/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.pl /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- MAYBE proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Left Termination of the query pattern qs(a,g) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) PrologToPiTRSProof [SOUND, 29 ms] (2) PiTRS (3) DependencyPairsProof [EQUIVALENT, 26 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 3 ms] (6) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [SOUND, 2 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) NonTerminationLoopProof [COMPLETE, 0 ms] (52) NO (53) PiDP (54) UsableRulesProof [EQUIVALENT, 0 ms] (55) PiDP (56) PiDPToQDPProof [SOUND, 0 ms] (57) QDP (58) PrologToPiTRSProof [SOUND, 26 ms] (59) PiTRS (60) DependencyPairsProof [EQUIVALENT, 17 ms] (61) PiDP (62) DependencyGraphProof [EQUIVALENT, 0 ms] (63) AND (64) PiDP (65) UsableRulesProof [EQUIVALENT, 0 ms] (66) PiDP (67) PiDPToQDPProof [SOUND, 1 ms] (68) QDP (69) QDPSizeChangeProof [EQUIVALENT, 0 ms] (70) YES (71) PiDP (72) UsableRulesProof [EQUIVALENT, 0 ms] (73) PiDP (74) PiDPToQDPProof [SOUND, 0 ms] (75) QDP (76) NonTerminationLoopProof [COMPLETE, 0 ms] (77) NO (78) PiDP (79) UsableRulesProof [EQUIVALENT, 0 ms] (80) PiDP (81) PiDPToQDPProof [SOUND, 0 ms] (82) QDP (83) QDPSizeChangeProof [EQUIVALENT, 0 ms] (84) YES (85) PiDP (86) UsableRulesProof [EQUIVALENT, 0 ms] (87) PiDP (88) PiDPToQDPProof [SOUND, 0 ms] (89) QDP (90) TransformationProof [SOUND, 0 ms] (91) QDP (92) TransformationProof [EQUIVALENT, 0 ms] (93) QDP (94) NonTerminationLoopProof [COMPLETE, 0 ms] (95) NO (96) PiDP (97) UsableRulesProof [EQUIVALENT, 0 ms] (98) PiDP (99) PiDPToQDPProof [SOUND, 0 ms] (100) QDP (101) NonTerminationLoopProof [COMPLETE, 0 ms] (102) NO (103) PiDP (104) UsableRulesProof [EQUIVALENT, 0 ms] (105) PiDP (106) PiDPToQDPProof [SOUND, 0 ms] (107) QDP (108) NonTerminationLoopProof [COMPLETE, 0 ms] (109) NO (110) PiDP (111) UsableRulesProof [EQUIVALENT, 0 ms] (112) PiDP (113) PiDPToQDPProof [SOUND, 0 ms] (114) QDP (115) PrologToTRSTransformerProof [SOUND, 76 ms] (116) QTRS (117) DependencyPairsProof [EQUIVALENT, 0 ms] (118) QDP (119) DependencyGraphProof [EQUIVALENT, 0 ms] (120) AND (121) QDP (122) UsableRulesProof [EQUIVALENT, 0 ms] (123) QDP (124) QDPSizeChangeProof [EQUIVALENT, 0 ms] (125) YES (126) QDP (127) UsableRulesProof [EQUIVALENT, 0 ms] (128) QDP (129) NonTerminationLoopProof [COMPLETE, 0 ms] (130) NO (131) QDP (132) UsableRulesProof [EQUIVALENT, 0 ms] (133) QDP (134) QDPSizeChangeProof [EQUIVALENT, 0 ms] (135) YES (136) QDP (137) NonTerminationLoopProof [COMPLETE, 0 ms] (138) NO (139) QDP (140) UsableRulesProof [EQUIVALENT, 0 ms] (141) QDP (142) NonTerminationLoopProof [COMPLETE, 0 ms] (143) NO (144) QDP (145) UsableRulesProof [EQUIVALENT, 0 ms] (146) QDP (147) NonTerminationLoopProof [COMPLETE, 0 ms] (148) NO (149) QDP (150) NonLoopProof [COMPLETE, 967 ms] (151) NO (152) PrologToIRSwTTransformerProof [SOUND, 133 ms] (153) AND (154) IRSwT (155) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (156) TRUE (157) IRSwT (158) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (159) TRUE (160) IRSwT (161) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (162) TRUE (163) IRSwT (164) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (165) TRUE (166) IRSwT (167) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (168) IRSwT (169) IntTRSCompressionProof [EQUIVALENT, 20 ms] (170) IRSwT (171) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (172) IRSwT (173) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms] (174) IRSwT (175) FilterProof [EQUIVALENT, 0 ms] (176) IntTRS (177) IntTRSPeriodicNontermProof [COMPLETE, 1 ms] (178) NO (179) IRSwT (180) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (181) IRSwT (182) IntTRSCompressionProof [EQUIVALENT, 0 ms] (183) IRSwT (184) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (185) IRSwT (186) IRSwTTerminationDigraphProof [EQUIVALENT, 1 ms] (187) IRSwT (188) FilterProof [EQUIVALENT, 0 ms] (189) IntTRS (190) IntTRSNonPeriodicNontermProof [COMPLETE, 0 ms] (191) NO (192) IRSwT (193) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (194) IRSwT (195) IntTRSCompressionProof [EQUIVALENT, 39 ms] (196) IRSwT (197) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (198) IRSwT (199) IRSwTTerminationDigraphProof [EQUIVALENT, 85 ms] (200) IRSwT (201) IntTRSCompressionProof [EQUIVALENT, 21 ms] (202) IRSwT (203) PrologToDTProblemTransformerProof [SOUND, 158 ms] (204) TRIPLES (205) UndefinedPredicateInTriplesTransformerProof [SOUND, 0 ms] (206) TRIPLES (207) TriplesToPiDPProof [SOUND, 181 ms] (208) PiDP (209) DependencyGraphProof [EQUIVALENT, 0 ms] (210) AND (211) PiDP (212) UsableRulesProof [EQUIVALENT, 0 ms] (213) PiDP (214) PiDPToQDPProof [SOUND, 0 ms] (215) QDP (216) QDPSizeChangeProof [EQUIVALENT, 0 ms] (217) YES (218) PiDP (219) UsableRulesProof [EQUIVALENT, 0 ms] (220) PiDP (221) PiDPToQDPProof [SOUND, 0 ms] (222) QDP (223) QDPSizeChangeProof [EQUIVALENT, 0 ms] (224) YES (225) PiDP (226) UsableRulesProof [EQUIVALENT, 0 ms] (227) PiDP (228) PiDPToQDPProof [SOUND, 0 ms] (229) QDP (230) QDPSizeChangeProof [EQUIVALENT, 0 ms] (231) YES (232) PiDP (233) UsableRulesProof [EQUIVALENT, 0 ms] (234) PiDP (235) PiDPToQDPProof [SOUND, 0 ms] (236) QDP (237) QDPSizeChangeProof [EQUIVALENT, 0 ms] (238) YES (239) PiDP (240) UsableRulesProof [EQUIVALENT, 0 ms] (241) PiDP (242) PiDPToQDPProof [EQUIVALENT, 0 ms] (243) QDP (244) QDPSizeChangeProof [EQUIVALENT, 0 ms] (245) YES (246) PiDP (247) UsableRulesProof [EQUIVALENT, 0 ms] (248) PiDP (249) PiDPToQDPProof [SOUND, 1 ms] (250) QDP (251) QDPSizeChangeProof [EQUIVALENT, 0 ms] (252) YES (253) PiDP (254) UsableRulesProof [EQUIVALENT, 0 ms] (255) PiDP (256) PiDPToQDPProof [SOUND, 0 ms] (257) QDP (258) QDPSizeChangeProof [EQUIVALENT, 0 ms] (259) YES (260) PiDP (261) UsableRulesProof [EQUIVALENT, 0 ms] (262) PiDP (263) PiDPToQDPProof [SOUND, 5 ms] (264) QDP (265) QDPOrderProof [EQUIVALENT, 72 ms] (266) QDP (267) DependencyGraphProof [EQUIVALENT, 0 ms] (268) TRUE (269) PiDP (270) UsableRulesProof [EQUIVALENT, 0 ms] (271) PiDP (272) PiDPToQDPProof [SOUND, 0 ms] (273) QDP (274) NonTerminationLoopProof [COMPLETE, 0 ms] (275) NO (276) PiDP (277) UsableRulesProof [EQUIVALENT, 0 ms] (278) PiDP (279) PiDPToQDPProof [SOUND, 0 ms] (280) QDP (281) QDPSizeChangeProof [EQUIVALENT, 0 ms] (282) YES (283) PiDP (284) UsableRulesProof [EQUIVALENT, 0 ms] (285) PiDP (286) PiDPToQDPProof [SOUND, 0 ms] (287) QDP (288) TransformationProof [SOUND, 0 ms] (289) QDP (290) TransformationProof [EQUIVALENT, 0 ms] (291) QDP (292) NonTerminationLoopProof [COMPLETE, 0 ms] (293) NO (294) PiDP (295) UsableRulesProof [EQUIVALENT, 0 ms] (296) PiDP (297) PiDPToQDPProof [SOUND, 0 ms] (298) QDP (299) PiDP (300) UsableRulesProof [EQUIVALENT, 0 ms] (301) PiDP (302) PiDP (303) UsableRulesProof [EQUIVALENT, 0 ms] (304) PiDP ---------------------------------------- (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 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 ---------------------------------------- (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 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) ---------------------------------------- (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 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 ---------------------------------------- (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 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 ---------------------------------------- (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 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 ---------------------------------------- (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 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 ---------------------------------------- (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 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 ---------------------------------------- (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 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 ---------------------------------------- (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 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 ---------------------------------------- (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) -> 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. ---------------------------------------- (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),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)))) ---------------------------------------- (34) 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. ---------------------------------------- (35) 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))) ---------------------------------------- (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) 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. ---------------------------------------- (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 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 ---------------------------------------- (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 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 ---------------------------------------- (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) 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_AAAA evaluates to t =PART_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 PART_IN_AAAA to PART_IN_AAAA. ---------------------------------------- (52) NO ---------------------------------------- (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: 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 ---------------------------------------- (54) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (55) 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 ---------------------------------------- (56) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (57) Obligation: Q DP problem: The TRS P consists of the following rules: QS_IN_AA -> U1_AA(part_in_aaaa) U1_AA(part_out_aaaa) -> U2_AA(qs_in_aa) U2_AA(qs_out_aa) -> QS_IN_AA U1_AA(part_out_aaaa) -> QS_IN_AA The TRS R consists of the following rules: part_in_aaaa -> U5_aaaa(less_in_aa) part_in_aaaa -> U7_aaaa(part_in_aaaa) part_in_aaaa -> part_out_aaaa qs_in_aa -> qs_out_aa qs_in_aa -> U1_aa(part_in_aaaa) U5_aaaa(less_out_aa(X)) -> U6_aaaa(part_in_gaaa(X)) U7_aaaa(part_out_aaaa) -> part_out_aaaa U1_aa(part_out_aaaa) -> U2_aa(qs_in_aa) less_in_aa -> less_out_aa(0) less_in_aa -> U9_aa(less_in_aa) U6_aaaa(part_out_gaaa) -> part_out_aaaa U2_aa(qs_out_aa) -> U3_aa(qs_in_aa) U9_aa(less_out_aa(X)) -> less_out_aa(s(X)) part_in_gaaa(X) -> U5_gaaa(X, less_in_ga(X)) part_in_gaaa(X) -> U7_gaaa(part_in_gaaa(X)) part_in_gaaa(X1) -> part_out_gaaa U3_aa(qs_out_aa) -> U4_aa(app_in_aaa) U5_gaaa(X, less_out_ga) -> U6_gaaa(part_in_gaaa(X)) U7_gaaa(part_out_gaaa) -> part_out_gaaa U4_aa(app_out_aaa) -> qs_out_aa less_in_ga(0) -> less_out_ga less_in_ga(s(X)) -> U9_ga(less_in_ga(X)) U6_gaaa(part_out_gaaa) -> part_out_gaaa app_in_aaa -> app_out_aaa app_in_aaa -> U8_aaa(app_in_aaa) U9_ga(less_out_ga) -> less_out_ga U8_aaa(app_out_aaa) -> app_out_aaa The set Q consists of the following terms: part_in_aaaa qs_in_aa U5_aaaa(x0) U7_aaaa(x0) U1_aa(x0) less_in_aa U6_aaaa(x0) U2_aa(x0) U9_aa(x0) part_in_gaaa(x0) U3_aa(x0) U5_gaaa(x0, x1) U7_gaaa(x0) U4_aa(x0) less_in_ga(x0) U6_gaaa(x0) app_in_aaa U9_ga(x0) U8_aaa(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (58) 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 ---------------------------------------- (59) 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) ---------------------------------------- (60) 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 ---------------------------------------- (61) 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 ---------------------------------------- (62) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 7 SCCs with 24 less nodes. ---------------------------------------- (63) Complex Obligation (AND) ---------------------------------------- (64) 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 ---------------------------------------- (65) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (66) 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 ---------------------------------------- (67) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (68) 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. ---------------------------------------- (69) 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 ---------------------------------------- (70) YES ---------------------------------------- (71) 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 ---------------------------------------- (72) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (73) 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 ---------------------------------------- (74) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (75) 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. ---------------------------------------- (76) 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. ---------------------------------------- (77) NO ---------------------------------------- (78) 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 ---------------------------------------- (79) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (80) 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 ---------------------------------------- (81) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (82) 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. ---------------------------------------- (83) 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 ---------------------------------------- (84) YES ---------------------------------------- (85) 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 ---------------------------------------- (86) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (87) 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 ---------------------------------------- (88) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (89) 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. ---------------------------------------- (90) 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)))) ---------------------------------------- (91) 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. ---------------------------------------- (92) 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))) ---------------------------------------- (93) 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. ---------------------------------------- (94) 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). ---------------------------------------- (95) NO ---------------------------------------- (96) 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 ---------------------------------------- (97) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (98) Obligation: Pi DP problem: The TRS P consists of the following rules: 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 ---------------------------------------- (99) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (100) Obligation: Q DP problem: The TRS P consists of the following rules: LESS_IN_AA -> LESS_IN_AA R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (101) 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. ---------------------------------------- (102) NO ---------------------------------------- (103) 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 ---------------------------------------- (104) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (105) Obligation: Pi DP problem: The TRS P consists of the following rules: 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 ---------------------------------------- (106) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (107) 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. ---------------------------------------- (108) 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_AAAA evaluates to t =PART_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 PART_IN_AAAA to