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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Left Termination of the query pattern qs(a,g) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) PrologToPiTRSProof [SOUND, 29 ms] (2)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, 28 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [SOUND, 12 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, 34 ms] (59) PiTRS (60) DependencyPairsProof [EQUIVALENT, 21 ms] (61) PiDP (62) DependencyGraphProof [EQUIVALENT, 0 ms] (63) AND (64) PiDP (65) UsableRulesProof [EQUIVALENT, 0 ms] (66) PiDP (67) PiDPToQDPProof [SOUND, 6 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, 3 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, 79 ms] (116) QTRS (117) DependencyPairsProof [EQUIVALENT, 10 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, 1103 ms] (151) NO (152) PrologToIRSwTTransformerProof [SOUND, 107 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, 0 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) IntTRSNonPeriodicNontermProof [COMPLETE, 0 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, 0 ms] (187) IRSwT (188) FilterProof [EQUIVALENT, 0 ms] (189) IntTRS (190) IntTRSPeriodicNontermProof [COMPLETE, 4 ms] (191) NO (192) IRSwT (193) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (194) IRSwT (195) IntTRSCompressionProof [EQUIVALENT, 36 ms] (196) IRSwT (197) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (198) IRSwT (199) IRSwTTerminationDigraphProof [EQUIVALENT, 89 ms] (200) IRSwT (201) IntTRSCompressionProof [EQUIVALENT, 21 ms] (202) IRSwT (203) PrologToDTProblemTransformerProof [SOUND, 179 ms] (204) TRIPLES (205) UndefinedPredicateInTriplesTransformerProof [SOUND, 0 ms] (206) TRIPLES (207) TriplesToPiDPProof [SOUND, 199 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, 0 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, 0 ms] (264) QDP (265) QDPQMonotonicMRRProof [EQUIVALENT, 79 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(x2) U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U9_aa(x1, x2, x3) = U9_aa(x3) U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U9_ga(x1, x2, x3) = U9_ga(x1, x3) U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x1, x6) U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x1, x6) part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa(x1) part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) qs_in_aa(x1, x2) = qs_in_aa qs_out_aa(x1, x2) = qs_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U4_aa(x1, x2, x3, x4) = U4_aa(x4) .(x1, x2) = .(x1, x2) app_in_aaa(x1, x2, x3) = app_in_aaa app_out_aaa(x1, x2, x3) = app_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4) app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2, x3) U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x4, x5) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: qs_in_ag([], []) -> qs_out_ag([], []) qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U5_aaaa(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U5_gaaa(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) qs_in_aa([], []) -> qs_out_aa([], []) qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) app_in_aaa([], X, X) -> app_out_aaa([], X, X) app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) app_in_aag([], X, X) -> app_out_aag([], X, X) app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ag(x1, x2) = qs_in_ag(x2) [] = [] qs_out_ag(x1, x2) = qs_out_ag(x2) U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U9_aa(x1, x2, x3) = U9_aa(x3) U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U9_ga(x1, x2, x3) = U9_ga(x1, x3) U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x1, x6) U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x1, x6) part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa(x1) part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) qs_in_aa(x1, x2) = qs_in_aa qs_out_aa(x1, x2) = qs_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U4_aa(x1, x2, x3, x4) = U4_aa(x4) .(x1, x2) = .(x1, x2) app_in_aaa(x1, x2, x3) = app_in_aaa app_out_aaa(x1, x2, x3) = app_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4) app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2, x3) U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x4, x5) ---------------------------------------- (3) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: Pi DP problem: The TRS P consists of the following rules: QS_IN_AG(.(X, Xs), Ys) -> U1_AG(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) QS_IN_AG(.(X, Xs), Ys) -> PART_IN_AAAA(X, Xs, Littles, Bigs) PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_AAAA(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> LESS_IN_AA(X, Y) LESS_IN_AA(s(X), s(Y)) -> U9_AA(X, Y, less_in_aa(X, Y)) LESS_IN_AA(s(X), s(Y)) -> LESS_IN_AA(X, Y) U5_AAAA(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> U6_AAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) U5_AAAA(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs) PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GAAA(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> LESS_IN_GA(X, Y) LESS_IN_GA(s(X), s(Y)) -> U9_GA(X, Y, less_in_ga(X, Y)) LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) U5_GAAA(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> U6_GAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) U5_GAAA(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs) PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_GAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_GAAA(X, Xs, Ls, Bs) PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_AAAA(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_AAAA(X, Xs, Ls, Bs) U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AG(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> QS_IN_AA(Littles, Ls) QS_IN_AA(.(X, Xs), Ys) -> U1_AA(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) QS_IN_AA(.(X, Xs), Ys) -> PART_IN_AAAA(X, Xs, Littles, Bigs) U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AA(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> QS_IN_AA(Littles, Ls) U2_AA(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_AA(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U2_AA(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> QS_IN_AA(Bigs, Bs) U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_AA(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> APP_IN_AAA(Ls, .(X, Bs), Ys) APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> U8_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAA(Xs, Ys, Zs) U2_AG(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_AG(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U2_AG(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> QS_IN_AA(Bigs, Bs) U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_AG(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> APP_IN_AAG(Ls, .(X, Bs), Ys) APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> U8_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs) The TRS R consists of the following rules: qs_in_ag([], []) -> qs_out_ag([], []) qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U5_aaaa(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U5_gaaa(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) qs_in_aa([], []) -> qs_out_aa([], []) qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) app_in_aaa([], X, X) -> app_out_aaa([], X, X) app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) app_in_aag([], X, X) -> app_out_aag([], X, X) app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ag(x1, x2) = qs_in_ag(x2) [] = [] qs_out_ag(x1, x2) = qs_out_ag(x2) U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U9_aa(x1, x2, x3) = U9_aa(x3) U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U9_ga(x1, x2, x3) = U9_ga(x1, x3) U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x1, x6) U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x1, x6) part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa(x1) part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) qs_in_aa(x1, x2) = qs_in_aa qs_out_aa(x1, x2) = qs_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U4_aa(x1, x2, x3, x4) = U4_aa(x4) .(x1, x2) = .(x1, x2) app_in_aaa(x1, x2, x3) = app_in_aaa app_out_aaa(x1, x2, x3) = app_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4) app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2, x3) U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x4, x5) QS_IN_AG(x1, x2) = QS_IN_AG(x2) U1_AG(x1, x2, x3, x4) = U1_AG(x3, x4) PART_IN_AAAA(x1, x2, x3, x4) = PART_IN_AAAA U5_AAAA(x1, x2, x3, x4, x5, x6) = U5_AAAA(x6) LESS_IN_AA(x1, x2) = LESS_IN_AA U9_AA(x1, x2, x3) = U9_AA(x3) U6_AAAA(x1, x2, x3, x4, x5, x6) = U6_AAAA(x6) PART_IN_GAAA(x1, x2, x3, x4) = PART_IN_GAAA(x1) U5_GAAA(x1, x2, x3, x4, x5, x6) = U5_GAAA(x1, x6) LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) U9_GA(x1, x2, x3) = U9_GA(x1, x3) U6_GAAA(x1, x2, x3, x4, x5, x6) = U6_GAAA(x1, x6) U7_GAAA(x1, x2, x3, x4, x5, x6) = U7_GAAA(x1, x6) U7_AAAA(x1, x2, x3, x4, x5, x6) = U7_AAAA(x6) U2_AG(x1, x2, x3, x4, x5) = U2_AG(x3, x5) QS_IN_AA(x1, x2) = QS_IN_AA U1_AA(x1, x2, x3, x4) = U1_AA(x4) U2_AA(x1, x2, x3, x4, x5) = U2_AA(x5) U3_AA(x1, x2, x3, x4, x5) = U3_AA(x5) U4_AA(x1, x2, x3, x4) = U4_AA(x4) APP_IN_AAA(x1, x2, x3) = APP_IN_AAA U8_AAA(x1, x2, x3, x4, x5) = U8_AAA(x5) U3_AG(x1, x2, x3, x4, x5) = U3_AG(x3, x5) U4_AG(x1, x2, x3, x4) = U4_AG(x3, x4) APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3) U8_AAG(x1, x2, x3, x4, x5) = U8_AAG(x1, x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: QS_IN_AG(.(X, Xs), Ys) -> U1_AG(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) QS_IN_AG(.(X, Xs), Ys) -> PART_IN_AAAA(X, Xs, Littles, Bigs) PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_AAAA(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> LESS_IN_AA(X, Y) LESS_IN_AA(s(X), s(Y)) -> U9_AA(X, Y, less_in_aa(X, Y)) LESS_IN_AA(s(X), s(Y)) -> LESS_IN_AA(X, Y) U5_AAAA(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> U6_AAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) U5_AAAA(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs) PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GAAA(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> LESS_IN_GA(X, Y) LESS_IN_GA(s(X), s(Y)) -> U9_GA(X, Y, less_in_ga(X, Y)) LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) U5_GAAA(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> U6_GAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) U5_GAAA(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs) PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_GAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_GAAA(X, Xs, Ls, Bs) PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_AAAA(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_AAAA(X, Xs, Ls, Bs) U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AG(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> QS_IN_AA(Littles, Ls) QS_IN_AA(.(X, Xs), Ys) -> U1_AA(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) QS_IN_AA(.(X, Xs), Ys) -> PART_IN_AAAA(X, Xs, Littles, Bigs) U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AA(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> QS_IN_AA(Littles, Ls) U2_AA(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_AA(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U2_AA(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> QS_IN_AA(Bigs, Bs) U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_AA(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> APP_IN_AAA(Ls, .(X, Bs), Ys) APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> U8_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAA(Xs, Ys, Zs) U2_AG(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_AG(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U2_AG(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> QS_IN_AA(Bigs, Bs) U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_AG(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> APP_IN_AAG(Ls, .(X, Bs), Ys) APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> U8_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs) The TRS R consists of the following rules: qs_in_ag([], []) -> qs_out_ag([], []) qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U5_aaaa(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U5_gaaa(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) qs_in_aa([], []) -> qs_out_aa([], []) qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) app_in_aaa([], X, X) -> app_out_aaa([], X, X) app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) app_in_aag([], X, X) -> app_out_aag([], X, X) app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ag(x1, x2) = qs_in_ag(x2) [] = [] qs_out_ag(x1, x2) = qs_out_ag(x2) U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U9_aa(x1, x2, x3) = U9_aa(x3) U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U9_ga(x1, x2, x3) = U9_ga(x1, x3) U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x1, x6) U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x1, x6) part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa(x1) part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) qs_in_aa(x1, x2) = qs_in_aa qs_out_aa(x1, x2) = qs_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U4_aa(x1, x2, x3, x4) = U4_aa(x4) .(x1, x2) = .(x1, x2) app_in_aaa(x1, x2, x3) = app_in_aaa app_out_aaa(x1, x2, x3) = app_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4) app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2, x3) U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x4, x5) QS_IN_AG(x1, x2) = QS_IN_AG(x2) U1_AG(x1, x2, x3, x4) = U1_AG(x3, x4) PART_IN_AAAA(x1, x2, x3, x4) = PART_IN_AAAA U5_AAAA(x1, x2, x3, x4, x5, x6) = U5_AAAA(x6) LESS_IN_AA(x1, x2) = LESS_IN_AA U9_AA(x1, x2, x3) = U9_AA(x3) U6_AAAA(x1, x2, x3, x4, x5, x6) = U6_AAAA(x6) PART_IN_GAAA(x1, x2, x3, x4) = PART_IN_GAAA(x1) U5_GAAA(x1, x2, x3, x4, x5, x6) = U5_GAAA(x1, x6) LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) U9_GA(x1, x2, x3) = U9_GA(x1, x3) U6_GAAA(x1, x2, x3, x4, x5, x6) = U6_GAAA(x1, x6) U7_GAAA(x1, x2, x3, x4, x5, x6) = U7_GAAA(x1, x6) U7_AAAA(x1, x2, x3, x4, x5, x6) = U7_AAAA(x6) U2_AG(x1, x2, x3, x4, x5) = U2_AG(x3, x5) QS_IN_AA(x1, x2) = QS_IN_AA U1_AA(x1, x2, x3, x4) = U1_AA(x4) U2_AA(x1, x2, x3, x4, x5) = U2_AA(x5) U3_AA(x1, x2, x3, x4, x5) = U3_AA(x5) U4_AA(x1, x2, x3, x4) = U4_AA(x4) APP_IN_AAA(x1, x2, x3) = APP_IN_AAA U8_AAA(x1, x2, x3, x4, x5) = U8_AAA(x5) U3_AG(x1, x2, x3, x4, x5) = U3_AG(x3, x5) U4_AG(x1, x2, x3, x4) = U4_AG(x3, x4) APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3) U8_AAG(x1, x2, x3, x4, x5) = U8_AAG(x1, x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 7 SCCs with 24 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs) The TRS R consists of the following rules: qs_in_ag([], []) -> qs_out_ag([], []) qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U5_aaaa(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U5_gaaa(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) qs_in_aa([], []) -> qs_out_aa([], []) qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) app_in_aaa([], X, X) -> app_out_aaa([], X, X) app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) app_in_aag([], X, X) -> app_out_aag([], X, X) app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ag(x1, x2) = qs_in_ag(x2) [] = [] qs_out_ag(x1, x2) = qs_out_ag(x2) U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U9_aa(x1, x2, x3) = U9_aa(x3) U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U9_ga(x1, x2, x3) = U9_ga(x1, x3) U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x1, x6) U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x1, x6) part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa(x1) part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) qs_in_aa(x1, x2) = qs_in_aa qs_out_aa(x1, x2) = qs_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U4_aa(x1, x2, x3, x4) = U4_aa(x4) .(x1, x2) = .(x1, x2) app_in_aaa(x1, x2, x3) = app_in_aaa app_out_aaa(x1, x2, x3) = app_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4) app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2, x3) U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x4, x5) APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (8) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (9) Obligation: Pi DP problem: The TRS P consists of the following rules: APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (10) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: APP_IN_AAG(.(X, Zs)) -> APP_IN_AAG(Zs) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (12) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *APP_IN_AAG(.(X, Zs)) -> APP_IN_AAG(Zs) The graph contains the following edges 1 > 1 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAA(Xs, Ys, Zs) The TRS R consists of the following rules: qs_in_ag([], []) -> qs_out_ag([], []) qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U5_aaaa(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U5_gaaa(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) qs_in_aa([], []) -> qs_out_aa([], []) qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) app_in_aaa([], X, X) -> app_out_aaa([], X, X) app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) app_in_aag([], X, X) -> app_out_aag([], X, X) app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ag(x1, x2) = qs_in_ag(x2) [] = [] qs_out_ag(x1, x2) = qs_out_ag(x2) U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U9_aa(x1, x2, x3) = U9_aa(x3) U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U9_ga(x1, x2, x3) = U9_ga(x1, x3) U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x1, x6) U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x1, x6) part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa(x1) part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) qs_in_aa(x1, x2) = qs_in_aa qs_out_aa(x1, x2) = qs_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U4_aa(x1, x2, x3, x4) = U4_aa(x4) .