/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.pl /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- MAYBE proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern pds(g,a) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) IntegerArithmeticTransformerProof [SOUND, 0 ms] (2) Prolog (3) CutEliminatorProof [SOUND, 0 ms] (4) Prolog (5) UnifyTransformerProof [EQUIVALENT, 0 ms] (6) Prolog (7) PrologToPiTRSProof [SOUND, 32 ms] (8) PiTRS (9) DependencyPairsProof [EQUIVALENT, 161 ms] (10) PiDP (11) DependencyGraphProof [EQUIVALENT, 0 ms] (12) AND (13) PiDP (14) UsableRulesProof [EQUIVALENT, 0 ms] (15) PiDP (16) PiDPToQDPProof [SOUND, 0 ms] (17) QDP (18) QDPSizeChangeProof [EQUIVALENT, 0 ms] (19) YES (20) PiDP (21) UsableRulesProof [EQUIVALENT, 0 ms] (22) PiDP (23) PiDPToQDPProof [EQUIVALENT, 0 ms] (24) QDP (25) QDPSizeChangeProof [EQUIVALENT, 0 ms] (26) YES (27) PiDP (28) UsableRulesProof [EQUIVALENT, 0 ms] (29) PiDP (30) PiDPToQDPProof [SOUND, 0 ms] (31) QDP (32) QDPSizeChangeProof [EQUIVALENT, 0 ms] (33) YES (34) PiDP (35) UsableRulesProof [EQUIVALENT, 0 ms] (36) PiDP (37) PiDPToQDPProof [EQUIVALENT, 0 ms] (38) QDP (39) QDPSizeChangeProof [EQUIVALENT, 0 ms] (40) YES (41) PiDP (42) UsableRulesProof [EQUIVALENT, 0 ms] (43) PiDP (44) PiDPToQDPProof [EQUIVALENT, 0 ms] (45) QDP (46) QDPSizeChangeProof [EQUIVALENT, 0 ms] (47) YES (48) PiDP (49) UsableRulesProof [EQUIVALENT, 0 ms] (50) PiDP (51) PiDPToQDPProof [SOUND, 0 ms] (52) QDP (53) QDPSizeChangeProof [EQUIVALENT, 0 ms] (54) YES (55) PiDP (56) UsableRulesProof [EQUIVALENT, 0 ms] (57) PiDP (58) PiDPToQDPProof [SOUND, 0 ms] (59) QDP (60) QDPSizeChangeProof [EQUIVALENT, 0 ms] (61) YES (62) PiDP (63) UsableRulesProof [EQUIVALENT, 0 ms] (64) PiDP (65) PiDPToQDPProof [SOUND, 0 ms] (66) QDP (67) QDPSizeChangeProof [EQUIVALENT, 0 ms] (68) YES (69) PiDP (70) UsableRulesProof [EQUIVALENT, 0 ms] (71) PiDP (72) PiDPToQDPProof [SOUND, 0 ms] (73) QDP (74) QDPSizeChangeProof [EQUIVALENT, 0 ms] (75) YES (76) PiDP (77) UsableRulesProof [EQUIVALENT, 0 ms] (78) PiDP (79) PiDPToQDPProof [SOUND, 0 ms] (80) QDP (81) QDPQMonotonicMRRProof [EQUIVALENT, 307 ms] (82) QDP (83) UsableRulesProof [EQUIVALENT, 1 ms] (84) QDP (85) QReductionProof [EQUIVALENT, 2 ms] (86) QDP (87) QDPOrderProof [EQUIVALENT, 80 ms] (88) QDP (89) DependencyGraphProof [EQUIVALENT, 0 ms] (90) TRUE (91) PiDP (92) UsableRulesProof [EQUIVALENT, 0 ms] (93) PiDP (94) PiDPToQDPProof [SOUND, 0 ms] (95) QDP (96) QDPSizeChangeProof [EQUIVALENT, 0 ms] (97) YES (98) PiDP (99) UsableRulesProof [EQUIVALENT, 0 ms] (100) PiDP (101) PiDPToQDPProof [SOUND, 0 ms] (102) QDP (103) PiDP (104) UsableRulesProof [EQUIVALENT, 0 ms] (105) PiDP (106) PiDP (107) UsableRulesProof [EQUIVALENT, 0 ms] (108) PiDP (109) PrologToPiTRSProof [SOUND, 47 ms] (110) PiTRS (111) DependencyPairsProof [EQUIVALENT, 113 ms] (112) PiDP (113) DependencyGraphProof [EQUIVALENT, 0 ms] (114) AND (115) PiDP (116) UsableRulesProof [EQUIVALENT, 0 ms] (117) PiDP (118) PiDPToQDPProof [SOUND, 0 ms] (119) QDP (120) QDPSizeChangeProof [EQUIVALENT, 0 ms] (121) YES (122) PiDP (123) UsableRulesProof [EQUIVALENT, 0 ms] (124) PiDP (125) PiDPToQDPProof [EQUIVALENT, 0 ms] (126) QDP (127) QDPSizeChangeProof [EQUIVALENT, 0 ms] (128) YES (129) PiDP (130) UsableRulesProof [EQUIVALENT, 0 ms] (131) PiDP (132) PiDPToQDPProof [SOUND, 0 ms] (133) QDP (134) QDPSizeChangeProof [EQUIVALENT, 0 ms] (135) YES (136) PiDP (137) UsableRulesProof [EQUIVALENT, 0 ms] (138) PiDP (139) PiDPToQDPProof [EQUIVALENT, 0 ms] (140) QDP (141) QDPSizeChangeProof [EQUIVALENT, 0 ms] (142) YES (143) PiDP (144) UsableRulesProof [EQUIVALENT, 0 ms] (145) PiDP (146) PiDPToQDPProof [EQUIVALENT, 0 ms] (147) QDP (148) QDPSizeChangeProof [EQUIVALENT, 0 ms] (149) YES (150) PiDP (151) UsableRulesProof [EQUIVALENT, 0 ms] (152) PiDP (153) PiDPToQDPProof [SOUND, 0 ms] (154) QDP (155) QDPSizeChangeProof [EQUIVALENT, 0 ms] (156) YES (157) PiDP (158) UsableRulesProof [EQUIVALENT, 0 ms] (159) PiDP (160) PiDPToQDPProof [SOUND, 0 ms] (161) QDP (162) QDPSizeChangeProof [EQUIVALENT, 0 ms] (163) YES (164) PiDP (165) UsableRulesProof [EQUIVALENT, 0 ms] (166) PiDP (167) PiDPToQDPProof [SOUND, 0 ms] (168) QDP (169) QDPSizeChangeProof [EQUIVALENT, 0 ms] (170) YES (171) PiDP (172) UsableRulesProof [EQUIVALENT, 0 ms] (173) PiDP (174) PiDPToQDPProof [SOUND, 0 ms] (175) QDP (176) QDPSizeChangeProof [EQUIVALENT, 0 ms] (177) YES (178) PiDP (179) UsableRulesProof [EQUIVALENT, 0 ms] (180) PiDP (181) PiDPToQDPProof [SOUND, 0 ms] (182) QDP (183) QDPQMonotonicMRRProof [EQUIVALENT, 443 ms] (184) QDP (185) UsableRulesProof [EQUIVALENT, 0 ms] (186) QDP (187) QReductionProof [EQUIVALENT, 0 ms] (188) QDP (189) QDPQMonotonicMRRProof [EQUIVALENT, 195 ms] (190) QDP (191) QDPQMonotonicMRRProof [EQUIVALENT, 181 ms] (192) QDP (193) DependencyGraphProof [EQUIVALENT, 0 ms] (194) TRUE (195) PiDP (196) UsableRulesProof [EQUIVALENT, 0 ms] (197) PiDP (198) PiDPToQDPProof [SOUND, 0 ms] (199) QDP (200) QDPSizeChangeProof [EQUIVALENT, 0 ms] (201) YES (202) PiDP (203) UsableRulesProof [EQUIVALENT, 0 ms] (204) PiDP (205) PiDPToQDPProof [SOUND, 0 ms] (206) QDP (207) TransformationProof [EQUIVALENT, 1 ms] (208) QDP (209) UsableRulesProof [EQUIVALENT, 0 ms] (210) QDP (211) QReductionProof [EQUIVALENT, 0 ms] (212) QDP (213) TransformationProof [EQUIVALENT, 0 ms] (214) QDP (215) QDPQMonotonicMRRProof [EQUIVALENT, 18 ms] (216) QDP (217) QDPQMonotonicMRRProof [EQUIVALENT, 11 ms] (218) QDP (219) QReductionProof [EQUIVALENT, 0 ms] (220) QDP (221) QDPQMonotonicMRRProof [EQUIVALENT, 10 ms] (222) QDP (223) QReductionProof [EQUIVALENT, 0 ms] (224) QDP (225) QDPQMonotonicMRRProof [EQUIVALENT, 12 ms] (226) QDP (227) UsableRulesProof [EQUIVALENT, 0 ms] (228) QDP (229) QReductionProof [EQUIVALENT, 0 ms] (230) QDP (231) PiDP (232) UsableRulesProof [EQUIVALENT, 0 ms] (233) PiDP (234) PiDP (235) UsableRulesProof [EQUIVALENT, 0 ms] (236) PiDP (237) CutEliminatorProof [SOUND, 0 ms] (238) Prolog (239) UndefinedPredicateHandlerProof [SOUND, 0 ms] (240) Prolog (241) PrologToPiTRSProof [SOUND, 39 ms] (242) PiTRS (243) DependencyPairsProof [EQUIVALENT, 84 ms] (244) PiDP (245) DependencyGraphProof [EQUIVALENT, 1 ms] (246) AND (247) PiDP (248) UsableRulesProof [EQUIVALENT, 0 ms] (249) PiDP (250) PiDPToQDPProof [SOUND, 23 ms] (251) QDP (252) TransformationProof [EQUIVALENT, 0 ms] (253) QDP (254) TransformationProof [EQUIVALENT, 0 ms] (255) QDP (256) UsableRulesProof [EQUIVALENT, 0 ms] (257) QDP (258) QReductionProof [EQUIVALENT, 0 ms] (259) QDP (260) NonTerminationLoopProof [COMPLETE, 4 ms] (261) NO (262) PiDP (263) UsableRulesProof [EQUIVALENT, 0 ms] (264) PiDP (265) PiDPToQDPProof [SOUND, 0 ms] (266) QDP (267) NonTerminationLoopProof [COMPLETE, 0 ms] (268) NO (269) PiDP (270) UsableRulesProof [EQUIVALENT, 0 ms] (271) PiDP (272) PiDPToQDPProof [SOUND, 0 ms] (273) QDP (274) TransformationProof [EQUIVALENT, 0 ms] (275) QDP (276) UsableRulesProof [EQUIVALENT, 0 ms] (277) QDP (278) QReductionProof [EQUIVALENT, 0 ms] (279) QDP (280) PiDP (281) UsableRulesProof [EQUIVALENT, 0 ms] (282) PiDP (283) PiDP (284) UsableRulesProof [EQUIVALENT, 0 ms] (285) PiDP (286) PiDP (287) UsableRulesProof [EQUIVALENT, 0 ms] (288) PiDP (289) PiDP (290) UsableRulesProof [EQUIVALENT, 0 ms] (291) PiDP (292) PrologToPiTRSProof [SOUND, 33 ms] (293) PiTRS (294) DependencyPairsProof [EQUIVALENT, 76 ms] (295) PiDP (296) DependencyGraphProof [EQUIVALENT, 0 ms] (297) AND (298) PiDP (299) UsableRulesProof [EQUIVALENT, 0 ms] (300) PiDP (301) PiDPToQDPProof [SOUND, 0 ms] (302) QDP (303) TransformationProof [EQUIVALENT, 0 ms] (304) QDP (305) TransformationProof [EQUIVALENT, 0 ms] (306) QDP (307) UsableRulesProof [EQUIVALENT, 0 ms] (308) QDP (309) QReductionProof [EQUIVALENT, 0 ms] (310) QDP (311) NonTerminationLoopProof [COMPLETE, 0 ms] (312) NO (313) PiDP (314) UsableRulesProof [EQUIVALENT, 0 ms] (315) PiDP (316) PiDPToQDPProof [SOUND, 0 ms] (317) QDP (318) NonTerminationLoopProof [COMPLETE, 0 ms] (319) NO (320) PiDP (321) UsableRulesProof [EQUIVALENT, 0 ms] (322) PiDP (323) PiDPToQDPProof [SOUND, 0 ms] (324) QDP (325) TransformationProof [EQUIVALENT, 0 ms] (326) QDP (327) UsableRulesProof [EQUIVALENT, 0 ms] (328) QDP (329) QReductionProof [EQUIVALENT, 0 ms] (330) QDP (331) PiDP (332) UsableRulesProof [EQUIVALENT, 0 ms] (333) PiDP (334) PiDP (335) UsableRulesProof [EQUIVALENT, 0 ms] (336) PiDP (337) PiDP (338) UsableRulesProof [EQUIVALENT, 0 ms] (339) PiDP (340) PiDP (341) UsableRulesProof [EQUIVALENT, 0 ms] (342) PiDP (343) PrologToDTProblemTransformerProof [SOUND, 750 ms] (344) TRIPLES (345) UndefinedPredicateInTriplesTransformerProof [SOUND, 0 ms] (346) TRIPLES (347) TriplesToPiDPProof [SOUND, 411 ms] (348) PiDP (349) DependencyGraphProof [EQUIVALENT, 0 ms] (350) AND (351) PiDP (352) UsableRulesProof [EQUIVALENT, 0 ms] (353) PiDP (354) PiDPToQDPProof [SOUND, 0 ms] (355) QDP (356) QDPSizeChangeProof [EQUIVALENT, 0 ms] (357) YES (358) PiDP (359) UsableRulesProof [EQUIVALENT, 0 ms] (360) PiDP (361) PiDPToQDPProof [SOUND, 0 ms] (362) QDP (363) QDPSizeChangeProof [EQUIVALENT, 0 ms] (364) YES (365) PiDP (366) UsableRulesProof [EQUIVALENT, 0 ms] (367) PiDP (368) PiDPToQDPProof [SOUND, 0 ms] (369) QDP (370) QDPSizeChangeProof [EQUIVALENT, 0 ms] (371) YES (372) PiDP (373) UsableRulesProof [EQUIVALENT, 0 ms] (374) PiDP (375) PiDPToQDPProof [SOUND, 0 ms] (376) QDP (377) QDPSizeChangeProof [EQUIVALENT, 0 ms] (378) YES (379) PiDP (380) UsableRulesProof [EQUIVALENT, 0 ms] (381) PiDP (382) PiDPToQDPProof [SOUND, 1 ms] (383) QDP (384) QDPQMonotonicMRRProof [EQUIVALENT, 372 ms] (385) QDP (386) QDPQMonotonicMRRProof [EQUIVALENT, 290 ms] (387) QDP (388) QDPQMonotonicMRRProof [EQUIVALENT, 411 ms] (389) QDP (390) QReductionProof [EQUIVALENT, 0 ms] (391) QDP (392) TransformationProof [EQUIVALENT, 0 ms] (393) QDP (394) UsableRulesProof [EQUIVALENT, 0 ms] (395) QDP (396) QReductionProof [EQUIVALENT, 0 ms] (397) QDP (398) QDPQMonotonicMRRProof [EQUIVALENT, 326 ms] (399) QDP (400) PiDP (401) UsableRulesProof [EQUIVALENT, 0 ms] (402) PiDP (403) PrologToIRSwTTransformerProof [SOUND, 4109 ms] (404) AND (405) IRSwT (406) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (407) TRUE (408) IRSwT (409) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (410) TRUE (411) IRSwT (412) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (413) TRUE (414) IRSwT (415) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (416) TRUE (417) IRSwT (418) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (419) TRUE (420) IRSwT (421) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (422) IRSwT (423) IntTRSCompressionProof [EQUIVALENT, 45 ms] (424) IRSwT (425) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (426) IRSwT (427) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms] (428) IRSwT (429) IntTRSCompressionProof [EQUIVALENT, 0 ms] (430) IRSwT (431) IRSwTToIntTRSProof [SOUND, 0 ms] (432) IRSwT (433) IntTRSNonPeriodicNontermProof [COMPLETE, 5 ms] (434) NO ---------------------------------------- (0) Obligation: Clauses: member(X, .(X, Xs)). member(X, .(X1, Xs)) :- member(X, Xs). iota(N, List) :- iota1(0, N, List). iota1(K, K, []) :- !. iota1(K, N, .(K, List)) :- ','(is(K1, +(K, 1)), iota1(K1, N, List)). dif([], X2, X3, [], []). dif(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) :- ','(is(D, -(Val, S)), ','(is(D2, -(Mod, D)), dif(Ss, Val, Mod, Ds, D2s))). rev([], L, L). rev(.(X, Xs), Y, L) :- rev(Xs, .(X, Y), L). mergedelete([], L, L). mergedelete(.(D, Ds), .(D, R), L2) :- mergedelete(Ds, R, L2). mergedelete(.(D, Ds), .(X, R), .(X, L2)) :- ','(>(D, X), mergedelete(.(D, Ds), R, L2)). check([], X4, L, L, X5) :- !. check(S, Choice, Old, L3, Modulus) :- ','(dif(S, Choice, Modulus, Ds, Dds), ','(mergedelete(Ds, Old, L2), ','(rev(Dds, [], Rds), ','(mergedelete(Rds, L2, L3), !)))). pds1([], X6, [], X7) :- !. pds1(Unused, List, .(Choice, Rest), Mod) :- ','(member(Choice, Unused), ','(check(List, Choice, Unused, U3, Mod), pds1(U3, .(Choice, List), Rest, Mod))). pds(Order, .(0, Ans)) :- ','(is(N, +(*(Order, +(Order, 1)), 1)), ','(iota(N, .(0, List)), pds1(List, .(0, []), Ans, N))). pdsbm(N) :- pds(N, .(0, .(1, X))). Query: pds(g,a) ---------------------------------------- (1) IntegerArithmeticTransformerProof (SOUND) Added definitions of predefined predicates [PROLOG]. ---------------------------------------- (2) Obligation: Clauses: member(X, .(X, Xs)). member(X, .(X1, Xs)) :- member(X, Xs). iota(N, List) :- iota1(zero, N, List). iota1(K, K, []) :- !. iota1(K, N, .(K, List)) :- ','(isPlus(K, succ(zero), U), ','(=(K1, U), iota1(K1, N, List))). dif([], X2, X3, [], []). dif(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) :- ','(isMinus(Val, S, U), ','(=(D, U), ','(isMinus(Mod, D, U), ','(=(D2, U), dif(Ss, Val, Mod, Ds, D2s))))). rev([], L, L). rev(.(X, Xs), Y, L) :- rev(Xs, .(X, Y), L). mergedelete([], L, L). mergedelete(.(D, Ds), .(D, R), L2) :- mergedelete(Ds, R, L2). mergedelete(.(D, Ds), .(X, R), .(X, L2)) :- ','(=(X1, D), ','(=(X2, X), ','(isGreater(X1, X2), mergedelete(.(D, Ds), R, L2)))). check([], X4, L, L, X5) :- !. check(S, Choice, Old, L3, Modulus) :- ','(dif(S, Choice, Modulus, Ds, Dds), ','(mergedelete(Ds, Old, L2), ','(rev(Dds, [], Rds), ','(mergedelete(Rds, L2, L3), !)))). pds1([], X6, [], X7) :- !. pds1(Unused, List, .(Choice, Rest), Mod) :- ','(member(Choice, Unused), ','(check(List, Choice, Unused, U3, Mod), pds1(U3, .(Choice, List), Rest, Mod))). pds(Order, .(zero, Ans)) :- ','(isPlus(Order, succ(zero), U), ','(isTimes(Order, U, U1), ','(isPlus(U1, succ(zero), U2), ','(=(N, U2), ','(iota(N, .(zero, List)), pds1(List, .(zero, []), Ans, N)))))). pdsbm(N) :- pds(N, .(zero, .(succ(zero), X))). isPlus(zero, X, X). isPlus(succ(X), zero, succ(X)). isPlus(succ(X), succ(Y), succ(succ(Z))) :- isPlus(X, Y, Z). isPlus(succ(X), pred(Y), Z) :- isPlus(X, Y, Z). isPlus(pred(X), zero, pred(X)). isPlus(pred(X), succ(Y), Z) :- isPlus(X, Y, Z). isPlus(pred(X), pred(Y), pred(pred(Z))) :- isPlus(X, Y, Z). isMinus(X, zero, X). isMinus(zero, succ(Y), pred(Z)) :- isMinus(zero, Y, Z). isMinus(zero, pred(Y), succ(Z)) :- isMinus(zero, Y, Z). isMinus(succ(X), succ(Y), Z) :- isMinus(X, Y, Z). isMinus(succ(X), pred(Y), succ(succ(Z))) :- isMinus(X, Y, Z). isMinus(pred(X), succ(Y), pred(pred(Z))) :- isMinus(X, Y, Z). isMinus(pred(X), pred(Y), Z) :- isMinus(X, Y, Z). isTimes(X, zero, zero). isTimes(zero, succ(Y), zero). isTimes(zero, pred(Y), zero). isTimes(succ(X), succ(Y), Z) :- ','(isTimes(succ(X), Y, A), isPlus(A, succ(X), Z)). isTimes(succ(X), pred(Y), Z) :- ','(isTimes(succ(X), Y, A), isMinus(A, succ(X), Z)). isTimes(pred(X), succ(Y), Z) :- ','(isTimes(pred(X), Y, A), isPlus(A, pred(X), Z)). isTimes(pred(X), pred(Y), Z) :- ','(isTimes(pred(X), Y, A), isMinus(A, pred(X), Z)). isDiv(zero, succ(Y), zero). isDiv(zero, pred(Y), zero). isDiv(succ(X), succ(Y), zero) :- isMinus(succ(X), succ(Y), pred(Z)). isDiv(succ(X), succ(Y), succ(Z)) :- ','(isMinus(succ(X), succ(Y), A), isDiv(A, succ(Y), Z)). isDiv(succ(X), pred(Y), Z) :- ','(isMinus(zero, pred(Y), A), ','(isDiv(succ(X), A, B), isMinus(zero, B, Z))). isDiv(pred(X), pred(Y), zero) :- isMinus(pred(X), pred(Y), succ(Z)). isDiv(pred(X), pred(Y), succ(Z)) :- ','(isMinus(pred(X), pred(Y), A), isDiv(A, pred(Y), Z)). isDiv(pred(X), succ(Y), Z) :- ','(isMinus(zero, pred(X), A), ','(isDiv(A, succ(Y), B), isMinus(zero, B, Z))). isModulo(X, Y, Z) :- ','(isDiv(X, Y, A), ','(isTimes(A, Y, B), isMinus(X, B, Z))). isGreater(succ(X), zero). isGreater(succ(X), pred(Y)). isGreater(succ(X), succ(Y)) :- isGreater(X, Y). isGreater(zero, pred(Y)). isGreater(pred(X), pred(Y)) :- isGreater(X, Y). Query: pds(g,a) ---------------------------------------- (3) CutEliminatorProof (SOUND) Eliminated all cuts by simply ignoring them[PROLOG]. ---------------------------------------- (4) Obligation: Clauses: member(X, .(X, Xs)). member(X, .(X1, Xs)) :- member(X, Xs). iota(N, List) :- iota1(zero, N, List). iota1(K, K, []). iota1(K, N, .(K, List)) :- ','(isPlus(K, succ(zero), U), ','(=(K1, U), iota1(K1, N, List))). dif([], X2, X3, [], []). dif(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) :- ','(isMinus(Val, S, U), ','(=(D, U), ','(isMinus(Mod, D, U), ','(=(D2, U), dif(Ss, Val, Mod, Ds, D2s))))). rev([], L, L). rev(.(X, Xs), Y, L) :- rev(Xs, .(X, Y), L). mergedelete([], L, L). mergedelete(.(D, Ds), .(D, R), L2) :- mergedelete(Ds, R, L2). mergedelete(.(D, Ds), .(X, R), .(X, L2)) :- ','(=(X1, D), ','(=(X2, X), ','(isGreater(X1, X2), mergedelete(.(D, Ds), R, L2)))). check([], X4, L, L, X5). check(S, Choice, Old, L3, Modulus) :- ','(dif(S, Choice, Modulus, Ds, Dds), ','(mergedelete(Ds, Old, L2), ','(rev(Dds, [], Rds), mergedelete(Rds, L2, L3)))). pds1([], X6, [], X7). pds1(Unused, List, .(Choice, Rest), Mod) :- ','(member(Choice, Unused), ','(check(List, Choice, Unused, U3, Mod), pds1(U3, .(Choice, List), Rest, Mod))). pds(Order, .(zero, Ans)) :- ','(isPlus(Order, succ(zero), U), ','(isTimes(Order, U, U1), ','(isPlus(U1, succ(zero), U2), ','(=(N, U2), ','(iota(N, .(zero, List)), pds1(List, .(zero, []), Ans, N)))))). pdsbm(N) :- pds(N, .(zero, .(succ(zero), X))). isPlus(zero, X, X). isPlus(succ(X), zero, succ(X)). isPlus(succ(X), succ(Y), succ(succ(Z))) :- isPlus(X, Y, Z). isPlus(succ(X), pred(Y), Z) :- isPlus(X, Y, Z). isPlus(pred(X), zero, pred(X)). isPlus(pred(X), succ(Y), Z) :- isPlus(X, Y, Z). isPlus(pred(X), pred(Y), pred(pred(Z))) :- isPlus(X, Y, Z). isMinus(X, zero, X). isMinus(zero, succ(Y), pred(Z)) :- isMinus(zero, Y, Z). isMinus(zero, pred(Y), succ(Z)) :- isMinus(zero, Y, Z). isMinus(succ(X), succ(Y), Z) :- isMinus(X, Y, Z). isMinus(succ(X), pred(Y), succ(succ(Z))) :- isMinus(X, Y, Z). isMinus(pred(X), succ(Y), pred(pred(Z))) :- isMinus(X, Y, Z). isMinus(pred(X), pred(Y), Z) :- isMinus(X, Y, Z). isTimes(X, zero, zero). isTimes(zero, succ(Y), zero). isTimes(zero, pred(Y), zero). isTimes(succ(X), succ(Y), Z) :- ','(isTimes(succ(X), Y, A), isPlus(A, succ(X), Z)). isTimes(succ(X), pred(Y), Z) :- ','(isTimes(succ(X), Y, A), isMinus(A, succ(X), Z)). isTimes(pred(X), succ(Y), Z) :- ','(isTimes(pred(X), Y, A), isPlus(A, pred(X), Z)). isTimes(pred(X), pred(Y), Z) :- ','(isTimes(pred(X), Y, A), isMinus(A, pred(X), Z)). isDiv(zero, succ(Y), zero). isDiv(zero, pred(Y), zero). isDiv(succ(X), succ(Y), zero) :- isMinus(succ(X), succ(Y), pred(Z)). isDiv(succ(X), succ(Y), succ(Z)) :- ','(isMinus(succ(X), succ(Y), A), isDiv(A, succ(Y), Z)). isDiv(succ(X), pred(Y), Z) :- ','(isMinus(zero, pred(Y), A), ','(isDiv(succ(X), A, B), isMinus(zero, B, Z))). isDiv(pred(X), pred(Y), zero) :- isMinus(pred(X), pred(Y), succ(Z)). isDiv(pred(X), pred(Y), succ(Z)) :- ','(isMinus(pred(X), pred(Y), A), isDiv(A, pred(Y), Z)). isDiv(pred(X), succ(Y), Z) :- ','(isMinus(zero, pred(X), A), ','(isDiv(A, succ(Y), B), isMinus(zero, B, Z))). isModulo(X, Y, Z) :- ','(isDiv(X, Y, A), ','(isTimes(A, Y, B), isMinus(X, B, Z))). isGreater(succ(X), zero). isGreater(succ(X), pred(Y)). isGreater(succ(X), succ(Y)) :- isGreater(X, Y). isGreater(zero, pred(Y)). isGreater(pred(X), pred(Y)) :- isGreater(X, Y). Query: pds(g,a) ---------------------------------------- (5) UnifyTransformerProof (EQUIVALENT) Added a fact for the built-in = predicate [PROLOG]. ---------------------------------------- (6) Obligation: Clauses: member(X, .(X, Xs)). member(X, .(X1, Xs)) :- member(X, Xs). iota(N, List) :- iota1(zero, N, List). iota1(K, K, []). iota1(K, N, .(K, List)) :- ','(isPlus(K, succ(zero), U), ','(=(K1, U), iota1(K1, N, List))). dif([], X2, X3, [], []). dif(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) :- ','(isMinus(Val, S, U), ','(=(D, U), ','(isMinus(Mod, D, U), ','(=(D2, U), dif(Ss, Val, Mod, Ds, D2s))))). rev([], L, L). rev(.(X, Xs), Y, L) :- rev(Xs, .(X, Y), L). mergedelete([], L, L). mergedelete(.(D, Ds), .(D, R), L2) :- mergedelete(Ds, R, L2). mergedelete(.(D, Ds), .(X, R), .(X, L2)) :- ','(=(X1, D), ','(=(X2, X), ','(isGreater(X1, X2), mergedelete(.(D, Ds), R, L2)))). check([], X4, L, L, X5). check(S, Choice, Old, L3, Modulus) :- ','(dif(S, Choice, Modulus, Ds, Dds), ','(mergedelete(Ds, Old, L2), ','(rev(Dds, [], Rds), mergedelete(Rds, L2, L3)))). pds1([], X6, [], X7). pds1(Unused, List, .(Choice, Rest), Mod) :- ','(member(Choice, Unused), ','(check(List, Choice, Unused, U3, Mod), pds1(U3, .(Choice, List), Rest, Mod))). pds(Order, .(zero, Ans)) :- ','(isPlus(Order, succ(zero), U), ','(isTimes(Order, U, U1), ','(isPlus(U1, succ(zero), U2), ','(=(N, U2), ','(iota(N, .(zero, List)), pds1(List, .(zero, []), Ans, N)))))). pdsbm(N) :- pds(N, .(zero, .(succ(zero), X))). isPlus(zero, X, X). isPlus(succ(X), zero, succ(X)). isPlus(succ(X), succ(Y), succ(succ(Z))) :- isPlus(X, Y, Z). isPlus(succ(X), pred(Y), Z) :- isPlus(X, Y, Z). isPlus(pred(X), zero, pred(X)). isPlus(pred(X), succ(Y), Z) :- isPlus(X, Y, Z). isPlus(pred(X), pred(Y), pred(pred(Z))) :- isPlus(X, Y, Z). isMinus(X, zero, X). isMinus(zero, succ(Y), pred(Z)) :- isMinus(zero, Y, Z). isMinus(zero, pred(Y), succ(Z)) :- isMinus(zero, Y, Z). isMinus(succ(X), succ(Y), Z) :- isMinus(X, Y, Z). isMinus(succ(X), pred(Y), succ(succ(Z))) :- isMinus(X, Y, Z). isMinus(pred(X), succ(Y), pred(pred(Z))) :- isMinus(X, Y, Z). isMinus(pred(X), pred(Y), Z) :- isMinus(X, Y, Z). isTimes(X, zero, zero). isTimes(zero, succ(Y), zero). isTimes(zero, pred(Y), zero). isTimes(succ(X), succ(Y), Z) :- ','(isTimes(succ(X), Y, A), isPlus(A, succ(X), Z)). isTimes(succ(X), pred(Y), Z) :- ','(isTimes(succ(X), Y, A), isMinus(A, succ(X), Z)). isTimes(pred(X), succ(Y), Z) :- ','(isTimes(pred(X), Y, A), isPlus(A, pred(X), Z)). isTimes(pred(X), pred(Y), Z) :- ','(isTimes(pred(X), Y, A), isMinus(A, pred(X), Z)). isDiv(zero, succ(Y), zero). isDiv(zero, pred(Y), zero). isDiv(succ(X), succ(Y), zero) :- isMinus(succ(X), succ(Y), pred(Z)). isDiv(succ(X), succ(Y), succ(Z)) :- ','(isMinus(succ(X), succ(Y), A), isDiv(A, succ(Y), Z)). isDiv(succ(X), pred(Y), Z) :- ','(isMinus(zero, pred(Y), A), ','(isDiv(succ(X), A, B), isMinus(zero, B, Z))). isDiv(pred(X), pred(Y), zero) :- isMinus(pred(X), pred(Y), succ(Z)). isDiv(pred(X), pred(Y), succ(Z)) :- ','(isMinus(pred(X), pred(Y), A), isDiv(A, pred(Y), Z)). isDiv(pred(X), succ(Y), Z) :- ','(isMinus(zero, pred(X), A), ','(isDiv(A, succ(Y), B), isMinus(zero, B, Z))). isModulo(X, Y, Z) :- ','(isDiv(X, Y, A), ','(isTimes(A, Y, B), isMinus(X, B, Z))). isGreater(succ(X), zero). isGreater(succ(X), pred(Y)). isGreater(succ(X), succ(Y)) :- isGreater(X, Y). isGreater(zero, pred(Y)). isGreater(pred(X), pred(Y)) :- isGreater(X, Y). =(X, X). Query: pds(g,a) ---------------------------------------- (7) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: pds_in_2: (b,f) isPlus_in_3: (b,b,f) isTimes_in_3: (b,b,f) isMinus_in_3: (b,b,f) (b,b,b) iota_in_2: (b,f) iota1_in_3: (b,b,f) pds1_in_4: (b,b,f,b) member_in_2: (f,b) check_in_5: (b,b,b,f,b) dif_in_5: (b,b,b,f,f) mergedelete_in_3: (b,b,f) isGreater_in_2: (b,b) rev_in_3: (b,b,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: pds_in_ga(Order, .(zero, Ans)) -> U24_ga(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) isPlus_in_gga(zero, X, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), zero, succ(X)) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), succ(Y), succ(succ(Z))) -> U31_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(succ(X), pred(Y), Z) -> U32_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), zero, pred(X)) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), succ(Y), Z) -> U33_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), pred(Y), pred(pred(Z))) -> U34_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) U34_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U33_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U32_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U31_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U24_ga(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_ga(Order, Ans, isTimes_in_gga(Order, U, U1)) isTimes_in_gga(X, zero, zero) -> isTimes_out_gga(X, zero, zero) isTimes_in_gga(zero, succ(Y), zero) -> isTimes_out_gga(zero, succ(Y), zero) isTimes_in_gga(zero, pred(Y), zero) -> isTimes_out_gga(zero, pred(Y), zero) isTimes_in_gga(succ(X), succ(Y), Z) -> U41_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(succ(X), pred(Y), Z) -> U43_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(pred(X), succ(Y), Z) -> U45_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) isTimes_in_gga(pred(X), pred(Y), Z) -> U47_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) U47_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_gga(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U48_gga(X, Y, Z, isMinus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), pred(Y), Z) U45_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_gga(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U46_gga(X, Y, Z, isPlus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), succ(Y), Z) U43_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_gga(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U44_gga(X, Y, Z, isMinus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), pred(Y), Z) U41_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_gga(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U42_gga(X, Y, Z, isPlus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), succ(Y), Z) U25_ga(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_ga(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U26_ga(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_ga(Order, Ans, =_in_ag(N, U2)) =_in_ag(X, X) -> =_out_ag(X, X) U27_ga(Order, Ans, =_out_ag(N, U2)) -> U28_ga(Order, Ans, N, iota_in_ga(N, .(zero, List))) iota_in_ga(N, List) -> U2_ga(N, List, iota1_in_gga(zero, N, List)) iota1_in_gga(K, K, []) -> iota1_out_gga(K, K, []) iota1_in_gga(K, N, .(K, List)) -> U3_gga(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_gga(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_gga(K, N, List, =_in_ag(K1, U)) U4_gga(K, N, List, =_out_ag(K1, U)) -> U5_gga(K, N, List, iota1_in_gga(K1, N, List)) U5_gga(K, N, List, iota1_out_gga(K1, N, List)) -> iota1_out_gga(K, N, .(K, List)) U2_ga(N, List, iota1_out_gga(zero, N, List)) -> iota_out_ga(N, List) U28_ga(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_ga(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) pds1_in_ggag([], X6, [], X7) -> pds1_out_ggag([], X6, [], X7) pds1_in_ggag(Unused, List, .(Choice, Rest), Mod) -> U21_ggag(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(X, .(X1, Xs)) -> U1_ag(X, X1, Xs, member_in_ag(X, Xs)) U1_ag(X, X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) U21_ggag(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_ggag(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) check_in_gggag([], X4, L, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, L3, Modulus) -> U17_gggag(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) dif_in_gggaa([], X2, X3, [], []) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U17_gggag(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) mergedelete_in_gga([], L, L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R), L2) -> U12_gga(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) mergedelete_in_gga(.(D, Ds), .(X, R), .(X, L2)) -> U13_gga(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_gga(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_gga(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U15_gga(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U16_gga(D, Ds, X, R, L2, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U12_gga(D, Ds, R, L2, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) rev_in_gga([], L, L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y, L) -> U11_gga(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) U11_gga(X, Xs, Y, L, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U22_ggag(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_ggag(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U23_ggag(Unused, List, Choice, Rest, Mod, pds1_out_ggag(U3, .(Choice, List), Rest, Mod)) -> pds1_out_ggag(Unused, List, .(Choice, Rest), Mod) U29_ga(Order, Ans, pds1_out_ggag(List, .(zero, []), Ans, N)) -> pds_out_ga(Order, .(zero, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U24_ga(x1, x2, x3) = U24_ga(x1, x3) isPlus_in_gga(x1, x2, x3) = isPlus_in_gga(x1, x2) zero = zero isPlus_out_gga(x1, x2, x3) = isPlus_out_gga(x3) succ(x1) = succ(x1) U31_gga(x1, x2, x3, x4) = U31_gga(x4) pred(x1) = pred(x1) U32_gga(x1, x2, x3, x4) = U32_gga(x4) U33_gga(x1, x2, x3, x4) = U33_gga(x4) U34_gga(x1, x2, x3, x4) = U34_gga(x4) U25_ga(x1, x2, x3) = U25_ga(x3) isTimes_in_gga(x1, x2, x3) = isTimes_in_gga(x1, x2) isTimes_out_gga(x1, x2, x3) = isTimes_out_gga(x3) U41_gga(x1, x2, x3, x4) = U41_gga(x1, x4) U43_gga(x1, x2, x3, x4) = U43_gga(x1, x4) U45_gga(x1, x2, x3, x4) = U45_gga(x1, x4) U47_gga(x1, x2, x3, x4) = U47_gga(x1, x4) U48_gga(x1, x2, x3, x4) = U48_gga(x4) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x3) U35_gga(x1, x2, x3) = U35_gga(x3) U36_gga(x1, x2, x3) = U36_gga(x3) U37_gga(x1, x2, x3, x4) = U37_gga(x4) U38_gga(x1, x2, x3, x4) = U38_gga(x4) U39_gga(x1, x2, x3, x4) = U39_gga(x4) U40_gga(x1, x2, x3, x4) = U40_gga(x4) U46_gga(x1, x2, x3, x4) = U46_gga(x4) U44_gga(x1, x2, x3, x4) = U44_gga(x4) U42_gga(x1, x2, x3, x4) = U42_gga(x4) U26_ga(x1, x2, x3) = U26_ga(x3) U27_ga(x1, x2, x3) = U27_ga(x3) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1) U28_ga(x1, x2, x3, x4) = U28_ga(x3, x4) iota_in_ga(x1, x2) = iota_in_ga(x1) U2_ga(x1, x2, x3) = U2_ga(x3) iota1_in_gga(x1, x2, x3) = iota1_in_gga(x1, x2) iota1_out_gga(x1, x2, x3) = iota1_out_gga(x3) U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x4) iota_out_ga(x1, x2) = iota_out_ga(x2) .(x1, x2) = .(x1, x2) U29_ga(x1, x2, x3) = U29_ga(x3) pds1_in_ggag(x1, x2, x3, x4) = pds1_in_ggag(x1, x2, x4) [] = [] pds1_out_ggag(x1, x2, x3, x4) = pds1_out_ggag(x3) U21_ggag(x1, x2, x3, x4, x5, x6) = U21_ggag(x1, x2, x5, x6) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1) U1_ag(x1, x2, x3, x4) = U1_ag(x4) U22_ggag(x1, x2, x3, x4, x5, x6) = U22_ggag(x2, x3, x5, x6) check_in_gggag(x1, x2, x3, x4, x5) = check_in_gggag(x1, x2, x3, x5) check_out_gggag(x1, x2, x3, x4, x5) = check_out_gggag(x4) U17_gggag(x1, x2, x3, x4, x5, x6) = U17_gggag(x3, x6) dif_in_gggaa(x1, x2, x3, x4, x5) = dif_in_gggaa(x1, x2, x3) dif_out_gggaa(x1, x2, x3, x4, x5) = dif_out_gggaa(x4, x5) U6_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gggaa(x2, x3, x4, x9) U7_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_gggaa(x2, x3, x4, x9, x10) U8_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_gggaa(x2, x3, x4, x5, x9, x10) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg U35_ggg(x1, x2, x3) = U35_ggg(x3) U36_ggg(x1, x2, x3) = U36_ggg(x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x4) U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_gggaa(x2, x3, x4, x5, x9) U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_gggaa(x5, x7, x9) U18_gggag(x1, x2, x3, x4, x5, x6, x7) = U18_gggag(x6, x7) mergedelete_in_gga(x1, x2, x3) = mergedelete_in_gga(x1, x2) mergedelete_out_gga(x1, x2, x3) = mergedelete_out_gga(x3) U12_gga(x1, x2, x3, x4, x5) = U12_gga(x5) U13_gga(x1, x2, x3, x4, x5, x6) = U13_gga(x1, x2, x3, x4, x6) U14_gga(x1, x2, x3, x4, x5, x6, x7) = U14_gga(x1, x2, x3, x4, x6, x7) U15_gga(x1, x2, x3, x4, x5, x6) = U15_gga(x1, x2, x3, x4, x6) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg U64_gg(x1, x2, x3) = U64_gg(x3) U65_gg(x1, x2, x3) = U65_gg(x3) U16_gga(x1, x2, x3, x4, x5, x6) = U16_gga(x3, x6) U19_gggag(x1, x2, x3, x4, x5, x6, x7) = U19_gggag(x6, x7) rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2) rev_out_gga(x1, x2, x3) = rev_out_gga(x3) U11_gga(x1, x2, x3, x4, x5) = U11_gga(x5) U20_gggag(x1, x2, x3, x4, x5, x6) = U20_gggag(x6) U23_ggag(x1, x2, x3, x4, x5, x6) = U23_ggag(x3, x6) pds_out_ga(x1, x2) = pds_out_ga(x2) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (8) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: pds_in_ga(Order, .(zero, Ans)) -> U24_ga(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) isPlus_in_gga(zero, X, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), zero, succ(X)) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), succ(Y), succ(succ(Z))) -> U31_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(succ(X), pred(Y), Z) -> U32_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), zero, pred(X)) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), succ(Y), Z) -> U33_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), pred(Y), pred(pred(Z))) -> U34_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) U34_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U33_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U32_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U31_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U24_ga(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_ga(Order, Ans, isTimes_in_gga(Order, U, U1)) isTimes_in_gga(X, zero, zero) -> isTimes_out_gga(X, zero, zero) isTimes_in_gga(zero, succ(Y), zero) -> isTimes_out_gga(zero, succ(Y), zero) isTimes_in_gga(zero, pred(Y), zero) -> isTimes_out_gga(zero, pred(Y), zero) isTimes_in_gga(succ(X), succ(Y), Z) -> U41_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(succ(X), pred(Y), Z) -> U43_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(pred(X), succ(Y), Z) -> U45_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) isTimes_in_gga(pred(X), pred(Y), Z) -> U47_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) U47_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_gga(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U48_gga(X, Y, Z, isMinus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), pred(Y), Z) U45_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_gga(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U46_gga(X, Y, Z, isPlus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), succ(Y), Z) U43_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_gga(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U44_gga(X, Y, Z, isMinus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), pred(Y), Z) U41_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_gga(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U42_gga(X, Y, Z, isPlus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), succ(Y), Z) U25_ga(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_ga(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U26_ga(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_ga(Order, Ans, =_in_ag(N, U2)) =_in_ag(X, X) -> =_out_ag(X, X) U27_ga(Order, Ans, =_out_ag(N, U2)) -> U28_ga(Order, Ans, N, iota_in_ga(N, .(zero, List))) iota_in_ga(N, List) -> U2_ga(N, List, iota1_in_gga(zero, N, List)) iota1_in_gga(K, K, []) -> iota1_out_gga(K, K, []) iota1_in_gga(K, N, .(K, List)) -> U3_gga(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_gga(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_gga(K, N, List, =_in_ag(K1, U)) U4_gga(K, N, List, =_out_ag(K1, U)) -> U5_gga(K, N, List, iota1_in_gga(K1, N, List)) U5_gga(K, N, List, iota1_out_gga(K1, N, List)) -> iota1_out_gga(K, N, .(K, List)) U2_ga(N, List, iota1_out_gga(zero, N, List)) -> iota_out_ga(N, List) U28_ga(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_ga(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) pds1_in_ggag([], X6, [], X7) -> pds1_out_ggag([], X6, [], X7) pds1_in_ggag(Unused, List, .(Choice, Rest), Mod) -> U21_ggag(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(X, .(X1, Xs)) -> U1_ag(X, X1, Xs, member_in_ag(X, Xs)) U1_ag(X, X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) U21_ggag(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_ggag(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) check_in_gggag([], X4, L, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, L3, Modulus) -> U17_gggag(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) dif_in_gggaa([], X2, X3, [], []) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U17_gggag(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) mergedelete_in_gga([], L, L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R), L2) -> U12_gga(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) mergedelete_in_gga(.(D, Ds), .(X, R), .(X, L2)) -> U13_gga(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_gga(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_gga(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U15_gga(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U16_gga(D, Ds, X, R, L2, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U12_gga(D, Ds, R, L2, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) rev_in_gga([], L, L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y, L) -> U11_gga(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) U11_gga(X, Xs, Y, L, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U22_ggag(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_ggag(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U23_ggag(Unused, List, Choice, Rest, Mod, pds1_out_ggag(U3, .(Choice, List), Rest, Mod)) -> pds1_out_ggag(Unused, List, .(Choice, Rest), Mod) U29_ga(Order, Ans, pds1_out_ggag(List, .(zero, []), Ans, N)) -> pds_out_ga(Order, .(zero, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U24_ga(x1, x2, x3) = U24_ga(x1, x3) isPlus_in_gga(x1, x2, x3) = isPlus_in_gga(x1, x2) zero = zero isPlus_out_gga(x1, x2, x3) = isPlus_out_gga(x3) succ(x1) = succ(x1) U31_gga(x1, x2, x3, x4) = U31_gga(x4) pred(x1) = pred(x1) U32_gga(x1, x2, x3, x4) = U32_gga(x4) U33_gga(x1, x2, x3, x4) = U33_gga(x4) U34_gga(x1, x2, x3, x4) = U34_gga(x4) U25_ga(x1, x2, x3) = U25_ga(x3) isTimes_in_gga(x1, x2, x3) = isTimes_in_gga(x1, x2) isTimes_out_gga(x1, x2, x3) = isTimes_out_gga(x3) U41_gga(x1, x2, x3, x4) = U41_gga(x1, x4) U43_gga(x1, x2, x3, x4) = U43_gga(x1, x4) U45_gga(x1, x2, x3, x4) = U45_gga(x1, x4) U47_gga(x1, x2, x3, x4) = U47_gga(x1, x4) U48_gga(x1, x2, x3, x4) = U48_gga(x4) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x3) U35_gga(x1, x2, x3) = U35_gga(x3) U36_gga(x1, x2, x3) = U36_gga(x3) U37_gga(x1, x2, x3, x4) = U37_gga(x4) U38_gga(x1, x2, x3, x4) = U38_gga(x4) U39_gga(x1, x2, x3, x4) = U39_gga(x4) U40_gga(x1, x2, x3, x4) = U40_gga(x4) U46_gga(x1, x2, x3, x4) = U46_gga(x4) U44_gga(x1, x2, x3, x4) = U44_gga(x4) U42_gga(x1, x2, x3, x4) = U42_gga(x4) U26_ga(x1, x2, x3) = U26_ga(x3) U27_ga(x1, x2, x3) = U27_ga(x3) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1) U28_ga(x1, x2, x3, x4) = U28_ga(x3, x4) iota_in_ga(x1, x2) = iota_in_ga(x1) U2_ga(x1, x2, x3) = U2_ga(x3) iota1_in_gga(x1, x2, x3) = iota1_in_gga(x1, x2) iota1_out_gga(x1, x2, x3) = iota1_out_gga(x3) U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x4) iota_out_ga(x1, x2) = iota_out_ga(x2) .(x1, x2) = .(x1, x2) U29_ga(x1, x2, x3) = U29_ga(x3) pds1_in_ggag(x1, x2, x3, x4) = pds1_in_ggag(x1, x2, x4) [] = [] pds1_out_ggag(x1, x2, x3, x4) = pds1_out_ggag(x3) U21_ggag(x1, x2, x3, x4, x5, x6) = U21_ggag(x1, x2, x5, x6) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1) U1_ag(x1, x2, x3, x4) = U1_ag(x4) U22_ggag(x1, x2, x3, x4, x5, x6) = U22_ggag(x2, x3, x5, x6) check_in_gggag(x1, x2, x3, x4, x5) = check_in_gggag(x1, x2, x3, x5) check_out_gggag(x1, x2, x3, x4, x5) = check_out_gggag(x4) U17_gggag(x1, x2, x3, x4, x5, x6) = U17_gggag(x3, x6) dif_in_gggaa(x1, x2, x3, x4, x5) = dif_in_gggaa(x1, x2, x3) dif_out_gggaa(x1, x2, x3, x4, x5) = dif_out_gggaa(x4, x5) U6_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gggaa(x2, x3, x4, x9) U7_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_gggaa(x2, x3, x4, x9, x10) U8_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_gggaa(x2, x3, x4, x5, x9, x10) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg U35_ggg(x1, x2, x3) = U35_ggg(x3) U36_ggg(x1, x2, x3) = U36_ggg(x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x4) U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_gggaa(x2, x3, x4, x5, x9) U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_gggaa(x5, x7, x9) U18_gggag(x1, x2, x3, x4, x5, x6, x7) = U18_gggag(x6, x7) mergedelete_in_gga(x1, x2, x3) = mergedelete_in_gga(x1, x2) mergedelete_out_gga(x1, x2, x3) = mergedelete_out_gga(x3) U12_gga(x1, x2, x3, x4, x5) = U12_gga(x5) U13_gga(x1, x2, x3, x4, x5, x6) = U13_gga(x1, x2, x3, x4, x6) U14_gga(x1, x2, x3, x4, x5, x6, x7) = U14_gga(x1, x2, x3, x4, x6, x7) U15_gga(x1, x2, x3, x4, x5, x6) = U15_gga(x1, x2, x3, x4, x6) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg U64_gg(x1, x2, x3) = U64_gg(x3) U65_gg(x1, x2, x3) = U65_gg(x3) U16_gga(x1, x2, x3, x4, x5, x6) = U16_gga(x3, x6) U19_gggag(x1, x2, x3, x4, x5, x6, x7) = U19_gggag(x6, x7) rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2) rev_out_gga(x1, x2, x3) = rev_out_gga(x3) U11_gga(x1, x2, x3, x4, x5) = U11_gga(x5) U20_gggag(x1, x2, x3, x4, x5, x6) = U20_gggag(x6) U23_ggag(x1, x2, x3, x4, x5, x6) = U23_ggag(x3, x6) pds_out_ga(x1, x2) = pds_out_ga(x2) ---------------------------------------- (9) 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: PDS_IN_GA(Order, .(zero, Ans)) -> U24_GA(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) PDS_IN_GA(Order, .(zero, Ans)) -> ISPLUS_IN_GGA(Order, succ(zero), U) ISPLUS_IN_GGA(succ(X), succ(Y), succ(succ(Z))) -> U31_GGA(X, Y, Z, isPlus_in_gga(X, Y, Z)) ISPLUS_IN_GGA(succ(X), succ(Y), succ(succ(Z))) -> ISPLUS_IN_GGA(X, Y, Z) ISPLUS_IN_GGA(succ(X), pred(Y), Z) -> U32_GGA(X, Y, Z, isPlus_in_gga(X, Y, Z)) ISPLUS_IN_GGA(succ(X), pred(Y), Z) -> ISPLUS_IN_GGA(X, Y, Z) ISPLUS_IN_GGA(pred(X), succ(Y), Z) -> U33_GGA(X, Y, Z, isPlus_in_gga(X, Y, Z)) ISPLUS_IN_GGA(pred(X), succ(Y), Z) -> ISPLUS_IN_GGA(X, Y, Z) ISPLUS_IN_GGA(pred(X), pred(Y), pred(pred(Z))) -> U34_GGA(X, Y, Z, isPlus_in_gga(X, Y, Z)) ISPLUS_IN_GGA(pred(X), pred(Y), pred(pred(Z))) -> ISPLUS_IN_GGA(X, Y, Z) U24_GA(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_GA(Order, Ans, isTimes_in_gga(Order, U, U1)) U24_GA(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> ISTIMES_IN_GGA(Order, U, U1) ISTIMES_IN_GGA(succ(X), succ(Y), Z) -> U41_GGA(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) ISTIMES_IN_GGA(succ(X), succ(Y), Z) -> ISTIMES_IN_GGA(succ(X), Y, A) ISTIMES_IN_GGA(succ(X), pred(Y), Z) -> U43_GGA(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) ISTIMES_IN_GGA(succ(X), pred(Y), Z) -> ISTIMES_IN_GGA(succ(X), Y, A) ISTIMES_IN_GGA(pred(X), succ(Y), Z) -> U45_GGA(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) ISTIMES_IN_GGA(pred(X), succ(Y), Z) -> ISTIMES_IN_GGA(pred(X), Y, A) ISTIMES_IN_GGA(pred(X), pred(Y), Z) -> U47_GGA(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) ISTIMES_IN_GGA(pred(X), pred(Y), Z) -> ISTIMES_IN_GGA(pred(X), Y, A) U47_GGA(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_GGA(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) U47_GGA(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> ISMINUS_IN_GGA(A, pred(X), Z) ISMINUS_IN_GGA(zero, succ(Y), pred(Z)) -> U35_GGA(Y, Z, isMinus_in_gga(zero, Y, Z)) ISMINUS_IN_GGA(zero, succ(Y), pred(Z)) -> ISMINUS_IN_GGA(zero, Y, Z) ISMINUS_IN_GGA(zero, pred(Y), succ(Z)) -> U36_GGA(Y, Z, isMinus_in_gga(zero, Y, Z)) ISMINUS_IN_GGA(zero, pred(Y), succ(Z)) -> ISMINUS_IN_GGA(zero, Y, Z) ISMINUS_IN_GGA(succ(X), succ(Y), Z) -> U37_GGA(X, Y, Z, isMinus_in_gga(X, Y, Z)) ISMINUS_IN_GGA(succ(X), succ(Y), Z) -> ISMINUS_IN_GGA(X, Y, Z) ISMINUS_IN_GGA(succ(X), pred(Y), succ(succ(Z))) -> U38_GGA(X, Y, Z, isMinus_in_gga(X, Y, Z)) ISMINUS_IN_GGA(succ(X), pred(Y), succ(succ(Z))) -> ISMINUS_IN_GGA(X, Y, Z) ISMINUS_IN_GGA(pred(X), succ(Y), pred(pred(Z))) -> U39_GGA(X, Y, Z, isMinus_in_gga(X, Y, Z)) ISMINUS_IN_GGA(pred(X), succ(Y), pred(pred(Z))) -> ISMINUS_IN_GGA(X, Y, Z) ISMINUS_IN_GGA(pred(X), pred(Y), Z) -> U40_GGA(X, Y, Z, isMinus_in_gga(X, Y, Z)) ISMINUS_IN_GGA(pred(X), pred(Y), Z) -> ISMINUS_IN_GGA(X, Y, Z) U45_GGA(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_GGA(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U45_GGA(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> ISPLUS_IN_GGA(A, pred(X), Z) U43_GGA(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_GGA(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U43_GGA(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> ISMINUS_IN_GGA(A, succ(X), Z) U41_GGA(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_GGA(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U41_GGA(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> ISPLUS_IN_GGA(A, succ(X), Z) U25_GA(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_GA(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U25_GA(Order, Ans, isTimes_out_gga(Order, U, U1)) -> ISPLUS_IN_GGA(U1, succ(zero), U2) U26_GA(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_GA(Order, Ans, =_in_ag(N, U2)) U26_GA(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> =_IN_AG(N, U2) U27_GA(Order, Ans, =_out_ag(N, U2)) -> U28_GA(Order, Ans, N, iota_in_ga(N, .(zero, List))) U27_GA(Order, Ans, =_out_ag(N, U2)) -> IOTA_IN_GA(N, .(zero, List)) IOTA_IN_GA(N, List) -> U2_GA(N, List, iota1_in_gga(zero, N, List)) IOTA_IN_GA(N, List) -> IOTA1_IN_GGA(zero, N, List) IOTA1_IN_GGA(K, N, .(K, List)) -> U3_GGA(K, N, List, isPlus_in_gga(K, succ(zero), U)) IOTA1_IN_GGA(K, N, .(K, List)) -> ISPLUS_IN_GGA(K, succ(zero), U) U3_GGA(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_GGA(K, N, List, =_in_ag(K1, U)) U3_GGA(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> =_IN_AG(K1, U) U4_GGA(K, N, List, =_out_ag(K1, U)) -> U5_GGA(K, N, List, iota1_in_gga(K1, N, List)) U4_GGA(K, N, List, =_out_ag(K1, U)) -> IOTA1_IN_GGA(K1, N, List) U28_GA(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_GA(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) U28_GA(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> PDS1_IN_GGAG(List, .(zero, []), Ans, N) PDS1_IN_GGAG(Unused, List, .(Choice, Rest), Mod) -> U21_GGAG(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) PDS1_IN_GGAG(Unused, List, .(Choice, Rest), Mod) -> MEMBER_IN_AG(Choice, Unused) MEMBER_IN_AG(X, .(X1, Xs)) -> U1_AG(X, X1, Xs, member_in_ag(X, Xs)) MEMBER_IN_AG(X, .(X1, Xs)) -> MEMBER_IN_AG(X, Xs) U21_GGAG(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_GGAG(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) U21_GGAG(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> CHECK_IN_GGGAG(List, Choice, Unused, U3, Mod) CHECK_IN_GGGAG(S, Choice, Old, L3, Modulus) -> U17_GGGAG(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) CHECK_IN_GGGAG(S, Choice, Old, L3, Modulus) -> DIF_IN_GGGAA(S, Choice, Modulus, Ds, Dds) DIF_IN_GGGAA(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) DIF_IN_GGGAA(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> ISMINUS_IN_GGA(Val, S, U) U6_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U6_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> =_IN_AG(D, U) U7_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) U7_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> ISMINUS_IN_GGG(Mod, D, U) ISMINUS_IN_GGG(zero, succ(Y), pred(Z)) -> U35_GGG(Y, Z, isMinus_in_ggg(zero, Y, Z)) ISMINUS_IN_GGG(zero, succ(Y), pred(Z)) -> ISMINUS_IN_GGG(zero, Y, Z) ISMINUS_IN_GGG(zero, pred(Y), succ(Z)) -> U36_GGG(Y, Z, isMinus_in_ggg(zero, Y, Z)) ISMINUS_IN_GGG(zero, pred(Y), succ(Z)) -> ISMINUS_IN_GGG(zero, Y, Z) ISMINUS_IN_GGG(succ(X), succ(Y), Z) -> U37_GGG(X, Y, Z, isMinus_in_ggg(X, Y, Z)) ISMINUS_IN_GGG(succ(X), succ(Y), Z) -> ISMINUS_IN_GGG(X, Y, Z) ISMINUS_IN_GGG(succ(X), pred(Y), succ(succ(Z))) -> U38_GGG(X, Y, Z, isMinus_in_ggg(X, Y, Z)) ISMINUS_IN_GGG(succ(X), pred(Y), succ(succ(Z))) -> ISMINUS_IN_GGG(X, Y, Z) ISMINUS_IN_GGG(pred(X), succ(Y), pred(pred(Z))) -> U39_GGG(X, Y, Z, isMinus_in_ggg(X, Y, Z)) ISMINUS_IN_GGG(pred(X), succ(Y), pred(pred(Z))) -> ISMINUS_IN_GGG(X, Y, Z) ISMINUS_IN_GGG(pred(X), pred(Y), Z) -> U40_GGG(X, Y, Z, isMinus_in_ggg(X, Y, Z)) ISMINUS_IN_GGG(pred(X), pred(Y), Z) -> ISMINUS_IN_GGG(X, Y, Z) U8_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U8_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> =_IN_AG(D2, U) U9_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U9_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> DIF_IN_GGGAA(Ss, Val, Mod, Ds, D2s) U17_GGGAG(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_GGGAG(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) U17_GGGAG(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> MERGEDELETE_IN_GGA(Ds, Old, L2) MERGEDELETE_IN_GGA(.(D, Ds), .(D, R), L2) -> U12_GGA(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) MERGEDELETE_IN_GGA(.(D, Ds), .(D, R), L2) -> MERGEDELETE_IN_GGA(Ds, R, L2) MERGEDELETE_IN_GGA(.(D, Ds), .(X, R), .(X, L2)) -> U13_GGA(D, Ds, X, R, L2, =_in_ag(X1, D)) MERGEDELETE_IN_GGA(.(D, Ds), .(X, R), .(X, L2)) -> =_IN_AG(X1, D) U13_GGA(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_GGA(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U13_GGA(D, Ds, X, R, L2, =_out_ag(X1, D)) -> =_IN_AG(X2, X) U14_GGA(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_GGA(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) U14_GGA(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> ISGREATER_IN_GG(X1, X2) ISGREATER_IN_GG(succ(X), succ(Y)) -> U64_GG(X, Y, isGreater_in_gg(X, Y)) ISGREATER_IN_GG(succ(X), succ(Y)) -> ISGREATER_IN_GG(X, Y) ISGREATER_IN_GG(pred(X), pred(Y)) -> U65_GG(X, Y, isGreater_in_gg(X, Y)) ISGREATER_IN_GG(pred(X), pred(Y)) -> ISGREATER_IN_GG(X, Y) U15_GGA(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_GGA(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U15_GGA(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> MERGEDELETE_IN_GGA(.(D, Ds), R, L2) U18_GGGAG(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_GGGAG(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) U18_GGGAG(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> REV_IN_GGA(Dds, [], Rds) REV_IN_GGA(.(X, Xs), Y, L) -> U11_GGA(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) REV_IN_GGA(.(X, Xs), Y, L) -> REV_IN_GGA(Xs, .(X, Y), L) U19_GGGAG(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_GGGAG(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U19_GGGAG(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> MERGEDELETE_IN_GGA(Rds, L2, L3) U22_GGAG(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_GGAG(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U22_GGAG(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> PDS1_IN_GGAG(U3, .(Choice, List), Rest, Mod) The TRS R consists of the following rules: pds_in_ga(Order, .(zero, Ans)) -> U24_ga(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) isPlus_in_gga(zero, X, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), zero, succ(X)) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), succ(Y), succ(succ(Z))) -> U31_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(succ(X), pred(Y), Z) -> U32_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), zero, pred(X)) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), succ(Y), Z) -> U33_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), pred(Y), pred(pred(Z))) -> U34_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) U34_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U33_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U32_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U31_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U24_ga(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_ga(Order, Ans, isTimes_in_gga(Order, U, U1)) isTimes_in_gga(X, zero, zero) -> isTimes_out_gga(X, zero, zero) isTimes_in_gga(zero, succ(Y), zero) -> isTimes_out_gga(zero, succ(Y), zero) isTimes_in_gga(zero, pred(Y), zero) -> isTimes_out_gga(zero, pred(Y), zero) isTimes_in_gga(succ(X), succ(Y), Z) -> U41_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(succ(X), pred(Y), Z) -> U43_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(pred(X), succ(Y), Z) -> U45_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) isTimes_in_gga(pred(X), pred(Y), Z) -> U47_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) U47_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_gga(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U48_gga(X, Y, Z, isMinus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), pred(Y), Z) U45_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_gga(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U46_gga(X, Y, Z, isPlus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), succ(Y), Z) U43_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_gga(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U44_gga(X, Y, Z, isMinus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), pred(Y), Z) U41_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_gga(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U42_gga(X, Y, Z, isPlus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), succ(Y), Z) U25_ga(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_ga(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U26_ga(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_ga(Order, Ans, =_in_ag(N, U2)) =_in_ag(X, X) -> =_out_ag(X, X) U27_ga(Order, Ans, =_out_ag(N, U2)) -> U28_ga(Order, Ans, N, iota_in_ga(N, .(zero, List))) iota_in_ga(N, List) -> U2_ga(N, List, iota1_in_gga(zero, N, List)) iota1_in_gga(K, K, []) -> iota1_out_gga(K, K, []) iota1_in_gga(K, N, .(K, List)) -> U3_gga(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_gga(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_gga(K, N, List, =_in_ag(K1, U)) U4_gga(K, N, List, =_out_ag(K1, U)) -> U5_gga(K, N, List, iota1_in_gga(K1, N, List)) U5_gga(K, N, List, iota1_out_gga(K1, N, List)) -> iota1_out_gga(K, N, .(K, List)) U2_ga(N, List, iota1_out_gga(zero, N, List)) -> iota_out_ga(N, List) U28_ga(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_ga(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) pds1_in_ggag([], X6, [], X7) -> pds1_out_ggag([], X6, [], X7) pds1_in_ggag(Unused, List, .(Choice, Rest), Mod) -> U21_ggag(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(X, .(X1, Xs)) -> U1_ag(X, X1, Xs, member_in_ag(X, Xs)) U1_ag(X, X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) U21_ggag(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_ggag(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) check_in_gggag([], X4, L, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, L3, Modulus) -> U17_gggag(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) dif_in_gggaa([], X2, X3, [], []) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U17_gggag(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) mergedelete_in_gga([], L, L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R), L2) -> U12_gga(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) mergedelete_in_gga(.(D, Ds), .(X, R), .(X, L2)) -> U13_gga(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_gga(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_gga(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U15_gga(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U16_gga(D, Ds, X, R, L2, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U12_gga(D, Ds, R, L2, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) rev_in_gga([], L, L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y, L) -> U11_gga(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) U11_gga(X, Xs, Y, L, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U22_ggag(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_ggag(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U23_ggag(Unused, List, Choice, Rest, Mod, pds1_out_ggag(U3, .(Choice, List), Rest, Mod)) -> pds1_out_ggag(Unused, List, .(Choice, Rest), Mod) U29_ga(Order, Ans, pds1_out_ggag(List, .(zero, []), Ans, N)) -> pds_out_ga(Order, .(zero, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U24_ga(x1, x2, x3) = U24_ga(x1, x3) isPlus_in_gga(x1, x2, x3) = isPlus_in_gga(x1, x2) zero = zero isPlus_out_gga(x1, x2, x3) = isPlus_out_gga(x3) succ(x1) = succ(x1) U31_gga(x1, x2, x3, x4) = U31_gga(x4) pred(x1) = pred(x1) U32_gga(x1, x2, x3, x4) = U32_gga(x4) U33_gga(x1, x2, x3, x4) = U33_gga(x4) U34_gga(x1, x2, x3, x4) = U34_gga(x4) U25_ga(x1, x2, x3) = U25_ga(x3) isTimes_in_gga(x1, x2, x3) = isTimes_in_gga(x1, x2) isTimes_out_gga(x1, x2, x3) = isTimes_out_gga(x3) U41_gga(x1, x2, x3, x4) = U41_gga(x1, x4) U43_gga(x1, x2, x3, x4) = U43_gga(x1, x4) U45_gga(x1, x2, x3, x4) = U45_gga(x1, x4) U47_gga(x1, x2, x3, x4) = U47_gga(x1, x4) U48_gga(x1, x2, x3, x4) = U48_gga(x4) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x3) U35_gga(x1, x2, x3) = U35_gga(x3) U36_gga(x1, x2, x3) = U36_gga(x3) U37_gga(x1, x2, x3, x4) = U37_gga(x4) U38_gga(x1, x2, x3, x4) = U38_gga(x4) U39_gga(x1, x2, x3, x4) = U39_gga(x4) U40_gga(x1, x2, x3, x4) = U40_gga(x4) U46_gga(x1, x2, x3, x4) = U46_gga(x4) U44_gga(x1, x2, x3, x4) = U44_gga(x4) U42_gga(x1, x2, x3, x4) = U42_gga(x4) U26_ga(x1, x2, x3) = U26_ga(x3) U27_ga(x1, x2, x3) = U27_ga(x3) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1) U28_ga(x1, x2, x3, x4) = U28_ga(x3, x4) iota_in_ga(x1, x2) = iota_in_ga(x1) U2_ga(x1, x2, x3) = U2_ga(x3) iota1_in_gga(x1, x2, x3) = iota1_in_gga(x1, x2) iota1_out_gga(x1, x2, x3) = iota1_out_gga(x3) U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x4) iota_out_ga(x1, x2) = iota_out_ga(x2) .(x1, x2) = .(x1, x2) U29_ga(x1, x2, x3) = U29_ga(x3) pds1_in_ggag(x1, x2, x3, x4) = pds1_in_ggag(x1, x2, x4) [] = [] pds1_out_ggag(x1, x2, x3, x4) = pds1_out_ggag(x3) U21_ggag(x1, x2, x3, x4, x5, x6) = U21_ggag(x1, x2, x5, x6) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1) U1_ag(x1, x2, x3, x4) = U1_ag(x4) U22_ggag(x1, x2, x3, x4, x5, x6) = U22_ggag(x2, x3, x5, x6) check_in_gggag(x1, x2, x3, x4, x5) = check_in_gggag(x1, x2, x3, x5) check_out_gggag(x1, x2, x3, x4, x5) = check_out_gggag(x4) U17_gggag(x1, x2, x3, x4, x5, x6) = U17_gggag(x3, x6) dif_in_gggaa(x1, x2, x3, x4, x5) = dif_in_gggaa(x1, x2, x3) dif_out_gggaa(x1, x2, x3, x4, x5) = dif_out_gggaa(x4, x5) U6_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gggaa(x2, x3, x4, x9) U7_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_gggaa(x2, x3, x4, x9, x10) U8_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_gggaa(x2, x3, x4, x5, x9, x10) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg U35_ggg(x1, x2, x3) = U35_ggg(x3) U36_ggg(x1, x2, x3) = U36_ggg(x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x4) U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_gggaa(x2, x3, x4, x5, x9) U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_gggaa(x5, x7, x9) U18_gggag(x1, x2, x3, x4, x5, x6, x7) = U18_gggag(x6, x7) mergedelete_in_gga(x1, x2, x3) = mergedelete_in_gga(x1, x2) mergedelete_out_gga(x1, x2, x3) = mergedelete_out_gga(x3) U12_gga(x1, x2, x3, x4, x5) = U12_gga(x5) U13_gga(x1, x2, x3, x4, x5, x6) = U13_gga(x1, x2, x3, x4, x6) U14_gga(x1, x2, x3, x4, x5, x6, x7) = U14_gga(x1, x2, x3, x4, x6, x7) U15_gga(x1, x2, x3, x4, x5, x6) = U15_gga(x1, x2, x3, x4, x6) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg U64_gg(x1, x2, x3) = U64_gg(x3) U65_gg(x1, x2, x3) = U65_gg(x3) U16_gga(x1, x2, x3, x4, x5, x6) = U16_gga(x3, x6) U19_gggag(x1, x2, x3, x4, x5, x6, x7) = U19_gggag(x6, x7) rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2) rev_out_gga(x1, x2, x3) = rev_out_gga(x3) U11_gga(x1, x2, x3, x4, x5) = U11_gga(x5) U20_gggag(x1, x2, x3, x4, x5, x6) = U20_gggag(x6) U23_ggag(x1, x2, x3, x4, x5, x6) = U23_ggag(x3, x6) pds_out_ga(x1, x2) = pds_out_ga(x2) PDS_IN_GA(x1, x2) = PDS_IN_GA(x1) U24_GA(x1, x2, x3) = U24_GA(x1, x3) ISPLUS_IN_GGA(x1, x2, x3) = ISPLUS_IN_GGA(x1, x2) U31_GGA(x1, x2, x3, x4) = U31_GGA(x4) U32_GGA(x1, x2, x3, x4) = U32_GGA(x4) U33_GGA(x1, x2, x3, x4) = U33_GGA(x4) U34_GGA(x1, x2, x3, x4) = U34_GGA(x4) U25_GA(x1, x2, x3) = U25_GA(x3) ISTIMES_IN_GGA(x1, x2, x3) = ISTIMES_IN_GGA(x1, x2) U41_GGA(x1, x2, x3, x4) = U41_GGA(x1, x4) U43_GGA(x1, x2, x3, x4) = U43_GGA(x1, x4) U45_GGA(x1, x2, x3, x4) = U45_GGA(x1, x4) U47_GGA(x1, x2, x3, x4) = U47_GGA(x1, x4) U48_GGA(x1, x2, x3, x4) = U48_GGA(x4) ISMINUS_IN_GGA(x1, x2, x3) = ISMINUS_IN_GGA(x1, x2) U35_GGA(x1, x2, x3) = U35_GGA(x3) U36_GGA(x1, x2, x3) = U36_GGA(x3) U37_GGA(x1, x2, x3, x4) = U37_GGA(x4) U38_GGA(x1, x2, x3, x4) = U38_GGA(x4) U39_GGA(x1, x2, x3, x4) = U39_GGA(x4) U40_GGA(x1, x2, x3, x4) = U40_GGA(x4) U46_GGA(x1, x2, x3, x4) = U46_GGA(x4) U44_GGA(x1, x2, x3, x4) = U44_GGA(x4) U42_GGA(x1, x2, x3, x4) = U42_GGA(x4) U26_GA(x1, x2, x3) = U26_GA(x3) U27_GA(x1, x2, x3) = U27_GA(x3) =_IN_AG(x1, x2) = =_IN_AG(x2) U28_GA(x1, x2, x3, x4) = U28_GA(x3, x4) IOTA_IN_GA(x1, x2) = IOTA_IN_GA(x1) U2_GA(x1, x2, x3) = U2_GA(x3) IOTA1_IN_GGA(x1, x2, x3) = IOTA1_IN_GGA(x1, x2) U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4) U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4) U5_GGA(x1, x2, x3, x4) = U5_GGA(x1, x4) U29_GA(x1, x2, x3) = U29_GA(x3) PDS1_IN_GGAG(x1, x2, x3, x4) = PDS1_IN_GGAG(x1, x2, x4) U21_GGAG(x1, x2, x3, x4, x5, x6) = U21_GGAG(x1, x2, x5, x6) MEMBER_IN_AG(x1, x2) = MEMBER_IN_AG(x2) U1_AG(x1, x2, x3, x4) = U1_AG(x4) U22_GGAG(x1, x2, x3, x4, x5, x6) = U22_GGAG(x2, x3, x5, x6) CHECK_IN_GGGAG(x1, x2, x3, x4, x5) = CHECK_IN_GGGAG(x1, x2, x3, x5) U17_GGGAG(x1, x2, x3, x4, x5, x6) = U17_GGGAG(x3, x6) DIF_IN_GGGAA(x1, x2, x3, x4, x5) = DIF_IN_GGGAA(x1, x2, x3) U6_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_GGGAA(x2, x3, x4, x9) U7_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_GGGAA(x2, x3, x4, x9, x10) U8_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_GGGAA(x2, x3, x4, x5, x9, x10) ISMINUS_IN_GGG(x1, x2, x3) = ISMINUS_IN_GGG(x1, x2, x3) U35_GGG(x1, x2, x3) = U35_GGG(x3) U36_GGG(x1, x2, x3) = U36_GGG(x3) U37_GGG(x1, x2, x3, x4) = U37_GGG(x4) U38_GGG(x1, x2, x3, x4) = U38_GGG(x4) U39_GGG(x1, x2, x3, x4) = U39_GGG(x4) U40_GGG(x1, x2, x3, x4) = U40_GGG(x4) U9_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_GGGAA(x2, x3, x4, x5, x9) U10_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_GGGAA(x5, x7, x9) U18_GGGAG(x1, x2, x3, x4, x5, x6, x7) = U18_GGGAG(x6, x7) MERGEDELETE_IN_GGA(x1, x2, x3) = MERGEDELETE_IN_GGA(x1, x2) U12_GGA(x1, x2, x3, x4, x5) = U12_GGA(x5) U13_GGA(x1, x2, x3, x4, x5, x6) = U13_GGA(x1, x2, x3, x4, x6) U14_GGA(x1, x2, x3, x4, x5, x6, x7) = U14_GGA(x1, x2, x3, x4, x6, x7) U15_GGA(x1, x2, x3, x4, x5, x6) = U15_GGA(x1, x2, x3, x4, x6) ISGREATER_IN_GG(x1, x2) = ISGREATER_IN_GG(x1, x2) U64_GG(x1, x2, x3) = U64_GG(x3) U65_GG(x1, x2, x3) = U65_GG(x3) U16_GGA(x1, x2, x3, x4, x5, x6) = U16_GGA(x3, x6) U19_GGGAG(x1, x2, x3, x4, x5, x6, x7) = U19_GGGAG(x6, x7) REV_IN_GGA(x1, x2, x3) = REV_IN_GGA(x1, x2) U11_GGA(x1, x2, x3, x4, x5) = U11_GGA(x5) U20_GGGAG(x1, x2, x3, x4, x5, x6) = U20_GGGAG(x6) U23_GGAG(x1, x2, x3, x4, x5, x6) = U23_GGAG(x3, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (10) Obligation: Pi DP problem: The TRS P consists of the following rules: PDS_IN_GA(Order, .(zero, Ans)) -> U24_GA(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) PDS_IN_GA(Order, .(zero, Ans)) -> ISPLUS_IN_GGA(Order, succ(zero), U) ISPLUS_IN_GGA(succ(X), succ(Y), succ(succ(Z))) -> U31_GGA(X, Y, Z, isPlus_in_gga(X, Y, Z)) ISPLUS_IN_GGA(succ(X), succ(Y), succ(succ(Z))) -> ISPLUS_IN_GGA(X, Y, Z) ISPLUS_IN_GGA(succ(X), pred(Y), Z) -> U32_GGA(X, Y, Z, isPlus_in_gga(X, Y, Z)) ISPLUS_IN_GGA(succ(X), pred(Y), Z) -> ISPLUS_IN_GGA(X, Y, Z) ISPLUS_IN_GGA(pred(X), succ(Y), Z) -> U33_GGA(X, Y, Z, isPlus_in_gga(X, Y, Z)) ISPLUS_IN_GGA(pred(X), succ(Y), Z) -> ISPLUS_IN_GGA(X, Y, Z) ISPLUS_IN_GGA(pred(X), pred(Y), pred(pred(Z))) -> U34_GGA(X, Y, Z, isPlus_in_gga(X, Y, Z)) ISPLUS_IN_GGA(pred(X), pred(Y), pred(pred(Z))) -> ISPLUS_IN_GGA(X, Y, Z) U24_GA(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_GA(Order, Ans, isTimes_in_gga(Order, U, U1)) U24_GA(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> ISTIMES_IN_GGA(Order, U, U1) ISTIMES_IN_GGA(succ(X), succ(Y), Z) -> U41_GGA(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) ISTIMES_IN_GGA(succ(X), succ(Y), Z) -> ISTIMES_IN_GGA(succ(X), Y, A) ISTIMES_IN_GGA(succ(X), pred(Y), Z) -> U43_GGA(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) ISTIMES_IN_GGA(succ(X), pred(Y), Z) -> ISTIMES_IN_GGA(succ(X), Y, A) ISTIMES_IN_GGA(pred(X), succ(Y), Z) -> U45_GGA(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) ISTIMES_IN_GGA(pred(X), succ(Y), Z) -> ISTIMES_IN_GGA(pred(X), Y, A) ISTIMES_IN_GGA(pred(X), pred(Y), Z) -> U47_GGA(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) ISTIMES_IN_GGA(pred(X), pred(Y), Z) -> ISTIMES_IN_GGA(pred(X), Y, A) U47_GGA(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_GGA(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) U47_GGA(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> ISMINUS_IN_GGA(A, pred(X), Z) ISMINUS_IN_GGA(zero, succ(Y), pred(Z)) -> U35_GGA(Y, Z, isMinus_in_gga(zero, Y, Z)) ISMINUS_IN_GGA(zero, succ(Y), pred(Z)) -> ISMINUS_IN_GGA(zero, Y, Z) ISMINUS_IN_GGA(zero, pred(Y), succ(Z)) -> U36_GGA(Y, Z, isMinus_in_gga(zero, Y, Z)) ISMINUS_IN_GGA(zero, pred(Y), succ(Z)) -> ISMINUS_IN_GGA(zero, Y, Z) ISMINUS_IN_GGA(succ(X), succ(Y), Z) -> U37_GGA(X, Y, Z, isMinus_in_gga(X, Y, Z)) ISMINUS_IN_GGA(succ(X), succ(Y), Z) -> ISMINUS_IN_GGA(X, Y, Z) ISMINUS_IN_GGA(succ(X), pred(Y), succ(succ(Z))) -> U38_GGA(X, Y, Z, isMinus_in_gga(X, Y, Z)) ISMINUS_IN_GGA(succ(X), pred(Y), succ(succ(Z))) -> ISMINUS_IN_GGA(X, Y, Z) ISMINUS_IN_GGA(pred(X), succ(Y), pred(pred(Z))) -> U39_GGA(X, Y, Z, isMinus_in_gga(X, Y, Z)) ISMINUS_IN_GGA(pred(X), succ(Y), pred(pred(Z))) -> ISMINUS_IN_GGA(X, Y, Z) ISMINUS_IN_GGA(pred(X), pred(Y), Z) -> U40_GGA(X, Y, Z, isMinus_in_gga(X, Y, Z)) ISMINUS_IN_GGA(pred(X), pred(Y), Z) -> ISMINUS_IN_GGA(X, Y, Z) U45_GGA(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_GGA(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U45_GGA(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> ISPLUS_IN_GGA(A, pred(X), Z) U43_GGA(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_GGA(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U43_GGA(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> ISMINUS_IN_GGA(A, succ(X), Z) U41_GGA(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_GGA(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U41_GGA(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> ISPLUS_IN_GGA(A, succ(X), Z) U25_GA(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_GA(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U25_GA(Order, Ans, isTimes_out_gga(Order, U, U1)) -> ISPLUS_IN_GGA(U1, succ(zero), U2) U26_GA(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_GA(Order, Ans, =_in_ag(N, U2)) U26_GA(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> =_IN_AG(N, U2) U27_GA(Order, Ans, =_out_ag(N, U2)) -> U28_GA(Order, Ans, N, iota_in_ga(N, .(zero, List))) U27_GA(Order, Ans, =_out_ag(N, U2)) -> IOTA_IN_GA(N, .(zero, List)) IOTA_IN_GA(N, List) -> U2_GA(N, List, iota1_in_gga(zero, N, List)) IOTA_IN_GA(N, List) -> IOTA1_IN_GGA(zero, N, List) IOTA1_IN_GGA(K, N, .(K, List)) -> U3_GGA(K, N, List, isPlus_in_gga(K, succ(zero), U)) IOTA1_IN_GGA(K, N, .(K, List)) -> ISPLUS_IN_GGA(K, succ(zero), U) U3_GGA(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_GGA(K, N, List, =_in_ag(K1, U)) U3_GGA(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> =_IN_AG(K1, U) U4_GGA(K, N, List, =_out_ag(K1, U)) -> U5_GGA(K, N, List, iota1_in_gga(K1, N, List)) U4_GGA(K, N, List, =_out_ag(K1, U)) -> IOTA1_IN_GGA(K1, N, List) U28_GA(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_GA(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) U28_GA(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> PDS1_IN_GGAG(List, .(zero, []), Ans, N) PDS1_IN_GGAG(Unused, List, .(Choice, Rest), Mod) -> U21_GGAG(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) PDS1_IN_GGAG(Unused, List, .(Choice, Rest), Mod) -> MEMBER_IN_AG(Choice, Unused) MEMBER_IN_AG(X, .(X1, Xs)) -> U1_AG(X, X1, Xs, member_in_ag(X, Xs)) MEMBER_IN_AG(X, .(X1, Xs)) -> MEMBER_IN_AG(X, Xs) U21_GGAG(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_GGAG(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) U21_GGAG(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> CHECK_IN_GGGAG(List, Choice, Unused, U3, Mod) CHECK_IN_GGGAG(S, Choice, Old, L3, Modulus) -> U17_GGGAG(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) CHECK_IN_GGGAG(S, Choice, Old, L3, Modulus) -> DIF_IN_GGGAA(S, Choice, Modulus, Ds, Dds) DIF_IN_GGGAA(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) DIF_IN_GGGAA(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> ISMINUS_IN_GGA(Val, S, U) U6_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U6_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> =_IN_AG(D, U) U7_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) U7_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> ISMINUS_IN_GGG(Mod, D, U) ISMINUS_IN_GGG(zero, succ(Y), pred(Z)) -> U35_GGG(Y, Z, isMinus_in_ggg(zero, Y, Z)) ISMINUS_IN_GGG(zero, succ(Y), pred(Z)) -> ISMINUS_IN_GGG(zero, Y, Z) ISMINUS_IN_GGG(zero, pred(Y), succ(Z)) -> U36_GGG(Y, Z, isMinus_in_ggg(zero, Y, Z)) ISMINUS_IN_GGG(zero, pred(Y), succ(Z)) -> ISMINUS_IN_GGG(zero, Y, Z) ISMINUS_IN_GGG(succ(X), succ(Y), Z) -> U37_GGG(X, Y, Z, isMinus_in_ggg(X, Y, Z)) ISMINUS_IN_GGG(succ(X), succ(Y), Z) -> ISMINUS_IN_GGG(X, Y, Z) ISMINUS_IN_GGG(succ(X), pred(Y), succ(succ(Z))) -> U38_GGG(X, Y, Z, isMinus_in_ggg(X, Y, Z)) ISMINUS_IN_GGG(succ(X), pred(Y), succ(succ(Z))) -> ISMINUS_IN_GGG(X, Y, Z) ISMINUS_IN_GGG(pred(X), succ(Y), pred(pred(Z))) -> U39_GGG(X, Y, Z, isMinus_in_ggg(X, Y, Z)) ISMINUS_IN_GGG(pred(X), succ(Y), pred(pred(Z))) -> ISMINUS_IN_GGG(X, Y, Z) ISMINUS_IN_GGG(pred(X), pred(Y), Z) -> U40_GGG(X, Y, Z, isMinus_in_ggg(X, Y, Z)) ISMINUS_IN_GGG(pred(X), pred(Y), Z) -> ISMINUS_IN_GGG(X, Y, Z) U8_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U8_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> =_IN_AG(D2, U) U9_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U9_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> DIF_IN_GGGAA(Ss, Val, Mod, Ds, D2s) U17_GGGAG(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_GGGAG(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) U17_GGGAG(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> MERGEDELETE_IN_GGA(Ds, Old, L2) MERGEDELETE_IN_GGA(.(D, Ds), .(D, R), L2) -> U12_GGA(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) MERGEDELETE_IN_GGA(.(D, Ds), .(D, R), L2) -> MERGEDELETE_IN_GGA(Ds, R, L2) MERGEDELETE_IN_GGA(.(D, Ds), .(X, R), .(X, L2)) -> U13_GGA(D, Ds, X, R, L2, =_in_ag(X1, D)) MERGEDELETE_IN_GGA(.(D, Ds), .(X, R), .(X, L2)) -> =_IN_AG(X1, D) U13_GGA(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_GGA(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U13_GGA(D, Ds, X, R, L2, =_out_ag(X1, D)) -> =_IN_AG(X2, X) U14_GGA(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_GGA(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) U14_GGA(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> ISGREATER_IN_GG(X1, X2) ISGREATER_IN_GG(succ(X), succ(Y)) -> U64_GG(X, Y, isGreater_in_gg(X, Y)) ISGREATER_IN_GG(succ(X), succ(Y)) -> ISGREATER_IN_GG(X, Y) ISGREATER_IN_GG(pred(X), pred(Y)) -> U65_GG(X, Y, isGreater_in_gg(X, Y)) ISGREATER_IN_GG(pred(X), pred(Y)) -> ISGREATER_IN_GG(X, Y) U15_GGA(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_GGA(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U15_GGA(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> MERGEDELETE_IN_GGA(.(D, Ds), R, L2) U18_GGGAG(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_GGGAG(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) U18_GGGAG(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> REV_IN_GGA(Dds, [], Rds) REV_IN_GGA(.(X, Xs), Y, L) -> U11_GGA(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) REV_IN_GGA(.(X, Xs), Y, L) -> REV_IN_GGA(Xs, .(X, Y), L) U19_GGGAG(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_GGGAG(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U19_GGGAG(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> MERGEDELETE_IN_GGA(Rds, L2, L3) U22_GGAG(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_GGAG(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U22_GGAG(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> PDS1_IN_GGAG(U3, .(Choice, List), Rest, Mod) The TRS R consists of the following rules: pds_in_ga(Order, .(zero, Ans)) -> U24_ga(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) isPlus_in_gga(zero, X, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), zero, succ(X)) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), succ(Y), succ(succ(Z))) -> U31_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(succ(X), pred(Y), Z) -> U32_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), zero, pred(X)) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), succ(Y), Z) -> U33_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), pred(Y), pred(pred(Z))) -> U34_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) U34_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U33_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U32_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U31_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U24_ga(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_ga(Order, Ans, isTimes_in_gga(Order, U, U1)) isTimes_in_gga(X, zero, zero) -> isTimes_out_gga(X, zero, zero) isTimes_in_gga(zero, succ(Y), zero) -> isTimes_out_gga(zero, succ(Y), zero) isTimes_in_gga(zero, pred(Y), zero) -> isTimes_out_gga(zero, pred(Y), zero) isTimes_in_gga(succ(X), succ(Y), Z) -> U41_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(succ(X), pred(Y), Z) -> U43_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(pred(X), succ(Y), Z) -> U45_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) isTimes_in_gga(pred(X), pred(Y), Z) -> U47_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) U47_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_gga(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U48_gga(X, Y, Z, isMinus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), pred(Y), Z) U45_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_gga(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U46_gga(X, Y, Z, isPlus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), succ(Y), Z) U43_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_gga(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U44_gga(X, Y, Z, isMinus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), pred(Y), Z) U41_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_gga(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U42_gga(X, Y, Z, isPlus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), succ(Y), Z) U25_ga(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_ga(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U26_ga(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_ga(Order, Ans, =_in_ag(N, U2)) =_in_ag(X, X) -> =_out_ag(X, X) U27_ga(Order, Ans, =_out_ag(N, U2)) -> U28_ga(Order, Ans, N, iota_in_ga(N, .(zero, List))) iota_in_ga(N, List) -> U2_ga(N, List, iota1_in_gga(zero, N, List)) iota1_in_gga(K, K, []) -> iota1_out_gga(K, K, []) iota1_in_gga(K, N, .(K, List)) -> U3_gga(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_gga(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_gga(K, N, List, =_in_ag(K1, U)) U4_gga(K, N, List, =_out_ag(K1, U)) -> U5_gga(K, N, List, iota1_in_gga(K1, N, List)) U5_gga(K, N, List, iota1_out_gga(K1, N, List)) -> iota1_out_gga(K, N, .(K, List)) U2_ga(N, List, iota1_out_gga(zero, N, List)) -> iota_out_ga(N, List) U28_ga(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_ga(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) pds1_in_ggag([], X6, [], X7) -> pds1_out_ggag([], X6, [], X7) pds1_in_ggag(Unused, List, .(Choice, Rest), Mod) -> U21_ggag(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(X, .(X1, Xs)) -> U1_ag(X, X1, Xs, member_in_ag(X, Xs)) U1_ag(X, X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) U21_ggag(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_ggag(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) check_in_gggag([], X4, L, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, L3, Modulus) -> U17_gggag(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) dif_in_gggaa([], X2, X3, [], []) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U17_gggag(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) mergedelete_in_gga([], L, L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R), L2) -> U12_gga(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) mergedelete_in_gga(.(D, Ds), .(X, R), .(X, L2)) -> U13_gga(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_gga(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_gga(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U15_gga(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U16_gga(D, Ds, X, R, L2, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U12_gga(D, Ds, R, L2, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) rev_in_gga([], L, L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y, L) -> U11_gga(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) U11_gga(X, Xs, Y, L, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U22_ggag(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_ggag(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U23_ggag(Unused, List, Choice, Rest, Mod, pds1_out_ggag(U3, .(Choice, List), Rest, Mod)) -> pds1_out_ggag(Unused, List, .(Choice, Rest), Mod) U29_ga(Order, Ans, pds1_out_ggag(List, .(zero, []), Ans, N)) -> pds_out_ga(Order, .(zero, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U24_ga(x1, x2, x3) = U24_ga(x1, x3) isPlus_in_gga(x1, x2, x3) = isPlus_in_gga(x1, x2) zero = zero isPlus_out_gga(x1, x2, x3) = isPlus_out_gga(x3) succ(x1) = succ(x1) U31_gga(x1, x2, x3, x4) = U31_gga(x4) pred(x1) = pred(x1) U32_gga(x1, x2, x3, x4) = U32_gga(x4) U33_gga(x1, x2, x3, x4) = U33_gga(x4) U34_gga(x1, x2, x3, x4) = U34_gga(x4) U25_ga(x1, x2, x3) = U25_ga(x3) isTimes_in_gga(x1, x2, x3) = isTimes_in_gga(x1, x2) isTimes_out_gga(x1, x2, x3) = isTimes_out_gga(x3) U41_gga(x1, x2, x3, x4) = U41_gga(x1, x4) U43_gga(x1, x2, x3, x4) = U43_gga(x1, x4) U45_gga(x1, x2, x3, x4) = U45_gga(x1, x4) U47_gga(x1, x2, x3, x4) = U47_gga(x1, x4) U48_gga(x1, x2, x3, x4) = U48_gga(x4) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x3) U35_gga(x1, x2, x3) = U35_gga(x3) U36_gga(x1, x2, x3) = U36_gga(x3) U37_gga(x1, x2, x3, x4) = U37_gga(x4) U38_gga(x1, x2, x3, x4) = U38_gga(x4) U39_gga(x1, x2, x3, x4) = U39_gga(x4) U40_gga(x1, x2, x3, x4) = U40_gga(x4) U46_gga(x1, x2, x3, x4) = U46_gga(x4) U44_gga(x1, x2, x3, x4) = U44_gga(x4) U42_gga(x1, x2, x3, x4) = U42_gga(x4) U26_ga(x1, x2, x3) = U26_ga(x3) U27_ga(x1, x2, x3) = U27_ga(x3) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1) U28_ga(x1, x2, x3, x4) = U28_ga(x3, x4) iota_in_ga(x1, x2) = iota_in_ga(x1) U2_ga(x1, x2, x3) = U2_ga(x3) iota1_in_gga(x1, x2, x3) = iota1_in_gga(x1, x2) iota1_out_gga(x1, x2, x3) = iota1_out_gga(x3) U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x4) iota_out_ga(x1, x2) = iota_out_ga(x2) .(x1, x2) = .(x1, x2) U29_ga(x1, x2, x3) = U29_ga(x3) pds1_in_ggag(x1, x2, x3, x4) = pds1_in_ggag(x1, x2, x4) [] = [] pds1_out_ggag(x1, x2, x3, x4) = pds1_out_ggag(x3) U21_ggag(x1, x2, x3, x4, x5, x6) = U21_ggag(x1, x2, x5, x6) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1) U1_ag(x1, x2, x3, x4) = U1_ag(x4) U22_ggag(x1, x2, x3, x4, x5, x6) = U22_ggag(x2, x3, x5, x6) check_in_gggag(x1, x2, x3, x4, x5) = check_in_gggag(x1, x2, x3, x5) check_out_gggag(x1, x2, x3, x4, x5) = check_out_gggag(x4) U17_gggag(x1, x2, x3, x4, x5, x6) = U17_gggag(x3, x6) dif_in_gggaa(x1, x2, x3, x4, x5) = dif_in_gggaa(x1, x2, x3) dif_out_gggaa(x1, x2, x3, x4, x5) = dif_out_gggaa(x4, x5) U6_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gggaa(x2, x3, x4, x9) U7_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_gggaa(x2, x3, x4, x9, x10) U8_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_gggaa(x2, x3, x4, x5, x9, x10) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg U35_ggg(x1, x2, x3) = U35_ggg(x3) U36_ggg(x1, x2, x3) = U36_ggg(x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x4) U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_gggaa(x2, x3, x4, x5, x9) U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_gggaa(x5, x7, x9) U18_gggag(x1, x2, x3, x4, x5, x6, x7) = U18_gggag(x6, x7) mergedelete_in_gga(x1, x2, x3) = mergedelete_in_gga(x1, x2) mergedelete_out_gga(x1, x2, x3) = mergedelete_out_gga(x3) U12_gga(x1, x2, x3, x4, x5) = U12_gga(x5) U13_gga(x1, x2, x3, x4, x5, x6) = U13_gga(x1, x2, x3, x4, x6) U14_gga(x1, x2, x3, x4, x5, x6, x7) = U14_gga(x1, x2, x3, x4, x6, x7) U15_gga(x1, x2, x3, x4, x5, x6) = U15_gga(x1, x2, x3, x4, x6) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg U64_gg(x1, x2, x3) = U64_gg(x3) U65_gg(x1, x2, x3) = U65_gg(x3) U16_gga(x1, x2, x3, x4, x5, x6) = U16_gga(x3, x6) U19_gggag(x1, x2, x3, x4, x5, x6, x7) = U19_gggag(x6, x7) rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2) rev_out_gga(x1, x2, x3) = rev_out_gga(x3) U11_gga(x1, x2, x3, x4, x5) = U11_gga(x5) U20_gggag(x1, x2, x3, x4, x5, x6) = U20_gggag(x6) U23_ggag(x1, x2, x3, x4, x5, x6) = U23_ggag(x3, x6) pds_out_ga(x1, x2) = pds_out_ga(x2) PDS_IN_GA(x1, x2) = PDS_IN_GA(x1) U24_GA(x1, x2, x3) = U24_GA(x1, x3) ISPLUS_IN_GGA(x1, x2, x3) = ISPLUS_IN_GGA(x1, x2) U31_GGA(x1, x2, x3, x4) = U31_GGA(x4) U32_GGA(x1, x2, x3, x4) = U32_GGA(x4) U33_GGA(x1, x2, x3, x4) = U33_GGA(x4) U34_GGA(x1, x2, x3, x4) = U34_GGA(x4) U25_GA(x1, x2, x3) = U25_GA(x3) ISTIMES_IN_GGA(x1, x2, x3) = ISTIMES_IN_GGA(x1, x2) U41_GGA(x1, x2, x3, x4) = U41_GGA(x1, x4) U43_GGA(x1, x2, x3, x4) = U43_GGA(x1, x4) U45_GGA(x1, x2, x3, x4) = U45_GGA(x1, x4) U47_GGA(x1, x2, x3, x4) = U47_GGA(x1, x4) U48_GGA(x1, x2, x3, x4) = U48_GGA(x4) ISMINUS_IN_GGA(x1, x2, x3) = ISMINUS_IN_GGA(x1, x2) U35_GGA(x1, x2, x3) = U35_GGA(x3) U36_GGA(x1, x2, x3) = U36_GGA(x3) U37_GGA(x1, x2, x3, x4) = U37_GGA(x4) U38_GGA(x1, x2, x3, x4) = U38_GGA(x4) U39_GGA(x1, x2, x3, x4) = U39_GGA(x4) U40_GGA(x1, x2, x3, x4) = U40_GGA(x4) U46_GGA(x1, x2, x3, x4) = U46_GGA(x4) U44_GGA(x1, x2, x3, x4) = U44_GGA(x4) U42_GGA(x1, x2, x3, x4) = U42_GGA(x4) U26_GA(x1, x2, x3) = U26_GA(x3) U27_GA(x1, x2, x3) = U27_GA(x3) =_IN_AG(x1, x2) = =_IN_AG(x2) U28_GA(x1, x2, x3, x4) = U28_GA(x3, x4) IOTA_IN_GA(x1, x2) = IOTA_IN_GA(x1) U2_GA(x1, x2, x3) = U2_GA(x3) IOTA1_IN_GGA(x1, x2, x3) = IOTA1_IN_GGA(x1, x2) U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4) U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4) U5_GGA(x1, x2, x3, x4) = U5_GGA(x1, x4) U29_GA(x1, x2, x3) = U29_GA(x3) PDS1_IN_GGAG(x1, x2, x3, x4) = PDS1_IN_GGAG(x1, x2, x4) U21_GGAG(x1, x2, x3, x4, x5, x6) = U21_GGAG(x1, x2, x5, x6) MEMBER_IN_AG(x1, x2) = MEMBER_IN_AG(x2) U1_AG(x1, x2, x3, x4) = U1_AG(x4) U22_GGAG(x1, x2, x3, x4, x5, x6) = U22_GGAG(x2, x3, x5, x6) CHECK_IN_GGGAG(x1, x2, x3, x4, x5) = CHECK_IN_GGGAG(x1, x2, x3, x5) U17_GGGAG(x1, x2, x3, x4, x5, x6) = U17_GGGAG(x3, x6) DIF_IN_GGGAA(x1, x2, x3, x4, x5) = DIF_IN_GGGAA(x1, x2, x3) U6_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_GGGAA(x2, x3, x4, x9) U7_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_GGGAA(x2, x3, x4, x9, x10) U8_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_GGGAA(x2, x3, x4, x5, x9, x10) ISMINUS_IN_GGG(x1, x2, x3) = ISMINUS_IN_GGG(x1, x2, x3) U35_GGG(x1, x2, x3) = U35_GGG(x3) U36_GGG(x1, x2, x3) = U36_GGG(x3) U37_GGG(x1, x2, x3, x4) = U37_GGG(x4) U38_GGG(x1, x2, x3, x4) = U38_GGG(x4) U39_GGG(x1, x2, x3, x4) = U39_GGG(x4) U40_GGG(x1, x2, x3, x4) = U40_GGG(x4) U9_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_GGGAA(x2, x3, x4, x5, x9) U10_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_GGGAA(x5, x7, x9) U18_GGGAG(x1, x2, x3, x4, x5, x6, x7) = U18_GGGAG(x6, x7) MERGEDELETE_IN_GGA(x1, x2, x3) = MERGEDELETE_IN_GGA(x1, x2) U12_GGA(x1, x2, x3, x4, x5) = U12_GGA(x5) U13_GGA(x1, x2, x3, x4, x5, x6) = U13_GGA(x1, x2, x3, x4, x6) U14_GGA(x1, x2, x3, x4, x5, x6, x7) = U14_GGA(x1, x2, x3, x4, x6, x7) U15_GGA(x1, x2, x3, x4, x5, x6) = U15_GGA(x1, x2, x3, x4, x6) ISGREATER_IN_GG(x1, x2) = ISGREATER_IN_GG(x1, x2) U64_GG(x1, x2, x3) = U64_GG(x3) U65_GG(x1, x2, x3) = U65_GG(x3) U16_GGA(x1, x2, x3, x4, x5, x6) = U16_GGA(x3, x6) U19_GGGAG(x1, x2, x3, x4, x5, x6, x7) = U19_GGGAG(x6, x7) REV_IN_GGA(x1, x2, x3) = REV_IN_GGA(x1, x2) U11_GGA(x1, x2, x3, x4, x5) = U11_GGA(x5) U20_GGGAG(x1, x2, x3, x4, x5, x6) = U20_GGGAG(x6) U23_GGAG(x1, x2, x3, x4, x5, x6) = U23_GGAG(x3, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (11) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 14 SCCs with 70 less nodes. ---------------------------------------- (12) Complex Obligation (AND) ---------------------------------------- (13) Obligation: Pi DP problem: The TRS P consists of the following rules: REV_IN_GGA(.(X, Xs), Y, L) -> REV_IN_GGA(Xs, .(X, Y), L) The TRS R consists of the following rules: pds_in_ga(Order, .(zero, Ans)) -> U24_ga(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) isPlus_in_gga(zero, X, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), zero, succ(X)) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), succ(Y), succ(succ(Z))) -> U31_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(succ(X), pred(Y), Z) -> U32_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), zero, pred(X)) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), succ(Y), Z) -> U33_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), pred(Y), pred(pred(Z))) -> U34_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) U34_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U33_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U32_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U31_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U24_ga(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_ga(Order, Ans, isTimes_in_gga(Order, U, U1)) isTimes_in_gga(X, zero, zero) -> isTimes_out_gga(X, zero, zero) isTimes_in_gga(zero, succ(Y), zero) -> isTimes_out_gga(zero, succ(Y), zero) isTimes_in_gga(zero, pred(Y), zero) -> isTimes_out_gga(zero, pred(Y), zero) isTimes_in_gga(succ(X), succ(Y), Z) -> U41_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(succ(X), pred(Y), Z) -> U43_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(pred(X), succ(Y), Z) -> U45_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) isTimes_in_gga(pred(X), pred(Y), Z) -> U47_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) U47_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_gga(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U48_gga(X, Y, Z, isMinus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), pred(Y), Z) U45_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_gga(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U46_gga(X, Y, Z, isPlus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), succ(Y), Z) U43_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_gga(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U44_gga(X, Y, Z, isMinus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), pred(Y), Z) U41_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_gga(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U42_gga(X, Y, Z, isPlus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), succ(Y), Z) U25_ga(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_ga(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U26_ga(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_ga(Order, Ans, =_in_ag(N, U2)) =_in_ag(X, X) -> =_out_ag(X, X) U27_ga(Order, Ans, =_out_ag(N, U2)) -> U28_ga(Order, Ans, N, iota_in_ga(N, .(zero, List))) iota_in_ga(N, List) -> U2_ga(N, List, iota1_in_gga(zero, N, List)) iota1_in_gga(K, K, []) -> iota1_out_gga(K, K, []) iota1_in_gga(K, N, .(K, List)) -> U3_gga(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_gga(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_gga(K, N, List, =_in_ag(K1, U)) U4_gga(K, N, List, =_out_ag(K1, U)) -> U5_gga(K, N, List, iota1_in_gga(K1, N, List)) U5_gga(K, N, List, iota1_out_gga(K1, N, List)) -> iota1_out_gga(K, N, .(K, List)) U2_ga(N, List, iota1_out_gga(zero, N, List)) -> iota_out_ga(N, List) U28_ga(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_ga(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) pds1_in_ggag([], X6, [], X7) -> pds1_out_ggag([], X6, [], X7) pds1_in_ggag(Unused, List, .(Choice, Rest), Mod) -> U21_ggag(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(X, .(X1, Xs)) -> U1_ag(X, X1, Xs, member_in_ag(X, Xs)) U1_ag(X, X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) U21_ggag(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_ggag(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) check_in_gggag([], X4, L, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, L3, Modulus) -> U17_gggag(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) dif_in_gggaa([], X2, X3, [], []) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U17_gggag(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) mergedelete_in_gga([], L, L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R), L2) -> U12_gga(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) mergedelete_in_gga(.(D, Ds), .(X, R), .(X, L2)) -> U13_gga(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_gga(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_gga(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U15_gga(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U16_gga(D, Ds, X, R, L2, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U12_gga(D, Ds, R, L2, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) rev_in_gga([], L, L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y, L) -> U11_gga(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) U11_gga(X, Xs, Y, L, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U22_ggag(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_ggag(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U23_ggag(Unused, List, Choice, Rest, Mod, pds1_out_ggag(U3, .(Choice, List), Rest, Mod)) -> pds1_out_ggag(Unused, List, .(Choice, Rest), Mod) U29_ga(Order, Ans, pds1_out_ggag(List, .(zero, []), Ans, N)) -> pds_out_ga(Order, .(zero, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U24_ga(x1, x2, x3) = U24_ga(x1, x3) isPlus_in_gga(x1, x2, x3) = isPlus_in_gga(x1, x2) zero = zero isPlus_out_gga(x1, x2, x3) = isPlus_out_gga(x3) succ(x1) = succ(x1) U31_gga(x1, x2, x3, x4) = U31_gga(x4) pred(x1) = pred(x1) U32_gga(x1, x2, x3, x4) = U32_gga(x4) U33_gga(x1, x2, x3, x4) = U33_gga(x4) U34_gga(x1, x2, x3, x4) = U34_gga(x4) U25_ga(x1, x2, x3) = U25_ga(x3) isTimes_in_gga(x1, x2, x3) = isTimes_in_gga(x1, x2) isTimes_out_gga(x1, x2, x3) = isTimes_out_gga(x3) U41_gga(x1, x2, x3, x4) = U41_gga(x1, x4) U43_gga(x1, x2, x3, x4) = U43_gga(x1, x4) U45_gga(x1, x2, x3, x4) = U45_gga(x1, x4) U47_gga(x1, x2, x3, x4) = U47_gga(x1, x4) U48_gga(x1, x2, x3, x4) = U48_gga(x4) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x3) U35_gga(x1, x2, x3) = U35_gga(x3) U36_gga(x1, x2, x3) = U36_gga(x3) U37_gga(x1, x2, x3, x4) = U37_gga(x4) U38_gga(x1, x2, x3, x4) = U38_gga(x4) U39_gga(x1, x2, x3, x4) = U39_gga(x4) U40_gga(x1, x2, x3, x4) = U40_gga(x4) U46_gga(x1, x2, x3, x4) = U46_gga(x4) U44_gga(x1, x2, x3, x4) = U44_gga(x4) U42_gga(x1, x2, x3, x4) = U42_gga(x4) U26_ga(x1, x2, x3) = U26_ga(x3) U27_ga(x1, x2, x3) = U27_ga(x3) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1) U28_ga(x1, x2, x3, x4) = U28_ga(x3, x4) iota_in_ga(x1, x2) = iota_in_ga(x1) U2_ga(x1, x2, x3) = U2_ga(x3) iota1_in_gga(x1, x2, x3) = iota1_in_gga(x1, x2) iota1_out_gga(x1, x2, x3) = iota1_out_gga(x3) U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x4) iota_out_ga(x1, x2) = iota_out_ga(x2) .(x1, x2) = .(x1, x2) U29_ga(x1, x2, x3) = U29_ga(x3) pds1_in_ggag(x1, x2, x3, x4) = pds1_in_ggag(x1, x2, x4) [] = [] pds1_out_ggag(x1, x2, x3, x4) = pds1_out_ggag(x3) U21_ggag(x1, x2, x3, x4, x5, x6) = U21_ggag(x1, x2, x5, x6) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1) U1_ag(x1, x2, x3, x4) = U1_ag(x4) U22_ggag(x1, x2, x3, x4, x5, x6) = U22_ggag(x2, x3, x5, x6) check_in_gggag(x1, x2, x3, x4, x5) = check_in_gggag(x1, x2, x3, x5) check_out_gggag(x1, x2, x3, x4, x5) = check_out_gggag(x4) U17_gggag(x1, x2, x3, x4, x5, x6) = U17_gggag(x3, x6) dif_in_gggaa(x1, x2, x3, x4, x5) = dif_in_gggaa(x1, x2, x3) dif_out_gggaa(x1, x2, x3, x4, x5) = dif_out_gggaa(x4, x5) U6_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gggaa(x2, x3, x4, x9) U7_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_gggaa(x2, x3, x4, x9, x10) U8_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_gggaa(x2, x3, x4, x5, x9, x10) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg U35_ggg(x1, x2, x3) = U35_ggg(x3) U36_ggg(x1, x2, x3) = U36_ggg(x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x4) U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_gggaa(x2, x3, x4, x5, x9) U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_gggaa(x5, x7, x9) U18_gggag(x1, x2, x3, x4, x5, x6, x7) = U18_gggag(x6, x7) mergedelete_in_gga(x1, x2, x3) = mergedelete_in_gga(x1, x2) mergedelete_out_gga(x1, x2, x3) = mergedelete_out_gga(x3) U12_gga(x1, x2, x3, x4, x5) = U12_gga(x5) U13_gga(x1, x2, x3, x4, x5, x6) = U13_gga(x1, x2, x3, x4, x6) U14_gga(x1, x2, x3, x4, x5, x6, x7) = U14_gga(x1, x2, x3, x4, x6, x7) U15_gga(x1, x2, x3, x4, x5, x6) = U15_gga(x1, x2, x3, x4, x6) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg U64_gg(x1, x2, x3) = U64_gg(x3) U65_gg(x1, x2, x3) = U65_gg(x3) U16_gga(x1, x2, x3, x4, x5, x6) = U16_gga(x3, x6) U19_gggag(x1, x2, x3, x4, x5, x6, x7) = U19_gggag(x6, x7) rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2) rev_out_gga(x1, x2, x3) = rev_out_gga(x3) U11_gga(x1, x2, x3, x4, x5) = U11_gga(x5) U20_gggag(x1, x2, x3, x4, x5, x6) = U20_gggag(x6) U23_ggag(x1, x2, x3, x4, x5, x6) = U23_ggag(x3, x6) pds_out_ga(x1, x2) = pds_out_ga(x2) REV_IN_GGA(x1, x2, x3) = REV_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (14) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (15) Obligation: Pi DP problem: The TRS P consists of the following rules: REV_IN_GGA(.(X, Xs), Y, L) -> REV_IN_GGA(Xs, .(X, Y), L) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) REV_IN_GGA(x1, x2, x3) = REV_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (16) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (17) Obligation: Q DP problem: The TRS P consists of the following rules: REV_IN_GGA(.(X, Xs), Y) -> REV_IN_GGA(Xs, .(X, Y)) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (18) 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: *REV_IN_GGA(.(X, Xs), Y) -> REV_IN_GGA(Xs, .(X, Y)) The graph contains the following edges 1 > 1 ---------------------------------------- (19) YES ---------------------------------------- (20) Obligation: Pi DP problem: The TRS P consists of the following rules: ISGREATER_IN_GG(pred(X), pred(Y)) -> ISGREATER_IN_GG(X, Y) ISGREATER_IN_GG(succ(X), succ(Y)) -> ISGREATER_IN_GG(X, Y) The TRS R consists of the following rules: pds_in_ga(Order, .(zero, Ans)) -> U24_ga(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) isPlus_in_gga(zero, X, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), zero, succ(X)) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), succ(Y), succ(succ(Z))) -> U31_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(succ(X), pred(Y), Z) -> U32_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), zero, pred(X)) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), succ(Y), Z) -> U33_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), pred(Y), pred(pred(Z))) -> U34_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) U34_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U33_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U32_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U31_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U24_ga(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_ga(Order, Ans, isTimes_in_gga(Order, U, U1)) isTimes_in_gga(X, zero, zero) -> isTimes_out_gga(X, zero, zero) isTimes_in_gga(zero, succ(Y), zero) -> isTimes_out_gga(zero, succ(Y), zero) isTimes_in_gga(zero, pred(Y), zero) -> isTimes_out_gga(zero, pred(Y), zero) isTimes_in_gga(succ(X), succ(Y), Z) -> U41_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(succ(X), pred(Y), Z) -> U43_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(pred(X), succ(Y), Z) -> U45_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) isTimes_in_gga(pred(X), pred(Y), Z) -> U47_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) U47_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_gga(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U48_gga(X, Y, Z, isMinus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), pred(Y), Z) U45_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_gga(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U46_gga(X, Y, Z, isPlus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), succ(Y), Z) U43_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_gga(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U44_gga(X, Y, Z, isMinus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), pred(Y), Z) U41_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_gga(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U42_gga(X, Y, Z, isPlus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), succ(Y), Z) U25_ga(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_ga(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U26_ga(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_ga(Order, Ans, =_in_ag(N, U2)) =_in_ag(X, X) -> =_out_ag(X, X) U27_ga(Order, Ans, =_out_ag(N, U2)) -> U28_ga(Order, Ans, N, iota_in_ga(N, .(zero, List))) iota_in_ga(N, List) -> U2_ga(N, List, iota1_in_gga(zero, N, List)) iota1_in_gga(K, K, []) -> iota1_out_gga(K, K, []) iota1_in_gga(K, N, .(K, List)) -> U3_gga(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_gga(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_gga(K, N, List, =_in_ag(K1, U)) U4_gga(K, N, List, =_out_ag(K1, U)) -> U5_gga(K, N, List, iota1_in_gga(K1, N, List)) U5_gga(K, N, List, iota1_out_gga(K1, N, List)) -> iota1_out_gga(K, N, .(K, List)) U2_ga(N, List, iota1_out_gga(zero, N, List)) -> iota_out_ga(N, List) U28_ga(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_ga(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) pds1_in_ggag([], X6, [], X7) -> pds1_out_ggag([], X6, [], X7) pds1_in_ggag(Unused, List, .(Choice, Rest), Mod) -> U21_ggag(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(X, .(X1, Xs)) -> U1_ag(X, X1, Xs, member_in_ag(X, Xs)) U1_ag(X, X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) U21_ggag(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_ggag(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) check_in_gggag([], X4, L, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, L3, Modulus) -> U17_gggag(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) dif_in_gggaa([], X2, X3, [], []) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U17_gggag(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) mergedelete_in_gga([], L, L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R), L2) -> U12_gga(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) mergedelete_in_gga(.(D, Ds), .(X, R), .(X, L2)) -> U13_gga(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_gga(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_gga(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U15_gga(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U16_gga(D, Ds, X, R, L2, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U12_gga(D, Ds, R, L2, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) rev_in_gga([], L, L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y, L) -> U11_gga(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) U11_gga(X, Xs, Y, L, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U22_ggag(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_ggag(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U23_ggag(Unused, List, Choice, Rest, Mod, pds1_out_ggag(U3, .(Choice, List), Rest, Mod)) -> pds1_out_ggag(Unused, List, .(Choice, Rest), Mod) U29_ga(Order, Ans, pds1_out_ggag(List, .(zero, []), Ans, N)) -> pds_out_ga(Order, .(zero, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U24_ga(x1, x2, x3) = U24_ga(x1, x3) isPlus_in_gga(x1, x2, x3) = isPlus_in_gga(x1, x2) zero = zero isPlus_out_gga(x1, x2, x3) = isPlus_out_gga(x3) succ(x1) = succ(x1) U31_gga(x1, x2, x3, x4) = U31_gga(x4) pred(x1) = pred(x1) U32_gga(x1, x2, x3, x4) = U32_gga(x4) U33_gga(x1, x2, x3, x4) = U33_gga(x4) U34_gga(x1, x2, x3, x4) = U34_gga(x4) U25_ga(x1, x2, x3) = U25_ga(x3) isTimes_in_gga(x1, x2, x3) = isTimes_in_gga(x1, x2) isTimes_out_gga(x1, x2, x3) = isTimes_out_gga(x3) U41_gga(x1, x2, x3, x4) = U41_gga(x1, x4) U43_gga(x1, x2, x3, x4) = U43_gga(x1, x4) U45_gga(x1, x2, x3, x4) = U45_gga(x1, x4) U47_gga(x1, x2, x3, x4) = U47_gga(x1, x4) U48_gga(x1, x2, x3, x4) = U48_gga(x4) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x3) U35_gga(x1, x2, x3) = U35_gga(x3) U36_gga(x1, x2, x3) = U36_gga(x3) U37_gga(x1, x2, x3, x4) = U37_gga(x4) U38_gga(x1, x2, x3, x4) = U38_gga(x4) U39_gga(x1, x2, x3, x4) = U39_gga(x4) U40_gga(x1, x2, x3, x4) = U40_gga(x4) U46_gga(x1, x2, x3, x4) = U46_gga(x4) U44_gga(x1, x2, x3, x4) = U44_gga(x4) U42_gga(x1, x2, x3, x4) = U42_gga(x4) U26_ga(x1, x2, x3) = U26_ga(x3) U27_ga(x1, x2, x3) = U27_ga(x3) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1) U28_ga(x1, x2, x3, x4) = U28_ga(x3, x4) iota_in_ga(x1, x2) = iota_in_ga(x1) U2_ga(x1, x2, x3) = U2_ga(x3) iota1_in_gga(x1, x2, x3) = iota1_in_gga(x1, x2) iota1_out_gga(x1, x2, x3) = iota1_out_gga(x3) U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x4) iota_out_ga(x1, x2) = iota_out_ga(x2) .(x1, x2) = .(x1, x2) U29_ga(x1, x2, x3) = U29_ga(x3) pds1_in_ggag(x1, x2, x3, x4) = pds1_in_ggag(x1, x2, x4) [] = [] pds1_out_ggag(x1, x2, x3, x4) = pds1_out_ggag(x3) U21_ggag(x1, x2, x3, x4, x5, x6) = U21_ggag(x1, x2, x5, x6) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1) U1_ag(x1, x2, x3, x4) = U1_ag(x4) U22_ggag(x1, x2, x3, x4, x5, x6) = U22_ggag(x2, x3, x5, x6) check_in_gggag(x1, x2, x3, x4, x5) = check_in_gggag(x1, x2, x3, x5) check_out_gggag(x1, x2, x3, x4, x5) = check_out_gggag(x4) U17_gggag(x1, x2, x3, x4, x5, x6) = U17_gggag(x3, x6) dif_in_gggaa(x1, x2, x3, x4, x5) = dif_in_gggaa(x1, x2, x3) dif_out_gggaa(x1, x2, x3, x4, x5) = dif_out_gggaa(x4, x5) U6_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gggaa(x2, x3, x4, x9) U7_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_gggaa(x2, x3, x4, x9, x10) U8_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_gggaa(x2, x3, x4, x5, x9, x10) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg U35_ggg(x1, x2, x3) = U35_ggg(x3) U36_ggg(x1, x2, x3) = U36_ggg(x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x4) U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_gggaa(x2, x3, x4, x5, x9) U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_gggaa(x5, x7, x9) U18_gggag(x1, x2, x3, x4, x5, x6, x7) = U18_gggag(x6, x7) mergedelete_in_gga(x1, x2, x3) = mergedelete_in_gga(x1, x2) mergedelete_out_gga(x1, x2, x3) = mergedelete_out_gga(x3) U12_gga(x1, x2, x3, x4, x5) = U12_gga(x5) U13_gga(x1, x2, x3, x4, x5, x6) = U13_gga(x1, x2, x3, x4, x6) U14_gga(x1, x2, x3, x4, x5, x6, x7) = U14_gga(x1, x2, x3, x4, x6, x7) U15_gga(x1, x2, x3, x4, x5, x6) = U15_gga(x1, x2, x3, x4, x6) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg U64_gg(x1, x2, x3) = U64_gg(x3) U65_gg(x1, x2, x3) = U65_gg(x3) U16_gga(x1, x2, x3, x4, x5, x6) = U16_gga(x3, x6) U19_gggag(x1, x2, x3, x4, x5, x6, x7) = U19_gggag(x6, x7) rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2) rev_out_gga(x1, x2, x3) = rev_out_gga(x3) U11_gga(x1, x2, x3, x4, x5) = U11_gga(x5) U20_gggag(x1, x2, x3, x4, x5, x6) = U20_gggag(x6) U23_ggag(x1, x2, x3, x4, x5, x6) = U23_ggag(x3, x6) pds_out_ga(x1, x2) = pds_out_ga(x2) ISGREATER_IN_GG(x1, x2) = ISGREATER_IN_GG(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (21) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (22) Obligation: Pi DP problem: The TRS P consists of the following rules: ISGREATER_IN_GG(pred(X), pred(Y)) -> ISGREATER_IN_GG(X, Y) ISGREATER_IN_GG(succ(X), succ(Y)) -> ISGREATER_IN_GG(X, Y) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (23) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (24) Obligation: Q DP problem: The TRS P consists of the following rules: ISGREATER_IN_GG(pred(X), pred(Y)) -> ISGREATER_IN_GG(X, Y) ISGREATER_IN_GG(succ(X), succ(Y)) -> ISGREATER_IN_GG(X, Y) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (25) 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: *ISGREATER_IN_GG(pred(X), pred(Y)) -> ISGREATER_IN_GG(X, Y) The graph contains the following edges 1 > 1, 2 > 2 *ISGREATER_IN_GG(succ(X), succ(Y)) -> ISGREATER_IN_GG(X, Y) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (26) YES ---------------------------------------- (27) Obligation: Pi DP problem: The TRS P consists of the following rules: MERGEDELETE_IN_GGA(.(D, Ds), .(X, R), .(X, L2)) -> U13_GGA(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_GGA(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_GGA(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_GGA(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_GGA(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) U15_GGA(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> MERGEDELETE_IN_GGA(.(D, Ds), R, L2) MERGEDELETE_IN_GGA(.(D, Ds), .(D, R), L2) -> MERGEDELETE_IN_GGA(Ds, R, L2) The TRS R consists of the following rules: pds_in_ga(Order, .(zero, Ans)) -> U24_ga(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) isPlus_in_gga(zero, X, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), zero, succ(X)) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), succ(Y), succ(succ(Z))) -> U31_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(succ(X), pred(Y), Z) -> U32_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), zero, pred(X)) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), succ(Y), Z) -> U33_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), pred(Y), pred(pred(Z))) -> U34_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) U34_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U33_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U32_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U31_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U24_ga(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_ga(Order, Ans, isTimes_in_gga(Order, U, U1)) isTimes_in_gga(X, zero, zero) -> isTimes_out_gga(X, zero, zero) isTimes_in_gga(zero, succ(Y), zero) -> isTimes_out_gga(zero, succ(Y), zero) isTimes_in_gga(zero, pred(Y), zero) -> isTimes_out_gga(zero, pred(Y), zero) isTimes_in_gga(succ(X), succ(Y), Z) -> U41_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(succ(X), pred(Y), Z) -> U43_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(pred(X), succ(Y), Z) -> U45_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) isTimes_in_gga(pred(X), pred(Y), Z) -> U47_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) U47_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_gga(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U48_gga(X, Y, Z, isMinus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), pred(Y), Z) U45_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_gga(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U46_gga(X, Y, Z, isPlus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), succ(Y), Z) U43_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_gga(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U44_gga(X, Y, Z, isMinus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), pred(Y), Z) U41_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_gga(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U42_gga(X, Y, Z, isPlus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), succ(Y), Z) U25_ga(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_ga(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U26_ga(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_ga(Order, Ans, =_in_ag(N, U2)) =_in_ag(X, X) -> =_out_ag(X, X) U27_ga(Order, Ans, =_out_ag(N, U2)) -> U28_ga(Order, Ans, N, iota_in_ga(N, .(zero, List))) iota_in_ga(N, List) -> U2_ga(N, List, iota1_in_gga(zero, N, List)) iota1_in_gga(K, K, []) -> iota1_out_gga(K, K, []) iota1_in_gga(K, N, .(K, List)) -> U3_gga(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_gga(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_gga(K, N, List, =_in_ag(K1, U)) U4_gga(K, N, List, =_out_ag(K1, U)) -> U5_gga(K, N, List, iota1_in_gga(K1, N, List)) U5_gga(K, N, List, iota1_out_gga(K1, N, List)) -> iota1_out_gga(K, N, .(K, List)) U2_ga(N, List, iota1_out_gga(zero, N, List)) -> iota_out_ga(N, List) U28_ga(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_ga(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) pds1_in_ggag([], X6, [], X7) -> pds1_out_ggag([], X6, [], X7) pds1_in_ggag(Unused, List, .(Choice, Rest), Mod) -> U21_ggag(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(X, .(X1, Xs)) -> U1_ag(X, X1, Xs, member_in_ag(X, Xs)) U1_ag(X, X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) U21_ggag(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_ggag(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) check_in_gggag([], X4, L, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, L3, Modulus) -> U17_gggag(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) dif_in_gggaa([], X2, X3, [], []) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U17_gggag(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) mergedelete_in_gga([], L, L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R), L2) -> U12_gga(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) mergedelete_in_gga(.(D, Ds), .(X, R), .(X, L2)) -> U13_gga(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_gga(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_gga(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U15_gga(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U16_gga(D, Ds, X, R, L2, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U12_gga(D, Ds, R, L2, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) rev_in_gga([], L, L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y, L) -> U11_gga(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) U11_gga(X, Xs, Y, L, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U22_ggag(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_ggag(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U23_ggag(Unused, List, Choice, Rest, Mod, pds1_out_ggag(U3, .(Choice, List), Rest, Mod)) -> pds1_out_ggag(Unused, List, .(Choice, Rest), Mod) U29_ga(Order, Ans, pds1_out_ggag(List, .(zero, []), Ans, N)) -> pds_out_ga(Order, .(zero, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U24_ga(x1, x2, x3) = U24_ga(x1, x3) isPlus_in_gga(x1, x2, x3) = isPlus_in_gga(x1, x2) zero = zero isPlus_out_gga(x1, x2, x3) = isPlus_out_gga(x3) succ(x1) = succ(x1) U31_gga(x1, x2, x3, x4) = U31_gga(x4) pred(x1) = pred(x1) U32_gga(x1, x2, x3, x4) = U32_gga(x4) U33_gga(x1, x2, x3, x4) = U33_gga(x4) U34_gga(x1, x2, x3, x4) = U34_gga(x4) U25_ga(x1, x2, x3) = U25_ga(x3) isTimes_in_gga(x1, x2, x3) = isTimes_in_gga(x1, x2) isTimes_out_gga(x1, x2, x3) = isTimes_out_gga(x3) U41_gga(x1, x2, x3, x4) = U41_gga(x1, x4) U43_gga(x1, x2, x3, x4) = U43_gga(x1, x4) U45_gga(x1, x2, x3, x4) = U45_gga(x1, x4) U47_gga(x1, x2, x3, x4) = U47_gga(x1, x4) U48_gga(x1, x2, x3, x4) = U48_gga(x4) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x3) U35_gga(x1, x2, x3) = U35_gga(x3) U36_gga(x1, x2, x3) = U36_gga(x3) U37_gga(x1, x2, x3, x4) = U37_gga(x4) U38_gga(x1, x2, x3, x4) = U38_gga(x4) U39_gga(x1, x2, x3, x4) = U39_gga(x4) U40_gga(x1, x2, x3, x4) = U40_gga(x4) U46_gga(x1, x2, x3, x4) = U46_gga(x4) U44_gga(x1, x2, x3, x4) = U44_gga(x4) U42_gga(x1, x2, x3, x4) = U42_gga(x4) U26_ga(x1, x2, x3) = U26_ga(x3) U27_ga(x1, x2, x3) = U27_ga(x3) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1) U28_ga(x1, x2, x3, x4) = U28_ga(x3, x4) iota_in_ga(x1, x2) = iota_in_ga(x1) U2_ga(x1, x2, x3) = U2_ga(x3) iota1_in_gga(x1, x2, x3) = iota1_in_gga(x1, x2) iota1_out_gga(x1, x2, x3) = iota1_out_gga(x3) U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x4) iota_out_ga(x1, x2) = iota_out_ga(x2) .(x1, x2) = .(x1, x2) U29_ga(x1, x2, x3) = U29_ga(x3) pds1_in_ggag(x1, x2, x3, x4) = pds1_in_ggag(x1, x2, x4) [] = [] pds1_out_ggag(x1, x2, x3, x4) = pds1_out_ggag(x3) U21_ggag(x1, x2, x3, x4, x5, x6) = U21_ggag(x1, x2, x5, x6) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1) U1_ag(x1, x2, x3, x4) = U1_ag(x4) U22_ggag(x1, x2, x3, x4, x5, x6) = U22_ggag(x2, x3, x5, x6) check_in_gggag(x1, x2, x3, x4, x5) = check_in_gggag(x1, x2, x3, x5) check_out_gggag(x1, x2, x3, x4, x5) = check_out_gggag(x4) U17_gggag(x1, x2, x3, x4, x5, x6) = U17_gggag(x3, x6) dif_in_gggaa(x1, x2, x3, x4, x5) = dif_in_gggaa(x1, x2, x3) dif_out_gggaa(x1, x2, x3, x4, x5) = dif_out_gggaa(x4, x5) U6_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gggaa(x2, x3, x4, x9) U7_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_gggaa(x2, x3, x4, x9, x10) U8_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_gggaa(x2, x3, x4, x5, x9, x10) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg U35_ggg(x1, x2, x3) = U35_ggg(x3) U36_ggg(x1, x2, x3) = U36_ggg(x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x4) U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_gggaa(x2, x3, x4, x5, x9) U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_gggaa(x5, x7, x9) U18_gggag(x1, x2, x3, x4, x5, x6, x7) = U18_gggag(x6, x7) mergedelete_in_gga(x1, x2, x3) = mergedelete_in_gga(x1, x2) mergedelete_out_gga(x1, x2, x3) = mergedelete_out_gga(x3) U12_gga(x1, x2, x3, x4, x5) = U12_gga(x5) U13_gga(x1, x2, x3, x4, x5, x6) = U13_gga(x1, x2, x3, x4, x6) U14_gga(x1, x2, x3, x4, x5, x6, x7) = U14_gga(x1, x2, x3, x4, x6, x7) U15_gga(x1, x2, x3, x4, x5, x6) = U15_gga(x1, x2, x3, x4, x6) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg U64_gg(x1, x2, x3) = U64_gg(x3) U65_gg(x1, x2, x3) = U65_gg(x3) U16_gga(x1, x2, x3, x4, x5, x6) = U16_gga(x3, x6) U19_gggag(x1, x2, x3, x4, x5, x6, x7) = U19_gggag(x6, x7) rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2) rev_out_gga(x1, x2, x3) = rev_out_gga(x3) U11_gga(x1, x2, x3, x4, x5) = U11_gga(x5) U20_gggag(x1, x2, x3, x4, x5, x6) = U20_gggag(x6) U23_ggag(x1, x2, x3, x4, x5, x6) = U23_ggag(x3, x6) pds_out_ga(x1, x2) = pds_out_ga(x2) MERGEDELETE_IN_GGA(x1, x2, x3) = MERGEDELETE_IN_GGA(x1, x2) U13_GGA(x1, x2, x3, x4, x5, x6) = U13_GGA(x1, x2, x3, x4, x6) U14_GGA(x1, x2, x3, x4, x5, x6, x7) = U14_GGA(x1, x2, x3, x4, x6, x7) U15_GGA(x1, x2, x3, x4, x5, x6) = U15_GGA(x1, x2, x3, x4, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (28) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (29) Obligation: Pi DP problem: The TRS P consists of the following rules: MERGEDELETE_IN_GGA(.(D, Ds), .(X, R), .(X, L2)) -> U13_GGA(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_GGA(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_GGA(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_GGA(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_GGA(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) U15_GGA(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> MERGEDELETE_IN_GGA(.(D, Ds), R, L2) MERGEDELETE_IN_GGA(.(D, Ds), .(D, R), L2) -> MERGEDELETE_IN_GGA(Ds, R, L2) The TRS R consists of the following rules: =_in_ag(X, X) -> =_out_ag(X, X) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) The argument filtering Pi contains the following mapping: zero = zero succ(x1) = succ(x1) pred(x1) = pred(x1) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1) .(x1, x2) = .(x1, x2) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg U64_gg(x1, x2, x3) = U64_gg(x3) U65_gg(x1, x2, x3) = U65_gg(x3) MERGEDELETE_IN_GGA(x1, x2, x3) = MERGEDELETE_IN_GGA(x1, x2) U13_GGA(x1, x2, x3, x4, x5, x6) = U13_GGA(x1, x2, x3, x4, x6) U14_GGA(x1, x2, x3, x4, x5, x6, x7) = U14_GGA(x1, x2, x3, x4, x6, x7) U15_GGA(x1, x2, x3, x4, x5, x6) = U15_GGA(x1, x2, x3, x4, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (30) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (31) Obligation: Q DP problem: The TRS P consists of the following rules: MERGEDELETE_IN_GGA(.(D, Ds), .(X, R)) -> U13_GGA(D, Ds, X, R, =_in_ag(D)) U13_GGA(D, Ds, X, R, =_out_ag(X1)) -> U14_GGA(D, Ds, X, R, X1, =_in_ag(X)) U14_GGA(D, Ds, X, R, X1, =_out_ag(X2)) -> U15_GGA(D, Ds, X, R, isGreater_in_gg(X1, X2)) U15_GGA(D, Ds, X, R, isGreater_out_gg) -> MERGEDELETE_IN_GGA(.(D, Ds), R) MERGEDELETE_IN_GGA(.(D, Ds), .(D, R)) -> MERGEDELETE_IN_GGA(Ds, R) The TRS R consists of the following rules: =_in_ag(X) -> =_out_ag(X) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(isGreater_in_gg(X, Y)) U64_gg(isGreater_out_gg) -> isGreater_out_gg U65_gg(isGreater_out_gg) -> isGreater_out_gg The set Q consists of the following terms: =_in_ag(x0) isGreater_in_gg(x0, x1) U64_gg(x0) U65_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (32) 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: *U13_GGA(D, Ds, X, R, =_out_ag(X1)) -> U14_GGA(D, Ds, X, R, X1, =_in_ag(X)) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 > 5 *U14_GGA(D, Ds, X, R, X1, =_out_ag(X2)) -> U15_GGA(D, Ds, X, R, isGreater_in_gg(X1, X2)) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4 *MERGEDELETE_IN_GGA(.(D, Ds), .(X, R)) -> U13_GGA(D, Ds, X, R, =_in_ag(D)) The graph contains the following edges 1 > 1, 1 > 2, 2 > 3, 2 > 4 *U15_GGA(D, Ds, X, R, isGreater_out_gg) -> MERGEDELETE_IN_GGA(.(D, Ds), R) The graph contains the following edges 4 >= 2 *MERGEDELETE_IN_GGA(.(D, Ds), .(D, R)) -> MERGEDELETE_IN_GGA(Ds, R) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (33) YES ---------------------------------------- (34) Obligation: Pi DP problem: The TRS P consists of the following rules: ISMINUS_IN_GGG(zero, pred(Y), succ(Z)) -> ISMINUS_IN_GGG(zero, Y, Z) ISMINUS_IN_GGG(zero, succ(Y), pred(Z)) -> ISMINUS_IN_GGG(zero, Y, Z) The TRS R consists of the following rules: pds_in_ga(Order, .(zero, Ans)) -> U24_ga(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) isPlus_in_gga(zero, X, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), zero, succ(X)) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), succ(Y), succ(succ(Z))) -> U31_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(succ(X), pred(Y), Z) -> U32_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), zero, pred(X)) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), succ(Y), Z) -> U33_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), pred(Y), pred(pred(Z))) -> U34_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) U34_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U33_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U32_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U31_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U24_ga(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_ga(Order, Ans, isTimes_in_gga(Order, U, U1)) isTimes_in_gga(X, zero, zero) -> isTimes_out_gga(X, zero, zero) isTimes_in_gga(zero, succ(Y), zero) -> isTimes_out_gga(zero, succ(Y), zero) isTimes_in_gga(zero, pred(Y), zero) -> isTimes_out_gga(zero, pred(Y), zero) isTimes_in_gga(succ(X), succ(Y), Z) -> U41_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(succ(X), pred(Y), Z) -> U43_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(pred(X), succ(Y), Z) -> U45_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) isTimes_in_gga(pred(X), pred(Y), Z) -> U47_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) U47_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_gga(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U48_gga(X, Y, Z, isMinus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), pred(Y), Z) U45_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_gga(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U46_gga(X, Y, Z, isPlus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), succ(Y), Z) U43_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_gga(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U44_gga(X, Y, Z, isMinus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), pred(Y), Z) U41_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_gga(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U42_gga(X, Y, Z, isPlus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), succ(Y), Z) U25_ga(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_ga(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U26_ga(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_ga(Order, Ans, =_in_ag(N, U2)) =_in_ag(X, X) -> =_out_ag(X, X) U27_ga(Order, Ans, =_out_ag(N, U2)) -> U28_ga(Order, Ans, N, iota_in_ga(N, .(zero, List))) iota_in_ga(N, List) -> U2_ga(N, List, iota1_in_gga(zero, N, List)) iota1_in_gga(K, K, []) -> iota1_out_gga(K, K, []) iota1_in_gga(K, N, .(K, List)) -> U3_gga(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_gga(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_gga(K, N, List, =_in_ag(K1, U)) U4_gga(K, N, List, =_out_ag(K1, U)) -> U5_gga(K, N, List, iota1_in_gga(K1, N, List)) U5_gga(K, N, List, iota1_out_gga(K1, N, List)) -> iota1_out_gga(K, N, .(K, List)) U2_ga(N, List, iota1_out_gga(zero, N, List)) -> iota_out_ga(N, List) U28_ga(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_ga(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) pds1_in_ggag([], X6, [], X7) -> pds1_out_ggag([], X6, [], X7) pds1_in_ggag(Unused, List, .(Choice, Rest), Mod) -> U21_ggag(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(X, .(X1, Xs)) -> U1_ag(X, X1, Xs, member_in_ag(X, Xs)) U1_ag(X, X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) U21_ggag(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_ggag(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) check_in_gggag([], X4, L, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, L3, Modulus) -> U17_gggag(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) dif_in_gggaa([], X2, X3, [], []) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U17_gggag(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) mergedelete_in_gga([], L, L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R), L2) -> U12_gga(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) mergedelete_in_gga(.(D, Ds), .(X, R), .(X, L2)) -> U13_gga(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_gga(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_gga(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U15_gga(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U16_gga(D, Ds, X, R, L2, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U12_gga(D, Ds, R, L2, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) rev_in_gga([], L, L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y, L) -> U11_gga(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) U11_gga(X, Xs, Y, L, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U22_ggag(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_ggag(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U23_ggag(Unused, List, Choice, Rest, Mod, pds1_out_ggag(U3, .(Choice, List), Rest, Mod)) -> pds1_out_ggag(Unused, List, .(Choice, Rest), Mod) U29_ga(Order, Ans, pds1_out_ggag(List, .(zero, []), Ans, N)) -> pds_out_ga(Order, .(zero, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U24_ga(x1, x2, x3) = U24_ga(x1, x3) isPlus_in_gga(x1, x2, x3) = isPlus_in_gga(x1, x2) zero = zero isPlus_out_gga(x1, x2, x3) = isPlus_out_gga(x3) succ(x1) = succ(x1) U31_gga(x1, x2, x3, x4) = U31_gga(x4) pred(x1) = pred(x1) U32_gga(x1, x2, x3, x4) = U32_gga(x4) U33_gga(x1, x2, x3, x4) = U33_gga(x4) U34_gga(x1, x2, x3, x4) = U34_gga(x4) U25_ga(x1, x2, x3) = U25_ga(x3) isTimes_in_gga(x1, x2, x3) = isTimes_in_gga(x1, x2) isTimes_out_gga(x1, x2, x3) = isTimes_out_gga(x3) U41_gga(x1, x2, x3, x4) = U41_gga(x1, x4) U43_gga(x1, x2, x3, x4) = U43_gga(x1, x4) U45_gga(x1, x2, x3, x4) = U45_gga(x1, x4) U47_gga(x1, x2, x3, x4) = U47_gga(x1, x4) U48_gga(x1, x2, x3, x4) = U48_gga(x4) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x3) U35_gga(x1, x2, x3) = U35_gga(x3) U36_gga(x1, x2, x3) = U36_gga(x3) U37_gga(x1, x2, x3, x4) = U37_gga(x4) U38_gga(x1, x2, x3, x4) = U38_gga(x4) U39_gga(x1, x2, x3, x4) = U39_gga(x4) U40_gga(x1, x2, x3, x4) = U40_gga(x4) U46_gga(x1, x2, x3, x4) = U46_gga(x4) U44_gga(x1, x2, x3, x4) = U44_gga(x4) U42_gga(x1, x2, x3, x4) = U42_gga(x4) U26_ga(x1, x2, x3) = U26_ga(x3) U27_ga(x1, x2, x3) = U27_ga(x3) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1) U28_ga(x1, x2, x3, x4) = U28_ga(x3, x4) iota_in_ga(x1, x2) = iota_in_ga(x1) U2_ga(x1, x2, x3) = U2_ga(x3) iota1_in_gga(x1, x2, x3) = iota1_in_gga(x1, x2) iota1_out_gga(x1, x2, x3) = iota1_out_gga(x3) U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x4) iota_out_ga(x1, x2) = iota_out_ga(x2) .(x1, x2) = .(x1, x2) U29_ga(x1, x2, x3) = U29_ga(x3) pds1_in_ggag(x1, x2, x3, x4) = pds1_in_ggag(x1, x2, x4) [] = [] pds1_out_ggag(x1, x2, x3, x4) = pds1_out_ggag(x3) U21_ggag(x1, x2, x3, x4, x5, x6) = U21_ggag(x1, x2, x5, x6) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1) U1_ag(x1, x2, x3, x4) = U1_ag(x4) U22_ggag(x1, x2, x3, x4, x5, x6) = U22_ggag(x2, x3, x5, x6) check_in_gggag(x1, x2, x3, x4, x5) = check_in_gggag(x1, x2, x3, x5) check_out_gggag(x1, x2, x3, x4, x5) = check_out_gggag(x4) U17_gggag(x1, x2, x3, x4, x5, x6) = U17_gggag(x3, x6) dif_in_gggaa(x1, x2, x3, x4, x5) = dif_in_gggaa(x1, x2, x3) dif_out_gggaa(x1, x2, x3, x4, x5) = dif_out_gggaa(x4, x5) U6_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gggaa(x2, x3, x4, x9) U7_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_gggaa(x2, x3, x4, x9, x10) U8_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_gggaa(x2, x3, x4, x5, x9, x10) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg U35_ggg(x1, x2, x3) = U35_ggg(x3) U36_ggg(x1, x2, x3) = U36_ggg(x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x4) U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_gggaa(x2, x3, x4, x5, x9) U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_gggaa(x5, x7, x9) U18_gggag(x1, x2, x3, x4, x5, x6, x7) = U18_gggag(x6, x7) mergedelete_in_gga(x1, x2, x3) = mergedelete_in_gga(x1, x2) mergedelete_out_gga(x1, x2, x3) = mergedelete_out_gga(x3) U12_gga(x1, x2, x3, x4, x5) = U12_gga(x5) U13_gga(x1, x2, x3, x4, x5, x6) = U13_gga(x1, x2, x3, x4, x6) U14_gga(x1, x2, x3, x4, x5, x6, x7) = U14_gga(x1, x2, x3, x4, x6, x7) U15_gga(x1, x2, x3, x4, x5, x6) = U15_gga(x1, x2, x3, x4, x6) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg U64_gg(x1, x2, x3) = U64_gg(x3) U65_gg(x1, x2, x3) = U65_gg(x3) U16_gga(x1, x2, x3, x4, x5, x6) = U16_gga(x3, x6) U19_gggag(x1, x2, x3, x4, x5, x6, x7) = U19_gggag(x6, x7) rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2) rev_out_gga(x1, x2, x3) = rev_out_gga(x3) U11_gga(x1, x2, x3, x4, x5) = U11_gga(x5) U20_gggag(x1, x2, x3, x4, x5, x6) = U20_gggag(x6) U23_ggag(x1, x2, x3, x4, x5, x6) = U23_ggag(x3, x6) pds_out_ga(x1, x2) = pds_out_ga(x2) ISMINUS_IN_GGG(x1, x2, x3) = ISMINUS_IN_GGG(x1, x2, x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (35) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (36) Obligation: Pi DP problem: The TRS P consists of the following rules: ISMINUS_IN_GGG(zero, pred(Y), succ(Z)) -> ISMINUS_IN_GGG(zero, Y, Z) ISMINUS_IN_GGG(zero, succ(Y), pred(Z)) -> ISMINUS_IN_GGG(zero, Y, Z) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (37) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (38) Obligation: Q DP problem: The TRS P consists of the following rules: ISMINUS_IN_GGG(zero, pred(Y), succ(Z)) -> ISMINUS_IN_GGG(zero, Y, Z) ISMINUS_IN_GGG(zero, succ(Y), pred(Z)) -> ISMINUS_IN_GGG(zero, Y, Z) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (39) 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: *ISMINUS_IN_GGG(zero, pred(Y), succ(Z)) -> ISMINUS_IN_GGG(zero, Y, Z) The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3 *ISMINUS_IN_GGG(zero, succ(Y), pred(Z)) -> ISMINUS_IN_GGG(zero, Y, Z) The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3 ---------------------------------------- (40) YES ---------------------------------------- (41) Obligation: Pi DP problem: The TRS P consists of the following rules: ISMINUS_IN_GGG(succ(X), pred(Y), succ(succ(Z))) -> ISMINUS_IN_GGG(X, Y, Z) ISMINUS_IN_GGG(succ(X), succ(Y), Z) -> ISMINUS_IN_GGG(X, Y, Z) ISMINUS_IN_GGG(pred(X), succ(Y), pred(pred(Z))) -> ISMINUS_IN_GGG(X, Y, Z) ISMINUS_IN_GGG(pred(X), pred(Y), Z) -> ISMINUS_IN_GGG(X, Y, Z) The TRS R consists of the following rules: pds_in_ga(Order, .(zero, Ans)) -> U24_ga(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) isPlus_in_gga(zero, X, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), zero, succ(X)) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), succ(Y), succ(succ(Z))) -> U31_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(succ(X), pred(Y), Z) -> U32_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), zero, pred(X)) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), succ(Y), Z) -> U33_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), pred(Y), pred(pred(Z))) -> U34_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) U34_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U33_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U32_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U31_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U24_ga(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_ga(Order, Ans, isTimes_in_gga(Order, U, U1)) isTimes_in_gga(X, zero, zero) -> isTimes_out_gga(X, zero, zero) isTimes_in_gga(zero, succ(Y), zero) -> isTimes_out_gga(zero, succ(Y), zero) isTimes_in_gga(zero, pred(Y), zero) -> isTimes_out_gga(zero, pred(Y), zero) isTimes_in_gga(succ(X), succ(Y), Z) -> U41_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(succ(X), pred(Y), Z) -> U43_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(pred(X), succ(Y), Z) -> U45_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) isTimes_in_gga(pred(X), pred(Y), Z) -> U47_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) U47_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_gga(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U48_gga(X, Y, Z, isMinus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), pred(Y), Z) U45_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_gga(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U46_gga(X, Y, Z, isPlus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), succ(Y), Z) U43_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_gga(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U44_gga(X, Y, Z, isMinus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), pred(Y), Z) U41_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_gga(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U42_gga(X, Y, Z, isPlus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), succ(Y), Z) U25_ga(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_ga(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U26_ga(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_ga(Order, Ans, =_in_ag(N, U2)) =_in_ag(X, X) -> =_out_ag(X, X) U27_ga(Order, Ans, =_out_ag(N, U2)) -> U28_ga(Order, Ans, N, iota_in_ga(N, .(zero, List))) iota_in_ga(N, List) -> U2_ga(N, List, iota1_in_gga(zero, N, List)) iota1_in_gga(K, K, []) -> iota1_out_gga(K, K, []) iota1_in_gga(K, N, .(K, List)) -> U3_gga(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_gga(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_gga(K, N, List, =_in_ag(K1, U)) U4_gga(K, N, List, =_out_ag(K1, U)) -> U5_gga(K, N, List, iota1_in_gga(K1, N, List)) U5_gga(K, N, List, iota1_out_gga(K1, N, List)) -> iota1_out_gga(K, N, .(K, List)) U2_ga(N, List, iota1_out_gga(zero, N, List)) -> iota_out_ga(N, List) U28_ga(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_ga(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) pds1_in_ggag([], X6, [], X7) -> pds1_out_ggag([], X6, [], X7) pds1_in_ggag(Unused, List, .(Choice, Rest), Mod) -> U21_ggag(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(X, .(X1, Xs)) -> U1_ag(X, X1, Xs, member_in_ag(X, Xs)) U1_ag(X, X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) U21_ggag(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_ggag(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) check_in_gggag([], X4, L, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, L3, Modulus) -> U17_gggag(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) dif_in_gggaa([], X2, X3, [], []) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U17_gggag(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) mergedelete_in_gga([], L, L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R), L2) -> U12_gga(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) mergedelete_in_gga(.(D, Ds), .(X, R), .(X, L2)) -> U13_gga(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_gga(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_gga(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U15_gga(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U16_gga(D, Ds, X, R, L2, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U12_gga(D, Ds, R, L2, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) rev_in_gga([], L, L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y, L) -> U11_gga(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) U11_gga(X, Xs, Y, L, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U22_ggag(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_ggag(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U23_ggag(Unused, List, Choice, Rest, Mod, pds1_out_ggag(U3, .(Choice, List), Rest, Mod)) -> pds1_out_ggag(Unused, List, .(Choice, Rest), Mod) U29_ga(Order, Ans, pds1_out_ggag(List, .(zero, []), Ans, N)) -> pds_out_ga(Order, .(zero, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U24_ga(x1, x2, x3) = U24_ga(x1, x3) isPlus_in_gga(x1, x2, x3) = isPlus_in_gga(x1, x2) zero = zero isPlus_out_gga(x1, x2, x3) = isPlus_out_gga(x3) succ(x1) = succ(x1) U31_gga(x1, x2, x3, x4) = U31_gga(x4) pred(x1) = pred(x1) U32_gga(x1, x2, x3, x4) = U32_gga(x4) U33_gga(x1, x2, x3, x4) = U33_gga(x4) U34_gga(x1, x2, x3, x4) = U34_gga(x4) U25_ga(x1, x2, x3) = U25_ga(x3) isTimes_in_gga(x1, x2, x3) = isTimes_in_gga(x1, x2) isTimes_out_gga(x1, x2, x3) = isTimes_out_gga(x3) U41_gga(x1, x2, x3, x4) = U41_gga(x1, x4) U43_gga(x1, x2, x3, x4) = U43_gga(x1, x4) U45_gga(x1, x2, x3, x4) = U45_gga(x1, x4) U47_gga(x1, x2, x3, x4) = U47_gga(x1, x4) U48_gga(x1, x2, x3, x4) = U48_gga(x4) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x3) U35_gga(x1, x2, x3) = U35_gga(x3) U36_gga(x1, x2, x3) = U36_gga(x3) U37_gga(x1, x2, x3, x4) = U37_gga(x4) U38_gga(x1, x2, x3, x4) = U38_gga(x4) U39_gga(x1, x2, x3, x4) = U39_gga(x4) U40_gga(x1, x2, x3, x4) = U40_gga(x4) U46_gga(x1, x2, x3, x4) = U46_gga(x4) U44_gga(x1, x2, x3, x4) = U44_gga(x4) U42_gga(x1, x2, x3, x4) = U42_gga(x4) U26_ga(x1, x2, x3) = U26_ga(x3) U27_ga(x1, x2, x3) = U27_ga(x3) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1) U28_ga(x1, x2, x3, x4) = U28_ga(x3, x4) iota_in_ga(x1, x2) = iota_in_ga(x1) U2_ga(x1, x2, x3) = U2_ga(x3) iota1_in_gga(x1, x2, x3) = iota1_in_gga(x1, x2) iota1_out_gga(x1, x2, x3) = iota1_out_gga(x3) U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x4) iota_out_ga(x1, x2) = iota_out_ga(x2) .(x1, x2) = .(x1, x2) U29_ga(x1, x2, x3) = U29_ga(x3) pds1_in_ggag(x1, x2, x3, x4) = pds1_in_ggag(x1, x2, x4) [] = [] pds1_out_ggag(x1, x2, x3, x4) = pds1_out_ggag(x3) U21_ggag(x1, x2, x3, x4, x5, x6) = U21_ggag(x1, x2, x5, x6) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1) U1_ag(x1, x2, x3, x4) = U1_ag(x4) U22_ggag(x1, x2, x3, x4, x5, x6) = U22_ggag(x2, x3, x5, x6) check_in_gggag(x1, x2, x3, x4, x5) = check_in_gggag(x1, x2, x3, x5) check_out_gggag(x1, x2, x3, x4, x5) = check_out_gggag(x4) U17_gggag(x1, x2, x3, x4, x5, x6) = U17_gggag(x3, x6) dif_in_gggaa(x1, x2, x3, x4, x5) = dif_in_gggaa(x1, x2, x3) dif_out_gggaa(x1, x2, x3, x4, x5) = dif_out_gggaa(x4, x5) U6_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gggaa(x2, x3, x4, x9) U7_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_gggaa(x2, x3, x4, x9, x10) U8_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_gggaa(x2, x3, x4, x5, x9, x10) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg U35_ggg(x1, x2, x3) = U35_ggg(x3) U36_ggg(x1, x2, x3) = U36_ggg(x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x4) U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_gggaa(x2, x3, x4, x5, x9) U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_gggaa(x5, x7, x9) U18_gggag(x1, x2, x3, x4, x5, x6, x7) = U18_gggag(x6, x7) mergedelete_in_gga(x1, x2, x3) = mergedelete_in_gga(x1, x2) mergedelete_out_gga(x1, x2, x3) = mergedelete_out_gga(x3) U12_gga(x1, x2, x3, x4, x5) = U12_gga(x5) U13_gga(x1, x2, x3, x4, x5, x6) = U13_gga(x1, x2, x3, x4, x6) U14_gga(x1, x2, x3, x4, x5, x6, x7) = U14_gga(x1, x2, x3, x4, x6, x7) U15_gga(x1, x2, x3, x4, x5, x6) = U15_gga(x1, x2, x3, x4, x6) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg U64_gg(x1, x2, x3) = U64_gg(x3) U65_gg(x1, x2, x3) = U65_gg(x3) U16_gga(x1, x2, x3, x4, x5, x6) = U16_gga(x3, x6) U19_gggag(x1, x2, x3, x4, x5, x6, x7) = U19_gggag(x6, x7) rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2) rev_out_gga(x1, x2, x3) = rev_out_gga(x3) U11_gga(x1, x2, x3, x4, x5) = U11_gga(x5) U20_gggag(x1, x2, x3, x4, x5, x6) = U20_gggag(x6) U23_ggag(x1, x2, x3, x4, x5, x6) = U23_ggag(x3, x6) pds_out_ga(x1, x2) = pds_out_ga(x2) ISMINUS_IN_GGG(x1, x2, x3) = ISMINUS_IN_GGG(x1, x2, x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (42) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (43) Obligation: Pi DP problem: The TRS P consists of the following rules: ISMINUS_IN_GGG(succ(X), pred(Y), succ(succ(Z))) -> ISMINUS_IN_GGG(X, Y, Z) ISMINUS_IN_GGG(succ(X), succ(Y), Z) -> ISMINUS_IN_GGG(X, Y, Z) ISMINUS_IN_GGG(pred(X), succ(Y), pred(pred(Z))) -> ISMINUS_IN_GGG(X, Y, Z) ISMINUS_IN_GGG(pred(X), pred(Y), Z) -> ISMINUS_IN_GGG(X, Y, Z) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (44) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (45) Obligation: Q DP problem: The TRS P consists of the following rules: ISMINUS_IN_GGG(succ(X), pred(Y), succ(succ(Z))) -> ISMINUS_IN_GGG(X, Y, Z) ISMINUS_IN_GGG(succ(X), succ(Y), Z) -> ISMINUS_IN_GGG(X, Y, Z) ISMINUS_IN_GGG(pred(X), succ(Y), pred(pred(Z))) -> ISMINUS_IN_GGG(X, Y, Z) ISMINUS_IN_GGG(pred(X), pred(Y), Z) -> ISMINUS_IN_GGG(X, Y, Z) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (46) 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: *ISMINUS_IN_GGG(succ(X), pred(Y), succ(succ(Z))) -> ISMINUS_IN_GGG(X, Y, Z) The graph contains the following edges 1 > 1, 2 > 2, 3 > 3 *ISMINUS_IN_GGG(succ(X), succ(Y), Z) -> ISMINUS_IN_GGG(X, Y, Z) The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3 *ISMINUS_IN_GGG(pred(X), succ(Y), pred(pred(Z))) -> ISMINUS_IN_GGG(X, Y, Z) The graph contains the following edges 1 > 1, 2 > 2, 3 > 3 *ISMINUS_IN_GGG(pred(X), pred(Y), Z) -> ISMINUS_IN_GGG(X, Y, Z) The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3 ---------------------------------------- (47) YES ---------------------------------------- (48) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER_IN_AG(X, .(X1, Xs)) -> MEMBER_IN_AG(X, Xs) The TRS R consists of the following rules: pds_in_ga(Order, .(zero, Ans)) -> U24_ga(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) isPlus_in_gga(zero, X, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), zero, succ(X)) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), succ(Y), succ(succ(Z))) -> U31_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(succ(X), pred(Y), Z) -> U32_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), zero, pred(X)) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), succ(Y), Z) -> U33_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), pred(Y), pred(pred(Z))) -> U34_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) U34_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U33_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U32_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U31_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U24_ga(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_ga(Order, Ans, isTimes_in_gga(Order, U, U1)) isTimes_in_gga(X, zero, zero) -> isTimes_out_gga(X, zero, zero) isTimes_in_gga(zero, succ(Y), zero) -> isTimes_out_gga(zero, succ(Y), zero) isTimes_in_gga(zero, pred(Y), zero) -> isTimes_out_gga(zero, pred(Y), zero) isTimes_in_gga(succ(X), succ(Y), Z) -> U41_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(succ(X), pred(Y), Z) -> U43_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(pred(X), succ(Y), Z) -> U45_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) isTimes_in_gga(pred(X), pred(Y), Z) -> U47_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) U47_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_gga(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U48_gga(X, Y, Z, isMinus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), pred(Y), Z) U45_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_gga(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U46_gga(X, Y, Z, isPlus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), succ(Y), Z) U43_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_gga(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U44_gga(X, Y, Z, isMinus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), pred(Y), Z) U41_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_gga(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U42_gga(X, Y, Z, isPlus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), succ(Y), Z) U25_ga(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_ga(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U26_ga(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_ga(Order, Ans, =_in_ag(N, U2)) =_in_ag(X, X) -> =_out_ag(X, X) U27_ga(Order, Ans, =_out_ag(N, U2)) -> U28_ga(Order, Ans, N, iota_in_ga(N, .(zero, List))) iota_in_ga(N, List) -> U2_ga(N, List, iota1_in_gga(zero, N, List)) iota1_in_gga(K, K, []) -> iota1_out_gga(K, K, []) iota1_in_gga(K, N, .(K, List)) -> U3_gga(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_gga(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_gga(K, N, List, =_in_ag(K1, U)) U4_gga(K, N, List, =_out_ag(K1, U)) -> U5_gga(K, N, List, iota1_in_gga(K1, N, List)) U5_gga(K, N, List, iota1_out_gga(K1, N, List)) -> iota1_out_gga(K, N, .(K, List)) U2_ga(N, List, iota1_out_gga(zero, N, List)) -> iota_out_ga(N, List) U28_ga(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_ga(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) pds1_in_ggag([], X6, [], X7) -> pds1_out_ggag([], X6, [], X7) pds1_in_ggag(Unused, List, .(Choice, Rest), Mod) -> U21_ggag(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(X, .(X1, Xs)) -> U1_ag(X, X1, Xs, member_in_ag(X, Xs)) U1_ag(X, X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) U21_ggag(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_ggag(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) check_in_gggag([], X4, L, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, L3, Modulus) -> U17_gggag(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) dif_in_gggaa([], X2, X3, [], []) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U17_gggag(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) mergedelete_in_gga([], L, L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R), L2) -> U12_gga(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) mergedelete_in_gga(.(D, Ds), .(X, R), .(X, L2)) -> U13_gga(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_gga(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_gga(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U15_gga(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U16_gga(D, Ds, X, R, L2, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U12_gga(D, Ds, R, L2, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) rev_in_gga([], L, L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y, L) -> U11_gga(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) U11_gga(X, Xs, Y, L, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U22_ggag(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_ggag(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U23_ggag(Unused, List, Choice, Rest, Mod, pds1_out_ggag(U3, .(Choice, List), Rest, Mod)) -> pds1_out_ggag(Unused, List, .(Choice, Rest), Mod) U29_ga(Order, Ans, pds1_out_ggag(List, .(zero, []), Ans, N)) -> pds_out_ga(Order, .(zero, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U24_ga(x1, x2, x3) = U24_ga(x1, x3) isPlus_in_gga(x1, x2, x3) = isPlus_in_gga(x1, x2) zero = zero isPlus_out_gga(x1, x2, x3) = isPlus_out_gga(x3) succ(x1) = succ(x1) U31_gga(x1, x2, x3, x4) = U31_gga(x4) pred(x1) = pred(x1) U32_gga(x1, x2, x3, x4) = U32_gga(x4) U33_gga(x1, x2, x3, x4) = U33_gga(x4) U34_gga(x1, x2, x3, x4) = U34_gga(x4) U25_ga(x1, x2, x3) = U25_ga(x3) isTimes_in_gga(x1, x2, x3) = isTimes_in_gga(x1, x2) isTimes_out_gga(x1, x2, x3) = isTimes_out_gga(x3) U41_gga(x1, x2, x3, x4) = U41_gga(x1, x4) U43_gga(x1, x2, x3, x4) = U43_gga(x1, x4) U45_gga(x1, x2, x3, x4) = U45_gga(x1, x4) U47_gga(x1, x2, x3, x4) = U47_gga(x1, x4) U48_gga(x1, x2, x3, x4) = U48_gga(x4) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x3) U35_gga(x1, x2, x3) = U35_gga(x3) U36_gga(x1, x2, x3) = U36_gga(x3) U37_gga(x1, x2, x3, x4) = U37_gga(x4) U38_gga(x1, x2, x3, x4) = U38_gga(x4) U39_gga(x1, x2, x3, x4) = U39_gga(x4) U40_gga(x1, x2, x3, x4) = U40_gga(x4) U46_gga(x1, x2, x3, x4) = U46_gga(x4) U44_gga(x1, x2, x3, x4) = U44_gga(x4) U42_gga(x1, x2, x3, x4) = U42_gga(x4) U26_ga(x1, x2, x3) = U26_ga(x3) U27_ga(x1, x2, x3) = U27_ga(x3) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1) U28_ga(x1, x2, x3, x4) = U28_ga(x3, x4) iota_in_ga(x1, x2) = iota_in_ga(x1) U2_ga(x1, x2, x3) = U2_ga(x3) iota1_in_gga(x1, x2, x3) = iota1_in_gga(x1, x2) iota1_out_gga(x1, x2, x3) = iota1_out_gga(x3) U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x4) iota_out_ga(x1, x2) = iota_out_ga(x2) .(x1, x2) = .(x1, x2) U29_ga(x1, x2, x3) = U29_ga(x3) pds1_in_ggag(x1, x2, x3, x4) = pds1_in_ggag(x1, x2, x4) [] = [] pds1_out_ggag(x1, x2, x3, x4) = pds1_out_ggag(x3) U21_ggag(x1, x2, x3, x4, x5, x6) = U21_ggag(x1, x2, x5, x6) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1) U1_ag(x1, x2, x3, x4) = U1_ag(x4) U22_ggag(x1, x2, x3, x4, x5, x6) = U22_ggag(x2, x3, x5, x6) check_in_gggag(x1, x2, x3, x4, x5) = check_in_gggag(x1, x2, x3, x5) check_out_gggag(x1, x2, x3, x4, x5) = check_out_gggag(x4) U17_gggag(x1, x2, x3, x4, x5, x6) = U17_gggag(x3, x6) dif_in_gggaa(x1, x2, x3, x4, x5) = dif_in_gggaa(x1, x2, x3) dif_out_gggaa(x1, x2, x3, x4, x5) = dif_out_gggaa(x4, x5) U6_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gggaa(x2, x3, x4, x9) U7_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_gggaa(x2, x3, x4, x9, x10) U8_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_gggaa(x2, x3, x4, x5, x9, x10) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg U35_ggg(x1, x2, x3) = U35_ggg(x3) U36_ggg(x1, x2, x3) = U36_ggg(x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x4) U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_gggaa(x2, x3, x4, x5, x9) U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_gggaa(x5, x7, x9) U18_gggag(x1, x2, x3, x4, x5, x6, x7) = U18_gggag(x6, x7) mergedelete_in_gga(x1, x2, x3) = mergedelete_in_gga(x1, x2) mergedelete_out_gga(x1, x2, x3) = mergedelete_out_gga(x3) U12_gga(x1, x2, x3, x4, x5) = U12_gga(x5) U13_gga(x1, x2, x3, x4, x5, x6) = U13_gga(x1, x2, x3, x4, x6) U14_gga(x1, x2, x3, x4, x5, x6, x7) = U14_gga(x1, x2, x3, x4, x6, x7) U15_gga(x1, x2, x3, x4, x5, x6) = U15_gga(x1, x2, x3, x4, x6) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg U64_gg(x1, x2, x3) = U64_gg(x3) U65_gg(x1, x2, x3) = U65_gg(x3) U16_gga(x1, x2, x3, x4, x5, x6) = U16_gga(x3, x6) U19_gggag(x1, x2, x3, x4, x5, x6, x7) = U19_gggag(x6, x7) rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2) rev_out_gga(x1, x2, x3) = rev_out_gga(x3) U11_gga(x1, x2, x3, x4, x5) = U11_gga(x5) U20_gggag(x1, x2, x3, x4, x5, x6) = U20_gggag(x6) U23_ggag(x1, x2, x3, x4, x5, x6) = U23_ggag(x3, x6) pds_out_ga(x1, x2) = pds_out_ga(x2) MEMBER_IN_AG(x1, x2) = MEMBER_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (49) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (50) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER_IN_AG(X, .(X1, 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 ---------------------------------------- (51) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (52) Obligation: Q DP problem: The TRS P consists of the following rules: MEMBER_IN_AG(.(X1, Xs)) -> MEMBER_IN_AG(Xs) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (53) 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(.(X1, Xs)) -> MEMBER_IN_AG(Xs) The graph contains the following edges 1 > 1 ---------------------------------------- (54) YES ---------------------------------------- (55) Obligation: Pi DP problem: The TRS P consists of the following rules: ISMINUS_IN_GGA(zero, pred(Y), succ(Z)) -> ISMINUS_IN_GGA(zero, Y, Z) ISMINUS_IN_GGA(zero, succ(Y), pred(Z)) -> ISMINUS_IN_GGA(zero, Y, Z) The TRS R consists of the following rules: pds_in_ga(Order, .(zero, Ans)) -> U24_ga(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) isPlus_in_gga(zero, X, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), zero, succ(X)) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), succ(Y), succ(succ(Z))) -> U31_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(succ(X), pred(Y), Z) -> U32_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), zero, pred(X)) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), succ(Y), Z) -> U33_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), pred(Y), pred(pred(Z))) -> U34_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) U34_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U33_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U32_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U31_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U24_ga(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_ga(Order, Ans, isTimes_in_gga(Order, U, U1)) isTimes_in_gga(X, zero, zero) -> isTimes_out_gga(X, zero, zero) isTimes_in_gga(zero, succ(Y), zero) -> isTimes_out_gga(zero, succ(Y), zero) isTimes_in_gga(zero, pred(Y), zero) -> isTimes_out_gga(zero, pred(Y), zero) isTimes_in_gga(succ(X), succ(Y), Z) -> U41_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(succ(X), pred(Y), Z) -> U43_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(pred(X), succ(Y), Z) -> U45_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) isTimes_in_gga(pred(X), pred(Y), Z) -> U47_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) U47_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_gga(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U48_gga(X, Y, Z, isMinus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), pred(Y), Z) U45_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_gga(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U46_gga(X, Y, Z, isPlus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), succ(Y), Z) U43_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_gga(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U44_gga(X, Y, Z, isMinus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), pred(Y), Z) U41_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_gga(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U42_gga(X, Y, Z, isPlus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), succ(Y), Z) U25_ga(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_ga(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U26_ga(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_ga(Order, Ans, =_in_ag(N, U2)) =_in_ag(X, X) -> =_out_ag(X, X) U27_ga(Order, Ans, =_out_ag(N, U2)) -> U28_ga(Order, Ans, N, iota_in_ga(N, .(zero, List))) iota_in_ga(N, List) -> U2_ga(N, List, iota1_in_gga(zero, N, List)) iota1_in_gga(K, K, []) -> iota1_out_gga(K, K, []) iota1_in_gga(K, N, .(K, List)) -> U3_gga(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_gga(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_gga(K, N, List, =_in_ag(K1, U)) U4_gga(K, N, List, =_out_ag(K1, U)) -> U5_gga(K, N, List, iota1_in_gga(K1, N, List)) U5_gga(K, N, List, iota1_out_gga(K1, N, List)) -> iota1_out_gga(K, N, .(K, List)) U2_ga(N, List, iota1_out_gga(zero, N, List)) -> iota_out_ga(N, List) U28_ga(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_ga(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) pds1_in_ggag([], X6, [], X7) -> pds1_out_ggag([], X6, [], X7) pds1_in_ggag(Unused, List, .(Choice, Rest), Mod) -> U21_ggag(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(X, .(X1, Xs)) -> U1_ag(X, X1, Xs, member_in_ag(X, Xs)) U1_ag(X, X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) U21_ggag(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_ggag(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) check_in_gggag([], X4, L, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, L3, Modulus) -> U17_gggag(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) dif_in_gggaa([], X2, X3, [], []) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U17_gggag(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) mergedelete_in_gga([], L, L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R), L2) -> U12_gga(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) mergedelete_in_gga(.(D, Ds), .(X, R), .(X, L2)) -> U13_gga(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_gga(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_gga(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U15_gga(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U16_gga(D, Ds, X, R, L2, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U12_gga(D, Ds, R, L2, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) rev_in_gga([], L, L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y, L) -> U11_gga(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) U11_gga(X, Xs, Y, L, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U22_ggag(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_ggag(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U23_ggag(Unused, List, Choice, Rest, Mod, pds1_out_ggag(U3, .(Choice, List), Rest, Mod)) -> pds1_out_ggag(Unused, List, .(Choice, Rest), Mod) U29_ga(Order, Ans, pds1_out_ggag(List, .(zero, []), Ans, N)) -> pds_out_ga(Order, .(zero, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U24_ga(x1, x2, x3) = U24_ga(x1, x3) isPlus_in_gga(x1, x2, x3) = isPlus_in_gga(x1, x2) zero = zero isPlus_out_gga(x1, x2, x3) = isPlus_out_gga(x3) succ(x1) = succ(x1) U31_gga(x1, x2, x3, x4) = U31_gga(x4) pred(x1) = pred(x1) U32_gga(x1, x2, x3, x4) = U32_gga(x4) U33_gga(x1, x2, x3, x4) = U33_gga(x4) U34_gga(x1, x2, x3, x4) = U34_gga(x4) U25_ga(x1, x2, x3) = U25_ga(x3) isTimes_in_gga(x1, x2, x3) = isTimes_in_gga(x1, x2) isTimes_out_gga(x1, x2, x3) = isTimes_out_gga(x3) U41_gga(x1, x2, x3, x4) = U41_gga(x1, x4) U43_gga(x1, x2, x3, x4) = U43_gga(x1, x4) U45_gga(x1, x2, x3, x4) = U45_gga(x1, x4) U47_gga(x1, x2, x3, x4) = U47_gga(x1, x4) U48_gga(x1, x2, x3, x4) = U48_gga(x4) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x3) U35_gga(x1, x2, x3) = U35_gga(x3) U36_gga(x1, x2, x3) = U36_gga(x3) U37_gga(x1, x2, x3, x4) = U37_gga(x4) U38_gga(x1, x2, x3, x4) = U38_gga(x4) U39_gga(x1, x2, x3, x4) = U39_gga(x4) U40_gga(x1, x2, x3, x4) = U40_gga(x4) U46_gga(x1, x2, x3, x4) = U46_gga(x4) U44_gga(x1, x2, x3, x4) = U44_gga(x4) U42_gga(x1, x2, x3, x4) = U42_gga(x4) U26_ga(x1, x2, x3) = U26_ga(x3) U27_ga(x1, x2, x3) = U27_ga(x3) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1) U28_ga(x1, x2, x3, x4) = U28_ga(x3, x4) iota_in_ga(x1, x2) = iota_in_ga(x1) U2_ga(x1, x2, x3) = U2_ga(x3) iota1_in_gga(x1, x2, x3) = iota1_in_gga(x1, x2) iota1_out_gga(x1, x2, x3) = iota1_out_gga(x3) U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x4) iota_out_ga(x1, x2) = iota_out_ga(x2) .(x1, x2) = .(x1, x2) U29_ga(x1, x2, x3) = U29_ga(x3) pds1_in_ggag(x1, x2, x3, x4) = pds1_in_ggag(x1, x2, x4) [] = [] pds1_out_ggag(x1, x2, x3, x4) = pds1_out_ggag(x3) U21_ggag(x1, x2, x3, x4, x5, x6) = U21_ggag(x1, x2, x5, x6) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1) U1_ag(x1, x2, x3, x4) = U1_ag(x4) U22_ggag(x1, x2, x3, x4, x5, x6) = U22_ggag(x2, x3, x5, x6) check_in_gggag(x1, x2, x3, x4, x5) = check_in_gggag(x1, x2, x3, x5) check_out_gggag(x1, x2, x3, x4, x5) = check_out_gggag(x4) U17_gggag(x1, x2, x3, x4, x5, x6) = U17_gggag(x3, x6) dif_in_gggaa(x1, x2, x3, x4, x5) = dif_in_gggaa(x1, x2, x3) dif_out_gggaa(x1, x2, x3, x4, x5) = dif_out_gggaa(x4, x5) U6_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gggaa(x2, x3, x4, x9) U7_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_gggaa(x2, x3, x4, x9, x10) U8_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_gggaa(x2, x3, x4, x5, x9, x10) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg U35_ggg(x1, x2, x3) = U35_ggg(x3) U36_ggg(x1, x2, x3) = U36_ggg(x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x4) U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_gggaa(x2, x3, x4, x5, x9) U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_gggaa(x5, x7, x9) U18_gggag(x1, x2, x3, x4, x5, x6, x7) = U18_gggag(x6, x7) mergedelete_in_gga(x1, x2, x3) = mergedelete_in_gga(x1, x2) mergedelete_out_gga(x1, x2, x3) = mergedelete_out_gga(x3) U12_gga(x1, x2, x3, x4, x5) = U12_gga(x5) U13_gga(x1, x2, x3, x4, x5, x6) = U13_gga(x1, x2, x3, x4, x6) U14_gga(x1, x2, x3, x4, x5, x6, x7) = U14_gga(x1, x2, x3, x4, x6, x7) U15_gga(x1, x2, x3, x4, x5, x6) = U15_gga(x1, x2, x3, x4, x6) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg U64_gg(x1, x2, x3) = U64_gg(x3) U65_gg(x1, x2, x3) = U65_gg(x3) U16_gga(x1, x2, x3, x4, x5, x6) = U16_gga(x3, x6) U19_gggag(x1, x2, x3, x4, x5, x6, x7) = U19_gggag(x6, x7) rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2) rev_out_gga(x1, x2, x3) = rev_out_gga(x3) U11_gga(x1, x2, x3, x4, x5) = U11_gga(x5) U20_gggag(x1, x2, x3, x4, x5, x6) = U20_gggag(x6) U23_ggag(x1, x2, x3, x4, x5, x6) = U23_ggag(x3, x6) pds_out_ga(x1, x2) = pds_out_ga(x2) ISMINUS_IN_GGA(x1, x2, x3) = ISMINUS_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (56) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (57) Obligation: Pi DP problem: The TRS P consists of the following rules: ISMINUS_IN_GGA(zero, pred(Y), succ(Z)) -> ISMINUS_IN_GGA(zero, Y, Z) ISMINUS_IN_GGA(zero, succ(Y), pred(Z)) -> ISMINUS_IN_GGA(zero, Y, Z) R is empty. The argument filtering Pi contains the following mapping: zero = zero succ(x1) = succ(x1) pred(x1) = pred(x1) ISMINUS_IN_GGA(x1, x2, x3) = ISMINUS_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (58) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (59) Obligation: Q DP problem: The TRS P consists of the following rules: ISMINUS_IN_GGA(zero, pred(Y)) -> ISMINUS_IN_GGA(zero, Y) ISMINUS_IN_GGA(zero, succ(Y)) -> ISMINUS_IN_GGA(zero, Y) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (60) 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: *ISMINUS_IN_GGA(zero, pred(Y)) -> ISMINUS_IN_GGA(zero, Y) The graph contains the following edges 1 >= 1, 2 > 2 *ISMINUS_IN_GGA(zero, succ(Y)) -> ISMINUS_IN_GGA(zero, Y) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (61) YES ---------------------------------------- (62) Obligation: Pi DP problem: The TRS P consists of the following rules: ISMINUS_IN_GGA(succ(X), pred(Y), succ(succ(Z))) -> ISMINUS_IN_GGA(X, Y, Z) ISMINUS_IN_GGA(succ(X), succ(Y), Z) -> ISMINUS_IN_GGA(X, Y, Z) ISMINUS_IN_GGA(pred(X), succ(Y), pred(pred(Z))) -> ISMINUS_IN_GGA(X, Y, Z) ISMINUS_IN_GGA(pred(X), pred(Y), Z) -> ISMINUS_IN_GGA(X, Y, Z) The TRS R consists of the following rules: pds_in_ga(Order, .(zero, Ans)) -> U24_ga(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) isPlus_in_gga(zero, X, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), zero, succ(X)) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), succ(Y), succ(succ(Z))) -> U31_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(succ(X), pred(Y), Z) -> U32_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), zero, pred(X)) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), succ(Y), Z) -> U33_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), pred(Y), pred(pred(Z))) -> U34_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) U34_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U33_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U32_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U31_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U24_ga(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_ga(Order, Ans, isTimes_in_gga(Order, U, U1)) isTimes_in_gga(X, zero, zero) -> isTimes_out_gga(X, zero, zero) isTimes_in_gga(zero, succ(Y), zero) -> isTimes_out_gga(zero, succ(Y), zero) isTimes_in_gga(zero, pred(Y), zero) -> isTimes_out_gga(zero, pred(Y), zero) isTimes_in_gga(succ(X), succ(Y), Z) -> U41_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(succ(X), pred(Y), Z) -> U43_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(pred(X), succ(Y), Z) -> U45_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) isTimes_in_gga(pred(X), pred(Y), Z) -> U47_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) U47_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_gga(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U48_gga(X, Y, Z, isMinus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), pred(Y), Z) U45_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_gga(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U46_gga(X, Y, Z, isPlus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), succ(Y), Z) U43_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_gga(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U44_gga(X, Y, Z, isMinus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), pred(Y), Z) U41_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_gga(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U42_gga(X, Y, Z, isPlus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), succ(Y), Z) U25_ga(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_ga(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U26_ga(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_ga(Order, Ans, =_in_ag(N, U2)) =_in_ag(X, X) -> =_out_ag(X, X) U27_ga(Order, Ans, =_out_ag(N, U2)) -> U28_ga(Order, Ans, N, iota_in_ga(N, .(zero, List))) iota_in_ga(N, List) -> U2_ga(N, List, iota1_in_gga(zero, N, List)) iota1_in_gga(K, K, []) -> iota1_out_gga(K, K, []) iota1_in_gga(K, N, .(K, List)) -> U3_gga(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_gga(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_gga(K, N, List, =_in_ag(K1, U)) U4_gga(K, N, List, =_out_ag(K1, U)) -> U5_gga(K, N, List, iota1_in_gga(K1, N, List)) U5_gga(K, N, List, iota1_out_gga(K1, N, List)) -> iota1_out_gga(K, N, .(K, List)) U2_ga(N, List, iota1_out_gga(zero, N, List)) -> iota_out_ga(N, List) U28_ga(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_ga(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) pds1_in_ggag([], X6, [], X7) -> pds1_out_ggag([], X6, [], X7) pds1_in_ggag(Unused, List, .(Choice, Rest), Mod) -> U21_ggag(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(X, .(X1, Xs)) -> U1_ag(X, X1, Xs, member_in_ag(X, Xs)) U1_ag(X, X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) U21_ggag(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_ggag(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) check_in_gggag([], X4, L, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, L3, Modulus) -> U17_gggag(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) dif_in_gggaa([], X2, X3, [], []) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U17_gggag(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) mergedelete_in_gga([], L, L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R), L2) -> U12_gga(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) mergedelete_in_gga(.(D, Ds), .(X, R), .(X, L2)) -> U13_gga(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_gga(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_gga(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U15_gga(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U16_gga(D, Ds, X, R, L2, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U12_gga(D, Ds, R, L2, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) rev_in_gga([], L, L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y, L) -> U11_gga(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) U11_gga(X, Xs, Y, L, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U22_ggag(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_ggag(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U23_ggag(Unused, List, Choice, Rest, Mod, pds1_out_ggag(U3, .(Choice, List), Rest, Mod)) -> pds1_out_ggag(Unused, List, .(Choice, Rest), Mod) U29_ga(Order, Ans, pds1_out_ggag(List, .(zero, []), Ans, N)) -> pds_out_ga(Order, .(zero, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U24_ga(x1, x2, x3) = U24_ga(x1, x3) isPlus_in_gga(x1, x2, x3) = isPlus_in_gga(x1, x2) zero = zero isPlus_out_gga(x1, x2, x3) = isPlus_out_gga(x3) succ(x1) = succ(x1) U31_gga(x1, x2, x3, x4) = U31_gga(x4) pred(x1) = pred(x1) U32_gga(x1, x2, x3, x4) = U32_gga(x4) U33_gga(x1, x2, x3, x4) = U33_gga(x4) U34_gga(x1, x2, x3, x4) = U34_gga(x4) U25_ga(x1, x2, x3) = U25_ga(x3) isTimes_in_gga(x1, x2, x3) = isTimes_in_gga(x1, x2) isTimes_out_gga(x1, x2, x3) = isTimes_out_gga(x3) U41_gga(x1, x2, x3, x4) = U41_gga(x1, x4) U43_gga(x1, x2, x3, x4) = U43_gga(x1, x4) U45_gga(x1, x2, x3, x4) = U45_gga(x1, x4) U47_gga(x1, x2, x3, x4) = U47_gga(x1, x4) U48_gga(x1, x2, x3, x4) = U48_gga(x4) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x3) U35_gga(x1, x2, x3) = U35_gga(x3) U36_gga(x1, x2, x3) = U36_gga(x3) U37_gga(x1, x2, x3, x4) = U37_gga(x4) U38_gga(x1, x2, x3, x4) = U38_gga(x4) U39_gga(x1, x2, x3, x4) = U39_gga(x4) U40_gga(x1, x2, x3, x4) = U40_gga(x4) U46_gga(x1, x2, x3, x4) = U46_gga(x4) U44_gga(x1, x2, x3, x4) = U44_gga(x4) U42_gga(x1, x2, x3, x4) = U42_gga(x4) U26_ga(x1, x2, x3) = U26_ga(x3) U27_ga(x1, x2, x3) = U27_ga(x3) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1) U28_ga(x1, x2, x3, x4) = U28_ga(x3, x4) iota_in_ga(x1, x2) = iota_in_ga(x1) U2_ga(x1, x2, x3) = U2_ga(x3) iota1_in_gga(x1, x2, x3) = iota1_in_gga(x1, x2) iota1_out_gga(x1, x2, x3) = iota1_out_gga(x3) U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x4) iota_out_ga(x1, x2) = iota_out_ga(x2) .(x1, x2) = .(x1, x2) U29_ga(x1, x2, x3) = U29_ga(x3) pds1_in_ggag(x1, x2, x3, x4) = pds1_in_ggag(x1, x2, x4) [] = [] pds1_out_ggag(x1, x2, x3, x4) = pds1_out_ggag(x3) U21_ggag(x1, x2, x3, x4, x5, x6) = U21_ggag(x1, x2, x5, x6) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1) U1_ag(x1, x2, x3, x4) = U1_ag(x4) U22_ggag(x1, x2, x3, x4, x5, x6) = U22_ggag(x2, x3, x5, x6) check_in_gggag(x1, x2, x3, x4, x5) = check_in_gggag(x1, x2, x3, x5) check_out_gggag(x1, x2, x3, x4, x5) = check_out_gggag(x4) U17_gggag(x1, x2, x3, x4, x5, x6) = U17_gggag(x3, x6) dif_in_gggaa(x1, x2, x3, x4, x5) = dif_in_gggaa(x1, x2, x3) dif_out_gggaa(x1, x2, x3, x4, x5) = dif_out_gggaa(x4, x5) U6_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gggaa(x2, x3, x4, x9) U7_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_gggaa(x2, x3, x4, x9, x10) U8_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_gggaa(x2, x3, x4, x5, x9, x10) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg U35_ggg(x1, x2, x3) = U35_ggg(x3) U36_ggg(x1, x2, x3) = U36_ggg(x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x4) U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_gggaa(x2, x3, x4, x5, x9) U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_gggaa(x5, x7, x9) U18_gggag(x1, x2, x3, x4, x5, x6, x7) = U18_gggag(x6, x7) mergedelete_in_gga(x1, x2, x3) = mergedelete_in_gga(x1, x2) mergedelete_out_gga(x1, x2, x3) = mergedelete_out_gga(x3) U12_gga(x1, x2, x3, x4, x5) = U12_gga(x5) U13_gga(x1, x2, x3, x4, x5, x6) = U13_gga(x1, x2, x3, x4, x6) U14_gga(x1, x2, x3, x4, x5, x6, x7) = U14_gga(x1, x2, x3, x4, x6, x7) U15_gga(x1, x2, x3, x4, x5, x6) = U15_gga(x1, x2, x3, x4, x6) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg U64_gg(x1, x2, x3) = U64_gg(x3) U65_gg(x1, x2, x3) = U65_gg(x3) U16_gga(x1, x2, x3, x4, x5, x6) = U16_gga(x3, x6) U19_gggag(x1, x2, x3, x4, x5, x6, x7) = U19_gggag(x6, x7) rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2) rev_out_gga(x1, x2, x3) = rev_out_gga(x3) U11_gga(x1, x2, x3, x4, x5) = U11_gga(x5) U20_gggag(x1, x2, x3, x4, x5, x6) = U20_gggag(x6) U23_ggag(x1, x2, x3, x4, x5, x6) = U23_ggag(x3, x6) pds_out_ga(x1, x2) = pds_out_ga(x2) ISMINUS_IN_GGA(x1, x2, x3) = ISMINUS_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (63) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (64) Obligation: Pi DP problem: The TRS P consists of the following rules: ISMINUS_IN_GGA(succ(X), pred(Y), succ(succ(Z))) -> ISMINUS_IN_GGA(X, Y, Z) ISMINUS_IN_GGA(succ(X), succ(Y), Z) -> ISMINUS_IN_GGA(X, Y, Z) ISMINUS_IN_GGA(pred(X), succ(Y), pred(pred(Z))) -> ISMINUS_IN_GGA(X, Y, Z) ISMINUS_IN_GGA(pred(X), pred(Y), Z) -> ISMINUS_IN_GGA(X, Y, Z) R is empty. The argument filtering Pi contains the following mapping: succ(x1) = succ(x1) pred(x1) = pred(x1) ISMINUS_IN_GGA(x1, x2, x3) = ISMINUS_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (65) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (66) Obligation: Q DP problem: The TRS P consists of the following rules: ISMINUS_IN_GGA(succ(X), pred(Y)) -> ISMINUS_IN_GGA(X, Y) ISMINUS_IN_GGA(succ(X), succ(Y)) -> ISMINUS_IN_GGA(X, Y) ISMINUS_IN_GGA(pred(X), succ(Y)) -> ISMINUS_IN_GGA(X, Y) ISMINUS_IN_GGA(pred(X), pred(Y)) -> ISMINUS_IN_GGA(X, Y) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (67) 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: *ISMINUS_IN_GGA(succ(X), pred(Y)) -> ISMINUS_IN_GGA(X, Y) The graph contains the following edges 1 > 1, 2 > 2 *ISMINUS_IN_GGA(succ(X), succ(Y)) -> ISMINUS_IN_GGA(X, Y) The graph contains the following edges 1 > 1, 2 > 2 *ISMINUS_IN_GGA(pred(X), succ(Y)) -> ISMINUS_IN_GGA(X, Y) The graph contains the following edges 1 > 1, 2 > 2 *ISMINUS_IN_GGA(pred(X), pred(Y)) -> ISMINUS_IN_GGA(X, Y) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (68) YES ---------------------------------------- (69) Obligation: Pi DP problem: The TRS P consists of the following rules: DIF_IN_GGGAA(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) U8_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> DIF_IN_GGGAA(Ss, Val, Mod, Ds, D2s) The TRS R consists of the following rules: pds_in_ga(Order, .(zero, Ans)) -> U24_ga(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) isPlus_in_gga(zero, X, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), zero, succ(X)) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), succ(Y), succ(succ(Z))) -> U31_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(succ(X), pred(Y), Z) -> U32_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), zero, pred(X)) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), succ(Y), Z) -> U33_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), pred(Y), pred(pred(Z))) -> U34_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) U34_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U33_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U32_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U31_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U24_ga(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_ga(Order, Ans, isTimes_in_gga(Order, U, U1)) isTimes_in_gga(X, zero, zero) -> isTimes_out_gga(X, zero, zero) isTimes_in_gga(zero, succ(Y), zero) -> isTimes_out_gga(zero, succ(Y), zero) isTimes_in_gga(zero, pred(Y), zero) -> isTimes_out_gga(zero, pred(Y), zero) isTimes_in_gga(succ(X), succ(Y), Z) -> U41_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(succ(X), pred(Y), Z) -> U43_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(pred(X), succ(Y), Z) -> U45_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) isTimes_in_gga(pred(X), pred(Y), Z) -> U47_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) U47_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_gga(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U48_gga(X, Y, Z, isMinus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), pred(Y), Z) U45_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_gga(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U46_gga(X, Y, Z, isPlus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), succ(Y), Z) U43_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_gga(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U44_gga(X, Y, Z, isMinus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), pred(Y), Z) U41_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_gga(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U42_gga(X, Y, Z, isPlus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), succ(Y), Z) U25_ga(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_ga(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U26_ga(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_ga(Order, Ans, =_in_ag(N, U2)) =_in_ag(X, X) -> =_out_ag(X, X) U27_ga(Order, Ans, =_out_ag(N, U2)) -> U28_ga(Order, Ans, N, iota_in_ga(N, .(zero, List))) iota_in_ga(N, List) -> U2_ga(N, List, iota1_in_gga(zero, N, List)) iota1_in_gga(K, K, []) -> iota1_out_gga(K, K, []) iota1_in_gga(K, N, .(K, List)) -> U3_gga(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_gga(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_gga(K, N, List, =_in_ag(K1, U)) U4_gga(K, N, List, =_out_ag(K1, U)) -> U5_gga(K, N, List, iota1_in_gga(K1, N, List)) U5_gga(K, N, List, iota1_out_gga(K1, N, List)) -> iota1_out_gga(K, N, .(K, List)) U2_ga(N, List, iota1_out_gga(zero, N, List)) -> iota_out_ga(N, List) U28_ga(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_ga(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) pds1_in_ggag([], X6, [], X7) -> pds1_out_ggag([], X6, [], X7) pds1_in_ggag(Unused, List, .(Choice, Rest), Mod) -> U21_ggag(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(X, .(X1, Xs)) -> U1_ag(X, X1, Xs, member_in_ag(X, Xs)) U1_ag(X, X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) U21_ggag(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_ggag(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) check_in_gggag([], X4, L, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, L3, Modulus) -> U17_gggag(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) dif_in_gggaa([], X2, X3, [], []) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U17_gggag(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) mergedelete_in_gga([], L, L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R), L2) -> U12_gga(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) mergedelete_in_gga(.(D, Ds), .(X, R), .(X, L2)) -> U13_gga(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_gga(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_gga(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U15_gga(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U16_gga(D, Ds, X, R, L2, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U12_gga(D, Ds, R, L2, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) rev_in_gga([], L, L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y, L) -> U11_gga(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) U11_gga(X, Xs, Y, L, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U22_ggag(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_ggag(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U23_ggag(Unused, List, Choice, Rest, Mod, pds1_out_ggag(U3, .(Choice, List), Rest, Mod)) -> pds1_out_ggag(Unused, List, .(Choice, Rest), Mod) U29_ga(Order, Ans, pds1_out_ggag(List, .(zero, []), Ans, N)) -> pds_out_ga(Order, .(zero, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U24_ga(x1, x2, x3) = U24_ga(x1, x3) isPlus_in_gga(x1, x2, x3) = isPlus_in_gga(x1, x2) zero = zero isPlus_out_gga(x1, x2, x3) = isPlus_out_gga(x3) succ(x1) = succ(x1) U31_gga(x1, x2, x3, x4) = U31_gga(x4) pred(x1) = pred(x1) U32_gga(x1, x2, x3, x4) = U32_gga(x4) U33_gga(x1, x2, x3, x4) = U33_gga(x4) U34_gga(x1, x2, x3, x4) = U34_gga(x4) U25_ga(x1, x2, x3) = U25_ga(x3) isTimes_in_gga(x1, x2, x3) = isTimes_in_gga(x1, x2) isTimes_out_gga(x1, x2, x3) = isTimes_out_gga(x3) U41_gga(x1, x2, x3, x4) = U41_gga(x1, x4) U43_gga(x1, x2, x3, x4) = U43_gga(x1, x4) U45_gga(x1, x2, x3, x4) = U45_gga(x1, x4) U47_gga(x1, x2, x3, x4) = U47_gga(x1, x4) U48_gga(x1, x2, x3, x4) = U48_gga(x4) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x3) U35_gga(x1, x2, x3) = U35_gga(x3) U36_gga(x1, x2, x3) = U36_gga(x3) U37_gga(x1, x2, x3, x4) = U37_gga(x4) U38_gga(x1, x2, x3, x4) = U38_gga(x4) U39_gga(x1, x2, x3, x4) = U39_gga(x4) U40_gga(x1, x2, x3, x4) = U40_gga(x4) U46_gga(x1, x2, x3, x4) = U46_gga(x4) U44_gga(x1, x2, x3, x4) = U44_gga(x4) U42_gga(x1, x2, x3, x4) = U42_gga(x4) U26_ga(x1, x2, x3) = U26_ga(x3) U27_ga(x1, x2, x3) = U27_ga(x3) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1) U28_ga(x1, x2, x3, x4) = U28_ga(x3, x4) iota_in_ga(x1, x2) = iota_in_ga(x1) U2_ga(x1, x2, x3) = U2_ga(x3) iota1_in_gga(x1, x2, x3) = iota1_in_gga(x1, x2) iota1_out_gga(x1, x2, x3) = iota1_out_gga(x3) U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x4) iota_out_ga(x1, x2) = iota_out_ga(x2) .(x1, x2) = .(x1, x2) U29_ga(x1, x2, x3) = U29_ga(x3) pds1_in_ggag(x1, x2, x3, x4) = pds1_in_ggag(x1, x2, x4) [] = [] pds1_out_ggag(x1, x2, x3, x4) = pds1_out_ggag(x3) U21_ggag(x1, x2, x3, x4, x5, x6) = U21_ggag(x1, x2, x5, x6) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1) U1_ag(x1, x2, x3, x4) = U1_ag(x4) U22_ggag(x1, x2, x3, x4, x5, x6) = U22_ggag(x2, x3, x5, x6) check_in_gggag(x1, x2, x3, x4, x5) = check_in_gggag(x1, x2, x3, x5) check_out_gggag(x1, x2, x3, x4, x5) = check_out_gggag(x4) U17_gggag(x1, x2, x3, x4, x5, x6) = U17_gggag(x3, x6) dif_in_gggaa(x1, x2, x3, x4, x5) = dif_in_gggaa(x1, x2, x3) dif_out_gggaa(x1, x2, x3, x4, x5) = dif_out_gggaa(x4, x5) U6_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gggaa(x2, x3, x4, x9) U7_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_gggaa(x2, x3, x4, x9, x10) U8_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_gggaa(x2, x3, x4, x5, x9, x10) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg U35_ggg(x1, x2, x3) = U35_ggg(x3) U36_ggg(x1, x2, x3) = U36_ggg(x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x4) U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_gggaa(x2, x3, x4, x5, x9) U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_gggaa(x5, x7, x9) U18_gggag(x1, x2, x3, x4, x5, x6, x7) = U18_gggag(x6, x7) mergedelete_in_gga(x1, x2, x3) = mergedelete_in_gga(x1, x2) mergedelete_out_gga(x1, x2, x3) = mergedelete_out_gga(x3) U12_gga(x1, x2, x3, x4, x5) = U12_gga(x5) U13_gga(x1, x2, x3, x4, x5, x6) = U13_gga(x1, x2, x3, x4, x6) U14_gga(x1, x2, x3, x4, x5, x6, x7) = U14_gga(x1, x2, x3, x4, x6, x7) U15_gga(x1, x2, x3, x4, x5, x6) = U15_gga(x1, x2, x3, x4, x6) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg U64_gg(x1, x2, x3) = U64_gg(x3) U65_gg(x1, x2, x3) = U65_gg(x3) U16_gga(x1, x2, x3, x4, x5, x6) = U16_gga(x3, x6) U19_gggag(x1, x2, x3, x4, x5, x6, x7) = U19_gggag(x6, x7) rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2) rev_out_gga(x1, x2, x3) = rev_out_gga(x3) U11_gga(x1, x2, x3, x4, x5) = U11_gga(x5) U20_gggag(x1, x2, x3, x4, x5, x6) = U20_gggag(x6) U23_ggag(x1, x2, x3, x4, x5, x6) = U23_ggag(x3, x6) pds_out_ga(x1, x2) = pds_out_ga(x2) DIF_IN_GGGAA(x1, x2, x3, x4, x5) = DIF_IN_GGGAA(x1, x2, x3) U6_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_GGGAA(x2, x3, x4, x9) U7_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_GGGAA(x2, x3, x4, x9, x10) U8_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_GGGAA(x2, x3, x4, x5, x9, x10) U9_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_GGGAA(x2, x3, x4, x5, x9) We have to consider all (P,R,Pi)-chains ---------------------------------------- (70) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (71) Obligation: Pi DP problem: The TRS P consists of the following rules: DIF_IN_GGGAA(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) U8_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> DIF_IN_GGGAA(Ss, Val, Mod, Ds, D2s) The TRS R consists of the following rules: isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) =_in_ag(X, X) -> =_out_ag(X, X) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) The argument filtering Pi contains the following mapping: zero = zero succ(x1) = succ(x1) pred(x1) = pred(x1) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x3) U35_gga(x1, x2, x3) = U35_gga(x3) U36_gga(x1, x2, x3) = U36_gga(x3) U37_gga(x1, x2, x3, x4) = U37_gga(x4) U38_gga(x1, x2, x3, x4) = U38_gga(x4) U39_gga(x1, x2, x3, x4) = U39_gga(x4) U40_gga(x1, x2, x3, x4) = U40_gga(x4) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1) .(x1, x2) = .(x1, x2) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg U35_ggg(x1, x2, x3) = U35_ggg(x3) U36_ggg(x1, x2, x3) = U36_ggg(x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x4) DIF_IN_GGGAA(x1, x2, x3, x4, x5) = DIF_IN_GGGAA(x1, x2, x3) U6_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_GGGAA(x2, x3, x4, x9) U7_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_GGGAA(x2, x3, x4, x9, x10) U8_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_GGGAA(x2, x3, x4, x5, x9, x10) U9_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_GGGAA(x2, x3, x4, x5, x9) We have to consider all (P,R,Pi)-chains ---------------------------------------- (72) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (73) Obligation: Q DP problem: The TRS P consists of the following rules: DIF_IN_GGGAA(.(S, Ss), Val, Mod) -> U6_GGGAA(Ss, Val, Mod, isMinus_in_gga(Val, S)) U6_GGGAA(Ss, Val, Mod, isMinus_out_gga(U)) -> U7_GGGAA(Ss, Val, Mod, U, =_in_ag(U)) U7_GGGAA(Ss, Val, Mod, U, =_out_ag(D)) -> U8_GGGAA(Ss, Val, Mod, D, U, isMinus_in_ggg(Mod, D, U)) U8_GGGAA(Ss, Val, Mod, D, U, isMinus_out_ggg) -> U9_GGGAA(Ss, Val, Mod, D, =_in_ag(U)) U9_GGGAA(Ss, Val, Mod, D, =_out_ag(D2)) -> DIF_IN_GGGAA(Ss, Val, Mod) The TRS R consists of the following rules: isMinus_in_gga(X, zero) -> isMinus_out_gga(X) isMinus_in_gga(zero, succ(Y)) -> U35_gga(isMinus_in_gga(zero, Y)) isMinus_in_gga(zero, pred(Y)) -> U36_gga(isMinus_in_gga(zero, Y)) isMinus_in_gga(succ(X), succ(Y)) -> U37_gga(isMinus_in_gga(X, Y)) isMinus_in_gga(succ(X), pred(Y)) -> U38_gga(isMinus_in_gga(X, Y)) isMinus_in_gga(pred(X), succ(Y)) -> U39_gga(isMinus_in_gga(X, Y)) isMinus_in_gga(pred(X), pred(Y)) -> U40_gga(isMinus_in_gga(X, Y)) =_in_ag(X) -> =_out_ag(X) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(isMinus_in_ggg(X, Y, Z)) U35_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(pred(Z)) U36_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(succ(Z)) U37_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(Z) U38_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(succ(succ(Z))) U39_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(pred(pred(Z))) U40_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(Z) U35_ggg(isMinus_out_ggg) -> isMinus_out_ggg U36_ggg(isMinus_out_ggg) -> isMinus_out_ggg U37_ggg(isMinus_out_ggg) -> isMinus_out_ggg U38_ggg(isMinus_out_ggg) -> isMinus_out_ggg U39_ggg(isMinus_out_ggg) -> isMinus_out_ggg U40_ggg(isMinus_out_ggg) -> isMinus_out_ggg The set Q consists of the following terms: isMinus_in_gga(x0, x1) =_in_ag(x0) isMinus_in_ggg(x0, x1, x2) U35_gga(x0) U36_gga(x0) U37_gga(x0) U38_gga(x0) U39_gga(x0) U40_gga(x0) U35_ggg(x0) U36_ggg(x0) U37_ggg(x0) U38_ggg(x0) U39_ggg(x0) U40_ggg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (74) 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: *U6_GGGAA(Ss, Val, Mod, isMinus_out_gga(U)) -> U7_GGGAA(Ss, Val, Mod, U, =_in_ag(U)) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4 *U9_GGGAA(Ss, Val, Mod, D, =_out_ag(D2)) -> DIF_IN_GGGAA(Ss, Val, Mod) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3 *U7_GGGAA(Ss, Val, Mod, U, =_out_ag(D)) -> U8_GGGAA(Ss, Val, Mod, D, U, isMinus_in_ggg(Mod, D, U)) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 5 > 4, 4 >= 5 *DIF_IN_GGGAA(.(S, Ss), Val, Mod) -> U6_GGGAA(Ss, Val, Mod, isMinus_in_gga(Val, S)) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U8_GGGAA(Ss, Val, Mod, D, U, isMinus_out_ggg) -> U9_GGGAA(Ss, Val, Mod, D, =_in_ag(U)) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4 ---------------------------------------- (75) YES ---------------------------------------- (76) Obligation: Pi DP problem: The TRS P consists of the following rules: U21_GGAG(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_GGAG(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) U22_GGAG(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> PDS1_IN_GGAG(U3, .(Choice, List), Rest, Mod) PDS1_IN_GGAG(Unused, List, .(Choice, Rest), Mod) -> U21_GGAG(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) The TRS R consists of the following rules: pds_in_ga(Order, .(zero, Ans)) -> U24_ga(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) isPlus_in_gga(zero, X, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), zero, succ(X)) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), succ(Y), succ(succ(Z))) -> U31_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(succ(X), pred(Y), Z) -> U32_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), zero, pred(X)) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), succ(Y), Z) -> U33_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), pred(Y), pred(pred(Z))) -> U34_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) U34_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U33_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U32_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U31_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U24_ga(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_ga(Order, Ans, isTimes_in_gga(Order, U, U1)) isTimes_in_gga(X, zero, zero) -> isTimes_out_gga(X, zero, zero) isTimes_in_gga(zero, succ(Y), zero) -> isTimes_out_gga(zero, succ(Y), zero) isTimes_in_gga(zero, pred(Y), zero) -> isTimes_out_gga(zero, pred(Y), zero) isTimes_in_gga(succ(X), succ(Y), Z) -> U41_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(succ(X), pred(Y), Z) -> U43_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(pred(X), succ(Y), Z) -> U45_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) isTimes_in_gga(pred(X), pred(Y), Z) -> U47_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) U47_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_gga(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U48_gga(X, Y, Z, isMinus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), pred(Y), Z) U45_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_gga(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U46_gga(X, Y, Z, isPlus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), succ(Y), Z) U43_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_gga(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U44_gga(X, Y, Z, isMinus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), pred(Y), Z) U41_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_gga(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U42_gga(X, Y, Z, isPlus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), succ(Y), Z) U25_ga(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_ga(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U26_ga(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_ga(Order, Ans, =_in_ag(N, U2)) =_in_ag(X, X) -> =_out_ag(X, X) U27_ga(Order, Ans, =_out_ag(N, U2)) -> U28_ga(Order, Ans, N, iota_in_ga(N, .(zero, List))) iota_in_ga(N, List) -> U2_ga(N, List, iota1_in_gga(zero, N, List)) iota1_in_gga(K, K, []) -> iota1_out_gga(K, K, []) iota1_in_gga(K, N, .(K, List)) -> U3_gga(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_gga(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_gga(K, N, List, =_in_ag(K1, U)) U4_gga(K, N, List, =_out_ag(K1, U)) -> U5_gga(K, N, List, iota1_in_gga(K1, N, List)) U5_gga(K, N, List, iota1_out_gga(K1, N, List)) -> iota1_out_gga(K, N, .(K, List)) U2_ga(N, List, iota1_out_gga(zero, N, List)) -> iota_out_ga(N, List) U28_ga(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_ga(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) pds1_in_ggag([], X6, [], X7) -> pds1_out_ggag([], X6, [], X7) pds1_in_ggag(Unused, List, .(Choice, Rest), Mod) -> U21_ggag(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(X, .(X1, Xs)) -> U1_ag(X, X1, Xs, member_in_ag(X, Xs)) U1_ag(X, X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) U21_ggag(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_ggag(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) check_in_gggag([], X4, L, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, L3, Modulus) -> U17_gggag(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) dif_in_gggaa([], X2, X3, [], []) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U17_gggag(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) mergedelete_in_gga([], L, L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R), L2) -> U12_gga(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) mergedelete_in_gga(.(D, Ds), .(X, R), .(X, L2)) -> U13_gga(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_gga(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_gga(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U15_gga(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U16_gga(D, Ds, X, R, L2, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U12_gga(D, Ds, R, L2, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) rev_in_gga([], L, L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y, L) -> U11_gga(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) U11_gga(X, Xs, Y, L, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U22_ggag(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_ggag(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U23_ggag(Unused, List, Choice, Rest, Mod, pds1_out_ggag(U3, .(Choice, List), Rest, Mod)) -> pds1_out_ggag(Unused, List, .(Choice, Rest), Mod) U29_ga(Order, Ans, pds1_out_ggag(List, .(zero, []), Ans, N)) -> pds_out_ga(Order, .(zero, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U24_ga(x1, x2, x3) = U24_ga(x1, x3) isPlus_in_gga(x1, x2, x3) = isPlus_in_gga(x1, x2) zero = zero isPlus_out_gga(x1, x2, x3) = isPlus_out_gga(x3) succ(x1) = succ(x1) U31_gga(x1, x2, x3, x4) = U31_gga(x4) pred(x1) = pred(x1) U32_gga(x1, x2, x3, x4) = U32_gga(x4) U33_gga(x1, x2, x3, x4) = U33_gga(x4) U34_gga(x1, x2, x3, x4) = U34_gga(x4) U25_ga(x1, x2, x3) = U25_ga(x3) isTimes_in_gga(x1, x2, x3) = isTimes_in_gga(x1, x2) isTimes_out_gga(x1, x2, x3) = isTimes_out_gga(x3) U41_gga(x1, x2, x3, x4) = U41_gga(x1, x4) U43_gga(x1, x2, x3, x4) = U43_gga(x1, x4) U45_gga(x1, x2, x3, x4) = U45_gga(x1, x4) U47_gga(x1, x2, x3, x4) = U47_gga(x1, x4) U48_gga(x1, x2, x3, x4) = U48_gga(x4) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x3) U35_gga(x1, x2, x3) = U35_gga(x3) U36_gga(x1, x2, x3) = U36_gga(x3) U37_gga(x1, x2, x3, x4) = U37_gga(x4) U38_gga(x1, x2, x3, x4) = U38_gga(x4) U39_gga(x1, x2, x3, x4) = U39_gga(x4) U40_gga(x1, x2, x3, x4) = U40_gga(x4) U46_gga(x1, x2, x3, x4) = U46_gga(x4) U44_gga(x1, x2, x3, x4) = U44_gga(x4) U42_gga(x1, x2, x3, x4) = U42_gga(x4) U26_ga(x1, x2, x3) = U26_ga(x3) U27_ga(x1, x2, x3) = U27_ga(x3) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1) U28_ga(x1, x2, x3, x4) = U28_ga(x3, x4) iota_in_ga(x1, x2) = iota_in_ga(x1) U2_ga(x1, x2, x3) = U2_ga(x3) iota1_in_gga(x1, x2, x3) = iota1_in_gga(x1, x2) iota1_out_gga(x1, x2, x3) = iota1_out_gga(x3) U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x4) iota_out_ga(x1, x2) = iota_out_ga(x2) .(x1, x2) = .