PART_IN_AAAA. ---------------------------------------- (109) NO ---------------------------------------- (110) 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 ---------------------------------------- (111) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (112) 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 ---------------------------------------- (113) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (114) Obligation: Q DP problem: The TRS P consists of the following rules: QS_IN_AA -> U1_AA(part_in_aaaa) U1_AA(part_out_aaaa) -> U2_AA(qs_in_aa) U2_AA(qs_out_aa) -> QS_IN_AA U1_AA(part_out_aaaa) -> QS_IN_AA The TRS R consists of the following rules: part_in_aaaa -> U5_aaaa(less_in_aa) part_in_aaaa -> U7_aaaa(part_in_aaaa) part_in_aaaa -> part_out_aaaa qs_in_aa -> qs_out_aa qs_in_aa -> U1_aa(part_in_aaaa) U5_aaaa(less_out_aa(X)) -> U6_aaaa(part_in_gaaa(X)) U7_aaaa(part_out_aaaa) -> part_out_aaaa U1_aa(part_out_aaaa) -> U2_aa(qs_in_aa) less_in_aa -> less_out_aa(0) less_in_aa -> U9_aa(less_in_aa) U6_aaaa(part_out_gaaa(X)) -> part_out_aaaa U2_aa(qs_out_aa) -> U3_aa(qs_in_aa) U9_aa(less_out_aa(X)) -> less_out_aa(s(X)) part_in_gaaa(X) -> U5_gaaa(X, less_in_ga(X)) part_in_gaaa(X) -> U7_gaaa(X, part_in_gaaa(X)) part_in_gaaa(X1) -> part_out_gaaa(X1) U3_aa(qs_out_aa) -> U4_aa(app_in_aaa) U5_gaaa(X, less_out_ga(X)) -> U6_gaaa(X, part_in_gaaa(X)) U7_gaaa(X, part_out_gaaa(X)) -> part_out_gaaa(X) U4_aa(app_out_aaa) -> qs_out_aa less_in_ga(0) -> less_out_ga(0) less_in_ga(s(X)) -> U9_ga(X, less_in_ga(X)) U6_gaaa(X, part_out_gaaa(X)) -> part_out_gaaa(X) app_in_aaa -> app_out_aaa app_in_aaa -> U8_aaa(app_in_aaa) U9_ga(X, less_out_ga(X)) -> less_out_ga(s(X)) U8_aaa(app_out_aaa) -> app_out_aaa The set Q consists of the following terms: part_in_aaaa qs_in_aa U5_aaaa(x0) U7_aaaa(x0) U1_aa(x0) less_in_aa U6_aaaa(x0) U2_aa(x0) U9_aa(x0) part_in_gaaa(x0) U3_aa(x0) U5_gaaa(x0, x1) U7_gaaa(x0, x1) U4_aa(x0) less_in_ga(x0) U6_gaaa(x0, x1) app_in_aaa U9_ga(x0, x1) U8_aaa(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (115) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 3, "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": { "46": { "goal": [ { "clause": 0, "scope": 1, "term": "(qs T1 T2)" }, { "clause": 1, "scope": 1, "term": "(qs T1 T2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "type": "Nodes", "394": { "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": [] } }, "396": { "goal": [{ "clause": 7, "scope": 3, "term": "(less T38 T39)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "990": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "397": { "goal": [{ "clause": 8, "scope": 3, "term": "(less T38 T39)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "991": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "233": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (qs T17 X18) (',' (qs T18 X19) (app X18 (. T19 X19) T11)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T11"], "free": [ "X18", "X19" ], "exprvars": [] } }, "950": { "goal": [{ "clause": -1, "scope": -1, "term": "(qs T163 X259)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X259"], "exprvars": [] } }, "994": { "goal": [{ "clause": -1, "scope": -1, "term": "(app T201 (. T202 T203) X297)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X297"], "exprvars": [] } }, "951": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (qs T170 X260) (app T169 (. T171 X260) X261))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X261", "X260" ], "exprvars": [] } }, "995": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "999": { "goal": [{ "clause": -1, "scope": -1, "term": "(qs T150 X19)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X19"], "exprvars": [] } }, "915": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "916": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "917": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "56": { "goal": [{ "clause": 0, "scope": 1, "term": "(qs T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "919": { "goal": [{ "clause": -1, "scope": -1, "term": "(qs T17 X18)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X18"], "exprvars": [] } }, "57": { "goal": [{ "clause": 1, "scope": 1, "term": "(qs T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "3": { "goal": [{ "clause": -1, "scope": -1, "term": "(qs T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "762": { "goal": [{ "clause": 7, "scope": 5, "term": "(less T77 T80)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T77"], "free": [], "exprvars": [] } }, "961": { "goal": [{ "clause": -1, "scope": -1, "term": "(qs T170 X260)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X260"], "exprvars": [] } }, "401": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "764": { "goal": [{ "clause": 8, "scope": 5, "term": "(less T77 T80)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T77"], "free": [], "exprvars": [] } }, "962": { "goal": [{ "clause": -1, "scope": -1, "term": "(app T173 (. T174 T172) X261)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X261"], "exprvars": [] } }, "402": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "403": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "920": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (qs T150 X19) (app T149 (. 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T174 T172) X261)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X261"], "exprvars": [] } }, "922": { "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": [] } }, "60": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "725": { "goal": [{ "clause": -1, "scope": -1, "term": "(part T77 T84 X126 X127)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T77"], "free": [ "X126", "X127" ], "exprvars": [] } }, "923": { "goal": [{ "clause": 0, "scope": 6, "term": "(qs T17 X18)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X18"], "exprvars": [] } }, "61": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "968": { "goal": [{ "clause": 5, "scope": 7, "term": "(app T173 (. 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T12 X19) T11))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T11"], "free": [ "X16", "X17", "X18", "X19" ], "exprvars": [] } }, "64": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "927": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "928": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "929": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1000": { "goal": [{ "clause": -1, "scope": -1, "term": "(app T209 (. T210 T208) T11)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T11"], "free": [], "exprvars": [] } }, "1042": { "goal": [], "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": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "374": { "goal": [{ "clause": 2, "scope": 2, "term": "(part T12 T13 X16 X17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X16", "X17" ], "exprvars": [] } }, "375": { "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": [] } }, "694": { "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": [] } }, "892": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1039": { "goal": [{ "clause": 6, "scope": 8, "term": "(app T209 (. T210 T208) T11)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T11"], "free": [], "exprvars": [] } }, "894": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1038": { "goal": [{ "clause": 5, "scope": 8, "term": "(app T209 (. T210 T208) T11)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T11"], "free": [], "exprvars": [] } }, "1037": { "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": [] } }, "698": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "852": { "goal": [{ "clause": -1, "scope": -1, "term": "(part T116 T119 X185 X186)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T116"], "free": [ "X185", "X186" ], "exprvars": [] } }, "896": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "776": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "810": { "goal": [{ "clause": 3, "scope": 4, "term": "(part T43 T44 X67 X68)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T43"], "free": [ "X67", "X68" ], "exprvars": [] } }, "898": { "goal": [{ "clause": 3, "scope": 2, "term": "(part T12 T13 X16 X17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X16", "X17" ], "exprvars": [] } }, "899": { "goal": [{ "clause": 4, "scope": 2, "term": "(part T12 T13 X16 X17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X16", "X17" ], "exprvars": [] } }, "779": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "812": { "goal": [{ "clause": 4, "scope": 4, "term": "(part T43 T44 X67 X68)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T43"], "free": [ "X67", "X68" ], "exprvars": [] } }, "856": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1055": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1054": { "goal": [{ "clause": -1, "scope": -1, "term": "(app T240 (. T241 T242) T239)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T239"], "free": [], "exprvars": [] } }, "382": { "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": [] } }, "384": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "781": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "386": { "goal": [{ "clause": -1, "scope": -1, "term": "(less T38 T39)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "387": { "goal": [{ "clause": -1, "scope": -1, "term": "(part T43 T44 X67 X68)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T43"], "free": [ "X67", "X68" ], "exprvars": [] } }, "223": { "goal": [{ "clause": -1, "scope": -1, "term": "(part T12 T13 X16 X17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X16", "X17" ], "exprvars": [] } }, "741": { "goal": [ { "clause": 7, "scope": 5, "term": "(less T77 T80)" }, { "clause": 8, "scope": 5, "term": "(less T77 T80)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T77"], "free": [], "exprvars": [] } }, "785": { "goal": [{ "clause": -1, "scope": -1, "term": "(less T96 T98)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T96"], "free": [], "exprvars": [] } }, "786": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "468": { "goal": [{ "clause": -1, "scope": -1, "term": "(less T58 T59)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "469": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "942": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (part T158 T159 X257 X258) (',' (qs X257 X259) (',' (qs X258 X260) (app X259 (. T158 X260) X261))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X261", "X257", "X258", "X259", "X260" ], "exprvars": [] } }, "943": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "504": { "goal": [ { "clause": 2, "scope": 4, "term": "(part T43 T44 X67 X68)" }, { "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": [] } }, "505": { "goal": [{ "clause": 2, "scope": 4, "term": "(part T43 T44 X67 X68)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T43"], "free": [ "X67", "X68" ], "exprvars": [] } }, "945": { "goal": [{ "clause": -1, "scope": -1, "term": "(part T158 T159 X257 X258)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X257", "X258" ], "exprvars": [] } }, "989": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "506": { "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": [] } }, "902": { "goal": [{ "clause": -1, "scope": -1, "term": "(part T141 T142 X230 X231)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X230", "X231" ], "exprvars": [] } }, "946": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (qs T163 X259) (',' (qs T164 X260) (app X259 (. T165 X260) X261)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X261", "X259", "X260" ], "exprvars": [] } }, "903": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 3, "to": 46, "label": "CASE" }, { "from": 46, "to": 56, "label": "PARALLEL" }, { "from": 46, "to": 57, "label": "PARALLEL" }, { "from": 56, "to": 60, "label": "EVAL with clause\nqs([], []).\nand substitutionT1 -> [],\nT2 -> []" }, { "from": 56, "to": 61, "label": "EVAL-BACKTRACK" }, { "from": 57, "to": 63, "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": 57, "to": 64, "label": "EVAL-BACKTRACK" }, { "from": 60, "to": 62, "label": "SUCCESS" }, { "from": 63, "to": 223, "label": "SPLIT 1" }, { "from": 63, "to": 233, "label": "SPLIT 2\nreplacements:X16 -> T17,\nX17 -> T18,\nT12 -> T19" }, { "from": 223, "to": 327, "label": "CASE" }, { "from": 233, "to": 919, "label": "SPLIT 1" }, { "from": 233, "to": 920, "label": "SPLIT 2\nreplacements:X18 -> T149,\nT18 -> T150,\nT19 -> T151" }, { "from": 327, "to": 374, "label": "PARALLEL" }, { "from": 327, "to": 375, "label": "PARALLEL" }, { "from": 374, "to": 382, "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": 374, "to": 384, "label": "EVAL-BACKTRACK" }, { "from": 375, "to": 898, "label": "PARALLEL" }, { "from": 375, "to": 899, "label": "PARALLEL" }, { "from": 382, "to": 386, "label": "SPLIT 1" }, { "from": 382, "to": 387, "label": "SPLIT 2\nnew knowledge:\nT43 is ground\nreplacements:T38 -> T43,\nT40 -> T44" }, { "from": 386, "to": 394, "label": "CASE" }, { "from": 387, "to": 504, "label": "CASE" }, { "from": 394, "to": 396, "label": "PARALLEL" }, { "from": 394, "to": 397, "label": "PARALLEL" }, { "from": 396, "to": 401, "label": "EVAL with clause\nless(0, s(X77)).\nand substitutionT38 -> 0,\nX77 -> T51,\nT39 -> s(T51)" }, { "from": 396, "to": 402, "label": "EVAL-BACKTRACK" }, { "from": 397, "to": 468, "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": 397, "to": 469, "label": "EVAL-BACKTRACK" }, { "from": 401, "to": 403, "label": "SUCCESS" }, { "from": 468, "to": 386, "label": "INSTANCE with matching:\nT38 -> T58\nT39 -> T59" }, { "from": 504, "to": 505, "label": "PARALLEL" }, { "from": 504, "to": 506, "label": "PARALLEL" }, { "from": 505, "to": 694, "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": 505, "to": 698, "label": "EVAL-BACKTRACK" }, { "from": 506, "to": 810, "label": "PARALLEL" }, { "from": 506, "to": 812, "label": "PARALLEL" }, { "from": 694, "to": 723, "label": "SPLIT 1" }, { "from": 694, "to": 725, "label": "SPLIT 2\nnew knowledge:\nT77 is ground\nreplacements:T81 -> T84" }, { "from": 723, "to": 741, "label": "CASE" }, { "from": 725, "to": 387, "label": "INSTANCE with matching:\nT43 -> T77\nT44 -> T84\nX67 -> X126\nX68 -> X127" }, { "from": 741, "to": 762, "label": "PARALLEL" }, { "from": 741, "to": 764, "label": "PARALLEL" }, { "from": 762, "to": 776, "label": "EVAL with clause\nless(0, s(X136)).\nand substitutionT77 -> 0,\nX136 -> T91,\nT80 -> s(T91)" }, { "from": 762, "to": 779, "label": "EVAL-BACKTRACK" }, { "from": 764, "to": 785, "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": 764, "to": 786, "label": "EVAL-BACKTRACK" }, { "from": 776, "to": 781, "label": "SUCCESS" }, { "from": 785, "to": 723, "label": "INSTANCE with matching:\nT77 -> T96\nT80 -> T98" }, { "from": 810, "to": 852, "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": 810, "to": 856, "label": "EVAL-BACKTRACK" }, { "from": 812, "to": 892, "label": "EVAL with clause\npart(X196, [], [], []).\nand substitutionT43 -> T125,\nX196 -> T125,\nT44 -> [],\nX67 -> [],\nX68 -> []" }, { "from": 812, "to": 894, "label": "EVAL-BACKTRACK" }, { "from": 852, "to": 387, "label": "INSTANCE with matching:\nT43 -> T116\nT44 -> T119\nX67 -> X185\nX68 -> X186" }, { "from": 892, "to": 896, "label": "SUCCESS" }, { "from": 898, "to": 902, "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": 898, "to": 903, "label": "EVAL-BACKTRACK" }, { "from": 899, "to": 915, "label": "EVAL with clause\npart(X241, [], [], []).\nand substitutionT12 -> T148,\nX241 -> T148,\nT13 -> [],\nX16 -> [],\nX17 -> []" }, { "from": 899, "to": 916, "label": "EVAL-BACKTRACK" }, { "from": 902, "to": 223, "label": "INSTANCE with matching:\nT12 -> T141\nT13 -> T142\nX16 -> X230\nX17 -> X231" }, { "from": 915, "to": 917, "label": "SUCCESS" }, { "from": 919, "to": 922, "label": "CASE" }, { "from": 920, "to": 999, "label": "SPLIT 1" }, { "from": 920, "to": 1000, "label": "SPLIT 2\nreplacements:X19 -> T208,\nT149 -> T209,\nT151 -> T210" }, { "from": 922, "to": 923, "label": "PARALLEL" }, { "from": 922, "to": 925, "label": "PARALLEL" }, { "from": 923, "to": 927, "label": "EVAL with clause\nqs([], []).