(x1, x2) = .(x1, x2) app_in_aaa(x1, x2, x3) = app_in_aaa app_out_aaa(x1, x2, x3) = app_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4) app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2, x3) U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x4, x5) APP_IN_AAA(x1, x2, x3) = APP_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (15) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAA(Xs, Ys, Zs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) APP_IN_AAA(x1, x2, x3) = APP_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: APP_IN_AAA -> APP_IN_AAA R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (19) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by semiunifying a rule from P directly. s = APP_IN_AAA evaluates to t =APP_IN_AAA Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence The DP semiunifies directly so there is only one rewrite step from APP_IN_AAA to APP_IN_AAA. ---------------------------------------- (20) NO ---------------------------------------- (21) Obligation: Pi DP problem: The TRS P consists of the following rules: LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) The TRS R consists of the following rules: qs_in_ag([], []) -> qs_out_ag([], []) qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U5_aaaa(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U5_gaaa(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) qs_in_aa([], []) -> qs_out_aa([], []) qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) app_in_aaa([], X, X) -> app_out_aaa([], X, X) app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) app_in_aag([], X, X) -> app_out_aag([], X, X) app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ag(x1, x2) = qs_in_ag(x2) [] = [] qs_out_ag(x1, x2) = qs_out_ag(x2) U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U9_aa(x1, x2, x3) = U9_aa(x3) U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U9_ga(x1, x2, x3) = U9_ga(x1, x3) U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x1, x6) U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x1, x6) part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa(x1) part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) qs_in_aa(x1, x2) = qs_in_aa qs_out_aa(x1, x2) = qs_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U4_aa(x1, x2, x3, x4) = U4_aa(x4) .(x1, x2) = .(x1, x2) app_in_aaa(x1, x2, x3) = app_in_aaa app_out_aaa(x1, x2, x3) = app_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4) app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2, x3) U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x4, x5) LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (22) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (23) Obligation: Pi DP problem: The TRS P consists of the following rules: LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (24) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (25) Obligation: Q DP problem: The TRS P consists of the following rules: LESS_IN_GA(s(X)) -> LESS_IN_GA(X) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (26) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *LESS_IN_GA(s(X)) -> LESS_IN_GA(X) The graph contains the following edges 1 > 1 ---------------------------------------- (27) YES ---------------------------------------- (28) Obligation: Pi DP problem: The TRS P consists of the following rules: U5_GAAA(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs) PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GAAA(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_GAAA(X, Xs, Ls, Bs) The TRS R consists of the following rules: qs_in_ag([], []) -> qs_out_ag([], []) qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U5_aaaa(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U5_gaaa(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) qs_in_aa([], []) -> qs_out_aa([], []) qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) app_in_aaa([], X, X) -> app_out_aaa([], X, X) app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) app_in_aag([], X, X) -> app_out_aag([], X, X) app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ag(x1, x2) = qs_in_ag(x2) [] = [] qs_out_ag(x1, x2) = qs_out_ag(x2) U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U9_aa(x1, x2, x3) = U9_aa(x3) U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U9_ga(x1, x2, x3) = U9_ga(x1, x3) U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x1, x6) U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x1, x6) part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa(x1) part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) qs_in_aa(x1, x2) = qs_in_aa qs_out_aa(x1, x2) = qs_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U4_aa(x1, x2, x3, x4) = U4_aa(x4) .(x1, x2) = .(x1, x2) app_in_aaa(x1, x2, x3) = app_in_aaa app_out_aaa(x1, x2, x3) = app_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4) app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2, x3) U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x4, x5) PART_IN_GAAA(x1, x2, x3, x4) = PART_IN_GAAA(x1) U5_GAAA(x1, x2, x3, x4, x5, x6) = U5_GAAA(x1, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (29) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (30) Obligation: Pi DP problem: The TRS P consists of the following rules: U5_GAAA(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs) PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GAAA(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_GAAA(X, Xs, Ls, Bs) The TRS R consists of the following rules: less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) The argument filtering Pi contains the following mapping: less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U9_ga(x1, x2, x3) = U9_ga(x1, x3) .(x1, x2) = .(x1, x2) PART_IN_GAAA(x1, x2, x3, x4) = PART_IN_GAAA(x1) U5_GAAA(x1, x2, x3, x4, x5, x6) = U5_GAAA(x1, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (31) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: U5_GAAA(X, less_out_ga(X)) -> PART_IN_GAAA(X) PART_IN_GAAA(X) -> U5_GAAA(X, less_in_ga(X)) PART_IN_GAAA(X) -> PART_IN_GAAA(X) The TRS R consists of the following rules: less_in_ga(0) -> less_out_ga(0) less_in_ga(s(X)) -> U9_ga(X, less_in_ga(X)) U9_ga(X, less_out_ga(X)) -> less_out_ga(s(X)) The set Q consists of the following terms: less_in_ga(x0) U9_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (33) TransformationProof (SOUND) By narrowing [LPAR04] the rule PART_IN_GAAA(X) -> U5_GAAA(X, less_in_ga(X)) at position [1] we obtained the following new rules [LPAR04]: (PART_IN_GAAA(0) -> U5_GAAA(0, less_out_ga(0)),PART_IN_GAAA(0) -> U5_GAAA(0, less_out_ga(0))) (PART_IN_GAAA(s(x0)) -> U5_GAAA(s(x0), U9_ga(x0, less_in_ga(x0))),PART_IN_GAAA(s(x0)) -> U5_GAAA(s(x0), U9_ga(x0, less_in_ga(x0)))) ---------------------------------------- (34) Obligation: Q DP problem: The TRS P consists of the following rules: U5_GAAA(X, less_out_ga(X)) -> PART_IN_GAAA(X) PART_IN_GAAA(X) -> PART_IN_GAAA(X) PART_IN_GAAA(0) -> U5_GAAA(0, less_out_ga(0)) PART_IN_GAAA(s(x0)) -> U5_GAAA(s(x0), U9_ga(x0, less_in_ga(x0))) The TRS R consists of the following rules: less_in_ga(0) -> less_out_ga(0) less_in_ga(s(X)) -> U9_ga(X, less_in_ga(X)) U9_ga(X, less_out_ga(X)) -> less_out_ga(s(X)) The set Q consists of the following terms: less_in_ga(x0) U9_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (35) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U5_GAAA(X, less_out_ga(X)) -> PART_IN_GAAA(X) we obtained the following new rules [LPAR04]: (U5_GAAA(0, less_out_ga(0)) -> PART_IN_GAAA(0),U5_GAAA(0, less_out_ga(0)) -> PART_IN_GAAA(0)) (U5_GAAA(s(z0), less_out_ga(s(z0))) -> PART_IN_GAAA(s(z0)),U5_GAAA(s(z0), less_out_ga(s(z0))) -> PART_IN_GAAA(s(z0))) ---------------------------------------- (36) Obligation: Q DP problem: The TRS P consists of the following rules: PART_IN_GAAA(X) -> PART_IN_GAAA(X) PART_IN_GAAA(0) -> U5_GAAA(0, less_out_ga(0)) PART_IN_GAAA(s(x0)) -> U5_GAAA(s(x0), U9_ga(x0, less_in_ga(x0))) U5_GAAA(0, less_out_ga(0)) -> PART_IN_GAAA(0) U5_GAAA(s(z0), less_out_ga(s(z0))) -> PART_IN_GAAA(s(z0)) The TRS R consists of the following rules: less_in_ga(0) -> less_out_ga(0) less_in_ga(s(X)) -> U9_ga(X, less_in_ga(X)) U9_ga(X, less_out_ga(X)) -> less_out_ga(s(X)) The set Q consists of the following terms: less_in_ga(x0) U9_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (37) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by semiunifying a rule from P directly. s = PART_IN_GAAA(X) evaluates to t =PART_IN_GAAA(X) Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence The DP semiunifies directly so there is only one rewrite step from PART_IN_GAAA(X) to PART_IN_GAAA(X). ---------------------------------------- (38) NO ---------------------------------------- (39) Obligation: Pi DP problem: The TRS P consists of the following rules: LESS_IN_AA(s(X), s(Y)) -> LESS_IN_AA(X, Y) The TRS R consists of the following rules: qs_in_ag([], []) -> qs_out_ag([], []) qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U5_aaaa(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U5_gaaa(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) qs_in_aa([], []) -> qs_out_aa([], []) qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) app_in_aaa([], X, X) -> app_out_aaa([], X, X) app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) app_in_aag([], X, X) -> app_out_aag([], X, X) app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ag(x1, x2) = qs_in_ag(x2) [] = [] qs_out_ag(x1, x2) = qs_out_ag(x2) U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U9_aa(x1, x2, x3) = U9_aa(x3) U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U9_ga(x1, x2, x3) = U9_ga(x1, x3) U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x1, x6) U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x1, x6) part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa(x1) part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) qs_in_aa(x1, x2) = qs_in_aa qs_out_aa(x1, x2) = qs_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U4_aa(x1, x2, x3, x4) = U4_aa(x4) .(x1, x2) = .(x1, x2) app_in_aaa(x1, x2, x3) = app_in_aaa app_out_aaa(x1, x2, x3) = app_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4) app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2, x3) U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x4, x5) LESS_IN_AA(x1, x2) = LESS_IN_AA We have to consider all (P,R,Pi)-chains ---------------------------------------- (40) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (41) Obligation: Pi DP problem: The TRS P consists of the following rules: LESS_IN_AA(s(X), s(Y)) -> LESS_IN_AA(X, Y) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) LESS_IN_AA(x1, x2) = LESS_IN_AA We have to consider all (P,R,Pi)-chains ---------------------------------------- (42) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (43) Obligation: Q DP problem: The TRS P consists of the following rules: LESS_IN_AA -> LESS_IN_AA R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (44) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by semiunifying a rule from P directly. s = LESS_IN_AA evaluates to t =LESS_IN_AA Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence The DP semiunifies directly so there is only one rewrite step from LESS_IN_AA to LESS_IN_AA. ---------------------------------------- (45) NO ---------------------------------------- (46) Obligation: Pi DP problem: The TRS P consists of the following rules: PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_AAAA(X, Xs, Ls, Bs) The TRS R consists of the following rules: qs_in_ag([], []) -> qs_out_ag([], []) qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U5_aaaa(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U5_gaaa(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) qs_in_aa([], []) -> qs_out_aa([], []) qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) app_in_aaa([], X, X) -> app_out_aaa([], X, X) app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) app_in_aag([], X, X) -> app_out_aag([], X, X) app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: qs_in_ag(x1, x2) = qs_in_ag(x2) [] = [] qs_out_ag(x1, x2) = qs_out_ag(x2) U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U9_aa(x1, x2, x3) = U9_aa(x3) U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U9_ga(x1, x2, x3) = U9_ga(x1, x3) U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x1, x6) U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x1, x6) part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa(x1) part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) qs_in_aa(x1, x2) = qs_in_aa qs_out_aa(x1, x2) = qs_out_aa U1_aa(x1, x2, x3, x4) = U1_aa(x4) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U4_aa(x1, x2, x3, x4) = U4_aa(x4) .(x1, x2) = .(x1, x2) app_in_aaa(x1, x2, x3) = app_in_aaa app_out_aaa(x1, x2, x3) = app_out_aaa U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4) app_in_aag(x1, x2, x3) = app_in_aag(x3) app_out_aag(x1, x2, x3) = app_out_aag(x1, x2, x3) U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x4, x5) PART_IN_AAAA(x1, x2, x3, x4) = PART_IN_AAAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (47) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (48) Obligation: Pi DP problem: The TRS P consists of the following rules: PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_AAAA(X, Xs, Ls, Bs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) PART_IN_AAAA(x1, x2, x3, x4) = PART_IN_AAAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (49) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (50) Obligation: Q DP problem: The TRS P consists of the following rules: PART_IN_AAAA -> PART_IN_AAAA R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (51) 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(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 ---------------------------------------- (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(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 ---------------------------------------- (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(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. ---------------------------------------- (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 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 ---------------------------------------- (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 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) ---------------------------------------- (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 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 ---------------------------------------- (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 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 ---------------------------------------- (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 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 ---------------------------------------- (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 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 ---------------------------------------- (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 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 ---------------------------------------- (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 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 ---------------------------------------- (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 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 ---------------------------------------- (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) -> 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. ---------------------------------------- (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),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)))) ---------------------------------------- (91) 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. ---------------------------------------- (92) 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))) ---------------------------------------- (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) 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. ---------------------------------------- (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 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 ---------------------------------------- (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 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 ---------------------------------------- (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 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 ---------------------------------------- (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 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 ---------------------------------------- (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) -> 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. ---------------------------------------- (115) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "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": { "type": "Nodes", "990": { "goal": [{ "clause": 0, "scope": 6, "term": "(qs T17 X18)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X18"], "exprvars": [] } }, "991": { "goal": [{ "clause": 1, "scope": 6, "term": "(qs T17 X18)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X18"], "exprvars": [] } }, "552": { "goal": [{ "clause": 7, "scope": 3, "term": "(less T38 T39)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "750": { "goal": [{ "clause": -1, "scope": -1, "term": "(less T96 T98)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T96"], "free": [], "exprvars": [] } }, "992": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "553": { "goal": [{ "clause": 8, "scope": 3, "term": "(less T38 T39)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "993": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "598": { "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": [] } }, "994": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "753": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "830": { "goal": [{ "clause": 4, "scope": 4, "term": "(part T43 T44 X67 X68)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T43"], "free": [ "X67", "X68" ], "exprvars": [] } }, "57": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "58": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "59": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1110": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1072": { "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": [] } }, "560": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1107": { "goal": [{ "clause": 5, "scope": 8, "term": "(app T209 (. 