53.13/15.17 MAYBE 53.13/15.19 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 53.13/15.19 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 53.13/15.19 53.13/15.19 53.13/15.19 Left Termination of the query pattern 53.13/15.19 53.13/15.19 qs(a,g) 53.13/15.19 53.13/15.19 w.r.t. the given Prolog program could not be shown: 53.13/15.19 53.13/15.19 (0) Prolog 53.13/15.19 (1) PrologToPiTRSProof [SOUND, 47 ms] 53.13/15.19 (2) PiTRS 53.13/15.19 (3) DependencyPairsProof [EQUIVALENT, 20 ms] 53.13/15.19 (4) PiDP 53.13/15.19 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 53.13/15.19 (6) AND 53.13/15.19 (7) PiDP 53.13/15.19 (8) UsableRulesProof [EQUIVALENT, 0 ms] 53.13/15.19 (9) PiDP 53.13/15.19 (10) PiDPToQDPProof [SOUND, 0 ms] 53.13/15.19 (11) QDP 53.13/15.19 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 53.13/15.19 (13) YES 53.13/15.19 (14) PiDP 53.13/15.19 (15) UsableRulesProof [EQUIVALENT, 0 ms] 53.13/15.19 (16) PiDP 53.13/15.19 (17) PiDPToQDPProof [SOUND, 0 ms] 53.13/15.19 (18) QDP 53.13/15.19 (19) NonTerminationLoopProof [COMPLETE, 0 ms] 53.13/15.19 (20) NO 53.13/15.19 (21) PiDP 53.13/15.19 (22) UsableRulesProof [EQUIVALENT, 0 ms] 53.13/15.19 (23) PiDP 53.13/15.19 (24) PiDPToQDPProof [SOUND, 0 ms] 53.13/15.19 (25) QDP 53.13/15.19 (26) QDPSizeChangeProof [EQUIVALENT, 0 ms] 53.13/15.19 (27) YES 53.13/15.19 (28) PiDP 53.13/15.19 (29) UsableRulesProof [EQUIVALENT, 0 ms] 53.13/15.19 (30) PiDP 53.13/15.19 (31) PiDPToQDPProof [SOUND, 0 ms] 53.13/15.19 (32) QDP 53.13/15.19 (33) TransformationProof [SOUND, 0 ms] 53.13/15.19 (34) QDP 53.13/15.19 (35) TransformationProof [EQUIVALENT, 0 ms] 53.13/15.19 (36) QDP 53.13/15.19 (37) NonTerminationLoopProof [COMPLETE, 0 ms] 53.13/15.19 (38) NO 53.13/15.19 (39) PiDP 53.13/15.19 (40) UsableRulesProof [EQUIVALENT, 0 ms] 53.13/15.19 (41) PiDP 53.13/15.19 (42) PiDPToQDPProof [SOUND, 0 ms] 53.13/15.19 (43) QDP 53.13/15.19 (44) PiDP 53.13/15.19 (45) UsableRulesProof [EQUIVALENT, 0 ms] 53.13/15.19 (46) PiDP 53.13/15.19 (47) PiDP 53.13/15.19 (48) UsableRulesProof [EQUIVALENT, 0 ms] 53.13/15.19 (49) PiDP 53.13/15.19 (50) PrologToPiTRSProof [SOUND, 23 ms] 53.13/15.19 (51) PiTRS 53.13/15.19 (52) DependencyPairsProof [EQUIVALENT, 29 ms] 53.13/15.19 (53) PiDP 53.13/15.19 (54) DependencyGraphProof [EQUIVALENT, 0 ms] 53.13/15.19 (55) AND 53.13/15.19 (56) PiDP 53.13/15.19 (57) UsableRulesProof [EQUIVALENT, 0 ms] 53.13/15.19 (58) PiDP 53.13/15.19 (59) PiDPToQDPProof [SOUND, 0 ms] 53.13/15.19 (60) QDP 53.13/15.19 (61) QDPSizeChangeProof [EQUIVALENT, 0 ms] 53.13/15.19 (62) YES 53.13/15.19 (63) PiDP 53.13/15.19 (64) UsableRulesProof [EQUIVALENT, 0 ms] 53.13/15.19 (65) PiDP 53.13/15.19 (66) PiDPToQDPProof [SOUND, 0 ms] 53.13/15.19 (67) QDP 53.13/15.19 (68) NonTerminationLoopProof [COMPLETE, 0 ms] 53.13/15.19 (69) NO 53.13/15.19 (70) PiDP 53.13/15.19 (71) UsableRulesProof [EQUIVALENT, 0 ms] 53.13/15.19 (72) PiDP 53.13/15.19 (73) PiDPToQDPProof [SOUND, 0 ms] 53.13/15.19 (74) QDP 53.13/15.19 (75) QDPSizeChangeProof [EQUIVALENT, 0 ms] 53.13/15.19 (76) YES 53.13/15.19 (77) PiDP 53.13/15.19 (78) UsableRulesProof [EQUIVALENT, 0 ms] 53.13/15.19 (79) PiDP 53.13/15.19 (80) PiDPToQDPProof [SOUND, 0 ms] 53.13/15.19 (81) QDP 53.13/15.19 (82) TransformationProof [SOUND, 0 ms] 53.13/15.19 (83) QDP 53.13/15.19 (84) TransformationProof [EQUIVALENT, 0 ms] 53.13/15.19 (85) QDP 53.13/15.19 (86) PiDP 53.13/15.19 (87) UsableRulesProof [EQUIVALENT, 0 ms] 53.13/15.19 (88) PiDP 53.13/15.19 (89) PiDP 53.13/15.19 (90) UsableRulesProof [EQUIVALENT, 0 ms] 53.13/15.19 (91) PiDP 53.13/15.19 (92) PiDP 53.13/15.19 (93) UsableRulesProof [EQUIVALENT, 0 ms] 53.13/15.19 (94) PiDP 53.13/15.19 (95) PrologToTRSTransformerProof [SOUND, 38 ms] 53.13/15.19 (96) QTRS 53.13/15.19 (97) DependencyPairsProof [EQUIVALENT, 0 ms] 53.13/15.19 (98) QDP 53.13/15.19 (99) DependencyGraphProof [EQUIVALENT, 0 ms] 53.13/15.19 (100) AND 53.13/15.19 (101) QDP 53.13/15.19 (102) UsableRulesProof [EQUIVALENT, 0 ms] 53.13/15.19 (103) QDP 53.13/15.19 (104) QDPSizeChangeProof [EQUIVALENT, 0 ms] 53.13/15.19 (105) YES 53.13/15.19 (106) QDP 53.13/15.19 (107) UsableRulesProof [EQUIVALENT, 0 ms] 53.13/15.19 (108) QDP 53.13/15.19 (109) NonTerminationLoopProof [COMPLETE, 0 ms] 53.13/15.19 (110) NO 53.13/15.19 (111) QDP 53.13/15.19 (112) UsableRulesProof [EQUIVALENT, 0 ms] 53.13/15.19 (113) QDP 53.13/15.19 (114) QDPSizeChangeProof [EQUIVALENT, 0 ms] 53.13/15.19 (115) YES 53.13/15.19 (116) QDP 53.13/15.19 (117) NonTerminationLoopProof [COMPLETE, 0 ms] 53.13/15.19 (118) NO 53.13/15.19 (119) QDP 53.13/15.19 (120) UsableRulesProof [EQUIVALENT, 0 ms] 53.13/15.19 (121) QDP 53.13/15.19 (122) NonTerminationLoopProof [COMPLETE, 0 ms] 53.13/15.19 (123) NO 53.13/15.19 (124) QDP 53.13/15.19 (125) UsableRulesProof [EQUIVALENT, 0 ms] 53.13/15.19 (126) QDP 53.13/15.19 (127) NonTerminationLoopProof [COMPLETE, 0 ms] 53.13/15.19 (128) NO 53.13/15.19 (129) QDP 53.13/15.19 (130) NonLoopProof [COMPLETE, 1097 ms] 53.13/15.19 (131) NO 53.13/15.19 (132) PrologToDTProblemTransformerProof [SOUND, 162 ms] 53.13/15.19 (133) TRIPLES 53.13/15.19 (134) UndefinedPredicateInTriplesTransformerProof [SOUND, 3 ms] 53.13/15.19 (135) TRIPLES 53.13/15.19 (136) TriplesToPiDPProof [SOUND, 185 ms] 53.13/15.19 (137) PiDP 53.13/15.19 (138) DependencyGraphProof [EQUIVALENT, 0 ms] 53.13/15.19 (139) AND 53.13/15.19 (140) PiDP 53.13/15.19 (141) UsableRulesProof [EQUIVALENT, 0 ms] 53.13/15.19 (142) PiDP 53.13/15.19 (143) PiDPToQDPProof [SOUND, 0 ms] 53.13/15.19 (144) QDP 53.13/15.19 (145) QDPSizeChangeProof [EQUIVALENT, 0 ms] 53.13/15.19 (146) YES 53.13/15.19 (147) PiDP 53.13/15.19 (148) UsableRulesProof [EQUIVALENT, 0 ms] 53.13/15.19 (149) PiDP 53.13/15.19 (150) PiDPToQDPProof [SOUND, 0 ms] 53.13/15.19 (151) QDP 53.13/15.19 (152) QDPSizeChangeProof [EQUIVALENT, 0 ms] 53.13/15.19 (153) YES 53.13/15.19 (154) PiDP 53.13/15.19 (155) UsableRulesProof [EQUIVALENT, 0 ms] 53.13/15.19 (156) PiDP 53.13/15.19 (157) PiDPToQDPProof [SOUND, 0 ms] 53.13/15.19 (158) QDP 53.13/15.19 (159) QDPSizeChangeProof [EQUIVALENT, 0 ms] 53.13/15.19 (160) YES 53.13/15.19 (161) PiDP 53.13/15.19 (162) UsableRulesProof [EQUIVALENT, 0 ms] 53.13/15.19 (163) PiDP 53.13/15.19 (164) PiDPToQDPProof [SOUND, 0 ms] 53.13/15.19 (165) QDP 53.13/15.19 (166) QDPSizeChangeProof [EQUIVALENT, 0 ms] 53.13/15.19 (167) YES 53.13/15.19 (168) PiDP 53.13/15.19 (169) UsableRulesProof [EQUIVALENT, 0 ms] 53.13/15.19 (170) PiDP 53.13/15.19 (171) PiDPToQDPProof [EQUIVALENT, 0 ms] 53.13/15.19 (172) QDP 53.13/15.19 (173) QDPSizeChangeProof [EQUIVALENT, 0 ms] 53.13/15.19 (174) YES 53.13/15.19 (175) PiDP 53.13/15.19 (176) UsableRulesProof [EQUIVALENT, 0 ms] 53.13/15.19 (177) PiDP 53.13/15.19 (178) PiDPToQDPProof [SOUND, 0 ms] 53.13/15.19 (179) QDP 53.13/15.19 (180) QDPSizeChangeProof [EQUIVALENT, 0 ms] 53.13/15.19 (181) YES 53.13/15.19 (182) PiDP 53.13/15.19 (183) UsableRulesProof [EQUIVALENT, 0 ms] 53.13/15.19 (184) PiDP 53.13/15.19 (185) PiDPToQDPProof [SOUND, 0 ms] 53.13/15.19 (186) QDP 53.13/15.19 (187) QDPSizeChangeProof [EQUIVALENT, 0 ms] 53.13/15.19 (188) YES 53.13/15.19 (189) PiDP 53.13/15.19 (190) UsableRulesProof [EQUIVALENT, 0 ms] 53.13/15.19 (191) PiDP 53.13/15.19 (192) PiDPToQDPProof [SOUND, 0 ms] 53.13/15.19 (193) QDP 53.13/15.19 (194) QDPOrderProof [EQUIVALENT, 0 ms] 53.13/15.19 (195) QDP 53.13/15.19 (196) DependencyGraphProof [EQUIVALENT, 0 ms] 53.13/15.19 (197) TRUE 53.13/15.19 (198) PiDP 53.13/15.19 (199) UsableRulesProof [EQUIVALENT, 0 ms] 53.13/15.19 (200) PiDP 53.13/15.19 (201) PiDPToQDPProof [SOUND, 0 ms] 53.13/15.19 (202) QDP 53.13/15.19 (203) NonTerminationLoopProof [COMPLETE, 0 ms] 53.13/15.19 (204) NO 53.13/15.19 (205) PiDP 53.13/15.19 (206) UsableRulesProof [EQUIVALENT, 0 ms] 53.13/15.19 (207) PiDP 53.13/15.19 (208) PiDPToQDPProof [SOUND, 0 ms] 53.13/15.19 (209) QDP 53.13/15.19 (210) QDPSizeChangeProof [EQUIVALENT, 0 ms] 53.13/15.19 (211) YES 53.13/15.19 (212) PiDP 53.13/15.19 (213) UsableRulesProof [EQUIVALENT, 0 ms] 53.13/15.19 (214) PiDP 53.13/15.19 (215) PiDPToQDPProof [SOUND, 0 ms] 53.13/15.19 (216) QDP 53.13/15.19 (217) TransformationProof [SOUND, 0 ms] 53.13/15.19 (218) QDP 53.13/15.19 (219) TransformationProof [EQUIVALENT, 0 ms] 53.13/15.19 (220) QDP 53.13/15.19 (221) PiDP 53.13/15.19 (222) UsableRulesProof [EQUIVALENT, 0 ms] 53.13/15.19 (223) PiDP 53.13/15.19 (224) PiDP 53.13/15.19 (225) UsableRulesProof [EQUIVALENT, 0 ms] 53.13/15.19 (226) PiDP 53.13/15.19 (227) PiDP 53.13/15.19 (228) UsableRulesProof [EQUIVALENT, 0 ms] 53.13/15.19 (229) PiDP 53.13/15.19 (230) PrologToIRSwTTransformerProof [SOUND, 109 ms] 53.13/15.19 (231) AND 53.13/15.19 (232) IRSwT 53.13/15.19 (233) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] 53.13/15.19 (234) TRUE 53.13/15.19 (235) IRSwT 53.13/15.19 (236) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] 53.13/15.19 (237) TRUE 53.13/15.19 (238) IRSwT 53.13/15.19 (239) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] 53.13/15.19 (240) TRUE 53.13/15.19 (241) IRSwT 53.13/15.19 (242) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] 53.13/15.19 (243) TRUE 53.13/15.19 (244) IRSwT 53.13/15.19 (245) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] 53.13/15.19 (246) IRSwT 53.13/15.19 (247) IntTRSCompressionProof [EQUIVALENT, 16 ms] 53.13/15.19 (248) IRSwT 53.13/15.19 (249) IRSFormatTransformerProof [EQUIVALENT, 0 ms] 53.13/15.19 (250) IRSwT 53.13/15.19 (251) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms] 53.13/15.19 (252) IRSwT 53.13/15.19 (253) FilterProof [EQUIVALENT, 0 ms] 53.13/15.19 (254) IntTRS 53.13/15.19 (255) IntTRSNonPeriodicNontermProof [COMPLETE, 0 ms] 53.13/15.19 (256) NO 53.13/15.19 (257) IRSwT 53.13/15.19 (258) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] 53.13/15.19 (259) IRSwT 53.13/15.19 (260) IntTRSCompressionProof [EQUIVALENT, 1 ms] 53.13/15.19 (261) IRSwT 53.13/15.19 (262) IRSFormatTransformerProof [EQUIVALENT, 0 ms] 53.13/15.19 (263) IRSwT 53.13/15.19 (264) IRSwTTerminationDigraphProof [EQUIVALENT, 2 ms] 53.13/15.19 (265) IRSwT 53.13/15.19 (266) FilterProof [EQUIVALENT, 0 ms] 53.13/15.19 (267) IntTRS 53.13/15.19 (268) IntTRSPeriodicNontermProof [COMPLETE, 0 ms] 53.13/15.19 (269) NO 53.13/15.19 (270) IRSwT 53.13/15.19 (271) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] 53.13/15.19 (272) IRSwT 53.13/15.19 (273) IntTRSCompressionProof [EQUIVALENT, 36 ms] 53.13/15.19 (274) IRSwT 53.13/15.19 (275) IRSFormatTransformerProof [EQUIVALENT, 0 ms] 53.13/15.19 (276) IRSwT 53.13/15.19 53.13/15.19 53.13/15.19 ---------------------------------------- 53.13/15.19 53.13/15.19 (0) 53.13/15.19 Obligation: 53.13/15.19 Clauses: 53.13/15.19 53.13/15.19 qs([], []). 53.13/15.19 qs(.(X, Xs), Ys) :- ','(part(X, Xs, Littles, Bigs), ','(qs(Littles, Ls), ','(qs(Bigs, Bs), app(Ls, .(X, Bs), Ys)))). 53.13/15.19 part(X, .(Y, Xs), .(Y, Ls), Bs) :- ','(less(X, Y), part(X, Xs, Ls, Bs)). 53.13/15.19 part(X, .(Y, Xs), Ls, .(Y, Bs)) :- part(X, Xs, Ls, Bs). 53.13/15.19 part(X1, [], [], []). 53.13/15.19 app([], X, X). 53.13/15.19 app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs). 53.13/15.19 less(0, s(X2)). 53.13/15.19 less(s(X), s(Y)) :- less(X, Y). 53.13/15.19 53.13/15.19 53.13/15.19 Query: qs(a,g) 53.13/15.19 ---------------------------------------- 53.13/15.19 53.13/15.19 (1) PrologToPiTRSProof (SOUND) 53.13/15.19 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 53.13/15.19 53.13/15.19 qs_in_2: (f,b) (f,f) 53.13/15.19 53.13/15.19 part_in_4: (f,f,f,f) (b,f,f,f) 53.13/15.19 53.13/15.19 less_in_2: (f,f) (b,f) 53.13/15.19 53.13/15.19 app_in_3: (f,f,f) (f,f,b) 53.13/15.19 53.13/15.19 Transforming Prolog into the following Term Rewriting System: 53.13/15.19 53.13/15.19 Pi-finite rewrite system: 53.13/15.19 The TRS R consists of the following rules: 53.13/15.19 53.13/15.19 qs_in_ag([], []) -> qs_out_ag([], []) 53.13/15.19 qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.19 part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) 53.13/15.19 less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) 53.13/15.19 less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) 53.13/15.19 U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) 53.13/15.19 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)) 53.13/15.19 part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) 53.13/15.19 less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) 53.13/15.19 less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) 53.13/15.19 U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 53.13/15.19 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)) 53.13/15.19 part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) 53.13/15.19 part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) 53.13/15.19 U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.13/15.19 U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.13/15.19 U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.13/15.19 part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) 53.13/15.19 part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) 53.13/15.19 U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.13/15.19 U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.19 qs_in_aa([], []) -> qs_out_aa([], []) 53.13/15.19 qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.19 U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.19 U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.19 U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) 53.13/15.19 app_in_aaa([], X, X) -> app_out_aaa([], X, X) 53.13/15.19 app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) 53.13/15.19 U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) 53.13/15.19 U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) 53.13/15.19 U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.19 U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) 53.13/15.19 app_in_aag([], X, X) -> app_out_aag([], X, X) 53.13/15.19 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 53.13/15.19 U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 53.13/15.19 U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) 53.13/15.19 53.13/15.19 The argument filtering Pi contains the following mapping: 53.13/15.19 qs_in_ag(x1, x2) = qs_in_ag(x2) 53.13/15.19 53.13/15.19 [] = [] 53.13/15.19 53.13/15.19 qs_out_ag(x1, x2) = qs_out_ag 53.13/15.19 53.13/15.19 U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) 53.13/15.19 53.13/15.19 part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa 53.13/15.19 53.13/15.19 U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) 53.13/15.19 53.13/15.19 less_in_aa(x1, x2) = less_in_aa 53.13/15.19 53.13/15.19 less_out_aa(x1, x2) = less_out_aa(x1) 53.13/15.19 53.13/15.19 U9_aa(x1, x2, x3) = U9_aa(x3) 53.13/15.19 53.13/15.19 U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) 53.13/15.19 53.13/15.19 part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) 53.13/15.19 53.13/15.19 U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) 53.13/15.19 53.13/15.19 less_in_ga(x1, x2) = less_in_ga(x1) 53.13/15.19 53.13/15.19 0 = 0 53.13/15.19 53.13/15.19 less_out_ga(x1, x2) = less_out_ga 53.13/15.19 53.13/15.19 s(x1) = s(x1) 53.13/15.19 53.13/15.19 U9_ga(x1, x2, x3) = U9_ga(x3) 53.13/15.19 53.13/15.19 U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x6) 53.13/15.19 53.13/15.19 U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x6) 53.13/15.19 53.13/15.19 part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa 53.13/15.19 53.13/15.19 part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa 53.13/15.19 53.13/15.19 U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) 53.13/15.19 53.13/15.19 U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) 53.13/15.19 53.13/15.19 qs_in_aa(x1, x2) = qs_in_aa 53.13/15.19 53.13/15.19 qs_out_aa(x1, x2) = qs_out_aa 53.13/15.19 53.13/15.19 U1_aa(x1, x2, x3, x4) = U1_aa(x4) 53.13/15.19 53.13/15.19 U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) 53.13/15.19 53.13/15.19 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) 53.13/15.19 53.13/15.19 U4_aa(x1, x2, x3, x4) = U4_aa(x4) 53.13/15.19 53.13/15.19 .