X624), X625))).\nand substitutionX622 -> T950,\nX623 -> T951,\nT710 -> .(T950, T951),\nT709 -> T943,\nT697 -> T944,\nT698 -> T945,\nT699 -> T946,\nT700 -> T947,\nT701 -> T948,\nX624 -> .(T943, .(T944, .(T945, .(T946, .(T947, .(T948, [])))))),\nT702 -> T949,\nX625 -> T949,\nT941 -> T950,\nT942 -> T951" }, { "from": 852, "to": 873, "label": "EVAL-BACKTRACK" }, { "from": 855, "to": 857, "label": "SUCCESS" }, { "from": 872, "to": 877, "label": "SPLIT 1" }, { "from": 872, "to": 878, "label": "SPLIT 2\nnew knowledge:\nT956 is ground\nT949 is ground\nreplacements:T950 -> T956,\nT951 -> T957" }, { "from": 877, "to": 235, "label": "INSTANCE with matching:\nT29 -> T950\nT28 -> T949" }, { "from": 878, "to": 883, "label": "SPLIT 1" }, { "from": 878, "to": 884, "label": "SPLIT 2\nnew knowledge:\nT956 is ground\nT943 is ground\nT944 is ground\nT945 is ground\nT946 is ground\nT947 is ground\nT948 is ground" }, { "from": 883, "to": 885, "label": "CASE" }, { "from": 884, "to": 975, "label": "CASE" }, { "from": 885, "to": 889, "label": "PARALLEL" }, { "from": 885, "to": 890, "label": "PARALLEL" }, { "from": 889, "to": 909, "label": "ONLY EVAL with clause\nnot_member(X654, X655) :- ','(member(X654, X655), ','(!_28, failure(a))).\nand substitutionT956 -> T1028,\nX654 -> T1028,\nT943 -> T1029,\nT944 -> T1030,\nT945 -> T1031,\nT946 -> T1032,\nT947 -> T1033,\nT948 -> T1034,\nX655 -> .(T1029, .(T1030, .(T1031, .(T1032, .(T1033, .(T1034, []))))))" }, { "from": 890, "to": 970, "label": "ONLY EVAL with clause\nnot_member(X695, X696).\nand substitutionT956 -> T1144,\nX695 -> T1144,\nT943 -> T1145,\nT944 -> T1146,\nT945 -> T1147,\nT946 -> T1148,\nT947 -> T1149,\nT948 -> T1150,\nX696 -> .(T1145, .(T1146, .(T1147, .(T1148, .(T1149, .(T1150, []))))))" }, { "from": 909, "to": 913, "label": "SPLIT 1" }, { "from": 909, "to": 914, "label": "SPLIT 2\nnew knowledge:\nT1028 is ground\nT1029 is ground\nT1030 is ground\nT1031 is ground\nT1032 is ground\nT1033 is ground\nT1034 is ground" }, { "from": 913, "to": 927, "label": "CASE" }, { "from": 914, "to": 959, "label": "CUT" }, { "from": 927, "to": 936, "label": "PARALLEL" }, { "from": 927, "to": 937, "label": "PARALLEL" }, { "from": 936, "to": 941, "label": "EVAL with clause\nmember(X672, .(X672, X673)).\nand substitutionT1028 -> T1083,\nX672 -> T1083,\nT1029 -> T1083,\nT1030 -> T1084,\nT1031 -> T1085,\nT1032 -> T1086,\nT1033 -> T1087,\nT1034 -> T1088,\nX673 -> .(T1084, .(T1085, .(T1086, .(T1087, .(T1088, [])))))" }, { "from": 936, "to": 942, "label": "EVAL-BACKTRACK" }, { "from": 937, "to": 955, "label": "ONLY EVAL with clause\nmember(X684, .(X685, X686)) :- member(X684, X686).\nand substitutionT1028 -> T1113,\nX684 -> T1113,\nT1029 -> T1114,\nX685 -> T1114,\nT1030 -> T1115,\nT1031 -> T1116,\nT1032 -> T1117,\nT1033 -> T1118,\nT1034 -> T1119,\nX686 -> .(T1115, .(T1116, .(T1117, .(T1118, .(T1119, [])))))" }, { "from": 941, "to": 943, "label": "SUCCESS" }, { "from": 955, "to": 820, "label": "INSTANCE with matching:\nT771 -> T1113\nT772 -> T1115\nT773 -> T1116\nT774 -> T1117\nT775 -> T1118\nT776 -> T1119" }, { "from": 959, "to": 962, "label": "CASE" }, { "from": 962, "to": 964, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 970, "to": 971, "label": "SUCCESS" }, { "from": 975, "to": 981, "label": "PARALLEL" }, { "from": 975, "to": 982, "label": "PARALLEL" }, { "from": 981, "to": 1001, "label": "EVAL with clause\nsubsetchecked([], X709, X710).\nand substitutionT957 -> [],\nT956 -> T1199,\nT943 -> T1200,\nT944 -> T1201,\nT945 -> T1202,\nT946 -> T1203,\nT947 -> T1204,\nT948 -> T1205,\nX709 -> .(T1199, .(T1200, .(T1201, .(T1202, .(T1203, .(T1204, .(T1205, []))))))),\nT949 -> T1206,\nX710 -> T1206" }, { "from": 981, "to": 1002, "label": "EVAL-BACKTRACK" }, { "from": 982, "to": 1016, "label": "EVAL with clause\nsubsetchecked(.(X719, X720), X721, X722) :- ','(member(X719, X722), ','(not_member(X719, X721), subsetchecked(X720, .(X719, X721), X722))).\nand substitutionX719 -> T1237,\nX720 -> T1238,\nT957 -> .(T1237, T1238),\nT956 -> T1229,\nT943 -> T1230,\nT944 -> T1231,\nT945 -> T1232,\nT946 -> T1233,\nT947 -> T1234,\nT948 -> T1235,\nX721 -> .(T1229, .(T1230, .(T1231, .(T1232, .(T1233, .(T1234, .(T1235, []))))))),\nT949 -> T1236,\nX722 -> T1236,\nT1227 -> T1237,\nT1228 -> T1238" }, { "from": 982, "to": 1017, "label": "EVAL-BACKTRACK" }, { "from": 1001, "to": 1003, "label": "SUCCESS" }, { "from": 1016, "to": 1018, "label": "GENERALIZATION\nT1243 <-- .(T1229, .(T1230, .(T1231, .(T1232, .(T1233, .(T1234, .(T1235, [])))))))\n\nNew Knowledge:\nT1243 is ground" }, { "from": 1018, "to": 1019, "label": "SPLIT 1" }, { "from": 1018, "to": 1020, "label": "SPLIT 2\nnew knowledge:\nT1248 is ground\nT1236 is ground\nreplacements:T1237 -> T1248,\nT1238 -> T1249" }, { "from": 1019, "to": 235, "label": "INSTANCE with matching:\nT29 -> T1237\nT28 -> T1236" }, { "from": 1020, "to": 1021, "label": "SPLIT 1" }, { "from": 1020, "to": 1022, "label": "SPLIT 2\nnew knowledge:\nT1248 is ground\nT1229 is ground\nT1230 is ground\nT1231 is ground\nT1232 is ground\nT1233 is ground\nT1234 is ground\nT1235 is ground" }, { "from": 1021, "to": 1023, "label": "CASE" }, { "from": 1022, "to": 1098, "label": "CASE" }, { "from": 1023, "to": 1027, "label": "PARALLEL" }, { "from": 1023, "to": 1028, "label": "PARALLEL" }, { "from": 1027, "to": 1037, "label": "ONLY EVAL with clause\nnot_member(X755, X756) :- ','(member(X755, X756), ','(!_32, failure(a))).\nand substitutionT1248 -> T1330,\nX755 -> T1330,\nT1229 -> T1331,\nT1230 -> T1332,\nT1231 -> T1333,\nT1232 -> T1334,\nT1233 -> T1335,\nT1234 -> T1336,\nT1235 -> T1337,\nX756 -> .(T1331, .(T1332, .(T1333, .(T1334, .(T1335, .(T1336, .(T1337, [])))))))" }, { "from": 1028, "to": 1096, "label": "ONLY EVAL with clause\nnot_member(X796, X797).\nand substitutionT1248 -> T1465,\nX796 -> T1465,\nT1229 -> T1466,\nT1230 -> T1467,\nT1231 -> T1468,\nT1232 -> T1469,\nT1233 -> T1470,\nT1234 -> T1471,\nT1235 -> T1472,\nX797 -> .(T1466, .(T1467, .(T1468, .(T1469, .(T1470, .(T1471, .(T1472, [])))))))" }, { "from": 1037, "to": 1038, "label": "SPLIT 1" }, { "from": 1037, "to": 1039, "label": "SPLIT 2\nnew knowledge:\nT1330 is ground\nT1331 is ground\nT1332 is ground\nT1333 is ground\nT1334 is ground\nT1335 is ground\nT1336 is ground\nT1337 is ground" }, { "from": 1038, "to": 1040, "label": "CASE" }, { "from": 1039, "to": 1091, "label": "CUT" }, { "from": 1040, "to": 1046, "label": "PARALLEL" }, { "from": 1040, "to": 1047, "label": "PARALLEL" }, { "from": 1046, "to": 1048, "label": "EVAL with clause\nmember(X773, .(X773, X774)).\nand substitutionT1330 -> T1394,\nX773 -> T1394,\nT1331 -> T1394,\nT1332 -> T1395,\nT1333 -> T1396,\nT1334 -> T1397,\nT1335 -> T1398,\nT1336 -> T1399,\nT1337 -> T1400,\nX774 -> .(T1395, .(T1396, .(T1397, .(T1398, .(T1399, .(T1400, []))))))" }, { "from": 1046, "to": 1049, "label": "EVAL-BACKTRACK" }, { "from": 1047, "to": 1090, "label": "ONLY EVAL with clause\nmember(X785, .(X786, X787)) :- member(X785, X787).\nand substitutionT1330 -> T1429,\nX785 -> T1429,\nT1331 -> T1430,\nX786 -> T1430,\nT1332 -> T1431,\nT1333 -> T1432,\nT1334 -> T1433,\nT1335 -> T1434,\nT1336 -> T1435,\nT1337 -> T1436,\nX787 -> .(T1431, .(T1432, .(T1433, .(T1434, .(T1435, .(T1436, []))))))" }, { "from": 1048, "to": 1050, "label": "SUCCESS" }, { "from": 1090, "to": 913, "label": "INSTANCE with matching:\nT1028 -> T1429\nT1029 -> T1431\nT1030 -> T1432\nT1031 -> T1433\nT1032 -> T1434\nT1033 -> T1435\nT1034 -> T1436" }, { "from": 1091, "to": 1092, "label": "CASE" }, { "from": 1092, "to": 1093, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 1096, "to": 1097, "label": "SUCCESS" }, { "from": 1098, "to": 1100, "label": "PARALLEL" }, { "from": 1098, "to": 1101, "label": "PARALLEL" }, { "from": 1100, "to": 1102, "label": "EVAL with clause\nsubsetchecked([], X810, X811).\nand substitutionT1249 -> [],\nT1248 -> T1491,\nT1243 -> T1492,\nX810 -> .(T1491, T1492),\nT1236 -> T1493,\nX811 -> T1493" }, { "from": 1100, "to": 1103, "label": "EVAL-BACKTRACK" }, { "from": 1101, "to": 1110, "label": "EVAL with clause\nsubsetchecked(.(X820, X821), X822, X823) :- ','(member(X820, X823), ','(not_member(X820, X822), subsetchecked(X821, .(X820, X822), X823))).\nand substitutionX820 -> T1509,\nX821 -> T1510,\nT1249 -> .(T1509, T1510),\nT1248 -> T1506,\nT1243 -> T1507,\nX822 -> .(T1506, T1507),\nT1236 -> T1508,\nX823 -> T1508,\nT1504 -> T1509,\nT1505 -> T1510" }, { "from": 1101, "to": 1111, "label": "EVAL-BACKTRACK" }, { "from": 1102, "to": 1104, "label": "SUCCESS" }, { "from": 1110, "to": 1115, "label": "SPLIT 1" }, { "from": 1110, "to": 1116, "label": "SPLIT 2\nnew knowledge:\nT1515 is ground\nT1508 is ground\nreplacements:T1509 -> T1515,\nT1510 -> T1516" }, { "from": 1115, "to": 235, "label": "INSTANCE with matching:\nT29 -> T1509\nT28 -> T1508" }, { "from": 1116, "to": 1144, "label": "SPLIT 1" }, { "from": 1116, "to": 1145, "label": "SPLIT 2\nnew knowledge:\nT1515 is ground\nT1506 is ground\nT1507 is ground" }, { "from": 1144, "to": 1146, "label": "CASE" }, { "from": 1145, "to": 1022, "label": "INSTANCE with matching:\nT1249 -> T1516\nT1248 -> T1515\nT1243 -> .(T1506, T1507)\nT1236 -> T1508" }, { "from": 1146, "to": 1147, "label": "PARALLEL" }, { "from": 1146, "to": 1148, "label": "PARALLEL" }, { "from": 1147, "to": 1153, "label": "ONLY EVAL with clause\nnot_member(X852, X853) :- ','(member(X852, X853), ','(!_36, failure(a))).\nand substitutionT1515 -> T1547,\nX852 -> T1547,\nT1506 -> T1548,\nT1507 -> T1549,\nX853 -> .(T1548, T1549)" }, { "from": 1148, "to": 1189, "label": "ONLY EVAL with clause\nnot_member(X916, X917).\nand substitutionT1515 -> T1614,\nX916 -> T1614,\nT1506 -> T1615,\nT1507 -> T1616,\nX917 -> .(T1615, T1616)" }, { "from": 1153, "to": 1154, "label": "SPLIT 1" }, { "from": 1153, "to": 1155, "label": "SPLIT 2\nnew knowledge:\nT1547 is ground\nT1548 is ground\nT1549 is ground" }, { "from": 1154, "to": 1156, "label": "CASE" }, { "from": 1155, "to": 1186, "label": "CUT" }, { "from": 1156, "to": 1157, "label": "PARALLEL" }, { "from": 1156, "to": 1158, "label": "PARALLEL" }, { "from": 1157, "to": 1159, "label": "EVAL with clause\nmember(X870, .(X870, X871)).\nand substitutionT1547 -> T1566,\nX870 -> T1566,\nT1548 -> T1566,\nT1549 -> T1567,\nX871 -> T1567" }, { "from": 1157, "to": 1160, "label": "EVAL-BACKTRACK" }, { "from": 1158, "to": 1164, "label": "ONLY EVAL with clause\nmember(X882, .(X883, X884)) :- member(X882, X884).\nand substitutionT1547 -> T1578,\nX882 -> T1578,\nT1548 -> T1579,\nX883 -> T1579,\nT1549 -> T1580,\nX884 -> T1580" }, { "from": 1159, "to": 1161, "label": "SUCCESS" }, { "from": 1164, "to": 1168, "label": "CASE" }, { "from": 1168, "to": 1169, "label": "PARALLEL" }, { "from": 1168, "to": 1170, "label": "PARALLEL" }, { "from": 1169, "to": 1174, "label": "EVAL with clause\nmember(X897, .(X897, X898)).\nand substitutionT1578 -> T1593,\nX897 -> T1593,\nX898 -> T1594,\nT1580 -> .(T1593, T1594)" }, { "from": 1169, "to": 1175, "label": "EVAL-BACKTRACK" }, { "from": 1170, "to": 1181, "label": "EVAL with clause\nmember(X905, .(X906, X907)) :- member(X905, X907).\nand substitutionT1578 -> T1601,\nX905 -> T1601,\nX906 -> T1602,\nX907 -> T1603,\nT1580 -> .(T1602, T1603)" }, { "from": 1170, "to": 1182, "label": "EVAL-BACKTRACK" }, { "from": 1174, "to": 1176, "label": "SUCCESS" }, { "from": 1181, "to": 1164, "label": "INSTANCE with matching:\nT1578 -> T1601\nT1580 -> T1603" }, { "from": 1186, "to": 1187, "label": "CASE" }, { "from": 1187, "to": 1188, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 1189, "to": 1190, "label": "SUCCESS" } ], "type": "Graph" } } ---------------------------------------- (193) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f3_in(T19) -> f3_out1([]) f3_in(T28) -> U1(f150_in(T28), T28) U1(f150_out1(T29, T30), T28) -> f3_out1(.(T29, T30)) f235_in(.(T49, T50)) -> f235_out1(T49) f235_in(.(T58, T59)) -> U2(f235_in(T59), .(T58, T59)) U2(f235_out1(T60), .(T58, T59)) -> f235_out1(T60) f445_in(T149, T149) -> f445_out1 f445_in(T156, T157) -> U3(f316_in(T156), T156, T157) U3(f316_out1, T156, T157) -> f445_out1 f537_in(T259, T259, T260) -> f537_out1 f537_in(T269, T270, T271) -> U4(f445_in(T269, T271), T269, T270, T271) U4(f445_out1, T269, T270, T271) -> f537_out1 f635_in(T405, T405, T406, T407) -> f635_out1 f635_in(T420, T421, T422, T423) -> U5(f537_in(T420, T422, T423), T420, T421, T422, T423) U5(f537_out1, T420, T421, T422, T423) -> f635_out1 f724_in(T591, T591, T592, T593, T594) -> f724_out1 f724_in(T611, T612, T613, T614, T615) -> U6(f635_in(T611, T613, T614, T615), T611, T612, T613, T614, T615) U6(f635_out1, T611, T612, T613, T614, T615) -> f724_out1 f820_in(T817, T817, T818, T819, T820, T821) -> f820_out1 f820_in(T842, T843, T844, T845, T846, T847) -> U7(f724_in(T842, T844, T845, T846, T847), T842, T843, T844, T845, T846, T847) U7(f724_out1, T842, T843, T844, T845, T846, T847) -> f820_out1 f913_in(T1083, T1083, T1084, T1085, T1086, T1087, T1088) -> f913_out1 f913_in(T1113, T1114, T1115, T1116, T1117, T1118, T1119) -> U8(f820_in(T1113, T1115, T1116, T1117, T1118, T1119), T1113, T1114, T1115, T1116, T1117, T1118, T1119) U8(f820_out1, T1113, T1114, T1115, T1116, T1117, T1118, T1119) -> f913_out1 f1022_in(T1491, T1492, T1493) -> f1022_out1([]) f1022_in(T1506, T1507, T1508) -> U9(f1110_in(T1508, T1506, T1507), T1506, T1507, T1508) U9(f1110_out1(T1509, T1510), T1506, T1507, T1508) -> f1022_out1(.(T1509, T1510)) f1164_in(T1593, .(T1593, T1594)) -> f1164_out1 f1164_in(T1601, .(T1602, T1603)) -> U10(f1164_in(T1601, T1603), T1601, .(T1602, T1603)) U10(f1164_out1, T1601, .(T1602, T1603)) -> f1164_out1 f289_in(T75) -> U11(f299_in(T75), T75) U11(f299_out1, T75) -> f289_out1 f289_in(T84) -> f289_out1 f290_in(T97, T98) -> f290_out1([]) f290_in(T109, T110) -> U12(f408_in(T110, T109), T109, T110) U12(f408_out1(T111, T112), T109, T110) -> f290_out1(.(T111, T112)) f412_in(T139, T140) -> U13(f439_in(T139, T140), T139, T140) U13(f439_out1, T139, T140) -> f412_out1 f412_in(T164, T165) -> f412_out1 f413_in(T184, T185, T186) -> f413_out1([]) f413_in(T199, T200, T201) -> U14(f505_in(T201, T199, T200), T199, T200, T201) U14(f505_out1(T202, T203), T199, T200, T201) -> f413_out1(.(T202, T203)) f517_in(T240, T241, T242) -> U15(f535_in(T240, T241, T242), T240, T241, T242) U15(f535_out1, T240, T241, T242) -> f517_out1 f517_in(T280, T281, T282) -> f517_out1 f518_in(T307, T308, T309, T310) -> f518_out1([]) f518_in(T325, T326, T327, T328) -> U16(f591_in(T328, T325, T326, T327), T325, T326, T327, T328) U16(f591_out1(T329, T330), T325, T326, T327, T328) -> f518_out1(.(T329, T330)) f596_in(T377, T378, T379, T380) -> U17(f630_in(T377, T378, T379, T380), T377, T378, T379, T380) U17(f630_out1, T377, T378, T379, T380) -> f596_out1 f596_in(T436, T437, T438, T439) -> f596_out1 f597_in(T470, T471, T472, T473, T474) -> f597_out1([]) f597_in(T491, T492, T493, T494, T495) -> U18(f681_in(T495, T491, T492, T493, T494), T491, T492, T493, T494, T495) U18(f681_out1(T496, T497), T491, T492, T493, T494, T495) -> f597_out1(.(T496, T497)) f694_in(T554, T555, T556, T557, T558) -> U19(f712_in(T554, T555, T556, T557, T558), T554, T555, T556, T557, T558) U19(f712_out1, T554, T555, T556, T557, T558) -> f694_out1 f694_in(T632, T633, T634, T635, T636) -> f694_out1 f695_in(T673, T674, T675, T676, T677, T678) -> f695_out1([]) f695_in(T697, T698, T699, T700, T701, T702) -> U20(f778_in(T702, T697, T698, T699, T700, T701), T697, T698, T699, T700, T701, T702) U20(f778_out1(T703, T704), T697, T698, T699, T700, T701, T702) -> f695_out1(.(T703, T704)) f785_in(T771, T772, T773, T774, T775, T776) -> U21(f817_in(T771, T772, T773, T774, T775, T776), T771, T772, T773, T774, T775, T776) U21(f817_out1, T771, T772, T773, T774, T775, T776) -> f785_out1 f785_in(T868, T869, T870, T871, T872, T873) -> f785_out1 f786_in(T916, T917, T918, T919, T920, T921, T922) -> f786_out1([]) f786_in(T943, T944, T945, T946, T947, T948, T949) -> U22(f872_in(T949, T943, T944, T945, T946, T947, T948), T943, T944, T945, T946, T947, T948, T949) U22(f872_out1(T950, T951), T943, T944, T945, T946, T947, T948, T949) -> f786_out1(.(T950, T951)) f883_in(T1028, T1029, T1030, T1031, T1032, T1033, T1034) -> U23(f909_in(T1028, T1029, T1030, T1031, T1032, T1033, T1034), T1028, T1029, T1030, T1031, T1032, T1033, T1034) U23(f909_out1, T1028, T1029, T1030, T1031, T1032, T1033, T1034) -> f883_out1 f883_in(T1144, T1145, T1146, T1147, T1148, T1149, T1150) -> f883_out1 f884_in(T1199, T1200, T1201, T1202, T1203, T1204, T1205, T1206) -> f884_out1([]) f884_in(T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1236) -> U24(f1018_in(T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235, .(T1229, .(T1230, .(T1231, .(T1232, .(T1233, .(T1234, .(T1235, [])))))))), T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1236) U24(f1018_out1(T1237, T1238), T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1236) -> f884_out1(.(T1237, T1238)) f1021_in(T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) -> U25(f1037_in(T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337), T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) U25(f1037_out1, T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) -> f1021_out1 f1021_in(T1465, T1466, T1467, T1468, T1469, T1470, T1471, T1472) -> f1021_out1 f1038_in(T1394, T1394, T1395, T1396, T1397, T1398, T1399, T1400) -> f1038_out1 f1038_in(T1429, T1430, T1431, T1432, T1433, T1434, T1435, T1436) -> U26(f913_in(T1429, T1431, T1432, T1433, T1434, T1435, T1436), T1429, T1430, T1431, T1432, T1433, T1434, T1435, T1436) U26(f913_out1, T1429, T1430, T1431, T1432, T1433, T1434, T1435, T1436) -> f1038_out1 f1144_in(T1547, T1548, T1549) -> U27(f1153_in(T1547, T1548, T1549), T1547, T1548, T1549) U27(f1153_out1, T1547, T1548, T1549) -> f1144_out1 f1144_in(T1614, T1615, T1616) -> f1144_out1 f1154_in(T1566, T1566, T1567) -> f1154_out1 f1154_in(T1578, T1579, T1580) -> U28(f1164_in(T1578, T1580), T1578, T1579, T1580) U28(f1164_out1, T1578, T1579, T1580) -> f1154_out1 f150_in(T28) -> U29(f235_in(T28), T28) U29(f235_out1(T35), T28) -> U30(f236_in(T35, T28), T28, T35) U30(f236_out1(T36), T28, T35) -> f150_out1(T35, T36) f236_in(T35, T28) -> U31(f289_in(T35), T35, T28) U31(f289_out1, T35, T28) -> U32(f290_in(T35, T28), T35, T28) U32(f290_out1(T36), T35, T28) -> f236_out1(T36) f299_in(T75) -> U33(f316_in(T75), T75) U33(f316_out1, T75) -> U34(f317_in, T75) U34(f317_out1, T75) -> f299_out1 f408_in(T110, T109) -> U35(f235_in(T110), T110, T109) U35(f235_out1(T117), T110, T109) -> U36(f411_in(T117, T109, T110), T110, T109, T117) U36(f411_out1(T118), T110, T109, T117) -> f408_out1(T117, T118) f411_in(T117, T109, T110) -> U37(f412_in(T117, T109), T117, T109, T110) U37(f412_out1, T117, T109, T110) -> U38(f413_in(T117, T109, T110), T117, T109, T110) U38(f413_out1(T118), T117, T109, T110) -> f411_out1(T118) f439_in(T139, T140) -> U39(f445_in(T139, T140), T139, T140) U39(f445_out1, T139, T140) -> U40(f446_in, T139, T140) U40(f446_out1, T139, T140) -> f439_out1 f505_in(T201, T199, T200) -> U41(f235_in(T201), T201, T199, T200) U41(f235_out1(T208), T201, T199, T200) -> U42(f510_in(T208, T199, T200, T201), T201, T199, T200, T208) U42(f510_out1(T209), T201, T199, T200, T208) -> f505_out1(T208, T209) f510_in(T208, T199, T200, T201) -> U43(f517_in(T208, T199, T200), T208, T199, T200, T201) U43(f517_out1, T208, T199, T200, T201) -> U44(f518_in(T208, T199, T200, T201), T208, T199, T200, T201) U44(f518_out1(T209), T208, T199, T200, T201) -> f510_out1(T209) f535_in(T240, T241, T242) -> U45(f537_in(T240, T241, T242), T240, T241, T242) U45(f537_out1, T240, T241, T242) -> U46(f538_in, T240, T241, T242) U46(f538_out1, T240, T241, T242) -> f535_out1 f591_in(T328, T325, T326, T327) -> U47(f235_in(T328), T328, T325, T326, T327) U47(f235_out1(T335), T328, T325, T326, T327) -> U48(f594_in(T335, T325, T326, T327, T328), T328, T325, T326, T327, T335) U48(f594_out1(T336), T328, T325, T326, T327, T335) -> f591_out1(T335, T336) f594_in(T335, T325, T326, T327, T328) -> U49(f596_in(T335, T325, T326, T327), T335, T325, T326, T327, T328) U49(f596_out1, T335, T325, T326, T327, T328) -> U50(f597_in(T335, T325, T326, T327, T328), T335, T325, T326, T327, T328) U50(f597_out1(T336), T335, T325, T326, T327, T328) -> f594_out1(T336) f630_in(T377, T378, T379, T380) -> U51(f635_in(T377, T378, T379, T380), T377, T378, T379, T380) U51(f635_out1, T377, T378, T379, T380) -> U52(f636_in, T377, T378, T379, T380) U52(f636_out1, T377, T378, T379, T380) -> f630_out1 f681_in(T495, T491, T492, T493, T494) -> U53(f235_in(T495), T495, T491, T492, T493, T494) U53(f235_out1(T502), T495, T491, T492, T493, T494) -> U54(f684_in(T502, T491, T492, T493, T494, T495), T495, T491, T492, T493, T494, T502) U54(f684_out1(T503), T495, T491, T492, T493, T494, T502) -> f681_out1(T502, T503) f684_in(T502, T491, T492, T493, T494, T495) -> U55(f694_in(T502, T491, T492, T493, T494), T502, T491, T492, T493, T494, T495) U55(f694_out1, T502, T491, T492, T493, T494, T495) -> U56(f695_in(T502, T491, T492, T493, T494, T495), T502, T491, T492, T493, T494, T495) U56(f695_out1(T503), T502, T491, T492, T493, T494, T495) -> f684_out1(T503) f712_in(T554, T555, T556, T557, T558) -> U57(f724_in(T554, T555, T556, T557, T558), T554, T555, T556, T557, T558) U57(f724_out1, T554, T555, T556, T557, T558) -> U58(f725_in, T554, T555, T556, T557, T558) U58(f725_out1, T554, T555, T556, T557, T558) -> f712_out1 f778_in(T702, T697, T698, T699, T700, T701) -> U59(f235_in(T702), T702, T697, T698, T699, T700, T701) U59(f235_out1(T709), T702, T697, T698, T699, T700, T701) -> U60(f782_in(T709, T697, T698, T699, T700, T701, T702), T702, T697, T698, T699, T700, T701, T709) U60(f782_out1(T710), T702, T697, T698, T699, T700, T701, T709) -> f778_out1(T709, T710) f782_in(T709, T697, T698, T699, T700, T701, T702) -> U61(f785_in(T709, T697, T698, T699, T700, T701), T709, T697, T698, T699, T700, T701, T702) U61(f785_out1, T709, T697, T698, T699, T700, T701, T702) -> U62(f786_in(T709, T697, T698, T699, T700, T701, T702), T709, T697, T698, T699, T700, T701, T702) U62(f786_out1(T710), T709, T697, T698, T699, T700, T701, T702) -> f782_out1(T710) f817_in(T771, T772, T773, T774, T775, T776) -> U63(f820_in(T771, T772, T773, T774, T775, T776), T771, T772, T773, T774, T775, T776) U63(f820_out1, T771, T772, T773, T774, T775, T776) -> U64(f821_in, T771, T772, T773, T774, T775, T776) U64(f821_out1, T771, T772, T773, T774, T775, T776) -> f817_out1 f872_in(T949, T943, T944, T945, T946, T947, T948) -> U65(f235_in(T949), T949, T943, T944, T945, T946, T947, T948) U65(f235_out1(T956), T949, T943, T944, T945, T946, T947, T948) -> U66(f878_in(T956, T943, T944, T945, T946, T947, T948, T949), T949, T943, T944, T945, T946, T947, T948, T956) U66(f878_out1(T957), T949, T943, T944, T945, T946, T947, T948, T956) -> f872_out1(T956, T957) f878_in(T956, T943, T944, T945, T946, T947, T948, T949) -> U67(f883_in(T956, T943, T944, T945, T946, T947, T948), T956, T943, T944, T945, T946, T947, T948, T949) U67(f883_out1, T956, T943, T944, T945, T946, T947, T948, T949) -> U68(f884_in(T956, T943, T944, T945, T946, T947, T948, T949), T956, T943, T944, T945, T946, T947, T948, T949) U68(f884_out1(T957), T956, T943, T944, T945, T946, T947, T948, T949) -> f878_out1(T957) f909_in(T1028, T1029, T1030, T1031, T1032, T1033, T1034) -> U69(f913_in(T1028, T1029, T1030, T1031, T1032, T1033, T1034), T1028, T1029, T1030, T1031, T1032, T1033, T1034) U69(f913_out1, T1028, T1029, T1030, T1031, T1032, T1033, T1034) -> U70(f914_in, T1028, T1029, T1030, T1031, T1032, T1033, T1034) U70(f914_out1, T1028, T1029, T1030, T1031, T1032, T1033, T1034) -> f909_out1 f1018_in(T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243) -> U71(f235_in(T1236), T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243) U71(f235_out1(T1248), T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243) -> U72(f1020_in(T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1236), T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1248) U72(f1020_out1(T1249), T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1248) -> f1018_out1(T1248, T1249) f1020_in(T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1236) -> U73(f1021_in(T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235), T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1236) U73(f1021_out1, T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1236) -> U74(f1022_in(T1248, T1243, T1236), T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1236) U74(f1022_out1(T1249), T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1236) -> f1020_out1(T1249) f1037_in(T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) -> U75(f1038_in(T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337), T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) U75(f1038_out1, T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) -> U76(f1039_in, T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) U76(f1039_out1, T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) -> f1037_out1 f1110_in(T1508, T1506, T1507) -> U77(f235_in(T1508), T1508, T1506, T1507) U77(f235_out1(T1515), T1508, T1506, T1507) -> U78(f1116_in(T1515, T1506, T1507, T1508), T1508, T1506, T1507, T1515) U78(f1116_out1(T1516), T1508, T1506, T1507, T1515) -> f1110_out1(T1515, T1516) f1116_in(T1515, T1506, T1507, T1508) -> U79(f1144_in(T1515, T1506, T1507), T1515, T1506, T1507, T1508) U79(f1144_out1, T1515, T1506, T1507, T1508) -> U80(f1022_in(T1515, .