(x1, x2) U29_ga(x1, x2, x3) = U29_ga(x3) pds1_in_ggag(x1, x2, x3, x4) = pds1_in_ggag(x1, x2, x4) [] = [] pds1_out_ggag(x1, x2, x3, x4) = pds1_out_ggag(x3) U21_ggag(x1, x2, x3, x4, x5, x6) = U21_ggag(x1, x2, x5, x6) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1) U1_ag(x1, x2, x3, x4) = U1_ag(x4) U22_ggag(x1, x2, x3, x4, x5, x6) = U22_ggag(x2, x3, x5, x6) check_in_gggag(x1, x2, x3, x4, x5) = check_in_gggag(x1, x2, x3, x5) check_out_gggag(x1, x2, x3, x4, x5) = check_out_gggag(x4) U17_gggag(x1, x2, x3, x4, x5, x6) = U17_gggag(x3, x6) dif_in_gggaa(x1, x2, x3, x4, x5) = dif_in_gggaa(x1, x2, x3) dif_out_gggaa(x1, x2, x3, x4, x5) = dif_out_gggaa(x4, x5) U6_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gggaa(x2, x3, x4, x9) U7_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_gggaa(x2, x3, x4, x9, x10) U8_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_gggaa(x2, x3, x4, x5, x9, x10) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg U35_ggg(x1, x2, x3) = U35_ggg(x3) U36_ggg(x1, x2, x3) = U36_ggg(x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x4) U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_gggaa(x2, x3, x4, x5, x9) U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_gggaa(x5, x7, x9) U18_gggag(x1, x2, x3, x4, x5, x6, x7) = U18_gggag(x6, x7) mergedelete_in_gga(x1, x2, x3) = mergedelete_in_gga(x1, x2) mergedelete_out_gga(x1, x2, x3) = mergedelete_out_gga(x3) U12_gga(x1, x2, x3, x4, x5) = U12_gga(x5) U13_gga(x1, x2, x3, x4, x5, x6) = U13_gga(x1, x2, x3, x4, x6) U14_gga(x1, x2, x3, x4, x5, x6, x7) = U14_gga(x1, x2, x3, x4, x6, x7) U15_gga(x1, x2, x3, x4, x5, x6) = U15_gga(x1, x2, x3, x4, x6) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg U64_gg(x1, x2, x3) = U64_gg(x3) U65_gg(x1, x2, x3) = U65_gg(x3) U16_gga(x1, x2, x3, x4, x5, x6) = U16_gga(x3, x6) U19_gggag(x1, x2, x3, x4, x5, x6, x7) = U19_gggag(x6, x7) rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2) rev_out_gga(x1, x2, x3) = rev_out_gga(x3) U11_gga(x1, x2, x3, x4, x5) = U11_gga(x5) U20_gggag(x1, x2, x3, x4, x5, x6) = U20_gggag(x6) U23_ggag(x1, x2, x3, x4, x5, x6) = U23_ggag(x3, x6) pds_out_ga(x1, x2) = pds_out_ga(x2) PDS1_IN_GGAG(x1, x2, x3, x4) = PDS1_IN_GGAG(x1, x2, x4) U21_GGAG(x1, x2, x3, x4, x5, x6) = U21_GGAG(x1, x2, x5, x6) U22_GGAG(x1, x2, x3, x4, x5, x6) = U22_GGAG(x2, x3, x5, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (77) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (78) Obligation: Pi DP problem: The TRS P consists of the following rules: U21_GGAG(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_GGAG(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) U22_GGAG(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> PDS1_IN_GGAG(U3, .(Choice, List), Rest, Mod) PDS1_IN_GGAG(Unused, List, .(Choice, Rest), Mod) -> U21_GGAG(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) The TRS R consists of the following rules: check_in_gggag([], X4, L, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, L3, Modulus) -> U17_gggag(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(X, .(X1, Xs)) -> U1_ag(X, X1, Xs, member_in_ag(X, Xs)) U17_gggag(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) U1_ag(X, X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) dif_in_gggaa([], X2, X3, [], []) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) mergedelete_in_gga([], L, L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R), L2) -> U12_gga(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) mergedelete_in_gga(.(D, Ds), .(X, R), .(X, L2)) -> U13_gga(D, Ds, X, R, L2, =_in_ag(X1, D)) U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) U12_gga(D, Ds, R, L2, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U13_gga(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) rev_in_gga([], L, L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y, L) -> U11_gga(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) =_in_ag(X, X) -> =_out_ag(X, X) U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U14_gga(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) U11_gga(X, Xs, Y, L, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U15_gga(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U16_gga(D, Ds, X, R, L2, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) The argument filtering Pi contains the following mapping: zero = zero succ(x1) = succ(x1) pred(x1) = pred(x1) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x3) U35_gga(x1, x2, x3) = U35_gga(x3) U36_gga(x1, x2, x3) = U36_gga(x3) U37_gga(x1, x2, x3, x4) = U37_gga(x4) U38_gga(x1, x2, x3, x4) = U38_gga(x4) U39_gga(x1, x2, x3, x4) = U39_gga(x4) U40_gga(x1, x2, x3, x4) = U40_gga(x4) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1) .(x1, x2) = .(x1, x2) [] = [] member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1) U1_ag(x1, x2, x3, x4) = U1_ag(x4) check_in_gggag(x1, x2, x3, x4, x5) = check_in_gggag(x1, x2, x3, x5) check_out_gggag(x1, x2, x3, x4, x5) = check_out_gggag(x4) U17_gggag(x1, x2, x3, x4, x5, x6) = U17_gggag(x3, x6) dif_in_gggaa(x1, x2, x3, x4, x5) = dif_in_gggaa(x1, x2, x3) dif_out_gggaa(x1, x2, x3, x4, x5) = dif_out_gggaa(x4, x5) U6_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gggaa(x2, x3, x4, x9) U7_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_gggaa(x2, x3, x4, x9, x10) U8_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_gggaa(x2, x3, x4, x5, x9, x10) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg U35_ggg(x1, x2, x3) = U35_ggg(x3) U36_ggg(x1, x2, x3) = U36_ggg(x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x4) U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_gggaa(x2, x3, x4, x5, x9) U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_gggaa(x5, x7, x9) U18_gggag(x1, x2, x3, x4, x5, x6, x7) = U18_gggag(x6, x7) mergedelete_in_gga(x1, x2, x3) = mergedelete_in_gga(x1, x2) mergedelete_out_gga(x1, x2, x3) = mergedelete_out_gga(x3) U12_gga(x1, x2, x3, x4, x5) = U12_gga(x5) U13_gga(x1, x2, x3, x4, x5, x6) = U13_gga(x1, x2, x3, x4, x6) U14_gga(x1, x2, x3, x4, x5, x6, x7) = U14_gga(x1, x2, x3, x4, x6, x7) U15_gga(x1, x2, x3, x4, x5, x6) = U15_gga(x1, x2, x3, x4, x6) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg U64_gg(x1, x2, x3) = U64_gg(x3) U65_gg(x1, x2, x3) = U65_gg(x3) U16_gga(x1, x2, x3, x4, x5, x6) = U16_gga(x3, x6) U19_gggag(x1, x2, x3, x4, x5, x6, x7) = U19_gggag(x6, x7) rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2) rev_out_gga(x1, x2, x3) = rev_out_gga(x3) U11_gga(x1, x2, x3, x4, x5) = U11_gga(x5) U20_gggag(x1, x2, x3, x4, x5, x6) = U20_gggag(x6) PDS1_IN_GGAG(x1, x2, x3, x4) = PDS1_IN_GGAG(x1, x2, x4) U21_GGAG(x1, x2, x3, x4, x5, x6) = U21_GGAG(x1, x2, x5, x6) U22_GGAG(x1, x2, x3, x4, x5, x6) = U22_GGAG(x2, x3, x5, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (79) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (80) Obligation: Q DP problem: The TRS P consists of the following rules: U21_GGAG(Unused, List, Mod, member_out_ag(Choice)) -> U22_GGAG(List, Choice, Mod, check_in_gggag(List, Choice, Unused, Mod)) U22_GGAG(List, Choice, Mod, check_out_gggag(U3)) -> PDS1_IN_GGAG(U3, .(Choice, List), Mod) PDS1_IN_GGAG(Unused, List, Mod) -> U21_GGAG(Unused, List, Mod, member_in_ag(Unused)) The TRS R consists of the following rules: check_in_gggag([], X4, L, X5) -> check_out_gggag(L) check_in_gggag(S, Choice, Old, Modulus) -> U17_gggag(Old, dif_in_gggaa(S, Choice, Modulus)) member_in_ag(.(X, Xs)) -> member_out_ag(X) member_in_ag(.(X1, Xs)) -> U1_ag(member_in_ag(Xs)) U17_gggag(Old, dif_out_gggaa(Ds, Dds)) -> U18_gggag(Dds, mergedelete_in_gga(Ds, Old)) U1_ag(member_out_ag(X)) -> member_out_ag(X) dif_in_gggaa([], X2, X3) -> dif_out_gggaa([], []) dif_in_gggaa(.(S, Ss), Val, Mod) -> U6_gggaa(Ss, Val, Mod, isMinus_in_gga(Val, S)) U18_gggag(Dds, mergedelete_out_gga(L2)) -> U19_gggag(L2, rev_in_gga(Dds, [])) U6_gggaa(Ss, Val, Mod, isMinus_out_gga(U)) -> U7_gggaa(Ss, Val, Mod, U, =_in_ag(U)) mergedelete_in_gga([], L) -> mergedelete_out_gga(L) mergedelete_in_gga(.(D, Ds), .(D, R)) -> U12_gga(mergedelete_in_gga(Ds, R)) mergedelete_in_gga(.(D, Ds), .(X, R)) -> U13_gga(D, Ds, X, R, =_in_ag(D)) U19_gggag(L2, rev_out_gga(Rds)) -> U20_gggag(mergedelete_in_gga(Rds, L2)) isMinus_in_gga(X, zero) -> isMinus_out_gga(X) isMinus_in_gga(zero, succ(Y)) -> U35_gga(isMinus_in_gga(zero, Y)) isMinus_in_gga(zero, pred(Y)) -> U36_gga(isMinus_in_gga(zero, Y)) isMinus_in_gga(succ(X), succ(Y)) -> U37_gga(isMinus_in_gga(X, Y)) isMinus_in_gga(succ(X), pred(Y)) -> U38_gga(isMinus_in_gga(X, Y)) isMinus_in_gga(pred(X), succ(Y)) -> U39_gga(isMinus_in_gga(X, Y)) isMinus_in_gga(pred(X), pred(Y)) -> U40_gga(isMinus_in_gga(X, Y)) U7_gggaa(Ss, Val, Mod, U, =_out_ag(D)) -> U8_gggaa(Ss, Val, Mod, D, U, isMinus_in_ggg(Mod, D, U)) U12_gga(mergedelete_out_gga(L2)) -> mergedelete_out_gga(L2) U13_gga(D, Ds, X, R, =_out_ag(X1)) -> U14_gga(D, Ds, X, R, X1, =_in_ag(X)) rev_in_gga([], L) -> rev_out_gga(L) rev_in_gga(.(X, Xs), Y) -> U11_gga(rev_in_gga(Xs, .(X, Y))) U20_gggag(mergedelete_out_gga(L3)) -> check_out_gggag(L3) U35_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(pred(Z)) U36_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(succ(Z)) U37_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(Z) U38_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(succ(succ(Z))) U39_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(pred(pred(Z))) U40_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(Z) =_in_ag(X) -> =_out_ag(X) U8_gggaa(Ss, Val, Mod, D, U, isMinus_out_ggg) -> U9_gggaa(Ss, Val, Mod, D, =_in_ag(U)) U14_gga(D, Ds, X, R, X1, =_out_ag(X2)) -> U15_gga(D, Ds, X, R, isGreater_in_gg(X1, X2)) U11_gga(rev_out_gga(L)) -> rev_out_gga(L) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(isMinus_in_ggg(X, Y, Z)) U9_gggaa(Ss, Val, Mod, D, =_out_ag(D2)) -> U10_gggaa(D, D2, dif_in_gggaa(Ss, Val, Mod)) U15_gga(D, Ds, X, R, isGreater_out_gg) -> U16_gga(X, mergedelete_in_gga(.(D, Ds), R)) U35_ggg(isMinus_out_ggg) -> isMinus_out_ggg U36_ggg(isMinus_out_ggg) -> isMinus_out_ggg U37_ggg(isMinus_out_ggg) -> isMinus_out_ggg U38_ggg(isMinus_out_ggg) -> isMinus_out_ggg U39_ggg(isMinus_out_ggg) -> isMinus_out_ggg U40_ggg(isMinus_out_ggg) -> isMinus_out_ggg U10_gggaa(D, D2, dif_out_gggaa(Ds, D2s)) -> dif_out_gggaa(.(D, Ds), .(D2, D2s)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(isGreater_in_gg(X, Y)) U16_gga(X, mergedelete_out_gga(L2)) -> mergedelete_out_gga(.(X, L2)) U64_gg(isGreater_out_gg) -> isGreater_out_gg U65_gg(isGreater_out_gg) -> isGreater_out_gg The set Q consists of the following terms: check_in_gggag(x0, x1, x2, x3) member_in_ag(x0) U17_gggag(x0, x1) U1_ag(x0) dif_in_gggaa(x0, x1, x2) U18_gggag(x0, x1) U6_gggaa(x0, x1, x2, x3) mergedelete_in_gga(x0, x1) U19_gggag(x0, x1) isMinus_in_gga(x0, x1) U7_gggaa(x0, x1, x2, x3, x4) U12_gga(x0) U13_gga(x0, x1, x2, x3, x4) rev_in_gga(x0, x1) U20_gggag(x0) U35_gga(x0) U36_gga(x0) U37_gga(x0) U38_gga(x0) U39_gga(x0) U40_gga(x0) =_in_ag(x0) U8_gggaa(x0, x1, x2, x3, x4, x5) U14_gga(x0, x1, x2, x3, x4, x5) U11_gga(x0) isMinus_in_ggg(x0, x1, x2) U9_gggaa(x0, x1, x2, x3, x4) U15_gga(x0, x1, x2, x3, x4) U35_ggg(x0) U36_ggg(x0) U37_ggg(x0) U38_ggg(x0) U39_ggg(x0) U40_ggg(x0) U10_gggaa(x0, x1, x2) isGreater_in_gg(x0, x1) U16_gga(x0, x1) U64_gg(x0) U65_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (81) 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: mergedelete_in_gga(.(D, Ds), .(D, R)) -> U12_gga(mergedelete_in_gga(Ds, R)) Used ordering: Polynomial interpretation [POLO]: POL(.(x_1, x_2)) = 1 + x_2 POL(=_in_ag(x_1)) = 0 POL(=_out_ag(x_1)) = 0 POL(PDS1_IN_GGAG(x_1, x_2, x_3)) = 2*x_1 + 2*x_3 POL(U10_gggaa(x_1, x_2, x_3)) = 0 POL(U11_gga(x_1)) = 0 POL(U12_gga(x_1)) = x_1 POL(U13_gga(x_1, x_2, x_3, x_4, x_5)) = 1 + x_4 POL(U14_gga(x_1, x_2, x_3, x_4, x_5, x_6)) = 1 + x_4 POL(U15_gga(x_1, x_2, x_3, x_4, x_5)) = 1 + x_4 POL(U16_gga(x_1, x_2)) = 1 + x_2 POL(U17_gggag(x_1, x_2)) = x_1 POL(U18_gggag(x_1, x_2)) = x_2 POL(U19_gggag(x_1, x_2)) = x_1 POL(U1_ag(x_1)) = 0 POL(U20_gggag(x_1)) = x_1 POL(U21_GGAG(x_1, x_2, x_3, x_4)) = 2*x_1 + 2*x_3 POL(U22_GGAG(x_1, x_2, x_3, x_4)) = 2*x_3 + 2*x_4 POL(U35_gga(x_1)) = 0 POL(U35_ggg(x_1)) = 0 POL(U36_gga(x_1)) = 0 POL(U36_ggg(x_1)) = 0 POL(U37_gga(x_1)) = 0 POL(U37_ggg(x_1)) = 0 POL(U38_gga(x_1)) = 0 POL(U38_ggg(x_1)) = 0 POL(U39_gga(x_1)) = 0 POL(U39_ggg(x_1)) = 0 POL(U40_gga(x_1)) = 0 POL(U40_ggg(x_1)) = 0 POL(U64_gg(x_1)) = 0 POL(U65_gg(x_1)) = 0 POL(U6_gggaa(x_1, x_2, x_3, x_4)) = 2*x_1 POL(U7_gggaa(x_1, x_2, x_3, x_4, x_5)) = 2*x_1 POL(U8_gggaa(x_1, x_2, x_3, x_4, x_5, x_6)) = x_1 POL(U9_gggaa(x_1, x_2, x_3, x_4, x_5)) = x_1 POL([]) = 0 POL(check_in_gggag(x_1, x_2, x_3, x_4)) = x_3 POL(check_out_gggag(x_1)) = x_1 POL(dif_in_gggaa(x_1, x_2, x_3)) = 2*x_1 + x_3 POL(dif_out_gggaa(x_1, x_2)) = 0 POL(isGreater_in_gg(x_1, x_2)) = 0 POL(isGreater_out_gg) = 0 POL(isMinus_in_gga(x_1, x_2)) = 0 POL(isMinus_in_ggg(x_1, x_2, x_3)) = 0 POL(isMinus_out_gga(x_1)) = 0 POL(isMinus_out_ggg) = 0 POL(member_in_ag(x_1)) = 0 POL(member_out_ag(x_1)) = 0 POL(mergedelete_in_gga(x_1, x_2)) = x_2 POL(mergedelete_out_gga(x_1)) = x_1 POL(pred(x_1)) = 0 POL(rev_in_gga(x_1, x_2)) = 0 POL(rev_out_gga(x_1)) = 0 POL(succ(x_1)) = 0 POL(zero) = 0 ---------------------------------------- (82) Obligation: Q DP problem: The TRS P consists of the following rules: U21_GGAG(Unused, List, Mod, member_out_ag(Choice)) -> U22_GGAG(List, Choice, Mod, check_in_gggag(List, Choice, Unused, Mod)) U22_GGAG(List, Choice, Mod, check_out_gggag(U3)) -> PDS1_IN_GGAG(U3, .(Choice, List), Mod) PDS1_IN_GGAG(Unused, List, Mod) -> U21_GGAG(Unused, List, Mod, member_in_ag(Unused)) The TRS R consists of the following rules: check_in_gggag([], X4, L, X5) -> check_out_gggag(L) check_in_gggag(S, Choice, Old, Modulus) -> U17_gggag(Old, dif_in_gggaa(S, Choice, Modulus)) member_in_ag(.(X, Xs)) -> member_out_ag(X) member_in_ag(.(X1, Xs)) -> U1_ag(member_in_ag(Xs)) U17_gggag(Old, dif_out_gggaa(Ds, Dds)) -> U18_gggag(Dds, mergedelete_in_gga(Ds, Old)) U1_ag(member_out_ag(X)) -> member_out_ag(X) dif_in_gggaa([], X2, X3) -> dif_out_gggaa([], []) dif_in_gggaa(.(S, Ss), Val, Mod) -> U6_gggaa(Ss, Val, Mod, isMinus_in_gga(Val, S)) U18_gggag(Dds, mergedelete_out_gga(L2)) -> U19_gggag(L2, rev_in_gga(Dds, [])) U6_gggaa(Ss, Val, Mod, isMinus_out_gga(U)) -> U7_gggaa(Ss, Val, Mod, U, =_in_ag(U)) mergedelete_in_gga([], L) -> mergedelete_out_gga(L) mergedelete_in_gga(.(D, Ds), .(X, R)) -> U13_gga(D, Ds, X, R, =_in_ag(D)) U19_gggag(L2, rev_out_gga(Rds)) -> U20_gggag(mergedelete_in_gga(Rds, L2)) isMinus_in_gga(X, zero) -> isMinus_out_gga(X) isMinus_in_gga(zero, succ(Y)) -> U35_gga(isMinus_in_gga(zero, Y)) isMinus_in_gga(zero, pred(Y)) -> U36_gga(isMinus_in_gga(zero, Y)) isMinus_in_gga(succ(X), succ(Y)) -> U37_gga(isMinus_in_gga(X, Y)) isMinus_in_gga(succ(X), pred(Y)) -> U38_gga(isMinus_in_gga(X, Y)) isMinus_in_gga(pred(X), succ(Y)) -> U39_gga(isMinus_in_gga(X, Y)) isMinus_in_gga(pred(X), pred(Y)) -> U40_gga(isMinus_in_gga(X, Y)) U7_gggaa(Ss, Val, Mod, U, =_out_ag(D)) -> U8_gggaa(Ss, Val, Mod, D, U, isMinus_in_ggg(Mod, D, U)) U12_gga(mergedelete_out_gga(L2)) -> mergedelete_out_gga(L2) U13_gga(D, Ds, X, R, =_out_ag(X1)) -> U14_gga(D, Ds, X, R, X1, =_in_ag(X)) rev_in_gga([], L) -> rev_out_gga(L) rev_in_gga(.(X, Xs), Y) -> U11_gga(rev_in_gga(Xs, .(X, Y))) U20_gggag(mergedelete_out_gga(L3)) -> check_out_gggag(L3) U35_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(pred(Z)) U36_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(succ(Z)) U37_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(Z) U38_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(succ(succ(Z))) U39_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(pred(pred(Z))) U40_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(Z) =_in_ag(X) -> =_out_ag(X) U8_gggaa(Ss, Val, Mod, D, U, isMinus_out_ggg) -> U9_gggaa(Ss, Val, Mod, D, =_in_ag(U)) U14_gga(D, Ds, X, R, X1, =_out_ag(X2)) -> U15_gga(D, Ds, X, R, isGreater_in_gg(X1, X2)) U11_gga(rev_out_gga(L)) -> rev_out_gga(L) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(isMinus_in_ggg(X, Y, Z)) U9_gggaa(Ss, Val, Mod, D, =_out_ag(D2)) -> U10_gggaa(D, D2, dif_in_gggaa(Ss, Val, Mod)) U15_gga(D, Ds, X, R, isGreater_out_gg) -> U16_gga(X, mergedelete_in_gga(.(D, Ds), R)) U35_ggg(isMinus_out_ggg) -> isMinus_out_ggg U36_ggg(isMinus_out_ggg) -> isMinus_out_ggg U37_ggg(isMinus_out_ggg) -> isMinus_out_ggg U38_ggg(isMinus_out_ggg) -> isMinus_out_ggg U39_ggg(isMinus_out_ggg) -> isMinus_out_ggg U40_ggg(isMinus_out_ggg) -> isMinus_out_ggg U10_gggaa(D, D2, dif_out_gggaa(Ds, D2s)) -> dif_out_gggaa(.(D, Ds), .(D2, D2s)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(isGreater_in_gg(X, Y)) U16_gga(X, mergedelete_out_gga(L2)) -> mergedelete_out_gga(.(X, L2)) U64_gg(isGreater_out_gg) -> isGreater_out_gg U65_gg(isGreater_out_gg) -> isGreater_out_gg The set Q consists of the following terms: check_in_gggag(x0, x1, x2, x3) member_in_ag(x0) U17_gggag(x0, x1) U1_ag(x0) dif_in_gggaa(x0, x1, x2) U18_gggag(x0, x1) U6_gggaa(x0, x1, x2, x3) mergedelete_in_gga(x0, x1) U19_gggag(x0, x1) isMinus_in_gga(x0, x1) U7_gggaa(x0, x1, x2, x3, x4) U12_gga(x0) U13_gga(x0, x1, x2, x3, x4) rev_in_gga(x0, x1) U20_gggag(x0) U35_gga(x0) U36_gga(x0) U37_gga(x0) U38_gga(x0) U39_gga(x0) U40_gga(x0) =_in_ag(x0) U8_gggaa(x0, x1, x2, x3, x4, x5) U14_gga(x0, x1, x2, x3, x4, x5) U11_gga(x0) isMinus_in_ggg(x0, x1, x2) U9_gggaa(x0, x1, x2, x3, x4) U15_gga(x0, x1, x2, x3, x4) U35_ggg(x0) U36_ggg(x0) U37_ggg(x0) U38_ggg(x0) U39_ggg(x0) U40_ggg(x0) U10_gggaa(x0, x1, x2) isGreater_in_gg(x0, x1) U16_gga(x0, x1) U64_gg(x0) U65_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (83) 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. ---------------------------------------- (84) Obligation: Q DP problem: The TRS P consists of the following rules: U21_GGAG(Unused, List, Mod, member_out_ag(Choice)) -> U22_GGAG(List, Choice, Mod, check_in_gggag(List, Choice, Unused, Mod)) U22_GGAG(List, Choice, Mod, check_out_gggag(U3)) -> PDS1_IN_GGAG(U3, .(Choice, List), Mod) PDS1_IN_GGAG(Unused, List, Mod) -> U21_GGAG(Unused, List, Mod, member_in_ag(Unused)) The TRS R consists of the following rules: member_in_ag(.(X, Xs)) -> member_out_ag(X) member_in_ag(.(X1, Xs)) -> U1_ag(member_in_ag(Xs)) U1_ag(member_out_ag(X)) -> member_out_ag(X) check_in_gggag([], X4, L, X5) -> check_out_gggag(L) check_in_gggag(S, Choice, Old, Modulus) -> U17_gggag(Old, dif_in_gggaa(S, Choice, Modulus)) dif_in_gggaa([], X2, X3) -> dif_out_gggaa([], []) dif_in_gggaa(.(S, Ss), Val, Mod) -> U6_gggaa(Ss, Val, Mod, isMinus_in_gga(Val, S)) U17_gggag(Old, dif_out_gggaa(Ds, Dds)) -> U18_gggag(Dds, mergedelete_in_gga(Ds, Old)) mergedelete_in_gga([], L) -> mergedelete_out_gga(L) mergedelete_in_gga(.(D, Ds), .(X, R)) -> U13_gga(D, Ds, X, R, =_in_ag(D)) U18_gggag(Dds, mergedelete_out_gga(L2)) -> U19_gggag(L2, rev_in_gga(Dds, [])) rev_in_gga([], L) -> rev_out_gga(L) rev_in_gga(.(X, Xs), Y) -> U11_gga(rev_in_gga(Xs, .(X, Y))) U19_gggag(L2, rev_out_gga(Rds)) -> U20_gggag(mergedelete_in_gga(Rds, L2)) U20_gggag(mergedelete_out_gga(L3)) -> check_out_gggag(L3) U11_gga(rev_out_gga(L)) -> rev_out_gga(L) =_in_ag(X) -> =_out_ag(X) U13_gga(D, Ds, X, R, =_out_ag(X1)) -> U14_gga(D, Ds, X, R, X1, =_in_ag(X)) U14_gga(D, Ds, X, R, X1, =_out_ag(X2)) -> U15_gga(D, Ds, X, R, isGreater_in_gg(X1, X2)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(isGreater_in_gg(X, Y)) U15_gga(D, Ds, X, R, isGreater_out_gg) -> U16_gga(X, mergedelete_in_gga(.(D, Ds), R)) U16_gga(X, mergedelete_out_gga(L2)) -> mergedelete_out_gga(.(X, L2)) U65_gg(isGreater_out_gg) -> isGreater_out_gg U64_gg(isGreater_out_gg) -> isGreater_out_gg isMinus_in_gga(X, zero) -> isMinus_out_gga(X) isMinus_in_gga(zero, succ(Y)) -> U35_gga(isMinus_in_gga(zero, Y)) isMinus_in_gga(zero, pred(Y)) -> U36_gga(isMinus_in_gga(zero, Y)) isMinus_in_gga(succ(X), succ(Y)) -> U37_gga(isMinus_in_gga(X, Y)) isMinus_in_gga(succ(X), pred(Y)) -> U38_gga(isMinus_in_gga(X, Y)) isMinus_in_gga(pred(X), succ(Y)) -> U39_gga(isMinus_in_gga(X, Y)) isMinus_in_gga(pred(X), pred(Y)) -> U40_gga(isMinus_in_gga(X, Y)) U6_gggaa(Ss, Val, Mod, isMinus_out_gga(U)) -> U7_gggaa(Ss, Val, Mod, U, =_in_ag(U)) U7_gggaa(Ss, Val, Mod, U, =_out_ag(D)) -> U8_gggaa(Ss, Val, Mod, D, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(isMinus_in_ggg(X, Y, Z)) U8_gggaa(Ss, Val, Mod, D, U, isMinus_out_ggg) -> U9_gggaa(Ss, Val, Mod, D, =_in_ag(U)) U9_gggaa(Ss, Val, Mod, D, =_out_ag(D2)) -> U10_gggaa(D, D2, dif_in_gggaa(Ss, Val, Mod)) U10_gggaa(D, D2, dif_out_gggaa(Ds, D2s)) -> dif_out_gggaa(.(D, Ds), .(D2, D2s)) U40_ggg(isMinus_out_ggg) -> isMinus_out_ggg U39_ggg(isMinus_out_ggg) -> isMinus_out_ggg U38_ggg(isMinus_out_ggg) -> isMinus_out_ggg U37_ggg(isMinus_out_ggg) -> isMinus_out_ggg U36_ggg(isMinus_out_ggg) -> isMinus_out_ggg U35_ggg(isMinus_out_ggg) -> isMinus_out_ggg U40_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(Z) U39_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(pred(pred(Z))) U38_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(succ(succ(Z))) U37_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(Z) U36_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(succ(Z)) U35_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(pred(Z)) The set Q consists of the following terms: check_in_gggag(x0, x1, x2, x3) member_in_ag(x0) U17_gggag(x0, x1) U1_ag(x0) dif_in_gggaa(x0, x1, x2) U18_gggag(x0, x1) U6_gggaa(x0, x1, x2, x3) mergedelete_in_gga(x0, x1) U19_gggag(x0, x1) isMinus_in_gga(x0, x1) U7_gggaa(x0, x1, x2, x3, x4) U12_gga(x0) U13_gga(x0, x1, x2, x3, x4) rev_in_gga(x0, x1) U20_gggag(x0) U35_gga(x0) U36_gga(x0) U37_gga(x0) U38_gga(x0) U39_gga(x0) U40_gga(x0) =_in_ag(x0) U8_gggaa(x0, x1, x2, x3, x4, x5) U14_gga(x0, x1, x2, x3, x4, x5) U11_gga(x0) isMinus_in_ggg(x0, x1, x2) U9_gggaa(x0, x1, x2, x3, x4) U15_gga(x0, x1, x2, x3, x4) U35_ggg(x0) U36_ggg(x0) U37_ggg(x0) U38_ggg(x0) U39_ggg(x0) U40_ggg(x0) U10_gggaa(x0, x1, x2) isGreater_in_gg(x0, x1) U16_gga(x0, x1) U64_gg(x0) U65_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (85) 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]. U12_gga(x0) ---------------------------------------- (86) Obligation: Q DP problem: The TRS P consists of the following rules: U21_GGAG(Unused, List, Mod, member_out_ag(Choice)) -> U22_GGAG(List, Choice, Mod, check_in_gggag(List, Choice, Unused, Mod)) U22_GGAG(List, Choice, Mod, check_out_gggag(U3)) -> PDS1_IN_GGAG(U3, .(Choice, List), Mod) PDS1_IN_GGAG(Unused, List, Mod) -> U21_GGAG(Unused, List, Mod, member_in_ag(Unused)) The TRS R consists of the following rules: member_in_ag(.(X, Xs)) -> member_out_ag(X) member_in_ag(.(X1, Xs)) -> U1_ag(member_in_ag(Xs)) U1_ag(member_out_ag(X)) -> member_out_ag(X) check_in_gggag([], X4, L, X5) -> check_out_gggag(L) check_in_gggag(S, Choice, Old, Modulus) -> U17_gggag(Old, dif_in_gggaa(S, Choice, Modulus)) dif_in_gggaa([], X2, X3) -> dif_out_gggaa([], []) dif_in_gggaa(.(S, Ss), Val, Mod) -> U6_gggaa(Ss, Val, Mod, isMinus_in_gga(Val, S)) U17_gggag(Old, dif_out_gggaa(Ds, Dds)) -> U18_gggag(Dds, mergedelete_in_gga(Ds, Old)) mergedelete_in_gga([], L) -> mergedelete_out_gga(L) mergedelete_in_gga(.(D, Ds), .(X, R)) -> U13_gga(D, Ds, X, R, =_in_ag(D)) U18_gggag(Dds, mergedelete_out_gga(L2)) -> U19_gggag(L2, rev_in_gga(Dds, [])) rev_in_gga([], L) -> rev_out_gga(L) rev_in_gga(.(X, Xs), Y) -> U11_gga(rev_in_gga(Xs, .(X, Y))) U19_gggag(L2, rev_out_gga(Rds)) -> U20_gggag(mergedelete_in_gga(Rds, L2)) U20_gggag(mergedelete_out_gga(L3)) -> check_out_gggag(L3) U11_gga(rev_out_gga(L)) -> rev_out_gga(L) =_in_ag(X) -> =_out_ag(X) U13_gga(D, Ds, X, R, =_out_ag(X1)) -> U14_gga(D, Ds, X, R, X1, =_in_ag(X)) U14_gga(D, Ds, X, R, X1, =_out_ag(X2)) -> U15_gga(D, Ds, X, R, isGreater_in_gg(X1, X2)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(isGreater_in_gg(X, Y)) U15_gga(D, Ds, X, R, isGreater_out_gg) -> U16_gga(X, mergedelete_in_gga(.(D, Ds), R)) U16_gga(X, mergedelete_out_gga(L2)) -> mergedelete_out_gga(.(X, L2)) U65_gg(isGreater_out_gg) -> isGreater_out_gg U64_gg(isGreater_out_gg) -> isGreater_out_gg isMinus_in_gga(X, zero) -> isMinus_out_gga(X) isMinus_in_gga(zero, succ(Y)) -> U35_gga(isMinus_in_gga(zero, Y)) isMinus_in_gga(zero, pred(Y)) -> U36_gga(isMinus_in_gga(zero, Y)) isMinus_in_gga(succ(X), succ(Y)) -> U37_gga(isMinus_in_gga(X, Y)) isMinus_in_gga(succ(X), pred(Y)) -> U38_gga(isMinus_in_gga(X, Y)) isMinus_in_gga(pred(X), succ(Y)) -> U39_gga(isMinus_in_gga(X, Y)) isMinus_in_gga(pred(X), pred(Y)) -> U40_gga(isMinus_in_gga(X, Y)) U6_gggaa(Ss, Val, Mod, isMinus_out_gga(U)) -> U7_gggaa(Ss, Val, Mod, U, =_in_ag(U)) U7_gggaa(Ss, Val, Mod, U, =_out_ag(D)) -> U8_gggaa(Ss, Val, Mod, D, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(isMinus_in_ggg(X, Y, Z)) U8_gggaa(Ss, Val, Mod, D, U, isMinus_out_ggg) -> U9_gggaa(Ss, Val, Mod, D, =_in_ag(U)) U9_gggaa(Ss, Val, Mod, D, =_out_ag(D2)) -> U10_gggaa(D, D2, dif_in_gggaa(Ss, Val, Mod)) U10_gggaa(D, D2, dif_out_gggaa(Ds, D2s)) -> dif_out_gggaa(.(D, Ds), .(D2, D2s)) U40_ggg(isMinus_out_ggg) -> isMinus_out_ggg U39_ggg(isMinus_out_ggg) -> isMinus_out_ggg U38_ggg(isMinus_out_ggg) -> isMinus_out_ggg U37_ggg(isMinus_out_ggg) -> isMinus_out_ggg U36_ggg(isMinus_out_ggg) -> isMinus_out_ggg U35_ggg(isMinus_out_ggg) -> isMinus_out_ggg U40_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(Z) U39_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(pred(pred(Z))) U38_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(succ(succ(Z))) U37_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(Z) U36_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(succ(Z)) U35_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(pred(Z)) The set Q consists of the following terms: check_in_gggag(x0, x1, x2, x3) member_in_ag(x0) U17_gggag(x0, x1) U1_ag(x0) dif_in_gggaa(x0, x1, x2) U18_gggag(x0, x1) U6_gggaa(x0, x1, x2, x3) mergedelete_in_gga(x0, x1) U19_gggag(x0, x1) isMinus_in_gga(x0, x1) U7_gggaa(x0, x1, x2, x3, x4) U13_gga(x0, x1, x2, x3, x4) rev_in_gga(x0, x1) U20_gggag(x0) U35_gga(x0) U36_gga(x0) U37_gga(x0) U38_gga(x0) U39_gga(x0) U40_gga(x0) =_in_ag(x0) U8_gggaa(x0, x1, x2, x3, x4, x5) U14_gga(x0, x1, x2, x3, x4, x5) U11_gga(x0) isMinus_in_ggg(x0, x1, x2) U9_gggaa(x0, x1, x2, x3, x4) U15_gga(x0, x1, x2, x3, x4) U35_ggg(x0) U36_ggg(x0) U37_ggg(x0) U38_ggg(x0) U39_ggg(x0) U40_ggg(x0) U10_gggaa(x0, x1, x2) isGreater_in_gg(x0, x1) U16_gga(x0, x1) U64_gg(x0) U65_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (87) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. U22_GGAG(List, Choice, Mod, check_out_gggag(U3)) -> PDS1_IN_GGAG(U3, .(Choice, List), Mod) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( U22_GGAG_4(x_1, ..., x_4) ) = x_4 POL( check_in_gggag_4(x_1, ..., x_4) ) = 2x_1 POL( [] ) = 2 POL( check_out_gggag_1(x_1) ) = 1 POL( U17_gggag_2(x_1, x_2) ) = 2x_2 POL( dif_in_gggaa_3(x_1, ..., x_3) ) = x_1 POL( U21_GGAG_4(x_1, ..., x_4) ) = 2x_2 POL( member_in_ag_1(x_1) ) = 0 POL( ._2(x_1, x_2) ) = 0 POL( member_out_ag_1(x_1) ) = 0 POL( U1_ag_1(x_1) ) = 2 POL( dif_out_gggaa_2(x_1, x_2) ) = x_1 POL( U6_gggaa_4(x_1, ..., x_4) ) = max{0, 2x_4 - 2} POL( isMinus_in_gga_2(x_1, x_2) ) = max{0, -2} POL( U18_gggag_2(x_1, x_2) ) = max{0, x_2 - 1} POL( mergedelete_in_gga_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( zero ) = 0 POL( isMinus_out_gga_1(x_1) ) = max{0, -2} POL( succ_1(x_1) ) = 2x_1 POL( U35_gga_1(x_1) ) = 0 POL( pred_1(x_1) ) = x_1 POL( U36_gga_1(x_1) ) = 0 POL( U37_gga_1(x_1) ) = max{0, -2} POL( U38_gga_1(x_1) ) = 0 POL( U39_gga_1(x_1) ) = max{0, -2} POL( U40_gga_1(x_1) ) = x_1 POL( U7_gggaa_5(x_1, ..., x_5) ) = max{0, -2} POL( =_in_ag_1(x_1) ) = 0 POL( U13_gga_5(x_1, ..., x_5) ) = max{0, -2} POL( U20_gggag_1(x_1) ) = 1 POL( U14_gga_6(x_1, ..., x_6) ) = max{0, -2} POL( U9_gggaa_5(x_1, ..., x_5) ) = max{0, -2} POL( =_out_ag_1(x_1) ) = 0 POL( U8_gggaa_6(x_1, ..., x_6) ) = max{0, -2} POL( isMinus_in_ggg_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2} POL( isMinus_out_ggg ) = 0 POL( U35_ggg_1(x_1) ) = 2 POL( U36_ggg_1(x_1) ) = 2 POL( U37_ggg_1(x_1) ) = x_1 + 2 POL( U38_ggg_1(x_1) ) = max{0, -2} POL( U39_ggg_1(x_1) ) = x_1 + 1 POL( U40_ggg_1(x_1) ) = 2x_1 + 2 POL( U10_gggaa_3(x_1, ..., x_3) ) = 0 POL( mergedelete_out_gga_1(x_1) ) = 2 POL( U19_gggag_2(x_1, x_2) ) = 1 POL( rev_in_gga_2(x_1, x_2) ) = max{0, -2} POL( rev_out_gga_1(x_1) ) = 0 POL( U11_gga_1(x_1) ) = max{0, -2} POL( U15_gga_5(x_1, ..., x_5) ) = max{0, -2} POL( isGreater_in_gg_2(x_1, x_2) ) = 0 POL( isGreater_out_gg ) = 2 POL( U64_gg_1(x_1) ) = max{0, -2} POL( U65_gg_1(x_1) ) = max{0, -2} POL( U16_gga_2(x_1, x_2) ) = x_2 POL( PDS1_IN_GGAG_3(x_1, ..., x_3) ) = 2x_2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: check_in_gggag([], X4, L, X5) -> check_out_gggag(L) check_in_gggag(S, Choice, Old, Modulus) -> U17_gggag(Old, dif_in_gggaa(S, Choice, Modulus)) dif_in_gggaa([], X2, X3) -> dif_out_gggaa([], []) dif_in_gggaa(.(S, Ss), Val, Mod) -> U6_gggaa(Ss, Val, Mod, isMinus_in_gga(Val, S)) U17_gggag(Old, dif_out_gggaa(Ds, Dds)) -> U18_gggag(Dds, mergedelete_in_gga(Ds, Old)) isMinus_in_gga(X, zero) -> isMinus_out_gga(X) isMinus_in_gga(zero, succ(Y)) -> U35_gga(isMinus_in_gga(zero, Y)) isMinus_in_gga(zero, pred(Y)) -> U36_gga(isMinus_in_gga(zero, Y)) isMinus_in_gga(succ(X), succ(Y)) -> U37_gga(isMinus_in_gga(X, Y)) isMinus_in_gga(succ(X), pred(Y)) -> U38_gga(isMinus_in_gga(X, Y)) isMinus_in_gga(pred(X), succ(Y)) -> U39_gga(isMinus_in_gga(X, Y)) isMinus_in_gga(pred(X), pred(Y)) -> U40_gga(isMinus_in_gga(X, Y)) U6_gggaa(Ss, Val, Mod, isMinus_out_gga(U)) -> U7_gggaa(Ss, Val, Mod, U, =_in_ag(U)) U7_gggaa(Ss, Val, Mod, U, =_out_ag(D)) -> U8_gggaa(Ss, Val, Mod, D, U, isMinus_in_ggg(Mod, D, U)) U8_gggaa(Ss, Val, Mod, D, U, isMinus_out_ggg) -> U9_gggaa(Ss, Val, Mod, D, =_in_ag(U)) U9_gggaa(Ss, Val, Mod, D, =_out_ag(D2)) -> U10_gggaa(D, D2, dif_in_gggaa(Ss, Val, Mod)) U10_gggaa(D, D2, dif_out_gggaa(Ds, D2s)) -> dif_out_gggaa(.(D, Ds), .(D2, D2s)) mergedelete_in_gga([], L) -> mergedelete_out_gga(L) mergedelete_in_gga(.(D, Ds), .(X, R)) -> U13_gga(D, Ds, X, R, =_in_ag(D)) U18_gggag(Dds, mergedelete_out_gga(L2)) -> U19_gggag(L2, rev_in_gga(Dds, [])) U19_gggag(L2, rev_out_gga(Rds)) -> U20_gggag(mergedelete_in_gga(Rds, L2)) U20_gggag(mergedelete_out_gga(L3)) -> check_out_gggag(L3) U13_gga(D, Ds, X, R, =_out_ag(X1)) -> U14_gga(D, Ds, X, R, X1, =_in_ag(X)) U14_gga(D, Ds, X, R, X1, =_out_ag(X2)) -> U15_gga(D, Ds, X, R, isGreater_in_gg(X1, X2)) U15_gga(D, Ds, X, R, isGreater_out_gg) -> U16_gga(X, mergedelete_in_gga(.(D, Ds), R)) U16_gga(X, mergedelete_out_gga(L2)) -> mergedelete_out_gga(.(X, L2)) U38_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(succ(succ(Z))) U37_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(Z) U39_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(pred(pred(Z))) U40_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(Z) U36_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(succ(Z)) U35_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(pred(Z)) ---------------------------------------- (88) Obligation: Q DP problem: The TRS P consists of the following rules: U21_GGAG(Unused, List, Mod, member_out_ag(Choice)) -> U22_GGAG(List, Choice, Mod, check_in_gggag(List, Choice, Unused, Mod)) PDS1_IN_GGAG(Unused, List, Mod) -> U21_GGAG(Unused, List, Mod, member_in_ag(Unused)) The TRS R consists of the following rules: member_in_ag(.(X, Xs)) -> member_out_ag(X) member_in_ag(.(X1, Xs)) -> U1_ag(member_in_ag(Xs)) U1_ag(member_out_ag(X)) -> member_out_ag(X) check_in_gggag([], X4, L, X5) -> check_out_gggag(L) check_in_gggag(S, Choice, Old, Modulus) -> U17_gggag(Old, dif_in_gggaa(S, Choice, Modulus)) dif_in_gggaa([], X2, X3) -> dif_out_gggaa([], []) dif_in_gggaa(.(S, Ss), Val, Mod) -> U6_gggaa(Ss, Val, Mod, isMinus_in_gga(Val, S)) U17_gggag(Old, dif_out_gggaa(Ds, Dds)) -> U18_gggag(Dds, mergedelete_in_gga(Ds, Old)) mergedelete_in_gga([], L) -> mergedelete_out_gga(L) mergedelete_in_gga(.(D, Ds), .(X, R)) -> U13_gga(D, Ds, X, R, =_in_ag(D)) U18_gggag(Dds, mergedelete_out_gga(L2)) -> U19_gggag(L2, rev_in_gga(Dds, [])) rev_in_gga([], L) -> rev_out_gga(L) rev_in_gga(.(X, Xs), Y) -> U11_gga(rev_in_gga(Xs, .(X, Y))) U19_gggag(L2, rev_out_gga(Rds)) -> U20_gggag(mergedelete_in_gga(Rds, L2)) U20_gggag(mergedelete_out_gga(L3)) -> check_out_gggag(L3) U11_gga(rev_out_gga(L)) -> rev_out_gga(L) =_in_ag(X) -> =_out_ag(X) U13_gga(D, Ds, X, R, =_out_ag(X1)) -> U14_gga(D, Ds, X, R, X1, =_in_ag(X)) U14_gga(D, Ds, X, R, X1, =_out_ag(X2)) -> U15_gga(D, Ds, X, R, isGreater_in_gg(X1, X2)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(isGreater_in_gg(X, Y)) U15_gga(D, Ds, X, R, isGreater_out_gg) -> U16_gga(X, mergedelete_in_gga(.(D, Ds), R)) U16_gga(X, mergedelete_out_gga(L2)) -> mergedelete_out_gga(.(X, L2)) U65_gg(isGreater_out_gg) -> isGreater_out_gg U64_gg(isGreater_out_gg) -> isGreater_out_gg isMinus_in_gga(X, zero) -> isMinus_out_gga(X) isMinus_in_gga(zero, succ(Y)) -> U35_gga(isMinus_in_gga(zero, Y)) isMinus_in_gga(zero, pred(Y)) -> U36_gga(isMinus_in_gga(zero, Y)) isMinus_in_gga(succ(X), succ(Y)) -> U37_gga(isMinus_in_gga(X, Y)) isMinus_in_gga(succ(X), pred(Y)) -> U38_gga(isMinus_in_gga(X, Y)) isMinus_in_gga(pred(X), succ(Y)) -> U39_gga(isMinus_in_gga(X, Y)) isMinus_in_gga(pred(X), pred(Y)) -> U40_gga(isMinus_in_gga(X, Y)) U6_gggaa(Ss, Val, Mod, isMinus_out_gga(U)) -> U7_gggaa(Ss, Val, Mod, U, =_in_ag(U)) U7_gggaa(Ss, Val, Mod, U, =_out_ag(D)) -> U8_gggaa(Ss, Val, Mod, D, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(isMinus_in_ggg(X, Y, Z)) U8_gggaa(Ss, Val, Mod, D, U, isMinus_out_ggg) -> U9_gggaa(Ss, Val, Mod, D, =_in_ag(U)) U9_gggaa(Ss, Val, Mod, D, =_out_ag(D2)) -> U10_gggaa(D, D2, dif_in_gggaa(Ss, Val, Mod)) U10_gggaa(D, D2, dif_out_gggaa(Ds, D2s)) -> dif_out_gggaa(.(D, Ds), .(D2, D2s)) U40_ggg(isMinus_out_ggg) -> isMinus_out_ggg U39_ggg(isMinus_out_ggg) -> isMinus_out_ggg U38_ggg(isMinus_out_ggg) -> isMinus_out_ggg U37_ggg(isMinus_out_ggg) -> isMinus_out_ggg U36_ggg(isMinus_out_ggg) -> isMinus_out_ggg U35_ggg(isMinus_out_ggg) -> isMinus_out_ggg U40_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(Z) U39_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(pred(pred(Z))) U38_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(succ(succ(Z))) U37_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(Z) U36_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(succ(Z)) U35_gga(isMinus_out_gga(Z)) -> isMinus_out_gga(pred(Z)) The set Q consists of the following terms: check_in_gggag(x0, x1, x2, x3) member_in_ag(x0) U17_gggag(x0, x1) U1_ag(x0) dif_in_gggaa(x0, x1, x2) U18_gggag(x0, x1) U6_gggaa(x0, x1, x2, x3) mergedelete_in_gga(x0, x1) U19_gggag(x0, x1) isMinus_in_gga(x0, x1) U7_gggaa(x0, x1, x2, x3, x4) U13_gga(x0, x1, x2, x3, x4) rev_in_gga(x0, x1) U20_gggag(x0) U35_gga(x0) U36_gga(x0) U37_gga(x0) U38_gga(x0) U39_gga(x0) U40_gga(x0) =_in_ag(x0) U8_gggaa(x0, x1, x2, x3, x4, x5) U14_gga(x0, x1, x2, x3, x4, x5) U11_gga(x0) isMinus_in_ggg(x0, x1, x2) U9_gggaa(x0, x1, x2, x3, x4) U15_gga(x0, x1, x2, x3, x4) U35_ggg(x0) U36_ggg(x0) U37_ggg(x0) U38_ggg(x0) U39_ggg(x0) U40_ggg(x0) U10_gggaa(x0, x1, x2) isGreater_in_gg(x0, x1) U16_gga(x0, x1) U64_gg(x0) U65_gg(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (89) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. ---------------------------------------- (90) TRUE ---------------------------------------- (91) Obligation: Pi DP problem: The TRS P consists of the following rules: ISPLUS_IN_GGA(succ(X), pred(Y), Z) -> ISPLUS_IN_GGA(X, Y, Z) ISPLUS_IN_GGA(succ(X), succ(Y), succ(succ(Z))) -> ISPLUS_IN_GGA(X, Y, Z) ISPLUS_IN_GGA(pred(X), succ(Y), Z) -> ISPLUS_IN_GGA(X, Y, Z) ISPLUS_IN_GGA(pred(X), pred(Y), pred(pred(Z))) -> ISPLUS_IN_GGA(X, Y, Z) The TRS R consists of the following rules: pds_in_ga(Order, .(zero, Ans)) -> U24_ga(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) isPlus_in_gga(zero, X, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), zero, succ(X)) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), succ(Y), succ(succ(Z))) -> U31_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(succ(X), pred(Y), Z) -> U32_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), zero, pred(X)) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), succ(Y), Z) -> U33_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), pred(Y), pred(pred(Z))) -> U34_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) U34_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U33_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U32_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U31_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U24_ga(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_ga(Order, Ans, isTimes_in_gga(Order, U, U1)) isTimes_in_gga(X, zero, zero) -> isTimes_out_gga(X, zero, zero) isTimes_in_gga(zero, succ(Y), zero) -> isTimes_out_gga(zero, succ(Y), zero) isTimes_in_gga(zero, pred(Y), zero) -> isTimes_out_gga(zero, pred(Y), zero) isTimes_in_gga(succ(X), succ(Y), Z) -> U41_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(succ(X), pred(Y), Z) -> U43_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(pred(X), succ(Y), Z) -> U45_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) isTimes_in_gga(pred(X), pred(Y), Z) -> U47_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) U47_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_gga(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U48_gga(X, Y, Z, isMinus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), pred(Y), Z) U45_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_gga(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U46_gga(X, Y, Z, isPlus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), succ(Y), Z) U43_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_gga(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U44_gga(X, Y, Z, isMinus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), pred(Y), Z) U41_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_gga(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U42_gga(X, Y, Z, isPlus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), succ(Y), Z) U25_ga(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_ga(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U26_ga(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_ga(Order, Ans, =_in_ag(N, U2)) =_in_ag(X, X) -> =_out_ag(X, X) U27_ga(Order, Ans, =_out_ag(N, U2)) -> U28_ga(Order, Ans, N, iota_in_ga(N, .(zero, List))) iota_in_ga(N, List) -> U2_ga(N, List, iota1_in_gga(zero, N, List)) iota1_in_gga(K, K, []) -> iota1_out_gga(K, K, []) iota1_in_gga(K, N, .(K, List)) -> U3_gga(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_gga(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_gga(K, N, List, =_in_ag(K1, U)) U4_gga(K, N, List, =_out_ag(K1, U)) -> U5_gga(K, N, List, iota1_in_gga(K1, N, List)) U5_gga(K, N, List, iota1_out_gga(K1, N, List)) -> iota1_out_gga(K, N, .(K, List)) U2_ga(N, List, iota1_out_gga(zero, N, List)) -> iota_out_ga(N, List) U28_ga(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_ga(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) pds1_in_ggag([], X6, [], X7) -> pds1_out_ggag([], X6, [], X7) pds1_in_ggag(Unused, List, .(Choice, Rest), Mod) -> U21_ggag(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(X, .(X1, Xs)) -> U1_ag(X, X1, Xs, member_in_ag(X, Xs)) U1_ag(X, X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) U21_ggag(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_ggag(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) check_in_gggag([], X4, L, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, L3, Modulus) -> U17_gggag(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) dif_in_gggaa([], X2, X3, [], []) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U17_gggag(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) mergedelete_in_gga([], L, L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R), L2) -> U12_gga(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) mergedelete_in_gga(.(D, Ds), .(X, R), .(X, L2)) -> U13_gga(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_gga(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_gga(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U15_gga(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U16_gga(D, Ds, X, R, L2, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U12_gga(D, Ds, R, L2, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) rev_in_gga([], L, L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y, L) -> U11_gga(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) U11_gga(X, Xs, Y, L, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U22_ggag(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_ggag(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U23_ggag(Unused, List, Choice, Rest, Mod, pds1_out_ggag(U3, .(Choice, List), Rest, Mod)) -> pds1_out_ggag(Unused, List, .(Choice, Rest), Mod) U29_ga(Order, Ans, pds1_out_ggag(List, .(zero, []), Ans, N)) -> pds_out_ga(Order, .(zero, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U24_ga(x1, x2, x3) = U24_ga(x1, x3) isPlus_in_gga(x1, x2, x3) = isPlus_in_gga(x1, x2) zero = zero isPlus_out_gga(x1, x2, x3) = isPlus_out_gga(x3) succ(x1) = succ(x1) U31_gga(x1, x2, x3, x4) = U31_gga(x4) pred(x1) = pred(x1) U32_gga(x1, x2, x3, x4) = U32_gga(x4) U33_gga(x1, x2, x3, x4) = U33_gga(x4) U34_gga(x1, x2, x3, x4) = U34_gga(x4) U25_ga(x1, x2, x3) = U25_ga(x3) isTimes_in_gga(x1, x2, x3) = isTimes_in_gga(x1, x2) isTimes_out_gga(x1, x2, x3) = isTimes_out_gga(x3) U41_gga(x1, x2, x3, x4) = U41_gga(x1, x4) U43_gga(x1, x2, x3, x4) = U43_gga(x1, x4) U45_gga(x1, x2, x3, x4) = U45_gga(x1, x4) U47_gga(x1, x2, x3, x4) = U47_gga(x1, x4) U48_gga(x1, x2, x3, x4) = U48_gga(x4) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x3) U35_gga(x1, x2, x3) = U35_gga(x3) U36_gga(x1, x2, x3) = U36_gga(x3) U37_gga(x1, x2, x3, x4) = U37_gga(x4) U38_gga(x1, x2, x3, x4) = U38_gga(x4) U39_gga(x1, x2, x3, x4) = U39_gga(x4) U40_gga(x1, x2, x3, x4) = U40_gga(x4) U46_gga(x1, x2, x3, x4) = U46_gga(x4) U44_gga(x1, x2, x3, x4) = U44_gga(x4) U42_gga(x1, x2, x3, x4) = U42_gga(x4) U26_ga(x1, x2, x3) = U26_ga(x3) U27_ga(x1, x2, x3) = U27_ga(x3) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1) U28_ga(x1, x2, x3, x4) = U28_ga(x3, x4) iota_in_ga(x1, x2) = iota_in_ga(x1) U2_ga(x1, x2, x3) = U2_ga(x3) iota1_in_gga(x1, x2, x3) = iota1_in_gga(x1, x2) iota1_out_gga(x1, x2, x3) = iota1_out_gga(x3) U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x4) iota_out_ga(x1, x2) = iota_out_ga(x2) .(x1, x2) = .(x1, x2) U29_ga(x1, x2, x3) = U29_ga(x3) pds1_in_ggag(x1, x2, x3, x4) = pds1_in_ggag(x1, x2, x4) [] = [] pds1_out_ggag(x1, x2, x3, x4) = pds1_out_ggag(x3) U21_ggag(x1, x2, x3, x4, x5, x6) = U21_ggag(x1, x2, x5, x6) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1) U1_ag(x1, x2, x3, x4) = U1_ag(x4) U22_ggag(x1, x2, x3, x4, x5, x6) = U22_ggag(x2, x3, x5, x6) check_in_gggag(x1, x2, x3, x4, x5) = check_in_gggag(x1, x2, x3, x5) check_out_gggag(x1, x2, x3, x4, x5) = check_out_gggag(x4) U17_gggag(x1, x2, x3, x4, x5, x6) = U17_gggag(x3, x6) dif_in_gggaa(x1, x2, x3, x4, x5) = dif_in_gggaa(x1, x2, x3) dif_out_gggaa(x1, x2, x3, x4, x5) = dif_out_gggaa(x4, x5) U6_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gggaa(x2, x3, x4, x9) U7_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_gggaa(x2, x3, x4, x9, x10) U8_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_gggaa(x2, x3, x4, x5, x9, x10) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg U35_ggg(x1, x2, x3) = U35_ggg(x3) U36_ggg(x1, x2, x3) = U36_ggg(x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x4) U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_gggaa(x2, x3, x4, x5, x9) U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_gggaa(x5, x7, x9) U18_gggag(x1, x2, x3, x4, x5, x6, x7) = U18_gggag(x6, x7) mergedelete_in_gga(x1, x2, x3) = mergedelete_in_gga(x1, x2) mergedelete_out_gga(x1, x2, x3) = mergedelete_out_gga(x3) U12_gga(x1, x2, x3, x4, x5) = U12_gga(x5) U13_gga(x1, x2, x3, x4, x5, x6) = U13_gga(x1, x2, x3, x4, x6) U14_gga(x1, x2, x3, x4, x5, x6, x7) = U14_gga(x1, x2, x3, x4, x6, x7) U15_gga(x1, x2, x3, x4, x5, x6) = U15_gga(x1, x2, x3, x4, x6) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg U64_gg(x1, x2, x3) = U64_gg(x3) U65_gg(x1, x2, x3) = U65_gg(x3) U16_gga(x1, x2, x3, x4, x5, x6) = U16_gga(x3, x6) U19_gggag(x1, x2, x3, x4, x5, x6, x7) = U19_gggag(x6, x7) rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2) rev_out_gga(x1, x2, x3) = rev_out_gga(x3) U11_gga(x1, x2, x3, x4, x5) = U11_gga(x5) U20_gggag(x1, x2, x3, x4, x5, x6) = U20_gggag(x6) U23_ggag(x1, x2, x3, x4, x5, x6) = U23_ggag(x3, x6) pds_out_ga(x1, x2) = pds_out_ga(x2) ISPLUS_IN_GGA(x1, x2, x3) = ISPLUS_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (92) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (93) Obligation: Pi DP problem: The TRS P consists of the following rules: ISPLUS_IN_GGA(succ(X), pred(Y), Z) -> ISPLUS_IN_GGA(X, Y, Z) ISPLUS_IN_GGA(succ(X), succ(Y), succ(succ(Z))) -> ISPLUS_IN_GGA(X, Y, Z) ISPLUS_IN_GGA(pred(X), succ(Y), Z) -> ISPLUS_IN_GGA(X, Y, Z) ISPLUS_IN_GGA(pred(X), pred(Y), pred(pred(Z))) -> ISPLUS_IN_GGA(X, Y, Z) R is empty. The argument filtering Pi contains the following mapping: succ(x1) = succ(x1) pred(x1) = pred(x1) ISPLUS_IN_GGA(x1, x2, x3) = ISPLUS_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (94) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (95) Obligation: Q DP problem: The TRS P consists of the following rules: ISPLUS_IN_GGA(succ(X), pred(Y)) -> ISPLUS_IN_GGA(X, Y) ISPLUS_IN_GGA(succ(X), succ(Y)) -> ISPLUS_IN_GGA(X, Y) ISPLUS_IN_GGA(pred(X), succ(Y)) -> ISPLUS_IN_GGA(X, Y) ISPLUS_IN_GGA(pred(X), pred(Y)) -> ISPLUS_IN_GGA(X, Y) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (96) 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: *ISPLUS_IN_GGA(succ(X), pred(Y)) -> ISPLUS_IN_GGA(X, Y) The graph contains the following edges 1 > 1, 2 > 2 *ISPLUS_IN_GGA(succ(X), succ(Y)) -> ISPLUS_IN_GGA(X, Y) The graph contains the following edges 1 > 1, 2 > 2 *ISPLUS_IN_GGA(pred(X), succ(Y)) -> ISPLUS_IN_GGA(X, Y) The graph contains the following edges 1 > 1, 2 > 2 *ISPLUS_IN_GGA(pred(X), pred(Y)) -> ISPLUS_IN_GGA(X, Y) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (97) YES ---------------------------------------- (98) Obligation: Pi DP problem: The TRS P consists of the following rules: IOTA1_IN_GGA(K, N, .(K, List)) -> U3_GGA(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_GGA(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_GGA(K, N, List, =_in_ag(K1, U)) U4_GGA(K, N, List, =_out_ag(K1, U)) -> IOTA1_IN_GGA(K1, N, List) The TRS R consists of the following rules: pds_in_ga(Order, .(zero, Ans)) -> U24_ga(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) isPlus_in_gga(zero, X, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), zero, succ(X)) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), succ(Y), succ(succ(Z))) -> U31_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(succ(X), pred(Y), Z) -> U32_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), zero, pred(X)) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), succ(Y), Z) -> U33_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), pred(Y), pred(pred(Z))) -> U34_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) U34_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U33_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U32_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U31_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U24_ga(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_ga(Order, Ans, isTimes_in_gga(Order, U, U1)) isTimes_in_gga(X, zero, zero) -> isTimes_out_gga(X, zero, zero) isTimes_in_gga(zero, succ(Y), zero) -> isTimes_out_gga(zero, succ(Y), zero) isTimes_in_gga(zero, pred(Y), zero) -> isTimes_out_gga(zero, pred(Y), zero) isTimes_in_gga(succ(X), succ(Y), Z) -> U41_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(succ(X), pred(Y), Z) -> U43_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(pred(X), succ(Y), Z) -> U45_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) isTimes_in_gga(pred(X), pred(Y), Z) -> U47_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) U47_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_gga(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U48_gga(X, Y, Z, isMinus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), pred(Y), Z) U45_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_gga(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U46_gga(X, Y, Z, isPlus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), succ(Y), Z) U43_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_gga(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U44_gga(X, Y, Z, isMinus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), pred(Y), Z) U41_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_gga(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U42_gga(X, Y, Z, isPlus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), succ(Y), Z) U25_ga(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_ga(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U26_ga(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_ga(Order, Ans, =_in_ag(N, U2)) =_in_ag(X, X) -> =_out_ag(X, X) U27_ga(Order, Ans, =_out_ag(N, U2)) -> U28_ga(Order, Ans, N, iota_in_ga(N, .(zero, List))) iota_in_ga(N, List) -> U2_ga(N, List, iota1_in_gga(zero, N, List)) iota1_in_gga(K, K, []) -> iota1_out_gga(K, K, []) iota1_in_gga(K, N, .(K, List)) -> U3_gga(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_gga(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_gga(K, N, List, =_in_ag(K1, U)) U4_gga(K, N, List, =_out_ag(K1, U)) -> U5_gga(K, N, List, iota1_in_gga(K1, N, List)) U5_gga(K, N, List, iota1_out_gga(K1, N, List)) -> iota1_out_gga(K, N, .(K, List)) U2_ga(N, List, iota1_out_gga(zero, N, List)) -> iota_out_ga(N, List) U28_ga(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_ga(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) pds1_in_ggag([], X6, [], X7) -> pds1_out_ggag([], X6, [], X7) pds1_in_ggag(Unused, List, .(Choice, Rest), Mod) -> U21_ggag(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(X, .(X1, Xs)) -> U1_ag(X, X1, Xs, member_in_ag(X, Xs)) U1_ag(X, X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) U21_ggag(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_ggag(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) check_in_gggag([], X4, L, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, L3, Modulus) -> U17_gggag(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) dif_in_gggaa([], X2, X3, [], []) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U17_gggag(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) mergedelete_in_gga([], L, L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R), L2) -> U12_gga(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) mergedelete_in_gga(.(D, Ds), .(X, R), .(X, L2)) -> U13_gga(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_gga(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_gga(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U15_gga(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U16_gga(D, Ds, X, R, L2, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U12_gga(D, Ds, R, L2, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) rev_in_gga([], L, L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y, L) -> U11_gga(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) U11_gga(X, Xs, Y, L, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U22_ggag(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_ggag(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U23_ggag(Unused, List, Choice, Rest, Mod, pds1_out_ggag(U3, .(Choice, List), Rest, Mod)) -> pds1_out_ggag(Unused, List, .(Choice, Rest), Mod) U29_ga(Order, Ans, pds1_out_ggag(List, .(zero, []), Ans, N)) -> pds_out_ga(Order, .(zero, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U24_ga(x1, x2, x3) = U24_ga(x1, x3) isPlus_in_gga(x1, x2, x3) = isPlus_in_gga(x1, x2) zero = zero isPlus_out_gga(x1, x2, x3) = isPlus_out_gga(x3) succ(x1) = succ(x1) U31_gga(x1, x2, x3, x4) = U31_gga(x4) pred(x1) = pred(x1) U32_gga(x1, x2, x3, x4) = U32_gga(x4) U33_gga(x1, x2, x3, x4) = U33_gga(x4) U34_gga(x1, x2, x3, x4) = U34_gga(x4) U25_ga(x1, x2, x3) = U25_ga(x3) isTimes_in_gga(x1, x2, x3) = isTimes_in_gga(x1, x2) isTimes_out_gga(x1, x2, x3) = isTimes_out_gga(x3) U41_gga(x1, x2, x3, x4) = U41_gga(x1, x4) U43_gga(x1, x2, x3, x4) = U43_gga(x1, x4) U45_gga(x1, x2, x3, x4) = U45_gga(x1, x4) U47_gga(x1, x2, x3, x4) = U47_gga(x1, x4) U48_gga(x1, x2, x3, x4) = U48_gga(x4) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x3) U35_gga(x1, x2, x3) = U35_gga(x3) U36_gga(x1, x2, x3) = U36_gga(x3) U37_gga(x1, x2, x3, x4) = U37_gga(x4) U38_gga(x1, x2, x3, x4) = U38_gga(x4) U39_gga(x1, x2, x3, x4) = U39_gga(x4) U40_gga(x1, x2, x3, x4) = U40_gga(x4) U46_gga(x1, x2, x3, x4) = U46_gga(x4) U44_gga(x1, x2, x3, x4) = U44_gga(x4) U42_gga(x1, x2, x3, x4) = U42_gga(x4) U26_ga(x1, x2, x3) = U26_ga(x3) U27_ga(x1, x2, x3) = U27_ga(x3) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1) U28_ga(x1, x2, x3, x4) = U28_ga(x3, x4) iota_in_ga(x1, x2) = iota_in_ga(x1) U2_ga(x1, x2, x3) = U2_ga(x3) iota1_in_gga(x1, x2, x3) = iota1_in_gga(x1, x2) iota1_out_gga(x1, x2, x3) = iota1_out_gga(x3) U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x4) iota_out_ga(x1, x2) = iota_out_ga(x2) .(x1, x2) = .(x1, x2) U29_ga(x1, x2, x3) = U29_ga(x3) pds1_in_ggag(x1, x2, x3, x4) = pds1_in_ggag(x1, x2, x4) [] = [] pds1_out_ggag(x1, x2, x3, x4) = pds1_out_ggag(x3) U21_ggag(x1, x2, x3, x4, x5, x6) = U21_ggag(x1, x2, x5, x6) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1) U1_ag(x1, x2, x3, x4) = U1_ag(x4) U22_ggag(x1, x2, x3, x4, x5, x6) = U22_ggag(x2, x3, x5, x6) check_in_gggag(x1, x2, x3, x4, x5) = check_in_gggag(x1, x2, x3, x5) check_out_gggag(x1, x2, x3, x4, x5) = check_out_gggag(x4) U17_gggag(x1, x2, x3, x4, x5, x6) = U17_gggag(x3, x6) dif_in_gggaa(x1, x2, x3, x4, x5) = dif_in_gggaa(x1, x2, x3) dif_out_gggaa(x1, x2, x3, x4, x5) = dif_out_gggaa(x4, x5) U6_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gggaa(x2, x3, x4, x9) U7_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_gggaa(x2, x3, x4, x9, x10) U8_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_gggaa(x2, x3, x4, x5, x9, x10) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg U35_ggg(x1, x2, x3) = U35_ggg(x3) U36_ggg(x1, x2, x3) = U36_ggg(x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x4) U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_gggaa(x2, x3, x4, x5, x9) U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_gggaa(x5, x7, x9) U18_gggag(x1, x2, x3, x4, x5, x6, x7) = U18_gggag(x6, x7) mergedelete_in_gga(x1, x2, x3) = mergedelete_in_gga(x1, x2) mergedelete_out_gga(x1, x2, x3) = mergedelete_out_gga(x3) U12_gga(x1, x2, x3, x4, x5) = U12_gga(x5) U13_gga(x1, x2, x3, x4, x5, x6) = U13_gga(x1, x2, x3, x4, x6) U14_gga(x1, x2, x3, x4, x5, x6, x7) = U14_gga(x1, x2, x3, x4, x6, x7) U15_gga(x1, x2, x3, x4, x5, x6) = U15_gga(x1, x2, x3, x4, x6) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg U64_gg(x1, x2, x3) = U64_gg(x3) U65_gg(x1, x2, x3) = U65_gg(x3) U16_gga(x1, x2, x3, x4, x5, x6) = U16_gga(x3, x6) U19_gggag(x1, x2, x3, x4, x5, x6, x7) = U19_gggag(x6, x7) rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2) rev_out_gga(x1, x2, x3) = rev_out_gga(x3) U11_gga(x1, x2, x3, x4, x5) = U11_gga(x5) U20_gggag(x1, x2, x3, x4, x5, x6) = U20_gggag(x6) U23_ggag(x1, x2, x3, x4, x5, x6) = U23_ggag(x3, x6) pds_out_ga(x1, x2) = pds_out_ga(x2) IOTA1_IN_GGA(x1, x2, x3) = IOTA1_IN_GGA(x1, x2) U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4) U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (99) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (100) Obligation: Pi DP problem: The TRS P consists of the following rules: IOTA1_IN_GGA(K, N, .(K, List)) -> U3_GGA(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_GGA(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_GGA(K, N, List, =_in_ag(K1, U)) U4_GGA(K, N, List, =_out_ag(K1, U)) -> IOTA1_IN_GGA(K1, N, List) The TRS R consists of the following rules: isPlus_in_gga(zero, X, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), succ(Y), succ(succ(Z))) -> U31_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), succ(Y), Z) -> U33_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) =_in_ag(X, X) -> =_out_ag(X, X) U31_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U33_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) isPlus_in_gga(succ(X), zero, succ(X)) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), pred(Y), Z) -> U32_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), zero, pred(X)) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), pred(Y), pred(pred(Z))) -> U34_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) U32_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U34_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) The argument filtering Pi contains the following mapping: isPlus_in_gga(x1, x2, x3) = isPlus_in_gga(x1, x2) zero = zero isPlus_out_gga(x1, x2, x3) = isPlus_out_gga(x3) succ(x1) = succ(x1) U31_gga(x1, x2, x3, x4) = U31_gga(x4) pred(x1) = pred(x1) U32_gga(x1, x2, x3, x4) = U32_gga(x4) U33_gga(x1, x2, x3, x4) = U33_gga(x4) U34_gga(x1, x2, x3, x4) = U34_gga(x4) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1) .(x1, x2) = .(x1, x2) IOTA1_IN_GGA(x1, x2, x3) = IOTA1_IN_GGA(x1, x2) U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4) U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (101) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (102) Obligation: Q DP problem: The TRS P consists of the following rules: IOTA1_IN_GGA(K, N) -> U3_GGA(K, N, isPlus_in_gga(K, succ(zero))) U3_GGA(K, N, isPlus_out_gga(U)) -> U4_GGA(K, N, =_in_ag(U)) U4_GGA(K, N, =_out_ag(K1)) -> IOTA1_IN_GGA(K1, N) The TRS R consists of the following rules: isPlus_in_gga(zero, X) -> isPlus_out_gga(X) isPlus_in_gga(succ(X), succ(Y)) -> U31_gga(isPlus_in_gga(X, Y)) isPlus_in_gga(pred(X), succ(Y)) -> U33_gga(isPlus_in_gga(X, Y)) =_in_ag(X) -> =_out_ag(X) U31_gga(isPlus_out_gga(Z)) -> isPlus_out_gga(succ(succ(Z))) U33_gga(isPlus_out_gga(Z)) -> isPlus_out_gga(Z) isPlus_in_gga(succ(X), zero) -> isPlus_out_gga(succ(X)) isPlus_in_gga(succ(X), pred(Y)) -> U32_gga(isPlus_in_gga(X, Y)) isPlus_in_gga(pred(X), zero) -> isPlus_out_gga(pred(X)) isPlus_in_gga(pred(X), pred(Y)) -> U34_gga(isPlus_in_gga(X, Y)) U32_gga(isPlus_out_gga(Z)) -> isPlus_out_gga(Z) U34_gga(isPlus_out_gga(Z)) -> isPlus_out_gga(pred(pred(Z))) The set Q consists of the following terms: isPlus_in_gga(x0, x1) =_in_ag(x0) U31_gga(x0) U33_gga(x0) U32_gga(x0) U34_gga(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (103) Obligation: Pi DP problem: The TRS P consists of the following rules: ISTIMES_IN_GGA(succ(X), pred(Y), Z) -> ISTIMES_IN_GGA(succ(X), Y, A) ISTIMES_IN_GGA(succ(X), succ(Y), Z) -> ISTIMES_IN_GGA(succ(X), Y, A) The TRS R consists of the following rules: pds_in_ga(Order, .(zero, Ans)) -> U24_ga(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) isPlus_in_gga(zero, X, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), zero, succ(X)) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), succ(Y), succ(succ(Z))) -> U31_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(succ(X), pred(Y), Z) -> U32_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), zero, pred(X)) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), succ(Y), Z) -> U33_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), pred(Y), pred(pred(Z))) -> U34_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) U34_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U33_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U32_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U31_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U24_ga(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_ga(Order, Ans, isTimes_in_gga(Order, U, U1)) isTimes_in_gga(X, zero, zero) -> isTimes_out_gga(X, zero, zero) isTimes_in_gga(zero, succ(Y), zero) -> isTimes_out_gga(zero, succ(Y), zero) isTimes_in_gga(zero, pred(Y), zero) -> isTimes_out_gga(zero, pred(Y), zero) isTimes_in_gga(succ(X), succ(Y), Z) -> U41_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(succ(X), pred(Y), Z) -> U43_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(pred(X), succ(Y), Z) -> U45_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) isTimes_in_gga(pred(X), pred(Y), Z) -> U47_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) U47_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_gga(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U48_gga(X, Y, Z, isMinus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), pred(Y), Z) U45_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_gga(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U46_gga(X, Y, Z, isPlus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), succ(Y), Z) U43_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_gga(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U44_gga(X, Y, Z, isMinus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), pred(Y), Z) U41_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_gga(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U42_gga(X, Y, Z, isPlus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), succ(Y), Z) U25_ga(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_ga(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U26_ga(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_ga(Order, Ans, =_in_ag(N, U2)) =_in_ag(X, X) -> =_out_ag(X, X) U27_ga(Order, Ans, =_out_ag(N, U2)) -> U28_ga(Order, Ans, N, iota_in_ga(N, .(zero, List))) iota_in_ga(N, List) -> U2_ga(N, List, iota1_in_gga(zero, N, List)) iota1_in_gga(K, K, []) -> iota1_out_gga(K, K, []) iota1_in_gga(K, N, .(K, List)) -> U3_gga(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_gga(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_gga(K, N, List, =_in_ag(K1, U)) U4_gga(K, N, List, =_out_ag(K1, U)) -> U5_gga(K, N, List, iota1_in_gga(K1, N, List)) U5_gga(K, N, List, iota1_out_gga(K1, N, List)) -> iota1_out_gga(K, N, .(K, List)) U2_ga(N, List, iota1_out_gga(zero, N, List)) -> iota_out_ga(N, List) U28_ga(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_ga(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) pds1_in_ggag([], X6, [], X7) -> pds1_out_ggag([], X6, [], X7) pds1_in_ggag(Unused, List, .(Choice, Rest), Mod) -> U21_ggag(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(X, .(X1, Xs)) -> U1_ag(X, X1, Xs, member_in_ag(X, Xs)) U1_ag(X, X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) U21_ggag(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_ggag(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) check_in_gggag([], X4, L, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, L3, Modulus) -> U17_gggag(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) dif_in_gggaa([], X2, X3, [], []) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U17_gggag(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) mergedelete_in_gga([], L, L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R), L2) -> U12_gga(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) mergedelete_in_gga(.(D, Ds), .(X, R), .(X, L2)) -> U13_gga(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_gga(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_gga(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U15_gga(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U16_gga(D, Ds, X, R, L2, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U12_gga(D, Ds, R, L2, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) rev_in_gga([], L, L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y, L) -> U11_gga(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) U11_gga(X, Xs, Y, L, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U22_ggag(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_ggag(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U23_ggag(Unused, List, Choice, Rest, Mod, pds1_out_ggag(U3, .(Choice, List), Rest, Mod)) -> pds1_out_ggag(Unused, List, .(Choice, Rest), Mod) U29_ga(Order, Ans, pds1_out_ggag(List, .(zero, []), Ans, N)) -> pds_out_ga(Order, .(zero, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U24_ga(x1, x2, x3) = U24_ga(x1, x3) isPlus_in_gga(x1, x2, x3) = isPlus_in_gga(x1, x2) zero = zero isPlus_out_gga(x1, x2, x3) = isPlus_out_gga(x3) succ(x1) = succ(x1) U31_gga(x1, x2, x3, x4) = U31_gga(x4) pred(x1) = pred(x1) U32_gga(x1, x2, x3, x4) = U32_gga(x4) U33_gga(x1, x2, x3, x4) = U33_gga(x4) U34_gga(x1, x2, x3, x4) = U34_gga(x4) U25_ga(x1, x2, x3) = U25_ga(x3) isTimes_in_gga(x1, x2, x3) = isTimes_in_gga(x1, x2) isTimes_out_gga(x1, x2, x3) = isTimes_out_gga(x3) U41_gga(x1, x2, x3, x4) = U41_gga(x1, x4) U43_gga(x1, x2, x3, x4) = U43_gga(x1, x4) U45_gga(x1, x2, x3, x4) = U45_gga(x1, x4) U47_gga(x1, x2, x3, x4) = U47_gga(x1, x4) U48_gga(x1, x2, x3, x4) = U48_gga(x4) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x3) U35_gga(x1, x2, x3) = U35_gga(x3) U36_gga(x1, x2, x3) = U36_gga(x3) U37_gga(x1, x2, x3, x4) = U37_gga(x4) U38_gga(x1, x2, x3, x4) = U38_gga(x4) U39_gga(x1, x2, x3, x4) = U39_gga(x4) U40_gga(x1, x2, x3, x4) = U40_gga(x4) U46_gga(x1, x2, x3, x4) = U46_gga(x4) U44_gga(x1, x2, x3, x4) = U44_gga(x4) U42_gga(x1, x2, x3, x4) = U42_gga(x4) U26_ga(x1, x2, x3) = U26_ga(x3) U27_ga(x1, x2, x3) = U27_ga(x3) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1) U28_ga(x1, x2, x3, x4) = U28_ga(x3, x4) iota_in_ga(x1, x2) = iota_in_ga(x1) U2_ga(x1, x2, x3) = U2_ga(x3) iota1_in_gga(x1, x2, x3) = iota1_in_gga(x1, x2) iota1_out_gga(x1, x2, x3) = iota1_out_gga(x3) U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x4) iota_out_ga(x1, x2) = iota_out_ga(x2) .(x1, x2) = .(x1, x2) U29_ga(x1, x2, x3) = U29_ga(x3) pds1_in_ggag(x1, x2, x3, x4) = pds1_in_ggag(x1, x2, x4) [] = [] pds1_out_ggag(x1, x2, x3, x4) = pds1_out_ggag(x3) U21_ggag(x1, x2, x3, x4, x5, x6) = U21_ggag(x1, x2, x5, x6) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1) U1_ag(x1, x2, x3, x4) = U1_ag(x4) U22_ggag(x1, x2, x3, x4, x5, x6) = U22_ggag(x2, x3, x5, x6) check_in_gggag(x1, x2, x3, x4, x5) = check_in_gggag(x1, x2, x3, x5) check_out_gggag(x1, x2, x3, x4, x5) = check_out_gggag(x4) U17_gggag(x1, x2, x3, x4, x5, x6) = U17_gggag(x3, x6) dif_in_gggaa(x1, x2, x3, x4, x5) = dif_in_gggaa(x1, x2, x3) dif_out_gggaa(x1, x2, x3, x4, x5) = dif_out_gggaa(x4, x5) U6_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gggaa(x2, x3, x4, x9) U7_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_gggaa(x2, x3, x4, x9, x10) U8_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_gggaa(x2, x3, x4, x5, x9, x10) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg U35_ggg(x1, x2, x3) = U35_ggg(x3) U36_ggg(x1, x2, x3) = U36_ggg(x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x4) U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_gggaa(x2, x3, x4, x5, x9) U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_gggaa(x5, x7, x9) U18_gggag(x1, x2, x3, x4, x5, x6, x7) = U18_gggag(x6, x7) mergedelete_in_gga(x1, x2, x3) = mergedelete_in_gga(x1, x2) mergedelete_out_gga(x1, x2, x3) = mergedelete_out_gga(x3) U12_gga(x1, x2, x3, x4, x5) = U12_gga(x5) U13_gga(x1, x2, x3, x4, x5, x6) = U13_gga(x1, x2, x3, x4, x6) U14_gga(x1, x2, x3, x4, x5, x6, x7) = U14_gga(x1, x2, x3, x4, x6, x7) U15_gga(x1, x2, x3, x4, x5, x6) = U15_gga(x1, x2, x3, x4, x6) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg U64_gg(x1, x2, x3) = U64_gg(x3) U65_gg(x1, x2, x3) = U65_gg(x3) U16_gga(x1, x2, x3, x4, x5, x6) = U16_gga(x3, x6) U19_gggag(x1, x2, x3, x4, x5, x6, x7) = U19_gggag(x6, x7) rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2) rev_out_gga(x1, x2, x3) = rev_out_gga(x3) U11_gga(x1, x2, x3, x4, x5) = U11_gga(x5) U20_gggag(x1, x2, x3, x4, x5, x6) = U20_gggag(x6) U23_ggag(x1, x2, x3, x4, x5, x6) = U23_ggag(x3, x6) pds_out_ga(x1, x2) = pds_out_ga(x2) ISTIMES_IN_GGA(x1, x2, x3) = ISTIMES_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (104) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (105) Obligation: Pi DP problem: The TRS P consists of the following rules: ISTIMES_IN_GGA(succ(X), pred(Y), Z) -> ISTIMES_IN_GGA(succ(X), Y, A) ISTIMES_IN_GGA(succ(X), succ(Y), Z) -> ISTIMES_IN_GGA(succ(X), Y, A) R is empty. The argument filtering Pi contains the following mapping: succ(x1) = succ(x1) pred(x1) = pred(x1) ISTIMES_IN_GGA(x1, x2, x3) = ISTIMES_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (106) Obligation: Pi DP problem: The TRS P consists of the following rules: ISTIMES_IN_GGA(pred(X), pred(Y), Z) -> ISTIMES_IN_GGA(pred(X), Y, A) ISTIMES_IN_GGA(pred(X), succ(Y), Z) -> ISTIMES_IN_GGA(pred(X), Y, A) The TRS R consists of the following rules: pds_in_ga(Order, .(zero, Ans)) -> U24_ga(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) isPlus_in_gga(zero, X, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), zero, succ(X)) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), succ(Y), succ(succ(Z))) -> U31_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(succ(X), pred(Y), Z) -> U32_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), zero, pred(X)) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), succ(Y), Z) -> U33_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), pred(Y), pred(pred(Z))) -> U34_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) U34_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U33_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U32_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U31_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U24_ga(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_ga(Order, Ans, isTimes_in_gga(Order, U, U1)) isTimes_in_gga(X, zero, zero) -> isTimes_out_gga(X, zero, zero) isTimes_in_gga(zero, succ(Y), zero) -> isTimes_out_gga(zero, succ(Y), zero) isTimes_in_gga(zero, pred(Y), zero) -> isTimes_out_gga(zero, pred(Y), zero) isTimes_in_gga(succ(X), succ(Y), Z) -> U41_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(succ(X), pred(Y), Z) -> U43_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(pred(X), succ(Y), Z) -> U45_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) isTimes_in_gga(pred(X), pred(Y), Z) -> U47_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) U47_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_gga(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U48_gga(X, Y, Z, isMinus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), pred(Y), Z) U45_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_gga(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U46_gga(X, Y, Z, isPlus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), succ(Y), Z) U43_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_gga(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U44_gga(X, Y, Z, isMinus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), pred(Y), Z) U41_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_gga(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U42_gga(X, Y, Z, isPlus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), succ(Y), Z) U25_ga(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_ga(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U26_ga(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_ga(Order, Ans, =_in_ag(N, U2)) =_in_ag(X, X) -> =_out_ag(X, X) U27_ga(Order, Ans, =_out_ag(N, U2)) -> U28_ga(Order, Ans, N, iota_in_ga(N, .(zero, List))) iota_in_ga(N, List) -> U2_ga(N, List, iota1_in_gga(zero, N, List)) iota1_in_gga(K, K, []) -> iota1_out_gga(K, K, []) iota1_in_gga(K, N, .(K, List)) -> U3_gga(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_gga(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_gga(K, N, List, =_in_ag(K1, U)) U4_gga(K, N, List, =_out_ag(K1, U)) -> U5_gga(K, N, List, iota1_in_gga(K1, N, List)) U5_gga(K, N, List, iota1_out_gga(K1, N, List)) -> iota1_out_gga(K, N, .(K, List)) U2_ga(N, List, iota1_out_gga(zero, N, List)) -> iota_out_ga(N, List) U28_ga(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_ga(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) pds1_in_ggag([], X6, [], X7) -> pds1_out_ggag([], X6, [], X7) pds1_in_ggag(Unused, List, .(Choice, Rest), Mod) -> U21_ggag(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(X, .(X1, Xs)) -> U1_ag(X, X1, Xs, member_in_ag(X, Xs)) U1_ag(X, X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) U21_ggag(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_ggag(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) check_in_gggag([], X4, L, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, L3, Modulus) -> U17_gggag(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) dif_in_gggaa([], X2, X3, [], []) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U17_gggag(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) mergedelete_in_gga([], L, L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R), L2) -> U12_gga(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) mergedelete_in_gga(.(D, Ds), .(X, R), .(X, L2)) -> U13_gga(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_gga(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_gga(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U15_gga(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U16_gga(D, Ds, X, R, L2, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U12_gga(D, Ds, R, L2, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) rev_in_gga([], L, L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y, L) -> U11_gga(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) U11_gga(X, Xs, Y, L, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U22_ggag(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_ggag(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U23_ggag(Unused, List, Choice, Rest, Mod, pds1_out_ggag(U3, .(Choice, List), Rest, Mod)) -> pds1_out_ggag(Unused, List, .(Choice, Rest), Mod) U29_ga(Order, Ans, pds1_out_ggag(List, .(zero, []), Ans, N)) -> pds_out_ga(Order, .(zero, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U24_ga(x1, x2, x3) = U24_ga(x1, x3) isPlus_in_gga(x1, x2, x3) = isPlus_in_gga(x1, x2) zero = zero isPlus_out_gga(x1, x2, x3) = isPlus_out_gga(x3) succ(x1) = succ(x1) U31_gga(x1, x2, x3, x4) = U31_gga(x4) pred(x1) = pred(x1) U32_gga(x1, x2, x3, x4) = U32_gga(x4) U33_gga(x1, x2, x3, x4) = U33_gga(x4) U34_gga(x1, x2, x3, x4) = U34_gga(x4) U25_ga(x1, x2, x3) = U25_ga(x3) isTimes_in_gga(x1, x2, x3) = isTimes_in_gga(x1, x2) isTimes_out_gga(x1, x2, x3) = isTimes_out_gga(x3) U41_gga(x1, x2, x3, x4) = U41_gga(x1, x4) U43_gga(x1, x2, x3, x4) = U43_gga(x1, x4) U45_gga(x1, x2, x3, x4) = U45_gga(x1, x4) U47_gga(x1, x2, x3, x4) = U47_gga(x1, x4) U48_gga(x1, x2, x3, x4) = U48_gga(x4) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x3) U35_gga(x1, x2, x3) = U35_gga(x3) U36_gga(x1, x2, x3) = U36_gga(x3) U37_gga(x1, x2, x3, x4) = U37_gga(x4) U38_gga(x1, x2, x3, x4) = U38_gga(x4) U39_gga(x1, x2, x3, x4) = U39_gga(x4) U40_gga(x1, x2, x3, x4) = U40_gga(x4) U46_gga(x1, x2, x3, x4) = U46_gga(x4) U44_gga(x1, x2, x3, x4) = U44_gga(x4) U42_gga(x1, x2, x3, x4) = U42_gga(x4) U26_ga(x1, x2, x3) = U26_ga(x3) U27_ga(x1, x2, x3) = U27_ga(x3) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1) U28_ga(x1, x2, x3, x4) = U28_ga(x3, x4) iota_in_ga(x1, x2) = iota_in_ga(x1) U2_ga(x1, x2, x3) = U2_ga(x3) iota1_in_gga(x1, x2, x3) = iota1_in_gga(x1, x2) iota1_out_gga(x1, x2, x3) = iota1_out_gga(x3) U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x4) iota_out_ga(x1, x2) = iota_out_ga(x2) .(x1, x2) = .(x1, x2) U29_ga(x1, x2, x3) = U29_ga(x3) pds1_in_ggag(x1, x2, x3, x4) = pds1_in_ggag(x1, x2, x4) [] = [] pds1_out_ggag(x1, x2, x3, x4) = pds1_out_ggag(x3) U21_ggag(x1, x2, x3, x4, x5, x6) = U21_ggag(x1, x2, x5, x6) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1) U1_ag(x1, x2, x3, x4) = U1_ag(x4) U22_ggag(x1, x2, x3, x4, x5, x6) = U22_ggag(x2, x3, x5, x6) check_in_gggag(x1, x2, x3, x4, x5) = check_in_gggag(x1, x2, x3, x5) check_out_gggag(x1, x2, x3, x4, x5) = check_out_gggag(x4) U17_gggag(x1, x2, x3, x4, x5, x6) = U17_gggag(x3, x6) dif_in_gggaa(x1, x2, x3, x4, x5) = dif_in_gggaa(x1, x2, x3) dif_out_gggaa(x1, x2, x3, x4, x5) = dif_out_gggaa(x4, x5) U6_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gggaa(x2, x3, x4, x9) U7_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_gggaa(x2, x3, x4, x9, x10) U8_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_gggaa(x2, x3, x4, x5, x9, x10) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg U35_ggg(x1, x2, x3) = U35_ggg(x3) U36_ggg(x1, x2, x3) = U36_ggg(x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x4) U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_gggaa(x2, x3, x4, x5, x9) U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_gggaa(x5, x7, x9) U18_gggag(x1, x2, x3, x4, x5, x6, x7) = U18_gggag(x6, x7) mergedelete_in_gga(x1, x2, x3) = mergedelete_in_gga(x1, x2) mergedelete_out_gga(x1, x2, x3) = mergedelete_out_gga(x3) U12_gga(x1, x2, x3, x4, x5) = U12_gga(x5) U13_gga(x1, x2, x3, x4, x5, x6) = U13_gga(x1, x2, x3, x4, x6) U14_gga(x1, x2, x3, x4, x5, x6, x7) = U14_gga(x1, x2, x3, x4, x6, x7) U15_gga(x1, x2, x3, x4, x5, x6) = U15_gga(x1, x2, x3, x4, x6) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg U64_gg(x1, x2, x3) = U64_gg(x3) U65_gg(x1, x2, x3) = U65_gg(x3) U16_gga(x1, x2, x3, x4, x5, x6) = U16_gga(x3, x6) U19_gggag(x1, x2, x3, x4, x5, x6, x7) = U19_gggag(x6, x7) rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2) rev_out_gga(x1, x2, x3) = rev_out_gga(x3) U11_gga(x1, x2, x3, x4, x5) = U11_gga(x5) U20_gggag(x1, x2, x3, x4, x5, x6) = U20_gggag(x6) U23_ggag(x1, x2, x3, x4, x5, x6) = U23_ggag(x3, x6) pds_out_ga(x1, x2) = pds_out_ga(x2) ISTIMES_IN_GGA(x1, x2, x3) = ISTIMES_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (107) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (108) Obligation: Pi DP problem: The TRS P consists of the following rules: ISTIMES_IN_GGA(pred(X), pred(Y), Z) -> ISTIMES_IN_GGA(pred(X), Y, A) ISTIMES_IN_GGA(pred(X), succ(Y), Z) -> ISTIMES_IN_GGA(pred(X), Y, A) R is empty. The argument filtering Pi contains the following mapping: succ(x1) = succ(x1) pred(x1) = pred(x1) ISTIMES_IN_GGA(x1, x2, x3) = ISTIMES_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (109) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: pds_in_2: (b,f) isPlus_in_3: (b,b,f) isTimes_in_3: (b,b,f) isMinus_in_3: (b,b,f) (b,b,b) iota_in_2: (b,f) iota1_in_3: (b,b,f) pds1_in_4: (b,b,f,b) member_in_2: (f,b) check_in_5: (b,b,b,f,b) dif_in_5: (b,b,b,f,f) mergedelete_in_3: (b,b,f) isGreater_in_2: (b,b) rev_in_3: (b,b,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: pds_in_ga(Order, .(zero, Ans)) -> U24_ga(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) isPlus_in_gga(zero, X, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), zero, succ(X)) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), succ(Y), succ(succ(Z))) -> U31_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(succ(X), pred(Y), Z) -> U32_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), zero, pred(X)) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), succ(Y), Z) -> U33_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), pred(Y), pred(pred(Z))) -> U34_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) U34_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U33_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U32_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U31_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U24_ga(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_ga(Order, Ans, isTimes_in_gga(Order, U, U1)) isTimes_in_gga(X, zero, zero) -> isTimes_out_gga(X, zero, zero) isTimes_in_gga(zero, succ(Y), zero) -> isTimes_out_gga(zero, succ(Y), zero) isTimes_in_gga(zero, pred(Y), zero) -> isTimes_out_gga(zero, pred(Y), zero) isTimes_in_gga(succ(X), succ(Y), Z) -> U41_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(succ(X), pred(Y), Z) -> U43_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(pred(X), succ(Y), Z) -> U45_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) isTimes_in_gga(pred(X), pred(Y), Z) -> U47_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) U47_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_gga(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U48_gga(X, Y, Z, isMinus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), pred(Y), Z) U45_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_gga(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U46_gga(X, Y, Z, isPlus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), succ(Y), Z) U43_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_gga(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U44_gga(X, Y, Z, isMinus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), pred(Y), Z) U41_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_gga(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U42_gga(X, Y, Z, isPlus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), succ(Y), Z) U25_ga(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_ga(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U26_ga(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_ga(Order, Ans, =_in_ag(N, U2)) =_in_ag(X, X) -> =_out_ag(X, X) U27_ga(Order, Ans, =_out_ag(N, U2)) -> U28_ga(Order, Ans, N, iota_in_ga(N, .(zero, List))) iota_in_ga(N, List) -> U2_ga(N, List, iota1_in_gga(zero, N, List)) iota1_in_gga(K, K, []) -> iota1_out_gga(K, K, []) iota1_in_gga(K, N, .(K, List)) -> U3_gga(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_gga(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_gga(K, N, List, =_in_ag(K1, U)) U4_gga(K, N, List, =_out_ag(K1, U)) -> U5_gga(K, N, List, iota1_in_gga(K1, N, List)) U5_gga(K, N, List, iota1_out_gga(K1, N, List)) -> iota1_out_gga(K, N, .(K, List)) U2_ga(N, List, iota1_out_gga(zero, N, List)) -> iota_out_ga(N, List) U28_ga(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_ga(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) pds1_in_ggag([], X6, [], X7) -> pds1_out_ggag([], X6, [], X7) pds1_in_ggag(Unused, List, .(Choice, Rest), Mod) -> U21_ggag(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(X, .(X1, Xs)) -> U1_ag(X, X1, Xs, member_in_ag(X, Xs)) U1_ag(X, X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) U21_ggag(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_ggag(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) check_in_gggag([], X4, L, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, L3, Modulus) -> U17_gggag(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) dif_in_gggaa([], X2, X3, [], []) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U17_gggag(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) mergedelete_in_gga([], L, L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R), L2) -> U12_gga(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) mergedelete_in_gga(.(D, Ds), .(X, R), .(X, L2)) -> U13_gga(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_gga(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_gga(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U15_gga(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U16_gga(D, Ds, X, R, L2, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U12_gga(D, Ds, R, L2, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) rev_in_gga([], L, L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y, L) -> U11_gga(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) U11_gga(X, Xs, Y, L, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U22_ggag(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_ggag(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U23_ggag(Unused, List, Choice, Rest, Mod, pds1_out_ggag(U3, .(Choice, List), Rest, Mod)) -> pds1_out_ggag(Unused, List, .(Choice, Rest), Mod) U29_ga(Order, Ans, pds1_out_ggag(List, .(zero, []), Ans, N)) -> pds_out_ga(Order, .(zero, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U24_ga(x1, x2, x3) = U24_ga(x1, x3) isPlus_in_gga(x1, x2, x3) = isPlus_in_gga(x1, x2) zero = zero isPlus_out_gga(x1, x2, x3) = isPlus_out_gga(x1, x2, x3) succ(x1) = succ(x1) U31_gga(x1, x2, x3, x4) = U31_gga(x1, x2, x4) pred(x1) = pred(x1) U32_gga(x1, x2, x3, x4) = U32_gga(x1, x2, x4) U33_gga(x1, x2, x3, x4) = U33_gga(x1, x2, x4) U34_gga(x1, x2, x3, x4) = U34_gga(x1, x2, x4) U25_ga(x1, x2, x3) = U25_ga(x1, x3) isTimes_in_gga(x1, x2, x3) = isTimes_in_gga(x1, x2) isTimes_out_gga(x1, x2, x3) = isTimes_out_gga(x1, x2, x3) U41_gga(x1, x2, x3, x4) = U41_gga(x1, x2, x4) U43_gga(x1, x2, x3, x4) = U43_gga(x1, x2, x4) U45_gga(x1, x2, x3, x4) = U45_gga(x1, x2, x4) U47_gga(x1, x2, x3, x4) = U47_gga(x1, x2, x4) U48_gga(x1, x2, x3, x4) = U48_gga(x1, x2, x4) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x1, x2, x3) U35_gga(x1, x2, x3) = U35_gga(x1, x3) U36_gga(x1, x2, x3) = U36_gga(x1, x3) U37_gga(x1, x2, x3, x4) = U37_gga(x1, x2, x4) U38_gga(x1, x2, x3, x4) = U38_gga(x1, x2, x4) U39_gga(x1, x2, x3, x4) = U39_gga(x1, x2, x4) U40_gga(x1, x2, x3, x4) = U40_gga(x1, x2, x4) U46_gga(x1, x2, x3, x4) = U46_gga(x1, x2, x4) U44_gga(x1, x2, x3, x4) = U44_gga(x1, x2, x4) U42_gga(x1, x2, x3, x4) = U42_gga(x1, x2, x4) U26_ga(x1, x2, x3) = U26_ga(x1, x3) U27_ga(x1, x2, x3) = U27_ga(x1, x3) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1, x2) U28_ga(x1, x2, x3, x4) = U28_ga(x1, x3, x4) iota_in_ga(x1, x2) = iota_in_ga(x1) U2_ga(x1, x2, x3) = U2_ga(x1, x3) iota1_in_gga(x1, x2, x3) = iota1_in_gga(x1, x2) iota1_out_gga(x1, x2, x3) = iota1_out_gga(x1, x2, x3) U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4) iota_out_ga(x1, x2) = iota_out_ga(x1, x2) .(x1, x2) = .(x1, x2) U29_ga(x1, x2, x3) = U29_ga(x1, x3) pds1_in_ggag(x1, x2, x3, x4) = pds1_in_ggag(x1, x2, x4) [] = [] pds1_out_ggag(x1, x2, x3, x4) = pds1_out_ggag(x1, x2, x3, x4) U21_ggag(x1, x2, x3, x4, x5, x6) = U21_ggag(x1, x2, x5, x6) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1, x2) U1_ag(x1, x2, x3, x4) = U1_ag(x2, x3, x4) U22_ggag(x1, x2, x3, x4, x5, x6) = U22_ggag(x1, x2, x3, x5, x6) check_in_gggag(x1, x2, x3, x4, x5) = check_in_gggag(x1, x2, x3, x5) check_out_gggag(x1, x2, x3, x4, x5) = check_out_gggag(x1, x2, x3, x4, x5) U17_gggag(x1, x2, x3, x4, x5, x6) = U17_gggag(x1, x2, x3, x5, x6) dif_in_gggaa(x1, x2, x3, x4, x5) = dif_in_gggaa(x1, x2, x3) dif_out_gggaa(x1, x2, x3, x4, x5) = dif_out_gggaa(x1, x2, x3, x4, x5) U6_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gggaa(x1, x2, x3, x4, x9) U7_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_gggaa(x1, x2, x3, x4, x9, x10) U8_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_gggaa(x1, x2, x3, x4, x5, x9, x10) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg(x1, x2, x3) U35_ggg(x1, x2, x3) = U35_ggg(x1, x2, x3) U36_ggg(x1, x2, x3) = U36_ggg(x1, x2, x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x1, x2, x3, x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x1, x2, x3, x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x1, x2, x3, x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x1, x2, x3, x4) U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_gggaa(x1, x2, x3, x4, x5, x9) U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_gggaa(x1, x2, x3, x4, x5, x7, x9) U18_gggag(x1, x2, x3, x4, x5, x6, x7) = U18_gggag(x1, x2, x3, x5, x6, x7) mergedelete_in_gga(x1, x2, x3) = mergedelete_in_gga(x1, x2) mergedelete_out_gga(x1, x2, x3) = mergedelete_out_gga(x1, x2, x3) U12_gga(x1, x2, x3, x4, x5) = U12_gga(x1, x2, x3, x5) U13_gga(x1, x2, x3, x4, x5, x6) = U13_gga(x1, x2, x3, x4, x6) U14_gga(x1, x2, x3, x4, x5, x6, x7) = U14_gga(x1, x2, x3, x4, x6, x7) U15_gga(x1, x2, x3, x4, x5, x6) = U15_gga(x1, x2, x3, x4, x6) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg(x1, x2) U64_gg(x1, x2, x3) = U64_gg(x1, x2, x3) U65_gg(x1, x2, x3) = U65_gg(x1, x2, x3) U16_gga(x1, x2, x3, x4, x5, x6) = U16_gga(x1, x2, x3, x4, x6) U19_gggag(x1, x2, x3, x4, x5, x6, x7) = U19_gggag(x1, x2, x3, x5, x6, x7) rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2) rev_out_gga(x1, x2, x3) = rev_out_gga(x1, x2, x3) U11_gga(x1, x2, x3, x4, x5) = U11_gga(x1, x2, x3, x5) U20_gggag(x1, x2, x3, x4, x5, x6) = U20_gggag(x1, x2, x3, x5, x6) U23_ggag(x1, x2, x3, x4, x5, x6) = U23_ggag(x1, x2, x3, x5, x6) pds_out_ga(x1, x2) = pds_out_ga(x1, x2) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (110) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: pds_in_ga(Order, .(zero, Ans)) -> U24_ga(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) isPlus_in_gga(zero, X, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), zero, succ(X)) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), succ(Y), succ(succ(Z))) -> U31_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(succ(X), pred(Y), Z) -> U32_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), zero, pred(X)) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), succ(Y), Z) -> U33_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), pred(Y), pred(pred(Z))) -> U34_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) U34_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U33_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U32_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U31_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U24_ga(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_ga(Order, Ans, isTimes_in_gga(Order, U, U1)) isTimes_in_gga(X, zero, zero) -> isTimes_out_gga(X, zero, zero) isTimes_in_gga(zero, succ(Y), zero) -> isTimes_out_gga(zero, succ(Y), zero) isTimes_in_gga(zero, pred(Y), zero) -> isTimes_out_gga(zero, pred(Y), zero) isTimes_in_gga(succ(X), succ(Y), Z) -> U41_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(succ(X), pred(Y), Z) -> U43_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(pred(X), succ(Y), Z) -> U45_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) isTimes_in_gga(pred(X), pred(Y), Z) -> U47_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) U47_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_gga(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U48_gga(X, Y, Z, isMinus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), pred(Y), Z) U45_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_gga(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U46_gga(X, Y, Z, isPlus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), succ(Y), Z) U43_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_gga(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U44_gga(X, Y, Z, isMinus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), pred(Y), Z) U41_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_gga(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U42_gga(X, Y, Z, isPlus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), succ(Y), Z) U25_ga(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_ga(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U26_ga(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_ga(Order, Ans, =_in_ag(N, U2)) =_in_ag(X, X) -> =_out_ag(X, X) U27_ga(Order, Ans, =_out_ag(N, U2)) -> U28_ga(Order, Ans, N, iota_in_ga(N, .(zero, List))) iota_in_ga(N, List) -> U2_ga(N, List, iota1_in_gga(zero, N, List)) iota1_in_gga(K, K, []) -> iota1_out_gga(K, K, []) iota1_in_gga(K, N, .(K, List)) -> U3_gga(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_gga(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_gga(K, N, List, =_in_ag(K1, U)) U4_gga(K, N, List, =_out_ag(K1, U)) -> U5_gga(K, N, List, iota1_in_gga(K1, N, List)) U5_gga(K, N, List, iota1_out_gga(K1, N, List)) -> iota1_out_gga(K, N, .(K, List)) U2_ga(N, List, iota1_out_gga(zero, N, List)) -> iota_out_ga(N, List) U28_ga(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_ga(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) pds1_in_ggag([], X6, [], X7) -> pds1_out_ggag([], X6, [], X7) pds1_in_ggag(Unused, List, .(Choice, Rest), Mod) -> U21_ggag(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(X, .(X1, Xs)) -> U1_ag(X, X1, Xs, member_in_ag(X, Xs)) U1_ag(X, X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) U21_ggag(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_ggag(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) check_in_gggag([], X4, L, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, L3, Modulus) -> U17_gggag(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) dif_in_gggaa([], X2, X3, [], []) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U17_gggag(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) mergedelete_in_gga([], L, L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R), L2) -> U12_gga(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) mergedelete_in_gga(.(D, Ds), .(X, R), .(X, L2)) -> U13_gga(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_gga(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_gga(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U15_gga(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U16_gga(D, Ds, X, R, L2, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U12_gga(D, Ds, R, L2, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) rev_in_gga([], L, L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y, L) -> U11_gga(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) U11_gga(X, Xs, Y, L, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U22_ggag(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_ggag(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U23_ggag(Unused, List, Choice, Rest, Mod, pds1_out_ggag(U3, .(Choice, List), Rest, Mod)) -> pds1_out_ggag(Unused, List, .(Choice, Rest), Mod) U29_ga(Order, Ans, pds1_out_ggag(List, .(zero, []), Ans, N)) -> pds_out_ga(Order, .(zero, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U24_ga(x1, x2, x3) = U24_ga(x1, x3) isPlus_in_gga(x1, x2, x3) = isPlus_in_gga(x1, x2) zero = zero isPlus_out_gga(x1, x2, x3) = isPlus_out_gga(x1, x2, x3) succ(x1) = succ(x1) U31_gga(x1, x2, x3, x4) = U31_gga(x1, x2, x4) pred(x1) = pred(x1) U32_gga(x1, x2, x3, x4) = U32_gga(x1, x2, x4) U33_gga(x1, x2, x3, x4) = U33_gga(x1, x2, x4) U34_gga(x1, x2, x3, x4) = U34_gga(x1, x2, x4) U25_ga(x1, x2, x3) = U25_ga(x1, x3) isTimes_in_gga(x1, x2, x3) = isTimes_in_gga(x1, x2) isTimes_out_gga(x1, x2, x3) = isTimes_out_gga(x1, x2, x3) U41_gga(x1, x2, x3, x4) = U41_gga(x1, x2, x4) U43_gga(x1, x2, x3, x4) = U43_gga(x1, x2, x4) U45_gga(x1, x2, x3, x4) = U45_gga(x1, x2, x4) U47_gga(x1, x2, x3, x4) = U47_gga(x1, x2, x4) U48_gga(x1, x2, x3, x4) = U48_gga(x1, x2, x4) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x1, x2, x3) U35_gga(x1, x2, x3) = U35_gga(x1, x3) U36_gga(x1, x2, x3) = U36_gga(x1, x3) U37_gga(x1, x2, x3, x4) = U37_gga(x1, x2, x4) U38_gga(x1, x2, x3, x4) = U38_gga(x1, x2, x4) U39_gga(x1, x2, x3, x4) = U39_gga(x1, x2, x4) U40_gga(x1, x2, x3, x4) = U40_gga(x1, x2, x4) U46_gga(x1, x2, x3, x4) = U46_gga(x1, x2, x4) U44_gga(x1, x2, x3, x4) = U44_gga(x1, x2, x4) U42_gga(x1, x2, x3, x4) = U42_gga(x1, x2, x4) U26_ga(x1, x2, x3) = U26_ga(x1, x3) U27_ga(x1, x2, x3) = U27_ga(x1, x3) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1, x2) U28_ga(x1, x2, x3, x4) = U28_ga(x1, x3, x4) iota_in_ga(x1, x2) = iota_in_ga(x1) U2_ga(x1, x2, x3) = U2_ga(x1, x3) iota1_in_gga(x1, x2, x3) = iota1_in_gga(x1, x2) iota1_out_gga(x1, x2, x3) = iota1_out_gga(x1, x2, x3) U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4) iota_out_ga(x1, x2) = iota_out_ga(x1, x2) .(x1, x2) = .(x1, x2) U29_ga(x1, x2, x3) = U29_ga(x1, x3) pds1_in_ggag(x1, x2, x3, x4) = pds1_in_ggag(x1, x2, x4) [] = [] pds1_out_ggag(x1, x2, x3, x4) = pds1_out_ggag(x1, x2, x3, x4) U21_ggag(x1, x2, x3, x4, x5, x6) = U21_ggag(x1, x2, x5, x6) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1, x2) U1_ag(x1, x2, x3, x4) = U1_ag(x2, x3, x4) U22_ggag(x1, x2, x3, x4, x5, x6) = U22_ggag(x1, x2, x3, x5, x6) check_in_gggag(x1, x2, x3, x4, x5) = check_in_gggag(x1, x2, x3, x5) check_out_gggag(x1, x2, x3, x4, x5) = check_out_gggag(x1, x2, x3, x4, x5) U17_gggag(x1, x2, x3, x4, x5, x6) = U17_gggag(x1, x2, x3, x5, x6) dif_in_gggaa(x1, x2, x3, x4, x5) = dif_in_gggaa(x1, x2, x3) dif_out_gggaa(x1, x2, x3, x4, x5) = dif_out_gggaa(x1, x2, x3, x4, x5) U6_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gggaa(x1, x2, x3, x4, x9) U7_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_gggaa(x1, x2, x3, x4, x9, x10) U8_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_gggaa(x1, x2, x3, x4, x5, x9, x10) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg(x1, x2, x3) U35_ggg(x1, x2, x3) = U35_ggg(x1, x2, x3) U36_ggg(x1, x2, x3) = U36_ggg(x1, x2, x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x1, x2, x3, x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x1, x2, x3, x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x1, x2, x3, x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x1, x2, x3, x4) U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_gggaa(x1, x2, x3, x4, x5, x9) U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_gggaa(x1, x2, x3, x4, x5, x7, x9) U18_gggag(x1, x2, x3, x4, x5, x6, x7) = U18_gggag(x1, x2, x3, x5, x6, x7) mergedelete_in_gga(x1, x2, x3) = mergedelete_in_gga(x1, x2) mergedelete_out_gga(x1, x2, x3) = mergedelete_out_gga(x1, x2, x3) U12_gga(x1, x2, x3, x4, x5) = U12_gga(x1, x2, x3, x5) U13_gga(x1, x2, x3, x4, x5, x6) = U13_gga(x1, x2, x3, x4, x6) U14_gga(x1, x2, x3, x4, x5, x6, x7) = U14_gga(x1, x2, x3, x4, x6, x7) U15_gga(x1, x2, x3, x4, x5, x6) = U15_gga(x1, x2, x3, x4, x6) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg(x1, x2) U64_gg(x1, x2, x3) = U64_gg(x1, x2, x3) U65_gg(x1, x2, x3) = U65_gg(x1, x2, x3) U16_gga(x1, x2, x3, x4, x5, x6) = U16_gga(x1, x2, x3, x4, x6) U19_gggag(x1, x2, x3, x4, x5, x6, x7) = U19_gggag(x1, x2, x3, x5, x6, x7) rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2) rev_out_gga(x1, x2, x3) = rev_out_gga(x1, x2, x3) U11_gga(x1, x2, x3, x4, x5) = U11_gga(x1, x2, x3, x5) U20_gggag(x1, x2, x3, x4, x5, x6) = U20_gggag(x1, x2, x3, x5, x6) U23_ggag(x1, x2, x3, x4, x5, x6) = U23_ggag(x1, x2, x3, x5, x6) pds_out_ga(x1, x2) = pds_out_ga(x1, x2) ---------------------------------------- (111) 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: PDS_IN_GA(Order, .(zero, Ans)) -> U24_GA(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) PDS_IN_GA(Order, .(zero, Ans)) -> ISPLUS_IN_GGA(Order, succ(zero), U) ISPLUS_IN_GGA(succ(X), succ(Y), succ(succ(Z))) -> U31_GGA(X, Y, Z, isPlus_in_gga(X, Y, Z)) ISPLUS_IN_GGA(succ(X), succ(Y), succ(succ(Z))) -> ISPLUS_IN_GGA(X, Y, Z) ISPLUS_IN_GGA(succ(X), pred(Y), Z) -> U32_GGA(X, Y, Z, isPlus_in_gga(X, Y, Z)) ISPLUS_IN_GGA(succ(X), pred(Y), Z) -> ISPLUS_IN_GGA(X, Y, Z) ISPLUS_IN_GGA(pred(X), succ(Y), Z) -> U33_GGA(X, Y, Z, isPlus_in_gga(X, Y, Z)) ISPLUS_IN_GGA(pred(X), succ(Y), Z) -> ISPLUS_IN_GGA(X, Y, Z) ISPLUS_IN_GGA(pred(X), pred(Y), pred(pred(Z))) -> U34_GGA(X, Y, Z, isPlus_in_gga(X, Y, Z)) ISPLUS_IN_GGA(pred(X), pred(Y), pred(pred(Z))) -> ISPLUS_IN_GGA(X, Y, Z) U24_GA(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_GA(Order, Ans, isTimes_in_gga(Order, U, U1)) U24_GA(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> ISTIMES_IN_GGA(Order, U, U1) ISTIMES_IN_GGA(succ(X), succ(Y), Z) -> U41_GGA(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) ISTIMES_IN_GGA(succ(X), succ(Y), Z) -> ISTIMES_IN_GGA(succ(X), Y, A) ISTIMES_IN_GGA(succ(X), pred(Y), Z) -> U43_GGA(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) ISTIMES_IN_GGA(succ(X), pred(Y), Z) -> ISTIMES_IN_GGA(succ(X), Y, A) ISTIMES_IN_GGA(pred(X), succ(Y), Z) -> U45_GGA(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) ISTIMES_IN_GGA(pred(X), succ(Y), Z) -> ISTIMES_IN_GGA(pred(X), Y, A) ISTIMES_IN_GGA(pred(X), pred(Y), Z) -> U47_GGA(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) ISTIMES_IN_GGA(pred(X), pred(Y), Z) -> ISTIMES_IN_GGA(pred(X), Y, A) U47_GGA(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_GGA(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) U47_GGA(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> ISMINUS_IN_GGA(A, pred(X), Z) ISMINUS_IN_GGA(zero, succ(Y), pred(Z)) -> U35_GGA(Y, Z, isMinus_in_gga(zero, Y, Z)) ISMINUS_IN_GGA(zero, succ(Y), pred(Z)) -> ISMINUS_IN_GGA(zero, Y, Z) ISMINUS_IN_GGA(zero, pred(Y), succ(Z)) -> U36_GGA(Y, Z, isMinus_in_gga(zero, Y, Z)) ISMINUS_IN_GGA(zero, pred(Y), succ(Z)) -> ISMINUS_IN_GGA(zero, Y, Z) ISMINUS_IN_GGA(succ(X), succ(Y), Z) -> U37_GGA(X, Y, Z, isMinus_in_gga(X, Y, Z)) ISMINUS_IN_GGA(succ(X), succ(Y), Z) -> ISMINUS_IN_GGA(X, Y, Z) ISMINUS_IN_GGA(succ(X), pred(Y), succ(succ(Z))) -> U38_GGA(X, Y, Z, isMinus_in_gga(X, Y, Z)) ISMINUS_IN_GGA(succ(X), pred(Y), succ(succ(Z))) -> ISMINUS_IN_GGA(X, Y, Z) ISMINUS_IN_GGA(pred(X), succ(Y), pred(pred(Z))) -> U39_GGA(X, Y, Z, isMinus_in_gga(X, Y, Z)) ISMINUS_IN_GGA(pred(X), succ(Y), pred(pred(Z))) -> ISMINUS_IN_GGA(X, Y, Z) ISMINUS_IN_GGA(pred(X), pred(Y), Z) -> U40_GGA(X, Y, Z, isMinus_in_gga(X, Y, Z)) ISMINUS_IN_GGA(pred(X), pred(Y), Z) -> ISMINUS_IN_GGA(X, Y, Z) U45_GGA(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_GGA(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U45_GGA(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> ISPLUS_IN_GGA(A, pred(X), Z) U43_GGA(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_GGA(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U43_GGA(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> ISMINUS_IN_GGA(A, succ(X), Z) U41_GGA(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_GGA(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U41_GGA(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> ISPLUS_IN_GGA(A, succ(X), Z) U25_GA(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_GA(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U25_GA(Order, Ans, isTimes_out_gga(Order, U, U1)) -> ISPLUS_IN_GGA(U1, succ(zero), U2) U26_GA(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_GA(Order, Ans, =_in_ag(N, U2)) U26_GA(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> =_IN_AG(N, U2) U27_GA(Order, Ans, =_out_ag(N, U2)) -> U28_GA(Order, Ans, N, iota_in_ga(N, .(zero, List))) U27_GA(Order, Ans, =_out_ag(N, U2)) -> IOTA_IN_GA(N, .(zero, List)) IOTA_IN_GA(N, List) -> U2_GA(N, List, iota1_in_gga(zero, N, List)) IOTA_IN_GA(N, List) -> IOTA1_IN_GGA(zero, N, List) IOTA1_IN_GGA(K, N, .(K, List)) -> U3_GGA(K, N, List, isPlus_in_gga(K, succ(zero), U)) IOTA1_IN_GGA(K, N, .(K, List)) -> ISPLUS_IN_GGA(K, succ(zero), U) U3_GGA(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_GGA(K, N, List, =_in_ag(K1, U)) U3_GGA(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> =_IN_AG(K1, U) U4_GGA(K, N, List, =_out_ag(K1, U)) -> U5_GGA(K, N, List, iota1_in_gga(K1, N, List)) U4_GGA(K, N, List, =_out_ag(K1, U)) -> IOTA1_IN_GGA(K1, N, List) U28_GA(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_GA(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) U28_GA(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> PDS1_IN_GGAG(List, .(zero, []), Ans, N) PDS1_IN_GGAG(Unused, List, .(Choice, Rest), Mod) -> U21_GGAG(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) PDS1_IN_GGAG(Unused, List, .(Choice, Rest), Mod) -> MEMBER_IN_AG(Choice, Unused) MEMBER_IN_AG(X, .(X1, Xs)) -> U1_AG(X, X1, Xs, member_in_ag(X, Xs)) MEMBER_IN_AG(X, .(X1, Xs)) -> MEMBER_IN_AG(X, Xs) U21_GGAG(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_GGAG(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) U21_GGAG(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> CHECK_IN_GGGAG(List, Choice, Unused, U3, Mod) CHECK_IN_GGGAG(S, Choice, Old, L3, Modulus) -> U17_GGGAG(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) CHECK_IN_GGGAG(S, Choice, Old, L3, Modulus) -> DIF_IN_GGGAA(S, Choice, Modulus, Ds, Dds) DIF_IN_GGGAA(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) DIF_IN_GGGAA(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> ISMINUS_IN_GGA(Val, S, U) U6_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U6_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> =_IN_AG(D, U) U7_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) U7_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> ISMINUS_IN_GGG(Mod, D, U) ISMINUS_IN_GGG(zero, succ(Y), pred(Z)) -> U35_GGG(Y, Z, isMinus_in_ggg(zero, Y, Z)) ISMINUS_IN_GGG(zero, succ(Y), pred(Z)) -> ISMINUS_IN_GGG(zero, Y, Z) ISMINUS_IN_GGG(zero, pred(Y), succ(Z)) -> U36_GGG(Y, Z, isMinus_in_ggg(zero, Y, Z)) ISMINUS_IN_GGG(zero, pred(Y), succ(Z)) -> ISMINUS_IN_GGG(zero, Y, Z) ISMINUS_IN_GGG(succ(X), succ(Y), Z) -> U37_GGG(X, Y, Z, isMinus_in_ggg(X, Y, Z)) ISMINUS_IN_GGG(succ(X), succ(Y), Z) -> ISMINUS_IN_GGG(X, Y, Z) ISMINUS_IN_GGG(succ(X), pred(Y), succ(succ(Z))) -> U38_GGG(X, Y, Z, isMinus_in_ggg(X, Y, Z)) ISMINUS_IN_GGG(succ(X), pred(Y), succ(succ(Z))) -> ISMINUS_IN_GGG(X, Y, Z) ISMINUS_IN_GGG(pred(X), succ(Y), pred(pred(Z))) -> U39_GGG(X, Y, Z, isMinus_in_ggg(X, Y, Z)) ISMINUS_IN_GGG(pred(X), succ(Y), pred(pred(Z))) -> ISMINUS_IN_GGG(X, Y, Z) ISMINUS_IN_GGG(pred(X), pred(Y), Z) -> U40_GGG(X, Y, Z, isMinus_in_ggg(X, Y, Z)) ISMINUS_IN_GGG(pred(X), pred(Y), Z) -> ISMINUS_IN_GGG(X, Y, Z) U8_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U8_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> =_IN_AG(D2, U) U9_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U9_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> DIF_IN_GGGAA(Ss, Val, Mod, Ds, D2s) U17_GGGAG(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_GGGAG(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) U17_GGGAG(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> MERGEDELETE_IN_GGA(Ds, Old, L2) MERGEDELETE_IN_GGA(.(D, Ds), .(D, R), L2) -> U12_GGA(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) MERGEDELETE_IN_GGA(.(D, Ds), .(D, R), L2) -> MERGEDELETE_IN_GGA(Ds, R, L2) MERGEDELETE_IN_GGA(.(D, Ds), .(X, R), .(X, L2)) -> U13_GGA(D, Ds, X, R, L2, =_in_ag(X1, D)) MERGEDELETE_IN_GGA(.(D, Ds), .(X, R), .(X, L2)) -> =_IN_AG(X1, D) U13_GGA(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_GGA(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U13_GGA(D, Ds, X, R, L2, =_out_ag(X1, D)) -> =_IN_AG(X2, X) U14_GGA(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_GGA(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) U14_GGA(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> ISGREATER_IN_GG(X1, X2) ISGREATER_IN_GG(succ(X), succ(Y)) -> U64_GG(X, Y, isGreater_in_gg(X, Y)) ISGREATER_IN_GG(succ(X), succ(Y)) -> ISGREATER_IN_GG(X, Y) ISGREATER_IN_GG(pred(X), pred(Y)) -> U65_GG(X, Y, isGreater_in_gg(X, Y)) ISGREATER_IN_GG(pred(X), pred(Y)) -> ISGREATER_IN_GG(X, Y) U15_GGA(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_GGA(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U15_GGA(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> MERGEDELETE_IN_GGA(.(D, Ds), R, L2) U18_GGGAG(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_GGGAG(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) U18_GGGAG(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> REV_IN_GGA(Dds, [], Rds) REV_IN_GGA(.(X, Xs), Y, L) -> U11_GGA(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) REV_IN_GGA(.(X, Xs), Y, L) -> REV_IN_GGA(Xs, .(X, Y), L) U19_GGGAG(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_GGGAG(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U19_GGGAG(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> MERGEDELETE_IN_GGA(Rds, L2, L3) U22_GGAG(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_GGAG(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U22_GGAG(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> PDS1_IN_GGAG(U3, .(Choice, List), Rest, Mod) The TRS R consists of the following rules: pds_in_ga(Order, .(zero, Ans)) -> U24_ga(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) isPlus_in_gga(zero, X, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), zero, succ(X)) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), succ(Y), succ(succ(Z))) -> U31_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(succ(X), pred(Y), Z) -> U32_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), zero, pred(X)) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), succ(Y), Z) -> U33_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), pred(Y), pred(pred(Z))) -> U34_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) U34_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U33_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U32_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U31_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U24_ga(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_ga(Order, Ans, isTimes_in_gga(Order, U, U1)) isTimes_in_gga(X, zero, zero) -> isTimes_out_gga(X, zero, zero) isTimes_in_gga(zero, succ(Y), zero) -> isTimes_out_gga(zero, succ(Y), zero) isTimes_in_gga(zero, pred(Y), zero) -> isTimes_out_gga(zero, pred(Y), zero) isTimes_in_gga(succ(X), succ(Y), Z) -> U41_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(succ(X), pred(Y), Z) -> U43_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(pred(X), succ(Y), Z) -> U45_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) isTimes_in_gga(pred(X), pred(Y), Z) -> U47_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) U47_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_gga(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U48_gga(X, Y, Z, isMinus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), pred(Y), Z) U45_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_gga(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U46_gga(X, Y, Z, isPlus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), succ(Y), Z) U43_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_gga(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U44_gga(X, Y, Z, isMinus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), pred(Y), Z) U41_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_gga(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U42_gga(X, Y, Z, isPlus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), succ(Y), Z) U25_ga(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_ga(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U26_ga(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_ga(Order, Ans, =_in_ag(N, U2)) =_in_ag(X, X) -> =_out_ag(X, X) U27_ga(Order, Ans, =_out_ag(N, U2)) -> U28_ga(Order, Ans, N, iota_in_ga(N, .(zero, List))) iota_in_ga(N, List) -> U2_ga(N, List, iota1_in_gga(zero, N, List)) iota1_in_gga(K, K, []) -> iota1_out_gga(K, K, []) iota1_in_gga(K, N, .(K, List)) -> U3_gga(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_gga(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_gga(K, N, List, =_in_ag(K1, U)) U4_gga(K, N, List, =_out_ag(K1, U)) -> U5_gga(K, N, List, iota1_in_gga(K1, N, List)) U5_gga(K, N, List, iota1_out_gga(K1, N, List)) -> iota1_out_gga(K, N, .(K, List)) U2_ga(N, List, iota1_out_gga(zero, N, List)) -> iota_out_ga(N, List) U28_ga(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_ga(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) pds1_in_ggag([], X6, [], X7) -> pds1_out_ggag([], X6, [], X7) pds1_in_ggag(Unused, List, .(Choice, Rest), Mod) -> U21_ggag(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(X, .(X1, Xs)) -> U1_ag(X, X1, Xs, member_in_ag(X, Xs)) U1_ag(X, X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) U21_ggag(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_ggag(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) check_in_gggag([], X4, L, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, L3, Modulus) -> U17_gggag(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) dif_in_gggaa([], X2, X3, [], []) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U17_gggag(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) mergedelete_in_gga([], L, L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R), L2) -> U12_gga(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) mergedelete_in_gga(.(D, Ds), .(X, R), .(X, L2)) -> U13_gga(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_gga(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_gga(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U15_gga(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U16_gga(D, Ds, X, R, L2, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U12_gga(D, Ds, R, L2, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) rev_in_gga([], L, L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y, L) -> U11_gga(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) U11_gga(X, Xs, Y, L, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U22_ggag(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_ggag(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U23_ggag(Unused, List, Choice, Rest, Mod, pds1_out_ggag(U3, .(Choice, List), Rest, Mod)) -> pds1_out_ggag(Unused, List, .(Choice, Rest), Mod) U29_ga(Order, Ans, pds1_out_ggag(List, .(zero, []), Ans, N)) -> pds_out_ga(Order, .(zero, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U24_ga(x1, x2, x3) = U24_ga(x1, x3) isPlus_in_gga(x1, x2, x3) = isPlus_in_gga(x1, x2) zero = zero isPlus_out_gga(x1, x2, x3) = isPlus_out_gga(x1, x2, x3) succ(x1) = succ(x1) U31_gga(x1, x2, x3, x4) = U31_gga(x1, x2, x4) pred(x1) = pred(x1) U32_gga(x1, x2, x3, x4) = U32_gga(x1, x2, x4) U33_gga(x1, x2, x3, x4) = U33_gga(x1, x2, x4) U34_gga(x1, x2, x3, x4) = U34_gga(x1, x2, x4) U25_ga(x1, x2, x3) = U25_ga(x1, x3) isTimes_in_gga(x1, x2, x3) = isTimes_in_gga(x1, x2) isTimes_out_gga(x1, x2, x3) = isTimes_out_gga(x1, x2, x3) U41_gga(x1, x2, x3, x4) = U41_gga(x1, x2, x4) U43_gga(x1, x2, x3, x4) = U43_gga(x1, x2, x4) U45_gga(x1, x2, x3, x4) = U45_gga(x1, x2, x4) U47_gga(x1, x2, x3, x4) = U47_gga(x1, x2, x4) U48_gga(x1, x2, x3, x4) = U48_gga(x1, x2, x4) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x1, x2, x3) U35_gga(x1, x2, x3) = U35_gga(x1, x3) U36_gga(x1, x2, x3) = U36_gga(x1, x3) U37_gga(x1, x2, x3, x4) = U37_gga(x1, x2, x4) U38_gga(x1, x2, x3, x4) = U38_gga(x1, x2, x4) U39_gga(x1, x2, x3, x4) = U39_gga(x1, x2, x4) U40_gga(x1, x2, x3, x4) = U40_gga(x1, x2, x4) U46_gga(x1, x2, x3, x4) = U46_gga(x1, x2, x4) U44_gga(x1, x2, x3, x4) = U44_gga(x1, x2, x4) U42_gga(x1, x2, x3, x4) = U42_gga(x1, x2, x4) U26_ga(x1, x2, x3) = U26_ga(x1, x3) U27_ga(x1, x2, x3) = U27_ga(x1, x3) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1, x2) U28_ga(x1, x2, x3, x4) = U28_ga(x1, x3, x4) iota_in_ga(x1, x2) = iota_in_ga(x1) U2_ga(x1, x2, x3) = U2_ga(x1, x3) iota1_in_gga(x1, x2, x3) = iota1_in_gga(x1, x2) iota1_out_gga(x1, x2, x3) = iota1_out_gga(x1, x2, x3) U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4) iota_out_ga(x1, x2) = iota_out_ga(x1, x2) .(x1, x2) = .(x1, x2) U29_ga(x1, x2, x3) = U29_ga(x1, x3) pds1_in_ggag(x1, x2, x3, x4) = pds1_in_ggag(x1, x2, x4) [] = [] pds1_out_ggag(x1, x2, x3, x4) = pds1_out_ggag(x1, x2, x3, x4) U21_ggag(x1, x2, x3, x4, x5, x6) = U21_ggag(x1, x2, x5, x6) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1, x2) U1_ag(x1, x2, x3, x4) = U1_ag(x2, x3, x4) U22_ggag(x1, x2, x3, x4, x5, x6) = U22_ggag(x1, x2, x3, x5, x6) check_in_gggag(x1, x2, x3, x4, x5) = check_in_gggag(x1, x2, x3, x5) check_out_gggag(x1, x2, x3, x4, x5) = check_out_gggag(x1, x2, x3, x4, x5) U17_gggag(x1, x2, x3, x4, x5, x6) = U17_gggag(x1, x2, x3, x5, x6) dif_in_gggaa(x1, x2, x3, x4, x5) = dif_in_gggaa(x1, x2, x3) dif_out_gggaa(x1, x2, x3, x4, x5) = dif_out_gggaa(x1, x2, x3, x4, x5) U6_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gggaa(x1, x2, x3, x4, x9) U7_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_gggaa(x1, x2, x3, x4, x9, x10) U8_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_gggaa(x1, x2, x3, x4, x5, x9, x10) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg(x1, x2, x3) U35_ggg(x1, x2, x3) = U35_ggg(x1, x2, x3) U36_ggg(x1, x2, x3) = U36_ggg(x1, x2, x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x1, x2, x3, x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x1, x2, x3, x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x1, x2, x3, x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x1, x2, x3, x4) U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_gggaa(x1, x2, x3, x4, x5, x9) U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_gggaa(x1, x2, x3, x4, x5, x7, x9) U18_gggag(x1, x2, x3, x4, x5, x6, x7) = U18_gggag(x1, x2, x3, x5, x6, x7) mergedelete_in_gga(x1, x2, x3) = mergedelete_in_gga(x1, x2) mergedelete_out_gga(x1, x2, x3) = mergedelete_out_gga(x1, x2, x3) U12_gga(x1, x2, x3, x4, x5) = U12_gga(x1, x2, x3, x5) U13_gga(x1, x2, x3, x4, x5, x6) = U13_gga(x1, x2, x3, x4, x6) U14_gga(x1, x2, x3, x4, x5, x6, x7) = U14_gga(x1, x2, x3, x4, x6, x7) U15_gga(x1, x2, x3, x4, x5, x6) = U15_gga(x1, x2, x3, x4, x6) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg(x1, x2) U64_gg(x1, x2, x3) = U64_gg(x1, x2, x3) U65_gg(x1, x2, x3) = U65_gg(x1, x2, x3) U16_gga(x1, x2, x3, x4, x5, x6) = U16_gga(x1, x2, x3, x4, x6) U19_gggag(x1, x2, x3, x4, x5, x6, x7) = U19_gggag(x1, x2, x3, x5, x6, x7) rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2) rev_out_gga(x1, x2, x3) = rev_out_gga(x1, x2, x3) U11_gga(x1, x2, x3, x4, x5) = U11_gga(x1, x2, x3, x5) U20_gggag(x1, x2, x3, x4, x5, x6) = U20_gggag(x1, x2, x3, x5, x6) U23_ggag(x1, x2, x3, x4, x5, x6) = U23_ggag(x1, x2, x3, x5, x6) pds_out_ga(x1, x2) = pds_out_ga(x1, x2) PDS_IN_GA(x1, x2) = PDS_IN_GA(x1) U24_GA(x1, x2, x3) = U24_GA(x1, x3) ISPLUS_IN_GGA(x1, x2, x3) = ISPLUS_IN_GGA(x1, x2) U31_GGA(x1, x2, x3, x4) = U31_GGA(x1, x2, x4) U32_GGA(x1, x2, x3, x4) = U32_GGA(x1, x2, x4) U33_GGA(x1, x2, x3, x4) = U33_GGA(x1, x2, x4) U34_GGA(x1, x2, x3, x4) = U34_GGA(x1, x2, x4) U25_GA(x1, x2, x3) = U25_GA(x1, x3) ISTIMES_IN_GGA(x1, x2, x3) = ISTIMES_IN_GGA(x1, x2) U41_GGA(x1, x2, x3, x4) = U41_GGA(x1, x2, x4) U43_GGA(x1, x2, x3, x4) = U43_GGA(x1, x2, x4) U45_GGA(x1, x2, x3, x4) = U45_GGA(x1, x2, x4) U47_GGA(x1, x2, x3, x4) = U47_GGA(x1, x2, x4) U48_GGA(x1, x2, x3, x4) = U48_GGA(x1, x2, x4) ISMINUS_IN_GGA(x1, x2, x3) = ISMINUS_IN_GGA(x1, x2) U35_GGA(x1, x2, x3) = U35_GGA(x1, x3) U36_GGA(x1, x2, x3) = U36_GGA(x1, x3) U37_GGA(x1, x2, x3, x4) = U37_GGA(x1, x2, x4) U38_GGA(x1, x2, x3, x4) = U38_GGA(x1, x2, x4) U39_GGA(x1, x2, x3, x4) = U39_GGA(x1, x2, x4) U40_GGA(x1, x2, x3, x4) = U40_GGA(x1, x2, x4) U46_GGA(x1, x2, x3, x4) = U46_GGA(x1, x2, x4) U44_GGA(x1, x2, x3, x4) = U44_GGA(x1, x2, x4) U42_GGA(x1, x2, x3, x4) = U42_GGA(x1, x2, x4) U26_GA(x1, x2, x3) = U26_GA(x1, x3) U27_GA(x1, x2, x3) = U27_GA(x1, x3) =_IN_AG(x1, x2) = =_IN_AG(x2) U28_GA(x1, x2, x3, x4) = U28_GA(x1, x3, x4) IOTA_IN_GA(x1, x2) = IOTA_IN_GA(x1) U2_GA(x1, x2, x3) = U2_GA(x1, x3) IOTA1_IN_GGA(x1, x2, x3) = IOTA1_IN_GGA(x1, x2) U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4) U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4) U5_GGA(x1, x2, x3, x4) = U5_GGA(x1, x2, x4) U29_GA(x1, x2, x3) = U29_GA(x1, x3) PDS1_IN_GGAG(x1, x2, x3, x4) = PDS1_IN_GGAG(x1, x2, x4) U21_GGAG(x1, x2, x3, x4, x5, x6) = U21_GGAG(x1, x2, x5, x6) MEMBER_IN_AG(x1, x2) = MEMBER_IN_AG(x2) U1_AG(x1, x2, x3, x4) = U1_AG(x2, x3, x4) U22_GGAG(x1, x2, x3, x4, x5, x6) = U22_GGAG(x1, x2, x3, x5, x6) CHECK_IN_GGGAG(x1, x2, x3, x4, x5) = CHECK_IN_GGGAG(x1, x2, x3, x5) U17_GGGAG(x1, x2, x3, x4, x5, x6) = U17_GGGAG(x1, x2, x3, x5, x6) DIF_IN_GGGAA(x1, x2, x3, x4, x5) = DIF_IN_GGGAA(x1, x2, x3) U6_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_GGGAA(x1, x2, x3, x4, x9) U7_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_GGGAA(x1, x2, x3, x4, x9, x10) U8_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_GGGAA(x1, x2, x3, x4, x5, x9, x10) ISMINUS_IN_GGG(x1, x2, x3) = ISMINUS_IN_GGG(x1, x2, x3) U35_GGG(x1, x2, x3) = U35_GGG(x1, x2, x3) U36_GGG(x1, x2, x3) = U36_GGG(x1, x2, x3) U37_GGG(x1, x2, x3, x4) = U37_GGG(x1, x2, x3, x4) U38_GGG(x1, x2, x3, x4) = U38_GGG(x1, x2, x3, x4) U39_GGG(x1, x2, x3, x4) = U39_GGG(x1, x2, x3, x4) U40_GGG(x1, x2, x3, x4) = U40_GGG(x1, x2, x3, x4) U9_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_GGGAA(x1, x2, x3, x4, x5, x9) U10_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_GGGAA(x1, x2, x3, x4, x5, x7, x9) U18_GGGAG(x1, x2, x3, x4, x5, x6, x7) = U18_GGGAG(x1, x2, x3, x5, x6, x7) MERGEDELETE_IN_GGA(x1, x2, x3) = MERGEDELETE_IN_GGA(x1, x2) U12_GGA(x1, x2, x3, x4, x5) = U12_GGA(x1, x2, x3, x5) U13_GGA(x1, x2, x3, x4, x5, x6) = U13_GGA(x1, x2, x3, x4, x6) U14_GGA(x1, x2, x3, x4, x5, x6, x7) = U14_GGA(x1, x2, x3, x4, x6, x7) U15_GGA(x1, x2, x3, x4, x5, x6) = U15_GGA(x1, x2, x3, x4, x6) ISGREATER_IN_GG(x1, x2) = ISGREATER_IN_GG(x1, x2) U64_GG(x1, x2, x3) = U64_GG(x1, x2, x3) U65_GG(x1, x2, x3) = U65_GG(x1, x2, x3) U16_GGA(x1, x2, x3, x4, x5, x6) = U16_GGA(x1, x2, x3, x4, x6) U19_GGGAG(x1, x2, x3, x4, x5, x6, x7) = U19_GGGAG(x1, x2, x3, x5, x6, x7) REV_IN_GGA(x1, x2, x3) = REV_IN_GGA(x1, x2) U11_GGA(x1, x2, x3, x4, x5) = U11_GGA(x1, x2, x3, x5) U20_GGGAG(x1, x2, x3, x4, x5, x6) = U20_GGGAG(x1, x2, x3, x5, x6) U23_GGAG(x1, x2, x3, x4, x5, x6) = U23_GGAG(x1, x2, x3, x5, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (112) Obligation: Pi DP problem: The TRS P consists of the following rules: PDS_IN_GA(Order, .(zero, Ans)) -> U24_GA(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) PDS_IN_GA(Order, .(zero, Ans)) -> ISPLUS_IN_GGA(Order, succ(zero), U) ISPLUS_IN_GGA(succ(X), succ(Y), succ(succ(Z))) -> U31_GGA(X, Y, Z, isPlus_in_gga(X, Y, Z)) ISPLUS_IN_GGA(succ(X), succ(Y), succ(succ(Z))) -> ISPLUS_IN_GGA(X, Y, Z) ISPLUS_IN_GGA(succ(X), pred(Y), Z) -> U32_GGA(X, Y, Z, isPlus_in_gga(X, Y, Z)) ISPLUS_IN_GGA(succ(X), pred(Y), Z) -> ISPLUS_IN_GGA(X, Y, Z) ISPLUS_IN_GGA(pred(X), succ(Y), Z) -> U33_GGA(X, Y, Z, isPlus_in_gga(X, Y, Z)) ISPLUS_IN_GGA(pred(X), succ(Y), Z) -> ISPLUS_IN_GGA(X, Y, Z) ISPLUS_IN_GGA(pred(X), pred(Y), pred(pred(Z))) -> U34_GGA(X, Y, Z, isPlus_in_gga(X, Y, Z)) ISPLUS_IN_GGA(pred(X), pred(Y), pred(pred(Z))) -> ISPLUS_IN_GGA(X, Y, Z) U24_GA(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_GA(Order, Ans, isTimes_in_gga(Order, U, U1)) U24_GA(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> ISTIMES_IN_GGA(Order, U, U1) ISTIMES_IN_GGA(succ(X), succ(Y), Z) -> U41_GGA(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) ISTIMES_IN_GGA(succ(X), succ(Y), Z) -> ISTIMES_IN_GGA(succ(X), Y, A) ISTIMES_IN_GGA(succ(X), pred(Y), Z) -> U43_GGA(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) ISTIMES_IN_GGA(succ(X), pred(Y), Z) -> ISTIMES_IN_GGA(succ(X), Y, A) ISTIMES_IN_GGA(pred(X), succ(Y), Z) -> U45_GGA(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) ISTIMES_IN_GGA(pred(X), succ(Y), Z) -> ISTIMES_IN_GGA(pred(X), Y, A) ISTIMES_IN_GGA(pred(X), pred(Y), Z) -> U47_GGA(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) ISTIMES_IN_GGA(pred(X), pred(Y), Z) -> ISTIMES_IN_GGA(pred(X), Y, A) U47_GGA(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_GGA(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) U47_GGA(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> ISMINUS_IN_GGA(A, pred(X), Z) ISMINUS_IN_GGA(zero, succ(Y), pred(Z)) -> U35_GGA(Y, Z, isMinus_in_gga(zero, Y, Z)) ISMINUS_IN_GGA(zero, succ(Y), pred(Z)) -> ISMINUS_IN_GGA(zero, Y, Z) ISMINUS_IN_GGA(zero, pred(Y), succ(Z)) -> U36_GGA(Y, Z, isMinus_in_gga(zero, Y, Z)) ISMINUS_IN_GGA(zero, pred(Y), succ(Z)) -> ISMINUS_IN_GGA(zero, Y, Z) ISMINUS_IN_GGA(succ(X), succ(Y), Z) -> U37_GGA(X, Y, Z, isMinus_in_gga(X, Y, Z)) ISMINUS_IN_GGA(succ(X), succ(Y), Z) -> ISMINUS_IN_GGA(X, Y, Z) ISMINUS_IN_GGA(succ(X), pred(Y), succ(succ(Z))) -> U38_GGA(X, Y, Z, isMinus_in_gga(X, Y, Z)) ISMINUS_IN_GGA(succ(X), pred(Y), succ(succ(Z))) -> ISMINUS_IN_GGA(X, Y, Z) ISMINUS_IN_GGA(pred(X), succ(Y), pred(pred(Z))) -> U39_GGA(X, Y, Z, isMinus_in_gga(X, Y, Z)) ISMINUS_IN_GGA(pred(X), succ(Y), pred(pred(Z))) -> ISMINUS_IN_GGA(X, Y, Z) ISMINUS_IN_GGA(pred(X), pred(Y), Z) -> U40_GGA(X, Y, Z, isMinus_in_gga(X, Y, Z)) ISMINUS_IN_GGA(pred(X), pred(Y), Z) -> ISMINUS_IN_GGA(X, Y, Z) U45_GGA(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_GGA(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U45_GGA(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> ISPLUS_IN_GGA(A, pred(X), Z) U43_GGA(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_GGA(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U43_GGA(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> ISMINUS_IN_GGA(A, succ(X), Z) U41_GGA(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_GGA(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U41_GGA(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> ISPLUS_IN_GGA(A, succ(X), Z) U25_GA(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_GA(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U25_GA(Order, Ans, isTimes_out_gga(Order, U, U1)) -> ISPLUS_IN_GGA(U1, succ(zero), U2) U26_GA(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_GA(Order, Ans, =_in_ag(N, U2)) U26_GA(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> =_IN_AG(N, U2) U27_GA(Order, Ans, =_out_ag(N, U2)) -> U28_GA(Order, Ans, N, iota_in_ga(N, .(zero, List))) U27_GA(Order, Ans, =_out_ag(N, U2)) -> IOTA_IN_GA(N, .(zero, List)) IOTA_IN_GA(N, List) -> U2_GA(N, List, iota1_in_gga(zero, N, List)) IOTA_IN_GA(N, List) -> IOTA1_IN_GGA(zero, N, List) IOTA1_IN_GGA(K, N, .(K, List)) -> U3_GGA(K, N, List, isPlus_in_gga(K, succ(zero), U)) IOTA1_IN_GGA(K, N, .(K, List)) -> ISPLUS_IN_GGA(K, succ(zero), U) U3_GGA(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_GGA(K, N, List, =_in_ag(K1, U)) U3_GGA(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> =_IN_AG(K1, U) U4_GGA(K, N, List, =_out_ag(K1, U)) -> U5_GGA(K, N, List, iota1_in_gga(K1, N, List)) U4_GGA(K, N, List, =_out_ag(K1, U)) -> IOTA1_IN_GGA(K1, N, List) U28_GA(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_GA(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) U28_GA(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> PDS1_IN_GGAG(List, .(zero, []), Ans, N) PDS1_IN_GGAG(Unused, List, .(Choice, Rest), Mod) -> U21_GGAG(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) PDS1_IN_GGAG(Unused, List, .(Choice, Rest), Mod) -> MEMBER_IN_AG(Choice, Unused) MEMBER_IN_AG(X, .(X1, Xs)) -> U1_AG(X, X1, Xs, member_in_ag(X, Xs)) MEMBER_IN_AG(X, .(X1, Xs)) -> MEMBER_IN_AG(X, Xs) U21_GGAG(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_GGAG(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) U21_GGAG(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> CHECK_IN_GGGAG(List, Choice, Unused, U3, Mod) CHECK_IN_GGGAG(S, Choice, Old, L3, Modulus) -> U17_GGGAG(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) CHECK_IN_GGGAG(S, Choice, Old, L3, Modulus) -> DIF_IN_GGGAA(S, Choice, Modulus, Ds, Dds) DIF_IN_GGGAA(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) DIF_IN_GGGAA(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> ISMINUS_IN_GGA(Val, S, U) U6_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U6_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> =_IN_AG(D, U) U7_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) U7_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> ISMINUS_IN_GGG(Mod, D, U) ISMINUS_IN_GGG(zero, succ(Y), pred(Z)) -> U35_GGG(Y, Z, isMinus_in_ggg(zero, Y, Z)) ISMINUS_IN_GGG(zero, succ(Y), pred(Z)) -> ISMINUS_IN_GGG(zero, Y, Z) ISMINUS_IN_GGG(zero, pred(Y), succ(Z)) -> U36_GGG(Y, Z, isMinus_in_ggg(zero, Y, Z)) ISMINUS_IN_GGG(zero, pred(Y), succ(Z)) -> ISMINUS_IN_GGG(zero, Y, Z) ISMINUS_IN_GGG(succ(X), succ(Y), Z) -> U37_GGG(X, Y, Z, isMinus_in_ggg(X, Y, Z)) ISMINUS_IN_GGG(succ(X), succ(Y), Z) -> ISMINUS_IN_GGG(X, Y, Z) ISMINUS_IN_GGG(succ(X), pred(Y), succ(succ(Z))) -> U38_GGG(X, Y, Z, isMinus_in_ggg(X, Y, Z)) ISMINUS_IN_GGG(succ(X), pred(Y), succ(succ(Z))) -> ISMINUS_IN_GGG(X, Y, Z) ISMINUS_IN_GGG(pred(X), succ(Y), pred(pred(Z))) -> U39_GGG(X, Y, Z, isMinus_in_ggg(X, Y, Z)) ISMINUS_IN_GGG(pred(X), succ(Y), pred(pred(Z))) -> ISMINUS_IN_GGG(X, Y, Z) ISMINUS_IN_GGG(pred(X), pred(Y), Z) -> U40_GGG(X, Y, Z, isMinus_in_ggg(X, Y, Z)) ISMINUS_IN_GGG(pred(X), pred(Y), Z) -> ISMINUS_IN_GGG(X, Y, Z) U8_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U8_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> =_IN_AG(D2, U) U9_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U9_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> DIF_IN_GGGAA(Ss, Val, Mod, Ds, D2s) U17_GGGAG(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_GGGAG(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) U17_GGGAG(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> MERGEDELETE_IN_GGA(Ds, Old, L2) MERGEDELETE_IN_GGA(.(D, Ds), .(D, R), L2) -> U12_GGA(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) MERGEDELETE_IN_GGA(.(D, Ds), .(D, R), L2) -> MERGEDELETE_IN_GGA(Ds, R, L2) MERGEDELETE_IN_GGA(.(D, Ds), .(X, R), .(X, L2)) -> U13_GGA(D, Ds, X, R, L2, =_in_ag(X1, D)) MERGEDELETE_IN_GGA(.(D, Ds), .(X, R), .(X, L2)) -> =_IN_AG(X1, D) U13_GGA(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_GGA(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U13_GGA(D, Ds, X, R, L2, =_out_ag(X1, D)) -> =_IN_AG(X2, X) U14_GGA(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_GGA(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) U14_GGA(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> ISGREATER_IN_GG(X1, X2) ISGREATER_IN_GG(succ(X), succ(Y)) -> U64_GG(X, Y, isGreater_in_gg(X, Y)) ISGREATER_IN_GG(succ(X), succ(Y)) -> ISGREATER_IN_GG(X, Y) ISGREATER_IN_GG(pred(X), pred(Y)) -> U65_GG(X, Y, isGreater_in_gg(X, Y)) ISGREATER_IN_GG(pred(X), pred(Y)) -> ISGREATER_IN_GG(X, Y) U15_GGA(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_GGA(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U15_GGA(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> MERGEDELETE_IN_GGA(.(D, Ds), R, L2) U18_GGGAG(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_GGGAG(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) U18_GGGAG(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> REV_IN_GGA(Dds, [], Rds) REV_IN_GGA(.(X, Xs), Y, L) -> U11_GGA(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) REV_IN_GGA(.(X, Xs), Y, L) -> REV_IN_GGA(Xs, .(X, Y), L) U19_GGGAG(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_GGGAG(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U19_GGGAG(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> MERGEDELETE_IN_GGA(Rds, L2, L3) U22_GGAG(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_GGAG(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U22_GGAG(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> PDS1_IN_GGAG(U3, .(Choice, List), Rest, Mod) The TRS R consists of the following rules: pds_in_ga(Order, .(zero, Ans)) -> U24_ga(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) isPlus_in_gga(zero, X, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), zero, succ(X)) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), succ(Y), succ(succ(Z))) -> U31_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(succ(X), pred(Y), Z) -> U32_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), zero, pred(X)) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), succ(Y), Z) -> U33_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), pred(Y), pred(pred(Z))) -> U34_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) U34_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U33_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U32_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U31_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U24_ga(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_ga(Order, Ans, isTimes_in_gga(Order, U, U1)) isTimes_in_gga(X, zero, zero) -> isTimes_out_gga(X, zero, zero) isTimes_in_gga(zero, succ(Y), zero) -> isTimes_out_gga(zero, succ(Y), zero) isTimes_in_gga(zero, pred(Y), zero) -> isTimes_out_gga(zero, pred(Y), zero) isTimes_in_gga(succ(X), succ(Y), Z) -> U41_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(succ(X), pred(Y), Z) -> U43_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(pred(X), succ(Y), Z) -> U45_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) isTimes_in_gga(pred(X), pred(Y), Z) -> U47_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) U47_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_gga(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U48_gga(X, Y, Z, isMinus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), pred(Y), Z) U45_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_gga(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U46_gga(X, Y, Z, isPlus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), succ(Y), Z) U43_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_gga(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U44_gga(X, Y, Z, isMinus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), pred(Y), Z) U41_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_gga(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U42_gga(X, Y, Z, isPlus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), succ(Y), Z) U25_ga(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_ga(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U26_ga(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_ga(Order, Ans, =_in_ag(N, U2)) =_in_ag(X, X) -> =_out_ag(X, X) U27_ga(Order, Ans, =_out_ag(N, U2)) -> U28_ga(Order, Ans, N, iota_in_ga(N, .(zero, List))) iota_in_ga(N, List) -> U2_ga(N, List, iota1_in_gga(zero, N, List)) iota1_in_gga(K, K, []) -> iota1_out_gga(K, K, []) iota1_in_gga(K, N, .(K, List)) -> U3_gga(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_gga(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_gga(K, N, List, =_in_ag(K1, U)) U4_gga(K, N, List, =_out_ag(K1, U)) -> U5_gga(K, N, List, iota1_in_gga(K1, N, List)) U5_gga(K, N, List, iota1_out_gga(K1, N, List)) -> iota1_out_gga(K, N, .(K, List)) U2_ga(N, List, iota1_out_gga(zero, N, List)) -> iota_out_ga(N, List) U28_ga(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_ga(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) pds1_in_ggag([], X6, [], X7) -> pds1_out_ggag([], X6, [], X7) pds1_in_ggag(Unused, List, .(Choice, Rest), Mod) -> U21_ggag(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(X, .(X1, Xs)) -> U1_ag(X, X1, Xs, member_in_ag(X, Xs)) U1_ag(X, X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) U21_ggag(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_ggag(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) check_in_gggag([], X4, L, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, L3, Modulus) -> U17_gggag(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) dif_in_gggaa([], X2, X3, [], []) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U17_gggag(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) mergedelete_in_gga([], L, L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R), L2) -> U12_gga(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) mergedelete_in_gga(.(D, Ds), .(X, R), .(X, L2)) -> U13_gga(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_gga(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_gga(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U15_gga(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U16_gga(D, Ds, X, R, L2, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U12_gga(D, Ds, R, L2, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) rev_in_gga([], L, L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y, L) -> U11_gga(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) U11_gga(X, Xs, Y, L, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U22_ggag(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_ggag(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U23_ggag(Unused, List, Choice, Rest, Mod, pds1_out_ggag(U3, .(Choice, List), Rest, Mod)) -> pds1_out_ggag(Unused, List, .(Choice, Rest), Mod) U29_ga(Order, Ans, pds1_out_ggag(List, .(zero, []), Ans, N)) -> pds_out_ga(Order, .(zero, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U24_ga(x1, x2, x3) = U24_ga(x1, x3) isPlus_in_gga(x1, x2, x3) = isPlus_in_gga(x1, x2) zero = zero isPlus_out_gga(x1, x2, x3) = isPlus_out_gga(x1, x2, x3) succ(x1) = succ(x1) U31_gga(x1, x2, x3, x4) = U31_gga(x1, x2, x4) pred(x1) = pred(x1) U32_gga(x1, x2, x3, x4) = U32_gga(x1, x2, x4) U33_gga(x1, x2, x3, x4) = U33_gga(x1, x2, x4) U34_gga(x1, x2, x3, x4) = U34_gga(x1, x2, x4) U25_ga(x1, x2, x3) = U25_ga(x1, x3) isTimes_in_gga(x1, x2, x3) = isTimes_in_gga(x1, x2) isTimes_out_gga(x1, x2, x3) = isTimes_out_gga(x1, x2, x3) U41_gga(x1, x2, x3, x4) = U41_gga(x1, x2, x4) U43_gga(x1, x2, x3, x4) = U43_gga(x1, x2, x4) U45_gga(x1, x2, x3, x4) = U45_gga(x1, x2, x4) U47_gga(x1, x2, x3, x4) = U47_gga(x1, x2, x4) U48_gga(x1, x2, x3, x4) = U48_gga(x1, x2, x4) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x1, x2, x3) U35_gga(x1, x2, x3) = U35_gga(x1, x3) U36_gga(x1, x2, x3) = U36_gga(x1, x3) U37_gga(x1, x2, x3, x4) = U37_gga(x1, x2, x4) U38_gga(x1, x2, x3, x4) = U38_gga(x1, x2, x4) U39_gga(x1, x2, x3, x4) = U39_gga(x1, x2, x4) U40_gga(x1, x2, x3, x4) = U40_gga(x1, x2, x4) U46_gga(x1, x2, x3, x4) = U46_gga(x1, x2, x4) U44_gga(x1, x2, x3, x4) = U44_gga(x1, x2, x4) U42_gga(x1, x2, x3, x4) = U42_gga(x1, x2, x4) U26_ga(x1, x2, x3) = U26_ga(x1, x3) U27_ga(x1, x2, x3) = U27_ga(x1, x3) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1, x2) U28_ga(x1, x2, x3, x4) = U28_ga(x1, x3, x4) iota_in_ga(x1, x2) = iota_in_ga(x1) U2_ga(x1, x2, x3) = U2_ga(x1, x3) iota1_in_gga(x1, x2, x3) = iota1_in_gga(x1, x2) iota1_out_gga(x1, x2, x3) = iota1_out_gga(x1, x2, x3) U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4) iota_out_ga(x1, x2) = iota_out_ga(x1, x2) .(x1, x2) = .(x1, x2) U29_ga(x1, x2, x3) = U29_ga(x1, x3) pds1_in_ggag(x1, x2, x3, x4) = pds1_in_ggag(x1, x2, x4) [] = [] pds1_out_ggag(x1, x2, x3, x4) = pds1_out_ggag(x1, x2, x3, x4) U21_ggag(x1, x2, x3, x4, x5, x6) = U21_ggag(x1, x2, x5, x6) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1, x2) U1_ag(x1, x2, x3, x4) = U1_ag(x2, x3, x4) U22_ggag(x1, x2, x3, x4, x5, x6) = U22_ggag(x1, x2, x3, x5, x6) check_in_gggag(x1, x2, x3, x4, x5) = check_in_gggag(x1, x2, x3, x5) check_out_gggag(x1, x2, x3, x4, x5) = check_out_gggag(x1, x2, x3, x4, x5) U17_gggag(x1, x2, x3, x4, x5, x6) = U17_gggag(x1, x2, x3, x5, x6) dif_in_gggaa(x1, x2, x3, x4, x5) = dif_in_gggaa(x1, x2, x3) dif_out_gggaa(x1, x2, x3, x4, x5) = dif_out_gggaa(x1, x2, x3, x4, x5) U6_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gggaa(x1, x2, x3, x4, x9) U7_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_gggaa(x1, x2, x3, x4, x9, x10) U8_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_gggaa(x1, x2, x3, x4, x5, x9, x10) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg(x1, x2, x3) U35_ggg(x1, x2, x3) = U35_ggg(x1, x2, x3) U36_ggg(x1, x2, x3) = U36_ggg(x1, x2, x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x1, x2, x3, x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x1, x2, x3, x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x1, x2, x3, x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x1, x2, x3, x4) U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_gggaa(x1, x2, x3, x4, x5, x9) U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_gggaa(x1, x2, x3, x4, x5, x7, x9) U18_gggag(x1, x2, x3, x4, x5, x6, x7) = U18_gggag(x1, x2, x3, x5, x6, x7) mergedelete_in_gga(x1, x2, x3) = mergedelete_in_gga(x1, x2) mergedelete_out_gga(x1, x2, x3) = mergedelete_out_gga(x1, x2, x3) U12_gga(x1, x2, x3, x4, x5) = U12_gga(x1, x2, x3, x5) U13_gga(x1, x2, x3, x4, x5, x6) = U13_gga(x1, x2, x3, x4, x6) U14_gga(x1, x2, x3, x4, x5, x6, x7) = U14_gga(x1, x2, x3, x4, x6, x7) U15_gga(x1, x2, x3, x4, x5, x6) = U15_gga(x1, x2, x3, x4, x6) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg(x1, x2) U64_gg(x1, x2, x3) = U64_gg(x1, x2, x3) U65_gg(x1, x2, x3) = U65_gg(x1, x2, x3) U16_gga(x1, x2, x3, x4, x5, x6) = U16_gga(x1, x2, x3, x4, x6) U19_gggag(x1, x2, x3, x4, x5, x6, x7) = U19_gggag(x1, x2, x3, x5, x6, x7) rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2) rev_out_gga(x1, x2, x3) = rev_out_gga(x1, x2, x3) U11_gga(x1, x2, x3, x4, x5) = U11_gga(x1, x2, x3, x5) U20_gggag(x1, x2, x3, x4, x5, x6) = U20_gggag(x1, x2, x3, x5, x6) U23_ggag(x1, x2, x3, x4, x5, x6) = U23_ggag(x1, x2, x3, x5, x6) pds_out_ga(x1, x2) = pds_out_ga(x1, x2) PDS_IN_GA(x1, x2) = PDS_IN_GA(x1) U24_GA(x1, x2, x3) = U24_GA(x1, x3) ISPLUS_IN_GGA(x1, x2, x3) = ISPLUS_IN_GGA(x1, x2) U31_GGA(x1, x2, x3, x4) = U31_GGA(x1, x2, x4) U32_GGA(x1, x2, x3, x4) = U32_GGA(x1, x2, x4) U33_GGA(x1, x2, x3, x4) = U33_GGA(x1, x2, x4) U34_GGA(x1, x2, x3, x4) = U34_GGA(x1, x2, x4) U25_GA(x1, x2, x3) = U25_GA(x1, x3) ISTIMES_IN_GGA(x1, x2, x3) = ISTIMES_IN_GGA(x1, x2) U41_GGA(x1, x2, x3, x4) = U41_GGA(x1, x2, x4) U43_GGA(x1, x2, x3, x4) = U43_GGA(x1, x2, x4) U45_GGA(x1, x2, x3, x4) = U45_GGA(x1, x2, x4) U47_GGA(x1, x2, x3, x4) = U47_GGA(x1, x2, x4) U48_GGA(x1, x2, x3, x4) = U48_GGA(x1, x2, x4) ISMINUS_IN_GGA(x1, x2, x3) = ISMINUS_IN_GGA(x1, x2) U35_GGA(x1, x2, x3) = U35_GGA(x1, x3) U36_GGA(x1, x2, x3) = U36_GGA(x1, x3) U37_GGA(x1, x2, x3, x4) = U37_GGA(x1, x2, x4) U38_GGA(x1, x2, x3, x4) = U38_GGA(x1, x2, x4) U39_GGA(x1, x2, x3, x4) = U39_GGA(x1, x2, x4) U40_GGA(x1, x2, x3, x4) = U40_GGA(x1, x2, x4) U46_GGA(x1, x2, x3, x4) = U46_GGA(x1, x2, x4) U44_GGA(x1, x2, x3, x4) = U44_GGA(x1, x2, x4) U42_GGA(x1, x2, x3, x4) = U42_GGA(x1, x2, x4) U26_GA(x1, x2, x3) = U26_GA(x1, x3) U27_GA(x1, x2, x3) = U27_GA(x1, x3) =_IN_AG(x1, x2) = =_IN_AG(x2) U28_GA(x1, x2, x3, x4) = U28_GA(x1, x3, x4) IOTA_IN_GA(x1, x2) = IOTA_IN_GA(x1) U2_GA(x1, x2, x3) = U2_GA(x1, x3) IOTA1_IN_GGA(x1, x2, x3) = IOTA1_IN_GGA(x1, x2) U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4) U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4) U5_GGA(x1, x2, x3, x4) = U5_GGA(x1, x2, x4) U29_GA(x1, x2, x3) = U29_GA(x1, x3) PDS1_IN_GGAG(x1, x2, x3, x4) = PDS1_IN_GGAG(x1, x2, x4) U21_GGAG(x1, x2, x3, x4, x5, x6) = U21_GGAG(x1, x2, x5, x6) MEMBER_IN_AG(x1, x2) = MEMBER_IN_AG(x2) U1_AG(x1, x2, x3, x4) = U1_AG(x2, x3, x4) U22_GGAG(x1, x2, x3, x4, x5, x6) = U22_GGAG(x1, x2, x3, x5, x6) CHECK_IN_GGGAG(x1, x2, x3, x4, x5) = CHECK_IN_GGGAG(x1, x2, x3, x5) U17_GGGAG(x1, x2, x3, x4, x5, x6) = U17_GGGAG(x1, x2, x3, x5, x6) DIF_IN_GGGAA(x1, x2, x3, x4, x5) = DIF_IN_GGGAA(x1, x2, x3) U6_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_GGGAA(x1, x2, x3, x4, x9) U7_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_GGGAA(x1, x2, x3, x4, x9, x10) U8_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_GGGAA(x1, x2, x3, x4, x5, x9, x10) ISMINUS_IN_GGG(x1, x2, x3) = ISMINUS_IN_GGG(x1, x2, x3) U35_GGG(x1, x2, x3) = U35_GGG(x1, x2, x3) U36_GGG(x1, x2, x3) = U36_GGG(x1, x2, x3) U37_GGG(x1, x2, x3, x4) = U37_GGG(x1, x2, x3, x4) U38_GGG(x1, x2, x3, x4) = U38_GGG(x1, x2, x3, x4) U39_GGG(x1, x2, x3, x4) = U39_GGG(x1, x2, x3, x4) U40_GGG(x1, x2, x3, x4) = U40_GGG(x1, x2, x3, x4) U9_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_GGGAA(x1, x2, x3, x4, x5, x9) U10_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_GGGAA(x1, x2, x3, x4, x5, x7, x9) U18_GGGAG(x1, x2, x3, x4, x5, x6, x7) = U18_GGGAG(x1, x2, x3, x5, x6, x7) MERGEDELETE_IN_GGA(x1, x2, x3) = MERGEDELETE_IN_GGA(x1, x2) U12_GGA(x1, x2, x3, x4, x5) = U12_GGA(x1, x2, x3, x5) U13_GGA(x1, x2, x3, x4, x5, x6) = U13_GGA(x1, x2, x3, x4, x6) U14_GGA(x1, x2, x3, x4, x5, x6, x7) = U14_GGA(x1, x2, x3, x4, x6, x7) U15_GGA(x1, x2, x3, x4, x5, x6) = U15_GGA(x1, x2, x3, x4, x6) ISGREATER_IN_GG(x1, x2) = ISGREATER_IN_GG(x1, x2) U64_GG(x1, x2, x3) = U64_GG(x1, x2, x3) U65_GG(x1, x2, x3) = U65_GG(x1, x2, x3) U16_GGA(x1, x2, x3, x4, x5, x6) = U16_GGA(x1, x2, x3, x4, x6) U19_GGGAG(x1, x2, x3, x4, x5, x6, x7) = U19_GGGAG(x1, x2, x3, x5, x6, x7) REV_IN_GGA(x1, x2, x3) = REV_IN_GGA(x1, x2) U11_GGA(x1, x2, x3, x4, x5) = U11_GGA(x1, x2, x3, x5) U20_GGGAG(x1, x2, x3, x4, x5, x6) = U20_GGGAG(x1, x2, x3, x5, x6) U23_GGAG(x1, x2, x3, x4, x5, x6) = U23_GGAG(x1, x2, x3, x5, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (113) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 14 SCCs with 70 less nodes. ---------------------------------------- (114) Complex Obligation (AND) ---------------------------------------- (115) Obligation: Pi DP problem: The TRS P consists of the following rules: REV_IN_GGA(.(X, Xs), Y, L) -> REV_IN_GGA(Xs, .(X, Y), L) The TRS R consists of the following rules: pds_in_ga(Order, .(zero, Ans)) -> U24_ga(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) isPlus_in_gga(zero, X, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), zero, succ(X)) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), succ(Y), succ(succ(Z))) -> U31_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(succ(X), pred(Y), Z) -> U32_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), zero, pred(X)) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), succ(Y), Z) -> U33_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), pred(Y), pred(pred(Z))) -> U34_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) U34_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U33_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U32_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U31_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U24_ga(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_ga(Order, Ans, isTimes_in_gga(Order, U, U1)) isTimes_in_gga(X, zero, zero) -> isTimes_out_gga(X, zero, zero) isTimes_in_gga(zero, succ(Y), zero) -> isTimes_out_gga(zero, succ(Y), zero) isTimes_in_gga(zero, pred(Y), zero) -> isTimes_out_gga(zero, pred(Y), zero) isTimes_in_gga(succ(X), succ(Y), Z) -> U41_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(succ(X), pred(Y), Z) -> U43_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(pred(X), succ(Y), Z) -> U45_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) isTimes_in_gga(pred(X), pred(Y), Z) -> U47_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) U47_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_gga(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U48_gga(X, Y, Z, isMinus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), pred(Y), Z) U45_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_gga(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U46_gga(X, Y, Z, isPlus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), succ(Y), Z) U43_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_gga(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U44_gga(X, Y, Z, isMinus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), pred(Y), Z) U41_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_gga(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U42_gga(X, Y, Z, isPlus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), succ(Y), Z) U25_ga(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_ga(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U26_ga(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_ga(Order, Ans, =_in_ag(N, U2)) =_in_ag(X, X) -> =_out_ag(X, X) U27_ga(Order, Ans, =_out_ag(N, U2)) -> U28_ga(Order, Ans, N, iota_in_ga(N, .(zero, List))) iota_in_ga(N, List) -> U2_ga(N, List, iota1_in_gga(zero, N, List)) iota1_in_gga(K, K, []) -> iota1_out_gga(K, K, []) iota1_in_gga(K, N, .(K, List)) -> U3_gga(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_gga(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_gga(K, N, List, =_in_ag(K1, U)) U4_gga(K, N, List, =_out_ag(K1, U)) -> U5_gga(K, N, List, iota1_in_gga(K1, N, List)) U5_gga(K, N, List, iota1_out_gga(K1, N, List)) -> iota1_out_gga(K, N, .(K, List)) U2_ga(N, List, iota1_out_gga(zero, N, List)) -> iota_out_ga(N, List) U28_ga(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_ga(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) pds1_in_ggag([], X6, [], X7) -> pds1_out_ggag([], X6, [], X7) pds1_in_ggag(Unused, List, .(Choice, Rest), Mod) -> U21_ggag(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(X, .(X1, Xs)) -> U1_ag(X, X1, Xs, member_in_ag(X, Xs)) U1_ag(X, X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) U21_ggag(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_ggag(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) check_in_gggag([], X4, L, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, L3, Modulus) -> U17_gggag(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) dif_in_gggaa([], X2, X3, [], []) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U17_gggag(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) mergedelete_in_gga([], L, L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R), L2) -> U12_gga(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) mergedelete_in_gga(.(D, Ds), .(X, R), .(X, L2)) -> U13_gga(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_gga(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_gga(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U15_gga(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U16_gga(D, Ds, X, R, L2, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U12_gga(D, Ds, R, L2, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) rev_in_gga([], L, L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y, L) -> U11_gga(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) U11_gga(X, Xs, Y, L, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U22_ggag(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_ggag(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U23_ggag(Unused, List, Choice, Rest, Mod, pds1_out_ggag(U3, .(Choice, List), Rest, Mod)) -> pds1_out_ggag(Unused, List, .(Choice, Rest), Mod) U29_ga(Order, Ans, pds1_out_ggag(List, .(zero, []), Ans, N)) -> pds_out_ga(Order, .(zero, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U24_ga(x1, x2, x3) = U24_ga(x1, x3) isPlus_in_gga(x1, x2, x3) = isPlus_in_gga(x1, x2) zero = zero isPlus_out_gga(x1, x2, x3) = isPlus_out_gga(x1, x2, x3) succ(x1) = succ(x1) U31_gga(x1, x2, x3, x4) = U31_gga(x1, x2, x4) pred(x1) = pred(x1) U32_gga(x1, x2, x3, x4) = U32_gga(x1, x2, x4) U33_gga(x1, x2, x3, x4) = U33_gga(x1, x2, x4) U34_gga(x1, x2, x3, x4) = U34_gga(x1, x2, x4) U25_ga(x1, x2, x3) = U25_ga(x1, x3) isTimes_in_gga(x1, x2, x3) = isTimes_in_gga(x1, x2) isTimes_out_gga(x1, x2, x3) = isTimes_out_gga(x1, x2, x3) U41_gga(x1, x2, x3, x4) = U41_gga(x1, x2, x4) U43_gga(x1, x2, x3, x4) = U43_gga(x1, x2, x4) U45_gga(x1, x2, x3, x4) = U45_gga(x1, x2, x4) U47_gga(x1, x2, x3, x4) = U47_gga(x1, x2, x4) U48_gga(x1, x2, x3, x4) = U48_gga(x1, x2, x4) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x1, x2, x3) U35_gga(x1, x2, x3) = U35_gga(x1, x3) U36_gga(x1, x2, x3) = U36_gga(x1, x3) U37_gga(x1, x2, x3, x4) = U37_gga(x1, x2, x4) U38_gga(x1, x2, x3, x4) = U38_gga(x1, x2, x4) U39_gga(x1, x2, x3, x4) = U39_gga(x1, x2, x4) U40_gga(x1, x2, x3, x4) = U40_gga(x1, x2, x4) U46_gga(x1, x2, x3, x4) = U46_gga(x1, x2, x4) U44_gga(x1, x2, x3, x4) = U44_gga(x1, x2, x4) U42_gga(x1, x2, x3, x4) = U42_gga(x1, x2, x4) U26_ga(x1, x2, x3) = U26_ga(x1, x3) U27_ga(x1, x2, x3) = U27_ga(x1, x3) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1, x2) U28_ga(x1, x2, x3, x4) = U28_ga(x1, x3, x4) iota_in_ga(x1, x2) = iota_in_ga(x1) U2_ga(x1, x2, x3) = U2_ga(x1, x3) iota1_in_gga(x1, x2, x3) = iota1_in_gga(x1, x2) iota1_out_gga(x1, x2, x3) = iota1_out_gga(x1, x2, x3) U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4) iota_out_ga(x1, x2) = iota_out_ga(x1, x2) .(x1, x2) = .(x1, x2) U29_ga(x1, x2, x3) = U29_ga(x1, x3) pds1_in_ggag(x1, x2, x3, x4) = pds1_in_ggag(x1, x2, x4) [] = [] pds1_out_ggag(x1, x2, x3, x4) = pds1_out_ggag(x1, x2, x3, x4) U21_ggag(x1, x2, x3, x4, x5, x6) = U21_ggag(x1, x2, x5, x6) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1, x2) U1_ag(x1, x2, x3, x4) = U1_ag(x2, x3, x4) U22_ggag(x1, x2, x3, x4, x5, x6) = U22_ggag(x1, x2, x3, x5, x6) check_in_gggag(x1, x2, x3, x4, x5) = check_in_gggag(x1, x2, x3, x5) check_out_gggag(x1, x2, x3, x4, x5) = check_out_gggag(x1, x2, x3, x4, x5) U17_gggag(x1, x2, x3, x4, x5, x6) = U17_gggag(x1, x2, x3, x5, x6) dif_in_gggaa(x1, x2, x3, x4, x5) = dif_in_gggaa(x1, x2, x3) dif_out_gggaa(x1, x2, x3, x4, x5) = dif_out_gggaa(x1, x2, x3, x4, x5) U6_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gggaa(x1, x2, x3, x4, x9) U7_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_gggaa(x1, x2, x3, x4, x9, x10) U8_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_gggaa(x1, x2, x3, x4, x5, x9, x10) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg(x1, x2, x3) U35_ggg(x1, x2, x3) = U35_ggg(x1, x2, x3) U36_ggg(x1, x2, x3) = U36_ggg(x1, x2, x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x1, x2, x3, x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x1, x2, x3, x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x1, x2, x3, x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x1, x2, x3, x4) U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_gggaa(x1, x2, x3, x4, x5, x9) U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_gggaa(x1, x2, x3, x4, x5, x7, x9) U18_gggag(x1, x2, x3, x4, x5, x6, x7) = U18_gggag(x1, x2, x3, x5, x6, x7) mergedelete_in_gga(x1, x2, x3) = mergedelete_in_gga(x1, x2) mergedelete_out_gga(x1, x2, x3) = mergedelete_out_gga(x1, x2, x3) U12_gga(x1, x2, x3, x4, x5) = U12_gga(x1, x2, x3, x5) U13_gga(x1, x2, x3, x4, x5, x6) = U13_gga(x1, x2, x3, x4, x6) U14_gga(x1, x2, x3, x4, x5, x6, x7) = U14_gga(x1, x2, x3, x4, x6, x7) U15_gga(x1, x2, x3, x4, x5, x6) = U15_gga(x1, x2, x3, x4, x6) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg(x1, x2) U64_gg(x1, x2, x3) = U64_gg(x1, x2, x3) U65_gg(x1, x2, x3) = U65_gg(x1, x2, x3) U16_gga(x1, x2, x3, x4, x5, x6) = U16_gga(x1, x2, x3, x4, x6) U19_gggag(x1, x2, x3, x4, x5, x6, x7) = U19_gggag(x1, x2, x3, x5, x6, x7) rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2) rev_out_gga(x1, x2, x3) = rev_out_gga(x1, x2, x3) U11_gga(x1, x2, x3, x4, x5) = U11_gga(x1, x2, x3, x5) U20_gggag(x1, x2, x3, x4, x5, x6) = U20_gggag(x1, x2, x3, x5, x6) U23_ggag(x1, x2, x3, x4, x5, x6) = U23_ggag(x1, x2, x3, x5, x6) pds_out_ga(x1, x2) = pds_out_ga(x1, x2) REV_IN_GGA(x1, x2, x3) = REV_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (116) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (117) Obligation: Pi DP problem: The TRS P consists of the following rules: REV_IN_GGA(.(X, Xs), Y, L) -> REV_IN_GGA(Xs, .(X, Y), L) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) REV_IN_GGA(x1, x2, x3) = REV_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (118) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (119) Obligation: Q DP problem: The TRS P consists of the following rules: REV_IN_GGA(.(X, Xs), Y) -> REV_IN_GGA(Xs, .(X, Y)) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (120) 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: *REV_IN_GGA(.(X, Xs), Y) -> REV_IN_GGA(Xs, .(X, Y)) The graph contains the following edges 1 > 1 ---------------------------------------- (121) YES ---------------------------------------- (122) Obligation: Pi DP problem: The TRS P consists of the following rules: ISGREATER_IN_GG(pred(X), pred(Y)) -> ISGREATER_IN_GG(X, Y) ISGREATER_IN_GG(succ(X), succ(Y)) -> ISGREATER_IN_GG(X, Y) The TRS R consists of the following rules: pds_in_ga(Order, .(zero, Ans)) -> U24_ga(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) isPlus_in_gga(zero, X, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), zero, succ(X)) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), succ(Y), succ(succ(Z))) -> U31_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(succ(X), pred(Y), Z) -> U32_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), zero, pred(X)) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), succ(Y), Z) -> U33_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), pred(Y), pred(pred(Z))) -> U34_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) U34_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U33_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U32_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U31_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U24_ga(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_ga(Order, Ans, isTimes_in_gga(Order, U, U1)) isTimes_in_gga(X, zero, zero) -> isTimes_out_gga(X, zero, zero) isTimes_in_gga(zero, succ(Y), zero) -> isTimes_out_gga(zero, succ(Y), zero) isTimes_in_gga(zero, pred(Y), zero) -> isTimes_out_gga(zero, pred(Y), zero) isTimes_in_gga(succ(X), succ(Y), Z) -> U41_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(succ(X), pred(Y), Z) -> U43_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(pred(X), succ(Y), Z) -> U45_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) isTimes_in_gga(pred(X), pred(Y), Z) -> U47_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) U47_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_gga(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U48_gga(X, Y, Z, isMinus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), pred(Y), Z) U45_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_gga(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U46_gga(X, Y, Z, isPlus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), succ(Y), Z) U43_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_gga(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U44_gga(X, Y, Z, isMinus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), pred(Y), Z) U41_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_gga(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U42_gga(X, Y, Z, isPlus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), succ(Y), Z) U25_ga(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_ga(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U26_ga(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_ga(Order, Ans, =_in_ag(N, U2)) =_in_ag(X, X) -> =_out_ag(X, X) U27_ga(Order, Ans, =_out_ag(N, U2)) -> U28_ga(Order, Ans, N, iota_in_ga(N, .(zero, List))) iota_in_ga(N, List) -> U2_ga(N, List, iota1_in_gga(zero, N, List)) iota1_in_gga(K, K, []) -> iota1_out_gga(K, K, []) iota1_in_gga(K, N, .(K, List)) -> U3_gga(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_gga(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_gga(K, N, List, =_in_ag(K1, U)) U4_gga(K, N, List, =_out_ag(K1, U)) -> U5_gga(K, N, List, iota1_in_gga(K1, N, List)) U5_gga(K, N, List, iota1_out_gga(K1, N, List)) -> iota1_out_gga(K, N, .(K, List)) U2_ga(N, List, iota1_out_gga(zero, N, List)) -> iota_out_ga(N, List) U28_ga(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_ga(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) pds1_in_ggag([], X6, [], X7) -> pds1_out_ggag([], X6, [], X7) pds1_in_ggag(Unused, List, .(Choice, Rest), Mod) -> U21_ggag(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(X, .(X1, Xs)) -> U1_ag(X, X1, Xs, member_in_ag(X, Xs)) U1_ag(X, X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) U21_ggag(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_ggag(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) check_in_gggag([], X4, L, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, L3, Modulus) -> U17_gggag(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) dif_in_gggaa([], X2, X3, [], []) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U17_gggag(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) mergedelete_in_gga([], L, L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R), L2) -> U12_gga(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) mergedelete_in_gga(.(D, Ds), .(X, R), .(X, L2)) -> U13_gga(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_gga(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_gga(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U15_gga(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U16_gga(D, Ds, X, R, L2, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U12_gga(D, Ds, R, L2, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) rev_in_gga([], L, L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y, L) -> U11_gga(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) U11_gga(X, Xs, Y, L, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U22_ggag(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_ggag(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U23_ggag(Unused, List, Choice, Rest, Mod, pds1_out_ggag(U3, .(Choice, List), Rest, Mod)) -> pds1_out_ggag(Unused, List, .(Choice, Rest), Mod) U29_ga(Order, Ans, pds1_out_ggag(List, .(zero, []), Ans, N)) -> pds_out_ga(Order, .(zero, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U24_ga(x1, x2, x3) = U24_ga(x1, x3) isPlus_in_gga(x1, x2, x3) = isPlus_in_gga(x1, x2) zero = zero isPlus_out_gga(x1, x2, x3) = isPlus_out_gga(x1, x2, x3) succ(x1) = succ(x1) U31_gga(x1, x2, x3, x4) = U31_gga(x1, x2, x4) pred(x1) = pred(x1) U32_gga(x1, x2, x3, x4) = U32_gga(x1, x2, x4) U33_gga(x1, x2, x3, x4) = U33_gga(x1, x2, x4) U34_gga(x1, x2, x3, x4) = U34_gga(x1, x2, x4) U25_ga(x1, x2, x3) = U25_ga(x1, x3) isTimes_in_gga(x1, x2, x3) = isTimes_in_gga(x1, x2) isTimes_out_gga(x1, x2, x3) = isTimes_out_gga(x1, x2, x3) U41_gga(x1, x2, x3, x4) = U41_gga(x1, x2, x4) U43_gga(x1, x2, x3, x4) = U43_gga(x1, x2, x4) U45_gga(x1, x2, x3, x4) = U45_gga(x1, x2, x4) U47_gga(x1, x2, x3, x4) = U47_gga(x1, x2, x4) U48_gga(x1, x2, x3, x4) = U48_gga(x1, x2, x4) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x1, x2, x3) U35_gga(x1, x2, x3) = U35_gga(x1, x3) U36_gga(x1, x2, x3) = U36_gga(x1, x3) U37_gga(x1, x2, x3, x4) = U37_gga(x1, x2, x4) U38_gga(x1, x2, x3, x4) = U38_gga(x1, x2, x4) U39_gga(x1, x2, x3, x4) = U39_gga(x1, x2, x4) U40_gga(x1, x2, x3, x4) = U40_gga(x1, x2, x4) U46_gga(x1, x2, x3, x4) = U46_gga(x1, x2, x4) U44_gga(x1, x2, x3, x4) = U44_gga(x1, x2, x4) U42_gga(x1, x2, x3, x4) = U42_gga(x1, x2, x4) U26_ga(x1, x2, x3) = U26_ga(x1, x3) U27_ga(x1, x2, x3) = U27_ga(x1, x3) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1, x2) U28_ga(x1, x2, x3, x4) = U28_ga(x1, x3, x4) iota_in_ga(x1, x2) = iota_in_ga(x1) U2_ga(x1, x2, x3) = U2_ga(x1, x3) iota1_in_gga(x1, x2, x3) = iota1_in_gga(x1, x2) iota1_out_gga(x1, x2, x3) = iota1_out_gga(x1, x2, x3) U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4) iota_out_ga(x1, x2) = iota_out_ga(x1, x2) .(x1, x2) = .(x1, x2) U29_ga(x1, x2, x3) = U29_ga(x1, x3) pds1_in_ggag(x1, x2, x3, x4) = pds1_in_ggag(x1, x2, x4) [] = [] pds1_out_ggag(x1, x2, x3, x4) = pds1_out_ggag(x1, x2, x3, x4) U21_ggag(x1, x2, x3, x4, x5, x6) = U21_ggag(x1, x2, x5, x6) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1, x2) U1_ag(x1, x2, x3, x4) = U1_ag(x2, x3, x4) U22_ggag(x1, x2, x3, x4, x5, x6) = U22_ggag(x1, x2, x3, x5, x6) check_in_gggag(x1, x2, x3, x4, x5) = check_in_gggag(x1, x2, x3, x5) check_out_gggag(x1, x2, x3, x4, x5) = check_out_gggag(x1, x2, x3, x4, x5) U17_gggag(x1, x2, x3, x4, x5, x6) = U17_gggag(x1, x2, x3, x5, x6) dif_in_gggaa(x1, x2, x3, x4, x5) = dif_in_gggaa(x1, x2, x3) dif_out_gggaa(x1, x2, x3, x4, x5) = dif_out_gggaa(x1, x2, x3, x4, x5) U6_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gggaa(x1, x2, x3, x4, x9) U7_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_gggaa(x1, x2, x3, x4, x9, x10) U8_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_gggaa(x1, x2, x3, x4, x5, x9, x10) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg(x1, x2, x3) U35_ggg(x1, x2, x3) = U35_ggg(x1, x2, x3) U36_ggg(x1, x2, x3) = U36_ggg(x1, x2, x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x1, x2, x3, x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x1, x2, x3, x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x1, x2, x3, x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x1, x2, x3, x4) U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_gggaa(x1, x2, x3, x4, x5, x9) U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_gggaa(x1, x2, x3, x4, x5, x7, x9) U18_gggag(x1, x2, x3, x4, x5, x6, x7) = U18_gggag(x1, x2, x3, x5, x6, x7) mergedelete_in_gga(x1, x2, x3) = mergedelete_in_gga(x1, x2) mergedelete_out_gga(x1, x2, x3) = mergedelete_out_gga(x1, x2, x3) U12_gga(x1, x2, x3, x4, x5) = U12_gga(x1, x2, x3, x5) U13_gga(x1, x2, x3, x4, x5, x6) = U13_gga(x1, x2, x3, x4, x6) U14_gga(x1, x2, x3, x4, x5, x6, x7) = U14_gga(x1, x2, x3, x4, x6, x7) U15_gga(x1, x2, x3, x4, x5, x6) = U15_gga(x1, x2, x3, x4, x6) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg(x1, x2) U64_gg(x1, x2, x3) = U64_gg(x1, x2, x3) U65_gg(x1, x2, x3) = U65_gg(x1, x2, x3) U16_gga(x1, x2, x3, x4, x5, x6) = U16_gga(x1, x2, x3, x4, x6) U19_gggag(x1, x2, x3, x4, x5, x6, x7) = U19_gggag(x1, x2, x3, x5, x6, x7) rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2) rev_out_gga(x1, x2, x3) = rev_out_gga(x1, x2, x3) U11_gga(x1, x2, x3, x4, x5) = U11_gga(x1, x2, x3, x5) U20_gggag(x1, x2, x3, x4, x5, x6) = U20_gggag(x1, x2, x3, x5, x6) U23_ggag(x1, x2, x3, x4, x5, x6) = U23_ggag(x1, x2, x3, x5, x6) pds_out_ga(x1, x2) = pds_out_ga(x1, x2) ISGREATER_IN_GG(x1, x2) = ISGREATER_IN_GG(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (123) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (124) Obligation: Pi DP problem: The TRS P consists of the following rules: ISGREATER_IN_GG(pred(X), pred(Y)) -> ISGREATER_IN_GG(X, Y) ISGREATER_IN_GG(succ(X), succ(Y)) -> ISGREATER_IN_GG(X, Y) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (125) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (126) Obligation: Q DP problem: The TRS P consists of the following rules: ISGREATER_IN_GG(pred(X), pred(Y)) -> ISGREATER_IN_GG(X, Y) ISGREATER_IN_GG(succ(X), succ(Y)) -> ISGREATER_IN_GG(X, Y) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (127) 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: *ISGREATER_IN_GG(pred(X), pred(Y)) -> ISGREATER_IN_GG(X, Y) The graph contains the following edges 1 > 1, 2 > 2 *ISGREATER_IN_GG(succ(X), succ(Y)) -> ISGREATER_IN_GG(X, Y) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (128) YES ---------------------------------------- (129) Obligation: Pi DP problem: The TRS P consists of the following rules: MERGEDELETE_IN_GGA(.(D, Ds), .(X, R), .(X, L2)) -> U13_GGA(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_GGA(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_GGA(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_GGA(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_GGA(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) U15_GGA(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> MERGEDELETE_IN_GGA(.(D, Ds), R, L2) MERGEDELETE_IN_GGA(.(D, Ds), .(D, R), L2) -> MERGEDELETE_IN_GGA(Ds, R, L2) The TRS R consists of the following rules: pds_in_ga(Order, .(zero, Ans)) -> U24_ga(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) isPlus_in_gga(zero, X, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), zero, succ(X)) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), succ(Y), succ(succ(Z))) -> U31_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(succ(X), pred(Y), Z) -> U32_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), zero, pred(X)) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), succ(Y), Z) -> U33_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), pred(Y), pred(pred(Z))) -> U34_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) U34_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U33_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U32_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U31_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U24_ga(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_ga(Order, Ans, isTimes_in_gga(Order, U, U1)) isTimes_in_gga(X, zero, zero) -> isTimes_out_gga(X, zero, zero) isTimes_in_gga(zero, succ(Y), zero) -> isTimes_out_gga(zero, succ(Y), zero) isTimes_in_gga(zero, pred(Y), zero) -> isTimes_out_gga(zero, pred(Y), zero) isTimes_in_gga(succ(X), succ(Y), Z) -> U41_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(succ(X), pred(Y), Z) -> U43_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(pred(X), succ(Y), Z) -> U45_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) isTimes_in_gga(pred(X), pred(Y), Z) -> U47_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) U47_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_gga(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U48_gga(X, Y, Z, isMinus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), pred(Y), Z) U45_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_gga(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U46_gga(X, Y, Z, isPlus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), succ(Y), Z) U43_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_gga(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U44_gga(X, Y, Z, isMinus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), pred(Y), Z) U41_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_gga(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U42_gga(X, Y, Z, isPlus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), succ(Y), Z) U25_ga(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_ga(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U26_ga(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_ga(Order, Ans, =_in_ag(N, U2)) =_in_ag(X, X) -> =_out_ag(X, X) U27_ga(Order, Ans, =_out_ag(N, U2)) -> U28_ga(Order, Ans, N, iota_in_ga(N, .(zero, List))) iota_in_ga(N, List) -> U2_ga(N, List, iota1_in_gga(zero, N, List)) iota1_in_gga(K, K, []) -> iota1_out_gga(K, K, []) iota1_in_gga(K, N, .(K, List)) -> U3_gga(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_gga(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_gga(K, N, List, =_in_ag(K1, U)) U4_gga(K, N, List, =_out_ag(K1, U)) -> U5_gga(K, N, List, iota1_in_gga(K1, N, List)) U5_gga(K, N, List, iota1_out_gga(K1, N, List)) -> iota1_out_gga(K, N, .(K, List)) U2_ga(N, List, iota1_out_gga(zero, N, List)) -> iota_out_ga(N, List) U28_ga(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_ga(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) pds1_in_ggag([], X6, [], X7) -> pds1_out_ggag([], X6, [], X7) pds1_in_ggag(Unused, List, .(Choice, Rest), Mod) -> U21_ggag(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(X, .(X1, Xs)) -> U1_ag(X, X1, Xs, member_in_ag(X, Xs)) U1_ag(X, X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) U21_ggag(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_ggag(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) check_in_gggag([], X4, L, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, L3, Modulus) -> U17_gggag(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) dif_in_gggaa([], X2, X3, [], []) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U17_gggag(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) mergedelete_in_gga([], L, L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R), L2) -> U12_gga(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) mergedelete_in_gga(.(D, Ds), .(X, R), .(X, L2)) -> U13_gga(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_gga(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_gga(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U15_gga(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U16_gga(D, Ds, X, R, L2, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U12_gga(D, Ds, R, L2, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) rev_in_gga([], L, L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y, L) -> U11_gga(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) U11_gga(X, Xs, Y, L, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U22_ggag(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_ggag(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U23_ggag(Unused, List, Choice, Rest, Mod, pds1_out_ggag(U3, .(Choice, List), Rest, Mod)) -> pds1_out_ggag(Unused, List, .(Choice, Rest), Mod) U29_ga(Order, Ans, pds1_out_ggag(List, .(zero, []), Ans, N)) -> pds_out_ga(Order, .(zero, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U24_ga(x1, x2, x3) = U24_ga(x1, x3) isPlus_in_gga(x1, x2, x3) = isPlus_in_gga(x1, x2) zero = zero isPlus_out_gga(x1, x2, x3) = isPlus_out_gga(x1, x2, x3) succ(x1) = succ(x1) U31_gga(x1, x2, x3, x4) = U31_gga(x1, x2, x4) pred(x1) = pred(x1) U32_gga(x1, x2, x3, x4) = U32_gga(x1, x2, x4) U33_gga(x1, x2, x3, x4) = U33_gga(x1, x2, x4) U34_gga(x1, x2, x3, x4) = U34_gga(x1, x2, x4) U25_ga(x1, x2, x3) = U25_ga(x1, x3) isTimes_in_gga(x1, x2, x3) = isTimes_in_gga(x1, x2) isTimes_out_gga(x1, x2, x3) = isTimes_out_gga(x1, x2, x3) U41_gga(x1, x2, x3, x4) = U41_gga(x1, x2, x4) U43_gga(x1, x2, x3, x4) = U43_gga(x1, x2, x4) U45_gga(x1, x2, x3, x4) = U45_gga(x1, x2, x4) U47_gga(x1, x2, x3, x4) = U47_gga(x1, x2, x4) U48_gga(x1, x2, x3, x4) = U48_gga(x1, x2, x4) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x1, x2, x3) U35_gga(x1, x2, x3) = U35_gga(x1, x3) U36_gga(x1, x2, x3) = U36_gga(x1, x3) U37_gga(x1, x2, x3, x4) = U37_gga(x1, x2, x4) U38_gga(x1, x2, x3, x4) = U38_gga(x1, x2, x4) U39_gga(x1, x2, x3, x4) = U39_gga(x1, x2, x4) U40_gga(x1, x2, x3, x4) = U40_gga(x1, x2, x4) U46_gga(x1, x2, x3, x4) = U46_gga(x1, x2, x4) U44_gga(x1, x2, x3, x4) = U44_gga(x1, x2, x4) U42_gga(x1, x2, x3, x4) = U42_gga(x1, x2, x4) U26_ga(x1, x2, x3) = U26_ga(x1, x3) U27_ga(x1, x2, x3) = U27_ga(x1, x3) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1, x2) U28_ga(x1, x2, x3, x4) = U28_ga(x1, x3, x4) iota_in_ga(x1, x2) = iota_in_ga(x1) U2_ga(x1, x2, x3) = U2_ga(x1, x3) iota1_in_gga(x1, x2, x3) = iota1_in_gga(x1, x2) iota1_out_gga(x1, x2, x3) = iota1_out_gga(x1, x2, x3) U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4) iota_out_ga(x1, x2) = iota_out_ga(x1, x2) .(x1, x2) = .(x1, x2) U29_ga(x1, x2, x3) = U29_ga(x1, x3) pds1_in_ggag(x1, x2, x3, x4) = pds1_in_ggag(x1, x2, x4) [] = [] pds1_out_ggag(x1, x2, x3, x4) = pds1_out_ggag(x1, x2, x3, x4) U21_ggag(x1, x2, x3, x4, x5, x6) = U21_ggag(x1, x2, x5, x6) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1, x2) U1_ag(x1, x2, x3, x4) = U1_ag(x2, x3, x4) U22_ggag(x1, x2, x3, x4, x5, x6) = U22_ggag(x1, x2, x3, x5, x6) check_in_gggag(x1, x2, x3, x4, x5) = check_in_gggag(x1, x2, x3, x5) check_out_gggag(x1, x2, x3, x4, x5) = check_out_gggag(x1, x2, x3, x4, x5) U17_gggag(x1, x2, x3, x4, x5, x6) = U17_gggag(x1, x2, x3, x5, x6) dif_in_gggaa(x1, x2, x3, x4, x5) = dif_in_gggaa(x1, x2, x3) dif_out_gggaa(x1, x2, x3, x4, x5) = dif_out_gggaa(x1, x2, x3, x4, x5) U6_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gggaa(x1, x2, x3, x4, x9) U7_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_gggaa(x1, x2, x3, x4, x9, x10) U8_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_gggaa(x1, x2, x3, x4, x5, x9, x10) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg(x1, x2, x3) U35_ggg(x1, x2, x3) = U35_ggg(x1, x2, x3) U36_ggg(x1, x2, x3) = U36_ggg(x1, x2, x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x1, x2, x3, x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x1, x2, x3, x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x1, x2, x3, x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x1, x2, x3, x4) U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_gggaa(x1, x2, x3, x4, x5, x9) U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_gggaa(x1, x2, x3, x4, x5, x7, x9) U18_gggag(x1, x2, x3, x4, x5, x6, x7) = U18_gggag(x1, x2, x3, x5, x6, x7) mergedelete_in_gga(x1, x2, x3) = mergedelete_in_gga(x1, x2) mergedelete_out_gga(x1, x2, x3) = mergedelete_out_gga(x1, x2, x3) U12_gga(x1, x2, x3, x4, x5) = U12_gga(x1, x2, x3, x5) U13_gga(x1, x2, x3, x4, x5, x6) = U13_gga(x1, x2, x3, x4, x6) U14_gga(x1, x2, x3, x4, x5, x6, x7) = U14_gga(x1, x2, x3, x4, x6, x7) U15_gga(x1, x2, x3, x4, x5, x6) = U15_gga(x1, x2, x3, x4, x6) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg(x1, x2) U64_gg(x1, x2, x3) = U64_gg(x1, x2, x3) U65_gg(x1, x2, x3) = U65_gg(x1, x2, x3) U16_gga(x1, x2, x3, x4, x5, x6) = U16_gga(x1, x2, x3, x4, x6) U19_gggag(x1, x2, x3, x4, x5, x6, x7) = U19_gggag(x1, x2, x3, x5, x6, x7) rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2) rev_out_gga(x1, x2, x3) = rev_out_gga(x1, x2, x3) U11_gga(x1, x2, x3, x4, x5) = U11_gga(x1, x2, x3, x5) U20_gggag(x1, x2, x3, x4, x5, x6) = U20_gggag(x1, x2, x3, x5, x6) U23_ggag(x1, x2, x3, x4, x5, x6) = U23_ggag(x1, x2, x3, x5, x6) pds_out_ga(x1, x2) = pds_out_ga(x1, x2) MERGEDELETE_IN_GGA(x1, x2, x3) = MERGEDELETE_IN_GGA(x1, x2) U13_GGA(x1, x2, x3, x4, x5, x6) = U13_GGA(x1, x2, x3, x4, x6) U14_GGA(x1, x2, x3, x4, x5, x6, x7) = U14_GGA(x1, x2, x3, x4, x6, x7) U15_GGA(x1, x2, x3, x4, x5, x6) = U15_GGA(x1, x2, x3, x4, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (130) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (131) Obligation: Pi DP problem: The TRS P consists of the following rules: MERGEDELETE_IN_GGA(.(D, Ds), .(X, R), .(X, L2)) -> U13_GGA(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_GGA(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_GGA(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_GGA(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_GGA(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) U15_GGA(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> MERGEDELETE_IN_GGA(.(D, Ds), R, L2) MERGEDELETE_IN_GGA(.(D, Ds), .(D, R), L2) -> MERGEDELETE_IN_GGA(Ds, R, L2) The TRS R consists of the following rules: =_in_ag(X, X) -> =_out_ag(X, X) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) The argument filtering Pi contains the following mapping: zero = zero succ(x1) = succ(x1) pred(x1) = pred(x1) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1, x2) .(x1, x2) = .(x1, x2) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg(x1, x2) U64_gg(x1, x2, x3) = U64_gg(x1, x2, x3) U65_gg(x1, x2, x3) = U65_gg(x1, x2, x3) MERGEDELETE_IN_GGA(x1, x2, x3) = MERGEDELETE_IN_GGA(x1, x2) U13_GGA(x1, x2, x3, x4, x5, x6) = U13_GGA(x1, x2, x3, x4, x6) U14_GGA(x1, x2, x3, x4, x5, x6, x7) = U14_GGA(x1, x2, x3, x4, x6, x7) U15_GGA(x1, x2, x3, x4, x5, x6) = U15_GGA(x1, x2, x3, x4, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (132) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (133) Obligation: Q DP problem: The TRS P consists of the following rules: MERGEDELETE_IN_GGA(.(D, Ds), .(X, R)) -> U13_GGA(D, Ds, X, R, =_in_ag(D)) U13_GGA(D, Ds, X, R, =_out_ag(X1, D)) -> U14_GGA(D, Ds, X, R, X1, =_in_ag(X)) U14_GGA(D, Ds, X, R, X1, =_out_ag(X2, X)) -> U15_GGA(D, Ds, X, R, isGreater_in_gg(X1, X2)) U15_GGA(D, Ds, X, R, isGreater_out_gg(X1, X2)) -> MERGEDELETE_IN_GGA(.(D, Ds), R) MERGEDELETE_IN_GGA(.(D, Ds), .(D, R)) -> MERGEDELETE_IN_GGA(Ds, R) The TRS R consists of the following rules: =_in_ag(X) -> =_out_ag(X, X) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) The set Q consists of the following terms: =_in_ag(x0) isGreater_in_gg(x0, x1) U64_gg(x0, x1, x2) U65_gg(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (134) 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: *U13_GGA(D, Ds, X, R, =_out_ag(X1, D)) -> U14_GGA(D, Ds, X, R, X1, =_in_ag(X)) The graph contains the following edges 1 >= 1, 5 > 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 > 5 *U14_GGA(D, Ds, X, R, X1, =_out_ag(X2, X)) -> U15_GGA(D, Ds, X, R, isGreater_in_gg(X1, X2)) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 6 > 3, 4 >= 4 *MERGEDELETE_IN_GGA(.(D, Ds), .(X, R)) -> U13_GGA(D, Ds, X, R, =_in_ag(D)) The graph contains the following edges 1 > 1, 1 > 2, 2 > 3, 2 > 4 *U15_GGA(D, Ds, X, R, isGreater_out_gg(X1, X2)) -> MERGEDELETE_IN_GGA(.(D, Ds), R) The graph contains the following edges 4 >= 2 *MERGEDELETE_IN_GGA(.(D, Ds), .(D, R)) -> MERGEDELETE_IN_GGA(Ds, R) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (135) YES ---------------------------------------- (136) Obligation: Pi DP problem: The TRS P consists of the following rules: ISMINUS_IN_GGG(zero, pred(Y), succ(Z)) -> ISMINUS_IN_GGG(zero, Y, Z) ISMINUS_IN_GGG(zero, succ(Y), pred(Z)) -> ISMINUS_IN_GGG(zero, Y, Z) The TRS R consists of the following rules: pds_in_ga(Order, .(zero, Ans)) -> U24_ga(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) isPlus_in_gga(zero, X, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), zero, succ(X)) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), succ(Y), succ(succ(Z))) -> U31_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(succ(X), pred(Y), Z) -> U32_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), zero, pred(X)) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), succ(Y), Z) -> U33_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), pred(Y), pred(pred(Z))) -> U34_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) U34_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U33_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U32_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U31_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U24_ga(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_ga(Order, Ans, isTimes_in_gga(Order, U, U1)) isTimes_in_gga(X, zero, zero) -> isTimes_out_gga(X, zero, zero) isTimes_in_gga(zero, succ(Y), zero) -> isTimes_out_gga(zero, succ(Y), zero) isTimes_in_gga(zero, pred(Y), zero) -> isTimes_out_gga(zero, pred(Y), zero) isTimes_in_gga(succ(X), succ(Y), Z) -> U41_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(succ(X), pred(Y), Z) -> U43_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(pred(X), succ(Y), Z) -> U45_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) isTimes_in_gga(pred(X), pred(Y), Z) -> U47_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) U47_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_gga(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U48_gga(X, Y, Z, isMinus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), pred(Y), Z) U45_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_gga(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U46_gga(X, Y, Z, isPlus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), succ(Y), Z) U43_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_gga(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U44_gga(X, Y, Z, isMinus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), pred(Y), Z) U41_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_gga(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U42_gga(X, Y, Z, isPlus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), succ(Y), Z) U25_ga(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_ga(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U26_ga(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_ga(Order, Ans, =_in_ag(N, U2)) =_in_ag(X, X) -> =_out_ag(X, X) U27_ga(Order, Ans, =_out_ag(N, U2)) -> U28_ga(Order, Ans, N, iota_in_ga(N, .(zero, List))) iota_in_ga(N, List) -> U2_ga(N, List, iota1_in_gga(zero, N, List)) iota1_in_gga(K, K, []) -> iota1_out_gga(K, K, []) iota1_in_gga(K, N, .(K, List)) -> U3_gga(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_gga(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_gga(K, N, List, =_in_ag(K1, U)) U4_gga(K, N, List, =_out_ag(K1, U)) -> U5_gga(K, N, List, iota1_in_gga(K1, N, List)) U5_gga(K, N, List, iota1_out_gga(K1, N, List)) -> iota1_out_gga(K, N, .(K, List)) U2_ga(N, List, iota1_out_gga(zero, N, List)) -> iota_out_ga(N, List) U28_ga(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_ga(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) pds1_in_ggag([], X6, [], X7) -> pds1_out_ggag([], X6, [], X7) pds1_in_ggag(Unused, List, .(Choice, Rest), Mod) -> U21_ggag(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(X, .(X1, Xs)) -> U1_ag(X, X1, Xs, member_in_ag(X, Xs)) U1_ag(X, X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) U21_ggag(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_ggag(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) check_in_gggag([], X4, L, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, L3, Modulus) -> U17_gggag(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) dif_in_gggaa([], X2, X3, [], []) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U17_gggag(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) mergedelete_in_gga([], L, L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R), L2) -> U12_gga(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) mergedelete_in_gga(.(D, Ds), .(X, R), .(X, L2)) -> U13_gga(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_gga(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_gga(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U15_gga(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U16_gga(D, Ds, X, R, L2, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U12_gga(D, Ds, R, L2, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) rev_in_gga([], L, L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y, L) -> U11_gga(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) U11_gga(X, Xs, Y, L, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U22_ggag(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_ggag(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U23_ggag(Unused, List, Choice, Rest, Mod, pds1_out_ggag(U3, .(Choice, List), Rest, Mod)) -> pds1_out_ggag(Unused, List, .(Choice, Rest), Mod) U29_ga(Order, Ans, pds1_out_ggag(List, .(zero, []), Ans, N)) -> pds_out_ga(Order, .(zero, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U24_ga(x1, x2, x3) = U24_ga(x1, x3) isPlus_in_gga(x1, x2, x3) = isPlus_in_gga(x1, x2) zero = zero isPlus_out_gga(x1, x2, x3) = isPlus_out_gga(x1, x2, x3) succ(x1) = succ(x1) U31_gga(x1, x2, x3, x4) = U31_gga(x1, x2, x4) pred(x1) = pred(x1) U32_gga(x1, x2, x3, x4) = U32_gga(x1, x2, x4) U33_gga(x1, x2, x3, x4) = U33_gga(x1, x2, x4) U34_gga(x1, x2, x3, x4) = U34_gga(x1, x2, x4) U25_ga(x1, x2, x3) = U25_ga(x1, x3) isTimes_in_gga(x1, x2, x3) = isTimes_in_gga(x1, x2) isTimes_out_gga(x1, x2, x3) = isTimes_out_gga(x1, x2, x3) U41_gga(x1, x2, x3, x4) = U41_gga(x1, x2, x4) U43_gga(x1, x2, x3, x4) = U43_gga(x1, x2, x4) U45_gga(x1, x2, x3, x4) = U45_gga(x1, x2, x4) U47_gga(x1, x2, x3, x4) = U47_gga(x1, x2, x4) U48_gga(x1, x2, x3, x4) = U48_gga(x1, x2, x4) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x1, x2, x3) U35_gga(x1, x2, x3) = U35_gga(x1, x3) U36_gga(x1, x2, x3) = U36_gga(x1, x3) U37_gga(x1, x2, x3, x4) = U37_gga(x1, x2, x4) U38_gga(x1, x2, x3, x4) = U38_gga(x1, x2, x4) U39_gga(x1, x2, x3, x4) = U39_gga(x1, x2, x4) U40_gga(x1, x2, x3, x4) = U40_gga(x1, x2, x4) U46_gga(x1, x2, x3, x4) = U46_gga(x1, x2, x4) U44_gga(x1, x2, x3, x4) = U44_gga(x1, x2, x4) U42_gga(x1, x2, x3, x4) = U42_gga(x1, x2, x4) U26_ga(x1, x2, x3) = U26_ga(x1, x3) U27_ga(x1, x2, x3) = U27_ga(x1, x3) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1, x2) U28_ga(x1, x2, x3, x4) = U28_ga(x1, x3, x4) iota_in_ga(x1, x2) = iota_in_ga(x1) U2_ga(x1, x2, x3) = U2_ga(x1, x3) iota1_in_gga(x1, x2, x3) = iota1_in_gga(x1, x2) iota1_out_gga(x1, x2, x3) = iota1_out_gga(x1, x2, x3) U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4) iota_out_ga(x1, x2) = iota_out_ga(x1, x2) .(x1, x2) = .(x1, x2) U29_ga(x1, x2, x3) = U29_ga(x1, x3) pds1_in_ggag(x1, x2, x3, x4) = pds1_in_ggag(x1, x2, x4) [] = [] pds1_out_ggag(x1, x2, x3, x4) = pds1_out_ggag(x1, x2, x3, x4) U21_ggag(x1, x2, x3, x4, x5, x6) = U21_ggag(x1, x2, x5, x6) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1, x2) U1_ag(x1, x2, x3, x4) = U1_ag(x2, x3, x4) U22_ggag(x1, x2, x3, x4, x5, x6) = U22_ggag(x1, x2, x3, x5, x6) check_in_gggag(x1, x2, x3, x4, x5) = check_in_gggag(x1, x2, x3, x5) check_out_gggag(x1, x2, x3, x4, x5) = check_out_gggag(x1, x2, x3, x4, x5) U17_gggag(x1, x2, x3, x4, x5, x6) = U17_gggag(x1, x2, x3, x5, x6) dif_in_gggaa(x1, x2, x3, x4, x5) = dif_in_gggaa(x1, x2, x3) dif_out_gggaa(x1, x2, x3, x4, x5) = dif_out_gggaa(x1, x2, x3, x4, x5) U6_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gggaa(x1, x2, x3, x4, x9) U7_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_gggaa(x1, x2, x3, x4, x9, x10) U8_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_gggaa(x1, x2, x3, x4, x5, x9, x10) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg(x1, x2, x3) U35_ggg(x1, x2, x3) = U35_ggg(x1, x2, x3) U36_ggg(x1, x2, x3) = U36_ggg(x1, x2, x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x1, x2, x3, x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x1, x2, x3, x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x1, x2, x3, x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x1, x2, x3, x4) U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_gggaa(x1, x2, x3, x4, x5, x9) U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_gggaa(x1, x2, x3, x4, x5, x7, x9) U18_gggag(x1, x2, x3, x4, x5, x6, x7) = U18_gggag(x1, x2, x3, x5, x6, x7) mergedelete_in_gga(x1, x2, x3) = mergedelete_in_gga(x1, x2) mergedelete_out_gga(x1, x2, x3) = mergedelete_out_gga(x1, x2, x3) U12_gga(x1, x2, x3, x4, x5) = U12_gga(x1, x2, x3, x5) U13_gga(x1, x2, x3, x4, x5, x6) = U13_gga(x1, x2, x3, x4, x6) U14_gga(x1, x2, x3, x4, x5, x6, x7) = U14_gga(x1, x2, x3, x4, x6, x7) U15_gga(x1, x2, x3, x4, x5, x6) = U15_gga(x1, x2, x3, x4, x6) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg(x1, x2) U64_gg(x1, x2, x3) = U64_gg(x1, x2, x3) U65_gg(x1, x2, x3) = U65_gg(x1, x2, x3) U16_gga(x1, x2, x3, x4, x5, x6) = U16_gga(x1, x2, x3, x4, x6) U19_gggag(x1, x2, x3, x4, x5, x6, x7) = U19_gggag(x1, x2, x3, x5, x6, x7) rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2) rev_out_gga(x1, x2, x3) = rev_out_gga(x1, x2, x3) U11_gga(x1, x2, x3, x4, x5) = U11_gga(x1, x2, x3, x5) U20_gggag(x1, x2, x3, x4, x5, x6) = U20_gggag(x1, x2, x3, x5, x6) U23_ggag(x1, x2, x3, x4, x5, x6) = U23_ggag(x1, x2, x3, x5, x6) pds_out_ga(x1, x2) = pds_out_ga(x1, x2) ISMINUS_IN_GGG(x1, x2, x3) = ISMINUS_IN_GGG(x1, x2, x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (137) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (138) Obligation: Pi DP problem: The TRS P consists of the following rules: ISMINUS_IN_GGG(zero, pred(Y), succ(Z)) -> ISMINUS_IN_GGG(zero, Y, Z) ISMINUS_IN_GGG(zero, succ(Y), pred(Z)) -> ISMINUS_IN_GGG(zero, Y, Z) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (139) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (140) Obligation: Q DP problem: The TRS P consists of the following rules: ISMINUS_IN_GGG(zero, pred(Y), succ(Z)) -> ISMINUS_IN_GGG(zero, Y, Z) ISMINUS_IN_GGG(zero, succ(Y), pred(Z)) -> ISMINUS_IN_GGG(zero, Y, Z) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (141) 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: *ISMINUS_IN_GGG(zero, pred(Y), succ(Z)) -> ISMINUS_IN_GGG(zero, Y, Z) The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3 *ISMINUS_IN_GGG(zero, succ(Y), pred(Z)) -> ISMINUS_IN_GGG(zero, Y, Z) The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3 ---------------------------------------- (142) YES ---------------------------------------- (143) Obligation: Pi DP problem: The TRS P consists of the following rules: ISMINUS_IN_GGG(succ(X), pred(Y), succ(succ(Z))) -> ISMINUS_IN_GGG(X, Y, Z) ISMINUS_IN_GGG(succ(X), succ(Y), Z) -> ISMINUS_IN_GGG(X, Y, Z) ISMINUS_IN_GGG(pred(X), succ(Y), pred(pred(Z))) -> ISMINUS_IN_GGG(X, Y, Z) ISMINUS_IN_GGG(pred(X), pred(Y), Z) -> ISMINUS_IN_GGG(X, Y, Z) The TRS R consists of the following rules: pds_in_ga(Order, .(zero, Ans)) -> U24_ga(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) isPlus_in_gga(zero, X, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), zero, succ(X)) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), succ(Y), succ(succ(Z))) -> U31_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(succ(X), pred(Y), Z) -> U32_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), zero, pred(X)) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), succ(Y), Z) -> U33_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), pred(Y), pred(pred(Z))) -> U34_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) U34_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U33_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U32_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U31_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U24_ga(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_ga(Order, Ans, isTimes_in_gga(Order, U, U1)) isTimes_in_gga(X, zero, zero) -> isTimes_out_gga(X, zero, zero) isTimes_in_gga(zero, succ(Y), zero) -> isTimes_out_gga(zero, succ(Y), zero) isTimes_in_gga(zero, pred(Y), zero) -> isTimes_out_gga(zero, pred(Y), zero) isTimes_in_gga(succ(X), succ(Y), Z) -> U41_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(succ(X), pred(Y), Z) -> U43_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(pred(X), succ(Y), Z) -> U45_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) isTimes_in_gga(pred(X), pred(Y), Z) -> U47_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) U47_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_gga(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U48_gga(X, Y, Z, isMinus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), pred(Y), Z) U45_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_gga(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U46_gga(X, Y, Z, isPlus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), succ(Y), Z) U43_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_gga(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U44_gga(X, Y, Z, isMinus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), pred(Y), Z) U41_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_gga(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U42_gga(X, Y, Z, isPlus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), succ(Y), Z) U25_ga(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_ga(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U26_ga(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_ga(Order, Ans, =_in_ag(N, U2)) =_in_ag(X, X) -> =_out_ag(X, X) U27_ga(Order, Ans, =_out_ag(N, U2)) -> U28_ga(Order, Ans, N, iota_in_ga(N, .(zero, List))) iota_in_ga(N, List) -> U2_ga(N, List, iota1_in_gga(zero, N, List)) iota1_in_gga(K, K, []) -> iota1_out_gga(K, K, []) iota1_in_gga(K, N, .(K, List)) -> U3_gga(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_gga(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_gga(K, N, List, =_in_ag(K1, U)) U4_gga(K, N, List, =_out_ag(K1, U)) -> U5_gga(K, N, List, iota1_in_gga(K1, N, List)) U5_gga(K, N, List, iota1_out_gga(K1, N, List)) -> iota1_out_gga(K, N, .(K, List)) U2_ga(N, List, iota1_out_gga(zero, N, List)) -> iota_out_ga(N, List) U28_ga(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_ga(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) pds1_in_ggag([], X6, [], X7) -> pds1_out_ggag([], X6, [], X7) pds1_in_ggag(Unused, List, .(Choice, Rest), Mod) -> U21_ggag(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(X, .