(x1, x2) = .(x1, x2) 53.13/15.19 53.13/15.19 app_in_aaa(x1, x2, x3) = app_in_aaa 53.13/15.19 53.13/15.19 app_out_aaa(x1, x2, x3) = app_out_aaa 53.13/15.19 53.13/15.19 U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) 53.13/15.19 53.13/15.19 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) 53.13/15.19 53.13/15.19 U4_ag(x1, x2, x3, x4) = U4_ag(x4) 53.13/15.19 53.13/15.19 app_in_aag(x1, x2, x3) = app_in_aag(x3) 53.13/15.19 53.13/15.19 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) 53.13/15.19 53.13/15.19 U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x5) 53.13/15.19 53.13/15.19 53.13/15.19 53.13/15.19 53.13/15.19 53.13/15.19 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 53.13/15.19 53.13/15.19 53.13/15.19 53.13/15.19 ---------------------------------------- 53.13/15.19 53.13/15.19 (2) 53.13/15.19 Obligation: 53.13/15.19 Pi-finite rewrite system: 53.13/15.19 The TRS R consists of the following rules: 53.13/15.19 53.13/15.19 qs_in_ag([], []) -> qs_out_ag([], []) 53.13/15.19 qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.19 part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) 53.13/15.19 less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) 53.13/15.19 less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) 53.13/15.19 U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) 53.13/15.19 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)) 53.13/15.19 part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) 53.13/15.19 less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) 53.13/15.19 less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) 53.13/15.19 U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 53.13/15.19 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)) 53.13/15.19 part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) 53.13/15.19 part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) 53.13/15.19 U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.13/15.19 U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.13/15.19 U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.13/15.19 part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) 53.13/15.19 part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) 53.13/15.19 U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.13/15.19 U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.19 qs_in_aa([], []) -> qs_out_aa([], []) 53.13/15.19 qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.19 U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.19 U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.19 U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) 53.13/15.19 app_in_aaa([], X, X) -> app_out_aaa([], X, X) 53.13/15.19 app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) 53.13/15.19 U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) 53.13/15.19 U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) 53.13/15.19 U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.19 U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) 53.13/15.19 app_in_aag([], X, X) -> app_out_aag([], X, X) 53.13/15.19 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 53.13/15.19 U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 53.13/15.19 U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) 53.13/15.19 53.13/15.19 The argument filtering Pi contains the following mapping: 53.13/15.19 qs_in_ag(x1, x2) = qs_in_ag(x2) 53.13/15.19 53.13/15.19 [] = [] 53.13/15.19 53.13/15.19 qs_out_ag(x1, x2) = qs_out_ag 53.13/15.19 53.13/15.19 U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) 53.13/15.19 53.13/15.19 part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa 53.13/15.19 53.13/15.19 U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) 53.13/15.19 53.13/15.19 less_in_aa(x1, x2) = less_in_aa 53.13/15.19 53.13/15.19 less_out_aa(x1, x2) = less_out_aa(x1) 53.13/15.19 53.13/15.19 U9_aa(x1, x2, x3) = U9_aa(x3) 53.13/15.19 53.13/15.19 U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) 53.13/15.19 53.13/15.19 part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) 53.13/15.19 53.13/15.19 U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) 53.13/15.19 53.13/15.19 less_in_ga(x1, x2) = less_in_ga(x1) 53.13/15.19 53.13/15.19 0 = 0 53.13/15.19 53.13/15.19 less_out_ga(x1, x2) = less_out_ga 53.13/15.19 53.13/15.19 s(x1) = s(x1) 53.13/15.19 53.13/15.19 U9_ga(x1, x2, x3) = U9_ga(x3) 53.13/15.19 53.13/15.19 U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x6) 53.13/15.19 53.13/15.19 U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x6) 53.13/15.19 53.13/15.19 part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa 53.13/15.19 53.13/15.19 part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa 53.13/15.19 53.13/15.19 U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) 53.13/15.19 53.13/15.19 U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) 53.13/15.19 53.13/15.19 qs_in_aa(x1, x2) = qs_in_aa 53.13/15.19 53.13/15.19 qs_out_aa(x1, x2) = qs_out_aa 53.13/15.19 53.13/15.19 U1_aa(x1, x2, x3, x4) = U1_aa(x4) 53.13/15.19 53.13/15.19 U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) 53.13/15.19 53.13/15.19 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) 53.13/15.19 53.13/15.19 U4_aa(x1, x2, x3, x4) = U4_aa(x4) 53.13/15.19 53.13/15.19 .(x1, x2) = .(x1, x2) 53.13/15.19 53.13/15.19 app_in_aaa(x1, x2, x3) = app_in_aaa 53.13/15.19 53.13/15.19 app_out_aaa(x1, x2, x3) = app_out_aaa 53.13/15.19 53.13/15.19 U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) 53.13/15.19 53.13/15.19 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) 53.13/15.19 53.13/15.19 U4_ag(x1, x2, x3, x4) = U4_ag(x4) 53.13/15.19 53.13/15.19 app_in_aag(x1, x2, x3) = app_in_aag(x3) 53.13/15.19 53.13/15.19 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) 53.13/15.19 53.13/15.19 U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x5) 53.13/15.19 53.13/15.19 53.13/15.19 53.13/15.19 ---------------------------------------- 53.13/15.19 53.13/15.19 (3) DependencyPairsProof (EQUIVALENT) 53.13/15.19 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 53.13/15.19 Pi DP problem: 53.13/15.19 The TRS P consists of the following rules: 53.13/15.19 53.13/15.19 QS_IN_AG(.(X, Xs), Ys) -> U1_AG(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.19 QS_IN_AG(.(X, Xs), Ys) -> PART_IN_AAAA(X, Xs, Littles, Bigs) 53.13/15.19 PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_AAAA(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) 53.13/15.19 PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> LESS_IN_AA(X, Y) 53.13/15.19 LESS_IN_AA(s(X), s(Y)) -> U9_AA(X, Y, less_in_aa(X, Y)) 53.13/15.19 LESS_IN_AA(s(X), s(Y)) -> LESS_IN_AA(X, Y) 53.13/15.19 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)) 53.13/15.19 U5_AAAA(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs) 53.13/15.19 PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GAAA(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) 53.13/15.19 PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> LESS_IN_GA(X, Y) 53.13/15.19 LESS_IN_GA(s(X), s(Y)) -> U9_GA(X, Y, less_in_ga(X, Y)) 53.13/15.19 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 53.13/15.19 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)) 53.13/15.19 U5_GAAA(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs) 53.13/15.19 PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_GAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) 53.13/15.19 PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_GAAA(X, Xs, Ls, Bs) 53.13/15.19 PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_AAAA(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) 53.13/15.19 PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_AAAA(X, Xs, Ls, Bs) 53.13/15.19 U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AG(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.19 U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> QS_IN_AA(Littles, Ls) 53.13/15.19 QS_IN_AA(.(X, Xs), Ys) -> U1_AA(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.19 QS_IN_AA(.(X, Xs), Ys) -> PART_IN_AAAA(X, Xs, Littles, Bigs) 53.13/15.19 U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AA(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.19 U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> QS_IN_AA(Littles, Ls) 53.13/15.19 U2_AA(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_AA(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.19 U2_AA(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> QS_IN_AA(Bigs, Bs) 53.13/15.19 U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_AA(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) 53.13/15.19 U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> APP_IN_AAA(Ls, .(X, Bs), Ys) 53.13/15.19 APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> U8_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) 53.13/15.19 APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAA(Xs, Ys, Zs) 53.13/15.19 U2_AG(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_AG(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.19 U2_AG(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> QS_IN_AA(Bigs, Bs) 53.13/15.19 U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_AG(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) 53.13/15.19 U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> APP_IN_AAG(Ls, .(X, Bs), Ys) 53.13/15.19 APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> U8_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 53.13/15.19 APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs) 53.13/15.19 53.13/15.19 The TRS R consists of the following rules: 53.13/15.19 53.13/15.19 qs_in_ag([], []) -> qs_out_ag([], []) 53.13/15.19 qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.19 part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) 53.13/15.19 less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) 53.13/15.19 less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) 53.13/15.19 U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) 53.13/15.19 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)) 53.13/15.19 part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) 53.13/15.19 less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) 53.13/15.19 less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) 53.13/15.19 U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 53.13/15.19 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)) 53.13/15.19 part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) 53.13/15.19 part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) 53.13/15.19 U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.13/15.19 U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.13/15.19 U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.13/15.19 part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) 53.13/15.19 part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) 53.13/15.19 U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.13/15.19 U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.19 qs_in_aa([], []) -> qs_out_aa([], []) 53.13/15.19 qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.19 U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.19 U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.19 U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) 53.13/15.19 app_in_aaa([], X, X) -> app_out_aaa([], X, X) 53.13/15.19 app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) 53.13/15.19 U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) 53.13/15.19 U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) 53.13/15.19 U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.19 U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) 53.13/15.19 app_in_aag([], X, X) -> app_out_aag([], X, X) 53.13/15.19 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 53.13/15.19 U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 53.13/15.19 U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) 53.13/15.19 53.13/15.19 The argument filtering Pi contains the following mapping: 53.13/15.19 qs_in_ag(x1, x2) = qs_in_ag(x2) 53.13/15.19 53.13/15.19 [] = [] 53.13/15.19 53.13/15.19 qs_out_ag(x1, x2) = qs_out_ag 53.13/15.19 53.13/15.19 U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) 53.13/15.19 53.13/15.19 part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa 53.13/15.19 53.13/15.19 U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) 53.13/15.19 53.13/15.19 less_in_aa(x1, x2) = less_in_aa 53.13/15.19 53.13/15.19 less_out_aa(x1, x2) = less_out_aa(x1) 53.13/15.19 53.13/15.19 U9_aa(x1, x2, x3) = U9_aa(x3) 53.13/15.19 53.13/15.19 U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) 53.13/15.19 53.13/15.19 part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) 53.13/15.19 53.13/15.19 U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) 53.13/15.19 53.13/15.19 less_in_ga(x1, x2) = less_in_ga(x1) 53.13/15.19 53.13/15.19 0 = 0 53.13/15.19 53.13/15.19 less_out_ga(x1, x2) = less_out_ga 53.13/15.19 53.13/15.19 s(x1) = s(x1) 53.13/15.19 53.13/15.19 U9_ga(x1, x2, x3) = U9_ga(x3) 53.13/15.19 53.13/15.19 U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x6) 53.13/15.19 53.13/15.19 U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x6) 53.13/15.19 53.13/15.19 part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa 53.13/15.19 53.13/15.19 part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa 53.13/15.19 53.13/15.19 U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) 53.13/15.19 53.13/15.19 U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) 53.13/15.19 53.13/15.19 qs_in_aa(x1, x2) = qs_in_aa 53.13/15.19 53.13/15.19 qs_out_aa(x1, x2) = qs_out_aa 53.13/15.19 53.13/15.19 U1_aa(x1, x2, x3, x4) = U1_aa(x4) 53.13/15.19 53.13/15.19 U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) 53.13/15.19 53.13/15.19 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) 53.13/15.19 53.13/15.19 U4_aa(x1, x2, x3, x4) = U4_aa(x4) 53.13/15.19 53.13/15.19 .