(T1506, T1507), T1508), T1515, T1506, T1507, T1508) U80(f1022_out1(T1516), T1515, T1506, T1507, T1508) -> f1116_out1(T1516) f1153_in(T1547, T1548, T1549) -> U81(f1154_in(T1547, T1548, T1549), T1547, T1548, T1549) U81(f1154_out1, T1547, T1548, T1549) -> U82(f1155_in, T1547, T1548, T1549) U82(f1155_out1, T1547, T1548, T1549) -> f1153_out1 Q is empty. ---------------------------------------- (194) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (195) Obligation: Q DP problem: The TRS P consists of the following rules: F3_IN(T28) -> U1^1(f150_in(T28), T28) F3_IN(T28) -> F150_IN(T28) F235_IN(.(T58, T59)) -> U2^1(f235_in(T59), .(T58, T59)) F235_IN(.(T58, T59)) -> F235_IN(T59) F445_IN(T156, T157) -> U3^1(f316_in(T156), T156, T157) F537_IN(T269, T270, T271) -> U4^1(f445_in(T269, T271), T269, T270, T271) F537_IN(T269, T270, T271) -> F445_IN(T269, T271) F635_IN(T420, T421, T422, T423) -> U5^1(f537_in(T420, T422, T423), T420, T421, T422, T423) F635_IN(T420, T421, T422, T423) -> F537_IN(T420, T422, T423) F724_IN(T611, T612, T613, T614, T615) -> U6^1(f635_in(T611, T613, T614, T615), T611, T612, T613, T614, T615) F724_IN(T611, T612, T613, T614, T615) -> F635_IN(T611, T613, T614, T615) F820_IN(T842, T843, T844, T845, T846, T847) -> U7^1(f724_in(T842, T844, T845, T846, T847), T842, T843, T844, T845, T846, T847) F820_IN(T842, T843, T844, T845, T846, T847) -> F724_IN(T842, T844, T845, T846, T847) F913_IN(T1113, T1114, T1115, T1116, T1117, T1118, T1119) -> U8^1(f820_in(T1113, T1115, T1116, T1117, T1118, T1119), T1113, T1114, T1115, T1116, T1117, T1118, T1119) F913_IN(T1113, T1114, T1115, T1116, T1117, T1118, T1119) -> F820_IN(T1113, T1115, T1116, T1117, T1118, T1119) F1022_IN(T1506, T1507, T1508) -> U9^1(f1110_in(T1508, T1506, T1507), T1506, T1507, T1508) F1022_IN(T1506, T1507, T1508) -> F1110_IN(T1508, T1506, T1507) F1164_IN(T1601, .(T1602, T1603)) -> U10^1(f1164_in(T1601, T1603), T1601, .(T1602, T1603)) F1164_IN(T1601, .(T1602, T1603)) -> F1164_IN(T1601, T1603) F289_IN(T75) -> U11^1(f299_in(T75), T75) F289_IN(T75) -> F299_IN(T75) F290_IN(T109, T110) -> U12^1(f408_in(T110, T109), T109, T110) F290_IN(T109, T110) -> F408_IN(T110, T109) F412_IN(T139, T140) -> U13^1(f439_in(T139, T140), T139, T140) F412_IN(T139, T140) -> F439_IN(T139, T140) F413_IN(T199, T200, T201) -> U14^1(f505_in(T201, T199, T200), T199, T200, T201) F413_IN(T199, T200, T201) -> F505_IN(T201, T199, T200) F517_IN(T240, T241, T242) -> U15^1(f535_in(T240, T241, T242), T240, T241, T242) F517_IN(T240, T241, T242) -> F535_IN(T240, T241, T242) F518_IN(T325, T326, T327, T328) -> U16^1(f591_in(T328, T325, T326, T327), T325, T326, T327, T328) F518_IN(T325, T326, T327, T328) -> F591_IN(T328, T325, T326, T327) F596_IN(T377, T378, T379, T380) -> U17^1(f630_in(T377, T378, T379, T380), T377, T378, T379, T380) F596_IN(T377, T378, T379, T380) -> F630_IN(T377, T378, T379, T380) F597_IN(T491, T492, T493, T494, T495) -> U18^1(f681_in(T495, T491, T492, T493, T494), T491, T492, T493, T494, T495) F597_IN(T491, T492, T493, T494, T495) -> F681_IN(T495, T491, T492, T493, T494) F694_IN(T554, T555, T556, T557, T558) -> U19^1(f712_in(T554, T555, T556, T557, T558), T554, T555, T556, T557, T558) F694_IN(T554, T555, T556, T557, T558) -> F712_IN(T554, T555, T556, T557, T558) F695_IN(T697, T698, T699, T700, T701, T702) -> U20^1(f778_in(T702, T697, T698, T699, T700, T701), T697, T698, T699, T700, T701, T702) F695_IN(T697, T698, T699, T700, T701, T702) -> F778_IN(T702, T697, T698, T699, T700, T701) F785_IN(T771, T772, T773, T774, T775, T776) -> U21^1(f817_in(T771, T772, T773, T774, T775, T776), T771, T772, T773, T774, T775, T776) F785_IN(T771, T772, T773, T774, T775, T776) -> F817_IN(T771, T772, T773, T774, T775, T776) F786_IN(T943, T944, T945, T946, T947, T948, T949) -> U22^1(f872_in(T949, T943, T944, T945, T946, T947, T948), T943, T944, T945, T946, T947, T948, T949) F786_IN(T943, T944, T945, T946, T947, T948, T949) -> F872_IN(T949, T943, T944, T945, T946, T947, T948) F883_IN(T1028, T1029, T1030, T1031, T1032, T1033, T1034) -> U23^1(f909_in(T1028, T1029, T1030, T1031, T1032, T1033, T1034), T1028, T1029, T1030, T1031, T1032, T1033, T1034) F883_IN(T1028, T1029, T1030, T1031, T1032, T1033, T1034) -> F909_IN(T1028, T1029, T1030, T1031, T1032, T1033, T1034) F884_IN(T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1236) -> U24^1(f1018_in(T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235, .(T1229, .(T1230, .(T1231, .(T1232, .(T1233, .(T1234, .(T1235, [])))))))), T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1236) F884_IN(T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1236) -> F1018_IN(T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235, .(T1229, .(T1230, .(T1231, .(T1232, .(T1233, .(T1234, .(T1235, [])))))))) F1021_IN(T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) -> U25^1(f1037_in(T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337), T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) F1021_IN(T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) -> F1037_IN(T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) F1038_IN(T1429, T1430, T1431, T1432, T1433, T1434, T1435, T1436) -> U26^1(f913_in(T1429, T1431, T1432, T1433, T1434, T1435, T1436), T1429, T1430, T1431, T1432, T1433, T1434, T1435, T1436) F1038_IN(T1429, T1430, T1431, T1432, T1433, T1434, T1435, T1436) -> F913_IN(T1429, T1431, T1432, T1433, T1434, T1435, T1436) F1144_IN(T1547, T1548, T1549) -> U27^1(f1153_in(T1547, T1548, T1549), T1547, T1548, T1549) F1144_IN(T1547, T1548, T1549) -> F1153_IN(T1547, T1548, T1549) F1154_IN(T1578, T1579, T1580) -> U28^1(f1164_in(T1578, T1580), T1578, T1579, T1580) F1154_IN(T1578, T1579, T1580) -> F1164_IN(T1578, T1580) F150_IN(T28) -> U29^1(f235_in(T28), T28) F150_IN(T28) -> F235_IN(T28) U29^1(f235_out1(T35), T28) -> U30^1(f236_in(T35, T28), T28, T35) U29^1(f235_out1(T35), T28) -> F236_IN(T35, T28) F236_IN(T35, T28) -> U31^1(f289_in(T35), T35, T28) F236_IN(T35, T28) -> F289_IN(T35) U31^1(f289_out1, T35, T28) -> U32^1(f290_in(T35, T28), T35, T28) U31^1(f289_out1, T35, T28) -> F290_IN(T35, T28) F299_IN(T75) -> U33^1(f316_in(T75), T75) U33^1(f316_out1, T75) -> U34^1(f317_in, T75) F408_IN(T110, T109) -> U35^1(f235_in(T110), T110, T109) F408_IN(T110, T109) -> F235_IN(T110) U35^1(f235_out1(T117), T110, T109) -> U36^1(f411_in(T117, T109, T110), T110, T109, T117) U35^1(f235_out1(T117), T110, T109) -> F411_IN(T117, T109, T110) F411_IN(T117, T109, T110) -> U37^1(f412_in(T117, T109), T117, T109, T110) F411_IN(T117, T109, T110) -> F412_IN(T117, T109) U37^1(f412_out1, T117, T109, T110) -> U38^1(f413_in(T117, T109, T110), T117, T109, T110) U37^1(f412_out1, T117, T109, T110) -> F413_IN(T117, T109, T110) F439_IN(T139, T140) -> U39^1(f445_in(T139, T140), T139, T140) F439_IN(T139, T140) -> F445_IN(T139, T140) U39^1(f445_out1, T139, T140) -> U40^1(f446_in, T139, T140) F505_IN(T201, T199, T200) -> U41^1(f235_in(T201), T201, T199, T200) F505_IN(T201, T199, T200) -> F235_IN(T201) U41^1(f235_out1(T208), T201, T199, T200) -> U42^1(f510_in(T208, T199, T200, T201), T201, T199, T200, T208) U41^1(f235_out1(T208), T201, T199, T200) -> F510_IN(T208, T199, T200, T201) F510_IN(T208, T199, T200, T201) -> U43^1(f517_in(T208, T199, T200), T208, T199, T200, T201) F510_IN(T208, T199, T200, T201) -> F517_IN(T208, T199, T200) U43^1(f517_out1, T208, T199, T200, T201) -> U44^1(f518_in(T208, T199, T200, T201), T208, T199, T200, T201) U43^1(f517_out1, T208, T199, T200, T201) -> F518_IN(T208, T199, T200, T201) F535_IN(T240, T241, T242) -> U45^1(f537_in(T240, T241, T242), T240, T241, T242) F535_IN(T240, T241, T242) -> F537_IN(T240, T241, T242) U45^1(f537_out1, T240, T241, T242) -> U46^1(f538_in, T240, T241, T242) F591_IN(T328, T325, T326, T327) -> U47^1(f235_in(T328), T328, T325, T326, T327) F591_IN(T328, T325, T326, T327) -> F235_IN(T328) U47^1(f235_out1(T335), T328, T325, T326, T327) -> U48^1(f594_in(T335, T325, T326, T327, T328), T328, T325, T326, T327, T335) U47^1(f235_out1(T335), T328, T325, T326, T327) -> F594_IN(T335, T325, T326, T327, T328) F594_IN(T335, T325, T326, T327, T328) -> U49^1(f596_in(T335, T325, T326, T327), T335, T325, T326, T327, T328) F594_IN(T335, T325, T326, T327, T328) -> F596_IN(T335, T325, T326, T327) U49^1(f596_out1, T335, T325, T326, T327, T328) -> U50^1(f597_in(T335, T325, T326, T327, T328), T335, T325, T326, T327, T328) U49^1(f596_out1, T335, T325, T326, T327, T328) -> F597_IN(T335, T325, T326, T327, T328) F630_IN(T377, T378, T379, T380) -> U51^1(f635_in(T377, T378, T379, T380), T377, T378, T379, T380) F630_IN(T377, T378, T379, T380) -> F635_IN(T377, T378, T379, T380) U51^1(f635_out1, T377, T378, T379, T380) -> U52^1(f636_in, T377, T378, T379, T380) F681_IN(T495, T491, T492, T493, T494) -> U53^1(f235_in(T495), T495, T491, T492, T493, T494) F681_IN(T495, T491, T492, T493, T494) -> F235_IN(T495) U53^1(f235_out1(T502), T495, T491, T492, T493, T494) -> U54^1(f684_in(T502, T491, T492, T493, T494, T495), T495, T491, T492, T493, T494, T502) U53^1(f235_out1(T502), T495, T491, T492, T493, T494) -> F684_IN(T502, T491, T492, T493, T494, T495) F684_IN(T502, T491, T492, T493, T494, T495) -> U55^1(f694_in(T502, T491, T492, T493, T494), T502, T491, T492, T493, T494, T495) F684_IN(T502, T491, T492, T493, T494, T495) -> F694_IN(T502, T491, T492, T493, T494) U55^1(f694_out1, T502, T491, T492, T493, T494, T495) -> U56^1(f695_in(T502, T491, T492, T493, T494, T495), T502, T491, T492, T493, T494, T495) U55^1(f694_out1, T502, T491, T492, T493, T494, T495) -> F695_IN(T502, T491, T492, T493, T494, T495) F712_IN(T554, T555, T556, T557, T558) -> U57^1(f724_in(T554, T555, T556, T557, T558), T554, T555, T556, T557, T558) F712_IN(T554, T555, T556, T557, T558) -> F724_IN(T554, T555, T556, T557, T558) U57^1(f724_out1, T554, T555, T556, T557, T558) -> U58^1(f725_in, T554, T555, T556, T557, T558) F778_IN(T702, T697, T698, T699, T700, T701) -> U59^1(f235_in(T702), T702, T697, T698, T699, T700, T701) F778_IN(T702, T697, T698, T699, T700, T701) -> F235_IN(T702) U59^1(f235_out1(T709), T702, T697, T698, T699, T700, T701) -> U60^1(f782_in(T709, T697, T698, T699, T700, T701, T702), T702, T697, T698, T699, T700, T701, T709) U59^1(f235_out1(T709), T702, T697, T698, T699, T700, T701) -> F782_IN(T709, T697, T698, T699, T700, T701, T702) F782_IN(T709, T697, T698, T699, T700, T701, T702) -> U61^1(f785_in(T709, T697, T698, T699, T700, T701), T709, T697, T698, T699, T700, T701, T702) F782_IN(T709, T697, T698, T699, T700, T701, T702) -> F785_IN(T709, T697, T698, T699, T700, T701) U61^1(f785_out1, T709, T697, T698, T699, T700, T701, T702) -> U62^1(f786_in(T709, T697, T698, T699, T700, T701, T702), T709, T697, T698, T699, T700, T701, T702) U61^1(f785_out1, T709, T697, T698, T699, T700, T701, T702) -> F786_IN(T709, T697, T698, T699, T700, T701, T702) F817_IN(T771, T772, T773, T774, T775, T776) -> U63^1(f820_in(T771, T772, T773, T774, T775, T776), T771, T772, T773, T774, T775, T776) F817_IN(T771, T772, T773, T774, T775, T776) -> F820_IN(T771, T772, T773, T774, T775, T776) U63^1(f820_out1, T771, T772, T773, T774, T775, T776) -> U64^1(f821_in, T771, T772, T773, T774, T775, T776) F872_IN(T949, T943, T944, T945, T946, T947, T948) -> U65^1(f235_in(T949), T949, T943, T944, T945, T946, T947, T948) F872_IN(T949, T943, T944, T945, T946, T947, T948) -> F235_IN(T949) U65^1(f235_out1(T956), T949, T943, T944, T945, T946, T947, T948) -> U66^1(f878_in(T956, T943, T944, T945, T946, T947, T948, T949), T949, T943, T944, T945, T946, T947, T948, T956) U65^1(f235_out1(T956), T949, T943, T944, T945, T946, T947, T948) -> F878_IN(T956, T943, T944, T945, T946, T947, T948, T949) F878_IN(T956, T943, T944, T945, T946, T947, T948, T949) -> U67^1(f883_in(T956, T943, T944, T945, T946, T947, T948), T956, T943, T944, T945, T946, T947, T948, T949) F878_IN(T956, T943, T944, T945, T946, T947, T948, T949) -> F883_IN(T956, T943, T944, T945, T946, T947, T948) U67^1(f883_out1, T956, T943, T944, T945, T946, T947, T948, T949) -> U68^1(f884_in(T956, T943, T944, T945, T946, T947, T948, T949), T956, T943, T944, T945, T946, T947, T948, T949) U67^1(f883_out1, T956, T943, T944, T945, T946, T947, T948, T949) -> F884_IN(T956, T943, T944, T945, T946, T947, T948, T949) F909_IN(T1028, T1029, T1030, T1031, T1032, T1033, T1034) -> U69^1(f913_in(T1028, T1029, T1030, T1031, T1032, T1033, T1034), T1028, T1029, T1030, T1031, T1032, T1033, T1034) F909_IN(T1028, T1029, T1030, T1031, T1032, T1033, T1034) -> F913_IN(T1028, T1029, T1030, T1031, T1032, T1033, T1034) U69^1(f913_out1, T1028, T1029, T1030, T1031, T1032, T1033, T1034) -> U70^1(f914_in, T1028, T1029, T1030, T1031, T1032, T1033, T1034) F1018_IN(T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243) -> U71^1(f235_in(T1236), T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243) F1018_IN(T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243) -> F235_IN(T1236) U71^1(f235_out1(T1248), T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243) -> U72^1(f1020_in(T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1236), T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1248) U71^1(f235_out1(T1248), T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243) -> F1020_IN(T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1236) F1020_IN(T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1236) -> U73^1(f1021_in(T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235), T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1236) F1020_IN(T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1236) -> F1021_IN(T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235) U73^1(f1021_out1, T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1236) -> U74^1(f1022_in(T1248, T1243, T1236), T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1236) U73^1(f1021_out1, T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1236) -> F1022_IN(T1248, T1243, T1236) F1037_IN(T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) -> U75^1(f1038_in(T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337), T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) F1037_IN(T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) -> F1038_IN(T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) U75^1(f1038_out1, T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) -> U76^1(f1039_in, T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) F1110_IN(T1508, T1506, T1507) -> U77^1(f235_in(T1508), T1508, T1506, T1507) F1110_IN(T1508, T1506, T1507) -> F235_IN(T1508) U77^1(f235_out1(T1515), T1508, T1506, T1507) -> U78^1(f1116_in(T1515, T1506, T1507, T1508), T1508, T1506, T1507, T1515) U77^1(f235_out1(T1515), T1508, T1506, T1507) -> F1116_IN(T1515, T1506, T1507, T1508) F1116_IN(T1515, T1506, T1507, T1508) -> U79^1(f1144_in(T1515, T1506, T1507), T1515, T1506, T1507, T1508) F1116_IN(T1515, T1506, T1507, T1508) -> F1144_IN(T1515, T1506, T1507) U79^1(f1144_out1, T1515, T1506, T1507, T1508) -> U80^1(f1022_in(T1515, .(T1506, T1507), T1508), T1515, T1506, T1507, T1508) U79^1(f1144_out1, T1515, T1506, T1507, T1508) -> F1022_IN(T1515, .(T1506, T1507), T1508) F1153_IN(T1547, T1548, T1549) -> U81^1(f1154_in(T1547, T1548, T1549), T1547, T1548, T1549) F1153_IN(T1547, T1548, T1549) -> F1154_IN(T1547, T1548, T1549) U81^1(f1154_out1, T1547, T1548, T1549) -> U82^1(f1155_in, T1547, T1548, T1549) The TRS R consists of the following rules: f3_in(T19) -> f3_out1([]) f3_in(T28) -> U1(f150_in(T28), T28) U1(f150_out1(T29, T30), T28) -> f3_out1(.(T29, T30)) f235_in(.(T49, T50)) -> f235_out1(T49) f235_in(.(T58, T59)) -> U2(f235_in(T59), .(T58, T59)) U2(f235_out1(T60), .(T58, T59)) -> f235_out1(T60) f445_in(T149, T149) -> f445_out1 f445_in(T156, T157) -> U3(f316_in(T156), T156, T157) U3(f316_out1, T156, T157) -> f445_out1 f537_in(T259, T259, T260) -> f537_out1 f537_in(T269, T270, T271) -> U4(f445_in(T269, T271), T269, T270, T271) U4(f445_out1, T269, T270, T271) -> f537_out1 f635_in(T405, T405, T406, T407) -> f635_out1 f635_in(T420, T421, T422, T423) -> U5(f537_in(T420, T422, T423), T420, T421, T422, T423) U5(f537_out1, T420, T421, T422, T423) -> f635_out1 f724_in(T591, T591, T592, T593, T594) -> f724_out1 f724_in(T611, T612, T613, T614, T615) -> U6(f635_in(T611, T613, T614, T615), T611, T612, T613, T614, T615) U6(f635_out1, T611, T612, T613, T614, T615) -> f724_out1 f820_in(T817, T817, T818, T819, T820, T821) -> f820_out1 f820_in(T842, T843, T844, T845, T846, T847) -> U7(f724_in(T842, T844, T845, T846, T847), T842, T843, T844, T845, T846, T847) U7(f724_out1, T842, T843, T844, T845, T846, T847) -> f820_out1 f913_in(T1083, T1083, T1084, T1085, T1086, T1087, T1088) -> f913_out1 f913_in(T1113, T1114, T1115, T1116, T1117, T1118, T1119) -> U8(f820_in(T1113, T1115, T1116, T1117, T1118, T1119), T1113, T1114, T1115, T1116, T1117, T1118, T1119) U8(f820_out1, T1113, T1114, T1115, T1116, T1117, T1118, T1119) -> f913_out1 f1022_in(T1491, T1492, T1493) -> f1022_out1([]) f1022_in(T1506, T1507, T1508) -> U9(f1110_in(T1508, T1506, T1507), T1506, T1507, T1508) U9(f1110_out1(T1509, T1510), T1506, T1507, T1508) -> f1022_out1(.(T1509, T1510)) f1164_in(T1593, .(T1593, T1594)) -> f1164_out1 f1164_in(T1601, .(T1602, T1603)) -> U10(f1164_in(T1601, T1603), T1601, .(T1602, T1603)) U10(f1164_out1, T1601, .(T1602, T1603)) -> f1164_out1 f289_in(T75) -> U11(f299_in(T75), T75) U11(f299_out1, T75) -> f289_out1 f289_in(T84) -> f289_out1 f290_in(T97, T98) -> f290_out1([]) f290_in(T109, T110) -> U12(f408_in(T110, T109), T109, T110) U12(f408_out1(T111, T112), T109, T110) -> f290_out1(.(T111, T112)) f412_in(T139, T140) -> U13(f439_in(T139, T140), T139, T140) U13(f439_out1, T139, T140) -> f412_out1 f412_in(T164, T165) -> f412_out1 f413_in(T184, T185, T186) -> f413_out1([]) f413_in(T199, T200, T201) -> U14(f505_in(T201, T199, T200), T199, T200, T201) U14(f505_out1(T202, T203), T199, T200, T201) -> f413_out1(.(T202, T203)) f517_in(T240, T241, T242) -> U15(f535_in(T240, T241, T242), T240, T241, T242) U15(f535_out1, T240, T241, T242) -> f517_out1 f517_in(T280, T281, T282) -> f517_out1 f518_in(T307, T308, T309, T310) -> f518_out1([]) f518_in(T325, T326, T327, T328) -> U16(f591_in(T328, T325, T326, T327), T325, T326, T327, T328) U16(f591_out1(T329, T330), T325, T326, T327, T328) -> f518_out1(.(T329, T330)) f596_in(T377, T378, T379, T380) -> U17(f630_in(T377, T378, T379, T380), T377, T378, T379, T380) U17(f630_out1, T377, T378, T379, T380) -> f596_out1 f596_in(T436, T437, T438, T439) -> f596_out1 f597_in(T470, T471, T472, T473, T474) -> f597_out1([]) f597_in(T491, T492, T493, T494, T495) -> U18(f681_in(T495, T491, T492, T493, T494), T491, T492, T493, T494, T495) U18(f681_out1(T496, T497), T491, T492, T493, T494, T495) -> f597_out1(.(T496, T497)) f694_in(T554, T555, T556, T557, T558) -> U19(f712_in(T554, T555, T556, T557, T558), T554, T555, T556, T557, T558) U19(f712_out1, T554, T555, T556, T557, T558) -> f694_out1 f694_in(T632, T633, T634, T635, T636) -> f694_out1 f695_in(T673, T674, T675, T676, T677, T678) -> f695_out1([]) f695_in(T697, T698, T699, T700, T701, T702) -> U20(f778_in(T702, T697, T698, T699, T700, T701), T697, T698, T699, T700, T701, T702) U20(f778_out1(T703, T704), T697, T698, T699, T700, T701, T702) -> f695_out1(.(T703, T704)) f785_in(T771, T772, T773, T774, T775, T776) -> U21(f817_in(T771, T772, T773, T774, T775, T776), T771, T772, T773, T774, T775, T776) U21(f817_out1, T771, T772, T773, T774, T775, T776) -> f785_out1 f785_in(T868, T869, T870, T871, T872, T873) -> f785_out1 f786_in(T916, T917, T918, T919, T920, T921, T922) -> f786_out1([]) f786_in(T943, T944, T945, T946, T947, T948, T949) -> U22(f872_in(T949, T943, T944, T945, T946, T947, T948), T943, T944, T945, T946, T947, T948, T949) U22(f872_out1(T950, T951), T943, T944, T945, T946, T947, T948, T949) -> f786_out1(.(T950, T951)) f883_in(T1028, T1029, T1030, T1031, T1032, T1033, T1034) -> U23(f909_in(T1028, T1029, T1030, T1031, T1032, T1033, T1034), T1028, T1029, T1030, T1031, T1032, T1033, T1034) U23(f909_out1, T1028, T1029, T1030, T1031, T1032, T1033, T1034) -> f883_out1 f883_in(T1144, T1145, T1146, T1147, T1148, T1149, T1150) -> f883_out1 f884_in(T1199, T1200, T1201, T1202, T1203, T1204, T1205, T1206) -> f884_out1([]) f884_in(T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1236) -> U24(f1018_in(T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235, .(T1229, .(T1230, .(T1231, .(T1232, .(T1233, .(T1234, .(T1235, [])))))))), T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1236) U24(f1018_out1(T1237, T1238), T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1236) -> f884_out1(.