(X1, Xs)) -> U1_ag(X, X1, Xs, member_in_ag(X, Xs)) U1_ag(X, X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) U21_ggag(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_ggag(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) check_in_gggag([], X4, L, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, L3, Modulus) -> U17_gggag(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) dif_in_gggaa([], X2, X3, [], []) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U17_gggag(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) mergedelete_in_gga([], L, L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R), L2) -> U12_gga(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) mergedelete_in_gga(.(D, Ds), .(X, R), .(X, L2)) -> U13_gga(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_gga(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_gga(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U15_gga(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U16_gga(D, Ds, X, R, L2, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U12_gga(D, Ds, R, L2, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) rev_in_gga([], L, L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y, L) -> U11_gga(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) U11_gga(X, Xs, Y, L, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U22_ggag(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_ggag(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U23_ggag(Unused, List, Choice, Rest, Mod, pds1_out_ggag(U3, .(Choice, List), Rest, Mod)) -> pds1_out_ggag(Unused, List, .(Choice, Rest), Mod) U29_ga(Order, Ans, pds1_out_ggag(List, .(zero, []), Ans, N)) -> pds_out_ga(Order, .(zero, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U24_ga(x1, x2, x3) = U24_ga(x1, x3) isPlus_in_gga(x1, x2, x3) = isPlus_in_gga(x1, x2) zero = zero isPlus_out_gga(x1, x2, x3) = isPlus_out_gga(x1, x2, x3) succ(x1) = succ(x1) U31_gga(x1, x2, x3, x4) = U31_gga(x1, x2, x4) pred(x1) = pred(x1) U32_gga(x1, x2, x3, x4) = U32_gga(x1, x2, x4) U33_gga(x1, x2, x3, x4) = U33_gga(x1, x2, x4) U34_gga(x1, x2, x3, x4) = U34_gga(x1, x2, x4) U25_ga(x1, x2, x3) = U25_ga(x1, x3) isTimes_in_gga(x1, x2, x3) = isTimes_in_gga(x1, x2) isTimes_out_gga(x1, x2, x3) = isTimes_out_gga(x1, x2, x3) U41_gga(x1, x2, x3, x4) = U41_gga(x1, x2, x4) U43_gga(x1, x2, x3, x4) = U43_gga(x1, x2, x4) U45_gga(x1, x2, x3, x4) = U45_gga(x1, x2, x4) U47_gga(x1, x2, x3, x4) = U47_gga(x1, x2, x4) U48_gga(x1, x2, x3, x4) = U48_gga(x1, x2, x4) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x1, x2, x3) U35_gga(x1, x2, x3) = U35_gga(x1, x3) U36_gga(x1, x2, x3) = U36_gga(x1, x3) U37_gga(x1, x2, x3, x4) = U37_gga(x1, x2, x4) U38_gga(x1, x2, x3, x4) = U38_gga(x1, x2, x4) U39_gga(x1, x2, x3, x4) = U39_gga(x1, x2, x4) U40_gga(x1, x2, x3, x4) = U40_gga(x1, x2, x4) U46_gga(x1, x2, x3, x4) = U46_gga(x1, x2, x4) U44_gga(x1, x2, x3, x4) = U44_gga(x1, x2, x4) U42_gga(x1, x2, x3, x4) = U42_gga(x1, x2, x4) U26_ga(x1, x2, x3) = U26_ga(x1, x3) U27_ga(x1, x2, x3) = U27_ga(x1, x3) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1, x2) U28_ga(x1, x2, x3, x4) = U28_ga(x1, x3, x4) iota_in_ga(x1, x2) = iota_in_ga(x1) U2_ga(x1, x2, x3) = U2_ga(x1, x3) iota1_in_gga(x1, x2, x3) = iota1_in_gga(x1, x2) iota1_out_gga(x1, x2, x3) = iota1_out_gga(x1, x2, x3) U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4) iota_out_ga(x1, x2) = iota_out_ga(x1, x2) .(x1, x2) = .(x1, x2) U29_ga(x1, x2, x3) = U29_ga(x1, x3) pds1_in_ggag(x1, x2, x3, x4) = pds1_in_ggag(x1, x2, x4) [] = [] pds1_out_ggag(x1, x2, x3, x4) = pds1_out_ggag(x1, x2, x3, x4) U21_ggag(x1, x2, x3, x4, x5, x6) = U21_ggag(x1, x2, x5, x6) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1, x2) U1_ag(x1, x2, x3, x4) = U1_ag(x2, x3, x4) U22_ggag(x1, x2, x3, x4, x5, x6) = U22_ggag(x1, x2, x3, x5, x6) check_in_gggag(x1, x2, x3, x4, x5) = check_in_gggag(x1, x2, x3, x5) check_out_gggag(x1, x2, x3, x4, x5) = check_out_gggag(x1, x2, x3, x4, x5) U17_gggag(x1, x2, x3, x4, x5, x6) = U17_gggag(x1, x2, x3, x5, x6) dif_in_gggaa(x1, x2, x3, x4, x5) = dif_in_gggaa(x1, x2, x3) dif_out_gggaa(x1, x2, x3, x4, x5) = dif_out_gggaa(x1, x2, x3, x4, x5) U6_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gggaa(x1, x2, x3, x4, x9) U7_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_gggaa(x1, x2, x3, x4, x9, x10) U8_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_gggaa(x1, x2, x3, x4, x5, x9, x10) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg(x1, x2, x3) U35_ggg(x1, x2, x3) = U35_ggg(x1, x2, x3) U36_ggg(x1, x2, x3) = U36_ggg(x1, x2, x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x1, x2, x3, x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x1, x2, x3, x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x1, x2, x3, x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x1, x2, x3, x4) U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_gggaa(x1, x2, x3, x4, x5, x9) U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_gggaa(x1, x2, x3, x4, x5, x7, x9) U18_gggag(x1, x2, x3, x4, x5, x6, x7) = U18_gggag(x1, x2, x3, x5, x6, x7) mergedelete_in_gga(x1, x2, x3) = mergedelete_in_gga(x1, x2) mergedelete_out_gga(x1, x2, x3) = mergedelete_out_gga(x1, x2, x3) U12_gga(x1, x2, x3, x4, x5) = U12_gga(x1, x2, x3, x5) U13_gga(x1, x2, x3, x4, x5, x6) = U13_gga(x1, x2, x3, x4, x6) U14_gga(x1, x2, x3, x4, x5, x6, x7) = U14_gga(x1, x2, x3, x4, x6, x7) U15_gga(x1, x2, x3, x4, x5, x6) = U15_gga(x1, x2, x3, x4, x6) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg(x1, x2) U64_gg(x1, x2, x3) = U64_gg(x1, x2, x3) U65_gg(x1, x2, x3) = U65_gg(x1, x2, x3) U16_gga(x1, x2, x3, x4, x5, x6) = U16_gga(x1, x2, x3, x4, x6) U19_gggag(x1, x2, x3, x4, x5, x6, x7) = U19_gggag(x1, x2, x3, x5, x6, x7) rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2) rev_out_gga(x1, x2, x3) = rev_out_gga(x1, x2, x3) U11_gga(x1, x2, x3, x4, x5) = U11_gga(x1, x2, x3, x5) U20_gggag(x1, x2, x3, x4, x5, x6) = U20_gggag(x1, x2, x3, x5, x6) U23_ggag(x1, x2, x3, x4, x5, x6) = U23_ggag(x1, x2, x3, x5, x6) pds_out_ga(x1, x2) = pds_out_ga(x1, x2) ISMINUS_IN_GGG(x1, x2, x3) = ISMINUS_IN_GGG(x1, x2, x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (144) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (145) Obligation: Pi DP problem: The TRS P consists of the following rules: ISMINUS_IN_GGG(succ(X), pred(Y), succ(succ(Z))) -> ISMINUS_IN_GGG(X, Y, Z) ISMINUS_IN_GGG(succ(X), succ(Y), Z) -> ISMINUS_IN_GGG(X, Y, Z) ISMINUS_IN_GGG(pred(X), succ(Y), pred(pred(Z))) -> ISMINUS_IN_GGG(X, Y, Z) ISMINUS_IN_GGG(pred(X), pred(Y), Z) -> ISMINUS_IN_GGG(X, Y, Z) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (146) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (147) Obligation: Q DP problem: The TRS P consists of the following rules: ISMINUS_IN_GGG(succ(X), pred(Y), succ(succ(Z))) -> ISMINUS_IN_GGG(X, Y, Z) ISMINUS_IN_GGG(succ(X), succ(Y), Z) -> ISMINUS_IN_GGG(X, Y, Z) ISMINUS_IN_GGG(pred(X), succ(Y), pred(pred(Z))) -> ISMINUS_IN_GGG(X, Y, Z) ISMINUS_IN_GGG(pred(X), pred(Y), Z) -> ISMINUS_IN_GGG(X, Y, Z) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (148) 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: *ISMINUS_IN_GGG(succ(X), pred(Y), succ(succ(Z))) -> ISMINUS_IN_GGG(X, Y, Z) The graph contains the following edges 1 > 1, 2 > 2, 3 > 3 *ISMINUS_IN_GGG(succ(X), succ(Y), Z) -> ISMINUS_IN_GGG(X, Y, Z) The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3 *ISMINUS_IN_GGG(pred(X), succ(Y), pred(pred(Z))) -> ISMINUS_IN_GGG(X, Y, Z) The graph contains the following edges 1 > 1, 2 > 2, 3 > 3 *ISMINUS_IN_GGG(pred(X), pred(Y), Z) -> ISMINUS_IN_GGG(X, Y, Z) The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3 ---------------------------------------- (149) YES ---------------------------------------- (150) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER_IN_AG(X, .(X1, Xs)) -> MEMBER_IN_AG(X, Xs) The TRS R consists of the following rules: pds_in_ga(Order, .(zero, Ans)) -> U24_ga(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) isPlus_in_gga(zero, X, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), zero, succ(X)) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), succ(Y), succ(succ(Z))) -> U31_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(succ(X), pred(Y), Z) -> U32_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), zero, pred(X)) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), succ(Y), Z) -> U33_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), pred(Y), pred(pred(Z))) -> U34_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) U34_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U33_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U32_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U31_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U24_ga(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_ga(Order, Ans, isTimes_in_gga(Order, U, U1)) isTimes_in_gga(X, zero, zero) -> isTimes_out_gga(X, zero, zero) isTimes_in_gga(zero, succ(Y), zero) -> isTimes_out_gga(zero, succ(Y), zero) isTimes_in_gga(zero, pred(Y), zero) -> isTimes_out_gga(zero, pred(Y), zero) isTimes_in_gga(succ(X), succ(Y), Z) -> U41_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(succ(X), pred(Y), Z) -> U43_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(pred(X), succ(Y), Z) -> U45_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) isTimes_in_gga(pred(X), pred(Y), Z) -> U47_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) U47_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_gga(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U48_gga(X, Y, Z, isMinus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), pred(Y), Z) U45_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_gga(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U46_gga(X, Y, Z, isPlus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), succ(Y), Z) U43_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_gga(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U44_gga(X, Y, Z, isMinus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), pred(Y), Z) U41_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_gga(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U42_gga(X, Y, Z, isPlus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), succ(Y), Z) U25_ga(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_ga(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U26_ga(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_ga(Order, Ans, =_in_ag(N, U2)) =_in_ag(X, X) -> =_out_ag(X, X) U27_ga(Order, Ans, =_out_ag(N, U2)) -> U28_ga(Order, Ans, N, iota_in_ga(N, .(zero, List))) iota_in_ga(N, List) -> U2_ga(N, List, iota1_in_gga(zero, N, List)) iota1_in_gga(K, K, []) -> iota1_out_gga(K, K, []) iota1_in_gga(K, N, .(K, List)) -> U3_gga(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_gga(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_gga(K, N, List, =_in_ag(K1, U)) U4_gga(K, N, List, =_out_ag(K1, U)) -> U5_gga(K, N, List, iota1_in_gga(K1, N, List)) U5_gga(K, N, List, iota1_out_gga(K1, N, List)) -> iota1_out_gga(K, N, .(K, List)) U2_ga(N, List, iota1_out_gga(zero, N, List)) -> iota_out_ga(N, List) U28_ga(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_ga(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) pds1_in_ggag([], X6, [], X7) -> pds1_out_ggag([], X6, [], X7) pds1_in_ggag(Unused, List, .(Choice, Rest), Mod) -> U21_ggag(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(X, .(X1, Xs)) -> U1_ag(X, X1, Xs, member_in_ag(X, Xs)) U1_ag(X, X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) U21_ggag(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_ggag(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) check_in_gggag([], X4, L, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, L3, Modulus) -> U17_gggag(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) dif_in_gggaa([], X2, X3, [], []) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U17_gggag(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) mergedelete_in_gga([], L, L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R), L2) -> U12_gga(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) mergedelete_in_gga(.(D, Ds), .(X, R), .(X, L2)) -> U13_gga(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_gga(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_gga(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U15_gga(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U16_gga(D, Ds, X, R, L2, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U12_gga(D, Ds, R, L2, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) rev_in_gga([], L, L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y, L) -> U11_gga(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) U11_gga(X, Xs, Y, L, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U22_ggag(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_ggag(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U23_ggag(Unused, List, Choice, Rest, Mod, pds1_out_ggag(U3, .(Choice, List), Rest, Mod)) -> pds1_out_ggag(Unused, List, .(Choice, Rest), Mod) U29_ga(Order, Ans, pds1_out_ggag(List, .(zero, []), Ans, N)) -> pds_out_ga(Order, .(zero, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U24_ga(x1, x2, x3) = U24_ga(x1, x3) isPlus_in_gga(x1, x2, x3) = isPlus_in_gga(x1, x2) zero = zero isPlus_out_gga(x1, x2, x3) = isPlus_out_gga(x1, x2, x3) succ(x1) = succ(x1) U31_gga(x1, x2, x3, x4) = U31_gga(x1, x2, x4) pred(x1) = pred(x1) U32_gga(x1, x2, x3, x4) = U32_gga(x1, x2, x4) U33_gga(x1, x2, x3, x4) = U33_gga(x1, x2, x4) U34_gga(x1, x2, x3, x4) = U34_gga(x1, x2, x4) U25_ga(x1, x2, x3) = U25_ga(x1, x3) isTimes_in_gga(x1, x2, x3) = isTimes_in_gga(x1, x2) isTimes_out_gga(x1, x2, x3) = isTimes_out_gga(x1, x2, x3) U41_gga(x1, x2, x3, x4) = U41_gga(x1, x2, x4) U43_gga(x1, x2, x3, x4) = U43_gga(x1, x2, x4) U45_gga(x1, x2, x3, x4) = U45_gga(x1, x2, x4) U47_gga(x1, x2, x3, x4) = U47_gga(x1, x2, x4) U48_gga(x1, x2, x3, x4) = U48_gga(x1, x2, x4) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x1, x2, x3) U35_gga(x1, x2, x3) = U35_gga(x1, x3) U36_gga(x1, x2, x3) = U36_gga(x1, x3) U37_gga(x1, x2, x3, x4) = U37_gga(x1, x2, x4) U38_gga(x1, x2, x3, x4) = U38_gga(x1, x2, x4) U39_gga(x1, x2, x3, x4) = U39_gga(x1, x2, x4) U40_gga(x1, x2, x3, x4) = U40_gga(x1, x2, x4) U46_gga(x1, x2, x3, x4) = U46_gga(x1, x2, x4) U44_gga(x1, x2, x3, x4) = U44_gga(x1, x2, x4) U42_gga(x1, x2, x3, x4) = U42_gga(x1, x2, x4) U26_ga(x1, x2, x3) = U26_ga(x1, x3) U27_ga(x1, x2, x3) = U27_ga(x1, x3) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1, x2) U28_ga(x1, x2, x3, x4) = U28_ga(x1, x3, x4) iota_in_ga(x1, x2) = iota_in_ga(x1) U2_ga(x1, x2, x3) = U2_ga(x1, x3) iota1_in_gga(x1, x2, x3) = iota1_in_gga(x1, x2) iota1_out_gga(x1, x2, x3) = iota1_out_gga(x1, x2, x3) U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4) iota_out_ga(x1, x2) = iota_out_ga(x1, x2) .(x1, x2) = .(x1, x2) U29_ga(x1, x2, x3) = U29_ga(x1, x3) pds1_in_ggag(x1, x2, x3, x4) = pds1_in_ggag(x1, x2, x4) [] = [] pds1_out_ggag(x1, x2, x3, x4) = pds1_out_ggag(x1, x2, x3, x4) U21_ggag(x1, x2, x3, x4, x5, x6) = U21_ggag(x1, x2, x5, x6) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1, x2) U1_ag(x1, x2, x3, x4) = U1_ag(x2, x3, x4) U22_ggag(x1, x2, x3, x4, x5, x6) = U22_ggag(x1, x2, x3, x5, x6) check_in_gggag(x1, x2, x3, x4, x5) = check_in_gggag(x1, x2, x3, x5) check_out_gggag(x1, x2, x3, x4, x5) = check_out_gggag(x1, x2, x3, x4, x5) U17_gggag(x1, x2, x3, x4, x5, x6) = U17_gggag(x1, x2, x3, x5, x6) dif_in_gggaa(x1, x2, x3, x4, x5) = dif_in_gggaa(x1, x2, x3) dif_out_gggaa(x1, x2, x3, x4, x5) = dif_out_gggaa(x1, x2, x3, x4, x5) U6_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gggaa(x1, x2, x3, x4, x9) U7_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_gggaa(x1, x2, x3, x4, x9, x10) U8_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_gggaa(x1, x2, x3, x4, x5, x9, x10) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg(x1, x2, x3) U35_ggg(x1, x2, x3) = U35_ggg(x1, x2, x3) U36_ggg(x1, x2, x3) = U36_ggg(x1, x2, x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x1, x2, x3, x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x1, x2, x3, x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x1, x2, x3, x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x1, x2, x3, x4) U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_gggaa(x1, x2, x3, x4, x5, x9) U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_gggaa(x1, x2, x3, x4, x5, x7, x9) U18_gggag(x1, x2, x3, x4, x5, x6, x7) = U18_gggag(x1, x2, x3, x5, x6, x7) mergedelete_in_gga(x1, x2, x3) = mergedelete_in_gga(x1, x2) mergedelete_out_gga(x1, x2, x3) = mergedelete_out_gga(x1, x2, x3) U12_gga(x1, x2, x3, x4, x5) = U12_gga(x1, x2, x3, x5) U13_gga(x1, x2, x3, x4, x5, x6) = U13_gga(x1, x2, x3, x4, x6) U14_gga(x1, x2, x3, x4, x5, x6, x7) = U14_gga(x1, x2, x3, x4, x6, x7) U15_gga(x1, x2, x3, x4, x5, x6) = U15_gga(x1, x2, x3, x4, x6) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg(x1, x2) U64_gg(x1, x2, x3) = U64_gg(x1, x2, x3) U65_gg(x1, x2, x3) = U65_gg(x1, x2, x3) U16_gga(x1, x2, x3, x4, x5, x6) = U16_gga(x1, x2, x3, x4, x6) U19_gggag(x1, x2, x3, x4, x5, x6, x7) = U19_gggag(x1, x2, x3, x5, x6, x7) rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2) rev_out_gga(x1, x2, x3) = rev_out_gga(x1, x2, x3) U11_gga(x1, x2, x3, x4, x5) = U11_gga(x1, x2, x3, x5) U20_gggag(x1, x2, x3, x4, x5, x6) = U20_gggag(x1, x2, x3, x5, x6) U23_ggag(x1, x2, x3, x4, x5, x6) = U23_ggag(x1, x2, x3, x5, x6) pds_out_ga(x1, x2) = pds_out_ga(x1, x2) MEMBER_IN_AG(x1, x2) = MEMBER_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (151) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (152) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER_IN_AG(X, .(X1, 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 ---------------------------------------- (153) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (154) Obligation: Q DP problem: The TRS P consists of the following rules: MEMBER_IN_AG(.(X1, Xs)) -> MEMBER_IN_AG(Xs) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (155) 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(.(X1, Xs)) -> MEMBER_IN_AG(Xs) The graph contains the following edges 1 > 1 ---------------------------------------- (156) YES ---------------------------------------- (157) Obligation: Pi DP problem: The TRS P consists of the following rules: ISMINUS_IN_GGA(zero, pred(Y), succ(Z)) -> ISMINUS_IN_GGA(zero, Y, Z) ISMINUS_IN_GGA(zero, succ(Y), pred(Z)) -> ISMINUS_IN_GGA(zero, Y, Z) The TRS R consists of the following rules: pds_in_ga(Order, .(zero, Ans)) -> U24_ga(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) isPlus_in_gga(zero, X, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), zero, succ(X)) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), succ(Y), succ(succ(Z))) -> U31_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(succ(X), pred(Y), Z) -> U32_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), zero, pred(X)) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), succ(Y), Z) -> U33_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), pred(Y), pred(pred(Z))) -> U34_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) U34_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U33_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U32_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U31_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U24_ga(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_ga(Order, Ans, isTimes_in_gga(Order, U, U1)) isTimes_in_gga(X, zero, zero) -> isTimes_out_gga(X, zero, zero) isTimes_in_gga(zero, succ(Y), zero) -> isTimes_out_gga(zero, succ(Y), zero) isTimes_in_gga(zero, pred(Y), zero) -> isTimes_out_gga(zero, pred(Y), zero) isTimes_in_gga(succ(X), succ(Y), Z) -> U41_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(succ(X), pred(Y), Z) -> U43_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(pred(X), succ(Y), Z) -> U45_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) isTimes_in_gga(pred(X), pred(Y), Z) -> U47_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) U47_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_gga(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U48_gga(X, Y, Z, isMinus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), pred(Y), Z) U45_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_gga(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U46_gga(X, Y, Z, isPlus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), succ(Y), Z) U43_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_gga(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U44_gga(X, Y, Z, isMinus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), pred(Y), Z) U41_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_gga(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U42_gga(X, Y, Z, isPlus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), succ(Y), Z) U25_ga(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_ga(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U26_ga(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_ga(Order, Ans, =_in_ag(N, U2)) =_in_ag(X, X) -> =_out_ag(X, X) U27_ga(Order, Ans, =_out_ag(N, U2)) -> U28_ga(Order, Ans, N, iota_in_ga(N, .(zero, List))) iota_in_ga(N, List) -> U2_ga(N, List, iota1_in_gga(zero, N, List)) iota1_in_gga(K, K, []) -> iota1_out_gga(K, K, []) iota1_in_gga(K, N, .(K, List)) -> U3_gga(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_gga(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_gga(K, N, List, =_in_ag(K1, U)) U4_gga(K, N, List, =_out_ag(K1, U)) -> U5_gga(K, N, List, iota1_in_gga(K1, N, List)) U5_gga(K, N, List, iota1_out_gga(K1, N, List)) -> iota1_out_gga(K, N, .(K, List)) U2_ga(N, List, iota1_out_gga(zero, N, List)) -> iota_out_ga(N, List) U28_ga(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_ga(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) pds1_in_ggag([], X6, [], X7) -> pds1_out_ggag([], X6, [], X7) pds1_in_ggag(Unused, List, .(Choice, Rest), Mod) -> U21_ggag(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(X, .(X1, Xs)) -> U1_ag(X, X1, Xs, member_in_ag(X, Xs)) U1_ag(X, X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) U21_ggag(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_ggag(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) check_in_gggag([], X4, L, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, L3, Modulus) -> U17_gggag(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) dif_in_gggaa([], X2, X3, [], []) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U17_gggag(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) mergedelete_in_gga([], L, L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R), L2) -> U12_gga(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) mergedelete_in_gga(.(D, Ds), .(X, R), .(X, L2)) -> U13_gga(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_gga(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_gga(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U15_gga(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U16_gga(D, Ds, X, R, L2, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U12_gga(D, Ds, R, L2, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) rev_in_gga([], L, L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y, L) -> U11_gga(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) U11_gga(X, Xs, Y, L, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U22_ggag(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_ggag(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U23_ggag(Unused, List, Choice, Rest, Mod, pds1_out_ggag(U3, .(Choice, List), Rest, Mod)) -> pds1_out_ggag(Unused, List, .(Choice, Rest), Mod) U29_ga(Order, Ans, pds1_out_ggag(List, .(zero, []), Ans, N)) -> pds_out_ga(Order, .(zero, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U24_ga(x1, x2, x3) = U24_ga(x1, x3) isPlus_in_gga(x1, x2, x3) = isPlus_in_gga(x1, x2) zero = zero isPlus_out_gga(x1, x2, x3) = isPlus_out_gga(x1, x2, x3) succ(x1) = succ(x1) U31_gga(x1, x2, x3, x4) = U31_gga(x1, x2, x4) pred(x1) = pred(x1) U32_gga(x1, x2, x3, x4) = U32_gga(x1, x2, x4) U33_gga(x1, x2, x3, x4) = U33_gga(x1, x2, x4) U34_gga(x1, x2, x3, x4) = U34_gga(x1, x2, x4) U25_ga(x1, x2, x3) = U25_ga(x1, x3) isTimes_in_gga(x1, x2, x3) = isTimes_in_gga(x1, x2) isTimes_out_gga(x1, x2, x3) = isTimes_out_gga(x1, x2, x3) U41_gga(x1, x2, x3, x4) = U41_gga(x1, x2, x4) U43_gga(x1, x2, x3, x4) = U43_gga(x1, x2, x4) U45_gga(x1, x2, x3, x4) = U45_gga(x1, x2, x4) U47_gga(x1, x2, x3, x4) = U47_gga(x1, x2, x4) U48_gga(x1, x2, x3, x4) = U48_gga(x1, x2, x4) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x1, x2, x3) U35_gga(x1, x2, x3) = U35_gga(x1, x3) U36_gga(x1, x2, x3) = U36_gga(x1, x3) U37_gga(x1, x2, x3, x4) = U37_gga(x1, x2, x4) U38_gga(x1, x2, x3, x4) = U38_gga(x1, x2, x4) U39_gga(x1, x2, x3, x4) = U39_gga(x1, x2, x4) U40_gga(x1, x2, x3, x4) = U40_gga(x1, x2, x4) U46_gga(x1, x2, x3, x4) = U46_gga(x1, x2, x4) U44_gga(x1, x2, x3, x4) = U44_gga(x1, x2, x4) U42_gga(x1, x2, x3, x4) = U42_gga(x1, x2, x4) U26_ga(x1, x2, x3) = U26_ga(x1, x3) U27_ga(x1, x2, x3) = U27_ga(x1, x3) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1, x2) U28_ga(x1, x2, x3, x4) = U28_ga(x1, x3, x4) iota_in_ga(x1, x2) = iota_in_ga(x1) U2_ga(x1, x2, x3) = U2_ga(x1, x3) iota1_in_gga(x1, x2, x3) = iota1_in_gga(x1, x2) iota1_out_gga(x1, x2, x3) = iota1_out_gga(x1, x2, x3) U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4) iota_out_ga(x1, x2) = iota_out_ga(x1, x2) .(x1, x2) = .(x1, x2) U29_ga(x1, x2, x3) = U29_ga(x1, x3) pds1_in_ggag(x1, x2, x3, x4) = pds1_in_ggag(x1, x2, x4) [] = [] pds1_out_ggag(x1, x2, x3, x4) = pds1_out_ggag(x1, x2, x3, x4) U21_ggag(x1, x2, x3, x4, x5, x6) = U21_ggag(x1, x2, x5, x6) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1, x2) U1_ag(x1, x2, x3, x4) = U1_ag(x2, x3, x4) U22_ggag(x1, x2, x3, x4, x5, x6) = U22_ggag(x1, x2, x3, x5, x6) check_in_gggag(x1, x2, x3, x4, x5) = check_in_gggag(x1, x2, x3, x5) check_out_gggag(x1, x2, x3, x4, x5) = check_out_gggag(x1, x2, x3, x4, x5) U17_gggag(x1, x2, x3, x4, x5, x6) = U17_gggag(x1, x2, x3, x5, x6) dif_in_gggaa(x1, x2, x3, x4, x5) = dif_in_gggaa(x1, x2, x3) dif_out_gggaa(x1, x2, x3, x4, x5) = dif_out_gggaa(x1, x2, x3, x4, x5) U6_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gggaa(x1, x2, x3, x4, x9) U7_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_gggaa(x1, x2, x3, x4, x9, x10) U8_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_gggaa(x1, x2, x3, x4, x5, x9, x10) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg(x1, x2, x3) U35_ggg(x1, x2, x3) = U35_ggg(x1, x2, x3) U36_ggg(x1, x2, x3) = U36_ggg(x1, x2, x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x1, x2, x3, x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x1, x2, x3, x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x1, x2, x3, x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x1, x2, x3, x4) U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_gggaa(x1, x2, x3, x4, x5, x9) U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_gggaa(x1, x2, x3, x4, x5, x7, x9) U18_gggag(x1, x2, x3, x4, x5, x6, x7) = U18_gggag(x1, x2, x3, x5, x6, x7) mergedelete_in_gga(x1, x2, x3) = mergedelete_in_gga(x1, x2) mergedelete_out_gga(x1, x2, x3) = mergedelete_out_gga(x1, x2, x3) U12_gga(x1, x2, x3, x4, x5) = U12_gga(x1, x2, x3, x5) U13_gga(x1, x2, x3, x4, x5, x6) = U13_gga(x1, x2, x3, x4, x6) U14_gga(x1, x2, x3, x4, x5, x6, x7) = U14_gga(x1, x2, x3, x4, x6, x7) U15_gga(x1, x2, x3, x4, x5, x6) = U15_gga(x1, x2, x3, x4, x6) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg(x1, x2) U64_gg(x1, x2, x3) = U64_gg(x1, x2, x3) U65_gg(x1, x2, x3) = U65_gg(x1, x2, x3) U16_gga(x1, x2, x3, x4, x5, x6) = U16_gga(x1, x2, x3, x4, x6) U19_gggag(x1, x2, x3, x4, x5, x6, x7) = U19_gggag(x1, x2, x3, x5, x6, x7) rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2) rev_out_gga(x1, x2, x3) = rev_out_gga(x1, x2, x3) U11_gga(x1, x2, x3, x4, x5) = U11_gga(x1, x2, x3, x5) U20_gggag(x1, x2, x3, x4, x5, x6) = U20_gggag(x1, x2, x3, x5, x6) U23_ggag(x1, x2, x3, x4, x5, x6) = U23_ggag(x1, x2, x3, x5, x6) pds_out_ga(x1, x2) = pds_out_ga(x1, x2) ISMINUS_IN_GGA(x1, x2, x3) = ISMINUS_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (158) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (159) Obligation: Pi DP problem: The TRS P consists of the following rules: ISMINUS_IN_GGA(zero, pred(Y), succ(Z)) -> ISMINUS_IN_GGA(zero, Y, Z) ISMINUS_IN_GGA(zero, succ(Y), pred(Z)) -> ISMINUS_IN_GGA(zero, Y, Z) R is empty. The argument filtering Pi contains the following mapping: zero = zero succ(x1) = succ(x1) pred(x1) = pred(x1) ISMINUS_IN_GGA(x1, x2, x3) = ISMINUS_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (160) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (161) Obligation: Q DP problem: The TRS P consists of the following rules: ISMINUS_IN_GGA(zero, pred(Y)) -> ISMINUS_IN_GGA(zero, Y) ISMINUS_IN_GGA(zero, succ(Y)) -> ISMINUS_IN_GGA(zero, Y) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (162) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *ISMINUS_IN_GGA(zero, pred(Y)) -> ISMINUS_IN_GGA(zero, Y) The graph contains the following edges 1 >= 1, 2 > 2 *ISMINUS_IN_GGA(zero, succ(Y)) -> ISMINUS_IN_GGA(zero, Y) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (163) YES ---------------------------------------- (164) Obligation: Pi DP problem: The TRS P consists of the following rules: ISMINUS_IN_GGA(succ(X), pred(Y), succ(succ(Z))) -> ISMINUS_IN_GGA(X, Y, Z) ISMINUS_IN_GGA(succ(X), succ(Y), Z) -> ISMINUS_IN_GGA(X, Y, Z) ISMINUS_IN_GGA(pred(X), succ(Y), pred(pred(Z))) -> ISMINUS_IN_GGA(X, Y, Z) ISMINUS_IN_GGA(pred(X), pred(Y), Z) -> ISMINUS_IN_GGA(X, Y, Z) The TRS R consists of the following rules: pds_in_ga(Order, .(zero, Ans)) -> U24_ga(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) isPlus_in_gga(zero, X, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), zero, succ(X)) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), succ(Y), succ(succ(Z))) -> U31_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(succ(X), pred(Y), Z) -> U32_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), zero, pred(X)) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), succ(Y), Z) -> U33_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), pred(Y), pred(pred(Z))) -> U34_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) U34_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U33_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U32_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U31_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U24_ga(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_ga(Order, Ans, isTimes_in_gga(Order, U, U1)) isTimes_in_gga(X, zero, zero) -> isTimes_out_gga(X, zero, zero) isTimes_in_gga(zero, succ(Y), zero) -> isTimes_out_gga(zero, succ(Y), zero) isTimes_in_gga(zero, pred(Y), zero) -> isTimes_out_gga(zero, pred(Y), zero) isTimes_in_gga(succ(X), succ(Y), Z) -> U41_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(succ(X), pred(Y), Z) -> U43_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(pred(X), succ(Y), Z) -> U45_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) isTimes_in_gga(pred(X), pred(Y), Z) -> U47_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) U47_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_gga(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U48_gga(X, Y, Z, isMinus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), pred(Y), Z) U45_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_gga(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U46_gga(X, Y, Z, isPlus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), succ(Y), Z) U43_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_gga(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U44_gga(X, Y, Z, isMinus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), pred(Y), Z) U41_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_gga(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U42_gga(X, Y, Z, isPlus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), succ(Y), Z) U25_ga(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_ga(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U26_ga(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_ga(Order, Ans, =_in_ag(N, U2)) =_in_ag(X, X) -> =_out_ag(X, X) U27_ga(Order, Ans, =_out_ag(N, U2)) -> U28_ga(Order, Ans, N, iota_in_ga(N, .(zero, List))) iota_in_ga(N, List) -> U2_ga(N, List, iota1_in_gga(zero, N, List)) iota1_in_gga(K, K, []) -> iota1_out_gga(K, K, []) iota1_in_gga(K, N, .(K, List)) -> U3_gga(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_gga(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_gga(K, N, List, =_in_ag(K1, U)) U4_gga(K, N, List, =_out_ag(K1, U)) -> U5_gga(K, N, List, iota1_in_gga(K1, N, List)) U5_gga(K, N, List, iota1_out_gga(K1, N, List)) -> iota1_out_gga(K, N, .(K, List)) U2_ga(N, List, iota1_out_gga(zero, N, List)) -> iota_out_ga(N, List) U28_ga(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_ga(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) pds1_in_ggag([], X6, [], X7) -> pds1_out_ggag([], X6, [], X7) pds1_in_ggag(Unused, List, .(Choice, Rest), Mod) -> U21_ggag(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(X, .(X1, Xs)) -> U1_ag(X, X1, Xs, member_in_ag(X, Xs)) U1_ag(X, X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) U21_ggag(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_ggag(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) check_in_gggag([], X4, L, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, L3, Modulus) -> U17_gggag(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) dif_in_gggaa([], X2, X3, [], []) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U17_gggag(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) mergedelete_in_gga([], L, L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R), L2) -> U12_gga(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) mergedelete_in_gga(.(D, Ds), .(X, R), .(X, L2)) -> U13_gga(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_gga(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_gga(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U15_gga(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U16_gga(D, Ds, X, R, L2, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U12_gga(D, Ds, R, L2, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) rev_in_gga([], L, L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y, L) -> U11_gga(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) U11_gga(X, Xs, Y, L, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U22_ggag(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_ggag(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U23_ggag(Unused, List, Choice, Rest, Mod, pds1_out_ggag(U3, .(Choice, List), Rest, Mod)) -> pds1_out_ggag(Unused, List, .(Choice, Rest), Mod) U29_ga(Order, Ans, pds1_out_ggag(List, .(zero, []), Ans, N)) -> pds_out_ga(Order, .(zero, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U24_ga(x1, x2, x3) = U24_ga(x1, x3) isPlus_in_gga(x1, x2, x3) = isPlus_in_gga(x1, x2) zero = zero isPlus_out_gga(x1, x2, x3) = isPlus_out_gga(x1, x2, x3) succ(x1) = succ(x1) U31_gga(x1, x2, x3, x4) = U31_gga(x1, x2, x4) pred(x1) = pred(x1) U32_gga(x1, x2, x3, x4) = U32_gga(x1, x2, x4) U33_gga(x1, x2, x3, x4) = U33_gga(x1, x2, x4) U34_gga(x1, x2, x3, x4) = U34_gga(x1, x2, x4) U25_ga(x1, x2, x3) = U25_ga(x1, x3) isTimes_in_gga(x1, x2, x3) = isTimes_in_gga(x1, x2) isTimes_out_gga(x1, x2, x3) = isTimes_out_gga(x1, x2, x3) U41_gga(x1, x2, x3, x4) = U41_gga(x1, x2, x4) U43_gga(x1, x2, x3, x4) = U43_gga(x1, x2, x4) U45_gga(x1, x2, x3, x4) = U45_gga(x1, x2, x4) U47_gga(x1, x2, x3, x4) = U47_gga(x1, x2, x4) U48_gga(x1, x2, x3, x4) = U48_gga(x1, x2, x4) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x1, x2, x3) U35_gga(x1, x2, x3) = U35_gga(x1, x3) U36_gga(x1, x2, x3) = U36_gga(x1, x3) U37_gga(x1, x2, x3, x4) = U37_gga(x1, x2, x4) U38_gga(x1, x2, x3, x4) = U38_gga(x1, x2, x4) U39_gga(x1, x2, x3, x4) = U39_gga(x1, x2, x4) U40_gga(x1, x2, x3, x4) = U40_gga(x1, x2, x4) U46_gga(x1, x2, x3, x4) = U46_gga(x1, x2, x4) U44_gga(x1, x2, x3, x4) = U44_gga(x1, x2, x4) U42_gga(x1, x2, x3, x4) = U42_gga(x1, x2, x4) U26_ga(x1, x2, x3) = U26_ga(x1, x3) U27_ga(x1, x2, x3) = U27_ga(x1, x3) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1, x2) U28_ga(x1, x2, x3, x4) = U28_ga(x1, x3, x4) iota_in_ga(x1, x2) = iota_in_ga(x1) U2_ga(x1, x2, x3) = U2_ga(x1, x3) iota1_in_gga(x1, x2, x3) = iota1_in_gga(x1, x2) iota1_out_gga(x1, x2, x3) = iota1_out_gga(x1, x2, x3) U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4) iota_out_ga(x1, x2) = iota_out_ga(x1, x2) .(x1, x2) = .(x1, x2) U29_ga(x1, x2, x3) = U29_ga(x1, x3) pds1_in_ggag(x1, x2, x3, x4) = pds1_in_ggag(x1, x2, x4) [] = [] pds1_out_ggag(x1, x2, x3, x4) = pds1_out_ggag(x1, x2, x3, x4) U21_ggag(x1, x2, x3, x4, x5, x6) = U21_ggag(x1, x2, x5, x6) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1, x2) U1_ag(x1, x2, x3, x4) = U1_ag(x2, x3, x4) U22_ggag(x1, x2, x3, x4, x5, x6) = U22_ggag(x1, x2, x3, x5, x6) check_in_gggag(x1, x2, x3, x4, x5) = check_in_gggag(x1, x2, x3, x5) check_out_gggag(x1, x2, x3, x4, x5) = check_out_gggag(x1, x2, x3, x4, x5) U17_gggag(x1, x2, x3, x4, x5, x6) = U17_gggag(x1, x2, x3, x5, x6) dif_in_gggaa(x1, x2, x3, x4, x5) = dif_in_gggaa(x1, x2, x3) dif_out_gggaa(x1, x2, x3, x4, x5) = dif_out_gggaa(x1, x2, x3, x4, x5) U6_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gggaa(x1, x2, x3, x4, x9) U7_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_gggaa(x1, x2, x3, x4, x9, x10) U8_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_gggaa(x1, x2, x3, x4, x5, x9, x10) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg(x1, x2, x3) U35_ggg(x1, x2, x3) = U35_ggg(x1, x2, x3) U36_ggg(x1, x2, x3) = U36_ggg(x1, x2, x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x1, x2, x3, x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x1, x2, x3, x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x1, x2, x3, x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x1, x2, x3, x4) U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_gggaa(x1, x2, x3, x4, x5, x9) U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_gggaa(x1, x2, x3, x4, x5, x7, x9) U18_gggag(x1, x2, x3, x4, x5, x6, x7) = U18_gggag(x1, x2, x3, x5, x6, x7) mergedelete_in_gga(x1, x2, x3) = mergedelete_in_gga(x1, x2) mergedelete_out_gga(x1, x2, x3) = mergedelete_out_gga(x1, x2, x3) U12_gga(x1, x2, x3, x4, x5) = U12_gga(x1, x2, x3, x5) U13_gga(x1, x2, x3, x4, x5, x6) = U13_gga(x1, x2, x3, x4, x6) U14_gga(x1, x2, x3, x4, x5, x6, x7) = U14_gga(x1, x2, x3, x4, x6, x7) U15_gga(x1, x2, x3, x4, x5, x6) = U15_gga(x1, x2, x3, x4, x6) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg(x1, x2) U64_gg(x1, x2, x3) = U64_gg(x1, x2, x3) U65_gg(x1, x2, x3) = U65_gg(x1, x2, x3) U16_gga(x1, x2, x3, x4, x5, x6) = U16_gga(x1, x2, x3, x4, x6) U19_gggag(x1, x2, x3, x4, x5, x6, x7) = U19_gggag(x1, x2, x3, x5, x6, x7) rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2) rev_out_gga(x1, x2, x3) = rev_out_gga(x1, x2, x3) U11_gga(x1, x2, x3, x4, x5) = U11_gga(x1, x2, x3, x5) U20_gggag(x1, x2, x3, x4, x5, x6) = U20_gggag(x1, x2, x3, x5, x6) U23_ggag(x1, x2, x3, x4, x5, x6) = U23_ggag(x1, x2, x3, x5, x6) pds_out_ga(x1, x2) = pds_out_ga(x1, x2) ISMINUS_IN_GGA(x1, x2, x3) = ISMINUS_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (165) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (166) Obligation: Pi DP problem: The TRS P consists of the following rules: ISMINUS_IN_GGA(succ(X), pred(Y), succ(succ(Z))) -> ISMINUS_IN_GGA(X, Y, Z) ISMINUS_IN_GGA(succ(X), succ(Y), Z) -> ISMINUS_IN_GGA(X, Y, Z) ISMINUS_IN_GGA(pred(X), succ(Y), pred(pred(Z))) -> ISMINUS_IN_GGA(X, Y, Z) ISMINUS_IN_GGA(pred(X), pred(Y), Z) -> ISMINUS_IN_GGA(X, Y, Z) R is empty. The argument filtering Pi contains the following mapping: succ(x1) = succ(x1) pred(x1) = pred(x1) ISMINUS_IN_GGA(x1, x2, x3) = ISMINUS_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (167) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (168) Obligation: Q DP problem: The TRS P consists of the following rules: ISMINUS_IN_GGA(succ(X), pred(Y)) -> ISMINUS_IN_GGA(X, Y) ISMINUS_IN_GGA(succ(X), succ(Y)) -> ISMINUS_IN_GGA(X, Y) ISMINUS_IN_GGA(pred(X), succ(Y)) -> ISMINUS_IN_GGA(X, Y) ISMINUS_IN_GGA(pred(X), pred(Y)) -> ISMINUS_IN_GGA(X, Y) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (169) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *ISMINUS_IN_GGA(succ(X), pred(Y)) -> ISMINUS_IN_GGA(X, Y) The graph contains the following edges 1 > 1, 2 > 2 *ISMINUS_IN_GGA(succ(X), succ(Y)) -> ISMINUS_IN_GGA(X, Y) The graph contains the following edges 1 > 1, 2 > 2 *ISMINUS_IN_GGA(pred(X), succ(Y)) -> ISMINUS_IN_GGA(X, Y) The graph contains the following edges 1 > 1, 2 > 2 *ISMINUS_IN_GGA(pred(X), pred(Y)) -> ISMINUS_IN_GGA(X, Y) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (170) YES ---------------------------------------- (171) Obligation: Pi DP problem: The TRS P consists of the following rules: DIF_IN_GGGAA(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) U8_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> DIF_IN_GGGAA(Ss, Val, Mod, Ds, D2s) The TRS R consists of the following rules: pds_in_ga(Order, .(zero, Ans)) -> U24_ga(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) isPlus_in_gga(zero, X, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), zero, succ(X)) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), succ(Y), succ(succ(Z))) -> U31_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(succ(X), pred(Y), Z) -> U32_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), zero, pred(X)) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), succ(Y), Z) -> U33_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), pred(Y), pred(pred(Z))) -> U34_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) U34_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U33_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U32_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U31_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U24_ga(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_ga(Order, Ans, isTimes_in_gga(Order, U, U1)) isTimes_in_gga(X, zero, zero) -> isTimes_out_gga(X, zero, zero) isTimes_in_gga(zero, succ(Y), zero) -> isTimes_out_gga(zero, succ(Y), zero) isTimes_in_gga(zero, pred(Y), zero) -> isTimes_out_gga(zero, pred(Y), zero) isTimes_in_gga(succ(X), succ(Y), Z) -> U41_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(succ(X), pred(Y), Z) -> U43_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(pred(X), succ(Y), Z) -> U45_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) isTimes_in_gga(pred(X), pred(Y), Z) -> U47_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) U47_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_gga(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U48_gga(X, Y, Z, isMinus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), pred(Y), Z) U45_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_gga(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U46_gga(X, Y, Z, isPlus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), succ(Y), Z) U43_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_gga(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U44_gga(X, Y, Z, isMinus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), pred(Y), Z) U41_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_gga(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U42_gga(X, Y, Z, isPlus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), succ(Y), Z) U25_ga(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_ga(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U26_ga(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_ga(Order, Ans, =_in_ag(N, U2)) =_in_ag(X, X) -> =_out_ag(X, X) U27_ga(Order, Ans, =_out_ag(N, U2)) -> U28_ga(Order, Ans, N, iota_in_ga(N, .(zero, List))) iota_in_ga(N, List) -> U2_ga(N, List, iota1_in_gga(zero, N, List)) iota1_in_gga(K, K, []) -> iota1_out_gga(K, K, []) iota1_in_gga(K, N, .(K, List)) -> U3_gga(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_gga(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_gga(K, N, List, =_in_ag(K1, U)) U4_gga(K, N, List, =_out_ag(K1, U)) -> U5_gga(K, N, List, iota1_in_gga(K1, N, List)) U5_gga(K, N, List, iota1_out_gga(K1, N, List)) -> iota1_out_gga(K, N, .(K, List)) U2_ga(N, List, iota1_out_gga(zero, N, List)) -> iota_out_ga(N, List) U28_ga(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_ga(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) pds1_in_ggag([], X6, [], X7) -> pds1_out_ggag([], X6, [], X7) pds1_in_ggag(Unused, List, .(Choice, Rest), Mod) -> U21_ggag(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(X, .(X1, Xs)) -> U1_ag(X, X1, Xs, member_in_ag(X, Xs)) U1_ag(X, X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) U21_ggag(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_ggag(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) check_in_gggag([], X4, L, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, L3, Modulus) -> U17_gggag(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) dif_in_gggaa([], X2, X3, [], []) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U17_gggag(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) mergedelete_in_gga([], L, L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R), L2) -> U12_gga(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) mergedelete_in_gga(.(D, Ds), .(X, R), .(X, L2)) -> U13_gga(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_gga(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_gga(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U15_gga(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U16_gga(D, Ds, X, R, L2, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U12_gga(D, Ds, R, L2, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) rev_in_gga([], L, L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y, L) -> U11_gga(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) U11_gga(X, Xs, Y, L, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U22_ggag(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_ggag(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U23_ggag(Unused, List, Choice, Rest, Mod, pds1_out_ggag(U3, .(Choice, List), Rest, Mod)) -> pds1_out_ggag(Unused, List, .(Choice, Rest), Mod) U29_ga(Order, Ans, pds1_out_ggag(List, .(zero, []), Ans, N)) -> pds_out_ga(Order, .(zero, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U24_ga(x1, x2, x3) = U24_ga(x1, x3) isPlus_in_gga(x1, x2, x3) = isPlus_in_gga(x1, x2) zero = zero isPlus_out_gga(x1, x2, x3) = isPlus_out_gga(x1, x2, x3) succ(x1) = succ(x1) U31_gga(x1, x2, x3, x4) = U31_gga(x1, x2, x4) pred(x1) = pred(x1) U32_gga(x1, x2, x3, x4) = U32_gga(x1, x2, x4) U33_gga(x1, x2, x3, x4) = U33_gga(x1, x2, x4) U34_gga(x1, x2, x3, x4) = U34_gga(x1, x2, x4) U25_ga(x1, x2, x3) = U25_ga(x1, x3) isTimes_in_gga(x1, x2, x3) = isTimes_in_gga(x1, x2) isTimes_out_gga(x1, x2, x3) = isTimes_out_gga(x1, x2, x3) U41_gga(x1, x2, x3, x4) = U41_gga(x1, x2, x4) U43_gga(x1, x2, x3, x4) = U43_gga(x1, x2, x4) U45_gga(x1, x2, x3, x4) = U45_gga(x1, x2, x4) U47_gga(x1, x2, x3, x4) = U47_gga(x1, x2, x4) U48_gga(x1, x2, x3, x4) = U48_gga(x1, x2, x4) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x1, x2, x3) U35_gga(x1, x2, x3) = U35_gga(x1, x3) U36_gga(x1, x2, x3) = U36_gga(x1, x3) U37_gga(x1, x2, x3, x4) = U37_gga(x1, x2, x4) U38_gga(x1, x2, x3, x4) = U38_gga(x1, x2, x4) U39_gga(x1, x2, x3, x4) = U39_gga(x1, x2, x4) U40_gga(x1, x2, x3, x4) = U40_gga(x1, x2, x4) U46_gga(x1, x2, x3, x4) = U46_gga(x1, x2, x4) U44_gga(x1, x2, x3, x4) = U44_gga(x1, x2, x4) U42_gga(x1, x2, x3, x4) = U42_gga(x1, x2, x4) U26_ga(x1, x2, x3) = U26_ga(x1, x3) U27_ga(x1, x2, x3) = U27_ga(x1, x3) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1, x2) U28_ga(x1, x2, x3, x4) = U28_ga(x1, x3, x4) iota_in_ga(x1, x2) = iota_in_ga(x1) U2_ga(x1, x2, x3) = U2_ga(x1, x3) iota1_in_gga(x1, x2, x3) = iota1_in_gga(x1, x2) iota1_out_gga(x1, x2, x3) = iota1_out_gga(x1, x2, x3) U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4) iota_out_ga(x1, x2) = iota_out_ga(x1, x2) .(x1, x2) = .(x1, x2) U29_ga(x1, x2, x3) = U29_ga(x1, x3) pds1_in_ggag(x1, x2, x3, x4) = pds1_in_ggag(x1, x2, x4) [] = [] pds1_out_ggag(x1, x2, x3, x4) = pds1_out_ggag(x1, x2, x3, x4) U21_ggag(x1, x2, x3, x4, x5, x6) = U21_ggag(x1, x2, x5, x6) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1, x2) U1_ag(x1, x2, x3, x4) = U1_ag(x2, x3, x4) U22_ggag(x1, x2, x3, x4, x5, x6) = U22_ggag(x1, x2, x3, x5, x6) check_in_gggag(x1, x2, x3, x4, x5) = check_in_gggag(x1, x2, x3, x5) check_out_gggag(x1, x2, x3, x4, x5) = check_out_gggag(x1, x2, x3, x4, x5) U17_gggag(x1, x2, x3, x4, x5, x6) = U17_gggag(x1, x2, x3, x5, x6) dif_in_gggaa(x1, x2, x3, x4, x5) = dif_in_gggaa(x1, x2, x3) dif_out_gggaa(x1, x2, x3, x4, x5) = dif_out_gggaa(x1, x2, x3, x4, x5) U6_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gggaa(x1, x2, x3, x4, x9) U7_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_gggaa(x1, x2, x3, x4, x9, x10) U8_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_gggaa(x1, x2, x3, x4, x5, x9, x10) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg(x1, x2, x3) U35_ggg(x1, x2, x3) = U35_ggg(x1, x2, x3) U36_ggg(x1, x2, x3) = U36_ggg(x1, x2, x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x1, x2, x3, x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x1, x2, x3, x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x1, x2, x3, x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x1, x2, x3, x4) U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_gggaa(x1, x2, x3, x4, x5, x9) U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_gggaa(x1, x2, x3, x4, x5, x7, x9) U18_gggag(x1, x2, x3, x4, x5, x6, x7) = U18_gggag(x1, x2, x3, x5, x6, x7) mergedelete_in_gga(x1, x2, x3) = mergedelete_in_gga(x1, x2) mergedelete_out_gga(x1, x2, x3) = mergedelete_out_gga(x1, x2, x3) U12_gga(x1, x2, x3, x4, x5) = U12_gga(x1, x2, x3, x5) U13_gga(x1, x2, x3, x4, x5, x6) = U13_gga(x1, x2, x3, x4, x6) U14_gga(x1, x2, x3, x4, x5, x6, x7) = U14_gga(x1, x2, x3, x4, x6, x7) U15_gga(x1, x2, x3, x4, x5, x6) = U15_gga(x1, x2, x3, x4, x6) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg(x1, x2) U64_gg(x1, x2, x3) = U64_gg(x1, x2, x3) U65_gg(x1, x2, x3) = U65_gg(x1, x2, x3) U16_gga(x1, x2, x3, x4, x5, x6) = U16_gga(x1, x2, x3, x4, x6) U19_gggag(x1, x2, x3, x4, x5, x6, x7) = U19_gggag(x1, x2, x3, x5, x6, x7) rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2) rev_out_gga(x1, x2, x3) = rev_out_gga(x1, x2, x3) U11_gga(x1, x2, x3, x4, x5) = U11_gga(x1, x2, x3, x5) U20_gggag(x1, x2, x3, x4, x5, x6) = U20_gggag(x1, x2, x3, x5, x6) U23_ggag(x1, x2, x3, x4, x5, x6) = U23_ggag(x1, x2, x3, x5, x6) pds_out_ga(x1, x2) = pds_out_ga(x1, x2) DIF_IN_GGGAA(x1, x2, x3, x4, x5) = DIF_IN_GGGAA(x1, x2, x3) U6_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_GGGAA(x1, x2, x3, x4, x9) U7_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_GGGAA(x1, x2, x3, x4, x9, x10) U8_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_GGGAA(x1, x2, x3, x4, x5, x9, x10) U9_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_GGGAA(x1, x2, x3, x4, x5, x9) We have to consider all (P,R,Pi)-chains ---------------------------------------- (172) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (173) Obligation: Pi DP problem: The TRS P consists of the following rules: DIF_IN_GGGAA(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) U8_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_GGGAA(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> DIF_IN_GGGAA(Ss, Val, Mod, Ds, D2s) The TRS R consists of the following rules: isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) =_in_ag(X, X) -> =_out_ag(X, X) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) The argument filtering Pi contains the following mapping: zero = zero succ(x1) = succ(x1) pred(x1) = pred(x1) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x1, x2, x3) U35_gga(x1, x2, x3) = U35_gga(x1, x3) U36_gga(x1, x2, x3) = U36_gga(x1, x3) U37_gga(x1, x2, x3, x4) = U37_gga(x1, x2, x4) U38_gga(x1, x2, x3, x4) = U38_gga(x1, x2, x4) U39_gga(x1, x2, x3, x4) = U39_gga(x1, x2, x4) U40_gga(x1, x2, x3, x4) = U40_gga(x1, x2, x4) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1, x2) .(x1, x2) = .(x1, x2) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg(x1, x2, x3) U35_ggg(x1, x2, x3) = U35_ggg(x1, x2, x3) U36_ggg(x1, x2, x3) = U36_ggg(x1, x2, x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x1, x2, x3, x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x1, x2, x3, x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x1, x2, x3, x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x1, x2, x3, x4) DIF_IN_GGGAA(x1, x2, x3, x4, x5) = DIF_IN_GGGAA(x1, x2, x3) U6_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_GGGAA(x1, x2, x3, x4, x9) U7_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_GGGAA(x1, x2, x3, x4, x9, x10) U8_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_GGGAA(x1, x2, x3, x4, x5, x9, x10) U9_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_GGGAA(x1, x2, x3, x4, x5, x9) We have to consider all (P,R,Pi)-chains ---------------------------------------- (174) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (175) Obligation: Q DP problem: The TRS P consists of the following rules: DIF_IN_GGGAA(.(S, Ss), Val, Mod) -> U6_GGGAA(S, Ss, Val, Mod, isMinus_in_gga(Val, S)) U6_GGGAA(S, Ss, Val, Mod, isMinus_out_gga(Val, S, U)) -> U7_GGGAA(S, Ss, Val, Mod, U, =_in_ag(U)) U7_GGGAA(S, Ss, Val, Mod, U, =_out_ag(D, U)) -> U8_GGGAA(S, Ss, Val, Mod, D, U, isMinus_in_ggg(Mod, D, U)) U8_GGGAA(S, Ss, Val, Mod, D, U, isMinus_out_ggg(Mod, D, U)) -> U9_GGGAA(S, Ss, Val, Mod, D, =_in_ag(U)) U9_GGGAA(S, Ss, Val, Mod, D, =_out_ag(D2, U)) -> DIF_IN_GGGAA(Ss, Val, Mod) The TRS R consists of the following rules: isMinus_in_gga(X, zero) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y)) -> U35_gga(Y, isMinus_in_gga(zero, Y)) isMinus_in_gga(zero, pred(Y)) -> U36_gga(Y, isMinus_in_gga(zero, Y)) isMinus_in_gga(succ(X), succ(Y)) -> U37_gga(X, Y, isMinus_in_gga(X, Y)) isMinus_in_gga(succ(X), pred(Y)) -> U38_gga(X, Y, isMinus_in_gga(X, Y)) isMinus_in_gga(pred(X), succ(Y)) -> U39_gga(X, Y, isMinus_in_gga(X, Y)) isMinus_in_gga(pred(X), pred(Y)) -> U40_gga(X, Y, isMinus_in_gga(X, Y)) =_in_ag(X) -> =_out_ag(X, X) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U35_gga(Y, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U36_gga(Y, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U37_gga(X, Y, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U38_gga(X, Y, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U39_gga(X, Y, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U40_gga(X, Y, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) The set Q consists of the following terms: isMinus_in_gga(x0, x1) =_in_ag(x0) isMinus_in_ggg(x0, x1, x2) U35_gga(x0, x1) U36_gga(x0, x1) U37_gga(x0, x1, x2) U38_gga(x0, x1, x2) U39_gga(x0, x1, x2) U40_gga(x0, x1, x2) U35_ggg(x0, x1, x2) U36_ggg(x0, x1, x2) U37_ggg(x0, x1, x2, x3) U38_ggg(x0, x1, x2, x3) U39_ggg(x0, x1, x2, x3) U40_ggg(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (176) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *U6_GGGAA(S, Ss, Val, Mod, isMinus_out_gga(Val, S, U)) -> U7_GGGAA(S, Ss, Val, Mod, U, =_in_ag(U)) The graph contains the following edges 1 >= 1, 5 > 1, 2 >= 2, 3 >= 3, 5 > 3, 4 >= 4, 5 > 5 *U9_GGGAA(S, Ss, Val, Mod, D, =_out_ag(D2, U)) -> DIF_IN_GGGAA(Ss, Val, Mod) The graph contains the following edges 2 >= 1, 3 >= 2, 4 >= 3 *U7_GGGAA(S, Ss, Val, Mod, U, =_out_ag(D, U)) -> U8_GGGAA(S, Ss, Val, Mod, D, U, isMinus_in_ggg(Mod, D, U)) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 6 > 5, 5 >= 6, 6 > 6 *DIF_IN_GGGAA(.(S, Ss), Val, Mod) -> U6_GGGAA(S, Ss, Val, Mod, isMinus_in_gga(Val, S)) The graph contains the following edges 1 > 1, 1 > 2, 2 >= 3, 3 >= 4 *U8_GGGAA(S, Ss, Val, Mod, D, U, isMinus_out_ggg(Mod, D, U)) -> U9_GGGAA(S, Ss, Val, Mod, D, =_in_ag(U)) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 7 > 4, 5 >= 5, 7 > 5 ---------------------------------------- (177) YES ---------------------------------------- (178) Obligation: Pi DP problem: The TRS P consists of the following rules: U21_GGAG(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_GGAG(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) U22_GGAG(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> PDS1_IN_GGAG(U3, .(Choice, List), Rest, Mod) PDS1_IN_GGAG(Unused, List, .(Choice, Rest), Mod) -> U21_GGAG(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) The TRS R consists of the following rules: pds_in_ga(Order, .(zero, Ans)) -> U24_ga(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) isPlus_in_gga(zero, X, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), zero, succ(X)) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), succ(Y), succ(succ(Z))) -> U31_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(succ(X), pred(Y), Z) -> U32_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), zero, pred(X)) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), succ(Y), Z) -> U33_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), pred(Y), pred(pred(Z))) -> U34_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) U34_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U33_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U32_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U31_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U24_ga(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_ga(Order, Ans, isTimes_in_gga(Order, U, U1)) isTimes_in_gga(X, zero, zero) -> isTimes_out_gga(X, zero, zero) isTimes_in_gga(zero, succ(Y), zero) -> isTimes_out_gga(zero, succ(Y), zero) isTimes_in_gga(zero, pred(Y), zero) -> isTimes_out_gga(zero, pred(Y), zero) isTimes_in_gga(succ(X), succ(Y), Z) -> U41_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(succ(X), pred(Y), Z) -> U43_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(pred(X), succ(Y), Z) -> U45_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) isTimes_in_gga(pred(X), pred(Y), Z) -> U47_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) U47_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_gga(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U48_gga(X, Y, Z, isMinus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), pred(Y), Z) U45_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_gga(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U46_gga(X, Y, Z, isPlus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), succ(Y), Z) U43_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_gga(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U44_gga(X, Y, Z, isMinus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), pred(Y), Z) U41_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_gga(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U42_gga(X, Y, Z, isPlus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), succ(Y), Z) U25_ga(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_ga(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U26_ga(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_ga(Order, Ans, =_in_ag(N, U2)) =_in_ag(X, X) -> =_out_ag(X, X) U27_ga(Order, Ans, =_out_ag(N, U2)) -> U28_ga(Order, Ans, N, iota_in_ga(N, .(zero, List))) iota_in_ga(N, List) -> U2_ga(N, List, iota1_in_gga(zero, N, List)) iota1_in_gga(K, K, []) -> iota1_out_gga(K, K, []) iota1_in_gga(K, N, .(K, List)) -> U3_gga(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_gga(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_gga(K, N, List, =_in_ag(K1, U)) U4_gga(K, N, List, =_out_ag(K1, U)) -> U5_gga(K, N, List, iota1_in_gga(K1, N, List)) U5_gga(K, N, List, iota1_out_gga(K1, N, List)) -> iota1_out_gga(K, N, .(K, List)) U2_ga(N, List, iota1_out_gga(zero, N, List)) -> iota_out_ga(N, List) U28_ga(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_ga(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) pds1_in_ggag([], X6, [], X7) -> pds1_out_ggag([], X6, [], X7) pds1_in_ggag(Unused, List, .(Choice, Rest), Mod) -> U21_ggag(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(X, .(X1, Xs)) -> U1_ag(X, X1, Xs, member_in_ag(X, Xs)) U1_ag(X, X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) U21_ggag(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_ggag(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) check_in_gggag([], X4, L, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, L3, Modulus) -> U17_gggag(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) dif_in_gggaa([], X2, X3, [], []) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U17_gggag(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) mergedelete_in_gga([], L, L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R), L2) -> U12_gga(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) mergedelete_in_gga(.(D, Ds), .(X, R), .(X, L2)) -> U13_gga(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_gga(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_gga(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U15_gga(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U16_gga(D, Ds, X, R, L2, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U12_gga(D, Ds, R, L2, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) rev_in_gga([], L, L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y, L) -> U11_gga(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) U11_gga(X, Xs, Y, L, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U22_ggag(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_ggag(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U23_ggag(Unused, List, Choice, Rest, Mod, pds1_out_ggag(U3, .(Choice, List), Rest, Mod)) -> pds1_out_ggag(Unused, List, .(Choice, Rest), Mod) U29_ga(Order, Ans, pds1_out_ggag(List, .(zero, []), Ans, N)) -> pds_out_ga(Order, .(zero, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U24_ga(x1, x2, x3) = U24_ga(x1, x3) isPlus_in_gga(x1, x2, x3) = isPlus_in_gga(x1, x2) zero = zero isPlus_out_gga(x1, x2, x3) = isPlus_out_gga(x1, x2, x3) succ(x1) = succ(x1) U31_gga(x1, x2, x3, x4) = U31_gga(x1, x2, x4) pred(x1) = pred(x1) U32_gga(x1, x2, x3, x4) = U32_gga(x1, x2, x4) U33_gga(x1, x2, x3, x4) = U33_gga(x1, x2, x4) U34_gga(x1, x2, x3, x4) = U34_gga(x1, x2, x4) U25_ga(x1, x2, x3) = U25_ga(x1, x3) isTimes_in_gga(x1, x2, x3) = isTimes_in_gga(x1, x2) isTimes_out_gga(x1, x2, x3) = isTimes_out_gga(x1, x2, x3) U41_gga(x1, x2, x3, x4) = U41_gga(x1, x2, x4) U43_gga(x1, x2, x3, x4) = U43_gga(x1, x2, x4) U45_gga(x1, x2, x3, x4) = U45_gga(x1, x2, x4) U47_gga(x1, x2, x3, x4) = U47_gga(x1, x2, x4) U48_gga(x1, x2, x3, x4) = U48_gga(x1, x2, x4) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x1, x2, x3) U35_gga(x1, x2, x3) = U35_gga(x1, x3) U36_gga(x1, x2, x3) = U36_gga(x1, x3) U37_gga(x1, x2, x3, x4) = U37_gga(x1, x2, x4) U38_gga(x1, x2, x3, x4) = U38_gga(x1, x2, x4) U39_gga(x1, x2, x3, x4) = U39_gga(x1, x2, x4) U40_gga(x1, x2, x3, x4) = U40_gga(x1, x2, x4) U46_gga(x1, x2, x3, x4) = U46_gga(x1, x2, x4) U44_gga(x1, x2, x3, x4) = U44_gga(x1, x2, x4) U42_gga(x1, x2, x3, x4) = U42_gga(x1, x2, x4) U26_ga(x1, x2, x3) = U26_ga(x1, x3) U27_ga(x1, x2, x3) = U27_ga(x1, x3) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1, x2) U28_ga(x1, x2, x3, x4) = U28_ga(x1, x3, x4) iota_in_ga(x1, x2) = iota_in_ga(x1) U2_ga(x1, x2, x3) = U2_ga(x1, x3) iota1_in_gga(x1, x2, x3) = iota1_in_gga(x1, x2) iota1_out_gga(x1, x2, x3) = iota1_out_gga(x1, x2, x3) U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4) iota_out_ga(x1, x2) = iota_out_ga(x1, x2) .(x1, x2) = .(x1, x2) U29_ga(x1, x2, x3) = U29_ga(x1, x3) pds1_in_ggag(x1, x2, x3, x4) = pds1_in_ggag(x1, x2, x4) [] = [] pds1_out_ggag(x1, x2, x3, x4) = pds1_out_ggag(x1, x2, x3, x4) U21_ggag(x1, x2, x3, x4, x5, x6) = U21_ggag(x1, x2, x5, x6) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1, x2) U1_ag(x1, x2, x3, x4) = U1_ag(x2, x3, x4) U22_ggag(x1, x2, x3, x4, x5, x6) = U22_ggag(x1, x2, x3, x5, x6) check_in_gggag(x1, x2, x3, x4, x5) = check_in_gggag(x1, x2, x3, x5) check_out_gggag(x1, x2, x3, x4, x5) = check_out_gggag(x1, x2, x3, x4, x5) U17_gggag(x1, x2, x3, x4, x5, x6) = U17_gggag(x1, x2, x3, x5, x6) dif_in_gggaa(x1, x2, x3, x4, x5) = dif_in_gggaa(x1, x2, x3) dif_out_gggaa(x1, x2, x3, x4, x5) = dif_out_gggaa(x1, x2, x3, x4, x5) U6_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gggaa(x1, x2, x3, x4, x9) U7_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_gggaa(x1, x2, x3, x4, x9, x10) U8_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_gggaa(x1, x2, x3, x4, x5, x9, x10) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg(x1, x2, x3) U35_ggg(x1, x2, x3) = U35_ggg(x1, x2, x3) U36_ggg(x1, x2, x3) = U36_ggg(x1, x2, x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x1, x2, x3, x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x1, x2, x3, x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x1, x2, x3, x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x1, x2, x3, x4) U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_gggaa(x1, x2, x3, x4, x5, x9) U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_gggaa(x1, x2, x3, x4, x5, x7, x9) U18_gggag(x1, x2, x3, x4, x5, x6, x7) = U18_gggag(x1, x2, x3, x5, x6, x7) mergedelete_in_gga(x1, x2, x3) = mergedelete_in_gga(x1, x2) mergedelete_out_gga(x1, x2, x3) = mergedelete_out_gga(x1, x2, x3) U12_gga(x1, x2, x3, x4, x5) = U12_gga(x1, x2, x3, x5) U13_gga(x1, x2, x3, x4, x5, x6) = U13_gga(x1, x2, x3, x4, x6) U14_gga(x1, x2, x3, x4, x5, x6, x7) = U14_gga(x1, x2, x3, x4, x6, x7) U15_gga(x1, x2, x3, x4, x5, x6) = U15_gga(x1, x2, x3, x4, x6) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg(x1, x2) U64_gg(x1, x2, x3) = U64_gg(x1, x2, x3) U65_gg(x1, x2, x3) = U65_gg(x1, x2, x3) U16_gga(x1, x2, x3, x4, x5, x6) = U16_gga(x1, x2, x3, x4, x6) U19_gggag(x1, x2, x3, x4, x5, x6, x7) = U19_gggag(x1, x2, x3, x5, x6, x7) rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2) rev_out_gga(x1, x2, x3) = rev_out_gga(x1, x2, x3) U11_gga(x1, x2, x3, x4, x5) = U11_gga(x1, x2, x3, x5) U20_gggag(x1, x2, x3, x4, x5, x6) = U20_gggag(x1, x2, x3, x5, x6) U23_ggag(x1, x2, x3, x4, x5, x6) = U23_ggag(x1, x2, x3, x5, x6) pds_out_ga(x1, x2) = pds_out_ga(x1, x2) PDS1_IN_GGAG(x1, x2, x3, x4) = PDS1_IN_GGAG(x1, x2, x4) U21_GGAG(x1, x2, x3, x4, x5, x6) = U21_GGAG(x1, x2, x5, x6) U22_GGAG(x1, x2, x3, x4, x5, x6) = U22_GGAG(x1, x2, x3, x5, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (179) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (180) Obligation: Pi DP problem: The TRS P consists of the following rules: U21_GGAG(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_GGAG(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) U22_GGAG(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> PDS1_IN_GGAG(U3, .(Choice, List), Rest, Mod) PDS1_IN_GGAG(Unused, List, .(Choice, Rest), Mod) -> U21_GGAG(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) The TRS R consists of the following rules: check_in_gggag([], X4, L, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, L3, Modulus) -> U17_gggag(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(X, .(X1, Xs)) -> U1_ag(X, X1, Xs, member_in_ag(X, Xs)) U17_gggag(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) U1_ag(X, X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) dif_in_gggaa([], X2, X3, [], []) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) mergedelete_in_gga([], L, L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R), L2) -> U12_gga(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) mergedelete_in_gga(.(D, Ds), .(X, R), .(X, L2)) -> U13_gga(D, Ds, X, R, L2, =_in_ag(X1, D)) U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) U12_gga(D, Ds, R, L2, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U13_gga(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) rev_in_gga([], L, L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y, L) -> U11_gga(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) =_in_ag(X, X) -> =_out_ag(X, X) U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U14_gga(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) U11_gga(X, Xs, Y, L, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U15_gga(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U16_gga(D, Ds, X, R, L2, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) The argument filtering Pi contains the following mapping: zero = zero succ(x1) = succ(x1) pred(x1) = pred(x1) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x1, x2, x3) U35_gga(x1, x2, x3) = U35_gga(x1, x3) U36_gga(x1, x2, x3) = U36_gga(x1, x3) U37_gga(x1, x2, x3, x4) = U37_gga(x1, x2, x4) U38_gga(x1, x2, x3, x4) = U38_gga(x1, x2, x4) U39_gga(x1, x2, x3, x4) = U39_gga(x1, x2, x4) U40_gga(x1, x2, x3, x4) = U40_gga(x1, x2, x4) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1, x2) .(x1, x2) = .(x1, x2) [] = [] member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1, x2) U1_ag(x1, x2, x3, x4) = U1_ag(x2, x3, x4) check_in_gggag(x1, x2, x3, x4, x5) = check_in_gggag(x1, x2, x3, x5) check_out_gggag(x1, x2, x3, x4, x5) = check_out_gggag(x1, x2, x3, x4, x5) U17_gggag(x1, x2, x3, x4, x5, x6) = U17_gggag(x1, x2, x3, x5, x6) dif_in_gggaa(x1, x2, x3, x4, x5) = dif_in_gggaa(x1, x2, x3) dif_out_gggaa(x1, x2, x3, x4, x5) = dif_out_gggaa(x1, x2, x3, x4, x5) U6_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gggaa(x1, x2, x3, x4, x9) U7_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_gggaa(x1, x2, x3, x4, x9, x10) U8_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_gggaa(x1, x2, x3, x4, x5, x9, x10) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg(x1, x2, x3) U35_ggg(x1, x2, x3) = U35_ggg(x1, x2, x3) U36_ggg(x1, x2, x3) = U36_ggg(x1, x2, x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x1, x2, x3, x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x1, x2, x3, x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x1, x2, x3, x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x1, x2, x3, x4) U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_gggaa(x1, x2, x3, x4, x5, x9) U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_gggaa(x1, x2, x3, x4, x5, x7, x9) U18_gggag(x1, x2, x3, x4, x5, x6, x7) = U18_gggag(x1, x2, x3, x5, x6, x7) mergedelete_in_gga(x1, x2, x3) = mergedelete_in_gga(x1, x2) mergedelete_out_gga(x1, x2, x3) = mergedelete_out_gga(x1, x2, x3) U12_gga(x1, x2, x3, x4, x5) = U12_gga(x1, x2, x3, x5) U13_gga(x1, x2, x3, x4, x5, x6) = U13_gga(x1, x2, x3, x4, x6) U14_gga(x1, x2, x3, x4, x5, x6, x7) = U14_gga(x1, x2, x3, x4, x6, x7) U15_gga(x1, x2, x3, x4, x5, x6) = U15_gga(x1, x2, x3, x4, x6) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg(x1, x2) U64_gg(x1, x2, x3) = U64_gg(x1, x2, x3) U65_gg(x1, x2, x3) = U65_gg(x1, x2, x3) U16_gga(x1, x2, x3, x4, x5, x6) = U16_gga(x1, x2, x3, x4, x6) U19_gggag(x1, x2, x3, x4, x5, x6, x7) = U19_gggag(x1, x2, x3, x5, x6, x7) rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2) rev_out_gga(x1, x2, x3) = rev_out_gga(x1, x2, x3) U11_gga(x1, x2, x3, x4, x5) = U11_gga(x1, x2, x3, x5) U20_gggag(x1, x2, x3, x4, x5, x6) = U20_gggag(x1, x2, x3, x5, x6) PDS1_IN_GGAG(x1, x2, x3, x4) = PDS1_IN_GGAG(x1, x2, x4) U21_GGAG(x1, x2, x3, x4, x5, x6) = U21_GGAG(x1, x2, x5, x6) U22_GGAG(x1, x2, x3, x4, x5, x6) = U22_GGAG(x1, x2, x3, x5, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (181) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (182) Obligation: Q DP problem: The TRS P consists of the following rules: U21_GGAG(Unused, List, Mod, member_out_ag(Choice, Unused)) -> U22_GGAG(Unused, List, Choice, Mod, check_in_gggag(List, Choice, Unused, Mod)) U22_GGAG(Unused, List, Choice, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> PDS1_IN_GGAG(U3, .(Choice, List), Mod) PDS1_IN_GGAG(Unused, List, Mod) -> U21_GGAG(Unused, List, Mod, member_in_ag(Unused)) The TRS R consists of the following rules: check_in_gggag([], X4, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, Modulus) -> U17_gggag(S, Choice, Old, Modulus, dif_in_gggaa(S, Choice, Modulus)) member_in_ag(.(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(.(X1, Xs)) -> U1_ag(X1, Xs, member_in_ag(Xs)) U17_gggag(S, Choice, Old, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, Modulus, Dds, mergedelete_in_gga(Ds, Old)) U1_ag(X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) dif_in_gggaa([], X2, X3) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod) -> U6_gggaa(S, Ss, Val, Mod, isMinus_in_gga(Val, S)) U18_gggag(S, Choice, Old, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, Modulus, L2, rev_in_gga(Dds, [])) U6_gggaa(S, Ss, Val, Mod, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, U, =_in_ag(U)) mergedelete_in_gga([], L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R)) -> U12_gga(D, Ds, R, mergedelete_in_gga(Ds, R)) mergedelete_in_gga(.(D, Ds), .(X, R)) -> U13_gga(D, Ds, X, R, =_in_ag(D)) U19_gggag(S, Choice, Old, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, Modulus, mergedelete_in_gga(Rds, L2)) isMinus_in_gga(X, zero) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y)) -> U35_gga(Y, isMinus_in_gga(zero, Y)) isMinus_in_gga(zero, pred(Y)) -> U36_gga(Y, isMinus_in_gga(zero, Y)) isMinus_in_gga(succ(X), succ(Y)) -> U37_gga(X, Y, isMinus_in_gga(X, Y)) isMinus_in_gga(succ(X), pred(Y)) -> U38_gga(X, Y, isMinus_in_gga(X, Y)) isMinus_in_gga(pred(X), succ(Y)) -> U39_gga(X, Y, isMinus_in_gga(X, Y)) isMinus_in_gga(pred(X), pred(Y)) -> U40_gga(X, Y, isMinus_in_gga(X, Y)) U7_gggaa(S, Ss, Val, Mod, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, U, isMinus_in_ggg(Mod, D, U)) U12_gga(D, Ds, R, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U13_gga(D, Ds, X, R, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, X1, =_in_ag(X)) rev_in_gga([], L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y) -> U11_gga(X, Xs, Y, rev_in_gga(Xs, .(X, Y))) U20_gggag(S, Choice, Old, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U35_gga(Y, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U36_gga(Y, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U37_gga(X, Y, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U38_gga(X, Y, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U39_gga(X, Y, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U40_gga(X, Y, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) =_in_ag(X) -> =_out_ag(X, X) U8_gggaa(S, Ss, Val, Mod, D, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, =_in_ag(U)) U14_gga(D, Ds, X, R, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, isGreater_in_gg(X1, X2)) U11_gga(X, Xs, Y, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U9_gggaa(S, Ss, Val, Mod, D, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, D2, dif_in_gggaa(Ss, Val, Mod)) U15_gga(D, Ds, X, R, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, mergedelete_in_gga(.(D, Ds), R)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U10_gggaa(S, Ss, Val, Mod, D, D2, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U16_gga(D, Ds, X, R, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) The set Q consists of the following terms: check_in_gggag(x0, x1, x2, x3) member_in_ag(x0) U17_gggag(x0, x1, x2, x3, x4) U1_ag(x0, x1, x2) dif_in_gggaa(x0, x1, x2) U18_gggag(x0, x1, x2, x3, x4, x5) U6_gggaa(x0, x1, x2, x3, x4) mergedelete_in_gga(x0, x1) U19_gggag(x0, x1, x2, x3, x4, x5) isMinus_in_gga(x0, x1) U7_gggaa(x0, x1, x2, x3, x4, x5) U12_gga(x0, x1, x2, x3) U13_gga(x0, x1, x2, x3, x4) rev_in_gga(x0, x1) U20_gggag(x0, x1, x2, x3, x4) U35_gga(x0, x1) U36_gga(x0, x1) U37_gga(x0, x1, x2) U38_gga(x0, x1, x2) U39_gga(x0, x1, x2) U40_gga(x0, x1, x2) =_in_ag(x0) U8_gggaa(x0, x1, x2, x3, x4, x5, x6) U14_gga(x0, x1, x2, x3, x4, x5) U11_gga(x0, x1, x2, x3) isMinus_in_ggg(x0, x1, x2) U9_gggaa(x0, x1, x2, x3, x4, x5) U15_gga(x0, x1, x2, x3, x4) U35_ggg(x0, x1, x2) U36_ggg(x0, x1, x2) U37_ggg(x0, x1, x2, x3) U38_ggg(x0, x1, x2, x3) U39_ggg(x0, x1, x2, x3) U40_ggg(x0, x1, x2, x3) U10_gggaa(x0, x1, x2, x3, x4, x5, x6) isGreater_in_gg(x0, x1) U16_gga(x0, x1, x2, x3, x4) U64_gg(x0, x1, x2) U65_gg(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (183) 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: U14_gga(D, Ds, X, R, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, isGreater_in_gg(X1, X2)) Used ordering: Polynomial interpretation [POLO]: POL(.(x_1, x_2)) = 2 + 2*x_2 POL(=_in_ag(x_1)) = 1 POL(=_out_ag(x_1, x_2)) = 1 POL(PDS1_IN_GGAG(x_1, x_2, x_3)) = 2*x_1 POL(U10_gggaa(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = 0 POL(U11_gga(x_1, x_2, x_3, x_4)) = 1 POL(U12_gga(x_1, x_2, x_3, x_4)) = 2 + 2*x_4 POL(U13_gga(x_1, x_2, x_3, x_4, x_5)) = 2*x_4 + 2*x_5 POL(U14_gga(x_1, x_2, x_3, x_4, x_5, x_6)) = 2*x_4 + 2*x_6 POL(U15_gga(x_1, x_2, x_3, x_4, x_5)) = 2*x_4 POL(U16_gga(x_1, x_2, x_3, x_4, x_5)) = 2*x_5 POL(U17_gggag(x_1, x_2, x_3, x_4, x_5)) = x_3 POL(U18_gggag(x_1, x_2, x_3, x_4, x_5, x_6)) = x_6 POL(U19_gggag(x_1, x_2, x_3, x_4, x_5, x_6)) = x_5 POL(U1_ag(x_1, x_2, x_3)) = 0 POL(U20_gggag(x_1, x_2, x_3, x_4, x_5)) = x_5 POL(U21_GGAG(x_1, x_2, x_3, x_4)) = 2*x_1 POL(U22_GGAG(x_1, x_2, x_3, x_4, x_5)) = 2*x_5 POL(U35_gga(x_1, x_2)) = 0 POL(U35_ggg(x_1, x_2, x_3)) = 0 POL(U36_gga(x_1, x_2)) = 0 POL(U36_ggg(x_1, x_2, x_3)) = 0 POL(U37_gga(x_1, x_2, x_3)) = 0 POL(U37_ggg(x_1, x_2, x_3, x_4)) = 0 POL(U38_gga(x_1, x_2, x_3)) = 0 POL(U38_ggg(x_1, x_2, x_3, x_4)) = 0 POL(U39_gga(x_1, x_2, x_3)) = 0 POL(U39_ggg(x_1, x_2, x_3, x_4)) = 0 POL(U40_gga(x_1, x_2, x_3)) = 0 POL(U40_ggg(x_1, x_2, x_3, x_4)) = 0 POL(U64_gg(x_1, x_2, x_3)) = 0 POL(U65_gg(x_1, x_2, x_3)) = 0 POL(U6_gggaa(x_1, x_2, x_3, x_4, x_5)) = 0 POL(U7_gggaa(x_1, x_2, x_3, x_4, x_5, x_6)) = 0 POL(U8_gggaa(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = 0 POL(U9_gggaa(x_1, x_2, x_3, x_4, x_5, x_6)) = 0 POL([]) = 0 POL(check_in_gggag(x_1, x_2, x_3, x_4)) = x_3 POL(check_out_gggag(x_1, x_2, x_3, x_4, x_5)) = x_4 POL(dif_in_gggaa(x_1, x_2, x_3)) = 0 POL(dif_out_gggaa(x_1, x_2, x_3, x_4, x_5)) = 0 POL(isGreater_in_gg(x_1, x_2)) = 0 POL(isGreater_out_gg(x_1, x_2)) = 0 POL(isMinus_in_gga(x_1, x_2)) = 0 POL(isMinus_in_ggg(x_1, x_2, x_3)) = 0 POL(isMinus_out_gga(x_1, x_2, x_3)) = 0 POL(isMinus_out_ggg(x_1, x_2, x_3)) = 0 POL(member_in_ag(x_1)) = 0 POL(member_out_ag(x_1, x_2)) = 0 POL(mergedelete_in_gga(x_1, x_2)) = x_2 POL(mergedelete_out_gga(x_1, x_2, x_3)) = x_1 + x_3 POL(pred(x_1)) = 0 POL(rev_in_gga(x_1, x_2)) = x_1 POL(rev_out_gga(x_1, x_2, x_3)) = 0 POL(succ(x_1)) = 0 POL(zero) = 0 ---------------------------------------- (184) Obligation: Q DP problem: The TRS P consists of the following rules: U21_GGAG(Unused, List, Mod, member_out_ag(Choice, Unused)) -> U22_GGAG(Unused, List, Choice, Mod, check_in_gggag(List, Choice, Unused, Mod)) U22_GGAG(Unused, List, Choice, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> PDS1_IN_GGAG(U3, .(Choice, List), Mod) PDS1_IN_GGAG(Unused, List, Mod) -> U21_GGAG(Unused, List, Mod, member_in_ag(Unused)) The TRS R consists of the following rules: check_in_gggag([], X4, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, Modulus) -> U17_gggag(S, Choice, Old, Modulus, dif_in_gggaa(S, Choice, Modulus)) member_in_ag(.(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(.(X1, Xs)) -> U1_ag(X1, Xs, member_in_ag(Xs)) U17_gggag(S, Choice, Old, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, Modulus, Dds, mergedelete_in_gga(Ds, Old)) U1_ag(X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) dif_in_gggaa([], X2, X3) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod) -> U6_gggaa(S, Ss, Val, Mod, isMinus_in_gga(Val, S)) U18_gggag(S, Choice, Old, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, Modulus, L2, rev_in_gga(Dds, [])) U6_gggaa(S, Ss, Val, Mod, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, U, =_in_ag(U)) mergedelete_in_gga([], L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R)) -> U12_gga(D, Ds, R, mergedelete_in_gga(Ds, R)) mergedelete_in_gga(.(D, Ds), .(X, R)) -> U13_gga(D, Ds, X, R, =_in_ag(D)) U19_gggag(S, Choice, Old, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, Modulus, mergedelete_in_gga(Rds, L2)) isMinus_in_gga(X, zero) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y)) -> U35_gga(Y, isMinus_in_gga(zero, Y)) isMinus_in_gga(zero, pred(Y)) -> U36_gga(Y, isMinus_in_gga(zero, Y)) isMinus_in_gga(succ(X), succ(Y)) -> U37_gga(X, Y, isMinus_in_gga(X, Y)) isMinus_in_gga(succ(X), pred(Y)) -> U38_gga(X, Y, isMinus_in_gga(X, Y)) isMinus_in_gga(pred(X), succ(Y)) -> U39_gga(X, Y, isMinus_in_gga(X, Y)) isMinus_in_gga(pred(X), pred(Y)) -> U40_gga(X, Y, isMinus_in_gga(X, Y)) U7_gggaa(S, Ss, Val, Mod, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, U, isMinus_in_ggg(Mod, D, U)) U12_gga(D, Ds, R, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U13_gga(D, Ds, X, R, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, X1, =_in_ag(X)) rev_in_gga([], L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y) -> U11_gga(X, Xs, Y, rev_in_gga(Xs, .(X, Y))) U20_gggag(S, Choice, Old, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U35_gga(Y, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U36_gga(Y, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U37_gga(X, Y, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U38_gga(X, Y, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U39_gga(X, Y, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U40_gga(X, Y, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) =_in_ag(X) -> =_out_ag(X, X) U8_gggaa(S, Ss, Val, Mod, D, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, =_in_ag(U)) U11_gga(X, Xs, Y, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U9_gggaa(S, Ss, Val, Mod, D, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, D2, dif_in_gggaa(Ss, Val, Mod)) U15_gga(D, Ds, X, R, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, mergedelete_in_gga(.(D, Ds), R)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U10_gggaa(S, Ss, Val, Mod, D, D2, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U16_gga(D, Ds, X, R, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) The set Q consists of the following terms: check_in_gggag(x0, x1, x2, x3) member_in_ag(x0) U17_gggag(x0, x1, x2, x3, x4) U1_ag(x0, x1, x2) dif_in_gggaa(x0, x1, x2) U18_gggag(x0, x1, x2, x3, x4, x5) U6_gggaa(x0, x1, x2, x3, x4) mergedelete_in_gga(x0, x1) U19_gggag(x0, x1, x2, x3, x4, x5) isMinus_in_gga(x0, x1) U7_gggaa(x0, x1, x2, x3, x4, x5) U12_gga(x0, x1, x2, x3) U13_gga(x0, x1, x2, x3, x4) rev_in_gga(x0, x1) U20_gggag(x0, x1, x2, x3, x4) U35_gga(x0, x1) U36_gga(x0, x1) U37_gga(x0, x1, x2) U38_gga(x0, x1, x2) U39_gga(x0, x1, x2) U40_gga(x0, x1, x2) =_in_ag(x0) U8_gggaa(x0, x1, x2, x3, x4, x5, x6) U14_gga(x0, x1, x2, x3, x4, x5) U11_gga(x0, x1, x2, x3) isMinus_in_ggg(x0, x1, x2) U9_gggaa(x0, x1, x2, x3, x4, x5) U15_gga(x0, x1, x2, x3, x4) U35_ggg(x0, x1, x2) U36_ggg(x0, x1, x2) U37_ggg(x0, x1, x2, x3) U38_ggg(x0, x1, x2, x3) U39_ggg(x0, x1, x2, x3) U40_ggg(x0, x1, x2, x3) U10_gggaa(x0, x1, x2, x3, x4, x5, x6) isGreater_in_gg(x0, x1) U16_gga(x0, x1, x2, x3, x4) U64_gg(x0, x1, x2) U65_gg(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (185) 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. ---------------------------------------- (186) Obligation: Q DP problem: The TRS P consists of the following rules: U21_GGAG(Unused, List, Mod, member_out_ag(Choice, Unused)) -> U22_GGAG(Unused, List, Choice, Mod, check_in_gggag(List, Choice, Unused, Mod)) U22_GGAG(Unused, List, Choice, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> PDS1_IN_GGAG(U3, .(Choice, List), Mod) PDS1_IN_GGAG(Unused, List, Mod) -> U21_GGAG(Unused, List, Mod, member_in_ag(Unused)) The TRS R consists of the following rules: member_in_ag(.(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(.(X1, Xs)) -> U1_ag(X1, Xs, member_in_ag(Xs)) U1_ag(X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) check_in_gggag([], X4, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, Modulus) -> U17_gggag(S, Choice, Old, Modulus, dif_in_gggaa(S, Choice, Modulus)) dif_in_gggaa([], X2, X3) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod) -> U6_gggaa(S, Ss, Val, Mod, isMinus_in_gga(Val, S)) U17_gggag(S, Choice, Old, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, Modulus, Dds, mergedelete_in_gga(Ds, Old)) mergedelete_in_gga([], L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R)) -> U12_gga(D, Ds, R, mergedelete_in_gga(Ds, R)) mergedelete_in_gga(.(D, Ds), .(X, R)) -> U13_gga(D, Ds, X, R, =_in_ag(D)) U18_gggag(S, Choice, Old, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, Modulus, L2, rev_in_gga(Dds, [])) rev_in_gga([], L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y) -> U11_gga(X, Xs, Y, rev_in_gga(Xs, .(X, Y))) U19_gggag(S, Choice, Old, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, Modulus, mergedelete_in_gga(Rds, L2)) U20_gggag(S, Choice, Old, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U11_gga(X, Xs, Y, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) =_in_ag(X) -> =_out_ag(X, X) U13_gga(D, Ds, X, R, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, X1, =_in_ag(X)) U12_gga(D, Ds, R, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) isMinus_in_gga(X, zero) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y)) -> U35_gga(Y, isMinus_in_gga(zero, Y)) isMinus_in_gga(zero, pred(Y)) -> U36_gga(Y, isMinus_in_gga(zero, Y)) isMinus_in_gga(succ(X), succ(Y)) -> U37_gga(X, Y, isMinus_in_gga(X, Y)) isMinus_in_gga(succ(X), pred(Y)) -> U38_gga(X, Y, isMinus_in_gga(X, Y)) isMinus_in_gga(pred(X), succ(Y)) -> U39_gga(X, Y, isMinus_in_gga(X, Y)) isMinus_in_gga(pred(X), pred(Y)) -> U40_gga(X, Y, isMinus_in_gga(X, Y)) U6_gggaa(S, Ss, Val, Mod, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, U, =_in_ag(U)) U7_gggaa(S, Ss, Val, Mod, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U8_gggaa(S, Ss, Val, Mod, D, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, =_in_ag(U)) U9_gggaa(S, Ss, Val, Mod, D, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, D2, dif_in_gggaa(Ss, Val, Mod)) U10_gggaa(S, Ss, Val, Mod, D, D2, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U40_gga(X, Y, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) The set Q consists of the following terms: check_in_gggag(x0, x1, x2, x3) member_in_ag(x0) U17_gggag(x0, x1, x2, x3, x4) U1_ag(x0, x1, x2) dif_in_gggaa(x0, x1, x2) U18_gggag(x0, x1, x2, x3, x4, x5) U6_gggaa(x0, x1, x2, x3, x4) mergedelete_in_gga(x0, x1) U19_gggag(x0, x1, x2, x3, x4, x5) isMinus_in_gga(x0, x1) U7_gggaa(x0, x1, x2, x3, x4, x5) U12_gga(x0, x1, x2, x3) U13_gga(x0, x1, x2, x3, x4) rev_in_gga(x0, x1) U20_gggag(x0, x1, x2, x3, x4) U35_gga(x0, x1) U36_gga(x0, x1) U37_gga(x0, x1, x2) U38_gga(x0, x1, x2) U39_gga(x0, x1, x2) U40_gga(x0, x1, x2) =_in_ag(x0) U8_gggaa(x0, x1, x2, x3, x4, x5, x6) U14_gga(x0, x1, x2, x3, x4, x5) U11_gga(x0, x1, x2, x3) isMinus_in_ggg(x0, x1, x2) U9_gggaa(x0, x1, x2, x3, x4, x5) U15_gga(x0, x1, x2, x3, x4) U35_ggg(x0, x1, x2) U36_ggg(x0, x1, x2) U37_ggg(x0, x1, x2, x3) U38_ggg(x0, x1, x2, x3) U39_ggg(x0, x1, x2, x3) U40_ggg(x0, x1, x2, x3) U10_gggaa(x0, x1, x2, x3, x4, x5, x6) isGreater_in_gg(x0, x1) U16_gga(x0, x1, x2, x3, x4) U64_gg(x0, x1, x2) U65_gg(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (187) 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]. U15_gga(x0, x1, x2, x3, x4) isGreater_in_gg(x0, x1) U16_gga(x0, x1, x2, x3, x4) U64_gg(x0, x1, x2) U65_gg(x0, x1, x2) ---------------------------------------- (188) Obligation: Q DP problem: The TRS P consists of the following rules: U21_GGAG(Unused, List, Mod, member_out_ag(Choice, Unused)) -> U22_GGAG(Unused, List, Choice, Mod, check_in_gggag(List, Choice, Unused, Mod)) U22_GGAG(Unused, List, Choice, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> PDS1_IN_GGAG(U3, .(Choice, List), Mod) PDS1_IN_GGAG(Unused, List, Mod) -> U21_GGAG(Unused, List, Mod, member_in_ag(Unused)) The TRS R consists of the following rules: member_in_ag(.(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(.(X1, Xs)) -> U1_ag(X1, Xs, member_in_ag(Xs)) U1_ag(X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) check_in_gggag([], X4, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, Modulus) -> U17_gggag(S, Choice, Old, Modulus, dif_in_gggaa(S, Choice, Modulus)) dif_in_gggaa([], X2, X3) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod) -> U6_gggaa(S, Ss, Val, Mod, isMinus_in_gga(Val, S)) U17_gggag(S, Choice, Old, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, Modulus, Dds, mergedelete_in_gga(Ds, Old)) mergedelete_in_gga([], L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R)) -> U12_gga(D, Ds, R, mergedelete_in_gga(Ds, R)) mergedelete_in_gga(.(D, Ds), .(X, R)) -> U13_gga(D, Ds, X, R, =_in_ag(D)) U18_gggag(S, Choice, Old, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, Modulus, L2, rev_in_gga(Dds, [])) rev_in_gga([], L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y) -> U11_gga(X, Xs, Y, rev_in_gga(Xs, .(X, Y))) U19_gggag(S, Choice, Old, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, Modulus, mergedelete_in_gga(Rds, L2)) U20_gggag(S, Choice, Old, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U11_gga(X, Xs, Y, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) =_in_ag(X) -> =_out_ag(X, X) U13_gga(D, Ds, X, R, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, X1, =_in_ag(X)) U12_gga(D, Ds, R, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) isMinus_in_gga(X, zero) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y)) -> U35_gga(Y, isMinus_in_gga(zero, Y)) isMinus_in_gga(zero, pred(Y)) -> U36_gga(Y, isMinus_in_gga(zero, Y)) isMinus_in_gga(succ(X), succ(Y)) -> U37_gga(X, Y, isMinus_in_gga(X, Y)) isMinus_in_gga(succ(X), pred(Y)) -> U38_gga(X, Y, isMinus_in_gga(X, Y)) isMinus_in_gga(pred(X), succ(Y)) -> U39_gga(X, Y, isMinus_in_gga(X, Y)) isMinus_in_gga(pred(X), pred(Y)) -> U40_gga(X, Y, isMinus_in_gga(X, Y)) U6_gggaa(S, Ss, Val, Mod, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, U, =_in_ag(U)) U7_gggaa(S, Ss, Val, Mod, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U8_gggaa(S, Ss, Val, Mod, D, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, =_in_ag(U)) U9_gggaa(S, Ss, Val, Mod, D, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, D2, dif_in_gggaa(Ss, Val, Mod)) U10_gggaa(S, Ss, Val, Mod, D, D2, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U40_gga(X, Y, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) The set Q consists of the following terms: check_in_gggag(x0, x1, x2, x3) member_in_ag(x0) U17_gggag(x0, x1, x2, x3, x4) U1_ag(x0, x1, x2) dif_in_gggaa(x0, x1, x2) U18_gggag(x0, x1, x2, x3, x4, x5) U6_gggaa(x0, x1, x2, x3, x4) mergedelete_in_gga(x0, x1) U19_gggag(x0, x1, x2, x3, x4, x5) isMinus_in_gga(x0, x1) U7_gggaa(x0, x1, x2, x3, x4, x5) U12_gga(x0, x1, x2, x3) U13_gga(x0, x1, x2, x3, x4) rev_in_gga(x0, x1) U20_gggag(x0, x1, x2, x3, x4) U35_gga(x0, x1) U36_gga(x0, x1) U37_gga(x0, x1, x2) U38_gga(x0, x1, x2) U39_gga(x0, x1, x2) U40_gga(x0, x1, x2) =_in_ag(x0) U8_gggaa(x0, x1, x2, x3, x4, x5, x6) U14_gga(x0, x1, x2, x3, x4, x5) U11_gga(x0, x1, x2, x3) isMinus_in_ggg(x0, x1, x2) U9_gggaa(x0, x1, x2, x3, x4, x5) U35_ggg(x0, x1, x2) U36_ggg(x0, x1, x2) U37_ggg(x0, x1, x2, x3) U38_ggg(x0, x1, x2, x3) U39_ggg(x0, x1, x2, x3) U40_ggg(x0, x1, x2, x3) U10_gggaa(x0, x1, x2, x3, x4, x5, x6) We have to consider all (P,Q,R)-chains. ---------------------------------------- (189) 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: U12_gga(D, Ds, R, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) Used ordering: Polynomial interpretation [POLO]: POL(.(x_1, x_2)) = 2 + 2*x_2 POL(=_in_ag(x_1)) = 0 POL(=_out_ag(x_1, x_2)) = 0 POL(PDS1_IN_GGAG(x_1, x_2, x_3)) = 2*x_1 POL(U10_gggaa(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = 0 POL(U11_gga(x_1, x_2, x_3, x_4)) = 0 POL(U12_gga(x_1, x_2, x_3, x_4)) = 2 + 2*x_4 POL(U13_gga(x_1, x_2, x_3, x_4, x_5)) = 2 + 2*x_4 POL(U14_gga(x_1, x_2, x_3, x_4, x_5, x_6)) = 2 + 2*x_4 POL(U17_gggag(x_1, x_2, x_3, x_4, x_5)) = x_3 POL(U18_gggag(x_1, x_2, x_3, x_4, x_5, x_6)) = x_6 POL(U19_gggag(x_1, x_2, x_3, x_4, x_5, x_6)) = x_5 POL(U1_ag(x_1, x_2, x_3)) = 0 POL(U20_gggag(x_1, x_2, x_3, x_4, x_5)) = x_5 POL(U21_GGAG(x_1, x_2, x_3, x_4)) = 2*x_1 POL(U22_GGAG(x_1, x_2, x_3, x_4, x_5)) = 2*x_5 POL(U35_gga(x_1, x_2)) = 0 POL(U35_ggg(x_1, x_2, x_3)) = 0 POL(U36_gga(x_1, x_2)) = 0 POL(U36_ggg(x_1, x_2, x_3)) = 0 POL(U37_gga(x_1, x_2, x_3)) = 0 POL(U37_ggg(x_1, x_2, x_3, x_4)) = 0 POL(U38_gga(x_1, x_2, x_3)) = 0 POL(U38_ggg(x_1, x_2, x_3, x_4)) = 0 POL(U39_gga(x_1, x_2, x_3)) = 0 POL(U39_ggg(x_1, x_2, x_3, x_4)) = 1 POL(U40_gga(x_1, x_2, x_3)) = 0 POL(U40_ggg(x_1, x_2, x_3, x_4)) = 0 POL(U6_gggaa(x_1, x_2, x_3, x_4, x_5)) = 0 POL(U7_gggaa(x_1, x_2, x_3, x_4, x_5, x_6)) = 0 POL(U8_gggaa(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = 0 POL(U9_gggaa(x_1, x_2, x_3, x_4, x_5, x_6)) = 0 POL([]) = 0 POL(check_in_gggag(x_1, x_2, x_3, x_4)) = x_3 POL(check_out_gggag(x_1, x_2, x_3, x_4, x_5)) = x_4 POL(dif_in_gggaa(x_1, x_2, x_3)) = 0 POL(dif_out_gggaa(x_1, x_2, x_3, x_4, x_5)) = 0 POL(isMinus_in_gga(x_1, x_2)) = 0 POL(isMinus_in_ggg(x_1, x_2, x_3)) = 2*x_2 + x_3 POL(isMinus_out_gga(x_1, x_2, x_3)) = 0 POL(isMinus_out_ggg(x_1, x_2, x_3)) = 0 POL(member_in_ag(x_1)) = 0 POL(member_out_ag(x_1, x_2)) = 0 POL(mergedelete_in_gga(x_1, x_2)) = x_2 POL(mergedelete_out_gga(x_1, x_2, x_3)) = x_3 POL(pred(x_1)) = 2 POL(rev_in_gga(x_1, x_2)) = 0 POL(rev_out_gga(x_1, x_2, x_3)) = 0 POL(succ(x_1)) = x_1 POL(zero) = 0 ---------------------------------------- (190) Obligation: Q DP problem: The TRS P consists of the following rules: U21_GGAG(Unused, List, Mod, member_out_ag(Choice, Unused)) -> U22_GGAG(Unused, List, Choice, Mod, check_in_gggag(List, Choice, Unused, Mod)) U22_GGAG(Unused, List, Choice, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> PDS1_IN_GGAG(U3, .(Choice, List), Mod) PDS1_IN_GGAG(Unused, List, Mod) -> U21_GGAG(Unused, List, Mod, member_in_ag(Unused)) The TRS R consists of the following rules: member_in_ag(.(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(.(X1, Xs)) -> U1_ag(X1, Xs, member_in_ag(Xs)) U1_ag(X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) check_in_gggag([], X4, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, Modulus) -> U17_gggag(S, Choice, Old, Modulus, dif_in_gggaa(S, Choice, Modulus)) dif_in_gggaa([], X2, X3) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod) -> U6_gggaa(S, Ss, Val, Mod, isMinus_in_gga(Val, S)) U17_gggag(S, Choice, Old, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, Modulus, Dds, mergedelete_in_gga(Ds, Old)) mergedelete_in_gga([], L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R)) -> U12_gga(D, Ds, R, mergedelete_in_gga(Ds, R)) mergedelete_in_gga(.(D, Ds), .(X, R)) -> U13_gga(D, Ds, X, R, =_in_ag(D)) U18_gggag(S, Choice, Old, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, Modulus, L2, rev_in_gga(Dds, [])) rev_in_gga([], L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y) -> U11_gga(X, Xs, Y, rev_in_gga(Xs, .(X, Y))) U19_gggag(S, Choice, Old, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, Modulus, mergedelete_in_gga(Rds, L2)) U20_gggag(S, Choice, Old, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U11_gga(X, Xs, Y, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) =_in_ag(X) -> =_out_ag(X, X) U13_gga(D, Ds, X, R, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, X1, =_in_ag(X)) isMinus_in_gga(X, zero) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y)) -> U35_gga(Y, isMinus_in_gga(zero, Y)) isMinus_in_gga(zero, pred(Y)) -> U36_gga(Y, isMinus_in_gga(zero, Y)) isMinus_in_gga(succ(X), succ(Y)) -> U37_gga(X, Y, isMinus_in_gga(X, Y)) isMinus_in_gga(succ(X), pred(Y)) -> U38_gga(X, Y, isMinus_in_gga(X, Y)) isMinus_in_gga(pred(X), succ(Y)) -> U39_gga(X, Y, isMinus_in_gga(X, Y)) isMinus_in_gga(pred(X), pred(Y)) -> U40_gga(X, Y, isMinus_in_gga(X, Y)) U6_gggaa(S, Ss, Val, Mod, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, U, =_in_ag(U)) U7_gggaa(S, Ss, Val, Mod, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U8_gggaa(S, Ss, Val, Mod, D, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, =_in_ag(U)) U9_gggaa(S, Ss, Val, Mod, D, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, D2, dif_in_gggaa(Ss, Val, Mod)) U10_gggaa(S, Ss, Val, Mod, D, D2, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U40_gga(X, Y, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) The set Q consists of the following terms: check_in_gggag(x0, x1, x2, x3) member_in_ag(x0) U17_gggag(x0, x1, x2, x3, x4) U1_ag(x0, x1, x2) dif_in_gggaa(x0, x1, x2) U18_gggag(x0, x1, x2, x3, x4, x5) U6_gggaa(x0, x1, x2, x3, x4) mergedelete_in_gga(x0, x1) U19_gggag(x0, x1, x2, x3, x4, x5) isMinus_in_gga(x0, x1) U7_gggaa(x0, x1, x2, x3, x4, x5) U12_gga(x0, x1, x2, x3) U13_gga(x0, x1, x2, x3, x4) rev_in_gga(x0, x1) U20_gggag(x0, x1, x2, x3, x4) U35_gga(x0, x1) U36_gga(x0, x1) U37_gga(x0, x1, x2) U38_gga(x0, x1, x2) U39_gga(x0, x1, x2) U40_gga(x0, x1, x2) =_in_ag(x0) U8_gggaa(x0, x1, x2, x3, x4, x5, x6) U14_gga(x0, x1, x2, x3, x4, x5) U11_gga(x0, x1, x2, x3) isMinus_in_ggg(x0, x1, x2) U9_gggaa(x0, x1, x2, x3, x4, x5) U35_ggg(x0, x1, x2) U36_ggg(x0, x1, x2) U37_ggg(x0, x1, x2, x3) U38_ggg(x0, x1, x2, x3) U39_ggg(x0, x1, x2, x3) U40_ggg(x0, x1, x2, x3) U10_gggaa(x0, x1, x2, x3, x4, x5, x6) We have to consider all (P,Q,R)-chains. ---------------------------------------- (191) 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: check_in_gggag([], X4, L, X5) -> check_out_gggag([], X4, L, L, X5) U19_gggag(S, Choice, Old, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, Modulus, mergedelete_in_gga(Rds, L2)) Used ordering: Polynomial interpretation [POLO]: POL(.(x_1, x_2)) = 0 POL(=_in_ag(x_1)) = 0 POL(=_out_ag(x_1, x_2)) = 0 POL(PDS1_IN_GGAG(x_1, x_2, x_3)) = 2*x_2 POL(U10_gggaa(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = 0 POL(U11_gga(x_1, x_2, x_3, x_4)) = 0 POL(U12_gga(x_1, x_2, x_3, x_4)) = 0 POL(U13_gga(x_1, x_2, x_3, x_4, x_5)) = 0 POL(U14_gga(x_1, x_2, x_3, x_4, x_5, x_6)) = 0 POL(U17_gggag(x_1, x_2, x_3, x_4, x_5)) = x_5 POL(U18_gggag(x_1, x_2, x_3, x_4, x_5, x_6)) = x_6 POL(U19_gggag(x_1, x_2, x_3, x_4, x_5, x_6)) = 1 POL(U1_ag(x_1, x_2, x_3)) = 0 POL(U20_gggag(x_1, x_2, x_3, x_4, x_5)) = 0 POL(U21_GGAG(x_1, x_2, x_3, x_4)) = 2*x_2 POL(U22_GGAG(x_1, x_2, x_3, x_4, x_5)) = 2*x_5 POL(U35_gga(x_1, x_2)) = 0 POL(U35_ggg(x_1, x_2, x_3)) = 0 POL(U36_gga(x_1, x_2)) = 0 POL(U36_ggg(x_1, x_2, x_3)) = 0 POL(U37_gga(x_1, x_2, x_3)) = 0 POL(U37_ggg(x_1, x_2, x_3, x_4)) = 0 POL(U38_gga(x_1, x_2, x_3)) = 0 POL(U38_ggg(x_1, x_2, x_3, x_4)) = 0 POL(U39_gga(x_1, x_2, x_3)) = 0 POL(U39_ggg(x_1, x_2, x_3, x_4)) = 0 POL(U40_gga(x_1, x_2, x_3)) = 0 POL(U40_ggg(x_1, x_2, x_3, x_4)) = 0 POL(U6_gggaa(x_1, x_2, x_3, x_4, x_5)) = 0 POL(U7_gggaa(x_1, x_2, x_3, x_4, x_5, x_6)) = 0 POL(U8_gggaa(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = 0 POL(U9_gggaa(x_1, x_2, x_3, x_4, x_5, x_6)) = 0 POL([]) = 1 POL(check_in_gggag(x_1, x_2, x_3, x_4)) = x_1 POL(check_out_gggag(x_1, x_2, x_3, x_4, x_5)) = 0 POL(dif_in_gggaa(x_1, x_2, x_3)) = x_1 POL(dif_out_gggaa(x_1, x_2, x_3, x_4, x_5)) = x_4 POL(isMinus_in_gga(x_1, x_2)) = 0 POL(isMinus_in_ggg(x_1, x_2, x_3)) = 0 POL(isMinus_out_gga(x_1, x_2, x_3)) = 0 POL(isMinus_out_ggg(x_1, x_2, x_3)) = 0 POL(member_in_ag(x_1)) = 0 POL(member_out_ag(x_1, x_2)) = 0 POL(mergedelete_in_gga(x_1, x_2)) = x_1 POL(mergedelete_out_gga(x_1, x_2, x_3)) = 1 POL(pred(x_1)) = 0 POL(rev_in_gga(x_1, x_2)) = 0 POL(rev_out_gga(x_1, x_2, x_3)) = 0 POL(succ(x_1)) = 0 POL(zero) = 0 ---------------------------------------- (192) Obligation: Q DP problem: The TRS P consists of the following rules: U21_GGAG(Unused, List, Mod, member_out_ag(Choice, Unused)) -> U22_GGAG(Unused, List, Choice, Mod, check_in_gggag(List, Choice, Unused, Mod)) U22_GGAG(Unused, List, Choice, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> PDS1_IN_GGAG(U3, .(Choice, List), Mod) PDS1_IN_GGAG(Unused, List, Mod) -> U21_GGAG(Unused, List, Mod, member_in_ag(Unused)) The TRS R consists of the following rules: member_in_ag(.(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(.(X1, Xs)) -> U1_ag(X1, Xs, member_in_ag(Xs)) U1_ag(X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) check_in_gggag(S, Choice, Old, Modulus) -> U17_gggag(S, Choice, Old, Modulus, dif_in_gggaa(S, Choice, Modulus)) dif_in_gggaa([], X2, X3) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod) -> U6_gggaa(S, Ss, Val, Mod, isMinus_in_gga(Val, S)) U17_gggag(S, Choice, Old, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, Modulus, Dds, mergedelete_in_gga(Ds, Old)) mergedelete_in_gga([], L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R)) -> U12_gga(D, Ds, R, mergedelete_in_gga(Ds, R)) mergedelete_in_gga(.(D, Ds), .(X, R)) -> U13_gga(D, Ds, X, R, =_in_ag(D)) U18_gggag(S, Choice, Old, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, Modulus, L2, rev_in_gga(Dds, [])) rev_in_gga([], L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y) -> U11_gga(X, Xs, Y, rev_in_gga(Xs, .(X, Y))) U20_gggag(S, Choice, Old, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U11_gga(X, Xs, Y, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) =_in_ag(X) -> =_out_ag(X, X) U13_gga(D, Ds, X, R, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, X1, =_in_ag(X)) isMinus_in_gga(X, zero) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y)) -> U35_gga(Y, isMinus_in_gga(zero, Y)) isMinus_in_gga(zero, pred(Y)) -> U36_gga(Y, isMinus_in_gga(zero, Y)) isMinus_in_gga(succ(X), succ(Y)) -> U37_gga(X, Y, isMinus_in_gga(X, Y)) isMinus_in_gga(succ(X), pred(Y)) -> U38_gga(X, Y, isMinus_in_gga(X, Y)) isMinus_in_gga(pred(X), succ(Y)) -> U39_gga(X, Y, isMinus_in_gga(X, Y)) isMinus_in_gga(pred(X), pred(Y)) -> U40_gga(X, Y, isMinus_in_gga(X, Y)) U6_gggaa(S, Ss, Val, Mod, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, U, =_in_ag(U)) U7_gggaa(S, Ss, Val, Mod, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U8_gggaa(S, Ss, Val, Mod, D, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, =_in_ag(U)) U9_gggaa(S, Ss, Val, Mod, D, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, D2, dif_in_gggaa(Ss, Val, Mod)) U10_gggaa(S, Ss, Val, Mod, D, D2, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U40_gga(X, Y, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) The set Q consists of the following terms: check_in_gggag(x0, x1, x2, x3) member_in_ag(x0) U17_gggag(x0, x1, x2, x3, x4) U1_ag(x0, x1, x2) dif_in_gggaa(x0, x1, x2) U18_gggag(x0, x1, x2, x3, x4, x5) U6_gggaa(x0, x1, x2, x3, x4) mergedelete_in_gga(x0, x1) U19_gggag(x0, x1, x2, x3, x4, x5) isMinus_in_gga(x0, x1) U7_gggaa(x0, x1, x2, x3, x4, x5) U12_gga(x0, x1, x2, x3) U13_gga(x0, x1, x2, x3, x4) rev_in_gga(x0, x1) U20_gggag(x0, x1, x2, x3, x4) U35_gga(x0, x1) U36_gga(x0, x1) U37_gga(x0, x1, x2) U38_gga(x0, x1, x2) U39_gga(x0, x1, x2) U40_gga(x0, x1, x2) =_in_ag(x0) U8_gggaa(x0, x1, x2, x3, x4, x5, x6) U14_gga(x0, x1, x2, x3, x4, x5) U11_gga(x0, x1, x2, x3) isMinus_in_ggg(x0, x1, x2) U9_gggaa(x0, x1, x2, x3, x4, x5) U35_ggg(x0, x1, x2) U36_ggg(x0, x1, x2) U37_ggg(x0, x1, x2, x3) U38_ggg(x0, x1, x2, x3) U39_ggg(x0, x1, x2, x3) U40_ggg(x0, x1, x2, x3) U10_gggaa(x0, x1, x2, x3, x4, x5, x6) We have to consider all (P,Q,R)-chains. ---------------------------------------- (193) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes. ---------------------------------------- (194) TRUE ---------------------------------------- (195) Obligation: Pi DP problem: The TRS P consists of the following rules: ISPLUS_IN_GGA(succ(X), pred(Y), Z) -> ISPLUS_IN_GGA(X, Y, Z) ISPLUS_IN_GGA(succ(X), succ(Y), succ(succ(Z))) -> ISPLUS_IN_GGA(X, Y, Z) ISPLUS_IN_GGA(pred(X), succ(Y), Z) -> ISPLUS_IN_GGA(X, Y, Z) ISPLUS_IN_GGA(pred(X), pred(Y), pred(pred(Z))) -> ISPLUS_IN_GGA(X, Y, Z) The TRS R consists of the following rules: pds_in_ga(Order, .(zero, Ans)) -> U24_ga(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) isPlus_in_gga(zero, X, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), zero, succ(X)) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), succ(Y), succ(succ(Z))) -> U31_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(succ(X), pred(Y), Z) -> U32_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), zero, pred(X)) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), succ(Y), Z) -> U33_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), pred(Y), pred(pred(Z))) -> U34_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) U34_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U33_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U32_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U31_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U24_ga(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_ga(Order, Ans, isTimes_in_gga(Order, U, U1)) isTimes_in_gga(X, zero, zero) -> isTimes_out_gga(X, zero, zero) isTimes_in_gga(zero, succ(Y), zero) -> isTimes_out_gga(zero, succ(Y), zero) isTimes_in_gga(zero, pred(Y), zero) -> isTimes_out_gga(zero, pred(Y), zero) isTimes_in_gga(succ(X), succ(Y), Z) -> U41_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(succ(X), pred(Y), Z) -> U43_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(pred(X), succ(Y), Z) -> U45_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) isTimes_in_gga(pred(X), pred(Y), Z) -> U47_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) U47_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_gga(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U48_gga(X, Y, Z, isMinus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), pred(Y), Z) U45_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_gga(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U46_gga(X, Y, Z, isPlus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), succ(Y), Z) U43_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_gga(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U44_gga(X, Y, Z, isMinus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), pred(Y), Z) U41_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_gga(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U42_gga(X, Y, Z, isPlus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), succ(Y), Z) U25_ga(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_ga(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U26_ga(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_ga(Order, Ans, =_in_ag(N, U2)) =_in_ag(X, X) -> =_out_ag(X, X) U27_ga(Order, Ans, =_out_ag(N, U2)) -> U28_ga(Order, Ans, N, iota_in_ga(N, .(zero, List))) iota_in_ga(N, List) -> U2_ga(N, List, iota1_in_gga(zero, N, List)) iota1_in_gga(K, K, []) -> iota1_out_gga(K, K, []) iota1_in_gga(K, N, .(K, List)) -> U3_gga(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_gga(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_gga(K, N, List, =_in_ag(K1, U)) U4_gga(K, N, List, =_out_ag(K1, U)) -> U5_gga(K, N, List, iota1_in_gga(K1, N, List)) U5_gga(K, N, List, iota1_out_gga(K1, N, List)) -> iota1_out_gga(K, N, .(K, List)) U2_ga(N, List, iota1_out_gga(zero, N, List)) -> iota_out_ga(N, List) U28_ga(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_ga(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) pds1_in_ggag([], X6, [], X7) -> pds1_out_ggag([], X6, [], X7) pds1_in_ggag(Unused, List, .(Choice, Rest), Mod) -> U21_ggag(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(X, .(X1, Xs)) -> U1_ag(X, X1, Xs, member_in_ag(X, Xs)) U1_ag(X, X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) U21_ggag(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_ggag(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) check_in_gggag([], X4, L, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, L3, Modulus) -> U17_gggag(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) dif_in_gggaa([], X2, X3, [], []) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U17_gggag(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) mergedelete_in_gga([], L, L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R), L2) -> U12_gga(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) mergedelete_in_gga(.(D, Ds), .(X, R), .(X, L2)) -> U13_gga(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_gga(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_gga(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U15_gga(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U16_gga(D, Ds, X, R, L2, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U12_gga(D, Ds, R, L2, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) rev_in_gga([], L, L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y, L) -> U11_gga(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) U11_gga(X, Xs, Y, L, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U22_ggag(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_ggag(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U23_ggag(Unused, List, Choice, Rest, Mod, pds1_out_ggag(U3, .(Choice, List), Rest, Mod)) -> pds1_out_ggag(Unused, List, .(Choice, Rest), Mod) U29_ga(Order, Ans, pds1_out_ggag(List, .(zero, []), Ans, N)) -> pds_out_ga(Order, .(zero, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U24_ga(x1, x2, x3) = U24_ga(x1, x3) isPlus_in_gga(x1, x2, x3) = isPlus_in_gga(x1, x2) zero = zero isPlus_out_gga(x1, x2, x3) = isPlus_out_gga(x1, x2, x3) succ(x1) = succ(x1) U31_gga(x1, x2, x3, x4) = U31_gga(x1, x2, x4) pred(x1) = pred(x1) U32_gga(x1, x2, x3, x4) = U32_gga(x1, x2, x4) U33_gga(x1, x2, x3, x4) = U33_gga(x1, x2, x4) U34_gga(x1, x2, x3, x4) = U34_gga(x1, x2, x4) U25_ga(x1, x2, x3) = U25_ga(x1, x3) isTimes_in_gga(x1, x2, x3) = isTimes_in_gga(x1, x2) isTimes_out_gga(x1, x2, x3) = isTimes_out_gga(x1, x2, x3) U41_gga(x1, x2, x3, x4) = U41_gga(x1, x2, x4) U43_gga(x1, x2, x3, x4) = U43_gga(x1, x2, x4) U45_gga(x1, x2, x3, x4) = U45_gga(x1, x2, x4) U47_gga(x1, x2, x3, x4) = U47_gga(x1, x2, x4) U48_gga(x1, x2, x3, x4) = U48_gga(x1, x2, x4) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x1, x2, x3) U35_gga(x1, x2, x3) = U35_gga(x1, x3) U36_gga(x1, x2, x3) = U36_gga(x1, x3) U37_gga(x1, x2, x3, x4) = U37_gga(x1, x2, x4) U38_gga(x1, x2, x3, x4) = U38_gga(x1, x2, x4) U39_gga(x1, x2, x3, x4) = U39_gga(x1, x2, x4) U40_gga(x1, x2, x3, x4) = U40_gga(x1, x2, x4) U46_gga(x1, x2, x3, x4) = U46_gga(x1, x2, x4) U44_gga(x1, x2, x3, x4) = U44_gga(x1, x2, x4) U42_gga(x1, x2, x3, x4) = U42_gga(x1, x2, x4) U26_ga(x1, x2, x3) = U26_ga(x1, x3) U27_ga(x1, x2, x3) = U27_ga(x1, x3) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1, x2) U28_ga(x1, x2, x3, x4) = U28_ga(x1, x3, x4) iota_in_ga(x1, x2) = iota_in_ga(x1) U2_ga(x1, x2, x3) = U2_ga(x1, x3) iota1_in_gga(x1, x2, x3) = iota1_in_gga(x1, x2) iota1_out_gga(x1, x2, x3) = iota1_out_gga(x1, x2, x3) U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4) iota_out_ga(x1, x2) = iota_out_ga(x1, x2) .(x1, x2) = .(x1, x2) U29_ga(x1, x2, x3) = U29_ga(x1, x3) pds1_in_ggag(x1, x2, x3, x4) = pds1_in_ggag(x1, x2, x4) [] = [] pds1_out_ggag(x1, x2, x3, x4) = pds1_out_ggag(x1, x2, x3, x4) U21_ggag(x1, x2, x3, x4, x5, x6) = U21_ggag(x1, x2, x5, x6) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1, x2) U1_ag(x1, x2, x3, x4) = U1_ag(x2, x3, x4) U22_ggag(x1, x2, x3, x4, x5, x6) = U22_ggag(x1, x2, x3, x5, x6) check_in_gggag(x1, x2, x3, x4, x5) = check_in_gggag(x1, x2, x3, x5) check_out_gggag(x1, x2, x3, x4, x5) = check_out_gggag(x1, x2, x3, x4, x5) U17_gggag(x1, x2, x3, x4, x5, x6) = U17_gggag(x1, x2, x3, x5, x6) dif_in_gggaa(x1, x2, x3, x4, x5) = dif_in_gggaa(x1, x2, x3) dif_out_gggaa(x1, x2, x3, x4, x5) = dif_out_gggaa(x1, x2, x3, x4, x5) U6_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gggaa(x1, x2, x3, x4, x9) U7_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_gggaa(x1, x2, x3, x4, x9, x10) U8_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_gggaa(x1, x2, x3, x4, x5, x9, x10) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg(x1, x2, x3) U35_ggg(x1, x2, x3) = U35_ggg(x1, x2, x3) U36_ggg(x1, x2, x3) = U36_ggg(x1, x2, x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x1, x2, x3, x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x1, x2, x3, x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x1, x2, x3, x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x1, x2, x3, x4) U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_gggaa(x1, x2, x3, x4, x5, x9) U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_gggaa(x1, x2, x3, x4, x5, x7, x9) U18_gggag(x1, x2, x3, x4, x5, x6, x7) = U18_gggag(x1, x2, x3, x5, x6, x7) mergedelete_in_gga(x1, x2, x3) = mergedelete_in_gga(x1, x2) mergedelete_out_gga(x1, x2, x3) = mergedelete_out_gga(x1, x2, x3) U12_gga(x1, x2, x3, x4, x5) = U12_gga(x1, x2, x3, x5) U13_gga(x1, x2, x3, x4, x5, x6) = U13_gga(x1, x2, x3, x4, x6) U14_gga(x1, x2, x3, x4, x5, x6, x7) = U14_gga(x1, x2, x3, x4, x6, x7) U15_gga(x1, x2, x3, x4, x5, x6) = U15_gga(x1, x2, x3, x4, x6) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg(x1, x2) U64_gg(x1, x2, x3) = U64_gg(x1, x2, x3) U65_gg(x1, x2, x3) = U65_gg(x1, x2, x3) U16_gga(x1, x2, x3, x4, x5, x6) = U16_gga(x1, x2, x3, x4, x6) U19_gggag(x1, x2, x3, x4, x5, x6, x7) = U19_gggag(x1, x2, x3, x5, x6, x7) rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2) rev_out_gga(x1, x2, x3) = rev_out_gga(x1, x2, x3) U11_gga(x1, x2, x3, x4, x5) = U11_gga(x1, x2, x3, x5) U20_gggag(x1, x2, x3, x4, x5, x6) = U20_gggag(x1, x2, x3, x5, x6) U23_ggag(x1, x2, x3, x4, x5, x6) = U23_ggag(x1, x2, x3, x5, x6) pds_out_ga(x1, x2) = pds_out_ga(x1, x2) ISPLUS_IN_GGA(x1, x2, x3) = ISPLUS_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (196) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (197) Obligation: Pi DP problem: The TRS P consists of the following rules: ISPLUS_IN_GGA(succ(X), pred(Y), Z) -> ISPLUS_IN_GGA(X, Y, Z) ISPLUS_IN_GGA(succ(X), succ(Y), succ(succ(Z))) -> ISPLUS_IN_GGA(X, Y, Z) ISPLUS_IN_GGA(pred(X), succ(Y), Z) -> ISPLUS_IN_GGA(X, Y, Z) ISPLUS_IN_GGA(pred(X), pred(Y), pred(pred(Z))) -> ISPLUS_IN_GGA(X, Y, Z) R is empty. The argument filtering Pi contains the following mapping: succ(x1) = succ(x1) pred(x1) = pred(x1) ISPLUS_IN_GGA(x1, x2, x3) = ISPLUS_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (198) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (199) Obligation: Q DP problem: The TRS P consists of the following rules: ISPLUS_IN_GGA(succ(X), pred(Y)) -> ISPLUS_IN_GGA(X, Y) ISPLUS_IN_GGA(succ(X), succ(Y)) -> ISPLUS_IN_GGA(X, Y) ISPLUS_IN_GGA(pred(X), succ(Y)) -> ISPLUS_IN_GGA(X, Y) ISPLUS_IN_GGA(pred(X), pred(Y)) -> ISPLUS_IN_GGA(X, Y) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (200) 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: *ISPLUS_IN_GGA(succ(X), pred(Y)) -> ISPLUS_IN_GGA(X, Y) The graph contains the following edges 1 > 1, 2 > 2 *ISPLUS_IN_GGA(succ(X), succ(Y)) -> ISPLUS_IN_GGA(X, Y) The graph contains the following edges 1 > 1, 2 > 2 *ISPLUS_IN_GGA(pred(X), succ(Y)) -> ISPLUS_IN_GGA(X, Y) The graph contains the following edges 1 > 1, 2 > 2 *ISPLUS_IN_GGA(pred(X), pred(Y)) -> ISPLUS_IN_GGA(X, Y) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (201) YES ---------------------------------------- (202) Obligation: Pi DP problem: The TRS P consists of the following rules: IOTA1_IN_GGA(K, N, .(K, List)) -> U3_GGA(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_GGA(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_GGA(K, N, List, =_in_ag(K1, U)) U4_GGA(K, N, List, =_out_ag(K1, U)) -> IOTA1_IN_GGA(K1, N, List) The TRS R consists of the following rules: pds_in_ga(Order, .(zero, Ans)) -> U24_ga(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) isPlus_in_gga(zero, X, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), zero, succ(X)) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), succ(Y), succ(succ(Z))) -> U31_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(succ(X), pred(Y), Z) -> U32_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), zero, pred(X)) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), succ(Y), Z) -> U33_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), pred(Y), pred(pred(Z))) -> U34_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) U34_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U33_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U32_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U31_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U24_ga(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_ga(Order, Ans, isTimes_in_gga(Order, U, U1)) isTimes_in_gga(X, zero, zero) -> isTimes_out_gga(X, zero, zero) isTimes_in_gga(zero, succ(Y), zero) -> isTimes_out_gga(zero, succ(Y), zero) isTimes_in_gga(zero, pred(Y), zero) -> isTimes_out_gga(zero, pred(Y), zero) isTimes_in_gga(succ(X), succ(Y), Z) -> U41_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(succ(X), pred(Y), Z) -> U43_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(pred(X), succ(Y), Z) -> U45_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) isTimes_in_gga(pred(X), pred(Y), Z) -> U47_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) U47_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_gga(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U48_gga(X, Y, Z, isMinus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), pred(Y), Z) U45_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_gga(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U46_gga(X, Y, Z, isPlus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), succ(Y), Z) U43_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_gga(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U44_gga(X, Y, Z, isMinus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), pred(Y), Z) U41_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_gga(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U42_gga(X, Y, Z, isPlus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), succ(Y), Z) U25_ga(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_ga(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U26_ga(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_ga(Order, Ans, =_in_ag(N, U2)) =_in_ag(X, X) -> =_out_ag(X, X) U27_ga(Order, Ans, =_out_ag(N, U2)) -> U28_ga(Order, Ans, N, iota_in_ga(N, .(zero, List))) iota_in_ga(N, List) -> U2_ga(N, List, iota1_in_gga(zero, N, List)) iota1_in_gga(K, K, []) -> iota1_out_gga(K, K, []) iota1_in_gga(K, N, .(K, List)) -> U3_gga(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_gga(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_gga(K, N, List, =_in_ag(K1, U)) U4_gga(K, N, List, =_out_ag(K1, U)) -> U5_gga(K, N, List, iota1_in_gga(K1, N, List)) U5_gga(K, N, List, iota1_out_gga(K1, N, List)) -> iota1_out_gga(K, N, .(K, List)) U2_ga(N, List, iota1_out_gga(zero, N, List)) -> iota_out_ga(N, List) U28_ga(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_ga(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) pds1_in_ggag([], X6, [], X7) -> pds1_out_ggag([], X6, [], X7) pds1_in_ggag(Unused, List, .(Choice, Rest), Mod) -> U21_ggag(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(X, .(X1, Xs)) -> U1_ag(X, X1, Xs, member_in_ag(X, Xs)) U1_ag(X, X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) U21_ggag(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_ggag(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) check_in_gggag([], X4, L, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, L3, Modulus) -> U17_gggag(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) dif_in_gggaa([], X2, X3, [], []) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U17_gggag(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) mergedelete_in_gga([], L, L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R), L2) -> U12_gga(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) mergedelete_in_gga(.(D, Ds), .(X, R), .(X, L2)) -> U13_gga(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_gga(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_gga(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U15_gga(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U16_gga(D, Ds, X, R, L2, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U12_gga(D, Ds, R, L2, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) rev_in_gga([], L, L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y, L) -> U11_gga(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) U11_gga(X, Xs, Y, L, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U22_ggag(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_ggag(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U23_ggag(Unused, List, Choice, Rest, Mod, pds1_out_ggag(U3, .(Choice, List), Rest, Mod)) -> pds1_out_ggag(Unused, List, .(Choice, Rest), Mod) U29_ga(Order, Ans, pds1_out_ggag(List, .(zero, []), Ans, N)) -> pds_out_ga(Order, .(zero, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U24_ga(x1, x2, x3) = U24_ga(x1, x3) isPlus_in_gga(x1, x2, x3) = isPlus_in_gga(x1, x2) zero = zero isPlus_out_gga(x1, x2, x3) = isPlus_out_gga(x1, x2, x3) succ(x1) = succ(x1) U31_gga(x1, x2, x3, x4) = U31_gga(x1, x2, x4) pred(x1) = pred(x1) U32_gga(x1, x2, x3, x4) = U32_gga(x1, x2, x4) U33_gga(x1, x2, x3, x4) = U33_gga(x1, x2, x4) U34_gga(x1, x2, x3, x4) = U34_gga(x1, x2, x4) U25_ga(x1, x2, x3) = U25_ga(x1, x3) isTimes_in_gga(x1, x2, x3) = isTimes_in_gga(x1, x2) isTimes_out_gga(x1, x2, x3) = isTimes_out_gga(x1, x2, x3) U41_gga(x1, x2, x3, x4) = U41_gga(x1, x2, x4) U43_gga(x1, x2, x3, x4) = U43_gga(x1, x2, x4) U45_gga(x1, x2, x3, x4) = U45_gga(x1, x2, x4) U47_gga(x1, x2, x3, x4) = U47_gga(x1, x2, x4) U48_gga(x1, x2, x3, x4) = U48_gga(x1, x2, x4) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x1, x2, x3) U35_gga(x1, x2, x3) = U35_gga(x1, x3) U36_gga(x1, x2, x3) = U36_gga(x1, x3) U37_gga(x1, x2, x3, x4) = U37_gga(x1, x2, x4) U38_gga(x1, x2, x3, x4) = U38_gga(x1, x2, x4) U39_gga(x1, x2, x3, x4) = U39_gga(x1, x2, x4) U40_gga(x1, x2, x3, x4) = U40_gga(x1, x2, x4) U46_gga(x1, x2, x3, x4) = U46_gga(x1, x2, x4) U44_gga(x1, x2, x3, x4) = U44_gga(x1, x2, x4) U42_gga(x1, x2, x3, x4) = U42_gga(x1, x2, x4) U26_ga(x1, x2, x3) = U26_ga(x1, x3) U27_ga(x1, x2, x3) = U27_ga(x1, x3) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1, x2) U28_ga(x1, x2, x3, x4) = U28_ga(x1, x3, x4) iota_in_ga(x1, x2) = iota_in_ga(x1) U2_ga(x1, x2, x3) = U2_ga(x1, x3) iota1_in_gga(x1, x2, x3) = iota1_in_gga(x1, x2) iota1_out_gga(x1, x2, x3) = iota1_out_gga(x1, x2, x3) U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4) iota_out_ga(x1, x2) = iota_out_ga(x1, x2) .(x1, x2) = .(x1, x2) U29_ga(x1, x2, x3) = U29_ga(x1, x3) pds1_in_ggag(x1, x2, x3, x4) = pds1_in_ggag(x1, x2, x4) [] = [] pds1_out_ggag(x1, x2, x3, x4) = pds1_out_ggag(x1, x2, x3, x4) U21_ggag(x1, x2, x3, x4, x5, x6) = U21_ggag(x1, x2, x5, x6) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1, x2) U1_ag(x1, x2, x3, x4) = U1_ag(x2, x3, x4) U22_ggag(x1, x2, x3, x4, x5, x6) = U22_ggag(x1, x2, x3, x5, x6) check_in_gggag(x1, x2, x3, x4, x5) = check_in_gggag(x1, x2, x3, x5) check_out_gggag(x1, x2, x3, x4, x5) = check_out_gggag(x1, x2, x3, x4, x5) U17_gggag(x1, x2, x3, x4, x5, x6) = U17_gggag(x1, x2, x3, x5, x6) dif_in_gggaa(x1, x2, x3, x4, x5) = dif_in_gggaa(x1, x2, x3) dif_out_gggaa(x1, x2, x3, x4, x5) = dif_out_gggaa(x1, x2, x3, x4, x5) U6_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gggaa(x1, x2, x3, x4, x9) U7_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_gggaa(x1, x2, x3, x4, x9, x10) U8_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_gggaa(x1, x2, x3, x4, x5, x9, x10) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg(x1, x2, x3) U35_ggg(x1, x2, x3) = U35_ggg(x1, x2, x3) U36_ggg(x1, x2, x3) = U36_ggg(x1, x2, x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x1, x2, x3, x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x1, x2, x3, x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x1, x2, x3, x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x1, x2, x3, x4) U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_gggaa(x1, x2, x3, x4, x5, x9) U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_gggaa(x1, x2, x3, x4, x5, x7, x9) U18_gggag(x1, x2, x3, x4, x5, x6, x7) = U18_gggag(x1, x2, x3, x5, x6, x7) mergedelete_in_gga(x1, x2, x3) = mergedelete_in_gga(x1, x2) mergedelete_out_gga(x1, x2, x3) = mergedelete_out_gga(x1, x2, x3) U12_gga(x1, x2, x3, x4, x5) = U12_gga(x1, x2, x3, x5) U13_gga(x1, x2, x3, x4, x5, x6) = U13_gga(x1, x2, x3, x4, x6) U14_gga(x1, x2, x3, x4, x5, x6, x7) = U14_gga(x1, x2, x3, x4, x6, x7) U15_gga(x1, x2, x3, x4, x5, x6) = U15_gga(x1, x2, x3, x4, x6) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg(x1, x2) U64_gg(x1, x2, x3) = U64_gg(x1, x2, x3) U65_gg(x1, x2, x3) = U65_gg(x1, x2, x3) U16_gga(x1, x2, x3, x4, x5, x6) = U16_gga(x1, x2, x3, x4, x6) U19_gggag(x1, x2, x3, x4, x5, x6, x7) = U19_gggag(x1, x2, x3, x5, x6, x7) rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2) rev_out_gga(x1, x2, x3) = rev_out_gga(x1, x2, x3) U11_gga(x1, x2, x3, x4, x5) = U11_gga(x1, x2, x3, x5) U20_gggag(x1, x2, x3, x4, x5, x6) = U20_gggag(x1, x2, x3, x5, x6) U23_ggag(x1, x2, x3, x4, x5, x6) = U23_ggag(x1, x2, x3, x5, x6) pds_out_ga(x1, x2) = pds_out_ga(x1, x2) IOTA1_IN_GGA(x1, x2, x3) = IOTA1_IN_GGA(x1, x2) U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4) U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (203) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (204) Obligation: Pi DP problem: The TRS P consists of the following rules: IOTA1_IN_GGA(K, N, .(K, List)) -> U3_GGA(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_GGA(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_GGA(K, N, List, =_in_ag(K1, U)) U4_GGA(K, N, List, =_out_ag(K1, U)) -> IOTA1_IN_GGA(K1, N, List) The TRS R consists of the following rules: isPlus_in_gga(zero, X, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), succ(Y), succ(succ(Z))) -> U31_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), succ(Y), Z) -> U33_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) =_in_ag(X, X) -> =_out_ag(X, X) U31_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U33_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) isPlus_in_gga(succ(X), zero, succ(X)) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), pred(Y), Z) -> U32_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), zero, pred(X)) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), pred(Y), pred(pred(Z))) -> U34_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) U32_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U34_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) The argument filtering Pi contains the following mapping: isPlus_in_gga(x1, x2, x3) = isPlus_in_gga(x1, x2) zero = zero isPlus_out_gga(x1, x2, x3) = isPlus_out_gga(x1, x2, x3) succ(x1) = succ(x1) U31_gga(x1, x2, x3, x4) = U31_gga(x1, x2, x4) pred(x1) = pred(x1) U32_gga(x1, x2, x3, x4) = U32_gga(x1, x2, x4) U33_gga(x1, x2, x3, x4) = U33_gga(x1, x2, x4) U34_gga(x1, x2, x3, x4) = U34_gga(x1, x2, x4) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1, x2) .(x1, x2) = .(x1, x2) IOTA1_IN_GGA(x1, x2, x3) = IOTA1_IN_GGA(x1, x2) U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4) U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (205) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (206) Obligation: Q DP problem: The TRS P consists of the following rules: IOTA1_IN_GGA(K, N) -> U3_GGA(K, N, isPlus_in_gga(K, succ(zero))) U3_GGA(K, N, isPlus_out_gga(K, succ(zero), U)) -> U4_GGA(K, N, =_in_ag(U)) U4_GGA(K, N, =_out_ag(K1, U)) -> IOTA1_IN_GGA(K1, N) The TRS R consists of the following rules: isPlus_in_gga(zero, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), succ(Y)) -> U31_gga(X, Y, isPlus_in_gga(X, Y)) isPlus_in_gga(pred(X), succ(Y)) -> U33_gga(X, Y, isPlus_in_gga(X, Y)) =_in_ag(X) -> =_out_ag(X, X) U31_gga(X, Y, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U33_gga(X, Y, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) isPlus_in_gga(succ(X), zero) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), pred(Y)) -> U32_gga(X, Y, isPlus_in_gga(X, Y)) isPlus_in_gga(pred(X), zero) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), pred(Y)) -> U34_gga(X, Y, isPlus_in_gga(X, Y)) U32_gga(X, Y, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U34_gga(X, Y, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) The set Q consists of the following terms: isPlus_in_gga(x0, x1) =_in_ag(x0) U31_gga(x0, x1, x2) U33_gga(x0, x1, x2) U32_gga(x0, x1, x2) U34_gga(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (207) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule U3_GGA(K, N, isPlus_out_gga(K, succ(zero), U)) -> U4_GGA(K, N, =_in_ag(U)) at position [2] we obtained the following new rules [LPAR04]: (U3_GGA(K, N, isPlus_out_gga(K, succ(zero), U)) -> U4_GGA(K, N, =_out_ag(U, U)),U3_GGA(K, N, isPlus_out_gga(K, succ(zero), U)) -> U4_GGA(K, N, =_out_ag(U, U))) ---------------------------------------- (208) Obligation: Q DP problem: The TRS P consists of the following rules: IOTA1_IN_GGA(K, N) -> U3_GGA(K, N, isPlus_in_gga(K, succ(zero))) U4_GGA(K, N, =_out_ag(K1, U)) -> IOTA1_IN_GGA(K1, N) U3_GGA(K, N, isPlus_out_gga(K, succ(zero), U)) -> U4_GGA(K, N, =_out_ag(U, U)) The TRS R consists of the following rules: isPlus_in_gga(zero, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), succ(Y)) -> U31_gga(X, Y, isPlus_in_gga(X, Y)) isPlus_in_gga(pred(X), succ(Y)) -> U33_gga(X, Y, isPlus_in_gga(X, Y)) =_in_ag(X) -> =_out_ag(X, X) U31_gga(X, Y, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U33_gga(X, Y, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) isPlus_in_gga(succ(X), zero) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), pred(Y)) -> U32_gga(X, Y, isPlus_in_gga(X, Y)) isPlus_in_gga(pred(X), zero) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), pred(Y)) -> U34_gga(X, Y, isPlus_in_gga(X, Y)) U32_gga(X, Y, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U34_gga(X, Y, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) The set Q consists of the following terms: isPlus_in_gga(x0, x1) =_in_ag(x0) U31_gga(x0, x1, x2) U33_gga(x0, x1, x2) U32_gga(x0, x1, x2) U34_gga(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (209) 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. ---------------------------------------- (210) Obligation: Q DP problem: The TRS P consists of the following rules: IOTA1_IN_GGA(K, N) -> U3_GGA(K, N, isPlus_in_gga(K, succ(zero))) U4_GGA(K, N, =_out_ag(K1, U)) -> IOTA1_IN_GGA(K1, N) U3_GGA(K, N, isPlus_out_gga(K, succ(zero), U)) -> U4_GGA(K, N, =_out_ag(U, U)) The TRS R consists of the following rules: isPlus_in_gga(zero, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), succ(Y)) -> U31_gga(X, Y, isPlus_in_gga(X, Y)) isPlus_in_gga(pred(X), succ(Y)) -> U33_gga(X, Y, isPlus_in_gga(X, Y)) isPlus_in_gga(succ(X), zero) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), pred(Y)) -> U32_gga(X, Y, isPlus_in_gga(X, Y)) isPlus_in_gga(pred(X), zero) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), pred(Y)) -> U34_gga(X, Y, isPlus_in_gga(X, Y)) U33_gga(X, Y, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U34_gga(X, Y, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U32_gga(X, Y, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U31_gga(X, Y, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) The set Q consists of the following terms: isPlus_in_gga(x0, x1) =_in_ag(x0) U31_gga(x0, x1, x2) U33_gga(x0, x1, x2) U32_gga(x0, x1, x2) U34_gga(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (211) 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]. =_in_ag(x0) ---------------------------------------- (212) Obligation: Q DP problem: The TRS P consists of the following rules: IOTA1_IN_GGA(K, N) -> U3_GGA(K, N, isPlus_in_gga(K, succ(zero))) U4_GGA(K, N, =_out_ag(K1, U)) -> IOTA1_IN_GGA(K1, N) U3_GGA(K, N, isPlus_out_gga(K, succ(zero), U)) -> U4_GGA(K, N, =_out_ag(U, U)) The TRS R consists of the following rules: isPlus_in_gga(zero, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), succ(Y)) -> U31_gga(X, Y, isPlus_in_gga(X, Y)) isPlus_in_gga(pred(X), succ(Y)) -> U33_gga(X, Y, isPlus_in_gga(X, Y)) isPlus_in_gga(succ(X), zero) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), pred(Y)) -> U32_gga(X, Y, isPlus_in_gga(X, Y)) isPlus_in_gga(pred(X), zero) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), pred(Y)) -> U34_gga(X, Y, isPlus_in_gga(X, Y)) U33_gga(X, Y, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U34_gga(X, Y, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U32_gga(X, Y, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U31_gga(X, Y, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) The set Q consists of the following terms: isPlus_in_gga(x0, x1) U31_gga(x0, x1, x2) U33_gga(x0, x1, x2) U32_gga(x0, x1, x2) U34_gga(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (213) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U4_GGA(K, N, =_out_ag(K1, U)) -> IOTA1_IN_GGA(K1, N) we obtained the following new rules [LPAR04]: (U4_GGA(z0, z1, =_out_ag(z2, z2)) -> IOTA1_IN_GGA(z2, z1),U4_GGA(z0, z1, =_out_ag(z2, z2)) -> IOTA1_IN_GGA(z2, z1)) ---------------------------------------- (214) Obligation: Q DP problem: The TRS P consists of the following rules: IOTA1_IN_GGA(K, N) -> U3_GGA(K, N, isPlus_in_gga(K, succ(zero))) U3_GGA(K, N, isPlus_out_gga(K, succ(zero), U)) -> U4_GGA(K, N, =_out_ag(U, U)) U4_GGA(z0, z1, =_out_ag(z2, z2)) -> IOTA1_IN_GGA(z2, z1) The TRS R consists of the following rules: isPlus_in_gga(zero, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), succ(Y)) -> U31_gga(X, Y, isPlus_in_gga(X, Y)) isPlus_in_gga(pred(X), succ(Y)) -> U33_gga(X, Y, isPlus_in_gga(X, Y)) isPlus_in_gga(succ(X), zero) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), pred(Y)) -> U32_gga(X, Y, isPlus_in_gga(X, Y)) isPlus_in_gga(pred(X), zero) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), pred(Y)) -> U34_gga(X, Y, isPlus_in_gga(X, Y)) U33_gga(X, Y, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U34_gga(X, Y, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U32_gga(X, Y, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U31_gga(X, Y, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) The set Q consists of the following terms: isPlus_in_gga(x0, x1) U31_gga(x0, x1, x2) U33_gga(x0, x1, x2) U32_gga(x0, x1, x2) U34_gga(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (215) 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: U34_gga(X, Y, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U32_gga(X, Y, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) Used ordering: Polynomial interpretation [POLO]: POL(=_out_ag(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 POL(IOTA1_IN_GGA(x_1, x_2)) = x_2 POL(U31_gga(x_1, x_2, x_3)) = 2*x_3 POL(U32_gga(x_1, x_2, x_3)) = 2 + x_3 POL(U33_gga(x_1, x_2, x_3)) = x_3 POL(U34_gga(x_1, x_2, x_3)) = 2 + x_3 POL(U3_GGA(x_1, x_2, x_3)) = x_2 + 2*x_3 POL(U4_GGA(x_1, x_2, x_3)) = x_2 POL(isPlus_in_gga(x_1, x_2)) = 2*x_2 POL(isPlus_out_gga(x_1, x_2, x_3)) = 0 POL(pred(x_1)) = 1 + x_1 POL(succ(x_1)) = 2*x_1 POL(zero) = 0 ---------------------------------------- (216) Obligation: Q DP problem: The TRS P consists of the following rules: IOTA1_IN_GGA(K, N) -> U3_GGA(K, N, isPlus_in_gga(K, succ(zero))) U3_GGA(K, N, isPlus_out_gga(K, succ(zero), U)) -> U4_GGA(K, N, =_out_ag(U, U)) U4_GGA(z0, z1, =_out_ag(z2, z2)) -> IOTA1_IN_GGA(z2, z1) The TRS R consists of the following rules: isPlus_in_gga(zero, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), succ(Y)) -> U31_gga(X, Y, isPlus_in_gga(X, Y)) isPlus_in_gga(pred(X), succ(Y)) -> U33_gga(X, Y, isPlus_in_gga(X, Y)) isPlus_in_gga(succ(X), zero) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), pred(Y)) -> U32_gga(X, Y, isPlus_in_gga(X, Y)) isPlus_in_gga(pred(X), zero) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), pred(Y)) -> U34_gga(X, Y, isPlus_in_gga(X, Y)) U33_gga(X, Y, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U31_gga(X, Y, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) The set Q consists of the following terms: isPlus_in_gga(x0, x1) U31_gga(x0, x1, x2) U33_gga(x0, x1, x2) U32_gga(x0, x1, x2) U34_gga(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (217) 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: isPlus_in_gga(pred(X), pred(Y)) -> U34_gga(X, Y, isPlus_in_gga(X, Y)) Used ordering: Polynomial interpretation [POLO]: POL(=_out_ag(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 POL(IOTA1_IN_GGA(x_1, x_2)) = x_2 POL(U31_gga(x_1, x_2, x_3)) = x_3 POL(U32_gga(x_1, x_2, x_3)) = 1 + x_3 POL(U33_gga(x_1, x_2, x_3)) = x_3 POL(U34_gga(x_1, x_2, x_3)) = x_3 POL(U3_GGA(x_1, x_2, x_3)) = x_2 + 2*x_3 POL(U4_GGA(x_1, x_2, x_3)) = x_2 POL(isPlus_in_gga(x_1, x_2)) = x_2 POL(isPlus_out_gga(x_1, x_2, x_3)) = 0 POL(pred(x_1)) = 1 + x_1 POL(succ(x_1)) = x_1 POL(zero) = 0 ---------------------------------------- (218) Obligation: Q DP problem: The TRS P consists of the following rules: IOTA1_IN_GGA(K, N) -> U3_GGA(K, N, isPlus_in_gga(K, succ(zero))) U3_GGA(K, N, isPlus_out_gga(K, succ(zero), U)) -> U4_GGA(K, N, =_out_ag(U, U)) U4_GGA(z0, z1, =_out_ag(z2, z2)) -> IOTA1_IN_GGA(z2, z1) The TRS R consists of the following rules: isPlus_in_gga(zero, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), succ(Y)) -> U31_gga(X, Y, isPlus_in_gga(X, Y)) isPlus_in_gga(pred(X), succ(Y)) -> U33_gga(X, Y, isPlus_in_gga(X, Y)) isPlus_in_gga(succ(X), zero) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), pred(Y)) -> U32_gga(X, Y, isPlus_in_gga(X, Y)) isPlus_in_gga(pred(X), zero) -> isPlus_out_gga(pred(X), zero, pred(X)) U33_gga(X, Y, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U31_gga(X, Y, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) The set Q consists of the following terms: isPlus_in_gga(x0, x1) U31_gga(x0, x1, x2) U33_gga(x0, x1, x2) U32_gga(x0, x1, x2) U34_gga(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (219) 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]. U34_gga(x0, x1, x2) ---------------------------------------- (220) Obligation: Q DP problem: The TRS P consists of the following rules: IOTA1_IN_GGA(K, N) -> U3_GGA(K, N, isPlus_in_gga(K, succ(zero))) U3_GGA(K, N, isPlus_out_gga(K, succ(zero), U)) -> U4_GGA(K, N, =_out_ag(U, U)) U4_GGA(z0, z1, =_out_ag(z2, z2)) -> IOTA1_IN_GGA(z2, z1) The TRS R consists of the following rules: isPlus_in_gga(zero, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), succ(Y)) -> U31_gga(X, Y, isPlus_in_gga(X, Y)) isPlus_in_gga(pred(X), succ(Y)) -> U33_gga(X, Y, isPlus_in_gga(X, Y)) isPlus_in_gga(succ(X), zero) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), pred(Y)) -> U32_gga(X, Y, isPlus_in_gga(X, Y)) isPlus_in_gga(pred(X), zero) -> isPlus_out_gga(pred(X), zero, pred(X)) U33_gga(X, Y, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U31_gga(X, Y, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) The set Q consists of the following terms: isPlus_in_gga(x0, x1) U31_gga(x0, x1, x2) U33_gga(x0, x1, x2) U32_gga(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (221) 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: isPlus_in_gga(succ(X), pred(Y)) -> U32_gga(X, Y, isPlus_in_gga(X, Y)) Used ordering: Polynomial interpretation [POLO]: POL(=_out_ag(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 POL(IOTA1_IN_GGA(x_1, x_2)) = x_2 POL(U31_gga(x_1, x_2, x_3)) = x_3 POL(U32_gga(x_1, x_2, x_3)) = x_3 POL(U33_gga(x_1, x_2, x_3)) = 2*x_3 POL(U3_GGA(x_1, x_2, x_3)) = x_2 + 2*x_3 POL(U4_GGA(x_1, x_2, x_3)) = x_2 POL(isPlus_in_gga(x_1, x_2)) = 2*x_2 POL(isPlus_out_gga(x_1, x_2, x_3)) = 0 POL(pred(x_1)) = 2 + x_1 POL(succ(x_1)) = 2*x_1 POL(zero) = 0 ---------------------------------------- (222) Obligation: Q DP problem: The TRS P consists of the following rules: IOTA1_IN_GGA(K, N) -> U3_GGA(K, N, isPlus_in_gga(K, succ(zero))) U3_GGA(K, N, isPlus_out_gga(K, succ(zero), U)) -> U4_GGA(K, N, =_out_ag(U, U)) U4_GGA(z0, z1, =_out_ag(z2, z2)) -> IOTA1_IN_GGA(z2, z1) The TRS R consists of the following rules: isPlus_in_gga(zero, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), succ(Y)) -> U31_gga(X, Y, isPlus_in_gga(X, Y)) isPlus_in_gga(pred(X), succ(Y)) -> U33_gga(X, Y, isPlus_in_gga(X, Y)) isPlus_in_gga(succ(X), zero) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(pred(X), zero) -> isPlus_out_gga(pred(X), zero, pred(X)) U33_gga(X, Y, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U31_gga(X, Y, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) The set Q consists of the following terms: isPlus_in_gga(x0, x1) U31_gga(x0, x1, x2) U33_gga(x0, x1, x2) U32_gga(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (223) 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]. U32_gga(x0, x1, x2) ---------------------------------------- (224) Obligation: Q DP problem: The TRS P consists of the following rules: IOTA1_IN_GGA(K, N) -> U3_GGA(K, N, isPlus_in_gga(K, succ(zero))) U3_GGA(K, N, isPlus_out_gga(K, succ(zero), U)) -> U4_GGA(K, N, =_out_ag(U, U)) U4_GGA(z0, z1, =_out_ag(z2, z2)) -> IOTA1_IN_GGA(z2, z1) The TRS R consists of the following rules: isPlus_in_gga(zero, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), succ(Y)) -> U31_gga(X, Y, isPlus_in_gga(X, Y)) isPlus_in_gga(pred(X), succ(Y)) -> U33_gga(X, Y, isPlus_in_gga(X, Y)) isPlus_in_gga(succ(X), zero) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(pred(X), zero) -> isPlus_out_gga(pred(X), zero, pred(X)) U33_gga(X, Y, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U31_gga(X, Y, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) The set Q consists of the following terms: isPlus_in_gga(x0, x1) U31_gga(x0, x1, x2) U33_gga(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (225) 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: isPlus_in_gga(pred(X), succ(Y)) -> U33_gga(X, Y, isPlus_in_gga(X, Y)) Used ordering: Polynomial interpretation [POLO]: POL(=_out_ag(x_1, x_2)) = 2*x_2 POL(IOTA1_IN_GGA(x_1, x_2)) = 2*x_1 + 2*x_2 POL(U31_gga(x_1, x_2, x_3)) = x_3 POL(U33_gga(x_1, x_2, x_3)) = x_1 + x_3 POL(U3_GGA(x_1, x_2, x_3)) = 2*x_2 + 2*x_3 POL(U4_GGA(x_1, x_2, x_3)) = 2*x_2 + x_3 POL(isPlus_in_gga(x_1, x_2)) = x_1 + 2*x_2 POL(isPlus_out_gga(x_1, x_2, x_3)) = x_3 POL(pred(x_1)) = 1 + 2*x_1 POL(succ(x_1)) = x_1 POL(zero) = 0 ---------------------------------------- (226) Obligation: Q DP problem: The TRS P consists of the following rules: IOTA1_IN_GGA(K, N) -> U3_GGA(K, N, isPlus_in_gga(K, succ(zero))) U3_GGA(K, N, isPlus_out_gga(K, succ(zero), U)) -> U4_GGA(K, N, =_out_ag(U, U)) U4_GGA(z0, z1, =_out_ag(z2, z2)) -> IOTA1_IN_GGA(z2, z1) The TRS R consists of the following rules: isPlus_in_gga(zero, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), succ(Y)) -> U31_gga(X, Y, isPlus_in_gga(X, Y)) isPlus_in_gga(succ(X), zero) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(pred(X), zero) -> isPlus_out_gga(pred(X), zero, pred(X)) U33_gga(X, Y, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U31_gga(X, Y, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) The set Q consists of the following terms: isPlus_in_gga(x0, x1) U31_gga(x0, x1, x2) U33_gga(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (227) 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. ---------------------------------------- (228) Obligation: Q DP problem: The TRS P consists of the following rules: IOTA1_IN_GGA(K, N) -> U3_GGA(K, N, isPlus_in_gga(K, succ(zero))) U3_GGA(K, N, isPlus_out_gga(K, succ(zero), U)) -> U4_GGA(K, N, =_out_ag(U, U)) U4_GGA(z0, z1, =_out_ag(z2, z2)) -> IOTA1_IN_GGA(z2, z1) The TRS R consists of the following rules: isPlus_in_gga(zero, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), succ(Y)) -> U31_gga(X, Y, isPlus_in_gga(X, Y)) isPlus_in_gga(succ(X), zero) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(pred(X), zero) -> isPlus_out_gga(pred(X), zero, pred(X)) U31_gga(X, Y, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) The set Q consists of the following terms: isPlus_in_gga(x0, x1) U31_gga(x0, x1, x2) U33_gga(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (229) 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]. U33_gga(x0, x1, x2) ---------------------------------------- (230) Obligation: Q DP problem: The TRS P consists of the following rules: IOTA1_IN_GGA(K, N) -> U3_GGA(K, N, isPlus_in_gga(K, succ(zero))) U3_GGA(K, N, isPlus_out_gga(K, succ(zero), U)) -> U4_GGA(K, N, =_out_ag(U, U)) U4_GGA(z0, z1, =_out_ag(z2, z2)) -> IOTA1_IN_GGA(z2, z1) The TRS R consists of the following rules: isPlus_in_gga(zero, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), succ(Y)) -> U31_gga(X, Y, isPlus_in_gga(X, Y)) isPlus_in_gga(succ(X), zero) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(pred(X), zero) -> isPlus_out_gga(pred(X), zero, pred(X)) U31_gga(X, Y, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) The set Q consists of the following terms: isPlus_in_gga(x0, x1) U31_gga(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (231) Obligation: Pi DP problem: The TRS P consists of the following rules: ISTIMES_IN_GGA(succ(X), pred(Y), Z) -> ISTIMES_IN_GGA(succ(X), Y, A) ISTIMES_IN_GGA(succ(X), succ(Y), Z) -> ISTIMES_IN_GGA(succ(X), Y, A) The TRS R consists of the following rules: pds_in_ga(Order, .(zero, Ans)) -> U24_ga(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) isPlus_in_gga(zero, X, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), zero, succ(X)) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), succ(Y), succ(succ(Z))) -> U31_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(succ(X), pred(Y), Z) -> U32_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), zero, pred(X)) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), succ(Y), Z) -> U33_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), pred(Y), pred(pred(Z))) -> U34_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) U34_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U33_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U32_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U31_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U24_ga(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_ga(Order, Ans, isTimes_in_gga(Order, U, U1)) isTimes_in_gga(X, zero, zero) -> isTimes_out_gga(X, zero, zero) isTimes_in_gga(zero, succ(Y), zero) -> isTimes_out_gga(zero, succ(Y), zero) isTimes_in_gga(zero, pred(Y), zero) -> isTimes_out_gga(zero, pred(Y), zero) isTimes_in_gga(succ(X), succ(Y), Z) -> U41_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(succ(X), pred(Y), Z) -> U43_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(pred(X), succ(Y), Z) -> U45_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) isTimes_in_gga(pred(X), pred(Y), Z) -> U47_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) U47_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_gga(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U48_gga(X, Y, Z, isMinus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), pred(Y), Z) U45_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_gga(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U46_gga(X, Y, Z, isPlus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), succ(Y), Z) U43_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_gga(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U44_gga(X, Y, Z, isMinus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), pred(Y), Z) U41_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_gga(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U42_gga(X, Y, Z, isPlus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), succ(Y), Z) U25_ga(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_ga(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U26_ga(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_ga(Order, Ans, =_in_ag(N, U2)) =_in_ag(X, X) -> =_out_ag(X, X) U27_ga(Order, Ans, =_out_ag(N, U2)) -> U28_ga(Order, Ans, N, iota_in_ga(N, .(zero, List))) iota_in_ga(N, List) -> U2_ga(N, List, iota1_in_gga(zero, N, List)) iota1_in_gga(K, K, []) -> iota1_out_gga(K, K, []) iota1_in_gga(K, N, .(K, List)) -> U3_gga(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_gga(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_gga(K, N, List, =_in_ag(K1, U)) U4_gga(K, N, List, =_out_ag(K1, U)) -> U5_gga(K, N, List, iota1_in_gga(K1, N, List)) U5_gga(K, N, List, iota1_out_gga(K1, N, List)) -> iota1_out_gga(K, N, .(K, List)) U2_ga(N, List, iota1_out_gga(zero, N, List)) -> iota_out_ga(N, List) U28_ga(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_ga(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) pds1_in_ggag([], X6, [], X7) -> pds1_out_ggag([], X6, [], X7) pds1_in_ggag(Unused, List, .(Choice, Rest), Mod) -> U21_ggag(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(X, .(X1, Xs)) -> U1_ag(X, X1, Xs, member_in_ag(X, Xs)) U1_ag(X, X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) U21_ggag(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_ggag(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) check_in_gggag([], X4, L, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, L3, Modulus) -> U17_gggag(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) dif_in_gggaa([], X2, X3, [], []) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U17_gggag(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) mergedelete_in_gga([], L, L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R), L2) -> U12_gga(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) mergedelete_in_gga(.(D, Ds), .(X, R), .(X, L2)) -> U13_gga(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_gga(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_gga(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U15_gga(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U16_gga(D, Ds, X, R, L2, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U12_gga(D, Ds, R, L2, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) rev_in_gga([], L, L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y, L) -> U11_gga(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) U11_gga(X, Xs, Y, L, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U22_ggag(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_ggag(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U23_ggag(Unused, List, Choice, Rest, Mod, pds1_out_ggag(U3, .(Choice, List), Rest, Mod)) -> pds1_out_ggag(Unused, List, .(Choice, Rest), Mod) U29_ga(Order, Ans, pds1_out_ggag(List, .(zero, []), Ans, N)) -> pds_out_ga(Order, .(zero, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U24_ga(x1, x2, x3) = U24_ga(x1, x3) isPlus_in_gga(x1, x2, x3) = isPlus_in_gga(x1, x2) zero = zero isPlus_out_gga(x1, x2, x3) = isPlus_out_gga(x1, x2, x3) succ(x1) = succ(x1) U31_gga(x1, x2, x3, x4) = U31_gga(x1, x2, x4) pred(x1) = pred(x1) U32_gga(x1, x2, x3, x4) = U32_gga(x1, x2, x4) U33_gga(x1, x2, x3, x4) = U33_gga(x1, x2, x4) U34_gga(x1, x2, x3, x4) = U34_gga(x1, x2, x4) U25_ga(x1, x2, x3) = U25_ga(x1, x3) isTimes_in_gga(x1, x2, x3) = isTimes_in_gga(x1, x2) isTimes_out_gga(x1, x2, x3) = isTimes_out_gga(x1, x2, x3) U41_gga(x1, x2, x3, x4) = U41_gga(x1, x2, x4) U43_gga(x1, x2, x3, x4) = U43_gga(x1, x2, x4) U45_gga(x1, x2, x3, x4) = U45_gga(x1, x2, x4) U47_gga(x1, x2, x3, x4) = U47_gga(x1, x2, x4) U48_gga(x1, x2, x3, x4) = U48_gga(x1, x2, x4) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x1, x2, x3) U35_gga(x1, x2, x3) = U35_gga(x1, x3) U36_gga(x1, x2, x3) = U36_gga(x1, x3) U37_gga(x1, x2, x3, x4) = U37_gga(x1, x2, x4) U38_gga(x1, x2, x3, x4) = U38_gga(x1, x2, x4) U39_gga(x1, x2, x3, x4) = U39_gga(x1, x2, x4) U40_gga(x1, x2, x3, x4) = U40_gga(x1, x2, x4) U46_gga(x1, x2, x3, x4) = U46_gga(x1, x2, x4) U44_gga(x1, x2, x3, x4) = U44_gga(x1, x2, x4) U42_gga(x1, x2, x3, x4) = U42_gga(x1, x2, x4) U26_ga(x1, x2, x3) = U26_ga(x1, x3) U27_ga(x1, x2, x3) = U27_ga(x1, x3) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1, x2) U28_ga(x1, x2, x3, x4) = U28_ga(x1, x3, x4) iota_in_ga(x1, x2) = iota_in_ga(x1) U2_ga(x1, x2, x3) = U2_ga(x1, x3) iota1_in_gga(x1, x2, x3) = iota1_in_gga(x1, x2) iota1_out_gga(x1, x2, x3) = iota1_out_gga(x1, x2, x3) U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4) iota_out_ga(x1, x2) = iota_out_ga(x1, x2) .(x1, x2) = .(x1, x2) U29_ga(x1, x2, x3) = U29_ga(x1, x3) pds1_in_ggag(x1, x2, x3, x4) = pds1_in_ggag(x1, x2, x4) [] = [] pds1_out_ggag(x1, x2, x3, x4) = pds1_out_ggag(x1, x2, x3, x4) U21_ggag(x1, x2, x3, x4, x5, x6) = U21_ggag(x1, x2, x5, x6) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1, x2) U1_ag(x1, x2, x3, x4) = U1_ag(x2, x3, x4) U22_ggag(x1, x2, x3, x4, x5, x6) = U22_ggag(x1, x2, x3, x5, x6) check_in_gggag(x1, x2, x3, x4, x5) = check_in_gggag(x1, x2, x3, x5) check_out_gggag(x1, x2, x3, x4, x5) = check_out_gggag(x1, x2, x3, x4, x5) U17_gggag(x1, x2, x3, x4, x5, x6) = U17_gggag(x1, x2, x3, x5, x6) dif_in_gggaa(x1, x2, x3, x4, x5) = dif_in_gggaa(x1, x2, x3) dif_out_gggaa(x1, x2, x3, x4, x5) = dif_out_gggaa(x1, x2, x3, x4, x5) U6_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gggaa(x1, x2, x3, x4, x9) U7_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_gggaa(x1, x2, x3, x4, x9, x10) U8_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_gggaa(x1, x2, x3, x4, x5, x9, x10) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg(x1, x2, x3) U35_ggg(x1, x2, x3) = U35_ggg(x1, x2, x3) U36_ggg(x1, x2, x3) = U36_ggg(x1, x2, x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x1, x2, x3, x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x1, x2, x3, x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x1, x2, x3, x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x1, x2, x3, x4) U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_gggaa(x1, x2, x3, x4, x5, x9) U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_gggaa(x1, x2, x3, x4, x5, x7, x9) U18_gggag(x1, x2, x3, x4, x5, x6, x7) = U18_gggag(x1, x2, x3, x5, x6, x7) mergedelete_in_gga(x1, x2, x3) = mergedelete_in_gga(x1, x2) mergedelete_out_gga(x1, x2, x3) = mergedelete_out_gga(x1, x2, x3) U12_gga(x1, x2, x3, x4, x5) = U12_gga(x1, x2, x3, x5) U13_gga(x1, x2, x3, x4, x5, x6) = U13_gga(x1, x2, x3, x4, x6) U14_gga(x1, x2, x3, x4, x5, x6, x7) = U14_gga(x1, x2, x3, x4, x6, x7) U15_gga(x1, x2, x3, x4, x5, x6) = U15_gga(x1, x2, x3, x4, x6) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg(x1, x2) U64_gg(x1, x2, x3) = U64_gg(x1, x2, x3) U65_gg(x1, x2, x3) = U65_gg(x1, x2, x3) U16_gga(x1, x2, x3, x4, x5, x6) = U16_gga(x1, x2, x3, x4, x6) U19_gggag(x1, x2, x3, x4, x5, x6, x7) = U19_gggag(x1, x2, x3, x5, x6, x7) rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2) rev_out_gga(x1, x2, x3) = rev_out_gga(x1, x2, x3) U11_gga(x1, x2, x3, x4, x5) = U11_gga(x1, x2, x3, x5) U20_gggag(x1, x2, x3, x4, x5, x6) = U20_gggag(x1, x2, x3, x5, x6) U23_ggag(x1, x2, x3, x4, x5, x6) = U23_ggag(x1, x2, x3, x5, x6) pds_out_ga(x1, x2) = pds_out_ga(x1, x2) ISTIMES_IN_GGA(x1, x2, x3) = ISTIMES_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (232) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (233) Obligation: Pi DP problem: The TRS P consists of the following rules: ISTIMES_IN_GGA(succ(X), pred(Y), Z) -> ISTIMES_IN_GGA(succ(X), Y, A) ISTIMES_IN_GGA(succ(X), succ(Y), Z) -> ISTIMES_IN_GGA(succ(X), Y, A) R is empty. The argument filtering Pi contains the following mapping: succ(x1) = succ(x1) pred(x1) = pred(x1) ISTIMES_IN_GGA(x1, x2, x3) = ISTIMES_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (234) Obligation: Pi DP problem: The TRS P consists of the following rules: ISTIMES_IN_GGA(pred(X), pred(Y), Z) -> ISTIMES_IN_GGA(pred(X), Y, A) ISTIMES_IN_GGA(pred(X), succ(Y), Z) -> ISTIMES_IN_GGA(pred(X), Y, A) The TRS R consists of the following rules: pds_in_ga(Order, .(zero, Ans)) -> U24_ga(Order, Ans, isPlus_in_gga(Order, succ(zero), U)) isPlus_in_gga(zero, X, X) -> isPlus_out_gga(zero, X, X) isPlus_in_gga(succ(X), zero, succ(X)) -> isPlus_out_gga(succ(X), zero, succ(X)) isPlus_in_gga(succ(X), succ(Y), succ(succ(Z))) -> U31_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(succ(X), pred(Y), Z) -> U32_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), zero, pred(X)) -> isPlus_out_gga(pred(X), zero, pred(X)) isPlus_in_gga(pred(X), succ(Y), Z) -> U33_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) isPlus_in_gga(pred(X), pred(Y), pred(pred(Z))) -> U34_gga(X, Y, Z, isPlus_in_gga(X, Y, Z)) U34_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), pred(Y), pred(pred(Z))) U33_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(pred(X), succ(Y), Z) U32_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), pred(Y), Z) U31_gga(X, Y, Z, isPlus_out_gga(X, Y, Z)) -> isPlus_out_gga(succ(X), succ(Y), succ(succ(Z))) U24_ga(Order, Ans, isPlus_out_gga(Order, succ(zero), U)) -> U25_ga(Order, Ans, isTimes_in_gga(Order, U, U1)) isTimes_in_gga(X, zero, zero) -> isTimes_out_gga(X, zero, zero) isTimes_in_gga(zero, succ(Y), zero) -> isTimes_out_gga(zero, succ(Y), zero) isTimes_in_gga(zero, pred(Y), zero) -> isTimes_out_gga(zero, pred(Y), zero) isTimes_in_gga(succ(X), succ(Y), Z) -> U41_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(succ(X), pred(Y), Z) -> U43_gga(X, Y, Z, isTimes_in_gga(succ(X), Y, A)) isTimes_in_gga(pred(X), succ(Y), Z) -> U45_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) isTimes_in_gga(pred(X), pred(Y), Z) -> U47_gga(X, Y, Z, isTimes_in_gga(pred(X), Y, A)) U47_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U48_gga(X, Y, Z, isMinus_in_gga(A, pred(X), Z)) isMinus_in_gga(X, zero, X) -> isMinus_out_gga(X, zero, X) isMinus_in_gga(zero, succ(Y), pred(Z)) -> U35_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(zero, pred(Y), succ(Z)) -> U36_gga(Y, Z, isMinus_in_gga(zero, Y, Z)) isMinus_in_gga(succ(X), succ(Y), Z) -> U37_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(succ(X), pred(Y), succ(succ(Z))) -> U38_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), succ(Y), pred(pred(Z))) -> U39_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) isMinus_in_gga(pred(X), pred(Y), Z) -> U40_gga(X, Y, Z, isMinus_in_gga(X, Y, Z)) U40_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), pred(Y), Z) U39_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(pred(X), succ(Y), pred(pred(Z))) U38_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), pred(Y), succ(succ(Z))) U37_gga(X, Y, Z, isMinus_out_gga(X, Y, Z)) -> isMinus_out_gga(succ(X), succ(Y), Z) U36_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, pred(Y), succ(Z)) U35_gga(Y, Z, isMinus_out_gga(zero, Y, Z)) -> isMinus_out_gga(zero, succ(Y), pred(Z)) U48_gga(X, Y, Z, isMinus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), pred(Y), Z) U45_gga(X, Y, Z, isTimes_out_gga(pred(X), Y, A)) -> U46_gga(X, Y, Z, isPlus_in_gga(A, pred(X), Z)) U46_gga(X, Y, Z, isPlus_out_gga(A, pred(X), Z)) -> isTimes_out_gga(pred(X), succ(Y), Z) U43_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U44_gga(X, Y, Z, isMinus_in_gga(A, succ(X), Z)) U44_gga(X, Y, Z, isMinus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), pred(Y), Z) U41_gga(X, Y, Z, isTimes_out_gga(succ(X), Y, A)) -> U42_gga(X, Y, Z, isPlus_in_gga(A, succ(X), Z)) U42_gga(X, Y, Z, isPlus_out_gga(A, succ(X), Z)) -> isTimes_out_gga(succ(X), succ(Y), Z) U25_ga(Order, Ans, isTimes_out_gga(Order, U, U1)) -> U26_ga(Order, Ans, isPlus_in_gga(U1, succ(zero), U2)) U26_ga(Order, Ans, isPlus_out_gga(U1, succ(zero), U2)) -> U27_ga(Order, Ans, =_in_ag(N, U2)) =_in_ag(X, X) -> =_out_ag(X, X) U27_ga(Order, Ans, =_out_ag(N, U2)) -> U28_ga(Order, Ans, N, iota_in_ga(N, .(zero, List))) iota_in_ga(N, List) -> U2_ga(N, List, iota1_in_gga(zero, N, List)) iota1_in_gga(K, K, []) -> iota1_out_gga(K, K, []) iota1_in_gga(K, N, .(K, List)) -> U3_gga(K, N, List, isPlus_in_gga(K, succ(zero), U)) U3_gga(K, N, List, isPlus_out_gga(K, succ(zero), U)) -> U4_gga(K, N, List, =_in_ag(K1, U)) U4_gga(K, N, List, =_out_ag(K1, U)) -> U5_gga(K, N, List, iota1_in_gga(K1, N, List)) U5_gga(K, N, List, iota1_out_gga(K1, N, List)) -> iota1_out_gga(K, N, .(K, List)) U2_ga(N, List, iota1_out_gga(zero, N, List)) -> iota_out_ga(N, List) U28_ga(Order, Ans, N, iota_out_ga(N, .(zero, List))) -> U29_ga(Order, Ans, pds1_in_ggag(List, .(zero, []), Ans, N)) pds1_in_ggag([], X6, [], X7) -> pds1_out_ggag([], X6, [], X7) pds1_in_ggag(Unused, List, .(Choice, Rest), Mod) -> U21_ggag(Unused, List, Choice, Rest, Mod, member_in_ag(Choice, Unused)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) member_in_ag(X, .(X1, Xs)) -> U1_ag(X, X1, Xs, member_in_ag(X, Xs)) U1_ag(X, X1, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X1, Xs)) U21_ggag(Unused, List, Choice, Rest, Mod, member_out_ag(Choice, Unused)) -> U22_ggag(Unused, List, Choice, Rest, Mod, check_in_gggag(List, Choice, Unused, U3, Mod)) check_in_gggag([], X4, L, L, X5) -> check_out_gggag([], X4, L, L, X5) check_in_gggag(S, Choice, Old, L3, Modulus) -> U17_gggag(S, Choice, Old, L3, Modulus, dif_in_gggaa(S, Choice, Modulus, Ds, Dds)) dif_in_gggaa([], X2, X3, [], []) -> dif_out_gggaa([], X2, X3, [], []) dif_in_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_in_gga(Val, S, U)) U6_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, isMinus_out_gga(Val, S, U)) -> U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_in_ag(D, U)) U7_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, =_out_ag(D, U)) -> U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_in_ggg(Mod, D, U)) isMinus_in_ggg(X, zero, X) -> isMinus_out_ggg(X, zero, X) isMinus_in_ggg(zero, succ(Y), pred(Z)) -> U35_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(zero, pred(Y), succ(Z)) -> U36_ggg(Y, Z, isMinus_in_ggg(zero, Y, Z)) isMinus_in_ggg(succ(X), succ(Y), Z) -> U37_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(succ(X), pred(Y), succ(succ(Z))) -> U38_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), succ(Y), pred(pred(Z))) -> U39_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) isMinus_in_ggg(pred(X), pred(Y), Z) -> U40_ggg(X, Y, Z, isMinus_in_ggg(X, Y, Z)) U40_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), pred(Y), Z) U39_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(pred(X), succ(Y), pred(pred(Z))) U38_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), pred(Y), succ(succ(Z))) U37_ggg(X, Y, Z, isMinus_out_ggg(X, Y, Z)) -> isMinus_out_ggg(succ(X), succ(Y), Z) U36_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, pred(Y), succ(Z)) U35_ggg(Y, Z, isMinus_out_ggg(zero, Y, Z)) -> isMinus_out_ggg(zero, succ(Y), pred(Z)) U8_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, U, isMinus_out_ggg(Mod, D, U)) -> U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_in_ag(D2, U)) U9_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, =_out_ag(D2, U)) -> U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gggaa(Ss, Val, Mod, Ds, D2s)) U10_gggaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gggaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gggaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U17_gggag(S, Choice, Old, L3, Modulus, dif_out_gggaa(S, Choice, Modulus, Ds, Dds)) -> U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_gga(Ds, Old, L2)) mergedelete_in_gga([], L, L) -> mergedelete_out_gga([], L, L) mergedelete_in_gga(.(D, Ds), .(D, R), L2) -> U12_gga(D, Ds, R, L2, mergedelete_in_gga(Ds, R, L2)) mergedelete_in_gga(.(D, Ds), .(X, R), .(X, L2)) -> U13_gga(D, Ds, X, R, L2, =_in_ag(X1, D)) U13_gga(D, Ds, X, R, L2, =_out_ag(X1, D)) -> U14_gga(D, Ds, X, R, L2, X1, =_in_ag(X2, X)) U14_gga(D, Ds, X, R, L2, X1, =_out_ag(X2, X)) -> U15_gga(D, Ds, X, R, L2, isGreater_in_gg(X1, X2)) isGreater_in_gg(succ(X), zero) -> isGreater_out_gg(succ(X), zero) isGreater_in_gg(succ(X), pred(Y)) -> isGreater_out_gg(succ(X), pred(Y)) isGreater_in_gg(succ(X), succ(Y)) -> U64_gg(X, Y, isGreater_in_gg(X, Y)) isGreater_in_gg(zero, pred(Y)) -> isGreater_out_gg(zero, pred(Y)) isGreater_in_gg(pred(X), pred(Y)) -> U65_gg(X, Y, isGreater_in_gg(X, Y)) U65_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(pred(X), pred(Y)) U64_gg(X, Y, isGreater_out_gg(X, Y)) -> isGreater_out_gg(succ(X), succ(Y)) U15_gga(D, Ds, X, R, L2, isGreater_out_gg(X1, X2)) -> U16_gga(D, Ds, X, R, L2, mergedelete_in_gga(.(D, Ds), R, L2)) U16_gga(D, Ds, X, R, L2, mergedelete_out_gga(.(D, Ds), R, L2)) -> mergedelete_out_gga(.(D, Ds), .(X, R), .(X, L2)) U12_gga(D, Ds, R, L2, mergedelete_out_gga(Ds, R, L2)) -> mergedelete_out_gga(.(D, Ds), .(D, R), L2) U18_gggag(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_gga(Ds, Old, L2)) -> U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_in_gga(Dds, [], Rds)) rev_in_gga([], L, L) -> rev_out_gga([], L, L) rev_in_gga(.(X, Xs), Y, L) -> U11_gga(X, Xs, Y, L, rev_in_gga(Xs, .(X, Y), L)) U11_gga(X, Xs, Y, L, rev_out_gga(Xs, .(X, Y), L)) -> rev_out_gga(.(X, Xs), Y, L) U19_gggag(S, Choice, Old, L3, Modulus, L2, rev_out_gga(Dds, [], Rds)) -> U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_in_gga(Rds, L2, L3)) U20_gggag(S, Choice, Old, L3, Modulus, mergedelete_out_gga(Rds, L2, L3)) -> check_out_gggag(S, Choice, Old, L3, Modulus) U22_ggag(Unused, List, Choice, Rest, Mod, check_out_gggag(List, Choice, Unused, U3, Mod)) -> U23_ggag(Unused, List, Choice, Rest, Mod, pds1_in_ggag(U3, .(Choice, List), Rest, Mod)) U23_ggag(Unused, List, Choice, Rest, Mod, pds1_out_ggag(U3, .(Choice, List), Rest, Mod)) -> pds1_out_ggag(Unused, List, .(Choice, Rest), Mod) U29_ga(Order, Ans, pds1_out_ggag(List, .(zero, []), Ans, N)) -> pds_out_ga(Order, .(zero, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U24_ga(x1, x2, x3) = U24_ga(x1, x3) isPlus_in_gga(x1, x2, x3) = isPlus_in_gga(x1, x2) zero = zero isPlus_out_gga(x1, x2, x3) = isPlus_out_gga(x1, x2, x3) succ(x1) = succ(x1) U31_gga(x1, x2, x3, x4) = U31_gga(x1, x2, x4) pred(x1) = pred(x1) U32_gga(x1, x2, x3, x4) = U32_gga(x1, x2, x4) U33_gga(x1, x2, x3, x4) = U33_gga(x1, x2, x4) U34_gga(x1, x2, x3, x4) = U34_gga(x1, x2, x4) U25_ga(x1, x2, x3) = U25_ga(x1, x3) isTimes_in_gga(x1, x2, x3) = isTimes_in_gga(x1, x2) isTimes_out_gga(x1, x2, x3) = isTimes_out_gga(x1, x2, x3) U41_gga(x1, x2, x3, x4) = U41_gga(x1, x2, x4) U43_gga(x1, x2, x3, x4) = U43_gga(x1, x2, x4) U45_gga(x1, x2, x3, x4) = U45_gga(x1, x2, x4) U47_gga(x1, x2, x3, x4) = U47_gga(x1, x2, x4) U48_gga(x1, x2, x3, x4) = U48_gga(x1, x2, x4) isMinus_in_gga(x1, x2, x3) = isMinus_in_gga(x1, x2) isMinus_out_gga(x1, x2, x3) = isMinus_out_gga(x1, x2, x3) U35_gga(x1, x2, x3) = U35_gga(x1, x3) U36_gga(x1, x2, x3) = U36_gga(x1, x3) U37_gga(x1, x2, x3, x4) = U37_gga(x1, x2, x4) U38_gga(x1, x2, x3, x4) = U38_gga(x1, x2, x4) U39_gga(x1, x2, x3, x4) = U39_gga(x1, x2, x4) U40_gga(x1, x2, x3, x4) = U40_gga(x1, x2, x4) U46_gga(x1, x2, x3, x4) = U46_gga(x1, x2, x4) U44_gga(x1, x2, x3, x4) = U44_gga(x1, x2, x4) U42_gga(x1, x2, x3, x4) = U42_gga(x1, x2, x4) U26_ga(x1, x2, x3) = U26_ga(x1, x3) U27_ga(x1, x2, x3) = U27_ga(x1, x3) =_in_ag(x1, x2) = =_in_ag(x2) =_out_ag(x1, x2) = =_out_ag(x1, x2) U28_ga(x1, x2, x3, x4) = U28_ga(x1, x3, x4) iota_in_ga(x1, x2) = iota_in_ga(x1) U2_ga(x1, x2, x3) = U2_ga(x1, x3) iota1_in_gga(x1, x2, x3) = iota1_in_gga(x1, x2) iota1_out_gga(x1, x2, x3) = iota1_out_gga(x1, x2, x3) U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4) iota_out_ga(x1, x2) = iota_out_ga(x1, x2) .(x1, x2) = .(x1, x2) U29_ga(x1, x2, x3) = U29_ga(x1, x3) pds1_in_ggag(x1, x2, x3, x4) = pds1_in_ggag(x1, x2, x4) [] = [] pds1_out_ggag(x1, x2, x3, x4) = pds1_out_ggag(x1, x2, x3, x4) U21_ggag(x1, x2, x3, x4, x5, x6) = U21_ggag(x1, x2, x5, x6) member_in_ag(x1, x2) = member_in_ag(x2) member_out_ag(x1, x2) = member_out_ag(x1, x2) U1_ag(x1, x2, x3, x4) = U1_ag(x2, x3, x4) U22_ggag(x1, x2, x3, x4, x5, x6) = U22_ggag(x1, x2, x3, x5, x6) check_in_gggag(x1, x2, x3, x4, x5) = check_in_gggag(x1, x2, x3, x5) check_out_gggag(x1, x2, x3, x4, x5) = check_out_gggag(x1, x2, x3, x4, x5) U17_gggag(x1, x2, x3, x4, x5, x6) = U17_gggag(x1, x2, x3, x5, x6) dif_in_gggaa(x1, x2, x3, x4, x5) = dif_in_gggaa(x1, x2, x3) dif_out_gggaa(x1, x2, x3, x4, x5) = dif_out_gggaa(x1, x2, x3, x4, x5) U6_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gggaa(x1, x2, x3, x4, x9) U7_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U7_gggaa(x1, x2, x3, x4, x9, x10) U8_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U8_gggaa(x1, x2, x3, x4, x5, x9, x10) isMinus_in_ggg(x1, x2, x3) = isMinus_in_ggg(x1, x2, x3) isMinus_out_ggg(x1, x2, x3) = isMinus_out_ggg(x1, x2, x3) U35_ggg(x1, x2, x3) = U35_ggg(x1, x2, x3) U36_ggg(x1, x2, x3) = U36_ggg(x1, x2, x3) U37_ggg(x1, x2, x3, x4) = U37_ggg(x1, x2, x3, x4) U38_ggg(x1, x2, x3, x4) = U38_ggg(x1, x2, x3, x4) U39_ggg(x1, x2, x3, x4) = U39_ggg(x1, x2, x3, x4) U40_ggg(x1, x2, x3, x4) = U40_ggg(x1, x2, x3, x4) U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_gggaa(x1, x2, x3, x4, x5, x9) U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U10_gggaa(x1, x2, x3, x4, x5, x7, x9) U18_gggag(x1, x2, x3, x4, x5, x6, x7) = U18_gggag(x1, x2, x3, x5, x6, x7) mergedelete_in_gga(x1, x2, x3) = mergedelete_in_gga(x1, x2) mergedelete_out_gga(x1, x2, x3) = mergedelete_out_gga(x1, x2, x3) U12_gga(x1, x2, x3, x4, x5) = U12_gga(x1, x2, x3, x5) U13_gga(x1, x2, x3, x4, x5, x6) = U13_gga(x1, x2, x3, x4, x6) U14_gga(x1, x2, x3, x4, x5, x6, x7) = U14_gga(x1, x2, x3, x4, x6, x7) U15_gga(x1, x2, x3, x4, x5, x6) = U15_gga(x1, x2, x3, x4, x6) isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2) isGreater_out_gg(x1, x2) = isGreater_out_gg(x1, x2) U64_gg(x1, x2, x3) = U64_gg(x1, x2, x3) U65_gg(x1, x2, x3) = U65_gg(x1, x2, x3) U16_gga(x1, x2, x3, x4, x5, x6) = U16_gga(x1, x2, x3, x4, x6) U19_gggag(x1, x2, x3, x4, x5, x6, x7) = U19_gggag(x1, x2, x3, x5, x6, x7) rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2) rev_out_gga(x1, x2, x3) = rev_out_gga(x1, x2, x3) U11_gga(x1, x2, x3, x4, x5) = U11_gga(x1, x2, x3, x5) U20_gggag(x1, x2, x3, x4, x5, x6) = U20_gggag(x1, x2, x3, x5, x6) U23_ggag(x1, x2, x3, x4, x5, x6) = U23_ggag(x1, x2, x3, x5, x6) pds_out_ga(x1, x2) = pds_out_ga(x1, x2) ISTIMES_IN_GGA(x1, x2, x3) = ISTIMES_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (235) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (236) Obligation: Pi DP problem: The TRS P consists of the following rules: ISTIMES_IN_GGA(pred(X), pred(Y), Z) -> ISTIMES_IN_GGA(pred(X), Y, A) ISTIMES_IN_GGA(pred(X), succ(Y), Z) -> ISTIMES_IN_GGA(pred(X), Y, A) R is empty. The argument filtering Pi contains the following mapping: succ(x1) = succ(x1) pred(x1) = pred(x1) ISTIMES_IN_GGA(x1, x2, x3) = ISTIMES_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (237) CutEliminatorProof (SOUND) Eliminated all cuts by simply ignoring them[PROLOG]. ---------------------------------------- (238) Obligation: Clauses: member(X, .(X, Xs)). member(X, .(X1, Xs)) :- member(X, Xs). iota(N, List) :- iota1(0, N, List). iota1(K, K, []). iota1(K, N, .(K, List)) :- ','(is(K1, +(K, 1)), iota1(K1, N, List)). dif([], X2, X3, [], []). dif(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) :- ','(is(D, -(Val, S)), ','(is(D2, -(Mod, D)), dif(Ss, Val, Mod, Ds, D2s))). rev([], L, L). rev(.(X, Xs), Y, L) :- rev(Xs, .(X, Y), L). mergedelete([], L, L). mergedelete(.(D, Ds), .(D, R), L2) :- mergedelete(Ds, R, L2). mergedelete(.(D, Ds), .(X, R), .(X, L2)) :- ','(>(D, X), mergedelete(.(D, Ds), R, L2)). check([], X4, L, L, X5). check(S, Choice, Old, L3, Modulus) :- ','(dif(S, Choice, Modulus, Ds, Dds), ','(mergedelete(Ds, Old, L2), ','(rev(Dds, [], Rds), mergedelete(Rds, L2, L3)))). pds1([], X6, [], X7). pds1(Unused, List, .(Choice, Rest), Mod) :- ','(member(Choice, Unused), ','(check(List, Choice, Unused, U3, Mod), pds1(U3, .(Choice, List), Rest, Mod))). pds(Order, .(0, Ans)) :- ','(is(N, +(*(Order, +(Order, 1)), 1)), ','(iota(N, .(0, List)), pds1(List, .(0, []), Ans, N))). pdsbm(N) :- pds(N, .(0, .(1, X))). Query: pds(g,a) ---------------------------------------- (239) UndefinedPredicateHandlerProof (SOUND) Added facts for all undefined predicates [PROLOG]. ---------------------------------------- (240) Obligation: Clauses: member(X, .(X, Xs)). member(X, .(X1, Xs)) :- member(X, Xs). iota(N, List) :- iota1(0, N, List). iota1(K, K, []). iota1(K, N, .(K, List)) :- ','(is(K1, +(K, 1)), iota1(K1, N, List)). dif([], X2, X3, [], []). dif(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) :- ','(is(D, -(Val, S)), ','(is(D2, -(Mod, D)), dif(Ss, Val, Mod, Ds, D2s))). rev([], L, L). rev(.(X, Xs), Y, L) :- rev(Xs, .(X, Y), L). mergedelete([], L, L). mergedelete(.(D, Ds), .(D, R), L2) :- mergedelete(Ds, R, L2). mergedelete(.(D, Ds), .(X, R), .(X, L2)) :- ','(>(D, X), mergedelete(.(D, Ds), R, L2)). check([], X4, L, L, X5). check(S, Choice, Old, L3, Modulus) :- ','(dif(S, Choice, Modulus, Ds, Dds), ','(mergedelete(Ds, Old, L2), ','(rev(Dds, [], Rds), mergedelete(Rds, L2, L3)))). pds1([], X6, [], X7). pds1(Unused, List, .(Choice, Rest), Mod) :- ','(member(Choice, Unused), ','(check(List, Choice, Unused, U3, Mod), pds1(U3, .(Choice, List), Rest, Mod))). pds(Order, .(0, Ans)) :- ','(is(N, +(*(Order, +(Order, 1)), 1)), ','(iota(N, .(0, List)), pds1(List, .(0, []), Ans, N))). pdsbm(N) :- pds(N, .(0, .(1, X))). is(X0, X1). >(X0, X1). Query: pds(g,a) ---------------------------------------- (241) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: pds_in_2: (b,f) iota_in_2: (f,f) iota1_in_3: (b,f,f) (f,f,f) pds1_in_4: (f,b,f,f) (f,f,f,f) member_in_2: (f,f) check_in_5: (b,f,f,f,f) (f,f,f,f,f) dif_in_5: (b,f,f,f,f) (f,f,f,f,f) mergedelete_in_3: (f,f,f) rev_in_3: (f,b,f) (f,f,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: pds_in_ga(Order, .(0, Ans)) -> U19_ga(Order, Ans, is_in_ag(N, +(*(Order, +(Order, 1)), 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U19_ga(Order, Ans, is_out_ag(N, +(*(Order, +(Order, 1)), 1))) -> U20_ga(Order, Ans, N, iota_in_aa(N, .(0, List))) iota_in_aa(N, List) -> U2_aa(N, List, iota1_in_gaa(0, N, List)) iota1_in_gaa(K, K, []) -> iota1_out_gaa(K, K, []) iota1_in_gaa(K, N, .(K, List)) -> U3_gaa(K, N, List, is_in_ag(K1, +(K, 1))) U3_gaa(K, N, List, is_out_ag(K1, +(K, 1))) -> U4_gaa(K, N, List, iota1_in_aaa(K1, N, List)) iota1_in_aaa(K, K, []) -> iota1_out_aaa(K, K, []) iota1_in_aaa(K, N, .(K, List)) -> U3_aaa(K, N, List, is_in_aa(K1, +(K, 1))) is_in_aa(X0, X1) -> is_out_aa(X0, X1) U3_aaa(K, N, List, is_out_aa(K1, +(K, 1))) -> U4_aaa(K, N, List, iota1_in_aaa(K1, N, List)) U4_aaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_aaa(K, N, .(K, List)) U4_gaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_gaa(K, N, .(K, List)) U2_aa(N, List, iota1_out_gaa(0, N, List)) -> iota_out_aa(N, List) U20_ga(Order, Ans, N, iota_out_aa(N, .(0, List))) -> U21_ga(Order, Ans, pds1_in_agaa(List, .(0, []), Ans, N)) pds1_in_agaa([], X6, [], X7) -> pds1_out_agaa([], X6, [], X7) pds1_in_agaa(Unused, List, .(Choice, Rest), Mod) -> U16_agaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) member_in_aa(X, .(X, Xs)) -> member_out_aa(X, .(X, Xs)) member_in_aa(X, .(X1, Xs)) -> U1_aa(X, X1, Xs, member_in_aa(X, Xs)) U1_aa(X, X1, Xs, member_out_aa(X, Xs)) -> member_out_aa(X, .(X1, Xs)) U16_agaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_agaa(Unused, List, Choice, Rest, Mod, check_in_gaaaa(List, Choice, Unused, U3, Mod)) check_in_gaaaa([], X4, L, L, X5) -> check_out_gaaaa([], X4, L, L, X5) check_in_gaaaa(S, Choice, Old, L3, Modulus) -> U12_gaaaa(S, Choice, Old, L3, Modulus, dif_in_gaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_gaaaa([], X2, X3, [], []) -> dif_out_gaaaa([], X2, X3, [], []) dif_in_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gaaaa(Ss, Val, Mod, Ds, D2s)) U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_gaaaa(S, Choice, Old, L3, Modulus, dif_out_gaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) mergedelete_in_aaa([], L, L) -> mergedelete_out_aaa([], L, L) mergedelete_in_aaa(.(D, Ds), .(D, R), L2) -> U9_aaa(D, Ds, R, L2, mergedelete_in_aaa(Ds, R, L2)) mergedelete_in_aaa(.(D, Ds), .(X, R), .(X, L2)) -> U10_aaa(D, Ds, X, R, L2, >_in_aa(D, X)) >_in_aa(X0, X1) -> >_out_aa(X0, X1) U10_aaa(D, Ds, X, R, L2, >_out_aa(D, X)) -> U11_aaa(D, Ds, X, R, L2, mergedelete_in_aaa(.(D, Ds), R, L2)) U11_aaa(D, Ds, X, R, L2, mergedelete_out_aaa(.(D, Ds), R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(X, R), .(X, L2)) U9_aaa(D, Ds, R, L2, mergedelete_out_aaa(Ds, R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(D, R), L2) U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) rev_in_aga([], L, L) -> rev_out_aga([], L, L) rev_in_aga(.(X, Xs), Y, L) -> U8_aga(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) rev_in_aaa([], L, L) -> rev_out_aaa([], L, L) rev_in_aaa(.(X, Xs), Y, L) -> U8_aaa(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) U8_aaa(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aaa(.(X, Xs), Y, L) U8_aga(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aga(.(X, Xs), Y, L) U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_gaaaa(S, Choice, Old, L3, Modulus) U17_agaa(Unused, List, Choice, Rest, Mod, check_out_gaaaa(List, Choice, Unused, U3, Mod)) -> U18_agaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) pds1_in_aaaa([], X6, [], X7) -> pds1_out_aaaa([], X6, [], X7) pds1_in_aaaa(Unused, List, .(Choice, Rest), Mod) -> U16_aaaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) U16_aaaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_aaaa(Unused, List, Choice, Rest, Mod, check_in_aaaaa(List, Choice, Unused, U3, Mod)) check_in_aaaaa([], X4, L, L, X5) -> check_out_aaaaa([], X4, L, L, X5) check_in_aaaaa(S, Choice, Old, L3, Modulus) -> U12_aaaaa(S, Choice, Old, L3, Modulus, dif_in_aaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_aaaaa([], X2, X3, [], []) -> dif_out_aaaaa([], X2, X3, [], []) dif_in_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_aaaaa(Ss, Val, Mod, Ds, D2s)) U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_aaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_aaaaa(S, Choice, Old, L3, Modulus, dif_out_aaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_aaaaa(S, Choice, Old, L3, Modulus) U17_aaaa(Unused, List, Choice, Rest, Mod, check_out_aaaaa(List, Choice, Unused, U3, Mod)) -> U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_aaaa(Unused, List, .(Choice, Rest), Mod) U18_agaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_agaa(Unused, List, .(Choice, Rest), Mod) U21_ga(Order, Ans, pds1_out_agaa(List, .(0, []), Ans, N)) -> pds_out_ga(Order, .(0, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U19_ga(x1, x2, x3) = U19_ga(x3) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag +(x1, x2) = +(x1, x2) *(x1, x2) = *(x1, x2) 1 = 1 U20_ga(x1, x2, x3, x4) = U20_ga(x4) iota_in_aa(x1, x2) = iota_in_aa U2_aa(x1, x2, x3) = U2_aa(x3) iota1_in_gaa(x1, x2, x3) = iota1_in_gaa(x1) iota1_out_gaa(x1, x2, x3) = iota1_out_gaa U3_gaa(x1, x2, x3, x4) = U3_gaa(x4) U4_gaa(x1, x2, x3, x4) = U4_gaa(x4) iota1_in_aaa(x1, x2, x3) = iota1_in_aaa iota1_out_aaa(x1, x2, x3) = iota1_out_aaa U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) is_in_aa(x1, x2) = is_in_aa is_out_aa(x1, x2) = is_out_aa U4_aaa(x1, x2, x3, x4) = U4_aaa(x4) 0 = 0 iota_out_aa(x1, x2) = iota_out_aa U21_ga(x1, x2, x3) = U21_ga(x3) pds1_in_agaa(x1, x2, x3, x4) = pds1_in_agaa(x2) pds1_out_agaa(x1, x2, x3, x4) = pds1_out_agaa U16_agaa(x1, x2, x3, x4, x5, x6) = U16_agaa(x2, x6) member_in_aa(x1, x2) = member_in_aa member_out_aa(x1, x2) = member_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U17_agaa(x1, x2, x3, x4, x5, x6) = U17_agaa(x6) check_in_gaaaa(x1, x2, x3, x4, x5) = check_in_gaaaa(x1) [] = [] check_out_gaaaa(x1, x2, x3, x4, x5) = check_out_gaaaa U12_gaaaa(x1, x2, x3, x4, x5, x6) = U12_gaaaa(x6) dif_in_gaaaa(x1, x2, x3, x4, x5) = dif_in_gaaaa(x1) dif_out_gaaaa(x1, x2, x3, x4, x5) = dif_out_gaaaa .(x1, x2) = .(x1, x2) U5_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_gaaaa(x2, x9) U6_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gaaaa(x2, x9) U7_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_gaaaa(x9) U13_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_gaaaa(x7) mergedelete_in_aaa(x1, x2, x3) = mergedelete_in_aaa mergedelete_out_aaa(x1, x2, x3) = mergedelete_out_aaa U9_aaa(x1, x2, x3, x4, x5) = U9_aaa(x5) U10_aaa(x1, x2, x3, x4, x5, x6) = U10_aaa(x6) >_in_aa(x1, x2) = >_in_aa >_out_aa(x1, x2) = >_out_aa U11_aaa(x1, x2, x3, x4, x5, x6) = U11_aaa(x6) U14_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_gaaaa(x7) rev_in_aga(x1, x2, x3) = rev_in_aga(x2) rev_out_aga(x1, x2, x3) = rev_out_aga U8_aga(x1, x2, x3, x4, x5) = U8_aga(x5) rev_in_aaa(x1, x2, x3) = rev_in_aaa rev_out_aaa(x1, x2, x3) = rev_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U15_gaaaa(x1, x2, x3, x4, x5, x6) = U15_gaaaa(x6) U18_agaa(x1, x2, x3, x4, x5, x6) = U18_agaa(x6) pds1_in_aaaa(x1, x2, x3, x4) = pds1_in_aaaa pds1_out_aaaa(x1, x2, x3, x4) = pds1_out_aaaa U16_aaaa(x1, x2, x3, x4, x5, x6) = U16_aaaa(x6) U17_aaaa(x1, x2, x3, x4, x5, x6) = U17_aaaa(x6) check_in_aaaaa(x1, x2, x3, x4, x5) = check_in_aaaaa check_out_aaaaa(x1, x2, x3, x4, x5) = check_out_aaaaa U12_aaaaa(x1, x2, x3, x4, x5, x6) = U12_aaaaa(x6) dif_in_aaaaa(x1, x2, x3, x4, x5) = dif_in_aaaaa dif_out_aaaaa(x1, x2, x3, x4, x5) = dif_out_aaaaa U5_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_aaaaa(x9) U6_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_aaaaa(x9) U7_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_aaaaa(x9) U13_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_aaaaa(x7) U14_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_aaaaa(x7) U15_aaaaa(x1, x2, x3, x4, x5, x6) = U15_aaaaa(x6) U18_aaaa(x1, x2, x3, x4, x5, x6) = U18_aaaa(x6) pds_out_ga(x1, x2) = pds_out_ga Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (242) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: pds_in_ga(Order, .(0, Ans)) -> U19_ga(Order, Ans, is_in_ag(N, +(*(Order, +(Order, 1)), 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U19_ga(Order, Ans, is_out_ag(N, +(*(Order, +(Order, 1)), 1))) -> U20_ga(Order, Ans, N, iota_in_aa(N, .(0, List))) iota_in_aa(N, List) -> U2_aa(N, List, iota1_in_gaa(0, N, List)) iota1_in_gaa(K, K, []) -> iota1_out_gaa(K, K, []) iota1_in_gaa(K, N, .(K, List)) -> U3_gaa(K, N, List, is_in_ag(K1, +(K, 1))) U3_gaa(K, N, List, is_out_ag(K1, +(K, 1))) -> U4_gaa(K, N, List, iota1_in_aaa(K1, N, List)) iota1_in_aaa(K, K, []) -> iota1_out_aaa(K, K, []) iota1_in_aaa(K, N, .(K, List)) -> U3_aaa(K, N, List, is_in_aa(K1, +(K, 1))) is_in_aa(X0, X1) -> is_out_aa(X0, X1) U3_aaa(K, N, List, is_out_aa(K1, +(K, 1))) -> U4_aaa(K, N, List, iota1_in_aaa(K1, N, List)) U4_aaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_aaa(K, N, .(K, List)) U4_gaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_gaa(K, N, .(K, List)) U2_aa(N, List, iota1_out_gaa(0, N, List)) -> iota_out_aa(N, List) U20_ga(Order, Ans, N, iota_out_aa(N, .(0, List))) -> U21_ga(Order, Ans, pds1_in_agaa(List, .(0, []), Ans, N)) pds1_in_agaa([], X6, [], X7) -> pds1_out_agaa([], X6, [], X7) pds1_in_agaa(Unused, List, .(Choice, Rest), Mod) -> U16_agaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) member_in_aa(X, .(X, Xs)) -> member_out_aa(X, .(X, Xs)) member_in_aa(X, .(X1, Xs)) -> U1_aa(X, X1, Xs, member_in_aa(X, Xs)) U1_aa(X, X1, Xs, member_out_aa(X, Xs)) -> member_out_aa(X, .(X1, Xs)) U16_agaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_agaa(Unused, List, Choice, Rest, Mod, check_in_gaaaa(List, Choice, Unused, U3, Mod)) check_in_gaaaa([], X4, L, L, X5) -> check_out_gaaaa([], X4, L, L, X5) check_in_gaaaa(S, Choice, Old, L3, Modulus) -> U12_gaaaa(S, Choice, Old, L3, Modulus, dif_in_gaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_gaaaa([], X2, X3, [], []) -> dif_out_gaaaa([], X2, X3, [], []) dif_in_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gaaaa(Ss, Val, Mod, Ds, D2s)) U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_gaaaa(S, Choice, Old, L3, Modulus, dif_out_gaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) mergedelete_in_aaa([], L, L) -> mergedelete_out_aaa([], L, L) mergedelete_in_aaa(.(D, Ds), .(D, R), L2) -> U9_aaa(D, Ds, R, L2, mergedelete_in_aaa(Ds, R, L2)) mergedelete_in_aaa(.(D, Ds), .(X, R), .(X, L2)) -> U10_aaa(D, Ds, X, R, L2, >_in_aa(D, X)) >_in_aa(X0, X1) -> >_out_aa(X0, X1) U10_aaa(D, Ds, X, R, L2, >_out_aa(D, X)) -> U11_aaa(D, Ds, X, R, L2, mergedelete_in_aaa(.(D, Ds), R, L2)) U11_aaa(D, Ds, X, R, L2, mergedelete_out_aaa(.(D, Ds), R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(X, R), .(X, L2)) U9_aaa(D, Ds, R, L2, mergedelete_out_aaa(Ds, R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(D, R), L2) U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) rev_in_aga([], L, L) -> rev_out_aga([], L, L) rev_in_aga(.(X, Xs), Y, L) -> U8_aga(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) rev_in_aaa([], L, L) -> rev_out_aaa([], L, L) rev_in_aaa(.(X, Xs), Y, L) -> U8_aaa(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) U8_aaa(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aaa(.(X, Xs), Y, L) U8_aga(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aga(.(X, Xs), Y, L) U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_gaaaa(S, Choice, Old, L3, Modulus) U17_agaa(Unused, List, Choice, Rest, Mod, check_out_gaaaa(List, Choice, Unused, U3, Mod)) -> U18_agaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) pds1_in_aaaa([], X6, [], X7) -> pds1_out_aaaa([], X6, [], X7) pds1_in_aaaa(Unused, List, .(Choice, Rest), Mod) -> U16_aaaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) U16_aaaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_aaaa(Unused, List, Choice, Rest, Mod, check_in_aaaaa(List, Choice, Unused, U3, Mod)) check_in_aaaaa([], X4, L, L, X5) -> check_out_aaaaa([], X4, L, L, X5) check_in_aaaaa(S, Choice, Old, L3, Modulus) -> U12_aaaaa(S, Choice, Old, L3, Modulus, dif_in_aaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_aaaaa([], X2, X3, [], []) -> dif_out_aaaaa([], X2, X3, [], []) dif_in_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_aaaaa(Ss, Val, Mod, Ds, D2s)) U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_aaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_aaaaa(S, Choice, Old, L3, Modulus, dif_out_aaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_aaaaa(S, Choice, Old, L3, Modulus) U17_aaaa(Unused, List, Choice, Rest, Mod, check_out_aaaaa(List, Choice, Unused, U3, Mod)) -> U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_aaaa(Unused, List, .(Choice, Rest), Mod) U18_agaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_agaa(Unused, List, .(Choice, Rest), Mod) U21_ga(Order, Ans, pds1_out_agaa(List, .(0, []), Ans, N)) -> pds_out_ga(Order, .(0, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U19_ga(x1, x2, x3) = U19_ga(x3) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag +(x1, x2) = +(x1, x2) *(x1, x2) = *(x1, x2) 1 = 1 U20_ga(x1, x2, x3, x4) = U20_ga(x4) iota_in_aa(x1, x2) = iota_in_aa U2_aa(x1, x2, x3) = U2_aa(x3) iota1_in_gaa(x1, x2, x3) = iota1_in_gaa(x1) iota1_out_gaa(x1, x2, x3) = iota1_out_gaa U3_gaa(x1, x2, x3, x4) = U3_gaa(x4) U4_gaa(x1, x2, x3, x4) = U4_gaa(x4) iota1_in_aaa(x1, x2, x3) = iota1_in_aaa iota1_out_aaa(x1, x2, x3) = iota1_out_aaa U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) is_in_aa(x1, x2) = is_in_aa is_out_aa(x1, x2) = is_out_aa U4_aaa(x1, x2, x3, x4) = U4_aaa(x4) 0 = 0 iota_out_aa(x1, x2) = iota_out_aa U21_ga(x1, x2, x3) = U21_ga(x3) pds1_in_agaa(x1, x2, x3, x4) = pds1_in_agaa(x2) pds1_out_agaa(x1, x2, x3, x4) = pds1_out_agaa U16_agaa(x1, x2, x3, x4, x5, x6) = U16_agaa(x2, x6) member_in_aa(x1, x2) = member_in_aa member_out_aa(x1, x2) = member_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U17_agaa(x1, x2, x3, x4, x5, x6) = U17_agaa(x6) check_in_gaaaa(x1, x2, x3, x4, x5) = check_in_gaaaa(x1) [] = [] check_out_gaaaa(x1, x2, x3, x4, x5) = check_out_gaaaa U12_gaaaa(x1, x2, x3, x4, x5, x6) = U12_gaaaa(x6) dif_in_gaaaa(x1, x2, x3, x4, x5) = dif_in_gaaaa(x1) dif_out_gaaaa(x1, x2, x3, x4, x5) = dif_out_gaaaa .(x1, x2) = .(x1, x2) U5_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_gaaaa(x2, x9) U6_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gaaaa(x2, x9) U7_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_gaaaa(x9) U13_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_gaaaa(x7) mergedelete_in_aaa(x1, x2, x3) = mergedelete_in_aaa mergedelete_out_aaa(x1, x2, x3) = mergedelete_out_aaa U9_aaa(x1, x2, x3, x4, x5) = U9_aaa(x5) U10_aaa(x1, x2, x3, x4, x5, x6) = U10_aaa(x6) >_in_aa(x1, x2) = >_in_aa >_out_aa(x1, x2) = >_out_aa U11_aaa(x1, x2, x3, x4, x5, x6) = U11_aaa(x6) U14_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_gaaaa(x7) rev_in_aga(x1, x2, x3) = rev_in_aga(x2) rev_out_aga(x1, x2, x3) = rev_out_aga U8_aga(x1, x2, x3, x4, x5) = U8_aga(x5) rev_in_aaa(x1, x2, x3) = rev_in_aaa rev_out_aaa(x1, x2, x3) = rev_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U15_gaaaa(x1, x2, x3, x4, x5, x6) = U15_gaaaa(x6) U18_agaa(x1, x2, x3, x4, x5, x6) = U18_agaa(x6) pds1_in_aaaa(x1, x2, x3, x4) = pds1_in_aaaa pds1_out_aaaa(x1, x2, x3, x4) = pds1_out_aaaa U16_aaaa(x1, x2, x3, x4, x5, x6) = U16_aaaa(x6) U17_aaaa(x1, x2, x3, x4, x5, x6) = U17_aaaa(x6) check_in_aaaaa(x1, x2, x3, x4, x5) = check_in_aaaaa check_out_aaaaa(x1, x2, x3, x4, x5) = check_out_aaaaa U12_aaaaa(x1, x2, x3, x4, x5, x6) = U12_aaaaa(x6) dif_in_aaaaa(x1, x2, x3, x4, x5) = dif_in_aaaaa dif_out_aaaaa(x1, x2, x3, x4, x5) = dif_out_aaaaa U5_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_aaaaa(x9) U6_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_aaaaa(x9) U7_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_aaaaa(x9) U13_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_aaaaa(x7) U14_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_aaaaa(x7) U15_aaaaa(x1, x2, x3, x4, x5, x6) = U15_aaaaa(x6) U18_aaaa(x1, x2, x3, x4, x5, x6) = U18_aaaa(x6) pds_out_ga(x1, x2) = pds_out_ga ---------------------------------------- (243) 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: PDS_IN_GA(Order, .(0, Ans)) -> U19_GA(Order, Ans, is_in_ag(N, +(*(Order, +(Order, 1)), 1))) PDS_IN_GA(Order, .(0, Ans)) -> IS_IN_AG(N, +(*(Order, +(Order, 1)), 1)) U19_GA(Order, Ans, is_out_ag(N, +(*(Order, +(Order, 1)), 1))) -> U20_GA(Order, Ans, N, iota_in_aa(N, .(0, List))) U19_GA(Order, Ans, is_out_ag(N, +(*(Order, +(Order, 1)), 1))) -> IOTA_IN_AA(N, .(0, List)) IOTA_IN_AA(N, List) -> U2_AA(N, List, iota1_in_gaa(0, N, List)) IOTA_IN_AA(N, List) -> IOTA1_IN_GAA(0, N, List) IOTA1_IN_GAA(K, N, .(K, List)) -> U3_GAA(K, N, List, is_in_ag(K1, +(K, 1))) IOTA1_IN_GAA(K, N, .(K, List)) -> IS_IN_AG(K1, +(K, 1)) U3_GAA(K, N, List, is_out_ag(K1, +(K, 1))) -> U4_GAA(K, N, List, iota1_in_aaa(K1, N, List)) U3_GAA(K, N, List, is_out_ag(K1, +(K, 1))) -> IOTA1_IN_AAA(K1, N, List) IOTA1_IN_AAA(K, N, .(K, List)) -> U3_AAA(K, N, List, is_in_aa(K1, +(K, 1))) IOTA1_IN_AAA(K, N, .(K, List)) -> IS_IN_AA(K1, +(K, 1)) U3_AAA(K, N, List, is_out_aa(K1, +(K, 1))) -> U4_AAA(K, N, List, iota1_in_aaa(K1, N, List)) U3_AAA(K, N, List, is_out_aa(K1, +(K, 1))) -> IOTA1_IN_AAA(K1, N, List) U20_GA(Order, Ans, N, iota_out_aa(N, .(0, List))) -> U21_GA(Order, Ans, pds1_in_agaa(List, .(0, []), Ans, N)) U20_GA(Order, Ans, N, iota_out_aa(N, .(0, List))) -> PDS1_IN_AGAA(List, .(0, []), Ans, N) PDS1_IN_AGAA(Unused, List, .(Choice, Rest), Mod) -> U16_AGAA(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) PDS1_IN_AGAA(Unused, List, .(Choice, Rest), Mod) -> MEMBER_IN_AA(Choice, Unused) MEMBER_IN_AA(X, .(X1, Xs)) -> U1_AA(X, X1, Xs, member_in_aa(X, Xs)) MEMBER_IN_AA(X, .(X1, Xs)) -> MEMBER_IN_AA(X, Xs) U16_AGAA(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_AGAA(Unused, List, Choice, Rest, Mod, check_in_gaaaa(List, Choice, Unused, U3, Mod)) U16_AGAA(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> CHECK_IN_GAAAA(List, Choice, Unused, U3, Mod) CHECK_IN_GAAAA(S, Choice, Old, L3, Modulus) -> U12_GAAAA(S, Choice, Old, L3, Modulus, dif_in_gaaaa(S, Choice, Modulus, Ds, Dds)) CHECK_IN_GAAAA(S, Choice, Old, L3, Modulus) -> DIF_IN_GAAAA(S, Choice, Modulus, Ds, Dds) DIF_IN_GAAAA(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) DIF_IN_GAAAA(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> IS_IN_AA(D, -(Val, S)) U5_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U5_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> IS_IN_AA(D2, -(Mod, D)) U6_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gaaaa(Ss, Val, Mod, Ds, D2s)) U6_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> DIF_IN_GAAAA(Ss, Val, Mod, Ds, D2s) U12_GAAAA(S, Choice, Old, L3, Modulus, dif_out_gaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_GAAAA(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) U12_GAAAA(S, Choice, Old, L3, Modulus, dif_out_gaaaa(S, Choice, Modulus, Ds, Dds)) -> MERGEDELETE_IN_AAA(Ds, Old, L2) MERGEDELETE_IN_AAA(.(D, Ds), .(D, R), L2) -> U9_AAA(D, Ds, R, L2, mergedelete_in_aaa(Ds, R, L2)) MERGEDELETE_IN_AAA(.(D, Ds), .(D, R), L2) -> MERGEDELETE_IN_AAA(Ds, R, L2) MERGEDELETE_IN_AAA(.(D, Ds), .(X, R), .(X, L2)) -> U10_AAA(D, Ds, X, R, L2, >_in_aa(D, X)) MERGEDELETE_IN_AAA(.(D, Ds), .(X, R), .(X, L2)) -> >_IN_AA(D, X) U10_AAA(D, Ds, X, R, L2, >_out_aa(D, X)) -> U11_AAA(D, Ds, X, R, L2, mergedelete_in_aaa(.(D, Ds), R, L2)) U10_AAA(D, Ds, X, R, L2, >_out_aa(D, X)) -> MERGEDELETE_IN_AAA(.