(x1, x2) = .(x1, x2) 53.13/15.19 53.13/15.19 app_in_aaa(x1, x2, x3) = app_in_aaa 53.13/15.19 53.13/15.19 app_out_aaa(x1, x2, x3) = app_out_aaa 53.13/15.19 53.13/15.19 U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) 53.13/15.19 53.13/15.19 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) 53.13/15.19 53.13/15.19 U4_ag(x1, x2, x3, x4) = U4_ag(x4) 53.13/15.19 53.13/15.19 app_in_aag(x1, x2, x3) = app_in_aag(x3) 53.13/15.19 53.13/15.19 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) 53.13/15.19 53.13/15.19 U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x5) 53.13/15.19 53.13/15.19 QS_IN_AG(x1, x2) = QS_IN_AG(x2) 53.13/15.19 53.13/15.19 U1_AG(x1, x2, x3, x4) = U1_AG(x3, x4) 53.13/15.19 53.13/15.19 PART_IN_AAAA(x1, x2, x3, x4) = PART_IN_AAAA 53.13/15.19 53.13/15.19 U5_AAAA(x1, x2, x3, x4, x5, x6) = U5_AAAA(x6) 53.13/15.19 53.13/15.19 LESS_IN_AA(x1, x2) = LESS_IN_AA 53.13/15.19 53.13/15.19 U9_AA(x1, x2, x3) = U9_AA(x3) 53.13/15.19 53.13/15.19 U6_AAAA(x1, x2, x3, x4, x5, x6) = U6_AAAA(x6) 53.13/15.19 53.13/15.19 PART_IN_GAAA(x1, x2, x3, x4) = PART_IN_GAAA(x1) 53.13/15.19 53.13/15.19 U5_GAAA(x1, x2, x3, x4, x5, x6) = U5_GAAA(x1, x6) 53.13/15.19 53.13/15.19 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 53.13/15.19 53.13/15.19 U9_GA(x1, x2, x3) = U9_GA(x3) 53.13/15.19 53.13/15.19 U6_GAAA(x1, x2, x3, x4, x5, x6) = U6_GAAA(x6) 53.13/15.19 53.13/15.19 U7_GAAA(x1, x2, x3, x4, x5, x6) = U7_GAAA(x6) 53.13/15.19 53.13/15.19 U7_AAAA(x1, x2, x3, x4, x5, x6) = U7_AAAA(x6) 53.13/15.19 53.13/15.19 U2_AG(x1, x2, x3, x4, x5) = U2_AG(x3, x5) 53.13/15.19 53.13/15.19 QS_IN_AA(x1, x2) = QS_IN_AA 53.13/15.19 53.13/15.19 U1_AA(x1, x2, x3, x4) = U1_AA(x4) 53.13/15.19 53.13/15.19 U2_AA(x1, x2, x3, x4, x5) = U2_AA(x5) 53.13/15.19 53.13/15.19 U3_AA(x1, x2, x3, x4, x5) = U3_AA(x5) 53.13/15.19 53.13/15.19 U4_AA(x1, x2, x3, x4) = U4_AA(x4) 53.13/15.19 53.13/15.19 APP_IN_AAA(x1, x2, x3) = APP_IN_AAA 53.13/15.19 53.13/15.19 U8_AAA(x1, x2, x3, x4, x5) = U8_AAA(x5) 53.13/15.19 53.13/15.19 U3_AG(x1, x2, x3, x4, x5) = U3_AG(x3, x5) 53.13/15.19 53.13/15.19 U4_AG(x1, x2, x3, x4) = U4_AG(x4) 53.13/15.19 53.13/15.19 APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3) 53.13/15.19 53.13/15.19 U8_AAG(x1, x2, x3, x4, x5) = U8_AAG(x1, x5) 53.13/15.19 53.13/15.19 53.13/15.19 We have to consider all (P,R,Pi)-chains 53.13/15.19 ---------------------------------------- 53.13/15.19 53.13/15.19 (4) 53.13/15.19 Obligation: 53.13/15.19 Pi DP problem: 53.13/15.19 The TRS P consists of the following rules: 53.13/15.19 53.13/15.19 QS_IN_AG(.(X, Xs), Ys) -> U1_AG(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.19 QS_IN_AG(.(X, Xs), Ys) -> PART_IN_AAAA(X, Xs, Littles, Bigs) 53.13/15.19 PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_AAAA(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) 53.13/15.19 PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> LESS_IN_AA(X, Y) 53.13/15.19 LESS_IN_AA(s(X), s(Y)) -> U9_AA(X, Y, less_in_aa(X, Y)) 53.13/15.19 LESS_IN_AA(s(X), s(Y)) -> LESS_IN_AA(X, Y) 53.13/15.19 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)) 53.13/15.19 U5_AAAA(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs) 53.13/15.19 PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GAAA(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) 53.13/15.19 PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> LESS_IN_GA(X, Y) 53.13/15.19 LESS_IN_GA(s(X), s(Y)) -> U9_GA(X, Y, less_in_ga(X, Y)) 53.13/15.19 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 53.13/15.19 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)) 53.13/15.19 U5_GAAA(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs) 53.13/15.19 PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_GAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) 53.13/15.19 PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_GAAA(X, Xs, Ls, Bs) 53.13/15.19 PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_AAAA(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) 53.13/15.19 PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_AAAA(X, Xs, Ls, Bs) 53.13/15.19 U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AG(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.19 U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> QS_IN_AA(Littles, Ls) 53.13/15.19 QS_IN_AA(.(X, Xs), Ys) -> U1_AA(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.19 QS_IN_AA(.(X, Xs), Ys) -> PART_IN_AAAA(X, Xs, Littles, Bigs) 53.13/15.19 U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AA(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.19 U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> QS_IN_AA(Littles, Ls) 53.13/15.19 U2_AA(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_AA(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.19 U2_AA(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> QS_IN_AA(Bigs, Bs) 53.13/15.19 U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_AA(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) 53.13/15.19 U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> APP_IN_AAA(Ls, .(X, Bs), Ys) 53.13/15.19 APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> U8_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) 53.13/15.19 APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAA(Xs, Ys, Zs) 53.13/15.19 U2_AG(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_AG(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.19 U2_AG(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> QS_IN_AA(Bigs, Bs) 53.13/15.19 U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_AG(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) 53.13/15.19 U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> APP_IN_AAG(Ls, .(X, Bs), Ys) 53.13/15.19 APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> U8_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 53.13/15.19 APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs) 53.13/15.19 53.13/15.19 The TRS R consists of the following rules: 53.13/15.19 53.13/15.19 qs_in_ag([], []) -> qs_out_ag([], []) 53.13/15.19 qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.19 part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) 53.13/15.19 less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) 53.13/15.19 less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) 53.13/15.19 U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) 53.13/15.19 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)) 53.13/15.19 part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) 53.13/15.19 less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) 53.13/15.19 less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) 53.13/15.19 U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 53.13/15.19 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)) 53.13/15.19 part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) 53.13/15.19 part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) 53.13/15.19 U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.13/15.19 U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.13/15.19 U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.13/15.19 part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) 53.13/15.19 part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) 53.13/15.19 U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.13/15.19 U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.19 qs_in_aa([], []) -> qs_out_aa([], []) 53.13/15.19 qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.19 U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.19 U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.19 U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) 53.13/15.19 app_in_aaa([], X, X) -> app_out_aaa([], X, X) 53.13/15.19 app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) 53.13/15.19 U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) 53.13/15.19 U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) 53.13/15.19 U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.19 U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) 53.13/15.19 app_in_aag([], X, X) -> app_out_aag([], X, X) 53.13/15.19 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 53.13/15.19 U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 53.13/15.19 U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) 53.13/15.19 53.13/15.19 The argument filtering Pi contains the following mapping: 53.13/15.19 qs_in_ag(x1, x2) = qs_in_ag(x2) 53.13/15.19 53.13/15.19 [] = [] 53.13/15.19 53.13/15.19 qs_out_ag(x1, x2) = qs_out_ag 53.13/15.19 53.13/15.19 U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) 53.13/15.19 53.13/15.19 part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa 53.13/15.19 53.13/15.19 U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) 53.13/15.19 53.13/15.19 less_in_aa(x1, x2) = less_in_aa 53.13/15.19 53.13/15.19 less_out_aa(x1, x2) = less_out_aa(x1) 53.13/15.19 53.13/15.19 U9_aa(x1, x2, x3) = U9_aa(x3) 53.13/15.19 53.13/15.19 U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) 53.13/15.19 53.13/15.19 part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) 53.13/15.19 53.13/15.19 U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) 53.13/15.19 53.13/15.19 less_in_ga(x1, x2) = less_in_ga(x1) 53.13/15.19 53.13/15.19 0 = 0 53.13/15.19 53.13/15.19 less_out_ga(x1, x2) = less_out_ga 53.13/15.19 53.13/15.19 s(x1) = s(x1) 53.13/15.19 53.13/15.19 U9_ga(x1, x2, x3) = U9_ga(x3) 53.13/15.19 53.13/15.19 U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x6) 53.13/15.19 53.13/15.19 U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x6) 53.13/15.19 53.13/15.19 part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa 53.13/15.19 53.13/15.19 part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa 53.13/15.19 53.13/15.19 U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) 53.13/15.19 53.13/15.19 U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) 53.13/15.19 53.13/15.19 qs_in_aa(x1, x2) = qs_in_aa 53.13/15.19 53.13/15.19 qs_out_aa(x1, x2) = qs_out_aa 53.13/15.19 53.13/15.19 U1_aa(x1, x2, x3, x4) = U1_aa(x4) 53.13/15.19 53.13/15.19 U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) 53.13/15.19 53.13/15.19 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) 53.13/15.19 53.13/15.19 U4_aa(x1, x2, x3, x4) = U4_aa(x4) 53.13/15.19 53.13/15.19 .(x1, x2) = .(x1, x2) 53.13/15.19 53.13/15.19 app_in_aaa(x1, x2, x3) = app_in_aaa 53.13/15.19 53.13/15.19 app_out_aaa(x1, x2, x3) = app_out_aaa 53.13/15.19 53.13/15.19 U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) 53.13/15.19 53.13/15.19 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) 53.13/15.19 53.13/15.19 U4_ag(x1, x2, x3, x4) = U4_ag(x4) 53.13/15.19 53.13/15.19 app_in_aag(x1, x2, x3) = app_in_aag(x3) 53.13/15.19 53.13/15.19 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) 53.13/15.19 53.13/15.19 U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x5) 53.13/15.19 53.13/15.19 QS_IN_AG(x1, x2) = QS_IN_AG(x2) 53.13/15.19 53.13/15.19 U1_AG(x1, x2, x3, x4) = U1_AG(x3, x4) 53.13/15.19 53.13/15.19 PART_IN_AAAA(x1, x2, x3, x4) = PART_IN_AAAA 53.13/15.19 53.13/15.19 U5_AAAA(x1, x2, x3, x4, x5, x6) = U5_AAAA(x6) 53.13/15.19 53.13/15.19 LESS_IN_AA(x1, x2) = LESS_IN_AA 53.13/15.19 53.13/15.19 U9_AA(x1, x2, x3) = U9_AA(x3) 53.13/15.19 53.13/15.19 U6_AAAA(x1, x2, x3, x4, x5, x6) = U6_AAAA(x6) 53.13/15.19 53.13/15.19 PART_IN_GAAA(x1, x2, x3, x4) = PART_IN_GAAA(x1) 53.13/15.19 53.13/15.19 U5_GAAA(x1, x2, x3, x4, x5, x6) = U5_GAAA(x1, x6) 53.13/15.19 53.13/15.19 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 53.13/15.19 53.13/15.19 U9_GA(x1, x2, x3) = U9_GA(x3) 53.13/15.19 53.13/15.19 U6_GAAA(x1, x2, x3, x4, x5, x6) = U6_GAAA(x6) 53.13/15.19 53.13/15.19 U7_GAAA(x1, x2, x3, x4, x5, x6) = U7_GAAA(x6) 53.13/15.19 53.13/15.19 U7_AAAA(x1, x2, x3, x4, x5, x6) = U7_AAAA(x6) 53.13/15.19 53.13/15.19 U2_AG(x1, x2, x3, x4, x5) = U2_AG(x3, x5) 53.13/15.19 53.13/15.19 QS_IN_AA(x1, x2) = QS_IN_AA 53.13/15.19 53.13/15.19 U1_AA(x1, x2, x3, x4) = U1_AA(x4) 53.13/15.19 53.13/15.19 U2_AA(x1, x2, x3, x4, x5) = U2_AA(x5) 53.13/15.19 53.13/15.19 U3_AA(x1, x2, x3, x4, x5) = U3_AA(x5) 53.13/15.19 53.13/15.19 U4_AA(x1, x2, x3, x4) = U4_AA(x4) 53.13/15.19 53.13/15.19 APP_IN_AAA(x1, x2, x3) = APP_IN_AAA 53.13/15.19 53.13/15.19 U8_AAA(x1, x2, x3, x4, x5) = U8_AAA(x5) 53.13/15.19 53.13/15.19 U3_AG(x1, x2, x3, x4, x5) = U3_AG(x3, x5) 53.13/15.19 53.13/15.19 U4_AG(x1, x2, x3, x4) = U4_AG(x4) 53.13/15.19 53.13/15.19 APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3) 53.13/15.19 53.13/15.19 U8_AAG(x1, x2, x3, x4, x5) = U8_AAG(x1, x5) 53.13/15.19 53.13/15.19 53.13/15.19 We have to consider all (P,R,Pi)-chains 53.13/15.19 ---------------------------------------- 53.13/15.19 53.13/15.19 (5) DependencyGraphProof (EQUIVALENT) 53.13/15.19 The approximation of the Dependency Graph [LOPSTR] contains 7 SCCs with 24 less nodes. 