(T1237, T1238)) f1021_in(T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) -> U25(f1037_in(T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337), T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) U25(f1037_out1, T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) -> f1021_out1 f1021_in(T1465, T1466, T1467, T1468, T1469, T1470, T1471, T1472) -> f1021_out1 f1038_in(T1394, T1394, T1395, T1396, T1397, T1398, T1399, T1400) -> f1038_out1 f1038_in(T1429, T1430, T1431, T1432, T1433, T1434, T1435, T1436) -> U26(f913_in(T1429, T1431, T1432, T1433, T1434, T1435, T1436), T1429, T1430, T1431, T1432, T1433, T1434, T1435, T1436) U26(f913_out1, T1429, T1430, T1431, T1432, T1433, T1434, T1435, T1436) -> f1038_out1 f1144_in(T1547, T1548, T1549) -> U27(f1153_in(T1547, T1548, T1549), T1547, T1548, T1549) U27(f1153_out1, T1547, T1548, T1549) -> f1144_out1 f1144_in(T1614, T1615, T1616) -> f1144_out1 f1154_in(T1566, T1566, T1567) -> f1154_out1 f1154_in(T1578, T1579, T1580) -> U28(f1164_in(T1578, T1580), T1578, T1579, T1580) U28(f1164_out1, T1578, T1579, T1580) -> f1154_out1 f150_in(T28) -> U29(f235_in(T28), T28) U29(f235_out1(T35), T28) -> U30(f236_in(T35, T28), T28, T35) U30(f236_out1(T36), T28, T35) -> f150_out1(T35, T36) f236_in(T35, T28) -> U31(f289_in(T35), T35, T28) U31(f289_out1, T35, T28) -> U32(f290_in(T35, T28), T35, T28) U32(f290_out1(T36), T35, T28) -> f236_out1(T36) f299_in(T75) -> U33(f316_in(T75), T75) U33(f316_out1, T75) -> U34(f317_in, T75) U34(f317_out1, T75) -> f299_out1 f408_in(T110, T109) -> U35(f235_in(T110), T110, T109) U35(f235_out1(T117), T110, T109) -> U36(f411_in(T117, T109, T110), T110, T109, T117) U36(f411_out1(T118), T110, T109, T117) -> f408_out1(T117, T118) f411_in(T117, T109, T110) -> U37(f412_in(T117, T109), T117, T109, T110) U37(f412_out1, T117, T109, T110) -> U38(f413_in(T117, T109, T110), T117, T109, T110) U38(f413_out1(T118), T117, T109, T110) -> f411_out1(T118) f439_in(T139, T140) -> U39(f445_in(T139, T140), T139, T140) U39(f445_out1, T139, T140) -> U40(f446_in, T139, T140) U40(f446_out1, T139, T140) -> f439_out1 f505_in(T201, T199, T200) -> U41(f235_in(T201), T201, T199, T200) U41(f235_out1(T208), T201, T199, T200) -> U42(f510_in(T208, T199, T200, T201), T201, T199, T200, T208) U42(f510_out1(T209), T201, T199, T200, T208) -> f505_out1(T208, T209) f510_in(T208, T199, T200, T201) -> U43(f517_in(T208, T199, T200), T208, T199, T200, T201) U43(f517_out1, T208, T199, T200, T201) -> U44(f518_in(T208, T199, T200, T201), T208, T199, T200, T201) U44(f518_out1(T209), T208, T199, T200, T201) -> f510_out1(T209) f535_in(T240, T241, T242) -> U45(f537_in(T240, T241, T242), T240, T241, T242) U45(f537_out1, T240, T241, T242) -> U46(f538_in, T240, T241, T242) U46(f538_out1, T240, T241, T242) -> f535_out1 f591_in(T328, T325, T326, T327) -> U47(f235_in(T328), T328, T325, T326, T327) U47(f235_out1(T335), T328, T325, T326, T327) -> U48(f594_in(T335, T325, T326, T327, T328), T328, T325, T326, T327, T335) U48(f594_out1(T336), T328, T325, T326, T327, T335) -> f591_out1(T335, T336) f594_in(T335, T325, T326, T327, T328) -> U49(f596_in(T335, T325, T326, T327), T335, T325, T326, T327, T328) U49(f596_out1, T335, T325, T326, T327, T328) -> U50(f597_in(T335, T325, T326, T327, T328), T335, T325, T326, T327, T328) U50(f597_out1(T336), T335, T325, T326, T327, T328) -> f594_out1(T336) f630_in(T377, T378, T379, T380) -> U51(f635_in(T377, T378, T379, T380), T377, T378, T379, T380) U51(f635_out1, T377, T378, T379, T380) -> U52(f636_in, T377, T378, T379, T380) U52(f636_out1, T377, T378, T379, T380) -> f630_out1 f681_in(T495, T491, T492, T493, T494) -> U53(f235_in(T495), T495, T491, T492, T493, T494) U53(f235_out1(T502), T495, T491, T492, T493, T494) -> U54(f684_in(T502, T491, T492, T493, T494, T495), T495, T491, T492, T493, T494, T502) U54(f684_out1(T503), T495, T491, T492, T493, T494, T502) -> f681_out1(T502, T503) f684_in(T502, T491, T492, T493, T494, T495) -> U55(f694_in(T502, T491, T492, T493, T494), T502, T491, T492, T493, T494, T495) U55(f694_out1, T502, T491, T492, T493, T494, T495) -> U56(f695_in(T502, T491, T492, T493, T494, T495), T502, T491, T492, T493, T494, T495) U56(f695_out1(T503), T502, T491, T492, T493, T494, T495) -> f684_out1(T503) f712_in(T554, T555, T556, T557, T558) -> U57(f724_in(T554, T555, T556, T557, T558), T554, T555, T556, T557, T558) U57(f724_out1, T554, T555, T556, T557, T558) -> U58(f725_in, T554, T555, T556, T557, T558) U58(f725_out1, T554, T555, T556, T557, T558) -> f712_out1 f778_in(T702, T697, T698, T699, T700, T701) -> U59(f235_in(T702), T702, T697, T698, T699, T700, T701) U59(f235_out1(T709), T702, T697, T698, T699, T700, T701) -> U60(f782_in(T709, T697, T698, T699, T700, T701, T702), T702, T697, T698, T699, T700, T701, T709) U60(f782_out1(T710), T702, T697, T698, T699, T700, T701, T709) -> f778_out1(T709, T710) f782_in(T709, T697, T698, T699, T700, T701, T702) -> U61(f785_in(T709, T697, T698, T699, T700, T701), T709, T697, T698, T699, T700, T701, T702) U61(f785_out1, T709, T697, T698, T699, T700, T701, T702) -> U62(f786_in(T709, T697, T698, T699, T700, T701, T702), T709, T697, T698, T699, T700, T701, T702) U62(f786_out1(T710), T709, T697, T698, T699, T700, T701, T702) -> f782_out1(T710) f817_in(T771, T772, T773, T774, T775, T776) -> U63(f820_in(T771, T772, T773, T774, T775, T776), T771, T772, T773, T774, T775, T776) U63(f820_out1, T771, T772, T773, T774, T775, T776) -> U64(f821_in, T771, T772, T773, T774, T775, T776) U64(f821_out1, T771, T772, T773, T774, T775, T776) -> f817_out1 f872_in(T949, T943, T944, T945, T946, T947, T948) -> U65(f235_in(T949), T949, T943, T944, T945, T946, T947, T948) U65(f235_out1(T956), T949, T943, T944, T945, T946, T947, T948) -> U66(f878_in(T956, T943, T944, T945, T946, T947, T948, T949), T949, T943, T944, T945, T946, T947, T948, T956) U66(f878_out1(T957), T949, T943, T944, T945, T946, T947, T948, T956) -> f872_out1(T956, T957) f878_in(T956, T943, T944, T945, T946, T947, T948, T949) -> U67(f883_in(T956, T943, T944, T945, T946, T947, T948), T956, T943, T944, T945, T946, T947, T948, T949) U67(f883_out1, T956, T943, T944, T945, T946, T947, T948, T949) -> U68(f884_in(T956, T943, T944, T945, T946, T947, T948, T949), T956, T943, T944, T945, T946, T947, T948, T949) U68(f884_out1(T957), T956, T943, T944, T945, T946, T947, T948, T949) -> f878_out1(T957) f909_in(T1028, T1029, T1030, T1031, T1032, T1033, T1034) -> U69(f913_in(T1028, T1029, T1030, T1031, T1032, T1033, T1034), T1028, T1029, T1030, T1031, T1032, T1033, T1034) U69(f913_out1, T1028, T1029, T1030, T1031, T1032, T1033, T1034) -> U70(f914_in, T1028, T1029, T1030, T1031, T1032, T1033, T1034) U70(f914_out1, T1028, T1029, T1030, T1031, T1032, T1033, T1034) -> f909_out1 f1018_in(T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243) -> U71(f235_in(T1236), T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243) U71(f235_out1(T1248), T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243) -> U72(f1020_in(T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1236), T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1248) U72(f1020_out1(T1249), T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1248) -> f1018_out1(T1248, T1249) f1020_in(T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1236) -> U73(f1021_in(T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235), T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1236) U73(f1021_out1, T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1236) -> U74(f1022_in(T1248, T1243, T1236), T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1236) U74(f1022_out1(T1249), T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1236) -> f1020_out1(T1249) f1037_in(T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) -> U75(f1038_in(T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337), T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) U75(f1038_out1, T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) -> U76(f1039_in, T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) U76(f1039_out1, T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) -> f1037_out1 f1110_in(T1508, T1506, T1507) -> U77(f235_in(T1508), T1508, T1506, T1507) U77(f235_out1(T1515), T1508, T1506, T1507) -> U78(f1116_in(T1515, T1506, T1507, T1508), T1508, T1506, T1507, T1515) U78(f1116_out1(T1516), T1508, T1506, T1507, T1515) -> f1110_out1(T1515, T1516) f1116_in(T1515, T1506, T1507, T1508) -> U79(f1144_in(T1515, T1506, T1507), T1515, T1506, T1507, T1508) U79(f1144_out1, T1515, T1506, T1507, T1508) -> U80(f1022_in(T1515, .(T1506, T1507), T1508), T1515, T1506, T1507, T1508) U80(f1022_out1(T1516), T1515, T1506, T1507, T1508) -> f1116_out1(T1516) f1153_in(T1547, T1548, T1549) -> U81(f1154_in(T1547, T1548, T1549), T1547, T1548, T1549) U81(f1154_out1, T1547, T1548, T1549) -> U82(f1155_in, T1547, T1548, T1549) U82(f1155_out1, T1547, T1548, T1549) -> f1153_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (196) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 146 less nodes. ---------------------------------------- (197) Complex Obligation (AND) ---------------------------------------- (198) Obligation: Q DP problem: The TRS P consists of the following rules: F1164_IN(T1601, .(T1602, T1603)) -> F1164_IN(T1601, T1603) The TRS R consists of the following rules: f3_in(T19) -> f3_out1([]) f3_in(T28) -> U1(f150_in(T28), T28) U1(f150_out1(T29, T30), T28) -> f3_out1(.(T29, T30)) f235_in(.(T49, T50)) -> f235_out1(T49) f235_in(.(T58, T59)) -> U2(f235_in(T59), .(T58, T59)) U2(f235_out1(T60), .(T58, T59)) -> f235_out1(T60) f445_in(T149, T149) -> f445_out1 f445_in(T156, T157) -> U3(f316_in(T156), T156, T157) U3(f316_out1, T156, T157) -> f445_out1 f537_in(T259, T259, T260) -> f537_out1 f537_in(T269, T270, T271) -> U4(f445_in(T269, T271), T269, T270, T271) U4(f445_out1, T269, T270, T271) -> f537_out1 f635_in(T405, T405, T406, T407) -> f635_out1 f635_in(T420, T421, T422, T423) -> U5(f537_in(T420, T422, T423), T420, T421, T422, T423) U5(f537_out1, T420, T421, T422, T423) -> f635_out1 f724_in(T591, T591, T592, T593, T594) -> f724_out1 f724_in(T611, T612, T613, T614, T615) -> U6(f635_in(T611, T613, T614, T615), T611, T612, T613, T614, T615) U6(f635_out1, T611, T612, T613, T614, T615) -> f724_out1 f820_in(T817, T817, T818, T819, T820, T821) -> f820_out1 f820_in(T842, T843, T844, T845, T846, T847) -> U7(f724_in(T842, T844, T845, T846, T847), T842, T843, T844, T845, T846, T847) U7(f724_out1, T842, T843, T844, T845, T846, T847) -> f820_out1 f913_in(T1083, T1083, T1084, T1085, T1086, T1087, T1088) -> f913_out1 f913_in(T1113, T1114, T1115, T1116, T1117, T1118, T1119) -> U8(f820_in(T1113, T1115, T1116, T1117, T1118, T1119), T1113, T1114, T1115, T1116, T1117, T1118, T1119) U8(f820_out1, T1113, T1114, T1115, T1116, T1117, T1118, T1119) -> f913_out1 f1022_in(T1491, T1492, T1493) -> f1022_out1([]) f1022_in(T1506, T1507, T1508) -> U9(f1110_in(T1508, T1506, T1507), T1506, T1507, T1508) U9(f1110_out1(T1509, T1510), T1506, T1507, T1508) -> f1022_out1(.(T1509, T1510)) f1164_in(T1593, .(T1593, T1594)) -> f1164_out1 f1164_in(T1601, .(T1602, T1603)) -> U10(f1164_in(T1601, T1603), T1601, .(T1602, T1603)) U10(f1164_out1, T1601, .(T1602, T1603)) -> f1164_out1 f289_in(T75) -> U11(f299_in(T75), T75) U11(f299_out1, T75) -> f289_out1 f289_in(T84) -> f289_out1 f290_in(T97, T98) -> f290_out1([]) f290_in(T109, T110) -> U12(f408_in(T110, T109), T109, T110) U12(f408_out1(T111, T112), T109, T110) -> f290_out1(.(T111, T112)) f412_in(T139, T140) -> U13(f439_in(T139, T140), T139, T140) U13(f439_out1, T139, T140) -> f412_out1 f412_in(T164, T165) -> f412_out1 f413_in(T184, T185, T186) -> f413_out1([]) f413_in(T199, T200, T201) -> U14(f505_in(T201, T199, T200), T199, T200, T201) U14(f505_out1(T202, T203), T199, T200, T201) -> f413_out1(.(T202, T203)) f517_in(T240, T241, T242) -> U15(f535_in(T240, T241, T242), T240, T241, T242) U15(f535_out1, T240, T241, T242) -> f517_out1 f517_in(T280, T281, T282) -> f517_out1 f518_in(T307, T308, T309, T310) -> f518_out1([]) f518_in(T325, T326, T327, T328) -> U16(f591_in(T328, T325, T326, T327), T325, T326, T327, T328) U16(f591_out1(T329, T330), T325, T326, T327, T328) -> f518_out1(.(T329, T330)) f596_in(T377, T378, T379, T380) -> U17(f630_in(T377, T378, T379, T380), T377, T378, T379, T380) U17(f630_out1, T377, T378, T379, T380) -> f596_out1 f596_in(T436, T437, T438, T439) -> f596_out1 f597_in(T470, T471, T472, T473, T474) -> f597_out1([]) f597_in(T491, T492, T493, T494, T495) -> U18(f681_in(T495, T491, T492, T493, T494), T491, T492, T493, T494, T495) U18(f681_out1(T496, T497), T491, T492, T493, T494, T495) -> f597_out1(.(T496, T497)) f694_in(T554, T555, T556, T557, T558) -> U19(f712_in(T554, T555, T556, T557, T558), T554, T555, T556, T557, T558) U19(f712_out1, T554, T555, T556, T557, T558) -> f694_out1 f694_in(T632, T633, T634, T635, T636) -> f694_out1 f695_in(T673, T674, T675, T676, T677, T678) -> f695_out1([]) f695_in(T697, T698, T699, T700, T701, T702) -> U20(f778_in(T702, T697, T698, T699, T700, T701), T697, T698, T699, T700, T701, T702) U20(f778_out1(T703, T704), T697, T698, T699, T700, T701, T702) -> f695_out1(.(T703, T704)) f785_in(T771, T772, T773, T774, T775, T776) -> U21(f817_in(T771, T772, T773, T774, T775, T776), T771, T772, T773, T774, T775, T776) U21(f817_out1, T771, T772, T773, T774, T775, T776) -> f785_out1 f785_in(T868, T869, T870, T871, T872, T873) -> f785_out1 f786_in(T916, T917, T918, T919, T920, T921, T922) -> f786_out1([]) f786_in(T943, T944, T945, T946, T947, T948, T949) -> U22(f872_in(T949, T943, T944, T945, T946, T947, T948), T943, T944, T945, T946, T947, T948, T949) U22(f872_out1(T950, T951), T943, T944, T945, T946, T947, T948, T949) -> f786_out1(.(T950, T951)) f883_in(T1028, T1029, T1030, T1031, T1032, T1033, T1034) -> U23(f909_in(T1028, T1029, T1030, T1031, T1032, T1033, T1034), T1028, T1029, T1030, T1031, T1032, T1033, T1034) U23(f909_out1, T1028, T1029, T1030, T1031, T1032, T1033, T1034) -> f883_out1 f883_in(T1144, T1145, T1146, T1147, T1148, T1149, T1150) -> f883_out1 f884_in(T1199, T1200, T1201, T1202, T1203, T1204, T1205, T1206) -> f884_out1([]) f884_in(T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1236) -> U24(f1018_in(T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235, .(T1229, .(T1230, .(T1231, .(T1232, .(T1233, .(T1234, .(T1235, [])))))))), T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1236) U24(f1018_out1(T1237, T1238), T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1236) -> f884_out1(.(T1237, T1238)) f1021_in(T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) -> U25(f1037_in(T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337), T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) U25(f1037_out1, T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) -> f1021_out1 f1021_in(T1465, T1466, T1467, T1468, T1469, T1470, T1471, T1472) -> f1021_out1 f1038_in(T1394, T1394, T1395, T1396, T1397, T1398, T1399, T1400) -> f1038_out1 f1038_in(T1429, T1430, T1431, T1432, T1433, T1434, T1435, T1436) -> U26(f913_in(T1429, T1431, T1432, T1433, T1434, T1435, T1436), T1429, T1430, T1431, T1432, T1433, T1434, T1435, T1436) U26(f913_out1, T1429, T1430, T1431, T1432, T1433, T1434, T1435, T1436) -> f1038_out1 f1144_in(T1547, T1548, T1549) -> U27(f1153_in(T1547, T1548, T1549), T1547, T1548, T1549) U27(f1153_out1, T1547, T1548, T1549) -> f1144_out1 f1144_in(T1614, T1615, T1616) -> f1144_out1 f1154_in(T1566, T1566, T1567) -> f1154_out1 f1154_in(T1578, T1579, T1580) -> U28(f1164_in(T1578, T1580), T1578, T1579, T1580) U28(f1164_out1, T1578, T1579, T1580) -> f1154_out1 f150_in(T28) -> U29(f235_in(T28), T28) U29(f235_out1(T35), T28) -> U30(f236_in(T35, T28), T28, T35) U30(f236_out1(T36), T28, T35) -> f150_out1(T35, T36) f236_in(T35, T28) -> U31(f289_in(T35), T35, T28) U31(f289_out1, T35, T28) -> U32(f290_in(T35, T28), T35, T28) U32(f290_out1(T36), T35, T28) -> f236_out1(T36) f299_in(T75) -> U33(f316_in(T75), T75) U33(f316_out1, T75) -> U34(f317_in, T75) U34(f317_out1, T75) -> f299_out1 f408_in(T110, T109) -> U35(f235_in(T110), T110, T109) U35(f235_out1(T117), T110, T109) -> U36(f411_in(T117, T109, T110), T110, T109, T117) U36(f411_out1(T118), T110, T109, T117) -> f408_out1(T117, T118) f411_in(T117, T109, T110) -> U37(f412_in(T117, T109), T117, T109, T110) U37(f412_out1, T117, T109, T110) -> U38(f413_in(T117, T109, T110), T117, T109, T110) U38(f413_out1(T118), T117, T109, T110) -> f411_out1(T118) f439_in(T139, T140) -> U39(f445_in(T139, T140), T139, T140) U39(f445_out1, T139, T140) -> U40(f446_in, T139, T140) U40(f446_out1, T139, T140) -> f439_out1 f505_in(T201, T199, T200) -> U41(f235_in(T201), T201, T199, T200) U41(f235_out1(T208), T201, T199, T200) -> U42(f510_in(T208, T199, T200, T201), T201, T199, T200, T208) U42(f510_out1(T209), T201, T199, T200, T208) -> f505_out1(T208, T209) f510_in(T208, T199, T200, T201) -> U43(f517_in(T208, T199, T200), T208, T199, T200, T201) U43(f517_out1, T208, T199, T200, T201) -> U44(f518_in(T208, T199, T200, T201), T208, T199, T200, T201) U44(f518_out1(T209), T208, T199, T200, T201) -> f510_out1(T209) f535_in(T240, T241, T242) -> U45(f537_in(T240, T241, T242), T240, T241, T242) U45(f537_out1, T240, T241, T242) -> U46(f538_in, T240, T241, T242) U46(f538_out1, T240, T241, T242) -> f535_out1 f591_in(T328, T325, T326, T327) -> U47(f235_in(T328), T328, T325, T326, T327) U47(f235_out1(T335), T328, T325, T326, T327) -> U48(f594_in(T335, T325, T326, T327, T328), T328, T325, T326, T327, T335) U48(f594_out1(T336), T328, T325, T326, T327, T335) -> f591_out1(T335, T336) f594_in(T335, T325, T326, T327, T328) -> U49(f596_in(T335, T325, T326, T327), T335, T325, T326, T327, T328) U49(f596_out1, T335, T325, T326, T327, T328) -> U50(f597_in(T335, T325, T326, T327, T328), T335, T325, T326, T327, T328) U50(f597_out1(T336), T335, T325, T326, T327, T328) -> f594_out1(T336) f630_in(T377, T378, T379, T380) -> U51(f635_in(T377, T378, T379, T380), T377, T378, T379, T380) U51(f635_out1, T377, T378, T379, T380) -> U52(f636_in, T377, T378, T379, T380) U52(f636_out1, T377, T378, T379, T380) -> f630_out1 f681_in(T495, T491, T492, T493, T494) -> U53(f235_in(T495), T495, T491, T492, T493, T494) U53(f235_out1(T502), T495, T491, T492, T493, T494) -> U54(f684_in(T502, T491, T492, T493, T494, T495), T495, T491, T492, T493, T494, T502) U54(f684_out1(T503), T495, T491, T492, T493, T494, T502) -> f681_out1(T502, T503) f684_in(T502, T491, T492, T493, T494, T495) -> U55(f694_in(T502, T491, T492, T493, T494), T502, T491, T492, T493, T494, T495) U55(f694_out1, T502, T491, T492, T493, T494, T495) -> U56(f695_in(T502, T491, T492, T493, T494, T495), T502, T491, T492, T493, T494, T495) U56(f695_out1(T503), T502, T491, T492, T493, T494, T495) -> f684_out1(T503) f712_in(T554, T555, T556, T557, T558) -> U57(f724_in(T554, T555, T556, T557, T558), T554, T555, T556, T557, T558) U57(f724_out1, T554, T555, T556, T557, T558) -> U58(f725_in, T554, T555, T556, T557, T558) U58(f725_out1, T554, T555, T556, T557, T558) -> f712_out1 f778_in(T702, T697, T698, T699, T700, T701) -> U59(f235_in(T702), T702, T697, T698, T699, T700, T701) U59(f235_out1(T709), T702, T697, T698, T699, T700, T701) -> U60(f782_in(T709, T697, T698, T699, T700, T701, T702), T702, T697, T698, T699, T700, T701, T709) U60(f782_out1(T710), T702, T697, T698, T699, T700, T701, T709) -> f778_out1(T709, T710) f782_in(T709, T697, T698, T699, T700, T701, T702) -> U61(f785_in(T709, T697, T698, T699, T700, T701), T709, T697, T698, T699, T700, T701, T702) U61(f785_out1, T709, T697, T698, T699, T700, T701, T702) -> U62(f786_in(T709, T697, T698, T699, T700, T701, T702), T709, T697, T698, T699, T700, T701, T702) U62(f786_out1(T710), T709, T697, T698, T699, T700, T701, T702) -> f782_out1(T710) f817_in(T771, T772, T773, T774, T775, T776) -> U63(f820_in(T771, T772, T773, T774, T775, T776), T771, T772, T773, T774, T775, T776) U63(f820_out1, T771, T772, T773, T774, T775, T776) -> U64(f821_in, T771, T772, T773, T774, T775, T776) U64(f821_out1, T771, T772, T773, T774, T775, T776) -> f817_out1 f872_in(T949, T943, T944, T945, T946, T947, T948) -> U65(f235_in(T949), T949, T943, T944, T945, T946, T947, T948) U65(f235_out1(T956), T949, T943, T944, T945, T946, T947, T948) -> U66(f878_in(T956, T943, T944, T945, T946, T947, T948, T949), T949, T943, T944, T945, T946, T947, T948, T956) U66(f878_out1(T957), T949, T943, T944, T945, T946, T947, T948, T956) -> f872_out1(T956, T957) f878_in(T956, T943, T944, T945, T946, T947, T948, T949) -> U67(f883_in(T956, T943, T944, T945, T946, T947, T948), T956, T943, T944, T945, T946, T947, T948, T949) U67(f883_out1, T956, T943, T944, T945, T946, T947, T948, T949) -> U68(f884_in(T956, T943, T944, T945, T946, T947, T948, T949), T956, T943, T944, T945, T946, T947, T948, T949) U68(f884_out1(T957), T956, T943, T944, T945, T946, T947, T948, T949) -> f878_out1(T957) f909_in(T1028, T1029, T1030, T1031, T1032, T1033, T1034) -> U69(f913_in(T1028, T1029, T1030, T1031, T1032, T1033, T1034), T1028, T1029, T1030, T1031, T1032, T1033, T1034) U69(f913_out1, T1028, T1029, T1030, T1031, T1032, T1033, T1034) -> U70(f914_in, T1028, T1029, T1030, T1031, T1032, T1033, T1034) U70(f914_out1, T1028, T1029, T1030, T1031, T1032, T1033, T1034) -> f909_out1 f1018_in(T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243) -> U71(f235_in(T1236), T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243) U71(f235_out1(T1248), T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243) -> U72(f1020_in(T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1236), T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1248) U72(f1020_out1(T1249), T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1248) -> f1018_out1(T1248, T1249) f1020_in(T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1236) -> U73(f1021_in(T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235), T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1236) U73(f1021_out1, T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1236) -> U74(f1022_in(T1248, T1243, T1236), T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1236) U74(f1022_out1(T1249), T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1236) -> f1020_out1(T1249) f1037_in(T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) -> U75(f1038_in(T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337), T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) U75(f1038_out1, T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) -> U76(f1039_in, T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) U76(f1039_out1, T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) -> f1037_out1 f1110_in(T1508, T1506, T1507) -> U77(f235_in(T1508), T1508, T1506, T1507) U77(f235_out1(T1515), T1508, T1506, T1507) -> U78(f1116_in(T1515, T1506, T1507, T1508), T1508, T1506, T1507, T1515) U78(f1116_out1(T1516), T1508, T1506, T1507, T1515) -> f1110_out1(T1515, T1516) f1116_in(T1515, T1506, T1507, T1508) -> U79(f1144_in(T1515, T1506, T1507), T1515, T1506, T1507, T1508) U79(f1144_out1, T1515, T1506, T1507, T1508) -> U80(f1022_in(T1515, .(T1506, T1507), T1508), T1515, T1506, T1507, T1508) U80(f1022_out1(T1516), T1515, T1506, T1507, T1508) -> f1116_out1(T1516) f1153_in(T1547, T1548, T1549) -> U81(f1154_in(T1547, T1548, T1549), T1547, T1548, T1549) U81(f1154_out1, T1547, T1548, T1549) -> U82(f1155_in, T1547, T1548, T1549) U82(f1155_out1, T1547, T1548, T1549) -> f1153_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (199) 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. ---------------------------------------- (200) Obligation: Q DP problem: The TRS P consists of the following rules: F1164_IN(T1601, .(T1602, T1603)) -> F1164_IN(T1601, T1603) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (201) 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: *F1164_IN(T1601, .