\nand substitutionT17 -> [],\nX18 -> []" }, { "from": 923, "to": 928, "label": "EVAL-BACKTRACK" }, { "from": 925, "to": 942, "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": 925, "to": 943, "label": "EVAL-BACKTRACK" }, { "from": 927, "to": 929, "label": "SUCCESS" }, { "from": 942, "to": 945, "label": "SPLIT 1" }, { "from": 942, "to": 946, "label": "SPLIT 2\nreplacements:X257 -> T163,\nX258 -> T164,\nT158 -> T165" }, { "from": 945, "to": 223, "label": "INSTANCE with matching:\nT12 -> T158\nT13 -> T159\nX16 -> X257\nX17 -> X258" }, { "from": 946, "to": 950, "label": "SPLIT 1" }, { "from": 946, "to": 951, "label": "SPLIT 2\nreplacements:X259 -> T169,\nT164 -> T170,\nT165 -> T171" }, { "from": 950, "to": 919, "label": "INSTANCE with matching:\nT17 -> T163\nX18 -> X259" }, { "from": 951, "to": 961, "label": "SPLIT 1" }, { "from": 951, "to": 962, "label": "SPLIT 2\nreplacements:X260 -> T172,\nT169 -> T173,\nT171 -> T174" }, { "from": 961, "to": 919, "label": "INSTANCE with matching:\nT17 -> T170\nX18 -> X260" }, { "from": 962, "to": 965, "label": "CASE" }, { "from": 965, "to": 968, "label": "PARALLEL" }, { "from": 965, "to": 969, "label": "PARALLEL" }, { "from": 968, "to": 989, "label": "EVAL with clause\napp([], X282, X282).\nand substitutionT173 -> [],\nT174 -> T187,\nT172 -> T188,\nX282 -> .(T187, T188),\nX261 -> .(T187, T188)" }, { "from": 968, "to": 990, "label": "EVAL-BACKTRACK" }, { "from": 969, "to": 994, "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": 969, "to": 995, "label": "EVAL-BACKTRACK" }, { "from": 989, "to": 991, "label": "SUCCESS" }, { "from": 994, "to": 962, "label": "INSTANCE with matching:\nT173 -> T201\nT174 -> T202\nT172 -> T203\nX261 -> X297" }, { "from": 999, "to": 919, "label": "INSTANCE with matching:\nT17 -> T150\nX18 -> X19" }, { "from": 1000, "to": 1037, "label": "CASE" }, { "from": 1037, "to": 1038, "label": "PARALLEL" }, { "from": 1037, "to": 1039, "label": "PARALLEL" }, { "from": 1038, "to": 1040, "label": "EVAL with clause\napp([], X306, X306).\nand substitutionT209 -> [],\nT210 -> T223,\nT208 -> T224,\nX306 -> .(T223, T224),\nT11 -> .(T223, T224)" }, { "from": 1038, "to": 1041, "label": "EVAL-BACKTRACK" }, { "from": 1039, "to": 1054, "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": 1039, "to": 1055, "label": "EVAL-BACKTRACK" }, { "from": 1040, "to": 1042, "label": "SUCCESS" }, { "from": 1054, "to": 1000, "label": "INSTANCE with matching:\nT209 -> T240\nT210 -> T241\nT208 -> T242\nT11 -> T239" } ], "type": "Graph" } } ---------------------------------------- (116) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f3_in([]) -> f3_out1 f3_in(T11) -> U1(f63_in(T11), T11) U1(f63_out1(X18, T12, X19), T11) -> f3_out1 f386_in -> f386_out1(0) f386_in -> U2(f386_in) U2(f386_out1(T58)) -> f386_out1(s(T58)) f387_in(T77) -> U3(f694_in(T77), T77) U3(f694_out1, T77) -> f387_out1 f387_in(T116) -> U4(f387_in(T116), T116) U4(f387_out1, T116) -> f387_out1 f387_in(T125) -> f387_out1 f723_in(0) -> f723_out1 f723_in(s(T96)) -> U5(f723_in(T96), s(T96)) U5(f723_out1, s(T96)) -> f723_out1 f223_in -> U6(f382_in) U6(f382_out1(T38)) -> f223_out1 f223_in -> U7(f223_in) U7(f223_out1) -> f223_out1 f223_in -> f223_out1 f919_in -> f919_out1 f919_in -> U8(f942_in) U8(f942_out1) -> f919_out1 f962_in -> f962_out1 f962_in -> U9(f962_in) U9(f962_out1) -> f962_out1 f1000_in(.(T223, T224)) -> f1000_out1([], T223, T224) f1000_in(.(T235, T239)) -> U10(f1000_in(T239), .(T235, T239)) U10(f1000_out1(T240, T241, T242), .(T235, T239)) -> f1000_out1(.(T235, T240), T241, T242) f63_in(T11) -> U11(f223_in, T11) U11(f223_out1, T11) -> U12(f233_in(T11), T11) U12(f233_out1(X18, T19, X19), T11) -> f63_out1(X18, T19, X19) f233_in(T11) -> U13(f919_in, T11) U13(f919_out1, T11) -> U14(f920_in(T11), T11) U14(f920_out1(T149, T151, X19), T11) -> f233_out1(T149, T151, X19) f382_in -> U15(f386_in) U15(f386_out1(T43)) -> U16(f387_in(T43), T43) U16(f387_out1, T43) -> f382_out1(T43) f694_in(T77) -> U17(f723_in(T77), T77) U17(f723_out1, T77) -> U18(f387_in(T77), T77) U18(f387_out1, T77) -> f694_out1 f920_in(T11) -> U19(f919_in, T11) U19(f919_out1, T11) -> U20(f1000_in(T11), T11) U20(f1000_out1(T209, T210, T208), T11) -> f920_out1(T209, T210, T208) f942_in -> U21(f223_in) U21(f223_out1) -> U22(f946_in) U22(f946_out1) -> f942_out1 f946_in -> U23(f919_in) U23(f919_out1) -> U24(f951_in) U24(f951_out1) -> f946_out1 f951_in -> U25(f919_in) U25(f919_out1) -> U26(f962_in) U26(f962_out1) -> f951_out1 Q is empty. ---------------------------------------- (117) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (118) Obligation: Q DP problem: The TRS P consists of the following rules: F3_IN(T11) -> U1^1(f63_in(T11), T11) F3_IN(T11) -> F63_IN(T11) F386_IN -> U2^1(f386_in) F386_IN -> F386_IN F387_IN(T77) -> U3^1(f694_in(T77), T77) F387_IN(T77) -> F694_IN(T77) F387_IN(T116) -> U4^1(f387_in(T116), T116) F387_IN(T116) -> F387_IN(T116) F723_IN(s(T96)) -> U5^1(f723_in(T96), s(T96)) F723_IN(s(T96)) -> F723_IN(T96) F223_IN -> U6^1(f382_in) F223_IN -> F382_IN F223_IN -> U7^1(f223_in) F223_IN -> F223_IN F919_IN -> U8^1(f942_in) F919_IN -> F942_IN F962_IN -> U9^1(f962_in) F962_IN -> F962_IN F1000_IN(.(T235, T239)) -> U10^1(f1000_in(T239), .(T235, T239)) F1000_IN(.(T235, T239)) -> F1000_IN(T239) F63_IN(T11) -> U11^1(f223_in, T11) F63_IN(T11) -> F223_IN U11^1(f223_out1, T11) -> U12^1(f233_in(T11), T11) U11^1(f223_out1, T11) -> F233_IN(T11) F233_IN(T11) -> U13^1(f919_in, T11) F233_IN(T11) -> F919_IN U13^1(f919_out1, T11) -> U14^1(f920_in(T11), T11) U13^1(f919_out1, T11) -> F920_IN(T11) F382_IN -> U15^1(f386_in) F382_IN -> F386_IN U15^1(f386_out1(T43)) -> U16^1(f387_in(T43), T43) U15^1(f386_out1(T43)) -> F387_IN(T43) F694_IN(T77) -> U17^1(f723_in(T77), T77) F694_IN(T77) -> F723_IN(T77) U17^1(f723_out1, T77) -> U18^1(f387_in(T77), T77) U17^1(f723_out1, T77) -> F387_IN(T77) F920_IN(T11) -> U19^1(f919_in, T11) F920_IN(T11) -> F919_IN U19^1(f919_out1, T11) -> U20^1(f1000_in(T11), T11) U19^1(f919_out1, T11) -> F1000_IN(T11) F942_IN -> U21^1(f223_in) F942_IN -> F223_IN U21^1(f223_out1) -> U22^1(f946_in) U21^1(f223_out1) -> F946_IN F946_IN -> U23^1(f919_in) F946_IN -> F919_IN U23^1(f919_out1) -> U24^1(f951_in) U23^1(f919_out1) -> F951_IN F951_IN -> U25^1(f919_in) F951_IN -> F919_IN U25^1(f919_out1) -> U26^1(f962_in) U25^1(f919_out1) -> F962_IN The TRS R consists of the following rules: f3_in([]) -> f3_out1 f3_in(T11) -> U1(f63_in(T11), T11) U1(f63_out1(X18, T12, X19), T11) -> f3_out1 f386_in -> f386_out1(0) f386_in -> U2(f386_in) U2(f386_out1(T58)) -> f386_out1(s(T58)) f387_in(T77) -> U3(f694_in(T77), T77) U3(f694_out1, T77) -> f387_out1 f387_in(T116) -> U4(f387_in(T116), T116) U4(f387_out1, T116) -> f387_out1 f387_in(T125) -> f387_out1 f723_in(0) -> f723_out1 f723_in(s(T96)) -> U5(f723_in(T96), s(T96)) U5(f723_out1, s(T96)) -> f723_out1 f223_in -> U6(f382_in) U6(f382_out1(T38)) -> f223_out1 f223_in -> U7(f223_in) U7(f223_out1) -> f223_out1 f223_in -> f223_out1 f919_in -> f919_out1 f919_in -> U8(f942_in) U8(f942_out1) -> f919_out1 f962_in -> f962_out1 f962_in -> U9(f962_in) U9(f962_out1) -> f962_out1 f1000_in(.(T223, T224)) -> f1000_out1([], T223, T224) f1000_in(.(T235, T239)) -> U10(f1000_in(T239), .(T235, T239)) U10(f1000_out1(T240, T241, T242), .(T235, T239)) -> f1000_out1(.(T235, T240), T241, T242) f63_in(T11) -> U11(f223_in, T11) U11(f223_out1, T11) -> U12(f233_in(T11), T11) U12(f233_out1(X18, T19, X19), T11) -> f63_out1(X18, T19, X19) f233_in(T11) -> U13(f919_in, T11) U13(f919_out1, T11) -> U14(f920_in(T11), T11) U14(f920_out1(T149, T151, X19), T11) -> f233_out1(T149, T151, X19) f382_in -> U15(f386_in) U15(f386_out1(T43)) -> U16(f387_in(T43), T43) U16(f387_out1, T43) -> f382_out1(T43) f694_in(T77) -> U17(f723_in(T77), T77) U17(f723_out1, T77) -> U18(f387_in(T77), T77) U18(f387_out1, T77) -> f694_out1 f920_in(T11) -> U19(f919_in, T11) U19(f919_out1, T11) -> U20(f1000_in(T11), T11) U20(f1000_out1(T209, T210, T208), T11) -> f920_out1(T209, T210, T208) f942_in -> U21(f223_in) U21(f223_out1) -> U22(f946_in) U22(f946_out1) -> f942_out1 f946_in -> U23(f919_in) U23(f919_out1) -> U24(f951_in) U24(f951_out1) -> f946_out1 f951_in -> U25(f919_in) U25(f919_out1) -> U26(f962_in) U26(f962_out1) -> f951_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (119) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 7 SCCs with 36 less nodes. ---------------------------------------- (120) Complex Obligation (AND) ---------------------------------------- (121) Obligation: Q DP problem: The TRS P consists of the following rules: F1000_IN(.(T235, T239)) -> F1000_IN(T239) The TRS R consists of the following rules: f3_in([]) -> f3_out1 f3_in(T11) -> U1(f63_in(T11), T11) U1(f63_out1(X18, T12, X19), T11) -> f3_out1 f386_in -> f386_out1(0) f386_in -> U2(f386_in) U2(f386_out1(T58)) -> f386_out1(s(T58)) f387_in(T77) -> U3(f694_in(T77), T77) U3(f694_out1, T77) -> f387_out1 f387_in(T116) -> U4(f387_in(T116), T116) U4(f387_out1, T116) -> f387_out1 f387_in(T125) -> f387_out1 f723_in(0) -> f723_out1 f723_in(s(T96)) -> U5(f723_in(T96), s(T96)) U5(f723_out1, s(T96)) -> f723_out1 f223_in -> U6(f382_in) U6(f382_out1(T38)) -> f223_out1 f223_in -> U7(f223_in) U7(f223_out1) -> f223_out1 f223_in -> f223_out1 f919_in -> f919_out1 f919_in -> U8(f942_in) U8(f942_out1) -> f919_out1 f962_in -> f962_out1 f962_in -> U9(f962_in) U9(f962_out1) -> f962_out1 f1000_in(.(T223, T224)) -> f1000_out1([], T223, T224) f1000_in(.(T235, T239)) -> U10(f1000_in(T239), .(T235, T239)) U10(f1000_out1(T240, T241, T242), .(T235, T239)) -> f1000_out1(.(T235, T240), T241, T242) f63_in(T11) -> U11(f223_in, T11) U11(f223_out1, T11) -> U12(f233_in(T11), T11) U12(f233_out1(X18, T19, X19), T11) -> f63_out1(X18, T19, X19) f233_in(T11) -> U13(f919_in, T11) U13(f919_out1, T11) -> U14(f920_in(T11), T11) U14(f920_out1(T149, T151, X19), T11) -> f233_out1(T149, T151, X19) f382_in -> U15(f386_in) U15(f386_out1(T43)) -> U16(f387_in(T43), T43) U16(f387_out1, T43) -> f382_out1(T43) f694_in(T77) -> U17(f723_in(T77), T77) U17(f723_out1, T77) -> U18(f387_in(T77), T77) U18(f387_out1, T77) -> f694_out1 f920_in(T11) -> U19(f919_in, T11) U19(f919_out1, T11) -> U20(f1000_in(T11), T11) U20(f1000_out1(T209, T210, T208), T11) -> f920_out1(T209, T210, T208) f942_in -> U21(f223_in) U21(f223_out1) -> U22(f946_in) U22(f946_out1) -> f942_out1 f946_in -> U23(f919_in) U23(f919_out1) -> U24(f951_in) U24(f951_out1) -> f946_out1 f951_in -> U25(f919_in) U25(f919_out1) -> U26(f962_in) U26(f962_out1) -> f951_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (122) 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. ---------------------------------------- (123) Obligation: Q DP problem: The TRS P consists of the following rules: F1000_IN(.(T235, T239)) -> F1000_IN(T239) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (124) 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: *F1000_IN(.(T235, T239)) -> F1000_IN(T239) The graph contains the following edges 1 > 1 ---------------------------------------- (125) YES ---------------------------------------- (126) Obligation: Q DP problem: The TRS P consists of the following rules: F962_IN -> F962_IN The TRS R consists of the following rules: f3_in([]) -> f3_out1 f3_in(T11) -> U1(f63_in(T11), T11) U1(f63_out1(X18, T12, X19), T11) -> f3_out1 f386_in -> f386_out1(0) f386_in -> U2(f386_in) U2(f386_out1(T58)) -> f386_out1(s(T58)) f387_in(T77) -> U3(f694_in(T77), T77) U3(f694_out1, T77) -> f387_out1 f387_in(T116) -> U4(f387_in(T116), T116) U4(f387_out1, T116) -> f387_out1 f387_in(T125) -> f387_out1 f723_in(0) -> f723_out1 f723_in(s(T96)) -> U5(f723_in(T96), s(T96)) U5(f723_out1, s(T96)) -> f723_out1 f223_in -> U6(f382_in) U6(f382_out1(T38)) -> f223_out1 f223_in -> U7(f223_in) U7(f223_out1) -> f223_out1 f223_in -> f223_out1 f919_in -> f919_out1 f919_in -> U8(f942_in) U8(f942_out1) -> f919_out1 f962_in -> f962_out1 f962_in -> U9(f962_in) U9(f962_out1) -> f962_out1 f1000_in(.(T223, T224)) -> f1000_out1([], T223, T224) f1000_in(.(T235, T239)) -> U10(f1000_in(T239), .(T235, T239)) U10(f1000_out1(T240, T241, T242), .(T235, T239)) -> f1000_out1(.(T235, T240), T241, T242) f63_in(T11) -> U11(f223_in, T11) U11(f223_out1, T11) -> U12(f233_in(T11), T11) U12(f233_out1(X18, T19, X19), T11) -> f63_out1(X18, T19, X19) f233_in(T11) -> U13(f919_in, T11) U13(f919_out1, T11) -> U14(f920_in(T11), T11) U14(f920_out1(T149, T151, X19), T11) -> f233_out1(T149, T151, X19) f382_in -> U15(f386_in) U15(f386_out1(T43)) -> U16(f387_in(T43), T43) U16(f387_out1, T43) -> f382_out1(T43) f694_in(T77) -> U17(f723_in(T77), T77) U17(f723_out1, T77) -> U18(f387_in(T77), T77) U18(f387_out1, T77) -> f694_out1 f920_in(T11) -> U19(f919_in, T11) U19(f919_out1, T11) -> U20(f1000_in(T11), T11) U20(f1000_out1(T209, T210, T208), T11) -> f920_out1(T209, T210, T208) f942_in -> U21(f223_in) U21(f223_out1) -> U22(f946_in) U22(f946_out1) -> f942_out1 f946_in -> U23(f919_in) U23(f919_out1) -> U24(f951_in) U24(f951_out1) -> f946_out1 f951_in -> U25(f919_in) U25(f919_out1) -> U26(f962_in) U26(f962_out1) -> f951_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (127) 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. ---------------------------------------- (128) Obligation: Q DP problem: The TRS P consists of the following rules: F962_IN -> F962_IN R is empty. 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 = F962_IN evaluates to t =F962_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 F962_IN to F962_IN. ---------------------------------------- (130) NO ---------------------------------------- (131) Obligation: Q DP problem: The TRS P consists of the following rules: F723_IN(s(T96)) -> F723_IN(T96) The TRS R consists of the following rules: f3_in([]) -> f3_out1 f3_in(T11) -> U1(f63_in(T11), T11) U1(f63_out1(X18, T12, X19), T11) -> f3_out1 f386_in -> f386_out1(0) f386_in -> U2(f386_in) U2(f386_out1(T58)) -> f386_out1(s(T58)) f387_in(T77) -> U3(f694_in(T77), T77) U3(f694_out1, T77) -> f387_out1 f387_in(T116) -> U4(f387_in(T116), T116) U4(f387_out1, T116) -> f387_out1 f387_in(T125) -> f387_out1 f723_in(0) -> f723_out1 f723_in(s(T96)) -> U5(f723_in(T96), s(T96)) U5(f723_out1, s(T96)) -> f723_out1 f223_in -> U6(f382_in) U6(f382_out1(T38)) -> f223_out1 f223_in -> U7(f223_in) U7(f223_out1) -> f223_out1 f223_in -> f223_out1 f919_in -> f919_out1 f919_in -> U8(f942_in) U8(f942_out1) -> f919_out1 f962_in -> f962_out1 f962_in -> U9(f962_in) U9(f962_out1) -> f962_out1 f1000_in(.(T223, T224)) -> f1000_out1([], T223, T224) f1000_in(.(T235, T239)) -> U10(f1000_in(T239), .(T235, T239)) U10(f1000_out1(T240, T241, T242), .(T235, T239)) -> f1000_out1(.(T235, T240), T241, T242) f63_in(T11) -> U11(f223_in, T11) U11(f223_out1, T11) -> U12(f233_in(T11), T11) U12(f233_out1(X18, T19, X19), T11) -> f63_out1(X18, T19, X19) f233_in(T11) -> U13(f919_in, T11) U13(f919_out1, T11) -> U14(f920_in(T11), T11) U14(f920_out1(T149, T151, X19), T11) -> f233_out1(T149, T151, X19) f382_in -> U15(f386_in) U15(f386_out1(T43)) -> U16(f387_in(T43), T43) U16(f387_out1, T43) -> f382_out1(T43) f694_in(T77) -> U17(f723_in(T77), T77) U17(f723_out1, T77) -> U18(f387_in(T77), T77) U18(f387_out1, T77) -> f694_out1 f920_in(T11) -> U19(f919_in, T11) U19(f919_out1, T11) -> U20(f1000_in(T11), T11) U20(f1000_out1(T209, T210, T208), T11) -> f920_out1(T209, T210, T208) f942_in -> U21(f223_in) U21(f223_out1) -> U22(f946_in) U22(f946_out1) -> f942_out1 f946_in -> U23(f919_in) U23(f919_out1) -> U24(f951_in) U24(f951_out1) -> f946_out1 f951_in -> U25(f919_in) U25(f919_out1) -> U26(f962_in) U26(f962_out1) -> f951_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: F723_IN(s(T96)) -> F723_IN(T96) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (134) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *F723_IN(s(T96)) -> F723_IN(T96) The graph contains the following edges 1 > 1 ---------------------------------------- (135) YES ---------------------------------------- (136) Obligation: Q DP problem: The TRS P consists of the following rules: F387_IN(T77) -> F694_IN(T77) F694_IN(T77) -> U17^1(f723_in(T77), T77) U17^1(f723_out1, T77) -> F387_IN(T77) F387_IN(T116) -> F387_IN(T116) The TRS R consists of the following rules: f3_in([]) -> f3_out1 f3_in(T11) -> U1(f63_in(T11), T11) U1(f63_out1(X18, T12, X19), T11) -> f3_out1 f386_in -> f386_out1(0) f386_in -> U2(f386_in) U2(f386_out1(T58)) -> f386_out1(s(T58)) f387_in(T77) -> U3(f694_in(T77), T77) U3(f694_out1, T77) -> f387_out1 f387_in(T116) -> U4(f387_in(T116), T116) U4(f387_out1, T116) -> f387_out1 f387_in(T125) -> f387_out1 f723_in(0) -> f723_out1 f723_in(s(T96)) -> U5(f723_in(T96), s(T96)) U5(f723_out1, s(T96)) -> f723_out1 f223_in -> U6(f382_in) U6(f382_out1(T38)) -> f223_out1 f223_in -> U7(f223_in) U7(f223_out1) -> f223_out1 f223_in -> f223_out1 f919_in -> f919_out1 f919_in -> U8(f942_in) U8(f942_out1) -> f919_out1 f962_in -> f962_out1 f962_in -> U9(f962_in) U9(f962_out1) -> f962_out1 f1000_in(.