53.13/15.19 ---------------------------------------- 53.13/15.19 53.13/15.19 (6) 53.13/15.19 Complex Obligation (AND) 53.13/15.19 53.13/15.19 ---------------------------------------- 53.13/15.19 53.13/15.19 (7) 53.13/15.19 Obligation: 53.13/15.19 Pi DP problem: 53.13/15.19 The TRS P consists of the following rules: 53.13/15.19 53.13/15.19 APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs) 53.13/15.19 53.13/15.19 The TRS R consists of the following rules: 53.13/15.19 53.13/15.19 qs_in_ag([], []) -> qs_out_ag([], []) 53.13/15.19 qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.19 part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) 53.13/15.19 less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) 53.13/15.19 less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) 53.13/15.19 U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) 53.13/15.19 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)) 53.13/15.19 part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) 53.13/15.19 less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) 53.13/15.19 less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) 53.13/15.19 U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 53.13/15.19 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)) 53.13/15.19 part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) 53.13/15.19 part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) 53.13/15.19 U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.13/15.19 U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.13/15.19 U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.13/15.19 part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) 53.13/15.19 part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) 53.13/15.19 U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.13/15.19 U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.19 qs_in_aa([], []) -> qs_out_aa([], []) 53.13/15.19 qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.19 U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.19 U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.19 U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) 53.13/15.19 app_in_aaa([], X, X) -> app_out_aaa([], X, X) 53.13/15.19 app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) 53.13/15.19 U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) 53.13/15.19 U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) 53.13/15.19 U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.19 U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) 53.13/15.19 app_in_aag([], X, X) -> app_out_aag([], X, X) 53.13/15.19 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 53.13/15.19 U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 53.13/15.19 U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) 53.13/15.19 53.13/15.19 The argument filtering Pi contains the following mapping: 53.13/15.19 qs_in_ag(x1, x2) = qs_in_ag(x2) 53.13/15.19 53.13/15.19 [] = [] 53.13/15.19 53.13/15.19 qs_out_ag(x1, x2) = qs_out_ag 53.13/15.19 53.13/15.19 U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) 53.13/15.19 53.13/15.19 part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa 53.13/15.19 53.13/15.19 U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) 53.13/15.19 53.13/15.19 less_in_aa(x1, x2) = less_in_aa 53.13/15.19 53.13/15.19 less_out_aa(x1, x2) = less_out_aa(x1) 53.13/15.19 53.13/15.19 U9_aa(x1, x2, x3) = U9_aa(x3) 53.13/15.19 53.13/15.19 U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) 53.13/15.19 53.13/15.19 part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) 53.13/15.19 53.13/15.19 U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) 53.13/15.19 53.13/15.19 less_in_ga(x1, x2) = less_in_ga(x1) 53.13/15.19 53.13/15.19 0 = 0 53.13/15.19 53.13/15.19 less_out_ga(x1, x2) = less_out_ga 53.13/15.19 53.13/15.19 s(x1) = s(x1) 53.13/15.19 53.13/15.19 U9_ga(x1, x2, x3) = U9_ga(x3) 53.13/15.19 53.13/15.19 U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x6) 53.13/15.19 53.13/15.19 U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x6) 53.13/15.19 53.13/15.19 part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa 53.13/15.19 53.13/15.19 part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa 53.13/15.19 53.13/15.19 U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) 53.13/15.19 53.13/15.19 U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) 53.13/15.19 53.13/15.19 qs_in_aa(x1, x2) = qs_in_aa 53.13/15.19 53.13/15.19 qs_out_aa(x1, x2) = qs_out_aa 53.13/15.19 53.13/15.19 U1_aa(x1, x2, x3, x4) = U1_aa(x4) 53.13/15.19 53.13/15.19 U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) 53.13/15.19 53.13/15.19 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) 53.13/15.19 53.13/15.19 U4_aa(x1, x2, x3, x4) = U4_aa(x4) 53.13/15.19 53.13/15.19 .(x1, x2) = .(x1, x2) 53.13/15.19 53.13/15.19 app_in_aaa(x1, x2, x3) = app_in_aaa 53.13/15.19 53.13/15.19 app_out_aaa(x1, x2, x3) = app_out_aaa 53.13/15.19 53.13/15.19 U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) 53.13/15.19 53.13/15.19 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) 53.13/15.19 53.13/15.19 U4_ag(x1, x2, x3, x4) = U4_ag(x4) 53.13/15.19 53.13/15.19 app_in_aag(x1, x2, x3) = app_in_aag(x3) 53.13/15.19 53.13/15.19 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) 53.13/15.19 53.13/15.19 U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x5) 53.13/15.19 53.13/15.19 APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3) 53.13/15.19 53.13/15.19 53.13/15.19 We have to consider all (P,R,Pi)-chains 53.13/15.19 ---------------------------------------- 53.13/15.19 53.13/15.19 (8) UsableRulesProof (EQUIVALENT) 53.13/15.19 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 53.13/15.19 ---------------------------------------- 53.13/15.19 53.13/15.19 (9) 53.13/15.19 Obligation: 53.13/15.19 Pi DP problem: 53.13/15.19 The TRS P consists of the following rules: 53.13/15.19 53.13/15.19 APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs) 53.13/15.19 53.13/15.19 R is empty. 53.13/15.19 The argument filtering Pi contains the following mapping: 53.13/15.19 .(x1, x2) = .(x1, x2) 53.13/15.19 53.13/15.19 APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3) 53.13/15.19 53.13/15.19 53.13/15.19 We have to consider all (P,R,Pi)-chains 53.13/15.19 ---------------------------------------- 53.13/15.19 53.13/15.19 (10) PiDPToQDPProof (SOUND) 53.13/15.19 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 53.13/15.19 ---------------------------------------- 53.13/15.19 53.13/15.19 (11) 53.13/15.19 Obligation: 53.13/15.19 Q DP problem: 53.13/15.19 The TRS P consists of the following rules: 53.13/15.19 53.13/15.19 APP_IN_AAG(.(X, Zs)) -> APP_IN_AAG(Zs) 53.13/15.19 53.13/15.19 R is empty. 53.13/15.19 Q is empty. 53.13/15.19 We have to consider all (P,Q,R)-chains. 53.13/15.19 ---------------------------------------- 53.13/15.19 53.13/15.19 (12) QDPSizeChangeProof (EQUIVALENT) 53.13/15.19 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. 53.13/15.19 53.13/15.19 From the DPs we obtained the following set of size-change graphs: 53.13/15.19 *APP_IN_AAG(.(X, Zs)) -> APP_IN_AAG(Zs) 53.13/15.19 The graph contains the following edges 1 > 1 53.13/15.19 53.13/15.19 53.13/15.19 ---------------------------------------- 53.13/15.19 53.13/15.19 (13) 53.13/15.19 YES 53.13/15.19 53.13/15.19 ---------------------------------------- 53.13/15.19 53.13/15.19 (14) 53.13/15.19 Obligation: 53.13/15.19 Pi DP problem: 53.13/15.19 The TRS P consists of the following rules: 53.13/15.19 53.13/15.19 APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAA(Xs, Ys, Zs) 53.13/15.19 53.13/15.19 The TRS R consists of the following rules: 53.13/15.19 53.13/15.19 qs_in_ag([], []) -> qs_out_ag([], []) 53.13/15.19 qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.19 part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) 53.13/15.19 less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) 53.13/15.19 less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) 53.13/15.19 U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) 53.13/15.19 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)) 53.13/15.19 part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) 53.13/15.19 less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) 53.13/15.19 less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) 53.13/15.19 U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 53.13/15.19 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)) 53.13/15.19 part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) 53.13/15.19 part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) 53.13/15.19 U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.13/15.19 U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.13/15.19 U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.13/15.19 part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) 53.13/15.19 part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) 53.13/15.19 U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.13/15.19 U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.19 qs_in_aa([], []) -> qs_out_aa([], []) 53.13/15.19 qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.19 U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.19 U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.19 U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) 53.13/15.19 app_in_aaa([], X, X) -> app_out_aaa([], X, X) 53.13/15.19 app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) 53.13/15.19 U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) 53.13/15.19 U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) 53.13/15.19 U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.19 U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) 53.13/15.19 app_in_aag([], X, X) -> app_out_aag([], X, X) 53.13/15.19 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 53.13/15.19 U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 53.13/15.19 U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) 53.13/15.19 53.13/15.19 The argument filtering Pi contains the following mapping: 53.13/15.19 qs_in_ag(x1, x2) = qs_in_ag(x2) 53.13/15.19 53.13/15.19 [] = [] 53.13/15.19 53.13/15.19 qs_out_ag(x1, x2) = qs_out_ag 53.13/15.19 53.13/15.19 U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) 53.13/15.19 53.13/15.19 part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa 53.13/15.19 53.13/15.19 U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) 53.13/15.19 53.13/15.19 less_in_aa(x1, x2) = less_in_aa 53.13/15.19 53.13/15.19 less_out_aa(x1, x2) = less_out_aa(x1) 53.13/15.19 53.13/15.19 U9_aa(x1, x2, x3) = U9_aa(x3) 53.13/15.19 53.13/15.19 U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) 53.13/15.19 53.13/15.19 part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) 53.13/15.19 53.13/15.19 U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) 53.13/15.19 53.13/15.19 less_in_ga(x1, x2) = less_in_ga(x1) 53.13/15.19 53.13/15.19 0 = 0 53.13/15.19 53.13/15.19 less_out_ga(x1, x2) = less_out_ga 53.13/15.19 53.13/15.19 s(x1) = s(x1) 53.13/15.19 53.13/15.19 U9_ga(x1, x2, x3) = U9_ga(x3) 53.13/15.19 53.13/15.19 U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x6) 53.13/15.19 53.13/15.19 U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x6) 53.13/15.19 53.13/15.19 part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa 53.13/15.19 53.13/15.19 part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa 53.13/15.19 53.13/15.19 U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) 53.13/15.19 53.13/15.19 U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) 53.13/15.19 53.13/15.19 qs_in_aa(x1, x2) = qs_in_aa 53.13/15.19 53.13/15.19 qs_out_aa(x1, x2) = qs_out_aa 53.13/15.19 53.13/15.19 U1_aa(x1, x2, x3, x4) = U1_aa(x4) 53.13/15.19 53.13/15.19 U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) 53.13/15.19 53.13/15.19 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) 53.13/15.19 53.13/15.19 U4_aa(x1, x2, x3, x4) = U4_aa(x4) 53.13/15.19 53.13/15.19 .(x1, x2) = .(x1, x2) 53.13/15.19 53.13/15.19 app_in_aaa(x1, x2, x3) = app_in_aaa 53.13/15.19 53.13/15.19 app_out_aaa(x1, x2, x3) = app_out_aaa 53.13/15.19 53.13/15.19 U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) 53.13/15.19 53.13/15.19 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) 53.13/15.19 53.13/15.19 U4_ag(x1, x2, x3, x4) = U4_ag(x4) 53.13/15.19 53.13/15.19 app_in_aag(x1, x2, x3) = app_in_aag(x3) 53.13/15.19 53.13/15.19 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) 53.13/15.19 53.13/15.19 U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x5) 53.13/15.19 53.13/15.19 APP_IN_AAA(x1, x2, x3) = APP_IN_AAA 53.13/15.19 53.13/15.19 53.13/15.19 We have to consider all (P,R,Pi)-chains 53.13/15.19 ---------------------------------------- 53.13/15.19 53.13/15.19 (15) UsableRulesProof (EQUIVALENT) 53.13/15.19 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 53.13/15.19 ---------------------------------------- 53.13/15.19 53.13/15.19 (16) 53.13/15.19 Obligation: 53.13/15.19 Pi DP problem: 53.13/15.19 The TRS P consists of the following rules: 53.13/15.19 53.13/15.19 APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAA(Xs, Ys, Zs) 53.13/15.19 53.13/15.19 R is empty. 53.13/15.19 The argument filtering Pi contains the following mapping: 53.13/15.19 .