/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.pl /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- MAYBE proof of /export/starexec/sandbox/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern qs(a,g) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) PrologToPiTRSProof [SOUND, 36 ms] (2) PiTRS (3) DependencyPairsProof [EQUIVALENT, 0 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [SOUND, 16 ms] (11) QDP (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] (13) YES (14) PiDP (15) UsableRulesProof [EQUIVALENT, 0 ms] (16) PiDP (17) PiDPToQDPProof [SOUND, 0 ms] (18) QDP (19) NonTerminationLoopProof [COMPLETE, 0 ms] (20) NO (21) PiDP (22) UsableRulesProof [EQUIVALENT, 0 ms] (23) PiDP (24) PiDPToQDPProof [SOUND, 0 ms] (25) QDP (26) QDPSizeChangeProof [EQUIVALENT, 0 ms] (27) YES (28) PiDP (29) UsableRulesProof [EQUIVALENT, 0 ms] (30) PiDP (31) PiDPToQDPProof [SOUND, 0 ms] (32) QDP (33) TransformationProof [SOUND, 0 ms] (34) QDP (35) TransformationProof [EQUIVALENT, 0 ms] (36) QDP (37) NonTerminationLoopProof [COMPLETE, 0 ms] (38) NO (39) PiDP (40) UsableRulesProof [EQUIVALENT, 0 ms] (41) PiDP (42) PiDPToQDPProof [SOUND, 0 ms] (43) QDP (44) NonTerminationLoopProof [COMPLETE, 0 ms] (45) NO (46) PiDP (47) UsableRulesProof [EQUIVALENT, 0 ms] (48) PiDP (49) PiDPToQDPProof [SOUND, 0 ms] (50) QDP (51) PiDP (52) UsableRulesProof [EQUIVALENT, 0 ms] (53) PiDP (54) PrologToPiTRSProof [SOUND, 33 ms] (55) PiTRS (56) DependencyPairsProof [EQUIVALENT, 16 ms] (57) PiDP (58) DependencyGraphProof [EQUIVALENT, 0 ms] (59) AND (60) PiDP (61) UsableRulesProof [EQUIVALENT, 0 ms] (62) PiDP (63) PiDPToQDPProof [SOUND, 1 ms] (64) QDP (65) QDPSizeChangeProof [EQUIVALENT, 0 ms] (66) YES (67) PiDP (68) UsableRulesProof [EQUIVALENT, 0 ms] (69) PiDP (70) PiDPToQDPProof [SOUND, 0 ms] (71) QDP (72) NonTerminationLoopProof [COMPLETE, 0 ms] (73) NO (74) PiDP (75) UsableRulesProof [EQUIVALENT, 0 ms] (76) PiDP (77) PiDPToQDPProof [SOUND, 0 ms] (78) QDP (79) QDPSizeChangeProof [EQUIVALENT, 0 ms] (80) YES (81) PiDP (82) UsableRulesProof [EQUIVALENT, 0 ms] (83) PiDP (84) PiDPToQDPProof [SOUND, 0 ms] (85) QDP (86) TransformationProof [SOUND, 0 ms] (87) QDP (88) TransformationProof [EQUIVALENT, 0 ms] (89) QDP (90) NonTerminationLoopProof [COMPLETE, 0 ms] (91) NO (92) PiDP (93) UsableRulesProof [EQUIVALENT, 0 ms] (94) PiDP (95) PiDPToQDPProof [SOUND, 0 ms] (96) QDP (97) NonTerminationLoopProof [COMPLETE, 0 ms] (98) NO (99) PiDP (100) UsableRulesProof [EQUIVALENT, 0 ms] (101) PiDP (102) PiDPToQDPProof [SOUND, 0 ms] (103) QDP (104) PiDP (105) UsableRulesProof [EQUIVALENT, 0 ms] (106) PiDP (107) PrologToTRSTransformerProof [SOUND, 81 ms] (108) QTRS (109) DependencyPairsProof [EQUIVALENT, 0 ms] (110) QDP (111) DependencyGraphProof [EQUIVALENT, 2 ms] (112) AND (113) QDP (114) UsableRulesProof [EQUIVALENT, 2 ms] (115) QDP (116) QDPSizeChangeProof [EQUIVALENT, 0 ms] (117) YES (118) QDP (119) UsableRulesProof [EQUIVALENT, 0 ms] (120) QDP (121) NonTerminationLoopProof [COMPLETE, 0 ms] (122) NO (123) QDP (124) UsableRulesProof [EQUIVALENT, 1 ms] (125) QDP (126) QDPSizeChangeProof [EQUIVALENT, 0 ms] (127) YES (128) QDP (129) NonTerminationLoopProof [COMPLETE, 0 ms] (130) NO (131) QDP (132) UsableRulesProof [EQUIVALENT, 0 ms] (133) QDP (134) NonTerminationLoopProof [COMPLETE, 0 ms] (135) NO (136) QDP (137) UsableRulesProof [EQUIVALENT, 0 ms] (138) QDP (139) NonTerminationLoopProof [COMPLETE, 0 ms] (140) NO (141) QDP (142) NonLoopProof [COMPLETE, 1084 ms] (143) NO (144) PrologToDTProblemTransformerProof [SOUND, 85 ms] (145) TRIPLES (146) UndefinedPredicateInTriplesTransformerProof [SOUND, 0 ms] (147) TRIPLES (148) TriplesToPiDPProof [SOUND, 172 ms] (149) PiDP (150) DependencyGraphProof [EQUIVALENT, 0 ms] (151) AND (152) PiDP (153) UsableRulesProof [EQUIVALENT, 0 ms] (154) PiDP (155) PiDPToQDPProof [SOUND, 0 ms] (156) QDP (157) QDPSizeChangeProof [EQUIVALENT, 0 ms] (158) YES (159) PiDP (160) UsableRulesProof [EQUIVALENT, 0 ms] (161) PiDP (162) PiDPToQDPProof [SOUND, 0 ms] (163) QDP (164) QDPSizeChangeProof [EQUIVALENT, 0 ms] (165) YES (166) PiDP (167) UsableRulesProof [EQUIVALENT, 0 ms] (168) PiDP (169) PiDPToQDPProof [SOUND, 0 ms] (170) QDP (171) QDPSizeChangeProof [EQUIVALENT, 0 ms] (172) YES (173) PiDP (174) UsableRulesProof [EQUIVALENT, 0 ms] (175) PiDP (176) PiDPToQDPProof [SOUND, 0 ms] (177) QDP (178) QDPSizeChangeProof [EQUIVALENT, 0 ms] (179) YES (180) PiDP (181) UsableRulesProof [EQUIVALENT, 0 ms] (182) PiDP (183) PiDPToQDPProof [EQUIVALENT, 0 ms] (184) QDP (185) QDPSizeChangeProof [EQUIVALENT, 0 ms] (186) YES (187) PiDP (188) UsableRulesProof [EQUIVALENT, 0 ms] (189) PiDP (190) PiDPToQDPProof [SOUND, 0 ms] (191) QDP (192) QDPSizeChangeProof [EQUIVALENT, 0 ms] (193) YES (194) PiDP (195) UsableRulesProof [EQUIVALENT, 0 ms] (196) PiDP (197) PiDPToQDPProof [SOUND, 0 ms] (198) QDP (199) QDPSizeChangeProof [EQUIVALENT, 0 ms] (200) YES (201) PiDP (202) UsableRulesProof [EQUIVALENT, 0 ms] (203) PiDP (204) PiDPToQDPProof [SOUND, 0 ms] (205) QDP (206) QDPQMonotonicMRRProof [EQUIVALENT, 21 ms] (207) QDP (208) DependencyGraphProof [EQUIVALENT, 0 ms] (209) TRUE (210) PiDP (211) UsableRulesProof [EQUIVALENT, 0 ms] (212) PiDP (213) PiDPToQDPProof [SOUND, 0 ms] (214) QDP (215) NonTerminationLoopProof [COMPLETE, 0 ms] (216) NO (217) PiDP (218) UsableRulesProof [EQUIVALENT, 0 ms] (219) PiDP (220) PiDPToQDPProof [SOUND, 0 ms] (221) QDP (222) QDPSizeChangeProof [EQUIVALENT, 0 ms] (223) YES (224) PiDP (225) UsableRulesProof [EQUIVALENT, 0 ms] (226) PiDP (227) PiDPToQDPProof [SOUND, 0 ms] (228) QDP (229) TransformationProof [SOUND, 0 ms] (230) QDP (231) TransformationProof [EQUIVALENT, 0 ms] (232) QDP (233) PiDP (234) UsableRulesProof [EQUIVALENT, 0 ms] (235) PiDP (236) PiDP (237) UsableRulesProof [EQUIVALENT, 0 ms] (238) PiDP (239) PiDP (240) UsableRulesProof [EQUIVALENT, 1 ms] (241) PiDP (242) PrologToIRSwTTransformerProof [SOUND, 126 ms] (243) AND (244) IRSwT (245) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (246) TRUE (247) IRSwT (248) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (249) TRUE (250) IRSwT (251) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (252) TRUE (253) IRSwT (254) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (255) TRUE (256) IRSwT (257) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (258) IRSwT (259) IntTRSCompressionProof [EQUIVALENT, 18 ms] (260) IRSwT (261) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (262) IRSwT (263) IRSwTTerminationDigraphProof [EQUIVALENT, 6 ms] (264) IRSwT (265) FilterProof [EQUIVALENT, 0 ms] (266) IntTRS (267) IntTRSNonPeriodicNontermProof [COMPLETE, 6 ms] (268) NO (269) IRSwT (270) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 1 ms] (271) IRSwT (272) IntTRSCompressionProof [EQUIVALENT, 0 ms] (273) IRSwT (274) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (275) IRSwT (276) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms] (277) IRSwT (278) FilterProof [EQUIVALENT, 0 ms] (279) IntTRS (280) IntTRSNonPeriodicNontermProof [COMPLETE, 3 ms] (281) NO (282) IRSwT (283) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (284) IRSwT (285) IntTRSCompressionProof [EQUIVALENT, 40 ms] (286) IRSwT (287) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (288) IRSwT (289) IRSwTTerminationDigraphProof [EQUIVALENT, 95 ms] (290) IRSwT (291) IntTRSCompressionProof [EQUIVALENT, 23 ms] (292) IRSwT ---------------------------------------- (0) Obligation: Clauses: qs([], []). qs(.(X, Xs), Ys) :- ','(part(X, Xs, Littles, Bigs), ','(qs(Littles, Ls), ','(qs(Bigs, Bs), app(Ls, .(X, Bs), Ys)))). part(X, .(Y, Xs), .(Y, Ls), Bs) :- ','(less(X, Y), part(X, Xs, Ls, Bs)). part(X, .(Y, Xs), Ls, .(Y, Bs)) :- part(X, Xs, Ls, Bs). part(X1, [], [], []). app([], X, X). app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs). less(0, s(X2)). less(s(X), s(Y)) :- less(X, Y). Query: qs(a,g) ---------------------------------------- (1) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: qs_in_2: (f,b) (f,f) part_in_4: (f,f,f,f) (b,f,f,f) less_in_2: (f,f) (b,f) app_in_3: (f,f,f) (f,f,b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: qs_in_ag([], []) -> qs_out_ag([], []) qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U5_aaaa(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U5_gaaa(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) qs_in_aa([], []) -> qs_out_aa([], []) qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) app_in_aaa([], X, X) -> app_out_aaa([], X, X) app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) app_in_aag([], X, X) -> app_out_aag([], X, X) app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ag(x1, x2) = qs_in_ag(x2) [] = [] qs_out_ag(x1, x2) = qs_out_ag(x2) U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U9_aa(x1, x2, x3) = U9_aa(x3) U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U9_ga(x1, x2, x3) = U9_ga(x1, x3) U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x1, x6) U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x1, x6) part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa(x1) part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) qs_in_aa(x1, x2) = qs_in_aa qs_out_aa(x1, x2) = qs_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U4_aa(x1, x2, x3, x4) = U4_aa(x4) .(x1, x2) = .(x1, x2) app_in_aaa(x1, x2, x3) = app_in_aaa app_out_aaa(x1, x2, x3) = app_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4) app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2, x3) U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x4, x5) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: qs_in_ag([], []) -> qs_out_ag([], []) qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U5_aaaa(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U5_gaaa(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) qs_in_aa([], []) -> qs_out_aa([], []) qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) app_in_aaa([], X, X) -> app_out_aaa([], X, X) app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) app_in_aag([], X, X) -> app_out_aag([], X, X) app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ag(x1, x2) = qs_in_ag(x2) [] = [] qs_out_ag(x1, x2) = qs_out_ag(x2) U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U9_aa(x1, x2, x3) = U9_aa(x3) U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U9_ga(x1, x2, x3) = U9_ga(x1, x3) U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x1, x6) U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x1, x6) part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa(x1) part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) qs_in_aa(x1, x2) = qs_in_aa qs_out_aa(x1, x2) = qs_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U4_aa(x1, x2, x3, x4) = U4_aa(x4) .(x1, x2) = .(x1, x2) app_in_aaa(x1, x2, x3) = app_in_aaa app_out_aaa(x1, x2, x3) = app_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4) app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2, x3) U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x4, x5) ---------------------------------------- (3) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: Pi DP problem: The TRS P consists of the following rules: QS_IN_AG(.(X, Xs), Ys) -> U1_AG(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) QS_IN_AG(.(X, Xs), Ys) -> PART_IN_AAAA(X, Xs, Littles, Bigs) PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_AAAA(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> LESS_IN_AA(X, Y) LESS_IN_AA(s(X), s(Y)) -> U9_AA(X, Y, less_in_aa(X, Y)) LESS_IN_AA(s(X), s(Y)) -> LESS_IN_AA(X, Y) U5_AAAA(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> U6_AAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) U5_AAAA(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs) PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GAAA(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> LESS_IN_GA(X, Y) LESS_IN_GA(s(X), s(Y)) -> U9_GA(X, Y, less_in_ga(X, Y)) LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) U5_GAAA(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> U6_GAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) U5_GAAA(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs) PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_GAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_GAAA(X, Xs, Ls, Bs) PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_AAAA(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_AAAA(X, Xs, Ls, Bs) U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AG(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> QS_IN_AA(Littles, Ls) QS_IN_AA(.(X, Xs), Ys) -> U1_AA(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) QS_IN_AA(.(X, Xs), Ys) -> PART_IN_AAAA(X, Xs, Littles, Bigs) U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AA(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> QS_IN_AA(Littles, Ls) U2_AA(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_AA(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U2_AA(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> QS_IN_AA(Bigs, Bs) U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_AA(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> APP_IN_AAA(Ls, .(X, Bs), Ys) APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> U8_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAA(Xs, Ys, Zs) U2_AG(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_AG(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U2_AG(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> QS_IN_AA(Bigs, Bs) U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_AG(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> APP_IN_AAG(Ls, .(X, Bs), Ys) APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> U8_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs) The TRS R consists of the following rules: qs_in_ag([], []) -> qs_out_ag([], []) qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U5_aaaa(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U5_gaaa(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) qs_in_aa([], []) -> qs_out_aa([], []) qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) app_in_aaa([], X, X) -> app_out_aaa([], X, X) app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) app_in_aag([], X, X) -> app_out_aag([], X, X) app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ag(x1, x2) = qs_in_ag(x2) [] = [] qs_out_ag(x1, x2) = qs_out_ag(x2) U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U9_aa(x1, x2, x3) = U9_aa(x3) U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U9_ga(x1, x2, x3) = U9_ga(x1, x3) U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x1, x6) U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x1, x6) part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa(x1) part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) qs_in_aa(x1, x2) = qs_in_aa qs_out_aa(x1, x2) = qs_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U4_aa(x1, x2, x3, x4) = U4_aa(x4) .(x1, x2) = .(x1, x2) app_in_aaa(x1, x2, x3) = app_in_aaa app_out_aaa(x1, x2, x3) = app_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4) app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2, x3) U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x4, x5) QS_IN_AG(x1, x2) = QS_IN_AG(x2) U1_AG(x1, x2, x3, x4) = U1_AG(x3, x4) PART_IN_AAAA(x1, x2, x3, x4) = PART_IN_AAAA U5_AAAA(x1, x2, x3, x4, x5, x6) = U5_AAAA(x6) LESS_IN_AA(x1, x2) = LESS_IN_AA U9_AA(x1, x2, x3) = U9_AA(x3) U6_AAAA(x1, x2, x3, x4, x5, x6) = U6_AAAA(x6) PART_IN_GAAA(x1, x2, x3, x4) = PART_IN_GAAA(x1) U5_GAAA(x1, x2, x3, x4, x5, x6) = U5_GAAA(x1, x6) LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) U9_GA(x1, x2, x3) = U9_GA(x1, x3) U6_GAAA(x1, x2, x3, x4, x5, x6) = U6_GAAA(x1, x6) U7_GAAA(x1, x2, x3, x4, x5, x6) = U7_GAAA(x1, x6) U7_AAAA(x1, x2, x3, x4, x5, x6) = U7_AAAA(x6) U2_AG(x1, x2, x3, x4, x5) = U2_AG(x3, x5) QS_IN_AA(x1, x2) = QS_IN_AA U1_AA(x1, x2, x3, x4) = U1_AA(x4) U2_AA(x1, x2, x3, x4, x5) = U2_AA(x5) U3_AA(x1, x2, x3, x4, x5) = U3_AA(x5) U4_AA(x1, x2, x3, x4) = U4_AA(x4) APP_IN_AAA(x1, x2, x3) = APP_IN_AAA U8_AAA(x1, x2, x3, x4, x5) = U8_AAA(x5) U3_AG(x1, x2, x3, x4, x5) = U3_AG(x3, x5) U4_AG(x1, x2, x3, x4) = U4_AG(x3, x4) APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3) U8_AAG(x1, x2, x3, x4, x5) = U8_AAG(x1, x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: QS_IN_AG(.(X, Xs), Ys) -> U1_AG(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) QS_IN_AG(.(X, Xs), Ys) -> PART_IN_AAAA(X, Xs, Littles, Bigs) PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_AAAA(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> LESS_IN_AA(X, Y) LESS_IN_AA(s(X), s(Y)) -> U9_AA(X, Y, less_in_aa(X, Y)) LESS_IN_AA(s(X), s(Y)) -> LESS_IN_AA(X, Y) U5_AAAA(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> U6_AAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) U5_AAAA(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs) PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GAAA(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> LESS_IN_GA(X, Y) LESS_IN_GA(s(X), s(Y)) -> U9_GA(X, Y, less_in_ga(X, Y)) LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) U5_GAAA(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> U6_GAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) U5_GAAA(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs) PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_GAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_GAAA(X, Xs, Ls, Bs) PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_AAAA(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_AAAA(X, Xs, Ls, Bs) U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AG(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> QS_IN_AA(Littles, Ls) QS_IN_AA(.(X, Xs), Ys) -> U1_AA(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) QS_IN_AA(.(X, Xs), Ys) -> PART_IN_AAAA(X, Xs, Littles, Bigs) U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AA(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> QS_IN_AA(Littles, Ls) U2_AA(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_AA(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U2_AA(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> QS_IN_AA(Bigs, Bs) U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_AA(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> APP_IN_AAA(Ls, .(X, Bs), Ys) APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> U8_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAA(Xs, Ys, Zs) U2_AG(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_AG(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U2_AG(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> QS_IN_AA(Bigs, Bs) U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_AG(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> APP_IN_AAG(Ls, .(X, Bs), Ys) APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> U8_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs) The TRS R consists of the following rules: qs_in_ag([], []) -> qs_out_ag([], []) qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U5_aaaa(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U5_gaaa(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) qs_in_aa([], []) -> qs_out_aa([], []) qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) app_in_aaa([], X, X) -> app_out_aaa([], X, X) app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) app_in_aag([], X, X) -> app_out_aag([], X, X) app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ag(x1, x2) = qs_in_ag(x2) [] = [] qs_out_ag(x1, x2) = qs_out_ag(x2) U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U9_aa(x1, x2, x3) = U9_aa(x3) U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U9_ga(x1, x2, x3) = U9_ga(x1, x3) U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x1, x6) U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x1, x6) part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa(x1) part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) qs_in_aa(x1, x2) = qs_in_aa qs_out_aa(x1, x2) = qs_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U4_aa(x1, x2, x3, x4) = U4_aa(x4) .(x1, x2) = .(x1, x2) app_in_aaa(x1, x2, x3) = app_in_aaa app_out_aaa(x1, x2, x3) = app_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4) app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2, x3) U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x4, x5) QS_IN_AG(x1, x2) = QS_IN_AG(x2) U1_AG(x1, x2, x3, x4) = U1_AG(x3, x4) PART_IN_AAAA(x1, x2, x3, x4) = PART_IN_AAAA U5_AAAA(x1, x2, x3, x4, x5, x6) = U5_AAAA(x6) LESS_IN_AA(x1, x2) = LESS_IN_AA U9_AA(x1, x2, x3) = U9_AA(x3) U6_AAAA(x1, x2, x3, x4, x5, x6) = U6_AAAA(x6) PART_IN_GAAA(x1, x2, x3, x4) = PART_IN_GAAA(x1) U5_GAAA(x1, x2, x3, x4, x5, x6) = U5_GAAA(x1, x6) LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) U9_GA(x1, x2, x3) = U9_GA(x1, x3) U6_GAAA(x1, x2, x3, x4, x5, x6) = U6_GAAA(x1, x6) U7_GAAA(x1, x2, x3, x4, x5, x6) = U7_GAAA(x1, x6) U7_AAAA(x1, x2, x3, x4, x5, x6) = U7_AAAA(x6) U2_AG(x1, x2, x3, x4, x5) = U2_AG(x3, x5) QS_IN_AA(x1, x2) = QS_IN_AA U1_AA(x1, x2, x3, x4) = U1_AA(x4) U2_AA(x1, x2, x3, x4, x5) = U2_AA(x5) U3_AA(x1, x2, x3, x4, x5) = U3_AA(x5) U4_AA(x1, x2, x3, x4) = U4_AA(x4) APP_IN_AAA(x1, x2, x3) = APP_IN_AAA U8_AAA(x1, x2, x3, x4, x5) = U8_AAA(x5) U3_AG(x1, x2, x3, x4, x5) = U3_AG(x3, x5) U4_AG(x1, x2, x3, x4) = U4_AG(x3, x4) APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3) U8_AAG(x1, x2, x3, x4, x5) = U8_AAG(x1, x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 7 SCCs with 24 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs) The TRS R consists of the following rules: qs_in_ag([], []) -> qs_out_ag([], []) qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U5_aaaa(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U5_gaaa(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) qs_in_aa([], []) -> qs_out_aa([], []) qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) app_in_aaa([], X, X) -> app_out_aaa([], X, X) app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) app_in_aag([], X, X) -> app_out_aag([], X, X) app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ag(x1, x2) = qs_in_ag(x2) [] = [] qs_out_ag(x1, x2) = qs_out_ag(x2) U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U9_aa(x1, x2, x3) = U9_aa(x3) U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U9_ga(x1, x2, x3) = U9_ga(x1, x3) U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x1, x6) U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x1, x6) part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa(x1) part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) qs_in_aa(x1, x2) = qs_in_aa qs_out_aa(x1, x2) = qs_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U4_aa(x1, x2, x3, x4) = U4_aa(x4) .(x1, x2) = .(x1, x2) app_in_aaa(x1, x2, x3) = app_in_aaa app_out_aaa(x1, x2, x3) = app_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4) app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2, x3) U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x4, x5) APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (8) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (9) Obligation: Pi DP problem: The TRS P consists of the following rules: APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (10) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: APP_IN_AAG(.(X, Zs)) -> APP_IN_AAG(Zs) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (12) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *APP_IN_AAG(.(X, Zs)) -> APP_IN_AAG(Zs) The graph contains the following edges 1 > 1 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAA(Xs, Ys, Zs) The TRS R consists of the following rules: qs_in_ag([], []) -> qs_out_ag([], []) qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U5_aaaa(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U5_gaaa(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) qs_in_aa([], []) -> qs_out_aa([], []) qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) app_in_aaa([], X, X) -> app_out_aaa([], X, X) app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) app_in_aag([], X, X) -> app_out_aag([], X, X) app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ag(x1, x2) = qs_in_ag(x2) [] = [] qs_out_ag(x1, x2) = qs_out_ag(x2) U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U9_aa(x1, x2, x3) = U9_aa(x3) U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U9_ga(x1, x2, x3) = U9_ga(x1, x3) U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x1, x6) U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x1, x6) part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa(x1) part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) qs_in_aa(x1, x2) = qs_in_aa qs_out_aa(x1, x2) = qs_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U4_aa(x1, x2, x3, x4) = U4_aa(x4) .(x1, x2) = .(x1, x2) app_in_aaa(x1, x2, x3) = app_in_aaa app_out_aaa(x1, x2, x3) = app_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4) app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2, x3) U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x4, x5) APP_IN_AAA(x1, x2, x3) = APP_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (15) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAA(Xs, Ys, Zs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) APP_IN_AAA(x1, x2, x3) = APP_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: APP_IN_AAA -> APP_IN_AAA R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (19) 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 = APP_IN_AAA evaluates to t =APP_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 APP_IN_AAA to APP_IN_AAA. ---------------------------------------- (20) NO ---------------------------------------- (21) Obligation: Pi DP problem: The TRS P consists of the following rules: LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) The TRS R consists of the following rules: qs_in_ag([], []) -> qs_out_ag([], []) qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U5_aaaa(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U5_gaaa(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) qs_in_aa([], []) -> qs_out_aa([], []) qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) app_in_aaa([], X, X) -> app_out_aaa([], X, X) app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) app_in_aag([], X, X) -> app_out_aag([], X, X) app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ag(x1, x2) = qs_in_ag(x2) [] = [] qs_out_ag(x1, x2) = qs_out_ag(x2) U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U9_aa(x1, x2, x3) = U9_aa(x3) U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U9_ga(x1, x2, x3) = U9_ga(x1, x3) U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x1, x6) U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x1, x6) part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa(x1) part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) qs_in_aa(x1, x2) = qs_in_aa qs_out_aa(x1, x2) = qs_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U4_aa(x1, x2, x3, x4) = U4_aa(x4) .(x1, x2) = .(x1, x2) app_in_aaa(x1, x2, x3) = app_in_aaa app_out_aaa(x1, x2, x3) = app_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4) app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2, x3) U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x4, x5) LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (22) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (23) Obligation: Pi DP problem: The TRS P consists of the following rules: LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (24) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (25) Obligation: Q DP problem: The TRS P consists of the following rules: LESS_IN_GA(s(X)) -> LESS_IN_GA(X) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (26) 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: *LESS_IN_GA(s(X)) -> LESS_IN_GA(X) The graph contains the following edges 1 > 1 ---------------------------------------- (27) YES ---------------------------------------- (28) Obligation: Pi DP problem: The TRS P consists of the following rules: U5_GAAA(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs) PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GAAA(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_GAAA(X, Xs, Ls, Bs) The TRS R consists of the following rules: qs_in_ag([], []) -> qs_out_ag([], []) qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U5_aaaa(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U5_gaaa(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) qs_in_aa([], []) -> qs_out_aa([], []) qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) app_in_aaa([], X, X) -> app_out_aaa([], X, X) app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) app_in_aag([], X, X) -> app_out_aag([], X, X) app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ag(x1, x2) = qs_in_ag(x2) [] = [] qs_out_ag(x1, x2) = qs_out_ag(x2) U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U9_aa(x1, x2, x3) = U9_aa(x3) U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U9_ga(x1, x2, x3) = U9_ga(x1, x3) U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x1, x6) U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x1, x6) part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa(x1) part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) qs_in_aa(x1, x2) = qs_in_aa qs_out_aa(x1, x2) = qs_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U4_aa(x1, x2, x3, x4) = U4_aa(x4) .(x1, x2) = .(x1, x2) app_in_aaa(x1, x2, x3) = app_in_aaa app_out_aaa(x1, x2, x3) = app_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4) app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2, x3) U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x4, x5) PART_IN_GAAA(x1, x2, x3, x4) = PART_IN_GAAA(x1) U5_GAAA(x1, x2, x3, x4, x5, x6) = U5_GAAA(x1, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (29) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (30) Obligation: Pi DP problem: The TRS P consists of the following rules: U5_GAAA(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs) PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GAAA(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_GAAA(X, Xs, Ls, Bs) The TRS R consists of the following rules: less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) The argument filtering Pi contains the following mapping: less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U9_ga(x1, x2, x3) = U9_ga(x1, x3) .(x1, x2) = .(x1, x2) PART_IN_GAAA(x1, x2, x3, x4) = PART_IN_GAAA(x1) U5_GAAA(x1, x2, x3, x4, x5, x6) = U5_GAAA(x1, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (31) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: U5_GAAA(X, less_out_ga(X)) -> PART_IN_GAAA(X) PART_IN_GAAA(X) -> U5_GAAA(X, less_in_ga(X)) PART_IN_GAAA(X) -> PART_IN_GAAA(X) The TRS R consists of the following rules: less_in_ga(0) -> less_out_ga(0) less_in_ga(s(X)) -> U9_ga(X, less_in_ga(X)) U9_ga(X, less_out_ga(X)) -> less_out_ga(s(X)) The set Q consists of the following terms: less_in_ga(x0) U9_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (33) TransformationProof (SOUND) By narrowing [LPAR04] the rule PART_IN_GAAA(X) -> U5_GAAA(X, less_in_ga(X)) at position [1] we obtained the following new rules [LPAR04]: (PART_IN_GAAA(0) -> U5_GAAA(0, less_out_ga(0)),PART_IN_GAAA(0) -> U5_GAAA(0, less_out_ga(0))) (PART_IN_GAAA(s(x0)) -> U5_GAAA(s(x0), U9_ga(x0, less_in_ga(x0))),PART_IN_GAAA(s(x0)) -> U5_GAAA(s(x0), U9_ga(x0, less_in_ga(x0)))) ---------------------------------------- (34) Obligation: Q DP problem: The TRS P consists of the following rules: U5_GAAA(X, less_out_ga(X)) -> PART_IN_GAAA(X) PART_IN_GAAA(X) -> PART_IN_GAAA(X) PART_IN_GAAA(0) -> U5_GAAA(0, less_out_ga(0)) PART_IN_GAAA(s(x0)) -> U5_GAAA(s(x0), U9_ga(x0, less_in_ga(x0))) The TRS R consists of the following rules: less_in_ga(0) -> less_out_ga(0) less_in_ga(s(X)) -> U9_ga(X, less_in_ga(X)) U9_ga(X, less_out_ga(X)) -> less_out_ga(s(X)) The set Q consists of the following terms: less_in_ga(x0) U9_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (35) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U5_GAAA(X, less_out_ga(X)) -> PART_IN_GAAA(X) we obtained the following new rules [LPAR04]: (U5_GAAA(0, less_out_ga(0)) -> PART_IN_GAAA(0),U5_GAAA(0, less_out_ga(0)) -> PART_IN_GAAA(0)) (U5_GAAA(s(z0), less_out_ga(s(z0))) -> PART_IN_GAAA(s(z0)),U5_GAAA(s(z0), less_out_ga(s(z0))) -> PART_IN_GAAA(s(z0))) ---------------------------------------- (36) Obligation: Q DP problem: The TRS P consists of the following rules: PART_IN_GAAA(X) -> PART_IN_GAAA(X) PART_IN_GAAA(0) -> U5_GAAA(0, less_out_ga(0)) PART_IN_GAAA(s(x0)) -> U5_GAAA(s(x0), U9_ga(x0, less_in_ga(x0))) U5_GAAA(0, less_out_ga(0)) -> PART_IN_GAAA(0) U5_GAAA(s(z0), less_out_ga(s(z0))) -> PART_IN_GAAA(s(z0)) The TRS R consists of the following rules: less_in_ga(0) -> less_out_ga(0) less_in_ga(s(X)) -> U9_ga(X, less_in_ga(X)) U9_ga(X, less_out_ga(X)) -> less_out_ga(s(X)) The set Q consists of the following terms: less_in_ga(x0) U9_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (37) 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 = PART_IN_GAAA(X) evaluates to t =PART_IN_GAAA(X) 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 PART_IN_GAAA(X) to PART_IN_GAAA(X). ---------------------------------------- (38) NO ---------------------------------------- (39) Obligation: Pi DP problem: The TRS P consists of the following rules: LESS_IN_AA(s(X), s(Y)) -> LESS_IN_AA(X, Y) The TRS R consists of the following rules: qs_in_ag([], []) -> qs_out_ag([], []) qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U5_aaaa(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U5_gaaa(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) qs_in_aa([], []) -> qs_out_aa([], []) qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) app_in_aaa([], X, X) -> app_out_aaa([], X, X) app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) app_in_aag([], X, X) -> app_out_aag([], X, X) app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ag(x1, x2) = qs_in_ag(x2) [] = [] qs_out_ag(x1, x2) = qs_out_ag(x2) U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U9_aa(x1, x2, x3) = U9_aa(x3) U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U9_ga(x1, x2, x3) = U9_ga(x1, x3) U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x1, x6) U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x1, x6) part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa(x1) part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) qs_in_aa(x1, x2) = qs_in_aa qs_out_aa(x1, x2) = qs_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U4_aa(x1, x2, x3, x4) = U4_aa(x4) .(x1, x2) = .(x1, x2) app_in_aaa(x1, x2, x3) = app_in_aaa app_out_aaa(x1, x2, x3) = app_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4) app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2, x3) U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x4, x5) LESS_IN_AA(x1, x2) = LESS_IN_AA We have to consider all (P,R,Pi)-chains ---------------------------------------- (40) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (41) Obligation: Pi DP problem: The TRS P consists of the following rules: LESS_IN_AA(s(X), s(Y)) -> LESS_IN_AA(X, Y) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) LESS_IN_AA(x1, x2) = LESS_IN_AA We have to consider all (P,R,Pi)-chains ---------------------------------------- (42) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (43) Obligation: Q DP problem: The TRS P consists of the following rules: LESS_IN_AA -> LESS_IN_AA R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (44) 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 = LESS_IN_AA evaluates to t =LESS_IN_AA 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 LESS_IN_AA to LESS_IN_AA. ---------------------------------------- (45) NO ---------------------------------------- (46) Obligation: Pi DP problem: The TRS P consists of the following rules: PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_AAAA(X, Xs, Ls, Bs) The TRS R consists of the following rules: qs_in_ag([], []) -> qs_out_ag([], []) qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U5_aaaa(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U5_gaaa(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) qs_in_aa([], []) -> qs_out_aa([], []) qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) app_in_aaa([], X, X) -> app_out_aaa([], X, X) app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) app_in_aag([], X, X) -> app_out_aag([], X, X) app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ag(x1, x2) = qs_in_ag(x2) [] = [] qs_out_ag(x1, x2) = qs_out_ag(x2) U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U9_aa(x1, x2, x3) = U9_aa(x3) U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U9_ga(x1, x2, x3) = U9_ga(x1, x3) U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x1, x6) U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x1, x6) part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa(x1) part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) qs_in_aa(x1, x2) = qs_in_aa qs_out_aa(x1, x2) = qs_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U4_aa(x1, x2, x3, x4) = U4_aa(x4) .(x1, x2) = .(x1, x2) app_in_aaa(x1, x2, x3) = app_in_aaa app_out_aaa(x1, x2, x3) = app_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4) app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2, x3) U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x4, x5) PART_IN_AAAA(x1, x2, x3, x4) = PART_IN_AAAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (47) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (48) Obligation: Pi DP problem: The TRS P consists of the following rules: PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_AAAA(X, Xs, Ls, Bs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) PART_IN_AAAA(x1, x2, x3, x4) = PART_IN_AAAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (49) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (50) Obligation: Q DP problem: The TRS P consists of the following rules: PART_IN_AAAA -> PART_IN_AAAA R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (51) Obligation: Pi DP problem: The TRS P consists of the following rules: QS_IN_AA(.(X, Xs), Ys) -> U1_AA(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AA(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U2_AA(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> QS_IN_AA(Bigs, Bs) U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> QS_IN_AA(Littles, Ls) The TRS R consists of the following rules: qs_in_ag([], []) -> qs_out_ag([], []) qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U5_aaaa(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U5_gaaa(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) qs_in_aa([], []) -> qs_out_aa([], []) qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) app_in_aaa([], X, X) -> app_out_aaa([], X, X) app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) app_in_aag([], X, X) -> app_out_aag([], X, X) app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ag(x1, x2) = qs_in_ag(x2) [] = [] qs_out_ag(x1, x2) = qs_out_ag(x2) U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U9_aa(x1, x2, x3) = U9_aa(x3) U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U9_ga(x1, x2, x3) = U9_ga(x1, x3) U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x1, x6) U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x1, x6) part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa(x1) part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) qs_in_aa(x1, x2) = qs_in_aa qs_out_aa(x1, x2) = qs_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U4_aa(x1, x2, x3, x4) = U4_aa(x4) .(x1, x2) = .(x1, x2) app_in_aaa(x1, x2, x3) = app_in_aaa app_out_aaa(x1, x2, x3) = app_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4) app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2, x3) U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x4, x5) QS_IN_AA(x1, x2) = QS_IN_AA U1_AA(x1, x2, x3, x4) = U1_AA(x4) U2_AA(x1, x2, x3, x4, x5) = U2_AA(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (52) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (53) Obligation: Pi DP problem: The TRS P consists of the following rules: QS_IN_AA(.(X, Xs), Ys) -> U1_AA(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AA(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U2_AA(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> QS_IN_AA(Bigs, Bs) U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> QS_IN_AA(Littles, Ls) The TRS R consists of the following rules: part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) qs_in_aa([], []) -> qs_out_aa([], []) qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) U5_aaaa(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) U5_gaaa(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) app_in_aaa([], X, X) -> app_out_aaa([], X, X) app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) The argument filtering Pi contains the following mapping: [] = [] part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U9_aa(x1, x2, x3) = U9_aa(x3) U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U9_ga(x1, x2, x3) = U9_ga(x1, x3) U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x1, x6) U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x1, x6) part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa(x1) part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) qs_in_aa(x1, x2) = qs_in_aa qs_out_aa(x1, x2) = qs_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U4_aa(x1, x2, x3, x4) = U4_aa(x4) .(x1, x2) = .(x1, x2) app_in_aaa(x1, x2, x3) = app_in_aaa app_out_aaa(x1, x2, x3) = app_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) QS_IN_AA(x1, x2) = QS_IN_AA U1_AA(x1, x2, x3, x4) = U1_AA(x4) U2_AA(x1, x2, x3, x4, x5) = U2_AA(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (54) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: qs_in_2: (f,b) (f,f) part_in_4: (f,f,f,f) (b,f,f,f) less_in_2: (f,f) (b,f) app_in_3: (f,f,f) (f,f,b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: qs_in_ag([], []) -> qs_out_ag([], []) qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U5_aaaa(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U5_gaaa(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) qs_in_aa([], []) -> qs_out_aa([], []) qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) app_in_aaa([], X, X) -> app_out_aaa([], X, X) app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) app_in_aag([], X, X) -> app_out_aag([], X, X) app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ag(x1, x2) = qs_in_ag(x2) [] = [] qs_out_ag(x1, x2) = qs_out_ag U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U9_aa(x1, x2, x3) = U9_aa(x3) U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga s(x1) = s(x1) U9_ga(x1, x2, x3) = U9_ga(x3) U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x6) U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x6) part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) qs_in_aa(x1, x2) = qs_in_aa qs_out_aa(x1, x2) = qs_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U4_aa(x1, x2, x3, x4) = U4_aa(x4) .(x1, x2) = .(x1, x2) app_in_aaa(x1, x2, x3) = app_in_aaa app_out_aaa(x1, x2, x3) = app_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) U4_ag(x1, x2, x3, x4) = U4_ag(x4) app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x5) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (55) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: qs_in_ag([], []) -> qs_out_ag([], []) qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U5_aaaa(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U5_gaaa(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) qs_in_aa([], []) -> qs_out_aa([], []) qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) app_in_aaa([], X, X) -> app_out_aaa([], X, X) app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) app_in_aag([], X, X) -> app_out_aag([], X, X) app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ag(x1, x2) = qs_in_ag(x2) [] = [] qs_out_ag(x1, x2) = qs_out_ag U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U9_aa(x1, x2, x3) = U9_aa(x3) U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga s(x1) = s(x1) U9_ga(x1, x2, x3) = U9_ga(x3) U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x6) U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x6) part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) qs_in_aa(x1, x2) = qs_in_aa qs_out_aa(x1, x2) = qs_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U4_aa(x1, x2, x3, x4) = U4_aa(x4) .(x1, x2) = .(x1, x2) app_in_aaa(x1, x2, x3) = app_in_aaa app_out_aaa(x1, x2, x3) = app_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) U4_ag(x1, x2, x3, x4) = U4_ag(x4) app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x5) ---------------------------------------- (56) 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: QS_IN_AG(.(X, Xs), Ys) -> U1_AG(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) QS_IN_AG(.(X, Xs), Ys) -> PART_IN_AAAA(X, Xs, Littles, Bigs) PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_AAAA(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> LESS_IN_AA(X, Y) LESS_IN_AA(s(X), s(Y)) -> U9_AA(X, Y, less_in_aa(X, Y)) LESS_IN_AA(s(X), s(Y)) -> LESS_IN_AA(X, Y) U5_AAAA(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> U6_AAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) U5_AAAA(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs) PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GAAA(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> LESS_IN_GA(X, Y) LESS_IN_GA(s(X), s(Y)) -> U9_GA(X, Y, less_in_ga(X, Y)) LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) U5_GAAA(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> U6_GAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) U5_GAAA(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs) PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_GAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_GAAA(X, Xs, Ls, Bs) PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_AAAA(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_AAAA(X, Xs, Ls, Bs) U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AG(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> QS_IN_AA(Littles, Ls) QS_IN_AA(.(X, Xs), Ys) -> U1_AA(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) QS_IN_AA(.(X, Xs), Ys) -> PART_IN_AAAA(X, Xs, Littles, Bigs) U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AA(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> QS_IN_AA(Littles, Ls) U2_AA(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_AA(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U2_AA(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> QS_IN_AA(Bigs, Bs) U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_AA(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> APP_IN_AAA(Ls, .(X, Bs), Ys) APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> U8_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAA(Xs, Ys, Zs) U2_AG(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_AG(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U2_AG(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> QS_IN_AA(Bigs, Bs) U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_AG(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> APP_IN_AAG(Ls, .(X, Bs), Ys) APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> U8_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs) The TRS R consists of the following rules: qs_in_ag([], []) -> qs_out_ag([], []) qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U5_aaaa(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U5_gaaa(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) qs_in_aa([], []) -> qs_out_aa([], []) qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) app_in_aaa([], X, X) -> app_out_aaa([], X, X) app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) app_in_aag([], X, X) -> app_out_aag([], X, X) app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ag(x1, x2) = qs_in_ag(x2) [] = [] qs_out_ag(x1, x2) = qs_out_ag U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U9_aa(x1, x2, x3) = U9_aa(x3) U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga s(x1) = s(x1) U9_ga(x1, x2, x3) = U9_ga(x3) U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x6) U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x6) part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) qs_in_aa(x1, x2) = qs_in_aa qs_out_aa(x1, x2) = qs_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U4_aa(x1, x2, x3, x4) = U4_aa(x4) .(x1, x2) = .