(T1602, T1603)) -> F1164_IN(T1601, T1603) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (202) YES ---------------------------------------- (203) Obligation: Q DP problem: The TRS P consists of the following rules: F235_IN(.(T58, T59)) -> F235_IN(T59) The TRS R consists of the following rules: f3_in(T19) -> f3_out1([]) f3_in(T28) -> U1(f150_in(T28), T28) U1(f150_out1(T29, T30), T28) -> f3_out1(.(T29, T30)) f235_in(.(T49, T50)) -> f235_out1(T49) f235_in(.(T58, T59)) -> U2(f235_in(T59), .(T58, T59)) U2(f235_out1(T60), .(T58, T59)) -> f235_out1(T60) f445_in(T149, T149) -> f445_out1 f445_in(T156, T157) -> U3(f316_in(T156), T156, T157) U3(f316_out1, T156, T157) -> f445_out1 f537_in(T259, T259, T260) -> f537_out1 f537_in(T269, T270, T271) -> U4(f445_in(T269, T271), T269, T270, T271) U4(f445_out1, T269, T270, T271) -> f537_out1 f635_in(T405, T405, T406, T407) -> f635_out1 f635_in(T420, T421, T422, T423) -> U5(f537_in(T420, T422, T423), T420, T421, T422, T423) U5(f537_out1, T420, T421, T422, T423) -> f635_out1 f724_in(T591, T591, T592, T593, T594) -> f724_out1 f724_in(T611, T612, T613, T614, T615) -> U6(f635_in(T611, T613, T614, T615), T611, T612, T613, T614, T615) U6(f635_out1, T611, T612, T613, T614, T615) -> f724_out1 f820_in(T817, T817, T818, T819, T820, T821) -> f820_out1 f820_in(T842, T843, T844, T845, T846, T847) -> U7(f724_in(T842, T844, T845, T846, T847), T842, T843, T844, T845, T846, T847) U7(f724_out1, T842, T843, T844, T845, T846, T847) -> f820_out1 f913_in(T1083, T1083, T1084, T1085, T1086, T1087, T1088) -> f913_out1 f913_in(T1113, T1114, T1115, T1116, T1117, T1118, T1119) -> U8(f820_in(T1113, T1115, T1116, T1117, T1118, T1119), T1113, T1114, T1115, T1116, T1117, T1118, T1119) U8(f820_out1, T1113, T1114, T1115, T1116, T1117, T1118, T1119) -> f913_out1 f1022_in(T1491, T1492, T1493) -> f1022_out1([]) f1022_in(T1506, T1507, T1508) -> U9(f1110_in(T1508, T1506, T1507), T1506, T1507, T1508) U9(f1110_out1(T1509, T1510), T1506, T1507, T1508) -> f1022_out1(.(T1509, T1510)) f1164_in(T1593, .(T1593, T1594)) -> f1164_out1 f1164_in(T1601, .(T1602, T1603)) -> U10(f1164_in(T1601, T1603), T1601, .(T1602, T1603)) U10(f1164_out1, T1601, .(T1602, T1603)) -> f1164_out1 f289_in(T75) -> U11(f299_in(T75), T75) U11(f299_out1, T75) -> f289_out1 f289_in(T84) -> f289_out1 f290_in(T97, T98) -> f290_out1([]) f290_in(T109, T110) -> U12(f408_in(T110, T109), T109, T110) U12(f408_out1(T111, T112), T109, T110) -> f290_out1(.(T111, T112)) f412_in(T139, T140) -> U13(f439_in(T139, T140), T139, T140) U13(f439_out1, T139, T140) -> f412_out1 f412_in(T164, T165) -> f412_out1 f413_in(T184, T185, T186) -> f413_out1([]) f413_in(T199, T200, T201) -> U14(f505_in(T201, T199, T200), T199, T200, T201) U14(f505_out1(T202, T203), T199, T200, T201) -> f413_out1(.(T202, T203)) f517_in(T240, T241, T242) -> U15(f535_in(T240, T241, T242), T240, T241, T242) U15(f535_out1, T240, T241, T242) -> f517_out1 f517_in(T280, T281, T282) -> f517_out1 f518_in(T307, T308, T309, T310) -> f518_out1([]) f518_in(T325, T326, T327, T328) -> U16(f591_in(T328, T325, T326, T327), T325, T326, T327, T328) U16(f591_out1(T329, T330), T325, T326, T327, T328) -> f518_out1(.(T329, T330)) f596_in(T377, T378, T379, T380) -> U17(f630_in(T377, T378, T379, T380), T377, T378, T379, T380) U17(f630_out1, T377, T378, T379, T380) -> f596_out1 f596_in(T436, T437, T438, T439) -> f596_out1 f597_in(T470, T471, T472, T473, T474) -> f597_out1([]) f597_in(T491, T492, T493, T494, T495) -> U18(f681_in(T495, T491, T492, T493, T494), T491, T492, T493, T494, T495) U18(f681_out1(T496, T497), T491, T492, T493, T494, T495) -> f597_out1(.(T496, T497)) f694_in(T554, T555, T556, T557, T558) -> U19(f712_in(T554, T555, T556, T557, T558), T554, T555, T556, T557, T558) U19(f712_out1, T554, T555, T556, T557, T558) -> f694_out1 f694_in(T632, T633, T634, T635, T636) -> f694_out1 f695_in(T673, T674, T675, T676, T677, T678) -> f695_out1([]) f695_in(T697, T698, T699, T700, T701, T702) -> U20(f778_in(T702, T697, T698, T699, T700, T701), T697, T698, T699, T700, T701, T702) U20(f778_out1(T703, T704), T697, T698, T699, T700, T701, T702) -> f695_out1(.(T703, T704)) f785_in(T771, T772, T773, T774, T775, T776) -> U21(f817_in(T771, T772, T773, T774, T775, T776), T771, T772, T773, T774, T775, T776) U21(f817_out1, T771, T772, T773, T774, T775, T776) -> f785_out1 f785_in(T868, T869, T870, T871, T872, T873) -> f785_out1 f786_in(T916, T917, T918, T919, T920, T921, T922) -> f786_out1([]) f786_in(T943, T944, T945, T946, T947, T948, T949) -> U22(f872_in(T949, T943, T944, T945, T946, T947, T948), T943, T944, T945, T946, T947, T948, T949) U22(f872_out1(T950, T951), T943, T944, T945, T946, T947, T948, T949) -> f786_out1(.(T950, T951)) f883_in(T1028, T1029, T1030, T1031, T1032, T1033, T1034) -> U23(f909_in(T1028, T1029, T1030, T1031, T1032, T1033, T1034), T1028, T1029, T1030, T1031, T1032, T1033, T1034) U23(f909_out1, T1028, T1029, T1030, T1031, T1032, T1033, T1034) -> f883_out1 f883_in(T1144, T1145, T1146, T1147, T1148, T1149, T1150) -> f883_out1 f884_in(T1199, T1200, T1201, T1202, T1203, T1204, T1205, T1206) -> f884_out1([]) f884_in(T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1236) -> U24(f1018_in(T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235, .(T1229, .(T1230, .(T1231, .(T1232, .(T1233, .(T1234, .(T1235, [])))))))), T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1236) U24(f1018_out1(T1237, T1238), T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1236) -> f884_out1(.(T1237, T1238)) f1021_in(T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) -> U25(f1037_in(T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337), T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) U25(f1037_out1, T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) -> f1021_out1 f1021_in(T1465, T1466, T1467, T1468, T1469, T1470, T1471, T1472) -> f1021_out1 f1038_in(T1394, T1394, T1395, T1396, T1397, T1398, T1399, T1400) -> f1038_out1 f1038_in(T1429, T1430, T1431, T1432, T1433, T1434, T1435, T1436) -> U26(f913_in(T1429, T1431, T1432, T1433, T1434, T1435, T1436), T1429, T1430, T1431, T1432, T1433, T1434, T1435, T1436) U26(f913_out1, T1429, T1430, T1431, T1432, T1433, T1434, T1435, T1436) -> f1038_out1 f1144_in(T1547, T1548, T1549) -> U27(f1153_in(T1547, T1548, T1549), T1547, T1548, T1549) U27(f1153_out1, T1547, T1548, T1549) -> f1144_out1 f1144_in(T1614, T1615, T1616) -> f1144_out1 f1154_in(T1566, T1566, T1567) -> f1154_out1 f1154_in(T1578, T1579, T1580) -> U28(f1164_in(T1578, T1580), T1578, T1579, T1580) U28(f1164_out1, T1578, T1579, T1580) -> f1154_out1 f150_in(T28) -> U29(f235_in(T28), T28) U29(f235_out1(T35), T28) -> U30(f236_in(T35, T28), T28, T35) U30(f236_out1(T36), T28, T35) -> f150_out1(T35, T36) f236_in(T35, T28) -> U31(f289_in(T35), T35, T28) U31(f289_out1, T35, T28) -> U32(f290_in(T35, T28), T35, T28) U32(f290_out1(T36), T35, T28) -> f236_out1(T36) f299_in(T75) -> U33(f316_in(T75), T75) U33(f316_out1, T75) -> U34(f317_in, T75) U34(f317_out1, T75) -> f299_out1 f408_in(T110, T109) -> U35(f235_in(T110), T110, T109) U35(f235_out1(T117), T110, T109) -> U36(f411_in(T117, T109, T110), T110, T109, T117) U36(f411_out1(T118), T110, T109, T117) -> f408_out1(T117, T118) f411_in(T117, T109, T110) -> U37(f412_in(T117, T109), T117, T109, T110) U37(f412_out1, T117, T109, T110) -> U38(f413_in(T117, T109, T110), T117, T109, T110) U38(f413_out1(T118), T117, T109, T110) -> f411_out1(T118) f439_in(T139, T140) -> U39(f445_in(T139, T140), T139, T140) U39(f445_out1, T139, T140) -> U40(f446_in, T139, T140) U40(f446_out1, T139, T140) -> f439_out1 f505_in(T201, T199, T200) -> U41(f235_in(T201), T201, T199, T200) U41(f235_out1(T208), T201, T199, T200) -> U42(f510_in(T208, T199, T200, T201), T201, T199, T200, T208) U42(f510_out1(T209), T201, T199, T200, T208) -> f505_out1(T208, T209) f510_in(T208, T199, T200, T201) -> U43(f517_in(T208, T199, T200), T208, T199, T200, T201) U43(f517_out1, T208, T199, T200, T201) -> U44(f518_in(T208, T199, T200, T201), T208, T199, T200, T201) U44(f518_out1(T209), T208, T199, T200, T201) -> f510_out1(T209) f535_in(T240, T241, T242) -> U45(f537_in(T240, T241, T242), T240, T241, T242) U45(f537_out1, T240, T241, T242) -> U46(f538_in, T240, T241, T242) U46(f538_out1, T240, T241, T242) -> f535_out1 f591_in(T328, T325, T326, T327) -> U47(f235_in(T328), T328, T325, T326, T327) U47(f235_out1(T335), T328, T325, T326, T327) -> U48(f594_in(T335, T325, T326, T327, T328), T328, T325, T326, T327, T335) U48(f594_out1(T336), T328, T325, T326, T327, T335) -> f591_out1(T335, T336) f594_in(T335, T325, T326, T327, T328) -> U49(f596_in(T335, T325, T326, T327), T335, T325, T326, T327, T328) U49(f596_out1, T335, T325, T326, T327, T328) -> U50(f597_in(T335, T325, T326, T327, T328), T335, T325, T326, T327, T328) U50(f597_out1(T336), T335, T325, T326, T327, T328) -> f594_out1(T336) f630_in(T377, T378, T379, T380) -> U51(f635_in(T377, T378, T379, T380), T377, T378, T379, T380) U51(f635_out1, T377, T378, T379, T380) -> U52(f636_in, T377, T378, T379, T380) U52(f636_out1, T377, T378, T379, T380) -> f630_out1 f681_in(T495, T491, T492, T493, T494) -> U53(f235_in(T495), T495, T491, T492, T493, T494) U53(f235_out1(T502), T495, T491, T492, T493, T494) -> U54(f684_in(T502, T491, T492, T493, T494, T495), T495, T491, T492, T493, T494, T502) U54(f684_out1(T503), T495, T491, T492, T493, T494, T502) -> f681_out1(T502, T503) f684_in(T502, T491, T492, T493, T494, T495) -> U55(f694_in(T502, T491, T492, T493, T494), T502, T491, T492, T493, T494, T495) U55(f694_out1, T502, T491, T492, T493, T494, T495) -> U56(f695_in(T502, T491, T492, T493, T494, T495), T502, T491, T492, T493, T494, T495) U56(f695_out1(T503), T502, T491, T492, T493, T494, T495) -> f684_out1(T503) f712_in(T554, T555, T556, T557, T558) -> U57(f724_in(T554, T555, T556, T557, T558), T554, T555, T556, T557, T558) U57(f724_out1, T554, T555, T556, T557, T558) -> U58(f725_in, T554, T555, T556, T557, T558) U58(f725_out1, T554, T555, T556, T557, T558) -> f712_out1 f778_in(T702, T697, T698, T699, T700, T701) -> U59(f235_in(T702), T702, T697, T698, T699, T700, T701) U59(f235_out1(T709), T702, T697, T698, T699, T700, T701) -> U60(f782_in(T709, T697, T698, T699, T700, T701, T702), T702, T697, T698, T699, T700, T701, T709) U60(f782_out1(T710), T702, T697, T698, T699, T700, T701, T709) -> f778_out1(T709, T710) f782_in(T709, T697, T698, T699, T700, T701, T702) -> U61(f785_in(T709, T697, T698, T699, T700, T701), T709, T697, T698, T699, T700, T701, T702) U61(f785_out1, T709, T697, T698, T699, T700, T701, T702) -> U62(f786_in(T709, T697, T698, T699, T700, T701, T702), T709, T697, T698, T699, T700, T701, T702) U62(f786_out1(T710), T709, T697, T698, T699, T700, T701, T702) -> f782_out1(T710) f817_in(T771, T772, T773, T774, T775, T776) -> U63(f820_in(T771, T772, T773, T774, T775, T776), T771, T772, T773, T774, T775, T776) U63(f820_out1, T771, T772, T773, T774, T775, T776) -> U64(f821_in, T771, T772, T773, T774, T775, T776) U64(f821_out1, T771, T772, T773, T774, T775, T776) -> f817_out1 f872_in(T949, T943, T944, T945, T946, T947, T948) -> U65(f235_in(T949), T949, T943, T944, T945, T946, T947, T948) U65(f235_out1(T956), T949, T943, T944, T945, T946, T947, T948) -> U66(f878_in(T956, T943, T944, T945, T946, T947, T948, T949), T949, T943, T944, T945, T946, T947, T948, T956) U66(f878_out1(T957), T949, T943, T944, T945, T946, T947, T948, T956) -> f872_out1(T956, T957) f878_in(T956, T943, T944, T945, T946, T947, T948, T949) -> U67(f883_in(T956, T943, T944, T945, T946, T947, T948), T956, T943, T944, T945, T946, T947, T948, T949) U67(f883_out1, T956, T943, T944, T945, T946, T947, T948, T949) -> U68(f884_in(T956, T943, T944, T945, T946, T947, T948, T949), T956, T943, T944, T945, T946, T947, T948, T949) U68(f884_out1(T957), T956, T943, T944, T945, T946, T947, T948, T949) -> f878_out1(T957) f909_in(T1028, T1029, T1030, T1031, T1032, T1033, T1034) -> U69(f913_in(T1028, T1029, T1030, T1031, T1032, T1033, T1034), T1028, T1029, T1030, T1031, T1032, T1033, T1034) U69(f913_out1, T1028, T1029, T1030, T1031, T1032, T1033, T1034) -> U70(f914_in, T1028, T1029, T1030, T1031, T1032, T1033, T1034) U70(f914_out1, T1028, T1029, T1030, T1031, T1032, T1033, T1034) -> f909_out1 f1018_in(T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243) -> U71(f235_in(T1236), T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243) U71(f235_out1(T1248), T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243) -> U72(f1020_in(T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1236), T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1248) U72(f1020_out1(T1249), T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1248) -> f1018_out1(T1248, T1249) f1020_in(T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1236) -> U73(f1021_in(T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235), T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1236) U73(f1021_out1, T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1236) -> U74(f1022_in(T1248, T1243, T1236), T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1236) U74(f1022_out1(T1249), T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1236) -> f1020_out1(T1249) f1037_in(T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) -> U75(f1038_in(T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337), T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) U75(f1038_out1, T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) -> U76(f1039_in, T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) U76(f1039_out1, T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) -> f1037_out1 f1110_in(T1508, T1506, T1507) -> U77(f235_in(T1508), T1508, T1506, T1507) U77(f235_out1(T1515), T1508, T1506, T1507) -> U78(f1116_in(T1515, T1506, T1507, T1508), T1508, T1506, T1507, T1515) U78(f1116_out1(T1516), T1508, T1506, T1507, T1515) -> f1110_out1(T1515, T1516) f1116_in(T1515, T1506, T1507, T1508) -> U79(f1144_in(T1515, T1506, T1507), T1515, T1506, T1507, T1508) U79(f1144_out1, T1515, T1506, T1507, T1508) -> U80(f1022_in(T1515, .(T1506, T1507), T1508), T1515, T1506, T1507, T1508) U80(f1022_out1(T1516), T1515, T1506, T1507, T1508) -> f1116_out1(T1516) f1153_in(T1547, T1548, T1549) -> U81(f1154_in(T1547, T1548, T1549), T1547, T1548, T1549) U81(f1154_out1, T1547, T1548, T1549) -> U82(f1155_in, T1547, T1548, T1549) U82(f1155_out1, T1547, T1548, T1549) -> f1153_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (204) 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. ---------------------------------------- (205) Obligation: Q DP problem: The TRS P consists of the following rules: F235_IN(.(T58, T59)) -> F235_IN(T59) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (206) 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: *F235_IN(.(T58, T59)) -> F235_IN(T59) The graph contains the following edges 1 > 1 ---------------------------------------- (207) YES ---------------------------------------- (208) Obligation: Q DP problem: The TRS P consists of the following rules: F1110_IN(T1508, T1506, T1507) -> U77^1(f235_in(T1508), T1508, T1506, T1507) U77^1(f235_out1(T1515), T1508, T1506, T1507) -> F1116_IN(T1515, T1506, T1507, T1508) F1116_IN(T1515, T1506, T1507, T1508) -> U79^1(f1144_in(T1515, T1506, T1507), T1515, T1506, T1507, T1508) U79^1(f1144_out1, T1515, T1506, T1507, T1508) -> F1022_IN(T1515, .(T1506, T1507), T1508) F1022_IN(T1506, T1507, T1508) -> F1110_IN(T1508, T1506, T1507) The TRS R consists of the following rules: f3_in(T19) -> f3_out1([]) f3_in(T28) -> U1(f150_in(T28), T28) U1(f150_out1(T29, T30), T28) -> f3_out1(.(T29, T30)) f235_in(.(T49, T50)) -> f235_out1(T49) f235_in(.(T58, T59)) -> U2(f235_in(T59), .(T58, T59)) U2(f235_out1(T60), .(T58, T59)) -> f235_out1(T60) f445_in(T149, T149) -> f445_out1 f445_in(T156, T157) -> U3(f316_in(T156), T156, T157) U3(f316_out1, T156, T157) -> f445_out1 f537_in(T259, T259, T260) -> f537_out1 f537_in(T269, T270, T271) -> U4(f445_in(T269, T271), T269, T270, T271) U4(f445_out1, T269, T270, T271) -> f537_out1 f635_in(T405, T405, T406, T407) -> f635_out1 f635_in(T420, T421, T422, T423) -> U5(f537_in(T420, T422, T423), T420, T421, T422, T423) U5(f537_out1, T420, T421, T422, T423) -> f635_out1 f724_in(T591, T591, T592, T593, T594) -> f724_out1 f724_in(T611, T612, T613, T614, T615) -> U6(f635_in(T611, T613, T614, T615), T611, T612, T613, T614, T615) U6(f635_out1, T611, T612, T613, T614, T615) -> f724_out1 f820_in(T817, T817, T818, T819, T820, T821) -> f820_out1 f820_in(T842, T843, T844, T845, T846, T847) -> U7(f724_in(T842, T844, T845, T846, T847), T842, T843, T844, T845, T846, T847) U7(f724_out1, T842, T843, T844, T845, T846, T847) -> f820_out1 f913_in(T1083, T1083, T1084, T1085, T1086, T1087, T1088) -> f913_out1 f913_in(T1113, T1114, T1115, T1116, T1117, T1118, T1119) -> U8(f820_in(T1113, T1115, T1116, T1117, T1118, T1119), T1113, T1114, T1115, T1116, T1117, T1118, T1119) U8(f820_out1, T1113, T1114, T1115, T1116, T1117, T1118, T1119) -> f913_out1 f1022_in(T1491, T1492, T1493) -> f1022_out1([]) f1022_in(T1506, T1507, T1508) -> U9(f1110_in(T1508, T1506, T1507), T1506, T1507, T1508) U9(f1110_out1(T1509, T1510), T1506, T1507, T1508) -> f1022_out1(.(T1509, T1510)) f1164_in(T1593, .(T1593, T1594)) -> f1164_out1 f1164_in(T1601, .(T1602, T1603)) -> U10(f1164_in(T1601, T1603), T1601, .(T1602, T1603)) U10(f1164_out1, T1601, .(T1602, T1603)) -> f1164_out1 f289_in(T75) -> U11(f299_in(T75), T75) U11(f299_out1, T75) -> f289_out1 f289_in(T84) -> f289_out1 f290_in(T97, T98) -> f290_out1([]) f290_in(T109, T110) -> U12(f408_in(T110, T109), T109, T110) U12(f408_out1(T111, T112), T109, T110) -> f290_out1(.(T111, T112)) f412_in(T139, T140) -> U13(f439_in(T139, T140), T139, T140) U13(f439_out1, T139, T140) -> f412_out1 f412_in(T164, T165) -> f412_out1 f413_in(T184, T185, T186) -> f413_out1([]) f413_in(T199, T200, T201) -> U14(f505_in(T201, T199, T200), T199, T200, T201) U14(f505_out1(T202, T203), T199, T200, T201) -> f413_out1(.(T202, T203)) f517_in(T240, T241, T242) -> U15(f535_in(T240, T241, T242), T240, T241, T242) U15(f535_out1, T240, T241, T242) -> f517_out1 f517_in(T280, T281, T282) -> f517_out1 f518_in(T307, T308, T309, T310) -> f518_out1([]) f518_in(T325, T326, T327, T328) -> U16(f591_in(T328, T325, T326, T327), T325, T326, T327, T328) U16(f591_out1(T329, T330), T325, T326, T327, T328) -> f518_out1(.(T329, T330)) f596_in(T377, T378, T379, T380) -> U17(f630_in(T377, T378, T379, T380), T377, T378, T379, T380) U17(f630_out1, T377, T378, T379, T380) -> f596_out1 f596_in(T436, T437, T438, T439) -> f596_out1 f597_in(T470, T471, T472, T473, T474) -> f597_out1([]) f597_in(T491, T492, T493, T494, T495) -> U18(f681_in(T495, T491, T492, T493, T494), T491, T492, T493, T494, T495) U18(f681_out1(T496, T497), T491, T492, T493, T494, T495) -> f597_out1(.(T496, T497)) f694_in(T554, T555, T556, T557, T558) -> U19(f712_in(T554, T555, T556, T557, T558), T554, T555, T556, T557, T558) U19(f712_out1, T554, T555, T556, T557, T558) -> f694_out1 f694_in(T632, T633, T634, T635, T636) -> f694_out1 f695_in(T673, T674, T675, T676, T677, T678) -> f695_out1([]) f695_in(T697, T698, T699, T700, T701, T702) -> U20(f778_in(T702, T697, T698, T699, T700, T701), T697, T698, T699, T700, T701, T702) U20(f778_out1(T703, T704), T697, T698, T699, T700, T701, T702) -> f695_out1(.(T703, T704)) f785_in(T771, T772, T773, T774, T775, T776) -> U21(f817_in(T771, T772, T773, T774, T775, T776), T771, T772, T773, T774, T775, T776) U21(f817_out1, T771, T772, T773, T774, T775, T776) -> f785_out1 f785_in(T868, T869, T870, T871, T872, T873) -> f785_out1 f786_in(T916, T917, T918, T919, T920, T921, T922) -> f786_out1([]) f786_in(T943, T944, T945, T946, T947, T948, T949) -> U22(f872_in(T949, T943, T944, T945, T946, T947, T948), T943, T944, T945, T946, T947, T948, T949) U22(f872_out1(T950, T951), T943, T944, T945, T946, T947, T948, T949) -> f786_out1(.(T950, T951)) f883_in(T1028, T1029, T1030, T1031, T1032, T1033, T1034) -> U23(f909_in(T1028, T1029, T1030, T1031, T1032, T1033, T1034), T1028, T1029, T1030, T1031, T1032, T1033, T1034) U23(f909_out1, T1028, T1029, T1030, T1031, T1032, T1033, T1034) -> f883_out1 f883_in(T1144, T1145, T1146, T1147, T1148, T1149, T1150) -> f883_out1 f884_in(T1199, T1200, T1201, T1202, T1203, T1204, T1205, T1206) -> f884_out1([]) f884_in(T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1236) -> U24(f1018_in(T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235, .(T1229, .(T1230, .(T1231, .(T1232, .(T1233, .(T1234, .(T1235, [])))))))), T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1236) U24(f1018_out1(T1237, T1238), T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1236) -> f884_out1(.