(T223, T224)) -> f1000_out1([], T223, T224) f1000_in(.(T235, T239)) -> U10(f1000_in(T239), .(T235, T239)) U10(f1000_out1(T240, T241, T242), .(T235, T239)) -> f1000_out1(.(T235, T240), T241, T242) f63_in(T11) -> U11(f223_in, T11) U11(f223_out1, T11) -> U12(f233_in(T11), T11) U12(f233_out1(X18, T19, X19), T11) -> f63_out1(X18, T19, X19) f233_in(T11) -> U13(f919_in, T11) U13(f919_out1, T11) -> U14(f920_in(T11), T11) U14(f920_out1(T149, T151, X19), T11) -> f233_out1(T149, T151, X19) f382_in -> U15(f386_in) U15(f386_out1(T43)) -> U16(f387_in(T43), T43) U16(f387_out1, T43) -> f382_out1(T43) f694_in(T77) -> U17(f723_in(T77), T77) U17(f723_out1, T77) -> U18(f387_in(T77), T77) U18(f387_out1, T77) -> f694_out1 f920_in(T11) -> U19(f919_in, T11) U19(f919_out1, T11) -> U20(f1000_in(T11), T11) U20(f1000_out1(T209, T210, T208), T11) -> f920_out1(T209, T210, T208) f942_in -> U21(f223_in) U21(f223_out1) -> U22(f946_in) U22(f946_out1) -> f942_out1 f946_in -> U23(f919_in) U23(f919_out1) -> U24(f951_in) U24(f951_out1) -> f946_out1 f951_in -> U25(f919_in) U25(f919_out1) -> U26(f962_in) U26(f962_out1) -> f951_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (137) 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 = F387_IN(T116) evaluates to t =F387_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 F387_IN(T116) to F387_IN(T116). ---------------------------------------- (138) NO ---------------------------------------- (139) Obligation: Q DP problem: The TRS P consists of the following rules: F386_IN -> F386_IN The TRS R consists of the following rules: f3_in([]) -> f3_out1 f3_in(T11) -> U1(f63_in(T11), T11) U1(f63_out1(X18, T12, X19), T11) -> f3_out1 f386_in -> f386_out1(0) f386_in -> U2(f386_in) U2(f386_out1(T58)) -> f386_out1(s(T58)) f387_in(T77) -> U3(f694_in(T77), T77) U3(f694_out1, T77) -> f387_out1 f387_in(T116) -> U4(f387_in(T116), T116) U4(f387_out1, T116) -> f387_out1 f387_in(T125) -> f387_out1 f723_in(0) -> f723_out1 f723_in(s(T96)) -> U5(f723_in(T96), s(T96)) U5(f723_out1, s(T96)) -> f723_out1 f223_in -> U6(f382_in) U6(f382_out1(T38)) -> f223_out1 f223_in -> U7(f223_in) U7(f223_out1) -> f223_out1 f223_in -> f223_out1 f919_in -> f919_out1 f919_in -> U8(f942_in) U8(f942_out1) -> f919_out1 f962_in -> f962_out1 f962_in -> U9(f962_in) U9(f962_out1) -> f962_out1 f1000_in(.(T223, T224)) -> f1000_out1([], T223, T224) f1000_in(.(T235, T239)) -> U10(f1000_in(T239), .(T235, T239)) U10(f1000_out1(T240, T241, T242), .(T235, T239)) -> f1000_out1(.(T235, T240), T241, T242) f63_in(T11) -> U11(f223_in, T11) U11(f223_out1, T11) -> U12(f233_in(T11), T11) U12(f233_out1(X18, T19, X19), T11) -> f63_out1(X18, T19, X19) f233_in(T11) -> U13(f919_in, T11) U13(f919_out1, T11) -> U14(f920_in(T11), T11) U14(f920_out1(T149, T151, X19), T11) -> f233_out1(T149, T151, X19) f382_in -> U15(f386_in) U15(f386_out1(T43)) -> U16(f387_in(T43), T43) U16(f387_out1, T43) -> f382_out1(T43) f694_in(T77) -> U17(f723_in(T77), T77) U17(f723_out1, T77) -> U18(f387_in(T77), T77) U18(f387_out1, T77) -> f694_out1 f920_in(T11) -> U19(f919_in, T11) U19(f919_out1, T11) -> U20(f1000_in(T11), T11) U20(f1000_out1(T209, T210, T208), T11) -> f920_out1(T209, T210, T208) f942_in -> U21(f223_in) U21(f223_out1) -> U22(f946_in) U22(f946_out1) -> f942_out1 f946_in -> U23(f919_in) U23(f919_out1) -> U24(f951_in) U24(f951_out1) -> f946_out1 f951_in -> U25(f919_in) U25(f919_out1) -> U26(f962_in) U26(f962_out1) -> f951_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (140) 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. ---------------------------------------- (141) Obligation: Q DP problem: The TRS P consists of the following rules: F386_IN -> F386_IN R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (142) 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 = F386_IN evaluates to t =F386_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 F386_IN to F386_IN. ---------------------------------------- (143) NO ---------------------------------------- (144) Obligation: Q DP problem: The TRS P consists of the following rules: F223_IN -> F223_IN The TRS R consists of the following rules: f3_in([]) -> f3_out1 f3_in(T11) -> U1(f63_in(T11), T11) U1(f63_out1(X18, T12, X19), T11) -> f3_out1 f386_in -> f386_out1(0) f386_in -> U2(f386_in) U2(f386_out1(T58)) -> f386_out1(s(T58)) f387_in(T77) -> U3(f694_in(T77), T77) U3(f694_out1, T77) -> f387_out1 f387_in(T116) -> U4(f387_in(T116), T116) U4(f387_out1, T116) -> f387_out1 f387_in(T125) -> f387_out1 f723_in(0) -> f723_out1 f723_in(s(T96)) -> U5(f723_in(T96), s(T96)) U5(f723_out1, s(T96)) -> f723_out1 f223_in -> U6(f382_in) U6(f382_out1(T38)) -> f223_out1 f223_in -> U7(f223_in) U7(f223_out1) -> f223_out1 f223_in -> f223_out1 f919_in -> f919_out1 f919_in -> U8(f942_in) U8(f942_out1) -> f919_out1 f962_in -> f962_out1 f962_in -> U9(f962_in) U9(f962_out1) -> f962_out1 f1000_in(.(T223, T224)) -> f1000_out1([], T223, T224) f1000_in(.(T235, T239)) -> U10(f1000_in(T239), .(T235, T239)) U10(f1000_out1(T240, T241, T242), .(T235, T239)) -> f1000_out1(.(T235, T240), T241, T242) f63_in(T11) -> U11(f223_in, T11) U11(f223_out1, T11) -> U12(f233_in(T11), T11) U12(f233_out1(X18, T19, X19), T11) -> f63_out1(X18, T19, X19) f233_in(T11) -> U13(f919_in, T11) U13(f919_out1, T11) -> U14(f920_in(T11), T11) U14(f920_out1(T149, T151, X19), T11) -> f233_out1(T149, T151, X19) f382_in -> U15(f386_in) U15(f386_out1(T43)) -> U16(f387_in(T43), T43) U16(f387_out1, T43) -> f382_out1(T43) f694_in(T77) -> U17(f723_in(T77), T77) U17(f723_out1, T77) -> U18(f387_in(T77), T77) U18(f387_out1, T77) -> f694_out1 f920_in(T11) -> U19(f919_in, T11) U19(f919_out1, T11) -> U20(f1000_in(T11), T11) U20(f1000_out1(T209, T210, T208), T11) -> f920_out1(T209, T210, T208) f942_in -> U21(f223_in) U21(f223_out1) -> U22(f946_in) U22(f946_out1) -> f942_out1 f946_in -> U23(f919_in) U23(f919_out1) -> U24(f951_in) U24(f951_out1) -> f946_out1 f951_in -> U25(f919_in) U25(f919_out1) -> U26(f962_in) U26(f962_out1) -> f951_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (145) 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. ---------------------------------------- (146) Obligation: Q DP problem: The TRS P consists of the following rules: F223_IN -> F223_IN R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (147) 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 = F223_IN evaluates to t =F223_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 F223_IN to F223_IN. ---------------------------------------- (148) NO ---------------------------------------- (149) Obligation: Q DP problem: The TRS P consists of the following rules: F942_IN -> U21^1(f223_in) U21^1(f223_out1) -> F946_IN F946_IN -> U23^1(f919_in) U23^1(f919_out1) -> F951_IN F951_IN -> F919_IN F919_IN -> F942_IN F946_IN -> F919_IN The TRS R consists of the following rules: f3_in([]) -> f3_out1 f3_in(T11) -> U1(f63_in(T11), T11) U1(f63_out1(X18, T12, X19), T11) -> f3_out1 f386_in -> f386_out1(0) f386_in -> U2(f386_in) U2(f386_out1(T58)) -> f386_out1(s(T58)) f387_in(T77) -> U3(f694_in(T77), T77) U3(f694_out1, T77) -> f387_out1 f387_in(T116) -> U4(f387_in(T116), T116) U4(f387_out1, T116) -> f387_out1 f387_in(T125) -> f387_out1 f723_in(0) -> f723_out1 f723_in(s(T96)) -> U5(f723_in(T96), s(T96)) U5(f723_out1, s(T96)) -> f723_out1 f223_in -> U6(f382_in) U6(f382_out1(T38)) -> f223_out1 f223_in -> U7(f223_in) U7(f223_out1) -> f223_out1 f223_in -> f223_out1 f919_in -> f919_out1 f919_in -> U8(f942_in) U8(f942_out1) -> f919_out1 f962_in -> f962_out1 f962_in -> U9(f962_in) U9(f962_out1) -> f962_out1 f1000_in(.(T223, T224)) -> f1000_out1([], T223, T224) f1000_in(.(T235, T239)) -> U10(f1000_in(T239), .(T235, T239)) U10(f1000_out1(T240, T241, T242), .(T235, T239)) -> f1000_out1(.(T235, T240), T241, T242) f63_in(T11) -> U11(f223_in, T11) U11(f223_out1, T11) -> U12(f233_in(T11), T11) U12(f233_out1(X18, T19, X19), T11) -> f63_out1(X18, T19, X19) f233_in(T11) -> U13(f919_in, T11) U13(f919_out1, T11) -> U14(f920_in(T11), T11) U14(f920_out1(T149, T151, X19), T11) -> f233_out1(T149, T151, X19) f382_in -> U15(f386_in) U15(f386_out1(T43)) -> U16(f387_in(T43), T43) U16(f387_out1, T43) -> f382_out1(T43) f694_in(T77) -> U17(f723_in(T77), T77) U17(f723_out1, T77) -> U18(f387_in(T77), T77) U18(f387_out1, T77) -> f694_out1 f920_in(T11) -> U19(f919_in, T11) U19(f919_out1, T11) -> U20(f1000_in(T11), T11) U20(f1000_out1(T209, T210, T208), T11) -> f920_out1(T209, T210, T208) f942_in -> U21(f223_in) U21(f223_out1) -> U22(f946_in) U22(f946_out1) -> f942_out1 f946_in -> U23(f919_in) U23(f919_out1) -> U24(f951_in) U24(f951_out1) -> f946_out1 f951_in -> U25(f919_in) U25(f919_out1) -> U26(f962_in) U26(f962_out1) -> f951_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (150) 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 F946_IN[ ]^n[ ] -> F946_IN[ ]^n[ ] This rule is correct for the QDP as the following derivation shows: F946_IN[ ]^n[ ] -> F946_IN[ ]^n[ ] by Narrowing at position: [] F946_IN[ ]^n[ ] -> F919_IN[ ]^n[ ] by Rule from TRS P F919_IN[ ]^n[ ] -> F946_IN[ ]^n[ ] by Narrowing at position: [] F919_IN[ ]^n[ ] -> F942_IN[ ]^n[ ] by Rule from TRS P F942_IN[ ]^n[ ] -> F946_IN[ ]^n[ ] by Narrowing at position: [] F942_IN[ ]^n[ ] -> U21^1(f223_out1)[ ]^n[ ] by Narrowing at position: [0] F942_IN[ ]^n[ ] -> U21^1(f223_in)[ ]^n[ ] by Rule from TRS P f223_in[ ]^n[ ] -> f223_out1[ ]^n[ ] by Rule from TRS R U21^1(f223_out1)[ ]^n[ ] -> F946_IN[ ]^n[ ] by Rule from TRS P ---------------------------------------- (151) NO ---------------------------------------- (152) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. 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T241 T242) T239)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T239"], "free": [], "exprvars": [] } }, "1108": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "24": { "goal": [ { "clause": 0, "scope": 1, "term": "(qs T1 T2)" }, { "clause": 1, "scope": 1, "term": "(qs T1 T2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "890": { "goal": [{ "clause": 7, "scope": 5, "term": "(less T77 T80)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T77"], "free": [], "exprvars": [] } }, "891": { "goal": [{ "clause": 8, "scope": 5, "term": "(less T77 T80)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T77"], "free": [], "exprvars": [] } }, "893": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "970": { "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": [] } }, "971": { "goal": [{ "clause": 0, "scope": 6, "term": "(qs T17 X18)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X18"], "exprvars": [] } }, "411": { "goal": [{ "clause": 7, "scope": 3, "term": "(less T38 T39)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "895": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "972": { "goal": [{ "clause": 1, "scope": 6, "term": "(qs T17 X18)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X18"], "exprvars": [] } }, "973": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "413": { "goal": [{ "clause": 8, "scope": 3, "term": "(less T38 T39)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "897": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "974": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "975": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "217": { "goal": [ { "clause": 2, "scope": 2, "term": "(part T12 T13 X16 X17)" }, { "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": [] } }, "976": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (part T158 T159 X257 X258) (',' (qs X257 X259) (',' (qs X258 X260) (app X259 (. T158 X260) X261))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X261", "X257", "X258", "X259", "X260" ], "exprvars": [] } }, "977": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "418": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "937": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "938": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "939": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1098": { "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": [] } }, "1053": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1097": { "goal": [{ "clause": -1, "scope": -1, "term": "(app T209 (. T210 T208) T11)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T11"], "free": [], "exprvars": [] } }, "1052": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1096": { "goal": [{ "clause": -1, "scope": -1, "term": "(qs T150 X19)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X19"], "exprvars": [] } }, "1051": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1093": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1092": { "goal": [{ "clause": -1, "scope": -1, "term": "(app T201 (. T202 T203) X297)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X297"], "exprvars": [] } }, "420": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "982": { "goal": [{ "clause": -1, "scope": -1, "term": "(part T158 T159 X257 X258)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X257", "X258" ], "exprvars": [] } }, "422": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "983": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (qs T163 X259) (',' (qs T164 X260) (app X259 (. T165 X260) X261)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X261", "X259", "X260" ], "exprvars": [] } }, "940": { "goal": [{ "clause": 3, "scope": 2, "term": "(part T12 T13 X16 X17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X16", "X17" ], "exprvars": [] } }, "941": { "goal": [{ "clause": 4, "scope": 2, "term": "(part T12 T13 X16 X17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X16", "X17" ], "exprvars": [] } }, "987": { "goal": [{ "clause": -1, "scope": -1, "term": "(qs T163 X259)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X259"], "exprvars": [] } }, "900": { "goal": [{ "clause": -1, "scope": -1, "term": "(less T96 T98)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T96"], "free": [], "exprvars": [] } }, "988": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (qs T170 X260) (app T169 (. T171 X260) X261))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X261", "X260" ], "exprvars": [] } }, "901": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 2, "to": 24, "label": "CASE" }, { "from": 24, "to": 51, "label": "PARALLEL" }, { "from": 24, "to": 52, "label": "PARALLEL" }, { "from": 51, "to": 53, "label": "EVAL with clause\nqs([], []).\nand substitutionT1 -> [],\nT2 -> []" }, { "from": 51, "to": 54, "label": "EVAL-BACKTRACK" }, { "from": 52, "to": 58, "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": 52, "to": 59, "label": "EVAL-BACKTRACK" }, { "from": 53, "to": 55, "label": "SUCCESS" }, { "from": 58, "to": 190, "label": "SPLIT 1" }, { "from": 58, "to": 192, "label": "SPLIT 2\nreplacements:X16 -> T17,\nX17 -> T18,\nT12 -> T19" }, { "from": 190, "to": 217, "label": "CASE" }, { "from": 192, "to": 966, "label": "SPLIT 1" }, { "from": 192, "to": 967, "label": "SPLIT 2\nreplacements:X18 -> T149,\nT18 -> T150,\nT19 -> T151" }, { "from": 217, "to": 230, "label": "PARALLEL" }, { "from": 217, "to": 237, "label": "PARALLEL" }, { "from": 230, "to": 368, "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": 230, "to": 400, "label": "EVAL-BACKTRACK" }, { "from": 237, "to": 940, "label": "PARALLEL" }, { "from": 237, "to": 941, "label": "PARALLEL" }, { "from": 368, "to": 404, "label": "SPLIT 1" }, { "from": 368, "to": 405, "label": "SPLIT 2\nnew knowledge:\nT43 is ground\nreplacements:T38 -> T43,\nT40 -> T44" }, { "from": 404, "to": 407, "label": "CASE" }, { "from": 405, "to": 470, "label": "CASE" }, { "from": 407, "to": 411, "label": "PARALLEL" }, { "from": 407, "to": 413, "label": "PARALLEL" }, { "from": 411, "to": 418, "label": "EVAL with clause\nless(0, s(X77)).