(x1, x2) app_in_aaa(x1, x2, x3) = app_in_aaa app_out_aaa(x1, x2, x3) = app_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) U4_ag(x1, x2, x3, x4) = U4_ag(x4) app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x5) QS_IN_AG(x1, x2) = QS_IN_AG(x2) U1_AG(x1, x2, x3, x4) = U1_AG(x3, x4) PART_IN_AAAA(x1, x2, x3, x4) = PART_IN_AAAA U5_AAAA(x1, x2, x3, x4, x5, x6) = U5_AAAA(x6) LESS_IN_AA(x1, x2) = LESS_IN_AA U9_AA(x1, x2, x3) = U9_AA(x3) U6_AAAA(x1, x2, x3, x4, x5, x6) = U6_AAAA(x6) PART_IN_GAAA(x1, x2, x3, x4) = PART_IN_GAAA(x1) U5_GAAA(x1, x2, x3, x4, x5, x6) = U5_GAAA(x1, x6) LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) U9_GA(x1, x2, x3) = U9_GA(x3) U6_GAAA(x1, x2, x3, x4, x5, x6) = U6_GAAA(x6) U7_GAAA(x1, x2, x3, x4, x5, x6) = U7_GAAA(x6) U7_AAAA(x1, x2, x3, x4, x5, x6) = U7_AAAA(x6) U2_AG(x1, x2, x3, x4, x5) = U2_AG(x3, x5) QS_IN_AA(x1, x2) = QS_IN_AA U1_AA(x1, x2, x3, x4) = U1_AA(x4) U2_AA(x1, x2, x3, x4, x5) = U2_AA(x5) U3_AA(x1, x2, x3, x4, x5) = U3_AA(x5) U4_AA(x1, x2, x3, x4) = U4_AA(x4) APP_IN_AAA(x1, x2, x3) = APP_IN_AAA U8_AAA(x1, x2, x3, x4, x5) = U8_AAA(x5) U3_AG(x1, x2, x3, x4, x5) = U3_AG(x3, x5) U4_AG(x1, x2, x3, x4) = U4_AG(x4) APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3) U8_AAG(x1, x2, x3, x4, x5) = U8_AAG(x1, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (57) Obligation: Pi DP problem: The TRS P consists of the following rules: QS_IN_AG(.(X, Xs), Ys) -> U1_AG(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) QS_IN_AG(.(X, Xs), Ys) -> PART_IN_AAAA(X, Xs, Littles, Bigs) PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_AAAA(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> LESS_IN_AA(X, Y) LESS_IN_AA(s(X), s(Y)) -> U9_AA(X, Y, less_in_aa(X, Y)) LESS_IN_AA(s(X), s(Y)) -> LESS_IN_AA(X, Y) U5_AAAA(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> U6_AAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) U5_AAAA(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs) PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GAAA(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> LESS_IN_GA(X, Y) LESS_IN_GA(s(X), s(Y)) -> U9_GA(X, Y, less_in_ga(X, Y)) LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) U5_GAAA(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> U6_GAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) U5_GAAA(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs) PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_GAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_GAAA(X, Xs, Ls, Bs) PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_AAAA(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_AAAA(X, Xs, Ls, Bs) U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AG(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> QS_IN_AA(Littles, Ls) QS_IN_AA(.(X, Xs), Ys) -> U1_AA(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) QS_IN_AA(.(X, Xs), Ys) -> PART_IN_AAAA(X, Xs, Littles, Bigs) U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AA(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> QS_IN_AA(Littles, Ls) U2_AA(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_AA(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U2_AA(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> QS_IN_AA(Bigs, Bs) U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_AA(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> APP_IN_AAA(Ls, .(X, Bs), Ys) APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> U8_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAA(Xs, Ys, Zs) U2_AG(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_AG(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U2_AG(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> QS_IN_AA(Bigs, Bs) U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_AG(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> APP_IN_AAG(Ls, .(X, Bs), Ys) APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> U8_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs) The TRS R consists of the following rules: qs_in_ag([], []) -> qs_out_ag([], []) qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U5_aaaa(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U5_gaaa(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) qs_in_aa([], []) -> qs_out_aa([], []) qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) app_in_aaa([], X, X) -> app_out_aaa([], X, X) app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) app_in_aag([], X, X) -> app_out_aag([], X, X) app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ag(x1, x2) = qs_in_ag(x2) [] = [] qs_out_ag(x1, x2) = qs_out_ag U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U9_aa(x1, x2, x3) = U9_aa(x3) U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga s(x1) = s(x1) U9_ga(x1, x2, x3) = U9_ga(x3) U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x6) U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x6) part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) qs_in_aa(x1, x2) = qs_in_aa qs_out_aa(x1, x2) = qs_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U4_aa(x1, x2, x3, x4) = U4_aa(x4) .(x1, x2) = .(x1, x2) app_in_aaa(x1, x2, x3) = app_in_aaa app_out_aaa(x1, x2, x3) = app_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) U4_ag(x1, x2, x3, x4) = U4_ag(x4) app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x5) QS_IN_AG(x1, x2) = QS_IN_AG(x2) U1_AG(x1, x2, x3, x4) = U1_AG(x3, x4) PART_IN_AAAA(x1, x2, x3, x4) = PART_IN_AAAA U5_AAAA(x1, x2, x3, x4, x5, x6) = U5_AAAA(x6) LESS_IN_AA(x1, x2) = LESS_IN_AA U9_AA(x1, x2, x3) = U9_AA(x3) U6_AAAA(x1, x2, x3, x4, x5, x6) = U6_AAAA(x6) PART_IN_GAAA(x1, x2, x3, x4) = PART_IN_GAAA(x1) U5_GAAA(x1, x2, x3, x4, x5, x6) = U5_GAAA(x1, x6) LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) U9_GA(x1, x2, x3) = U9_GA(x3) U6_GAAA(x1, x2, x3, x4, x5, x6) = U6_GAAA(x6) U7_GAAA(x1, x2, x3, x4, x5, x6) = U7_GAAA(x6) U7_AAAA(x1, x2, x3, x4, x5, x6) = U7_AAAA(x6) U2_AG(x1, x2, x3, x4, x5) = U2_AG(x3, x5) QS_IN_AA(x1, x2) = QS_IN_AA U1_AA(x1, x2, x3, x4) = U1_AA(x4) U2_AA(x1, x2, x3, x4, x5) = U2_AA(x5) U3_AA(x1, x2, x3, x4, x5) = U3_AA(x5) U4_AA(x1, x2, x3, x4) = U4_AA(x4) APP_IN_AAA(x1, x2, x3) = APP_IN_AAA U8_AAA(x1, x2, x3, x4, x5) = U8_AAA(x5) U3_AG(x1, x2, x3, x4, x5) = U3_AG(x3, x5) U4_AG(x1, x2, x3, x4) = U4_AG(x4) APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3) U8_AAG(x1, x2, x3, x4, x5) = U8_AAG(x1, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (58) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 7 SCCs with 24 less nodes. ---------------------------------------- (59) Complex Obligation (AND) ---------------------------------------- (60) Obligation: Pi DP problem: The TRS P consists of the following rules: APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs) The TRS R consists of the following rules: qs_in_ag([], []) -> qs_out_ag([], []) qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U5_aaaa(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U5_gaaa(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) qs_in_aa([], []) -> qs_out_aa([], []) qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) app_in_aaa([], X, X) -> app_out_aaa([], X, X) app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) app_in_aag([], X, X) -> app_out_aag([], X, X) app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ag(x1, x2) = qs_in_ag(x2) [] = [] qs_out_ag(x1, x2) = qs_out_ag U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U9_aa(x1, x2, x3) = U9_aa(x3) U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga s(x1) = s(x1) U9_ga(x1, x2, x3) = U9_ga(x3) U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x6) U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x6) part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) qs_in_aa(x1, x2) = qs_in_aa qs_out_aa(x1, x2) = qs_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U4_aa(x1, x2, x3, x4) = U4_aa(x4) .(x1, x2) = .(x1, x2) app_in_aaa(x1, x2, x3) = app_in_aaa app_out_aaa(x1, x2, x3) = app_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) U4_ag(x1, x2, x3, x4) = U4_ag(x4) app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x5) APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (61) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (62) Obligation: Pi DP problem: The TRS P consists of the following rules: APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (63) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (64) Obligation: Q DP problem: The TRS P consists of the following rules: APP_IN_AAG(.(X, Zs)) -> APP_IN_AAG(Zs) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (65) 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: *APP_IN_AAG(.(X, Zs)) -> APP_IN_AAG(Zs) The graph contains the following edges 1 > 1 ---------------------------------------- (66) YES ---------------------------------------- (67) Obligation: Pi DP problem: The TRS P consists of the following rules: APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAA(Xs, Ys, Zs) The TRS R consists of the following rules: qs_in_ag([], []) -> qs_out_ag([], []) qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U5_aaaa(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U5_gaaa(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) qs_in_aa([], []) -> qs_out_aa([], []) qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) app_in_aaa([], X, X) -> app_out_aaa([], X, X) app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) app_in_aag([], X, X) -> app_out_aag([], X, X) app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ag(x1, x2) = qs_in_ag(x2) [] = [] qs_out_ag(x1, x2) = qs_out_ag U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U9_aa(x1, x2, x3) = U9_aa(x3) U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga s(x1) = s(x1) U9_ga(x1, x2, x3) = U9_ga(x3) U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x6) U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x6) part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) qs_in_aa(x1, x2) = qs_in_aa qs_out_aa(x1, x2) = qs_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U4_aa(x1, x2, x3, x4) = U4_aa(x4) .(x1, x2) = .(x1, x2) app_in_aaa(x1, x2, x3) = app_in_aaa app_out_aaa(x1, x2, x3) = app_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) U4_ag(x1, x2, x3, x4) = U4_ag(x4) app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x5) APP_IN_AAA(x1, x2, x3) = APP_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (68) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (69) Obligation: Pi DP problem: The TRS P consists of the following rules: APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAA(Xs, Ys, Zs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) APP_IN_AAA(x1, x2, x3) = APP_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (70) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (71) Obligation: Q DP problem: The TRS P consists of the following rules: APP_IN_AAA -> APP_IN_AAA R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (72) 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 = APP_IN_AAA evaluates to t =APP_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 APP_IN_AAA to APP_IN_AAA. ---------------------------------------- (73) NO ---------------------------------------- (74) Obligation: Pi DP problem: The TRS P consists of the following rules: LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) The TRS R consists of the following rules: qs_in_ag([], []) -> qs_out_ag([], []) qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U5_aaaa(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U5_gaaa(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) qs_in_aa([], []) -> qs_out_aa([], []) qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) app_in_aaa([], X, X) -> app_out_aaa([], X, X) app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) app_in_aag([], X, X) -> app_out_aag([], X, X) app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ag(x1, x2) = qs_in_ag(x2) [] = [] qs_out_ag(x1, x2) = qs_out_ag U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U9_aa(x1, x2, x3) = U9_aa(x3) U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga s(x1) = s(x1) U9_ga(x1, x2, x3) = U9_ga(x3) U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x6) U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x6) part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) qs_in_aa(x1, x2) = qs_in_aa qs_out_aa(x1, x2) = qs_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U4_aa(x1, x2, x3, x4) = U4_aa(x4) .(x1, x2) = .(x1, x2) app_in_aaa(x1, x2, x3) = app_in_aaa app_out_aaa(x1, x2, x3) = app_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) U4_ag(x1, x2, x3, x4) = U4_ag(x4) app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x5) LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (75) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (76) Obligation: Pi DP problem: The TRS P consists of the following rules: LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (77) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (78) Obligation: Q DP problem: The TRS P consists of the following rules: LESS_IN_GA(s(X)) -> LESS_IN_GA(X) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (79) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *LESS_IN_GA(s(X)) -> LESS_IN_GA(X) The graph contains the following edges 1 > 1 ---------------------------------------- (80) YES ---------------------------------------- (81) Obligation: Pi DP problem: The TRS P consists of the following rules: U5_GAAA(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs) PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GAAA(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_GAAA(X, Xs, Ls, Bs) The TRS R consists of the following rules: qs_in_ag([], []) -> qs_out_ag([], []) qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U5_aaaa(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U5_gaaa(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) qs_in_aa([], []) -> qs_out_aa([], []) qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) app_in_aaa([], X, X) -> app_out_aaa([], X, X) app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) app_in_aag([], X, X) -> app_out_aag([], X, X) app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ag(x1, x2) = qs_in_ag(x2) [] = [] qs_out_ag(x1, x2) = qs_out_ag U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U9_aa(x1, x2, x3) = U9_aa(x3) U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga s(x1) = s(x1) U9_ga(x1, x2, x3) = U9_ga(x3) U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x6) U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x6) part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) qs_in_aa(x1, x2) = qs_in_aa qs_out_aa(x1, x2) = qs_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U4_aa(x1, x2, x3, x4) = U4_aa(x4) .(x1, x2) = .(x1, x2) app_in_aaa(x1, x2, x3) = app_in_aaa app_out_aaa(x1, x2, x3) = app_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) U4_ag(x1, x2, x3, x4) = U4_ag(x4) app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x5) PART_IN_GAAA(x1, x2, x3, x4) = PART_IN_GAAA(x1) U5_GAAA(x1, x2, x3, x4, x5, x6) = U5_GAAA(x1, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (82) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (83) Obligation: Pi DP problem: The TRS P consists of the following rules: U5_GAAA(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs) PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GAAA(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_GAAA(X, Xs, Ls, Bs) The TRS R consists of the following rules: less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) The argument filtering Pi contains the following mapping: less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga s(x1) = s(x1) U9_ga(x1, x2, x3) = U9_ga(x3) .(x1, x2) = .(x1, x2) PART_IN_GAAA(x1, x2, x3, x4) = PART_IN_GAAA(x1) U5_GAAA(x1, x2, x3, x4, x5, x6) = U5_GAAA(x1, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (84) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (85) Obligation: Q DP problem: The TRS P consists of the following rules: U5_GAAA(X, less_out_ga) -> PART_IN_GAAA(X) PART_IN_GAAA(X) -> U5_GAAA(X, less_in_ga(X)) PART_IN_GAAA(X) -> PART_IN_GAAA(X) The TRS R consists of the following rules: less_in_ga(0) -> less_out_ga less_in_ga(s(X)) -> U9_ga(less_in_ga(X)) U9_ga(less_out_ga) -> less_out_ga The set Q consists of the following terms: less_in_ga(x0) U9_ga(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (86) TransformationProof (SOUND) By narrowing [LPAR04] the rule PART_IN_GAAA(X) -> U5_GAAA(X, less_in_ga(X)) at position [1] we obtained the following new rules [LPAR04]: (PART_IN_GAAA(0) -> U5_GAAA(0, less_out_ga),PART_IN_GAAA(0) -> U5_GAAA(0, less_out_ga)) (PART_IN_GAAA(s(x0)) -> U5_GAAA(s(x0), U9_ga(less_in_ga(x0))),PART_IN_GAAA(s(x0)) -> U5_GAAA(s(x0), U9_ga(less_in_ga(x0)))) ---------------------------------------- (87) Obligation: Q DP problem: The TRS P consists of the following rules: U5_GAAA(X, less_out_ga) -> PART_IN_GAAA(X) PART_IN_GAAA(X) -> PART_IN_GAAA(X) PART_IN_GAAA(0) -> U5_GAAA(0, less_out_ga) PART_IN_GAAA(s(x0)) -> U5_GAAA(s(x0), U9_ga(less_in_ga(x0))) The TRS R consists of the following rules: less_in_ga(0) -> less_out_ga less_in_ga(s(X)) -> U9_ga(less_in_ga(X)) U9_ga(less_out_ga) -> less_out_ga The set Q consists of the following terms: less_in_ga(x0) U9_ga(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (88) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U5_GAAA(X, less_out_ga) -> PART_IN_GAAA(X) we obtained the following new rules [LPAR04]: (U5_GAAA(0, less_out_ga) -> PART_IN_GAAA(0),U5_GAAA(0, less_out_ga) -> PART_IN_GAAA(0)) (U5_GAAA(s(z0), less_out_ga) -> PART_IN_GAAA(s(z0)),U5_GAAA(s(z0), less_out_ga) -> PART_IN_GAAA(s(z0))) ---------------------------------------- (89) Obligation: Q DP problem: The TRS P consists of the following rules: PART_IN_GAAA(X) -> PART_IN_GAAA(X) PART_IN_GAAA(0) -> U5_GAAA(0, less_out_ga) PART_IN_GAAA(s(x0)) -> U5_GAAA(s(x0), U9_ga(less_in_ga(x0))) U5_GAAA(0, less_out_ga) -> PART_IN_GAAA(0) U5_GAAA(s(z0), less_out_ga) -> PART_IN_GAAA(s(z0)) The TRS R consists of the following rules: less_in_ga(0) -> less_out_ga less_in_ga(s(X)) -> U9_ga(less_in_ga(X)) U9_ga(less_out_ga) -> less_out_ga The set Q consists of the following terms: less_in_ga(x0) U9_ga(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (90) 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 = PART_IN_GAAA(X) evaluates to t =PART_IN_GAAA(X) 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 PART_IN_GAAA(X) to PART_IN_GAAA(X). ---------------------------------------- (91) NO ---------------------------------------- (92) Obligation: Pi DP problem: The TRS P consists of the following rules: LESS_IN_AA(s(X), s(Y)) -> LESS_IN_AA(X, Y) The TRS R consists of the following rules: qs_in_ag([], []) -> qs_out_ag([], []) qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U5_aaaa(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U5_gaaa(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) qs_in_aa([], []) -> qs_out_aa([], []) qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) app_in_aaa([], X, X) -> app_out_aaa([], X, X) app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) app_in_aag([], X, X) -> app_out_aag([], X, X) app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ag(x1, x2) = qs_in_ag(x2) [] = [] qs_out_ag(x1, x2) = qs_out_ag U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U9_aa(x1, x2, x3) = U9_aa(x3) U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga s(x1) = s(x1) U9_ga(x1, x2, x3) = U9_ga(x3) U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x6) U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x6) part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) qs_in_aa(x1, x2) = qs_in_aa qs_out_aa(x1, x2) = qs_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U4_aa(x1, x2, x3, x4) = U4_aa(x4) .(x1, x2) = .(x1, x2) app_in_aaa(x1, x2, x3) = app_in_aaa app_out_aaa(x1, x2, x3) = app_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) U4_ag(x1, x2, x3, x4) = U4_ag(x4) app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x5) LESS_IN_AA(x1, x2) = LESS_IN_AA We have to consider all (P,R,Pi)-chains ---------------------------------------- (93) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (94) Obligation: Pi DP problem: The TRS P consists of the following rules: LESS_IN_AA(s(X), s(Y)) -> LESS_IN_AA(X, Y) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) LESS_IN_AA(x1, x2) = LESS_IN_AA We have to consider all (P,R,Pi)-chains ---------------------------------------- (95) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (96) Obligation: Q DP problem: The TRS P consists of the following rules: LESS_IN_AA -> LESS_IN_AA R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (97) 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 = LESS_IN_AA evaluates to t =LESS_IN_AA 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 LESS_IN_AA to LESS_IN_AA. ---------------------------------------- (98) NO ---------------------------------------- (99) Obligation: Pi DP problem: The TRS P consists of the following rules: PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_AAAA(X, Xs, Ls, Bs) The TRS R consists of the following rules: qs_in_ag([], []) -> qs_out_ag([], []) qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U5_aaaa(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U5_gaaa(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) qs_in_aa([], []) -> qs_out_aa([], []) qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) app_in_aaa([], X, X) -> app_out_aaa([], X, X) app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) app_in_aag([], X, X) -> app_out_aag([], X, X) app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ag(x1, x2) = qs_in_ag(x2) [] = [] qs_out_ag(x1, x2) = qs_out_ag U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U9_aa(x1, x2, x3) = U9_aa(x3) U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga s(x1) = s(x1) U9_ga(x1, x2, x3) = U9_ga(x3) U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x6) U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x6) part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) qs_in_aa(x1, x2) = qs_in_aa qs_out_aa(x1, x2) = qs_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U4_aa(x1, x2, x3, x4) = U4_aa(x4) .(x1, x2) = .(x1, x2) app_in_aaa(x1, x2, x3) = app_in_aaa app_out_aaa(x1, x2, x3) = app_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) U4_ag(x1, x2, x3, x4) = U4_ag(x4) app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x5) PART_IN_AAAA(x1, x2, x3, x4) = PART_IN_AAAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (100) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (101) Obligation: Pi DP problem: The TRS P consists of the following rules: PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_AAAA(X, Xs, Ls, Bs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) PART_IN_AAAA(x1, x2, x3, x4) = PART_IN_AAAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (102) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (103) Obligation: Q DP problem: The TRS P consists of the following rules: PART_IN_AAAA -> PART_IN_AAAA R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (104) Obligation: Pi DP problem: The TRS P consists of the following rules: QS_IN_AA(.(X, Xs), Ys) -> U1_AA(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AA(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U2_AA(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> QS_IN_AA(Bigs, Bs) U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> QS_IN_AA(Littles, Ls) The TRS R consists of the following rules: qs_in_ag([], []) -> qs_out_ag([], []) qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U5_aaaa(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U5_gaaa(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) qs_in_aa([], []) -> qs_out_aa([], []) qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) app_in_aaa([], X, X) -> app_out_aaa([], X, X) app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) app_in_aag([], X, X) -> app_out_aag([], X, X) app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ag(x1, x2) = qs_in_ag(x2) [] = [] qs_out_ag(x1, x2) = qs_out_ag U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U9_aa(x1, x2, x3) = U9_aa(x3) U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga s(x1) = s(x1) U9_ga(x1, x2, x3) = U9_ga(x3) U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x6) U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x6) part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) qs_in_aa(x1, x2) = qs_in_aa qs_out_aa(x1, x2) = qs_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U4_aa(x1, x2, x3, x4) = U4_aa(x4) .(x1, x2) = .(x1, x2) app_in_aaa(x1, x2, x3) = app_in_aaa app_out_aaa(x1, x2, x3) = app_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) U4_ag(x1, x2, x3, x4) = U4_ag(x4) app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x5) QS_IN_AA(x1, x2) = QS_IN_AA U1_AA(x1, x2, x3, x4) = U1_AA(x4) U2_AA(x1, x2, x3, x4, x5) = U2_AA(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (105) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (106) Obligation: Pi DP problem: The TRS P consists of the following rules: QS_IN_AA(.(X, Xs), Ys) -> U1_AA(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AA(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U2_AA(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> QS_IN_AA(Bigs, Bs) U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> QS_IN_AA(Littles, Ls) The TRS R consists of the following rules: part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) qs_in_aa([], []) -> qs_out_aa([], []) qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) U5_aaaa(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) U5_gaaa(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) app_in_aaa([], X, X) -> app_out_aaa([], X, X) app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) The argument filtering Pi contains the following mapping: [] = [] part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U9_aa(x1, x2, x3) = U9_aa(x3) U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga s(x1) = s(x1) U9_ga(x1, x2, x3) = U9_ga(x3) U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x6) U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x6) part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) qs_in_aa(x1, x2) = qs_in_aa qs_out_aa(x1, x2) = qs_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U4_aa(x1, x2, x3, x4) = U4_aa(x4) .(x1, x2) = .(x1, x2) app_in_aaa(x1, x2, x3) = app_in_aaa app_out_aaa(x1, x2, x3) = app_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) QS_IN_AA(x1, x2) = QS_IN_AA U1_AA(x1, x2, x3, x4) = U1_AA(x4) U2_AA(x1, x2, x3, x4, x5) = U2_AA(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (107) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 2, "program": { "directives": [], "clauses": [ [ "(qs ([]) ([]))", null ], [ "(qs (. X Xs) Ys)", "(',' (part X Xs Littles Bigs) (',' (qs Littles Ls) (',' (qs Bigs Bs) (app Ls (. X Bs) Ys))))" ], [ "(part X (. Y Xs) (. Y Ls) Bs)", "(',' (less X Y) (part X Xs Ls Bs))" ], [ "(part X (. Y Xs) Ls (. Y Bs))", "(part X Xs Ls Bs)" ], [ "(part X1 ([]) ([]) ([]))", null ], [ "(app ([]) X X)", null ], [ "(app (. X Xs) Ys (. X Zs))", "(app Xs Ys Zs)" ], [ "(less (0) (s X2))", null ], [ "(less (s X) (s Y))", "(less X Y)" ] ] }, "graph": { "nodes": { "907": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "1062": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1061": { "goal": [{ "clause": -1, "scope": -1, "term": "(app T240 (. T241 T242) T239)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T239"], "free": [], "exprvars": [] } }, "1060": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "792": { "goal": [{ "clause": -1, "scope": -1, "term": "(less T77 T80)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T77"], "free": [], "exprvars": [] } }, "233": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (less T38 T39) (part T38 T40 X67 X68))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X67", "X68" ], "exprvars": [] } }, "794": { "goal": [{ "clause": -1, "scope": -1, "term": "(part T77 T84 X126 X127)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T77"], "free": [ "X126", "X127" ], "exprvars": [] } }, "234": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "950": { "goal": [{ "clause": 5, "scope": 7, "term": "(app T173 (. T174 T172) X261)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X261"], "exprvars": [] } }, "1059": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "830": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "951": { "goal": [{ "clause": 6, "scope": 7, "term": "(app T173 (. T174 T172) X261)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X261"], "exprvars": [] } }, "1058": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "237": { "goal": [{ "clause": -1, "scope": -1, "term": "(less T38 T39)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "479": { "goal": [{ "clause": -1, "scope": -1, "term": "(less T58 T59)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "952": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1057": { "goal": [{ "clause": 6, "scope": 8, "term": "(app T209 (. T210 T208) T11)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T11"], "free": [], "exprvars": [] } }, "238": { "goal": [{ "clause": -1, "scope": -1, "term": "(part T43 T44 X67 X68)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T43"], "free": [ "X67", "X68" ], "exprvars": [] } }, "832": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "953": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1056": { "goal": [{ "clause": 5, "scope": 8, "term": "(app T209 (. T210 T208) T11)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T11"], "free": [], "exprvars": [] } }, "954": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "834": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "955": { "goal": [{ "clause": -1, "scope": -1, "term": "(app T201 (. T202 T203) X297)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X297"], "exprvars": [] } }, "956": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "759": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (less T77 T80) (part T77 T81 X126 X127))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T77"], "free": [ "X126", "X127" ], "exprvars": [] } }, "913": { "goal": [{ "clause": 3, "scope": 2, "term": "(part T12 T13 X16 X17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X16", "X17" ], "exprvars": [] } }, "914": { "goal": [{ "clause": 4, "scope": 2, "term": "(part T12 T13 X16 X17)" }], "kb": 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T174 T172) X261)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X261"], "exprvars": [] } }, "906": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 2, "to": 6, "label": "CASE" }, { "from": 6, "to": 9, "label": "PARALLEL" }, { "from": 6, "to": 10, "label": "PARALLEL" }, { "from": 9, "to": 11, "label": "EVAL with clause\nqs([], []).\nand substitutionT1 -> [],\nT2 -> []" }, { "from": 9, "to": 60, "label": "EVAL-BACKTRACK" }, { "from": 10, "to": 62, "label": "EVAL with clause\nqs(.(X13, X14), X15) :- ','(part(X13, X14, X16, X17), ','(qs(X16, X18), ','(qs(X17, X19), app(X18, .(X13, X19), X15)))).\nand substitutionX13 -> T12,\nX14 -> T13,\nT1 -> .(T12, T13),\nT2 -> T11,\nX15 -> T11,\nT9 -> T12,\nT10 -> T13" }, { "from": 10, "to": 63, "label": "EVAL-BACKTRACK" }, { "from": 11, "to": 61, "label": "SUCCESS" }, { "from": 62, "to": 182, "label": "SPLIT 1" }, { "from": 62, "to": 185, "label": "SPLIT 2\nreplacements:X16 -> T17,\nX17 -> T18,\nT12 -> T19" }, { "from": 182, "to": 216, "label": "CASE" }, { "from": 185, "to": 925, "label": "SPLIT 1" }, { "from": 185, "to": 926, "label": "SPLIT 2\nreplacements:X18 -> T149,\nT18 -> T150,\nT19 -> T151" }, { "from": 216, "to": 217, "label": "PARALLEL" }, { "from": 216, "to": 218, "label": "PARALLEL" }, { "from": 217, "to": 233, "label": "EVAL with clause\npart(X62, .(X63, X64), .(X63, X65), X66) :- ','(less(X62, X63), part(X62, X64, X65, X66)).\nand substitutionT12 -> T38,\nX62 -> T38,\nX63 -> T39,\nX64 -> T40,\nT13 -> .(T39, T40),\nX65 -> X67,\nX16 -> .(T39, X67),\nX17 -> X68,\nX66 -> X68,\nT35 -> T38,\nT36 -> T39,\nT37 -> T40" }, { "from": 217, "to": 234, "label": "EVAL-BACKTRACK" }, { "from": 218, "to": 913, "label": "PARALLEL" }, { "from": 218, "to": 914, "label": "PARALLEL" }, { "from": 233, "to": 237, "label": "SPLIT 1" }, { "from": 233, "to": 238, "label": "SPLIT 2\nnew knowledge:\nT43 is ground\nreplacements:T38 -> T43,\nT40 -> T44" }, { "from": 237, "to": 240, "label": "CASE" }, { "from": 238, "to": 488, "label": "CASE" }, { "from": 240, "to": 241, "label": "PARALLEL" }, { "from": 240, "to": 242, "label": "PARALLEL" }, { "from": 241, "to": 243, "label": "EVAL with clause\nless(0, s(X77)).\nand substitutionT38 -> 0,\nX77 -> T51,\nT39 -> s(T51)" }, { "from": 241, "to": 244, "label": "EVAL-BACKTRACK" }, { "from": 242, "to": 479, "label": "EVAL with clause\nless(s(X82), s(X83)) :- less(X82, X83).\nand substitutionX82 -> T58,\nT38 -> s(T58),\nX83 -> T59,\nT39 -> s(T59),\nT56 -> T58,\nT57 -> T59" }, { "from": 242, "to": 480, "label": "EVAL-BACKTRACK" }, { "from": 243, "to": 245, "label": "SUCCESS" }, { "from": 479, "to": 237, "label": "INSTANCE with matching:\nT38 -> T58\nT39 -> T59" }, { "from": 488, "to": 687, "label": "PARALLEL" }, { "from": 488, "to": 691, "label": "PARALLEL" }, { "from": 687, "to": 759, "label": "EVAL with clause\npart(X121, .(X122, X123), .(X122, X124), X125) :- ','(less(X121, X122), part(X121, X123, X124, X125)).\nand substitutionT43 -> T77,\nX121 -> T77,\nX122 -> T80,\nX123 -> T81,\nT44 -> .(T80, T81),\nX124 -> X126,\nX67 -> .(T80, X126),\nX68 -> X127,\nX125 -> X127,\nT78 -> T80,\nT79 -> T81" }, { "from": 687, "to": 768, "label": "EVAL-BACKTRACK" }, { "from": 691, "to": 883, "label": "PARALLEL" }, { "from": 691, "to": 884, "label": "PARALLEL" }, { "from": 759, "to": 792, "label": "SPLIT 1" }, { "from": 759, "to": 794, "label": "SPLIT 2\nnew knowledge:\nT77 is ground\nreplacements:T81 -> T84" }, { "from": 792, "to": 809, "label": "CASE" }, { "from": 794, "to": 238, "label": "INSTANCE with matching:\nT43 -> T77\nT44 -> T84\nX67 -> X126\nX68 -> X127" }, { "from": 809, "to": 816, "label": "PARALLEL" }, { "from": 809, "to": 818, "label": "PARALLEL" }, { "from": 816, "to": 830, "label": "EVAL with clause\nless(0, s(X136)).\nand substitutionT77 -> 0,\nX136 -> T91,\nT80 -> s(T91)" }, { "from": 816, "to": 832, "label": "EVAL-BACKTRACK" }, { "from": 818, "to": 850, "label": "EVAL with clause\nless(s(X141), s(X142)) :- less(X141, X142).\nand substitutionX141 -> T96,\nT77 -> s(T96),\nX142 -> T98,\nT80 -> s(T98),\nT97 -> T98" }, { "from": 818, "to": 854, "label": "EVAL-BACKTRACK" }, { "from": 830, "to": 834, "label": "SUCCESS" }, { "from": 850, "to": 792, "label": "INSTANCE with matching:\nT77 -> T96\nT80 -> T98" }, { "from": 883, "to": 903, "label": "EVAL with clause\npart(X180, .(X181, X182), X183, .(X181, X184)) :- part(X180, X182, X183, X184).\nand substitutionT43 -> T116,\nX180 -> T116,\nX181 -> T117,\nX182 -> T119,\nT44 -> .(T117, T119),\nX67 -> X185,\nX183 -> X185,\nX184 -> X186,\nX68 -> .(T117, X186),\nT118 -> T119" }, { "from": 883, "to": 904, "label": "EVAL-BACKTRACK" }, { "from": 884, "to": 905, "label": "EVAL with clause\npart(X196, [], [], []).\nand substitutionT43 -> T125,\nX196 -> T125,\nT44 -> [],\nX67 -> [],\nX68 -> []" }, { "from": 884, "to": 906, "label": "EVAL-BACKTRACK" }, { "from": 903, "to": 238, "label": "INSTANCE with matching:\nT43 -> T116\nT44 -> T119\nX67 -> X185\nX68 -> X186" }, { "from": 905, "to": 907, "label": "SUCCESS" }, { "from": 913, "to": 920, "label": "EVAL with clause\npart(X225, .(X226, X227), X228, .(X226, X229)) :- part(X225, X227, X228, X229).\nand substitutionT12 -> T141,\nX225 -> T141,\nX226 -> T139,\nX227 -> T142,\nT13 -> .(T139, T142),\nX16 -> X230,\nX228 -> X230,\nX229 -> X231,\nX17 -> .(T139, X231),\nT138 -> T141,\nT140 -> T142" }, { "from": 913, "to": 921, "label": "EVAL-BACKTRACK" }, { "from": 914, "to": 922, "label": "EVAL with clause\npart(X241, [], [], []).\nand substitutionT12 -> T148,\nX241 -> T148,\nT13 -> [],\nX16 -> [],\nX17 -> []" }, { "from": 914, "to": 923, "label": "EVAL-BACKTRACK" }, { "from": 920, "to": 182, "label": "INSTANCE with matching:\nT12 -> T141\nT13 -> T142\nX16 -> X230\nX17 -> X231" }, { "from": 922, "to": 924, "label": "SUCCESS" }, { "from": 925, "to": 927, "label": "CASE" }, { "from": 926, "to": 1053, "label": "SPLIT 1" }, { "from": 926, "to": 1054, "label": "SPLIT 2\nreplacements:X19 -> T208,\nT149 -> T209,\nT151 -> T210" }, { "from": 927, "to": 928, "label": "PARALLEL" }, { "from": 927, "to": 929, "label": "PARALLEL" }, { "from": 928, "to": 930, "label": "EVAL with clause\nqs([], []).\nand substitutionT17 -> [],\nX18 -> []" }, { "from": 928, "to": 931, "label": "EVAL-BACKTRACK" }, { "from": 929, "to": 934, "label": "EVAL with clause\nqs(.(X254, X255), X256) :- ','(part(X254, X255, X257, X258), ','(qs(X257, X259), ','(qs(X258, X260), app(X259, .(X254, X260), X256)))).\nand substitutionX254 -> T158,\nX255 -> T159,\nT17 -> .(T158, T159),\nX18 -> X261,\nX256 -> X261,\nT156 -> T158,\nT157 -> T159" }, { "from": 929, "to": 935, "label": "EVAL-BACKTRACK" }, { "from": 930, "to": 932, "label": "SUCCESS" }, { "from": 934, "to": 939, "label": "SPLIT 1" }, { "from": 934, "to": 940, "label": "SPLIT 2\nreplacements:X257 -> T163,\nX258 -> T164,\nT158 -> T165" }, { "from": 939, "to": 182, "label": "INSTANCE with matching:\nT12 -> T158\nT13 -> T159\nX16 -> X257\nX17 -> X258" }, { "from": 940, "to": 943, "label": "SPLIT 1" }, { "from": 940, "to": 944, "label": "SPLIT 2\nreplacements:X259 -> T169,\nT164 -> T170,\nT165 -> T171" }, { "from": 943, "to": 925, "label": "INSTANCE with matching:\nT17 -> T163\nX18 -> X259" }, { "from": 944, "to": 947, "label": "SPLIT 1" }, { "from": 944, "to": 948, "label": "SPLIT 2\nreplacements:X260 -> T172,\nT169 -> T173,\nT171 -> T174" }, { "from": 947, "to": 925, "label": "INSTANCE with matching:\nT17 -> T170\nX18 -> X260" }, { "from": 948, "to": 949, "label": "CASE" }, { "from": 949, "to": 950, "label": "PARALLEL" }, { "from": 949, "to": 951, "label": "PARALLEL" }, { "from": 950, "to": 952, "label": "EVAL with clause\napp([], X282, X282).\nand substitutionT173 -> [],\nT174 -> T187,\nT172 -> T188,\nX282 -> .(T187, T188),\nX261 -> .(T187, T188)" }, { "from": 950, "to": 953, "label": "EVAL-BACKTRACK" }, { "from": 951, "to": 955, "label": "EVAL with clause\napp(.(X293, X294), X295, .(X293, X296)) :- app(X294, X295, X296).\nand substitutionX293 -> T197,\nX294 -> T201,\nT173 -> .(T197, T201),\nT174 -> T202,\nT172 -> T203,\nX295 -> .(T202, T203),\nX296 -> X297,\nX261 -> .(T197, X297),\nT198 -> T201,\nT199 -> T202,\nT200 -> T203" }, { "from": 951, "to": 956, "label": "EVAL-BACKTRACK" }, { "from": 952, "to": 954, "label": "SUCCESS" }, { "from": 955, "to": 948, "label": "INSTANCE with matching:\nT173 -> T201\nT174 -> T202\nT172 -> T203\nX261 -> X297" }, { "from": 1053, "to": 925, "label": "INSTANCE with matching:\nT17 -> T150\nX18 -> X19" }, { "from": 1054, "to": 1055, "label": "CASE" }, { "from": 1055, "to": 1056, "label": "PARALLEL" }, { "from": 1055, "to": 1057, "label": "PARALLEL" }, { "from": 1056, "to": 1058, "label": "EVAL with clause\napp([], X306, X306).\nand substitutionT209 -> [],\nT210 -> T223,\nT208 -> T224,\nX306 -> .(T223, T224),\nT11 -> .(T223, T224)" }, { "from": 1056, "to": 1059, "label": "EVAL-BACKTRACK" }, { "from": 1057, "to": 1061, "label": "EVAL with clause\napp(.(X315, X316), X317, .(X315, X318)) :- app(X316, X317, X318).\nand substitutionX315 -> T235,\nX316 -> T240,\nT209 -> .(T235, T240),\nT210 -> T241,\nT208 -> T242,\nX317 -> .(T241, T242),\nX318 -> T239,\nT11 -> .(T235, T239),\nT236 -> T240,\nT237 -> T241,\nT238 -> T242" }, { "from": 1057, "to": 1062, "label": "EVAL-BACKTRACK" }, { "from": 1058, "to": 1060, "label": "SUCCESS" }, { "from": 1061, "to": 1054, "label": "INSTANCE with matching:\nT209 -> T240\nT210 -> T241\nT208 -> T242\nT11 -> T239" } ], "type": "Graph" } } ---------------------------------------- (108) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f2_in([]) -> f2_out1 f2_in(T11) -> U1(f62_in(T11), T11) U1(f62_out1(X18, T12, X19), T11) -> f2_out1 f237_in -> f237_out1(0) f237_in -> U2(f237_in) U2(f237_out1(T58)) -> f237_out1(s(T58)) f238_in(T77) -> U3(f759_in(T77), T77) U3(f759_out1, T77) -> f238_out1 f238_in(T116) -> U4(f238_in(T116), T116) U4(f238_out1, T116) -> f238_out1 f238_in(T125) -> f238_out1 f792_in(0) -> f792_out1 f792_in(s(T96)) -> U5(f792_in(T96), s(T96)) U5(f792_out1, s(T96)) -> f792_out1 f182_in -> U6(f233_in) U6(f233_out1(T38)) -> f182_out1 f182_in -> U7(f182_in) U7(f182_out1) -> f182_out1 f182_in -> f182_out1 f925_in -> f925_out1 f925_in -> U8(f934_in) U8(f934_out1) -> f925_out1 f948_in -> f948_out1 f948_in -> U9(f948_in) U9(f948_out1) -> f948_out1 f1054_in(.(T223, T224)) -> f1054_out1([], T223, T224) f1054_in(.(T235, T239)) -> U10(f1054_in(T239), .(T235, T239)) U10(f1054_out1(T240, T241, T242), .(T235, T239)) -> f1054_out1(.(T235, T240), T241, T242) f62_in(T11) -> U11(f182_in, T11) U11(f182_out1, T11) -> U12(f185_in(T11), T11) U12(f185_out1(X18, T19, X19), T11) -> f62_out1(X18, T19, X19) f185_in(T11) -> U13(f925_in, T11) U13(f925_out1, T11) -> U14(f926_in(T11), T11) U14(f926_out1(T149, T151, X19), T11) -> f185_out1(T149, T151, X19) f233_in -> U15(f237_in) U15(f237_out1(T43)) -> U16(f238_in(T43), T43) U16(f238_out1, T43) -> f233_out1(T43) f759_in(T77) -> U17(f792_in(T77), T77) U17(f792_out1, T77) -> U18(f238_in(T77), T77) U18(f238_out1, T77) -> f759_out1 f926_in(T11) -> U19(f925_in, T11) U19(f925_out1, T11) -> U20(f1054_in(T11), T11) U20(f1054_out1(T209, T210, T208), T11) -> f926_out1(T209, T210, T208) f934_in -> U21(f182_in) U21(f182_out1) -> U22(f940_in) U22(f940_out1) -> f934_out1 f940_in -> U23(f925_in) U23(f925_out1) -> U24(f944_in) U24(f944_out1) -> f940_out1 f944_in -> U25(f925_in) U25(f925_out1) -> U26(f948_in) U26(f948_out1) -> f944_out1 Q is empty. ---------------------------------------- (109) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (110) Obligation: Q DP problem: The TRS P consists of the following rules: F2_IN(T11) -> U1^1(f62_in(T11), T11) F2_IN(T11) -> F62_IN(T11) F237_IN -> U2^1(f237_in) F237_IN -> F237_IN F238_IN(T77) -> U3^1(f759_in(T77), T77) F238_IN(T77) -> F759_IN(T77) F238_IN(T116) -> U4^1(f238_in(T116), T116) F238_IN(T116) -> F238_IN(T116) F792_IN(s(T96)) -> U5^1(f792_in(T96), s(T96)) F792_IN(s(T96)) -> F792_IN(T96) F182_IN -> U6^1(f233_in) F182_IN -> F233_IN F182_IN -> U7^1(f182_in) F182_IN -> F182_IN F925_IN -> U8^1(f934_in) F925_IN -> F934_IN F948_IN -> U9^1(f948_in) F948_IN -> F948_IN F1054_IN(.(T235, T239)) -> U10^1(f1054_in(T239), .(T235, T239)) F1054_IN(.(T235, T239)) -> F1054_IN(T239) F62_IN(T11) -> U11^1(f182_in, T11) F62_IN(T11) -> F182_IN U11^1(f182_out1, T11) -> U12^1(f185_in(T11), T11) U11^1(f182_out1, T11) -> F185_IN(T11) F185_IN(T11) -> U13^1(f925_in, T11) F185_IN(T11) -> F925_IN U13^1(f925_out1, T11) -> U14^1(f926_in(T11), T11) U13^1(f925_out1, T11) -> F926_IN(T11) F233_IN -> U15^1(f237_in) F233_IN -> F237_IN U15^1(f237_out1(T43)) -> U16^1(f238_in(T43), T43) U15^1(f237_out1(T43)) -> F238_IN(T43) F759_IN(T77) -> U17^1(f792_in(T77), T77) F759_IN(T77) -> F792_IN(T77) U17^1(f792_out1, T77) -> U18^1(f238_in(T77), T77) U17^1(f792_out1, T77) -> F238_IN(T77) F926_IN(T11) -> U19^1(f925_in, T11) F926_IN(T11) -> F925_IN U19^1(f925_out1, T11) -> U20^1(f1054_in(T11), T11) U19^1(f925_out1, T11) -> F1054_IN(T11) F934_IN -> U21^1(f182_in) F934_IN -> F182_IN U21^1(f182_out1) -> U22^1(f940_in) U21^1(f182_out1) -> F940_IN F940_IN -> U23^1(f925_in) F940_IN -> F925_IN U23^1(f925_out1) -> U24^1(f944_in) U23^1(f925_out1) -> F944_IN F944_IN -> U25^1(f925_in) F944_IN -> F925_IN U25^1(f925_out1) -> U26^1(f948_in) U25^1(f925_out1) -> F948_IN The TRS R consists of the following rules: f2_in([]) -> f2_out1 f2_in(T11) -> U1(f62_in(T11), T11) U1(f62_out1(X18, T12, X19), T11) -> f2_out1 f237_in -> f237_out1(0) f237_in -> U2(f237_in) U2(f237_out1(T58)) -> f237_out1(s(T58)) f238_in(T77) -> U3(f759_in(T77), T77) U3(f759_out1, T77) -> f238_out1 f238_in(T116) -> U4(f238_in(T116), T116) U4(f238_out1, T116) -> f238_out1 f238_in(T125) -> f238_out1 f792_in(0) -> f792_out1 f792_in(s(T96)) -> U5(f792_in(T96), s(T96)) U5(f792_out1, s(T96)) -> f792_out1 f182_in -> U6(f233_in) U6(f233_out1(T38)) -> f182_out1 f182_in -> U7(f182_in) U7(f182_out1) -> f182_out1 f182_in -> f182_out1 f925_in -> f925_out1 f925_in -> U8(f934_in) U8(f934_out1) -> f925_out1 f948_in -> f948_out1 f948_in -> U9(f948_in) U9(f948_out1) -> f948_out1 f1054_in(.(T223, T224)) -> f1054_out1([], T223, T224) f1054_in(.(T235, T239)) -> U10(f1054_in(T239), .(T235, T239)) U10(f1054_out1(T240, T241, T242), .(T235, T239)) -> f1054_out1(.(T235, T240), T241, T242) f62_in(T11) -> U11(f182_in, T11) U11(f182_out1, T11) -> U12(f185_in(T11), T11) U12(f185_out1(X18, T19, X19), T11) -> f62_out1(X18, T19, X19) f185_in(T11) -> U13(f925_in, T11) U13(f925_out1, T11) -> U14(f926_in(T11), T11) U14(f926_out1(T149, T151, X19), T11) -> f185_out1(T149, T151, X19) f233_in -> U15(f237_in) U15(f237_out1(T43)) -> U16(f238_in(T43), T43) U16(f238_out1, T43) -> f233_out1(T43) f759_in(T77) -> U17(f792_in(T77), T77) U17(f792_out1, T77) -> U18(f238_in(T77), T77) U18(f238_out1, T77) -> f759_out1 f926_in(T11) -> U19(f925_in, T11) U19(f925_out1, T11) -> U20(f1054_in(T11), T11) U20(f1054_out1(T209, T210, T208), T11) -> f926_out1(T209, T210, T208) f934_in -> U21(f182_in) U21(f182_out1) -> U22(f940_in) U22(f940_out1) -> f934_out1 f940_in -> U23(f925_in) U23(f925_out1) -> U24(f944_in) U24(f944_out1) -> f940_out1 f944_in -> U25(f925_in) U25(f925_out1) -> U26(f948_in) U26(f948_out1) -> f944_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (111) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 7 SCCs with 36 less nodes. ---------------------------------------- (112) Complex Obligation (AND) ---------------------------------------- (113) Obligation: Q DP problem: The TRS P consists of the following rules: F1054_IN(.(T235, T239)) -> F1054_IN(T239) The TRS R consists of the following rules: f2_in([]) -> f2_out1 f2_in(T11) -> U1(f62_in(T11), T11) U1(f62_out1(X18, T12, X19), T11) -> f2_out1 f237_in -> f237_out1(0) f237_in -> U2(f237_in) U2(f237_out1(T58)) -> f237_out1(s(T58)) f238_in(T77) -> U3(f759_in(T77), T77) U3(f759_out1, T77) -> f238_out1 f238_in(T116) -> U4(f238_in(T116), T116) U4(f238_out1, T116) -> f238_out1 f238_in(T125) -> f238_out1 f792_in(0) -> f792_out1 f792_in(s(T96)) -> U5(f792_in(T96), s(T96)) U5(f792_out1, s(T96)) -> f792_out1 f182_in -> U6(f233_in) U6(f233_out1(T38)) -> f182_out1 f182_in -> U7(f182_in) U7(f182_out1) -> f182_out1 f182_in -> f182_out1 f925_in -> f925_out1 f925_in -> U8(f934_in) U8(f934_out1) -> f925_out1 f948_in -> f948_out1 f948_in -> U9(f948_in) U9(f948_out1) -> f948_out1 f1054_in(.(T223, T224)) -> f1054_out1([], T223, T224) f1054_in(.(T235, T239)) -> U10(f1054_in(T239), .(T235, T239)) U10(f1054_out1(T240, T241, T242), .(T235, T239)) -> f1054_out1(.(T235, T240), T241, T242) f62_in(T11) -> U11(f182_in, T11) U11(f182_out1, T11) -> U12(f185_in(T11), T11) U12(f185_out1(X18, T19, X19), T11) -> f62_out1(X18, T19, X19) f185_in(T11) -> U13(f925_in, T11) U13(f925_out1, T11) -> U14(f926_in(T11), T11) U14(f926_out1(T149, T151, X19), T11) -> f185_out1(T149, T151, X19) f233_in -> U15(f237_in) U15(f237_out1(T43)) -> U16(f238_in(T43), T43) U16(f238_out1, T43) -> f233_out1(T43) f759_in(T77) -> U17(f792_in(T77), T77) U17(f792_out1, T77) -> U18(f238_in(T77), T77) U18(f238_out1, T77) -> f759_out1 f926_in(T11) -> U19(f925_in, T11) U19(f925_out1, T11) -> U20(f1054_in(T11), T11) U20(f1054_out1(T209, T210, T208), T11) -> f926_out1(T209, T210, T208) f934_in -> U21(f182_in) U21(f182_out1) -> U22(f940_in) U22(f940_out1) -> f934_out1 f940_in -> U23(f925_in) U23(f925_out1) -> U24(f944_in) U24(f944_out1) -> f940_out1 f944_in -> U25(f925_in) U25(f925_out1) -> U26(f948_in) U26(f948_out1) -> f944_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (114) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (115) Obligation: Q DP problem: The TRS P consists of the following rules: F1054_IN(.(T235, T239)) -> F1054_IN(T239) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (116) 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: *F1054_IN(.(T235, T239)) -> F1054_IN(T239) The graph contains the following edges 1 > 1 ---------------------------------------- (117) YES ---------------------------------------- (118) Obligation: Q DP problem: The TRS P consists of the following rules: F948_IN -> F948_IN The TRS R consists of the following rules: f2_in([]) -> f2_out1 f2_in(T11) -> U1(f62_in(T11), T11) U1(f62_out1(X18, T12, X19), T11) -> f2_out1 f237_in -> f237_out1(0) f237_in -> U2(f237_in) U2(f237_out1(T58)) -> f237_out1(s(T58)) f238_in(T77) -> U3(f759_in(T77), T77) U3(f759_out1, T77) -> f238_out1 f238_in(T116) -> U4(f238_in(T116), T116) U4(f238_out1, T116) -> f238_out1 f238_in(T125) -> f238_out1 f792_in(0) -> f792_out1 f792_in(s(T96)) -> U5(f792_in(T96), s(T96)) U5(f792_out1, s(T96)) -> f792_out1 f182_in -> U6(f233_in) U6(f233_out1(T38)) -> f182_out1 f182_in -> U7(f182_in) U7(f182_out1) -> f182_out1 f182_in -> f182_out1 f925_in -> f925_out1 f925_in -> U8(f934_in) U8(f934_out1) -> f925_out1 f948_in -> f948_out1 f948_in -> U9(f948_in) U9(f948_out1) -> f948_out1 f1054_in(.