(T1237, T1238)) f1021_in(T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) -> U25(f1037_in(T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337), T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) U25(f1037_out1, T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) -> f1021_out1 f1021_in(T1465, T1466, T1467, T1468, T1469, T1470, T1471, T1472) -> f1021_out1 f1038_in(T1394, T1394, T1395, T1396, T1397, T1398, T1399, T1400) -> f1038_out1 f1038_in(T1429, T1430, T1431, T1432, T1433, T1434, T1435, T1436) -> U26(f913_in(T1429, T1431, T1432, T1433, T1434, T1435, T1436), T1429, T1430, T1431, T1432, T1433, T1434, T1435, T1436) U26(f913_out1, T1429, T1430, T1431, T1432, T1433, T1434, T1435, T1436) -> f1038_out1 f1144_in(T1547, T1548, T1549) -> U27(f1153_in(T1547, T1548, T1549), T1547, T1548, T1549) U27(f1153_out1, T1547, T1548, T1549) -> f1144_out1 f1144_in(T1614, T1615, T1616) -> f1144_out1 f1154_in(T1566, T1566, T1567) -> f1154_out1 f1154_in(T1578, T1579, T1580) -> U28(f1164_in(T1578, T1580), T1578, T1579, T1580) U28(f1164_out1, T1578, T1579, T1580) -> f1154_out1 f150_in(T28) -> U29(f235_in(T28), T28) U29(f235_out1(T35), T28) -> U30(f236_in(T35, T28), T28, T35) U30(f236_out1(T36), T28, T35) -> f150_out1(T35, T36) f236_in(T35, T28) -> U31(f289_in(T35), T35, T28) U31(f289_out1, T35, T28) -> U32(f290_in(T35, T28), T35, T28) U32(f290_out1(T36), T35, T28) -> f236_out1(T36) f299_in(T75) -> U33(f316_in(T75), T75) U33(f316_out1, T75) -> U34(f317_in, T75) U34(f317_out1, T75) -> f299_out1 f408_in(T110, T109) -> U35(f235_in(T110), T110, T109) U35(f235_out1(T117), T110, T109) -> U36(f411_in(T117, T109, T110), T110, T109, T117) U36(f411_out1(T118), T110, T109, T117) -> f408_out1(T117, T118) f411_in(T117, T109, T110) -> U37(f412_in(T117, T109), T117, T109, T110) U37(f412_out1, T117, T109, T110) -> U38(f413_in(T117, T109, T110), T117, T109, T110) U38(f413_out1(T118), T117, T109, T110) -> f411_out1(T118) f439_in(T139, T140) -> U39(f445_in(T139, T140), T139, T140) U39(f445_out1, T139, T140) -> U40(f446_in, T139, T140) U40(f446_out1, T139, T140) -> f439_out1 f505_in(T201, T199, T200) -> U41(f235_in(T201), T201, T199, T200) U41(f235_out1(T208), T201, T199, T200) -> U42(f510_in(T208, T199, T200, T201), T201, T199, T200, T208) U42(f510_out1(T209), T201, T199, T200, T208) -> f505_out1(T208, T209) f510_in(T208, T199, T200, T201) -> U43(f517_in(T208, T199, T200), T208, T199, T200, T201) U43(f517_out1, T208, T199, T200, T201) -> U44(f518_in(T208, T199, T200, T201), T208, T199, T200, T201) U44(f518_out1(T209), T208, T199, T200, T201) -> f510_out1(T209) f535_in(T240, T241, T242) -> U45(f537_in(T240, T241, T242), T240, T241, T242) U45(f537_out1, T240, T241, T242) -> U46(f538_in, T240, T241, T242) U46(f538_out1, T240, T241, T242) -> f535_out1 f591_in(T328, T325, T326, T327) -> U47(f235_in(T328), T328, T325, T326, T327) U47(f235_out1(T335), T328, T325, T326, T327) -> U48(f594_in(T335, T325, T326, T327, T328), T328, T325, T326, T327, T335) U48(f594_out1(T336), T328, T325, T326, T327, T335) -> f591_out1(T335, T336) f594_in(T335, T325, T326, T327, T328) -> U49(f596_in(T335, T325, T326, T327), T335, T325, T326, T327, T328) U49(f596_out1, T335, T325, T326, T327, T328) -> U50(f597_in(T335, T325, T326, T327, T328), T335, T325, T326, T327, T328) U50(f597_out1(T336), T335, T325, T326, T327, T328) -> f594_out1(T336) f630_in(T377, T378, T379, T380) -> U51(f635_in(T377, T378, T379, T380), T377, T378, T379, T380) U51(f635_out1, T377, T378, T379, T380) -> U52(f636_in, T377, T378, T379, T380) U52(f636_out1, T377, T378, T379, T380) -> f630_out1 f681_in(T495, T491, T492, T493, T494) -> U53(f235_in(T495), T495, T491, T492, T493, T494) U53(f235_out1(T502), T495, T491, T492, T493, T494) -> U54(f684_in(T502, T491, T492, T493, T494, T495), T495, T491, T492, T493, T494, T502) U54(f684_out1(T503), T495, T491, T492, T493, T494, T502) -> f681_out1(T502, T503) f684_in(T502, T491, T492, T493, T494, T495) -> U55(f694_in(T502, T491, T492, T493, T494), T502, T491, T492, T493, T494, T495) U55(f694_out1, T502, T491, T492, T493, T494, T495) -> U56(f695_in(T502, T491, T492, T493, T494, T495), T502, T491, T492, T493, T494, T495) U56(f695_out1(T503), T502, T491, T492, T493, T494, T495) -> f684_out1(T503) f712_in(T554, T555, T556, T557, T558) -> U57(f724_in(T554, T555, T556, T557, T558), T554, T555, T556, T557, T558) U57(f724_out1, T554, T555, T556, T557, T558) -> U58(f725_in, T554, T555, T556, T557, T558) U58(f725_out1, T554, T555, T556, T557, T558) -> f712_out1 f778_in(T702, T697, T698, T699, T700, T701) -> U59(f235_in(T702), T702, T697, T698, T699, T700, T701) U59(f235_out1(T709), T702, T697, T698, T699, T700, T701) -> U60(f782_in(T709, T697, T698, T699, T700, T701, T702), T702, T697, T698, T699, T700, T701, T709) U60(f782_out1(T710), T702, T697, T698, T699, T700, T701, T709) -> f778_out1(T709, T710) f782_in(T709, T697, T698, T699, T700, T701, T702) -> U61(f785_in(T709, T697, T698, T699, T700, T701), T709, T697, T698, T699, T700, T701, T702) U61(f785_out1, T709, T697, T698, T699, T700, T701, T702) -> U62(f786_in(T709, T697, T698, T699, T700, T701, T702), T709, T697, T698, T699, T700, T701, T702) U62(f786_out1(T710), T709, T697, T698, T699, T700, T701, T702) -> f782_out1(T710) f817_in(T771, T772, T773, T774, T775, T776) -> U63(f820_in(T771, T772, T773, T774, T775, T776), T771, T772, T773, T774, T775, T776) U63(f820_out1, T771, T772, T773, T774, T775, T776) -> U64(f821_in, T771, T772, T773, T774, T775, T776) U64(f821_out1, T771, T772, T773, T774, T775, T776) -> f817_out1 f872_in(T949, T943, T944, T945, T946, T947, T948) -> U65(f235_in(T949), T949, T943, T944, T945, T946, T947, T948) U65(f235_out1(T956), T949, T943, T944, T945, T946, T947, T948) -> U66(f878_in(T956, T943, T944, T945, T946, T947, T948, T949), T949, T943, T944, T945, T946, T947, T948, T956) U66(f878_out1(T957), T949, T943, T944, T945, T946, T947, T948, T956) -> f872_out1(T956, T957) f878_in(T956, T943, T944, T945, T946, T947, T948, T949) -> U67(f883_in(T956, T943, T944, T945, T946, T947, T948), T956, T943, T944, T945, T946, T947, T948, T949) U67(f883_out1, T956, T943, T944, T945, T946, T947, T948, T949) -> U68(f884_in(T956, T943, T944, T945, T946, T947, T948, T949), T956, T943, T944, T945, T946, T947, T948, T949) U68(f884_out1(T957), T956, T943, T944, T945, T946, T947, T948, T949) -> f878_out1(T957) f909_in(T1028, T1029, T1030, T1031, T1032, T1033, T1034) -> U69(f913_in(T1028, T1029, T1030, T1031, T1032, T1033, T1034), T1028, T1029, T1030, T1031, T1032, T1033, T1034) U69(f913_out1, T1028, T1029, T1030, T1031, T1032, T1033, T1034) -> U70(f914_in, T1028, T1029, T1030, T1031, T1032, T1033, T1034) U70(f914_out1, T1028, T1029, T1030, T1031, T1032, T1033, T1034) -> f909_out1 f1018_in(T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243) -> U71(f235_in(T1236), T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243) U71(f235_out1(T1248), T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243) -> U72(f1020_in(T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1236), T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1248) U72(f1020_out1(T1249), T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1248) -> f1018_out1(T1248, T1249) f1020_in(T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1236) -> U73(f1021_in(T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235), T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1236) U73(f1021_out1, T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1236) -> U74(f1022_in(T1248, T1243, T1236), T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1236) U74(f1022_out1(T1249), T1248, T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1243, T1236) -> f1020_out1(T1249) f1037_in(T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) -> U75(f1038_in(T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337), T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) U75(f1038_out1, T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) -> U76(f1039_in, T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) U76(f1039_out1, T1330, T1331, T1332, T1333, T1334, T1335, T1336, T1337) -> f1037_out1 f1110_in(T1508, T1506, T1507) -> U77(f235_in(T1508), T1508, T1506, T1507) U77(f235_out1(T1515), T1508, T1506, T1507) -> U78(f1116_in(T1515, T1506, T1507, T1508), T1508, T1506, T1507, T1515) U78(f1116_out1(T1516), T1508, T1506, T1507, T1515) -> f1110_out1(T1515, T1516) f1116_in(T1515, T1506, T1507, T1508) -> U79(f1144_in(T1515, T1506, T1507), T1515, T1506, T1507, T1508) U79(f1144_out1, T1515, T1506, T1507, T1508) -> U80(f1022_in(T1515, .(T1506, T1507), T1508), T1515, T1506, T1507, T1508) U80(f1022_out1(T1516), T1515, T1506, T1507, T1508) -> f1116_out1(T1516) f1153_in(T1547, T1548, T1549) -> U81(f1154_in(T1547, T1548, T1549), T1547, T1548, T1549) U81(f1154_out1, T1547, T1548, T1549) -> U82(f1155_in, T1547, T1548, T1549) U82(f1155_out1, T1547, T1548, T1549) -> f1153_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (209) NonLoopProof (COMPLETE) By Theorem 8 [NONLOOP] we deduce infiniteness of the QDP. We apply the theorem with m = 1, b = 0, σ' = [ ], and μ' = [x0 / x3, x2 / .(x0, x2)] on the rule F1022_IN(x3, .(x0, x2), .(x3, x1))[ ]^n[ ] -> F1022_IN(x3, .(x0, x2), .(x3, x1))[ ]^n[x0 / x3, x2 / .(x0, x2)] This rule is correct for the QDP as the following derivation shows: F1022_IN(x3, .(x0, x2), .(x3, x1))[ ]^n[ ] -> F1022_IN(x3, .(x0, x2), .(x3, x1))[ ]^n[x0 / x3, x2 / .(x0, x2)] by Equivalency by Simplifying Mu with mu1: [x0 / x3, x2 / .(x0, x2)] mu2: [ ] intermediate steps: Instantiate mu - Instantiation F1022_IN(x2, x1, .(y0, y1))[ ]^n[ ] -> F1022_IN(y0, .(x2, x1), .(y0, y1))[ ]^n[ ] by Narrowing at position: [] intermediate steps: Instantiation - Instantiation - Instantiation F1022_IN(T1506, T1507, T1508)[ ]^n[ ] -> F1110_IN(T1508, T1506, T1507)[ ]^n[ ] by Rule from TRS P intermediate steps: Instantiation - Instantiation - Instantiation - Instantiation F1110_IN(.(x1, x3), x2, x0)[ ]^n[ ] -> F1022_IN(x1, .(x2, x0), .(x1, x3))[ ]^n[ ] by Narrowing at position: [] intermediate steps: Instantiation - Instantiation F1110_IN(.(x1, x3), x2, x0)[ ]^n[ ] -> U79^1(f1144_out1, x1, x2, x0, .(x1, x3))[ ]^n[ ] by Narrowing at position: [0] intermediate steps: Instantiation - Instantiation F1110_IN(.(x1, x3), x2, x0)[ ]^n[ ] -> U79^1(f1144_in(x1, x2, x0), x1, x2, x0, .(x1, x3))[ ]^n[ ] by Narrowing at position: [] intermediate steps: Instantiation - Instantiation F1110_IN(.(x1, x3), x2, x0)[ ]^n[ ] -> F1116_IN(x1, x2, x0, .(x1, x3))[ ]^n[ ] by Narrowing at position: [] intermediate steps: Instantiation - Instantiation F1110_IN(.(y0, y1), x2, x1)[ ]^n[ ] -> U77^1(f235_out1(y0), .(y0, y1), x2, x1)[ ]^n[ ] by Narrowing at position: [0] intermediate steps: Instantiation - Instantiation - Instantiation F1110_IN(T1508, T1506, T1507)[ ]^n[ ] -> U77^1(f235_in(T1508), T1508, T1506, T1507)[ ]^n[ ] by Rule from TRS P intermediate steps: Instantiation - Instantiation f235_in(.(T49, T50))[ ]^n[ ] -> f235_out1(T49)[ ]^n[ ] by Rule from TRS R intermediate steps: Instantiation - Instantiation - Instantiation - Instantiation - Instantiation - Instantiation U77^1(f235_out1(T1515), T1508, T1506, T1507)[ ]^n[ ] -> F1116_IN(T1515, T1506, T1507, T1508)[ ]^n[ ] by Rule from TRS P intermediate steps: Instantiation - Instantiation - Instantiation - Instantiation - Instantiation - Instantiation F1116_IN(T1515, T1506, T1507, T1508)[ ]^n[ ] -> U79^1(f1144_in(T1515, T1506, T1507), T1515, T1506, T1507, T1508)[ ]^n[ ] by Rule from TRS P intermediate steps: Instantiation - Instantiation - Instantiation - Instantiation - Instantiation f1144_in(T1614, T1615, T1616)[ ]^n[ ] -> f1144_out1[ ]^n[ ] by Rule from TRS R intermediate steps: Instantiation - Instantiation - Instantiation - Instantiation - Instantiation - Instantiation U79^1(f1144_out1, T1515, T1506, T1507, T1508)[ ]^n[ ] -> F1022_IN(T1515, .(T1506, T1507), T1508)[ ]^n[ ] by Rule from TRS P ---------------------------------------- (210) NO ---------------------------------------- (211) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. 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T109 ([]))) T110))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T109", "T110", "T117" ], "free": [], "exprvars": [] } }, "1206": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1327": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1205": { "goal": [{ "clause": 4, "scope": 38, "term": "(member T1578 T1580)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1578", "T1580" ], "free": [], "exprvars": [] } }, "1326": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1204": { "goal": [{ "clause": 3, "scope": 38, "term": "(member T1578 T1580)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1578", "T1580" ], "free": [], "exprvars": [] } }, "586": { "goal": [{ "clause": -1, "scope": -1, "term": "(not_member T208 (. T199 (. T200 ([]))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T199", "T200", "T208" ], "free": [], "exprvars": [] } }, "1203": { "goal": [ { "clause": 3, "scope": 38, "term": "(member T1578 T1580)" }, { "clause": 4, "scope": 38, "term": "(member T1578 T1580)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1578", "T1580" ], "free": [], "exprvars": [] } }, "587": { "goal": [{ "clause": -1, "scope": -1, "term": "(subsetchecked T209 (. T208 (. T199 (. T200 ([])))) T201)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T199", "T200", "T201", "T208" ], "free": [], "exprvars": [] } }, "1202": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T1578 T1580)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1578", "T1580" ], "free": [], "exprvars": [] } }, "1201": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1200": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "469": { "goal": [{ "clause": -1, "scope": -1, "term": "(not_member T117 (. T109 ([])))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T109", "T117" ], "free": [], "exprvars": [] } }, "900": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T950 T949)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T949"], "free": [], "exprvars": [] } }, "901": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (not_member T956 (. T943 (. T944 (. T945 (. T946 (. T947 (. T948 ([])))))))) (subsetchecked T957 (. T956 (. T943 (. T944 (. T945 (. T946 (. T947 (. T948 ([])))))))) T949))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T943", "T944", "T945", "T946", "T947", "T948", "T949", "T956" ], "free": [], "exprvars": [] } }, "1209": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T1601 T1603)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1601", "T1603" ], "free": [], "exprvars": [] } }, "1208": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1207": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 2, "to": 140, "label": "CASE" }, { "from": 140, "to": 152, "label": "ONLY EVAL with clause\nsubset(X15, X16) :- subsetchecked(X15, [], X16).\nand substitutionT1 -> T12,\nX15 -> T12,\nT2 -> T11,\nX16 -> T11,\nT10 -> T12" }, { "from": 152, "to": 153, "label": "CASE" }, { "from": 153, "to": 162, "label": "PARALLEL" }, { "from": 153, "to": 163, "label": "PARALLEL" }, { "from": 162, "to": 284, "label": "EVAL with clause\nsubsetchecked([], X29, X30).\nand substitutionT12 -> [],\nX29 -> [],\nT11 -> T19,\nX30 -> T19" }, { "from": 162, "to": 285, "label": "EVAL-BACKTRACK" }, { "from": 163, "to": 287, "label": "EVAL with clause\nsubsetchecked(.(X39, X40), X41, X42) :- ','(member(X39, X42), ','(not_member(X39, X41), subsetchecked(X40, .(X39, X41), X42))).\nand substitutionX39 -> T29,\nX40 -> T30,\nT12 -> .(T29, T30),\nX41 -> [],\nT11 -> T28,\nX42 -> T28,\nT26 -> T29,\nT27 -> T30" }, { "from": 163, "to": 288, "label": "EVAL-BACKTRACK" }, { "from": 284, "to": 286, "label": "SUCCESS" }, { "from": 287, "to": 300, "label": "SPLIT 1" }, { "from": 287, "to": 302, "label": "SPLIT 2\nnew knowledge:\nT35 is ground\nT28 is ground\nreplacements:T29 -> T35,\nT30 -> T36" }, { "from": 300, "to": 305, "label": "CASE" }, { "from": 302, "to": 320, "label": "SPLIT 1" }, { "from": 302, "to": 321, "label": "SPLIT 2\nnew knowledge:\nT35 is ground" }, { "from": 305, "to": 306, "label": "PARALLEL" }, { "from": 305, "to": 307, "label": "PARALLEL" }, { "from": 306, "to": 308, "label": "EVAL with clause\nmember(X59, .(X59, X60)).\nand substitutionT29 -> T49,\nX59 -> T49,\nX60 -> T50,\nT28 -> .(T49, T50)" }, { "from": 306, "to": 311, "label": "EVAL-BACKTRACK" }, { "from": 307, "to": 314, "label": "EVAL with clause\nmember(X67, .(X68, X69)) :- member(X67, X69).\nand substitutionT29 -> T60,\nX67 -> T60,\nX68 -> T58,\nX69 -> T59,\nT28 -> .(T58, T59),\nT57 -> T60" }, { "from": 307, "to": 315, "label": "EVAL-BACKTRACK" }, { "from": 308, "to": 312, "label": "SUCCESS" }, { "from": 314, "to": 300, "label": "INSTANCE with matching:\nT29 -> T60\nT28 -> T59" }, { "from": 320, "to": 324, "label": "CASE" }, { "from": 321, "to": 440, "label": "CASE" }, { "from": 324, "to": 333, "label": "PARALLEL" }, { "from": 324, "to": 335, "label": "PARALLEL" }, { "from": 333, "to": 378, "label": "ONLY EVAL with clause\nnot_member(X94, X95) :- ','(member(X94, X95), ','(!_4, failure(a))).\nand substitutionT35 -> T75,\nX94 -> T75,\nX95 -> []" }, { "from": 335, "to": 437, "label": "ONLY EVAL with clause\nnot_member(X113, X114).\nand substitutionT35 -> T84,\nX113 -> T84,\nX114 -> []" }, { "from": 378, "to": 394, "label": "SPLIT 1" }, { "from": 378, "to": 395, "label": "SPLIT 2\nnew knowledge:\nT75 is ground" }, { "from": 394, "to": 406, "label": "CASE" }, { "from": 395, "to": 416, "label": "CUT" }, { "from": 406, "to": 407, "label": "BACKTRACK\nfor clause: member(X, .(X, X3))because of non-unification" }, { "from": 407, "to": 415, "label": "BACKTRACK\nfor clause: member(X, .(X4, Xs)) :- member(X, Xs)because of non-unification" }, { "from": 416, "to": 433, "label": "CASE" }, { "from": 433, "to": 435, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 437, "to": 438, "label": "SUCCESS" }, { "from": 440, "to": 441, "label": "PARALLEL" }, { "from": 440, "to": 442, "label": "PARALLEL" }, { "from": 441, "to": 447, "label": "EVAL with clause\nsubsetchecked([], X127, X128).\nand substitutionT36 -> [],\nT35 -> T97,\nX127 -> .(T97, []),\nT28 -> T98,\nX128 -> T98" }, { "from": 441, "to": 448, "label": "EVAL-BACKTRACK" }, { "from": 442, "to": 454, "label": "EVAL with clause\nsubsetchecked(.(X137, X138), X139, X140) :- ','(member(X137, X140), ','(not_member(X137, X139), subsetchecked(X138, .(X137, X139), X140))).\nand substitutionX137 -> T111,\nX138 -> T112,\nT36 -> .(T111, T112),\nT35 -> T109,\nX139 -> .(T109, []),\nT28 -> T110,\nX140 -> T110,\nT107 -> T111,\nT108 -> T112" }, { "from": 442, "to": 455, "label": "EVAL-BACKTRACK" }, { "from": 447, "to": 449, "label": "SUCCESS" }, { "from": 454, "to": 461, "label": "SPLIT 1" }, { "from": 454, "to": 462, "label": "SPLIT 2\nnew knowledge:\nT117 is ground\nT110 is ground\nreplacements:T111 -> T117,\nT112 -> T118" }, { "from": 461, "to": 300, "label": "INSTANCE with matching:\nT29 -> T111\nT28 -> T110" }, { "from": 462, "to": 469, "label": "SPLIT 1" }, { "from": 462, "to": 470, "label": "SPLIT 2\nnew knowledge:\nT117 is ground\nT109 is ground" }, { "from": 469, "to": 474, "label": "CASE" }, { "from": 470, "to": 533, "label": "CASE" }, { "from": 474, "to": 480, "label": "PARALLEL" }, { "from": 474, "to": 481, "label": "PARALLEL" }, { "from": 480, "to": 483, "label": "ONLY EVAL with clause\nnot_member(X169, X170) :- ','(member(X169, X170), ','(!_8, failure(a))).\nand substitutionT117 -> T139,\nX169 -> T139,\nT109 -> T140,\nX170 -> .(T140, [])" }, { "from": 481, "to": 529, "label": "ONLY EVAL with clause\nnot_member(X210, X211).\nand substitutionT117 -> T164,\nX210 -> T164,\nT109 -> T165,\nX211 -> .(T165, [])" }, { "from": 483, "to": 507, "label": "SPLIT 1" }, { "from": 483, "to": 508, "label": "SPLIT 2\nnew knowledge:\nT139 is ground\nT140 is ground" }, { "from": 507, "to": 511, "label": "CASE" }, { "from": 508, "to": 524, "label": "CUT" }, { "from": 511, "to": 512, "label": "PARALLEL" }, { "from": 511, "to": 513, "label": "PARALLEL" }, { "from": 512, "to": 514, "label": "EVAL with clause\nmember(X187, .(X187, X188)).\nand substitutionT139 -> T149,\nX187 -> T149,\nT140 -> T149,\nX188 -> []" }, { "from": 512, "to": 515, "label": "EVAL-BACKTRACK" }, { "from": 513, "to": 520, "label": "ONLY EVAL with clause\nmember(X199, .(X200, X201)) :- member(X199, X201).\nand substitutionT139 -> T156,\nX199 -> T156,\nT140 -> T157,\nX200 -> T157,\nX201 -> []" }, { "from": 514, "to": 516, "label": "SUCCESS" }, { "from": 520, "to": 394, "label": "INSTANCE with matching:\nT75 -> T156" }, { "from": 524, "to": 525, "label": "CASE" }, { "from": 525, "to": 526, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 529, "to": 530, "label": "SUCCESS" }, { "from": 533, "to": 550, "label": "PARALLEL" }, { "from": 533, "to": 551, "label": "PARALLEL" }, { "from": 550, "to": 552, "label": "EVAL with clause\nsubsetchecked([], X224, X225).\nand substitutionT118 -> [],\nT117 -> T184,\nT109 -> T185,\nX224 -> .(T184, .(T185, [])),\nT110 -> T186,\nX225 -> T186" }, { "from": 550, "to": 553, "label": "EVAL-BACKTRACK" }, { "from": 551, "to": 556, "label": "EVAL with clause\nsubsetchecked(.(X234, X235), X236, X237) :- ','(member(X234, X237), ','(not_member(X234, X236), subsetchecked(X235, .(X234, X236), X237))).\nand substitutionX234 -> T202,\nX235 -> T203,\nT118 -> .(T202, T203),\nT117 -> T199,\nT109 -> T200,\nX236 -> .(T199, .(T200, [])),\nT110 -> T201,\nX237 -> T201,\nT197 -> T202,\nT198 -> T203" }, { "from": 551, "to": 557, "label": "EVAL-BACKTRACK" }, { "from": 552, "to": 554, "label": "SUCCESS" }, { "from": 556, "to": 560, "label": "SPLIT 1" }, { "from": 556, "to": 561, "label": "SPLIT 2\nnew knowledge:\nT208 is ground\nT201 is ground\nreplacements:T202 -> T208,\nT203 -> T209" }, { "from": 560, "to": 300, "label": "INSTANCE with matching:\nT29 -> T202\nT28 -> T201" }, { "from": 561, "to": 586, "label": "SPLIT 1" }, { "from": 561, "to": 587, "label": "SPLIT 2\nnew knowledge:\nT208 is ground\nT199 is ground\nT200 is ground" }, { "from": 586, "to": 595, "label": "CASE" }, { "from": 587, "to": 619, "label": "CASE" }, { "from": 595, "to": 598, "label": "PARALLEL" }, { "from": 595, "to": 600, "label": "PARALLEL" }, { "from": 598, "to": 601, "label": "ONLY EVAL with clause\nnot_member(X266, X267) :- ','(member(X266, X267), ','(!_12, failure(a))).\nand substitutionT208 -> T240,\nX266 -> T240,\nT199 -> T241,\nT200 -> T242,\nX267 -> .(T241, .(T242, []))" }, { "from": 600, "to": 616, "label": "ONLY EVAL with clause\nnot_member(X307, X308).\nand substitutionT208 -> T280,\nX307 -> T280,\nT199 -> T281,\nT200 -> T282,\nX308 -> .(T281, .(T282, []))" }, { "from": 601, "to": 602, "label": "SPLIT 1" }, { "from": 601, "to": 603, "label": "SPLIT 2\nnew knowledge:\nT240 is ground\nT241 is ground\nT242 is ground" }, { "from": 602, "to": 604, "label": "CASE" }, { "from": 603, "to": 611, "label": "CUT" }, { "from": 604, "to": 605, "label": "PARALLEL" }, { "from": 604, "to": 606, "label": "PARALLEL" }, { "from": 605, "to": 607, "label": "EVAL with clause\nmember(X284, .(X284, X285)).\nand substitutionT240 -> T259,\nX284 -> T259,\nT241 -> T259,\nT242 -> T260,\nX285 -> .(T260, [])" }, { "from": 605, "to": 608, "label": "EVAL-BACKTRACK" }, { "from": 606, "to": 610, "label": "ONLY EVAL with clause\nmember(X296, .(X297, X298)) :- member(X296, X298).\nand substitutionT240 -> T269,\nX296 -> T269,\nT241 -> T270,\nX297 -> T270,\nT242 -> T271,\nX298 -> .(T271, [])" }, { "from": 607, "to": 609, "label": "SUCCESS" }, { "from": 610, "to": 507, "label": "INSTANCE with matching:\nT139 -> T269\nT140 -> T271" }, { "from": 611, "to": 612, "label": "CASE" }, { "from": 612, "to": 613, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 616, "to": 617, "label": "SUCCESS" }, { "from": 619, "to": 620, "label": "PARALLEL" }, { "from": 619, "to": 621, "label": "PARALLEL" }, { "from": 620, "to": 626, "label": "EVAL with clause\nsubsetchecked([], X321, X322).\nand substitutionT209 -> [],\nT208 -> T307,\nT199 -> T308,\nT200 -> T309,\nX321 -> .(T307, .(T308, .(T309, []))),\nT201 -> T310,\nX322 -> T310" }, { "from": 620, "to": 627, "label": "EVAL-BACKTRACK" }, { "from": 621, "to": 631, "label": "EVAL with clause\nsubsetchecked(.(X331, X332), X333, X334) :- ','(member(X331, X334), ','(not_member(X331, X333), subsetchecked(X332, .(X331, X333), X334))).\nand substitutionX331 -> T329,\nX332 -> T330,\nT209 -> .(T329, T330),\nT208 -> T325,\nT199 -> T326,\nT200 -> T327,\nX333 -> .(T325, .(T326, .(T327, []))),\nT201 -> T328,\nX334 -> T328,\nT323 -> T329,\nT324 -> T330" }, { "from": 621, "to": 632, "label": "EVAL-BACKTRACK" }, { "from": 626, "to": 628, "label": "SUCCESS" }, { "from": 631, "to": 637, "label": "SPLIT 1" }, { "from": 631, "to": 638, "label": "SPLIT 2\nnew knowledge:\nT335 is ground\nT328 is ground\nreplacements:T329 -> T335,\nT330 -> T336" }, { "from": 637, "to": 300, "label": "INSTANCE with matching:\nT29 -> T329\nT28 -> T328" }, { "from": 638, "to": 647, "label": "SPLIT 1" }, { "from": 638, "to": 648, "label": "SPLIT 2\nnew knowledge:\nT335 is ground\nT325 is ground\nT326 is ground\nT327 is ground" }, { "from": 647, "to": 652, "label": "CASE" }, { "from": 648, "to": 707, "label": "CASE" }, { "from": 652, "to": 656, "label": "PARALLEL" }, { "from": 652, "to": 657, "label": "PARALLEL" }, { "from": 656, "to": 659, "label": "ONLY EVAL with clause\nnot_member(X363, X364) :- ','(member(X363, X364), ','(!_16, failure(a))).\nand substitutionT335 -> T377,\nX363 -> T377,\nT325 -> T378,\nT326 -> T379,\nT327 -> T380,\nX364 -> .(T378, .(T379, .(T380, [])))" }, { "from": 657, "to": 702, "label": "ONLY EVAL with clause\nnot_member(X404, X405).\nand substitutionT335 -> T436,\nX404 -> T436,\nT325 -> T437,\nT326 -> T438,\nT327 -> T439,\nX405 -> .