(x1, x2) = .(x1, x2) 53.13/15.19 53.13/15.19 APP_IN_AAA(x1, x2, x3) = APP_IN_AAA 53.13/15.19 53.13/15.19 53.13/15.19 We have to consider all (P,R,Pi)-chains 53.13/15.19 ---------------------------------------- 53.13/15.19 53.13/15.19 (17) PiDPToQDPProof (SOUND) 53.13/15.19 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 53.13/15.19 ---------------------------------------- 53.13/15.19 53.13/15.19 (18) 53.13/15.19 Obligation: 53.13/15.19 Q DP problem: 53.13/15.19 The TRS P consists of the following rules: 53.13/15.19 53.13/15.19 APP_IN_AAA -> APP_IN_AAA 53.13/15.19 53.13/15.19 R is empty. 53.13/15.19 Q is empty. 53.13/15.19 We have to consider all (P,Q,R)-chains. 53.13/15.19 ---------------------------------------- 53.13/15.19 53.13/15.19 (19) NonTerminationLoopProof (COMPLETE) 53.13/15.19 We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. 53.13/15.19 Found a loop by semiunifying a rule from P directly. 53.13/15.19 53.13/15.19 s = APP_IN_AAA evaluates to t =APP_IN_AAA 53.13/15.19 53.13/15.19 Thus s starts an infinite chain as s semiunifies with t with the following substitutions: 53.13/15.19 * Matcher: [ ] 53.13/15.19 * Semiunifier: [ ] 53.13/15.19 53.13/15.19 -------------------------------------------------------------------------------- 53.13/15.19 Rewriting sequence 53.13/15.19 53.13/15.19 The DP semiunifies directly so there is only one rewrite step from APP_IN_AAA to APP_IN_AAA. 53.13/15.19 53.13/15.19 53.13/15.19 53.13/15.19 53.13/15.19 ---------------------------------------- 53.13/15.19 53.13/15.19 (20) 53.13/15.19 NO 53.13/15.19 53.13/15.19 ---------------------------------------- 53.13/15.19 53.13/15.19 (21) 53.13/15.19 Obligation: 53.13/15.19 Pi DP problem: 53.13/15.19 The TRS P consists of the following rules: 53.13/15.19 53.13/15.19 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 53.13/15.19 53.13/15.19 The TRS R consists of the following rules: 53.13/15.19 53.13/15.19 qs_in_ag([], []) -> qs_out_ag([], []) 53.13/15.19 qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.19 part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) 53.13/15.19 less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) 53.13/15.19 less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) 53.13/15.19 U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) 53.13/15.19 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)) 53.13/15.19 part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) 53.13/15.19 less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) 53.13/15.19 less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) 53.13/15.19 U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 53.13/15.19 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)) 53.13/15.19 part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) 53.13/15.19 part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) 53.13/15.19 U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.13/15.19 U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.13/15.19 U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.13/15.19 part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) 53.13/15.19 part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) 53.13/15.19 U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.13/15.19 U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.19 qs_in_aa([], []) -> qs_out_aa([], []) 53.13/15.19 qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.19 U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.19 U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.19 U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) 53.13/15.19 app_in_aaa([], X, X) -> app_out_aaa([], X, X) 53.13/15.19 app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) 53.13/15.19 U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) 53.13/15.19 U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) 53.13/15.19 U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.19 U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) 53.13/15.19 app_in_aag([], X, X) -> app_out_aag([], X, X) 53.13/15.19 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 53.13/15.19 U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 53.13/15.19 U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) 53.13/15.19 53.13/15.19 The argument filtering Pi contains the following mapping: 53.13/15.19 qs_in_ag(x1, x2) = qs_in_ag(x2) 53.13/15.19 53.13/15.19 [] = [] 53.13/15.19 53.13/15.19 qs_out_ag(x1, x2) = qs_out_ag 53.13/15.19 53.13/15.19 U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) 53.13/15.19 53.13/15.19 part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa 53.13/15.19 53.13/15.19 U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) 53.13/15.19 53.13/15.19 less_in_aa(x1, x2) = less_in_aa 53.13/15.19 53.13/15.19 less_out_aa(x1, x2) = less_out_aa(x1) 53.13/15.19 53.13/15.19 U9_aa(x1, x2, x3) = U9_aa(x3) 53.13/15.19 53.13/15.19 U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) 53.13/15.19 53.13/15.19 part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) 53.13/15.19 53.13/15.19 U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) 53.13/15.19 53.13/15.19 less_in_ga(x1, x2) = less_in_ga(x1) 53.13/15.19 53.13/15.19 0 = 0 53.13/15.19 53.13/15.19 less_out_ga(x1, x2) = less_out_ga 53.13/15.19 53.13/15.19 s(x1) = s(x1) 53.13/15.19 53.13/15.19 U9_ga(x1, x2, x3) = U9_ga(x3) 53.13/15.19 53.13/15.19 U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x6) 53.13/15.19 53.13/15.19 U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x6) 53.13/15.19 53.13/15.19 part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa 53.13/15.19 53.13/15.19 part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa 53.13/15.19 53.13/15.19 U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) 53.13/15.19 53.13/15.19 U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) 53.13/15.19 53.13/15.19 qs_in_aa(x1, x2) = qs_in_aa 53.13/15.19 53.13/15.19 qs_out_aa(x1, x2) = qs_out_aa 53.13/15.19 53.13/15.19 U1_aa(x1, x2, x3, x4) = U1_aa(x4) 53.13/15.19 53.13/15.19 U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) 53.13/15.19 53.13/15.19 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) 53.13/15.19 53.13/15.19 U4_aa(x1, x2, x3, x4) = U4_aa(x4) 53.13/15.19 53.13/15.19 .(x1, x2) = .(x1, x2) 53.13/15.19 53.13/15.19 app_in_aaa(x1, x2, x3) = app_in_aaa 53.13/15.19 53.13/15.19 app_out_aaa(x1, x2, x3) = app_out_aaa 53.13/15.19 53.13/15.19 U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) 53.13/15.19 53.13/15.19 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) 53.13/15.19 53.13/15.19 U4_ag(x1, x2, x3, x4) = U4_ag(x4) 53.13/15.19 53.13/15.19 app_in_aag(x1, x2, x3) = app_in_aag(x3) 53.13/15.19 53.13/15.19 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) 53.13/15.19 53.13/15.19 U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x5) 53.13/15.19 53.13/15.19 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 53.13/15.19 53.13/15.19 53.13/15.19 We have to consider all (P,R,Pi)-chains 53.13/15.19 ---------------------------------------- 53.13/15.19 53.13/15.19 (22) UsableRulesProof (EQUIVALENT) 53.13/15.19 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 53.13/15.19 ---------------------------------------- 53.13/15.19 53.13/15.19 (23) 53.13/15.19 Obligation: 53.13/15.19 Pi DP problem: 53.13/15.19 The TRS P consists of the following rules: 53.13/15.19 53.13/15.19 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 53.13/15.19 53.13/15.19 R is empty. 53.13/15.19 The argument filtering Pi contains the following mapping: 53.13/15.19 s(x1) = s(x1) 53.13/15.19 53.13/15.19 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 53.13/15.19 53.13/15.19 53.13/15.19 We have to consider all (P,R,Pi)-chains 53.13/15.19 ---------------------------------------- 53.13/15.19 53.13/15.19 (24) PiDPToQDPProof (SOUND) 53.13/15.19 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 53.13/15.19 ---------------------------------------- 53.13/15.19 53.13/15.19 (25) 53.13/15.19 Obligation: 53.13/15.19 Q DP problem: 53.13/15.19 The TRS P consists of the following rules: 53.13/15.19 53.13/15.19 LESS_IN_GA(s(X)) -> LESS_IN_GA(X) 53.13/15.19 53.13/15.19 R is empty. 53.13/15.19 Q is empty. 53.13/15.19 We have to consider all (P,Q,R)-chains. 53.13/15.19 ---------------------------------------- 53.13/15.19 53.13/15.19 (26) QDPSizeChangeProof (EQUIVALENT) 53.13/15.19 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. 53.13/15.19 53.13/15.19 From the DPs we obtained the following set of size-change graphs: 53.13/15.19 *LESS_IN_GA(s(X)) -> LESS_IN_GA(X) 53.13/15.19 The graph contains the following edges 1 > 1 53.13/15.19 53.13/15.19 53.13/15.19 ---------------------------------------- 53.13/15.19 53.13/15.19 (27) 53.13/15.19 YES 53.13/15.19 53.13/15.19 ---------------------------------------- 53.13/15.19 53.13/15.19 (28) 53.13/15.19 Obligation: 53.13/15.19 Pi DP problem: 53.13/15.19 The TRS P consists of the following rules: 53.13/15.19 53.13/15.19 U5_GAAA(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs) 53.13/15.19 PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GAAA(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) 53.13/15.19 PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_GAAA(X, Xs, Ls, Bs) 53.13/15.19 53.13/15.19 The TRS R consists of the following rules: 53.13/15.19 53.13/15.19 qs_in_ag([], []) -> qs_out_ag([], []) 53.13/15.19 qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.19 part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) 53.13/15.19 less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) 53.13/15.19 less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) 53.13/15.19 U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) 53.13/15.19 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)) 53.13/15.19 part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) 53.13/15.19 less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) 53.13/15.19 less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) 53.13/15.19 U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 53.13/15.19 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)) 53.13/15.19 part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) 53.13/15.19 part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) 53.13/15.19 U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.13/15.19 U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.13/15.19 U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.13/15.19 part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) 53.13/15.19 part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) 53.13/15.19 U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.13/15.19 U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.19 qs_in_aa([], []) -> qs_out_aa([], []) 53.13/15.19 qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.19 U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.19 U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.19 U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) 53.13/15.19 app_in_aaa([], X, X) -> app_out_aaa([], X, X) 53.13/15.19 app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) 53.13/15.19 U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) 53.13/15.19 U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) 53.13/15.19 U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.19 U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) 53.13/15.19 app_in_aag([], X, X) -> app_out_aag([], X, X) 53.13/15.19 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 53.13/15.19 U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 53.13/15.19 U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) 53.13/15.19 53.13/15.19 The argument filtering Pi contains the following mapping: 53.13/15.19 qs_in_ag(x1, x2) = qs_in_ag(x2) 53.13/15.19 53.13/15.19 [] = [] 53.13/15.19 53.13/15.19 qs_out_ag(x1, x2) = qs_out_ag 53.13/15.19 53.13/15.19 U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) 53.13/15.19 53.13/15.19 part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa 53.13/15.19 53.13/15.19 U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) 53.13/15.19 53.13/15.19 less_in_aa(x1, x2) = less_in_aa 53.13/15.19 53.13/15.19 less_out_aa(x1, x2) = less_out_aa(x1) 53.13/15.19 53.13/15.19 U9_aa(x1, x2, x3) = U9_aa(x3) 53.13/15.19 53.13/15.19 U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) 53.13/15.19 53.13/15.19 part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) 53.13/15.19 53.13/15.19 U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) 53.13/15.19 53.13/15.19 less_in_ga(x1, x2) = less_in_ga(x1) 53.13/15.19 53.13/15.19 0 = 0 53.13/15.19 53.13/15.19 less_out_ga(x1, x2) = less_out_ga 53.13/15.19 53.13/15.19 s(x1) = s(x1) 53.13/15.19 53.13/15.19 U9_ga(x1, x2, x3) = U9_ga(x3) 53.13/15.19 53.13/15.19 U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x6) 53.13/15.19 53.13/15.19 U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x6) 53.13/15.19 53.13/15.19 part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa 53.13/15.19 53.13/15.19 part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa 53.13/15.19 53.13/15.19 U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) 53.13/15.19 53.13/15.19 U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) 53.13/15.19 53.13/15.19 qs_in_aa(x1, x2) = qs_in_aa 53.13/15.19 53.13/15.19 qs_out_aa(x1, x2) = qs_out_aa 53.13/15.19 53.13/15.19 U1_aa(x1, x2, x3, x4) = U1_aa(x4) 53.13/15.19 53.13/15.19 U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) 53.13/15.19 53.13/15.19 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) 53.13/15.19 53.13/15.19 U4_aa(x1, x2, x3, x4) = U4_aa(x4) 53.13/15.19 53.13/15.19 .(x1, x2) = .