(D, Ds), R, L2) U13_GAAAA(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_GAAAA(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) U13_GAAAA(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> REV_IN_AGA(Dds, [], Rds) REV_IN_AGA(.(X, Xs), Y, L) -> U8_AGA(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) REV_IN_AGA(.(X, Xs), Y, L) -> REV_IN_AAA(Xs, .(X, Y), L) REV_IN_AAA(.(X, Xs), Y, L) -> U8_AAA(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) REV_IN_AAA(.(X, Xs), Y, L) -> REV_IN_AAA(Xs, .(X, Y), L) U14_GAAAA(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_GAAAA(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U14_GAAAA(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> MERGEDELETE_IN_AAA(Rds, L2, L3) U17_AGAA(Unused, List, Choice, Rest, Mod, check_out_gaaaa(List, Choice, Unused, U3, Mod)) -> U18_AGAA(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) U17_AGAA(Unused, List, Choice, Rest, Mod, check_out_gaaaa(List, Choice, Unused, U3, Mod)) -> PDS1_IN_AAAA(U3, .(Choice, List), Rest, Mod) PDS1_IN_AAAA(Unused, List, .(Choice, Rest), Mod) -> U16_AAAA(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) PDS1_IN_AAAA(Unused, List, .(Choice, Rest), Mod) -> MEMBER_IN_AA(Choice, Unused) U16_AAAA(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_AAAA(Unused, List, Choice, Rest, Mod, check_in_aaaaa(List, Choice, Unused, U3, Mod)) U16_AAAA(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> CHECK_IN_AAAAA(List, Choice, Unused, U3, Mod) CHECK_IN_AAAAA(S, Choice, Old, L3, Modulus) -> U12_AAAAA(S, Choice, Old, L3, Modulus, dif_in_aaaaa(S, Choice, Modulus, Ds, Dds)) CHECK_IN_AAAAA(S, Choice, Old, L3, Modulus) -> DIF_IN_AAAAA(S, Choice, Modulus, Ds, Dds) DIF_IN_AAAAA(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) DIF_IN_AAAAA(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> IS_IN_AA(D, -(Val, S)) U5_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U5_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> IS_IN_AA(D2, -(Mod, D)) U6_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_aaaaa(Ss, Val, Mod, Ds, D2s)) U6_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> DIF_IN_AAAAA(Ss, Val, Mod, Ds, D2s) U12_AAAAA(S, Choice, Old, L3, Modulus, dif_out_aaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_AAAAA(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) U12_AAAAA(S, Choice, Old, L3, Modulus, dif_out_aaaaa(S, Choice, Modulus, Ds, Dds)) -> MERGEDELETE_IN_AAA(Ds, Old, L2) U13_AAAAA(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_AAAAA(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) U13_AAAAA(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> REV_IN_AGA(Dds, [], Rds) U14_AAAAA(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_AAAAA(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U14_AAAAA(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> MERGEDELETE_IN_AAA(Rds, L2, L3) U17_AAAA(Unused, List, Choice, Rest, Mod, check_out_aaaaa(List, Choice, Unused, U3, Mod)) -> U18_AAAA(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) U17_AAAA(Unused, List, Choice, Rest, Mod, check_out_aaaaa(List, Choice, Unused, U3, Mod)) -> PDS1_IN_AAAA(U3, .(Choice, List), Rest, Mod) The TRS R consists of the following rules: pds_in_ga(Order, .(0, Ans)) -> U19_ga(Order, Ans, is_in_ag(N, +(*(Order, +(Order, 1)), 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U19_ga(Order, Ans, is_out_ag(N, +(*(Order, +(Order, 1)), 1))) -> U20_ga(Order, Ans, N, iota_in_aa(N, .(0, List))) iota_in_aa(N, List) -> U2_aa(N, List, iota1_in_gaa(0, N, List)) iota1_in_gaa(K, K, []) -> iota1_out_gaa(K, K, []) iota1_in_gaa(K, N, .(K, List)) -> U3_gaa(K, N, List, is_in_ag(K1, +(K, 1))) U3_gaa(K, N, List, is_out_ag(K1, +(K, 1))) -> U4_gaa(K, N, List, iota1_in_aaa(K1, N, List)) iota1_in_aaa(K, K, []) -> iota1_out_aaa(K, K, []) iota1_in_aaa(K, N, .(K, List)) -> U3_aaa(K, N, List, is_in_aa(K1, +(K, 1))) is_in_aa(X0, X1) -> is_out_aa(X0, X1) U3_aaa(K, N, List, is_out_aa(K1, +(K, 1))) -> U4_aaa(K, N, List, iota1_in_aaa(K1, N, List)) U4_aaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_aaa(K, N, .(K, List)) U4_gaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_gaa(K, N, .(K, List)) U2_aa(N, List, iota1_out_gaa(0, N, List)) -> iota_out_aa(N, List) U20_ga(Order, Ans, N, iota_out_aa(N, .(0, List))) -> U21_ga(Order, Ans, pds1_in_agaa(List, .(0, []), Ans, N)) pds1_in_agaa([], X6, [], X7) -> pds1_out_agaa([], X6, [], X7) pds1_in_agaa(Unused, List, .(Choice, Rest), Mod) -> U16_agaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) member_in_aa(X, .(X, Xs)) -> member_out_aa(X, .(X, Xs)) member_in_aa(X, .(X1, Xs)) -> U1_aa(X, X1, Xs, member_in_aa(X, Xs)) U1_aa(X, X1, Xs, member_out_aa(X, Xs)) -> member_out_aa(X, .(X1, Xs)) U16_agaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_agaa(Unused, List, Choice, Rest, Mod, check_in_gaaaa(List, Choice, Unused, U3, Mod)) check_in_gaaaa([], X4, L, L, X5) -> check_out_gaaaa([], X4, L, L, X5) check_in_gaaaa(S, Choice, Old, L3, Modulus) -> U12_gaaaa(S, Choice, Old, L3, Modulus, dif_in_gaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_gaaaa([], X2, X3, [], []) -> dif_out_gaaaa([], X2, X3, [], []) dif_in_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gaaaa(Ss, Val, Mod, Ds, D2s)) U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_gaaaa(S, Choice, Old, L3, Modulus, dif_out_gaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) mergedelete_in_aaa([], L, L) -> mergedelete_out_aaa([], L, L) mergedelete_in_aaa(.(D, Ds), .(D, R), L2) -> U9_aaa(D, Ds, R, L2, mergedelete_in_aaa(Ds, R, L2)) mergedelete_in_aaa(.(D, Ds), .(X, R), .(X, L2)) -> U10_aaa(D, Ds, X, R, L2, >_in_aa(D, X)) >_in_aa(X0, X1) -> >_out_aa(X0, X1) U10_aaa(D, Ds, X, R, L2, >_out_aa(D, X)) -> U11_aaa(D, Ds, X, R, L2, mergedelete_in_aaa(.(D, Ds), R, L2)) U11_aaa(D, Ds, X, R, L2, mergedelete_out_aaa(.(D, Ds), R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(X, R), .(X, L2)) U9_aaa(D, Ds, R, L2, mergedelete_out_aaa(Ds, R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(D, R), L2) U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) rev_in_aga([], L, L) -> rev_out_aga([], L, L) rev_in_aga(.(X, Xs), Y, L) -> U8_aga(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) rev_in_aaa([], L, L) -> rev_out_aaa([], L, L) rev_in_aaa(.(X, Xs), Y, L) -> U8_aaa(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) U8_aaa(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aaa(.(X, Xs), Y, L) U8_aga(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aga(.(X, Xs), Y, L) U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_gaaaa(S, Choice, Old, L3, Modulus) U17_agaa(Unused, List, Choice, Rest, Mod, check_out_gaaaa(List, Choice, Unused, U3, Mod)) -> U18_agaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) pds1_in_aaaa([], X6, [], X7) -> pds1_out_aaaa([], X6, [], X7) pds1_in_aaaa(Unused, List, .(Choice, Rest), Mod) -> U16_aaaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) U16_aaaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_aaaa(Unused, List, Choice, Rest, Mod, check_in_aaaaa(List, Choice, Unused, U3, Mod)) check_in_aaaaa([], X4, L, L, X5) -> check_out_aaaaa([], X4, L, L, X5) check_in_aaaaa(S, Choice, Old, L3, Modulus) -> U12_aaaaa(S, Choice, Old, L3, Modulus, dif_in_aaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_aaaaa([], X2, X3, [], []) -> dif_out_aaaaa([], X2, X3, [], []) dif_in_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_aaaaa(Ss, Val, Mod, Ds, D2s)) U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_aaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_aaaaa(S, Choice, Old, L3, Modulus, dif_out_aaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_aaaaa(S, Choice, Old, L3, Modulus) U17_aaaa(Unused, List, Choice, Rest, Mod, check_out_aaaaa(List, Choice, Unused, U3, Mod)) -> U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_aaaa(Unused, List, .(Choice, Rest), Mod) U18_agaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_agaa(Unused, List, .(Choice, Rest), Mod) U21_ga(Order, Ans, pds1_out_agaa(List, .(0, []), Ans, N)) -> pds_out_ga(Order, .(0, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U19_ga(x1, x2, x3) = U19_ga(x3) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag +(x1, x2) = +(x1, x2) *(x1, x2) = *(x1, x2) 1 = 1 U20_ga(x1, x2, x3, x4) = U20_ga(x4) iota_in_aa(x1, x2) = iota_in_aa U2_aa(x1, x2, x3) = U2_aa(x3) iota1_in_gaa(x1, x2, x3) = iota1_in_gaa(x1) iota1_out_gaa(x1, x2, x3) = iota1_out_gaa U3_gaa(x1, x2, x3, x4) = U3_gaa(x4) U4_gaa(x1, x2, x3, x4) = U4_gaa(x4) iota1_in_aaa(x1, x2, x3) = iota1_in_aaa iota1_out_aaa(x1, x2, x3) = iota1_out_aaa U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) is_in_aa(x1, x2) = is_in_aa is_out_aa(x1, x2) = is_out_aa U4_aaa(x1, x2, x3, x4) = U4_aaa(x4) 0 = 0 iota_out_aa(x1, x2) = iota_out_aa U21_ga(x1, x2, x3) = U21_ga(x3) pds1_in_agaa(x1, x2, x3, x4) = pds1_in_agaa(x2) pds1_out_agaa(x1, x2, x3, x4) = pds1_out_agaa U16_agaa(x1, x2, x3, x4, x5, x6) = U16_agaa(x2, x6) member_in_aa(x1, x2) = member_in_aa member_out_aa(x1, x2) = member_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U17_agaa(x1, x2, x3, x4, x5, x6) = U17_agaa(x6) check_in_gaaaa(x1, x2, x3, x4, x5) = check_in_gaaaa(x1) [] = [] check_out_gaaaa(x1, x2, x3, x4, x5) = check_out_gaaaa U12_gaaaa(x1, x2, x3, x4, x5, x6) = U12_gaaaa(x6) dif_in_gaaaa(x1, x2, x3, x4, x5) = dif_in_gaaaa(x1) dif_out_gaaaa(x1, x2, x3, x4, x5) = dif_out_gaaaa .(x1, x2) = .(x1, x2) U5_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_gaaaa(x2, x9) U6_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gaaaa(x2, x9) U7_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_gaaaa(x9) U13_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_gaaaa(x7) mergedelete_in_aaa(x1, x2, x3) = mergedelete_in_aaa mergedelete_out_aaa(x1, x2, x3) = mergedelete_out_aaa U9_aaa(x1, x2, x3, x4, x5) = U9_aaa(x5) U10_aaa(x1, x2, x3, x4, x5, x6) = U10_aaa(x6) >_in_aa(x1, x2) = >_in_aa >_out_aa(x1, x2) = >_out_aa U11_aaa(x1, x2, x3, x4, x5, x6) = U11_aaa(x6) U14_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_gaaaa(x7) rev_in_aga(x1, x2, x3) = rev_in_aga(x2) rev_out_aga(x1, x2, x3) = rev_out_aga U8_aga(x1, x2, x3, x4, x5) = U8_aga(x5) rev_in_aaa(x1, x2, x3) = rev_in_aaa rev_out_aaa(x1, x2, x3) = rev_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U15_gaaaa(x1, x2, x3, x4, x5, x6) = U15_gaaaa(x6) U18_agaa(x1, x2, x3, x4, x5, x6) = U18_agaa(x6) pds1_in_aaaa(x1, x2, x3, x4) = pds1_in_aaaa pds1_out_aaaa(x1, x2, x3, x4) = pds1_out_aaaa U16_aaaa(x1, x2, x3, x4, x5, x6) = U16_aaaa(x6) U17_aaaa(x1, x2, x3, x4, x5, x6) = U17_aaaa(x6) check_in_aaaaa(x1, x2, x3, x4, x5) = check_in_aaaaa check_out_aaaaa(x1, x2, x3, x4, x5) = check_out_aaaaa U12_aaaaa(x1, x2, x3, x4, x5, x6) = U12_aaaaa(x6) dif_in_aaaaa(x1, x2, x3, x4, x5) = dif_in_aaaaa dif_out_aaaaa(x1, x2, x3, x4, x5) = dif_out_aaaaa U5_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_aaaaa(x9) U6_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_aaaaa(x9) U7_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_aaaaa(x9) U13_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_aaaaa(x7) U14_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_aaaaa(x7) U15_aaaaa(x1, x2, x3, x4, x5, x6) = U15_aaaaa(x6) U18_aaaa(x1, x2, x3, x4, x5, x6) = U18_aaaa(x6) pds_out_ga(x1, x2) = pds_out_ga PDS_IN_GA(x1, x2) = PDS_IN_GA(x1) U19_GA(x1, x2, x3) = U19_GA(x3) IS_IN_AG(x1, x2) = IS_IN_AG(x2) U20_GA(x1, x2, x3, x4) = U20_GA(x4) IOTA_IN_AA(x1, x2) = IOTA_IN_AA U2_AA(x1, x2, x3) = U2_AA(x3) IOTA1_IN_GAA(x1, x2, x3) = IOTA1_IN_GAA(x1) U3_GAA(x1, x2, x3, x4) = U3_GAA(x4) U4_GAA(x1, x2, x3, x4) = U4_GAA(x4) IOTA1_IN_AAA(x1, x2, x3) = IOTA1_IN_AAA U3_AAA(x1, x2, x3, x4) = U3_AAA(x4) IS_IN_AA(x1, x2) = IS_IN_AA U4_AAA(x1, x2, x3, x4) = U4_AAA(x4) U21_GA(x1, x2, x3) = U21_GA(x3) PDS1_IN_AGAA(x1, x2, x3, x4) = PDS1_IN_AGAA(x2) U16_AGAA(x1, x2, x3, x4, x5, x6) = U16_AGAA(x2, x6) MEMBER_IN_AA(x1, x2) = MEMBER_IN_AA U1_AA(x1, x2, x3, x4) = U1_AA(x4) U17_AGAA(x1, x2, x3, x4, x5, x6) = U17_AGAA(x6) CHECK_IN_GAAAA(x1, x2, x3, x4, x5) = CHECK_IN_GAAAA(x1) U12_GAAAA(x1, x2, x3, x4, x5, x6) = U12_GAAAA(x6) DIF_IN_GAAAA(x1, x2, x3, x4, x5) = DIF_IN_GAAAA(x1) U5_GAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_GAAAA(x2, x9) U6_GAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_GAAAA(x2, x9) U7_GAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_GAAAA(x9) U13_GAAAA(x1, x2, x3, x4, x5, x6, x7) = U13_GAAAA(x7) MERGEDELETE_IN_AAA(x1, x2, x3) = MERGEDELETE_IN_AAA U9_AAA(x1, x2, x3, x4, x5) = U9_AAA(x5) U10_AAA(x1, x2, x3, x4, x5, x6) = U10_AAA(x6) >_IN_AA(x1, x2) = >_IN_AA U11_AAA(x1, x2, x3, x4, x5, x6) = U11_AAA(x6) U14_GAAAA(x1, x2, x3, x4, x5, x6, x7) = U14_GAAAA(x7) REV_IN_AGA(x1, x2, x3) = REV_IN_AGA(x2) U8_AGA(x1, x2, x3, x4, x5) = U8_AGA(x5) REV_IN_AAA(x1, x2, x3) = REV_IN_AAA U8_AAA(x1, x2, x3, x4, x5) = U8_AAA(x5) U15_GAAAA(x1, x2, x3, x4, x5, x6) = U15_GAAAA(x6) U18_AGAA(x1, x2, x3, x4, x5, x6) = U18_AGAA(x6) PDS1_IN_AAAA(x1, x2, x3, x4) = PDS1_IN_AAAA U16_AAAA(x1, x2, x3, x4, x5, x6) = U16_AAAA(x6) U17_AAAA(x1, x2, x3, x4, x5, x6) = U17_AAAA(x6) CHECK_IN_AAAAA(x1, x2, x3, x4, x5) = CHECK_IN_AAAAA U12_AAAAA(x1, x2, x3, x4, x5, x6) = U12_AAAAA(x6) DIF_IN_AAAAA(x1, x2, x3, x4, x5) = DIF_IN_AAAAA U5_AAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_AAAAA(x9) U6_AAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_AAAAA(x9) U7_AAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_AAAAA(x9) U13_AAAAA(x1, x2, x3, x4, x5, x6, x7) = U13_AAAAA(x7) U14_AAAAA(x1, x2, x3, x4, x5, x6, x7) = U14_AAAAA(x7) U15_AAAAA(x1, x2, x3, x4, x5, x6) = U15_AAAAA(x6) U18_AAAA(x1, x2, x3, x4, x5, x6) = U18_AAAA(x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (244) Obligation: Pi DP problem: The TRS P consists of the following rules: PDS_IN_GA(Order, .(0, Ans)) -> U19_GA(Order, Ans, is_in_ag(N, +(*(Order, +(Order, 1)), 1))) PDS_IN_GA(Order, .(0, Ans)) -> IS_IN_AG(N, +(*(Order, +(Order, 1)), 1)) U19_GA(Order, Ans, is_out_ag(N, +(*(Order, +(Order, 1)), 1))) -> U20_GA(Order, Ans, N, iota_in_aa(N, .(0, List))) U19_GA(Order, Ans, is_out_ag(N, +(*(Order, +(Order, 1)), 1))) -> IOTA_IN_AA(N, .(0, List)) IOTA_IN_AA(N, List) -> U2_AA(N, List, iota1_in_gaa(0, N, List)) IOTA_IN_AA(N, List) -> IOTA1_IN_GAA(0, N, List) IOTA1_IN_GAA(K, N, .(K, List)) -> U3_GAA(K, N, List, is_in_ag(K1, +(K, 1))) IOTA1_IN_GAA(K, N, .(K, List)) -> IS_IN_AG(K1, +(K, 1)) U3_GAA(K, N, List, is_out_ag(K1, +(K, 1))) -> U4_GAA(K, N, List, iota1_in_aaa(K1, N, List)) U3_GAA(K, N, List, is_out_ag(K1, +(K, 1))) -> IOTA1_IN_AAA(K1, N, List) IOTA1_IN_AAA(K, N, .(K, List)) -> U3_AAA(K, N, List, is_in_aa(K1, +(K, 1))) IOTA1_IN_AAA(K, N, .(K, List)) -> IS_IN_AA(K1, +(K, 1)) U3_AAA(K, N, List, is_out_aa(K1, +(K, 1))) -> U4_AAA(K, N, List, iota1_in_aaa(K1, N, List)) U3_AAA(K, N, List, is_out_aa(K1, +(K, 1))) -> IOTA1_IN_AAA(K1, N, List) U20_GA(Order, Ans, N, iota_out_aa(N, .(0, List))) -> U21_GA(Order, Ans, pds1_in_agaa(List, .(0, []), Ans, N)) U20_GA(Order, Ans, N, iota_out_aa(N, .(0, List))) -> PDS1_IN_AGAA(List, .(0, []), Ans, N) PDS1_IN_AGAA(Unused, List, .(Choice, Rest), Mod) -> U16_AGAA(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) PDS1_IN_AGAA(Unused, List, .(Choice, Rest), Mod) -> MEMBER_IN_AA(Choice, Unused) MEMBER_IN_AA(X, .(X1, Xs)) -> U1_AA(X, X1, Xs, member_in_aa(X, Xs)) MEMBER_IN_AA(X, .(X1, Xs)) -> MEMBER_IN_AA(X, Xs) U16_AGAA(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_AGAA(Unused, List, Choice, Rest, Mod, check_in_gaaaa(List, Choice, Unused, U3, Mod)) U16_AGAA(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> CHECK_IN_GAAAA(List, Choice, Unused, U3, Mod) CHECK_IN_GAAAA(S, Choice, Old, L3, Modulus) -> U12_GAAAA(S, Choice, Old, L3, Modulus, dif_in_gaaaa(S, Choice, Modulus, Ds, Dds)) CHECK_IN_GAAAA(S, Choice, Old, L3, Modulus) -> DIF_IN_GAAAA(S, Choice, Modulus, Ds, Dds) DIF_IN_GAAAA(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) DIF_IN_GAAAA(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> IS_IN_AA(D, -(Val, S)) U5_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U5_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> IS_IN_AA(D2, -(Mod, D)) U6_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gaaaa(Ss, Val, Mod, Ds, D2s)) U6_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> DIF_IN_GAAAA(Ss, Val, Mod, Ds, D2s) U12_GAAAA(S, Choice, Old, L3, Modulus, dif_out_gaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_GAAAA(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) U12_GAAAA(S, Choice, Old, L3, Modulus, dif_out_gaaaa(S, Choice, Modulus, Ds, Dds)) -> MERGEDELETE_IN_AAA(Ds, Old, L2) MERGEDELETE_IN_AAA(.(D, Ds), .(D, R), L2) -> U9_AAA(D, Ds, R, L2, mergedelete_in_aaa(Ds, R, L2)) MERGEDELETE_IN_AAA(.(D, Ds), .(D, R), L2) -> MERGEDELETE_IN_AAA(Ds, R, L2) MERGEDELETE_IN_AAA(.(D, Ds), .(X, R), .(X, L2)) -> U10_AAA(D, Ds, X, R, L2, >_in_aa(D, X)) MERGEDELETE_IN_AAA(.(D, Ds), .(X, R), .(X, L2)) -> >_IN_AA(D, X) U10_AAA(D, Ds, X, R, L2, >_out_aa(D, X)) -> U11_AAA(D, Ds, X, R, L2, mergedelete_in_aaa(.(D, Ds), R, L2)) U10_AAA(D, Ds, X, R, L2, >_out_aa(D, X)) -> MERGEDELETE_IN_AAA(.(D, Ds), R, L2) U13_GAAAA(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_GAAAA(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) U13_GAAAA(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> REV_IN_AGA(Dds, [], Rds) REV_IN_AGA(.(X, Xs), Y, L) -> U8_AGA(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) REV_IN_AGA(.(X, Xs), Y, L) -> REV_IN_AAA(Xs, .(X, Y), L) REV_IN_AAA(.(X, Xs), Y, L) -> U8_AAA(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) REV_IN_AAA(.(X, Xs), Y, L) -> REV_IN_AAA(Xs, .(X, Y), L) U14_GAAAA(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_GAAAA(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U14_GAAAA(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> MERGEDELETE_IN_AAA(Rds, L2, L3) U17_AGAA(Unused, List, Choice, Rest, Mod, check_out_gaaaa(List, Choice, Unused, U3, Mod)) -> U18_AGAA(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) U17_AGAA(Unused, List, Choice, Rest, Mod, check_out_gaaaa(List, Choice, Unused, U3, Mod)) -> PDS1_IN_AAAA(U3, .(Choice, List), Rest, Mod) PDS1_IN_AAAA(Unused, List, .(Choice, Rest), Mod) -> U16_AAAA(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) PDS1_IN_AAAA(Unused, List, .(Choice, Rest), Mod) -> MEMBER_IN_AA(Choice, Unused) U16_AAAA(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_AAAA(Unused, List, Choice, Rest, Mod, check_in_aaaaa(List, Choice, Unused, U3, Mod)) U16_AAAA(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> CHECK_IN_AAAAA(List, Choice, Unused, U3, Mod) CHECK_IN_AAAAA(S, Choice, Old, L3, Modulus) -> U12_AAAAA(S, Choice, Old, L3, Modulus, dif_in_aaaaa(S, Choice, Modulus, Ds, Dds)) CHECK_IN_AAAAA(S, Choice, Old, L3, Modulus) -> DIF_IN_AAAAA(S, Choice, Modulus, Ds, Dds) DIF_IN_AAAAA(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) DIF_IN_AAAAA(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> IS_IN_AA(D, -(Val, S)) U5_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U5_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> IS_IN_AA(D2, -(Mod, D)) U6_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_aaaaa(Ss, Val, Mod, Ds, D2s)) U6_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> DIF_IN_AAAAA(Ss, Val, Mod, Ds, D2s) U12_AAAAA(S, Choice, Old, L3, Modulus, dif_out_aaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_AAAAA(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) U12_AAAAA(S, Choice, Old, L3, Modulus, dif_out_aaaaa(S, Choice, Modulus, Ds, Dds)) -> MERGEDELETE_IN_AAA(Ds, Old, L2) U13_AAAAA(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_AAAAA(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) U13_AAAAA(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> REV_IN_AGA(Dds, [], Rds) U14_AAAAA(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_AAAAA(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U14_AAAAA(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> MERGEDELETE_IN_AAA(Rds, L2, L3) U17_AAAA(Unused, List, Choice, Rest, Mod, check_out_aaaaa(List, Choice, Unused, U3, Mod)) -> U18_AAAA(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) U17_AAAA(Unused, List, Choice, Rest, Mod, check_out_aaaaa(List, Choice, Unused, U3, Mod)) -> PDS1_IN_AAAA(U3, .(Choice, List), Rest, Mod) The TRS R consists of the following rules: pds_in_ga(Order, .(0, Ans)) -> U19_ga(Order, Ans, is_in_ag(N, +(*(Order, +(Order, 1)), 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U19_ga(Order, Ans, is_out_ag(N, +(*(Order, +(Order, 1)), 1))) -> U20_ga(Order, Ans, N, iota_in_aa(N, .(0, List))) iota_in_aa(N, List) -> U2_aa(N, List, iota1_in_gaa(0, N, List)) iota1_in_gaa(K, K, []) -> iota1_out_gaa(K, K, []) iota1_in_gaa(K, N, .(K, List)) -> U3_gaa(K, N, List, is_in_ag(K1, +(K, 1))) U3_gaa(K, N, List, is_out_ag(K1, +(K, 1))) -> U4_gaa(K, N, List, iota1_in_aaa(K1, N, List)) iota1_in_aaa(K, K, []) -> iota1_out_aaa(K, K, []) iota1_in_aaa(K, N, .(K, List)) -> U3_aaa(K, N, List, is_in_aa(K1, +(K, 1))) is_in_aa(X0, X1) -> is_out_aa(X0, X1) U3_aaa(K, N, List, is_out_aa(K1, +(K, 1))) -> U4_aaa(K, N, List, iota1_in_aaa(K1, N, List)) U4_aaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_aaa(K, N, .(K, List)) U4_gaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_gaa(K, N, .(K, List)) U2_aa(N, List, iota1_out_gaa(0, N, List)) -> iota_out_aa(N, List) U20_ga(Order, Ans, N, iota_out_aa(N, .(0, List))) -> U21_ga(Order, Ans, pds1_in_agaa(List, .(0, []), Ans, N)) pds1_in_agaa([], X6, [], X7) -> pds1_out_agaa([], X6, [], X7) pds1_in_agaa(Unused, List, .(Choice, Rest), Mod) -> U16_agaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) member_in_aa(X, .(X, Xs)) -> member_out_aa(X, .(X, Xs)) member_in_aa(X, .(X1, Xs)) -> U1_aa(X, X1, Xs, member_in_aa(X, Xs)) U1_aa(X, X1, Xs, member_out_aa(X, Xs)) -> member_out_aa(X, .(X1, Xs)) U16_agaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_agaa(Unused, List, Choice, Rest, Mod, check_in_gaaaa(List, Choice, Unused, U3, Mod)) check_in_gaaaa([], X4, L, L, X5) -> check_out_gaaaa([], X4, L, L, X5) check_in_gaaaa(S, Choice, Old, L3, Modulus) -> U12_gaaaa(S, Choice, Old, L3, Modulus, dif_in_gaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_gaaaa([], X2, X3, [], []) -> dif_out_gaaaa([], X2, X3, [], []) dif_in_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gaaaa(Ss, Val, Mod, Ds, D2s)) U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_gaaaa(S, Choice, Old, L3, Modulus, dif_out_gaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) mergedelete_in_aaa([], L, L) -> mergedelete_out_aaa([], L, L) mergedelete_in_aaa(.(D, Ds), .(D, R), L2) -> U9_aaa(D, Ds, R, L2, mergedelete_in_aaa(Ds, R, L2)) mergedelete_in_aaa(.(D, Ds), .(X, R), .(X, L2)) -> U10_aaa(D, Ds, X, R, L2, >_in_aa(D, X)) >_in_aa(X0, X1) -> >_out_aa(X0, X1) U10_aaa(D, Ds, X, R, L2, >_out_aa(D, X)) -> U11_aaa(D, Ds, X, R, L2, mergedelete_in_aaa(.(D, Ds), R, L2)) U11_aaa(D, Ds, X, R, L2, mergedelete_out_aaa(.(D, Ds), R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(X, R), .(X, L2)) U9_aaa(D, Ds, R, L2, mergedelete_out_aaa(Ds, R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(D, R), L2) U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) rev_in_aga([], L, L) -> rev_out_aga([], L, L) rev_in_aga(.(X, Xs), Y, L) -> U8_aga(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) rev_in_aaa([], L, L) -> rev_out_aaa([], L, L) rev_in_aaa(.(X, Xs), Y, L) -> U8_aaa(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) U8_aaa(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aaa(.(X, Xs), Y, L) U8_aga(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aga(.(X, Xs), Y, L) U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_gaaaa(S, Choice, Old, L3, Modulus) U17_agaa(Unused, List, Choice, Rest, Mod, check_out_gaaaa(List, Choice, Unused, U3, Mod)) -> U18_agaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) pds1_in_aaaa([], X6, [], X7) -> pds1_out_aaaa([], X6, [], X7) pds1_in_aaaa(Unused, List, .(Choice, Rest), Mod) -> U16_aaaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) U16_aaaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_aaaa(Unused, List, Choice, Rest, Mod, check_in_aaaaa(List, Choice, Unused, U3, Mod)) check_in_aaaaa([], X4, L, L, X5) -> check_out_aaaaa([], X4, L, L, X5) check_in_aaaaa(S, Choice, Old, L3, Modulus) -> U12_aaaaa(S, Choice, Old, L3, Modulus, dif_in_aaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_aaaaa([], X2, X3, [], []) -> dif_out_aaaaa([], X2, X3, [], []) dif_in_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_aaaaa(Ss, Val, Mod, Ds, D2s)) U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_aaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_aaaaa(S, Choice, Old, L3, Modulus, dif_out_aaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_aaaaa(S, Choice, Old, L3, Modulus) U17_aaaa(Unused, List, Choice, Rest, Mod, check_out_aaaaa(List, Choice, Unused, U3, Mod)) -> U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_aaaa(Unused, List, .(Choice, Rest), Mod) U18_agaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_agaa(Unused, List, .(Choice, Rest), Mod) U21_ga(Order, Ans, pds1_out_agaa(List, .(0, []), Ans, N)) -> pds_out_ga(Order, .(0, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U19_ga(x1, x2, x3) = U19_ga(x3) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag +(x1, x2) = +(x1, x2) *(x1, x2) = *(x1, x2) 1 = 1 U20_ga(x1, x2, x3, x4) = U20_ga(x4) iota_in_aa(x1, x2) = iota_in_aa U2_aa(x1, x2, x3) = U2_aa(x3) iota1_in_gaa(x1, x2, x3) = iota1_in_gaa(x1) iota1_out_gaa(x1, x2, x3) = iota1_out_gaa U3_gaa(x1, x2, x3, x4) = U3_gaa(x4) U4_gaa(x1, x2, x3, x4) = U4_gaa(x4) iota1_in_aaa(x1, x2, x3) = iota1_in_aaa iota1_out_aaa(x1, x2, x3) = iota1_out_aaa U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) is_in_aa(x1, x2) = is_in_aa is_out_aa(x1, x2) = is_out_aa U4_aaa(x1, x2, x3, x4) = U4_aaa(x4) 0 = 0 iota_out_aa(x1, x2) = iota_out_aa U21_ga(x1, x2, x3) = U21_ga(x3) pds1_in_agaa(x1, x2, x3, x4) = pds1_in_agaa(x2) pds1_out_agaa(x1, x2, x3, x4) = pds1_out_agaa U16_agaa(x1, x2, x3, x4, x5, x6) = U16_agaa(x2, x6) member_in_aa(x1, x2) = member_in_aa member_out_aa(x1, x2) = member_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U17_agaa(x1, x2, x3, x4, x5, x6) = U17_agaa(x6) check_in_gaaaa(x1, x2, x3, x4, x5) = check_in_gaaaa(x1) [] = [] check_out_gaaaa(x1, x2, x3, x4, x5) = check_out_gaaaa U12_gaaaa(x1, x2, x3, x4, x5, x6) = U12_gaaaa(x6) dif_in_gaaaa(x1, x2, x3, x4, x5) = dif_in_gaaaa(x1) dif_out_gaaaa(x1, x2, x3, x4, x5) = dif_out_gaaaa .(x1, x2) = .(x1, x2) U5_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_gaaaa(x2, x9) U6_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gaaaa(x2, x9) U7_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_gaaaa(x9) U13_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_gaaaa(x7) mergedelete_in_aaa(x1, x2, x3) = mergedelete_in_aaa mergedelete_out_aaa(x1, x2, x3) = mergedelete_out_aaa U9_aaa(x1, x2, x3, x4, x5) = U9_aaa(x5) U10_aaa(x1, x2, x3, x4, x5, x6) = U10_aaa(x6) >_in_aa(x1, x2) = >_in_aa' >_out_aa(x1, x2) = >_out_aa U11_aaa(x1, x2, x3, x4, x5, x6) = U11_aaa(x6) U14_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_gaaaa(x7) rev_in_aga(x1, x2, x3) = rev_in_aga(x2) rev_out_aga(x1, x2, x3) = rev_out_aga U8_aga(x1, x2, x3, x4, x5) = U8_aga(x5) rev_in_aaa(x1, x2, x3) = rev_in_aaa rev_out_aaa(x1, x2, x3) = rev_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U15_gaaaa(x1, x2, x3, x4, x5, x6) = U15_gaaaa(x6) U18_agaa(x1, x2, x3, x4, x5, x6) = U18_agaa(x6) pds1_in_aaaa(x1, x2, x3, x4) = pds1_in_aaaa pds1_out_aaaa(x1, x2, x3, x4) = pds1_out_aaaa U16_aaaa(x1, x2, x3, x4, x5, x6) = U16_aaaa(x6) U17_aaaa(x1, x2, x3, x4, x5, x6) = U17_aaaa(x6) check_in_aaaaa(x1, x2, x3, x4, x5) = check_in_aaaaa check_out_aaaaa(x1, x2, x3, x4, x5) = check_out_aaaaa U12_aaaaa(x1, x2, x3, x4, x5, x6) = U12_aaaaa(x6) dif_in_aaaaa(x1, x2, x3, x4, x5) = dif_in_aaaaa dif_out_aaaaa(x1, x2, x3, x4, x5) = dif_out_aaaaa U5_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_aaaaa(x9) U6_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_aaaaa(x9) U7_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_aaaaa(x9) U13_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_aaaaa(x7) U14_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_aaaaa(x7) U15_aaaaa(x1, x2, x3, x4, x5, x6) = U15_aaaaa(x6) U18_aaaa(x1, x2, x3, x4, x5, x6) = U18_aaaa(x6) pds_out_ga(x1, x2) = pds_out_ga PDS_IN_GA(x1, x2) = PDS_IN_GA(x1) U19_GA(x1, x2, x3) = U19_GA(x3) IS_IN_AG(x1, x2) = IS_IN_AG(x2) U20_GA(x1, x2, x3, x4) = U20_GA(x4) IOTA_IN_AA(x1, x2) = IOTA_IN_AA U2_AA(x1, x2, x3) = U2_AA(x3) IOTA1_IN_GAA(x1, x2, x3) = IOTA1_IN_GAA(x1) U3_GAA(x1, x2, x3, x4) = U3_GAA(x4) U4_GAA(x1, x2, x3, x4) = U4_GAA(x4) IOTA1_IN_AAA(x1, x2, x3) = IOTA1_IN_AAA U3_AAA(x1, x2, x3, x4) = U3_AAA(x4) IS_IN_AA(x1, x2) = IS_IN_AA U4_AAA(x1, x2, x3, x4) = U4_AAA(x4) U21_GA(x1, x2, x3) = U21_GA(x3) PDS1_IN_AGAA(x1, x2, x3, x4) = PDS1_IN_AGAA(x2) U16_AGAA(x1, x2, x3, x4, x5, x6) = U16_AGAA(x2, x6) MEMBER_IN_AA(x1, x2) = MEMBER_IN_AA U1_AA(x1, x2, x3, x4) = U1_AA(x4) U17_AGAA(x1, x2, x3, x4, x5, x6) = U17_AGAA(x6) CHECK_IN_GAAAA(x1, x2, x3, x4, x5) = CHECK_IN_GAAAA(x1) U12_GAAAA(x1, x2, x3, x4, x5, x6) = U12_GAAAA(x6) DIF_IN_GAAAA(x1, x2, x3, x4, x5) = DIF_IN_GAAAA(x1) U5_GAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_GAAAA(x2, x9) U6_GAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_GAAAA(x2, x9) U7_GAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_GAAAA(x9) U13_GAAAA(x1, x2, x3, x4, x5, x6, x7) = U13_GAAAA(x7) MERGEDELETE_IN_AAA(x1, x2, x3) = MERGEDELETE_IN_AAA U9_AAA(x1, x2, x3, x4, x5) = U9_AAA(x5) U10_AAA(x1, x2, x3, x4, x5, x6) = U10_AAA(x6) >_IN_AA(x1, x2) = >_IN_AA U11_AAA(x1, x2, x3, x4, x5, x6) = U11_AAA(x6) U14_GAAAA(x1, x2, x3, x4, x5, x6, x7) = U14_GAAAA(x7) REV_IN_AGA(x1, x2, x3) = REV_IN_AGA(x2) U8_AGA(x1, x2, x3, x4, x5) = U8_AGA(x5) REV_IN_AAA(x1, x2, x3) = REV_IN_AAA U8_AAA(x1, x2, x3, x4, x5) = U8_AAA(x5) U15_GAAAA(x1, x2, x3, x4, x5, x6) = U15_GAAAA(x6) U18_AGAA(x1, x2, x3, x4, x5, x6) = U18_AGAA(x6) PDS1_IN_AAAA(x1, x2, x3, x4) = PDS1_IN_AAAA U16_AAAA(x1, x2, x3, x4, x5, x6) = U16_AAAA(x6) U17_AAAA(x1, x2, x3, x4, x5, x6) = U17_AAAA(x6) CHECK_IN_AAAAA(x1, x2, x3, x4, x5) = CHECK_IN_AAAAA U12_AAAAA(x1, x2, x3, x4, x5, x6) = U12_AAAAA(x6) DIF_IN_AAAAA(x1, x2, x3, x4, x5) = DIF_IN_AAAAA U5_AAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_AAAAA(x9) U6_AAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_AAAAA(x9) U7_AAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_AAAAA(x9) U13_AAAAA(x1, x2, x3, x4, x5, x6, x7) = U13_AAAAA(x7) U14_AAAAA(x1, x2, x3, x4, x5, x6, x7) = U14_AAAAA(x7) U15_AAAAA(x1, x2, x3, x4, x5, x6) = U15_AAAAA(x6) U18_AAAA(x1, x2, x3, x4, x5, x6) = U18_AAAA(x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (245) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 7 SCCs with 52 less nodes. ---------------------------------------- (246) Complex Obligation (AND) ---------------------------------------- (247) Obligation: Pi DP problem: The TRS P consists of the following rules: U5_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> DIF_IN_AAAAA(Ss, Val, Mod, Ds, D2s) DIF_IN_AAAAA(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) The TRS R consists of the following rules: pds_in_ga(Order, .(0, Ans)) -> U19_ga(Order, Ans, is_in_ag(N, +(*(Order, +(Order, 1)), 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U19_ga(Order, Ans, is_out_ag(N, +(*(Order, +(Order, 1)), 1))) -> U20_ga(Order, Ans, N, iota_in_aa(N, .(0, List))) iota_in_aa(N, List) -> U2_aa(N, List, iota1_in_gaa(0, N, List)) iota1_in_gaa(K, K, []) -> iota1_out_gaa(K, K, []) iota1_in_gaa(K, N, .(K, List)) -> U3_gaa(K, N, List, is_in_ag(K1, +(K, 1))) U3_gaa(K, N, List, is_out_ag(K1, +(K, 1))) -> U4_gaa(K, N, List, iota1_in_aaa(K1, N, List)) iota1_in_aaa(K, K, []) -> iota1_out_aaa(K, K, []) iota1_in_aaa(K, N, .(K, List)) -> U3_aaa(K, N, List, is_in_aa(K1, +(K, 1))) is_in_aa(X0, X1) -> is_out_aa(X0, X1) U3_aaa(K, N, List, is_out_aa(K1, +(K, 1))) -> U4_aaa(K, N, List, iota1_in_aaa(K1, N, List)) U4_aaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_aaa(K, N, .(K, List)) U4_gaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_gaa(K, N, .(K, List)) U2_aa(N, List, iota1_out_gaa(0, N, List)) -> iota_out_aa(N, List) U20_ga(Order, Ans, N, iota_out_aa(N, .(0, List))) -> U21_ga(Order, Ans, pds1_in_agaa(List, .(0, []), Ans, N)) pds1_in_agaa([], X6, [], X7) -> pds1_out_agaa([], X6, [], X7) pds1_in_agaa(Unused, List, .(Choice, Rest), Mod) -> U16_agaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) member_in_aa(X, .(X, Xs)) -> member_out_aa(X, .(X, Xs)) member_in_aa(X, .(X1, Xs)) -> U1_aa(X, X1, Xs, member_in_aa(X, Xs)) U1_aa(X, X1, Xs, member_out_aa(X, Xs)) -> member_out_aa(X, .(X1, Xs)) U16_agaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_agaa(Unused, List, Choice, Rest, Mod, check_in_gaaaa(List, Choice, Unused, U3, Mod)) check_in_gaaaa([], X4, L, L, X5) -> check_out_gaaaa([], X4, L, L, X5) check_in_gaaaa(S, Choice, Old, L3, Modulus) -> U12_gaaaa(S, Choice, Old, L3, Modulus, dif_in_gaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_gaaaa([], X2, X3, [], []) -> dif_out_gaaaa([], X2, X3, [], []) dif_in_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gaaaa(Ss, Val, Mod, Ds, D2s)) U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_gaaaa(S, Choice, Old, L3, Modulus, dif_out_gaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) mergedelete_in_aaa([], L, L) -> mergedelete_out_aaa([], L, L) mergedelete_in_aaa(.(D, Ds), .(D, R), L2) -> U9_aaa(D, Ds, R, L2, mergedelete_in_aaa(Ds, R, L2)) mergedelete_in_aaa(.(D, Ds), .(X, R), .(X, L2)) -> U10_aaa(D, Ds, X, R, L2, >_in_aa(D, X)) >_in_aa(X0, X1) -> >_out_aa(X0, X1) U10_aaa(D, Ds, X, R, L2, >_out_aa(D, X)) -> U11_aaa(D, Ds, X, R, L2, mergedelete_in_aaa(.(D, Ds), R, L2)) U11_aaa(D, Ds, X, R, L2, mergedelete_out_aaa(.(D, Ds), R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(X, R), .(X, L2)) U9_aaa(D, Ds, R, L2, mergedelete_out_aaa(Ds, R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(D, R), L2) U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) rev_in_aga([], L, L) -> rev_out_aga([], L, L) rev_in_aga(.(X, Xs), Y, L) -> U8_aga(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) rev_in_aaa([], L, L) -> rev_out_aaa([], L, L) rev_in_aaa(.(X, Xs), Y, L) -> U8_aaa(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) U8_aaa(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aaa(.(X, Xs), Y, L) U8_aga(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aga(.(X, Xs), Y, L) U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_gaaaa(S, Choice, Old, L3, Modulus) U17_agaa(Unused, List, Choice, Rest, Mod, check_out_gaaaa(List, Choice, Unused, U3, Mod)) -> U18_agaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) pds1_in_aaaa([], X6, [], X7) -> pds1_out_aaaa([], X6, [], X7) pds1_in_aaaa(Unused, List, .(Choice, Rest), Mod) -> U16_aaaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) U16_aaaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_aaaa(Unused, List, Choice, Rest, Mod, check_in_aaaaa(List, Choice, Unused, U3, Mod)) check_in_aaaaa([], X4, L, L, X5) -> check_out_aaaaa([], X4, L, L, X5) check_in_aaaaa(S, Choice, Old, L3, Modulus) -> U12_aaaaa(S, Choice, Old, L3, Modulus, dif_in_aaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_aaaaa([], X2, X3, [], []) -> dif_out_aaaaa([], X2, X3, [], []) dif_in_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_aaaaa(Ss, Val, Mod, Ds, D2s)) U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_aaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_aaaaa(S, Choice, Old, L3, Modulus, dif_out_aaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_aaaaa(S, Choice, Old, L3, Modulus) U17_aaaa(Unused, List, Choice, Rest, Mod, check_out_aaaaa(List, Choice, Unused, U3, Mod)) -> U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_aaaa(Unused, List, .(Choice, Rest), Mod) U18_agaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_agaa(Unused, List, .(Choice, Rest), Mod) U21_ga(Order, Ans, pds1_out_agaa(List, .(0, []), Ans, N)) -> pds_out_ga(Order, .(0, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U19_ga(x1, x2, x3) = U19_ga(x3) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag +(x1, x2) = +(x1, x2) *(x1, x2) = *(x1, x2) 1 = 1 U20_ga(x1, x2, x3, x4) = U20_ga(x4) iota_in_aa(x1, x2) = iota_in_aa U2_aa(x1, x2, x3) = U2_aa(x3) iota1_in_gaa(x1, x2, x3) = iota1_in_gaa(x1) iota1_out_gaa(x1, x2, x3) = iota1_out_gaa U3_gaa(x1, x2, x3, x4) = U3_gaa(x4) U4_gaa(x1, x2, x3, x4) = U4_gaa(x4) iota1_in_aaa(x1, x2, x3) = iota1_in_aaa iota1_out_aaa(x1, x2, x3) = iota1_out_aaa U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) is_in_aa(x1, x2) = is_in_aa is_out_aa(x1, x2) = is_out_aa U4_aaa(x1, x2, x3, x4) = U4_aaa(x4) 0 = 0 iota_out_aa(x1, x2) = iota_out_aa U21_ga(x1, x2, x3) = U21_ga(x3) pds1_in_agaa(x1, x2, x3, x4) = pds1_in_agaa(x2) pds1_out_agaa(x1, x2, x3, x4) = pds1_out_agaa U16_agaa(x1, x2, x3, x4, x5, x6) = U16_agaa(x2, x6) member_in_aa(x1, x2) = member_in_aa member_out_aa(x1, x2) = member_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U17_agaa(x1, x2, x3, x4, x5, x6) = U17_agaa(x6) check_in_gaaaa(x1, x2, x3, x4, x5) = check_in_gaaaa(x1) [] = [] check_out_gaaaa(x1, x2, x3, x4, x5) = check_out_gaaaa U12_gaaaa(x1, x2, x3, x4, x5, x6) = U12_gaaaa(x6) dif_in_gaaaa(x1, x2, x3, x4, x5) = dif_in_gaaaa(x1) dif_out_gaaaa(x1, x2, x3, x4, x5) = dif_out_gaaaa .(x1, x2) = .(x1, x2) U5_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_gaaaa(x2, x9) U6_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gaaaa(x2, x9) U7_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_gaaaa(x9) U13_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_gaaaa(x7) mergedelete_in_aaa(x1, x2, x3) = mergedelete_in_aaa mergedelete_out_aaa(x1, x2, x3) = mergedelete_out_aaa U9_aaa(x1, x2, x3, x4, x5) = U9_aaa(x5) U10_aaa(x1, x2, x3, x4, x5, x6) = U10_aaa(x6) >_in_aa(x1, x2) = >_in_aa >_out_aa(x1, x2) = >_out_aa U11_aaa(x1, x2, x3, x4, x5, x6) = U11_aaa(x6) U14_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_gaaaa(x7) rev_in_aga(x1, x2, x3) = rev_in_aga(x2) rev_out_aga(x1, x2, x3) = rev_out_aga U8_aga(x1, x2, x3, x4, x5) = U8_aga(x5) rev_in_aaa(x1, x2, x3) = rev_in_aaa rev_out_aaa(x1, x2, x3) = rev_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U15_gaaaa(x1, x2, x3, x4, x5, x6) = U15_gaaaa(x6) U18_agaa(x1, x2, x3, x4, x5, x6) = U18_agaa(x6) pds1_in_aaaa(x1, x2, x3, x4) = pds1_in_aaaa pds1_out_aaaa(x1, x2, x3, x4) = pds1_out_aaaa U16_aaaa(x1, x2, x3, x4, x5, x6) = U16_aaaa(x6) U17_aaaa(x1, x2, x3, x4, x5, x6) = U17_aaaa(x6) check_in_aaaaa(x1, x2, x3, x4, x5) = check_in_aaaaa check_out_aaaaa(x1, x2, x3, x4, x5) = check_out_aaaaa U12_aaaaa(x1, x2, x3, x4, x5, x6) = U12_aaaaa(x6) dif_in_aaaaa(x1, x2, x3, x4, x5) = dif_in_aaaaa dif_out_aaaaa(x1, x2, x3, x4, x5) = dif_out_aaaaa U5_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_aaaaa(x9) U6_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_aaaaa(x9) U7_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_aaaaa(x9) U13_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_aaaaa(x7) U14_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_aaaaa(x7) U15_aaaaa(x1, x2, x3, x4, x5, x6) = U15_aaaaa(x6) U18_aaaa(x1, x2, x3, x4, x5, x6) = U18_aaaa(x6) pds_out_ga(x1, x2) = pds_out_ga DIF_IN_AAAAA(x1, x2, x3, x4, x5) = DIF_IN_AAAAA U5_AAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_AAAAA(x9) U6_AAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_AAAAA(x9) We have to consider all (P,R,Pi)-chains ---------------------------------------- (248) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (249) Obligation: Pi DP problem: The TRS P consists of the following rules: U5_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> DIF_IN_AAAAA(Ss, Val, Mod, Ds, D2s) DIF_IN_AAAAA(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) The TRS R consists of the following rules: is_in_aa(X0, X1) -> is_out_aa(X0, X1) The argument filtering Pi contains the following mapping: is_in_aa(x1, x2) = is_in_aa is_out_aa(x1, x2) = is_out_aa .(x1, x2) = .(x1, x2) DIF_IN_AAAAA(x1, x2, x3, x4, x5) = DIF_IN_AAAAA U5_AAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_AAAAA(x9) U6_AAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_AAAAA(x9) We have to consider all (P,R,Pi)-chains ---------------------------------------- (250) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (251) Obligation: Q DP problem: The TRS P consists of the following rules: U5_AAAAA(is_out_aa) -> U6_AAAAA(is_in_aa) U6_AAAAA(is_out_aa) -> DIF_IN_AAAAA DIF_IN_AAAAA -> U5_AAAAA(is_in_aa) The TRS R consists of the following rules: is_in_aa -> is_out_aa The set Q consists of the following terms: is_in_aa We have to consider all (P,Q,R)-chains. ---------------------------------------- (252) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule U5_AAAAA(is_out_aa) -> U6_AAAAA(is_in_aa) at position [0] we obtained the following new rules [LPAR04]: (U5_AAAAA(is_out_aa) -> U6_AAAAA(is_out_aa),U5_AAAAA(is_out_aa) -> U6_AAAAA(is_out_aa)) ---------------------------------------- (253) Obligation: Q DP problem: The TRS P consists of the following rules: U6_AAAAA(is_out_aa) -> DIF_IN_AAAAA DIF_IN_AAAAA -> U5_AAAAA(is_in_aa) U5_AAAAA(is_out_aa) -> U6_AAAAA(is_out_aa) The TRS R consists of the following rules: is_in_aa -> is_out_aa The set Q consists of the following terms: is_in_aa We have to consider all (P,Q,R)-chains. ---------------------------------------- (254) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule DIF_IN_AAAAA -> U5_AAAAA(is_in_aa) at position [0] we obtained the following new rules [LPAR04]: (DIF_IN_AAAAA -> U5_AAAAA(is_out_aa),DIF_IN_AAAAA -> U5_AAAAA(is_out_aa)) ---------------------------------------- (255) Obligation: Q DP problem: The TRS P consists of the following rules: U6_AAAAA(is_out_aa) -> DIF_IN_AAAAA U5_AAAAA(is_out_aa) -> U6_AAAAA(is_out_aa) DIF_IN_AAAAA -> U5_AAAAA(is_out_aa) The TRS R consists of the following rules: is_in_aa -> is_out_aa The set Q consists of the following terms: is_in_aa We have to consider all (P,Q,R)-chains. ---------------------------------------- (256) 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. ---------------------------------------- (257) Obligation: Q DP problem: The TRS P consists of the following rules: U6_AAAAA(is_out_aa) -> DIF_IN_AAAAA U5_AAAAA(is_out_aa) -> U6_AAAAA(is_out_aa) DIF_IN_AAAAA -> U5_AAAAA(is_out_aa) R is empty. The set Q consists of the following terms: is_in_aa We have to consider all (P,Q,R)-chains. ---------------------------------------- (258) 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]. is_in_aa ---------------------------------------- (259) Obligation: Q DP problem: The TRS P consists of the following rules: U6_AAAAA(is_out_aa) -> DIF_IN_AAAAA U5_AAAAA(is_out_aa) -> U6_AAAAA(is_out_aa) DIF_IN_AAAAA -> U5_AAAAA(is_out_aa) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (260) 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 = DIF_IN_AAAAA evaluates to t =DIF_IN_AAAAA Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence DIF_IN_AAAAA -> U5_AAAAA(is_out_aa) with rule DIF_IN_AAAAA -> U5_AAAAA(is_out_aa) at position [] and matcher [ ] U5_AAAAA(is_out_aa) -> U6_AAAAA(is_out_aa) with rule U5_AAAAA(is_out_aa) -> U6_AAAAA(is_out_aa) at position [] and matcher [ ] U6_AAAAA(is_out_aa) -> DIF_IN_AAAAA with rule U6_AAAAA(is_out_aa) -> DIF_IN_AAAAA 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. ---------------------------------------- (261) NO ---------------------------------------- (262) Obligation: Pi DP problem: The TRS P consists of the following rules: REV_IN_AAA(.(X, Xs), Y, L) -> REV_IN_AAA(Xs, .(X, Y), L) The TRS R consists of the following rules: pds_in_ga(Order, .(0, Ans)) -> U19_ga(Order, Ans, is_in_ag(N, +(*(Order, +(Order, 1)), 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U19_ga(Order, Ans, is_out_ag(N, +(*(Order, +(Order, 1)), 1))) -> U20_ga(Order, Ans, N, iota_in_aa(N, .(0, List))) iota_in_aa(N, List) -> U2_aa(N, List, iota1_in_gaa(0, N, List)) iota1_in_gaa(K, K, []) -> iota1_out_gaa(K, K, []) iota1_in_gaa(K, N, .(K, List)) -> U3_gaa(K, N, List, is_in_ag(K1, +(K, 1))) U3_gaa(K, N, List, is_out_ag(K1, +(K, 1))) -> U4_gaa(K, N, List, iota1_in_aaa(K1, N, List)) iota1_in_aaa(K, K, []) -> iota1_out_aaa(K, K, []) iota1_in_aaa(K, N, .(K, List)) -> U3_aaa(K, N, List, is_in_aa(K1, +(K, 1))) is_in_aa(X0, X1) -> is_out_aa(X0, X1) U3_aaa(K, N, List, is_out_aa(K1, +(K, 1))) -> U4_aaa(K, N, List, iota1_in_aaa(K1, N, List)) U4_aaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_aaa(K, N, .(K, List)) U4_gaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_gaa(K, N, .(K, List)) U2_aa(N, List, iota1_out_gaa(0, N, List)) -> iota_out_aa(N, List) U20_ga(Order, Ans, N, iota_out_aa(N, .(0, List))) -> U21_ga(Order, Ans, pds1_in_agaa(List, .(0, []), Ans, N)) pds1_in_agaa([], X6, [], X7) -> pds1_out_agaa([], X6, [], X7) pds1_in_agaa(Unused, List, .(Choice, Rest), Mod) -> U16_agaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) member_in_aa(X, .(X, Xs)) -> member_out_aa(X, .(X, Xs)) member_in_aa(X, .(X1, Xs)) -> U1_aa(X, X1, Xs, member_in_aa(X, Xs)) U1_aa(X, X1, Xs, member_out_aa(X, Xs)) -> member_out_aa(X, .(X1, Xs)) U16_agaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_agaa(Unused, List, Choice, Rest, Mod, check_in_gaaaa(List, Choice, Unused, U3, Mod)) check_in_gaaaa([], X4, L, L, X5) -> check_out_gaaaa([], X4, L, L, X5) check_in_gaaaa(S, Choice, Old, L3, Modulus) -> U12_gaaaa(S, Choice, Old, L3, Modulus, dif_in_gaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_gaaaa([], X2, X3, [], []) -> dif_out_gaaaa([], X2, X3, [], []) dif_in_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gaaaa(Ss, Val, Mod, Ds, D2s)) U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_gaaaa(S, Choice, Old, L3, Modulus, dif_out_gaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) mergedelete_in_aaa([], L, L) -> mergedelete_out_aaa([], L, L) mergedelete_in_aaa(.(D, Ds), .(D, R), L2) -> U9_aaa(D, Ds, R, L2, mergedelete_in_aaa(Ds, R, L2)) mergedelete_in_aaa(.(D, Ds), .(X, R), .(X, L2)) -> U10_aaa(D, Ds, X, R, L2, >_in_aa(D, X)) >_in_aa(X0, X1) -> >_out_aa(X0, X1) U10_aaa(D, Ds, X, R, L2, >_out_aa(D, X)) -> U11_aaa(D, Ds, X, R, L2, mergedelete_in_aaa(.(D, Ds), R, L2)) U11_aaa(D, Ds, X, R, L2, mergedelete_out_aaa(.(D, Ds), R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(X, R), .(X, L2)) U9_aaa(D, Ds, R, L2, mergedelete_out_aaa(Ds, R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(D, R), L2) U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) rev_in_aga([], L, L) -> rev_out_aga([], L, L) rev_in_aga(.(X, Xs), Y, L) -> U8_aga(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) rev_in_aaa([], L, L) -> rev_out_aaa([], L, L) rev_in_aaa(.(X, Xs), Y, L) -> U8_aaa(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) U8_aaa(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aaa(.(X, Xs), Y, L) U8_aga(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aga(.(X, Xs), Y, L) U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_gaaaa(S, Choice, Old, L3, Modulus) U17_agaa(Unused, List, Choice, Rest, Mod, check_out_gaaaa(List, Choice, Unused, U3, Mod)) -> U18_agaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) pds1_in_aaaa([], X6, [], X7) -> pds1_out_aaaa([], X6, [], X7) pds1_in_aaaa(Unused, List, .(Choice, Rest), Mod) -> U16_aaaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) U16_aaaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_aaaa(Unused, List, Choice, Rest, Mod, check_in_aaaaa(List, Choice, Unused, U3, Mod)) check_in_aaaaa([], X4, L, L, X5) -> check_out_aaaaa([], X4, L, L, X5) check_in_aaaaa(S, Choice, Old, L3, Modulus) -> U12_aaaaa(S, Choice, Old, L3, Modulus, dif_in_aaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_aaaaa([], X2, X3, [], []) -> dif_out_aaaaa([], X2, X3, [], []) dif_in_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_aaaaa(Ss, Val, Mod, Ds, D2s)) U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_aaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_aaaaa(S, Choice, Old, L3, Modulus, dif_out_aaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_aaaaa(S, Choice, Old, L3, Modulus) U17_aaaa(Unused, List, Choice, Rest, Mod, check_out_aaaaa(List, Choice, Unused, U3, Mod)) -> U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_aaaa(Unused, List, .(Choice, Rest), Mod) U18_agaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_agaa(Unused, List, .(Choice, Rest), Mod) U21_ga(Order, Ans, pds1_out_agaa(List, .(0, []), Ans, N)) -> pds_out_ga(Order, .(0, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U19_ga(x1, x2, x3) = U19_ga(x3) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag +(x1, x2) = +(x1, x2) *(x1, x2) = *(x1, x2) 1 = 1 U20_ga(x1, x2, x3, x4) = U20_ga(x4) iota_in_aa(x1, x2) = iota_in_aa U2_aa(x1, x2, x3) = U2_aa(x3) iota1_in_gaa(x1, x2, x3) = iota1_in_gaa(x1) iota1_out_gaa(x1, x2, x3) = iota1_out_gaa U3_gaa(x1, x2, x3, x4) = U3_gaa(x4) U4_gaa(x1, x2, x3, x4) = U4_gaa(x4) iota1_in_aaa(x1, x2, x3) = iota1_in_aaa iota1_out_aaa(x1, x2, x3) = iota1_out_aaa U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) is_in_aa(x1, x2) = is_in_aa is_out_aa(x1, x2) = is_out_aa U4_aaa(x1, x2, x3, x4) = U4_aaa(x4) 0 = 0 iota_out_aa(x1, x2) = iota_out_aa U21_ga(x1, x2, x3) = U21_ga(x3) pds1_in_agaa(x1, x2, x3, x4) = pds1_in_agaa(x2) pds1_out_agaa(x1, x2, x3, x4) = pds1_out_agaa U16_agaa(x1, x2, x3, x4, x5, x6) = U16_agaa(x2, x6) member_in_aa(x1, x2) = member_in_aa member_out_aa(x1, x2) = member_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U17_agaa(x1, x2, x3, x4, x5, x6) = U17_agaa(x6) check_in_gaaaa(x1, x2, x3, x4, x5) = check_in_gaaaa(x1) [] = [] check_out_gaaaa(x1, x2, x3, x4, x5) = check_out_gaaaa U12_gaaaa(x1, x2, x3, x4, x5, x6) = U12_gaaaa(x6) dif_in_gaaaa(x1, x2, x3, x4, x5) = dif_in_gaaaa(x1) dif_out_gaaaa(x1, x2, x3, x4, x5) = dif_out_gaaaa .(x1, x2) = .(x1, x2) U5_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_gaaaa(x2, x9) U6_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gaaaa(x2, x9) U7_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_gaaaa(x9) U13_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_gaaaa(x7) mergedelete_in_aaa(x1, x2, x3) = mergedelete_in_aaa mergedelete_out_aaa(x1, x2, x3) = mergedelete_out_aaa U9_aaa(x1, x2, x3, x4, x5) = U9_aaa(x5) U10_aaa(x1, x2, x3, x4, x5, x6) = U10_aaa(x6) >_in_aa(x1, x2) = >_in_aa >_out_aa(x1, x2) = >_out_aa U11_aaa(x1, x2, x3, x4, x5, x6) = U11_aaa(x6) U14_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_gaaaa(x7) rev_in_aga(x1, x2, x3) = rev_in_aga(x2) rev_out_aga(x1, x2, x3) = rev_out_aga U8_aga(x1, x2, x3, x4, x5) = U8_aga(x5) rev_in_aaa(x1, x2, x3) = rev_in_aaa rev_out_aaa(x1, x2, x3) = rev_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U15_gaaaa(x1, x2, x3, x4, x5, x6) = U15_gaaaa(x6) U18_agaa(x1, x2, x3, x4, x5, x6) = U18_agaa(x6) pds1_in_aaaa(x1, x2, x3, x4) = pds1_in_aaaa pds1_out_aaaa(x1, x2, x3, x4) = pds1_out_aaaa U16_aaaa(x1, x2, x3, x4, x5, x6) = U16_aaaa(x6) U17_aaaa(x1, x2, x3, x4, x5, x6) = U17_aaaa(x6) check_in_aaaaa(x1, x2, x3, x4, x5) = check_in_aaaaa check_out_aaaaa(x1, x2, x3, x4, x5) = check_out_aaaaa U12_aaaaa(x1, x2, x3, x4, x5, x6) = U12_aaaaa(x6) dif_in_aaaaa(x1, x2, x3, x4, x5) = dif_in_aaaaa dif_out_aaaaa(x1, x2, x3, x4, x5) = dif_out_aaaaa U5_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_aaaaa(x9) U6_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_aaaaa(x9) U7_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_aaaaa(x9) U13_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_aaaaa(x7) U14_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_aaaaa(x7) U15_aaaaa(x1, x2, x3, x4, x5, x6) = U15_aaaaa(x6) U18_aaaa(x1, x2, x3, x4, x5, x6) = U18_aaaa(x6) pds_out_ga(x1, x2) = pds_out_ga REV_IN_AAA(x1, x2, x3) = REV_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (263) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (264) Obligation: Pi DP problem: The TRS P consists of the following rules: REV_IN_AAA(.(X, Xs), Y, L) -> REV_IN_AAA(Xs, .(X, Y), L) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) REV_IN_AAA(x1, x2, x3) = REV_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (265) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (266) Obligation: Q DP problem: The TRS P consists of the following rules: REV_IN_AAA -> REV_IN_AAA R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (267) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by semiunifying a rule from P directly. s = REV_IN_AAA evaluates to t =REV_IN_AAA Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence The DP semiunifies directly so there is only one rewrite step from REV_IN_AAA to REV_IN_AAA. ---------------------------------------- (268) NO ---------------------------------------- (269) Obligation: Pi DP problem: The TRS P consists of the following rules: MERGEDELETE_IN_AAA(.(D, Ds), .(X, R), .(X, L2)) -> U10_AAA(D, Ds, X, R, L2, >_in_aa(D, X)) U10_AAA(D, Ds, X, R, L2, >_out_aa(D, X)) -> MERGEDELETE_IN_AAA(.(D, Ds), R, L2) MERGEDELETE_IN_AAA(.(D, Ds), .(D, R), L2) -> MERGEDELETE_IN_AAA(Ds, R, L2) The TRS R consists of the following rules: pds_in_ga(Order, .(0, Ans)) -> U19_ga(Order, Ans, is_in_ag(N, +(*(Order, +(Order, 1)), 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U19_ga(Order, Ans, is_out_ag(N, +(*(Order, +(Order, 1)), 1))) -> U20_ga(Order, Ans, N, iota_in_aa(N, .(0, List))) iota_in_aa(N, List) -> U2_aa(N, List, iota1_in_gaa(0, N, List)) iota1_in_gaa(K, K, []) -> iota1_out_gaa(K, K, []) iota1_in_gaa(K, N, .(K, List)) -> U3_gaa(K, N, List, is_in_ag(K1, +(K, 1))) U3_gaa(K, N, List, is_out_ag(K1, +(K, 1))) -> U4_gaa(K, N, List, iota1_in_aaa(K1, N, List)) iota1_in_aaa(K, K, []) -> iota1_out_aaa(K, K, []) iota1_in_aaa(K, N, .(K, List)) -> U3_aaa(K, N, List, is_in_aa(K1, +(K, 1))) is_in_aa(X0, X1) -> is_out_aa(X0, X1) U3_aaa(K, N, List, is_out_aa(K1, +(K, 1))) -> U4_aaa(K, N, List, iota1_in_aaa(K1, N, List)) U4_aaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_aaa(K, N, .(K, List)) U4_gaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_gaa(K, N, .(K, List)) U2_aa(N, List, iota1_out_gaa(0, N, List)) -> iota_out_aa(N, List) U20_ga(Order, Ans, N, iota_out_aa(N, .(0, List))) -> U21_ga(Order, Ans, pds1_in_agaa(List, .(0, []), Ans, N)) pds1_in_agaa([], X6, [], X7) -> pds1_out_agaa([], X6, [], X7) pds1_in_agaa(Unused, List, .(Choice, Rest), Mod) -> U16_agaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) member_in_aa(X, .(X, Xs)) -> member_out_aa(X, .(X, Xs)) member_in_aa(X, .(X1, Xs)) -> U1_aa(X, X1, Xs, member_in_aa(X, Xs)) U1_aa(X, X1, Xs, member_out_aa(X, Xs)) -> member_out_aa(X, .(X1, Xs)) U16_agaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_agaa(Unused, List, Choice, Rest, Mod, check_in_gaaaa(List, Choice, Unused, U3, Mod)) check_in_gaaaa([], X4, L, L, X5) -> check_out_gaaaa([], X4, L, L, X5) check_in_gaaaa(S, Choice, Old, L3, Modulus) -> U12_gaaaa(S, Choice, Old, L3, Modulus, dif_in_gaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_gaaaa([], X2, X3, [], []) -> dif_out_gaaaa([], X2, X3, [], []) dif_in_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gaaaa(Ss, Val, Mod, Ds, D2s)) U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_gaaaa(S, Choice, Old, L3, Modulus, dif_out_gaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) mergedelete_in_aaa([], L, L) -> mergedelete_out_aaa([], L, L) mergedelete_in_aaa(.(D, Ds), .(D, R), L2) -> U9_aaa(D, Ds, R, L2, mergedelete_in_aaa(Ds, R, L2)) mergedelete_in_aaa(.(D, Ds), .(X, R), .(X, L2)) -> U10_aaa(D, Ds, X, R, L2, >_in_aa(D, X)) >_in_aa(X0, X1) -> >_out_aa(X0, X1) U10_aaa(D, Ds, X, R, L2, >_out_aa(D, X)) -> U11_aaa(D, Ds, X, R, L2, mergedelete_in_aaa(.(D, Ds), R, L2)) U11_aaa(D, Ds, X, R, L2, mergedelete_out_aaa(.(D, Ds), R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(X, R), .(X, L2)) U9_aaa(D, Ds, R, L2, mergedelete_out_aaa(Ds, R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(D, R), L2) U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) rev_in_aga([], L, L) -> rev_out_aga([], L, L) rev_in_aga(.(X, Xs), Y, L) -> U8_aga(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) rev_in_aaa([], L, L) -> rev_out_aaa([], L, L) rev_in_aaa(.(X, Xs), Y, L) -> U8_aaa(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) U8_aaa(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aaa(.(X, Xs), Y, L) U8_aga(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aga(.(X, Xs), Y, L) U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_gaaaa(S, Choice, Old, L3, Modulus) U17_agaa(Unused, List, Choice, Rest, Mod, check_out_gaaaa(List, Choice, Unused, U3, Mod)) -> U18_agaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) pds1_in_aaaa([], X6, [], X7) -> pds1_out_aaaa([], X6, [], X7) pds1_in_aaaa(Unused, List, .(Choice, Rest), Mod) -> U16_aaaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) U16_aaaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_aaaa(Unused, List, Choice, Rest, Mod, check_in_aaaaa(List, Choice, Unused, U3, Mod)) check_in_aaaaa([], X4, L, L, X5) -> check_out_aaaaa([], X4, L, L, X5) check_in_aaaaa(S, Choice, Old, L3, Modulus) -> U12_aaaaa(S, Choice, Old, L3, Modulus, dif_in_aaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_aaaaa([], X2, X3, [], []) -> dif_out_aaaaa([], X2, X3, [], []) dif_in_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_aaaaa(Ss, Val, Mod, Ds, D2s)) U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_aaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_aaaaa(S, Choice, Old, L3, Modulus, dif_out_aaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_aaaaa(S, Choice, Old, L3, Modulus) U17_aaaa(Unused, List, Choice, Rest, Mod, check_out_aaaaa(List, Choice, Unused, U3, Mod)) -> U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_aaaa(Unused, List, .(Choice, Rest), Mod) U18_agaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_agaa(Unused, List, .(Choice, Rest), Mod) U21_ga(Order, Ans, pds1_out_agaa(List, .(0, []), Ans, N)) -> pds_out_ga(Order, .(0, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U19_ga(x1, x2, x3) = U19_ga(x3) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag +(x1, x2) = +(x1, x2) *(x1, x2) = *(x1, x2) 1 = 1 U20_ga(x1, x2, x3, x4) = U20_ga(x4) iota_in_aa(x1, x2) = iota_in_aa U2_aa(x1, x2, x3) = U2_aa(x3) iota1_in_gaa(x1, x2, x3) = iota1_in_gaa(x1) iota1_out_gaa(x1, x2, x3) = iota1_out_gaa U3_gaa(x1, x2, x3, x4) = U3_gaa(x4) U4_gaa(x1, x2, x3, x4) = U4_gaa(x4) iota1_in_aaa(x1, x2, x3) = iota1_in_aaa iota1_out_aaa(x1, x2, x3) = iota1_out_aaa U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) is_in_aa(x1, x2) = is_in_aa is_out_aa(x1, x2) = is_out_aa U4_aaa(x1, x2, x3, x4) = U4_aaa(x4) 0 = 0 iota_out_aa(x1, x2) = iota_out_aa U21_ga(x1, x2, x3) = U21_ga(x3) pds1_in_agaa(x1, x2, x3, x4) = pds1_in_agaa(x2) pds1_out_agaa(x1, x2, x3, x4) = pds1_out_agaa U16_agaa(x1, x2, x3, x4, x5, x6) = U16_agaa(x2, x6) member_in_aa(x1, x2) = member_in_aa member_out_aa(x1, x2) = member_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U17_agaa(x1, x2, x3, x4, x5, x6) = U17_agaa(x6) check_in_gaaaa(x1, x2, x3, x4, x5) = check_in_gaaaa(x1) [] = [] check_out_gaaaa(x1, x2, x3, x4, x5) = check_out_gaaaa U12_gaaaa(x1, x2, x3, x4, x5, x6) = U12_gaaaa(x6) dif_in_gaaaa(x1, x2, x3, x4, x5) = dif_in_gaaaa(x1) dif_out_gaaaa(x1, x2, x3, x4, x5) = dif_out_gaaaa .(x1, x2) = .(x1, x2) U5_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_gaaaa(x2, x9) U6_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gaaaa(x2, x9) U7_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_gaaaa(x9) U13_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_gaaaa(x7) mergedelete_in_aaa(x1, x2, x3) = mergedelete_in_aaa mergedelete_out_aaa(x1, x2, x3) = mergedelete_out_aaa U9_aaa(x1, x2, x3, x4, x5) = U9_aaa(x5) U10_aaa(x1, x2, x3, x4, x5, x6) = U10_aaa(x6) >_in_aa(x1, x2) = >_in_aa >_out_aa(x1, x2) = >_out_aa U11_aaa(x1, x2, x3, x4, x5, x6) = U11_aaa(x6) U14_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_gaaaa(x7) rev_in_aga(x1, x2, x3) = rev_in_aga(x2) rev_out_aga(x1, x2, x3) = rev_out_aga U8_aga(x1, x2, x3, x4, x5) = U8_aga(x5) rev_in_aaa(x1, x2, x3) = rev_in_aaa rev_out_aaa(x1, x2, x3) = rev_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U15_gaaaa(x1, x2, x3, x4, x5, x6) = U15_gaaaa(x6) U18_agaa(x1, x2, x3, x4, x5, x6) = U18_agaa(x6) pds1_in_aaaa(x1, x2, x3, x4) = pds1_in_aaaa pds1_out_aaaa(x1, x2, x3, x4) = pds1_out_aaaa U16_aaaa(x1, x2, x3, x4, x5, x6) = U16_aaaa(x6) U17_aaaa(x1, x2, x3, x4, x5, x6) = U17_aaaa(x6) check_in_aaaaa(x1, x2, x3, x4, x5) = check_in_aaaaa check_out_aaaaa(x1, x2, x3, x4, x5) = check_out_aaaaa U12_aaaaa(x1, x2, x3, x4, x5, x6) = U12_aaaaa(x6) dif_in_aaaaa(x1, x2, x3, x4, x5) = dif_in_aaaaa dif_out_aaaaa(x1, x2, x3, x4, x5) = dif_out_aaaaa U5_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_aaaaa(x9) U6_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_aaaaa(x9) U7_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_aaaaa(x9) U13_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_aaaaa(x7) U14_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_aaaaa(x7) U15_aaaaa(x1, x2, x3, x4, x5, x6) = U15_aaaaa(x6) U18_aaaa(x1, x2, x3, x4, x5, x6) = U18_aaaa(x6) pds_out_ga(x1, x2) = pds_out_ga MERGEDELETE_IN_AAA(x1, x2, x3) = MERGEDELETE_IN_AAA U10_AAA(x1, x2, x3, x4, x5, x6) = U10_AAA(x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (270) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (271) Obligation: Pi DP problem: The TRS P consists of the following rules: MERGEDELETE_IN_AAA(.(D, Ds), .(X, R), .(X, L2)) -> U10_AAA(D, Ds, X, R, L2, >_in_aa(D, X)) U10_AAA(D, Ds, X, R, L2, >_out_aa(D, X)) -> MERGEDELETE_IN_AAA(.(D, Ds), R, L2) MERGEDELETE_IN_AAA(.(D, Ds), .(D, R), L2) -> MERGEDELETE_IN_AAA(Ds, R, L2) The TRS R consists of the following rules: >_in_aa(X0, X1) -> >_out_aa(X0, X1) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) >_in_aa(x1, x2) = >_in_aa >_out_aa(x1, x2) = >_out_aa MERGEDELETE_IN_AAA(x1, x2, x3) = MERGEDELETE_IN_AAA U10_AAA(x1, x2, x3, x4, x5, x6) = U10_AAA(x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (272) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (273) Obligation: Q DP problem: The TRS P consists of the following rules: MERGEDELETE_IN_AAA -> U10_AAA(>_in_aa) U10_AAA(>_out_aa) -> MERGEDELETE_IN_AAA MERGEDELETE_IN_AAA -> MERGEDELETE_IN_AAA The TRS R consists of the following rules: >_in_aa -> >_out_aa The set Q consists of the following terms: >_in_aa We have to consider all (P,Q,R)-chains. ---------------------------------------- (274) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule MERGEDELETE_IN_AAA -> U10_AAA(>_in_aa) at position [0] we obtained the following new rules [LPAR04]: (MERGEDELETE_IN_AAA -> U10_AAA(>_out_aa),MERGEDELETE_IN_AAA -> U10_AAA(>_out_aa)) ---------------------------------------- (275) Obligation: Q DP problem: The TRS P consists of the following rules: U10_AAA(>_out_aa) -> MERGEDELETE_IN_AAA MERGEDELETE_IN_AAA -> MERGEDELETE_IN_AAA MERGEDELETE_IN_AAA -> U10_AAA(>_out_aa) The TRS R consists of the following rules: >_in_aa -> >_out_aa The set Q consists of the following terms: >_in_aa We have to consider all (P,Q,R)-chains. ---------------------------------------- (276) 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. ---------------------------------------- (277) Obligation: Q DP problem: The TRS P consists of the following rules: U10_AAA(>_out_aa) -> MERGEDELETE_IN_AAA MERGEDELETE_IN_AAA -> MERGEDELETE_IN_AAA MERGEDELETE_IN_AAA -> U10_AAA(>_out_aa) R is empty. The set Q consists of the following terms: >_in_aa We have to consider all (P,Q,R)-chains. ---------------------------------------- (278) 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]. >_in_aa ---------------------------------------- (279) Obligation: Q DP problem: The TRS P consists of the following rules: U10_AAA(>_out_aa) -> MERGEDELETE_IN_AAA MERGEDELETE_IN_AAA -> MERGEDELETE_IN_AAA MERGEDELETE_IN_AAA -> U10_AAA(>_out_aa) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (280) Obligation: Pi DP problem: The TRS P consists of the following rules: U5_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> DIF_IN_GAAAA(Ss, Val, Mod, Ds, D2s) DIF_IN_GAAAA(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) The TRS R consists of the following rules: pds_in_ga(Order, .(0, Ans)) -> U19_ga(Order, Ans, is_in_ag(N, +(*(Order, +(Order, 1)), 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U19_ga(Order, Ans, is_out_ag(N, +(*(Order, +(Order, 1)), 1))) -> U20_ga(Order, Ans, N, iota_in_aa(N, .(0, List))) iota_in_aa(N, List) -> U2_aa(N, List, iota1_in_gaa(0, N, List)) iota1_in_gaa(K, K, []) -> iota1_out_gaa(K, K, []) iota1_in_gaa(K, N, .(K, List)) -> U3_gaa(K, N, List, is_in_ag(K1, +(K, 1))) U3_gaa(K, N, List, is_out_ag(K1, +(K, 1))) -> U4_gaa(K, N, List, iota1_in_aaa(K1, N, List)) iota1_in_aaa(K, K, []) -> iota1_out_aaa(K, K, []) iota1_in_aaa(K, N, .(K, List)) -> U3_aaa(K, N, List, is_in_aa(K1, +(K, 1))) is_in_aa(X0, X1) -> is_out_aa(X0, X1) U3_aaa(K, N, List, is_out_aa(K1, +(K, 1))) -> U4_aaa(K, N, List, iota1_in_aaa(K1, N, List)) U4_aaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_aaa(K, N, .(K, List)) U4_gaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_gaa(K, N, .(K, List)) U2_aa(N, List, iota1_out_gaa(0, N, List)) -> iota_out_aa(N, List) U20_ga(Order, Ans, N, iota_out_aa(N, .(0, List))) -> U21_ga(Order, Ans, pds1_in_agaa(List, .(0, []), Ans, N)) pds1_in_agaa([], X6, [], X7) -> pds1_out_agaa([], X6, [], X7) pds1_in_agaa(Unused, List, .(Choice, Rest), Mod) -> U16_agaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) member_in_aa(X, .(X, Xs)) -> member_out_aa(X, .(X, Xs)) member_in_aa(X, .(X1, Xs)) -> U1_aa(X, X1, Xs, member_in_aa(X, Xs)) U1_aa(X, X1, Xs, member_out_aa(X, Xs)) -> member_out_aa(X, .(X1, Xs)) U16_agaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_agaa(Unused, List, Choice, Rest, Mod, check_in_gaaaa(List, Choice, Unused, U3, Mod)) check_in_gaaaa([], X4, L, L, X5) -> check_out_gaaaa([], X4, L, L, X5) check_in_gaaaa(S, Choice, Old, L3, Modulus) -> U12_gaaaa(S, Choice, Old, L3, Modulus, dif_in_gaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_gaaaa([], X2, X3, [], []) -> dif_out_gaaaa([], X2, X3, [], []) dif_in_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gaaaa(Ss, Val, Mod, Ds, D2s)) U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_gaaaa(S, Choice, Old, L3, Modulus, dif_out_gaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) mergedelete_in_aaa([], L, L) -> mergedelete_out_aaa([], L, L) mergedelete_in_aaa(.(D, Ds), .(D, R), L2) -> U9_aaa(D, Ds, R, L2, mergedelete_in_aaa(Ds, R, L2)) mergedelete_in_aaa(.(D, Ds), .(X, R), .(X, L2)) -> U10_aaa(D, Ds, X, R, L2, >_in_aa(D, X)) >_in_aa(X0, X1) -> >_out_aa(X0, X1) U10_aaa(D, Ds, X, R, L2, >_out_aa(D, X)) -> U11_aaa(D, Ds, X, R, L2, mergedelete_in_aaa(.(D, Ds), R, L2)) U11_aaa(D, Ds, X, R, L2, mergedelete_out_aaa(.(D, Ds), R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(X, R), .(X, L2)) U9_aaa(D, Ds, R, L2, mergedelete_out_aaa(Ds, R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(D, R), L2) U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) rev_in_aga([], L, L) -> rev_out_aga([], L, L) rev_in_aga(.(X, Xs), Y, L) -> U8_aga(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) rev_in_aaa([], L, L) -> rev_out_aaa([], L, L) rev_in_aaa(.(X, Xs), Y, L) -> U8_aaa(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) U8_aaa(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aaa(.(X, Xs), Y, L) U8_aga(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aga(.(X, Xs), Y, L) U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_gaaaa(S, Choice, Old, L3, Modulus) U17_agaa(Unused, List, Choice, Rest, Mod, check_out_gaaaa(List, Choice, Unused, U3, Mod)) -> U18_agaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) pds1_in_aaaa([], X6, [], X7) -> pds1_out_aaaa([], X6, [], X7) pds1_in_aaaa(Unused, List, .(Choice, Rest), Mod) -> U16_aaaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) U16_aaaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_aaaa(Unused, List, Choice, Rest, Mod, check_in_aaaaa(List, Choice, Unused, U3, Mod)) check_in_aaaaa([], X4, L, L, X5) -> check_out_aaaaa([], X4, L, L, X5) check_in_aaaaa(S, Choice, Old, L3, Modulus) -> U12_aaaaa(S, Choice, Old, L3, Modulus, dif_in_aaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_aaaaa([], X2, X3, [], []) -> dif_out_aaaaa([], X2, X3, [], []) dif_in_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_aaaaa(Ss, Val, Mod, Ds, D2s)) U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_aaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_aaaaa(S, Choice, Old, L3, Modulus, dif_out_aaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_aaaaa(S, Choice, Old, L3, Modulus) U17_aaaa(Unused, List, Choice, Rest, Mod, check_out_aaaaa(List, Choice, Unused, U3, Mod)) -> U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_aaaa(Unused, List, .(Choice, Rest), Mod) U18_agaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_agaa(Unused, List, .(Choice, Rest), Mod) U21_ga(Order, Ans, pds1_out_agaa(List, .(0, []), Ans, N)) -> pds_out_ga(Order, .(0, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U19_ga(x1, x2, x3) = U19_ga(x3) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag +(x1, x2) = +(x1, x2) *(x1, x2) = *(x1, x2) 1 = 1 U20_ga(x1, x2, x3, x4) = U20_ga(x4) iota_in_aa(x1, x2) = iota_in_aa U2_aa(x1, x2, x3) = U2_aa(x3) iota1_in_gaa(x1, x2, x3) = iota1_in_gaa(x1) iota1_out_gaa(x1, x2, x3) = iota1_out_gaa U3_gaa(x1, x2, x3, x4) = U3_gaa(x4) U4_gaa(x1, x2, x3, x4) = U4_gaa(x4) iota1_in_aaa(x1, x2, x3) = iota1_in_aaa iota1_out_aaa(x1, x2, x3) = iota1_out_aaa U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) is_in_aa(x1, x2) = is_in_aa is_out_aa(x1, x2) = is_out_aa U4_aaa(x1, x2, x3, x4) = U4_aaa(x4) 0 = 0 iota_out_aa(x1, x2) = iota_out_aa U21_ga(x1, x2, x3) = U21_ga(x3) pds1_in_agaa(x1, x2, x3, x4) = pds1_in_agaa(x2) pds1_out_agaa(x1, x2, x3, x4) = pds1_out_agaa U16_agaa(x1, x2, x3, x4, x5, x6) = U16_agaa(x2, x6) member_in_aa(x1, x2) = member_in_aa member_out_aa(x1, x2) = member_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U17_agaa(x1, x2, x3, x4, x5, x6) = U17_agaa(x6) check_in_gaaaa(x1, x2, x3, x4, x5) = check_in_gaaaa(x1) [] = [] check_out_gaaaa(x1, x2, x3, x4, x5) = check_out_gaaaa U12_gaaaa(x1, x2, x3, x4, x5, x6) = U12_gaaaa(x6) dif_in_gaaaa(x1, x2, x3, x4, x5) = dif_in_gaaaa(x1) dif_out_gaaaa(x1, x2, x3, x4, x5) = dif_out_gaaaa .(x1, x2) = .(x1, x2) U5_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_gaaaa(x2, x9) U6_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gaaaa(x2, x9) U7_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_gaaaa(x9) U13_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_gaaaa(x7) mergedelete_in_aaa(x1, x2, x3) = mergedelete_in_aaa mergedelete_out_aaa(x1, x2, x3) = mergedelete_out_aaa U9_aaa(x1, x2, x3, x4, x5) = U9_aaa(x5) U10_aaa(x1, x2, x3, x4, x5, x6) = U10_aaa(x6) >_in_aa(x1, x2) = >_in_aa >_out_aa(x1, x2) = >_out_aa U11_aaa(x1, x2, x3, x4, x5, x6) = U11_aaa(x6) U14_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_gaaaa(x7) rev_in_aga(x1, x2, x3) = rev_in_aga(x2) rev_out_aga(x1, x2, x3) = rev_out_aga U8_aga(x1, x2, x3, x4, x5) = U8_aga(x5) rev_in_aaa(x1, x2, x3) = rev_in_aaa rev_out_aaa(x1, x2, x3) = rev_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U15_gaaaa(x1, x2, x3, x4, x5, x6) = U15_gaaaa(x6) U18_agaa(x1, x2, x3, x4, x5, x6) = U18_agaa(x6) pds1_in_aaaa(x1, x2, x3, x4) = pds1_in_aaaa pds1_out_aaaa(x1, x2, x3, x4) = pds1_out_aaaa U16_aaaa(x1, x2, x3, x4, x5, x6) = U16_aaaa(x6) U17_aaaa(x1, x2, x3, x4, x5, x6) = U17_aaaa(x6) check_in_aaaaa(x1, x2, x3, x4, x5) = check_in_aaaaa check_out_aaaaa(x1, x2, x3, x4, x5) = check_out_aaaaa U12_aaaaa(x1, x2, x3, x4, x5, x6) = U12_aaaaa(x6) dif_in_aaaaa(x1, x2, x3, x4, x5) = dif_in_aaaaa dif_out_aaaaa(x1, x2, x3, x4, x5) = dif_out_aaaaa U5_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_aaaaa(x9) U6_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_aaaaa(x9) U7_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_aaaaa(x9) U13_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_aaaaa(x7) U14_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_aaaaa(x7) U15_aaaaa(x1, x2, x3, x4, x5, x6) = U15_aaaaa(x6) U18_aaaa(x1, x2, x3, x4, x5, x6) = U18_aaaa(x6) pds_out_ga(x1, x2) = pds_out_ga DIF_IN_GAAAA(x1, x2, x3, x4, x5) = DIF_IN_GAAAA(x1) U5_GAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_GAAAA(x2, x9) U6_GAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_GAAAA(x2, x9) We have to consider all (P,R,Pi)-chains ---------------------------------------- (281) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (282) Obligation: Pi DP problem: The TRS P consists of the following rules: U5_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> DIF_IN_GAAAA(Ss, Val, Mod, Ds, D2s) DIF_IN_GAAAA(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) The TRS R consists of the following rules: is_in_aa(X0, X1) -> is_out_aa(X0, X1) The argument filtering Pi contains the following mapping: is_in_aa(x1, x2) = is_in_aa is_out_aa(x1, x2) = is_out_aa .(x1, x2) = .(x1, x2) DIF_IN_GAAAA(x1, x2, x3, x4, x5) = DIF_IN_GAAAA(x1) U5_GAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_GAAAA(x2, x9) U6_GAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_GAAAA(x2, x9) We have to consider all (P,R,Pi)-chains ---------------------------------------- (283) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER_IN_AA(X, .(X1, Xs)) -> MEMBER_IN_AA(X, Xs) The TRS R consists of the following rules: pds_in_ga(Order, .(0, Ans)) -> U19_ga(Order, Ans, is_in_ag(N, +(*(Order, +(Order, 1)), 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U19_ga(Order, Ans, is_out_ag(N, +(*(Order, +(Order, 1)), 1))) -> U20_ga(Order, Ans, N, iota_in_aa(N, .(0, List))) iota_in_aa(N, List) -> U2_aa(N, List, iota1_in_gaa(0, N, List)) iota1_in_gaa(K, K, []) -> iota1_out_gaa(K, K, []) iota1_in_gaa(K, N, .(K, List)) -> U3_gaa(K, N, List, is_in_ag(K1, +(K, 1))) U3_gaa(K, N, List, is_out_ag(K1, +(K, 1))) -> U4_gaa(K, N, List, iota1_in_aaa(K1, N, List)) iota1_in_aaa(K, K, []) -> iota1_out_aaa(K, K, []) iota1_in_aaa(K, N, .(K, List)) -> U3_aaa(K, N, List, is_in_aa(K1, +(K, 1))) is_in_aa(X0, X1) -> is_out_aa(X0, X1) U3_aaa(K, N, List, is_out_aa(K1, +(K, 1))) -> U4_aaa(K, N, List, iota1_in_aaa(K1, N, List)) U4_aaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_aaa(K, N, .(K, List)) U4_gaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_gaa(K, N, .(K, List)) U2_aa(N, List, iota1_out_gaa(0, N, List)) -> iota_out_aa(N, List) U20_ga(Order, Ans, N, iota_out_aa(N, .(0, List))) -> U21_ga(Order, Ans, pds1_in_agaa(List, .(0, []), Ans, N)) pds1_in_agaa([], X6, [], X7) -> pds1_out_agaa([], X6, [], X7) pds1_in_agaa(Unused, List, .(Choice, Rest), Mod) -> U16_agaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) member_in_aa(X, .(X, Xs)) -> member_out_aa(X, .(X, Xs)) member_in_aa(X, .(X1, Xs)) -> U1_aa(X, X1, Xs, member_in_aa(X, Xs)) U1_aa(X, X1, Xs, member_out_aa(X, Xs)) -> member_out_aa(X, .(X1, Xs)) U16_agaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_agaa(Unused, List, Choice, Rest, Mod, check_in_gaaaa(List, Choice, Unused, U3, Mod)) check_in_gaaaa([], X4, L, L, X5) -> check_out_gaaaa([], X4, L, L, X5) check_in_gaaaa(S, Choice, Old, L3, Modulus) -> U12_gaaaa(S, Choice, Old, L3, Modulus, dif_in_gaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_gaaaa([], X2, X3, [], []) -> dif_out_gaaaa([], X2, X3, [], []) dif_in_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gaaaa(Ss, Val, Mod, Ds, D2s)) U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_gaaaa(S, Choice, Old, L3, Modulus, dif_out_gaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) mergedelete_in_aaa([], L, L) -> mergedelete_out_aaa([], L, L) mergedelete_in_aaa(.(D, Ds), .(D, R), L2) -> U9_aaa(D, Ds, R, L2, mergedelete_in_aaa(Ds, R, L2)) mergedelete_in_aaa(.(D, Ds), .(X, R), .(X, L2)) -> U10_aaa(D, Ds, X, R, L2, >_in_aa(D, X)) >_in_aa(X0, X1) -> >_out_aa(X0, X1) U10_aaa(D, Ds, X, R, L2, >_out_aa(D, X)) -> U11_aaa(D, Ds, X, R, L2, mergedelete_in_aaa(.(D, Ds), R, L2)) U11_aaa(D, Ds, X, R, L2, mergedelete_out_aaa(.(D, Ds), R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(X, R), .(X, L2)) U9_aaa(D, Ds, R, L2, mergedelete_out_aaa(Ds, R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(D, R), L2) U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) rev_in_aga([], L, L) -> rev_out_aga([], L, L) rev_in_aga(.(X, Xs), Y, L) -> U8_aga(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) rev_in_aaa([], L, L) -> rev_out_aaa([], L, L) rev_in_aaa(.(X, Xs), Y, L) -> U8_aaa(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) U8_aaa(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aaa(.(X, Xs), Y, L) U8_aga(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aga(.(X, Xs), Y, L) U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_gaaaa(S, Choice, Old, L3, Modulus) U17_agaa(Unused, List, Choice, Rest, Mod, check_out_gaaaa(List, Choice, Unused, U3, Mod)) -> U18_agaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) pds1_in_aaaa([], X6, [], X7) -> pds1_out_aaaa([], X6, [], X7) pds1_in_aaaa(Unused, List, .(Choice, Rest), Mod) -> U16_aaaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) U16_aaaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_aaaa(Unused, List, Choice, Rest, Mod, check_in_aaaaa(List, Choice, Unused, U3, Mod)) check_in_aaaaa([], X4, L, L, X5) -> check_out_aaaaa([], X4, L, L, X5) check_in_aaaaa(S, Choice, Old, L3, Modulus) -> U12_aaaaa(S, Choice, Old, L3, Modulus, dif_in_aaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_aaaaa([], X2, X3, [], []) -> dif_out_aaaaa([], X2, X3, [], []) dif_in_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_aaaaa(Ss, Val, Mod, Ds, D2s)) U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_aaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_aaaaa(S, Choice, Old, L3, Modulus, dif_out_aaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_aaaaa(S, Choice, Old, L3, Modulus) U17_aaaa(Unused, List, Choice, Rest, Mod, check_out_aaaaa(List, Choice, Unused, U3, Mod)) -> U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_aaaa(Unused, List, .(Choice, Rest), Mod) U18_agaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_agaa(Unused, List, .(Choice, Rest), Mod) U21_ga(Order, Ans, pds1_out_agaa(List, .(0, []), Ans, N)) -> pds_out_ga(Order, .(0, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U19_ga(x1, x2, x3) = U19_ga(x3) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag +(x1, x2) = +(x1, x2) *(x1, x2) = *(x1, x2) 1 = 1 U20_ga(x1, x2, x3, x4) = U20_ga(x4) iota_in_aa(x1, x2) = iota_in_aa U2_aa(x1, x2, x3) = U2_aa(x3) iota1_in_gaa(x1, x2, x3) = iota1_in_gaa(x1) iota1_out_gaa(x1, x2, x3) = iota1_out_gaa U3_gaa(x1, x2, x3, x4) = U3_gaa(x4) U4_gaa(x1, x2, x3, x4) = U4_gaa(x4) iota1_in_aaa(x1, x2, x3) = iota1_in_aaa iota1_out_aaa(x1, x2, x3) = iota1_out_aaa U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) is_in_aa(x1, x2) = is_in_aa is_out_aa(x1, x2) = is_out_aa U4_aaa(x1, x2, x3, x4) = U4_aaa(x4) 0 = 0 iota_out_aa(x1, x2) = iota_out_aa U21_ga(x1, x2, x3) = U21_ga(x3) pds1_in_agaa(x1, x2, x3, x4) = pds1_in_agaa(x2) pds1_out_agaa(x1, x2, x3, x4) = pds1_out_agaa U16_agaa(x1, x2, x3, x4, x5, x6) = U16_agaa(x2, x6) member_in_aa(x1, x2) = member_in_aa member_out_aa(x1, x2) = member_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U17_agaa(x1, x2, x3, x4, x5, x6) = U17_agaa(x6) check_in_gaaaa(x1, x2, x3, x4, x5) = check_in_gaaaa(x1) [] = [] check_out_gaaaa(x1, x2, x3, x4, x5) = check_out_gaaaa U12_gaaaa(x1, x2, x3, x4, x5, x6) = U12_gaaaa(x6) dif_in_gaaaa(x1, x2, x3, x4, x5) = dif_in_gaaaa(x1) dif_out_gaaaa(x1, x2, x3, x4, x5) = dif_out_gaaaa .(x1, x2) = .(x1, x2) U5_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_gaaaa(x2, x9) U6_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gaaaa(x2, x9) U7_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_gaaaa(x9) U13_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_gaaaa(x7) mergedelete_in_aaa(x1, x2, x3) = mergedelete_in_aaa mergedelete_out_aaa(x1, x2, x3) = mergedelete_out_aaa U9_aaa(x1, x2, x3, x4, x5) = U9_aaa(x5) U10_aaa(x1, x2, x3, x4, x5, x6) = U10_aaa(x6) >_in_aa(x1, x2) = >_in_aa >_out_aa(x1, x2) = >_out_aa U11_aaa(x1, x2, x3, x4, x5, x6) = U11_aaa(x6) U14_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_gaaaa(x7) rev_in_aga(x1, x2, x3) = rev_in_aga(x2) rev_out_aga(x1, x2, x3) = rev_out_aga U8_aga(x1, x2, x3, x4, x5) = U8_aga(x5) rev_in_aaa(x1, x2, x3) = rev_in_aaa rev_out_aaa(x1, x2, x3) = rev_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U15_gaaaa(x1, x2, x3, x4, x5, x6) = U15_gaaaa(x6) U18_agaa(x1, x2, x3, x4, x5, x6) = U18_agaa(x6) pds1_in_aaaa(x1, x2, x3, x4) = pds1_in_aaaa pds1_out_aaaa(x1, x2, x3, x4) = pds1_out_aaaa U16_aaaa(x1, x2, x3, x4, x5, x6) = U16_aaaa(x6) U17_aaaa(x1, x2, x3, x4, x5, x6) = U17_aaaa(x6) check_in_aaaaa(x1, x2, x3, x4, x5) = check_in_aaaaa check_out_aaaaa(x1, x2, x3, x4, x5) = check_out_aaaaa U12_aaaaa(x1, x2, x3, x4, x5, x6) = U12_aaaaa(x6) dif_in_aaaaa(x1, x2, x3, x4, x5) = dif_in_aaaaa dif_out_aaaaa(x1, x2, x3, x4, x5) = dif_out_aaaaa U5_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_aaaaa(x9) U6_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_aaaaa(x9) U7_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_aaaaa(x9) U13_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_aaaaa(x7) U14_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_aaaaa(x7) U15_aaaaa(x1, x2, x3, x4, x5, x6) = U15_aaaaa(x6) U18_aaaa(x1, x2, x3, x4, x5, x6) = U18_aaaa(x6) pds_out_ga(x1, x2) = pds_out_ga MEMBER_IN_AA(x1, x2) = MEMBER_IN_AA We have to consider all (P,R,Pi)-chains ---------------------------------------- (284) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (285) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER_IN_AA(X, .(X1, Xs)) -> MEMBER_IN_AA(X, Xs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) MEMBER_IN_AA(x1, x2) = MEMBER_IN_AA We have to consider all (P,R,Pi)-chains ---------------------------------------- (286) Obligation: Pi DP problem: The TRS P consists of the following rules: PDS1_IN_AAAA(Unused, List, .(Choice, Rest), Mod) -> U16_AAAA(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) U16_AAAA(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_AAAA(Unused, List, Choice, Rest, Mod, check_in_aaaaa(List, Choice, Unused, U3, Mod)) U17_AAAA(Unused, List, Choice, Rest, Mod, check_out_aaaaa(List, Choice, Unused, U3, Mod)) -> PDS1_IN_AAAA(U3, .(Choice, List), Rest, Mod) The TRS R consists of the following rules: pds_in_ga(Order, .(0, Ans)) -> U19_ga(Order, Ans, is_in_ag(N, +(*(Order, +(Order, 1)), 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U19_ga(Order, Ans, is_out_ag(N, +(*(Order, +(Order, 1)), 1))) -> U20_ga(Order, Ans, N, iota_in_aa(N, .(0, List))) iota_in_aa(N, List) -> U2_aa(N, List, iota1_in_gaa(0, N, List)) iota1_in_gaa(K, K, []) -> iota1_out_gaa(K, K, []) iota1_in_gaa(K, N, .(K, List)) -> U3_gaa(K, N, List, is_in_ag(K1, +(K, 1))) U3_gaa(K, N, List, is_out_ag(K1, +(K, 1))) -> U4_gaa(K, N, List, iota1_in_aaa(K1, N, List)) iota1_in_aaa(K, K, []) -> iota1_out_aaa(K, K, []) iota1_in_aaa(K, N, .(K, List)) -> U3_aaa(K, N, List, is_in_aa(K1, +(K, 1))) is_in_aa(X0, X1) -> is_out_aa(X0, X1) U3_aaa(K, N, List, is_out_aa(K1, +(K, 1))) -> U4_aaa(K, N, List, iota1_in_aaa(K1, N, List)) U4_aaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_aaa(K, N, .(K, List)) U4_gaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_gaa(K, N, .(K, List)) U2_aa(N, List, iota1_out_gaa(0, N, List)) -> iota_out_aa(N, List) U20_ga(Order, Ans, N, iota_out_aa(N, .(0, List))) -> U21_ga(Order, Ans, pds1_in_agaa(List, .(0, []), Ans, N)) pds1_in_agaa([], X6, [], X7) -> pds1_out_agaa([], X6, [], X7) pds1_in_agaa(Unused, List, .(Choice, Rest), Mod) -> U16_agaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) member_in_aa(X, .(X, Xs)) -> member_out_aa(X, .(X, Xs)) member_in_aa(X, .(X1, Xs)) -> U1_aa(X, X1, Xs, member_in_aa(X, Xs)) U1_aa(X, X1, Xs, member_out_aa(X, Xs)) -> member_out_aa(X, .(X1, Xs)) U16_agaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_agaa(Unused, List, Choice, Rest, Mod, check_in_gaaaa(List, Choice, Unused, U3, Mod)) check_in_gaaaa([], X4, L, L, X5) -> check_out_gaaaa([], X4, L, L, X5) check_in_gaaaa(S, Choice, Old, L3, Modulus) -> U12_gaaaa(S, Choice, Old, L3, Modulus, dif_in_gaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_gaaaa([], X2, X3, [], []) -> dif_out_gaaaa([], X2, X3, [], []) dif_in_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gaaaa(Ss, Val, Mod, Ds, D2s)) U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_gaaaa(S, Choice, Old, L3, Modulus, dif_out_gaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) mergedelete_in_aaa([], L, L) -> mergedelete_out_aaa([], L, L) mergedelete_in_aaa(.(D, Ds), .(D, R), L2) -> U9_aaa(D, Ds, R, L2, mergedelete_in_aaa(Ds, R, L2)) mergedelete_in_aaa(.(D, Ds), .(X, R), .(X, L2)) -> U10_aaa(D, Ds, X, R, L2, >_in_aa(D, X)) >_in_aa(X0, X1) -> >_out_aa(X0, X1) U10_aaa(D, Ds, X, R, L2, >_out_aa(D, X)) -> U11_aaa(D, Ds, X, R, L2, mergedelete_in_aaa(.(D, Ds), R, L2)) U11_aaa(D, Ds, X, R, L2, mergedelete_out_aaa(.(D, Ds), R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(X, R), .(X, L2)) U9_aaa(D, Ds, R, L2, mergedelete_out_aaa(Ds, R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(D, R), L2) U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) rev_in_aga([], L, L) -> rev_out_aga([], L, L) rev_in_aga(.(X, Xs), Y, L) -> U8_aga(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) rev_in_aaa([], L, L) -> rev_out_aaa([], L, L) rev_in_aaa(.(X, Xs), Y, L) -> U8_aaa(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) U8_aaa(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aaa(.(X, Xs), Y, L) U8_aga(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aga(.(X, Xs), Y, L) U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_gaaaa(S, Choice, Old, L3, Modulus) U17_agaa(Unused, List, Choice, Rest, Mod, check_out_gaaaa(List, Choice, Unused, U3, Mod)) -> U18_agaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) pds1_in_aaaa([], X6, [], X7) -> pds1_out_aaaa([], X6, [], X7) pds1_in_aaaa(Unused, List, .(Choice, Rest), Mod) -> U16_aaaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) U16_aaaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_aaaa(Unused, List, Choice, Rest, Mod, check_in_aaaaa(List, Choice, Unused, U3, Mod)) check_in_aaaaa([], X4, L, L, X5) -> check_out_aaaaa([], X4, L, L, X5) check_in_aaaaa(S, Choice, Old, L3, Modulus) -> U12_aaaaa(S, Choice, Old, L3, Modulus, dif_in_aaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_aaaaa([], X2, X3, [], []) -> dif_out_aaaaa([], X2, X3, [], []) dif_in_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_aaaaa(Ss, Val, Mod, Ds, D2s)) U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_aaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_aaaaa(S, Choice, Old, L3, Modulus, dif_out_aaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_aaaaa(S, Choice, Old, L3, Modulus) U17_aaaa(Unused, List, Choice, Rest, Mod, check_out_aaaaa(List, Choice, Unused, U3, Mod)) -> U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_aaaa(Unused, List, .(Choice, Rest), Mod) U18_agaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_agaa(Unused, List, .(Choice, Rest), Mod) U21_ga(Order, Ans, pds1_out_agaa(List, .(0, []), Ans, N)) -> pds_out_ga(Order, .(0, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U19_ga(x1, x2, x3) = U19_ga(x3) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag +(x1, x2) = +(x1, x2) *(x1, x2) = *(x1, x2) 1 = 1 U20_ga(x1, x2, x3, x4) = U20_ga(x4) iota_in_aa(x1, x2) = iota_in_aa U2_aa(x1, x2, x3) = U2_aa(x3) iota1_in_gaa(x1, x2, x3) = iota1_in_gaa(x1) iota1_out_gaa(x1, x2, x3) = iota1_out_gaa U3_gaa(x1, x2, x3, x4) = U3_gaa(x4) U4_gaa(x1, x2, x3, x4) = U4_gaa(x4) iota1_in_aaa(x1, x2, x3) = iota1_in_aaa iota1_out_aaa(x1, x2, x3) = iota1_out_aaa U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) is_in_aa(x1, x2) = is_in_aa is_out_aa(x1, x2) = is_out_aa U4_aaa(x1, x2, x3, x4) = U4_aaa(x4) 0 = 0 iota_out_aa(x1, x2) = iota_out_aa U21_ga(x1, x2, x3) = U21_ga(x3) pds1_in_agaa(x1, x2, x3, x4) = pds1_in_agaa(x2) pds1_out_agaa(x1, x2, x3, x4) = pds1_out_agaa U16_agaa(x1, x2, x3, x4, x5, x6) = U16_agaa(x2, x6) member_in_aa(x1, x2) = member_in_aa member_out_aa(x1, x2) = member_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U17_agaa(x1, x2, x3, x4, x5, x6) = U17_agaa(x6) check_in_gaaaa(x1, x2, x3, x4, x5) = check_in_gaaaa(x1) [] = [] check_out_gaaaa(x1, x2, x3, x4, x5) = check_out_gaaaa U12_gaaaa(x1, x2, x3, x4, x5, x6) = U12_gaaaa(x6) dif_in_gaaaa(x1, x2, x3, x4, x5) = dif_in_gaaaa(x1) dif_out_gaaaa(x1, x2, x3, x4, x5) = dif_out_gaaaa .(x1, x2) = .(x1, x2) U5_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_gaaaa(x2, x9) U6_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gaaaa(x2, x9) U7_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_gaaaa(x9) U13_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_gaaaa(x7) mergedelete_in_aaa(x1, x2, x3) = mergedelete_in_aaa mergedelete_out_aaa(x1, x2, x3) = mergedelete_out_aaa U9_aaa(x1, x2, x3, x4, x5) = U9_aaa(x5) U10_aaa(x1, x2, x3, x4, x5, x6) = U10_aaa(x6) >_in_aa(x1, x2) = >_in_aa >_out_aa(x1, x2) = >_out_aa U11_aaa(x1, x2, x3, x4, x5, x6) = U11_aaa(x6) U14_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_gaaaa(x7) rev_in_aga(x1, x2, x3) = rev_in_aga(x2) rev_out_aga(x1, x2, x3) = rev_out_aga U8_aga(x1, x2, x3, x4, x5) = U8_aga(x5) rev_in_aaa(x1, x2, x3) = rev_in_aaa rev_out_aaa(x1, x2, x3) = rev_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U15_gaaaa(x1, x2, x3, x4, x5, x6) = U15_gaaaa(x6) U18_agaa(x1, x2, x3, x4, x5, x6) = U18_agaa(x6) pds1_in_aaaa(x1, x2, x3, x4) = pds1_in_aaaa pds1_out_aaaa(x1, x2, x3, x4) = pds1_out_aaaa U16_aaaa(x1, x2, x3, x4, x5, x6) = U16_aaaa(x6) U17_aaaa(x1, x2, x3, x4, x5, x6) = U17_aaaa(x6) check_in_aaaaa(x1, x2, x3, x4, x5) = check_in_aaaaa check_out_aaaaa(x1, x2, x3, x4, x5) = check_out_aaaaa U12_aaaaa(x1, x2, x3, x4, x5, x6) = U12_aaaaa(x6) dif_in_aaaaa(x1, x2, x3, x4, x5) = dif_in_aaaaa dif_out_aaaaa(x1, x2, x3, x4, x5) = dif_out_aaaaa U5_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_aaaaa(x9) U6_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_aaaaa(x9) U7_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_aaaaa(x9) U13_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_aaaaa(x7) U14_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_aaaaa(x7) U15_aaaaa(x1, x2, x3, x4, x5, x6) = U15_aaaaa(x6) U18_aaaa(x1, x2, x3, x4, x5, x6) = U18_aaaa(x6) pds_out_ga(x1, x2) = pds_out_ga PDS1_IN_AAAA(x1, x2, x3, x4) = PDS1_IN_AAAA U16_AAAA(x1, x2, x3, x4, x5, x6) = U16_AAAA(x6) U17_AAAA(x1, x2, x3, x4, x5, x6) = U17_AAAA(x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (287) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (288) Obligation: Pi DP problem: The TRS P consists of the following rules: PDS1_IN_AAAA(Unused, List, .(Choice, Rest), Mod) -> U16_AAAA(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) U16_AAAA(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_AAAA(Unused, List, Choice, Rest, Mod, check_in_aaaaa(List, Choice, Unused, U3, Mod)) U17_AAAA(Unused, List, Choice, Rest, Mod, check_out_aaaaa(List, Choice, Unused, U3, Mod)) -> PDS1_IN_AAAA(U3, .(Choice, List), Rest, Mod) The TRS R consists of the following rules: member_in_aa(X, .(X, Xs)) -> member_out_aa(X, .(X, Xs)) member_in_aa(X, .(X1, Xs)) -> U1_aa(X, X1, Xs, member_in_aa(X, Xs)) check_in_aaaaa([], X4, L, L, X5) -> check_out_aaaaa([], X4, L, L, X5) check_in_aaaaa(S, Choice, Old, L3, Modulus) -> U12_aaaaa(S, Choice, Old, L3, Modulus, dif_in_aaaaa(S, Choice, Modulus, Ds, Dds)) U1_aa(X, X1, Xs, member_out_aa(X, Xs)) -> member_out_aa(X, .(X1, Xs)) U12_aaaaa(S, Choice, Old, L3, Modulus, dif_out_aaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) dif_in_aaaaa([], X2, X3, [], []) -> dif_out_aaaaa([], X2, X3, [], []) dif_in_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) mergedelete_in_aaa([], L, L) -> mergedelete_out_aaa([], L, L) mergedelete_in_aaa(.(D, Ds), .(D, R), L2) -> U9_aaa(D, Ds, R, L2, mergedelete_in_aaa(Ds, R, L2)) mergedelete_in_aaa(.(D, Ds), .(X, R), .(X, L2)) -> U10_aaa(D, Ds, X, R, L2, >_in_aa(D, X)) U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) is_in_aa(X0, X1) -> is_out_aa(X0, X1) U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_aaaaa(Ss, Val, Mod, Ds, D2s)) U9_aaa(D, Ds, R, L2, mergedelete_out_aaa(Ds, R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(D, R), L2) U10_aaa(D, Ds, X, R, L2, >_out_aa(D, X)) -> U11_aaa(D, Ds, X, R, L2, mergedelete_in_aaa(.(D, Ds), R, L2)) rev_in_aga([], L, L) -> rev_out_aga([], L, L) rev_in_aga(.(X, Xs), Y, L) -> U8_aga(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_aaaaa(S, Choice, Old, L3, Modulus) U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_aaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) >_in_aa(X0, X1) -> >_out_aa(X0, X1) U11_aaa(D, Ds, X, R, L2, mergedelete_out_aaa(.(D, Ds), R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(X, R), .(X, L2)) U8_aga(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aga(.(X, Xs), Y, L) rev_in_aaa([], L, L) -> rev_out_aaa([], L, L) rev_in_aaa(.(X, Xs), Y, L) -> U8_aaa(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) U8_aaa(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aaa(.(X, Xs), Y, L) The argument filtering Pi contains the following mapping: is_in_aa(x1, x2) = is_in_aa is_out_aa(x1, x2) = is_out_aa member_in_aa(x1, x2) = member_in_aa member_out_aa(x1, x2) = member_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) [] = [] .(x1, x2) = .(x1, x2) mergedelete_in_aaa(x1, x2, x3) = mergedelete_in_aaa mergedelete_out_aaa(x1, x2, x3) = mergedelete_out_aaa U9_aaa(x1, x2, x3, x4, x5) = U9_aaa(x5) U10_aaa(x1, x2, x3, x4, x5, x6) = U10_aaa(x6) >_in_aa(x1, x2) = >_in_aa >_out_aa(x1, x2) = >_out_aa U11_aaa(x1, x2, x3, x4, x5, x6) = U11_aaa(x6) rev_in_aga(x1, x2, x3) = rev_in_aga(x2) rev_out_aga(x1, x2, x3) = rev_out_aga U8_aga(x1, x2, x3, x4, x5) = U8_aga(x5) rev_in_aaa(x1, x2, x3) = rev_in_aaa rev_out_aaa(x1, x2, x3) = rev_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) check_in_aaaaa(x1, x2, x3, x4, x5) = check_in_aaaaa check_out_aaaaa(x1, x2, x3, x4, x5) = check_out_aaaaa U12_aaaaa(x1, x2, x3, x4, x5, x6) = U12_aaaaa(x6) dif_in_aaaaa(x1, x2, x3, x4, x5) = dif_in_aaaaa dif_out_aaaaa(x1, x2, x3, x4, x5) = dif_out_aaaaa U5_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_aaaaa(x9) U6_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_aaaaa(x9) U7_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_aaaaa(x9) U13_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_aaaaa(x7) U14_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_aaaaa(x7) U15_aaaaa(x1, x2, x3, x4, x5, x6) = U15_aaaaa(x6) PDS1_IN_AAAA(x1, x2, x3, x4) = PDS1_IN_AAAA U16_AAAA(x1, x2, x3, x4, x5, x6) = U16_AAAA(x6) U17_AAAA(x1, x2, x3, x4, x5, x6) = U17_AAAA(x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (289) Obligation: Pi DP problem: The TRS P consists of the following rules: U3_AAA(K, N, List, is_out_aa(K1, +(K, 1))) -> IOTA1_IN_AAA(K1, N, List) IOTA1_IN_AAA(K, N, .(K, List)) -> U3_AAA(K, N, List, is_in_aa(K1, +(K, 1))) The TRS R consists of the following rules: pds_in_ga(Order, .(0, Ans)) -> U19_ga(Order, Ans, is_in_ag(N, +(*(Order, +(Order, 1)), 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U19_ga(Order, Ans, is_out_ag(N, +(*(Order, +(Order, 1)), 1))) -> U20_ga(Order, Ans, N, iota_in_aa(N, .(0, List))) iota_in_aa(N, List) -> U2_aa(N, List, iota1_in_gaa(0, N, List)) iota1_in_gaa(K, K, []) -> iota1_out_gaa(K, K, []) iota1_in_gaa(K, N, .(K, List)) -> U3_gaa(K, N, List, is_in_ag(K1, +(K, 1))) U3_gaa(K, N, List, is_out_ag(K1, +(K, 1))) -> U4_gaa(K, N, List, iota1_in_aaa(K1, N, List)) iota1_in_aaa(K, K, []) -> iota1_out_aaa(K, K, []) iota1_in_aaa(K, N, .(K, List)) -> U3_aaa(K, N, List, is_in_aa(K1, +(K, 1))) is_in_aa(X0, X1) -> is_out_aa(X0, X1) U3_aaa(K, N, List, is_out_aa(K1, +(K, 1))) -> U4_aaa(K, N, List, iota1_in_aaa(K1, N, List)) U4_aaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_aaa(K, N, .(K, List)) U4_gaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_gaa(K, N, .(K, List)) U2_aa(N, List, iota1_out_gaa(0, N, List)) -> iota_out_aa(N, List) U20_ga(Order, Ans, N, iota_out_aa(N, .(0, List))) -> U21_ga(Order, Ans, pds1_in_agaa(List, .(0, []), Ans, N)) pds1_in_agaa([], X6, [], X7) -> pds1_out_agaa([], X6, [], X7) pds1_in_agaa(Unused, List, .(Choice, Rest), Mod) -> U16_agaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) member_in_aa(X, .(X, Xs)) -> member_out_aa(X, .(X, Xs)) member_in_aa(X, .(X1, Xs)) -> U1_aa(X, X1, Xs, member_in_aa(X, Xs)) U1_aa(X, X1, Xs, member_out_aa(X, Xs)) -> member_out_aa(X, .(X1, Xs)) U16_agaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_agaa(Unused, List, Choice, Rest, Mod, check_in_gaaaa(List, Choice, Unused, U3, Mod)) check_in_gaaaa([], X4, L, L, X5) -> check_out_gaaaa([], X4, L, L, X5) check_in_gaaaa(S, Choice, Old, L3, Modulus) -> U12_gaaaa(S, Choice, Old, L3, Modulus, dif_in_gaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_gaaaa([], X2, X3, [], []) -> dif_out_gaaaa([], X2, X3, [], []) dif_in_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gaaaa(Ss, Val, Mod, Ds, D2s)) U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_gaaaa(S, Choice, Old, L3, Modulus, dif_out_gaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) mergedelete_in_aaa([], L, L) -> mergedelete_out_aaa([], L, L) mergedelete_in_aaa(.(D, Ds), .(D, R), L2) -> U9_aaa(D, Ds, R, L2, mergedelete_in_aaa(Ds, R, L2)) mergedelete_in_aaa(.(D, Ds), .(X, R), .(X, L2)) -> U10_aaa(D, Ds, X, R, L2, >_in_aa(D, X)) >_in_aa(X0, X1) -> >_out_aa(X0, X1) U10_aaa(D, Ds, X, R, L2, >_out_aa(D, X)) -> U11_aaa(D, Ds, X, R, L2, mergedelete_in_aaa(.(D, Ds), R, L2)) U11_aaa(D, Ds, X, R, L2, mergedelete_out_aaa(.(D, Ds), R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(X, R), .(X, L2)) U9_aaa(D, Ds, R, L2, mergedelete_out_aaa(Ds, R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(D, R), L2) U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) rev_in_aga([], L, L) -> rev_out_aga([], L, L) rev_in_aga(.(X, Xs), Y, L) -> U8_aga(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) rev_in_aaa([], L, L) -> rev_out_aaa([], L, L) rev_in_aaa(.(X, Xs), Y, L) -> U8_aaa(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) U8_aaa(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aaa(.(X, Xs), Y, L) U8_aga(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aga(.(X, Xs), Y, L) U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_gaaaa(S, Choice, Old, L3, Modulus) U17_agaa(Unused, List, Choice, Rest, Mod, check_out_gaaaa(List, Choice, Unused, U3, Mod)) -> U18_agaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) pds1_in_aaaa([], X6, [], X7) -> pds1_out_aaaa([], X6, [], X7) pds1_in_aaaa(Unused, List, .(Choice, Rest), Mod) -> U16_aaaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) U16_aaaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_aaaa(Unused, List, Choice, Rest, Mod, check_in_aaaaa(List, Choice, Unused, U3, Mod)) check_in_aaaaa([], X4, L, L, X5) -> check_out_aaaaa([], X4, L, L, X5) check_in_aaaaa(S, Choice, Old, L3, Modulus) -> U12_aaaaa(S, Choice, Old, L3, Modulus, dif_in_aaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_aaaaa([], X2, X3, [], []) -> dif_out_aaaaa([], X2, X3, [], []) dif_in_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_aaaaa(Ss, Val, Mod, Ds, D2s)) U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_aaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_aaaaa(S, Choice, Old, L3, Modulus, dif_out_aaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_aaaaa(S, Choice, Old, L3, Modulus) U17_aaaa(Unused, List, Choice, Rest, Mod, check_out_aaaaa(List, Choice, Unused, U3, Mod)) -> U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_aaaa(Unused, List, .(Choice, Rest), Mod) U18_agaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_agaa(Unused, List, .(Choice, Rest), Mod) U21_ga(Order, Ans, pds1_out_agaa(List, .(0, []), Ans, N)) -> pds_out_ga(Order, .(0, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U19_ga(x1, x2, x3) = U19_ga(x3) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag +(x1, x2) = +(x1, x2) *(x1, x2) = *(x1, x2) 1 = 1 U20_ga(x1, x2, x3, x4) = U20_ga(x4) iota_in_aa(x1, x2) = iota_in_aa U2_aa(x1, x2, x3) = U2_aa(x3) iota1_in_gaa(x1, x2, x3) = iota1_in_gaa(x1) iota1_out_gaa(x1, x2, x3) = iota1_out_gaa U3_gaa(x1, x2, x3, x4) = U3_gaa(x4) U4_gaa(x1, x2, x3, x4) = U4_gaa(x4) iota1_in_aaa(x1, x2, x3) = iota1_in_aaa iota1_out_aaa(x1, x2, x3) = iota1_out_aaa U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) is_in_aa(x1, x2) = is_in_aa is_out_aa(x1, x2) = is_out_aa U4_aaa(x1, x2, x3, x4) = U4_aaa(x4) 0 = 0 iota_out_aa(x1, x2) = iota_out_aa U21_ga(x1, x2, x3) = U21_ga(x3) pds1_in_agaa(x1, x2, x3, x4) = pds1_in_agaa(x2) pds1_out_agaa(x1, x2, x3, x4) = pds1_out_agaa U16_agaa(x1, x2, x3, x4, x5, x6) = U16_agaa(x2, x6) member_in_aa(x1, x2) = member_in_aa member_out_aa(x1, x2) = member_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U17_agaa(x1, x2, x3, x4, x5, x6) = U17_agaa(x6) check_in_gaaaa(x1, x2, x3, x4, x5) = check_in_gaaaa(x1) [] = [] check_out_gaaaa(x1, x2, x3, x4, x5) = check_out_gaaaa U12_gaaaa(x1, x2, x3, x4, x5, x6) = U12_gaaaa(x6) dif_in_gaaaa(x1, x2, x3, x4, x5) = dif_in_gaaaa(x1) dif_out_gaaaa(x1, x2, x3, x4, x5) = dif_out_gaaaa .(x1, x2) = .(x1, x2) U5_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_gaaaa(x2, x9) U6_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gaaaa(x2, x9) U7_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_gaaaa(x9) U13_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_gaaaa(x7) mergedelete_in_aaa(x1, x2, x3) = mergedelete_in_aaa mergedelete_out_aaa(x1, x2, x3) = mergedelete_out_aaa U9_aaa(x1, x2, x3, x4, x5) = U9_aaa(x5) U10_aaa(x1, x2, x3, x4, x5, x6) = U10_aaa(x6) >_in_aa(x1, x2) = >_in_aa >_out_aa(x1, x2) = >_out_aa U11_aaa(x1, x2, x3, x4, x5, x6) = U11_aaa(x6) U14_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_gaaaa(x7) rev_in_aga(x1, x2, x3) = rev_in_aga(x2) rev_out_aga(x1, x2, x3) = rev_out_aga U8_aga(x1, x2, x3, x4, x5) = U8_aga(x5) rev_in_aaa(x1, x2, x3) = rev_in_aaa rev_out_aaa(x1, x2, x3) = rev_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U15_gaaaa(x1, x2, x3, x4, x5, x6) = U15_gaaaa(x6) U18_agaa(x1, x2, x3, x4, x5, x6) = U18_agaa(x6) pds1_in_aaaa(x1, x2, x3, x4) = pds1_in_aaaa pds1_out_aaaa(x1, x2, x3, x4) = pds1_out_aaaa U16_aaaa(x1, x2, x3, x4, x5, x6) = U16_aaaa(x6) U17_aaaa(x1, x2, x3, x4, x5, x6) = U17_aaaa(x6) check_in_aaaaa(x1, x2, x3, x4, x5) = check_in_aaaaa check_out_aaaaa(x1, x2, x3, x4, x5) = check_out_aaaaa U12_aaaaa(x1, x2, x3, x4, x5, x6) = U12_aaaaa(x6) dif_in_aaaaa(x1, x2, x3, x4, x5) = dif_in_aaaaa dif_out_aaaaa(x1, x2, x3, x4, x5) = dif_out_aaaaa U5_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_aaaaa(x9) U6_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_aaaaa(x9) U7_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_aaaaa(x9) U13_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_aaaaa(x7) U14_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_aaaaa(x7) U15_aaaaa(x1, x2, x3, x4, x5, x6) = U15_aaaaa(x6) U18_aaaa(x1, x2, x3, x4, x5, x6) = U18_aaaa(x6) pds_out_ga(x1, x2) = pds_out_ga IOTA1_IN_AAA(x1, x2, x3) = IOTA1_IN_AAA U3_AAA(x1, x2, x3, x4) = U3_AAA(x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (290) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (291) Obligation: Pi DP problem: The TRS P consists of the following rules: U3_AAA(K, N, List, is_out_aa(K1, +(K, 1))) -> IOTA1_IN_AAA(K1, N, List) IOTA1_IN_AAA(K, N, .(K, List)) -> U3_AAA(K, N, List, is_in_aa(K1, +(K, 1))) The TRS R consists of the following rules: is_in_aa(X0, X1) -> is_out_aa(X0, X1) The argument filtering Pi contains the following mapping: +(x1, x2) = +(x1, x2) 1 = 1 is_in_aa(x1, x2) = is_in_aa is_out_aa(x1, x2) = is_out_aa .(x1, x2) = .(x1, x2) IOTA1_IN_AAA(x1, x2, x3) = IOTA1_IN_AAA U3_AAA(x1, x2, x3, x4) = U3_AAA(x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (292) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: pds_in_2: (b,f) iota_in_2: (f,f) iota1_in_3: (b,f,f) (f,f,f) pds1_in_4: (f,b,f,f) (f,f,f,f) member_in_2: (f,f) check_in_5: (b,f,f,f,f) (f,f,f,f,f) dif_in_5: (b,f,f,f,f) (f,f,f,f,f) mergedelete_in_3: (f,f,f) rev_in_3: (f,b,f) (f,f,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: pds_in_ga(Order, .(0, Ans)) -> U19_ga(Order, Ans, is_in_ag(N, +(*(Order, +(Order, 1)), 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U19_ga(Order, Ans, is_out_ag(N, +(*(Order, +(Order, 1)), 1))) -> U20_ga(Order, Ans, N, iota_in_aa(N, .(0, List))) iota_in_aa(N, List) -> U2_aa(N, List, iota1_in_gaa(0, N, List)) iota1_in_gaa(K, K, []) -> iota1_out_gaa(K, K, []) iota1_in_gaa(K, N, .(K, List)) -> U3_gaa(K, N, List, is_in_ag(K1, +(K, 1))) U3_gaa(K, N, List, is_out_ag(K1, +(K, 1))) -> U4_gaa(K, N, List, iota1_in_aaa(K1, N, List)) iota1_in_aaa(K, K, []) -> iota1_out_aaa(K, K, []) iota1_in_aaa(K, N, .(K, List)) -> U3_aaa(K, N, List, is_in_aa(K1, +(K, 1))) is_in_aa(X0, X1) -> is_out_aa(X0, X1) U3_aaa(K, N, List, is_out_aa(K1, +(K, 1))) -> U4_aaa(K, N, List, iota1_in_aaa(K1, N, List)) U4_aaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_aaa(K, N, .(K, List)) U4_gaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_gaa(K, N, .(K, List)) U2_aa(N, List, iota1_out_gaa(0, N, List)) -> iota_out_aa(N, List) U20_ga(Order, Ans, N, iota_out_aa(N, .(0, List))) -> U21_ga(Order, Ans, pds1_in_agaa(List, .(0, []), Ans, N)) pds1_in_agaa([], X6, [], X7) -> pds1_out_agaa([], X6, [], X7) pds1_in_agaa(Unused, List, .(Choice, Rest), Mod) -> U16_agaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) member_in_aa(X, .(X, Xs)) -> member_out_aa(X, .(X, Xs)) member_in_aa(X, .(X1, Xs)) -> U1_aa(X, X1, Xs, member_in_aa(X, Xs)) U1_aa(X, X1, Xs, member_out_aa(X, Xs)) -> member_out_aa(X, .(X1, Xs)) U16_agaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_agaa(Unused, List, Choice, Rest, Mod, check_in_gaaaa(List, Choice, Unused, U3, Mod)) check_in_gaaaa([], X4, L, L, X5) -> check_out_gaaaa([], X4, L, L, X5) check_in_gaaaa(S, Choice, Old, L3, Modulus) -> U12_gaaaa(S, Choice, Old, L3, Modulus, dif_in_gaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_gaaaa([], X2, X3, [], []) -> dif_out_gaaaa([], X2, X3, [], []) dif_in_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gaaaa(Ss, Val, Mod, Ds, D2s)) U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_gaaaa(S, Choice, Old, L3, Modulus, dif_out_gaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) mergedelete_in_aaa([], L, L) -> mergedelete_out_aaa([], L, L) mergedelete_in_aaa(.(D, Ds), .(D, R), L2) -> U9_aaa(D, Ds, R, L2, mergedelete_in_aaa(Ds, R, L2)) mergedelete_in_aaa(.(D, Ds), .(X, R), .(X, L2)) -> U10_aaa(D, Ds, X, R, L2, >_in_aa(D, X)) >_in_aa(X0, X1) -> >_out_aa(X0, X1) U10_aaa(D, Ds, X, R, L2, >_out_aa(D, X)) -> U11_aaa(D, Ds, X, R, L2, mergedelete_in_aaa(.(D, Ds), R, L2)) U11_aaa(D, Ds, X, R, L2, mergedelete_out_aaa(.(D, Ds), R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(X, R), .(X, L2)) U9_aaa(D, Ds, R, L2, mergedelete_out_aaa(Ds, R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(D, R), L2) U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) rev_in_aga([], L, L) -> rev_out_aga([], L, L) rev_in_aga(.(X, Xs), Y, L) -> U8_aga(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) rev_in_aaa([], L, L) -> rev_out_aaa([], L, L) rev_in_aaa(.(X, Xs), Y, L) -> U8_aaa(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) U8_aaa(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aaa(.(X, Xs), Y, L) U8_aga(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aga(.(X, Xs), Y, L) U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_gaaaa(S, Choice, Old, L3, Modulus) U17_agaa(Unused, List, Choice, Rest, Mod, check_out_gaaaa(List, Choice, Unused, U3, Mod)) -> U18_agaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) pds1_in_aaaa([], X6, [], X7) -> pds1_out_aaaa([], X6, [], X7) pds1_in_aaaa(Unused, List, .(Choice, Rest), Mod) -> U16_aaaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) U16_aaaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_aaaa(Unused, List, Choice, Rest, Mod, check_in_aaaaa(List, Choice, Unused, U3, Mod)) check_in_aaaaa([], X4, L, L, X5) -> check_out_aaaaa([], X4, L, L, X5) check_in_aaaaa(S, Choice, Old, L3, Modulus) -> U12_aaaaa(S, Choice, Old, L3, Modulus, dif_in_aaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_aaaaa([], X2, X3, [], []) -> dif_out_aaaaa([], X2, X3, [], []) dif_in_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_aaaaa(Ss, Val, Mod, Ds, D2s)) U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_aaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_aaaaa(S, Choice, Old, L3, Modulus, dif_out_aaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_aaaaa(S, Choice, Old, L3, Modulus) U17_aaaa(Unused, List, Choice, Rest, Mod, check_out_aaaaa(List, Choice, Unused, U3, Mod)) -> U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_aaaa(Unused, List, .(Choice, Rest), Mod) U18_agaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_agaa(Unused, List, .(Choice, Rest), Mod) U21_ga(Order, Ans, pds1_out_agaa(List, .(0, []), Ans, N)) -> pds_out_ga(Order, .(0, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U19_ga(x1, x2, x3) = U19_ga(x1, x3) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag(x2) +(x1, x2) = +(x1, x2) *(x1, x2) = *(x1, x2) 1 = 1 U20_ga(x1, x2, x3, x4) = U20_ga(x1, x4) iota_in_aa(x1, x2) = iota_in_aa U2_aa(x1, x2, x3) = U2_aa(x3) iota1_in_gaa(x1, x2, x3) = iota1_in_gaa(x1) iota1_out_gaa(x1, x2, x3) = iota1_out_gaa(x1) U3_gaa(x1, x2, x3, x4) = U3_gaa(x1, x4) U4_gaa(x1, x2, x3, x4) = U4_gaa(x1, x4) iota1_in_aaa(x1, x2, x3) = iota1_in_aaa iota1_out_aaa(x1, x2, x3) = iota1_out_aaa U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) is_in_aa(x1, x2) = is_in_aa is_out_aa(x1, x2) = is_out_aa U4_aaa(x1, x2, x3, x4) = U4_aaa(x4) 0 = 0 iota_out_aa(x1, x2) = iota_out_aa U21_ga(x1, x2, x3) = U21_ga(x1, x3) pds1_in_agaa(x1, x2, x3, x4) = pds1_in_agaa(x2) pds1_out_agaa(x1, x2, x3, x4) = pds1_out_agaa(x2) U16_agaa(x1, x2, x3, x4, x5, x6) = U16_agaa(x2, x6) member_in_aa(x1, x2) = member_in_aa member_out_aa(x1, x2) = member_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U17_agaa(x1, x2, x3, x4, x5, x6) = U17_agaa(x2, x6) check_in_gaaaa(x1, x2, x3, x4, x5) = check_in_gaaaa(x1) [] = [] check_out_gaaaa(x1, x2, x3, x4, x5) = check_out_gaaaa(x1) U12_gaaaa(x1, x2, x3, x4, x5, x6) = U12_gaaaa(x1, x6) dif_in_gaaaa(x1, x2, x3, x4, x5) = dif_in_gaaaa(x1) dif_out_gaaaa(x1, x2, x3, x4, x5) = dif_out_gaaaa(x1) .(x1, x2) = .(x1, x2) U5_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_gaaaa(x1, x2, x9) U6_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gaaaa(x1, x2, x9) U7_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_gaaaa(x1, x2, x9) U13_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_gaaaa(x1, x7) mergedelete_in_aaa(x1, x2, x3) = mergedelete_in_aaa mergedelete_out_aaa(x1, x2, x3) = mergedelete_out_aaa U9_aaa(x1, x2, x3, x4, x5) = U9_aaa(x5) U10_aaa(x1, x2, x3, x4, x5, x6) = U10_aaa(x6) >_in_aa(x1, x2) = >_in_aa >_out_aa(x1, x2) = >_out_aa U11_aaa(x1, x2, x3, x4, x5, x6) = U11_aaa(x6) U14_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_gaaaa(x1, x7) rev_in_aga(x1, x2, x3) = rev_in_aga(x2) rev_out_aga(x1, x2, x3) = rev_out_aga(x2) U8_aga(x1, x2, x3, x4, x5) = U8_aga(x3, x5) rev_in_aaa(x1, x2, x3) = rev_in_aaa rev_out_aaa(x1, x2, x3) = rev_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U15_gaaaa(x1, x2, x3, x4, x5, x6) = U15_gaaaa(x1, x6) U18_agaa(x1, x2, x3, x4, x5, x6) = U18_agaa(x2, x6) pds1_in_aaaa(x1, x2, x3, x4) = pds1_in_aaaa pds1_out_aaaa(x1, x2, x3, x4) = pds1_out_aaaa U16_aaaa(x1, x2, x3, x4, x5, x6) = U16_aaaa(x6) U17_aaaa(x1, x2, x3, x4, x5, x6) = U17_aaaa(x6) check_in_aaaaa(x1, x2, x3, x4, x5) = check_in_aaaaa check_out_aaaaa(x1, x2, x3, x4, x5) = check_out_aaaaa U12_aaaaa(x1, x2, x3, x4, x5, x6) = U12_aaaaa(x6) dif_in_aaaaa(x1, x2, x3, x4, x5) = dif_in_aaaaa dif_out_aaaaa(x1, x2, x3, x4, x5) = dif_out_aaaaa U5_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_aaaaa(x9) U6_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_aaaaa(x9) U7_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_aaaaa(x9) U13_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_aaaaa(x7) U14_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_aaaaa(x7) U15_aaaaa(x1, x2, x3, x4, x5, x6) = U15_aaaaa(x6) U18_aaaa(x1, x2, x3, x4, x5, x6) = U18_aaaa(x6) pds_out_ga(x1, x2) = pds_out_ga(x1) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (293) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: pds_in_ga(Order, .(0, Ans)) -> U19_ga(Order, Ans, is_in_ag(N, +(*(Order, +(Order, 1)), 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U19_ga(Order, Ans, is_out_ag(N, +(*(Order, +(Order, 1)), 1))) -> U20_ga(Order, Ans, N, iota_in_aa(N, .(0, List))) iota_in_aa(N, List) -> U2_aa(N, List, iota1_in_gaa(0, N, List)) iota1_in_gaa(K, K, []) -> iota1_out_gaa(K, K, []) iota1_in_gaa(K, N, .(K, List)) -> U3_gaa(K, N, List, is_in_ag(K1, +(K, 1))) U3_gaa(K, N, List, is_out_ag(K1, +(K, 1))) -> U4_gaa(K, N, List, iota1_in_aaa(K1, N, List)) iota1_in_aaa(K, K, []) -> iota1_out_aaa(K, K, []) iota1_in_aaa(K, N, .(K, List)) -> U3_aaa(K, N, List, is_in_aa(K1, +(K, 1))) is_in_aa(X0, X1) -> is_out_aa(X0, X1) U3_aaa(K, N, List, is_out_aa(K1, +(K, 1))) -> U4_aaa(K, N, List, iota1_in_aaa(K1, N, List)) U4_aaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_aaa(K, N, .(K, List)) U4_gaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_gaa(K, N, .(K, List)) U2_aa(N, List, iota1_out_gaa(0, N, List)) -> iota_out_aa(N, List) U20_ga(Order, Ans, N, iota_out_aa(N, .(0, List))) -> U21_ga(Order, Ans, pds1_in_agaa(List, .(0, []), Ans, N)) pds1_in_agaa([], X6, [], X7) -> pds1_out_agaa([], X6, [], X7) pds1_in_agaa(Unused, List, .(Choice, Rest), Mod) -> U16_agaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) member_in_aa(X, .(X, Xs)) -> member_out_aa(X, .(X, Xs)) member_in_aa(X, .(X1, Xs)) -> U1_aa(X, X1, Xs, member_in_aa(X, Xs)) U1_aa(X, X1, Xs, member_out_aa(X, Xs)) -> member_out_aa(X, .(X1, Xs)) U16_agaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_agaa(Unused, List, Choice, Rest, Mod, check_in_gaaaa(List, Choice, Unused, U3, Mod)) check_in_gaaaa([], X4, L, L, X5) -> check_out_gaaaa([], X4, L, L, X5) check_in_gaaaa(S, Choice, Old, L3, Modulus) -> U12_gaaaa(S, Choice, Old, L3, Modulus, dif_in_gaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_gaaaa([], X2, X3, [], []) -> dif_out_gaaaa([], X2, X3, [], []) dif_in_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gaaaa(Ss, Val, Mod, Ds, D2s)) U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_gaaaa(S, Choice, Old, L3, Modulus, dif_out_gaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) mergedelete_in_aaa([], L, L) -> mergedelete_out_aaa([], L, L) mergedelete_in_aaa(.(D, Ds), .(D, R), L2) -> U9_aaa(D, Ds, R, L2, mergedelete_in_aaa(Ds, R, L2)) mergedelete_in_aaa(.(D, Ds), .(X, R), .(X, L2)) -> U10_aaa(D, Ds, X, R, L2, >_in_aa(D, X)) >_in_aa(X0, X1) -> >_out_aa(X0, X1) U10_aaa(D, Ds, X, R, L2, >_out_aa(D, X)) -> U11_aaa(D, Ds, X, R, L2, mergedelete_in_aaa(.(D, Ds), R, L2)) U11_aaa(D, Ds, X, R, L2, mergedelete_out_aaa(.(D, Ds), R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(X, R), .(X, L2)) U9_aaa(D, Ds, R, L2, mergedelete_out_aaa(Ds, R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(D, R), L2) U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) rev_in_aga([], L, L) -> rev_out_aga([], L, L) rev_in_aga(.(X, Xs), Y, L) -> U8_aga(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) rev_in_aaa([], L, L) -> rev_out_aaa([], L, L) rev_in_aaa(.(X, Xs), Y, L) -> U8_aaa(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) U8_aaa(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aaa(.(X, Xs), Y, L) U8_aga(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aga(.(X, Xs), Y, L) U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_gaaaa(S, Choice, Old, L3, Modulus) U17_agaa(Unused, List, Choice, Rest, Mod, check_out_gaaaa(List, Choice, Unused, U3, Mod)) -> U18_agaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) pds1_in_aaaa([], X6, [], X7) -> pds1_out_aaaa([], X6, [], X7) pds1_in_aaaa(Unused, List, .(Choice, Rest), Mod) -> U16_aaaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) U16_aaaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_aaaa(Unused, List, Choice, Rest, Mod, check_in_aaaaa(List, Choice, Unused, U3, Mod)) check_in_aaaaa([], X4, L, L, X5) -> check_out_aaaaa([], X4, L, L, X5) check_in_aaaaa(S, Choice, Old, L3, Modulus) -> U12_aaaaa(S, Choice, Old, L3, Modulus, dif_in_aaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_aaaaa([], X2, X3, [], []) -> dif_out_aaaaa([], X2, X3, [], []) dif_in_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_aaaaa(Ss, Val, Mod, Ds, D2s)) U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_aaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_aaaaa(S, Choice, Old, L3, Modulus, dif_out_aaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_aaaaa(S, Choice, Old, L3, Modulus) U17_aaaa(Unused, List, Choice, Rest, Mod, check_out_aaaaa(List, Choice, Unused, U3, Mod)) -> U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_aaaa(Unused, List, .(Choice, Rest), Mod) U18_agaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_agaa(Unused, List, .(Choice, Rest), Mod) U21_ga(Order, Ans, pds1_out_agaa(List, .(0, []), Ans, N)) -> pds_out_ga(Order, .(0, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U19_ga(x1, x2, x3) = U19_ga(x1, x3) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag(x2) +(x1, x2) = +(x1, x2) *(x1, x2) = *(x1, x2) 1 = 1 U20_ga(x1, x2, x3, x4) = U20_ga(x1, x4) iota_in_aa(x1, x2) = iota_in_aa U2_aa(x1, x2, x3) = U2_aa(x3) iota1_in_gaa(x1, x2, x3) = iota1_in_gaa(x1) iota1_out_gaa(x1, x2, x3) = iota1_out_gaa(x1) U3_gaa(x1, x2, x3, x4) = U3_gaa(x1, x4) U4_gaa(x1, x2, x3, x4) = U4_gaa(x1, x4) iota1_in_aaa(x1, x2, x3) = iota1_in_aaa iota1_out_aaa(x1, x2, x3) = iota1_out_aaa U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) is_in_aa(x1, x2) = is_in_aa is_out_aa(x1, x2) = is_out_aa U4_aaa(x1, x2, x3, x4) = U4_aaa(x4) 0 = 0 iota_out_aa(x1, x2) = iota_out_aa U21_ga(x1, x2, x3) = U21_ga(x1, x3) pds1_in_agaa(x1, x2, x3, x4) = pds1_in_agaa(x2) pds1_out_agaa(x1, x2, x3, x4) = pds1_out_agaa(x2) U16_agaa(x1, x2, x3, x4, x5, x6) = U16_agaa(x2, x6) member_in_aa(x1, x2) = member_in_aa member_out_aa(x1, x2) = member_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U17_agaa(x1, x2, x3, x4, x5, x6) = U17_agaa(x2, x6) check_in_gaaaa(x1, x2, x3, x4, x5) = check_in_gaaaa(x1) [] = [] check_out_gaaaa(x1, x2, x3, x4, x5) = check_out_gaaaa(x1) U12_gaaaa(x1, x2, x3, x4, x5, x6) = U12_gaaaa(x1, x6) dif_in_gaaaa(x1, x2, x3, x4, x5) = dif_in_gaaaa(x1) dif_out_gaaaa(x1, x2, x3, x4, x5) = dif_out_gaaaa(x1) .(x1, x2) = .(x1, x2) U5_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_gaaaa(x1, x2, x9) U6_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gaaaa(x1, x2, x9) U7_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_gaaaa(x1, x2, x9) U13_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_gaaaa(x1, x7) mergedelete_in_aaa(x1, x2, x3) = mergedelete_in_aaa mergedelete_out_aaa(x1, x2, x3) = mergedelete_out_aaa U9_aaa(x1, x2, x3, x4, x5) = U9_aaa(x5) U10_aaa(x1, x2, x3, x4, x5, x6) = U10_aaa(x6) >_in_aa(x1, x2) = >_in_aa >_out_aa(x1, x2) = >_out_aa U11_aaa(x1, x2, x3, x4, x5, x6) = U11_aaa(x6) U14_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_gaaaa(x1, x7) rev_in_aga(x1, x2, x3) = rev_in_aga(x2) rev_out_aga(x1, x2, x3) = rev_out_aga(x2) U8_aga(x1, x2, x3, x4, x5) = U8_aga(x3, x5) rev_in_aaa(x1, x2, x3) = rev_in_aaa rev_out_aaa(x1, x2, x3) = rev_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U15_gaaaa(x1, x2, x3, x4, x5, x6) = U15_gaaaa(x1, x6) U18_agaa(x1, x2, x3, x4, x5, x6) = U18_agaa(x2, x6) pds1_in_aaaa(x1, x2, x3, x4) = pds1_in_aaaa pds1_out_aaaa(x1, x2, x3, x4) = pds1_out_aaaa U16_aaaa(x1, x2, x3, x4, x5, x6) = U16_aaaa(x6) U17_aaaa(x1, x2, x3, x4, x5, x6) = U17_aaaa(x6) check_in_aaaaa(x1, x2, x3, x4, x5) = check_in_aaaaa check_out_aaaaa(x1, x2, x3, x4, x5) = check_out_aaaaa U12_aaaaa(x1, x2, x3, x4, x5, x6) = U12_aaaaa(x6) dif_in_aaaaa(x1, x2, x3, x4, x5) = dif_in_aaaaa dif_out_aaaaa(x1, x2, x3, x4, x5) = dif_out_aaaaa U5_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_aaaaa(x9) U6_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_aaaaa(x9) U7_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_aaaaa(x9) U13_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_aaaaa(x7) U14_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_aaaaa(x7) U15_aaaaa(x1, x2, x3, x4, x5, x6) = U15_aaaaa(x6) U18_aaaa(x1, x2, x3, x4, x5, x6) = U18_aaaa(x6) pds_out_ga(x1, x2) = pds_out_ga(x1) ---------------------------------------- (294) 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: PDS_IN_GA(Order, .(0, Ans)) -> U19_GA(Order, Ans, is_in_ag(N, +(*(Order, +(Order, 1)), 1))) PDS_IN_GA(Order, .(0, Ans)) -> IS_IN_AG(N, +(*(Order, +(Order, 1)), 1)) U19_GA(Order, Ans, is_out_ag(N, +(*(Order, +(Order, 1)), 1))) -> U20_GA(Order, Ans, N, iota_in_aa(N, .(0, List))) U19_GA(Order, Ans, is_out_ag(N, +(*(Order, +(Order, 1)), 1))) -> IOTA_IN_AA(N, .(0, List)) IOTA_IN_AA(N, List) -> U2_AA(N, List, iota1_in_gaa(0, N, List)) IOTA_IN_AA(N, List) -> IOTA1_IN_GAA(0, N, List) IOTA1_IN_GAA(K, N, .(K, List)) -> U3_GAA(K, N, List, is_in_ag(K1, +(K, 1))) IOTA1_IN_GAA(K, N, .(K, List)) -> IS_IN_AG(K1, +(K, 1)) U3_GAA(K, N, List, is_out_ag(K1, +(K, 1))) -> U4_GAA(K, N, List, iota1_in_aaa(K1, N, List)) U3_GAA(K, N, List, is_out_ag(K1, +(K, 1))) -> IOTA1_IN_AAA(K1, N, List) IOTA1_IN_AAA(K, N, .(K, List)) -> U3_AAA(K, N, List, is_in_aa(K1, +(K, 1))) IOTA1_IN_AAA(K, N, .(K, List)) -> IS_IN_AA(K1, +(K, 1)) U3_AAA(K, N, List, is_out_aa(K1, +(K, 1))) -> U4_AAA(K, N, List, iota1_in_aaa(K1, N, List)) U3_AAA(K, N, List, is_out_aa(K1, +(K, 1))) -> IOTA1_IN_AAA(K1, N, List) U20_GA(Order, Ans, N, iota_out_aa(N, .(0, List))) -> U21_GA(Order, Ans, pds1_in_agaa(List, .(0, []), Ans, N)) U20_GA(Order, Ans, N, iota_out_aa(N, .(0, List))) -> PDS1_IN_AGAA(List, .(0, []), Ans, N) PDS1_IN_AGAA(Unused, List, .(Choice, Rest), Mod) -> U16_AGAA(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) PDS1_IN_AGAA(Unused, List, .(Choice, Rest), Mod) -> MEMBER_IN_AA(Choice, Unused) MEMBER_IN_AA(X, .(X1, Xs)) -> U1_AA(X, X1, Xs, member_in_aa(X, Xs)) MEMBER_IN_AA(X, .(X1, Xs)) -> MEMBER_IN_AA(X, Xs) U16_AGAA(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_AGAA(Unused, List, Choice, Rest, Mod, check_in_gaaaa(List, Choice, Unused, U3, Mod)) U16_AGAA(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> CHECK_IN_GAAAA(List, Choice, Unused, U3, Mod) CHECK_IN_GAAAA(S, Choice, Old, L3, Modulus) -> U12_GAAAA(S, Choice, Old, L3, Modulus, dif_in_gaaaa(S, Choice, Modulus, Ds, Dds)) CHECK_IN_GAAAA(S, Choice, Old, L3, Modulus) -> DIF_IN_GAAAA(S, Choice, Modulus, Ds, Dds) DIF_IN_GAAAA(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) DIF_IN_GAAAA(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> IS_IN_AA(D, -(Val, S)) U5_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U5_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> IS_IN_AA(D2, -(Mod, D)) U6_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gaaaa(Ss, Val, Mod, Ds, D2s)) U6_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> DIF_IN_GAAAA(Ss, Val, Mod, Ds, D2s) U12_GAAAA(S, Choice, Old, L3, Modulus, dif_out_gaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_GAAAA(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) U12_GAAAA(S, Choice, Old, L3, Modulus, dif_out_gaaaa(S, Choice, Modulus, Ds, Dds)) -> MERGEDELETE_IN_AAA(Ds, Old, L2) MERGEDELETE_IN_AAA(.(D, Ds), .(D, R), L2) -> U9_AAA(D, Ds, R, L2, mergedelete_in_aaa(Ds, R, L2)) MERGEDELETE_IN_AAA(.(D, Ds), .(D, R), L2) -> MERGEDELETE_IN_AAA(Ds, R, L2) MERGEDELETE_IN_AAA(.(D, Ds), .(X, R), .(X, L2)) -> U10_AAA(D, Ds, X, R, L2, >_in_aa(D, X)) MERGEDELETE_IN_AAA(.(D, Ds), .(X, R), .(X, L2)) -> >_IN_AA(D, X) U10_AAA(D, Ds, X, R, L2, >_out_aa(D, X)) -> U11_AAA(D, Ds, X, R, L2, mergedelete_in_aaa(.(D, Ds), R, L2)) U10_AAA(D, Ds, X, R, L2, >_out_aa(D, X)) -> MERGEDELETE_IN_AAA(.(D, Ds), R, L2) U13_GAAAA(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_GAAAA(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) U13_GAAAA(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> REV_IN_AGA(Dds, [], Rds) REV_IN_AGA(.(X, Xs), Y, L) -> U8_AGA(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) REV_IN_AGA(.(X, Xs), Y, L) -> REV_IN_AAA(Xs, .(X, Y), L) REV_IN_AAA(.(X, Xs), Y, L) -> U8_AAA(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) REV_IN_AAA(.(X, Xs), Y, L) -> REV_IN_AAA(Xs, .(X, Y), L) U14_GAAAA(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_GAAAA(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U14_GAAAA(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> MERGEDELETE_IN_AAA(Rds, L2, L3) U17_AGAA(Unused, List, Choice, Rest, Mod, check_out_gaaaa(List, Choice, Unused, U3, Mod)) -> U18_AGAA(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) U17_AGAA(Unused, List, Choice, Rest, Mod, check_out_gaaaa(List, Choice, Unused, U3, Mod)) -> PDS1_IN_AAAA(U3, .(Choice, List), Rest, Mod) PDS1_IN_AAAA(Unused, List, .(Choice, Rest), Mod) -> U16_AAAA(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) PDS1_IN_AAAA(Unused, List, .(Choice, Rest), Mod) -> MEMBER_IN_AA(Choice, Unused) U16_AAAA(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_AAAA(Unused, List, Choice, Rest, Mod, check_in_aaaaa(List, Choice, Unused, U3, Mod)) U16_AAAA(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> CHECK_IN_AAAAA(List, Choice, Unused, U3, Mod) CHECK_IN_AAAAA(S, Choice, Old, L3, Modulus) -> U12_AAAAA(S, Choice, Old, L3, Modulus, dif_in_aaaaa(S, Choice, Modulus, Ds, Dds)) CHECK_IN_AAAAA(S, Choice, Old, L3, Modulus) -> DIF_IN_AAAAA(S, Choice, Modulus, Ds, Dds) DIF_IN_AAAAA(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) DIF_IN_AAAAA(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> IS_IN_AA(D, -(Val, S)) U5_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U5_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> IS_IN_AA(D2, -(Mod, D)) U6_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_aaaaa(Ss, Val, Mod, Ds, D2s)) U6_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> DIF_IN_AAAAA(Ss, Val, Mod, Ds, D2s) U12_AAAAA(S, Choice, Old, L3, Modulus, dif_out_aaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_AAAAA(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) U12_AAAAA(S, Choice, Old, L3, Modulus, dif_out_aaaaa(S, Choice, Modulus, Ds, Dds)) -> MERGEDELETE_IN_AAA(Ds, Old, L2) U13_AAAAA(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_AAAAA(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) U13_AAAAA(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> REV_IN_AGA(Dds, [], Rds) U14_AAAAA(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_AAAAA(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U14_AAAAA(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> MERGEDELETE_IN_AAA(Rds, L2, L3) U17_AAAA(Unused, List, Choice, Rest, Mod, check_out_aaaaa(List, Choice, Unused, U3, Mod)) -> U18_AAAA(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) U17_AAAA(Unused, List, Choice, Rest, Mod, check_out_aaaaa(List, Choice, Unused, U3, Mod)) -> PDS1_IN_AAAA(U3, .(Choice, List), Rest, Mod) The TRS R consists of the following rules: pds_in_ga(Order, .(0, Ans)) -> U19_ga(Order, Ans, is_in_ag(N, +(*(Order, +(Order, 1)), 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U19_ga(Order, Ans, is_out_ag(N, +(*(Order, +(Order, 1)), 1))) -> U20_ga(Order, Ans, N, iota_in_aa(N, .(0, List))) iota_in_aa(N, List) -> U2_aa(N, List, iota1_in_gaa(0, N, List)) iota1_in_gaa(K, K, []) -> iota1_out_gaa(K, K, []) iota1_in_gaa(K, N, .(K, List)) -> U3_gaa(K, N, List, is_in_ag(K1, +(K, 1))) U3_gaa(K, N, List, is_out_ag(K1, +(K, 1))) -> U4_gaa(K, N, List, iota1_in_aaa(K1, N, List)) iota1_in_aaa(K, K, []) -> iota1_out_aaa(K, K, []) iota1_in_aaa(K, N, .(K, List)) -> U3_aaa(K, N, List, is_in_aa(K1, +(K, 1))) is_in_aa(X0, X1) -> is_out_aa(X0, X1) U3_aaa(K, N, List, is_out_aa(K1, +(K, 1))) -> U4_aaa(K, N, List, iota1_in_aaa(K1, N, List)) U4_aaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_aaa(K, N, .(K, List)) U4_gaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_gaa(K, N, .(K, List)) U2_aa(N, List, iota1_out_gaa(0, N, List)) -> iota_out_aa(N, List) U20_ga(Order, Ans, N, iota_out_aa(N, .(0, List))) -> U21_ga(Order, Ans, pds1_in_agaa(List, .(0, []), Ans, N)) pds1_in_agaa([], X6, [], X7) -> pds1_out_agaa([], X6, [], X7) pds1_in_agaa(Unused, List, .(Choice, Rest), Mod) -> U16_agaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) member_in_aa(X, .(X, Xs)) -> member_out_aa(X, .(X, Xs)) member_in_aa(X, .(X1, Xs)) -> U1_aa(X, X1, Xs, member_in_aa(X, Xs)) U1_aa(X, X1, Xs, member_out_aa(X, Xs)) -> member_out_aa(X, .(X1, Xs)) U16_agaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_agaa(Unused, List, Choice, Rest, Mod, check_in_gaaaa(List, Choice, Unused, U3, Mod)) check_in_gaaaa([], X4, L, L, X5) -> check_out_gaaaa([], X4, L, L, X5) check_in_gaaaa(S, Choice, Old, L3, Modulus) -> U12_gaaaa(S, Choice, Old, L3, Modulus, dif_in_gaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_gaaaa([], X2, X3, [], []) -> dif_out_gaaaa([], X2, X3, [], []) dif_in_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gaaaa(Ss, Val, Mod, Ds, D2s)) U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_gaaaa(S, Choice, Old, L3, Modulus, dif_out_gaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) mergedelete_in_aaa([], L, L) -> mergedelete_out_aaa([], L, L) mergedelete_in_aaa(.(D, Ds), .(D, R), L2) -> U9_aaa(D, Ds, R, L2, mergedelete_in_aaa(Ds, R, L2)) mergedelete_in_aaa(.(D, Ds), .(X, R), .(X, L2)) -> U10_aaa(D, Ds, X, R, L2, >_in_aa(D, X)) >_in_aa(X0, X1) -> >_out_aa(X0, X1) U10_aaa(D, Ds, X, R, L2, >_out_aa(D, X)) -> U11_aaa(D, Ds, X, R, L2, mergedelete_in_aaa(.(D, Ds), R, L2)) U11_aaa(D, Ds, X, R, L2, mergedelete_out_aaa(.(D, Ds), R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(X, R), .(X, L2)) U9_aaa(D, Ds, R, L2, mergedelete_out_aaa(Ds, R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(D, R), L2) U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) rev_in_aga([], L, L) -> rev_out_aga([], L, L) rev_in_aga(.(X, Xs), Y, L) -> U8_aga(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) rev_in_aaa([], L, L) -> rev_out_aaa([], L, L) rev_in_aaa(.(X, Xs), Y, L) -> U8_aaa(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) U8_aaa(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aaa(.(X, Xs), Y, L) U8_aga(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aga(.(X, Xs), Y, L) U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_gaaaa(S, Choice, Old, L3, Modulus) U17_agaa(Unused, List, Choice, Rest, Mod, check_out_gaaaa(List, Choice, Unused, U3, Mod)) -> U18_agaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) pds1_in_aaaa([], X6, [], X7) -> pds1_out_aaaa([], X6, [], X7) pds1_in_aaaa(Unused, List, .(Choice, Rest), Mod) -> U16_aaaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) U16_aaaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_aaaa(Unused, List, Choice, Rest, Mod, check_in_aaaaa(List, Choice, Unused, U3, Mod)) check_in_aaaaa([], X4, L, L, X5) -> check_out_aaaaa([], X4, L, L, X5) check_in_aaaaa(S, Choice, Old, L3, Modulus) -> U12_aaaaa(S, Choice, Old, L3, Modulus, dif_in_aaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_aaaaa([], X2, X3, [], []) -> dif_out_aaaaa([], X2, X3, [], []) dif_in_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_aaaaa(Ss, Val, Mod, Ds, D2s)) U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_aaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_aaaaa(S, Choice, Old, L3, Modulus, dif_out_aaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_aaaaa(S, Choice, Old, L3, Modulus) U17_aaaa(Unused, List, Choice, Rest, Mod, check_out_aaaaa(List, Choice, Unused, U3, Mod)) -> U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_aaaa(Unused, List, .(Choice, Rest), Mod) U18_agaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_agaa(Unused, List, .(Choice, Rest), Mod) U21_ga(Order, Ans, pds1_out_agaa(List, .(0, []), Ans, N)) -> pds_out_ga(Order, .(0, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U19_ga(x1, x2, x3) = U19_ga(x1, x3) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag(x2) +(x1, x2) = +(x1, x2) *(x1, x2) = *(x1, x2) 1 = 1 U20_ga(x1, x2, x3, x4) = U20_ga(x1, x4) iota_in_aa(x1, x2) = iota_in_aa U2_aa(x1, x2, x3) = U2_aa(x3) iota1_in_gaa(x1, x2, x3) = iota1_in_gaa(x1) iota1_out_gaa(x1, x2, x3) = iota1_out_gaa(x1) U3_gaa(x1, x2, x3, x4) = U3_gaa(x1, x4) U4_gaa(x1, x2, x3, x4) = U4_gaa(x1, x4) iota1_in_aaa(x1, x2, x3) = iota1_in_aaa iota1_out_aaa(x1, x2, x3) = iota1_out_aaa U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) is_in_aa(x1, x2) = is_in_aa is_out_aa(x1, x2) = is_out_aa U4_aaa(x1, x2, x3, x4) = U4_aaa(x4) 0 = 0 iota_out_aa(x1, x2) = iota_out_aa U21_ga(x1, x2, x3) = U21_ga(x1, x3) pds1_in_agaa(x1, x2, x3, x4) = pds1_in_agaa(x2) pds1_out_agaa(x1, x2, x3, x4) = pds1_out_agaa(x2) U16_agaa(x1, x2, x3, x4, x5, x6) = U16_agaa(x2, x6) member_in_aa(x1, x2) = member_in_aa member_out_aa(x1, x2) = member_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U17_agaa(x1, x2, x3, x4, x5, x6) = U17_agaa(x2, x6) check_in_gaaaa(x1, x2, x3, x4, x5) = check_in_gaaaa(x1) [] = [] check_out_gaaaa(x1, x2, x3, x4, x5) = check_out_gaaaa(x1) U12_gaaaa(x1, x2, x3, x4, x5, x6) = U12_gaaaa(x1, x6) dif_in_gaaaa(x1, x2, x3, x4, x5) = dif_in_gaaaa(x1) dif_out_gaaaa(x1, x2, x3, x4, x5) = dif_out_gaaaa(x1) .(x1, x2) = .(x1, x2) U5_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_gaaaa(x1, x2, x9) U6_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gaaaa(x1, x2, x9) U7_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_gaaaa(x1, x2, x9) U13_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_gaaaa(x1, x7) mergedelete_in_aaa(x1, x2, x3) = mergedelete_in_aaa mergedelete_out_aaa(x1, x2, x3) = mergedelete_out_aaa U9_aaa(x1, x2, x3, x4, x5) = U9_aaa(x5) U10_aaa(x1, x2, x3, x4, x5, x6) = U10_aaa(x6) >_in_aa(x1, x2) = >_in_aa >_out_aa(x1, x2) = >_out_aa U11_aaa(x1, x2, x3, x4, x5, x6) = U11_aaa(x6) U14_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_gaaaa(x1, x7) rev_in_aga(x1, x2, x3) = rev_in_aga(x2) rev_out_aga(x1, x2, x3) = rev_out_aga(x2) U8_aga(x1, x2, x3, x4, x5) = U8_aga(x3, x5) rev_in_aaa(x1, x2, x3) = rev_in_aaa rev_out_aaa(x1, x2, x3) = rev_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U15_gaaaa(x1, x2, x3, x4, x5, x6) = U15_gaaaa(x1, x6) U18_agaa(x1, x2, x3, x4, x5, x6) = U18_agaa(x2, x6) pds1_in_aaaa(x1, x2, x3, x4) = pds1_in_aaaa pds1_out_aaaa(x1, x2, x3, x4) = pds1_out_aaaa U16_aaaa(x1, x2, x3, x4, x5, x6) = U16_aaaa(x6) U17_aaaa(x1, x2, x3, x4, x5, x6) = U17_aaaa(x6) check_in_aaaaa(x1, x2, x3, x4, x5) = check_in_aaaaa check_out_aaaaa(x1, x2, x3, x4, x5) = check_out_aaaaa U12_aaaaa(x1, x2, x3, x4, x5, x6) = U12_aaaaa(x6) dif_in_aaaaa(x1, x2, x3, x4, x5) = dif_in_aaaaa dif_out_aaaaa(x1, x2, x3, x4, x5) = dif_out_aaaaa U5_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_aaaaa(x9) U6_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_aaaaa(x9) U7_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_aaaaa(x9) U13_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_aaaaa(x7) U14_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_aaaaa(x7) U15_aaaaa(x1, x2, x3, x4, x5, x6) = U15_aaaaa(x6) U18_aaaa(x1, x2, x3, x4, x5, x6) = U18_aaaa(x6) pds_out_ga(x1, x2) = pds_out_ga(x1) PDS_IN_GA(x1, x2) = PDS_IN_GA(x1) U19_GA(x1, x2, x3) = U19_GA(x1, x3) IS_IN_AG(x1, x2) = IS_IN_AG(x2) U20_GA(x1, x2, x3, x4) = U20_GA(x1, x4) IOTA_IN_AA(x1, x2) = IOTA_IN_AA U2_AA(x1, x2, x3) = U2_AA(x3) IOTA1_IN_GAA(x1, x2, x3) = IOTA1_IN_GAA(x1) U3_GAA(x1, x2, x3, x4) = U3_GAA(x1, x4) U4_GAA(x1, x2, x3, x4) = U4_GAA(x1, x4) IOTA1_IN_AAA(x1, x2, x3) = IOTA1_IN_AAA U3_AAA(x1, x2, x3, x4) = U3_AAA(x4) IS_IN_AA(x1, x2) = IS_IN_AA U4_AAA(x1, x2, x3, x4) = U4_AAA(x4) U21_GA(x1, x2, x3) = U21_GA(x1, x3) PDS1_IN_AGAA(x1, x2, x3, x4) = PDS1_IN_AGAA(x2) U16_AGAA(x1, x2, x3, x4, x5, x6) = U16_AGAA(x2, x6) MEMBER_IN_AA(x1, x2) = MEMBER_IN_AA U1_AA(x1, x2, x3, x4) = U1_AA(x4) U17_AGAA(x1, x2, x3, x4, x5, x6) = U17_AGAA(x2, x6) CHECK_IN_GAAAA(x1, x2, x3, x4, x5) = CHECK_IN_GAAAA(x1) U12_GAAAA(x1, x2, x3, x4, x5, x6) = U12_GAAAA(x1, x6) DIF_IN_GAAAA(x1, x2, x3, x4, x5) = DIF_IN_GAAAA(x1) U5_GAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_GAAAA(x1, x2, x9) U6_GAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_GAAAA(x1, x2, x9) U7_GAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_GAAAA(x1, x2, x9) U13_GAAAA(x1, x2, x3, x4, x5, x6, x7) = U13_GAAAA(x1, x7) MERGEDELETE_IN_AAA(x1, x2, x3) = MERGEDELETE_IN_AAA U9_AAA(x1, x2, x3, x4, x5) = U9_AAA(x5) U10_AAA(x1, x2, x3, x4, x5, x6) = U10_AAA(x6) >_IN_AA(x1, x2) = >_IN_AA U11_AAA(x1, x2, x3, x4, x5, x6) = U11_AAA(x6) U14_GAAAA(x1, x2, x3, x4, x5, x6, x7) = U14_GAAAA(x1, x7) REV_IN_AGA(x1, x2, x3) = REV_IN_AGA(x2) U8_AGA(x1, x2, x3, x4, x5) = U8_AGA(x3, x5) REV_IN_AAA(x1, x2, x3) = REV_IN_AAA U8_AAA(x1, x2, x3, x4, x5) = U8_AAA(x5) U15_GAAAA(x1, x2, x3, x4, x5, x6) = U15_GAAAA(x1, x6) U18_AGAA(x1, x2, x3, x4, x5, x6) = U18_AGAA(x2, x6) PDS1_IN_AAAA(x1, x2, x3, x4) = PDS1_IN_AAAA U16_AAAA(x1, x2, x3, x4, x5, x6) = U16_AAAA(x6) U17_AAAA(x1, x2, x3, x4, x5, x6) = U17_AAAA(x6) CHECK_IN_AAAAA(x1, x2, x3, x4, x5) = CHECK_IN_AAAAA U12_AAAAA(x1, x2, x3, x4, x5, x6) = U12_AAAAA(x6) DIF_IN_AAAAA(x1, x2, x3, x4, x5) = DIF_IN_AAAAA U5_AAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_AAAAA(x9) U6_AAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_AAAAA(x9) U7_AAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_AAAAA(x9) U13_AAAAA(x1, x2, x3, x4, x5, x6, x7) = U13_AAAAA(x7) U14_AAAAA(x1, x2, x3, x4, x5, x6, x7) = U14_AAAAA(x7) U15_AAAAA(x1, x2, x3, x4, x5, x6) = U15_AAAAA(x6) U18_AAAA(x1, x2, x3, x4, x5, x6) = U18_AAAA(x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (295) Obligation: Pi DP problem: The TRS P consists of the following rules: PDS_IN_GA(Order, .(0, Ans)) -> U19_GA(Order, Ans, is_in_ag(N, +(*(Order, +(Order, 1)), 1))) PDS_IN_GA(Order, .(0, Ans)) -> IS_IN_AG(N, +(*(Order, +(Order, 1)), 1)) U19_GA(Order, Ans, is_out_ag(N, +(*(Order, +(Order, 1)), 1))) -> U20_GA(Order, Ans, N, iota_in_aa(N, .(0, List))) U19_GA(Order, Ans, is_out_ag(N, +(*(Order, +(Order, 1)), 1))) -> IOTA_IN_AA(N, .(0, List)) IOTA_IN_AA(N, List) -> U2_AA(N, List, iota1_in_gaa(0, N, List)) IOTA_IN_AA(N, List) -> IOTA1_IN_GAA(0, N, List) IOTA1_IN_GAA(K, N, .(K, List)) -> U3_GAA(K, N, List, is_in_ag(K1, +(K, 1))) IOTA1_IN_GAA(K, N, .(K, List)) -> IS_IN_AG(K1, +(K, 1)) U3_GAA(K, N, List, is_out_ag(K1, +(K, 1))) -> U4_GAA(K, N, List, iota1_in_aaa(K1, N, List)) U3_GAA(K, N, List, is_out_ag(K1, +(K, 1))) -> IOTA1_IN_AAA(K1, N, List) IOTA1_IN_AAA(K, N, .(K, List)) -> U3_AAA(K, N, List, is_in_aa(K1, +(K, 1))) IOTA1_IN_AAA(K, N, .(K, List)) -> IS_IN_AA(K1, +(K, 1)) U3_AAA(K, N, List, is_out_aa(K1, +(K, 1))) -> U4_AAA(K, N, List, iota1_in_aaa(K1, N, List)) U3_AAA(K, N, List, is_out_aa(K1, +(K, 1))) -> IOTA1_IN_AAA(K1, N, List) U20_GA(Order, Ans, N, iota_out_aa(N, .(0, List))) -> U21_GA(Order, Ans, pds1_in_agaa(List, .(0, []), Ans, N)) U20_GA(Order, Ans, N, iota_out_aa(N, .(0, List))) -> PDS1_IN_AGAA(List, .(0, []), Ans, N) PDS1_IN_AGAA(Unused, List, .(Choice, Rest), Mod) -> U16_AGAA(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) PDS1_IN_AGAA(Unused, List, .(Choice, Rest), Mod) -> MEMBER_IN_AA(Choice, Unused) MEMBER_IN_AA(X, .(X1, Xs)) -> U1_AA(X, X1, Xs, member_in_aa(X, Xs)) MEMBER_IN_AA(X, .(X1, Xs)) -> MEMBER_IN_AA(X, Xs) U16_AGAA(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_AGAA(Unused, List, Choice, Rest, Mod, check_in_gaaaa(List, Choice, Unused, U3, Mod)) U16_AGAA(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> CHECK_IN_GAAAA(List, Choice, Unused, U3, Mod) CHECK_IN_GAAAA(S, Choice, Old, L3, Modulus) -> U12_GAAAA(S, Choice, Old, L3, Modulus, dif_in_gaaaa(S, Choice, Modulus, Ds, Dds)) CHECK_IN_GAAAA(S, Choice, Old, L3, Modulus) -> DIF_IN_GAAAA(S, Choice, Modulus, Ds, Dds) DIF_IN_GAAAA(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) DIF_IN_GAAAA(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> IS_IN_AA(D, -(Val, S)) U5_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U5_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> IS_IN_AA(D2, -(Mod, D)) U6_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gaaaa(Ss, Val, Mod, Ds, D2s)) U6_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> DIF_IN_GAAAA(Ss, Val, Mod, Ds, D2s) U12_GAAAA(S, Choice, Old, L3, Modulus, dif_out_gaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_GAAAA(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) U12_GAAAA(S, Choice, Old, L3, Modulus, dif_out_gaaaa(S, Choice, Modulus, Ds, Dds)) -> MERGEDELETE_IN_AAA(Ds, Old, L2) MERGEDELETE_IN_AAA(.(D, Ds), .(D, R), L2) -> U9_AAA(D, Ds, R, L2, mergedelete_in_aaa(Ds, R, L2)) MERGEDELETE_IN_AAA(.(D, Ds), .(D, R), L2) -> MERGEDELETE_IN_AAA(Ds, R, L2) MERGEDELETE_IN_AAA(.(D, Ds), .(X, R), .(X, L2)) -> U10_AAA(D, Ds, X, R, L2, >_in_aa(D, X)) MERGEDELETE_IN_AAA(.(D, Ds), .(X, R), .(X, L2)) -> >_IN_AA(D, X) U10_AAA(D, Ds, X, R, L2, >_out_aa(D, X)) -> U11_AAA(D, Ds, X, R, L2, mergedelete_in_aaa(.(D, Ds), R, L2)) U10_AAA(D, Ds, X, R, L2, >_out_aa(D, X)) -> MERGEDELETE_IN_AAA(.(D, Ds), R, L2) U13_GAAAA(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_GAAAA(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) U13_GAAAA(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> REV_IN_AGA(Dds, [], Rds) REV_IN_AGA(.(X, Xs), Y, L) -> U8_AGA(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) REV_IN_AGA(.(X, Xs), Y, L) -> REV_IN_AAA(Xs, .(X, Y), L) REV_IN_AAA(.(X, Xs), Y, L) -> U8_AAA(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) REV_IN_AAA(.(X, Xs), Y, L) -> REV_IN_AAA(Xs, .(X, Y), L) U14_GAAAA(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_GAAAA(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U14_GAAAA(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> MERGEDELETE_IN_AAA(Rds, L2, L3) U17_AGAA(Unused, List, Choice, Rest, Mod, check_out_gaaaa(List, Choice, Unused, U3, Mod)) -> U18_AGAA(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) U17_AGAA(Unused, List, Choice, Rest, Mod, check_out_gaaaa(List, Choice, Unused, U3, Mod)) -> PDS1_IN_AAAA(U3, .(Choice, List), Rest, Mod) PDS1_IN_AAAA(Unused, List, .(Choice, Rest), Mod) -> U16_AAAA(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) PDS1_IN_AAAA(Unused, List, .(Choice, Rest), Mod) -> MEMBER_IN_AA(Choice, Unused) U16_AAAA(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_AAAA(Unused, List, Choice, Rest, Mod, check_in_aaaaa(List, Choice, Unused, U3, Mod)) U16_AAAA(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> CHECK_IN_AAAAA(List, Choice, Unused, U3, Mod) CHECK_IN_AAAAA(S, Choice, Old, L3, Modulus) -> U12_AAAAA(S, Choice, Old, L3, Modulus, dif_in_aaaaa(S, Choice, Modulus, Ds, Dds)) CHECK_IN_AAAAA(S, Choice, Old, L3, Modulus) -> DIF_IN_AAAAA(S, Choice, Modulus, Ds, Dds) DIF_IN_AAAAA(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) DIF_IN_AAAAA(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> IS_IN_AA(D, -(Val, S)) U5_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U5_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> IS_IN_AA(D2, -(Mod, D)) U6_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_aaaaa(Ss, Val, Mod, Ds, D2s)) U6_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> DIF_IN_AAAAA(Ss, Val, Mod, Ds, D2s) U12_AAAAA(S, Choice, Old, L3, Modulus, dif_out_aaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_AAAAA(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) U12_AAAAA(S, Choice, Old, L3, Modulus, dif_out_aaaaa(S, Choice, Modulus, Ds, Dds)) -> MERGEDELETE_IN_AAA(Ds, Old, L2) U13_AAAAA(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_AAAAA(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) U13_AAAAA(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> REV_IN_AGA(Dds, [], Rds) U14_AAAAA(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_AAAAA(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U14_AAAAA(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> MERGEDELETE_IN_AAA(Rds, L2, L3) U17_AAAA(Unused, List, Choice, Rest, Mod, check_out_aaaaa(List, Choice, Unused, U3, Mod)) -> U18_AAAA(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) U17_AAAA(Unused, List, Choice, Rest, Mod, check_out_aaaaa(List, Choice, Unused, U3, Mod)) -> PDS1_IN_AAAA(U3, .(Choice, List), Rest, Mod) The TRS R consists of the following rules: pds_in_ga(Order, .(0, Ans)) -> U19_ga(Order, Ans, is_in_ag(N, +(*(Order, +(Order, 1)), 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U19_ga(Order, Ans, is_out_ag(N, +(*(Order, +(Order, 1)), 1))) -> U20_ga(Order, Ans, N, iota_in_aa(N, .(0, List))) iota_in_aa(N, List) -> U2_aa(N, List, iota1_in_gaa(0, N, List)) iota1_in_gaa(K, K, []) -> iota1_out_gaa(K, K, []) iota1_in_gaa(K, N, .(K, List)) -> U3_gaa(K, N, List, is_in_ag(K1, +(K, 1))) U3_gaa(K, N, List, is_out_ag(K1, +(K, 1))) -> U4_gaa(K, N, List, iota1_in_aaa(K1, N, List)) iota1_in_aaa(K, K, []) -> iota1_out_aaa(K, K, []) iota1_in_aaa(K, N, .(K, List)) -> U3_aaa(K, N, List, is_in_aa(K1, +(K, 1))) is_in_aa(X0, X1) -> is_out_aa(X0, X1) U3_aaa(K, N, List, is_out_aa(K1, +(K, 1))) -> U4_aaa(K, N, List, iota1_in_aaa(K1, N, List)) U4_aaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_aaa(K, N, .(K, List)) U4_gaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_gaa(K, N, .(K, List)) U2_aa(N, List, iota1_out_gaa(0, N, List)) -> iota_out_aa(N, List) U20_ga(Order, Ans, N, iota_out_aa(N, .(0, List))) -> U21_ga(Order, Ans, pds1_in_agaa(List, .(0, []), Ans, N)) pds1_in_agaa([], X6, [], X7) -> pds1_out_agaa([], X6, [], X7) pds1_in_agaa(Unused, List, .(Choice, Rest), Mod) -> U16_agaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) member_in_aa(X, .(X, Xs)) -> member_out_aa(X, .(X, Xs)) member_in_aa(X, .(X1, Xs)) -> U1_aa(X, X1, Xs, member_in_aa(X, Xs)) U1_aa(X, X1, Xs, member_out_aa(X, Xs)) -> member_out_aa(X, .(X1, Xs)) U16_agaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_agaa(Unused, List, Choice, Rest, Mod, check_in_gaaaa(List, Choice, Unused, U3, Mod)) check_in_gaaaa([], X4, L, L, X5) -> check_out_gaaaa([], X4, L, L, X5) check_in_gaaaa(S, Choice, Old, L3, Modulus) -> U12_gaaaa(S, Choice, Old, L3, Modulus, dif_in_gaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_gaaaa([], X2, X3, [], []) -> dif_out_gaaaa([], X2, X3, [], []) dif_in_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gaaaa(Ss, Val, Mod, Ds, D2s)) U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_gaaaa(S, Choice, Old, L3, Modulus, dif_out_gaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) mergedelete_in_aaa([], L, L) -> mergedelete_out_aaa([], L, L) mergedelete_in_aaa(.(D, Ds), .(D, R), L2) -> U9_aaa(D, Ds, R, L2, mergedelete_in_aaa(Ds, R, L2)) mergedelete_in_aaa(.(D, Ds), .(X, R), .(X, L2)) -> U10_aaa(D, Ds, X, R, L2, >_in_aa(D, X)) >_in_aa(X0, X1) -> >_out_aa(X0, X1) U10_aaa(D, Ds, X, R, L2, >_out_aa(D, X)) -> U11_aaa(D, Ds, X, R, L2, mergedelete_in_aaa(.(D, Ds), R, L2)) U11_aaa(D, Ds, X, R, L2, mergedelete_out_aaa(.(D, Ds), R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(X, R), .(X, L2)) U9_aaa(D, Ds, R, L2, mergedelete_out_aaa(Ds, R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(D, R), L2) U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) rev_in_aga([], L, L) -> rev_out_aga([], L, L) rev_in_aga(.(X, Xs), Y, L) -> U8_aga(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) rev_in_aaa([], L, L) -> rev_out_aaa([], L, L) rev_in_aaa(.(X, Xs), Y, L) -> U8_aaa(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) U8_aaa(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aaa(.(X, Xs), Y, L) U8_aga(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aga(.(X, Xs), Y, L) U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_gaaaa(S, Choice, Old, L3, Modulus) U17_agaa(Unused, List, Choice, Rest, Mod, check_out_gaaaa(List, Choice, Unused, U3, Mod)) -> U18_agaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) pds1_in_aaaa([], X6, [], X7) -> pds1_out_aaaa([], X6, [], X7) pds1_in_aaaa(Unused, List, .(Choice, Rest), Mod) -> U16_aaaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) U16_aaaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_aaaa(Unused, List, Choice, Rest, Mod, check_in_aaaaa(List, Choice, Unused, U3, Mod)) check_in_aaaaa([], X4, L, L, X5) -> check_out_aaaaa([], X4, L, L, X5) check_in_aaaaa(S, Choice, Old, L3, Modulus) -> U12_aaaaa(S, Choice, Old, L3, Modulus, dif_in_aaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_aaaaa([], X2, X3, [], []) -> dif_out_aaaaa([], X2, X3, [], []) dif_in_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_aaaaa(Ss, Val, Mod, Ds, D2s)) U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_aaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_aaaaa(S, Choice, Old, L3, Modulus, dif_out_aaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_aaaaa(S, Choice, Old, L3, Modulus) U17_aaaa(Unused, List, Choice, Rest, Mod, check_out_aaaaa(List, Choice, Unused, U3, Mod)) -> U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_aaaa(Unused, List, .(Choice, Rest), Mod) U18_agaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_agaa(Unused, List, .(Choice, Rest), Mod) U21_ga(Order, Ans, pds1_out_agaa(List, .(0, []), Ans, N)) -> pds_out_ga(Order, .(0, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U19_ga(x1, x2, x3) = U19_ga(x1, x3) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag(x2) +(x1, x2) = +(x1, x2) *(x1, x2) = *(x1, x2) 1 = 1 U20_ga(x1, x2, x3, x4) = U20_ga(x1, x4) iota_in_aa(x1, x2) = iota_in_aa U2_aa(x1, x2, x3) = U2_aa(x3) iota1_in_gaa(x1, x2, x3) = iota1_in_gaa(x1) iota1_out_gaa(x1, x2, x3) = iota1_out_gaa(x1) U3_gaa(x1, x2, x3, x4) = U3_gaa(x1, x4) U4_gaa(x1, x2, x3, x4) = U4_gaa(x1, x4) iota1_in_aaa(x1, x2, x3) = iota1_in_aaa iota1_out_aaa(x1, x2, x3) = iota1_out_aaa U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) is_in_aa(x1, x2) = is_in_aa is_out_aa(x1, x2) = is_out_aa U4_aaa(x1, x2, x3, x4) = U4_aaa(x4) 0 = 0 iota_out_aa(x1, x2) = iota_out_aa U21_ga(x1, x2, x3) = U21_ga(x1, x3) pds1_in_agaa(x1, x2, x3, x4) = pds1_in_agaa(x2) pds1_out_agaa(x1, x2, x3, x4) = pds1_out_agaa(x2) U16_agaa(x1, x2, x3, x4, x5, x6) = U16_agaa(x2, x6) member_in_aa(x1, x2) = member_in_aa member_out_aa(x1, x2) = member_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U17_agaa(x1, x2, x3, x4, x5, x6) = U17_agaa(x2, x6) check_in_gaaaa(x1, x2, x3, x4, x5) = check_in_gaaaa(x1) [] = [] check_out_gaaaa(x1, x2, x3, x4, x5) = check_out_gaaaa(x1) U12_gaaaa(x1, x2, x3, x4, x5, x6) = U12_gaaaa(x1, x6) dif_in_gaaaa(x1, x2, x3, x4, x5) = dif_in_gaaaa(x1) dif_out_gaaaa(x1, x2, x3, x4, x5) = dif_out_gaaaa(x1) .(x1, x2) = .(x1, x2) U5_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_gaaaa(x1, x2, x9) U6_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gaaaa(x1, x2, x9) U7_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_gaaaa(x1, x2, x9) U13_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_gaaaa(x1, x7) mergedelete_in_aaa(x1, x2, x3) = mergedelete_in_aaa mergedelete_out_aaa(x1, x2, x3) = mergedelete_out_aaa U9_aaa(x1, x2, x3, x4, x5) = U9_aaa(x5) U10_aaa(x1, x2, x3, x4, x5, x6) = U10_aaa(x6) >_in_aa(x1, x2) = >_in_aa >_out_aa(x1, x2) = >_out_aa U11_aaa(x1, x2, x3, x4, x5, x6) = U11_aaa(x6) U14_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_gaaaa(x1, x7) rev_in_aga(x1, x2, x3) = rev_in_aga(x2) rev_out_aga(x1, x2, x3) = rev_out_aga(x2) U8_aga(x1, x2, x3, x4, x5) = U8_aga(x3, x5) rev_in_aaa(x1, x2, x3) = rev_in_aaa rev_out_aaa(x1, x2, x3) = rev_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U15_gaaaa(x1, x2, x3, x4, x5, x6) = U15_gaaaa(x1, x6) U18_agaa(x1, x2, x3, x4, x5, x6) = U18_agaa(x2, x6) pds1_in_aaaa(x1, x2, x3, x4) = pds1_in_aaaa pds1_out_aaaa(x1, x2, x3, x4) = pds1_out_aaaa U16_aaaa(x1, x2, x3, x4, x5, x6) = U16_aaaa(x6) U17_aaaa(x1, x2, x3, x4, x5, x6) = U17_aaaa(x6) check_in_aaaaa(x1, x2, x3, x4, x5) = check_in_aaaaa check_out_aaaaa(x1, x2, x3, x4, x5) = check_out_aaaaa U12_aaaaa(x1, x2, x3, x4, x5, x6) = U12_aaaaa(x6) dif_in_aaaaa(x1, x2, x3, x4, x5) = dif_in_aaaaa dif_out_aaaaa(x1, x2, x3, x4, x5) = dif_out_aaaaa U5_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_aaaaa(x9) U6_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_aaaaa(x9) U7_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_aaaaa(x9) U13_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_aaaaa(x7) U14_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_aaaaa(x7) U15_aaaaa(x1, x2, x3, x4, x5, x6) = U15_aaaaa(x6) U18_aaaa(x1, x2, x3, x4, x5, x6) = U18_aaaa(x6) pds_out_ga(x1, x2) = pds_out_ga(x1) PDS_IN_GA(x1, x2) = PDS_IN_GA(x1) U19_GA(x1, x2, x3) = U19_GA(x1, x3) IS_IN_AG(x1, x2) = IS_IN_AG(x2) U20_GA(x1, x2, x3, x4) = U20_GA(x1, x4) IOTA_IN_AA(x1, x2) = IOTA_IN_AA U2_AA(x1, x2, x3) = U2_AA(x3) IOTA1_IN_GAA(x1, x2, x3) = IOTA1_IN_GAA(x1) U3_GAA(x1, x2, x3, x4) = U3_GAA(x1, x4) U4_GAA(x1, x2, x3, x4) = U4_GAA(x1, x4) IOTA1_IN_AAA(x1, x2, x3) = IOTA1_IN_AAA U3_AAA(x1, x2, x3, x4) = U3_AAA(x4) IS_IN_AA(x1, x2) = IS_IN_AA U4_AAA(x1, x2, x3, x4) = U4_AAA(x4) U21_GA(x1, x2, x3) = U21_GA(x1, x3) PDS1_IN_AGAA(x1, x2, x3, x4) = PDS1_IN_AGAA(x2) U16_AGAA(x1, x2, x3, x4, x5, x6) = U16_AGAA(x2, x6) MEMBER_IN_AA(x1, x2) = MEMBER_IN_AA U1_AA(x1, x2, x3, x4) = U1_AA(x4) U17_AGAA(x1, x2, x3, x4, x5, x6) = U17_AGAA(x2, x6) CHECK_IN_GAAAA(x1, x2, x3, x4, x5) = CHECK_IN_GAAAA(x1) U12_GAAAA(x1, x2, x3, x4, x5, x6) = U12_GAAAA(x1, x6) DIF_IN_GAAAA(x1, x2, x3, x4, x5) = DIF_IN_GAAAA(x1) U5_GAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_GAAAA(x1, x2, x9) U6_GAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_GAAAA(x1, x2, x9) U7_GAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_GAAAA(x1, x2, x9) U13_GAAAA(x1, x2, x3, x4, x5, x6, x7) = U13_GAAAA(x1, x7) MERGEDELETE_IN_AAA(x1, x2, x3) = MERGEDELETE_IN_AAA U9_AAA(x1, x2, x3, x4, x5) = U9_AAA(x5) U10_AAA(x1, x2, x3, x4, x5, x6) = U10_AAA(x6) >_IN_AA(x1, x2) = >_IN_AA U11_AAA(x1, x2, x3, x4, x5, x6) = U11_AAA(x6) U14_GAAAA(x1, x2, x3, x4, x5, x6, x7) = U14_GAAAA(x1, x7) REV_IN_AGA(x1, x2, x3) = REV_IN_AGA(x2) U8_AGA(x1, x2, x3, x4, x5) = U8_AGA(x3, x5) REV_IN_AAA(x1, x2, x3) = REV_IN_AAA U8_AAA(x1, x2, x3, x4, x5) = U8_AAA(x5) U15_GAAAA(x1, x2, x3, x4, x5, x6) = U15_GAAAA(x1, x6) U18_AGAA(x1, x2, x3, x4, x5, x6) = U18_AGAA(x2, x6) PDS1_IN_AAAA(x1, x2, x3, x4) = PDS1_IN_AAAA U16_AAAA(x1, x2, x3, x4, x5, x6) = U16_AAAA(x6) U17_AAAA(x1, x2, x3, x4, x5, x6) = U17_AAAA(x6) CHECK_IN_AAAAA(x1, x2, x3, x4, x5) = CHECK_IN_AAAAA U12_AAAAA(x1, x2, x3, x4, x5, x6) = U12_AAAAA(x6) DIF_IN_AAAAA(x1, x2, x3, x4, x5) = DIF_IN_AAAAA U5_AAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_AAAAA(x9) U6_AAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_AAAAA(x9) U7_AAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_AAAAA(x9) U13_AAAAA(x1, x2, x3, x4, x5, x6, x7) = U13_AAAAA(x7) U14_AAAAA(x1, x2, x3, x4, x5, x6, x7) = U14_AAAAA(x7) U15_AAAAA(x1, x2, x3, x4, x5, x6) = U15_AAAAA(x6) U18_AAAA(x1, x2, x3, x4, x5, x6) = U18_AAAA(x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (296) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 7 SCCs with 52 less nodes. ---------------------------------------- (297) Complex Obligation (AND) ---------------------------------------- (298) Obligation: Pi DP problem: The TRS P consists of the following rules: U5_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> DIF_IN_AAAAA(Ss, Val, Mod, Ds, D2s) DIF_IN_AAAAA(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) The TRS R consists of the following rules: pds_in_ga(Order, .(0, Ans)) -> U19_ga(Order, Ans, is_in_ag(N, +(*(Order, +(Order, 1)), 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U19_ga(Order, Ans, is_out_ag(N, +(*(Order, +(Order, 1)), 1))) -> U20_ga(Order, Ans, N, iota_in_aa(N, .(0, List))) iota_in_aa(N, List) -> U2_aa(N, List, iota1_in_gaa(0, N, List)) iota1_in_gaa(K, K, []) -> iota1_out_gaa(K, K, []) iota1_in_gaa(K, N, .(K, List)) -> U3_gaa(K, N, List, is_in_ag(K1, +(K, 1))) U3_gaa(K, N, List, is_out_ag(K1, +(K, 1))) -> U4_gaa(K, N, List, iota1_in_aaa(K1, N, List)) iota1_in_aaa(K, K, []) -> iota1_out_aaa(K, K, []) iota1_in_aaa(K, N, .(K, List)) -> U3_aaa(K, N, List, is_in_aa(K1, +(K, 1))) is_in_aa(X0, X1) -> is_out_aa(X0, X1) U3_aaa(K, N, List, is_out_aa(K1, +(K, 1))) -> U4_aaa(K, N, List, iota1_in_aaa(K1, N, List)) U4_aaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_aaa(K, N, .(K, List)) U4_gaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_gaa(K, N, .(K, List)) U2_aa(N, List, iota1_out_gaa(0, N, List)) -> iota_out_aa(N, List) U20_ga(Order, Ans, N, iota_out_aa(N, .(0, List))) -> U21_ga(Order, Ans, pds1_in_agaa(List, .(0, []), Ans, N)) pds1_in_agaa([], X6, [], X7) -> pds1_out_agaa([], X6, [], X7) pds1_in_agaa(Unused, List, .(Choice, Rest), Mod) -> U16_agaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) member_in_aa(X, .(X, Xs)) -> member_out_aa(X, .(X, Xs)) member_in_aa(X, .(X1, Xs)) -> U1_aa(X, X1, Xs, member_in_aa(X, Xs)) U1_aa(X, X1, Xs, member_out_aa(X, Xs)) -> member_out_aa(X, .(X1, Xs)) U16_agaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_agaa(Unused, List, Choice, Rest, Mod, check_in_gaaaa(List, Choice, Unused, U3, Mod)) check_in_gaaaa([], X4, L, L, X5) -> check_out_gaaaa([], X4, L, L, X5) check_in_gaaaa(S, Choice, Old, L3, Modulus) -> U12_gaaaa(S, Choice, Old, L3, Modulus, dif_in_gaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_gaaaa([], X2, X3, [], []) -> dif_out_gaaaa([], X2, X3, [], []) dif_in_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gaaaa(Ss, Val, Mod, Ds, D2s)) U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_gaaaa(S, Choice, Old, L3, Modulus, dif_out_gaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) mergedelete_in_aaa([], L, L) -> mergedelete_out_aaa([], L, L) mergedelete_in_aaa(.(D, Ds), .(D, R), L2) -> U9_aaa(D, Ds, R, L2, mergedelete_in_aaa(Ds, R, L2)) mergedelete_in_aaa(.(D, Ds), .(X, R), .(X, L2)) -> U10_aaa(D, Ds, X, R, L2, >_in_aa(D, X)) >_in_aa(X0, X1) -> >_out_aa(X0, X1) U10_aaa(D, Ds, X, R, L2, >_out_aa(D, X)) -> U11_aaa(D, Ds, X, R, L2, mergedelete_in_aaa(.(D, Ds), R, L2)) U11_aaa(D, Ds, X, R, L2, mergedelete_out_aaa(.(D, Ds), R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(X, R), .(X, L2)) U9_aaa(D, Ds, R, L2, mergedelete_out_aaa(Ds, R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(D, R), L2) U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) rev_in_aga([], L, L) -> rev_out_aga([], L, L) rev_in_aga(.(X, Xs), Y, L) -> U8_aga(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) rev_in_aaa([], L, L) -> rev_out_aaa([], L, L) rev_in_aaa(.(X, Xs), Y, L) -> U8_aaa(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) U8_aaa(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aaa(.(X, Xs), Y, L) U8_aga(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aga(.(X, Xs), Y, L) U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_gaaaa(S, Choice, Old, L3, Modulus) U17_agaa(Unused, List, Choice, Rest, Mod, check_out_gaaaa(List, Choice, Unused, U3, Mod)) -> U18_agaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) pds1_in_aaaa([], X6, [], X7) -> pds1_out_aaaa([], X6, [], X7) pds1_in_aaaa(Unused, List, .(Choice, Rest), Mod) -> U16_aaaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) U16_aaaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_aaaa(Unused, List, Choice, Rest, Mod, check_in_aaaaa(List, Choice, Unused, U3, Mod)) check_in_aaaaa([], X4, L, L, X5) -> check_out_aaaaa([], X4, L, L, X5) check_in_aaaaa(S, Choice, Old, L3, Modulus) -> U12_aaaaa(S, Choice, Old, L3, Modulus, dif_in_aaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_aaaaa([], X2, X3, [], []) -> dif_out_aaaaa([], X2, X3, [], []) dif_in_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_aaaaa(Ss, Val, Mod, Ds, D2s)) U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_aaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_aaaaa(S, Choice, Old, L3, Modulus, dif_out_aaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_aaaaa(S, Choice, Old, L3, Modulus) U17_aaaa(Unused, List, Choice, Rest, Mod, check_out_aaaaa(List, Choice, Unused, U3, Mod)) -> U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_aaaa(Unused, List, .(Choice, Rest), Mod) U18_agaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_agaa(Unused, List, .(Choice, Rest), Mod) U21_ga(Order, Ans, pds1_out_agaa(List, .(0, []), Ans, N)) -> pds_out_ga(Order, .(0, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U19_ga(x1, x2, x3) = U19_ga(x1, x3) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag(x2) +(x1, x2) = +(x1, x2) *(x1, x2) = *(x1, x2) 1 = 1 U20_ga(x1, x2, x3, x4) = U20_ga(x1, x4) iota_in_aa(x1, x2) = iota_in_aa U2_aa(x1, x2, x3) = U2_aa(x3) iota1_in_gaa(x1, x2, x3) = iota1_in_gaa(x1) iota1_out_gaa(x1, x2, x3) = iota1_out_gaa(x1) U3_gaa(x1, x2, x3, x4) = U3_gaa(x1, x4) U4_gaa(x1, x2, x3, x4) = U4_gaa(x1, x4) iota1_in_aaa(x1, x2, x3) = iota1_in_aaa iota1_out_aaa(x1, x2, x3) = iota1_out_aaa U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) is_in_aa(x1, x2) = is_in_aa is_out_aa(x1, x2) = is_out_aa U4_aaa(x1, x2, x3, x4) = U4_aaa(x4) 0 = 0 iota_out_aa(x1, x2) = iota_out_aa U21_ga(x1, x2, x3) = U21_ga(x1, x3) pds1_in_agaa(x1, x2, x3, x4) = pds1_in_agaa(x2) pds1_out_agaa(x1, x2, x3, x4) = pds1_out_agaa(x2) U16_agaa(x1, x2, x3, x4, x5, x6) = U16_agaa(x2, x6) member_in_aa(x1, x2) = member_in_aa member_out_aa(x1, x2) = member_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U17_agaa(x1, x2, x3, x4, x5, x6) = U17_agaa(x2, x6) check_in_gaaaa(x1, x2, x3, x4, x5) = check_in_gaaaa(x1) [] = [] check_out_gaaaa(x1, x2, x3, x4, x5) = check_out_gaaaa(x1) U12_gaaaa(x1, x2, x3, x4, x5, x6) = U12_gaaaa(x1, x6) dif_in_gaaaa(x1, x2, x3, x4, x5) = dif_in_gaaaa(x1) dif_out_gaaaa(x1, x2, x3, x4, x5) = dif_out_gaaaa(x1) .(x1, x2) = .(x1, x2) U5_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_gaaaa(x1, x2, x9) U6_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gaaaa(x1, x2, x9) U7_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_gaaaa(x1, x2, x9) U13_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_gaaaa(x1, x7) mergedelete_in_aaa(x1, x2, x3) = mergedelete_in_aaa mergedelete_out_aaa(x1, x2, x3) = mergedelete_out_aaa U9_aaa(x1, x2, x3, x4, x5) = U9_aaa(x5) U10_aaa(x1, x2, x3, x4, x5, x6) = U10_aaa(x6) >_in_aa(x1, x2) = >_in_aa >_out_aa(x1, x2) = >_out_aa U11_aaa(x1, x2, x3, x4, x5, x6) = U11_aaa(x6) U14_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_gaaaa(x1, x7) rev_in_aga(x1, x2, x3) = rev_in_aga(x2) rev_out_aga(x1, x2, x3) = rev_out_aga(x2) U8_aga(x1, x2, x3, x4, x5) = U8_aga(x3, x5) rev_in_aaa(x1, x2, x3) = rev_in_aaa rev_out_aaa(x1, x2, x3) = rev_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U15_gaaaa(x1, x2, x3, x4, x5, x6) = U15_gaaaa(x1, x6) U18_agaa(x1, x2, x3, x4, x5, x6) = U18_agaa(x2, x6) pds1_in_aaaa(x1, x2, x3, x4) = pds1_in_aaaa pds1_out_aaaa(x1, x2, x3, x4) = pds1_out_aaaa U16_aaaa(x1, x2, x3, x4, x5, x6) = U16_aaaa(x6) U17_aaaa(x1, x2, x3, x4, x5, x6) = U17_aaaa(x6) check_in_aaaaa(x1, x2, x3, x4, x5) = check_in_aaaaa check_out_aaaaa(x1, x2, x3, x4, x5) = check_out_aaaaa U12_aaaaa(x1, x2, x3, x4, x5, x6) = U12_aaaaa(x6) dif_in_aaaaa(x1, x2, x3, x4, x5) = dif_in_aaaaa dif_out_aaaaa(x1, x2, x3, x4, x5) = dif_out_aaaaa U5_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_aaaaa(x9) U6_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_aaaaa(x9) U7_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_aaaaa(x9) U13_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_aaaaa(x7) U14_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_aaaaa(x7) U15_aaaaa(x1, x2, x3, x4, x5, x6) = U15_aaaaa(x6) U18_aaaa(x1, x2, x3, x4, x5, x6) = U18_aaaa(x6) pds_out_ga(x1, x2) = pds_out_ga(x1) DIF_IN_AAAAA(x1, x2, x3, x4, x5) = DIF_IN_AAAAA U5_AAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_AAAAA(x9) U6_AAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_AAAAA(x9) We have to consider all (P,R,Pi)-chains ---------------------------------------- (299) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (300) Obligation: Pi DP problem: The TRS P consists of the following rules: U5_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> DIF_IN_AAAAA(Ss, Val, Mod, Ds, D2s) DIF_IN_AAAAA(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_AAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) The TRS R consists of the following rules: is_in_aa(X0, X1) -> is_out_aa(X0, X1) The argument filtering Pi contains the following mapping: is_in_aa(x1, x2) = is_in_aa is_out_aa(x1, x2) = is_out_aa .(x1, x2) = .(x1, x2) DIF_IN_AAAAA(x1, x2, x3, x4, x5) = DIF_IN_AAAAA U5_AAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_AAAAA(x9) U6_AAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_AAAAA(x9) We have to consider all (P,R,Pi)-chains ---------------------------------------- (301) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (302) Obligation: Q DP problem: The TRS P consists of the following rules: U5_AAAAA(is_out_aa) -> U6_AAAAA(is_in_aa) U6_AAAAA(is_out_aa) -> DIF_IN_AAAAA DIF_IN_AAAAA -> U5_AAAAA(is_in_aa) The TRS R consists of the following rules: is_in_aa -> is_out_aa The set Q consists of the following terms: is_in_aa We have to consider all (P,Q,R)-chains. ---------------------------------------- (303) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule U5_AAAAA(is_out_aa) -> U6_AAAAA(is_in_aa) at position [0] we obtained the following new rules [LPAR04]: (U5_AAAAA(is_out_aa) -> U6_AAAAA(is_out_aa),U5_AAAAA(is_out_aa) -> U6_AAAAA(is_out_aa)) ---------------------------------------- (304) Obligation: Q DP problem: The TRS P consists of the following rules: U6_AAAAA(is_out_aa) -> DIF_IN_AAAAA DIF_IN_AAAAA -> U5_AAAAA(is_in_aa) U5_AAAAA(is_out_aa) -> U6_AAAAA(is_out_aa) The TRS R consists of the following rules: is_in_aa -> is_out_aa The set Q consists of the following terms: is_in_aa We have to consider all (P,Q,R)-chains. ---------------------------------------- (305) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule DIF_IN_AAAAA -> U5_AAAAA(is_in_aa) at position [0] we obtained the following new rules [LPAR04]: (DIF_IN_AAAAA -> U5_AAAAA(is_out_aa),DIF_IN_AAAAA -> U5_AAAAA(is_out_aa)) ---------------------------------------- (306) Obligation: Q DP problem: The TRS P consists of the following rules: U6_AAAAA(is_out_aa) -> DIF_IN_AAAAA U5_AAAAA(is_out_aa) -> U6_AAAAA(is_out_aa) DIF_IN_AAAAA -> U5_AAAAA(is_out_aa) The TRS R consists of the following rules: is_in_aa -> is_out_aa The set Q consists of the following terms: is_in_aa We have to consider all (P,Q,R)-chains. ---------------------------------------- (307) 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. ---------------------------------------- (308) Obligation: Q DP problem: The TRS P consists of the following rules: U6_AAAAA(is_out_aa) -> DIF_IN_AAAAA U5_AAAAA(is_out_aa) -> U6_AAAAA(is_out_aa) DIF_IN_AAAAA -> U5_AAAAA(is_out_aa) R is empty. The set Q consists of the following terms: is_in_aa We have to consider all (P,Q,R)-chains. ---------------------------------------- (309) 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]. is_in_aa ---------------------------------------- (310) Obligation: Q DP problem: The TRS P consists of the following rules: U6_AAAAA(is_out_aa) -> DIF_IN_AAAAA U5_AAAAA(is_out_aa) -> U6_AAAAA(is_out_aa) DIF_IN_AAAAA -> U5_AAAAA(is_out_aa) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (311) 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 = DIF_IN_AAAAA evaluates to t =DIF_IN_AAAAA Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence DIF_IN_AAAAA -> U5_AAAAA(is_out_aa) with rule DIF_IN_AAAAA -> U5_AAAAA(is_out_aa) at position [] and matcher [ ] U5_AAAAA(is_out_aa) -> U6_AAAAA(is_out_aa) with rule U5_AAAAA(is_out_aa) -> U6_AAAAA(is_out_aa) at position [] and matcher [ ] U6_AAAAA(is_out_aa) -> DIF_IN_AAAAA with rule U6_AAAAA(is_out_aa) -> DIF_IN_AAAAA 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. ---------------------------------------- (312) NO ---------------------------------------- (313) Obligation: Pi DP problem: The TRS P consists of the following rules: REV_IN_AAA(.(X, Xs), Y, L) -> REV_IN_AAA(Xs, .(X, Y), L) The TRS R consists of the following rules: pds_in_ga(Order, .(0, Ans)) -> U19_ga(Order, Ans, is_in_ag(N, +(*(Order, +(Order, 1)), 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U19_ga(Order, Ans, is_out_ag(N, +(*(Order, +(Order, 1)), 1))) -> U20_ga(Order, Ans, N, iota_in_aa(N, .(0, List))) iota_in_aa(N, List) -> U2_aa(N, List, iota1_in_gaa(0, N, List)) iota1_in_gaa(K, K, []) -> iota1_out_gaa(K, K, []) iota1_in_gaa(K, N, .(K, List)) -> U3_gaa(K, N, List, is_in_ag(K1, +(K, 1))) U3_gaa(K, N, List, is_out_ag(K1, +(K, 1))) -> U4_gaa(K, N, List, iota1_in_aaa(K1, N, List)) iota1_in_aaa(K, K, []) -> iota1_out_aaa(K, K, []) iota1_in_aaa(K, N, .(K, List)) -> U3_aaa(K, N, List, is_in_aa(K1, +(K, 1))) is_in_aa(X0, X1) -> is_out_aa(X0, X1) U3_aaa(K, N, List, is_out_aa(K1, +(K, 1))) -> U4_aaa(K, N, List, iota1_in_aaa(K1, N, List)) U4_aaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_aaa(K, N, .(K, List)) U4_gaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_gaa(K, N, .(K, List)) U2_aa(N, List, iota1_out_gaa(0, N, List)) -> iota_out_aa(N, List) U20_ga(Order, Ans, N, iota_out_aa(N, .(0, List))) -> U21_ga(Order, Ans, pds1_in_agaa(List, .(0, []), Ans, N)) pds1_in_agaa([], X6, [], X7) -> pds1_out_agaa([], X6, [], X7) pds1_in_agaa(Unused, List, .(Choice, Rest), Mod) -> U16_agaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) member_in_aa(X, .(X, Xs)) -> member_out_aa(X, .(X, Xs)) member_in_aa(X, .(X1, Xs)) -> U1_aa(X, X1, Xs, member_in_aa(X, Xs)) U1_aa(X, X1, Xs, member_out_aa(X, Xs)) -> member_out_aa(X, .(X1, Xs)) U16_agaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_agaa(Unused, List, Choice, Rest, Mod, check_in_gaaaa(List, Choice, Unused, U3, Mod)) check_in_gaaaa([], X4, L, L, X5) -> check_out_gaaaa([], X4, L, L, X5) check_in_gaaaa(S, Choice, Old, L3, Modulus) -> U12_gaaaa(S, Choice, Old, L3, Modulus, dif_in_gaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_gaaaa([], X2, X3, [], []) -> dif_out_gaaaa([], X2, X3, [], []) dif_in_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gaaaa(Ss, Val, Mod, Ds, D2s)) U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_gaaaa(S, Choice, Old, L3, Modulus, dif_out_gaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) mergedelete_in_aaa([], L, L) -> mergedelete_out_aaa([], L, L) mergedelete_in_aaa(.(D, Ds), .(D, R), L2) -> U9_aaa(D, Ds, R, L2, mergedelete_in_aaa(Ds, R, L2)) mergedelete_in_aaa(.(D, Ds), .(X, R), .(X, L2)) -> U10_aaa(D, Ds, X, R, L2, >_in_aa(D, X)) >_in_aa(X0, X1) -> >_out_aa(X0, X1) U10_aaa(D, Ds, X, R, L2, >_out_aa(D, X)) -> U11_aaa(D, Ds, X, R, L2, mergedelete_in_aaa(.(D, Ds), R, L2)) U11_aaa(D, Ds, X, R, L2, mergedelete_out_aaa(.(D, Ds), R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(X, R), .(X, L2)) U9_aaa(D, Ds, R, L2, mergedelete_out_aaa(Ds, R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(D, R), L2) U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) rev_in_aga([], L, L) -> rev_out_aga([], L, L) rev_in_aga(.(X, Xs), Y, L) -> U8_aga(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) rev_in_aaa([], L, L) -> rev_out_aaa([], L, L) rev_in_aaa(.(X, Xs), Y, L) -> U8_aaa(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) U8_aaa(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aaa(.(X, Xs), Y, L) U8_aga(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aga(.(X, Xs), Y, L) U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_gaaaa(S, Choice, Old, L3, Modulus) U17_agaa(Unused, List, Choice, Rest, Mod, check_out_gaaaa(List, Choice, Unused, U3, Mod)) -> U18_agaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) pds1_in_aaaa([], X6, [], X7) -> pds1_out_aaaa([], X6, [], X7) pds1_in_aaaa(Unused, List, .(Choice, Rest), Mod) -> U16_aaaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) U16_aaaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_aaaa(Unused, List, Choice, Rest, Mod, check_in_aaaaa(List, Choice, Unused, U3, Mod)) check_in_aaaaa([], X4, L, L, X5) -> check_out_aaaaa([], X4, L, L, X5) check_in_aaaaa(S, Choice, Old, L3, Modulus) -> U12_aaaaa(S, Choice, Old, L3, Modulus, dif_in_aaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_aaaaa([], X2, X3, [], []) -> dif_out_aaaaa([], X2, X3, [], []) dif_in_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_aaaaa(Ss, Val, Mod, Ds, D2s)) U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_aaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_aaaaa(S, Choice, Old, L3, Modulus, dif_out_aaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_aaaaa(S, Choice, Old, L3, Modulus) U17_aaaa(Unused, List, Choice, Rest, Mod, check_out_aaaaa(List, Choice, Unused, U3, Mod)) -> U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_aaaa(Unused, List, .(Choice, Rest), Mod) U18_agaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_agaa(Unused, List, .(Choice, Rest), Mod) U21_ga(Order, Ans, pds1_out_agaa(List, .(0, []), Ans, N)) -> pds_out_ga(Order, .(0, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U19_ga(x1, x2, x3) = U19_ga(x1, x3) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag(x2) +(x1, x2) = +(x1, x2) *(x1, x2) = *(x1, x2) 1 = 1 U20_ga(x1, x2, x3, x4) = U20_ga(x1, x4) iota_in_aa(x1, x2) = iota_in_aa U2_aa(x1, x2, x3) = U2_aa(x3) iota1_in_gaa(x1, x2, x3) = iota1_in_gaa(x1) iota1_out_gaa(x1, x2, x3) = iota1_out_gaa(x1) U3_gaa(x1, x2, x3, x4) = U3_gaa(x1, x4) U4_gaa(x1, x2, x3, x4) = U4_gaa(x1, x4) iota1_in_aaa(x1, x2, x3) = iota1_in_aaa iota1_out_aaa(x1, x2, x3) = iota1_out_aaa U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) is_in_aa(x1, x2) = is_in_aa is_out_aa(x1, x2) = is_out_aa U4_aaa(x1, x2, x3, x4) = U4_aaa(x4) 0 = 0 iota_out_aa(x1, x2) = iota_out_aa U21_ga(x1, x2, x3) = U21_ga(x1, x3) pds1_in_agaa(x1, x2, x3, x4) = pds1_in_agaa(x2) pds1_out_agaa(x1, x2, x3, x4) = pds1_out_agaa(x2) U16_agaa(x1, x2, x3, x4, x5, x6) = U16_agaa(x2, x6) member_in_aa(x1, x2) = member_in_aa member_out_aa(x1, x2) = member_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U17_agaa(x1, x2, x3, x4, x5, x6) = U17_agaa(x2, x6) check_in_gaaaa(x1, x2, x3, x4, x5) = check_in_gaaaa(x1) [] = [] check_out_gaaaa(x1, x2, x3, x4, x5) = check_out_gaaaa(x1) U12_gaaaa(x1, x2, x3, x4, x5, x6) = U12_gaaaa(x1, x6) dif_in_gaaaa(x1, x2, x3, x4, x5) = dif_in_gaaaa(x1) dif_out_gaaaa(x1, x2, x3, x4, x5) = dif_out_gaaaa(x1) .(x1, x2) = .(x1, x2) U5_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_gaaaa(x1, x2, x9) U6_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gaaaa(x1, x2, x9) U7_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_gaaaa(x1, x2, x9) U13_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_gaaaa(x1, x7) mergedelete_in_aaa(x1, x2, x3) = mergedelete_in_aaa mergedelete_out_aaa(x1, x2, x3) = mergedelete_out_aaa U9_aaa(x1, x2, x3, x4, x5) = U9_aaa(x5) U10_aaa(x1, x2, x3, x4, x5, x6) = U10_aaa(x6) >_in_aa(x1, x2) = >_in_aa >_out_aa(x1, x2) = >_out_aa U11_aaa(x1, x2, x3, x4, x5, x6) = U11_aaa(x6) U14_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_gaaaa(x1, x7) rev_in_aga(x1, x2, x3) = rev_in_aga(x2) rev_out_aga(x1, x2, x3) = rev_out_aga(x2) U8_aga(x1, x2, x3, x4, x5) = U8_aga(x3, x5) rev_in_aaa(x1, x2, x3) = rev_in_aaa rev_out_aaa(x1, x2, x3) = rev_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U15_gaaaa(x1, x2, x3, x4, x5, x6) = U15_gaaaa(x1, x6) U18_agaa(x1, x2, x3, x4, x5, x6) = U18_agaa(x2, x6) pds1_in_aaaa(x1, x2, x3, x4) = pds1_in_aaaa pds1_out_aaaa(x1, x2, x3, x4) = pds1_out_aaaa U16_aaaa(x1, x2, x3, x4, x5, x6) = U16_aaaa(x6) U17_aaaa(x1, x2, x3, x4, x5, x6) = U17_aaaa(x6) check_in_aaaaa(x1, x2, x3, x4, x5) = check_in_aaaaa check_out_aaaaa(x1, x2, x3, x4, x5) = check_out_aaaaa U12_aaaaa(x1, x2, x3, x4, x5, x6) = U12_aaaaa(x6) dif_in_aaaaa(x1, x2, x3, x4, x5) = dif_in_aaaaa dif_out_aaaaa(x1, x2, x3, x4, x5) = dif_out_aaaaa U5_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_aaaaa(x9) U6_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_aaaaa(x9) U7_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_aaaaa(x9) U13_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_aaaaa(x7) U14_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_aaaaa(x7) U15_aaaaa(x1, x2, x3, x4, x5, x6) = U15_aaaaa(x6) U18_aaaa(x1, x2, x3, x4, x5, x6) = U18_aaaa(x6) pds_out_ga(x1, x2) = pds_out_ga(x1) REV_IN_AAA(x1, x2, x3) = REV_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (314) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (315) Obligation: Pi DP problem: The TRS P consists of the following rules: REV_IN_AAA(.(X, Xs), Y, L) -> REV_IN_AAA(Xs, .(X, Y), L) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) REV_IN_AAA(x1, x2, x3) = REV_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (316) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (317) Obligation: Q DP problem: The TRS P consists of the following rules: REV_IN_AAA -> REV_IN_AAA R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (318) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by semiunifying a rule from P directly. s = REV_IN_AAA evaluates to t =REV_IN_AAA Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence The DP semiunifies directly so there is only one rewrite step from REV_IN_AAA to REV_IN_AAA. ---------------------------------------- (319) NO ---------------------------------------- (320) Obligation: Pi DP problem: The TRS P consists of the following rules: MERGEDELETE_IN_AAA(.(D, Ds), .(X, R), .(X, L2)) -> U10_AAA(D, Ds, X, R, L2, >_in_aa(D, X)) U10_AAA(D, Ds, X, R, L2, >_out_aa(D, X)) -> MERGEDELETE_IN_AAA(.(D, Ds), R, L2) MERGEDELETE_IN_AAA(.(D, Ds), .(D, R), L2) -> MERGEDELETE_IN_AAA(Ds, R, L2) The TRS R consists of the following rules: pds_in_ga(Order, .(0, Ans)) -> U19_ga(Order, Ans, is_in_ag(N, +(*(Order, +(Order, 1)), 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U19_ga(Order, Ans, is_out_ag(N, +(*(Order, +(Order, 1)), 1))) -> U20_ga(Order, Ans, N, iota_in_aa(N, .(0, List))) iota_in_aa(N, List) -> U2_aa(N, List, iota1_in_gaa(0, N, List)) iota1_in_gaa(K, K, []) -> iota1_out_gaa(K, K, []) iota1_in_gaa(K, N, .(K, List)) -> U3_gaa(K, N, List, is_in_ag(K1, +(K, 1))) U3_gaa(K, N, List, is_out_ag(K1, +(K, 1))) -> U4_gaa(K, N, List, iota1_in_aaa(K1, N, List)) iota1_in_aaa(K, K, []) -> iota1_out_aaa(K, K, []) iota1_in_aaa(K, N, .(K, List)) -> U3_aaa(K, N, List, is_in_aa(K1, +(K, 1))) is_in_aa(X0, X1) -> is_out_aa(X0, X1) U3_aaa(K, N, List, is_out_aa(K1, +(K, 1))) -> U4_aaa(K, N, List, iota1_in_aaa(K1, N, List)) U4_aaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_aaa(K, N, .(K, List)) U4_gaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_gaa(K, N, .(K, List)) U2_aa(N, List, iota1_out_gaa(0, N, List)) -> iota_out_aa(N, List) U20_ga(Order, Ans, N, iota_out_aa(N, .(0, List))) -> U21_ga(Order, Ans, pds1_in_agaa(List, .(0, []), Ans, N)) pds1_in_agaa([], X6, [], X7) -> pds1_out_agaa([], X6, [], X7) pds1_in_agaa(Unused, List, .(Choice, Rest), Mod) -> U16_agaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) member_in_aa(X, .(X, Xs)) -> member_out_aa(X, .(X, Xs)) member_in_aa(X, .(X1, Xs)) -> U1_aa(X, X1, Xs, member_in_aa(X, Xs)) U1_aa(X, X1, Xs, member_out_aa(X, Xs)) -> member_out_aa(X, .(X1, Xs)) U16_agaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_agaa(Unused, List, Choice, Rest, Mod, check_in_gaaaa(List, Choice, Unused, U3, Mod)) check_in_gaaaa([], X4, L, L, X5) -> check_out_gaaaa([], X4, L, L, X5) check_in_gaaaa(S, Choice, Old, L3, Modulus) -> U12_gaaaa(S, Choice, Old, L3, Modulus, dif_in_gaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_gaaaa([], X2, X3, [], []) -> dif_out_gaaaa([], X2, X3, [], []) dif_in_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gaaaa(Ss, Val, Mod, Ds, D2s)) U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_gaaaa(S, Choice, Old, L3, Modulus, dif_out_gaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) mergedelete_in_aaa([], L, L) -> mergedelete_out_aaa([], L, L) mergedelete_in_aaa(.(D, Ds), .(D, R), L2) -> U9_aaa(D, Ds, R, L2, mergedelete_in_aaa(Ds, R, L2)) mergedelete_in_aaa(.(D, Ds), .(X, R), .(X, L2)) -> U10_aaa(D, Ds, X, R, L2, >_in_aa(D, X)) >_in_aa(X0, X1) -> >_out_aa(X0, X1) U10_aaa(D, Ds, X, R, L2, >_out_aa(D, X)) -> U11_aaa(D, Ds, X, R, L2, mergedelete_in_aaa(.(D, Ds), R, L2)) U11_aaa(D, Ds, X, R, L2, mergedelete_out_aaa(.(D, Ds), R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(X, R), .(X, L2)) U9_aaa(D, Ds, R, L2, mergedelete_out_aaa(Ds, R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(D, R), L2) U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) rev_in_aga([], L, L) -> rev_out_aga([], L, L) rev_in_aga(.(X, Xs), Y, L) -> U8_aga(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) rev_in_aaa([], L, L) -> rev_out_aaa([], L, L) rev_in_aaa(.(X, Xs), Y, L) -> U8_aaa(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) U8_aaa(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aaa(.(X, Xs), Y, L) U8_aga(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aga(.(X, Xs), Y, L) U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_gaaaa(S, Choice, Old, L3, Modulus) U17_agaa(Unused, List, Choice, Rest, Mod, check_out_gaaaa(List, Choice, Unused, U3, Mod)) -> U18_agaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) pds1_in_aaaa([], X6, [], X7) -> pds1_out_aaaa([], X6, [], X7) pds1_in_aaaa(Unused, List, .(Choice, Rest), Mod) -> U16_aaaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) U16_aaaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_aaaa(Unused, List, Choice, Rest, Mod, check_in_aaaaa(List, Choice, Unused, U3, Mod)) check_in_aaaaa([], X4, L, L, X5) -> check_out_aaaaa([], X4, L, L, X5) check_in_aaaaa(S, Choice, Old, L3, Modulus) -> U12_aaaaa(S, Choice, Old, L3, Modulus, dif_in_aaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_aaaaa([], X2, X3, [], []) -> dif_out_aaaaa([], X2, X3, [], []) dif_in_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_aaaaa(Ss, Val, Mod, Ds, D2s)) U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_aaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_aaaaa(S, Choice, Old, L3, Modulus, dif_out_aaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_aaaaa(S, Choice, Old, L3, Modulus) U17_aaaa(Unused, List, Choice, Rest, Mod, check_out_aaaaa(List, Choice, Unused, U3, Mod)) -> U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_aaaa(Unused, List, .(Choice, Rest), Mod) U18_agaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_agaa(Unused, List, .(Choice, Rest), Mod) U21_ga(Order, Ans, pds1_out_agaa(List, .(0, []), Ans, N)) -> pds_out_ga(Order, .(0, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U19_ga(x1, x2, x3) = U19_ga(x1, x3) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag(x2) +(x1, x2) = +(x1, x2) *(x1, x2) = *(x1, x2) 1 = 1 U20_ga(x1, x2, x3, x4) = U20_ga(x1, x4) iota_in_aa(x1, x2) = iota_in_aa U2_aa(x1, x2, x3) = U2_aa(x3) iota1_in_gaa(x1, x2, x3) = iota1_in_gaa(x1) iota1_out_gaa(x1, x2, x3) = iota1_out_gaa(x1) U3_gaa(x1, x2, x3, x4) = U3_gaa(x1, x4) U4_gaa(x1, x2, x3, x4) = U4_gaa(x1, x4) iota1_in_aaa(x1, x2, x3) = iota1_in_aaa iota1_out_aaa(x1, x2, x3) = iota1_out_aaa U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) is_in_aa(x1, x2) = is_in_aa is_out_aa(x1, x2) = is_out_aa U4_aaa(x1, x2, x3, x4) = U4_aaa(x4) 0 = 0 iota_out_aa(x1, x2) = iota_out_aa U21_ga(x1, x2, x3) = U21_ga(x1, x3) pds1_in_agaa(x1, x2, x3, x4) = pds1_in_agaa(x2) pds1_out_agaa(x1, x2, x3, x4) = pds1_out_agaa(x2) U16_agaa(x1, x2, x3, x4, x5, x6) = U16_agaa(x2, x6) member_in_aa(x1, x2) = member_in_aa member_out_aa(x1, x2) = member_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U17_agaa(x1, x2, x3, x4, x5, x6) = U17_agaa(x2, x6) check_in_gaaaa(x1, x2, x3, x4, x5) = check_in_gaaaa(x1) [] = [] check_out_gaaaa(x1, x2, x3, x4, x5) = check_out_gaaaa(x1) U12_gaaaa(x1, x2, x3, x4, x5, x6) = U12_gaaaa(x1, x6) dif_in_gaaaa(x1, x2, x3, x4, x5) = dif_in_gaaaa(x1) dif_out_gaaaa(x1, x2, x3, x4, x5) = dif_out_gaaaa(x1) .(x1, x2) = .(x1, x2) U5_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_gaaaa(x1, x2, x9) U6_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gaaaa(x1, x2, x9) U7_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_gaaaa(x1, x2, x9) U13_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_gaaaa(x1, x7) mergedelete_in_aaa(x1, x2, x3) = mergedelete_in_aaa mergedelete_out_aaa(x1, x2, x3) = mergedelete_out_aaa U9_aaa(x1, x2, x3, x4, x5) = U9_aaa(x5) U10_aaa(x1, x2, x3, x4, x5, x6) = U10_aaa(x6) >_in_aa(x1, x2) = >_in_aa >_out_aa(x1, x2) = >_out_aa U11_aaa(x1, x2, x3, x4, x5, x6) = U11_aaa(x6) U14_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_gaaaa(x1, x7) rev_in_aga(x1, x2, x3) = rev_in_aga(x2) rev_out_aga(x1, x2, x3) = rev_out_aga(x2) U8_aga(x1, x2, x3, x4, x5) = U8_aga(x3, x5) rev_in_aaa(x1, x2, x3) = rev_in_aaa rev_out_aaa(x1, x2, x3) = rev_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U15_gaaaa(x1, x2, x3, x4, x5, x6) = U15_gaaaa(x1, x6) U18_agaa(x1, x2, x3, x4, x5, x6) = U18_agaa(x2, x6) pds1_in_aaaa(x1, x2, x3, x4) = pds1_in_aaaa pds1_out_aaaa(x1, x2, x3, x4) = pds1_out_aaaa U16_aaaa(x1, x2, x3, x4, x5, x6) = U16_aaaa(x6) U17_aaaa(x1, x2, x3, x4, x5, x6) = U17_aaaa(x6) check_in_aaaaa(x1, x2, x3, x4, x5) = check_in_aaaaa check_out_aaaaa(x1, x2, x3, x4, x5) = check_out_aaaaa U12_aaaaa(x1, x2, x3, x4, x5, x6) = U12_aaaaa(x6) dif_in_aaaaa(x1, x2, x3, x4, x5) = dif_in_aaaaa dif_out_aaaaa(x1, x2, x3, x4, x5) = dif_out_aaaaa U5_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_aaaaa(x9) U6_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_aaaaa(x9) U7_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_aaaaa(x9) U13_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_aaaaa(x7) U14_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_aaaaa(x7) U15_aaaaa(x1, x2, x3, x4, x5, x6) = U15_aaaaa(x6) U18_aaaa(x1, x2, x3, x4, x5, x6) = U18_aaaa(x6) pds_out_ga(x1, x2) = pds_out_ga(x1) MERGEDELETE_IN_AAA(x1, x2, x3) = MERGEDELETE_IN_AAA U10_AAA(x1, x2, x3, x4, x5, x6) = U10_AAA(x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (321) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (322) Obligation: Pi DP problem: The TRS P consists of the following rules: MERGEDELETE_IN_AAA(.(D, Ds), .(X, R), .(X, L2)) -> U10_AAA(D, Ds, X, R, L2, >_in_aa(D, X)) U10_AAA(D, Ds, X, R, L2, >_out_aa(D, X)) -> MERGEDELETE_IN_AAA(.(D, Ds), R, L2) MERGEDELETE_IN_AAA(.(D, Ds), .(D, R), L2) -> MERGEDELETE_IN_AAA(Ds, R, L2) The TRS R consists of the following rules: >_in_aa(X0, X1) -> >_out_aa(X0, X1) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) >_in_aa(x1, x2) = >_in_aa >_out_aa(x1, x2) = >_out_aa MERGEDELETE_IN_AAA(x1, x2, x3) = MERGEDELETE_IN_AAA U10_AAA(x1, x2, x3, x4, x5, x6) = U10_AAA(x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (323) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (324) Obligation: Q DP problem: The TRS P consists of the following rules: MERGEDELETE_IN_AAA -> U10_AAA(>_in_aa) U10_AAA(>_out_aa) -> MERGEDELETE_IN_AAA MERGEDELETE_IN_AAA -> MERGEDELETE_IN_AAA The TRS R consists of the following rules: >_in_aa -> >_out_aa The set Q consists of the following terms: >_in_aa We have to consider all (P,Q,R)-chains. ---------------------------------------- (325) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule MERGEDELETE_IN_AAA -> U10_AAA(>_in_aa) at position [0] we obtained the following new rules [LPAR04]: (MERGEDELETE_IN_AAA -> U10_AAA(>_out_aa),MERGEDELETE_IN_AAA -> U10_AAA(>_out_aa)) ---------------------------------------- (326) Obligation: Q DP problem: The TRS P consists of the following rules: U10_AAA(>_out_aa) -> MERGEDELETE_IN_AAA MERGEDELETE_IN_AAA -> MERGEDELETE_IN_AAA MERGEDELETE_IN_AAA -> U10_AAA(>_out_aa) The TRS R consists of the following rules: >_in_aa -> >_out_aa The set Q consists of the following terms: >_in_aa We have to consider all (P,Q,R)-chains. ---------------------------------------- (327) 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. ---------------------------------------- (328) Obligation: Q DP problem: The TRS P consists of the following rules: U10_AAA(>_out_aa) -> MERGEDELETE_IN_AAA MERGEDELETE_IN_AAA -> MERGEDELETE_IN_AAA MERGEDELETE_IN_AAA -> U10_AAA(>_out_aa) R is empty. The set Q consists of the following terms: >_in_aa We have to consider all (P,Q,R)-chains. ---------------------------------------- (329) 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]. >_in_aa ---------------------------------------- (330) Obligation: Q DP problem: The TRS P consists of the following rules: U10_AAA(>_out_aa) -> MERGEDELETE_IN_AAA MERGEDELETE_IN_AAA -> MERGEDELETE_IN_AAA MERGEDELETE_IN_AAA -> U10_AAA(>_out_aa) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (331) Obligation: Pi DP problem: The TRS P consists of the following rules: U5_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> DIF_IN_GAAAA(Ss, Val, Mod, Ds, D2s) DIF_IN_GAAAA(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) The TRS R consists of the following rules: pds_in_ga(Order, .(0, Ans)) -> U19_ga(Order, Ans, is_in_ag(N, +(*(Order, +(Order, 1)), 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U19_ga(Order, Ans, is_out_ag(N, +(*(Order, +(Order, 1)), 1))) -> U20_ga(Order, Ans, N, iota_in_aa(N, .(0, List))) iota_in_aa(N, List) -> U2_aa(N, List, iota1_in_gaa(0, N, List)) iota1_in_gaa(K, K, []) -> iota1_out_gaa(K, K, []) iota1_in_gaa(K, N, .(K, List)) -> U3_gaa(K, N, List, is_in_ag(K1, +(K, 1))) U3_gaa(K, N, List, is_out_ag(K1, +(K, 1))) -> U4_gaa(K, N, List, iota1_in_aaa(K1, N, List)) iota1_in_aaa(K, K, []) -> iota1_out_aaa(K, K, []) iota1_in_aaa(K, N, .(K, List)) -> U3_aaa(K, N, List, is_in_aa(K1, +(K, 1))) is_in_aa(X0, X1) -> is_out_aa(X0, X1) U3_aaa(K, N, List, is_out_aa(K1, +(K, 1))) -> U4_aaa(K, N, List, iota1_in_aaa(K1, N, List)) U4_aaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_aaa(K, N, .(K, List)) U4_gaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_gaa(K, N, .(K, List)) U2_aa(N, List, iota1_out_gaa(0, N, List)) -> iota_out_aa(N, List) U20_ga(Order, Ans, N, iota_out_aa(N, .(0, List))) -> U21_ga(Order, Ans, pds1_in_agaa(List, .(0, []), Ans, N)) pds1_in_agaa([], X6, [], X7) -> pds1_out_agaa([], X6, [], X7) pds1_in_agaa(Unused, List, .(Choice, Rest), Mod) -> U16_agaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) member_in_aa(X, .(X, Xs)) -> member_out_aa(X, .(X, Xs)) member_in_aa(X, .(X1, Xs)) -> U1_aa(X, X1, Xs, member_in_aa(X, Xs)) U1_aa(X, X1, Xs, member_out_aa(X, Xs)) -> member_out_aa(X, .(X1, Xs)) U16_agaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_agaa(Unused, List, Choice, Rest, Mod, check_in_gaaaa(List, Choice, Unused, U3, Mod)) check_in_gaaaa([], X4, L, L, X5) -> check_out_gaaaa([], X4, L, L, X5) check_in_gaaaa(S, Choice, Old, L3, Modulus) -> U12_gaaaa(S, Choice, Old, L3, Modulus, dif_in_gaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_gaaaa([], X2, X3, [], []) -> dif_out_gaaaa([], X2, X3, [], []) dif_in_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gaaaa(Ss, Val, Mod, Ds, D2s)) U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_gaaaa(S, Choice, Old, L3, Modulus, dif_out_gaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) mergedelete_in_aaa([], L, L) -> mergedelete_out_aaa([], L, L) mergedelete_in_aaa(.(D, Ds), .(D, R), L2) -> U9_aaa(D, Ds, R, L2, mergedelete_in_aaa(Ds, R, L2)) mergedelete_in_aaa(.(D, Ds), .(X, R), .(X, L2)) -> U10_aaa(D, Ds, X, R, L2, >_in_aa(D, X)) >_in_aa(X0, X1) -> >_out_aa(X0, X1) U10_aaa(D, Ds, X, R, L2, >_out_aa(D, X)) -> U11_aaa(D, Ds, X, R, L2, mergedelete_in_aaa(.(D, Ds), R, L2)) U11_aaa(D, Ds, X, R, L2, mergedelete_out_aaa(.(D, Ds), R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(X, R), .(X, L2)) U9_aaa(D, Ds, R, L2, mergedelete_out_aaa(Ds, R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(D, R), L2) U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) rev_in_aga([], L, L) -> rev_out_aga([], L, L) rev_in_aga(.(X, Xs), Y, L) -> U8_aga(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) rev_in_aaa([], L, L) -> rev_out_aaa([], L, L) rev_in_aaa(.(X, Xs), Y, L) -> U8_aaa(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) U8_aaa(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aaa(.(X, Xs), Y, L) U8_aga(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aga(.(X, Xs), Y, L) U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_gaaaa(S, Choice, Old, L3, Modulus) U17_agaa(Unused, List, Choice, Rest, Mod, check_out_gaaaa(List, Choice, Unused, U3, Mod)) -> U18_agaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) pds1_in_aaaa([], X6, [], X7) -> pds1_out_aaaa([], X6, [], X7) pds1_in_aaaa(Unused, List, .