\nand substitutionT38 -> 0,\nX77 -> T51,\nT39 -> s(T51)" }, { "from": 411, "to": 420, "label": "EVAL-BACKTRACK" }, { "from": 413, "to": 430, "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": 413, "to": 433, "label": "EVAL-BACKTRACK" }, { "from": 418, "to": 422, "label": "SUCCESS" }, { "from": 430, "to": 404, "label": "INSTANCE with matching:\nT38 -> T58\nT39 -> T59" }, { "from": 470, "to": 598, "label": "PARALLEL" }, { "from": 470, "to": 599, "label": "PARALLEL" }, { "from": 598, "to": 600, "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": 598, "to": 601, "label": "EVAL-BACKTRACK" }, { "from": 599, "to": 913, "label": "PARALLEL" }, { "from": 599, "to": 914, "label": "PARALLEL" }, { "from": 600, "to": 602, "label": "SPLIT 1" }, { "from": 600, "to": 603, "label": "SPLIT 2\nnew knowledge:\nT77 is ground\nreplacements:T81 -> T84" }, { "from": 602, "to": 606, "label": "CASE" }, { "from": 603, "to": 405, "label": "INSTANCE with matching:\nT43 -> T77\nT44 -> T84\nX67 -> X126\nX68 -> X127" }, { "from": 606, "to": 890, "label": "PARALLEL" }, { "from": 606, "to": 891, "label": "PARALLEL" }, { "from": 890, "to": 893, "label": "EVAL with clause\nless(0, s(X136)).\nand substitutionT77 -> 0,\nX136 -> T91,\nT80 -> s(T91)" }, { "from": 890, "to": 895, "label": "EVAL-BACKTRACK" }, { "from": 891, "to": 900, "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": 891, "to": 901, "label": "EVAL-BACKTRACK" }, { "from": 893, "to": 897, "label": "SUCCESS" }, { "from": 900, "to": 602, "label": "INSTANCE with matching:\nT77 -> T96\nT80 -> T98" }, { "from": 913, "to": 918, "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": 913, "to": 921, "label": "EVAL-BACKTRACK" }, { "from": 914, "to": 937, "label": "EVAL with clause\npart(X196, [], [], []).\nand substitutionT43 -> T125,\nX196 -> T125,\nT44 -> [],\nX67 -> [],\nX68 -> []" }, { "from": 914, "to": 938, "label": "EVAL-BACKTRACK" }, { "from": 918, "to": 405, "label": "INSTANCE with matching:\nT43 -> T116\nT44 -> T119\nX67 -> X185\nX68 -> X186" }, { "from": 937, "to": 939, "label": "SUCCESS" }, { "from": 940, "to": 958, "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": 940, "to": 959, "label": "EVAL-BACKTRACK" }, { "from": 941, "to": 960, "label": "EVAL with clause\npart(X241, [], [], []).\nand substitutionT12 -> T148,\nX241 -> T148,\nT13 -> [],\nX16 -> [],\nX17 -> []" }, { "from": 941, "to": 963, "label": "EVAL-BACKTRACK" }, { "from": 958, "to": 190, "label": "INSTANCE with matching:\nT12 -> T141\nT13 -> T142\nX16 -> X230\nX17 -> X231" }, { "from": 960, "to": 964, "label": "SUCCESS" }, { "from": 966, "to": 970, "label": "CASE" }, { "from": 967, "to": 1096, "label": "SPLIT 1" }, { "from": 967, "to": 1097, "label": "SPLIT 2\nreplacements:X19 -> T208,\nT149 -> T209,\nT151 -> T210" }, { "from": 970, "to": 971, "label": "PARALLEL" }, { "from": 970, "to": 972, "label": "PARALLEL" }, { "from": 971, "to": 973, "label": "EVAL with clause\nqs([], []).\nand substitutionT17 -> [],\nX18 -> []" }, { "from": 971, "to": 974, "label": "EVAL-BACKTRACK" }, { "from": 972, "to": 976, "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": 972, "to": 977, "label": "EVAL-BACKTRACK" }, { "from": 973, "to": 975, "label": "SUCCESS" }, { "from": 976, "to": 982, "label": "SPLIT 1" }, { "from": 976, "to": 983, "label": "SPLIT 2\nreplacements:X257 -> T163,\nX258 -> T164,\nT158 -> T165" }, { "from": 982, "to": 190, "label": "INSTANCE with matching:\nT12 -> T158\nT13 -> T159\nX16 -> X257\nX17 -> X258" }, { "from": 983, "to": 987, "label": "SPLIT 1" }, { "from": 983, "to": 988, "label": "SPLIT 2\nreplacements:X259 -> T169,\nT164 -> T170,\nT165 -> T171" }, { "from": 987, "to": 966, "label": "INSTANCE with matching:\nT17 -> T163\nX18 -> X259" }, { "from": 988, "to": 992, "label": "SPLIT 1" }, { "from": 988, "to": 993, "label": "SPLIT 2\nreplacements:X260 -> T172,\nT169 -> T173,\nT171 -> T174" }, { "from": 992, "to": 966, "label": "INSTANCE with matching:\nT17 -> T170\nX18 -> X260" }, { "from": 993, "to": 996, "label": "CASE" }, { "from": 996, "to": 997, "label": "PARALLEL" }, { "from": 996, "to": 998, "label": "PARALLEL" }, { "from": 997, "to": 1051, "label": "EVAL with clause\napp([], X282, X282).\nand substitutionT173 -> [],\nT174 -> T187,\nT172 -> T188,\nX282 -> .(T187, T188),\nX261 -> .(T187, T188)" }, { "from": 997, "to": 1052, "label": "EVAL-BACKTRACK" }, { "from": 998, "to": 1092, "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": 998, "to": 1093, "label": "EVAL-BACKTRACK" }, { "from": 1051, "to": 1053, "label": "SUCCESS" }, { "from": 1092, "to": 993, "label": "INSTANCE with matching:\nT173 -> T201\nT174 -> T202\nT172 -> T203\nX261 -> X297" }, { "from": 1096, "to": 966, "label": "INSTANCE with matching:\nT17 -> T150\nX18 -> X19" }, { "from": 1097, "to": 1098, "label": "CASE" }, { "from": 1098, "to": 1104, "label": "PARALLEL" }, { "from": 1098, "to": 1105, "label": "PARALLEL" }, { "from": 1104, "to": 1106, "label": "EVAL with clause\napp([], X306, X306).\nand substitutionT209 -> [],\nT210 -> T223,\nT208 -> T224,\nX306 -> .(T223, T224),\nT11 -> .(T223, T224)" }, { "from": 1104, "to": 1107, "label": "EVAL-BACKTRACK" }, { "from": 1105, "to": 1109, "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": 1105, "to": 1110, "label": "EVAL-BACKTRACK" }, { "from": 1106, "to": 1108, "label": "SUCCESS" }, { "from": 1109, "to": 1097, "label": "INSTANCE with matching:\nT209 -> T240\nT210 -> T241\nT208 -> T242\nT11 -> T239" } ], "type": "Graph" } } ---------------------------------------- (153) Complex Obligation (AND) ---------------------------------------- (154) Obligation: Rules: f1097_in(T11) -> f1098_in(T11) :|: TRUE f1098_out(x) -> f1097_out(x) :|: TRUE f1104_out(x1) -> f1098_out(x1) :|: TRUE f1098_in(x2) -> f1105_in(x2) :|: TRUE f1098_in(x3) -> f1104_in(x3) :|: TRUE f1105_out(x4) -> f1098_out(x4) :|: TRUE f1105_in(.(T235, T239)) -> f1109_in(T239) :|: TRUE f1105_in(x5) -> f1110_in :|: TRUE f1109_out(x6) -> f1105_out(.(x7, x6)) :|: TRUE f1110_out -> f1105_out(x8) :|: TRUE f1109_in(x9) -> f1097_in(x9) :|: TRUE f1097_out(x10) -> f1109_out(x10) :|: TRUE f24_out(T2) -> f2_out(T2) :|: TRUE f2_in(x11) -> f24_in(x11) :|: TRUE f24_in(x12) -> f52_in(x12) :|: TRUE f52_out(x13) -> f24_out(x13) :|: TRUE f51_out(x14) -> f24_out(x14) :|: TRUE f24_in(x15) -> f51_in(x15) :|: TRUE f52_in(x16) -> f59_in :|: TRUE f59_out -> f52_out(x17) :|: TRUE f58_out(x18) -> f52_out(x18) :|: TRUE f52_in(x19) -> f58_in(x19) :|: TRUE f190_out -> f192_in(x20) :|: TRUE f192_out(x21) -> f58_out(x21) :|: TRUE f58_in(x22) -> f190_in :|: TRUE f967_out(x23) -> f192_out(x23) :|: TRUE f966_out -> f967_in(x24) :|: TRUE f192_in(x25) -> f966_in :|: TRUE f1096_out -> f1097_in(x26) :|: TRUE f1097_out(x27) -> f967_out(x27) :|: TRUE f967_in(x28) -> f1096_in :|: TRUE Start term: f2_in(T2) ---------------------------------------- (155) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (156) TRUE ---------------------------------------- (157) Obligation: Rules: f993_in -> f996_in :|: TRUE f996_out -> f993_out :|: TRUE f1092_in -> f993_in :|: TRUE f993_out -> f1092_out :|: TRUE f997_out -> f996_out :|: TRUE f996_in -> f997_in :|: TRUE f998_out -> f996_out :|: TRUE f996_in -> f998_in :|: TRUE f1093_out -> f998_out :|: TRUE f1092_out -> f998_out :|: TRUE f998_in -> f1093_in :|: TRUE f998_in -> f1092_in :|: TRUE f24_out(T2) -> f2_out(T2) :|: TRUE f2_in(x) -> f24_in(x) :|: TRUE f24_in(x1) -> f52_in(x1) :|: TRUE f52_out(x2) -> f24_out(x2) :|: TRUE f51_out(x3) -> f24_out(x3) :|: TRUE f24_in(x4) -> f51_in(x4) :|: TRUE f52_in(x5) -> f59_in :|: TRUE f59_out -> f52_out(x6) :|: TRUE f58_out(T11) -> f52_out(T11) :|: TRUE f52_in(x7) -> f58_in(x7) :|: TRUE f190_out -> f192_in(x8) :|: TRUE f192_out(x9) -> f58_out(x9) :|: TRUE f58_in(x10) -> f190_in :|: TRUE f967_out(x11) -> f192_out(x11) :|: TRUE f966_out -> f967_in(x12) :|: TRUE f192_in(x13) -> f966_in :|: TRUE f1096_out -> f1097_in(x14) :|: TRUE f1097_out(x15) -> f967_out(x15) :|: TRUE f967_in(x16) -> f1096_in :|: TRUE f1096_in -> f966_in :|: TRUE f966_out -> f1096_out :|: TRUE f966_in -> f970_in :|: TRUE f970_out -> f966_out :|: TRUE f970_in -> f971_in :|: TRUE f972_out -> f970_out :|: TRUE f970_in -> f972_in :|: TRUE f971_out -> f970_out :|: TRUE f972_in -> f977_in :|: TRUE f976_out -> f972_out :|: TRUE f977_out -> f972_out :|: TRUE f972_in -> f976_in :|: TRUE f983_out -> f976_out :|: TRUE f982_out -> f983_in :|: TRUE f976_in -> f982_in :|: TRUE f983_in -> f987_in :|: TRUE f987_out -> f988_in :|: TRUE f988_out -> f983_out :|: TRUE f988_in -> f992_in :|: TRUE f992_out -> f993_in :|: TRUE f993_out -> f988_out :|: TRUE Start term: f2_in(T2) ---------------------------------------- (158) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (159) TRUE ---------------------------------------- (160) Obligation: Rules: f891_in(T77) -> f901_in :|: TRUE f891_in(s(T96)) -> f900_in(T96) :|: TRUE f901_out -> f891_out(x) :|: TRUE f900_out(x1) -> f891_out(s(x1)) :|: TRUE f890_out(x2) -> f606_out(x2) :|: TRUE f606_in(x3) -> f890_in(x3) :|: TRUE f606_in(x4) -> f891_in(x4) :|: TRUE f891_out(x5) -> f606_out(x5) :|: TRUE f602_out(x6) -> f900_out(x6) :|: TRUE f900_in(x7) -> f602_in(x7) :|: TRUE f602_in(x8) -> f606_in(x8) :|: TRUE f606_out(x9) -> f602_out(x9) :|: TRUE f24_out(T2) -> f2_out(T2) :|: TRUE f2_in(x10) -> f24_in(x10) :|: TRUE f24_in(x11) -> f52_in(x11) :|: TRUE f52_out(x12) -> f24_out(x12) :|: TRUE f51_out(x13) -> f24_out(x13) :|: TRUE f24_in(x14) -> f51_in(x14) :|: TRUE f52_in(x15) -> f59_in :|: TRUE f59_out -> f52_out(x16) :|: TRUE f58_out(T11) -> f52_out(T11) :|: TRUE f52_in(x17) -> f58_in(x17) :|: TRUE f190_out -> f192_in(x18) :|: TRUE f192_out(x19) -> f58_out(x19) :|: TRUE f58_in(x20) -> f190_in :|: TRUE f967_out(x21) -> f192_out(x21) :|: TRUE f966_out -> f967_in(x22) :|: TRUE f192_in(x23) -> f966_in :|: TRUE f966_in -> f970_in :|: TRUE f970_out -> f966_out :|: TRUE f970_in -> f971_in :|: TRUE f972_out -> f970_out :|: TRUE f970_in -> f972_in :|: TRUE f971_out -> f970_out :|: TRUE f972_in -> f977_in :|: TRUE f976_out -> f972_out :|: TRUE f977_out -> f972_out :|: TRUE f972_in -> f976_in :|: TRUE f983_out -> f976_out :|: TRUE f982_out -> f983_in :|: TRUE f976_in -> f982_in :|: TRUE f982_in -> f190_in :|: TRUE f190_out -> f982_out :|: TRUE f190_in -> f217_in :|: TRUE f217_out -> f190_out :|: TRUE f217_in -> f230_in :|: TRUE f217_in -> f237_in :|: TRUE f237_out -> f217_out :|: TRUE f230_out -> f217_out :|: TRUE f230_in -> f400_in :|: TRUE f368_out -> f230_out :|: TRUE f230_in -> f368_in :|: TRUE f400_out -> f230_out :|: TRUE f368_in -> f404_in :|: TRUE f404_out -> f405_in(T43) :|: TRUE f405_out(x24) -> f368_out :|: TRUE f470_out(x25) -> f405_out(x25) :|: TRUE f405_in(x26) -> f470_in(x26) :|: TRUE f598_out(x27) -> f470_out(x27) :|: TRUE f470_in(x28) -> f598_in(x28) :|: TRUE f470_in(x29) -> f599_in(x29) :|: TRUE f599_out(x30) -> f470_out(x30) :|: TRUE f598_in(x31) -> f601_in :|: TRUE f598_in(x32) -> f600_in(x32) :|: TRUE f601_out -> f598_out(x33) :|: TRUE f600_out(x34) -> f598_out(x34) :|: TRUE f603_out(x35) -> f600_out(x35) :|: TRUE f600_in(x36) -> f602_in(x36) :|: TRUE f602_out(x37) -> f603_in(x37) :|: TRUE f1096_out -> f1097_in(x38) :|: TRUE f1097_out(x39) -> f967_out(x39) :|: TRUE f967_in(x40) -> f1096_in :|: TRUE f1096_in -> f966_in :|: TRUE f966_out -> f1096_out :|: TRUE Start term: f2_in(T2) ---------------------------------------- (161) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (162) TRUE ---------------------------------------- (163) Obligation: Rules: f603_out(T77) -> f600_out(T77) :|: TRUE f600_in(x) -> f602_in(x) :|: TRUE f602_out(x1) -> f603_in(x1) :|: TRUE f602_out(T96) -> f900_out(T96) :|: TRUE f900_in(x2) -> f602_in(x2) :|: TRUE f603_in(x3) -> f405_in(x3) :|: TRUE f405_out(x4) -> f603_out(x4) :|: TRUE f893_out -> f890_out(0) :|: TRUE f895_out -> f890_out(x5) :|: TRUE f890_in(x6) -> f895_in :|: TRUE f890_in(0) -> f893_in :|: TRUE f598_out(T43) -> f470_out(T43) :|: TRUE f470_in(x7) -> f598_in(x7) :|: TRUE f470_in(x8) -> f599_in(x8) :|: TRUE f599_out(x9) -> f470_out(x9) :|: TRUE f470_out(x10) -> f405_out(x10) :|: TRUE f405_in(x11) -> f470_in(x11) :|: TRUE f891_in(x12) -> f901_in :|: TRUE f891_in(s(x13)) -> f900_in(x13) :|: TRUE f901_out -> f891_out(x14) :|: TRUE f900_out(x15) -> f891_out(s(x15)) :|: TRUE f921_out -> f913_out(x16) :|: TRUE f918_out(T116) -> f913_out(T116) :|: TRUE f913_in(x17) -> f918_in(x17) :|: TRUE f913_in(x18) -> f921_in :|: TRUE f598_in(x19) -> f601_in :|: TRUE f598_in(x20) -> f600_in(x20) :|: TRUE f601_out -> f598_out(x21) :|: TRUE f600_out(x22) -> f598_out(x22) :|: TRUE f599_in(x23) -> f914_in(x23) :|: TRUE f914_out(x24) -> f599_out(x24) :|: TRUE f913_out(x25) -> f599_out(x25) :|: TRUE f599_in(x26) -> f913_in(x26) :|: TRUE f890_out(x27) -> f606_out(x27) :|: TRUE f606_in(x28) -> f890_in(x28) :|: TRUE f606_in(x29) -> f891_in(x29) :|: TRUE f891_out(x30) -> f606_out(x30) :|: TRUE f602_in(x31) -> f606_in(x31) :|: TRUE f606_out(x32) -> f602_out(x32) :|: TRUE f893_in -> f893_out :|: TRUE f918_in(x33) -> f405_in(x33) :|: TRUE f405_out(x34) -> f918_out(x34) :|: TRUE f24_out(T2) -> f2_out(T2) :|: TRUE f2_in(x35) -> f24_in(x35) :|: TRUE f24_in(x36) -> f52_in(x36) :|: TRUE f52_out(x37) -> f24_out(x37) :|: TRUE f51_out(x38) -> f24_out(x38) :|: TRUE f24_in(x39) -> f51_in(x39) :|: TRUE f52_in(x40) -> f59_in :|: TRUE f59_out -> f52_out(x41) :|: TRUE f58_out(T11) -> f52_out(T11) :|: TRUE f52_in(x42) -> f58_in(x42) :|: TRUE f190_out -> f192_in(x43) :|: TRUE f192_out(x44) -> f58_out(x44) :|: TRUE f58_in(x45) -> f190_in :|: TRUE f190_in -> f217_in :|: TRUE f217_out -> f190_out :|: TRUE f217_in -> f230_in :|: TRUE f217_in -> f237_in :|: TRUE f237_out -> f217_out :|: TRUE f230_out -> f217_out :|: TRUE f230_in -> f400_in :|: TRUE f368_out -> f230_out :|: TRUE f230_in -> f368_in :|: TRUE f400_out -> f230_out :|: TRUE f368_in -> f404_in :|: TRUE f404_out -> f405_in(x46) :|: TRUE f405_out(x47) -> f368_out :|: TRUE f967_out(x48) -> f192_out(x48) :|: TRUE f966_out -> f967_in(x49) :|: TRUE f192_in(x50) -> f966_in :|: TRUE f1096_out -> f1097_in(x51) :|: TRUE f1097_out(x52) -> f967_out(x52) :|: TRUE f967_in(x53) -> f1096_in :|: TRUE f1096_in -> f966_in :|: TRUE f966_out -> f1096_out :|: TRUE f966_in -> f970_in :|: TRUE f970_out -> f966_out :|: TRUE f970_in -> f971_in :|: TRUE f972_out -> f970_out :|: TRUE f970_in -> f972_in :|: TRUE f971_out -> f970_out :|: TRUE f972_in -> f977_in :|: TRUE f976_out -> f972_out :|: TRUE f977_out -> f972_out :|: TRUE f972_in -> f976_in :|: TRUE f983_out -> f976_out :|: TRUE f982_out -> f983_in :|: TRUE f976_in -> f982_in :|: TRUE f982_in -> f190_in :|: TRUE f190_out -> f982_out :|: TRUE Start term: f2_in(T2) ---------------------------------------- (164) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (165) TRUE ---------------------------------------- (166) Obligation: Rules: f407_out -> f404_out :|: TRUE f404_in -> f407_in :|: TRUE f413_in -> f433_in :|: TRUE f413_in -> f430_in :|: TRUE f430_out -> f413_out :|: TRUE f433_out -> f413_out :|: TRUE f407_in -> f411_in :|: TRUE f413_out -> f407_out :|: TRUE f407_in -> f413_in :|: TRUE f411_out -> f407_out :|: TRUE f430_in -> f404_in :|: TRUE f404_out -> f430_out :|: TRUE f24_out(T2) -> f2_out(T2) :|: TRUE f2_in(x) -> f24_in(x) :|: TRUE f24_in(x1) -> f52_in(x1) :|: TRUE f52_out(x2) -> f24_out(x2) :|: TRUE f51_out(x3) -> f24_out(x3) :|: TRUE f24_in(x4) -> f51_in(x4) :|: TRUE f52_in(x5) -> f59_in :|: TRUE f59_out -> f52_out(x6) :|: TRUE f58_out(T11) -> f52_out(T11) :|: TRUE f52_in(x7) -> f58_in(x7) :|: TRUE f190_out -> f192_in(x8) :|: TRUE f192_out(x9) -> f58_out(x9) :|: TRUE f58_in(x10) -> f190_in :|: TRUE f190_in -> f217_in :|: TRUE f217_out -> f190_out :|: TRUE f217_in -> f230_in :|: TRUE f217_in -> f237_in :|: TRUE f237_out -> f217_out :|: TRUE f230_out -> f217_out :|: TRUE f230_in -> f400_in :|: TRUE f368_out -> f230_out :|: TRUE f230_in -> f368_in :|: TRUE f400_out -> f230_out :|: TRUE f368_in -> f404_in :|: TRUE f404_out -> f405_in(T43) :|: TRUE f405_out(x11) -> f368_out :|: TRUE f967_out(x12) -> f192_out(x12) :|: TRUE f966_out -> f967_in(x13) :|: TRUE f192_in(x14) -> f966_in :|: TRUE f1096_out -> f1097_in(x15) :|: TRUE f1097_out(x16) -> f967_out(x16) :|: TRUE f967_in(x17) -> f1096_in :|: TRUE f1096_in -> f966_in :|: TRUE f966_out -> f1096_out :|: TRUE f966_in -> f970_in :|: TRUE f970_out -> f966_out :|: TRUE f970_in -> f971_in :|: TRUE f972_out -> f970_out :|: TRUE f970_in -> f972_in :|: TRUE f971_out -> f970_out :|: TRUE f972_in -> f977_in :|: TRUE f976_out -> f972_out :|: TRUE f977_out -> f972_out :|: TRUE f972_in -> f976_in :|: TRUE f983_out -> f976_out :|: TRUE f982_out -> f983_in :|: TRUE f976_in -> f982_in :|: TRUE f982_in -> f190_in :|: TRUE f190_out -> f982_out :|: TRUE Start term: f2_in(T2) ---------------------------------------- (167) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f404_in -> f407_in :|: TRUE f413_in -> f430_in :|: TRUE f407_in -> f413_in :|: TRUE f430_in -> f404_in :|: TRUE ---------------------------------------- (168) Obligation: Rules: f404_in -> f407_in :|: TRUE f413_in -> f430_in :|: TRUE f407_in -> f413_in :|: TRUE f430_in -> f404_in :|: TRUE ---------------------------------------- (169) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (170) Obligation: Rules: f404_in -> f404_in :|: TRUE ---------------------------------------- (171) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (172) Obligation: Rules: f404_in -> f404_in :|: TRUE ---------------------------------------- (173) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f404_in -> f404_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (174) Obligation: Termination digraph: Nodes: (1) f404_in -> f404_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (175) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f404_in() Replaced non-predefined constructor symbols by 0. ---------------------------------------- (176) Obligation: Rules: f404_in -> f404_in :|: TRUE ---------------------------------------- (177) IntTRSPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc) -> f(1) :|: pc = 1 && TRUE Witness term starting non-terminating reduction: f(1) ---------------------------------------- (178) NO ---------------------------------------- (179) Obligation: Rules: f940_in -> f959_in :|: TRUE f940_in -> f958_in :|: TRUE f958_out -> f940_out :|: TRUE f959_out -> f940_out :|: TRUE f958_in -> f190_in :|: TRUE f190_out -> f958_out :|: TRUE f190_in -> f217_in :|: TRUE f217_out -> f190_out :|: TRUE f217_in -> f230_in :|: TRUE f217_in -> f237_in :|: TRUE f237_out -> f217_out :|: TRUE f230_out -> f217_out :|: TRUE f941_out -> f237_out :|: TRUE f940_out -> f237_out :|: TRUE f237_in -> f940_in :|: TRUE f237_in -> f941_in :|: TRUE f24_out(T2) -> f2_out(T2) :|: TRUE f2_in(x) -> f24_in(x) :|: TRUE f24_in(x1) -> f52_in(x1) :|: TRUE f52_out(x2) -> f24_out(x2) :|: TRUE f51_out(x3) -> f24_out(x3) :|: TRUE f24_in(x4) -> f51_in(x4) :|: TRUE f52_in(x5) -> f59_in :|: TRUE f59_out -> f52_out(x6) :|: TRUE f58_out(T11) -> f52_out(T11) :|: TRUE f52_in(x7) -> f58_in(x7) :|: TRUE f190_out -> f192_in(x8) :|: TRUE f192_out(x9) -> f58_out(x9) :|: TRUE f58_in(x10) -> f190_in :|: TRUE f967_out(x11) -> f192_out(x11) :|: TRUE f966_out -> f967_in(x12) :|: TRUE f192_in(x13) -> f966_in :|: TRUE f966_in -> f970_in :|: TRUE f970_out -> f966_out :|: TRUE f970_in -> f971_in :|: TRUE f972_out -> f970_out :|: TRUE f970_in -> f972_in :|: TRUE f971_out -> f970_out :|: TRUE f972_in -> f977_in :|: TRUE f976_out -> f972_out :|: TRUE f977_out -> f972_out :|: TRUE f972_in -> f976_in :|: TRUE f983_out -> f976_out :|: TRUE f982_out -> f983_in :|: TRUE f976_in -> f982_in :|: TRUE f982_in -> f190_in :|: TRUE f190_out -> f982_out :|: TRUE f1096_out -> f1097_in(x14) :|: TRUE f1097_out(x15) -> f967_out(x15) :|: TRUE f967_in(x16) -> f1096_in :|: TRUE f1096_in -> f966_in :|: TRUE f966_out -> f1096_out :|: TRUE Start term: f2_in(T2) ---------------------------------------- (180) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f940_in -> f958_in :|: TRUE f958_in -> f190_in :|: TRUE f190_in -> f217_in :|: TRUE f217_in -> f237_in :|: TRUE f237_in -> f940_in :|: TRUE ---------------------------------------- (181) Obligation: Rules: f940_in -> f958_in :|: TRUE f958_in -> f190_in :|: TRUE f190_in -> f217_in :|: TRUE f217_in -> f237_in :|: TRUE f237_in -> f940_in :|: TRUE ---------------------------------------- (182) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (183) Obligation: Rules: f940_in -> f940_in :|: TRUE ---------------------------------------- (184) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (185) Obligation: Rules: f940_in -> f940_in :|: TRUE ---------------------------------------- (186) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f940_in -> f940_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (187) Obligation: Termination digraph: Nodes: (1) f940_in -> f940_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (188) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f940_in() Replaced non-predefined constructor symbols by 0. ---------------------------------------- (189) Obligation: Rules: f940_in -> f940_in :|: TRUE ---------------------------------------- (190) 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)) ---------------------------------------- (191) NO ---------------------------------------- (192) Obligation: Rules: f1092_in -> f993_in :|: TRUE f993_out -> f1092_out :|: TRUE f1051_out -> f997_out :|: TRUE f1052_out -> f997_out :|: TRUE f997_in -> f1052_in :|: TRUE f997_in -> f1051_in :|: TRUE f602_out(T96) -> f900_out(T96) :|: TRUE f900_in(x) -> f602_in(x) :|: TRUE f430_in -> f404_in :|: TRUE f404_out -> f430_out :|: TRUE f603_in(T77) -> f405_in(T77) :|: TRUE f405_out(x1) -> f603_out(x1) :|: TRUE f1093_out -> f998_out :|: TRUE f1092_out -> f998_out :|: TRUE f998_in -> f1093_in :|: TRUE f998_in -> f1092_in :|: TRUE f993_in -> f996_in :|: TRUE f996_out -> f993_out :|: TRUE f983_in -> f987_in :|: TRUE f987_out -> f988_in :|: TRUE f988_out -> f983_out :|: TRUE f997_out -> f996_out :|: TRUE f996_in -> f997_in :|: TRUE f998_out -> f996_out :|: TRUE f996_in -> f998_in :|: TRUE f921_out -> f913_out(T43) :|: TRUE f918_out(T116) -> f913_out(T116) :|: TRUE f913_in(x2) -> f918_in(x2) :|: TRUE f913_in(x3) -> f921_in :|: TRUE f890_out(x4) -> f606_out(x4) :|: TRUE f606_in(x5) -> f890_in(x5) :|: TRUE f606_in(x6) -> f891_in(x6) :|: TRUE f891_out(x7) -> f606_out(x7) :|: TRUE f972_in -> f977_in :|: TRUE f976_out -> f972_out :|: TRUE f977_out -> f972_out :|: TRUE f972_in -> f976_in :|: TRUE f603_out(x8) -> f600_out(x8) :|: TRUE f600_in(x9) -> f602_in(x9) :|: TRUE f602_out(x10) -> f603_in(x10) :|: TRUE f982_in -> f190_in :|: TRUE f190_out -> f982_out :|: TRUE f937_in -> f937_out :|: TRUE f470_out(x11) -> f405_out(x11) :|: TRUE f405_in(x12) -> f470_in(x12) :|: TRUE f368_in -> f404_in :|: TRUE f404_out -> f405_in(x13) :|: TRUE f405_out(x14) -> f368_out :|: TRUE f958_in -> f190_in :|: TRUE f190_out -> f958_out :|: TRUE f190_in -> f217_in :|: TRUE f217_out -> f190_out :|: TRUE f960_in -> f960_out :|: TRUE f941_out -> f237_out :|: TRUE f940_out -> f237_out :|: TRUE f237_in -> f940_in :|: TRUE f237_in -> f941_in :|: TRUE f893_in -> f893_out :|: TRUE f918_in(x15) -> f405_in(x15) :|: TRUE f405_out(x16) -> f918_out(x16) :|: TRUE f941_in -> f963_in :|: TRUE f963_out -> f941_out :|: TRUE f960_out -> f941_out :|: TRUE f941_in -> f960_in :|: TRUE f413_in -> f433_in :|: TRUE f413_in -> f430_in :|: TRUE f430_out -> f413_out :|: TRUE f433_out -> f413_out :|: TRUE f983_out -> f976_out :|: TRUE f982_out -> f983_in :|: TRUE f976_in -> f982_in :|: TRUE f966_out -> f992_out :|: TRUE f992_in -> f966_in :|: TRUE f893_out -> f890_out(0) :|: TRUE f895_out -> f890_out(x17) :|: TRUE f890_in(x18) -> f895_in :|: TRUE f890_in(0) -> f893_in :|: TRUE f411_in -> f418_in :|: TRUE f418_out -> f411_out :|: TRUE f420_out -> f411_out :|: TRUE f411_in -> f420_in :|: TRUE f598_in(x19) -> f601_in :|: TRUE f598_in(x20) -> f600_in(x20) :|: TRUE f601_out -> f598_out(x21) :|: TRUE f600_out(x22) -> f598_out(x22) :|: TRUE f970_in -> f971_in :|: TRUE f972_out -> f970_out :|: TRUE f970_in -> f972_in :|: TRUE f971_out -> f970_out :|: TRUE f988_in -> f992_in :|: TRUE f992_out -> f993_in :|: TRUE f993_out -> f988_out :|: TRUE f599_in(x23) -> f914_in(x23) :|: TRUE f914_out(x24) -> f599_out(x24) :|: TRUE f913_out(x25) -> f599_out(x25) :|: TRUE f599_in(x26) -> f913_in(x26) :|: TRUE f602_in(x27) -> f606_in(x27) :|: TRUE f606_out(x28) -> f602_out(x28) :|: TRUE f966_in -> f970_in :|: TRUE f970_out -> f966_out :|: TRUE f1051_in -> f1051_out :|: TRUE f407_out -> f404_out :|: TRUE f404_in -> f407_in :|: TRUE f407_in -> f411_in :|: TRUE f413_out -> f407_out :|: TRUE f407_in -> f413_in :|: TRUE f411_out -> f407_out :|: TRUE f217_in -> f230_in :|: TRUE f217_in -> f237_in :|: TRUE f237_out -> f217_out :|: TRUE f230_out -> f217_out :|: TRUE f940_in -> f959_in :|: TRUE f940_in -> f958_in :|: TRUE f958_out -> f940_out :|: TRUE f959_out -> f940_out :|: TRUE f598_out(x29) -> f470_out(x29) :|: TRUE f470_in(x30) -> f598_in(x30) :|: TRUE f470_in(x31) -> f599_in(x31) :|: TRUE f599_out(x32) -> f470_out(x32) :|: TRUE f891_in(x33) -> f901_in :|: TRUE f891_in(s(x34)) -> f900_in(x34) :|: TRUE f901_out -> f891_out(x35) :|: TRUE f900_out(x36) -> f891_out(s(x36)) :|: TRUE f230_in -> f400_in :|: TRUE f368_out -> f230_out :|: TRUE f230_in -> f368_in :|: TRUE f400_out -> f230_out :|: TRUE f987_in -> f966_in :|: TRUE f966_out -> f987_out :|: TRUE f914_in(T125) -> f937_in :|: TRUE f914_in(x37) -> f938_in :|: TRUE f938_out -> f914_out(x38) :|: TRUE f937_out -> f914_out(x39) :|: TRUE f418_in -> f418_out :|: TRUE f24_out(T2) -> f2_out(T2) :|: TRUE f2_in(x40) -> f24_in(x40) :|: TRUE f24_in(x41) -> f52_in(x41) :|: TRUE f52_out(x42) -> f24_out(x42) :|: TRUE f51_out(x43) -> f24_out(x43) :|: TRUE f24_in(x44) -> f51_in(x44) :|: TRUE f52_in(x45) -> f59_in :|: TRUE f59_out -> f52_out(x46) :|: TRUE f58_out(T11) -> f52_out(T11) :|: TRUE f52_in(x47) -> f58_in(x47) :|: TRUE f190_out -> f192_in(x48) :|: TRUE f192_out(x49) -> f58_out(x49) :|: TRUE f58_in(x50) -> f190_in :|: TRUE f967_out(x51) -> f192_out(x51) :|: TRUE f966_out -> f967_in(x52) :|: TRUE f192_in(x53) -> f966_in :|: TRUE f1096_out -> f1097_in(x54) :|: TRUE f1097_out(x55) -> f967_out(x55) :|: TRUE f967_in(x56) -> f1096_in :|: TRUE f1096_in -> f966_in :|: TRUE f966_out -> f1096_out :|: TRUE Start term: f2_in(T2) ---------------------------------------- (193) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f602_out(T96) -> f900_out(T96) :|: TRUE f900_in(x) -> f602_in(x) :|: TRUE f430_in -> f404_in :|: TRUE f404_out -> f430_out :|: TRUE f603_in(T77) -> f405_in(T77) :|: TRUE f405_out(x1) -> f603_out(x1) :|: TRUE f983_in -> f987_in :|: TRUE f918_out(T116) -> f913_out(T116) :|: TRUE f913_in(x2) -> f918_in(x2) :|: TRUE f890_out(x4) -> f606_out(x4) :|: TRUE f606_in(x5) -> f890_in(x5) :|: TRUE f606_in(x6) -> f891_in(x6) :|: TRUE f891_out(x7) -> f606_out(x7) :|: TRUE f972_in -> f976_in :|: TRUE f603_out(x8) -> f600_out(x8) :|: TRUE f600_in(x9) -> f602_in(x9) :|: TRUE f602_out(x10) -> f603_in(x10) :|: TRUE f982_in -> f190_in :|: TRUE f190_out -> f982_out :|: TRUE f937_in -> f937_out :|: TRUE f470_out(x11) -> f405_out(x11) :|: TRUE f405_in(x12) -> f470_in(x12) :|: TRUE f368_in -> f404_in :|: TRUE f404_out -> f405_in(x13) :|: TRUE f405_out(x14) -> f368_out :|: TRUE f958_in -> f190_in :|: TRUE f190_out -> f958_out :|: TRUE f190_in -> f217_in :|: TRUE f217_out -> f190_out :|: TRUE f960_in -> f960_out :|: TRUE f941_out -> f237_out :|: TRUE f940_out -> f237_out :|: TRUE f237_in -> f940_in :|: TRUE f237_in -> f941_in :|: TRUE f893_in -> f893_out :|: TRUE f918_in(x15) -> f405_in(x15) :|: TRUE f405_out(x16) -> f918_out(x16) :|: TRUE f960_out -> f941_out :|: TRUE f941_in -> f960_in :|: TRUE f413_in -> f430_in :|: TRUE f430_out -> f413_out :|: TRUE f982_out -> f983_in :|: TRUE f976_in -> f982_in :|: TRUE f893_out -> f890_out(0) :|: TRUE f890_in(0) -> f893_in :|: TRUE f411_in -> f418_in :|: TRUE f418_out -> f411_out :|: TRUE f598_in(x20) -> f600_in(x20) :|: TRUE f600_out(x22) -> f598_out(x22) :|: TRUE f970_in -> f972_in :|: TRUE f599_in(x23) -> f914_in(x23) :|: TRUE f914_out(x24) -> f599_out(x24) :|: TRUE f913_out(x25) -> f599_out(x25) :|: TRUE f599_in(x26) -> f913_in(x26) :|: TRUE f602_in(x27) -> f606_in(x27) :|: TRUE f606_out(x28) -> f602_out(x28) :|: TRUE f966_in -> f970_in :|: TRUE f407_out -> f404_out :|: TRUE f404_in -> f407_in :|: TRUE f407_in -> f411_in :|: TRUE f413_out -> f407_out :|: TRUE f407_in -> f413_in :|: TRUE f411_out -> f407_out :|: TRUE f217_in -> f230_in :|: TRUE f217_in -> f237_in :|: TRUE f237_out -> f217_out :|: TRUE f230_out -> f217_out :|: TRUE f940_in -> f958_in :|: TRUE f958_out -> f940_out :|: TRUE f598_out(x29) -> f470_out(x29) :|: TRUE f470_in(x30) -> f598_in(x30) :|: TRUE f470_in(x31) -> f599_in(x31) :|: TRUE f599_out(x32) -> f470_out(x32) :|: TRUE f891_in(s(x34)) -> f900_in(x34) :|: TRUE f900_out(x36) -> f891_out(s(x36)) :|: TRUE f368_out -> f230_out :|: TRUE f230_in -> f368_in :|: TRUE f987_in -> f966_in :|: TRUE f914_in(T125) -> f937_in :|: TRUE f937_out -> f914_out(x39) :|: TRUE f418_in -> f418_out :|: TRUE f190_out -> f192_in(x48) :|: TRUE f192_in(x53) -> f966_in :|: TRUE ---------------------------------------- (194) Obligation: Rules: f602_out(T96) -> f900_out(T96) :|: TRUE f900_in(x) -> f602_in(x) :|: TRUE f430_in -> f404_in :|: TRUE f404_out -> f430_out :|: TRUE f603_in(T77) -> f405_in(T77) :|: TRUE f405_out(x1) -> f603_out(x1) :|: TRUE f983_in -> f987_in :|: TRUE f918_out(T116) -> f913_out(T116) :|: TRUE f913_in(x2) -> f918_in(x2) :|: TRUE f890_out(x4) -> f606_out(x4) :|: TRUE f606_in(x5) -> f890_in(x5) :|: TRUE f606_in(x6) -> f891_in(x6) :|: TRUE f891_out(x7) -> f606_out(x7) :|: TRUE f972_in -> f976_in :|: TRUE f603_out(x8) -> f600_out(x8) :|: TRUE f600_in(x9) -> f602_in(x9) :|: TRUE f602_out(x10) -> f603_in(x10) :|: TRUE f982_in -> f190_in :|: TRUE f190_out -> f982_out :|: TRUE f937_in -> f937_out :|: TRUE f470_out(x11) -> f405_out(x11) :|: TRUE f405_in(x12) -> f470_in(x12) :|: TRUE f368_in -> f404_in :|: TRUE f404_out -> f405_in(x13) :|: TRUE f405_out(x14) -> f368_out :|: TRUE f958_in -> f190_in :|: TRUE f190_out -> f958_out :|: TRUE f190_in -> f217_in :|: TRUE f217_out -> f190_out :|: TRUE f960_in -> f960_out :|: TRUE f941_out -> f237_out :|: TRUE f940_out -> f237_out :|: TRUE f237_in -> f940_in :|: TRUE f237_in -> f941_in :|: TRUE f893_in -> f893_out :|: TRUE f918_in(x15) -> f405_in(x15) :|: TRUE f405_out(x16) -> f918_out(x16) :|: TRUE f960_out -> f941_out :|: TRUE f941_in -> f960_in :|: TRUE f413_in -> f430_in :|: TRUE f430_out -> f413_out :|: TRUE f982_out -> f983_in :|: TRUE f976_in -> f982_in :|: TRUE f893_out -> f890_out(0) :|: TRUE f890_in(0) -> f893_in :|: TRUE f411_in -> f418_in :|: TRUE f418_out -> f411_out :|: TRUE f598_in(x20) -> f600_in(x20) :|: TRUE f600_out(x22) -> f598_out(x22) :|: TRUE f970_in -> f972_in :|: TRUE f599_in(x23) -> f914_in(x23) :|: TRUE f914_out(x24) -> f599_out(x24) :|: TRUE f913_out(x25) -> f599_out(x25) :|: TRUE f599_in(x26) -> f913_in(x26) :|: TRUE f602_in(x27) -> f606_in(x27) :|: TRUE f606_out(x28) -> f602_out(x28) :|: TRUE f966_in -> f970_in :|: TRUE f407_out -> f404_out :|: TRUE f404_in -> f407_in :|: TRUE f407_in -> f411_in :|: TRUE f413_out -> f407_out :|: TRUE f407_in -> f413_in :|: TRUE f411_out -> f407_out :|: TRUE f217_in -> f230_in :|: TRUE f217_in -> f237_in :|: TRUE f237_out -> f217_out :|: TRUE f230_out -> f217_out :|: TRUE f940_in -> f958_in :|: TRUE f958_out -> f940_out :|: TRUE f598_out(x29) -> f470_out(x29) :|: TRUE f470_in(x30) -> f598_in(x30) :|: TRUE f470_in(x31) -> f599_in(x31) :|: TRUE f599_out(x32) -> f470_out(x32) :|: TRUE f891_in(s(x34)) -> f900_in(x34) :|: TRUE f900_out(x36) -> f891_out(s(x36)) :|: TRUE f368_out -> f230_out :|: TRUE f230_in -> f368_in :|: TRUE f987_in -> f966_in :|: TRUE f914_in(T125) -> f937_in :|: TRUE f937_out -> f914_out(x39) :|: TRUE f418_in -> f418_out :|: TRUE f190_out -> f192_in(x48) :|: TRUE f192_in(x53) -> f966_in :|: TRUE ---------------------------------------- (195) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (196) Obligation: Rules: f470_out(x11:0) -> f470_out(x11:0) :|: TRUE f217_out -> f217_in :|: TRUE f217_in -> f217_out :|: TRUE f470_in(x31:0) -> f470_in(x31:0) :|: TRUE f217_out -> f217_out :|: TRUE f606_in(s(x34:0)) -> f606_in(x34:0) :|: TRUE f407_out -> f470_in(x13:0) :|: TRUE f470_in(x30:0) -> f606_in(x30:0) :|: TRUE f470_out(x) -> f217_out :|: TRUE f602_out(T96:0) -> f602_out(s(T96:0)) :|: TRUE f407_in -> f407_out :|: TRUE f407_in -> f407_in :|: TRUE f470_in(x1) -> f470_out(x2) :|: TRUE f407_out -> f407_out :|: TRUE f602_out(x10:0) -> f470_in(x10:0) :|: TRUE f606_in(cons_0) -> f602_out(0) :|: TRUE && cons_0 = 0 f217_in -> f217_in :|: TRUE f217_in -> f407_in :|: TRUE ---------------------------------------- (197) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (198) Obligation: Rules: f470_out(x11:0) -> f470_out(x11:0) :|: TRUE f217_out -> f217_in :|: TRUE f217_in -> f217_out :|: TRUE f470_in(x31:0) -> f470_in(x31:0) :|: TRUE f217_out -> f217_out :|: TRUE f606_in(s(x34:0)) -> f606_in(x34:0) :|: TRUE f407_out -> f470_in(x13:0) :|: TRUE f470_in(x30:0) -> f606_in(x30:0) :|: TRUE f470_out(x) -> f217_out :|: TRUE f602_out(T96:0) -> f602_out(s(T96:0)) :|: TRUE f407_in -> f407_out :|: TRUE f407_in -> f407_in :|: TRUE f470_in(x1) -> f470_out(x2) :|: TRUE f407_out -> f407_out :|: TRUE f602_out(x10:0) -> f470_in(x10:0) :|: TRUE f606_in(cons_0) -> f602_out(0) :|: TRUE && cons_0 = 0 f217_in -> f217_in :|: TRUE f217_in -> f407_in :|: TRUE ---------------------------------------- (199) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f470_out(x11:0) -> f470_out(x11:0) :|: TRUE (2) f217_out -> f217_in :|: TRUE (3) f217_in -> f217_out :|: TRUE (4) f470_in(x31:0) -> f470_in(x31:0) :|: TRUE (5) f217_out -> f217_out :|: TRUE (6) f606_in(s(x34:0)) -> f606_in(x34:0) :|: TRUE (7) f407_out -> f470_in(x13:0) :|: TRUE (8) f470_in(x30:0) -> f606_in(x30:0) :|: TRUE (9) f470_out(x) -> f217_out :|: TRUE (10) f602_out(T96:0) -> f602_out(s(T96:0)) :|: TRUE (11) f407_in -> f407_out :|: TRUE (12) f407_in -> f407_in :|: TRUE (13) f470_in(x1) -> f470_out(x2) :|: TRUE (14) f407_out -> f407_out :|: TRUE (15) f602_out(x10:0) -> f470_in(x10:0) :|: TRUE (16) f606_in(cons_0) -> f602_out(0) :|: TRUE && cons_0 = 0 (17) f217_in -> f217_in :|: TRUE (18) f217_in -> f407_in :|: TRUE Arcs: (1) -> (1), (9) (2) -> (3), (17), (18) (3) -> (2), (5) (4) -> (4), (8), (13) (5) -> (2), (5) (6) -> (6), (16) (7) -> (4), (8), (13) (8) -> (6), (16) (9) -> (2), (5) (10) -> (10), (15) (11) -> (7), (14) (12) -> (11), (12) (13) -> (1), (9) (14) -> (7), (14) (15) -> (4), (8), (13) (16) -> (10), (15) (17) -> (3), (17), (18) (18) -> (11), (12) This digraph is fully evaluated! ---------------------------------------- (200) Obligation: Termination digraph: Nodes: (1) f470_out(x11:0) -> f470_out(x11:0) :|: TRUE (2) f470_in(x1) -> f470_out(x2) :|: TRUE (3) f470_in(x31:0) -> f470_in(x31:0) :|: TRUE (4) f602_out(x10:0) -> f470_in(x10:0) :|: TRUE (5) f602_out(T96:0) -> f602_out(s(T96:0)) :|: TRUE (6) f606_in(cons_0) -> f602_out(0) :|: TRUE && cons_0 = 0 (7) f606_in(s(x34:0)) -> f606_in(x34:0) :|: TRUE (8) f470_in(x30:0) -> f606_in(x30:0) :|: TRUE (9) f407_out -> f470_in(x13:0) :|: TRUE (10) f407_out -> f407_out :|: TRUE (11) f407_in -> f407_out :|: TRUE (12) f407_in -> f407_in :|: TRUE (13) f217_in -> f407_in :|: TRUE (14) f217_out -> f217_in :|: TRUE (15) f217_out -> f217_out :|: TRUE (16) f470_out(x) -> f217_out :|: TRUE (17) f217_in -> f217_out :|: TRUE (18) f217_in -> f217_in :|: TRUE Arcs: (1) -> (1), (16) (2) -> (1), (16) (3) -> (2), (3), (8) (4) -> (2), (3), (8) (5) -> (4), (5) (6) -> (4), (5) (7) -> (6), (7) (8) -> (6), (7) (9) -> (2), (3), (8) (10) -> (9), (10) (11) -> (9), (10) (12) -> (11), (12) (13) -> (11), (12) (14) -> (13), (17), (18) (15) -> (14), (15) (16) -> (14), (15) (17) -> (14), (15) (18) -> (13), (17), (18) This digraph is fully evaluated! ---------------------------------------- (201) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (202) Obligation: Rules: f217_out -> f217_out :|: TRUE f407_in -> f407_in :|: TRUE f606_in(s(x34:0:0)) -> f606_in(x34:0:0) :|: TRUE f217_out -> f217_in :|: TRUE f217_in -> f217_out :|: TRUE f470_in(x30:0:0) -> f606_in(x30:0:0) :|: TRUE f470_out(x11:0:0) -> f470_out(x11:0:0) :|: TRUE f407_out -> f470_in(x13:0:0) :|: TRUE f470_out(x:0) -> f217_out :|: TRUE f602_out(x10:0:0) -> f470_in(x10:0:0) :|: TRUE f217_in -> f407_in :|: TRUE f602_out(T96:0:0) -> f602_out(s(T96:0:0)) :|: TRUE f217_in -> f217_in :|: TRUE f407_in -> f407_out :|: TRUE f470_in(x1:0) -> f470_out(x2:0) :|: TRUE f470_in(x31:0:0) -> f470_in(x31:0:0) :|: TRUE f407_out -> f407_out :|: TRUE f606_in(cons_0) -> f602_out(0) :|: TRUE && cons_0 = 0 ---------------------------------------- (203) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "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": { "907": { "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": [] } }, "908": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "909": { "goal": [{ "clause": -1, "scope": -1, "term": "(less T70 T73)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T70"], "free": [], "exprvars": [] } }, "1187": { "goal": [{ "clause": -1, "scope": -1, "term": "(app T483 (. T484 T485) T482)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T482"], "free": [], "exprvars": [] } }, "1186": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1185": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1184": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "1183": { "goal": [{ "clause": 6, "scope": 14, "term": "(app T451 (. T452 T450) T305)" }], "kb": { "nonunifying": [[ "(qs T453 T305)", "(qs ([]) ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T305"], "free": [], "exprvars": [] } }, "1182": { "goal": [{ "clause": 5, "scope": 14, "term": "(app T451 (. T452 T450) T305)" }], "kb": { "nonunifying": [[ "(qs T453 T305)", "(qs ([]) ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T305"], "free": [], "exprvars": [] } }, "1181": { "goal": [ { "clause": 5, "scope": 14, "term": "(app T451 (. T452 T450) T305)" }, { "clause": 6, "scope": 14, "term": "(app T451 (. T452 T450) T305)" } ], "kb": { "nonunifying": [[ "(qs T453 T305)", "(qs ([]) ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T305"], "free": [], "exprvars": [] } }, "1180": { "goal": [{ "clause": -1, "scope": -1, "term": "(app T451 (. T452 T450) T305)" }], "kb": { "nonunifying": [[ "(qs T453 T305)", "(qs ([]) ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T305"], "free": [], "exprvars": [] } }, "1179": { "goal": [{ "clause": -1, "scope": -1, "term": "(qs (. T446 T447) X438)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X438"], "exprvars": [] } }, "1178": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (qs (. T446 T447) X438) (app T445 (. T448 X438) T305))" }], "kb": { "nonunifying": [[ "(qs T449 T305)", "(qs ([]) ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T305"], "free": ["X438"], "exprvars": [] } }, "1177": { "goal": [{ "clause": -1, "scope": -1, "term": "(qs T437 X437)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X437"], "exprvars": [] } }, "910": { "goal": [{ "clause": -1, "scope": -1, "term": "(part T70 T77 X112 X113)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T70"], "free": [ "X112", "X113" ], "exprvars": [] } }, "911": { "goal": [{ "clause": 3, "scope": 4, "term": "(part T29 T30 X46 X47)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T29"], "free": [ "X46", "X47" ], "exprvars": [] } }, "912": { "goal": [{ "clause": 4, "scope": 4, "term": "(part T29 T30 X46 X47)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T29"], "free": [ "X46", "X47" ], "exprvars": [] } }, "1190": { "goal": [{ "clause": 5, "scope": 15, "term": "(app T483 (. T484 T485) T482)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T482"], "free": [], "exprvars": [] } }, "1198": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1197": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (qs ([]) X437) (',' (qs ([]) X438) (app X437 (. T525 X438) T305)))" }], "kb": { "nonunifying": [[ "(qs T1 T305)", "(qs ([]) ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T305"], "free": [ "X437", "X438" ], "exprvars": [] } }, "1196": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1195": { "goal": [{ "clause": -1, "scope": -1, "term": "(app T515 (. T516 T517) T514)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T514"], "free": [], "exprvars": [] } }, "1194": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1193": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1192": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1191": { "goal": [{ "clause": 6, "scope": 15, "term": "(app T483 (. T484 T485) T482)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T482"], "free": [], "exprvars": [] } }, "1": { "goal": [{ "clause": -1, "scope": -1, "term": "(qs T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "1103": { "goal": [{ "clause": 6, "scope": 8, "term": "(app T194 (. T195 T193) X196)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X196"], "exprvars": [] } }, "1102": { "goal": [{ "clause": 5, "scope": 8, "term": "(app T194 (. T195 T193) X196)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X196"], "exprvars": [] } }, "6": { "goal": [ { "clause": 0, "scope": 1, "term": "(qs T1 T2)" }, { "clause": 1, "scope": 1, "term": "(qs T1 T2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "1101": { "goal": [ { "clause": 5, "scope": 8, "term": "(app T194 (. T195 T193) X196)" }, { "clause": 6, "scope": 8, "term": "(app T194 (. T195 T193) X196)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X196"], "exprvars": [] } }, "1189": { "goal": [ { "clause": 5, "scope": 15, "term": "(app T483 (. T484 T485) T482)" }, { "clause": 6, "scope": 15, "term": "(app T483 (. T484 T485) T482)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T482"], "free": [], "exprvars": [] } }, "1100": { "goal": [{ "clause": -1, "scope": -1, "term": "(app T194 (. T195 T193) X196)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X196"], "exprvars": [] } }, "1188": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "924": { "goal": [{ "clause": -1, "scope": -1, "term": "(part T95 T98 X158 X159)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T95"], "free": [ "X158", "X159" ], "exprvars": [] } }, "926": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1121": { "goal": [{ "clause": 3, "scope": 2, "term": "(',' (part T7 T8 X9 X10) (',' (qs X9 X11) (',' (qs X10 X12) (app X11 (. 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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": [] } }, "1050": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1171": { "goal": [{ "clause": 3, "scope": 11, "term": "(',' (part T306 T307 X435 X436) (',' (qs X435 X437) (',' (qs X436 X438) (app X437 (. 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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": [] } }, "984": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1047": { "goal": [{ "clause": 1, "scope": 7, "term": "(qs T123 X194)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X194"], "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": [] } }, "985": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1046": { "goal": [{ "clause": 0, "scope": 7, "term": "(qs T123 X194)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X194"], "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": [] } }, "986": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1045": { "goal": [ { "clause": 0, "scope": 7, "term": "(qs T123 X194)" }, { "clause": 1, "scope": 7, "term": "(qs T123 X194)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X194"], "exprvars": [] } }, "1166": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "904": { "goal": [ { "clause": 2, "scope": 4, "term": "(part T29 T30 X46 X47)" }, { "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": [] } }, "905": { "goal": [{ "clause": 2, "scope": 4, "term": "(part T29 T30 X46 X47)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T29"], "free": [ "X46", "X47" ], "exprvars": [] } }, "906": { "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": [] } } }, "edges": [ { "from": 1, "to": 6, "label": "CASE" }, { "from": 6, "to": 44, "label": "EVAL with clause\nqs([], []).\nand substitutionT1 -> [],\nT2 -> []" }, { "from": 6, "to": 45, "label": "EVAL-BACKTRACK" }, { "from": 44, "to": 47, "label": "SUCCESS" }, { "from": 45, "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": 45, "to": 1141, "label": "EVAL-BACKTRACK" }, { "from": 47, "to": 48, "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": 47, "to": 49, "label": "EVAL-BACKTRACK" }, { "from": 48, "to": 50, "label": "CASE" }, { "from": 50, "to": 377, "label": "PARALLEL" }, { "from": 50, "to": 379, "label": "PARALLEL" }, { "from": 377, "to": 383, "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": 377, "to": 385, "label": "EVAL-BACKTRACK" }, { "from": 379, "to": 1121, "label": "PARALLEL" }, { "from": 379, "to": 1122, "label": "PARALLEL" }, { "from": 383, "to": 388, "label": "SPLIT 1" }, { "from": 383, "to": 390, "label": "SPLIT 2\nnew knowledge:\nT29 is ground\nreplacements:T24 -> T29,\nT26 -> T30,\nT25 -> T31" }, { "from": 388, "to": 395, "label": "CASE" }, { "from": 390, "to": 604, "label": "SPLIT 1" }, { "from": 390, "to": 605, "label": "SPLIT 2\nnew knowledge:\nT29 is ground\nreplacements:X46 -> T52,\nX47 -> T53,\nT31 -> T54" }, { "from": 395, "to": 398, "label": "PARALLEL" }, { "from": 395, "to": 399, "label": "PARALLEL" }, { "from": 398, "to": 437, "label": "EVAL with clause\nless(0, s(X56)).\nand substitutionT24 -> 0,\nX56 -> T38,\nT25 -> s(T38)" }, { "from": 398, "to": 438, "label": "EVAL-BACKTRACK" }, { "from": 399, "to": 444, "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": 399, "to": 447, "label": "EVAL-BACKTRACK" }, { "from": 437, "to": 439, "label": "SUCCESS" }, { "from": 444, "to": 388, "label": "INSTANCE with matching:\nT24 -> T45\nT25 -> T46" }, { "from": 604, "to": 904, "label": "CASE" }, { "from": 605, "to": 933, "label": "SPLIT 1" }, { "from": 605, "to": 934, "label": "SPLIT 2\nreplacements:X11 -> T105,\nT53 -> T106" }, { "from": 904, "to": 905, "label": "PARALLEL" }, { "from": 904, "to": 906, "label": "PARALLEL" }, { "from": 905, "to": 907, "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": 905, "to": 908, "label": "EVAL-BACKTRACK" }, { "from": 906, "to": 911, "label": "PARALLEL" }, { "from": 906, "to": 912, "label": "PARALLEL" }, { "from": 907, "to": 909, "label": "SPLIT 1" }, { "from": 907, "to": 910, "label": "SPLIT 2\nnew knowledge:\nT70 is ground\nreplacements:T74 -> T77" }, { "from": 909, "to": 388, "label": "INSTANCE with matching:\nT24 -> T70\nT25 -> T73" }, { "from": 910, "to": 604, "label": "INSTANCE with matching:\nT29 -> T70\nT30 -> T77\nX46 -> X112\nX47 -> X113" }, { "from": 911, "to": 924, "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": 911, "to": 926, "label": "EVAL-BACKTRACK" }, { "from": 912, "to": 930, "label": "EVAL with clause\npart(X169, [], [], []).