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T174 T172) X261)" }, { "clause": 6, "scope": 7, "term": "(app T173 (. T174 T172) X261)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X261"], "exprvars": [] } }, "1087": { "goal": [{ "clause": -1, "scope": -1, "term": "(app T173 (. T174 T172) X261)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X261"], "exprvars": [] } }, "1086": { "goal": [{ "clause": -1, "scope": -1, "term": "(qs T170 X260)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X260"], "exprvars": [] } }, "1083": { "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": [] } }, "1082": { "goal": [{ "clause": -1, "scope": -1, "term": "(qs T163 X259)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X259"], "exprvars": [] } }, "1081": { "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": [] } }, "1117": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "574": { "goal": [{ "clause": -1, "scope": -1, "term": "(less T58 T59)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1116": { "goal": [{ "clause": -1, "scope": -1, "term": "(app T240 (. T241 T242) T239)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T239"], "free": [], "exprvars": [] } }, "577": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "732": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1079": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "733": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1111": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "734": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "537": { "goal": [{ "clause": -1, "scope": -1, "term": "(less T38 T39)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "538": { "goal": [{ "clause": -1, "scope": -1, "term": "(part T43 T44 X67 X68)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T43"], "free": [ "X67", "X68" ], "exprvars": [] } }, "934": { "goal": [{ "clause": 3, "scope": 2, "term": "(part T12 T13 X16 X17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X16", "X17" ], "exprvars": [] } }, "978": { "goal": [{ "clause": -1, "scope": -1, "term": "(part T141 T142 X230 X231)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X230", "X231" ], "exprvars": [] } }, "935": { "goal": [{ "clause": 4, "scope": 2, "term": "(part T12 T13 X16 X17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X16", "X17" ], "exprvars": [] } }, "618": { "goal": [{ "clause": 2, "scope": 4, "term": "(part T43 T44 X67 X68)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T43"], "free": [ "X67", "X68" ], "exprvars": [] } }, "619": { "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": [] } }, "1090": { "goal": [{ "clause": 6, "scope": 7, "term": "(app T173 (. T174 T172) X261)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X261"], "exprvars": [] } }, "1097": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1096": { "goal": [{ "clause": -1, "scope": -1, "term": "(app T201 (. T202 T203) X297)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X297"], "exprvars": [] } }, "1095": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1094": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1093": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "981": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "982": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "620": { "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": [] } }, "983": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "621": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "984": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "622": { "goal": [{ "clause": -1, "scope": -1, "term": "(less T77 T80)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T77"], "free": [], "exprvars": [] } }, "985": { "goal": [{ "clause": -1, "scope": -1, "term": "(qs T17 X18)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X18"], "exprvars": [] } }, "623": { "goal": [{ "clause": -1, "scope": -1, "term": "(part T77 T84 X126 X127)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T77"], "free": [ "X126", "X127" ], "exprvars": [] } }, "986": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (qs T150 X19) (app T149 (. T151 X19) T11))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T11"], "free": ["X19"], "exprvars": [] } }, "1089": { "goal": [{ "clause": 5, "scope": 7, "term": "(app T173 (. T174 T172) X261)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X261"], "exprvars": [] } }, "503": { "goal": [{ "clause": -1, "scope": -1, "term": "(part T12 T13 X16 X17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X16", "X17" ], "exprvars": [] } }, "547": { "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": [] } }, "624": { "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": [] } }, "504": { "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": [] } }, "625": { "goal": [{ "clause": 7, "scope": 5, "term": "(less T77 T80)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T77"], "free": [], "exprvars": [] } }, "505": { "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": [] } }, "626": { "goal": [{ "clause": 8, "scope": 5, "term": "(less T77 T80)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T77"], "free": [], "exprvars": [] } }, "989": { "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": [] } }, "506": { "goal": [{ "clause": 2, "scope": 2, "term": "(part T12 T13 X16 X17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X16", "X17" ], "exprvars": [] } }, "507": { "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": [] } }, "829": { "goal": [{ "clause": 3, "scope": 4, "term": "(part T43 T44 X67 X68)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T43"], "free": [ "X67", "X68" ], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 8, "label": "CASE" }, { "from": 8, "to": 28, "label": "PARALLEL" }, { "from": 8, "to": 29, "label": "PARALLEL" }, { "from": 28, "to": 57, "label": "EVAL with clause\nqs([], []).\nand substitutionT1 -> [],\nT2 -> []" }, { "from": 28, "to": 58, "label": "EVAL-BACKTRACK" }, { "from": 29, "to": 200, "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": 29, "to": 205, "label": "EVAL-BACKTRACK" }, { "from": 57, "to": 59, "label": "SUCCESS" }, { "from": 200, "to": 503, "label": "SPLIT 1" }, { "from": 200, "to": 504, "label": "SPLIT 2\nreplacements:X16 -> T17,\nX17 -> T18,\nT12 -> T19" }, { "from": 503, "to": 505, "label": "CASE" }, { "from": 504, "to": 985, "label": "SPLIT 1" }, { "from": 504, "to": 986, "label": "SPLIT 2\nreplacements:X18 -> T149,\nT18 -> T150,\nT19 -> T151" }, { "from": 505, "to": 506, "label": "PARALLEL" }, { "from": 505, "to": 507, "label": "PARALLEL" }, { "from": 506, "to": 524, "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": 506, "to": 527, "label": "EVAL-BACKTRACK" }, { "from": 507, "to": 934, "label": "PARALLEL" }, { "from": 507, "to": 935, "label": "PARALLEL" }, { "from": 524, "to": 537, "label": "SPLIT 1" }, { "from": 524, "to": 538, "label": "SPLIT 2\nnew knowledge:\nT43 is ground\nreplacements:T38 -> T43,\nT40 -> T44" }, { "from": 537, "to": 547, "label": "CASE" }, { "from": 538, "to": 598, "label": "CASE" }, { "from": 547, "to": 552, "label": "PARALLEL" }, { "from": 547, "to": 553, "label": "PARALLEL" }, { "from": 552, "to": 560, "label": "EVAL with clause\nless(0, s(X77)).\nand substitutionT38 -> 0,\nX77 -> T51,\nT39 -> s(T51)" }, { "from": 552, "to": 562, "label": "EVAL-BACKTRACK" }, { "from": 553, "to": 574, "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": 553, "to": 577, "label": "EVAL-BACKTRACK" }, { "from": 560, "to": 563, "label": "SUCCESS" }, { "from": 574, "to": 537, "label": "INSTANCE with matching:\nT38 -> T58\nT39 -> T59" }, { "from": 598, "to": 618, "label": "PARALLEL" }, { "from": 598, "to": 619, "label": "PARALLEL" }, { "from": 618, "to": 620, "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": 618, "to": 621, "label": "EVAL-BACKTRACK" }, { "from": 619, "to": 829, "label": "PARALLEL" }, { "from": 619, "to": 830, "label": "PARALLEL" }, { "from": 620, "to": 622, "label": "SPLIT 1" }, { "from": 620, "to": 623, "label": "SPLIT 2\nnew knowledge:\nT77 is ground\nreplacements:T81 -> T84" }, { "from": 622, "to": 624, "label": "CASE" }, { "from": 623, "to": 538, "label": "INSTANCE with matching:\nT43 -> T77\nT44 -> T84\nX67 -> X126\nX68 -> X127" }, { "from": 624, "to": 625, "label": "PARALLEL" }, { "from": 624, "to": 626, "label": "PARALLEL" }, { "from": 625, "to": 732, "label": "EVAL with clause\nless(0, s(X136)).\nand substitutionT77 -> 0,\nX136 -> T91,\nT80 -> s(T91)" }, { "from": 625, "to": 733, "label": "EVAL-BACKTRACK" }, { "from": 626, "to": 750, "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": 626, "to": 753, "label": "EVAL-BACKTRACK" }, { "from": 732, "to": 734, "label": "SUCCESS" }, { "from": 750, "to": 622, "label": "INSTANCE with matching:\nT77 -> T96\nT80 -> T98" }, { "from": 829, "to": 881, "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": 829, "to": 882, "label": "EVAL-BACKTRACK" }, { "from": 830, "to": 886, "label": "EVAL with clause\npart(X196, [], [], []).\nand substitutionT43 -> T125,\nX196 -> T125,\nT44 -> [],\nX67 -> [],\nX68 -> []" }, { "from": 830, "to": 887, "label": "EVAL-BACKTRACK" }, { "from": 881, "to": 538, "label": "INSTANCE with matching:\nT43 -> T116\nT44 -> T119\nX67 -> X185\nX68 -> X186" }, { "from": 886, "to": 888, "label": "SUCCESS" }, { "from": 934, "to": 978, "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": 934, "to": 981, "label": "EVAL-BACKTRACK" }, { "from": 935, "to": 982, "label": "EVAL with clause\npart(X241, [], [], []).\nand substitutionT12 -> T148,\nX241 -> T148,\nT13 -> [],\nX16 -> [],\nX17 -> []" }, { "from": 935, "to": 983, "label": "EVAL-BACKTRACK" }, { "from": 978, "to": 503, "label": "INSTANCE with matching:\nT12 -> T141\nT13 -> T142\nX16 -> X230\nX17 -> X231" }, { "from": 982, "to": 984, "label": "SUCCESS" }, { "from": 985, "to": 989, "label": "CASE" }, { "from": 986, "to": 1104, "label": "SPLIT 1" }, { "from": 986, "to": 1105, "label": "SPLIT 2\nreplacements:X19 -> T208,\nT149 -> T209,\nT151 -> T210" }, { "from": 989, "to": 990, "label": "PARALLEL" }, { "from": 989, "to": 991, "label": "PARALLEL" }, { "from": 990, "to": 992, "label": "EVAL with clause\nqs([], []).\nand substitutionT17 -> [],\nX18 -> []" }, { "from": 990, "to": 993, "label": "EVAL-BACKTRACK" }, { "from": 991, "to": 1072, "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": 991, "to": 1079, "label": "EVAL-BACKTRACK" }, { "from": 992, "to": 994, "label": "SUCCESS" }, { "from": 1072, "to": 1080, "label": "SPLIT 1" }, { "from": 1072, "to": 1081, "label": "SPLIT 2\nreplacements:X257 -> T163,\nX258 -> T164,\nT158 -> T165" }, { "from": 1080, "to": 503, "label": "INSTANCE with matching:\nT12 -> T158\nT13 -> T159\nX16 -> X257\nX17 -> X258" }, { "from": 1081, "to": 1082, "label": "SPLIT 1" }, { "from": 1081, "to": 1083, "label": "SPLIT 2\nreplacements:X259 -> T169,\nT164 -> T170,\nT165 -> T171" }, { "from": 1082, "to": 985, "label": "INSTANCE with matching:\nT17 -> T163\nX18 -> X259" }, { "from": 1083, "to": 1086, "label": "SPLIT 1" }, { "from": 1083, "to": 1087, "label": "SPLIT 2\nreplacements:X260 -> T172,\nT169 -> T173,\nT171 -> T174" }, { "from": 1086, "to": 985, "label": "INSTANCE with matching:\nT17 -> T170\nX18 -> X260" }, { "from": 1087, "to": 1088, "label": "CASE" }, { "from": 1088, "to": 1089, "label": "PARALLEL" }, { "from": 1088, "to": 1090, "label": "PARALLEL" }, { "from": 1089, "to": 1093, "label": "EVAL with clause\napp([], X282, X282).\nand substitutionT173 -> [],\nT174 -> T187,\nT172 -> T188,\nX282 -> .(T187, T188),\nX261 -> .(T187, T188)" }, { "from": 1089, "to": 1094, "label": "EVAL-BACKTRACK" }, { "from": 1090, "to": 1096, "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": 1090, "to": 1097, "label": "EVAL-BACKTRACK" }, { "from": 1093, "to": 1095, "label": "SUCCESS" }, { "from": 1096, "to": 1087, "label": "INSTANCE with matching:\nT173 -> T201\nT174 -> T202\nT172 -> T203\nX261 -> X297" }, { "from": 1104, "to": 985, "label": "INSTANCE with matching:\nT17 -> T150\nX18 -> X19" }, { "from": 1105, "to": 1106, "label": "CASE" }, { "from": 1106, "to": 1107, "label": "PARALLEL" }, { "from": 1106, "to": 1108, "label": "PARALLEL" }, { "from": 1107, "to": 1109, "label": "EVAL with clause\napp([], X306, X306).\nand substitutionT209 -> [],\nT210 -> T223,\nT208 -> T224,\nX306 -> .(T223, T224),\nT11 -> .(T223, T224)" }, { "from": 1107, "to": 1110, "label": "EVAL-BACKTRACK" }, { "from": 1108, "to": 1116, "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": 1108, "to": 1117, "label": "EVAL-BACKTRACK" }, { "from": 1109, "to": 1111, "label": "SUCCESS" }, { "from": 1116, "to": 1105, "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: f1_in([]) -> f1_out1 f1_in(T11) -> U1(f200_in(T11), T11) U1(f200_out1(X18, T12, X19), T11) -> f1_out1 f537_in -> f537_out1(0) f537_in -> U2(f537_in) U2(f537_out1(T58)) -> f537_out1(s(T58)) f538_in(T77) -> U3(f620_in(T77), T77) U3(f620_out1, T77) -> f538_out1 f538_in(T116) -> U4(f538_in(T116), T116) U4(f538_out1, T116) -> f538_out1 f538_in(T125) -> f538_out1 f622_in(0) -> f622_out1 f622_in(s(T96)) -> U5(f622_in(T96), s(T96)) U5(f622_out1, s(T96)) -> f622_out1 f503_in -> U6(f524_in) U6(f524_out1(T38)) -> f503_out1 f503_in -> U7(f503_in) U7(f503_out1) -> f503_out1 f503_in -> f503_out1 f985_in -> f985_out1 f985_in -> U8(f1072_in) U8(f1072_out1) -> f985_out1 f1087_in -> f1087_out1 f1087_in -> U9(f1087_in) U9(f1087_out1) -> f1087_out1 f1105_in(.(T223, T224)) -> f1105_out1([], T223, T224) f1105_in(.(T235, T239)) -> U10(f1105_in(T239), .(T235, T239)) U10(f1105_out1(T240, T241, T242), .(T235, T239)) -> f1105_out1(.(T235, T240), T241, T242) f200_in(T11) -> U11(f503_in, T11) U11(f503_out1, T11) -> U12(f504_in(T11), T11) U12(f504_out1(X18, T19, X19), T11) -> f200_out1(X18, T19, X19) f504_in(T11) -> U13(f985_in, T11) U13(f985_out1, T11) -> U14(f986_in(T11), T11) U14(f986_out1(T149, T151, X19), T11) -> f504_out1(T149, T151, X19) f524_in -> U15(f537_in) U15(f537_out1(T43)) -> U16(f538_in(T43), T43) U16(f538_out1, T43) -> f524_out1(T43) f620_in(T77) -> U17(f622_in(T77), T77) U17(f622_out1, T77) -> U18(f538_in(T77), T77) U18(f538_out1, T77) -> f620_out1 f986_in(T11) -> U19(f985_in, T11) U19(f985_out1, T11) -> U20(f1105_in(T11), T11) U20(f1105_out1(T209, T210, T208), T11) -> f986_out1(T209, T210, T208) f1072_in -> U21(f503_in) U21(f503_out1) -> U22(f1081_in) U22(f1081_out1) -> f1072_out1 f1081_in -> U23(f985_in) U23(f985_out1) -> U24(f1083_in) U24(f1083_out1) -> f1081_out1 f1083_in -> U25(f985_in) U25(f985_out1) -> U26(f1087_in) U26(f1087_out1) -> f1083_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: F1_IN(T11) -> U1^1(f200_in(T11), T11) F1_IN(T11) -> F200_IN(T11) F537_IN -> U2^1(f537_in) F537_IN -> F537_IN F538_IN(T77) -> U3^1(f620_in(T77), T77) F538_IN(T77) -> F620_IN(T77) F538_IN(T116) -> U4^1(f538_in(T116), T116) F538_IN(T116) -> F538_IN(T116) F622_IN(s(T96)) -> U5^1(f622_in(T96), s(T96)) F622_IN(s(T96)) -> F622_IN(T96) F503_IN -> U6^1(f524_in) F503_IN -> F524_IN F503_IN -> U7^1(f503_in) F503_IN -> F503_IN F985_IN -> U8^1(f1072_in) F985_IN -> F1072_IN F1087_IN -> U9^1(f1087_in) F1087_IN -> F1087_IN F1105_IN(.(T235, T239)) -> U10^1(f1105_in(T239), .(T235, T239)) F1105_IN(.(T235, T239)) -> F1105_IN(T239) F200_IN(T11) -> U11^1(f503_in, T11) F200_IN(T11) -> F503_IN U11^1(f503_out1, T11) -> U12^1(f504_in(T11), T11) U11^1(f503_out1, T11) -> F504_IN(T11) F504_IN(T11) -> U13^1(f985_in, T11) F504_IN(T11) -> F985_IN U13^1(f985_out1, T11) -> U14^1(f986_in(T11), T11) U13^1(f985_out1, T11) -> F986_IN(T11) F524_IN -> U15^1(f537_in) F524_IN -> F537_IN U15^1(f537_out1(T43)) -> U16^1(f538_in(T43), T43) U15^1(f537_out1(T43)) -> F538_IN(T43) F620_IN(T77) -> U17^1(f622_in(T77), T77) F620_IN(T77) -> F622_IN(T77) U17^1(f622_out1, T77) -> U18^1(f538_in(T77), T77) U17^1(f622_out1, T77) -> F538_IN(T77) F986_IN(T11) -> U19^1(f985_in, T11) F986_IN(T11) -> F985_IN U19^1(f985_out1, T11) -> U20^1(f1105_in(T11), T11) U19^1(f985_out1, T11) -> F1105_IN(T11) F1072_IN -> U21^1(f503_in) F1072_IN -> F503_IN U21^1(f503_out1) -> U22^1(f1081_in) U21^1(f503_out1) -> F1081_IN F1081_IN -> U23^1(f985_in) F1081_IN -> F985_IN U23^1(f985_out1) -> U24^1(f1083_in) U23^1(f985_out1) -> F1083_IN F1083_IN -> U25^1(f985_in) F1083_IN -> F985_IN U25^1(f985_out1) -> U26^1(f1087_in) U25^1(f985_out1) -> F1087_IN The TRS R consists of the following rules: f1_in([]) -> f1_out1 f1_in(T11) -> U1(f200_in(T11), T11) U1(f200_out1(X18, T12, X19), T11) -> f1_out1 f537_in -> f537_out1(0) f537_in -> U2(f537_in) U2(f537_out1(T58)) -> f537_out1(s(T58)) f538_in(T77) -> U3(f620_in(T77), T77) U3(f620_out1, T77) -> f538_out1 f538_in(T116) -> U4(f538_in(T116), T116) U4(f538_out1, T116) -> f538_out1 f538_in(T125) -> f538_out1 f622_in(0) -> f622_out1 f622_in(s(T96)) -> U5(f622_in(T96), s(T96)) U5(f622_out1, s(T96)) -> f622_out1 f503_in -> U6(f524_in) U6(f524_out1(T38)) -> f503_out1 f503_in -> U7(f503_in) U7(f503_out1) -> f503_out1 f503_in -> f503_out1 f985_in -> f985_out1 f985_in -> U8(f1072_in) U8(f1072_out1) -> f985_out1 f1087_in -> f1087_out1 f1087_in -> U9(f1087_in) U9(f1087_out1) -> f1087_out1 f1105_in(.