(Choice, Rest), Mod) -> U16_aaaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) U16_aaaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_aaaa(Unused, List, Choice, Rest, Mod, check_in_aaaaa(List, Choice, Unused, U3, Mod)) check_in_aaaaa([], X4, L, L, X5) -> check_out_aaaaa([], X4, L, L, X5) check_in_aaaaa(S, Choice, Old, L3, Modulus) -> U12_aaaaa(S, Choice, Old, L3, Modulus, dif_in_aaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_aaaaa([], X2, X3, [], []) -> dif_out_aaaaa([], X2, X3, [], []) dif_in_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_aaaaa(Ss, Val, Mod, Ds, D2s)) U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_aaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_aaaaa(S, Choice, Old, L3, Modulus, dif_out_aaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_aaaaa(S, Choice, Old, L3, Modulus) U17_aaaa(Unused, List, Choice, Rest, Mod, check_out_aaaaa(List, Choice, Unused, U3, Mod)) -> U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_aaaa(Unused, List, .(Choice, Rest), Mod) U18_agaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_agaa(Unused, List, .(Choice, Rest), Mod) U21_ga(Order, Ans, pds1_out_agaa(List, .(0, []), Ans, N)) -> pds_out_ga(Order, .(0, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U19_ga(x1, x2, x3) = U19_ga(x1, x3) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag(x2) +(x1, x2) = +(x1, x2) *(x1, x2) = *(x1, x2) 1 = 1 U20_ga(x1, x2, x3, x4) = U20_ga(x1, x4) iota_in_aa(x1, x2) = iota_in_aa U2_aa(x1, x2, x3) = U2_aa(x3) iota1_in_gaa(x1, x2, x3) = iota1_in_gaa(x1) iota1_out_gaa(x1, x2, x3) = iota1_out_gaa(x1) U3_gaa(x1, x2, x3, x4) = U3_gaa(x1, x4) U4_gaa(x1, x2, x3, x4) = U4_gaa(x1, x4) iota1_in_aaa(x1, x2, x3) = iota1_in_aaa iota1_out_aaa(x1, x2, x3) = iota1_out_aaa U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) is_in_aa(x1, x2) = is_in_aa is_out_aa(x1, x2) = is_out_aa U4_aaa(x1, x2, x3, x4) = U4_aaa(x4) 0 = 0 iota_out_aa(x1, x2) = iota_out_aa U21_ga(x1, x2, x3) = U21_ga(x1, x3) pds1_in_agaa(x1, x2, x3, x4) = pds1_in_agaa(x2) pds1_out_agaa(x1, x2, x3, x4) = pds1_out_agaa(x2) U16_agaa(x1, x2, x3, x4, x5, x6) = U16_agaa(x2, x6) member_in_aa(x1, x2) = member_in_aa member_out_aa(x1, x2) = member_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U17_agaa(x1, x2, x3, x4, x5, x6) = U17_agaa(x2, x6) check_in_gaaaa(x1, x2, x3, x4, x5) = check_in_gaaaa(x1) [] = [] check_out_gaaaa(x1, x2, x3, x4, x5) = check_out_gaaaa(x1) U12_gaaaa(x1, x2, x3, x4, x5, x6) = U12_gaaaa(x1, x6) dif_in_gaaaa(x1, x2, x3, x4, x5) = dif_in_gaaaa(x1) dif_out_gaaaa(x1, x2, x3, x4, x5) = dif_out_gaaaa(x1) .(x1, x2) = .(x1, x2) U5_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_gaaaa(x1, x2, x9) U6_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gaaaa(x1, x2, x9) U7_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_gaaaa(x1, x2, x9) U13_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_gaaaa(x1, x7) mergedelete_in_aaa(x1, x2, x3) = mergedelete_in_aaa mergedelete_out_aaa(x1, x2, x3) = mergedelete_out_aaa U9_aaa(x1, x2, x3, x4, x5) = U9_aaa(x5) U10_aaa(x1, x2, x3, x4, x5, x6) = U10_aaa(x6) >_in_aa(x1, x2) = >_in_aa >_out_aa(x1, x2) = >_out_aa U11_aaa(x1, x2, x3, x4, x5, x6) = U11_aaa(x6) U14_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_gaaaa(x1, x7) rev_in_aga(x1, x2, x3) = rev_in_aga(x2) rev_out_aga(x1, x2, x3) = rev_out_aga(x2) U8_aga(x1, x2, x3, x4, x5) = U8_aga(x3, x5) rev_in_aaa(x1, x2, x3) = rev_in_aaa rev_out_aaa(x1, x2, x3) = rev_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U15_gaaaa(x1, x2, x3, x4, x5, x6) = U15_gaaaa(x1, x6) U18_agaa(x1, x2, x3, x4, x5, x6) = U18_agaa(x2, x6) pds1_in_aaaa(x1, x2, x3, x4) = pds1_in_aaaa pds1_out_aaaa(x1, x2, x3, x4) = pds1_out_aaaa U16_aaaa(x1, x2, x3, x4, x5, x6) = U16_aaaa(x6) U17_aaaa(x1, x2, x3, x4, x5, x6) = U17_aaaa(x6) check_in_aaaaa(x1, x2, x3, x4, x5) = check_in_aaaaa check_out_aaaaa(x1, x2, x3, x4, x5) = check_out_aaaaa U12_aaaaa(x1, x2, x3, x4, x5, x6) = U12_aaaaa(x6) dif_in_aaaaa(x1, x2, x3, x4, x5) = dif_in_aaaaa dif_out_aaaaa(x1, x2, x3, x4, x5) = dif_out_aaaaa U5_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_aaaaa(x9) U6_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_aaaaa(x9) U7_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_aaaaa(x9) U13_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_aaaaa(x7) U14_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_aaaaa(x7) U15_aaaaa(x1, x2, x3, x4, x5, x6) = U15_aaaaa(x6) U18_aaaa(x1, x2, x3, x4, x5, x6) = U18_aaaa(x6) pds_out_ga(x1, x2) = pds_out_ga(x1) DIF_IN_GAAAA(x1, x2, x3, x4, x5) = DIF_IN_GAAAA(x1) U5_GAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_GAAAA(x1, x2, x9) U6_GAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_GAAAA(x1, x2, x9) We have to consider all (P,R,Pi)-chains ---------------------------------------- (332) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (333) Obligation: Pi DP problem: The TRS P consists of the following rules: U5_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> DIF_IN_GAAAA(Ss, Val, Mod, Ds, D2s) DIF_IN_GAAAA(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_GAAAA(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) The TRS R consists of the following rules: is_in_aa(X0, X1) -> is_out_aa(X0, X1) The argument filtering Pi contains the following mapping: is_in_aa(x1, x2) = is_in_aa is_out_aa(x1, x2) = is_out_aa .(x1, x2) = .(x1, x2) DIF_IN_GAAAA(x1, x2, x3, x4, x5) = DIF_IN_GAAAA(x1) U5_GAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_GAAAA(x1, x2, x9) U6_GAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_GAAAA(x1, x2, x9) We have to consider all (P,R,Pi)-chains ---------------------------------------- (334) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER_IN_AA(X, .(X1, Xs)) -> MEMBER_IN_AA(X, Xs) The TRS R consists of the following rules: pds_in_ga(Order, .(0, Ans)) -> U19_ga(Order, Ans, is_in_ag(N, +(*(Order, +(Order, 1)), 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U19_ga(Order, Ans, is_out_ag(N, +(*(Order, +(Order, 1)), 1))) -> U20_ga(Order, Ans, N, iota_in_aa(N, .(0, List))) iota_in_aa(N, List) -> U2_aa(N, List, iota1_in_gaa(0, N, List)) iota1_in_gaa(K, K, []) -> iota1_out_gaa(K, K, []) iota1_in_gaa(K, N, .(K, List)) -> U3_gaa(K, N, List, is_in_ag(K1, +(K, 1))) U3_gaa(K, N, List, is_out_ag(K1, +(K, 1))) -> U4_gaa(K, N, List, iota1_in_aaa(K1, N, List)) iota1_in_aaa(K, K, []) -> iota1_out_aaa(K, K, []) iota1_in_aaa(K, N, .(K, List)) -> U3_aaa(K, N, List, is_in_aa(K1, +(K, 1))) is_in_aa(X0, X1) -> is_out_aa(X0, X1) U3_aaa(K, N, List, is_out_aa(K1, +(K, 1))) -> U4_aaa(K, N, List, iota1_in_aaa(K1, N, List)) U4_aaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_aaa(K, N, .(K, List)) U4_gaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_gaa(K, N, .(K, List)) U2_aa(N, List, iota1_out_gaa(0, N, List)) -> iota_out_aa(N, List) U20_ga(Order, Ans, N, iota_out_aa(N, .(0, List))) -> U21_ga(Order, Ans, pds1_in_agaa(List, .(0, []), Ans, N)) pds1_in_agaa([], X6, [], X7) -> pds1_out_agaa([], X6, [], X7) pds1_in_agaa(Unused, List, .(Choice, Rest), Mod) -> U16_agaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) member_in_aa(X, .(X, Xs)) -> member_out_aa(X, .(X, Xs)) member_in_aa(X, .(X1, Xs)) -> U1_aa(X, X1, Xs, member_in_aa(X, Xs)) U1_aa(X, X1, Xs, member_out_aa(X, Xs)) -> member_out_aa(X, .(X1, Xs)) U16_agaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_agaa(Unused, List, Choice, Rest, Mod, check_in_gaaaa(List, Choice, Unused, U3, Mod)) check_in_gaaaa([], X4, L, L, X5) -> check_out_gaaaa([], X4, L, L, X5) check_in_gaaaa(S, Choice, Old, L3, Modulus) -> U12_gaaaa(S, Choice, Old, L3, Modulus, dif_in_gaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_gaaaa([], X2, X3, [], []) -> dif_out_gaaaa([], X2, X3, [], []) dif_in_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gaaaa(Ss, Val, Mod, Ds, D2s)) U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_gaaaa(S, Choice, Old, L3, Modulus, dif_out_gaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) mergedelete_in_aaa([], L, L) -> mergedelete_out_aaa([], L, L) mergedelete_in_aaa(.(D, Ds), .(D, R), L2) -> U9_aaa(D, Ds, R, L2, mergedelete_in_aaa(Ds, R, L2)) mergedelete_in_aaa(.(D, Ds), .(X, R), .(X, L2)) -> U10_aaa(D, Ds, X, R, L2, >_in_aa(D, X)) >_in_aa(X0, X1) -> >_out_aa(X0, X1) U10_aaa(D, Ds, X, R, L2, >_out_aa(D, X)) -> U11_aaa(D, Ds, X, R, L2, mergedelete_in_aaa(.(D, Ds), R, L2)) U11_aaa(D, Ds, X, R, L2, mergedelete_out_aaa(.(D, Ds), R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(X, R), .(X, L2)) U9_aaa(D, Ds, R, L2, mergedelete_out_aaa(Ds, R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(D, R), L2) U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) rev_in_aga([], L, L) -> rev_out_aga([], L, L) rev_in_aga(.(X, Xs), Y, L) -> U8_aga(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) rev_in_aaa([], L, L) -> rev_out_aaa([], L, L) rev_in_aaa(.(X, Xs), Y, L) -> U8_aaa(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) U8_aaa(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aaa(.(X, Xs), Y, L) U8_aga(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aga(.(X, Xs), Y, L) U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_gaaaa(S, Choice, Old, L3, Modulus) U17_agaa(Unused, List, Choice, Rest, Mod, check_out_gaaaa(List, Choice, Unused, U3, Mod)) -> U18_agaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) pds1_in_aaaa([], X6, [], X7) -> pds1_out_aaaa([], X6, [], X7) pds1_in_aaaa(Unused, List, .(Choice, Rest), Mod) -> U16_aaaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) U16_aaaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_aaaa(Unused, List, Choice, Rest, Mod, check_in_aaaaa(List, Choice, Unused, U3, Mod)) check_in_aaaaa([], X4, L, L, X5) -> check_out_aaaaa([], X4, L, L, X5) check_in_aaaaa(S, Choice, Old, L3, Modulus) -> U12_aaaaa(S, Choice, Old, L3, Modulus, dif_in_aaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_aaaaa([], X2, X3, [], []) -> dif_out_aaaaa([], X2, X3, [], []) dif_in_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_aaaaa(Ss, Val, Mod, Ds, D2s)) U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_aaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_aaaaa(S, Choice, Old, L3, Modulus, dif_out_aaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_aaaaa(S, Choice, Old, L3, Modulus) U17_aaaa(Unused, List, Choice, Rest, Mod, check_out_aaaaa(List, Choice, Unused, U3, Mod)) -> U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_aaaa(Unused, List, .(Choice, Rest), Mod) U18_agaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_agaa(Unused, List, .(Choice, Rest), Mod) U21_ga(Order, Ans, pds1_out_agaa(List, .(0, []), Ans, N)) -> pds_out_ga(Order, .(0, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U19_ga(x1, x2, x3) = U19_ga(x1, x3) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag(x2) +(x1, x2) = +(x1, x2) *(x1, x2) = *(x1, x2) 1 = 1 U20_ga(x1, x2, x3, x4) = U20_ga(x1, x4) iota_in_aa(x1, x2) = iota_in_aa U2_aa(x1, x2, x3) = U2_aa(x3) iota1_in_gaa(x1, x2, x3) = iota1_in_gaa(x1) iota1_out_gaa(x1, x2, x3) = iota1_out_gaa(x1) U3_gaa(x1, x2, x3, x4) = U3_gaa(x1, x4) U4_gaa(x1, x2, x3, x4) = U4_gaa(x1, x4) iota1_in_aaa(x1, x2, x3) = iota1_in_aaa iota1_out_aaa(x1, x2, x3) = iota1_out_aaa U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) is_in_aa(x1, x2) = is_in_aa is_out_aa(x1, x2) = is_out_aa U4_aaa(x1, x2, x3, x4) = U4_aaa(x4) 0 = 0 iota_out_aa(x1, x2) = iota_out_aa U21_ga(x1, x2, x3) = U21_ga(x1, x3) pds1_in_agaa(x1, x2, x3, x4) = pds1_in_agaa(x2) pds1_out_agaa(x1, x2, x3, x4) = pds1_out_agaa(x2) U16_agaa(x1, x2, x3, x4, x5, x6) = U16_agaa(x2, x6) member_in_aa(x1, x2) = member_in_aa member_out_aa(x1, x2) = member_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U17_agaa(x1, x2, x3, x4, x5, x6) = U17_agaa(x2, x6) check_in_gaaaa(x1, x2, x3, x4, x5) = check_in_gaaaa(x1) [] = [] check_out_gaaaa(x1, x2, x3, x4, x5) = check_out_gaaaa(x1) U12_gaaaa(x1, x2, x3, x4, x5, x6) = U12_gaaaa(x1, x6) dif_in_gaaaa(x1, x2, x3, x4, x5) = dif_in_gaaaa(x1) dif_out_gaaaa(x1, x2, x3, x4, x5) = dif_out_gaaaa(x1) .(x1, x2) = .(x1, x2) U5_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_gaaaa(x1, x2, x9) U6_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gaaaa(x1, x2, x9) U7_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_gaaaa(x1, x2, x9) U13_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_gaaaa(x1, x7) mergedelete_in_aaa(x1, x2, x3) = mergedelete_in_aaa mergedelete_out_aaa(x1, x2, x3) = mergedelete_out_aaa U9_aaa(x1, x2, x3, x4, x5) = U9_aaa(x5) U10_aaa(x1, x2, x3, x4, x5, x6) = U10_aaa(x6) >_in_aa(x1, x2) = >_in_aa >_out_aa(x1, x2) = >_out_aa U11_aaa(x1, x2, x3, x4, x5, x6) = U11_aaa(x6) U14_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_gaaaa(x1, x7) rev_in_aga(x1, x2, x3) = rev_in_aga(x2) rev_out_aga(x1, x2, x3) = rev_out_aga(x2) U8_aga(x1, x2, x3, x4, x5) = U8_aga(x3, x5) rev_in_aaa(x1, x2, x3) = rev_in_aaa rev_out_aaa(x1, x2, x3) = rev_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U15_gaaaa(x1, x2, x3, x4, x5, x6) = U15_gaaaa(x1, x6) U18_agaa(x1, x2, x3, x4, x5, x6) = U18_agaa(x2, x6) pds1_in_aaaa(x1, x2, x3, x4) = pds1_in_aaaa pds1_out_aaaa(x1, x2, x3, x4) = pds1_out_aaaa U16_aaaa(x1, x2, x3, x4, x5, x6) = U16_aaaa(x6) U17_aaaa(x1, x2, x3, x4, x5, x6) = U17_aaaa(x6) check_in_aaaaa(x1, x2, x3, x4, x5) = check_in_aaaaa check_out_aaaaa(x1, x2, x3, x4, x5) = check_out_aaaaa U12_aaaaa(x1, x2, x3, x4, x5, x6) = U12_aaaaa(x6) dif_in_aaaaa(x1, x2, x3, x4, x5) = dif_in_aaaaa dif_out_aaaaa(x1, x2, x3, x4, x5) = dif_out_aaaaa U5_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_aaaaa(x9) U6_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_aaaaa(x9) U7_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_aaaaa(x9) U13_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_aaaaa(x7) U14_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_aaaaa(x7) U15_aaaaa(x1, x2, x3, x4, x5, x6) = U15_aaaaa(x6) U18_aaaa(x1, x2, x3, x4, x5, x6) = U18_aaaa(x6) pds_out_ga(x1, x2) = pds_out_ga(x1) MEMBER_IN_AA(x1, x2) = MEMBER_IN_AA We have to consider all (P,R,Pi)-chains ---------------------------------------- (335) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (336) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER_IN_AA(X, .(X1, Xs)) -> MEMBER_IN_AA(X, Xs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) MEMBER_IN_AA(x1, x2) = MEMBER_IN_AA We have to consider all (P,R,Pi)-chains ---------------------------------------- (337) Obligation: Pi DP problem: The TRS P consists of the following rules: PDS1_IN_AAAA(Unused, List, .(Choice, Rest), Mod) -> U16_AAAA(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) U16_AAAA(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_AAAA(Unused, List, Choice, Rest, Mod, check_in_aaaaa(List, Choice, Unused, U3, Mod)) U17_AAAA(Unused, List, Choice, Rest, Mod, check_out_aaaaa(List, Choice, Unused, U3, Mod)) -> PDS1_IN_AAAA(U3, .(Choice, List), Rest, Mod) The TRS R consists of the following rules: pds_in_ga(Order, .(0, Ans)) -> U19_ga(Order, Ans, is_in_ag(N, +(*(Order, +(Order, 1)), 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U19_ga(Order, Ans, is_out_ag(N, +(*(Order, +(Order, 1)), 1))) -> U20_ga(Order, Ans, N, iota_in_aa(N, .(0, List))) iota_in_aa(N, List) -> U2_aa(N, List, iota1_in_gaa(0, N, List)) iota1_in_gaa(K, K, []) -> iota1_out_gaa(K, K, []) iota1_in_gaa(K, N, .(K, List)) -> U3_gaa(K, N, List, is_in_ag(K1, +(K, 1))) U3_gaa(K, N, List, is_out_ag(K1, +(K, 1))) -> U4_gaa(K, N, List, iota1_in_aaa(K1, N, List)) iota1_in_aaa(K, K, []) -> iota1_out_aaa(K, K, []) iota1_in_aaa(K, N, .(K, List)) -> U3_aaa(K, N, List, is_in_aa(K1, +(K, 1))) is_in_aa(X0, X1) -> is_out_aa(X0, X1) U3_aaa(K, N, List, is_out_aa(K1, +(K, 1))) -> U4_aaa(K, N, List, iota1_in_aaa(K1, N, List)) U4_aaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_aaa(K, N, .(K, List)) U4_gaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_gaa(K, N, .(K, List)) U2_aa(N, List, iota1_out_gaa(0, N, List)) -> iota_out_aa(N, List) U20_ga(Order, Ans, N, iota_out_aa(N, .(0, List))) -> U21_ga(Order, Ans, pds1_in_agaa(List, .(0, []), Ans, N)) pds1_in_agaa([], X6, [], X7) -> pds1_out_agaa([], X6, [], X7) pds1_in_agaa(Unused, List, .(Choice, Rest), Mod) -> U16_agaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) member_in_aa(X, .(X, Xs)) -> member_out_aa(X, .(X, Xs)) member_in_aa(X, .(X1, Xs)) -> U1_aa(X, X1, Xs, member_in_aa(X, Xs)) U1_aa(X, X1, Xs, member_out_aa(X, Xs)) -> member_out_aa(X, .(X1, Xs)) U16_agaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_agaa(Unused, List, Choice, Rest, Mod, check_in_gaaaa(List, Choice, Unused, U3, Mod)) check_in_gaaaa([], X4, L, L, X5) -> check_out_gaaaa([], X4, L, L, X5) check_in_gaaaa(S, Choice, Old, L3, Modulus) -> U12_gaaaa(S, Choice, Old, L3, Modulus, dif_in_gaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_gaaaa([], X2, X3, [], []) -> dif_out_gaaaa([], X2, X3, [], []) dif_in_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gaaaa(Ss, Val, Mod, Ds, D2s)) U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_gaaaa(S, Choice, Old, L3, Modulus, dif_out_gaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) mergedelete_in_aaa([], L, L) -> mergedelete_out_aaa([], L, L) mergedelete_in_aaa(.(D, Ds), .(D, R), L2) -> U9_aaa(D, Ds, R, L2, mergedelete_in_aaa(Ds, R, L2)) mergedelete_in_aaa(.(D, Ds), .(X, R), .(X, L2)) -> U10_aaa(D, Ds, X, R, L2, >_in_aa(D, X)) >_in_aa(X0, X1) -> >_out_aa(X0, X1) U10_aaa(D, Ds, X, R, L2, >_out_aa(D, X)) -> U11_aaa(D, Ds, X, R, L2, mergedelete_in_aaa(.(D, Ds), R, L2)) U11_aaa(D, Ds, X, R, L2, mergedelete_out_aaa(.(D, Ds), R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(X, R), .(X, L2)) U9_aaa(D, Ds, R, L2, mergedelete_out_aaa(Ds, R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(D, R), L2) U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) rev_in_aga([], L, L) -> rev_out_aga([], L, L) rev_in_aga(.(X, Xs), Y, L) -> U8_aga(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) rev_in_aaa([], L, L) -> rev_out_aaa([], L, L) rev_in_aaa(.(X, Xs), Y, L) -> U8_aaa(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) U8_aaa(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aaa(.(X, Xs), Y, L) U8_aga(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aga(.(X, Xs), Y, L) U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_gaaaa(S, Choice, Old, L3, Modulus) U17_agaa(Unused, List, Choice, Rest, Mod, check_out_gaaaa(List, Choice, Unused, U3, Mod)) -> U18_agaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) pds1_in_aaaa([], X6, [], X7) -> pds1_out_aaaa([], X6, [], X7) pds1_in_aaaa(Unused, List, .(Choice, Rest), Mod) -> U16_aaaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) U16_aaaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_aaaa(Unused, List, Choice, Rest, Mod, check_in_aaaaa(List, Choice, Unused, U3, Mod)) check_in_aaaaa([], X4, L, L, X5) -> check_out_aaaaa([], X4, L, L, X5) check_in_aaaaa(S, Choice, Old, L3, Modulus) -> U12_aaaaa(S, Choice, Old, L3, Modulus, dif_in_aaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_aaaaa([], X2, X3, [], []) -> dif_out_aaaaa([], X2, X3, [], []) dif_in_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_aaaaa(Ss, Val, Mod, Ds, D2s)) U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_aaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_aaaaa(S, Choice, Old, L3, Modulus, dif_out_aaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_aaaaa(S, Choice, Old, L3, Modulus) U17_aaaa(Unused, List, Choice, Rest, Mod, check_out_aaaaa(List, Choice, Unused, U3, Mod)) -> U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_aaaa(Unused, List, .(Choice, Rest), Mod) U18_agaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_agaa(Unused, List, .(Choice, Rest), Mod) U21_ga(Order, Ans, pds1_out_agaa(List, .(0, []), Ans, N)) -> pds_out_ga(Order, .(0, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U19_ga(x1, x2, x3) = U19_ga(x1, x3) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag(x2) +(x1, x2) = +(x1, x2) *(x1, x2) = *(x1, x2) 1 = 1 U20_ga(x1, x2, x3, x4) = U20_ga(x1, x4) iota_in_aa(x1, x2) = iota_in_aa U2_aa(x1, x2, x3) = U2_aa(x3) iota1_in_gaa(x1, x2, x3) = iota1_in_gaa(x1) iota1_out_gaa(x1, x2, x3) = iota1_out_gaa(x1) U3_gaa(x1, x2, x3, x4) = U3_gaa(x1, x4) U4_gaa(x1, x2, x3, x4) = U4_gaa(x1, x4) iota1_in_aaa(x1, x2, x3) = iota1_in_aaa iota1_out_aaa(x1, x2, x3) = iota1_out_aaa U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) is_in_aa(x1, x2) = is_in_aa is_out_aa(x1, x2) = is_out_aa U4_aaa(x1, x2, x3, x4) = U4_aaa(x4) 0 = 0 iota_out_aa(x1, x2) = iota_out_aa U21_ga(x1, x2, x3) = U21_ga(x1, x3) pds1_in_agaa(x1, x2, x3, x4) = pds1_in_agaa(x2) pds1_out_agaa(x1, x2, x3, x4) = pds1_out_agaa(x2) U16_agaa(x1, x2, x3, x4, x5, x6) = U16_agaa(x2, x6) member_in_aa(x1, x2) = member_in_aa member_out_aa(x1, x2) = member_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U17_agaa(x1, x2, x3, x4, x5, x6) = U17_agaa(x2, x6) check_in_gaaaa(x1, x2, x3, x4, x5) = check_in_gaaaa(x1) [] = [] check_out_gaaaa(x1, x2, x3, x4, x5) = check_out_gaaaa(x1) U12_gaaaa(x1, x2, x3, x4, x5, x6) = U12_gaaaa(x1, x6) dif_in_gaaaa(x1, x2, x3, x4, x5) = dif_in_gaaaa(x1) dif_out_gaaaa(x1, x2, x3, x4, x5) = dif_out_gaaaa(x1) .(x1, x2) = .(x1, x2) U5_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_gaaaa(x1, x2, x9) U6_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gaaaa(x1, x2, x9) U7_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_gaaaa(x1, x2, x9) U13_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_gaaaa(x1, x7) mergedelete_in_aaa(x1, x2, x3) = mergedelete_in_aaa mergedelete_out_aaa(x1, x2, x3) = mergedelete_out_aaa U9_aaa(x1, x2, x3, x4, x5) = U9_aaa(x5) U10_aaa(x1, x2, x3, x4, x5, x6) = U10_aaa(x6) >_in_aa(x1, x2) = >_in_aa >_out_aa(x1, x2) = >_out_aa U11_aaa(x1, x2, x3, x4, x5, x6) = U11_aaa(x6) U14_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_gaaaa(x1, x7) rev_in_aga(x1, x2, x3) = rev_in_aga(x2) rev_out_aga(x1, x2, x3) = rev_out_aga(x2) U8_aga(x1, x2, x3, x4, x5) = U8_aga(x3, x5) rev_in_aaa(x1, x2, x3) = rev_in_aaa rev_out_aaa(x1, x2, x3) = rev_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U15_gaaaa(x1, x2, x3, x4, x5, x6) = U15_gaaaa(x1, x6) U18_agaa(x1, x2, x3, x4, x5, x6) = U18_agaa(x2, x6) pds1_in_aaaa(x1, x2, x3, x4) = pds1_in_aaaa pds1_out_aaaa(x1, x2, x3, x4) = pds1_out_aaaa U16_aaaa(x1, x2, x3, x4, x5, x6) = U16_aaaa(x6) U17_aaaa(x1, x2, x3, x4, x5, x6) = U17_aaaa(x6) check_in_aaaaa(x1, x2, x3, x4, x5) = check_in_aaaaa check_out_aaaaa(x1, x2, x3, x4, x5) = check_out_aaaaa U12_aaaaa(x1, x2, x3, x4, x5, x6) = U12_aaaaa(x6) dif_in_aaaaa(x1, x2, x3, x4, x5) = dif_in_aaaaa dif_out_aaaaa(x1, x2, x3, x4, x5) = dif_out_aaaaa U5_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_aaaaa(x9) U6_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_aaaaa(x9) U7_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_aaaaa(x9) U13_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_aaaaa(x7) U14_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_aaaaa(x7) U15_aaaaa(x1, x2, x3, x4, x5, x6) = U15_aaaaa(x6) U18_aaaa(x1, x2, x3, x4, x5, x6) = U18_aaaa(x6) pds_out_ga(x1, x2) = pds_out_ga(x1) PDS1_IN_AAAA(x1, x2, x3, x4) = PDS1_IN_AAAA U16_AAAA(x1, x2, x3, x4, x5, x6) = U16_AAAA(x6) U17_AAAA(x1, x2, x3, x4, x5, x6) = U17_AAAA(x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (338) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (339) Obligation: Pi DP problem: The TRS P consists of the following rules: PDS1_IN_AAAA(Unused, List, .(Choice, Rest), Mod) -> U16_AAAA(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) U16_AAAA(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_AAAA(Unused, List, Choice, Rest, Mod, check_in_aaaaa(List, Choice, Unused, U3, Mod)) U17_AAAA(Unused, List, Choice, Rest, Mod, check_out_aaaaa(List, Choice, Unused, U3, Mod)) -> PDS1_IN_AAAA(U3, .(Choice, List), Rest, Mod) The TRS R consists of the following rules: member_in_aa(X, .(X, Xs)) -> member_out_aa(X, .(X, Xs)) member_in_aa(X, .(X1, Xs)) -> U1_aa(X, X1, Xs, member_in_aa(X, Xs)) check_in_aaaaa([], X4, L, L, X5) -> check_out_aaaaa([], X4, L, L, X5) check_in_aaaaa(S, Choice, Old, L3, Modulus) -> U12_aaaaa(S, Choice, Old, L3, Modulus, dif_in_aaaaa(S, Choice, Modulus, Ds, Dds)) U1_aa(X, X1, Xs, member_out_aa(X, Xs)) -> member_out_aa(X, .(X1, Xs)) U12_aaaaa(S, Choice, Old, L3, Modulus, dif_out_aaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) dif_in_aaaaa([], X2, X3, [], []) -> dif_out_aaaaa([], X2, X3, [], []) dif_in_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) mergedelete_in_aaa([], L, L) -> mergedelete_out_aaa([], L, L) mergedelete_in_aaa(.(D, Ds), .(D, R), L2) -> U9_aaa(D, Ds, R, L2, mergedelete_in_aaa(Ds, R, L2)) mergedelete_in_aaa(.(D, Ds), .(X, R), .(X, L2)) -> U10_aaa(D, Ds, X, R, L2, >_in_aa(D, X)) U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) is_in_aa(X0, X1) -> is_out_aa(X0, X1) U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_aaaaa(Ss, Val, Mod, Ds, D2s)) U9_aaa(D, Ds, R, L2, mergedelete_out_aaa(Ds, R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(D, R), L2) U10_aaa(D, Ds, X, R, L2, >_out_aa(D, X)) -> U11_aaa(D, Ds, X, R, L2, mergedelete_in_aaa(.(D, Ds), R, L2)) rev_in_aga([], L, L) -> rev_out_aga([], L, L) rev_in_aga(.(X, Xs), Y, L) -> U8_aga(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_aaaaa(S, Choice, Old, L3, Modulus) U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_aaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) >_in_aa(X0, X1) -> >_out_aa(X0, X1) U11_aaa(D, Ds, X, R, L2, mergedelete_out_aaa(.(D, Ds), R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(X, R), .(X, L2)) U8_aga(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aga(.(X, Xs), Y, L) rev_in_aaa([], L, L) -> rev_out_aaa([], L, L) rev_in_aaa(.(X, Xs), Y, L) -> U8_aaa(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) U8_aaa(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aaa(.(X, Xs), Y, L) The argument filtering Pi contains the following mapping: is_in_aa(x1, x2) = is_in_aa is_out_aa(x1, x2) = is_out_aa member_in_aa(x1, x2) = member_in_aa member_out_aa(x1, x2) = member_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) [] = [] .(x1, x2) = .(x1, x2) mergedelete_in_aaa(x1, x2, x3) = mergedelete_in_aaa mergedelete_out_aaa(x1, x2, x3) = mergedelete_out_aaa U9_aaa(x1, x2, x3, x4, x5) = U9_aaa(x5) U10_aaa(x1, x2, x3, x4, x5, x6) = U10_aaa(x6) >_in_aa(x1, x2) = >_in_aa >_out_aa(x1, x2) = >_out_aa U11_aaa(x1, x2, x3, x4, x5, x6) = U11_aaa(x6) rev_in_aga(x1, x2, x3) = rev_in_aga(x2) rev_out_aga(x1, x2, x3) = rev_out_aga(x2) U8_aga(x1, x2, x3, x4, x5) = U8_aga(x3, x5) rev_in_aaa(x1, x2, x3) = rev_in_aaa rev_out_aaa(x1, x2, x3) = rev_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) check_in_aaaaa(x1, x2, x3, x4, x5) = check_in_aaaaa check_out_aaaaa(x1, x2, x3, x4, x5) = check_out_aaaaa U12_aaaaa(x1, x2, x3, x4, x5, x6) = U12_aaaaa(x6) dif_in_aaaaa(x1, x2, x3, x4, x5) = dif_in_aaaaa dif_out_aaaaa(x1, x2, x3, x4, x5) = dif_out_aaaaa U5_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_aaaaa(x9) U6_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_aaaaa(x9) U7_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_aaaaa(x9) U13_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_aaaaa(x7) U14_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_aaaaa(x7) U15_aaaaa(x1, x2, x3, x4, x5, x6) = U15_aaaaa(x6) PDS1_IN_AAAA(x1, x2, x3, x4) = PDS1_IN_AAAA U16_AAAA(x1, x2, x3, x4, x5, x6) = U16_AAAA(x6) U17_AAAA(x1, x2, x3, x4, x5, x6) = U17_AAAA(x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (340) Obligation: Pi DP problem: The TRS P consists of the following rules: U3_AAA(K, N, List, is_out_aa(K1, +(K, 1))) -> IOTA1_IN_AAA(K1, N, List) IOTA1_IN_AAA(K, N, .(K, List)) -> U3_AAA(K, N, List, is_in_aa(K1, +(K, 1))) The TRS R consists of the following rules: pds_in_ga(Order, .(0, Ans)) -> U19_ga(Order, Ans, is_in_ag(N, +(*(Order, +(Order, 1)), 1))) is_in_ag(X0, X1) -> is_out_ag(X0, X1) U19_ga(Order, Ans, is_out_ag(N, +(*(Order, +(Order, 1)), 1))) -> U20_ga(Order, Ans, N, iota_in_aa(N, .(0, List))) iota_in_aa(N, List) -> U2_aa(N, List, iota1_in_gaa(0, N, List)) iota1_in_gaa(K, K, []) -> iota1_out_gaa(K, K, []) iota1_in_gaa(K, N, .(K, List)) -> U3_gaa(K, N, List, is_in_ag(K1, +(K, 1))) U3_gaa(K, N, List, is_out_ag(K1, +(K, 1))) -> U4_gaa(K, N, List, iota1_in_aaa(K1, N, List)) iota1_in_aaa(K, K, []) -> iota1_out_aaa(K, K, []) iota1_in_aaa(K, N, .(K, List)) -> U3_aaa(K, N, List, is_in_aa(K1, +(K, 1))) is_in_aa(X0, X1) -> is_out_aa(X0, X1) U3_aaa(K, N, List, is_out_aa(K1, +(K, 1))) -> U4_aaa(K, N, List, iota1_in_aaa(K1, N, List)) U4_aaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_aaa(K, N, .(K, List)) U4_gaa(K, N, List, iota1_out_aaa(K1, N, List)) -> iota1_out_gaa(K, N, .(K, List)) U2_aa(N, List, iota1_out_gaa(0, N, List)) -> iota_out_aa(N, List) U20_ga(Order, Ans, N, iota_out_aa(N, .(0, List))) -> U21_ga(Order, Ans, pds1_in_agaa(List, .(0, []), Ans, N)) pds1_in_agaa([], X6, [], X7) -> pds1_out_agaa([], X6, [], X7) pds1_in_agaa(Unused, List, .(Choice, Rest), Mod) -> U16_agaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) member_in_aa(X, .(X, Xs)) -> member_out_aa(X, .(X, Xs)) member_in_aa(X, .(X1, Xs)) -> U1_aa(X, X1, Xs, member_in_aa(X, Xs)) U1_aa(X, X1, Xs, member_out_aa(X, Xs)) -> member_out_aa(X, .(X1, Xs)) U16_agaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_agaa(Unused, List, Choice, Rest, Mod, check_in_gaaaa(List, Choice, Unused, U3, Mod)) check_in_gaaaa([], X4, L, L, X5) -> check_out_gaaaa([], X4, L, L, X5) check_in_gaaaa(S, Choice, Old, L3, Modulus) -> U12_gaaaa(S, Choice, Old, L3, Modulus, dif_in_gaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_gaaaa([], X2, X3, [], []) -> dif_out_gaaaa([], X2, X3, [], []) dif_in_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_gaaaa(Ss, Val, Mod, Ds, D2s)) U7_gaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_gaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_gaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_gaaaa(S, Choice, Old, L3, Modulus, dif_out_gaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) mergedelete_in_aaa([], L, L) -> mergedelete_out_aaa([], L, L) mergedelete_in_aaa(.(D, Ds), .(D, R), L2) -> U9_aaa(D, Ds, R, L2, mergedelete_in_aaa(Ds, R, L2)) mergedelete_in_aaa(.(D, Ds), .(X, R), .(X, L2)) -> U10_aaa(D, Ds, X, R, L2, >_in_aa(D, X)) >_in_aa(X0, X1) -> >_out_aa(X0, X1) U10_aaa(D, Ds, X, R, L2, >_out_aa(D, X)) -> U11_aaa(D, Ds, X, R, L2, mergedelete_in_aaa(.(D, Ds), R, L2)) U11_aaa(D, Ds, X, R, L2, mergedelete_out_aaa(.(D, Ds), R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(X, R), .(X, L2)) U9_aaa(D, Ds, R, L2, mergedelete_out_aaa(Ds, R, L2)) -> mergedelete_out_aaa(.(D, Ds), .(D, R), L2) U13_gaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) rev_in_aga([], L, L) -> rev_out_aga([], L, L) rev_in_aga(.(X, Xs), Y, L) -> U8_aga(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) rev_in_aaa([], L, L) -> rev_out_aaa([], L, L) rev_in_aaa(.(X, Xs), Y, L) -> U8_aaa(X, Xs, Y, L, rev_in_aaa(Xs, .(X, Y), L)) U8_aaa(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aaa(.(X, Xs), Y, L) U8_aga(X, Xs, Y, L, rev_out_aaa(Xs, .(X, Y), L)) -> rev_out_aga(.(X, Xs), Y, L) U14_gaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_gaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_gaaaa(S, Choice, Old, L3, Modulus) U17_agaa(Unused, List, Choice, Rest, Mod, check_out_gaaaa(List, Choice, Unused, U3, Mod)) -> U18_agaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) pds1_in_aaaa([], X6, [], X7) -> pds1_out_aaaa([], X6, [], X7) pds1_in_aaaa(Unused, List, .(Choice, Rest), Mod) -> U16_aaaa(Unused, List, Choice, Rest, Mod, member_in_aa(Choice, Unused)) U16_aaaa(Unused, List, Choice, Rest, Mod, member_out_aa(Choice, Unused)) -> U17_aaaa(Unused, List, Choice, Rest, Mod, check_in_aaaaa(List, Choice, Unused, U3, Mod)) check_in_aaaaa([], X4, L, L, X5) -> check_out_aaaaa([], X4, L, L, X5) check_in_aaaaa(S, Choice, Old, L3, Modulus) -> U12_aaaaa(S, Choice, Old, L3, Modulus, dif_in_aaaaa(S, Choice, Modulus, Ds, Dds)) dif_in_aaaaa([], X2, X3, [], []) -> dif_out_aaaaa([], X2, X3, [], []) dif_in_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) -> U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D, -(Val, S))) U5_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D, -(Val, S))) -> U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_in_aa(D2, -(Mod, D))) U6_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, is_out_aa(D2, -(Mod, D))) -> U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_in_aaaaa(Ss, Val, Mod, Ds, D2s)) U7_aaaaa(S, Ss, Val, Mod, D, Ds, D2, D2s, dif_out_aaaaa(Ss, Val, Mod, Ds, D2s)) -> dif_out_aaaaa(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) U12_aaaaa(S, Choice, Old, L3, Modulus, dif_out_aaaaa(S, Choice, Modulus, Ds, Dds)) -> U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_in_aaa(Ds, Old, L2)) U13_aaaaa(S, Choice, Old, L3, Modulus, Dds, mergedelete_out_aaa(Ds, Old, L2)) -> U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_in_aga(Dds, [], Rds)) U14_aaaaa(S, Choice, Old, L3, Modulus, L2, rev_out_aga(Dds, [], Rds)) -> U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_in_aaa(Rds, L2, L3)) U15_aaaaa(S, Choice, Old, L3, Modulus, mergedelete_out_aaa(Rds, L2, L3)) -> check_out_aaaaa(S, Choice, Old, L3, Modulus) U17_aaaa(Unused, List, Choice, Rest, Mod, check_out_aaaaa(List, Choice, Unused, U3, Mod)) -> U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_in_aaaa(U3, .(Choice, List), Rest, Mod)) U18_aaaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_aaaa(Unused, List, .(Choice, Rest), Mod) U18_agaa(Unused, List, Choice, Rest, Mod, pds1_out_aaaa(U3, .(Choice, List), Rest, Mod)) -> pds1_out_agaa(Unused, List, .(Choice, Rest), Mod) U21_ga(Order, Ans, pds1_out_agaa(List, .(0, []), Ans, N)) -> pds_out_ga(Order, .(0, Ans)) The argument filtering Pi contains the following mapping: pds_in_ga(x1, x2) = pds_in_ga(x1) U19_ga(x1, x2, x3) = U19_ga(x1, x3) is_in_ag(x1, x2) = is_in_ag(x2) is_out_ag(x1, x2) = is_out_ag(x2) +(x1, x2) = +(x1, x2) *(x1, x2) = *(x1, x2) 1 = 1 U20_ga(x1, x2, x3, x4) = U20_ga(x1, x4) iota_in_aa(x1, x2) = iota_in_aa U2_aa(x1, x2, x3) = U2_aa(x3) iota1_in_gaa(x1, x2, x3) = iota1_in_gaa(x1) iota1_out_gaa(x1, x2, x3) = iota1_out_gaa(x1) U3_gaa(x1, x2, x3, x4) = U3_gaa(x1, x4) U4_gaa(x1, x2, x3, x4) = U4_gaa(x1, x4) iota1_in_aaa(x1, x2, x3) = iota1_in_aaa iota1_out_aaa(x1, x2, x3) = iota1_out_aaa U3_aaa(x1, x2, x3, x4) = U3_aaa(x4) is_in_aa(x1, x2) = is_in_aa is_out_aa(x1, x2) = is_out_aa U4_aaa(x1, x2, x3, x4) = U4_aaa(x4) 0 = 0 iota_out_aa(x1, x2) = iota_out_aa U21_ga(x1, x2, x3) = U21_ga(x1, x3) pds1_in_agaa(x1, x2, x3, x4) = pds1_in_agaa(x2) pds1_out_agaa(x1, x2, x3, x4) = pds1_out_agaa(x2) U16_agaa(x1, x2, x3, x4, x5, x6) = U16_agaa(x2, x6) member_in_aa(x1, x2) = member_in_aa member_out_aa(x1, x2) = member_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U17_agaa(x1, x2, x3, x4, x5, x6) = U17_agaa(x2, x6) check_in_gaaaa(x1, x2, x3, x4, x5) = check_in_gaaaa(x1) [] = [] check_out_gaaaa(x1, x2, x3, x4, x5) = check_out_gaaaa(x1) U12_gaaaa(x1, x2, x3, x4, x5, x6) = U12_gaaaa(x1, x6) dif_in_gaaaa(x1, x2, x3, x4, x5) = dif_in_gaaaa(x1) dif_out_gaaaa(x1, x2, x3, x4, x5) = dif_out_gaaaa(x1) .(x1, x2) = .(x1, x2) U5_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_gaaaa(x1, x2, x9) U6_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_gaaaa(x1, x2, x9) U7_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_gaaaa(x1, x2, x9) U13_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_gaaaa(x1, x7) mergedelete_in_aaa(x1, x2, x3) = mergedelete_in_aaa mergedelete_out_aaa(x1, x2, x3) = mergedelete_out_aaa U9_aaa(x1, x2, x3, x4, x5) = U9_aaa(x5) U10_aaa(x1, x2, x3, x4, x5, x6) = U10_aaa(x6) >_in_aa(x1, x2) = >_in_aa >_out_aa(x1, x2) = >_out_aa U11_aaa(x1, x2, x3, x4, x5, x6) = U11_aaa(x6) U14_gaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_gaaaa(x1, x7) rev_in_aga(x1, x2, x3) = rev_in_aga(x2) rev_out_aga(x1, x2, x3) = rev_out_aga(x2) U8_aga(x1, x2, x3, x4, x5) = U8_aga(x3, x5) rev_in_aaa(x1, x2, x3) = rev_in_aaa rev_out_aaa(x1, x2, x3) = rev_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U15_gaaaa(x1, x2, x3, x4, x5, x6) = U15_gaaaa(x1, x6) U18_agaa(x1, x2, x3, x4, x5, x6) = U18_agaa(x2, x6) pds1_in_aaaa(x1, x2, x3, x4) = pds1_in_aaaa pds1_out_aaaa(x1, x2, x3, x4) = pds1_out_aaaa U16_aaaa(x1, x2, x3, x4, x5, x6) = U16_aaaa(x6) U17_aaaa(x1, x2, x3, x4, x5, x6) = U17_aaaa(x6) check_in_aaaaa(x1, x2, x3, x4, x5) = check_in_aaaaa check_out_aaaaa(x1, x2, x3, x4, x5) = check_out_aaaaa U12_aaaaa(x1, x2, x3, x4, x5, x6) = U12_aaaaa(x6) dif_in_aaaaa(x1, x2, x3, x4, x5) = dif_in_aaaaa dif_out_aaaaa(x1, x2, x3, x4, x5) = dif_out_aaaaa U5_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_aaaaa(x9) U6_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_aaaaa(x9) U7_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_aaaaa(x9) U13_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U13_aaaaa(x7) U14_aaaaa(x1, x2, x3, x4, x5, x6, x7) = U14_aaaaa(x7) U15_aaaaa(x1, x2, x3, x4, x5, x6) = U15_aaaaa(x6) U18_aaaa(x1, x2, x3, x4, x5, x6) = U18_aaaa(x6) pds_out_ga(x1, x2) = pds_out_ga(x1) IOTA1_IN_AAA(x1, x2, x3) = IOTA1_IN_AAA U3_AAA(x1, x2, x3, x4) = U3_AAA(x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (341) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (342) Obligation: Pi DP problem: The TRS P consists of the following rules: U3_AAA(K, N, List, is_out_aa(K1, +(K, 1))) -> IOTA1_IN_AAA(K1, N, List) IOTA1_IN_AAA(K, N, .(K, List)) -> U3_AAA(K, N, List, is_in_aa(K1, +(K, 1))) The TRS R consists of the following rules: is_in_aa(X0, X1) -> is_out_aa(X0, X1) The argument filtering Pi contains the following mapping: +(x1, x2) = +(x1, x2) 1 = 1 is_in_aa(x1, x2) = is_in_aa is_out_aa(x1, x2) = is_out_aa .(x1, x2) = .(x1, x2) IOTA1_IN_AAA(x1, x2, x3) = IOTA1_IN_AAA U3_AAA(x1, x2, x3, x4) = U3_AAA(x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (343) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 2, "program": { "directives": [], "clauses": [ [ "(member X (. X Xs))", null ], [ "(member X (. X1 Xs))", "(member X Xs)" ], [ "(iota N List)", "(iota1 (0) N List)" ], [ "(iota1 K K ([]))", "(!)" ], [ "(iota1 K N (. K List))", "(',' (is K1 (+ K (1))) (iota1 K1 N List))" ], [ "(dif ([]) X2 X3 ([]) ([]))", null ], [ "(dif (. S Ss) Val Mod (. D Ds) (. D2 D2s))", "(',' (is D (- Val S)) (',' (is D2 (- Mod D)) (dif Ss Val Mod Ds D2s)))" ], [ "(rev ([]) L L)", null ], [ "(rev (. X Xs) Y L)", "(rev Xs (. X Y) L)" ], [ "(mergedelete ([]) L L)", null ], [ "(mergedelete (. D Ds) (. D R) L2)", "(mergedelete Ds R L2)" ], [ "(mergedelete (. D Ds) (. X R) (. X L2))", "(',' (> D X) (mergedelete (. D Ds) R L2))" ], [ "(check ([]) X4 L L X5)", "(!)" ], [ "(check S Choice Old L3 Modulus)", "(',' (dif S Choice Modulus Ds Dds) (',' (mergedelete Ds Old L2) (',' (rev Dds ([]) Rds) (',' (mergedelete Rds L2 L3) (!)))))" ], [ "(pds1 ([]) X6 ([]) X7)", "(!)" ], [ "(pds1 Unused List (. 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"X969", "X1102", "X1103", "X1236", "X1237", "X1374", "X1375", "X1483", "X1485" ], "exprvars": [ "T1333", "T1334", "T1329", "T1331", "T1332" ] } }, "4836": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "5804": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [ { "lhs": { "name": "T1333", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1331", "type": "PlainIntegerVariable" }, { "name": "T1329", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "name": "T1334", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1332", "type": "PlainIntegerVariable" }, { "name": "T1333", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "arguments": [ { "name": "T1347", "type": "PlainIntegerVariable" }, { "name": "T1329", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1331", "type": "PlainIntegerVariable" }, { "name": "T1329", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "arguments": [ { "name": "T1348", "type": "PlainIntegerVariable" }, { "name": "T1333", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1332", "type": "PlainIntegerVariable" }, { "name": "T1333", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" } ] }, "ground": [], "free": [], "exprvars": [ "T1333", "T1334", "T1329", "T1347", "T1331", "T1348", "T1332" ] } } }, "edges": [ { "from": 2, "to": 6, "label": "CASE" }, { "from": 6, "to": 13, "label": "EVAL with clause\npds(X10, .(0, X11)) :- ','(is(X12, +(*(X10, +(X10, 1)), 1)), ','(iota(X12, .(0, X13)), pds1(X13, .(0, []), X11, X12))).\nand substitutionT1 -> T5,\nX10 -> T5,\nX11 -> T7,\nT2 -> .(0, T7),\nT6 -> T7" }, { "from": 6, "to": 14, "label": "EVAL-BACKTRACK" }, { "from": 13, "to": 15, "label": "IS ERROR" }, { "from": 13, "to": 18, "label": "\nX12 -> T8" }, { "from": 18, "to": 19, "label": "CASE" }, { "from": 19, "to": 20, "label": "ONLY EVAL with clause\niota(X21, X22) :- iota1(0, X21, X22).\nand substitutionT8 -> T11,\nX21 -> T11,\nX13 -> X23,\nX22 -> .(0, X23)" }, { "from": 20, "to": 25, "label": "SPLIT 1" }, { "from": 20, "to": 27, "label": "SPLIT 2\nnew knowledge:\nT11 is ground\nT12 is ground\nreplacements:X23 -> T12" }, { "from": 25, "to": 28, "label": "CASE" }, { "from": 27, "to": 4637, "label": "CASE" }, { "from": 28, "to": 29, "label": "BACKTRACK\nfor clause: iota1(K, K, []) :- !because of non-unification" }, { "from": 29, "to": 33, "label": "ONLY EVAL with clause\niota1(X36, X37, .(X36, X38)) :- ','(is(X39, +(X36, 1)), iota1(X39, X37, X38)).\nand substitutionX36 -> 0,\nT11 -> T15,\nX37 -> T15,\nX23 -> X40,\nX38 -> X40" }, { "from": 33, "to": 34, "label": "\nX39 -> T16" }, { "from": 34, "to": 37, "label": "CASE" }, { "from": 37, "to": 39, "label": "EVAL with clause\niota1(X44, X44, []) :- !_4.\nand substitutionT16 -> T20,\nX44 -> T20,\nT15 -> T20,\nX40 -> []" }, { "from": 37, "to": 42, "label": "EVAL-BACKTRACK" }, { "from": 39, "to": 44, "label": "CUT" }, { "from": 42, "to": 50, "label": "ONLY EVAL with clause\niota1(X54, X55, .(X54, X56)) :- ','(is(X57, +(X54, 1)), iota1(X57, X55, X56)).\nand substitutionT16 -> T25,\nX54 -> T25,\nT15 -> T26,\nX55 -> T26,\nX56 -> X58,\nX40 -> .(T25, X58)" }, { "from": 44, "to": 49, "label": "SUCCESS" }, { "from": 50, "to": 51, "label": "\nX57 -> T27" }, { "from": 51, "to": 34, "label": "INSTANCE with matching:\nT16 -> T27\nT15 -> T26\nX40 -> X58\nX39 -> X44" }, { "from": 4637, "to": 4638, "label": "EVAL with clause\npds1([], X66, [], X67) :- !_5.\nand substitutionT12 -> [],\nX66 -> .(0, []),\nT7 -> [],\nT11 -> T32,\nX67 -> T32" }, { "from": 4637, "to": 4639, "label": "EVAL-BACKTRACK" }, { "from": 4638, "to": 4640, "label": "CUT" }, { "from": 4639, "to": 4642, "label": "EVAL with clause\npds1(X79, X80, .(X81, X82), X83) :- ','(member(X81, X79), ','(check(X80, X81, X79, X84, X83), pds1(X84, .(X81, X80), X82, X83))).\nand substitutionT12 -> T41,\nX79 -> T41,\nX80 -> .(0, []),\nX81 -> T45,\nX82 -> T46,\nT7 -> .(T45, T46),\nT11 -> T44,\nX83 -> T44,\nT42 -> T45,\nT43 -> T46" }, { "from": 4639, "to": 4643, "label": "EVAL-BACKTRACK" }, { "from": 4640, "to": 4641, "label": "SUCCESS" }, { "from": 4642, "to": 4644, "label": "SPLIT 1" }, { "from": 4642, "to": 4645, "label": "SPLIT 2\nnew knowledge:\nT51 is ground\nT41 is ground\nreplacements:T45 -> T51,\nT46 -> T52,\nT7 -> T53" }, { "from": 4644, "to": 4646, "label": "CASE" }, { "from": 4645, "to": 4662, "label": "SPLIT 1" }, { "from": 4645, "to": 4663, "label": "SPLIT 2\nnew knowledge:\nT51 is ground\nT41 is ground\nT106 is ground\nT44 is ground\nreplacements:X84 -> T106" }, { "from": 4646, "to": 4647, "label": "PARALLEL" }, { "from": 4646, "to": 4648, "label": "PARALLEL" }, { "from": 4647, "to": 4649, "label": "EVAL with clause\nmember(X101, .(X101, X102)).\nand substitutionT45 -> T66,\nX101 -> T66,\nX102 -> T67,\nT41 -> .(T66, T67)" }, { "from": 4647, "to": 4650, "label": "EVAL-BACKTRACK" }, { "from": 4648, "to": 4652, "label": "EVAL with clause\nmember(X109, .(X110, X111)) :- member(X109, X111).\nand substitutionT45 -> T77,\nX109 -> T77,\nX110 -> T75,\nX111 -> T76,\nT41 -> .(T75, T76),\nT74 -> T77" }, { "from": 4648, "to": 4653, "label": "EVAL-BACKTRACK" }, { "from": 4649, "to": 4651, "label": "SUCCESS" }, { "from": 4652, "to": 4654, "label": "CASE" }, { "from": 4654, "to": 4655, "label": "PARALLEL" }, { "from": 4654, "to": 4656, "label": "PARALLEL" }, { "from": 4655, "to": 4657, "label": "EVAL with clause\nmember(X124, .(X124, X125)).\nand substitutionT77 -> T90,\nX124 -> T90,\nX125 -> T91,\nT76 -> .(T90, T91)" }, { "from": 4655, "to": 4658, "label": "EVAL-BACKTRACK" }, { "from": 4656, "to": 4660, "label": "EVAL with clause\nmember(X132, .(X133, X134)) :- member(X132, X134).\nand substitutionT77 -> T101,\nX132 -> T101,\nX133 -> T99,\nX134 -> T100,\nT76 -> .(T99, T100),\nT98 -> T101" }, { "from": 4656, "to": 4661, "label": "EVAL-BACKTRACK" }, { "from": 4657, "to": 4659, "label": "SUCCESS" }, { "from": 4660, "to": 4652, "label": "INSTANCE with matching:\nT77 -> T101\nT76 -> T100" }, { "from": 4662, "to": 4664, "label": "CASE" }, { "from": 4663, "to": 5131, "label": "CASE" }, { "from": 4664, "to": 4665, "label": "BACKTRACK\nfor clause: check([], X4, L, L, X5) :- !because of non-unification" }, { "from": 4665, "to": 4779, "label": "ONLY EVAL with clause\ncheck(X168, X169, X170, X171, X172) :- ','(dif(X168, X169, X172, X173, X174), ','(mergedelete(X173, X170, X175), ','(rev(X174, [], X176), ','(mergedelete(X176, X175, X171), !_8)))).\nand substitutionX168 -> .(0, []),\nT51 -> T113,\nX169 -> T113,\nT41 -> T114,\nX170 -> T114,\nX84 -> X177,\nX171 -> X177,\nT44 -> T115,\nX172 -> T115" }, { "from": 4779, "to": 4780, "label": "SPLIT 1" }, { "from": 4779, "to": 4781, "label": "SPLIT 2\nnew knowledge:\nT113 is ground\nT115 is ground\nT116 is ground\nT117 is ground\nreplacements:X173 -> T116,\nX174 -> T117" }, { "from": 4780, "to": 4782, "label": "CASE" }, { "from": 4781, "to": 4795, "label": "SPLIT 1" }, { "from": 4781, "to": 4796, "label": "SPLIT 2\nnew knowledge:\nT116 is ground\nT114 is ground\nT140 is ground\nreplacements:X175 -> T140" }, { "from": 4782, "to": 4783, "label": "BACKTRACK\nfor clause: dif([], X2, X3, [], [])because of non-unification" }, { "from": 4783, "to": 4784, "label": "ONLY EVAL with clause\ndif(.(X212, X213), X214, X215, .(X216, X217), .(X218, X219)) :- ','(is(X216, -(X214, X212)), ','(is(X218, -(X215, X216)), dif(X213, X214, X215, X217, X219))).\nand substitutionX212 -> 0,\nX213 -> [],\nT113 -> T122,\nX214 -> T122,\nT115 -> T123,\nX215 -> T123,\nX216 -> X220,\nX217 -> X221,\nX173 -> .(X220, X221),\nX218 -> X222,\nX219 -> X223,\nX174 -> .(X222, X223)" }, { "from": 4784, "to": 4785, "label": "IS ERROR" }, { "from": 4784, "to": 4786, "label": "\nX220 -> T124" }, { "from": 4786, "to": 4787, "label": "IS ERROR" }, { "from": 4786, "to": 4788, "label": "\nX222 -> T125" }, { "from": 4788, "to": 4789, "label": "CASE" }, { "from": 4789, "to": 4790, "label": "PARALLEL" }, { "from": 4789, "to": 4791, "label": "PARALLEL" }, { "from": 4790, "to": 4792, "label": "ONLY EVAL with clause\ndif([], X234, X235, [], []).\nand substitutionT122 -> T136,\nX234 -> T136,\nT123 -> T137,\nX235 -> T137,\nX221 -> [],\nX223 -> []" }, { "from": 4791, "to": 4794, "label": "BACKTRACK\nfor clause: dif(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) :- ','(is(D, -(Val, S)), ','(is(D2, -(Mod, D)), dif(Ss, Val, Mod, Ds, D2s)))because of non-unification" }, { "from": 4792, "to": 4793, "label": "SUCCESS" }, { "from": 4795, "to": 4797, "label": "CASE" }, { "from": 4796, "to": 4967, "label": "SPLIT 1" }, { "from": 4796, "to": 4968, "label": "SPLIT 2\nnew knowledge:\nT117 is ground\nT211 is ground\nreplacements:X176 -> T211" }, { "from": 4797, "to": 4798, "label": "PARALLEL" }, { "from": 4797, "to": 4799, "label": "PARALLEL" }, { "from": 4798, "to": 4800, "label": "EVAL with clause\nmergedelete([], X252, X252).\nand substitutionT116 -> [],\nT114 -> T147,\nX252 -> T147,\nX175 -> T147" }, { "from": 4798, "to": 4801, "label": "EVAL-BACKTRACK" }, { "from": 4799, "to": 4803, "label": "PARALLEL" }, { "from": 4799, "to": 4804, "label": "PARALLEL" }, { "from": 4800, "to": 4802, "label": "SUCCESS" }, { "from": 4803, "to": 4805, "label": "EVAL with clause\nmergedelete(.(X273, X274), .(X273, X275), X276) :- mergedelete(X274, X275, X276).\nand substitutionX273 -> T160,\nX274 -> T161,\nT116 -> .(T160, T161),\nX275 -> T162,\nT114 -> .(T160, T162),\nX175 -> X277,\nX276 -> X277" }, { "from": 4803, "to": 4806, "label": "EVAL-BACKTRACK" }, { "from": 4804, "to": 4899, "label": "EVAL with clause\nmergedelete(.(X344, X345), .(X346, X347), .(X346, X348)) :- ','(>(X344, X346), mergedelete(.(X344, X345), X347, X348)).\nand substitutionX344 -> T207,\nX345 -> T208,\nT116 -> .(T207, T208),\nX346 -> T209,\nX347 -> T210,\nT114 -> .(T209, T210),\nX348 -> X349,\nX175 -> .(T209, X349)" }, { "from": 4804, "to": 4900, "label": "EVAL-BACKTRACK" }, { "from": 4805, "to": 4807, "label": "CASE" }, { "from": 4807, "to": 4808, "label": "PARALLEL" }, { "from": 4807, "to": 4809, "label": "PARALLEL" }, { "from": 4808, "to": 4810, "label": "EVAL with clause\nmergedelete([], X284, X284).\nand substitutionT161 -> [],\nT162 -> T169,\nX284 -> T169,\nX277 -> T169" }, { "from": 4808, "to": 4811, "label": "EVAL-BACKTRACK" }, { "from": 4809, "to": 4813, "label": "PARALLEL" }, { "from": 4809, "to": 4814, "label": "PARALLEL" }, { "from": 4810, "to": 4812, "label": "SUCCESS" }, { "from": 4813, "to": 4815, "label": "EVAL with clause\nmergedelete(.(X305, X306), .(X305, X307), X308) :- mergedelete(X306, X307, X308).\nand substitutionX305 -> T182,\nX306 -> T183,\nT161 -> .(T182, T183),\nX307 -> T184,\nT162 -> .(T182, T184),\nX277 -> X309,\nX308 -> X309" }, { "from": 4813, "to": 4816, "label": "EVAL-BACKTRACK" }, { "from": 4814, "to": 4834, "label": "EVAL with clause\nmergedelete(.(X324, X325), .(X326, X327), .(X326, X328)) :- ','(>(X324, X326), mergedelete(.(X324, X325), X327, X328)).\nand substitutionX324 -> T195,\nX325 -> T196,\nT161 -> .(T195, T196),\nX326 -> T197,\nX327 -> T198,\nT162 -> .(T197, T198),\nX328 -> X329,\nX277 -> .(T197, X329)" }, { "from": 4814, "to": 4835, "label": "EVAL-BACKTRACK" }, { "from": 4815, "to": 4805, "label": "INSTANCE with matching:\nT161 -> T183\nT162 -> T184\nX277 -> X309" }, { "from": 4834, "to": 4836, "label": "IS ERROR" }, { "from": 4834, "to": 4897, "label": "ARITHCOMP SUCCESS" }, { "from": 4834, "to": 4898, "label": "ARITHCOMP FAIL" }, { "from": 4897, "to": 4805, "label": "INSTANCE with matching:\nT161 -> .(T195, T196)\nT162 -> T198\nX277 -> X329" }, { "from": 4899, "to": 4901, "label": "IS ERROR" }, { "from": 4899, "to": 4965, "label": "ARITHCOMP SUCCESS" }, { "from": 4899, "to": 4966, "label": "ARITHCOMP FAIL" }, { "from": 4965, "to": 4805, "label": "INSTANCE with matching:\nT161 -> .(T207, T208)\nT162 -> T210\nX277 -> X349" }, { "from": 4967, "to": 4970, "label": "CASE" }, { "from": 4968, "to": 5127, "label": "SPLIT 1" }, { "from": 4968, "to": 5128, "label": "SPLIT 2\nnew knowledge:\nT211 is ground\nT140 is ground\nT591 is ground\nreplacements:X177 -> T591" }, { "from": 4970, "to": 4971, "label": "PARALLEL" }, { "from": 4970, "to": 4972, "label": "PARALLEL" }, { "from": 4971, "to": 4973, "label": "EVAL with clause\nrev([], X360, X360).\nand substitutionT117 -> [],\nX360 -> [],\nX176 -> []" }, { "from": 4971, "to": 4974, "label": "EVAL-BACKTRACK" }, { "from": 4972, "to": 4976, "label": "EVAL with clause\nrev(.(X371, X372), X373, X374) :- rev(X372, .(X371, X373), X374).\nand substitutionX371 -> T216,\nX372 -> T217,\nT117 -> .(T216, T217),\nX373 -> [],\nX176 -> X375,\nX374 -> X375" }, { "from": 4972, "to": 4977, "label": "EVAL-BACKTRACK" }, { "from": 4973, "to": 4975, "label": "SUCCESS" }, { "from": 4976, "to": 4978, "label": "CASE" }, { "from": 4978, "to": 4979, "label": "PARALLEL" }, { "from": 4978, "to": 4980, "label": "PARALLEL" }, { "from": 4979, "to": 4981, "label": "EVAL with clause\nrev([], X382, X382).\nand substitutionT217 -> [],\nT216 -> T224,\nX382 -> .(T224, []),\nX375 -> .(T224, [])" }, { "from": 4979, "to": 4982, "label": "EVAL-BACKTRACK" }, { "from": 4980, "to": 4984, "label": "EVAL with clause\nrev(.(X393, X394), X395, X396) :- rev(X394, .(X393, X395), X396).\nand substitutionX393 -> T231,\nX394 -> T232,\nT217 -> .(T231, T232),\nT216 -> T233,\nX395 -> .(T233, []),\nX375 -> X397,\nX396 -> X397" }, { "from": 4980, "to": 4985, "label": "EVAL-BACKTRACK" }, { "from": 4981, "to": 4983, "label": "SUCCESS" }, { "from": 4984, "to": 4986, "label": "CASE" }, { "from": 4986, "to": 4987, "label": "PARALLEL" }, { "from": 4986, "to": 4988, "label": "PARALLEL" }, { "from": 4987, "to": 4989, "label": "EVAL with clause\nrev([], X404, X404).\nand substitutionT232 -> [],\nT231 -> T246,\nT233 -> T247,\nX404 -> .(T246, .(T247, [])),\nX397 -> .(T246, .(T247, []))" }, { "from": 4987, "to": 4990, "label": "EVAL-BACKTRACK" }, { "from": 4988, "to": 4996, "label": "EVAL with clause\nrev(.(X415, X416), X417, X418) :- rev(X416, .(X415, X417), X418).\nand substitutionX415 -> T256,\nX416 -> T257,\nT232 -> .(T256, T257),\nT231 -> T258,\nT233 -> T259,\nX417 -> .(T258, .(T259, [])),\nX397 -> X419,\nX418 -> X419" }, { "from": 4988, "to": 4997, "label": "EVAL-BACKTRACK" }, { "from": 4989, "to": 4991, "label": "SUCCESS" }, { "from": 4996, "to": 5004, "label": "CASE" }, { "from": 5004, "to": 5005, "label": "PARALLEL" }, { "from": 5004, "to": 5006, "label": "PARALLEL" }, { "from": 5005, "to": 5007, "label": "EVAL with clause\nrev([], X426, X426).\nand substitutionT257 -> [],\nT256 -> T278,\nT258 -> T279,\nT259 -> T280,\nX426 -> .(T278, .(T279, .(T280, []))),\nX419 -> .(T278, .(T279, .(T280, [])))" }, { "from": 5005, "to": 5008, "label": "EVAL-BACKTRACK" }, { "from": 5006, "to": 5018, "label": "EVAL with clause\nrev(.(X437, X438), X439, X440) :- rev(X438, .(X437, X439), X440).\nand substitutionX437 -> T291,\nX438 -> T292,\nT257 -> .(T291, T292),\nT256 -> T293,\nT258 -> T294,\nT259 -> T295,\nX439 -> .(T293, .(T294, .(T295, []))),\nX419 -> X441,\nX440 -> X441" }, { "from": 5006, "to": 5019, "label": "EVAL-BACKTRACK" }, { "from": 5007, "to": 5009, "label": "SUCCESS" }, { "from": 5018, "to": 5021, "label": "CASE" }, { "from": 5021, "to": 5023, "label": "PARALLEL" }, { "from": 5021, "to": 5024, "label": "PARALLEL" }, { "from": 5023, "to": 5028, "label": "EVAL with clause\nrev([], X448, X448).\nand substitutionT292 -> [],\nT291 -> T320,\nT293 -> T321,\nT294 -> T322,\nT295 -> T323,\nX448 -> .(T320, .(T321, .(T322, .(T323, [])))),\nX441 -> .(T320, .(T321, .(T322, .(T323, []))))" }, { "from": 5023, "to": 5029, "label": "EVAL-BACKTRACK" }, { "from": 5024, "to": 5036, "label": "EVAL with clause\nrev(.(X459, X460), X461, X462) :- rev(X460, .(X459, X461), X462).\nand substitutionX459 -> T336,\nX460 -> T337,\nT292 -> .(T336, T337),\nT291 -> T338,\nT293 -> T339,\nT294 -> T340,\nT295 -> T341,\nX461 -> .(T338, .(T339, .(T340, .(T341, [])))),\nX441 -> X463,\nX462 -> X463" }, { "from": 5024, "to": 5037, "label": "EVAL-BACKTRACK" }, { "from": 5028, "to": 5030, "label": "SUCCESS" }, { "from": 5036, "to": 5039, "label": "CASE" }, { "from": 5039, "to": 5042, "label": "PARALLEL" }, { "from": 5039, "to": 5043, "label": "PARALLEL" }, { "from": 5042, "to": 5047, "label": "EVAL with clause\nrev([], X470, X470).\nand substitutionT337 -> [],\nT336 -> T372,\nT338 -> T373,\nT339 -> T374,\nT340 -> T375,\nT341 -> T376,\nX470 -> .(T372, .(T373, .(T374, .(T375, .(T376, []))))),\nX463 -> .(T372, .(T373, .(T374, .(T375, .(T376, [])))))" }, { "from": 5042, "to": 5048, "label": "EVAL-BACKTRACK" }, { "from": 5043, "to": 5060, "label": "EVAL with clause\nrev(.(X481, X482), X483, X484) :- rev(X482, .(X481, X483), X484).\nand substitutionX481 -> T391,\nX482 -> T392,\nT337 -> .(T391, T392),\nT336 -> T393,\nT338 -> T394,\nT339 -> T395,\nT340 -> T396,\nT341 -> T397,\nX483 -> .(T393, .(T394, .(T395, .(T396, .(T397, []))))),\nX463 -> X485,\nX484 -> X485" }, { "from": 5043, "to": 5061, "label": "EVAL-BACKTRACK" }, { "from": 5047, "to": 5049, "label": "SUCCESS" }, { "from": 5060, "to": 5062, "label": "CASE" }, { "from": 5062, "to": 5063, "label": "PARALLEL" }, { "from": 5062, "to": 5064, "label": "PARALLEL" }, { "from": 5063, "to": 5068, "label": "EVAL with clause\nrev([], X492, X492).\nand substitutionT392 -> [],\nT391 -> T434,\nT393 -> T435,\nT394 -> T436,\nT395 -> T437,\nT396 -> T438,\nT397 -> T439,\nX492 -> .(T434, .(T435, .(T436, .(T437, .(T438, .(T439, [])))))),\nX485 -> .(T434, .(T435, .(T436, .(T437, .(T438, .(T439, []))))))" }, { "from": 5063, "to": 5069, "label": "EVAL-BACKTRACK" }, { "from": 5064, "to": 5084, "label": "EVAL with clause\nrev(.(X503, X504), X505, X506) :- rev(X504, .(X503, X505), X506).\nand substitutionX503 -> T456,\nX504 -> T457,\nT392 -> .(T456, T457),\nT391 -> T458,\nT393 -> T459,\nT394 -> T460,\nT395 -> T461,\nT396 -> T462,\nT397 -> T463,\nX505 -> .(T458, .(T459, .(T460, .(T461, .(T462, .(T463, [])))))),\nX485 -> X507,\nX506 -> X507" }, { "from": 5064, "to": 5085, "label": "EVAL-BACKTRACK" }, { "from": 5068, "to": 5070, "label": "SUCCESS" }, { "from": 5084, "to": 5097, "label": "CASE" }, { "from": 5097, "to": 5098, "label": "PARALLEL" }, { "from": 5097, "to": 5099, "label": "PARALLEL" }, { "from": 5098, "to": 5100, "label": "EVAL with clause\nrev([], X514, X514).\nand substitutionT457 -> [],\nT456 -> T506,\nT458 -> T507,\nT459 -> T508,\nT460 -> T509,\nT461 -> T510,\nT462 -> T511,\nT463 -> T512,\nX514 -> .(T506, .(T507, .(T508, .(T509, .(T510, .(T511, .(T512, []))))))),\nX507 -> .(T506, .(T507, .(T508, .(T509, .(T510, .(T511, .(T512, [])))))))" }, { "from": 5098, "to": 5101, "label": "EVAL-BACKTRACK" }, { "from": 5099, "to": 5103, "label": "EVAL with clause\nrev(.(X525, X526), X527, X528) :- rev(X526, .(X525, X527), X528).\nand substitutionX525 -> T531,\nX526 -> T532,\nT457 -> .(T531, T532),\nT456 -> T533,\nT458 -> T534,\nT459 -> T535,\nT460 -> T536,\nT461 -> T537,\nT462 -> T538,\nT463 -> T539,\nX527 -> .(T533, .(T534, .(T535, .(T536, .(T537, .(T538, .(T539, []))))))),\nX507 -> X529,\nX528 -> X529" }, { "from": 5099, "to": 5104, "label": "EVAL-BACKTRACK" }, { "from": 5100, "to": 5102, "label": "SUCCESS" }, { "from": 5103, "to": 5105, "label": "GENERALIZATION\nT556 <-- .(T533, .(T534, .(T535, .(T536, .(T537, .(T538, .(T539, [])))))))\n\nNew Knowledge:\nT556 is ground" }, { "from": 5105, "to": 5112, "label": "CASE" }, { "from": 5112, "to": 5113, "label": "PARALLEL" }, { "from": 5112, "to": 5114, "label": "PARALLEL" }, { "from": 5113, "to": 5116, "label": "EVAL with clause\nrev([], X538, X538).\nand substitutionT532 -> [],\nT531 -> T569,\nT556 -> T570,\nX538 -> .(T569, T570),\nX529 -> .(T569, T570)" }, { "from": 5113, "to": 5117, "label": "EVAL-BACKTRACK" }, { "from": 5114, "to": 5125, "label": "EVAL with clause\nrev(.(X549, X550), X551, X552) :- rev(X550, .(X549, X551), X552).\nand substitutionX549 -> T579,\nX550 -> T580,\nT532 -> .(T579, T580),\nT531 -> T581,\nT556 -> T582,\nX551 -> .(T581, T582),\nX529 -> X553,\nX552 -> X553" }, { "from": 5114, "to": 5126, "label": "EVAL-BACKTRACK" }, { "from": 5116, "to": 5118, "label": "SUCCESS" }, { "from": 5125, "to": 5105, "label": "INSTANCE with matching:\nT532 -> T580\nT531 -> T579\nT556 -> .(T581, T582)\nX529 -> X553" }, { "from": 5127, "to": 4805, "label": "INSTANCE with matching:\nT161 -> T211\nT162 -> T140\nX277 -> X177" }, { "from": 5128, "to": 5129, "label": "CUT" }, { "from": 5129, "to": 5130, "label": "SUCCESS" }, { "from": 5131, "to": 5136, "label": "EVAL with clause\npds1([], X566, [], X567) :- !_22.\nand substitutionT106 -> [],\nT51 -> T600,\nX566 -> .(T600, .(0, [])),\nT52 -> [],\nT44 -> T601,\nX567 -> T601" }, { "from": 5131, "to": 5137, "label": "EVAL-BACKTRACK" }, { "from": 5136, "to": 5138, "label": "CUT" }, { "from": 5137, "to": 5157, "label": "EVAL with clause\npds1(X579, X580, .(X581, X582), X583) :- ','(member(X581, X579), ','(check(X580, X581, X579, X584, X583), pds1(X584, .(X581, X580), X582, X583))).\nand substitutionT106 -> T612,\nX579 -> T612,\nT51 -> T613,\nX580 -> .(T613, .(0, [])),\nX581 -> T617,\nX582 -> T618,\nT52 -> .(T617, T618),\nT44 -> T616,\nX583 -> T616,\nT614 -> T617,\nT615 -> T618" }, { "from": 5137, "to": 5158, "label": "EVAL-BACKTRACK" }, { "from": 5138, "to": 5139, "label": "SUCCESS" }, { "from": 5157, "to": 5163, "label": "SPLIT 1" }, { "from": 5157, "to": 5165, "label": "SPLIT 2\nnew knowledge:\nT623 is ground\nT612 is ground\nreplacements:T617 -> T623,\nT618 -> T624,\nT53 -> T625,\nT52 -> T626" }, { "from": 5163, "to": 4652, "label": "INSTANCE with matching:\nT77 -> T617\nT76 -> T612" }, { "from": 5165, "to": 5169, "label": "SPLIT 1" }, { "from": 5165, "to": 5170, "label": "SPLIT 2\nnew knowledge:\nT613 is ground\nT623 is ground\nT612 is ground\nT631 is ground\nT616 is ground\nreplacements:X584 -> T631" }, { "from": 5169, "to": 5175, "label": "CASE" }, { "from": 5170, "to": 5256, "label": "CASE" }, { "from": 5175, "to": 5176, "label": "BACKTRACK\nfor clause: check([], X4, L, L, X5) :- !because of non-unification" }, { "from": 5176, "to": 5184, "label": "ONLY EVAL with clause\ncheck(X622, X623, X624, X625, X626) :- ','(dif(X622, X623, X626, X627, X628), ','(mergedelete(X627, X624, X629), ','(rev(X628, [], X630), ','(mergedelete(X630, X629, X625), !_23)))).\nand substitutionT613 -> T640,\nX622 -> .(T640, .(0, [])),\nT623 -> T641,\nX623 -> T641,\nT612 -> T642,\nX624 -> T642,\nX584 -> X631,\nX625 -> X631,\nT616 -> T643,\nX626 -> T643" }, { "from": 5184, "to": 5185, "label": "SPLIT 1" }, { "from": 5184, "to": 5186, "label": "SPLIT 2\nnew knowledge:\nT640 is ground\nT641 is ground\nT643 is ground\nT644 is ground\nT645 is ground\nreplacements:X627 -> T644,\nX628 -> T645" }, { "from": 5185, "to": 5191, "label": "CASE" }, { "from": 5186, "to": 5234, "label": "SPLIT 1" }, { "from": 5186, "to": 5235, "label": "SPLIT 2\nnew knowledge:\nT644 is ground\nT642 is ground\nT659 is ground\nreplacements:X629 -> T659" }, { "from": 5191, "to": 5192, "label": "BACKTRACK\nfor clause: dif([], X2, X3, [], [])because of non-unification" }, { "from": 5192, "to": 5215, "label": "ONLY EVAL with clause\ndif(.(X666, X667), X668, X669, .(X670, X671), .(X672, X673)) :- ','(is(X670, -(X668, X666)), ','(is(X672, -(X669, X670)), dif(X667, X668, X669, X671, X673))).\nand substitutionT640 -> T652,\nX666 -> T652,\nX667 -> .(0, []),\nT641 -> T653,\nX668 -> T653,\nT643 -> T654,\nX669 -> T654,\nX670 -> X674,\nX671 -> X675,\nX627 -> .(X674, X675),\nX672 -> X676,\nX673 -> X677,\nX628 -> .(X676, X677)" }, { "from": 5215, "to": 5216, "label": "IS ERROR" }, { "from": 5215, "to": 5217, "label": "\nX674 -> T655" }, { "from": 5217, "to": 5225, "label": "IS ERROR" }, { "from": 5217, "to": 5226, "label": "\nX676 -> T656" }, { "from": 5226, "to": 4780, "label": "INSTANCE with matching:\nT113 -> T653\nT115 -> T654\nX173 -> X675\nX174 -> X677\nT114 -> T41\nT53 -> T625" }, { "from": 5234, "to": 4805, "label": "INSTANCE with matching:\nT161 -> T644\nT162 -> T642\nX277 -> X629" }, { "from": 5235, "to": 5237, "label": "SPLIT 1" }, { "from": 5235, "to": 5238, "label": "SPLIT 2\nnew knowledge:\nT645 is ground\nT662 is ground\nreplacements:X630 -> T662" }, { "from": 5237, "to": 4967, "label": "INSTANCE with matching:\nT117 -> T645\nX176 -> X630" }, { "from": 5238, "to": 5242, "label": "SPLIT 1" }, { "from": 5238, "to": 5243, "label": "SPLIT 2\nnew knowledge:\nT662 is ground\nT659 is ground\nT665 is ground\nreplacements:X631 -> T665" }, { "from": 5242, "to": 4805, "label": "INSTANCE with matching:\nT161 -> T662\nT162 -> T659\nX277 -> X631" }, { "from": 5243, "to": 5246, "label": "CUT" }, { "from": 5246, "to": 5247, "label": "SUCCESS" }, { "from": 5256, "to": 5263, "label": "EVAL with clause\npds1([], X700, [], X701) :- !_25.\nand substitutionT631 -> [],\nT623 -> T677,\nT613 -> T678,\nX700 -> .(T677, .(T678, .(0, []))),\nT624 -> [],\nT616 -> T679,\nX701 -> T679" }, { "from": 5256, "to": 5264, "label": "EVAL-BACKTRACK" }, { "from": 5263, "to": 5265, "label": "CUT" }, { "from": 5264, "to": 5275, "label": "EVAL with clause\npds1(X713, X714, .(X715, X716), X717) :- ','(member(X715, X713), ','(check(X714, X715, X713, X718, X717), pds1(X718, .(X715, X714), X716, X717))).\nand substitutionT631 -> T692,\nX713 -> T692,\nT623 -> T693,\nT613 -> T694,\nX714 -> .(T693, .(T694, .(0, []))),\nX715 -> T698,\nX716 -> T699,\nT624 -> .(T698, T699),\nT616 -> T697,\nX717 -> T697,\nT695 -> T698,\nT696 -> T699" }, { "from": 5264, "to": 5276, "label": "EVAL-BACKTRACK" }, { "from": 5265, "to": 5266, "label": "SUCCESS" }, { "from": 5275, "to": 5289, "label": "SPLIT 1" }, { "from": 5275, "to": 5290, "label": "SPLIT 2\nnew knowledge:\nT704 is ground\nT692 is ground\nreplacements:T698 -> T704,\nT699 -> T705,\nT625 -> T706,\nT626 -> T707,\nT624 -> T708" }, { "from": 5289, "to": 4652, "label": "INSTANCE with matching:\nT77 -> T698\nT76 -> T692" }, { "from": 5290, "to": 5295, "label": "SPLIT 1" }, { "from": 5290, "to": 5297, "label": "SPLIT 2\nnew knowledge:\nT693 is ground\nT694 is ground\nT704 is ground\nT692 is ground\nT713 is ground\nT697 is ground\nreplacements:X718 -> T713" }, { "from": 5295, "to": 5301, "label": "CASE" }, { "from": 5297, "to": 5352, "label": "CASE" }, { "from": 5301, "to": 5302, "label": "BACKTRACK\nfor clause: check([], X4, L, L, X5) :- !because of non-unification" }, { "from": 5302, "to": 5308, "label": "ONLY EVAL with clause\ncheck(X756, X757, X758, X759, X760) :- ','(dif(X756, X757, X760, X761, X762), ','(mergedelete(X761, X758, X763), ','(rev(X762, [], X764), ','(mergedelete(X764, X763, X759), !_26)))).\nand substitutionT693 -> T724,\nT694 -> T725,\nX756 -> .(T724, .(T725, .(0, []))),\nT704 -> T726,\nX757 -> T726,\nT692 -> T727,\nX758 -> T727,\nX718 -> X765,\nX759 -> X765,\nT697 -> T728,\nX760 -> T728" }, { "from": 5308, "to": 5316, "label": "SPLIT 1" }, { "from": 5308, "to": 5317, "label": "SPLIT 2\nnew knowledge:\nT724 is ground\nT725 is ground\nT726 is ground\nT728 is ground\nT729 is ground\nT730 is ground\nreplacements:X761 -> T729,\nX762 -> T730" }, { "from": 5316, "to": 5320, "label": "CASE" }, { "from": 5317, "to": 5334, "label": "SPLIT 1" }, { "from": 5317, "to": 5335, "label": "SPLIT 2\nnew knowledge:\nT729 is ground\nT727 is ground\nT747 is ground\nreplacements:X763 -> T747" }, { "from": 5320, "to": 5321, "label": "BACKTRACK\nfor clause: dif([], X2, X3, [], [])because of non-unification" }, { "from": 5321, "to": 5324, "label": "ONLY EVAL with clause\ndif(.(X800, X801), X802, X803, .(X804, X805), .(X806, X807)) :- ','(is(X804, -(X802, X800)), ','(is(X806, -(X803, X804)), dif(X801, X802, X803, X805, X807))).\nand substitutionT724 -> T739,\nX800 -> T739,\nT725 -> T740,\nX801 -> .(T740, .(0, [])),\nT726 -> T741,\nX802 -> T741,\nT728 -> T742,\nX803 -> T742,\nX804 -> X808,\nX805 -> X809,\nX761 -> .(X808, X809),\nX806 -> X810,\nX807 -> X811,\nX762 -> .(X810, X811)" }, { "from": 5324, "to": 5325, "label": "IS ERROR" }, { "from": 5324, "to": 5326, "label": "\nX808 -> T743" }, { "from": 5326, "to": 5327, "label": "IS ERROR" }, { "from": 5326, "to": 5328, "label": "\nX810 -> T744" }, { "from": 5328, "to": 5185, "label": "INSTANCE with matching:\nT640 -> T740\nT641 -> T741\nT643 -> T742\nX627 -> X809\nX628 -> X811\nT625 -> T706\nT642 -> T612\nT626 -> T707" }, { "from": 5334, "to": 4805, "label": "INSTANCE with matching:\nT161 -> T729\nT162 -> T727\nX277 -> X763" }, { "from": 5335, "to": 5338, "label": "SPLIT 1" }, { "from": 5335, "to": 5339, "label": "SPLIT 2\nnew knowledge:\nT730 is ground\nT750 is ground\nreplacements:X764 -> T750" }, { "from": 5338, "to": 4967, "label": "INSTANCE with matching:\nT117 -> T730\nX176 -> X764" }, { "from": 5339, "to": 5342, "label": "SPLIT 1" }, { "from": 5339, "to": 5343, "label": "SPLIT 2\nnew knowledge:\nT750 is ground\nT747 is ground\nT753 is ground\nreplacements:X765 -> T753" }, { "from": 5342, "to": 4805, "label": "INSTANCE with matching:\nT161 -> T750\nT162 -> T747\nX277 -> X765" }, { "from": 5343, "to": 5349, "label": "CUT" }, { "from": 5349, "to": 5350, "label": "SUCCESS" }, { "from": 5352, "to": 5353, "label": "EVAL with clause\npds1([], X834, [], X835) :- !_28.\nand substitutionT713 -> [],\nT704 -> T768,\nT693 -> T769,\nT694 -> T770,\nX834 -> .(T768, .(T769, .(T770, .(0, [])))),\nT705 -> [],\nT697 -> T771,\nX835 -> T771" }, { "from": 5352, "to": 5354, "label": "EVAL-BACKTRACK" }, { "from": 5353, "to": 5355, "label": "CUT" }, { "from": 5354, "to": 5357, "label": "EVAL with clause\npds1(X847, X848, .(X849, X850), X851) :- ','(member(X849, X847), ','(check(X848, X849, X847, X852, X851), pds1(X852, .(X849, X848), X850, X851))).\nand substitutionT713 -> T786,\nX847 -> T786,\nT704 -> T787,\nT693 -> T788,\nT694 -> T789,\nX848 -> .(T787, .(T788, .(T789, .(0, [])))),\nX849 -> T793,\nX850 -> T794,\nT705 -> .(T793, T794),\nT697 -> T792,\nX851 -> T792,\nT790 -> T793,\nT791 -> T794" }, { "from": 5354, "to": 5358, "label": "EVAL-BACKTRACK" }, { "from": 5355, "to": 5356, "label": "SUCCESS" }, { "from": 5357, "to": 5361, "label": "SPLIT 1" }, { "from": 5357, "to": 5362, "label": "SPLIT 2\nnew knowledge:\nT799 is ground\nT786 is ground\nreplacements:T793 -> T799,\nT794 -> T800,\nT706 -> T801,\nT707 -> T802,\nT708 -> T803,\nT705 -> T804" }, { "from": 5361, "to": 4652, "label": "INSTANCE with matching:\nT77 -> T793\nT76 -> T786" }, { "from": 5362, "to": 5366, "label": "SPLIT 1" }, { "from": 5362, "to": 5367, "label": "SPLIT 2\nnew knowledge:\nT787 is ground\nT788 is ground\nT789 is ground\nT799 is ground\nT786 is ground\nT809 is ground\nT792 is ground\nreplacements:X852 -> T809" }, { "from": 5366, "to": 5376, "label": "CASE" }, { "from": 5367, "to": 5445, "label": "CASE" }, { "from": 5376, "to": 5377, "label": "BACKTRACK\nfor clause: check([], X4, L, L, X5) :- !because of non-unification" }, { "from": 5377, "to": 5384, "label": "ONLY EVAL with clause\ncheck(X890, X891, X892, X893, X894) :- ','(dif(X890, X891, X894, X895, X896), ','(mergedelete(X895, X892, X897), ','(rev(X896, [], X898), ','(mergedelete(X898, X897, X893), !_29)))).\nand substitutionT787 -> T822,\nT788 -> T823,\nT789 -> T824,\nX890 -> .(T822, .(T823, .(T824, .(0, [])))),\nT799 -> T825,\nX891 -> T825,\nT786 -> T826,\nX892 -> T826,\nX852 -> X899,\nX893 -> X899,\nT792 -> T827,\nX894 -> T827" }, { "from": 5384, "to": 5391, "label": "SPLIT 1" }, { "from": 5384, "to": 5392, "label": "SPLIT 2\nnew knowledge:\nT822 is ground\nT823 is ground\nT824 is ground\nT825 is ground\nT827 is ground\nT828 is ground\nT829 is ground\nreplacements:X895 -> T828,\nX896 -> T829" }, { "from": 5391, "to": 5393, "label": "CASE" }, { "from": 5392, "to": 5416, "label": "SPLIT 1" }, { "from": 5392, "to": 5417, "label": "SPLIT 2\nnew knowledge:\nT828 is ground\nT826 is ground\nT849 is ground\nreplacements:X897 -> T849" }, { "from": 5393, "to": 5397, "label": "BACKTRACK\nfor clause: dif([], X2, X3, [], [])because of non-unification" }, { "from": 5397, "to": 5411, "label": "ONLY EVAL with clause\ndif(.(X934, X935), X936, X937, .(X938, X939), .(X940, X941)) :- ','(is(X938, -(X936, X934)), ','(is(X940, -(X937, X938)), dif(X935, X936, X937, X939, X941))).\nand substitutionT822 -> T840,\nX934 -> T840,\nT823 -> T841,\nT824 -> T842,\nX935 -> .(T841, .(T842, .(0, []))),\nT825 -> T843,\nX936 -> T843,\nT827 -> T844,\nX937 -> T844,\nX938 -> X942,\nX939 -> X943,\nX895 -> .(X942, X943),\nX940 -> X944,\nX941 -> X945,\nX896 -> .(X944, X945)" }, { "from": 5411, "to": 5412, "label": "IS ERROR" }, { "from": 5411, "to": 5413, "label": "\nX942 -> T845" }, { "from": 5413, "to": 5414, "label": "IS ERROR" }, { "from": 5413, "to": 5415, "label": "\nX944 -> T846" }, { "from": 5415, "to": 5316, "label": "INSTANCE with matching:\nT724 -> T841\nT725 -> T842\nT726 -> T843\nT728 -> T844\nX761 -> X943\nX762 -> X945\nT706 -> T801\nT707 -> T802\nT727 -> T692\nT708 -> T803" }, { "from": 5416, "to": 4805, "label": "INSTANCE with matching:\nT161 -> T828\nT162 -> T826\nX277 -> X897" }, { "from": 5417, "to": 5439, "label": "SPLIT 1" }, { "from": 5417, "to": 5440, "label": "SPLIT 2\nnew knowledge:\nT829 is ground\nT852 is ground\nreplacements:X898 -> T852" }, { "from": 5439, "to": 4967, "label": "INSTANCE with matching:\nT117 -> T829\nX176 -> X898" }, { "from": 5440, "to": 5441, "label": "SPLIT 1" }, { "from": 5440, "to": 5442, "label": "SPLIT 2\nnew knowledge:\nT852 is ground\nT849 is ground\nT855 is ground\nreplacements:X899 -> T855" }, { "from": 5441, "to": 4805, "label": "INSTANCE with matching:\nT161 -> T852\nT162 -> T849\nX277 -> X899" }, { "from": 5442, "to": 5443, "label": "CUT" }, { "from": 5443, "to": 5444, "label": "SUCCESS" }, { "from": 5445, "to": 5451, "label": "EVAL with clause\npds1([], X968, [], X969) :- !_31.\nand substitutionT809 -> [],\nT799 -> T873,\nT787 -> T874,\nT788 -> T875,\nT789 -> T876,\nX968 -> .(T873, .(T874, .(T875, .(T876, .(0, []))))),\nT800 -> [],\nT792 -> T877,\nX969 -> T877" }, { "from": 5445, "to": 5452, "label": "EVAL-BACKTRACK" }, { "from": 5451, "to": 5453, "label": "CUT" }, { "from": 5452, "to": 5482, "label": "EVAL with clause\npds1(X981, X982, .(X983, X984), X985) :- ','(member(X983, X981), ','(check(X982, X983, X981, X986, X985), pds1(X986, .(X983, X982), X984, X985))).\nand substitutionT809 -> T894,\nX981 -> T894,\nT799 -> T895,\nT787 -> T896,\nT788 -> T897,\nT789 -> T898,\nX982 -> .(T895, .(T896, .(T897, .(T898, .(0, []))))),\nX983 -> T902,\nX984 -> T903,\nT800 -> .(T902, T903),\nT792 -> T901,\nX985 -> T901,\nT899 -> T902,\nT900 -> T903" }, { "from": 5452, "to": 5483, "label": "EVAL-BACKTRACK" }, { "from": 5453, "to": 5454, "label": "SUCCESS" }, { "from": 5482, "to": 5498, "label": "SPLIT 1" }, { "from": 5482, "to": 5499, "label": "SPLIT 2\nnew knowledge:\nT908 is ground\nT894 is ground\nreplacements:T902 -> T908,\nT903 -> T909,\nT801 -> T910,\nT802 -> T911,\nT803 -> T912,\nT804 -> T913,\nT800 -> T914" }, { "from": 5498, "to": 4652, "label": "INSTANCE with matching:\nT77 -> T902\nT76 -> T894" }, { "from": 5499, "to": 5504, "label": "SPLIT 1" }, { "from": 5499, "to": 5505, "label": "SPLIT 2\nnew knowledge:\nT895 is ground\nT896 is ground\nT897 is ground\nT898 is ground\nT908 is ground\nT894 is ground\nT919 is ground\nT901 is ground\nreplacements:X986 -> T919" }, { "from": 5504, "to": 5515, "label": "CASE" }, { "from": 5505, "to": 5577, "label": "CASE" }, { "from": 5515, "to": 5516, "label": "BACKTRACK\nfor clause: check([], X4, L, L, X5) :- !because of non-unification" }, { "from": 5516, "to": 5523, "label": "ONLY EVAL with clause\ncheck(X1024, X1025, X1026, X1027, X1028) :- ','(dif(X1024, X1025, X1028, X1029, X1030), ','(mergedelete(X1029, X1026, X1031), ','(rev(X1030, [], X1032), ','(mergedelete(X1032, X1031, X1027), !_32)))).\nand substitutionT895 -> T934,\nT896 -> T935,\nT897 -> T936,\nT898 -> T937,\nX1024 -> .(T934, .(T935, .(T936, .(T937, .(0, []))))),\nT908 -> T938,\nX1025 -> T938,\nT894 -> T939,\nX1026 -> T939,\nX986 -> X1033,\nX1027 -> X1033,\nT901 -> T940,\nX1028 -> T940" }, { "from": 5523, "to": 5525, "label": "SPLIT 1" }, { "from": 5523, "to": 5526, "label": "SPLIT 2\nnew knowledge:\nT934 is ground\nT935 is ground\nT936 is ground\nT937 is ground\nT938 is ground\nT940 is ground\nT941 is ground\nT942 is ground\nreplacements:X1029 -> T941,\nX1030 -> T942" }, { "from": 5525, "to": 5529, "label": "CASE" }, { "from": 5526, "to": 5552, "label": "SPLIT 1" }, { "from": 5526, "to": 5554, "label": "SPLIT 2\nnew knowledge:\nT941 is ground\nT939 is ground\nT965 is ground\nreplacements:X1031 -> T965" }, { "from": 5529, "to": 5530, "label": "BACKTRACK\nfor clause: dif([], X2, X3, [], [])because of non-unification" }, { "from": 5530, "to": 5545, "label": "ONLY EVAL with clause\ndif(.(X1068, X1069), X1070, X1071, .(X1072, X1073), .(X1074, X1075)) :- ','(is(X1072, -(X1070, X1068)), ','(is(X1074, -(X1071, X1072)), dif(X1069, X1070, X1071, X1073, X1075))).\nand substitutionT934 -> T955,\nX1068 -> T955,\nT935 -> T956,\nT936 -> T957,\nT937 -> T958,\nX1069 -> .(T956, .(T957, .(T958, .(0, [])))),\nT938 -> T959,\nX1070 -> T959,\nT940 -> T960,\nX1071 -> T960,\nX1072 -> X1076,\nX1073 -> X1077,\nX1029 -> .(X1076, X1077),\nX1074 -> X1078,\nX1075 -> X1079,\nX1030 -> .(X1078, X1079)" }, { "from": 5545, "to": 5546, "label": "IS ERROR" }, { "from": 5545, "to": 5547, "label": "\nX1076 -> T961" }, { "from": 5547, "to": 5548, "label": "IS ERROR" }, { "from": 5547, "to": 5549, "label": "\nX1078 -> T962" }, { "from": 5549, "to": 5391, "label": "INSTANCE with matching:\nT822 -> T956\nT823 -> T957\nT824 -> T958\nT825 -> T959\nT827 -> T960\nX895 -> X1077\nX896 -> X1079\nT801 -> T910\nT802 -> T911\nT803 -> T912\nT826 -> T786\nT804 -> T913" }, { "from": 5552, "to": 4805, "label": "INSTANCE with matching:\nT161 -> T941\nT162 -> T939\nX277 -> X1031" }, { "from": 5554, "to": 5556, "label": "SPLIT 1" }, { "from": 5554, "to": 5557, "label": "SPLIT 2\nnew knowledge:\nT942 is ground\nT968 is ground\nreplacements:X1032 -> T968" }, { "from": 5556, "to": 4967, "label": "INSTANCE with matching:\nT117 -> T942\nX176 -> X1032" }, { "from": 5557, "to": 5571, "label": "SPLIT 1" }, { "from": 5557, "to": 5572, "label": "SPLIT 2\nnew knowledge:\nT968 is ground\nT965 is ground\nT971 is ground\nreplacements:X1033 -> T971" }, { "from": 5571, "to": 4805, "label": "INSTANCE with matching:\nT161 -> T968\nT162 -> T965\nX277 -> X1033" }, { "from": 5572, "to": 5573, "label": "CUT" }, { "from": 5573, "to": 5574, "label": "SUCCESS" }, { "from": 5577, "to": 5582, "label": "EVAL with clause\npds1([], X1102, [], X1103) :- !_34.\nand substitutionT919 -> [],\nT908 -> T992,\nT895 -> T993,\nT896 -> T994,\nT897 -> T995,\nT898 -> T996,\nX1102 -> .(T992, .(T993, .(T994, .(T995, .(T996, .(0, [])))))),\nT909 -> [],\nT901 -> T997,\nX1103 -> T997" }, { "from": 5577, "to": 5583, "label": "EVAL-BACKTRACK" }, { "from": 5582, "to": 5584, "label": "CUT" }, { "from": 5583, "to": 5593, "label": "EVAL with clause\npds1(X1115, X1116, .(X1117, X1118), X1119) :- ','(member(X1117, X1115), ','(check(X1116, X1117, X1115, X1120, X1119), pds1(X1120, .(X1117, X1116), X1118, X1119))).\nand substitutionT919 -> T1016,\nX1115 -> T1016,\nT908 -> T1017,\nT895 -> T1018,\nT896 -> T1019,\nT897 -> T1020,\nT898 -> T1021,\nX1116 -> .(T1017, .(T1018, .(T1019, .(T1020, .(T1021, .(0, [])))))),\nX1117 -> T1025,\nX1118 -> T1026,\nT909 -> .(T1025, T1026),\nT901 -> T1024,\nX1119 -> T1024,\nT1022 -> T1025,\nT1023 -> T1026" }, { "from": 5583, "to": 5594, "label": "EVAL-BACKTRACK" }, { "from": 5584, "to": 5585, "label": "SUCCESS" }, { "from": 5593, "to": 5595, "label": "SPLIT 1" }, { "from": 5593, "to": 5597, "label": "SPLIT 2\nnew knowledge:\nT1031 is ground\nT1016 is ground\nreplacements:T1025 -> T1031,\nT1026 -> T1032,\nT910 -> T1033,\nT911 -> T1034,\nT912 -> T1035,\nT913 -> T1036,\nT914 -> T1037,\nT909 -> T1038" }, { "from": 5595, "to": 4652, "label": "INSTANCE with matching:\nT77 -> T1025\nT76 -> T1016" }, { "from": 5597, "to": 5600, "label": "SPLIT 1" }, { "from": 5597, "to": 5601, "label": "SPLIT 2\nnew knowledge:\nT1017 is ground\nT1018 is ground\nT1019 is ground\nT1020 is ground\nT1021 is ground\nT1031 is ground\nT1016 is ground\nT1043 is ground\nT1024 is ground\nreplacements:X1120 -> T1043" }, { "from": 5600, "to": 5604, "label": "CASE" }, { "from": 5601, "to": 5650, "label": "CASE" }, { "from": 5604, "to": 5605, "label": "BACKTRACK\nfor clause: check([], X4, L, L, X5) :- !because of non-unification" }, { "from": 5605, "to": 5610, "label": "ONLY EVAL with clause\ncheck(X1158, X1159, X1160, X1161, X1162) :- ','(dif(X1158, X1159, X1162, X1163, X1164), ','(mergedelete(X1163, X1160, X1165), ','(rev(X1164, [], X1166), ','(mergedelete(X1166, X1165, X1161), !_35)))).\nand substitutionT1017 -> T1060,\nT1018 -> T1061,\nT1019 -> T1062,\nT1020 -> T1063,\nT1021 -> T1064,\nX1158 -> .(T1060, .(T1061, .(T1062, .(T1063, .(T1064, .(0, [])))))),\nT1031 -> T1065,\nX1159 -> T1065,\nT1016 -> T1066,\nX1160 -> T1066,\nX1120 -> X1167,\nX1161 -> X1167,\nT1024 -> T1067,\nX1162 -> T1067" }, { "from": 5610, "to": 5616, "label": "SPLIT 1" }, { "from": 5610, "to": 5617, "label": "SPLIT 2\nnew knowledge:\nT1060 is ground\nT1061 is ground\nT1062 is ground\nT1063 is ground\nT1064 is ground\nT1065 is ground\nT1067 is ground\nT1068 is ground\nT1069 is ground\nreplacements:X1163 -> T1068,\nX1164 -> T1069" }, { "from": 5616, "to": 5619, "label": "CASE" }, { "from": 5617, "to": 5636, "label": "SPLIT 1" }, { "from": 5617, "to": 5637, "label": "SPLIT 2\nnew knowledge:\nT1068 is ground\nT1066 is ground\nT1095 is ground\nreplacements:X1165 -> T1095" }, { "from": 5619, "to": 5621, "label": "BACKTRACK\nfor clause: dif([], X2, X3, [], [])because of non-unification" }, { "from": 5621, "to": 5628, "label": "ONLY EVAL with clause\ndif(.(X1202, X1203), X1204, X1205, .(X1206, X1207), .(X1208, X1209)) :- ','(is(X1206, -(X1204, X1202)), ','(is(X1208, -(X1205, X1206)), dif(X1203, X1204, X1205, X1207, X1209))).\nand substitutionT1060 -> T1084,\nX1202 -> T1084,\nT1061 -> T1085,\nT1062 -> T1086,\nT1063 -> T1087,\nT1064 -> T1088,\nX1203 -> .(T1085, .(T1086, .(T1087, .(T1088, .(0, []))))),\nT1065 -> T1089,\nX1204 -> T1089,\nT1067 -> T1090,\nX1205 -> T1090,\nX1206 -> X1210,\nX1207 -> X1211,\nX1163 -> .(X1210, X1211),\nX1208 -> X1212,\nX1209 -> X1213,\nX1164 -> .(X1212, X1213)" }, { "from": 5628, "to": 5629, "label": "IS ERROR" }, { "from": 5628, "to": 5630, "label": "\nX1210 -> T1091" }, { "from": 5630, "to": 5632, "label": "IS ERROR" }, { "from": 5630, "to": 5633, "label": "\nX1212 -> T1092" }, { "from": 5633, "to": 5525, "label": "INSTANCE with matching:\nT934 -> T1085\nT935 -> T1086\nT936 -> T1087\nT937 -> T1088\nT938 -> T1089\nT940 -> T1090\nX1029 -> X1211\nX1030 -> X1213\nT910 -> T1033\nT911 -> T1034\nT912 -> T1035\nT913 -> T1036\nT939 -> T894\nT914 -> T1037" }, { "from": 5636, "to": 4805, "label": "INSTANCE with matching:\nT161 -> T1068\nT162 -> T1066\nX277 -> X1165" }, { "from": 5637, "to": 5640, "label": "SPLIT 1" }, { "from": 5637, "to": 5641, "label": "SPLIT 2\nnew knowledge:\nT1069 is ground\nT1098 is ground\nreplacements:X1166 -> T1098" }, { "from": 5640, "to": 4967, "label": "INSTANCE with matching:\nT117 -> T1069\nX176 -> X1166" }, { "from": 5641, "to": 5643, "label": "SPLIT 1" }, { "from": 5641, "to": 5644, "label": "SPLIT 2\nnew knowledge:\nT1098 is ground\nT1095 is ground\nT1101 is ground\nreplacements:X1167 -> T1101" }, { "from": 5643, "to": 4805, "label": "INSTANCE with matching:\nT161 -> T1098\nT162 -> T1095\nX277 -> X1167" }, { "from": 5644, "to": 5645, "label": "CUT" }, { "from": 5645, "to": 5646, "label": "SUCCESS" }, { "from": 5650, "to": 5654, "label": "EVAL with clause\npds1([], X1236, [], X1237) :- !_37.\nand substitutionT1043 -> [],\nT1031 -> T1125,\nT1017 -> T1126,\nT1018 -> T1127,\nT1019 -> T1128,\nT1020 -> T1129,\nT1021 -> T1130,\nX1236 -> .(T1125, .(T1126, .(T1127, .(T1128, .(T1129, .(T1130, .(0, []))))))),\nT1032 -> [],\nT1024 -> T1131,\nX1237 -> T1131" }, { "from": 5650, "to": 5656, "label": "EVAL-BACKTRACK" }, { "from": 5654, "to": 5657, "label": "CUT" }, { "from": 5656, "to": 5663, "label": "EVAL with clause\npds1(X1249, X1250, .(X1251, X1252), X1253) :- ','(member(X1251, X1249), ','(check(X1250, X1251, X1249, X1254, X1253), pds1(X1254, .(X1251, X1250), X1252, X1253))).\nand substitutionT1043 -> T1152,\nX1249 -> T1152,\nT1031 -> T1153,\nT1017 -> T1154,\nT1018 -> T1155,\nT1019 -> T1156,\nT1020 -> T1157,\nT1021 -> T1158,\nX1250 -> .(T1153, .(T1154, .(T1155, .(T1156, .(T1157, .(T1158, .(0, []))))))),\nX1251 -> T1162,\nX1252 -> T1163,\nT1032 -> .(T1162, T1163),\nT1024 -> T1161,\nX1253 -> T1161,\nT1159 -> T1162,\nT1160 -> T1163" }, { "from": 5656, "to": 5665, "label": "EVAL-BACKTRACK" }, { "from": 5657, "to": 5658, "label": "SUCCESS" }, { "from": 5663, "to": 5666, "label": "GENERALIZATION\nT1168 <-- .(T1153, .(T1154, .(T1155, .(T1156, .(T1157, .(T1158, .(0, [])))))))\n\nNew Knowledge:\nT1168 is ground" }, { "from": 5666, "to": 5672, "label": "SPLIT 1" }, { "from": 5666, "to": 5673, "label": "SPLIT 2\nnew knowledge:\nT1173 is ground\nT1152 is ground\nreplacements:T1162 -> T1173,\nT1163 -> T1174,\nT1033 -> T1175,\nT1034 -> T1176,\nT1035 -> T1177,\nT1036 -> T1178,\nT1037 -> T1179,\nT1038 -> T1180,\nT1032 -> T1181" }, { "from": 5672, "to": 4652, "label": "INSTANCE with matching:\nT77 -> T1162\nT76 -> T1152" }, { "from": 5673, "to": 5681, "label": "SPLIT 1" }, { "from": 5673, "to": 5682, "label": "SPLIT 2\nnew knowledge:\nT1153 is ground\nT1154 is ground\nT1155 is ground\nT1156 is ground\nT1157 is ground\nT1158 is ground\nT1173 is ground\nT1152 is ground\nT1186 is ground\nT1161 is ground\nreplacements:X1254 -> T1186" }, { "from": 5681, "to": 5688, "label": "CASE" }, { "from": 5682, "to": 5735, "label": "CASE" }, { "from": 5688, "to": 5690, "label": "BACKTRACK\nfor clause: check([], X4, L, L, X5) :- !because of non-unification" }, { "from": 5690, "to": 5692, "label": "ONLY EVAL with clause\ncheck(X1296, X1297, X1298, X1299, X1300) :- ','(dif(X1296, X1297, X1300, X1301, X1302), ','(mergedelete(X1301, X1298, X1303), ','(rev(X1302, [], X1304), ','(mergedelete(X1304, X1303, X1299), !_38)))).\nand substitutionT1153 -> T1205,\nT1154 -> T1206,\nT1155 -> T1207,\nT1156 -> T1208,\nT1157 -> T1209,\nT1158 -> T1210,\nX1296 -> .(T1205, .(T1206, .(T1207, .(T1208, .(T1209, .(T1210, .(0, []))))))),\nT1173 -> T1211,\nX1297 -> T1211,\nT1152 -> T1212,\nX1298 -> T1212,\nX1254 -> X1305,\nX1299 -> X1305,\nT1161 -> T1213,\nX1300 -> T1213" }, { "from": 5692, "to": 5698, "label": "SPLIT 1" }, { "from": 5692, "to": 5699, "label": "SPLIT 2\nnew knowledge:\nT1205 is ground\nT1206 is ground\nT1207 is ground\nT1208 is ground\nT1209 is ground\nT1210 is ground\nT1211 is ground\nT1213 is ground\nT1214 is ground\nT1215 is ground\nreplacements:X1301 -> T1214,\nX1302 -> T1215" }, { "from": 5698, "to": 5701, "label": "CASE" }, { "from": 5699, "to": 5722, "label": "SPLIT 1" }, { "from": 5699, "to": 5723, "label": "SPLIT 2\nnew knowledge:\nT1214 is ground\nT1212 is ground\nT1244 is ground\nreplacements:X1303 -> T1244" }, { "from": 5701, "to": 5702, "label": "BACKTRACK\nfor clause: dif([], X2, X3, [], [])because of non-unification" }, { "from": 5702, "to": 5708, "label": "ONLY EVAL with clause\ndif(.(X1340, X1341), X1342, X1343, .(X1344, X1345), .(X1346, X1347)) :- ','(is(X1344, -(X1342, X1340)), ','(is(X1346, -(X1343, X1344)), dif(X1341, X1342, X1343, X1345, X1347))).\nand substitutionT1205 -> T1232,\nX1340 -> T1232,\nT1206 -> T1233,\nT1207 -> T1234,\nT1208 -> T1235,\nT1209 -> T1236,\nT1210 -> T1237,\nX1341 -> .(T1233, .(T1234, .(T1235, .(T1236, .(T1237, .(0, [])))))),\nT1211 -> T1238,\nX1342 -> T1238,\nT1213 -> T1239,\nX1343 -> T1239,\nX1344 -> X1348,\nX1345 -> X1349,\nX1301 -> .(X1348, X1349),\nX1346 -> X1350,\nX1347 -> X1351,\nX1302 -> .(X1350, X1351)" }, { "from": 5708, "to": 5709, "label": "IS ERROR" }, { "from": 5708, "to": 5710, "label": "\nX1348 -> T1240" }, { "from": 5710, "to": 5711, "label": "IS ERROR" }, { "from": 5710, "to": 5712, "label": "\nX1350 -> T1241" }, { "from": 5712, "to": 5616, "label": "INSTANCE with matching:\nT1060 -> T1233\nT1061 -> T1234\nT1062 -> T1235\nT1063 -> T1236\nT1064 -> T1237\nT1065 -> T1238\nT1067 -> T1239\nX1163 -> X1349\nX1164 -> X1351\nT1033 -> T1175\nT1034 -> T1176\nT1035 -> T1177\nT1036 -> T1178\nT1037 -> T1179\nT1066 -> T1016\nT1038 -> T1180" }, { "from": 5722, "to": 4805, "label": "INSTANCE with matching:\nT161 -> T1214\nT162 -> T1212\nX277 -> X1303" }, { "from": 5723, "to": 5724, "label": "SPLIT 1" }, { "from": 5723, "to": 5725, "label": "SPLIT 2\nnew knowledge:\nT1215 is ground\nT1247 is ground\nreplacements:X1304 -> T1247" }, { "from": 5724, "to": 4967, "label": "INSTANCE with matching:\nT117 -> T1215\nX176 -> X1304" }, { "from": 5725, "to": 5729, "label": "SPLIT 1" }, { "from": 5725, "to": 5730, "label": "SPLIT 2\nnew knowledge:\nT1247 is ground\nT1244 is ground\nT1250 is ground\nreplacements:X1305 -> T1250" }, { "from": 5729, "to": 4805, "label": "INSTANCE with matching:\nT161 -> T1247\nT162 -> T1244\nX277 -> X1305" }, { "from": 5730, "to": 5731, "label": "CUT" }, { "from": 5731, "to": 5732, "label": "SUCCESS" }, { "from": 5735, "to": 5737, "label": "EVAL with clause\npds1([], X1374, [], X1375) :- !_40.\nand substitutionT1186 -> [],\nT1173 -> T1262,\nT1168 -> T1263,\nX1374 -> .(T1262, T1263),\nT1174 -> [],\nT1161 -> T1264,\nX1375 -> T1264" }, { "from": 5735, "to": 5739, "label": "EVAL-BACKTRACK" }, { "from": 5737, "to": 5742, "label": "CUT" }, { "from": 5739, "to": 5747, "label": "EVAL with clause\npds1(X1387, X1388, .(X1389, X1390), X1391) :- ','(member(X1389, X1387), ','(check(X1388, X1389, X1387, X1392, X1391), pds1(X1392, .(X1389, X1388), X1390, X1391))).\nand substitutionT1186 -> T1277,\nX1387 -> T1277,\nT1173 -> T1278,\nT1168 -> T1279,\nX1388 -> .(T1278, T1279),\nX1389 -> T1283,\nX1390 -> T1284,\nT1174 -> .(T1283, T1284),\nT1161 -> T1282,\nX1391 -> T1282,\nT1280 -> T1283,\nT1281 -> T1284" }, { "from": 5739, "to": 5749, "label": "EVAL-BACKTRACK" }, { "from": 5742, "to": 5743, "label": "SUCCESS" }, { "from": 5747, "to": 5752, "label": "SPLIT 1" }, { "from": 5747, "to": 5753, "label": "SPLIT 2\nnew knowledge:\nT1289 is ground\nT1277 is ground\nreplacements:T1283 -> T1289,\nT1284 -> T1290,\nT1175 -> T1291,\nT1176 -> T1292,\nT1177 -> T1293,\nT1178 -> T1294,\nT1179 -> T1295,\nT1180 -> T1296,\nT1181 -> T1297,\nT1174 -> T1298" }, { "from": 5752, "to": 4652, "label": "INSTANCE with matching:\nT77 -> T1283\nT76 -> T1277" }, { "from": 5753, "to": 5759, "label": "SPLIT 1" }, { "from": 5753, "to": 5760, "label": "SPLIT 2\nnew knowledge:\nT1278 is ground\nT1279 is ground\nT1289 is ground\nT1277 is ground\nT1303 is ground\nT1282 is ground\nreplacements:X1392 -> T1303" }, { "from": 5759, "to": 5761, "label": "CASE" }, { "from": 5760, "to": 5682, "label": "INSTANCE with matching:\nT1186 -> T1303\nT1173 -> T1289\nT1168 -> .(T1278, T1279)\nT1174 -> T1290\nT1161 -> T1282\nT1175 -> T1291\nT1176 -> T1292\nT1177 -> T1293\nT1178 -> T1294\nT1179 -> T1295\nT1180 -> T1296\nT1181 -> T1297" }, { "from": 5761, "to": 5764, "label": "BACKTRACK\nfor clause: check([], X4, L, L, X5) :- !because of non-unification" }, { "from": 5764, "to": 5774, "label": "ONLY EVAL with clause\ncheck(X1430, X1431, X1432, X1433, X1434) :- ','(dif(X1430, X1431, X1434, X1435, X1436), ','(mergedelete(X1435, X1432, X1437), ','(rev(X1436, [], X1438), ','(mergedelete(X1438, X1437, X1433), !_41)))).\nand substitutionT1278 -> T1314,\nT1279 -> T1315,\nX1430 -> .(T1314, T1315),\nT1289 -> T1316,\nX1431 -> T1316,\nT1277 -> T1317,\nX1432 -> T1317,\nX1392 -> X1439,\nX1433 -> X1439,\nT1282 -> T1318,\nX1434 -> T1318" }, { "from": 5774, "to": 5775, "label": "SPLIT 1" }, { "from": 5774, "to": 5776, "label": "SPLIT 2\nnew knowledge:\nT1314 is ground\nT1315 is ground\nT1316 is ground\nT1318 is ground\nT1319 is ground\nT1320 is ground\nreplacements:X1435 -> T1319,\nX1436 -> T1320" }, { "from": 5775, "to": 5777, "label": "CASE" }, { "from": 5776, "to": 6072, "label": "SPLIT 1" }, { "from": 5776, "to": 6073, "label": "SPLIT 2\nnew knowledge:\nT1319 is ground\nT1317 is ground\nT1570 is ground\nreplacements:X1437 -> T1570" }, { "from": 5777, "to": 5778, "label": "BACKTRACK\nfor clause: dif([], X2, X3, [], [])because of non-unification" }, { "from": 5778, "to": 5779, "label": "ONLY EVAL with clause\ndif(.(X1474, X1475), X1476, X1477, .(X1478, X1479), .(X1480, X1481)) :- ','(is(X1478, -(X1476, X1474)), ','(is(X1480, -(X1477, X1478)), dif(X1475, X1476, X1477, X1479, X1481))).\nand substitutionT1314 -> T1329,\nX1474 -> T1329,\nT1315 -> T1330,\nX1475 -> T1330,\nT1316 -> T1331,\nX1476 -> T1331,\nT1318 -> T1332,\nX1477 -> T1332,\nX1478 -> X1482,\nX1479 -> X1483,\nX1435 -> .(X1482, X1483),\nX1480 -> X1484,\nX1481 -> X1485,\nX1436 -> .(X1484, X1485)" }, { "from": 5779, "to": 5780, "label": "IS ERROR" }, { "from": 5779, "to": 5781, "label": "\nX1482 -> T1333" }, { "from": 5781, "to": 5782, "label": "IS ERROR" }, { "from": 5781, "to": 5783, "label": "\nX1484 -> T1334" }, { "from": 5783, "to": 5796, "label": "CASE" }, { "from": 5796, "to": 5800, "label": "PARALLEL" }, { "from": 5796, "to": 5803, "label": "PARALLEL" }, { "from": 5800, "to": 5804, "label": "EVAL with clause\ndif([], X1498, X1499, [], []).\nand substitutionT1330 -> [],\nT1331 -> T1347,\nX1498 -> T1347,\nT1332 -> T1348,\nX1499 -> T1348,\nX1483 -> [],\nX1485 -> []" }, { "from": 5800, "to": 5805, "label": "EVAL-BACKTRACK" }, { "from": 5803, "to": 5807, "label": "EVAL with clause\ndif(.(X1524, X1525), X1526, X1527, .(X1528, X1529), .(X1530, X1531)) :- ','(is(X1528, -(X1526, X1524)), ','(is(X1530, -(X1527, X1528)), dif(X1525, X1526, X1527, X1529, X1531))).\nand substitutionX1524 -> T1357,\nX1525 -> T1358,\nT1330 -> .(T1357, T1358),\nT1331 -> T1359,\nX1526 -> T1359,\nT1332 -> T1360,\nX1527 -> T1360,\nX1528 -> X1532,\nX1529 -> X1533,\nX1483 -> .(X1532, X1533),\nX1530 -> X1534,\nX1531 -> X1535,\nX1485 -> .(X1534, X1535)" }, { "from": 5803, "to": 5808, "label": "EVAL-BACKTRACK" }, { "from": 5804, "to": 5806, "label": "SUCCESS" }, { "from": 5807, "to": 5809, "label": "IS ERROR" }, { "from": 5807, "to": 5810, "label": "\nX1532 -> T1361" }, { "from": 5810, "to": 5811, "label": "\nX1534 -> T1362" }, { "from": 5811, "to": 5812, "label": "CASE" }, { "from": 5812, "to": 5813, "label": "PARALLEL" }, { "from": 5812, "to": 5814, "label": "PARALLEL" }, { "from": 5813, "to": 5815, "label": "EVAL with clause\ndif([], X1548, X1549, [], []).\nand substitutionT1358 -> [],\nT1359 -> T1375,\nX1548 -> T1375,\nT1360 -> T1376,\nX1549 -> T1376,\nX1533 -> [],\nX1535 -> []" }, { "from": 5813, "to": 5816, "label": "EVAL-BACKTRACK" }, { "from": 5814, "to": 5823, "label": "EVAL with clause\ndif(.(X1574, X1575), X1576, X1577, .(X1578, X1579), .(X1580, X1581)) :- ','(is(X1578, -(X1576, X1574)), ','(is(X1580, -(X1577, X1578)), dif(X1575, X1576, X1577, X1579, X1581))).\nand substitutionX1574 -> T1385,\nX1575 -> T1386,\nT1358 -> .(T1385, T1386),\nT1359 -> T1387,\nX1576 -> T1387,\nT1360 -> T1388,\nX1577 -> T1388,\nX1578 -> X1582,\nX1579 -> X1583,\nX1533 -> .(X1582, X1583),\nX1580 -> X1584,\nX1581 -> X1585,\nX1535 -> .(X1584, X1585)" }, { "from": 5814, "to": 5824, "label": "EVAL-BACKTRACK" }, { "from": 5815, "to": 5817, "label": "SUCCESS" }, { "from": 5823, "to": 5825, "label": "IS ERROR" }, { "from": 5823, "to": 5826, "label": "\nX1582 -> T1389" }, { "from": 5826, "to": 5829, "label": "\nX1584 -> T1390" }, { "from": 5829, "to": 5831, "label": "CASE" }, { "from": 5831, "to": 5836, "label": "PARALLEL" }, { "from": 5831, "to": 5837, "label": "PARALLEL" }, { "from": 5836, "to": 5843, "label": "EVAL with clause\ndif([], X1598, X1599, [], []).\nand substitutionT1386 -> [],\nT1387 -> T1403,\nX1598 -> T1403,\nT1388 -> T1404,\nX1599 -> T1404,\nX1583 -> [],\nX1585 -> []" }, { "from": 5836, "to": 5844, "label": "EVAL-BACKTRACK" }, { "from": 5837, "to": 5853, "label": "EVAL with clause\ndif(.(X1624, X1625), X1626, X1627, .(X1628, X1629), .(X1630, X1631)) :- ','(is(X1628, -(X1626, X1624)), ','(is(X1630, -(X1627, X1628)), dif(X1625, X1626, X1627, X1629, X1631))).\nand substitutionX1624 -> T1413,\nX1625 -> T1414,\nT1386 -> .(T1413, T1414),\nT1387 -> T1415,\nX1626 -> T1415,\nT1388 -> T1416,\nX1627 -> T1416,\nX1628 -> X1632,\nX1629 -> X1633,\nX1583 -> .(X1632, X1633),\nX1630 -> X1634,\nX1631 -> X1635,\nX1585 -> .(X1634, X1635)" }, { "from": 5837, "to": 5854, "label": "EVAL-BACKTRACK" }, { "from": 5843, "to": 5845, "label": "SUCCESS" }, { "from": 5853, "to": 5855, "label": "IS ERROR" }, { "from": 5853, "to": 5856, "label": "\nX1632 -> T1417" }, { "from": 5856, "to": 5857, "label": "\nX1634 -> T1418" }, { "from": 5857, "to": 5864, "label": "CASE" }, { "from": 5864, "to": 5866, "label": "PARALLEL" }, { "from": 5864, "to": 5867, "label": "PARALLEL" }, { "from": 5866, "to": 5868, "label": "EVAL with clause\ndif([], X1648, X1649, [], []).\nand substitutionT1414 -> [],\nT1415 -> T1431,\nX1648 -> T1431,\nT1416 -> T1432,\nX1649 -> T1432,\nX1633 -> [],\nX1635 -> []" }, { "from": 5866, "to": 5869, "label": "EVAL-BACKTRACK" }, { "from": 5867, "to": 5893, "label": "EVAL with clause\ndif(.(X1674, X1675), X1676, X1677, .(X1678, X1679), .(X1680, X1681)) :- ','(is(X1678, -(X1676, X1674)), ','(is(X1680, -(X1677, X1678)), dif(X1675, X1676, X1677, X1679, X1681))).\nand substitutionX1674 -> T1441,\nX1675 -> T1442,\nT1414 -> .(T1441, T1442),\nT1415 -> T1443,\nX1676 -> T1443,\nT1416 -> T1444,\nX1677 -> T1444,\nX1678 -> X1682,\nX1679 -> X1683,\nX1633 -> .(X1682, X1683),\nX1680 -> X1684,\nX1681 -> X1685,\nX1635 -> .(X1684, X1685)" }, { "from": 5867, "to": 5894, "label": "EVAL-BACKTRACK" }, { "from": 5868, "to": 5870, "label": "SUCCESS" }, { "from": 5893, "to": 5895, "label": "IS ERROR" }, { "from": 5893, "to": 5896, "label": "\nX1682 -> T1445" }, { "from": 5896, "to": 5897, "label": "\nX1684 -> T1446" }, { "from": 5897, "to": 5903, "label": "CASE" }, { "from": 5903, "to": 5904, "label": "PARALLEL" }, { "from": 5903, "to": 5905, "label": "PARALLEL" }, { "from": 5904, "to": 5906, "label": "EVAL with clause\ndif([], X1698, X1699, [], []).\nand substitutionT1442 -> [],\nT1443 -> T1459,\nX1698 -> T1459,\nT1444 -> T1460,\nX1699 -> T1460,\nX1683 -> [],\nX1685 -> []" }, { "from": 5904, "to": 5907, "label": "EVAL-BACKTRACK" }, { "from": 5905, "to": 5952, "label": "EVAL with clause\ndif(.(X1724, X1725), X1726, X1727, .(X1728, X1729), .(X1730, X1731)) :- ','(is(X1728, -(X1726, X1724)), ','(is(X1730, -(X1727, X1728)), dif(X1725, X1726, X1727, X1729, X1731))).\nand substitutionX1724 -> T1469,\nX1725 -> T1470,\nT1442 -> .(T1469, T1470),\nT1443 -> T1471,\nX1726 -> T1471,\nT1444 -> T1472,\nX1727 -> T1472,\nX1728 -> X1732,\nX1729 -> X1733,\nX1683 -> .(X1732, X1733),\nX1730 -> X1734,\nX1731 -> X1735,\nX1685 -> .(X1734, X1735)" }, { "from": 5905, "to": 5957, "label": "EVAL-BACKTRACK" }, { "from": 5906, "to": 5908, "label": "SUCCESS" }, { "from": 5952, "to": 5958, "label": "IS ERROR" }, { "from": 5952, "to": 5959, "label": "\nX1732 -> T1473" }, { "from": 5959, "to": 5960, "label": "\nX1734 -> T1474" }, { "from": 5960, "to": 5963, "label": "CASE" }, { "from": 5963, "to": 5964, "label": "PARALLEL" }, { "from": 5963, "to": 5965, "label": "PARALLEL" }, { "from": 5964, "to": 5976, "label": "EVAL with clause\ndif([], X1748, X1749, [], []).\nand substitutionT1470 -> [],\nT1471 -> T1487,\nX1748 -> T1487,\nT1472 -> T1488,\nX1749 -> T1488,\nX1733 -> [],\nX1735 -> []" }, { "from": 5964, "to": 5978, "label": "EVAL-BACKTRACK" }, { "from": 5965, "to": 5985, "label": "EVAL with clause\ndif(.(X1774, X1775), X1776, X1777, .(X1778, X1779), .(X1780, X1781)) :- ','(is(X1778, -(X1776, X1774)), ','(is(X1780, -(X1777, X1778)), dif(X1775, X1776, X1777, X1779, X1781))).\nand substitutionX1774 -> T1497,\nX1775 -> T1498,\nT1470 -> .(T1497, T1498),\nT1471 -> T1499,\nX1776 -> T1499,\nT1472 -> T1500,\nX1777 -> T1500,\nX1778 -> X1782,\nX1779 -> X1783,\nX1733 -> .(X1782, X1783),\nX1780 -> X1784,\nX1781 -> X1785,\nX1735 -> .(X1784, X1785)" }, { "from": 5965, "to": 5986, "label": "EVAL-BACKTRACK" }, { "from": 5976, "to": 5979, "label": "SUCCESS" }, { "from": 5985, "to": 5987, "label": "IS ERROR" }, { "from": 5985, "to": 5988, "label": "\nX1782 -> T1501" }, { "from": 5988, "to": 5989, "label": "\nX1784 -> T1502" }, { "from": 5989, "to": 5995, "label": "CASE" }, { "from": 5995, "to": 5998, "label": "PARALLEL" }, { "from": 5995, "to": 5999, "label": "PARALLEL" }, { "from": 5998, "to": 6003, "label": "EVAL with clause\ndif([], X1798, X1799, [], []).\nand substitutionT1498 -> [],\nT1499 -> T1515,\nX1798 -> T1515,\nT1500 -> T1516,\nX1799 -> T1516,\nX1783 -> [],\nX1785 -> []" }, { "from": 5998, "to": 6004, "label": "EVAL-BACKTRACK" }, { "from": 5999, "to": 6024, "label": "EVAL with clause\ndif(.(X1824, X1825), X1826, X1827, .(X1828, X1829), .(X1830, X1831)) :- ','(is(X1828, -(X1826, X1824)), ','(is(X1830, -(X1827, X1828)), dif(X1825, X1826, X1827, X1829, X1831))).\nand substitutionX1824 -> T1525,\nX1825 -> T1526,\nT1498 -> .(T1525, T1526),\nT1499 -> T1527,\nX1826 -> T1527,\nT1500 -> T1528,\nX1827 -> T1528,\nX1828 -> X1832,\nX1829 -> X1833,\nX1783 -> .(X1832, X1833),\nX1830 -> X1834,\nX1831 -> X1835,\nX1785 -> .(X1834, X1835)" }, { "from": 5999, "to": 6025, "label": "EVAL-BACKTRACK" }, { "from": 6003, "to": 6006, "label": "SUCCESS" }, { "from": 6024, "to": 6026, "label": "IS ERROR" }, { "from": 6024, "to": 6027, "label": "\nX1832 -> T1529" }, { "from": 6027, "to": 6028, "label": "\nX1834 -> T1530" }, { "from": 6028, "to": 6029, "label": "GENERALIZATION\nT1535 <-- .(T1357, .(T1385, .(T1413, .(T1441, .(T1469, .(T1497, .(T1525, T1526)))))))\n\nNew Knowledge:\nT1535 is ground" }, { "from": 6029, "to": 6030, "label": "CASE" }, { "from": 6030, "to": 6045, "label": "PARALLEL" }, { "from": 6030, "to": 6046, "label": "PARALLEL" }, { "from": 6045, "to": 6048, "label": "EVAL with clause\ndif([], X1852, X1853, [], []).\nand substitutionT1526 -> [],\nT1527 -> T1548,\nX1852 -> T1548,\nT1528 -> T1549,\nX1853 -> T1549,\nX1833 -> [],\nX1835 -> []" }, { "from": 6045, "to": 6050, "label": "EVAL-BACKTRACK" }, { "from": 6046, "to": 6062, "label": "EVAL with clause\ndif(.(X1878, X1879), X1880, X1881, .(X1882, X1883), .(X1884, X1885)) :- ','(is(X1882, -(X1880, X1878)), ','(is(X1884, -(X1881, X1882)), dif(X1879, X1880, X1881, X1883, X1885))).\nand substitutionX1878 -> T1558,\nX1879 -> T1559,\nT1526 -> .(T1558, T1559),\nT1527 -> T1560,\nX1880 -> T1560,\nT1528 -> T1561,\nX1881 -> T1561,\nX1882 -> X1886,\nX1883 -> X1887,\nX1833 -> .(X1886, X1887),\nX1884 -> X1888,\nX1885 -> X1889,\nX1835 -> .(X1888, X1889)" }, { "from": 6046, "to": 6063, "label": "EVAL-BACKTRACK" }, { "from": 6048, "to": 6051, "label": "SUCCESS" }, { "from": 6062, "to": 6064, "label": "IS ERROR" }, { "from": 6062, "to": 6065, "label": "\nX1886 -> T1562" }, { "from": 6065, "to": 6066, "label": "IS ERROR" }, { "from": 6065, "to": 6069, "label": "\nX1888 -> T1563" }, { "from": 6069, "to": 6029, "label": "INSTANCE with matching:\nT1526 -> T1559\nT1527 -> T1560\nT1528 -> T1561\nX1833 -> X1887\nX1835 -> X1889" }, { "from": 6072, "to": 4805, "label": "INSTANCE with matching:\nT161 -> T1319\nT162 -> T1317\nX277 -> X1437" }, { "from": 6073, "to": 6074, "label": "SPLIT 1" }, { "from": 6073, "to": 6075, "label": "SPLIT 2\nnew knowledge:\nT1320 is ground\nT1573 is ground\nreplacements:X1438 -> T1573" }, { "from": 6074, "to": 4967, "label": "INSTANCE with matching:\nT117 -> T1320\nX176 -> X1438" }, { "from": 6075, "to": 6076, "label": "SPLIT 1" }, { "from": 6075, "to": 6077, "label": "SPLIT 2\nnew knowledge:\nT1573 is ground\nT1570 is ground\nT1576 is ground\nreplacements:X1439 -> T1576" }, { "from": 6076, "to": 4805, "label": "INSTANCE with matching:\nT161 -> T1573\nT162 -> T1570\nX277 -> X1439" }, { "from": 6077, "to": 6078, "label": "CUT" }, { "from": 6078, "to": 6079, "label": "SUCCESS" } ], "type": "Graph" } } ---------------------------------------- (344) Obligation: Triples: iota1A(X1, X2, .(X1, X3)) :- iota1A(X4, X2, X3). memberB(X1, .(X2, X3)) :- memberB(X1, X3). mergedeleteC(.(X1, X2), .(X1, X3), X4) :- mergedeleteC(X2, X3, X4). mergedeleteC(.(X1, X2), .(X3, X4), .(X3, X5)) :- mergedeleteC(.(X1, X2), X4, X5). revD(.(X1, X2), X3, X4, X5) :- revD(X2, X1, .(X3, X4), X5). revF(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) :- revD(X9, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))), X10). difG(X1, X2, X3, .(X4, X5), .(X6, X7)) :- difE(X2, X3, X5, X7). difH(X1, X2, X3, X4, .(X5, X6), .(X7, X8)) :- difG(X2, X3, X4, X6, X8). difI(X1, X2, X3, X4, X5, .(X6, X7), .(X8, X9)) :- difH(X2, X3, X4, X5, X7, X9). difJ(X1, X2, X3, X4, X5, X6, .(X7, X8), .(X9, X10)) :- difI(X2, X3, X4, X5, X6, X8, X10). difK(X1, X2, X3, X4, X5, X6, X7, .(X8, X9), .(X10, X11)) :- difJ(X2, X3, X4, X5, X6, X7, X9, X11). pds1L(X1, X2, X3, .(X4, X5), X6) :- memberB(X4, X1). pds1L(X1, X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, .(X9, X10))))))), .(X11, X12), X13) :- ','(membercB(X11, X1), difN(X10, X11, X13, X14, X15)). pds1L(X1, X2, X3, .(X4, X5), X6) :- ','(membercB(X4, X1), ','(difcN1(X2, X3, X4, X6, X7, X8), mergedeleteC(X7, X1, X9))). pds1L(X1, X2, X3, .(X4, X5), X6) :- ','(membercB(X4, X1), ','(difcN1(X2, X3, X4, X6, X7, X8), ','(mergedeletecC(X7, X1, X9), revF(X8, X10)))). pds1L(X1, X2, X3, .(X4, X5), X6) :- ','(membercB(X4, X1), ','(difcN1(X2, X3, X4, X6, X7, X8), ','(mergedeletecC(X7, X1, X9), ','(revcF(X8, X10), mergedeleteC(X10, X9, X11))))). pds1L(X1, X2, X3, .(X4, X5), X6) :- ','(membercB(X4, X1), ','(checkcM(X2, X3, X4, X1, X7, X6), pds1L(X7, X4, .(X2, X3), X5, X6))). difN(.(X1, X2), X3, X4, .(X5, X6), .(X7, X8)) :- difN(X2, X3, X4, X6, X8). pO(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12) :- memberB(X1, X2). pO(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12) :- ','(membercB(X1, X2), difK(X4, X5, X6, X7, X8, X1, X10, X13, X14)). pO(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12) :- ','(membercB(X1, X2), ','(difcZ(X3, X4, X5, X6, X7, X8, X1, X10, X13, X14), mergedeleteC(X13, X2, X15))). pO(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12) :- ','(membercB(X1, X2), ','(difcZ(X3, X4, X5, X6, X7, X8, X1, X10, X13, X14), ','(mergedeletecC(X13, X2, X15), revF(X14, X16)))). pO(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12) :- ','(membercB(X1, X2), ','(difcZ(X3, X4, X5, X6, X7, X8, X1, X10, X13, X14), ','(mergedeletecC(X13, X2, X15), ','(revcF(X14, X16), mergedeleteC(X16, X15, X9))))). pO(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12) :- ','(membercB(X1, X2), ','(checkcP(X3, X4, X5, X6, X7, X8, X1, X2, X9, X10), pds1L(X9, X1, X11, X12, X10))). pdsN2(X1, .(0, X2)) :- iota1A(X3, X4, X5). pdsN2(X1, .(0, .(X2, X3))) :- ','(iota1cQ(X4, .(X5, X6)), memberB(X2, X6)). pdsN2(X1, .(0, .(X2, X3))) :- ','(iota1cQ(X4, X5), ','(membercR(X2, X5), difE(X2, X4, X6, X7))). pdsN2(X1, .(0, .(X2, X3))) :- ','(iota1cQ(X4, .(X5, X6)), ','(membercR(X2, .(X5, X6)), ','(difcE(X2, X4, .(X5, X7), X8), mergedeleteC(X7, X6, X9)))). pdsN2(X1, .(0, .(X2, X3))) :- ','(iota1cQ(X4, .(X5, X6)), ','(membercR(X2, .(X5, X6)), ','(difcE(X2, X4, .(X7, X8), X9), mergedeleteC(.(X7, X8), X6, X10)))). pdsN2(X1, .(0, .(X2, X3))) :- ','(iota1cQ(X4, X5), ','(membercR(X2, X5), ','(difcE(X2, X4, X6, X7), ','(mergedeletecT(X6, X5, X8), revF(X7, X9))))). pdsN2(X1, .(0, .(X2, X3))) :- ','(iota1cQ(X4, X5), ','(membercR(X2, X5), ','(difcE(X2, X4, X6, X7), ','(mergedeletecT(X6, X5, X8), ','(revcF(X7, X9), mergedeleteC(X9, X8, X10)))))). pdsN2(X1, .(0, .(X2, .(X3, X4)))) :- ','(iota1cQ(X5, X6), ','(membercR(X2, X6), ','(checkcS(X2, X6, X7, X5), memberB(X3, X7)))). pdsN2(X1, .(0, .(X2, .(X3, X4)))) :- ','(iota1cQ(X5, X6), ','(membercR(X2, X6), ','(checkcS(X2, X6, X7, X5), ','(membercB(X3, X7), difG(X2, X3, X5, X8, X9))))). pdsN2(X1, .(0, .(X2, .(X3, X4)))) :- ','(iota1cQ(X5, X6), ','(membercR(X2, X6), ','(checkcS(X2, X6, X7, X5), ','(membercB(X3, X7), ','(difcG(X2, X3, X5, X8, X9), mergedeleteC(X8, X7, X10)))))). pdsN2(X1, .(0, .(X2, .(X3, X4)))) :- ','(iota1cQ(X5, X6), ','(membercR(X2, X6), ','(checkcS(X2, X6, X7, X5), ','(membercB(X3, X7), ','(difcG(X2, X3, X5, X8, X9), ','(mergedeletecC(X8, X7, X10), revF(X9, X11))))))). pdsN2(X1, .(0, .(X2, .(X3, X4)))) :- ','(iota1cQ(X5, X6), ','(membercR(X2, X6), ','(checkcS(X2, X6, X7, X5), ','(membercB(X3, X7), ','(difcG(X2, X3, X5, X8, X9), ','(mergedeletecC(X8, X7, X10), ','(revcF(X9, X11), mergedeleteC(X11, X10, X12)))))))). pdsN2(X1, .(0, .(X2, .(X3, .(X4, X5))))) :- ','(iota1cQ(X6, X7), ','(membercR(X2, X7), ','(checkcS(X2, X7, X8, X6), ','(membercB(X3, X8), ','(checkcU(X2, X3, X8, X9, X6), memberB(X4, X9)))))). pdsN2(X1, .(0, .(X2, .(X3, .(X4, X5))))) :- ','(iota1cQ(X6, X7), ','(membercR(X2, X7), ','(checkcS(X2, X7, X8, X6), ','(membercB(X3, X8), ','(checkcU(X2, X3, X8, X9, X6), ','(membercB(X4, X9), difH(X3, X2, X4, X6, X10, X11))))))). pdsN2(X1, .(0, .(X2, .(X3, .(X4, X5))))) :- ','(iota1cQ(X6, X7), ','(membercR(X2, X7), ','(checkcS(X2, X7, X8, X6), ','(membercB(X3, X8), ','(checkcU(X2, X3, X8, X9, X6), ','(membercB(X4, X9), ','(difcH(X3, X2, X4, X6, X10, X11), mergedeleteC(X10, X9, X12)))))))). pdsN2(X1, .(0, .(X2, .(X3, .(X4, X5))))) :- ','(iota1cQ(X6, X7), ','(membercR(X2, X7), ','(checkcS(X2, X7, X8, X6), ','(membercB(X3, X8), ','(checkcU(X2, X3, X8, X9, X6), ','(membercB(X4, X9), ','(difcH(X3, X2, X4, X6, X10, X11), ','(mergedeletecC(X10, X9, X12), revF(X11, X13))))))))). pdsN2(X1, .(0, .(X2, .(X3, .(X4, X5))))) :- ','(iota1cQ(X6, X7), ','(membercR(X2, X7), ','(checkcS(X2, X7, X8, X6), ','(membercB(X3, X8), ','(checkcU(X2, X3, X8, X9, X6), ','(membercB(X4, X9), ','(difcH(X3, X2, X4, X6, X10, X11), ','(mergedeletecC(X10, X9, X12), ','(revcF(X11, X13), mergedeleteC(X13, X12, X14)))))))))). pdsN2(X1, .(0, .(X2, .(X3, .(X4, .(X5, X6)))))) :- ','(iota1cQ(X7, X8), ','(membercR(X2, X8), ','(checkcS(X2, X8, X9, X7), ','(membercB(X3, X9), ','(checkcU(X2, X3, X9, X10, X7), ','(membercB(X4, X10), ','(checkcV(X3, X2, X4, X10, X11, X7), memberB(X5, X11)))))))). pdsN2(X1, .(0, .(X2, .(X3, .(X4, .(X5, X6)))))) :- ','(iota1cQ(X7, X8), ','(membercR(X2, X8), ','(checkcS(X2, X8, X9, X7), ','(membercB(X3, X9), ','(checkcU(X2, X3, X9, X10, X7), ','(membercB(X4, X10), ','(checkcV(X3, X2, X4, X10, X11, X7), ','(membercB(X5, X11), difI(X4, X3, X2, X5, X7, X12, X13))))))))). pdsN2(X1, .(0, .(X2, .(X3, .(X4, .(X5, X6)))))) :- ','(iota1cQ(X7, X8), ','(membercR(X2, X8), ','(checkcS(X2, X8, X9, X7), ','(membercB(X3, X9), ','(checkcU(X2, X3, X9, X10, X7), ','(membercB(X4, X10), ','(checkcV(X3, X2, X4, X10, X11, X7), ','(membercB(X5, X11), ','(difcI(X4, X3, X2, X5, X7, X12, X13), mergedeleteC(X12, X11, X14)))))))))). pdsN2(X1, .(0, .(X2, .(X3, .(X4, .(X5, X6)))))) :- ','(iota1cQ(X7, X8), ','(membercR(X2, X8), ','(checkcS(X2, X8, X9, X7), ','(membercB(X3, X9), ','(checkcU(X2, X3, X9, X10, X7), ','(membercB(X4, X10), ','(checkcV(X3, X2, X4, X10, X11, X7), ','(membercB(X5, X11), ','(difcI(X4, X3, X2, X5, X7, X12, X13), ','(mergedeletecC(X12, X11, X14), revF(X13, X15))))))))))). pdsN2(X1, .(0, .(X2, .(X3, .(X4, .(X5, X6)))))) :- ','(iota1cQ(X7, X8), ','(membercR(X2, X8), ','(checkcS(X2, X8, X9, X7), ','(membercB(X3, X9), ','(checkcU(X2, X3, X9, X10, X7), ','(membercB(X4, X10), ','(checkcV(X3, X2, X4, X10, X11, X7), ','(membercB(X5, X11), ','(difcI(X4, X3, X2, X5, X7, X12, X13), ','(mergedeletecC(X12, X11, X14), ','(revcF(X13, X15), mergedeleteC(X15, X14, X16)))))))))))). pdsN2(X1, .(0, .(X2, .(X3, .(X4, .(X5, .(X6, X7))))))) :- ','(iota1cQ(X8, X9), ','(membercR(X2, X9), ','(checkcS(X2, X9, X10, X8), ','(membercB(X3, X10), ','(checkcU(X2, X3, X10, X11, X8), ','(membercB(X4, X11), ','(checkcV(X3, X2, X4, X11, X12, X8), ','(membercB(X5, X12), ','(checkcW(X4, X3, X2, X5, X12, X13, X8), memberB(X6, X13)))))))))). pdsN2(X1, .(0, .(X2, .(X3, .(X4, .(X5, .(X6, X7))))))) :- ','(iota1cQ(X8, X9), ','(membercR(X2, X9), ','(checkcS(X2, X9, X10, X8), ','(membercB(X3, X10), ','(checkcU(X2, X3, X10, X11, X8), ','(membercB(X4, X11), ','(checkcV(X3, X2, X4, X11, X12, X8), ','(membercB(X5, X12), ','(checkcW(X4, X3, X2, X5, X12, X13, X8), ','(membercB(X6, X13), difJ(X5, X4, X3, X2, X6, X8, X14, X15))))))))))). pdsN2(X1, .(0, .(X2, .(X3, .(X4, .(X5, .(X6, X7))))))) :- ','(iota1cQ(X8, X9), ','(membercR(X2, X9), ','(checkcS(X2, X9, X10, X8), ','(membercB(X3, X10), ','(checkcU(X2, X3, X10, X11, X8), ','(membercB(X4, X11), ','(checkcV(X3, X2, X4, X11, X12, X8), ','(membercB(X5, X12), ','(checkcW(X4, X3, X2, X5, X12, X13, X8), ','(membercB(X6, X13), ','(difcJ(X5, X4, X3, X2, X6, X8, X14, X15), mergedeleteC(X14, X13, X16)))))))))))). pdsN2(X1, .(0, .(X2, .(X3, .(X4, .(X5, .(X6, X7))))))) :- ','(iota1cQ(X8, X9), ','(membercR(X2, X9), ','(checkcS(X2, X9, X10, X8), ','(membercB(X3, X10), ','(checkcU(X2, X3, X10, X11, X8), ','(membercB(X4, X11), ','(checkcV(X3, X2, X4, X11, X12, X8), ','(membercB(X5, X12), ','(checkcW(X4, X3, X2, X5, X12, X13, X8), ','(membercB(X6, X13), ','(difcJ(X5, X4, X3, X2, X6, X8, X14, X15), ','(mergedeletecC(X14, X13, X16), revF(X15, X17))))))))))))). pdsN2(X1, .(0, .(X2, .(X3, .(X4, .(X5, .(X6, X7))))))) :- ','(iota1cQ(X8, X9), ','(membercR(X2, X9), ','(checkcS(X2, X9, X10, X8), ','(membercB(X3, X10), ','(checkcU(X2, X3, X10, X11, X8), ','(membercB(X4, X11), ','(checkcV(X3, X2, X4, X11, X12, X8), ','(membercB(X5, X12), ','(checkcW(X4, X3, X2, X5, X12, X13, X8), ','(membercB(X6, X13), ','(difcJ(X5, X4, X3, X2, X6, X8, X14, X15), ','(mergedeletecC(X14, X13, X16), ','(revcF(X15, X17), mergedeleteC(X17, X16, X18)))))))))))))). pdsN2(X1, .(0, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, X8)))))))) :- ','(iota1cQ(X9, X10), ','(membercR(X2, X10), ','(checkcS(X2, X10, X11, X9), ','(membercB(X3, X11), ','(checkcU(X2, X3, X11, X12, X9), ','(membercB(X4, X12), ','(checkcV(X3, X2, X4, X12, X13, X9), ','(membercB(X5, X13), ','(checkcW(X4, X3, X2, X5, X13, X14, X9), ','(membercB(X6, X14), ','(checkcX(X5, X4, X3, X2, X6, X14, X15, X9), memberB(X7, X15)))))))))))). pdsN2(X1, .(0, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, X8)))))))) :- ','(iota1cQ(X9, X10), ','(membercR(X2, X10), ','(checkcS(X2, X10, X11, X9), ','(membercB(X3, X11), ','(checkcU(X2, X3, X11, X12, X9), ','(membercB(X4, X12), ','(checkcV(X3, X2, X4, X12, X13, X9), ','(membercB(X5, X13), ','(checkcW(X4, X3, X2, X5, X13, X14, X9), ','(membercB(X6, X14), ','(checkcX(X5, X4, X3, X2, X6, X14, X15, X9), ','(membercB(X7, X15), difK(X6, X5, X4, X3, X2, X7, X9, X16, X17))))))))))))). pdsN2(X1, .(0, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, X8)))))))) :- ','(iota1cQ(X9, X10), ','(membercR(X2, X10), ','(checkcS(X2, X10, X11, X9), ','(membercB(X3, X11), ','(checkcU(X2, X3, X11, X12, X9), ','(membercB(X4, X12), ','(checkcV(X3, X2, X4, X12, X13, X9), ','(membercB(X5, X13), ','(checkcW(X4, X3, X2, X5, X13, X14, X9), ','(membercB(X6, X14), ','(checkcX(X5, X4, X3, X2, X6, X14, X15, X9), ','(membercB(X7, X15), ','(difcK(X6, X5, X4, X3, X2, X7, X9, X16, X17), mergedeleteC(X16, X15, X18)))))))))))))). pdsN2(X1, .(0, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, X8)))))))) :- ','(iota1cQ(X9, X10), ','(membercR(X2, X10), ','(checkcS(X2, X10, X11, X9), ','(membercB(X3, X11), ','(checkcU(X2, X3, X11, X12, X9), ','(membercB(X4, X12), ','(checkcV(X3, X2, X4, X12, X13, X9), ','(membercB(X5, X13), ','(checkcW(X4, X3, X2, X5, X13, X14, X9), ','(membercB(X6, X14), ','(checkcX(X5, X4, X3, X2, X6, X14, X15, X9), ','(membercB(X7, X15), ','(difcK(X6, X5, X4, X3, X2, X7, X9, X16, X17), ','(mergedeletecC(X16, X15, X18), revF(X17, X19))))))))))))))). pdsN2(X1, .(0, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, X8)))))))) :- ','(iota1cQ(X9, X10), ','(membercR(X2, X10), ','(checkcS(X2, X10, X11, X9), ','(membercB(X3, X11), ','(checkcU(X2, X3, X11, X12, X9), ','(membercB(X4, X12), ','(checkcV(X3, X2, X4, X12, X13, X9), ','(membercB(X5, X13), ','(checkcW(X4, X3, X2, X5, X13, X14, X9), ','(membercB(X6, X14), ','(checkcX(X5, X4, X3, X2, X6, X14, X15, X9), ','(membercB(X7, X15), ','(difcK(X6, X5, X4, X3, X2, X7, X9, X16, X17), ','(mergedeletecC(X16, X15, X18), ','(revcF(X17, X19), mergedeleteC(X19, X18, X20)))))))))))))))). pdsN2(X1, .(0, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9))))))))) :- ','(iota1cQ(X10, X11), ','(membercR(X2, X11), ','(checkcS(X2, X11, X12, X10), ','(membercB(X3, X12), ','(checkcU(X2, X3, X12, X13, X10), ','(membercB(X4, X13), ','(checkcV(X3, X2, X4, X13, X14, X10), ','(membercB(X5, X14), ','(checkcW(X4, X3, X2, X5, X14, X15, X10), ','(membercB(X6, X15), ','(checkcX(X5, X4, X3, X2, X6, X15, X16, X10), ','(membercB(X7, X16), ','(checkcY(X6, X5, X4, X3, X2, X7, X16, X17, X10), pO(X8, X17, X7, X6, X5, X4, X3, X2, X18, X10, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(0, []))))))), X9)))))))))))))). Clauses: iota1cA(X1, X1, []). iota1cA(X1, X2, .(X1, X3)) :- iota1cA(X4, X2, X3). membercB(X1, .(X1, X2)). membercB(X1, .(X2, X3)) :- membercB(X1, X3). mergedeletecC([], X1, X1). mergedeletecC(.(X1, X2), .(X1, X3), X4) :- mergedeletecC(X2, X3, X4). mergedeletecC(.(X1, X2), .(X3, X4), .(X3, X5)) :- mergedeletecC(.(X1, X2), X4, X5). revcD([], X1, X2, .(X1, X2)). revcD(.(X1, X2), X3, X4, X5) :- revcD(X2, X1, .(X3, X4), X5). difcE(X1, X2, .(X3, []), .(X4, [])). revcF([], []). revcF(.(X1, []), .(X1, [])). revcF(.(X1, .(X2, [])), .(X2, .(X1, []))). revcF(.(X1, .(X2, .(X3, []))), .(X3, .(X2, .(X1, [])))). revcF(.(X1, .(X2, .(X3, .(X4, [])))), .(X4, .(X3, .(X2, .(X1, []))))). revcF(.(X1, .(X2, .(X3, .(X4, .(X5, []))))), .(X5, .(X4, .(X3, .(X2, .(X1, [])))))). revcF(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, [])))))), .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))). revcF(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, []))))))), .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, [])))))))). revcF(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) :- revcD(X9, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))), X10). difcG(X1, X2, X3, .(X4, X5), .(X6, X7)) :- difcE(X2, X3, X5, X7). difcH(X1, X2, X3, X4, .(X5, X6), .(X7, X8)) :- difcG(X2, X3, X4, X6, X8). difcI(X1, X2, X3, X4, X5, .(X6, X7), .(X8, X9)) :- difcH(X2, X3, X4, X5, X7, X9). difcJ(X1, X2, X3, X4, X5, X6, .(X7, X8), .(X9, X10)) :- difcI(X2, X3, X4, X5, X6, X8, X10). difcK(X1, X2, X3, X4, X5, X6, X7, .(X8, X9), .(X10, X11)) :- difcJ(X2, X3, X4, X5, X6, X7, X9, X11). pds1cL([], X1, X2, [], X3). pds1cL(X1, X2, X3, .(X4, X5), X6) :- ','(membercB(X4, X1), ','(checkcM(X2, X3, X4, X1, X7, X6), pds1cL(X7, X4, .(X2, X3), X5, X6))). difcN([], X1, X2, [], []). difcN(.(X1, X2), X3, X4, .(X5, X6), .(X7, X8)) :- difcN(X2, X3, X4, X6, X8). qcO(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12) :- ','(membercB(X1, X2), ','(checkcP(X3, X4, X5, X6, X7, X8, X1, X2, X9, X10), pds1cL(X9, X1, X11, X12, X10))). iota1cQ(X1, X2) :- iota1cA(X3, X1, X2). membercR(X1, .(X1, X2)). membercR(X1, .(X2, X3)) :- membercB(X1, X3). checkcS(X1, X2, X3, X4) :- ','(difcE(X1, X4, X5, X6), ','(mergedeletecT(X5, X2, X7), ','(revcF(X6, X8), mergedeletecC(X8, X7, X3)))). mergedeletecT([], X1, X1). mergedeletecT(.(X1, X2), .(X1, X3), X4) :- mergedeletecC(X2, X3, X4). mergedeletecT(.(X1, X2), .(X3, X4), .(X3, X5)) :- mergedeletecC(.(X1, X2), X4, X5). checkcU(X1, X2, X3, X4, X5) :- ','(difcG(X1, X2, X5, X6, X7), ','(mergedeletecC(X6, X3, X8), ','(revcF(X7, X9), mergedeletecC(X9, X8, X4)))). checkcV(X1, X2, X3, X4, X5, X6) :- ','(difcH(X1, X2, X3, X6, X7, X8), ','(mergedeletecC(X7, X4, X9), ','(revcF(X8, X10), mergedeletecC(X10, X9, X5)))). checkcW(X1, X2, X3, X4, X5, X6, X7) :- ','(difcI(X1, X2, X3, X4, X7, X8, X9), ','(mergedeletecC(X8, X5, X10), ','(revcF(X9, X11), mergedeletecC(X11, X10, X6)))). checkcX(X1, X2, X3, X4, X5, X6, X7, X8) :- ','(difcJ(X1, X2, X3, X4, X5, X8, X9, X10), ','(mergedeletecC(X9, X6, X11), ','(revcF(X10, X12), mergedeletecC(X12, X11, X7)))). checkcY(X1, X2, X3, X4, X5, X6, X7, X8, X9) :- ','(difcK(X1, X2, X3, X4, X5, X6, X9, X10, X11), ','(mergedeletecC(X10, X7, X12), ','(revcF(X11, X13), mergedeletecC(X13, X12, X8)))). checkcP(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10) :- ','(difcZ(X1, X2, X3, X4, X5, X6, X7, X10, X11, X12), ','(mergedeletecC(X11, X8, X13), ','(revcF(X12, X14), mergedeletecC(X14, X13, X9)))). difcZ(X1, X2, X3, X4, X5, X6, X7, X8, .(X9, X10), .(X11, X12)) :- difcK(X2, X3, X4, X5, X6, X7, X8, X10, X12). checkcM(X1, X2, X3, X4, X5, X6) :- ','(difcN1(X1, X2, X3, X6, X7, X8), ','(mergedeletecC(X7, X4, X9), ','(revcF(X8, X10), mergedeletecC(X10, X9, X5)))). difcN1(X1, [], X2, X3, .(X4, []), .(X5, [])). difcN1(X1, .(X2, []), X3, X4, .(X5, .(X6, [])), .(X7, .(X8, []))). difcN1(X1, .(X2, .(X3, [])), X4, X5, .(X6, .(X7, .(X8, []))), .(X9, .(X10, .(X11, [])))). difcN1(X1, .(X2, .(X3, .(X4, []))), X5, X6, .(X7, .(X8, .(X9, .(X10, [])))), .(X11, .(X12, .(X13, .(X14, []))))). difcN1(X1, .(X2, .(X3, .(X4, .(X5, [])))), X6, X7, .(X8, .(X9, .(X10, .(X11, .(X12, []))))), .(X13, .(X14, .(X15, .(X16, .(X17, [])))))). difcN1(X1, .(X2, .(X3, .(X4, .(X5, .(X6, []))))), X7, X8, .(X9, .(X10, .(X11, .(X12, .(X13, .(X14, [])))))), .(X15, .(X16, .(X17, .(X18, .(X19, .(X20, []))))))). difcN1(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, [])))))), X8, X9, .(X10, .(X11, .(X12, .(X13, .(X14, .(X15, .(X16, []))))))), .(X17, .(X18, .(X19, .(X20, .(X21, .(X22, .(X23, [])))))))). difcN1(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9))))))), X10, X11, .(X12, .(X13, .(X14, .(X15, .(X16, .(X17, .(X18, .(X19, X20)))))))), .(X21, .(X22, .(X23, .(X24, .(X25, .(X26, .(X27, .(X28, X29))))))))) :- difcN(X9, X10, X11, X20, X29). Afs: pdsN2(x1, x2) = pdsN2(x1) ---------------------------------------- (345) UndefinedPredicateInTriplesTransformerProof (SOUND) Deleted triples and predicates having undefined goals [DT09]. ---------------------------------------- (346) Obligation: Triples: iota1A(X1, X2, .(X1, X3)) :- iota1A(X4, X2, X3). memberB(X1, .(X2, X3)) :- memberB(X1, X3). mergedeleteC(.(X1, X2), .(X1, X3), X4) :- mergedeleteC(X2, X3, X4). mergedeleteC(.(X1, X2), .(X3, X4), .(X3, X5)) :- mergedeleteC(.(X1, X2), X4, X5). revD(.(X1, X2), X3, X4, X5) :- revD(X2, X1, .(X3, X4), X5). revF(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) :- revD(X9, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))), X10). pds1L(X1, X2, X3, .(X4, X5), X6) :- memberB(X4, X1). pds1L(X1, X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, .(X9, X10))))))), .(X11, X12), X13) :- ','(membercB(X11, X1), difN(X10, X11, X13, X14, X15)). pds1L(X1, X2, X3, .(X4, X5), X6) :- ','(membercB(X4, X1), ','(difcN1(X2, X3, X4, X6, X7, X8), mergedeleteC(X7, X1, X9))). pds1L(X1, X2, X3, .(X4, X5), X6) :- ','(membercB(X4, X1), ','(difcN1(X2, X3, X4, X6, X7, X8), ','(mergedeletecC(X7, X1, X9), revF(X8, X10)))). pds1L(X1, X2, X3, .(X4, X5), X6) :- ','(membercB(X4, X1), ','(difcN1(X2, X3, X4, X6, X7, X8), ','(mergedeletecC(X7, X1, X9), ','(revcF(X8, X10), mergedeleteC(X10, X9, X11))))). pds1L(X1, X2, X3, .(X4, X5), X6) :- ','(membercB(X4, X1), ','(checkcM(X2, X3, X4, X1, X7, X6), pds1L(X7, X4, .(X2, X3), X5, X6))). difN(.(X1, X2), X3, X4, .(X5, X6), .(X7, X8)) :- difN(X2, X3, X4, X6, X8). pO(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12) :- memberB(X1, X2). pO(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12) :- ','(membercB(X1, X2), ','(difcZ(X3, X4, X5, X6, X7, X8, X1, X10, X13, X14), mergedeleteC(X13, X2, X15))). pO(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12) :- ','(membercB(X1, X2), ','(difcZ(X3, X4, X5, X6, X7, X8, X1, X10, X13, X14), ','(mergedeletecC(X13, X2, X15), revF(X14, X16)))). pO(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12) :- ','(membercB(X1, X2), ','(difcZ(X3, X4, X5, X6, X7, X8, X1, X10, X13, X14), ','(mergedeletecC(X13, X2, X15), ','(revcF(X14, X16), mergedeleteC(X16, X15, X9))))). pO(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12) :- ','(membercB(X1, X2), ','(checkcP(X3, X4, X5, X6, X7, X8, X1, X2, X9, X10), pds1L(X9, X1, X11, X12, X10))). pdsN2(X1, .(0, X2)) :- iota1A(X3, X4, X5). pdsN2(X1, .(0, .(X2, X3))) :- ','(iota1cQ(X4, .(X5, X6)), memberB(X2, X6)). pdsN2(X1, .(0, .(X2, X3))) :- ','(iota1cQ(X4, .(X5, X6)), ','(membercR(X2, .(X5, X6)), ','(difcE(X2, X4, .(X5, X7), X8), mergedeleteC(X7, X6, X9)))). pdsN2(X1, .(0, .(X2, X3))) :- ','(iota1cQ(X4, .(X5, X6)), ','(membercR(X2, .(X5, X6)), ','(difcE(X2, X4, .(X7, X8), X9), mergedeleteC(.(X7, X8), X6, X10)))). pdsN2(X1, .(0, .(X2, X3))) :- ','(iota1cQ(X4, X5), ','(membercR(X2, X5), ','(difcE(X2, X4, X6, X7), ','(mergedeletecT(X6, X5, X8), revF(X7, X9))))). pdsN2(X1, .(0, .(X2, X3))) :- ','(iota1cQ(X4, X5), ','(membercR(X2, X5), ','(difcE(X2, X4, X6, X7), ','(mergedeletecT(X6, X5, X8), ','(revcF(X7, X9), mergedeleteC(X9, X8, X10)))))). pdsN2(X1, .(0, .(X2, .(X3, X4)))) :- ','(iota1cQ(X5, X6), ','(membercR(X2, X6), ','(checkcS(X2, X6, X7, X5), memberB(X3, X7)))). pdsN2(X1, .(0, .(X2, .(X3, X4)))) :- ','(iota1cQ(X5, X6), ','(membercR(X2, X6), ','(checkcS(X2, X6, X7, X5), ','(membercB(X3, X7), ','(difcG(X2, X3, X5, X8, X9), mergedeleteC(X8, X7, X10)))))). pdsN2(X1, .(0, .(X2, .(X3, X4)))) :- ','(iota1cQ(X5, X6), ','(membercR(X2, X6), ','(checkcS(X2, X6, X7, X5), ','(membercB(X3, X7), ','(difcG(X2, X3, X5, X8, X9), ','(mergedeletecC(X8, X7, X10), revF(X9, X11))))))). pdsN2(X1, .(0, .(X2, .(X3, X4)))) :- ','(iota1cQ(X5, X6), ','(membercR(X2, X6), ','(checkcS(X2, X6, X7, X5), ','(membercB(X3, X7), ','(difcG(X2, X3, X5, X8, X9), ','(mergedeletecC(X8, X7, X10), ','(revcF(X9, X11), mergedeleteC(X11, X10, X12)))))))). pdsN2(X1, .(0, .(X2, .(X3, .(X4, X5))))) :- ','(iota1cQ(X6, X7), ','(membercR(X2, X7), ','(checkcS(X2, X7, X8, X6), ','(membercB(X3, X8), ','(checkcU(X2, X3, X8, X9, X6), memberB(X4, X9)))))). pdsN2(X1, .(0, .(X2, .(X3, .(X4, X5))))) :- ','(iota1cQ(X6, X7), ','(membercR(X2, X7), ','(checkcS(X2, X7, X8, X6), ','(membercB(X3, X8), ','(checkcU(X2, X3, X8, X9, X6), ','(membercB(X4, X9), ','(difcH(X3, X2, X4, X6, X10, X11), mergedeleteC(X10, X9, X12)))))))). pdsN2(X1, .(0, .(X2, .(X3, .(X4, X5))))) :- ','(iota1cQ(X6, X7), ','(membercR(X2, X7), ','(checkcS(X2, X7, X8, X6), ','(membercB(X3, X8), ','(checkcU(X2, X3, X8, X9, X6), ','(membercB(X4, X9), ','(difcH(X3, X2, X4, X6, X10, X11), ','(mergedeletecC(X10, X9, X12), revF(X11, X13))))))))). pdsN2(X1, .(0, .(X2, .(X3, .(X4, X5))))) :- ','(iota1cQ(X6, X7), ','(membercR(X2, X7), ','(checkcS(X2, X7, X8, X6), ','(membercB(X3, X8), ','(checkcU(X2, X3, X8, X9, X6), ','(membercB(X4, X9), ','(difcH(X3, X2, X4, X6, X10, X11), ','(mergedeletecC(X10, X9, X12), ','(revcF(X11, X13), mergedeleteC(X13, X12, X14)))))))))). pdsN2(X1, .(0, .(X2, .(X3, .(X4, .(X5, X6)))))) :- ','(iota1cQ(X7, X8), ','(membercR(X2, X8), ','(checkcS(X2, X8, X9, X7), ','(membercB(X3, X9), ','(checkcU(X2, X3, X9, X10, X7), ','(membercB(X4, X10), ','(checkcV(X3, X2, X4, X10, X11, X7), memberB(X5, X11)))))))). pdsN2(X1, .(0, .(X2, .(X3, .(X4, .(X5, X6)))))) :- ','(iota1cQ(X7, X8), ','(membercR(X2, X8), ','(checkcS(X2, X8, X9, X7), ','(membercB(X3, X9), ','(checkcU(X2, X3, X9, X10, X7), ','(membercB(X4, X10), ','(checkcV(X3, X2, X4, X10, X11, X7), ','(membercB(X5, X11), ','(difcI(X4, X3, X2, X5, X7, X12, X13), mergedeleteC(X12, X11, X14)))))))))). pdsN2(X1, .(0, .(X2, .(X3, .(X4, .(X5, X6)))))) :- ','(iota1cQ(X7, X8), ','(membercR(X2, X8), ','(checkcS(X2, X8, X9, X7), ','(membercB(X3, X9), ','(checkcU(X2, X3, X9, X10, X7), ','(membercB(X4, X10), ','(checkcV(X3, X2, X4, X10, X11, X7), ','(membercB(X5, X11), ','(difcI(X4, X3, X2, X5, X7, X12, X13), ','(mergedeletecC(X12, X11, X14), revF(X13, X15))))))))))). pdsN2(X1, .(0, .(X2, .(X3, .(X4, .(X5, X6)))))) :- ','(iota1cQ(X7, X8), ','(membercR(X2, X8), ','(checkcS(X2, X8, X9, X7), ','(membercB(X3, X9), ','(checkcU(X2, X3, X9, X10, X7), ','(membercB(X4, X10), ','(checkcV(X3, X2, X4, X10, X11, X7), ','(membercB(X5, X11), ','(difcI(X4, X3, X2, X5, X7, X12, X13), ','(mergedeletecC(X12, X11, X14), ','(revcF(X13, X15), mergedeleteC(X15, X14, X16)))))))))))). pdsN2(X1, .(0, .(X2, .(X3, .(X4, .(X5, .(X6, X7))))))) :- ','(iota1cQ(X8, X9), ','(membercR(X2, X9), ','(checkcS(X2, X9, X10, X8), ','(membercB(X3, X10), ','(checkcU(X2, X3, X10, X11, X8), ','(membercB(X4, X11), ','(checkcV(X3, X2, X4, X11, X12, X8), ','(membercB(X5, X12), ','(checkcW(X4, X3, X2, X5, X12, X13, X8), memberB(X6, X13)))))))))). pdsN2(X1, .(0, .(X2, .(X3, .(X4, .(X5, .(X6, X7))))))) :- ','(iota1cQ(X8, X9), ','(membercR(X2, X9), ','(checkcS(X2, X9, X10, X8), ','(membercB(X3, X10), ','(checkcU(X2, X3, X10, X11, X8), ','(membercB(X4, X11), ','(checkcV(X3, X2, X4, X11, X12, X8), ','(membercB(X5, X12), ','(checkcW(X4, X3, X2, X5, X12, X13, X8), ','(membercB(X6, X13), ','(difcJ(X5, X4, X3, X2, X6, X8, X14, X15), mergedeleteC(X14, X13, X16)))))))))))). pdsN2(X1, .(0, .(X2, .(X3, .(X4, .(X5, .(X6, X7))))))) :- ','(iota1cQ(X8, X9), ','(membercR(X2, X9), ','(checkcS(X2, X9, X10, X8), ','(membercB(X3, X10), ','(checkcU(X2, X3, X10, X11, X8), ','(membercB(X4, X11), ','(checkcV(X3, X2, X4, X11, X12, X8), ','(membercB(X5, X12), ','(checkcW(X4, X3, X2, X5, X12, X13, X8), ','(membercB(X6, X13), ','(difcJ(X5, X4, X3, X2, X6, X8, X14, X15), ','(mergedeletecC(X14, X13, X16), revF(X15, X17))))))))))))). pdsN2(X1, .(0, .(X2, .(X3, .(X4, .(X5, .(X6, X7))))))) :- ','(iota1cQ(X8, X9), ','(membercR(X2, X9), ','(checkcS(X2, X9, X10, X8), ','(membercB(X3, X10), ','(checkcU(X2, X3, X10, X11, X8), ','(membercB(X4, X11), ','(checkcV(X3, X2, X4, X11, X12, X8), ','(membercB(X5, X12), ','(checkcW(X4, X3, X2, X5, X12, X13, X8), ','(membercB(X6, X13), ','(difcJ(X5, X4, X3, X2, X6, X8, X14, X15), ','(mergedeletecC(X14, X13, X16), ','(revcF(X15, X17), mergedeleteC(X17, X16, X18)))))))))))))). pdsN2(X1, .(0, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, X8)))))))) :- ','(iota1cQ(X9, X10), ','(membercR(X2, X10), ','(checkcS(X2, X10, X11, X9), ','(membercB(X3, X11), ','(checkcU(X2, X3, X11, X12, X9), ','(membercB(X4, X12), ','(checkcV(X3, X2, X4, X12, X13, X9), ','(membercB(X5, X13), ','(checkcW(X4, X3, X2, X5, X13, X14, X9), ','(membercB(X6, X14), ','(checkcX(X5, X4, X3, X2, X6, X14, X15, X9), memberB(X7, X15)))))))))))). pdsN2(X1, .(0, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, X8)))))))) :- ','(iota1cQ(X9, X10), ','(membercR(X2, X10), ','(checkcS(X2, X10, X11, X9), ','(membercB(X3, X11), ','(checkcU(X2, X3, X11, X12, X9), ','(membercB(X4, X12), ','(checkcV(X3, X2, X4, X12, X13, X9), ','(membercB(X5, X13), ','(checkcW(X4, X3, X2, X5, X13, X14, X9), ','(membercB(X6, X14), ','(checkcX(X5, X4, X3, X2, X6, X14, X15, X9), ','(membercB(X7, X15), ','(difcK(X6, X5, X4, X3, X2, X7, X9, X16, X17), mergedeleteC(X16, X15, X18)))))))))))))). pdsN2(X1, .(0, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, X8)))))))) :- ','(iota1cQ(X9, X10), ','(membercR(X2, X10), ','(checkcS(X2, X10, X11, X9), ','(membercB(X3, X11), ','(checkcU(X2, X3, X11, X12, X9), ','(membercB(X4, X12), ','(checkcV(X3, X2, X4, X12, X13, X9), ','(membercB(X5, X13), ','(checkcW(X4, X3, X2, X5, X13, X14, X9), ','(membercB(X6, X14), ','(checkcX(X5, X4, X3, X2, X6, X14, X15, X9), ','(membercB(X7, X15), ','(difcK(X6, X5, X4, X3, X2, X7, X9, X16, X17), ','(mergedeletecC(X16, X15, X18), revF(X17, X19))))))))))))))). pdsN2(X1, .(0, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, X8)))))))) :- ','(iota1cQ(X9, X10), ','(membercR(X2, X10), ','(checkcS(X2, X10, X11, X9), ','(membercB(X3, X11), ','(checkcU(X2, X3, X11, X12, X9), ','(membercB(X4, X12), ','(checkcV(X3, X2, X4, X12, X13, X9), ','(membercB(X5, X13), ','(checkcW(X4, X3, X2, X5, X13, X14, X9), ','(membercB(X6, X14), ','(checkcX(X5, X4, X3, X2, X6, X14, X15, X9), ','(membercB(X7, X15), ','(difcK(X6, X5, X4, X3, X2, X7, X9, X16, X17), ','(mergedeletecC(X16, X15, X18), ','(revcF(X17, X19), mergedeleteC(X19, X18, X20)))))))))))))))). pdsN2(X1, .(0, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9))))))))) :- ','(iota1cQ(X10, X11), ','(membercR(X2, X11), ','(checkcS(X2, X11, X12, X10), ','(membercB(X3, X12), ','(checkcU(X2, X3, X12, X13, X10), ','(membercB(X4, X13), ','(checkcV(X3, X2, X4, X13, X14, X10), ','(membercB(X5, X14), ','(checkcW(X4, X3, X2, X5, X14, X15, X10), ','(membercB(X6, X15), ','(checkcX(X5, X4, X3, X2, X6, X15, X16, X10), ','(membercB(X7, X16), ','(checkcY(X6, X5, X4, X3, X2, X7, X16, X17, X10), pO(X8, X17, X7, X6, X5, X4, X3, X2, X18, X10, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(0, []))))))), X9)))))))))))))). Clauses: iota1cA(X1, X1, []). iota1cA(X1, X2, .(X1, X3)) :- iota1cA(X4, X2, X3). membercB(X1, .(X1, X2)). membercB(X1, .(X2, X3)) :- membercB(X1, X3). mergedeletecC([], X1, X1). mergedeletecC(.(X1, X2), .(X1, X3), X4) :- mergedeletecC(X2, X3, X4). mergedeletecC(.(X1, X2), .(X3, X4), .(X3, X5)) :- mergedeletecC(.(X1, X2), X4, X5). revcD([], X1, X2, .(X1, X2)). revcD(.(X1, X2), X3, X4, X5) :- revcD(X2, X1, .(X3, X4), X5). difcE(X1, X2, .(X3, []), .(X4, [])). revcF([], []). revcF(.(X1, []), .(X1, [])). revcF(.(X1, .(X2, [])), .(X2, .(X1, []))). revcF(.(X1, .(X2, .(X3, []))), .(X3, .(X2, .(X1, [])))). revcF(.(X1, .(X2, .(X3, .(X4, [])))), .(X4, .(X3, .(X2, .(X1, []))))). revcF(.(X1, .(X2, .(X3, .(X4, .(X5, []))))), .(X5, .(X4, .(X3, .(X2, .(X1, [])))))). revcF(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, [])))))), .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))). revcF(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, []))))))), .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, [])))))))). revcF(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) :- revcD(X9, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))), X10). difcG(X1, X2, X3, .(X4, X5), .(X6, X7)) :- difcE(X2, X3, X5, X7). difcH(X1, X2, X3, X4, .(X5, X6), .(X7, X8)) :- difcG(X2, X3, X4, X6, X8). difcI(X1, X2, X3, X4, X5, .(X6, X7), .(X8, X9)) :- difcH(X2, X3, X4, X5, X7, X9). difcJ(X1, X2, X3, X4, X5, X6, .(X7, X8), .(X9, X10)) :- difcI(X2, X3, X4, X5, X6, X8, X10). difcK(X1, X2, X3, X4, X5, X6, X7, .(X8, X9), .(X10, X11)) :- difcJ(X2, X3, X4, X5, X6, X7, X9, X11). pds1cL([], X1, X2, [], X3). pds1cL(X1, X2, X3, .(X4, X5), X6) :- ','(membercB(X4, X1), ','(checkcM(X2, X3, X4, X1, X7, X6), pds1cL(X7, X4, .(X2, X3), X5, X6))). difcN([], X1, X2, [], []). difcN(.(X1, X2), X3, X4, .(X5, X6), .(X7, X8)) :- difcN(X2, X3, X4, X6, X8). qcO(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12) :- ','(membercB(X1, X2), ','(checkcP(X3, X4, X5, X6, X7, X8, X1, X2, X9, X10), pds1cL(X9, X1, X11, X12, X10))). iota1cQ(X1, X2) :- iota1cA(X3, X1, X2). membercR(X1, .(X1, X2)). membercR(X1, .(X2, X3)) :- membercB(X1, X3). checkcS(X1, X2, X3, X4) :- ','(difcE(X1, X4, X5, X6), ','(mergedeletecT(X5, X2, X7), ','(revcF(X6, X8), mergedeletecC(X8, X7, X3)))). mergedeletecT([], X1, X1). mergedeletecT(.(X1, X2), .(X1, X3), X4) :- mergedeletecC(X2, X3, X4). mergedeletecT(.(X1, X2), .(X3, X4), .(X3, X5)) :- mergedeletecC(.(X1, X2), X4, X5). checkcU(X1, X2, X3, X4, X5) :- ','(difcG(X1, X2, X5, X6, X7), ','(mergedeletecC(X6, X3, X8), ','(revcF(X7, X9), mergedeletecC(X9, X8, X4)))). checkcV(X1, X2, X3, X4, X5, X6) :- ','(difcH(X1, X2, X3, X6, X7, X8), ','(mergedeletecC(X7, X4, X9), ','(revcF(X8, X10), mergedeletecC(X10, X9, X5)))). checkcW(X1, X2, X3, X4, X5, X6, X7) :- ','(difcI(X1, X2, X3, X4, X7, X8, X9), ','(mergedeletecC(X8, X5, X10), ','(revcF(X9, X11), mergedeletecC(X11, X10, X6)))). checkcX(X1, X2, X3, X4, X5, X6, X7, X8) :- ','(difcJ(X1, X2, X3, X4, X5, X8, X9, X10), ','(mergedeletecC(X9, X6, X11), ','(revcF(X10, X12), mergedeletecC(X12, X11, X7)))). checkcY(X1, X2, X3, X4, X5, X6, X7, X8, X9) :- ','(difcK(X1, X2, X3, X4, X5, X6, X9, X10, X11), ','(mergedeletecC(X10, X7, X12), ','(revcF(X11, X13), mergedeletecC(X13, X12, X8)))). checkcP(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10) :- ','(difcZ(X1, X2, X3, X4, X5, X6, X7, X10, X11, X12), ','(mergedeletecC(X11, X8, X13), ','(revcF(X12, X14), mergedeletecC(X14, X13, X9)))). difcZ(X1, X2, X3, X4, X5, X6, X7, X8, .(X9, X10), .(X11, X12)) :- difcK(X2, X3, X4, X5, X6, X7, X8, X10, X12). checkcM(X1, X2, X3, X4, X5, X6) :- ','(difcN1(X1, X2, X3, X6, X7, X8), ','(mergedeletecC(X7, X4, X9), ','(revcF(X8, X10), mergedeletecC(X10, X9, X5)))). difcN1(X1, [], X2, X3, .(X4, []), .(X5, [])). difcN1(X1, .(X2, []), X3, X4, .(X5, .(X6, [])), .(X7, .(X8, []))). difcN1(X1, .(X2, .(X3, [])), X4, X5, .(X6, .(X7, .(X8, []))), .(X9, .(X10, .(X11, [])))). difcN1(X1, .(X2, .(X3, .(X4, []))), X5, X6, .(X7, .(X8, .(X9, .(X10, [])))), .(X11, .(X12, .(X13, .(X14, []))))). difcN1(X1, .(X2, .(X3, .(X4, .(X5, [])))), X6, X7, .(X8, .(X9, .(X10, .(X11, .(X12, []))))), .(X13, .(X14, .(X15, .(X16, .(X17, [])))))). difcN1(X1, .(X2, .(X3, .(X4, .(X5, .(X6, []))))), X7, X8, .(X9, .(X10, .(X11, .(X12, .(X13, .(X14, [])))))), .(X15, .(X16, .(X17, .(X18, .(X19, .(X20, []))))))). difcN1(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, [])))))), X8, X9, .(X10, .(X11, .(X12, .(X13, .(X14, .(X15, .(X16, []))))))), .(X17, .(X18, .(X19, .(X20, .(X21, .(X22, .(X23, [])))))))). difcN1(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9))))))), X10, X11, .(X12, .(X13, .(X14, .(X15, .(X16, .(X17, .(X18, .(X19, X20)))))))), .(X21, .(X22, .(X23, .(X24, .(X25, .(X26, .(X27, .(X28, X29))))))))) :- difcN(X9, X10, X11, X20, X29). Afs: pdsN2(x1, x2) = pdsN2(x1) ---------------------------------------- (347) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: pdsN2_in_2: (b,f) iota1A_in_3: (f,f,f) iota1cQ_in_2: (f,f) iota1cA_in_3: (f,f,f) memberB_in_2: (f,b) membercR_in_2: (f,b) membercB_in_2: (f,b) mergedeleteC_in_3: (b,b,f) mergedeletecT_in_3: (b,b,f) mergedeletecC_in_3: (b,b,f) revF_in_2: (b,f) revD_in_4: (b,f,b,f) revcF_in_2: (b,f) revcD_in_4: (b,f,b,f) checkcS_in_4: (f,b,f,f) difcG_in_5: (f,f,f,f,f) checkcU_in_5: (f,f,b,f,f) difcH_in_6: (f,f,f,f,f,f) checkcV_in_6: (f,f,f,b,f,f) difcI_in_7: (f,f,f,f,f,f,f) checkcW_in_7: (f,f,f,f,b,f,f) difcJ_in_8: (f,f,f,f,f,f,f,f) checkcX_in_8: (f,f,f,f,f,b,f,f) difcK_in_9: (f,f,f,f,f,f,f,f,f) checkcY_in_9: (f,f,f,f,f,f,b,f,f) pO_in_12: (f,b,f,f,f,f,f,f,f,f,b,f) difcZ_in_10: (f,f,f,f,f,f,f,f,f,f) checkcP_in_10: (f,f,f,f,f,f,f,b,f,f) pds1L_in_5: (b,f,b,f,f) difN_in_5: (b,f,f,f,f) difcN1_in_6: (f,b,f,f,f,f) difcN_in_5: (b,f,f,f,f) checkcM_in_6: (f,b,f,b,f,f) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: PDSN2_IN_GA(X1, .(0, X2)) -> U30_GA(X1, X2, iota1A_in_aaa(X3, X4, X5)) PDSN2_IN_GA(X1, .(0, X2)) -> IOTA1A_IN_AAA(X3, X4, X5) IOTA1A_IN_AAA(X1, X2, .(X1, X3)) -> U1_AAA(X1, X2, X3, iota1A_in_aaa(X4, X2, X3)) IOTA1A_IN_AAA(X1, X2, .(X1, X3)) -> IOTA1A_IN_AAA(X4, X2, X3) PDSN2_IN_GA(X1, .(0, .(X2, X3))) -> U31_GA(X1, X2, X3, iota1cQ_in_aa(X4, .(X5, X6))) U31_GA(X1, X2, X3, iota1cQ_out_aa(X4, .(X5, X6))) -> U32_GA(X1, X2, X3, memberB_in_ag(X2, X6)) U31_GA(X1, X2, X3, iota1cQ_out_aa(X4, .(X5, X6))) -> MEMBERB_IN_AG(X2, X6) MEMBERB_IN_AG(X1, .(X2, X3)) -> U2_AG(X1, X2, X3, memberB_in_ag(X1, X3)) MEMBERB_IN_AG(X1, .(X2, X3)) -> MEMBERB_IN_AG(X1, X3) U31_GA(X1, X2, X3, iota1cQ_out_aa(X4, .(X5, X6))) -> U33_GA(X1, X2, X3, X4, X5, X6, membercR_in_ag(X2, .(X5, X6))) U33_GA(X1, X2, X3, X4, X5, X6, membercR_out_ag(X2, .(X5, X6))) -> U34_GA(X1, X2, X3, X6, difcE_in_aaaa(X2, X4, .(X5, X7), X8)) U34_GA(X1, X2, X3, X6, difcE_out_aaaa(X2, X4, .(X5, X7), X8)) -> U35_GA(X1, X2, X3, mergedeleteC_in_gga(X7, X6, X9)) U34_GA(X1, X2, X3, X6, difcE_out_aaaa(X2, X4, .(X5, X7), X8)) -> MERGEDELETEC_IN_GGA(X7, X6, X9) MERGEDELETEC_IN_GGA(.(X1, X2), .(X1, X3), X4) -> U3_GGA(X1, X2, X3, X4, mergedeleteC_in_gga(X2, X3, X4)) MERGEDELETEC_IN_GGA(.(X1, X2), .(X1, X3), X4) -> MERGEDELETEC_IN_GGA(X2, X3, X4) MERGEDELETEC_IN_GGA(.(X1, X2), .(X3, X4), .(X3, X5)) -> U4_GGA(X1, X2, X3, X4, X5, mergedeleteC_in_gga(.(X1, X2), X4, X5)) MERGEDELETEC_IN_GGA(.(X1, X2), .(X3, X4), .(X3, X5)) -> MERGEDELETEC_IN_GGA(.(X1, X2), X4, X5) U33_GA(X1, X2, X3, X4, X5, X6, membercR_out_ag(X2, .(X5, X6))) -> U36_GA(X1, X2, X3, X6, difcE_in_aaaa(X2, X4, .(X7, X8), X9)) U36_GA(X1, X2, X3, X6, difcE_out_aaaa(X2, X4, .(X7, X8), X9)) -> U37_GA(X1, X2, X3, mergedeleteC_in_gga(.(X7, X8), X6, X10)) U36_GA(X1, X2, X3, X6, difcE_out_aaaa(X2, X4, .(X7, X8), X9)) -> MERGEDELETEC_IN_GGA(.(X7, X8), X6, X10) PDSN2_IN_GA(X1, .(0, .(X2, X3))) -> U38_GA(X1, X2, X3, iota1cQ_in_aa(X4, X5)) U38_GA(X1, X2, X3, iota1cQ_out_aa(X4, X5)) -> U39_GA(X1, X2, X3, X4, X5, membercR_in_ag(X2, X5)) U39_GA(X1, X2, X3, X4, X5, membercR_out_ag(X2, X5)) -> U40_GA(X1, X2, X3, X5, difcE_in_aaaa(X2, X4, X6, X7)) U40_GA(X1, X2, X3, X5, difcE_out_aaaa(X2, X4, X6, X7)) -> U41_GA(X1, X2, X3, X7, mergedeletecT_in_gga(X6, X5, X8)) U41_GA(X1, X2, X3, X7, mergedeletecT_out_gga(X6, X5, X8)) -> U42_GA(X1, X2, X3, revF_in_ga(X7, X9)) U41_GA(X1, X2, X3, X7, mergedeletecT_out_gga(X6, X5, X8)) -> REVF_IN_GA(X7, X9) REVF_IN_GA(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) -> U6_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revD_in_gaga(X9, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))), X10)) REVF_IN_GA(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) -> REVD_IN_GAGA(X9, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))), X10) REVD_IN_GAGA(.(X1, X2), X3, X4, X5) -> U5_GAGA(X1, X2, X3, X4, X5, revD_in_gaga(X2, X1, .(X3, X4), X5)) REVD_IN_GAGA(.(X1, X2), X3, X4, X5) -> REVD_IN_GAGA(X2, X1, .(X3, X4), X5) U41_GA(X1, X2, X3, X7, mergedeletecT_out_gga(X6, X5, X8)) -> U43_GA(X1, X2, X3, X8, revcF_in_ga(X7, X9)) U43_GA(X1, X2, X3, X8, revcF_out_ga(X7, X9)) -> U44_GA(X1, X2, X3, mergedeleteC_in_gga(X9, X8, X10)) U43_GA(X1, X2, X3, X8, revcF_out_ga(X7, X9)) -> MERGEDELETEC_IN_GGA(X9, X8, X10) PDSN2_IN_GA(X1, .(0, .(X2, .(X3, X4)))) -> U45_GA(X1, X2, X3, X4, iota1cQ_in_aa(X5, X6)) U45_GA(X1, X2, X3, X4, iota1cQ_out_aa(X5, X6)) -> U46_GA(X1, X2, X3, X4, X5, X6, membercR_in_ag(X2, X6)) U46_GA(X1, X2, X3, X4, X5, X6, membercR_out_ag(X2, X6)) -> U47_GA(X1, X2, X3, X4, X5, checkcS_in_agaa(X2, X6, X7, X5)) U47_GA(X1, X2, X3, X4, X5, checkcS_out_agaa(X2, X6, X7, X5)) -> U48_GA(X1, X2, X3, X4, memberB_in_ag(X3, X7)) U47_GA(X1, X2, X3, X4, X5, checkcS_out_agaa(X2, X6, X7, X5)) -> MEMBERB_IN_AG(X3, X7) U47_GA(X1, X2, X3, X4, X5, checkcS_out_agaa(X2, X6, X7, X5)) -> U49_GA(X1, X2, X3, X4, X5, X7, membercB_in_ag(X3, X7)) U49_GA(X1, X2, X3, X4, X5, X7, membercB_out_ag(X3, X7)) -> U50_GA(X1, X2, X3, X4, X7, difcG_in_aaaaa(X2, X3, X5, X8, X9)) U50_GA(X1, X2, X3, X4, X7, difcG_out_aaaaa(X2, X3, X5, X8, X9)) -> U51_GA(X1, X2, X3, X4, mergedeleteC_in_gga(X8, X7, X10)) U50_GA(X1, X2, X3, X4, X7, difcG_out_aaaaa(X2, X3, X5, X8, X9)) -> MERGEDELETEC_IN_GGA(X8, X7, X10) U50_GA(X1, X2, X3, X4, X7, difcG_out_aaaaa(X2, X3, X5, X8, X9)) -> U52_GA(X1, X2, X3, X4, X9, mergedeletecC_in_gga(X8, X7, X10)) U52_GA(X1, X2, X3, X4, X9, mergedeletecC_out_gga(X8, X7, X10)) -> U53_GA(X1, X2, X3, X4, revF_in_ga(X9, X11)) U52_GA(X1, X2, X3, X4, X9, mergedeletecC_out_gga(X8, X7, X10)) -> REVF_IN_GA(X9, X11) U52_GA(X1, X2, X3, X4, X9, mergedeletecC_out_gga(X8, X7, X10)) -> U54_GA(X1, X2, X3, X4, X10, revcF_in_ga(X9, X11)) U54_GA(X1, X2, X3, X4, X10, revcF_out_ga(X9, X11)) -> U55_GA(X1, X2, X3, X4, mergedeleteC_in_gga(X11, X10, X12)) U54_GA(X1, X2, X3, X4, X10, revcF_out_ga(X9, X11)) -> MERGEDELETEC_IN_GGA(X11, X10, X12) PDSN2_IN_GA(X1, .(0, .(X2, .(X3, .(X4, X5))))) -> U56_GA(X1, X2, X3, X4, X5, iota1cQ_in_aa(X6, X7)) U56_GA(X1, X2, X3, X4, X5, iota1cQ_out_aa(X6, X7)) -> U57_GA(X1, X2, X3, X4, X5, X6, X7, membercR_in_ag(X2, X7)) U57_GA(X1, X2, X3, X4, X5, X6, X7, membercR_out_ag(X2, X7)) -> U58_GA(X1, X2, X3, X4, X5, X6, checkcS_in_agaa(X2, X7, X8, X6)) U58_GA(X1, X2, X3, X4, X5, X6, checkcS_out_agaa(X2, X7, X8, X6)) -> U59_GA(X1, X2, X3, X4, X5, X6, X8, membercB_in_ag(X3, X8)) U59_GA(X1, X2, X3, X4, X5, X6, X8, membercB_out_ag(X3, X8)) -> U60_GA(X1, X2, X3, X4, X5, X6, checkcU_in_aagaa(X2, X3, X8, X9, X6)) U60_GA(X1, X2, X3, X4, X5, X6, checkcU_out_aagaa(X2, X3, X8, X9, X6)) -> U61_GA(X1, X2, X3, X4, X5, memberB_in_ag(X4, X9)) U60_GA(X1, X2, X3, X4, X5, X6, checkcU_out_aagaa(X2, X3, X8, X9, X6)) -> MEMBERB_IN_AG(X4, X9) U60_GA(X1, X2, X3, X4, X5, X6, checkcU_out_aagaa(X2, X3, X8, X9, X6)) -> U62_GA(X1, X2, X3, X4, X5, X6, X9, membercB_in_ag(X4, X9)) U62_GA(X1, X2, X3, X4, X5, X6, X9, membercB_out_ag(X4, X9)) -> U63_GA(X1, X2, X3, X4, X5, X9, difcH_in_aaaaaa(X3, X2, X4, X6, X10, X11)) U63_GA(X1, X2, X3, X4, X5, X9, difcH_out_aaaaaa(X3, X2, X4, X6, X10, X11)) -> U64_GA(X1, X2, X3, X4, X5, mergedeleteC_in_gga(X10, X9, X12)) U63_GA(X1, X2, X3, X4, X5, X9, difcH_out_aaaaaa(X3, X2, X4, X6, X10, X11)) -> MERGEDELETEC_IN_GGA(X10, X9, X12) U63_GA(X1, X2, X3, X4, X5, X9, difcH_out_aaaaaa(X3, X2, X4, X6, X10, X11)) -> U65_GA(X1, X2, X3, X4, X5, X11, mergedeletecC_in_gga(X10, X9, X12)) U65_GA(X1, X2, X3, X4, X5, X11, mergedeletecC_out_gga(X10, X9, X12)) -> U66_GA(X1, X2, X3, X4, X5, revF_in_ga(X11, X13)) U65_GA(X1, X2, X3, X4, X5, X11, mergedeletecC_out_gga(X10, X9, X12)) -> REVF_IN_GA(X11, X13) U65_GA(X1, X2, X3, X4, X5, X11, mergedeletecC_out_gga(X10, X9, X12)) -> U67_GA(X1, X2, X3, X4, X5, X12, revcF_in_ga(X11, X13)) U67_GA(X1, X2, X3, X4, X5, X12, revcF_out_ga(X11, X13)) -> U68_GA(X1, X2, X3, X4, X5, mergedeleteC_in_gga(X13, X12, X14)) U67_GA(X1, X2, X3, X4, X5, X12, revcF_out_ga(X11, X13)) -> MERGEDELETEC_IN_GGA(X13, X12, X14) PDSN2_IN_GA(X1, .(0, .(X2, .(X3, .(X4, .(X5, X6)))))) -> U69_GA(X1, X2, X3, X4, X5, X6, iota1cQ_in_aa(X7, X8)) U69_GA(X1, X2, X3, X4, X5, X6, iota1cQ_out_aa(X7, X8)) -> U70_GA(X1, X2, X3, X4, X5, X6, X7, X8, membercR_in_ag(X2, X8)) U70_GA(X1, X2, X3, X4, X5, X6, X7, X8, membercR_out_ag(X2, X8)) -> U71_GA(X1, X2, X3, X4, X5, X6, X7, checkcS_in_agaa(X2, X8, X9, X7)) U71_GA(X1, X2, X3, X4, X5, X6, X7, checkcS_out_agaa(X2, X8, X9, X7)) -> U72_GA(X1, X2, X3, X4, X5, X6, X7, X9, membercB_in_ag(X3, X9)) U72_GA(X1, X2, X3, X4, X5, X6, X7, X9, membercB_out_ag(X3, X9)) -> U73_GA(X1, X2, X3, X4, X5, X6, X7, checkcU_in_aagaa(X2, X3, X9, X10, X7)) U73_GA(X1, X2, X3, X4, X5, X6, X7, checkcU_out_aagaa(X2, X3, X9, X10, X7)) -> U74_GA(X1, X2, X3, X4, X5, X6, X7, X10, membercB_in_ag(X4, X10)) U74_GA(X1, X2, X3, X4, X5, X6, X7, X10, membercB_out_ag(X4, X10)) -> U75_GA(X1, X2, X3, X4, X5, X6, X7, checkcV_in_aaagaa(X3, X2, X4, X10, X11, X7)) U75_GA(X1, X2, X3, X4, X5, X6, X7, checkcV_out_aaagaa(X3, X2, X4, X10, X11, X7)) -> U76_GA(X1, X2, X3, X4, X5, X6, memberB_in_ag(X5, X11)) U75_GA(X1, X2, X3, X4, X5, X6, X7, checkcV_out_aaagaa(X3, X2, X4, X10, X11, X7)) -> MEMBERB_IN_AG(X5, X11) U75_GA(X1, X2, X3, X4, X5, X6, X7, checkcV_out_aaagaa(X3, X2, X4, X10, X11, X7)) -> U77_GA(X1, X2, X3, X4, X5, X6, X7, X11, membercB_in_ag(X5, X11)) U77_GA(X1, X2, X3, X4, X5, X6, X7, X11, membercB_out_ag(X5, X11)) -> U78_GA(X1, X2, X3, X4, X5, X6, X11, difcI_in_aaaaaaa(X4, X3, X2, X5, X7, X12, X13)) U78_GA(X1, X2, X3, X4, X5, X6, X11, difcI_out_aaaaaaa(X4, X3, X2, X5, X7, X12, X13)) -> U79_GA(X1, X2, X3, X4, X5, X6, mergedeleteC_in_gga(X12, X11, X14)) U78_GA(X1, X2, X3, X4, X5, X6, X11, difcI_out_aaaaaaa(X4, X3, X2, X5, X7, X12, X13)) -> MERGEDELETEC_IN_GGA(X12, X11, X14) U78_GA(X1, X2, X3, X4, X5, X6, X11, difcI_out_aaaaaaa(X4, X3, X2, X5, X7, X12, X13)) -> U80_GA(X1, X2, X3, X4, X5, X6, X13, mergedeletecC_in_gga(X12, X11, X14)) U80_GA(X1, X2, X3, X4, X5, X6, X13, mergedeletecC_out_gga(X12, X11, X14)) -> U81_GA(X1, X2, X3, X4, X5, X6, revF_in_ga(X13, X15)) U80_GA(X1, X2, X3, X4, X5, X6, X13, mergedeletecC_out_gga(X12, X11, X14)) -> REVF_IN_GA(X13, X15) U80_GA(X1, X2, X3, X4, X5, X6, X13, mergedeletecC_out_gga(X12, X11, X14)) -> U82_GA(X1, X2, X3, X4, X5, X6, X14, revcF_in_ga(X13, X15)) U82_GA(X1, X2, X3, X4, X5, X6, X14, revcF_out_ga(X13, X15)) -> U83_GA(X1, X2, X3, X4, X5, X6, mergedeleteC_in_gga(X15, X14, X16)) U82_GA(X1, X2, X3, X4, X5, X6, X14, revcF_out_ga(X13, X15)) -> MERGEDELETEC_IN_GGA(X15, X14, X16) PDSN2_IN_GA(X1, .(0, .(X2, .(X3, .(X4, .(X5, .(X6, X7))))))) -> U84_GA(X1, X2, X3, X4, X5, X6, X7, iota1cQ_in_aa(X8, X9)) U84_GA(X1, X2, X3, X4, X5, X6, X7, iota1cQ_out_aa(X8, X9)) -> U85_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, membercR_in_ag(X2, X9)) U85_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, membercR_out_ag(X2, X9)) -> U86_GA(X1, X2, X3, X4, X5, X6, X7, X8, checkcS_in_agaa(X2, X9, X10, X8)) U86_GA(X1, X2, X3, X4, X5, X6, X7, X8, checkcS_out_agaa(X2, X9, X10, X8)) -> U87_GA(X1, X2, X3, X4, X5, X6, X7, X8, X10, membercB_in_ag(X3, X10)) U87_GA(X1, X2, X3, X4, X5, X6, X7, X8, X10, membercB_out_ag(X3, X10)) -> U88_GA(X1, X2, X3, X4, X5, X6, X7, X8, checkcU_in_aagaa(X2, X3, X10, X11, X8)) U88_GA(X1, X2, X3, X4, X5, X6, X7, X8, checkcU_out_aagaa(X2, X3, X10, X11, X8)) -> U89_GA(X1, X2, X3, X4, X5, X6, X7, X8, X11, membercB_in_ag(X4, X11)) U89_GA(X1, X2, X3, X4, X5, X6, X7, X8, X11, membercB_out_ag(X4, X11)) -> U90_GA(X1, X2, X3, X4, X5, X6, X7, X8, checkcV_in_aaagaa(X3, X2, X4, X11, X12, X8)) U90_GA(X1, X2, X3, X4, X5, X6, X7, X8, checkcV_out_aaagaa(X3, X2, X4, X11, X12, X8)) -> U91_GA(X1, X2, X3, X4, X5, X6, X7, X8, X12, membercB_in_ag(X5, X12)) U91_GA(X1, X2, X3, X4, X5, X6, X7, X8, X12, membercB_out_ag(X5, X12)) -> U92_GA(X1, X2, X3, X4, X5, X6, X7, X8, checkcW_in_aaaagaa(X4, X3, X2, X5, X12, X13, X8)) U92_GA(X1, X2, X3, X4, X5, X6, X7, X8, checkcW_out_aaaagaa(X4, X3, X2, X5, X12, X13, X8)) -> U93_GA(X1, X2, X3, X4, X5, X6, X7, memberB_in_ag(X6, X13)) U92_GA(X1, X2, X3, X4, X5, X6, X7, X8, checkcW_out_aaaagaa(X4, X3, X2, X5, X12, X13, X8)) -> MEMBERB_IN_AG(X6, X13) U92_GA(X1, X2, X3, X4, X5, X6, X7, X8, checkcW_out_aaaagaa(X4, X3, X2, X5, X12, X13, X8)) -> U94_GA(X1, X2, X3, X4, X5, X6, X7, X8, X13, membercB_in_ag(X6, X13)) U94_GA(X1, X2, X3, X4, X5, X6, X7, X8, X13, membercB_out_ag(X6, X13)) -> U95_GA(X1, X2, X3, X4, X5, X6, X7, X13, difcJ_in_aaaaaaaa(X5, X4, X3, X2, X6, X8, X14, X15)) U95_GA(X1, X2, X3, X4, X5, X6, X7, X13, difcJ_out_aaaaaaaa(X5, X4, X3, X2, X6, X8, X14, X15)) -> U96_GA(X1, X2, X3, X4, X5, X6, X7, mergedeleteC_in_gga(X14, X13, X16)) U95_GA(X1, X2, X3, X4, X5, X6, X7, X13, difcJ_out_aaaaaaaa(X5, X4, X3, X2, X6, X8, X14, X15)) -> MERGEDELETEC_IN_GGA(X14, X13, X16) U95_GA(X1, X2, X3, X4, X5, X6, X7, X13, difcJ_out_aaaaaaaa(X5, X4, X3, X2, X6, X8, X14, X15)) -> U97_GA(X1, X2, X3, X4, X5, X6, X7, X15, mergedeletecC_in_gga(X14, X13, X16)) U97_GA(X1, X2, X3, X4, X5, X6, X7, X15, mergedeletecC_out_gga(X14, X13, X16)) -> U98_GA(X1, X2, X3, X4, X5, X6, X7, revF_in_ga(X15, X17)) U97_GA(X1, X2, X3, X4, X5, X6, X7, X15, mergedeletecC_out_gga(X14, X13, X16)) -> REVF_IN_GA(X15, X17) U97_GA(X1, X2, X3, X4, X5, X6, X7, X15, mergedeletecC_out_gga(X14, X13, X16)) -> U99_GA(X1, X2, X3, X4, X5, X6, X7, X16, revcF_in_ga(X15, X17)) U99_GA(X1, X2, X3, X4, X5, X6, X7, X16, revcF_out_ga(X15, X17)) -> U100_GA(X1, X2, X3, X4, X5, X6, X7, mergedeleteC_in_gga(X17, X16, X18)) U99_GA(X1, X2, X3, X4, X5, X6, X7, X16, revcF_out_ga(X15, X17)) -> MERGEDELETEC_IN_GGA(X17, X16, X18) PDSN2_IN_GA(X1, .(0, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, X8)))))))) -> U101_GA(X1, X2, X3, X4, X5, X6, X7, X8, iota1cQ_in_aa(X9, X10)) U101_GA(X1, X2, X3, X4, X5, X6, X7, X8, iota1cQ_out_aa(X9, X10)) -> U102_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, membercR_in_ag(X2, X10)) U102_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, membercR_out_ag(X2, X10)) -> U103_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, checkcS_in_agaa(X2, X10, X11, X9)) U103_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, checkcS_out_agaa(X2, X10, X11, X9)) -> U104_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X11, membercB_in_ag(X3, X11)) U104_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X11, membercB_out_ag(X3, X11)) -> U105_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, checkcU_in_aagaa(X2, X3, X11, X12, X9)) U105_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, checkcU_out_aagaa(X2, X3, X11, X12, X9)) -> U106_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X12, membercB_in_ag(X4, X12)) U106_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X12, membercB_out_ag(X4, X12)) -> U107_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, checkcV_in_aaagaa(X3, X2, X4, X12, X13, X9)) U107_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, checkcV_out_aaagaa(X3, X2, X4, X12, X13, X9)) -> U108_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X13, membercB_in_ag(X5, X13)) U108_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X13, membercB_out_ag(X5, X13)) -> U109_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, checkcW_in_aaaagaa(X4, X3, X2, X5, X13, X14, X9)) U109_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, checkcW_out_aaaagaa(X4, X3, X2, X5, X13, X14, X9)) -> U110_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X14, membercB_in_ag(X6, X14)) U110_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X14, membercB_out_ag(X6, X14)) -> U111_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, checkcX_in_aaaaagaa(X5, X4, X3, X2, X6, X14, X15, X9)) U111_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, checkcX_out_aaaaagaa(X5, X4, X3, X2, X6, X14, X15, X9)) -> U112_GA(X1, X2, X3, X4, X5, X6, X7, X8, memberB_in_ag(X7, X15)) U111_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, checkcX_out_aaaaagaa(X5, X4, X3, X2, X6, X14, X15, X9)) -> MEMBERB_IN_AG(X7, X15) U111_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, checkcX_out_aaaaagaa(X5, X4, X3, X2, X6, X14, X15, X9)) -> U113_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X15, membercB_in_ag(X7, X15)) U113_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X15, membercB_out_ag(X7, X15)) -> U114_GA(X1, X2, X3, X4, X5, X6, X7, X8, X15, difcK_in_aaaaaaaaa(X6, X5, X4, X3, X2, X7, X9, X16, X17)) U114_GA(X1, X2, X3, X4, X5, X6, X7, X8, X15, difcK_out_aaaaaaaaa(X6, X5, X4, X3, X2, X7, X9, X16, X17)) -> U115_GA(X1, X2, X3, X4, X5, X6, X7, X8, mergedeleteC_in_gga(X16, X15, X18)) U114_GA(X1, X2, X3, X4, X5, X6, X7, X8, X15, difcK_out_aaaaaaaaa(X6, X5, X4, X3, X2, X7, X9, X16, X17)) -> MERGEDELETEC_IN_GGA(X16, X15, X18) U114_GA(X1, X2, X3, X4, X5, X6, X7, X8, X15, difcK_out_aaaaaaaaa(X6, X5, X4, X3, X2, X7, X9, X16, X17)) -> U116_GA(X1, X2, X3, X4, X5, X6, X7, X8, X17, mergedeletecC_in_gga(X16, X15, X18)) U116_GA(X1, X2, X3, X4, X5, X6, X7, X8, X17, mergedeletecC_out_gga(X16, X15, X18)) -> U117_GA(X1, X2, X3, X4, X5, X6, X7, X8, revF_in_ga(X17, X19)) U116_GA(X1, X2, X3, X4, X5, X6, X7, X8, X17, mergedeletecC_out_gga(X16, X15, X18)) -> REVF_IN_GA(X17, X19) U116_GA(X1, X2, X3, X4, X5, X6, X7, X8, X17, mergedeletecC_out_gga(X16, X15, X18)) -> U118_GA(X1, X2, X3, X4, X5, X6, X7, X8, X18, revcF_in_ga(X17, X19)) U118_GA(X1, X2, X3, X4, X5, X6, X7, X8, X18, revcF_out_ga(X17, X19)) -> U119_GA(X1, X2, X3, X4, X5, X6, X7, X8, mergedeleteC_in_gga(X19, X18, X20)) U118_GA(X1, X2, X3, X4, X5, X6, X7, X8, X18, revcF_out_ga(X17, X19)) -> MERGEDELETEC_IN_GGA(X19, X18, X20) PDSN2_IN_GA(X1, .(0, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9))))))))) -> U120_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, iota1cQ_in_aa(X10, X11)) U120_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, iota1cQ_out_aa(X10, X11)) -> U121_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, membercR_in_ag(X2, X11)) U121_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, membercR_out_ag(X2, X11)) -> U122_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, checkcS_in_agaa(X2, X11, X12, X10)) U122_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, checkcS_out_agaa(X2, X11, X12, X10)) -> U123_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X12, membercB_in_ag(X3, X12)) U123_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X12, membercB_out_ag(X3, X12)) -> U124_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, checkcU_in_aagaa(X2, X3, X12, X13, X10)) U124_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, checkcU_out_aagaa(X2, X3, X12, X13, X10)) -> U125_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X13, membercB_in_ag(X4, X13)) U125_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X13, membercB_out_ag(X4, X13)) -> U126_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, checkcV_in_aaagaa(X3, X2, X4, X13, X14, X10)) U126_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, checkcV_out_aaagaa(X3, X2, X4, X13, X14, X10)) -> U127_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X14, membercB_in_ag(X5, X14)) U127_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X14, membercB_out_ag(X5, X14)) -> U128_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, checkcW_in_aaaagaa(X4, X3, X2, X5, X14, X15, X10)) U128_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, checkcW_out_aaaagaa(X4, X3, X2, X5, X14, X15, X10)) -> U129_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X15, membercB_in_ag(X6, X15)) U129_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X15, membercB_out_ag(X6, X15)) -> U130_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, checkcX_in_aaaaagaa(X5, X4, X3, X2, X6, X15, X16, X10)) U130_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, checkcX_out_aaaaagaa(X5, X4, X3, X2, X6, X15, X16, X10)) -> U131_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X16, membercB_in_ag(X7, X16)) U131_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X16, membercB_out_ag(X7, X16)) -> U132_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, checkcY_in_aaaaaagaa(X6, X5, X4, X3, X2, X7, X16, X17, X10)) U132_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, checkcY_out_aaaaaagaa(X6, X5, X4, X3, X2, X7, X16, X17, X10)) -> U133_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, pO_in_agaaaaaaaaga(X8, X17, X7, X6, X5, X4, X3, X2, X18, X10, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(0, []))))))), X9)) U132_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, checkcY_out_aaaaaagaa(X6, X5, X4, X3, X2, X7, X16, X17, X10)) -> PO_IN_AGAAAAAAAAGA(X8, X17, X7, X6, X5, X4, X3, X2, X18, X10, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(0, []))))))), X9) PO_IN_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12) -> U20_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, memberB_in_ag(X1, X2)) PO_IN_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12) -> MEMBERB_IN_AG(X1, X2) PO_IN_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12) -> U21_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, membercB_in_ag(X1, X2)) U21_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, membercB_out_ag(X1, X2)) -> U22_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, difcZ_in_aaaaaaaaaa(X3, X4, X5, X6, X7, X8, X1, X10, X13, X14)) U22_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, difcZ_out_aaaaaaaaaa(X3, X4, X5, X6, X7, X8, X1, X10, X13, X14)) -> U23_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, mergedeleteC_in_gga(X13, X2, X15)) U22_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, difcZ_out_aaaaaaaaaa(X3, X4, X5, X6, X7, X8, X1, X10, X13, X14)) -> MERGEDELETEC_IN_GGA(X13, X2, X15) U22_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, difcZ_out_aaaaaaaaaa(X3, X4, X5, X6, X7, X8, X1, X10, X13, X14)) -> U24_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X14, mergedeletecC_in_gga(X13, X2, X15)) U24_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X14, mergedeletecC_out_gga(X13, X2, X15)) -> U25_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, revF_in_ga(X14, X16)) U24_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X14, mergedeletecC_out_gga(X13, X2, X15)) -> REVF_IN_GA(X14, X16) U24_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X14, mergedeletecC_out_gga(X13, X2, X15)) -> U26_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X15, revcF_in_ga(X14, X16)) U26_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X15, revcF_out_ga(X14, X16)) -> U27_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, mergedeleteC_in_gga(X16, X15, X9)) U26_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X15, revcF_out_ga(X14, X16)) -> MERGEDELETEC_IN_GGA(X16, X15, X9) U21_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, membercB_out_ag(X1, X2)) -> U28_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, checkcP_in_aaaaaaagaa(X3, X4, X5, X6, X7, X8, X1, X2, X9, X10)) U28_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, checkcP_out_aaaaaaagaa(X3, X4, X5, X6, X7, X8, X1, X2, X9, X10)) -> U29_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, pds1L_in_gagaa(X9, X1, X11, X12, X10)) U28_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, checkcP_out_aaaaaaagaa(X3, X4, X5, X6, X7, X8, X1, X2, X9, X10)) -> PDS1L_IN_GAGAA(X9, X1, X11, X12, X10) PDS1L_IN_GAGAA(X1, X2, X3, .(X4, X5), X6) -> U7_GAGAA(X1, X2, X3, X4, X5, X6, memberB_in_ag(X4, X1)) PDS1L_IN_GAGAA(X1, X2, X3, .(X4, X5), X6) -> MEMBERB_IN_AG(X4, X1) PDS1L_IN_GAGAA(X1, X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, .(X9, X10))))))), .(X11, X12), X13) -> U8_GAGAA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, membercB_in_ag(X11, X1)) U8_GAGAA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, membercB_out_ag(X11, X1)) -> U9_GAGAA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, difN_in_gaaaa(X10, X11, X13, X14, X15)) U8_GAGAA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, membercB_out_ag(X11, X1)) -> DIFN_IN_GAAAA(X10, X11, X13, X14, X15) DIFN_IN_GAAAA(.(X1, X2), X3, X4, .(X5, X6), .(X7, X8)) -> U19_GAAAA(X1, X2, X3, X4, X5, X6, X7, X8, difN_in_gaaaa(X2, X3, X4, X6, X8)) DIFN_IN_GAAAA(.(X1, X2), X3, X4, .(X5, X6), .(X7, X8)) -> DIFN_IN_GAAAA(X2, X3, X4, X6, X8) PDS1L_IN_GAGAA(X1, X2, X3, .(X4, X5), X6) -> U10_GAGAA(X1, X2, X3, X4, X5, X6, membercB_in_ag(X4, X1)) U10_GAGAA(X1, X2, X3, X4, X5, X6, membercB_out_ag(X4, X1)) -> U11_GAGAA(X1, X2, X3, X4, X5, X6, difcN1_in_agaaaa(X2, X3, X4, X6, X7, X8)) U11_GAGAA(X1, X2, X3, X4, X5, X6, difcN1_out_agaaaa(X2, X3, X4, X6, X7, X8)) -> U12_GAGAA(X1, X2, X3, X4, X5, X6, mergedeleteC_in_gga(X7, X1, X9)) U11_GAGAA(X1, X2, X3, X4, X5, X6, difcN1_out_agaaaa(X2, X3, X4, X6, X7, X8)) -> MERGEDELETEC_IN_GGA(X7, X1, X9) U11_GAGAA(X1, X2, X3, X4, X5, X6, difcN1_out_agaaaa(X2, X3, X4, X6, X7, X8)) -> U13_GAGAA(X1, X2, X3, X4, X5, X6, X8, mergedeletecC_in_gga(X7, X1, X9)) U13_GAGAA(X1, X2, X3, X4, X5, X6, X8, mergedeletecC_out_gga(X7, X1, X9)) -> U14_GAGAA(X1, X2, X3, X4, X5, X6, revF_in_ga(X8, X10)) U13_GAGAA(X1, X2, X3, X4, X5, X6, X8, mergedeletecC_out_gga(X7, X1, X9)) -> REVF_IN_GA(X8, X10) U13_GAGAA(X1, X2, X3, X4, X5, X6, X8, mergedeletecC_out_gga(X7, X1, X9)) -> U15_GAGAA(X1, X2, X3, X4, X5, X6, X9, revcF_in_ga(X8, X10)) U15_GAGAA(X1, X2, X3, X4, X5, X6, X9, revcF_out_ga(X8, X10)) -> U16_GAGAA(X1, X2, X3, X4, X5, X6, mergedeleteC_in_gga(X10, X9, X11)) U15_GAGAA(X1, X2, X3, X4, X5, X6, X9, revcF_out_ga(X8, X10)) -> MERGEDELETEC_IN_GGA(X10, X9, X11) U10_GAGAA(X1, X2, X3, X4, X5, X6, membercB_out_ag(X4, X1)) -> U17_GAGAA(X1, X2, X3, X4, X5, X6, checkcM_in_agagaa(X2, X3, X4, X1, X7, X6)) U17_GAGAA(X1, X2, X3, X4, X5, X6, checkcM_out_agagaa(X2, X3, X4, X1, X7, X6)) -> U18_GAGAA(X1, X2, X3, X4, X5, X6, pds1L_in_gagaa(X7, X4, .(X2, X3), X5, X6)) U17_GAGAA(X1, X2, X3, X4, X5, X6, checkcM_out_agagaa(X2, X3, X4, X1, X7, X6)) -> PDS1L_IN_GAGAA(X7, X4, .(X2, X3), X5, X6) The TRS R consists of the following rules: iota1cQ_in_aa(X1, X2) -> U153_aa(X1, X2, iota1cA_in_aaa(X3, X1, X2)) iota1cA_in_aaa(X1, X1, []) -> iota1cA_out_aaa(X1, X1, []) iota1cA_in_aaa(X1, X2, .(X1, X3)) -> U135_aaa(X1, X2, X3, iota1cA_in_aaa(X4, X2, X3)) U135_aaa(X1, X2, X3, iota1cA_out_aaa(X4, X2, X3)) -> iota1cA_out_aaa(X1, X2, .(X1, X3)) U153_aa(X1, X2, iota1cA_out_aaa(X3, X1, X2)) -> iota1cQ_out_aa(X1, X2) membercR_in_ag(X1, .(X1, X2)) -> membercR_out_ag(X1, .(X1, X2)) membercR_in_ag(X1, .(X2, X3)) -> U154_ag(X1, X2, X3, membercB_in_ag(X1, X3)) membercB_in_ag(X1, .(X1, X2)) -> membercB_out_ag(X1, .(X1, X2)) membercB_in_ag(X1, .(X2, X3)) -> U136_ag(X1, X2, X3, membercB_in_ag(X1, X3)) U136_ag(X1, X2, X3, membercB_out_ag(X1, X3)) -> membercB_out_ag(X1, .(X2, X3)) U154_ag(X1, X2, X3, membercB_out_ag(X1, X3)) -> membercR_out_ag(X1, .(X2, X3)) difcE_in_aaaa(X1, X2, .(X3, []), .(X4, [])) -> difcE_out_aaaa(X1, X2, .(X3, []), .(X4, [])) mergedeletecT_in_gga([], X1, X1) -> mergedeletecT_out_gga([], X1, X1) mergedeletecT_in_gga(.(X1, X2), .(X1, X3), X4) -> U159_gga(X1, X2, X3, X4, mergedeletecC_in_gga(X2, X3, X4)) mergedeletecC_in_gga([], X1, X1) -> mergedeletecC_out_gga([], X1, X1) mergedeletecC_in_gga(.(X1, X2), .(X1, X3), X4) -> U137_gga(X1, X2, X3, X4, mergedeletecC_in_gga(X2, X3, X4)) mergedeletecC_in_gga(.(X1, X2), .(X3, X4), .(X3, X5)) -> U138_gga(X1, X2, X3, X4, X5, mergedeletecC_in_gga(.(X1, X2), X4, X5)) U138_gga(X1, X2, X3, X4, X5, mergedeletecC_out_gga(.(X1, X2), X4, X5)) -> mergedeletecC_out_gga(.(X1, X2), .(X3, X4), .(X3, X5)) U137_gga(X1, X2, X3, X4, mergedeletecC_out_gga(X2, X3, X4)) -> mergedeletecC_out_gga(.(X1, X2), .(X1, X3), X4) U159_gga(X1, X2, X3, X4, mergedeletecC_out_gga(X2, X3, X4)) -> mergedeletecT_out_gga(.(X1, X2), .(X1, X3), X4) mergedeletecT_in_gga(.(X1, X2), .(X3, X4), .(X3, X5)) -> U160_gga(X1, X2, X3, X4, X5, mergedeletecC_in_gga(.(X1, X2), X4, X5)) U160_gga(X1, X2, X3, X4, X5, mergedeletecC_out_gga(.(X1, X2), X4, X5)) -> mergedeletecT_out_gga(.(X1, X2), .(X3, X4), .(X3, X5)) revcF_in_ga([], []) -> revcF_out_ga([], []) revcF_in_ga(.(X1, []), .(X1, [])) -> revcF_out_ga(.(X1, []), .(X1, [])) revcF_in_ga(.(X1, .(X2, [])), .(X2, .(X1, []))) -> revcF_out_ga(.(X1, .(X2, [])), .(X2, .(X1, []))) revcF_in_ga(.(X1, .(X2, .(X3, []))), .(X3, .(X2, .(X1, [])))) -> revcF_out_ga(.(X1, .(X2, .(X3, []))), .(X3, .(X2, .(X1, [])))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, [])))), .(X4, .(X3, .(X2, .(X1, []))))) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, [])))), .(X4, .(X3, .(X2, .(X1, []))))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, .(X5, []))))), .(X5, .(X4, .(X3, .(X2, .(X1, [])))))) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, .(X5, []))))), .(X5, .(X4, .(X3, .(X2, .(X1, [])))))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, [])))))), .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, [])))))), .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, []))))))), .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, [])))))))) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, []))))))), .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, [])))))))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) -> U140_ga(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revcD_in_gaga(X9, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))), X10)) revcD_in_gaga([], X1, X2, .(X1, X2)) -> revcD_out_gaga([], X1, X2, .(X1, X2)) revcD_in_gaga(.(X1, X2), X3, X4, X5) -> U139_gaga(X1, X2, X3, X4, X5, revcD_in_gaga(X2, X1, .(X3, X4), X5)) U139_gaga(X1, X2, X3, X4, X5, revcD_out_gaga(X2, X1, .(X3, X4), X5)) -> revcD_out_gaga(.(X1, X2), X3, X4, X5) U140_ga(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revcD_out_gaga(X9, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))), X10)) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) checkcS_in_agaa(X1, X2, X3, X4) -> U155_agaa(X1, X2, X3, X4, difcE_in_aaaa(X1, X4, X5, X6)) U155_agaa(X1, X2, X3, X4, difcE_out_aaaa(X1, X4, X5, X6)) -> U156_agaa(X1, X2, X3, X4, X5, X6, mergedeletecT_in_gga(X5, X2, X7)) U156_agaa(X1, X2, X3, X4, X5, X6, mergedeletecT_out_gga(X5, X2, X7)) -> U157_agaa(X1, X2, X3, X4, X5, X6, X7, revcF_in_ga(X6, X8)) U157_agaa(X1, X2, X3, X4, X5, X6, X7, revcF_out_ga(X6, X8)) -> U158_agaa(X1, X2, X3, X4, mergedeletecC_in_gga(X8, X7, X3)) U158_agaa(X1, X2, X3, X4, mergedeletecC_out_gga(X8, X7, X3)) -> checkcS_out_agaa(X1, X2, X3, X4) difcG_in_aaaaa(X1, X2, X3, .(X4, X5), .(X6, X7)) -> U141_aaaaa(X1, X2, X3, X4, X5, X6, X7, difcE_in_aaaa(X2, X3, X5, X7)) U141_aaaaa(X1, X2, X3, X4, X5, X6, X7, difcE_out_aaaa(X2, X3, X5, X7)) -> difcG_out_aaaaa(X1, X2, X3, .(X4, X5), .(X6, X7)) checkcU_in_aagaa(X1, X2, X3, X4, X5) -> U161_aagaa(X1, X2, X3, X4, X5, difcG_in_aaaaa(X1, X2, X5, X6, X7)) U161_aagaa(X1, X2, X3, X4, X5, difcG_out_aaaaa(X1, X2, X5, X6, X7)) -> U162_aagaa(X1, X2, X3, X4, X5, X6, X7, mergedeletecC_in_gga(X6, X3, X8)) U162_aagaa(X1, X2, X3, X4, X5, X6, X7, mergedeletecC_out_gga(X6, X3, X8)) -> U163_aagaa(X1, X2, X3, X4, X5, X6, X7, X8, revcF_in_ga(X7, X9)) U163_aagaa(X1, X2, X3, X4, X5, X6, X7, X8, revcF_out_ga(X7, X9)) -> U164_aagaa(X1, X2, X3, X4, X5, mergedeletecC_in_gga(X9, X8, X4)) U164_aagaa(X1, X2, X3, X4, X5, mergedeletecC_out_gga(X9, X8, X4)) -> checkcU_out_aagaa(X1, X2, X3, X4, X5) difcH_in_aaaaaa(X1, X2, X3, X4, .(X5, X6), .(X7, X8)) -> U142_aaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, difcG_in_aaaaa(X2, X3, X4, X6, X8)) U142_aaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, difcG_out_aaaaa(X2, X3, X4, X6, X8)) -> difcH_out_aaaaaa(X1, X2, X3, X4, .(X5, X6), .(X7, X8)) checkcV_in_aaagaa(X1, X2, X3, X4, X5, X6) -> U165_aaagaa(X1, X2, X3, X4, X5, X6, difcH_in_aaaaaa(X1, X2, X3, X6, X7, X8)) U165_aaagaa(X1, X2, X3, X4, X5, X6, difcH_out_aaaaaa(X1, X2, X3, X6, X7, X8)) -> U166_aaagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_in_gga(X7, X4, X9)) U166_aaagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_out_gga(X7, X4, X9)) -> U167_aaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, revcF_in_ga(X8, X10)) U167_aaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, revcF_out_ga(X8, X10)) -> U168_aaagaa(X1, X2, X3, X4, X5, X6, mergedeletecC_in_gga(X10, X9, X5)) U168_aaagaa(X1, X2, X3, X4, X5, X6, mergedeletecC_out_gga(X10, X9, X5)) -> checkcV_out_aaagaa(X1, X2, X3, X4, X5, X6) difcI_in_aaaaaaa(X1, X2, X3, X4, X5, .(X6, X7), .(X8, X9)) -> U143_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, difcH_in_aaaaaa(X2, X3, X4, X5, X7, X9)) U143_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, difcH_out_aaaaaa(X2, X3, X4, X5, X7, X9)) -> difcI_out_aaaaaaa(X1, X2, X3, X4, X5, .(X6, X7), .(X8, X9)) checkcW_in_aaaagaa(X1, X2, X3, X4, X5, X6, X7) -> U169_aaaagaa(X1, X2, X3, X4, X5, X6, X7, difcI_in_aaaaaaa(X1, X2, X3, X4, X7, X8, X9)) U169_aaaagaa(X1, X2, X3, X4, X5, X6, X7, difcI_out_aaaaaaa(X1, X2, X3, X4, X7, X8, X9)) -> U170_aaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, mergedeletecC_in_gga(X8, X5, X10)) U170_aaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, mergedeletecC_out_gga(X8, X5, X10)) -> U171_aaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revcF_in_ga(X9, X11)) U171_aaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revcF_out_ga(X9, X11)) -> U172_aaaagaa(X1, X2, X3, X4, X5, X6, X7, mergedeletecC_in_gga(X11, X10, X6)) U172_aaaagaa(X1, X2, X3, X4, X5, X6, X7, mergedeletecC_out_gga(X11, X10, X6)) -> checkcW_out_aaaagaa(X1, X2, X3, X4, X5, X6, X7) difcJ_in_aaaaaaaa(X1, X2, X3, X4, X5, X6, .(X7, X8), .(X9, X10)) -> U144_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, difcI_in_aaaaaaa(X2, X3, X4, X5, X6, X8, X10)) U144_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, difcI_out_aaaaaaa(X2, X3, X4, X5, X6, X8, X10)) -> difcJ_out_aaaaaaaa(X1, X2, X3, X4, X5, X6, .(X7, X8), .(X9, X10)) checkcX_in_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8) -> U173_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, difcJ_in_aaaaaaaa(X1, X2, X3, X4, X5, X8, X9, X10)) U173_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, difcJ_out_aaaaaaaa(X1, X2, X3, X4, X5, X8, X9, X10)) -> U174_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, mergedeletecC_in_gga(X9, X6, X11)) U174_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, mergedeletecC_out_gga(X9, X6, X11)) -> U175_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, revcF_in_ga(X10, X12)) U175_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, revcF_out_ga(X10, X12)) -> U176_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_in_gga(X12, X11, X7)) U176_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_out_gga(X12, X11, X7)) -> checkcX_out_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8) difcK_in_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, .(X8, X9), .(X10, X11)) -> U145_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, difcJ_in_aaaaaaaa(X2, X3, X4, X5, X6, X7, X9, X11)) U145_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, difcJ_out_aaaaaaaa(X2, X3, X4, X5, X6, X7, X9, X11)) -> difcK_out_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, .(X8, X9), .(X10, X11)) checkcY_in_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9) -> U177_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, difcK_in_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X9, X10, X11)) U177_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, difcK_out_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X9, X10, X11)) -> U178_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, mergedeletecC_in_gga(X10, X7, X12)) U178_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, mergedeletecC_out_gga(X10, X7, X12)) -> U179_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, revcF_in_ga(X11, X13)) U179_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, revcF_out_ga(X11, X13)) -> U180_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, mergedeletecC_in_gga(X13, X12, X8)) U180_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, mergedeletecC_out_gga(X13, X12, X8)) -> checkcY_out_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9) difcZ_in_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, .(X9, X10), .(X11, X12)) -> U185_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, difcK_in_aaaaaaaaa(X2, X3, X4, X5, X6, X7, X8, X10, X12)) U185_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, difcK_out_aaaaaaaaa(X2, X3, X4, X5, X6, X7, X8, X10, X12)) -> difcZ_out_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, .(X9, X10), .(X11, X12)) checkcP_in_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10) -> U181_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, difcZ_in_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X10, X11, X12)) U181_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, difcZ_out_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X10, X11, X12)) -> U182_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, mergedeletecC_in_gga(X11, X8, X13)) U182_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, mergedeletecC_out_gga(X11, X8, X13)) -> U183_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, revcF_in_ga(X12, X14)) U183_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, revcF_out_ga(X12, X14)) -> U184_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, mergedeletecC_in_gga(X14, X13, X9)) U184_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, mergedeletecC_out_gga(X14, X13, X9)) -> checkcP_out_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10) difcN1_in_agaaaa(X1, [], X2, X3, .(X4, []), .(X5, [])) -> difcN1_out_agaaaa(X1, [], X2, X3, .(X4, []), .(X5, [])) difcN1_in_agaaaa(X1, .(X2, []), X3, X4, .(X5, .(X6, [])), .(X7, .(X8, []))) -> difcN1_out_agaaaa(X1, .(X2, []), X3, X4, .(X5, .(X6, [])), .(X7, .(X8, []))) difcN1_in_agaaaa(X1, .(X2, .(X3, [])), X4, X5, .(X6, .(X7, .(X8, []))), .(X9, .(X10, .(X11, [])))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, [])), X4, X5, .(X6, .(X7, .(X8, []))), .(X9, .(X10, .(X11, [])))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, []))), X5, X6, .(X7, .(X8, .(X9, .(X10, [])))), .(X11, .(X12, .(X13, .(X14, []))))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, []))), X5, X6, .(X7, .(X8, .(X9, .(X10, [])))), .(X11, .(X12, .(X13, .(X14, []))))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, [])))), X6, X7, .(X8, .(X9, .(X10, .(X11, .(X12, []))))), .(X13, .(X14, .(X15, .(X16, .(X17, [])))))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, [])))), X6, X7, .(X8, .(X9, .(X10, .(X11, .(X12, []))))), .(X13, .(X14, .(X15, .(X16, .(X17, [])))))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, []))))), X7, X8, .(X9, .(X10, .(X11, .(X12, .(X13, .(X14, [])))))), .(X15, .(X16, .(X17, .(X18, .(X19, .(X20, []))))))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, []))))), X7, X8, .(X9, .(X10, .(X11, .(X12, .(X13, .(X14, [])))))), .(X15, .(X16, .(X17, .(X18, .(X19, .(X20, []))))))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, [])))))), X8, X9, .(X10, .(X11, .(X12, .(X13, .(X14, .(X15, .(X16, []))))))), .(X17, .(X18, .(X19, .(X20, .(X21, .(X22, .(X23, [])))))))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, [])))))), X8, X9, .(X10, .(X11, .(X12, .(X13, .(X14, .(X15, .(X16, []))))))), .(X17, .(X18, .(X19, .(X20, .(X21, .(X22, .(X23, [])))))))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9))))))), X10, X11, .(X12, .(X13, .(X14, .(X15, .(X16, .(X17, .(X18, .(X19, X20)))))))), .(X21, .(X22, .(X23, .(X24, .(X25, .(X26, .(X27, .(X28, X29))))))))) -> U190_agaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, X21, X22, X23, X24, X25, X26, X27, X28, X29, difcN_in_gaaaa(X9, X10, X11, X20, X29)) difcN_in_gaaaa([], X1, X2, [], []) -> difcN_out_gaaaa([], X1, X2, [], []) difcN_in_gaaaa(.(X1, X2), X3, X4, .(X5, X6), .(X7, X8)) -> U149_gaaaa(X1, X2, X3, X4, X5, X6, X7, X8, difcN_in_gaaaa(X2, X3, X4, X6, X8)) U149_gaaaa(X1, X2, X3, X4, X5, X6, X7, X8, difcN_out_gaaaa(X2, X3, X4, X6, X8)) -> difcN_out_gaaaa(.(X1, X2), X3, X4, .(X5, X6), .(X7, X8)) U190_agaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, X21, X22, X23, X24, X25, X26, X27, X28, X29, difcN_out_gaaaa(X9, X10, X11, X20, X29)) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9))))))), X10, X11, .(X12, .(X13, .(X14, .(X15, .(X16, .(X17, .(X18, .(X19, X20)))))))), .(X21, .(X22, .(X23, .(X24, .(X25, .(X26, .(X27, .(X28, X29))))))))) checkcM_in_agagaa(X1, X2, X3, X4, X5, X6) -> U186_agagaa(X1, X2, X3, X4, X5, X6, difcN1_in_agaaaa(X1, X2, X3, X6, X7, X8)) U186_agagaa(X1, X2, X3, X4, X5, X6, difcN1_out_agaaaa(X1, X2, X3, X6, X7, X8)) -> U187_agagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_in_gga(X7, X4, X9)) U187_agagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_out_gga(X7, X4, X9)) -> U188_agagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, revcF_in_ga(X8, X10)) U188_agagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, revcF_out_ga(X8, X10)) -> U189_agagaa(X1, X2, X3, X4, X5, X6, mergedeletecC_in_gga(X10, X9, X5)) U189_agagaa(X1, X2, X3, X4, X5, X6, mergedeletecC_out_gga(X10, X9, X5)) -> checkcM_out_agagaa(X1, X2, X3, X4, X5, X6) The argument filtering Pi contains the following mapping: iota1A_in_aaa(x1, x2, x3) = iota1A_in_aaa .(x1, x2) = .(x2) iota1cQ_in_aa(x1, x2) = iota1cQ_in_aa U153_aa(x1, x2, x3) = U153_aa(x3) iota1cA_in_aaa(x1, x2, x3) = iota1cA_in_aaa iota1cA_out_aaa(x1, x2, x3) = iota1cA_out_aaa(x3) U135_aaa(x1, x2, x3, x4) = U135_aaa(x4) iota1cQ_out_aa(x1, x2) = iota1cQ_out_aa(x2) memberB_in_ag(x1, x2) = memberB_in_ag(x2) membercR_in_ag(x1, x2) = membercR_in_ag(x2) membercR_out_ag(x1, x2) = membercR_out_ag(x2) U154_ag(x1, x2, x3, x4) = U154_ag(x3, x4) membercB_in_ag(x1, x2) = membercB_in_ag(x2) membercB_out_ag(x1, x2) = membercB_out_ag(x2) U136_ag(x1, x2, x3, x4) = U136_ag(x3, x4) difcE_in_aaaa(x1, x2, x3, x4) = difcE_in_aaaa difcE_out_aaaa(x1, x2, x3, x4) = difcE_out_aaaa(x3, x4) mergedeleteC_in_gga(x1, x2, x3) = mergedeleteC_in_gga(x1, x2) mergedeletecT_in_gga(x1, x2, x3) = mergedeletecT_in_gga(x1, x2) [] = [] mergedeletecT_out_gga(x1, x2, x3) = mergedeletecT_out_gga(x1, x2, x3) U159_gga(x1, x2, x3, x4, x5) = U159_gga(x2, x3, x5) mergedeletecC_in_gga(x1, x2, x3) = mergedeletecC_in_gga(x1, x2) mergedeletecC_out_gga(x1, x2, x3) = mergedeletecC_out_gga(x1, x2, x3) U137_gga(x1, x2, x3, x4, x5) = U137_gga(x2, x3, x5) U138_gga(x1, x2, x3, x4, x5, x6) = U138_gga(x2, x4, x6) U160_gga(x1, x2, x3, x4, x5, x6) = U160_gga(x2, x4, x6) revF_in_ga(x1, x2) = revF_in_ga(x1) revD_in_gaga(x1, x2, x3, x4) = revD_in_gaga(x1, x3) revcF_in_ga(x1, x2) = revcF_in_ga(x1) revcF_out_ga(x1, x2) = revcF_out_ga(x1, x2) U140_ga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U140_ga(x9, x11) revcD_in_gaga(x1, x2, x3, x4) = revcD_in_gaga(x1, x3) revcD_out_gaga(x1, x2, x3, x4) = revcD_out_gaga(x1, x3, x4) U139_gaga(x1, x2, x3, x4, x5, x6) = U139_gaga(x2, x4, x6) checkcS_in_agaa(x1, x2, x3, x4) = checkcS_in_agaa(x2) U155_agaa(x1, x2, x3, x4, x5) = U155_agaa(x2, x5) U156_agaa(x1, x2, x3, x4, x5, x6, x7) = U156_agaa(x2, x6, x7) U157_agaa(x1, x2, x3, x4, x5, x6, x7, x8) = U157_agaa(x2, x7, x8) U158_agaa(x1, x2, x3, x4, x5) = U158_agaa(x2, x5) checkcS_out_agaa(x1, x2, x3, x4) = checkcS_out_agaa(x2, x3) difcG_in_aaaaa(x1, x2, x3, x4, x5) = difcG_in_aaaaa U141_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U141_aaaaa(x8) difcG_out_aaaaa(x1, x2, x3, x4, x5) = difcG_out_aaaaa(x4, x5) checkcU_in_aagaa(x1, x2, x3, x4, x5) = checkcU_in_aagaa(x3) U161_aagaa(x1, x2, x3, x4, x5, x6) = U161_aagaa(x3, x6) U162_aagaa(x1, x2, x3, x4, x5, x6, x7, x8) = U162_aagaa(x3, x7, x8) U163_aagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U163_aagaa(x3, x8, x9) U164_aagaa(x1, x2, x3, x4, x5, x6) = U164_aagaa(x3, x6) checkcU_out_aagaa(x1, x2, x3, x4, x5) = checkcU_out_aagaa(x3, x4) difcH_in_aaaaaa(x1, x2, x3, x4, x5, x6) = difcH_in_aaaaaa U142_aaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U142_aaaaaa(x9) difcH_out_aaaaaa(x1, x2, x3, x4, x5, x6) = difcH_out_aaaaaa(x5, x6) checkcV_in_aaagaa(x1, x2, x3, x4, x5, x6) = checkcV_in_aaagaa(x4) U165_aaagaa(x1, x2, x3, x4, x5, x6, x7) = U165_aaagaa(x4, x7) U166_aaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U166_aaagaa(x4, x8, x9) U167_aaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U167_aaagaa(x4, x9, x10) U168_aaagaa(x1, x2, x3, x4, x5, x6, x7) = U168_aaagaa(x4, x7) checkcV_out_aaagaa(x1, x2, x3, x4, x5, x6) = checkcV_out_aaagaa(x4, x5) difcI_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = difcI_in_aaaaaaa U143_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U143_aaaaaaa(x10) difcI_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = difcI_out_aaaaaaa(x6, x7) checkcW_in_aaaagaa(x1, x2, x3, x4, x5, x6, x7) = checkcW_in_aaaagaa(x5) U169_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8) = U169_aaaagaa(x5, x8) U170_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U170_aaaagaa(x5, x9, x10) U171_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U171_aaaagaa(x5, x10, x11) U172_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8) = U172_aaaagaa(x5, x8) checkcW_out_aaaagaa(x1, x2, x3, x4, x5, x6, x7) = checkcW_out_aaaagaa(x5, x6) difcJ_in_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = difcJ_in_aaaaaaaa U144_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U144_aaaaaaaa(x11) difcJ_out_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = difcJ_out_aaaaaaaa(x7, x8) checkcX_in_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8) = checkcX_in_aaaaagaa(x6) U173_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U173_aaaaagaa(x6, x9) U174_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U174_aaaaagaa(x6, x10, x11) U175_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U175_aaaaagaa(x6, x11, x12) U176_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U176_aaaaagaa(x6, x9) checkcX_out_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8) = checkcX_out_aaaaagaa(x6, x7) difcK_in_aaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = difcK_in_aaaaaaaaa U145_aaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U145_aaaaaaaaa(x12) difcK_out_aaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = difcK_out_aaaaaaaaa(x8, x9) checkcY_in_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = checkcY_in_aaaaaagaa(x7) U177_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U177_aaaaaagaa(x7, x10) U178_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U178_aaaaaagaa(x7, x11, x12) U179_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U179_aaaaaagaa(x7, x12, x13) U180_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U180_aaaaaagaa(x7, x10) checkcY_out_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = checkcY_out_aaaaaagaa(x7, x8) pO_in_agaaaaaaaaga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = pO_in_agaaaaaaaaga(x2, x11) difcZ_in_aaaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = difcZ_in_aaaaaaaaaa U185_aaaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U185_aaaaaaaaaa(x13) difcZ_out_aaaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = difcZ_out_aaaaaaaaaa(x9, x10) checkcP_in_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = checkcP_in_aaaaaaagaa(x8) U181_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U181_aaaaaaagaa(x8, x11) U182_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U182_aaaaaaagaa(x8, x12, x13) U183_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14) = U183_aaaaaaagaa(x8, x13, x14) U184_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U184_aaaaaaagaa(x8, x11) checkcP_out_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = checkcP_out_aaaaaaagaa(x8, x9) pds1L_in_gagaa(x1, x2, x3, x4, x5) = pds1L_in_gagaa(x1, x3) difN_in_gaaaa(x1, x2, x3, x4, x5) = difN_in_gaaaa(x1) difcN1_in_agaaaa(x1, x2, x3, x4, x5, x6) = difcN1_in_agaaaa(x2) difcN1_out_agaaaa(x1, x2, x3, x4, x5, x6) = difcN1_out_agaaaa(x2, x5, x6) U190_agaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30) = U190_agaaaa(x9, x30) difcN_in_gaaaa(x1, x2, x3, x4, x5) = difcN_in_gaaaa(x1) difcN_out_gaaaa(x1, x2, x3, x4, x5) = difcN_out_gaaaa(x1, x4, x5) U149_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U149_gaaaa(x2, x9) checkcM_in_agagaa(x1, x2, x3, x4, x5, x6) = checkcM_in_agagaa(x2, x4) U186_agagaa(x1, x2, x3, x4, x5, x6, x7) = U186_agagaa(x2, x4, x7) U187_agagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U187_agagaa(x2, x4, x8, x9) U188_agagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U188_agagaa(x2, x4, x9, x10) U189_agagaa(x1, x2, x3, x4, x5, x6, x7) = U189_agagaa(x2, x4, x7) checkcM_out_agagaa(x1, x2, x3, x4, x5, x6) = checkcM_out_agagaa(x2, x4, x5) PDSN2_IN_GA(x1, x2) = PDSN2_IN_GA(x1) U30_GA(x1, x2, x3) = U30_GA(x1, x3) IOTA1A_IN_AAA(x1, x2, x3) = IOTA1A_IN_AAA U1_AAA(x1, x2, x3, x4) = U1_AAA(x4) U31_GA(x1, x2, x3, x4) = U31_GA(x1, x4) U32_GA(x1, x2, x3, x4) = U32_GA(x1, x4) MEMBERB_IN_AG(x1, x2) = MEMBERB_IN_AG(x2) U2_AG(x1, x2, x3, x4) = U2_AG(x3, x4) U33_GA(x1, x2, x3, x4, x5, x6, x7) = U33_GA(x1, x6, x7) U34_GA(x1, x2, x3, x4, x5) = U34_GA(x1, x4, x5) U35_GA(x1, x2, x3, x4) = U35_GA(x1, x4) MERGEDELETEC_IN_GGA(x1, x2, x3) = MERGEDELETEC_IN_GGA(x1, x2) U3_GGA(x1, x2, x3, x4, x5) = U3_GGA(x2, x3, x5) U4_GGA(x1, x2, x3, x4, x5, x6) = U4_GGA(x2, x4, x6) U36_GA(x1, x2, x3, x4, x5) = U36_GA(x1, x4, x5) U37_GA(x1, x2, x3, x4) = U37_GA(x1, x4) U38_GA(x1, x2, x3, x4) = U38_GA(x1, x4) U39_GA(x1, x2, x3, x4, x5, x6) = U39_GA(x1, x5, x6) U40_GA(x1, x2, x3, x4, x5) = U40_GA(x1, x4, x5) U41_GA(x1, x2, x3, x4, x5) = U41_GA(x1, x4, x5) U42_GA(x1, x2, x3, x4) = U42_GA(x1, x4) REVF_IN_GA(x1, x2) = REVF_IN_GA(x1) U6_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U6_GA(x9, x11) REVD_IN_GAGA(x1, x2, x3, x4) = REVD_IN_GAGA(x1, x3) U5_GAGA(x1, x2, x3, x4, x5, x6) = U5_GAGA(x2, x4, x6) U43_GA(x1, x2, x3, x4, x5) = U43_GA(x1, x4, x5) U44_GA(x1, x2, x3, x4) = U44_GA(x1, x4) U45_GA(x1, x2, x3, x4, x5) = U45_GA(x1, x5) U46_GA(x1, x2, x3, x4, x5, x6, x7) = U46_GA(x1, x6, x7) U47_GA(x1, x2, x3, x4, x5, x6) = U47_GA(x1, x6) U48_GA(x1, x2, x3, x4, x5) = U48_GA(x1, x5) U49_GA(x1, x2, x3, x4, x5, x6, x7) = U49_GA(x1, x6, x7) U50_GA(x1, x2, x3, x4, x5, x6) = U50_GA(x1, x5, x6) U51_GA(x1, x2, x3, x4, x5) = U51_GA(x1, x5) U52_GA(x1, x2, x3, x4, x5, x6) = U52_GA(x1, x5, x6) U53_GA(x1, x2, x3, x4, x5) = U53_GA(x1, x5) U54_GA(x1, x2, x3, x4, x5, x6) = U54_GA(x1, x5, x6) U55_GA(x1, x2, x3, x4, x5) = U55_GA(x1, x5) U56_GA(x1, x2, x3, x4, x5, x6) = U56_GA(x1, x6) U57_GA(x1, x2, x3, x4, x5, x6, x7, x8) = U57_GA(x1, x7, x8) U58_GA(x1, x2, x3, x4, x5, x6, x7) = U58_GA(x1, x7) U59_GA(x1, x2, x3, x4, x5, x6, x7, x8) = U59_GA(x1, x7, x8) U60_GA(x1, x2, x3, x4, x5, x6, x7) = U60_GA(x1, x7) U61_GA(x1, x2, x3, x4, x5, x6) = U61_GA(x1, x6) U62_GA(x1, x2, x3, x4, x5, x6, x7, x8) = U62_GA(x1, x7, x8) U63_GA(x1, x2, x3, x4, x5, x6, x7) = U63_GA(x1, x6, x7) U64_GA(x1, x2, x3, x4, x5, x6) = U64_GA(x1, x6) U65_GA(x1, x2, x3, x4, x5, x6, x7) = U65_GA(x1, x6, x7) U66_GA(x1, x2, x3, x4, x5, x6) = U66_GA(x1, x6) U67_GA(x1, x2, x3, x4, x5, x6, x7) = U67_GA(x1, x6, x7) U68_GA(x1, x2, x3, x4, x5, x6) = U68_GA(x1, x6) U69_GA(x1, x2, x3, x4, x5, x6, x7) = U69_GA(x1, x7) U70_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U70_GA(x1, x8, x9) U71_GA(x1, x2, x3, x4, x5, x6, x7, x8) = U71_GA(x1, x8) U72_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U72_GA(x1, x8, x9) U73_GA(x1, x2, x3, x4, x5, x6, x7, x8) = U73_GA(x1, x8) U74_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U74_GA(x1, x8, x9) U75_GA(x1, x2, x3, x4, x5, x6, x7, x8) = U75_GA(x1, x8) U76_GA(x1, x2, x3, x4, x5, x6, x7) = U76_GA(x1, x7) U77_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U77_GA(x1, x8, x9) U78_GA(x1, x2, x3, x4, x5, x6, x7, x8) = U78_GA(x1, x7, x8) U79_GA(x1, x2, x3, x4, x5, x6, x7) = U79_GA(x1, x7) U80_GA(x1, x2, x3, x4, x5, x6, x7, x8) = U80_GA(x1, x7, x8) U81_GA(x1, x2, x3, x4, x5, x6, x7) = U81_GA(x1, x7) U82_GA(x1, x2, x3, x4, x5, x6, x7, x8) = U82_GA(x1, x7, x8) U83_GA(x1, x2, x3, x4, x5, x6, x7) = U83_GA(x1, x7) U84_GA(x1, x2, x3, x4, x5, x6, x7, x8) = U84_GA(x1, x8) U85_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U85_GA(x1, x9, x10) U86_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U86_GA(x1, x9) U87_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U87_GA(x1, x9, x10) U88_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U88_GA(x1, x9) U89_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U89_GA(x1, x9, x10) U90_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U90_GA(x1, x9) U91_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U91_GA(x1, x9, x10) U92_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U92_GA(x1, x9) U93_GA(x1, x2, x3, x4, x5, x6, x7, x8) = U93_GA(x1, x8) U94_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U94_GA(x1, x9, x10) U95_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U95_GA(x1, x8, x9) U96_GA(x1, x2, x3, x4, x5, x6, x7, x8) = U96_GA(x1, x8) U97_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U97_GA(x1, x8, x9) U98_GA(x1, x2, x3, x4, x5, x6, x7, x8) = U98_GA(x1, x8) U99_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U99_GA(x1, x8, x9) U100_GA(x1, x2, x3, x4, x5, x6, x7, x8) = U100_GA(x1, x8) U101_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U101_GA(x1, x9) U102_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U102_GA(x1, x10, x11) U103_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U103_GA(x1, x10) U104_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U104_GA(x1, x10, x11) U105_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U105_GA(x1, x10) U106_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U106_GA(x1, x10, x11) U107_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U107_GA(x1, x10) U108_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U108_GA(x1, x10, x11) U109_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U109_GA(x1, x10) U110_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U110_GA(x1, x10, x11) U111_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U111_GA(x1, x10) U112_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U112_GA(x1, x9) U113_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U113_GA(x1, x10, x11) U114_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U114_GA(x1, x9, x10) U115_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U115_GA(x1, x9) U116_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U116_GA(x1, x9, x10) U117_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U117_GA(x1, x9) U118_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U118_GA(x1, x9, x10) U119_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U119_GA(x1, x9) U120_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U120_GA(x1, x10) U121_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U121_GA(x1, x11, x12) U122_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U122_GA(x1, x11) U123_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U123_GA(x1, x11, x12) U124_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U124_GA(x1, x11) U125_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U125_GA(x1, x11, x12) U126_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U126_GA(x1, x11) U127_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U127_GA(x1, x11, x12) U128_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U128_GA(x1, x11) U129_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U129_GA(x1, x11, x12) U130_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U130_GA(x1, x11) U131_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U131_GA(x1, x11, x12) U132_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U132_GA(x1, x11) U133_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U133_GA(x1, x10) PO_IN_AGAAAAAAAAGA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = PO_IN_AGAAAAAAAAGA(x2, x11) U20_AGAAAAAAAAGA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U20_AGAAAAAAAAGA(x2, x11, x13) U21_AGAAAAAAAAGA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U21_AGAAAAAAAAGA(x2, x11, x13) U22_AGAAAAAAAAGA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U22_AGAAAAAAAAGA(x2, x11, x13) U23_AGAAAAAAAAGA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U23_AGAAAAAAAAGA(x2, x11, x13) U24_AGAAAAAAAAGA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14) = U24_AGAAAAAAAAGA(x2, x11, x13, x14) U25_AGAAAAAAAAGA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U25_AGAAAAAAAAGA(x2, x11, x13) U26_AGAAAAAAAAGA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14) = U26_AGAAAAAAAAGA(x2, x11, x13, x14) U27_AGAAAAAAAAGA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U27_AGAAAAAAAAGA(x2, x11, x13) U28_AGAAAAAAAAGA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U28_AGAAAAAAAAGA(x2, x11, x13) U29_AGAAAAAAAAGA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U29_AGAAAAAAAAGA(x2, x11, x13) PDS1L_IN_GAGAA(x1, x2, x3, x4, x5) = PDS1L_IN_GAGAA(x1, x3) U7_GAGAA(x1, x2, x3, x4, x5, x6, x7) = U7_GAGAA(x1, x3, x7) U8_GAGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14) = U8_GAGAA(x1, x10, x14) U9_GAGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14) = U9_GAGAA(x1, x10, x14) DIFN_IN_GAAAA(x1, x2, x3, x4, x5) = DIFN_IN_GAAAA(x1) U19_GAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U19_GAAAA(x2, x9) U10_GAGAA(x1, x2, x3, x4, x5, x6, x7) = U10_GAGAA(x1, x3, x7) U11_GAGAA(x1, x2, x3, x4, x5, x6, x7) = U11_GAGAA(x1, x3, x7) U12_GAGAA(x1, x2, x3, x4, x5, x6, x7) = U12_GAGAA(x1, x3, x7) U13_GAGAA(x1, x2, x3, x4, x5, x6, x7, x8) = U13_GAGAA(x1, x3, x7, x8) U14_GAGAA(x1, x2, x3, x4, x5, x6, x7) = U14_GAGAA(x1, x3, x7) U15_GAGAA(x1, x2, x3, x4, x5, x6, x7, x8) = U15_GAGAA(x1, x3, x7, x8) U16_GAGAA(x1, x2, x3, x4, x5, x6, x7) = U16_GAGAA(x1, x3, x7) U17_GAGAA(x1, x2, x3, x4, x5, x6, x7) = U17_GAGAA(x1, x3, x7) U18_GAGAA(x1, x2, x3, x4, x5, x6, x7) = U18_GAGAA(x1, x3, x7) We have to consider all (P,R,Pi)-chains Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES ---------------------------------------- (348) Obligation: Pi DP problem: The TRS P consists of the following rules: PDSN2_IN_GA(X1, .