(T437, .(T438, .(T439, [])))" }, { "from": 659, "to": 685, "label": "SPLIT 1" }, { "from": 659, "to": 686, "label": "SPLIT 2\nnew knowledge:\nT377 is ground\nT378 is ground\nT379 is ground\nT380 is ground" }, { "from": 685, "to": 687, "label": "CASE" }, { "from": 686, "to": 698, "label": "CUT" }, { "from": 687, "to": 688, "label": "PARALLEL" }, { "from": 687, "to": 689, "label": "PARALLEL" }, { "from": 688, "to": 690, "label": "EVAL with clause\nmember(X381, .(X381, X382)).\nand substitutionT377 -> T405,\nX381 -> T405,\nT378 -> T405,\nT379 -> T406,\nT380 -> T407,\nX382 -> .(T406, .(T407, []))" }, { "from": 688, "to": 691, "label": "EVAL-BACKTRACK" }, { "from": 689, "to": 693, "label": "ONLY EVAL with clause\nmember(X393, .(X394, X395)) :- member(X393, X395).\nand substitutionT377 -> T420,\nX393 -> T420,\nT378 -> T421,\nX394 -> T421,\nT379 -> T422,\nT380 -> T423,\nX395 -> .(T422, .(T423, []))" }, { "from": 690, "to": 692, "label": "SUCCESS" }, { "from": 693, "to": 602, "label": "INSTANCE with matching:\nT240 -> T420\nT241 -> T422\nT242 -> T423" }, { "from": 698, "to": 700, "label": "CASE" }, { "from": 700, "to": 701, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 702, "to": 703, "label": "SUCCESS" }, { "from": 707, "to": 708, "label": "PARALLEL" }, { "from": 707, "to": 709, "label": "PARALLEL" }, { "from": 708, "to": 716, "label": "EVAL with clause\nsubsetchecked([], X418, X419).\nand substitutionT336 -> [],\nT335 -> T470,\nT325 -> T471,\nT326 -> T472,\nT327 -> T473,\nX418 -> .(T470, .(T471, .(T472, .(T473, [])))),\nT328 -> T474,\nX419 -> T474" }, { "from": 708, "to": 717, "label": "EVAL-BACKTRACK" }, { "from": 709, "to": 721, "label": "EVAL with clause\nsubsetchecked(.(X428, X429), X430, X431) :- ','(member(X428, X431), ','(not_member(X428, X430), subsetchecked(X429, .(X428, X430), X431))).\nand substitutionX428 -> T496,\nX429 -> T497,\nT336 -> .(T496, T497),\nT335 -> T491,\nT325 -> T492,\nT326 -> T493,\nT327 -> T494,\nX430 -> .(T491, .(T492, .(T493, .(T494, [])))),\nT328 -> T495,\nX431 -> T495,\nT489 -> T496,\nT490 -> T497" }, { "from": 709, "to": 722, "label": "EVAL-BACKTRACK" }, { "from": 716, "to": 718, "label": "SUCCESS" }, { "from": 721, "to": 728, "label": "SPLIT 1" }, { "from": 721, "to": 729, "label": "SPLIT 2\nnew knowledge:\nT502 is ground\nT495 is ground\nreplacements:T496 -> T502,\nT497 -> T503" }, { "from": 728, "to": 300, "label": "INSTANCE with matching:\nT29 -> T496\nT28 -> T495" }, { "from": 729, "to": 750, "label": "SPLIT 1" }, { "from": 729, "to": 751, "label": "SPLIT 2\nnew knowledge:\nT502 is ground\nT491 is ground\nT492 is ground\nT493 is ground\nT494 is ground" }, { "from": 750, "to": 752, "label": "CASE" }, { "from": 751, "to": 807, "label": "CASE" }, { "from": 752, "to": 754, "label": "PARALLEL" }, { "from": 752, "to": 755, "label": "PARALLEL" }, { "from": 754, "to": 761, "label": "ONLY EVAL with clause\nnot_member(X460, X461) :- ','(member(X460, X461), ','(!_20, failure(a))).\nand substitutionT502 -> T554,\nX460 -> T554,\nT491 -> T555,\nT492 -> T556,\nT493 -> T557,\nT494 -> T558,\nX461 -> .(T555, .(T556, .(T557, .(T558, []))))" }, { "from": 755, "to": 802, "label": "ONLY EVAL with clause\nnot_member(X501, X502).\nand substitutionT502 -> T632,\nX501 -> T632,\nT491 -> T633,\nT492 -> T634,\nT493 -> T635,\nT494 -> T636,\nX502 -> .(T633, .(T634, .(T635, .(T636, []))))" }, { "from": 761, "to": 762, "label": "SPLIT 1" }, { "from": 761, "to": 764, "label": "SPLIT 2\nnew knowledge:\nT554 is ground\nT555 is ground\nT556 is ground\nT557 is ground\nT558 is ground" }, { "from": 762, "to": 766, "label": "CASE" }, { "from": 764, "to": 799, "label": "CUT" }, { "from": 766, "to": 768, "label": "PARALLEL" }, { "from": 766, "to": 769, "label": "PARALLEL" }, { "from": 768, "to": 773, "label": "EVAL with clause\nmember(X478, .(X478, X479)).\nand substitutionT554 -> T591,\nX478 -> T591,\nT555 -> T591,\nT556 -> T592,\nT557 -> T593,\nT558 -> T594,\nX479 -> .(T592, .(T593, .(T594, [])))" }, { "from": 768, "to": 774, "label": "EVAL-BACKTRACK" }, { "from": 769, "to": 798, "label": "ONLY EVAL with clause\nmember(X490, .(X491, X492)) :- member(X490, X492).\nand substitutionT554 -> T611,\nX490 -> T611,\nT555 -> T612,\nX491 -> T612,\nT556 -> T613,\nT557 -> T614,\nT558 -> T615,\nX492 -> .(T613, .(T614, .(T615, [])))" }, { "from": 773, "to": 775, "label": "SUCCESS" }, { "from": 798, "to": 685, "label": "INSTANCE with matching:\nT377 -> T611\nT378 -> T613\nT379 -> T614\nT380 -> T615" }, { "from": 799, "to": 800, "label": "CASE" }, { "from": 800, "to": 801, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 802, "to": 803, "label": "SUCCESS" }, { "from": 807, "to": 808, "label": "PARALLEL" }, { "from": 807, "to": 809, "label": "PARALLEL" }, { "from": 808, "to": 810, "label": "EVAL with clause\nsubsetchecked([], X515, X516).\nand substitutionT503 -> [],\nT502 -> T673,\nT491 -> T674,\nT492 -> T675,\nT493 -> T676,\nT494 -> T677,\nX515 -> .(T673, .(T674, .(T675, .(T676, .(T677, []))))),\nT495 -> T678,\nX516 -> T678" }, { "from": 808, "to": 811, "label": "EVAL-BACKTRACK" }, { "from": 809, "to": 828, "label": "EVAL with clause\nsubsetchecked(.(X525, X526), X527, X528) :- ','(member(X525, X528), ','(not_member(X525, X527), subsetchecked(X526, .(X525, X527), X528))).\nand substitutionX525 -> T703,\nX526 -> T704,\nT503 -> .(T703, T704),\nT502 -> T697,\nT491 -> T698,\nT492 -> T699,\nT493 -> T700,\nT494 -> T701,\nX527 -> .(T697, .(T698, .(T699, .(T700, .(T701, []))))),\nT495 -> T702,\nX528 -> T702,\nT695 -> T703,\nT696 -> T704" }, { "from": 809, "to": 829, "label": "EVAL-BACKTRACK" }, { "from": 810, "to": 812, "label": "SUCCESS" }, { "from": 828, "to": 830, "label": "SPLIT 1" }, { "from": 828, "to": 831, "label": "SPLIT 2\nnew knowledge:\nT709 is ground\nT702 is ground\nreplacements:T703 -> T709,\nT704 -> T710" }, { "from": 830, "to": 300, "label": "INSTANCE with matching:\nT29 -> T703\nT28 -> T702" }, { "from": 831, "to": 832, "label": "SPLIT 1" }, { "from": 831, "to": 848, "label": "SPLIT 2\nnew knowledge:\nT709 is ground\nT697 is ground\nT698 is ground\nT699 is ground\nT700 is ground\nT701 is ground" }, { "from": 832, "to": 849, "label": "CASE" }, { "from": 848, "to": 888, "label": "CASE" }, { "from": 849, "to": 853, "label": "PARALLEL" }, { "from": 849, "to": 854, "label": "PARALLEL" }, { "from": 853, "to": 860, "label": "ONLY EVAL with clause\nnot_member(X557, X558) :- ','(member(X557, X558), ','(!_24, failure(a))).\nand substitutionT709 -> T771,\nX557 -> T771,\nT697 -> T772,\nT698 -> T773,\nT699 -> T774,\nT700 -> T775,\nT701 -> T776,\nX558 -> .(T772, .(T773, .(T774, .(T775, .(T776, [])))))" }, { "from": 854, "to": 886, "label": "ONLY EVAL with clause\nnot_member(X598, X599).\nand substitutionT709 -> T868,\nX598 -> T868,\nT697 -> T869,\nT698 -> T870,\nT699 -> T871,\nT700 -> T872,\nT701 -> T873,\nX599 -> .(T869, .(T870, .(T871, .(T872, .(T873, [])))))" }, { "from": 860, "to": 861, "label": "SPLIT 1" }, { "from": 860, "to": 862, "label": "SPLIT 2\nnew knowledge:\nT771 is ground\nT772 is ground\nT773 is ground\nT774 is ground\nT775 is ground\nT776 is ground" }, { "from": 861, "to": 864, "label": "CASE" }, { "from": 862, "to": 880, "label": "CUT" }, { "from": 864, "to": 867, "label": "PARALLEL" }, { "from": 864, "to": 868, "label": "PARALLEL" }, { "from": 867, "to": 869, "label": "EVAL with clause\nmember(X575, .(X575, X576)).\nand substitutionT771 -> T817,\nX575 -> T817,\nT772 -> T817,\nT773 -> T818,\nT774 -> T819,\nT775 -> T820,\nT776 -> T821,\nX576 -> .(T818, .(T819, .(T820, .(T821, []))))" }, { "from": 867, "to": 870, "label": "EVAL-BACKTRACK" }, { "from": 868, "to": 879, "label": "ONLY EVAL with clause\nmember(X587, .(X588, X589)) :- member(X587, X589).\nand substitutionT771 -> T842,\nX587 -> T842,\nT772 -> T843,\nX588 -> T843,\nT773 -> T844,\nT774 -> T845,\nT775 -> T846,\nT776 -> T847,\nX589 -> .(T844, .(T845, .(T846, .(T847, []))))" }, { "from": 869, "to": 871, "label": "SUCCESS" }, { "from": 879, "to": 762, "label": "INSTANCE with matching:\nT554 -> T842\nT555 -> T844\nT556 -> T845\nT557 -> T846\nT558 -> T847" }, { "from": 880, "to": 881, "label": "CASE" }, { "from": 881, "to": 882, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 886, "to": 887, "label": "SUCCESS" }, { "from": 888, "to": 891, "label": "PARALLEL" }, { "from": 888, "to": 892, "label": "PARALLEL" }, { "from": 891, "to": 893, "label": "EVAL with clause\nsubsetchecked([], X612, X613).\nand substitutionT710 -> [],\nT709 -> T916,\nT697 -> T917,\nT698 -> T918,\nT699 -> T919,\nT700 -> T920,\nT701 -> T921,\nX612 -> .(T916, .(T917, .(T918, .(T919, .(T920, .(T921, [])))))),\nT702 -> T922,\nX613 -> T922" }, { "from": 891, "to": 894, "label": "EVAL-BACKTRACK" }, { "from": 892, "to": 898, "label": "EVAL with clause\nsubsetchecked(.(X622, X623), X624, X625) :- ','(member(X622, X625), ','(not_member(X622, X624), subsetchecked(X623, .(X622, X624), X625))).\nand substitutionX622 -> T950,\nX623 -> T951,\nT710 -> .(T950, T951),\nT709 -> T943,\nT697 -> T944,\nT698 -> T945,\nT699 -> T946,\nT700 -> T947,\nT701 -> T948,\nX624 -> .(T943, .(T944, .(T945, .(T946, .(T947, .(T948, [])))))),\nT702 -> T949,\nX625 -> T949,\nT941 -> T950,\nT942 -> T951" }, { "from": 892, "to": 899, "label": "EVAL-BACKTRACK" }, { "from": 893, "to": 895, "label": "SUCCESS" }, { "from": 898, "to": 900, "label": "SPLIT 1" }, { "from": 898, "to": 901, "label": "SPLIT 2\nnew knowledge:\nT956 is ground\nT949 is ground\nreplacements:T950 -> T956,\nT951 -> T957" }, { "from": 900, "to": 300, "label": "INSTANCE with matching:\nT29 -> T950\nT28 -> T949" }, { "from": 901, "to": 907, "label": "SPLIT 1" }, { "from": 901, "to": 908, "label": "SPLIT 2\nnew knowledge:\nT956 is ground\nT943 is ground\nT944 is ground\nT945 is ground\nT946 is ground\nT947 is ground\nT948 is ground" }, { "from": 907, "to": 911, "label": "CASE" }, { "from": 908, "to": 999, "label": "CASE" }, { "from": 911, "to": 938, "label": "PARALLEL" }, { "from": 911, "to": 939, "label": "PARALLEL" }, { "from": 938, "to": 944, "label": "ONLY EVAL with clause\nnot_member(X654, X655) :- ','(member(X654, X655), ','(!_28, failure(a))).\nand substitutionT956 -> T1028,\nX654 -> T1028,\nT943 -> T1029,\nT944 -> T1030,\nT945 -> T1031,\nT946 -> T1032,\nT947 -> T1033,\nT948 -> T1034,\nX655 -> .(T1029, .(T1030, .(T1031, .(T1032, .(T1033, .(T1034, []))))))" }, { "from": 939, "to": 994, "label": "ONLY EVAL with clause\nnot_member(X695, X696).\nand substitutionT956 -> T1144,\nX695 -> T1144,\nT943 -> T1145,\nT944 -> T1146,\nT945 -> T1147,\nT946 -> T1148,\nT947 -> T1149,\nT948 -> T1150,\nX696 -> .(T1145, .(T1146, .(T1147, .(T1148, .(T1149, .(T1150, []))))))" }, { "from": 944, "to": 947, "label": "SPLIT 1" }, { "from": 944, "to": 948, "label": "SPLIT 2\nnew knowledge:\nT1028 is ground\nT1029 is ground\nT1030 is ground\nT1031 is ground\nT1032 is ground\nT1033 is ground\nT1034 is ground" }, { "from": 947, "to": 952, "label": "CASE" }, { "from": 948, "to": 976, "label": "CUT" }, { "from": 952, "to": 953, "label": "PARALLEL" }, { "from": 952, "to": 954, "label": "PARALLEL" }, { "from": 953, "to": 958, "label": "EVAL with clause\nmember(X672, .(X672, X673)).\nand substitutionT1028 -> T1083,\nX672 -> T1083,\nT1029 -> T1083,\nT1030 -> T1084,\nT1031 -> T1085,\nT1032 -> T1086,\nT1033 -> T1087,\nT1034 -> T1088,\nX673 -> .(T1084, .(T1085, .(T1086, .(T1087, .(T1088, [])))))" }, { "from": 953, "to": 960, "label": "EVAL-BACKTRACK" }, { "from": 954, "to": 972, "label": "ONLY EVAL with clause\nmember(X684, .(X685, X686)) :- member(X684, X686).\nand substitutionT1028 -> T1113,\nX684 -> T1113,\nT1029 -> T1114,\nX685 -> T1114,\nT1030 -> T1115,\nT1031 -> T1116,\nT1032 -> T1117,\nT1033 -> T1118,\nT1034 -> T1119,\nX686 -> .(T1115, .(T1116, .(T1117, .(T1118, .(T1119, [])))))" }, { "from": 958, "to": 963, "label": "SUCCESS" }, { "from": 972, "to": 861, "label": "INSTANCE with matching:\nT771 -> T1113\nT772 -> T1115\nT773 -> T1116\nT774 -> T1117\nT775 -> T1118\nT776 -> T1119" }, { "from": 976, "to": 977, "label": "CASE" }, { "from": 977, "to": 978, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 994, "to": 995, "label": "SUCCESS" }, { "from": 999, "to": 1009, "label": "PARALLEL" }, { "from": 999, "to": 1010, "label": "PARALLEL" }, { "from": 1009, "to": 1011, "label": "EVAL with clause\nsubsetchecked([], X709, X710).\nand substitutionT957 -> [],\nT956 -> T1199,\nT943 -> T1200,\nT944 -> T1201,\nT945 -> T1202,\nT946 -> T1203,\nT947 -> T1204,\nT948 -> T1205,\nX709 -> .(T1199, .(T1200, .(T1201, .(T1202, .(T1203, .(T1204, .(T1205, []))))))),\nT949 -> T1206,\nX710 -> T1206" }, { "from": 1009, "to": 1012, "label": "EVAL-BACKTRACK" }, { "from": 1010, "to": 1024, "label": "EVAL with clause\nsubsetchecked(.(X719, X720), X721, X722) :- ','(member(X719, X722), ','(not_member(X719, X721), subsetchecked(X720, .(X719, X721), X722))).\nand substitutionX719 -> T1237,\nX720 -> T1238,\nT957 -> .(T1237, T1238),\nT956 -> T1229,\nT943 -> T1230,\nT944 -> T1231,\nT945 -> T1232,\nT946 -> T1233,\nT947 -> T1234,\nT948 -> T1235,\nX721 -> .(T1229, .(T1230, .(T1231, .(T1232, .(T1233, .(T1234, .(T1235, []))))))),\nT949 -> T1236,\nX722 -> T1236,\nT1227 -> T1237,\nT1228 -> T1238" }, { "from": 1010, "to": 1025, "label": "EVAL-BACKTRACK" }, { "from": 1011, "to": 1013, "label": "SUCCESS" }, { "from": 1024, "to": 1026, "label": "GENERALIZATION\nT1243 <-- .(T1229, .(T1230, .(T1231, .(T1232, .(T1233, .(T1234, .(T1235, [])))))))\n\nNew Knowledge:\nT1243 is ground" }, { "from": 1026, "to": 1035, "label": "SPLIT 1" }, { "from": 1026, "to": 1036, "label": "SPLIT 2\nnew knowledge:\nT1248 is ground\nT1236 is ground\nreplacements:T1237 -> T1248,\nT1238 -> T1249" }, { "from": 1035, "to": 300, "label": "INSTANCE with matching:\nT29 -> T1237\nT28 -> T1236" }, { "from": 1036, "to": 1041, "label": "SPLIT 1" }, { "from": 1036, "to": 1042, "label": "SPLIT 2\nnew knowledge:\nT1248 is ground\nT1229 is ground\nT1230 is ground\nT1231 is ground\nT1232 is ground\nT1233 is ground\nT1234 is ground\nT1235 is ground" }, { "from": 1041, "to": 1051, "label": "CASE" }, { "from": 1042, "to": 1165, "label": "CASE" }, { "from": 1051, "to": 1094, "label": "PARALLEL" }, { "from": 1051, "to": 1095, "label": "PARALLEL" }, { "from": 1094, "to": 1099, "label": "ONLY EVAL with clause\nnot_member(X755, X756) :- ','(member(X755, X756), ','(!_32, failure(a))).\nand substitutionT1248 -> T1330,\nX755 -> T1330,\nT1229 -> T1331,\nT1230 -> T1332,\nT1231 -> T1333,\nT1232 -> T1334,\nT1233 -> T1335,\nT1234 -> T1336,\nT1235 -> T1337,\nX756 -> .(T1331, .(T1332, .(T1333, .(T1334, .(T1335, .(T1336, .(T1337, [])))))))" }, { "from": 1095, "to": 1162, "label": "ONLY EVAL with clause\nnot_member(X796, X797).\nand substitutionT1248 -> T1465,\nX796 -> T1465,\nT1229 -> T1466,\nT1230 -> T1467,\nT1231 -> T1468,\nT1232 -> T1469,\nT1233 -> T1470,\nT1234 -> T1471,\nT1235 -> T1472,\nX797 -> .(T1466, .(T1467, .(T1468, .(T1469, .(T1470, .(T1471, .(T1472, [])))))))" }, { "from": 1099, "to": 1105, "label": "SPLIT 1" }, { "from": 1099, "to": 1106, "label": "SPLIT 2\nnew knowledge:\nT1330 is ground\nT1331 is ground\nT1332 is ground\nT1333 is ground\nT1334 is ground\nT1335 is ground\nT1336 is ground\nT1337 is ground" }, { "from": 1105, "to": 1107, "label": "CASE" }, { "from": 1106, "to": 1150, "label": "CUT" }, { "from": 1107, "to": 1108, "label": "PARALLEL" }, { "from": 1107, "to": 1109, "label": "PARALLEL" }, { "from": 1108, "to": 1141, "label": "EVAL with clause\nmember(X773, .(X773, X774)).\nand substitutionT1330 -> T1394,\nX773 -> T1394,\nT1331 -> T1394,\nT1332 -> T1395,\nT1333 -> T1396,\nT1334 -> T1397,\nT1335 -> T1398,\nT1336 -> T1399,\nT1337 -> T1400,\nX774 -> .(T1395, .(T1396, .(T1397, .(T1398, .(T1399, .(T1400, []))))))" }, { "from": 1108, "to": 1142, "label": "EVAL-BACKTRACK" }, { "from": 1109, "to": 1149, "label": "ONLY EVAL with clause\nmember(X785, .(X786, X787)) :- member(X785, X787).\nand substitutionT1330 -> T1429,\nX785 -> T1429,\nT1331 -> T1430,\nX786 -> T1430,\nT1332 -> T1431,\nT1333 -> T1432,\nT1334 -> T1433,\nT1335 -> T1434,\nT1336 -> T1435,\nT1337 -> T1436,\nX787 -> .(T1431, .(T1432, .(T1433, .(T1434, .(T1435, .(T1436, []))))))" }, { "from": 1141, "to": 1143, "label": "SUCCESS" }, { "from": 1149, "to": 947, "label": "INSTANCE with matching:\nT1028 -> T1429\nT1029 -> T1431\nT1030 -> T1432\nT1031 -> T1433\nT1032 -> T1434\nT1033 -> T1435\nT1034 -> T1436" }, { "from": 1150, "to": 1151, "label": "CASE" }, { "from": 1151, "to": 1152, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 1162, "to": 1163, "label": "SUCCESS" }, { "from": 1165, "to": 1166, "label": "PARALLEL" }, { "from": 1165, "to": 1167, "label": "PARALLEL" }, { "from": 1166, "to": 1171, "label": "EVAL with clause\nsubsetchecked([], X810, X811).\nand substitutionT1249 -> [],\nT1248 -> T1491,\nT1243 -> T1492,\nX810 -> .(T1491, T1492),\nT1236 -> T1493,\nX811 -> T1493" }, { "from": 1166, "to": 1172, "label": "EVAL-BACKTRACK" }, { "from": 1167, "to": 1177, "label": "EVAL with clause\nsubsetchecked(.(X820, X821), X822, X823) :- ','(member(X820, X823), ','(not_member(X820, X822), subsetchecked(X821, .(X820, X822), X823))).\nand substitutionX820 -> T1509,\nX821 -> T1510,\nT1249 -> .(T1509, T1510),\nT1248 -> T1506,\nT1243 -> T1507,\nX822 -> .(T1506, T1507),\nT1236 -> T1508,\nX823 -> T1508,\nT1504 -> T1509,\nT1505 -> T1510" }, { "from": 1167, "to": 1178, "label": "EVAL-BACKTRACK" }, { "from": 1171, "to": 1173, "label": "SUCCESS" }, { "from": 1177, "to": 1179, "label": "SPLIT 1" }, { "from": 1177, "to": 1180, "label": "SPLIT 2\nnew knowledge:\nT1515 is ground\nT1508 is ground\nreplacements:T1509 -> T1515,\nT1510 -> T1516" }, { "from": 1179, "to": 300, "label": "INSTANCE with matching:\nT29 -> T1509\nT28 -> T1508" }, { "from": 1180, "to": 1183, "label": "SPLIT 1" }, { "from": 1180, "to": 1184, "label": "SPLIT 2\nnew knowledge:\nT1515 is ground\nT1506 is ground\nT1507 is ground" }, { "from": 1183, "to": 1185, "label": "CASE" }, { "from": 1184, "to": 1042, "label": "INSTANCE with matching:\nT1249 -> T1516\nT1248 -> T1515\nT1243 -> .(T1506, T1507)\nT1236 -> T1508" }, { "from": 1185, "to": 1191, "label": "PARALLEL" }, { "from": 1185, "to": 1192, "label": "PARALLEL" }, { "from": 1191, "to": 1193, "label": "ONLY EVAL with clause\nnot_member(X852, X853) :- ','(member(X852, X853), ','(!_36, failure(a))).\nand substitutionT1515 -> T1547,\nX852 -> T1547,\nT1506 -> T1548,\nT1507 -> T1549,\nX853 -> .(T1548, T1549)" }, { "from": 1192, "to": 1326, "label": "ONLY EVAL with clause\nnot_member(X916, X917).\nand substitutionT1515 -> T1614,\nX916 -> T1614,\nT1506 -> T1615,\nT1507 -> T1616,\nX917 -> .(T1615, T1616)" }, { "from": 1193, "to": 1194, "label": "SPLIT 1" }, { "from": 1193, "to": 1195, "label": "SPLIT 2\nnew knowledge:\nT1547 is ground\nT1548 is ground\nT1549 is ground" }, { "from": 1194, "to": 1196, "label": "CASE" }, { "from": 1195, "to": 1211, "label": "CUT" }, { "from": 1196, "to": 1197, "label": "PARALLEL" }, { "from": 1196, "to": 1198, "label": "PARALLEL" }, { "from": 1197, "to": 1199, "label": "EVAL with clause\nmember(X870, .(X870, X871)).\nand substitutionT1547 -> T1566,\nX870 -> T1566,\nT1548 -> T1566,\nT1549 -> T1567,\nX871 -> T1567" }, { "from": 1197, "to": 1200, "label": "EVAL-BACKTRACK" }, { "from": 1198, "to": 1202, "label": "ONLY EVAL with clause\nmember(X882, .(X883, X884)) :- member(X882, X884).\nand substitutionT1547 -> T1578,\nX882 -> T1578,\nT1548 -> T1579,\nX883 -> T1579,\nT1549 -> T1580,\nX884 -> T1580" }, { "from": 1199, "to": 1201, "label": "SUCCESS" }, { "from": 1202, "to": 1203, "label": "CASE" }, { "from": 1203, "to": 1204, "label": "PARALLEL" }, { "from": 1203, "to": 1205, "label": "PARALLEL" }, { "from": 1204, "to": 1206, "label": "EVAL with clause\nmember(X897, .(X897, X898)).\nand substitutionT1578 -> T1593,\nX897 -> T1593,\nX898 -> T1594,\nT1580 -> .(T1593, T1594)" }, { "from": 1204, "to": 1207, "label": "EVAL-BACKTRACK" }, { "from": 1205, "to": 1209, "label": "EVAL with clause\nmember(X905, .(X906, X907)) :- member(X905, X907).\nand substitutionT1578 -> T1601,\nX905 -> T1601,\nX906 -> T1602,\nX907 -> T1603,\nT1580 -> .(T1602, T1603)" }, { "from": 1205, "to": 1210, "label": "EVAL-BACKTRACK" }, { "from": 1206, "to": 1208, "label": "SUCCESS" }, { "from": 1209, "to": 1202, "label": "INSTANCE with matching:\nT1578 -> T1601\nT1580 -> T1603" }, { "from": 1211, "to": 1212, "label": "CASE" }, { "from": 1212, "to": 1213, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 1326, "to": 1327, "label": "SUCCESS" } ], "type": "Graph" } } ---------------------------------------- (212) Complex Obligation (AND) ---------------------------------------- (213) Obligation: Rules: f1203_in(T1578, T1580) -> f1204_in(T1578, T1580) :|: TRUE f1204_out(x, x1) -> f1203_out(x, x1) :|: TRUE f1203_in(x2, x3) -> f1205_in(x2, x3) :|: TRUE f1205_out(x4, x5) -> f1203_out(x4, x5) :|: TRUE f1202_in(x6, x7) -> f1203_in(x6, x7) :|: TRUE f1203_out(x8, x9) -> f1202_out(x8, x9) :|: TRUE f1210_out -> f1205_out(x10, x11) :|: TRUE f1209_out(T1601, T1603) -> f1205_out(T1601, .(T1602, T1603)) :|: TRUE f1205_in(x12, .