\nand substitutionT29 -> T104,\nX169 -> T104,\nT30 -> [],\nX46 -> [],\nX47 -> []" }, { "from": 912, "to": 931, "label": "EVAL-BACKTRACK" }, { "from": 924, "to": 604, "label": "INSTANCE with matching:\nT29 -> T95\nT30 -> T98\nX46 -> X158\nX47 -> X159" }, { "from": 930, "to": 932, "label": "SUCCESS" }, { "from": 933, "to": 935, "label": "CASE" }, { "from": 934, "to": 1116, "label": "SPLIT 1" }, { "from": 934, "to": 1117, "label": "SPLIT 2\nreplacements:X12 -> T229,\nT105 -> T230" }, { "from": 935, "to": 936, "label": "BACKTRACK\nfor clause: qs([], [])because of non-unification" }, { "from": 936, "to": 944, "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": 944, "to": 947, "label": "SPLIT 1" }, { "from": 944, "to": 948, "label": "SPLIT 2\nreplacements:X192 -> T123,\nX193 -> T124,\nT118 -> T125" }, { "from": 947, "to": 949, "label": "CASE" }, { "from": 948, "to": 1043, "label": "SPLIT 1" }, { "from": 948, "to": 1044, "label": "SPLIT 2\nreplacements:X194 -> T179,\nT124 -> T180,\nT125 -> T181" }, { "from": 949, "to": 952, "label": "PARALLEL" }, { "from": 949, "to": 953, "label": "PARALLEL" }, { "from": 952, "to": 954, "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": 952, "to": 955, "label": "EVAL-BACKTRACK" }, { "from": 953, "to": 978, "label": "PARALLEL" }, { "from": 953, "to": 979, "label": "PARALLEL" }, { "from": 954, "to": 956, "label": "SPLIT 1" }, { "from": 954, "to": 957, "label": "SPLIT 2\nnew knowledge:\nT149 is ground\nreplacements:T144 -> T149,\nT146 -> T150" }, { "from": 956, "to": 388, "label": "INSTANCE with matching:\nT24 -> T144\nT25 -> T145" }, { "from": 957, "to": 604, "label": "INSTANCE with matching:\nT29 -> T149\nT30 -> T150\nX46 -> X244\nX47 -> X245" }, { "from": 978, "to": 980, "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": 978, "to": 981, "label": "EVAL-BACKTRACK" }, { "from": 979, "to": 984, "label": "EVAL with clause\npart(X301, [], [], []).\nand substitutionT118 -> T178,\nX301 -> T178,\nT119 -> [],\nX192 -> [],\nX193 -> []" }, { "from": 979, "to": 985, "label": "EVAL-BACKTRACK" }, { "from": 980, "to": 947, "label": "INSTANCE with matching:\nT118 -> T171\nT119 -> T172\nX192 -> X290\nX193 -> X291" }, { "from": 984, "to": 986, "label": "SUCCESS" }, { "from": 1043, "to": 1045, "label": "CASE" }, { "from": 1044, "to": 1099, "label": "SPLIT 1" }, { "from": 1044, "to": 1100, "label": "SPLIT 2\nreplacements:X195 -> T193,\nT179 -> T194,\nT181 -> T195" }, { "from": 1045, "to": 1046, "label": "PARALLEL" }, { "from": 1045, "to": 1047, "label": "PARALLEL" }, { "from": 1046, "to": 1048, "label": "EVAL with clause\nqs([], []).\nand substitutionT123 -> [],\nX194 -> []" }, { "from": 1046, "to": 1049, "label": "EVAL-BACKTRACK" }, { "from": 1047, "to": 1094, "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": 1047, "to": 1095, "label": "EVAL-BACKTRACK" }, { "from": 1048, "to": 1050, "label": "SUCCESS" }, { "from": 1094, "to": 944, "label": "INSTANCE with matching:\nT118 -> T188\nT119 -> T189\nX192 -> X317\nX193 -> X318\nX194 -> X319\nX195 -> X320\nX196 -> X321" }, { "from": 1099, "to": 1043, "label": "INSTANCE with matching:\nT123 -> T180\nX194 -> X195" }, { "from": 1100, "to": 1101, "label": "CASE" }, { "from": 1101, "to": 1102, "label": "PARALLEL" }, { "from": 1101, "to": 1103, "label": "PARALLEL" }, { "from": 1102, "to": 1111, "label": "EVAL with clause\napp([], X335, X335).\nand substitutionT194 -> [],\nT195 -> T208,\nT193 -> T209,\nX335 -> .(T208, T209),\nX196 -> .(T208, T209)" }, { "from": 1102, "to": 1112, "label": "EVAL-BACKTRACK" }, { "from": 1103, "to": 1114, "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": 1103, "to": 1115, "label": "EVAL-BACKTRACK" }, { "from": 1111, "to": 1113, "label": "SUCCESS" }, { "from": 1114, "to": 1100, "label": "INSTANCE with matching:\nT194 -> T222\nT195 -> T223\nT193 -> T224\nX196 -> X350" }, { "from": 1116, "to": 1043, "label": "INSTANCE with matching:\nT123 -> T106\nX194 -> X12" }, { "from": 1117, "to": 1118, "label": "CASE" }, { "from": 1118, "to": 1119, "label": "BACKTRACK\nfor clause: app([], X, X)because of non-unification" }, { "from": 1119, "to": 1120, "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": 947, "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": 1043, "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": 933, "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": 1043, "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": 1043, "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": 388, "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": 604, "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": 933, "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": 1043, "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": 947, "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": 1043, "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": 933, "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": 1043, "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": 1043, "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" } } ---------------------------------------- (204) 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) ---------------------------------------- (205) UndefinedPredicateInTriplesTransformerProof (SOUND) Deleted triples and predicates having undefined goals [DT09]. ---------------------------------------- (206) 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) ---------------------------------------- (207) 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 ---------------------------------------- (208) 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 ---------------------------------------- (209) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 14 SCCs with 110 less nodes. ---------------------------------------- (210) Complex Obligation (AND) ---------------------------------------- (211) 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 ---------------------------------------- (212) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (213) 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 ---------------------------------------- (214) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (215) 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. ---------------------------------------- (216) 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 ---------------------------------------- (217) YES ---------------------------------------- (218) 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 ---------------------------------------- (219) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (220) 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 ---------------------------------------- (221) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (222) 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. ---------------------------------------- (223) 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 ---------------------------------------- (224) YES ---------------------------------------- (225) 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 ---------------------------------------- (226) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (227) 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 ---------------------------------------- (228) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (229) 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. ---------------------------------------- (230) 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 ---------------------------------------- (231) YES ---------------------------------------- (232) 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 ---------------------------------------- (233) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (234) 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 ---------------------------------------- (235) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (236) 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. ---------------------------------------- (237) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *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 ---------------------------------------- (238) YES ---------------------------------------- (239) 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 ---------------------------------------- (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: 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 ---------------------------------------- (242) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (243) 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. ---------------------------------------- (244) 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 ---------------------------------------- (245) YES ---------------------------------------- (246) 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 ---------------------------------------- (247) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (248) 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 ---------------------------------------- (249) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (250) 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. ---------------------------------------- (251) 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 ---------------------------------------- (252) YES ---------------------------------------- (253) 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 ---------------------------------------- (254) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (255) 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 ---------------------------------------- (256) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (257) 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. ---------------------------------------- (258) 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 ---------------------------------------- (259) YES ---------------------------------------- (260) 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 ---------------------------------------- (261) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (262) 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 ---------------------------------------- (263) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (264) 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. ---------------------------------------- (265) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 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)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( U11_GGAAAAA_3(x_1, ..., x_3) ) = x_3 + 2 POL( U13_GGAAAAA_4(x_1, ..., x_4) ) = 2x_3 POL( U70_ggaaaaa_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2} POL( partcC_in_ggaa_2(x_1, x_2) ) = x_1 + 2x_2 POL( ._2(x_1, x_2) ) = x_1 + x_2 + 1 POL( U67_ggaa_4(x_1, ..., x_4) ) = 2x_3 + x_4 + 2 POL( lesscA_in_gg_2(x_1, x_2) ) = x_1 + x_2 POL( U69_ggaa_4(x_1, ..., x_4) ) = 2x_2 + x_4 + 2 POL( [] ) = 0 POL( partcC_out_ggaa_4(x_1, ..., x_4) ) = max{0, 2x_3 + 2x_4 - 1} POL( qscF_in_ga_1(x_1) ) = 2 POL( qscF_out_ga_2(x_1, x_2) ) = max{0, x_2 - 2} POL( U74_ga_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 - 2} POL( qcD_in_ggaaaaa_2(x_1, x_2) ) = 2x_1 + 2 POL( U71_ggaaaaa_5(x_1, ..., x_5) ) = 2x_1 + x_2 + 2x_3 + 2x_4 + 2x_5 + 2 POL( qcD_out_ggaaaaa_7(x_1, ..., x_7) ) = x_1 + 2x_4 + 2x_6 + 2x_7 + 2 POL( U72_ggaaaaa_6(x_1, ..., x_6) ) = max{0, x_3 - 2} POL( U73_ggaaaaa_7(x_1, ..., x_7) ) = max{0, 2x_4 + 2x_5 + 2x_6 - 2} POL( appcE_in_ggga_3(x_1, ..., x_3) ) = x_2 POL( U64_ggaa_4(x_1, ..., x_4) ) = x_2 + x_3 + 1 POL( U68_ggaa_4(x_1, ..., x_4) ) = 2x_2 + 2x_4 + 1 POL( 0 ) = 2 POL( s_1(x_1) ) = 1 POL( lesscA_out_gg_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 1} POL( U63_gg_3(x_1, ..., x_3) ) = 2 POL( partcB_in_ggaa_2(x_1, x_2) ) = x_2 POL( U66_ggaa_4(x_1, ..., x_4) ) = x_2 + x_4 + 1 POL( partcB_out_ggaa_4(x_1, ..., x_4) ) = x_3 + x_4 POL( U65_ggaa_4(x_1, ..., x_4) ) = x_2 + x_4 + 1 POL( appcE_out_ggga_4(x_1, ..., x_4) ) = max{0, 2x_3 - 2} POL( U75_ggga_5(x_1, ..., x_5) ) = 2x_1 + 2x_2 + x_3 + 2 POL( QSF_IN_GA_1(x_1) ) = 2x_1 POL( PD_IN_GGAAAAA_2(x_1, x_2) ) = x_1 + 2x_2 + 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 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, [], [], []) U69_ggaa(X1, X2, X3, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 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)) U67_ggaa(X1, X2, X3, lesscA_out_gg(X1, X2)) -> U68_ggaa(X1, X2, X3, partcB_in_ggaa(X1, X3)) 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, [], [], []) U68_ggaa(X1, X2, X3, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) U64_ggaa(X1, X2, X3, lesscA_out_gg(X1, X2)) -> U65_ggaa(X1, X2, X3, partcB_in_ggaa(X1, X3)) U65_ggaa(X1, X2, X3, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) U66_ggaa(X1, X2, X3, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) ---------------------------------------- (266) 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)) 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. ---------------------------------------- (267) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes. ---------------------------------------- (268) TRUE ---------------------------------------- (269) 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 ---------------------------------------- (270) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (271) Obligation: Pi DP problem: The TRS P consists of the following rules: 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 ---------------------------------------- (272) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (273) Obligation: Q DP problem: The TRS P consists of the following rules: APPE_IN_AAAA -> APPE_IN_AAAA R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (274) 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. ---------------------------------------- (275) NO ---------------------------------------- (276) 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 ---------------------------------------- (277) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (278) 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 ---------------------------------------- (279) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (280) 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. ---------------------------------------- (281) 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 ---------------------------------------- (282) YES ---------------------------------------- (283) 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 ---------------------------------------- (284) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (285) Obligation: Pi DP problem: The TRS P consists of the following rules: 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 ---------------------------------------- (286) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (287) 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. ---------------------------------------- (288) 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)))) ---------------------------------------- (289) 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. ---------------------------------------- (290) 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))) ---------------------------------------- (291) 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. ---------------------------------------- (292) 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 = PARTB_IN_GAAA(X1) evaluates to t =PARTB_IN_GAAA(X1) 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 PARTB_IN_GAAA(X1) to PARTB_IN_GAAA(X1). ---------------------------------------- (293) NO ---------------------------------------- (294) 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 ---------------------------------------- (295) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (296) 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 ---------------------------------------- (297) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (298) Obligation: Q DP problem: The TRS P consists of the following rules: LESSA_IN_AA -> LESSA_IN_AA R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (299) 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 ---------------------------------------- (300) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (301) 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 ---------------------------------------- (302) 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 ---------------------------------------- (303) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (304) 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