(T223, T224)) -> f1054_out1([], T223, T224) f1054_in(.(T235, T239)) -> U10(f1054_in(T239), .(T235, T239)) U10(f1054_out1(T240, T241, T242), .(T235, T239)) -> f1054_out1(.(T235, T240), T241, T242) f62_in(T11) -> U11(f182_in, T11) U11(f182_out1, T11) -> U12(f185_in(T11), T11) U12(f185_out1(X18, T19, X19), T11) -> f62_out1(X18, T19, X19) f185_in(T11) -> U13(f925_in, T11) U13(f925_out1, T11) -> U14(f926_in(T11), T11) U14(f926_out1(T149, T151, X19), T11) -> f185_out1(T149, T151, X19) f233_in -> U15(f237_in) U15(f237_out1(T43)) -> U16(f238_in(T43), T43) U16(f238_out1, T43) -> f233_out1(T43) f759_in(T77) -> U17(f792_in(T77), T77) U17(f792_out1, T77) -> U18(f238_in(T77), T77) U18(f238_out1, T77) -> f759_out1 f926_in(T11) -> U19(f925_in, T11) U19(f925_out1, T11) -> U20(f1054_in(T11), T11) U20(f1054_out1(T209, T210, T208), T11) -> f926_out1(T209, T210, T208) f934_in -> U21(f182_in) U21(f182_out1) -> U22(f940_in) U22(f940_out1) -> f934_out1 f940_in -> U23(f925_in) U23(f925_out1) -> U24(f944_in) U24(f944_out1) -> f940_out1 f944_in -> U25(f925_in) U25(f925_out1) -> U26(f948_in) U26(f948_out1) -> f944_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (119) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (120) Obligation: Q DP problem: The TRS P consists of the following rules: F948_IN -> F948_IN R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (121) 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 = F948_IN evaluates to t =F948_IN 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 F948_IN to F948_IN. ---------------------------------------- (122) NO ---------------------------------------- (123) Obligation: Q DP problem: The TRS P consists of the following rules: F792_IN(s(T96)) -> F792_IN(T96) The TRS R consists of the following rules: f2_in([]) -> f2_out1 f2_in(T11) -> U1(f62_in(T11), T11) U1(f62_out1(X18, T12, X19), T11) -> f2_out1 f237_in -> f237_out1(0) f237_in -> U2(f237_in) U2(f237_out1(T58)) -> f237_out1(s(T58)) f238_in(T77) -> U3(f759_in(T77), T77) U3(f759_out1, T77) -> f238_out1 f238_in(T116) -> U4(f238_in(T116), T116) U4(f238_out1, T116) -> f238_out1 f238_in(T125) -> f238_out1 f792_in(0) -> f792_out1 f792_in(s(T96)) -> U5(f792_in(T96), s(T96)) U5(f792_out1, s(T96)) -> f792_out1 f182_in -> U6(f233_in) U6(f233_out1(T38)) -> f182_out1 f182_in -> U7(f182_in) U7(f182_out1) -> f182_out1 f182_in -> f182_out1 f925_in -> f925_out1 f925_in -> U8(f934_in) U8(f934_out1) -> f925_out1 f948_in -> f948_out1 f948_in -> U9(f948_in) U9(f948_out1) -> f948_out1 f1054_in(.(T223, T224)) -> f1054_out1([], T223, T224) f1054_in(.(T235, T239)) -> U10(f1054_in(T239), .(T235, T239)) U10(f1054_out1(T240, T241, T242), .(T235, T239)) -> f1054_out1(.(T235, T240), T241, T242) f62_in(T11) -> U11(f182_in, T11) U11(f182_out1, T11) -> U12(f185_in(T11), T11) U12(f185_out1(X18, T19, X19), T11) -> f62_out1(X18, T19, X19) f185_in(T11) -> U13(f925_in, T11) U13(f925_out1, T11) -> U14(f926_in(T11), T11) U14(f926_out1(T149, T151, X19), T11) -> f185_out1(T149, T151, X19) f233_in -> U15(f237_in) U15(f237_out1(T43)) -> U16(f238_in(T43), T43) U16(f238_out1, T43) -> f233_out1(T43) f759_in(T77) -> U17(f792_in(T77), T77) U17(f792_out1, T77) -> U18(f238_in(T77), T77) U18(f238_out1, T77) -> f759_out1 f926_in(T11) -> U19(f925_in, T11) U19(f925_out1, T11) -> U20(f1054_in(T11), T11) U20(f1054_out1(T209, T210, T208), T11) -> f926_out1(T209, T210, T208) f934_in -> U21(f182_in) U21(f182_out1) -> U22(f940_in) U22(f940_out1) -> f934_out1 f940_in -> U23(f925_in) U23(f925_out1) -> U24(f944_in) U24(f944_out1) -> f940_out1 f944_in -> U25(f925_in) U25(f925_out1) -> U26(f948_in) U26(f948_out1) -> f944_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (124) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (125) Obligation: Q DP problem: The TRS P consists of the following rules: F792_IN(s(T96)) -> F792_IN(T96) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (126) 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: *F792_IN(s(T96)) -> F792_IN(T96) The graph contains the following edges 1 > 1 ---------------------------------------- (127) YES ---------------------------------------- (128) Obligation: Q DP problem: The TRS P consists of the following rules: F238_IN(T77) -> F759_IN(T77) F759_IN(T77) -> U17^1(f792_in(T77), T77) U17^1(f792_out1, T77) -> F238_IN(T77) F238_IN(T116) -> F238_IN(T116) The TRS R consists of the following rules: f2_in([]) -> f2_out1 f2_in(T11) -> U1(f62_in(T11), T11) U1(f62_out1(X18, T12, X19), T11) -> f2_out1 f237_in -> f237_out1(0) f237_in -> U2(f237_in) U2(f237_out1(T58)) -> f237_out1(s(T58)) f238_in(T77) -> U3(f759_in(T77), T77) U3(f759_out1, T77) -> f238_out1 f238_in(T116) -> U4(f238_in(T116), T116) U4(f238_out1, T116) -> f238_out1 f238_in(T125) -> f238_out1 f792_in(0) -> f792_out1 f792_in(s(T96)) -> U5(f792_in(T96), s(T96)) U5(f792_out1, s(T96)) -> f792_out1 f182_in -> U6(f233_in) U6(f233_out1(T38)) -> f182_out1 f182_in -> U7(f182_in) U7(f182_out1) -> f182_out1 f182_in -> f182_out1 f925_in -> f925_out1 f925_in -> U8(f934_in) U8(f934_out1) -> f925_out1 f948_in -> f948_out1 f948_in -> U9(f948_in) U9(f948_out1) -> f948_out1 f1054_in(.(T223, T224)) -> f1054_out1([], T223, T224) f1054_in(.(T235, T239)) -> U10(f1054_in(T239), .(T235, T239)) U10(f1054_out1(T240, T241, T242), .(T235, T239)) -> f1054_out1(.(T235, T240), T241, T242) f62_in(T11) -> U11(f182_in, T11) U11(f182_out1, T11) -> U12(f185_in(T11), T11) U12(f185_out1(X18, T19, X19), T11) -> f62_out1(X18, T19, X19) f185_in(T11) -> U13(f925_in, T11) U13(f925_out1, T11) -> U14(f926_in(T11), T11) U14(f926_out1(T149, T151, X19), T11) -> f185_out1(T149, T151, X19) f233_in -> U15(f237_in) U15(f237_out1(T43)) -> U16(f238_in(T43), T43) U16(f238_out1, T43) -> f233_out1(T43) f759_in(T77) -> U17(f792_in(T77), T77) U17(f792_out1, T77) -> U18(f238_in(T77), T77) U18(f238_out1, T77) -> f759_out1 f926_in(T11) -> U19(f925_in, T11) U19(f925_out1, T11) -> U20(f1054_in(T11), T11) U20(f1054_out1(T209, T210, T208), T11) -> f926_out1(T209, T210, T208) f934_in -> U21(f182_in) U21(f182_out1) -> U22(f940_in) U22(f940_out1) -> f934_out1 f940_in -> U23(f925_in) U23(f925_out1) -> U24(f944_in) U24(f944_out1) -> f940_out1 f944_in -> U25(f925_in) U25(f925_out1) -> U26(f948_in) U26(f948_out1) -> f944_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (129) 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 = F238_IN(T116) evaluates to t =F238_IN(T116) 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 F238_IN(T116) to F238_IN(T116). ---------------------------------------- (130) NO ---------------------------------------- (131) Obligation: Q DP problem: The TRS P consists of the following rules: F237_IN -> F237_IN The TRS R consists of the following rules: f2_in([]) -> f2_out1 f2_in(T11) -> U1(f62_in(T11), T11) U1(f62_out1(X18, T12, X19), T11) -> f2_out1 f237_in -> f237_out1(0) f237_in -> U2(f237_in) U2(f237_out1(T58)) -> f237_out1(s(T58)) f238_in(T77) -> U3(f759_in(T77), T77) U3(f759_out1, T77) -> f238_out1 f238_in(T116) -> U4(f238_in(T116), T116) U4(f238_out1, T116) -> f238_out1 f238_in(T125) -> f238_out1 f792_in(0) -> f792_out1 f792_in(s(T96)) -> U5(f792_in(T96), s(T96)) U5(f792_out1, s(T96)) -> f792_out1 f182_in -> U6(f233_in) U6(f233_out1(T38)) -> f182_out1 f182_in -> U7(f182_in) U7(f182_out1) -> f182_out1 f182_in -> f182_out1 f925_in -> f925_out1 f925_in -> U8(f934_in) U8(f934_out1) -> f925_out1 f948_in -> f948_out1 f948_in -> U9(f948_in) U9(f948_out1) -> f948_out1 f1054_in(.(T223, T224)) -> f1054_out1([], T223, T224) f1054_in(.(T235, T239)) -> U10(f1054_in(T239), .(T235, T239)) U10(f1054_out1(T240, T241, T242), .(T235, T239)) -> f1054_out1(.(T235, T240), T241, T242) f62_in(T11) -> U11(f182_in, T11) U11(f182_out1, T11) -> U12(f185_in(T11), T11) U12(f185_out1(X18, T19, X19), T11) -> f62_out1(X18, T19, X19) f185_in(T11) -> U13(f925_in, T11) U13(f925_out1, T11) -> U14(f926_in(T11), T11) U14(f926_out1(T149, T151, X19), T11) -> f185_out1(T149, T151, X19) f233_in -> U15(f237_in) U15(f237_out1(T43)) -> U16(f238_in(T43), T43) U16(f238_out1, T43) -> f233_out1(T43) f759_in(T77) -> U17(f792_in(T77), T77) U17(f792_out1, T77) -> U18(f238_in(T77), T77) U18(f238_out1, T77) -> f759_out1 f926_in(T11) -> U19(f925_in, T11) U19(f925_out1, T11) -> U20(f1054_in(T11), T11) U20(f1054_out1(T209, T210, T208), T11) -> f926_out1(T209, T210, T208) f934_in -> U21(f182_in) U21(f182_out1) -> U22(f940_in) U22(f940_out1) -> f934_out1 f940_in -> U23(f925_in) U23(f925_out1) -> U24(f944_in) U24(f944_out1) -> f940_out1 f944_in -> U25(f925_in) U25(f925_out1) -> U26(f948_in) U26(f948_out1) -> f944_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (132) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (133) Obligation: Q DP problem: The TRS P consists of the following rules: F237_IN -> F237_IN R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (134) 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 = F237_IN evaluates to t =F237_IN 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 F237_IN to F237_IN. ---------------------------------------- (135) NO ---------------------------------------- (136) Obligation: Q DP problem: The TRS P consists of the following rules: F182_IN -> F182_IN The TRS R consists of the following rules: f2_in([]) -> f2_out1 f2_in(T11) -> U1(f62_in(T11), T11) U1(f62_out1(X18, T12, X19), T11) -> f2_out1 f237_in -> f237_out1(0) f237_in -> U2(f237_in) U2(f237_out1(T58)) -> f237_out1(s(T58)) f238_in(T77) -> U3(f759_in(T77), T77) U3(f759_out1, T77) -> f238_out1 f238_in(T116) -> U4(f238_in(T116), T116) U4(f238_out1, T116) -> f238_out1 f238_in(T125) -> f238_out1 f792_in(0) -> f792_out1 f792_in(s(T96)) -> U5(f792_in(T96), s(T96)) U5(f792_out1, s(T96)) -> f792_out1 f182_in -> U6(f233_in) U6(f233_out1(T38)) -> f182_out1 f182_in -> U7(f182_in) U7(f182_out1) -> f182_out1 f182_in -> f182_out1 f925_in -> f925_out1 f925_in -> U8(f934_in) U8(f934_out1) -> f925_out1 f948_in -> f948_out1 f948_in -> U9(f948_in) U9(f948_out1) -> f948_out1 f1054_in(.(T223, T224)) -> f1054_out1([], T223, T224) f1054_in(.(T235, T239)) -> U10(f1054_in(T239), .(T235, T239)) U10(f1054_out1(T240, T241, T242), .(T235, T239)) -> f1054_out1(.(T235, T240), T241, T242) f62_in(T11) -> U11(f182_in, T11) U11(f182_out1, T11) -> U12(f185_in(T11), T11) U12(f185_out1(X18, T19, X19), T11) -> f62_out1(X18, T19, X19) f185_in(T11) -> U13(f925_in, T11) U13(f925_out1, T11) -> U14(f926_in(T11), T11) U14(f926_out1(T149, T151, X19), T11) -> f185_out1(T149, T151, X19) f233_in -> U15(f237_in) U15(f237_out1(T43)) -> U16(f238_in(T43), T43) U16(f238_out1, T43) -> f233_out1(T43) f759_in(T77) -> U17(f792_in(T77), T77) U17(f792_out1, T77) -> U18(f238_in(T77), T77) U18(f238_out1, T77) -> f759_out1 f926_in(T11) -> U19(f925_in, T11) U19(f925_out1, T11) -> U20(f1054_in(T11), T11) U20(f1054_out1(T209, T210, T208), T11) -> f926_out1(T209, T210, T208) f934_in -> U21(f182_in) U21(f182_out1) -> U22(f940_in) U22(f940_out1) -> f934_out1 f940_in -> U23(f925_in) U23(f925_out1) -> U24(f944_in) U24(f944_out1) -> f940_out1 f944_in -> U25(f925_in) U25(f925_out1) -> U26(f948_in) U26(f948_out1) -> f944_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (137) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (138) Obligation: Q DP problem: The TRS P consists of the following rules: F182_IN -> F182_IN R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (139) 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 = F182_IN evaluates to t =F182_IN 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 F182_IN to F182_IN. ---------------------------------------- (140) NO ---------------------------------------- (141) Obligation: Q DP problem: The TRS P consists of the following rules: F934_IN -> U21^1(f182_in) U21^1(f182_out1) -> F940_IN F940_IN -> U23^1(f925_in) U23^1(f925_out1) -> F944_IN F944_IN -> F925_IN F925_IN -> F934_IN F940_IN -> F925_IN The TRS R consists of the following rules: f2_in([]) -> f2_out1 f2_in(T11) -> U1(f62_in(T11), T11) U1(f62_out1(X18, T12, X19), T11) -> f2_out1 f237_in -> f237_out1(0) f237_in -> U2(f237_in) U2(f237_out1(T58)) -> f237_out1(s(T58)) f238_in(T77) -> U3(f759_in(T77), T77) U3(f759_out1, T77) -> f238_out1 f238_in(T116) -> U4(f238_in(T116), T116) U4(f238_out1, T116) -> f238_out1 f238_in(T125) -> f238_out1 f792_in(0) -> f792_out1 f792_in(s(T96)) -> U5(f792_in(T96), s(T96)) U5(f792_out1, s(T96)) -> f792_out1 f182_in -> U6(f233_in) U6(f233_out1(T38)) -> f182_out1 f182_in -> U7(f182_in) U7(f182_out1) -> f182_out1 f182_in -> f182_out1 f925_in -> f925_out1 f925_in -> U8(f934_in) U8(f934_out1) -> f925_out1 f948_in -> f948_out1 f948_in -> U9(f948_in) U9(f948_out1) -> f948_out1 f1054_in(.(T223, T224)) -> f1054_out1([], T223, T224) f1054_in(.(T235, T239)) -> U10(f1054_in(T239), .(T235, T239)) U10(f1054_out1(T240, T241, T242), .(T235, T239)) -> f1054_out1(.(T235, T240), T241, T242) f62_in(T11) -> U11(f182_in, T11) U11(f182_out1, T11) -> U12(f185_in(T11), T11) U12(f185_out1(X18, T19, X19), T11) -> f62_out1(X18, T19, X19) f185_in(T11) -> U13(f925_in, T11) U13(f925_out1, T11) -> U14(f926_in(T11), T11) U14(f926_out1(T149, T151, X19), T11) -> f185_out1(T149, T151, X19) f233_in -> U15(f237_in) U15(f237_out1(T43)) -> U16(f238_in(T43), T43) U16(f238_out1, T43) -> f233_out1(T43) f759_in(T77) -> U17(f792_in(T77), T77) U17(f792_out1, T77) -> U18(f238_in(T77), T77) U18(f238_out1, T77) -> f759_out1 f926_in(T11) -> U19(f925_in, T11) U19(f925_out1, T11) -> U20(f1054_in(T11), T11) U20(f1054_out1(T209, T210, T208), T11) -> f926_out1(T209, T210, T208) f934_in -> U21(f182_in) U21(f182_out1) -> U22(f940_in) U22(f940_out1) -> f934_out1 f940_in -> U23(f925_in) U23(f925_out1) -> U24(f944_in) U24(f944_out1) -> f940_out1 f944_in -> U25(f925_in) U25(f925_out1) -> U26(f948_in) U26(f948_out1) -> f944_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (142) NonLoopProof (COMPLETE) By Theorem 8 [NONLOOP] we deduce infiniteness of the QDP. We apply the theorem with m = 1, b = 0, σ' = [ ], and μ' = [ ] on the rule F940_IN[ ]^n[ ] -> F940_IN[ ]^n[ ] This rule is correct for the QDP as the following derivation shows: F940_IN[ ]^n[ ] -> F940_IN[ ]^n[ ] by Narrowing at position: [] F940_IN[ ]^n[ ] -> F925_IN[ ]^n[ ] by Rule from TRS P F925_IN[ ]^n[ ] -> F940_IN[ ]^n[ ] by Narrowing at position: [] F925_IN[ ]^n[ ] -> F934_IN[ ]^n[ ] by Rule from TRS P F934_IN[ ]^n[ ] -> F940_IN[ ]^n[ ] by Narrowing at position: [] F934_IN[ ]^n[ ] -> U21^1(f182_out1)[ ]^n[ ] by Narrowing at position: [0] F934_IN[ ]^n[ ] -> U21^1(f182_in)[ ]^n[ ] by Rule from TRS P f182_in[ ]^n[ ] -> f182_out1[ ]^n[ ] by Rule from TRS R U21^1(f182_out1)[ ]^n[ ] -> F940_IN[ ]^n[ ] by Rule from TRS P ---------------------------------------- (143) NO ---------------------------------------- (144) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 4, "program": { "directives": [], "clauses": [ [ "(qs ([]) ([]))", null ], [ "(qs (. 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T289 X12) ([]))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X11", "X12" ], "exprvars": [] } }, "1133": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "438": { "goal": [{ "clause": 2, "scope": 4, "term": "(part T29 T30 X46 X47)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T29"], "free": [ "X46", "X47" ], "exprvars": [] } }, "58": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (',' (less T24 T25) (part T24 T26 X46 X47)) (',' (qs (. T25 X46) X11) (',' (qs X47 X12) (app X11 (. T24 X12) ([])))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X11", "X12", "X46", "X47" ], "exprvars": [] } }, "59": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1154": { "goal": [{ "clause": -1, "scope": -1, "term": "(app T348 (. T328 T347) T305)" }], "kb": { "nonunifying": [[ "(qs T349 T305)", "(qs ([]) ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T305", "T328" ], "free": [], "exprvars": [] } }, "1153": { "goal": [{ "clause": -1, "scope": -1, "term": "(qs T345 X438)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X438"], "exprvars": [] } }, "1152": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (qs T345 X438) (app T344 (. T328 X438) T305))" }], "kb": { "nonunifying": [[ "(qs T346 T305)", "(qs ([]) ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T305", "T328" ], "free": ["X438"], "exprvars": [] } }, "1151": { "goal": [{ "clause": -1, "scope": -1, "term": "(qs (. T339 T337) X437)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X437"], "exprvars": [] } }, "1150": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (qs (. T339 T337) X437) (',' (qs T338 X438) (app X437 (. T328 X438) T305)))" }], "kb": { "nonunifying": [[ "(qs T340 T305)", "(qs ([]) ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T305", "T328" ], "free": [ "X437", "X438" ], "exprvars": [] } }, "440": { "goal": [ { "clause": 3, "scope": 4, "term": "(part T29 T30 X46 X47)" }, { "clause": 4, "scope": 4, "term": "(part T29 T30 X46 X47)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T29"], "free": [ "X46", "X47" ], "exprvars": [] } }, "1149": { "goal": [{ "clause": -1, "scope": -1, "term": "(part T328 T329 X472 X473)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T328"], "free": [ "X472", "X473" ], "exprvars": [] } }, "1148": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (part T328 T329 X472 X473) (',' (qs (. T330 X472) X437) (',' (qs X473 X438) (app X437 (. T328 X438) T305))))" }], "kb": { "nonunifying": [[ "(qs T331 T305)", "(qs ([]) ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T305", "T328" ], "free": [ "X437", "X438", "X472", "X473" ], "exprvars": [] } }, "1147": { "goal": [{ "clause": -1, "scope": -1, "term": "(less T323 T324)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1146": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1145": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (',' (less T323 T324) (part T323 T325 X472 X473)) (',' (qs (. T324 X472) X437) (',' (qs X473 X438) (app X437 (. T323 X438) T305))))" }], "kb": { "nonunifying": [[ "(qs T1 T305)", "(qs ([]) ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T305"], "free": [ "X437", "X438", "X472", "X473" ], "exprvars": [] } }, "1144": { "goal": [ { "clause": 3, "scope": 11, "term": "(',' (part T306 T307 X435 X436) (',' (qs X435 X437) (',' (qs X436 X438) (app X437 (. T306 X438) T305))))" }, { "clause": 4, "scope": 11, "term": "(',' (part T306 T307 X435 X436) (',' (qs X435 X437) (',' (qs X436 X438) (app X437 (. T306 X438) T305))))" } ], "kb": { "nonunifying": [[ "(qs T1 T305)", "(qs ([]) ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T305"], "free": [ "X435", "X436", "X437", "X438" ], "exprvars": [] } }, "1165": { "goal": [{ "clause": 6, "scope": 13, "term": "(app T379 (. T376 T380) T378)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T376", "T378" ], "free": [], "exprvars": [] } }, "1164": { "goal": [{ "clause": 5, "scope": 13, "term": "(app T379 (. T376 T380) T378)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T376", "T378" ], "free": [], "exprvars": [] } }, "1163": { "goal": [ { "clause": 5, "scope": 13, "term": "(app T379 (. T376 T380) T378)" }, { "clause": 6, "scope": 13, "term": "(app T379 (. T376 T380) T378)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T376", "T378" ], "free": [], "exprvars": [] } }, "1162": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1161": { "goal": [{ "clause": -1, "scope": -1, "term": "(app T379 (. T376 T380) T378)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T376", "T378" ], "free": [], "exprvars": [] } }, "1160": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1159": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "455": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (less T70 T73) (part T70 T74 X112 X113))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T70"], "free": [ "X112", "X113" ], "exprvars": [] } }, "1158": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1157": { "goal": [{ "clause": 6, "scope": 12, "term": "(app T348 (. T328 T347) T305)" }], "kb": { "nonunifying": [[ "(qs T349 T305)", "(qs ([]) ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T305", "T328" ], "free": [], "exprvars": [] } }, "1156": { "goal": [{ "clause": 5, "scope": 12, "term": "(app T348 (. T328 T347) T305)" }], "kb": { "nonunifying": [[ "(qs T349 T305)", "(qs ([]) ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T305", "T328" ], "free": [], "exprvars": [] } }, "458": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1155": { "goal": [ { "clause": 5, "scope": 12, "term": "(app T348 (. T328 T347) T305)" }, { "clause": 6, "scope": 12, "term": "(app T348 (. T328 T347) T305)" } ], "kb": { "nonunifying": [[ "(qs T349 T305)", "(qs ([]) ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T305", "T328" ], "free": [], "exprvars": [] } }, "1176": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (qs T437 X437) (',' (qs (. T439 T438) X438) (app X437 (. T440 X438) T305)))" }], "kb": { "nonunifying": [[ "(qs T441 T305)", "(qs ([]) ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T305"], "free": [ "X437", "X438" ], "exprvars": [] } }, "1175": { "goal": [{ "clause": -1, "scope": -1, "term": "(part T431 T432 X565 X566)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X565", "X566" ], "exprvars": [] } }, "1174": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1173": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (part T431 T432 X565 X566) (',' (qs X565 X437) (',' (qs (. T433 X566) X438) (app X437 (. T431 X438) T305))))" }], "kb": { "nonunifying": [[ "(qs T1 T305)", "(qs ([]) ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T305"], "free": [ "X437", "X438", "X565", "X566" ], "exprvars": [] } }, "1172": { "goal": [{ "clause": 4, "scope": 11, "term": "(',' (part T306 T307 X435 X436) (',' (qs X435 X437) (',' (qs X436 X438) (app X437 (. T306 X438) T305))))" }], "kb": { "nonunifying": [[ "(qs T1 T305)", "(qs ([]) ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T305"], "free": [ "X435", "X436", "X437", "X438" ], "exprvars": [] } }, "1171": { "goal": [{ "clause": 3, "scope": 11, "term": "(',' (part T306 T307 X435 X436) (',' (qs X435 X437) (',' (qs X436 X438) (app X437 (. T306 X438) T305))))" }], "kb": { "nonunifying": [[ "(qs T1 T305)", "(qs ([]) ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T305"], "free": [ "X435", "X436", "X437", "X438" ], "exprvars": [] } }, "1170": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "224": { "goal": [{ "clause": -1, "scope": -1, "term": "(less T24 T25)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1169": { "goal": [{ "clause": -1, "scope": -1, "term": "(app T410 (. T407 T411) T409)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T407", "T409" ], "free": [], "exprvars": [] } }, "1202": { "goal": [{ "clause": -1, "scope": -1, "term": "(app T530 (. T525 T535) T305)" }], "kb": { "nonunifying": [[ "(qs T1 T305)", "(qs ([]) ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T305", "T530", "T535" ], "free": [], "exprvars": [] } }, "225": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (part T29 T30 X46 X47) (',' (qs (. T31 X46) X11) (',' (qs X47 X12) (app X11 (. T29 X12) ([])))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T29"], "free": [ "X11", "X12", "X46", "X47" ], "exprvars": [] } }, "1168": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1201": { "goal": [{ "clause": -1, "scope": -1, "term": "(qs ([]) X438)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X438"], "exprvars": [] } }, "1167": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1200": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (qs ([]) X438) (app T530 (. T525 X438) T305))" }], "kb": { "nonunifying": [[ "(qs T1 T305)", "(qs ([]) ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T305", "T530" ], "free": ["X438"], "exprvars": [] } }, "227": { "goal": [ { "clause": 7, "scope": 3, "term": "(less T24 T25)" }, { "clause": 8, "scope": 3, "term": "(less T24 T25)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1166": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "900": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (qs T106 X12) (app T105 (. T29 X12) ([])))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T29"], "free": ["X12"], "exprvars": [] } }, "901": { "goal": [ { "clause": 0, "scope": 5, "term": "(qs (. T54 T52) X11)" }, { "clause": 1, "scope": 5, "term": "(qs (. T54 T52) X11)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X11"], "exprvars": [] } }, "902": { "goal": [{ "clause": 1, "scope": 5, "term": "(qs (. T54 T52) X11)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X11"], "exprvars": [] } } }, "edges": [ { "from": 4, "to": 5, "label": "CASE" }, { "from": 5, "to": 12, "label": "EVAL with clause\nqs([], []).\nand substitutionT1 -> [],\nT2 -> []" }, { "from": 5, "to": 13, "label": "EVAL-BACKTRACK" }, { "from": 12, "to": 14, "label": "SUCCESS" }, { "from": 13, "to": 1140, "label": "EVAL with clause\nqs(.(X432, X433), X434) :- ','(part(X432, X433, X435, X436), ','(qs(X435, X437), ','(qs(X436, X438), app(X437, .(X432, X438), X434)))).\nand substitutionX432 -> T306,\nX433 -> T307,\nT1 -> .(T306, T307),\nT2 -> T305,\nX434 -> T305,\nT303 -> T306,\nT304 -> T307" }, { "from": 13, "to": 1141, "label": "EVAL-BACKTRACK" }, { "from": 14, "to": 18, "label": "EVAL with clause\nqs(.(X6, X7), X8) :- ','(part(X6, X7, X9, X10), ','(qs(X9, X11), ','(qs(X10, X12), app(X11, .(X6, X12), X8)))).\nand substitutionX6 -> T7,\nX7 -> T8,\nT1 -> .(T7, T8),\nX8 -> [],\nT5 -> T7,\nT6 -> T8" }, { "from": 14, "to": 30, "label": "EVAL-BACKTRACK" }, { "from": 18, "to": 36, "label": "CASE" }, { "from": 36, "to": 37, "label": "PARALLEL" }, { "from": 36, "to": 38, "label": "PARALLEL" }, { "from": 37, "to": 58, "label": "EVAL with clause\npart(X41, .(X42, X43), .(X42, X44), X45) :- ','(less(X41, X42), part(X41, X43, X44, X45)).\nand substitutionT7 -> T24,\nX41 -> T24,\nX42 -> T25,\nX43 -> T26,\nT8 -> .(T25, T26),\nX44 -> X46,\nX9 -> .(T25, X46),\nX10 -> X47,\nX45 -> X47,\nT21 -> T24,\nT22 -> T25,\nT23 -> T26" }, { "from": 37, "to": 59, "label": "EVAL-BACKTRACK" }, { "from": 38, "to": 1121, "label": "PARALLEL" }, { "from": 38, "to": 1122, "label": "PARALLEL" }, { "from": 58, "to": 224, "label": "SPLIT 1" }, { "from": 58, "to": 225, "label": "SPLIT 2\nnew knowledge:\nT29 is ground\nreplacements:T24 -> T29,\nT26 -> T30,\nT25 -> T31" }, { "from": 224, "to": 227, "label": "CASE" }, { "from": 225, "to": 418, "label": "SPLIT 1" }, { "from": 225, "to": 420, "label": "SPLIT 2\nnew knowledge:\nT29 is ground\nreplacements:X46 -> T52,\nX47 -> T53,\nT31 -> T54" }, { "from": 227, "to": 231, "label": "PARALLEL" }, { "from": 227, "to": 232, "label": "PARALLEL" }, { "from": 231, "to": 246, "label": "EVAL with clause\nless(0, s(X56)).\nand substitutionT24 -> 0,\nX56 -> T38,\nT25 -> s(T38)" }, { "from": 231, "to": 249, "label": "EVAL-BACKTRACK" }, { "from": 232, "to": 305, "label": "EVAL with clause\nless(s(X61), s(X62)) :- less(X61, X62).\nand substitutionX61 -> T45,\nT24 -> s(T45),\nX62 -> T46,\nT25 -> s(T46),\nT43 -> T45,\nT44 -> T46" }, { "from": 232, "to": 314, "label": "EVAL-BACKTRACK" }, { "from": 246, "to": 256, "label": "SUCCESS" }, { "from": 305, "to": 224, "label": "INSTANCE with matching:\nT24 -> T45\nT25 -> T46" }, { "from": 418, "to": 429, "label": "CASE" }, { "from": 420, "to": 899, "label": "SPLIT 1" }, { "from": 420, "to": 900, "label": "SPLIT 2\nreplacements:X11 -> T105,\nT53 -> T106" }, { "from": 429, "to": 438, "label": "PARALLEL" }, { "from": 429, "to": 440, "label": "PARALLEL" }, { "from": 438, "to": 455, "label": "EVAL with clause\npart(X107, .(X108, X109), .(X108, X110), X111) :- ','(less(X107, X108), part(X107, X109, X110, X111)).\nand substitutionT29 -> T70,\nX107 -> T70,\nX108 -> T73,\nX109 -> T74,\nT30 -> .(T73, T74),\nX110 -> X112,\nX46 -> .(T73, X112),\nX47 -> X113,\nX111 -> X113,\nT71 -> T73,\nT72 -> T74" }, { "from": 438, "to": 458, "label": "EVAL-BACKTRACK" }, { "from": 440, "to": 887, "label": "PARALLEL" }, { "from": 440, "to": 888, "label": "PARALLEL" }, { "from": 455, "to": 474, "label": "SPLIT 1" }, { "from": 455, "to": 475, "label": "SPLIT 2\nnew knowledge:\nT70 is ground\nreplacements:T74 -> T77" }, { "from": 474, "to": 224, "label": "INSTANCE with matching:\nT24 -> T70\nT25 -> T73" }, { "from": 475, "to": 418, "label": "INSTANCE with matching:\nT29 -> T70\nT30 -> T77\nX46 -> X112\nX47 -> X113" }, { "from": 887, "to": 891, "label": "EVAL with clause\npart(X153, .(X154, X155), X156, .(X154, X157)) :- part(X153, X155, X156, X157).\nand substitutionT29 -> T95,\nX153 -> T95,\nX154 -> T96,\nX155 -> T98,\nT30 -> .(T96, T98),\nX46 -> X158,\nX156 -> X158,\nX157 -> X159,\nX47 -> .(T96, X159),\nT97 -> T98" }, { "from": 887, "to": 893, "label": "EVAL-BACKTRACK" }, { "from": 888, "to": 894, "label": "EVAL with clause\npart(X169, [], [], []).\nand substitutionT29 -> T104,\nX169 -> T104,\nT30 -> [],\nX46 -> [],\nX47 -> []" }, { "from": 888, "to": 895, "label": "EVAL-BACKTRACK" }, { "from": 891, "to": 418, "label": "INSTANCE with matching:\nT29 -> T95\nT30 -> T98\nX46 -> X158\nX47 -> X159" }, { "from": 894, "to": 896, "label": "SUCCESS" }, { "from": 899, "to": 901, "label": "CASE" }, { "from": 900, "to": 1104, "label": "SPLIT 1" }, { "from": 900, "to": 1105, "label": "SPLIT 2\nreplacements:X12 -> T229,\nT105 -> T230" }, { "from": 901, "to": 902, "label": "BACKTRACK\nfor clause: qs([], [])because of non-unification" }, { "from": 902, "to": 933, "label": "ONLY EVAL with clause\nqs(.(X189, X190), X191) :- ','(part(X189, X190, X192, X193), ','(qs(X192, X194), ','(qs(X193, X195), app(X194, .(X189, X195), X191)))).\nand substitutionT54 -> T118,\nX189 -> T118,\nT52 -> T119,\nX190 -> T119,\nX11 -> X196,\nX191 -> X196,\nT116 -> T118,\nT117 -> T119" }, { "from": 933, "to": 936, "label": "SPLIT 1" }, { "from": 933, "to": 937, "label": "SPLIT 2\nreplacements:X192 -> T123,\nX193 -> T124,\nT118 -> T125" }, { "from": 936, "to": 938, "label": "CASE" }, { "from": 937, "to": 1084, "label": "SPLIT 1" }, { "from": 937, "to": 1085, "label": "SPLIT 2\nreplacements:X194 -> T179,\nT124 -> T180,\nT125 -> T181" }, { "from": 938, "to": 941, "label": "PARALLEL" }, { "from": 938, "to": 942, "label": "PARALLEL" }, { "from": 941, "to": 993, "label": "EVAL with clause\npart(X239, .(X240, X241), .(X240, X242), X243) :- ','(less(X239, X240), part(X239, X241, X242, X243)).\nand substitutionT118 -> T144,\nX239 -> T144,\nX240 -> T145,\nX241 -> T146,\nT119 -> .(T145, T146),\nX242 -> X244,\nX192 -> .(T145, X244),\nX193 -> X245,\nX243 -> X245,\nT141 -> T144,\nT142 -> T145,\nT143 -> T146" }, { "from": 941, "to": 994, "label": "EVAL-BACKTRACK" }, { "from": 942, "to": 1063, "label": "PARALLEL" }, { "from": 942, "to": 1064, "label": "PARALLEL" }, { "from": 993, "to": 995, "label": "SPLIT 1" }, { "from": 993, "to": 996, "label": "SPLIT 2\nnew knowledge:\nT149 is ground\nreplacements:T144 -> T149,\nT146 -> T150" }, { "from": 995, "to": 224, "label": "INSTANCE with matching:\nT24 -> T144\nT25 -> T145" }, { "from": 996, "to": 418, "label": "INSTANCE with matching:\nT29 -> T149\nT30 -> T150\nX46 -> X244\nX47 -> X245" }, { "from": 1063, "to": 1079, "label": "EVAL with clause\npart(X285, .(X286, X287), X288, .(X286, X289)) :- part(X285, X287, X288, X289).\nand substitutionT118 -> T171,\nX285 -> T171,\nX286 -> T169,\nX287 -> T172,\nT119 -> .(T169, T172),\nX192 -> X290,\nX288 -> X290,\nX289 -> X291,\nX193 -> .(T169, X291),\nT168 -> T171,\nT170 -> T172" }, { "from": 1063, "to": 1080, "label": "EVAL-BACKTRACK" }, { "from": 1064, "to": 1081, "label": "EVAL with clause\npart(X301, [], [], []).\nand substitutionT118 -> T178,\nX301 -> T178,\nT119 -> [],\nX192 -> [],\nX193 -> []" }, { "from": 1064, "to": 1082, "label": "EVAL-BACKTRACK" }, { "from": 1079, "to": 936, "label": "INSTANCE with matching:\nT118 -> T171\nT119 -> T172\nX192 -> X290\nX193 -> X291" }, { "from": 1081, "to": 1083, "label": "SUCCESS" }, { "from": 1084, "to": 1086, "label": "CASE" }, { "from": 1085, "to": 1094, "label": "SPLIT 1" }, { "from": 1085, "to": 1095, "label": "SPLIT 2\nreplacements:X195 -> T193,\nT179 -> T194,\nT181 -> T195" }, { "from": 1086, "to": 1087, "label": "PARALLEL" }, { "from": 1086, "to": 1088, "label": "PARALLEL" }, { "from": 1087, "to": 1089, "label": "EVAL with clause\nqs([], []).\nand substitutionT123 -> [],\nX194 -> []" }, { "from": 1087, "to": 1090, "label": "EVAL-BACKTRACK" }, { "from": 1088, "to": 1092, "label": "EVAL with clause\nqs(.(X314, X315), X316) :- ','(part(X314, X315, X317, X318), ','(qs(X317, X319), ','(qs(X318, X320), app(X319, .(X314, X320), X316)))).\nand substitutionX314 -> T188,\nX315 -> T189,\nT123 -> .(T188, T189),\nX194 -> X321,\nX316 -> X321,\nT186 -> T188,\nT187 -> T189" }, { "from": 1088, "to": 1093, "label": "EVAL-BACKTRACK" }, { "from": 1089, "to": 1091, "label": "SUCCESS" }, { "from": 1092, "to": 933, "label": "INSTANCE with matching:\nT118 -> T188\nT119 -> T189\nX192 -> X317\nX193 -> X318\nX194 -> X319\nX195 -> X320\nX196 -> X321" }, { "from": 1094, "to": 1084, "label": "INSTANCE with matching:\nT123 -> T180\nX194 -> X195" }, { "from": 1095, "to": 1096, "label": "CASE" }, { "from": 1096, "to": 1097, "label": "PARALLEL" }, { "from": 1096, "to": 1098, "label": "PARALLEL" }, { "from": 1097, "to": 1099, "label": "EVAL with clause\napp([], X335, X335).\nand substitutionT194 -> [],\nT195 -> T208,\nT193 -> T209,\nX335 -> .(T208, T209),\nX196 -> .(T208, T209)" }, { "from": 1097, "to": 1100, "label": "EVAL-BACKTRACK" }, { "from": 1098, "to": 1102, "label": "EVAL with clause\napp(.(X346, X347), X348, .(X346, X349)) :- app(X347, X348, X349).\nand substitutionX346 -> T218,\nX347 -> T222,\nT194 -> .(T218, T222),\nT195 -> T223,\nT193 -> T224,\nX348 -> .(T223, T224),\nX349 -> X350,\nX196 -> .(T218, X350),\nT219 -> T222,\nT220 -> T223,\nT221 -> T224" }, { "from": 1098, "to": 1103, "label": "EVAL-BACKTRACK" }, { "from": 1099, "to": 1101, "label": "SUCCESS" }, { "from": 1102, "to": 1095, "label": "INSTANCE with matching:\nT194 -> T222\nT195 -> T223\nT193 -> T224\nX196 -> X350" }, { "from": 1104, "to": 1084, "label": "INSTANCE with matching:\nT123 -> T106\nX194 -> X12" }, { "from": 1105, "to": 1106, "label": "CASE" }, { "from": 1106, "to": 1107, "label": "BACKTRACK\nfor clause: app([], X, X)because of non-unification" }, { "from": 1107, "to": 1108, "label": "BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification" }, { "from": 1121, "to": 1123, "label": "EVAL with clause\npart(X388, .(X389, X390), X391, .(X389, X392)) :- part(X388, X390, X391, X392).\nand substitutionT7 -> T256,\nX388 -> T256,\nX389 -> T258,\nX390 -> T257,\nT8 -> .(T258, T257),\nX9 -> X393,\nX391 -> X393,\nX392 -> X394,\nX10 -> .(T258, X394),\nT253 -> T256,\nT255 -> T257,\nT254 -> T258" }, { "from": 1121, "to": 1124, "label": "EVAL-BACKTRACK" }, { "from": 1122, "to": 1134, "label": "EVAL with clause\npart(X418, [], [], []).\nand substitutionT7 -> T289,\nX418 -> T289,\nT8 -> [],\nX9 -> [],\nX10 -> [],\nT288 -> T289" }, { "from": 1122, "to": 1135, "label": "EVAL-BACKTRACK" }, { "from": 1123, "to": 1125, "label": "SPLIT 1" }, { "from": 1123, "to": 1126, "label": "SPLIT 2\nreplacements:X393 -> T262,\nX394 -> T263,\nT258 -> T264,\nT256 -> T265" }, { "from": 1125, "to": 936, "label": "INSTANCE with matching:\nT118 -> T256\nT119 -> T257\nX192 -> X393\nX193 -> X394" }, { "from": 1126, "to": 1127, "label": "SPLIT 1" }, { "from": 1126, "to": 1128, "label": "SPLIT 2\nreplacements:X11 -> T269,\nT264 -> T270,\nT263 -> T271,\nT265 -> T272" }, { "from": 1127, "to": 1084, "label": "INSTANCE with matching:\nT123 -> T262\nX194 -> X11" }, { "from": 1128, "to": 1129, "label": "SPLIT 1" }, { "from": 1128, "to": 1130, "label": "SPLIT 2\nreplacements:X12 -> T273,\nT269 -> T274,\nT272 -> T275" }, { "from": 1129, "to": 899, "label": "INSTANCE with matching:\nT54 -> T270\nT52 -> T271\nX11 -> X12" }, { "from": 1130, "to": 1131, "label": "CASE" }, { "from": 1131, "to": 1132, "label": "BACKTRACK\nfor clause: app([], X, X)because of non-unification" }, { "from": 1132, "to": 1133, "label": "BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification" }, { "from": 1134, "to": 1136, "label": "SPLIT 1" }, { "from": 1134, "to": 1137, "label": "SPLIT 2\nnew knowledge:\nT292 is ground\nreplacements:X11 -> T292" }, { "from": 1136, "to": 1084, "label": "INSTANCE with matching:\nT123 -> []\nX194 -> X11" }, { "from": 1137, "to": 1138, "label": "SPLIT 1" }, { "from": 1137, "to": 1139, "label": "SPLIT 2\nnew knowledge:\nT295 is ground\nreplacements:X12 -> T295" }, { "from": 1138, "to": 1084, "label": "INSTANCE with matching:\nT123 -> []\nX194 -> X12" }, { "from": 1139, "to": 1130, "label": "INSTANCE with matching:\nT274 -> T292\nT275 -> T289\nT273 -> T295" }, { "from": 1140, "to": 1142, "label": "CASE" }, { "from": 1142, "to": 1143, "label": "PARALLEL" }, { "from": 1142, "to": 1144, "label": "PARALLEL" }, { "from": 1143, "to": 1145, "label": "EVAL with clause\npart(X467, .(X468, X469), .(X468, X470), X471) :- ','(less(X467, X468), part(X467, X469, X470, X471)).\nand substitutionT306 -> T323,\nX467 -> T323,\nX468 -> T324,\nX469 -> T325,\nT307 -> .(T324, T325),\nX470 -> X472,\nX435 -> .(T324, X472),\nX436 -> X473,\nX471 -> X473,\nT320 -> T323,\nT321 -> T324,\nT322 -> T325" }, { "from": 1143, "to": 1146, "label": "EVAL-BACKTRACK" }, { "from": 1144, "to": 1171, "label": "PARALLEL" }, { "from": 1144, "to": 1172, "label": "PARALLEL" }, { "from": 1145, "to": 1147, "label": "SPLIT 1" }, { "from": 1145, "to": 1148, "label": "SPLIT 2\nnew knowledge:\nT328 is ground\nreplacements:T323 -> T328,\nT325 -> T329,\nT324 -> T330,\nT1 -> T331" }, { "from": 1147, "to": 224, "label": "INSTANCE with matching:\nT24 -> T323\nT25 -> T324" }, { "from": 1148, "to": 1149, "label": "SPLIT 1" }, { "from": 1148, "to": 1150, "label": "SPLIT 2\nnew knowledge:\nT328 is ground\nreplacements:X472 -> T337,\nX473 -> T338,\nT330 -> T339,\nT331 -> T340" }, { "from": 1149, "to": 418, "label": "INSTANCE with matching:\nT29 -> T328\nT30 -> T329\nX46 -> X472\nX47 -> X473" }, { "from": 1150, "to": 1151, "label": "SPLIT 1" }, { "from": 1150, "to": 1152, "label": "SPLIT 2\nreplacements:X437 -> T344,\nT338 -> T345,\nT340 -> T346" }, { "from": 1151, "to": 899, "label": "INSTANCE with matching:\nT54 -> T339\nT52 -> T337\nX11 -> X437" }, { "from": 1152, "to": 1153, "label": "SPLIT 1" }, { "from": 1152, "to": 1154, "label": "SPLIT 2\nreplacements:X438 -> T347,\nT344 -> T348,\nT346 -> T349" }, { "from": 1153, "to": 1084, "label": "INSTANCE with matching:\nT123 -> T345\nX194 -> X438" }, { "from": 1154, "to": 1155, "label": "CASE" }, { "from": 1155, "to": 1156, "label": "PARALLEL" }, { "from": 1155, "to": 1157, "label": "PARALLEL" }, { "from": 1156, "to": 1158, "label": "EVAL with clause\napp([], X498, X498).\nand substitutionT348 -> [],\nT328 -> T362,\nT347 -> T363,\nX498 -> .(T362, T363),\nT305 -> .(T362, T363)" }, { "from": 1156, "to": 1159, "label": "EVAL-BACKTRACK" }, { "from": 1157, "to": 1161, "label": "EVAL with clause\napp(.(X507, X508), X509, .(X507, X510)) :- app(X508, X509, X510).\nand substitutionX507 -> T374,\nX508 -> T379,\nT348 -> .(T374, T379),\nT328 -> T376,\nT347 -> T380,\nX509 -> .(T376, T380),\nX510 -> T378,\nT305 -> .(T374, T378),\nT375 -> T379,\nT377 -> T380" }, { "from": 1157, "to": 1162, "label": "EVAL-BACKTRACK" }, { "from": 1158, "to": 1160, "label": "SUCCESS" }, { "from": 1161, "to": 1163, "label": "CASE" }, { "from": 1163, "to": 1164, "label": "PARALLEL" }, { "from": 1163, "to": 1165, "label": "PARALLEL" }, { "from": 1164, "to": 1166, "label": "EVAL with clause\napp([], X517, X517).\nand substitutionT379 -> [],\nT376 -> T393,\nT380 -> T394,\nX517 -> .(T393, T394),\nT378 -> .(T393, T394)" }, { "from": 1164, "to": 1167, "label": "EVAL-BACKTRACK" }, { "from": 1165, "to": 1169, "label": "EVAL with clause\napp(.(X526, X527), X528, .(X526, X529)) :- app(X527, X528, X529).\nand substitutionX526 -> T405,\nX527 -> T410,\nT379 -> .(T405, T410),\nT376 -> T407,\nT380 -> T411,\nX528 -> .(T407, T411),\nX529 -> T409,\nT378 -> .(T405, T409),\nT406 -> T410,\nT408 -> T411" }, { "from": 1165, "to": 1170, "label": "EVAL-BACKTRACK" }, { "from": 1166, "to": 1168, "label": "SUCCESS" }, { "from": 1169, "to": 1161, "label": "INSTANCE with matching:\nT379 -> T410\nT376 -> T407\nT380 -> T411\nT378 -> T409" }, { "from": 1171, "to": 1173, "label": "EVAL with clause\npart(X560, .(X561, X562), X563, .(X561, X564)) :- part(X560, X562, X563, X564).\nand substitutionT306 -> T431,\nX560 -> T431,\nX561 -> T433,\nX562 -> T432,\nT307 -> .(T433, T432),\nX435 -> X565,\nX563 -> X565,\nX564 -> X566,\nX436 -> .(T433, X566),\nT428 -> T431,\nT430 -> T432,\nT429 -> T433" }, { "from": 1171, "to": 1174, "label": "EVAL-BACKTRACK" }, { "from": 1172, "to": 1197, "label": "EVAL with clause\npart(X623, [], [], []).\nand substitutionT306 -> T525,\nX623 -> T525,\nT307 -> [],\nX435 -> [],\nX436 -> [],\nT524 -> T525" }, { "from": 1172, "to": 1198, "label": "EVAL-BACKTRACK" }, { "from": 1173, "to": 1175, "label": "SPLIT 1" }, { "from": 1173, "to": 1176, "label": "SPLIT 2\nreplacements:X565 -> T437,\nX566 -> T438,\nT433 -> T439,\nT431 -> T440,\nT1 -> T441" }, { "from": 1175, "to": 936, "label": "INSTANCE with matching:\nT118 -> T431\nT119 -> T432\nX192 -> X565\nX193 -> X566" }, { "from": 1176, "to": 1177, "label": "SPLIT 1" }, { "from": 1176, "to": 1178, "label": "SPLIT 2\nreplacements:X437 -> T445,\nT439 -> T446,\nT438 -> T447,\nT440 -> T448,\nT441 -> T449" }, { "from": 1177, "to": 1084, "label": "INSTANCE with matching:\nT123 -> T437\nX194 -> X437" }, { "from": 1178, "to": 1179, "label": "SPLIT 1" }, { "from": 1178, "to": 1180, "label": "SPLIT 2\nreplacements:X438 -> T450,\nT445 -> T451,\nT448 -> T452,\nT449 -> T453" }, { "from": 1179, "to": 899, "label": "INSTANCE with matching:\nT54 -> T446\nT52 -> T447\nX11 -> X438" }, { "from": 1180, "to": 1181, "label": "CASE" }, { "from": 1181, "to": 1182, "label": "PARALLEL" }, { "from": 1181, "to": 1183, "label": "PARALLEL" }, { "from": 1182, "to": 1184, "label": "EVAL with clause\napp([], X587, X587).\nand substitutionT451 -> [],\nT452 -> T466,\nT450 -> T467,\nX587 -> .(T466, T467),\nT305 -> .(T466, T467)" }, { "from": 1182, "to": 1185, "label": "EVAL-BACKTRACK" }, { "from": 1183, "to": 1187, "label": "EVAL with clause\napp(.(X596, X597), X598, .(X596, X599)) :- app(X597, X598, X599).\nand substitutionX596 -> T478,\nX597 -> T483,\nT451 -> .(T478, T483),\nT452 -> T484,\nT450 -> T485,\nX598 -> .(T484, T485),\nX599 -> T482,\nT305 -> .(T478, T482),\nT479 -> T483,\nT480 -> T484,\nT481 -> T485" }, { "from": 1183, "to": 1188, "label": "EVAL-BACKTRACK" }, { "from": 1184, "to": 1186, "label": "SUCCESS" }, { "from": 1187, "to": 1189, "label": "CASE" }, { "from": 1189, "to": 1190, "label": "PARALLEL" }, { "from": 1189, "to": 1191, "label": "PARALLEL" }, { "from": 1190, "to": 1192, "label": "EVAL with clause\napp([], X606, X606).\nand substitutionT483 -> [],\nT484 -> T498,\nT485 -> T499,\nX606 -> .(T498, T499),\nT482 -> .(T498, T499)" }, { "from": 1190, "to": 1193, "label": "EVAL-BACKTRACK" }, { "from": 1191, "to": 1195, "label": "EVAL with clause\napp(.(X615, X616), X617, .(X615, X618)) :- app(X616, X617, X618).\nand substitutionX615 -> T510,\nX616 -> T515,\nT483 -> .(T510, T515),\nT484 -> T516,\nT485 -> T517,\nX617 -> .(T516, T517),\nX618 -> T514,\nT482 -> .(T510, T514),\nT511 -> T515,\nT512 -> T516,\nT513 -> T517" }, { "from": 1191, "to": 1196, "label": "EVAL-BACKTRACK" }, { "from": 1192, "to": 1194, "label": "SUCCESS" }, { "from": 1195, "to": 1187, "label": "INSTANCE with matching:\nT483 -> T515\nT484 -> T516\nT485 -> T517\nT482 -> T514" }, { "from": 1197, "to": 1199, "label": "SPLIT 1" }, { "from": 1197, "to": 1200, "label": "SPLIT 2\nnew knowledge:\nT530 is ground\nreplacements:X437 -> T530" }, { "from": 1199, "to": 1084, "label": "INSTANCE with matching:\nT123 -> []\nX194 -> X437" }, { "from": 1200, "to": 1201, "label": "SPLIT 1" }, { "from": 1200, "to": 1202, "label": "SPLIT 2\nnew knowledge:\nT535 is ground\nreplacements:X438 -> T535" }, { "from": 1201, "to": 1084, "label": "INSTANCE with matching:\nT123 -> []\nX194 -> X438" }, { "from": 1202, "to": 1180, "label": "INSTANCE with matching:\nT451 -> T530\nT452 -> T525\nT450 -> T535\nT453 -> T1" } ], "type": "Graph" } } ---------------------------------------- (145) Obligation: Triples: lessA(s(X1), s(X2)) :- lessA(X1, X2). partB(X1, .(X2, X3), .(X2, X4), X5) :- lessA(X1, X2). partB(X1, .(X2, X3), .(X2, X4), X5) :- ','(lesscA(X1, X2), partB(X1, X3, X4, X5)). partB(X1, .(X2, X3), X4, .(X2, X5)) :- partB(X1, X3, X4, X5). partC(X1, .(X2, X3), .(X2, X4), X5) :- lessA(X1, X2). partC(X1, .(X2, X3), .(X2, X4), X5) :- ','(lesscA(X1, X2), partB(X1, X3, X4, X5)). partC(X1, .(X2, X3), X4, .(X2, X5)) :- partC(X1, X3, X4, X5). pD(X1, X2, X3, X4, X5, X6, X7) :- partC(X1, X2, X3, X4). pD(X1, X2, X3, X4, X5, X6, X7) :- ','(partcC(X1, X2, X3, X4), qsF(X3, X5)). pD(X1, X2, X3, X4, X5, X6, X7) :- ','(partcC(X1, X2, X3, X4), ','(qscF(X3, X5), qsF(X4, X6))). pD(X1, X2, X3, X4, X5, X6, X7) :- ','(partcC(X1, X2, X3, X4), ','(qscF(X3, X5), ','(qscF(X4, X6), appE(X5, X1, X6, X7)))). qsF(.(X1, X2), X3) :- pD(X1, X2, X4, X5, X6, X7, X3). appE(.(X1, X2), X3, X4, .(X1, X5)) :- appE(X2, X3, X4, X5). qsG(X1, X2, X3) :- pD(X1, X2, X4, X5, X6, X7, X3). appH(.(X1, X2), X3, X4, .(X1, X5)) :- appH(X2, X3, X4, X5). appI(.(X1, X2), X3, X4, .(X1, X5)) :- appI(X2, X3, X4, X5). appJ(.(X1, X2), X3, X4, .(X1, X5)) :- appI(X2, X3, X4, X5). qsK(.(X1, .(X2, X3)), []) :- lessA(X1, X2). qsK(.(X1, .(X2, X3)), []) :- ','(lesscA(X1, X2), partB(X1, X3, X4, X5)). qsK(.(X1, .(X2, X3)), []) :- ','(lesscA(X1, X2), ','(partcB(X1, X3, X4, X5), qsG(X2, X4, X6))). qsK(.(X1, .(X2, X3)), []) :- ','(lesscA(X1, X2), ','(partcB(X1, X3, X4, X5), ','(qscG(X2, X4, X6), qsF(X5, X7)))). qsK(.(X1, .(X2, X3)), []) :- partC(X1, X3, X4, X5). qsK(.(X1, .(X2, X3)), []) :- ','(partcC(X1, X3, X4, X5), qsF(X4, X6)). qsK(.(X1, .(X2, X3)), []) :- ','(partcC(X1, X3, X4, X5), ','(qscF(X4, X6), qsG(X2, X5, X7))). qsK(.(X1, .(X2, X3)), []) :- ','(partcC(X1, X3, X4, X5), ','(qscF(X4, X6), ','(qscG(X2, X5, X7), appL(X6, X1, X7)))). qsK(.(X1, []), []) :- qsF([], X2). qsK(.(X1, []), []) :- ','(qscF([], X2), qsF([], X3)). qsK(.(X1, []), []) :- ','(qscF([], X2), ','(qscF([], X3), appL(X2, X1, X3))). qsK(.(X1, .(X2, X3)), X4) :- lessA(X1, X2). qsK(.(X1, .(X2, X3)), X4) :- ','(lesscA(X1, X2), partB(X1, X3, X5, X6)). qsK(.(X1, .(X2, X3)), X4) :- ','(lesscA(X1, X2), ','(partcB(X1, X3, X5, X6), qsG(X2, X5, X7))). qsK(.(X1, .(X2, X3)), X4) :- ','(lesscA(X1, X2), ','(partcB(X1, X3, X5, X6), ','(qscG(X2, X5, X7), qsF(X6, X8)))). qsK(.(X1, .(X2, X3)), .(X4, X5)) :- ','(lesscA(X1, X2), ','(partcB(X1, X3, X6, X7), ','(qscG(X2, X6, .(X4, X8)), ','(qscF(X7, X9), appH(X8, X1, X9, X5))))). qsK(.(X1, .(X2, X3)), X4) :- partC(X1, X3, X5, X6). qsK(.(X1, .(X2, X3)), X4) :- ','(partcC(X1, X3, X5, X6), qsF(X5, X7)). qsK(.(X1, .(X2, X3)), X4) :- ','(partcC(X1, X3, X5, X6), ','(qscF(X5, X7), qsG(X2, X6, X8))). qsK(.(X1, .(X2, X3)), X4) :- ','(partcC(X1, X3, X5, X6), ','(qscF(X5, X7), ','(qscG(X2, X6, X8), appJ(X7, X1, X8, X4)))). qsK(.(X1, []), X2) :- qsF([], X3). qsK(.(X1, []), X2) :- ','(qscF([], X3), qsF([], X4)). qsK(.(X1, []), X2) :- ','(qscF([], X3), ','(qscF([], X4), appJ(X3, X1, X4, X2))). Clauses: lesscA(0, s(X1)). lesscA(s(X1), s(X2)) :- lesscA(X1, X2). partcB(X1, .(X2, X3), .(X2, X4), X5) :- ','(lesscA(X1, X2), partcB(X1, X3, X4, X5)). partcB(X1, .(X2, X3), X4, .(X2, X5)) :- partcB(X1, X3, X4, X5). partcB(X1, [], [], []). partcC(X1, .(X2, X3), .(X2, X4), X5) :- ','(lesscA(X1, X2), partcB(X1, X3, X4, X5)). partcC(X1, .(X2, X3), X4, .(X2, X5)) :- partcC(X1, X3, X4, X5). partcC(X1, [], [], []). qcD(X1, X2, X3, X4, X5, X6, X7) :- ','(partcC(X1, X2, X3, X4), ','(qscF(X3, X5), ','(qscF(X4, X6), appcE(X5, X1, X6, X7)))). qscF([], []). qscF(.(X1, X2), X3) :- qcD(X1, X2, X4, X5, X6, X7, X3). appcE([], X1, X2, .(X1, X2)). appcE(.(X1, X2), X3, X4, .(X1, X5)) :- appcE(X2, X3, X4, X5). qscG(X1, X2, X3) :- qcD(X1, X2, X4, X5, X6, X7, X3). appcH([], X1, X2, .(X1, X2)). appcH(.(X1, X2), X3, X4, .(X1, X5)) :- appcH(X2, X3, X4, X5). appcI([], X1, X2, .(X1, X2)). appcI(.(X1, X2), X3, X4, .(X1, X5)) :- appcI(X2, X3, X4, X5). appcJ([], X1, X2, .(X1, X2)). appcJ(.(X1, X2), X3, X4, .(X1, X5)) :- appcI(X2, X3, X4, X5). Afs: qsK(x1, x2) = qsK(x2) ---------------------------------------- (146) UndefinedPredicateInTriplesTransformerProof (SOUND) Deleted triples and predicates having undefined goals [DT09]. ---------------------------------------- (147) Obligation: Triples: lessA(s(X1), s(X2)) :- lessA(X1, X2). partB(X1, .(X2, X3), .(X2, X4), X5) :- lessA(X1, X2). partB(X1, .(X2, X3), .(X2, X4), X5) :- ','(lesscA(X1, X2), partB(X1, X3, X4, X5)). partB(X1, .(X2, X3), X4, .(X2, X5)) :- partB(X1, X3, X4, X5). partC(X1, .(X2, X3), .(X2, X4), X5) :- lessA(X1, X2). partC(X1, .(X2, X3), .(X2, X4), X5) :- ','(lesscA(X1, X2), partB(X1, X3, X4, X5)). partC(X1, .(X2, X3), X4, .(X2, X5)) :- partC(X1, X3, X4, X5). pD(X1, X2, X3, X4, X5, X6, X7) :- partC(X1, X2, X3, X4). pD(X1, X2, X3, X4, X5, X6, X7) :- ','(partcC(X1, X2, X3, X4), qsF(X3, X5)). pD(X1, X2, X3, X4, X5, X6, X7) :- ','(partcC(X1, X2, X3, X4), ','(qscF(X3, X5), qsF(X4, X6))). pD(X1, X2, X3, X4, X5, X6, X7) :- ','(partcC(X1, X2, X3, X4), ','(qscF(X3, X5), ','(qscF(X4, X6), appE(X5, X1, X6, X7)))). qsF(.