(x13, x14)) -> f1209_in(x12, x14) :|: TRUE f1205_in(x15, x16) -> f1210_in :|: TRUE f1209_in(x17, x18) -> f1202_in(x17, x18) :|: TRUE f1202_out(x19, x20) -> f1209_out(x19, x20) :|: TRUE f2_in(T2) -> f140_in(T2) :|: TRUE f140_out(x21) -> f2_out(x21) :|: TRUE f152_out(T11) -> f140_out(T11) :|: TRUE f140_in(x22) -> f152_in(x22) :|: TRUE f152_in(x23) -> f153_in(x23) :|: TRUE f153_out(x24) -> f152_out(x24) :|: TRUE f153_in(x25) -> f163_in(x25) :|: TRUE f162_out(x26) -> f153_out(x26) :|: TRUE f153_in(x27) -> f162_in(x27) :|: TRUE f163_out(x28) -> f153_out(x28) :|: TRUE f163_in(x29) -> f288_in :|: TRUE f288_out -> f163_out(x30) :|: TRUE f163_in(T28) -> f287_in(T28) :|: TRUE f287_out(x31) -> f163_out(x31) :|: TRUE f302_out(x32, x33) -> f287_out(x33) :|: TRUE f300_out(x34) -> f302_in(x35, x34) :|: TRUE f287_in(x36) -> f300_in(x36) :|: TRUE f302_in(x37, x38) -> f320_in(x37) :|: TRUE f320_out(x39) -> f321_in(x39, x40) :|: TRUE f321_out(x41, x42) -> f302_out(x41, x42) :|: TRUE f321_in(x43, x44) -> f440_in(x43, x44) :|: TRUE f440_out(x45, x46) -> f321_out(x45, x46) :|: TRUE f442_out(x47, x48) -> f440_out(x47, x48) :|: TRUE f440_in(x49, x50) -> f441_in(x49, x50) :|: TRUE f441_out(x51, x52) -> f440_out(x51, x52) :|: TRUE f440_in(x53, x54) -> f442_in(x53, x54) :|: TRUE f454_out(T110, T109) -> f442_out(T109, T110) :|: TRUE f455_out -> f442_out(x55, x56) :|: TRUE f442_in(x57, x58) -> f455_in :|: TRUE f442_in(x59, x60) -> f454_in(x60, x59) :|: TRUE f454_in(x61, x62) -> f461_in(x61) :|: TRUE f461_out(x63) -> f462_in(x64, x65, x63) :|: TRUE f462_out(x66, x67, x68) -> f454_out(x68, x67) :|: TRUE f462_in(x69, x70, x71) -> f469_in(x69, x70) :|: TRUE f469_out(x72, x73) -> f470_in(x72, x73, x74) :|: TRUE f470_out(x75, x76, x77) -> f462_out(x75, x76, x77) :|: TRUE f470_in(x78, x79, x80) -> f533_in(x78, x79, x80) :|: TRUE f533_out(x81, x82, x83) -> f470_out(x81, x82, x83) :|: TRUE f550_out(x84, x85, x86) -> f533_out(x84, x85, x86) :|: TRUE f551_out(x87, x88, x89) -> f533_out(x87, x88, x89) :|: TRUE f533_in(x90, x91, x92) -> f551_in(x90, x91, x92) :|: TRUE f533_in(x93, x94, x95) -> f550_in(x93, x94, x95) :|: TRUE f557_out -> f551_out(x96, x97, x98) :|: TRUE f556_out(T201, T199, T200) -> f551_out(T199, T200, T201) :|: TRUE f551_in(x99, x100, x101) -> f556_in(x101, x99, x100) :|: TRUE f551_in(x102, x103, x104) -> f557_in :|: TRUE f561_out(x105, x106, x107, x108) -> f556_out(x108, x106, x107) :|: TRUE f556_in(x109, x110, x111) -> f560_in(x109) :|: TRUE f560_out(x112) -> f561_in(x113, x114, x115, x112) :|: TRUE f586_out(x116, x117, x118) -> f587_in(x116, x117, x118, x119) :|: TRUE f587_out(x120, x121, x122, x123) -> f561_out(x120, x121, x122, x123) :|: TRUE f561_in(x124, x125, x126, x127) -> f586_in(x124, x125, x126) :|: TRUE f619_out(x128, x129, x130, x131) -> f587_out(x128, x129, x130, x131) :|: TRUE f587_in(x132, x133, x134, x135) -> f619_in(x132, x133, x134, x135) :|: TRUE f619_in(x136, x137, x138, x139) -> f620_in(x136, x137, x138, x139) :|: TRUE f619_in(x140, x141, x142, x143) -> f621_in(x140, x141, x142, x143) :|: TRUE f621_out(x144, x145, x146, x147) -> f619_out(x144, x145, x146, x147) :|: TRUE f620_out(x148, x149, x150, x151) -> f619_out(x148, x149, x150, x151) :|: TRUE f621_in(x152, x153, x154, x155) -> f632_in :|: TRUE f621_in(T325, T326, T327, T328) -> f631_in(T328, T325, T326, T327) :|: TRUE f632_out -> f621_out(x156, x157, x158, x159) :|: TRUE f631_out(x160, x161, x162, x163) -> f621_out(x161, x162, x163, x160) :|: TRUE f631_in(x164, x165, x166, x167) -> f637_in(x164) :|: TRUE f637_out(x168) -> f638_in(x169, x170, x171, x172, x168) :|: TRUE f638_out(x173, x174, x175, x176, x177) -> f631_out(x177, x174, x175, x176) :|: TRUE f648_out(x178, x179, x180, x181, x182) -> f638_out(x178, x179, x180, x181, x182) :|: TRUE f647_out(x183, x184, x185, x186) -> f648_in(x183, x184, x185, x186, x187) :|: TRUE f638_in(x188, x189, x190, x191, x192) -> f647_in(x188, x189, x190, x191) :|: TRUE f648_in(x193, x194, x195, x196, x197) -> f707_in(x193, x194, x195, x196, x197) :|: TRUE f707_out(x198, x199, x200, x201, x202) -> f648_out(x198, x199, x200, x201, x202) :|: TRUE f708_out(x203, x204, x205, x206, x207) -> f707_out(x203, x204, x205, x206, x207) :|: TRUE f709_out(x208, x209, x210, x211, x212) -> f707_out(x208, x209, x210, x211, x212) :|: TRUE f707_in(x213, x214, x215, x216, x217) -> f709_in(x213, x214, x215, x216, x217) :|: TRUE f707_in(x218, x219, x220, x221, x222) -> f708_in(x218, x219, x220, x221, x222) :|: TRUE f721_out(T495, T491, T492, T493, T494) -> f709_out(T491, T492, T493, T494, T495) :|: TRUE f709_in(x223, x224, x225, x226, x227) -> f721_in(x227, x223, x224, x225, x226) :|: TRUE f722_out -> f709_out(x228, x229, x230, x231, x232) :|: TRUE f709_in(x233, x234, x235, x236, x237) -> f722_in :|: TRUE f721_in(x238, x239, x240, x241, x242) -> f728_in(x238) :|: TRUE f729_out(x243, x244, x245, x246, x247, x248) -> f721_out(x248, x244, x245, x246, x247) :|: TRUE f728_out(x249) -> f729_in(x250, x251, x252, x253, x254, x249) :|: TRUE f729_in(x255, x256, x257, x258, x259, x260) -> f750_in(x255, x256, x257, x258, x259) :|: TRUE f750_out(x261, x262, x263, x264, x265) -> f751_in(x261, x262, x263, x264, x265, x266) :|: TRUE f751_out(x267, x268, x269, x270, x271, x272) -> f729_out(x267, x268, x269, x270, x271, x272) :|: TRUE f751_in(x273, x274, x275, x276, x277, x278) -> f807_in(x273, x274, x275, x276, x277, x278) :|: TRUE f807_out(x279, x280, x281, x282, x283, x284) -> f751_out(x279, x280, x281, x282, x283, x284) :|: TRUE f808_out(x285, x286, x287, x288, x289, x290) -> f807_out(x285, x286, x287, x288, x289, x290) :|: TRUE f807_in(x291, x292, x293, x294, x295, x296) -> f809_in(x291, x292, x293, x294, x295, x296) :|: TRUE f809_out(x297, x298, x299, x300, x301, x302) -> f807_out(x297, x298, x299, x300, x301, x302) :|: TRUE f807_in(x303, x304, x305, x306, x307, x308) -> f808_in(x303, x304, x305, x306, x307, x308) :|: TRUE f829_out -> f809_out(x309, x310, x311, x312, x313, x314) :|: TRUE f809_in(T697, T698, T699, T700, T701, T702) -> f828_in(T702, T697, T698, T699, T700, T701) :|: TRUE f828_out(x315, x316, x317, x318, x319, x320) -> f809_out(x316, x317, x318, x319, x320, x315) :|: TRUE f809_in(x321, x322, x323, x324, x325, x326) -> f829_in :|: TRUE f831_out(x327, x328, x329, x330, x331, x332, x333) -> f828_out(x333, x328, x329, x330, x331, x332) :|: TRUE f830_out(x334) -> f831_in(x335, x336, x337, x338, x339, x340, x334) :|: TRUE f828_in(x341, x342, x343, x344, x345, x346) -> f830_in(x341) :|: TRUE f832_out(x347, x348, x349, x350, x351, x352) -> f848_in(x347, x348, x349, x350, x351, x352, x353) :|: TRUE f848_out(x354, x355, x356, x357, x358, x359, x360) -> f831_out(x354, x355, x356, x357, x358, x359, x360) :|: TRUE f831_in(x361, x362, x363, x364, x365, x366, x367) -> f832_in(x361, x362, x363, x364, x365, x366) :|: TRUE f848_in(x368, x369, x370, x371, x372, x373, x374) -> f888_in(x368, x369, x370, x371, x372, x373, x374) :|: TRUE f888_out(x375, x376, x377, x378, x379, x380, x381) -> f848_out(x375, x376, x377, x378, x379, x380, x381) :|: TRUE f891_out(x382, x383, x384, x385, x386, x387, x388) -> f888_out(x382, x383, x384, x385, x386, x387, x388) :|: TRUE f892_out(x389, x390, x391, x392, x393, x394, x395) -> f888_out(x389, x390, x391, x392, x393, x394, x395) :|: TRUE f888_in(x396, x397, x398, x399, x400, x401, x402) -> f891_in(x396, x397, x398, x399, x400, x401, x402) :|: TRUE f888_in(x403, x404, x405, x406, x407, x408, x409) -> f892_in(x403, x404, x405, x406, x407, x408, x409) :|: TRUE f899_out -> f892_out(x410, x411, x412, x413, x414, x415, x416) :|: TRUE f892_in(T943, T944, T945, T946, T947, T948, T949) -> f898_in(T949, T943, T944, T945, T946, T947, T948) :|: TRUE f892_in(x417, x418, x419, x420, x421, x422, x423) -> f899_in :|: TRUE f898_out(x424, x425, x426, x427, x428, x429, x430) -> f892_out(x425, x426, x427, x428, x429, x430, x424) :|: TRUE f898_in(x431, x432, x433, x434, x435, x436, x437) -> f900_in(x431) :|: TRUE f900_out(x438) -> f901_in(x439, x440, x441, x442, x443, x444, x445, x438) :|: TRUE f901_out(x446, x447, x448, x449, x450, x451, x452, x453) -> f898_out(x453, x447, x448, x449, x450, x451, x452) :|: TRUE f908_out(x454, x455, x456, x457, x458, x459, x460, x461) -> f901_out(x454, x455, x456, x457, x458, x459, x460, x461) :|: TRUE f901_in(x462, x463, x464, x465, x466, x467, x468, x469) -> f907_in(x462, x463, x464, x465, x466, x467, x468) :|: TRUE f907_out(x470, x471, x472, x473, x474, x475, x476) -> f908_in(x470, x471, x472, x473, x474, x475, x476, x477) :|: TRUE f908_in(x478, x479, x480, x481, x482, x483, x484, x485) -> f999_in(x478, x479, x480, x481, x482, x483, x484, x485) :|: TRUE f999_out(x486, x487, x488, x489, x490, x491, x492, x493) -> f908_out(x486, x487, x488, x489, x490, x491, x492, x493) :|: TRUE f1009_out(x494, x495, x496, x497, x498, x499, x500, x501) -> f999_out(x494, x495, x496, x497, x498, x499, x500, x501) :|: TRUE f999_in(x502, x503, x504, x505, x506, x507, x508, x509) -> f1009_in(x502, x503, x504, x505, x506, x507, x508, x509) :|: TRUE f1010_out(x510, x511, x512, x513, x514, x515, x516, x517) -> f999_out(x510, x511, x512, x513, x514, x515, x516, x517) :|: TRUE f999_in(x518, x519, x520, x521, x522, x523, x524, x525) -> f1010_in(x518, x519, x520, x521, x522, x523, x524, x525) :|: TRUE f1024_out(T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235) -> f1010_out(T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1236) :|: TRUE f1010_in(x526, x527, x528, x529, x530, x531, x532, x533) -> f1024_in(x533, x526, x527, x528, x529, x530, x531, x532) :|: TRUE f1010_in(x534, x535, x536, x537, x538, x539, x540, x541) -> f1025_in :|: TRUE f1025_out -> f1010_out(x542, x543, x544, x545, x546, x547, x548, x549) :|: TRUE f1024_in(x550, x551, x552, x553, x554, x555, x556, x557) -> f1026_in(x550, x551, x552, x553, x554, x555, x556, x557, .(x551, .(x552, .(x553, .(x554, .(x555, .(x556, .(x557, [])))))))) :|: TRUE f1026_out(x558, x559, x560, x561, x562, x563, x564, x565, .(x559, .(x560, .(x561, .(x562, .(x563, .(x564, .(x565, [])))))))) -> f1024_out(x558, x559, x560, x561, x562, x563, x564, x565) :|: TRUE f1035_out(x566) -> f1036_in(x567, x568, x569, x570, x571, x572, x573, x574, x575, x566) :|: TRUE f1036_out(x576, x577, x578, x579, x580, x581, x582, x583, x584, x585) -> f1026_out(x585, x577, x578, x579, x580, x581, x582, x583, x584) :|: TRUE f1026_in(x586, x587, x588, x589, x590, x591, x592, x593, x594) -> f1035_in(x586) :|: TRUE f1041_out(x595, x596, x597, x598, x599, x600, x601, x602) -> f1042_in(x595, x603, x604) :|: TRUE f1042_out(x605, x606, x607) -> f1036_out(x605, x608, x609, x610, x611, x612, x613, x614, x606, x607) :|: TRUE f1036_in(x615, x616, x617, x618, x619, x620, x621, x622, x623, x624) -> f1041_in(x615, x616, x617, x618, x619, x620, x621, x622) :|: TRUE f1042_in(x625, x626, x627) -> f1165_in(x625, x626, x627) :|: TRUE f1165_out(x628, x629, x630) -> f1042_out(x628, x629, x630) :|: TRUE f1166_out(x631, x632, x633) -> f1165_out(x631, x632, x633) :|: TRUE f1165_in(x634, x635, x636) -> f1166_in(x634, x635, x636) :|: TRUE f1165_in(x637, x638, x639) -> f1167_in(x637, x638, x639) :|: TRUE f1167_out(x640, x641, x642) -> f1165_out(x640, x641, x642) :|: TRUE f1178_out -> f1167_out(x643, x644, x645) :|: TRUE f1167_in(T1506, T1507, T1508) -> f1177_in(T1508, T1506, T1507) :|: TRUE f1167_in(x646, x647, x648) -> f1178_in :|: TRUE f1177_out(x649, x650, x651) -> f1167_out(x650, x651, x649) :|: TRUE f1180_out(x652, x653, x654, x655) -> f1177_out(x655, x653, x654) :|: TRUE f1179_out(x656) -> f1180_in(x657, x658, x659, x656) :|: TRUE f1177_in(x660, x661, x662) -> f1179_in(x660) :|: TRUE f1183_out(x663, x664, x665) -> f1184_in(x663, x664, x665, x666) :|: TRUE f1184_out(x667, x668, x669, x670) -> f1180_out(x667, x668, x669, x670) :|: TRUE f1180_in(x671, x672, x673, x674) -> f1183_in(x671, x672, x673) :|: TRUE f1185_out(x675, x676, x677) -> f1183_out(x675, x676, x677) :|: TRUE f1183_in(x678, x679, x680) -> f1185_in(x678, x679, x680) :|: TRUE f1185_in(x681, x682, x683) -> f1191_in(x681, x682, x683) :|: TRUE f1191_out(x684, x685, x686) -> f1185_out(x684, x685, x686) :|: TRUE f1185_in(x687, x688, x689) -> f1192_in(x687, x688, x689) :|: TRUE f1192_out(x690, x691, x692) -> f1185_out(x690, x691, x692) :|: TRUE f1191_in(T1547, T1548, T1549) -> f1193_in(T1547, T1548, T1549) :|: TRUE f1193_out(x693, x694, x695) -> f1191_out(x693, x694, x695) :|: TRUE f1193_in(x696, x697, x698) -> f1194_in(x696, x697, x698) :|: TRUE f1194_out(x699, x700, x701) -> f1195_in :|: TRUE f1195_out -> f1193_out(x702, x703, x704) :|: TRUE f1196_out(x705, x706, x707) -> f1194_out(x705, x706, x707) :|: TRUE f1194_in(x708, x709, x710) -> f1196_in(x708, x709, x710) :|: TRUE f1197_out(x711, x712, x713) -> f1196_out(x711, x712, x713) :|: TRUE f1196_in(x714, x715, x716) -> f1198_in(x714, x715, x716) :|: TRUE f1196_in(x717, x718, x719) -> f1197_in(x717, x718, x719) :|: TRUE f1198_out(x720, x721, x722) -> f1196_out(x720, x721, x722) :|: TRUE f1198_in(x723, x724, x725) -> f1202_in(x723, x725) :|: TRUE f1202_out(x726, x727) -> f1198_out(x726, x728, x727) :|: TRUE Start term: f2_in(T2) ---------------------------------------- (214) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (215) TRUE ---------------------------------------- (216) Obligation: Rules: f307_in(.(T58, T59)) -> f314_in(T59) :|: TRUE f314_out(x) -> f307_out(.(x1, x)) :|: TRUE f315_out -> f307_out(T28) :|: TRUE f307_in(x2) -> f315_in :|: TRUE f305_out(x3) -> f300_out(x3) :|: TRUE f300_in(x4) -> f305_in(x4) :|: TRUE f307_out(x5) -> f305_out(x5) :|: TRUE f305_in(x6) -> f306_in(x6) :|: TRUE f305_in(x7) -> f307_in(x7) :|: TRUE f306_out(x8) -> f305_out(x8) :|: TRUE f300_out(x9) -> f314_out(x9) :|: TRUE f314_in(x10) -> f300_in(x10) :|: TRUE f2_in(T2) -> f140_in(T2) :|: TRUE f140_out(x11) -> f2_out(x11) :|: TRUE f152_out(T11) -> f140_out(T11) :|: TRUE f140_in(x12) -> f152_in(x12) :|: TRUE f152_in(x13) -> f153_in(x13) :|: TRUE f153_out(x14) -> f152_out(x14) :|: TRUE f153_in(x15) -> f163_in(x15) :|: TRUE f162_out(x16) -> f153_out(x16) :|: TRUE f153_in(x17) -> f162_in(x17) :|: TRUE f163_out(x18) -> f153_out(x18) :|: TRUE f163_in(x19) -> f288_in :|: TRUE f288_out -> f163_out(x20) :|: TRUE f163_in(x21) -> f287_in(x21) :|: TRUE f287_out(x22) -> f163_out(x22) :|: TRUE f302_out(x23, x24) -> f287_out(x24) :|: TRUE f300_out(x25) -> f302_in(x26, x25) :|: TRUE f287_in(x27) -> f300_in(x27) :|: TRUE f302_in(x28, x29) -> f320_in(x28) :|: TRUE f320_out(x30) -> f321_in(x30, x31) :|: TRUE f321_out(x32, x33) -> f302_out(x32, x33) :|: TRUE f321_in(x34, x35) -> f440_in(x34, x35) :|: TRUE f440_out(x36, x37) -> f321_out(x36, x37) :|: TRUE f442_out(x38, x39) -> f440_out(x38, x39) :|: TRUE f440_in(x40, x41) -> f441_in(x40, x41) :|: TRUE f441_out(x42, x43) -> f440_out(x42, x43) :|: TRUE f440_in(x44, x45) -> f442_in(x44, x45) :|: TRUE f454_out(T110, T109) -> f442_out(T109, T110) :|: TRUE f455_out -> f442_out(x46, x47) :|: TRUE f442_in(x48, x49) -> f455_in :|: TRUE f442_in(x50, x51) -> f454_in(x51, x50) :|: TRUE f454_in(x52, x53) -> f461_in(x52) :|: TRUE f461_out(x54) -> f462_in(x55, x56, x54) :|: TRUE f462_out(x57, x58, x59) -> f454_out(x59, x58) :|: TRUE f462_in(x60, x61, x62) -> f469_in(x60, x61) :|: TRUE f469_out(x63, x64) -> f470_in(x63, x64, x65) :|: TRUE f470_out(x66, x67, x68) -> f462_out(x66, x67, x68) :|: TRUE f470_in(x69, x70, x71) -> f533_in(x69, x70, x71) :|: TRUE f533_out(x72, x73, x74) -> f470_out(x72, x73, x74) :|: TRUE f550_out(x75, x76, x77) -> f533_out(x75, x76, x77) :|: TRUE f551_out(x78, x79, x80) -> f533_out(x78, x79, x80) :|: TRUE f533_in(x81, x82, x83) -> f551_in(x81, x82, x83) :|: TRUE f533_in(x84, x85, x86) -> f550_in(x84, x85, x86) :|: TRUE f557_out -> f551_out(x87, x88, x89) :|: TRUE f556_out(T201, T199, T200) -> f551_out(T199, T200, T201) :|: TRUE f551_in(x90, x91, x92) -> f556_in(x92, x90, x91) :|: TRUE f551_in(x93, x94, x95) -> f557_in :|: TRUE f561_out(x96, x97, x98, x99) -> f556_out(x99, x97, x98) :|: TRUE f556_in(x100, x101, x102) -> f560_in(x100) :|: TRUE f560_out(x103) -> f561_in(x104, x105, x106, x103) :|: TRUE f586_out(x107, x108, x109) -> f587_in(x107, x108, x109, x110) :|: TRUE f587_out(x111, x112, x113, x114) -> f561_out(x111, x112, x113, x114) :|: TRUE f561_in(x115, x116, x117, x118) -> f586_in(x115, x116, x117) :|: TRUE f619_out(x119, x120, x121, x122) -> f587_out(x119, x120, x121, x122) :|: TRUE f587_in(x123, x124, x125, x126) -> f619_in(x123, x124, x125, x126) :|: TRUE f619_in(x127, x128, x129, x130) -> f620_in(x127, x128, x129, x130) :|: TRUE f619_in(x131, x132, x133, x134) -> f621_in(x131, x132, x133, x134) :|: TRUE f621_out(x135, x136, x137, x138) -> f619_out(x135, x136, x137, x138) :|: TRUE f620_out(x139, x140, x141, x142) -> f619_out(x139, x140, x141, x142) :|: TRUE f621_in(x143, x144, x145, x146) -> f632_in :|: TRUE f621_in(T325, T326, T327, T328) -> f631_in(T328, T325, T326, T327) :|: TRUE f632_out -> f621_out(x147, x148, x149, x150) :|: TRUE f631_out(x151, x152, x153, x154) -> f621_out(x152, x153, x154, x151) :|: TRUE f631_in(x155, x156, x157, x158) -> f637_in(x155) :|: TRUE f637_out(x159) -> f638_in(x160, x161, x162, x163, x159) :|: TRUE f638_out(x164, x165, x166, x167, x168) -> f631_out(x168, x165, x166, x167) :|: TRUE f648_out(x169, x170, x171, x172, x173) -> f638_out(x169, x170, x171, x172, x173) :|: TRUE f647_out(x174, x175, x176, x177) -> f648_in(x174, x175, x176, x177, x178) :|: TRUE f638_in(x179, x180, x181, x182, x183) -> f647_in(x179, x180, x181, x182) :|: TRUE f648_in(x184, x185, x186, x187, x188) -> f707_in(x184, x185, x186, x187, x188) :|: TRUE f707_out(x189, x190, x191, x192, x193) -> f648_out(x189, x190, x191, x192, x193) :|: TRUE f708_out(x194, x195, x196, x197, x198) -> f707_out(x194, x195, x196, x197, x198) :|: TRUE f709_out(x199, x200, x201, x202, x203) -> f707_out(x199, x200, x201, x202, x203) :|: TRUE f707_in(x204, x205, x206, x207, x208) -> f709_in(x204, x205, x206, x207, x208) :|: TRUE f707_in(x209, x210, x211, x212, x213) -> f708_in(x209, x210, x211, x212, x213) :|: TRUE f721_out(T495, T491, T492, T493, T494) -> f709_out(T491, T492, T493, T494, T495) :|: TRUE f709_in(x214, x215, x216, x217, x218) -> f721_in(x218, x214, x215, x216, x217) :|: TRUE f722_out -> f709_out(x219, x220, x221, x222, x223) :|: TRUE f709_in(x224, x225, x226, x227, x228) -> f722_in :|: TRUE f721_in(x229, x230, x231, x232, x233) -> f728_in(x229) :|: TRUE f729_out(x234, x235, x236, x237, x238, x239) -> f721_out(x239, x235, x236, x237, x238) :|: TRUE f728_out(x240) -> f729_in(x241, x242, x243, x244, x245, x240) :|: TRUE f729_in(x246, x247, x248, x249, x250, x251) -> f750_in(x246, x247, x248, x249, x250) :|: TRUE f750_out(x252, x253, x254, x255, x256) -> f751_in(x252, x253, x254, x255, x256, x257) :|: TRUE f751_out(x258, x259, x260, x261, x262, x263) -> f729_out(x258, x259, x260, x261, x262, x263) :|: TRUE f751_in(x264, x265, x266, x267, x268, x269) -> f807_in(x264, x265, x266, x267, x268, x269) :|: TRUE f807_out(x270, x271, x272, x273, x274, x275) -> f751_out(x270, x271, x272, x273, x274, x275) :|: TRUE f808_out(x276, x277, x278, x279, x280, x281) -> f807_out(x276, x277, x278, x279, x280, x281) :|: TRUE f807_in(x282, x283, x284, x285, x286, x287) -> f809_in(x282, x283, x284, x285, x286, x287) :|: TRUE f809_out(x288, x289, x290, x291, x292, x293) -> f807_out(x288, x289, x290, x291, x292, x293) :|: TRUE f807_in(x294, x295, x296, x297, x298, x299) -> f808_in(x294, x295, x296, x297, x298, x299) :|: TRUE f829_out -> f809_out(x300, x301, x302, x303, x304, x305) :|: TRUE f809_in(T697, T698, T699, T700, T701, T702) -> f828_in(T702, T697, T698, T699, T700, T701) :|: TRUE f828_out(x306, x307, x308, x309, x310, x311) -> f809_out(x307, x308, x309, x310, x311, x306) :|: TRUE f809_in(x312, x313, x314, x315, x316, x317) -> f829_in :|: TRUE f831_out(x318, x319, x320, x321, x322, x323, x324) -> f828_out(x324, x319, x320, x321, x322, x323) :|: TRUE f830_out(x325) -> f831_in(x326, x327, x328, x329, x330, x331, x325) :|: TRUE f828_in(x332, x333, x334, x335, x336, x337) -> f830_in(x332) :|: TRUE f832_out(x338, x339, x340, x341, x342, x343) -> f848_in(x338, x339, x340, x341, x342, x343, x344) :|: TRUE f848_out(x345, x346, x347, x348, x349, x350, x351) -> f831_out(x345, x346, x347, x348, x349, x350, x351) :|: TRUE f831_in(x352, x353, x354, x355, x356, x357, x358) -> f832_in(x352, x353, x354, x355, x356, x357) :|: TRUE f848_in(x359, x360, x361, x362, x363, x364, x365) -> f888_in(x359, x360, x361, x362, x363, x364, x365) :|: TRUE f888_out(x366, x367, x368, x369, x370, x371, x372) -> f848_out(x366, x367, x368, x369, x370, x371, x372) :|: TRUE f891_out(x373, x374, x375, x376, x377, x378, x379) -> f888_out(x373, x374, x375, x376, x377, x378, x379) :|: TRUE f892_out(x380, x381, x382, x383, x384, x385, x386) -> f888_out(x380, x381, x382, x383, x384, x385, x386) :|: TRUE f888_in(x387, x388, x389, x390, x391, x392, x393) -> f891_in(x387, x388, x389, x390, x391, x392, x393) :|: TRUE f888_in(x394, x395, x396, x397, x398, x399, x400) -> f892_in(x394, x395, x396, x397, x398, x399, x400) :|: TRUE f899_out -> f892_out(x401, x402, x403, x404, x405, x406, x407) :|: TRUE f892_in(T943, T944, T945, T946, T947, T948, T949) -> f898_in(T949, T943, T944, T945, T946, T947, T948) :|: TRUE f892_in(x408, x409, x410, x411, x412, x413, x414) -> f899_in :|: TRUE f898_out(x415, x416, x417, x418, x419, x420, x421) -> f892_out(x416, x417, x418, x419, x420, x421, x415) :|: TRUE f898_in(x422, x423, x424, x425, x426, x427, x428) -> f900_in(x422) :|: TRUE f900_out(x429) -> f901_in(x430, x431, x432, x433, x434, x435, x436, x429) :|: TRUE f901_out(x437, x438, x439, x440, x441, x442, x443, x444) -> f898_out(x444, x438, x439, x440, x441, x442, x443) :|: TRUE f908_out(x445, x446, x447, x448, x449, x450, x451, x452) -> f901_out(x445, x446, x447, x448, x449, x450, x451, x452) :|: TRUE f901_in(x453, x454, x455, x456, x457, x458, x459, x460) -> f907_in(x453, x454, x455, x456, x457, x458, x459) :|: TRUE f907_out(x461, x462, x463, x464, x465, x466, x467) -> f908_in(x461, x462, x463, x464, x465, x466, x467, x468) :|: TRUE f908_in(x469, x470, x471, x472, x473, x474, x475, x476) -> f999_in(x469, x470, x471, x472, x473, x474, x475, x476) :|: TRUE f999_out(x477, x478, x479, x480, x481, x482, x483, x484) -> f908_out(x477, x478, x479, x480, x481, x482, x483, x484) :|: TRUE f1009_out(x485, x486, x487, x488, x489, x490, x491, x492) -> f999_out(x485, x486, x487, x488, x489, x490, x491, x492) :|: TRUE f999_in(x493, x494, x495, x496, x497, x498, x499, x500) -> f1009_in(x493, x494, x495, x496, x497, x498, x499, x500) :|: TRUE f1010_out(x501, x502, x503, x504, x505, x506, x507, x508) -> f999_out(x501, x502, x503, x504, x505, x506, x507, x508) :|: TRUE f999_in(x509, x510, x511, x512, x513, x514, x515, x516) -> f1010_in(x509, x510, x511, x512, x513, x514, x515, x516) :|: TRUE f1024_out(T1236, T1229, T1230, T1231, T1232, T1233, T1234, T1235) -> f1010_out(T1229, T1230, T1231, T1232, T1233, T1234, T1235, T1236) :|: TRUE f1010_in(x517, x518, x519, x520, x521, x522, x523, x524) -> f1024_in(x524, x517, x518, x519, x520, x521, x522, x523) :|: TRUE f1010_in(x525, x526, x527, x528, x529, x530, x531, x532) -> f1025_in :|: TRUE f1025_out -> f1010_out(x533, x534, x535, x536, x537, x538, x539, x540) :|: TRUE f1024_in(x541, x542, x543, x544, x545, x546, x547, x548) -> f1026_in(x541, x542, x543, x544, x545, x546, x547, x548, .(x542, .(x543, .(x544, .(x545, .(x546, .(x547, .(x548, [])))))))) :|: TRUE f1026_out(x549, x550, x551, x552, x553, x554, x555, x556, .(x550, .(x551, .(x552, .(x553, .(x554, .(x555, .(x556, [])))))))) -> f1024_out(x549, x550, x551, x552, x553, x554, x555, x556) :|: TRUE f1035_out(x557) -> f1036_in(x558, x559, x560, x561, x562, x563, x564, x565, x566, x557) :|: TRUE f1036_out(x567, x568, x569, x570, x571, x572, x573, x574, x575, x576) -> f1026_out(x576, x568, x569, x570, x571, x572, x573, x574, x575) :|: TRUE f1026_in(x577, x578, x579, x580, x581, x582, x583, x584, x585) -> f1035_in(x577) :|: TRUE f1041_out(x586, x587, x588, x589, x590, x591, x592, x593) -> f1042_in(x586, x594, x595) :|: TRUE f1042_out(x596, x597, x598) -> f1036_out(x596, x599, x600, x601, x602, x603, x604, x605, x597, x598) :|: TRUE f1036_in(x606, x607, x608, x609, x610, x611, x612, x613, x614, x615) -> f1041_in(x606, x607, x608, x609, x610, x611, x612, x613) :|: TRUE f1042_in(x616, x617, x618) -> f1165_in(x616, x617, x618) :|: TRUE f1165_out(x619, x620, x621) -> f1042_out(x619, x620, x621) :|: TRUE f1166_out(x622, x623, x624) -> f1165_out(x622, x623, x624) :|: TRUE f1165_in(x625, x626, x627) -> f1166_in(x625, x626, x627) :|: TRUE f1165_in(x628, x629, x630) -> f1167_in(x628, x629, x630) :|: TRUE f1167_out(x631, x632, x633) -> f1165_out(x631, x632, x633) :|: TRUE f1178_out -> f1167_out(x634, x635, x636) :|: TRUE f1167_in(T1506, T1507, T1508) -> f1177_in(T1508, T1506, T1507) :|: TRUE f1167_in(x637, x638, x639) -> f1178_in :|: TRUE f1177_out(x640, x641, x642) -> f1167_out(x641, x642, x640) :|: TRUE f1180_out(x643, x644, x645, x646) -> f1177_out(x646, x644, x645) :|: TRUE f1179_out(x647) -> f1180_in(x648, x649, x650, x647) :|: TRUE f1177_in(x651, x652, x653) -> f1179_in(x651) :|: TRUE f1179_in(x654) -> f300_in(x654) :|: TRUE f300_out(x655) -> f1179_out(x655) :|: TRUE f300_out(x656) -> f900_out(x656) :|: TRUE f900_in(x657) -> f300_in(x657) :|: TRUE f728_in(x658) -> f300_in(x658) :|: TRUE f300_out(x659) -> f728_out(x659) :|: TRUE f300_out(x660) -> f830_out(x660) :|: TRUE f830_in(x661) -> f300_in(x661) :|: TRUE f1035_in(x662) -> f300_in(x662) :|: TRUE f300_out(x663) -> f1035_out(x663) :|: TRUE f560_in(x664) -> f300_in(x664) :|: TRUE f300_out(x665) -> f560_out(x665) :|: TRUE f300_out(x666) -> f637_out(x666) :|: TRUE f637_in(x667) -> f300_in(x667) :|: TRUE f461_in(x668) -> f300_in(x668) :|: TRUE f300_out(x669) -> f461_out(x669) :|: TRUE Start term: f2_in(T2) ---------------------------------------- (217) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f307_in(.