(T223, T224)) -> f1105_out1([], T223, T224) f1105_in(.(T235, T239)) -> U10(f1105_in(T239), .(T235, T239)) U10(f1105_out1(T240, T241, T242), .(T235, T239)) -> f1105_out1(.(T235, T240), T241, T242) f200_in(T11) -> U11(f503_in, T11) U11(f503_out1, T11) -> U12(f504_in(T11), T11) U12(f504_out1(X18, T19, X19), T11) -> f200_out1(X18, T19, X19) f504_in(T11) -> U13(f985_in, T11) U13(f985_out1, T11) -> U14(f986_in(T11), T11) U14(f986_out1(T149, T151, X19), T11) -> f504_out1(T149, T151, X19) f524_in -> U15(f537_in) U15(f537_out1(T43)) -> U16(f538_in(T43), T43) U16(f538_out1, T43) -> f524_out1(T43) f620_in(T77) -> U17(f622_in(T77), T77) U17(f622_out1, T77) -> U18(f538_in(T77), T77) U18(f538_out1, T77) -> f620_out1 f986_in(T11) -> U19(f985_in, T11) U19(f985_out1, T11) -> U20(f1105_in(T11), T11) U20(f1105_out1(T209, T210, T208), T11) -> f986_out1(T209, T210, T208) f1072_in -> U21(f503_in) U21(f503_out1) -> U22(f1081_in) U22(f1081_out1) -> f1072_out1 f1081_in -> U23(f985_in) U23(f985_out1) -> U24(f1083_in) U24(f1083_out1) -> f1081_out1 f1083_in -> U25(f985_in) U25(f985_out1) -> U26(f1087_in) U26(f1087_out1) -> f1083_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: F1105_IN(.(T235, T239)) -> F1105_IN(T239) The TRS R consists of the following rules: f1_in([]) -> f1_out1 f1_in(T11) -> U1(f200_in(T11), T11) U1(f200_out1(X18, T12, X19), T11) -> f1_out1 f537_in -> f537_out1(0) f537_in -> U2(f537_in) U2(f537_out1(T58)) -> f537_out1(s(T58)) f538_in(T77) -> U3(f620_in(T77), T77) U3(f620_out1, T77) -> f538_out1 f538_in(T116) -> U4(f538_in(T116), T116) U4(f538_out1, T116) -> f538_out1 f538_in(T125) -> f538_out1 f622_in(0) -> f622_out1 f622_in(s(T96)) -> U5(f622_in(T96), s(T96)) U5(f622_out1, s(T96)) -> f622_out1 f503_in -> U6(f524_in) U6(f524_out1(T38)) -> f503_out1 f503_in -> U7(f503_in) U7(f503_out1) -> f503_out1 f503_in -> f503_out1 f985_in -> f985_out1 f985_in -> U8(f1072_in) U8(f1072_out1) -> f985_out1 f1087_in -> f1087_out1 f1087_in -> U9(f1087_in) U9(f1087_out1) -> f1087_out1 f1105_in(.(T223, T224)) -> f1105_out1([], T223, T224) f1105_in(.(T235, T239)) -> U10(f1105_in(T239), .(T235, T239)) U10(f1105_out1(T240, T241, T242), .(T235, T239)) -> f1105_out1(.(T235, T240), T241, T242) f200_in(T11) -> U11(f503_in, T11) U11(f503_out1, T11) -> U12(f504_in(T11), T11) U12(f504_out1(X18, T19, X19), T11) -> f200_out1(X18, T19, X19) f504_in(T11) -> U13(f985_in, T11) U13(f985_out1, T11) -> U14(f986_in(T11), T11) U14(f986_out1(T149, T151, X19), T11) -> f504_out1(T149, T151, X19) f524_in -> U15(f537_in) U15(f537_out1(T43)) -> U16(f538_in(T43), T43) U16(f538_out1, T43) -> f524_out1(T43) f620_in(T77) -> U17(f622_in(T77), T77) U17(f622_out1, T77) -> U18(f538_in(T77), T77) U18(f538_out1, T77) -> f620_out1 f986_in(T11) -> U19(f985_in, T11) U19(f985_out1, T11) -> U20(f1105_in(T11), T11) U20(f1105_out1(T209, T210, T208), T11) -> f986_out1(T209, T210, T208) f1072_in -> U21(f503_in) U21(f503_out1) -> U22(f1081_in) U22(f1081_out1) -> f1072_out1 f1081_in -> U23(f985_in) U23(f985_out1) -> U24(f1083_in) U24(f1083_out1) -> f1081_out1 f1083_in -> U25(f985_in) U25(f985_out1) -> U26(f1087_in) U26(f1087_out1) -> f1083_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: F1105_IN(.(T235, T239)) -> F1105_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: *F1105_IN(.(T235, T239)) -> F1105_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: F1087_IN -> F1087_IN The TRS R consists of the following rules: f1_in([]) -> f1_out1 f1_in(T11) -> U1(f200_in(T11), T11) U1(f200_out1(X18, T12, X19), T11) -> f1_out1 f537_in -> f537_out1(0) f537_in -> U2(f537_in) U2(f537_out1(T58)) -> f537_out1(s(T58)) f538_in(T77) -> U3(f620_in(T77), T77) U3(f620_out1, T77) -> f538_out1 f538_in(T116) -> U4(f538_in(T116), T116) U4(f538_out1, T116) -> f538_out1 f538_in(T125) -> f538_out1 f622_in(0) -> f622_out1 f622_in(s(T96)) -> U5(f622_in(T96), s(T96)) U5(f622_out1, s(T96)) -> f622_out1 f503_in -> U6(f524_in) U6(f524_out1(T38)) -> f503_out1 f503_in -> U7(f503_in) U7(f503_out1) -> f503_out1 f503_in -> f503_out1 f985_in -> f985_out1 f985_in -> U8(f1072_in) U8(f1072_out1) -> f985_out1 f1087_in -> f1087_out1 f1087_in -> U9(f1087_in) U9(f1087_out1) -> f1087_out1 f1105_in(.(T223, T224)) -> f1105_out1([], T223, T224) f1105_in(.(T235, T239)) -> U10(f1105_in(T239), .(T235, T239)) U10(f1105_out1(T240, T241, T242), .(T235, T239)) -> f1105_out1(.(T235, T240), T241, T242) f200_in(T11) -> U11(f503_in, T11) U11(f503_out1, T11) -> U12(f504_in(T11), T11) U12(f504_out1(X18, T19, X19), T11) -> f200_out1(X18, T19, X19) f504_in(T11) -> U13(f985_in, T11) U13(f985_out1, T11) -> U14(f986_in(T11), T11) U14(f986_out1(T149, T151, X19), T11) -> f504_out1(T149, T151, X19) f524_in -> U15(f537_in) U15(f537_out1(T43)) -> U16(f538_in(T43), T43) U16(f538_out1, T43) -> f524_out1(T43) f620_in(T77) -> U17(f622_in(T77), T77) U17(f622_out1, T77) -> U18(f538_in(T77), T77) U18(f538_out1, T77) -> f620_out1 f986_in(T11) -> U19(f985_in, T11) U19(f985_out1, T11) -> U20(f1105_in(T11), T11) U20(f1105_out1(T209, T210, T208), T11) -> f986_out1(T209, T210, T208) f1072_in -> U21(f503_in) U21(f503_out1) -> U22(f1081_in) U22(f1081_out1) -> f1072_out1 f1081_in -> U23(f985_in) U23(f985_out1) -> U24(f1083_in) U24(f1083_out1) -> f1081_out1 f1083_in -> U25(f985_in) U25(f985_out1) -> U26(f1087_in) U26(f1087_out1) -> f1083_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: F1087_IN -> F1087_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 = F1087_IN evaluates to t =F1087_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 F1087_IN to F1087_IN. ---------------------------------------- (130) NO ---------------------------------------- (131) Obligation: Q DP problem: The TRS P consists of the following rules: F622_IN(s(T96)) -> F622_IN(T96) The TRS R consists of the following rules: f1_in([]) -> f1_out1 f1_in(T11) -> U1(f200_in(T11), T11) U1(f200_out1(X18, T12, X19), T11) -> f1_out1 f537_in -> f537_out1(0) f537_in -> U2(f537_in) U2(f537_out1(T58)) -> f537_out1(s(T58)) f538_in(T77) -> U3(f620_in(T77), T77) U3(f620_out1, T77) -> f538_out1 f538_in(T116) -> U4(f538_in(T116), T116) U4(f538_out1, T116) -> f538_out1 f538_in(T125) -> f538_out1 f622_in(0) -> f622_out1 f622_in(s(T96)) -> U5(f622_in(T96), s(T96)) U5(f622_out1, s(T96)) -> f622_out1 f503_in -> U6(f524_in) U6(f524_out1(T38)) -> f503_out1 f503_in -> U7(f503_in) U7(f503_out1) -> f503_out1 f503_in -> f503_out1 f985_in -> f985_out1 f985_in -> U8(f1072_in) U8(f1072_out1) -> f985_out1 f1087_in -> f1087_out1 f1087_in -> U9(f1087_in) U9(f1087_out1) -> f1087_out1 f1105_in(.(T223, T224)) -> f1105_out1([], T223, T224) f1105_in(.(T235, T239)) -> U10(f1105_in(T239), .(T235, T239)) U10(f1105_out1(T240, T241, T242), .(T235, T239)) -> f1105_out1(.(T235, T240), T241, T242) f200_in(T11) -> U11(f503_in, T11) U11(f503_out1, T11) -> U12(f504_in(T11), T11) U12(f504_out1(X18, T19, X19), T11) -> f200_out1(X18, T19, X19) f504_in(T11) -> U13(f985_in, T11) U13(f985_out1, T11) -> U14(f986_in(T11), T11) U14(f986_out1(T149, T151, X19), T11) -> f504_out1(T149, T151, X19) f524_in -> U15(f537_in) U15(f537_out1(T43)) -> U16(f538_in(T43), T43) U16(f538_out1, T43) -> f524_out1(T43) f620_in(T77) -> U17(f622_in(T77), T77) U17(f622_out1, T77) -> U18(f538_in(T77), T77) U18(f538_out1, T77) -> f620_out1 f986_in(T11) -> U19(f985_in, T11) U19(f985_out1, T11) -> U20(f1105_in(T11), T11) U20(f1105_out1(T209, T210, T208), T11) -> f986_out1(T209, T210, T208) f1072_in -> U21(f503_in) U21(f503_out1) -> U22(f1081_in) U22(f1081_out1) -> f1072_out1 f1081_in -> U23(f985_in) U23(f985_out1) -> U24(f1083_in) U24(f1083_out1) -> f1081_out1 f1083_in -> U25(f985_in) U25(f985_out1) -> U26(f1087_in) U26(f1087_out1) -> f1083_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: F622_IN(s(T96)) -> F622_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: *F622_IN(s(T96)) -> F622_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: F538_IN(T77) -> F620_IN(T77) F620_IN(T77) -> U17^1(f622_in(T77), T77) U17^1(f622_out1, T77) -> F538_IN(T77) F538_IN(T116) -> F538_IN(T116) The TRS R consists of the following rules: f1_in([]) -> f1_out1 f1_in(T11) -> U1(f200_in(T11), T11) U1(f200_out1(X18, T12, X19), T11) -> f1_out1 f537_in -> f537_out1(0) f537_in -> U2(f537_in) U2(f537_out1(T58)) -> f537_out1(s(T58)) f538_in(T77) -> U3(f620_in(T77), T77) U3(f620_out1, T77) -> f538_out1 f538_in(T116) -> U4(f538_in(T116), T116) U4(f538_out1, T116) -> f538_out1 f538_in(T125) -> f538_out1 f622_in(0) -> f622_out1 f622_in(s(T96)) -> U5(f622_in(T96), s(T96)) U5(f622_out1, s(T96)) -> f622_out1 f503_in -> U6(f524_in) U6(f524_out1(T38)) -> f503_out1 f503_in -> U7(f503_in) U7(f503_out1) -> f503_out1 f503_in -> f503_out1 f985_in -> f985_out1 f985_in -> U8(f1072_in) U8(f1072_out1) -> f985_out1 f1087_in -> f1087_out1 f1087_in -> U9(f1087_in) U9(f1087_out1) -> f1087_out1 f1105_in(.(T223, T224)) -> f1105_out1([], T223, T224) f1105_in(.(T235, T239)) -> U10(f1105_in(T239), .(T235, T239)) U10(f1105_out1(T240, T241, T242), .(T235, T239)) -> f1105_out1(.(T235, T240), T241, T242) f200_in(T11) -> U11(f503_in, T11) U11(f503_out1, T11) -> U12(f504_in(T11), T11) U12(f504_out1(X18, T19, X19), T11) -> f200_out1(X18, T19, X19) f504_in(T11) -> U13(f985_in, T11) U13(f985_out1, T11) -> U14(f986_in(T11), T11) U14(f986_out1(T149, T151, X19), T11) -> f504_out1(T149, T151, X19) f524_in -> U15(f537_in) U15(f537_out1(T43)) -> U16(f538_in(T43), T43) U16(f538_out1, T43) -> f524_out1(T43) f620_in(T77) -> U17(f622_in(T77), T77) U17(f622_out1, T77) -> U18(f538_in(T77), T77) U18(f538_out1, T77) -> f620_out1 f986_in(T11) -> U19(f985_in, T11) U19(f985_out1, T11) -> U20(f1105_in(T11), T11) U20(f1105_out1(T209, T210, T208), T11) -> f986_out1(T209, T210, T208) f1072_in -> U21(f503_in) U21(f503_out1) -> U22(f1081_in) U22(f1081_out1) -> f1072_out1 f1081_in -> U23(f985_in) U23(f985_out1) -> U24(f1083_in) U24(f1083_out1) -> f1081_out1 f1083_in -> U25(f985_in) U25(f985_out1) -> U26(f1087_in) U26(f1087_out1) -> f1083_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 = F538_IN(T116) evaluates to t =F538_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 F538_IN(T116) to F538_IN(T116). ---------------------------------------- (138) NO ---------------------------------------- (139) Obligation: Q DP problem: The TRS P consists of the following rules: F537_IN -> F537_IN The TRS R consists of the following rules: f1_in([]) -> f1_out1 f1_in(T11) -> U1(f200_in(T11), T11) U1(f200_out1(X18, T12, X19), T11) -> f1_out1 f537_in -> f537_out1(0) f537_in -> U2(f537_in) U2(f537_out1(T58)) -> f537_out1(s(T58)) f538_in(T77) -> U3(f620_in(T77), T77) U3(f620_out1, T77) -> f538_out1 f538_in(T116) -> U4(f538_in(T116), T116) U4(f538_out1, T116) -> f538_out1 f538_in(T125) -> f538_out1 f622_in(0) -> f622_out1 f622_in(s(T96)) -> U5(f622_in(T96), s(T96)) U5(f622_out1, s(T96)) -> f622_out1 f503_in -> U6(f524_in) U6(f524_out1(T38)) -> f503_out1 f503_in -> U7(f503_in) U7(f503_out1) -> f503_out1 f503_in -> f503_out1 f985_in -> f985_out1 f985_in -> U8(f1072_in) U8(f1072_out1) -> f985_out1 f1087_in -> f1087_out1 f1087_in -> U9(f1087_in) U9(f1087_out1) -> f1087_out1 f1105_in(.(T223, T224)) -> f1105_out1([], T223, T224) f1105_in(.(T235, T239)) -> U10(f1105_in(T239), .(T235, T239)) U10(f1105_out1(T240, T241, T242), .(T235, T239)) -> f1105_out1(.(T235, T240), T241, T242) f200_in(T11) -> U11(f503_in, T11) U11(f503_out1, T11) -> U12(f504_in(T11), T11) U12(f504_out1(X18, T19, X19), T11) -> f200_out1(X18, T19, X19) f504_in(T11) -> U13(f985_in, T11) U13(f985_out1, T11) -> U14(f986_in(T11), T11) U14(f986_out1(T149, T151, X19), T11) -> f504_out1(T149, T151, X19) f524_in -> U15(f537_in) U15(f537_out1(T43)) -> U16(f538_in(T43), T43) U16(f538_out1, T43) -> f524_out1(T43) f620_in(T77) -> U17(f622_in(T77), T77) U17(f622_out1, T77) -> U18(f538_in(T77), T77) U18(f538_out1, T77) -> f620_out1 f986_in(T11) -> U19(f985_in, T11) U19(f985_out1, T11) -> U20(f1105_in(T11), T11) U20(f1105_out1(T209, T210, T208), T11) -> f986_out1(T209, T210, T208) f1072_in -> U21(f503_in) U21(f503_out1) -> U22(f1081_in) U22(f1081_out1) -> f1072_out1 f1081_in -> U23(f985_in) U23(f985_out1) -> U24(f1083_in) U24(f1083_out1) -> f1081_out1 f1083_in -> U25(f985_in) U25(f985_out1) -> U26(f1087_in) U26(f1087_out1) -> f1083_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: F537_IN -> F537_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 = F537_IN evaluates to t =F537_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 F537_IN to F537_IN. ---------------------------------------- (143) NO ---------------------------------------- (144) Obligation: Q DP problem: The TRS P consists of the following rules: F503_IN -> F503_IN The TRS R consists of the following rules: f1_in([]) -> f1_out1 f1_in(T11) -> U1(f200_in(T11), T11) U1(f200_out1(X18, T12, X19), T11) -> f1_out1 f537_in -> f537_out1(0) f537_in -> U2(f537_in) U2(f537_out1(T58)) -> f537_out1(s(T58)) f538_in(T77) -> U3(f620_in(T77), T77) U3(f620_out1, T77) -> f538_out1 f538_in(T116) -> U4(f538_in(T116), T116) U4(f538_out1, T116) -> f538_out1 f538_in(T125) -> f538_out1 f622_in(0) -> f622_out1 f622_in(s(T96)) -> U5(f622_in(T96), s(T96)) U5(f622_out1, s(T96)) -> f622_out1 f503_in -> U6(f524_in) U6(f524_out1(T38)) -> f503_out1 f503_in -> U7(f503_in) U7(f503_out1) -> f503_out1 f503_in -> f503_out1 f985_in -> f985_out1 f985_in -> U8(f1072_in) U8(f1072_out1) -> f985_out1 f1087_in -> f1087_out1 f1087_in -> U9(f1087_in) U9(f1087_out1) -> f1087_out1 f1105_in(.(T223, T224)) -> f1105_out1([], T223, T224) f1105_in(.(T235, T239)) -> U10(f1105_in(T239), .(T235, T239)) U10(f1105_out1(T240, T241, T242), .(T235, T239)) -> f1105_out1(.(T235, T240), T241, T242) f200_in(T11) -> U11(f503_in, T11) U11(f503_out1, T11) -> U12(f504_in(T11), T11) U12(f504_out1(X18, T19, X19), T11) -> f200_out1(X18, T19, X19) f504_in(T11) -> U13(f985_in, T11) U13(f985_out1, T11) -> U14(f986_in(T11), T11) U14(f986_out1(T149, T151, X19), T11) -> f504_out1(T149, T151, X19) f524_in -> U15(f537_in) U15(f537_out1(T43)) -> U16(f538_in(T43), T43) U16(f538_out1, T43) -> f524_out1(T43) f620_in(T77) -> U17(f622_in(T77), T77) U17(f622_out1, T77) -> U18(f538_in(T77), T77) U18(f538_out1, T77) -> f620_out1 f986_in(T11) -> U19(f985_in, T11) U19(f985_out1, T11) -> U20(f1105_in(T11), T11) U20(f1105_out1(T209, T210, T208), T11) -> f986_out1(T209, T210, T208) f1072_in -> U21(f503_in) U21(f503_out1) -> U22(f1081_in) U22(f1081_out1) -> f1072_out1 f1081_in -> U23(f985_in) U23(f985_out1) -> U24(f1083_in) U24(f1083_out1) -> f1081_out1 f1083_in -> U25(f985_in) U25(f985_out1) -> U26(f1087_in) U26(f1087_out1) -> f1083_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: F503_IN -> F503_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 = F503_IN evaluates to t =F503_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 F503_IN to F503_IN. ---------------------------------------- (148) NO ---------------------------------------- (149) Obligation: Q DP problem: The TRS P consists of the following rules: F1072_IN -> U21^1(f503_in) U21^1(f503_out1) -> F1081_IN F1081_IN -> U23^1(f985_in) U23^1(f985_out1) -> F1083_IN F1083_IN -> F985_IN F985_IN -> F1072_IN F1081_IN -> F985_IN The TRS R consists of the following rules: f1_in([]) -> f1_out1 f1_in(T11) -> U1(f200_in(T11), T11) U1(f200_out1(X18, T12, X19), T11) -> f1_out1 f537_in -> f537_out1(0) f537_in -> U2(f537_in) U2(f537_out1(T58)) -> f537_out1(s(T58)) f538_in(T77) -> U3(f620_in(T77), T77) U3(f620_out1, T77) -> f538_out1 f538_in(T116) -> U4(f538_in(T116), T116) U4(f538_out1, T116) -> f538_out1 f538_in(T125) -> f538_out1 f622_in(0) -> f622_out1 f622_in(s(T96)) -> U5(f622_in(T96), s(T96)) U5(f622_out1, s(T96)) -> f622_out1 f503_in -> U6(f524_in) U6(f524_out1(T38)) -> f503_out1 f503_in -> U7(f503_in) U7(f503_out1) -> f503_out1 f503_in -> f503_out1 f985_in -> f985_out1 f985_in -> U8(f1072_in) U8(f1072_out1) -> f985_out1 f1087_in -> f1087_out1 f1087_in -> U9(f1087_in) U9(f1087_out1) -> f1087_out1 f1105_in(.(T223, T224)) -> f1105_out1([], T223, T224) f1105_in(.(T235, T239)) -> U10(f1105_in(T239), .(T235, T239)) U10(f1105_out1(T240, T241, T242), .(T235, T239)) -> f1105_out1(.