(x1, x2) 53.13/15.19 53.13/15.19 app_in_aaa(x1, x2, x3) = app_in_aaa 53.13/15.19 53.13/15.19 app_out_aaa(x1, x2, x3) = app_out_aaa 53.13/15.19 53.13/15.19 U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) 53.13/15.19 53.13/15.19 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) 53.13/15.19 53.13/15.19 U4_ag(x1, x2, x3, x4) = U4_ag(x4) 53.13/15.19 53.13/15.19 app_in_aag(x1, x2, x3) = app_in_aag(x3) 53.13/15.19 53.13/15.19 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) 53.13/15.19 53.13/15.19 U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x5) 53.13/15.19 53.13/15.19 PART_IN_GAAA(x1, x2, x3, x4) = PART_IN_GAAA(x1) 53.13/15.19 53.13/15.19 U5_GAAA(x1, x2, x3, x4, x5, x6) = U5_GAAA(x1, x6) 53.13/15.19 53.13/15.19 53.13/15.19 We have to consider all (P,R,Pi)-chains 53.13/15.19 ---------------------------------------- 53.13/15.19 53.13/15.19 (29) UsableRulesProof (EQUIVALENT) 53.13/15.19 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 53.13/15.19 ---------------------------------------- 53.13/15.19 53.13/15.19 (30) 53.13/15.19 Obligation: 53.13/15.19 Pi DP problem: 53.13/15.19 The TRS P consists of the following rules: 53.13/15.19 53.13/15.19 U5_GAAA(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs) 53.13/15.19 PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GAAA(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) 53.13/15.19 PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_GAAA(X, Xs, Ls, Bs) 53.13/15.19 53.13/15.19 The TRS R consists of the following rules: 53.13/15.19 53.13/15.19 less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) 53.13/15.19 less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) 53.13/15.19 U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 53.13/15.19 53.13/15.19 The argument filtering Pi contains the following mapping: 53.13/15.19 less_in_ga(x1, x2) = less_in_ga(x1) 53.13/15.19 53.13/15.19 0 = 0 53.13/15.19 53.13/15.19 less_out_ga(x1, x2) = less_out_ga 53.13/15.19 53.13/15.19 s(x1) = s(x1) 53.13/15.19 53.13/15.19 U9_ga(x1, x2, x3) = U9_ga(x3) 53.13/15.19 53.13/15.19 .(x1, x2) = .(x1, x2) 53.13/15.19 53.13/15.19 PART_IN_GAAA(x1, x2, x3, x4) = PART_IN_GAAA(x1) 53.13/15.19 53.13/15.19 U5_GAAA(x1, x2, x3, x4, x5, x6) = U5_GAAA(x1, x6) 53.13/15.19 53.13/15.19 53.13/15.19 We have to consider all (P,R,Pi)-chains 53.13/15.19 ---------------------------------------- 53.13/15.19 53.13/15.19 (31) PiDPToQDPProof (SOUND) 53.13/15.19 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 53.13/15.19 ---------------------------------------- 53.13/15.19 53.13/15.19 (32) 53.13/15.19 Obligation: 53.13/15.19 Q DP problem: 53.13/15.19 The TRS P consists of the following rules: 53.13/15.19 53.13/15.19 U5_GAAA(X, less_out_ga) -> PART_IN_GAAA(X) 53.13/15.19 PART_IN_GAAA(X) -> U5_GAAA(X, less_in_ga(X)) 53.13/15.19 PART_IN_GAAA(X) -> PART_IN_GAAA(X) 53.13/15.19 53.13/15.19 The TRS R consists of the following rules: 53.13/15.19 53.13/15.19 less_in_ga(0) -> less_out_ga 53.13/15.19 less_in_ga(s(X)) -> U9_ga(less_in_ga(X)) 53.13/15.19 U9_ga(less_out_ga) -> less_out_ga 53.13/15.19 53.13/15.19 The set Q consists of the following terms: 53.13/15.19 53.13/15.19 less_in_ga(x0) 53.13/15.19 U9_ga(x0) 53.13/15.19 53.13/15.19 We have to consider all (P,Q,R)-chains. 53.13/15.19 ---------------------------------------- 53.13/15.19 53.13/15.19 (33) TransformationProof (SOUND) 53.13/15.19 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]: 53.13/15.19 53.13/15.19 (PART_IN_GAAA(0) -> U5_GAAA(0, less_out_ga),PART_IN_GAAA(0) -> U5_GAAA(0, less_out_ga)) 53.13/15.19 (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)))) 53.13/15.19 53.13/15.19 53.13/15.19 ---------------------------------------- 53.13/15.19 53.13/15.19 (34) 53.13/15.19 Obligation: 53.13/15.19 Q DP problem: 53.13/15.19 The TRS P consists of the following rules: 53.13/15.19 53.13/15.19 U5_GAAA(X, less_out_ga) -> PART_IN_GAAA(X) 53.13/15.19 PART_IN_GAAA(X) -> PART_IN_GAAA(X) 53.13/15.19 PART_IN_GAAA(0) -> U5_GAAA(0, less_out_ga) 53.13/15.19 PART_IN_GAAA(s(x0)) -> U5_GAAA(s(x0), U9_ga(less_in_ga(x0))) 53.13/15.19 53.13/15.19 The TRS R consists of the following rules: 53.13/15.19 53.13/15.19 less_in_ga(0) -> less_out_ga 53.13/15.19 less_in_ga(s(X)) -> U9_ga(less_in_ga(X)) 53.13/15.19 U9_ga(less_out_ga) -> less_out_ga 53.13/15.19 53.13/15.19 The set Q consists of the following terms: 53.13/15.19 53.13/15.19 less_in_ga(x0) 53.13/15.19 U9_ga(x0) 53.13/15.19 53.13/15.19 We have to consider all (P,Q,R)-chains. 53.13/15.19 ---------------------------------------- 53.13/15.19 53.13/15.19 (35) TransformationProof (EQUIVALENT) 53.13/15.19 By instantiating [LPAR04] the rule U5_GAAA(X, less_out_ga) -> PART_IN_GAAA(X) we obtained the following new rules [LPAR04]: 53.13/15.19 53.13/15.19 (U5_GAAA(0, less_out_ga) -> PART_IN_GAAA(0),U5_GAAA(0, less_out_ga) -> PART_IN_GAAA(0)) 53.13/15.19 (U5_GAAA(s(z0), less_out_ga) -> PART_IN_GAAA(s(z0)),U5_GAAA(s(z0), less_out_ga) -> PART_IN_GAAA(s(z0))) 53.13/15.19 53.13/15.19 53.13/15.19 ---------------------------------------- 53.13/15.19 53.13/15.19 (36) 53.13/15.19 Obligation: 53.13/15.19 Q DP problem: 53.13/15.19 The TRS P consists of the following rules: 53.13/15.19 53.13/15.19 PART_IN_GAAA(X) -> PART_IN_GAAA(X) 53.13/15.19 PART_IN_GAAA(0) -> U5_GAAA(0, less_out_ga) 53.13/15.19 PART_IN_GAAA(s(x0)) -> U5_GAAA(s(x0), U9_ga(less_in_ga(x0))) 53.13/15.19 U5_GAAA(0, less_out_ga) -> PART_IN_GAAA(0) 53.13/15.19 U5_GAAA(s(z0), less_out_ga) -> PART_IN_GAAA(s(z0)) 53.13/15.19 53.13/15.19 The TRS R consists of the following rules: 53.13/15.19 53.13/15.19 less_in_ga(0) -> less_out_ga 53.13/15.19 less_in_ga(s(X)) -> U9_ga(less_in_ga(X)) 53.13/15.19 U9_ga(less_out_ga) -> less_out_ga 53.13/15.19 53.13/15.19 The set Q consists of the following terms: 53.13/15.19 53.13/15.19 less_in_ga(x0) 53.13/15.19 U9_ga(x0) 53.13/15.19 53.13/15.19 We have to consider all (P,Q,R)-chains. 53.13/15.19 ---------------------------------------- 53.13/15.19 53.13/15.19 (37) NonTerminationLoopProof (COMPLETE) 53.13/15.19 We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. 53.13/15.19 Found a loop by semiunifying a rule from P directly. 53.13/15.19 53.13/15.19 s = PART_IN_GAAA(X) evaluates to t =PART_IN_GAAA(X) 53.13/15.19 53.13/15.19 Thus s starts an infinite chain as s semiunifies with t with the following substitutions: 53.13/15.19 * Matcher: [ ] 53.13/15.19 * Semiunifier: [ ] 53.13/15.19 53.13/15.19 -------------------------------------------------------------------------------- 53.13/15.19 Rewriting sequence 53.13/15.19 53.13/15.19 The DP semiunifies directly so there is only one rewrite step from PART_IN_GAAA(X) to PART_IN_GAAA(X). 53.13/15.20 53.13/15.20 53.13/15.20 53.13/15.20 53.13/15.20 ---------------------------------------- 53.13/15.20 53.13/15.20 (38) 53.13/15.20 NO 53.13/15.20 53.13/15.20 ---------------------------------------- 53.13/15.20 53.13/15.20 (39) 53.13/15.20 Obligation: 53.13/15.20 Pi DP problem: 53.13/15.20 The TRS P consists of the following rules: 53.13/15.20 53.13/15.20 LESS_IN_AA(s(X), s(Y)) -> LESS_IN_AA(X, Y) 53.13/15.20 53.13/15.20 The TRS R consists of the following rules: 53.13/15.20 53.13/15.20 qs_in_ag([], []) -> qs_out_ag([], []) 53.13/15.20 qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.20 part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) 53.13/15.20 less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) 53.13/15.20 less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) 53.13/15.20 U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) 53.13/15.20 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)) 53.13/15.20 part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) 53.13/15.20 less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) 53.13/15.20 less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) 53.13/15.20 U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 53.13/15.20 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)) 53.13/15.20 part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) 53.13/15.20 part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) 53.13/15.20 U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.13/15.20 U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.13/15.20 U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.13/15.20 part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) 53.13/15.20 part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) 53.13/15.20 U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.13/15.20 U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.20 qs_in_aa([], []) -> qs_out_aa([], []) 53.13/15.20 qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.20 U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.20 U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.20 U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) 53.13/15.20 app_in_aaa([], X, X) -> app_out_aaa([], X, X) 53.13/15.20 app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) 53.13/15.20 U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) 53.13/15.20 U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) 53.13/15.20 U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.20 U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) 53.13/15.20 app_in_aag([], X, X) -> app_out_aag([], X, X) 53.13/15.20 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 53.13/15.20 U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 53.13/15.20 U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) 53.13/15.20 53.13/15.20 The argument filtering Pi contains the following mapping: 53.13/15.20 qs_in_ag(x1, x2) = qs_in_ag(x2) 53.13/15.20 53.13/15.20 [] = [] 53.13/15.20 53.13/15.20 qs_out_ag(x1, x2) = qs_out_ag 53.13/15.20 53.13/15.20 U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) 53.13/15.20 53.13/15.20 part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa 53.13/15.20 53.13/15.20 U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) 53.13/15.20 53.13/15.20 less_in_aa(x1, x2) = less_in_aa 53.13/15.20 53.13/15.20 less_out_aa(x1, x2) = less_out_aa(x1) 53.13/15.20 53.13/15.20 U9_aa(x1, x2, x3) = U9_aa(x3) 53.13/15.20 53.13/15.20 U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) 53.13/15.20 53.13/15.20 part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) 53.13/15.20 53.13/15.20 U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) 53.13/15.20 53.13/15.20 less_in_ga(x1, x2) = less_in_ga(x1) 53.13/15.20 53.13/15.20 0 = 0 53.13/15.20 53.13/15.20 less_out_ga(x1, x2) = less_out_ga 53.13/15.20 53.13/15.20 s(x1) = s(x1) 53.13/15.20 53.13/15.20 U9_ga(x1, x2, x3) = U9_ga(x3) 53.13/15.20 53.13/15.20 U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x6) 53.13/15.20 53.13/15.20 U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x6) 53.13/15.20 53.13/15.20 part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa 53.13/15.20 53.13/15.20 part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa 53.13/15.20 53.13/15.20 U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) 53.13/15.20 53.13/15.20 U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) 53.13/15.20 53.13/15.20 qs_in_aa(x1, x2) = qs_in_aa 53.13/15.20 53.13/15.20 qs_out_aa(x1, x2) = qs_out_aa 53.13/15.20 53.13/15.20 U1_aa(x1, x2, x3, x4) = U1_aa(x4) 53.13/15.20 53.13/15.20 U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) 53.13/15.20 53.13/15.20 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) 53.13/15.20 53.13/15.20 U4_aa(x1, x2, x3, x4) = U4_aa(x4) 53.13/15.20 53.13/15.20 .(x1, x2) = .(x1, x2) 53.13/15.20 53.13/15.20 app_in_aaa(x1, x2, x3) = app_in_aaa 53.13/15.20 53.13/15.20 app_out_aaa(x1, x2, x3) = app_out_aaa 53.13/15.20 53.13/15.20 U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) 53.13/15.20 53.13/15.20 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) 53.13/15.20 53.13/15.20 U4_ag(x1, x2, x3, x4) = U4_ag(x4) 53.13/15.20 53.13/15.20 app_in_aag(x1, x2, x3) = app_in_aag(x3) 53.13/15.20 53.13/15.20 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) 53.13/15.20 53.13/15.20 U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x5) 53.13/15.20 53.13/15.20 LESS_IN_AA(x1, x2) = LESS_IN_AA 53.13/15.20 53.13/15.20 53.13/15.20 We have to consider all (P,R,Pi)-chains 53.13/15.20 ---------------------------------------- 53.13/15.20 53.13/15.20 (40) UsableRulesProof (EQUIVALENT) 53.13/15.20 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 53.13/15.20 ---------------------------------------- 53.13/15.20 53.13/15.20 (41) 53.13/15.20 Obligation: 53.13/15.20 Pi DP problem: 53.13/15.20 The TRS P consists of the following rules: 53.13/15.20 53.13/15.20 LESS_IN_AA(s(X), s(Y)) -> LESS_IN_AA(X, Y) 53.13/15.20 53.13/15.20 R is empty. 53.13/15.20 The argument filtering Pi contains the following mapping: 53.13/15.20 s(x1) = s(x1) 53.13/15.20 53.13/15.20 LESS_IN_AA(x1, x2) = LESS_IN_AA 53.13/15.20 53.13/15.20 53.13/15.20 We have to consider all (P,R,Pi)-chains 53.13/15.20 ---------------------------------------- 53.13/15.20 53.13/15.20 (42) PiDPToQDPProof (SOUND) 53.13/15.20 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 53.13/15.20 ---------------------------------------- 53.13/15.20 53.13/15.20 (43) 53.13/15.20 Obligation: 53.13/15.20 Q DP problem: 53.13/15.20 The TRS P consists of the following rules: 53.13/15.20 53.13/15.20 LESS_IN_AA -> LESS_IN_AA 53.13/15.20 53.13/15.20 R is empty. 53.13/15.20 Q is empty. 53.13/15.20 We have to consider all (P,Q,R)-chains. 