(X1, X2), X3) :- pD(X1, X2, X4, X5, X6, X7, X3). appE(.(X1, X2), X3, X4, .(X1, X5)) :- appE(X2, X3, X4, X5). qsG(X1, X2, X3) :- pD(X1, X2, X4, X5, X6, X7, X3). appH(.(X1, X2), X3, X4, .(X1, X5)) :- appH(X2, X3, X4, X5). appI(.(X1, X2), X3, X4, .(X1, X5)) :- appI(X2, X3, X4, X5). appJ(.(X1, X2), X3, X4, .(X1, X5)) :- appI(X2, X3, X4, X5). qsK(.(X1, .(X2, X3)), []) :- lessA(X1, X2). qsK(.(X1, .(X2, X3)), []) :- ','(lesscA(X1, X2), partB(X1, X3, X4, X5)). qsK(.(X1, .(X2, X3)), []) :- ','(lesscA(X1, X2), ','(partcB(X1, X3, X4, X5), qsG(X2, X4, X6))). qsK(.(X1, .(X2, X3)), []) :- ','(lesscA(X1, X2), ','(partcB(X1, X3, X4, X5), ','(qscG(X2, X4, X6), qsF(X5, X7)))). qsK(.(X1, .(X2, X3)), []) :- partC(X1, X3, X4, X5). qsK(.(X1, .(X2, X3)), []) :- ','(partcC(X1, X3, X4, X5), qsF(X4, X6)). qsK(.(X1, .(X2, X3)), []) :- ','(partcC(X1, X3, X4, X5), ','(qscF(X4, X6), qsG(X2, X5, X7))). qsK(.(X1, []), []) :- qsF([], X2). qsK(.(X1, []), []) :- ','(qscF([], X2), qsF([], X3)). qsK(.(X1, .(X2, X3)), X4) :- lessA(X1, X2). qsK(.(X1, .(X2, X3)), X4) :- ','(lesscA(X1, X2), partB(X1, X3, X5, X6)). qsK(.(X1, .(X2, X3)), X4) :- ','(lesscA(X1, X2), ','(partcB(X1, X3, X5, X6), qsG(X2, X5, X7))). qsK(.(X1, .(X2, X3)), X4) :- ','(lesscA(X1, X2), ','(partcB(X1, X3, X5, X6), ','(qscG(X2, X5, X7), qsF(X6, X8)))). qsK(.(X1, .(X2, X3)), .(X4, X5)) :- ','(lesscA(X1, X2), ','(partcB(X1, X3, X6, X7), ','(qscG(X2, X6, .(X4, X8)), ','(qscF(X7, X9), appH(X8, X1, X9, X5))))). qsK(.(X1, .(X2, X3)), X4) :- partC(X1, X3, X5, X6). qsK(.(X1, .(X2, X3)), X4) :- ','(partcC(X1, X3, X5, X6), qsF(X5, X7)). qsK(.(X1, .(X2, X3)), X4) :- ','(partcC(X1, X3, X5, X6), ','(qscF(X5, X7), qsG(X2, X6, X8))). qsK(.(X1, .(X2, X3)), X4) :- ','(partcC(X1, X3, X5, X6), ','(qscF(X5, X7), ','(qscG(X2, X6, X8), appJ(X7, X1, X8, X4)))). qsK(.(X1, []), X2) :- qsF([], X3). qsK(.(X1, []), X2) :- ','(qscF([], X3), qsF([], X4)). qsK(.(X1, []), X2) :- ','(qscF([], X3), ','(qscF([], X4), appJ(X3, X1, X4, X2))). Clauses: lesscA(0, s(X1)). lesscA(s(X1), s(X2)) :- lesscA(X1, X2). partcB(X1, .(X2, X3), .(X2, X4), X5) :- ','(lesscA(X1, X2), partcB(X1, X3, X4, X5)). partcB(X1, .(X2, X3), X4, .(X2, X5)) :- partcB(X1, X3, X4, X5). partcB(X1, [], [], []). partcC(X1, .(X2, X3), .(X2, X4), X5) :- ','(lesscA(X1, X2), partcB(X1, X3, X4, X5)). partcC(X1, .(X2, X3), X4, .(X2, X5)) :- partcC(X1, X3, X4, X5). partcC(X1, [], [], []). qcD(X1, X2, X3, X4, X5, X6, X7) :- ','(partcC(X1, X2, X3, X4), ','(qscF(X3, X5), ','(qscF(X4, X6), appcE(X5, X1, X6, X7)))). qscF([], []). qscF(.(X1, X2), X3) :- qcD(X1, X2, X4, X5, X6, X7, X3). appcE([], X1, X2, .(X1, X2)). appcE(.(X1, X2), X3, X4, .(X1, X5)) :- appcE(X2, X3, X4, X5). qscG(X1, X2, X3) :- qcD(X1, X2, X4, X5, X6, X7, X3). appcH([], X1, X2, .(X1, X2)). appcH(.(X1, X2), X3, X4, .(X1, X5)) :- appcH(X2, X3, X4, X5). appcI([], X1, X2, .(X1, X2)). appcI(.(X1, X2), X3, X4, .(X1, X5)) :- appcI(X2, X3, X4, X5). appcJ([], X1, X2, .(X1, X2)). appcJ(.(X1, X2), X3, X4, .(X1, X5)) :- appcI(X2, X3, X4, X5). Afs: qsK(x1, x2) = qsK(x2) ---------------------------------------- (148) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: qsK_in_2: (f,b) lessA_in_2: (f,f) (b,f) (b,b) lesscA_in_2: (f,f) (b,f) (b,b) partB_in_4: (b,f,f,f) (b,b,f,f) partcB_in_4: (b,f,f,f) (b,b,f,f) qsG_in_3: (f,f,f) pD_in_7: (f,f,f,f,f,f,f) (b,b,f,f,f,f,f) partC_in_4: (f,f,f,f) (b,b,f,f) partcC_in_4: (f,f,f,f) (b,b,f,f) qsF_in_2: (f,f) (b,f) qscF_in_2: (f,f) (b,f) qcD_in_7: (f,f,f,f,f,f,f) (b,b,f,f,f,f,f) appcE_in_4: (f,f,f,f) (b,b,b,f) appE_in_4: (f,f,f,f) (b,b,b,f) qscG_in_3: (f,f,f) appH_in_4: (f,b,f,b) appJ_in_4: (f,f,f,b) (b,f,b,b) appI_in_4: (f,f,f,b) (b,f,b,b) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: QSK_IN_AG(.(X1, .(X2, X3)), []) -> U23_AG(X1, X2, X3, lessA_in_aa(X1, X2)) QSK_IN_AG(.(X1, .(X2, X3)), []) -> LESSA_IN_AA(X1, X2) LESSA_IN_AA(s(X1), s(X2)) -> U1_AA(X1, X2, lessA_in_aa(X1, X2)) LESSA_IN_AA(s(X1), s(X2)) -> LESSA_IN_AA(X1, X2) QSK_IN_AG(.(X1, .(X2, X3)), []) -> U24_AG(X1, X2, X3, lesscA_in_aa(X1, X2)) U24_AG(X1, X2, X3, lesscA_out_aa(X1, X2)) -> U25_AG(X1, X2, X3, partB_in_gaaa(X1, X3, X4, X5)) U24_AG(X1, X2, X3, lesscA_out_aa(X1, X2)) -> PARTB_IN_GAAA(X1, X3, X4, X5) PARTB_IN_GAAA(X1, .(X2, X3), .(X2, X4), X5) -> U2_GAAA(X1, X2, X3, X4, X5, lessA_in_ga(X1, X2)) PARTB_IN_GAAA(X1, .(X2, X3), .(X2, X4), X5) -> LESSA_IN_GA(X1, X2) LESSA_IN_GA(s(X1), s(X2)) -> U1_GA(X1, X2, lessA_in_ga(X1, X2)) LESSA_IN_GA(s(X1), s(X2)) -> LESSA_IN_GA(X1, X2) PARTB_IN_GAAA(X1, .(X2, X3), .(X2, X4), X5) -> U3_GAAA(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) U3_GAAA(X1, X2, X3, X4, X5, lesscA_out_ga(X1, X2)) -> U4_GAAA(X1, X2, X3, X4, X5, partB_in_gaaa(X1, X3, X4, X5)) U3_GAAA(X1, X2, X3, X4, X5, lesscA_out_ga(X1, X2)) -> PARTB_IN_GAAA(X1, X3, X4, X5) PARTB_IN_GAAA(X1, .(X2, X3), X4, .(X2, X5)) -> U5_GAAA(X1, X2, X3, X4, X5, partB_in_gaaa(X1, X3, X4, X5)) PARTB_IN_GAAA(X1, .(X2, X3), X4, .(X2, X5)) -> PARTB_IN_GAAA(X1, X3, X4, X5) U24_AG(X1, X2, X3, lesscA_out_aa(X1, X2)) -> U26_AG(X1, X2, X3, partcB_in_gaaa(X1, X3, X4, X5)) U26_AG(X1, X2, X3, partcB_out_gaaa(X1, X3, X4, X5)) -> U27_AG(X1, X2, X3, qsG_in_aaa(X2, X4, X6)) U26_AG(X1, X2, X3, partcB_out_gaaa(X1, X3, X4, X5)) -> QSG_IN_AAA(X2, X4, X6) QSG_IN_AAA(X1, X2, X3) -> U19_AAA(X1, X2, X3, pD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) QSG_IN_AAA(X1, X2, X3) -> PD_IN_AAAAAAA(X1, X2, X4, X5, X6, X7, X3) PD_IN_AAAAAAA(X1, X2, X3, X4, X5, X6, X7) -> U10_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, partC_in_aaaa(X1, X2, X3, X4)) PD_IN_AAAAAAA(X1, X2, X3, X4, X5, X6, X7) -> PARTC_IN_AAAA(X1, X2, X3, X4) PARTC_IN_AAAA(X1, .(X2, X3), .(X2, X4), X5) -> U6_AAAA(X1, X2, X3, X4, X5, lessA_in_aa(X1, X2)) PARTC_IN_AAAA(X1, .(X2, X3), .(X2, X4), X5) -> LESSA_IN_AA(X1, X2) PARTC_IN_AAAA(X1, .(X2, X3), .(X2, X4), X5) -> U7_AAAA(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) U7_AAAA(X1, X2, X3, X4, X5, lesscA_out_aa(X1, X2)) -> U8_AAAA(X1, X2, X3, X4, X5, partB_in_gaaa(X1, X3, X4, X5)) U7_AAAA(X1, X2, X3, X4, X5, lesscA_out_aa(X1, X2)) -> PARTB_IN_GAAA(X1, X3, X4, X5) PARTC_IN_AAAA(X1, .(X2, X3), X4, .(X2, X5)) -> U9_AAAA(X1, X2, X3, X4, X5, partC_in_aaaa(X1, X3, X4, X5)) PARTC_IN_AAAA(X1, .(X2, X3), X4, .(X2, X5)) -> PARTC_IN_AAAA(X1, X3, X4, X5) PD_IN_AAAAAAA(X1, X2, X3, X4, X5, X6, X7) -> U11_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, partcC_in_aaaa(X1, X2, X3, X4)) U11_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, partcC_out_aaaa(X1, X2, X3, X4)) -> U12_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, qsF_in_aa(X3, X5)) U11_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, partcC_out_aaaa(X1, X2, X3, X4)) -> QSF_IN_AA(X3, X5) QSF_IN_AA(.(X1, X2), X3) -> U17_AA(X1, X2, X3, pD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) QSF_IN_AA(.(X1, X2), X3) -> PD_IN_AAAAAAA(X1, X2, X4, X5, X6, X7, X3) U11_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, partcC_out_aaaa(X1, X2, X3, X4)) -> U13_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X3, X5)) U13_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X3, X5)) -> U14_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, qsF_in_aa(X4, X6)) U13_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X3, X5)) -> QSF_IN_AA(X4, X6) U13_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X3, X5)) -> U15_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X4, X6)) U15_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X4, X6)) -> U16_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, appE_in_aaaa(X5, X1, X6, X7)) U15_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X4, X6)) -> APPE_IN_AAAA(X5, X1, X6, X7) APPE_IN_AAAA(.(X1, X2), X3, X4, .(X1, X5)) -> U18_AAAA(X1, X2, X3, X4, X5, appE_in_aaaa(X2, X3, X4, X5)) APPE_IN_AAAA(.(X1, X2), X3, X4, .(X1, X5)) -> APPE_IN_AAAA(X2, X3, X4, X5) U26_AG(X1, X2, X3, partcB_out_gaaa(X1, X3, X4, X5)) -> U28_AG(X1, X2, X3, X5, qscG_in_aaa(X2, X4, X6)) U28_AG(X1, X2, X3, X5, qscG_out_aaa(X2, X4, X6)) -> U29_AG(X1, X2, X3, qsF_in_aa(X5, X7)) U28_AG(X1, X2, X3, X5, qscG_out_aaa(X2, X4, X6)) -> QSF_IN_AA(X5, X7) QSK_IN_AG(.(X1, .(X2, X3)), []) -> U30_AG(X1, X2, X3, partC_in_aaaa(X1, X3, X4, X5)) QSK_IN_AG(.(X1, .(X2, X3)), []) -> PARTC_IN_AAAA(X1, X3, X4, X5) QSK_IN_AG(.(X1, .(X2, X3)), []) -> U31_AG(X1, X2, X3, partcC_in_aaaa(X1, X3, X4, X5)) U31_AG(X1, X2, X3, partcC_out_aaaa(X1, X3, X4, X5)) -> U32_AG(X1, X2, X3, qsF_in_aa(X4, X6)) U31_AG(X1, X2, X3, partcC_out_aaaa(X1, X3, X4, X5)) -> QSF_IN_AA(X4, X6) U31_AG(X1, X2, X3, partcC_out_aaaa(X1, X3, X4, X5)) -> U33_AG(X1, X2, X3, X5, qscF_in_aa(X4, X6)) U33_AG(X1, X2, X3, X5, qscF_out_aa(X4, X6)) -> U34_AG(X1, X2, X3, qsG_in_aaa(X2, X5, X7)) U33_AG(X1, X2, X3, X5, qscF_out_aa(X4, X6)) -> QSG_IN_AAA(X2, X5, X7) QSK_IN_AG(.(X1, []), []) -> U35_AG(X1, qsF_in_ga([], X2)) QSK_IN_AG(.(X1, []), []) -> QSF_IN_GA([], X2) QSF_IN_GA(.(X1, X2), X3) -> U17_GA(X1, X2, X3, pD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) QSF_IN_GA(.(X1, X2), X3) -> PD_IN_GGAAAAA(X1, X2, X4, X5, X6, X7, X3) PD_IN_GGAAAAA(X1, X2, X3, X4, X5, X6, X7) -> U10_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, partC_in_ggaa(X1, X2, X3, X4)) PD_IN_GGAAAAA(X1, X2, X3, X4, X5, X6, X7) -> PARTC_IN_GGAA(X1, X2, X3, X4) PARTC_IN_GGAA(X1, .(X2, X3), .(X2, X4), X5) -> U6_GGAA(X1, X2, X3, X4, X5, lessA_in_gg(X1, X2)) PARTC_IN_GGAA(X1, .(X2, X3), .(X2, X4), X5) -> LESSA_IN_GG(X1, X2) LESSA_IN_GG(s(X1), s(X2)) -> U1_GG(X1, X2, lessA_in_gg(X1, X2)) LESSA_IN_GG(s(X1), s(X2)) -> LESSA_IN_GG(X1, X2) PARTC_IN_GGAA(X1, .(X2, X3), .(X2, X4), X5) -> U7_GGAA(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) U7_GGAA(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> U8_GGAA(X1, X2, X3, X4, X5, partB_in_ggaa(X1, X3, X4, X5)) U7_GGAA(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> PARTB_IN_GGAA(X1, X3, X4, X5) PARTB_IN_GGAA(X1, .(X2, X3), .(X2, X4), X5) -> U2_GGAA(X1, X2, X3, X4, X5, lessA_in_gg(X1, X2)) PARTB_IN_GGAA(X1, .(X2, X3), .(X2, X4), X5) -> LESSA_IN_GG(X1, X2) PARTB_IN_GGAA(X1, .(X2, X3), .(X2, X4), X5) -> U3_GGAA(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) U3_GGAA(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> U4_GGAA(X1, X2, X3, X4, X5, partB_in_ggaa(X1, X3, X4, X5)) U3_GGAA(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> PARTB_IN_GGAA(X1, X3, X4, X5) PARTB_IN_GGAA(X1, .(X2, X3), X4, .(X2, X5)) -> U5_GGAA(X1, X2, X3, X4, X5, partB_in_ggaa(X1, X3, X4, X5)) PARTB_IN_GGAA(X1, .(X2, X3), X4, .(X2, X5)) -> PARTB_IN_GGAA(X1, X3, X4, X5) PARTC_IN_GGAA(X1, .(X2, X3), X4, .(X2, X5)) -> U9_GGAA(X1, X2, X3, X4, X5, partC_in_ggaa(X1, X3, X4, X5)) PARTC_IN_GGAA(X1, .(X2, X3), X4, .(X2, X5)) -> PARTC_IN_GGAA(X1, X3, X4, X5) PD_IN_GGAAAAA(X1, X2, X3, X4, X5, X6, X7) -> U11_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, partcC_in_ggaa(X1, X2, X3, X4)) U11_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, partcC_out_ggaa(X1, X2, X3, X4)) -> U12_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qsF_in_ga(X3, X5)) U11_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, partcC_out_ggaa(X1, X2, X3, X4)) -> QSF_IN_GA(X3, X5) U11_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, partcC_out_ggaa(X1, X2, X3, X4)) -> U13_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5)) U13_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U14_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qsF_in_ga(X4, X6)) U13_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> QSF_IN_GA(X4, X6) U13_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U15_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6)) U15_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U16_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, appE_in_ggga(X5, X1, X6, X7)) U15_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> APPE_IN_GGGA(X5, X1, X6, X7) APPE_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) -> U18_GGGA(X1, X2, X3, X4, X5, appE_in_ggga(X2, X3, X4, X5)) APPE_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) -> APPE_IN_GGGA(X2, X3, X4, X5) QSK_IN_AG(.(X1, []), []) -> U36_AG(X1, qscF_in_ga([], X2)) U36_AG(X1, qscF_out_ga([], X2)) -> U37_AG(X1, qsF_in_ga([], X3)) U36_AG(X1, qscF_out_ga([], X2)) -> QSF_IN_GA([], X3) QSK_IN_AG(.(X1, .(X2, X3)), X4) -> U38_AG(X1, X2, X3, X4, lessA_in_aa(X1, X2)) QSK_IN_AG(.(X1, .(X2, X3)), X4) -> LESSA_IN_AA(X1, X2) QSK_IN_AG(.(X1, .(X2, X3)), X4) -> U39_AG(X1, X2, X3, X4, lesscA_in_aa(X1, X2)) U39_AG(X1, X2, X3, X4, lesscA_out_aa(X1, X2)) -> U40_AG(X1, X2, X3, X4, partB_in_gaaa(X1, X3, X5, X6)) U39_AG(X1, X2, X3, X4, lesscA_out_aa(X1, X2)) -> PARTB_IN_GAAA(X1, X3, X5, X6) U39_AG(X1, X2, X3, X4, lesscA_out_aa(X1, X2)) -> U41_AG(X1, X2, X3, X4, partcB_in_gaaa(X1, X3, X5, X6)) U41_AG(X1, X2, X3, X4, partcB_out_gaaa(X1, X3, X5, X6)) -> U42_AG(X1, X2, X3, X4, qsG_in_aaa(X2, X5, X7)) U41_AG(X1, X2, X3, X4, partcB_out_gaaa(X1, X3, X5, X6)) -> QSG_IN_AAA(X2, X5, X7) U41_AG(X1, X2, X3, X4, partcB_out_gaaa(X1, X3, X5, X6)) -> U43_AG(X1, X2, X3, X4, X6, qscG_in_aaa(X2, X5, X7)) U43_AG(X1, X2, X3, X4, X6, qscG_out_aaa(X2, X5, X7)) -> U44_AG(X1, X2, X3, X4, qsF_in_aa(X6, X8)) U43_AG(X1, X2, X3, X4, X6, qscG_out_aaa(X2, X5, X7)) -> QSF_IN_AA(X6, X8) QSK_IN_AG(.(X1, .(X2, X3)), .(X4, X5)) -> U45_AG(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) U45_AG(X1, X2, X3, X4, X5, lesscA_out_aa(X1, X2)) -> U46_AG(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X6, X7)) U46_AG(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X6, X7)) -> U47_AG(X1, X2, X3, X4, X5, X7, qscG_in_aaa(X2, X6, .(X4, X8))) U47_AG(X1, X2, X3, X4, X5, X7, qscG_out_aaa(X2, X6, .(X4, X8))) -> U48_AG(X1, X2, X3, X4, X5, X8, qscF_in_aa(X7, X9)) U48_AG(X1, X2, X3, X4, X5, X8, qscF_out_aa(X7, X9)) -> U49_AG(X1, X2, X3, X4, X5, appH_in_agag(X8, X1, X9, X5)) U48_AG(X1, X2, X3, X4, X5, X8, qscF_out_aa(X7, X9)) -> APPH_IN_AGAG(X8, X1, X9, X5) APPH_IN_AGAG(.(X1, X2), X3, X4, .(X1, X5)) -> U20_AGAG(X1, X2, X3, X4, X5, appH_in_agag(X2, X3, X4, X5)) APPH_IN_AGAG(.(X1, X2), X3, X4, .(X1, X5)) -> APPH_IN_AGAG(X2, X3, X4, X5) QSK_IN_AG(.(X1, .(X2, X3)), X4) -> U50_AG(X1, X2, X3, X4, partC_in_aaaa(X1, X3, X5, X6)) QSK_IN_AG(.(X1, .(X2, X3)), X4) -> PARTC_IN_AAAA(X1, X3, X5, X6) QSK_IN_AG(.(X1, .(X2, X3)), X4) -> U51_AG(X1, X2, X3, X4, partcC_in_aaaa(X1, X3, X5, X6)) U51_AG(X1, X2, X3, X4, partcC_out_aaaa(X1, X3, X5, X6)) -> U52_AG(X1, X2, X3, X4, qsF_in_aa(X5, X7)) U51_AG(X1, X2, X3, X4, partcC_out_aaaa(X1, X3, X5, X6)) -> QSF_IN_AA(X5, X7) U51_AG(X1, X2, X3, X4, partcC_out_aaaa(X1, X3, X5, X6)) -> U53_AG(X1, X2, X3, X4, X6, qscF_in_aa(X5, X7)) U53_AG(X1, X2, X3, X4, X6, qscF_out_aa(X5, X7)) -> U54_AG(X1, X2, X3, X4, qsG_in_aaa(X2, X6, X8)) U53_AG(X1, X2, X3, X4, X6, qscF_out_aa(X5, X7)) -> QSG_IN_AAA(X2, X6, X8) U53_AG(X1, X2, X3, X4, X6, qscF_out_aa(X5, X7)) -> U55_AG(X1, X2, X3, X4, X7, qscG_in_aaa(X2, X6, X8)) U55_AG(X1, X2, X3, X4, X7, qscG_out_aaa(X2, X6, X8)) -> U56_AG(X1, X2, X3, X4, appJ_in_aaag(X7, X1, X8, X4)) U55_AG(X1, X2, X3, X4, X7, qscG_out_aaa(X2, X6, X8)) -> APPJ_IN_AAAG(X7, X1, X8, X4) APPJ_IN_AAAG(.(X1, X2), X3, X4, .(X1, X5)) -> U22_AAAG(X1, X2, X3, X4, X5, appI_in_aaag(X2, X3, X4, X5)) APPJ_IN_AAAG(.(X1, X2), X3, X4, .(X1, X5)) -> APPI_IN_AAAG(X2, X3, X4, X5) APPI_IN_AAAG(.(X1, X2), X3, X4, .(X1, X5)) -> U21_AAAG(X1, X2, X3, X4, X5, appI_in_aaag(X2, X3, X4, X5)) APPI_IN_AAAG(.(X1, X2), X3, X4, .(X1, X5)) -> APPI_IN_AAAG(X2, X3, X4, X5) QSK_IN_AG(.(X1, []), X2) -> U57_AG(X1, X2, qsF_in_ga([], X3)) QSK_IN_AG(.(X1, []), X2) -> QSF_IN_GA([], X3) QSK_IN_AG(.(X1, []), X2) -> U58_AG(X1, X2, qscF_in_ga([], X3)) U58_AG(X1, X2, qscF_out_ga([], X3)) -> U59_AG(X1, X2, qsF_in_ga([], X4)) U58_AG(X1, X2, qscF_out_ga([], X3)) -> QSF_IN_GA([], X4) U58_AG(X1, X2, qscF_out_ga([], X3)) -> U60_AG(X1, X2, X3, qscF_in_ga([], X4)) U60_AG(X1, X2, X3, qscF_out_ga([], X4)) -> U61_AG(X1, X2, appJ_in_gagg(X3, X1, X4, X2)) U60_AG(X1, X2, X3, qscF_out_ga([], X4)) -> APPJ_IN_GAGG(X3, X1, X4, X2) APPJ_IN_GAGG(.(X1, X2), X3, X4, .(X1, X5)) -> U22_GAGG(X1, X2, X3, X4, X5, appI_in_gagg(X2, X3, X4, X5)) APPJ_IN_GAGG(.(X1, X2), X3, X4, .(X1, X5)) -> APPI_IN_GAGG(X2, X3, X4, X5) APPI_IN_GAGG(.(X1, X2), X3, X4, .(X1, X5)) -> U21_GAGG(X1, X2, X3, X4, X5, appI_in_gagg(X2, X3, X4, X5)) APPI_IN_GAGG(.(X1, X2), X3, X4, .(X1, X5)) -> APPI_IN_GAGG(X2, X3, X4, X5) The TRS R consists of the following rules: lesscA_in_aa(0, s(X1)) -> lesscA_out_aa(0, s(X1)) lesscA_in_aa(s(X1), s(X2)) -> U63_aa(X1, X2, lesscA_in_aa(X1, X2)) U63_aa(X1, X2, lesscA_out_aa(X1, X2)) -> lesscA_out_aa(s(X1), s(X2)) lesscA_in_ga(0, s(X1)) -> lesscA_out_ga(0, s(X1)) lesscA_in_ga(s(X1), s(X2)) -> U63_ga(X1, X2, lesscA_in_ga(X1, X2)) U63_ga(X1, X2, lesscA_out_ga(X1, X2)) -> lesscA_out_ga(s(X1), s(X2)) partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_gaaa(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) U64_gaaa(X1, X2, X3, X4, X5, lesscA_out_ga(X1, X2)) -> U65_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], []) U66_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5)) U65_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_aaaa(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) U67_aaaa(X1, X2, X3, X4, X5, lesscA_out_aa(X1, X2)) -> U68_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) U68_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_aaaa(X1, X2, X3, X4, X5, partcC_in_aaaa(X1, X3, X4, X5)) partcC_in_aaaa(X1, [], [], []) -> partcC_out_aaaa(X1, [], [], []) U69_aaaa(X1, X2, X3, X4, X5, partcC_out_aaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), X4, .(X2, X5)) qscF_in_aa([], []) -> qscF_out_aa([], []) qscF_in_aa(.(X1, X2), X3) -> U74_aa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) qcD_in_aaaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U70_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_in_aaaa(X1, X2, X3, X4)) U70_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_out_aaaa(X1, X2, X3, X4)) -> U71_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X3, X5)) U71_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X3, X5)) -> U72_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X4, X6)) U72_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X4, X6)) -> U73_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_aaaa(X5, X1, X6, X7)) appcE_in_aaaa([], X1, X2, .(X1, X2)) -> appcE_out_aaaa([], X1, X2, .(X1, X2)) appcE_in_aaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U75_aaaa(X1, X2, X3, X4, X5, appcE_in_aaaa(X2, X3, X4, X5)) U75_aaaa(X1, X2, X3, X4, X5, appcE_out_aaaa(X2, X3, X4, X5)) -> appcE_out_aaaa(.(X1, X2), X3, X4, .(X1, X5)) U73_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_aaaa(X5, X1, X6, X7)) -> qcD_out_aaaaaaa(X1, X2, X3, X4, X5, X6, X7) U74_aa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_aa(.(X1, X2), X3) qscG_in_aaa(X1, X2, X3) -> U76_aaa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) U76_aaa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscG_out_aaa(X1, X2, X3) lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) partcC_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) U67_ggaa(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> U68_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) U64_ggaa(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> U65_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], []) U66_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) U65_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) U68_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_ggaa(X1, X2, X3, X4, X5, partcC_in_ggaa(X1, X3, X4, X5)) partcC_in_ggaa(X1, [], [], []) -> partcC_out_ggaa(X1, [], [], []) U69_ggaa(X1, X2, X3, X4, X5, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) qscF_in_ga([], []) -> qscF_out_ga([], []) qscF_in_ga(.(X1, X2), X3) -> U74_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) qcD_in_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U70_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_in_ggaa(X1, X2, X3, X4)) U70_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_out_ggaa(X1, X2, X3, X4)) -> U71_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5)) U71_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U72_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6)) U72_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U73_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_ggga(X5, X1, X6, X7)) appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2)) appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U75_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5)) U75_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5)) U73_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) U74_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3) The argument filtering Pi contains the following mapping: [] = [] lessA_in_aa(x1, x2) = lessA_in_aa lesscA_in_aa(x1, x2) = lesscA_in_aa lesscA_out_aa(x1, x2) = lesscA_out_aa(x1) U63_aa(x1, x2, x3) = U63_aa(x3) partB_in_gaaa(x1, x2, x3, x4) = partB_in_gaaa(x1) lessA_in_ga(x1, x2) = lessA_in_ga(x1) s(x1) = s(x1) lesscA_in_ga(x1, x2) = lesscA_in_ga(x1) 0 = 0 lesscA_out_ga(x1, x2) = lesscA_out_ga(x1) U63_ga(x1, x2, x3) = U63_ga(x1, x3) partcB_in_gaaa(x1, x2, x3, x4) = partcB_in_gaaa(x1) U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6) U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x6) U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6) partcB_out_gaaa(x1, x2, x3, x4) = partcB_out_gaaa(x1) qsG_in_aaa(x1, x2, x3) = qsG_in_aaa pD_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = pD_in_aaaaaaa partC_in_aaaa(x1, x2, x3, x4) = partC_in_aaaa partcC_in_aaaa(x1, x2, x3, x4) = partcC_in_aaaa U67_aaaa(x1, x2, x3, x4, x5, x6) = U67_aaaa(x6) U68_aaaa(x1, x2, x3, x4, x5, x6) = U68_aaaa(x6) partcC_out_aaaa(x1, x2, x3, x4) = partcC_out_aaaa U69_aaaa(x1, x2, x3, x4, x5, x6) = U69_aaaa(x6) qsF_in_aa(x1, x2) = qsF_in_aa qscF_in_aa(x1, x2) = qscF_in_aa qscF_out_aa(x1, x2) = qscF_out_aa U74_aa(x1, x2, x3, x4) = U74_aa(x4) qcD_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_aaaaaaa U70_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_aaaaaaa(x8) U71_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_aaaaaaa(x8) U72_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_aaaaaaa(x8) U73_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_aaaaaaa(x8) .(x1, x2) = .(x1, x2) qcD_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_aaaaaaa appcE_in_aaaa(x1, x2, x3, x4) = appcE_in_aaaa appcE_out_aaaa(x1, x2, x3, x4) = appcE_out_aaaa U75_aaaa(x1, x2, x3, x4, x5, x6) = U75_aaaa(x6) appE_in_aaaa(x1, x2, x3, x4) = appE_in_aaaa qscG_in_aaa(x1, x2, x3) = qscG_in_aaa U76_aaa(x1, x2, x3, x4) = U76_aaa(x4) qscG_out_aaa(x1, x2, x3) = qscG_out_aaa qsF_in_ga(x1, x2) = qsF_in_ga(x1) pD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = pD_in_ggaaaaa(x1, x2) partC_in_ggaa(x1, x2, x3, x4) = partC_in_ggaa(x1, x2) lessA_in_gg(x1, x2) = lessA_in_gg(x1, x2) lesscA_in_gg(x1, x2) = lesscA_in_gg(x1, x2) lesscA_out_gg(x1, x2) = lesscA_out_gg(x1, x2) U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3) partB_in_ggaa(x1, x2, x3, x4) = partB_in_ggaa(x1, x2) partcC_in_ggaa(x1, x2, x3, x4) = partcC_in_ggaa(x1, x2) U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6) U68_ggaa(x1, x2, x3, x4, x5, x6) = U68_ggaa(x1, x2, x3, x6) partcB_in_ggaa(x1, x2, x3, x4) = partcB_in_ggaa(x1, x2) U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6) U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6) U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6) partcB_out_ggaa(x1, x2, x3, x4) = partcB_out_ggaa(x1, x2, x3, x4) partcC_out_ggaa(x1, x2, x3, x4) = partcC_out_ggaa(x1, x2, x3, x4) U69_ggaa(x1, x2, x3, x4, x5, x6) = U69_ggaa(x1, x2, x3, x6) qscF_in_ga(x1, x2) = qscF_in_ga(x1) qscF_out_ga(x1, x2) = qscF_out_ga(x1, x2) U74_ga(x1, x2, x3, x4) = U74_ga(x1, x2, x4) qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_ggaaaaa(x1, x2) U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x8) U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x8) U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x8) U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x8) appcE_in_ggga(x1, x2, x3, x4) = appcE_in_ggga(x1, x2, x3) appcE_out_ggga(x1, x2, x3, x4) = appcE_out_ggga(x1, x2, x3, x4) U75_ggga(x1, x2, x3, x4, x5, x6) = U75_ggga(x1, x2, x3, x4, x6) qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) appE_in_ggga(x1, x2, x3, x4) = appE_in_ggga(x1, x2, x3) appH_in_agag(x1, x2, x3, x4) = appH_in_agag(x2, x4) appJ_in_aaag(x1, x2, x3, x4) = appJ_in_aaag(x4) appI_in_aaag(x1, x2, x3, x4) = appI_in_aaag(x4) appJ_in_gagg(x1, x2, x3, x4) = appJ_in_gagg(x1, x3, x4) appI_in_gagg(x1, x2, x3, x4) = appI_in_gagg(x1, x3, x4) QSK_IN_AG(x1, x2) = QSK_IN_AG(x2) U23_AG(x1, x2, x3, x4) = U23_AG(x4) LESSA_IN_AA(x1, x2) = LESSA_IN_AA U1_AA(x1, x2, x3) = U1_AA(x3) U24_AG(x1, x2, x3, x4) = U24_AG(x4) U25_AG(x1, x2, x3, x4) = U25_AG(x4) PARTB_IN_GAAA(x1, x2, x3, x4) = PARTB_IN_GAAA(x1) U2_GAAA(x1, x2, x3, x4, x5, x6) = U2_GAAA(x1, x6) LESSA_IN_GA(x1, x2) = LESSA_IN_GA(x1) U1_GA(x1, x2, x3) = U1_GA(x1, x3) U3_GAAA(x1, x2, x3, x4, x5, x6) = U3_GAAA(x1, x6) U4_GAAA(x1, x2, x3, x4, x5, x6) = U4_GAAA(x1, x6) U5_GAAA(x1, x2, x3, x4, x5, x6) = U5_GAAA(x1, x6) U26_AG(x1, x2, x3, x4) = U26_AG(x4) U27_AG(x1, x2, x3, x4) = U27_AG(x4) QSG_IN_AAA(x1, x2, x3) = QSG_IN_AAA U19_AAA(x1, x2, x3, x4) = U19_AAA(x4) PD_IN_AAAAAAA(x1, x2, x3, x4, x5, x6, x7) = PD_IN_AAAAAAA U10_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U10_AAAAAAA(x8) PARTC_IN_AAAA(x1, x2, x3, x4) = PARTC_IN_AAAA U6_AAAA(x1, x2, x3, x4, x5, x6) = U6_AAAA(x6) U7_AAAA(x1, x2, x3, x4, x5, x6) = U7_AAAA(x6) U8_AAAA(x1, x2, x3, x4, x5, x6) = U8_AAAA(x1, x6) U9_AAAA(x1, x2, x3, x4, x5, x6) = U9_AAAA(x6) U11_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U11_AAAAAAA(x8) U12_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U12_AAAAAAA(x8) QSF_IN_AA(x1, x2) = QSF_IN_AA U17_AA(x1, x2, x3, x4) = U17_AA(x4) U13_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U13_AAAAAAA(x8) U14_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U14_AAAAAAA(x8) U15_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U15_AAAAAAA(x8) U16_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U16_AAAAAAA(x8) APPE_IN_AAAA(x1, x2, x3, x4) = APPE_IN_AAAA U18_AAAA(x1, x2, x3, x4, x5, x6) = U18_AAAA(x6) U28_AG(x1, x2, x3, x4, x5) = U28_AG(x5) U29_AG(x1, x2, x3, x4) = U29_AG(x4) U30_AG(x1, x2, x3, x4) = U30_AG(x4) U31_AG(x1, x2, x3, x4) = U31_AG(x4) U32_AG(x1, x2, x3, x4) = U32_AG(x4) U33_AG(x1, x2, x3, x4, x5) = U33_AG(x5) U34_AG(x1, x2, x3, x4) = U34_AG(x4) U35_AG(x1, x2) = U35_AG(x2) QSF_IN_GA(x1, x2) = QSF_IN_GA(x1) U17_GA(x1, x2, x3, x4) = U17_GA(x1, x2, x4) PD_IN_GGAAAAA(x1, x2, x3, x4, x5, x6, x7) = PD_IN_GGAAAAA(x1, x2) U10_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U10_GGAAAAA(x1, x2, x8) PARTC_IN_GGAA(x1, x2, x3, x4) = PARTC_IN_GGAA(x1, x2) U6_GGAA(x1, x2, x3, x4, x5, x6) = U6_GGAA(x1, x2, x3, x6) LESSA_IN_GG(x1, x2) = LESSA_IN_GG(x1, x2) U1_GG(x1, x2, x3) = U1_GG(x1, x2, x3) U7_GGAA(x1, x2, x3, x4, x5, x6) = U7_GGAA(x1, x2, x3, x6) U8_GGAA(x1, x2, x3, x4, x5, x6) = U8_GGAA(x1, x2, x3, x6) PARTB_IN_GGAA(x1, x2, x3, x4) = PARTB_IN_GGAA(x1, x2) U2_GGAA(x1, x2, x3, x4, x5, x6) = U2_GGAA(x1, x2, x3, x6) U3_GGAA(x1, x2, x3, x4, x5, x6) = U3_GGAA(x1, x2, x3, x6) U4_GGAA(x1, x2, x3, x4, x5, x6) = U4_GGAA(x1, x2, x3, x6) U5_GGAA(x1, x2, x3, x4, x5, x6) = U5_GGAA(x1, x2, x3, x6) U9_GGAA(x1, x2, x3, x4, x5, x6) = U9_GGAA(x1, x2, x3, x6) U11_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U11_GGAAAAA(x1, x2, x8) U12_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U12_GGAAAAA(x1, x2, x8) U13_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U13_GGAAAAA(x1, x2, x4, x8) U14_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U14_GGAAAAA(x1, x2, x8) U15_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U15_GGAAAAA(x1, x2, x5, x8) U16_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U16_GGAAAAA(x1, x2, x8) APPE_IN_GGGA(x1, x2, x3, x4) = APPE_IN_GGGA(x1, x2, x3) U18_GGGA(x1, x2, x3, x4, x5, x6) = U18_GGGA(x1, x2, x3, x4, x6) U36_AG(x1, x2) = U36_AG(x2) U37_AG(x1, x2) = U37_AG(x2) U38_AG(x1, x2, x3, x4, x5) = U38_AG(x4, x5) U39_AG(x1, x2, x3, x4, x5) = U39_AG(x4, x5) U40_AG(x1, x2, x3, x4, x5) = U40_AG(x4, x5) U41_AG(x1, x2, x3, x4, x5) = U41_AG(x4, x5) U42_AG(x1, x2, x3, x4, x5) = U42_AG(x4, x5) U43_AG(x1, x2, x3, x4, x5, x6) = U43_AG(x4, x6) U44_AG(x1, x2, x3, x4, x5) = U44_AG(x4, x5) U45_AG(x1, x2, x3, x4, x5, x6) = U45_AG(x4, x5, x6) U46_AG(x1, x2, x3, x4, x5, x6) = U46_AG(x1, x4, x5, x6) U47_AG(x1, x2, x3, x4, x5, x6, x7) = U47_AG(x1, x4, x5, x7) U48_AG(x1, x2, x3, x4, x5, x6, x7) = U48_AG(x1, x4, x5, x7) U49_AG(x1, x2, x3, x4, x5, x6) = U49_AG(x4, x5, x6) APPH_IN_AGAG(x1, x2, x3, x4) = APPH_IN_AGAG(x2, x4) U20_AGAG(x1, x2, x3, x4, x5, x6) = U20_AGAG(x1, x3, x5, x6) U50_AG(x1, x2, x3, x4, x5) = U50_AG(x4, x5) U51_AG(x1, x2, x3, x4, x5) = U51_AG(x4, x5) U52_AG(x1, x2, x3, x4, x5) = U52_AG(x4, x5) U53_AG(x1, x2, x3, x4, x5, x6) = U53_AG(x4, x6) U54_AG(x1, x2, x3, x4, x5) = U54_AG(x4, x5) U55_AG(x1, x2, x3, x4, x5, x6) = U55_AG(x4, x6) U56_AG(x1, x2, x3, x4, x5) = U56_AG(x4, x5) APPJ_IN_AAAG(x1, x2, x3, x4) = APPJ_IN_AAAG(x4) U22_AAAG(x1, x2, x3, x4, x5, x6) = U22_AAAG(x1, x5, x6) APPI_IN_AAAG(x1, x2, x3, x4) = APPI_IN_AAAG(x4) U21_AAAG(x1, x2, x3, x4, x5, x6) = U21_AAAG(x1, x5, x6) U57_AG(x1, x2, x3) = U57_AG(x2, x3) U58_AG(x1, x2, x3) = U58_AG(x2, x3) U59_AG(x1, x2, x3) = U59_AG(x2, x3) U60_AG(x1, x2, x3, x4) = U60_AG(x2, x3, x4) U61_AG(x1, x2, x3) = U61_AG(x2, x3) APPJ_IN_GAGG(x1, x2, x3, x4) = APPJ_IN_GAGG(x1, x3, x4) U22_GAGG(x1, x2, x3, x4, x5, x6) = U22_GAGG(x1, x2, x4, x5, x6) APPI_IN_GAGG(x1, x2, x3, x4) = APPI_IN_GAGG(x1, x3, x4) U21_GAGG(x1, x2, x3, x4, x5, x6) = U21_GAGG(x1, x2, x4, x5, x6) We have to consider all (P,R,Pi)-chains Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES ---------------------------------------- (149) Obligation: Pi DP problem: The TRS P consists of the following rules: QSK_IN_AG(.(X1, .(X2, X3)), []) -> U23_AG(X1, X2, X3, lessA_in_aa(X1, X2)) QSK_IN_AG(.(X1, .(X2, X3)), []) -> LESSA_IN_AA(X1, X2) LESSA_IN_AA(s(X1), s(X2)) -> U1_AA(X1, X2, lessA_in_aa(X1, X2)) LESSA_IN_AA(s(X1), s(X2)) -> LESSA_IN_AA(X1, X2) QSK_IN_AG(.(X1, .(X2, X3)), []) -> U24_AG(X1, X2, X3, lesscA_in_aa(X1, X2)) U24_AG(X1, X2, X3, lesscA_out_aa(X1, X2)) -> U25_AG(X1, X2, X3, partB_in_gaaa(X1, X3, X4, X5)) U24_AG(X1, X2, X3, lesscA_out_aa(X1, X2)) -> PARTB_IN_GAAA(X1, X3, X4, X5) PARTB_IN_GAAA(X1, .(X2, X3), .(X2, X4), X5) -> U2_GAAA(X1, X2, X3, X4, X5, lessA_in_ga(X1, X2)) PARTB_IN_GAAA(X1, .(X2, X3), .(X2, X4), X5) -> LESSA_IN_GA(X1, X2) LESSA_IN_GA(s(X1), s(X2)) -> U1_GA(X1, X2, lessA_in_ga(X1, X2)) LESSA_IN_GA(s(X1), s(X2)) -> LESSA_IN_GA(X1, X2) PARTB_IN_GAAA(X1, .(X2, X3), .(X2, X4), X5) -> U3_GAAA(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) U3_GAAA(X1, X2, X3, X4, X5, lesscA_out_ga(X1, X2)) -> U4_GAAA(X1, X2, X3, X4, X5, partB_in_gaaa(X1, X3, X4, X5)) U3_GAAA(X1, X2, X3, X4, X5, lesscA_out_ga(X1, X2)) -> PARTB_IN_GAAA(X1, X3, X4, X5) PARTB_IN_GAAA(X1, .(X2, X3), X4, .(X2, X5)) -> U5_GAAA(X1, X2, X3, X4, X5, partB_in_gaaa(X1, X3, X4, X5)) PARTB_IN_GAAA(X1, .(X2, X3), X4, .(X2, X5)) -> PARTB_IN_GAAA(X1, X3, X4, X5) U24_AG(X1, X2, X3, lesscA_out_aa(X1, X2)) -> U26_AG(X1, X2, X3, partcB_in_gaaa(X1, X3, X4, X5)) U26_AG(X1, X2, X3, partcB_out_gaaa(X1, X3, X4, X5)) -> U27_AG(X1, X2, X3, qsG_in_aaa(X2, X4, X6)) U26_AG(X1, X2, X3, partcB_out_gaaa(X1, X3, X4, X5)) -> QSG_IN_AAA(X2, X4, X6) QSG_IN_AAA(X1, X2, X3) -> U19_AAA(X1, X2, X3, pD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) QSG_IN_AAA(X1, X2, X3) -> PD_IN_AAAAAAA(X1, X2, X4, X5, X6, X7, X3) PD_IN_AAAAAAA(X1, X2, X3, X4, X5, X6, X7) -> U10_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, partC_in_aaaa(X1, X2, X3, X4)) PD_IN_AAAAAAA(X1, X2, X3, X4, X5, X6, X7) -> PARTC_IN_AAAA(X1, X2, X3, X4) PARTC_IN_AAAA(X1, .(X2, X3), .(X2, X4), X5) -> U6_AAAA(X1, X2, X3, X4, X5, lessA_in_aa(X1, X2)) PARTC_IN_AAAA(X1, .(X2, X3), .(X2, X4), X5) -> LESSA_IN_AA(X1, X2) PARTC_IN_AAAA(X1, .(X2, X3), .(X2, X4), X5) -> U7_AAAA(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) U7_AAAA(X1, X2, X3, X4, X5, lesscA_out_aa(X1, X2)) -> U8_AAAA(X1, X2, X3, X4, X5, partB_in_gaaa(X1, X3, X4, X5)) U7_AAAA(X1, X2, X3, X4, X5, lesscA_out_aa(X1, X2)) -> PARTB_IN_GAAA(X1, X3, X4, X5) PARTC_IN_AAAA(X1, .(X2, X3), X4, .(X2, X5)) -> U9_AAAA(X1, X2, X3, X4, X5, partC_in_aaaa(X1, X3, X4, X5)) PARTC_IN_AAAA(X1, .(X2, X3), X4, .(X2, X5)) -> PARTC_IN_AAAA(X1, X3, X4, X5) PD_IN_AAAAAAA(X1, X2, X3, X4, X5, X6, X7) -> U11_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, partcC_in_aaaa(X1, X2, X3, X4)) U11_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, partcC_out_aaaa(X1, X2, X3, X4)) -> U12_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, qsF_in_aa(X3, X5)) U11_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, partcC_out_aaaa(X1, X2, X3, X4)) -> QSF_IN_AA(X3, X5) QSF_IN_AA(.(X1, X2), X3) -> U17_AA(X1, X2, X3, pD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) QSF_IN_AA(.(X1, X2), X3) -> PD_IN_AAAAAAA(X1, X2, X4, X5, X6, X7, X3) U11_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, partcC_out_aaaa(X1, X2, X3, X4)) -> U13_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X3, X5)) U13_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X3, X5)) -> U14_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, qsF_in_aa(X4, X6)) U13_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X3, X5)) -> QSF_IN_AA(X4, X6) U13_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X3, X5)) -> U15_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X4, X6)) U15_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X4, X6)) -> U16_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, appE_in_aaaa(X5, X1, X6, X7)) U15_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X4, X6)) -> APPE_IN_AAAA(X5, X1, X6, X7) APPE_IN_AAAA(.(X1, X2), X3, X4, .(X1, X5)) -> U18_AAAA(X1, X2, X3, X4, X5, appE_in_aaaa(X2, X3, X4, X5)) APPE_IN_AAAA(.(X1, X2), X3, X4, .(X1, X5)) -> APPE_IN_AAAA(X2, X3, X4, X5) U26_AG(X1, X2, X3, partcB_out_gaaa(X1, X3, X4, X5)) -> U28_AG(X1, X2, X3, X5, qscG_in_aaa(X2, X4, X6)) U28_AG(X1, X2, X3, X5, qscG_out_aaa(X2, X4, X6)) -> U29_AG(X1, X2, X3, qsF_in_aa(X5, X7)) U28_AG(X1, X2, X3, X5, qscG_out_aaa(X2, X4, X6)) -> QSF_IN_AA(X5, X7) QSK_IN_AG(.(X1, .(X2, X3)), []) -> U30_AG(X1, X2, X3, partC_in_aaaa(X1, X3, X4, X5)) QSK_IN_AG(.(X1, .(X2, X3)), []) -> PARTC_IN_AAAA(X1, X3, X4, X5) QSK_IN_AG(.(X1, .(X2, X3)), []) -> U31_AG(X1, X2, X3, partcC_in_aaaa(X1, X3, X4, X5)) U31_AG(X1, X2, X3, partcC_out_aaaa(X1, X3, X4, X5)) -> U32_AG(X1, X2, X3, qsF_in_aa(X4, X6)) U31_AG(X1, X2, X3, partcC_out_aaaa(X1, X3, X4, X5)) -> QSF_IN_AA(X4, X6) U31_AG(X1, X2, X3, partcC_out_aaaa(X1, X3, X4, X5)) -> U33_AG(X1, X2, X3, X5, qscF_in_aa(X4, X6)) U33_AG(X1, X2, X3, X5, qscF_out_aa(X4, X6)) -> U34_AG(X1, X2, X3, qsG_in_aaa(X2, X5, X7)) U33_AG(X1, X2, X3, X5, qscF_out_aa(X4, X6)) -> QSG_IN_AAA(X2, X5, X7) QSK_IN_AG(.(X1, []), []) -> U35_AG(X1, qsF_in_ga([], X2)) QSK_IN_AG(.(X1, []), []) -> QSF_IN_GA([], X2) QSF_IN_GA(.(X1, X2), X3) -> U17_GA(X1, X2, X3, pD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) QSF_IN_GA(.(X1, X2), X3) -> PD_IN_GGAAAAA(X1, X2, X4, X5, X6, X7, X3) PD_IN_GGAAAAA(X1, X2, X3, X4, X5, X6, X7) -> U10_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, partC_in_ggaa(X1, X2, X3, X4)) PD_IN_GGAAAAA(X1, X2, X3, X4, X5, X6, X7) -> PARTC_IN_GGAA(X1, X2, X3, X4) PARTC_IN_GGAA(X1, .(X2, X3), .(X2, X4), X5) -> U6_GGAA(X1, X2, X3, X4, X5, lessA_in_gg(X1, X2)) PARTC_IN_GGAA(X1, .(X2, X3), .(X2, X4), X5) -> LESSA_IN_GG(X1, X2) LESSA_IN_GG(s(X1), s(X2)) -> U1_GG(X1, X2, lessA_in_gg(X1, X2)) LESSA_IN_GG(s(X1), s(X2)) -> LESSA_IN_GG(X1, X2) PARTC_IN_GGAA(X1, .(X2, X3), .(X2, X4), X5) -> U7_GGAA(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) U7_GGAA(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> U8_GGAA(X1, X2, X3, X4, X5, partB_in_ggaa(X1, X3, X4, X5)) U7_GGAA(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> PARTB_IN_GGAA(X1, X3, X4, X5) PARTB_IN_GGAA(X1, .(X2, X3), .(X2, X4), X5) -> U2_GGAA(X1, X2, X3, X4, X5, lessA_in_gg(X1, X2)) PARTB_IN_GGAA(X1, .(X2, X3), .(X2, X4), X5) -> LESSA_IN_GG(X1, X2) PARTB_IN_GGAA(X1, .(X2, X3), .(X2, X4), X5) -> U3_GGAA(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) U3_GGAA(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> U4_GGAA(X1, X2, X3, X4, X5, partB_in_ggaa(X1, X3, X4, X5)) U3_GGAA(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> PARTB_IN_GGAA(X1, X3, X4, X5) PARTB_IN_GGAA(X1, .(X2, X3), X4, .(X2, X5)) -> U5_GGAA(X1, X2, X3, X4, X5, partB_in_ggaa(X1, X3, X4, X5)) PARTB_IN_GGAA(X1, .(X2, X3), X4, .(X2, X5)) -> PARTB_IN_GGAA(X1, X3, X4, X5) PARTC_IN_GGAA(X1, .(X2, X3), X4, .(X2, X5)) -> U9_GGAA(X1, X2, X3, X4, X5, partC_in_ggaa(X1, X3, X4, X5)) PARTC_IN_GGAA(X1, .(X2, X3), X4, .(X2, X5)) -> PARTC_IN_GGAA(X1, X3, X4, X5) PD_IN_GGAAAAA(X1, X2, X3, X4, X5, X6, X7) -> U11_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, partcC_in_ggaa(X1, X2, X3, X4)) U11_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, partcC_out_ggaa(X1, X2, X3, X4)) -> U12_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qsF_in_ga(X3, X5)) U11_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, partcC_out_ggaa(X1, X2, X3, X4)) -> QSF_IN_GA(X3, X5) U11_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, partcC_out_ggaa(X1, X2, X3, X4)) -> U13_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5)) U13_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U14_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qsF_in_ga(X4, X6)) U13_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> QSF_IN_GA(X4, X6) U13_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U15_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6)) U15_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U16_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, appE_in_ggga(X5, X1, X6, X7)) U15_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> APPE_IN_GGGA(X5, X1, X6, X7) APPE_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) -> U18_GGGA(X1, X2, X3, X4, X5, appE_in_ggga(X2, X3, X4, X5)) APPE_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) -> APPE_IN_GGGA(X2, X3, X4, X5) QSK_IN_AG(.(X1, []), []) -> U36_AG(X1, qscF_in_ga([], X2)) U36_AG(X1, qscF_out_ga([], X2)) -> U37_AG(X1, qsF_in_ga([], X3)) U36_AG(X1, qscF_out_ga([], X2)) -> QSF_IN_GA([], X3) QSK_IN_AG(.(X1, .(X2, X3)), X4) -> U38_AG(X1, X2, X3, X4, lessA_in_aa(X1, X2)) QSK_IN_AG(.(X1, .(X2, X3)), X4) -> LESSA_IN_AA(X1, X2) QSK_IN_AG(.(X1, .(X2, X3)), X4) -> U39_AG(X1, X2, X3, X4, lesscA_in_aa(X1, X2)) U39_AG(X1, X2, X3, X4, lesscA_out_aa(X1, X2)) -> U40_AG(X1, X2, X3, X4, partB_in_gaaa(X1, X3, X5, X6)) U39_AG(X1, X2, X3, X4, lesscA_out_aa(X1, X2)) -> PARTB_IN_GAAA(X1, X3, X5, X6) U39_AG(X1, X2, X3, X4, lesscA_out_aa(X1, X2)) -> U41_AG(X1, X2, X3, X4, partcB_in_gaaa(X1, X3, X5, X6)) U41_AG(X1, X2, X3, X4, partcB_out_gaaa(X1, X3, X5, X6)) -> U42_AG(X1, X2, X3, X4, qsG_in_aaa(X2, X5, X7)) U41_AG(X1, X2, X3, X4, partcB_out_gaaa(X1, X3, X5, X6)) -> QSG_IN_AAA(X2, X5, X7) U41_AG(X1, X2, X3, X4, partcB_out_gaaa(X1, X3, X5, X6)) -> U43_AG(X1, X2, X3, X4, X6, qscG_in_aaa(X2, X5, X7)) U43_AG(X1, X2, X3, X4, X6, qscG_out_aaa(X2, X5, X7)) -> U44_AG(X1, X2, X3, X4, qsF_in_aa(X6, X8)) U43_AG(X1, X2, X3, X4, X6, qscG_out_aaa(X2, X5, X7)) -> QSF_IN_AA(X6, X8) QSK_IN_AG(.(X1, .(X2, X3)), .(X4, X5)) -> U45_AG(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) U45_AG(X1, X2, X3, X4, X5, lesscA_out_aa(X1, X2)) -> U46_AG(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X6, X7)) U46_AG(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X6, X7)) -> U47_AG(X1, X2, X3, X4, X5, X7, qscG_in_aaa(X2, X6, .(X4, X8))) U47_AG(X1, X2, X3, X4, X5, X7, qscG_out_aaa(X2, X6, .(X4, X8))) -> U48_AG(X1, X2, X3, X4, X5, X8, qscF_in_aa(X7, X9)) U48_AG(X1, X2, X3, X4, X5, X8, qscF_out_aa(X7, X9)) -> U49_AG(X1, X2, X3, X4, X5, appH_in_agag(X8, X1, X9, X5)) U48_AG(X1, X2, X3, X4, X5, X8, qscF_out_aa(X7, X9)) -> APPH_IN_AGAG(X8, X1, X9, X5) APPH_IN_AGAG(.(X1, X2), X3, X4, .(X1, X5)) -> U20_AGAG(X1, X2, X3, X4, X5, appH_in_agag(X2, X3, X4, X5)) APPH_IN_AGAG(.(X1, X2), X3, X4, .(X1, X5)) -> APPH_IN_AGAG(X2, X3, X4, X5) QSK_IN_AG(.(X1, .(X2, X3)), X4) -> U50_AG(X1, X2, X3, X4, partC_in_aaaa(X1, X3, X5, X6)) QSK_IN_AG(.(X1, .(X2, X3)), X4) -> PARTC_IN_AAAA(X1, X3, X5, X6) QSK_IN_AG(.(X1, .(X2, X3)), X4) -> U51_AG(X1, X2, X3, X4, partcC_in_aaaa(X1, X3, X5, X6)) U51_AG(X1, X2, X3, X4, partcC_out_aaaa(X1, X3, X5, X6)) -> U52_AG(X1, X2, X3, X4, qsF_in_aa(X5, X7)) U51_AG(X1, X2, X3, X4, partcC_out_aaaa(X1, X3, X5, X6)) -> QSF_IN_AA(X5, X7) U51_AG(X1, X2, X3, X4, partcC_out_aaaa(X1, X3, X5, X6)) -> U53_AG(X1, X2, X3, X4, X6, qscF_in_aa(X5, X7)) U53_AG(X1, X2, X3, X4, X6, qscF_out_aa(X5, X7)) -> U54_AG(X1, X2, X3, X4, qsG_in_aaa(X2, X6, X8)) U53_AG(X1, X2, X3, X4, X6, qscF_out_aa(X5, X7)) -> QSG_IN_AAA(X2, X6, X8) U53_AG(X1, X2, X3, X4, X6, qscF_out_aa(X5, X7)) -> U55_AG(X1, X2, X3, X4, X7, qscG_in_aaa(X2, X6, X8)) U55_AG(X1, X2, X3, X4, X7, qscG_out_aaa(X2, X6, X8)) -> U56_AG(X1, X2, X3, X4, appJ_in_aaag(X7, X1, X8, X4)) U55_AG(X1, X2, X3, X4, X7, qscG_out_aaa(X2, X6, X8)) -> APPJ_IN_AAAG(X7, X1, X8, X4) APPJ_IN_AAAG(.(X1, X2), X3, X4, .(X1, X5)) -> U22_AAAG(X1, X2, X3, X4, X5, appI_in_aaag(X2, X3, X4, X5)) APPJ_IN_AAAG(.(X1, X2), X3, X4, .(X1, X5)) -> APPI_IN_AAAG(X2, X3, X4, X5) APPI_IN_AAAG(.(X1, X2), X3, X4, .(X1, X5)) -> U21_AAAG(X1, X2, X3, X4, X5, appI_in_aaag(X2, X3, X4, X5)) APPI_IN_AAAG(.(X1, X2), X3, X4, .(X1, X5)) -> APPI_IN_AAAG(X2, X3, X4, X5) QSK_IN_AG(.(X1, []), X2) -> U57_AG(X1, X2, qsF_in_ga([], X3)) QSK_IN_AG(.(X1, []), X2) -> QSF_IN_GA([], X3) QSK_IN_AG(.(X1, []), X2) -> U58_AG(X1, X2, qscF_in_ga([], X3)) U58_AG(X1, X2, qscF_out_ga([], X3)) -> U59_AG(X1, X2, qsF_in_ga([], X4)) U58_AG(X1, X2, qscF_out_ga([], X3)) -> QSF_IN_GA([], X4) U58_AG(X1, X2, qscF_out_ga([], X3)) -> U60_AG(X1, X2, X3, qscF_in_ga([], X4)) U60_AG(X1, X2, X3, qscF_out_ga([], X4)) -> U61_AG(X1, X2, appJ_in_gagg(X3, X1, X4, X2)) U60_AG(X1, X2, X3, qscF_out_ga([], X4)) -> APPJ_IN_GAGG(X3, X1, X4, X2) APPJ_IN_GAGG(.(X1, X2), X3, X4, .(X1, X5)) -> U22_GAGG(X1, X2, X3, X4, X5, appI_in_gagg(X2, X3, X4, X5)) APPJ_IN_GAGG(.(X1, X2), X3, X4, .(X1, X5)) -> APPI_IN_GAGG(X2, X3, X4, X5) APPI_IN_GAGG(.(X1, X2), X3, X4, .(X1, X5)) -> U21_GAGG(X1, X2, X3, X4, X5, appI_in_gagg(X2, X3, X4, X5)) APPI_IN_GAGG(.(X1, X2), X3, X4, .(X1, X5)) -> APPI_IN_GAGG(X2, X3, X4, X5) The TRS R consists of the following rules: lesscA_in_aa(0, s(X1)) -> lesscA_out_aa(0, s(X1)) lesscA_in_aa(s(X1), s(X2)) -> U63_aa(X1, X2, lesscA_in_aa(X1, X2)) U63_aa(X1, X2, lesscA_out_aa(X1, X2)) -> lesscA_out_aa(s(X1), s(X2)) lesscA_in_ga(0, s(X1)) -> lesscA_out_ga(0, s(X1)) lesscA_in_ga(s(X1), s(X2)) -> U63_ga(X1, X2, lesscA_in_ga(X1, X2)) U63_ga(X1, X2, lesscA_out_ga(X1, X2)) -> lesscA_out_ga(s(X1), s(X2)) partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_gaaa(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) U64_gaaa(X1, X2, X3, X4, X5, lesscA_out_ga(X1, X2)) -> U65_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], []) U66_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5)) U65_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_aaaa(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) U67_aaaa(X1, X2, X3, X4, X5, lesscA_out_aa(X1, X2)) -> U68_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) U68_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_aaaa(X1, X2, X3, X4, X5, partcC_in_aaaa(X1, X3, X4, X5)) partcC_in_aaaa(X1, [], [], []) -> partcC_out_aaaa(X1, [], [], []) U69_aaaa(X1, X2, X3, X4, X5, partcC_out_aaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), X4, .(X2, X5)) qscF_in_aa([], []) -> qscF_out_aa([], []) qscF_in_aa(.(X1, X2), X3) -> U74_aa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) qcD_in_aaaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U70_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_in_aaaa(X1, X2, X3, X4)) U70_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_out_aaaa(X1, X2, X3, X4)) -> U71_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X3, X5)) U71_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X3, X5)) -> U72_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X4, X6)) U72_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X4, X6)) -> U73_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_aaaa(X5, X1, X6, X7)) appcE_in_aaaa([], X1, X2, .(X1, X2)) -> appcE_out_aaaa([], X1, X2, .(X1, X2)) appcE_in_aaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U75_aaaa(X1, X2, X3, X4, X5, appcE_in_aaaa(X2, X3, X4, X5)) U75_aaaa(X1, X2, X3, X4, X5, appcE_out_aaaa(X2, X3, X4, X5)) -> appcE_out_aaaa(.(X1, X2), X3, X4, .(X1, X5)) U73_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_aaaa(X5, X1, X6, X7)) -> qcD_out_aaaaaaa(X1, X2, X3, X4, X5, X6, X7) U74_aa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_aa(.(X1, X2), X3) qscG_in_aaa(X1, X2, X3) -> U76_aaa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) U76_aaa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscG_out_aaa(X1, X2, X3) lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) partcC_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) U67_ggaa(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> U68_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) U64_ggaa(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> U65_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], []) U66_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) U65_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) U68_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_ggaa(X1, X2, X3, X4, X5, partcC_in_ggaa(X1, X3, X4, X5)) partcC_in_ggaa(X1, [], [], []) -> partcC_out_ggaa(X1, [], [], []) U69_ggaa(X1, X2, X3, X4, X5, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) qscF_in_ga([], []) -> qscF_out_ga([], []) qscF_in_ga(.(X1, X2), X3) -> U74_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) qcD_in_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U70_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_in_ggaa(X1, X2, X3, X4)) U70_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_out_ggaa(X1, X2, X3, X4)) -> U71_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5)) U71_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U72_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6)) U72_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U73_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_ggga(X5, X1, X6, X7)) appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2)) appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U75_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5)) U75_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5)) U73_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) U74_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3) The argument filtering Pi contains the following mapping: [] = [] lessA_in_aa(x1, x2) = lessA_in_aa lesscA_in_aa(x1, x2) = lesscA_in_aa lesscA_out_aa(x1, x2) = lesscA_out_aa(x1) U63_aa(x1, x2, x3) = U63_aa(x3) partB_in_gaaa(x1, x2, x3, x4) = partB_in_gaaa(x1) lessA_in_ga(x1, x2) = lessA_in_ga(x1) s(x1) = s(x1) lesscA_in_ga(x1, x2) = lesscA_in_ga(x1) 0 = 0 lesscA_out_ga(x1, x2) = lesscA_out_ga(x1) U63_ga(x1, x2, x3) = U63_ga(x1, x3) partcB_in_gaaa(x1, x2, x3, x4) = partcB_in_gaaa(x1) U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6) U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x6) U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6) partcB_out_gaaa(x1, x2, x3, x4) = partcB_out_gaaa(x1) qsG_in_aaa(x1, x2, x3) = qsG_in_aaa pD_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = pD_in_aaaaaaa partC_in_aaaa(x1, x2, x3, x4) = partC_in_aaaa partcC_in_aaaa(x1, x2, x3, x4) = partcC_in_aaaa U67_aaaa(x1, x2, x3, x4, x5, x6) = U67_aaaa(x6) U68_aaaa(x1, x2, x3, x4, x5, x6) = U68_aaaa(x6) partcC_out_aaaa(x1, x2, x3, x4) = partcC_out_aaaa U69_aaaa(x1, x2, x3, x4, x5, x6) = U69_aaaa(x6) qsF_in_aa(x1, x2) = qsF_in_aa qscF_in_aa(x1, x2) = qscF_in_aa qscF_out_aa(x1, x2) = qscF_out_aa U74_aa(x1, x2, x3, x4) = U74_aa(x4) qcD_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_aaaaaaa U70_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_aaaaaaa(x8) U71_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_aaaaaaa(x8) U72_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_aaaaaaa(x8) U73_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_aaaaaaa(x8) .(x1, x2) = .