(0, X2)) -> U30_GA(X1, X2, iota1A_in_aaa(X3, X4, X5)) PDSN2_IN_GA(X1, .(0, X2)) -> IOTA1A_IN_AAA(X3, X4, X5) IOTA1A_IN_AAA(X1, X2, .(X1, X3)) -> U1_AAA(X1, X2, X3, iota1A_in_aaa(X4, X2, X3)) IOTA1A_IN_AAA(X1, X2, .(X1, X3)) -> IOTA1A_IN_AAA(X4, X2, X3) PDSN2_IN_GA(X1, .(0, .(X2, X3))) -> U31_GA(X1, X2, X3, iota1cQ_in_aa(X4, .(X5, X6))) U31_GA(X1, X2, X3, iota1cQ_out_aa(X4, .(X5, X6))) -> U32_GA(X1, X2, X3, memberB_in_ag(X2, X6)) U31_GA(X1, X2, X3, iota1cQ_out_aa(X4, .(X5, X6))) -> MEMBERB_IN_AG(X2, X6) MEMBERB_IN_AG(X1, .(X2, X3)) -> U2_AG(X1, X2, X3, memberB_in_ag(X1, X3)) MEMBERB_IN_AG(X1, .(X2, X3)) -> MEMBERB_IN_AG(X1, X3) U31_GA(X1, X2, X3, iota1cQ_out_aa(X4, .(X5, X6))) -> U33_GA(X1, X2, X3, X4, X5, X6, membercR_in_ag(X2, .(X5, X6))) U33_GA(X1, X2, X3, X4, X5, X6, membercR_out_ag(X2, .(X5, X6))) -> U34_GA(X1, X2, X3, X6, difcE_in_aaaa(X2, X4, .(X5, X7), X8)) U34_GA(X1, X2, X3, X6, difcE_out_aaaa(X2, X4, .(X5, X7), X8)) -> U35_GA(X1, X2, X3, mergedeleteC_in_gga(X7, X6, X9)) U34_GA(X1, X2, X3, X6, difcE_out_aaaa(X2, X4, .(X5, X7), X8)) -> MERGEDELETEC_IN_GGA(X7, X6, X9) MERGEDELETEC_IN_GGA(.(X1, X2), .(X1, X3), X4) -> U3_GGA(X1, X2, X3, X4, mergedeleteC_in_gga(X2, X3, X4)) MERGEDELETEC_IN_GGA(.(X1, X2), .(X1, X3), X4) -> MERGEDELETEC_IN_GGA(X2, X3, X4) MERGEDELETEC_IN_GGA(.(X1, X2), .(X3, X4), .(X3, X5)) -> U4_GGA(X1, X2, X3, X4, X5, mergedeleteC_in_gga(.(X1, X2), X4, X5)) MERGEDELETEC_IN_GGA(.(X1, X2), .(X3, X4), .(X3, X5)) -> MERGEDELETEC_IN_GGA(.(X1, X2), X4, X5) U33_GA(X1, X2, X3, X4, X5, X6, membercR_out_ag(X2, .(X5, X6))) -> U36_GA(X1, X2, X3, X6, difcE_in_aaaa(X2, X4, .(X7, X8), X9)) U36_GA(X1, X2, X3, X6, difcE_out_aaaa(X2, X4, .(X7, X8), X9)) -> U37_GA(X1, X2, X3, mergedeleteC_in_gga(.(X7, X8), X6, X10)) U36_GA(X1, X2, X3, X6, difcE_out_aaaa(X2, X4, .(X7, X8), X9)) -> MERGEDELETEC_IN_GGA(.(X7, X8), X6, X10) PDSN2_IN_GA(X1, .(0, .(X2, X3))) -> U38_GA(X1, X2, X3, iota1cQ_in_aa(X4, X5)) U38_GA(X1, X2, X3, iota1cQ_out_aa(X4, X5)) -> U39_GA(X1, X2, X3, X4, X5, membercR_in_ag(X2, X5)) U39_GA(X1, X2, X3, X4, X5, membercR_out_ag(X2, X5)) -> U40_GA(X1, X2, X3, X5, difcE_in_aaaa(X2, X4, X6, X7)) U40_GA(X1, X2, X3, X5, difcE_out_aaaa(X2, X4, X6, X7)) -> U41_GA(X1, X2, X3, X7, mergedeletecT_in_gga(X6, X5, X8)) U41_GA(X1, X2, X3, X7, mergedeletecT_out_gga(X6, X5, X8)) -> U42_GA(X1, X2, X3, revF_in_ga(X7, X9)) U41_GA(X1, X2, X3, X7, mergedeletecT_out_gga(X6, X5, X8)) -> REVF_IN_GA(X7, X9) REVF_IN_GA(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) -> U6_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revD_in_gaga(X9, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))), X10)) REVF_IN_GA(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) -> REVD_IN_GAGA(X9, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))), X10) REVD_IN_GAGA(.(X1, X2), X3, X4, X5) -> U5_GAGA(X1, X2, X3, X4, X5, revD_in_gaga(X2, X1, .(X3, X4), X5)) REVD_IN_GAGA(.(X1, X2), X3, X4, X5) -> REVD_IN_GAGA(X2, X1, .(X3, X4), X5) U41_GA(X1, X2, X3, X7, mergedeletecT_out_gga(X6, X5, X8)) -> U43_GA(X1, X2, X3, X8, revcF_in_ga(X7, X9)) U43_GA(X1, X2, X3, X8, revcF_out_ga(X7, X9)) -> U44_GA(X1, X2, X3, mergedeleteC_in_gga(X9, X8, X10)) U43_GA(X1, X2, X3, X8, revcF_out_ga(X7, X9)) -> MERGEDELETEC_IN_GGA(X9, X8, X10) PDSN2_IN_GA(X1, .(0, .(X2, .(X3, X4)))) -> U45_GA(X1, X2, X3, X4, iota1cQ_in_aa(X5, X6)) U45_GA(X1, X2, X3, X4, iota1cQ_out_aa(X5, X6)) -> U46_GA(X1, X2, X3, X4, X5, X6, membercR_in_ag(X2, X6)) U46_GA(X1, X2, X3, X4, X5, X6, membercR_out_ag(X2, X6)) -> U47_GA(X1, X2, X3, X4, X5, checkcS_in_agaa(X2, X6, X7, X5)) U47_GA(X1, X2, X3, X4, X5, checkcS_out_agaa(X2, X6, X7, X5)) -> U48_GA(X1, X2, X3, X4, memberB_in_ag(X3, X7)) U47_GA(X1, X2, X3, X4, X5, checkcS_out_agaa(X2, X6, X7, X5)) -> MEMBERB_IN_AG(X3, X7) U47_GA(X1, X2, X3, X4, X5, checkcS_out_agaa(X2, X6, X7, X5)) -> U49_GA(X1, X2, X3, X4, X5, X7, membercB_in_ag(X3, X7)) U49_GA(X1, X2, X3, X4, X5, X7, membercB_out_ag(X3, X7)) -> U50_GA(X1, X2, X3, X4, X7, difcG_in_aaaaa(X2, X3, X5, X8, X9)) U50_GA(X1, X2, X3, X4, X7, difcG_out_aaaaa(X2, X3, X5, X8, X9)) -> U51_GA(X1, X2, X3, X4, mergedeleteC_in_gga(X8, X7, X10)) U50_GA(X1, X2, X3, X4, X7, difcG_out_aaaaa(X2, X3, X5, X8, X9)) -> MERGEDELETEC_IN_GGA(X8, X7, X10) U50_GA(X1, X2, X3, X4, X7, difcG_out_aaaaa(X2, X3, X5, X8, X9)) -> U52_GA(X1, X2, X3, X4, X9, mergedeletecC_in_gga(X8, X7, X10)) U52_GA(X1, X2, X3, X4, X9, mergedeletecC_out_gga(X8, X7, X10)) -> U53_GA(X1, X2, X3, X4, revF_in_ga(X9, X11)) U52_GA(X1, X2, X3, X4, X9, mergedeletecC_out_gga(X8, X7, X10)) -> REVF_IN_GA(X9, X11) U52_GA(X1, X2, X3, X4, X9, mergedeletecC_out_gga(X8, X7, X10)) -> U54_GA(X1, X2, X3, X4, X10, revcF_in_ga(X9, X11)) U54_GA(X1, X2, X3, X4, X10, revcF_out_ga(X9, X11)) -> U55_GA(X1, X2, X3, X4, mergedeleteC_in_gga(X11, X10, X12)) U54_GA(X1, X2, X3, X4, X10, revcF_out_ga(X9, X11)) -> MERGEDELETEC_IN_GGA(X11, X10, X12) PDSN2_IN_GA(X1, .(0, .(X2, .(X3, .(X4, X5))))) -> U56_GA(X1, X2, X3, X4, X5, iota1cQ_in_aa(X6, X7)) U56_GA(X1, X2, X3, X4, X5, iota1cQ_out_aa(X6, X7)) -> U57_GA(X1, X2, X3, X4, X5, X6, X7, membercR_in_ag(X2, X7)) U57_GA(X1, X2, X3, X4, X5, X6, X7, membercR_out_ag(X2, X7)) -> U58_GA(X1, X2, X3, X4, X5, X6, checkcS_in_agaa(X2, X7, X8, X6)) U58_GA(X1, X2, X3, X4, X5, X6, checkcS_out_agaa(X2, X7, X8, X6)) -> U59_GA(X1, X2, X3, X4, X5, X6, X8, membercB_in_ag(X3, X8)) U59_GA(X1, X2, X3, X4, X5, X6, X8, membercB_out_ag(X3, X8)) -> U60_GA(X1, X2, X3, X4, X5, X6, checkcU_in_aagaa(X2, X3, X8, X9, X6)) U60_GA(X1, X2, X3, X4, X5, X6, checkcU_out_aagaa(X2, X3, X8, X9, X6)) -> U61_GA(X1, X2, X3, X4, X5, memberB_in_ag(X4, X9)) U60_GA(X1, X2, X3, X4, X5, X6, checkcU_out_aagaa(X2, X3, X8, X9, X6)) -> MEMBERB_IN_AG(X4, X9) U60_GA(X1, X2, X3, X4, X5, X6, checkcU_out_aagaa(X2, X3, X8, X9, X6)) -> U62_GA(X1, X2, X3, X4, X5, X6, X9, membercB_in_ag(X4, X9)) U62_GA(X1, X2, X3, X4, X5, X6, X9, membercB_out_ag(X4, X9)) -> U63_GA(X1, X2, X3, X4, X5, X9, difcH_in_aaaaaa(X3, X2, X4, X6, X10, X11)) U63_GA(X1, X2, X3, X4, X5, X9, difcH_out_aaaaaa(X3, X2, X4, X6, X10, X11)) -> U64_GA(X1, X2, X3, X4, X5, mergedeleteC_in_gga(X10, X9, X12)) U63_GA(X1, X2, X3, X4, X5, X9, difcH_out_aaaaaa(X3, X2, X4, X6, X10, X11)) -> MERGEDELETEC_IN_GGA(X10, X9, X12) U63_GA(X1, X2, X3, X4, X5, X9, difcH_out_aaaaaa(X3, X2, X4, X6, X10, X11)) -> U65_GA(X1, X2, X3, X4, X5, X11, mergedeletecC_in_gga(X10, X9, X12)) U65_GA(X1, X2, X3, X4, X5, X11, mergedeletecC_out_gga(X10, X9, X12)) -> U66_GA(X1, X2, X3, X4, X5, revF_in_ga(X11, X13)) U65_GA(X1, X2, X3, X4, X5, X11, mergedeletecC_out_gga(X10, X9, X12)) -> REVF_IN_GA(X11, X13) U65_GA(X1, X2, X3, X4, X5, X11, mergedeletecC_out_gga(X10, X9, X12)) -> U67_GA(X1, X2, X3, X4, X5, X12, revcF_in_ga(X11, X13)) U67_GA(X1, X2, X3, X4, X5, X12, revcF_out_ga(X11, X13)) -> U68_GA(X1, X2, X3, X4, X5, mergedeleteC_in_gga(X13, X12, X14)) U67_GA(X1, X2, X3, X4, X5, X12, revcF_out_ga(X11, X13)) -> MERGEDELETEC_IN_GGA(X13, X12, X14) PDSN2_IN_GA(X1, .(0, .(X2, .(X3, .(X4, .(X5, X6)))))) -> U69_GA(X1, X2, X3, X4, X5, X6, iota1cQ_in_aa(X7, X8)) U69_GA(X1, X2, X3, X4, X5, X6, iota1cQ_out_aa(X7, X8)) -> U70_GA(X1, X2, X3, X4, X5, X6, X7, X8, membercR_in_ag(X2, X8)) U70_GA(X1, X2, X3, X4, X5, X6, X7, X8, membercR_out_ag(X2, X8)) -> U71_GA(X1, X2, X3, X4, X5, X6, X7, checkcS_in_agaa(X2, X8, X9, X7)) U71_GA(X1, X2, X3, X4, X5, X6, X7, checkcS_out_agaa(X2, X8, X9, X7)) -> U72_GA(X1, X2, X3, X4, X5, X6, X7, X9, membercB_in_ag(X3, X9)) U72_GA(X1, X2, X3, X4, X5, X6, X7, X9, membercB_out_ag(X3, X9)) -> U73_GA(X1, X2, X3, X4, X5, X6, X7, checkcU_in_aagaa(X2, X3, X9, X10, X7)) U73_GA(X1, X2, X3, X4, X5, X6, X7, checkcU_out_aagaa(X2, X3, X9, X10, X7)) -> U74_GA(X1, X2, X3, X4, X5, X6, X7, X10, membercB_in_ag(X4, X10)) U74_GA(X1, X2, X3, X4, X5, X6, X7, X10, membercB_out_ag(X4, X10)) -> U75_GA(X1, X2, X3, X4, X5, X6, X7, checkcV_in_aaagaa(X3, X2, X4, X10, X11, X7)) U75_GA(X1, X2, X3, X4, X5, X6, X7, checkcV_out_aaagaa(X3, X2, X4, X10, X11, X7)) -> U76_GA(X1, X2, X3, X4, X5, X6, memberB_in_ag(X5, X11)) U75_GA(X1, X2, X3, X4, X5, X6, X7, checkcV_out_aaagaa(X3, X2, X4, X10, X11, X7)) -> MEMBERB_IN_AG(X5, X11) U75_GA(X1, X2, X3, X4, X5, X6, X7, checkcV_out_aaagaa(X3, X2, X4, X10, X11, X7)) -> U77_GA(X1, X2, X3, X4, X5, X6, X7, X11, membercB_in_ag(X5, X11)) U77_GA(X1, X2, X3, X4, X5, X6, X7, X11, membercB_out_ag(X5, X11)) -> U78_GA(X1, X2, X3, X4, X5, X6, X11, difcI_in_aaaaaaa(X4, X3, X2, X5, X7, X12, X13)) U78_GA(X1, X2, X3, X4, X5, X6, X11, difcI_out_aaaaaaa(X4, X3, X2, X5, X7, X12, X13)) -> U79_GA(X1, X2, X3, X4, X5, X6, mergedeleteC_in_gga(X12, X11, X14)) U78_GA(X1, X2, X3, X4, X5, X6, X11, difcI_out_aaaaaaa(X4, X3, X2, X5, X7, X12, X13)) -> MERGEDELETEC_IN_GGA(X12, X11, X14) U78_GA(X1, X2, X3, X4, X5, X6, X11, difcI_out_aaaaaaa(X4, X3, X2, X5, X7, X12, X13)) -> U80_GA(X1, X2, X3, X4, X5, X6, X13, mergedeletecC_in_gga(X12, X11, X14)) U80_GA(X1, X2, X3, X4, X5, X6, X13, mergedeletecC_out_gga(X12, X11, X14)) -> U81_GA(X1, X2, X3, X4, X5, X6, revF_in_ga(X13, X15)) U80_GA(X1, X2, X3, X4, X5, X6, X13, mergedeletecC_out_gga(X12, X11, X14)) -> REVF_IN_GA(X13, X15) U80_GA(X1, X2, X3, X4, X5, X6, X13, mergedeletecC_out_gga(X12, X11, X14)) -> U82_GA(X1, X2, X3, X4, X5, X6, X14, revcF_in_ga(X13, X15)) U82_GA(X1, X2, X3, X4, X5, X6, X14, revcF_out_ga(X13, X15)) -> U83_GA(X1, X2, X3, X4, X5, X6, mergedeleteC_in_gga(X15, X14, X16)) U82_GA(X1, X2, X3, X4, X5, X6, X14, revcF_out_ga(X13, X15)) -> MERGEDELETEC_IN_GGA(X15, X14, X16) PDSN2_IN_GA(X1, .(0, .(X2, .(X3, .(X4, .(X5, .(X6, X7))))))) -> U84_GA(X1, X2, X3, X4, X5, X6, X7, iota1cQ_in_aa(X8, X9)) U84_GA(X1, X2, X3, X4, X5, X6, X7, iota1cQ_out_aa(X8, X9)) -> U85_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, membercR_in_ag(X2, X9)) U85_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, membercR_out_ag(X2, X9)) -> U86_GA(X1, X2, X3, X4, X5, X6, X7, X8, checkcS_in_agaa(X2, X9, X10, X8)) U86_GA(X1, X2, X3, X4, X5, X6, X7, X8, checkcS_out_agaa(X2, X9, X10, X8)) -> U87_GA(X1, X2, X3, X4, X5, X6, X7, X8, X10, membercB_in_ag(X3, X10)) U87_GA(X1, X2, X3, X4, X5, X6, X7, X8, X10, membercB_out_ag(X3, X10)) -> U88_GA(X1, X2, X3, X4, X5, X6, X7, X8, checkcU_in_aagaa(X2, X3, X10, X11, X8)) U88_GA(X1, X2, X3, X4, X5, X6, X7, X8, checkcU_out_aagaa(X2, X3, X10, X11, X8)) -> U89_GA(X1, X2, X3, X4, X5, X6, X7, X8, X11, membercB_in_ag(X4, X11)) U89_GA(X1, X2, X3, X4, X5, X6, X7, X8, X11, membercB_out_ag(X4, X11)) -> U90_GA(X1, X2, X3, X4, X5, X6, X7, X8, checkcV_in_aaagaa(X3, X2, X4, X11, X12, X8)) U90_GA(X1, X2, X3, X4, X5, X6, X7, X8, checkcV_out_aaagaa(X3, X2, X4, X11, X12, X8)) -> U91_GA(X1, X2, X3, X4, X5, X6, X7, X8, X12, membercB_in_ag(X5, X12)) U91_GA(X1, X2, X3, X4, X5, X6, X7, X8, X12, membercB_out_ag(X5, X12)) -> U92_GA(X1, X2, X3, X4, X5, X6, X7, X8, checkcW_in_aaaagaa(X4, X3, X2, X5, X12, X13, X8)) U92_GA(X1, X2, X3, X4, X5, X6, X7, X8, checkcW_out_aaaagaa(X4, X3, X2, X5, X12, X13, X8)) -> U93_GA(X1, X2, X3, X4, X5, X6, X7, memberB_in_ag(X6, X13)) U92_GA(X1, X2, X3, X4, X5, X6, X7, X8, checkcW_out_aaaagaa(X4, X3, X2, X5, X12, X13, X8)) -> MEMBERB_IN_AG(X6, X13) U92_GA(X1, X2, X3, X4, X5, X6, X7, X8, checkcW_out_aaaagaa(X4, X3, X2, X5, X12, X13, X8)) -> U94_GA(X1, X2, X3, X4, X5, X6, X7, X8, X13, membercB_in_ag(X6, X13)) U94_GA(X1, X2, X3, X4, X5, X6, X7, X8, X13, membercB_out_ag(X6, X13)) -> U95_GA(X1, X2, X3, X4, X5, X6, X7, X13, difcJ_in_aaaaaaaa(X5, X4, X3, X2, X6, X8, X14, X15)) U95_GA(X1, X2, X3, X4, X5, X6, X7, X13, difcJ_out_aaaaaaaa(X5, X4, X3, X2, X6, X8, X14, X15)) -> U96_GA(X1, X2, X3, X4, X5, X6, X7, mergedeleteC_in_gga(X14, X13, X16)) U95_GA(X1, X2, X3, X4, X5, X6, X7, X13, difcJ_out_aaaaaaaa(X5, X4, X3, X2, X6, X8, X14, X15)) -> MERGEDELETEC_IN_GGA(X14, X13, X16) U95_GA(X1, X2, X3, X4, X5, X6, X7, X13, difcJ_out_aaaaaaaa(X5, X4, X3, X2, X6, X8, X14, X15)) -> U97_GA(X1, X2, X3, X4, X5, X6, X7, X15, mergedeletecC_in_gga(X14, X13, X16)) U97_GA(X1, X2, X3, X4, X5, X6, X7, X15, mergedeletecC_out_gga(X14, X13, X16)) -> U98_GA(X1, X2, X3, X4, X5, X6, X7, revF_in_ga(X15, X17)) U97_GA(X1, X2, X3, X4, X5, X6, X7, X15, mergedeletecC_out_gga(X14, X13, X16)) -> REVF_IN_GA(X15, X17) U97_GA(X1, X2, X3, X4, X5, X6, X7, X15, mergedeletecC_out_gga(X14, X13, X16)) -> U99_GA(X1, X2, X3, X4, X5, X6, X7, X16, revcF_in_ga(X15, X17)) U99_GA(X1, X2, X3, X4, X5, X6, X7, X16, revcF_out_ga(X15, X17)) -> U100_GA(X1, X2, X3, X4, X5, X6, X7, mergedeleteC_in_gga(X17, X16, X18)) U99_GA(X1, X2, X3, X4, X5, X6, X7, X16, revcF_out_ga(X15, X17)) -> MERGEDELETEC_IN_GGA(X17, X16, X18) PDSN2_IN_GA(X1, .(0, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, X8)))))))) -> U101_GA(X1, X2, X3, X4, X5, X6, X7, X8, iota1cQ_in_aa(X9, X10)) U101_GA(X1, X2, X3, X4, X5, X6, X7, X8, iota1cQ_out_aa(X9, X10)) -> U102_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, membercR_in_ag(X2, X10)) U102_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, membercR_out_ag(X2, X10)) -> U103_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, checkcS_in_agaa(X2, X10, X11, X9)) U103_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, checkcS_out_agaa(X2, X10, X11, X9)) -> U104_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X11, membercB_in_ag(X3, X11)) U104_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X11, membercB_out_ag(X3, X11)) -> U105_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, checkcU_in_aagaa(X2, X3, X11, X12, X9)) U105_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, checkcU_out_aagaa(X2, X3, X11, X12, X9)) -> U106_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X12, membercB_in_ag(X4, X12)) U106_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X12, membercB_out_ag(X4, X12)) -> U107_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, checkcV_in_aaagaa(X3, X2, X4, X12, X13, X9)) U107_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, checkcV_out_aaagaa(X3, X2, X4, X12, X13, X9)) -> U108_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X13, membercB_in_ag(X5, X13)) U108_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X13, membercB_out_ag(X5, X13)) -> U109_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, checkcW_in_aaaagaa(X4, X3, X2, X5, X13, X14, X9)) U109_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, checkcW_out_aaaagaa(X4, X3, X2, X5, X13, X14, X9)) -> U110_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X14, membercB_in_ag(X6, X14)) U110_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X14, membercB_out_ag(X6, X14)) -> U111_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, checkcX_in_aaaaagaa(X5, X4, X3, X2, X6, X14, X15, X9)) U111_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, checkcX_out_aaaaagaa(X5, X4, X3, X2, X6, X14, X15, X9)) -> U112_GA(X1, X2, X3, X4, X5, X6, X7, X8, memberB_in_ag(X7, X15)) U111_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, checkcX_out_aaaaagaa(X5, X4, X3, X2, X6, X14, X15, X9)) -> MEMBERB_IN_AG(X7, X15) U111_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, checkcX_out_aaaaagaa(X5, X4, X3, X2, X6, X14, X15, X9)) -> U113_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X15, membercB_in_ag(X7, X15)) U113_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X15, membercB_out_ag(X7, X15)) -> U114_GA(X1, X2, X3, X4, X5, X6, X7, X8, X15, difcK_in_aaaaaaaaa(X6, X5, X4, X3, X2, X7, X9, X16, X17)) U114_GA(X1, X2, X3, X4, X5, X6, X7, X8, X15, difcK_out_aaaaaaaaa(X6, X5, X4, X3, X2, X7, X9, X16, X17)) -> U115_GA(X1, X2, X3, X4, X5, X6, X7, X8, mergedeleteC_in_gga(X16, X15, X18)) U114_GA(X1, X2, X3, X4, X5, X6, X7, X8, X15, difcK_out_aaaaaaaaa(X6, X5, X4, X3, X2, X7, X9, X16, X17)) -> MERGEDELETEC_IN_GGA(X16, X15, X18) U114_GA(X1, X2, X3, X4, X5, X6, X7, X8, X15, difcK_out_aaaaaaaaa(X6, X5, X4, X3, X2, X7, X9, X16, X17)) -> U116_GA(X1, X2, X3, X4, X5, X6, X7, X8, X17, mergedeletecC_in_gga(X16, X15, X18)) U116_GA(X1, X2, X3, X4, X5, X6, X7, X8, X17, mergedeletecC_out_gga(X16, X15, X18)) -> U117_GA(X1, X2, X3, X4, X5, X6, X7, X8, revF_in_ga(X17, X19)) U116_GA(X1, X2, X3, X4, X5, X6, X7, X8, X17, mergedeletecC_out_gga(X16, X15, X18)) -> REVF_IN_GA(X17, X19) U116_GA(X1, X2, X3, X4, X5, X6, X7, X8, X17, mergedeletecC_out_gga(X16, X15, X18)) -> U118_GA(X1, X2, X3, X4, X5, X6, X7, X8, X18, revcF_in_ga(X17, X19)) U118_GA(X1, X2, X3, X4, X5, X6, X7, X8, X18, revcF_out_ga(X17, X19)) -> U119_GA(X1, X2, X3, X4, X5, X6, X7, X8, mergedeleteC_in_gga(X19, X18, X20)) U118_GA(X1, X2, X3, X4, X5, X6, X7, X8, X18, revcF_out_ga(X17, X19)) -> MERGEDELETEC_IN_GGA(X19, X18, X20) PDSN2_IN_GA(X1, .(0, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9))))))))) -> U120_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, iota1cQ_in_aa(X10, X11)) U120_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, iota1cQ_out_aa(X10, X11)) -> U121_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, membercR_in_ag(X2, X11)) U121_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, membercR_out_ag(X2, X11)) -> U122_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, checkcS_in_agaa(X2, X11, X12, X10)) U122_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, checkcS_out_agaa(X2, X11, X12, X10)) -> U123_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X12, membercB_in_ag(X3, X12)) U123_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X12, membercB_out_ag(X3, X12)) -> U124_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, checkcU_in_aagaa(X2, X3, X12, X13, X10)) U124_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, checkcU_out_aagaa(X2, X3, X12, X13, X10)) -> U125_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X13, membercB_in_ag(X4, X13)) U125_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X13, membercB_out_ag(X4, X13)) -> U126_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, checkcV_in_aaagaa(X3, X2, X4, X13, X14, X10)) U126_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, checkcV_out_aaagaa(X3, X2, X4, X13, X14, X10)) -> U127_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X14, membercB_in_ag(X5, X14)) U127_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X14, membercB_out_ag(X5, X14)) -> U128_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, checkcW_in_aaaagaa(X4, X3, X2, X5, X14, X15, X10)) U128_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, checkcW_out_aaaagaa(X4, X3, X2, X5, X14, X15, X10)) -> U129_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X15, membercB_in_ag(X6, X15)) U129_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X15, membercB_out_ag(X6, X15)) -> U130_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, checkcX_in_aaaaagaa(X5, X4, X3, X2, X6, X15, X16, X10)) U130_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, checkcX_out_aaaaagaa(X5, X4, X3, X2, X6, X15, X16, X10)) -> U131_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X16, membercB_in_ag(X7, X16)) U131_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X16, membercB_out_ag(X7, X16)) -> U132_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, checkcY_in_aaaaaagaa(X6, X5, X4, X3, X2, X7, X16, X17, X10)) U132_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, checkcY_out_aaaaaagaa(X6, X5, X4, X3, X2, X7, X16, X17, X10)) -> U133_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, pO_in_agaaaaaaaaga(X8, X17, X7, X6, X5, X4, X3, X2, X18, X10, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(0, []))))))), X9)) U132_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, checkcY_out_aaaaaagaa(X6, X5, X4, X3, X2, X7, X16, X17, X10)) -> PO_IN_AGAAAAAAAAGA(X8, X17, X7, X6, X5, X4, X3, X2, X18, X10, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(0, []))))))), X9) PO_IN_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12) -> U20_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, memberB_in_ag(X1, X2)) PO_IN_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12) -> MEMBERB_IN_AG(X1, X2) PO_IN_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12) -> U21_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, membercB_in_ag(X1, X2)) U21_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, membercB_out_ag(X1, X2)) -> U22_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, difcZ_in_aaaaaaaaaa(X3, X4, X5, X6, X7, X8, X1, X10, X13, X14)) U22_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, difcZ_out_aaaaaaaaaa(X3, X4, X5, X6, X7, X8, X1, X10, X13, X14)) -> U23_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, mergedeleteC_in_gga(X13, X2, X15)) U22_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, difcZ_out_aaaaaaaaaa(X3, X4, X5, X6, X7, X8, X1, X10, X13, X14)) -> MERGEDELETEC_IN_GGA(X13, X2, X15) U22_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, difcZ_out_aaaaaaaaaa(X3, X4, X5, X6, X7, X8, X1, X10, X13, X14)) -> U24_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X14, mergedeletecC_in_gga(X13, X2, X15)) U24_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X14, mergedeletecC_out_gga(X13, X2, X15)) -> U25_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, revF_in_ga(X14, X16)) U24_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X14, mergedeletecC_out_gga(X13, X2, X15)) -> REVF_IN_GA(X14, X16) U24_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X14, mergedeletecC_out_gga(X13, X2, X15)) -> U26_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X15, revcF_in_ga(X14, X16)) U26_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X15, revcF_out_ga(X14, X16)) -> U27_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, mergedeleteC_in_gga(X16, X15, X9)) U26_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X15, revcF_out_ga(X14, X16)) -> MERGEDELETEC_IN_GGA(X16, X15, X9) U21_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, membercB_out_ag(X1, X2)) -> U28_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, checkcP_in_aaaaaaagaa(X3, X4, X5, X6, X7, X8, X1, X2, X9, X10)) U28_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, checkcP_out_aaaaaaagaa(X3, X4, X5, X6, X7, X8, X1, X2, X9, X10)) -> U29_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, pds1L_in_gagaa(X9, X1, X11, X12, X10)) U28_AGAAAAAAAAGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, checkcP_out_aaaaaaagaa(X3, X4, X5, X6, X7, X8, X1, X2, X9, X10)) -> PDS1L_IN_GAGAA(X9, X1, X11, X12, X10) PDS1L_IN_GAGAA(X1, X2, X3, .(X4, X5), X6) -> U7_GAGAA(X1, X2, X3, X4, X5, X6, memberB_in_ag(X4, X1)) PDS1L_IN_GAGAA(X1, X2, X3, .(X4, X5), X6) -> MEMBERB_IN_AG(X4, X1) PDS1L_IN_GAGAA(X1, X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, .(X9, X10))))))), .(X11, X12), X13) -> U8_GAGAA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, membercB_in_ag(X11, X1)) U8_GAGAA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, membercB_out_ag(X11, X1)) -> U9_GAGAA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, difN_in_gaaaa(X10, X11, X13, X14, X15)) U8_GAGAA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, membercB_out_ag(X11, X1)) -> DIFN_IN_GAAAA(X10, X11, X13, X14, X15) DIFN_IN_GAAAA(.(X1, X2), X3, X4, .(X5, X6), .(X7, X8)) -> U19_GAAAA(X1, X2, X3, X4, X5, X6, X7, X8, difN_in_gaaaa(X2, X3, X4, X6, X8)) DIFN_IN_GAAAA(.(X1, X2), X3, X4, .(X5, X6), .(X7, X8)) -> DIFN_IN_GAAAA(X2, X3, X4, X6, X8) PDS1L_IN_GAGAA(X1, X2, X3, .(X4, X5), X6) -> U10_GAGAA(X1, X2, X3, X4, X5, X6, membercB_in_ag(X4, X1)) U10_GAGAA(X1, X2, X3, X4, X5, X6, membercB_out_ag(X4, X1)) -> U11_GAGAA(X1, X2, X3, X4, X5, X6, difcN1_in_agaaaa(X2, X3, X4, X6, X7, X8)) U11_GAGAA(X1, X2, X3, X4, X5, X6, difcN1_out_agaaaa(X2, X3, X4, X6, X7, X8)) -> U12_GAGAA(X1, X2, X3, X4, X5, X6, mergedeleteC_in_gga(X7, X1, X9)) U11_GAGAA(X1, X2, X3, X4, X5, X6, difcN1_out_agaaaa(X2, X3, X4, X6, X7, X8)) -> MERGEDELETEC_IN_GGA(X7, X1, X9) U11_GAGAA(X1, X2, X3, X4, X5, X6, difcN1_out_agaaaa(X2, X3, X4, X6, X7, X8)) -> U13_GAGAA(X1, X2, X3, X4, X5, X6, X8, mergedeletecC_in_gga(X7, X1, X9)) U13_GAGAA(X1, X2, X3, X4, X5, X6, X8, mergedeletecC_out_gga(X7, X1, X9)) -> U14_GAGAA(X1, X2, X3, X4, X5, X6, revF_in_ga(X8, X10)) U13_GAGAA(X1, X2, X3, X4, X5, X6, X8, mergedeletecC_out_gga(X7, X1, X9)) -> REVF_IN_GA(X8, X10) U13_GAGAA(X1, X2, X3, X4, X5, X6, X8, mergedeletecC_out_gga(X7, X1, X9)) -> U15_GAGAA(X1, X2, X3, X4, X5, X6, X9, revcF_in_ga(X8, X10)) U15_GAGAA(X1, X2, X3, X4, X5, X6, X9, revcF_out_ga(X8, X10)) -> U16_GAGAA(X1, X2, X3, X4, X5, X6, mergedeleteC_in_gga(X10, X9, X11)) U15_GAGAA(X1, X2, X3, X4, X5, X6, X9, revcF_out_ga(X8, X10)) -> MERGEDELETEC_IN_GGA(X10, X9, X11) U10_GAGAA(X1, X2, X3, X4, X5, X6, membercB_out_ag(X4, X1)) -> U17_GAGAA(X1, X2, X3, X4, X5, X6, checkcM_in_agagaa(X2, X3, X4, X1, X7, X6)) U17_GAGAA(X1, X2, X3, X4, X5, X6, checkcM_out_agagaa(X2, X3, X4, X1, X7, X6)) -> U18_GAGAA(X1, X2, X3, X4, X5, X6, pds1L_in_gagaa(X7, X4, .(X2, X3), X5, X6)) U17_GAGAA(X1, X2, X3, X4, X5, X6, checkcM_out_agagaa(X2, X3, X4, X1, X7, X6)) -> PDS1L_IN_GAGAA(X7, X4, .(X2, X3), X5, X6) The TRS R consists of the following rules: iota1cQ_in_aa(X1, X2) -> U153_aa(X1, X2, iota1cA_in_aaa(X3, X1, X2)) iota1cA_in_aaa(X1, X1, []) -> iota1cA_out_aaa(X1, X1, []) iota1cA_in_aaa(X1, X2, .(X1, X3)) -> U135_aaa(X1, X2, X3, iota1cA_in_aaa(X4, X2, X3)) U135_aaa(X1, X2, X3, iota1cA_out_aaa(X4, X2, X3)) -> iota1cA_out_aaa(X1, X2, .(X1, X3)) U153_aa(X1, X2, iota1cA_out_aaa(X3, X1, X2)) -> iota1cQ_out_aa(X1, X2) membercR_in_ag(X1, .(X1, X2)) -> membercR_out_ag(X1, .(X1, X2)) membercR_in_ag(X1, .(X2, X3)) -> U154_ag(X1, X2, X3, membercB_in_ag(X1, X3)) membercB_in_ag(X1, .(X1, X2)) -> membercB_out_ag(X1, .(X1, X2)) membercB_in_ag(X1, .(X2, X3)) -> U136_ag(X1, X2, X3, membercB_in_ag(X1, X3)) U136_ag(X1, X2, X3, membercB_out_ag(X1, X3)) -> membercB_out_ag(X1, .(X2, X3)) U154_ag(X1, X2, X3, membercB_out_ag(X1, X3)) -> membercR_out_ag(X1, .(X2, X3)) difcE_in_aaaa(X1, X2, .(X3, []), .(X4, [])) -> difcE_out_aaaa(X1, X2, .(X3, []), .(X4, [])) mergedeletecT_in_gga([], X1, X1) -> mergedeletecT_out_gga([], X1, X1) mergedeletecT_in_gga(.(X1, X2), .(X1, X3), X4) -> U159_gga(X1, X2, X3, X4, mergedeletecC_in_gga(X2, X3, X4)) mergedeletecC_in_gga([], X1, X1) -> mergedeletecC_out_gga([], X1, X1) mergedeletecC_in_gga(.(X1, X2), .(X1, X3), X4) -> U137_gga(X1, X2, X3, X4, mergedeletecC_in_gga(X2, X3, X4)) mergedeletecC_in_gga(.(X1, X2), .(X3, X4), .(X3, X5)) -> U138_gga(X1, X2, X3, X4, X5, mergedeletecC_in_gga(.(X1, X2), X4, X5)) U138_gga(X1, X2, X3, X4, X5, mergedeletecC_out_gga(.(X1, X2), X4, X5)) -> mergedeletecC_out_gga(.(X1, X2), .(X3, X4), .(X3, X5)) U137_gga(X1, X2, X3, X4, mergedeletecC_out_gga(X2, X3, X4)) -> mergedeletecC_out_gga(.(X1, X2), .(X1, X3), X4) U159_gga(X1, X2, X3, X4, mergedeletecC_out_gga(X2, X3, X4)) -> mergedeletecT_out_gga(.(X1, X2), .(X1, X3), X4) mergedeletecT_in_gga(.(X1, X2), .(X3, X4), .(X3, X5)) -> U160_gga(X1, X2, X3, X4, X5, mergedeletecC_in_gga(.(X1, X2), X4, X5)) U160_gga(X1, X2, X3, X4, X5, mergedeletecC_out_gga(.(X1, X2), X4, X5)) -> mergedeletecT_out_gga(.(X1, X2), .(X3, X4), .(X3, X5)) revcF_in_ga([], []) -> revcF_out_ga([], []) revcF_in_ga(.(X1, []), .(X1, [])) -> revcF_out_ga(.(X1, []), .(X1, [])) revcF_in_ga(.(X1, .(X2, [])), .(X2, .(X1, []))) -> revcF_out_ga(.(X1, .(X2, [])), .(X2, .(X1, []))) revcF_in_ga(.(X1, .(X2, .(X3, []))), .(X3, .(X2, .(X1, [])))) -> revcF_out_ga(.(X1, .(X2, .(X3, []))), .(X3, .(X2, .(X1, [])))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, [])))), .(X4, .(X3, .(X2, .(X1, []))))) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, [])))), .(X4, .(X3, .(X2, .(X1, []))))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, .(X5, []))))), .(X5, .(X4, .(X3, .(X2, .(X1, [])))))) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, .(X5, []))))), .(X5, .(X4, .(X3, .(X2, .(X1, [])))))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, [])))))), .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, [])))))), .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, []))))))), .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, [])))))))) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, []))))))), .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, [])))))))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) -> U140_ga(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revcD_in_gaga(X9, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))), X10)) revcD_in_gaga([], X1, X2, .(X1, X2)) -> revcD_out_gaga([], X1, X2, .(X1, X2)) revcD_in_gaga(.(X1, X2), X3, X4, X5) -> U139_gaga(X1, X2, X3, X4, X5, revcD_in_gaga(X2, X1, .(X3, X4), X5)) U139_gaga(X1, X2, X3, X4, X5, revcD_out_gaga(X2, X1, .(X3, X4), X5)) -> revcD_out_gaga(.(X1, X2), X3, X4, X5) U140_ga(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revcD_out_gaga(X9, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))), X10)) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) checkcS_in_agaa(X1, X2, X3, X4) -> U155_agaa(X1, X2, X3, X4, difcE_in_aaaa(X1, X4, X5, X6)) U155_agaa(X1, X2, X3, X4, difcE_out_aaaa(X1, X4, X5, X6)) -> U156_agaa(X1, X2, X3, X4, X5, X6, mergedeletecT_in_gga(X5, X2, X7)) U156_agaa(X1, X2, X3, X4, X5, X6, mergedeletecT_out_gga(X5, X2, X7)) -> U157_agaa(X1, X2, X3, X4, X5, X6, X7, revcF_in_ga(X6, X8)) U157_agaa(X1, X2, X3, X4, X5, X6, X7, revcF_out_ga(X6, X8)) -> U158_agaa(X1, X2, X3, X4, mergedeletecC_in_gga(X8, X7, X3)) U158_agaa(X1, X2, X3, X4, mergedeletecC_out_gga(X8, X7, X3)) -> checkcS_out_agaa(X1, X2, X3, X4) difcG_in_aaaaa(X1, X2, X3, .(X4, X5), .(X6, X7)) -> U141_aaaaa(X1, X2, X3, X4, X5, X6, X7, difcE_in_aaaa(X2, X3, X5, X7)) U141_aaaaa(X1, X2, X3, X4, X5, X6, X7, difcE_out_aaaa(X2, X3, X5, X7)) -> difcG_out_aaaaa(X1, X2, X3, .(X4, X5), .(X6, X7)) checkcU_in_aagaa(X1, X2, X3, X4, X5) -> U161_aagaa(X1, X2, X3, X4, X5, difcG_in_aaaaa(X1, X2, X5, X6, X7)) U161_aagaa(X1, X2, X3, X4, X5, difcG_out_aaaaa(X1, X2, X5, X6, X7)) -> U162_aagaa(X1, X2, X3, X4, X5, X6, X7, mergedeletecC_in_gga(X6, X3, X8)) U162_aagaa(X1, X2, X3, X4, X5, X6, X7, mergedeletecC_out_gga(X6, X3, X8)) -> U163_aagaa(X1, X2, X3, X4, X5, X6, X7, X8, revcF_in_ga(X7, X9)) U163_aagaa(X1, X2, X3, X4, X5, X6, X7, X8, revcF_out_ga(X7, X9)) -> U164_aagaa(X1, X2, X3, X4, X5, mergedeletecC_in_gga(X9, X8, X4)) U164_aagaa(X1, X2, X3, X4, X5, mergedeletecC_out_gga(X9, X8, X4)) -> checkcU_out_aagaa(X1, X2, X3, X4, X5) difcH_in_aaaaaa(X1, X2, X3, X4, .(X5, X6), .(X7, X8)) -> U142_aaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, difcG_in_aaaaa(X2, X3, X4, X6, X8)) U142_aaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, difcG_out_aaaaa(X2, X3, X4, X6, X8)) -> difcH_out_aaaaaa(X1, X2, X3, X4, .(X5, X6), .(X7, X8)) checkcV_in_aaagaa(X1, X2, X3, X4, X5, X6) -> U165_aaagaa(X1, X2, X3, X4, X5, X6, difcH_in_aaaaaa(X1, X2, X3, X6, X7, X8)) U165_aaagaa(X1, X2, X3, X4, X5, X6, difcH_out_aaaaaa(X1, X2, X3, X6, X7, X8)) -> U166_aaagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_in_gga(X7, X4, X9)) U166_aaagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_out_gga(X7, X4, X9)) -> U167_aaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, revcF_in_ga(X8, X10)) U167_aaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, revcF_out_ga(X8, X10)) -> U168_aaagaa(X1, X2, X3, X4, X5, X6, mergedeletecC_in_gga(X10, X9, X5)) U168_aaagaa(X1, X2, X3, X4, X5, X6, mergedeletecC_out_gga(X10, X9, X5)) -> checkcV_out_aaagaa(X1, X2, X3, X4, X5, X6) difcI_in_aaaaaaa(X1, X2, X3, X4, X5, .(X6, X7), .(X8, X9)) -> U143_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, difcH_in_aaaaaa(X2, X3, X4, X5, X7, X9)) U143_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, difcH_out_aaaaaa(X2, X3, X4, X5, X7, X9)) -> difcI_out_aaaaaaa(X1, X2, X3, X4, X5, .(X6, X7), .(X8, X9)) checkcW_in_aaaagaa(X1, X2, X3, X4, X5, X6, X7) -> U169_aaaagaa(X1, X2, X3, X4, X5, X6, X7, difcI_in_aaaaaaa(X1, X2, X3, X4, X7, X8, X9)) U169_aaaagaa(X1, X2, X3, X4, X5, X6, X7, difcI_out_aaaaaaa(X1, X2, X3, X4, X7, X8, X9)) -> U170_aaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, mergedeletecC_in_gga(X8, X5, X10)) U170_aaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, mergedeletecC_out_gga(X8, X5, X10)) -> U171_aaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revcF_in_ga(X9, X11)) U171_aaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revcF_out_ga(X9, X11)) -> U172_aaaagaa(X1, X2, X3, X4, X5, X6, X7, mergedeletecC_in_gga(X11, X10, X6)) U172_aaaagaa(X1, X2, X3, X4, X5, X6, X7, mergedeletecC_out_gga(X11, X10, X6)) -> checkcW_out_aaaagaa(X1, X2, X3, X4, X5, X6, X7) difcJ_in_aaaaaaaa(X1, X2, X3, X4, X5, X6, .(X7, X8), .(X9, X10)) -> U144_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, difcI_in_aaaaaaa(X2, X3, X4, X5, X6, X8, X10)) U144_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, difcI_out_aaaaaaa(X2, X3, X4, X5, X6, X8, X10)) -> difcJ_out_aaaaaaaa(X1, X2, X3, X4, X5, X6, .(X7, X8), .(X9, X10)) checkcX_in_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8) -> U173_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, difcJ_in_aaaaaaaa(X1, X2, X3, X4, X5, X8, X9, X10)) U173_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, difcJ_out_aaaaaaaa(X1, X2, X3, X4, X5, X8, X9, X10)) -> U174_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, mergedeletecC_in_gga(X9, X6, X11)) U174_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, mergedeletecC_out_gga(X9, X6, X11)) -> U175_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, revcF_in_ga(X10, X12)) U175_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, revcF_out_ga(X10, X12)) -> U176_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_in_gga(X12, X11, X7)) U176_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_out_gga(X12, X11, X7)) -> checkcX_out_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8) difcK_in_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, .(X8, X9), .(X10, X11)) -> U145_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, difcJ_in_aaaaaaaa(X2, X3, X4, X5, X6, X7, X9, X11)) U145_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, difcJ_out_aaaaaaaa(X2, X3, X4, X5, X6, X7, X9, X11)) -> difcK_out_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, .(X8, X9), .(X10, X11)) checkcY_in_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9) -> U177_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, difcK_in_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X9, X10, X11)) U177_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, difcK_out_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X9, X10, X11)) -> U178_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, mergedeletecC_in_gga(X10, X7, X12)) U178_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, mergedeletecC_out_gga(X10, X7, X12)) -> U179_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, revcF_in_ga(X11, X13)) U179_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, revcF_out_ga(X11, X13)) -> U180_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, mergedeletecC_in_gga(X13, X12, X8)) U180_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, mergedeletecC_out_gga(X13, X12, X8)) -> checkcY_out_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9) difcZ_in_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, .(X9, X10), .(X11, X12)) -> U185_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, difcK_in_aaaaaaaaa(X2, X3, X4, X5, X6, X7, X8, X10, X12)) U185_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, difcK_out_aaaaaaaaa(X2, X3, X4, X5, X6, X7, X8, X10, X12)) -> difcZ_out_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, .(X9, X10), .(X11, X12)) checkcP_in_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10) -> U181_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, difcZ_in_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X10, X11, X12)) U181_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, difcZ_out_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X10, X11, X12)) -> U182_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, mergedeletecC_in_gga(X11, X8, X13)) U182_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, mergedeletecC_out_gga(X11, X8, X13)) -> U183_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, revcF_in_ga(X12, X14)) U183_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, revcF_out_ga(X12, X14)) -> U184_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, mergedeletecC_in_gga(X14, X13, X9)) U184_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, mergedeletecC_out_gga(X14, X13, X9)) -> checkcP_out_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10) difcN1_in_agaaaa(X1, [], X2, X3, .(X4, []), .(X5, [])) -> difcN1_out_agaaaa(X1, [], X2, X3, .(X4, []), .(X5, [])) difcN1_in_agaaaa(X1, .(X2, []), X3, X4, .(X5, .(X6, [])), .(X7, .(X8, []))) -> difcN1_out_agaaaa(X1, .(X2, []), X3, X4, .(X5, .(X6, [])), .(X7, .(X8, []))) difcN1_in_agaaaa(X1, .(X2, .(X3, [])), X4, X5, .(X6, .(X7, .(X8, []))), .(X9, .(X10, .(X11, [])))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, [])), X4, X5, .(X6, .(X7, .(X8, []))), .(X9, .(X10, .(X11, [])))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, []))), X5, X6, .(X7, .(X8, .(X9, .(X10, [])))), .(X11, .(X12, .(X13, .(X14, []))))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, []))), X5, X6, .(X7, .(X8, .(X9, .(X10, [])))), .(X11, .(X12, .(X13, .(X14, []))))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, [])))), X6, X7, .(X8, .(X9, .(X10, .(X11, .(X12, []))))), .(X13, .(X14, .(X15, .(X16, .(X17, [])))))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, [])))), X6, X7, .(X8, .(X9, .(X10, .(X11, .(X12, []))))), .(X13, .(X14, .(X15, .(X16, .(X17, [])))))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, []))))), X7, X8, .(X9, .(X10, .(X11, .(X12, .(X13, .(X14, [])))))), .(X15, .(X16, .(X17, .(X18, .(X19, .(X20, []))))))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, []))))), X7, X8, .(X9, .(X10, .(X11, .(X12, .(X13, .(X14, [])))))), .(X15, .(X16, .(X17, .(X18, .(X19, .(X20, []))))))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, [])))))), X8, X9, .(X10, .(X11, .(X12, .(X13, .(X14, .(X15, .(X16, []))))))), .(X17, .(X18, .(X19, .(X20, .(X21, .(X22, .(X23, [])))))))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, [])))))), X8, X9, .(X10, .(X11, .(X12, .(X13, .(X14, .(X15, .(X16, []))))))), .(X17, .(X18, .(X19, .(X20, .(X21, .(X22, .(X23, [])))))))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9))))))), X10, X11, .(X12, .(X13, .(X14, .(X15, .(X16, .(X17, .(X18, .(X19, X20)))))))), .(X21, .(X22, .(X23, .(X24, .(X25, .(X26, .(X27, .(X28, X29))))))))) -> U190_agaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, X21, X22, X23, X24, X25, X26, X27, X28, X29, difcN_in_gaaaa(X9, X10, X11, X20, X29)) difcN_in_gaaaa([], X1, X2, [], []) -> difcN_out_gaaaa([], X1, X2, [], []) difcN_in_gaaaa(.(X1, X2), X3, X4, .(X5, X6), .(X7, X8)) -> U149_gaaaa(X1, X2, X3, X4, X5, X6, X7, X8, difcN_in_gaaaa(X2, X3, X4, X6, X8)) U149_gaaaa(X1, X2, X3, X4, X5, X6, X7, X8, difcN_out_gaaaa(X2, X3, X4, X6, X8)) -> difcN_out_gaaaa(.(X1, X2), X3, X4, .(X5, X6), .(X7, X8)) U190_agaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, X21, X22, X23, X24, X25, X26, X27, X28, X29, difcN_out_gaaaa(X9, X10, X11, X20, X29)) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9))))))), X10, X11, .(X12, .(X13, .(X14, .(X15, .(X16, .(X17, .(X18, .(X19, X20)))))))), .(X21, .(X22, .(X23, .(X24, .(X25, .(X26, .(X27, .(X28, X29))))))))) checkcM_in_agagaa(X1, X2, X3, X4, X5, X6) -> U186_agagaa(X1, X2, X3, X4, X5, X6, difcN1_in_agaaaa(X1, X2, X3, X6, X7, X8)) U186_agagaa(X1, X2, X3, X4, X5, X6, difcN1_out_agaaaa(X1, X2, X3, X6, X7, X8)) -> U187_agagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_in_gga(X7, X4, X9)) U187_agagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_out_gga(X7, X4, X9)) -> U188_agagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, revcF_in_ga(X8, X10)) U188_agagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, revcF_out_ga(X8, X10)) -> U189_agagaa(X1, X2, X3, X4, X5, X6, mergedeletecC_in_gga(X10, X9, X5)) U189_agagaa(X1, X2, X3, X4, X5, X6, mergedeletecC_out_gga(X10, X9, X5)) -> checkcM_out_agagaa(X1, X2, X3, X4, X5, X6) The argument filtering Pi contains the following mapping: iota1A_in_aaa(x1, x2, x3) = iota1A_in_aaa .(x1, x2) = .(x2) iota1cQ_in_aa(x1, x2) = iota1cQ_in_aa U153_aa(x1, x2, x3) = U153_aa(x3) iota1cA_in_aaa(x1, x2, x3) = iota1cA_in_aaa iota1cA_out_aaa(x1, x2, x3) = iota1cA_out_aaa(x3) U135_aaa(x1, x2, x3, x4) = U135_aaa(x4) iota1cQ_out_aa(x1, x2) = iota1cQ_out_aa(x2) memberB_in_ag(x1, x2) = memberB_in_ag(x2) membercR_in_ag(x1, x2) = membercR_in_ag(x2) membercR_out_ag(x1, x2) = membercR_out_ag(x2) U154_ag(x1, x2, x3, x4) = U154_ag(x3, x4) membercB_in_ag(x1, x2) = membercB_in_ag(x2) membercB_out_ag(x1, x2) = membercB_out_ag(x2) U136_ag(x1, x2, x3, x4) = U136_ag(x3, x4) difcE_in_aaaa(x1, x2, x3, x4) = difcE_in_aaaa difcE_out_aaaa(x1, x2, x3, x4) = difcE_out_aaaa(x3, x4) mergedeleteC_in_gga(x1, x2, x3) = mergedeleteC_in_gga(x1, x2) mergedeletecT_in_gga(x1, x2, x3) = mergedeletecT_in_gga(x1, x2) [] = [] mergedeletecT_out_gga(x1, x2, x3) = mergedeletecT_out_gga(x1, x2, x3) U159_gga(x1, x2, x3, x4, x5) = U159_gga(x2, x3, x5) mergedeletecC_in_gga(x1, x2, x3) = mergedeletecC_in_gga(x1, x2) mergedeletecC_out_gga(x1, x2, x3) = mergedeletecC_out_gga(x1, x2, x3) U137_gga(x1, x2, x3, x4, x5) = U137_gga(x2, x3, x5) U138_gga(x1, x2, x3, x4, x5, x6) = U138_gga(x2, x4, x6) U160_gga(x1, x2, x3, x4, x5, x6) = U160_gga(x2, x4, x6) revF_in_ga(x1, x2) = revF_in_ga(x1) revD_in_gaga(x1, x2, x3, x4) = revD_in_gaga(x1, x3) revcF_in_ga(x1, x2) = revcF_in_ga(x1) revcF_out_ga(x1, x2) = revcF_out_ga(x1, x2) U140_ga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U140_ga(x9, x11) revcD_in_gaga(x1, x2, x3, x4) = revcD_in_gaga(x1, x3) revcD_out_gaga(x1, x2, x3, x4) = revcD_out_gaga(x1, x3, x4) U139_gaga(x1, x2, x3, x4, x5, x6) = U139_gaga(x2, x4, x6) checkcS_in_agaa(x1, x2, x3, x4) = checkcS_in_agaa(x2) U155_agaa(x1, x2, x3, x4, x5) = U155_agaa(x2, x5) U156_agaa(x1, x2, x3, x4, x5, x6, x7) = U156_agaa(x2, x6, x7) U157_agaa(x1, x2, x3, x4, x5, x6, x7, x8) = U157_agaa(x2, x7, x8) U158_agaa(x1, x2, x3, x4, x5) = U158_agaa(x2, x5) checkcS_out_agaa(x1, x2, x3, x4) = checkcS_out_agaa(x2, x3) difcG_in_aaaaa(x1, x2, x3, x4, x5) = difcG_in_aaaaa U141_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U141_aaaaa(x8) difcG_out_aaaaa(x1, x2, x3, x4, x5) = difcG_out_aaaaa(x4, x5) checkcU_in_aagaa(x1, x2, x3, x4, x5) = checkcU_in_aagaa(x3) U161_aagaa(x1, x2, x3, x4, x5, x6) = U161_aagaa(x3, x6) U162_aagaa(x1, x2, x3, x4, x5, x6, x7, x8) = U162_aagaa(x3, x7, x8) U163_aagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U163_aagaa(x3, x8, x9) U164_aagaa(x1, x2, x3, x4, x5, x6) = U164_aagaa(x3, x6) checkcU_out_aagaa(x1, x2, x3, x4, x5) = checkcU_out_aagaa(x3, x4) difcH_in_aaaaaa(x1, x2, x3, x4, x5, x6) = difcH_in_aaaaaa U142_aaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U142_aaaaaa(x9) difcH_out_aaaaaa(x1, x2, x3, x4, x5, x6) = difcH_out_aaaaaa(x5, x6) checkcV_in_aaagaa(x1, x2, x3, x4, x5, x6) = checkcV_in_aaagaa(x4) U165_aaagaa(x1, x2, x3, x4, x5, x6, x7) = U165_aaagaa(x4, x7) U166_aaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U166_aaagaa(x4, x8, x9) U167_aaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U167_aaagaa(x4, x9, x10) U168_aaagaa(x1, x2, x3, x4, x5, x6, x7) = U168_aaagaa(x4, x7) checkcV_out_aaagaa(x1, x2, x3, x4, x5, x6) = checkcV_out_aaagaa(x4, x5) difcI_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = difcI_in_aaaaaaa U143_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U143_aaaaaaa(x10) difcI_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = difcI_out_aaaaaaa(x6, x7) checkcW_in_aaaagaa(x1, x2, x3, x4, x5, x6, x7) = checkcW_in_aaaagaa(x5) U169_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8) = U169_aaaagaa(x5, x8) U170_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U170_aaaagaa(x5, x9, x10) U171_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U171_aaaagaa(x5, x10, x11) U172_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8) = U172_aaaagaa(x5, x8) checkcW_out_aaaagaa(x1, x2, x3, x4, x5, x6, x7) = checkcW_out_aaaagaa(x5, x6) difcJ_in_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = difcJ_in_aaaaaaaa U144_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U144_aaaaaaaa(x11) difcJ_out_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = difcJ_out_aaaaaaaa(x7, x8) checkcX_in_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8) = checkcX_in_aaaaagaa(x6) U173_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U173_aaaaagaa(x6, x9) U174_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U174_aaaaagaa(x6, x10, x11) U175_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U175_aaaaagaa(x6, x11, x12) U176_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U176_aaaaagaa(x6, x9) checkcX_out_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8) = checkcX_out_aaaaagaa(x6, x7) difcK_in_aaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = difcK_in_aaaaaaaaa U145_aaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U145_aaaaaaaaa(x12) difcK_out_aaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = difcK_out_aaaaaaaaa(x8, x9) checkcY_in_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = checkcY_in_aaaaaagaa(x7) U177_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U177_aaaaaagaa(x7, x10) U178_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U178_aaaaaagaa(x7, x11, x12) U179_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U179_aaaaaagaa(x7, x12, x13) U180_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U180_aaaaaagaa(x7, x10) checkcY_out_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = checkcY_out_aaaaaagaa(x7, x8) pO_in_agaaaaaaaaga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = pO_in_agaaaaaaaaga(x2, x11) difcZ_in_aaaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = difcZ_in_aaaaaaaaaa U185_aaaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U185_aaaaaaaaaa(x13) difcZ_out_aaaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = difcZ_out_aaaaaaaaaa(x9, x10) checkcP_in_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = checkcP_in_aaaaaaagaa(x8) U181_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U181_aaaaaaagaa(x8, x11) U182_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U182_aaaaaaagaa(x8, x12, x13) U183_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14) = U183_aaaaaaagaa(x8, x13, x14) U184_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U184_aaaaaaagaa(x8, x11) checkcP_out_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = checkcP_out_aaaaaaagaa(x8, x9) pds1L_in_gagaa(x1, x2, x3, x4, x5) = pds1L_in_gagaa(x1, x3) difN_in_gaaaa(x1, x2, x3, x4, x5) = difN_in_gaaaa(x1) difcN1_in_agaaaa(x1, x2, x3, x4, x5, x6) = difcN1_in_agaaaa(x2) difcN1_out_agaaaa(x1, x2, x3, x4, x5, x6) = difcN1_out_agaaaa(x2, x5, x6) U190_agaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30) = U190_agaaaa(x9, x30) difcN_in_gaaaa(x1, x2, x3, x4, x5) = difcN_in_gaaaa(x1) difcN_out_gaaaa(x1, x2, x3, x4, x5) = difcN_out_gaaaa(x1, x4, x5) U149_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U149_gaaaa(x2, x9) checkcM_in_agagaa(x1, x2, x3, x4, x5, x6) = checkcM_in_agagaa(x2, x4) U186_agagaa(x1, x2, x3, x4, x5, x6, x7) = U186_agagaa(x2, x4, x7) U187_agagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U187_agagaa(x2, x4, x8, x9) U188_agagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U188_agagaa(x2, x4, x9, x10) U189_agagaa(x1, x2, x3, x4, x5, x6, x7) = U189_agagaa(x2, x4, x7) checkcM_out_agagaa(x1, x2, x3, x4, x5, x6) = checkcM_out_agagaa(x2, x4, x5) PDSN2_IN_GA(x1, x2) = PDSN2_IN_GA(x1) U30_GA(x1, x2, x3) = U30_GA(x1, x3) IOTA1A_IN_AAA(x1, x2, x3) = IOTA1A_IN_AAA U1_AAA(x1, x2, x3, x4) = U1_AAA(x4) U31_GA(x1, x2, x3, x4) = U31_GA(x1, x4) U32_GA(x1, x2, x3, x4) = U32_GA(x1, x4) MEMBERB_IN_AG(x1, x2) = MEMBERB_IN_AG(x2) U2_AG(x1, x2, x3, x4) = U2_AG(x3, x4) U33_GA(x1, x2, x3, x4, x5, x6, x7) = U33_GA(x1, x6, x7) U34_GA(x1, x2, x3, x4, x5) = U34_GA(x1, x4, x5) U35_GA(x1, x2, x3, x4) = U35_GA(x1, x4) MERGEDELETEC_IN_GGA(x1, x2, x3) = MERGEDELETEC_IN_GGA(x1, x2) U3_GGA(x1, x2, x3, x4, x5) = U3_GGA(x2, x3, x5) U4_GGA(x1, x2, x3, x4, x5, x6) = U4_GGA(x2, x4, x6) U36_GA(x1, x2, x3, x4, x5) = U36_GA(x1, x4, x5) U37_GA(x1, x2, x3, x4) = U37_GA(x1, x4) U38_GA(x1, x2, x3, x4) = U38_GA(x1, x4) U39_GA(x1, x2, x3, x4, x5, x6) = U39_GA(x1, x5, x6) U40_GA(x1, x2, x3, x4, x5) = U40_GA(x1, x4, x5) U41_GA(x1, x2, x3, x4, x5) = U41_GA(x1, x4, x5) U42_GA(x1, x2, x3, x4) = U42_GA(x1, x4) REVF_IN_GA(x1, x2) = REVF_IN_GA(x1) U6_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U6_GA(x9, x11) REVD_IN_GAGA(x1, x2, x3, x4) = REVD_IN_GAGA(x1, x3) U5_GAGA(x1, x2, x3, x4, x5, x6) = U5_GAGA(x2, x4, x6) U43_GA(x1, x2, x3, x4, x5) = U43_GA(x1, x4, x5) U44_GA(x1, x2, x3, x4) = U44_GA(x1, x4) U45_GA(x1, x2, x3, x4, x5) = U45_GA(x1, x5) U46_GA(x1, x2, x3, x4, x5, x6, x7) = U46_GA(x1, x6, x7) U47_GA(x1, x2, x3, x4, x5, x6) = U47_GA(x1, x6) U48_GA(x1, x2, x3, x4, x5) = U48_GA(x1, x5) U49_GA(x1, x2, x3, x4, x5, x6, x7) = U49_GA(x1, x6, x7) U50_GA(x1, x2, x3, x4, x5, x6) = U50_GA(x1, x5, x6) U51_GA(x1, x2, x3, x4, x5) = U51_GA(x1, x5) U52_GA(x1, x2, x3, x4, x5, x6) = U52_GA(x1, x5, x6) U53_GA(x1, x2, x3, x4, x5) = U53_GA(x1, x5) U54_GA(x1, x2, x3, x4, x5, x6) = U54_GA(x1, x5, x6) U55_GA(x1, x2, x3, x4, x5) = U55_GA(x1, x5) U56_GA(x1, x2, x3, x4, x5, x6) = U56_GA(x1, x6) U57_GA(x1, x2, x3, x4, x5, x6, x7, x8) = U57_GA(x1, x7, x8) U58_GA(x1, x2, x3, x4, x5, x6, x7) = U58_GA(x1, x7) U59_GA(x1, x2, x3, x4, x5, x6, x7, x8) = U59_GA(x1, x7, x8) U60_GA(x1, x2, x3, x4, x5, x6, x7) = U60_GA(x1, x7) U61_GA(x1, x2, x3, x4, x5, x6) = U61_GA(x1, x6) U62_GA(x1, x2, x3, x4, x5, x6, x7, x8) = U62_GA(x1, x7, x8) U63_GA(x1, x2, x3, x4, x5, x6, x7) = U63_GA(x1, x6, x7) U64_GA(x1, x2, x3, x4, x5, x6) = U64_GA(x1, x6) U65_GA(x1, x2, x3, x4, x5, x6, x7) = U65_GA(x1, x6, x7) U66_GA(x1, x2, x3, x4, x5, x6) = U66_GA(x1, x6) U67_GA(x1, x2, x3, x4, x5, x6, x7) = U67_GA(x1, x6, x7) U68_GA(x1, x2, x3, x4, x5, x6) = U68_GA(x1, x6) U69_GA(x1, x2, x3, x4, x5, x6, x7) = U69_GA(x1, x7) U70_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U70_GA(x1, x8, x9) U71_GA(x1, x2, x3, x4, x5, x6, x7, x8) = U71_GA(x1, x8) U72_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U72_GA(x1, x8, x9) U73_GA(x1, x2, x3, x4, x5, x6, x7, x8) = U73_GA(x1, x8) U74_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U74_GA(x1, x8, x9) U75_GA(x1, x2, x3, x4, x5, x6, x7, x8) = U75_GA(x1, x8) U76_GA(x1, x2, x3, x4, x5, x6, x7) = U76_GA(x1, x7) U77_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U77_GA(x1, x8, x9) U78_GA(x1, x2, x3, x4, x5, x6, x7, x8) = U78_GA(x1, x7, x8) U79_GA(x1, x2, x3, x4, x5, x6, x7) = U79_GA(x1, x7) U80_GA(x1, x2, x3, x4, x5, x6, x7, x8) = U80_GA(x1, x7, x8) U81_GA(x1, x2, x3, x4, x5, x6, x7) = U81_GA(x1, x7) U82_GA(x1, x2, x3, x4, x5, x6, x7, x8) = U82_GA(x1, x7, x8) U83_GA(x1, x2, x3, x4, x5, x6, x7) = U83_GA(x1, x7) U84_GA(x1, x2, x3, x4, x5, x6, x7, x8) = U84_GA(x1, x8) U85_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U85_GA(x1, x9, x10) U86_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U86_GA(x1, x9) U87_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U87_GA(x1, x9, x10) U88_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U88_GA(x1, x9) U89_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U89_GA(x1, x9, x10) U90_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U90_GA(x1, x9) U91_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U91_GA(x1, x9, x10) U92_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U92_GA(x1, x9) U93_GA(x1, x2, x3, x4, x5, x6, x7, x8) = U93_GA(x1, x8) U94_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U94_GA(x1, x9, x10) U95_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U95_GA(x1, x8, x9) U96_GA(x1, x2, x3, x4, x5, x6, x7, x8) = U96_GA(x1, x8) U97_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U97_GA(x1, x8, x9) U98_GA(x1, x2, x3, x4, x5, x6, x7, x8) = U98_GA(x1, x8) U99_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U99_GA(x1, x8, x9) U100_GA(x1, x2, x3, x4, x5, x6, x7, x8) = U100_GA(x1, x8) U101_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U101_GA(x1, x9) U102_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U102_GA(x1, x10, x11) U103_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U103_GA(x1, x10) U104_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U104_GA(x1, x10, x11) U105_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U105_GA(x1, x10) U106_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U106_GA(x1, x10, x11) U107_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U107_GA(x1, x10) U108_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U108_GA(x1, x10, x11) U109_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U109_GA(x1, x10) U110_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U110_GA(x1, x10, x11) U111_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U111_GA(x1, x10) U112_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U112_GA(x1, x9) U113_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U113_GA(x1, x10, x11) U114_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U114_GA(x1, x9, x10) U115_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U115_GA(x1, x9) U116_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U116_GA(x1, x9, x10) U117_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U117_GA(x1, x9) U118_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U118_GA(x1, x9, x10) U119_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U119_GA(x1, x9) U120_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U120_GA(x1, x10) U121_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U121_GA(x1, x11, x12) U122_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U122_GA(x1, x11) U123_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U123_GA(x1, x11, x12) U124_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U124_GA(x1, x11) U125_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U125_GA(x1, x11, x12) U126_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U126_GA(x1, x11) U127_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U127_GA(x1, x11, x12) U128_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U128_GA(x1, x11) U129_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U129_GA(x1, x11, x12) U130_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U130_GA(x1, x11) U131_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U131_GA(x1, x11, x12) U132_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U132_GA(x1, x11) U133_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U133_GA(x1, x10) PO_IN_AGAAAAAAAAGA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = PO_IN_AGAAAAAAAAGA(x2, x11) U20_AGAAAAAAAAGA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U20_AGAAAAAAAAGA(x2, x11, x13) U21_AGAAAAAAAAGA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U21_AGAAAAAAAAGA(x2, x11, x13) U22_AGAAAAAAAAGA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U22_AGAAAAAAAAGA(x2, x11, x13) U23_AGAAAAAAAAGA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U23_AGAAAAAAAAGA(x2, x11, x13) U24_AGAAAAAAAAGA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14) = U24_AGAAAAAAAAGA(x2, x11, x13, x14) U25_AGAAAAAAAAGA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U25_AGAAAAAAAAGA(x2, x11, x13) U26_AGAAAAAAAAGA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14) = U26_AGAAAAAAAAGA(x2, x11, x13, x14) U27_AGAAAAAAAAGA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U27_AGAAAAAAAAGA(x2, x11, x13) U28_AGAAAAAAAAGA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U28_AGAAAAAAAAGA(x2, x11, x13) U29_AGAAAAAAAAGA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U29_AGAAAAAAAAGA(x2, x11, x13) PDS1L_IN_GAGAA(x1, x2, x3, x4, x5) = PDS1L_IN_GAGAA(x1, x3) U7_GAGAA(x1, x2, x3, x4, x5, x6, x7) = U7_GAGAA(x1, x3, x7) U8_GAGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14) = U8_GAGAA(x1, x10, x14) U9_GAGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14) = U9_GAGAA(x1, x10, x14) DIFN_IN_GAAAA(x1, x2, x3, x4, x5) = DIFN_IN_GAAAA(x1) U19_GAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U19_GAAAA(x2, x9) U10_GAGAA(x1, x2, x3, x4, x5, x6, x7) = U10_GAGAA(x1, x3, x7) U11_GAGAA(x1, x2, x3, x4, x5, x6, x7) = U11_GAGAA(x1, x3, x7) U12_GAGAA(x1, x2, x3, x4, x5, x6, x7) = U12_GAGAA(x1, x3, x7) U13_GAGAA(x1, x2, x3, x4, x5, x6, x7, x8) = U13_GAGAA(x1, x3, x7, x8) U14_GAGAA(x1, x2, x3, x4, x5, x6, x7) = U14_GAGAA(x1, x3, x7) U15_GAGAA(x1, x2, x3, x4, x5, x6, x7, x8) = U15_GAGAA(x1, x3, x7, x8) U16_GAGAA(x1, x2, x3, x4, x5, x6, x7) = U16_GAGAA(x1, x3, x7) U17_GAGAA(x1, x2, x3, x4, x5, x6, x7) = U17_GAGAA(x1, x3, x7) U18_GAGAA(x1, x2, x3, x4, x5, x6, x7) = U18_GAGAA(x1, x3, x7) We have to consider all (P,R,Pi)-chains ---------------------------------------- (349) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 169 less nodes. ---------------------------------------- (350) Complex Obligation (AND) ---------------------------------------- (351) Obligation: Pi DP problem: The TRS P consists of the following rules: DIFN_IN_GAAAA(.(X1, X2), X3, X4, .(X5, X6), .(X7, X8)) -> DIFN_IN_GAAAA(X2, X3, X4, X6, X8) The TRS R consists of the following rules: iota1cQ_in_aa(X1, X2) -> U153_aa(X1, X2, iota1cA_in_aaa(X3, X1, X2)) iota1cA_in_aaa(X1, X1, []) -> iota1cA_out_aaa(X1, X1, []) iota1cA_in_aaa(X1, X2, .(X1, X3)) -> U135_aaa(X1, X2, X3, iota1cA_in_aaa(X4, X2, X3)) U135_aaa(X1, X2, X3, iota1cA_out_aaa(X4, X2, X3)) -> iota1cA_out_aaa(X1, X2, .(X1, X3)) U153_aa(X1, X2, iota1cA_out_aaa(X3, X1, X2)) -> iota1cQ_out_aa(X1, X2) membercR_in_ag(X1, .(X1, X2)) -> membercR_out_ag(X1, .(X1, X2)) membercR_in_ag(X1, .(X2, X3)) -> U154_ag(X1, X2, X3, membercB_in_ag(X1, X3)) membercB_in_ag(X1, .(X1, X2)) -> membercB_out_ag(X1, .(X1, X2)) membercB_in_ag(X1, .(X2, X3)) -> U136_ag(X1, X2, X3, membercB_in_ag(X1, X3)) U136_ag(X1, X2, X3, membercB_out_ag(X1, X3)) -> membercB_out_ag(X1, .(X2, X3)) U154_ag(X1, X2, X3, membercB_out_ag(X1, X3)) -> membercR_out_ag(X1, .(X2, X3)) difcE_in_aaaa(X1, X2, .(X3, []), .(X4, [])) -> difcE_out_aaaa(X1, X2, .(X3, []), .(X4, [])) mergedeletecT_in_gga([], X1, X1) -> mergedeletecT_out_gga([], X1, X1) mergedeletecT_in_gga(.(X1, X2), .(X1, X3), X4) -> U159_gga(X1, X2, X3, X4, mergedeletecC_in_gga(X2, X3, X4)) mergedeletecC_in_gga([], X1, X1) -> mergedeletecC_out_gga([], X1, X1) mergedeletecC_in_gga(.(X1, X2), .(X1, X3), X4) -> U137_gga(X1, X2, X3, X4, mergedeletecC_in_gga(X2, X3, X4)) mergedeletecC_in_gga(.(X1, X2), .(X3, X4), .(X3, X5)) -> U138_gga(X1, X2, X3, X4, X5, mergedeletecC_in_gga(.(X1, X2), X4, X5)) U138_gga(X1, X2, X3, X4, X5, mergedeletecC_out_gga(.(X1, X2), X4, X5)) -> mergedeletecC_out_gga(.(X1, X2), .(X3, X4), .(X3, X5)) U137_gga(X1, X2, X3, X4, mergedeletecC_out_gga(X2, X3, X4)) -> mergedeletecC_out_gga(.(X1, X2), .(X1, X3), X4) U159_gga(X1, X2, X3, X4, mergedeletecC_out_gga(X2, X3, X4)) -> mergedeletecT_out_gga(.(X1, X2), .(X1, X3), X4) mergedeletecT_in_gga(.(X1, X2), .(X3, X4), .(X3, X5)) -> U160_gga(X1, X2, X3, X4, X5, mergedeletecC_in_gga(.(X1, X2), X4, X5)) U160_gga(X1, X2, X3, X4, X5, mergedeletecC_out_gga(.(X1, X2), X4, X5)) -> mergedeletecT_out_gga(.(X1, X2), .(X3, X4), .(X3, X5)) revcF_in_ga([], []) -> revcF_out_ga([], []) revcF_in_ga(.(X1, []), .(X1, [])) -> revcF_out_ga(.(X1, []), .(X1, [])) revcF_in_ga(.(X1, .(X2, [])), .(X2, .(X1, []))) -> revcF_out_ga(.(X1, .(X2, [])), .(X2, .(X1, []))) revcF_in_ga(.(X1, .(X2, .(X3, []))), .(X3, .(X2, .(X1, [])))) -> revcF_out_ga(.(X1, .(X2, .(X3, []))), .(X3, .(X2, .(X1, [])))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, [])))), .(X4, .(X3, .(X2, .(X1, []))))) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, [])))), .(X4, .(X3, .(X2, .(X1, []))))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, .(X5, []))))), .(X5, .(X4, .(X3, .(X2, .(X1, [])))))) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, .(X5, []))))), .(X5, .(X4, .(X3, .(X2, .(X1, [])))))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, [])))))), .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, [])))))), .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, []))))))), .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, [])))))))) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, []))))))), .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, [])))))))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) -> U140_ga(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revcD_in_gaga(X9, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))), X10)) revcD_in_gaga([], X1, X2, .(X1, X2)) -> revcD_out_gaga([], X1, X2, .(X1, X2)) revcD_in_gaga(.(X1, X2), X3, X4, X5) -> U139_gaga(X1, X2, X3, X4, X5, revcD_in_gaga(X2, X1, .(X3, X4), X5)) U139_gaga(X1, X2, X3, X4, X5, revcD_out_gaga(X2, X1, .(X3, X4), X5)) -> revcD_out_gaga(.(X1, X2), X3, X4, X5) U140_ga(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revcD_out_gaga(X9, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))), X10)) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) checkcS_in_agaa(X1, X2, X3, X4) -> U155_agaa(X1, X2, X3, X4, difcE_in_aaaa(X1, X4, X5, X6)) U155_agaa(X1, X2, X3, X4, difcE_out_aaaa(X1, X4, X5, X6)) -> U156_agaa(X1, X2, X3, X4, X5, X6, mergedeletecT_in_gga(X5, X2, X7)) U156_agaa(X1, X2, X3, X4, X5, X6, mergedeletecT_out_gga(X5, X2, X7)) -> U157_agaa(X1, X2, X3, X4, X5, X6, X7, revcF_in_ga(X6, X8)) U157_agaa(X1, X2, X3, X4, X5, X6, X7, revcF_out_ga(X6, X8)) -> U158_agaa(X1, X2, X3, X4, mergedeletecC_in_gga(X8, X7, X3)) U158_agaa(X1, X2, X3, X4, mergedeletecC_out_gga(X8, X7, X3)) -> checkcS_out_agaa(X1, X2, X3, X4) difcG_in_aaaaa(X1, X2, X3, .(X4, X5), .(X6, X7)) -> U141_aaaaa(X1, X2, X3, X4, X5, X6, X7, difcE_in_aaaa(X2, X3, X5, X7)) U141_aaaaa(X1, X2, X3, X4, X5, X6, X7, difcE_out_aaaa(X2, X3, X5, X7)) -> difcG_out_aaaaa(X1, X2, X3, .(X4, X5), .(X6, X7)) checkcU_in_aagaa(X1, X2, X3, X4, X5) -> U161_aagaa(X1, X2, X3, X4, X5, difcG_in_aaaaa(X1, X2, X5, X6, X7)) U161_aagaa(X1, X2, X3, X4, X5, difcG_out_aaaaa(X1, X2, X5, X6, X7)) -> U162_aagaa(X1, X2, X3, X4, X5, X6, X7, mergedeletecC_in_gga(X6, X3, X8)) U162_aagaa(X1, X2, X3, X4, X5, X6, X7, mergedeletecC_out_gga(X6, X3, X8)) -> U163_aagaa(X1, X2, X3, X4, X5, X6, X7, X8, revcF_in_ga(X7, X9)) U163_aagaa(X1, X2, X3, X4, X5, X6, X7, X8, revcF_out_ga(X7, X9)) -> U164_aagaa(X1, X2, X3, X4, X5, mergedeletecC_in_gga(X9, X8, X4)) U164_aagaa(X1, X2, X3, X4, X5, mergedeletecC_out_gga(X9, X8, X4)) -> checkcU_out_aagaa(X1, X2, X3, X4, X5) difcH_in_aaaaaa(X1, X2, X3, X4, .(X5, X6), .(X7, X8)) -> U142_aaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, difcG_in_aaaaa(X2, X3, X4, X6, X8)) U142_aaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, difcG_out_aaaaa(X2, X3, X4, X6, X8)) -> difcH_out_aaaaaa(X1, X2, X3, X4, .(X5, X6), .(X7, X8)) checkcV_in_aaagaa(X1, X2, X3, X4, X5, X6) -> U165_aaagaa(X1, X2, X3, X4, X5, X6, difcH_in_aaaaaa(X1, X2, X3, X6, X7, X8)) U165_aaagaa(X1, X2, X3, X4, X5, X6, difcH_out_aaaaaa(X1, X2, X3, X6, X7, X8)) -> U166_aaagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_in_gga(X7, X4, X9)) U166_aaagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_out_gga(X7, X4, X9)) -> U167_aaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, revcF_in_ga(X8, X10)) U167_aaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, revcF_out_ga(X8, X10)) -> U168_aaagaa(X1, X2, X3, X4, X5, X6, mergedeletecC_in_gga(X10, X9, X5)) U168_aaagaa(X1, X2, X3, X4, X5, X6, mergedeletecC_out_gga(X10, X9, X5)) -> checkcV_out_aaagaa(X1, X2, X3, X4, X5, X6) difcI_in_aaaaaaa(X1, X2, X3, X4, X5, .(X6, X7), .(X8, X9)) -> U143_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, difcH_in_aaaaaa(X2, X3, X4, X5, X7, X9)) U143_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, difcH_out_aaaaaa(X2, X3, X4, X5, X7, X9)) -> difcI_out_aaaaaaa(X1, X2, X3, X4, X5, .(X6, X7), .(X8, X9)) checkcW_in_aaaagaa(X1, X2, X3, X4, X5, X6, X7) -> U169_aaaagaa(X1, X2, X3, X4, X5, X6, X7, difcI_in_aaaaaaa(X1, X2, X3, X4, X7, X8, X9)) U169_aaaagaa(X1, X2, X3, X4, X5, X6, X7, difcI_out_aaaaaaa(X1, X2, X3, X4, X7, X8, X9)) -> U170_aaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, mergedeletecC_in_gga(X8, X5, X10)) U170_aaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, mergedeletecC_out_gga(X8, X5, X10)) -> U171_aaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revcF_in_ga(X9, X11)) U171_aaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revcF_out_ga(X9, X11)) -> U172_aaaagaa(X1, X2, X3, X4, X5, X6, X7, mergedeletecC_in_gga(X11, X10, X6)) U172_aaaagaa(X1, X2, X3, X4, X5, X6, X7, mergedeletecC_out_gga(X11, X10, X6)) -> checkcW_out_aaaagaa(X1, X2, X3, X4, X5, X6, X7) difcJ_in_aaaaaaaa(X1, X2, X3, X4, X5, X6, .(X7, X8), .(X9, X10)) -> U144_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, difcI_in_aaaaaaa(X2, X3, X4, X5, X6, X8, X10)) U144_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, difcI_out_aaaaaaa(X2, X3, X4, X5, X6, X8, X10)) -> difcJ_out_aaaaaaaa(X1, X2, X3, X4, X5, X6, .(X7, X8), .(X9, X10)) checkcX_in_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8) -> U173_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, difcJ_in_aaaaaaaa(X1, X2, X3, X4, X5, X8, X9, X10)) U173_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, difcJ_out_aaaaaaaa(X1, X2, X3, X4, X5, X8, X9, X10)) -> U174_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, mergedeletecC_in_gga(X9, X6, X11)) U174_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, mergedeletecC_out_gga(X9, X6, X11)) -> U175_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, revcF_in_ga(X10, X12)) U175_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, revcF_out_ga(X10, X12)) -> U176_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_in_gga(X12, X11, X7)) U176_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_out_gga(X12, X11, X7)) -> checkcX_out_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8) difcK_in_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, .(X8, X9), .(X10, X11)) -> U145_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, difcJ_in_aaaaaaaa(X2, X3, X4, X5, X6, X7, X9, X11)) U145_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, difcJ_out_aaaaaaaa(X2, X3, X4, X5, X6, X7, X9, X11)) -> difcK_out_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, .(X8, X9), .(X10, X11)) checkcY_in_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9) -> U177_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, difcK_in_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X9, X10, X11)) U177_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, difcK_out_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X9, X10, X11)) -> U178_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, mergedeletecC_in_gga(X10, X7, X12)) U178_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, mergedeletecC_out_gga(X10, X7, X12)) -> U179_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, revcF_in_ga(X11, X13)) U179_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, revcF_out_ga(X11, X13)) -> U180_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, mergedeletecC_in_gga(X13, X12, X8)) U180_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, mergedeletecC_out_gga(X13, X12, X8)) -> checkcY_out_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9) difcZ_in_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, .(X9, X10), .(X11, X12)) -> U185_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, difcK_in_aaaaaaaaa(X2, X3, X4, X5, X6, X7, X8, X10, X12)) U185_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, difcK_out_aaaaaaaaa(X2, X3, X4, X5, X6, X7, X8, X10, X12)) -> difcZ_out_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, .(X9, X10), .(X11, X12)) checkcP_in_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10) -> U181_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, difcZ_in_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X10, X11, X12)) U181_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, difcZ_out_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X10, X11, X12)) -> U182_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, mergedeletecC_in_gga(X11, X8, X13)) U182_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, mergedeletecC_out_gga(X11, X8, X13)) -> U183_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, revcF_in_ga(X12, X14)) U183_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, revcF_out_ga(X12, X14)) -> U184_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, mergedeletecC_in_gga(X14, X13, X9)) U184_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, mergedeletecC_out_gga(X14, X13, X9)) -> checkcP_out_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10) difcN1_in_agaaaa(X1, [], X2, X3, .(X4, []), .(X5, [])) -> difcN1_out_agaaaa(X1, [], X2, X3, .(X4, []), .(X5, [])) difcN1_in_agaaaa(X1, .(X2, []), X3, X4, .(X5, .(X6, [])), .(X7, .(X8, []))) -> difcN1_out_agaaaa(X1, .(X2, []), X3, X4, .(X5, .(X6, [])), .(X7, .(X8, []))) difcN1_in_agaaaa(X1, .(X2, .(X3, [])), X4, X5, .(X6, .(X7, .(X8, []))), .(X9, .(X10, .(X11, [])))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, [])), X4, X5, .(X6, .(X7, .(X8, []))), .(X9, .(X10, .(X11, [])))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, []))), X5, X6, .(X7, .(X8, .(X9, .(X10, [])))), .(X11, .(X12, .(X13, .(X14, []))))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, []))), X5, X6, .(X7, .(X8, .(X9, .(X10, [])))), .(X11, .(X12, .(X13, .(X14, []))))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, [])))), X6, X7, .(X8, .(X9, .(X10, .(X11, .(X12, []))))), .(X13, .(X14, .(X15, .(X16, .(X17, [])))))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, [])))), X6, X7, .(X8, .(X9, .(X10, .(X11, .(X12, []))))), .(X13, .(X14, .(X15, .(X16, .(X17, [])))))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, []))))), X7, X8, .(X9, .(X10, .(X11, .(X12, .(X13, .(X14, [])))))), .(X15, .(X16, .(X17, .(X18, .(X19, .(X20, []))))))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, []))))), X7, X8, .(X9, .(X10, .(X11, .(X12, .(X13, .(X14, [])))))), .(X15, .(X16, .(X17, .(X18, .(X19, .(X20, []))))))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, [])))))), X8, X9, .(X10, .(X11, .(X12, .(X13, .(X14, .(X15, .(X16, []))))))), .(X17, .(X18, .(X19, .(X20, .(X21, .(X22, .(X23, [])))))))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, [])))))), X8, X9, .(X10, .(X11, .(X12, .(X13, .(X14, .(X15, .(X16, []))))))), .(X17, .(X18, .(X19, .(X20, .(X21, .(X22, .(X23, [])))))))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9))))))), X10, X11, .(X12, .(X13, .(X14, .(X15, .(X16, .(X17, .(X18, .(X19, X20)))))))), .(X21, .(X22, .(X23, .(X24, .(X25, .(X26, .(X27, .(X28, X29))))))))) -> U190_agaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, X21, X22, X23, X24, X25, X26, X27, X28, X29, difcN_in_gaaaa(X9, X10, X11, X20, X29)) difcN_in_gaaaa([], X1, X2, [], []) -> difcN_out_gaaaa([], X1, X2, [], []) difcN_in_gaaaa(.(X1, X2), X3, X4, .(X5, X6), .(X7, X8)) -> U149_gaaaa(X1, X2, X3, X4, X5, X6, X7, X8, difcN_in_gaaaa(X2, X3, X4, X6, X8)) U149_gaaaa(X1, X2, X3, X4, X5, X6, X7, X8, difcN_out_gaaaa(X2, X3, X4, X6, X8)) -> difcN_out_gaaaa(.(X1, X2), X3, X4, .(X5, X6), .(X7, X8)) U190_agaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, X21, X22, X23, X24, X25, X26, X27, X28, X29, difcN_out_gaaaa(X9, X10, X11, X20, X29)) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9))))))), X10, X11, .(X12, .(X13, .(X14, .(X15, .(X16, .(X17, .(X18, .(X19, X20)))))))), .(X21, .(X22, .(X23, .(X24, .(X25, .(X26, .(X27, .(X28, X29))))))))) checkcM_in_agagaa(X1, X2, X3, X4, X5, X6) -> U186_agagaa(X1, X2, X3, X4, X5, X6, difcN1_in_agaaaa(X1, X2, X3, X6, X7, X8)) U186_agagaa(X1, X2, X3, X4, X5, X6, difcN1_out_agaaaa(X1, X2, X3, X6, X7, X8)) -> U187_agagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_in_gga(X7, X4, X9)) U187_agagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_out_gga(X7, X4, X9)) -> U188_agagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, revcF_in_ga(X8, X10)) U188_agagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, revcF_out_ga(X8, X10)) -> U189_agagaa(X1, X2, X3, X4, X5, X6, mergedeletecC_in_gga(X10, X9, X5)) U189_agagaa(X1, X2, X3, X4, X5, X6, mergedeletecC_out_gga(X10, X9, X5)) -> checkcM_out_agagaa(X1, X2, X3, X4, X5, X6) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) iota1cQ_in_aa(x1, x2) = iota1cQ_in_aa U153_aa(x1, x2, x3) = U153_aa(x3) iota1cA_in_aaa(x1, x2, x3) = iota1cA_in_aaa iota1cA_out_aaa(x1, x2, x3) = iota1cA_out_aaa(x3) U135_aaa(x1, x2, x3, x4) = U135_aaa(x4) iota1cQ_out_aa(x1, x2) = iota1cQ_out_aa(x2) membercR_in_ag(x1, x2) = membercR_in_ag(x2) membercR_out_ag(x1, x2) = membercR_out_ag(x2) U154_ag(x1, x2, x3, x4) = U154_ag(x3, x4) membercB_in_ag(x1, x2) = membercB_in_ag(x2) membercB_out_ag(x1, x2) = membercB_out_ag(x2) U136_ag(x1, x2, x3, x4) = U136_ag(x3, x4) difcE_in_aaaa(x1, x2, x3, x4) = difcE_in_aaaa difcE_out_aaaa(x1, x2, x3, x4) = difcE_out_aaaa(x3, x4) mergedeletecT_in_gga(x1, x2, x3) = mergedeletecT_in_gga(x1, x2) [] = [] mergedeletecT_out_gga(x1, x2, x3) = mergedeletecT_out_gga(x1, x2, x3) U159_gga(x1, x2, x3, x4, x5) = U159_gga(x2, x3, x5) mergedeletecC_in_gga(x1, x2, x3) = mergedeletecC_in_gga(x1, x2) mergedeletecC_out_gga(x1, x2, x3) = mergedeletecC_out_gga(x1, x2, x3) U137_gga(x1, x2, x3, x4, x5) = U137_gga(x2, x3, x5) U138_gga(x1, x2, x3, x4, x5, x6) = U138_gga(x2, x4, x6) U160_gga(x1, x2, x3, x4, x5, x6) = U160_gga(x2, x4, x6) revcF_in_ga(x1, x2) = revcF_in_ga(x1) revcF_out_ga(x1, x2) = revcF_out_ga(x1, x2) U140_ga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U140_ga(x9, x11) revcD_in_gaga(x1, x2, x3, x4) = revcD_in_gaga(x1, x3) revcD_out_gaga(x1, x2, x3, x4) = revcD_out_gaga(x1, x3, x4) U139_gaga(x1, x2, x3, x4, x5, x6) = U139_gaga(x2, x4, x6) checkcS_in_agaa(x1, x2, x3, x4) = checkcS_in_agaa(x2) U155_agaa(x1, x2, x3, x4, x5) = U155_agaa(x2, x5) U156_agaa(x1, x2, x3, x4, x5, x6, x7) = U156_agaa(x2, x6, x7) U157_agaa(x1, x2, x3, x4, x5, x6, x7, x8) = U157_agaa(x2, x7, x8) U158_agaa(x1, x2, x3, x4, x5) = U158_agaa(x2, x5) checkcS_out_agaa(x1, x2, x3, x4) = checkcS_out_agaa(x2, x3) difcG_in_aaaaa(x1, x2, x3, x4, x5) = difcG_in_aaaaa U141_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U141_aaaaa(x8) difcG_out_aaaaa(x1, x2, x3, x4, x5) = difcG_out_aaaaa(x4, x5) checkcU_in_aagaa(x1, x2, x3, x4, x5) = checkcU_in_aagaa(x3) U161_aagaa(x1, x2, x3, x4, x5, x6) = U161_aagaa(x3, x6) U162_aagaa(x1, x2, x3, x4, x5, x6, x7, x8) = U162_aagaa(x3, x7, x8) U163_aagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U163_aagaa(x3, x8, x9) U164_aagaa(x1, x2, x3, x4, x5, x6) = U164_aagaa(x3, x6) checkcU_out_aagaa(x1, x2, x3, x4, x5) = checkcU_out_aagaa(x3, x4) difcH_in_aaaaaa(x1, x2, x3, x4, x5, x6) = difcH_in_aaaaaa U142_aaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U142_aaaaaa(x9) difcH_out_aaaaaa(x1, x2, x3, x4, x5, x6) = difcH_out_aaaaaa(x5, x6) checkcV_in_aaagaa(x1, x2, x3, x4, x5, x6) = checkcV_in_aaagaa(x4) U165_aaagaa(x1, x2, x3, x4, x5, x6, x7) = U165_aaagaa(x4, x7) U166_aaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U166_aaagaa(x4, x8, x9) U167_aaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U167_aaagaa(x4, x9, x10) U168_aaagaa(x1, x2, x3, x4, x5, x6, x7) = U168_aaagaa(x4, x7) checkcV_out_aaagaa(x1, x2, x3, x4, x5, x6) = checkcV_out_aaagaa(x4, x5) difcI_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = difcI_in_aaaaaaa U143_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U143_aaaaaaa(x10) difcI_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = difcI_out_aaaaaaa(x6, x7) checkcW_in_aaaagaa(x1, x2, x3, x4, x5, x6, x7) = checkcW_in_aaaagaa(x5) U169_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8) = U169_aaaagaa(x5, x8) U170_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U170_aaaagaa(x5, x9, x10) U171_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U171_aaaagaa(x5, x10, x11) U172_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8) = U172_aaaagaa(x5, x8) checkcW_out_aaaagaa(x1, x2, x3, x4, x5, x6, x7) = checkcW_out_aaaagaa(x5, x6) difcJ_in_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = difcJ_in_aaaaaaaa U144_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U144_aaaaaaaa(x11) difcJ_out_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = difcJ_out_aaaaaaaa(x7, x8) checkcX_in_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8) = checkcX_in_aaaaagaa(x6) U173_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U173_aaaaagaa(x6, x9) U174_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U174_aaaaagaa(x6, x10, x11) U175_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U175_aaaaagaa(x6, x11, x12) U176_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U176_aaaaagaa(x6, x9) checkcX_out_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8) = checkcX_out_aaaaagaa(x6, x7) difcK_in_aaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = difcK_in_aaaaaaaaa U145_aaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U145_aaaaaaaaa(x12) difcK_out_aaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = difcK_out_aaaaaaaaa(x8, x9) checkcY_in_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = checkcY_in_aaaaaagaa(x7) U177_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U177_aaaaaagaa(x7, x10) U178_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U178_aaaaaagaa(x7, x11, x12) U179_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U179_aaaaaagaa(x7, x12, x13) U180_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U180_aaaaaagaa(x7, x10) checkcY_out_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = checkcY_out_aaaaaagaa(x7, x8) difcZ_in_aaaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = difcZ_in_aaaaaaaaaa U185_aaaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U185_aaaaaaaaaa(x13) difcZ_out_aaaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = difcZ_out_aaaaaaaaaa(x9, x10) checkcP_in_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = checkcP_in_aaaaaaagaa(x8) U181_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U181_aaaaaaagaa(x8, x11) U182_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U182_aaaaaaagaa(x8, x12, x13) U183_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14) = U183_aaaaaaagaa(x8, x13, x14) U184_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U184_aaaaaaagaa(x8, x11) checkcP_out_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = checkcP_out_aaaaaaagaa(x8, x9) difcN1_in_agaaaa(x1, x2, x3, x4, x5, x6) = difcN1_in_agaaaa(x2) difcN1_out_agaaaa(x1, x2, x3, x4, x5, x6) = difcN1_out_agaaaa(x2, x5, x6) U190_agaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30) = U190_agaaaa(x9, x30) difcN_in_gaaaa(x1, x2, x3, x4, x5) = difcN_in_gaaaa(x1) difcN_out_gaaaa(x1, x2, x3, x4, x5) = difcN_out_gaaaa(x1, x4, x5) U149_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U149_gaaaa(x2, x9) checkcM_in_agagaa(x1, x2, x3, x4, x5, x6) = checkcM_in_agagaa(x2, x4) U186_agagaa(x1, x2, x3, x4, x5, x6, x7) = U186_agagaa(x2, x4, x7) U187_agagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U187_agagaa(x2, x4, x8, x9) U188_agagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U188_agagaa(x2, x4, x9, x10) U189_agagaa(x1, x2, x3, x4, x5, x6, x7) = U189_agagaa(x2, x4, x7) checkcM_out_agagaa(x1, x2, x3, x4, x5, x6) = checkcM_out_agagaa(x2, x4, x5) DIFN_IN_GAAAA(x1, x2, x3, x4, x5) = DIFN_IN_GAAAA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (352) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (353) Obligation: Pi DP problem: The TRS P consists of the following rules: DIFN_IN_GAAAA(.(X1, X2), X3, X4, .(X5, X6), .(X7, X8)) -> DIFN_IN_GAAAA(X2, X3, X4, X6, X8) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) DIFN_IN_GAAAA(x1, x2, x3, x4, x5) = DIFN_IN_GAAAA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (354) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (355) Obligation: Q DP problem: The TRS P consists of the following rules: DIFN_IN_GAAAA(.(X2)) -> DIFN_IN_GAAAA(X2) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (356) 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: *DIFN_IN_GAAAA(.(X2)) -> DIFN_IN_GAAAA(X2) The graph contains the following edges 1 > 1 ---------------------------------------- (357) YES ---------------------------------------- (358) Obligation: Pi DP problem: The TRS P consists of the following rules: REVD_IN_GAGA(.(X1, X2), X3, X4, X5) -> REVD_IN_GAGA(X2, X1, .(X3, X4), X5) The TRS R consists of the following rules: iota1cQ_in_aa(X1, X2) -> U153_aa(X1, X2, iota1cA_in_aaa(X3, X1, X2)) iota1cA_in_aaa(X1, X1, []) -> iota1cA_out_aaa(X1, X1, []) iota1cA_in_aaa(X1, X2, .(X1, X3)) -> U135_aaa(X1, X2, X3, iota1cA_in_aaa(X4, X2, X3)) U135_aaa(X1, X2, X3, iota1cA_out_aaa(X4, X2, X3)) -> iota1cA_out_aaa(X1, X2, .(X1, X3)) U153_aa(X1, X2, iota1cA_out_aaa(X3, X1, X2)) -> iota1cQ_out_aa(X1, X2) membercR_in_ag(X1, .(X1, X2)) -> membercR_out_ag(X1, .(X1, X2)) membercR_in_ag(X1, .(X2, X3)) -> U154_ag(X1, X2, X3, membercB_in_ag(X1, X3)) membercB_in_ag(X1, .(X1, X2)) -> membercB_out_ag(X1, .(X1, X2)) membercB_in_ag(X1, .(X2, X3)) -> U136_ag(X1, X2, X3, membercB_in_ag(X1, X3)) U136_ag(X1, X2, X3, membercB_out_ag(X1, X3)) -> membercB_out_ag(X1, .(X2, X3)) U154_ag(X1, X2, X3, membercB_out_ag(X1, X3)) -> membercR_out_ag(X1, .(X2, X3)) difcE_in_aaaa(X1, X2, .(X3, []), .(X4, [])) -> difcE_out_aaaa(X1, X2, .(X3, []), .(X4, [])) mergedeletecT_in_gga([], X1, X1) -> mergedeletecT_out_gga([], X1, X1) mergedeletecT_in_gga(.(X1, X2), .(X1, X3), X4) -> U159_gga(X1, X2, X3, X4, mergedeletecC_in_gga(X2, X3, X4)) mergedeletecC_in_gga([], X1, X1) -> mergedeletecC_out_gga([], X1, X1) mergedeletecC_in_gga(.(X1, X2), .(X1, X3), X4) -> U137_gga(X1, X2, X3, X4, mergedeletecC_in_gga(X2, X3, X4)) mergedeletecC_in_gga(.(X1, X2), .(X3, X4), .(X3, X5)) -> U138_gga(X1, X2, X3, X4, X5, mergedeletecC_in_gga(.(X1, X2), X4, X5)) U138_gga(X1, X2, X3, X4, X5, mergedeletecC_out_gga(.(X1, X2), X4, X5)) -> mergedeletecC_out_gga(.(X1, X2), .(X3, X4), .(X3, X5)) U137_gga(X1, X2, X3, X4, mergedeletecC_out_gga(X2, X3, X4)) -> mergedeletecC_out_gga(.(X1, X2), .(X1, X3), X4) U159_gga(X1, X2, X3, X4, mergedeletecC_out_gga(X2, X3, X4)) -> mergedeletecT_out_gga(.(X1, X2), .(X1, X3), X4) mergedeletecT_in_gga(.(X1, X2), .(X3, X4), .(X3, X5)) -> U160_gga(X1, X2, X3, X4, X5, mergedeletecC_in_gga(.(X1, X2), X4, X5)) U160_gga(X1, X2, X3, X4, X5, mergedeletecC_out_gga(.(X1, X2), X4, X5)) -> mergedeletecT_out_gga(.(X1, X2), .(X3, X4), .(X3, X5)) revcF_in_ga([], []) -> revcF_out_ga([], []) revcF_in_ga(.(X1, []), .(X1, [])) -> revcF_out_ga(.(X1, []), .(X1, [])) revcF_in_ga(.(X1, .(X2, [])), .(X2, .(X1, []))) -> revcF_out_ga(.(X1, .(X2, [])), .(X2, .(X1, []))) revcF_in_ga(.(X1, .(X2, .(X3, []))), .(X3, .(X2, .(X1, [])))) -> revcF_out_ga(.(X1, .(X2, .(X3, []))), .(X3, .(X2, .(X1, [])))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, [])))), .(X4, .(X3, .(X2, .(X1, []))))) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, [])))), .(X4, .(X3, .(X2, .(X1, []))))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, .(X5, []))))), .(X5, .(X4, .(X3, .(X2, .(X1, [])))))) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, .(X5, []))))), .(X5, .(X4, .(X3, .(X2, .(X1, [])))))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, [])))))), .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, [])))))), .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, []))))))), .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, [])))))))) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, []))))))), .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, [])))))))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) -> U140_ga(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revcD_in_gaga(X9, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))), X10)) revcD_in_gaga([], X1, X2, .(X1, X2)) -> revcD_out_gaga([], X1, X2, .(X1, X2)) revcD_in_gaga(.(X1, X2), X3, X4, X5) -> U139_gaga(X1, X2, X3, X4, X5, revcD_in_gaga(X2, X1, .(X3, X4), X5)) U139_gaga(X1, X2, X3, X4, X5, revcD_out_gaga(X2, X1, .(X3, X4), X5)) -> revcD_out_gaga(.(X1, X2), X3, X4, X5) U140_ga(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revcD_out_gaga(X9, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))), X10)) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) checkcS_in_agaa(X1, X2, X3, X4) -> U155_agaa(X1, X2, X3, X4, difcE_in_aaaa(X1, X4, X5, X6)) U155_agaa(X1, X2, X3, X4, difcE_out_aaaa(X1, X4, X5, X6)) -> U156_agaa(X1, X2, X3, X4, X5, X6, mergedeletecT_in_gga(X5, X2, X7)) U156_agaa(X1, X2, X3, X4, X5, X6, mergedeletecT_out_gga(X5, X2, X7)) -> U157_agaa(X1, X2, X3, X4, X5, X6, X7, revcF_in_ga(X6, X8)) U157_agaa(X1, X2, X3, X4, X5, X6, X7, revcF_out_ga(X6, X8)) -> U158_agaa(X1, X2, X3, X4, mergedeletecC_in_gga(X8, X7, X3)) U158_agaa(X1, X2, X3, X4, mergedeletecC_out_gga(X8, X7, X3)) -> checkcS_out_agaa(X1, X2, X3, X4) difcG_in_aaaaa(X1, X2, X3, .(X4, X5), .(X6, X7)) -> U141_aaaaa(X1, X2, X3, X4, X5, X6, X7, difcE_in_aaaa(X2, X3, X5, X7)) U141_aaaaa(X1, X2, X3, X4, X5, X6, X7, difcE_out_aaaa(X2, X3, X5, X7)) -> difcG_out_aaaaa(X1, X2, X3, .(X4, X5), .(X6, X7)) checkcU_in_aagaa(X1, X2, X3, X4, X5) -> U161_aagaa(X1, X2, X3, X4, X5, difcG_in_aaaaa(X1, X2, X5, X6, X7)) U161_aagaa(X1, X2, X3, X4, X5, difcG_out_aaaaa(X1, X2, X5, X6, X7)) -> U162_aagaa(X1, X2, X3, X4, X5, X6, X7, mergedeletecC_in_gga(X6, X3, X8)) U162_aagaa(X1, X2, X3, X4, X5, X6, X7, mergedeletecC_out_gga(X6, X3, X8)) -> U163_aagaa(X1, X2, X3, X4, X5, X6, X7, X8, revcF_in_ga(X7, X9)) U163_aagaa(X1, X2, X3, X4, X5, X6, X7, X8, revcF_out_ga(X7, X9)) -> U164_aagaa(X1, X2, X3, X4, X5, mergedeletecC_in_gga(X9, X8, X4)) U164_aagaa(X1, X2, X3, X4, X5, mergedeletecC_out_gga(X9, X8, X4)) -> checkcU_out_aagaa(X1, X2, X3, X4, X5) difcH_in_aaaaaa(X1, X2, X3, X4, .(X5, X6), .(X7, X8)) -> U142_aaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, difcG_in_aaaaa(X2, X3, X4, X6, X8)) U142_aaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, difcG_out_aaaaa(X2, X3, X4, X6, X8)) -> difcH_out_aaaaaa(X1, X2, X3, X4, .(X5, X6), .(X7, X8)) checkcV_in_aaagaa(X1, X2, X3, X4, X5, X6) -> U165_aaagaa(X1, X2, X3, X4, X5, X6, difcH_in_aaaaaa(X1, X2, X3, X6, X7, X8)) U165_aaagaa(X1, X2, X3, X4, X5, X6, difcH_out_aaaaaa(X1, X2, X3, X6, X7, X8)) -> U166_aaagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_in_gga(X7, X4, X9)) U166_aaagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_out_gga(X7, X4, X9)) -> U167_aaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, revcF_in_ga(X8, X10)) U167_aaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, revcF_out_ga(X8, X10)) -> U168_aaagaa(X1, X2, X3, X4, X5, X6, mergedeletecC_in_gga(X10, X9, X5)) U168_aaagaa(X1, X2, X3, X4, X5, X6, mergedeletecC_out_gga(X10, X9, X5)) -> checkcV_out_aaagaa(X1, X2, X3, X4, X5, X6) difcI_in_aaaaaaa(X1, X2, X3, X4, X5, .(X6, X7), .(X8, X9)) -> U143_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, difcH_in_aaaaaa(X2, X3, X4, X5, X7, X9)) U143_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, difcH_out_aaaaaa(X2, X3, X4, X5, X7, X9)) -> difcI_out_aaaaaaa(X1, X2, X3, X4, X5, .(X6, X7), .(X8, X9)) checkcW_in_aaaagaa(X1, X2, X3, X4, X5, X6, X7) -> U169_aaaagaa(X1, X2, X3, X4, X5, X6, X7, difcI_in_aaaaaaa(X1, X2, X3, X4, X7, X8, X9)) U169_aaaagaa(X1, X2, X3, X4, X5, X6, X7, difcI_out_aaaaaaa(X1, X2, X3, X4, X7, X8, X9)) -> U170_aaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, mergedeletecC_in_gga(X8, X5, X10)) U170_aaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, mergedeletecC_out_gga(X8, X5, X10)) -> U171_aaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revcF_in_ga(X9, X11)) U171_aaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revcF_out_ga(X9, X11)) -> U172_aaaagaa(X1, X2, X3, X4, X5, X6, X7, mergedeletecC_in_gga(X11, X10, X6)) U172_aaaagaa(X1, X2, X3, X4, X5, X6, X7, mergedeletecC_out_gga(X11, X10, X6)) -> checkcW_out_aaaagaa(X1, X2, X3, X4, X5, X6, X7) difcJ_in_aaaaaaaa(X1, X2, X3, X4, X5, X6, .(X7, X8), .(X9, X10)) -> U144_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, difcI_in_aaaaaaa(X2, X3, X4, X5, X6, X8, X10)) U144_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, difcI_out_aaaaaaa(X2, X3, X4, X5, X6, X8, X10)) -> difcJ_out_aaaaaaaa(X1, X2, X3, X4, X5, X6, .(X7, X8), .(X9, X10)) checkcX_in_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8) -> U173_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, difcJ_in_aaaaaaaa(X1, X2, X3, X4, X5, X8, X9, X10)) U173_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, difcJ_out_aaaaaaaa(X1, X2, X3, X4, X5, X8, X9, X10)) -> U174_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, mergedeletecC_in_gga(X9, X6, X11)) U174_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, mergedeletecC_out_gga(X9, X6, X11)) -> U175_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, revcF_in_ga(X10, X12)) U175_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, revcF_out_ga(X10, X12)) -> U176_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_in_gga(X12, X11, X7)) U176_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_out_gga(X12, X11, X7)) -> checkcX_out_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8) difcK_in_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, .(X8, X9), .(X10, X11)) -> U145_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, difcJ_in_aaaaaaaa(X2, X3, X4, X5, X6, X7, X9, X11)) U145_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, difcJ_out_aaaaaaaa(X2, X3, X4, X5, X6, X7, X9, X11)) -> difcK_out_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, .(X8, X9), .(X10, X11)) checkcY_in_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9) -> U177_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, difcK_in_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X9, X10, X11)) U177_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, difcK_out_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X9, X10, X11)) -> U178_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, mergedeletecC_in_gga(X10, X7, X12)) U178_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, mergedeletecC_out_gga(X10, X7, X12)) -> U179_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, revcF_in_ga(X11, X13)) U179_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, revcF_out_ga(X11, X13)) -> U180_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, mergedeletecC_in_gga(X13, X12, X8)) U180_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, mergedeletecC_out_gga(X13, X12, X8)) -> checkcY_out_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9) difcZ_in_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, .(X9, X10), .(X11, X12)) -> U185_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, difcK_in_aaaaaaaaa(X2, X3, X4, X5, X6, X7, X8, X10, X12)) U185_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, difcK_out_aaaaaaaaa(X2, X3, X4, X5, X6, X7, X8, X10, X12)) -> difcZ_out_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, .(X9, X10), .(X11, X12)) checkcP_in_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10) -> U181_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, difcZ_in_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X10, X11, X12)) U181_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, difcZ_out_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X10, X11, X12)) -> U182_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, mergedeletecC_in_gga(X11, X8, X13)) U182_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, mergedeletecC_out_gga(X11, X8, X13)) -> U183_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, revcF_in_ga(X12, X14)) U183_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, revcF_out_ga(X12, X14)) -> U184_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, mergedeletecC_in_gga(X14, X13, X9)) U184_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, mergedeletecC_out_gga(X14, X13, X9)) -> checkcP_out_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10) difcN1_in_agaaaa(X1, [], X2, X3, .(X4, []), .(X5, [])) -> difcN1_out_agaaaa(X1, [], X2, X3, .(X4, []), .(X5, [])) difcN1_in_agaaaa(X1, .(X2, []), X3, X4, .(X5, .(X6, [])), .(X7, .(X8, []))) -> difcN1_out_agaaaa(X1, .(X2, []), X3, X4, .(X5, .(X6, [])), .(X7, .(X8, []))) difcN1_in_agaaaa(X1, .(X2, .(X3, [])), X4, X5, .(X6, .(X7, .(X8, []))), .(X9, .(X10, .(X11, [])))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, [])), X4, X5, .(X6, .(X7, .(X8, []))), .(X9, .(X10, .(X11, [])))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, []))), X5, X6, .(X7, .(X8, .(X9, .(X10, [])))), .(X11, .(X12, .(X13, .(X14, []))))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, []))), X5, X6, .(X7, .(X8, .(X9, .(X10, [])))), .(X11, .(X12, .(X13, .(X14, []))))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, [])))), X6, X7, .(X8, .(X9, .(X10, .(X11, .(X12, []))))), .(X13, .(X14, .(X15, .(X16, .(X17, [])))))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, [])))), X6, X7, .(X8, .(X9, .(X10, .(X11, .(X12, []))))), .(X13, .(X14, .(X15, .(X16, .(X17, [])))))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, []))))), X7, X8, .(X9, .(X10, .(X11, .(X12, .(X13, .(X14, [])))))), .(X15, .(X16, .(X17, .(X18, .(X19, .(X20, []))))))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, []))))), X7, X8, .(X9, .(X10, .(X11, .(X12, .(X13, .(X14, [])))))), .(X15, .(X16, .(X17, .(X18, .(X19, .(X20, []))))))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, [])))))), X8, X9, .(X10, .(X11, .(X12, .(X13, .(X14, .(X15, .(X16, []))))))), .(X17, .(X18, .(X19, .(X20, .(X21, .(X22, .(X23, [])))))))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, [])))))), X8, X9, .(X10, .(X11, .(X12, .(X13, .(X14, .(X15, .(X16, []))))))), .(X17, .(X18, .(X19, .(X20, .(X21, .(X22, .(X23, [])))))))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9))))))), X10, X11, .(X12, .(X13, .(X14, .(X15, .(X16, .(X17, .(X18, .(X19, X20)))))))), .(X21, .(X22, .(X23, .(X24, .(X25, .(X26, .(X27, .(X28, X29))))))))) -> U190_agaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, X21, X22, X23, X24, X25, X26, X27, X28, X29, difcN_in_gaaaa(X9, X10, X11, X20, X29)) difcN_in_gaaaa([], X1, X2, [], []) -> difcN_out_gaaaa([], X1, X2, [], []) difcN_in_gaaaa(.(X1, X2), X3, X4, .(X5, X6), .(X7, X8)) -> U149_gaaaa(X1, X2, X3, X4, X5, X6, X7, X8, difcN_in_gaaaa(X2, X3, X4, X6, X8)) U149_gaaaa(X1, X2, X3, X4, X5, X6, X7, X8, difcN_out_gaaaa(X2, X3, X4, X6, X8)) -> difcN_out_gaaaa(.(X1, X2), X3, X4, .(X5, X6), .(X7, X8)) U190_agaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, X21, X22, X23, X24, X25, X26, X27, X28, X29, difcN_out_gaaaa(X9, X10, X11, X20, X29)) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9))))))), X10, X11, .(X12, .(X13, .(X14, .(X15, .(X16, .(X17, .(X18, .(X19, X20)))))))), .(X21, .(X22, .(X23, .(X24, .(X25, .(X26, .(X27, .(X28, X29))))))))) checkcM_in_agagaa(X1, X2, X3, X4, X5, X6) -> U186_agagaa(X1, X2, X3, X4, X5, X6, difcN1_in_agaaaa(X1, X2, X3, X6, X7, X8)) U186_agagaa(X1, X2, X3, X4, X5, X6, difcN1_out_agaaaa(X1, X2, X3, X6, X7, X8)) -> U187_agagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_in_gga(X7, X4, X9)) U187_agagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_out_gga(X7, X4, X9)) -> U188_agagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, revcF_in_ga(X8, X10)) U188_agagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, revcF_out_ga(X8, X10)) -> U189_agagaa(X1, X2, X3, X4, X5, X6, mergedeletecC_in_gga(X10, X9, X5)) U189_agagaa(X1, X2, X3, X4, X5, X6, mergedeletecC_out_gga(X10, X9, X5)) -> checkcM_out_agagaa(X1, X2, X3, X4, X5, X6) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) iota1cQ_in_aa(x1, x2) = iota1cQ_in_aa U153_aa(x1, x2, x3) = U153_aa(x3) iota1cA_in_aaa(x1, x2, x3) = iota1cA_in_aaa iota1cA_out_aaa(x1, x2, x3) = iota1cA_out_aaa(x3) U135_aaa(x1, x2, x3, x4) = U135_aaa(x4) iota1cQ_out_aa(x1, x2) = iota1cQ_out_aa(x2) membercR_in_ag(x1, x2) = membercR_in_ag(x2) membercR_out_ag(x1, x2) = membercR_out_ag(x2) U154_ag(x1, x2, x3, x4) = U154_ag(x3, x4) membercB_in_ag(x1, x2) = membercB_in_ag(x2) membercB_out_ag(x1, x2) = membercB_out_ag(x2) U136_ag(x1, x2, x3, x4) = U136_ag(x3, x4) difcE_in_aaaa(x1, x2, x3, x4) = difcE_in_aaaa difcE_out_aaaa(x1, x2, x3, x4) = difcE_out_aaaa(x3, x4) mergedeletecT_in_gga(x1, x2, x3) = mergedeletecT_in_gga(x1, x2) [] = [] mergedeletecT_out_gga(x1, x2, x3) = mergedeletecT_out_gga(x1, x2, x3) U159_gga(x1, x2, x3, x4, x5) = U159_gga(x2, x3, x5) mergedeletecC_in_gga(x1, x2, x3) = mergedeletecC_in_gga(x1, x2) mergedeletecC_out_gga(x1, x2, x3) = mergedeletecC_out_gga(x1, x2, x3) U137_gga(x1, x2, x3, x4, x5) = U137_gga(x2, x3, x5) U138_gga(x1, x2, x3, x4, x5, x6) = U138_gga(x2, x4, x6) U160_gga(x1, x2, x3, x4, x5, x6) = U160_gga(x2, x4, x6) revcF_in_ga(x1, x2) = revcF_in_ga(x1) revcF_out_ga(x1, x2) = revcF_out_ga(x1, x2) U140_ga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U140_ga(x9, x11) revcD_in_gaga(x1, x2, x3, x4) = revcD_in_gaga(x1, x3) revcD_out_gaga(x1, x2, x3, x4) = revcD_out_gaga(x1, x3, x4) U139_gaga(x1, x2, x3, x4, x5, x6) = U139_gaga(x2, x4, x6) checkcS_in_agaa(x1, x2, x3, x4) = checkcS_in_agaa(x2) U155_agaa(x1, x2, x3, x4, x5) = U155_agaa(x2, x5) U156_agaa(x1, x2, x3, x4, x5, x6, x7) = U156_agaa(x2, x6, x7) U157_agaa(x1, x2, x3, x4, x5, x6, x7, x8) = U157_agaa(x2, x7, x8) U158_agaa(x1, x2, x3, x4, x5) = U158_agaa(x2, x5) checkcS_out_agaa(x1, x2, x3, x4) = checkcS_out_agaa(x2, x3) difcG_in_aaaaa(x1, x2, x3, x4, x5) = difcG_in_aaaaa U141_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U141_aaaaa(x8) difcG_out_aaaaa(x1, x2, x3, x4, x5) = difcG_out_aaaaa(x4, x5) checkcU_in_aagaa(x1, x2, x3, x4, x5) = checkcU_in_aagaa(x3) U161_aagaa(x1, x2, x3, x4, x5, x6) = U161_aagaa(x3, x6) U162_aagaa(x1, x2, x3, x4, x5, x6, x7, x8) = U162_aagaa(x3, x7, x8) U163_aagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U163_aagaa(x3, x8, x9) U164_aagaa(x1, x2, x3, x4, x5, x6) = U164_aagaa(x3, x6) checkcU_out_aagaa(x1, x2, x3, x4, x5) = checkcU_out_aagaa(x3, x4) difcH_in_aaaaaa(x1, x2, x3, x4, x5, x6) = difcH_in_aaaaaa U142_aaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U142_aaaaaa(x9) difcH_out_aaaaaa(x1, x2, x3, x4, x5, x6) = difcH_out_aaaaaa(x5, x6) checkcV_in_aaagaa(x1, x2, x3, x4, x5, x6) = checkcV_in_aaagaa(x4) U165_aaagaa(x1, x2, x3, x4, x5, x6, x7) = U165_aaagaa(x4, x7) U166_aaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U166_aaagaa(x4, x8, x9) U167_aaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U167_aaagaa(x4, x9, x10) U168_aaagaa(x1, x2, x3, x4, x5, x6, x7) = U168_aaagaa(x4, x7) checkcV_out_aaagaa(x1, x2, x3, x4, x5, x6) = checkcV_out_aaagaa(x4, x5) difcI_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = difcI_in_aaaaaaa U143_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U143_aaaaaaa(x10) difcI_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = difcI_out_aaaaaaa(x6, x7) checkcW_in_aaaagaa(x1, x2, x3, x4, x5, x6, x7) = checkcW_in_aaaagaa(x5) U169_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8) = U169_aaaagaa(x5, x8) U170_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U170_aaaagaa(x5, x9, x10) U171_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U171_aaaagaa(x5, x10, x11) U172_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8) = U172_aaaagaa(x5, x8) checkcW_out_aaaagaa(x1, x2, x3, x4, x5, x6, x7) = checkcW_out_aaaagaa(x5, x6) difcJ_in_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = difcJ_in_aaaaaaaa U144_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U144_aaaaaaaa(x11) difcJ_out_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = difcJ_out_aaaaaaaa(x7, x8) checkcX_in_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8) = checkcX_in_aaaaagaa(x6) U173_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U173_aaaaagaa(x6, x9) U174_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U174_aaaaagaa(x6, x10, x11) U175_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U175_aaaaagaa(x6, x11, x12) U176_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U176_aaaaagaa(x6, x9) checkcX_out_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8) = checkcX_out_aaaaagaa(x6, x7) difcK_in_aaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = difcK_in_aaaaaaaaa U145_aaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U145_aaaaaaaaa(x12) difcK_out_aaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = difcK_out_aaaaaaaaa(x8, x9) checkcY_in_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = checkcY_in_aaaaaagaa(x7) U177_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U177_aaaaaagaa(x7, x10) U178_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U178_aaaaaagaa(x7, x11, x12) U179_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U179_aaaaaagaa(x7, x12, x13) U180_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U180_aaaaaagaa(x7, x10) checkcY_out_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = checkcY_out_aaaaaagaa(x7, x8) difcZ_in_aaaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = difcZ_in_aaaaaaaaaa U185_aaaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U185_aaaaaaaaaa(x13) difcZ_out_aaaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = difcZ_out_aaaaaaaaaa(x9, x10) checkcP_in_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = checkcP_in_aaaaaaagaa(x8) U181_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U181_aaaaaaagaa(x8, x11) U182_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U182_aaaaaaagaa(x8, x12, x13) U183_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14) = U183_aaaaaaagaa(x8, x13, x14) U184_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U184_aaaaaaagaa(x8, x11) checkcP_out_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = checkcP_out_aaaaaaagaa(x8, x9) difcN1_in_agaaaa(x1, x2, x3, x4, x5, x6) = difcN1_in_agaaaa(x2) difcN1_out_agaaaa(x1, x2, x3, x4, x5, x6) = difcN1_out_agaaaa(x2, x5, x6) U190_agaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30) = U190_agaaaa(x9, x30) difcN_in_gaaaa(x1, x2, x3, x4, x5) = difcN_in_gaaaa(x1) difcN_out_gaaaa(x1, x2, x3, x4, x5) = difcN_out_gaaaa(x1, x4, x5) U149_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U149_gaaaa(x2, x9) checkcM_in_agagaa(x1, x2, x3, x4, x5, x6) = checkcM_in_agagaa(x2, x4) U186_agagaa(x1, x2, x3, x4, x5, x6, x7) = U186_agagaa(x2, x4, x7) U187_agagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U187_agagaa(x2, x4, x8, x9) U188_agagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U188_agagaa(x2, x4, x9, x10) U189_agagaa(x1, x2, x3, x4, x5, x6, x7) = U189_agagaa(x2, x4, x7) checkcM_out_agagaa(x1, x2, x3, x4, x5, x6) = checkcM_out_agagaa(x2, x4, x5) REVD_IN_GAGA(x1, x2, x3, x4) = REVD_IN_GAGA(x1, x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (359) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (360) Obligation: Pi DP problem: The TRS P consists of the following rules: REVD_IN_GAGA(.(X1, X2), X3, X4, X5) -> REVD_IN_GAGA(X2, X1, .(X3, X4), X5) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) REVD_IN_GAGA(x1, x2, x3, x4) = REVD_IN_GAGA(x1, x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (361) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (362) Obligation: Q DP problem: The TRS P consists of the following rules: REVD_IN_GAGA(.(X2), X4) -> REVD_IN_GAGA(X2, .(X4)) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (363) 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: *REVD_IN_GAGA(.(X2), X4) -> REVD_IN_GAGA(X2, .(X4)) The graph contains the following edges 1 > 1 ---------------------------------------- (364) YES ---------------------------------------- (365) Obligation: Pi DP problem: The TRS P consists of the following rules: MERGEDELETEC_IN_GGA(.(X1, X2), .(X3, X4), .(X3, X5)) -> MERGEDELETEC_IN_GGA(.(X1, X2), X4, X5) MERGEDELETEC_IN_GGA(.(X1, X2), .(X1, X3), X4) -> MERGEDELETEC_IN_GGA(X2, X3, X4) The TRS R consists of the following rules: iota1cQ_in_aa(X1, X2) -> U153_aa(X1, X2, iota1cA_in_aaa(X3, X1, X2)) iota1cA_in_aaa(X1, X1, []) -> iota1cA_out_aaa(X1, X1, []) iota1cA_in_aaa(X1, X2, .(X1, X3)) -> U135_aaa(X1, X2, X3, iota1cA_in_aaa(X4, X2, X3)) U135_aaa(X1, X2, X3, iota1cA_out_aaa(X4, X2, X3)) -> iota1cA_out_aaa(X1, X2, .(X1, X3)) U153_aa(X1, X2, iota1cA_out_aaa(X3, X1, X2)) -> iota1cQ_out_aa(X1, X2) membercR_in_ag(X1, .(X1, X2)) -> membercR_out_ag(X1, .(X1, X2)) membercR_in_ag(X1, .(X2, X3)) -> U154_ag(X1, X2, X3, membercB_in_ag(X1, X3)) membercB_in_ag(X1, .(X1, X2)) -> membercB_out_ag(X1, .(X1, X2)) membercB_in_ag(X1, .(X2, X3)) -> U136_ag(X1, X2, X3, membercB_in_ag(X1, X3)) U136_ag(X1, X2, X3, membercB_out_ag(X1, X3)) -> membercB_out_ag(X1, .(X2, X3)) U154_ag(X1, X2, X3, membercB_out_ag(X1, X3)) -> membercR_out_ag(X1, .(X2, X3)) difcE_in_aaaa(X1, X2, .(X3, []), .(X4, [])) -> difcE_out_aaaa(X1, X2, .(X3, []), .(X4, [])) mergedeletecT_in_gga([], X1, X1) -> mergedeletecT_out_gga([], X1, X1) mergedeletecT_in_gga(.(X1, X2), .(X1, X3), X4) -> U159_gga(X1, X2, X3, X4, mergedeletecC_in_gga(X2, X3, X4)) mergedeletecC_in_gga([], X1, X1) -> mergedeletecC_out_gga([], X1, X1) mergedeletecC_in_gga(.(X1, X2), .(X1, X3), X4) -> U137_gga(X1, X2, X3, X4, mergedeletecC_in_gga(X2, X3, X4)) mergedeletecC_in_gga(.(X1, X2), .(X3, X4), .(X3, X5)) -> U138_gga(X1, X2, X3, X4, X5, mergedeletecC_in_gga(.(X1, X2), X4, X5)) U138_gga(X1, X2, X3, X4, X5, mergedeletecC_out_gga(.(X1, X2), X4, X5)) -> mergedeletecC_out_gga(.(X1, X2), .(X3, X4), .(X3, X5)) U137_gga(X1, X2, X3, X4, mergedeletecC_out_gga(X2, X3, X4)) -> mergedeletecC_out_gga(.(X1, X2), .(X1, X3), X4) U159_gga(X1, X2, X3, X4, mergedeletecC_out_gga(X2, X3, X4)) -> mergedeletecT_out_gga(.(X1, X2), .(X1, X3), X4) mergedeletecT_in_gga(.(X1, X2), .(X3, X4), .(X3, X5)) -> U160_gga(X1, X2, X3, X4, X5, mergedeletecC_in_gga(.(X1, X2), X4, X5)) U160_gga(X1, X2, X3, X4, X5, mergedeletecC_out_gga(.(X1, X2), X4, X5)) -> mergedeletecT_out_gga(.(X1, X2), .(X3, X4), .(X3, X5)) revcF_in_ga([], []) -> revcF_out_ga([], []) revcF_in_ga(.(X1, []), .(X1, [])) -> revcF_out_ga(.(X1, []), .(X1, [])) revcF_in_ga(.(X1, .(X2, [])), .(X2, .(X1, []))) -> revcF_out_ga(.(X1, .(X2, [])), .(X2, .(X1, []))) revcF_in_ga(.(X1, .(X2, .(X3, []))), .(X3, .(X2, .(X1, [])))) -> revcF_out_ga(.(X1, .(X2, .(X3, []))), .(X3, .(X2, .(X1, [])))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, [])))), .(X4, .(X3, .(X2, .(X1, []))))) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, [])))), .(X4, .(X3, .(X2, .(X1, []))))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, .(X5, []))))), .(X5, .(X4, .(X3, .(X2, .(X1, [])))))) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, .(X5, []))))), .(X5, .(X4, .(X3, .(X2, .(X1, [])))))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, [])))))), .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, [])))))), .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, []))))))), .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, [])))))))) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, []))))))), .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, [])))))))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) -> U140_ga(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revcD_in_gaga(X9, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))), X10)) revcD_in_gaga([], X1, X2, .(X1, X2)) -> revcD_out_gaga([], X1, X2, .(X1, X2)) revcD_in_gaga(.(X1, X2), X3, X4, X5) -> U139_gaga(X1, X2, X3, X4, X5, revcD_in_gaga(X2, X1, .(X3, X4), X5)) U139_gaga(X1, X2, X3, X4, X5, revcD_out_gaga(X2, X1, .(X3, X4), X5)) -> revcD_out_gaga(.(X1, X2), X3, X4, X5) U140_ga(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revcD_out_gaga(X9, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))), X10)) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) checkcS_in_agaa(X1, X2, X3, X4) -> U155_agaa(X1, X2, X3, X4, difcE_in_aaaa(X1, X4, X5, X6)) U155_agaa(X1, X2, X3, X4, difcE_out_aaaa(X1, X4, X5, X6)) -> U156_agaa(X1, X2, X3, X4, X5, X6, mergedeletecT_in_gga(X5, X2, X7)) U156_agaa(X1, X2, X3, X4, X5, X6, mergedeletecT_out_gga(X5, X2, X7)) -> U157_agaa(X1, X2, X3, X4, X5, X6, X7, revcF_in_ga(X6, X8)) U157_agaa(X1, X2, X3, X4, X5, X6, X7, revcF_out_ga(X6, X8)) -> U158_agaa(X1, X2, X3, X4, mergedeletecC_in_gga(X8, X7, X3)) U158_agaa(X1, X2, X3, X4, mergedeletecC_out_gga(X8, X7, X3)) -> checkcS_out_agaa(X1, X2, X3, X4) difcG_in_aaaaa(X1, X2, X3, .(X4, X5), .(X6, X7)) -> U141_aaaaa(X1, X2, X3, X4, X5, X6, X7, difcE_in_aaaa(X2, X3, X5, X7)) U141_aaaaa(X1, X2, X3, X4, X5, X6, X7, difcE_out_aaaa(X2, X3, X5, X7)) -> difcG_out_aaaaa(X1, X2, X3, .(X4, X5), .(X6, X7)) checkcU_in_aagaa(X1, X2, X3, X4, X5) -> U161_aagaa(X1, X2, X3, X4, X5, difcG_in_aaaaa(X1, X2, X5, X6, X7)) U161_aagaa(X1, X2, X3, X4, X5, difcG_out_aaaaa(X1, X2, X5, X6, X7)) -> U162_aagaa(X1, X2, X3, X4, X5, X6, X7, mergedeletecC_in_gga(X6, X3, X8)) U162_aagaa(X1, X2, X3, X4, X5, X6, X7, mergedeletecC_out_gga(X6, X3, X8)) -> U163_aagaa(X1, X2, X3, X4, X5, X6, X7, X8, revcF_in_ga(X7, X9)) U163_aagaa(X1, X2, X3, X4, X5, X6, X7, X8, revcF_out_ga(X7, X9)) -> U164_aagaa(X1, X2, X3, X4, X5, mergedeletecC_in_gga(X9, X8, X4)) U164_aagaa(X1, X2, X3, X4, X5, mergedeletecC_out_gga(X9, X8, X4)) -> checkcU_out_aagaa(X1, X2, X3, X4, X5) difcH_in_aaaaaa(X1, X2, X3, X4, .(X5, X6), .(X7, X8)) -> U142_aaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, difcG_in_aaaaa(X2, X3, X4, X6, X8)) U142_aaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, difcG_out_aaaaa(X2, X3, X4, X6, X8)) -> difcH_out_aaaaaa(X1, X2, X3, X4, .(X5, X6), .(X7, X8)) checkcV_in_aaagaa(X1, X2, X3, X4, X5, X6) -> U165_aaagaa(X1, X2, X3, X4, X5, X6, difcH_in_aaaaaa(X1, X2, X3, X6, X7, X8)) U165_aaagaa(X1, X2, X3, X4, X5, X6, difcH_out_aaaaaa(X1, X2, X3, X6, X7, X8)) -> U166_aaagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_in_gga(X7, X4, X9)) U166_aaagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_out_gga(X7, X4, X9)) -> U167_aaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, revcF_in_ga(X8, X10)) U167_aaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, revcF_out_ga(X8, X10)) -> U168_aaagaa(X1, X2, X3, X4, X5, X6, mergedeletecC_in_gga(X10, X9, X5)) U168_aaagaa(X1, X2, X3, X4, X5, X6, mergedeletecC_out_gga(X10, X9, X5)) -> checkcV_out_aaagaa(X1, X2, X3, X4, X5, X6) difcI_in_aaaaaaa(X1, X2, X3, X4, X5, .(X6, X7), .(X8, X9)) -> U143_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, difcH_in_aaaaaa(X2, X3, X4, X5, X7, X9)) U143_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, difcH_out_aaaaaa(X2, X3, X4, X5, X7, X9)) -> difcI_out_aaaaaaa(X1, X2, X3, X4, X5, .(X6, X7), .(X8, X9)) checkcW_in_aaaagaa(X1, X2, X3, X4, X5, X6, X7) -> U169_aaaagaa(X1, X2, X3, X4, X5, X6, X7, difcI_in_aaaaaaa(X1, X2, X3, X4, X7, X8, X9)) U169_aaaagaa(X1, X2, X3, X4, X5, X6, X7, difcI_out_aaaaaaa(X1, X2, X3, X4, X7, X8, X9)) -> U170_aaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, mergedeletecC_in_gga(X8, X5, X10)) U170_aaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, mergedeletecC_out_gga(X8, X5, X10)) -> U171_aaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revcF_in_ga(X9, X11)) U171_aaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revcF_out_ga(X9, X11)) -> U172_aaaagaa(X1, X2, X3, X4, X5, X6, X7, mergedeletecC_in_gga(X11, X10, X6)) U172_aaaagaa(X1, X2, X3, X4, X5, X6, X7, mergedeletecC_out_gga(X11, X10, X6)) -> checkcW_out_aaaagaa(X1, X2, X3, X4, X5, X6, X7) difcJ_in_aaaaaaaa(X1, X2, X3, X4, X5, X6, .(X7, X8), .(X9, X10)) -> U144_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, difcI_in_aaaaaaa(X2, X3, X4, X5, X6, X8, X10)) U144_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, difcI_out_aaaaaaa(X2, X3, X4, X5, X6, X8, X10)) -> difcJ_out_aaaaaaaa(X1, X2, X3, X4, X5, X6, .(X7, X8), .(X9, X10)) checkcX_in_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8) -> U173_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, difcJ_in_aaaaaaaa(X1, X2, X3, X4, X5, X8, X9, X10)) U173_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, difcJ_out_aaaaaaaa(X1, X2, X3, X4, X5, X8, X9, X10)) -> U174_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, mergedeletecC_in_gga(X9, X6, X11)) U174_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, mergedeletecC_out_gga(X9, X6, X11)) -> U175_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, revcF_in_ga(X10, X12)) U175_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, revcF_out_ga(X10, X12)) -> U176_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_in_gga(X12, X11, X7)) U176_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_out_gga(X12, X11, X7)) -> checkcX_out_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8) difcK_in_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, .(X8, X9), .(X10, X11)) -> U145_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, difcJ_in_aaaaaaaa(X2, X3, X4, X5, X6, X7, X9, X11)) U145_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, difcJ_out_aaaaaaaa(X2, X3, X4, X5, X6, X7, X9, X11)) -> difcK_out_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, .(X8, X9), .(X10, X11)) checkcY_in_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9) -> U177_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, difcK_in_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X9, X10, X11)) U177_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, difcK_out_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X9, X10, X11)) -> U178_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, mergedeletecC_in_gga(X10, X7, X12)) U178_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, mergedeletecC_out_gga(X10, X7, X12)) -> U179_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, revcF_in_ga(X11, X13)) U179_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, revcF_out_ga(X11, X13)) -> U180_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, mergedeletecC_in_gga(X13, X12, X8)) U180_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, mergedeletecC_out_gga(X13, X12, X8)) -> checkcY_out_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9) difcZ_in_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, .(X9, X10), .(X11, X12)) -> U185_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, difcK_in_aaaaaaaaa(X2, X3, X4, X5, X6, X7, X8, X10, X12)) U185_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, difcK_out_aaaaaaaaa(X2, X3, X4, X5, X6, X7, X8, X10, X12)) -> difcZ_out_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, .(X9, X10), .(X11, X12)) checkcP_in_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10) -> U181_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, difcZ_in_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X10, X11, X12)) U181_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, difcZ_out_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X10, X11, X12)) -> U182_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, mergedeletecC_in_gga(X11, X8, X13)) U182_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, mergedeletecC_out_gga(X11, X8, X13)) -> U183_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, revcF_in_ga(X12, X14)) U183_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, revcF_out_ga(X12, X14)) -> U184_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, mergedeletecC_in_gga(X14, X13, X9)) U184_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, mergedeletecC_out_gga(X14, X13, X9)) -> checkcP_out_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10) difcN1_in_agaaaa(X1, [], X2, X3, .(X4, []), .(X5, [])) -> difcN1_out_agaaaa(X1, [], X2, X3, .(X4, []), .(X5, [])) difcN1_in_agaaaa(X1, .(X2, []), X3, X4, .(X5, .(X6, [])), .(X7, .(X8, []))) -> difcN1_out_agaaaa(X1, .(X2, []), X3, X4, .(X5, .(X6, [])), .(X7, .(X8, []))) difcN1_in_agaaaa(X1, .(X2, .(X3, [])), X4, X5, .(X6, .(X7, .(X8, []))), .(X9, .(X10, .(X11, [])))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, [])), X4, X5, .(X6, .(X7, .(X8, []))), .(X9, .(X10, .(X11, [])))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, []))), X5, X6, .(X7, .(X8, .(X9, .(X10, [])))), .(X11, .(X12, .(X13, .(X14, []))))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, []))), X5, X6, .(X7, .(X8, .(X9, .(X10, [])))), .(X11, .(X12, .(X13, .(X14, []))))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, [])))), X6, X7, .(X8, .(X9, .(X10, .(X11, .(X12, []))))), .(X13, .(X14, .(X15, .(X16, .(X17, [])))))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, [])))), X6, X7, .(X8, .(X9, .(X10, .(X11, .(X12, []))))), .(X13, .(X14, .(X15, .(X16, .(X17, [])))))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, []))))), X7, X8, .(X9, .(X10, .(X11, .(X12, .(X13, .(X14, [])))))), .(X15, .(X16, .(X17, .(X18, .(X19, .(X20, []))))))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, []))))), X7, X8, .(X9, .(X10, .(X11, .(X12, .(X13, .(X14, [])))))), .(X15, .(X16, .(X17, .(X18, .(X19, .(X20, []))))))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, [])))))), X8, X9, .(X10, .(X11, .(X12, .(X13, .(X14, .(X15, .(X16, []))))))), .(X17, .(X18, .(X19, .(X20, .(X21, .(X22, .(X23, [])))))))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, [])))))), X8, X9, .(X10, .(X11, .(X12, .(X13, .(X14, .(X15, .(X16, []))))))), .(X17, .(X18, .(X19, .(X20, .(X21, .(X22, .(X23, [])))))))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9))))))), X10, X11, .(X12, .(X13, .(X14, .(X15, .(X16, .(X17, .(X18, .(X19, X20)))))))), .(X21, .(X22, .(X23, .(X24, .(X25, .(X26, .(X27, .(X28, X29))))))))) -> U190_agaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, X21, X22, X23, X24, X25, X26, X27, X28, X29, difcN_in_gaaaa(X9, X10, X11, X20, X29)) difcN_in_gaaaa([], X1, X2, [], []) -> difcN_out_gaaaa([], X1, X2, [], []) difcN_in_gaaaa(.(X1, X2), X3, X4, .(X5, X6), .(X7, X8)) -> U149_gaaaa(X1, X2, X3, X4, X5, X6, X7, X8, difcN_in_gaaaa(X2, X3, X4, X6, X8)) U149_gaaaa(X1, X2, X3, X4, X5, X6, X7, X8, difcN_out_gaaaa(X2, X3, X4, X6, X8)) -> difcN_out_gaaaa(.(X1, X2), X3, X4, .(X5, X6), .(X7, X8)) U190_agaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, X21, X22, X23, X24, X25, X26, X27, X28, X29, difcN_out_gaaaa(X9, X10, X11, X20, X29)) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9))))))), X10, X11, .(X12, .(X13, .(X14, .(X15, .(X16, .(X17, .(X18, .(X19, X20)))))))), .(X21, .(X22, .(X23, .(X24, .(X25, .(X26, .(X27, .(X28, X29))))))))) checkcM_in_agagaa(X1, X2, X3, X4, X5, X6) -> U186_agagaa(X1, X2, X3, X4, X5, X6, difcN1_in_agaaaa(X1, X2, X3, X6, X7, X8)) U186_agagaa(X1, X2, X3, X4, X5, X6, difcN1_out_agaaaa(X1, X2, X3, X6, X7, X8)) -> U187_agagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_in_gga(X7, X4, X9)) U187_agagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_out_gga(X7, X4, X9)) -> U188_agagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, revcF_in_ga(X8, X10)) U188_agagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, revcF_out_ga(X8, X10)) -> U189_agagaa(X1, X2, X3, X4, X5, X6, mergedeletecC_in_gga(X10, X9, X5)) U189_agagaa(X1, X2, X3, X4, X5, X6, mergedeletecC_out_gga(X10, X9, X5)) -> checkcM_out_agagaa(X1, X2, X3, X4, X5, X6) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) iota1cQ_in_aa(x1, x2) = iota1cQ_in_aa U153_aa(x1, x2, x3) = U153_aa(x3) iota1cA_in_aaa(x1, x2, x3) = iota1cA_in_aaa iota1cA_out_aaa(x1, x2, x3) = iota1cA_out_aaa(x3) U135_aaa(x1, x2, x3, x4) = U135_aaa(x4) iota1cQ_out_aa(x1, x2) = iota1cQ_out_aa(x2) membercR_in_ag(x1, x2) = membercR_in_ag(x2) membercR_out_ag(x1, x2) = membercR_out_ag(x2) U154_ag(x1, x2, x3, x4) = U154_ag(x3, x4) membercB_in_ag(x1, x2) = membercB_in_ag(x2) membercB_out_ag(x1, x2) = membercB_out_ag(x2) U136_ag(x1, x2, x3, x4) = U136_ag(x3, x4) difcE_in_aaaa(x1, x2, x3, x4) = difcE_in_aaaa difcE_out_aaaa(x1, x2, x3, x4) = difcE_out_aaaa(x3, x4) mergedeletecT_in_gga(x1, x2, x3) = mergedeletecT_in_gga(x1, x2) [] = [] mergedeletecT_out_gga(x1, x2, x3) = mergedeletecT_out_gga(x1, x2, x3) U159_gga(x1, x2, x3, x4, x5) = U159_gga(x2, x3, x5) mergedeletecC_in_gga(x1, x2, x3) = mergedeletecC_in_gga(x1, x2) mergedeletecC_out_gga(x1, x2, x3) = mergedeletecC_out_gga(x1, x2, x3) U137_gga(x1, x2, x3, x4, x5) = U137_gga(x2, x3, x5) U138_gga(x1, x2, x3, x4, x5, x6) = U138_gga(x2, x4, x6) U160_gga(x1, x2, x3, x4, x5, x6) = U160_gga(x2, x4, x6) revcF_in_ga(x1, x2) = revcF_in_ga(x1) revcF_out_ga(x1, x2) = revcF_out_ga(x1, x2) U140_ga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U140_ga(x9, x11) revcD_in_gaga(x1, x2, x3, x4) = revcD_in_gaga(x1, x3) revcD_out_gaga(x1, x2, x3, x4) = revcD_out_gaga(x1, x3, x4) U139_gaga(x1, x2, x3, x4, x5, x6) = U139_gaga(x2, x4, x6) checkcS_in_agaa(x1, x2, x3, x4) = checkcS_in_agaa(x2) U155_agaa(x1, x2, x3, x4, x5) = U155_agaa(x2, x5) U156_agaa(x1, x2, x3, x4, x5, x6, x7) = U156_agaa(x2, x6, x7) U157_agaa(x1, x2, x3, x4, x5, x6, x7, x8) = U157_agaa(x2, x7, x8) U158_agaa(x1, x2, x3, x4, x5) = U158_agaa(x2, x5) checkcS_out_agaa(x1, x2, x3, x4) = checkcS_out_agaa(x2, x3) difcG_in_aaaaa(x1, x2, x3, x4, x5) = difcG_in_aaaaa U141_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U141_aaaaa(x8) difcG_out_aaaaa(x1, x2, x3, x4, x5) = difcG_out_aaaaa(x4, x5) checkcU_in_aagaa(x1, x2, x3, x4, x5) = checkcU_in_aagaa(x3) U161_aagaa(x1, x2, x3, x4, x5, x6) = U161_aagaa(x3, x6) U162_aagaa(x1, x2, x3, x4, x5, x6, x7, x8) = U162_aagaa(x3, x7, x8) U163_aagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U163_aagaa(x3, x8, x9) U164_aagaa(x1, x2, x3, x4, x5, x6) = U164_aagaa(x3, x6) checkcU_out_aagaa(x1, x2, x3, x4, x5) = checkcU_out_aagaa(x3, x4) difcH_in_aaaaaa(x1, x2, x3, x4, x5, x6) = difcH_in_aaaaaa U142_aaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U142_aaaaaa(x9) difcH_out_aaaaaa(x1, x2, x3, x4, x5, x6) = difcH_out_aaaaaa(x5, x6) checkcV_in_aaagaa(x1, x2, x3, x4, x5, x6) = checkcV_in_aaagaa(x4) U165_aaagaa(x1, x2, x3, x4, x5, x6, x7) = U165_aaagaa(x4, x7) U166_aaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U166_aaagaa(x4, x8, x9) U167_aaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U167_aaagaa(x4, x9, x10) U168_aaagaa(x1, x2, x3, x4, x5, x6, x7) = U168_aaagaa(x4, x7) checkcV_out_aaagaa(x1, x2, x3, x4, x5, x6) = checkcV_out_aaagaa(x4, x5) difcI_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = difcI_in_aaaaaaa U143_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U143_aaaaaaa(x10) difcI_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = difcI_out_aaaaaaa(x6, x7) checkcW_in_aaaagaa(x1, x2, x3, x4, x5, x6, x7) = checkcW_in_aaaagaa(x5) U169_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8) = U169_aaaagaa(x5, x8) U170_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U170_aaaagaa(x5, x9, x10) U171_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U171_aaaagaa(x5, x10, x11) U172_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8) = U172_aaaagaa(x5, x8) checkcW_out_aaaagaa(x1, x2, x3, x4, x5, x6, x7) = checkcW_out_aaaagaa(x5, x6) difcJ_in_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = difcJ_in_aaaaaaaa U144_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U144_aaaaaaaa(x11) difcJ_out_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = difcJ_out_aaaaaaaa(x7, x8) checkcX_in_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8) = checkcX_in_aaaaagaa(x6) U173_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U173_aaaaagaa(x6, x9) U174_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U174_aaaaagaa(x6, x10, x11) U175_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U175_aaaaagaa(x6, x11, x12) U176_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U176_aaaaagaa(x6, x9) checkcX_out_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8) = checkcX_out_aaaaagaa(x6, x7) difcK_in_aaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = difcK_in_aaaaaaaaa U145_aaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U145_aaaaaaaaa(x12) difcK_out_aaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = difcK_out_aaaaaaaaa(x8, x9) checkcY_in_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = checkcY_in_aaaaaagaa(x7) U177_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U177_aaaaaagaa(x7, x10) U178_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U178_aaaaaagaa(x7, x11, x12) U179_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U179_aaaaaagaa(x7, x12, x13) U180_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U180_aaaaaagaa(x7, x10) checkcY_out_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = checkcY_out_aaaaaagaa(x7, x8) difcZ_in_aaaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = difcZ_in_aaaaaaaaaa U185_aaaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U185_aaaaaaaaaa(x13) difcZ_out_aaaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = difcZ_out_aaaaaaaaaa(x9, x10) checkcP_in_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = checkcP_in_aaaaaaagaa(x8) U181_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U181_aaaaaaagaa(x8, x11) U182_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U182_aaaaaaagaa(x8, x12, x13) U183_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14) = U183_aaaaaaagaa(x8, x13, x14) U184_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U184_aaaaaaagaa(x8, x11) checkcP_out_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = checkcP_out_aaaaaaagaa(x8, x9) difcN1_in_agaaaa(x1, x2, x3, x4, x5, x6) = difcN1_in_agaaaa(x2) difcN1_out_agaaaa(x1, x2, x3, x4, x5, x6) = difcN1_out_agaaaa(x2, x5, x6) U190_agaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30) = U190_agaaaa(x9, x30) difcN_in_gaaaa(x1, x2, x3, x4, x5) = difcN_in_gaaaa(x1) difcN_out_gaaaa(x1, x2, x3, x4, x5) = difcN_out_gaaaa(x1, x4, x5) U149_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U149_gaaaa(x2, x9) checkcM_in_agagaa(x1, x2, x3, x4, x5, x6) = checkcM_in_agagaa(x2, x4) U186_agagaa(x1, x2, x3, x4, x5, x6, x7) = U186_agagaa(x2, x4, x7) U187_agagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U187_agagaa(x2, x4, x8, x9) U188_agagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U188_agagaa(x2, x4, x9, x10) U189_agagaa(x1, x2, x3, x4, x5, x6, x7) = U189_agagaa(x2, x4, x7) checkcM_out_agagaa(x1, x2, x3, x4, x5, x6) = checkcM_out_agagaa(x2, x4, x5) MERGEDELETEC_IN_GGA(x1, x2, x3) = MERGEDELETEC_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (366) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (367) Obligation: Pi DP problem: The TRS P consists of the following rules: MERGEDELETEC_IN_GGA(.(X1, X2), .(X3, X4), .(X3, X5)) -> MERGEDELETEC_IN_GGA(.(X1, X2), X4, X5) MERGEDELETEC_IN_GGA(.(X1, X2), .(X1, X3), X4) -> MERGEDELETEC_IN_GGA(X2, X3, X4) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) MERGEDELETEC_IN_GGA(x1, x2, x3) = MERGEDELETEC_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (368) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (369) Obligation: Q DP problem: The TRS P consists of the following rules: MERGEDELETEC_IN_GGA(.(X2), .(X4)) -> MERGEDELETEC_IN_GGA(.(X2), X4) MERGEDELETEC_IN_GGA(.(X2), .(X3)) -> MERGEDELETEC_IN_GGA(X2, X3) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (370) 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: *MERGEDELETEC_IN_GGA(.(X2), .(X4)) -> MERGEDELETEC_IN_GGA(.(X2), X4) The graph contains the following edges 1 >= 1, 2 > 2 *MERGEDELETEC_IN_GGA(.(X2), .(X3)) -> MERGEDELETEC_IN_GGA(X2, X3) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (371) YES ---------------------------------------- (372) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBERB_IN_AG(X1, .(X2, X3)) -> MEMBERB_IN_AG(X1, X3) The TRS R consists of the following rules: iota1cQ_in_aa(X1, X2) -> U153_aa(X1, X2, iota1cA_in_aaa(X3, X1, X2)) iota1cA_in_aaa(X1, X1, []) -> iota1cA_out_aaa(X1, X1, []) iota1cA_in_aaa(X1, X2, .(X1, X3)) -> U135_aaa(X1, X2, X3, iota1cA_in_aaa(X4, X2, X3)) U135_aaa(X1, X2, X3, iota1cA_out_aaa(X4, X2, X3)) -> iota1cA_out_aaa(X1, X2, .(X1, X3)) U153_aa(X1, X2, iota1cA_out_aaa(X3, X1, X2)) -> iota1cQ_out_aa(X1, X2) membercR_in_ag(X1, .(X1, X2)) -> membercR_out_ag(X1, .(X1, X2)) membercR_in_ag(X1, .(X2, X3)) -> U154_ag(X1, X2, X3, membercB_in_ag(X1, X3)) membercB_in_ag(X1, .(X1, X2)) -> membercB_out_ag(X1, .(X1, X2)) membercB_in_ag(X1, .(X2, X3)) -> U136_ag(X1, X2, X3, membercB_in_ag(X1, X3)) U136_ag(X1, X2, X3, membercB_out_ag(X1, X3)) -> membercB_out_ag(X1, .(X2, X3)) U154_ag(X1, X2, X3, membercB_out_ag(X1, X3)) -> membercR_out_ag(X1, .(X2, X3)) difcE_in_aaaa(X1, X2, .(X3, []), .(X4, [])) -> difcE_out_aaaa(X1, X2, .(X3, []), .(X4, [])) mergedeletecT_in_gga([], X1, X1) -> mergedeletecT_out_gga([], X1, X1) mergedeletecT_in_gga(.(X1, X2), .(X1, X3), X4) -> U159_gga(X1, X2, X3, X4, mergedeletecC_in_gga(X2, X3, X4)) mergedeletecC_in_gga([], X1, X1) -> mergedeletecC_out_gga([], X1, X1) mergedeletecC_in_gga(.(X1, X2), .(X1, X3), X4) -> U137_gga(X1, X2, X3, X4, mergedeletecC_in_gga(X2, X3, X4)) mergedeletecC_in_gga(.(X1, X2), .(X3, X4), .(X3, X5)) -> U138_gga(X1, X2, X3, X4, X5, mergedeletecC_in_gga(.(X1, X2), X4, X5)) U138_gga(X1, X2, X3, X4, X5, mergedeletecC_out_gga(.(X1, X2), X4, X5)) -> mergedeletecC_out_gga(.(X1, X2), .(X3, X4), .(X3, X5)) U137_gga(X1, X2, X3, X4, mergedeletecC_out_gga(X2, X3, X4)) -> mergedeletecC_out_gga(.(X1, X2), .(X1, X3), X4) U159_gga(X1, X2, X3, X4, mergedeletecC_out_gga(X2, X3, X4)) -> mergedeletecT_out_gga(.(X1, X2), .(X1, X3), X4) mergedeletecT_in_gga(.(X1, X2), .(X3, X4), .(X3, X5)) -> U160_gga(X1, X2, X3, X4, X5, mergedeletecC_in_gga(.(X1, X2), X4, X5)) U160_gga(X1, X2, X3, X4, X5, mergedeletecC_out_gga(.(X1, X2), X4, X5)) -> mergedeletecT_out_gga(.(X1, X2), .(X3, X4), .(X3, X5)) revcF_in_ga([], []) -> revcF_out_ga([], []) revcF_in_ga(.(X1, []), .(X1, [])) -> revcF_out_ga(.(X1, []), .(X1, [])) revcF_in_ga(.(X1, .(X2, [])), .(X2, .(X1, []))) -> revcF_out_ga(.(X1, .(X2, [])), .(X2, .(X1, []))) revcF_in_ga(.(X1, .(X2, .(X3, []))), .(X3, .(X2, .(X1, [])))) -> revcF_out_ga(.(X1, .(X2, .(X3, []))), .(X3, .(X2, .(X1, [])))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, [])))), .(X4, .(X3, .(X2, .(X1, []))))) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, [])))), .(X4, .(X3, .(X2, .(X1, []))))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, .(X5, []))))), .(X5, .(X4, .(X3, .(X2, .(X1, [])))))) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, .(X5, []))))), .(X5, .(X4, .(X3, .(X2, .(X1, [])))))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, [])))))), .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, [])))))), .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, []))))))), .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, [])))))))) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, []))))))), .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, [])))))))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) -> U140_ga(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revcD_in_gaga(X9, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))), X10)) revcD_in_gaga([], X1, X2, .(X1, X2)) -> revcD_out_gaga([], X1, X2, .(X1, X2)) revcD_in_gaga(.(X1, X2), X3, X4, X5) -> U139_gaga(X1, X2, X3, X4, X5, revcD_in_gaga(X2, X1, .(X3, X4), X5)) U139_gaga(X1, X2, X3, X4, X5, revcD_out_gaga(X2, X1, .(X3, X4), X5)) -> revcD_out_gaga(.(X1, X2), X3, X4, X5) U140_ga(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revcD_out_gaga(X9, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))), X10)) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) checkcS_in_agaa(X1, X2, X3, X4) -> U155_agaa(X1, X2, X3, X4, difcE_in_aaaa(X1, X4, X5, X6)) U155_agaa(X1, X2, X3, X4, difcE_out_aaaa(X1, X4, X5, X6)) -> U156_agaa(X1, X2, X3, X4, X5, X6, mergedeletecT_in_gga(X5, X2, X7)) U156_agaa(X1, X2, X3, X4, X5, X6, mergedeletecT_out_gga(X5, X2, X7)) -> U157_agaa(X1, X2, X3, X4, X5, X6, X7, revcF_in_ga(X6, X8)) U157_agaa(X1, X2, X3, X4, X5, X6, X7, revcF_out_ga(X6, X8)) -> U158_agaa(X1, X2, X3, X4, mergedeletecC_in_gga(X8, X7, X3)) U158_agaa(X1, X2, X3, X4, mergedeletecC_out_gga(X8, X7, X3)) -> checkcS_out_agaa(X1, X2, X3, X4) difcG_in_aaaaa(X1, X2, X3, .(X4, X5), .(X6, X7)) -> U141_aaaaa(X1, X2, X3, X4, X5, X6, X7, difcE_in_aaaa(X2, X3, X5, X7)) U141_aaaaa(X1, X2, X3, X4, X5, X6, X7, difcE_out_aaaa(X2, X3, X5, X7)) -> difcG_out_aaaaa(X1, X2, X3, .(X4, X5), .(X6, X7)) checkcU_in_aagaa(X1, X2, X3, X4, X5) -> U161_aagaa(X1, X2, X3, X4, X5, difcG_in_aaaaa(X1, X2, X5, X6, X7)) U161_aagaa(X1, X2, X3, X4, X5, difcG_out_aaaaa(X1, X2, X5, X6, X7)) -> U162_aagaa(X1, X2, X3, X4, X5, X6, X7, mergedeletecC_in_gga(X6, X3, X8)) U162_aagaa(X1, X2, X3, X4, X5, X6, X7, mergedeletecC_out_gga(X6, X3, X8)) -> U163_aagaa(X1, X2, X3, X4, X5, X6, X7, X8, revcF_in_ga(X7, X9)) U163_aagaa(X1, X2, X3, X4, X5, X6, X7, X8, revcF_out_ga(X7, X9)) -> U164_aagaa(X1, X2, X3, X4, X5, mergedeletecC_in_gga(X9, X8, X4)) U164_aagaa(X1, X2, X3, X4, X5, mergedeletecC_out_gga(X9, X8, X4)) -> checkcU_out_aagaa(X1, X2, X3, X4, X5) difcH_in_aaaaaa(X1, X2, X3, X4, .(X5, X6), .(X7, X8)) -> U142_aaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, difcG_in_aaaaa(X2, X3, X4, X6, X8)) U142_aaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, difcG_out_aaaaa(X2, X3, X4, X6, X8)) -> difcH_out_aaaaaa(X1, X2, X3, X4, .(X5, X6), .(X7, X8)) checkcV_in_aaagaa(X1, X2, X3, X4, X5, X6) -> U165_aaagaa(X1, X2, X3, X4, X5, X6, difcH_in_aaaaaa(X1, X2, X3, X6, X7, X8)) U165_aaagaa(X1, X2, X3, X4, X5, X6, difcH_out_aaaaaa(X1, X2, X3, X6, X7, X8)) -> U166_aaagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_in_gga(X7, X4, X9)) U166_aaagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_out_gga(X7, X4, X9)) -> U167_aaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, revcF_in_ga(X8, X10)) U167_aaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, revcF_out_ga(X8, X10)) -> U168_aaagaa(X1, X2, X3, X4, X5, X6, mergedeletecC_in_gga(X10, X9, X5)) U168_aaagaa(X1, X2, X3, X4, X5, X6, mergedeletecC_out_gga(X10, X9, X5)) -> checkcV_out_aaagaa(X1, X2, X3, X4, X5, X6) difcI_in_aaaaaaa(X1, X2, X3, X4, X5, .(X6, X7), .(X8, X9)) -> U143_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, difcH_in_aaaaaa(X2, X3, X4, X5, X7, X9)) U143_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, difcH_out_aaaaaa(X2, X3, X4, X5, X7, X9)) -> difcI_out_aaaaaaa(X1, X2, X3, X4, X5, .(X6, X7), .(X8, X9)) checkcW_in_aaaagaa(X1, X2, X3, X4, X5, X6, X7) -> U169_aaaagaa(X1, X2, X3, X4, X5, X6, X7, difcI_in_aaaaaaa(X1, X2, X3, X4, X7, X8, X9)) U169_aaaagaa(X1, X2, X3, X4, X5, X6, X7, difcI_out_aaaaaaa(X1, X2, X3, X4, X7, X8, X9)) -> U170_aaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, mergedeletecC_in_gga(X8, X5, X10)) U170_aaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, mergedeletecC_out_gga(X8, X5, X10)) -> U171_aaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revcF_in_ga(X9, X11)) U171_aaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revcF_out_ga(X9, X11)) -> U172_aaaagaa(X1, X2, X3, X4, X5, X6, X7, mergedeletecC_in_gga(X11, X10, X6)) U172_aaaagaa(X1, X2, X3, X4, X5, X6, X7, mergedeletecC_out_gga(X11, X10, X6)) -> checkcW_out_aaaagaa(X1, X2, X3, X4, X5, X6, X7) difcJ_in_aaaaaaaa(X1, X2, X3, X4, X5, X6, .(X7, X8), .(X9, X10)) -> U144_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, difcI_in_aaaaaaa(X2, X3, X4, X5, X6, X8, X10)) U144_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, difcI_out_aaaaaaa(X2, X3, X4, X5, X6, X8, X10)) -> difcJ_out_aaaaaaaa(X1, X2, X3, X4, X5, X6, .(X7, X8), .(X9, X10)) checkcX_in_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8) -> U173_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, difcJ_in_aaaaaaaa(X1, X2, X3, X4, X5, X8, X9, X10)) U173_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, difcJ_out_aaaaaaaa(X1, X2, X3, X4, X5, X8, X9, X10)) -> U174_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, mergedeletecC_in_gga(X9, X6, X11)) U174_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, mergedeletecC_out_gga(X9, X6, X11)) -> U175_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, revcF_in_ga(X10, X12)) U175_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, revcF_out_ga(X10, X12)) -> U176_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_in_gga(X12, X11, X7)) U176_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_out_gga(X12, X11, X7)) -> checkcX_out_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8) difcK_in_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, .(X8, X9), .(X10, X11)) -> U145_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, difcJ_in_aaaaaaaa(X2, X3, X4, X5, X6, X7, X9, X11)) U145_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, difcJ_out_aaaaaaaa(X2, X3, X4, X5, X6, X7, X9, X11)) -> difcK_out_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, .(X8, X9), .(X10, X11)) checkcY_in_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9) -> U177_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, difcK_in_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X9, X10, X11)) U177_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, difcK_out_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X9, X10, X11)) -> U178_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, mergedeletecC_in_gga(X10, X7, X12)) U178_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, mergedeletecC_out_gga(X10, X7, X12)) -> U179_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, revcF_in_ga(X11, X13)) U179_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, revcF_out_ga(X11, X13)) -> U180_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, mergedeletecC_in_gga(X13, X12, X8)) U180_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, mergedeletecC_out_gga(X13, X12, X8)) -> checkcY_out_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9) difcZ_in_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, .(X9, X10), .(X11, X12)) -> U185_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, difcK_in_aaaaaaaaa(X2, X3, X4, X5, X6, X7, X8, X10, X12)) U185_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, difcK_out_aaaaaaaaa(X2, X3, X4, X5, X6, X7, X8, X10, X12)) -> difcZ_out_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, .(X9, X10), .(X11, X12)) checkcP_in_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10) -> U181_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, difcZ_in_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X10, X11, X12)) U181_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, difcZ_out_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X10, X11, X12)) -> U182_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, mergedeletecC_in_gga(X11, X8, X13)) U182_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, mergedeletecC_out_gga(X11, X8, X13)) -> U183_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, revcF_in_ga(X12, X14)) U183_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, revcF_out_ga(X12, X14)) -> U184_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, mergedeletecC_in_gga(X14, X13, X9)) U184_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, mergedeletecC_out_gga(X14, X13, X9)) -> checkcP_out_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10) difcN1_in_agaaaa(X1, [], X2, X3, .(X4, []), .(X5, [])) -> difcN1_out_agaaaa(X1, [], X2, X3, .(X4, []), .(X5, [])) difcN1_in_agaaaa(X1, .(X2, []), X3, X4, .(X5, .(X6, [])), .(X7, .(X8, []))) -> difcN1_out_agaaaa(X1, .(X2, []), X3, X4, .(X5, .(X6, [])), .(X7, .(X8, []))) difcN1_in_agaaaa(X1, .(X2, .(X3, [])), X4, X5, .(X6, .(X7, .(X8, []))), .(X9, .(X10, .(X11, [])))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, [])), X4, X5, .(X6, .(X7, .(X8, []))), .(X9, .(X10, .(X11, [])))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, []))), X5, X6, .(X7, .(X8, .(X9, .(X10, [])))), .(X11, .(X12, .(X13, .(X14, []))))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, []))), X5, X6, .(X7, .(X8, .(X9, .(X10, [])))), .(X11, .(X12, .(X13, .(X14, []))))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, [])))), X6, X7, .(X8, .(X9, .(X10, .(X11, .(X12, []))))), .(X13, .(X14, .(X15, .(X16, .(X17, [])))))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, [])))), X6, X7, .(X8, .(X9, .(X10, .(X11, .(X12, []))))), .(X13, .(X14, .(X15, .(X16, .(X17, [])))))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, []))))), X7, X8, .(X9, .(X10, .(X11, .(X12, .(X13, .(X14, [])))))), .(X15, .(X16, .(X17, .(X18, .(X19, .(X20, []))))))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, []))))), X7, X8, .(X9, .(X10, .(X11, .(X12, .(X13, .(X14, [])))))), .(X15, .(X16, .(X17, .(X18, .(X19, .(X20, []))))))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, [])))))), X8, X9, .(X10, .(X11, .(X12, .(X13, .(X14, .(X15, .(X16, []))))))), .(X17, .(X18, .(X19, .(X20, .(X21, .(X22, .(X23, [])))))))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, [])))))), X8, X9, .(X10, .(X11, .(X12, .(X13, .(X14, .(X15, .(X16, []))))))), .(X17, .(X18, .(X19, .(X20, .(X21, .(X22, .(X23, [])))))))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9))))))), X10, X11, .(X12, .(X13, .(X14, .(X15, .(X16, .(X17, .(X18, .(X19, X20)))))))), .(X21, .(X22, .(X23, .(X24, .(X25, .(X26, .(X27, .(X28, X29))))))))) -> U190_agaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, X21, X22, X23, X24, X25, X26, X27, X28, X29, difcN_in_gaaaa(X9, X10, X11, X20, X29)) difcN_in_gaaaa([], X1, X2, [], []) -> difcN_out_gaaaa([], X1, X2, [], []) difcN_in_gaaaa(.(X1, X2), X3, X4, .(X5, X6), .(X7, X8)) -> U149_gaaaa(X1, X2, X3, X4, X5, X6, X7, X8, difcN_in_gaaaa(X2, X3, X4, X6, X8)) U149_gaaaa(X1, X2, X3, X4, X5, X6, X7, X8, difcN_out_gaaaa(X2, X3, X4, X6, X8)) -> difcN_out_gaaaa(.(X1, X2), X3, X4, .(X5, X6), .(X7, X8)) U190_agaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, X21, X22, X23, X24, X25, X26, X27, X28, X29, difcN_out_gaaaa(X9, X10, X11, X20, X29)) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9))))))), X10, X11, .(X12, .(X13, .(X14, .(X15, .(X16, .(X17, .(X18, .(X19, X20)))))))), .(X21, .(X22, .(X23, .(X24, .(X25, .(X26, .(X27, .(X28, X29))))))))) checkcM_in_agagaa(X1, X2, X3, X4, X5, X6) -> U186_agagaa(X1, X2, X3, X4, X5, X6, difcN1_in_agaaaa(X1, X2, X3, X6, X7, X8)) U186_agagaa(X1, X2, X3, X4, X5, X6, difcN1_out_agaaaa(X1, X2, X3, X6, X7, X8)) -> U187_agagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_in_gga(X7, X4, X9)) U187_agagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_out_gga(X7, X4, X9)) -> U188_agagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, revcF_in_ga(X8, X10)) U188_agagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, revcF_out_ga(X8, X10)) -> U189_agagaa(X1, X2, X3, X4, X5, X6, mergedeletecC_in_gga(X10, X9, X5)) U189_agagaa(X1, X2, X3, X4, X5, X6, mergedeletecC_out_gga(X10, X9, X5)) -> checkcM_out_agagaa(X1, X2, X3, X4, X5, X6) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) iota1cQ_in_aa(x1, x2) = iota1cQ_in_aa U153_aa(x1, x2, x3) = U153_aa(x3) iota1cA_in_aaa(x1, x2, x3) = iota1cA_in_aaa iota1cA_out_aaa(x1, x2, x3) = iota1cA_out_aaa(x3) U135_aaa(x1, x2, x3, x4) = U135_aaa(x4) iota1cQ_out_aa(x1, x2) = iota1cQ_out_aa(x2) membercR_in_ag(x1, x2) = membercR_in_ag(x2) membercR_out_ag(x1, x2) = membercR_out_ag(x2) U154_ag(x1, x2, x3, x4) = U154_ag(x3, x4) membercB_in_ag(x1, x2) = membercB_in_ag(x2) membercB_out_ag(x1, x2) = membercB_out_ag(x2) U136_ag(x1, x2, x3, x4) = U136_ag(x3, x4) difcE_in_aaaa(x1, x2, x3, x4) = difcE_in_aaaa difcE_out_aaaa(x1, x2, x3, x4) = difcE_out_aaaa(x3, x4) mergedeletecT_in_gga(x1, x2, x3) = mergedeletecT_in_gga(x1, x2) [] = [] mergedeletecT_out_gga(x1, x2, x3) = mergedeletecT_out_gga(x1, x2, x3) U159_gga(x1, x2, x3, x4, x5) = U159_gga(x2, x3, x5) mergedeletecC_in_gga(x1, x2, x3) = mergedeletecC_in_gga(x1, x2) mergedeletecC_out_gga(x1, x2, x3) = mergedeletecC_out_gga(x1, x2, x3) U137_gga(x1, x2, x3, x4, x5) = U137_gga(x2, x3, x5) U138_gga(x1, x2, x3, x4, x5, x6) = U138_gga(x2, x4, x6) U160_gga(x1, x2, x3, x4, x5, x6) = U160_gga(x2, x4, x6) revcF_in_ga(x1, x2) = revcF_in_ga(x1) revcF_out_ga(x1, x2) = revcF_out_ga(x1, x2) U140_ga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U140_ga(x9, x11) revcD_in_gaga(x1, x2, x3, x4) = revcD_in_gaga(x1, x3) revcD_out_gaga(x1, x2, x3, x4) = revcD_out_gaga(x1, x3, x4) U139_gaga(x1, x2, x3, x4, x5, x6) = U139_gaga(x2, x4, x6) checkcS_in_agaa(x1, x2, x3, x4) = checkcS_in_agaa(x2) U155_agaa(x1, x2, x3, x4, x5) = U155_agaa(x2, x5) U156_agaa(x1, x2, x3, x4, x5, x6, x7) = U156_agaa(x2, x6, x7) U157_agaa(x1, x2, x3, x4, x5, x6, x7, x8) = U157_agaa(x2, x7, x8) U158_agaa(x1, x2, x3, x4, x5) = U158_agaa(x2, x5) checkcS_out_agaa(x1, x2, x3, x4) = checkcS_out_agaa(x2, x3) difcG_in_aaaaa(x1, x2, x3, x4, x5) = difcG_in_aaaaa U141_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U141_aaaaa(x8) difcG_out_aaaaa(x1, x2, x3, x4, x5) = difcG_out_aaaaa(x4, x5) checkcU_in_aagaa(x1, x2, x3, x4, x5) = checkcU_in_aagaa(x3) U161_aagaa(x1, x2, x3, x4, x5, x6) = U161_aagaa(x3, x6) U162_aagaa(x1, x2, x3, x4, x5, x6, x7, x8) = U162_aagaa(x3, x7, x8) U163_aagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U163_aagaa(x3, x8, x9) U164_aagaa(x1, x2, x3, x4, x5, x6) = U164_aagaa(x3, x6) checkcU_out_aagaa(x1, x2, x3, x4, x5) = checkcU_out_aagaa(x3, x4) difcH_in_aaaaaa(x1, x2, x3, x4, x5, x6) = difcH_in_aaaaaa U142_aaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U142_aaaaaa(x9) difcH_out_aaaaaa(x1, x2, x3, x4, x5, x6) = difcH_out_aaaaaa(x5, x6) checkcV_in_aaagaa(x1, x2, x3, x4, x5, x6) = checkcV_in_aaagaa(x4) U165_aaagaa(x1, x2, x3, x4, x5, x6, x7) = U165_aaagaa(x4, x7) U166_aaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U166_aaagaa(x4, x8, x9) U167_aaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U167_aaagaa(x4, x9, x10) U168_aaagaa(x1, x2, x3, x4, x5, x6, x7) = U168_aaagaa(x4, x7) checkcV_out_aaagaa(x1, x2, x3, x4, x5, x6) = checkcV_out_aaagaa(x4, x5) difcI_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = difcI_in_aaaaaaa U143_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U143_aaaaaaa(x10) difcI_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = difcI_out_aaaaaaa(x6, x7) checkcW_in_aaaagaa(x1, x2, x3, x4, x5, x6, x7) = checkcW_in_aaaagaa(x5) U169_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8) = U169_aaaagaa(x5, x8) U170_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U170_aaaagaa(x5, x9, x10) U171_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U171_aaaagaa(x5, x10, x11) U172_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8) = U172_aaaagaa(x5, x8) checkcW_out_aaaagaa(x1, x2, x3, x4, x5, x6, x7) = checkcW_out_aaaagaa(x5, x6) difcJ_in_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = difcJ_in_aaaaaaaa U144_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U144_aaaaaaaa(x11) difcJ_out_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = difcJ_out_aaaaaaaa(x7, x8) checkcX_in_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8) = checkcX_in_aaaaagaa(x6) U173_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U173_aaaaagaa(x6, x9) U174_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U174_aaaaagaa(x6, x10, x11) U175_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U175_aaaaagaa(x6, x11, x12) U176_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U176_aaaaagaa(x6, x9) checkcX_out_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8) = checkcX_out_aaaaagaa(x6, x7) difcK_in_aaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = difcK_in_aaaaaaaaa U145_aaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U145_aaaaaaaaa(x12) difcK_out_aaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = difcK_out_aaaaaaaaa(x8, x9) checkcY_in_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = checkcY_in_aaaaaagaa(x7) U177_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U177_aaaaaagaa(x7, x10) U178_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U178_aaaaaagaa(x7, x11, x12) U179_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U179_aaaaaagaa(x7, x12, x13) U180_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U180_aaaaaagaa(x7, x10) checkcY_out_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = checkcY_out_aaaaaagaa(x7, x8) difcZ_in_aaaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = difcZ_in_aaaaaaaaaa U185_aaaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U185_aaaaaaaaaa(x13) difcZ_out_aaaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = difcZ_out_aaaaaaaaaa(x9, x10) checkcP_in_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = checkcP_in_aaaaaaagaa(x8) U181_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U181_aaaaaaagaa(x8, x11) U182_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U182_aaaaaaagaa(x8, x12, x13) U183_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14) = U183_aaaaaaagaa(x8, x13, x14) U184_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U184_aaaaaaagaa(x8, x11) checkcP_out_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = checkcP_out_aaaaaaagaa(x8, x9) difcN1_in_agaaaa(x1, x2, x3, x4, x5, x6) = difcN1_in_agaaaa(x2) difcN1_out_agaaaa(x1, x2, x3, x4, x5, x6) = difcN1_out_agaaaa(x2, x5, x6) U190_agaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30) = U190_agaaaa(x9, x30) difcN_in_gaaaa(x1, x2, x3, x4, x5) = difcN_in_gaaaa(x1) difcN_out_gaaaa(x1, x2, x3, x4, x5) = difcN_out_gaaaa(x1, x4, x5) U149_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U149_gaaaa(x2, x9) checkcM_in_agagaa(x1, x2, x3, x4, x5, x6) = checkcM_in_agagaa(x2, x4) U186_agagaa(x1, x2, x3, x4, x5, x6, x7) = U186_agagaa(x2, x4, x7) U187_agagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U187_agagaa(x2, x4, x8, x9) U188_agagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U188_agagaa(x2, x4, x9, x10) U189_agagaa(x1, x2, x3, x4, x5, x6, x7) = U189_agagaa(x2, x4, x7) checkcM_out_agagaa(x1, x2, x3, x4, x5, x6) = checkcM_out_agagaa(x2, x4, x5) MEMBERB_IN_AG(x1, x2) = MEMBERB_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (373) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (374) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBERB_IN_AG(X1, .(X2, X3)) -> MEMBERB_IN_AG(X1, X3) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) MEMBERB_IN_AG(x1, x2) = MEMBERB_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (375) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (376) Obligation: Q DP problem: The TRS P consists of the following rules: MEMBERB_IN_AG(.(X3)) -> MEMBERB_IN_AG(X3) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (377) 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: *MEMBERB_IN_AG(.(X3)) -> MEMBERB_IN_AG(X3) The graph contains the following edges 1 > 1 ---------------------------------------- (378) YES ---------------------------------------- (379) Obligation: Pi DP problem: The TRS P consists of the following rules: PDS1L_IN_GAGAA(X1, X2, X3, .(X4, X5), X6) -> U10_GAGAA(X1, X2, X3, X4, X5, X6, membercB_in_ag(X4, X1)) U10_GAGAA(X1, X2, X3, X4, X5, X6, membercB_out_ag(X4, X1)) -> U17_GAGAA(X1, X2, X3, X4, X5, X6, checkcM_in_agagaa(X2, X3, X4, X1, X7, X6)) U17_GAGAA(X1, X2, X3, X4, X5, X6, checkcM_out_agagaa(X2, X3, X4, X1, X7, X6)) -> PDS1L_IN_GAGAA(X7, X4, .(X2, X3), X5, X6) The TRS R consists of the following rules: iota1cQ_in_aa(X1, X2) -> U153_aa(X1, X2, iota1cA_in_aaa(X3, X1, X2)) iota1cA_in_aaa(X1, X1, []) -> iota1cA_out_aaa(X1, X1, []) iota1cA_in_aaa(X1, X2, .(X1, X3)) -> U135_aaa(X1, X2, X3, iota1cA_in_aaa(X4, X2, X3)) U135_aaa(X1, X2, X3, iota1cA_out_aaa(X4, X2, X3)) -> iota1cA_out_aaa(X1, X2, .(X1, X3)) U153_aa(X1, X2, iota1cA_out_aaa(X3, X1, X2)) -> iota1cQ_out_aa(X1, X2) membercR_in_ag(X1, .(X1, X2)) -> membercR_out_ag(X1, .(X1, X2)) membercR_in_ag(X1, .(X2, X3)) -> U154_ag(X1, X2, X3, membercB_in_ag(X1, X3)) membercB_in_ag(X1, .(X1, X2)) -> membercB_out_ag(X1, .(X1, X2)) membercB_in_ag(X1, .(X2, X3)) -> U136_ag(X1, X2, X3, membercB_in_ag(X1, X3)) U136_ag(X1, X2, X3, membercB_out_ag(X1, X3)) -> membercB_out_ag(X1, .(X2, X3)) U154_ag(X1, X2, X3, membercB_out_ag(X1, X3)) -> membercR_out_ag(X1, .(X2, X3)) difcE_in_aaaa(X1, X2, .(X3, []), .(X4, [])) -> difcE_out_aaaa(X1, X2, .(X3, []), .(X4, [])) mergedeletecT_in_gga([], X1, X1) -> mergedeletecT_out_gga([], X1, X1) mergedeletecT_in_gga(.(X1, X2), .(X1, X3), X4) -> U159_gga(X1, X2, X3, X4, mergedeletecC_in_gga(X2, X3, X4)) mergedeletecC_in_gga([], X1, X1) -> mergedeletecC_out_gga([], X1, X1) mergedeletecC_in_gga(.(X1, X2), .(X1, X3), X4) -> U137_gga(X1, X2, X3, X4, mergedeletecC_in_gga(X2, X3, X4)) mergedeletecC_in_gga(.(X1, X2), .(X3, X4), .(X3, X5)) -> U138_gga(X1, X2, X3, X4, X5, mergedeletecC_in_gga(.(X1, X2), X4, X5)) U138_gga(X1, X2, X3, X4, X5, mergedeletecC_out_gga(.(X1, X2), X4, X5)) -> mergedeletecC_out_gga(.(X1, X2), .(X3, X4), .(X3, X5)) U137_gga(X1, X2, X3, X4, mergedeletecC_out_gga(X2, X3, X4)) -> mergedeletecC_out_gga(.(X1, X2), .(X1, X3), X4) U159_gga(X1, X2, X3, X4, mergedeletecC_out_gga(X2, X3, X4)) -> mergedeletecT_out_gga(.(X1, X2), .(X1, X3), X4) mergedeletecT_in_gga(.(X1, X2), .(X3, X4), .(X3, X5)) -> U160_gga(X1, X2, X3, X4, X5, mergedeletecC_in_gga(.(X1, X2), X4, X5)) U160_gga(X1, X2, X3, X4, X5, mergedeletecC_out_gga(.(X1, X2), X4, X5)) -> mergedeletecT_out_gga(.(X1, X2), .(X3, X4), .(X3, X5)) revcF_in_ga([], []) -> revcF_out_ga([], []) revcF_in_ga(.(X1, []), .(X1, [])) -> revcF_out_ga(.(X1, []), .(X1, [])) revcF_in_ga(.(X1, .(X2, [])), .(X2, .(X1, []))) -> revcF_out_ga(.(X1, .(X2, [])), .(X2, .(X1, []))) revcF_in_ga(.(X1, .(X2, .(X3, []))), .(X3, .(X2, .(X1, [])))) -> revcF_out_ga(.(X1, .(X2, .(X3, []))), .(X3, .(X2, .(X1, [])))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, [])))), .(X4, .(X3, .(X2, .(X1, []))))) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, [])))), .(X4, .(X3, .(X2, .(X1, []))))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, .(X5, []))))), .(X5, .(X4, .(X3, .(X2, .(X1, [])))))) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, .(X5, []))))), .(X5, .(X4, .(X3, .(X2, .(X1, [])))))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, [])))))), .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, [])))))), .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, []))))))), .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, [])))))))) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, []))))))), .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, [])))))))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) -> U140_ga(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revcD_in_gaga(X9, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))), X10)) revcD_in_gaga([], X1, X2, .(X1, X2)) -> revcD_out_gaga([], X1, X2, .(X1, X2)) revcD_in_gaga(.(X1, X2), X3, X4, X5) -> U139_gaga(X1, X2, X3, X4, X5, revcD_in_gaga(X2, X1, .(X3, X4), X5)) U139_gaga(X1, X2, X3, X4, X5, revcD_out_gaga(X2, X1, .(X3, X4), X5)) -> revcD_out_gaga(.(X1, X2), X3, X4, X5) U140_ga(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revcD_out_gaga(X9, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))), X10)) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) checkcS_in_agaa(X1, X2, X3, X4) -> U155_agaa(X1, X2, X3, X4, difcE_in_aaaa(X1, X4, X5, X6)) U155_agaa(X1, X2, X3, X4, difcE_out_aaaa(X1, X4, X5, X6)) -> U156_agaa(X1, X2, X3, X4, X5, X6, mergedeletecT_in_gga(X5, X2, X7)) U156_agaa(X1, X2, X3, X4, X5, X6, mergedeletecT_out_gga(X5, X2, X7)) -> U157_agaa(X1, X2, X3, X4, X5, X6, X7, revcF_in_ga(X6, X8)) U157_agaa(X1, X2, X3, X4, X5, X6, X7, revcF_out_ga(X6, X8)) -> U158_agaa(X1, X2, X3, X4, mergedeletecC_in_gga(X8, X7, X3)) U158_agaa(X1, X2, X3, X4, mergedeletecC_out_gga(X8, X7, X3)) -> checkcS_out_agaa(X1, X2, X3, X4) difcG_in_aaaaa(X1, X2, X3, .(X4, X5), .(X6, X7)) -> U141_aaaaa(X1, X2, X3, X4, X5, X6, X7, difcE_in_aaaa(X2, X3, X5, X7)) U141_aaaaa(X1, X2, X3, X4, X5, X6, X7, difcE_out_aaaa(X2, X3, X5, X7)) -> difcG_out_aaaaa(X1, X2, X3, .(X4, X5), .(X6, X7)) checkcU_in_aagaa(X1, X2, X3, X4, X5) -> U161_aagaa(X1, X2, X3, X4, X5, difcG_in_aaaaa(X1, X2, X5, X6, X7)) U161_aagaa(X1, X2, X3, X4, X5, difcG_out_aaaaa(X1, X2, X5, X6, X7)) -> U162_aagaa(X1, X2, X3, X4, X5, X6, X7, mergedeletecC_in_gga(X6, X3, X8)) U162_aagaa(X1, X2, X3, X4, X5, X6, X7, mergedeletecC_out_gga(X6, X3, X8)) -> U163_aagaa(X1, X2, X3, X4, X5, X6, X7, X8, revcF_in_ga(X7, X9)) U163_aagaa(X1, X2, X3, X4, X5, X6, X7, X8, revcF_out_ga(X7, X9)) -> U164_aagaa(X1, X2, X3, X4, X5, mergedeletecC_in_gga(X9, X8, X4)) U164_aagaa(X1, X2, X3, X4, X5, mergedeletecC_out_gga(X9, X8, X4)) -> checkcU_out_aagaa(X1, X2, X3, X4, X5) difcH_in_aaaaaa(X1, X2, X3, X4, .(X5, X6), .(X7, X8)) -> U142_aaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, difcG_in_aaaaa(X2, X3, X4, X6, X8)) U142_aaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, difcG_out_aaaaa(X2, X3, X4, X6, X8)) -> difcH_out_aaaaaa(X1, X2, X3, X4, .(X5, X6), .(X7, X8)) checkcV_in_aaagaa(X1, X2, X3, X4, X5, X6) -> U165_aaagaa(X1, X2, X3, X4, X5, X6, difcH_in_aaaaaa(X1, X2, X3, X6, X7, X8)) U165_aaagaa(X1, X2, X3, X4, X5, X6, difcH_out_aaaaaa(X1, X2, X3, X6, X7, X8)) -> U166_aaagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_in_gga(X7, X4, X9)) U166_aaagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_out_gga(X7, X4, X9)) -> U167_aaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, revcF_in_ga(X8, X10)) U167_aaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, revcF_out_ga(X8, X10)) -> U168_aaagaa(X1, X2, X3, X4, X5, X6, mergedeletecC_in_gga(X10, X9, X5)) U168_aaagaa(X1, X2, X3, X4, X5, X6, mergedeletecC_out_gga(X10, X9, X5)) -> checkcV_out_aaagaa(X1, X2, X3, X4, X5, X6) difcI_in_aaaaaaa(X1, X2, X3, X4, X5, .(X6, X7), .(X8, X9)) -> U143_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, difcH_in_aaaaaa(X2, X3, X4, X5, X7, X9)) U143_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, difcH_out_aaaaaa(X2, X3, X4, X5, X7, X9)) -> difcI_out_aaaaaaa(X1, X2, X3, X4, X5, .(X6, X7), .(X8, X9)) checkcW_in_aaaagaa(X1, X2, X3, X4, X5, X6, X7) -> U169_aaaagaa(X1, X2, X3, X4, X5, X6, X7, difcI_in_aaaaaaa(X1, X2, X3, X4, X7, X8, X9)) U169_aaaagaa(X1, X2, X3, X4, X5, X6, X7, difcI_out_aaaaaaa(X1, X2, X3, X4, X7, X8, X9)) -> U170_aaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, mergedeletecC_in_gga(X8, X5, X10)) U170_aaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, mergedeletecC_out_gga(X8, X5, X10)) -> U171_aaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revcF_in_ga(X9, X11)) U171_aaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revcF_out_ga(X9, X11)) -> U172_aaaagaa(X1, X2, X3, X4, X5, X6, X7, mergedeletecC_in_gga(X11, X10, X6)) U172_aaaagaa(X1, X2, X3, X4, X5, X6, X7, mergedeletecC_out_gga(X11, X10, X6)) -> checkcW_out_aaaagaa(X1, X2, X3, X4, X5, X6, X7) difcJ_in_aaaaaaaa(X1, X2, X3, X4, X5, X6, .(X7, X8), .(X9, X10)) -> U144_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, difcI_in_aaaaaaa(X2, X3, X4, X5, X6, X8, X10)) U144_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, difcI_out_aaaaaaa(X2, X3, X4, X5, X6, X8, X10)) -> difcJ_out_aaaaaaaa(X1, X2, X3, X4, X5, X6, .(X7, X8), .(X9, X10)) checkcX_in_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8) -> U173_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, difcJ_in_aaaaaaaa(X1, X2, X3, X4, X5, X8, X9, X10)) U173_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, difcJ_out_aaaaaaaa(X1, X2, X3, X4, X5, X8, X9, X10)) -> U174_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, mergedeletecC_in_gga(X9, X6, X11)) U174_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, mergedeletecC_out_gga(X9, X6, X11)) -> U175_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, revcF_in_ga(X10, X12)) U175_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, revcF_out_ga(X10, X12)) -> U176_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_in_gga(X12, X11, X7)) U176_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_out_gga(X12, X11, X7)) -> checkcX_out_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8) difcK_in_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, .(X8, X9), .(X10, X11)) -> U145_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, difcJ_in_aaaaaaaa(X2, X3, X4, X5, X6, X7, X9, X11)) U145_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, difcJ_out_aaaaaaaa(X2, X3, X4, X5, X6, X7, X9, X11)) -> difcK_out_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, .(X8, X9), .(X10, X11)) checkcY_in_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9) -> U177_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, difcK_in_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X9, X10, X11)) U177_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, difcK_out_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X9, X10, X11)) -> U178_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, mergedeletecC_in_gga(X10, X7, X12)) U178_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, mergedeletecC_out_gga(X10, X7, X12)) -> U179_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, revcF_in_ga(X11, X13)) U179_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, revcF_out_ga(X11, X13)) -> U180_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, mergedeletecC_in_gga(X13, X12, X8)) U180_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, mergedeletecC_out_gga(X13, X12, X8)) -> checkcY_out_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9) difcZ_in_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, .(X9, X10), .(X11, X12)) -> U185_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, difcK_in_aaaaaaaaa(X2, X3, X4, X5, X6, X7, X8, X10, X12)) U185_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, difcK_out_aaaaaaaaa(X2, X3, X4, X5, X6, X7, X8, X10, X12)) -> difcZ_out_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, .(X9, X10), .(X11, X12)) checkcP_in_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10) -> U181_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, difcZ_in_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X10, X11, X12)) U181_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, difcZ_out_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X10, X11, X12)) -> U182_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, mergedeletecC_in_gga(X11, X8, X13)) U182_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, mergedeletecC_out_gga(X11, X8, X13)) -> U183_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, revcF_in_ga(X12, X14)) U183_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, revcF_out_ga(X12, X14)) -> U184_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, mergedeletecC_in_gga(X14, X13, X9)) U184_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, mergedeletecC_out_gga(X14, X13, X9)) -> checkcP_out_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10) difcN1_in_agaaaa(X1, [], X2, X3, .(X4, []), .(X5, [])) -> difcN1_out_agaaaa(X1, [], X2, X3, .(X4, []), .(X5, [])) difcN1_in_agaaaa(X1, .(X2, []), X3, X4, .(X5, .(X6, [])), .(X7, .(X8, []))) -> difcN1_out_agaaaa(X1, .(X2, []), X3, X4, .(X5, .(X6, [])), .(X7, .(X8, []))) difcN1_in_agaaaa(X1, .(X2, .(X3, [])), X4, X5, .(X6, .(X7, .(X8, []))), .(X9, .(X10, .(X11, [])))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, [])), X4, X5, .(X6, .(X7, .(X8, []))), .(X9, .(X10, .(X11, [])))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, []))), X5, X6, .(X7, .(X8, .(X9, .(X10, [])))), .(X11, .(X12, .(X13, .(X14, []))))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, []))), X5, X6, .(X7, .(X8, .(X9, .(X10, [])))), .(X11, .(X12, .(X13, .(X14, []))))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, [])))), X6, X7, .(X8, .(X9, .(X10, .(X11, .(X12, []))))), .(X13, .(X14, .(X15, .(X16, .(X17, [])))))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, [])))), X6, X7, .(X8, .(X9, .(X10, .(X11, .(X12, []))))), .(X13, .(X14, .(X15, .(X16, .(X17, [])))))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, []))))), X7, X8, .(X9, .(X10, .(X11, .(X12, .(X13, .(X14, [])))))), .(X15, .(X16, .(X17, .(X18, .(X19, .(X20, []))))))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, []))))), X7, X8, .(X9, .(X10, .(X11, .(X12, .(X13, .(X14, [])))))), .(X15, .(X16, .(X17, .(X18, .(X19, .(X20, []))))))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, [])))))), X8, X9, .(X10, .(X11, .(X12, .(X13, .(X14, .(X15, .(X16, []))))))), .(X17, .(X18, .(X19, .(X20, .(X21, .(X22, .(X23, [])))))))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, [])))))), X8, X9, .(X10, .(X11, .(X12, .(X13, .(X14, .(X15, .(X16, []))))))), .(X17, .(X18, .(X19, .(X20, .(X21, .(X22, .(X23, [])))))))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9))))))), X10, X11, .(X12, .(X13, .(X14, .(X15, .(X16, .(X17, .(X18, .(X19, X20)))))))), .(X21, .(X22, .(X23, .(X24, .(X25, .(X26, .(X27, .(X28, X29))))))))) -> U190_agaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, X21, X22, X23, X24, X25, X26, X27, X28, X29, difcN_in_gaaaa(X9, X10, X11, X20, X29)) difcN_in_gaaaa([], X1, X2, [], []) -> difcN_out_gaaaa([], X1, X2, [], []) difcN_in_gaaaa(.(X1, X2), X3, X4, .(X5, X6), .(X7, X8)) -> U149_gaaaa(X1, X2, X3, X4, X5, X6, X7, X8, difcN_in_gaaaa(X2, X3, X4, X6, X8)) U149_gaaaa(X1, X2, X3, X4, X5, X6, X7, X8, difcN_out_gaaaa(X2, X3, X4, X6, X8)) -> difcN_out_gaaaa(.(X1, X2), X3, X4, .(X5, X6), .(X7, X8)) U190_agaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, X21, X22, X23, X24, X25, X26, X27, X28, X29, difcN_out_gaaaa(X9, X10, X11, X20, X29)) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9))))))), X10, X11, .(X12, .(X13, .(X14, .(X15, .(X16, .(X17, .(X18, .(X19, X20)))))))), .(X21, .(X22, .(X23, .(X24, .(X25, .(X26, .(X27, .(X28, X29))))))))) checkcM_in_agagaa(X1, X2, X3, X4, X5, X6) -> U186_agagaa(X1, X2, X3, X4, X5, X6, difcN1_in_agaaaa(X1, X2, X3, X6, X7, X8)) U186_agagaa(X1, X2, X3, X4, X5, X6, difcN1_out_agaaaa(X1, X2, X3, X6, X7, X8)) -> U187_agagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_in_gga(X7, X4, X9)) U187_agagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_out_gga(X7, X4, X9)) -> U188_agagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, revcF_in_ga(X8, X10)) U188_agagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, revcF_out_ga(X8, X10)) -> U189_agagaa(X1, X2, X3, X4, X5, X6, mergedeletecC_in_gga(X10, X9, X5)) U189_agagaa(X1, X2, X3, X4, X5, X6, mergedeletecC_out_gga(X10, X9, X5)) -> checkcM_out_agagaa(X1, X2, X3, X4, X5, X6) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) iota1cQ_in_aa(x1, x2) = iota1cQ_in_aa U153_aa(x1, x2, x3) = U153_aa(x3) iota1cA_in_aaa(x1, x2, x3) = iota1cA_in_aaa iota1cA_out_aaa(x1, x2, x3) = iota1cA_out_aaa(x3) U135_aaa(x1, x2, x3, x4) = U135_aaa(x4) iota1cQ_out_aa(x1, x2) = iota1cQ_out_aa(x2) membercR_in_ag(x1, x2) = membercR_in_ag(x2) membercR_out_ag(x1, x2) = membercR_out_ag(x2) U154_ag(x1, x2, x3, x4) = U154_ag(x3, x4) membercB_in_ag(x1, x2) = membercB_in_ag(x2) membercB_out_ag(x1, x2) = membercB_out_ag(x2) U136_ag(x1, x2, x3, x4) = U136_ag(x3, x4) difcE_in_aaaa(x1, x2, x3, x4) = difcE_in_aaaa difcE_out_aaaa(x1, x2, x3, x4) = difcE_out_aaaa(x3, x4) mergedeletecT_in_gga(x1, x2, x3) = mergedeletecT_in_gga(x1, x2) [] = [] mergedeletecT_out_gga(x1, x2, x3) = mergedeletecT_out_gga(x1, x2, x3) U159_gga(x1, x2, x3, x4, x5) = U159_gga(x2, x3, x5) mergedeletecC_in_gga(x1, x2, x3) = mergedeletecC_in_gga(x1, x2) mergedeletecC_out_gga(x1, x2, x3) = mergedeletecC_out_gga(x1, x2, x3) U137_gga(x1, x2, x3, x4, x5) = U137_gga(x2, x3, x5) U138_gga(x1, x2, x3, x4, x5, x6) = U138_gga(x2, x4, x6) U160_gga(x1, x2, x3, x4, x5, x6) = U160_gga(x2, x4, x6) revcF_in_ga(x1, x2) = revcF_in_ga(x1) revcF_out_ga(x1, x2) = revcF_out_ga(x1, x2) U140_ga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U140_ga(x9, x11) revcD_in_gaga(x1, x2, x3, x4) = revcD_in_gaga(x1, x3) revcD_out_gaga(x1, x2, x3, x4) = revcD_out_gaga(x1, x3, x4) U139_gaga(x1, x2, x3, x4, x5, x6) = U139_gaga(x2, x4, x6) checkcS_in_agaa(x1, x2, x3, x4) = checkcS_in_agaa(x2) U155_agaa(x1, x2, x3, x4, x5) = U155_agaa(x2, x5) U156_agaa(x1, x2, x3, x4, x5, x6, x7) = U156_agaa(x2, x6, x7) U157_agaa(x1, x2, x3, x4, x5, x6, x7, x8) = U157_agaa(x2, x7, x8) U158_agaa(x1, x2, x3, x4, x5) = U158_agaa(x2, x5) checkcS_out_agaa(x1, x2, x3, x4) = checkcS_out_agaa(x2, x3) difcG_in_aaaaa(x1, x2, x3, x4, x5) = difcG_in_aaaaa U141_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U141_aaaaa(x8) difcG_out_aaaaa(x1, x2, x3, x4, x5) = difcG_out_aaaaa(x4, x5) checkcU_in_aagaa(x1, x2, x3, x4, x5) = checkcU_in_aagaa(x3) U161_aagaa(x1, x2, x3, x4, x5, x6) = U161_aagaa(x3, x6) U162_aagaa(x1, x2, x3, x4, x5, x6, x7, x8) = U162_aagaa(x3, x7, x8) U163_aagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U163_aagaa(x3, x8, x9) U164_aagaa(x1, x2, x3, x4, x5, x6) = U164_aagaa(x3, x6) checkcU_out_aagaa(x1, x2, x3, x4, x5) = checkcU_out_aagaa(x3, x4) difcH_in_aaaaaa(x1, x2, x3, x4, x5, x6) = difcH_in_aaaaaa U142_aaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U142_aaaaaa(x9) difcH_out_aaaaaa(x1, x2, x3, x4, x5, x6) = difcH_out_aaaaaa(x5, x6) checkcV_in_aaagaa(x1, x2, x3, x4, x5, x6) = checkcV_in_aaagaa(x4) U165_aaagaa(x1, x2, x3, x4, x5, x6, x7) = U165_aaagaa(x4, x7) U166_aaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U166_aaagaa(x4, x8, x9) U167_aaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U167_aaagaa(x4, x9, x10) U168_aaagaa(x1, x2, x3, x4, x5, x6, x7) = U168_aaagaa(x4, x7) checkcV_out_aaagaa(x1, x2, x3, x4, x5, x6) = checkcV_out_aaagaa(x4, x5) difcI_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = difcI_in_aaaaaaa U143_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U143_aaaaaaa(x10) difcI_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = difcI_out_aaaaaaa(x6, x7) checkcW_in_aaaagaa(x1, x2, x3, x4, x5, x6, x7) = checkcW_in_aaaagaa(x5) U169_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8) = U169_aaaagaa(x5, x8) U170_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U170_aaaagaa(x5, x9, x10) U171_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U171_aaaagaa(x5, x10, x11) U172_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8) = U172_aaaagaa(x5, x8) checkcW_out_aaaagaa(x1, x2, x3, x4, x5, x6, x7) = checkcW_out_aaaagaa(x5, x6) difcJ_in_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = difcJ_in_aaaaaaaa U144_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U144_aaaaaaaa(x11) difcJ_out_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = difcJ_out_aaaaaaaa(x7, x8) checkcX_in_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8) = checkcX_in_aaaaagaa(x6) U173_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U173_aaaaagaa(x6, x9) U174_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U174_aaaaagaa(x6, x10, x11) U175_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U175_aaaaagaa(x6, x11, x12) U176_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U176_aaaaagaa(x6, x9) checkcX_out_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8) = checkcX_out_aaaaagaa(x6, x7) difcK_in_aaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = difcK_in_aaaaaaaaa U145_aaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U145_aaaaaaaaa(x12) difcK_out_aaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = difcK_out_aaaaaaaaa(x8, x9) checkcY_in_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = checkcY_in_aaaaaagaa(x7) U177_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U177_aaaaaagaa(x7, x10) U178_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U178_aaaaaagaa(x7, x11, x12) U179_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U179_aaaaaagaa(x7, x12, x13) U180_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U180_aaaaaagaa(x7, x10) checkcY_out_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = checkcY_out_aaaaaagaa(x7, x8) difcZ_in_aaaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = difcZ_in_aaaaaaaaaa U185_aaaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U185_aaaaaaaaaa(x13) difcZ_out_aaaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = difcZ_out_aaaaaaaaaa(x9, x10) checkcP_in_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = checkcP_in_aaaaaaagaa(x8) U181_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U181_aaaaaaagaa(x8, x11) U182_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U182_aaaaaaagaa(x8, x12, x13) U183_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14) = U183_aaaaaaagaa(x8, x13, x14) U184_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U184_aaaaaaagaa(x8, x11) checkcP_out_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = checkcP_out_aaaaaaagaa(x8, x9) difcN1_in_agaaaa(x1, x2, x3, x4, x5, x6) = difcN1_in_agaaaa(x2) difcN1_out_agaaaa(x1, x2, x3, x4, x5, x6) = difcN1_out_agaaaa(x2, x5, x6) U190_agaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30) = U190_agaaaa(x9, x30) difcN_in_gaaaa(x1, x2, x3, x4, x5) = difcN_in_gaaaa(x1) difcN_out_gaaaa(x1, x2, x3, x4, x5) = difcN_out_gaaaa(x1, x4, x5) U149_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U149_gaaaa(x2, x9) checkcM_in_agagaa(x1, x2, x3, x4, x5, x6) = checkcM_in_agagaa(x2, x4) U186_agagaa(x1, x2, x3, x4, x5, x6, x7) = U186_agagaa(x2, x4, x7) U187_agagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U187_agagaa(x2, x4, x8, x9) U188_agagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U188_agagaa(x2, x4, x9, x10) U189_agagaa(x1, x2, x3, x4, x5, x6, x7) = U189_agagaa(x2, x4, x7) checkcM_out_agagaa(x1, x2, x3, x4, x5, x6) = checkcM_out_agagaa(x2, x4, x5) PDS1L_IN_GAGAA(x1, x2, x3, x4, x5) = PDS1L_IN_GAGAA(x1, x3) U10_GAGAA(x1, x2, x3, x4, x5, x6, x7) = U10_GAGAA(x1, x3, x7) U17_GAGAA(x1, x2, x3, x4, x5, x6, x7) = U17_GAGAA(x1, x3, x7) We have to consider all (P,R,Pi)-chains ---------------------------------------- (380) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (381) Obligation: Pi DP problem: The TRS P consists of the following rules: PDS1L_IN_GAGAA(X1, X2, X3, .(X4, X5), X6) -> U10_GAGAA(X1, X2, X3, X4, X5, X6, membercB_in_ag(X4, X1)) U10_GAGAA(X1, X2, X3, X4, X5, X6, membercB_out_ag(X4, X1)) -> U17_GAGAA(X1, X2, X3, X4, X5, X6, checkcM_in_agagaa(X2, X3, X4, X1, X7, X6)) U17_GAGAA(X1, X2, X3, X4, X5, X6, checkcM_out_agagaa(X2, X3, X4, X1, X7, X6)) -> PDS1L_IN_GAGAA(X7, X4, .(X2, X3), X5, X6) The TRS R consists of the following rules: membercB_in_ag(X1, .(X1, X2)) -> membercB_out_ag(X1, .(X1, X2)) membercB_in_ag(X1, .(X2, X3)) -> U136_ag(X1, X2, X3, membercB_in_ag(X1, X3)) checkcM_in_agagaa(X1, X2, X3, X4, X5, X6) -> U186_agagaa(X1, X2, X3, X4, X5, X6, difcN1_in_agaaaa(X1, X2, X3, X6, X7, X8)) U136_ag(X1, X2, X3, membercB_out_ag(X1, X3)) -> membercB_out_ag(X1, .(X2, X3)) U186_agagaa(X1, X2, X3, X4, X5, X6, difcN1_out_agaaaa(X1, X2, X3, X6, X7, X8)) -> U187_agagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_in_gga(X7, X4, X9)) difcN1_in_agaaaa(X1, [], X2, X3, .(X4, []), .(X5, [])) -> difcN1_out_agaaaa(X1, [], X2, X3, .(X4, []), .(X5, [])) difcN1_in_agaaaa(X1, .(X2, []), X3, X4, .(X5, .(X6, [])), .(X7, .(X8, []))) -> difcN1_out_agaaaa(X1, .(X2, []), X3, X4, .(X5, .(X6, [])), .(X7, .(X8, []))) difcN1_in_agaaaa(X1, .(X2, .(X3, [])), X4, X5, .(X6, .(X7, .(X8, []))), .(X9, .(X10, .(X11, [])))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, [])), X4, X5, .(X6, .(X7, .(X8, []))), .(X9, .(X10, .(X11, [])))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, []))), X5, X6, .(X7, .(X8, .(X9, .(X10, [])))), .(X11, .(X12, .(X13, .(X14, []))))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, []))), X5, X6, .(X7, .(X8, .(X9, .(X10, [])))), .(X11, .(X12, .(X13, .(X14, []))))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, [])))), X6, X7, .(X8, .(X9, .(X10, .(X11, .(X12, []))))), .(X13, .(X14, .(X15, .(X16, .(X17, [])))))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, [])))), X6, X7, .(X8, .(X9, .(X10, .(X11, .(X12, []))))), .(X13, .(X14, .(X15, .(X16, .(X17, [])))))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, []))))), X7, X8, .(X9, .(X10, .(X11, .(X12, .(X13, .(X14, [])))))), .(X15, .(X16, .(X17, .(X18, .(X19, .(X20, []))))))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, []))))), X7, X8, .(X9, .(X10, .(X11, .(X12, .(X13, .(X14, [])))))), .(X15, .(X16, .(X17, .(X18, .(X19, .(X20, []))))))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, [])))))), X8, X9, .(X10, .(X11, .(X12, .(X13, .(X14, .(X15, .(X16, []))))))), .(X17, .(X18, .(X19, .(X20, .(X21, .(X22, .(X23, [])))))))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, [])))))), X8, X9, .(X10, .(X11, .(X12, .(X13, .(X14, .(X15, .(X16, []))))))), .(X17, .(X18, .(X19, .(X20, .(X21, .(X22, .(X23, [])))))))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9))))))), X10, X11, .(X12, .(X13, .(X14, .(X15, .(X16, .(X17, .(X18, .(X19, X20)))))))), .(X21, .(X22, .(X23, .(X24, .(X25, .(X26, .(X27, .(X28, X29))))))))) -> U190_agaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, X21, X22, X23, X24, X25, X26, X27, X28, X29, difcN_in_gaaaa(X9, X10, X11, X20, X29)) U187_agagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_out_gga(X7, X4, X9)) -> U188_agagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, revcF_in_ga(X8, X10)) U190_agaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, X21, X22, X23, X24, X25, X26, X27, X28, X29, difcN_out_gaaaa(X9, X10, X11, X20, X29)) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9))))))), X10, X11, .(X12, .(X13, .(X14, .(X15, .(X16, .(X17, .(X18, .(X19, X20)))))))), .(X21, .(X22, .(X23, .(X24, .(X25, .(X26, .(X27, .(X28, X29))))))))) mergedeletecC_in_gga([], X1, X1) -> mergedeletecC_out_gga([], X1, X1) mergedeletecC_in_gga(.(X1, X2), .(X1, X3), X4) -> U137_gga(X1, X2, X3, X4, mergedeletecC_in_gga(X2, X3, X4)) mergedeletecC_in_gga(.(X1, X2), .(X3, X4), .(X3, X5)) -> U138_gga(X1, X2, X3, X4, X5, mergedeletecC_in_gga(.(X1, X2), X4, X5)) U188_agagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, revcF_out_ga(X8, X10)) -> U189_agagaa(X1, X2, X3, X4, X5, X6, mergedeletecC_in_gga(X10, X9, X5)) difcN_in_gaaaa([], X1, X2, [], []) -> difcN_out_gaaaa([], X1, X2, [], []) difcN_in_gaaaa(.(X1, X2), X3, X4, .(X5, X6), .(X7, X8)) -> U149_gaaaa(X1, X2, X3, X4, X5, X6, X7, X8, difcN_in_gaaaa(X2, X3, X4, X6, X8)) U137_gga(X1, X2, X3, X4, mergedeletecC_out_gga(X2, X3, X4)) -> mergedeletecC_out_gga(.(X1, X2), .(X1, X3), X4) U138_gga(X1, X2, X3, X4, X5, mergedeletecC_out_gga(.(X1, X2), X4, X5)) -> mergedeletecC_out_gga(.(X1, X2), .(X3, X4), .(X3, X5)) revcF_in_ga([], []) -> revcF_out_ga([], []) revcF_in_ga(.(X1, []), .(X1, [])) -> revcF_out_ga(.(X1, []), .(X1, [])) revcF_in_ga(.(X1, .(X2, [])), .(X2, .(X1, []))) -> revcF_out_ga(.(X1, .(X2, [])), .(X2, .(X1, []))) revcF_in_ga(.(X1, .(X2, .(X3, []))), .(X3, .(X2, .(X1, [])))) -> revcF_out_ga(.(X1, .(X2, .(X3, []))), .(X3, .(X2, .(X1, [])))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, [])))), .(X4, .(X3, .(X2, .(X1, []))))) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, [])))), .(X4, .(X3, .(X2, .(X1, []))))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, .(X5, []))))), .(X5, .(X4, .(X3, .(X2, .(X1, [])))))) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, .(X5, []))))), .(X5, .(X4, .(X3, .(X2, .(X1, [])))))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, [])))))), .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, [])))))), .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, []))))))), .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, [])))))))) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, []))))))), .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, [])))))))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) -> U140_ga(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revcD_in_gaga(X9, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))), X10)) U189_agagaa(X1, X2, X3, X4, X5, X6, mergedeletecC_out_gga(X10, X9, X5)) -> checkcM_out_agagaa(X1, X2, X3, X4, X5, X6) U149_gaaaa(X1, X2, X3, X4, X5, X6, X7, X8, difcN_out_gaaaa(X2, X3, X4, X6, X8)) -> difcN_out_gaaaa(.(X1, X2), X3, X4, .(X5, X6), .(X7, X8)) U140_ga(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revcD_out_gaga(X9, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))), X10)) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) revcD_in_gaga([], X1, X2, .(X1, X2)) -> revcD_out_gaga([], X1, X2, .(X1, X2)) revcD_in_gaga(.(X1, X2), X3, X4, X5) -> U139_gaga(X1, X2, X3, X4, X5, revcD_in_gaga(X2, X1, .(X3, X4), X5)) U139_gaga(X1, X2, X3, X4, X5, revcD_out_gaga(X2, X1, .(X3, X4), X5)) -> revcD_out_gaga(.(X1, X2), X3, X4, X5) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) membercB_in_ag(x1, x2) = membercB_in_ag(x2) membercB_out_ag(x1, x2) = membercB_out_ag(x2) U136_ag(x1, x2, x3, x4) = U136_ag(x3, x4) [] = [] mergedeletecC_in_gga(x1, x2, x3) = mergedeletecC_in_gga(x1, x2) mergedeletecC_out_gga(x1, x2, x3) = mergedeletecC_out_gga(x1, x2, x3) U137_gga(x1, x2, x3, x4, x5) = U137_gga(x2, x3, x5) U138_gga(x1, x2, x3, x4, x5, x6) = U138_gga(x2, x4, x6) revcF_in_ga(x1, x2) = revcF_in_ga(x1) revcF_out_ga(x1, x2) = revcF_out_ga(x1, x2) U140_ga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U140_ga(x9, x11) revcD_in_gaga(x1, x2, x3, x4) = revcD_in_gaga(x1, x3) revcD_out_gaga(x1, x2, x3, x4) = revcD_out_gaga(x1, x3, x4) U139_gaga(x1, x2, x3, x4, x5, x6) = U139_gaga(x2, x4, x6) difcN1_in_agaaaa(x1, x2, x3, x4, x5, x6) = difcN1_in_agaaaa(x2) difcN1_out_agaaaa(x1, x2, x3, x4, x5, x6) = difcN1_out_agaaaa(x2, x5, x6) U190_agaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30) = U190_agaaaa(x9, x30) difcN_in_gaaaa(x1, x2, x3, x4, x5) = difcN_in_gaaaa(x1) difcN_out_gaaaa(x1, x2, x3, x4, x5) = difcN_out_gaaaa(x1, x4, x5) U149_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U149_gaaaa(x2, x9) checkcM_in_agagaa(x1, x2, x3, x4, x5, x6) = checkcM_in_agagaa(x2, x4) U186_agagaa(x1, x2, x3, x4, x5, x6, x7) = U186_agagaa(x2, x4, x7) U187_agagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U187_agagaa(x2, x4, x8, x9) U188_agagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U188_agagaa(x2, x4, x9, x10) U189_agagaa(x1, x2, x3, x4, x5, x6, x7) = U189_agagaa(x2, x4, x7) checkcM_out_agagaa(x1, x2, x3, x4, x5, x6) = checkcM_out_agagaa(x2, x4, x5) PDS1L_IN_GAGAA(x1, x2, x3, x4, x5) = PDS1L_IN_GAGAA(x1, x3) U10_GAGAA(x1, x2, x3, x4, x5, x6, x7) = U10_GAGAA(x1, x3, x7) U17_GAGAA(x1, x2, x3, x4, x5, x6, x7) = U17_GAGAA(x1, x3, x7) We have to consider all (P,R,Pi)-chains ---------------------------------------- (382) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (383) Obligation: Q DP problem: The TRS P consists of the following rules: PDS1L_IN_GAGAA(X1, X3) -> U10_GAGAA(X1, X3, membercB_in_ag(X1)) U10_GAGAA(X1, X3, membercB_out_ag(X1)) -> U17_GAGAA(X1, X3, checkcM_in_agagaa(X3, X1)) U17_GAGAA(X1, X3, checkcM_out_agagaa(X3, X1, X7)) -> PDS1L_IN_GAGAA(X7, .(X3)) The TRS R consists of the following rules: membercB_in_ag(.(X2)) -> membercB_out_ag(.(X2)) membercB_in_ag(.(X3)) -> U136_ag(X3, membercB_in_ag(X3)) checkcM_in_agagaa(X2, X4) -> U186_agagaa(X2, X4, difcN1_in_agaaaa(X2)) U136_ag(X3, membercB_out_ag(X3)) -> membercB_out_ag(.(X3)) U186_agagaa(X2, X4, difcN1_out_agaaaa(X2, X7, X8)) -> U187_agagaa(X2, X4, X8, mergedeletecC_in_gga(X7, X4)) difcN1_in_agaaaa([]) -> difcN1_out_agaaaa([], .([]), .([])) difcN1_in_agaaaa(.([])) -> difcN1_out_agaaaa(.([]), .(.([])), .(.([]))) difcN1_in_agaaaa(.(.([]))) -> difcN1_out_agaaaa(.(.([])), .(.(.([]))), .(.(.([])))) difcN1_in_agaaaa(.(.(.([])))) -> difcN1_out_agaaaa(.(.(.([]))), .(.(.(.([])))), .(.(.(.([]))))) difcN1_in_agaaaa(.(.(.(.([]))))) -> difcN1_out_agaaaa(.(.(.(.([])))), .(.(.(.(.([]))))), .(.(.(.(.([])))))) difcN1_in_agaaaa(.(.(.(.(.([])))))) -> difcN1_out_agaaaa(.(.(.(.(.([]))))), .(.(.(.(.(.([])))))), .(.(.(.(.(.([]))))))) difcN1_in_agaaaa(.(.(.(.(.(.([]))))))) -> difcN1_out_agaaaa(.(.(.(.(.(.([])))))), .(.(.(.(.(.(.([]))))))), .(.(.(.(.(.(.([])))))))) difcN1_in_agaaaa(.(.(.(.(.(.(.(X9)))))))) -> U190_agaaaa(X9, difcN_in_gaaaa(X9)) U187_agagaa(X2, X4, X8, mergedeletecC_out_gga(X7, X4, X9)) -> U188_agagaa(X2, X4, X9, revcF_in_ga(X8)) U190_agaaaa(X9, difcN_out_gaaaa(X9, X20, X29)) -> difcN1_out_agaaaa(.(.(.(.(.(.(.(X9))))))), .(.(.(.(.(.(.(.(X20)))))))), .(.(.(.(.(.(.(.(X29))))))))) mergedeletecC_in_gga([], X1) -> mergedeletecC_out_gga([], X1, X1) mergedeletecC_in_gga(.(X2), .(X3)) -> U137_gga(X2, X3, mergedeletecC_in_gga(X2, X3)) mergedeletecC_in_gga(.(X2), .(X4)) -> U138_gga(X2, X4, mergedeletecC_in_gga(.(X2), X4)) U188_agagaa(X2, X4, X9, revcF_out_ga(X8, X10)) -> U189_agagaa(X2, X4, mergedeletecC_in_gga(X10, X9)) difcN_in_gaaaa([]) -> difcN_out_gaaaa([], [], []) difcN_in_gaaaa(.(X2)) -> U149_gaaaa(X2, difcN_in_gaaaa(X2)) U137_gga(X2, X3, mergedeletecC_out_gga(X2, X3, X4)) -> mergedeletecC_out_gga(.(X2), .(X3), X4) U138_gga(X2, X4, mergedeletecC_out_gga(.(X2), X4, X5)) -> mergedeletecC_out_gga(.(X2), .(X4), .(X5)) revcF_in_ga([]) -> revcF_out_ga([], []) revcF_in_ga(.([])) -> revcF_out_ga(.([]), .([])) revcF_in_ga(.(.([]))) -> revcF_out_ga(.(.([])), .(.([]))) revcF_in_ga(.(.(.([])))) -> revcF_out_ga(.(.(.([]))), .(.(.([])))) revcF_in_ga(.(.(.(.([]))))) -> revcF_out_ga(.(.(.(.([])))), .(.(.(.([]))))) revcF_in_ga(.(.(.(.(.([])))))) -> revcF_out_ga(.(.(.(.(.([]))))), .(.(.(.(.([])))))) revcF_in_ga(.(.(.(.(.(.([]))))))) -> revcF_out_ga(.(.(.(.(.(.([])))))), .(.(.(.(.(.([]))))))) revcF_in_ga(.(.(.(.(.(.(.([])))))))) -> revcF_out_ga(.(.(.(.(.(.(.([]))))))), .(.(.(.(.(.(.([])))))))) revcF_in_ga(.(.(.(.(.(.(.(.(X9))))))))) -> U140_ga(X9, revcD_in_gaga(X9, .(.(.(.(.(.(.([]))))))))) U189_agagaa(X2, X4, mergedeletecC_out_gga(X10, X9, X5)) -> checkcM_out_agagaa(X2, X4, X5) U149_gaaaa(X2, difcN_out_gaaaa(X2, X6, X8)) -> difcN_out_gaaaa(.(X2), .(X6), .(X8)) U140_ga(X9, revcD_out_gaga(X9, .(.(.(.(.(.(.([]))))))), X10)) -> revcF_out_ga(.(.(.(.(.(.(.(.(X9)))))))), X10) revcD_in_gaga([], X2) -> revcD_out_gaga([], X2, .(X2)) revcD_in_gaga(.(X2), X4) -> U139_gaga(X2, X4, revcD_in_gaga(X2, .(X4))) U139_gaga(X2, X4, revcD_out_gaga(X2, .(X4), X5)) -> revcD_out_gaga(.(X2), X4, X5) The set Q consists of the following terms: membercB_in_ag(x0) checkcM_in_agagaa(x0, x1) U136_ag(x0, x1) U186_agagaa(x0, x1, x2) difcN1_in_agaaaa(x0) U187_agagaa(x0, x1, x2, x3) U190_agaaaa(x0, x1) mergedeletecC_in_gga(x0, x1) U188_agagaa(x0, x1, x2, x3) difcN_in_gaaaa(x0) U137_gga(x0, x1, x2) U138_gga(x0, x1, x2) revcF_in_ga(x0) U189_agagaa(x0, x1, x2) U149_gaaaa(x0, x1) U140_ga(x0, x1) revcD_in_gaga(x0, x1) U139_gaga(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (384) 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: difcN1_in_agaaaa([]) -> difcN1_out_agaaaa([], .([]), .([])) Used ordering: Polynomial interpretation [POLO]: POL(.(x_1)) = 0 POL(PDS1L_IN_GAGAA(x_1, x_2)) = 2*x_2 POL(U10_GAGAA(x_1, x_2, x_3)) = 2*x_2 POL(U136_ag(x_1, x_2)) = 0 POL(U137_gga(x_1, x_2, x_3)) = 0 POL(U138_gga(x_1, x_2, x_3)) = 0 POL(U139_gaga(x_1, x_2, x_3)) = 0 POL(U140_ga(x_1, x_2)) = 0 POL(U149_gaaaa(x_1, x_2)) = 0 POL(U17_GAGAA(x_1, x_2, x_3)) = x_3 POL(U186_agagaa(x_1, x_2, x_3)) = 2*x_3 POL(U187_agagaa(x_1, x_2, x_3, x_4)) = 0 POL(U188_agagaa(x_1, x_2, x_3, x_4)) = 0 POL(U189_agagaa(x_1, x_2, x_3)) = 0 POL(U190_agaaaa(x_1, x_2)) = 0 POL([]) = 2 POL(checkcM_in_agagaa(x_1, x_2)) = 2*x_1 POL(checkcM_out_agagaa(x_1, x_2, x_3)) = 0 POL(difcN1_in_agaaaa(x_1)) = x_1 POL(difcN1_out_agaaaa(x_1, x_2, x_3)) = 0 POL(difcN_in_gaaaa(x_1)) = 0 POL(difcN_out_gaaaa(x_1, x_2, x_3)) = 0 POL(membercB_in_ag(x_1)) = 0 POL(membercB_out_ag(x_1)) = 0 POL(mergedeletecC_in_gga(x_1, x_2)) = 0 POL(mergedeletecC_out_gga(x_1, x_2, x_3)) = 0 POL(revcD_in_gaga(x_1, x_2)) = 0 POL(revcD_out_gaga(x_1, x_2, x_3)) = 0 POL(revcF_in_ga(x_1)) = 2*x_1 POL(revcF_out_ga(x_1, x_2)) = 0 ---------------------------------------- (385) Obligation: Q DP problem: The TRS P consists of the following rules: PDS1L_IN_GAGAA(X1, X3) -> U10_GAGAA(X1, X3, membercB_in_ag(X1)) U10_GAGAA(X1, X3, membercB_out_ag(X1)) -> U17_GAGAA(X1, X3, checkcM_in_agagaa(X3, X1)) U17_GAGAA(X1, X3, checkcM_out_agagaa(X3, X1, X7)) -> PDS1L_IN_GAGAA(X7, .(X3)) The TRS R consists of the following rules: membercB_in_ag(.(X2)) -> membercB_out_ag(.(X2)) membercB_in_ag(.(X3)) -> U136_ag(X3, membercB_in_ag(X3)) checkcM_in_agagaa(X2, X4) -> U186_agagaa(X2, X4, difcN1_in_agaaaa(X2)) U136_ag(X3, membercB_out_ag(X3)) -> membercB_out_ag(.(X3)) U186_agagaa(X2, X4, difcN1_out_agaaaa(X2, X7, X8)) -> U187_agagaa(X2, X4, X8, mergedeletecC_in_gga(X7, X4)) difcN1_in_agaaaa(.([])) -> difcN1_out_agaaaa(.([]), .(.([])), .(.([]))) difcN1_in_agaaaa(.(.([]))) -> difcN1_out_agaaaa(.(.([])), .(.(.([]))), .(.(.([])))) difcN1_in_agaaaa(.(.(.([])))) -> difcN1_out_agaaaa(.(.(.([]))), .(.(.(.([])))), .(.(.(.([]))))) difcN1_in_agaaaa(.(.(.(.([]))))) -> difcN1_out_agaaaa(.(.(.(.([])))), .(.(.(.(.([]))))), .(.(.(.(.([])))))) difcN1_in_agaaaa(.(.(.(.(.([])))))) -> difcN1_out_agaaaa(.(.(.(.(.([]))))), .(.(.(.(.(.([])))))), .(.(.(.(.(.([]))))))) difcN1_in_agaaaa(.(.(.(.(.(.([]))))))) -> difcN1_out_agaaaa(.(.(.(.(.(.([])))))), .(.(.(.(.(.(.([]))))))), .(.(.(.(.(.(.([])))))))) difcN1_in_agaaaa(.(.(.(.(.(.(.(X9)))))))) -> U190_agaaaa(X9, difcN_in_gaaaa(X9)) U187_agagaa(X2, X4, X8, mergedeletecC_out_gga(X7, X4, X9)) -> U188_agagaa(X2, X4, X9, revcF_in_ga(X8)) U190_agaaaa(X9, difcN_out_gaaaa(X9, X20, X29)) -> difcN1_out_agaaaa(.(.(.(.(.(.(.(X9))))))), .(.(.(.(.(.(.(.(X20)))))))), .(.(.(.(.(.(.(.(X29))))))))) mergedeletecC_in_gga([], X1) -> mergedeletecC_out_gga([], X1, X1) mergedeletecC_in_gga(.(X2), .(X3)) -> U137_gga(X2, X3, mergedeletecC_in_gga(X2, X3)) mergedeletecC_in_gga(.(X2), .(X4)) -> U138_gga(X2, X4, mergedeletecC_in_gga(.(X2), X4)) U188_agagaa(X2, X4, X9, revcF_out_ga(X8, X10)) -> U189_agagaa(X2, X4, mergedeletecC_in_gga(X10, X9)) difcN_in_gaaaa([]) -> difcN_out_gaaaa([], [], []) difcN_in_gaaaa(.(X2)) -> U149_gaaaa(X2, difcN_in_gaaaa(X2)) U137_gga(X2, X3, mergedeletecC_out_gga(X2, X3, X4)) -> mergedeletecC_out_gga(.(X2), .(X3), X4) U138_gga(X2, X4, mergedeletecC_out_gga(.(X2), X4, X5)) -> mergedeletecC_out_gga(.(X2), .(X4), .(X5)) revcF_in_ga([]) -> revcF_out_ga([], []) revcF_in_ga(.([])) -> revcF_out_ga(.([]), .([])) revcF_in_ga(.(.([]))) -> revcF_out_ga(.(.([])), .(.([]))) revcF_in_ga(.(.(.([])))) -> revcF_out_ga(.(.(.([]))), .(.(.([])))) revcF_in_ga(.(.(.(.([]))))) -> revcF_out_ga(.(.(.(.([])))), .(.(.(.([]))))) revcF_in_ga(.(.(.(.(.([])))))) -> revcF_out_ga(.(.(.(.(.([]))))), .(.(.(.(.([])))))) revcF_in_ga(.(.(.(.(.(.([]))))))) -> revcF_out_ga(.(.(.(.(.(.([])))))), .(.(.(.(.(.([]))))))) revcF_in_ga(.(.(.(.(.(.(.([])))))))) -> revcF_out_ga(.(.(.(.(.(.(.([]))))))), .(.(.(.(.(.(.([])))))))) revcF_in_ga(.(.(.(.(.(.(.(.(X9))))))))) -> U140_ga(X9, revcD_in_gaga(X9, .(.(.(.(.(.(.([]))))))))) U189_agagaa(X2, X4, mergedeletecC_out_gga(X10, X9, X5)) -> checkcM_out_agagaa(X2, X4, X5) U149_gaaaa(X2, difcN_out_gaaaa(X2, X6, X8)) -> difcN_out_gaaaa(.(X2), .(X6), .(X8)) U140_ga(X9, revcD_out_gaga(X9, .(.(.(.(.(.(.([]))))))), X10)) -> revcF_out_ga(.(.(.(.(.(.(.(.(X9)))))))), X10) revcD_in_gaga([], X2) -> revcD_out_gaga([], X2, .(X2)) revcD_in_gaga(.(X2), X4) -> U139_gaga(X2, X4, revcD_in_gaga(X2, .(X4))) U139_gaga(X2, X4, revcD_out_gaga(X2, .(X4), X5)) -> revcD_out_gaga(.(X2), X4, X5) The set Q consists of the following terms: membercB_in_ag(x0) checkcM_in_agagaa(x0, x1) U136_ag(x0, x1) U186_agagaa(x0, x1, x2) difcN1_in_agaaaa(x0) U187_agagaa(x0, x1, x2, x3) U190_agaaaa(x0, x1) mergedeletecC_in_gga(x0, x1) U188_agagaa(x0, x1, x2, x3) difcN_in_gaaaa(x0) U137_gga(x0, x1, x2) U138_gga(x0, x1, x2) revcF_in_ga(x0) U189_agagaa(x0, x1, x2) U149_gaaaa(x0, x1) U140_ga(x0, x1) revcD_in_gaga(x0, x1) U139_gaga(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (386) 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: U138_gga(X2, X4, mergedeletecC_out_gga(.(X2), X4, X5)) -> mergedeletecC_out_gga(.(X2), .(X4), .(X5)) Used ordering: Polynomial interpretation [POLO]: POL(.(x_1)) = 2 + 2*x_1 POL(PDS1L_IN_GAGAA(x_1, x_2)) = x_1 POL(U10_GAGAA(x_1, x_2, x_3)) = x_1 POL(U136_ag(x_1, x_2)) = 0 POL(U137_gga(x_1, x_2, x_3)) = 2 + 2*x_3 POL(U138_gga(x_1, x_2, x_3)) = 2 + 2*x_3 POL(U139_gaga(x_1, x_2, x_3)) = 0 POL(U140_ga(x_1, x_2)) = 0 POL(U149_gaaaa(x_1, x_2)) = 0 POL(U17_GAGAA(x_1, x_2, x_3)) = x_3 POL(U186_agagaa(x_1, x_2, x_3)) = x_2 POL(U187_agagaa(x_1, x_2, x_3, x_4)) = x_4 POL(U188_agagaa(x_1, x_2, x_3, x_4)) = x_3 POL(U189_agagaa(x_1, x_2, x_3)) = x_3 POL(U190_agaaaa(x_1, x_2)) = 0 POL([]) = 0 POL(checkcM_in_agagaa(x_1, x_2)) = x_2 POL(checkcM_out_agagaa(x_1, x_2, x_3)) = x_3 POL(difcN1_in_agaaaa(x_1)) = 1 POL(difcN1_out_agaaaa(x_1, x_2, x_3)) = 0 POL(difcN_in_gaaaa(x_1)) = 0 POL(difcN_out_gaaaa(x_1, x_2, x_3)) = 0 POL(membercB_in_ag(x_1)) = 0 POL(membercB_out_ag(x_1)) = 0 POL(mergedeletecC_in_gga(x_1, x_2)) = x_2 POL(mergedeletecC_out_gga(x_1, x_2, x_3)) = x_1 + x_3 POL(revcD_in_gaga(x_1, x_2)) = 0 POL(revcD_out_gaga(x_1, x_2, x_3)) = 0 POL(revcF_in_ga(x_1)) = 0 POL(revcF_out_ga(x_1, x_2)) = 0 ---------------------------------------- (387) Obligation: Q DP problem: The TRS P consists of the following rules: PDS1L_IN_GAGAA(X1, X3) -> U10_GAGAA(X1, X3, membercB_in_ag(X1)) U10_GAGAA(X1, X3, membercB_out_ag(X1)) -> U17_GAGAA(X1, X3, checkcM_in_agagaa(X3, X1)) U17_GAGAA(X1, X3, checkcM_out_agagaa(X3, X1, X7)) -> PDS1L_IN_GAGAA(X7, .(X3)) The TRS R consists of the following rules: membercB_in_ag(.(X2)) -> membercB_out_ag(.(X2)) membercB_in_ag(.(X3)) -> U136_ag(X3, membercB_in_ag(X3)) checkcM_in_agagaa(X2, X4) -> U186_agagaa(X2, X4, difcN1_in_agaaaa(X2)) U136_ag(X3, membercB_out_ag(X3)) -> membercB_out_ag(.(X3)) U186_agagaa(X2, X4, difcN1_out_agaaaa(X2, X7, X8)) -> U187_agagaa(X2, X4, X8, mergedeletecC_in_gga(X7, X4)) difcN1_in_agaaaa(.([])) -> difcN1_out_agaaaa(.([]), .(.([])), .(.([]))) difcN1_in_agaaaa(.(.([]))) -> difcN1_out_agaaaa(.(.([])), .(.(.([]))), .(.(.([])))) difcN1_in_agaaaa(.(.(.([])))) -> difcN1_out_agaaaa(.(.(.([]))), .(.(.(.([])))), .(.(.(.([]))))) difcN1_in_agaaaa(.