53.13/15.20 ---------------------------------------- 53.13/15.20 53.13/15.20 (44) 53.13/15.20 Obligation: 53.13/15.20 Pi DP problem: 53.13/15.20 The TRS P consists of the following rules: 53.13/15.20 53.13/15.20 PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_AAAA(X, Xs, Ls, Bs) 53.13/15.20 53.13/15.20 The TRS R consists of the following rules: 53.13/15.20 53.13/15.20 qs_in_ag([], []) -> qs_out_ag([], []) 53.13/15.20 qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.20 part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) 53.13/15.20 less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) 53.13/15.20 less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) 53.13/15.20 U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) 53.13/15.20 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)) 53.13/15.20 part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) 53.13/15.20 less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) 53.13/15.20 less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) 53.13/15.20 U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 53.13/15.20 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)) 53.13/15.20 part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) 53.13/15.20 part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) 53.13/15.20 U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.13/15.20 U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.13/15.20 U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.13/15.20 part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) 53.13/15.20 part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) 53.13/15.20 U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.13/15.20 U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.20 qs_in_aa([], []) -> qs_out_aa([], []) 53.13/15.20 qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.20 U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.20 U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.20 U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) 53.13/15.20 app_in_aaa([], X, X) -> app_out_aaa([], X, X) 53.13/15.20 app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) 53.13/15.20 U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) 53.13/15.20 U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) 53.13/15.20 U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.20 U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) 53.13/15.20 app_in_aag([], X, X) -> app_out_aag([], X, X) 53.13/15.20 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 53.13/15.20 U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 53.13/15.20 U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) 53.13/15.20 53.13/15.20 The argument filtering Pi contains the following mapping: 53.13/15.20 qs_in_ag(x1, x2) = qs_in_ag(x2) 53.13/15.20 53.13/15.20 [] = [] 53.13/15.20 53.13/15.20 qs_out_ag(x1, x2) = qs_out_ag 53.13/15.20 53.13/15.20 U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) 53.13/15.20 53.13/15.20 part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa 53.13/15.20 53.13/15.20 U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) 53.13/15.20 53.13/15.20 less_in_aa(x1, x2) = less_in_aa 53.13/15.20 53.13/15.20 less_out_aa(x1, x2) = less_out_aa(x1) 53.13/15.20 53.13/15.20 U9_aa(x1, x2, x3) = U9_aa(x3) 53.13/15.20 53.13/15.20 U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) 53.13/15.20 53.13/15.20 part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) 53.13/15.20 53.13/15.20 U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) 53.13/15.20 53.13/15.20 less_in_ga(x1, x2) = less_in_ga(x1) 53.13/15.20 53.13/15.20 0 = 0 53.13/15.20 53.13/15.20 less_out_ga(x1, x2) = less_out_ga 53.13/15.20 53.13/15.20 s(x1) = s(x1) 53.13/15.20 53.13/15.20 U9_ga(x1, x2, x3) = U9_ga(x3) 53.13/15.20 53.13/15.20 U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x6) 53.13/15.20 53.13/15.20 U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x6) 53.13/15.20 53.13/15.20 part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa 53.13/15.20 53.13/15.20 part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa 53.13/15.20 53.13/15.20 U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) 53.13/15.20 53.13/15.20 U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) 53.13/15.20 53.13/15.20 qs_in_aa(x1, x2) = qs_in_aa 53.13/15.20 53.13/15.20 qs_out_aa(x1, x2) = qs_out_aa 53.13/15.20 53.13/15.20 U1_aa(x1, x2, x3, x4) = U1_aa(x4) 53.13/15.20 53.13/15.20 U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) 53.13/15.20 53.13/15.20 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) 53.13/15.20 53.13/15.20 U4_aa(x1, x2, x3, x4) = U4_aa(x4) 53.13/15.20 53.13/15.20 .(x1, x2) = .(x1, x2) 53.13/15.20 53.13/15.20 app_in_aaa(x1, x2, x3) = app_in_aaa 53.13/15.20 53.13/15.20 app_out_aaa(x1, x2, x3) = app_out_aaa 53.13/15.20 53.13/15.20 U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) 53.13/15.20 53.13/15.20 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) 53.13/15.20 53.13/15.20 U4_ag(x1, x2, x3, x4) = U4_ag(x4) 53.13/15.20 53.13/15.20 app_in_aag(x1, x2, x3) = app_in_aag(x3) 53.13/15.20 53.13/15.20 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) 53.13/15.20 53.13/15.20 U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x5) 53.13/15.20 53.13/15.20 PART_IN_AAAA(x1, x2, x3, x4) = PART_IN_AAAA 53.13/15.20 53.13/15.20 53.13/15.20 We have to consider all (P,R,Pi)-chains 53.13/15.20 ---------------------------------------- 53.13/15.20 53.13/15.20 (45) UsableRulesProof (EQUIVALENT) 53.13/15.20 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 53.13/15.20 ---------------------------------------- 53.13/15.20 53.13/15.20 (46) 53.13/15.20 Obligation: 53.13/15.20 Pi DP problem: 53.13/15.20 The TRS P consists of the following rules: 53.13/15.20 53.13/15.20 PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_AAAA(X, Xs, Ls, Bs) 53.13/15.20 53.13/15.20 R is empty. 53.13/15.20 The argument filtering Pi contains the following mapping: 53.13/15.20 .(x1, x2) = .(x1, x2) 53.13/15.20 53.13/15.20 PART_IN_AAAA(x1, x2, x3, x4) = PART_IN_AAAA 53.13/15.20 53.13/15.20 53.13/15.20 We have to consider all (P,R,Pi)-chains 53.13/15.20 ---------------------------------------- 53.13/15.20 53.13/15.20 (47) 53.13/15.20 Obligation: 53.13/15.20 Pi DP problem: 53.13/15.20 The TRS P consists of the following rules: 53.13/15.20 53.13/15.20 QS_IN_AA(.(X, Xs), Ys) -> U1_AA(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.20 U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AA(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.20 U2_AA(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> QS_IN_AA(Bigs, Bs) 53.13/15.20 U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> QS_IN_AA(Littles, Ls) 53.13/15.20 53.13/15.20 The TRS R consists of the following rules: 53.13/15.20 53.13/15.20 qs_in_ag([], []) -> qs_out_ag([], []) 53.13/15.20 qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.20 part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) 53.13/15.20 less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) 53.13/15.20 less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) 53.13/15.20 U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) 53.13/15.20 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)) 53.13/15.20 part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) 53.13/15.20 less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) 53.13/15.20 less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) 53.13/15.20 U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 53.13/15.20 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)) 53.13/15.20 part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) 53.13/15.20 part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) 53.13/15.20 U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.13/15.20 U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.13/15.20 U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.13/15.20 part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) 53.13/15.20 part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) 53.13/15.20 U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.13/15.20 U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.20 qs_in_aa([], []) -> qs_out_aa([], []) 53.13/15.20 qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.20 U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.20 U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.20 U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) 53.13/15.20 app_in_aaa([], X, X) -> app_out_aaa([], X, X) 53.13/15.20 app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) 53.13/15.20 U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) 53.13/15.20 U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) 53.13/15.20 U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.20 U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) 53.13/15.20 app_in_aag([], X, X) -> app_out_aag([], X, X) 53.13/15.20 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 53.13/15.20 U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 53.13/15.20 U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) 53.13/15.20 53.13/15.20 The argument filtering Pi contains the following mapping: 53.13/15.20 qs_in_ag(x1, x2) = qs_in_ag(x2) 53.13/15.20 53.13/15.20 [] = [] 53.13/15.20 53.13/15.20 qs_out_ag(x1, x2) = qs_out_ag 53.13/15.20 53.13/15.20 U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) 53.13/15.20 53.13/15.20 part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa 53.13/15.20 53.13/15.20 U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) 53.13/15.20 53.13/15.20 less_in_aa(x1, x2) = less_in_aa 53.13/15.20 53.13/15.20 less_out_aa(x1, x2) = less_out_aa(x1) 53.13/15.20 53.13/15.20 U9_aa(x1, x2, x3) = U9_aa(x3) 53.13/15.20 53.13/15.20 U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) 53.13/15.20 53.13/15.20 part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) 53.13/15.20 53.13/15.20 U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) 53.13/15.20 53.13/15.20 less_in_ga(x1, x2) = less_in_ga(x1) 53.13/15.20 53.13/15.20 0 = 0 53.13/15.20 53.13/15.20 less_out_ga(x1, x2) = less_out_ga 53.13/15.20 53.13/15.20 s(x1) = s(x1) 53.13/15.20 53.13/15.20 U9_ga(x1, x2, x3) = U9_ga(x3) 53.13/15.20 53.13/15.20 U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x6) 53.13/15.20 53.13/15.20 U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x6) 53.13/15.20 53.13/15.20 part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa 53.13/15.20 53.13/15.20 part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa 53.13/15.20 53.13/15.20 U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) 53.13/15.20 53.13/15.20 U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) 53.13/15.20 53.13/15.20 qs_in_aa(x1, x2) = qs_in_aa 53.13/15.20 53.13/15.20 qs_out_aa(x1, x2) = qs_out_aa 53.13/15.20 53.13/15.20 U1_aa(x1, x2, x3, x4) = U1_aa(x4) 53.13/15.20 53.13/15.20 U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) 53.13/15.20 53.13/15.20 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) 53.13/15.20 53.13/15.20 U4_aa(x1, x2, x3, x4) = U4_aa(x4) 53.13/15.20 53.13/15.20 .(x1, x2) = .(x1, x2) 53.13/15.20 53.13/15.20 app_in_aaa(x1, x2, x3) = app_in_aaa 53.13/15.20 53.13/15.20 app_out_aaa(x1, x2, x3) = app_out_aaa 53.13/15.20 53.13/15.20 U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) 53.13/15.20 53.13/15.20 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) 53.13/15.20 53.13/15.20 U4_ag(x1, x2, x3, x4) = U4_ag(x4) 53.13/15.20 53.13/15.20 app_in_aag(x1, x2, x3) = app_in_aag(x3) 53.13/15.20 53.13/15.20 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2) 53.13/15.20 53.13/15.20 U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x5) 53.13/15.20 53.13/15.20 QS_IN_AA(x1, x2) = QS_IN_AA 53.13/15.20 53.13/15.20 U1_AA(x1, x2, x3, x4) = U1_AA(x4) 53.13/15.20 53.13/15.20 U2_AA(x1, x2, x3, x4, x5) = U2_AA(x5) 53.13/15.20 53.13/15.20 53.13/15.20 We have to consider all (P,R,Pi)-chains 53.13/15.20 ---------------------------------------- 53.13/15.20 53.13/15.20 (48) UsableRulesProof (EQUIVALENT) 53.13/15.20 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 53.13/15.20 ---------------------------------------- 53.13/15.20 53.13/15.20 (49) 53.13/15.20 Obligation: 53.13/15.20 Pi DP problem: 53.13/15.20 The TRS P consists of the following rules: 53.13/15.20 53.13/15.20 QS_IN_AA(.(X, Xs), Ys) -> U1_AA(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.20 U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AA(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.20 U2_AA(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> QS_IN_AA(Bigs, Bs) 53.13/15.20 U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> QS_IN_AA(Littles, Ls) 53.13/15.20 53.13/15.20 The TRS R consists of the following rules: 53.13/15.20 53.13/15.20 part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) 53.13/15.20 part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) 53.13/15.20 part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) 53.13/15.20 qs_in_aa([], []) -> qs_out_aa([], []) 53.13/15.20 qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.20 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)) 53.13/15.20 U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.13/15.20 U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.20 less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) 53.13/15.20 less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) 53.13/15.20 U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.13/15.20 U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.20 U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) 53.13/15.20 part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) 53.13/15.20 part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) 53.13/15.20 part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) 53.13/15.20 U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) 53.13/15.20 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)) 53.13/15.20 U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.13/15.20 U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) 53.13/15.20 less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) 53.13/15.20 less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) 53.13/15.20 U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.13/15.20 app_in_aaa([], X, X) -> app_out_aaa([], X, X) 53.13/15.20 app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) 53.13/15.20 U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 53.13/15.20 U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) 53.13/15.20 53.13/15.20 The argument filtering Pi contains the following mapping: 53.13/15.20 [] = [] 53.13/15.20 53.13/15.20 part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa 53.13/15.20 53.13/15.20 U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) 53.13/15.20 53.13/15.20 less_in_aa(x1, x2) = less_in_aa 53.13/15.20 53.13/15.20 less_out_aa(x1, x2) = less_out_aa(x1) 53.13/15.20 53.13/15.20 U9_aa(x1, x2, x3) = U9_aa(x3) 53.13/15.20 53.13/15.20 U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) 53.13/15.20 53.13/15.20 part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) 53.13/15.20 53.13/15.20 U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) 53.13/15.20 53.13/15.20 less_in_ga(x1, x2) = less_in_ga(x1) 53.13/15.20 53.13/15.20 0 = 0 53.13/15.20 53.13/15.20 less_out_ga(x1, x2) = less_out_ga 53.13/15.20 53.13/15.20 s(x1) = s(x1) 53.13/15.20 53.13/15.20 U9_ga(x1, x2, x3) = U9_ga(x3) 53.13/15.20 53.13/15.20 U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x6) 53.13/15.20 53.13/15.20 U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x6) 53.13/15.20 53.13/15.20 part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa 53.13/15.20 53.13/15.20 part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa 53.13/15.20 53.13/15.20 U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) 53.13/15.20 53.13/15.20 qs_in_aa(x1, x2) = qs_in_aa 53.13/15.20 53.13/15.20 qs_out_aa(x1, x2) = qs_out_aa 53.13/15.20 53.13/15.20 U1_aa(x1, x2, x3, x4) = U1_aa(x4) 53.13/15.20 53.13/15.20 U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) 53.13/15.20 53.13/15.20 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) 53.13/15.20 53.13/15.20 U4_aa(x1, x2, x3, x4) = U4_aa(x4) 53.13/15.20 53.13/15.20 .