(x1, x2) qcD_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_aaaaaaa appcE_in_aaaa(x1, x2, x3, x4) = appcE_in_aaaa appcE_out_aaaa(x1, x2, x3, x4) = appcE_out_aaaa U75_aaaa(x1, x2, x3, x4, x5, x6) = U75_aaaa(x6) appE_in_aaaa(x1, x2, x3, x4) = appE_in_aaaa qscG_in_aaa(x1, x2, x3) = qscG_in_aaa U76_aaa(x1, x2, x3, x4) = U76_aaa(x4) qscG_out_aaa(x1, x2, x3) = qscG_out_aaa qsF_in_ga(x1, x2) = qsF_in_ga(x1) pD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = pD_in_ggaaaaa(x1, x2) partC_in_ggaa(x1, x2, x3, x4) = partC_in_ggaa(x1, x2) lessA_in_gg(x1, x2) = lessA_in_gg(x1, x2) lesscA_in_gg(x1, x2) = lesscA_in_gg(x1, x2) lesscA_out_gg(x1, x2) = lesscA_out_gg(x1, x2) U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3) partB_in_ggaa(x1, x2, x3, x4) = partB_in_ggaa(x1, x2) partcC_in_ggaa(x1, x2, x3, x4) = partcC_in_ggaa(x1, x2) U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6) U68_ggaa(x1, x2, x3, x4, x5, x6) = U68_ggaa(x1, x2, x3, x6) partcB_in_ggaa(x1, x2, x3, x4) = partcB_in_ggaa(x1, x2) U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6) U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6) U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6) partcB_out_ggaa(x1, x2, x3, x4) = partcB_out_ggaa(x1, x2, x3, x4) partcC_out_ggaa(x1, x2, x3, x4) = partcC_out_ggaa(x1, x2, x3, x4) U69_ggaa(x1, x2, x3, x4, x5, x6) = U69_ggaa(x1, x2, x3, x6) qscF_in_ga(x1, x2) = qscF_in_ga(x1) qscF_out_ga(x1, x2) = qscF_out_ga(x1, x2) U74_ga(x1, x2, x3, x4) = U74_ga(x1, x2, x4) qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_ggaaaaa(x1, x2) U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x8) U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x8) U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x8) U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x8) appcE_in_ggga(x1, x2, x3, x4) = appcE_in_ggga(x1, x2, x3) appcE_out_ggga(x1, x2, x3, x4) = appcE_out_ggga(x1, x2, x3, x4) U75_ggga(x1, x2, x3, x4, x5, x6) = U75_ggga(x1, x2, x3, x4, x6) qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) appE_in_ggga(x1, x2, x3, x4) = appE_in_ggga(x1, x2, x3) appH_in_agag(x1, x2, x3, x4) = appH_in_agag(x2, x4) appJ_in_aaag(x1, x2, x3, x4) = appJ_in_aaag(x4) appI_in_aaag(x1, x2, x3, x4) = appI_in_aaag(x4) appJ_in_gagg(x1, x2, x3, x4) = appJ_in_gagg(x1, x3, x4) appI_in_gagg(x1, x2, x3, x4) = appI_in_gagg(x1, x3, x4) QSK_IN_AG(x1, x2) = QSK_IN_AG(x2) U23_AG(x1, x2, x3, x4) = U23_AG(x4) LESSA_IN_AA(x1, x2) = LESSA_IN_AA U1_AA(x1, x2, x3) = U1_AA(x3) U24_AG(x1, x2, x3, x4) = U24_AG(x4) U25_AG(x1, x2, x3, x4) = U25_AG(x4) PARTB_IN_GAAA(x1, x2, x3, x4) = PARTB_IN_GAAA(x1) U2_GAAA(x1, x2, x3, x4, x5, x6) = U2_GAAA(x1, x6) LESSA_IN_GA(x1, x2) = LESSA_IN_GA(x1) U1_GA(x1, x2, x3) = U1_GA(x1, x3) U3_GAAA(x1, x2, x3, x4, x5, x6) = U3_GAAA(x1, x6) U4_GAAA(x1, x2, x3, x4, x5, x6) = U4_GAAA(x1, x6) U5_GAAA(x1, x2, x3, x4, x5, x6) = U5_GAAA(x1, x6) U26_AG(x1, x2, x3, x4) = U26_AG(x4) U27_AG(x1, x2, x3, x4) = U27_AG(x4) QSG_IN_AAA(x1, x2, x3) = QSG_IN_AAA U19_AAA(x1, x2, x3, x4) = U19_AAA(x4) PD_IN_AAAAAAA(x1, x2, x3, x4, x5, x6, x7) = PD_IN_AAAAAAA U10_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U10_AAAAAAA(x8) PARTC_IN_AAAA(x1, x2, x3, x4) = PARTC_IN_AAAA U6_AAAA(x1, x2, x3, x4, x5, x6) = U6_AAAA(x6) U7_AAAA(x1, x2, x3, x4, x5, x6) = U7_AAAA(x6) U8_AAAA(x1, x2, x3, x4, x5, x6) = U8_AAAA(x1, x6) U9_AAAA(x1, x2, x3, x4, x5, x6) = U9_AAAA(x6) U11_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U11_AAAAAAA(x8) U12_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U12_AAAAAAA(x8) QSF_IN_AA(x1, x2) = QSF_IN_AA U17_AA(x1, x2, x3, x4) = U17_AA(x4) U13_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U13_AAAAAAA(x8) U14_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U14_AAAAAAA(x8) U15_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U15_AAAAAAA(x8) U16_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U16_AAAAAAA(x8) APPE_IN_AAAA(x1, x2, x3, x4) = APPE_IN_AAAA U18_AAAA(x1, x2, x3, x4, x5, x6) = U18_AAAA(x6) U28_AG(x1, x2, x3, x4, x5) = U28_AG(x5) U29_AG(x1, x2, x3, x4) = U29_AG(x4) U30_AG(x1, x2, x3, x4) = U30_AG(x4) U31_AG(x1, x2, x3, x4) = U31_AG(x4) U32_AG(x1, x2, x3, x4) = U32_AG(x4) U33_AG(x1, x2, x3, x4, x5) = U33_AG(x5) U34_AG(x1, x2, x3, x4) = U34_AG(x4) U35_AG(x1, x2) = U35_AG(x2) QSF_IN_GA(x1, x2) = QSF_IN_GA(x1) U17_GA(x1, x2, x3, x4) = U17_GA(x1, x2, x4) PD_IN_GGAAAAA(x1, x2, x3, x4, x5, x6, x7) = PD_IN_GGAAAAA(x1, x2) U10_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U10_GGAAAAA(x1, x2, x8) PARTC_IN_GGAA(x1, x2, x3, x4) = PARTC_IN_GGAA(x1, x2) U6_GGAA(x1, x2, x3, x4, x5, x6) = U6_GGAA(x1, x2, x3, x6) LESSA_IN_GG(x1, x2) = LESSA_IN_GG(x1, x2) U1_GG(x1, x2, x3) = U1_GG(x1, x2, x3) U7_GGAA(x1, x2, x3, x4, x5, x6) = U7_GGAA(x1, x2, x3, x6) U8_GGAA(x1, x2, x3, x4, x5, x6) = U8_GGAA(x1, x2, x3, x6) PARTB_IN_GGAA(x1, x2, x3, x4) = PARTB_IN_GGAA(x1, x2) U2_GGAA(x1, x2, x3, x4, x5, x6) = U2_GGAA(x1, x2, x3, x6) U3_GGAA(x1, x2, x3, x4, x5, x6) = U3_GGAA(x1, x2, x3, x6) U4_GGAA(x1, x2, x3, x4, x5, x6) = U4_GGAA(x1, x2, x3, x6) U5_GGAA(x1, x2, x3, x4, x5, x6) = U5_GGAA(x1, x2, x3, x6) U9_GGAA(x1, x2, x3, x4, x5, x6) = U9_GGAA(x1, x2, x3, x6) U11_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U11_GGAAAAA(x1, x2, x8) U12_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U12_GGAAAAA(x1, x2, x8) U13_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U13_GGAAAAA(x1, x2, x4, x8) U14_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U14_GGAAAAA(x1, x2, x8) U15_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U15_GGAAAAA(x1, x2, x5, x8) U16_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U16_GGAAAAA(x1, x2, x8) APPE_IN_GGGA(x1, x2, x3, x4) = APPE_IN_GGGA(x1, x2, x3) U18_GGGA(x1, x2, x3, x4, x5, x6) = U18_GGGA(x1, x2, x3, x4, x6) U36_AG(x1, x2) = U36_AG(x2) U37_AG(x1, x2) = U37_AG(x2) U38_AG(x1, x2, x3, x4, x5) = U38_AG(x4, x5) U39_AG(x1, x2, x3, x4, x5) = U39_AG(x4, x5) U40_AG(x1, x2, x3, x4, x5) = U40_AG(x4, x5) U41_AG(x1, x2, x3, x4, x5) = U41_AG(x4, x5) U42_AG(x1, x2, x3, x4, x5) = U42_AG(x4, x5) U43_AG(x1, x2, x3, x4, x5, x6) = U43_AG(x4, x6) U44_AG(x1, x2, x3, x4, x5) = U44_AG(x4, x5) U45_AG(x1, x2, x3, x4, x5, x6) = U45_AG(x4, x5, x6) U46_AG(x1, x2, x3, x4, x5, x6) = U46_AG(x1, x4, x5, x6) U47_AG(x1, x2, x3, x4, x5, x6, x7) = U47_AG(x1, x4, x5, x7) U48_AG(x1, x2, x3, x4, x5, x6, x7) = U48_AG(x1, x4, x5, x7) U49_AG(x1, x2, x3, x4, x5, x6) = U49_AG(x4, x5, x6) APPH_IN_AGAG(x1, x2, x3, x4) = APPH_IN_AGAG(x2, x4) U20_AGAG(x1, x2, x3, x4, x5, x6) = U20_AGAG(x1, x3, x5, x6) U50_AG(x1, x2, x3, x4, x5) = U50_AG(x4, x5) U51_AG(x1, x2, x3, x4, x5) = U51_AG(x4, x5) U52_AG(x1, x2, x3, x4, x5) = U52_AG(x4, x5) U53_AG(x1, x2, x3, x4, x5, x6) = U53_AG(x4, x6) U54_AG(x1, x2, x3, x4, x5) = U54_AG(x4, x5) U55_AG(x1, x2, x3, x4, x5, x6) = U55_AG(x4, x6) U56_AG(x1, x2, x3, x4, x5) = U56_AG(x4, x5) APPJ_IN_AAAG(x1, x2, x3, x4) = APPJ_IN_AAAG(x4) U22_AAAG(x1, x2, x3, x4, x5, x6) = U22_AAAG(x1, x5, x6) APPI_IN_AAAG(x1, x2, x3, x4) = APPI_IN_AAAG(x4) U21_AAAG(x1, x2, x3, x4, x5, x6) = U21_AAAG(x1, x5, x6) U57_AG(x1, x2, x3) = U57_AG(x2, x3) U58_AG(x1, x2, x3) = U58_AG(x2, x3) U59_AG(x1, x2, x3) = U59_AG(x2, x3) U60_AG(x1, x2, x3, x4) = U60_AG(x2, x3, x4) U61_AG(x1, x2, x3) = U61_AG(x2, x3) APPJ_IN_GAGG(x1, x2, x3, x4) = APPJ_IN_GAGG(x1, x3, x4) U22_GAGG(x1, x2, x3, x4, x5, x6) = U22_GAGG(x1, x2, x4, x5, x6) APPI_IN_GAGG(x1, x2, x3, x4) = APPI_IN_GAGG(x1, x3, x4) U21_GAGG(x1, x2, x3, x4, x5, x6) = U21_GAGG(x1, x2, x4, x5, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (150) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 14 SCCs with 110 less nodes. ---------------------------------------- (151) Complex Obligation (AND) ---------------------------------------- (152) Obligation: Pi DP problem: The TRS P consists of the following rules: APPI_IN_GAGG(.(X1, X2), X3, X4, .(X1, X5)) -> APPI_IN_GAGG(X2, X3, X4, X5) The TRS R consists of the following rules: lesscA_in_aa(0, s(X1)) -> lesscA_out_aa(0, s(X1)) lesscA_in_aa(s(X1), s(X2)) -> U63_aa(X1, X2, lesscA_in_aa(X1, X2)) U63_aa(X1, X2, lesscA_out_aa(X1, X2)) -> lesscA_out_aa(s(X1), s(X2)) lesscA_in_ga(0, s(X1)) -> lesscA_out_ga(0, s(X1)) lesscA_in_ga(s(X1), s(X2)) -> U63_ga(X1, X2, lesscA_in_ga(X1, X2)) U63_ga(X1, X2, lesscA_out_ga(X1, X2)) -> lesscA_out_ga(s(X1), s(X2)) partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_gaaa(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) U64_gaaa(X1, X2, X3, X4, X5, lesscA_out_ga(X1, X2)) -> U65_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], []) U66_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5)) U65_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_aaaa(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) U67_aaaa(X1, X2, X3, X4, X5, lesscA_out_aa(X1, X2)) -> U68_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) U68_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_aaaa(X1, X2, X3, X4, X5, partcC_in_aaaa(X1, X3, X4, X5)) partcC_in_aaaa(X1, [], [], []) -> partcC_out_aaaa(X1, [], [], []) U69_aaaa(X1, X2, X3, X4, X5, partcC_out_aaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), X4, .(X2, X5)) qscF_in_aa([], []) -> qscF_out_aa([], []) qscF_in_aa(.(X1, X2), X3) -> U74_aa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) qcD_in_aaaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U70_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_in_aaaa(X1, X2, X3, X4)) U70_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_out_aaaa(X1, X2, X3, X4)) -> U71_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X3, X5)) U71_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X3, X5)) -> U72_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X4, X6)) U72_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X4, X6)) -> U73_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_aaaa(X5, X1, X6, X7)) appcE_in_aaaa([], X1, X2, .(X1, X2)) -> appcE_out_aaaa([], X1, X2, .(X1, X2)) appcE_in_aaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U75_aaaa(X1, X2, X3, X4, X5, appcE_in_aaaa(X2, X3, X4, X5)) U75_aaaa(X1, X2, X3, X4, X5, appcE_out_aaaa(X2, X3, X4, X5)) -> appcE_out_aaaa(.(X1, X2), X3, X4, .(X1, X5)) U73_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_aaaa(X5, X1, X6, X7)) -> qcD_out_aaaaaaa(X1, X2, X3, X4, X5, X6, X7) U74_aa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_aa(.(X1, X2), X3) qscG_in_aaa(X1, X2, X3) -> U76_aaa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) U76_aaa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscG_out_aaa(X1, X2, X3) lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) partcC_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) U67_ggaa(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> U68_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) U64_ggaa(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> U65_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], []) U66_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) U65_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) U68_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_ggaa(X1, X2, X3, X4, X5, partcC_in_ggaa(X1, X3, X4, X5)) partcC_in_ggaa(X1, [], [], []) -> partcC_out_ggaa(X1, [], [], []) U69_ggaa(X1, X2, X3, X4, X5, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) qscF_in_ga([], []) -> qscF_out_ga([], []) qscF_in_ga(.(X1, X2), X3) -> U74_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) qcD_in_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U70_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_in_ggaa(X1, X2, X3, X4)) U70_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_out_ggaa(X1, X2, X3, X4)) -> U71_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5)) U71_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U72_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6)) U72_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U73_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_ggga(X5, X1, X6, X7)) appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2)) appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U75_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5)) U75_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5)) U73_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) U74_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3) The argument filtering Pi contains the following mapping: [] = [] lesscA_in_aa(x1, x2) = lesscA_in_aa lesscA_out_aa(x1, x2) = lesscA_out_aa(x1) U63_aa(x1, x2, x3) = U63_aa(x3) s(x1) = s(x1) lesscA_in_ga(x1, x2) = lesscA_in_ga(x1) 0 = 0 lesscA_out_ga(x1, x2) = lesscA_out_ga(x1) U63_ga(x1, x2, x3) = U63_ga(x1, x3) partcB_in_gaaa(x1, x2, x3, x4) = partcB_in_gaaa(x1) U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6) U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x6) U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6) partcB_out_gaaa(x1, x2, x3, x4) = partcB_out_gaaa(x1) partcC_in_aaaa(x1, x2, x3, x4) = partcC_in_aaaa U67_aaaa(x1, x2, x3, x4, x5, x6) = U67_aaaa(x6) U68_aaaa(x1, x2, x3, x4, x5, x6) = U68_aaaa(x6) partcC_out_aaaa(x1, x2, x3, x4) = partcC_out_aaaa U69_aaaa(x1, x2, x3, x4, x5, x6) = U69_aaaa(x6) qscF_in_aa(x1, x2) = qscF_in_aa qscF_out_aa(x1, x2) = qscF_out_aa U74_aa(x1, x2, x3, x4) = U74_aa(x4) qcD_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_aaaaaaa U70_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_aaaaaaa(x8) U71_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_aaaaaaa(x8) U72_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_aaaaaaa(x8) U73_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_aaaaaaa(x8) .(x1, x2) = .(x1, x2) qcD_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_aaaaaaa appcE_in_aaaa(x1, x2, x3, x4) = appcE_in_aaaa appcE_out_aaaa(x1, x2, x3, x4) = appcE_out_aaaa U75_aaaa(x1, x2, x3, x4, x5, x6) = U75_aaaa(x6) qscG_in_aaa(x1, x2, x3) = qscG_in_aaa U76_aaa(x1, x2, x3, x4) = U76_aaa(x4) qscG_out_aaa(x1, x2, x3) = qscG_out_aaa lesscA_in_gg(x1, x2) = lesscA_in_gg(x1, x2) lesscA_out_gg(x1, x2) = lesscA_out_gg(x1, x2) U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3) partcC_in_ggaa(x1, x2, x3, x4) = partcC_in_ggaa(x1, x2) U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6) U68_ggaa(x1, x2, x3, x4, x5, x6) = U68_ggaa(x1, x2, x3, x6) partcB_in_ggaa(x1, x2, x3, x4) = partcB_in_ggaa(x1, x2) U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6) U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6) U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6) partcB_out_ggaa(x1, x2, x3, x4) = partcB_out_ggaa(x1, x2, x3, x4) partcC_out_ggaa(x1, x2, x3, x4) = partcC_out_ggaa(x1, x2, x3, x4) U69_ggaa(x1, x2, x3, x4, x5, x6) = U69_ggaa(x1, x2, x3, x6) qscF_in_ga(x1, x2) = qscF_in_ga(x1) qscF_out_ga(x1, x2) = qscF_out_ga(x1, x2) U74_ga(x1, x2, x3, x4) = U74_ga(x1, x2, x4) qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_ggaaaaa(x1, x2) U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x8) U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x8) U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x8) U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x8) appcE_in_ggga(x1, x2, x3, x4) = appcE_in_ggga(x1, x2, x3) appcE_out_ggga(x1, x2, x3, x4) = appcE_out_ggga(x1, x2, x3, x4) U75_ggga(x1, x2, x3, x4, x5, x6) = U75_ggga(x1, x2, x3, x4, x6) qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) APPI_IN_GAGG(x1, x2, x3, x4) = APPI_IN_GAGG(x1, x3, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (153) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (154) Obligation: Pi DP problem: The TRS P consists of the following rules: APPI_IN_GAGG(.(X1, X2), X3, X4, .(X1, X5)) -> APPI_IN_GAGG(X2, X3, X4, X5) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) APPI_IN_GAGG(x1, x2, x3, x4) = APPI_IN_GAGG(x1, x3, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (155) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (156) Obligation: Q DP problem: The TRS P consists of the following rules: APPI_IN_GAGG(.(X1, X2), X4, .(X1, X5)) -> APPI_IN_GAGG(X2, X4, X5) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (157) 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: *APPI_IN_GAGG(.(X1, X2), X4, .(X1, X5)) -> APPI_IN_GAGG(X2, X4, X5) The graph contains the following edges 1 > 1, 2 >= 2, 3 > 3 ---------------------------------------- (158) YES ---------------------------------------- (159) Obligation: Pi DP problem: The TRS P consists of the following rules: APPI_IN_AAAG(.(X1, X2), X3, X4, .(X1, X5)) -> APPI_IN_AAAG(X2, X3, X4, X5) The TRS R consists of the following rules: lesscA_in_aa(0, s(X1)) -> lesscA_out_aa(0, s(X1)) lesscA_in_aa(s(X1), s(X2)) -> U63_aa(X1, X2, lesscA_in_aa(X1, X2)) U63_aa(X1, X2, lesscA_out_aa(X1, X2)) -> lesscA_out_aa(s(X1), s(X2)) lesscA_in_ga(0, s(X1)) -> lesscA_out_ga(0, s(X1)) lesscA_in_ga(s(X1), s(X2)) -> U63_ga(X1, X2, lesscA_in_ga(X1, X2)) U63_ga(X1, X2, lesscA_out_ga(X1, X2)) -> lesscA_out_ga(s(X1), s(X2)) partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_gaaa(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) U64_gaaa(X1, X2, X3, X4, X5, lesscA_out_ga(X1, X2)) -> U65_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], []) U66_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5)) U65_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_aaaa(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) U67_aaaa(X1, X2, X3, X4, X5, lesscA_out_aa(X1, X2)) -> U68_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) U68_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_aaaa(X1, X2, X3, X4, X5, partcC_in_aaaa(X1, X3, X4, X5)) partcC_in_aaaa(X1, [], [], []) -> partcC_out_aaaa(X1, [], [], []) U69_aaaa(X1, X2, X3, X4, X5, partcC_out_aaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), X4, .(X2, X5)) qscF_in_aa([], []) -> qscF_out_aa([], []) qscF_in_aa(.(X1, X2), X3) -> U74_aa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) qcD_in_aaaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U70_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_in_aaaa(X1, X2, X3, X4)) U70_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_out_aaaa(X1, X2, X3, X4)) -> U71_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X3, X5)) U71_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X3, X5)) -> U72_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X4, X6)) U72_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X4, X6)) -> U73_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_aaaa(X5, X1, X6, X7)) appcE_in_aaaa([], X1, X2, .(X1, X2)) -> appcE_out_aaaa([], X1, X2, .(X1, X2)) appcE_in_aaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U75_aaaa(X1, X2, X3, X4, X5, appcE_in_aaaa(X2, X3, X4, X5)) U75_aaaa(X1, X2, X3, X4, X5, appcE_out_aaaa(X2, X3, X4, X5)) -> appcE_out_aaaa(.(X1, X2), X3, X4, .(X1, X5)) U73_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_aaaa(X5, X1, X6, X7)) -> qcD_out_aaaaaaa(X1, X2, X3, X4, X5, X6, X7) U74_aa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_aa(.(X1, X2), X3) qscG_in_aaa(X1, X2, X3) -> U76_aaa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) U76_aaa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscG_out_aaa(X1, X2, X3) lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) partcC_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) U67_ggaa(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> U68_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) U64_ggaa(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> U65_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], []) U66_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) U65_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) U68_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_ggaa(X1, X2, X3, X4, X5, partcC_in_ggaa(X1, X3, X4, X5)) partcC_in_ggaa(X1, [], [], []) -> partcC_out_ggaa(X1, [], [], []) U69_ggaa(X1, X2, X3, X4, X5, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) qscF_in_ga([], []) -> qscF_out_ga([], []) qscF_in_ga(.(X1, X2), X3) -> U74_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) qcD_in_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U70_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_in_ggaa(X1, X2, X3, X4)) U70_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_out_ggaa(X1, X2, X3, X4)) -> U71_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5)) U71_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U72_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6)) U72_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U73_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_ggga(X5, X1, X6, X7)) appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2)) appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U75_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5)) U75_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5)) U73_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) U74_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3) The argument filtering Pi contains the following mapping: [] = [] lesscA_in_aa(x1, x2) = lesscA_in_aa lesscA_out_aa(x1, x2) = lesscA_out_aa(x1) U63_aa(x1, x2, x3) = U63_aa(x3) s(x1) = s(x1) lesscA_in_ga(x1, x2) = lesscA_in_ga(x1) 0 = 0 lesscA_out_ga(x1, x2) = lesscA_out_ga(x1) U63_ga(x1, x2, x3) = U63_ga(x1, x3) partcB_in_gaaa(x1, x2, x3, x4) = partcB_in_gaaa(x1) U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6) U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x6) U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6) partcB_out_gaaa(x1, x2, x3, x4) = partcB_out_gaaa(x1) partcC_in_aaaa(x1, x2, x3, x4) = partcC_in_aaaa U67_aaaa(x1, x2, x3, x4, x5, x6) = U67_aaaa(x6) U68_aaaa(x1, x2, x3, x4, x5, x6) = U68_aaaa(x6) partcC_out_aaaa(x1, x2, x3, x4) = partcC_out_aaaa U69_aaaa(x1, x2, x3, x4, x5, x6) = U69_aaaa(x6) qscF_in_aa(x1, x2) = qscF_in_aa qscF_out_aa(x1, x2) = qscF_out_aa U74_aa(x1, x2, x3, x4) = U74_aa(x4) qcD_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_aaaaaaa U70_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_aaaaaaa(x8) U71_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_aaaaaaa(x8) U72_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_aaaaaaa(x8) U73_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_aaaaaaa(x8) .(x1, x2) = .(x1, x2) qcD_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_aaaaaaa appcE_in_aaaa(x1, x2, x3, x4) = appcE_in_aaaa appcE_out_aaaa(x1, x2, x3, x4) = appcE_out_aaaa U75_aaaa(x1, x2, x3, x4, x5, x6) = U75_aaaa(x6) qscG_in_aaa(x1, x2, x3) = qscG_in_aaa U76_aaa(x1, x2, x3, x4) = U76_aaa(x4) qscG_out_aaa(x1, x2, x3) = qscG_out_aaa lesscA_in_gg(x1, x2) = lesscA_in_gg(x1, x2) lesscA_out_gg(x1, x2) = lesscA_out_gg(x1, x2) U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3) partcC_in_ggaa(x1, x2, x3, x4) = partcC_in_ggaa(x1, x2) U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6) U68_ggaa(x1, x2, x3, x4, x5, x6) = U68_ggaa(x1, x2, x3, x6) partcB_in_ggaa(x1, x2, x3, x4) = partcB_in_ggaa(x1, x2) U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6) U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6) U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6) partcB_out_ggaa(x1, x2, x3, x4) = partcB_out_ggaa(x1, x2, x3, x4) partcC_out_ggaa(x1, x2, x3, x4) = partcC_out_ggaa(x1, x2, x3, x4) U69_ggaa(x1, x2, x3, x4, x5, x6) = U69_ggaa(x1, x2, x3, x6) qscF_in_ga(x1, x2) = qscF_in_ga(x1) qscF_out_ga(x1, x2) = qscF_out_ga(x1, x2) U74_ga(x1, x2, x3, x4) = U74_ga(x1, x2, x4) qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_ggaaaaa(x1, x2) U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x8) U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x8) U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x8) U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x8) appcE_in_ggga(x1, x2, x3, x4) = appcE_in_ggga(x1, x2, x3) appcE_out_ggga(x1, x2, x3, x4) = appcE_out_ggga(x1, x2, x3, x4) U75_ggga(x1, x2, x3, x4, x5, x6) = U75_ggga(x1, x2, x3, x4, x6) qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) APPI_IN_AAAG(x1, x2, x3, x4) = APPI_IN_AAAG(x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (160) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (161) Obligation: Pi DP problem: The TRS P consists of the following rules: APPI_IN_AAAG(.(X1, X2), X3, X4, .(X1, X5)) -> APPI_IN_AAAG(X2, X3, X4, X5) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) APPI_IN_AAAG(x1, x2, x3, x4) = APPI_IN_AAAG(x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (162) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (163) Obligation: Q DP problem: The TRS P consists of the following rules: APPI_IN_AAAG(.(X1, X5)) -> APPI_IN_AAAG(X5) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (164) 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: *APPI_IN_AAAG(.(X1, X5)) -> APPI_IN_AAAG(X5) The graph contains the following edges 1 > 1 ---------------------------------------- (165) YES ---------------------------------------- (166) Obligation: Pi DP problem: The TRS P consists of the following rules: APPH_IN_AGAG(.(X1, X2), X3, X4, .(X1, X5)) -> APPH_IN_AGAG(X2, X3, X4, X5) The TRS R consists of the following rules: lesscA_in_aa(0, s(X1)) -> lesscA_out_aa(0, s(X1)) lesscA_in_aa(s(X1), s(X2)) -> U63_aa(X1, X2, lesscA_in_aa(X1, X2)) U63_aa(X1, X2, lesscA_out_aa(X1, X2)) -> lesscA_out_aa(s(X1), s(X2)) lesscA_in_ga(0, s(X1)) -> lesscA_out_ga(0, s(X1)) lesscA_in_ga(s(X1), s(X2)) -> U63_ga(X1, X2, lesscA_in_ga(X1, X2)) U63_ga(X1, X2, lesscA_out_ga(X1, X2)) -> lesscA_out_ga(s(X1), s(X2)) partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_gaaa(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) U64_gaaa(X1, X2, X3, X4, X5, lesscA_out_ga(X1, X2)) -> U65_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], []) U66_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5)) U65_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_aaaa(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) U67_aaaa(X1, X2, X3, X4, X5, lesscA_out_aa(X1, X2)) -> U68_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) U68_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_aaaa(X1, X2, X3, X4, X5, partcC_in_aaaa(X1, X3, X4, X5)) partcC_in_aaaa(X1, [], [], []) -> partcC_out_aaaa(X1, [], [], []) U69_aaaa(X1, X2, X3, X4, X5, partcC_out_aaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), X4, .(X2, X5)) qscF_in_aa([], []) -> qscF_out_aa([], []) qscF_in_aa(.(X1, X2), X3) -> U74_aa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) qcD_in_aaaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U70_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_in_aaaa(X1, X2, X3, X4)) U70_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_out_aaaa(X1, X2, X3, X4)) -> U71_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X3, X5)) U71_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X3, X5)) -> U72_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X4, X6)) U72_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X4, X6)) -> U73_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_aaaa(X5, X1, X6, X7)) appcE_in_aaaa([], X1, X2, .(X1, X2)) -> appcE_out_aaaa([], X1, X2, .(X1, X2)) appcE_in_aaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U75_aaaa(X1, X2, X3, X4, X5, appcE_in_aaaa(X2, X3, X4, X5)) U75_aaaa(X1, X2, X3, X4, X5, appcE_out_aaaa(X2, X3, X4, X5)) -> appcE_out_aaaa(.(X1, X2), X3, X4, .(X1, X5)) U73_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_aaaa(X5, X1, X6, X7)) -> qcD_out_aaaaaaa(X1, X2, X3, X4, X5, X6, X7) U74_aa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_aa(.(X1, X2), X3) qscG_in_aaa(X1, X2, X3) -> U76_aaa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) U76_aaa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscG_out_aaa(X1, X2, X3) lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) partcC_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) U67_ggaa(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> U68_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) U64_ggaa(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> U65_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], []) U66_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) U65_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) U68_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_ggaa(X1, X2, X3, X4, X5, partcC_in_ggaa(X1, X3, X4, X5)) partcC_in_ggaa(X1, [], [], []) -> partcC_out_ggaa(X1, [], [], []) U69_ggaa(X1, X2, X3, X4, X5, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) qscF_in_ga([], []) -> qscF_out_ga([], []) qscF_in_ga(.(X1, X2), X3) -> U74_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) qcD_in_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U70_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_in_ggaa(X1, X2, X3, X4)) U70_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_out_ggaa(X1, X2, X3, X4)) -> U71_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5)) U71_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U72_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6)) U72_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U73_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_ggga(X5, X1, X6, X7)) appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2)) appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U75_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5)) U75_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5)) U73_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) U74_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3) The argument filtering Pi contains the following mapping: [] = [] lesscA_in_aa(x1, x2) = lesscA_in_aa lesscA_out_aa(x1, x2) = lesscA_out_aa(x1) U63_aa(x1, x2, x3) = U63_aa(x3) s(x1) = s(x1) lesscA_in_ga(x1, x2) = lesscA_in_ga(x1) 0 = 0 lesscA_out_ga(x1, x2) = lesscA_out_ga(x1) U63_ga(x1, x2, x3) = U63_ga(x1, x3) partcB_in_gaaa(x1, x2, x3, x4) = partcB_in_gaaa(x1) U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6) U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x6) U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6) partcB_out_gaaa(x1, x2, x3, x4) = partcB_out_gaaa(x1) partcC_in_aaaa(x1, x2, x3, x4) = partcC_in_aaaa U67_aaaa(x1, x2, x3, x4, x5, x6) = U67_aaaa(x6) U68_aaaa(x1, x2, x3, x4, x5, x6) = U68_aaaa(x6) partcC_out_aaaa(x1, x2, x3, x4) = partcC_out_aaaa U69_aaaa(x1, x2, x3, x4, x5, x6) = U69_aaaa(x6) qscF_in_aa(x1, x2) = qscF_in_aa qscF_out_aa(x1, x2) = qscF_out_aa U74_aa(x1, x2, x3, x4) = U74_aa(x4) qcD_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_aaaaaaa U70_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_aaaaaaa(x8) U71_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_aaaaaaa(x8) U72_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_aaaaaaa(x8) U73_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_aaaaaaa(x8) .(x1, x2) = .(x1, x2) qcD_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_aaaaaaa appcE_in_aaaa(x1, x2, x3, x4) = appcE_in_aaaa appcE_out_aaaa(x1, x2, x3, x4) = appcE_out_aaaa U75_aaaa(x1, x2, x3, x4, x5, x6) = U75_aaaa(x6) qscG_in_aaa(x1, x2, x3) = qscG_in_aaa U76_aaa(x1, x2, x3, x4) = U76_aaa(x4) qscG_out_aaa(x1, x2, x3) = qscG_out_aaa lesscA_in_gg(x1, x2) = lesscA_in_gg(x1, x2) lesscA_out_gg(x1, x2) = lesscA_out_gg(x1, x2) U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3) partcC_in_ggaa(x1, x2, x3, x4) = partcC_in_ggaa(x1, x2) U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6) U68_ggaa(x1, x2, x3, x4, x5, x6) = U68_ggaa(x1, x2, x3, x6) partcB_in_ggaa(x1, x2, x3, x4) = partcB_in_ggaa(x1, x2) U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6) U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6) U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6) partcB_out_ggaa(x1, x2, x3, x4) = partcB_out_ggaa(x1, x2, x3, x4) partcC_out_ggaa(x1, x2, x3, x4) = partcC_out_ggaa(x1, x2, x3, x4) U69_ggaa(x1, x2, x3, x4, x5, x6) = U69_ggaa(x1, x2, x3, x6) qscF_in_ga(x1, x2) = qscF_in_ga(x1) qscF_out_ga(x1, x2) = qscF_out_ga(x1, x2) U74_ga(x1, x2, x3, x4) = U74_ga(x1, x2, x4) qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_ggaaaaa(x1, x2) U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x8) U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x8) U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x8) U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x8) appcE_in_ggga(x1, x2, x3, x4) = appcE_in_ggga(x1, x2, x3) appcE_out_ggga(x1, x2, x3, x4) = appcE_out_ggga(x1, x2, x3, x4) U75_ggga(x1, x2, x3, x4, x5, x6) = U75_ggga(x1, x2, x3, x4, x6) qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) APPH_IN_AGAG(x1, x2, x3, x4) = APPH_IN_AGAG(x2, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (167) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (168) Obligation: Pi DP problem: The TRS P consists of the following rules: APPH_IN_AGAG(.(X1, X2), X3, X4, .(X1, X5)) -> APPH_IN_AGAG(X2, X3, X4, X5) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) APPH_IN_AGAG(x1, x2, x3, x4) = APPH_IN_AGAG(x2, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (169) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (170) Obligation: Q DP problem: The TRS P consists of the following rules: APPH_IN_AGAG(X3, .(X1, X5)) -> APPH_IN_AGAG(X3, X5) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (171) 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: *APPH_IN_AGAG(X3, .(X1, X5)) -> APPH_IN_AGAG(X3, X5) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (172) YES ---------------------------------------- (173) Obligation: Pi DP problem: The TRS P consists of the following rules: APPE_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) -> APPE_IN_GGGA(X2, X3, X4, X5) The TRS R consists of the following rules: lesscA_in_aa(0, s(X1)) -> lesscA_out_aa(0, s(X1)) lesscA_in_aa(s(X1), s(X2)) -> U63_aa(X1, X2, lesscA_in_aa(X1, X2)) U63_aa(X1, X2, lesscA_out_aa(X1, X2)) -> lesscA_out_aa(s(X1), s(X2)) lesscA_in_ga(0, s(X1)) -> lesscA_out_ga(0, s(X1)) lesscA_in_ga(s(X1), s(X2)) -> U63_ga(X1, X2, lesscA_in_ga(X1, X2)) U63_ga(X1, X2, lesscA_out_ga(X1, X2)) -> lesscA_out_ga(s(X1), s(X2)) partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_gaaa(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) U64_gaaa(X1, X2, X3, X4, X5, lesscA_out_ga(X1, X2)) -> U65_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], []) U66_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5)) U65_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_aaaa(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) U67_aaaa(X1, X2, X3, X4, X5, lesscA_out_aa(X1, X2)) -> U68_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) U68_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_aaaa(X1, X2, X3, X4, X5, partcC_in_aaaa(X1, X3, X4, X5)) partcC_in_aaaa(X1, [], [], []) -> partcC_out_aaaa(X1, [], [], []) U69_aaaa(X1, X2, X3, X4, X5, partcC_out_aaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), X4, .(X2, X5)) qscF_in_aa([], []) -> qscF_out_aa([], []) qscF_in_aa(.(X1, X2), X3) -> U74_aa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) qcD_in_aaaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U70_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_in_aaaa(X1, X2, X3, X4)) U70_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_out_aaaa(X1, X2, X3, X4)) -> U71_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X3, X5)) U71_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X3, X5)) -> U72_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X4, X6)) U72_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X4, X6)) -> U73_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_aaaa(X5, X1, X6, X7)) appcE_in_aaaa([], X1, X2, .(X1, X2)) -> appcE_out_aaaa([], X1, X2, .(X1, X2)) appcE_in_aaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U75_aaaa(X1, X2, X3, X4, X5, appcE_in_aaaa(X2, X3, X4, X5)) U75_aaaa(X1, X2, X3, X4, X5, appcE_out_aaaa(X2, X3, X4, X5)) -> appcE_out_aaaa(.(X1, X2), X3, X4, .(X1, X5)) U73_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_aaaa(X5, X1, X6, X7)) -> qcD_out_aaaaaaa(X1, X2, X3, X4, X5, X6, X7) U74_aa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_aa(.(X1, X2), X3) qscG_in_aaa(X1, X2, X3) -> U76_aaa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) U76_aaa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscG_out_aaa(X1, X2, X3) lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) partcC_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) U67_ggaa(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> U68_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) U64_ggaa(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> U65_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], []) U66_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) U65_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) U68_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_ggaa(X1, X2, X3, X4, X5, partcC_in_ggaa(X1, X3, X4, X5)) partcC_in_ggaa(X1, [], [], []) -> partcC_out_ggaa(X1, [], [], []) U69_ggaa(X1, X2, X3, X4, X5, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) qscF_in_ga([], []) -> qscF_out_ga([], []) qscF_in_ga(.(X1, X2), X3) -> U74_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) qcD_in_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U70_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_in_ggaa(X1, X2, X3, X4)) U70_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_out_ggaa(X1, X2, X3, X4)) -> U71_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5)) U71_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U72_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6)) U72_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U73_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_ggga(X5, X1, X6, X7)) appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2)) appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U75_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5)) U75_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5)) U73_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) U74_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3) The argument filtering Pi contains the following mapping: [] = [] lesscA_in_aa(x1, x2) = lesscA_in_aa lesscA_out_aa(x1, x2) = lesscA_out_aa(x1) U63_aa(x1, x2, x3) = U63_aa(x3) s(x1) = s(x1) lesscA_in_ga(x1, x2) = lesscA_in_ga(x1) 0 = 0 lesscA_out_ga(x1, x2) = lesscA_out_ga(x1) U63_ga(x1, x2, x3) = U63_ga(x1, x3) partcB_in_gaaa(x1, x2, x3, x4) = partcB_in_gaaa(x1) U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6) U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x6) U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6) partcB_out_gaaa(x1, x2, x3, x4) = partcB_out_gaaa(x1) partcC_in_aaaa(x1, x2, x3, x4) = partcC_in_aaaa U67_aaaa(x1, x2, x3, x4, x5, x6) = U67_aaaa(x6) U68_aaaa(x1, x2, x3, x4, x5, x6) = U68_aaaa(x6) partcC_out_aaaa(x1, x2, x3, x4) = partcC_out_aaaa U69_aaaa(x1, x2, x3, x4, x5, x6) = U69_aaaa(x6) qscF_in_aa(x1, x2) = qscF_in_aa qscF_out_aa(x1, x2) = qscF_out_aa U74_aa(x1, x2, x3, x4) = U74_aa(x4) qcD_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_aaaaaaa U70_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_aaaaaaa(x8) U71_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_aaaaaaa(x8) U72_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_aaaaaaa(x8) U73_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_aaaaaaa(x8) .(x1, x2) = .(x1, x2) qcD_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_aaaaaaa appcE_in_aaaa(x1, x2, x3, x4) = appcE_in_aaaa appcE_out_aaaa(x1, x2, x3, x4) = appcE_out_aaaa U75_aaaa(x1, x2, x3, x4, x5, x6) = U75_aaaa(x6) qscG_in_aaa(x1, x2, x3) = qscG_in_aaa U76_aaa(x1, x2, x3, x4) = U76_aaa(x4) qscG_out_aaa(x1, x2, x3) = qscG_out_aaa lesscA_in_gg(x1, x2) = lesscA_in_gg(x1, x2) lesscA_out_gg(x1, x2) = lesscA_out_gg(x1, x2) U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3) partcC_in_ggaa(x1, x2, x3, x4) = partcC_in_ggaa(x1, x2) U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6) U68_ggaa(x1, x2, x3, x4, x5, x6) = U68_ggaa(x1, x2, x3, x6) partcB_in_ggaa(x1, x2, x3, x4) = partcB_in_ggaa(x1, x2) U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6) U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6) U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6) partcB_out_ggaa(x1, x2, x3, x4) = partcB_out_ggaa(x1, x2, x3, x4) partcC_out_ggaa(x1, x2, x3, x4) = partcC_out_ggaa(x1, x2, x3, x4) U69_ggaa(x1, x2, x3, x4, x5, x6) = U69_ggaa(x1, x2, x3, x6) qscF_in_ga(x1, x2) = qscF_in_ga(x1) qscF_out_ga(x1, x2) = qscF_out_ga(x1, x2) U74_ga(x1, x2, x3, x4) = U74_ga(x1, x2, x4) qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_ggaaaaa(x1, x2) U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x8) U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x8) U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x8) U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x8) appcE_in_ggga(x1, x2, x3, x4) = appcE_in_ggga(x1, x2, x3) appcE_out_ggga(x1, x2, x3, x4) = appcE_out_ggga(x1, x2, x3, x4) U75_ggga(x1, x2, x3, x4, x5, x6) = U75_ggga(x1, x2, x3, x4, x6) qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) APPE_IN_GGGA(x1, x2, x3, x4) = APPE_IN_GGGA(x1, x2, x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (174) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (175) Obligation: Pi DP problem: The TRS P consists of the following rules: APPE_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) -> APPE_IN_GGGA(X2, X3, X4, X5) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) APPE_IN_GGGA(x1, x2, x3, x4) = APPE_IN_GGGA(x1, x2, x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (176) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (177) Obligation: Q DP problem: The TRS P consists of the following rules: APPE_IN_GGGA(.(X1, X2), X3, X4) -> APPE_IN_GGGA(X2, X3, X4) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (178) 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: *APPE_IN_GGGA(.(X1, X2), X3, X4) -> APPE_IN_GGGA(X2, X3, X4) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 ---------------------------------------- (179) YES ---------------------------------------- (180) Obligation: Pi DP problem: The TRS P consists of the following rules: LESSA_IN_GG(s(X1), s(X2)) -> LESSA_IN_GG(X1, X2) The TRS R consists of the following rules: lesscA_in_aa(0, s(X1)) -> lesscA_out_aa(0, s(X1)) lesscA_in_aa(s(X1), s(X2)) -> U63_aa(X1, X2, lesscA_in_aa(X1, X2)) U63_aa(X1, X2, lesscA_out_aa(X1, X2)) -> lesscA_out_aa(s(X1), s(X2)) lesscA_in_ga(0, s(X1)) -> lesscA_out_ga(0, s(X1)) lesscA_in_ga(s(X1), s(X2)) -> U63_ga(X1, X2, lesscA_in_ga(X1, X2)) U63_ga(X1, X2, lesscA_out_ga(X1, X2)) -> lesscA_out_ga(s(X1), s(X2)) partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_gaaa(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) U64_gaaa(X1, X2, X3, X4, X5, lesscA_out_ga(X1, X2)) -> U65_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], []) U66_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5)) U65_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_aaaa(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) U67_aaaa(X1, X2, X3, X4, X5, lesscA_out_aa(X1, X2)) -> U68_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) U68_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_aaaa(X1, X2, X3, X4, X5, partcC_in_aaaa(X1, X3, X4, X5)) partcC_in_aaaa(X1, [], [], []) -> partcC_out_aaaa(X1, [], [], []) U69_aaaa(X1, X2, X3, X4, X5, partcC_out_aaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), X4, .(X2, X5)) qscF_in_aa([], []) -> qscF_out_aa([], []) qscF_in_aa(.(X1, X2), X3) -> U74_aa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) qcD_in_aaaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U70_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_in_aaaa(X1, X2, X3, X4)) U70_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_out_aaaa(X1, X2, X3, X4)) -> U71_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X3, X5)) U71_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X3, X5)) -> U72_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X4, X6)) U72_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X4, X6)) -> U73_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_aaaa(X5, X1, X6, X7)) appcE_in_aaaa([], X1, X2, .(X1, X2)) -> appcE_out_aaaa([], X1, X2, .(X1, X2)) appcE_in_aaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U75_aaaa(X1, X2, X3, X4, X5, appcE_in_aaaa(X2, X3, X4, X5)) U75_aaaa(X1, X2, X3, X4, X5, appcE_out_aaaa(X2, X3, X4, X5)) -> appcE_out_aaaa(.(X1, X2), X3, X4, .(X1, X5)) U73_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_aaaa(X5, X1, X6, X7)) -> qcD_out_aaaaaaa(X1, X2, X3, X4, X5, X6, X7) U74_aa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_aa(.(X1, X2), X3) qscG_in_aaa(X1, X2, X3) -> U76_aaa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) U76_aaa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscG_out_aaa(X1, X2, X3) lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) partcC_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) U67_ggaa(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> U68_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) U64_ggaa(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> U65_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], []) U66_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) U65_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) U68_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_ggaa(X1, X2, X3, X4, X5, partcC_in_ggaa(X1, X3, X4, X5)) partcC_in_ggaa(X1, [], [], []) -> partcC_out_ggaa(X1, [], [], []) U69_ggaa(X1, X2, X3, X4, X5, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) qscF_in_ga([], []) -> qscF_out_ga([], []) qscF_in_ga(.(X1, X2), X3) -> U74_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) qcD_in_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U70_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_in_ggaa(X1, X2, X3, X4)) U70_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_out_ggaa(X1, X2, X3, X4)) -> U71_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5)) U71_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U72_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6)) U72_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U73_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_ggga(X5, X1, X6, X7)) appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2)) appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U75_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5)) U75_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5)) U73_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) U74_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3) The argument filtering Pi contains the following mapping: [] = [] lesscA_in_aa(x1, x2) = lesscA_in_aa lesscA_out_aa(x1, x2) = lesscA_out_aa(x1) U63_aa(x1, x2, x3) = U63_aa(x3) s(x1) = s(x1) lesscA_in_ga(x1, x2) = lesscA_in_ga(x1) 0 = 0 lesscA_out_ga(x1, x2) = lesscA_out_ga(x1) U63_ga(x1, x2, x3) = U63_ga(x1, x3) partcB_in_gaaa(x1, x2, x3, x4) = partcB_in_gaaa(x1) U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6) U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x6) U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6) partcB_out_gaaa(x1, x2, x3, x4) = partcB_out_gaaa(x1) partcC_in_aaaa(x1, x2, x3, x4) = partcC_in_aaaa U67_aaaa(x1, x2, x3, x4, x5, x6) = U67_aaaa(x6) U68_aaaa(x1, x2, x3, x4, x5, x6) = U68_aaaa(x6) partcC_out_aaaa(x1, x2, x3, x4) = partcC_out_aaaa U69_aaaa(x1, x2, x3, x4, x5, x6) = U69_aaaa(x6) qscF_in_aa(x1, x2) = qscF_in_aa qscF_out_aa(x1, x2) = qscF_out_aa U74_aa(x1, x2, x3, x4) = U74_aa(x4) qcD_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_aaaaaaa U70_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_aaaaaaa(x8) U71_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_aaaaaaa(x8) U72_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_aaaaaaa(x8) U73_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_aaaaaaa(x8) .(x1, x2) = .(x1, x2) qcD_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_aaaaaaa appcE_in_aaaa(x1, x2, x3, x4) = appcE_in_aaaa appcE_out_aaaa(x1, x2, x3, x4) = appcE_out_aaaa U75_aaaa(x1, x2, x3, x4, x5, x6) = U75_aaaa(x6) qscG_in_aaa(x1, x2, x3) = qscG_in_aaa U76_aaa(x1, x2, x3, x4) = U76_aaa(x4) qscG_out_aaa(x1, x2, x3) = qscG_out_aaa lesscA_in_gg(x1, x2) = lesscA_in_gg(x1, x2) lesscA_out_gg(x1, x2) = lesscA_out_gg(x1, x2) U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3) partcC_in_ggaa(x1, x2, x3, x4) = partcC_in_ggaa(x1, x2) U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6) U68_ggaa(x1, x2, x3, x4, x5, x6) = U68_ggaa(x1, x2, x3, x6) partcB_in_ggaa(x1, x2, x3, x4) = partcB_in_ggaa(x1, x2) U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6) U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6) U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6) partcB_out_ggaa(x1, x2, x3, x4) = partcB_out_ggaa(x1, x2, x3, x4) partcC_out_ggaa(x1, x2, x3, x4) = partcC_out_ggaa(x1, x2, x3, x4) U69_ggaa(x1, x2, x3, x4, x5, x6) = U69_ggaa(x1, x2, x3, x6) qscF_in_ga(x1, x2) = qscF_in_ga(x1) qscF_out_ga(x1, x2) = qscF_out_ga(x1, x2) U74_ga(x1, x2, x3, x4) = U74_ga(x1, x2, x4) qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_ggaaaaa(x1, x2) U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x8) U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x8) U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x8) U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x8) appcE_in_ggga(x1, x2, x3, x4) = appcE_in_ggga(x1, x2, x3) appcE_out_ggga(x1, x2, x3, x4) = appcE_out_ggga(x1, x2, x3, x4) U75_ggga(x1, x2, x3, x4, x5, x6) = U75_ggga(x1, x2, x3, x4, x6) qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) LESSA_IN_GG(x1, x2) = LESSA_IN_GG(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (181) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (182) Obligation: Pi DP problem: The TRS P consists of the following rules: LESSA_IN_GG(s(X1), s(X2)) -> LESSA_IN_GG(X1, X2) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (183) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (184) Obligation: Q DP problem: The TRS P consists of the following rules: LESSA_IN_GG(s(X1), s(X2)) -> LESSA_IN_GG(X1, X2) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (185) 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: *LESSA_IN_GG(s(X1), s(X2)) -> LESSA_IN_GG(X1, X2) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (186) YES ---------------------------------------- (187) Obligation: Pi DP problem: The TRS P consists of the following rules: PARTB_IN_GGAA(X1, .