(T58, T59)) -> f314_in(T59) :|: TRUE f300_in(x4) -> f305_in(x4) :|: TRUE f305_in(x7) -> f307_in(x7) :|: TRUE f314_in(x10) -> f300_in(x10) :|: TRUE ---------------------------------------- (218) Obligation: Rules: f307_in(.(T58, T59)) -> f314_in(T59) :|: TRUE f300_in(x4) -> f305_in(x4) :|: TRUE f305_in(x7) -> f307_in(x7) :|: TRUE f314_in(x10) -> f300_in(x10) :|: TRUE ---------------------------------------- (219) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (220) Obligation: Rules: f300_in(.(T58:0, T59:0)) -> f300_in(T59:0) :|: TRUE ---------------------------------------- (221) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (222) Obligation: Rules: f300_in(.(T58:0, T59:0)) -> f300_in(T59:0) :|: TRUE ---------------------------------------- (223) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f300_in(.(T58:0, T59:0)) -> f300_in(T59:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (224) Obligation: Termination digraph: Nodes: (1) f300_in(.(T58:0, T59:0)) -> f300_in(T59:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (225) IntTRSUnneededArgumentFilterProof (EQUIVALENT) Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements: .(x1, x2) -> .(x2) ---------------------------------------- (226) Obligation: Rules: f300_in(.(T59:0)) -> f300_in(T59:0) :|: TRUE ---------------------------------------- (227) TempFilterProof (SOUND) Used the following sort dictionary for filtering: f300_in(VARIABLE) .(VARIABLE) Removed predefined arithmetic. ---------------------------------------- (228) Obligation: Rules: f300_in(.(T59:0)) -> f300_in(T59:0) ---------------------------------------- (229) IRSwTToQDPProof (SOUND) Removed the integers and created a QDP-Problem. ---------------------------------------- (230) Obligation: Q DP problem: The TRS P consists of the following rules: f300_in(.(T59:0)) -> f300_in(T59:0) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (231) 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: *f300_in(.(T59:0)) -> f300_in(T59:0) The graph contains the following edges 1 > 1 ---------------------------------------- (232) YES ---------------------------------------- (233) Obligation: Rules: f1213_out -> f1212_out :|: TRUE f1212_in -> f1213_in :|: TRUE f1192_in(T1614, T1615, T1616) -> f1326_in :|: TRUE f1326_out -> f1192_out(x, x1, x2) :|: TRUE f1200_out -> f1197_out(T1547, T1548, T1549) :|: TRUE f1199_out -> f1197_out(T1566, T1566, T1567) :|: TRUE f1197_in(x3, x4, x5) -> f1200_in :|: TRUE f1197_in(x6, x6, x7) -> f1199_in :|: TRUE f1193_in(x8, x9, x10) -> f1194_in(x8, x9, x10) :|: TRUE f1194_out(x11, x12, x13) -> f1195_in :|: TRUE f1195_out -> f1193_out(x14, x15, x16) :|: TRUE f1209_in(T1601, T1603) -> f1202_in(T1601, T1603) :|: TRUE f1202_out(x17, x18) -> f1209_out(x17, x18) :|: TRUE f307_in(.(T58, T59)) -> f314_in(T59) :|: TRUE f314_out(x19) -> f307_out(.(x20, x19)) :|: TRUE f315_out -> f307_out(T28) :|: TRUE f307_in(x21) -> f315_in :|: TRUE f1042_in(T1248, T1243, T1236) -> f1165_in(T1248, T1243, T1236) :|: TRUE f1165_out(x22, x23, x24) -> f1042_out(x22, x23, x24) :|: TRUE f1202_in(T1578, T1580) -> f1203_in(T1578, T1580) :|: TRUE f1203_out(x25, x26) -> f1202_out(x25, x26) :|: TRUE f1206_out -> f1204_out(T1593, .(T1593, T1594)) :|: TRUE f1207_out -> f1204_out(x27, x28) :|: TRUE f1204_in(x29, x30) -> f1207_in :|: TRUE f1204_in(x31, .(x31, x32)) -> f1206_in :|: TRUE f1179_in(T1508) -> f300_in(T1508) :|: TRUE f300_out(x33) -> f1179_out(x33) :|: TRUE f308_out -> f306_out(.(T49, T50)) :|: TRUE f306_in(x34) -> f311_in :|: TRUE f311_out -> f306_out(x35) :|: TRUE f306_in(.(x36, x37)) -> f308_in :|: TRUE f1191_in(x38, x39, x40) -> f1193_in(x38, x39, x40) :|: TRUE f1193_out(x41, x42, x43) -> f1191_out(x41, x42, x43) :|: TRUE f1183_out(x44, x45, x46) -> f1184_in(x44, x45, x46, x47) :|: TRUE f1184_out(x48, x49, x50, x51) -> f1180_out(x48, x49, x50, x51) :|: TRUE f1180_in(x52, x53, x54, x55) -> f1183_in(x52, x53, x54) :|: TRUE f1184_in(x56, x57, x58, x59) -> f1042_in(x56, .(x57, x58), x59) :|: TRUE f1042_out(x60, .(x61, x62), x63) -> f1184_out(x60, x61, x62, x63) :|: TRUE f1178_out -> f1167_out(x64, x65, x66) :|: TRUE f1167_in(x67, x68, x69) -> f1177_in(x69, x67, x68) :|: TRUE f1167_in(x70, x71, x72) -> f1178_in :|: TRUE f1177_out(x73, x74, x75) -> f1167_out(x74, x75, x73) :|: TRUE f305_out(x76) -> f300_out(x76) :|: TRUE f300_in(x77) -> f305_in(x77) :|: TRUE f1197_out(x78, x79, x80) -> f1196_out(x78, x79, x80) :|: TRUE f1196_in(x81, x82, x83) -> f1198_in(x81, x82, x83) :|: TRUE f1196_in(x84, x85, x86) -> f1197_in(x84, x85, x86) :|: TRUE f1198_out(x87, x88, x89) -> f1196_out(x87, x88, x89) :|: TRUE f1206_in -> f1206_out :|: TRUE f1212_out -> f1211_out :|: TRUE f1211_in -> f1212_in :|: TRUE f1210_out -> f1205_out(x90, x91) :|: TRUE f1209_out(x92, x93) -> f1205_out(x92, .(x94, x93)) :|: TRUE f1205_in(x95, .(x96, x97)) -> f1209_in(x95, x97) :|: TRUE f1205_in(x98, x99) -> f1210_in :|: TRUE f300_out(x100) -> f314_out(x100) :|: TRUE f314_in(x101) -> f300_in(x101) :|: TRUE f1199_in -> f1199_out :|: TRUE f1185_out(T1515, T1506, T1507) -> f1183_out(T1515, T1506, T1507) :|: TRUE f1183_in(x102, x103, x104) -> f1185_in(x102, x103, x104) :|: TRUE f1185_in(x105, x106, x107) -> f1191_in(x105, x106, x107) :|: TRUE f1191_out(x108, x109, x110) -> f1185_out(x108, x109, x110) :|: TRUE f1185_in(x111, x112, x113) -> f1192_in(x111, x112, x113) :|: TRUE f1192_out(x114, x115, x116) -> f1185_out(x114, x115, x116) :|: TRUE f1195_in -> f1211_in :|: TRUE f1211_out -> f1195_out :|: TRUE f1203_in(x117, x118) -> f1204_in(x117, x118) :|: TRUE f1204_out(x119, x120) -> f1203_out(x119, x120) :|: TRUE f1203_in(x121, x122) -> f1205_in(x121, x122) :|: TRUE f1205_out(x123, x124) -> f1203_out(x123, x124) :|: TRUE f1198_in(x125, x126, x127) -> f1202_in(x125, x127) :|: TRUE f1202_out(x128, x129) -> f1198_out(x128, x130, x129) :|: TRUE f308_in -> f308_out :|: TRUE f307_out(x131) -> f305_out(x131) :|: TRUE f305_in(x132) -> f306_in(x132) :|: TRUE f305_in(x133) -> f307_in(x133) :|: TRUE f306_out(x134) -> f305_out(x134) :|: TRUE f1180_out(x135, x136, x137, x138) -> f1177_out(x138, x136, x137) :|: TRUE f1179_out(x139) -> f1180_in(x140, x141, x142, x139) :|: TRUE f1177_in(x143, x144, x145) -> f1179_in(x143) :|: TRUE f1326_in -> f1326_out :|: TRUE f1166_out(x146, x147, x148) -> f1165_out(x146, x147, x148) :|: TRUE f1165_in(x149, x150, x151) -> f1166_in(x149, x150, x151) :|: TRUE f1165_in(x152, x153, x154) -> f1167_in(x152, x153, x154) :|: TRUE f1167_out(x155, x156, x157) -> f1165_out(x155, x156, x157) :|: TRUE f1196_out(x158, x159, x160) -> f1194_out(x158, x159, x160) :|: TRUE f1194_in(x161, x162, x163) -> f1196_in(x161, x162, x163) :|: TRUE f2_in(T2) -> f140_in(T2) :|: TRUE f140_out(x164) -> f2_out(x164) :|: TRUE f152_out(T11) -> f140_out(T11) :|: TRUE f140_in(x165) -> f152_in(x165) :|: TRUE f152_in(x166) -> f153_in(x166) :|: TRUE f153_out(x167) -> f152_out(x167) :|: TRUE f153_in(x168) -> f163_in(x168) :|: TRUE f162_out(x169) -> f153_out(x169) :|: TRUE f153_in(x170) -> f162_in(x170) :|: TRUE f163_out(x171) -> f153_out(x171) :|: TRUE f163_in(x172) -> f288_in :|: TRUE f288_out -> f163_out(x173) :|: TRUE f163_in(x174) -> f287_in(x174) :|: TRUE f287_out(x175) -> f163_out(x175) :|: TRUE f302_out(x176, x177) -> f287_out(x177) :|: TRUE f300_out(x178) -> f302_in(x179, x178) :|: TRUE f287_in(x180) -> f300_in(x180) :|: TRUE f302_in(x181, x182) -> f320_in(x181) :|: TRUE f320_out(x183) -> f321_in(x183, x184) :|: TRUE f321_out(x185, x186) -> f302_out(x185, x186) :|: TRUE f321_in(x187, x188) -> f440_in(x187, x188) :|: TRUE f440_out(x189, x190) -> f321_out(x189, x190) :|: TRUE f442_out(x191, x192) -> f440_out(x191, x192) :|: TRUE f440_in(x193, x194) -> f441_in(x193, x194) :|: TRUE f441_out(x195, x196) -> f440_out(x195, x196) :|: TRUE f440_in(x197, x198) -> f442_in(x197, x198) :|: TRUE f454_out(T110, T109) -> f442_out(T109, T110) :|: TRUE f455_out -> f442_out(x199, x200) :|: TRUE f442_in(x201, x202) -> f455_in :|: TRUE f442_in(x203, x204) -> f454_in(x204, x203) :|: TRUE f454_in(x205, x206) -> f461_in(x205) :|: TRUE f461_out(x207) -> f462_in(x208, x209, x207) :|: TRUE f462_out(x210, x211, x212) -> f454_out(x212, x211) :|: TRUE f462_in(x213, x214, x215) -> f469_in(x213, x214) :|: TRUE f469_out(x216, x217) -> f470_in(x216, x217, x218) :|: TRUE f470_out(x219, x220, x221) -> f462_out(x219, x220, x221) :|: TRUE f470_in(x222, x223, x224) -> f533_in(x222, x223, x224) :|: TRUE f533_out(x225, x226, x227) -> f470_out(x225, x226, x227) :|: TRUE f550_out(x228, x229, x230) -> f533_out(x228, x229, x230) :|: TRUE f551_out(x231, x232, x233) -> f533_out(x231, x232, x233) :|: TRUE f533_in(x234, x235, x236) -> f551_in(x234, x235, x236) :|: TRUE f533_in(x237, x238, x239) -> f550_in(x237, x238, x239) :|: TRUE f557_out -> f551_out(x240, x241, x242) :|: TRUE f556_out(T201, T199, T200) -> f551_out(T199, T200, T201) :|: TRUE f551_in(x243, x244, x245) -> f556_in(x245, x243, x244) :|: TRUE f551_in(x246, x247, x248) -> f557_in :|: TRUE f561_out(x249, x250, x251, x252) -> f556_out(x252, x250, x251) :|: TRUE f556_in(x253, x254, x255) -> f560_in(x253) :|: TRUE f560_out(x256) -> f561_in(x257, x258, x259, x256) :|: TRUE f586_out(x260, x261, x262) -> f587_in(x260, x261, x262, x263) :|: TRUE f587_out(x264, x265, x266, x267) -> f561_out(x264, x265, x266, x267) :|: TRUE f561_in(x268, x269, x270, x271) -> f586_in(x268, x269, x270) :|: TRUE f619_out(x272, x273, x274, x275) -> f587_out(x272, x273, x274, x275) :|: TRUE f587_in(x276, x277, x278, x279) -> f619_in(x276, x277, x278, x279) :|: TRUE f619_in(x280, x281, x282, x283) -> f620_in(x280, x281, x282, x283) :|: TRUE f619_in(x284, x285, x286, x287) -> f621_in(x284, x285, x286, x287) :|: TRUE f621_out(x288, x289, x290, x291) -> f619_out(x288, x289, x290, x291) :|: TRUE f620_out(x292, x293, x294, x295) -> f619_out(x292, x293, x294, x295) :|: TRUE f621_in(x296, x297, x298, x299) -> f632_in :|: TRUE f621_in(T325, T326, T327, T328) -> f631_in(T328, T325, T326, T327) :|: TRUE f632_out -> f621_out(x300, x301, x302, x303) :|: TRUE f631_out(x304, x305, x306, x307) -> f621_out(x305, x306, x307, x304) :|: TRUE f631_in(x308, x309, x310, x311) -> f637_in(x308) :|: TRUE f637_out(x312) -> f638_in(x313, x314, x315, x316, x312) :|: TRUE f638_out(x317, x318, x319, x320, x321) -> f631_out(x321, x318, x319, x320) :|: TRUE f648_out(x322, x323, x324, x325, x326) -> f638_out(x322, x323, x324, x325, x326) :|: TRUE f647_out(x327, x328, x329, x330) -> f648_in(x327, x328, x329, x330, x331) :|: TRUE f638_in(x332, x333, x334, x335, x336) -> f647_in(x332, x333, x334, x335) :|: TRUE f648_in(x337, x338, x339, x340, x341) -> f707_in(x337, x338, x339, x340, x341) :|: TRUE f707_out(x342, x343, x344, x345, x346) -> f648_out(x342, x343, x344, x345, x346) :|: TRUE f708_out(x347, x348, x349, x350, x351) -> f707_out(x347, x348, x349, x350, x351) :|: TRUE f709_out(x352, x353, x354, x355, x356) -> f707_out(x352, x353, x354, x355, x356) :|: TRUE f707_in(x357, x358, x359, x360, x361) -> f709_in(x357, x358, x359, x360, x361) :|: TRUE f707_in(x362, x363, x364, x365, x366) -> f708_in(x362, x363, x364, x365, x366) :|: TRUE f721_out(T495, T491, T492, T493, T494) -> f709_out(T491, T492, T493, T494, T495) :|: TRUE f709_in(x367, x368, x369, x370, x371) -> f721_in(x371, x367, x368, x369, x370) :|: TRUE f722_out -> f709_out(x372, x373, x374, x375, x376) :|: TRUE f709_in(x377, x378, x379, x380, x381) -> f722_in :|: TRUE f721_in(x382, x383, x384, x385, x386) -> f728_in(x382) :|: TRUE f729_out(x387, x388, x389, x390, x391, x392) -> f721_out(x392, x388, x389, x390, x391) :|: TRUE f728_out(x393) -> f729_in(x394, x395, x396, x397, x398, x393) :|: TRUE f729_in(x399, x400, x401, x402, x403, x404) -> f750_in(x399, x400, x401, x402, x403) :|: TRUE f750_out(x405, x406, x407, x408, x409) -> f751_in(x405, x406, x407, x408, x409, x410) :|: TRUE f751_out(x411, x412, x413, x414, x415, x416) -> f729_out(x411, x412, x413, x414, x415, x416) :|: TRUE f751_in(x417, x418, x419, x420, x421, x422) -> f807_in(x417, x418, x419, x420, x421, x422) :|: TRUE f807_out(x423, x424, x425, x426, x427, x428) -> f751_out(x423, x424, x425, x426, x427, x428) :|: TRUE f808_out(x429, x430, x431, x432, x433, x434) -> f807_out(x429, x430, x431, x432, x433, x434) :|: TRUE f807_in(x435, x436, x437, x438, x439, x440) -> f809_in(x435, x436, x437, x438, x439, x440) :|: TRUE f809_out(x441, x442, x443, x444, x445, x446) -> f807_out(x441, x442, x443, x444, x445, x446) :|: TRUE f807_in(x447, x448, x449, x450, x451, x452) -> f808_in(x447, x448, x449, x450, x451, x452) :|: TRUE f829_out -> f809_out(x453, x454, x455, x456, x457, x458) :|: TRUE f809_in(T697, T698, T699, T700, T701, T702) -> f828_in(T702, T697, T698, T699, T700, T701) :|: TRUE f828_out(x459, x460, x461, x462, x463, x464) -> f809_out(x460, x461, x462, x463, x464, x459) :|: TRUE f809_in(x465, x466, x467, x468, x469, x470) -> f829_in :|: TRUE f831_out(x471, x472, x473, x474, x475, x476, x477) -> f828_out(x477, x472, x473, x474, x475, x476) :|: TRUE f830_out(x478) -> f831_in(x479, x480, x481, x482, x483, x484, x478) :|: TRUE f828_in(x485, x486, x487, x488, x489, x490) -> f830_in(x485) :|: TRUE f832_out(x491, x492, x493, x494, x495, x496) -> f848_in(x491, x492, x493, x494, x495, x496, x497) :|: TRUE f848_out(x498, x499, x500, x501, x502, x503, x504) -> f831_out(x498, x499, x500, x501, x502, x503, x504) :|: TRUE f831_in(x505, x506, x507, x508, x509, x510, x511) -> f832_in(x505, x506, x507, x508, x509, x510) :|: TRUE f848_in(x512, x513, x514, x515, x516, x517, x518) -> f888_in(x512, x513, x514, x515, x516, x517, x518) :|: TRUE f888_out(x519, x520, x521, x522, x523, x524, x525) -> f848_out(x519, x520, x521, x522, x523, x524, x525) :|: TRUE f891_out(x526, x527, x528, x529, x530, x531, x532) -> f888_out(x526, x527, x528, x529, x530, x531, x532) :|: TRUE f892_out(x533, x534, x535, x536, x537, x538, x539) -> f888_out(x533, x534, x535, x536, x537, x538, x539) :|: TRUE f888_in(x540, x541, x542, x543, x544, x545, x546) -> f891_in(x540, x541, x542, x543, x544, x545, x546) :|: TRUE f888_in(x547, x548, x549, x550, x551, x552, x553) -> f892_in(x547, x548, x549, x550, x551, x552, x553) :|: TRUE f899_out -> f892_out(x554, x555, x556, x557, x558, x559, x560) :|: TRUE f892_in(T943, T944, T945, T946, T947, T948, T949) -> f898_in(T949, T943, T944, T945, T946, T947, T948) :|: TRUE f892_in(x561, x562, x563, x564, x565, x566, x567) -> f899_in :|: TRUE f898_out(x568, x569, x570, x571, x572, x573, x574) -> f892_out(x569, x570, x571, x572, x573, x574, x568) :|: TRUE f898_in(x575, x576, x577, x578, x579, x580, x581) -> f900_in(x575) :|: TRUE f900_out(x582) -> f901_in(x583, x584, x585, x586, x587, x588, x589, x582) :|: TRUE f901_out(x590, x591, x592, x593, x594, x595, x596, x597) -> f898_out(x597, x591, x592, x593, x594, x595, x596) :|: TRUE f908_out(x598, x599, x600, x601, x602, x603, x604, x605) -> f901_out(x598, x599, x600, x601, x602, x603, x604, x605) :|: TRUE f901_in(x606, x607, x608, x609, x610, x611, x612, x613) -> f907_in(x606, x607, x608, x609, x610, x611, x612) :|: TRUE f907_out(x614, x615, x616, x617, x618, x619, x620) -> f908_in(x614, x615, x616, x617, x618, x619, x620, x621) :|: TRUE f908_in(x622, x623, x624, x625, x626, x627, x628, x629) -> f999_in(x622, x623, x624, x625, x626, x627, x628, x629) :|: TRUE f999_out(x630, x631, x632, x633, x634, x635, x636, x637) -> f908_out(x630, x631, x632, x633, x634, x635, x636, x637) :|: TRUE f1009_out(x638, x639, x640, x641, x642, x643, x644, x645) -> f999_out(x638, x639, x640, x641, x642, x643, x644, x645) :|: TRUE f999_in(x646, x647, x648, x649, x650, x651, x652, x653) -> f1009_in(x646, x647, x648, x649, x650, x651, x652, x653) :|: TRUE f1010_out(x654, x655, x656, x657, x658, x659, x660, x661) -> f999_out(x654, x655, x656, x657, x658, x659, x660, x661) :|: TRUE f999_in(x662, x663, x664, x665, x666, x667, x668, x669) -> f1010_in(x662, x663, x664, x665, x666, x667, x668, x669) :|: TRUE f1024_out(x670, x671, x672, x673, x674, x675, x676, x677) -> f1010_out(x671, x672, x673, x674, x675, x676, x677, x670) :|: TRUE f1010_in(x678, x679, x680, x681, x682, x683, x684, x685) -> f1024_in(x685, x678, x679, x680, x681, x682, x683, x684) :|: TRUE f1010_in(x686, x687, x688, x689, x690, x691, x692, x693) -> f1025_in :|: TRUE f1025_out -> f1010_out(x694, x695, x696, x697, x698, x699, x700, x701) :|: TRUE f1024_in(x702, x703, x704, x705, x706, x707, x708, x709) -> f1026_in(x702, x703, x704, x705, x706, x707, x708, x709, .(x703, .(x704, .(x705, .(x706, .(x707, .(x708, .(x709, [])))))))) :|: TRUE f1026_out(x710, x711, x712, x713, x714, x715, x716, x717, .(x711, .(x712, .(x713, .(x714, .(x715, .(x716, .(x717, [])))))))) -> f1024_out(x710, x711, x712, x713, x714, x715, x716, x717) :|: TRUE f1035_out(x718) -> f1036_in(x719, x720, x721, x722, x723, x724, x725, x726, x727, x718) :|: TRUE f1036_out(x728, x729, x730, x731, x732, x733, x734, x735, x736, x737) -> f1026_out(x737, x729, x730, x731, x732, x733, x734, x735, x736) :|: TRUE f1026_in(x738, x739, x740, x741, x742, x743, x744, x745, x746) -> f1035_in(x738) :|: TRUE f1041_out(x747, x748, x749, x750, x751, x752, x753, x754) -> f1042_in(x747, x755, x756) :|: TRUE f1042_out(x757, x758, x759) -> f1036_out(x757, x760, x761, x762, x763, x764, x765, x766, x758, x759) :|: TRUE f1036_in(x767, x768, x769, x770, x771, x772, x773, x774, x775, x776) -> f1041_in(x767, x768, x769, x770, x771, x772, x773, x774) :|: TRUE Start term: f2_in(T2) ---------------------------------------- (234) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f1202_out(x17, x18) -> f1209_out(x17, x18) :|: TRUE f1203_out(x25, x26) -> f1202_out(x25, x26) :|: TRUE f1209_out(x92, x93) -> f1205_out(x92, .(x94, x93)) :|: TRUE f1205_out(x123, x124) -> f1203_out(x123, x124) :|: TRUE intTRSProblem: f1209_in(T1601, T1603) -> f1202_in(T1601, T1603) :|: TRUE f1202_in(T1578, T1580) -> f1203_in(T1578, T1580) :|: TRUE f1205_in(x95, .(x96, x97)) -> f1209_in(x95, x97) :|: TRUE f1203_in(x121, x122) -> f1205_in(x121, x122) :|: TRUE intTRSProblem: f1192_in(T1614, T1615, T1616) -> f1326_in :|: TRUE f1326_out -> f1192_out(x, x1, x2) :|: TRUE f307_in(.(T58, T59)) -> f314_in(T59) :|: TRUE f314_out(x19) -> f307_out(.(x20, x19)) :|: TRUE f1042_in(T1248, T1243, T1236) -> f1165_in(T1248, T1243, T1236) :|: TRUE f1179_in(T1508) -> f300_in(T1508) :|: TRUE f300_out(x33) -> f1179_out(x33) :|: TRUE f308_out -> f306_out(.(T49, T50)) :|: TRUE f306_in(.(x36, x37)) -> f308_in :|: TRUE f1183_out(x44, x45, x46) -> f1184_in(x44, x45, x46, x47) :|: TRUE f1180_in(x52, x53, x54, x55) -> f1183_in(x52, x53, x54) :|: TRUE f1184_in(x56, x57, x58, x59) -> f1042_in(x56, .(x57, x58), x59) :|: TRUE f1167_in(x67, x68, x69) -> f1177_in(x69, x67, x68) :|: TRUE f305_out(x76) -> f300_out(x76) :|: TRUE f300_in(x77) -> f305_in(x77) :|: TRUE f300_out(x100) -> f314_out(x100) :|: TRUE f314_in(x101) -> f300_in(x101) :|: TRUE f1185_out(T1515, T1506, T1507) -> f1183_out(T1515, T1506, T1507) :|: TRUE f1183_in(x102, x103, x104) -> f1185_in(x102, x103, x104) :|: TRUE f1185_in(x111, x112, x113) -> f1192_in(x111, x112, x113) :|: TRUE f1192_out(x114, x115, x116) -> f1185_out(x114, x115, x116) :|: TRUE f308_in -> f308_out :|: TRUE f307_out(x131) -> f305_out(x131) :|: TRUE f305_in(x132) -> f306_in(x132) :|: TRUE f305_in(x133) -> f307_in(x133) :|: TRUE f306_out(x134) -> f305_out(x134) :|: TRUE f1179_out(x139) -> f1180_in(x140, x141, x142, x139) :|: TRUE f1177_in(x143, x144, x145) -> f1179_in(x143) :|: TRUE f1326_in -> f1326_out :|: TRUE f1165_in(x152, x153, x154) -> f1167_in(x152, x153, x154) :|: TRUE ---------------------------------------- (235) Complex Obligation (AND) ---------------------------------------- (236) Obligation: Rules: f1202_out(x17, x18) -> f1209_out(x17, x18) :|: TRUE f1203_out(x25, x26) -> f1202_out(x25, x26) :|: TRUE f1209_out(x92, x93) -> f1205_out(x92, .(x94, x93)) :|: TRUE f1205_out(x123, x124) -> f1203_out(x123, x124) :|: TRUE ---------------------------------------- (237) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (238) Obligation: Rules: f1203_out(x25:0, x26:0) -> f1203_out(x25:0, .(x94:0, x26:0)) :|: TRUE ---------------------------------------- (239) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (240) Obligation: Rules: f1203_out(x25:0, x26:0) -> f1203_out(x25:0, .(x94:0, x26:0)) :|: TRUE ---------------------------------------- (241) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f1203_out(x25:0, x26:0) -> f1203_out(x25:0, .(x94:0, x26:0)) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (242) Obligation: Termination digraph: Nodes: (1) f1203_out(x25:0, x26:0) -> f1203_out(x25:0, .(x94:0, x26:0)) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (243) IRSwTToIntTRSProof (SOUND) Applied path-length measure to transform intTRS with terms to intTRS. ---------------------------------------- (244) Obligation: Rules: f1203_out(x, x1) -> f1203_out(x, .(x2, x1)) :|: TRUE ---------------------------------------- (245) Obligation: Rules: f1209_in(T1601, T1603) -> f1202_in(T1601, T1603) :|: TRUE f1202_in(T1578, T1580) -> f1203_in(T1578, T1580) :|: TRUE f1205_in(x95, .(x96, x97)) -> f1209_in(x95, x97) :|: TRUE f1203_in(x121, x122) -> f1205_in(x121, x122) :|: TRUE ---------------------------------------- (246) Obligation: Rules: f1192_in(T1614, T1615, T1616) -> f1326_in :|: TRUE f1326_out -> f1192_out(x, x1, x2) :|: TRUE f307_in(.(T58, T59)) -> f314_in(T59) :|: TRUE f314_out(x19) -> f307_out(.(x20, x19)) :|: TRUE f1042_in(T1248, T1243, T1236) -> f1165_in(T1248, T1243, T1236) :|: TRUE f1179_in(T1508) -> f300_in(T1508) :|: TRUE f300_out(x33) -> f1179_out(x33) :|: TRUE f308_out -> f306_out(.(T49, T50)) :|: TRUE f306_in(.(x36, x37)) -> f308_in :|: TRUE f1183_out(x44, x45, x46) -> f1184_in(x44, x45, x46, x47) :|: TRUE f1180_in(x52, x53, x54, x55) -> f1183_in(x52, x53, x54) :|: TRUE f1184_in(x56, x57, x58, x59) -> f1042_in(x56, .(x57, x58), x59) :|: TRUE f1167_in(x67, x68, x69) -> f1177_in(x69, x67, x68) :|: TRUE f305_out(x76) -> f300_out(x76) :|: TRUE f300_in(x77) -> f305_in(x77) :|: TRUE f300_out(x100) -> f314_out(x100) :|: TRUE f314_in(x101) -> f300_in(x101) :|: TRUE f1185_out(T1515, T1506, T1507) -> f1183_out(T1515, T1506, T1507) :|: TRUE f1183_in(x102, x103, x104) -> f1185_in(x102, x103, x104) :|: TRUE f1185_in(x111, x112, x113) -> f1192_in(x111, x112, x113) :|: TRUE f1192_out(x114, x115, x116) -> f1185_out(x114, x115, x116) :|: TRUE f308_in -> f308_out :|: TRUE f307_out(x131) -> f305_out(x131) :|: TRUE f305_in(x132) -> f306_in(x132) :|: TRUE f305_in(x133) -> f307_in(x133) :|: TRUE f306_out(x134) -> f305_out(x134) :|: TRUE f1179_out(x139) -> f1180_in(x140, x141, x142, x139) :|: TRUE f1177_in(x143, x144, x145) -> f1179_in(x143) :|: TRUE f1326_in -> f1326_out :|: TRUE f1165_in(x152, x153, x154) -> f1167_in(x152, x153, x154) :|: TRUE