(T235, T240), T241, T242) f200_in(T11) -> U11(f503_in, T11) U11(f503_out1, T11) -> U12(f504_in(T11), T11) U12(f504_out1(X18, T19, X19), T11) -> f200_out1(X18, T19, X19) f504_in(T11) -> U13(f985_in, T11) U13(f985_out1, T11) -> U14(f986_in(T11), T11) U14(f986_out1(T149, T151, X19), T11) -> f504_out1(T149, T151, X19) f524_in -> U15(f537_in) U15(f537_out1(T43)) -> U16(f538_in(T43), T43) U16(f538_out1, T43) -> f524_out1(T43) f620_in(T77) -> U17(f622_in(T77), T77) U17(f622_out1, T77) -> U18(f538_in(T77), T77) U18(f538_out1, T77) -> f620_out1 f986_in(T11) -> U19(f985_in, T11) U19(f985_out1, T11) -> U20(f1105_in(T11), T11) U20(f1105_out1(T209, T210, T208), T11) -> f986_out1(T209, T210, T208) f1072_in -> U21(f503_in) U21(f503_out1) -> U22(f1081_in) U22(f1081_out1) -> f1072_out1 f1081_in -> U23(f985_in) U23(f985_out1) -> U24(f1083_in) U24(f1083_out1) -> f1081_out1 f1083_in -> U25(f985_in) U25(f985_out1) -> U26(f1087_in) U26(f1087_out1) -> f1083_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 F1081_IN[ ]^n[ ] -> F1081_IN[ ]^n[ ] This rule is correct for the QDP as the following derivation shows: F1081_IN[ ]^n[ ] -> F1081_IN[ ]^n[ ] by Narrowing at position: [] F1081_IN[ ]^n[ ] -> F985_IN[ ]^n[ ] by Rule from TRS P F985_IN[ ]^n[ ] -> F1081_IN[ ]^n[ ] by Narrowing at position: [] F985_IN[ ]^n[ ] -> F1072_IN[ ]^n[ ] by Rule from TRS P F1072_IN[ ]^n[ ] -> F1081_IN[ ]^n[ ] by Narrowing at position: [] F1072_IN[ ]^n[ ] -> U21^1(f503_out1)[ ]^n[ ] by Narrowing at position: [0] F1072_IN[ ]^n[ ] -> U21^1(f503_in)[ ]^n[ ] by Rule from TRS P f503_in[ ]^n[ ] -> f503_out1[ ]^n[ ] by Rule from TRS R U21^1(f503_out1)[ ]^n[ ] -> F1081_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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T210 T208) T11)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T11"], "free": [], "exprvars": [] } }, "62": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "969": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "927": { "goal": [{ "clause": -1, "scope": -1, "term": "(part T141 T142 X230 X231)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X230", "X231" ], "exprvars": [] } }, "928": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "22": { "goal": [{ "clause": 1, "scope": 1, "term": "(qs T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "929": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "450": { "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": [] } }, "692": { "goal": [{ "clause": -1, "scope": -1, "term": "(less T96 T98)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T96"], "free": [], "exprvars": [] } }, "890": { "goal": [{ "clause": 4, "scope": 2, "term": "(part T12 T13 X16 X17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X16", "X17" ], "exprvars": [] } }, "693": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "970": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "410": { "goal": [{ "clause": 2, "scope": 2, "term": "(part T12 T13 X16 X17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X16", "X17" ], "exprvars": [] } }, "971": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "411": { "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": [] } }, "455": { "goal": [{ "clause": 7, "scope": 3, "term": "(less T38 T39)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "972": { "goal": [{ "clause": -1, "scope": -1, "term": "(app T240 (. T241 T242) T239)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T239"], "free": [], "exprvars": [] } }, "852": { "goal": [{ "clause": 3, "scope": 4, "term": "(part T43 T44 X67 X68)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T43"], "free": [ "X67", "X68" ], "exprvars": [] } }, "973": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "457": { "goal": [{ "clause": 8, "scope": 3, "term": "(less T38 T39)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "853": { "goal": [{ "clause": 4, "scope": 4, "term": "(part T43 T44 X67 X68)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T43"], "free": [ "X67", "X68" ], "exprvars": [] } }, "930": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "931": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "932": { "goal": [{ "clause": -1, "scope": -1, "term": "(qs T17 X18)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X18"], "exprvars": [] } }, "933": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (qs T150 X19) (app T149 (. T151 X19) T11))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T11"], "free": ["X19"], "exprvars": [] } }, "936": { "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": [] } }, "937": { "goal": [{ "clause": 0, "scope": 6, "term": "(qs T17 X18)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X18"], "exprvars": [] } }, "938": { "goal": [{ "clause": 1, "scope": 6, "term": "(qs T17 X18)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X18"], "exprvars": [] } }, "939": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "464": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "467": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "940": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "468": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "941": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "943": { "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": [] } }, "944": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "429": { "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": [] } }, "947": { "goal": [{ "clause": -1, "scope": -1, "term": "(part T158 T159 X257 X258)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X257", "X258" ], "exprvars": [] } }, "948": { "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": [] } } }, "edges": [ { "from": 3, "to": 4, "label": "CASE" }, { "from": 4, "to": 18, "label": "PARALLEL" }, { "from": 4, "to": 22, "label": "PARALLEL" }, { "from": 18, "to": 52, "label": "EVAL with clause\nqs([], []).\nand substitutionT1 -> [],\nT2 -> []" }, { "from": 18, "to": 53, "label": "EVAL-BACKTRACK" }, { "from": 22, "to": 61, "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": 22, "to": 62, "label": "EVAL-BACKTRACK" }, { "from": 52, "to": 54, "label": "SUCCESS" }, { "from": 61, "to": 391, "label": "SPLIT 1" }, { "from": 61, "to": 392, "label": "SPLIT 2\nreplacements:X16 -> T17,\nX17 -> T18,\nT12 -> T19" }, { "from": 391, "to": 399, "label": "CASE" }, { "from": 392, "to": 932, "label": "SPLIT 1" }, { "from": 392, "to": 933, "label": "SPLIT 2\nreplacements:X18 -> T149,\nT18 -> T150,\nT19 -> T151" }, { "from": 399, "to": 410, "label": "PARALLEL" }, { "from": 399, "to": 411, "label": "PARALLEL" }, { "from": 410, "to": 429, "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": 410, "to": 432, "label": "EVAL-BACKTRACK" }, { "from": 411, "to": 889, "label": "PARALLEL" }, { "from": 411, "to": 890, "label": "PARALLEL" }, { "from": 429, "to": 442, "label": "SPLIT 1" }, { "from": 429, "to": 443, "label": "SPLIT 2\nnew knowledge:\nT43 is ground\nreplacements:T38 -> T43,\nT40 -> T44" }, { "from": 442, "to": 450, "label": "CASE" }, { "from": 443, "to": 672, "label": "CASE" }, { "from": 450, "to": 455, "label": "PARALLEL" }, { "from": 450, "to": 457, "label": "PARALLEL" }, { "from": 455, "to": 464, "label": "EVAL with clause\nless(0, s(X77)).\nand substitutionT38 -> 0,\nX77 -> T51,\nT39 -> s(T51)" }, { "from": 455, "to": 467, "label": "EVAL-BACKTRACK" }, { "from": 457, "to": 479, "label": "EVAL with clause\nless(s(X82), s(X83)) :- less(X82, X83).\nand substitutionX82 -> T58,\nT38 -> s(T58),\nX83 -> T59,\nT39 -> s(T59),\nT56 -> T58,\nT57 -> T59" }, { "from": 457, "to": 481, "label": "EVAL-BACKTRACK" }, { "from": 464, "to": 468, "label": "SUCCESS" }, { "from": 479, "to": 442, "label": "INSTANCE with matching:\nT38 -> T58\nT39 -> T59" }, { "from": 672, "to": 673, "label": "PARALLEL" }, { "from": 672, "to": 674, "label": "PARALLEL" }, { "from": 673, "to": 677, "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": 673, "to": 678, "label": "EVAL-BACKTRACK" }, { "from": 674, "to": 852, "label": "PARALLEL" }, { "from": 674, "to": 853, "label": "PARALLEL" }, { "from": 677, "to": 680, "label": "SPLIT 1" }, { "from": 677, "to": 681, "label": "SPLIT 2\nnew knowledge:\nT77 is ground\nreplacements:T81 -> T84" }, { "from": 680, "to": 683, "label": "CASE" }, { "from": 681, "to": 443, "label": "INSTANCE with matching:\nT43 -> T77\nT44 -> T84\nX67 -> X126\nX68 -> X127" }, { "from": 683, "to": 684, "label": "PARALLEL" }, { "from": 683, "to": 685, "label": "PARALLEL" }, { "from": 684, "to": 686, "label": "EVAL with clause\nless(0, s(X136)).\nand substitutionT77 -> 0,\nX136 -> T91,\nT80 -> s(T91)" }, { "from": 684, "to": 687, "label": "EVAL-BACKTRACK" }, { "from": 685, "to": 692, "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": 685, "to": 693, "label": "EVAL-BACKTRACK" }, { "from": 686, "to": 689, "label": "SUCCESS" }, { "from": 692, "to": 680, "label": "INSTANCE with matching:\nT77 -> T96\nT80 -> T98" }, { "from": 852, "to": 879, "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": 852, "to": 880, "label": "EVAL-BACKTRACK" }, { "from": 853, "to": 883, "label": "EVAL with clause\npart(X196, [], [], []).\nand substitutionT43 -> T125,\nX196 -> T125,\nT44 -> [],\nX67 -> [],\nX68 -> []" }, { "from": 853, "to": 884, "label": "EVAL-BACKTRACK" }, { "from": 879, "to": 443, "label": "INSTANCE with matching:\nT43 -> T116\nT44 -> T119\nX67 -> X185\nX68 -> X186" }, { "from": 883, "to": 885, "label": "SUCCESS" }, { "from": 889, "to": 927, "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": 889, "to": 928, "label": "EVAL-BACKTRACK" }, { "from": 890, "to": 929, "label": "EVAL with clause\npart(X241, [], [], []).\nand substitutionT12 -> T148,\nX241 -> T148,\nT13 -> [],\nX16 -> [],\nX17 -> []" }, { "from": 890, "to": 930, "label": "EVAL-BACKTRACK" }, { "from": 927, "to": 391, "label": "INSTANCE with matching:\nT12 -> T141\nT13 -> T142\nX16 -> X230\nX17 -> X231" }, { "from": 929, "to": 931, "label": "SUCCESS" }, { "from": 932, "to": 936, "label": "CASE" }, { "from": 933, "to": 964, "label": "SPLIT 1" }, { "from": 933, "to": 965, "label": "SPLIT 2\nreplacements:X19 -> T208,\nT149 -> T209,\nT151 -> T210" }, { "from": 936, "to": 937, "label": "PARALLEL" }, { "from": 936, "to": 938, "label": "PARALLEL" }, { "from": 937, "to": 939, "label": "EVAL with clause\nqs([], []).\nand substitutionT17 -> [],\nX18 -> []" }, { "from": 937, "to": 940, "label": "EVAL-BACKTRACK" }, { "from": 938, "to": 943, "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": 938, "to": 944, "label": "EVAL-BACKTRACK" }, { "from": 939, "to": 941, "label": "SUCCESS" }, { "from": 943, "to": 947, "label": "SPLIT 1" }, { "from": 943, "to": 948, "label": "SPLIT 2\nreplacements:X257 -> T163,\nX258 -> T164,\nT158 -> T165" }, { "from": 947, "to": 391, "label": "INSTANCE with matching:\nT12 -> T158\nT13 -> T159\nX16 -> X257\nX17 -> X258" }, { "from": 948, "to": 952, "label": "SPLIT 1" }, { "from": 948, "to": 953, "label": "SPLIT 2\nreplacements:X259 -> T169,\nT164 -> T170,\nT165 -> T171" }, { "from": 952, "to": 932, "label": "INSTANCE with matching:\nT17 -> T163\nX18 -> X259" }, { "from": 953, "to": 954, "label": "SPLIT 1" }, { "from": 953, "to": 955, "label": "SPLIT 2\nreplacements:X260 -> T172,\nT169 -> T173,\nT171 -> T174" }, { "from": 954, "to": 932, "label": "INSTANCE with matching:\nT17 -> T170\nX18 -> X260" }, { "from": 955, "to": 956, "label": "CASE" }, { "from": 956, "to": 957, "label": "PARALLEL" }, { "from": 956, "to": 958, "label": "PARALLEL" }, { "from": 957, "to": 959, "label": "EVAL with clause\napp([], X282, X282).\nand substitutionT173 -> [],\nT174 -> T187,\nT172 -> T188,\nX282 -> .(T187, T188),\nX261 -> .(T187, T188)" }, { "from": 957, "to": 960, "label": "EVAL-BACKTRACK" }, { "from": 958, "to": 962, "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": 958, "to": 963, "label": "EVAL-BACKTRACK" }, { "from": 959, "to": 961, "label": "SUCCESS" }, { "from": 962, "to": 955, "label": "INSTANCE with matching:\nT173 -> T201\nT174 -> T202\nT172 -> T203\nX261 -> X297" }, { "from": 964, "to": 932, "label": "INSTANCE with matching:\nT17 -> T150\nX18 -> X19" }, { "from": 965, "to": 966, "label": "CASE" }, { "from": 966, "to": 967, "label": "PARALLEL" }, { "from": 966, "to": 968, "label": "PARALLEL" }, { "from": 967, "to": 969, "label": "EVAL with clause\napp([], X306, X306).\nand substitutionT209 -> [],\nT210 -> T223,\nT208 -> T224,\nX306 -> .(T223, T224),\nT11 -> .(T223, T224)" }, { "from": 967, "to": 970, "label": "EVAL-BACKTRACK" }, { "from": 968, "to": 972, "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": 968, "to": 973, "label": "EVAL-BACKTRACK" }, { "from": 969, "to": 971, "label": "SUCCESS" }, { "from": 972, "to": 965, "label": "INSTANCE with matching:\nT209 -> T240\nT210 -> T241\nT208 -> T242\nT11 -> T239" } ], "type": "Graph" } } ---------------------------------------- (153) Complex Obligation (AND) ---------------------------------------- (154) Obligation: Rules: f972_out(T239) -> f968_out(.(T235, T239)) :|: TRUE f968_in(.(x, x1)) -> f972_in(x1) :|: TRUE f968_in(T11) -> f973_in :|: TRUE f973_out -> f968_out(x2) :|: TRUE f965_in(x3) -> f966_in(x3) :|: TRUE f966_out(x4) -> f965_out(x4) :|: TRUE f966_in(x5) -> f967_in(x5) :|: TRUE f966_in(x6) -> f968_in(x6) :|: TRUE f968_out(x7) -> f966_out(x7) :|: TRUE f967_out(x8) -> f966_out(x8) :|: TRUE f965_out(x9) -> f972_out(x9) :|: TRUE f972_in(x10) -> f965_in(x10) :|: TRUE f4_out(T2) -> f3_out(T2) :|: TRUE f3_in(x11) -> f4_in(x11) :|: TRUE f4_in(x12) -> f18_in(x12) :|: TRUE f4_in(x13) -> f22_in(x13) :|: TRUE f18_out(x14) -> f4_out(x14) :|: TRUE f22_out(x15) -> f4_out(x15) :|: TRUE f22_in(x16) -> f61_in(x16) :|: TRUE f22_in(x17) -> f62_in :|: TRUE f62_out -> f22_out(x18) :|: TRUE f61_out(x19) -> f22_out(x19) :|: TRUE f61_in(x20) -> f391_in :|: TRUE f392_out(x21) -> f61_out(x21) :|: TRUE f391_out -> f392_in(x22) :|: TRUE f933_out(x23) -> f392_out(x23) :|: TRUE f932_out -> f933_in(x24) :|: TRUE f392_in(x25) -> f932_in :|: TRUE f933_in(x26) -> f964_in :|: TRUE f964_out -> f965_in(x27) :|: TRUE f965_out(x28) -> f933_out(x28) :|: TRUE Start term: f3_in(T2) ---------------------------------------- (155) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (156) TRUE ---------------------------------------- (157) Obligation: Rules: f958_in -> f963_in :|: TRUE f962_out -> f958_out :|: TRUE f958_in -> f962_in :|: TRUE f963_out -> f958_out :|: TRUE f956_out -> f955_out :|: TRUE f955_in -> f956_in :|: TRUE f962_in -> f955_in :|: TRUE f955_out -> f962_out :|: TRUE f957_out -> f956_out :|: TRUE f956_in -> f958_in :|: TRUE f958_out -> f956_out :|: TRUE f956_in -> f957_in :|: TRUE f4_out(T2) -> f3_out(T2) :|: TRUE f3_in(x) -> f4_in(x) :|: TRUE f4_in(x1) -> f18_in(x1) :|: TRUE f4_in(x2) -> f22_in(x2) :|: TRUE f18_out(x3) -> f4_out(x3) :|: TRUE f22_out(x4) -> f4_out(x4) :|: TRUE f22_in(T11) -> f61_in(T11) :|: TRUE f22_in(x5) -> f62_in :|: TRUE f62_out -> f22_out(x6) :|: TRUE f61_out(x7) -> f22_out(x7) :|: TRUE f61_in(x8) -> f391_in :|: TRUE f392_out(x9) -> f61_out(x9) :|: TRUE f391_out -> f392_in(x10) :|: TRUE f933_out(x11) -> f392_out(x11) :|: TRUE f932_out -> f933_in(x12) :|: TRUE f392_in(x13) -> f932_in :|: TRUE f932_in -> f936_in :|: TRUE f936_out -> f932_out :|: TRUE f938_out -> f936_out :|: TRUE f937_out -> f936_out :|: TRUE f936_in -> f938_in :|: TRUE f936_in -> f937_in :|: TRUE f938_in -> f944_in :|: TRUE f944_out -> f938_out :|: TRUE f943_out -> f938_out :|: TRUE f938_in -> f943_in :|: TRUE f947_out -> f948_in :|: TRUE f943_in -> f947_in :|: TRUE f948_out -> f943_out :|: TRUE f953_out -> f948_out :|: TRUE f952_out -> f953_in :|: TRUE f948_in -> f952_in :|: TRUE f953_in -> f954_in :|: TRUE f955_out -> f953_out :|: TRUE f954_out -> f955_in :|: TRUE f933_in(x14) -> f964_in :|: TRUE f964_out -> f965_in(x15) :|: TRUE f965_out(x16) -> f933_out(x16) :|: TRUE f964_in -> f932_in :|: TRUE f932_out -> f964_out :|: TRUE Start term: f3_in(T2) ---------------------------------------- (158) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (159) TRUE ---------------------------------------- (160) Obligation: Rules: f683_in(T77) -> f684_in(T77) :|: TRUE f685_out(x) -> f683_out(x) :|: TRUE f683_in(x1) -> f685_in(x1) :|: TRUE f684_out(x2) -> f683_out(x2) :|: TRUE f685_in(s(T96)) -> f692_in(T96) :|: TRUE f692_out(x3) -> f685_out(s(x3)) :|: TRUE f685_in(x4) -> f693_in :|: TRUE f693_out -> f685_out(x5) :|: TRUE f680_out(x6) -> f692_out(x6) :|: TRUE f692_in(x7) -> f680_in(x7) :|: TRUE f683_out(x8) -> f680_out(x8) :|: TRUE f680_in(x9) -> f683_in(x9) :|: TRUE f4_out(T2) -> f3_out(T2) :|: TRUE f3_in(x10) -> f4_in(x10) :|: TRUE f4_in(x11) -> f18_in(x11) :|: TRUE f4_in(x12) -> f22_in(x12) :|: TRUE f18_out(x13) -> f4_out(x13) :|: TRUE f22_out(x14) -> f4_out(x14) :|: TRUE f22_in(T11) -> f61_in(T11) :|: TRUE f22_in(x15) -> f62_in :|: TRUE f62_out -> f22_out(x16) :|: TRUE f61_out(x17) -> f22_out(x17) :|: TRUE f61_in(x18) -> f391_in :|: TRUE f392_out(x19) -> f61_out(x19) :|: TRUE f391_out -> f392_in(x20) :|: TRUE f933_out(x21) -> f392_out(x21) :|: TRUE f932_out -> f933_in(x22) :|: TRUE f392_in(x23) -> f932_in :|: TRUE f932_in -> f936_in :|: TRUE f936_out -> f932_out :|: TRUE f938_out -> f936_out :|: TRUE f937_out -> f936_out :|: TRUE f936_in -> f938_in :|: TRUE f936_in -> f937_in :|: TRUE f938_in -> f944_in :|: TRUE f944_out -> f938_out :|: TRUE f943_out -> f938_out :|: TRUE f938_in -> f943_in :|: TRUE f947_out -> f948_in :|: TRUE f943_in -> f947_in :|: TRUE f948_out -> f943_out :|: TRUE f391_out -> f947_out :|: TRUE f947_in -> f391_in :|: TRUE f399_out -> f391_out :|: TRUE f391_in -> f399_in :|: TRUE f399_in -> f411_in :|: TRUE f411_out -> f399_out :|: TRUE f399_in -> f410_in :|: TRUE f410_out -> f399_out :|: TRUE f410_in -> f432_in :|: TRUE f410_in -> f429_in :|: TRUE f432_out -> f410_out :|: TRUE f429_out -> f410_out :|: TRUE f443_out(T43) -> f429_out :|: TRUE f429_in -> f442_in :|: TRUE f442_out -> f443_in(x24) :|: TRUE f672_out(x25) -> f443_out(x25) :|: TRUE f443_in(x26) -> f672_in(x26) :|: TRUE f672_in(x27) -> f673_in(x27) :|: TRUE f674_out(x28) -> f672_out(x28) :|: TRUE f673_out(x29) -> f672_out(x29) :|: TRUE f672_in(x30) -> f674_in(x30) :|: TRUE f673_in(x31) -> f678_in :|: TRUE f678_out -> f673_out(x32) :|: TRUE f677_out(x33) -> f673_out(x33) :|: TRUE f673_in(x34) -> f677_in(x34) :|: TRUE f681_out(x35) -> f677_out(x35) :|: TRUE f677_in(x36) -> f680_in(x36) :|: TRUE f680_out(x37) -> f681_in(x37) :|: TRUE f933_in(x38) -> f964_in :|: TRUE f964_out -> f965_in(x39) :|: TRUE f965_out(x40) -> f933_out(x40) :|: TRUE f964_in -> f932_in :|: TRUE f932_out -> f964_out :|: TRUE Start term: f3_in(T2) ---------------------------------------- (161) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (162) TRUE ---------------------------------------- (163) Obligation: Rules: f683_in(T77) -> f684_in(T77) :|: TRUE f685_out(x) -> f683_out(x) :|: TRUE f683_in(x1) -> f685_in(x1) :|: TRUE f684_out(x2) -> f683_out(x2) :|: TRUE f672_in(T43) -> f673_in(T43) :|: TRUE f674_out(x3) -> f672_out(x3) :|: TRUE f673_out(x4) -> f672_out(x4) :|: TRUE f672_in(x5) -> f674_in(x5) :|: TRUE f673_in(x6) -> f678_in :|: TRUE f678_out -> f673_out(x7) :|: TRUE f677_out(x8) -> f673_out(x8) :|: TRUE f673_in(x9) -> f677_in(x9) :|: TRUE f852_in(T116) -> f879_in(T116) :|: TRUE f879_out(x10) -> f852_out(x10) :|: TRUE f880_out -> f852_out(x11) :|: TRUE f852_in(x12) -> f880_in :|: TRUE f443_out(x13) -> f681_out(x13) :|: TRUE f681_in(x14) -> f443_in(x14) :|: TRUE f879_in(x15) -> f443_in(x15) :|: TRUE f443_out(x16) -> f879_out(x16) :|: TRUE f674_in(x17) -> f852_in(x17) :|: TRUE f674_in(x18) -> f853_in(x18) :|: TRUE f853_out(x19) -> f674_out(x19) :|: TRUE f852_out(x20) -> f674_out(x20) :|: TRUE f672_out(x21) -> f443_out(x21) :|: TRUE f443_in(x22) -> f672_in(x22) :|: TRUE f687_out -> f684_out(x23) :|: TRUE f684_in(0) -> f686_in :|: TRUE f684_in(x24) -> f687_in :|: TRUE f686_out -> f684_out(0) :|: TRUE f681_out(x25) -> f677_out(x25) :|: TRUE f677_in(x26) -> f680_in(x26) :|: TRUE f680_out(x27) -> f681_in(x27) :|: TRUE f686_in -> f686_out :|: TRUE f685_in(s(T96)) -> f692_in(T96) :|: TRUE f692_out(x28) -> f685_out(s(x28)) :|: TRUE f685_in(x29) -> f693_in :|: TRUE f693_out -> f685_out(x30) :|: TRUE f683_out(x31) -> f680_out(x31) :|: TRUE f680_in(x32) -> f683_in(x32) :|: TRUE f680_out(x33) -> f692_out(x33) :|: TRUE f692_in(x34) -> f680_in(x34) :|: TRUE f4_out(T2) -> f3_out(T2) :|: TRUE f3_in(x35) -> f4_in(x35) :|: TRUE f4_in(x36) -> f18_in(x36) :|: TRUE f4_in(x37) -> f22_in(x37) :|: TRUE f18_out(x38) -> f4_out(x38) :|: TRUE f22_out(x39) -> f4_out(x39) :|: TRUE f22_in(T11) -> f61_in(T11) :|: TRUE f22_in(x40) -> f62_in :|: TRUE f62_out -> f22_out(x41) :|: TRUE f61_out(x42) -> f22_out(x42) :|: TRUE f61_in(x43) -> f391_in :|: TRUE f392_out(x44) -> f61_out(x44) :|: TRUE f391_out -> f392_in(x45) :|: TRUE f399_out -> f391_out :|: TRUE f391_in -> f399_in :|: TRUE f399_in -> f411_in :|: TRUE f411_out -> f399_out :|: TRUE f399_in -> f410_in :|: TRUE f410_out -> f399_out :|: TRUE f410_in -> f432_in :|: TRUE f410_in -> f429_in :|: TRUE f432_out -> f410_out :|: TRUE f429_out -> f410_out :|: TRUE f443_out(x46) -> f429_out :|: TRUE f429_in -> f442_in :|: TRUE f442_out -> f443_in(x47) :|: TRUE f933_out(x48) -> f392_out(x48) :|: TRUE f932_out -> f933_in(x49) :|: TRUE f392_in(x50) -> f932_in :|: TRUE f932_in -> f936_in :|: TRUE f936_out -> f932_out :|: TRUE f938_out -> f936_out :|: TRUE f937_out -> f936_out :|: TRUE f936_in -> f938_in :|: TRUE f936_in -> f937_in :|: TRUE f938_in -> f944_in :|: TRUE f944_out -> f938_out :|: TRUE f943_out -> f938_out :|: TRUE f938_in -> f943_in :|: TRUE f947_out -> f948_in :|: TRUE f943_in -> f947_in :|: TRUE f948_out -> f943_out :|: TRUE f391_out -> f947_out :|: TRUE f947_in -> f391_in :|: TRUE f933_in(x51) -> f964_in :|: TRUE f964_out -> f965_in(x52) :|: TRUE f965_out(x53) -> f933_out(x53) :|: TRUE f964_in -> f932_in :|: TRUE f932_out -> f964_out :|: TRUE Start term: f3_in(T2) ---------------------------------------- (164) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (165) TRUE ---------------------------------------- (166) Obligation: Rules: f481_out -> f457_out :|: TRUE f457_in -> f481_in :|: TRUE f479_out -> f457_out :|: TRUE f457_in -> f479_in :|: TRUE f442_out -> f479_out :|: TRUE f479_in -> f442_in :|: TRUE f450_out -> f442_out :|: TRUE f442_in -> f450_in :|: TRUE f457_out -> f450_out :|: TRUE f455_out -> f450_out :|: TRUE f450_in -> f457_in :|: TRUE f450_in -> f455_in :|: TRUE f4_out(T2) -> f3_out(T2) :|: TRUE f3_in(x) -> f4_in(x) :|: TRUE f4_in(x1) -> f18_in(x1) :|: TRUE f4_in(x2) -> f22_in(x2) :|: TRUE f18_out(x3) -> f4_out(x3) :|: TRUE f22_out(x4) -> f4_out(x4) :|: TRUE f22_in(T11) -> f61_in(T11) :|: TRUE f22_in(x5) -> f62_in :|: TRUE f62_out -> f22_out(x6) :|: TRUE f61_out(x7) -> f22_out(x7) :|: TRUE f61_in(x8) -> f391_in :|: TRUE f392_out(x9) -> f61_out(x9) :|: TRUE f391_out -> f392_in(x10) :|: TRUE f933_out(x11) -> f392_out(x11) :|: TRUE f932_out -> f933_in(x12) :|: TRUE f392_in(x13) -> f932_in :|: TRUE f933_in(x14) -> f964_in :|: TRUE f964_out -> f965_in(x15) :|: TRUE f965_out(x16) -> f933_out(x16) :|: TRUE f964_in -> f932_in :|: TRUE f932_out -> f964_out :|: TRUE f932_in -> f936_in :|: TRUE f936_out -> f932_out :|: TRUE f938_out -> f936_out :|: TRUE f937_out -> f936_out :|: TRUE f936_in -> f938_in :|: TRUE f936_in -> f937_in :|: TRUE f938_in -> f944_in :|: TRUE f944_out -> f938_out :|: TRUE f943_out -> f938_out :|: TRUE f938_in -> f943_in :|: TRUE f947_out -> f948_in :|: TRUE f943_in -> f947_in :|: TRUE f948_out -> f943_out :|: TRUE f391_out -> f947_out :|: TRUE f947_in -> f391_in :|: TRUE f399_out -> f391_out :|: TRUE f391_in -> f399_in :|: TRUE f399_in -> f411_in :|: TRUE f411_out -> f399_out :|: TRUE f399_in -> f410_in :|: TRUE f410_out -> f399_out :|: TRUE f410_in -> f432_in :|: TRUE f410_in -> f429_in :|: TRUE f432_out -> f410_out :|: TRUE f429_out -> f410_out :|: TRUE f443_out(T43) -> f429_out :|: TRUE f429_in -> f442_in :|: TRUE f442_out -> f443_in(x17) :|: TRUE Start term: f3_in(T2) ---------------------------------------- (167) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f457_in -> f479_in :|: TRUE f479_in -> f442_in :|: TRUE f442_in -> f450_in :|: TRUE f450_in -> f457_in :|: TRUE ---------------------------------------- (168) Obligation: Rules: f457_in -> f479_in :|: TRUE f479_in -> f442_in :|: TRUE f442_in -> f450_in :|: TRUE f450_in -> f457_in :|: TRUE ---------------------------------------- (169) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (170) Obligation: Rules: f457_in -> f457_in :|: TRUE ---------------------------------------- (171) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (172) Obligation: Rules: f457_in -> f457_in :|: TRUE ---------------------------------------- (173) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f457_in -> f457_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (174) Obligation: Termination digraph: Nodes: (1) f457_in -> f457_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (175) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f457_in() Replaced non-predefined constructor symbols by 0. ---------------------------------------- (176) Obligation: Rules: f457_in -> f457_in :|: TRUE ---------------------------------------- (177) 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)) ---------------------------------------- (178) NO ---------------------------------------- (179) Obligation: Rules: f399_in -> f411_in :|: TRUE f411_out -> f399_out :|: TRUE f399_in -> f410_in :|: TRUE f410_out -> f399_out :|: TRUE f890_out -> f411_out :|: TRUE f411_in -> f889_in :|: TRUE f889_out -> f411_out :|: TRUE f411_in -> f890_in :|: TRUE f889_in -> f928_in :|: TRUE f928_out -> f889_out :|: TRUE f889_in -> f927_in :|: TRUE f927_out -> f889_out :|: TRUE f399_out -> f391_out :|: TRUE f391_in -> f399_in :|: TRUE f391_out -> f927_out :|: TRUE f927_in -> f391_in :|: TRUE f4_out(T2) -> f3_out(T2) :|: TRUE f3_in(x) -> f4_in(x) :|: TRUE f4_in(x1) -> f18_in(x1) :|: TRUE f4_in(x2) -> f22_in(x2) :|: TRUE f18_out(x3) -> f4_out(x3) :|: TRUE f22_out(x4) -> f4_out(x4) :|: TRUE f22_in(T11) -> f61_in(T11) :|: TRUE f22_in(x5) -> f62_in :|: TRUE f62_out -> f22_out(x6) :|: TRUE f61_out(x7) -> f22_out(x7) :|: TRUE f61_in(x8) -> f391_in :|: TRUE f392_out(x9) -> f61_out(x9) :|: TRUE f391_out -> f392_in(x10) :|: TRUE f933_out(x11) -> f392_out(x11) :|: TRUE f932_out -> f933_in(x12) :|: TRUE f392_in(x13) -> f932_in :|: TRUE f932_in -> f936_in :|: TRUE f936_out -> f932_out :|: TRUE f938_out -> f936_out :|: TRUE f937_out -> f936_out :|: TRUE f936_in -> f938_in :|: TRUE f936_in -> f937_in :|: TRUE f938_in -> f944_in :|: TRUE f944_out -> f938_out :|: TRUE f943_out -> f938_out :|: TRUE f938_in -> f943_in :|: TRUE f947_out -> f948_in :|: TRUE f943_in -> f947_in :|: TRUE f948_out -> f943_out :|: TRUE f391_out -> f947_out :|: TRUE f947_in -> f391_in :|: TRUE f933_in(x14) -> f964_in :|: TRUE f964_out -> f965_in(x15) :|: TRUE f965_out(x16) -> f933_out(x16) :|: TRUE f964_in -> f932_in :|: TRUE f932_out -> f964_out :|: TRUE Start term: f3_in(T2) ---------------------------------------- (180) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f399_in -> f411_in :|: TRUE f411_in -> f889_in :|: TRUE f889_in -> f927_in :|: TRUE f391_in -> f399_in :|: TRUE f927_in -> f391_in :|: TRUE ---------------------------------------- (181) Obligation: Rules: f399_in -> f411_in :|: TRUE f411_in -> f889_in :|: TRUE f889_in -> f927_in :|: TRUE f391_in -> f399_in :|: TRUE f927_in -> f391_in :|: TRUE ---------------------------------------- (182) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (183) Obligation: Rules: f391_in -> f391_in :|: TRUE ---------------------------------------- (184) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (185) Obligation: Rules: f391_in -> f391_in :|: TRUE ---------------------------------------- (186) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f391_in -> f391_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (187) Obligation: Termination digraph: Nodes: (1) f391_in -> f391_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (188) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f391_in() Replaced non-predefined constructor symbols by 0. ---------------------------------------- (189) Obligation: Rules: f391_in -> f391_in :|: TRUE ---------------------------------------- (190) IntTRSPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc) -> f(1) :|: pc = 1 && TRUE Witness term starting non-terminating reduction: f(1) ---------------------------------------- (191) NO ---------------------------------------- (192) Obligation: Rules: f410_in -> f432_in :|: TRUE f410_in -> f429_in :|: TRUE f432_out -> f410_out :|: TRUE f429_out -> f410_out :|: TRUE f391_out -> f947_out :|: TRUE f947_in -> f391_in :|: TRUE f962_in -> f955_in :|: TRUE f955_out -> f962_out :|: TRUE f929_in -> f929_out :|: TRUE f956_out -> f955_out :|: TRUE f955_in -> f956_in :|: TRUE f442_out -> f479_out :|: TRUE f479_in -> f442_in :|: TRUE f952_in -> f932_in :|: TRUE f932_out -> f952_out :|: TRUE f890_in -> f929_in :|: TRUE f930_out -> f890_out :|: TRUE f929_out -> f890_out :|: TRUE f890_in -> f930_in :|: TRUE f938_out -> f936_out :|: TRUE f937_out -> f936_out :|: TRUE f936_in -> f938_in :|: TRUE f936_in -> f937_in :|: TRUE f680_out(T96) -> f692_out(T96) :|: TRUE f692_in(x) -> f680_in(x) :|: TRUE f443_out(T43) -> f429_out :|: TRUE f429_in -> f442_in :|: TRUE f442_out -> f443_in(x1) :|: TRUE f672_in(x2) -> f673_in(x2) :|: TRUE f674_out(x3) -> f672_out(x3) :|: TRUE f673_out(x4) -> f672_out(x4) :|: TRUE f672_in(x5) -> f674_in(x5) :|: TRUE f958_in -> f963_in :|: TRUE f962_out -> f958_out :|: TRUE f958_in -> f962_in :|: TRUE f963_out -> f958_out :|: TRUE f953_out -> f948_out :|: TRUE f952_out -> f953_in :|: TRUE f948_in -> f952_in :|: TRUE f673_in(x6) -> f678_in :|: TRUE f678_out -> f673_out(x7) :|: TRUE f677_out(T77) -> f673_out(T77) :|: TRUE f673_in(x8) -> f677_in(x8) :|: TRUE f450_out -> f442_out :|: TRUE f442_in -> f450_in :|: TRUE f947_out -> f948_in :|: TRUE f943_in -> f947_in :|: TRUE f948_out -> f943_out :|: TRUE f399_out -> f391_out :|: TRUE f391_in -> f399_in :|: TRUE f443_out(x9) -> f681_out(x9) :|: TRUE f681_in(x10) -> f443_in(x10) :|: TRUE f399_in -> f411_in :|: TRUE f411_out -> f399_out :|: TRUE f399_in -> f410_in :|: TRUE f410_out -> f399_out :|: TRUE f672_out(x11) -> f443_out(x11) :|: TRUE f443_in(x12) -> f672_in(x12) :|: TRUE f932_out -> f954_out :|: TRUE f954_in -> f932_in :|: TRUE f889_in -> f928_in :|: TRUE f928_out -> f889_out :|: TRUE f889_in -> f927_in :|: TRUE f927_out -> f889_out :|: TRUE f932_in -> f936_in :|: TRUE f936_out -> f932_out :|: TRUE f683_out(x13) -> f680_out(x13) :|: TRUE f680_in(x14) -> f683_in(x14) :|: TRUE f883_in -> f883_out :|: TRUE f853_in(T125) -> f883_in :|: TRUE f883_out -> f853_out(x15) :|: TRUE f884_out -> f853_out(x16) :|: TRUE f853_in(x17) -> f884_in :|: TRUE f879_in(T116) -> f443_in(T116) :|: TRUE f443_out(x18) -> f879_out(x18) :|: TRUE f481_out -> f457_out :|: TRUE f457_in -> f481_in :|: TRUE f479_out -> f457_out :|: TRUE f457_in -> f479_in :|: TRUE f674_in(x19) -> f852_in(x19) :|: TRUE f674_in(x20) -> f853_in(x20) :|: TRUE f853_out(x21) -> f674_out(x21) :|: TRUE f852_out(x22) -> f674_out(x22) :|: TRUE f467_out -> f455_out :|: TRUE f455_in -> f467_in :|: TRUE f455_in -> f464_in :|: TRUE f464_out -> f455_out :|: TRUE f890_out -> f411_out :|: TRUE f411_in -> f889_in :|: TRUE f889_out -> f411_out :|: TRUE f411_in -> f890_in :|: TRUE f687_out -> f684_out(x23) :|: TRUE f684_in(0) -> f686_in :|: TRUE f684_in(x24) -> f687_in :|: TRUE f686_out -> f684_out(0) :|: TRUE f959_in -> f959_out :|: TRUE f464_in -> f464_out :|: TRUE f683_in(x25) -> f684_in(x25) :|: TRUE f685_out(x26) -> f683_out(x26) :|: TRUE f683_in(x27) -> f685_in(x27) :|: TRUE f684_out(x28) -> f683_out(x28) :|: TRUE f957_in -> f959_in :|: TRUE f960_out -> f957_out :|: TRUE f959_out -> f957_out :|: TRUE f957_in -> f960_in :|: TRUE f938_in -> f944_in :|: TRUE f944_out -> f938_out :|: TRUE f943_out -> f938_out :|: TRUE f938_in -> f943_in :|: TRUE f852_in(x29) -> f879_in(x29) :|: TRUE f879_out(x30) -> f852_out(x30) :|: TRUE f880_out -> f852_out(x31) :|: TRUE f852_in(x32) -> f880_in :|: TRUE f957_out -> f956_out :|: TRUE f956_in -> f958_in :|: TRUE f958_out -> f956_out :|: TRUE f956_in -> f957_in :|: TRUE f391_out -> f927_out :|: TRUE f927_in -> f391_in :|: TRUE f457_out -> f450_out :|: TRUE f455_out -> f450_out :|: TRUE f450_in -> f457_in :|: TRUE f450_in -> f455_in :|: TRUE f953_in -> f954_in :|: TRUE f955_out -> f953_out :|: TRUE f954_out -> f955_in :|: TRUE f681_out(x33) -> f677_out(x33) :|: TRUE f677_in(x34) -> f680_in(x34) :|: TRUE f680_out(x35) -> f681_in(x35) :|: TRUE f686_in -> f686_out :|: TRUE f685_in(s(x36)) -> f692_in(x36) :|: TRUE f692_out(x37) -> f685_out(s(x37)) :|: TRUE f685_in(x38) -> f693_in :|: TRUE f693_out -> f685_out(x39) :|: TRUE f4_out(T2) -> f3_out(T2) :|: TRUE f3_in(x40) -> f4_in(x40) :|: TRUE f4_in(x41) -> f18_in(x41) :|: TRUE f4_in(x42) -> f22_in(x42) :|: TRUE f18_out(x43) -> f4_out(x43) :|: TRUE f22_out(x44) -> f4_out(x44) :|: TRUE f22_in(T11) -> f61_in(T11) :|: TRUE f22_in(x45) -> f62_in :|: TRUE f62_out -> f22_out(x46) :|: TRUE f61_out(x47) -> f22_out(x47) :|: TRUE f61_in(x48) -> f391_in :|: TRUE f392_out(x49) -> f61_out(x49) :|: TRUE f391_out -> f392_in(x50) :|: TRUE f933_out(x51) -> f392_out(x51) :|: TRUE f932_out -> f933_in(x52) :|: TRUE f392_in(x53) -> f932_in :|: TRUE f933_in(x54) -> f964_in :|: TRUE f964_out -> f965_in(x55) :|: TRUE f965_out(x56) -> f933_out(x56) :|: TRUE f964_in -> f932_in :|: TRUE f932_out -> f964_out :|: TRUE Start term: f3_in(T2) ---------------------------------------- (193) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f410_in -> f429_in :|: TRUE f429_out -> f410_out :|: TRUE f391_out -> f947_out :|: TRUE f947_in -> f391_in :|: TRUE f929_in -> f929_out :|: TRUE f442_out -> f479_out :|: TRUE f479_in -> f442_in :|: TRUE f952_in -> f932_in :|: TRUE f890_in -> f929_in :|: TRUE f929_out -> f890_out :|: TRUE f936_in -> f938_in :|: TRUE f680_out(T96) -> f692_out(T96) :|: TRUE f692_in(x) -> f680_in(x) :|: TRUE f443_out(T43) -> f429_out :|: TRUE f429_in -> f442_in :|: TRUE f442_out -> f443_in(x1) :|: TRUE f672_in(x2) -> f673_in(x2) :|: TRUE f674_out(x3) -> f672_out(x3) :|: TRUE f673_out(x4) -> f672_out(x4) :|: TRUE f672_in(x5) -> f674_in(x5) :|: TRUE f948_in -> f952_in :|: TRUE f677_out(T77) -> f673_out(T77) :|: TRUE f673_in(x8) -> f677_in(x8) :|: TRUE f450_out -> f442_out :|: TRUE f442_in -> f450_in :|: TRUE f947_out -> f948_in :|: TRUE f943_in -> f947_in :|: TRUE f399_out -> f391_out :|: TRUE f391_in -> f399_in :|: TRUE f443_out(x9) -> f681_out(x9) :|: TRUE f681_in(x10) -> f443_in(x10) :|: TRUE f399_in -> f411_in :|: TRUE f411_out -> f399_out :|: TRUE f399_in -> f410_in :|: TRUE f410_out -> f399_out :|: TRUE f672_out(x11) -> f443_out(x11) :|: TRUE f443_in(x12) -> f672_in(x12) :|: TRUE f889_in -> f927_in :|: TRUE f927_out -> f889_out :|: TRUE f932_in -> f936_in :|: TRUE f683_out(x13) -> f680_out(x13) :|: TRUE f680_in(x14) -> f683_in(x14) :|: TRUE f883_in -> f883_out :|: TRUE f853_in(T125) -> f883_in :|: TRUE f883_out -> f853_out(x15) :|: TRUE f879_in(T116) -> f443_in(T116) :|: TRUE f443_out(x18) -> f879_out(x18) :|: TRUE f479_out -> f457_out :|: TRUE f457_in -> f479_in :|: TRUE f674_in(x19) -> f852_in(x19) :|: TRUE f674_in(x20) -> f853_in(x20) :|: TRUE f853_out(x21) -> f674_out(x21) :|: TRUE f852_out(x22) -> f674_out(x22) :|: TRUE f455_in -> f464_in :|: TRUE f464_out -> f455_out :|: TRUE f890_out -> f411_out :|: TRUE f411_in -> f889_in :|: TRUE f889_out -> f411_out :|: TRUE f411_in -> f890_in :|: TRUE f684_in(0) -> f686_in :|: TRUE f686_out -> f684_out(0) :|: TRUE f464_in -> f464_out :|: TRUE f683_in(x25) -> f684_in(x25) :|: TRUE f685_out(x26) -> f683_out(x26) :|: TRUE f683_in(x27) -> f685_in(x27) :|: TRUE f684_out(x28) -> f683_out(x28) :|: TRUE f938_in -> f943_in :|: TRUE f852_in(x29) -> f879_in(x29) :|: TRUE f879_out(x30) -> f852_out(x30) :|: TRUE f391_out -> f927_out :|: TRUE f927_in -> f391_in :|: TRUE f457_out -> f450_out :|: TRUE f455_out -> f450_out :|: TRUE f450_in -> f457_in :|: TRUE f450_in -> f455_in :|: TRUE f681_out(x33) -> f677_out(x33) :|: TRUE f677_in(x34) -> f680_in(x34) :|: TRUE f680_out(x35) -> f681_in(x35) :|: TRUE f686_in -> f686_out :|: TRUE f685_in(s(x36)) -> f692_in(x36) :|: TRUE f692_out(x37) -> f685_out(s(x37)) :|: TRUE f391_out -> f392_in(x50) :|: TRUE f392_in(x53) -> f932_in :|: TRUE ---------------------------------------- (194) Obligation: Rules: f410_in -> f429_in :|: TRUE f429_out -> f410_out :|: TRUE f391_out -> f947_out :|: TRUE f947_in -> f391_in :|: TRUE f929_in -> f929_out :|: TRUE f442_out -> f479_out :|: TRUE f479_in -> f442_in :|: TRUE f952_in -> f932_in :|: TRUE f890_in -> f929_in :|: TRUE f929_out -> f890_out :|: TRUE f936_in -> f938_in :|: TRUE f680_out(T96) -> f692_out(T96) :|: TRUE f692_in(x) -> f680_in(x) :|: TRUE f443_out(T43) -> f429_out :|: TRUE f429_in -> f442_in :|: TRUE f442_out -> f443_in(x1) :|: TRUE f672_in(x2) -> f673_in(x2) :|: TRUE f674_out(x3) -> f672_out(x3) :|: TRUE f673_out(x4) -> f672_out(x4) :|: TRUE f672_in(x5) -> f674_in(x5) :|: TRUE f948_in -> f952_in :|: TRUE f677_out(T77) -> f673_out(T77) :|: TRUE f673_in(x8) -> f677_in(x8) :|: TRUE f450_out -> f442_out :|: TRUE f442_in -> f450_in :|: TRUE f947_out -> f948_in :|: TRUE f943_in -> f947_in :|: TRUE f399_out -> f391_out :|: TRUE f391_in -> f399_in :|: TRUE f443_out(x9) -> f681_out(x9) :|: TRUE f681_in(x10) -> f443_in(x10) :|: TRUE f399_in -> f411_in :|: TRUE f411_out -> f399_out :|: TRUE f399_in -> f410_in :|: TRUE f410_out -> f399_out :|: TRUE f672_out(x11) -> f443_out(x11) :|: TRUE f443_in(x12) -> f672_in(x12) :|: TRUE f889_in -> f927_in :|: TRUE f927_out -> f889_out :|: TRUE f932_in -> f936_in :|: TRUE f683_out(x13) -> f680_out(x13) :|: TRUE f680_in(x14) -> f683_in(x14) :|: TRUE f883_in -> f883_out :|: TRUE f853_in(T125) -> f883_in :|: TRUE f883_out -> f853_out(x15) :|: TRUE f879_in(T116) -> f443_in(T116) :|: TRUE f443_out(x18) -> f879_out(x18) :|: TRUE f479_out -> f457_out :|: TRUE f457_in -> f479_in :|: TRUE f674_in(x19) -> f852_in(x19) :|: TRUE f674_in(x20) -> f853_in(x20) :|: TRUE f853_out(x21) -> f674_out(x21) :|: TRUE f852_out(x22) -> f674_out(x22) :|: TRUE f455_in -> f464_in :|: TRUE f464_out -> f455_out :|: TRUE f890_out -> f411_out :|: TRUE f411_in -> f889_in :|: TRUE f889_out -> f411_out :|: TRUE f411_in -> f890_in :|: TRUE f684_in(0) -> f686_in :|: TRUE f686_out -> f684_out(0) :|: TRUE f464_in -> f464_out :|: TRUE f683_in(x25) -> f684_in(x25) :|: TRUE f685_out(x26) -> f683_out(x26) :|: TRUE f683_in(x27) -> f685_in(x27) :|: TRUE f684_out(x28) -> f683_out(x28) :|: TRUE f938_in -> f943_in :|: TRUE f852_in(x29) -> f879_in(x29) :|: TRUE f879_out(x30) -> f852_out(x30) :|: TRUE f391_out -> f927_out :|: TRUE f927_in -> f391_in :|: TRUE f457_out -> f450_out :|: TRUE f455_out -> f450_out :|: TRUE f450_in -> f457_in :|: TRUE f450_in -> f455_in :|: TRUE f681_out(x33) -> f677_out(x33) :|: TRUE f677_in(x34) -> f680_in(x34) :|: TRUE f680_out(x35) -> f681_in(x35) :|: TRUE f686_in -> f686_out :|: TRUE f685_in(s(x36)) -> f692_in(x36) :|: TRUE f692_out(x37) -> f685_out(s(x37)) :|: TRUE f391_out -> f392_in(x50) :|: TRUE f392_in(x53) -> f932_in :|: TRUE ---------------------------------------- (195) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (196) Obligation: Rules: f399_out -> f399_out :|: TRUE f683_out(x13:0) -> f683_out(s(x13:0)) :|: TRUE f443_out(T43:0) -> f399_out :|: TRUE f450_in -> f450_out :|: TRUE f683_in(cons_0) -> f683_out(0) :|: TRUE && cons_0 = 0 f672_in(x2:0) -> f683_in(x2:0) :|: TRUE f399_in -> f399_in :|: TRUE f672_in(x5:0) -> f443_out(x15:0) :|: TRUE f683_out(x) -> f672_in(x) :|: TRUE f450_out -> f672_in(x1:0) :|: TRUE f399_out -> f399_in :|: TRUE f450_out -> f450_out :|: TRUE f672_in(x1) -> f672_in(x1) :|: TRUE f399_in -> f399_out :|: TRUE f683_in(s(x36:0)) -> f683_in(x36:0) :|: TRUE f399_in -> f450_in :|: TRUE f443_out(x18:0) -> f443_out(x18:0) :|: TRUE f450_in -> f450_in :|: TRUE ---------------------------------------- (197) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (198) Obligation: Rules: f399_out -> f399_out :|: TRUE f683_out(x13:0) -> f683_out(s(x13:0)) :|: TRUE f443_out(T43:0) -> f399_out :|: TRUE f450_in -> f450_out :|: TRUE f683_in(cons_0) -> f683_out(0) :|: TRUE && cons_0 = 0 f672_in(x2:0) -> f683_in(x2:0) :|: TRUE f399_in -> f399_in :|: TRUE f672_in(x5:0) -> f443_out(x15:0) :|: TRUE f683_out(x) -> f672_in(x) :|: TRUE f450_out -> f672_in(x1:0) :|: TRUE f399_out -> f399_in :|: TRUE f450_out -> f450_out :|: TRUE f672_in(x1) -> f672_in(x1) :|: TRUE f399_in -> f399_out :|: TRUE f683_in(s(x36:0)) -> f683_in(x36:0) :|: TRUE f399_in -> f450_in :|: TRUE f443_out(x18:0) -> f443_out(x18:0) :|: TRUE f450_in -> f450_in :|: TRUE ---------------------------------------- (199) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f399_out -> f399_out :|: TRUE (2) f683_out(x13:0) -> f683_out(s(x13:0)) :|: TRUE (3) f443_out(T43:0) -> f399_out :|: TRUE (4) f450_in -> f450_out :|: TRUE (5) f683_in(cons_0) -> f683_out(0) :|: TRUE && cons_0 = 0 (6) f672_in(x2:0) -> f683_in(x2:0) :|: TRUE (7) f399_in -> f399_in :|: TRUE (8) f672_in(x5:0) -> f443_out(x15:0) :|: TRUE (9) f683_out(x) -> f672_in(x) :|: TRUE (10) f450_out -> f672_in(x1:0) :|: TRUE (11) f399_out -> f399_in :|: TRUE (12) f450_out -> f450_out :|: TRUE (13) f672_in(x1) -> f672_in(x1) :|: TRUE (14) f399_in -> f399_out :|: TRUE (15) f683_in(s(x36:0)) -> f683_in(x36:0) :|: TRUE (16) f399_in -> f450_in :|: TRUE (17) f443_out(x18:0) -> f443_out(x18:0) :|: TRUE (18) f450_in -> f450_in :|: TRUE Arcs: (1) -> (1), (11) (2) -> (2), (9) (3) -> (1), (11) (4) -> (10), (12) (5) -> (2), (9) (6) -> (5), (15) (7) -> (7), (14), (16) (8) -> (3), (17) (9) -> (6), (8), (13) (10) -> (6), (8), (13) (11) -> (7), (14), (16) (12) -> (10), (12) (13) -> (6), (8), (13) (14) -> (1), (11) (15) -> (5), (15) (16) -> (4), (18) (17) -> (3), (17) (18) -> (4), (18) This digraph is fully evaluated! ---------------------------------------- (200) Obligation: Termination digraph: Nodes: (1) f399_out -> f399_out :|: TRUE (2) f443_out(T43:0) -> f399_out :|: TRUE (3) f443_out(x18:0) -> f443_out(x18:0) :|: TRUE (4) f672_in(x5:0) -> f443_out(x15:0) :|: TRUE (5) f683_out(x) -> f672_in(x) :|: TRUE (6) f683_out(x13:0) -> f683_out(s(x13:0)) :|: TRUE (7) f683_in(cons_0) -> f683_out(0) :|: TRUE && cons_0 = 0 (8) f683_in(s(x36:0)) -> f683_in(x36:0) :|: TRUE (9) f672_in(x2:0) -> f683_in(x2:0) :|: TRUE (10) f672_in(x1) -> f672_in(x1) :|: TRUE (11) f450_out -> f672_in(x1:0) :|: TRUE (12) f450_out -> f450_out :|: TRUE (13) f450_in -> f450_out :|: TRUE (14) f450_in -> f450_in :|: TRUE (15) f399_in -> f450_in :|: TRUE (16) f399_in -> f399_in :|: TRUE (17) f399_out -> f399_in :|: TRUE (18) f399_in -> f399_out :|: TRUE Arcs: (1) -> (1), (17) (2) -> (1), (17) (3) -> (2), (3) (4) -> (2), (3) (5) -> (4), (9), (10) (6) -> (5), (6) (7) -> (5), (6) (8) -> (7), (8) (9) -> (7), (8) (10) -> (4), (9), (10) (11) -> (4), (9), (10) (12) -> (11), (12) (13) -> (11), (12) (14) -> (13), (14) (15) -> (13), (14) (16) -> (15), (16), (18) (17) -> (15), (16), (18) (18) -> (1), (17) This digraph is fully evaluated! ---------------------------------------- (201) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (202) Obligation: Rules: f450_out -> f672_in(x1:0:0) :|: TRUE f399_out -> f399_out :|: TRUE f450_out -> f450_out :|: TRUE f672_in(x5:0:0) -> f443_out(x15:0:0) :|: TRUE f399_in -> f399_in :|: TRUE f399_out -> f399_in :|: TRUE f683_in(s(x36:0:0)) -> f683_in(x36:0:0) :|: TRUE f399_in -> f399_out :|: TRUE f683_in(cons_0) -> f683_out(0) :|: TRUE && cons_0 = 0 f450_in -> f450_out :|: TRUE f672_in(x1:0) -> f672_in(x1:0) :|: TRUE f443_out(T43:0:0) -> f399_out :|: TRUE f683_out(x13:0:0) -> f683_out(s(x13:0:0)) :|: TRUE f683_out(x:0) -> f672_in(x:0) :|: TRUE f443_out(x18:0:0) -> f443_out(x18:0:0) :|: TRUE f399_in -> f450_in :|: TRUE f672_in(x2:0:0) -> f683_in(x2:0:0) :|: TRUE f450_in -> f450_in :|: TRUE ---------------------------------------- (203) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 2, "program": { "directives": [], "clauses": [ [ "(qs ([]) ([]))", null ], [ "(qs (. X Xs) Ys)", "(',' (part X Xs Littles Bigs) (',' (qs Littles Ls) (',' (qs Bigs Bs) (app Ls (. X Bs) Ys))))" ], [ "(part X (. Y Xs) (. Y Ls) Bs)", "(',' (less X Y) (part X Xs Ls Bs))" ], [ "(part X (. Y Xs) Ls (. Y Bs))", "(part X Xs Ls Bs)" ], [ "(part X1 ([]) ([]) ([]))", null ], [ "(app ([]) X X)", null ], [ "(app (. X Xs) Ys (. X Zs))", "(app Xs Ys Zs)" ], [ "(less (0) (s X2))", null ], [ "(less (s X) (s Y))", "(less X Y)" ] ] }, "graph": { "nodes": { "type": "Nodes", "471": { "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": [] } }, "114": { "goal": [ { "clause": 3, "scope": 2, "term": "(',' (part T7 T8 X9 X10) (',' (qs X9 X11) (',' (qs X10 X12) (app X11 (. T7 X12) ([])))))" }, { "clause": 4, "scope": 2, "term": "(',' (part T7 T8 X9 X10) (',' (qs X9 X11) (',' (qs X10 X12) (app X11 (. T7 X12) ([])))))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X9", "X10", "X11", "X12" ], "exprvars": [] } }, "995": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1212": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1333": { "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": [] } }, "996": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "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": [] } }, "1211": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1332": { "goal": [{ "clause": -1, "scope": -1, "term": "(qs ([]) X438)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X438"], "exprvars": [] } }, "997": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1177": { "goal": [{ "clause": -1, "scope": -1, "term": "(qs T437 X437)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X437"], "exprvars": [] } }, "1210": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1077": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1076": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1075": { "goal": [{ "clause": 1, "scope": 7, "term": "(qs T123 X194)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X194"], "exprvars": [] } }, "1074": { "goal": [{ "clause": 0, "scope": 7, "term": "(qs T123 X194)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X194"], "exprvars": [] } }, "1073": { "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": [] } }, "1071": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (qs T180 X195) (app T179 (. T181 X195) X196))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X196", "X195" ], "exprvars": [] } }, "1070": { "goal": [{ "clause": -1, "scope": -1, "term": "(qs T123 X194)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X194"], "exprvars": [] } }, "2": { "goal": [{ "clause": -1, "scope": -1, "term": "(qs T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "487": { "goal": [{ "clause": 2, "scope": 4, "term": "(part T29 T30 X46 X47)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T29"], "free": [ "X46", "X47" ], "exprvars": [] } }, "1103": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "5": { "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": [] } }, "489": { "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": [] } }, "1102": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "644": { "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": [] } }, "1101": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1100": { "goal": [{ "clause": 6, "scope": 8, "term": "(app T194 (. T195 T193) X196)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X196"], "exprvars": [] } }, "648": { "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 (. T7 X12) ([])))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X9", "X10", "X11", "X12" ], "exprvars": [] } }, "1120": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1085": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1084": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (part T188 T189 X317 X318) (',' (qs X317 X319) (',' (qs X318 X320) (app X319 (. T188 X320) X321))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X321", "X317", "X318", "X319", "X320" ], "exprvars": [] } }, "373": { "goal": [ { "clause": 7, "scope": 3, "term": "(less T24 T25)" }, { "clause": 8, "scope": 3, "term": "(less T24 T25)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1118": { "goal": [ { "clause": 5, "scope": 9, "term": "(app T230 (. T29 T229) ([]))" }, { "clause": 6, "scope": 9, "term": "(app T230 (. 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T306 X438) T305))))" } ], "kb": { "nonunifying": [[ "(qs T1 T305)", "(qs ([]) ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T305"], "free": [ "X435", "X436", "X437", "X438" ], "exprvars": [] } }, "1265": { "goal": [{ "clause": -1, "scope": -1, "term": "(app T515 (. T516 T517) T514)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T514"], "free": [], "exprvars": [] } }, "327": { "goal": [{ "clause": -1, "scope": -1, "term": "(less T24 T25)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "60": { "goal": [ { "clause": 2, "scope": 2, "term": "(',' (part T7 T8 X9 X10) (',' (qs X9 X11) (',' (qs X10 X12) (app X11 (. T7 X12) ([])))))" }, { "clause": 3, "scope": 2, "term": "(',' (part T7 T8 X9 X10) (',' (qs X9 X11) (',' (qs X10 X12) (app X11 (. T7 X12) ([])))))" }, { "clause": 4, "scope": 2, "term": "(',' (part T7 T8 X9 X10) (',' (qs X9 X11) (',' (qs X10 X12) (app X11 (. T7 X12) ([])))))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X9", "X10", "X11", "X12" ], "exprvars": [] } }, "1165": { "goal": [{ "clause": 6, "scope": 13, "term": "(app T379 (. T376 T380) T378)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T376", "T378" ], "free": [], "exprvars": [] } }, "1164": { "goal": [{ "clause": 5, "scope": 13, "term": "(app T379 (. T376 T380) T378)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T376", "T378" ], "free": [], "exprvars": [] } }, "1163": { "goal": [ { "clause": 5, "scope": 13, "term": "(app T379 (. T376 T380) T378)" }, { "clause": 6, "scope": 13, "term": "(app T379 (. T376 T380) T378)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T376", "T378" ], "free": [], "exprvars": [] } }, "1162": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1161": { "goal": [{ "clause": -1, "scope": -1, "term": "(app T379 (. T376 T380) T378)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T376", "T378" ], "free": [], "exprvars": [] } }, "690": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1160": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "691": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "210": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (',' (less T24 T25) (part T24 T26 X46 X47)) (',' (qs (. T25 X46) X11) (',' (qs X47 X12) (app X11 (. T24 X12) ([])))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X11", "X12", "X46", "X47" ], "exprvars": [] } }, "694": { "goal": [{ "clause": -1, "scope": -1, "term": "(qs (. T54 T52) X11)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X11"], "exprvars": [] } }, "332": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (part T29 T30 X46 X47) (',' (qs (. T31 X46) X11) (',' (qs X47 X12) (app X11 (. T29 X12) ([])))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T29"], "free": [ "X11", "X12", "X46", "X47" ], "exprvars": [] } }, "453": { "goal": [{ "clause": -1, "scope": -1, "term": "(part T29 T30 X46 X47)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T29"], "free": [ "X46", "X47" ], "exprvars": [] } }, "695": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (qs T106 X12) (app T105 (. T29 X12) ([])))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T29"], "free": ["X12"], "exprvars": [] } }, "696": { "goal": [ { "clause": 0, "scope": 5, "term": "(qs (. T54 T52) X11)" }, { "clause": 1, "scope": 5, "term": "(qs (. T54 T52) X11)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X11"], "exprvars": [] } }, "1159": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "697": { "goal": [{ "clause": 1, "scope": 5, "term": "(qs (. T54 T52) X11)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X11"], "exprvars": [] } }, "1158": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "456": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (qs (. T54 T52) X11) (',' (qs T53 X12) (app X11 (. T29 X12) ([]))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T29"], "free": [ "X11", "X12" ], "exprvars": [] } }, "1157": { "goal": [{ "clause": 6, "scope": 12, "term": "(app T348 (. T328 T347) T305)" }], "kb": { "nonunifying": [[ "(qs T349 T305)", "(qs ([]) ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T305", "T328" ], "free": [], "exprvars": [] } }, "974": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (less T144 T145) (part T144 T146 X244 X245))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X244", "X245" ], "exprvars": [] } }, "1156": { "goal": [{ "clause": 5, "scope": 12, "term": "(app T348 (. T328 T347) T305)" }], "kb": { "nonunifying": [[ "(qs T349 T305)", "(qs ([]) ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T305", "T328" ], "free": [], "exprvars": [] } }, "975": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1155": { "goal": [ { "clause": 5, "scope": 12, "term": "(app T348 (. T328 T347) T305)" }, { "clause": 6, "scope": 12, "term": "(app T348 (. T328 T347) T305)" } ], "kb": { "nonunifying": [[ "(qs T349 T305)", "(qs ([]) ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T305", "T328" ], "free": [], "exprvars": [] } }, "976": { "goal": [{ "clause": -1, "scope": -1, "term": "(less T144 T145)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "218": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "977": { "goal": [{ "clause": -1, "scope": -1, "term": "(part T149 T150 X244 X245)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T149"], "free": [ "X244", "X245" ], "exprvars": [] } }, "979": { "goal": [{ "clause": 3, "scope": 6, "term": "(part T118 T119 X192 