(X2, X3), .(X2, X4), X5) -> U3_GGAA(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) U3_GGAA(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> PARTB_IN_GGAA(X1, X3, X4, X5) PARTB_IN_GGAA(X1, .(X2, X3), X4, .(X2, X5)) -> PARTB_IN_GGAA(X1, X3, X4, X5) The TRS R consists of the following rules: lesscA_in_aa(0, s(X1)) -> lesscA_out_aa(0, s(X1)) lesscA_in_aa(s(X1), s(X2)) -> U63_aa(X1, X2, lesscA_in_aa(X1, X2)) U63_aa(X1, X2, lesscA_out_aa(X1, X2)) -> lesscA_out_aa(s(X1), s(X2)) lesscA_in_ga(0, s(X1)) -> lesscA_out_ga(0, s(X1)) lesscA_in_ga(s(X1), s(X2)) -> U63_ga(X1, X2, lesscA_in_ga(X1, X2)) U63_ga(X1, X2, lesscA_out_ga(X1, X2)) -> lesscA_out_ga(s(X1), s(X2)) partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_gaaa(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) U64_gaaa(X1, X2, X3, X4, X5, lesscA_out_ga(X1, X2)) -> U65_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], []) U66_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5)) U65_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_aaaa(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) U67_aaaa(X1, X2, X3, X4, X5, lesscA_out_aa(X1, X2)) -> U68_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) U68_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_aaaa(X1, X2, X3, X4, X5, partcC_in_aaaa(X1, X3, X4, X5)) partcC_in_aaaa(X1, [], [], []) -> partcC_out_aaaa(X1, [], [], []) U69_aaaa(X1, X2, X3, X4, X5, partcC_out_aaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), X4, .(X2, X5)) qscF_in_aa([], []) -> qscF_out_aa([], []) qscF_in_aa(.(X1, X2), X3) -> U74_aa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) qcD_in_aaaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U70_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_in_aaaa(X1, X2, X3, X4)) U70_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_out_aaaa(X1, X2, X3, X4)) -> U71_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X3, X5)) U71_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X3, X5)) -> U72_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X4, X6)) U72_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X4, X6)) -> U73_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_aaaa(X5, X1, X6, X7)) appcE_in_aaaa([], X1, X2, .(X1, X2)) -> appcE_out_aaaa([], X1, X2, .(X1, X2)) appcE_in_aaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U75_aaaa(X1, X2, X3, X4, X5, appcE_in_aaaa(X2, X3, X4, X5)) U75_aaaa(X1, X2, X3, X4, X5, appcE_out_aaaa(X2, X3, X4, X5)) -> appcE_out_aaaa(.(X1, X2), X3, X4, .(X1, X5)) U73_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_aaaa(X5, X1, X6, X7)) -> qcD_out_aaaaaaa(X1, X2, X3, X4, X5, X6, X7) U74_aa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_aa(.(X1, X2), X3) qscG_in_aaa(X1, X2, X3) -> U76_aaa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) U76_aaa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscG_out_aaa(X1, X2, X3) lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) partcC_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) U67_ggaa(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> U68_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) U64_ggaa(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> U65_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], []) U66_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) U65_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) U68_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_ggaa(X1, X2, X3, X4, X5, partcC_in_ggaa(X1, X3, X4, X5)) partcC_in_ggaa(X1, [], [], []) -> partcC_out_ggaa(X1, [], [], []) U69_ggaa(X1, X2, X3, X4, X5, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) qscF_in_ga([], []) -> qscF_out_ga([], []) qscF_in_ga(.(X1, X2), X3) -> U74_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) qcD_in_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U70_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_in_ggaa(X1, X2, X3, X4)) U70_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_out_ggaa(X1, X2, X3, X4)) -> U71_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5)) U71_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U72_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6)) U72_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U73_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_ggga(X5, X1, X6, X7)) appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2)) appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U75_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5)) U75_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5)) U73_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) U74_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3) The argument filtering Pi contains the following mapping: [] = [] lesscA_in_aa(x1, x2) = lesscA_in_aa lesscA_out_aa(x1, x2) = lesscA_out_aa(x1) U63_aa(x1, x2, x3) = U63_aa(x3) s(x1) = s(x1) lesscA_in_ga(x1, x2) = lesscA_in_ga(x1) 0 = 0 lesscA_out_ga(x1, x2) = lesscA_out_ga(x1) U63_ga(x1, x2, x3) = U63_ga(x1, x3) partcB_in_gaaa(x1, x2, x3, x4) = partcB_in_gaaa(x1) U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6) U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x6) U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6) partcB_out_gaaa(x1, x2, x3, x4) = partcB_out_gaaa(x1) partcC_in_aaaa(x1, x2, x3, x4) = partcC_in_aaaa U67_aaaa(x1, x2, x3, x4, x5, x6) = U67_aaaa(x6) U68_aaaa(x1, x2, x3, x4, x5, x6) = U68_aaaa(x6) partcC_out_aaaa(x1, x2, x3, x4) = partcC_out_aaaa U69_aaaa(x1, x2, x3, x4, x5, x6) = U69_aaaa(x6) qscF_in_aa(x1, x2) = qscF_in_aa qscF_out_aa(x1, x2) = qscF_out_aa U74_aa(x1, x2, x3, x4) = U74_aa(x4) qcD_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_aaaaaaa U70_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_aaaaaaa(x8) U71_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_aaaaaaa(x8) U72_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_aaaaaaa(x8) U73_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_aaaaaaa(x8) .(x1, x2) = .(x1, x2) qcD_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_aaaaaaa appcE_in_aaaa(x1, x2, x3, x4) = appcE_in_aaaa appcE_out_aaaa(x1, x2, x3, x4) = appcE_out_aaaa U75_aaaa(x1, x2, x3, x4, x5, x6) = U75_aaaa(x6) qscG_in_aaa(x1, x2, x3) = qscG_in_aaa U76_aaa(x1, x2, x3, x4) = U76_aaa(x4) qscG_out_aaa(x1, x2, x3) = qscG_out_aaa lesscA_in_gg(x1, x2) = lesscA_in_gg(x1, x2) lesscA_out_gg(x1, x2) = lesscA_out_gg(x1, x2) U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3) partcC_in_ggaa(x1, x2, x3, x4) = partcC_in_ggaa(x1, x2) U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6) U68_ggaa(x1, x2, x3, x4, x5, x6) = U68_ggaa(x1, x2, x3, x6) partcB_in_ggaa(x1, x2, x3, x4) = partcB_in_ggaa(x1, x2) U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6) U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6) U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6) partcB_out_ggaa(x1, x2, x3, x4) = partcB_out_ggaa(x1, x2, x3, x4) partcC_out_ggaa(x1, x2, x3, x4) = partcC_out_ggaa(x1, x2, x3, x4) U69_ggaa(x1, x2, x3, x4, x5, x6) = U69_ggaa(x1, x2, x3, x6) qscF_in_ga(x1, x2) = qscF_in_ga(x1) qscF_out_ga(x1, x2) = qscF_out_ga(x1, x2) U74_ga(x1, x2, x3, x4) = U74_ga(x1, x2, x4) qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_ggaaaaa(x1, x2) U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x8) U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x8) U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x8) U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x8) appcE_in_ggga(x1, x2, x3, x4) = appcE_in_ggga(x1, x2, x3) appcE_out_ggga(x1, x2, x3, x4) = appcE_out_ggga(x1, x2, x3, x4) U75_ggga(x1, x2, x3, x4, x5, x6) = U75_ggga(x1, x2, x3, x4, x6) qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) PARTB_IN_GGAA(x1, x2, x3, x4) = PARTB_IN_GGAA(x1, x2) U3_GGAA(x1, x2, x3, x4, x5, x6) = U3_GGAA(x1, x2, x3, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (188) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (189) Obligation: Pi DP problem: The TRS P consists of the following rules: PARTB_IN_GGAA(X1, .(X2, X3), .(X2, X4), X5) -> U3_GGAA(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) U3_GGAA(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> PARTB_IN_GGAA(X1, X3, X4, X5) PARTB_IN_GGAA(X1, .(X2, X3), X4, .(X2, X5)) -> PARTB_IN_GGAA(X1, X3, X4, X5) The TRS R consists of the following rules: lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) The argument filtering Pi contains the following mapping: s(x1) = s(x1) 0 = 0 .(x1, x2) = .(x1, x2) lesscA_in_gg(x1, x2) = lesscA_in_gg(x1, x2) lesscA_out_gg(x1, x2) = lesscA_out_gg(x1, x2) U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3) PARTB_IN_GGAA(x1, x2, x3, x4) = PARTB_IN_GGAA(x1, x2) U3_GGAA(x1, x2, x3, x4, x5, x6) = U3_GGAA(x1, x2, x3, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (190) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (191) Obligation: Q DP problem: The TRS P consists of the following rules: PARTB_IN_GGAA(X1, .(X2, X3)) -> U3_GGAA(X1, X2, X3, lesscA_in_gg(X1, X2)) U3_GGAA(X1, X2, X3, lesscA_out_gg(X1, X2)) -> PARTB_IN_GGAA(X1, X3) PARTB_IN_GGAA(X1, .(X2, X3)) -> PARTB_IN_GGAA(X1, X3) The TRS R consists of the following rules: lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) The set Q consists of the following terms: lesscA_in_gg(x0, x1) U63_gg(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (192) 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: *U3_GGAA(X1, X2, X3, lesscA_out_gg(X1, X2)) -> PARTB_IN_GGAA(X1, X3) The graph contains the following edges 1 >= 1, 4 > 1, 3 >= 2 *PARTB_IN_GGAA(X1, .(X2, X3)) -> PARTB_IN_GGAA(X1, X3) The graph contains the following edges 1 >= 1, 2 > 2 *PARTB_IN_GGAA(X1, .(X2, X3)) -> U3_GGAA(X1, X2, X3, lesscA_in_gg(X1, X2)) The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3 ---------------------------------------- (193) YES ---------------------------------------- (194) Obligation: Pi DP problem: The TRS P consists of the following rules: PARTC_IN_GGAA(X1, .(X2, X3), X4, .(X2, X5)) -> PARTC_IN_GGAA(X1, X3, X4, X5) The TRS R consists of the following rules: lesscA_in_aa(0, s(X1)) -> lesscA_out_aa(0, s(X1)) lesscA_in_aa(s(X1), s(X2)) -> U63_aa(X1, X2, lesscA_in_aa(X1, X2)) U63_aa(X1, X2, lesscA_out_aa(X1, X2)) -> lesscA_out_aa(s(X1), s(X2)) lesscA_in_ga(0, s(X1)) -> lesscA_out_ga(0, s(X1)) lesscA_in_ga(s(X1), s(X2)) -> U63_ga(X1, X2, lesscA_in_ga(X1, X2)) U63_ga(X1, X2, lesscA_out_ga(X1, X2)) -> lesscA_out_ga(s(X1), s(X2)) partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_gaaa(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) U64_gaaa(X1, X2, X3, X4, X5, lesscA_out_ga(X1, X2)) -> U65_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], []) U66_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5)) U65_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_aaaa(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) U67_aaaa(X1, X2, X3, X4, X5, lesscA_out_aa(X1, X2)) -> U68_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) U68_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_aaaa(X1, X2, X3, X4, X5, partcC_in_aaaa(X1, X3, X4, X5)) partcC_in_aaaa(X1, [], [], []) -> partcC_out_aaaa(X1, [], [], []) U69_aaaa(X1, X2, X3, X4, X5, partcC_out_aaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), X4, .(X2, X5)) qscF_in_aa([], []) -> qscF_out_aa([], []) qscF_in_aa(.(X1, X2), X3) -> U74_aa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) qcD_in_aaaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U70_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_in_aaaa(X1, X2, X3, X4)) U70_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_out_aaaa(X1, X2, X3, X4)) -> U71_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X3, X5)) U71_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X3, X5)) -> U72_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X4, X6)) U72_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X4, X6)) -> U73_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_aaaa(X5, X1, X6, X7)) appcE_in_aaaa([], X1, X2, .(X1, X2)) -> appcE_out_aaaa([], X1, X2, .(X1, X2)) appcE_in_aaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U75_aaaa(X1, X2, X3, X4, X5, appcE_in_aaaa(X2, X3, X4, X5)) U75_aaaa(X1, X2, X3, X4, X5, appcE_out_aaaa(X2, X3, X4, X5)) -> appcE_out_aaaa(.(X1, X2), X3, X4, .(X1, X5)) U73_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_aaaa(X5, X1, X6, X7)) -> qcD_out_aaaaaaa(X1, X2, X3, X4, X5, X6, X7) U74_aa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_aa(.(X1, X2), X3) qscG_in_aaa(X1, X2, X3) -> U76_aaa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) U76_aaa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscG_out_aaa(X1, X2, X3) lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) partcC_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) U67_ggaa(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> U68_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) U64_ggaa(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> U65_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], []) U66_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) U65_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) U68_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_ggaa(X1, X2, X3, X4, X5, partcC_in_ggaa(X1, X3, X4, X5)) partcC_in_ggaa(X1, [], [], []) -> partcC_out_ggaa(X1, [], [], []) U69_ggaa(X1, X2, X3, X4, X5, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) qscF_in_ga([], []) -> qscF_out_ga([], []) qscF_in_ga(.(X1, X2), X3) -> U74_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) qcD_in_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U70_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_in_ggaa(X1, X2, X3, X4)) U70_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_out_ggaa(X1, X2, X3, X4)) -> U71_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5)) U71_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U72_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6)) U72_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U73_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_ggga(X5, X1, X6, X7)) appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2)) appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U75_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5)) U75_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5)) U73_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) U74_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3) The argument filtering Pi contains the following mapping: [] = [] lesscA_in_aa(x1, x2) = lesscA_in_aa lesscA_out_aa(x1, x2) = lesscA_out_aa(x1) U63_aa(x1, x2, x3) = U63_aa(x3) s(x1) = s(x1) lesscA_in_ga(x1, x2) = lesscA_in_ga(x1) 0 = 0 lesscA_out_ga(x1, x2) = lesscA_out_ga(x1) U63_ga(x1, x2, x3) = U63_ga(x1, x3) partcB_in_gaaa(x1, x2, x3, x4) = partcB_in_gaaa(x1) U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6) U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x6) U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6) partcB_out_gaaa(x1, x2, x3, x4) = partcB_out_gaaa(x1) partcC_in_aaaa(x1, x2, x3, x4) = partcC_in_aaaa U67_aaaa(x1, x2, x3, x4, x5, x6) = U67_aaaa(x6) U68_aaaa(x1, x2, x3, x4, x5, x6) = U68_aaaa(x6) partcC_out_aaaa(x1, x2, x3, x4) = partcC_out_aaaa U69_aaaa(x1, x2, x3, x4, x5, x6) = U69_aaaa(x6) qscF_in_aa(x1, x2) = qscF_in_aa qscF_out_aa(x1, x2) = qscF_out_aa U74_aa(x1, x2, x3, x4) = U74_aa(x4) qcD_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_aaaaaaa U70_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_aaaaaaa(x8) U71_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_aaaaaaa(x8) U72_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_aaaaaaa(x8) U73_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_aaaaaaa(x8) .(x1, x2) = .(x1, x2) qcD_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_aaaaaaa appcE_in_aaaa(x1, x2, x3, x4) = appcE_in_aaaa appcE_out_aaaa(x1, x2, x3, x4) = appcE_out_aaaa U75_aaaa(x1, x2, x3, x4, x5, x6) = U75_aaaa(x6) qscG_in_aaa(x1, x2, x3) = qscG_in_aaa U76_aaa(x1, x2, x3, x4) = U76_aaa(x4) qscG_out_aaa(x1, x2, x3) = qscG_out_aaa lesscA_in_gg(x1, x2) = lesscA_in_gg(x1, x2) lesscA_out_gg(x1, x2) = lesscA_out_gg(x1, x2) U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3) partcC_in_ggaa(x1, x2, x3, x4) = partcC_in_ggaa(x1, x2) U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6) U68_ggaa(x1, x2, x3, x4, x5, x6) = U68_ggaa(x1, x2, x3, x6) partcB_in_ggaa(x1, x2, x3, x4) = partcB_in_ggaa(x1, x2) U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6) U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6) U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6) partcB_out_ggaa(x1, x2, x3, x4) = partcB_out_ggaa(x1, x2, x3, x4) partcC_out_ggaa(x1, x2, x3, x4) = partcC_out_ggaa(x1, x2, x3, x4) U69_ggaa(x1, x2, x3, x4, x5, x6) = U69_ggaa(x1, x2, x3, x6) qscF_in_ga(x1, x2) = qscF_in_ga(x1) qscF_out_ga(x1, x2) = qscF_out_ga(x1, x2) U74_ga(x1, x2, x3, x4) = U74_ga(x1, x2, x4) qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_ggaaaaa(x1, x2) U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x8) U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x8) U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x8) U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x8) appcE_in_ggga(x1, x2, x3, x4) = appcE_in_ggga(x1, x2, x3) appcE_out_ggga(x1, x2, x3, x4) = appcE_out_ggga(x1, x2, x3, x4) U75_ggga(x1, x2, x3, x4, x5, x6) = U75_ggga(x1, x2, x3, x4, x6) qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) PARTC_IN_GGAA(x1, x2, x3, x4) = PARTC_IN_GGAA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (195) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (196) Obligation: Pi DP problem: The TRS P consists of the following rules: PARTC_IN_GGAA(X1, .(X2, X3), X4, .(X2, X5)) -> PARTC_IN_GGAA(X1, X3, X4, X5) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) PARTC_IN_GGAA(x1, x2, x3, x4) = PARTC_IN_GGAA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (197) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (198) Obligation: Q DP problem: The TRS P consists of the following rules: PARTC_IN_GGAA(X1, .(X2, X3)) -> PARTC_IN_GGAA(X1, X3) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (199) 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: *PARTC_IN_GGAA(X1, .(X2, X3)) -> PARTC_IN_GGAA(X1, X3) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (200) YES ---------------------------------------- (201) Obligation: Pi DP problem: The TRS P consists of the following rules: QSF_IN_GA(.(X1, X2), X3) -> PD_IN_GGAAAAA(X1, X2, X4, X5, X6, X7, X3) PD_IN_GGAAAAA(X1, X2, X3, X4, X5, X6, X7) -> U11_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, partcC_in_ggaa(X1, X2, X3, X4)) U11_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, partcC_out_ggaa(X1, X2, X3, X4)) -> QSF_IN_GA(X3, X5) U11_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, partcC_out_ggaa(X1, X2, X3, X4)) -> U13_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5)) U13_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> QSF_IN_GA(X4, X6) The TRS R consists of the following rules: lesscA_in_aa(0, s(X1)) -> lesscA_out_aa(0, s(X1)) lesscA_in_aa(s(X1), s(X2)) -> U63_aa(X1, X2, lesscA_in_aa(X1, X2)) U63_aa(X1, X2, lesscA_out_aa(X1, X2)) -> lesscA_out_aa(s(X1), s(X2)) lesscA_in_ga(0, s(X1)) -> lesscA_out_ga(0, s(X1)) lesscA_in_ga(s(X1), s(X2)) -> U63_ga(X1, X2, lesscA_in_ga(X1, X2)) U63_ga(X1, X2, lesscA_out_ga(X1, X2)) -> lesscA_out_ga(s(X1), s(X2)) partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_gaaa(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) U64_gaaa(X1, X2, X3, X4, X5, lesscA_out_ga(X1, X2)) -> U65_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], []) U66_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5)) U65_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_aaaa(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) U67_aaaa(X1, X2, X3, X4, X5, lesscA_out_aa(X1, X2)) -> U68_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) U68_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_aaaa(X1, X2, X3, X4, X5, partcC_in_aaaa(X1, X3, X4, X5)) partcC_in_aaaa(X1, [], [], []) -> partcC_out_aaaa(X1, [], [], []) U69_aaaa(X1, X2, X3, X4, X5, partcC_out_aaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), X4, .(X2, X5)) qscF_in_aa([], []) -> qscF_out_aa([], []) qscF_in_aa(.(X1, X2), X3) -> U74_aa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) qcD_in_aaaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U70_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_in_aaaa(X1, X2, X3, X4)) U70_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_out_aaaa(X1, X2, X3, X4)) -> U71_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X3, X5)) U71_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X3, X5)) -> U72_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X4, X6)) U72_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X4, X6)) -> U73_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_aaaa(X5, X1, X6, X7)) appcE_in_aaaa([], X1, X2, .(X1, X2)) -> appcE_out_aaaa([], X1, X2, .(X1, X2)) appcE_in_aaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U75_aaaa(X1, X2, X3, X4, X5, appcE_in_aaaa(X2, X3, X4, X5)) U75_aaaa(X1, X2, X3, X4, X5, appcE_out_aaaa(X2, X3, X4, X5)) -> appcE_out_aaaa(.(X1, X2), X3, X4, .(X1, X5)) U73_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_aaaa(X5, X1, X6, X7)) -> qcD_out_aaaaaaa(X1, X2, X3, X4, X5, X6, X7) U74_aa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_aa(.(X1, X2), X3) qscG_in_aaa(X1, X2, X3) -> U76_aaa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) U76_aaa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscG_out_aaa(X1, X2, X3) lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) partcC_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) U67_ggaa(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> U68_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) U64_ggaa(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> U65_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], []) U66_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) U65_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) U68_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_ggaa(X1, X2, X3, X4, X5, partcC_in_ggaa(X1, X3, X4, X5)) partcC_in_ggaa(X1, [], [], []) -> partcC_out_ggaa(X1, [], [], []) U69_ggaa(X1, X2, X3, X4, X5, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) qscF_in_ga([], []) -> qscF_out_ga([], []) qscF_in_ga(.(X1, X2), X3) -> U74_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) qcD_in_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U70_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_in_ggaa(X1, X2, X3, X4)) U70_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_out_ggaa(X1, X2, X3, X4)) -> U71_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5)) U71_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U72_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6)) U72_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U73_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_ggga(X5, X1, X6, X7)) appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2)) appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U75_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5)) U75_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5)) U73_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) U74_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3) The argument filtering Pi contains the following mapping: [] = [] lesscA_in_aa(x1, x2) = lesscA_in_aa lesscA_out_aa(x1, x2) = lesscA_out_aa(x1) U63_aa(x1, x2, x3) = U63_aa(x3) s(x1) = s(x1) lesscA_in_ga(x1, x2) = lesscA_in_ga(x1) 0 = 0 lesscA_out_ga(x1, x2) = lesscA_out_ga(x1) U63_ga(x1, x2, x3) = U63_ga(x1, x3) partcB_in_gaaa(x1, x2, x3, x4) = partcB_in_gaaa(x1) U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6) U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x6) U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6) partcB_out_gaaa(x1, x2, x3, x4) = partcB_out_gaaa(x1) partcC_in_aaaa(x1, x2, x3, x4) = partcC_in_aaaa U67_aaaa(x1, x2, x3, x4, x5, x6) = U67_aaaa(x6) U68_aaaa(x1, x2, x3, x4, x5, x6) = U68_aaaa(x6) partcC_out_aaaa(x1, x2, x3, x4) = partcC_out_aaaa U69_aaaa(x1, x2, x3, x4, x5, x6) = U69_aaaa(x6) qscF_in_aa(x1, x2) = qscF_in_aa qscF_out_aa(x1, x2) = qscF_out_aa U74_aa(x1, x2, x3, x4) = U74_aa(x4) qcD_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_aaaaaaa U70_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_aaaaaaa(x8) U71_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_aaaaaaa(x8) U72_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_aaaaaaa(x8) U73_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_aaaaaaa(x8) .(x1, x2) = .(x1, x2) qcD_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_aaaaaaa appcE_in_aaaa(x1, x2, x3, x4) = appcE_in_aaaa appcE_out_aaaa(x1, x2, x3, x4) = appcE_out_aaaa U75_aaaa(x1, x2, x3, x4, x5, x6) = U75_aaaa(x6) qscG_in_aaa(x1, x2, x3) = qscG_in_aaa U76_aaa(x1, x2, x3, x4) = U76_aaa(x4) qscG_out_aaa(x1, x2, x3) = qscG_out_aaa lesscA_in_gg(x1, x2) = lesscA_in_gg(x1, x2) lesscA_out_gg(x1, x2) = lesscA_out_gg(x1, x2) U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3) partcC_in_ggaa(x1, x2, x3, x4) = partcC_in_ggaa(x1, x2) U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6) U68_ggaa(x1, x2, x3, x4, x5, x6) = U68_ggaa(x1, x2, x3, x6) partcB_in_ggaa(x1, x2, x3, x4) = partcB_in_ggaa(x1, x2) U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6) U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6) U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6) partcB_out_ggaa(x1, x2, x3, x4) = partcB_out_ggaa(x1, x2, x3, x4) partcC_out_ggaa(x1, x2, x3, x4) = partcC_out_ggaa(x1, x2, x3, x4) U69_ggaa(x1, x2, x3, x4, x5, x6) = U69_ggaa(x1, x2, x3, x6) qscF_in_ga(x1, x2) = qscF_in_ga(x1) qscF_out_ga(x1, x2) = qscF_out_ga(x1, x2) U74_ga(x1, x2, x3, x4) = U74_ga(x1, x2, x4) qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_ggaaaaa(x1, x2) U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x8) U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x8) U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x8) U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x8) appcE_in_ggga(x1, x2, x3, x4) = appcE_in_ggga(x1, x2, x3) appcE_out_ggga(x1, x2, x3, x4) = appcE_out_ggga(x1, x2, x3, x4) U75_ggga(x1, x2, x3, x4, x5, x6) = U75_ggga(x1, x2, x3, x4, x6) qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) QSF_IN_GA(x1, x2) = QSF_IN_GA(x1) PD_IN_GGAAAAA(x1, x2, x3, x4, x5, x6, x7) = PD_IN_GGAAAAA(x1, x2) U11_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U11_GGAAAAA(x1, x2, x8) U13_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U13_GGAAAAA(x1, x2, x4, x8) We have to consider all (P,R,Pi)-chains ---------------------------------------- (202) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (203) Obligation: Pi DP problem: The TRS P consists of the following rules: QSF_IN_GA(.(X1, X2), X3) -> PD_IN_GGAAAAA(X1, X2, X4, X5, X6, X7, X3) PD_IN_GGAAAAA(X1, X2, X3, X4, X5, X6, X7) -> U11_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, partcC_in_ggaa(X1, X2, X3, X4)) U11_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, partcC_out_ggaa(X1, X2, X3, X4)) -> QSF_IN_GA(X3, X5) U11_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, partcC_out_ggaa(X1, X2, X3, X4)) -> U13_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5)) U13_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> QSF_IN_GA(X4, X6) The TRS R consists of the following rules: partcC_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) partcC_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_ggaa(X1, X2, X3, X4, X5, partcC_in_ggaa(X1, X3, X4, X5)) partcC_in_ggaa(X1, [], [], []) -> partcC_out_ggaa(X1, [], [], []) qscF_in_ga([], []) -> qscF_out_ga([], []) qscF_in_ga(.(X1, X2), X3) -> U74_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) U67_ggaa(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> U68_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) U69_ggaa(X1, X2, X3, X4, X5, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) U74_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3) lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) U68_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) qcD_in_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U70_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_in_ggaa(X1, X2, X3, X4)) U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], []) U70_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_out_ggaa(X1, X2, X3, X4)) -> U71_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5)) U64_ggaa(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> U65_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) U66_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) U71_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U72_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6)) U65_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) U72_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U73_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_ggga(X5, X1, X6, X7)) U73_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2)) appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U75_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5)) U75_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5)) The argument filtering Pi contains the following mapping: [] = [] s(x1) = s(x1) 0 = 0 .(x1, x2) = .(x1, x2) lesscA_in_gg(x1, x2) = lesscA_in_gg(x1, x2) lesscA_out_gg(x1, x2) = lesscA_out_gg(x1, x2) U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3) partcC_in_ggaa(x1, x2, x3, x4) = partcC_in_ggaa(x1, x2) U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6) U68_ggaa(x1, x2, x3, x4, x5, x6) = U68_ggaa(x1, x2, x3, x6) partcB_in_ggaa(x1, x2, x3, x4) = partcB_in_ggaa(x1, x2) U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6) U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6) U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6) partcB_out_ggaa(x1, x2, x3, x4) = partcB_out_ggaa(x1, x2, x3, x4) partcC_out_ggaa(x1, x2, x3, x4) = partcC_out_ggaa(x1, x2, x3, x4) U69_ggaa(x1, x2, x3, x4, x5, x6) = U69_ggaa(x1, x2, x3, x6) qscF_in_ga(x1, x2) = qscF_in_ga(x1) qscF_out_ga(x1, x2) = qscF_out_ga(x1, x2) U74_ga(x1, x2, x3, x4) = U74_ga(x1, x2, x4) qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_ggaaaaa(x1, x2) U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x8) U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x8) U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x8) U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x8) appcE_in_ggga(x1, x2, x3, x4) = appcE_in_ggga(x1, x2, x3) appcE_out_ggga(x1, x2, x3, x4) = appcE_out_ggga(x1, x2, x3, x4) U75_ggga(x1, x2, x3, x4, x5, x6) = U75_ggga(x1, x2, x3, x4, x6) qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) QSF_IN_GA(x1, x2) = QSF_IN_GA(x1) PD_IN_GGAAAAA(x1, x2, x3, x4, x5, x6, x7) = PD_IN_GGAAAAA(x1, x2) U11_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U11_GGAAAAA(x1, x2, x8) U13_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U13_GGAAAAA(x1, x2, x4, x8) We have to consider all (P,R,Pi)-chains ---------------------------------------- (204) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (205) Obligation: Q DP problem: The TRS P consists of the following rules: QSF_IN_GA(.(X1, X2)) -> PD_IN_GGAAAAA(X1, X2) PD_IN_GGAAAAA(X1, X2) -> U11_GGAAAAA(X1, X2, partcC_in_ggaa(X1, X2)) U11_GGAAAAA(X1, X2, partcC_out_ggaa(X1, X2, X3, X4)) -> QSF_IN_GA(X3) U11_GGAAAAA(X1, X2, partcC_out_ggaa(X1, X2, X3, X4)) -> U13_GGAAAAA(X1, X2, X4, qscF_in_ga(X3)) U13_GGAAAAA(X1, X2, X4, qscF_out_ga(X3, X5)) -> QSF_IN_GA(X4) The TRS R consists of the following rules: partcC_in_ggaa(X1, .(X2, X3)) -> U67_ggaa(X1, X2, X3, lesscA_in_gg(X1, X2)) partcC_in_ggaa(X1, .(X2, X3)) -> U69_ggaa(X1, X2, X3, partcC_in_ggaa(X1, X3)) partcC_in_ggaa(X1, []) -> partcC_out_ggaa(X1, [], [], []) qscF_in_ga([]) -> qscF_out_ga([], []) qscF_in_ga(.(X1, X2)) -> U74_ga(X1, X2, qcD_in_ggaaaaa(X1, X2)) U67_ggaa(X1, X2, X3, lesscA_out_gg(X1, X2)) -> U68_ggaa(X1, X2, X3, partcB_in_ggaa(X1, X3)) U69_ggaa(X1, X2, X3, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) U74_ga(X1, X2, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3) lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) U68_ggaa(X1, X2, X3, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) qcD_in_ggaaaaa(X1, X2) -> U70_ggaaaaa(X1, X2, partcC_in_ggaa(X1, X2)) U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) partcB_in_ggaa(X1, .(X2, X3)) -> U64_ggaa(X1, X2, X3, lesscA_in_gg(X1, X2)) partcB_in_ggaa(X1, .(X2, X3)) -> U66_ggaa(X1, X2, X3, partcB_in_ggaa(X1, X3)) partcB_in_ggaa(X1, []) -> partcB_out_ggaa(X1, [], [], []) U70_ggaaaaa(X1, X2, partcC_out_ggaa(X1, X2, X3, X4)) -> U71_ggaaaaa(X1, X2, X3, X4, qscF_in_ga(X3)) U64_ggaa(X1, X2, X3, lesscA_out_gg(X1, X2)) -> U65_ggaa(X1, X2, X3, partcB_in_ggaa(X1, X3)) U66_ggaa(X1, X2, X3, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) U71_ggaaaaa(X1, X2, X3, X4, qscF_out_ga(X3, X5)) -> U72_ggaaaaa(X1, X2, X3, X4, X5, qscF_in_ga(X4)) U65_ggaa(X1, X2, X3, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) U72_ggaaaaa(X1, X2, X3, X4, X5, qscF_out_ga(X4, X6)) -> U73_ggaaaaa(X1, X2, X3, X4, X5, X6, appcE_in_ggga(X5, X1, X6)) U73_ggaaaaa(X1, X2, X3, X4, X5, X6, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) appcE_in_ggga([], X1, X2) -> appcE_out_ggga([], X1, X2, .(X1, X2)) appcE_in_ggga(.(X1, X2), X3, X4) -> U75_ggga(X1, X2, X3, X4, appcE_in_ggga(X2, X3, X4)) U75_ggga(X1, X2, X3, X4, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5)) The set Q consists of the following terms: partcC_in_ggaa(x0, x1) qscF_in_ga(x0) U67_ggaa(x0, x1, x2, x3) U69_ggaa(x0, x1, x2, x3) U74_ga(x0, x1, x2) lesscA_in_gg(x0, x1) U68_ggaa(x0, x1, x2, x3) qcD_in_ggaaaaa(x0, x1) U63_gg(x0, x1, x2) partcB_in_ggaa(x0, x1) U70_ggaaaaa(x0, x1, x2) U64_ggaa(x0, x1, x2, x3) U66_ggaa(x0, x1, x2, x3) U71_ggaaaaa(x0, x1, x2, x3, x4) U65_ggaa(x0, x1, x2, x3) U72_ggaaaaa(x0, x1, x2, x3, x4, x5) U73_ggaaaaa(x0, x1, x2, x3, x4, x5, x6) appcE_in_ggga(x0, x1, x2) U75_ggga(x0, x1, x2, x3, x4) We have to consider all (P,Q,R)-chains. ---------------------------------------- (206) 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 dependency pairs: QSF_IN_GA(.(X1, X2)) -> PD_IN_GGAAAAA(X1, X2) U11_GGAAAAA(X1, X2, partcC_out_ggaa(X1, X2, X3, X4)) -> QSF_IN_GA(X3) U13_GGAAAAA(X1, X2, X4, qscF_out_ga(X3, X5)) -> QSF_IN_GA(X4) Used ordering: Polynomial interpretation [POLO]: POL(.(x_1, x_2)) = 2 + 2*x_2 POL(0) = 0 POL(PD_IN_GGAAAAA(x_1, x_2)) = 1 + 2*x_2 POL(QSF_IN_GA(x_1)) = x_1 POL(U11_GGAAAAA(x_1, x_2, x_3)) = 1 + 2*x_3 POL(U13_GGAAAAA(x_1, x_2, x_3, x_4)) = 1 + 2*x_3 POL(U63_gg(x_1, x_2, x_3)) = 0 POL(U64_ggaa(x_1, x_2, x_3, x_4)) = 2 + 2*x_3 POL(U65_ggaa(x_1, x_2, x_3, x_4)) = 2 + 2*x_4 POL(U66_ggaa(x_1, x_2, x_3, x_4)) = 2 + 2*x_4 POL(U67_ggaa(x_1, x_2, x_3, x_4)) = 2 + 2*x_3 POL(U68_ggaa(x_1, x_2, x_3, x_4)) = 2 + 2*x_4 POL(U69_ggaa(x_1, x_2, x_3, x_4)) = 2 + 2*x_4 POL(U70_ggaaaaa(x_1, x_2, x_3)) = 1 POL(U71_ggaaaaa(x_1, x_2, x_3, x_4, x_5)) = 0 POL(U72_ggaaaaa(x_1, x_2, x_3, x_4, x_5, x_6)) = 0 POL(U73_ggaaaaa(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = 0 POL(U74_ga(x_1, x_2, x_3)) = 0 POL(U75_ggga(x_1, x_2, x_3, x_4, x_5)) = 0 POL([]) = 0 POL(appcE_in_ggga(x_1, x_2, x_3)) = 1 POL(appcE_out_ggga(x_1, x_2, x_3, x_4)) = 0 POL(lesscA_in_gg(x_1, x_2)) = 0 POL(lesscA_out_gg(x_1, x_2)) = 0 POL(partcB_in_ggaa(x_1, x_2)) = x_2 POL(partcB_out_ggaa(x_1, x_2, x_3, x_4)) = x_3 + x_4 POL(partcC_in_ggaa(x_1, x_2)) = x_2 POL(partcC_out_ggaa(x_1, x_2, x_3, x_4)) = x_3 + x_4 POL(qcD_in_ggaaaaa(x_1, x_2)) = 2 POL(qcD_out_ggaaaaa(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = 0 POL(qscF_in_ga(x_1)) = 0 POL(qscF_out_ga(x_1, x_2)) = 0 POL(s(x_1)) = 0 ---------------------------------------- (207) Obligation: Q DP problem: The TRS P consists of the following rules: PD_IN_GGAAAAA(X1, X2) -> U11_GGAAAAA(X1, X2, partcC_in_ggaa(X1, X2)) U11_GGAAAAA(X1, X2, partcC_out_ggaa(X1, X2, X3, X4)) -> U13_GGAAAAA(X1, X2, X4, qscF_in_ga(X3)) The TRS R consists of the following rules: partcC_in_ggaa(X1, .(X2, X3)) -> U67_ggaa(X1, X2, X3, lesscA_in_gg(X1, X2)) partcC_in_ggaa(X1, .(X2, X3)) -> U69_ggaa(X1, X2, X3, partcC_in_ggaa(X1, X3)) partcC_in_ggaa(X1, []) -> partcC_out_ggaa(X1, [], [], []) qscF_in_ga([]) -> qscF_out_ga([], []) qscF_in_ga(.(X1, X2)) -> U74_ga(X1, X2, qcD_in_ggaaaaa(X1, X2)) U67_ggaa(X1, X2, X3, lesscA_out_gg(X1, X2)) -> U68_ggaa(X1, X2, X3, partcB_in_ggaa(X1, X3)) U69_ggaa(X1, X2, X3, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) U74_ga(X1, X2, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3) lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) U68_ggaa(X1, X2, X3, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) qcD_in_ggaaaaa(X1, X2) -> U70_ggaaaaa(X1, X2, partcC_in_ggaa(X1, X2)) U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) partcB_in_ggaa(X1, .(X2, X3)) -> U64_ggaa(X1, X2, X3, lesscA_in_gg(X1, X2)) partcB_in_ggaa(X1, .(X2, X3)) -> U66_ggaa(X1, X2, X3, partcB_in_ggaa(X1, X3)) partcB_in_ggaa(X1, []) -> partcB_out_ggaa(X1, [], [], []) U70_ggaaaaa(X1, X2, partcC_out_ggaa(X1, X2, X3, X4)) -> U71_ggaaaaa(X1, X2, X3, X4, qscF_in_ga(X3)) U64_ggaa(X1, X2, X3, lesscA_out_gg(X1, X2)) -> U65_ggaa(X1, X2, X3, partcB_in_ggaa(X1, X3)) U66_ggaa(X1, X2, X3, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) U71_ggaaaaa(X1, X2, X3, X4, qscF_out_ga(X3, X5)) -> U72_ggaaaaa(X1, X2, X3, X4, X5, qscF_in_ga(X4)) U65_ggaa(X1, X2, X3, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) U72_ggaaaaa(X1, X2, X3, X4, X5, qscF_out_ga(X4, X6)) -> U73_ggaaaaa(X1, X2, X3, X4, X5, X6, appcE_in_ggga(X5, X1, X6)) U73_ggaaaaa(X1, X2, X3, X4, X5, X6, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) appcE_in_ggga([], X1, X2) -> appcE_out_ggga([], X1, X2, .(X1, X2)) appcE_in_ggga(.(X1, X2), X3, X4) -> U75_ggga(X1, X2, X3, X4, appcE_in_ggga(X2, X3, X4)) U75_ggga(X1, X2, X3, X4, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5)) The set Q consists of the following terms: partcC_in_ggaa(x0, x1) qscF_in_ga(x0) U67_ggaa(x0, x1, x2, x3) U69_ggaa(x0, x1, x2, x3) U74_ga(x0, x1, x2) lesscA_in_gg(x0, x1) U68_ggaa(x0, x1, x2, x3) qcD_in_ggaaaaa(x0, x1) U63_gg(x0, x1, x2) partcB_in_ggaa(x0, x1) U70_ggaaaaa(x0, x1, x2) U64_ggaa(x0, x1, x2, x3) U66_ggaa(x0, x1, x2, x3) U71_ggaaaaa(x0, x1, x2, x3, x4) U65_ggaa(x0, x1, x2, x3) U72_ggaaaaa(x0, x1, x2, x3, x4, x5) U73_ggaaaaa(x0, x1, x2, x3, x4, x5, x6) appcE_in_ggga(x0, x1, x2) U75_ggga(x0, x1, x2, x3, x4) We have to consider all (P,Q,R)-chains. ---------------------------------------- (208) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. ---------------------------------------- (209) TRUE ---------------------------------------- (210) Obligation: Pi DP problem: The TRS P consists of the following rules: APPE_IN_AAAA(.(X1, X2), X3, X4, .(X1, X5)) -> APPE_IN_AAAA(X2, X3, X4, X5) The TRS R consists of the following rules: lesscA_in_aa(0, s(X1)) -> lesscA_out_aa(0, s(X1)) lesscA_in_aa(s(X1), s(X2)) -> U63_aa(X1, X2, lesscA_in_aa(X1, X2)) U63_aa(X1, X2, lesscA_out_aa(X1, X2)) -> lesscA_out_aa(s(X1), s(X2)) lesscA_in_ga(0, s(X1)) -> lesscA_out_ga(0, s(X1)) lesscA_in_ga(s(X1), s(X2)) -> U63_ga(X1, X2, lesscA_in_ga(X1, X2)) U63_ga(X1, X2, lesscA_out_ga(X1, X2)) -> lesscA_out_ga(s(X1), s(X2)) partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_gaaa(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) U64_gaaa(X1, X2, X3, X4, X5, lesscA_out_ga(X1, X2)) -> U65_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], []) U66_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5)) U65_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_aaaa(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) U67_aaaa(X1, X2, X3, X4, X5, lesscA_out_aa(X1, X2)) -> U68_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) U68_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_aaaa(X1, X2, X3, X4, X5, partcC_in_aaaa(X1, X3, X4, X5)) partcC_in_aaaa(X1, [], [], []) -> partcC_out_aaaa(X1, [], [], []) U69_aaaa(X1, X2, X3, X4, X5, partcC_out_aaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), X4, .(X2, X5)) qscF_in_aa([], []) -> qscF_out_aa([], []) qscF_in_aa(.(X1, X2), X3) -> U74_aa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) qcD_in_aaaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U70_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_in_aaaa(X1, X2, X3, X4)) U70_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_out_aaaa(X1, X2, X3, X4)) -> U71_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X3, X5)) U71_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X3, X5)) -> U72_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X4, X6)) U72_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X4, X6)) -> U73_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_aaaa(X5, X1, X6, X7)) appcE_in_aaaa([], X1, X2, .(X1, X2)) -> appcE_out_aaaa([], X1, X2, .(X1, X2)) appcE_in_aaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U75_aaaa(X1, X2, X3, X4, X5, appcE_in_aaaa(X2, X3, X4, X5)) U75_aaaa(X1, X2, X3, X4, X5, appcE_out_aaaa(X2, X3, X4, X5)) -> appcE_out_aaaa(.(X1, X2), X3, X4, .(X1, X5)) U73_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_aaaa(X5, X1, X6, X7)) -> qcD_out_aaaaaaa(X1, X2, X3, X4, X5, X6, X7) U74_aa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_aa(.(X1, X2), X3) qscG_in_aaa(X1, X2, X3) -> U76_aaa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) U76_aaa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscG_out_aaa(X1, X2, X3) lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) partcC_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) U67_ggaa(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> U68_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) U64_ggaa(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> U65_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], []) U66_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) U65_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) U68_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_ggaa(X1, X2, X3, X4, X5, partcC_in_ggaa(X1, X3, X4, X5)) partcC_in_ggaa(X1, [], [], []) -> partcC_out_ggaa(X1, [], [], []) U69_ggaa(X1, X2, X3, X4, X5, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) qscF_in_ga([], []) -> qscF_out_ga([], []) qscF_in_ga(.(X1, X2), X3) -> U74_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) qcD_in_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U70_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_in_ggaa(X1, X2, X3, X4)) U70_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_out_ggaa(X1, X2, X3, X4)) -> U71_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5)) U71_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U72_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6)) U72_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U73_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_ggga(X5, X1, X6, X7)) appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2)) appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U75_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5)) U75_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5)) U73_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) U74_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3) The argument filtering Pi contains the following mapping: [] = [] lesscA_in_aa(x1, x2) = lesscA_in_aa lesscA_out_aa(x1, x2) = lesscA_out_aa(x1) U63_aa(x1, x2, x3) = U63_aa(x3) s(x1) = s(x1) lesscA_in_ga(x1, x2) = lesscA_in_ga(x1) 0 = 0 lesscA_out_ga(x1, x2) = lesscA_out_ga(x1) U63_ga(x1, x2, x3) = U63_ga(x1, x3) partcB_in_gaaa(x1, x2, x3, x4) = partcB_in_gaaa(x1) U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6) U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x6) U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6) partcB_out_gaaa(x1, x2, x3, x4) = partcB_out_gaaa(x1) partcC_in_aaaa(x1, x2, x3, x4) = partcC_in_aaaa U67_aaaa(x1, x2, x3, x4, x5, x6) = U67_aaaa(x6) U68_aaaa(x1, x2, x3, x4, x5, x6) = U68_aaaa(x6) partcC_out_aaaa(x1, x2, x3, x4) = partcC_out_aaaa U69_aaaa(x1, x2, x3, x4, x5, x6) = U69_aaaa(x6) qscF_in_aa(x1, x2) = qscF_in_aa qscF_out_aa(x1, x2) = qscF_out_aa U74_aa(x1, x2, x3, x4) = U74_aa(x4) qcD_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_aaaaaaa U70_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_aaaaaaa(x8) U71_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_aaaaaaa(x8) U72_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_aaaaaaa(x8) U73_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_aaaaaaa(x8) .(x1, x2) = .(x1, x2) qcD_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_aaaaaaa appcE_in_aaaa(x1, x2, x3, x4) = appcE_in_aaaa appcE_out_aaaa(x1, x2, x3, x4) = appcE_out_aaaa U75_aaaa(x1, x2, x3, x4, x5, x6) = U75_aaaa(x6) qscG_in_aaa(x1, x2, x3) = qscG_in_aaa U76_aaa(x1, x2, x3, x4) = U76_aaa(x4) qscG_out_aaa(x1, x2, x3) = qscG_out_aaa lesscA_in_gg(x1, x2) = lesscA_in_gg(x1, x2) lesscA_out_gg(x1, x2) = lesscA_out_gg(x1, x2) U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3) partcC_in_ggaa(x1, x2, x3, x4) = partcC_in_ggaa(x1, x2) U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6) U68_ggaa(x1, x2, x3, x4, x5, x6) = U68_ggaa(x1, x2, x3, x6) partcB_in_ggaa(x1, x2, x3, x4) = partcB_in_ggaa(x1, x2) U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6) U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6) U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6) partcB_out_ggaa(x1, x2, x3, x4) = partcB_out_ggaa(x1, x2, x3, x4) partcC_out_ggaa(x1, x2, x3, x4) = partcC_out_ggaa(x1, x2, x3, x4) U69_ggaa(x1, x2, x3, x4, x5, x6) = U69_ggaa(x1, x2, x3, x6) qscF_in_ga(x1, x2) = qscF_in_ga(x1) qscF_out_ga(x1, x2) = qscF_out_ga(x1, x2) U74_ga(x1, x2, x3, x4) = U74_ga(x1, x2, x4) qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_ggaaaaa(x1, x2) U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x8) U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x8) U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x8) U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x8) appcE_in_ggga(x1, x2, x3, x4) = appcE_in_ggga(x1, x2, x3) appcE_out_ggga(x1, x2, x3, x4) = appcE_out_ggga(x1, x2, x3, x4) U75_ggga(x1, x2, x3, x4, x5, x6) = U75_ggga(x1, x2, x3, x4, x6) qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) APPE_IN_AAAA(x1, x2, x3, x4) = APPE_IN_AAAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (211) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (212) Obligation: Pi DP problem: The TRS P consists of the following rules: APPE_IN_AAAA(.(X1, X2), X3, X4, .(X1, X5)) -> APPE_IN_AAAA(X2, X3, X4, X5) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) APPE_IN_AAAA(x1, x2, x3, x4) = APPE_IN_AAAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (213) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (214) Obligation: Q DP problem: The TRS P consists of the following rules: APPE_IN_AAAA -> APPE_IN_AAAA R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (215) 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 = APPE_IN_AAAA evaluates to t =APPE_IN_AAAA 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 APPE_IN_AAAA to APPE_IN_AAAA. ---------------------------------------- (216) NO ---------------------------------------- (217) Obligation: Pi DP problem: The TRS P consists of the following rules: LESSA_IN_GA(s(X1), s(X2)) -> LESSA_IN_GA(X1, X2) The TRS R consists of the following rules: lesscA_in_aa(0, s(X1)) -> lesscA_out_aa(0, s(X1)) lesscA_in_aa(s(X1), s(X2)) -> U63_aa(X1, X2, lesscA_in_aa(X1, X2)) U63_aa(X1, X2, lesscA_out_aa(X1, X2)) -> lesscA_out_aa(s(X1), s(X2)) lesscA_in_ga(0, s(X1)) -> lesscA_out_ga(0, s(X1)) lesscA_in_ga(s(X1), s(X2)) -> U63_ga(X1, X2, lesscA_in_ga(X1, X2)) U63_ga(X1, X2, lesscA_out_ga(X1, X2)) -> lesscA_out_ga(s(X1), s(X2)) partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_gaaa(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) U64_gaaa(X1, X2, X3, X4, X5, lesscA_out_ga(X1, X2)) -> U65_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], []) U66_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5)) U65_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_aaaa(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) U67_aaaa(X1, X2, X3, X4, X5, lesscA_out_aa(X1, X2)) -> U68_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) U68_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_aaaa(X1, X2, X3, X4, X5, partcC_in_aaaa(X1, X3, X4, X5)) partcC_in_aaaa(X1, [], [], []) -> partcC_out_aaaa(X1, [], [], []) U69_aaaa(X1, X2, X3, X4, X5, partcC_out_aaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), X4, .