(.(.(.([]))))) -> difcN1_out_agaaaa(.(.(.(.([])))), .(.(.(.(.([]))))), .(.(.(.(.([])))))) difcN1_in_agaaaa(.(.(.(.(.([])))))) -> difcN1_out_agaaaa(.(.(.(.(.([]))))), .(.(.(.(.(.([])))))), .(.(.(.(.(.([]))))))) difcN1_in_agaaaa(.(.(.(.(.(.([]))))))) -> difcN1_out_agaaaa(.(.(.(.(.(.([])))))), .(.(.(.(.(.(.([]))))))), .(.(.(.(.(.(.([])))))))) difcN1_in_agaaaa(.(.(.(.(.(.(.(X9)))))))) -> U190_agaaaa(X9, difcN_in_gaaaa(X9)) U187_agagaa(X2, X4, X8, mergedeletecC_out_gga(X7, X4, X9)) -> U188_agagaa(X2, X4, X9, revcF_in_ga(X8)) U190_agaaaa(X9, difcN_out_gaaaa(X9, X20, X29)) -> difcN1_out_agaaaa(.(.(.(.(.(.(.(X9))))))), .(.(.(.(.(.(.(.(X20)))))))), .(.(.(.(.(.(.(.(X29))))))))) mergedeletecC_in_gga([], X1) -> mergedeletecC_out_gga([], X1, X1) mergedeletecC_in_gga(.(X2), .(X3)) -> U137_gga(X2, X3, mergedeletecC_in_gga(X2, X3)) mergedeletecC_in_gga(.(X2), .(X4)) -> U138_gga(X2, X4, mergedeletecC_in_gga(.(X2), X4)) U188_agagaa(X2, X4, X9, revcF_out_ga(X8, X10)) -> U189_agagaa(X2, X4, mergedeletecC_in_gga(X10, X9)) difcN_in_gaaaa([]) -> difcN_out_gaaaa([], [], []) difcN_in_gaaaa(.(X2)) -> U149_gaaaa(X2, difcN_in_gaaaa(X2)) U137_gga(X2, X3, mergedeletecC_out_gga(X2, X3, X4)) -> mergedeletecC_out_gga(.(X2), .(X3), X4) revcF_in_ga([]) -> revcF_out_ga([], []) revcF_in_ga(.([])) -> revcF_out_ga(.([]), .([])) revcF_in_ga(.(.([]))) -> revcF_out_ga(.(.([])), .(.([]))) revcF_in_ga(.(.(.([])))) -> revcF_out_ga(.(.(.([]))), .(.(.([])))) revcF_in_ga(.(.(.(.([]))))) -> revcF_out_ga(.(.(.(.([])))), .(.(.(.([]))))) revcF_in_ga(.(.(.(.(.([])))))) -> revcF_out_ga(.(.(.(.(.([]))))), .(.(.(.(.([])))))) revcF_in_ga(.(.(.(.(.(.([]))))))) -> revcF_out_ga(.(.(.(.(.(.([])))))), .(.(.(.(.(.([]))))))) revcF_in_ga(.(.(.(.(.(.(.([])))))))) -> revcF_out_ga(.(.(.(.(.(.(.([]))))))), .(.(.(.(.(.(.([])))))))) revcF_in_ga(.(.(.(.(.(.(.(.(X9))))))))) -> U140_ga(X9, revcD_in_gaga(X9, .(.(.(.(.(.(.([]))))))))) U189_agagaa(X2, X4, mergedeletecC_out_gga(X10, X9, X5)) -> checkcM_out_agagaa(X2, X4, X5) U149_gaaaa(X2, difcN_out_gaaaa(X2, X6, X8)) -> difcN_out_gaaaa(.(X2), .(X6), .(X8)) U140_ga(X9, revcD_out_gaga(X9, .(.(.(.(.(.(.([]))))))), X10)) -> revcF_out_ga(.(.(.(.(.(.(.(.(X9)))))))), X10) revcD_in_gaga([], X2) -> revcD_out_gaga([], X2, .(X2)) revcD_in_gaga(.(X2), X4) -> U139_gaga(X2, X4, revcD_in_gaga(X2, .(X4))) U139_gaga(X2, X4, revcD_out_gaga(X2, .(X4), X5)) -> revcD_out_gaga(.(X2), X4, X5) The set Q consists of the following terms: membercB_in_ag(x0) checkcM_in_agagaa(x0, x1) U136_ag(x0, x1) U186_agagaa(x0, x1, x2) difcN1_in_agaaaa(x0) U187_agagaa(x0, x1, x2, x3) U190_agaaaa(x0, x1) mergedeletecC_in_gga(x0, x1) U188_agagaa(x0, x1, x2, x3) difcN_in_gaaaa(x0) U137_gga(x0, x1, x2) U138_gga(x0, x1, x2) revcF_in_ga(x0) U189_agagaa(x0, x1, x2) U149_gaaaa(x0, x1) U140_ga(x0, x1) revcD_in_gaga(x0, x1) U139_gaga(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (388) 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: mergedeletecC_in_gga(.(X2), .(X4)) -> U138_gga(X2, X4, mergedeletecC_in_gga(.(X2), X4)) Used ordering: Polynomial interpretation [POLO]: POL(.(x_1)) = 2 + x_1 POL(PDS1L_IN_GAGAA(x_1, x_2)) = 2*x_1 POL(U10_GAGAA(x_1, x_2, x_3)) = 2*x_1 POL(U136_ag(x_1, x_2)) = 0 POL(U137_gga(x_1, x_2, x_3)) = 2 + x_3 POL(U138_gga(x_1, x_2, x_3)) = x_3 POL(U139_gaga(x_1, x_2, x_3)) = 0 POL(U140_ga(x_1, x_2)) = 0 POL(U149_gaaaa(x_1, x_2)) = 0 POL(U17_GAGAA(x_1, x_2, x_3)) = 2*x_3 POL(U186_agagaa(x_1, x_2, x_3)) = x_2 POL(U187_agagaa(x_1, x_2, x_3, x_4)) = x_4 POL(U188_agagaa(x_1, x_2, x_3, x_4)) = x_3 POL(U189_agagaa(x_1, x_2, x_3)) = x_3 POL(U190_agaaaa(x_1, x_2)) = 0 POL([]) = 0 POL(checkcM_in_agagaa(x_1, x_2)) = x_2 POL(checkcM_out_agagaa(x_1, x_2, x_3)) = x_3 POL(difcN1_in_agaaaa(x_1)) = 1 POL(difcN1_out_agaaaa(x_1, x_2, x_3)) = 0 POL(difcN_in_gaaaa(x_1)) = 0 POL(difcN_out_gaaaa(x_1, x_2, x_3)) = 0 POL(membercB_in_ag(x_1)) = 0 POL(membercB_out_ag(x_1)) = 0 POL(mergedeletecC_in_gga(x_1, x_2)) = x_2 POL(mergedeletecC_out_gga(x_1, x_2, x_3)) = x_1 + x_3 POL(revcD_in_gaga(x_1, x_2)) = 0 POL(revcD_out_gaga(x_1, x_2, x_3)) = 0 POL(revcF_in_ga(x_1)) = x_1 POL(revcF_out_ga(x_1, x_2)) = 0 ---------------------------------------- (389) Obligation: Q DP problem: The TRS P consists of the following rules: PDS1L_IN_GAGAA(X1, X3) -> U10_GAGAA(X1, X3, membercB_in_ag(X1)) U10_GAGAA(X1, X3, membercB_out_ag(X1)) -> U17_GAGAA(X1, X3, checkcM_in_agagaa(X3, X1)) U17_GAGAA(X1, X3, checkcM_out_agagaa(X3, X1, X7)) -> PDS1L_IN_GAGAA(X7, .(X3)) The TRS R consists of the following rules: membercB_in_ag(.(X2)) -> membercB_out_ag(.(X2)) membercB_in_ag(.(X3)) -> U136_ag(X3, membercB_in_ag(X3)) checkcM_in_agagaa(X2, X4) -> U186_agagaa(X2, X4, difcN1_in_agaaaa(X2)) U136_ag(X3, membercB_out_ag(X3)) -> membercB_out_ag(.(X3)) U186_agagaa(X2, X4, difcN1_out_agaaaa(X2, X7, X8)) -> U187_agagaa(X2, X4, X8, mergedeletecC_in_gga(X7, X4)) difcN1_in_agaaaa(.([])) -> difcN1_out_agaaaa(.([]), .(.([])), .(.([]))) difcN1_in_agaaaa(.(.([]))) -> difcN1_out_agaaaa(.(.([])), .(.(.([]))), .(.(.([])))) difcN1_in_agaaaa(.(.(.([])))) -> difcN1_out_agaaaa(.(.(.([]))), .(.(.(.([])))), .(.(.(.([]))))) difcN1_in_agaaaa(.(.(.(.([]))))) -> difcN1_out_agaaaa(.(.(.(.([])))), .(.(.(.(.([]))))), .(.(.(.(.([])))))) difcN1_in_agaaaa(.(.(.(.(.([])))))) -> difcN1_out_agaaaa(.(.(.(.(.([]))))), .(.(.(.(.(.([])))))), .(.(.(.(.(.([]))))))) difcN1_in_agaaaa(.(.(.(.(.(.([]))))))) -> difcN1_out_agaaaa(.(.(.(.(.(.([])))))), .(.(.(.(.(.(.([]))))))), .(.(.(.(.(.(.([])))))))) difcN1_in_agaaaa(.(.(.(.(.(.(.(X9)))))))) -> U190_agaaaa(X9, difcN_in_gaaaa(X9)) U187_agagaa(X2, X4, X8, mergedeletecC_out_gga(X7, X4, X9)) -> U188_agagaa(X2, X4, X9, revcF_in_ga(X8)) U190_agaaaa(X9, difcN_out_gaaaa(X9, X20, X29)) -> difcN1_out_agaaaa(.(.(.(.(.(.(.(X9))))))), .(.(.(.(.(.(.(.(X20)))))))), .(.(.(.(.(.(.(.(X29))))))))) mergedeletecC_in_gga([], X1) -> mergedeletecC_out_gga([], X1, X1) mergedeletecC_in_gga(.(X2), .(X3)) -> U137_gga(X2, X3, mergedeletecC_in_gga(X2, X3)) U188_agagaa(X2, X4, X9, revcF_out_ga(X8, X10)) -> U189_agagaa(X2, X4, mergedeletecC_in_gga(X10, X9)) difcN_in_gaaaa([]) -> difcN_out_gaaaa([], [], []) difcN_in_gaaaa(.(X2)) -> U149_gaaaa(X2, difcN_in_gaaaa(X2)) U137_gga(X2, X3, mergedeletecC_out_gga(X2, X3, X4)) -> mergedeletecC_out_gga(.(X2), .(X3), X4) revcF_in_ga([]) -> revcF_out_ga([], []) revcF_in_ga(.([])) -> revcF_out_ga(.([]), .([])) revcF_in_ga(.(.([]))) -> revcF_out_ga(.(.([])), .(.([]))) revcF_in_ga(.(.(.([])))) -> revcF_out_ga(.(.(.([]))), .(.(.([])))) revcF_in_ga(.(.(.(.([]))))) -> revcF_out_ga(.(.(.(.([])))), .(.(.(.([]))))) revcF_in_ga(.(.(.(.(.([])))))) -> revcF_out_ga(.(.(.(.(.([]))))), .(.(.(.(.([])))))) revcF_in_ga(.(.(.(.(.(.([]))))))) -> revcF_out_ga(.(.(.(.(.(.([])))))), .(.(.(.(.(.([]))))))) revcF_in_ga(.(.(.(.(.(.(.([])))))))) -> revcF_out_ga(.(.(.(.(.(.(.([]))))))), .(.(.(.(.(.(.([])))))))) revcF_in_ga(.(.(.(.(.(.(.(.(X9))))))))) -> U140_ga(X9, revcD_in_gaga(X9, .(.(.(.(.(.(.([]))))))))) U189_agagaa(X2, X4, mergedeletecC_out_gga(X10, X9, X5)) -> checkcM_out_agagaa(X2, X4, X5) U149_gaaaa(X2, difcN_out_gaaaa(X2, X6, X8)) -> difcN_out_gaaaa(.(X2), .(X6), .(X8)) U140_ga(X9, revcD_out_gaga(X9, .(.(.(.(.(.(.([]))))))), X10)) -> revcF_out_ga(.(.(.(.(.(.(.(.(X9)))))))), X10) revcD_in_gaga([], X2) -> revcD_out_gaga([], X2, .(X2)) revcD_in_gaga(.(X2), X4) -> U139_gaga(X2, X4, revcD_in_gaga(X2, .(X4))) U139_gaga(X2, X4, revcD_out_gaga(X2, .(X4), X5)) -> revcD_out_gaga(.(X2), X4, X5) The set Q consists of the following terms: membercB_in_ag(x0) checkcM_in_agagaa(x0, x1) U136_ag(x0, x1) U186_agagaa(x0, x1, x2) difcN1_in_agaaaa(x0) U187_agagaa(x0, x1, x2, x3) U190_agaaaa(x0, x1) mergedeletecC_in_gga(x0, x1) U188_agagaa(x0, x1, x2, x3) difcN_in_gaaaa(x0) U137_gga(x0, x1, x2) U138_gga(x0, x1, x2) revcF_in_ga(x0) U189_agagaa(x0, x1, x2) U149_gaaaa(x0, x1) U140_ga(x0, x1) revcD_in_gaga(x0, x1) U139_gaga(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (390) 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]. U138_gga(x0, x1, x2) ---------------------------------------- (391) Obligation: Q DP problem: The TRS P consists of the following rules: PDS1L_IN_GAGAA(X1, X3) -> U10_GAGAA(X1, X3, membercB_in_ag(X1)) U10_GAGAA(X1, X3, membercB_out_ag(X1)) -> U17_GAGAA(X1, X3, checkcM_in_agagaa(X3, X1)) U17_GAGAA(X1, X3, checkcM_out_agagaa(X3, X1, X7)) -> PDS1L_IN_GAGAA(X7, .(X3)) The TRS R consists of the following rules: membercB_in_ag(.(X2)) -> membercB_out_ag(.(X2)) membercB_in_ag(.(X3)) -> U136_ag(X3, membercB_in_ag(X3)) checkcM_in_agagaa(X2, X4) -> U186_agagaa(X2, X4, difcN1_in_agaaaa(X2)) U136_ag(X3, membercB_out_ag(X3)) -> membercB_out_ag(.(X3)) U186_agagaa(X2, X4, difcN1_out_agaaaa(X2, X7, X8)) -> U187_agagaa(X2, X4, X8, mergedeletecC_in_gga(X7, X4)) difcN1_in_agaaaa(.([])) -> difcN1_out_agaaaa(.([]), .(.([])), .(.([]))) difcN1_in_agaaaa(.(.([]))) -> difcN1_out_agaaaa(.(.([])), .(.(.([]))), .(.(.([])))) difcN1_in_agaaaa(.(.(.([])))) -> difcN1_out_agaaaa(.(.(.([]))), .(.(.(.([])))), .(.(.(.([]))))) difcN1_in_agaaaa(.(.(.(.([]))))) -> difcN1_out_agaaaa(.(.(.(.([])))), .(.(.(.(.([]))))), .(.(.(.(.([])))))) difcN1_in_agaaaa(.(.(.(.(.([])))))) -> difcN1_out_agaaaa(.(.(.(.(.([]))))), .(.(.(.(.(.([])))))), .(.(.(.(.(.([]))))))) difcN1_in_agaaaa(.(.(.(.(.(.([]))))))) -> difcN1_out_agaaaa(.(.(.(.(.(.([])))))), .(.(.(.(.(.(.([]))))))), .(.(.(.(.(.(.([])))))))) difcN1_in_agaaaa(.(.(.(.(.(.(.(X9)))))))) -> U190_agaaaa(X9, difcN_in_gaaaa(X9)) U187_agagaa(X2, X4, X8, mergedeletecC_out_gga(X7, X4, X9)) -> U188_agagaa(X2, X4, X9, revcF_in_ga(X8)) U190_agaaaa(X9, difcN_out_gaaaa(X9, X20, X29)) -> difcN1_out_agaaaa(.(.(.(.(.(.(.(X9))))))), .(.(.(.(.(.(.(.(X20)))))))), .(.(.(.(.(.(.(.(X29))))))))) mergedeletecC_in_gga([], X1) -> mergedeletecC_out_gga([], X1, X1) mergedeletecC_in_gga(.(X2), .(X3)) -> U137_gga(X2, X3, mergedeletecC_in_gga(X2, X3)) U188_agagaa(X2, X4, X9, revcF_out_ga(X8, X10)) -> U189_agagaa(X2, X4, mergedeletecC_in_gga(X10, X9)) difcN_in_gaaaa([]) -> difcN_out_gaaaa([], [], []) difcN_in_gaaaa(.(X2)) -> U149_gaaaa(X2, difcN_in_gaaaa(X2)) U137_gga(X2, X3, mergedeletecC_out_gga(X2, X3, X4)) -> mergedeletecC_out_gga(.(X2), .(X3), X4) revcF_in_ga([]) -> revcF_out_ga([], []) revcF_in_ga(.([])) -> revcF_out_ga(.([]), .([])) revcF_in_ga(.(.([]))) -> revcF_out_ga(.(.([])), .(.([]))) revcF_in_ga(.(.(.([])))) -> revcF_out_ga(.(.(.([]))), .(.(.([])))) revcF_in_ga(.(.(.(.([]))))) -> revcF_out_ga(.(.(.(.([])))), .(.(.(.([]))))) revcF_in_ga(.(.(.(.(.([])))))) -> revcF_out_ga(.(.(.(.(.([]))))), .(.(.(.(.([])))))) revcF_in_ga(.(.(.(.(.(.([]))))))) -> revcF_out_ga(.(.(.(.(.(.([])))))), .(.(.(.(.(.([]))))))) revcF_in_ga(.(.(.(.(.(.(.([])))))))) -> revcF_out_ga(.(.(.(.(.(.(.([]))))))), .(.(.(.(.(.(.([])))))))) revcF_in_ga(.(.(.(.(.(.(.(.(X9))))))))) -> U140_ga(X9, revcD_in_gaga(X9, .(.(.(.(.(.(.([]))))))))) U189_agagaa(X2, X4, mergedeletecC_out_gga(X10, X9, X5)) -> checkcM_out_agagaa(X2, X4, X5) U149_gaaaa(X2, difcN_out_gaaaa(X2, X6, X8)) -> difcN_out_gaaaa(.(X2), .(X6), .(X8)) U140_ga(X9, revcD_out_gaga(X9, .(.(.(.(.(.(.([]))))))), X10)) -> revcF_out_ga(.(.(.(.(.(.(.(.(X9)))))))), X10) revcD_in_gaga([], X2) -> revcD_out_gaga([], X2, .(X2)) revcD_in_gaga(.(X2), X4) -> U139_gaga(X2, X4, revcD_in_gaga(X2, .(X4))) U139_gaga(X2, X4, revcD_out_gaga(X2, .(X4), X5)) -> revcD_out_gaga(.(X2), X4, X5) The set Q consists of the following terms: membercB_in_ag(x0) checkcM_in_agagaa(x0, x1) U136_ag(x0, x1) U186_agagaa(x0, x1, x2) difcN1_in_agaaaa(x0) U187_agagaa(x0, x1, x2, x3) U190_agaaaa(x0, x1) mergedeletecC_in_gga(x0, x1) U188_agagaa(x0, x1, x2, x3) difcN_in_gaaaa(x0) U137_gga(x0, x1, x2) revcF_in_ga(x0) U189_agagaa(x0, x1, x2) U149_gaaaa(x0, x1) U140_ga(x0, x1) revcD_in_gaga(x0, x1) U139_gaga(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (392) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule U10_GAGAA(X1, X3, membercB_out_ag(X1)) -> U17_GAGAA(X1, X3, checkcM_in_agagaa(X3, X1)) at position [2] we obtained the following new rules [LPAR04]: (U10_GAGAA(X1, X3, membercB_out_ag(X1)) -> U17_GAGAA(X1, X3, U186_agagaa(X3, X1, difcN1_in_agaaaa(X3))),U10_GAGAA(X1, X3, membercB_out_ag(X1)) -> U17_GAGAA(X1, X3, U186_agagaa(X3, X1, difcN1_in_agaaaa(X3)))) ---------------------------------------- (393) Obligation: Q DP problem: The TRS P consists of the following rules: PDS1L_IN_GAGAA(X1, X3) -> U10_GAGAA(X1, X3, membercB_in_ag(X1)) U17_GAGAA(X1, X3, checkcM_out_agagaa(X3, X1, X7)) -> PDS1L_IN_GAGAA(X7, .(X3)) U10_GAGAA(X1, X3, membercB_out_ag(X1)) -> U17_GAGAA(X1, X3, U186_agagaa(X3, X1, difcN1_in_agaaaa(X3))) The TRS R consists of the following rules: membercB_in_ag(.(X2)) -> membercB_out_ag(.(X2)) membercB_in_ag(.(X3)) -> U136_ag(X3, membercB_in_ag(X3)) checkcM_in_agagaa(X2, X4) -> U186_agagaa(X2, X4, difcN1_in_agaaaa(X2)) U136_ag(X3, membercB_out_ag(X3)) -> membercB_out_ag(.(X3)) U186_agagaa(X2, X4, difcN1_out_agaaaa(X2, X7, X8)) -> U187_agagaa(X2, X4, X8, mergedeletecC_in_gga(X7, X4)) difcN1_in_agaaaa(.([])) -> difcN1_out_agaaaa(.([]), .(.([])), .(.([]))) difcN1_in_agaaaa(.(.([]))) -> difcN1_out_agaaaa(.(.([])), .(.(.([]))), .(.(.([])))) difcN1_in_agaaaa(.(.(.([])))) -> difcN1_out_agaaaa(.(.(.([]))), .(.(.(.([])))), .(.(.(.([]))))) difcN1_in_agaaaa(.(.(.(.([]))))) -> difcN1_out_agaaaa(.(.(.(.([])))), .(.(.(.(.([]))))), .(.(.(.(.([])))))) difcN1_in_agaaaa(.(.(.(.(.([])))))) -> difcN1_out_agaaaa(.(.(.(.(.([]))))), .(.(.(.(.(.([])))))), .(.(.(.(.(.([]))))))) difcN1_in_agaaaa(.(.(.(.(.(.([]))))))) -> difcN1_out_agaaaa(.(.(.(.(.(.([])))))), .(.(.(.(.(.(.([]))))))), .(.(.(.(.(.(.([])))))))) difcN1_in_agaaaa(.(.(.(.(.(.(.(X9)))))))) -> U190_agaaaa(X9, difcN_in_gaaaa(X9)) U187_agagaa(X2, X4, X8, mergedeletecC_out_gga(X7, X4, X9)) -> U188_agagaa(X2, X4, X9, revcF_in_ga(X8)) U190_agaaaa(X9, difcN_out_gaaaa(X9, X20, X29)) -> difcN1_out_agaaaa(.(.(.(.(.(.(.(X9))))))), .(.(.(.(.(.(.(.(X20)))))))), .(.(.(.(.(.(.(.(X29))))))))) mergedeletecC_in_gga([], X1) -> mergedeletecC_out_gga([], X1, X1) mergedeletecC_in_gga(.(X2), .(X3)) -> U137_gga(X2, X3, mergedeletecC_in_gga(X2, X3)) U188_agagaa(X2, X4, X9, revcF_out_ga(X8, X10)) -> U189_agagaa(X2, X4, mergedeletecC_in_gga(X10, X9)) difcN_in_gaaaa([]) -> difcN_out_gaaaa([], [], []) difcN_in_gaaaa(.(X2)) -> U149_gaaaa(X2, difcN_in_gaaaa(X2)) U137_gga(X2, X3, mergedeletecC_out_gga(X2, X3, X4)) -> mergedeletecC_out_gga(.(X2), .(X3), X4) revcF_in_ga([]) -> revcF_out_ga([], []) revcF_in_ga(.([])) -> revcF_out_ga(.([]), .([])) revcF_in_ga(.(.([]))) -> revcF_out_ga(.(.([])), .(.([]))) revcF_in_ga(.(.(.([])))) -> revcF_out_ga(.(.(.([]))), .(.(.([])))) revcF_in_ga(.(.(.(.([]))))) -> revcF_out_ga(.(.(.(.([])))), .(.(.(.([]))))) revcF_in_ga(.(.(.(.(.([])))))) -> revcF_out_ga(.(.(.(.(.([]))))), .(.(.(.(.([])))))) revcF_in_ga(.(.(.(.(.(.([]))))))) -> revcF_out_ga(.(.(.(.(.(.([])))))), .(.(.(.(.(.([]))))))) revcF_in_ga(.(.(.(.(.(.(.([])))))))) -> revcF_out_ga(.(.(.(.(.(.(.([]))))))), .(.(.(.(.(.(.([])))))))) revcF_in_ga(.(.(.(.(.(.(.(.(X9))))))))) -> U140_ga(X9, revcD_in_gaga(X9, .(.(.(.(.(.(.([]))))))))) U189_agagaa(X2, X4, mergedeletecC_out_gga(X10, X9, X5)) -> checkcM_out_agagaa(X2, X4, X5) U149_gaaaa(X2, difcN_out_gaaaa(X2, X6, X8)) -> difcN_out_gaaaa(.(X2), .(X6), .(X8)) U140_ga(X9, revcD_out_gaga(X9, .(.(.(.(.(.(.([]))))))), X10)) -> revcF_out_ga(.(.(.(.(.(.(.(.(X9)))))))), X10) revcD_in_gaga([], X2) -> revcD_out_gaga([], X2, .(X2)) revcD_in_gaga(.(X2), X4) -> U139_gaga(X2, X4, revcD_in_gaga(X2, .(X4))) U139_gaga(X2, X4, revcD_out_gaga(X2, .(X4), X5)) -> revcD_out_gaga(.(X2), X4, X5) The set Q consists of the following terms: membercB_in_ag(x0) checkcM_in_agagaa(x0, x1) U136_ag(x0, x1) U186_agagaa(x0, x1, x2) difcN1_in_agaaaa(x0) U187_agagaa(x0, x1, x2, x3) U190_agaaaa(x0, x1) mergedeletecC_in_gga(x0, x1) U188_agagaa(x0, x1, x2, x3) difcN_in_gaaaa(x0) U137_gga(x0, x1, x2) revcF_in_ga(x0) U189_agagaa(x0, x1, x2) U149_gaaaa(x0, x1) U140_ga(x0, x1) revcD_in_gaga(x0, x1) U139_gaga(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (394) 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. ---------------------------------------- (395) Obligation: Q DP problem: The TRS P consists of the following rules: PDS1L_IN_GAGAA(X1, X3) -> U10_GAGAA(X1, X3, membercB_in_ag(X1)) U17_GAGAA(X1, X3, checkcM_out_agagaa(X3, X1, X7)) -> PDS1L_IN_GAGAA(X7, .(X3)) U10_GAGAA(X1, X3, membercB_out_ag(X1)) -> U17_GAGAA(X1, X3, U186_agagaa(X3, X1, difcN1_in_agaaaa(X3))) The TRS R consists of the following rules: difcN1_in_agaaaa(.([])) -> difcN1_out_agaaaa(.([]), .(.([])), .(.([]))) difcN1_in_agaaaa(.(.([]))) -> difcN1_out_agaaaa(.(.([])), .(.(.([]))), .(.(.([])))) difcN1_in_agaaaa(.(.(.([])))) -> difcN1_out_agaaaa(.(.(.([]))), .(.(.(.([])))), .(.(.(.([]))))) difcN1_in_agaaaa(.(.(.(.([]))))) -> difcN1_out_agaaaa(.(.(.(.([])))), .(.(.(.(.([]))))), .(.(.(.(.([])))))) difcN1_in_agaaaa(.(.(.(.(.([])))))) -> difcN1_out_agaaaa(.(.(.(.(.([]))))), .(.(.(.(.(.([])))))), .(.(.(.(.(.([]))))))) difcN1_in_agaaaa(.(.(.(.(.(.([]))))))) -> difcN1_out_agaaaa(.(.(.(.(.(.([])))))), .(.(.(.(.(.(.([]))))))), .(.(.(.(.(.(.([])))))))) difcN1_in_agaaaa(.(.(.(.(.(.(.(X9)))))))) -> U190_agaaaa(X9, difcN_in_gaaaa(X9)) U186_agagaa(X2, X4, difcN1_out_agaaaa(X2, X7, X8)) -> U187_agagaa(X2, X4, X8, mergedeletecC_in_gga(X7, X4)) mergedeletecC_in_gga([], X1) -> mergedeletecC_out_gga([], X1, X1) mergedeletecC_in_gga(.(X2), .(X3)) -> U137_gga(X2, X3, mergedeletecC_in_gga(X2, X3)) U187_agagaa(X2, X4, X8, mergedeletecC_out_gga(X7, X4, X9)) -> U188_agagaa(X2, X4, X9, revcF_in_ga(X8)) revcF_in_ga([]) -> revcF_out_ga([], []) revcF_in_ga(.([])) -> revcF_out_ga(.([]), .([])) revcF_in_ga(.(.([]))) -> revcF_out_ga(.(.([])), .(.([]))) revcF_in_ga(.(.(.([])))) -> revcF_out_ga(.(.(.([]))), .(.(.([])))) revcF_in_ga(.(.(.(.([]))))) -> revcF_out_ga(.(.(.(.([])))), .(.(.(.([]))))) revcF_in_ga(.(.(.(.(.([])))))) -> revcF_out_ga(.(.(.(.(.([]))))), .(.(.(.(.([])))))) revcF_in_ga(.(.(.(.(.(.([]))))))) -> revcF_out_ga(.(.(.(.(.(.([])))))), .(.(.(.(.(.([]))))))) revcF_in_ga(.(.(.(.(.(.(.([])))))))) -> revcF_out_ga(.(.(.(.(.(.(.([]))))))), .(.(.(.(.(.(.([])))))))) revcF_in_ga(.(.(.(.(.(.(.(.(X9))))))))) -> U140_ga(X9, revcD_in_gaga(X9, .(.(.(.(.(.(.([]))))))))) U188_agagaa(X2, X4, X9, revcF_out_ga(X8, X10)) -> U189_agagaa(X2, X4, mergedeletecC_in_gga(X10, X9)) U189_agagaa(X2, X4, mergedeletecC_out_gga(X10, X9, X5)) -> checkcM_out_agagaa(X2, X4, X5) revcD_in_gaga([], X2) -> revcD_out_gaga([], X2, .(X2)) revcD_in_gaga(.(X2), X4) -> U139_gaga(X2, X4, revcD_in_gaga(X2, .(X4))) U140_ga(X9, revcD_out_gaga(X9, .(.(.(.(.(.(.([]))))))), X10)) -> revcF_out_ga(.(.(.(.(.(.(.(.(X9)))))))), X10) U139_gaga(X2, X4, revcD_out_gaga(X2, .(X4), X5)) -> revcD_out_gaga(.(X2), X4, X5) U137_gga(X2, X3, mergedeletecC_out_gga(X2, X3, X4)) -> mergedeletecC_out_gga(.(X2), .(X3), X4) difcN_in_gaaaa([]) -> difcN_out_gaaaa([], [], []) difcN_in_gaaaa(.(X2)) -> U149_gaaaa(X2, difcN_in_gaaaa(X2)) U190_agaaaa(X9, difcN_out_gaaaa(X9, X20, X29)) -> difcN1_out_agaaaa(.(.(.(.(.(.(.(X9))))))), .(.(.(.(.(.(.(.(X20)))))))), .(.(.(.(.(.(.(.(X29))))))))) U149_gaaaa(X2, difcN_out_gaaaa(X2, X6, X8)) -> difcN_out_gaaaa(.(X2), .(X6), .(X8)) membercB_in_ag(.(X2)) -> membercB_out_ag(.(X2)) membercB_in_ag(.(X3)) -> U136_ag(X3, membercB_in_ag(X3)) U136_ag(X3, membercB_out_ag(X3)) -> membercB_out_ag(.(X3)) The set Q consists of the following terms: membercB_in_ag(x0) checkcM_in_agagaa(x0, x1) U136_ag(x0, x1) U186_agagaa(x0, x1, x2) difcN1_in_agaaaa(x0) U187_agagaa(x0, x1, x2, x3) U190_agaaaa(x0, x1) mergedeletecC_in_gga(x0, x1) U188_agagaa(x0, x1, x2, x3) difcN_in_gaaaa(x0) U137_gga(x0, x1, x2) revcF_in_ga(x0) U189_agagaa(x0, x1, x2) U149_gaaaa(x0, x1) U140_ga(x0, x1) revcD_in_gaga(x0, x1) U139_gaga(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (396) 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]. checkcM_in_agagaa(x0, x1) ---------------------------------------- (397) Obligation: Q DP problem: The TRS P consists of the following rules: PDS1L_IN_GAGAA(X1, X3) -> U10_GAGAA(X1, X3, membercB_in_ag(X1)) U17_GAGAA(X1, X3, checkcM_out_agagaa(X3, X1, X7)) -> PDS1L_IN_GAGAA(X7, .(X3)) U10_GAGAA(X1, X3, membercB_out_ag(X1)) -> U17_GAGAA(X1, X3, U186_agagaa(X3, X1, difcN1_in_agaaaa(X3))) The TRS R consists of the following rules: difcN1_in_agaaaa(.([])) -> difcN1_out_agaaaa(.([]), .(.([])), .(.([]))) difcN1_in_agaaaa(.(.([]))) -> difcN1_out_agaaaa(.(.([])), .(.(.([]))), .(.(.([])))) difcN1_in_agaaaa(.(.(.([])))) -> difcN1_out_agaaaa(.(.(.([]))), .(.(.(.([])))), .(.(.(.([]))))) difcN1_in_agaaaa(.(.(.(.([]))))) -> difcN1_out_agaaaa(.(.(.(.([])))), .(.(.(.(.([]))))), .(.(.(.(.([])))))) difcN1_in_agaaaa(.(.(.(.(.([])))))) -> difcN1_out_agaaaa(.(.(.(.(.([]))))), .(.(.(.(.(.([])))))), .(.(.(.(.(.([]))))))) difcN1_in_agaaaa(.(.(.(.(.(.([]))))))) -> difcN1_out_agaaaa(.(.(.(.(.(.([])))))), .(.(.(.(.(.(.([]))))))), .(.(.(.(.(.(.([])))))))) difcN1_in_agaaaa(.(.(.(.(.(.(.(X9)))))))) -> U190_agaaaa(X9, difcN_in_gaaaa(X9)) U186_agagaa(X2, X4, difcN1_out_agaaaa(X2, X7, X8)) -> U187_agagaa(X2, X4, X8, mergedeletecC_in_gga(X7, X4)) mergedeletecC_in_gga([], X1) -> mergedeletecC_out_gga([], X1, X1) mergedeletecC_in_gga(.(X2), .(X3)) -> U137_gga(X2, X3, mergedeletecC_in_gga(X2, X3)) U187_agagaa(X2, X4, X8, mergedeletecC_out_gga(X7, X4, X9)) -> U188_agagaa(X2, X4, X9, revcF_in_ga(X8)) revcF_in_ga([]) -> revcF_out_ga([], []) revcF_in_ga(.([])) -> revcF_out_ga(.([]), .([])) revcF_in_ga(.(.([]))) -> revcF_out_ga(.(.([])), .(.([]))) revcF_in_ga(.(.(.([])))) -> revcF_out_ga(.(.(.([]))), .(.(.([])))) revcF_in_ga(.(.(.(.([]))))) -> revcF_out_ga(.(.(.(.([])))), .(.(.(.([]))))) revcF_in_ga(.(.(.(.(.([])))))) -> revcF_out_ga(.(.(.(.(.([]))))), .(.(.(.(.([])))))) revcF_in_ga(.(.(.(.(.(.([]))))))) -> revcF_out_ga(.(.(.(.(.(.([])))))), .(.(.(.(.(.([]))))))) revcF_in_ga(.(.(.(.(.(.(.([])))))))) -> revcF_out_ga(.(.(.(.(.(.(.([]))))))), .(.(.(.(.(.(.([])))))))) revcF_in_ga(.(.(.(.(.(.(.(.(X9))))))))) -> U140_ga(X9, revcD_in_gaga(X9, .(.(.(.(.(.(.([]))))))))) U188_agagaa(X2, X4, X9, revcF_out_ga(X8, X10)) -> U189_agagaa(X2, X4, mergedeletecC_in_gga(X10, X9)) U189_agagaa(X2, X4, mergedeletecC_out_gga(X10, X9, X5)) -> checkcM_out_agagaa(X2, X4, X5) revcD_in_gaga([], X2) -> revcD_out_gaga([], X2, .(X2)) revcD_in_gaga(.(X2), X4) -> U139_gaga(X2, X4, revcD_in_gaga(X2, .(X4))) U140_ga(X9, revcD_out_gaga(X9, .(.(.(.(.(.(.([]))))))), X10)) -> revcF_out_ga(.(.(.(.(.(.(.(.(X9)))))))), X10) U139_gaga(X2, X4, revcD_out_gaga(X2, .(X4), X5)) -> revcD_out_gaga(.(X2), X4, X5) U137_gga(X2, X3, mergedeletecC_out_gga(X2, X3, X4)) -> mergedeletecC_out_gga(.(X2), .(X3), X4) difcN_in_gaaaa([]) -> difcN_out_gaaaa([], [], []) difcN_in_gaaaa(.(X2)) -> U149_gaaaa(X2, difcN_in_gaaaa(X2)) U190_agaaaa(X9, difcN_out_gaaaa(X9, X20, X29)) -> difcN1_out_agaaaa(.(.(.(.(.(.(.(X9))))))), .(.(.(.(.(.(.(.(X20)))))))), .(.(.(.(.(.(.(.(X29))))))))) U149_gaaaa(X2, difcN_out_gaaaa(X2, X6, X8)) -> difcN_out_gaaaa(.(X2), .(X6), .(X8)) membercB_in_ag(.(X2)) -> membercB_out_ag(.(X2)) membercB_in_ag(.(X3)) -> U136_ag(X3, membercB_in_ag(X3)) U136_ag(X3, membercB_out_ag(X3)) -> membercB_out_ag(.(X3)) The set Q consists of the following terms: membercB_in_ag(x0) U136_ag(x0, x1) U186_agagaa(x0, x1, x2) difcN1_in_agaaaa(x0) U187_agagaa(x0, x1, x2, x3) U190_agaaaa(x0, x1) mergedeletecC_in_gga(x0, x1) U188_agagaa(x0, x1, x2, x3) difcN_in_gaaaa(x0) U137_gga(x0, x1, x2) revcF_in_ga(x0) U189_agagaa(x0, x1, x2) U149_gaaaa(x0, x1) U140_ga(x0, x1) revcD_in_gaga(x0, x1) U139_gaga(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (398) 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: U137_gga(X2, X3, mergedeletecC_out_gga(X2, X3, X4)) -> mergedeletecC_out_gga(.(X2), .(X3), X4) Used ordering: Polynomial interpretation [POLO]: POL(.(x_1)) = 2 + 2*x_1 POL(PDS1L_IN_GAGAA(x_1, x_2)) = 2*x_1 POL(U10_GAGAA(x_1, x_2, x_3)) = 2*x_1 POL(U136_ag(x_1, x_2)) = 0 POL(U137_gga(x_1, x_2, x_3)) = 2 + x_3 POL(U139_gaga(x_1, x_2, x_3)) = 0 POL(U140_ga(x_1, x_2)) = 0 POL(U149_gaaaa(x_1, x_2)) = 0 POL(U17_GAGAA(x_1, x_2, x_3)) = x_3 POL(U186_agagaa(x_1, x_2, x_3)) = 2*x_2 POL(U187_agagaa(x_1, x_2, x_3, x_4)) = 2*x_4 POL(U188_agagaa(x_1, x_2, x_3, x_4)) = 2*x_3 POL(U189_agagaa(x_1, x_2, x_3)) = 2*x_3 POL(U190_agaaaa(x_1, x_2)) = 0 POL([]) = 0 POL(checkcM_out_agagaa(x_1, x_2, x_3)) = 2*x_3 POL(difcN1_in_agaaaa(x_1)) = 0 POL(difcN1_out_agaaaa(x_1, x_2, x_3)) = 0 POL(difcN_in_gaaaa(x_1)) = 0 POL(difcN_out_gaaaa(x_1, x_2, x_3)) = 0 POL(membercB_in_ag(x_1)) = 0 POL(membercB_out_ag(x_1)) = 0 POL(mergedeletecC_in_gga(x_1, x_2)) = x_2 POL(mergedeletecC_out_gga(x_1, x_2, x_3)) = x_3 POL(revcD_in_gaga(x_1, x_2)) = 0 POL(revcD_out_gaga(x_1, x_2, x_3)) = 0 POL(revcF_in_ga(x_1)) = 0 POL(revcF_out_ga(x_1, x_2)) = 0 ---------------------------------------- (399) Obligation: Q DP problem: The TRS P consists of the following rules: PDS1L_IN_GAGAA(X1, X3) -> U10_GAGAA(X1, X3, membercB_in_ag(X1)) U17_GAGAA(X1, X3, checkcM_out_agagaa(X3, X1, X7)) -> PDS1L_IN_GAGAA(X7, .(X3)) U10_GAGAA(X1, X3, membercB_out_ag(X1)) -> U17_GAGAA(X1, X3, U186_agagaa(X3, X1, difcN1_in_agaaaa(X3))) The TRS R consists of the following rules: difcN1_in_agaaaa(.([])) -> difcN1_out_agaaaa(.([]), .(.([])), .(.([]))) difcN1_in_agaaaa(.(.([]))) -> difcN1_out_agaaaa(.(.([])), .(.(.([]))), .(.(.([])))) difcN1_in_agaaaa(.(.(.([])))) -> difcN1_out_agaaaa(.(.(.([]))), .(.(.(.([])))), .(.(.(.([]))))) difcN1_in_agaaaa(.(.(.(.([]))))) -> difcN1_out_agaaaa(.(.(.(.([])))), .(.(.(.(.([]))))), .(.(.(.(.([])))))) difcN1_in_agaaaa(.(.(.(.(.([])))))) -> difcN1_out_agaaaa(.(.(.(.(.([]))))), .(.(.(.(.(.([])))))), .(.(.(.(.(.([]))))))) difcN1_in_agaaaa(.(.(.(.(.(.([]))))))) -> difcN1_out_agaaaa(.(.(.(.(.(.([])))))), .(.(.(.(.(.(.([]))))))), .(.(.(.(.(.(.([])))))))) difcN1_in_agaaaa(.(.(.(.(.(.(.(X9)))))))) -> U190_agaaaa(X9, difcN_in_gaaaa(X9)) U186_agagaa(X2, X4, difcN1_out_agaaaa(X2, X7, X8)) -> U187_agagaa(X2, X4, X8, mergedeletecC_in_gga(X7, X4)) mergedeletecC_in_gga([], X1) -> mergedeletecC_out_gga([], X1, X1) mergedeletecC_in_gga(.(X2), .(X3)) -> U137_gga(X2, X3, mergedeletecC_in_gga(X2, X3)) U187_agagaa(X2, X4, X8, mergedeletecC_out_gga(X7, X4, X9)) -> U188_agagaa(X2, X4, X9, revcF_in_ga(X8)) revcF_in_ga([]) -> revcF_out_ga([], []) revcF_in_ga(.([])) -> revcF_out_ga(.([]), .([])) revcF_in_ga(.(.([]))) -> revcF_out_ga(.(.([])), .(.([]))) revcF_in_ga(.(.(.([])))) -> revcF_out_ga(.(.(.([]))), .(.(.([])))) revcF_in_ga(.(.(.(.([]))))) -> revcF_out_ga(.(.(.(.([])))), .(.(.(.([]))))) revcF_in_ga(.(.(.(.(.([])))))) -> revcF_out_ga(.(.(.(.(.([]))))), .(.(.(.(.([])))))) revcF_in_ga(.(.(.(.(.(.([]))))))) -> revcF_out_ga(.(.(.(.(.(.([])))))), .(.(.(.(.(.([]))))))) revcF_in_ga(.(.(.(.(.(.(.([])))))))) -> revcF_out_ga(.(.(.(.(.(.(.([]))))))), .(.(.(.(.(.(.([])))))))) revcF_in_ga(.(.(.(.(.(.(.(.(X9))))))))) -> U140_ga(X9, revcD_in_gaga(X9, .(.(.(.(.(.(.([]))))))))) U188_agagaa(X2, X4, X9, revcF_out_ga(X8, X10)) -> U189_agagaa(X2, X4, mergedeletecC_in_gga(X10, X9)) U189_agagaa(X2, X4, mergedeletecC_out_gga(X10, X9, X5)) -> checkcM_out_agagaa(X2, X4, X5) revcD_in_gaga([], X2) -> revcD_out_gaga([], X2, .(X2)) revcD_in_gaga(.(X2), X4) -> U139_gaga(X2, X4, revcD_in_gaga(X2, .(X4))) U140_ga(X9, revcD_out_gaga(X9, .(.(.(.(.(.(.([]))))))), X10)) -> revcF_out_ga(.(.(.(.(.(.(.(.(X9)))))))), X10) U139_gaga(X2, X4, revcD_out_gaga(X2, .(X4), X5)) -> revcD_out_gaga(.(X2), X4, X5) difcN_in_gaaaa([]) -> difcN_out_gaaaa([], [], []) difcN_in_gaaaa(.(X2)) -> U149_gaaaa(X2, difcN_in_gaaaa(X2)) U190_agaaaa(X9, difcN_out_gaaaa(X9, X20, X29)) -> difcN1_out_agaaaa(.(.(.(.(.(.(.(X9))))))), .(.(.(.(.(.(.(.(X20)))))))), .(.(.(.(.(.(.(.(X29))))))))) U149_gaaaa(X2, difcN_out_gaaaa(X2, X6, X8)) -> difcN_out_gaaaa(.(X2), .(X6), .(X8)) membercB_in_ag(.(X2)) -> membercB_out_ag(.(X2)) membercB_in_ag(.(X3)) -> U136_ag(X3, membercB_in_ag(X3)) U136_ag(X3, membercB_out_ag(X3)) -> membercB_out_ag(.(X3)) The set Q consists of the following terms: membercB_in_ag(x0) U136_ag(x0, x1) U186_agagaa(x0, x1, x2) difcN1_in_agaaaa(x0) U187_agagaa(x0, x1, x2, x3) U190_agaaaa(x0, x1) mergedeletecC_in_gga(x0, x1) U188_agagaa(x0, x1, x2, x3) difcN_in_gaaaa(x0) U137_gga(x0, x1, x2) revcF_in_ga(x0) U189_agagaa(x0, x1, x2) U149_gaaaa(x0, x1) U140_ga(x0, x1) revcD_in_gaga(x0, x1) U139_gaga(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (400) Obligation: Pi DP problem: The TRS P consists of the following rules: IOTA1A_IN_AAA(X1, X2, .(X1, X3)) -> IOTA1A_IN_AAA(X4, X2, X3) The TRS R consists of the following rules: iota1cQ_in_aa(X1, X2) -> U153_aa(X1, X2, iota1cA_in_aaa(X3, X1, X2)) iota1cA_in_aaa(X1, X1, []) -> iota1cA_out_aaa(X1, X1, []) iota1cA_in_aaa(X1, X2, .(X1, X3)) -> U135_aaa(X1, X2, X3, iota1cA_in_aaa(X4, X2, X3)) U135_aaa(X1, X2, X3, iota1cA_out_aaa(X4, X2, X3)) -> iota1cA_out_aaa(X1, X2, .(X1, X3)) U153_aa(X1, X2, iota1cA_out_aaa(X3, X1, X2)) -> iota1cQ_out_aa(X1, X2) membercR_in_ag(X1, .(X1, X2)) -> membercR_out_ag(X1, .(X1, X2)) membercR_in_ag(X1, .(X2, X3)) -> U154_ag(X1, X2, X3, membercB_in_ag(X1, X3)) membercB_in_ag(X1, .(X1, X2)) -> membercB_out_ag(X1, .(X1, X2)) membercB_in_ag(X1, .(X2, X3)) -> U136_ag(X1, X2, X3, membercB_in_ag(X1, X3)) U136_ag(X1, X2, X3, membercB_out_ag(X1, X3)) -> membercB_out_ag(X1, .(X2, X3)) U154_ag(X1, X2, X3, membercB_out_ag(X1, X3)) -> membercR_out_ag(X1, .(X2, X3)) difcE_in_aaaa(X1, X2, .(X3, []), .(X4, [])) -> difcE_out_aaaa(X1, X2, .(X3, []), .(X4, [])) mergedeletecT_in_gga([], X1, X1) -> mergedeletecT_out_gga([], X1, X1) mergedeletecT_in_gga(.(X1, X2), .(X1, X3), X4) -> U159_gga(X1, X2, X3, X4, mergedeletecC_in_gga(X2, X3, X4)) mergedeletecC_in_gga([], X1, X1) -> mergedeletecC_out_gga([], X1, X1) mergedeletecC_in_gga(.(X1, X2), .(X1, X3), X4) -> U137_gga(X1, X2, X3, X4, mergedeletecC_in_gga(X2, X3, X4)) mergedeletecC_in_gga(.(X1, X2), .(X3, X4), .(X3, X5)) -> U138_gga(X1, X2, X3, X4, X5, mergedeletecC_in_gga(.(X1, X2), X4, X5)) U138_gga(X1, X2, X3, X4, X5, mergedeletecC_out_gga(.(X1, X2), X4, X5)) -> mergedeletecC_out_gga(.(X1, X2), .(X3, X4), .(X3, X5)) U137_gga(X1, X2, X3, X4, mergedeletecC_out_gga(X2, X3, X4)) -> mergedeletecC_out_gga(.(X1, X2), .(X1, X3), X4) U159_gga(X1, X2, X3, X4, mergedeletecC_out_gga(X2, X3, X4)) -> mergedeletecT_out_gga(.(X1, X2), .(X1, X3), X4) mergedeletecT_in_gga(.(X1, X2), .(X3, X4), .(X3, X5)) -> U160_gga(X1, X2, X3, X4, X5, mergedeletecC_in_gga(.(X1, X2), X4, X5)) U160_gga(X1, X2, X3, X4, X5, mergedeletecC_out_gga(.(X1, X2), X4, X5)) -> mergedeletecT_out_gga(.(X1, X2), .(X3, X4), .(X3, X5)) revcF_in_ga([], []) -> revcF_out_ga([], []) revcF_in_ga(.(X1, []), .(X1, [])) -> revcF_out_ga(.(X1, []), .(X1, [])) revcF_in_ga(.(X1, .(X2, [])), .(X2, .(X1, []))) -> revcF_out_ga(.(X1, .(X2, [])), .(X2, .(X1, []))) revcF_in_ga(.(X1, .(X2, .(X3, []))), .(X3, .(X2, .(X1, [])))) -> revcF_out_ga(.(X1, .(X2, .(X3, []))), .(X3, .(X2, .(X1, [])))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, [])))), .(X4, .(X3, .(X2, .(X1, []))))) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, [])))), .(X4, .(X3, .(X2, .(X1, []))))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, .(X5, []))))), .(X5, .(X4, .(X3, .(X2, .(X1, [])))))) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, .(X5, []))))), .(X5, .(X4, .(X3, .(X2, .(X1, [])))))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, [])))))), .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, [])))))), .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, []))))))), .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, [])))))))) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, []))))))), .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, [])))))))) revcF_in_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) -> U140_ga(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revcD_in_gaga(X9, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))), X10)) revcD_in_gaga([], X1, X2, .(X1, X2)) -> revcD_out_gaga([], X1, X2, .(X1, X2)) revcD_in_gaga(.(X1, X2), X3, X4, X5) -> U139_gaga(X1, X2, X3, X4, X5, revcD_in_gaga(X2, X1, .(X3, X4), X5)) U139_gaga(X1, X2, X3, X4, X5, revcD_out_gaga(X2, X1, .(X3, X4), X5)) -> revcD_out_gaga(.(X1, X2), X3, X4, X5) U140_ga(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revcD_out_gaga(X9, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))), X10)) -> revcF_out_ga(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) checkcS_in_agaa(X1, X2, X3, X4) -> U155_agaa(X1, X2, X3, X4, difcE_in_aaaa(X1, X4, X5, X6)) U155_agaa(X1, X2, X3, X4, difcE_out_aaaa(X1, X4, X5, X6)) -> U156_agaa(X1, X2, X3, X4, X5, X6, mergedeletecT_in_gga(X5, X2, X7)) U156_agaa(X1, X2, X3, X4, X5, X6, mergedeletecT_out_gga(X5, X2, X7)) -> U157_agaa(X1, X2, X3, X4, X5, X6, X7, revcF_in_ga(X6, X8)) U157_agaa(X1, X2, X3, X4, X5, X6, X7, revcF_out_ga(X6, X8)) -> U158_agaa(X1, X2, X3, X4, mergedeletecC_in_gga(X8, X7, X3)) U158_agaa(X1, X2, X3, X4, mergedeletecC_out_gga(X8, X7, X3)) -> checkcS_out_agaa(X1, X2, X3, X4) difcG_in_aaaaa(X1, X2, X3, .(X4, X5), .(X6, X7)) -> U141_aaaaa(X1, X2, X3, X4, X5, X6, X7, difcE_in_aaaa(X2, X3, X5, X7)) U141_aaaaa(X1, X2, X3, X4, X5, X6, X7, difcE_out_aaaa(X2, X3, X5, X7)) -> difcG_out_aaaaa(X1, X2, X3, .(X4, X5), .(X6, X7)) checkcU_in_aagaa(X1, X2, X3, X4, X5) -> U161_aagaa(X1, X2, X3, X4, X5, difcG_in_aaaaa(X1, X2, X5, X6, X7)) U161_aagaa(X1, X2, X3, X4, X5, difcG_out_aaaaa(X1, X2, X5, X6, X7)) -> U162_aagaa(X1, X2, X3, X4, X5, X6, X7, mergedeletecC_in_gga(X6, X3, X8)) U162_aagaa(X1, X2, X3, X4, X5, X6, X7, mergedeletecC_out_gga(X6, X3, X8)) -> U163_aagaa(X1, X2, X3, X4, X5, X6, X7, X8, revcF_in_ga(X7, X9)) U163_aagaa(X1, X2, X3, X4, X5, X6, X7, X8, revcF_out_ga(X7, X9)) -> U164_aagaa(X1, X2, X3, X4, X5, mergedeletecC_in_gga(X9, X8, X4)) U164_aagaa(X1, X2, X3, X4, X5, mergedeletecC_out_gga(X9, X8, X4)) -> checkcU_out_aagaa(X1, X2, X3, X4, X5) difcH_in_aaaaaa(X1, X2, X3, X4, .(X5, X6), .(X7, X8)) -> U142_aaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, difcG_in_aaaaa(X2, X3, X4, X6, X8)) U142_aaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, difcG_out_aaaaa(X2, X3, X4, X6, X8)) -> difcH_out_aaaaaa(X1, X2, X3, X4, .(X5, X6), .(X7, X8)) checkcV_in_aaagaa(X1, X2, X3, X4, X5, X6) -> U165_aaagaa(X1, X2, X3, X4, X5, X6, difcH_in_aaaaaa(X1, X2, X3, X6, X7, X8)) U165_aaagaa(X1, X2, X3, X4, X5, X6, difcH_out_aaaaaa(X1, X2, X3, X6, X7, X8)) -> U166_aaagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_in_gga(X7, X4, X9)) U166_aaagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_out_gga(X7, X4, X9)) -> U167_aaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, revcF_in_ga(X8, X10)) U167_aaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, revcF_out_ga(X8, X10)) -> U168_aaagaa(X1, X2, X3, X4, X5, X6, mergedeletecC_in_gga(X10, X9, X5)) U168_aaagaa(X1, X2, X3, X4, X5, X6, mergedeletecC_out_gga(X10, X9, X5)) -> checkcV_out_aaagaa(X1, X2, X3, X4, X5, X6) difcI_in_aaaaaaa(X1, X2, X3, X4, X5, .(X6, X7), .(X8, X9)) -> U143_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, difcH_in_aaaaaa(X2, X3, X4, X5, X7, X9)) U143_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, difcH_out_aaaaaa(X2, X3, X4, X5, X7, X9)) -> difcI_out_aaaaaaa(X1, X2, X3, X4, X5, .(X6, X7), .(X8, X9)) checkcW_in_aaaagaa(X1, X2, X3, X4, X5, X6, X7) -> U169_aaaagaa(X1, X2, X3, X4, X5, X6, X7, difcI_in_aaaaaaa(X1, X2, X3, X4, X7, X8, X9)) U169_aaaagaa(X1, X2, X3, X4, X5, X6, X7, difcI_out_aaaaaaa(X1, X2, X3, X4, X7, X8, X9)) -> U170_aaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, mergedeletecC_in_gga(X8, X5, X10)) U170_aaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, mergedeletecC_out_gga(X8, X5, X10)) -> U171_aaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revcF_in_ga(X9, X11)) U171_aaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revcF_out_ga(X9, X11)) -> U172_aaaagaa(X1, X2, X3, X4, X5, X6, X7, mergedeletecC_in_gga(X11, X10, X6)) U172_aaaagaa(X1, X2, X3, X4, X5, X6, X7, mergedeletecC_out_gga(X11, X10, X6)) -> checkcW_out_aaaagaa(X1, X2, X3, X4, X5, X6, X7) difcJ_in_aaaaaaaa(X1, X2, X3, X4, X5, X6, .(X7, X8), .(X9, X10)) -> U144_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, difcI_in_aaaaaaa(X2, X3, X4, X5, X6, X8, X10)) U144_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, difcI_out_aaaaaaa(X2, X3, X4, X5, X6, X8, X10)) -> difcJ_out_aaaaaaaa(X1, X2, X3, X4, X5, X6, .(X7, X8), .(X9, X10)) checkcX_in_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8) -> U173_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, difcJ_in_aaaaaaaa(X1, X2, X3, X4, X5, X8, X9, X10)) U173_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, difcJ_out_aaaaaaaa(X1, X2, X3, X4, X5, X8, X9, X10)) -> U174_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, mergedeletecC_in_gga(X9, X6, X11)) U174_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, mergedeletecC_out_gga(X9, X6, X11)) -> U175_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, revcF_in_ga(X10, X12)) U175_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, revcF_out_ga(X10, X12)) -> U176_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_in_gga(X12, X11, X7)) U176_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_out_gga(X12, X11, X7)) -> checkcX_out_aaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8) difcK_in_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, .(X8, X9), .(X10, X11)) -> U145_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, difcJ_in_aaaaaaaa(X2, X3, X4, X5, X6, X7, X9, X11)) U145_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, difcJ_out_aaaaaaaa(X2, X3, X4, X5, X6, X7, X9, X11)) -> difcK_out_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, .(X8, X9), .(X10, X11)) checkcY_in_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9) -> U177_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, difcK_in_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X9, X10, X11)) U177_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, difcK_out_aaaaaaaaa(X1, X2, X3, X4, X5, X6, X9, X10, X11)) -> U178_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, mergedeletecC_in_gga(X10, X7, X12)) U178_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, mergedeletecC_out_gga(X10, X7, X12)) -> U179_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, revcF_in_ga(X11, X13)) U179_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, revcF_out_ga(X11, X13)) -> U180_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, mergedeletecC_in_gga(X13, X12, X8)) U180_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, mergedeletecC_out_gga(X13, X12, X8)) -> checkcY_out_aaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9) difcZ_in_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, .(X9, X10), .(X11, X12)) -> U185_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, difcK_in_aaaaaaaaa(X2, X3, X4, X5, X6, X7, X8, X10, X12)) U185_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, difcK_out_aaaaaaaaa(X2, X3, X4, X5, X6, X7, X8, X10, X12)) -> difcZ_out_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, .(X9, X10), .(X11, X12)) checkcP_in_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10) -> U181_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, difcZ_in_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X10, X11, X12)) U181_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, difcZ_out_aaaaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X10, X11, X12)) -> U182_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, mergedeletecC_in_gga(X11, X8, X13)) U182_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, mergedeletecC_out_gga(X11, X8, X13)) -> U183_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, revcF_in_ga(X12, X14)) U183_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, revcF_out_ga(X12, X14)) -> U184_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, mergedeletecC_in_gga(X14, X13, X9)) U184_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, mergedeletecC_out_gga(X14, X13, X9)) -> checkcP_out_aaaaaaagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10) difcN1_in_agaaaa(X1, [], X2, X3, .(X4, []), .(X5, [])) -> difcN1_out_agaaaa(X1, [], X2, X3, .(X4, []), .(X5, [])) difcN1_in_agaaaa(X1, .(X2, []), X3, X4, .(X5, .(X6, [])), .(X7, .(X8, []))) -> difcN1_out_agaaaa(X1, .(X2, []), X3, X4, .(X5, .(X6, [])), .(X7, .(X8, []))) difcN1_in_agaaaa(X1, .(X2, .(X3, [])), X4, X5, .(X6, .(X7, .(X8, []))), .(X9, .(X10, .(X11, [])))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, [])), X4, X5, .(X6, .(X7, .(X8, []))), .(X9, .(X10, .(X11, [])))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, []))), X5, X6, .(X7, .(X8, .(X9, .(X10, [])))), .(X11, .(X12, .(X13, .(X14, []))))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, []))), X5, X6, .(X7, .(X8, .(X9, .(X10, [])))), .(X11, .(X12, .(X13, .(X14, []))))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, [])))), X6, X7, .(X8, .(X9, .(X10, .(X11, .(X12, []))))), .(X13, .(X14, .(X15, .(X16, .(X17, [])))))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, [])))), X6, X7, .(X8, .(X9, .(X10, .(X11, .(X12, []))))), .(X13, .(X14, .(X15, .(X16, .(X17, [])))))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, []))))), X7, X8, .(X9, .(X10, .(X11, .(X12, .(X13, .(X14, [])))))), .(X15, .(X16, .(X17, .(X18, .(X19, .(X20, []))))))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, []))))), X7, X8, .(X9, .(X10, .(X11, .(X12, .(X13, .(X14, [])))))), .(X15, .(X16, .(X17, .(X18, .(X19, .(X20, []))))))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, [])))))), X8, X9, .(X10, .(X11, .(X12, .(X13, .(X14, .(X15, .(X16, []))))))), .(X17, .(X18, .(X19, .(X20, .(X21, .(X22, .(X23, [])))))))) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, [])))))), X8, X9, .(X10, .(X11, .(X12, .(X13, .(X14, .(X15, .(X16, []))))))), .(X17, .(X18, .(X19, .(X20, .(X21, .(X22, .(X23, [])))))))) difcN1_in_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9))))))), X10, X11, .(X12, .(X13, .(X14, .(X15, .(X16, .(X17, .(X18, .(X19, X20)))))))), .(X21, .(X22, .(X23, .(X24, .(X25, .(X26, .(X27, .(X28, X29))))))))) -> U190_agaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, X21, X22, X23, X24, X25, X26, X27, X28, X29, difcN_in_gaaaa(X9, X10, X11, X20, X29)) difcN_in_gaaaa([], X1, X2, [], []) -> difcN_out_gaaaa([], X1, X2, [], []) difcN_in_gaaaa(.(X1, X2), X3, X4, .(X5, X6), .(X7, X8)) -> U149_gaaaa(X1, X2, X3, X4, X5, X6, X7, X8, difcN_in_gaaaa(X2, X3, X4, X6, X8)) U149_gaaaa(X1, X2, X3, X4, X5, X6, X7, X8, difcN_out_gaaaa(X2, X3, X4, X6, X8)) -> difcN_out_gaaaa(.(X1, X2), X3, X4, .(X5, X6), .(X7, X8)) U190_agaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, X21, X22, X23, X24, X25, X26, X27, X28, X29, difcN_out_gaaaa(X9, X10, X11, X20, X29)) -> difcN1_out_agaaaa(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9))))))), X10, X11, .(X12, .(X13, .(X14, .(X15, .(X16, .(X17, .(X18, .(X19, X20)))))))), .(X21, .(X22, .(X23, .(X24, .(X25, .(X26, .(X27, .(X28, X29))))))))) checkcM_in_agagaa(X1, X2, X3, X4, X5, X6) -> U186_agagaa(X1, X2, X3, X4, X5, X6, difcN1_in_agaaaa(X1, X2, X3, X6, X7, X8)) U186_agagaa(X1, X2, X3, X4, X5, X6, difcN1_out_agaaaa(X1, X2, X3, X6, X7, X8)) -> U187_agagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_in_gga(X7, X4, X9)) U187_agagaa(X1, X2, X3, X4, X5, X6, X7, X8, mergedeletecC_out_gga(X7, X4, X9)) -> U188_agagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, revcF_in_ga(X8, X10)) U188_agagaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, revcF_out_ga(X8, X10)) -> U189_agagaa(X1, X2, X3, X4, X5, X6, mergedeletecC_in_gga(X10, X9, X5)) U189_agagaa(X1, X2, X3, X4, X5, X6, mergedeletecC_out_gga(X10, X9, X5)) -> checkcM_out_agagaa(X1, X2, X3, X4, X5, X6) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) iota1cQ_in_aa(x1, x2) = iota1cQ_in_aa U153_aa(x1, x2, x3) = U153_aa(x3) iota1cA_in_aaa(x1, x2, x3) = iota1cA_in_aaa iota1cA_out_aaa(x1, x2, x3) = iota1cA_out_aaa(x3) U135_aaa(x1, x2, x3, x4) = U135_aaa(x4) iota1cQ_out_aa(x1, x2) = iota1cQ_out_aa(x2) membercR_in_ag(x1, x2) = membercR_in_ag(x2) membercR_out_ag(x1, x2) = membercR_out_ag(x2) U154_ag(x1, x2, x3, x4) = U154_ag(x3, x4) membercB_in_ag(x1, x2) = membercB_in_ag(x2) membercB_out_ag(x1, x2) = membercB_out_ag(x2) U136_ag(x1, x2, x3, x4) = U136_ag(x3, x4) difcE_in_aaaa(x1, x2, x3, x4) = difcE_in_aaaa difcE_out_aaaa(x1, x2, x3, x4) = difcE_out_aaaa(x3, x4) mergedeletecT_in_gga(x1, x2, x3) = mergedeletecT_in_gga(x1, x2) [] = [] mergedeletecT_out_gga(x1, x2, x3) = mergedeletecT_out_gga(x1, x2, x3) U159_gga(x1, x2, x3, x4, x5) = U159_gga(x2, x3, x5) mergedeletecC_in_gga(x1, x2, x3) = mergedeletecC_in_gga(x1, x2) mergedeletecC_out_gga(x1, x2, x3) = mergedeletecC_out_gga(x1, x2, x3) U137_gga(x1, x2, x3, x4, x5) = U137_gga(x2, x3, x5) U138_gga(x1, x2, x3, x4, x5, x6) = U138_gga(x2, x4, x6) U160_gga(x1, x2, x3, x4, x5, x6) = U160_gga(x2, x4, x6) revcF_in_ga(x1, x2) = revcF_in_ga(x1) revcF_out_ga(x1, x2) = revcF_out_ga(x1, x2) U140_ga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U140_ga(x9, x11) revcD_in_gaga(x1, x2, x3, x4) = revcD_in_gaga(x1, x3) revcD_out_gaga(x1, x2, x3, x4) = revcD_out_gaga(x1, x3, x4) U139_gaga(x1, x2, x3, x4, x5, x6) = U139_gaga(x2, x4, x6) checkcS_in_agaa(x1, x2, x3, x4) = checkcS_in_agaa(x2) U155_agaa(x1, x2, x3, x4, x5) = U155_agaa(x2, x5) U156_agaa(x1, x2, x3, x4, x5, x6, x7) = U156_agaa(x2, x6, x7) U157_agaa(x1, x2, x3, x4, x5, x6, x7, x8) = U157_agaa(x2, x7, x8) U158_agaa(x1, x2, x3, x4, x5) = U158_agaa(x2, x5) checkcS_out_agaa(x1, x2, x3, x4) = checkcS_out_agaa(x2, x3) difcG_in_aaaaa(x1, x2, x3, x4, x5) = difcG_in_aaaaa U141_aaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U141_aaaaa(x8) difcG_out_aaaaa(x1, x2, x3, x4, x5) = difcG_out_aaaaa(x4, x5) checkcU_in_aagaa(x1, x2, x3, x4, x5) = checkcU_in_aagaa(x3) U161_aagaa(x1, x2, x3, x4, x5, x6) = U161_aagaa(x3, x6) U162_aagaa(x1, x2, x3, x4, x5, x6, x7, x8) = U162_aagaa(x3, x7, x8) U163_aagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U163_aagaa(x3, x8, x9) U164_aagaa(x1, x2, x3, x4, x5, x6) = U164_aagaa(x3, x6) checkcU_out_aagaa(x1, x2, x3, x4, x5) = checkcU_out_aagaa(x3, x4) difcH_in_aaaaaa(x1, x2, x3, x4, x5, x6) = difcH_in_aaaaaa U142_aaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U142_aaaaaa(x9) difcH_out_aaaaaa(x1, x2, x3, x4, x5, x6) = difcH_out_aaaaaa(x5, x6) checkcV_in_aaagaa(x1, x2, x3, x4, x5, x6) = checkcV_in_aaagaa(x4) U165_aaagaa(x1, x2, x3, x4, x5, x6, x7) = U165_aaagaa(x4, x7) U166_aaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U166_aaagaa(x4, x8, x9) U167_aaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U167_aaagaa(x4, x9, x10) U168_aaagaa(x1, x2, x3, x4, x5, x6, x7) = U168_aaagaa(x4, x7) checkcV_out_aaagaa(x1, x2, x3, x4, x5, x6) = checkcV_out_aaagaa(x4, x5) difcI_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = difcI_in_aaaaaaa U143_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U143_aaaaaaa(x10) difcI_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = difcI_out_aaaaaaa(x6, x7) checkcW_in_aaaagaa(x1, x2, x3, x4, x5, x6, x7) = checkcW_in_aaaagaa(x5) U169_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8) = U169_aaaagaa(x5, x8) U170_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U170_aaaagaa(x5, x9, x10) U171_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U171_aaaagaa(x5, x10, x11) U172_aaaagaa(x1, x2, x3, x4, x5, x6, x7, x8) = U172_aaaagaa(x5, x8) checkcW_out_aaaagaa(x1, x2, x3, x4, x5, x6, x7) = checkcW_out_aaaagaa(x5, x6) difcJ_in_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = difcJ_in_aaaaaaaa U144_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U144_aaaaaaaa(x11) difcJ_out_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = difcJ_out_aaaaaaaa(x7, x8) checkcX_in_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8) = checkcX_in_aaaaagaa(x6) U173_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U173_aaaaagaa(x6, x9) U174_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U174_aaaaagaa(x6, x10, x11) U175_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U175_aaaaagaa(x6, x11, x12) U176_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U176_aaaaagaa(x6, x9) checkcX_out_aaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8) = checkcX_out_aaaaagaa(x6, x7) difcK_in_aaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = difcK_in_aaaaaaaaa U145_aaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U145_aaaaaaaaa(x12) difcK_out_aaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = difcK_out_aaaaaaaaa(x8, x9) checkcY_in_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = checkcY_in_aaaaaagaa(x7) U177_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U177_aaaaaagaa(x7, x10) U178_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12) = U178_aaaaaagaa(x7, x11, x12) U179_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U179_aaaaaagaa(x7, x12, x13) U180_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U180_aaaaaagaa(x7, x10) checkcY_out_aaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = checkcY_out_aaaaaagaa(x7, x8) difcZ_in_aaaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = difcZ_in_aaaaaaaaaa U185_aaaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U185_aaaaaaaaaa(x13) difcZ_out_aaaaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = difcZ_out_aaaaaaaaaa(x9, x10) checkcP_in_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = checkcP_in_aaaaaaagaa(x8) U181_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U181_aaaaaaagaa(x8, x11) U182_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13) = U182_aaaaaaagaa(x8, x12, x13) U183_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14) = U183_aaaaaaagaa(x8, x13, x14) U184_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U184_aaaaaaagaa(x8, x11) checkcP_out_aaaaaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = checkcP_out_aaaaaaagaa(x8, x9) difcN1_in_agaaaa(x1, x2, x3, x4, x5, x6) = difcN1_in_agaaaa(x2) difcN1_out_agaaaa(x1, x2, x3, x4, x5, x6) = difcN1_out_agaaaa(x2, x5, x6) U190_agaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30) = U190_agaaaa(x9, x30) difcN_in_gaaaa(x1, x2, x3, x4, x5) = difcN_in_gaaaa(x1) difcN_out_gaaaa(x1, x2, x3, x4, x5) = difcN_out_gaaaa(x1, x4, x5) U149_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U149_gaaaa(x2, x9) checkcM_in_agagaa(x1, x2, x3, x4, x5, x6) = checkcM_in_agagaa(x2, x4) U186_agagaa(x1, x2, x3, x4, x5, x6, x7) = U186_agagaa(x2, x4, x7) U187_agagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U187_agagaa(x2, x4, x8, x9) U188_agagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U188_agagaa(x2, x4, x9, x10) U189_agagaa(x1, x2, x3, x4, x5, x6, x7) = U189_agagaa(x2, x4, x7) checkcM_out_agagaa(x1, x2, x3, x4, x5, x6) = checkcM_out_agagaa(x2, x4, x5) IOTA1A_IN_AAA(x1, x2, x3) = IOTA1A_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (401) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (402) Obligation: Pi DP problem: The TRS P consists of the following rules: IOTA1A_IN_AAA(X1, X2, .(X1, X3)) -> IOTA1A_IN_AAA(X4, X2, X3) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) IOTA1A_IN_AAA(x1, x2, x3) = IOTA1A_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (403) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 1, "program": { "directives": [], "clauses": [ [ "(member X (. X Xs))", null ], [ "(member X (. X1 Xs))", "(member X Xs)" ], [ "(iota N List)", "(iota1 (0) N List)" ], [ "(iota1 K K ([]))", "(!)" ], [ "(iota1 K N (. K List))", "(',' (is K1 (+ K (1))) (iota1 K1 N List))" ], [ "(dif ([]) X2 X3 ([]) ([]))", null ], [ "(dif (. S Ss) Val Mod (. D Ds) (. D2 D2s))", "(',' (is D (- Val S)) (',' (is D2 (- Mod D)) (dif Ss Val Mod Ds D2s)))" ], [ "(rev ([]) L L)", null ], [ "(rev (. X Xs) Y L)", "(rev Xs (. X Y) L)" ], [ "(mergedelete ([]) L L)", null ], [ "(mergedelete (. D Ds) (. D R) L2)", "(mergedelete Ds R L2)" ], [ "(mergedelete (. D Ds) (. X R) (. X L2))", "(',' (> D X) (mergedelete (. D Ds) R L2))" ], [ "(check ([]) X4 L L X5)", "(!)" ], [ "(check S Choice Old L3 Modulus)", "(',' (dif S Choice Modulus Ds Dds) (',' (mergedelete Ds Old L2) (',' (rev Dds ([]) Rds) (',' (mergedelete Rds L2 L3) (!)))))" ], [ "(pds1 ([]) X6 ([]) X7)", "(!)" ], [ "(pds1 Unused List (. Choice Rest) Mod)", "(',' (member Choice Unused) (',' (check List Choice Unused U3 Mod) (pds1 U3 (. Choice List) Rest Mod)))" ], [ "(pds Order (. (0) Ans))", "(',' (is N (+ (* Order (+ Order (1))) (1))) (',' (iota N (. (0) List)) (pds1 List (. (0) ([])) Ans N)))" ], [ "(pdsbm N)", "(pds N (. (0) (. (1) X)))" ] ] }, "graph": { "nodes": { "4969": { "goal": [ { "clause": 7, "scope": 20, "term": "(rev T461 (. T460 (. T462 (. T463 (. T464 (. T465 (. T466 (. T467 ([])))))))) X516)" }, { "clause": 8, "scope": 20, "term": "(rev T461 (. T460 (. T462 (. T463 (. T464 (. T465 (. T466 (. T467 ([])))))))) X516)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T460", "T461", "T462", "T463", "T464", "T465", "T466", "T467" ], "free": ["X516"], "exprvars": [] } }, "5939": { "goal": [ { "clause": 5, "scope": 45, "term": "(dif T1390 T1391 T1392 X1592 X1594)" }, { "clause": 6, "scope": 45, "term": "(dif T1390 T1391 T1392 X1592 X1594)" } ], "kb": { "nonunifying": [ [ "(pds1 T45 (. (0) ([])) T1295 T1392)", "(pds1 ([]) X75 ([]) X76)" ], [ "(pds1 T616 (. T1162 (. (0) ([]))) T1296 T1392)", "(pds1 ([]) X575 ([]) X576)" ], [ "(pds1 T696 (. T1161 (. T1162 (. (0) ([])))) T1297 T1392)", "(pds1 ([]) X709 ([]) X710)" ], [ "(pds1 T790 (. T1160 (. T1161 (. T1162 (. (0) ([]))))) T1298 T1392)", "(pds1 ([]) X843 ([]) X844)" ], [ "(pds1 T898 (. T1159 (. T1160 (. T1161 (. T1162 (. (0) ([])))))) T1299 T1392)", "(pds1 ([]) X977 ([]) X978)" ], [ "(pds1 T1020 (. T1158 (. T1159 (. T1160 (. T1161 (. T1162 (. (0) ([]))))))) T1300 T1392)", "(pds1 ([]) X1111 ([]) X1112)" ], [ "(pds1 T1156 (. T1157 (. T1158 (. T1159 (. T1160 (. T1161 (. T1162 (. (0) ([])))))))) T1301 T1392)", "(pds1 ([]) X1245 ([]) X1246)" ], [ "(pds1 T1321 (. T1333 (. T1361 (. T1389 T1390))) T1302 T1392)", "(pds1 ([]) X1383 ([]) X1384)" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [ { "lhs": { "name": "T1337", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1335", "type": "PlainIntegerVariable" }, { "name": "T1333", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "name": "T1338", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1336", "type": "PlainIntegerVariable" }, { "name": "T1337", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "arguments": [ { "name": "T1363", "type": "PlainIntegerVariable" }, { "name": "T1333", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1335", "type": "PlainIntegerVariable" }, { "name": "T1333", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "arguments": [ { "name": "T1364", "type": "PlainIntegerVariable" }, { "name": "T1337", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1336", "type": "PlainIntegerVariable" }, { "name": "T1337", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "name": "T1365", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1363", "type": "PlainIntegerVariable" }, { "name": "T1361", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "name": "T1366", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1364", "type": "PlainIntegerVariable" }, { "name": "T1365", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "arguments": [ { "name": "T1335", "type": "PlainIntegerVariable" }, { "name": "T1333", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1335", "type": "PlainIntegerVariable" }, { "name": "T1333", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "arguments": [ { "name": "T1336", "type": "PlainIntegerVariable" }, { "name": "T1337", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1336", "type": "PlainIntegerVariable" }, { "name": "T1337", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "arguments": [ { "name": "T1335", "type": "PlainIntegerVariable" }, { "name": "T1333", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1335", "type": "PlainIntegerVariable" }, { "name": "T1333", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "arguments": [ { "name": "T1336", "type": "PlainIntegerVariable" }, { "name": "T1337", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1336", "type": "PlainIntegerVariable" }, { "name": "T1337", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "arguments": [ { "name": "T1391", "type": "PlainIntegerVariable" }, { "name": "T1361", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1363", "type": "PlainIntegerVariable" }, { "name": "T1361", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "arguments": [ { "name": "T1392", "type": "PlainIntegerVariable" }, { "name": "T1365", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1364", "type": "PlainIntegerVariable" }, { "name": "T1365", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "name": "T1393", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1391", "type": "PlainIntegerVariable" }, { "name": "T1389", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "name": "T1394", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1392", "type": "PlainIntegerVariable" }, { "name": "T1393", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" } ] }, "ground": [ "T616", "T1156", "T1160", "T1389", "T1159", "T1390", "T696", "T790", "T1020", "T1157", "T1392", "T1161", "T1333", "T1361", "T45", "T1158", "T1391", "T898", "T1162", "T1321" ], "free": [ "X75", "X76", "X575", "X576", "X709", "X710", "X843", "X844", "X977", "X978", "X1111", "X1112", "X1245", "X1246", "X1383", "X1384", "X1592", "X1594" ], "exprvars": [ "T1393", "T1389", "T1336", "T1364", "T1392", "T1338", "T1333", "T1366", "T1361", "T1394", "T1335", "T1363", "T1391", "T1337", "T1365" ] } }, "4840": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "4961": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "4841": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "4962": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "4963": { "goal": [{ "clause": -1, "scope": -1, "term": "(rev T461 (. 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"operation": "-" }, "operation": "=" }, { "lhs": { "arguments": [ { "name": "T1364", "type": "PlainIntegerVariable" }, { "name": "T1337", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1336", "type": "PlainIntegerVariable" }, { "name": "T1337", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "name": "T1365", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1363", "type": "PlainIntegerVariable" }, { "name": "T1361", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "name": "T1366", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1364", "type": "PlainIntegerVariable" }, { "name": "T1365", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "arguments": [ { "name": "T1335", "type": "PlainIntegerVariable" }, { "name": "T1333", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1335", "type": "PlainIntegerVariable" }, { "name": "T1333", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "arguments": [ { "name": "T1336", "type": "PlainIntegerVariable" }, { "name": "T1337", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1336", "type": "PlainIntegerVariable" }, { "name": "T1337", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "arguments": [ { "name": "T1335", "type": "PlainIntegerVariable" }, { "name": "T1333", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1335", "type": "PlainIntegerVariable" }, { "name": "T1333", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "arguments": [ { "name": "T1336", "type": "PlainIntegerVariable" }, { "name": "T1337", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1336", "type": "PlainIntegerVariable" }, { "name": "T1337", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "arguments": [ { "name": "T1379", "type": "PlainIntegerVariable" }, { "name": "T1361", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1363", "type": "PlainIntegerVariable" }, { "name": "T1361", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "arguments": [ { "name": "T1380", "type": "PlainIntegerVariable" }, { "name": "T1365", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1364", "type": "PlainIntegerVariable" }, { "name": "T1365", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" } ] }, "ground": [], "free": [], "exprvars": [ "T1338", "T1333", "T1366", "T1361", "T1335", "T1379", "T1363", "T1380", "T1336", "T1364", "T1337", "T1365" ] } }, "4955": { "goal": [{ "clause": -1, "scope": -1, "term": "(rev T396 (. T395 (. T397 (. T398 (. T399 (. T400 (. T401 ([]))))))) X494)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T395", "T396", "T397", "T398", "T399", "T400", "T401" ], "free": ["X494"], "exprvars": [] } }, "5923": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [ { "lhs": { "name": "T1337", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1335", "type": "PlainIntegerVariable" }, { "name": "T1333", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "name": "T1338", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1336", "type": "PlainIntegerVariable" }, { "name": "T1337", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "arguments": [ { "name": "T1363", "type": "PlainIntegerVariable" }, { "name": "T1333", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1335", "type": "PlainIntegerVariable" }, { "name": "T1333", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "arguments": [ { "name": "T1364", "type": "PlainIntegerVariable" }, { "name": "T1337", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1336", "type": "PlainIntegerVariable" }, { "name": "T1337", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "name": "T1365", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1363", "type": "PlainIntegerVariable" }, { "name": "T1361", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "name": "T1366", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1364", "type": "PlainIntegerVariable" }, { "name": "T1365", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" } ] }, "ground": [], "free": [], "exprvars": [ "T1338", "T1333", "T1366", "T1361", "T1335", "T1363", "T1336", "T1364", "T1337", "T1365" ] } }, "4956": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "5924": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [ { "lhs": { "name": "T1337", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1335", "type": "PlainIntegerVariable" }, { "name": "T1333", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "name": "T1338", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1336", "type": "PlainIntegerVariable" }, { "name": "T1337", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "arguments": [ { "name": "T1363", "type": "PlainIntegerVariable" }, { "name": "T1333", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1335", "type": "PlainIntegerVariable" }, { "name": "T1333", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "arguments": [ { "name": "T1364", "type": "PlainIntegerVariable" }, { "name": "T1337", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1336", "type": "PlainIntegerVariable" }, { "name": "T1337", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "name": "T1365", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1363", "type": "PlainIntegerVariable" }, { "name": "T1361", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "name": "T1366", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1364", "type": "PlainIntegerVariable" }, { "name": "T1365", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "arguments": [ { "name": "T1335", "type": "PlainIntegerVariable" }, { "name": "T1333", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1335", "type": "PlainIntegerVariable" }, { "name": "T1333", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "arguments": [ { "name": "T1336", "type": "PlainIntegerVariable" }, { "name": "T1337", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1336", "type": "PlainIntegerVariable" }, { "name": "T1337", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "arguments": [ { "name": "T1335", "type": "PlainIntegerVariable" }, { "name": "T1333", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1335", "type": "PlainIntegerVariable" }, { "name": "T1333", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "arguments": [ { "name": "T1336", "type": "PlainIntegerVariable" }, { "name": "T1337", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1336", "type": "PlainIntegerVariable" }, { "name": "T1337", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "arguments": [ { "name": "T1379", "type": "PlainIntegerVariable" }, { "name": "T1361", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1363", "type": "PlainIntegerVariable" }, { "name": "T1361", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "arguments": [ { "name": "T1380", "type": "PlainIntegerVariable" }, { "name": "T1365", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T1364", "type": "PlainIntegerVariable" }, { "name": "T1365", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" } ] }, "ground": [], "free": [], "exprvars": [ "T1338", "T1333", "T1366", "T1361", "T1335", "T1379", "T1363", "T1380", "T1336", "T1364", "T1337", "T1365" ] } }, "4957": { "goal": [ { "clause": 7, "scope": 19, "term": "(rev T396 (. T395 (. T397 (. T398 (. T399 (. T400 (. T401 ([]))))))) X494)" }, { "clause": 8, "scope": 19, "term": "(rev T396 (. T395 (. T397 (. T398 (. T399 (. T400 (. T401 ([]))))))) X494)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T395", "T396", "T397", "T398", "T399", "T400", "T401" ], "free": ["X494"], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 4, "label": "CASE" }, { "from": 4, "to": 10, "label": "EVAL with clause\npds(X14, .(0, X15)) :- ','(is(X16, +(*(X14, +(X14, 1)), 1)), ','(iota(X16, .(0, X17)), pds1(X17, .(0, []), X15, X16))).\nand substitutionT1 -> T7,\nX14 -> T7,\nX15 -> T9,\nT2 -> .(0, T9),\nT8 -> T9" }, { "from": 4, "to": 11, "label": "EVAL-BACKTRACK" }, { "from": 10, "to": 12, "label": "IS ERROR" }, { "from": 10, "to": 16, "label": "\nX16 -> T10" }, { "from": 16, "to": 21, "label": "SPLIT 1" }, { "from": 16, "to": 24, "label": "SPLIT 2\nnew knowledge:\nT10 is ground\nT12 is ground\nreplacements:X17 -> T12" }, { "from": 21, "to": 48, "label": "CASE" }, { "from": 24, "to": 4324, "label": "CASE" }, { "from": 48, "to": 52, "label": "ONLY EVAL with clause\niota(X31, X32) :- iota1(0, X31, X32).\nand substitutionT10 -> T16,\nX31 -> T16,\nX17 -> X33,\nX32 -> .(0, X33)" }, { "from": 52, "to": 53, "label": "CASE" }, { "from": 53, "to": 54, "label": "BACKTRACK\nfor clause: iota1(K, K, []) :- !because of non-unification" }, { "from": 54, "to": 55, "label": "ONLY EVAL with clause\niota1(X45, X46, .(X45, X47)) :- ','(is(X48, +(X45, 1)), iota1(X48, X46, X47)).\nand substitutionX45 -> 0,\nT16 -> T19,\nX46 -> T19,\nX33 -> X49,\nX47 -> X49" }, { "from": 55, "to": 56, "label": "\nX48 -> T20" }, { "from": 56, "to": 57, "label": "CASE" }, { "from": 57, "to": 58, "label": "EVAL with clause\niota1(X53, X53, []) :- !_4.\nand substitutionT20 -> T24,\nX53 -> T24,\nT19 -> T24,\nX49 -> []" }, { "from": 57, "to": 59, "label": "EVAL-BACKTRACK" }, { "from": 58, "to": 60, "label": "CUT" }, { "from": 59, "to": 62, "label": "ONLY EVAL with clause\niota1(X63, X64, .(X63, X65)) :- ','(is(X66, +(X63, 1)), iota1(X66, X64, X65)).\nand substitutionT20 -> T29,\nX63 -> T29,\nT19 -> T30,\nX64 -> T30,\nX65 -> X67,\nX49 -> .(T29, X67)" }, { "from": 60, "to": 61, "label": "SUCCESS" }, { "from": 62, "to": 63, "label": "\nX66 -> T31" }, { "from": 63, "to": 56, "label": "INSTANCE with matching:\nT20 -> T31\nT19 -> T30\nX49 -> X67\nX48 -> X53" }, { "from": 4324, "to": 4325, "label": "EVAL with clause\npds1([], X75, [], X76) :- !_5.\nand substitutionT12 -> [],\nX75 -> .(0, []),\nT9 -> [],\nT10 -> T36,\nX76 -> T36" }, { "from": 4324, "to": 4326, "label": "EVAL-BACKTRACK" }, { "from": 4325, "to": 4327, "label": "CUT" }, { "from": 4326, "to": 4397, "label": "EVAL with clause\npds1(X88, X89, .(X90, X91), X92) :- ','(member(X90, X88), ','(check(X89, X90, X88, X93, X92), pds1(X93, .(X90, X89), X91, X92))).\nand substitutionT12 -> T45,\nX88 -> T45,\nX89 -> .(0, []),\nX90 -> T49,\nX91 -> T50,\nT9 -> .(T49, T50),\nT10 -> T48,\nX92 -> T48,\nT46 -> T49,\nT47 -> T50" }, { "from": 4326, "to": 4398, "label": "EVAL-BACKTRACK" }, { "from": 4327, "to": 4328, "label": "SUCCESS" }, { "from": 4397, "to": 4399, "label": "SPLIT 1" }, { "from": 4397, "to": 4400, "label": "SPLIT 2\nnew knowledge:\nT55 is ground\nT45 is ground\nreplacements:T49 -> T55,\nT50 -> T56,\nT9 -> T57" }, { "from": 4399, "to": 4401, "label": "CASE" }, { "from": 4400, "to": 4417, "label": "SPLIT 1" }, { "from": 4400, "to": 4418, "label": "SPLIT 2\nnew knowledge:\nT55 is ground\nT45 is ground\nT110 is ground\nT48 is ground\nreplacements:X93 -> T110" }, { "from": 4401, "to": 4402, "label": "PARALLEL" }, { "from": 4401, "to": 4403, "label": "PARALLEL" }, { "from": 4402, "to": 4404, "label": "EVAL with clause\nmember(X110, .(X110, X111)).\nand substitutionT49 -> T70,\nX110 -> T70,\nX111 -> T71,\nT45 -> .(T70, T71)" }, { "from": 4402, "to": 4405, "label": "EVAL-BACKTRACK" }, { "from": 4403, "to": 4407, "label": "EVAL with clause\nmember(X118, .(X119, X120)) :- member(X118, X120).\nand substitutionT49 -> T81,\nX118 -> T81,\nX119 -> T79,\nX120 -> T80,\nT45 -> .(T79, T80),\nT78 -> T81" }, { "from": 4403, "to": 4408, "label": "EVAL-BACKTRACK" }, { "from": 4404, "to": 4406, "label": "SUCCESS" }, { "from": 4407, "to": 4409, "label": "CASE" }, { "from": 4409, "to": 4410, "label": "PARALLEL" }, { "from": 4409, "to": 4411, "label": "PARALLEL" }, { "from": 4410, "to": 4412, "label": "EVAL with clause\nmember(X133, .(X133, X134)).\nand substitutionT81 -> T94,\nX133 -> T94,\nX134 -> T95,\nT80 -> .(T94, T95)" }, { "from": 4410, "to": 4413, "label": "EVAL-BACKTRACK" }, { "from": 4411, "to": 4415, "label": "EVAL with clause\nmember(X141, .(X142, X143)) :- member(X141, X143).\nand substitutionT81 -> T105,\nX141 -> T105,\nX142 -> T103,\nX143 -> T104,\nT80 -> .(T103, T104),\nT102 -> T105" }, { "from": 4411, "to": 4416, "label": "EVAL-BACKTRACK" }, { "from": 4412, "to": 4414, "label": "SUCCESS" }, { "from": 4415, "to": 4407, "label": "INSTANCE with matching:\nT81 -> T105\nT80 -> T104" }, { "from": 4417, "to": 4419, "label": "CASE" }, { "from": 4418, "to": 5096, "label": "CASE" }, { "from": 4419, "to": 4420, "label": "BACKTRACK\nfor clause: check([], X4, L, L, X5) :- !because of non-unification" }, { "from": 4420, "to": 4421, "label": "ONLY EVAL with clause\ncheck(X177, X178, X179, X180, X181) :- ','(dif(X177, X178, X181, X182, X183), ','(mergedelete(X182, X179, X184), ','(rev(X183, [], X185), ','(mergedelete(X185, X184, X180), !_8)))).\nand substitutionX177 -> .(0, []),\nT55 -> T117,\nX178 -> T117,\nT45 -> T118,\nX179 -> T118,\nX93 -> X186,\nX180 -> X186,\nT48 -> T119,\nX181 -> T119" }, { "from": 4421, "to": 4422, "label": "SPLIT 1" }, { "from": 4421, "to": 4423, "label": "SPLIT 2\nnew knowledge:\nT117 is ground\nT119 is ground\nT120 is ground\nT121 is ground\nreplacements:X182 -> T120,\nX183 -> T121" }, { "from": 4422, "to": 4424, "label": "CASE" }, { "from": 4423, "to": 4437, "label": "SPLIT 1" }, { "from": 4423, "to": 4438, "label": "SPLIT 2\nnew knowledge:\nT120 is ground\nT118 is ground\nT144 is ground\nreplacements:X184 -> T144" }, { "from": 4424, "to": 4425, "label": "BACKTRACK\nfor clause: dif([], X2, X3, [], [])because of non-unification" }, { "from": 4425, "to": 4426, "label": "ONLY EVAL with clause\ndif(.(X221, X222), X223, X224, .(X225, X226), .(X227, X228)) :- ','(is(X225, -(X223, X221)), ','(is(X227, -(X224, X225)), dif(X222, X223, X224, X226, X228))).\nand substitutionX221 -> 0,\nX222 -> [],\nT117 -> T126,\nX223 -> T126,\nT119 -> T127,\nX224 -> T127,\nX225 -> X229,\nX226 -> X230,\nX182 -> .(X229, X230),\nX227 -> X231,\nX228 -> X232,\nX183 -> .(X231, X232)" }, { "from": 4426, "to": 4427, "label": "IS ERROR" }, { "from": 4426, "to": 4428, "label": "\nX229 -> T128" }, { "from": 4428, "to": 4429, "label": "IS ERROR" }, { "from": 4428, "to": 4430, "label": "\nX231 -> T129" }, { "from": 4430, "to": 4431, "label": "CASE" }, { "from": 4431, "to": 4432, "label": "PARALLEL" }, { "from": 4431, "to": 4433, "label": "PARALLEL" }, { "from": 4432, "to": 4434, "label": "ONLY EVAL with clause\ndif([], X243, X244, [], []).\nand substitutionT126 -> T140,\nX243 -> T140,\nT127 -> T141,\nX244 -> T141,\nX230 -> [],\nX232 -> []" }, { "from": 4433, "to": 4436, "label": "BACKTRACK\nfor clause: dif(.(S, Ss), Val, Mod, .(D, Ds), .(D2, D2s)) :- ','(is(D, -(Val, S)), ','(is(D2, -(Mod, D)), dif(Ss, Val, Mod, Ds, D2s)))because of non-unification" }, { "from": 4434, "to": 4435, "label": "SUCCESS" }, { "from": 4437, "to": 4439, "label": "CASE" }, { "from": 4438, "to": 4902, "label": "SPLIT 1" }, { "from": 4438, "to": 4903, "label": "SPLIT 2\nnew knowledge:\nT121 is ground\nT215 is ground\nreplacements:X185 -> T215" }, { "from": 4439, "to": 4440, "label": "PARALLEL" }, { "from": 4439, "to": 4441, "label": "PARALLEL" }, { "from": 4440, "to": 4442, "label": "EVAL with clause\nmergedelete([], X261, X261).\nand substitutionT120 -> [],\nT118 -> T151,\nX261 -> T151,\nX184 -> T151" }, { "from": 4440, "to": 4443, "label": "EVAL-BACKTRACK" }, { "from": 4441, "to": 4445, "label": "PARALLEL" }, { "from": 4441, "to": 4446, "label": "PARALLEL" }, { "from": 4442, "to": 4444, "label": "SUCCESS" }, { "from": 4445, "to": 4447, "label": "EVAL with clause\nmergedelete(.(X282, X283), .(X282, X284), X285) :- mergedelete(X283, X284, X285).\nand substitutionX282 -> T164,\nX283 -> T165,\nT120 -> .(T164, T165),\nX284 -> T166,\nT118 -> .(T164, T166),\nX184 -> X286,\nX285 -> X286" }, { "from": 4445, "to": 4448, "label": "EVAL-BACKTRACK" }, { "from": 4446, "to": 4839, "label": "EVAL with clause\nmergedelete(.(X353, X354), .(X355, X356), .(X355, X357)) :- ','(>(X353, X355), mergedelete(.(X353, X354), X356, X357)).\nand substitutionX353 -> T211,\nX354 -> T212,\nT120 -> .(T211, T212),\nX355 -> T213,\nX356 -> T214,\nT118 -> .(T213, T214),\nX357 -> X358,\nX184 -> .(T213, X358)" }, { "from": 4446, "to": 4840, "label": "EVAL-BACKTRACK" }, { "from": 4447, "to": 4449, "label": "CASE" }, { "from": 4449, "to": 4450, "label": "PARALLEL" }, { "from": 4449, "to": 4451, "label": "PARALLEL" }, { "from": 4450, "to": 4452, "label": "EVAL with clause\nmergedelete([], X293, X293).\nand substitutionT165 -> [],\nT166 -> T173,\nX293 -> T173,\nX286 -> T173" }, { "from": 4450, "to": 4453, "label": "EVAL-BACKTRACK" }, { "from": 4451, "to": 4455, "label": "PARALLEL" }, { "from": 4451, "to": 4456, "label": "PARALLEL" }, { "from": 4452, "to": 4454, "label": "SUCCESS" }, { "from": 4455, "to": 4525, "label": "EVAL with clause\nmergedelete(.(X314, X315), .(X314, X316), X317) :- mergedelete(X315, X316, X317).\nand substitutionX314 -> T186,\nX315 -> T187,\nT165 -> .(T186, T187),\nX316 -> T188,\nT166 -> .(T186, T188),\nX286 -> X318,\nX317 -> X318" }, { "from": 4455, "to": 4526, "label": "EVAL-BACKTRACK" }, { "from": 4456, "to": 4666, "label": "EVAL with clause\nmergedelete(.(X333, X334), .(X335, X336), .(X335, X337)) :- ','(>(X333, X335), mergedelete(.(X333, X334), X336, X337)).\nand substitutionX333 -> T199,\nX334 -> T200,\nT165 -> .(T199, T200),\nX335 -> T201,\nX336 -> T202,\nT166 -> .(T201, T202),\nX337 -> X338,\nX286 -> .(T201, X338)" }, { "from": 4456, "to": 4667, "label": "EVAL-BACKTRACK" }, { "from": 4525, "to": 4447, "label": "INSTANCE with matching:\nT165 -> T187\nT166 -> T188\nX286 -> X318" }, { "from": 4666, "to": 4668, "label": "IS ERROR" }, { "from": 4666, "to": 4837, "label": "ARITHCOMP SUCCESS" }, { "from": 4666, "to": 4838, "label": "ARITHCOMP FAIL" }, { "from": 4837, "to": 4447, "label": "INSTANCE with matching:\nT165 -> .(T199, T200)\nT166 -> T202\nX286 -> X338" }, { "from": 4839, "to": 4841, "label": "IS ERROR" }, { "from": 4839, "to": 4895, "label": "ARITHCOMP SUCCESS" }, { "from": 4839, "to": 4896, "label": "ARITHCOMP FAIL" }, { "from": 4895, "to": 4447, "label": "INSTANCE with matching:\nT165 -> .(T211, T212)\nT166 -> T214\nX286 -> X358" }, { "from": 4902, "to": 4904, "label": "CASE" }, { "from": 4903, "to": 5087, "label": "SPLIT 1" }, { "from": 4903, "to": 5090, "label": "SPLIT 2\nnew knowledge:\nT215 is ground\nT144 is ground\nT595 is ground\nreplacements:X186 -> T595" }, { "from": 4904, "to": 4905, "label": "PARALLEL" }, { "from": 4904, "to": 4906, "label": "PARALLEL" }, { "from": 4905, "to": 4907, "label": "EVAL with clause\nrev([], X369, X369).\nand substitutionT121 -> [],\nX369 -> [],\nX185 -> []" }, { "from": 4905, "to": 4908, "label": "EVAL-BACKTRACK" }, { "from": 4906, "to": 4910, "label": "EVAL with clause\nrev(.(X380, X381), X382, X383) :- rev(X381, .(X380, X382), X383).\nand substitutionX380 -> T220,\nX381 -> T221,\nT121 -> .(T220, T221),\nX382 -> [],\nX185 -> X384,\nX383 -> X384" }, { "from": 4906, "to": 4911, "label": "EVAL-BACKTRACK" }, { "from": 4907, "to": 4909, "label": "SUCCESS" }, { "from": 4910, "to": 4912, "label": "CASE" }, { "from": 4912, "to": 4913, "label": "PARALLEL" }, { "from": 4912, "to": 4914, "label": "PARALLEL" }, { "from": 4913, "to": 4915, "label": "EVAL with clause\nrev([], X391, X391).\nand substitutionT221 -> [],\nT220 -> T228,\nX391 -> .(T228, []),\nX384 -> .(T228, [])" }, { "from": 4913, "to": 4916, "label": "EVAL-BACKTRACK" }, { "from": 4914, "to": 4918, "label": "EVAL with clause\nrev(.(X402, X403), X404, X405) :- rev(X403, .(X402, X404), X405).\nand substitutionX402 -> T235,\nX403 -> T236,\nT221 -> .(T235, T236),\nT220 -> T237,\nX404 -> .(T237, []),\nX384 -> X406,\nX405 -> X406" }, { "from": 4914, "to": 4919, "label": "EVAL-BACKTRACK" }, { "from": 4915, "to": 4917, "label": "SUCCESS" }, { "from": 4918, "to": 4920, "label": "CASE" }, { "from": 4920, "to": 4921, "label": "PARALLEL" }, { "from": 4920, "to": 4922, "label": "PARALLEL" }, { "from": 4921, "to": 4923, "label": "EVAL with clause\nrev([], X413, X413).\nand substitutionT236 -> [],\nT235 -> T250,\nT237 -> T251,\nX413 -> .(T250, .(T251, [])),\nX406 -> .(T250, .(T251, []))" }, { "from": 4921, "to": 4924, "label": "EVAL-BACKTRACK" }, { "from": 4922, "to": 4926, "label": "EVAL with clause\nrev(.(X424, X425), X426, X427) :- rev(X425, .(X424, X426), X427).\nand substitutionX424 -> T260,\nX425 -> T261,\nT236 -> .(T260, T261),\nT235 -> T262,\nT237 -> T263,\nX426 -> .(T262, .(T263, [])),\nX406 -> X428,\nX427 -> X428" }, { "from": 4922, "to": 4927, "label": "EVAL-BACKTRACK" }, { "from": 4923, "to": 4925, "label": "SUCCESS" }, { "from": 4926, "to": 4928, "label": "CASE" }, { "from": 4928, "to": 4929, "label": "PARALLEL" }, { "from": 4928, "to": 4930, "label": "PARALLEL" }, { "from": 4929, "to": 4931, "label": "EVAL with clause\nrev([], X435, X435).\nand substitutionT261 -> [],\nT260 -> T282,\nT262 -> T283,\nT263 -> T284,\nX435 -> .(T282, .(T283, .(T284, []))),\nX428 -> .(T282, .(T283, .(T284, [])))" }, { "from": 4929, "to": 4932, "label": "EVAL-BACKTRACK" }, { "from": 4930, "to": 4934, "label": "EVAL with clause\nrev(.(X446, X447), X448, X449) :- rev(X447, .(X446, X448), X449).\nand substitutionX446 -> T295,\nX447 -> T296,\nT261 -> .(T295, T296),\nT260 -> T297,\nT262 -> T298,\nT263 -> T299,\nX448 -> .(T297, .(T298, .(T299, []))),\nX428 -> X450,\nX449 -> X450" }, { "from": 4930, "to": 4935, "label": "EVAL-BACKTRACK" }, { "from": 4931, "to": 4933, "label": "SUCCESS" }, { "from": 4934, "to": 4936, "label": "CASE" }, { "from": 4936, "to": 4937, "label": "PARALLEL" }, { "from": 4936, "to": 4938, "label": "PARALLEL" }, { "from": 4937, "to": 4939, "label": "EVAL with clause\nrev([], X457, X457).\nand substitutionT296 -> [],\nT295 -> T324,\nT297 -> T325,\nT298 -> T326,\nT299 -> T327,\nX457 -> .(T324, .(T325, .(T326, .(T327, [])))),\nX450 -> .(T324, .(T325, .(T326, .(T327, []))))" }, { "from": 4937, "to": 4940, "label": "EVAL-BACKTRACK" }, { "from": 4938, "to": 4942, "label": "EVAL with clause\nrev(.(X468, X469), X470, X471) :- rev(X469, .(X468, X470), X471).\nand substitutionX468 -> T340,\nX469 -> T341,\nT296 -> .(T340, T341),\nT295 -> T342,\nT297 -> T343,\nT298 -> T344,\nT299 -> T345,\nX470 -> .(T342, .(T343, .(T344, .(T345, [])))),\nX450 -> X472,\nX471 -> X472" }, { "from": 4938, "to": 4943, "label": "EVAL-BACKTRACK" }, { "from": 4939, "to": 4941, "label": "SUCCESS" }, { "from": 4942, "to": 4944, "label": "CASE" }, { "from": 4944, "to": 4947, "label": "PARALLEL" }, { "from": 4944, "to": 4948, "label": "PARALLEL" }, { "from": 4947, "to": 4952, "label": "EVAL with clause\nrev([], X479, X479).\nand substitutionT341 -> [],\nT340 -> T376,\nT342 -> T377,\nT343 -> T378,\nT344 -> T379,\nT345 -> T380,\nX479 -> .(T376, .(T377, .(T378, .(T379, .(T380, []))))),\nX472 -> .(T376, .(T377, .(T378, .(T379, .(T380, [])))))" }, { "from": 4947, "to": 4953, "label": "EVAL-BACKTRACK" }, { "from": 4948, "to": 4955, "label": "EVAL with clause\nrev(.(X490, X491), X492, X493) :- rev(X491, .(X490, X492), X493).\nand substitutionX490 -> T395,\nX491 -> T396,\nT341 -> .(T395, T396),\nT340 -> T397,\nT342 -> T398,\nT343 -> T399,\nT344 -> T400,\nT345 -> T401,\nX492 -> .(T397, .(T398, .(T399, .(T400, .(T401, []))))),\nX472 -> X494,\nX493 -> X494" }, { "from": 4948, "to": 4956, "label": "EVAL-BACKTRACK" }, { "from": 4952, "to": 4954, "label": "SUCCESS" }, { "from": 4955, "to": 4957, "label": "CASE" }, { "from": 4957, "to": 4958, "label": "PARALLEL" }, { "from": 4957, "to": 4959, "label": "PARALLEL" }, { "from": 4958, "to": 4960, "label": "EVAL with clause\nrev([], X501, X501).\nand substitutionT396 -> [],\nT395 -> T438,\nT397 -> T439,\nT398 -> T440,\nT399 -> T441,\nT400 -> T442,\nT401 -> T443,\nX501 -> .(T438, .(T439, .(T440, .(T441, .(T442, .(T443, [])))))),\nX494 -> .(T438, .(T439, .(T440, .(T441, .(T442, .(T443, []))))))" }, { "from": 4958, "to": 4961, "label": "EVAL-BACKTRACK" }, { "from": 4959, "to": 4963, "label": "EVAL with clause\nrev(.(X512, X513), X514, X515) :- rev(X513, .(X512, X514), X515).\nand substitutionX512 -> T460,\nX513 -> T461,\nT396 -> .(T460, T461),\nT395 -> T462,\nT397 -> T463,\nT398 -> T464,\nT399 -> T465,\nT400 -> T466,\nT401 -> T467,\nX514 -> .(T462, .(T463, .(T464, .(T465, .(T466, .(T467, [])))))),\nX494 -> X516,\nX515 -> X516" }, { "from": 4959, "to": 4964, "label": "EVAL-BACKTRACK" }, { "from": 4960, "to": 4962, "label": "SUCCESS" }, { "from": 4963, "to": 4969, "label": "CASE" }, { "from": 4969, "to": 5052, "label": "PARALLEL" }, { "from": 4969, "to": 5053, "label": "PARALLEL" }, { "from": 5052, "to": 5054, "label": "EVAL with clause\nrev([], X523, X523).\nand substitutionT461 -> [],\nT460 -> T510,\nT462 -> T511,\nT463 -> T512,\nT464 -> T513,\nT465 -> T514,\nT466 -> T515,\nT467 -> T516,\nX523 -> .(T510, .(T511, .(T512, .(T513, .(T514, .(T515, .(T516, []))))))),\nX516 -> .(T510, .(T511, .(T512, .(T513, .(T514, .(T515, .(T516, [])))))))" }, { "from": 5052, "to": 5055, "label": "EVAL-BACKTRACK" }, { "from": 5053, "to": 5057, "label": "EVAL with clause\nrev(.(X534, X535), X536, X537) :- rev(X535, .(X534, X536), X537).\nand substitutionX534 -> T535,\nX535 -> T536,\nT461 -> .(T535, T536),\nT460 -> T537,\nT462 -> T538,\nT463 -> T539,\nT464 -> T540,\nT465 -> T541,\nT466 -> T542,\nT467 -> T543,\nX536 -> .(T537, .(T538, .(T539, .(T540, .(T541, .(T542, .(T543, []))))))),\nX516 -> X538,\nX537 -> X538" }, { "from": 5053, "to": 5058, "label": "EVAL-BACKTRACK" }, { "from": 5054, "to": 5056, "label": "SUCCESS" }, { "from": 5057, "to": 5059, "label": "GENERALIZATION\nT560 <-- .(T537, .(T538, .(T539, .(T540, .(T541, .(T542, .(T543, [])))))))\n\nNew Knowledge:\nT560 is ground" }, { "from": 5059, "to": 5076, "label": "CASE" }, { "from": 5076, "to": 5077, "label": "PARALLEL" }, { "from": 5076, "to": 5078, "label": "PARALLEL" }, { "from": 5077, "to": 5079, "label": "EVAL with clause\nrev([], X547, X547).\nand substitutionT536 -> [],\nT535 -> T573,\nT560 -> T574,\nX547 -> .(T573, T574),\nX538 -> .(T573, T574)" }, { "from": 5077, "to": 5080, "label": "EVAL-BACKTRACK" }, { "from": 5078, "to": 5082, "label": "EVAL with clause\nrev(.(X558, X559), X560, X561) :- rev(X559, .(X558, X560), X561).\nand substitutionX558 -> T583,\nX559 -> T584,\nT536 -> .(T583, T584),\nT535 -> T585,\nT560 -> T586,\nX560 -> .(T585, T586),\nX538 -> X562,\nX561 -> X562" }, { "from": 5078, "to": 5083, "label": "EVAL-BACKTRACK" }, { "from": 5079, "to": 5081, "label": "SUCCESS" }, { "from": 5082, "to": 5059, "label": "INSTANCE with matching:\nT536 -> T584\nT535 -> T583\nT560 -> .(T585, T586)\nX538 -> X562" }, { "from": 5087, "to": 4447, "label": "INSTANCE with matching:\nT165 -> T215\nT166 -> T144\nX286 -> X186" }, { "from": 5090, "to": 5091, "label": "CUT" }, { "from": 5091, "to": 5092, "label": "SUCCESS" }, { "from": 5096, "to": 5106, "label": "EVAL with clause\npds1([], X575, [], X576) :- !_22.\nand substitutionT110 -> [],\nT55 -> T604,\nX575 -> .(T604, .(0, [])),\nT56 -> [],\nT48 -> T605,\nX576 -> T605" }, { "from": 5096, "to": 5107, "label": "EVAL-BACKTRACK" }, { "from": 5106, "to": 5108, "label": "CUT" }, { "from": 5107, "to": 5121, "label": "EVAL with clause\npds1(X588, X589, .(X590, X591), X592) :- ','(member(X590, X588), ','(check(X589, X590, X588, X593, X592), pds1(X593, .(X590, X589), X591, X592))).\nand substitutionT110 -> T616,\nX588 -> T616,\nT55 -> T617,\nX589 -> .(T617, .(0, [])),\nX590 -> T621,\nX591 -> T622,\nT56 -> .(T621, T622),\nT48 -> T620,\nX592 -> T620,\nT618 -> T621,\nT619 -> T622" }, { "from": 5107, "to": 5122, "label": "EVAL-BACKTRACK" }, { "from": 5108, "to": 5109, "label": "SUCCESS" }, { "from": 5121, "to": 5123, "label": "SPLIT 1" }, { "from": 5121, "to": 5124, "label": "SPLIT 2\nnew knowledge:\nT627 is ground\nT616 is ground\nreplacements:T621 -> T627,\nT622 -> T628,\nT57 -> T629,\nT56 -> T630" }, { "from": 5123, "to": 4407, "label": "INSTANCE with matching:\nT81 -> T621\nT80 -> T616" }, { "from": 5124, "to": 5132, "label": "SPLIT 1" }, { "from": 5124, "to": 5133, "label": "SPLIT 2\nnew knowledge:\nT617 is ground\nT627 is ground\nT616 is ground\nT635 is ground\nT620 is ground\nreplacements:X593 -> T635" }, { "from": 5132, "to": 5134, "label": "CASE" }, { "from": 5133, "to": 5167, "label": "CASE" }, { "from": 5134, "to": 5135, "label": "BACKTRACK\nfor clause: check([], X4, L, L, X5) :- !because of non-unification" }, { "from": 5135, "to": 5143, "label": "ONLY EVAL with clause\ncheck(X631, X632, X633, X634, X635) :- ','(dif(X631, X632, X635, X636, X637), ','(mergedelete(X636, X633, X638), ','(rev(X637, [], X639), ','(mergedelete(X639, X638, X634), !_23)))).\nand substitutionT617 -> T644,\nX631 -> .(T644, .(0, [])),\nT627 -> T645,\nX632 -> T645,\nT616 -> T646,\nX633 -> T646,\nX593 -> X640,\nX634 -> X640,\nT620 -> T647,\nX635 -> T647" }, { "from": 5143, "to": 5144, "label": "SPLIT 1" }, { "from": 5143, "to": 5145, "label": "SPLIT 2\nnew knowledge:\nT644 is ground\nT645 is ground\nT647 is ground\nT648 is ground\nT649 is ground\nreplacements:X636 -> T648,\nX637 -> T649" }, { "from": 5144, "to": 5146, "label": "CASE" }, { "from": 5145, "to": 5153, "label": "SPLIT 1" }, { "from": 5145, "to": 5154, "label": "SPLIT 2\nnew knowledge:\nT648 is ground\nT646 is ground\nT663 is ground\nreplacements:X638 -> T663" }, { "from": 5146, "to": 5147, "label": "BACKTRACK\nfor clause: dif([], X2, X3, [], [])because of non-unification" }, { "from": 5147, "to": 5148, "label": "ONLY EVAL with clause\ndif(.(X675, X676), X677, X678, .(X679, X680), .(X681, X682)) :- ','(is(X679, -(X677, X675)), ','(is(X681, -(X678, X679)), dif(X676, X677, X678, X680, X682))).\nand substitutionT644 -> T656,\nX675 -> T656,\nX676 -> .(0, []),\nT645 -> T657,\nX677 -> T657,\nT647 -> T658,\nX678 -> T658,\nX679 -> X683,\nX680 -> X684,\nX636 -> .(X683, X684),\nX681 -> X685,\nX682 -> X686,\nX637 -> .(X685, X686)" }, { "from": 5148, "to": 5149, "label": "IS ERROR" }, { "from": 5148, "to": 5150, "label": "\nX683 -> T659" }, { "from": 5150, "to": 5151, "label": "IS ERROR" }, { "from": 5150, "to": 5152, "label": "\nX685 -> T660" }, { "from": 5152, "to": 4422, "label": "INSTANCE with matching:\nT117 -> T657\nT119 -> T658\nX182 -> X684\nX183 -> X686\nT118 -> T45\nT57 -> T629" }, { "from": 5153, "to": 4447, "label": "INSTANCE with matching:\nT165 -> T648\nT166 -> T646\nX286 -> X638" }, { "from": 5154, "to": 5155, "label": "SPLIT 1" }, { "from": 5154, "to": 5156, "label": "SPLIT 2\nnew knowledge:\nT649 is ground\nT666 is ground\nreplacements:X639 -> T666" }, { "from": 5155, "to": 4902, "label": "INSTANCE with matching:\nT121 -> T649\nX185 -> X639" }, { "from": 5156, "to": 5159, "label": "SPLIT 1" }, { "from": 5156, "to": 5160, "label": "SPLIT 2\nnew knowledge:\nT666 is ground\nT663 is ground\nT669 is ground\nreplacements:X640 -> T669" }, { "from": 5159, "to": 4447, "label": "INSTANCE with matching:\nT165 -> T666\nT166 -> T663\nX286 -> X640" }, { "from": 5160, "to": 5161, "label": "CUT" }, { "from": 5161, "to": 5162, "label": "SUCCESS" }, { "from": 5167, "to": 5171, "label": "EVAL with clause\npds1([], X709, [], X710) :- !_25.\nand substitutionT635 -> [],\nT627 -> T681,\nT617 -> T682,\nX709 -> .(T681, .(T682, .(0, []))),\nT628 -> [],\nT620 -> T683,\nX710 -> T683" }, { "from": 5167, "to": 5172, "label": "EVAL-BACKTRACK" }, { "from": 5171, "to": 5173, "label": "CUT" }, { "from": 5172, "to": 5182, "label": "EVAL with clause\npds1(X722, X723, .(X724, X725), X726) :- ','(member(X724, X722), ','(check(X723, X724, X722, X727, X726), pds1(X727, .(X724, X723), X725, X726))).\nand substitutionT635 -> T696,\nX722 -> T696,\nT627 -> T697,\nT617 -> T698,\nX723 -> .(T697, .(T698, .(0, []))),\nX724 -> T702,\nX725 -> T703,\nT628 -> .(T702, T703),\nT620 -> T701,\nX726 -> T701,\nT699 -> T702,\nT700 -> T703" }, { "from": 5172, "to": 5183, "label": "EVAL-BACKTRACK" }, { "from": 5173, "to": 5174, "label": "SUCCESS" }, { "from": 5182, "to": 5189, "label": "SPLIT 1" }, { "from": 5182, "to": 5190, "label": "SPLIT 2\nnew knowledge:\nT708 is ground\nT696 is ground\nreplacements:T702 -> T708,\nT703 -> T709,\nT629 -> T710,\nT630 -> T711,\nT628 -> T712" }, { "from": 5189, "to": 4407, "label": "INSTANCE with matching:\nT81 -> T702\nT80 -> T696" }, { "from": 5190, "to": 5194, "label": "SPLIT 1" }, { "from": 5190, "to": 5195, "label": "SPLIT 2\nnew knowledge:\nT697 is ground\nT698 is ground\nT708 is ground\nT696 is ground\nT717 is ground\nT701 is ground\nreplacements:X727 -> T717" }, { "from": 5194, "to": 5196, "label": "CASE" }, { "from": 5195, "to": 5288, "label": "CASE" }, { "from": 5196, "to": 5197, "label": "BACKTRACK\nfor clause: check([], X4, L, L, X5) :- !because of non-unification" }, { "from": 5197, "to": 5204, "label": "ONLY EVAL with clause\ncheck(X765, X766, X767, X768, X769) :- ','(dif(X765, X766, X769, X770, X771), ','(mergedelete(X770, X767, X772), ','(rev(X771, [], X773), ','(mergedelete(X773, X772, X768), !_26)))).\nand substitutionT697 -> T728,\nT698 -> T729,\nX765 -> .(T728, .(T729, .(0, []))),\nT708 -> T730,\nX766 -> T730,\nT696 -> T731,\nX767 -> T731,\nX727 -> X774,\nX768 -> X774,\nT701 -> T732,\nX769 -> T732" }, { "from": 5204, "to": 5207, "label": "SPLIT 1" }, { "from": 5204, "to": 5208, "label": "SPLIT 2\nnew knowledge:\nT728 is ground\nT729 is ground\nT730 is ground\nT732 is ground\nT733 is ground\nT734 is ground\nreplacements:X770 -> T733,\nX771 -> T734" }, { "from": 5207, "to": 5209, "label": "CASE" }, { "from": 5208, "to": 5244, "label": "SPLIT 1" }, { "from": 5208, "to": 5245, "label": "SPLIT 2\nnew knowledge:\nT733 is ground\nT731 is ground\nT751 is ground\nreplacements:X772 -> T751" }, { "from": 5209, "to": 5210, "label": "BACKTRACK\nfor clause: dif([], X2, X3, [], [])because of non-unification" }, { "from": 5210, "to": 5227, "label": "ONLY EVAL with clause\ndif(.(X809, X810), X811, X812, .(X813, X814), .(X815, X816)) :- ','(is(X813, -(X811, X809)), ','(is(X815, -(X812, X813)), dif(X810, X811, X812, X814, X816))).\nand substitutionT728 -> T743,\nX809 -> T743,\nT729 -> T744,\nX810 -> .(T744, .(0, [])),\nT730 -> T745,\nX811 -> T745,\nT732 -> T746,\nX812 -> T746,\nX813 -> X817,\nX814 -> X818,\nX770 -> .(X817, X818),\nX815 -> X819,\nX816 -> X820,\nX771 -> .(X819, X820)" }, { "from": 5227, "to": 5228, "label": "IS ERROR" }, { "from": 5227, "to": 5229, "label": "\nX817 -> T747" }, { "from": 5229, "to": 5230, "label": "IS ERROR" }, { "from": 5229, "to": 5231, "label": "\nX819 -> T748" }, { "from": 5231, "to": 5144, "label": "INSTANCE with matching:\nT644 -> T744\nT645 -> T745\nT647 -> T746\nX636 -> X818\nX637 -> X820\nT629 -> T710\nT646 -> T616\nT630 -> T711" }, { "from": 5244, "to": 4447, "label": "INSTANCE with matching:\nT165 -> T733\nT166 -> T731\nX286 -> X772" }, { "from": 5245, "to": 5253, "label": "SPLIT 1" }, { "from": 5245, "to": 5254, "label": "SPLIT 2\nnew knowledge:\nT734 is ground\nT754 is ground\nreplacements:X773 -> T754" }, { "from": 5253, "to": 4902, "label": "INSTANCE with matching:\nT121 -> T734\nX185 -> X773" }, { "from": 5254, "to": 5261, "label": "SPLIT 1" }, { "from": 5254, "to": 5262, "label": "SPLIT 2\nnew knowledge:\nT754 is ground\nT751 is ground\nT757 is ground\nreplacements:X774 -> T757" }, { "from": 5261, "to": 4447, "label": "INSTANCE with matching:\nT165 -> T754\nT166 -> T751\nX286 -> X774" }, { "from": 5262, "to": 5267, "label": "CUT" }, { "from": 5267, "to": 5268, "label": "SUCCESS" }, { "from": 5288, "to": 5294, "label": "EVAL with clause\npds1([], X843, [], X844) :- !_28.\nand substitutionT717 -> [],\nT708 -> T772,\nT697 -> T773,\nT698 -> T774,\nX843 -> .(T772, .(T773, .(T774, .(0, [])))),\nT709 -> [],\nT701 -> T775,\nX844 -> T775" }, { "from": 5288, "to": 5298, "label": "EVAL-BACKTRACK" }, { "from": 5294, "to": 5299, "label": "CUT" }, { "from": 5298, "to": 5314, "label": "EVAL with clause\npds1(X856, X857, .(X858, X859), X860) :- ','(member(X858, X856), ','(check(X857, X858, X856, X861, X860), pds1(X861, .(X858, X857), X859, X860))).\nand substitutionT717 -> T790,\nX856 -> T790,\nT708 -> T791,\nT697 -> T792,\nT698 -> T793,\nX857 -> .(T791, .(T792, .(T793, .(0, [])))),\nX858 -> T797,\nX859 -> T798,\nT709 -> .(T797, T798),\nT701 -> T796,\nX860 -> T796,\nT794 -> T797,\nT795 -> T798" }, { "from": 5298, "to": 5315, "label": "EVAL-BACKTRACK" }, { "from": 5299, "to": 5300, "label": "SUCCESS" }, { "from": 5314, "to": 5333, "label": "SPLIT 1" }, { "from": 5314, "to": 5336, "label": "SPLIT 2\nnew knowledge:\nT803 is ground\nT790 is ground\nreplacements:T797 -> T803,\nT798 -> T804,\nT710 -> T805,\nT711 -> T806,\nT712 -> T807,\nT709 -> T808" }, { "from": 5333, "to": 4407, "label": "INSTANCE with matching:\nT81 -> T797\nT80 -> T790" }, { "from": 5336, "to": 5340, "label": "SPLIT 1" }, { "from": 5336, "to": 5341, "label": "SPLIT 2\nnew knowledge:\nT791 is ground\nT792 is ground\nT793 is ground\nT803 is ground\nT790 is ground\nT813 is ground\nT796 is ground\nreplacements:X861 -> T813" }, { "from": 5340, "to": 5348, "label": "CASE" }, { "from": 5341, "to": 5408, "label": "CASE" }, { "from": 5348, "to": 5351, "label": "BACKTRACK\nfor clause: check([], X4, L, L, X5) :- !because of non-unification" }, { "from": 5351, "to": 5363, "label": "ONLY EVAL with clause\ncheck(X899, X900, X901, X902, X903) :- ','(dif(X899, X900, X903, X904, X905), ','(mergedelete(X904, X901, X906), ','(rev(X905, [], X907), ','(mergedelete(X907, X906, X902), !_29)))).\nand substitutionT791 -> T826,\nT792 -> T827,\nT793 -> T828,\nX899 -> .(T826, .(T827, .(T828, .(0, [])))),\nT803 -> T829,\nX900 -> T829,\nT790 -> T830,\nX901 -> T830,\nX861 -> X908,\nX902 -> X908,\nT796 -> T831,\nX903 -> T831" }, { "from": 5363, "to": 5368, "label": "SPLIT 1" }, { "from": 5363, "to": 5369, "label": "SPLIT 2\nnew knowledge:\nT826 is ground\nT827 is ground\nT828 is ground\nT829 is ground\nT831 is ground\nT832 is ground\nT833 is ground\nreplacements:X904 -> T832,\nX905 -> T833" }, { "from": 5368, "to": 5371, "label": "CASE" }, { "from": 5369, "to": 5389, "label": "SPLIT 1" }, { "from": 5369, "to": 5390, "label": "SPLIT 2\nnew knowledge:\nT832 is ground\nT830 is ground\nT853 is ground\nreplacements:X906 -> T853" }, { "from": 5371, "to": 5373, "label": "BACKTRACK\nfor clause: dif([], X2, X3, [], [])because of non-unification" }, { "from": 5373, "to": 5379, "label": "ONLY EVAL with clause\ndif(.(X943, X944), X945, X946, .(X947, X948), .(X949, X950)) :- ','(is(X947, -(X945, X943)), ','(is(X949, -(X946, X947)), dif(X944, X945, X946, X948, X950))).\nand substitutionT826 -> T844,\nX943 -> T844,\nT827 -> T845,\nT828 -> T846,\nX944 -> .(T845, .(T846, .(0, []))),\nT829 -> T847,\nX945 -> T847,\nT831 -> T848,\nX946 -> T848,\nX947 -> X951,\nX948 -> X952,\nX904 -> .(X951, X952),\nX949 -> X953,\nX950 -> X954,\nX905 -> .(X953, X954)" }, { "from": 5379, "to": 5380, "label": "IS ERROR" }, { "from": 5379, "to": 5381, "label": "\nX951 -> T849" }, { "from": 5381, "to": 5382, "label": "IS ERROR" }, { "from": 5381, "to": 5383, "label": "\nX953 -> T850" }, { "from": 5383, "to": 5207, "label": "INSTANCE with matching:\nT728 -> T845\nT729 -> T846\nT730 -> T847\nT732 -> T848\nX770 -> X952\nX771 -> X954\nT710 -> T805\nT711 -> T806\nT731 -> T696\nT712 -> T807" }, { "from": 5389, "to": 4447, "label": "INSTANCE with matching:\nT165 -> T832\nT166 -> T830\nX286 -> X906" }, { "from": 5390, "to": 5395, "label": "SPLIT 1" }, { "from": 5390, "to": 5396, "label": "SPLIT 2\nnew knowledge:\nT833 is ground\nT856 is ground\nreplacements:X907 -> T856" }, { "from": 5395, "to": 4902, "label": "INSTANCE with matching:\nT121 -> T833\nX185 -> X907" }, { "from": 5396, "to": 5398, "label": "SPLIT 1" }, { "from": 5396, "to": 5399, "label": "SPLIT 2\nnew knowledge:\nT856 is ground\nT853 is ground\nT859 is ground\nreplacements:X908 -> T859" }, { "from": 5398, "to": 4447, "label": "INSTANCE with matching:\nT165 -> T856\nT166 -> T853\nX286 -> X908" }, { "from": 5399, "to": 5402, "label": "CUT" }, { "from": 5402, "to": 5403, "label": "SUCCESS" }, { "from": 5408, "to": 5418, "label": "EVAL with clause\npds1([], X977, [], X978) :- !_31.\nand substitutionT813 -> [],\nT803 -> T877,\nT791 -> T878,\nT792 -> T879,\nT793 -> T880,\nX977 -> .(T877, .(T878, .(T879, .(T880, .(0, []))))),\nT804 -> [],\nT796 -> T881,\nX978 -> T881" }, { "from": 5408, "to": 5419, "label": "EVAL-BACKTRACK" }, { "from": 5418, "to": 5420, "label": "CUT" }, { "from": 5419, "to": 5422, "label": "EVAL with clause\npds1(X990, X991, .(X992, X993), X994) :- ','(member(X992, X990), ','(check(X991, X992, X990, X995, X994), pds1(X995, .(X992, X991), X993, X994))).\nand substitutionT813 -> T898,\nX990 -> T898,\nT803 -> T899,\nT791 -> T900,\nT792 -> T901,\nT793 -> T902,\nX991 -> .(T899, .(T900, .(T901, .(T902, .(0, []))))),\nX992 -> T906,\nX993 -> T907,\nT804 -> .(T906, T907),\nT796 -> T905,\nX994 -> T905,\nT903 -> T906,\nT904 -> T907" }, { "from": 5419, "to": 5423, "label": "EVAL-BACKTRACK" }, { "from": 5420, "to": 5421, "label": "SUCCESS" }, { "from": 5422, "to": 5424, "label": "SPLIT 1" }, { "from": 5422, "to": 5425, "label": "SPLIT 2\nnew knowledge:\nT912 is ground\nT898 is ground\nreplacements:T906 -> T912,\nT907 -> T913,\nT805 -> T914,\nT806 -> T915,\nT807 -> T916,\nT808 -> T917,\nT804 -> T918" }, { "from": 5424, "to": 4407, "label": "INSTANCE with matching:\nT81 -> T906\nT80 -> T898" }, { "from": 5425, "to": 5426, "label": "SPLIT 1" }, { "from": 5425, "to": 5427, "label": "SPLIT 2\nnew knowledge:\nT899 is ground\nT900 is ground\nT901 is ground\nT902 is ground\nT912 is ground\nT898 is ground\nT923 is ground\nT905 is ground\nreplacements:X995 -> T923" }, { "from": 5426, "to": 5428, "label": "CASE" }, { "from": 5427, "to": 5539, "label": "CASE" }, { "from": 5428, "to": 5429, "label": "BACKTRACK\nfor clause: check([], X4, L, L, X5) :- !because of non-unification" }, { "from": 5429, "to": 5472, "label": "ONLY EVAL with clause\ncheck(X1033, X1034, X1035, X1036, X1037) :- ','(dif(X1033, X1034, X1037, X1038, X1039), ','(mergedelete(X1038, X1035, X1040), ','(rev(X1039, [], X1041), ','(mergedelete(X1041, X1040, X1036), !_32)))).\nand substitutionT899 -> T938,\nT900 -> T939,\nT901 -> T940,\nT902 -> T941,\nX1033 -> .(T938, .(T939, .(T940, .(T941, .(0, []))))),\nT912 -> T942,\nX1034 -> T942,\nT898 -> T943,\nX1035 -> T943,\nX995 -> X1042,\nX1036 -> X1042,\nT905 -> T944,\nX1037 -> T944" }, { "from": 5472, "to": 5473, "label": "SPLIT 1" }, { "from": 5472, "to": 5474, "label": "SPLIT 2\nnew knowledge:\nT938 is ground\nT939 is ground\nT940 is ground\nT941 is ground\nT942 is ground\nT944 is ground\nT945 is ground\nT946 is ground\nreplacements:X1038 -> T945,\nX1039 -> T946" }, { "from": 5473, "to": 5475, "label": "CASE" }, { "from": 5474, "to": 5513, "label": "SPLIT 1" }, { "from": 5474, "to": 5514, "label": "SPLIT 2\nnew knowledge:\nT945 is ground\nT943 is ground\nT969 is ground\nreplacements:X1040 -> T969" }, { "from": 5475, "to": 5476, "label": "BACKTRACK\nfor clause: dif([], X2, X3, [], [])because of non-unification" }, { "from": 5476, "to": 5506, "label": "ONLY EVAL with clause\ndif(.(X1077, X1078), X1079, X1080, .(X1081, X1082), .(X1083, X1084)) :- ','(is(X1081, -(X1079, X1077)), ','(is(X1083, -(X1080, X1081)), dif(X1078, X1079, X1080, X1082, X1084))).\nand substitutionT938 -> T959,\nX1077 -> T959,\nT939 -> T960,\nT940 -> T961,\nT941 -> T962,\nX1078 -> .(T960, .(T961, .(T962, .(0, [])))),\nT942 -> T963,\nX1079 -> T963,\nT944 -> T964,\nX1080 -> T964,\nX1081 -> X1085,\nX1082 -> X1086,\nX1038 -> .(X1085, X1086),\nX1083 -> X1087,\nX1084 -> X1088,\nX1039 -> .(X1087, X1088)" }, { "from": 5506, "to": 5507, "label": "IS ERROR" }, { "from": 5506, "to": 5508, "label": "\nX1085 -> T965" }, { "from": 5508, "to": 5509, "label": "IS ERROR" }, { "from": 5508, "to": 5510, "label": "\nX1087 -> T966" }, { "from": 5510, "to": 5368, "label": "INSTANCE with matching:\nT826 -> T960\nT827 -> T961\nT828 -> T962\nT829 -> T963\nT831 -> T964\nX904 -> X1086\nX905 -> X1088\nT805 -> T914\nT806 -> T915\nT807 -> T916\nT830 -> T790\nT808 -> T917" }, { "from": 5513, "to": 4447, "label": "INSTANCE with matching:\nT165 -> T945\nT166 -> T943\nX286 -> X1040" }, { "from": 5514, "to": 5532, "label": "SPLIT 1" }, { "from": 5514, "to": 5533, "label": "SPLIT 2\nnew knowledge:\nT946 is ground\nT972 is ground\nreplacements:X1041 -> T972" }, { "from": 5532, "to": 4902, "label": "INSTANCE with matching:\nT121 -> T946\nX185 -> X1041" }, { "from": 5533, "to": 5535, "label": "SPLIT 1" }, { "from": 5533, "to": 5536, "label": "SPLIT 2\nnew knowledge:\nT972 is ground\nT969 is ground\nT975 is ground\nreplacements:X1042 -> T975" }, { "from": 5535, "to": 4447, "label": "INSTANCE with matching:\nT165 -> T972\nT166 -> T969\nX286 -> X1042" }, { "from": 5536, "to": 5537, "label": "CUT" }, { "from": 5537, "to": 5538, "label": "SUCCESS" }, { "from": 5539, "to": 5563, "label": "EVAL with clause\npds1([], X1111, [], X1112) :- !_34.\nand substitutionT923 -> [],\nT912 -> T996,\nT899 -> T997,\nT900 -> T998,\nT901 -> T999,\nT902 -> T1000,\nX1111 -> .(T996, .(T997, .(T998, .(T999, .(T1000, .(0, [])))))),\nT913 -> [],\nT905 -> T1001,\nX1112 -> T1001" }, { "from": 5539, "to": 5564, "label": "EVAL-BACKTRACK" }, { "from": 5563, "to": 5565, "label": "CUT" }, { "from": 5564, "to": 5581, "label": "EVAL with clause\npds1(X1124, X1125, .(X1126, X1127), X1128) :- ','(member(X1126, X1124), ','(check(X1125, X1126, X1124, X1129, X1128), pds1(X1129, .(X1126, X1125), X1127, X1128))).\nand substitutionT923 -> T1020,\nX1124 -> T1020,\nT912 -> T1021,\nT899 -> T1022,\nT900 -> T1023,\nT901 -> T1024,\nT902 -> T1025,\nX1125 -> .(T1021, .(T1022, .(T1023, .(T1024, .(T1025, .(0, [])))))),\nX1126 -> T1029,\nX1127 -> T1030,\nT913 -> .(T1029, T1030),\nT905 -> T1028,\nX1128 -> T1028,\nT1026 -> T1029,\nT1027 -> T1030" }, { "from": 5564, "to": 5586, "label": "EVAL-BACKTRACK" }, { "from": 5565, "to": 5566, "label": "SUCCESS" }, { "from": 5581, "to": 5591, "label": "SPLIT 1" }, { "from": 5581, "to": 5592, "label": "SPLIT 2\nnew knowledge:\nT1035 is ground\nT1020 is ground\nreplacements:T1029 -> T1035,\nT1030 -> T1036,\nT914 -> T1037,\nT915 -> T1038,\nT916 -> T1039,\nT917 -> T1040,\nT918 -> T1041,\nT913 -> T1042" }, { "from": 5591, "to": 4407, "label": "INSTANCE with matching:\nT81 -> T1029\nT80 -> T1020" }, { "from": 5592, "to": 5606, "label": "SPLIT 1" }, { "from": 5592, "to": 5607, "label": "SPLIT 2\nnew knowledge:\nT1021 is ground\nT1022 is ground\nT1023 is ground\nT1024 is ground\nT1025 is ground\nT1035 is ground\nT1020 is ground\nT1047 is ground\nT1028 is ground\nreplacements:X1129 -> T1047" }, { "from": 5606, "to": 5608, "label": "CASE" }, { "from": 5607, "to": 5691, "label": "CASE" }, { "from": 5608, "to": 5609, "label": "BACKTRACK\nfor clause: check([], X4, L, L, X5) :- !because of non-unification" }, { "from": 5609, "to": 5642, "label": "ONLY EVAL with clause\ncheck(X1167, X1168, X1169, X1170, X1171) :- ','(dif(X1167, X1168, X1171, X1172, X1173), ','(mergedelete(X1172, X1169, X1174), ','(rev(X1173, [], X1175), ','(mergedelete(X1175, X1174, X1170), !_35)))).\nand substitutionT1021 -> T1064,\nT1022 -> T1065,\nT1023 -> T1066,\nT1024 -> T1067,\nT1025 -> T1068,\nX1167 -> .(T1064, .(T1065, .(T1066, .(T1067, .(T1068, .(0, [])))))),\nT1035 -> T1069,\nX1168 -> T1069,\nT1020 -> T1070,\nX1169 -> T1070,\nX1129 -> X1176,\nX1170 -> X1176,\nT1028 -> T1071,\nX1171 -> T1071" }, { "from": 5642, "to": 5647, "label": "SPLIT 1" }, { "from": 5642, "to": 5648, "label": "SPLIT 2\nnew knowledge:\nT1064 is ground\nT1065 is ground\nT1066 is ground\nT1067 is ground\nT1068 is ground\nT1069 is ground\nT1071 is ground\nT1072 is ground\nT1073 is ground\nreplacements:X1172 -> T1072,\nX1173 -> T1073" }, { "from": 5647, "to": 5652, "label": "CASE" }, { "from": 5648, "to": 5675, "label": "SPLIT 1" }, { "from": 5648, "to": 5676, "label": "SPLIT 2\nnew knowledge:\nT1072 is ground\nT1070 is ground\nT1099 is ground\nreplacements:X1174 -> T1099" }, { "from": 5652, "to": 5653, "label": "BACKTRACK\nfor clause: dif([], X2, X3, [], [])because of non-unification" }, { "from": 5653, "to": 5667, "label": "ONLY EVAL with clause\ndif(.(X1211, X1212), X1213, X1214, .(X1215, X1216), .(X1217, X1218)) :- ','(is(X1215, -(X1213, X1211)), ','(is(X1217, -(X1214, X1215)), dif(X1212, X1213, X1214, X1216, X1218))).\nand substitutionT1064 -> T1088,\nX1211 -> T1088,\nT1065 -> T1089,\nT1066 -> T1090,\nT1067 -> T1091,\nT1068 -> T1092,\nX1212 -> .(T1089, .(T1090, .(T1091, .(T1092, .(0, []))))),\nT1069 -> T1093,\nX1213 -> T1093,\nT1071 -> T1094,\nX1214 -> T1094,\nX1215 -> X1219,\nX1216 -> X1220,\nX1172 -> .(X1219, X1220),\nX1217 -> X1221,\nX1218 -> X1222,\nX1173 -> .(X1221, X1222)" }, { "from": 5667, "to": 5668, "label": "IS ERROR" }, { "from": 5667, "to": 5669, "label": "\nX1219 -> T1095" }, { "from": 5669, "to": 5670, "label": "IS ERROR" }, { "from": 5669, "to": 5671, "label": "\nX1221 -> T1096" }, { "from": 5671, "to": 5473, "label": "INSTANCE with matching:\nT938 -> T1089\nT939 -> T1090\nT940 -> T1091\nT941 -> T1092\nT942 -> T1093\nT944 -> T1094\nX1038 -> X1220\nX1039 -> X1222\nT914 -> T1037\nT915 -> T1038\nT916 -> T1039\nT917 -> T1040\nT943 -> T898\nT918 -> T1041" }, { "from": 5675, "to": 4447, "label": "INSTANCE with matching:\nT165 -> T1072\nT166 -> T1070\nX286 -> X1174" }, { "from": 5676, "to": 5679, "label": "SPLIT 1" }, { "from": 5676, "to": 5680, "label": "SPLIT 2\nnew knowledge:\nT1073 is ground\nT1102 is ground\nreplacements:X1175 -> T1102" }, { "from": 5679, "to": 4902, "label": "INSTANCE with matching:\nT121 -> T1073\nX185 -> X1175" }, { "from": 5680, "to": 5683, "label": "SPLIT 1" }, { "from": 5680, "to": 5684, "label": "SPLIT 2\nnew knowledge:\nT1102 is ground\nT1099 is ground\nT1105 is ground\nreplacements:X1176 -> T1105" }, { "from": 5683, "to": 4447, "label": "INSTANCE with matching:\nT165 -> T1102\nT166 -> T1099\nX286 -> X1176" }, { "from": 5684, "to": 5685, "label": "CUT" }, { "from": 5685, "to": 5686, "label": "SUCCESS" }, { "from": 5691, "to": 5716, "label": "EVAL with clause\npds1([], X1245, [], X1246) :- !_37.\nand substitutionT1047 -> [],\nT1035 -> T1129,\nT1021 -> T1130,\nT1022 -> T1131,\nT1023 -> T1132,\nT1024 -> T1133,\nT1025 -> T1134,\nX1245 -> .(T1129, .(T1130, .(T1131, .(T1132, .(T1133, .(T1134, .(0, []))))))),\nT1036 -> [],\nT1028 -> T1135,\nX1246 -> T1135" }, { "from": 5691, "to": 5719, "label": "EVAL-BACKTRACK" }, { "from": 5716, "to": 5720, "label": "CUT" }, { "from": 5719, "to": 5733, "label": "EVAL with clause\npds1(X1258, X1259, .(X1260, X1261), X1262) :- ','(member(X1260, X1258), ','(check(X1259, X1260, X1258, X1263, X1262), pds1(X1263, .(X1260, X1259), X1261, X1262))).\nand substitutionT1047 -> T1156,\nX1258 -> T1156,\nT1035 -> T1157,\nT1021 -> T1158,\nT1022 -> T1159,\nT1023 -> T1160,\nT1024 -> T1161,\nT1025 -> T1162,\nX1259 -> .(T1157, .(T1158, .(T1159, .(T1160, .(T1161, .(T1162, .(0, []))))))),\nX1260 -> T1166,\nX1261 -> T1167,\nT1036 -> .(T1166, T1167),\nT1028 -> T1165,\nX1262 -> T1165,\nT1163 -> T1166,\nT1164 -> T1167" }, { "from": 5719, "to": 5734, "label": "EVAL-BACKTRACK" }, { "from": 5720, "to": 5721, "label": "SUCCESS" }, { "from": 5733, "to": 5736, "label": "GENERALIZATION\nT1172 <-- .(T1157, .(T1158, .(T1159, .(T1160, .(T1161, .(T1162, .(0, [])))))))\n\nNew Knowledge:\nT1172 is ground" }, { "from": 5736, "to": 5744, "label": "SPLIT 1" }, { "from": 5736, "to": 5745, "label": "SPLIT 2\nnew knowledge:\nT1177 is ground\nT1156 is ground\nreplacements:T1166 -> T1177,\nT1167 -> T1178,\nT1037 -> T1179,\nT1038 -> T1180,\nT1039 -> T1181,\nT1040 -> T1182,\nT1041 -> T1183,\nT1042 -> T1184,\nT1036 -> T1185" }, { "from": 5744, "to": 4407, "label": "INSTANCE with matching:\nT81 -> T1166\nT80 -> T1156" }, { "from": 5745, "to": 5746, "label": "SPLIT 1" }, { "from": 5745, "to": 5748, "label": "SPLIT 2\nnew knowledge:\nT1157 is ground\nT1158 is ground\nT1159 is ground\nT1160 is ground\nT1161 is ground\nT1162 is ground\nT1177 is ground\nT1156 is ground\nT1190 is ground\nT1165 is ground\nreplacements:X1263 -> T1190" }, { "from": 5746, "to": 5750, "label": "CASE" }, { "from": 5748, "to": 5830, "label": "CASE" }, { "from": 5750, "to": 5751, "label": "BACKTRACK\nfor clause: check([], X4, L, L, X5) :- !because of non-unification" }, { "from": 5751, "to": 5762, "label": "ONLY EVAL with clause\ncheck(X1305, X1306, X1307, X1308, X1309) :- ','(dif(X1305, X1306, X1309, X1310, X1311), ','(mergedelete(X1310, X1307, X1312), ','(rev(X1311, [], X1313), ','(mergedelete(X1313, X1312, X1308), !_38)))).\nand substitutionT1157 -> T1209,\nT1158 -> T1210,\nT1159 -> T1211,\nT1160 -> T1212,\nT1161 -> T1213,\nT1162 -> T1214,\nX1305 -> .(T1209, .(T1210, .(T1211, .(T1212, .(T1213, .(T1214, .(0, []))))))),\nT1177 -> T1215,\nX1306 -> T1215,\nT1156 -> T1216,\nX1307 -> T1216,\nX1263 -> X1314,\nX1308 -> X1314,\nT1165 -> T1217,\nX1309 -> T1217" }, { "from": 5762, "to": 5765, "label": "SPLIT 1" }, { "from": 5762, "to": 5767, "label": "SPLIT 2\nnew knowledge:\nT1209 is ground\nT1210 is ground\nT1211 is ground\nT1212 is ground\nT1213 is ground\nT1214 is ground\nT1215 is ground\nT1217 is ground\nT1218 is ground\nT1219 is ground\nreplacements:X1310 -> T1218,\nX1311 -> T1219" }, { "from": 5765, "to": 5769, "label": "CASE" }, { "from": 5767, "to": 5794, "label": "SPLIT 1" }, { "from": 5767, "to": 5795, "label": "SPLIT 2\nnew knowledge:\nT1218 is ground\nT1216 is ground\nT1248 is ground\nreplacements:X1312 -> T1248" }, { "from": 5769, "to": 5770, "label": "BACKTRACK\nfor clause: dif([], X2, X3, [], [])because of non-unification" }, { "from": 5770, "to": 5789, "label": "ONLY EVAL with clause\ndif(.(X1349, X1350), X1351, X1352, .(X1353, X1354), .(X1355, X1356)) :- ','(is(X1353, -(X1351, X1349)), ','(is(X1355, -(X1352, X1353)), dif(X1350, X1351, X1352, X1354, X1356))).\nand substitutionT1209 -> T1236,\nX1349 -> T1236,\nT1210 -> T1237,\nT1211 -> T1238,\nT1212 -> T1239,\nT1213 -> T1240,\nT1214 -> T1241,\nX1350 -> .(T1237, .(T1238, .(T1239, .(T1240, .(T1241, .(0, [])))))),\nT1215 -> T1242,\nX1351 -> T1242,\nT1217 -> T1243,\nX1352 -> T1243,\nX1353 -> X1357,\nX1354 -> X1358,\nX1310 -> .(X1357, X1358),\nX1355 -> X1359,\nX1356 -> X1360,\nX1311 -> .(X1359, X1360)" }, { "from": 5789, "to": 5790, "label": "IS ERROR" }, { "from": 5789, "to": 5791, "label": "\nX1357 -> T1244" }, { "from": 5791, "to": 5792, "label": "IS ERROR" }, { "from": 5791, "to": 5793, "label": "\nX1359 -> T1245" }, { "from": 5793, "to": 5647, "label": "INSTANCE with matching:\nT1064 -> T1237\nT1065 -> T1238\nT1066 -> T1239\nT1067 -> T1240\nT1068 -> T1241\nT1069 -> T1242\nT1071 -> T1243\nX1172 -> X1358\nX1173 -> X1360\nT1037 -> T1179\nT1038 -> T1180\nT1039 -> T1181\nT1040 -> T1182\nT1041 -> T1183\nT1070 -> T1020\nT1042 -> T1184" }, { "from": 5794, "to": 4447, "label": "INSTANCE with matching:\nT165 -> T1218\nT166 -> T1216\nX286 -> X1312" }, { "from": 5795, "to": 5798, "label": "SPLIT 1" }, { "from": 5795, "to": 5799, "label": "SPLIT 2\nnew knowledge:\nT1219 is ground\nT1251 is ground\nreplacements:X1313 -> T1251" }, { "from": 5798, "to": 4902, "label": "INSTANCE with matching:\nT121 -> T1219\nX185 -> X1313" }, { "from": 5799, "to": 5821, "label": "SPLIT 1" }, { "from": 5799, "to": 5822, "label": "SPLIT 2\nnew knowledge:\nT1251 is ground\nT1248 is ground\nT1254 is ground\nreplacements:X1314 -> T1254" }, { "from": 5821, "to": 4447, "label": "INSTANCE with matching:\nT165 -> T1251\nT166 -> T1248\nX286 -> X1314" }, { "from": 5822, "to": 5827, "label": "CUT" }, { "from": 5827, "to": 5828, "label": "SUCCESS" }, { "from": 5830, "to": 5832, "label": "EVAL with clause\npds1([], X1383, [], X1384) :- !_40.\nand substitutionT1190 -> [],\nT1177 -> T1266,\nT1172 -> T1267,\nX1383 -> .(T1266, T1267),\nT1178 -> [],\nT1165 -> T1268,\nX1384 -> T1268" }, { "from": 5830, "to": 5833, "label": "EVAL-BACKTRACK" }, { "from": 5832, "to": 5834, "label": "CUT" }, { "from": 5833, "to": 5846, "label": "EVAL with clause\npds1(X1396, X1397, .(X1398, X1399), X1400) :- ','(member(X1398, X1396), ','(check(X1397, X1398, X1396, X1401, X1400), pds1(X1401, .(X1398, X1397), X1399, X1400))).\nand substitutionT1190 -> T1281,\nX1396 -> T1281,\nT1177 -> T1282,\nT1172 -> T1283,\nX1397 -> .(T1282, T1283),\nX1398 -> T1287,\nX1399 -> T1288,\nT1178 -> .(T1287, T1288),\nT1165 -> T1286,\nX1400 -> T1286,\nT1284 -> T1287,\nT1285 -> T1288" }, { "from": 5833, "to": 5847, "label": "EVAL-BACKTRACK" }, { "from": 5834, "to": 5835, "label": "SUCCESS" }, { "from": 5846, "to": 5851, "label": "SPLIT 1" }, { "from": 5846, "to": 5852, "label": "SPLIT 2\nnew knowledge:\nT1293 is ground\nT1281 is ground\nreplacements:T1287 -> T1293,\nT1288 -> T1294,\nT1179 -> T1295,\nT1180 -> T1296,\nT1181 -> T1297,\nT1182 -> T1298,\nT1183 -> T1299,\nT1184 -> T1300,\nT1185 -> T1301,\nT1178 -> T1302" }, { "from": 5851, "to": 4407, "label": "INSTANCE with matching:\nT81 -> T1287\nT80 -> T1281" }, { "from": 5852, "to": 5861, "label": "SPLIT 1" }, { "from": 5852, "to": 5862, "label": "SPLIT 2\nnew knowledge:\nT1282 is ground\nT1283 is ground\nT1293 is ground\nT1281 is ground\nT1307 is ground\nT1286 is ground\nreplacements:X1401 -> T1307" }, { "from": 5861, "to": 5863, "label": "CASE" }, { "from": 5862, "to": 5748, "label": "INSTANCE with matching:\nT1190 -> T1307\nT1177 -> T1293\nT1172 -> .(T1282, T1283)\nT1178 -> T1294\nT1165 -> T1286\nT1179 -> T1295\nT1180 -> T1296\nT1181 -> T1297\nT1182 -> T1298\nT1183 -> T1299\nT1184 -> T1300\nT1185 -> T1301" }, { "from": 5863, "to": 5865, "label": "BACKTRACK\nfor clause: check([], X4, L, L, X5) :- !because of non-unification" }, { "from": 5865, "to": 5876, "label": "ONLY EVAL with clause\ncheck(X1439, X1440, X1441, X1442, X1443) :- ','(dif(X1439, X1440, X1443, X1444, X1445), ','(mergedelete(X1444, X1441, X1446), ','(rev(X1445, [], X1447), ','(mergedelete(X1447, X1446, X1442), !_41)))).\nand substitutionT1282 -> T1318,\nT1283 -> T1319,\nX1439 -> .(T1318, T1319),\nT1293 -> T1320,\nX1440 -> T1320,\nT1281 -> T1321,\nX1441 -> T1321,\nX1401 -> X1448,\nX1442 -> X1448,\nT1286 -> T1322,\nX1443 -> T1322" }, { "from": 5876, "to": 5877, "label": "SPLIT 1" }, { "from": 5876, "to": 5878, "label": "SPLIT 2\nnew knowledge:\nT1318 is ground\nT1319 is ground\nT1320 is ground\nT1322 is ground\nT1323 is ground\nT1324 is ground\nreplacements:X1444 -> T1323,\nX1445 -> T1324" }, { "from": 5877, "to": 5879, "label": "CASE" }, { "from": 5878, "to": 6104, "label": "SPLIT 1" }, { "from": 5878, "to": 6105, "label": "SPLIT 2\nnew knowledge:\nT1323 is ground\nT1321 is ground\nT1574 is ground\nreplacements:X1446 -> T1574" }, { "from": 5879, "to": 5880, "label": "BACKTRACK\nfor clause: dif([], X2, X3, [], [])because of non-unification" }, { "from": 5880, "to": 5887, "label": "ONLY EVAL with clause\ndif(.(X1483, X1484), X1485, X1486, .(X1487, X1488), .(X1489, X1490)) :- ','(is(X1487, -(X1485, X1483)), ','(is(X1489, -(X1486, X1487)), dif(X1484, X1485, X1486, X1488, X1490))).\nand substitutionT1318 -> T1333,\nX1483 -> T1333,\nT1319 -> T1334,\nX1484 -> T1334,\nT1320 -> T1335,\nX1485 -> T1335,\nT1322 -> T1336,\nX1486 -> T1336,\nX1487 -> X1491,\nX1488 -> X1492,\nX1444 -> .(X1491, X1492),\nX1489 -> X1493,\nX1490 -> X1494,\nX1445 -> .(X1493, X1494)" }, { "from": 5887, "to": 5888, "label": "IS ERROR" }, { "from": 5887, "to": 5889, "label": "\nX1491 -> T1337" }, { "from": 5889, "to": 5890, "label": "IS ERROR" }, { "from": 5889, "to": 5891, "label": "\nX1493 -> T1338" }, { "from": 5891, "to": 5892, "label": "CASE" }, { "from": 5892, "to": 5898, "label": "PARALLEL" }, { "from": 5892, "to": 5899, "label": "PARALLEL" }, { "from": 5898, "to": 5900, "label": "EVAL with clause\ndif([], X1507, X1508, [], []).\nand substitutionT1334 -> [],\nT1335 -> T1351,\nX1507 -> T1351,\nT1336 -> T1352,\nX1508 -> T1352,\nX1492 -> [],\nX1494 -> []" }, { "from": 5898, "to": 5901, "label": "EVAL-BACKTRACK" }, { "from": 5899, "to": 5909, "label": "EVAL with clause\ndif(.(X1533, X1534), X1535, X1536, .(X1537, X1538), .(X1539, X1540)) :- ','(is(X1537, -(X1535, X1533)), ','(is(X1539, -(X1536, X1537)), dif(X1534, X1535, X1536, X1538, X1540))).\nand substitutionX1533 -> T1361,\nX1534 -> T1362,\nT1334 -> .(T1361, T1362),\nT1335 -> T1363,\nX1535 -> T1363,\nT1336 -> T1364,\nX1536 -> T1364,\nX1537 -> X1541,\nX1538 -> X1542,\nX1492 -> .(X1541, X1542),\nX1539 -> X1543,\nX1540 -> X1544,\nX1494 -> .(X1543, X1544)" }, { "from": 5899, "to": 5910, "label": "EVAL-BACKTRACK" }, { "from": 5900, "to": 5902, "label": "SUCCESS" }, { "from": 5909, "to": 5911, "label": "IS ERROR" }, { "from": 5909, "to": 5912, "label": "\nX1541 -> T1365" }, { "from": 5912, "to": 5913, "label": "\nX1543 -> T1366" }, { "from": 5913, "to": 5914, "label": "CASE" }, { "from": 5914, "to": 5920, "label": "PARALLEL" }, { "from": 5914, "to": 5921, "label": "PARALLEL" }, { "from": 5920, "to": 5922, "label": "EVAL with clause\ndif([], X1557, X1558, [], []).\nand substitutionT1362 -> [],\nT1363 -> T1379,\nX1557 -> T1379,\nT1364 -> T1380,\nX1558 -> T1380,\nX1542 -> [],\nX1544 -> []" }, { "from": 5920, "to": 5923, "label": "EVAL-BACKTRACK" }, { "from": 5921, "to": 5931, "label": "EVAL with clause\ndif(.(X1583, X1584), X1585, X1586, .(X1587, X1588), .(X1589, X1590)) :- ','(is(X1587, -(X1585, X1583)), ','(is(X1589, -(X1586, X1587)), dif(X1584, X1585, X1586, X1588, X1590))).\nand substitutionX1583 -> T1389,\nX1584 -> T1390,\nT1362 -> .(T1389, T1390),\nT1363 -> T1391,\nX1585 -> T1391,\nT1364 -> T1392,\nX1586 -> T1392,\nX1587 -> X1591,\nX1588 -> X1592,\nX1542 -> .(X1591, X1592),\nX1589 -> X1593,\nX1590 -> X1594,\nX1544 -> .(X1593, X1594)" }, { "from": 5921, "to": 5932, "label": "EVAL-BACKTRACK" }, { "from": 5922, "to": 5924, "label": "SUCCESS" }, { "from": 5931, "to": 5933, "label": "IS ERROR" }, { "from": 5931, "to": 5934, "label": "\nX1591 -> T1393" }, { "from": 5934, "to": 5935, "label": "\nX1593 -> T1394" }, { "from": 5935, "to": 5939, "label": "CASE" }, { "from": 5939, "to": 5944, "label": "PARALLEL" }, { "from": 5939, "to": 5945, "label": "PARALLEL" }, { "from": 5944, "to": 5946, "label": "EVAL with clause\ndif([], X1607, X1608, [], []).\nand substitutionT1390 -> [],\nT1391 -> T1407,\nX1607 -> T1407,\nT1392 -> T1408,\nX1608 -> T1408,\nX1592 -> [],\nX1594 -> []" }, { "from": 5944, "to": 5947, "label": "EVAL-BACKTRACK" }, { "from": 5945, "to": 5966, "label": "EVAL with clause\ndif(.(X1633, X1634), X1635, X1636, .(X1637, X1638), .(X1639, X1640)) :- ','(is(X1637, -(X1635, X1633)), ','(is(X1639, -(X1636, X1637)), dif(X1634, X1635, X1636, X1638, X1640))).\nand substitutionX1633 -> T1417,\nX1634 -> T1418,\nT1390 -> .(T1417, T1418),\nT1391 -> T1419,\nX1635 -> T1419,\nT1392 -> T1420,\nX1636 -> T1420,\nX1637 -> X1641,\nX1638 -> X1642,\nX1592 -> .(X1641, X1642),\nX1639 -> X1643,\nX1640 -> X1644,\nX1594 -> .(X1643, X1644)" }, { "from": 5945, "to": 5967, "label": "EVAL-BACKTRACK" }, { "from": 5946, "to": 5948, "label": "SUCCESS" }, { "from": 5966, "to": 5968, "label": "IS ERROR" }, { "from": 5966, "to": 5969, "label": "\nX1641 -> T1421" }, { "from": 5969, "to": 5970, "label": "\nX1643 -> T1422" }, { "from": 5970, "to": 5977, "label": "CASE" }, { "from": 5977, "to": 5980, "label": "PARALLEL" }, { "from": 5977, "to": 5981, "label": "PARALLEL" }, { "from": 5980, "to": 5982, "label": "EVAL with clause\ndif([], X1657, X1658, [], []).\nand substitutionT1418 -> [],\nT1419 -> T1435,\nX1657 -> T1435,\nT1420 -> T1436,\nX1658 -> T1436,\nX1642 -> [],\nX1644 -> []" }, { "from": 5980, "to": 5983, "label": "EVAL-BACKTRACK" }, { "from": 5981, "to": 5990, "label": "EVAL with clause\ndif(.(X1683, X1684), X1685, X1686, .(X1687, X1688), .(X1689, X1690)) :- ','(is(X1687, -(X1685, X1683)), ','(is(X1689, -(X1686, X1687)), dif(X1684, X1685, X1686, X1688, X1690))).\nand substitutionX1683 -> T1445,\nX1684 -> T1446,\nT1418 -> .(T1445, T1446),\nT1419 -> T1447,\nX1685 -> T1447,\nT1420 -> T1448,\nX1686 -> T1448,\nX1687 -> X1691,\nX1688 -> X1692,\nX1642 -> .(X1691, X1692),\nX1689 -> X1693,\nX1690 -> X1694,\nX1644 -> .(X1693, X1694)" }, { "from": 5981, "to": 5991, "label": "EVAL-BACKTRACK" }, { "from": 5982, "to": 5984, "label": "SUCCESS" }, { "from": 5990, "to": 5992, "label": "IS ERROR" }, { "from": 5990, "to": 5993, "label": "\nX1691 -> T1449" }, { "from": 5993, "to": 5994, "label": "\nX1693 -> T1450" }, { "from": 5994, "to": 6001, "label": "CASE" }, { "from": 6001, "to": 6002, "label": "PARALLEL" }, { "from": 6001, "to": 6005, "label": "PARALLEL" }, { "from": 6002, "to": 6008, "label": "EVAL with clause\ndif([], X1707, X1708, [], []).\nand substitutionT1446 -> [],\nT1447 -> T1463,\nX1707 -> T1463,\nT1448 -> T1464,\nX1708 -> T1464,\nX1692 -> [],\nX1694 -> []" }, { "from": 6002, "to": 6009, "label": "EVAL-BACKTRACK" }, { "from": 6005, "to": 6016, "label": "EVAL with clause\ndif(.(X1733, X1734), X1735, X1736, .(X1737, X1738), .(X1739, X1740)) :- ','(is(X1737, -(X1735, X1733)), ','(is(X1739, -(X1736, X1737)), dif(X1734, X1735, X1736, X1738, X1740))).\nand substitutionX1733 -> T1473,\nX1734 -> T1474,\nT1446 -> .(T1473, T1474),\nT1447 -> T1475,\nX1735 -> T1475,\nT1448 -> T1476,\nX1736 -> T1476,\nX1737 -> X1741,\nX1738 -> X1742,\nX1692 -> .(X1741, X1742),\nX1739 -> X1743,\nX1740 -> X1744,\nX1694 -> .(X1743, X1744)" }, { "from": 6005, "to": 6017, "label": "EVAL-BACKTRACK" }, { "from": 6008, "to": 6010, "label": "SUCCESS" }, { "from": 6016, "to": 6018, "label": "IS ERROR" }, { "from": 6016, "to": 6019, "label": "\nX1741 -> T1477" }, { "from": 6019, "to": 6020, "label": "\nX1743 -> T1478" }, { "from": 6020, "to": 6021, "label": "CASE" }, { "from": 6021, "to": 6022, "label": "PARALLEL" }, { "from": 6021, "to": 6023, "label": "PARALLEL" }, { "from": 6022, "to": 6031, "label": "EVAL with clause\ndif([], X1757, X1758, [], []).\nand substitutionT1474 -> [],\nT1475 -> T1491,\nX1757 -> T1491,\nT1476 -> T1492,\nX1758 -> T1492,\nX1742 -> [],\nX1744 -> []" }, { "from": 6022, "to": 6032, "label": "EVAL-BACKTRACK" }, { "from": 6023, "to": 6040, "label": "EVAL with clause\ndif(.(X1783, X1784), X1785, X1786, .(X1787, X1788), .(X1789, X1790)) :- ','(is(X1787, -(X1785, X1783)), ','(is(X1789, -(X1786, X1787)), dif(X1784, X1785, X1786, X1788, X1790))).\nand substitutionX1783 -> T1501,\nX1784 -> T1502,\nT1474 -> .(T1501, T1502),\nT1475 -> T1503,\nX1785 -> T1503,\nT1476 -> T1504,\nX1786 -> T1504,\nX1787 -> X1791,\nX1788 -> X1792,\nX1742 -> .(X1791, X1792),\nX1789 -> X1793,\nX1790 -> X1794,\nX1744 -> .(X1793, X1794)" }, { "from": 6023, "to": 6041, "label": "EVAL-BACKTRACK" }, { "from": 6031, "to": 6033, "label": "SUCCESS" }, { "from": 6040, "to": 6042, "label": "IS ERROR" }, { "from": 6040, "to": 6043, "label": "\nX1791 -> T1505" }, { "from": 6043, "to": 6044, "label": "\nX1793 -> T1506" }, { "from": 6044, "to": 6080, "label": "CASE" }, { "from": 6080, "to": 6081, "label": "PARALLEL" }, { "from": 6080, "to": 6082, "label": "PARALLEL" }, { "from": 6081, "to": 6083, "label": "EVAL with clause\ndif([], X1807, X1808, [], []).\nand substitutionT1502 -> [],\nT1503 -> T1519,\nX1807 -> T1519,\nT1504 -> T1520,\nX1808 -> T1520,\nX1792 -> [],\nX1794 -> []" }, { "from": 6081, "to": 6084, "label": "EVAL-BACKTRACK" }, { "from": 6082, "to": 6086, "label": "EVAL with clause\ndif(.(X1833, X1834), X1835, X1836, .(X1837, X1838), .(X1839, X1840)) :- ','(is(X1837, -(X1835, X1833)), ','(is(X1839, -(X1836, X1837)), dif(X1834, X1835, X1836, X1838, X1840))).\nand substitutionX1833 -> T1529,\nX1834 -> T1530,\nT1502 -> .(T1529, T1530),\nT1503 -> T1531,\nX1835 -> T1531,\nT1504 -> T1532,\nX1836 -> T1532,\nX1837 -> X1841,\nX1838 -> X1842,\nX1792 -> .(X1841, X1842),\nX1839 -> X1843,\nX1840 -> X1844,\nX1794 -> .(X1843, X1844)" }, { "from": 6082, "to": 6087, "label": "EVAL-BACKTRACK" }, { "from": 6083, "to": 6085, "label": "SUCCESS" }, { "from": 6086, "to": 6088, "label": "IS ERROR" }, { "from": 6086, "to": 6089, "label": "\nX1841 -> T1533" }, { "from": 6089, "to": 6090, "label": "\nX1843 -> T1534" }, { "from": 6090, "to": 6091, "label": "GENERALIZATION\nT1539 <-- .(T1361, .(T1389, .(T1417, .(T1445, .(T1473, .(T1501, .(T1529, T1530)))))))\n\nNew Knowledge:\nT1539 is ground" }, { "from": 6091, "to": 6092, "label": "CASE" }, { "from": 6092, "to": 6093, "label": "PARALLEL" }, { "from": 6092, "to": 6094, "label": "PARALLEL" }, { "from": 6093, "to": 6095, "label": "EVAL with clause\ndif([], X1861, X1862, [], []).\nand substitutionT1530 -> [],\nT1531 -> T1552,\nX1861 -> T1552,\nT1532 -> T1553,\nX1862 -> T1553,\nX1842 -> [],\nX1844 -> []" }, { "from": 6093, "to": 6096, "label": "EVAL-BACKTRACK" }, { "from": 6094, "to": 6098, "label": "EVAL with clause\ndif(.(X1887, X1888), X1889, X1890, .(X1891, X1892), .(X1893, X1894)) :- ','(is(X1891, -(X1889, X1887)), ','(is(X1893, -(X1890, X1891)), dif(X1888, X1889, X1890, X1892, X1894))).\nand substitutionX1887 -> T1562,\nX1888 -> T1563,\nT1530 -> .(T1562, T1563),\nT1531 -> T1564,\nX1889 -> T1564,\nT1532 -> T1565,\nX1890 -> T1565,\nX1891 -> X1895,\nX1892 -> X1896,\nX1842 -> .(X1895, X1896),\nX1893 -> X1897,\nX1894 -> X1898,\nX1844 -> .(X1897, X1898)" }, { "from": 6094, "to": 6099, "label": "EVAL-BACKTRACK" }, { "from": 6095, "to": 6097, "label": "SUCCESS" }, { "from": 6098, "to": 6100, "label": "IS ERROR" }, { "from": 6098, "to": 6101, "label": "\nX1895 -> T1566" }, { "from": 6101, "to": 6102, "label": "IS ERROR" }, { "from": 6101, "to": 6103, "label": "\nX1897 -> T1567" }, { "from": 6103, "to": 6091, "label": "INSTANCE with matching:\nT1530 -> T1563\nT1531 -> T1564\nT1532 -> T1565\nX1842 -> X1896\nX1844 -> X1898" }, { "from": 6104, "to": 4447, "label": "INSTANCE with matching:\nT165 -> T1323\nT166 -> T1321\nX286 -> X1446" }, { "from": 6105, "to": 6106, "label": "SPLIT 1" }, { "from": 6105, "to": 6107, "label": "SPLIT 2\nnew knowledge:\nT1324 is ground\nT1577 is ground\nreplacements:X1447 -> T1577" }, { "from": 6106, "to": 4902, "label": "INSTANCE with matching:\nT121 -> T1324\nX185 -> X1447" }, { "from": 6107, "to": 6108, "label": "SPLIT 1" }, { "from": 6107, "to": 6109, "label": "SPLIT 2\nnew knowledge:\nT1577 is ground\nT1574 is ground\nT1580 is ground\nreplacements:X1448 -> T1580" }, { "from": 6108, "to": 4447, "label": "INSTANCE with matching:\nT165 -> T1577\nT166 -> T1574\nX286 -> X1448" }, { "from": 6109, "to": 6110, "label": "CUT" }, { "from": 6110, "to": 6111, "label": "SUCCESS" } ], "type": "Graph" } } ---------------------------------------- (404) Complex Obligation (AND) ---------------------------------------- (405) Obligation: Rules: f6098_in(T1564, T1562, T1565, T1563, T1156, T1159, T696, T45, T898, T1162, T616, T1160, T790, T1020, T1157, T1161, T1333, T1158, T1321, T1539) -> f6101_in(T1565, T1566, T1563, T1564, T1156, T616, T1160, T1159, T696, T1539, T790, T1020, T1157, T1562, T1161, T1333, T45, T1158, T898, T1162, T1321) :|: T1566 = T1564 - T1562 f6101_out(x, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20) -> f6098_out(x3, x13, x, x2, x4, x7, x8, x16, x18, x19, x5, x6, x10, x11, x12, x14, x15, x17, x20, x9) :|: TRUE f6100_out -> f6098_out(x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40) :|: TRUE f6098_in(x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60) -> f6100_in :|: TRUE f6103_in(x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82) -> f6091_in(x61, x62, x63, x64, x67, x68, x77, x80, x81, x65, x66, x71, x72, x73, x75, x76, x79, x82, x69) :|: TRUE f6091_out(x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101) -> f6103_out(x83, x84, x85, x86, x92, x93, x87, x88, x101, x102, x94, x95, x96, x103, x97, x98, x89, x104, x99, x90, x91, x100) :|: TRUE f6092_in(x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123) -> f6094_in(x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123) :|: TRUE f6094_out(x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142) -> f6092_out(x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142) :|: TRUE f6092_in(x143, x144, x145, x146, x147, x148, x149, x150, x151, x152, x153, x154, x155, x156, x157, x158, x159, x160, x161) -> f6093_in(x143, x144, x145, x146, x147, x148, x149, x150, x151, x152, x153, x154, x155, x156, x157, x158, x159, x160, x161) :|: TRUE f6093_out(x162, x163, x164, x165, x166, x167, x168, x169, x170, x171, x172, x173, x174, x175, x176, x177, x178, x179, x180) -> f6092_out(x162, x163, x164, x165, x166, x167, x168, x169, x170, x171, x172, x173, x174, x175, x176, x177, x178, x179, x180) :|: TRUE f6092_out(x181, x182, x183, x184, x185, x186, x187, x188, x189, x190, x191, x192, x193, x194, x195, x196, x197, x198, x199) -> f6091_out(x181, x182, x183, x184, x185, x186, x187, x188, x189, x190, x191, x192, x193, x194, x195, x196, x197, x198, x199) :|: TRUE f6091_in(x200, x201, x202, x203, x204, x205, x206, x207, x208, x209, x210, x211, x212, x213, x214, x215, x216, x217, x218) -> f6092_in(x200, x201, x202, x203, x204, x205, x206, x207, x208, x209, x210, x211, x212, x213, x214, x215, x216, x217, x218) :|: TRUE f6102_out -> f6101_out(x219, x220, x221, x222, x223, x224, x225, x226, x227, x228, x229, x230, x231, x232, x233, x234, x235, x236, x237, x238, x239) :|: TRUE f6101_in(x240, x241, x242, x243, x244, x245, x246, x247, x248, x249, x250, x251, x252, x253, x254, x255, x256, x257, x258, x259, x260) -> f6102_in :|: TRUE f6103_out(x261, x262, x263, x264, x265, x266, x267, x268, x269, x270, x271, x272, x273, x274, x275, x276, x277, x278, x279, x280, x281, x282) -> f6101_out(x263, x278, x261, x262, x264, x265, x266, x267, x268, x269, x271, x272, x273, x274, x275, x276, x277, x279, x280, x281, x282) :|: TRUE f6101_in(x283, x284, x285, x286, x287, x288, x289, x290, x291, x292, x293, x294, x295, x296, x297, x298, x299, x300, x301, x302, x303) -> f6103_in(x285, x286, x283, x287, x288, x289, x290, x291, x292, x304, x293, x294, x295, x296, x297, x298, x299, x284, x300, x301, x302, x303) :|: x304 = x283 - x284 f6094_in(.(x305, x306), x307, x308, x309, x310, x311, x312, x313, x314, x315, x316, x317, x318, x319, x320, x321, x322, x323, x324) -> f6098_in(x307, x305, x308, x306, x309, x310, x311, x312, x313, x314, x315, x316, x317, x318, x319, x320, x321, x322, x323, x324) :|: TRUE f6094_in(x325, x326, x327, x328, x329, x330, x331, x332, x333, x334, x335, x336, x337, x338, x339, x340, x341, x342, x343) -> f6099_in :|: TRUE f6099_out -> f6094_out(x344, x345, x346, x347, x348, x349, x350, x351, x352, x353, x354, x355, x356, x357, x358, x359, x360, x361, x362) :|: TRUE f6098_out(x363, x364, x365, x366, x367, x368, x369, x370, x371, x372, x373, x374, x375, x376, x377, x378, x379, x380, x381, x382) -> f6094_out(.(x364, x366), x363, x365, x367, x368, x369, x370, x371, x372, x373, x374, x375, x376, x377, x378, x379, x380, x381, x382) :|: TRUE f4_out(T1) -> f1_out(T1) :|: TRUE f1_in(x383) -> f4_in(x383) :|: TRUE f4_in(T7) -> f10_in(T7) :|: TRUE f11_out -> f4_out(x384) :|: TRUE f10_out(x385) -> f4_out(x385) :|: TRUE f4_in(x386) -> f11_in :|: TRUE f10_in(x387) -> f16_in(x388, x387) :|: x388 = x387 * (x387 + 1) + 1 f10_in(x389) -> f12_in :|: TRUE f16_out(x390, x391) -> f10_out(x391) :|: TRUE f12_out -> f10_out(x392) :|: TRUE f16_in(x393, x394) -> f21_in(x393) :|: TRUE f24_out(x395, x396) -> f16_out(x396, x397) :|: TRUE f21_out(T10) -> f24_in(T12, T10) :|: TRUE f4324_out(x398, x399) -> f24_out(x398, x399) :|: TRUE f24_in(x400, x401) -> f4324_in(x400, x401) :|: TRUE f4324_in(x402, x403) -> f4326_in(x402, x403) :|: TRUE f4326_out(x404, x405) -> f4324_out(x404, x405) :|: TRUE f4325_out(T36) -> f4324_out([], T36) :|: TRUE f4324_in([], x406) -> f4325_in(x406) :|: TRUE f4326_in(x407, x408) -> f4397_in(x407, x408) :|: TRUE f4326_in(x409, x410) -> f4398_in :|: TRUE f4397_out(x411, x412) -> f4326_out(x411, x412) :|: TRUE f4398_out -> f4326_out(x413, x414) :|: TRUE f4400_out(x415, x416, x417) -> f4397_out(x416, x417) :|: TRUE f4397_in(x418, x419) -> f4399_in(x418, x419) :|: TRUE f4399_out(x420, x421) -> f4400_in(x422, x420, x421) :|: TRUE f4418_out(x423, x424, x425, x426) -> f4400_out(x424, x426, x425) :|: TRUE f4400_in(x427, x428, x429) -> f4417_in(x427, x428, x429) :|: TRUE f4417_out(x430, x431, x432) -> f4418_in(x433, x430, x432, x431) :|: TRUE f5096_out(x434, x435, x436, x437) -> f4418_out(x434, x435, x436, x437) :|: TRUE f4418_in(x438, x439, x440, x441) -> f5096_in(x438, x439, x440, x441) :|: TRUE f5107_out(x442, x443, x444, x445) -> f5096_out(x442, x443, x444, x445) :|: TRUE f5106_out(x446, x447, x448) -> f5096_out([], x446, x447, x448) :|: TRUE f5096_in([], x449, x450, x451) -> f5106_in(x449, x450, x451) :|: TRUE f5096_in(x452, x453, x454, x455) -> f5107_in(x452, x453, x454, x455) :|: TRUE f5121_out(x456, x457, x458, x459) -> f5107_out(x456, x457, x458, x459) :|: TRUE f5122_out -> f5107_out(x460, x461, x462, x463) :|: TRUE f5107_in(x464, x465, x466, x467) -> f5121_in(x464, x465, x466, x467) :|: TRUE f5107_in(x468, x469, x470, x471) -> f5122_in :|: TRUE f5123_out(x472, x473, x474) -> f5124_in(x473, x475, x472, x474, x476) :|: TRUE f5121_in(x477, x478, x479, x480) -> f5123_in(x477, x478, x479) :|: TRUE f5124_out(x481, x482, x483, x484, x485) -> f5121_out(x483, x481, x484, x485) :|: TRUE f5124_in(x486, x487, x488, x489, x490) -> f5132_in(x486, x487, x488, x489, x490) :|: TRUE f5133_out(x491, x492, x493, x494, x495, x496) -> f5124_out(x493, x492, x496, x494, x495) :|: TRUE f5132_out(x497, x498, x499, x500, x501) -> f5133_in(x502, x498, x497, x500, x501, x499) :|: TRUE f5133_in(x503, x504, x505, x506, x507, x508) -> f5167_in(x503, x504, x505, x506, x507, x508) :|: TRUE f5167_out(x509, x510, x511, x512, x513, x514) -> f5133_out(x509, x510, x511, x512, x513, x514) :|: TRUE f5171_out(x515, x516, x517, x518, x519) -> f5167_out([], x515, x516, x517, x518, x519) :|: TRUE f5167_in([], x520, x521, x522, x523, x524) -> f5171_in(x520, x521, x522, x523, x524) :|: TRUE f5167_in(x525, x526, x527, x528, x529, x530) -> f5172_in(x525, x526, x527, x528, x529, x530) :|: TRUE f5172_out(x531, x532, x533, x534, x535, x536) -> f5167_out(x531, x532, x533, x534, x535, x536) :|: TRUE f5182_out(x537, x538, x539, x540, x541, x542) -> f5172_out(x537, x538, x539, x540, x541, x542) :|: TRUE f5172_in(x543, x544, x545, x546, x547, x548) -> f5183_in :|: TRUE f5183_out -> f5172_out(x549, x550, x551, x552, x553, x554) :|: TRUE f5172_in(x555, x556, x557, x558, x559, x560) -> f5182_in(x555, x556, x557, x558, x559, x560) :|: TRUE f5189_out(x561, x562, x563, x564) -> f5190_in(x562, x563, x565, x561, x564, x566, x567) :|: TRUE f5190_out(x568, x569, x570, x571, x572, x573, x574) -> f5182_out(x571, x568, x569, x572, x573, x574) :|: TRUE f5182_in(x575, x576, x577, x578, x579, x580) -> f5189_in(x575, x576, x577, x578) :|: TRUE f5195_out(x581, x582, x583, x584, x585, x586, x587, x588) -> f5190_out(x583, x584, x582, x588, x585, x586, x587) :|: TRUE f5190_in(x589, x590, x591, x592, x593, x594, x595) -> f5194_in(x589, x590, x591, x592, x593, x594, x595) :|: TRUE f5194_out(x596, x597, x598, x599, x600, x601, x602) -> f5195_in(x603, x598, x596, x597, x600, x601, x602, x599) :|: TRUE f5195_in(x604, x605, x606, x607, x608, x609, x610, x611) -> f5288_in(x604, x605, x606, x607, x608, x609, x610, x611) :|: TRUE f5288_out(x612, x613, x614, x615, x616, x617, x618, x619) -> f5195_out(x612, x613, x614, x615, x616, x617, x618, x619) :|: TRUE f5288_in([], x620, x621, x622, x623, x624, x625, x626) -> f5294_in(x620, x621, x622, x623, x624, x625, x626) :|: TRUE f5294_out(x627, x628, x629, x630, x631, x632, x633) -> f5288_out([], x627, x628, x629, x630, x631, x632, x633) :|: TRUE f5298_out(x634, x635, x636, x637, x638, x639, x640, x641) -> f5288_out(x634, x635, x636, x637, x638, x639, x640, x641) :|: TRUE f5288_in(x642, x643, x644, x645, x646, x647, x648, x649) -> f5298_in(x642, x643, x644, x645, x646, x647, x648, x649) :|: TRUE f5298_in(x650, x651, x652, x653, x654, x655, x656, x657) -> f5314_in(x650, x651, x652, x653, x654, x655, x656, x657) :|: TRUE f5298_in(x658, x659, x660, x661, x662, x663, x664, x665) -> f5315_in :|: TRUE f5315_out -> f5298_out(x666, x667, x668, x669, x670, x671, x672, x673) :|: TRUE f5314_out(x674, x675, x676, x677, x678, x679, x680, x681) -> f5298_out(x674, x675, x676, x677, x678, x679, x680, x681) :|: TRUE f5336_out(x682, x683, x684, x685, x686, x687, x688, x689, x690) -> f5314_out(x686, x682, x683, x684, x687, x688, x689, x690) :|: TRUE f5314_in(x691, x692, x693, x694, x695, x696, x697, x698) -> f5333_in(x691, x692, x693, x694, x695) :|: TRUE f5333_out(x699, x700, x701, x702, x703) -> f5336_in(x700, x701, x702, x704, x699, x703, x705, x706, x707) :|: TRUE f5336_in(x708, x709, x710, x711, x712, x713, x714, x715, x716) -> f5340_in(x708, x709, x710, x711, x712, x713, x714, x715, x716) :|: TRUE f5341_out(x717, x718, x719, x720, x721, x722, x723, x724, x725, x726) -> f5336_out(x719, x720, x721, x718, x726, x722, x723, x724, x725) :|: TRUE f5340_out(x727, x728, x729, x730, x731, x732, x733, x734, x735) -> f5341_in(x736, x730, x727, x728, x729, x732, x733, x734, x735, x731) :|: TRUE f5408_out(x737, x738, x739, x740, x741, x742, x743, x744, x745, x746) -> f5341_out(x737, x738, x739, x740, x741, x742, x743, x744, x745, x746) :|: TRUE f5341_in(x747, x748, x749, x750, x751, x752, x753, x754, x755, x756) -> f5408_in(x747, x748, x749, x750, x751, x752, x753, x754, x755, x756) :|: TRUE f5419_out(x757, x758, x759, x760, x761, x762, x763, x764, x765, x766) -> f5408_out(x757, x758, x759, x760, x761, x762, x763, x764, x765, x766) :|: TRUE f5418_out(x767, x768, x769, x770, x771, x772, x773, x774, x775) -> f5408_out([], x767, x768, x769, x770, x771, x772, x773, x774, x775) :|: TRUE f5408_in([], x776, x777, x778, x779, x780, x781, x782, x783, x784) -> f5418_in(x776, x777, x778, x779, x780, x781, x782, x783, x784) :|: TRUE f5408_in(x785, x786, x787, x788, x789, x790, x791, x792, x793, x794) -> f5419_in(x785, x786, x787, x788, x789, x790, x791, x792, x793, x794) :|: TRUE f5422_out(x795, x796, x797, x798, x799, x800, x801, x802, x803, x804) -> f5419_out(x795, x796, x797, x798, x799, x800, x801, x802, x803, x804) :|: TRUE f5419_in(x805, x806, x807, x808, x809, x810, x811, x812, x813, x814) -> f5422_in(x805, x806, x807, x808, x809, x810, x811, x812, x813, x814) :|: TRUE f5419_in(x815, x816, x817, x818, x819, x820, x821, x822, x823, x824) -> f5423_in :|: TRUE f5423_out -> f5419_out(x825, x826, x827, x828, x829, x830, x831, x832, x833, x834) :|: TRUE f5422_in(x835, x836, x837, x838, x839, x840, x841, x842, x843, x844) -> f5424_in(x835, x836, x837, x838, x839, x840) :|: TRUE f5424_out(x845, x846, x847, x848, x849, x850) -> f5425_in(x846, x847, x848, x849, x851, x845, x850, x852, x853, x854, x855) :|: TRUE f5425_out(x856, x857, x858, x859, x860, x861, x862, x863, x864, x865, x866) -> f5422_out(x861, x856, x857, x858, x859, x862, x863, x864, x865, x866) :|: TRUE f5426_out(x867, x868, x869, x870, x871, x872, x873, x874, x875, x876, x877) -> f5427_in(x878, x871, x867, x868, x869, x870, x873, x874, x875, x876, x877, x872) :|: TRUE f5427_out(x879, x880, x881, x882, x883, x884, x885, x886, x887, x888, x889, x890) -> f5425_out(x881, x882, x883, x884, x880, x890, x885, x886, x887, x888, x889) :|: TRUE f5425_in(x891, x892, x893, x894, x895, x896, x897, x898, x899, x900, x901) -> f5426_in(x891, x892, x893, x894, x895, x896, x897, x898, x899, x900, x901) :|: TRUE f5427_in(x902, x903, x904, x905, x906, x907, x908, x909, x910, x911, x912, x913) -> f5539_in(x902, x903, x904, x905, x906, x907, x908, x909, x910, x911, x912, x913) :|: TRUE f5539_out(x914, x915, x916, x917, x918, x919, x920, x921, x922, x923, x924, x925) -> f5427_out(x914, x915, x916, x917, x918, x919, x920, x921, x922, x923, x924, x925) :|: TRUE f5563_out(x926, x927, x928, x929, x930, x931, x932, x933, x934, x935, x936) -> f5539_out([], x926, x927, x928, x929, x930, x931, x932, x933, x934, x935, x936) :|: TRUE f5539_in([], x937, x938, x939, x940, x941, x942, x943, x944, x945, x946, x947) -> f5563_in(x937, x938, x939, x940, x941, x942, x943, x944, x945, x946, x947) :|: TRUE f5539_in(x948, x949, x950, x951, x952, x953, x954, x955, x956, x957, x958, x959) -> f5564_in(x948, x949, x950, x951, x952, x953, x954, x955, x956, x957, x958, x959) :|: TRUE f5564_out(x960, x961, x962, x963, x964, x965, x966, x967, x968, x969, x970, x971) -> f5539_out(x960, x961, x962, x963, x964, x965, x966, x967, x968, x969, x970, x971) :|: TRUE f5586_out -> f5564_out(x972, x973, x974, x975, x976, x977, x978, x979, x980, x981, x982, x983) :|: TRUE f5564_in(x984, x985, x986, x987, x988, x989, x990, x991, x992, x993, x994, x995) -> f5581_in(x984, x985, x986, x987, x988, x989, x990, x991, x992, x993, x994, x995) :|: TRUE f5581_out(x996, x997, x998, x999, x1000, x1001, x1002, x1003, x1004, x1005, x1006, x1007) -> f5564_out(x996, x997, x998, x999, x1000, x1001, x1002, x1003, x1004, x1005, x1006, x1007) :|: TRUE f5564_in(x1008, x1009, x1010, x1011, x1012, x1013, x1014, x1015, x1016, x1017, x1018, x1019) -> f5586_in :|: TRUE f5591_out(x1020, x1021, x1022, x1023, x1024, x1025, x1026) -> f5592_in(x1021, x1022, x1023, x1024, x1025, x1027, x1020, x1026, x1028, x1029, x1030, x1031, x1032) :|: TRUE f5592_out(x1033, x1034, x1035, x1036, x1037, x1038, x1039, x1040, x1041, x1042, x1043, x1044, x1045) -> f5581_out(x1039, x1033, x1034, x1035, x1036, x1037, x1040, x1041, x1042, x1043, x1044, x1045) :|: TRUE f5581_in(x1046, x1047, x1048, x1049, x1050, x1051, x1052, x1053, x1054, x1055, x1056, x1057) -> f5591_in(x1046, x1047, x1048, x1049, x1050, x1051, x1052) :|: TRUE f5606_out(x1058, x1059, x1060, x1061, x1062, x1063, x1064, x1065, x1066, x1067, x1068, x1069, x1070) -> f5607_in(x1071, x1063, x1058, x1059, x1060, x1061, x1062, x1065, x1066, x1067, x1068, x1069, x1070, x1064) :|: TRUE f5607_out(x1072, x1073, x1074, x1075, x1076, x1077, x1078, x1079, x1080, x1081, x1082, x1083, x1084, x1085) -> f5592_out(x1074, x1075, x1076, x1077, x1078, x1073, x1085, x1079, x1080, x1081, x1082, x1083, x1084) :|: TRUE f5592_in(x1086, x1087, x1088, x1089, x1090, x1091, x1092, x1093, x1094, x1095, x1096, x1097, x1098) -> f5606_in(x1086, x1087, x1088, x1089, x1090, x1091, x1092, x1093, x1094, x1095, x1096, x1097, x1098) :|: TRUE f5691_out(x1099, x1100, x1101, x1102, x1103, x1104, x1105, x1106, x1107, x1108, x1109, x1110, x1111, x1112) -> f5607_out(x1099, x1100, x1101, x1102, x1103, x1104, x1105, x1106, x1107, x1108, x1109, x1110, x1111, x1112) :|: TRUE f5607_in(x1113, x1114, x1115, x1116, x1117, x1118, x1119, x1120, x1121, x1122, x1123, x1124, x1125, x1126) -> f5691_in(x1113, x1114, x1115, x1116, x1117, x1118, x1119, x1120, x1121, x1122, x1123, x1124, x1125, x1126) :|: TRUE f5716_out(x1127, x1128, x1129, x1130, x1131, x1132, x1133, x1134, x1135, x1136, x1137, x1138, x1139) -> f5691_out([], x1127, x1128, x1129, x1130, x1131, x1132, x1133, x1134, x1135, x1136, x1137, x1138, x1139) :|: TRUE f5719_out(x1140, x1141, x1142, x1143, x1144, x1145, x1146, x1147, x1148, x1149, x1150, x1151, x1152, x1153) -> f5691_out(x1140, x1141, x1142, x1143, x1144, x1145, x1146, x1147, x1148, x1149, x1150, x1151, x1152, x1153) :|: TRUE f5691_in([], x1154, x1155, x1156, x1157, x1158, x1159, x1160, x1161, x1162, x1163, x1164, x1165, x1166) -> f5716_in(x1154, x1155, x1156, x1157, x1158, x1159, x1160, x1161, x1162, x1163, x1164, x1165, x1166) :|: TRUE f5691_in(x1167, x1168, x1169, x1170, x1171, x1172, x1173, x1174, x1175, x1176, x1177, x1178, x1179, x1180) -> f5719_in(x1167, x1168, x1169, x1170, x1171, x1172, x1173, x1174, x1175, x1176, x1177, x1178, x1179, x1180) :|: TRUE f5719_in(x1181, x1182, x1183, x1184, x1185, x1186, x1187, x1188, x1189, x1190, x1191, x1192, x1193, x1194) -> f5733_in(x1181, x1182, x1183, x1184, x1185, x1186, x1187, x1188, x1189, x1190, x1191, x1192, x1193, x1194) :|: TRUE f5733_out(x1195, x1196, x1197, x1198, x1199, x1200, x1201, x1202, x1203, x1204, x1205, x1206, x1207, x1208) -> f5719_out(x1195, x1196, x1197, x1198, x1199, x1200, x1201, x1202, x1203, x1204, x1205, x1206, x1207, x1208) :|: TRUE f5734_out -> f5719_out(x1209, x1210, x1211, x1212, x1213, x1214, x1215, x1216, x1217, x1218, x1219, x1220, x1221, x1222) :|: TRUE f5719_in(x1223, x1224, x1225, x1226, x1227, x1228, x1229, x1230, x1231, x1232, x1233, x1234, x1235, x1236) -> f5734_in :|: TRUE f5736_out(x1237, x1238, x1239, x1240, x1241, x1242, x1243, x1244, .(x1238, .(x1239, .(x1240, .(x1241, .(x1242, .(x1243, .(0, []))))))), x1245, x1246, x1247, x1248, x1249, x1250) -> f5733_out(x1237, x1238, x1239, x1240, x1241, x1242, x1243, x1244, x1245, x1246, x1247, x1248, x1249, x1250) :|: TRUE f5733_in(x1251, x1252, x1253, x1254, x1255, x1256, x1257, x1258, x1259, x1260, x1261, x1262, x1263, x1264) -> f5736_in(x1251, x1252, x1253, x1254, x1255, x1256, x1257, x1258, .(x1252, .(x1253, .(x1254, .(x1255, .(x1256, .(x1257, .(0, []))))))), x1259, x1260, x1261, x1262, x1263, x1264) :|: TRUE f5745_out(x1265, x1266, x1267, x1268, x1269, x1270, x1271, x1272, x1273, x1274, x1275, x1276, x1277, x1278, x1279, x1280) -> f5736_out(x1272, x1265, x1266, x1267, x1268, x1269, x1270, x1273, x1274, x1275, x1276, x1277, x1278, x1279, x1280) :|: TRUE f5736_in(x1281, x1282, x1283, x1284, x1285, x1286, x1287, x1288, x1289, x1290, x1291, x1292, x1293, x1294, x1295) -> f5744_in(x1281, x1282, x1283, x1284, x1285, x1286, x1287, x1288) :|: TRUE f5744_out(x1296, x1297, x1298, x1299, x1300, x1301, x1302, x1303) -> f5745_in(x1297, x1298, x1299, x1300, x1301, x1302, x1304, x1296, x1303, x1305, x1306, x1307, x1308, x1309, x1310, x1311) :|: TRUE f5746_out(x1312, x1313, x1314, x1315, x1316, x1317, x1318, x1319, x1320, x1321, x1322, x1323, x1324, x1325, x1326) -> f5748_in(x1327, x1318, x1328, x1320, x1321, x1322, x1323, x1324, x1325, x1326, x1319, x1312, x1313, x1314, x1315, x1316, x1317) :|: TRUE f5745_in(x1329, x1330, x1331, x1332, x1333, x1334, x1335, x1336, x1337, x1338, x1339, x1340, x1341, x1342, x1343, x1344) -> f5746_in(x1329, x1330, x1331, x1332, x1333, x1334, x1335, x1336, x1337, x1339, x1340, x1341, x1342, x1343, x1344) :|: TRUE f5748_out(x1345, x1346, x1347, x1348, x1349, x1350, x1351, x1352, x1353, x1354, x1355, x1356, x1357, x1358, x1359, x1360, x1361) -> f5745_out(x1356, x1357, x1358, x1359, x1360, x1361, x1346, x1355, x1348, x1347, x1349, x1350, x1351, x1352, x1353, x1354) :|: TRUE f5748_in(x1362, x1363, x1364, x1365, x1366, x1367, x1368, x1369, x1370, x1371, x1372, x1373, x1374, x1375, x1376, x1377, x1378) -> f5830_in(x1362, x1363, x1364, x1365, x1366, x1367, x1368, x1369, x1370, x1371, x1372, x1373, x1374, x1375, x1376, x1377, x1378) :|: TRUE f5830_out(x1379, x1380, x1381, x1382, x1383, x1384, x1385, x1386, x1387, x1388, x1389, x1390, x1391, x1392, x1393, x1394, x1395) -> f5748_out(x1379, x1380, x1381, x1382, x1383, x1384, x1385, x1386, x1387, x1388, x1389, x1390, x1391, x1392, x1393, x1394, x1395) :|: TRUE f5832_out(x1396, x1397, x1398, x1399, x1400, x1401, x1402, x1403, x1404, x1405, x1406, x1407, x1408, x1409, x1410, x1411) -> f5830_out([], x1396, x1397, x1398, x1399, x1400, x1401, x1402, x1403, x1404, x1405, x1406, x1407, x1408, x1409, x1410, x1411) :|: TRUE f5833_out(x1412, x1413, x1414, x1415, x1416, x1417, x1418, x1419, x1420, x1421, x1422, x1423, x1424, x1425, x1426, x1427, x1428) -> f5830_out(x1412, x1413, x1414, x1415, x1416, x1417, x1418, x1419, x1420, x1421, x1422, x1423, x1424, x1425, x1426, x1427, x1428) :|: TRUE f5830_in(x1429, x1430, x1431, x1432, x1433, x1434, x1435, x1436, x1437, x1438, x1439, x1440, x1441, x1442, x1443, x1444, x1445) -> f5833_in(x1429, x1430, x1431, x1432, x1433, x1434, x1435, x1436, x1437, x1438, x1439, x1440, x1441, x1442, x1443, x1444, x1445) :|: TRUE f5830_in([], x1446, x1447, x1448, x1449, x1450, x1451, x1452, x1453, x1454, x1455, x1456, x1457, x1458, x1459, x1460, x1461) -> f5832_in(x1446, x1447, x1448, x1449, x1450, x1451, x1452, x1453, x1454, x1455, x1456, x1457, x1458, x1459, x1460, x1461) :|: TRUE f5846_out(x1462, x1463, x1464, x1465, x1466, x1467, x1468, x1469, x1470, x1471, x1472, x1473, x1474, x1475, x1476, x1477, x1478) -> f5833_out(x1462, x1463, x1464, x1465, x1466, x1467, x1468, x1469, x1470, x1471, x1472, x1473, x1474, x1475, x1476, x1477, x1478) :|: TRUE f5833_in(x1479, x1480, x1481, x1482, x1483, x1484, x1485, x1486, x1487, x1488, x1489, x1490, x1491, x1492, x1493, x1494, x1495) -> f5846_in(x1479, x1480, x1481, x1482, x1483, x1484, x1485, x1486, x1487, x1488, x1489, x1490, x1491, x1492, x1493, x1494, x1495) :|: TRUE f5833_in(x1496, x1497, x1498, x1499, x1500, x1501, x1502, x1503, x1504, x1505, x1506, x1507, x1508, x1509, x1510, x1511, x1512) -> f5847_in :|: TRUE f5847_out -> f5833_out(x1513, x1514, x1515, x1516, x1517, x1518, x1519, x1520, x1521, x1522, x1523, x1524, x1525, x1526, x1527, x1528, x1529) :|: TRUE f5846_in(x1530, x1531, x1532, x1533, x1534, x1535, x1536, x1537, x1538, x1539, x1540, x1541, x1542, x1543, x1544, x1545, x1546) -> f5851_in(x1530, x1531, x1532, x1533) :|: TRUE f5851_out(x1547, x1548, x1549, x1550) -> f5852_in(x1548, x1549, x1551, x1547, x1550, x1552, x1553, x1554, x1555, x1556, x1557, x1558, x1559, x1560, x1561, x1562, x1563, x1564) :|: TRUE f5852_out(x1565, x1566, x1567, x1568, x1569, x1570, x1571, x1572, x1573, x1574, x1575, x1576, x1577, x1578, x1579, x1580, x1581, x1582) -> f5846_out(x1568, x1565, x1566, x1569, x1570, x1571, x1572, x1573, x1574, x1575, x1576, x1577, x1578, x1579, x1580, x1581, x1582) :|: TRUE f5862_out(x1583, x1584, x1585, x1586, x1587, x1588, x1589, x1590, x1591, x1592, x1593, x1594, x1595, x1596, x1597, x1598, x1599, x1600, x1601) -> f5852_out(x1585, x1586, x1584, x1601, x1587, x1588, x1589, x1590, x1591, x1592, x1593, x1594, x1595, x1596, x1597, x1598, x1599, x1600) :|: TRUE f5861_out(x1602, x1603, x1604, x1605, x1606, x1607, x1608, x1609, x1610, x1611, x1612, x1613, x1614, x1615, x1616, x1617, x1618, x1619) -> f5862_in(x1620, x1604, x1602, x1603, x1606, x1607, x1608, x1609, x1610, x1611, x1612, x1613, x1614, x1615, x1616, x1617, x1618, x1619, x1605) :|: TRUE f5852_in(x1621, x1622, x1623, x1624, x1625, x1626, x1627, x1628, x1629, x1630, x1631, x1632, x1633, x1634, x1635, x1636, x1637, x1638) -> f5861_in(x1621, x1622, x1623, x1624, x1625, x1626, x1627, x1628, x1629, x1630, x1631, x1632, x1633, x1634, x1635, x1636, x1637, x1638) :|: TRUE f5861_in(x1639, x1640, x1641, x1642, x1643, x1644, x1645, x1646, x1647, x1648, x1649, x1650, x1651, x1652, x1653, x1654, x1655, x1656) -> f5863_in(x1639, x1640, x1641, x1642, x1643, x1644, x1645, x1646, x1647, x1648, x1649, x1650, x1651, x1652, x1653, x1654, x1655, x1656) :|: TRUE f5863_out(x1657, x1658, x1659, x1660, x1661, x1662, x1663, x1664, x1665, x1666, x1667, x1668, x1669, x1670, x1671, x1672, x1673, x1674) -> f5861_out(x1657, x1658, x1659, x1660, x1661, x1662, x1663, x1664, x1665, x1666, x1667, x1668, x1669, x1670, x1671, x1672, x1673, x1674) :|: TRUE f5865_out(x1675, x1676, x1677, x1678, x1679, x1680, x1681, x1682, x1683, x1684, x1685, x1686, x1687, x1688, x1689, x1690, x1691, x1692) -> f5863_out(x1675, x1676, x1677, x1678, x1679, x1680, x1681, x1682, x1683, x1684, x1685, x1686, x1687, x1688, x1689, x1690, x1691, x1692) :|: TRUE f5863_in(x1693, x1694, x1695, x1696, x1697, x1698, x1699, x1700, x1701, x1702, x1703, x1704, x1705, x1706, x1707, x1708, x1709, x1710) -> f5865_in(x1693, x1694, x1695, x1696, x1697, x1698, x1699, x1700, x1701, x1702, x1703, x1704, x1705, x1706, x1707, x1708, x1709, x1710) :|: TRUE f5865_in(x1711, x1712, x1713, x1714, x1715, x1716, x1717, x1718, x1719, x1720, x1721, x1722, x1723, x1724, x1725, x1726, x1727, x1728) -> f5876_in(x1711, x1712, x1713, x1715, x1714, x1716, x1717, x1718, x1719, x1720, x1721, x1722, x1723, x1724, x1725, x1726, x1727, x1728) :|: TRUE f5876_out(x1729, x1730, x1731, x1732, x1733, x1734, x1735, x1736, x1737, x1738, x1739, x1740, x1741, x1742, x1743, x1744, x1745, x1746) -> f5865_out(x1729, x1730, x1731, x1733, x1732, x1734, x1735, x1736, x1737, x1738, x1739, x1740, x1741, x1742, x1743, x1744, x1745, x1746) :|: TRUE f5878_out(x1747, x1748, x1749, x1750, x1751, x1752) -> f5876_out(x1750, x1751, x1753, x1752, x1748, x1754, x1755, x1756, x1757, x1758, x1759, x1760, x1761, x1762, x1763, x1764, x1765, x1766) :|: TRUE f5876_in(x1767, x1768, x1769, x1770, x1771, x1772, x1773, x1774, x1775, x1776, x1777, x1778, x1779, x1780, x1781, x1782, x1783, x1784) -> f5877_in(x1767, x1768, x1769, x1770, x1772, x1773, x1774, x1775, x1776, x1777, x1778, x1779, x1780, x1781, x1782, x1783, x1784, x1771) :|: TRUE f5877_out(x1785, x1786, x1787, x1788, x1789, x1790, x1791, x1792, x1793, x1794, x1795, x1796, x1797, x1798, x1799, x1800, x1801, x1802) -> f5878_in(x1803, x1802, x1804, x1785, x1786, x1788) :|: TRUE f5879_out(x1805, x1806, x1807, x1808, x1809, x1810, x1811, x1812, x1813, x1814, x1815, x1816, x1817, x1818, x1819, x1820, x1821, x1822) -> f5877_out(x1805, x1806, x1807, x1808, x1809, x1810, x1811, x1812, x1813, x1814, x1815, x1816, x1817, x1818, x1819, x1820, x1821, x1822) :|: TRUE f5877_in(x1823, x1824, x1825, x1826, x1827, x1828, x1829, x1830, x1831, x1832, x1833, x1834, x1835, x1836, x1837, x1838, x1839, x1840) -> f5879_in(x1823, x1824, x1825, x1826, x1827, x1828, x1829, x1830, x1831, x1832, x1833, x1834, x1835, x1836, x1837, x1838, x1839, x1840) :|: TRUE f5880_out(x1841, x1842, x1843, x1844, x1845, x1846, x1847, x1848, x1849, x1850, x1851, x1852, x1853, x1854, x1855, x1856, x1857, x1858) -> f5879_out(x1841, x1842, x1843, x1844, x1845, x1846, x1847, x1848, x1849, x1850, x1851, x1852, x1853, x1854, x1855, x1856, x1857, x1858) :|: TRUE f5879_in(x1859, x1860, x1861, x1862, x1863, x1864, x1865, x1866, x1867, x1868, x1869, x1870, x1871, x1872, x1873, x1874, x1875, x1876) -> f5880_in(x1859, x1860, x1861, x1862, x1863, x1864, x1865, x1866, x1867, x1868, x1869, x1870, x1871, x1872, x1873, x1874, x1875, x1876) :|: TRUE f5880_in(x1877, x1878, x1879, x1880, x1881, x1882, x1883, x1884, x1885, x1886, x1887, x1888, x1889, x1890, x1891, x1892, x1893, x1894) -> f5887_in(x1879, x1877, x1880, x1878, x1881, x1882, x1883, x1884, x1885, x1886, x1887, x1888, x1889, x1890, x1891, x1892, x1893, x1894) :|: TRUE f5887_out(x1895, x1896, x1897, x1898, x1899, x1900, x1901, x1902, x1903, x1904, x1905, x1906, x1907, x1908, x1909, x1910, x1911, x1912) -> f5880_out(x1896, x1898, x1895, x1897, x1899, x1900, x1901, x1902, x1903, x1904, x1905, x1906, x1907, x1908, x1909, x1910, x1911, x1912) :|: TRUE f5889_out(x1913, x1914, x1915, x1916, x1917, x1918, x1919, x1920, x1921, x1922, x1923, x1924, x1925, x1926, x1927, x1928, x1929, x1930, x1931) -> f5887_out(x1916, x1926, x1913, x1915, x1927, x1917, x1921, x1922, x1929, x1923, x1918, x1924, x1928, x1920, x1919, x1925, x1930, x1931) :|: TRUE f5888_out -> f5887_out(x1932, x1933, x1934, x1935, x1936, x1937, x1938, x1939, x1940, x1941, x1942, x1943, x1944, x1945, x1946, x1947, x1948, x1949) :|: TRUE f5887_in(x1950, x1951, x1952, x1953, x1954, x1955, x1956, x1957, x1958, x1959, x1960, x1961, x1962, x1963, x1964, x1965, x1966, x1967) -> f5888_in :|: TRUE f5887_in(x1968, x1969, x1970, x1971, x1972, x1973, x1974, x1975, x1976, x1977, x1978, x1979, x1980, x1981, x1982, x1983, x1984, x1985) -> f5889_in(x1970, x1986, x1971, x1968, x1973, x1978, x1982, x1981, x1974, x1975, x1977, x1979, x1983, x1969, x1972, x1980, x1976, x1984, x1985) :|: x1986 = x1968 - x1969 f5889_in(x1987, x1988, x1989, x1990, x1991, x1992, x1993, x1994, x1995, x1996, x1997, x1998, x1999, x2000, x2001, x2002, x2003, x2004, x2005) -> f5890_in :|: TRUE f5890_out -> f5889_out(x2006, x2007, x2008, x2009, x2010, x2011, x2012, x2013, x2014, x2015, x2016, x2017, x2018, x2019, x2020, x2021, x2022, x2023, x2024) :|: TRUE f5891_out(x2025, x2026, x2027, x2028, x2029, x2030, x2031, x2032, x2033, x2034, x2035, x2036, x2037, x2038, x2039, x2040, x2041, x2042, x2043, x2044) -> f5889_out(x2027, x2042, x2025, x2026, x2028, x2029, x2030, x2031, x2032, x2033, x2034, x2035, x2037, x2038, x2039, x2040, x2041, x2043, x2044) :|: TRUE f5889_in(x2045, x2046, x2047, x2048, x2049, x2050, x2051, x2052, x2053, x2054, x2055, x2056, x2057, x2058, x2059, x2060, x2061, x2062, x2063) -> f5891_in(x2047, x2048, x2045, x2049, x2050, x2051, x2052, x2053, x2054, x2055, x2056, x2064, x2057, x2058, x2059, x2060, x2061, x2046, x2062, x2063) :|: x2064 = x2045 - x2046 f5891_in(x2065, x2066, x2067, x2068, x2069, x2070, x2071, x2072, x2073, x2074, x2075, x2076, x2077, x2078, x2079, x2080, x2081, x2082, x2083, x2084) -> f5892_in(x2065, x2066, x2067, x2068, x2069, x2070, x2071, x2072, x2073, x2074, x2075, x2077, x2078, x2079, x2080, x2081, x2083, x2084) :|: TRUE f5892_out(x2085, x2086, x2087, x2088, x2089, x2090, x2091, x2092, x2093, x2094, x2095, x2096, x2097, x2098, x2099, x2100, x2101, x2102) -> f5891_out(x2085, x2086, x2087, x2088, x2089, x2090, x2091, x2092, x2093, x2094, x2095, x2103, x2096, x2097, x2098, x2099, x2100, x2104, x2101, x2102) :|: TRUE f5892_in(x2105, x2106, x2107, x2108, x2109, x2110, x2111, x2112, x2113, x2114, x2115, x2116, x2117, x2118, x2119, x2120, x2121, x2122) -> f5898_in(x2105, x2106, x2107, x2108, x2109, x2110, x2111, x2112, x2113, x2114, x2115, x2116, x2117, x2118, x2119, x2120, x2121, x2122) :|: TRUE f5899_out(x2123, x2124, x2125, x2126, x2127, x2128, x2129, x2130, x2131, x2132, x2133, x2134, x2135, x2136, x2137, x2138, x2139, x2140) -> f5892_out(x2123, x2124, x2125, x2126, x2127, x2128, x2129, x2130, x2131, x2132, x2133, x2134, x2135, x2136, x2137, x2138, x2139, x2140) :|: TRUE f5898_out(x2141, x2142, x2143, x2144, x2145, x2146, x2147, x2148, x2149, x2150, x2151, x2152, x2153, x2154, x2155, x2156, x2157, x2158) -> f5892_out(x2141, x2142, x2143, x2144, x2145, x2146, x2147, x2148, x2149, x2150, x2151, x2152, x2153, x2154, x2155, x2156, x2157, x2158) :|: TRUE f5892_in(x2159, x2160, x2161, x2162, x2163, x2164, x2165, x2166, x2167, x2168, x2169, x2170, x2171, x2172, x2173, x2174, x2175, x2176) -> f5899_in(x2159, x2160, x2161, x2162, x2163, x2164, x2165, x2166, x2167, x2168, x2169, x2170, x2171, x2172, x2173, x2174, x2175, x2176) :|: TRUE f5899_in(x2177, x2178, x2179, x2180, x2181, x2182, x2183, x2184, x2185, x2186, x2187, x2188, x2189, x2190, x2191, x2192, x2193, x2194) -> f5910_in :|: TRUE f5910_out -> f5899_out(x2195, x2196, x2197, x2198, x2199, x2200, x2201, x2202, x2203, x2204, x2205, x2206, x2207, x2208, x2209, x2210, x2211, x2212) :|: TRUE f5899_in(.(x2213, x2214), x2215, x2216, x2217, x2218, x2219, x2220, x2221, x2222, x2223, x2224, x2225, x2226, x2227, x2228, x2229, x2230, x2231) -> f5909_in(x2215, x2213, x2216, x2214, x2217, x2218, x2219, x2220, x2221, x2222, x2223, x2224, x2225, x2226, x2227, x2228, x2229, x2230, x2231) :|: TRUE f5909_out(x2232, x2233, x2234, x2235, x2236, x2237, x2238, x2239, x2240, x2241, x2242, x2243, x2244, x2245, x2246, x2247, x2248, x2249, x2250) -> f5899_out(.(x2233, x2235), x2232, x2234, x2236, x2237, x2238, x2239, x2240, x2241, x2242, x2243, x2244, x2245, x2246, x2247, x2248, x2249, x2250) :|: TRUE f5912_out(x2251, x2252, x2253, x2254, x2255, x2256, x2257, x2258, x2259, x2260, x2261, x2262, x2263, x2264, x2265, x2266, x2267, x2268, x2269, x2270) -> f5909_out(x2254, x2265, x2251, x2253, x2255, x2256, x2257, x2258, x2259, x2260, x2261, x2262, x2263, x2264, x2266, x2267, x2268, x2269, x2270) :|: TRUE f5911_out -> f5909_out(x2271, x2272, x2273, x2274, x2275, x2276, x2277, x2278, x2279, x2280, x2281, x2282, x2283, x2284, x2285, x2286, x2287, x2288, x2289) :|: TRUE f5909_in(x2290, x2291, x2292, x2293, x2294, x2295, x2296, x2297, x2298, x2299, x2300, x2301, x2302, x2303, x2304, x2305, x2306, x2307, x2308) -> f5912_in(x2292, x2309, x2293, x2290, x2294, x2295, x2296, x2297, x2298, x2299, x2300, x2301, x2302, x2303, x2291, x2304, x2305, x2306, x2307, x2308) :|: x2309 = x2290 - x2291 f5909_in(x2310, x2311, x2312, x2313, x2314, x2315, x2316, x2317, x2318, x2319, x2320, x2321, x2322, x2323, x2324, x2325, x2326, x2327, x2328) -> f5911_in :|: TRUE f5913_out(x2329, x2330, x2331, x2332, x2333, x2334, x2335, x2336, x2337, x2338, x2339, x2340, x2341, x2342, x2343, x2344, x2345, x2346, x2347, x2348, x2349) -> f5912_out(x2331, x2349, x2329, x2330, x2332, x2333, x2334, x2335, x2336, x2337, x2338, x2339, x2340, x2341, x2343, x2344, x2345, x2346, x2347, x2348) :|: TRUE f5912_in(x2350, x2351, x2352, x2353, x2354, x2355, x2356, x2357, x2358, x2359, x2360, x2361, x2362, x2363, x2364, x2365, x2366, x2367, x2368, x2369) -> f5913_in(x2352, x2353, x2350, x2354, x2355, x2356, x2357, x2358, x2359, x2360, x2361, x2362, x2363, x2370, x2364, x2365, x2366, x2367, x2368, x2369, x2351) :|: x2370 = x2350 - x2351 f5914_out(x2371, x2372, x2373, x2374, x2375, x2376, x2377, x2378, x2379, x2380, x2381, x2382, x2383, x2384, x2385, x2386, x2387, x2388, x2389) -> f5913_out(x2371, x2372, x2373, x2374, x2375, x2376, x2377, x2378, x2379, x2380, x2381, x2382, x2383, x2390, x2384, x2385, x2386, x2387, x2388, x2389, x2391) :|: TRUE f5913_in(x2392, x2393, x2394, x2395, x2396, x2397, x2398, x2399, x2400, x2401, x2402, x2403, x2404, x2405, x2406, x2407, x2408, x2409, x2410, x2411, x2412) -> f5914_in(x2392, x2393, x2394, x2395, x2396, x2397, x2398, x2399, x2400, x2401, x2402, x2403, x2404, x2406, x2407, x2408, x2409, x2410, x2411) :|: TRUE f5914_in(x2413, x2414, x2415, x2416, x2417, x2418, x2419, x2420, x2421, x2422, x2423, x2424, x2425, x2426, x2427, x2428, x2429, x2430, x2431) -> f5920_in(x2413, x2414, x2415, x2416, x2417, x2418, x2419, x2420, x2421, x2422, x2423, x2424, x2425, x2426, x2427, x2428, x2429, x2430, x2431) :|: TRUE f5921_out(x2432, x2433, x2434, x2435, x2436, x2437, x2438, x2439, x2440, x2441, x2442, x2443, x2444, x2445, x2446, x2447, x2448, x2449, x2450) -> f5914_out(x2432, x2433, x2434, x2435, x2436, x2437, x2438, x2439, x2440, x2441, x2442, x2443, x2444, x2445, x2446, x2447, x2448, x2449, x2450) :|: TRUE f5920_out(x2451, x2452, x2453, x2454, x2455, x2456, x2457, x2458, x2459, x2460, x2461, x2462, x2463, x2464, x2465, x2466, x2467, x2468, x2469) -> f5914_out(x2451, x2452, x2453, x2454, x2455, x2456, x2457, x2458, x2459, x2460, x2461, x2462, x2463, x2464, x2465, x2466, x2467, x2468, x2469) :|: TRUE f5914_in(x2470, x2471, x2472, x2473, x2474, x2475, x2476, x2477, x2478, x2479, x2480, x2481, x2482, x2483, x2484, x2485, x2486, x2487, x2488) -> f5921_in(x2470, x2471, x2472, x2473, x2474, x2475, x2476, x2477, x2478, x2479, x2480, x2481, x2482, x2483, x2484, x2485, x2486, x2487, x2488) :|: TRUE f5921_in(x2489, x2490, x2491, x2492, x2493, x2494, x2495, x2496, x2497, x2498, x2499, x2500, x2501, x2502, x2503, x2504, x2505, x2506, x2507) -> f5932_in :|: TRUE f5921_in(.(x2508, x2509), x2510, x2511, x2512, x2513, x2514, x2515, x2516, x2517, x2518, x2519, x2520, x2521, x2522, x2523, x2524, x2525, x2526, x2527) -> f5931_in(x2510, x2508, x2511, x2509, x2512, x2513, x2514, x2515, x2516, x2517, x2518, x2519, x2520, x2521, x2522, x2523, x2524, x2525, x2526, x2527) :|: TRUE f5931_out(x2528, x2529, x2530, x2531, x2532, x2533, x2534, x2535, x2536, x2537, x2538, x2539, x2540, x2541, x2542, x2543, x2544, x2545, x2546, x2547) -> f5921_out(.(x2529, x2531), x2528, x2530, x2532, x2533, x2534, x2535, x2536, x2537, x2538, x2539, x2540, x2541, x2542, x2543, x2544, x2545, x2546, x2547) :|: TRUE f5932_out -> f5921_out(x2548, x2549, x2550, x2551, x2552, x2553, x2554, x2555, x2556, x2557, x2558, x2559, x2560, x2561, x2562, x2563, x2564, x2565, x2566) :|: TRUE f5931_in(x2567, x2568, x2569, x2570, x2571, x2572, x2573, x2574, x2575, x2576, x2577, x2578, x2579, x2580, x2581, x2582, x2583, x2584, x2585, x2586) -> f5933_in :|: TRUE f5933_out -> f5931_out(x2587, x2588, x2589, x2590, x2591, x2592, x2593, x2594, x2595, x2596, x2597, x2598, x2599, x2600, x2601, x2602, x2603, x2604, x2605, x2606) :|: TRUE f5931_in(x2607, x2608, x2609, x2610, x2611, x2612, x2613, x2614, x2615, x2616, x2617, x2618, x2619, x2620, x2621, x2622, x2623, x2624, x2625, x2626) -> f5934_in(x2609, x2627, x2610, x2607, x2611, x2612, x2613, x2608, x2614, x2615, x2616, x2617, x2618, x2619, x2620, x2621, x2622, x2623, x2624, x2625, x2626) :|: x2627 = x2607 - x2608 f5934_out(x2628, x2629, x2630, x2631, x2632, x2633, x2634, x2635, x2636, x2637, x2638, x2639, x2640, x2641, x2642, x2643, x2644, x2645, x2646, x2647, x2648) -> f5931_out(x2631, x2635, x2628, x2630, x2632, x2633, x2634, x2636, x2637, x2638, x2639, x2640, x2641, x2642, x2643, x2644, x2645, x2646, x2647, x2648) :|: TRUE f5934_in(x2649, x2650, x2651, x2652, x2653, x2654, x2655, x2656, x2657, x2658, x2659, x2660, x2661, x2662, x2663, x2664, x2665, x2666, x2667, x2668, x2669) -> f5935_in(x2651, x2652, x2649, x2653, x2654, x2650, x2655, x2656, x2657, x2658, x2659, x2660, x2661, x2662, x2663, x2664, x2670, x2665, x2666, x2667, x2668, x2669) :|: x2670 = x2649 - x2650 f5935_out(x2671, x2672, x2673, x2674, x2675, x2676, x2677, x2678, x2679, x2680, x2681, x2682, x2683, x2684, x2685, x2686, x2687, x2688, x2689, x2690, x2691, x2692) -> f5934_out(x2673, x2676, x2671, x2672, x2674, x2675, x2677, x2678, x2679, x2680, x2681, x2682, x2683, x2684, x2685, x2686, x2688, x2689, x2690, x2691, x2692) :|: TRUE f5935_in(x2693, x2694, x2695, x2696, x2697, x2698, x2699, x2700, x2701, x2702, x2703, x2704, x2705, x2706, x2707, x2708, x2709, x2710, x2711, x2712, x2713, x2714) -> f5939_in(x2693, x2694, x2695, x2696, x2697, x2699, x2700, x2701, x2702, x2703, x2704, x2705, x2706, x2707, x2708, x2710, x2711, x2712, x2713, x2714) :|: TRUE f5939_out(x2715, x2716, x2717, x2718, x2719, x2720, x2721, x2722, x2723, x2724, x2725, x2726, x2727, x2728, x2729, x2730, x2731, x2732, x2733, x2734) -> f5935_out(x2715, x2716, x2717, x2718, x2719, x2735, x2720, x2721, x2722, x2723, x2724, x2725, x2726, x2727, x2728, x2729, x2736, x2730, x2731, x2732, x2733, x2734) :|: TRUE f5944_out(x2737, x2738, x2739, x2740, x2741, x2742, x2743, x2744, x2745, x2746, x2747, x2748, x2749, x2750, x2751, x2752, x2753, x2754, x2755, x2756) -> f5939_out(x2737, x2738, x2739, x2740, x2741, x2742, x2743, x2744, x2745, x2746, x2747, x2748, x2749, x2750, x2751, x2752, x2753, x2754, x2755, x2756) :|: TRUE f5945_out(x2757, x2758, x2759, x2760, x2761, x2762, x2763, x2764, x2765, x2766, x2767, x2768, x2769, x2770, x2771, x2772, x2773, x2774, x2775, x2776) -> f5939_out(x2757, x2758, x2759, x2760, x2761, x2762, x2763, x2764, x2765, x2766, x2767, x2768, x2769, x2770, x2771, x2772, x2773, x2774, x2775, x2776) :|: TRUE f5939_in(x2777, x2778, x2779, x2780, x2781, x2782, x2783, x2784, x2785, x2786, x2787, x2788, x2789, x2790, x2791, x2792, x2793, x2794, x2795, x2796) -> f5945_in(x2777, x2778, x2779, x2780, x2781, x2782, x2783, x2784, x2785, x2786, x2787, x2788, x2789, x2790, x2791, x2792, x2793, x2794, x2795, x2796) :|: TRUE f5939_in(x2797, x2798, x2799, x2800, x2801, x2802, x2803, x2804, x2805, x2806, x2807, x2808, x2809, x2810, x2811, x2812, x2813, x2814, x2815, x2816) -> f5944_in(x2797, x2798, x2799, x2800, x2801, x2802, x2803, x2804, x2805, x2806, x2807, x2808, x2809, x2810, x2811, x2812, x2813, x2814, x2815, x2816) :|: TRUE f5966_out(x2817, x2818, x2819, x2820, x2821, x2822, x2823, x2824, x2825, x2826, x2827, x2828, x2829, x2830, x2831, x2832, x2833, x2834, x2835, x2836, x2837) -> f5945_out(.(x2818, x2820), x2817, x2819, x2821, x2822, x2823, x2824, x2825, x2826, x2827, x2828, x2829, x2830, x2831, x2832, x2833, x2834, x2835, x2836, x2837) :|: TRUE f5967_out -> f5945_out(x2838, x2839, x2840, x2841, x2842, x2843, x2844, x2845, x2846, x2847, x2848, x2849, x2850, x2851, x2852, x2853, x2854, x2855, x2856, x2857) :|: TRUE f5945_in(.(x2858, x2859), x2860, x2861, x2862, x2863, x2864, x2865, x2866, x2867, x2868, x2869, x2870, x2871, x2872, x2873, x2874, x2875, x2876, x2877, x2878) -> f5966_in(x2860, x2858, x2861, x2859, x2862, x2863, x2864, x2865, x2866, x2867, x2868, x2869, x2870, x2871, x2872, x2873, x2874, x2875, x2876, x2877, x2878) :|: TRUE f5945_in(x2879, x2880, x2881, x2882, x2883, x2884, x2885, x2886, x2887, x2888, x2889, x2890, x2891, x2892, x2893, x2894, x2895, x2896, x2897, x2898) -> f5967_in :|: TRUE f5966_in(x2899, x2900, x2901, x2902, x2903, x2904, x2905, x2906, x2907, x2908, x2909, x2910, x2911, x2912, x2913, x2914, x2915, x2916, x2917, x2918, x2919) -> f5969_in(x2901, x2920, x2902, x2899, x2903, x2904, x2905, x2906, x2907, x2908, x2909, x2910, x2911, x2912, x2913, x2914, x2900, x2915, x2916, x2917, x2918, x2919) :|: x2920 = x2899 - x2900 f5968_out -> f5966_out(x2921, x2922, x2923, x2924, x2925, x2926, x2927, x2928, x2929, x2930, x2931, x2932, x2933, x2934, x2935, x2936, x2937, x2938, x2939, x2940, x2941) :|: TRUE f5966_in(x2942, x2943, x2944, x2945, x2946, x2947, x2948, x2949, x2950, x2951, x2952, x2953, x2954, x2955, x2956, x2957, x2958, x2959, x2960, x2961, x2962) -> f5968_in :|: TRUE f5969_out(x2963, x2964, x2965, x2966, x2967, x2968, x2969, x2970, x2971, x2972, x2973, x2974, x2975, x2976, x2977, x2978, x2979, x2980, x2981, x2982, x2983, x2984) -> f5966_out(x2966, x2979, x2963, x2965, x2967, x2968, x2969, x2970, x2971, x2972, x2973, x2974, x2975, x2976, x2977, x2978, x2980, x2981, x2982, x2983, x2984) :|: TRUE f5969_in(x2985, x2986, x2987, x2988, x2989, x2990, x2991, x2992, x2993, x2994, x2995, x2996, x2997, x2998, x2999, x3000, x3001, x3002, x3003, x3004, x3005, x3006) -> f5970_in(x2987, x2988, x2985, x2989, x2990, x2991, x2992, x3007, x2993, x2994, x2995, x2996, x2997, x2998, x2999, x2986, x3000, x3001, x3002, x3003, x3004, x3005, x3006) :|: x3007 = x2985 - x2986 f5970_out(x3008, x3009, x3010, x3011, x3012, x3013, x3014, x3015, x3016, x3017, x3018, x3019, x3020, x3021, x3022, x3023, x3024, x3025, x3026, x3027, x3028, x3029, x3030) -> f5969_out(x3010, x3023, x3008, x3009, x3011, x3012, x3013, x3014, x3016, x3017, x3018, x3019, x3020, x3021, x3022, x3024, x3025, x3026, x3027, x3028, x3029, x3030) :|: TRUE f5977_out(x3031, x3032, x3033, x3034, x3035, x3036, x3037, x3038, x3039, x3040, x3041, x3042, x3043, x3044, x3045, x3046, x3047, x3048, x3049, x3050, x3051) -> f5970_out(x3031, x3032, x3033, x3034, x3035, x3036, x3037, x3052, x3038, x3039, x3040, x3041, x3042, x3043, x3044, x3053, x3045, x3046, x3047, x3048, x3049, x3050, x3051) :|: TRUE f5970_in(x3054, x3055, x3056, x3057, x3058, x3059, x3060, x3061, x3062, x3063, x3064, x3065, x3066, x3067, x3068, x3069, x3070, x3071, x3072, x3073, x3074, x3075, x3076) -> f5977_in(x3054, x3055, x3056, x3057, x3058, x3059, x3060, x3062, x3063, x3064, x3065, x3066, x3067, x3068, x3070, x3071, x3072, x3073, x3074, x3075, x3076) :|: TRUE f5977_in(x3077, x3078, x3079, x3080, x3081, x3082, x3083, x3084, x3085, x3086, x3087, x3088, x3089, x3090, x3091, x3092, x3093, x3094, x3095, x3096, x3097) -> f5980_in(x3077, x3078, x3079, x3080, x3081, x3082, x3083, x3084, x3085, x3086, x3087, x3088, x3089, x3090, x3091, x3092, x3093, x3094, x3095, x3096, x3097) :|: TRUE f5977_in(x3098, x3099, x3100, x3101, x3102, x3103, x3104, x3105, x3106, x3107, x3108, x3109, x3110, x3111, x3112, x3113, x3114, x3115, x3116, x3117, x3118) -> f5981_in(x3098, x3099, x3100, x3101, x3102, x3103, x3104, x3105, x3106, x3107, x3108, x3109, x3110, x3111, x3112, x3113, x3114, x3115, x3116, x3117, x3118) :|: TRUE f5980_out(x3119, x3120, x3121, x3122, x3123, x3124, x3125, x3126, x3127, x3128, x3129, x3130, x3131, x3132, x3133, x3134, x3135, x3136, x3137, x3138, x3139) -> f5977_out(x3119, x3120, x3121, x3122, x3123, x3124, x3125, x3126, x3127, x3128, x3129, x3130, x3131, x3132, x3133, x3134, x3135, x3136, x3137, x3138, x3139) :|: TRUE f5981_out(x3140, x3141, x3142, x3143, x3144, x3145, x3146, x3147, x3148, x3149, x3150, x3151, x3152, x3153, x3154, x3155, x3156, x3157, x3158, x3159, x3160) -> f5977_out(x3140, x3141, x3142, x3143, x3144, x3145, x3146, x3147, x3148, x3149, x3150, x3151, x3152, x3153, x3154, x3155, x3156, x3157, x3158, x3159, x3160) :|: TRUE f5981_in(.(x3161, x3162), x3163, x3164, x3165, x3166, x3167, x3168, x3169, x3170, x3171, x3172, x3173, x3174, x3175, x3176, x3177, x3178, x3179, x3180, x3181, x3182) -> f5990_in(x3163, x3161, x3164, x3162, x3165, x3166, x3167, x3168, x3169, x3170, x3171, x3172, x3173, x3174, x3175, x3176, x3177, x3178, x3179, x3180, x3181, x3182) :|: TRUE f5990_out(x3183, x3184, x3185, x3186, x3187, x3188, x3189, x3190, x3191, x3192, x3193, x3194, x3195, x3196, x3197, x3198, x3199, x3200, x3201, x3202, x3203, x3204) -> f5981_out(.(x3184, x3186), x3183, x3185, x3187, x3188, x3189, x3190, x3191, x3192, x3193, x3194, x3195, x3196, x3197, x3198, x3199, x3200, x3201, x3202, x3203, x3204) :|: TRUE f5991_out -> f5981_out(x3205, x3206, x3207, x3208, x3209, x3210, x3211, x3212, x3213, x3214, x3215, x3216, x3217, x3218, x3219, x3220, x3221, x3222, x3223, x3224, x3225) :|: TRUE f5981_in(x3226, x3227, x3228, x3229, x3230, x3231, x3232, x3233, x3234, x3235, x3236, x3237, x3238, x3239, x3240, x3241, x3242, x3243, x3244, x3245, x3246) -> f5991_in :|: TRUE f5993_out(x3247, x3248, x3249, x3250, x3251, x3252, x3253, x3254, x3255, x3256, x3257, x3258, x3259, x3260, x3261, x3262, x3263, x3264, x3265, x3266, x3267, x3268, x3269) -> f5990_out(x3250, x3265, x3247, x3249, x3251, x3252, x3253, x3254, x3255, x3256, x3257, x3258, x3259, x3260, x3261, x3262, x3263, x3264, x3266, x3267, x3268, x3269) :|: TRUE f5992_out -> f5990_out(x3270, x3271, x3272, x3273, x3274, x3275, x3276, x3277, x3278, x3279, x3280, x3281, x3282, x3283, x3284, x3285, x3286, x3287, x3288, x3289, x3290, x3291) :|: TRUE f5990_in(x3292, x3293, x3294, x3295, x3296, x3297, x3298, x3299, x3300, x3301, x3302, x3303, x3304, x3305, x3306, x3307, x3308, x3309, x3310, x3311, x3312, x3313) -> f5993_in(x3294, x3314, x3295, x3292, x3296, x3297, x3298, x3299, x3300, x3301, x3302, x3303, x3304, x3305, x3306, x3307, x3308, x3309, x3293, x3310, x3311, x3312, x3313) :|: x3314 = x3292 - x3293 f5990_in(x3315, x3316, x3317, x3318, x3319, x3320, x3321, x3322, x3323, x3324, x3325, x3326, x3327, x3328, x3329, x3330, x3331, x3332, x3333, x3334, x3335, x3336) -> f5992_in :|: TRUE f5993_in(x3337, x3338, x3339, x3340, x3341, x3342, x3343, x3344, x3345, x3346, x3347, x3348, x3349, x3350, x3351, x3352, x3353, x3354, x3355, x3356, x3357, x3358, x3359) -> f5994_in(x3339, x3340, x3337, x3341, x3342, x3338, x3343, x3344, x3360, x3345, x3346, x3347, x3348, x3349, x3350, x3351, x3352, x3353, x3354, x3355, x3356, x3357, x3358, x3359) :|: x3360 = x3337 - x3338 f5994_out(x3361, x3362, x3363, x3364, x3365, x3366, x3367, x3368, x3369, x3370, x3371, x3372, x3373, x3374, x3375, x3376, x3377, x3378, x3379, x3380, x3381, x3382, x3383, x3384) -> f5993_out(x3363, x3366, x3361, x3362, x3364, x3365, x3367, x3368, x3370, x3371, x3372, x3373, x3374, x3375, x3376, x3377, x3378, x3379, x3380, x3381, x3382, x3383, x3384) :|: TRUE f5994_in(x3385, x3386, x3387, x3388, x3389, x3390, x3391, x3392, x3393, x3394, x3395, x3396, x3397, x3398, x3399, x3400, x3401, x3402, x3403, x3404, x3405, x3406, x3407, x3408) -> f6001_in(x3385, x3386, x3387, x3388, x3389, x3391, x3392, x3394, x3395, x3396, x3397, x3398, x3399, x3400, x3401, x3402, x3403, x3404, x3405, x3406, x3407, x3408) :|: TRUE f6001_out(x3409, x3410, x3411, x3412, x3413, x3414, x3415, x3416, x3417, x3418, x3419, x3420, x3421, x3422, x3423, x3424, x3425, x3426, x3427, x3428, x3429, x3430) -> f5994_out(x3409, x3410, x3411, x3412, x3413, x3431, x3414, x3415, x3432, x3416, x3417, x3418, x3419, x3420, x3421, x3422, x3423, x3424, x3425, x3426, x3427, x3428, x3429, x3430) :|: TRUE f6001_in(x3433, x3434, x3435, x3436, x3437, x3438, x3439, x3440, x3441, x3442, x3443, x3444, x3445, x3446, x3447, x3448, x3449, x3450, x3451, x3452, x3453, x3454) -> f6005_in(x3433, x3434, x3435, x3436, x3437, x3438, x3439, x3440, x3441, x3442, x3443, x3444, x3445, x3446, x3447, x3448, x3449, x3450, x3451, x3452, x3453, x3454) :|: TRUE f6001_in(x3455, x3456, x3457, x3458, x3459, x3460, x3461, x3462, x3463, x3464, x3465, x3466, x3467, x3468, x3469, x3470, x3471, x3472, x3473, x3474, x3475, x3476) -> f6002_in(x3455, x3456, x3457, x3458, x3459, x3460, x3461, x3462, x3463, x3464, x3465, x3466, x3467, x3468, x3469, x3470, x3471, x3472, x3473, x3474, x3475, x3476) :|: TRUE f6002_out(x3477, x3478, x3479, x3480, x3481, x3482, x3483, x3484, x3485, x3486, x3487, x3488, x3489, x3490, x3491, x3492, x3493, x3494, x3495, x3496, x3497, x3498) -> f6001_out(x3477, x3478, x3479, x3480, x3481, x3482, x3483, x3484, x3485, x3486, x3487, x3488, x3489, x3490, x3491, x3492, x3493, x3494, x3495, x3496, x3497, x3498) :|: TRUE f6005_out(x3499, x3500, x3501, x3502, x3503, x3504, x3505, x3506, x3507, x3508, x3509, x3510, x3511, x3512, x3513, x3514, x3515, x3516, x3517, x3518, x3519, x3520) -> f6001_out(x3499, x3500, x3501, x3502, x3503, x3504, x3505, x3506, x3507, x3508, x3509, x3510, x3511, x3512, x3513, x3514, x3515, x3516, x3517, x3518, x3519, x3520) :|: TRUE f6005_in(.(x3521, x3522), x3523, x3524, x3525, x3526, x3527, x3528, x3529, x3530, x3531, x3532, x3533, x3534, x3535, x3536, x3537, x3538, x3539, x3540, x3541, x3542, x3543) -> f6016_in(x3523, x3521, x3524, x3522, x3525, x3526, x3527, x3528, x3529, x3530, x3531, x3532, x3533, x3534, x3535, x3536, x3537, x3538, x3539, x3540, x3541, x3542, x3543) :|: TRUE f6016_out(x3544, x3545, x3546, x3547, x3548, x3549, x3550, x3551, x3552, x3553, x3554, x3555, x3556, x3557, x3558, x3559, x3560, x3561, x3562, x3563, x3564, x3565, x3566) -> f6005_out(.(x3545, x3547), x3544, x3546, x3548, x3549, x3550, x3551, x3552, x3553, x3554, x3555, x3556, x3557, x3558, x3559, x3560, x3561, x3562, x3563, x3564, x3565, x3566) :|: TRUE f6017_out -> f6005_out(x3567, x3568, x3569, x3570, x3571, x3572, x3573, x3574, x3575, x3576, x3577, x3578, x3579, x3580, x3581, x3582, x3583, x3584, x3585, x3586, x3587, x3588) :|: TRUE f6005_in(x3589, x3590, x3591, x3592, x3593, x3594, x3595, x3596, x3597, x3598, x3599, x3600, x3601, x3602, x3603, x3604, x3605, x3606, x3607, x3608, x3609, x3610) -> f6017_in :|: TRUE f6019_out(x3611, x3612, x3613, x3614, x3615, x3616, x3617, x3618, x3619, x3620, x3621, x3622, x3623, x3624, x3625, x3626, x3627, x3628, x3629, x3630, x3631, x3632, x3633, x3634) -> f6016_out(x3614, x3630, x3611, x3613, x3615, x3616, x3617, x3618, x3619, x3620, x3621, x3622, x3623, x3624, x3625, x3626, x3627, x3628, x3629, x3631, x3632, x3633, x3634) :|: TRUE f6016_in(x3635, x3636, x3637, x3638, x3639, x3640, x3641, x3642, x3643, x3644, x3645, x3646, x3647, x3648, x3649, x3650, x3651, x3652, x3653, x3654, x3655, x3656, x3657) -> f6018_in :|: TRUE f6016_in(x3658, x3659, x3660, x3661, x3662, x3663, x3664, x3665, x3666, x3667, x3668, x3669, x3670, x3671, x3672, x3673, x3674, x3675, x3676, x3677, x3678, x3679, x3680) -> f6019_in(x3660, x3681, x3661, x3658, x3662, x3663, x3664, x3665, x3666, x3667, x3668, x3669, x3670, x3671, x3672, x3673, x3674, x3675, x3676, x3659, x3677, x3678, x3679, x3680) :|: x3681 = x3658 - x3659 f6018_out -> f6016_out(x3682, x3683, x3684, x3685, x3686, x3687, x3688, x3689, x3690, x3691, x3692, x3693, x3694, x3695, x3696, x3697, x3698, x3699, x3700, x3701, x3702, x3703, x3704) :|: TRUE f6020_out(x3705, x3706, x3707, x3708, x3709, x3710, x3711, x3712, x3713, x3714, x3715, x3716, x3717, x3718, x3719, x3720, x3721, x3722, x3723, x3724, x3725, x3726, x3727, x3728, x3729) -> f6019_out(x3707, x3718, x3705, x3706, x3716, x3708, x3717, x3709, x3710, x3711, x3719, x3720, x3721, x3722, x3723, x3724, x3725, x3712, x3726, x3727, x3728, x3714, x3715, x3729) :|: TRUE f6019_in(x3730, x3731, x3732, x3733, x3734, x3735, x3736, x3737, x3738, x3739, x3740, x3741, x3742, x3743, x3744, x3745, x3746, x3747, x3748, x3749, x3750, x3751, x3752, x3753) -> f6020_in(x3732, x3733, x3730, x3735, x3737, x3738, x3739, x3747, x3754, x3751, x3752, x3734, x3736, x3731, x3740, x3741, x3742, x3743, x3744, x3745, x3746, x3748, x3749, x3750, x3753) :|: x3754 = x3730 - x3731 f6021_out(x3755, x3756, x3757, x3758, x3759, x3760, x3761, x3762, x3763, x3764, x3765, x3766, x3767, x3768, x3769, x3770, x3771, x3772, x3773, x3774, x3775, x3776, x3777) -> f6020_out(x3755, x3756, x3757, x3758, x3759, x3760, x3761, x3762, x3778, x3763, x3764, x3765, x3766, x3779, x3767, x3768, x3769, x3770, x3771, x3772, x3773, x3774, x3775, x3776, x3777) :|: TRUE f6020_in(x3780, x3781, x3782, x3783, x3784, x3785, x3786, x3787, x3788, x3789, x3790, x3791, x3792, x3793, x3794, x3795, x3796, x3797, x3798, x3799, x3800, x3801, x3802, x3803, x3804) -> f6021_in(x3780, x3781, x3782, x3783, x3784, x3785, x3786, x3787, x3789, x3790, x3791, x3792, x3794, x3795, x3796, x3797, x3798, x3799, x3800, x3801, x3802, x3803, x3804) :|: TRUE f6021_in(x3805, x3806, x3807, x3808, x3809, x3810, x3811, x3812, x3813, x3814, x3815, x3816, x3817, x3818, x3819, x3820, x3821, x3822, x3823, x3824, x3825, x3826, x3827) -> f6023_in(x3805, x3806, x3807, x3808, x3809, x3810, x3811, x3812, x3813, x3814, x3815, x3816, x3817, x3818, x3819, x3820, x3821, x3822, x3823, x3824, x3825, x3826, x3827) :|: TRUE f6022_out(x3828, x3829, x3830, x3831, x3832, x3833, x3834, x3835, x3836, x3837, x3838, x3839, x3840, x3841, x3842, x3843, x3844, x3845, x3846, x3847, x3848, x3849, x3850) -> f6021_out(x3828, x3829, x3830, x3831, x3832, x3833, x3834, x3835, x3836, x3837, x3838, x3839, x3840, x3841, x3842, x3843, x3844, x3845, x3846, x3847, x3848, x3849, x3850) :|: TRUE f6023_out(x3851, x3852, x3853, x3854, x3855, x3856, x3857, x3858, x3859, x3860, x3861, x3862, x3863, x3864, x3865, x3866, x3867, x3868, x3869, x3870, x3871, x3872, x3873) -> f6021_out(x3851, x3852, x3853, x3854, x3855, x3856, x3857, x3858, x3859, x3860, x3861, x3862, x3863, x3864, x3865, x3866, x3867, x3868, x3869, x3870, x3871, x3872, x3873) :|: TRUE f6021_in(x3874, x3875, x3876, x3877, x3878, x3879, x3880, x3881, x3882, x3883, x3884, x3885, x3886, x3887, x3888, x3889, x3890, x3891, x3892, x3893, x3894, x3895, x3896) -> f6022_in(x3874, x3875, x3876, x3877, x3878, x3879, x3880, x3881, x3882, x3883, x3884, x3885, x3886, x3887, x3888, x3889, x3890, x3891, x3892, x3893, x3894, x3895, x3896) :|: TRUE f6041_out -> f6023_out(x3897, x3898, x3899, x3900, x3901, x3902, x3903, x3904, x3905, x3906, x3907, x3908, x3909, x3910, x3911, x3912, x3913, x3914, x3915, x3916, x3917, x3918, x3919) :|: TRUE f6023_in(x3920, x3921, x3922, x3923, x3924, x3925, x3926, x3927, x3928, x3929, x3930, x3931, x3932, x3933, x3934, x3935, x3936, x3937, x3938, x3939, x3940, x3941, x3942) -> f6041_in :|: TRUE f6023_in(.(x3943, x3944), x3945, x3946, x3947, x3948, x3949, x3950, x3951, x3952, x3953, x3954, x3955, x3956, x3957, x3958, x3959, x3960, x3961, x3962, x3963, x3964, x3965, x3966) -> f6040_in(x3945, x3943, x3946, x3944, x3947, x3948, x3949, x3950, x3951, x3952, x3953, x3954, x3955, x3956, x3957, x3958, x3959, x3960, x3961, x3962, x3963, x3964, x3965, x3966) :|: TRUE f6040_out(x3967, x3968, x3969, x3970, x3971, x3972, x3973, x3974, x3975, x3976, x3977, x3978, x3979, x3980, x3981, x3982, x3983, x3984, x3985, x3986, x3987, x3988, x3989, x3990) -> f6023_out(.(x3968, x3970), x3967, x3969, x3971, x3972, x3973, x3974, x3975, x3976, x3977, x3978, x3979, x3980, x3981, x3982, x3983, x3984, x3985, x3986, x3987, x3988, x3989, x3990) :|: TRUE f6043_out(x3991, x3992, x3993, x3994, x3995, x3996, x3997, x3998, x3999, x4000, x4001, x4002, x4003, x4004, x4005, x4006, x4007, x4008, x4009, x4010, x4011, x4012, x4013, x4014, x4015) -> f6040_out(x3994, x4005, x3991, x3993, x3995, x3996, x3997, x3998, x3999, x4000, x4001, x4002, x4003, x4004, x4006, x4007, x4008, x4009, x4010, x4011, x4012, x4013, x4014, x4015) :|: TRUE f6040_in(x4016, x4017, x4018, x4019, x4020, x4021, x4022, x4023, x4024, x4025, x4026, x4027, x4028, x4029, x4030, x4031, x4032, x4033, x4034, x4035, x4036, x4037, x4038, x4039) -> f6042_in :|: TRUE f6042_out -> f6040_out(x4040, x4041, x4042, x4043, x4044, x4045, x4046, x4047, x4048, x4049, x4050, x4051, x4052, x4053, x4054, x4055, x4056, x4057, x4058, x4059, x4060, x4061, x4062, x4063) :|: TRUE f6040_in(x4064, x4065, x4066, x4067, x4068, x4069, x4070, x4071, x4072, x4073, x4074, x4075, x4076, x4077, x4078, x4079, x4080, x4081, x4082, x4083, x4084, x4085, x4086, x4087) -> f6043_in(x4066, x4088, x4067, x4064, x4068, x4069, x4070, x4071, x4072, x4073, x4074, x4075, x4076, x4077, x4065, x4078, x4079, x4080, x4081, x4082, x4083, x4084, x4085, x4086, x4087) :|: x4088 = x4064 - x4065 f6044_out(x4089, x4090, x4091, x4092, x4093, x4094, x4095, x4096, x4097, x4098, x4099, x4100, x4101, x4102, x4103, x4104, x4105, x4106, x4107, x4108, x4109, x4110, x4111, x4112, x4113, x4114) -> f6043_out(x4091, x4097, x4089, x4090, x4092, x4093, x4094, x4095, x4098, x4099, x4100, x4101, x4102, x4103, x4104, x4105, x4106, x4107, x4108, x4109, x4110, x4111, x4112, x4113, x4114) :|: TRUE f6043_in(x4115, x4116, x4117, x4118, x4119, x4120, x4121, x4122, x4123, x4124, x4125, x4126, x4127, x4128, x4129, x4130, x4131, x4132, x4133, x4134, x4135, x4136, x4137, x4138, x4139) -> f6044_in(x4117, x4118, x4115, x4119, x4120, x4121, x4122, x4140, x4116, x4123, x4124, x4125, x4126, x4127, x4128, x4129, x4130, x4131, x4132, x4133, x4134, x4135, x4136, x4137, x4138, x4139) :|: x4140 = x4115 - x4116 f6080_out(x4141, x4142, x4143, x4144, x4145, x4146, x4147, x4148, x4149, x4150, x4151, x4152, x4153, x4154, x4155, x4156, x4157, x4158, x4159, x4160, x4161, x4162, x4163, x4164) -> f6044_out(x4141, x4142, x4143, x4144, x4145, x4146, x4147, x4165, x4166, x4148, x4149, x4150, x4151, x4152, x4153, x4154, x4155, x4156, x4157, x4158, x4159, x4160, x4161, x4162, x4163, x4164) :|: TRUE f6044_in(x4167, x4168, x4169, x4170, x4171, x4172, x4173, x4174, x4175, x4176, x4177, x4178, x4179, x4180, x4181, x4182, x4183, x4184, x4185, x4186, x4187, x4188, x4189, x4190, x4191, x4192) -> f6080_in(x4167, x4168, x4169, x4170, x4171, x4172, x4173, x4176, x4177, x4178, x4179, x4180, x4181, x4182, x4183, x4184, x4185, x4186, x4187, x4188, x4189, x4190, x4191, x4192) :|: TRUE f6081_out(x4193, x4194, x4195, x4196, x4197, x4198, x4199, x4200, x4201, x4202, x4203, x4204, x4205, x4206, x4207, x4208, x4209, x4210, x4211, x4212, x4213, x4214, x4215, x4216) -> f6080_out(x4193, x4194, x4195, x4196, x4197, x4198, x4199, x4200, x4201, x4202, x4203, x4204, x4205, x4206, x4207, x4208, x4209, x4210, x4211, x4212, x4213, x4214, x4215, x4216) :|: TRUE f6080_in(x4217, x4218, x4219, x4220, x4221, x4222, x4223, x4224, x4225, x4226, x4227, x4228, x4229, x4230, x4231, x4232, x4233, x4234, x4235, x4236, x4237, x4238, x4239, x4240) -> f6082_in(x4217, x4218, x4219, x4220, x4221, x4222, x4223, x4224, x4225, x4226, x4227, x4228, x4229, x4230, x4231, x4232, x4233, x4234, x4235, x4236, x4237, x4238, x4239, x4240) :|: TRUE f6082_out(x4241, x4242, x4243, x4244, x4245, x4246, x4247, x4248, x4249, x4250, x4251, x4252, x4253, x4254, x4255, x4256, x4257, x4258, x4259, x4260, x4261, x4262, x4263, x4264) -> f6080_out(x4241, x4242, x4243, x4244, x4245, x4246, x4247, x4248, x4249, x4250, x4251, x4252, x4253, x4254, x4255, x4256, x4257, x4258, x4259, x4260, x4261, x4262, x4263, x4264) :|: TRUE f6080_in(x4265, x4266, x4267, x4268, x4269, x4270, x4271, x4272, x4273, x4274, x4275, x4276, x4277, x4278, x4279, x4280, x4281, x4282, x4283, x4284, x4285, x4286, x4287, x4288) -> f6081_in(x4265, x4266, x4267, x4268, x4269, x4270, x4271, x4272, x4273, x4274, x4275, x4276, x4277, x4278, x4279, x4280, x4281, x4282, x4283, x4284, x4285, x4286, x4287, x4288) :|: TRUE f6082_in(.(x4289, x4290), x4291, x4292, x4293, x4294, x4295, x4296, x4297, x4298, x4299, x4300, x4301, x4302, x4303, x4304, x4305, x4306, x4307, x4308, x4309, x4310, x4311, x4312, x4313) -> f6086_in(x4291, x4289, x4292, x4290, x4293, x4294, x4295, x4296, x4297, x4298, x4299, x4300, x4301, x4302, x4303, x4304, x4305, x4306, x4307, x4308, x4309, x4310, x4311, x4312, x4313) :|: TRUE f6082_in(x4314, x4315, x4316, x4317, x4318, x4319, x4320, x4321, x4322, x4323, x4324, x4325, x4326, x4327, x4328, x4329, x4330, x4331, x4332, x4333, x4334, x4335, x4336, x4337) -> f6087_in :|: TRUE f6087_out -> f6082_out(x4338, x4339, x4340, x4341, x4342, x4343, x4344, x4345, x4346, x4347, x4348, x4349, x4350, x4351, x4352, x4353, x4354, x4355, x4356, x4357, x4358, x4359, x4360, x4361) :|: TRUE f6086_out(x4362, x4363, x4364, x4365, x4366, x4367, x4368, x4369, x4370, x4371, x4372, x4373, x4374, x4375, x4376, x4377, x4378, x4379, x4380, x4381, x4382, x4383, x4384, x4385, x4386) -> f6082_out(.(x4363, x4365), x4362, x4364, x4366, x4367, x4368, x4369, x4370, x4371, x4372, x4373, x4374, x4375, x4376, x4377, x4378, x4379, x4380, x4381, x4382, x4383, x4384, x4385, x4386) :|: TRUE f6086_in(x4387, x4388, x4389, x4390, x4391, x4392, x4393, x4394, x4395, x4396, x4397, x4398, x4399, x4400, x4401, x4402, x4403, x4404, x4405, x4406, x4407, x4408, x4409, x4410, x4411) -> f6089_in(x4389, x4412, x4390, x4387, x4391, x4392, x4393, x4394, x4395, x4396, x4397, x4398, x4399, x4400, x4401, x4402, x4403, x4404, x4405, x4406, x4407, x4408, x4409, x4410, x4388, x4411) :|: x4412 = x4387 - x4388 f6086_in(x4413, x4414, x4415, x4416, x4417, x4418, x4419, x4420, x4421, x4422, x4423, x4424, x4425, x4426, x4427, x4428, x4429, x4430, x4431, x4432, x4433, x4434, x4435, x4436, x4437) -> f6088_in :|: TRUE f6089_out(x4438, x4439, x4440, x4441, x4442, x4443, x4444, x4445, x4446, x4447, x4448, x4449, x4450, x4451, x4452, x4453, x4454, x4455, x4456, x4457, x4458, x4459, x4460, x4461, x4462, x4463) -> f6086_out(x4441, x4462, x4438, x4440, x4442, x4443, x4444, x4445, x4446, x4447, x4448, x4449, x4450, x4451, x4452, x4453, x4454, x4455, x4456, x4457, x4458, x4459, x4460, x4461, x4463) :|: TRUE f6088_out -> f6086_out(x4464, x4465, x4466, x4467, x4468, x4469, x4470, x4471, x4472, x4473, x4474, x4475, x4476, x4477, x4478, x4479, x4480, x4481, x4482, x4483, x4484, x4485, x4486, x4487, x4488) :|: TRUE f6089_in(x4489, x4490, x4491, x4492, x4493, x4494, x4495, x4496, x4497, x4498, x4499, x4500, x4501, x4502, x4503, x4504, x4505, x4506, x4507, x4508, x4509, x4510, x4511, x4512, x4513, x4514) -> f6090_in(x4491, x4492, x4489, x4493, x4494, x4495, x4496, x4515, x4497, x4490, x4498, x4499, x4500, x4501, x4502, x4503, x4504, x4505, x4506, x4507, x4508, x4509, x4510, x4511, x4512, x4513, x4514) :|: x4515 = x4489 - x4490 f6090_out(x4516, x4517, x4518, x4519, x4520, x4521, x4522, x4523, x4524, x4525, x4526, x4527, x4528, x4529, x4530, x4531, x4532, x4533, x4534, x4535, x4536, x4537, x4538, x4539, x4540, x4541, x4542) -> f6089_out(x4518, x4525, x4516, x4517, x4519, x4520, x4521, x4522, x4524, x4526, x4527, x4528, x4529, x4530, x4531, x4532, x4533, x4534, x4535, x4536, x4537, x4538, x4539, x4540, x4541, x4542) :|: TRUE f6091_out(x4543, x4544, x4545, x4546, x4547, x4548, x4549, x4550, x4551, x4552, x4553, x4554, x4555, x4556, x4557, x4558, x4559, x4560, .(x4561, .(x4562, .(x4563, .(x4564, .(x4565, .(x4566, .(x4567, x4543)))))))) -> f6090_out(x4543, x4544, x4545, x4546, x4562, x4547, x4548, x4568, x4549, x4569, x4550, x4551, x4552, x4553, x4554, x4566, x4555, x4556, x4557, x4558, x4561, x4563, x4564, x4565, x4559, x4567, x4560) :|: TRUE f6090_in(x4570, x4571, x4572, x4573, x4574, x4575, x4576, x4577, x4578, x4579, x4580, x4581, x4582, x4583, x4584, x4585, x4586, x4587, x4588, x4589, x4590, x4591, x4592, x4593, x4594, x4595, x4596) -> f6091_in(x4570, x4571, x4572, x4573, x4575, x4576, x4578, x4580, x4581, x4582, x4583, x4584, x4586, x4587, x4588, x4589, x4594, x4596, .(x4590, .(x4574, .(x4591, .(x4592, .(x4593, .(x4585, .(x4595, x4570)))))))) :|: TRUE Start term: f1_in(T1) ---------------------------------------- (406) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (407) TRUE ---------------------------------------- (408) Obligation: Rules: f5082_out(T584, T583, T585, T586) -> f5078_out(.(T583, T584), T585, T586) :|: TRUE f5078_in(T536, T535, T560) -> f5083_in :|: TRUE f5078_in(.(x, x1), x2, x3) -> f5082_in(x1, x, x2, x3) :|: TRUE f5083_out -> f5078_out(x4, x5, x6) :|: TRUE f5059_in(x7, x8, x9) -> f5076_in(x7, x8, x9) :|: TRUE f5076_out(x10, x11, x12) -> f5059_out(x10, x11, x12) :|: TRUE f5077_out(x13, x14, x15) -> f5076_out(x13, x14, x15) :|: TRUE f5076_in(x16, x17, x18) -> f5078_in(x16, x17, x18) :|: TRUE f5076_in(x19, x20, x21) -> f5077_in(x19, x20, x21) :|: TRUE f5078_out(x22, x23, x24) -> f5076_out(x22, x23, x24) :|: TRUE f5082_in(x25, x26, x27, x28) -> f5059_in(x25, x26, .(x27, x28)) :|: TRUE f5059_out(x29, x30, .