X193)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X192", "X193" ], "exprvars": [] } }, "1176": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (qs T437 X437) (',' (qs (. T439 T438) X438) (app X437 (. T440 X438) T305)))" }], "kb": { "nonunifying": [[ "(qs T441 T305)", "(qs ([]) ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T305"], "free": [ "X437", "X438" ], "exprvars": [] } }, "1175": { "goal": [{ "clause": -1, "scope": -1, "term": "(part T431 T432 X565 X566)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X565", "X566" ], "exprvars": [] } }, "1174": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1173": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (part T431 T432 X565 X566) (',' (qs X565 X437) (',' (qs (. T433 X566) X438) (app X437 (. T431 X438) T305))))" }], "kb": { "nonunifying": [[ "(qs T1 T305)", "(qs ([]) ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T305"], "free": [ "X437", "X438", "X565", "X566" ], "exprvars": [] } }, "1172": { "goal": [{ "clause": 4, "scope": 11, "term": "(',' (part T306 T307 X435 X436) (',' (qs X435 X437) (',' (qs X436 X438) (app X437 (. T306 X438) T305))))" }], "kb": { "nonunifying": [[ "(qs T1 T305)", "(qs ([]) ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T305"], "free": [ "X435", "X436", "X437", "X438" ], "exprvars": [] } }, "1171": { "goal": [{ "clause": 3, "scope": 11, "term": "(',' (part T306 T307 X435 X436) (',' (qs X435 X437) (',' (qs X436 X438) (app X437 (. T306 X438) T305))))" }], "kb": { "nonunifying": [[ "(qs T1 T305)", "(qs ([]) ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T305"], "free": [ "X435", "X436", "X437", "X438" ], "exprvars": [] } }, "1170": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1206": { "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": [] } }, "980": { "goal": [{ "clause": 4, "scope": 6, "term": "(part T118 T119 X192 X193)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X192", "X193" ], "exprvars": [] } }, "1205": { "goal": [{ "clause": -1, "scope": -1, "term": "(qs (. T446 T447) X438)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X438"], "exprvars": [] } }, "1169": { "goal": [{ "clause": -1, "scope": -1, "term": "(app T410 (. T407 T411) T409)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T407", "T409" ], "free": [], "exprvars": [] } }, "1168": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "105": { "goal": [{ "clause": 2, "scope": 2, "term": "(',' (part T7 T8 X9 X10) (',' (qs X9 X11) (',' (qs X10 X12) (app X11 (. T7 X12) ([])))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X9", "X10", "X11", "X12" ], "exprvars": [] } }, "1167": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1166": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "987": { "goal": [{ "clause": -1, "scope": -1, "term": "(part T171 T172 X290 X291)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X290", "X291" ], "exprvars": [] } }, "988": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1209": { "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": [] } }, "1208": { "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": [] } }, "1207": { "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": [] } } }, "edges": [ { "from": 2, "to": 5, "label": "CASE" }, { "from": 5, "to": 49, "label": "EVAL with clause\nqs([], []).\nand substitutionT1 -> [],\nT2 -> []" }, { "from": 5, "to": 50, "label": "EVAL-BACKTRACK" }, { "from": 49, "to": 51, "label": "SUCCESS" }, { "from": 50, "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": 50, "to": 1141, "label": "EVAL-BACKTRACK" }, { "from": 51, "to": 55, "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": 51, "to": 56, "label": "EVAL-BACKTRACK" }, { "from": 55, "to": 60, "label": "CASE" }, { "from": 60, "to": 105, "label": "PARALLEL" }, { "from": 60, "to": 114, "label": "PARALLEL" }, { "from": 105, "to": 210, "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": 105, "to": 218, "label": "EVAL-BACKTRACK" }, { "from": 114, "to": 1121, "label": "PARALLEL" }, { "from": 114, "to": 1122, "label": "PARALLEL" }, { "from": 210, "to": 327, "label": "SPLIT 1" }, { "from": 210, "to": 332, "label": "SPLIT 2\nnew knowledge:\nT29 is ground\nreplacements:T24 -> T29,\nT26 -> T30,\nT25 -> T31" }, { "from": 327, "to": 373, "label": "CASE" }, { "from": 332, "to": 453, "label": "SPLIT 1" }, { "from": 332, "to": 456, "label": "SPLIT 2\nnew knowledge:\nT29 is ground\nreplacements:X46 -> T52,\nX47 -> T53,\nT31 -> T54" }, { "from": 373, "to": 376, "label": "PARALLEL" }, { "from": 373, "to": 377, "label": "PARALLEL" }, { "from": 376, "to": 378, "label": "EVAL with clause\nless(0, s(X56)).\nand substitutionT24 -> 0,\nX56 -> T38,\nT25 -> s(T38)" }, { "from": 376, "to": 379, "label": "EVAL-BACKTRACK" }, { "from": 377, "to": 385, "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": 377, "to": 388, "label": "EVAL-BACKTRACK" }, { "from": 378, "to": 380, "label": "SUCCESS" }, { "from": 385, "to": 327, "label": "INSTANCE with matching:\nT24 -> T45\nT25 -> T46" }, { "from": 453, "to": 471, "label": "CASE" }, { "from": 456, "to": 694, "label": "SPLIT 1" }, { "from": 456, "to": 695, "label": "SPLIT 2\nreplacements:X11 -> T105,\nT53 -> T106" }, { "from": 471, "to": 487, "label": "PARALLEL" }, { "from": 471, "to": 489, "label": "PARALLEL" }, { "from": 487, "to": 644, "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": 487, "to": 648, "label": "EVAL-BACKTRACK" }, { "from": 489, "to": 675, "label": "PARALLEL" }, { "from": 489, "to": 676, "label": "PARALLEL" }, { "from": 644, "to": 668, "label": "SPLIT 1" }, { "from": 644, "to": 671, "label": "SPLIT 2\nnew knowledge:\nT70 is ground\nreplacements:T74 -> T77" }, { "from": 668, "to": 327, "label": "INSTANCE with matching:\nT24 -> T70\nT25 -> T73" }, { "from": 671, "to": 453, "label": "INSTANCE with matching:\nT29 -> T70\nT30 -> T77\nX46 -> X112\nX47 -> X113" }, { "from": 675, "to": 679, "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": 675, "to": 682, "label": "EVAL-BACKTRACK" }, { "from": 676, "to": 688, "label": "EVAL with clause\npart(X169, [], [], []).\nand substitutionT29 -> T104,\nX169 -> T104,\nT30 -> [],\nX46 -> [],\nX47 -> []" }, { "from": 676, "to": 690, "label": "EVAL-BACKTRACK" }, { "from": 679, "to": 453, "label": "INSTANCE with matching:\nT29 -> T95\nT30 -> T98\nX46 -> X158\nX47 -> X159" }, { "from": 688, "to": 691, "label": "SUCCESS" }, { "from": 694, "to": 696, "label": "CASE" }, { "from": 695, "to": 1114, "label": "SPLIT 1" }, { "from": 695, "to": 1115, "label": "SPLIT 2\nreplacements:X12 -> T229,\nT105 -> T230" }, { "from": 696, "to": 697, "label": "BACKTRACK\nfor clause: qs([], [])because of non-unification" }, { "from": 697, "to": 942, "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": 942, "to": 945, "label": "SPLIT 1" }, { "from": 942, "to": 946, "label": "SPLIT 2\nreplacements:X192 -> T123,\nX193 -> T124,\nT118 -> T125" }, { "from": 945, "to": 949, "label": "CASE" }, { "from": 946, "to": 1070, "label": "SPLIT 1" }, { "from": 946, "to": 1071, "label": "SPLIT 2\nreplacements:X194 -> T179,\nT124 -> T180,\nT125 -> T181" }, { "from": 949, "to": 950, "label": "PARALLEL" }, { "from": 949, "to": 951, "label": "PARALLEL" }, { "from": 950, "to": 974, "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": 950, "to": 975, "label": "EVAL-BACKTRACK" }, { "from": 951, "to": 979, "label": "PARALLEL" }, { "from": 951, "to": 980, "label": "PARALLEL" }, { "from": 974, "to": 976, "label": "SPLIT 1" }, { "from": 974, "to": 977, "label": "SPLIT 2\nnew knowledge:\nT149 is ground\nreplacements:T144 -> T149,\nT146 -> T150" }, { "from": 976, "to": 327, "label": "INSTANCE with matching:\nT24 -> T144\nT25 -> T145" }, { "from": 977, "to": 453, "label": "INSTANCE with matching:\nT29 -> T149\nT30 -> T150\nX46 -> X244\nX47 -> X245" }, { "from": 979, "to": 987, "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": 979, "to": 988, "label": "EVAL-BACKTRACK" }, { "from": 980, "to": 995, "label": "EVAL with clause\npart(X301, [], [], []).\nand substitutionT118 -> T178,\nX301 -> T178,\nT119 -> [],\nX192 -> [],\nX193 -> []" }, { "from": 980, "to": 996, "label": "EVAL-BACKTRACK" }, { "from": 987, "to": 945, "label": "INSTANCE with matching:\nT118 -> T171\nT119 -> T172\nX192 -> X290\nX193 -> X291" }, { "from": 995, "to": 997, "label": "SUCCESS" }, { "from": 1070, "to": 1073, "label": "CASE" }, { "from": 1071, "to": 1091, "label": "SPLIT 1" }, { "from": 1071, "to": 1092, "label": "SPLIT 2\nreplacements:X195 -> T193,\nT179 -> T194,\nT181 -> T195" }, { "from": 1073, "to": 1074, "label": "PARALLEL" }, { "from": 1073, "to": 1075, "label": "PARALLEL" }, { "from": 1074, "to": 1076, "label": "EVAL with clause\nqs([], []).\nand substitutionT123 -> [],\nX194 -> []" }, { "from": 1074, "to": 1077, "label": "EVAL-BACKTRACK" }, { "from": 1075, "to": 1084, "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": 1075, "to": 1085, "label": "EVAL-BACKTRACK" }, { "from": 1076, "to": 1078, "label": "SUCCESS" }, { "from": 1084, "to": 942, "label": "INSTANCE with matching:\nT118 -> T188\nT119 -> T189\nX192 -> X317\nX193 -> X318\nX194 -> X319\nX195 -> X320\nX196 -> X321" }, { "from": 1091, "to": 1070, "label": "INSTANCE with matching:\nT123 -> T180\nX194 -> X195" }, { "from": 1092, "to": 1098, "label": "CASE" }, { "from": 1098, "to": 1099, "label": "PARALLEL" }, { "from": 1098, "to": 1100, "label": "PARALLEL" }, { "from": 1099, "to": 1101, "label": "EVAL with clause\napp([], X335, X335).\nand substitutionT194 -> [],\nT195 -> T208,\nT193 -> T209,\nX335 -> .(T208, T209),\nX196 -> .(T208, T209)" }, { "from": 1099, "to": 1102, "label": "EVAL-BACKTRACK" }, { "from": 1100, "to": 1112, "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": 1100, "to": 1113, "label": "EVAL-BACKTRACK" }, { "from": 1101, "to": 1103, "label": "SUCCESS" }, { "from": 1112, "to": 1092, "label": "INSTANCE with matching:\nT194 -> T222\nT195 -> T223\nT193 -> T224\nX196 -> X350" }, { "from": 1114, "to": 1070, "label": "INSTANCE with matching:\nT123 -> T106\nX194 -> X12" }, { "from": 1115, "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": 945, "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": 1070, "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": 694, "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": 1070, "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": 1070, "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": 327, "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": 453, "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": 694, "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": 1070, "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": 1267, "label": "EVAL with clause\npart(X623, [], [], []).\nand substitutionT306 -> T525,\nX623 -> T525,\nT307 -> [],\nX435 -> [],\nX436 -> [],\nT524 -> T525" }, { "from": 1172, "to": 1268, "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": 945, "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": 1070, "label": "INSTANCE with matching:\nT123 -> T437\nX194 -> X437" }, { "from": 1178, "to": 1205, "label": "SPLIT 1" }, { "from": 1178, "to": 1206, "label": "SPLIT 2\nreplacements:X438 -> T450,\nT445 -> T451,\nT448 -> T452,\nT449 -> T453" }, { "from": 1205, "to": 694, "label": "INSTANCE with matching:\nT54 -> T446\nT52 -> T447\nX11 -> X438" }, { "from": 1206, "to": 1207, "label": "CASE" }, { "from": 1207, "to": 1208, "label": "PARALLEL" }, { "from": 1207, "to": 1209, "label": "PARALLEL" }, { "from": 1208, "to": 1210, "label": "EVAL with clause\napp([], X587, X587).\nand substitutionT451 -> [],\nT452 -> T466,\nT450 -> T467,\nX587 -> .(T466, T467),\nT305 -> .(T466, T467)" }, { "from": 1208, "to": 1211, "label": "EVAL-BACKTRACK" }, { "from": 1209, "to": 1257, "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": 1209, "to": 1258, "label": "EVAL-BACKTRACK" }, { "from": 1210, "to": 1212, "label": "SUCCESS" }, { "from": 1257, "to": 1259, "label": "CASE" }, { "from": 1259, "to": 1260, "label": "PARALLEL" }, { "from": 1259, "to": 1261, "label": "PARALLEL" }, { "from": 1260, "to": 1262, "label": "EVAL with clause\napp([], X606, X606).\nand substitutionT483 -> [],\nT484 -> T498,\nT485 -> T499,\nX606 -> .(T498, T499),\nT482 -> .(T498, T499)" }, { "from": 1260, "to": 1263, "label": "EVAL-BACKTRACK" }, { "from": 1261, "to": 1265, "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": 1261, "to": 1266, "label": "EVAL-BACKTRACK" }, { "from": 1262, "to": 1264, "label": "SUCCESS" }, { "from": 1265, "to": 1257, "label": "INSTANCE with matching:\nT483 -> T515\nT484 -> T516\nT485 -> T517\nT482 -> T514" }, { "from": 1267, "to": 1269, "label": "SPLIT 1" }, { "from": 1267, "to": 1270, "label": "SPLIT 2\nnew knowledge:\nT530 is ground\nreplacements:X437 -> T530" }, { "from": 1269, "to": 1070, "label": "INSTANCE with matching:\nT123 -> []\nX194 -> X437" }, { "from": 1270, "to": 1332, "label": "SPLIT 1" }, { "from": 1270, "to": 1333, "label": "SPLIT 2\nnew knowledge:\nT535 is ground\nreplacements:X438 -> T535" }, { "from": 1332, "to": 1070, "label": "INSTANCE with matching:\nT123 -> []\nX194 -> X438" }, { "from": 1333, "to": 1206, "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) QDPQMonotonicMRRProof (EQUIVALENT) By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain. Strictly oriented dependency pairs: QSF_IN_GA(.(X1, X2)) -> PD_IN_GGAAAAA(X1, X2) U11_GGAAAAA(X1, X2, partcC_out_ggaa(X1, X2, X3, X4)) -> QSF_IN_GA(X3) U13_GGAAAAA(X1, X2, X4, qscF_out_ga(X3, X5)) -> QSF_IN_GA(X4) Used ordering: Polynomial interpretation [POLO]: POL(.(x_1, x_2)) = 2 + 2*x_2 POL(0) = 0 POL(PD_IN_GGAAAAA(x_1, x_2)) = 1 + 2*x_2 POL(QSF_IN_GA(x_1)) = x_1 POL(U11_GGAAAAA(x_1, x_2, x_3)) = 1 + 2*x_3 POL(U13_GGAAAAA(x_1, x_2, x_3, x_4)) = 1 + 2*x_3 POL(U63_gg(x_1, x_2, x_3)) = 0 POL(U64_ggaa(x_1, x_2, x_3, x_4)) = 2 + 2*x_3 POL(U65_ggaa(x_1, x_2, x_3, x_4)) = 2 + 2*x_4 POL(U66_ggaa(x_1, x_2, x_3, x_4)) = 2 + 2*x_4 POL(U67_ggaa(x_1, x_2, x_3, x_4)) = 2 + 2*x_3 POL(U68_ggaa(x_1, x_2, x_3, x_4)) = 2 + 2*x_4 POL(U69_ggaa(x_1, x_2, x_3, x_4)) = 2 + 2*x_4 POL(U70_ggaaaaa(x_1, x_2, x_3)) = 1 POL(U71_ggaaaaa(x_1, x_2, x_3, x_4, x_5)) = 0 POL(U72_ggaaaaa(x_1, x_2, x_3, x_4, x_5, x_6)) = 0 POL(U73_ggaaaaa(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = 0 POL(U74_ga(x_1, x_2, x_3)) = 0 POL(U75_ggga(x_1, x_2, x_3, x_4, x_5)) = 0 POL([]) = 0 POL(appcE_in_ggga(x_1, x_2, x_3)) = 1 POL(appcE_out_ggga(x_1, x_2, x_3, x_4)) = 0 POL(lesscA_in_gg(x_1, x_2)) = 0 POL(lesscA_out_gg(x_1, x_2)) = 0 POL(partcB_in_ggaa(x_1, x_2)) = x_2 POL(partcB_out_ggaa(x_1, x_2, x_3, x_4)) = x_3 + x_4 POL(partcC_in_ggaa(x_1, x_2)) = x_2 POL(partcC_out_ggaa(x_1, x_2, x_3, x_4)) = x_3 + x_4 POL(qcD_in_ggaaaaa(x_1, x_2)) = 2 POL(qcD_out_ggaaaaa(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = 0 POL(qscF_in_ga(x_1)) = 0 POL(qscF_out_ga(x_1, x_2)) = 0 POL(s(x_1)) = 0 ---------------------------------------- (266) Obligation: Q DP problem: The TRS P consists of the following rules: PD_IN_GGAAAAA(X1, X2) -> U11_GGAAAAA(X1, X2, partcC_in_ggaa(X1, X2)) U11_GGAAAAA(X1, X2, partcC_out_ggaa(X1, X2, X3, X4)) -> U13_GGAAAAA(X1, X2, X4, qscF_in_ga(X3)) The TRS R consists of the following rules: partcC_in_ggaa(X1, .(X2, X3)) -> U67_ggaa(X1, X2, X3, lesscA_in_gg(X1, X2)) partcC_in_ggaa(X1, .(X2, X3)) -> U69_ggaa(X1, X2, X3, partcC_in_ggaa(X1, X3)) partcC_in_ggaa(X1, []) -> partcC_out_ggaa(X1, [], [], []) qscF_in_ga([]) -> qscF_out_ga([], []) qscF_in_ga(.(X1, X2)) -> U74_ga(X1, X2, qcD_in_ggaaaaa(X1, X2)) U67_ggaa(X1, X2, X3, lesscA_out_gg(X1, X2)) -> U68_ggaa(X1, X2, X3, partcB_in_ggaa(X1, X3)) U69_ggaa(X1, X2, X3, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) U74_ga(X1, X2, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3) lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) U68_ggaa(X1, X2, X3, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) qcD_in_ggaaaaa(X1, X2) -> U70_ggaaaaa(X1, X2, partcC_in_ggaa(X1, X2)) U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) partcB_in_ggaa(X1, .(X2, X3)) -> U64_ggaa(X1, X2, X3, lesscA_in_gg(X1, X2)) partcB_in_ggaa(X1, .(X2, X3)) -> U66_ggaa(X1, X2, X3, partcB_in_ggaa(X1, X3)) partcB_in_ggaa(X1, []) -> partcB_out_ggaa(X1, [], [], []) U70_ggaaaaa(X1, X2, partcC_out_ggaa(X1, X2, X3, X4)) -> U71_ggaaaaa(X1, X2, X3, X4, qscF_in_ga(X3)) U64_ggaa(X1, X2, X3, lesscA_out_gg(X1, X2)) -> U65_ggaa(X1, X2, X3, partcB_in_ggaa(X1, X3)) U66_ggaa(X1, X2, X3, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) U71_ggaaaaa(X1, X2, X3, X4, qscF_out_ga(X3, X5)) -> U72_ggaaaaa(X1, X2, X3, X4, X5, qscF_in_ga(X4)) U65_ggaa(X1, X2, X3, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) U72_ggaaaaa(X1, X2, X3, X4, X5, qscF_out_ga(X4, X6)) -> U73_ggaaaaa(X1, X2, X3, X4, X5, X6, appcE_in_ggga(X5, X1, X6)) U73_ggaaaaa(X1, X2, X3, X4, X5, X6, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) appcE_in_ggga([], X1, X2) -> appcE_out_ggga([], X1, X2, .(X1, X2)) appcE_in_ggga(.(X1, X2), X3, X4) -> U75_ggga(X1, X2, X3, X4, appcE_in_ggga(X2, X3, X4)) U75_ggga(X1, X2, X3, X4, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5)) The set Q consists of the following terms: partcC_in_ggaa(x0, x1) qscF_in_ga(x0) U67_ggaa(x0, x1, x2, x3) U69_ggaa(x0, x1, x2, x3) U74_ga(x0, x1, x2) lesscA_in_gg(x0, x1) U68_ggaa(x0, x1, x2, x3) qcD_in_ggaaaaa(x0, x1) U63_gg(x0, x1, x2) partcB_in_ggaa(x0, x1) U70_ggaaaaa(x0, x1, x2) U64_ggaa(x0, x1, x2, x3) U66_ggaa(x0, x1, x2, x3) U71_ggaaaaa(x0, x1, x2, x3, x4) U65_ggaa(x0, x1, x2, x3) U72_ggaaaaa(x0, x1, x2, x3, x4, x5) U73_ggaaaaa(x0, x1, x2, x3, x4, x5, x6) appcE_in_ggga(x0, x1, x2) U75_ggga(x0, x1, x2, x3, x4) We have to consider all (P,Q,R)-chains. ---------------------------------------- (267) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 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