(X2, X5)) qscF_in_aa([], []) -> qscF_out_aa([], []) qscF_in_aa(.(X1, X2), X3) -> U74_aa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) qcD_in_aaaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U70_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_in_aaaa(X1, X2, X3, X4)) U70_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_out_aaaa(X1, X2, X3, X4)) -> U71_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X3, X5)) U71_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X3, X5)) -> U72_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X4, X6)) U72_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X4, X6)) -> U73_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_aaaa(X5, X1, X6, X7)) appcE_in_aaaa([], X1, X2, .(X1, X2)) -> appcE_out_aaaa([], X1, X2, .(X1, X2)) appcE_in_aaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U75_aaaa(X1, X2, X3, X4, X5, appcE_in_aaaa(X2, X3, X4, X5)) U75_aaaa(X1, X2, X3, X4, X5, appcE_out_aaaa(X2, X3, X4, X5)) -> appcE_out_aaaa(.(X1, X2), X3, X4, .(X1, X5)) U73_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_aaaa(X5, X1, X6, X7)) -> qcD_out_aaaaaaa(X1, X2, X3, X4, X5, X6, X7) U74_aa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_aa(.(X1, X2), X3) qscG_in_aaa(X1, X2, X3) -> U76_aaa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) U76_aaa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscG_out_aaa(X1, X2, X3) lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) partcC_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) U67_ggaa(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> U68_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) U64_ggaa(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> U65_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], []) U66_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) U65_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) U68_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_ggaa(X1, X2, X3, X4, X5, partcC_in_ggaa(X1, X3, X4, X5)) partcC_in_ggaa(X1, [], [], []) -> partcC_out_ggaa(X1, [], [], []) U69_ggaa(X1, X2, X3, X4, X5, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) qscF_in_ga([], []) -> qscF_out_ga([], []) qscF_in_ga(.(X1, X2), X3) -> U74_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) qcD_in_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U70_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_in_ggaa(X1, X2, X3, X4)) U70_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_out_ggaa(X1, X2, X3, X4)) -> U71_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5)) U71_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U72_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6)) U72_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U73_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_ggga(X5, X1, X6, X7)) appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2)) appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U75_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5)) U75_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5)) U73_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) U74_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3) The argument filtering Pi contains the following mapping: [] = [] lesscA_in_aa(x1, x2) = lesscA_in_aa lesscA_out_aa(x1, x2) = lesscA_out_aa(x1) U63_aa(x1, x2, x3) = U63_aa(x3) s(x1) = s(x1) lesscA_in_ga(x1, x2) = lesscA_in_ga(x1) 0 = 0 lesscA_out_ga(x1, x2) = lesscA_out_ga(x1) U63_ga(x1, x2, x3) = U63_ga(x1, x3) partcB_in_gaaa(x1, x2, x3, x4) = partcB_in_gaaa(x1) U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6) U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x6) U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6) partcB_out_gaaa(x1, x2, x3, x4) = partcB_out_gaaa(x1) partcC_in_aaaa(x1, x2, x3, x4) = partcC_in_aaaa U67_aaaa(x1, x2, x3, x4, x5, x6) = U67_aaaa(x6) U68_aaaa(x1, x2, x3, x4, x5, x6) = U68_aaaa(x6) partcC_out_aaaa(x1, x2, x3, x4) = partcC_out_aaaa U69_aaaa(x1, x2, x3, x4, x5, x6) = U69_aaaa(x6) qscF_in_aa(x1, x2) = qscF_in_aa qscF_out_aa(x1, x2) = qscF_out_aa U74_aa(x1, x2, x3, x4) = U74_aa(x4) qcD_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_aaaaaaa U70_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_aaaaaaa(x8) U71_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_aaaaaaa(x8) U72_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_aaaaaaa(x8) U73_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_aaaaaaa(x8) .(x1, x2) = .(x1, x2) qcD_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_aaaaaaa appcE_in_aaaa(x1, x2, x3, x4) = appcE_in_aaaa appcE_out_aaaa(x1, x2, x3, x4) = appcE_out_aaaa U75_aaaa(x1, x2, x3, x4, x5, x6) = U75_aaaa(x6) qscG_in_aaa(x1, x2, x3) = qscG_in_aaa U76_aaa(x1, x2, x3, x4) = U76_aaa(x4) qscG_out_aaa(x1, x2, x3) = qscG_out_aaa lesscA_in_gg(x1, x2) = lesscA_in_gg(x1, x2) lesscA_out_gg(x1, x2) = lesscA_out_gg(x1, x2) U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3) partcC_in_ggaa(x1, x2, x3, x4) = partcC_in_ggaa(x1, x2) U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6) U68_ggaa(x1, x2, x3, x4, x5, x6) = U68_ggaa(x1, x2, x3, x6) partcB_in_ggaa(x1, x2, x3, x4) = partcB_in_ggaa(x1, x2) U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6) U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6) U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6) partcB_out_ggaa(x1, x2, x3, x4) = partcB_out_ggaa(x1, x2, x3, x4) partcC_out_ggaa(x1, x2, x3, x4) = partcC_out_ggaa(x1, x2, x3, x4) U69_ggaa(x1, x2, x3, x4, x5, x6) = U69_ggaa(x1, x2, x3, x6) qscF_in_ga(x1, x2) = qscF_in_ga(x1) qscF_out_ga(x1, x2) = qscF_out_ga(x1, x2) U74_ga(x1, x2, x3, x4) = U74_ga(x1, x2, x4) qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_ggaaaaa(x1, x2) U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x8) U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x8) U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x8) U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x8) appcE_in_ggga(x1, x2, x3, x4) = appcE_in_ggga(x1, x2, x3) appcE_out_ggga(x1, x2, x3, x4) = appcE_out_ggga(x1, x2, x3, x4) U75_ggga(x1, x2, x3, x4, x5, x6) = U75_ggga(x1, x2, x3, x4, x6) qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) LESSA_IN_GA(x1, x2) = LESSA_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (218) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (219) Obligation: Pi DP problem: The TRS P consists of the following rules: LESSA_IN_GA(s(X1), s(X2)) -> LESSA_IN_GA(X1, X2) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) LESSA_IN_GA(x1, x2) = LESSA_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (220) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (221) Obligation: Q DP problem: The TRS P consists of the following rules: LESSA_IN_GA(s(X1)) -> LESSA_IN_GA(X1) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (222) 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: *LESSA_IN_GA(s(X1)) -> LESSA_IN_GA(X1) The graph contains the following edges 1 > 1 ---------------------------------------- (223) YES ---------------------------------------- (224) Obligation: Pi DP problem: The TRS P consists of the following rules: PARTB_IN_GAAA(X1, .(X2, X3), .(X2, X4), X5) -> U3_GAAA(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) U3_GAAA(X1, X2, X3, X4, X5, lesscA_out_ga(X1, X2)) -> PARTB_IN_GAAA(X1, X3, X4, X5) PARTB_IN_GAAA(X1, .(X2, X3), X4, .(X2, X5)) -> PARTB_IN_GAAA(X1, X3, X4, X5) The TRS R consists of the following rules: lesscA_in_aa(0, s(X1)) -> lesscA_out_aa(0, s(X1)) lesscA_in_aa(s(X1), s(X2)) -> U63_aa(X1, X2, lesscA_in_aa(X1, X2)) U63_aa(X1, X2, lesscA_out_aa(X1, X2)) -> lesscA_out_aa(s(X1), s(X2)) lesscA_in_ga(0, s(X1)) -> lesscA_out_ga(0, s(X1)) lesscA_in_ga(s(X1), s(X2)) -> U63_ga(X1, X2, lesscA_in_ga(X1, X2)) U63_ga(X1, X2, lesscA_out_ga(X1, X2)) -> lesscA_out_ga(s(X1), s(X2)) partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_gaaa(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) U64_gaaa(X1, X2, X3, X4, X5, lesscA_out_ga(X1, X2)) -> U65_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], []) U66_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5)) U65_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_aaaa(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) U67_aaaa(X1, X2, X3, X4, X5, lesscA_out_aa(X1, X2)) -> U68_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) U68_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_aaaa(X1, X2, X3, X4, X5, partcC_in_aaaa(X1, X3, X4, X5)) partcC_in_aaaa(X1, [], [], []) -> partcC_out_aaaa(X1, [], [], []) U69_aaaa(X1, X2, X3, X4, X5, partcC_out_aaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), X4, .(X2, X5)) qscF_in_aa([], []) -> qscF_out_aa([], []) qscF_in_aa(.(X1, X2), X3) -> U74_aa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) qcD_in_aaaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U70_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_in_aaaa(X1, X2, X3, X4)) U70_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_out_aaaa(X1, X2, X3, X4)) -> U71_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X3, X5)) U71_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X3, X5)) -> U72_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X4, X6)) U72_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X4, X6)) -> U73_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_aaaa(X5, X1, X6, X7)) appcE_in_aaaa([], X1, X2, .(X1, X2)) -> appcE_out_aaaa([], X1, X2, .(X1, X2)) appcE_in_aaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U75_aaaa(X1, X2, X3, X4, X5, appcE_in_aaaa(X2, X3, X4, X5)) U75_aaaa(X1, X2, X3, X4, X5, appcE_out_aaaa(X2, X3, X4, X5)) -> appcE_out_aaaa(.(X1, X2), X3, X4, .(X1, X5)) U73_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_aaaa(X5, X1, X6, X7)) -> qcD_out_aaaaaaa(X1, X2, X3, X4, X5, X6, X7) U74_aa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_aa(.(X1, X2), X3) qscG_in_aaa(X1, X2, X3) -> U76_aaa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) U76_aaa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscG_out_aaa(X1, X2, X3) lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) partcC_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) U67_ggaa(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> U68_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) U64_ggaa(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> U65_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], []) U66_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) U65_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) U68_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_ggaa(X1, X2, X3, X4, X5, partcC_in_ggaa(X1, X3, X4, X5)) partcC_in_ggaa(X1, [], [], []) -> partcC_out_ggaa(X1, [], [], []) U69_ggaa(X1, X2, X3, X4, X5, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) qscF_in_ga([], []) -> qscF_out_ga([], []) qscF_in_ga(.(X1, X2), X3) -> U74_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) qcD_in_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U70_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_in_ggaa(X1, X2, X3, X4)) U70_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_out_ggaa(X1, X2, X3, X4)) -> U71_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5)) U71_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U72_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6)) U72_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U73_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_ggga(X5, X1, X6, X7)) appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2)) appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U75_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5)) U75_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5)) U73_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) U74_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3) The argument filtering Pi contains the following mapping: [] = [] lesscA_in_aa(x1, x2) = lesscA_in_aa lesscA_out_aa(x1, x2) = lesscA_out_aa(x1) U63_aa(x1, x2, x3) = U63_aa(x3) s(x1) = s(x1) lesscA_in_ga(x1, x2) = lesscA_in_ga(x1) 0 = 0 lesscA_out_ga(x1, x2) = lesscA_out_ga(x1) U63_ga(x1, x2, x3) = U63_ga(x1, x3) partcB_in_gaaa(x1, x2, x3, x4) = partcB_in_gaaa(x1) U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6) U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x6) U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6) partcB_out_gaaa(x1, x2, x3, x4) = partcB_out_gaaa(x1) partcC_in_aaaa(x1, x2, x3, x4) = partcC_in_aaaa U67_aaaa(x1, x2, x3, x4, x5, x6) = U67_aaaa(x6) U68_aaaa(x1, x2, x3, x4, x5, x6) = U68_aaaa(x6) partcC_out_aaaa(x1, x2, x3, x4) = partcC_out_aaaa U69_aaaa(x1, x2, x3, x4, x5, x6) = U69_aaaa(x6) qscF_in_aa(x1, x2) = qscF_in_aa qscF_out_aa(x1, x2) = qscF_out_aa U74_aa(x1, x2, x3, x4) = U74_aa(x4) qcD_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_aaaaaaa U70_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_aaaaaaa(x8) U71_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_aaaaaaa(x8) U72_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_aaaaaaa(x8) U73_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_aaaaaaa(x8) .(x1, x2) = .(x1, x2) qcD_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_aaaaaaa appcE_in_aaaa(x1, x2, x3, x4) = appcE_in_aaaa appcE_out_aaaa(x1, x2, x3, x4) = appcE_out_aaaa U75_aaaa(x1, x2, x3, x4, x5, x6) = U75_aaaa(x6) qscG_in_aaa(x1, x2, x3) = qscG_in_aaa U76_aaa(x1, x2, x3, x4) = U76_aaa(x4) qscG_out_aaa(x1, x2, x3) = qscG_out_aaa lesscA_in_gg(x1, x2) = lesscA_in_gg(x1, x2) lesscA_out_gg(x1, x2) = lesscA_out_gg(x1, x2) U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3) partcC_in_ggaa(x1, x2, x3, x4) = partcC_in_ggaa(x1, x2) U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6) U68_ggaa(x1, x2, x3, x4, x5, x6) = U68_ggaa(x1, x2, x3, x6) partcB_in_ggaa(x1, x2, x3, x4) = partcB_in_ggaa(x1, x2) U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6) U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6) U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6) partcB_out_ggaa(x1, x2, x3, x4) = partcB_out_ggaa(x1, x2, x3, x4) partcC_out_ggaa(x1, x2, x3, x4) = partcC_out_ggaa(x1, x2, x3, x4) U69_ggaa(x1, x2, x3, x4, x5, x6) = U69_ggaa(x1, x2, x3, x6) qscF_in_ga(x1, x2) = qscF_in_ga(x1) qscF_out_ga(x1, x2) = qscF_out_ga(x1, x2) U74_ga(x1, x2, x3, x4) = U74_ga(x1, x2, x4) qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_ggaaaaa(x1, x2) U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x8) U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x8) U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x8) U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x8) appcE_in_ggga(x1, x2, x3, x4) = appcE_in_ggga(x1, x2, x3) appcE_out_ggga(x1, x2, x3, x4) = appcE_out_ggga(x1, x2, x3, x4) U75_ggga(x1, x2, x3, x4, x5, x6) = U75_ggga(x1, x2, x3, x4, x6) qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) PARTB_IN_GAAA(x1, x2, x3, x4) = PARTB_IN_GAAA(x1) U3_GAAA(x1, x2, x3, x4, x5, x6) = U3_GAAA(x1, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (225) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (226) Obligation: Pi DP problem: The TRS P consists of the following rules: PARTB_IN_GAAA(X1, .(X2, X3), .(X2, X4), X5) -> U3_GAAA(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) U3_GAAA(X1, X2, X3, X4, X5, lesscA_out_ga(X1, X2)) -> PARTB_IN_GAAA(X1, X3, X4, X5) PARTB_IN_GAAA(X1, .(X2, X3), X4, .(X2, X5)) -> PARTB_IN_GAAA(X1, X3, X4, X5) The TRS R consists of the following rules: lesscA_in_ga(0, s(X1)) -> lesscA_out_ga(0, s(X1)) lesscA_in_ga(s(X1), s(X2)) -> U63_ga(X1, X2, lesscA_in_ga(X1, X2)) U63_ga(X1, X2, lesscA_out_ga(X1, X2)) -> lesscA_out_ga(s(X1), s(X2)) The argument filtering Pi contains the following mapping: s(x1) = s(x1) lesscA_in_ga(x1, x2) = lesscA_in_ga(x1) 0 = 0 lesscA_out_ga(x1, x2) = lesscA_out_ga(x1) U63_ga(x1, x2, x3) = U63_ga(x1, x3) .(x1, x2) = .(x1, x2) PARTB_IN_GAAA(x1, x2, x3, x4) = PARTB_IN_GAAA(x1) U3_GAAA(x1, x2, x3, x4, x5, x6) = U3_GAAA(x1, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (227) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (228) Obligation: Q DP problem: The TRS P consists of the following rules: PARTB_IN_GAAA(X1) -> U3_GAAA(X1, lesscA_in_ga(X1)) U3_GAAA(X1, lesscA_out_ga(X1)) -> PARTB_IN_GAAA(X1) PARTB_IN_GAAA(X1) -> PARTB_IN_GAAA(X1) The TRS R consists of the following rules: lesscA_in_ga(0) -> lesscA_out_ga(0) lesscA_in_ga(s(X1)) -> U63_ga(X1, lesscA_in_ga(X1)) U63_ga(X1, lesscA_out_ga(X1)) -> lesscA_out_ga(s(X1)) The set Q consists of the following terms: lesscA_in_ga(x0) U63_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (229) TransformationProof (SOUND) By narrowing [LPAR04] the rule PARTB_IN_GAAA(X1) -> U3_GAAA(X1, lesscA_in_ga(X1)) at position [1] we obtained the following new rules [LPAR04]: (PARTB_IN_GAAA(0) -> U3_GAAA(0, lesscA_out_ga(0)),PARTB_IN_GAAA(0) -> U3_GAAA(0, lesscA_out_ga(0))) (PARTB_IN_GAAA(s(x0)) -> U3_GAAA(s(x0), U63_ga(x0, lesscA_in_ga(x0))),PARTB_IN_GAAA(s(x0)) -> U3_GAAA(s(x0), U63_ga(x0, lesscA_in_ga(x0)))) ---------------------------------------- (230) Obligation: Q DP problem: The TRS P consists of the following rules: U3_GAAA(X1, lesscA_out_ga(X1)) -> PARTB_IN_GAAA(X1) PARTB_IN_GAAA(X1) -> PARTB_IN_GAAA(X1) PARTB_IN_GAAA(0) -> U3_GAAA(0, lesscA_out_ga(0)) PARTB_IN_GAAA(s(x0)) -> U3_GAAA(s(x0), U63_ga(x0, lesscA_in_ga(x0))) The TRS R consists of the following rules: lesscA_in_ga(0) -> lesscA_out_ga(0) lesscA_in_ga(s(X1)) -> U63_ga(X1, lesscA_in_ga(X1)) U63_ga(X1, lesscA_out_ga(X1)) -> lesscA_out_ga(s(X1)) The set Q consists of the following terms: lesscA_in_ga(x0) U63_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (231) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U3_GAAA(X1, lesscA_out_ga(X1)) -> PARTB_IN_GAAA(X1) we obtained the following new rules [LPAR04]: (U3_GAAA(0, lesscA_out_ga(0)) -> PARTB_IN_GAAA(0),U3_GAAA(0, lesscA_out_ga(0)) -> PARTB_IN_GAAA(0)) (U3_GAAA(s(z0), lesscA_out_ga(s(z0))) -> PARTB_IN_GAAA(s(z0)),U3_GAAA(s(z0), lesscA_out_ga(s(z0))) -> PARTB_IN_GAAA(s(z0))) ---------------------------------------- (232) Obligation: Q DP problem: The TRS P consists of the following rules: PARTB_IN_GAAA(X1) -> PARTB_IN_GAAA(X1) PARTB_IN_GAAA(0) -> U3_GAAA(0, lesscA_out_ga(0)) PARTB_IN_GAAA(s(x0)) -> U3_GAAA(s(x0), U63_ga(x0, lesscA_in_ga(x0))) U3_GAAA(0, lesscA_out_ga(0)) -> PARTB_IN_GAAA(0) U3_GAAA(s(z0), lesscA_out_ga(s(z0))) -> PARTB_IN_GAAA(s(z0)) The TRS R consists of the following rules: lesscA_in_ga(0) -> lesscA_out_ga(0) lesscA_in_ga(s(X1)) -> U63_ga(X1, lesscA_in_ga(X1)) U63_ga(X1, lesscA_out_ga(X1)) -> lesscA_out_ga(s(X1)) The set Q consists of the following terms: lesscA_in_ga(x0) U63_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (233) Obligation: Pi DP problem: The TRS P consists of the following rules: LESSA_IN_AA(s(X1), s(X2)) -> LESSA_IN_AA(X1, X2) The TRS R consists of the following rules: lesscA_in_aa(0, s(X1)) -> lesscA_out_aa(0, s(X1)) lesscA_in_aa(s(X1), s(X2)) -> U63_aa(X1, X2, lesscA_in_aa(X1, X2)) U63_aa(X1, X2, lesscA_out_aa(X1, X2)) -> lesscA_out_aa(s(X1), s(X2)) lesscA_in_ga(0, s(X1)) -> lesscA_out_ga(0, s(X1)) lesscA_in_ga(s(X1), s(X2)) -> U63_ga(X1, X2, lesscA_in_ga(X1, X2)) U63_ga(X1, X2, lesscA_out_ga(X1, X2)) -> lesscA_out_ga(s(X1), s(X2)) partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_gaaa(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) U64_gaaa(X1, X2, X3, X4, X5, lesscA_out_ga(X1, X2)) -> U65_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], []) U66_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5)) U65_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_aaaa(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) U67_aaaa(X1, X2, X3, X4, X5, lesscA_out_aa(X1, X2)) -> U68_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) U68_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_aaaa(X1, X2, X3, X4, X5, partcC_in_aaaa(X1, X3, X4, X5)) partcC_in_aaaa(X1, [], [], []) -> partcC_out_aaaa(X1, [], [], []) U69_aaaa(X1, X2, X3, X4, X5, partcC_out_aaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), X4, .(X2, X5)) qscF_in_aa([], []) -> qscF_out_aa([], []) qscF_in_aa(.(X1, X2), X3) -> U74_aa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) qcD_in_aaaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U70_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_in_aaaa(X1, X2, X3, X4)) U70_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_out_aaaa(X1, X2, X3, X4)) -> U71_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X3, X5)) U71_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X3, X5)) -> U72_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X4, X6)) U72_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X4, X6)) -> U73_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_aaaa(X5, X1, X6, X7)) appcE_in_aaaa([], X1, X2, .(X1, X2)) -> appcE_out_aaaa([], X1, X2, .(X1, X2)) appcE_in_aaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U75_aaaa(X1, X2, X3, X4, X5, appcE_in_aaaa(X2, X3, X4, X5)) U75_aaaa(X1, X2, X3, X4, X5, appcE_out_aaaa(X2, X3, X4, X5)) -> appcE_out_aaaa(.(X1, X2), X3, X4, .(X1, X5)) U73_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_aaaa(X5, X1, X6, X7)) -> qcD_out_aaaaaaa(X1, X2, X3, X4, X5, X6, X7) U74_aa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_aa(.(X1, X2), X3) qscG_in_aaa(X1, X2, X3) -> U76_aaa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) U76_aaa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscG_out_aaa(X1, X2, X3) lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) partcC_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) U67_ggaa(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> U68_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) U64_ggaa(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> U65_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], []) U66_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) U65_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) U68_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_ggaa(X1, X2, X3, X4, X5, partcC_in_ggaa(X1, X3, X4, X5)) partcC_in_ggaa(X1, [], [], []) -> partcC_out_ggaa(X1, [], [], []) U69_ggaa(X1, X2, X3, X4, X5, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) qscF_in_ga([], []) -> qscF_out_ga([], []) qscF_in_ga(.(X1, X2), X3) -> U74_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) qcD_in_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U70_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_in_ggaa(X1, X2, X3, X4)) U70_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_out_ggaa(X1, X2, X3, X4)) -> U71_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5)) U71_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U72_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6)) U72_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U73_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_ggga(X5, X1, X6, X7)) appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2)) appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U75_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5)) U75_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5)) U73_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) U74_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3) The argument filtering Pi contains the following mapping: [] = [] lesscA_in_aa(x1, x2) = lesscA_in_aa lesscA_out_aa(x1, x2) = lesscA_out_aa(x1) U63_aa(x1, x2, x3) = U63_aa(x3) s(x1) = s(x1) lesscA_in_ga(x1, x2) = lesscA_in_ga(x1) 0 = 0 lesscA_out_ga(x1, x2) = lesscA_out_ga(x1) U63_ga(x1, x2, x3) = U63_ga(x1, x3) partcB_in_gaaa(x1, x2, x3, x4) = partcB_in_gaaa(x1) U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6) U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x6) U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6) partcB_out_gaaa(x1, x2, x3, x4) = partcB_out_gaaa(x1) partcC_in_aaaa(x1, x2, x3, x4) = partcC_in_aaaa U67_aaaa(x1, x2, x3, x4, x5, x6) = U67_aaaa(x6) U68_aaaa(x1, x2, x3, x4, x5, x6) = U68_aaaa(x6) partcC_out_aaaa(x1, x2, x3, x4) = partcC_out_aaaa U69_aaaa(x1, x2, x3, x4, x5, x6) = U69_aaaa(x6) qscF_in_aa(x1, x2) = qscF_in_aa qscF_out_aa(x1, x2) = qscF_out_aa U74_aa(x1, x2, x3, x4) = U74_aa(x4) qcD_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_aaaaaaa U70_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_aaaaaaa(x8) U71_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_aaaaaaa(x8) U72_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_aaaaaaa(x8) U73_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_aaaaaaa(x8) .(x1, x2) = .(x1, x2) qcD_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_aaaaaaa appcE_in_aaaa(x1, x2, x3, x4) = appcE_in_aaaa appcE_out_aaaa(x1, x2, x3, x4) = appcE_out_aaaa U75_aaaa(x1, x2, x3, x4, x5, x6) = U75_aaaa(x6) qscG_in_aaa(x1, x2, x3) = qscG_in_aaa U76_aaa(x1, x2, x3, x4) = U76_aaa(x4) qscG_out_aaa(x1, x2, x3) = qscG_out_aaa lesscA_in_gg(x1, x2) = lesscA_in_gg(x1, x2) lesscA_out_gg(x1, x2) = lesscA_out_gg(x1, x2) U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3) partcC_in_ggaa(x1, x2, x3, x4) = partcC_in_ggaa(x1, x2) U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6) U68_ggaa(x1, x2, x3, x4, x5, x6) = U68_ggaa(x1, x2, x3, x6) partcB_in_ggaa(x1, x2, x3, x4) = partcB_in_ggaa(x1, x2) U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6) U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6) U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6) partcB_out_ggaa(x1, x2, x3, x4) = partcB_out_ggaa(x1, x2, x3, x4) partcC_out_ggaa(x1, x2, x3, x4) = partcC_out_ggaa(x1, x2, x3, x4) U69_ggaa(x1, x2, x3, x4, x5, x6) = U69_ggaa(x1, x2, x3, x6) qscF_in_ga(x1, x2) = qscF_in_ga(x1) qscF_out_ga(x1, x2) = qscF_out_ga(x1, x2) U74_ga(x1, x2, x3, x4) = U74_ga(x1, x2, x4) qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_ggaaaaa(x1, x2) U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x8) U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x8) U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x8) U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x8) appcE_in_ggga(x1, x2, x3, x4) = appcE_in_ggga(x1, x2, x3) appcE_out_ggga(x1, x2, x3, x4) = appcE_out_ggga(x1, x2, x3, x4) U75_ggga(x1, x2, x3, x4, x5, x6) = U75_ggga(x1, x2, x3, x4, x6) qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) LESSA_IN_AA(x1, x2) = LESSA_IN_AA We have to consider all (P,R,Pi)-chains ---------------------------------------- (234) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (235) Obligation: Pi DP problem: The TRS P consists of the following rules: LESSA_IN_AA(s(X1), s(X2)) -> LESSA_IN_AA(X1, X2) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) LESSA_IN_AA(x1, x2) = LESSA_IN_AA We have to consider all (P,R,Pi)-chains ---------------------------------------- (236) Obligation: Pi DP problem: The TRS P consists of the following rules: PARTC_IN_AAAA(X1, .(X2, X3), X4, .(X2, X5)) -> PARTC_IN_AAAA(X1, X3, X4, X5) The TRS R consists of the following rules: lesscA_in_aa(0, s(X1)) -> lesscA_out_aa(0, s(X1)) lesscA_in_aa(s(X1), s(X2)) -> U63_aa(X1, X2, lesscA_in_aa(X1, X2)) U63_aa(X1, X2, lesscA_out_aa(X1, X2)) -> lesscA_out_aa(s(X1), s(X2)) lesscA_in_ga(0, s(X1)) -> lesscA_out_ga(0, s(X1)) lesscA_in_ga(s(X1), s(X2)) -> U63_ga(X1, X2, lesscA_in_ga(X1, X2)) U63_ga(X1, X2, lesscA_out_ga(X1, X2)) -> lesscA_out_ga(s(X1), s(X2)) partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_gaaa(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) U64_gaaa(X1, X2, X3, X4, X5, lesscA_out_ga(X1, X2)) -> U65_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], []) U66_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5)) U65_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_aaaa(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) U67_aaaa(X1, X2, X3, X4, X5, lesscA_out_aa(X1, X2)) -> U68_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) U68_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_aaaa(X1, X2, X3, X4, X5, partcC_in_aaaa(X1, X3, X4, X5)) partcC_in_aaaa(X1, [], [], []) -> partcC_out_aaaa(X1, [], [], []) U69_aaaa(X1, X2, X3, X4, X5, partcC_out_aaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), X4, .(X2, X5)) qscF_in_aa([], []) -> qscF_out_aa([], []) qscF_in_aa(.(X1, X2), X3) -> U74_aa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) qcD_in_aaaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U70_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_in_aaaa(X1, X2, X3, X4)) U70_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_out_aaaa(X1, X2, X3, X4)) -> U71_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X3, X5)) U71_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X3, X5)) -> U72_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X4, X6)) U72_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X4, X6)) -> U73_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_aaaa(X5, X1, X6, X7)) appcE_in_aaaa([], X1, X2, .(X1, X2)) -> appcE_out_aaaa([], X1, X2, .(X1, X2)) appcE_in_aaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U75_aaaa(X1, X2, X3, X4, X5, appcE_in_aaaa(X2, X3, X4, X5)) U75_aaaa(X1, X2, X3, X4, X5, appcE_out_aaaa(X2, X3, X4, X5)) -> appcE_out_aaaa(.(X1, X2), X3, X4, .(X1, X5)) U73_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_aaaa(X5, X1, X6, X7)) -> qcD_out_aaaaaaa(X1, X2, X3, X4, X5, X6, X7) U74_aa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_aa(.(X1, X2), X3) qscG_in_aaa(X1, X2, X3) -> U76_aaa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) U76_aaa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscG_out_aaa(X1, X2, X3) lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) partcC_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) U67_ggaa(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> U68_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) U64_ggaa(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> U65_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], []) U66_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) U65_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) U68_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_ggaa(X1, X2, X3, X4, X5, partcC_in_ggaa(X1, X3, X4, X5)) partcC_in_ggaa(X1, [], [], []) -> partcC_out_ggaa(X1, [], [], []) U69_ggaa(X1, X2, X3, X4, X5, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) qscF_in_ga([], []) -> qscF_out_ga([], []) qscF_in_ga(.(X1, X2), X3) -> U74_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) qcD_in_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U70_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_in_ggaa(X1, X2, X3, X4)) U70_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_out_ggaa(X1, X2, X3, X4)) -> U71_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5)) U71_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U72_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6)) U72_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U73_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_ggga(X5, X1, X6, X7)) appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2)) appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U75_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5)) U75_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5)) U73_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) U74_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3) The argument filtering Pi contains the following mapping: [] = [] lesscA_in_aa(x1, x2) = lesscA_in_aa lesscA_out_aa(x1, x2) = lesscA_out_aa(x1) U63_aa(x1, x2, x3) = U63_aa(x3) s(x1) = s(x1) lesscA_in_ga(x1, x2) = lesscA_in_ga(x1) 0 = 0 lesscA_out_ga(x1, x2) = lesscA_out_ga(x1) U63_ga(x1, x2, x3) = U63_ga(x1, x3) partcB_in_gaaa(x1, x2, x3, x4) = partcB_in_gaaa(x1) U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6) U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x6) U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6) partcB_out_gaaa(x1, x2, x3, x4) = partcB_out_gaaa(x1) partcC_in_aaaa(x1, x2, x3, x4) = partcC_in_aaaa U67_aaaa(x1, x2, x3, x4, x5, x6) = U67_aaaa(x6) U68_aaaa(x1, x2, x3, x4, x5, x6) = U68_aaaa(x6) partcC_out_aaaa(x1, x2, x3, x4) = partcC_out_aaaa U69_aaaa(x1, x2, x3, x4, x5, x6) = U69_aaaa(x6) qscF_in_aa(x1, x2) = qscF_in_aa qscF_out_aa(x1, x2) = qscF_out_aa U74_aa(x1, x2, x3, x4) = U74_aa(x4) qcD_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_aaaaaaa U70_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_aaaaaaa(x8) U71_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_aaaaaaa(x8) U72_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_aaaaaaa(x8) U73_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_aaaaaaa(x8) .(x1, x2) = .(x1, x2) qcD_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_aaaaaaa appcE_in_aaaa(x1, x2, x3, x4) = appcE_in_aaaa appcE_out_aaaa(x1, x2, x3, x4) = appcE_out_aaaa U75_aaaa(x1, x2, x3, x4, x5, x6) = U75_aaaa(x6) qscG_in_aaa(x1, x2, x3) = qscG_in_aaa U76_aaa(x1, x2, x3, x4) = U76_aaa(x4) qscG_out_aaa(x1, x2, x3) = qscG_out_aaa lesscA_in_gg(x1, x2) = lesscA_in_gg(x1, x2) lesscA_out_gg(x1, x2) = lesscA_out_gg(x1, x2) U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3) partcC_in_ggaa(x1, x2, x3, x4) = partcC_in_ggaa(x1, x2) U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6) U68_ggaa(x1, x2, x3, x4, x5, x6) = U68_ggaa(x1, x2, x3, x6) partcB_in_ggaa(x1, x2, x3, x4) = partcB_in_ggaa(x1, x2) U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6) U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6) U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6) partcB_out_ggaa(x1, x2, x3, x4) = partcB_out_ggaa(x1, x2, x3, x4) partcC_out_ggaa(x1, x2, x3, x4) = partcC_out_ggaa(x1, x2, x3, x4) U69_ggaa(x1, x2, x3, x4, x5, x6) = U69_ggaa(x1, x2, x3, x6) qscF_in_ga(x1, x2) = qscF_in_ga(x1) qscF_out_ga(x1, x2) = qscF_out_ga(x1, x2) U74_ga(x1, x2, x3, x4) = U74_ga(x1, x2, x4) qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_ggaaaaa(x1, x2) U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x8) U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x8) U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x8) U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x8) appcE_in_ggga(x1, x2, x3, x4) = appcE_in_ggga(x1, x2, x3) appcE_out_ggga(x1, x2, x3, x4) = appcE_out_ggga(x1, x2, x3, x4) U75_ggga(x1, x2, x3, x4, x5, x6) = U75_ggga(x1, x2, x3, x4, x6) qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) PARTC_IN_AAAA(x1, x2, x3, x4) = PARTC_IN_AAAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (237) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (238) Obligation: Pi DP problem: The TRS P consists of the following rules: PARTC_IN_AAAA(X1, .(X2, X3), X4, .(X2, X5)) -> PARTC_IN_AAAA(X1, X3, X4, X5) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) PARTC_IN_AAAA(x1, x2, x3, x4) = PARTC_IN_AAAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (239) Obligation: Pi DP problem: The TRS P consists of the following rules: PD_IN_AAAAAAA(X1, X2, X3, X4, X5, X6, X7) -> U11_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, partcC_in_aaaa(X1, X2, X3, X4)) U11_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, partcC_out_aaaa(X1, X2, X3, X4)) -> QSF_IN_AA(X3, X5) QSF_IN_AA(.(X1, X2), X3) -> PD_IN_AAAAAAA(X1, X2, X4, X5, X6, X7, X3) U11_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, partcC_out_aaaa(X1, X2, X3, X4)) -> U13_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X3, X5)) U13_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X3, X5)) -> QSF_IN_AA(X4, X6) The TRS R consists of the following rules: lesscA_in_aa(0, s(X1)) -> lesscA_out_aa(0, s(X1)) lesscA_in_aa(s(X1), s(X2)) -> U63_aa(X1, X2, lesscA_in_aa(X1, X2)) U63_aa(X1, X2, lesscA_out_aa(X1, X2)) -> lesscA_out_aa(s(X1), s(X2)) lesscA_in_ga(0, s(X1)) -> lesscA_out_ga(0, s(X1)) lesscA_in_ga(s(X1), s(X2)) -> U63_ga(X1, X2, lesscA_in_ga(X1, X2)) U63_ga(X1, X2, lesscA_out_ga(X1, X2)) -> lesscA_out_ga(s(X1), s(X2)) partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_gaaa(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) U64_gaaa(X1, X2, X3, X4, X5, lesscA_out_ga(X1, X2)) -> U65_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], []) U66_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5)) U65_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_aaaa(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) U67_aaaa(X1, X2, X3, X4, X5, lesscA_out_aa(X1, X2)) -> U68_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) U68_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_aaaa(X1, X2, X3, X4, X5, partcC_in_aaaa(X1, X3, X4, X5)) partcC_in_aaaa(X1, [], [], []) -> partcC_out_aaaa(X1, [], [], []) U69_aaaa(X1, X2, X3, X4, X5, partcC_out_aaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), X4, .(X2, X5)) qscF_in_aa([], []) -> qscF_out_aa([], []) qscF_in_aa(.(X1, X2), X3) -> U74_aa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) qcD_in_aaaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U70_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_in_aaaa(X1, X2, X3, X4)) U70_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_out_aaaa(X1, X2, X3, X4)) -> U71_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X3, X5)) U71_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X3, X5)) -> U72_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X4, X6)) U72_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X4, X6)) -> U73_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_aaaa(X5, X1, X6, X7)) appcE_in_aaaa([], X1, X2, .(X1, X2)) -> appcE_out_aaaa([], X1, X2, .(X1, X2)) appcE_in_aaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U75_aaaa(X1, X2, X3, X4, X5, appcE_in_aaaa(X2, X3, X4, X5)) U75_aaaa(X1, X2, X3, X4, X5, appcE_out_aaaa(X2, X3, X4, X5)) -> appcE_out_aaaa(.(X1, X2), X3, X4, .(X1, X5)) U73_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_aaaa(X5, X1, X6, X7)) -> qcD_out_aaaaaaa(X1, X2, X3, X4, X5, X6, X7) U74_aa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_aa(.(X1, X2), X3) qscG_in_aaa(X1, X2, X3) -> U76_aaa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) U76_aaa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscG_out_aaa(X1, X2, X3) lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) partcC_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) U67_ggaa(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> U68_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) U64_ggaa(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> U65_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], []) U66_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) U65_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) U68_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) partcC_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_ggaa(X1, X2, X3, X4, X5, partcC_in_ggaa(X1, X3, X4, X5)) partcC_in_ggaa(X1, [], [], []) -> partcC_out_ggaa(X1, [], [], []) U69_ggaa(X1, X2, X3, X4, X5, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) qscF_in_ga([], []) -> qscF_out_ga([], []) qscF_in_ga(.(X1, X2), X3) -> U74_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) qcD_in_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U70_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_in_ggaa(X1, X2, X3, X4)) U70_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_out_ggaa(X1, X2, X3, X4)) -> U71_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5)) U71_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U72_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6)) U72_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U73_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_ggga(X5, X1, X6, X7)) appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2)) appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U75_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5)) U75_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5)) U73_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) U74_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3) The argument filtering Pi contains the following mapping: [] = [] lesscA_in_aa(x1, x2) = lesscA_in_aa lesscA_out_aa(x1, x2) = lesscA_out_aa(x1) U63_aa(x1, x2, x3) = U63_aa(x3) s(x1) = s(x1) lesscA_in_ga(x1, x2) = lesscA_in_ga(x1) 0 = 0 lesscA_out_ga(x1, x2) = lesscA_out_ga(x1) U63_ga(x1, x2, x3) = U63_ga(x1, x3) partcB_in_gaaa(x1, x2, x3, x4) = partcB_in_gaaa(x1) U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6) U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x6) U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6) partcB_out_gaaa(x1, x2, x3, x4) = partcB_out_gaaa(x1) partcC_in_aaaa(x1, x2, x3, x4) = partcC_in_aaaa U67_aaaa(x1, x2, x3, x4, x5, x6) = U67_aaaa(x6) U68_aaaa(x1, x2, x3, x4, x5, x6) = U68_aaaa(x6) partcC_out_aaaa(x1, x2, x3, x4) = partcC_out_aaaa U69_aaaa(x1, x2, x3, x4, x5, x6) = U69_aaaa(x6) qscF_in_aa(x1, x2) = qscF_in_aa qscF_out_aa(x1, x2) = qscF_out_aa U74_aa(x1, x2, x3, x4) = U74_aa(x4) qcD_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_aaaaaaa U70_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_aaaaaaa(x8) U71_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_aaaaaaa(x8) U72_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_aaaaaaa(x8) U73_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_aaaaaaa(x8) .(x1, x2) = .(x1, x2) qcD_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_aaaaaaa appcE_in_aaaa(x1, x2, x3, x4) = appcE_in_aaaa appcE_out_aaaa(x1, x2, x3, x4) = appcE_out_aaaa U75_aaaa(x1, x2, x3, x4, x5, x6) = U75_aaaa(x6) qscG_in_aaa(x1, x2, x3) = qscG_in_aaa U76_aaa(x1, x2, x3, x4) = U76_aaa(x4) qscG_out_aaa(x1, x2, x3) = qscG_out_aaa lesscA_in_gg(x1, x2) = lesscA_in_gg(x1, x2) lesscA_out_gg(x1, x2) = lesscA_out_gg(x1, x2) U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3) partcC_in_ggaa(x1, x2, x3, x4) = partcC_in_ggaa(x1, x2) U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6) U68_ggaa(x1, x2, x3, x4, x5, x6) = U68_ggaa(x1, x2, x3, x6) partcB_in_ggaa(x1, x2, x3, x4) = partcB_in_ggaa(x1, x2) U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6) U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6) U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6) partcB_out_ggaa(x1, x2, x3, x4) = partcB_out_ggaa(x1, x2, x3, x4) partcC_out_ggaa(x1, x2, x3, x4) = partcC_out_ggaa(x1, x2, x3, x4) U69_ggaa(x1, x2, x3, x4, x5, x6) = U69_ggaa(x1, x2, x3, x6) qscF_in_ga(x1, x2) = qscF_in_ga(x1) qscF_out_ga(x1, x2) = qscF_out_ga(x1, x2) U74_ga(x1, x2, x3, x4) = U74_ga(x1, x2, x4) qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_ggaaaaa(x1, x2) U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x8) U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x8) U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x8) U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x8) appcE_in_ggga(x1, x2, x3, x4) = appcE_in_ggga(x1, x2, x3) appcE_out_ggga(x1, x2, x3, x4) = appcE_out_ggga(x1, x2, x3, x4) U75_ggga(x1, x2, x3, x4, x5, x6) = U75_ggga(x1, x2, x3, x4, x6) qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) PD_IN_AAAAAAA(x1, x2, x3, x4, x5, x6, x7) = PD_IN_AAAAAAA U11_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U11_AAAAAAA(x8) QSF_IN_AA(x1, x2) = QSF_IN_AA U13_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U13_AAAAAAA(x8) We have to consider all (P,R,Pi)-chains ---------------------------------------- (240) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (241) Obligation: Pi DP problem: The TRS P consists of the following rules: PD_IN_AAAAAAA(X1, X2, X3, X4, X5, X6, X7) -> U11_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, partcC_in_aaaa(X1, X2, X3, X4)) U11_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, partcC_out_aaaa(X1, X2, X3, X4)) -> QSF_IN_AA(X3, X5) QSF_IN_AA(.(X1, X2), X3) -> PD_IN_AAAAAAA(X1, X2, X4, X5, X6, X7, X3) U11_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, partcC_out_aaaa(X1, X2, X3, X4)) -> U13_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X3, X5)) U13_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X3, X5)) -> QSF_IN_AA(X4, X6) The TRS R consists of the following rules: partcC_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_aaaa(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) partcC_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_aaaa(X1, X2, X3, X4, X5, partcC_in_aaaa(X1, X3, X4, X5)) partcC_in_aaaa(X1, [], [], []) -> partcC_out_aaaa(X1, [], [], []) qscF_in_aa([], []) -> qscF_out_aa([], []) qscF_in_aa(.(X1, X2), X3) -> U74_aa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) U67_aaaa(X1, X2, X3, X4, X5, lesscA_out_aa(X1, X2)) -> U68_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) U69_aaaa(X1, X2, X3, X4, X5, partcC_out_aaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), X4, .(X2, X5)) U74_aa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_aa(.(X1, X2), X3) lesscA_in_aa(0, s(X1)) -> lesscA_out_aa(0, s(X1)) lesscA_in_aa(s(X1), s(X2)) -> U63_aa(X1, X2, lesscA_in_aa(X1, X2)) U68_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), .(X2, X4), X5) qcD_in_aaaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U70_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_in_aaaa(X1, X2, X3, X4)) U63_aa(X1, X2, lesscA_out_aa(X1, X2)) -> lesscA_out_aa(s(X1), s(X2)) partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_gaaa(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], []) U70_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, partcC_out_aaaa(X1, X2, X3, X4)) -> U71_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X3, X5)) U64_gaaa(X1, X2, X3, X4, X5, lesscA_out_ga(X1, X2)) -> U65_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) U66_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5)) U71_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X3, X5)) -> U72_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_aa(X4, X6)) lesscA_in_ga(0, s(X1)) -> lesscA_out_ga(0, s(X1)) lesscA_in_ga(s(X1), s(X2)) -> U63_ga(X1, X2, lesscA_in_ga(X1, X2)) U65_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5) U72_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X4, X6)) -> U73_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_aaaa(X5, X1, X6, X7)) U63_ga(X1, X2, lesscA_out_ga(X1, X2)) -> lesscA_out_ga(s(X1), s(X2)) U73_aaaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_aaaa(X5, X1, X6, X7)) -> qcD_out_aaaaaaa(X1, X2, X3, X4, X5, X6, X7) appcE_in_aaaa([], X1, X2, .(X1, X2)) -> appcE_out_aaaa([], X1, X2, .(X1, X2)) appcE_in_aaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U75_aaaa(X1, X2, X3, X4, X5, appcE_in_aaaa(X2, X3, X4, X5)) U75_aaaa(X1, X2, X3, X4, X5, appcE_out_aaaa(X2, X3, X4, X5)) -> appcE_out_aaaa(.(X1, X2), X3, X4, .(X1, X5)) The argument filtering Pi contains the following mapping: [] = [] lesscA_in_aa(x1, x2) = lesscA_in_aa lesscA_out_aa(x1, x2) = lesscA_out_aa(x1) U63_aa(x1, x2, x3) = U63_aa(x3) s(x1) = s(x1) lesscA_in_ga(x1, x2) = lesscA_in_ga(x1) 0 = 0 lesscA_out_ga(x1, x2) = lesscA_out_ga(x1) U63_ga(x1, x2, x3) = U63_ga(x1, x3) partcB_in_gaaa(x1, x2, x3, x4) = partcB_in_gaaa(x1) U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6) U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x6) U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6) partcB_out_gaaa(x1, x2, x3, x4) = partcB_out_gaaa(x1) partcC_in_aaaa(x1, x2, x3, x4) = partcC_in_aaaa U67_aaaa(x1, x2, x3, x4, x5, x6) = U67_aaaa(x6) U68_aaaa(x1, x2, x3, x4, x5, x6) = U68_aaaa(x6) partcC_out_aaaa(x1, x2, x3, x4) = partcC_out_aaaa U69_aaaa(x1, x2, x3, x4, x5, x6) = U69_aaaa(x6) qscF_in_aa(x1, x2) = qscF_in_aa qscF_out_aa(x1, x2) = qscF_out_aa U74_aa(x1, x2, x3, x4) = U74_aa(x4) qcD_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_aaaaaaa U70_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_aaaaaaa(x8) U71_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_aaaaaaa(x8) U72_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_aaaaaaa(x8) U73_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_aaaaaaa(x8) .(x1, x2) = .