(x31, x32)) -> f5082_out(x29, x30, x31, x32) :|: TRUE f4_out(T1) -> f1_out(T1) :|: TRUE f1_in(x33) -> f4_in(x33) :|: TRUE f4_in(T7) -> f10_in(T7) :|: TRUE f11_out -> f4_out(x34) :|: TRUE f10_out(x35) -> f4_out(x35) :|: TRUE f4_in(x36) -> f11_in :|: TRUE f10_in(x37) -> f16_in(x38, x37) :|: x38 = x37 * (x37 + 1) + 1 f10_in(x39) -> f12_in :|: TRUE f16_out(x40, x41) -> f10_out(x41) :|: TRUE f12_out -> f10_out(x42) :|: TRUE f16_in(x43, x44) -> f21_in(x43) :|: TRUE f24_out(x45, x46) -> f16_out(x46, x47) :|: TRUE f21_out(T10) -> f24_in(T12, T10) :|: TRUE f4324_out(x48, x49) -> f24_out(x48, x49) :|: TRUE f24_in(x50, x51) -> f4324_in(x50, x51) :|: TRUE f4324_in(x52, x53) -> f4326_in(x52, x53) :|: TRUE f4326_out(x54, x55) -> f4324_out(x54, x55) :|: TRUE f4325_out(T36) -> f4324_out([], T36) :|: TRUE f4324_in([], x56) -> f4325_in(x56) :|: TRUE f4326_in(T45, T48) -> f4397_in(T45, T48) :|: TRUE f4326_in(x57, x58) -> f4398_in :|: TRUE f4397_out(x59, x60) -> f4326_out(x59, x60) :|: TRUE f4398_out -> f4326_out(x61, x62) :|: TRUE f4400_out(x63, x64, x65) -> f4397_out(x64, x65) :|: TRUE f4397_in(x66, x67) -> f4399_in(x66, x67) :|: TRUE f4399_out(x68, x69) -> f4400_in(x70, x68, x69) :|: TRUE f4418_out(x71, x72, x73, x74) -> f4400_out(x72, x74, x73) :|: TRUE f4400_in(x75, x76, x77) -> f4417_in(x75, x76, x77) :|: TRUE f4417_out(x78, x79, x80) -> f4418_in(x81, x78, x80, x79) :|: TRUE f5096_out(x82, x83, x84, x85) -> f4418_out(x82, x83, x84, x85) :|: TRUE f4418_in(x86, x87, x88, x89) -> f5096_in(x86, x87, x88, x89) :|: TRUE f5107_out(x90, x91, x92, x93) -> f5096_out(x90, x91, x92, x93) :|: TRUE f5106_out(x94, x95, x96) -> f5096_out([], x94, x95, x96) :|: TRUE f5096_in([], x97, x98, x99) -> f5106_in(x97, x98, x99) :|: TRUE f5096_in(x100, x101, x102, x103) -> f5107_in(x100, x101, x102, x103) :|: TRUE f5121_out(x104, x105, x106, x107) -> f5107_out(x104, x105, x106, x107) :|: TRUE f5122_out -> f5107_out(x108, x109, x110, x111) :|: TRUE f5107_in(x112, x113, x114, x115) -> f5121_in(x112, x113, x114, x115) :|: TRUE f5107_in(x116, x117, x118, x119) -> f5122_in :|: TRUE f5123_out(x120, x121, x122) -> f5124_in(x121, x123, x120, x122, x124) :|: TRUE f5121_in(x125, x126, x127, x128) -> f5123_in(x125, x126, x127) :|: TRUE f5124_out(x129, x130, x131, x132, x133) -> f5121_out(x131, x129, x132, x133) :|: TRUE f5124_in(x134, x135, x136, x137, x138) -> f5132_in(x134, x135, x136, x137, x138) :|: TRUE f5133_out(x139, x140, x141, x142, x143, x144) -> f5124_out(x141, x140, x144, x142, x143) :|: TRUE f5132_out(x145, x146, x147, x148, x149) -> f5133_in(x150, x146, x145, x148, x149, x147) :|: TRUE f5132_in(x151, x152, x153, x154, x155) -> f5134_in(x151, x152, x153, x154, x155) :|: TRUE f5134_out(x156, x157, x158, x159, x160) -> f5132_out(x156, x157, x158, x159, x160) :|: TRUE f5134_in(x161, x162, x163, x164, x165) -> f5135_in(x161, x162, x163, x164, x165) :|: TRUE f5135_out(x166, x167, x168, x169, x170) -> f5134_out(x166, x167, x168, x169, x170) :|: TRUE f5135_in(x171, x172, x173, x174, x175) -> f5143_in(x171, x172, x174, x173, x175) :|: TRUE f5143_out(x176, x177, x178, x179, x180) -> f5135_out(x176, x177, x179, x178, x180) :|: TRUE f5145_out(x181, x182, x183, x184, x185) -> f5143_out(x184, x186, x185, x182, x187) :|: TRUE f5144_out(x188, x189, x190, x191, x192) -> f5145_in(x193, x192, x194, x188, x190) :|: TRUE f5143_in(x195, x196, x197, x198, x199) -> f5144_in(x195, x196, x197, x199, x198) :|: TRUE f5145_in(T648, T646, T649, T644, T647) -> f5153_in(T648, T646, T644, T647) :|: TRUE f5153_out(x200, x201, x202, x203) -> f5154_in(x204, x205) :|: TRUE f5154_out(x206, x207) -> f5145_out(x208, x209, x206, x210, x211) :|: TRUE f5156_out(x212, x213) -> f5154_out(x214, x213) :|: TRUE f5155_out(x215) -> f5156_in(x216, x217) :|: TRUE f5154_in(x218, x219) -> f5155_in(x218) :|: TRUE f4902_out(x220) -> f5155_out(x220) :|: TRUE f5155_in(x221) -> f4902_in(x221) :|: TRUE f4902_in(T121) -> f4904_in(T121) :|: TRUE f4904_out(x222) -> f4902_out(x222) :|: TRUE f4905_out(x223) -> f4904_out(x223) :|: TRUE f4906_out(x224) -> f4904_out(x224) :|: TRUE f4904_in(x225) -> f4906_in(x225) :|: TRUE f4904_in(x226) -> f4905_in(x226) :|: TRUE f4911_out -> f4906_out(x227) :|: TRUE f4910_out(T221, T220) -> f4906_out(.(T220, T221)) :|: TRUE f4906_in(.(x228, x229)) -> f4910_in(x229, x228) :|: TRUE f4906_in(x230) -> f4911_in :|: TRUE f4910_in(x231, x232) -> f4912_in(x231, x232) :|: TRUE f4912_out(x233, x234) -> f4910_out(x233, x234) :|: TRUE f4913_out(x235, x236) -> f4912_out(x235, x236) :|: TRUE f4914_out(x237, x238) -> f4912_out(x237, x238) :|: TRUE f4912_in(x239, x240) -> f4914_in(x239, x240) :|: TRUE f4912_in(x241, x242) -> f4913_in(x241, x242) :|: TRUE f4914_in(x243, x244) -> f4919_in :|: TRUE f4919_out -> f4914_out(x245, x246) :|: TRUE f4918_out(T236, T235, T237) -> f4914_out(.(T235, T236), T237) :|: TRUE f4914_in(.(x247, x248), x249) -> f4918_in(x248, x247, x249) :|: TRUE f4918_in(x250, x251, x252) -> f4920_in(x250, x251, x252) :|: TRUE f4920_out(x253, x254, x255) -> f4918_out(x253, x254, x255) :|: TRUE f4920_in(x256, x257, x258) -> f4922_in(x256, x257, x258) :|: TRUE f4920_in(x259, x260, x261) -> f4921_in(x259, x260, x261) :|: TRUE f4921_out(x262, x263, x264) -> f4920_out(x262, x263, x264) :|: TRUE f4922_out(x265, x266, x267) -> f4920_out(x265, x266, x267) :|: TRUE f4927_out -> f4922_out(x268, x269, x270) :|: TRUE f4922_in(x271, x272, x273) -> f4927_in :|: TRUE f4922_in(.(T260, T261), T262, T263) -> f4926_in(T261, T260, T262, T263) :|: TRUE f4926_out(x274, x275, x276, x277) -> f4922_out(.(x275, x274), x276, x277) :|: TRUE f4928_out(x278, x279, x280, x281) -> f4926_out(x278, x279, x280, x281) :|: TRUE f4926_in(x282, x283, x284, x285) -> f4928_in(x282, x283, x284, x285) :|: TRUE f4930_out(x286, x287, x288, x289) -> f4928_out(x286, x287, x288, x289) :|: TRUE f4928_in(x290, x291, x292, x293) -> f4929_in(x290, x291, x292, x293) :|: TRUE f4929_out(x294, x295, x296, x297) -> f4928_out(x294, x295, x296, x297) :|: TRUE f4928_in(x298, x299, x300, x301) -> f4930_in(x298, x299, x300, x301) :|: TRUE f4934_out(T296, T295, T297, T298, T299) -> f4930_out(.(T295, T296), T297, T298, T299) :|: TRUE f4930_in(x302, x303, x304, x305) -> f4935_in :|: TRUE f4935_out -> f4930_out(x306, x307, x308, x309) :|: TRUE f4930_in(.(x310, x311), x312, x313, x314) -> f4934_in(x311, x310, x312, x313, x314) :|: TRUE f4936_out(x315, x316, x317, x318, x319) -> f4934_out(x315, x316, x317, x318, x319) :|: TRUE f4934_in(x320, x321, x322, x323, x324) -> f4936_in(x320, x321, x322, x323, x324) :|: TRUE f4936_in(x325, x326, x327, x328, x329) -> f4937_in(x325, x326, x327, x328, x329) :|: TRUE f4936_in(x330, x331, x332, x333, x334) -> f4938_in(x330, x331, x332, x333, x334) :|: TRUE f4938_out(x335, x336, x337, x338, x339) -> f4936_out(x335, x336, x337, x338, x339) :|: TRUE f4937_out(x340, x341, x342, x343, x344) -> f4936_out(x340, x341, x342, x343, x344) :|: TRUE f4938_in(x345, x346, x347, x348, x349) -> f4943_in :|: TRUE f4938_in(.(T340, T341), T342, T343, T344, T345) -> f4942_in(T341, T340, T342, T343, T344, T345) :|: TRUE f4943_out -> f4938_out(x350, x351, x352, x353, x354) :|: TRUE f4942_out(x355, x356, x357, x358, x359, x360) -> f4938_out(.(x356, x355), x357, x358, x359, x360) :|: TRUE f4944_out(x361, x362, x363, x364, x365, x366) -> f4942_out(x361, x362, x363, x364, x365, x366) :|: TRUE f4942_in(x367, x368, x369, x370, x371, x372) -> f4944_in(x367, x368, x369, x370, x371, x372) :|: TRUE f4944_in(x373, x374, x375, x376, x377, x378) -> f4947_in(x373, x374, x375, x376, x377, x378) :|: TRUE f4944_in(x379, x380, x381, x382, x383, x384) -> f4948_in(x379, x380, x381, x382, x383, x384) :|: TRUE f4948_out(x385, x386, x387, x388, x389, x390) -> f4944_out(x385, x386, x387, x388, x389, x390) :|: TRUE f4947_out(x391, x392, x393, x394, x395, x396) -> f4944_out(x391, x392, x393, x394, x395, x396) :|: TRUE f4956_out -> f4948_out(x397, x398, x399, x400, x401, x402) :|: TRUE f4948_in(x403, x404, x405, x406, x407, x408) -> f4956_in :|: TRUE f4948_in(.(T395, T396), T397, T398, T399, T400, T401) -> f4955_in(T396, T395, T397, T398, T399, T400, T401) :|: TRUE f4955_out(x409, x410, x411, x412, x413, x414, x415) -> f4948_out(.(x410, x409), x411, x412, x413, x414, x415) :|: TRUE f4957_out(x416, x417, x418, x419, x420, x421, x422) -> f4955_out(x416, x417, x418, x419, x420, x421, x422) :|: TRUE f4955_in(x423, x424, x425, x426, x427, x428, x429) -> f4957_in(x423, x424, x425, x426, x427, x428, x429) :|: TRUE f4957_in(x430, x431, x432, x433, x434, x435, x436) -> f4958_in(x430, x431, x432, x433, x434, x435, x436) :|: TRUE f4958_out(x437, x438, x439, x440, x441, x442, x443) -> f4957_out(x437, x438, x439, x440, x441, x442, x443) :|: TRUE f4957_in(x444, x445, x446, x447, x448, x449, x450) -> f4959_in(x444, x445, x446, x447, x448, x449, x450) :|: TRUE f4959_out(x451, x452, x453, x454, x455, x456, x457) -> f4957_out(x451, x452, x453, x454, x455, x456, x457) :|: TRUE f4964_out -> f4959_out(x458, x459, x460, x461, x462, x463, x464) :|: TRUE f4959_in(x465, x466, x467, x468, x469, x470, x471) -> f4964_in :|: TRUE f4959_in(.(T460, T461), T462, T463, T464, T465, T466, T467) -> f4963_in(T461, T460, T462, T463, T464, T465, T466, T467) :|: TRUE f4963_out(x472, x473, x474, x475, x476, x477, x478, x479) -> f4959_out(.(x473, x472), x474, x475, x476, x477, x478, x479) :|: TRUE f4969_out(x480, x481, x482, x483, x484, x485, x486, x487) -> f4963_out(x480, x481, x482, x483, x484, x485, x486, x487) :|: TRUE f4963_in(x488, x489, x490, x491, x492, x493, x494, x495) -> f4969_in(x488, x489, x490, x491, x492, x493, x494, x495) :|: TRUE f4969_in(x496, x497, x498, x499, x500, x501, x502, x503) -> f5052_in(x496, x497, x498, x499, x500, x501, x502, x503) :|: TRUE f5052_out(x504, x505, x506, x507, x508, x509, x510, x511) -> f4969_out(x504, x505, x506, x507, x508, x509, x510, x511) :|: TRUE f5053_out(x512, x513, x514, x515, x516, x517, x518, x519) -> f4969_out(x512, x513, x514, x515, x516, x517, x518, x519) :|: TRUE f4969_in(x520, x521, x522, x523, x524, x525, x526, x527) -> f5053_in(x520, x521, x522, x523, x524, x525, x526, x527) :|: TRUE f5053_in(.(x528, x529), x530, x531, x532, x533, x534, x535, x536) -> f5057_in(x529, x528, x530, x531, x532, x533, x534, x535, x536) :|: TRUE f5053_in(x537, x538, x539, x540, x541, x542, x543, x544) -> f5058_in :|: TRUE f5057_out(x545, x546, x547, x548, x549, x550, x551, x552, x553) -> f5053_out(.(x546, x545), x547, x548, x549, x550, x551, x552, x553) :|: TRUE f5058_out -> f5053_out(x554, x555, x556, x557, x558, x559, x560, x561) :|: TRUE f5059_out(x562, x563, .(x564, .(x565, .(x566, .(x567, .(x568, .(x569, .(x570, [])))))))) -> f5057_out(x562, x563, x564, x565, x566, x567, x568, x569, x570) :|: TRUE f5057_in(x571, x572, x573, x574, x575, x576, x577, x578, x579) -> f5059_in(x571, x572, .(x573, .(x574, .(x575, .(x576, .(x577, .(x578, .(x579, [])))))))) :|: TRUE f5133_in(x580, x581, x582, x583, x584, x585) -> f5167_in(x580, x581, x582, x583, x584, x585) :|: TRUE f5167_out(x586, x587, x588, x589, x590, x591) -> f5133_out(x586, x587, x588, x589, x590, x591) :|: TRUE f5171_out(x592, x593, x594, x595, x596) -> f5167_out([], x592, x593, x594, x595, x596) :|: TRUE f5167_in([], x597, x598, x599, x600, x601) -> f5171_in(x597, x598, x599, x600, x601) :|: TRUE f5167_in(x602, x603, x604, x605, x606, x607) -> f5172_in(x602, x603, x604, x605, x606, x607) :|: TRUE f5172_out(x608, x609, x610, x611, x612, x613) -> f5167_out(x608, x609, x610, x611, x612, x613) :|: TRUE f5182_out(x614, x615, x616, x617, x618, x619) -> f5172_out(x614, x615, x616, x617, x618, x619) :|: TRUE f5172_in(x620, x621, x622, x623, x624, x625) -> f5183_in :|: TRUE f5183_out -> f5172_out(x626, x627, x628, x629, x630, x631) :|: TRUE f5172_in(x632, x633, x634, x635, x636, x637) -> f5182_in(x632, x633, x634, x635, x636, x637) :|: TRUE f5189_out(x638, x639, x640, x641) -> f5190_in(x639, x640, x642, x638, x641, x643, x644) :|: TRUE f5190_out(x645, x646, x647, x648, x649, x650, x651) -> f5182_out(x648, x645, x646, x649, x650, x651) :|: TRUE f5182_in(x652, x653, x654, x655, x656, x657) -> f5189_in(x652, x653, x654, x655) :|: TRUE f5195_out(x658, x659, x660, x661, x662, x663, x664, x665) -> f5190_out(x660, x661, x659, x665, x662, x663, x664) :|: TRUE f5190_in(x666, x667, x668, x669, x670, x671, x672) -> f5194_in(x666, x667, x668, x669, x670, x671, x672) :|: TRUE f5194_out(x673, x674, x675, x676, x677, x678, x679) -> f5195_in(x680, x675, x673, x674, x677, x678, x679, x676) :|: TRUE f5194_in(x681, x682, x683, x684, x685, x686, x687) -> f5196_in(x681, x682, x683, x684, x685, x686, x687) :|: TRUE f5196_out(x688, x689, x690, x691, x692, x693, x694) -> f5194_out(x688, x689, x690, x691, x692, x693, x694) :|: TRUE f5197_out(x695, x696, x697, x698, x699, x700, x701) -> f5196_out(x695, x696, x697, x698, x699, x700, x701) :|: TRUE f5196_in(x702, x703, x704, x705, x706, x707, x708) -> f5197_in(x702, x703, x704, x705, x706, x707, x708) :|: TRUE f5204_out(x709, x710, x711, x712, x713, x714, x715) -> f5197_out(x709, x710, x711, x713, x712, x714, x715) :|: TRUE f5197_in(x716, x717, x718, x719, x720, x721, x722) -> f5204_in(x716, x717, x718, x720, x719, x721, x722) :|: TRUE f5208_out(x723, x724, x725, x726, x727, x728) -> f5204_out(x726, x727, x729, x728, x724, x730, x731) :|: TRUE f5204_in(x732, x733, x734, x735, x736, x737, x738) -> f5207_in(x732, x733, x734, x735, x737, x738, x736) :|: TRUE f5207_out(x739, x740, x741, x742, x743, x744, x745) -> f5208_in(x746, x745, x747, x739, x740, x742) :|: TRUE f5245_out(T734, T751) -> f5208_out(T733, T731, T734, T728, T729, T732) :|: TRUE f5244_out(x748, x749, x750, x751, x752) -> f5245_in(x753, x754) :|: TRUE f5208_in(x755, x756, x757, x758, x759, x760) -> f5244_in(x755, x756, x758, x759, x760) :|: TRUE f5253_out(x761) -> f5254_in(x762, x763) :|: TRUE f5245_in(x764, x765) -> f5253_in(x764) :|: TRUE f5254_out(x766, x767) -> f5245_out(x768, x767) :|: TRUE f4902_out(x769) -> f5253_out(x769) :|: TRUE f5253_in(x770) -> f4902_in(x770) :|: TRUE f5195_in(x771, x772, x773, x774, x775, x776, x777, x778) -> f5288_in(x771, x772, x773, x774, x775, x776, x777, x778) :|: TRUE f5288_out(x779, x780, x781, x782, x783, x784, x785, x786) -> f5195_out(x779, x780, x781, x782, x783, x784, x785, x786) :|: TRUE f5288_in([], x787, x788, x789, x790, x791, x792, x793) -> f5294_in(x787, x788, x789, x790, x791, x792, x793) :|: TRUE f5294_out(x794, x795, x796, x797, x798, x799, x800) -> f5288_out([], x794, x795, x796, x797, x798, x799, x800) :|: TRUE f5298_out(x801, x802, x803, x804, x805, x806, x807, x808) -> f5288_out(x801, x802, x803, x804, x805, x806, x807, x808) :|: TRUE f5288_in(x809, x810, x811, x812, x813, x814, x815, x816) -> f5298_in(x809, x810, x811, x812, x813, x814, x815, x816) :|: TRUE f5298_in(x817, x818, x819, x820, x821, x822, x823, x824) -> f5314_in(x817, x818, x819, x820, x821, x822, x823, x824) :|: TRUE f5298_in(x825, x826, x827, x828, x829, x830, x831, x832) -> f5315_in :|: TRUE f5315_out -> f5298_out(x833, x834, x835, x836, x837, x838, x839, x840) :|: TRUE f5314_out(x841, x842, x843, x844, x845, x846, x847, x848) -> f5298_out(x841, x842, x843, x844, x845, x846, x847, x848) :|: TRUE f5336_out(x849, x850, x851, x852, x853, x854, x855, x856, x857) -> f5314_out(x853, x849, x850, x851, x854, x855, x856, x857) :|: TRUE f5314_in(x858, x859, x860, x861, x862, x863, x864, x865) -> f5333_in(x858, x859, x860, x861, x862) :|: TRUE f5333_out(x866, x867, x868, x869, x870) -> f5336_in(x867, x868, x869, x871, x866, x870, x872, x873, x874) :|: TRUE f5336_in(x875, x876, x877, x878, x879, x880, x881, x882, x883) -> f5340_in(x875, x876, x877, x878, x879, x880, x881, x882, x883) :|: TRUE f5341_out(x884, x885, x886, x887, x888, x889, x890, x891, x892, x893) -> f5336_out(x886, x887, x888, x885, x893, x889, x890, x891, x892) :|: TRUE f5340_out(x894, x895, x896, x897, x898, x899, x900, x901, x902) -> f5341_in(x903, x897, x894, x895, x896, x899, x900, x901, x902, x898) :|: TRUE f5408_out(x904, x905, x906, x907, x908, x909, x910, x911, x912, x913) -> f5341_out(x904, x905, x906, x907, x908, x909, x910, x911, x912, x913) :|: TRUE f5341_in(x914, x915, x916, x917, x918, x919, x920, x921, x922, x923) -> f5408_in(x914, x915, x916, x917, x918, x919, x920, x921, x922, x923) :|: TRUE f5419_out(x924, x925, x926, x927, x928, x929, x930, x931, x932, x933) -> f5408_out(x924, x925, x926, x927, x928, x929, x930, x931, x932, x933) :|: TRUE f5418_out(x934, x935, x936, x937, x938, x939, x940, x941, x942) -> f5408_out([], x934, x935, x936, x937, x938, x939, x940, x941, x942) :|: TRUE f5408_in([], x943, x944, x945, x946, x947, x948, x949, x950, x951) -> f5418_in(x943, x944, x945, x946, x947, x948, x949, x950, x951) :|: TRUE f5408_in(x952, x953, x954, x955, x956, x957, x958, x959, x960, x961) -> f5419_in(x952, x953, x954, x955, x956, x957, x958, x959, x960, x961) :|: TRUE f5422_out(x962, x963, x964, x965, x966, x967, x968, x969, x970, x971) -> f5419_out(x962, x963, x964, x965, x966, x967, x968, x969, x970, x971) :|: TRUE f5419_in(x972, x973, x974, x975, x976, x977, x978, x979, x980, x981) -> f5422_in(x972, x973, x974, x975, x976, x977, x978, x979, x980, x981) :|: TRUE f5419_in(x982, x983, x984, x985, x986, x987, x988, x989, x990, x991) -> f5423_in :|: TRUE f5423_out -> f5419_out(x992, x993, x994, x995, x996, x997, x998, x999, x1000, x1001) :|: TRUE f5422_in(x1002, x1003, x1004, x1005, x1006, x1007, x1008, x1009, x1010, x1011) -> f5424_in(x1002, x1003, x1004, x1005, x1006, x1007) :|: TRUE f5424_out(x1012, x1013, x1014, x1015, x1016, x1017) -> f5425_in(x1013, x1014, x1015, x1016, x1018, x1012, x1017, x1019, x1020, x1021, x1022) :|: TRUE f5425_out(x1023, x1024, x1025, x1026, x1027, x1028, x1029, x1030, x1031, x1032, x1033) -> f5422_out(x1028, x1023, x1024, x1025, x1026, x1029, x1030, x1031, x1032, x1033) :|: TRUE f5426_out(x1034, x1035, x1036, x1037, x1038, x1039, x1040, x1041, x1042, x1043, x1044) -> f5427_in(x1045, x1038, x1034, x1035, x1036, x1037, x1040, x1041, x1042, x1043, x1044, x1039) :|: TRUE f5427_out(x1046, x1047, x1048, x1049, x1050, x1051, x1052, x1053, x1054, x1055, x1056, x1057) -> f5425_out(x1048, x1049, x1050, x1051, x1047, x1057, x1052, x1053, x1054, x1055, x1056) :|: TRUE f5425_in(x1058, x1059, x1060, x1061, x1062, x1063, x1064, x1065, x1066, x1067, x1068) -> f5426_in(x1058, x1059, x1060, x1061, x1062, x1063, x1064, x1065, x1066, x1067, x1068) :|: TRUE f5427_in(x1069, x1070, x1071, x1072, x1073, x1074, x1075, x1076, x1077, x1078, x1079, x1080) -> f5539_in(x1069, x1070, x1071, x1072, x1073, x1074, x1075, x1076, x1077, x1078, x1079, x1080) :|: TRUE f5539_out(x1081, x1082, x1083, x1084, x1085, x1086, x1087, x1088, x1089, x1090, x1091, x1092) -> f5427_out(x1081, x1082, x1083, x1084, x1085, x1086, x1087, x1088, x1089, x1090, x1091, x1092) :|: TRUE f5563_out(x1093, x1094, x1095, x1096, x1097, x1098, x1099, x1100, x1101, x1102, x1103) -> f5539_out([], x1093, x1094, x1095, x1096, x1097, x1098, x1099, x1100, x1101, x1102, x1103) :|: TRUE f5539_in([], x1104, x1105, x1106, x1107, x1108, x1109, x1110, x1111, x1112, x1113, x1114) -> f5563_in(x1104, x1105, x1106, x1107, x1108, x1109, x1110, x1111, x1112, x1113, x1114) :|: TRUE f5539_in(x1115, x1116, x1117, x1118, x1119, x1120, x1121, x1122, x1123, x1124, x1125, x1126) -> f5564_in(x1115, x1116, x1117, x1118, x1119, x1120, x1121, x1122, x1123, x1124, x1125, x1126) :|: TRUE f5564_out(x1127, x1128, x1129, x1130, x1131, x1132, x1133, x1134, x1135, x1136, x1137, x1138) -> f5539_out(x1127, x1128, x1129, x1130, x1131, x1132, x1133, x1134, x1135, x1136, x1137, x1138) :|: TRUE f5586_out -> f5564_out(x1139, x1140, x1141, x1142, x1143, x1144, x1145, x1146, x1147, x1148, x1149, x1150) :|: TRUE f5564_in(x1151, x1152, x1153, x1154, x1155, x1156, x1157, x1158, x1159, x1160, x1161, x1162) -> f5581_in(x1151, x1152, x1153, x1154, x1155, x1156, x1157, x1158, x1159, x1160, x1161, x1162) :|: TRUE f5581_out(x1163, x1164, x1165, x1166, x1167, x1168, x1169, x1170, x1171, x1172, x1173, x1174) -> f5564_out(x1163, x1164, x1165, x1166, x1167, x1168, x1169, x1170, x1171, x1172, x1173, x1174) :|: TRUE f5564_in(x1175, x1176, x1177, x1178, x1179, x1180, x1181, x1182, x1183, x1184, x1185, x1186) -> f5586_in :|: TRUE f5591_out(x1187, x1188, x1189, x1190, x1191, x1192, x1193) -> f5592_in(x1188, x1189, x1190, x1191, x1192, x1194, x1187, x1193, x1195, x1196, x1197, x1198, x1199) :|: TRUE f5592_out(x1200, x1201, x1202, x1203, x1204, x1205, x1206, x1207, x1208, x1209, x1210, x1211, x1212) -> f5581_out(x1206, x1200, x1201, x1202, x1203, x1204, x1207, x1208, x1209, x1210, x1211, x1212) :|: TRUE f5581_in(x1213, x1214, x1215, x1216, x1217, x1218, x1219, x1220, x1221, x1222, x1223, x1224) -> f5591_in(x1213, x1214, x1215, x1216, x1217, x1218, x1219) :|: TRUE f5606_out(x1225, x1226, x1227, x1228, x1229, x1230, x1231, x1232, x1233, x1234, x1235, x1236, x1237) -> f5607_in(x1238, x1230, x1225, x1226, x1227, x1228, x1229, x1232, x1233, x1234, x1235, x1236, x1237, x1231) :|: TRUE f5607_out(x1239, x1240, x1241, x1242, x1243, x1244, x1245, x1246, x1247, x1248, x1249, x1250, x1251, x1252) -> f5592_out(x1241, x1242, x1243, x1244, x1245, x1240, x1252, x1246, x1247, x1248, x1249, x1250, x1251) :|: TRUE f5592_in(x1253, x1254, x1255, x1256, x1257, x1258, x1259, x1260, x1261, x1262, x1263, x1264, x1265) -> f5606_in(x1253, x1254, x1255, x1256, x1257, x1258, x1259, x1260, x1261, x1262, x1263, x1264, x1265) :|: TRUE f5691_out(x1266, x1267, x1268, x1269, x1270, x1271, x1272, x1273, x1274, x1275, x1276, x1277, x1278, x1279) -> f5607_out(x1266, x1267, x1268, x1269, x1270, x1271, x1272, x1273, x1274, x1275, x1276, x1277, x1278, x1279) :|: TRUE f5607_in(x1280, x1281, x1282, x1283, x1284, x1285, x1286, x1287, x1288, x1289, x1290, x1291, x1292, x1293) -> f5691_in(x1280, x1281, x1282, x1283, x1284, x1285, x1286, x1287, x1288, x1289, x1290, x1291, x1292, x1293) :|: TRUE f5716_out(x1294, x1295, x1296, x1297, x1298, x1299, x1300, x1301, x1302, x1303, x1304, x1305, x1306) -> f5691_out([], x1294, x1295, x1296, x1297, x1298, x1299, x1300, x1301, x1302, x1303, x1304, x1305, x1306) :|: TRUE f5719_out(x1307, x1308, x1309, x1310, x1311, x1312, x1313, x1314, x1315, x1316, x1317, x1318, x1319, x1320) -> f5691_out(x1307, x1308, x1309, x1310, x1311, x1312, x1313, x1314, x1315, x1316, x1317, x1318, x1319, x1320) :|: TRUE f5691_in([], x1321, x1322, x1323, x1324, x1325, x1326, x1327, x1328, x1329, x1330, x1331, x1332, x1333) -> f5716_in(x1321, x1322, x1323, x1324, x1325, x1326, x1327, x1328, x1329, x1330, x1331, x1332, x1333) :|: TRUE f5691_in(x1334, x1335, x1336, x1337, x1338, x1339, x1340, x1341, x1342, x1343, x1344, x1345, x1346, x1347) -> f5719_in(x1334, x1335, x1336, x1337, x1338, x1339, x1340, x1341, x1342, x1343, x1344, x1345, x1346, x1347) :|: TRUE f5719_in(x1348, x1349, x1350, x1351, x1352, x1353, x1354, x1355, x1356, x1357, x1358, x1359, x1360, x1361) -> f5733_in(x1348, x1349, x1350, x1351, x1352, x1353, x1354, x1355, x1356, x1357, x1358, x1359, x1360, x1361) :|: TRUE f5733_out(x1362, x1363, x1364, x1365, x1366, x1367, x1368, x1369, x1370, x1371, x1372, x1373, x1374, x1375) -> f5719_out(x1362, x1363, x1364, x1365, x1366, x1367, x1368, x1369, x1370, x1371, x1372, x1373, x1374, x1375) :|: TRUE f5734_out -> f5719_out(x1376, x1377, x1378, x1379, x1380, x1381, x1382, x1383, x1384, x1385, x1386, x1387, x1388, x1389) :|: TRUE f5719_in(x1390, x1391, x1392, x1393, x1394, x1395, x1396, x1397, x1398, x1399, x1400, x1401, x1402, x1403) -> f5734_in :|: TRUE f5736_out(x1404, x1405, x1406, x1407, x1408, x1409, x1410, x1411, .(x1405, .(x1406, .(x1407, .(x1408, .(x1409, .(x1410, .(0, []))))))), x1412, x1413, x1414, x1415, x1416, x1417) -> f5733_out(x1404, x1405, x1406, x1407, x1408, x1409, x1410, x1411, x1412, x1413, x1414, x1415, x1416, x1417) :|: TRUE f5733_in(x1418, x1419, x1420, x1421, x1422, x1423, x1424, x1425, x1426, x1427, x1428, x1429, x1430, x1431) -> f5736_in(x1418, x1419, x1420, x1421, x1422, x1423, x1424, x1425, .(x1419, .(x1420, .(x1421, .(x1422, .(x1423, .(x1424, .(0, []))))))), x1426, x1427, x1428, x1429, x1430, x1431) :|: TRUE f5745_out(x1432, x1433, x1434, x1435, x1436, x1437, x1438, x1439, x1440, x1441, x1442, x1443, x1444, x1445, x1446, x1447) -> f5736_out(x1439, x1432, x1433, x1434, x1435, x1436, x1437, x1440, x1441, x1442, x1443, x1444, x1445, x1446, x1447) :|: TRUE f5736_in(x1448, x1449, x1450, x1451, x1452, x1453, x1454, x1455, x1456, x1457, x1458, x1459, x1460, x1461, x1462) -> f5744_in(x1448, x1449, x1450, x1451, x1452, x1453, x1454, x1455) :|: TRUE f5744_out(x1463, x1464, x1465, x1466, x1467, x1468, x1469, x1470) -> f5745_in(x1464, x1465, x1466, x1467, x1468, x1469, x1471, x1463, x1470, x1472, x1473, x1474, x1475, x1476, x1477, x1478) :|: TRUE f5746_out(x1479, x1480, x1481, x1482, x1483, x1484, x1485, x1486, x1487, x1488, x1489, x1490, x1491, x1492, x1493) -> f5748_in(x1494, x1485, x1495, x1487, x1488, x1489, x1490, x1491, x1492, x1493, x1486, x1479, x1480, x1481, x1482, x1483, x1484) :|: TRUE f5745_in(x1496, x1497, x1498, x1499, x1500, x1501, x1502, x1503, x1504, x1505, x1506, x1507, x1508, x1509, x1510, x1511) -> f5746_in(x1496, x1497, x1498, x1499, x1500, x1501, x1502, x1503, x1504, x1506, x1507, x1508, x1509, x1510, x1511) :|: TRUE f5748_out(x1512, x1513, x1514, x1515, x1516, x1517, x1518, x1519, x1520, x1521, x1522, x1523, x1524, x1525, x1526, x1527, x1528) -> f5745_out(x1523, x1524, x1525, x1526, x1527, x1528, x1513, x1522, x1515, x1514, x1516, x1517, x1518, x1519, x1520, x1521) :|: TRUE f5750_out(x1529, x1530, x1531, x1532, x1533, x1534, x1535, x1536, x1537, x1538, x1539, x1540, x1541, x1542, x1543) -> f5746_out(x1529, x1530, x1531, x1532, x1533, x1534, x1535, x1536, x1537, x1538, x1539, x1540, x1541, x1542, x1543) :|: TRUE f5746_in(x1544, x1545, x1546, x1547, x1548, x1549, x1550, x1551, x1552, x1553, x1554, x1555, x1556, x1557, x1558) -> f5750_in(x1544, x1545, x1546, x1547, x1548, x1549, x1550, x1551, x1552, x1553, x1554, x1555, x1556, x1557, x1558) :|: TRUE f5750_in(x1559, x1560, x1561, x1562, x1563, x1564, x1565, x1566, x1567, x1568, x1569, x1570, x1571, x1572, x1573) -> f5751_in(x1559, x1560, x1561, x1562, x1563, x1564, x1565, x1566, x1567, x1568, x1569, x1570, x1571, x1572, x1573) :|: TRUE f5751_out(x1574, x1575, x1576, x1577, x1578, x1579, x1580, x1581, x1582, x1583, x1584, x1585, x1586, x1587, x1588) -> f5750_out(x1574, x1575, x1576, x1577, x1578, x1579, x1580, x1581, x1582, x1583, x1584, x1585, x1586, x1587, x1588) :|: TRUE f5751_in(x1589, x1590, x1591, x1592, x1593, x1594, x1595, x1596, x1597, x1598, x1599, x1600, x1601, x1602, x1603) -> f5762_in(x1589, x1590, x1591, x1592, x1593, x1594, x1595, x1597, x1596, x1598, x1599, x1600, x1601, x1602, x1603) :|: TRUE f5762_out(x1604, x1605, x1606, x1607, x1608, x1609, x1610, x1611, x1612, x1613, x1614, x1615, x1616, x1617, x1618) -> f5751_out(x1604, x1605, x1606, x1607, x1608, x1609, x1610, x1612, x1611, x1613, x1614, x1615, x1616, x1617, x1618) :|: TRUE f5762_in(x1619, x1620, x1621, x1622, x1623, x1624, x1625, x1626, x1627, x1628, x1629, x1630, x1631, x1632, x1633) -> f5765_in(x1619, x1620, x1621, x1622, x1623, x1624, x1625, x1626, x1628, x1629, x1630, x1631, x1632, x1633, x1627) :|: TRUE f5765_out(x1634, x1635, x1636, x1637, x1638, x1639, x1640, x1641, x1642, x1643, x1644, x1645, x1646, x1647, x1648) -> f5767_in(x1649, x1648, x1650, x1634, x1635, x1636, x1637, x1638, x1639, x1641) :|: TRUE f5767_out(x1651, x1652, x1653, x1654, x1655, x1656, x1657, x1658, x1659, x1660) -> f5762_out(x1654, x1655, x1656, x1657, x1658, x1659, x1661, x1660, x1652, x1662, x1663, x1664, x1665, x1666, x1667) :|: TRUE f5794_out(T1218, T1216, T1209, T1210, T1211, T1212, T1213, T1214, T1217) -> f5795_in(T1219, T1248) :|: TRUE f5767_in(x1668, x1669, x1670, x1671, x1672, x1673, x1674, x1675, x1676, x1677) -> f5794_in(x1668, x1669, x1671, x1672, x1673, x1674, x1675, x1676, x1677) :|: TRUE f5795_out(x1678, x1679) -> f5767_out(x1680, x1681, x1678, x1682, x1683, x1684, x1685, x1686, x1687, x1688) :|: TRUE f5795_in(x1689, x1690) -> f5798_in(x1689) :|: TRUE f5799_out(x1691, x1692) -> f5795_out(x1693, x1692) :|: TRUE f5798_out(x1694) -> f5799_in(x1695, x1696) :|: TRUE f4902_out(x1697) -> f5798_out(x1697) :|: TRUE f5798_in(x1698) -> f4902_in(x1698) :|: TRUE f5748_in(x1699, x1700, x1701, x1702, x1703, x1704, x1705, x1706, x1707, x1708, x1709, x1710, x1711, x1712, x1713, x1714, x1715) -> f5830_in(x1699, x1700, x1701, x1702, x1703, x1704, x1705, x1706, x1707, x1708, x1709, x1710, x1711, x1712, x1713, x1714, x1715) :|: TRUE f5830_out(x1716, x1717, x1718, x1719, x1720, x1721, x1722, x1723, x1724, x1725, x1726, x1727, x1728, x1729, x1730, x1731, x1732) -> f5748_out(x1716, x1717, x1718, x1719, x1720, x1721, x1722, x1723, x1724, x1725, x1726, x1727, x1728, x1729, x1730, x1731, x1732) :|: TRUE f5832_out(x1733, x1734, x1735, x1736, x1737, x1738, x1739, x1740, x1741, x1742, x1743, x1744, x1745, x1746, x1747, x1748) -> f5830_out([], x1733, x1734, x1735, x1736, x1737, x1738, x1739, x1740, x1741, x1742, x1743, x1744, x1745, x1746, x1747, x1748) :|: TRUE f5833_out(x1749, x1750, x1751, x1752, x1753, x1754, x1755, x1756, x1757, x1758, x1759, x1760, x1761, x1762, x1763, x1764, x1765) -> f5830_out(x1749, x1750, x1751, x1752, x1753, x1754, x1755, x1756, x1757, x1758, x1759, x1760, x1761, x1762, x1763, x1764, x1765) :|: TRUE f5830_in(x1766, x1767, x1768, x1769, x1770, x1771, x1772, x1773, x1774, x1775, x1776, x1777, x1778, x1779, x1780, x1781, x1782) -> f5833_in(x1766, x1767, x1768, x1769, x1770, x1771, x1772, x1773, x1774, x1775, x1776, x1777, x1778, x1779, x1780, x1781, x1782) :|: TRUE f5830_in([], x1783, x1784, x1785, x1786, x1787, x1788, x1789, x1790, x1791, x1792, x1793, x1794, x1795, x1796, x1797, x1798) -> f5832_in(x1783, x1784, x1785, x1786, x1787, x1788, x1789, x1790, x1791, x1792, x1793, x1794, x1795, x1796, x1797, x1798) :|: TRUE f5846_out(x1799, x1800, x1801, x1802, x1803, x1804, x1805, x1806, x1807, x1808, x1809, x1810, x1811, x1812, x1813, x1814, x1815) -> f5833_out(x1799, x1800, x1801, x1802, x1803, x1804, x1805, x1806, x1807, x1808, x1809, x1810, x1811, x1812, x1813, x1814, x1815) :|: TRUE f5833_in(x1816, x1817, x1818, x1819, x1820, x1821, x1822, x1823, x1824, x1825, x1826, x1827, x1828, x1829, x1830, x1831, x1832) -> f5846_in(x1816, x1817, x1818, x1819, x1820, x1821, x1822, x1823, x1824, x1825, x1826, x1827, x1828, x1829, x1830, x1831, x1832) :|: TRUE f5833_in(x1833, x1834, x1835, x1836, x1837, x1838, x1839, x1840, x1841, x1842, x1843, x1844, x1845, x1846, x1847, x1848, x1849) -> f5847_in :|: TRUE f5847_out -> f5833_out(x1850, x1851, x1852, x1853, x1854, x1855, x1856, x1857, x1858, x1859, x1860, x1861, x1862, x1863, x1864, x1865, x1866) :|: TRUE f5846_in(x1867, x1868, x1869, x1870, x1871, x1872, x1873, x1874, x1875, x1876, x1877, x1878, x1879, x1880, x1881, x1882, x1883) -> f5851_in(x1867, x1868, x1869, x1870) :|: TRUE f5851_out(x1884, x1885, x1886, x1887) -> f5852_in(x1885, x1886, x1888, x1884, x1887, x1889, x1890, x1891, x1892, x1893, x1894, x1895, x1896, x1897, x1898, x1899, x1900, x1901) :|: TRUE f5852_out(x1902, x1903, x1904, x1905, x1906, x1907, x1908, x1909, x1910, x1911, x1912, x1913, x1914, x1915, x1916, x1917, x1918, x1919) -> f5846_out(x1905, x1902, x1903, x1906, x1907, x1908, x1909, x1910, x1911, x1912, x1913, x1914, x1915, x1916, x1917, x1918, x1919) :|: TRUE f5862_out(x1920, x1921, x1922, x1923, x1924, x1925, x1926, x1927, x1928, x1929, x1930, x1931, x1932, x1933, x1934, x1935, x1936, x1937, x1938) -> f5852_out(x1922, x1923, x1921, x1938, x1924, x1925, x1926, x1927, x1928, x1929, x1930, x1931, x1932, x1933, x1934, x1935, x1936, x1937) :|: TRUE f5861_out(x1939, x1940, x1941, x1942, x1943, x1944, x1945, x1946, x1947, x1948, x1949, x1950, x1951, x1952, x1953, x1954, x1955, x1956) -> f5862_in(x1957, x1941, x1939, x1940, x1943, x1944, x1945, x1946, x1947, x1948, x1949, x1950, x1951, x1952, x1953, x1954, x1955, x1956, x1942) :|: TRUE f5852_in(x1958, x1959, x1960, x1961, x1962, x1963, x1964, x1965, x1966, x1967, x1968, x1969, x1970, x1971, x1972, x1973, x1974, x1975) -> f5861_in(x1958, x1959, x1960, x1961, x1962, x1963, x1964, x1965, x1966, x1967, x1968, x1969, x1970, x1971, x1972, x1973, x1974, x1975) :|: TRUE f5861_in(x1976, x1977, x1978, x1979, x1980, x1981, x1982, x1983, x1984, x1985, x1986, x1987, x1988, x1989, x1990, x1991, x1992, x1993) -> f5863_in(x1976, x1977, x1978, x1979, x1980, x1981, x1982, x1983, x1984, x1985, x1986, x1987, x1988, x1989, x1990, x1991, x1992, x1993) :|: TRUE f5863_out(x1994, x1995, x1996, x1997, x1998, x1999, x2000, x2001, x2002, x2003, x2004, x2005, x2006, x2007, x2008, x2009, x2010, x2011) -> f5861_out(x1994, x1995, x1996, x1997, x1998, x1999, x2000, x2001, x2002, x2003, x2004, x2005, x2006, x2007, x2008, x2009, x2010, x2011) :|: TRUE f5865_out(x2012, x2013, x2014, x2015, x2016, x2017, x2018, x2019, x2020, x2021, x2022, x2023, x2024, x2025, x2026, x2027, x2028, x2029) -> f5863_out(x2012, x2013, x2014, x2015, x2016, x2017, x2018, x2019, x2020, x2021, x2022, x2023, x2024, x2025, x2026, x2027, x2028, x2029) :|: TRUE f5863_in(x2030, x2031, x2032, x2033, x2034, x2035, x2036, x2037, x2038, x2039, x2040, x2041, x2042, x2043, x2044, x2045, x2046, x2047) -> f5865_in(x2030, x2031, x2032, x2033, x2034, x2035, x2036, x2037, x2038, x2039, x2040, x2041, x2042, x2043, x2044, x2045, x2046, x2047) :|: TRUE f5865_in(x2048, x2049, x2050, x2051, x2052, x2053, x2054, x2055, x2056, x2057, x2058, x2059, x2060, x2061, x2062, x2063, x2064, x2065) -> f5876_in(x2048, x2049, x2050, x2052, x2051, x2053, x2054, x2055, x2056, x2057, x2058, x2059, x2060, x2061, x2062, x2063, x2064, x2065) :|: TRUE f5876_out(x2066, x2067, x2068, x2069, x2070, x2071, x2072, x2073, x2074, x2075, x2076, x2077, x2078, x2079, x2080, x2081, x2082, x2083) -> f5865_out(x2066, x2067, x2068, x2070, x2069, x2071, x2072, x2073, x2074, x2075, x2076, x2077, x2078, x2079, x2080, x2081, x2082, x2083) :|: TRUE f5878_out(x2084, x2085, x2086, x2087, x2088, x2089) -> f5876_out(x2087, x2088, x2090, x2089, x2085, x2091, x2092, x2093, x2094, x2095, x2096, x2097, x2098, x2099, x2100, x2101, x2102, x2103) :|: TRUE f5876_in(x2104, x2105, x2106, x2107, x2108, x2109, x2110, x2111, x2112, x2113, x2114, x2115, x2116, x2117, x2118, x2119, x2120, x2121) -> f5877_in(x2104, x2105, x2106, x2107, x2109, x2110, x2111, x2112, x2113, x2114, x2115, x2116, x2117, x2118, x2119, x2120, x2121, x2108) :|: TRUE f5877_out(x2122, x2123, x2124, x2125, x2126, x2127, x2128, x2129, x2130, x2131, x2132, x2133, x2134, x2135, x2136, x2137, x2138, x2139) -> f5878_in(x2140, x2139, x2141, x2122, x2123, x2125) :|: TRUE f6105_out(T1324, T1574) -> f5878_out(T1323, T1321, T1324, T1318, T1319, T1322) :|: TRUE f6104_out(x2142, x2143, x2144, x2145, x2146) -> f6105_in(x2147, x2148) :|: TRUE f5878_in(x2149, x2150, x2151, x2152, x2153, x2154) -> f6104_in(x2149, x2150, x2152, x2153, x2154) :|: TRUE f6107_out(x2155, x2156) -> f6105_out(x2157, x2156) :|: TRUE f6105_in(x2158, x2159) -> f6106_in(x2158) :|: TRUE f6106_out(x2160) -> f6107_in(x2161, x2162) :|: TRUE f4902_out(x2163) -> f6106_out(x2163) :|: TRUE f6106_in(x2164) -> f4902_in(x2164) :|: TRUE f4417_in(x2165, x2166, x2167) -> f4419_in(x2165, x2166, x2167) :|: TRUE f4419_out(x2168, x2169, x2170) -> f4417_out(x2168, x2169, x2170) :|: TRUE f4419_in(x2171, x2172, x2173) -> f4420_in(x2171, x2172, x2173) :|: TRUE f4420_out(x2174, x2175, x2176) -> f4419_out(x2174, x2175, x2176) :|: TRUE f4420_in(T117, T118, T119) -> f4421_in(T117, T119, T118) :|: TRUE f4421_out(x2177, x2178, x2179) -> f4420_out(x2177, x2179, x2178) :|: TRUE f4423_out(x2180, x2181, x2182, x2183) -> f4421_out(x2184, x2183, x2181) :|: TRUE f4421_in(x2185, x2186, x2187) -> f4422_in(x2185, x2186, x2187) :|: TRUE f4422_out(x2188, x2189, x2190) -> f4423_in(x2191, x2190, x2192, x2189) :|: TRUE f4438_out(x2193, x2194) -> f4423_out(x2195, x2196, x2193, x2197) :|: TRUE f4437_out(x2198, x2199, x2200) -> f4438_in(x2201, x2202) :|: TRUE f4423_in(x2203, x2204, x2205, x2206) -> f4437_in(x2203, x2204, x2206) :|: TRUE f4438_in(x2207, x2208) -> f4902_in(x2207) :|: TRUE f4902_out(x2209) -> f4903_in(x2210, x2211) :|: TRUE f4903_out(x2212, x2213) -> f4438_out(x2214, x2213) :|: TRUE f5348_out(x2215, x2216, x2217, x2218, x2219, x2220, x2221, x2222, x2223) -> f5340_out(x2215, x2216, x2217, x2218, x2219, x2220, x2221, x2222, x2223) :|: TRUE f5340_in(x2224, x2225, x2226, x2227, x2228, x2229, x2230, x2231, x2232) -> f5348_in(x2224, x2225, x2226, x2227, x2228, x2229, x2230, x2231, x2232) :|: TRUE f5351_out(x2233, x2234, x2235, x2236, x2237, x2238, x2239, x2240, x2241) -> f5348_out(x2233, x2234, x2235, x2236, x2237, x2238, x2239, x2240, x2241) :|: TRUE f5348_in(x2242, x2243, x2244, x2245, x2246, x2247, x2248, x2249, x2250) -> f5351_in(x2242, x2243, x2244, x2245, x2246, x2247, x2248, x2249, x2250) :|: TRUE f5363_out(x2251, x2252, x2253, x2254, x2255, x2256, x2257, x2258, x2259) -> f5351_out(x2251, x2252, x2253, x2254, x2256, x2255, x2257, x2258, x2259) :|: TRUE f5351_in(x2260, x2261, x2262, x2263, x2264, x2265, x2266, x2267, x2268) -> f5363_in(x2260, x2261, x2262, x2263, x2265, x2264, x2266, x2267, x2268) :|: TRUE f5363_in(x2269, x2270, x2271, x2272, x2273, x2274, x2275, x2276, x2277) -> f5368_in(x2269, x2270, x2271, x2272, x2273, x2275, x2276, x2277, x2274) :|: TRUE f5368_out(x2278, x2279, x2280, x2281, x2282, x2283, x2284, x2285, x2286) -> f5369_in(x2287, x2286, x2288, x2278, x2279, x2280, x2282) :|: TRUE f5369_out(x2289, x2290, x2291, x2292, x2293, x2294, x2295) -> f5363_out(x2292, x2293, x2294, x2296, x2295, x2290, x2297, x2298, x2299) :|: TRUE f5390_out(T833, T853) -> f5369_out(T832, T830, T833, T826, T827, T828, T831) :|: TRUE f5389_out(x2300, x2301, x2302, x2303, x2304, x2305) -> f5390_in(x2306, x2307) :|: TRUE f5369_in(x2308, x2309, x2310, x2311, x2312, x2313, x2314) -> f5389_in(x2308, x2309, x2311, x2312, x2313, x2314) :|: TRUE f5395_out(x2315) -> f5396_in(x2316, x2317) :|: TRUE f5390_in(x2318, x2319) -> f5395_in(x2318) :|: TRUE f5396_out(x2320, x2321) -> f5390_out(x2322, x2321) :|: TRUE f4902_out(x2323) -> f5395_out(x2323) :|: TRUE f5395_in(x2324) -> f4902_in(x2324) :|: TRUE f5426_in(x2325, x2326, x2327, x2328, x2329, x2330, x2331, x2332, x2333, x2334, x2335) -> f5428_in(x2325, x2326, x2327, x2328, x2329, x2330, x2331, x2332, x2333, x2334, x2335) :|: TRUE f5428_out(x2336, x2337, x2338, x2339, x2340, x2341, x2342, x2343, x2344, x2345, x2346) -> f5426_out(x2336, x2337, x2338, x2339, x2340, x2341, x2342, x2343, x2344, x2345, x2346) :|: TRUE f5428_in(x2347, x2348, x2349, x2350, x2351, x2352, x2353, x2354, x2355, x2356, x2357) -> f5429_in(x2347, x2348, x2349, x2350, x2351, x2352, x2353, x2354, x2355, x2356, x2357) :|: TRUE f5429_out(x2358, x2359, x2360, x2361, x2362, x2363, x2364, x2365, x2366, x2367, x2368) -> f5428_out(x2358, x2359, x2360, x2361, x2362, x2363, x2364, x2365, x2366, x2367, x2368) :|: TRUE f5472_out(x2369, x2370, x2371, x2372, x2373, x2374, x2375, x2376, x2377, x2378, x2379) -> f5429_out(x2369, x2370, x2371, x2372, x2373, x2375, x2374, x2376, x2377, x2378, x2379) :|: TRUE f5429_in(x2380, x2381, x2382, x2383, x2384, x2385, x2386, x2387, x2388, x2389, x2390) -> f5472_in(x2380, x2381, x2382, x2383, x2384, x2386, x2385, x2387, x2388, x2389, x2390) :|: TRUE f5473_out(x2391, x2392, x2393, x2394, x2395, x2396, x2397, x2398, x2399, x2400, x2401) -> f5474_in(x2402, x2401, x2403, x2391, x2392, x2393, x2394, x2396) :|: TRUE f5474_out(x2404, x2405, x2406, x2407, x2408, x2409, x2410, x2411) -> f5472_out(x2407, x2408, x2409, x2410, x2412, x2411, x2405, x2413, x2414, x2415, x2416) :|: TRUE f5472_in(x2417, x2418, x2419, x2420, x2421, x2422, x2423, x2424, x2425, x2426, x2427) -> f5473_in(x2417, x2418, x2419, x2420, x2421, x2422, x2424, x2425, x2426, x2427, x2423) :|: TRUE f5474_in(T945, T943, T946, T938, T939, T940, T941, T944) -> f5513_in(T945, T943, T938, T939, T940, T941, T944) :|: TRUE f5514_out(x2428, x2429) -> f5474_out(x2430, x2431, x2428, x2432, x2433, x2434, x2435, x2436) :|: TRUE f5513_out(x2437, x2438, x2439, x2440, x2441, x2442, x2443) -> f5514_in(x2444, x2445) :|: TRUE f5532_out(x2446) -> f5533_in(x2447, x2448) :|: TRUE f5514_in(x2449, x2450) -> f5532_in(x2449) :|: TRUE f5533_out(x2451, x2452) -> f5514_out(x2453, x2452) :|: TRUE f4902_out(x2454) -> f5532_out(x2454) :|: TRUE f5532_in(x2455) -> f4902_in(x2455) :|: TRUE f5608_out(x2456, x2457, x2458, x2459, x2460, x2461, x2462, x2463, x2464, x2465, x2466, x2467, x2468) -> f5606_out(x2456, x2457, x2458, x2459, x2460, x2461, x2462, x2463, x2464, x2465, x2466, x2467, x2468) :|: TRUE f5606_in(x2469, x2470, x2471, x2472, x2473, x2474, x2475, x2476, x2477, x2478, x2479, x2480, x2481) -> f5608_in(x2469, x2470, x2471, x2472, x2473, x2474, x2475, x2476, x2477, x2478, x2479, x2480, x2481) :|: TRUE f5609_out(x2482, x2483, x2484, x2485, x2486, x2487, x2488, x2489, x2490, x2491, x2492, x2493, x2494) -> f5608_out(x2482, x2483, x2484, x2485, x2486, x2487, x2488, x2489, x2490, x2491, x2492, x2493, x2494) :|: TRUE f5608_in(x2495, x2496, x2497, x2498, x2499, x2500, x2501, x2502, x2503, x2504, x2505, x2506, x2507) -> f5609_in(x2495, x2496, x2497, x2498, x2499, x2500, x2501, x2502, x2503, x2504, x2505, x2506, x2507) :|: TRUE f5609_in(x2508, x2509, x2510, x2511, x2512, x2513, x2514, x2515, x2516, x2517, x2518, x2519, x2520) -> f5642_in(x2508, x2509, x2510, x2511, x2512, x2513, x2515, x2514, x2516, x2517, x2518, x2519, x2520) :|: TRUE f5642_out(x2521, x2522, x2523, x2524, x2525, x2526, x2527, x2528, x2529, x2530, x2531, x2532, x2533) -> f5609_out(x2521, x2522, x2523, x2524, x2525, x2526, x2528, x2527, x2529, x2530, x2531, x2532, x2533) :|: TRUE f5642_in(x2534, x2535, x2536, x2537, x2538, x2539, x2540, x2541, x2542, x2543, x2544, x2545, x2546) -> f5647_in(x2534, x2535, x2536, x2537, x2538, x2539, x2540, x2542, x2543, x2544, x2545, x2546, x2541) :|: TRUE f5648_out(x2547, x2548, x2549, x2550, x2551, x2552, x2553, x2554, x2555) -> f5642_out(x2550, x2551, x2552, x2553, x2554, x2556, x2555, x2548, x2557, x2558, x2559, x2560, x2561) :|: TRUE f5647_out(x2562, x2563, x2564, x2565, x2566, x2567, x2568, x2569, x2570, x2571, x2572, x2573, x2574) -> f5648_in(x2575, x2574, x2576, x2562, x2563, x2564, x2565, x2566, x2568) :|: TRUE f5676_out(T1073, T1099) -> f5648_out(T1072, T1070, T1073, T1064, T1065, T1066, T1067, T1068, T1071) :|: TRUE f5648_in(x2577, x2578, x2579, x2580, x2581, x2582, x2583, x2584, x2585) -> f5675_in(x2577, x2578, x2580, x2581, x2582, x2583, x2584, x2585) :|: TRUE f5675_out(x2586, x2587, x2588, x2589, x2590, x2591, x2592, x2593) -> f5676_in(x2594, x2595) :|: TRUE f5679_out(x2596) -> f5680_in(x2597, x2598) :|: TRUE f5676_in(x2599, x2600) -> f5679_in(x2599) :|: TRUE f5680_out(x2601, x2602) -> f5676_out(x2603, x2602) :|: TRUE f4902_out(x2604) -> f5679_out(x2604) :|: TRUE f5679_in(x2605) -> f4902_in(x2605) :|: TRUE Start term: f1_in(T1) ---------------------------------------- (409) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (410) TRUE ---------------------------------------- (411) Obligation: Rules: f4456_in(.(T199, T200), .(T201, T202)) -> f4666_in(T199, T201, T200, T202) :|: TRUE f4666_out(x, x1, x2, x3) -> f4456_out(.(x, x2), .(x1, x3)) :|: TRUE f4456_in(T165, T166) -> f4667_in :|: TRUE f4667_out -> f4456_out(x4, x5) :|: TRUE f4666_in(x6, x7, x8, x9) -> f4837_in(x6, x8, x9, x7) :|: x6 > x7 f4838_out(x10, x11, x12, x13) -> f4666_out(x11, x13, x12, x10) :|: x11 <= x13 f4668_out -> f4666_out(x14, x15, x16, x17) :|: TRUE f4666_in(x18, x19, x20, x21) -> f4668_in :|: TRUE f4837_out(x22, x23, x24, x25) -> f4666_out(x22, x25, x23, x24) :|: x22 > x25 f4666_in(x26, x27, x28, x29) -> f4838_in(x29, x26, x28, x27) :|: x26 <= x27 f4455_in(x30, x31) -> f4526_in :|: TRUE f4455_in(.(T186, T187), .(T186, T188)) -> f4525_in(T187, T188) :|: TRUE f4525_out(x32, x33) -> f4455_out(.(x34, x32), .(x34, x33)) :|: TRUE f4526_out -> f4455_out(x35, x36) :|: TRUE f4451_in(x37, x38) -> f4456_in(x37, x38) :|: TRUE f4455_out(x39, x40) -> f4451_out(x39, x40) :|: TRUE f4451_in(x41, x42) -> f4455_in(x41, x42) :|: TRUE f4456_out(x43, x44) -> f4451_out(x43, x44) :|: TRUE f4447_out(x45, x46) -> f4525_out(x45, x46) :|: TRUE f4525_in(x47, x48) -> f4447_in(x47, x48) :|: TRUE f4451_out(x49, x50) -> f4449_out(x49, x50) :|: TRUE f4449_in(x51, x52) -> f4450_in(x51, x52) :|: TRUE f4450_out(x53, x54) -> f4449_out(x53, x54) :|: TRUE f4449_in(x55, x56) -> f4451_in(x55, x56) :|: TRUE f4449_out(x57, x58) -> f4447_out(x57, x58) :|: TRUE f4447_in(x59, x60) -> f4449_in(x59, x60) :|: TRUE f4447_out(.(x61, x62), x63) -> f4837_out(x61, x62, x63, x64) :|: TRUE f4837_in(x65, x66, x67, x68) -> f4447_in(.(x65, x66), x67) :|: TRUE f4_out(T1) -> f1_out(T1) :|: TRUE f1_in(x69) -> f4_in(x69) :|: TRUE f4_in(T7) -> f10_in(T7) :|: TRUE f11_out -> f4_out(x70) :|: TRUE f10_out(x71) -> f4_out(x71) :|: TRUE f4_in(x72) -> f11_in :|: TRUE f10_in(x73) -> f16_in(x74, x73) :|: x74 = x73 * (x73 + 1) + 1 f10_in(x75) -> f12_in :|: TRUE f16_out(x76, x77) -> f10_out(x77) :|: TRUE f12_out -> f10_out(x78) :|: TRUE f16_in(x79, x80) -> f21_in(x79) :|: TRUE f24_out(x81, x82) -> f16_out(x82, x83) :|: TRUE f21_out(T10) -> f24_in(T12, T10) :|: TRUE f4324_out(x84, x85) -> f24_out(x84, x85) :|: TRUE f24_in(x86, x87) -> f4324_in(x86, x87) :|: TRUE f4324_in(x88, x89) -> f4326_in(x88, x89) :|: TRUE f4326_out(x90, x91) -> f4324_out(x90, x91) :|: TRUE f4325_out(T36) -> f4324_out([], T36) :|: TRUE f4324_in([], x92) -> f4325_in(x92) :|: TRUE f4326_in(T45, T48) -> f4397_in(T45, T48) :|: TRUE f4326_in(x93, x94) -> f4398_in :|: TRUE f4397_out(x95, x96) -> f4326_out(x95, x96) :|: TRUE f4398_out -> f4326_out(x97, x98) :|: TRUE f4400_out(x99, x100, x101) -> f4397_out(x100, x101) :|: TRUE f4397_in(x102, x103) -> f4399_in(x102, x103) :|: TRUE f4399_out(x104, x105) -> f4400_in(x106, x104, x105) :|: TRUE f4418_out(x107, x108, x109, x110) -> f4400_out(x108, x110, x109) :|: TRUE f4400_in(x111, x112, x113) -> f4417_in(x111, x112, x113) :|: TRUE f4417_out(x114, x115, x116) -> f4418_in(x117, x114, x116, x115) :|: TRUE f5096_out(x118, x119, x120, x121) -> f4418_out(x118, x119, x120, x121) :|: TRUE f4418_in(x122, x123, x124, x125) -> f5096_in(x122, x123, x124, x125) :|: TRUE f5107_out(x126, x127, x128, x129) -> f5096_out(x126, x127, x128, x129) :|: TRUE f5106_out(x130, x131, x132) -> f5096_out([], x130, x131, x132) :|: TRUE f5096_in([], x133, x134, x135) -> f5106_in(x133, x134, x135) :|: TRUE f5096_in(x136, x137, x138, x139) -> f5107_in(x136, x137, x138, x139) :|: TRUE f5121_out(x140, x141, x142, x143) -> f5107_out(x140, x141, x142, x143) :|: TRUE f5122_out -> f5107_out(x144, x145, x146, x147) :|: TRUE f5107_in(x148, x149, x150, x151) -> f5121_in(x148, x149, x150, x151) :|: TRUE f5107_in(x152, x153, x154, x155) -> f5122_in :|: TRUE f5123_out(x156, x157, x158) -> f5124_in(x157, x159, x156, x158, x160) :|: TRUE f5121_in(x161, x162, x163, x164) -> f5123_in(x161, x162, x163) :|: TRUE f5124_out(x165, x166, x167, x168, x169) -> f5121_out(x167, x165, x168, x169) :|: TRUE f5124_in(x170, x171, x172, x173, x174) -> f5132_in(x170, x171, x172, x173, x174) :|: TRUE f5133_out(x175, x176, x177, x178, x179, x180) -> f5124_out(x177, x176, x180, x178, x179) :|: TRUE f5132_out(x181, x182, x183, x184, x185) -> f5133_in(x186, x182, x181, x184, x185, x183) :|: TRUE f5133_in(x187, x188, x189, x190, x191, x192) -> f5167_in(x187, x188, x189, x190, x191, x192) :|: TRUE f5167_out(x193, x194, x195, x196, x197, x198) -> f5133_out(x193, x194, x195, x196, x197, x198) :|: TRUE f5171_out(x199, x200, x201, x202, x203) -> f5167_out([], x199, x200, x201, x202, x203) :|: TRUE f5167_in([], x204, x205, x206, x207, x208) -> f5171_in(x204, x205, x206, x207, x208) :|: TRUE f5167_in(x209, x210, x211, x212, x213, x214) -> f5172_in(x209, x210, x211, x212, x213, x214) :|: TRUE f5172_out(x215, x216, x217, x218, x219, x220) -> f5167_out(x215, x216, x217, x218, x219, x220) :|: TRUE f5182_out(x221, x222, x223, x224, x225, x226) -> f5172_out(x221, x222, x223, x224, x225, x226) :|: TRUE f5172_in(x227, x228, x229, x230, x231, x232) -> f5183_in :|: TRUE f5183_out -> f5172_out(x233, x234, x235, x236, x237, x238) :|: TRUE f5172_in(x239, x240, x241, x242, x243, x244) -> f5182_in(x239, x240, x241, x242, x243, x244) :|: TRUE f5189_out(x245, x246, x247, x248) -> f5190_in(x246, x247, x249, x245, x248, x250, x251) :|: TRUE f5190_out(x252, x253, x254, x255, x256, x257, x258) -> f5182_out(x255, x252, x253, x256, x257, x258) :|: TRUE f5182_in(x259, x260, x261, x262, x263, x264) -> f5189_in(x259, x260, x261, x262) :|: TRUE f5195_out(x265, x266, x267, x268, x269, x270, x271, x272) -> f5190_out(x267, x268, x266, x272, x269, x270, x271) :|: TRUE f5190_in(x273, x274, x275, x276, x277, x278, x279) -> f5194_in(x273, x274, x275, x276, x277, x278, x279) :|: TRUE f5194_out(x280, x281, x282, x283, x284, x285, x286) -> f5195_in(x287, x282, x280, x281, x284, x285, x286, x283) :|: TRUE f5195_in(x288, x289, x290, x291, x292, x293, x294, x295) -> f5288_in(x288, x289, x290, x291, x292, x293, x294, x295) :|: TRUE f5288_out(x296, x297, x298, x299, x300, x301, x302, x303) -> f5195_out(x296, x297, x298, x299, x300, x301, x302, x303) :|: TRUE f5288_in([], x304, x305, x306, x307, x308, x309, x310) -> f5294_in(x304, x305, x306, x307, x308, x309, x310) :|: TRUE f5294_out(x311, x312, x313, x314, x315, x316, x317) -> f5288_out([], x311, x312, x313, x314, x315, x316, x317) :|: TRUE f5298_out(x318, x319, x320, x321, x322, x323, x324, x325) -> f5288_out(x318, x319, x320, x321, x322, x323, x324, x325) :|: TRUE f5288_in(x326, x327, x328, x329, x330, x331, x332, x333) -> f5298_in(x326, x327, x328, x329, x330, x331, x332, x333) :|: TRUE f5298_in(x334, x335, x336, x337, x338, x339, x340, x341) -> f5314_in(x334, x335, x336, x337, x338, x339, x340, x341) :|: TRUE f5298_in(x342, x343, x344, x345, x346, x347, x348, x349) -> f5315_in :|: TRUE f5315_out -> f5298_out(x350, x351, x352, x353, x354, x355, x356, x357) :|: TRUE f5314_out(x358, x359, x360, x361, x362, x363, x364, x365) -> f5298_out(x358, x359, x360, x361, x362, x363, x364, x365) :|: TRUE f5336_out(x366, x367, x368, x369, x370, x371, x372, x373, x374) -> f5314_out(x370, x366, x367, x368, x371, x372, x373, x374) :|: TRUE f5314_in(x375, x376, x377, x378, x379, x380, x381, x382) -> f5333_in(x375, x376, x377, x378, x379) :|: TRUE f5333_out(x383, x384, x385, x386, x387) -> f5336_in(x384, x385, x386, x388, x383, x387, x389, x390, x391) :|: TRUE f5336_in(x392, x393, x394, x395, x396, x397, x398, x399, x400) -> f5340_in(x392, x393, x394, x395, x396, x397, x398, x399, x400) :|: TRUE f5341_out(x401, x402, x403, x404, x405, x406, x407, x408, x409, x410) -> f5336_out(x403, x404, x405, x402, x410, x406, x407, x408, x409) :|: TRUE f5340_out(x411, x412, x413, x414, x415, x416, x417, x418, x419) -> f5341_in(x420, x414, x411, x412, x413, x416, x417, x418, x419, x415) :|: TRUE f5408_out(x421, x422, x423, x424, x425, x426, x427, x428, x429, x430) -> f5341_out(x421, x422, x423, x424, x425, x426, x427, x428, x429, x430) :|: TRUE f5341_in(x431, x432, x433, x434, x435, x436, x437, x438, x439, x440) -> f5408_in(x431, x432, x433, x434, x435, x436, x437, x438, x439, x440) :|: TRUE f5419_out(x441, x442, x443, x444, x445, x446, x447, x448, x449, x450) -> f5408_out(x441, x442, x443, x444, x445, x446, x447, x448, x449, x450) :|: TRUE f5418_out(x451, x452, x453, x454, x455, x456, x457, x458, x459) -> f5408_out([], x451, x452, x453, x454, x455, x456, x457, x458, x459) :|: TRUE f5408_in([], x460, x461, x462, x463, x464, x465, x466, x467, x468) -> f5418_in(x460, x461, x462, x463, x464, x465, x466, x467, x468) :|: TRUE f5408_in(x469, x470, x471, x472, x473, x474, x475, x476, x477, x478) -> f5419_in(x469, x470, x471, x472, x473, x474, x475, x476, x477, x478) :|: TRUE f5422_out(x479, x480, x481, x482, x483, x484, x485, x486, x487, x488) -> f5419_out(x479, x480, x481, x482, x483, x484, x485, x486, x487, x488) :|: TRUE f5419_in(x489, x490, x491, x492, x493, x494, x495, x496, x497, x498) -> f5422_in(x489, x490, x491, x492, x493, x494, x495, x496, x497, x498) :|: TRUE f5419_in(x499, x500, x501, x502, x503, x504, x505, x506, x507, x508) -> f5423_in :|: TRUE f5423_out -> f5419_out(x509, x510, x511, x512, x513, x514, x515, x516, x517, x518) :|: TRUE f5422_in(x519, x520, x521, x522, x523, x524, x525, x526, x527, x528) -> f5424_in(x519, x520, x521, x522, x523, x524) :|: TRUE f5424_out(x529, x530, x531, x532, x533, x534) -> f5425_in(x530, x531, x532, x533, x535, x529, x534, x536, x537, x538, x539) :|: TRUE f5425_out(x540, x541, x542, x543, x544, x545, x546, x547, x548, x549, x550) -> f5422_out(x545, x540, x541, x542, x543, x546, x547, x548, x549, x550) :|: TRUE f5426_out(x551, x552, x553, x554, x555, x556, x557, x558, x559, x560, x561) -> f5427_in(x562, x555, x551, x552, x553, x554, x557, x558, x559, x560, x561, x556) :|: TRUE f5427_out(x563, x564, x565, x566, x567, x568, x569, x570, x571, x572, x573, x574) -> f5425_out(x565, x566, x567, x568, x564, x574, x569, x570, x571, x572, x573) :|: TRUE f5425_in(x575, x576, x577, x578, x579, x580, x581, x582, x583, x584, x585) -> f5426_in(x575, x576, x577, x578, x579, x580, x581, x582, x583, x584, x585) :|: TRUE f5427_in(x586, x587, x588, x589, x590, x591, x592, x593, x594, x595, x596, x597) -> f5539_in(x586, x587, x588, x589, x590, x591, x592, x593, x594, x595, x596, x597) :|: TRUE f5539_out(x598, x599, x600, x601, x602, x603, x604, x605, x606, x607, x608, x609) -> f5427_out(x598, x599, x600, x601, x602, x603, x604, x605, x606, x607, x608, x609) :|: TRUE f5563_out(x610, x611, x612, x613, x614, x615, x616, x617, x618, x619, x620) -> f5539_out([], x610, x611, x612, x613, x614, x615, x616, x617, x618, x619, x620) :|: TRUE f5539_in([], x621, x622, x623, x624, x625, x626, x627, x628, x629, x630, x631) -> f5563_in(x621, x622, x623, x624, x625, x626, x627, x628, x629, x630, x631) :|: TRUE f5539_in(x632, x633, x634, x635, x636, x637, x638, x639, x640, x641, x642, x643) -> f5564_in(x632, x633, x634, x635, x636, x637, x638, x639, x640, x641, x642, x643) :|: TRUE f5564_out(x644, x645, x646, x647, x648, x649, x650, x651, x652, x653, x654, x655) -> f5539_out(x644, x645, x646, x647, x648, x649, x650, x651, x652, x653, x654, x655) :|: TRUE f5586_out -> f5564_out(x656, x657, x658, x659, x660, x661, x662, x663, x664, x665, x666, x667) :|: TRUE f5564_in(x668, x669, x670, x671, x672, x673, x674, x675, x676, x677, x678, x679) -> f5581_in(x668, x669, x670, x671, x672, x673, x674, x675, x676, x677, x678, x679) :|: TRUE f5581_out(x680, x681, x682, x683, x684, x685, x686, x687, x688, x689, x690, x691) -> f5564_out(x680, x681, x682, x683, x684, x685, x686, x687, x688, x689, x690, x691) :|: TRUE f5564_in(x692, x693, x694, x695, x696, x697, x698, x699, x700, x701, x702, x703) -> f5586_in :|: TRUE f5591_out(x704, x705, x706, x707, x708, x709, x710) -> f5592_in(x705, x706, x707, x708, x709, x711, x704, x710, x712, x713, x714, x715, x716) :|: TRUE f5592_out(x717, x718, x719, x720, x721, x722, x723, x724, x725, x726, x727, x728, x729) -> f5581_out(x723, x717, x718, x719, x720, x721, x724, x725, x726, x727, x728, x729) :|: TRUE f5581_in(x730, x731, x732, x733, x734, x735, x736, x737, x738, x739, x740, x741) -> f5591_in(x730, x731, x732, x733, x734, x735, x736) :|: TRUE f5606_out(x742, x743, x744, x745, x746, x747, x748, x749, x750, x751, x752, x753, x754) -> f5607_in(x755, x747, x742, x743, x744, x745, x746, x749, x750, x751, x752, x753, x754, x748) :|: TRUE f5607_out(x756, x757, x758, x759, x760, x761, x762, x763, x764, x765, x766, x767, x768, x769) -> f5592_out(x758, x759, x760, x761, x762, x757, x769, x763, x764, x765, x766, x767, x768) :|: TRUE f5592_in(x770, x771, x772, x773, x774, x775, x776, x777, x778, x779, x780, x781, x782) -> f5606_in(x770, x771, x772, x773, x774, x775, x776, x777, x778, x779, x780, x781, x782) :|: TRUE f5608_out(x783, x784, x785, x786, x787, x788, x789, x790, x791, x792, x793, x794, x795) -> f5606_out(x783, x784, x785, x786, x787, x788, x789, x790, x791, x792, x793, x794, x795) :|: TRUE f5606_in(x796, x797, x798, x799, x800, x801, x802, x803, x804, x805, x806, x807, x808) -> f5608_in(x796, x797, x798, x799, x800, x801, x802, x803, x804, x805, x806, x807, x808) :|: TRUE f5609_out(x809, x810, x811, x812, x813, x814, x815, x816, x817, x818, x819, x820, x821) -> f5608_out(x809, x810, x811, x812, x813, x814, x815, x816, x817, x818, x819, x820, x821) :|: TRUE f5608_in(x822, x823, x824, x825, x826, x827, x828, x829, x830, x831, x832, x833, x834) -> f5609_in(x822, x823, x824, x825, x826, x827, x828, x829, x830, x831, x832, x833, x834) :|: TRUE f5609_in(x835, x836, x837, x838, x839, x840, x841, x842, x843, x844, x845, x846, x847) -> f5642_in(x835, x836, x837, x838, x839, x840, x842, x841, x843, x844, x845, x846, x847) :|: TRUE f5642_out(x848, x849, x850, x851, x852, x853, x854, x855, x856, x857, x858, x859, x860) -> f5609_out(x848, x849, x850, x851, x852, x853, x855, x854, x856, x857, x858, x859, x860) :|: TRUE f5642_in(x861, x862, x863, x864, x865, x866, x867, x868, x869, x870, x871, x872, x873) -> f5647_in(x861, x862, x863, x864, x865, x866, x867, x869, x870, x871, x872, x873, x868) :|: TRUE f5648_out(x874, x875, x876, x877, x878, x879, x880, x881, x882) -> f5642_out(x877, x878, x879, x880, x881, x883, x882, x875, x884, x885, x886, x887, x888) :|: TRUE f5647_out(x889, x890, x891, x892, x893, x894, x895, x896, x897, x898, x899, x900, x901) -> f5648_in(x902, x901, x903, x889, x890, x891, x892, x893, x895) :|: TRUE f5676_out(T1073, T1099) -> f5648_out(T1072, T1070, T1073, T1064, T1065, T1066, T1067, T1068, T1071) :|: TRUE f5648_in(x904, x905, x906, x907, x908, x909, x910, x911, x912) -> f5675_in(x904, x905, x907, x908, x909, x910, x911, x912) :|: TRUE f5675_out(x913, x914, x915, x916, x917, x918, x919, x920) -> f5676_in(x921, x922) :|: TRUE f5679_out(x923) -> f5680_in(x924, x925) :|: TRUE f5676_in(x926, x927) -> f5679_in(x926) :|: TRUE f5680_out(x928, x929) -> f5676_out(x930, x929) :|: TRUE f5680_in(x931, x932) -> f5683_in(x931, x932) :|: TRUE f5684_out -> f5680_out(x933, x934) :|: TRUE f5683_out(x935, x936) -> f5684_in :|: TRUE f4447_out(x937, x938) -> f5683_out(x937, x938) :|: TRUE f5683_in(x939, x940) -> f4447_in(x939, x940) :|: TRUE f5132_in(x941, x942, x943, x944, x945) -> f5134_in(x941, x942, x943, x944, x945) :|: TRUE f5134_out(x946, x947, x948, x949, x950) -> f5132_out(x946, x947, x948, x949, x950) :|: TRUE f5134_in(x951, x952, x953, x954, x955) -> f5135_in(x951, x952, x953, x954, x955) :|: TRUE f5135_out(x956, x957, x958, x959, x960) -> f5134_out(x956, x957, x958, x959, x960) :|: TRUE f5135_in(x961, x962, x963, x964, x965) -> f5143_in(x961, x962, x964, x963, x965) :|: TRUE f5143_out(x966, x967, x968, x969, x970) -> f5135_out(x966, x967, x969, x968, x970) :|: TRUE f5145_out(x971, x972, x973, x974, x975) -> f5143_out(x974, x976, x975, x972, x977) :|: TRUE f5144_out(x978, x979, x980, x981, x982) -> f5145_in(x983, x982, x984, x978, x980) :|: TRUE f5143_in(x985, x986, x987, x988, x989) -> f5144_in(x985, x986, x987, x989, x988) :|: TRUE f5145_in(T648, T646, T649, T644, T647) -> f5153_in(T648, T646, T644, T647) :|: TRUE f5153_out(x990, x991, x992, x993) -> f5154_in(x994, x995) :|: TRUE f5154_out(x996, x997) -> f5145_out(x998, x999, x996, x1000, x1001) :|: TRUE f5156_out(x1002, x1003) -> f5154_out(x1004, x1003) :|: TRUE f5155_out(x1005) -> f5156_in(x1006, x1007) :|: TRUE f5154_in(x1008, x1009) -> f5155_in(x1008) :|: TRUE f5159_out(T666, T663) -> f5160_in :|: TRUE f5156_in(x1010, x1011) -> f5159_in(x1010, x1011) :|: TRUE f5160_out -> f5156_out(x1012, x1013) :|: TRUE f5159_in(x1014, x1015) -> f4447_in(x1014, x1015) :|: TRUE f4447_out(x1016, x1017) -> f5159_out(x1016, x1017) :|: TRUE f5426_in(x1018, x1019, x1020, x1021, x1022, x1023, x1024, x1025, x1026, x1027, x1028) -> f5428_in(x1018, x1019, x1020, x1021, x1022, x1023, x1024, x1025, x1026, x1027, x1028) :|: TRUE f5428_out(x1029, x1030, x1031, x1032, x1033, x1034, x1035, x1036, x1037, x1038, x1039) -> f5426_out(x1029, x1030, x1031, x1032, x1033, x1034, x1035, x1036, x1037, x1038, x1039) :|: TRUE f5428_in(x1040, x1041, x1042, x1043, x1044, x1045, x1046, x1047, x1048, x1049, x1050) -> f5429_in(x1040, x1041, x1042, x1043, x1044, x1045, x1046, x1047, x1048, x1049, x1050) :|: TRUE f5429_out(x1051, x1052, x1053, x1054, x1055, x1056, x1057, x1058, x1059, x1060, x1061) -> f5428_out(x1051, x1052, x1053, x1054, x1055, x1056, x1057, x1058, x1059, x1060, x1061) :|: TRUE f5472_out(x1062, x1063, x1064, x1065, x1066, x1067, x1068, x1069, x1070, x1071, x1072) -> f5429_out(x1062, x1063, x1064, x1065, x1066, x1068, x1067, x1069, x1070, x1071, x1072) :|: TRUE f5429_in(x1073, x1074, x1075, x1076, x1077, x1078, x1079, x1080, x1081, x1082, x1083) -> f5472_in(x1073, x1074, x1075, x1076, x1077, x1079, x1078, x1080, x1081, x1082, x1083) :|: TRUE f5473_out(x1084, x1085, x1086, x1087, x1088, x1089, x1090, x1091, x1092, x1093, x1094) -> f5474_in(x1095, x1094, x1096, x1084, x1085, x1086, x1087, x1089) :|: TRUE f5474_out(x1097, x1098, x1099, x1100, x1101, x1102, x1103, x1104) -> f5472_out(x1100, x1101, x1102, x1103, x1105, x1104, x1098, x1106, x1107, x1108, x1109) :|: TRUE f5472_in(x1110, x1111, x1112, x1113, x1114, x1115, x1116, x1117, x1118, x1119, x1120) -> f5473_in(x1110, x1111, x1112, x1113, x1114, x1115, x1117, x1118, x1119, x1120, x1116) :|: TRUE f5474_in(T945, T943, T946, T938, T939, T940, T941, T944) -> f5513_in(T945, T943, T938, T939, T940, T941, T944) :|: TRUE f5514_out(x1121, x1122) -> f5474_out(x1123, x1124, x1121, x1125, x1126, x1127, x1128, x1129) :|: TRUE f5513_out(x1130, x1131, x1132, x1133, x1134, x1135, x1136) -> f5514_in(x1137, x1138) :|: TRUE f5532_out(x1139) -> f5533_in(x1140, x1141) :|: TRUE f5514_in(x1142, x1143) -> f5532_in(x1142) :|: TRUE f5533_out(x1144, x1145) -> f5514_out(x1146, x1145) :|: TRUE f5536_out -> f5533_out(T972, T969) :|: TRUE f5533_in(x1147, x1148) -> f5535_in(x1147, x1148) :|: TRUE f5535_out(x1149, x1150) -> f5536_in :|: TRUE f4447_out(x1151, x1152) -> f5535_out(x1151, x1152) :|: TRUE f5535_in(x1153, x1154) -> f4447_in(x1153, x1154) :|: TRUE f5194_in(x1155, x1156, x1157, x1158, x1159, x1160, x1161) -> f5196_in(x1155, x1156, x1157, x1158, x1159, x1160, x1161) :|: TRUE f5196_out(x1162, x1163, x1164, x1165, x1166, x1167, x1168) -> f5194_out(x1162, x1163, x1164, x1165, x1166, x1167, x1168) :|: TRUE f5197_out(x1169, x1170, x1171, x1172, x1173, x1174, x1175) -> f5196_out(x1169, x1170, x1171, x1172, x1173, x1174, x1175) :|: TRUE f5196_in(x1176, x1177, x1178, x1179, x1180, x1181, x1182) -> f5197_in(x1176, x1177, x1178, x1179, x1180, x1181, x1182) :|: TRUE f5204_out(x1183, x1184, x1185, x1186, x1187, x1188, x1189) -> f5197_out(x1183, x1184, x1185, x1187, x1186, x1188, x1189) :|: TRUE f5197_in(x1190, x1191, x1192, x1193, x1194, x1195, x1196) -> f5204_in(x1190, x1191, x1192, x1194, x1193, x1195, x1196) :|: TRUE f5208_out(x1197, x1198, x1199, x1200, x1201, x1202) -> f5204_out(x1200, x1201, x1203, x1202, x1198, x1204, x1205) :|: TRUE f5204_in(x1206, x1207, x1208, x1209, x1210, x1211, x1212) -> f5207_in(x1206, x1207, x1208, x1209, x1211, x1212, x1210) :|: TRUE f5207_out(x1213, x1214, x1215, x1216, x1217, x1218, x1219) -> f5208_in(x1220, x1219, x1221, x1213, x1214, x1216) :|: TRUE f5245_out(T734, T751) -> f5208_out(T733, T731, T734, T728, T729, T732) :|: TRUE f5244_out(x1222, x1223, x1224, x1225, x1226) -> f5245_in(x1227, x1228) :|: TRUE f5208_in(x1229, x1230, x1231, x1232, x1233, x1234) -> f5244_in(x1229, x1230, x1232, x1233, x1234) :|: TRUE f5253_out(x1235) -> f5254_in(x1236, x1237) :|: TRUE f5245_in(x1238, x1239) -> f5253_in(x1238) :|: TRUE f5254_out(x1240, x1241) -> f5245_out(x1242, x1241) :|: TRUE f5262_out -> f5254_out(x1243, x1244) :|: TRUE f5254_in(x1245, x1246) -> f5261_in(x1245, x1246) :|: TRUE f5261_out(x1247, x1248) -> f5262_in :|: TRUE f4447_out(x1249, x1250) -> f5261_out(x1249, x1250) :|: TRUE f5261_in(x1251, x1252) -> f4447_in(x1251, x1252) :|: TRUE f5348_out(x1253, x1254, x1255, x1256, x1257, x1258, x1259, x1260, x1261) -> f5340_out(x1253, x1254, x1255, x1256, x1257, x1258, x1259, x1260, x1261) :|: TRUE f5340_in(x1262, x1263, x1264, x1265, x1266, x1267, x1268, x1269, x1270) -> f5348_in(x1262, x1263, x1264, x1265, x1266, x1267, x1268, x1269, x1270) :|: TRUE f5351_out(x1271, x1272, x1273, x1274, x1275, x1276, x1277, x1278, x1279) -> f5348_out(x1271, x1272, x1273, x1274, x1275, x1276, x1277, x1278, x1279) :|: TRUE f5348_in(x1280, x1281, x1282, x1283, x1284, x1285, x1286, x1287, x1288) -> f5351_in(x1280, x1281, x1282, x1283, x1284, x1285, x1286, x1287, x1288) :|: TRUE f5363_out(x1289, x1290, x1291, x1292, x1293, x1294, x1295, x1296, x1297) -> f5351_out(x1289, x1290, x1291, x1292, x1294, x1293, x1295, x1296, x1297) :|: TRUE f5351_in(x1298, x1299, x1300, x1301, x1302, x1303, x1304, x1305, x1306) -> f5363_in(x1298, x1299, x1300, x1301, x1303, x1302, x1304, x1305, x1306) :|: TRUE f5363_in(x1307, x1308, x1309, x1310, x1311, x1312, x1313, x1314, x1315) -> f5368_in(x1307, x1308, x1309, x1310, x1311, x1313, x1314, x1315, x1312) :|: TRUE f5368_out(x1316, x1317, x1318, x1319, x1320, x1321, x1322, x1323, x1324) -> f5369_in(x1325, x1324, x1326, x1316, x1317, x1318, x1320) :|: TRUE f5369_out(x1327, x1328, x1329, x1330, x1331, x1332, x1333) -> f5363_out(x1330, x1331, x1332, x1334, x1333, x1328, x1335, x1336, x1337) :|: TRUE f5390_out(T833, T853) -> f5369_out(T832, T830, T833, T826, T827, T828, T831) :|: TRUE f5389_out(x1338, x1339, x1340, x1341, x1342, x1343) -> f5390_in(x1344, x1345) :|: TRUE f5369_in(x1346, x1347, x1348, x1349, x1350, x1351, x1352) -> f5389_in(x1346, x1347, x1349, x1350, x1351, x1352) :|: TRUE f5395_out(x1353) -> f5396_in(x1354, x1355) :|: TRUE f5390_in(x1356, x1357) -> f5395_in(x1356) :|: TRUE f5396_out(x1358, x1359) -> f5390_out(x1360, x1359) :|: TRUE f5399_out -> f5396_out(x1361, x1362) :|: TRUE f5396_in(x1363, x1364) -> f5398_in(x1363, x1364) :|: TRUE f5398_out(x1365, x1366) -> f5399_in :|: TRUE f4447_out(x1367, x1368) -> f5398_out(x1367, x1368) :|: TRUE f5398_in(x1369, x1370) -> f4447_in(x1369, x1370) :|: TRUE f4417_in(x1371, x1372, x1373) -> f4419_in(x1371, x1372, x1373) :|: TRUE f4419_out(x1374, x1375, x1376) -> f4417_out(x1374, x1375, x1376) :|: TRUE f4419_in(x1377, x1378, x1379) -> f4420_in(x1377, x1378, x1379) :|: TRUE f4420_out(x1380, x1381, x1382) -> f4419_out(x1380, x1381, x1382) :|: TRUE f4420_in(T117, T118, T119) -> f4421_in(T117, T119, T118) :|: TRUE f4421_out(x1383, x1384, x1385) -> f4420_out(x1383, x1385, x1384) :|: TRUE f4423_out(x1386, x1387, x1388, x1389) -> f4421_out(x1390, x1389, x1387) :|: TRUE f4421_in(x1391, x1392, x1393) -> f4422_in(x1391, x1392, x1393) :|: TRUE f4422_out(x1394, x1395, x1396) -> f4423_in(x1397, x1396, x1398, x1395) :|: TRUE f4438_out(x1399, x1400) -> f4423_out(x1401, x1402, x1399, x1403) :|: TRUE f4437_out(x1404, x1405, x1406) -> f4438_in(x1407, x1408) :|: TRUE f4423_in(x1409, x1410, x1411, x1412) -> f4437_in(x1409, x1410, x1412) :|: TRUE f4438_in(T121, T144) -> f4902_in(T121) :|: TRUE f4902_out(x1413) -> f4903_in(x1414, x1415) :|: TRUE f4903_out(x1416, x1417) -> f4438_out(x1418, x1417) :|: TRUE f5090_out -> f4903_out(x1419, x1420) :|: TRUE f4903_in(x1421, x1422) -> f5087_in(x1421, x1422) :|: TRUE f5087_out(x1423, x1424) -> f5090_in :|: TRUE f5087_in(x1425, x1426) -> f4447_in(x1425, x1426) :|: TRUE f4447_out(x1427, x1428) -> f5087_out(x1427, x1428) :|: TRUE f4447_out(x1429, x1430) -> f5244_out(x1429, x1430, x1431, x1432, x1433) :|: TRUE f5244_in(x1434, x1435, x1436, x1437, x1438) -> f4447_in(x1434, x1435) :|: TRUE f5691_out(x1439, x1440, x1441, x1442, x1443, x1444, x1445, x1446, x1447, x1448, x1449, x1450, x1451, x1452) -> f5607_out(x1439, x1440, x1441, x1442, x1443, x1444, x1445, x1446, x1447, x1448, x1449, x1450, x1451, x1452) :|: TRUE f5607_in(x1453, x1454, x1455, x1456, x1457, x1458, x1459, x1460, x1461, x1462, x1463, x1464, x1465, x1466) -> f5691_in(x1453, x1454, x1455, x1456, x1457, x1458, x1459, x1460, x1461, x1462, x1463, x1464, x1465, x1466) :|: TRUE f5716_out(x1467, x1468, x1469, x1470, x1471, x1472, x1473, x1474, x1475, x1476, x1477, x1478, x1479) -> f5691_out([], x1467, x1468, x1469, x1470, x1471, x1472, x1473, x1474, x1475, x1476, x1477, x1478, x1479) :|: TRUE f5719_out(x1480, x1481, x1482, x1483, x1484, x1485, x1486, x1487, x1488, x1489, x1490, x1491, x1492, x1493) -> f5691_out(x1480, x1481, x1482, x1483, x1484, x1485, x1486, x1487, x1488, x1489, x1490, x1491, x1492, x1493) :|: TRUE f5691_in([], x1494, x1495, x1496, x1497, x1498, x1499, x1500, x1501, x1502, x1503, x1504, x1505, x1506) -> f5716_in(x1494, x1495, x1496, x1497, x1498, x1499, x1500, x1501, x1502, x1503, x1504, x1505, x1506) :|: TRUE f5691_in(x1507, x1508, x1509, x1510, x1511, x1512, x1513, x1514, x1515, x1516, x1517, x1518, x1519, x1520) -> f5719_in(x1507, x1508, x1509, x1510, x1511, x1512, x1513, x1514, x1515, x1516, x1517, x1518, x1519, x1520) :|: TRUE f5719_in(x1521, x1522, x1523, x1524, x1525, x1526, x1527, x1528, x1529, x1530, x1531, x1532, x1533, x1534) -> f5733_in(x1521, x1522, x1523, x1524, x1525, x1526, x1527, x1528, x1529, x1530, x1531, x1532, x1533, x1534) :|: TRUE f5733_out(x1535, x1536, x1537, x1538, x1539, x1540, x1541, x1542, x1543, x1544, x1545, x1546, x1547, x1548) -> f5719_out(x1535, x1536, x1537, x1538, x1539, x1540, x1541, x1542, x1543, x1544, x1545, x1546, x1547, x1548) :|: TRUE f5734_out -> f5719_out(x1549, x1550, x1551, x1552, x1553, x1554, x1555, x1556, x1557, x1558, x1559, x1560, x1561, x1562) :|: TRUE f5719_in(x1563, x1564, x1565, x1566, x1567, x1568, x1569, x1570, x1571, x1572, x1573, x1574, x1575, x1576) -> f5734_in :|: TRUE f5736_out(x1577, x1578, x1579, x1580, x1581, x1582, x1583, x1584, .(x1578, .(x1579, .(x1580, .(x1581, .(x1582, .(x1583, .(0, []))))))), x1585, x1586, x1587, x1588, x1589, x1590) -> f5733_out(x1577, x1578, x1579, x1580, x1581, x1582, x1583, x1584, x1585, x1586, x1587, x1588, x1589, x1590) :|: TRUE f5733_in(x1591, x1592, x1593, x1594, x1595, x1596, x1597, x1598, x1599, x1600, x1601, x1602, x1603, x1604) -> f5736_in(x1591, x1592, x1593, x1594, x1595, x1596, x1597, x1598, .(x1592, .(x1593, .(x1594, .(x1595, .(x1596, .(x1597, .(0, []))))))), x1599, x1600, x1601, x1602, x1603, x1604) :|: TRUE f5745_out(x1605, x1606, x1607, x1608, x1609, x1610, x1611, x1612, x1613, x1614, x1615, x1616, x1617, x1618, x1619, x1620) -> f5736_out(x1612, x1605, x1606, x1607, x1608, x1609, x1610, x1613, x1614, x1615, x1616, x1617, x1618, x1619, x1620) :|: TRUE f5736_in(x1621, x1622, x1623, x1624, x1625, x1626, x1627, x1628, x1629, x1630, x1631, x1632, x1633, x1634, x1635) -> f5744_in(x1621, x1622, x1623, x1624, x1625, x1626, x1627, x1628) :|: TRUE f5744_out(x1636, x1637, x1638, x1639, x1640, x1641, x1642, x1643) -> f5745_in(x1637, x1638, x1639, x1640, x1641, x1642, x1644, x1636, x1643, x1645, x1646, x1647, x1648, x1649, x1650, x1651) :|: TRUE f5746_out(x1652, x1653, x1654, x1655, x1656, x1657, x1658, x1659, x1660, x1661, x1662, x1663, x1664, x1665, x1666) -> f5748_in(x1667, x1658, x1668, x1660, x1661, x1662, x1663, x1664, x1665, x1666, x1659, x1652, x1653, x1654, x1655, x1656, x1657) :|: TRUE f5745_in(x1669, x1670, x1671, x1672, x1673, x1674, x1675, x1676, x1677, x1678, x1679, x1680, x1681, x1682, x1683, x1684) -> f5746_in(x1669, x1670, x1671, x1672, x1673, x1674, x1675, x1676, x1677, x1679, x1680, x1681, x1682, x1683, x1684) :|: TRUE f5748_out(x1685, x1686, x1687, x1688, x1689, x1690, x1691, x1692, x1693, x1694, x1695, x1696, x1697, x1698, x1699, x1700, x1701) -> f5745_out(x1696, x1697, x1698, x1699, x1700, x1701, x1686, x1695, x1688, x1687, x1689, x1690, x1691, x1692, x1693, x1694) :|: TRUE f5750_out(x1702, x1703, x1704, x1705, x1706, x1707, x1708, x1709, x1710, x1711, x1712, x1713, x1714, x1715, x1716) -> f5746_out(x1702, x1703, x1704, x1705, x1706, x1707, x1708, x1709, x1710, x1711, x1712, x1713, x1714, x1715, x1716) :|: TRUE f5746_in(x1717, x1718, x1719, x1720, x1721, x1722, x1723, x1724, x1725, x1726, x1727, x1728, x1729, x1730, x1731) -> f5750_in(x1717, x1718, x1719, x1720, x1721, x1722, x1723, x1724, x1725, x1726, x1727, x1728, x1729, x1730, x1731) :|: TRUE f5750_in(x1732, x1733, x1734, x1735, x1736, x1737, x1738, x1739, x1740, x1741, x1742, x1743, x1744, x1745, x1746) -> f5751_in(x1732, x1733, x1734, x1735, x1736, x1737, x1738, x1739, x1740, x1741, x1742, x1743, x1744, x1745, x1746) :|: TRUE f5751_out(x1747, x1748, x1749, x1750, x1751, x1752, x1753, x1754, x1755, x1756, x1757, x1758, x1759, x1760, x1761) -> f5750_out(x1747, x1748, x1749, x1750, x1751, x1752, x1753, x1754, x1755, x1756, x1757, x1758, x1759, x1760, x1761) :|: TRUE f5751_in(x1762, x1763, x1764, x1765, x1766, x1767, x1768, x1769, x1770, x1771, x1772, x1773, x1774, x1775, x1776) -> f5762_in(x1762, x1763, x1764, x1765, x1766, x1767, x1768, x1770, x1769, x1771, x1772, x1773, x1774, x1775, x1776) :|: TRUE f5762_out(x1777, x1778, x1779, x1780, x1781, x1782, x1783, x1784, x1785, x1786, x1787, x1788, x1789, x1790, x1791) -> f5751_out(x1777, x1778, x1779, x1780, x1781, x1782, x1783, x1785, x1784, x1786, x1787, x1788, x1789, x1790, x1791) :|: TRUE f5762_in(x1792, x1793, x1794, x1795, x1796, x1797, x1798, x1799, x1800, x1801, x1802, x1803, x1804, x1805, x1806) -> f5765_in(x1792, x1793, x1794, x1795, x1796, x1797, x1798, x1799, x1801, x1802, x1803, x1804, x1805, x1806, x1800) :|: TRUE f5765_out(x1807, x1808, x1809, x1810, x1811, x1812, x1813, x1814, x1815, x1816, x1817, x1818, x1819, x1820, x1821) -> f5767_in(x1822, x1821, x1823, x1807, x1808, x1809, x1810, x1811, x1812, x1814) :|: TRUE f5767_out(x1824, x1825, x1826, x1827, x1828, x1829, x1830, x1831, x1832, x1833) -> f5762_out(x1827, x1828, x1829, x1830, x1831, x1832, x1834, x1833, x1825, x1835, x1836, x1837, x1838, x1839, x1840) :|: TRUE f5794_out(T1218, T1216, T1209, T1210, T1211, T1212, T1213, T1214, T1217) -> f5795_in(T1219, T1248) :|: TRUE f5767_in(x1841, x1842, x1843, x1844, x1845, x1846, x1847, x1848, x1849, x1850) -> f5794_in(x1841, x1842, x1844, x1845, x1846, x1847, x1848, x1849, x1850) :|: TRUE f5795_out(x1851, x1852) -> f5767_out(x1853, x1854, x1851, x1855, x1856, x1857, x1858, x1859, x1860, x1861) :|: TRUE f4447_out(x1862, x1863) -> f5794_out(x1862, x1863, x1864, x1865, x1866, x1867, x1868, x1869, x1870) :|: TRUE f5794_in(x1871, x1872, x1873, x1874, x1875, x1876, x1877, x1878, x1879) -> f4447_in(x1871, x1872) :|: TRUE f5795_in(x1880, x1881) -> f5798_in(x1880) :|: TRUE f5799_out(x1882, x1883) -> f5795_out(x1884, x1883) :|: TRUE f5798_out(x1885) -> f5799_in(x1886, x1887) :|: TRUE f5799_in(x1888, x1889) -> f5821_in(x1888, x1889) :|: TRUE f5821_out(x1890, x1891) -> f5822_in :|: TRUE f5822_out -> f5799_out(x1892, x1893) :|: TRUE f5821_in(x1894, x1895) -> f4447_in(x1894, x1895) :|: TRUE f4447_out(x1896, x1897) -> f5821_out(x1896, x1897) :|: TRUE f5748_in(x1898, x1899, x1900, x1901, x1902, x1903, x1904, x1905, x1906, x1907, x1908, x1909, x1910, x1911, x1912, x1913, x1914) -> f5830_in(x1898, x1899, x1900, x1901, x1902, x1903, x1904, x1905, x1906, x1907, x1908, x1909, x1910, x1911, x1912, x1913, x1914) :|: TRUE f5830_out(x1915, x1916, x1917, x1918, x1919, x1920, x1921, x1922, x1923, x1924, x1925, x1926, x1927, x1928, x1929, x1930, x1931) -> f5748_out(x1915, x1916, x1917, x1918, x1919, x1920, x1921, x1922, x1923, x1924, x1925, x1926, x1927, x1928, x1929, x1930, x1931) :|: TRUE f5832_out(x1932, x1933, x1934, x1935, x1936, x1937, x1938, x1939, x1940, x1941, x1942, x1943, x1944, x1945, x1946, x1947) -> f5830_out([], x1932, x1933, x1934, x1935, x1936, x1937, x1938, x1939, x1940, x1941, x1942, x1943, x1944, x1945, x1946, x1947) :|: TRUE f5833_out(x1948, x1949, x1950, x1951, x1952, x1953, x1954, x1955, x1956, x1957, x1958, x1959, x1960, x1961, x1962, x1963, x1964) -> f5830_out(x1948, x1949, x1950, x1951, x1952, x1953, x1954, x1955, x1956, x1957, x1958, x1959, x1960, x1961, x1962, x1963, x1964) :|: TRUE f5830_in(x1965, x1966, x1967, x1968, x1969, x1970, x1971, x1972, x1973, x1974, x1975, x1976, x1977, x1978, x1979, x1980, x1981) -> f5833_in(x1965, x1966, x1967, x1968, x1969, x1970, x1971, x1972, x1973, x1974, x1975, x1976, x1977, x1978, x1979, x1980, x1981) :|: TRUE f5830_in([], x1982, x1983, x1984, x1985, x1986, x1987, x1988, x1989, x1990, x1991, x1992, x1993, x1994, x1995, x1996, x1997) -> f5832_in(x1982, x1983, x1984, x1985, x1986, x1987, x1988, x1989, x1990, x1991, x1992, x1993, x1994, x1995, x1996, x1997) :|: TRUE f5846_out(x1998, x1999, x2000, x2001, x2002, x2003, x2004, x2005, x2006, x2007, x2008, x2009, x2010, x2011, x2012, x2013, x2014) -> f5833_out(x1998, x1999, x2000, x2001, x2002, x2003, x2004, x2005, x2006, x2007, x2008, x2009, x2010, x2011, x2012, x2013, x2014) :|: TRUE f5833_in(x2015, x2016, x2017, x2018, x2019, x2020, x2021, x2022, x2023, x2024, x2025, x2026, x2027, x2028, x2029, x2030, x2031) -> f5846_in(x2015, x2016, x2017, x2018, x2019, x2020, x2021, x2022, x2023, x2024, x2025, x2026, x2027, x2028, x2029, x2030, x2031) :|: TRUE f5833_in(x2032, x2033, x2034, x2035, x2036, x2037, x2038, x2039, x2040, x2041, x2042, x2043, x2044, x2045, x2046, x2047, x2048) -> f5847_in :|: TRUE f5847_out -> f5833_out(x2049, x2050, x2051, x2052, x2053, x2054, x2055, x2056, x2057, x2058, x2059, x2060, x2061, x2062, x2063, x2064, x2065) :|: TRUE f5846_in(x2066, x2067, x2068, x2069, x2070, x2071, x2072, x2073, x2074, x2075, x2076, x2077, x2078, x2079, x2080, x2081, x2082) -> f5851_in(x2066, x2067, x2068, x2069) :|: TRUE f5851_out(x2083, x2084, x2085, x2086) -> f5852_in(x2084, x2085, x2087, x2083, x2086, x2088, x2089, x2090, x2091, x2092, x2093, x2094, x2095, x2096, x2097, x2098, x2099, x2100) :|: TRUE f5852_out(x2101, x2102, x2103, x2104, x2105, x2106, x2107, x2108, x2109, x2110, x2111, x2112, x2113, x2114, x2115, x2116, x2117, x2118) -> f5846_out(x2104, x2101, x2102, x2105, x2106, x2107, x2108, x2109, x2110, x2111, x2112, x2113, x2114, x2115, x2116, x2117, x2118) :|: TRUE f5862_out(x2119, x2120, x2121, x2122, x2123, x2124, x2125, x2126, x2127, x2128, x2129, x2130, x2131, x2132, x2133, x2134, x2135, x2136, x2137) -> f5852_out(x2121, x2122, x2120, x2137, x2123, x2124, x2125, x2126, x2127, x2128, x2129, x2130, x2131, x2132, x2133, x2134, x2135, x2136) :|: TRUE f5861_out(x2138, x2139, x2140, x2141, x2142, x2143, x2144, x2145, x2146, x2147, x2148, x2149, x2150, x2151, x2152, x2153, x2154, x2155) -> f5862_in(x2156, x2140, x2138, x2139, x2142, x2143, x2144, x2145, x2146, x2147, x2148, x2149, x2150, x2151, x2152, x2153, x2154, x2155, x2141) :|: TRUE f5852_in(x2157, x2158, x2159, x2160, x2161, x2162, x2163, x2164, x2165, x2166, x2167, x2168, x2169, x2170, x2171, x2172, x2173, x2174) -> f5861_in(x2157, x2158, x2159, x2160, x2161, x2162, x2163, x2164, x2165, x2166, x2167, x2168, x2169, x2170, x2171, x2172, x2173, x2174) :|: TRUE f5861_in(x2175, x2176, x2177, x2178, x2179, x2180, x2181, x2182, x2183, x2184, x2185, x2186, x2187, x2188, x2189, x2190, x2191, x2192) -> f5863_in(x2175, x2176, x2177, x2178, x2179, x2180, x2181, x2182, x2183, x2184, x2185, x2186, x2187, x2188, x2189, x2190, x2191, x2192) :|: TRUE f5863_out(x2193, x2194, x2195, x2196, x2197, x2198, x2199, x2200, x2201, x2202, x2203, x2204, x2205, x2206, x2207, x2208, x2209, x2210) -> f5861_out(x2193, x2194, x2195, x2196, x2197, x2198, x2199, x2200, x2201, x2202, x2203, x2204, x2205, x2206, x2207, x2208, x2209, x2210) :|: TRUE f5865_out(x2211, x2212, x2213, x2214, x2215, x2216, x2217, x2218, x2219, x2220, x2221, x2222, x2223, x2224, x2225, x2226, x2227, x2228) -> f5863_out(x2211, x2212, x2213, x2214, x2215, x2216, x2217, x2218, x2219, x2220, x2221, x2222, x2223, x2224, x2225, x2226, x2227, x2228) :|: TRUE f5863_in(x2229, x2230, x2231, x2232, x2233, x2234, x2235, x2236, x2237, x2238, x2239, x2240, x2241, x2242, x2243, x2244, x2245, x2246) -> f5865_in(x2229, x2230, x2231, x2232, x2233, x2234, x2235, x2236, x2237, x2238, x2239, x2240, x2241, x2242, x2243, x2244, x2245, x2246) :|: TRUE f5865_in(x2247, x2248, x2249, x2250, x2251, x2252, x2253, x2254, x2255, x2256, x2257, x2258, x2259, x2260, x2261, x2262, x2263, x2264) -> f5876_in(x2247, x2248, x2249, x2251, x2250, x2252, x2253, x2254, x2255, x2256, x2257, x2258, x2259, x2260, x2261, x2262, x2263, x2264) :|: TRUE f5876_out(x2265, x2266, x2267, x2268, x2269, x2270, x2271, x2272, x2273, x2274, x2275, x2276, x2277, x2278, x2279, x2280, x2281, x2282) -> f5865_out(x2265, x2266, x2267, x2269, x2268, x2270, x2271, x2272, x2273, x2274, x2275, x2276, x2277, x2278, x2279, x2280, x2281, x2282) :|: TRUE f5878_out(x2283, x2284, x2285, x2286, x2287, x2288) -> f5876_out(x2286, x2287, x2289, x2288, x2284, x2290, x2291, x2292, x2293, x2294, x2295, x2296, x2297, x2298, x2299, x2300, x2301, x2302) :|: TRUE f5876_in(x2303, x2304, x2305, x2306, x2307, x2308, x2309, x2310, x2311, x2312, x2313, x2314, x2315, x2316, x2317, x2318, x2319, x2320) -> f5877_in(x2303, x2304, x2305, x2306, x2308, x2309, x2310, x2311, x2312, x2313, x2314, x2315, x2316, x2317, x2318, x2319, x2320, x2307) :|: TRUE f5877_out(x2321, x2322, x2323, x2324, x2325, x2326, x2327, x2328, x2329, x2330, x2331, x2332, x2333, x2334, x2335, x2336, x2337, x2338) -> f5878_in(x2339, x2338, x2340, x2321, x2322, x2324) :|: TRUE f6105_out(T1324, T1574) -> f5878_out(T1323, T1321, T1324, T1318, T1319, T1322) :|: TRUE f6104_out(x2341, x2342, x2343, x2344, x2345) -> f6105_in(x2346, x2347) :|: TRUE f5878_in(x2348, x2349, x2350, x2351, x2352, x2353) -> f6104_in(x2348, x2349, x2351, x2352, x2353) :|: TRUE f6107_out(x2354, x2355) -> f6105_out(x2356, x2355) :|: TRUE f6105_in(x2357, x2358) -> f6106_in(x2357) :|: TRUE f6106_out(x2359) -> f6107_in(x2360, x2361) :|: TRUE f6107_in(x2362, x2363) -> f6108_in(x2362, x2363) :|: TRUE f6108_out(x2364, x2365) -> f6109_in :|: TRUE f6109_out -> f6107_out(x2366, x2367) :|: TRUE f4447_out(x2368, x2369) -> f6108_out(x2368, x2369) :|: TRUE f6108_in(x2370, x2371) -> f4447_in(x2370, x2371) :|: TRUE f5675_in(x2372, x2373, x2374, x2375, x2376, x2377, x2378, x2379) -> f4447_in(x2372, x2373) :|: TRUE f4447_out(x2380, x2381) -> f5675_out(x2380, x2381, x2382, x2383, x2384, x2385, x2386, x2387) :|: TRUE f5389_in(x2388, x2389, x2390, x2391, x2392, x2393) -> f4447_in(x2388, x2389) :|: TRUE f4447_out(x2394, x2395) -> f5389_out(x2394, x2395, x2396, x2397, x2398, x2399) :|: TRUE f4447_out(x2400, x2401) -> f6104_out(x2400, x2401, x2402, x2403, x2404) :|: TRUE f6104_in(x2405, x2406, x2407, x2408, x2409) -> f4447_in(x2405, x2406) :|: TRUE f4439_out(x2410, x2411, x2412) -> f4437_out(x2410, x2411, x2412) :|: TRUE f4437_in(x2413, x2414, x2415) -> f4439_in(x2413, x2414, x2415) :|: TRUE f4440_out(x2416, x2417, x2418) -> f4439_out(x2416, x2417, x2418) :|: TRUE f4441_out(x2419, x2420, x2421) -> f4439_out(x2419, x2420, x2421) :|: TRUE f4439_in(x2422, x2423, x2424) -> f4441_in(x2422, x2423, x2424) :|: TRUE f4439_in(x2425, x2426, x2427) -> f4440_in(x2425, x2426, x2427) :|: TRUE f4446_out(x2428, x2429, x2430) -> f4441_out(x2428, x2429, x2430) :|: TRUE f4441_in(x2431, x2432, x2433) -> f4446_in(x2431, x2432, x2433) :|: TRUE f4441_in(x2434, x2435, x2436) -> f4445_in(x2434, x2435, x2436) :|: TRUE f4445_out(x2437, x2438, x2439) -> f4441_out(x2437, x2438, x2439) :|: TRUE f4447_out(x2440, x2441) -> f4445_out(.(x2442, x2440), .(x2442, x2441), x2443) :|: TRUE f4448_out -> f4445_out(x2444, x2445, x2446) :|: TRUE f4445_in(x2447, x2448, x2449) -> f4448_in :|: TRUE f4445_in(.(x2450, x2451), .(x2450, x2452), x2453) -> f4447_in(x2451, x2452) :|: TRUE f5153_in(x2454, x2455, x2456, x2457) -> f4447_in(x2454, x2455) :|: TRUE f4447_out(x2458, x2459) -> f5153_out(x2458, x2459, x2460, x2461) :|: TRUE f5513_in(x2462, x2463, x2464, x2465, x2466, x2467, x2468) -> f4447_in(x2462, x2463) :|: TRUE f4447_out(x2469, x2470) -> f5513_out(x2469, x2470, x2471, x2472, x2473, x2474, x2475) :|: TRUE f4446_in(.(x2476, x2477), .(x2478, x2479), x2480) -> f4839_in(x2476, x2478, x2477, x2479) :|: TRUE f4446_in(x2481, x2482, x2483) -> f4840_in :|: TRUE f4840_out -> f4446_out(x2484, x2485, x2486) :|: TRUE f4839_out(x2487, x2488, x2489, x2490) -> f4446_out(.(x2487, x2489), .(x2488, x2490), x2491) :|: TRUE f4841_out -> f4839_out(T211, T213, T212, T214) :|: TRUE f4839_in(x2492, x2493, x2494, x2495) -> f4895_in(x2492, x2494, x2495, x2493) :|: x2492 > x2493 f4839_in(x2496, x2497, x2498, x2499) -> f4841_in :|: TRUE f4896_out(x2500, x2501, x2502, x2503) -> f4839_out(x2502, x2501, x2503, x2500) :|: x2502 <= x2501 f4895_out(x2504, x2505, x2506, x2507) -> f4839_out(x2504, x2507, x2505, x2506) :|: x2504 > x2507 f4839_in(x2508, x2509, x2510, x2511) -> f4896_in(x2511, x2509, x2508, x2510) :|: x2508 <= x2509 f4895_in(x2512, x2513, x2514, x2515) -> f4447_in(.(x2512, x2513), x2514) :|: TRUE f4447_out(.(x2516, x2517), x2518) -> f4895_out(x2516, x2517, x2518, x2519) :|: TRUE Start term: f1_in(T1) ---------------------------------------- (412) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (413) TRUE ---------------------------------------- (414) Obligation: Rules: f4407_out(T104) -> f4415_out(T104) :|: TRUE f4415_in(x) -> f4407_in(x) :|: TRUE f4407_in(T80) -> f4409_in(T80) :|: TRUE f4409_out(x1) -> f4407_out(x1) :|: TRUE f4411_in(.(x2, x3)) -> f4415_in(x3) :|: TRUE f4415_out(x4) -> f4411_out(.(x5, x4)) :|: TRUE f4411_in(x6) -> f4416_in :|: TRUE f4416_out -> f4411_out(x7) :|: TRUE f4409_in(x8) -> f4411_in(x8) :|: TRUE f4411_out(x9) -> f4409_out(x9) :|: TRUE f4410_out(x10) -> f4409_out(x10) :|: TRUE f4409_in(x11) -> f4410_in(x11) :|: TRUE f4_out(T1) -> f1_out(T1) :|: TRUE f1_in(x12) -> f4_in(x12) :|: TRUE f4_in(T7) -> f10_in(T7) :|: TRUE f11_out -> f4_out(x13) :|: TRUE f10_out(x14) -> f4_out(x14) :|: TRUE f4_in(x15) -> f11_in :|: TRUE f10_in(x16) -> f16_in(x17, x16) :|: x17 = x16 * (x16 + 1) + 1 f10_in(x18) -> f12_in :|: TRUE f16_out(x19, x20) -> f10_out(x20) :|: TRUE f12_out -> f10_out(x21) :|: TRUE f16_in(x22, x23) -> f21_in(x22) :|: TRUE f24_out(x24, x25) -> f16_out(x25, x26) :|: TRUE f21_out(T10) -> f24_in(T12, T10) :|: TRUE f4324_out(x27, x28) -> f24_out(x27, x28) :|: TRUE f24_in(x29, x30) -> f4324_in(x29, x30) :|: TRUE f4324_in(x31, x32) -> f4326_in(x31, x32) :|: TRUE f4326_out(x33, x34) -> f4324_out(x33, x34) :|: TRUE f4325_out(T36) -> f4324_out([], T36) :|: TRUE f4324_in([], x35) -> f4325_in(x35) :|: TRUE f4326_in(T45, T48) -> f4397_in(T45, T48) :|: TRUE f4326_in(x36, x37) -> f4398_in :|: TRUE f4397_out(x38, x39) -> f4326_out(x38, x39) :|: TRUE f4398_out -> f4326_out(x40, x41) :|: TRUE f4400_out(x42, x43, x44) -> f4397_out(x43, x44) :|: TRUE f4397_in(x45, x46) -> f4399_in(x45, x46) :|: TRUE f4399_out(x47, x48) -> f4400_in(x49, x47, x48) :|: TRUE f4418_out(x50, x51, x52, x53) -> f4400_out(x51, x53, x52) :|: TRUE f4400_in(x54, x55, x56) -> f4417_in(x54, x55, x56) :|: TRUE f4417_out(x57, x58, x59) -> f4418_in(x60, x57, x59, x58) :|: TRUE f5096_out(x61, x62, x63, x64) -> f4418_out(x61, x62, x63, x64) :|: TRUE f4418_in(x65, x66, x67, x68) -> f5096_in(x65, x66, x67, x68) :|: TRUE f5107_out(x69, x70, x71, x72) -> f5096_out(x69, x70, x71, x72) :|: TRUE f5106_out(x73, x74, x75) -> f5096_out([], x73, x74, x75) :|: TRUE f5096_in([], x76, x77, x78) -> f5106_in(x76, x77, x78) :|: TRUE f5096_in(x79, x80, x81, x82) -> f5107_in(x79, x80, x81, x82) :|: TRUE f5121_out(x83, x84, x85, x86) -> f5107_out(x83, x84, x85, x86) :|: TRUE f5122_out -> f5107_out(x87, x88, x89, x90) :|: TRUE f5107_in(x91, x92, x93, x94) -> f5121_in(x91, x92, x93, x94) :|: TRUE f5107_in(x95, x96, x97, x98) -> f5122_in :|: TRUE f5123_out(x99, x100, x101) -> f5124_in(x100, x102, x99, x101, x103) :|: TRUE f5121_in(x104, x105, x106, x107) -> f5123_in(x104, x105, x106) :|: TRUE f5124_out(x108, x109, x110, x111, x112) -> f5121_out(x110, x108, x111, x112) :|: TRUE f5124_in(x113, x114, x115, x116, x117) -> f5132_in(x113, x114, x115, x116, x117) :|: TRUE f5133_out(x118, x119, x120, x121, x122, x123) -> f5124_out(x120, x119, x123, x121, x122) :|: TRUE f5132_out(x124, x125, x126, x127, x128) -> f5133_in(x129, x125, x124, x127, x128, x126) :|: TRUE f5133_in(x130, x131, x132, x133, x134, x135) -> f5167_in(x130, x131, x132, x133, x134, x135) :|: TRUE f5167_out(x136, x137, x138, x139, x140, x141) -> f5133_out(x136, x137, x138, x139, x140, x141) :|: TRUE f5171_out(x142, x143, x144, x145, x146) -> f5167_out([], x142, x143, x144, x145, x146) :|: TRUE f5167_in([], x147, x148, x149, x150, x151) -> f5171_in(x147, x148, x149, x150, x151) :|: TRUE f5167_in(x152, x153, x154, x155, x156, x157) -> f5172_in(x152, x153, x154, x155, x156, x157) :|: TRUE f5172_out(x158, x159, x160, x161, x162, x163) -> f5167_out(x158, x159, x160, x161, x162, x163) :|: TRUE f5182_out(x164, x165, x166, x167, x168, x169) -> f5172_out(x164, x165, x166, x167, x168, x169) :|: TRUE f5172_in(x170, x171, x172, x173, x174, x175) -> f5183_in :|: TRUE f5183_out -> f5172_out(x176, x177, x178, x179, x180, x181) :|: TRUE f5172_in(x182, x183, x184, x185, x186, x187) -> f5182_in(x182, x183, x184, x185, x186, x187) :|: TRUE f5189_out(x188, x189, x190, x191) -> f5190_in(x189, x190, x192, x188, x191, x193, x194) :|: TRUE f5190_out(x195, x196, x197, x198, x199, x200, x201) -> f5182_out(x198, x195, x196, x199, x200, x201) :|: TRUE f5182_in(x202, x203, x204, x205, x206, x207) -> f5189_in(x202, x203, x204, x205) :|: TRUE f5195_out(x208, x209, x210, x211, x212, x213, x214, x215) -> f5190_out(x210, x211, x209, x215, x212, x213, x214) :|: TRUE f5190_in(x216, x217, x218, x219, x220, x221, x222) -> f5194_in(x216, x217, x218, x219, x220, x221, x222) :|: TRUE f5194_out(x223, x224, x225, x226, x227, x228, x229) -> f5195_in(x230, x225, x223, x224, x227, x228, x229, x226) :|: TRUE f5195_in(x231, x232, x233, x234, x235, x236, x237, x238) -> f5288_in(x231, x232, x233, x234, x235, x236, x237, x238) :|: TRUE f5288_out(x239, x240, x241, x242, x243, x244, x245, x246) -> f5195_out(x239, x240, x241, x242, x243, x244, x245, x246) :|: TRUE f5288_in([], x247, x248, x249, x250, x251, x252, x253) -> f5294_in(x247, x248, x249, x250, x251, x252, x253) :|: TRUE f5294_out(x254, x255, x256, x257, x258, x259, x260) -> f5288_out([], x254, x255, x256, x257, x258, x259, x260) :|: TRUE f5298_out(x261, x262, x263, x264, x265, x266, x267, x268) -> f5288_out(x261, x262, x263, x264, x265, x266, x267, x268) :|: TRUE f5288_in(x269, x270, x271, x272, x273, x274, x275, x276) -> f5298_in(x269, x270, x271, x272, x273, x274, x275, x276) :|: TRUE f5298_in(x277, x278, x279, x280, x281, x282, x283, x284) -> f5314_in(x277, x278, x279, x280, x281, x282, x283, x284) :|: TRUE f5298_in(x285, x286, x287, x288, x289, x290, x291, x292) -> f5315_in :|: TRUE f5315_out -> f5298_out(x293, x294, x295, x296, x297, x298, x299, x300) :|: TRUE f5314_out(x301, x302, x303, x304, x305, x306, x307, x308) -> f5298_out(x301, x302, x303, x304, x305, x306, x307, x308) :|: TRUE f5336_out(x309, x310, x311, x312, x313, x314, x315, x316, x317) -> f5314_out(x313, x309, x310, x311, x314, x315, x316, x317) :|: TRUE f5314_in(x318, x319, x320, x321, x322, x323, x324, x325) -> f5333_in(x318, x319, x320, x321, x322) :|: TRUE f5333_out(x326, x327, x328, x329, x330) -> f5336_in(x327, x328, x329, x331, x326, x330, x332, x333, x334) :|: TRUE f5336_in(x335, x336, x337, x338, x339, x340, x341, x342, x343) -> f5340_in(x335, x336, x337, x338, x339, x340, x341, x342, x343) :|: TRUE f5341_out(x344, x345, x346, x347, x348, x349, x350, x351, x352, x353) -> f5336_out(x346, x347, x348, x345, x353, x349, x350, x351, x352) :|: TRUE f5340_out(x354, x355, x356, x357, x358, x359, x360, x361, x362) -> f5341_in(x363, x357, x354, x355, x356, x359, x360, x361, x362, x358) :|: TRUE f5408_out(x364, x365, x366, x367, x368, x369, x370, x371, x372, x373) -> f5341_out(x364, x365, x366, x367, x368, x369, x370, x371, x372, x373) :|: TRUE f5341_in(x374, x375, x376, x377, x378, x379, x380, x381, x382, x383) -> f5408_in(x374, x375, x376, x377, x378, x379, x380, x381, x382, x383) :|: TRUE f5419_out(x384, x385, x386, x387, x388, x389, x390, x391, x392, x393) -> f5408_out(x384, x385, x386, x387, x388, x389, x390, x391, x392, x393) :|: TRUE f5418_out(x394, x395, x396, x397, x398, x399, x400, x401, x402) -> f5408_out([], x394, x395, x396, x397, x398, x399, x400, x401, x402) :|: TRUE f5408_in([], x403, x404, x405, x406, x407, x408, x409, x410, x411) -> f5418_in(x403, x404, x405, x406, x407, x408, x409, x410, x411) :|: TRUE f5408_in(x412, x413, x414, x415, x416, x417, x418, x419, x420, x421) -> f5419_in(x412, x413, x414, x415, x416, x417, x418, x419, x420, x421) :|: TRUE f5422_out(x422, x423, x424, x425, x426, x427, x428, x429, x430, x431) -> f5419_out(x422, x423, x424, x425, x426, x427, x428, x429, x430, x431) :|: TRUE f5419_in(x432, x433, x434, x435, x436, x437, x438, x439, x440, x441) -> f5422_in(x432, x433, x434, x435, x436, x437, x438, x439, x440, x441) :|: TRUE f5419_in(x442, x443, x444, x445, x446, x447, x448, x449, x450, x451) -> f5423_in :|: TRUE f5423_out -> f5419_out(x452, x453, x454, x455, x456, x457, x458, x459, x460, x461) :|: TRUE f5422_in(x462, x463, x464, x465, x466, x467, x468, x469, x470, x471) -> f5424_in(x462, x463, x464, x465, x466, x467) :|: TRUE f5424_out(x472, x473, x474, x475, x476, x477) -> f5425_in(x473, x474, x475, x476, x478, x472, x477, x479, x480, x481, x482) :|: TRUE f5425_out(x483, x484, x485, x486, x487, x488, x489, x490, x491, x492, x493) -> f5422_out(x488, x483, x484, x485, x486, x489, x490, x491, x492, x493) :|: TRUE f5426_out(x494, x495, x496, x497, x498, x499, x500, x501, x502, x503, x504) -> f5427_in(x505, x498, x494, x495, x496, x497, x500, x501, x502, x503, x504, x499) :|: TRUE f5427_out(x506, x507, x508, x509, x510, x511, x512, x513, x514, x515, x516, x517) -> f5425_out(x508, x509, x510, x511, x507, x517, x512, x513, x514, x515, x516) :|: TRUE f5425_in(x518, x519, x520, x521, x522, x523, x524, x525, x526, x527, x528) -> f5426_in(x518, x519, x520, x521, x522, x523, x524, x525, x526, x527, x528) :|: TRUE f5427_in(x529, x530, x531, x532, x533, x534, x535, x536, x537, x538, x539, x540) -> f5539_in(x529, x530, x531, x532, x533, x534, x535, x536, x537, x538, x539, x540) :|: TRUE f5539_out(x541, x542, x543, x544, x545, x546, x547, x548, x549, x550, x551, x552) -> f5427_out(x541, x542, x543, x544, x545, x546, x547, x548, x549, x550, x551, x552) :|: TRUE f5563_out(x553, x554, x555, x556, x557, x558, x559, x560, x561, x562, x563) -> f5539_out([], x553, x554, x555, x556, x557, x558, x559, x560, x561, x562, x563) :|: TRUE f5539_in([], x564, x565, x566, x567, x568, x569, x570, x571, x572, x573, x574) -> f5563_in(x564, x565, x566, x567, x568, x569, x570, x571, x572, x573, x574) :|: TRUE f5539_in(x575, x576, x577, x578, x579, x580, x581, x582, x583, x584, x585, x586) -> f5564_in(x575, x576, x577, x578, x579, x580, x581, x582, x583, x584, x585, x586) :|: TRUE f5564_out(x587, x588, x589, x590, x591, x592, x593, x594, x595, x596, x597, x598) -> f5539_out(x587, x588, x589, x590, x591, x592, x593, x594, x595, x596, x597, x598) :|: TRUE f5586_out -> f5564_out(x599, x600, x601, x602, x603, x604, x605, x606, x607, x608, x609, x610) :|: TRUE f5564_in(x611, x612, x613, x614, x615, x616, x617, x618, x619, x620, x621, x622) -> f5581_in(x611, x612, x613, x614, x615, x616, x617, x618, x619, x620, x621, x622) :|: TRUE f5581_out(x623, x624, x625, x626, x627, x628, x629, x630, x631, x632, x633, x634) -> f5564_out(x623, x624, x625, x626, x627, x628, x629, x630, x631, x632, x633, x634) :|: TRUE f5564_in(x635, x636, x637, x638, x639, x640, x641, x642, x643, x644, x645, x646) -> f5586_in :|: TRUE f5591_out(x647, x648, x649, x650, x651, x652, x653) -> f5592_in(x648, x649, x650, x651, x652, x654, x647, x653, x655, x656, x657, x658, x659) :|: TRUE f5592_out(x660, x661, x662, x663, x664, x665, x666, x667, x668, x669, x670, x671, x672) -> f5581_out(x666, x660, x661, x662, x663, x664, x667, x668, x669, x670, x671, x672) :|: TRUE f5581_in(x673, x674, x675, x676, x677, x678, x679, x680, x681, x682, x683, x684) -> f5591_in(x673, x674, x675, x676, x677, x678, x679) :|: TRUE f5591_in(T1020, T1021, T1022, T1023, T1024, T1025, T1028) -> f4407_in(T1020) :|: TRUE f4407_out(x685) -> f5591_out(x685, x686, x687, x688, x689, x690, x691) :|: TRUE f4407_out(T696) -> f5189_out(T696, T697, T698, T701) :|: TRUE f5189_in(x692, x693, x694, x695) -> f4407_in(x692) :|: TRUE f4407_out(T898) -> f5424_out(T898, T899, T900, T901, T902, T905) :|: TRUE f5424_in(x696, x697, x698, x699, x700, x701) -> f4407_in(x696) :|: TRUE f4401_out(x702, x703) -> f4399_out(x702, x703) :|: TRUE f4399_in(x704, x705) -> f4401_in(x704, x705) :|: TRUE f4403_out(x706, x707) -> f4401_out(x706, x707) :|: TRUE f4401_in(x708, x709) -> f4402_in(x708, x709) :|: TRUE f4402_out(x710, x711) -> f4401_out(x710, x711) :|: TRUE f4401_in(x712, x713) -> f4403_in(x712, x713) :|: TRUE f4403_in(.(x714, x715), x716) -> f4407_in(x715) :|: TRUE f4408_out -> f4403_out(x717, x718) :|: TRUE f4403_in(x719, x720) -> f4408_in :|: TRUE f4407_out(x721) -> f4403_out(.(x722, x721), x723) :|: TRUE f5606_out(x724, x725, x726, x727, x728, x729, x730, x731, x732, x733, x734, x735, x736) -> f5607_in(x737, x729, x724, x725, x726, x727, x728, x731, x732, x733, x734, x735, x736, x730) :|: TRUE f5607_out(x738, x739, x740, x741, x742, x743, x744, x745, x746, x747, x748, x749, x750, x751) -> f5592_out(x740, x741, x742, x743, x744, x739, x751, x745, x746, x747, x748, x749, x750) :|: TRUE f5592_in(x752, x753, x754, x755, x756, x757, x758, x759, x760, x761, x762, x763, x764) -> f5606_in(x752, x753, x754, x755, x756, x757, x758, x759, x760, x761, x762, x763, x764) :|: TRUE f5691_out(x765, x766, x767, x768, x769, x770, x771, x772, x773, x774, x775, x776, x777, x778) -> f5607_out(x765, x766, x767, x768, x769, x770, x771, x772, x773, x774, x775, x776, x777, x778) :|: TRUE f5607_in(x779, x780, x781, x782, x783, x784, x785, x786, x787, x788, x789, x790, x791, x792) -> f5691_in(x779, x780, x781, x782, x783, x784, x785, x786, x787, x788, x789, x790, x791, x792) :|: TRUE f5716_out(x793, x794, x795, x796, x797, x798, x799, x800, x801, x802, x803, x804, x805) -> f5691_out([], x793, x794, x795, x796, x797, x798, x799, x800, x801, x802, x803, x804, x805) :|: TRUE f5719_out(x806, x807, x808, x809, x810, x811, x812, x813, x814, x815, x816, x817, x818, x819) -> f5691_out(x806, x807, x808, x809, x810, x811, x812, x813, x814, x815, x816, x817, x818, x819) :|: TRUE f5691_in([], x820, x821, x822, x823, x824, x825, x826, x827, x828, x829, x830, x831, x832) -> f5716_in(x820, x821, x822, x823, x824, x825, x826, x827, x828, x829, x830, x831, x832) :|: TRUE f5691_in(x833, x834, x835, x836, x837, x838, x839, x840, x841, x842, x843, x844, x845, x846) -> f5719_in(x833, x834, x835, x836, x837, x838, x839, x840, x841, x842, x843, x844, x845, x846) :|: TRUE f5719_in(x847, x848, x849, x850, x851, x852, x853, x854, x855, x856, x857, x858, x859, x860) -> f5733_in(x847, x848, x849, x850, x851, x852, x853, x854, x855, x856, x857, x858, x859, x860) :|: TRUE f5733_out(x861, x862, x863, x864, x865, x866, x867, x868, x869, x870, x871, x872, x873, x874) -> f5719_out(x861, x862, x863, x864, x865, x866, x867, x868, x869, x870, x871, x872, x873, x874) :|: TRUE f5734_out -> f5719_out(x875, x876, x877, x878, x879, x880, x881, x882, x883, x884, x885, x886, x887, x888) :|: TRUE f5719_in(x889, x890, x891, x892, x893, x894, x895, x896, x897, x898, x899, x900, x901, x902) -> f5734_in :|: TRUE f5736_out(x903, x904, x905, x906, x907, x908, x909, x910, .(x904, .(x905, .(x906, .(x907, .(x908, .(x909, .(0, []))))))), x911, x912, x913, x914, x915, x916) -> f5733_out(x903, x904, x905, x906, x907, x908, x909, x910, x911, x912, x913, x914, x915, x916) :|: TRUE f5733_in(x917, x918, x919, x920, x921, x922, x923, x924, x925, x926, x927, x928, x929, x930) -> f5736_in(x917, x918, x919, x920, x921, x922, x923, x924, .(x918, .(x919, .(x920, .(x921, .(x922, .(x923, .(0, []))))))), x925, x926, x927, x928, x929, x930) :|: TRUE f5745_out(x931, x932, x933, x934, x935, x936, x937, x938, x939, x940, x941, x942, x943, x944, x945, x946) -> f5736_out(x938, x931, x932, x933, x934, x935, x936, x939, x940, x941, x942, x943, x944, x945, x946) :|: TRUE f5736_in(x947, x948, x949, x950, x951, x952, x953, x954, x955, x956, x957, x958, x959, x960, x961) -> f5744_in(x947, x948, x949, x950, x951, x952, x953, x954) :|: TRUE f5744_out(x962, x963, x964, x965, x966, x967, x968, x969) -> f5745_in(x963, x964, x965, x966, x967, x968, x970, x962, x969, x971, x972, x973, x974, x975, x976, x977) :|: TRUE f4407_out(T1156) -> f5744_out(T1156, T1157, T1158, T1159, T1160, T1161, T1162, T1165) :|: TRUE f5744_in(x978, x979, x980, x981, x982, x983, x984, x985) -> f4407_in(x978) :|: TRUE f5746_out(x986, x987, x988, x989, x990, x991, x992, x993, x994, x995, x996, x997, x998, x999, x1000) -> f5748_in(x1001, x992, x1002, x994, x995, x996, x997, x998, x999, x1000, x993, x986, x987, x988, x989, x990, x991) :|: TRUE f5745_in(x1003, x1004, x1005, x1006, x1007, x1008, x1009, x1010, x1011, x1012, x1013, x1014, x1015, x1016, x1017, x1018) -> f5746_in(x1003, x1004, x1005, x1006, x1007, x1008, x1009, x1010, x1011, x1013, x1014, x1015, x1016, x1017, x1018) :|: TRUE f5748_out(x1019, x1020, x1021, x1022, x1023, x1024, x1025, x1026, x1027, x1028, x1029, x1030, x1031, x1032, x1033, x1034, x1035) -> f5745_out(x1030, x1031, x1032, x1033, x1034, x1035, x1020, x1029, x1022, x1021, x1023, x1024, x1025, x1026, x1027, x1028) :|: TRUE f5748_in(x1036, x1037, x1038, x1039, x1040, x1041, x1042, x1043, x1044, x1045, x1046, x1047, x1048, x1049, x1050, x1051, x1052) -> f5830_in(x1036, x1037, x1038, x1039, x1040, x1041, x1042, x1043, x1044, x1045, x1046, x1047, x1048, x1049, x1050, x1051, x1052) :|: TRUE f5830_out(x1053, x1054, x1055, x1056, x1057, x1058, x1059, x1060, x1061, x1062, x1063, x1064, x1065, x1066, x1067, x1068, x1069) -> f5748_out(x1053, x1054, x1055, x1056, x1057, x1058, x1059, x1060, x1061, x1062, x1063, x1064, x1065, x1066, x1067, x1068, x1069) :|: TRUE f5832_out(x1070, x1071, x1072, x1073, x1074, x1075, x1076, x1077, x1078, x1079, x1080, x1081, x1082, x1083, x1084, x1085) -> f5830_out([], x1070, x1071, x1072, x1073, x1074, x1075, x1076, x1077, x1078, x1079, x1080, x1081, x1082, x1083, x1084, x1085) :|: TRUE f5833_out(x1086, x1087, x1088, x1089, x1090, x1091, x1092, x1093, x1094, x1095, x1096, x1097, x1098, x1099, x1100, x1101, x1102) -> f5830_out(x1086, x1087, x1088, x1089, x1090, x1091, x1092, x1093, x1094, x1095, x1096, x1097, x1098, x1099, x1100, x1101, x1102) :|: TRUE f5830_in(x1103, x1104, x1105, x1106, x1107, x1108, x1109, x1110, x1111, x1112, x1113, x1114, x1115, x1116, x1117, x1118, x1119) -> f5833_in(x1103, x1104, x1105, x1106, x1107, x1108, x1109, x1110, x1111, x1112, x1113, x1114, x1115, x1116, x1117, x1118, x1119) :|: TRUE f5830_in([], x1120, x1121, x1122, x1123, x1124, x1125, x1126, x1127, x1128, x1129, x1130, x1131, x1132, x1133, x1134, x1135) -> f5832_in(x1120, x1121, x1122, x1123, x1124, x1125, x1126, x1127, x1128, x1129, x1130, x1131, x1132, x1133, x1134, x1135) :|: TRUE f5846_out(x1136, x1137, x1138, x1139, x1140, x1141, x1142, x1143, x1144, x1145, x1146, x1147, x1148, x1149, x1150, x1151, x1152) -> f5833_out(x1136, x1137, x1138, x1139, x1140, x1141, x1142, x1143, x1144, x1145, x1146, x1147, x1148, x1149, x1150, x1151, x1152) :|: TRUE f5833_in(x1153, x1154, x1155, x1156, x1157, x1158, x1159, x1160, x1161, x1162, x1163, x1164, x1165, x1166, x1167, x1168, x1169) -> f5846_in(x1153, x1154, x1155, x1156, x1157, x1158, x1159, x1160, x1161, x1162, x1163, x1164, x1165, x1166, x1167, x1168, x1169) :|: TRUE f5833_in(x1170, x1171, x1172, x1173, x1174, x1175, x1176, x1177, x1178, x1179, x1180, x1181, x1182, x1183, x1184, x1185, x1186) -> f5847_in :|: TRUE f5847_out -> f5833_out(x1187, x1188, x1189, x1190, x1191, x1192, x1193, x1194, x1195, x1196, x1197, x1198, x1199, x1200, x1201, x1202, x1203) :|: TRUE f5846_in(x1204, x1205, x1206, x1207, x1208, x1209, x1210, x1211, x1212, x1213, x1214, x1215, x1216, x1217, x1218, x1219, x1220) -> f5851_in(x1204, x1205, x1206, x1207) :|: TRUE f5851_out(x1221, x1222, x1223, x1224) -> f5852_in(x1222, x1223, x1225, x1221, x1224, x1226, x1227, x1228, x1229, x1230, x1231, x1232, x1233, x1234, x1235, x1236, x1237, x1238) :|: TRUE f5852_out(x1239, x1240, x1241, x1242, x1243, x1244, x1245, x1246, x1247, x1248, x1249, x1250, x1251, x1252, x1253, x1254, x1255, x1256) -> f5846_out(x1242, x1239, x1240, x1243, x1244, x1245, x1246, x1247, x1248, x1249, x1250, x1251, x1252, x1253, x1254, x1255, x1256) :|: TRUE f4407_out(T1281) -> f5851_out(T1281, T1282, T1283, T1286) :|: TRUE f5851_in(x1257, x1258, x1259, x1260) -> f4407_in(x1257) :|: TRUE f4407_out(T790) -> f5333_out(T790, T791, T792, T793, T796) :|: TRUE f5333_in(x1261, x1262, x1263, x1264, x1265) -> f4407_in(x1261) :|: TRUE f4407_out(T616) -> f5123_out(T616, T617, T620) :|: TRUE f5123_in(x1266, x1267, x1268) -> f4407_in(x1266) :|: TRUE Start term: f1_in(T1) ---------------------------------------- (415) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (416) TRUE ---------------------------------------- (417) Obligation: Rules: f4455_in(T165, T166) -> f4526_in :|: TRUE f4455_in(.(T186, T187), .(T186, T188)) -> f4525_in(T187, T188) :|: TRUE f4525_out(x, x1) -> f4455_out(.(x2, x), .(x2, x1)) :|: TRUE f4526_out -> f4455_out(x3, x4) :|: TRUE f4407_in(T80) -> f4409_in(T80) :|: TRUE f4409_out(x5) -> f4407_out(x5) :|: TRUE f5966_out(T1419, T1417, T1420, T1418, T616, T1156, T1160, T1389, T1159, T696, T790, T1020, T1157, T1161, T1333, T1361, T45, T1158, T898, T1162, T1321) -> f5945_out(.(T1417, T1418), T1419, T1420, T616, T1156, T1160, T1389, T1159, T696, T790, T1020, T1157, T1161, T1333, T1361, T45, T1158, T898, T1162, T1321) :|: TRUE f5967_out -> f5945_out(x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25) :|: TRUE f5945_in(.(x26, x27), x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46) -> f5966_in(x28, x26, x29, x27, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46) :|: TRUE f5945_in(x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66) -> f5967_in :|: TRUE f5054_in -> f5054_out :|: TRUE f5900_out -> f5898_out([], x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83) :|: TRUE f5901_out -> f5898_out(x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101) :|: TRUE f5898_in(x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119) -> f5901_in :|: TRUE f5898_in([], x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136) -> f5900_in :|: TRUE f6044_out(x137, x138, x139, x140, x141, x142, x143, x144, x145, x146, x147, x148, x149, x150, x151, x152, x153, x154, x155, x156, x157, x158, x159, x160, x161, x162) -> f6043_out(x139, x145, x137, x138, x140, x141, x142, x143, x146, x147, x148, x149, x150, x151, x152, x153, x154, x155, x156, x157, x158, x159, x160, x161, x162) :|: TRUE f6043_in(x163, x164, x165, x166, x167, x168, x169, x170, x171, x172, x173, x174, x175, x176, x177, x178, x179, x180, x181, x182, x183, x184, x185, x186, x187) -> f6044_in(x165, x166, x163, x167, x168, x169, x170, x188, x164, x171, x172, x173, x174, x175, x176, x177, x178, x179, x180, x181, x182, x183, x184, x185, x186, x187) :|: x188 = x163 - x164 f6020_out(x189, x190, x191, x192, x193, x194, x195, x196, x197, x198, x199, x200, x201, x202, x203, x204, x205, x206, x207, x208, x209, x210, x211, x212, x213) -> f6019_out(x191, x202, x189, x190, x200, x192, x201, x193, x194, x195, x203, x204, x205, x206, x207, x208, x209, x196, x210, x211, x212, x198, x199, x213) :|: TRUE f6019_in(x214, x215, x216, x217, x218, x219, x220, x221, x222, x223, x224, x225, x226, x227, x228, x229, x230, x231, x232, x233, x234, x235, x236, x237) -> f6020_in(x216, x217, x214, x219, x221, x222, x223, x231, x238, x235, x236, x218, x220, x215, x224, x225, x226, x227, x228, x229, x230, x232, x233, x234, x237) :|: x238 = x214 - x215 f5079_out -> f5077_out([], T573, T574) :|: TRUE f5080_out -> f5077_out(T536, T535, T560) :|: TRUE f5077_in(x239, x240, x241) -> f5080_in :|: TRUE f5077_in([], x242, x243) -> f5079_in :|: TRUE f5082_in(T584, T583, T585, T586) -> f5059_in(T584, T583, .(T585, T586)) :|: TRUE f5059_out(x244, x245, .(x246, x247)) -> f5082_out(x244, x245, x246, x247) :|: TRUE f4666_in(T199, T201, T200, T202) -> f4837_in(T199, T200, T202, T201) :|: T199 > T201 f4838_out(x248, x249, x250, x251) -> f4666_out(x249, x251, x250, x248) :|: x249 <= x251 f4668_out -> f4666_out(x252, x253, x254, x255) :|: TRUE f4666_in(x256, x257, x258, x259) -> f4668_in :|: TRUE f4837_out(x260, x261, x262, x263) -> f4666_out(x260, x263, x261, x262) :|: x260 > x263 f4666_in(x264, x265, x266, x267) -> f4838_in(x267, x264, x266, x265) :|: x264 <= x265 f6001_in(x268, x269, x270, x271, x272, x273, x274, x275, x276, x277, x278, x279, x280, x281, x282, x283, x284, x285, x286, x287, x288, x289) -> f6005_in(x268, x269, x270, x271, x272, x273, x274, x275, x276, x277, x278, x279, x280, x281, x282, x283, x284, x285, x286, x287, x288, x289) :|: TRUE f6001_in(x290, x291, x292, x293, x294, x295, x296, x297, x298, x299, x300, x301, x302, x303, x304, x305, x306, x307, x308, x309, x310, x311) -> f6002_in(x290, x291, x292, x293, x294, x295, x296, x297, x298, x299, x300, x301, x302, x303, x304, x305, x306, x307, x308, x309, x310, x311) :|: TRUE f6002_out(x312, x313, x314, x315, x316, x317, x318, x319, x320, x321, x322, x323, x324, x325, x326, x327, x328, x329, x330, x331, x332, x333) -> f6001_out(x312, x313, x314, x315, x316, x317, x318, x319, x320, x321, x322, x323, x324, x325, x326, x327, x328, x329, x330, x331, x332, x333) :|: TRUE f6005_out(x334, x335, x336, x337, x338, x339, x340, x341, x342, x343, x344, x345, x346, x347, x348, x349, x350, x351, x352, x353, x354, x355) -> f6001_out(x334, x335, x336, x337, x338, x339, x340, x341, x342, x343, x344, x345, x346, x347, x348, x349, x350, x351, x352, x353, x354, x355) :|: TRUE f5993_in(x356, x357, x358, x359, x360, x361, x362, x363, x364, x365, x366, x367, x368, x369, x370, x371, x372, x373, x374, x375, x376, x377, x378) -> f5994_in(x358, x359, x356, x360, x361, x357, x362, x363, x379, x364, x365, x366, x367, x368, x369, x370, x371, x372, x373, x374, x375, x376, x377, x378) :|: x379 = x356 - x357 f5994_out(x380, x381, x382, x383, x384, x385, x386, x387, x388, x389, x390, x391, x392, x393, x394, x395, x396, x397, x398, x399, x400, x401, x402, x403) -> f5993_out(x382, x385, x380, x381, x383, x384, x386, x387, x389, x390, x391, x392, x393, x394, x395, x396, x397, x398, x399, x400, x401, x402, x403) :|: TRUE f6021_out(x404, x405, x406, x407, x408, x409, x410, x411, x412, x413, x414, x415, x416, x417, x418, x419, x420, x421, x422, x423, x424, x425, x426) -> f6020_out(x404, x405, x406, x407, x408, x409, x410, x411, x427, x412, x413, x414, x415, x428, x416, x417, x418, x419, x420, x421, x422, x423, x424, x425, x426) :|: TRUE f6020_in(x429, x430, x431, x432, x433, x434, x435, x436, x437, x438, x439, x440, x441, x442, x443, x444, x445, x446, x447, x448, x449, x450, x451, x452, x453) -> f6021_in(x429, x430, x431, x432, x433, x434, x435, x436, x438, x439, x440, x441, x443, x444, x445, x446, x447, x448, x449, x450, x451, x452, x453) :|: TRUE f6008_in -> f6008_out :|: TRUE f4932_out -> f4929_out(T261, T260, T262, T263) :|: TRUE f4929_in(x454, x455, x456, x457) -> f4932_in :|: TRUE f4929_in([], T282, T283, T284) -> f4931_in :|: TRUE f4931_out -> f4929_out([], x458, x459, x460) :|: TRUE f4938_in(T296, T295, T297, T298, T299) -> f4943_in :|: TRUE f4938_in(.(T340, T341), T342, T343, T344, T345) -> f4942_in(T341, T340, T342, T343, T344, T345) :|: TRUE f4943_out -> f4938_out(x461, x462, x463, x464, x465) :|: TRUE f4942_out(x466, x467, x468, x469, x470, x471) -> f4938_out(.(x467, x466), x468, x469, x470, x471) :|: TRUE f4921_in([], T250, T251) -> f4923_in :|: TRUE f4921_in(T236, T235, T237) -> f4924_in :|: TRUE f4923_out -> f4921_out([], x472, x473) :|: TRUE f4924_out -> f4921_out(x474, x475, x476) :|: TRUE f5982_in -> f5982_out :|: TRUE f5993_out(x477, x478, x479, x480, x481, x482, x483, x484, x485, x486, x487, x488, x489, x490, x491, x492, x493, x494, x495, x496, x497, x498, x499) -> f5990_out(x480, x495, x477, x479, x481, x482, x483, x484, x485, x486, x487, x488, x489, x490, x491, x492, x493, x494, x496, x497, x498, x499) :|: TRUE f5992_out -> f5990_out(x500, x501, x502, x503, x504, x505, x506, x507, x508, x509, x510, x511, x512, x513, x514, x515, x516, x517, x518, x519, x520, x521) :|: TRUE f5990_in(x522, x523, x524, x525, x526, x527, x528, x529, x530, x531, x532, x533, x534, x535, x536, x537, x538, x539, x540, x541, x542, x543) -> f5993_in(x524, x544, x525, x522, x526, x527, x528, x529, x530, x531, x532, x533, x534, x535, x536, x537, x538, x539, x523, x540, x541, x542, x543) :|: x544 = x522 - x523 f5990_in(x545, x546, x547, x548, x549, x550, x551, x552, x553, x554, x555, x556, x557, x558, x559, x560, x561, x562, x563, x564, x565, x566) -> f5992_in :|: TRUE f5079_in -> f5079_out :|: TRUE f4456_in(.(x567, x568), .(x569, x570)) -> f4666_in(x567, x569, x568, x570) :|: TRUE f4666_out(x571, x572, x573, x574) -> f4456_out(.(x571, x573), .(x572, x574)) :|: TRUE f4456_in(x575, x576) -> f4667_in :|: TRUE f4667_out -> f4456_out(x577, x578) :|: TRUE f5880_out(x579, x580, x581, x582, x583, x584, x585, x586, x587, x588, x589, x590, x591, x592, x593, x594, x595, x596) -> f5879_out(x579, x580, x581, x582, x583, x584, x585, x586, x587, x588, x589, x590, x591, x592, x593, x594, x595, x596) :|: TRUE f5879_in(x597, x598, x599, x600, x601, x602, x603, x604, x605, x606, x607, x608, x609, x610, x611, x612, x613, x614) -> f5880_in(x597, x598, x599, x600, x601, x602, x603, x604, x605, x606, x607, x608, x609, x610, x611, x612, x613, x614) :|: TRUE f6081_in(x615, x616, x617, x618, x619, x620, x621, x622, x623, x624, x625, x626, x627, x628, x629, x630, x631, x632, x633, x634, x635, x636, x637, x638) -> f6084_in :|: TRUE f6084_out -> f6081_out(x639, x640, x641, x642, x643, x644, x645, x646, x647, x648, x649, x650, x651, x652, x653, x654, x655, x656, x657, x658, x659, x660, x661, x662) :|: TRUE f6081_in([], x663, x664, x665, x666, x667, x668, x669, x670, x671, x672, x673, x674, x675, x676, x677, x678, x679, x680, x681, x682, x683, x684, x685) -> f6083_in :|: TRUE f6083_out -> f6081_out([], x686, x687, x688, x689, x690, x691, x692, x693, x694, x695, x696, x697, x698, x699, x700, x701, x702, x703, x704, x705, x706, x707, x708) :|: TRUE f5059_in(x709, x710, x711) -> f5076_in(x709, x710, x711) :|: TRUE f5076_out(x712, x713, x714) -> f5059_out(x712, x713, x714) :|: TRUE f4930_out(x715, x716, x717, x718) -> f4928_out(x715, x716, x717, x718) :|: TRUE f4928_in(x719, x720, x721, x722) -> f4929_in(x719, x720, x721, x722) :|: TRUE f4929_out(x723, x724, x725, x726) -> f4928_out(x723, x724, x725, x726) :|: TRUE f4928_in(x727, x728, x729, x730) -> f4930_in(x727, x728, x729, x730) :|: TRUE f5077_out(x731, x732, x733) -> f5076_out(x731, x732, x733) :|: TRUE f5076_in(x734, x735, x736) -> f5078_in(x734, x735, x736) :|: TRUE f5076_in(x737, x738, x739) -> f5077_in(x737, x738, x739) :|: TRUE f5078_out(x740, x741, x742) -> f5076_out(x740, x741, x742) :|: TRUE f6080_out(x743, x744, x745, x746, x747, x748, x749, x750, x751, x752, x753, x754, x755, x756, x757, x758, x759, x760, x761, x762, x763, x764, x765, x766) -> f6044_out(x743, x744, x745, x746, x747, x748, x749, x767, x768, x750, x751, x752, x753, x754, x755, x756, x757, x758, x759, x760, x761, x762, x763, x764, x765, x766) :|: TRUE f6044_in(x769, x770, x771, x772, x773, x774, x775, x776, x777, x778, x779, x780, x781, x782, x783, x784, x785, x786, x787, x788, x789, x790, x791, x792, x793, x794) -> f6080_in(x769, x770, x771, x772, x773, x774, x775, x778, x779, x780, x781, x782, x783, x784, x785, x786, x787, x788, x789, x790, x791, x792, x793, x794) :|: TRUE f6031_in -> f6031_out :|: TRUE f5980_in(x795, x796, x797, x798, x799, x800, x801, x802, x803, x804, x805, x806, x807, x808, x809, x810, x811, x812, x813, x814, x815) -> f5983_in :|: TRUE f5980_in([], x816, x817, x818, x819, x820, x821, x822, x823, x824, x825, x826, x827, x828, x829, x830, x831, x832, x833, x834, x835) -> f5982_in :|: TRUE f5982_out -> f5980_out([], x836, x837, x838, x839, x840, x841, x842, x843, x844, x845, x846, x847, x848, x849, x850, x851, x852, x853, x854, x855) :|: TRUE f5983_out -> f5980_out(x856, x857, x858, x859, x860, x861, x862, x863, x864, x865, x866, x867, x868, x869, x870, x871, x872, x873, x874, x875, x876) :|: TRUE f4934_out(x877, x878, x879, x880, x881) -> f4930_out(.(x878, x877), x879, x880, x881) :|: TRUE f4930_in(x882, x883, x884, x885) -> f4935_in :|: TRUE f4935_out -> f4930_out(x886, x887, x888, x889) :|: TRUE f4930_in(.(x890, x891), x892, x893, x894) -> f4934_in(x891, x890, x892, x893, x894) :|: TRUE f4451_in(x895, x896) -> f4456_in(x895, x896) :|: TRUE f4455_out(x897, x898) -> f4451_out(x897, x898) :|: TRUE f4451_in(x899, x900) -> f4455_in(x899, x900) :|: TRUE f4456_out(x901, x902) -> f4451_out(x901, x902) :|: TRUE f5931_in(x903, x904, x905, x906, x907, x908, x909, x910, x911, x912, x913, x914, x915, x916, x917, x918, x919, x920, x921, x922) -> f5933_in :|: TRUE f5933_out -> f5931_out(x923, x924, x925, x926, x927, x928, x929, x930, x931, x932, x933, x934, x935, x936, x937, x938, x939, x940, x941, x942) :|: TRUE f5931_in(x943, x944, x945, x946, x947, x948, x949, x950, x951, x952, x953, x954, x955, x956, x957, x958, x959, x960, x961, x962) -> f5934_in(x945, x963, x946, x943, x947, x948, x949, x944, x950, x951, x952, x953, x954, x955, x956, x957, x958, x959, x960, x961, x962) :|: x963 = x943 - x944 f5934_out(x964, x965, x966, x967, x968, x969, x970, x971, x972, x973, x974, x975, x976, x977, x978, x979, x980, x981, x982, x983, x984) -> f5931_out(x967, x971, x964, x966, x968, x969, x970, x972, x973, x974, x975, x976, x977, x978, x979, x980, x981, x982, x983, x984) :|: TRUE f4957_in(T396, T395, T397, T398, T399, T400, T401) -> f4958_in(T396, T395, T397, T398, T399, T400, T401) :|: TRUE f4958_out(x985, x986, x987, x988, x989, x990, x991) -> f4957_out(x985, x986, x987, x988, x989, x990, x991) :|: TRUE f4957_in(x992, x993, x994, x995, x996, x997, x998) -> f4959_in(x992, x993, x994, x995, x996, x997, x998) :|: TRUE f4959_out(x999, x1000, x1001, x1002, x1003, x1004, x1005) -> f4957_out(x999, x1000, x1001, x1002, x1003, x1004, x1005) :|: TRUE f4939_in -> f4939_out :|: TRUE f5994_in(x1006, x1007, x1008, x1009, x1010, x1011, x1012, x1013, x1014, x1015, x1016, x1017, x1018, x1019, x1020, x1021, x1022, x1023, x1024, x1025, x1026, x1027, x1028, x1029) -> f6001_in(x1006, x1007, x1008, x1009, x1010, x1012, x1013, x1015, x1016, x1017, x1018, x1019, x1020, x1021, x1022, x1023, x1024, x1025, x1026, x1027, x1028, x1029) :|: TRUE f6001_out(x1030, x1031, x1032, x1033, x1034, x1035, x1036, x1037, x1038, x1039, x1040, x1041, x1042, x1043, x1044, x1045, x1046, x1047, x1048, x1049, x1050, x1051) -> f5994_out(x1030, x1031, x1032, x1033, x1034, x1052, x1035, x1036, x1053, x1037, x1038, x1039, x1040, x1041, x1042, x1043, x1044, x1045, x1046, x1047, x1048, x1049, x1050, x1051) :|: TRUE f5846_out(x1054, x1055, x1056, x1057, x1058, x1059, x1060, x1061, x1062, x1063, x1064, x1065, x1066, x1067, x1068, x1069, x1070) -> f5833_out(x1054, x1055, x1056, x1057, x1058, x1059, x1060, x1061, x1062, x1063, x1064, x1065, x1066, x1067, x1068, x1069, x1070) :|: TRUE f5833_in(x1071, x1072, x1073, x1074, x1075, x1076, x1077, x1078, x1079, x1080, x1081, x1082, x1083, x1084, x1085, x1086, x1087) -> f5846_in(x1071, x1072, x1073, x1074, x1075, x1076, x1077, x1078, x1079, x1080, x1081, x1082, x1083, x1084, x1085, x1086, x1087) :|: TRUE f5833_in(x1088, x1089, x1090, x1091, x1092, x1093, x1094, x1095, x1096, x1097, x1098, x1099, x1100, x1101, x1102, x1103, x1104) -> f5847_in :|: TRUE f5847_out -> f5833_out(x1105, x1106, x1107, x1108, x1109, x1110, x1111, x1112, x1113, x1114, x1115, x1116, x1117, x1118, x1119, x1120, x1121) :|: TRUE f6092_in(x1122, x1123, x1124, x1125, x1126, x1127, x1128, x1129, x1130, x1131, x1132, x1133, x1134, x1135, x1136, x1137, x1138, x1139, x1140) -> f6094_in(x1122, x1123, x1124, x1125, x1126, x1127, x1128, x1129, x1130, x1131, x1132, x1133, x1134, x1135, x1136, x1137, x1138, x1139, x1140) :|: TRUE f6094_out(x1141, x1142, x1143, x1144, x1145, x1146, x1147, x1148, x1149, x1150, x1151, x1152, x1153, x1154, x1155, x1156, x1157, x1158, x1159) -> f6092_out(x1141, x1142, x1143, x1144, x1145, x1146, x1147, x1148, x1149, x1150, x1151, x1152, x1153, x1154, x1155, x1156, x1157, x1158, x1159) :|: TRUE f6092_in(x1160, x1161, x1162, x1163, x1164, x1165, x1166, x1167, x1168, x1169, x1170, x1171, x1172, x1173, x1174, x1175, x1176, x1177, x1178) -> f6093_in(x1160, x1161, x1162, x1163, x1164, x1165, x1166, x1167, x1168, x1169, x1170, x1171, x1172, x1173, x1174, x1175, x1176, x1177, x1178) :|: TRUE f6093_out(x1179, x1180, x1181, x1182, x1183, x1184, x1185, x1186, x1187, x1188, x1189, x1190, x1191, x1192, x1193, x1194, x1195, x1196, x1197) -> f6092_out(x1179, x1180, x1181, x1182, x1183, x1184, x1185, x1186, x1187, x1188, x1189, x1190, x1191, x1192, x1193, x1194, x1195, x1196, x1197) :|: TRUE f4902_in(T121) -> f4904_in(T121) :|: TRUE f4904_out(x1198) -> f4902_out(x1198) :|: TRUE f5969_in(x1199, x1200, x1201, x1202, x1203, x1204, x1205, x1206, x1207, x1208, x1209, x1210, x1211, x1212, x1213, x1214, x1215, x1216, x1217, x1218, x1219, x1220) -> f5970_in(x1201, x1202, x1199, x1203, x1204, x1205, x1206, x1221, x1207, x1208, x1209, x1210, x1211, x1212, x1213, x1200, x1214, x1215, x1216, x1217, x1218, x1219, x1220) :|: x1221 = x1199 - x1200 f5970_out(x1222, x1223, x1224, x1225, x1226, x1227, x1228, x1229, x1230, x1231, x1232, x1233, x1234, x1235, x1236, x1237, x1238, x1239, x1240, x1241, x1242, x1243, x1244) -> f5969_out(x1224, x1237, x1222, x1223, x1225, x1226, x1227, x1228, x1230, x1231, x1232, x1233, x1234, x1235, x1236, x1238, x1239, x1240, x1241, x1242, x1243, x1244) :|: TRUE f4947_in([], T376, T377, T378, T379, T380) -> f4952_in :|: TRUE f4947_in(x1245, x1246, x1247, x1248, x1249, x1250) -> f4953_in :|: TRUE f4953_out -> f4947_out(x1251, x1252, x1253, x1254, x1255, x1256) :|: TRUE f4952_out -> f4947_out([], x1257, x1258, x1259, x1260, x1261) :|: TRUE f4911_out -> f4906_out(x1262) :|: TRUE f4910_out(T221, T220) -> f4906_out(.(T220, T221)) :|: TRUE f4906_in(.(x1263, x1264)) -> f4910_in(x1264, x1263) :|: TRUE f4906_in(x1265) -> f4911_in :|: TRUE f6043_out(x1266, x1267, x1268, x1269, x1270, x1271, x1272, x1273, x1274, x1275, x1276, x1277, x1278, x1279, x1280, x1281, x1282, x1283, x1284, x1285, x1286, x1287, x1288, x1289, x1290) -> f6040_out(x1269, x1280, x1266, x1268, x1270, x1271, x1272, x1273, x1274, x1275, x1276, x1277, x1278, x1279, x1281, x1282, x1283, x1284, x1285, x1286, x1287, x1288, x1289, x1290) :|: TRUE f6040_in(x1291, x1292, x1293, x1294, x1295, x1296, x1297, x1298, x1299, x1300, x1301, x1302, x1303, x1304, x1305, x1306, x1307, x1308, x1309, x1310, x1311, x1312, x1313, x1314) -> f6042_in :|: TRUE f6042_out -> f6040_out(x1315, x1316, x1317, x1318, x1319, x1320, x1321, x1322, x1323, x1324, x1325, x1326, x1327, x1328, x1329, x1330, x1331, x1332, x1333, x1334, x1335, x1336, x1337, x1338) :|: TRUE f6040_in(x1339, x1340, x1341, x1342, x1343, x1344, x1345, x1346, x1347, x1348, x1349, x1350, x1351, x1352, x1353, x1354, x1355, x1356, x1357, x1358, x1359, x1360, x1361, x1362) -> f6043_in(x1341, x1363, x1342, x1339, x1343, x1344, x1345, x1346, x1347, x1348, x1349, x1350, x1351, x1352, x1340, x1353, x1354, x1355, x1356, x1357, x1358, x1359, x1360, x1361, x1362) :|: x1363 = x1339 - x1340 f5748_out(x1364, x1365, .(x1366, x1367), x1368, x1369, x1370, x1371, x1372, x1373, x1374, x1375, x1376, x1377, x1378, x1379, x1380, x1381) -> f5862_out(x1364, x1365, x1366, x1367, x1368, x1369, x1370, x1371, x1372, x1373, x1374, x1375, x1376, x1377, x1378, x1379, x1380, x1381, x1382) :|: TRUE f5862_in(x1383, x1384, x1385, x1386, x1387, x1388, x1389, x1390, x1391, x1392, x1393, x1394, x1395, x1396, x1397, x1398, x1399, x1400, x1401) -> f5748_in(x1383, x1384, .(x1385, x1386), x1387, x1388, x1389, x1390, x1391, x1392, x1393, x1394, x1395, x1396, x1397, x1398, x1399, x1400) :|: TRUE f4960_in -> f4960_out :|: TRUE f6109_in -> f6110_in :|: TRUE f6110_out -> f6109_out :|: TRUE f6110_in -> f6110_out :|: TRUE f4907_in -> f4907_out :|: TRUE f4407_out(T104) -> f4415_out(T104) :|: TRUE f4415_in(x1402) -> f4407_in(x1402) :|: TRUE f4957_out(x1403, x1404, x1405, x1406, x1407, x1408, x1409) -> f4955_out(x1403, x1404, x1405, x1406, x1407, x1408, x1409) :|: TRUE f4955_in(x1410, x1411, x1412, x1413, x1414, x1415, x1416) -> f4957_in(x1410, x1411, x1412, x1413, x1414, x1415, x1416) :|: TRUE f6081_out(x1417, x1418, x1419, x1420, x1421, x1422, x1423, x1424, x1425, x1426, x1427, x1428, x1429, x1430, x1431, x1432, x1433, x1434, x1435, x1436, x1437, x1438, x1439, x1440) -> f6080_out(x1417, x1418, x1419, x1420, x1421, x1422, x1423, x1424, x1425, x1426, x1427, x1428, x1429, x1430, x1431, x1432, x1433, x1434, x1435, x1436, x1437, x1438, x1439, x1440) :|: TRUE f6080_in(x1441, x1442, x1443, x1444, x1445, x1446, x1447, x1448, x1449, x1450, x1451, x1452, x1453, x1454, x1455, x1456, x1457, x1458, x1459, x1460, x1461, x1462, x1463, x1464) -> f6082_in(x1441, x1442, x1443, x1444, x1445, x1446, x1447, x1448, x1449, x1450, x1451, x1452, x1453, x1454, x1455, x1456, x1457, x1458, x1459, x1460, x1461, x1462, x1463, x1464) :|: TRUE f6082_out(x1465, x1466, x1467, x1468, x1469, x1470, x1471, x1472, x1473, x1474, x1475, x1476, x1477, x1478, x1479, x1480, x1481, x1482, x1483, x1484, x1485, x1486, x1487, x1488) -> f6080_out(x1465, x1466, x1467, x1468, x1469, x1470, x1471, x1472, x1473, x1474, x1475, x1476, x1477, x1478, x1479, x1480, x1481, x1482, x1483, x1484, x1485, x1486, x1487, x1488) :|: TRUE f6080_in(x1489, x1490, x1491, x1492, x1493, x1494, x1495, x1496, x1497, x1498, x1499, x1500, x1501, x1502, x1503, x1504, x1505, x1506, x1507, x1508, x1509, x1510, x1511, x1512) -> f6081_in(x1489, x1490, x1491, x1492, x1493, x1494, x1495, x1496, x1497, x1498, x1499, x1500, x1501, x1502, x1503, x1504, x1505, x1506, x1507, x1508, x1509, x1510, x1511, x1512) :|: TRUE f5846_in(x1513, x1514, x1515, x1516, x1517, x1518, x1519, x1520, x1521, x1522, x1523, x1524, x1525, x1526, x1527, x1528, x1529) -> f5851_in(x1513, x1514, x1515, x1516) :|: TRUE f5851_out(x1530, x1531, x1532, x1533) -> f5852_in(x1531, x1532, x1534, x1530, x1533, x1535, x1536, x1537, x1538, x1539, x1540, x1541, x1542, x1543, x1544, x1545, x1546, x1547) :|: TRUE f5852_out(x1548, x1549, x1550, x1551, x1552, x1553, x1554, x1555, x1556, x1557, x1558, x1559, x1560, x1561, x1562, x1563, x1564, x1565) -> f5846_out(x1551, x1548, x1549, x1552, x1553, x1554, x1555, x1556, x1557, x1558, x1559, x1560, x1561, x1562, x1563, x1564, x1565) :|: TRUE f4407_out(T1281) -> f5851_out(T1281, T1282, T1283, T1286) :|: TRUE f5851_in(x1566, x1567, x1568, x1569) -> f4407_in(x1566) :|: TRUE f5865_in(x1570, x1571, x1572, x1573, x1574, x1575, x1576, x1577, x1578, x1579, x1580, x1581, x1582, x1583, x1584, x1585, x1586, x1587) -> f5876_in(x1570, x1571, x1572, x1574, x1573, x1575, x1576, x1577, x1578, x1579, x1580, x1581, x1582, x1583, x1584, x1585, x1586, x1587) :|: TRUE f5876_out(x1588, x1589, x1590, x1591, x1592, x1593, x1594, x1595, x1596, x1597, x1598, x1599, x1600, x1601, x1602, x1603, x1604, x1605) -> f5865_out(x1588, x1589, x1590, x1592, x1591, x1593, x1594, x1595, x1596, x1597, x1598, x1599, x1600, x1601, x1602, x1603, x1604, x1605) :|: TRUE f4936_out(x1606, x1607, x1608, x1609, x1610) -> f4934_out(x1606, x1607, x1608, x1609, x1610) :|: TRUE f4934_in(x1611, x1612, x1613, x1614, x1615) -> f4936_in(x1611, x1612, x1613, x1614, x1615) :|: TRUE f5880_in(x1616, x1617, x1618, x1619, x1620, x1621, x1622, x1623, x1624, x1625, x1626, x1627, x1628, x1629, x1630, x1631, x1632, x1633) -> f5887_in(x1618, x1616, x1619, x1617, x1620, x1621, x1622, x1623, x1624, x1625, x1626, x1627, x1628, x1629, x1630, x1631, x1632, x1633) :|: TRUE f5887_out(x1634, x1635, x1636, x1637, x1638, x1639, x1640, x1641, x1642, x1643, x1644, x1645, x1646, x1647, x1648, x1649, x1650, x1651) -> f5880_out(x1635, x1637, x1634, x1636, x1638, x1639, x1640, x1641, x1642, x1643, x1644, x1645, x1646, x1647, x1648, x1649, x1650, x1651) :|: TRUE f6098_in(x1652, x1653, x1654, x1655, x1656, x1657, x1658, x1659, x1660, x1661, x1662, x1663, x1664, x1665, x1666, x1667, x1668, x1669, x1670, x1671) -> f6101_in(x1654, x1672, x1655, x1652, x1656, x1662, x1663, x1657, x1658, x1671, x1664, x1665, x1666, x1653, x1667, x1668, x1659, x1669, x1660, x1661, x1670) :|: x1672 = x1652 - x1653 f6101_out(x1673, x1674, x1675, x1676, x1677, x1678, x1679, x1680, x1681, x1682, x1683, x1684, x1685, x1686, x1687, x1688, x1689, x1690, x1691, x1692, x1693) -> f6098_out(x1676, x1686, x1673, x1675, x1677, x1680, x1681, x1689, x1691, x1692, x1678, x1679, x1683, x1684, x1685, x1687, x1688, x1690, x1693, x1682) :|: TRUE f6100_out -> f6098_out(x1694, x1695, x1696, x1697, x1698, x1699, x1700, x1701, x1702, x1703, x1704, x1705, x1706, x1707, x1708, x1709, x1710, x1711, x1712, x1713) :|: TRUE f6098_in(x1714, x1715, x1716, x1717, x1718, x1719, x1720, x1721, x1722, x1723, x1724, x1725, x1726, x1727, x1728, x1729, x1730, x1731, x1732, x1733) -> f6100_in :|: TRUE f4969_out(T461, T460, T462, T463, T464, T465, T466, T467) -> f4963_out(T461, T460, T462, T463, T464, T465, T466, T467) :|: TRUE f4963_in(x1734, x1735, x1736, x1737, x1738, x1739, x1740, x1741) -> f4969_in(x1734, x1735, x1736, x1737, x1738, x1739, x1740, x1741) :|: TRUE f4914_in(x1742, x1743) -> f4919_in :|: TRUE f4919_out -> f4914_out(x1744, x1745) :|: TRUE f4918_out(x1746, x1747, x1748) -> f4914_out(.(x1747, x1746), x1748) :|: TRUE f4914_in(.(x1749, x1750), x1751) -> f4918_in(x1750, x1749, x1751) :|: TRUE f4944_out(x1752, x1753, x1754, x1755, x1756, x1757) -> f4942_out(x1752, x1753, x1754, x1755, x1756, x1757) :|: TRUE f4942_in(x1758, x1759, x1760, x1761, x1762, x1763) -> f4944_in(x1758, x1759, x1760, x1761, x1762, x1763) :|: TRUE f5946_in -> f5946_out :|: TRUE f4920_in(x1764, x1765, x1766) -> f4922_in(x1764, x1765, x1766) :|: TRUE f4920_in(x1767, x1768, x1769) -> f4921_in(x1767, x1768, x1769) :|: TRUE f4921_out(x1770, x1771, x1772) -> f4920_out(x1770, x1771, x1772) :|: TRUE f4922_out(x1773, x1774, x1775) -> f4920_out(x1773, x1774, x1775) :|: TRUE f4960_out -> f4958_out([], T438, T439, T440, T441, T442, T443) :|: TRUE f4961_out -> f4958_out(x1776, x1777, x1778, x1779, x1780, x1781, x1782) :|: TRUE f4958_in(x1783, x1784, x1785, x1786, x1787, x1788, x1789) -> f4961_in :|: TRUE f4958_in([], x1790, x1791, x1792, x1793, x1794, x1795) -> f4960_in :|: TRUE f4969_in(x1796, x1797, x1798, x1799, x1800, x1801, x1802, x1803) -> f5052_in(x1796, x1797, x1798, x1799, x1800, x1801, x1802, x1803) :|: TRUE f5052_out(x1804, x1805, x1806, x1807, x1808, x1809, x1810, x1811) -> f4969_out(x1804, x1805, x1806, x1807, x1808, x1809, x1810, x1811) :|: TRUE f5053_out(x1812, x1813, x1814, x1815, x1816, x1817, x1818, x1819) -> f4969_out(x1812, x1813, x1814, x1815, x1816, x1817, x1818, x1819) :|: TRUE f4969_in(x1820, x1821, x1822, x1823, x1824, x1825, x1826, x1827) -> f5053_in(x1820, x1821, x1822, x1823, x1824, x1825, x1826, x1827) :|: TRUE f6031_out -> f6022_out([], x1828, x1829, x1830, x1831, x1832, x1833, x1834, x1835, x1836, x1837, x1838, x1839, x1840, x1841, x1842, x1843, x1844, x1845, x1846, x1847, x1848, x1849) :|: TRUE f6022_in([], x1850, x1851, x1852, x1853, x1854, x1855, x1856, x1857, x1858, x1859, x1860, x1861, x1862, x1863, x1864, x1865, x1866, x1867, x1868, x1869, x1870, x1871) -> f6031_in :|: TRUE f6022_in(x1872, x1873, x1874, x1875, x1876, x1877, x1878, x1879, x1880, x1881, x1882, x1883, x1884, x1885, x1886, x1887, x1888, x1889, x1890, x1891, x1892, x1893, x1894) -> f6032_in :|: TRUE f6032_out -> f6022_out(x1895, x1896, x1897, x1898, x1899, x1900, x1901, x1902, x1903, x1904, x1905, x1906, x1907, x1908, x1909, x1910, x1911, x1912, x1913, x1914, x1915, x1916, x1917) :|: TRUE f6091_out(x1918, x1919, x1920, x1921, x1922, x1923, x1924, x1925, x1926, x1927, x1928, x1929, x1930, x1931, x1932, x1933, x1934, x1935, .(x1936, .(x1937, .(x1938, .(x1939, .(x1940, .(x1941, .(x1942, x1918)))))))) -> f6090_out(x1918, x1919, x1920, x1921, x1937, x1922, x1923, x1943, x1924, x1944, x1925, x1926, x1927, x1928, x1929, x1941, x1930, x1931, x1932, x1933, x1936, x1938, x1939, x1940, x1934, x1942, x1935) :|: TRUE f6090_in(x1945, x1946, x1947, x1948, x1949, x1950, x1951, x1952, x1953, x1954, x1955, x1956, x1957, x1958, x1959, x1960, x1961, x1962, x1963, x1964, x1965, x1966, x1967, x1968, x1969, x1970, x1971) -> f6091_in(x1945, x1946, x1947, x1948, x1950, x1951, x1953, x1955, x1956, x1957, x1958, x1959, x1961, x1962, x1963, x1964, x1969, x1971, .(x1965, .(x1949, .(x1966, .(x1967, .(x1968, .(x1960, .(x1970, x1945)))))))) :|: TRUE f5977_out(x1972, x1973, x1974, x1975, x1976, x1977, x1978, x1979, x1980, x1981, x1982, x1983, x1984, x1985, x1986, x1987, x1988, x1989, x1990, x1991, x1992) -> f5970_out(x1972, x1973, x1974, x1975, x1976, x1977, x1978, x1993, x1979, x1980, x1981, x1982, x1983, x1984, x1985, x1994, x1986, x1987, x1988, x1989, x1990, x1991, x1992) :|: TRUE f5970_in(x1995, x1996, x1997, x1998, x1999, x2000, x2001, x2002, x2003, x2004, x2005, x2006, x2007, x2008, x2009, x2010, x2011, x2012, x2013, x2014, x2015, x2016, x2017) -> f5977_in(x1995, x1996, x1997, x1998, x1999, x2000, x2001, x2003, x2004, x2005, x2006, x2007, x2008, x2009, x2011, x2012, x2013, x2014, x2015, x2016, x2017) :|: TRUE f5879_out(x2018, x2019, x2020, x2021, x2022, x2023, x2024, x2025, x2026, x2027, x2028, x2029, x2030, x2031, x2032, x2033, x2034, x2035) -> f5877_out(x2018, x2019, x2020, x2021, x2022, x2023, x2024, x2025, x2026, x2027, x2028, x2029, x2030, x2031, x2032, x2033, x2034, x2035) :|: TRUE f5877_in(x2036, x2037, x2038, x2039, x2040, x2041, x2042, x2043, x2044, x2045, x2046, x2047, x2048, x2049, x2050, x2051, x2052, x2053) -> f5879_in(x2036, x2037, x2038, x2039, x2040, x2041, x2042, x2043, x2044, x2045, x2046, x2047, x2048, x2049, x2050, x2051, x2052, x2053) :|: TRUE f6089_in(x2054, x2055, x2056, x2057, x2058, x2059, x2060, x2061, x2062, x2063, x2064, x2065, x2066, x2067, x2068, x2069, x2070, x2071, x2072, x2073, x2074, x2075, x2076, x2077, x2078, x2079) -> f6090_in(x2056, x2057, x2054, x2058, x2059, x2060, x2061, x2080, x2062, x2055, x2063, x2064, x2065, x2066, x2067, x2068, x2069, x2070, x2071, x2072, x2073, x2074, x2075, x2076, x2077, x2078, x2079) :|: x2080 = x2054 - x2055 f6090_out(x2081, x2082, x2083, x2084, x2085, x2086, x2087, x2088, x2089, x2090, x2091, x2092, x2093, x2094, x2095, x2096, x2097, x2098, x2099, x2100, x2101, x2102, x2103, x2104, x2105, x2106, x2107) -> f6089_out(x2083, x2090, x2081, x2082, x2084, x2085, x2086, x2087, x2089, x2091, x2092, x2093, x2094, x2095, x2096, x2097, x2098, x2099, x2100, x2101, x2102, x2103, x2104, x2105, x2106, x2107) :|: TRUE f6103_in(x2108, x2109, x2110, x2111, x2112, x2113, x2114, x2115, x2116, x2117, x2118, x2119, x2120, x2121, x2122, x2123, x2124, x2125, x2126, x2127, x2128, x2129) -> f6091_in(x2108, x2109, x2110, x2111, x2114, x2115, x2124, x2127, x2128, x2112, x2113, x2118, x2119, x2120, x2122, x2123, x2126, x2129, x2116) :|: TRUE f6091_out(x2130, x2131, x2132, x2133, x2134, x2135, x2136, x2137, x2138, x2139, x2140, x2141, x2142, x2143, x2144, x2145, x2146, x2147, x2148) -> f6103_out(x2130, x2131, x2132, x2133, x2139, x2140, x2134, x2135, x2148, x2149, x2141, x2142, x2143, x2150, x2144, x2145, x2136, x2151, x2146, x2137, x2138, x2147) :|: TRUE f5748_in(x2152, x2153, x2154, x2155, x2156, x2157, x2158, x2159, x2160, x2161, x2162, x2163, x2164, x2165, x2166, x2167, x2168) -> f5830_in(x2152, x2153, x2154, x2155, x2156, x2157, x2158, x2159, x2160, x2161, x2162, x2163, x2164, x2165, x2166, x2167, x2168) :|: TRUE f5830_out(x2169, x2170, x2171, x2172, x2173, x2174, x2175, x2176, x2177, x2178, x2179, x2180, x2181, x2182, x2183, x2184, x2185) -> f5748_out(x2169, x2170, x2171, x2172, x2173, x2174, x2175, x2176, x2177, x2178, x2179, x2180, x2181, x2182, x2183, x2184, x2185) :|: TRUE f5865_out(x2186, x2187, x2188, x2189, x2190, x2191, x2192, x2193, x2194, x2195, x2196, x2197, x2198, x2199, x2200, x2201, x2202, x2203) -> f5863_out(x2186, x2187, x2188, x2189, x2190, x2191, x2192, x2193, x2194, x2195, x2196, x2197, x2198, x2199, x2200, x2201, x2202, x2203) :|: TRUE f5863_in(x2204, x2205, x2206, x2207, x2208, x2209, x2210, x2211, x2212, x2213, x2214, x2215, x2216, x2217, x2218, x2219, x2220, x2221) -> f5865_in(x2204, x2205, x2206, x2207, x2208, x2209, x2210, x2211, x2212, x2213, x2214, x2215, x2216, x2217, x2218, x2219, x2220, x2221) :|: TRUE f5892_in(x2222, x2223, x2224, x2225, x2226, x2227, x2228, x2229, x2230, x2231, x2232, x2233, x2234, x2235, x2236, x2237, x2238, x2239) -> f5898_in(x2222, x2223, x2224, x2225, x2226, x2227, x2228, x2229, x2230, x2231, x2232, x2233, x2234, x2235, x2236, x2237, x2238, x2239) :|: TRUE f5899_out(x2240, x2241, x2242, x2243, x2244, x2245, x2246, x2247, x2248, x2249, x2250, x2251, x2252, x2253, x2254, x2255, x2256, x2257) -> f5892_out(x2240, x2241, x2242, x2243, x2244, x2245, x2246, x2247, x2248, x2249, x2250, x2251, x2252, x2253, x2254, x2255, x2256, x2257) :|: TRUE f5898_out(x2258, x2259, x2260, x2261, x2262, x2263, x2264, x2265, x2266, x2267, x2268, x2269, x2270, x2271, x2272, x2273, x2274, x2275) -> f5892_out(x2258, x2259, x2260, x2261, x2262, x2263, x2264, x2265, x2266, x2267, x2268, x2269, x2270, x2271, x2272, x2273, x2274, x2275) :|: TRUE f5892_in(x2276, x2277, x2278, x2279, x2280, x2281, x2282, x2283, x2284, x2285, x2286, x2287, x2288, x2289, x2290, x2291, x2292, x2293) -> f5899_in(x2276, x2277, x2278, x2279, x2280, x2281, x2282, x2283, x2284, x2285, x2286, x2287, x2288, x2289, x2290, x2291, x2292, x2293) :|: TRUE f5053_in(.(x2294, x2295), x2296, x2297, x2298, x2299, x2300, x2301, x2302) -> f5057_in(x2295, x2294, x2296, x2297, x2298, x2299, x2300, x2301, x2302) :|: TRUE f5053_in(x2303, x2304, x2305, x2306, x2307, x2308, x2309, x2310) -> f5058_in :|: TRUE f5057_out(x2311, x2312, x2313, x2314, x2315, x2316, x2317, x2318, x2319) -> f5053_out(.(x2312, x2311), x2313, x2314, x2315, x2316, x2317, x2318, x2319) :|: TRUE f5058_out -> f5053_out(x2320, x2321, x2322, x2323, x2324, x2325, x2326, x2327) :|: TRUE f5082_out(x2328, x2329, x2330, x2331) -> f5078_out(.(x2329, x2328), x2330, x2331) :|: TRUE f5078_in(x2332, x2333, x2334) -> f5083_in :|: TRUE f5078_in(.(x2335, x2336), x2337, x2338) -> f5082_in(x2336, x2335, x2337, x2338) :|: TRUE f5083_out -> f5078_out(x2339, x2340, x2341) :|: TRUE f5899_in(x2342, x2343, x2344, x2345, x2346, x2347, x2348, x2349, x2350, x2351, x2352, x2353, x2354, x2355, x2356, x2357, x2358, x2359) -> f5910_in :|: TRUE f5910_out -> f5899_out(x2360, x2361, x2362, x2363, x2364, x2365, x2366, x2367, x2368, x2369, x2370, x2371, x2372, x2373, x2374, x2375, x2376, x2377) :|: TRUE f5899_in(.(x2378, x2379), x2380, x2381, x2382, x2383, x2384, x2385, x2386, x2387, x2388, x2389, x2390, x2391, x2392, x2393, x2394, x2395, x2396) -> f5909_in(x2380, x2378, x2381, x2379, x2382, x2383, x2384, x2385, x2386, x2387, x2388, x2389, x2390, x2391, x2392, x2393, x2394, x2395, x2396) :|: TRUE f5909_out(x2397, x2398, x2399, x2400, x2401, x2402, x2403, x2404, x2405, x2406, x2407, x2408, x2409, x2410, x2411, x2412, x2413, x2414, x2415) -> f5899_out(.(x2398, x2400), x2397, x2399, x2401, x2402, x2403, x2404, x2405, x2406, x2407, x2408, x2409, x2410, x2411, x2412, x2413, x2414, x2415) :|: TRUE f5966_in(x2416, x2417, x2418, x2419, x2420, x2421, x2422, x2423, x2424, x2425, x2426, x2427, x2428, x2429, x2430, x2431, x2432, x2433, x2434, x2435, x2436) -> f5969_in(x2418, x2437, x2419, x2416, x2420, x2421, x2422, x2423, x2424, x2425, x2426, x2427, x2428, x2429, x2430, x2431, x2417, x2432, x2433, x2434, x2435, x2436) :|: x2437 = x2416 - x2417 f5968_out -> f5966_out(x2438, x2439, x2440, x2441, x2442, x2443, x2444, x2445, x2446, x2447, x2448, x2449, x2450, x2451, x2452, x2453, x2454, x2455, x2456, x2457, x2458) :|: TRUE f5966_in(x2459, x2460, x2461, x2462, x2463, x2464, x2465, x2466, x2467, x2468, x2469, x2470, x2471, x2472, x2473, x2474, x2475, x2476, x2477, x2478, x2479) -> f5968_in :|: TRUE f5969_out(x2480, x2481, x2482, x2483, x2484, x2485, x2486, x2487, x2488, x2489, x2490, x2491, x2492, x2493, x2494, x2495, x2496, x2497, x2498, x2499, x2500, x2501) -> f5966_out(x2483, x2496, x2480, x2482, x2484, x2485, x2486, x2487, x2488, x2489, x2490, x2491, x2492, x2493, x2494, x2495, x2497, x2498, x2499, x2500, x2501) :|: TRUE f5935_in(x2502, x2503, x2504, x2505, x2506, x2507, x2508, x2509, x2510, x2511, x2512, x2513, x2514, x2515, x2516, x2517, x2518, x2519, x2520, x2521, x2522, x2523) -> f5939_in(x2502, x2503, x2504, x2505, x2506, x2508, x2509, x2510, x2511, x2512, x2513, x2514, x2515, x2516, x2517, x2519, x2520, x2521, x2522, x2523) :|: TRUE f5939_out(x2524, x2525, x2526, x2527, x2528, x2529, x2530, x2531, x2532, x2533, x2534, x2535, x2536, x2537, x2538, x2539, x2540, x2541, x2542, x2543) -> f5935_out(x2524, x2525, x2526, x2527, x2528, x2544, x2529, x2530, x2531, x2532, x2533, x2534, x2535, x2536, x2537, x2538, x2545, x2539, x2540, x2541, x2542, x2543) :|: TRUE f4905_out(x2546) -> f4904_out(x2546) :|: TRUE f4906_out(x2547) -> f4904_out(x2547) :|: TRUE f4904_in(x2548) -> f4906_in(x2548) :|: TRUE f4904_in(x2549) -> f4905_in(x2549) :|: TRUE f4412_in -> f4412_out :|: TRUE f4936_in(x2550, x2551, x2552, x2553, x2554) -> f4937_in(x2550, x2551, x2552, x2553, x2554) :|: TRUE f4936_in(x2555, x2556, x2557, x2558, x2559) -> f4938_in(x2555, x2556, x2557, x2558, x2559) :|: TRUE f4938_out(x2560, x2561, x2562, x2563, x2564) -> f4936_out(x2560, x2561, x2562, x2563, x2564) :|: TRUE f4937_out(x2565, x2566, x2567, x2568, x2569) -> f4936_out(x2565, x2566, x2567, x2568, x2569) :|: TRUE f4913_out(x2570, x2571) -> f4912_out(x2570, x2571) :|: TRUE f4914_out(x2572, x2573) -> f4912_out(x2572, x2573) :|: TRUE f4912_in(x2574, x2575) -> f4914_in(x2574, x2575) :|: TRUE f4912_in(x2576, x2577) -> f4913_in(x2576, x2577) :|: TRUE f5052_in(x2578, x2579, x2580, x2581, x2582, x2583, x2584, x2585) -> f5055_in :|: TRUE f5052_in([], T510, T511, T512, T513, T514, T515, T516) -> f5054_in :|: TRUE f5055_out -> f5052_out(x2586, x2587, x2588, x2589, x2590, x2591, x2592, x2593) :|: TRUE f5054_out -> f5052_out([], x2594, x2595, x2596, x2597, x2598, x2599, x2600) :|: TRUE f4964_out -> f4959_out(x2601, x2602, x2603, x2604, x2605, x2606, x2607) :|: TRUE f4959_in(x2608, x2609, x2610, x2611, x2612, x2613, x2614) -> f4964_in :|: TRUE f4959_in(.(x2615, x2616), x2617, x2618, x2619, x2620, x2621, x2622) -> f4963_in(x2616, x2615, x2617, x2618, x2619, x2620, x2621, x2622) :|: TRUE f4963_out(x2623, x2624, x2625, x2626, x2627, x2628, x2629, x2630) -> f4959_out(.(x2624, x2623), x2625, x2626, x2627, x2628, x2629, x2630) :|: TRUE f6002_in([], x2631, x2632, x2633, x2634, x2635, x2636, x2637, x2638, x2639, x2640, x2641, x2642, x2643, x2644, x2645, x2646, x2647, x2648, x2649, x2650, x2651) -> f6008_in :|: TRUE f6008_out -> f6002_out([], x2652, x2653, x2654, x2655, x2656, x2657, x2658, x2659, x2660, x2661, x2662, x2663, x2664, x2665, x2666, x2667, x2668, x2669, x2670, x2671, x2672) :|: TRUE f6002_in(x2673, x2674, x2675, x2676, x2677, x2678, x2679, x2680, x2681, x2682, x2683, x2684, x2685, x2686, x2687, x2688, x2689, x2690, x2691, x2692, x2693, x2694) -> f6009_in :|: TRUE f6009_out -> f6002_out(x2695, x2696, x2697, x2698, x2699, x2700, x2701, x2702, x2703, x2704, x2705, x2706, x2707, x2708, x2709, x2710, x2711, x2712, x2713, x2714, x2715, x2716) :|: TRUE f6093_in([], x2717, x2718, x2719, x2720, x2721, x2722, x2723, x2724, x2725, x2726, x2727, x2728, x2729, x2730, x2731, x2732, x2733, x2734) -> f6095_in :|: TRUE f6093_in(x2735, x2736, x2737, x2738, x2739, x2740, x2741, x2742, x2743, x2744, x2745, x2746, x2747, x2748, x2749, x2750, x2751, x2752, x2753) -> f6096_in :|: TRUE f6095_out -> f6093_out([], x2754, x2755, x2756, x2757, x2758, x2759, x2760, x2761, x2762, x2763, x2764, x2765, x2766, x2767, x2768, x2769, x2770, x2771) :|: TRUE f6096_out -> f6093_out(x2772, x2773, x2774, x2775, x2776, x2777, x2778, x2779, x2780, x2781, x2782, x2783, x2784, x2785, x2786, x2787, x2788, x2789, x2790) :|: TRUE f6094_in(.(x2791, x2792), x2793, x2794, x2795, x2796, x2797, x2798, x2799, x2800, x2801, x2802, x2803, x2804, x2805, x2806, x2807, x2808, x2809, x2810) -> f6098_in(x2793, x2791, x2794, x2792, x2795, x2796, x2797, x2798, x2799, x2800, x2801, x2802, x2803, x2804, x2805, x2806, x2807, x2808, x2809, x2810) :|: TRUE f6094_in(x2811, x2812, x2813, x2814, x2815, x2816, x2817, x2818, x2819, x2820, x2821, x2822, x2823, x2824, x2825, x2826, x2827, x2828, x2829) -> f6099_in :|: TRUE f6099_out -> f6094_out(x2830, x2831, x2832, x2833, x2834, x2835, x2836, x2837, x2838, x2839, x2840, x2841, x2842, x2843, x2844, x2845, x2846, x2847, x2848) :|: TRUE f6098_out(x2849, x2850, x2851, x2852, x2853, x2854, x2855, x2856, x2857, x2858, x2859, x2860, x2861, x2862, x2863, x2864, x2865, x2866, x2867, x2868) -> f6094_out(.(x2850, x2852), x2849, x2851, x2853, x2854, x2855, x2856, x2857, x2858, x2859, x2860, x2861, x2862, x2863, x2864, x2865, x2866, x2867, x2868) :|: TRUE f4923_in -> f4923_out :|: TRUE f5889_in(x2869, x2870, x2871, x2872, x2873, x2874, x2875, x2876, x2877, x2878, x2879, x2880, x2881, x2882, x2883, x2884, x2885, x2886, x2887) -> f5890_in :|: TRUE f5890_out -> f5889_out(x2888, x2889, x2890, x2891, x2892, x2893, x2894, x2895, x2896, x2897, x2898, x2899, x2900, x2901, x2902, x2903, x2904, x2905, x2906) :|: TRUE f5891_out(x2907, x2908, x2909, x2910, x2911, x2912, x2913, x2914, x2915, x2916, x2917, x2918, x2919, x2920, x2921, x2922, x2923, x2924, x2925, x2926) -> f5889_out(x2909, x2924, x2907, x2908, x2910, x2911, x2912, x2913, x2914, x2915, x2916, x2917, x2919, x2920, x2921, x2922, x2923, x2925, x2926) :|: TRUE f5889_in(x2927, x2928, x2929, x2930, x2931, x2932, x2933, x2934, x2935, x2936, x2937, x2938, x2939, x2940, x2941, x2942, x2943, x2944, x2945) -> f5891_in(x2929, x2930, x2927, x2931, x2932, x2933, x2934, x2935, x2936, x2937, x2938, x2946, x2939, x2940, x2941, x2942, x2943, x2928, x2944, x2945) :|: x2946 = x2927 - x2928 f5900_in -> f5900_out :|: TRUE f4447_out(T1577, T1574) -> f6108_out(T1577, T1574) :|: TRUE f6108_in(x2947, x2948) -> f4447_in(x2947, x2948) :|: TRUE f6092_out(x2949, x2950, x2951, x2952, x2953, x2954, x2955, x2956, x2957, x2958, x2959, x2960, x2961, x2962, x2963, x2964, x2965, x2966, x2967) -> f6091_out(x2949, x2950, x2951, x2952, x2953, x2954, x2955, x2956, x2957, x2958, x2959, x2960, x2961, x2962, x2963, x2964, x2965, x2966, x2967) :|: TRUE f6091_in(x2968, x2969, x2970, x2971, x2972, x2973, x2974, x2975, x2976, x2977, x2978, x2979, x2980, x2981, x2982, x2983, x2984, x2985, x2986) -> f6092_in(x2968, x2969, x2970, x2971, x2972, x2973, x2974, x2975, x2976, x2977, x2978, x2979, x2980, x2981, x2982, x2983, x2984, x2985, x2986) :|: TRUE f6086_in(x2987, x2988, x2989, x2990, x2991, x2992, x2993, x2994, x2995, x2996, x2997, x2998, x2999, x3000, x3001, x3002, x3003, x3004, x3005, x3006, x3007, x3008, x3009, x3010, x3011) -> f6089_in(x2989, x3012, x2990, x2987, x2991, x2992, x2993, x2994, x2995, x2996, x2997, x2998, x2999, x3000, x3001, x3002, x3003, x3004, x3005, x3006, x3007, x3008, x3009, x3010, x2988, x3011) :|: x3012 = x2987 - x2988 f6086_in(x3013, x3014, x3015, x3016, x3017, x3018, x3019, x3020, x3021, x3022, x3023, x3024, x3025, x3026, x3027, x3028, x3029, x3030, x3031, x3032, x3033, x3034, x3035, x3036, x3037) -> f6088_in :|: TRUE f6089_out(x3038, x3039, x3040, x3041, x3042, x3043, x3044, x3045, x3046, x3047, x3048, x3049, x3050, x3051, x3052, x3053, x3054, x3055, x3056, x3057, x3058, x3059, x3060, x3061, x3062, x3063) -> f6086_out(x3041, x3062, x3038, x3040, x3042, x3043, x3044, x3045, x3046, x3047, x3048, x3049, x3050, x3051, x3052, x3053, x3054, x3055, x3056, x3057, x3058, x3059, x3060, x3061, x3063) :|: TRUE f6088_out -> f6086_out(x3064, x3065, x3066, x3067, x3068, x3069, x3070, x3071, x3072, x3073, x3074, x3075, x3076, x3077, x3078, x3079, x3080, x3081, x3082, x3083, x3084, x3085, x3086, x3087, x3088) :|: TRUE f6019_out(x3089, x3090, x3091, x3092, x3093, x3094, x3095, x3096, x3097, x3098, x3099, x3100, x3101, x3102, x3103, x3104, x3105, x3106, x3107, x3108, x3109, x3110, x3111, x3112) -> f6016_out(x3092, x3108, x3089, x3091, x3093, x3094, x3095, x3096, x3097, x3098, x3099, x3100, x3101, x3102, x3103, x3104, x3105, x3106, x3107, x3109, x3110, x3111, x3112) :|: TRUE f6016_in(x3113, x3114, x3115, x3116, x3117, x3118, x3119, x3120, x3121, x3122, x3123, x3124, x3125, x3126, x3127, x3128, x3129, x3130, x3131, x3132, x3133, x3134, x3135) -> f6018_in :|: TRUE f6016_in(x3136, x3137, x3138, x3139, x3140, x3141, x3142, x3143, x3144, x3145, x3146, x3147, x3148, x3149, x3150, x3151, x3152, x3153, x3154, x3155, x3156, x3157, x3158) -> f6019_in(x3138, x3159, x3139, x3136, x3140, x3141, x3142, x3143, x3144, x3145, x3146, x3147, x3148, x3149, x3150, x3151, x3152, x3153, x3154, x3137, x3155, x3156, x3157, x3158) :|: x3159 = x3136 - x3137 f6018_out -> f6016_out(x3160, x3161, x3162, x3163, x3164, x3165, x3166, x3167, x3168, x3169, x3170, x3171, x3172, x3173, x3174, x3175, x3176, x3177, x3178, x3179, x3180, x3181, x3182) :|: TRUE f5862_out(x3183, x3184, x3185, x3186, x3187, x3188, x3189, x3190, x3191, x3192, x3193, x3194, x3195, x3196, x3197, x3198, x3199, x3200, x3201) -> f5852_out(x3185, x3186, x3184, x3201, x3187, x3188, x3189, x3190, x3191, x3192, x3193, x3194, x3195, x3196, x3197, x3198, x3199, x3200) :|: TRUE f5861_out(x3202, x3203, x3204, x3205, x3206, x3207, x3208, x3209, x3210, x3211, x3212, x3213, x3214, x3215, x3216, x3217, x3218, x3219) -> f5862_in(x3220, x3204, x3202, x3203, x3206, x3207, x3208, x3209, x3210, x3211, x3212, x3213, x3214, x3215, x3216, x3217, x3218, x3219, x3205) :|: TRUE f5852_in(x3221, x3222, x3223, x3224, x3225, x3226, x3227, x3228, x3229, x3230, x3231, x3232, x3233, x3234, x3235, x3236, x3237, x3238) -> f5861_in(x3221, x3222, x3223, x3224, x3225, x3226, x3227, x3228, x3229, x3230, x3231, x3232, x3233, x3234, x3235, x3236, x3237, x3238) :|: TRUE f6021_in(x3239, x3240, x3241, x3242, x3243, x3244, x3245, x3246, x3247, x3248, x3249, x3250, x3251, x3252, x3253, x3254, x3255, x3256, x3257, x3258, x3259, x3260, x3261) -> f6023_in(x3239, x3240, x3241, x3242, x3243, x3244, x3245, x3246, x3247, x3248, x3249, x3250, x3251, x3252, x3253, x3254, x3255, x3256, x3257, x3258, x3259, x3260, x3261) :|: TRUE f6022_out(x3262, x3263, x3264, x3265, x3266, x3267, x3268, x3269, x3270, x3271, x3272, x3273, x3274, x3275, x3276, x3277, x3278, x3279, x3280, x3281, x3282, x3283, x3284) -> f6021_out(x3262, x3263, x3264, x3265, x3266, x3267, x3268, x3269, x3270, x3271, x3272, x3273, x3274, x3275, x3276, x3277, x3278, x3279, x3280, x3281, x3282, x3283, x3284) :|: TRUE f6023_out(x3285, x3286, x3287, x3288, x3289, x3290, x3291, x3292, x3293, x3294, x3295, x3296, x3297, x3298, x3299, x3300, x3301, x3302, x3303, x3304, x3305, x3306, x3307) -> f6021_out(x3285, x3286, x3287, x3288, x3289, x3290, x3291, x3292, x3293, x3294, x3295, x3296, x3297, x3298, x3299, x3300, x3301, x3302, x3303, x3304, x3305, x3306, x3307) :|: TRUE f6021_in(x3308, x3309, x3310, x3311, x3312, x3313, x3314, x3315, x3316, x3317, x3318, x3319, x3320, x3321, x3322, x3323, x3324, x3325, x3326, x3327, x3328, x3329, x3330) -> f6022_in(x3308, x3309, x3310, x3311, x3312, x3313, x3314, x3315, x3316, x3317, x3318, x3319, x3320, x3321, x3322, x3323, x3324, x3325, x3326, x3327, x3328, x3329, x3330) :|: TRUE f6083_in -> f6083_out :|: TRUE f5944_in([], x3331, x3332, x3333, x3334, x3335, x3336, x3337, x3338, x3339, x3340, x3341, x3342, x3343, x3344, x3345, x3346, x3347, x3348, x3349) -> f5946_in :|: TRUE f5947_out -> f5944_out(x3350, x3351, x3352, x3353, x3354, x3355, x3356, x3357, x3358, x3359, x3360, x3361, x3362, x3363, x3364, x3365, x3366, x3367, x3368, x3369) :|: TRUE f5946_out -> f5944_out([], x3370, x3371, x3372, x3373, x3374, x3375, x3376, x3377, x3378, x3379, x3380, x3381, x3382, x3383, x3384, x3385, x3386, x3387, x3388) :|: TRUE f5944_in(x3389, x3390, x3391, x3392, x3393, x3394, x3395, x3396, x3397, x3398, x3399, x3400, x3401, x3402, x3403, x3404, x3405, x3406, x3407, x3408) -> f5947_in :|: TRUE f4910_in(x3409, x3410) -> f4912_in(x3409, x3410) :|: TRUE f4912_out(x3411, x3412) -> f4910_out(x3411, x3412) :|: TRUE f4937_in([], T324, T325, T326, T327) -> f4939_in :|: TRUE f4937_in(x3413, x3414, x3415, x3416, x3417) -> f4940_in :|: TRUE f4939_out -> f4937_out([], x3418, x3419, x3420, x3421) :|: TRUE f4940_out -> f4937_out(x3422, x3423, x3424, x3425, x3426) :|: TRUE f5934_in(x3427, x3428, x3429, x3430, x3431, x3432, x3433, x3434, x3435, x3436, x3437, x3438, x3439, x3440, x3441, x3442, x3443, x3444, x3445, x3446, x3447) -> f5935_in(x3429, x3430, x3427, x3431, x3432, x3428, x3433, x3434, x3435, x3436, x3437, x3438, x3439, x3440, x3441, x3442, x3448, x3443, x3444, x3445, x3446, x3447) :|: x3448 = x3427 - x3428 f5935_out(x3449, x3450, x3451, x3452, x3453, x3454, x3455, x3456, x3457, x3458, x3459, x3460, x3461, x3462, x3463, x3464, x3465, x3466, x3467, x3468, x3469, x3470) -> f5934_out(x3451, x3454, x3449, x3450, x3452, x3453, x3455, x3456, x3457, x3458, x3459, x3460, x3461, x3462, x3463, x3464, x3466, x3467, x3468, x3469, x3470) :|: TRUE f4409_in(x3471) -> f4411_in(x3471) :|: TRUE f4411_out(x3472) -> f4409_out(x3472) :|: TRUE f4410_out(x3473) -> f4409_out(x3473) :|: TRUE f4409_in(x3474) -> f4410_in(x3474) :|: TRUE f4410_in(.(T94, T95)) -> f4412_in :|: TRUE f4410_in(x3475) -> f4413_in :|: TRUE f4413_out -> f4410_out(x3476) :|: TRUE f4412_out -> f4410_out(.(x3477, x3478)) :|: TRUE f4452_in -> f4452_out :|: TRUE f5912_out(x3479, x3480, x3481, x3482, x3483, x3484, x3485, x3486, x3487, x3488, x3489, x3490, x3491, x3492, x3493, x3494, x3495, x3496, x3497, x3498) -> f5909_out(x3482, x3493, x3479, x3481, x3483, x3484, x3485, x3486, x3487, x3488, x3489, x3490, x3491, x3492, x3494, x3495, x3496, x3497, x3498) :|: TRUE f5911_out -> f5909_out(x3499, x3500, x3501, x3502, x3503, x3504, x3505, x3506, x3507, x3508, x3509, x3510, x3511, x3512, x3513, x3514, x3515, x3516, x3517) :|: TRUE f5909_in(x3518, x3519, x3520, x3521, x3522, x3523, x3524, x3525, x3526, x3527, x3528, x3529, x3530, x3531, x3532, x3533, x3534, x3535, x3536) -> f5912_in(x3520, x3537, x3521, x3518, x3522, x3523, x3524, x3525, x3526, x3527, x3528, x3529, x3530, x3531, x3519, x3532, x3533, x3534, x3535, x3536) :|: x3537 = x3518 - x3519 f5909_in(x3538, x3539, x3540, x3541, x3542, x3543, x3544, x3545, x3546, x3547, x3548, x3549, x3550, x3551, x3552, x3553, x3554, x3555, x3556) -> f5911_in :|: TRUE f5920_in([], x3557, x3558, x3559, x3560, x3561, x3562, x3563, x3564, x3565, x3566, x3567, x3568, x3569, x3570, x3571, x3572, x3573, x3574) -> f5922_in :|: TRUE f5922_out -> f5920_out([], x3575, x3576, x3577, x3578, x3579, x3580, x3581, x3582, x3583, x3584, x3585, x3586, x3587, x3588, x3589, x3590, x3591, x3592) :|: TRUE f5923_out -> f5920_out(x3593, x3594, x3595, x3596, x3597, x3598, x3599, x3600, x3601, x3602, x3603, x3604, x3605, x3606, x3607, x3608, x3609, x3610, x3611) :|: TRUE f5920_in(x3612, x3613, x3614, x3615, x3616, x3617, x3618, x3619, x3620, x3621, x3622, x3623, x3624, x3625, x3626, x3627, x3628, x3629, x3630) -> f5923_in :|: TRUE f5981_in(.(x3631, x3632), x3633, x3634, x3635, x3636, x3637, x3638, x3639, x3640, x3641, x3642, x3643, x3644, x3645, x3646, x3647, x3648, x3649, x3650, x3651, x3652) -> f5990_in(x3633, x3631, x3634, x3632, x3635, x3636, x3637, x3638, x3639, x3640, x3641, x3642, x3643, x3644, x3645, x3646, x3647, x3648, x3649, x3650, x3651, x3652) :|: TRUE f5990_out(x3653, x3654, x3655, x3656, x3657, x3658, x3659, x3660, x3661, x3662, x3663, x3664, x3665, x3666, x3667, x3668, x3669, x3670, x3671, x3672, x3673, x3674) -> f5981_out(.(x3654, x3656), x3653, x3655, x3657, x3658, x3659, x3660, x3661, x3662, x3663, x3664, x3665, x3666, x3667, x3668, x3669, x3670, x3671, x3672, x3673, x3674) :|: TRUE f5991_out -> f5981_out(x3675, x3676, x3677, x3678, x3679, x3680, x3681, x3682, x3683, x3684, x3685, x3686, x3687, x3688, x3689, x3690, x3691, x3692, x3693, x3694, x3695) :|: TRUE f5981_in(x3696, x3697, x3698, x3699, x3700, x3701, x3702, x3703, x3704, x3705, x3706, x3707, x3708, x3709, x3710, x3711, x3712, x3713, x3714, x3715, x3716) -> f5991_in :|: TRUE f6102_out -> f6101_out(x3717, x3718, x3719, x3720, x3721, x3722, x3723, x3724, x3725, x3726, x3727, x3728, x3729, x3730, x3731, x3732, x3733, x3734, x3735, x3736, x3737) :|: TRUE f6101_in(x3738, x3739, x3740, x3741, x3742, x3743, x3744, x3745, x3746, x3747, x3748, x3749, x3750, x3751, x3752, x3753, x3754, x3755, x3756, x3757, x3758) -> f6102_in :|: TRUE f6103_out(x3759, x3760, x3761, x3762, x3763, x3764, x3765, x3766, x3767, x3768, x3769, x3770, x3771, x3772, x3773, x3774, x3775, x3776, x3777, x3778, x3779, x3780) -> f6101_out(x3761, x3776, x3759, x3760, x3762, x3763, x3764, x3765, x3766, x3767, x3769, x3770, x3771, x3772, x3773, x3774, x3775, x3777, x3778, x3779, x3780) :|: TRUE f6101_in(x3781, x3782, x3783, x3784, x3785, x3786, x3787, x3788, x3789, x3790, x3791, x3792, x3793, x3794, x3795, x3796, x3797, x3798, x3799, x3800, x3801) -> f6103_in(x3783, x3784, x3781, x3785, x3786, x3787, x3788, x3789, x3790, x3802, x3791, x3792, x3793, x3794, x3795, x3796, x3797, x3782, x3798, x3799, x3800, x3801) :|: x3802 = x3781 - x3782 f4447_out(.(x3803, x3804), x3805) -> f4837_out(x3803, x3804, x3805, x3806) :|: TRUE f4837_in(x3807, x3808, x3809, x3810) -> f4447_in(.(x3807, x3808), x3809) :|: TRUE f4918_in(x3811, x3812, x3813) -> f4920_in(x3811, x3812, x3813) :|: TRUE f4920_out(x3814, x3815, x3816) -> f4918_out(x3814, x3815, x3816) :|: TRUE f6105_out(x3817, x3818) -> f5878_out(x3819, x3820, x3817, x3821, x3822, x3823) :|: TRUE f6104_out(x3824, x3825, x3826, x3827, x3828) -> f6105_in(x3829, x3830) :|: TRUE f5878_in(x3831, x3832, x3833, x3834, x3835, x3836) -> f6104_in(x3831, x3832, x3834, x3835, x3836) :|: TRUE f4931_in -> f4931_out :|: TRUE f5913_out(x3837, x3838, x3839, x3840, x3841, x3842, x3843, x3844, x3845, x3846, x3847, x3848, x3849, x3850, x3851, x3852, x3853, x3854, x3855, x3856, x3857) -> f5912_out(x3839, x3857, x3837, x3838, x3840, x3841, x3842, x3843, x3844, x3845, x3846, x3847, x3848, x3849, x3851, x3852, x3853, x3854, x3855, x3856) :|: TRUE f5912_in(x3858, x3859, x3860, x3861, x3862, x3863, x3864, x3865, x3866, x3867, x3868, x3869, x3870, x3871, x3872, x3873, x3874, x3875, x3876, x3877) -> f5913_in(x3860, x3861, x3858, x3862, x3863, x3864, x3865, x3866, x3867, x3868, x3869, x3870, x3871, x3878, x3872, x3873, x3874, x3875, x3876, x3877, x3859) :|: x3878 = x3858 - x3859 f5889_out(x3879, x3880, x3881, x3882, x3883, x3884, x3885, x3886, x3887, x3888, x3889, x3890, x3891, x3892, x3893, x3894, x3895, x3896, x3897) -> f5887_out(x3882, x3892, x3879, x3881, x3893, x3883, x3887, x3888, x3895, x3889, x3884, x3890, x3894, x3886, x3885, x3891, x3896, x3897) :|: TRUE f5888_out -> f5887_out(x3898, x3899, x3900, x3901, x3902, x3903, x3904, x3905, x3906, x3907, x3908, x3909, x3910, x3911, x3912, x3913, x3914, x3915) :|: TRUE f5887_in(x3916, x3917, x3918, x3919, x3920, x3921, x3922, x3923, x3924, x3925, x3926, x3927, x3928, x3929, x3930, x3931, x3932, x3933) -> f5888_in :|: TRUE f5887_in(x3934, x3935, x3936, x3937, x3938, x3939, x3940, x3941, x3942, x3943, x3944, x3945, x3946, x3947, x3948, x3949, x3950, x3951) -> f5889_in(x3936, x3952, x3937, x3934, x3939, x3944, x3948, x3947, x3940, x3941, x3943, x3945, x3949, x3935, x3938, x3946, x3942, x3950, x3951) :|: x3952 = x3934 - x3935 f4411_in(.(x3953, x3954)) -> f4415_in(x3954) :|: TRUE f4415_out(x3955) -> f4411_out(.(x3956, x3955)) :|: TRUE f4411_in(x3957) -> f4416_in :|: TRUE f4416_out -> f4411_out(x3958) :|: TRUE f5921_in(x3959, x3960, x3961, x3962, x3963, x3964, x3965, x3966, x3967, x3968, x3969, x3970, x3971, x3972, x3973, x3974, x3975, x3976, x3977) -> f5932_in :|: TRUE f5921_in(.(x3978, x3979), x3980, x3981, x3982, x3983, x3984, x3985, x3986, x3987, x3988, x3989, x3990, x3991, x3992, x3993, x3994, x3995, x3996, x3997) -> f5931_in(x3980, x3978, x3981, x3979, x3982, x3983, x3984, x3985, x3986, x3987, x3988, x3989, x3990, x3991, x3992, x3993, x3994, x3995, x3996, x3997) :|: TRUE f5931_out(x3998, x3999, x4000, x4001, x4002, x4003, x4004, x4005, x4006, x4007, x4008, x4009, x4010, x4011, x4012, x4013, x4014, x4015, x4016, x4017) -> f5921_out(.(x3999, x4001), x3998, x4000, x4002, x4003, x4004, x4005, x4006, x4007, x4008, x4009, x4010, x4011, x4012, x4013, x4014, x4015, x4016, x4017) :|: TRUE f5932_out -> f5921_out(x4018, x4019, x4020, x4021, x4022, x4023, x4024, x4025, x4026, x4027, x4028, x4029, x4030, x4031, x4032, x4033, x4034, x4035, x4036) :|: TRUE f4449_out(x4037, x4038) -> f4447_out(x4037, x4038) :|: TRUE f4447_in(x4039, x4040) -> f4449_in(x4039, x4040) :|: TRUE f4927_out -> f4922_out(x4041, x4042, x4043) :|: TRUE f4922_in(x4044, x4045, x4046) -> f4927_in :|: TRUE f4922_in(.(x4047, x4048), x4049, x4050) -> f4926_in(x4048, x4047, x4049, x4050) :|: TRUE f4926_out(x4051, x4052, x4053, x4054) -> f4922_out(.(x4052, x4051), x4053, x4054) :|: TRUE f5878_out(x4055, x4056, x4057, x4058, x4059, x4060) -> f5876_out(x4058, x4059, x4061, x4060, x4056, x4062, x4063, x4064, x4065, x4066, x4067, x4068, x4069, x4070, x4071, x4072, x4073, x4074) :|: TRUE f5876_in(x4075, x4076, x4077, x4078, x4079, x4080, x4081, x4082, x4083, x4084, x4085, x4086, x4087, x4088, x4089, x4090, x4091, x4092) -> f5877_in(x4075, x4076, x4077, x4078, x4080, x4081, x4082, x4083, x4084, x4085, x4086, x4087, x4088, x4089, x4090, x4091, x4092, x4079) :|: TRUE f5877_out(x4093, x4094, x4095, x4096, x4097, x4098, x4099, x4100, x4101, x4102, x4103, x4104, x4105, x4106, x4107, x4108, x4109, x4110) -> f5878_in(x4111, x4110, x4112, x4093, x4094, x4096) :|: TRUE f6107_in(x4113, x4114) -> f6108_in(x4113, x4114) :|: TRUE f6108_out(x4115, x4116) -> f6109_in :|: TRUE f6109_out -> f6107_out(x4117, x4118) :|: TRUE f5914_out(x4119, x4120, x4121, x4122, x4123, x4124, x4125, x4126, x4127, x4128, x4129, x4130, x4131, x4132, x4133, x4134, x4135, x4136, x4137) -> f5913_out(x4119, x4120, x4121, x4122, x4123, x4124, x4125, x4126, x4127, x4128, x4129, x4130, x4131, x4138, x4132, x4133, x4134, x4135, x4136, x4137, x4139) :|: TRUE f5913_in(x4140, x4141, x4142, x4143, x4144, x4145, x4146, x4147, x4148, x4149, x4150, x4151, x4152, x4153, x4154, x4155, x4156, x4157, x4158, x4159, x4160) -> f5914_in(x4140, x4141, x4142, x4143, x4144, x4145, x4146, x4147, x4148, x4149, x4150, x4151, x4152, x4154, x4155, x4156, x4157, x4158, x4159) :|: TRUE f4452_out -> f4450_out([], T173) :|: TRUE f4453_out -> f4450_out(x4161, x4162) :|: TRUE f4450_in(x4163, x4164) -> f4453_in :|: TRUE f4450_in([], x4165) -> f4452_in :|: TRUE f6041_out -> f6023_out(x4166, x4167, x4168, x4169, x4170, x4171, x4172, x4173, x4174, x4175, x4176, x4177, x4178, x4179, x4180, x4181, x4182, x4183, x4184, x4185, x4186, x4187, x4188) :|: TRUE f6023_in(x4189, x4190, x4191, x4192, x4193, x4194, x4195, x4196, x4197, x4198, x4199, x4200, x4201, x4202, x4203, x4204, x4205, x4206, x4207, x4208, x4209, x4210, x4211) -> f6041_in :|: TRUE f6023_in(.(x4212, x4213), x4214, x4215, x4216, x4217, x4218, x4219, x4220, x4221, x4222, x4223, x4224, x4225, x4226, x4227, x4228, x4229, x4230, x4231, x4232, x4233, x4234, x4235) -> f6040_in(x4214, x4212, x4215, x4213, x4216, x4217, x4218, x4219, x4220, x4221, x4222, x4223, x4224, x4225, x4226, x4227, x4228, x4229, x4230, x4231, x4232, x4233, x4234, x4235) :|: TRUE f6040_out(x4236, x4237, x4238, x4239, x4240, x4241, x4242, x4243, x4244, x4245, x4246, x4247, x4248, x4249, x4250, x4251, x4252, x4253, x4254, x4255, x4256, x4257, x4258, x4259) -> f6023_out(.(x4237, x4239), x4236, x4238, x4240, x4241, x4242, x4243, x4244, x4245, x4246, x4247, x4248, x4249, x4250, x4251, x4252, x4253, x4254, x4255, x4256, x4257, x4258, x4259) :|: TRUE f5977_in(x4260, x4261, x4262, x4263, x4264, x4265, x4266, x4267, x4268, x4269, x4270, x4271, x4272, x4273, x4274, x4275, x4276, x4277, x4278, x4279, x4280) -> f5980_in(x4260, x4261, x4262, x4263, x4264, x4265, x4266, x4267, x4268, x4269, x4270, x4271, x4272, x4273, x4274, x4275, x4276, x4277, x4278, x4279, x4280) :|: TRUE f5977_in(x4281, x4282, x4283, x4284, x4285, x4286, x4287, x4288, x4289, x4290, x4291, x4292, x4293, x4294, x4295, x4296, x4297, x4298, x4299, x4300, x4301) -> f5981_in(x4281, x4282, x4283, x4284, x4285, x4286, x4287, x4288, x4289, x4290, x4291, x4292, x4293, x4294, x4295, x4296, x4297, x4298, x4299, x4300, x4301) :|: TRUE f5980_out(x4302, x4303, x4304, x4305, x4306, x4307, x4308, x4309, x4310, x4311, x4312, x4313, x4314, x4315, x4316, x4317, x4318, x4319, x4320, x4321, x4322) -> f5977_out(x4302, x4303, x4304, x4305, x4306, x4307, x4308, x4309, x4310, x4311, x4312, x4313, x4314, x4315, x4316, x4317, x4318, x4319, x4320, x4321, x4322) :|: TRUE f5981_out(x4323, x4324, x4325, x4326, x4327, x4328, x4329, x4330, x4331, x4332, x4333, x4334, x4335, x4336, x4337, x4338, x4339, x4340, x4341, x4342, x4343) -> f5977_out(x4323, x4324, x4325, x4326, x4327, x4328, x4329, x4330, x4331, x4332, x4333, x4334, x4335, x4336, x4337, x4338, x4339, x4340, x4341, x4342, x4343) :|: TRUE f4915_in -> f4915_out :|: TRUE f5914_in(x4344, x4345, x4346, x4347, x4348, x4349, x4350, x4351, x4352, x4353, x4354, x4355, x4356, x4357, x4358, x4359, x4360, x4361, x4362) -> f5920_in(x4344, x4345, x4346, x4347, x4348, x4349, x4350, x4351, x4352, x4353, x4354, x4355, x4356, x4357, x4358, x4359, x4360, x4361, x4362) :|: TRUE f5921_out(x4363, x4364, x4365, x4366, x4367, x4368, x4369, x4370, x4371, x4372, x4373, x4374, x4375, x4376, x4377, x4378, x4379, x4380, x4381) -> f5914_out(x4363, x4364, x4365, x4366, x4367, x4368, x4369, x4370, x4371, x4372, x4373, x4374, x4375, x4376, x4377, x4378, x4379, x4380, x4381) :|: TRUE f5920_out(x4382, x4383, x4384, x4385, x4386, x4387, x4388, x4389, x4390, x4391, x4392, x4393, x4394, x4395, x4396, x4397, x4398, x4399, x4400) -> f5914_out(x4382, x4383, x4384, x4385, x4386, x4387, x4388, x4389, x4390, x4391, x4392, x4393, x4394, x4395, x4396, x4397, x4398, x4399, x4400) :|: TRUE f5914_in(x4401, x4402, x4403, x4404, x4405, x4406, x4407, x4408, x4409, x4410, x4411, x4412, x4413, x4414, x4415, x4416, x4417, x4418, x4419) -> f5921_in(x4401, x4402, x4403, x4404, x4405, x4406, x4407, x4408, x4409, x4410, x4411, x4412, x4413, x4414, x4415, x4416, x4417, x4418, x4419) :|: TRUE f4956_out -> f4948_out(x4420, x4421, x4422, x4423, x4424, x4425) :|: TRUE f4948_in(x4426, x4427, x4428, x4429, x4430, x4431) -> f4956_in :|: TRUE f4948_in(.(x4432, x4433), x4434, x4435, x4436, x4437, x4438) -> f4955_in(x4433, x4432, x4434, x4435, x4436, x4437, x4438) :|: TRUE f4955_out(x4439, x4440, x4441, x4442, x4443, x4444, x4445) -> f4948_out(.(x4440, x4439), x4441, x4442, x4443, x4444, x4445) :|: TRUE f4915_out -> f4913_out([], T228) :|: TRUE f4913_in(x4446, x4447) -> f4916_in :|: TRUE f4916_out -> f4913_out(x4448, x4449) :|: TRUE f4913_in([], x4450) -> f4915_in :|: TRUE f4952_in -> f4952_out :|: TRUE f5944_out(x4451, x4452, x4453, x4454, x4455, x4456, x4457, x4458, x4459, x4460, x4461, x4462, x4463, x4464, x4465, x4466, x4467, x4468, x4469, x4470) -> f5939_out(x4451, x4452, x4453, x4454, x4455, x4456, x4457, x4458, x4459, x4460, x4461, x4462, x4463, x4464, x4465, x4466, x4467, x4468, x4469, x4470) :|: TRUE f5945_out(x4471, x4472, x4473, x4474, x4475, x4476, x4477, x4478, x4479, x4480, x4481, x4482, x4483, x4484, x4485, x4486, x4487, x4488, x4489, x4490) -> f5939_out(x4471, x4472, x4473, x4474, x4475, x4476, x4477, x4478, x4479, x4480, x4481, x4482, x4483, x4484, x4485, x4486, x4487, x4488, x4489, x4490) :|: TRUE f5939_in(x4491, x4492, x4493, x4494, x4495, x4496, x4497, x4498, x4499, x4500, x4501, x4502, x4503, x4504, x4505, x4506, x4507, x4508, x4509, x4510) -> f5945_in(x4491, x4492, x4493, x4494, x4495, x4496, x4497, x4498, x4499, x4500, x4501, x4502, x4503, x4504, x4505, x4506, x4507, x4508, x4509, x4510) :|: TRUE f5939_in(x4511, x4512, x4513, x4514, x4515, x4516, x4517, x4518, x4519, x4520, x4521, x4522, x4523, x4524, x4525, x4526, x4527, x4528, x4529, x4530) -> f5944_in(x4511, x4512, x4513, x4514, x4515, x4516, x4517, x4518, x4519, x4520, x4521, x4522, x4523, x4524, x4525, x4526, x4527, x4528, x4529, x4530) :|: TRUE f5059_out(x4531, x4532, .(x4533, .(x4534, .(x4535, .(x4536, .(x4537, .(x4538, .(x4539, [])))))))) -> f5057_out(x4531, x4532, x4533, x4534, x4535, x4536, x4537, x4538, x4539) :|: TRUE f5057_in(x4540, x4541, x4542, x4543, x4544, x4545, x4546, x4547, x4548) -> f5059_in(x4540, x4541, .(x4542, .(x4543, .(x4544, .(x4545, .(x4546, .(x4547, .(x4548, [])))))))) :|: TRUE f4944_in(x4549, x4550, x4551, x4552, x4553, x4554) -> f4947_in(x4549, x4550, x4551, x4552, x4553, x4554) :|: TRUE f4944_in(x4555, x4556, x4557, x4558, x4559, x4560) -> f4948_in(x4555, x4556, x4557, x4558, x4559, x4560) :|: TRUE f4948_out(x4561, x4562, x4563, x4564, x4565, x4566) -> f4944_out(x4561, x4562, x4563, x4564, x4565, x4566) :|: TRUE f4947_out(x4567, x4568, x4569, x4570, x4571, x4572) -> f4944_out(x4567, x4568, x4569, x4570, x4571, x4572) :|: TRUE f5861_in(x4573, x4574, x4575, x4576, x4577, x4578, x4579, x4580, x4581, x4582, x4583, x4584, x4585, x4586, x4587, x4588, x4589, x4590) -> f5863_in(x4573, x4574, x4575, x4576, x4577, x4578, x4579, x4580, x4581, x4582, x4583, x4584, x4585, x4586, x4587, x4588, x4589, x4590) :|: TRUE f5863_out(x4591, x4592, x4593, x4594, x4595, x4596, x4597, x4598, x4599, x4600, x4601, x4602, x4603, x4604, x4605, x4606, x4607, x4608) -> f5861_out(x4591, x4592, x4593, x4594, x4595, x4596, x4597, x4598, x4599, x4600, x4601, x4602, x4603, x4604, x4605, x4606, x4607, x4608) :|: TRUE f6107_out(x4609, x4610) -> f6105_out(x4611, x4610) :|: TRUE f6105_in(x4612, x4613) -> f6106_in(x4612) :|: TRUE f6106_out(x4614) -> f6107_in(x4615, x4616) :|: TRUE f4447_out(x4617, x4618) -> f4525_out(x4617, x4618) :|: TRUE f4525_in(x4619, x4620) -> f4447_in(x4619, x4620) :|: TRUE f6005_in(.(x4621, x4622), x4623, x4624, x4625, x4626, x4627, x4628, x4629, x4630, x4631, x4632, x4633, x4634, x4635, x4636, x4637, x4638, x4639, x4640, x4641, x4642, x4643) -> f6016_in(x4623, x4621, x4624, x4622, x4625, x4626, x4627, x4628, x4629, x4630, x4631, x4632, x4633, x4634, x4635, x4636, x4637, x4638, x4639, x4640, x4641, x4642, x4643) :|: TRUE f6016_out(x4644, x4645, x4646, x4647, x4648, x4649, x4650, x4651, x4652, x4653, x4654, x4655, x4656, x4657, x4658, x4659, x4660, x4661, x4662, x4663, x4664, x4665, x4666) -> f6005_out(.(x4645, x4647), x4644, x4646, x4648, x4649, x4650, x4651, x4652, x4653, x4654, x4655, x4656, x4657, x4658, x4659, x4660, x4661, x4662, x4663, x4664, x4665, x4666) :|: TRUE f6017_out -> f6005_out(x4667, x4668, x4669, x4670, x4671, x4672, x4673, x4674, x4675, x4676, x4677, x4678, x4679, x4680, x4681, x4682, x4683, x4684, x4685, x4686, x4687, x4688) :|: TRUE f6005_in(x4689, x4690, x4691, x4692, x4693, x4694, x4695, x4696, x4697, x4698, x4699, x4700, x4701, x4702, x4703, x4704, x4705, x4706, x4707, x4708, x4709, x4710) -> f6017_in :|: TRUE f4451_out(x4711, x4712) -> f4449_out(x4711, x4712) :|: TRUE f4449_in(x4713, x4714) -> f4450_in(x4713, x4714) :|: TRUE f4450_out(x4715, x4716) -> f4449_out(x4715, x4716) :|: TRUE f4449_in(x4717, x4718) -> f4451_in(x4717, x4718) :|: TRUE f5891_in(x4719, x4720, x4721, x4722, x4723, x4724, x4725, x4726, x4727, x4728, x4729, x4730, x4731, x4732, x4733, x4734, x4735, x4736, x4737, x4738) -> f5892_in(x4719, x4720, x4721, x4722, x4723, x4724, x4725, x4726, x4727, x4728, x4729, x4731, x4732, x4733, x4734, x4735, x4737, x4738) :|: TRUE f5892_out(x4739, x4740, x4741, x4742, x4743, x4744, x4745, x4746, x4747, x4748, x4749, x4750, x4751, x4752, x4753, x4754, x4755, x4756) -> f5891_out(x4739, x4740, x4741, x4742, x4743, x4744, x4745, x4746, x4747, x4748, x4749, x4757, x4750, x4751, x4752, x4753, x4754, x4758, x4755, x4756) :|: TRUE f6082_in(.(x4759, x4760), x4761, x4762, x4763, x4764, x4765, x4766, x4767, x4768, x4769, x4770, x4771, x4772, x4773, x4774, x4775, x4776, x4777, x4778, x4779, x4780, x4781, x4782, x4783) -> f6086_in(x4761, x4759, x4762, x4760, x4763, x4764, x4765, x4766, x4767, x4768, x4769, x4770, x4771, x4772, x4773, x4774, x4775, x4776, x4777, x4778, x4779, x4780, x4781, x4782, x4783) :|: TRUE f6082_in(x4784, x4785, x4786, x4787, x4788, x4789, x4790, x4791, x4792, x4793, x4794, x4795, x4796, x4797, x4798, x4799, x4800, x4801, x4802, x4803, x4804, x4805, x4806, x4807) -> f6087_in :|: TRUE f6087_out -> f6082_out(x4808, x4809, x4810, x4811, x4812, x4813, x4814, x4815, x4816, x4817, x4818, x4819, x4820, x4821, x4822, x4823, x4824, x4825, x4826, x4827, x4828, x4829, x4830, x4831) :|: TRUE f6086_out(x4832, x4833, x4834, x4835, x4836, x4837, x4838, x4839, x4840, x4841, x4842, x4843, x4844, x4845, x4846, x4847, x4848, x4849, x4850, x4851, x4852, x4853, x4854, x4855, x4856) -> f6082_out(.(x4833, x4835), x4832, x4834, x4836, x4837, x4838, x4839, x4840, x4841, x4842, x4843, x4844, x4845, x4846, x4847, x4848, x4849, x4850, x4851, x4852, x4853, x4854, x4855, x4856) :|: TRUE f4928_out(x4857, x4858, x4859, x4860) -> f4926_out(x4857, x4858, x4859, x4860) :|: TRUE f4926_in(x4861, x4862, x4863, x4864) -> f4928_in(x4861, x4862, x4863, x4864) :|: TRUE f4908_out -> f4905_out(x4865) :|: TRUE f4905_in(x4866) -> f4908_in :|: TRUE f4905_in([]) -> f4907_in :|: TRUE f4907_out -> f4905_out([]) :|: TRUE f5922_in -> f5922_out :|: TRUE f4447_out(x4867, x4868) -> f6104_out(x4867, x4868, x4869, x4870, x4871) :|: TRUE f6104_in(x4872, x4873, x4874, x4875, x4876) -> f4447_in(x4872, x4873) :|: TRUE f5832_out(x4877, x4878, x4879, x4880, x4881, x4882, x4883, x4884, x4885, x4886, x4887, x4888, x4889, x4890, x4891, x4892) -> f5830_out([], x4877, x4878, x4879, x4880, x4881, x4882, x4883, x4884, x4885, x4886, x4887, x4888, x4889, x4890, x4891, x4892) :|: TRUE f5833_out(x4893, x4894, x4895, x4896, x4897, x4898, x4899, x4900, x4901, x4902, x4903, x4904, x4905, x4906, x4907, x4908, x4909) -> f5830_out(x4893, x4894, x4895, x4896, x4897, x4898, x4899, x4900, x4901, x4902, x4903, x4904, x4905, x4906, x4907, x4908, x4909) :|: TRUE f5830_in(x4910, x4911, x4912, x4913, x4914, x4915, x4916, x4917, x4918, x4919, x4920, x4921, x4922, x4923, x4924, x4925, x4926) -> f5833_in(x4910, x4911, x4912, x4913, x4914, x4915, x4916, x4917, x4918, x4919, x4920, x4921, x4922, x4923, x4924, x4925, x4926) :|: TRUE f5830_in([], x4927, x4928, x4929, x4930, x4931, x4932, x4933, x4934, x4935, x4936, x4937, x4938, x4939, x4940, x4941, x4942) -> f5832_in(x4927, x4928, x4929, x4930, x4931, x4932, x4933, x4934, x4935, x4936, x4937, x4938, x4939, x4940, x4941, x4942) :|: TRUE f6095_in -> f6095_out :|: TRUE f4902_out(T1324) -> f6106_out(T1324) :|: TRUE f6106_in(x4943) -> f4902_in(x4943) :|: TRUE f4_out(T1) -> f1_out(T1) :|: TRUE f1_in(x4944) -> f4_in(x4944) :|: TRUE f4_in(T7) -> f10_in(T7) :|: TRUE f11_out -> f4_out(x4945) :|: TRUE f10_out(x4946) -> f4_out(x4946) :|: TRUE f4_in(x4947) -> f11_in :|: TRUE f10_in(x4948) -> f16_in(x4949, x4948) :|: x4949 = x4948 * (x4948 + 1) + 1 f10_in(x4950) -> f12_in :|: TRUE f16_out(x4951, x4952) -> f10_out(x4952) :|: TRUE f12_out -> f10_out(x4953) :|: TRUE f16_in(x4954, x4955) -> f21_in(x4954) :|: TRUE f24_out(x4956, x4957) -> f16_out(x4957, x4958) :|: TRUE f21_out(T10) -> f24_in(T12, T10) :|: TRUE f4324_out(x4959, x4960) -> f24_out(x4959, x4960) :|: TRUE f24_in(x4961, x4962) -> f4324_in(x4961, x4962) :|: TRUE f4324_in(x4963, x4964) -> f4326_in(x4963, x4964) :|: TRUE f4326_out(x4965, x4966) -> f4324_out(x4965, x4966) :|: TRUE f4325_out(T36) -> f4324_out([], T36) :|: TRUE f4324_in([], x4967) -> f4325_in(x4967) :|: TRUE f4326_in(x4968, x4969) -> f4397_in(x4968, x4969) :|: TRUE f4326_in(x4970, x4971) -> f4398_in :|: TRUE f4397_out(x4972, x4973) -> f4326_out(x4972, x4973) :|: TRUE f4398_out -> f4326_out(x4974, x4975) :|: TRUE f4400_out(x4976, x4977, x4978) -> f4397_out(x4977, x4978) :|: TRUE f4397_in(x4979, x4980) -> f4399_in(x4979, x4980) :|: TRUE f4399_out(x4981, x4982) -> f4400_in(x4983, x4981, x4982) :|: TRUE f4418_out(x4984, x4985, x4986, x4987) -> f4400_out(x4985, x4987, x4986) :|: TRUE f4400_in(x4988, x4989, x4990) -> f4417_in(x4988, x4989, x4990) :|: TRUE f4417_out(x4991, x4992, x4993) -> f4418_in(x4994, x4991, x4993, x4992) :|: TRUE f5096_out(x4995, x4996, x4997, x4998) -> f4418_out(x4995, x4996, x4997, x4998) :|: TRUE f4418_in(x4999, x5000, x5001, x5002) -> f5096_in(x4999, x5000, x5001, x5002) :|: TRUE f5107_out(x5003, x5004, x5005, x5006) -> f5096_out(x5003, x5004, x5005, x5006) :|: TRUE f5106_out(x5007, x5008, x5009) -> f5096_out([], x5007, x5008, x5009) :|: TRUE f5096_in([], x5010, x5011, x5012) -> f5106_in(x5010, x5011, x5012) :|: TRUE f5096_in(x5013, x5014, x5015, x5016) -> f5107_in(x5013, x5014, x5015, x5016) :|: TRUE f5121_out(x5017, x5018, x5019, x5020) -> f5107_out(x5017, x5018, x5019, x5020) :|: TRUE f5122_out -> f5107_out(x5021, x5022, x5023, x5024) :|: TRUE f5107_in(x5025, x5026, x5027, x5028) -> f5121_in(x5025, x5026, x5027, x5028) :|: TRUE f5107_in(x5029, x5030, x5031, x5032) -> f5122_in :|: TRUE f5123_out(x5033, x5034, x5035) -> f5124_in(x5034, x5036, x5033, x5035, x5037) :|: TRUE f5121_in(x5038, x5039, x5040, x5041) -> f5123_in(x5038, x5039, x5040) :|: TRUE f5124_out(x5042, x5043, x5044, x5045, x5046) -> f5121_out(x5044, x5042, x5045, x5046) :|: TRUE f5124_in(x5047, x5048, x5049, x5050, x5051) -> f5132_in(x5047, x5048, x5049, x5050, x5051) :|: TRUE f5133_out(x5052, x5053, x5054, x5055, x5056, x5057) -> f5124_out(x5054, x5053, x5057, x5055, x5056) :|: TRUE f5132_out(x5058, x5059, x5060, x5061, x5062) -> f5133_in(x5063, x5059, x5058, x5061, x5062, x5060) :|: TRUE f5133_in(x5064, x5065, x5066, x5067, x5068, x5069) -> f5167_in(x5064, x5065, x5066, x5067, x5068, x5069) :|: TRUE f5167_out(x5070, x5071, x5072, x5073, x5074, x5075) -> f5133_out(x5070, x5071, x5072, x5073, x5074, x5075) :|: TRUE f5171_out(x5076, x5077, x5078, x5079, x5080) -> f5167_out([], x5076, x5077, x5078, x5079, x5080) :|: TRUE f5167_in([], x5081, x5082, x5083, x5084, x5085) -> f5171_in(x5081, x5082, x5083, x5084, x5085) :|: TRUE f5167_in(x5086, x5087, x5088, x5089, x5090, x5091) -> f5172_in(x5086, x5087, x5088, x5089, x5090, x5091) :|: TRUE f5172_out(x5092, x5093, x5094, x5095, x5096, x5097) -> f5167_out(x5092, x5093, x5094, x5095, x5096, x5097) :|: TRUE f5182_out(x5098, x5099, x5100, x5101, x5102, x5103) -> f5172_out(x5098, x5099, x5100, x5101, x5102, x5103) :|: TRUE f5172_in(x5104, x5105, x5106, x5107, x5108, x5109) -> f5183_in :|: TRUE f5183_out -> f5172_out(x5110, x5111, x5112, x5113, x5114, x5115) :|: TRUE f5172_in(x5116, x5117, x5118, x5119, x5120, x5121) -> f5182_in(x5116, x5117, x5118, x5119, x5120, x5121) :|: TRUE f5189_out(x5122, x5123, x5124, x5125) -> f5190_in(x5123, x5124, x5126, x5122, x5125, x5127, x5128) :|: TRUE f5190_out(x5129, x5130, x5131, x5132, x5133, x5134, x5135) -> f5182_out(x5132, x5129, x5130, x5133, x5134, x5135) :|: TRUE f5182_in(x5136, x5137, x5138, x5139, x5140, x5141) -> f5189_in(x5136, x5137, x5138, x5139) :|: TRUE f5195_out(x5142, x5143, x5144, x5145, x5146, x5147, x5148, x5149) -> f5190_out(x5144, x5145, x5143, x5149, x5146, x5147, x5148) :|: TRUE f5190_in(x5150, x5151, x5152, x5153, x5154, x5155, x5156) -> f5194_in(x5150, x5151, x5152, x5153, x5154, x5155, x5156) :|: TRUE f5194_out(x5157, x5158, x5159, x5160, x5161, x5162, x5163) -> f5195_in(x5164, x5159, x5157, x5158, x5161, x5162, x5163, x5160) :|: TRUE f5195_in(x5165, x5166, x5167, x5168, x5169, x5170, x5171, x5172) -> f5288_in(x5165, x5166, x5167, x5168, x5169, x5170, x5171, x5172) :|: TRUE f5288_out(x5173, x5174, x5175, x5176, x5177, x5178, x5179, x5180) -> f5195_out(x5173, x5174, x5175, x5176, x5177, x5178, x5179, x5180) :|: TRUE f5288_in([], x5181, x5182, x5183, x5184, x5185, x5186, x5187) -> f5294_in(x5181, x5182, x5183, x5184, x5185, x5186, x5187) :|: TRUE f5294_out(x5188, x5189, x5190, x5191, x5192, x5193, x5194) -> f5288_out([], x5188, x5189, x5190, x5191, x5192, x5193, x5194) :|: TRUE f5298_out(x5195, x5196, x5197, x5198, x5199, x5200, x5201, x5202) -> f5288_out(x5195, x5196, x5197, x5198, x5199, x5200, x5201, x5202) :|: TRUE f5288_in(x5203, x5204, x5205, x5206, x5207, x5208, x5209, x5210) -> f5298_in(x5203, x5204, x5205, x5206, x5207, x5208, x5209, x5210) :|: TRUE f5298_in(x5211, x5212, x5213, x5214, x5215, x5216, x5217, x5218) -> f5314_in(x5211, x5212, x5213, x5214, x5215, x5216, x5217, x5218) :|: TRUE f5298_in(x5219, x5220, x5221, x5222, x5223, x5224, x5225, x5226) -> f5315_in :|: TRUE f5315_out -> f5298_out(x5227, x5228, x5229, x5230, x5231, x5232, x5233, x5234) :|: TRUE f5314_out(x5235, x5236, x5237, x5238, x5239, x5240, x5241, x5242) -> f5298_out(x5235, x5236, x5237, x5238, x5239, x5240, x5241, x5242) :|: TRUE f5336_out(x5243, x5244, x5245, x5246, x5247, x5248, x5249, x5250, x5251) -> f5314_out(x5247, x5243, x5244, x5245, x5248, x5249, x5250, x5251) :|: TRUE f5314_in(x5252, x5253, x5254, x5255, x5256, x5257, x5258, x5259) -> f5333_in(x5252, x5253, x5254, x5255, x5256) :|: TRUE f5333_out(x5260, x5261, x5262, x5263, x5264) -> f5336_in(x5261, x5262, x5263, x5265, x5260, x5264, x5266, x5267, x5268) :|: TRUE f5336_in(x5269, x5270, x5271, x5272, x5273, x5274, x5275, x5276, x5277) -> f5340_in(x5269, x5270, x5271, x5272, x5273, x5274, x5275, x5276, x5277) :|: TRUE f5341_out(x5278, x5279, x5280, x5281, x5282, x5283, x5284, x5285, x5286, x5287) -> f5336_out(x5280, x5281, x5282, x5279, x5287, x5283, x5284, x5285, x5286) :|: TRUE f5340_out(x5288, x5289, x5290, x5291, x5292, x5293, x5294, x5295, x5296) -> f5341_in(x5297, x5291, x5288, x5289, x5290, x5293, x5294, x5295, x5296, x5292) :|: TRUE f5408_out(x5298, x5299, x5300, x5301, x5302, x5303, x5304, x5305, x5306, x5307) -> f5341_out(x5298, x5299, x5300, x5301, x5302, x5303, x5304, x5305, x5306, x5307) :|: TRUE f5341_in(x5308, x5309, x5310, x5311, x5312, x5313, x5314, x5315, x5316, x5317) -> f5408_in(x5308, x5309, x5310, x5311, x5312, x5313, x5314, x5315, x5316, x5317) :|: TRUE f5419_out(x5318, x5319, x5320, x5321, x5322, x5323, x5324, x5325, x5326, x5327) -> f5408_out(x5318, x5319, x5320, x5321, x5322, x5323, x5324, x5325, x5326, x5327) :|: TRUE f5418_out(x5328, x5329, x5330, x5331, x5332, x5333, x5334, x5335, x5336) -> f5408_out([], x5328, x5329, x5330, x5331, x5332, x5333, x5334, x5335, x5336) :|: TRUE f5408_in([], x5337, x5338, x5339, x5340, x5341, x5342, x5343, x5344, x5345) -> f5418_in(x5337, x5338, x5339, x5340, x5341, x5342, x5343, x5344, x5345) :|: TRUE f5408_in(x5346, x5347, x5348, x5349, x5350, x5351, x5352, x5353, x5354, x5355) -> f5419_in(x5346, x5347, x5348, x5349, x5350, x5351, x5352, x5353, x5354, x5355) :|: TRUE f5422_out(x5356, x5357, x5358, x5359, x5360, x5361, x5362, x5363, x5364, x5365) -> f5419_out(x5356, x5357, x5358, x5359, x5360, x5361, x5362, x5363, x5364, x5365) :|: TRUE f5419_in(x5366, x5367, x5368, x5369, x5370, x5371, x5372, x5373, x5374, x5375) -> f5422_in(x5366, x5367, x5368, x5369, x5370, x5371, x5372, x5373, x5374, x5375) :|: TRUE f5419_in(x5376, x5377, x5378, x5379, x5380, x5381, x5382, x5383, x5384, x5385) -> f5423_in :|: TRUE f5423_out -> f5419_out(x5386, x5387, x5388, x5389, x5390, x5391, x5392, x5393, x5394, x5395) :|: TRUE f5422_in(x5396, x5397, x5398, x5399, x5400, x5401, x5402, x5403, x5404, x5405) -> f5424_in(x5396, x5397, x5398, x5399, x5400, x5401) :|: TRUE f5424_out(x5406, x5407, x5408, x5409, x5410, x5411) -> f5425_in(x5407, x5408, x5409, x5410, x5412, x5406, x5411, x5413, x5414, x5415, x5416) :|: TRUE f5425_out(x5417, x5418, x5419, x5420, x5421, x5422, x5423, x5424, x5425, x5426, x5427) -> f5422_out(x5422, x5417, x5418, x5419, x5420, x5423, x5424, x5425, x5426, x5427) :|: TRUE f5426_out(x5428, x5429, x5430, x5431, x5432, x5433, x5434, x5435, x5436, x5437, x5438) -> f5427_in(x5439, x5432, x5428, x5429, x5430, x5431, x5434, x5435, x5436, x5437, x5438, x5433) :|: TRUE f5427_out(x5440, x5441, x5442, x5443, x5444, x5445, x5446, x5447, x5448, x5449, x5450, x5451) -> f5425_out(x5442, x5443, x5444, x5445, x5441, x5451, x5446, x5447, x5448, x5449, x5450) :|: TRUE f5425_in(x5452, x5453, x5454, x5455, x5456, x5457, x5458, x5459, x5460, x5461, x5462) -> f5426_in(x5452, x5453, x5454, x5455, x5456, x5457, x5458, x5459, x5460, x5461, x5462) :|: TRUE f5427_in(x5463, x5464, x5465, x5466, x5467, x5468, x5469, x5470, x5471, x5472, x5473, x5474) -> f5539_in(x5463, x5464, x5465, x5466, x5467, x5468, x5469, x5470, x5471, x5472, x5473, x5474) :|: TRUE f5539_out(x5475, x5476, x5477, x5478, x5479, x5480, x5481, x5482, x5483, x5484, x5485, x5486) -> f5427_out(x5475, x5476, x5477, x5478, x5479, x5480, x5481, x5482, x5483, x5484, x5485, x5486) :|: TRUE f5563_out(x5487, x5488, x5489, x5490, x5491, x5492, x5493, x5494, x5495, x5496, x5497) -> f5539_out([], x5487, x5488, x5489, x5490, x5491, x5492, x5493, x5494, x5495, x5496, x5497) :|: TRUE f5539_in([], x5498, x5499, x5500, x5501, x5502, x5503, x5504, x5505, x5506, x5507, x5508) -> f5563_in(x5498, x5499, x5500, x5501, x5502, x5503, x5504, x5505, x5506, x5507, x5508) :|: TRUE f5539_in(x5509, x5510, x5511, x5512, x5513, x5514, x5515, x5516, x5517, x5518, x5519, x5520) -> f5564_in(x5509, x5510, x5511, x5512, x5513, x5514, x5515, x5516, x5517, x5518, x5519, x5520) :|: TRUE f5564_out(x5521, x5522, x5523, x5524, x5525, x5526, x5527, x5528, x5529, x5530, x5531, x5532) -> f5539_out(x5521, x5522, x5523, x5524, x5525, x5526, x5527, x5528, x5529, x5530, x5531, x5532) :|: TRUE f5586_out -> f5564_out(x5533, x5534, x5535, x5536, x5537, x5538, x5539, x5540, x5541, x5542, x5543, x5544) :|: TRUE f5564_in(x5545, x5546, x5547, x5548, x5549, x5550, x5551, x5552, x5553, x5554, x5555, x5556) -> f5581_in(x5545, x5546, x5547, x5548, x5549, x5550, x5551, x5552, x5553, x5554, x5555, x5556) :|: TRUE f5581_out(x5557, x5558, x5559, x5560, x5561, x5562, x5563, x5564, x5565, x5566, x5567, x5568) -> f5564_out(x5557, x5558, x5559, x5560, x5561, x5562, x5563, x5564, x5565, x5566, x5567, x5568) :|: TRUE f5564_in(x5569, x5570, x5571, x5572, x5573, x5574, x5575, x5576, x5577, x5578, x5579, x5580) -> f5586_in :|: TRUE f5591_out(x5581, x5582, x5583, x5584, x5585, x5586, x5587) -> f5592_in(x5582, x5583, x5584, x5585, x5586, x5588, x5581, x5587, x5589, x5590, x5591, x5592, x5593) :|: TRUE f5592_out(x5594, x5595, x5596, x5597, x5598, x5599, x5600, x5601, x5602, x5603, x5604, x5605, x5606) -> f5581_out(x5600, x5594, x5595, x5596, x5597, x5598, x5601, x5602, x5603, x5604, x5605, x5606) :|: TRUE f5581_in(x5607, x5608, x5609, x5610, x5611, x5612, x5613, x5614, x5615, x5616, x5617, x5618) -> f5591_in(x5607, x5608, x5609, x5610, x5611, x5612, x5613) :|: TRUE f5606_out(x5619, x5620, x5621, x5622, x5623, x5624, x5625, x5626, x5627, x5628, x5629, x5630, x5631) -> f5607_in(x5632, x5624, x5619, x5620, x5621, x5622, x5623, x5626, x5627, x5628, x5629, x5630, x5631, x5625) :|: TRUE f5607_out(x5633, x5634, x5635, x5636, x5637, x5638, x5639, x5640, x5641, x5642, x5643, x5644, x5645, x5646) -> f5592_out(x5635, x5636, x5637, x5638, x5639, x5634, x5646, x5640, x5641, x5642, x5643, x5644, x5645) :|: TRUE f5592_in(x5647, x5648, x5649, x5650, x5651, x5652, x5653, x5654, x5655, x5656, x5657, x5658, x5659) -> f5606_in(x5647, x5648, x5649, x5650, x5651, x5652, x5653, x5654, x5655, x5656, x5657, x5658, x5659) :|: TRUE f5691_out(x5660, x5661, x5662, x5663, x5664, x5665, x5666, x5667, x5668, x5669, x5670, x5671, x5672, x5673) -> f5607_out(x5660, x5661, x5662, x5663, x5664, x5665, x5666, x5667, x5668, x5669, x5670, x5671, x5672, x5673) :|: TRUE f5607_in(x5674, x5675, x5676, x5677, x5678, x5679, x5680, x5681, x5682, x5683, x5684, x5685, x5686, x5687) -> f5691_in(x5674, x5675, x5676, x5677, x5678, x5679, x5680, x5681, x5682, x5683, x5684, x5685, x5686, x5687) :|: TRUE f5716_out(x5688, x5689, x5690, x5691, x5692, x5693, x5694, x5695, x5696, x5697, x5698, x5699, x5700) -> f5691_out([], x5688, x5689, x5690, x5691, x5692, x5693, x5694, x5695, x5696, x5697, x5698, x5699, x5700) :|: TRUE f5719_out(x5701, x5702, x5703, x5704, x5705, x5706, x5707, x5708, x5709, x5710, x5711, x5712, x5713, x5714) -> f5691_out(x5701, x5702, x5703, x5704, x5705, x5706, x5707, x5708, x5709, x5710, x5711, x5712, x5713, x5714) :|: TRUE f5691_in([], x5715, x5716, x5717, x5718, x5719, x5720, x5721, x5722, x5723, x5724, x5725, x5726, x5727) -> f5716_in(x5715, x5716, x5717, x5718, x5719, x5720, x5721, x5722, x5723, x5724, x5725, x5726, x5727) :|: TRUE f5691_in(x5728, x5729, x5730, x5731, x5732, x5733, x5734, x5735, x5736, x5737, x5738, x5739, x5740, x5741) -> f5719_in(x5728, x5729, x5730, x5731, x5732, x5733, x5734, x5735, x5736, x5737, x5738, x5739, x5740, x5741) :|: TRUE f5719_in(x5742, x5743, x5744, x5745, x5746, x5747, x5748, x5749, x5750, x5751, x5752, x5753, x5754, x5755) -> f5733_in(x5742, x5743, x5744, x5745, x5746, x5747, x5748, x5749, x5750, x5751, x5752, x5753, x5754, x5755) :|: TRUE f5733_out(x5756, x5757, x5758, x5759, x5760, x5761, x5762, x5763, x5764, x5765, x5766, x5767, x5768, x5769) -> f5719_out(x5756, x5757, x5758, x5759, x5760, x5761, x5762, x5763, x5764, x5765, x5766, x5767, x5768, x5769) :|: TRUE f5734_out -> f5719_out(x5770, x5771, x5772, x5773, x5774, x5775, x5776, x5777, x5778, x5779, x5780, x5781, x5782, x5783) :|: TRUE f5719_in(x5784, x5785, x5786, x5787, x5788, x5789, x5790, x5791, x5792, x5793, x5794, x5795, x5796, x5797) -> f5734_in :|: TRUE f5736_out(x5798, x5799, x5800, x5801, x5802, x5803, x5804, x5805, .(x5799, .(x5800, .(x5801, .(x5802, .(x5803, .(x5804, .(0, []))))))), x5806, x5807, x5808, x5809, x5810, x5811) -> f5733_out(x5798, x5799, x5800, x5801, x5802, x5803, x5804, x5805, x5806, x5807, x5808, x5809, x5810, x5811) :|: TRUE f5733_in(x5812, x5813, x5814, x5815, x5816, x5817, x5818, x5819, x5820, x5821, x5822, x5823, x5824, x5825) -> f5736_in(x5812, x5813, x5814, x5815, x5816, x5817, x5818, x5819, .(x5813, .(x5814, .(x5815, .(x5816, .(x5817, .(x5818, .(0, []))))))), x5820, x5821, x5822, x5823, x5824, x5825) :|: TRUE f5745_out(x5826, x5827, x5828, x5829, x5830, x5831, x5832, x5833, x5834, x5835, x5836, x5837, x5838, x5839, x5840, x5841) -> f5736_out(x5833, x5826, x5827, x5828, x5829, x5830, x5831, x5834, x5835, x5836, x5837, x5838, x5839, x5840, x5841) :|: TRUE f5736_in(x5842, x5843, x5844, x5845, x5846, x5847, x5848, x5849, x5850, x5851, x5852, x5853, x5854, x5855, x5856) -> f5744_in(x5842, x5843, x5844, x5845, x5846, x5847, x5848, x5849) :|: TRUE f5744_out(x5857, x5858, x5859, x5860, x5861, x5862, x5863, x5864) -> f5745_in(x5858, x5859, x5860, x5861, x5862, x5863, x5865, x5857, x5864, x5866, x5867, x5868, x5869, x5870, x5871, x5872) :|: TRUE f5746_out(x5873, x5874, x5875, x5876, x5877, x5878, x5879, x5880, x5881, x5882, x5883, x5884, x5885, x5886, x5887) -> f5748_in(x5888, x5879, x5889, x5881, x5882, x5883, x5884, x5885, x5886, x5887, x5880, x5873, x5874, x5875, x5876, x5877, x5878) :|: TRUE f5745_in(x5890, x5891, x5892, x5893, x5894, x5895, x5896, x5897, x5898, x5899, x5900, x5901, x5902, x5903, x5904, x5905) -> f5746_in(x5890, x5891, x5892, x5893, x5894, x5895, x5896, x5897, x5898, x5900, x5901, x5902, x5903, x5904, x5905) :|: TRUE f5748_out(x5906, x5907, x5908, x5909, x5910, x5911, x5912, x5913, x5914, x5915, x5916, x5917, x5918, x5919, x5920, x5921, x5922) -> f5745_out(x5917, x5918, x5919, x5920, x5921, x5922, x5907, x5916, x5909, x5908, x5910, x5911, x5912, x5913, x5914, x5915) :|: TRUE Start term: f1_in(T1) ---------------------------------------- (418) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (419) TRUE ---------------------------------------- (420) Obligation: Rules: f57_in(T24, T24) -> f58_in(T24) :|: TRUE f57_in(T20, T19) -> f59_in(T20, T19) :|: TRUE f58_out(x) -> f57_out(x, x) :|: TRUE f59_out(x1, x2) -> f57_out(x1, x2) :|: TRUE f56_out(T31, T30) -> f63_out(T31, T30, T29) :|: TRUE f63_in(x3, x4, x5) -> f56_in(x3, x4) :|: TRUE f62_out(x6, x7) -> f59_out(x6, x7) :|: TRUE f59_in(x8, x9) -> f62_in(x8, x9) :|: TRUE f56_in(x10, x11) -> f57_in(x10, x11) :|: TRUE f57_out(x12, x13) -> f56_out(x12, x13) :|: TRUE f62_in(x14, x15) -> f63_in(x16, x15, x14) :|: x16 = x14 + 1 f63_out(x17, x18, x19) -> f62_out(x19, x18) :|: TRUE f4_out(T1) -> f1_out(T1) :|: TRUE f1_in(x20) -> f4_in(x20) :|: TRUE f4_in(T7) -> f10_in(T7) :|: TRUE f11_out -> f4_out(x21) :|: TRUE f10_out(x22) -> f4_out(x22) :|: TRUE f4_in(x23) -> f11_in :|: TRUE f10_in(x24) -> f16_in(x25, x24) :|: x25 = x24 * (x24 + 1) + 1 f10_in(x26) -> f12_in :|: TRUE f16_out(x27, x28) -> f10_out(x28) :|: TRUE f12_out -> f10_out(x29) :|: TRUE f16_in(x30, x31) -> f21_in(x30) :|: TRUE f24_out(x32, x33) -> f16_out(x33, x34) :|: TRUE f21_out(T10) -> f24_in(T12, T10) :|: TRUE f21_in(x35) -> f48_in(x35) :|: TRUE f48_out(x36) -> f21_out(x36) :|: TRUE f48_in(T16) -> f52_in(T16) :|: TRUE f52_out(x37) -> f48_out(x37) :|: TRUE f53_out(x38) -> f52_out(x38) :|: TRUE f52_in(x39) -> f53_in(x39) :|: TRUE f53_in(x40) -> f54_in(x40) :|: TRUE f54_out(x41) -> f53_out(x41) :|: TRUE f55_out(x42) -> f54_out(x42) :|: TRUE f54_in(x43) -> f55_in(x43) :|: TRUE f55_in(x44) -> f56_in(x45, x44) :|: x45 = 0 + 1 f56_out(x46, x47) -> f55_out(x47) :|: TRUE Start term: f1_in(T1) ---------------------------------------- (421) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f57_in(T20, T19) -> f59_in(T20, T19) :|: TRUE f63_in(x3, x4, x5) -> f56_in(x3, x4) :|: TRUE f59_in(x8, x9) -> f62_in(x8, x9) :|: TRUE f56_in(x10, x11) -> f57_in(x10, x11) :|: TRUE f62_in(x14, x15) -> f63_in(x16, x15, x14) :|: x16 = x14 + 1 ---------------------------------------- (422) Obligation: Rules: f57_in(T20, T19) -> f59_in(T20, T19) :|: TRUE f63_in(x3, x4, x5) -> f56_in(x3, x4) :|: TRUE f59_in(x8, x9) -> f62_in(x8, x9) :|: TRUE f56_in(x10, x11) -> f57_in(x10, x11) :|: TRUE f62_in(x14, x15) -> f63_in(x16, x15, x14) :|: x16 = x14 + 1 ---------------------------------------- (423) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (424) Obligation: Rules: f63_in(x3:0, x4:0, x5:0) -> f63_in(x3:0 + 1, x4:0, x3:0) :|: TRUE ---------------------------------------- (425) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (426) Obligation: Rules: f63_in(x3:0, x4:0, x5:0) -> f63_in(arith, x4:0, x3:0) :|: TRUE && arith = x3:0 + 1 ---------------------------------------- (427) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f63_in(x3:0, x4:0, x5:0) -> f63_in(arith, x4:0, x3:0) :|: TRUE && arith = x3:0 + 1 Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (428) Obligation: Termination digraph: Nodes: (1) f63_in(x3:0, x4:0, x5:0) -> f63_in(arith, x4:0, x3:0) :|: TRUE && arith = x3:0 + 1 Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (429) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (430) Obligation: Rules: f63_in(x3:0:0, x4:0:0, x5:0:0) -> f63_in(x3:0:0 + 1, x4:0:0, x3:0:0) :|: TRUE ---------------------------------------- (431) IRSwTToIntTRSProof (SOUND) Applied path-length measure to transform intTRS with terms to intTRS. ---------------------------------------- (432) Obligation: Rules: f63_in(x, x1, x2) -> f63_in(x + 1, x1, x) :|: TRUE ---------------------------------------- (433) IntTRSNonPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc, x, x1, x2) -> f(1, x + 1, x1, x) :|: pc = 1 && TRUE Proved unsatisfiability of the following formula, indicating that the system is never left after entering: (((run2_0 = ((1 * 1)) and run2_1 = ((run1_1 * 1) + (1 * 1)) and run2_2 = ((run1_2 * 1)) and run2_3 = ((run1_1 * 1))) and (((run1_0 * 1)) = ((1 * 1)) and T)) and !(((run2_0 * 1)) = ((1 * 1)) and T)) Proved satisfiability of the following formula, indicating that the system is entered at least once: ((run2_0 = ((1 * 1)) and run2_1 = ((run1_1 * 1) + (1 * 1)) and run2_2 = ((run1_2 * 1)) and run2_3 = ((run1_1 * 1))) and (((run1_0 * 1)) = ((1 * 1)) and T)) ---------------------------------------- (434) NO