(x1, x2) qcD_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_aaaaaaa appcE_in_aaaa(x1, x2, x3, x4) = appcE_in_aaaa appcE_out_aaaa(x1, x2, x3, x4) = appcE_out_aaaa U75_aaaa(x1, x2, x3, x4, x5, x6) = U75_aaaa(x6) PD_IN_AAAAAAA(x1, x2, x3, x4, x5, x6, x7) = PD_IN_AAAAAAA U11_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U11_AAAAAAA(x8) QSF_IN_AA(x1, x2) = QSF_IN_AA U13_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U13_AAAAAAA(x8) We have to consider all (P,R,Pi)-chains ---------------------------------------- (242) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 1, "program": { "directives": [], "clauses": [ [ "(qs ([]) ([]))", null ], [ "(qs (. X Xs) Ys)", "(',' (part X Xs Littles Bigs) (',' (qs Littles Ls) (',' (qs Bigs Bs) (app Ls (. X Bs) Ys))))" ], [ "(part X (. Y Xs) (. Y Ls) Bs)", "(',' (less X Y) (part X Xs Ls Bs))" ], [ "(part X (. Y Xs) Ls (. Y Bs))", "(part X Xs Ls Bs)" ], [ "(part X1 ([]) ([]) ([]))", null ], [ "(app ([]) X X)", null ], [ "(app (. X Xs) Ys (. X Zs))", "(app Xs Ys Zs)" ], [ "(less (0) (s X2))", null ], [ "(less (s X) (s Y))", "(less X Y)" ] ] }, "graph": { "nodes": { "908": { "goal": [{ "clause": -1, "scope": -1, "term": "(less T77 T80)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T77"], "free": [], "exprvars": [] } }, "909": { "goal": [{ "clause": -1, "scope": -1, "term": "(part T77 T84 X126 X127)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T77"], "free": [ "X126", "X127" ], "exprvars": [] } }, "1066": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1065": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (part T158 T159 X257 X258) (',' (qs X257 X259) (',' (qs X258 X260) (app X259 (. 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T202 T203) X297)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X297"], "exprvars": [] } }, "1044": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1043": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1120": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1042": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1041": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1040": { "goal": [{ "clause": -1, "scope": -1, "term": "(part T141 T142 X230 X231)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X230", "X231" ], "exprvars": [] } }, "890": { "goal": [{ "clause": 2, "scope": 4, "term": "(part T43 T44 X67 X68)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T43"], "free": [ "X67", "X68" ], "exprvars": [] } }, "1118": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "892": { "goal": [ { "clause": 3, "scope": 4, "term": "(part T43 T44 X67 X68)" }, { "clause": 4, "scope": 4, "term": "(part T43 T44 X67 X68)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T43"], "free": [ "X67", "X68" ], "exprvars": [] } }, "1117": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "497": { "goal": [{ "clause": 2, "scope": 2, "term": "(part T12 T13 X16 X17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X16", "X17" ], "exprvars": [] } }, "530": { "goal": [{ "clause": -1, "scope": -1, "term": "(part T43 T44 X67 X68)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T43"], "free": [ "X67", "X68" ], "exprvars": [] } }, "1039": { "goal": [{ "clause": 4, "scope": 2, "term": "(part T12 T13 X16 X17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X16", "X17" ], "exprvars": [] } }, "1116": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1038": { "goal": [{ "clause": 3, "scope": 2, "term": "(part T12 T13 X16 X17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X16", "X17" ], "exprvars": [] } }, "1115": { "goal": [{ "clause": 6, "scope": 8, "term": "(app T209 (. T210 T208) T11)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T11"], "free": [], "exprvars": [] } }, "499": { "goal": [ { "clause": 3, "scope": 2, "term": "(part T12 T13 X16 X17)" }, { "clause": 4, "scope": 2, "term": "(part T12 T13 X16 X17)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X16", "X17" ], "exprvars": [] } }, "1037": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1114": { "goal": [{ "clause": 5, "scope": 8, "term": "(app T209 (. T210 T208) T11)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T11"], "free": [], "exprvars": [] } }, "1036": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1113": { "goal": [ { "clause": 5, "scope": 8, "term": "(app T209 (. T210 T208) T11)" }, { "clause": 6, "scope": 8, "term": "(app T209 (. T210 T208) T11)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T11"], "free": [], "exprvars": [] } }, "897": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (less T77 T80) (part T77 T81 X126 X127))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T77"], "free": [ "X126", "X127" ], "exprvars": [] } }, "1035": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1112": { "goal": [{ "clause": -1, "scope": -1, "term": "(app T209 (. T210 T208) T11)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T11"], "free": [], "exprvars": [] } }, "898": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1034": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1078": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1111": { "goal": [{ "clause": -1, "scope": -1, "term": "(qs T150 X19)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X19"], "exprvars": [] } }, "536": { "goal": [ { "clause": 7, "scope": 3, "term": "(less T38 T39)" }, { "clause": 8, "scope": 3, "term": "(less T38 T39)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "219": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1119": { "goal": [{ "clause": -1, "scope": -1, "term": "(app T240 (. T241 T242) T239)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T239"], "free": [], "exprvars": [] } }, "1052": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1051": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1050": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "220": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "221": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "540": { "goal": [{ "clause": 7, "scope": 3, "term": "(less T38 T39)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "222": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (part T12 T13 X16 X17) (',' (qs X16 X18) (',' (qs X17 X19) (app X18 (. T12 X19) T11))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T11"], "free": [ "X16", "X17", "X18", "X19" ], "exprvars": [] } }, "223": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "542": { "goal": [{ "clause": 8, "scope": 3, "term": "(less T38 T39)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1049": { "goal": [{ "clause": 1, "scope": 6, "term": "(qs T17 X18)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X18"], "exprvars": [] } }, "1048": { "goal": [{ "clause": 0, "scope": 6, "term": "(qs T17 X18)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X18"], "exprvars": [] } }, "1047": { "goal": [ { "clause": 0, "scope": 6, "term": "(qs T17 X18)" }, { "clause": 1, "scope": 6, "term": "(qs T17 X18)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X18"], "exprvars": [] } }, "1046": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (qs T150 X19) (app T149 (. T151 X19) T11))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T11"], "free": ["X19"], "exprvars": [] } }, "1045": { "goal": [{ "clause": -1, "scope": -1, "term": "(qs T17 X18)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X18"], "exprvars": [] } }, "548": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "945": { "goal": [{ "clause": 3, "scope": 4, "term": "(part T43 T44 X67 X68)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T43"], "free": [ "X67", "X68" ], "exprvars": [] } }, "946": { "goal": [{ "clause": 4, "scope": 4, "term": "(part T43 T44 X67 X68)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T43"], "free": [ "X67", "X68" ], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 3, "label": "CASE" }, { "from": 3, "to": 7, "label": "PARALLEL" }, { "from": 3, "to": 8, "label": "PARALLEL" }, { "from": 7, "to": 219, "label": "EVAL with clause\nqs([], []).\nand substitutionT1 -> [],\nT2 -> []" }, { "from": 7, "to": 220, "label": "EVAL-BACKTRACK" }, { "from": 8, "to": 222, "label": "EVAL with clause\nqs(.(X13, X14), X15) :- ','(part(X13, X14, X16, X17), ','(qs(X16, X18), ','(qs(X17, X19), app(X18, .(X13, X19), X15)))).\nand substitutionX13 -> T12,\nX14 -> T13,\nT1 -> .(T12, T13),\nT2 -> T11,\nX15 -> T11,\nT9 -> T12,\nT10 -> T13" }, { "from": 8, "to": 223, "label": "EVAL-BACKTRACK" }, { "from": 219, "to": 221, "label": "SUCCESS" }, { "from": 222, "to": 235, "label": "SPLIT 1" }, { "from": 222, "to": 236, "label": "SPLIT 2\nreplacements:X16 -> T17,\nX17 -> T18,\nT12 -> T19" }, { "from": 235, "to": 239, "label": "CASE" }, { "from": 236, "to": 1045, "label": "SPLIT 1" }, { "from": 236, "to": 1046, "label": "SPLIT 2\nreplacements:X18 -> T149,\nT18 -> T150,\nT19 -> T151" }, { "from": 239, "to": 497, "label": "PARALLEL" }, { "from": 239, "to": 499, "label": "PARALLEL" }, { "from": 497, "to": 516, "label": "EVAL with clause\npart(X62, .(X63, X64), .(X63, X65), X66) :- ','(less(X62, X63), part(X62, X64, X65, X66)).\nand substitutionT12 -> T38,\nX62 -> T38,\nX63 -> T39,\nX64 -> T40,\nT13 -> .(T39, T40),\nX65 -> X67,\nX16 -> .(T39, X67),\nX17 -> X68,\nX66 -> X68,\nT35 -> T38,\nT36 -> T39,\nT37 -> T40" }, { "from": 497, "to": 519, "label": "EVAL-BACKTRACK" }, { "from": 499, "to": 1038, "label": "PARALLEL" }, { "from": 499, "to": 1039, "label": "PARALLEL" }, { "from": 516, "to": 529, "label": "SPLIT 1" }, { "from": 516, "to": 530, "label": "SPLIT 2\nnew knowledge:\nT43 is ground\nreplacements:T38 -> T43,\nT40 -> T44" }, { "from": 529, "to": 536, "label": "CASE" }, { "from": 530, "to": 889, "label": "CASE" }, { "from": 536, "to": 540, "label": "PARALLEL" }, { "from": 536, "to": 542, "label": "PARALLEL" }, { "from": 540, "to": 548, "label": "EVAL with clause\nless(0, s(X77)).\nand substitutionT38 -> 0,\nX77 -> T51,\nT39 -> s(T51)" }, { "from": 540, "to": 550, "label": "EVAL-BACKTRACK" }, { "from": 542, "to": 885, "label": "EVAL with clause\nless(s(X82), s(X83)) :- less(X82, X83).\nand substitutionX82 -> T58,\nT38 -> s(T58),\nX83 -> T59,\nT39 -> s(T59),\nT56 -> T58,\nT57 -> T59" }, { "from": 542, "to": 886, "label": "EVAL-BACKTRACK" }, { "from": 548, "to": 551, "label": "SUCCESS" }, { "from": 885, "to": 529, "label": "INSTANCE with matching:\nT38 -> T58\nT39 -> T59" }, { "from": 889, "to": 890, "label": "PARALLEL" }, { "from": 889, "to": 892, "label": "PARALLEL" }, { "from": 890, "to": 897, "label": "EVAL with clause\npart(X121, .(X122, X123), .(X122, X124), X125) :- ','(less(X121, X122), part(X121, X123, X124, X125)).\nand substitutionT43 -> T77,\nX121 -> T77,\nX122 -> T80,\nX123 -> T81,\nT44 -> .(T80, T81),\nX124 -> X126,\nX67 -> .(T80, X126),\nX68 -> X127,\nX125 -> X127,\nT78 -> T80,\nT79 -> T81" }, { "from": 890, "to": 898, "label": "EVAL-BACKTRACK" }, { "from": 892, "to": 945, "label": "PARALLEL" }, { "from": 892, "to": 946, "label": "PARALLEL" }, { "from": 897, "to": 908, "label": "SPLIT 1" }, { "from": 897, "to": 909, "label": "SPLIT 2\nnew knowledge:\nT77 is ground\nreplacements:T81 -> T84" }, { "from": 908, "to": 910, "label": "CASE" }, { "from": 909, "to": 530, "label": "INSTANCE with matching:\nT43 -> T77\nT44 -> T84\nX67 -> X126\nX68 -> X127" }, { "from": 910, "to": 911, "label": "PARALLEL" }, { "from": 910, "to": 912, "label": "PARALLEL" }, { "from": 911, "to": 915, "label": "EVAL with clause\nless(0, s(X136)).\nand substitutionT77 -> 0,\nX136 -> T91,\nT80 -> s(T91)" }, { "from": 911, "to": 916, "label": "EVAL-BACKTRACK" }, { "from": 912, "to": 918, "label": "EVAL with clause\nless(s(X141), s(X142)) :- less(X141, X142).\nand substitutionX141 -> T96,\nT77 -> s(T96),\nX142 -> T98,\nT80 -> s(T98),\nT97 -> T98" }, { "from": 912, "to": 919, "label": "EVAL-BACKTRACK" }, { "from": 915, "to": 917, "label": "SUCCESS" }, { "from": 918, "to": 908, "label": "INSTANCE with matching:\nT77 -> T96\nT80 -> T98" }, { "from": 945, "to": 1033, "label": "EVAL with clause\npart(X180, .(X181, X182), X183, .(X181, X184)) :- part(X180, X182, X183, X184).\nand substitutionT43 -> T116,\nX180 -> T116,\nX181 -> T117,\nX182 -> T119,\nT44 -> .(T117, T119),\nX67 -> X185,\nX183 -> X185,\nX184 -> X186,\nX68 -> .(T117, X186),\nT118 -> T119" }, { "from": 945, "to": 1034, "label": "EVAL-BACKTRACK" }, { "from": 946, "to": 1035, "label": "EVAL with clause\npart(X196, [], [], []).\nand substitutionT43 -> T125,\nX196 -> T125,\nT44 -> [],\nX67 -> [],\nX68 -> []" }, { "from": 946, "to": 1036, "label": "EVAL-BACKTRACK" }, { "from": 1033, "to": 530, "label": "INSTANCE with matching:\nT43 -> T116\nT44 -> T119\nX67 -> X185\nX68 -> X186" }, { "from": 1035, "to": 1037, "label": "SUCCESS" }, { "from": 1038, "to": 1040, "label": "EVAL with clause\npart(X225, .(X226, X227), X228, .(X226, X229)) :- part(X225, X227, X228, X229).\nand substitutionT12 -> T141,\nX225 -> T141,\nX226 -> T139,\nX227 -> T142,\nT13 -> .(T139, T142),\nX16 -> X230,\nX228 -> X230,\nX229 -> X231,\nX17 -> .(T139, X231),\nT138 -> T141,\nT140 -> T142" }, { "from": 1038, "to": 1041, "label": "EVAL-BACKTRACK" }, { "from": 1039, "to": 1042, "label": "EVAL with clause\npart(X241, [], [], []).\nand substitutionT12 -> T148,\nX241 -> T148,\nT13 -> [],\nX16 -> [],\nX17 -> []" }, { "from": 1039, "to": 1043, "label": "EVAL-BACKTRACK" }, { "from": 1040, "to": 235, "label": "INSTANCE with matching:\nT12 -> T141\nT13 -> T142\nX16 -> X230\nX17 -> X231" }, { "from": 1042, "to": 1044, "label": "SUCCESS" }, { "from": 1045, "to": 1047, "label": "CASE" }, { "from": 1046, "to": 1111, "label": "SPLIT 1" }, { "from": 1046, "to": 1112, "label": "SPLIT 2\nreplacements:X19 -> T208,\nT149 -> T209,\nT151 -> T210" }, { "from": 1047, "to": 1048, "label": "PARALLEL" }, { "from": 1047, "to": 1049, "label": "PARALLEL" }, { "from": 1048, "to": 1050, "label": "EVAL with clause\nqs([], []).\nand substitutionT17 -> [],\nX18 -> []" }, { "from": 1048, "to": 1051, "label": "EVAL-BACKTRACK" }, { "from": 1049, "to": 1065, "label": "EVAL with clause\nqs(.(X254, X255), X256) :- ','(part(X254, X255, X257, X258), ','(qs(X257, X259), ','(qs(X258, X260), app(X259, .(X254, X260), X256)))).\nand substitutionX254 -> T158,\nX255 -> T159,\nT17 -> .(T158, T159),\nX18 -> X261,\nX256 -> X261,\nT156 -> T158,\nT157 -> T159" }, { "from": 1049, "to": 1066, "label": "EVAL-BACKTRACK" }, { "from": 1050, "to": 1052, "label": "SUCCESS" }, { "from": 1065, "to": 1067, "label": "SPLIT 1" }, { "from": 1065, "to": 1068, "label": "SPLIT 2\nreplacements:X257 -> T163,\nX258 -> T164,\nT158 -> T165" }, { "from": 1067, "to": 235, "label": "INSTANCE with matching:\nT12 -> T158\nT13 -> T159\nX16 -> X257\nX17 -> X258" }, { "from": 1068, "to": 1069, "label": "SPLIT 1" }, { "from": 1068, "to": 1070, "label": "SPLIT 2\nreplacements:X259 -> T169,\nT164 -> T170,\nT165 -> T171" }, { "from": 1069, "to": 1045, "label": "INSTANCE with matching:\nT17 -> T163\nX18 -> X259" }, { "from": 1070, "to": 1071, "label": "SPLIT 1" }, { "from": 1070, "to": 1072, "label": "SPLIT 2\nreplacements:X260 -> T172,\nT169 -> T173,\nT171 -> T174" }, { "from": 1071, "to": 1045, "label": "INSTANCE with matching:\nT17 -> T170\nX18 -> X260" }, { "from": 1072, "to": 1073, "label": "CASE" }, { "from": 1073, "to": 1074, "label": "PARALLEL" }, { "from": 1073, "to": 1075, "label": "PARALLEL" }, { "from": 1074, "to": 1076, "label": "EVAL with clause\napp([], X282, X282).\nand substitutionT173 -> [],\nT174 -> T187,\nT172 -> T188,\nX282 -> .(T187, T188),\nX261 -> .(T187, T188)" }, { "from": 1074, "to": 1077, "label": "EVAL-BACKTRACK" }, { "from": 1075, "to": 1109, "label": "EVAL with clause\napp(.(X293, X294), X295, .(X293, X296)) :- app(X294, X295, X296).\nand substitutionX293 -> T197,\nX294 -> T201,\nT173 -> .(T197, T201),\nT174 -> T202,\nT172 -> T203,\nX295 -> .(T202, T203),\nX296 -> X297,\nX261 -> .(T197, X297),\nT198 -> T201,\nT199 -> T202,\nT200 -> T203" }, { "from": 1075, "to": 1110, "label": "EVAL-BACKTRACK" }, { "from": 1076, "to": 1078, "label": "SUCCESS" }, { "from": 1109, "to": 1072, "label": "INSTANCE with matching:\nT173 -> T201\nT174 -> T202\nT172 -> T203\nX261 -> X297" }, { "from": 1111, "to": 1045, "label": "INSTANCE with matching:\nT17 -> T150\nX18 -> X19" }, { "from": 1112, "to": 1113, "label": "CASE" }, { "from": 1113, "to": 1114, "label": "PARALLEL" }, { "from": 1113, "to": 1115, "label": "PARALLEL" }, { "from": 1114, "to": 1116, "label": "EVAL with clause\napp([], X306, X306).\nand substitutionT209 -> [],\nT210 -> T223,\nT208 -> T224,\nX306 -> .(T223, T224),\nT11 -> .(T223, T224)" }, { "from": 1114, "to": 1117, "label": "EVAL-BACKTRACK" }, { "from": 1115, "to": 1119, "label": "EVAL with clause\napp(.(X315, X316), X317, .(X315, X318)) :- app(X316, X317, X318).\nand substitutionX315 -> T235,\nX316 -> T240,\nT209 -> .(T235, T240),\nT210 -> T241,\nT208 -> T242,\nX317 -> .(T241, T242),\nX318 -> T239,\nT11 -> .(T235, T239),\nT236 -> T240,\nT237 -> T241,\nT238 -> T242" }, { "from": 1115, "to": 1120, "label": "EVAL-BACKTRACK" }, { "from": 1116, "to": 1118, "label": "SUCCESS" }, { "from": 1119, "to": 1112, "label": "INSTANCE with matching:\nT209 -> T240\nT210 -> T241\nT208 -> T242\nT11 -> T239" } ], "type": "Graph" } } ---------------------------------------- (243) Complex Obligation (AND) ---------------------------------------- (244) Obligation: Rules: f1115_out(T11) -> f1113_out(T11) :|: TRUE f1113_in(x) -> f1115_in(x) :|: TRUE f1114_out(x1) -> f1113_out(x1) :|: TRUE f1113_in(x2) -> f1114_in(x2) :|: TRUE f1115_in(x3) -> f1120_in :|: TRUE f1120_out -> f1115_out(x4) :|: TRUE f1119_out(T239) -> f1115_out(.(T235, T239)) :|: TRUE f1115_in(.(x5, x6)) -> f1119_in(x6) :|: TRUE f1112_in(x7) -> f1113_in(x7) :|: TRUE f1113_out(x8) -> f1112_out(x8) :|: TRUE f1112_out(x9) -> f1119_out(x9) :|: TRUE f1119_in(x10) -> f1112_in(x10) :|: TRUE f1_in(T2) -> f3_in(T2) :|: TRUE f3_out(x11) -> f1_out(x11) :|: TRUE f8_out(x12) -> f3_out(x12) :|: TRUE f3_in(x13) -> f8_in(x13) :|: TRUE f3_in(x14) -> f7_in(x14) :|: TRUE f7_out(x15) -> f3_out(x15) :|: TRUE f8_in(x16) -> f223_in :|: TRUE f223_out -> f8_out(x17) :|: TRUE f222_out(x18) -> f8_out(x18) :|: TRUE f8_in(x19) -> f222_in(x19) :|: TRUE f222_in(x20) -> f235_in :|: TRUE f235_out -> f236_in(x21) :|: TRUE f236_out(x22) -> f222_out(x22) :|: TRUE f1046_out(x23) -> f236_out(x23) :|: TRUE f236_in(x24) -> f1045_in :|: TRUE f1045_out -> f1046_in(x25) :|: TRUE f1112_out(x26) -> f1046_out(x26) :|: TRUE f1111_out -> f1112_in(x27) :|: TRUE f1046_in(x28) -> f1111_in :|: TRUE Start term: f1_in(T2) ---------------------------------------- (245) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (246) TRUE ---------------------------------------- (247) Obligation: Rules: f1110_out -> f1075_out :|: TRUE f1075_in -> f1110_in :|: TRUE f1109_out -> f1075_out :|: TRUE f1075_in -> f1109_in :|: TRUE f1073_out -> f1072_out :|: TRUE f1072_in -> f1073_in :|: TRUE f1073_in -> f1074_in :|: TRUE f1075_out -> f1073_out :|: TRUE f1074_out -> f1073_out :|: TRUE f1073_in -> f1075_in :|: TRUE f1109_in -> f1072_in :|: TRUE f1072_out -> f1109_out :|: TRUE f1_in(T2) -> f3_in(T2) :|: TRUE f3_out(x) -> f1_out(x) :|: TRUE f8_out(x1) -> f3_out(x1) :|: TRUE f3_in(x2) -> f8_in(x2) :|: TRUE f3_in(x3) -> f7_in(x3) :|: TRUE f7_out(x4) -> f3_out(x4) :|: TRUE f8_in(x5) -> f223_in :|: TRUE f223_out -> f8_out(x6) :|: TRUE f222_out(T11) -> f8_out(T11) :|: TRUE f8_in(x7) -> f222_in(x7) :|: TRUE f222_in(x8) -> f235_in :|: TRUE f235_out -> f236_in(x9) :|: TRUE f236_out(x10) -> f222_out(x10) :|: TRUE f1046_out(x11) -> f236_out(x11) :|: TRUE f236_in(x12) -> f1045_in :|: TRUE f1045_out -> f1046_in(x13) :|: TRUE f1112_out(x14) -> f1046_out(x14) :|: TRUE f1111_out -> f1112_in(x15) :|: TRUE f1046_in(x16) -> f1111_in :|: TRUE f1111_in -> f1045_in :|: TRUE f1045_out -> f1111_out :|: TRUE f1045_in -> f1047_in :|: TRUE f1047_out -> f1045_out :|: TRUE f1048_out -> f1047_out :|: TRUE f1047_in -> f1048_in :|: TRUE f1049_out -> f1047_out :|: TRUE f1047_in -> f1049_in :|: TRUE f1065_out -> f1049_out :|: TRUE f1066_out -> f1049_out :|: TRUE f1049_in -> f1066_in :|: TRUE f1049_in -> f1065_in :|: TRUE f1068_out -> f1065_out :|: TRUE f1065_in -> f1067_in :|: TRUE f1067_out -> f1068_in :|: TRUE f1070_out -> f1068_out :|: TRUE f1069_out -> f1070_in :|: TRUE f1068_in -> f1069_in :|: TRUE f1072_out -> f1070_out :|: TRUE f1071_out -> f1072_in :|: TRUE f1070_in -> f1071_in :|: TRUE Start term: f1_in(T2) ---------------------------------------- (248) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (249) TRUE ---------------------------------------- (250) Obligation: Rules: f910_in(T77) -> f911_in(T77) :|: TRUE f912_out(x) -> f910_out(x) :|: TRUE f910_in(x1) -> f912_in(x1) :|: TRUE f911_out(x2) -> f910_out(x2) :|: TRUE f908_out(T96) -> f918_out(T96) :|: TRUE f918_in(x3) -> f908_in(x3) :|: TRUE f919_out -> f912_out(x4) :|: TRUE f912_in(x5) -> f919_in :|: TRUE f912_in(s(x6)) -> f918_in(x6) :|: TRUE f918_out(x7) -> f912_out(s(x7)) :|: TRUE f910_out(x8) -> f908_out(x8) :|: TRUE f908_in(x9) -> f910_in(x9) :|: TRUE f1_in(T2) -> f3_in(T2) :|: TRUE f3_out(x10) -> f1_out(x10) :|: TRUE f8_out(x11) -> f3_out(x11) :|: TRUE f3_in(x12) -> f8_in(x12) :|: TRUE f3_in(x13) -> f7_in(x13) :|: TRUE f7_out(x14) -> f3_out(x14) :|: TRUE f8_in(x15) -> f223_in :|: TRUE f223_out -> f8_out(x16) :|: TRUE f222_out(T11) -> f8_out(T11) :|: TRUE f8_in(x17) -> f222_in(x17) :|: TRUE f222_in(x18) -> f235_in :|: TRUE f235_out -> f236_in(x19) :|: TRUE f236_out(x20) -> f222_out(x20) :|: TRUE f239_out -> f235_out :|: TRUE f235_in -> f239_in :|: TRUE f497_out -> f239_out :|: TRUE f239_in -> f497_in :|: TRUE f239_in -> f499_in :|: TRUE f499_out -> f239_out :|: TRUE f497_in -> f519_in :|: TRUE f516_out -> f497_out :|: TRUE f497_in -> f516_in :|: TRUE f519_out -> f497_out :|: TRUE f529_out -> f530_in(T43) :|: TRUE f530_out(x21) -> f516_out :|: TRUE f516_in -> f529_in :|: TRUE f530_in(x22) -> f889_in(x22) :|: TRUE f889_out(x23) -> f530_out(x23) :|: TRUE f889_in(x24) -> f890_in(x24) :|: TRUE f890_out(x25) -> f889_out(x25) :|: TRUE f889_in(x26) -> f892_in(x26) :|: TRUE f892_out(x27) -> f889_out(x27) :|: TRUE f897_out(x28) -> f890_out(x28) :|: TRUE f898_out -> f890_out(x29) :|: TRUE f890_in(x30) -> f897_in(x30) :|: TRUE f890_in(x31) -> f898_in :|: TRUE f909_out(x32) -> f897_out(x32) :|: TRUE f897_in(x33) -> f908_in(x33) :|: TRUE f908_out(x34) -> f909_in(x34) :|: TRUE f1046_out(x35) -> f236_out(x35) :|: TRUE f236_in(x36) -> f1045_in :|: TRUE f1045_out -> f1046_in(x37) :|: TRUE f1112_out(x38) -> f1046_out(x38) :|: TRUE f1111_out -> f1112_in(x39) :|: TRUE f1046_in(x40) -> f1111_in :|: TRUE f1111_in -> f1045_in :|: TRUE f1045_out -> f1111_out :|: TRUE f1045_in -> f1047_in :|: TRUE f1047_out -> f1045_out :|: TRUE f1048_out -> f1047_out :|: TRUE f1047_in -> f1048_in :|: TRUE f1049_out -> f1047_out :|: TRUE f1047_in -> f1049_in :|: TRUE f1065_out -> f1049_out :|: TRUE f1066_out -> f1049_out :|: TRUE f1049_in -> f1066_in :|: TRUE f1049_in -> f1065_in :|: TRUE f1068_out -> f1065_out :|: TRUE f1065_in -> f1067_in :|: TRUE f1067_out -> f1068_in :|: TRUE f235_out -> f1067_out :|: TRUE f1067_in -> f235_in :|: TRUE Start term: f1_in(T2) ---------------------------------------- (251) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (252) TRUE ---------------------------------------- (253) Obligation: Rules: f908_out(T96) -> f918_out(T96) :|: TRUE f918_in(x) -> f908_in(x) :|: TRUE f919_out -> f912_out(T77) :|: TRUE f912_in(x1) -> f919_in :|: TRUE f912_in(s(x2)) -> f918_in(x2) :|: TRUE f918_out(x3) -> f912_out(s(x3)) :|: TRUE f910_out(x4) -> f908_out(x4) :|: TRUE f908_in(x5) -> f910_in(x5) :|: TRUE f911_in(0) -> f915_in :|: TRUE f916_out -> f911_out(x6) :|: TRUE f911_in(x7) -> f916_in :|: TRUE f915_out -> f911_out(0) :|: TRUE f946_out(T43) -> f892_out(T43) :|: TRUE f945_out(x8) -> f892_out(x8) :|: TRUE f892_in(x9) -> f946_in(x9) :|: TRUE f892_in(x10) -> f945_in(x10) :|: TRUE f530_out(x11) -> f909_out(x11) :|: TRUE f909_in(x12) -> f530_in(x12) :|: TRUE f1034_out -> f945_out(x13) :|: TRUE f1033_out(T116) -> f945_out(T116) :|: TRUE f945_in(x14) -> f1033_in(x14) :|: TRUE f945_in(x15) -> f1034_in :|: TRUE f897_out(x16) -> f890_out(x16) :|: TRUE f898_out -> f890_out(x17) :|: TRUE f890_in(x18) -> f897_in(x18) :|: TRUE f890_in(x19) -> f898_in :|: TRUE f910_in(x20) -> f911_in(x20) :|: TRUE f912_out(x21) -> f910_out(x21) :|: TRUE f910_in(x22) -> f912_in(x22) :|: TRUE f911_out(x23) -> f910_out(x23) :|: TRUE f915_in -> f915_out :|: TRUE f909_out(x24) -> f897_out(x24) :|: TRUE f897_in(x25) -> f908_in(x25) :|: TRUE f908_out(x26) -> f909_in(x26) :|: TRUE f530_in(x27) -> f889_in(x27) :|: TRUE f889_out(x28) -> f530_out(x28) :|: TRUE f889_in(x29) -> f890_in(x29) :|: TRUE f890_out(x30) -> f889_out(x30) :|: TRUE f889_in(x31) -> f892_in(x31) :|: TRUE f892_out(x32) -> f889_out(x32) :|: TRUE f530_out(x33) -> f1033_out(x33) :|: TRUE f1033_in(x34) -> f530_in(x34) :|: TRUE f1_in(T2) -> f3_in(T2) :|: TRUE f3_out(x35) -> f1_out(x35) :|: TRUE f8_out(x36) -> f3_out(x36) :|: TRUE f3_in(x37) -> f8_in(x37) :|: TRUE f3_in(x38) -> f7_in(x38) :|: TRUE f7_out(x39) -> f3_out(x39) :|: TRUE f8_in(x40) -> f223_in :|: TRUE f223_out -> f8_out(x41) :|: TRUE f222_out(T11) -> f8_out(T11) :|: TRUE f8_in(x42) -> f222_in(x42) :|: TRUE f222_in(x43) -> f235_in :|: TRUE f235_out -> f236_in(x44) :|: TRUE f236_out(x45) -> f222_out(x45) :|: TRUE f239_out -> f235_out :|: TRUE f235_in -> f239_in :|: TRUE f497_out -> f239_out :|: TRUE f239_in -> f497_in :|: TRUE f239_in -> f499_in :|: TRUE f499_out -> f239_out :|: TRUE f497_in -> f519_in :|: TRUE f516_out -> f497_out :|: TRUE f497_in -> f516_in :|: TRUE f519_out -> f497_out :|: TRUE f529_out -> f530_in(x46) :|: TRUE f530_out(x47) -> f516_out :|: TRUE f516_in -> f529_in :|: TRUE f1046_out(x48) -> f236_out(x48) :|: TRUE f236_in(x49) -> f1045_in :|: TRUE f1045_out -> f1046_in(x50) :|: TRUE f1112_out(x51) -> f1046_out(x51) :|: TRUE f1111_out -> f1112_in(x52) :|: TRUE f1046_in(x53) -> f1111_in :|: TRUE f1111_in -> f1045_in :|: TRUE f1045_out -> f1111_out :|: TRUE f1045_in -> f1047_in :|: TRUE f1047_out -> f1045_out :|: TRUE f1048_out -> f1047_out :|: TRUE f1047_in -> f1048_in :|: TRUE f1049_out -> f1047_out :|: TRUE f1047_in -> f1049_in :|: TRUE f1065_out -> f1049_out :|: TRUE f1066_out -> f1049_out :|: TRUE f1049_in -> f1066_in :|: TRUE f1049_in -> f1065_in :|: TRUE f1068_out -> f1065_out :|: TRUE f1065_in -> f1067_in :|: TRUE f1067_out -> f1068_in :|: TRUE f235_out -> f1067_out :|: TRUE f1067_in -> f235_in :|: TRUE Start term: f1_in(T2) ---------------------------------------- (254) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (255) TRUE ---------------------------------------- (256) Obligation: Rules: f536_out -> f529_out :|: TRUE f529_in -> f536_in :|: TRUE f542_in -> f885_in :|: TRUE f542_in -> f886_in :|: TRUE f885_out -> f542_out :|: TRUE f886_out -> f542_out :|: TRUE f540_out -> f536_out :|: TRUE f536_in -> f542_in :|: TRUE f542_out -> f536_out :|: TRUE f536_in -> f540_in :|: TRUE f529_out -> f885_out :|: TRUE f885_in -> f529_in :|: TRUE f1_in(T2) -> f3_in(T2) :|: TRUE f3_out(x) -> f1_out(x) :|: TRUE f8_out(x1) -> f3_out(x1) :|: TRUE f3_in(x2) -> f8_in(x2) :|: TRUE f3_in(x3) -> f7_in(x3) :|: TRUE f7_out(x4) -> f3_out(x4) :|: TRUE f8_in(x5) -> f223_in :|: TRUE f223_out -> f8_out(x6) :|: TRUE f222_out(T11) -> f8_out(T11) :|: TRUE f8_in(x7) -> f222_in(x7) :|: TRUE f222_in(x8) -> f235_in :|: TRUE f235_out -> f236_in(x9) :|: TRUE f236_out(x10) -> f222_out(x10) :|: TRUE f1046_out(x11) -> f236_out(x11) :|: TRUE f236_in(x12) -> f1045_in :|: TRUE f1045_out -> f1046_in(x13) :|: TRUE f1045_in -> f1047_in :|: TRUE f1047_out -> f1045_out :|: TRUE f1048_out -> f1047_out :|: TRUE f1047_in -> f1048_in :|: TRUE f1049_out -> f1047_out :|: TRUE f1047_in -> f1049_in :|: TRUE f1065_out -> f1049_out :|: TRUE f1066_out -> f1049_out :|: TRUE f1049_in -> f1066_in :|: TRUE f1049_in -> f1065_in :|: TRUE f1068_out -> f1065_out :|: TRUE f1065_in -> f1067_in :|: TRUE f1067_out -> f1068_in :|: TRUE f235_out -> f1067_out :|: TRUE f1067_in -> f235_in :|: TRUE f239_out -> f235_out :|: TRUE f235_in -> f239_in :|: TRUE f497_out -> f239_out :|: TRUE f239_in -> f497_in :|: TRUE f239_in -> f499_in :|: TRUE f499_out -> f239_out :|: TRUE f497_in -> f519_in :|: TRUE f516_out -> f497_out :|: TRUE f497_in -> f516_in :|: TRUE f519_out -> f497_out :|: TRUE f529_out -> f530_in(T43) :|: TRUE f530_out(x14) -> f516_out :|: TRUE f516_in -> f529_in :|: TRUE f1112_out(x15) -> f1046_out(x15) :|: TRUE f1111_out -> f1112_in(x16) :|: TRUE f1046_in(x17) -> f1111_in :|: TRUE f1111_in -> f1045_in :|: TRUE f1045_out -> f1111_out :|: TRUE Start term: f1_in(T2) ---------------------------------------- (257) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f529_in -> f536_in :|: TRUE f542_in -> f885_in :|: TRUE f536_in -> f542_in :|: TRUE f885_in -> f529_in :|: TRUE ---------------------------------------- (258) Obligation: Rules: f529_in -> f536_in :|: TRUE f542_in -> f885_in :|: TRUE f536_in -> f542_in :|: TRUE f885_in -> f529_in :|: TRUE ---------------------------------------- (259) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (260) Obligation: Rules: f529_in -> f529_in :|: TRUE ---------------------------------------- (261) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (262) Obligation: Rules: f529_in -> f529_in :|: TRUE ---------------------------------------- (263) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f529_in -> f529_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (264) Obligation: Termination digraph: Nodes: (1) f529_in -> f529_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (265) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f529_in() Replaced non-predefined constructor symbols by 0. ---------------------------------------- (266) Obligation: Rules: f529_in -> f529_in :|: TRUE ---------------------------------------- (267) IntTRSNonPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc) -> f(1) :|: pc = 1 && TRUE Proved unsatisfiability of the following formula, indicating that the system is never left after entering: ((run2_0 = ((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 (((run1_0 * 1)) = ((1 * 1)) and T)) ---------------------------------------- (268) NO ---------------------------------------- (269) Obligation: Rules: f1040_in -> f235_in :|: TRUE f235_out -> f1040_out :|: TRUE f239_out -> f235_out :|: TRUE f235_in -> f239_in :|: TRUE f499_in -> f1038_in :|: TRUE f1038_out -> f499_out :|: TRUE f499_in -> f1039_in :|: TRUE f1039_out -> f499_out :|: TRUE f497_out -> f239_out :|: TRUE f239_in -> f497_in :|: TRUE f239_in -> f499_in :|: TRUE f499_out -> f239_out :|: TRUE f1038_in -> f1040_in :|: TRUE f1038_in -> f1041_in :|: TRUE f1041_out -> f1038_out :|: TRUE f1040_out -> f1038_out :|: TRUE f1_in(T2) -> f3_in(T2) :|: TRUE f3_out(x) -> f1_out(x) :|: TRUE f8_out(x1) -> f3_out(x1) :|: TRUE f3_in(x2) -> f8_in(x2) :|: TRUE f3_in(x3) -> f7_in(x3) :|: TRUE f7_out(x4) -> f3_out(x4) :|: TRUE f8_in(x5) -> f223_in :|: TRUE f223_out -> f8_out(x6) :|: TRUE f222_out(T11) -> f8_out(T11) :|: TRUE f8_in(x7) -> f222_in(x7) :|: TRUE f222_in(x8) -> f235_in :|: TRUE f235_out -> f236_in(x9) :|: TRUE f236_out(x10) -> f222_out(x10) :|: TRUE f1046_out(x11) -> f236_out(x11) :|: TRUE f236_in(x12) -> f1045_in :|: TRUE f1045_out -> f1046_in(x13) :|: TRUE f1045_in -> f1047_in :|: TRUE f1047_out -> f1045_out :|: TRUE f1048_out -> f1047_out :|: TRUE f1047_in -> f1048_in :|: TRUE f1049_out -> f1047_out :|: TRUE f1047_in -> f1049_in :|: TRUE f1065_out -> f1049_out :|: TRUE f1066_out -> f1049_out :|: TRUE f1049_in -> f1066_in :|: TRUE f1049_in -> f1065_in :|: TRUE f1068_out -> f1065_out :|: TRUE f1065_in -> f1067_in :|: TRUE f1067_out -> f1068_in :|: TRUE f235_out -> f1067_out :|: TRUE f1067_in -> f235_in :|: TRUE f1112_out(x14) -> f1046_out(x14) :|: TRUE f1111_out -> f1112_in(x15) :|: TRUE f1046_in(x16) -> f1111_in :|: TRUE f1111_in -> f1045_in :|: TRUE f1045_out -> f1111_out :|: TRUE Start term: f1_in(T2) ---------------------------------------- (270) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f1040_in -> f235_in :|: TRUE f235_in -> f239_in :|: TRUE f499_in -> f1038_in :|: TRUE f239_in -> f499_in :|: TRUE f1038_in -> f1040_in :|: TRUE ---------------------------------------- (271) Obligation: Rules: f1040_in -> f235_in :|: TRUE f235_in -> f239_in :|: TRUE f499_in -> f1038_in :|: TRUE f239_in -> f499_in :|: TRUE f1038_in -> f1040_in :|: TRUE ---------------------------------------- (272) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (273) Obligation: Rules: f1040_in -> f1040_in :|: TRUE ---------------------------------------- (274) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (275) Obligation: Rules: f1040_in -> f1040_in :|: TRUE ---------------------------------------- (276) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f1040_in -> f1040_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (277) Obligation: Termination digraph: Nodes: (1) f1040_in -> f1040_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (278) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f1040_in() Replaced non-predefined constructor symbols by 0. ---------------------------------------- (279) Obligation: Rules: f1040_in -> f1040_in :|: TRUE ---------------------------------------- (280) IntTRSNonPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc) -> f(1) :|: pc = 1 && TRUE Proved unsatisfiability of the following formula, indicating that the system is never left after entering: ((run2_0 = ((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 (((run1_0 * 1)) = ((1 * 1)) and T)) ---------------------------------------- (281) NO ---------------------------------------- (282) Obligation: Rules: f1042_in -> f1042_out :|: TRUE f1073_in -> f1074_in :|: TRUE f1075_out -> f1073_out :|: TRUE f1074_out -> f1073_out :|: TRUE f1073_in -> f1075_in :|: TRUE f1109_in -> f1072_in :|: TRUE f1072_out -> f1109_out :|: TRUE f911_in(0) -> f915_in :|: TRUE f916_out -> f911_out(T77) :|: TRUE f911_in(x) -> f916_in :|: TRUE f915_out -> f911_out(0) :|: TRUE f946_out(T43) -> f892_out(T43) :|: TRUE f945_out(x1) -> f892_out(x1) :|: TRUE f892_in(x2) -> f946_in(x2) :|: TRUE f892_in(x3) -> f945_in(x3) :|: TRUE f1045_in -> f1047_in :|: TRUE f1047_out -> f1045_out :|: TRUE f1035_in -> f1035_out :|: TRUE f1038_in -> f1040_in :|: TRUE f1038_in -> f1041_in :|: TRUE f1041_out -> f1038_out :|: TRUE f1040_out -> f1038_out :|: TRUE f1043_out -> f1039_out :|: TRUE f1042_out -> f1039_out :|: TRUE f1039_in -> f1042_in :|: TRUE f1039_in -> f1043_in :|: TRUE f497_in -> f519_in :|: TRUE f516_out -> f497_out :|: TRUE f497_in -> f516_in :|: TRUE f519_out -> f497_out :|: TRUE f235_out -> f1067_out :|: TRUE f1067_in -> f235_in :|: TRUE f1040_in -> f235_in :|: TRUE f235_out -> f1040_out :|: TRUE f499_in -> f1038_in :|: TRUE f1038_out -> f499_out :|: TRUE f499_in -> f1039_in :|: TRUE f1039_out -> f499_out :|: TRUE f1048_out -> f1047_out :|: TRUE f1047_in -> f1048_in :|: TRUE f1049_out -> f1047_out :|: TRUE f1047_in -> f1049_in :|: TRUE f497_out -> f239_out :|: TRUE f239_in -> f497_in :|: TRUE f239_in -> f499_in :|: TRUE f499_out -> f239_out :|: TRUE f1034_out -> f945_out(x4) :|: TRUE f1033_out(T116) -> f945_out(T116) :|: TRUE f945_in(x5) -> f1033_in(x5) :|: TRUE f945_in(x6) -> f1034_in :|: TRUE f540_out -> f536_out :|: TRUE f536_in -> f542_in :|: TRUE f542_out -> f536_out :|: TRUE f536_in -> f540_in :|: TRUE f897_out(x7) -> f890_out(x7) :|: TRUE f898_out -> f890_out(x8) :|: TRUE f890_in(x9) -> f897_in(x9) :|: TRUE f890_in(x10) -> f898_in :|: TRUE f1071_in -> f1045_in :|: TRUE f1045_out -> f1071_out :|: TRUE f1065_out -> f1049_out :|: TRUE f1066_out -> f1049_out :|: TRUE f1049_in -> f1066_in :|: TRUE f1049_in -> f1065_in :|: TRUE f1076_in -> f1076_out :|: TRUE f530_in(x11) -> f889_in(x11) :|: TRUE f889_out(x12) -> f530_out(x12) :|: TRUE f889_in(x13) -> f890_in(x13) :|: TRUE f890_out(x14) -> f889_out(x14) :|: TRUE f889_in(x15) -> f892_in(x15) :|: TRUE f892_out(x16) -> f889_out(x16) :|: TRUE f908_out(T96) -> f918_out(T96) :|: TRUE f918_in(x17) -> f908_in(x17) :|: TRUE f919_out -> f912_out(x18) :|: TRUE f912_in(x19) -> f919_in :|: TRUE f912_in(s(x20)) -> f918_in(x20) :|: TRUE f918_out(x21) -> f912_out(s(x21)) :|: TRUE f530_out(x22) -> f909_out(x22) :|: TRUE f909_in(x23) -> f530_in(x23) :|: TRUE f910_in(x24) -> f911_in(x24) :|: TRUE f912_out(x25) -> f910_out(x25) :|: TRUE f910_in(x26) -> f912_in(x26) :|: TRUE f911_out(x27) -> f910_out(x27) :|: TRUE f529_out -> f530_in(x28) :|: TRUE f530_out(x29) -> f516_out :|: TRUE f516_in -> f529_in :|: TRUE f239_out -> f235_out :|: TRUE f235_in -> f239_in :|: TRUE f915_in -> f915_out :|: TRUE f542_in -> f885_in :|: TRUE f542_in -> f886_in :|: TRUE f885_out -> f542_out :|: TRUE f886_out -> f542_out :|: TRUE f1073_out -> f1072_out :|: TRUE f1072_in -> f1073_in :|: TRUE f1070_out -> f1068_out :|: TRUE f1069_out -> f1070_in :|: TRUE f1068_in -> f1069_in :|: TRUE f946_in(x30) -> f1036_in :|: TRUE f1035_out -> f946_out(T125) :|: TRUE f1036_out -> f946_out(x31) :|: TRUE f946_in(x32) -> f1035_in :|: TRUE f1072_out -> f1070_out :|: TRUE f1071_out -> f1072_in :|: TRUE f1070_in -> f1071_in :|: TRUE f530_out(x33) -> f1033_out(x33) :|: TRUE f1033_in(x34) -> f530_in(x34) :|: TRUE f540_in -> f548_in :|: TRUE f550_out -> f540_out :|: TRUE f548_out -> f540_out :|: TRUE f540_in -> f550_in :|: TRUE f910_out(x35) -> f908_out(x35) :|: TRUE f908_in(x36) -> f910_in(x36) :|: TRUE f529_out -> f885_out :|: TRUE f885_in -> f529_in :|: TRUE f1068_out -> f1065_out :|: TRUE f1065_in -> f1067_in :|: TRUE f1067_out -> f1068_in :|: TRUE f536_out -> f529_out :|: TRUE f529_in -> f536_in :|: TRUE f1110_out -> f1075_out :|: TRUE f1075_in -> f1110_in :|: TRUE f1109_out -> f1075_out :|: TRUE f1075_in -> f1109_in :|: TRUE f909_out(x37) -> f897_out(x37) :|: TRUE f897_in(x38) -> f908_in(x38) :|: TRUE f908_out(x39) -> f909_in(x39) :|: TRUE f1074_in -> f1077_in :|: TRUE f1074_in -> f1076_in :|: TRUE f1077_out -> f1074_out :|: TRUE f1076_out -> f1074_out :|: TRUE f548_in -> f548_out :|: TRUE f1069_in -> f1045_in :|: TRUE f1045_out -> f1069_out :|: TRUE f1_in(T2) -> f3_in(T2) :|: TRUE f3_out(x40) -> f1_out(x40) :|: TRUE f8_out(x41) -> f3_out(x41) :|: TRUE f3_in(x42) -> f8_in(x42) :|: TRUE f3_in(x43) -> f7_in(x43) :|: TRUE f7_out(x44) -> f3_out(x44) :|: TRUE f8_in(x45) -> f223_in :|: TRUE f223_out -> f8_out(x46) :|: TRUE f222_out(T11) -> f8_out(T11) :|: TRUE f8_in(x47) -> f222_in(x47) :|: TRUE f222_in(x48) -> f235_in :|: TRUE f235_out -> f236_in(x49) :|: TRUE f236_out(x50) -> f222_out(x50) :|: TRUE f1046_out(x51) -> f236_out(x51) :|: TRUE f236_in(x52) -> f1045_in :|: TRUE f1045_out -> f1046_in(x53) :|: TRUE f1112_out(x54) -> f1046_out(x54) :|: TRUE f1111_out -> f1112_in(x55) :|: TRUE f1046_in(x56) -> f1111_in :|: TRUE f1111_in -> f1045_in :|: TRUE f1045_out -> f1111_out :|: TRUE Start term: f1_in(T2) ---------------------------------------- (283) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f1042_in -> f1042_out :|: TRUE f911_in(0) -> f915_in :|: TRUE f915_out -> f911_out(0) :|: TRUE f946_out(T43) -> f892_out(T43) :|: TRUE f945_out(x1) -> f892_out(x1) :|: TRUE f892_in(x2) -> f946_in(x2) :|: TRUE f892_in(x3) -> f945_in(x3) :|: TRUE f1045_in -> f1047_in :|: TRUE f1035_in -> f1035_out :|: TRUE f1038_in -> f1040_in :|: TRUE f1040_out -> f1038_out :|: TRUE f1042_out -> f1039_out :|: TRUE f1039_in -> f1042_in :|: TRUE f516_out -> f497_out :|: TRUE f497_in -> f516_in :|: TRUE f235_out -> f1067_out :|: TRUE f1067_in -> f235_in :|: TRUE f1040_in -> f235_in :|: TRUE f235_out -> f1040_out :|: TRUE f499_in -> f1038_in :|: TRUE f1038_out -> f499_out :|: TRUE f499_in -> f1039_in :|: TRUE f1039_out -> f499_out :|: TRUE f1047_in -> f1049_in :|: TRUE f497_out -> f239_out :|: TRUE f239_in -> f497_in :|: TRUE f239_in -> f499_in :|: TRUE f499_out -> f239_out :|: TRUE f1033_out(T116) -> f945_out(T116) :|: TRUE f945_in(x5) -> f1033_in(x5) :|: TRUE f540_out -> f536_out :|: TRUE f536_in -> f542_in :|: TRUE f542_out -> f536_out :|: TRUE f536_in -> f540_in :|: TRUE f897_out(x7) -> f890_out(x7) :|: TRUE f890_in(x9) -> f897_in(x9) :|: TRUE f1049_in -> f1065_in :|: TRUE f530_in(x11) -> f889_in(x11) :|: TRUE f889_out(x12) -> f530_out(x12) :|: TRUE f889_in(x13) -> f890_in(x13) :|: TRUE f890_out(x14) -> f889_out(x14) :|: TRUE f889_in(x15) -> f892_in(x15) :|: TRUE f892_out(x16) -> f889_out(x16) :|: TRUE f908_out(T96) -> f918_out(T96) :|: TRUE f918_in(x17) -> f908_in(x17) :|: TRUE f912_in(s(x20)) -> f918_in(x20) :|: TRUE f918_out(x21) -> f912_out(s(x21)) :|: TRUE f530_out(x22) -> f909_out(x22) :|: TRUE f909_in(x23) -> f530_in(x23) :|: TRUE f910_in(x24) -> f911_in(x24) :|: TRUE f912_out(x25) -> f910_out(x25) :|: TRUE f910_in(x26) -> f912_in(x26) :|: TRUE f911_out(x27) -> f910_out(x27) :|: TRUE f529_out -> f530_in(x28) :|: TRUE f530_out(x29) -> f516_out :|: TRUE f516_in -> f529_in :|: TRUE f239_out -> f235_out :|: TRUE f235_in -> f239_in :|: TRUE f915_in -> f915_out :|: TRUE f542_in -> f885_in :|: TRUE f885_out -> f542_out :|: TRUE f1068_in -> f1069_in :|: TRUE f1035_out -> f946_out(T125) :|: TRUE f946_in(x32) -> f1035_in :|: TRUE f530_out(x33) -> f1033_out(x33) :|: TRUE f1033_in(x34) -> f530_in(x34) :|: TRUE f540_in -> f548_in :|: TRUE f548_out -> f540_out :|: TRUE f910_out(x35) -> f908_out(x35) :|: TRUE f908_in(x36) -> f910_in(x36) :|: TRUE f529_out -> f885_out :|: TRUE f885_in -> f529_in :|: TRUE f1065_in -> f1067_in :|: TRUE f1067_out -> f1068_in :|: TRUE f536_out -> f529_out :|: TRUE f529_in -> f536_in :|: TRUE f909_out(x37) -> f897_out(x37) :|: TRUE f897_in(x38) -> f908_in(x38) :|: TRUE f908_out(x39) -> f909_in(x39) :|: TRUE f548_in -> f548_out :|: TRUE f1069_in -> f1045_in :|: TRUE f235_out -> f236_in(x49) :|: TRUE f236_in(x52) -> f1045_in :|: TRUE ---------------------------------------- (284) Obligation: Rules: f1042_in -> f1042_out :|: TRUE f911_in(0) -> f915_in :|: TRUE f915_out -> f911_out(0) :|: TRUE f946_out(T43) -> f892_out(T43) :|: TRUE f945_out(x1) -> f892_out(x1) :|: TRUE f892_in(x2) -> f946_in(x2) :|: TRUE f892_in(x3) -> f945_in(x3) :|: TRUE f1045_in -> f1047_in :|: TRUE f1035_in -> f1035_out :|: TRUE f1038_in -> f1040_in :|: TRUE f1040_out -> f1038_out :|: TRUE f1042_out -> f1039_out :|: TRUE f1039_in -> f1042_in :|: TRUE f516_out -> f497_out :|: TRUE f497_in -> f516_in :|: TRUE f235_out -> f1067_out :|: TRUE f1067_in -> f235_in :|: TRUE f1040_in -> f235_in :|: TRUE f235_out -> f1040_out :|: TRUE f499_in -> f1038_in :|: TRUE f1038_out -> f499_out :|: TRUE f499_in -> f1039_in :|: TRUE f1039_out -> f499_out :|: TRUE f1047_in -> f1049_in :|: TRUE f497_out -> f239_out :|: TRUE f239_in -> f497_in :|: TRUE f239_in -> f499_in :|: TRUE f499_out -> f239_out :|: TRUE f1033_out(T116) -> f945_out(T116) :|: TRUE f945_in(x5) -> f1033_in(x5) :|: TRUE f540_out -> f536_out :|: TRUE f536_in -> f542_in :|: TRUE f542_out -> f536_out :|: TRUE f536_in -> f540_in :|: TRUE f897_out(x7) -> f890_out(x7) :|: TRUE f890_in(x9) -> f897_in(x9) :|: TRUE f1049_in -> f1065_in :|: TRUE f530_in(x11) -> f889_in(x11) :|: TRUE f889_out(x12) -> f530_out(x12) :|: TRUE f889_in(x13) -> f890_in(x13) :|: TRUE f890_out(x14) -> f889_out(x14) :|: TRUE f889_in(x15) -> f892_in(x15) :|: TRUE f892_out(x16) -> f889_out(x16) :|: TRUE f908_out(T96) -> f918_out(T96) :|: TRUE f918_in(x17) -> f908_in(x17) :|: TRUE f912_in(s(x20)) -> f918_in(x20) :|: TRUE f918_out(x21) -> f912_out(s(x21)) :|: TRUE f530_out(x22) -> f909_out(x22) :|: TRUE f909_in(x23) -> f530_in(x23) :|: TRUE f910_in(x24) -> f911_in(x24) :|: TRUE f912_out(x25) -> f910_out(x25) :|: TRUE f910_in(x26) -> f912_in(x26) :|: TRUE f911_out(x27) -> f910_out(x27) :|: TRUE f529_out -> f530_in(x28) :|: TRUE f530_out(x29) -> f516_out :|: TRUE f516_in -> f529_in :|: TRUE f239_out -> f235_out :|: TRUE f235_in -> f239_in :|: TRUE f915_in -> f915_out :|: TRUE f542_in -> f885_in :|: TRUE f885_out -> f542_out :|: TRUE f1068_in -> f1069_in :|: TRUE f1035_out -> f946_out(T125) :|: TRUE f946_in(x32) -> f1035_in :|: TRUE f530_out(x33) -> f1033_out(x33) :|: TRUE f1033_in(x34) -> f530_in(x34) :|: TRUE f540_in -> f548_in :|: TRUE f548_out -> f540_out :|: TRUE f910_out(x35) -> f908_out(x35) :|: TRUE f908_in(x36) -> f910_in(x36) :|: TRUE f529_out -> f885_out :|: TRUE f885_in -> f529_in :|: TRUE f1065_in -> f1067_in :|: TRUE f1067_out -> f1068_in :|: TRUE f536_out -> f529_out :|: TRUE f529_in -> f536_in :|: TRUE f909_out(x37) -> f897_out(x37) :|: TRUE f897_in(x38) -> f908_in(x38) :|: TRUE f908_out(x39) -> f909_in(x39) :|: TRUE f548_in -> f548_out :|: TRUE f1069_in -> f1045_in :|: TRUE f235_out -> f236_in(x49) :|: TRUE f236_in(x52) -> f1045_in :|: TRUE ---------------------------------------- (285) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (286) Obligation: Rules: f908_out(x39:0) -> f530_in(x39:0) :|: TRUE f910_in(s(x20:0)) -> f910_in(x20:0) :|: TRUE f910_in(cons_0) -> f908_out(0) :|: TRUE && cons_0 = 0 f239_in -> f235_out :|: TRUE f908_out(T96:0) -> f908_out(s(T96:0)) :|: TRUE f235_out -> f239_in :|: TRUE f529_out -> f530_in(x28:0) :|: TRUE f536_in -> f529_out :|: TRUE f889_out(x12:0) -> f235_out :|: TRUE f235_out -> f235_out :|: TRUE f536_in -> f536_in :|: TRUE f239_in -> f239_in :|: TRUE f530_in(x11:0) -> f530_in(x11:0) :|: TRUE f530_in(x) -> f889_out(x1) :|: TRUE f239_in -> f536_in :|: TRUE f530_in(x2) -> f910_in(x2) :|: TRUE f889_out(x3) -> f889_out(x3) :|: TRUE f529_out -> f529_out :|: TRUE ---------------------------------------- (287) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (288) Obligation: Rules: f908_out(x39:0) -> f530_in(x39:0) :|: TRUE f910_in(s(x20:0)) -> f910_in(x20:0) :|: TRUE f910_in(cons_0) -> f908_out(0) :|: TRUE && cons_0 = 0 f239_in -> f235_out :|: TRUE f908_out(T96:0) -> f908_out(s(T96:0)) :|: TRUE f235_out -> f239_in :|: TRUE f529_out -> f530_in(x28:0) :|: TRUE f536_in -> f529_out :|: TRUE f889_out(x12:0) -> f235_out :|: TRUE f235_out -> f235_out :|: TRUE f536_in -> f536_in :|: TRUE f239_in -> f239_in :|: TRUE f530_in(x11:0) -> f530_in(x11:0) :|: TRUE f530_in(x) -> f889_out(x1) :|: TRUE f239_in -> f536_in :|: TRUE f530_in(x2) -> f910_in(x2) :|: TRUE f889_out(x3) -> f889_out(x3) :|: TRUE f529_out -> f529_out :|: TRUE ---------------------------------------- (289) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f908_out(x39:0) -> f530_in(x39:0) :|: TRUE (2) f910_in(s(x20:0)) -> f910_in(x20:0) :|: TRUE (3) f910_in(cons_0) -> f908_out(0) :|: TRUE && cons_0 = 0 (4) f239_in -> f235_out :|: TRUE (5) f908_out(T96:0) -> f908_out(s(T96:0)) :|: TRUE (6) f235_out -> f239_in :|: TRUE (7) f529_out -> f530_in(x28:0) :|: TRUE (8) f536_in -> f529_out :|: TRUE (9) f889_out(x12:0) -> f235_out :|: TRUE (10) f235_out -> f235_out :|: TRUE (11) f536_in -> f536_in :|: TRUE (12) f239_in -> f239_in :|: TRUE (13) f530_in(x11:0) -> f530_in(x11:0) :|: TRUE (14) f530_in(x) -> f889_out(x1) :|: TRUE (15) f239_in -> f536_in :|: TRUE (16) f530_in(x2) -> f910_in(x2) :|: TRUE (17) f889_out(x3) -> f889_out(x3) :|: TRUE (18) f529_out -> f529_out :|: TRUE Arcs: (1) -> (13), (14), (16) (2) -> (2), (3) (3) -> (1), (5) (4) -> (6), (10) (5) -> (1), (5) (6) -> (4), (12), (15) (7) -> (13), (14), (16) (8) -> (7), (18) (9) -> (6), (10) (10) -> (6), (10) (11) -> (8), (11) (12) -> (4), (12), (15) (13) -> (13), (14), (16) (14) -> (9), (17) (15) -> (8), (11) (16) -> (2), (3) (17) -> (9), (17) (18) -> (7), (18) This digraph is fully evaluated! ---------------------------------------- (290) Obligation: Termination digraph: Nodes: (1) f908_out(x39:0) -> f530_in(x39:0) :|: TRUE (2) f908_out(T96:0) -> f908_out(s(T96:0)) :|: TRUE (3) f910_in(cons_0) -> f908_out(0) :|: TRUE && cons_0 = 0 (4) f910_in(s(x20:0)) -> f910_in(x20:0) :|: TRUE (5) f530_in(x2) -> f910_in(x2) :|: TRUE (6) f529_out -> f530_in(x28:0) :|: TRUE (7) f529_out -> f529_out :|: TRUE (8) f536_in -> f529_out :|: TRUE (9) f536_in -> f536_in :|: TRUE (10) f239_in -> f536_in :|: TRUE (11) f235_out -> f239_in :|: TRUE (12) f235_out -> f235_out :|: TRUE (13) f889_out(x12:0) -> f235_out :|: TRUE (14) f889_out(x3) -> f889_out(x3) :|: TRUE (15) f530_in(x) -> f889_out(x1) :|: TRUE (16) f530_in(x11:0) -> f530_in(x11:0) :|: TRUE (17) f239_in -> f235_out :|: TRUE (18) f239_in -> f239_in :|: TRUE Arcs: (1) -> (5), (15), (16) (2) -> (1), (2) (3) -> (1), (2) (4) -> (3), (4) (5) -> (3), (4) (6) -> (5), (15), (16) (7) -> (6), (7) (8) -> (6), (7) (9) -> (8), (9) (10) -> (8), (9) (11) -> (10), (17), (18) (12) -> (11), (12) (13) -> (11), (12) (14) -> (13), (14) (15) -> (13), (14) (16) -> (5), (15), (16) (17) -> (11), (12) (18) -> (10), (17), (18) This digraph is fully evaluated! ---------------------------------------- (291) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (292) Obligation: Rules: f529_out -> f529_out :|: TRUE f530_in(x:0) -> f889_out(x1:0) :|: TRUE f536_in -> f536_in :|: TRUE f908_out(x39:0:0) -> f530_in(x39:0:0) :|: TRUE f908_out(T96:0:0) -> f908_out(s(T96:0:0)) :|: TRUE f536_in -> f529_out :|: TRUE f910_in(cons_0) -> f908_out(0) :|: TRUE && cons_0 = 0 f235_out -> f235_out :|: TRUE f239_in -> f235_out :|: TRUE f529_out -> f530_in(x28:0:0) :|: TRUE f235_out -> f239_in :|: TRUE f239_in -> f239_in :|: TRUE f530_in(x2:0) -> f910_in(x2:0) :|: TRUE f239_in -> f536_in :|: TRUE f910_in(s(x20:0:0)) -> f910_in(x20:0:0) :|: TRUE f889_out(x3:0) -> f889_out(x3:0) :|: TRUE f530_in(x11:0:0) -> f530_in(x11:0:0) :|: TRUE f889_out(x12:0:0) -> f235_out :|: TRUE