(x1, x2) = .(x1, x2) 53.13/15.20 53.13/15.20 app_in_aaa(x1, x2, x3) = app_in_aaa 53.13/15.20 53.13/15.20 app_out_aaa(x1, x2, x3) = app_out_aaa 53.13/15.20 53.13/15.20 U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) 53.13/15.20 53.13/15.20 QS_IN_AA(x1, x2) = QS_IN_AA 53.13/15.20 53.13/15.20 U1_AA(x1, x2, x3, x4) = U1_AA(x4) 53.13/15.20 53.13/15.20 U2_AA(x1, x2, x3, x4, x5) = U2_AA(x5) 53.13/15.20 53.13/15.20 53.13/15.20 We have to consider all (P,R,Pi)-chains 53.13/15.20 ---------------------------------------- 53.13/15.20 53.13/15.20 (50) PrologToPiTRSProof (SOUND) 53.13/15.20 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 53.13/15.20 53.13/15.20 qs_in_2: (f,b) (f,f) 53.13/15.20 53.13/15.20 part_in_4: (f,f,f,f) (b,f,f,f) 53.13/15.20 53.13/15.20 less_in_2: (f,f) (b,f) 53.13/15.20 53.13/15.20 app_in_3: (f,f,f) (f,f,b) 53.13/15.20 53.13/15.20 Transforming Prolog into the following Term Rewriting System: 53.13/15.20 53.13/15.20 Pi-finite rewrite system: 53.13/15.20 The TRS R consists of the following rules: 53.13/15.20 53.13/15.20 qs_in_ag([], []) -> qs_out_ag([], []) 53.13/15.20 qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.20 part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) 53.13/15.20 less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) 53.13/15.20 less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) 53.13/15.20 U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) 53.13/15.20 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)) 53.13/15.20 part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) 53.13/15.20 less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) 53.13/15.20 less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) 53.13/15.20 U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 53.13/15.20 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)) 53.13/15.20 part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) 53.13/15.20 part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) 53.13/15.20 U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.13/15.20 U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.13/15.20 U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.13/15.20 part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) 53.13/15.20 part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) 53.13/15.20 U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.13/15.20 U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.20 qs_in_aa([], []) -> qs_out_aa([], []) 53.13/15.20 qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.20 U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.20 U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.20 U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) 53.13/15.20 app_in_aaa([], X, X) -> app_out_aaa([], X, X) 53.13/15.20 app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) 53.13/15.20 U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) 53.13/15.20 U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) 53.13/15.20 U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.20 U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) 53.13/15.20 app_in_aag([], X, X) -> app_out_aag([], X, X) 53.13/15.20 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 53.13/15.20 U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 53.13/15.20 U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) 53.13/15.20 53.13/15.20 The argument filtering Pi contains the following mapping: 53.13/15.20 qs_in_ag(x1, x2) = qs_in_ag(x2) 53.13/15.20 53.13/15.20 [] = [] 53.13/15.20 53.13/15.20 qs_out_ag(x1, x2) = qs_out_ag(x2) 53.13/15.20 53.13/15.20 U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) 53.13/15.20 53.13/15.20 part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa 53.13/15.20 53.13/15.20 U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) 53.13/15.20 53.13/15.20 less_in_aa(x1, x2) = less_in_aa 53.13/15.20 53.13/15.20 less_out_aa(x1, x2) = less_out_aa(x1) 53.13/15.20 53.13/15.20 U9_aa(x1, x2, x3) = U9_aa(x3) 53.13/15.20 53.13/15.20 U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) 53.13/15.20 53.13/15.20 part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) 53.13/15.20 53.13/15.20 U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) 53.13/15.20 53.13/15.20 less_in_ga(x1, x2) = less_in_ga(x1) 53.13/15.20 53.13/15.20 0 = 0 53.13/15.20 53.13/15.20 less_out_ga(x1, x2) = less_out_ga(x1) 53.13/15.20 53.13/15.20 s(x1) = s(x1) 53.13/15.20 53.13/15.20 U9_ga(x1, x2, x3) = U9_ga(x1, x3) 53.13/15.20 53.13/15.20 U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x1, x6) 53.13/15.20 53.13/15.20 U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x1, x6) 53.13/15.20 53.13/15.20 part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa(x1) 53.13/15.20 53.13/15.20 part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa 53.13/15.20 53.13/15.20 U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) 53.13/15.20 53.13/15.20 U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) 53.13/15.20 53.13/15.20 qs_in_aa(x1, x2) = qs_in_aa 53.13/15.20 53.13/15.20 qs_out_aa(x1, x2) = qs_out_aa 53.13/15.20 53.13/15.20 U1_aa(x1, x2, x3, x4) = U1_aa(x4) 53.13/15.20 53.13/15.20 U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) 53.13/15.20 53.13/15.20 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) 53.13/15.20 53.13/15.20 U4_aa(x1, x2, x3, x4) = U4_aa(x4) 53.13/15.20 53.13/15.20 .(x1, x2) = .(x1, x2) 53.13/15.20 53.13/15.20 app_in_aaa(x1, x2, x3) = app_in_aaa 53.13/15.20 53.13/15.20 app_out_aaa(x1, x2, x3) = app_out_aaa 53.13/15.20 53.13/15.20 U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) 53.13/15.20 53.13/15.20 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) 53.13/15.20 53.13/15.20 U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4) 53.13/15.20 53.13/15.20 app_in_aag(x1, x2, x3) = app_in_aag(x3) 53.13/15.20 53.13/15.20 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2, x3) 53.13/15.20 53.13/15.20 U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x4, x5) 53.13/15.20 53.13/15.20 53.13/15.20 53.13/15.20 53.13/15.20 53.13/15.20 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 53.13/15.20 53.13/15.20 53.13/15.20 53.13/15.20 ---------------------------------------- 53.13/15.20 53.13/15.20 (51) 53.13/15.20 Obligation: 53.13/15.20 Pi-finite rewrite system: 53.13/15.20 The TRS R consists of the following rules: 53.13/15.20 53.13/15.20 qs_in_ag([], []) -> qs_out_ag([], []) 53.13/15.20 qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.20 part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) 53.13/15.20 less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) 53.13/15.20 less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) 53.13/15.20 U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) 53.13/15.20 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)) 53.13/15.20 part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) 53.13/15.20 less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) 53.13/15.20 less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) 53.13/15.20 U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 53.13/15.20 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)) 53.13/15.20 part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) 53.13/15.20 part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) 53.13/15.20 U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.13/15.20 U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.13/15.20 U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.13/15.20 part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) 53.13/15.20 part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) 53.13/15.20 U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.13/15.20 U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.20 qs_in_aa([], []) -> qs_out_aa([], []) 53.13/15.20 qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.20 U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.20 U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.20 U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) 53.13/15.20 app_in_aaa([], X, X) -> app_out_aaa([], X, X) 53.13/15.20 app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) 53.13/15.20 U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) 53.13/15.20 U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) 53.13/15.20 U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.20 U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) 53.13/15.20 app_in_aag([], X, X) -> app_out_aag([], X, X) 53.13/15.20 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 53.13/15.20 U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 53.13/15.20 U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) 53.13/15.20 53.13/15.20 The argument filtering Pi contains the following mapping: 53.13/15.20 qs_in_ag(x1, x2) = qs_in_ag(x2) 53.13/15.20 53.13/15.20 [] = [] 53.13/15.20 53.13/15.20 qs_out_ag(x1, x2) = qs_out_ag(x2) 53.13/15.20 53.13/15.20 U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) 53.13/15.20 53.13/15.20 part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa 53.13/15.20 53.13/15.20 U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) 53.13/15.20 53.13/15.20 less_in_aa(x1, x2) = less_in_aa 53.13/15.20 53.13/15.20 less_out_aa(x1, x2) = less_out_aa(x1) 53.13/15.20 53.13/15.20 U9_aa(x1, x2, x3) = U9_aa(x3) 53.13/15.20 53.13/15.20 U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) 53.13/15.20 53.13/15.20 part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) 53.13/15.20 53.13/15.20 U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) 53.13/15.20 53.13/15.20 less_in_ga(x1, x2) = less_in_ga(x1) 53.13/15.20 53.13/15.20 0 = 0 53.13/15.20 53.13/15.20 less_out_ga(x1, x2) = less_out_ga(x1) 53.13/15.20 53.13/15.20 s(x1) = s(x1) 53.13/15.20 53.13/15.20 U9_ga(x1, x2, x3) = U9_ga(x1, x3) 53.13/15.20 53.13/15.20 U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x1, x6) 53.13/15.20 53.13/15.20 U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x1, x6) 53.13/15.20 53.13/15.20 part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa(x1) 53.13/15.20 53.13/15.20 part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa 53.13/15.20 53.13/15.20 U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) 53.13/15.20 53.13/15.20 U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) 53.13/15.20 53.13/15.20 qs_in_aa(x1, x2) = qs_in_aa 53.13/15.20 53.13/15.20 qs_out_aa(x1, x2) = qs_out_aa 53.13/15.20 53.13/15.20 U1_aa(x1, x2, x3, x4) = U1_aa(x4) 53.13/15.20 53.13/15.20 U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) 53.13/15.20 53.13/15.20 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) 53.13/15.20 53.13/15.20 U4_aa(x1, x2, x3, x4) = U4_aa(x4) 53.13/15.20 53.13/15.20 .(x1, x2) = .(x1, x2) 53.13/15.20 53.13/15.20 app_in_aaa(x1, x2, x3) = app_in_aaa 53.13/15.20 53.13/15.20 app_out_aaa(x1, x2, x3) = app_out_aaa 53.13/15.20 53.13/15.20 U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) 53.13/15.20 53.13/15.20 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) 53.13/15.20 53.13/15.20 U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4) 53.13/15.20 53.13/15.20 app_in_aag(x1, x2, x3) = app_in_aag(x3) 53.13/15.20 53.13/15.20 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2, x3) 53.13/15.20 53.13/15.20 U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x4, x5) 53.13/15.20 53.13/15.20 53.13/15.20 53.13/15.20 ---------------------------------------- 53.13/15.20 53.13/15.20 (52) DependencyPairsProof (EQUIVALENT) 53.13/15.20 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 53.13/15.20 Pi DP problem: 53.13/15.20 The TRS P consists of the following rules: 53.13/15.20 53.13/15.20 QS_IN_AG(.(X, Xs), Ys) -> U1_AG(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.20 QS_IN_AG(.(X, Xs), Ys) -> PART_IN_AAAA(X, Xs, Littles, Bigs) 53.13/15.20 PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_AAAA(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) 53.13/15.20 PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> LESS_IN_AA(X, Y) 53.13/15.20 LESS_IN_AA(s(X), s(Y)) -> U9_AA(X, Y, less_in_aa(X, Y)) 53.13/15.20 LESS_IN_AA(s(X), s(Y)) -> LESS_IN_AA(X, Y) 53.13/15.20 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)) 53.13/15.20 U5_AAAA(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs) 53.13/15.20 PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GAAA(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) 53.13/15.20 PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> LESS_IN_GA(X, Y) 53.13/15.20 LESS_IN_GA(s(X), s(Y)) -> U9_GA(X, Y, less_in_ga(X, Y)) 53.13/15.20 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 53.13/15.20 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)) 53.13/15.20 U5_GAAA(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs) 53.13/15.20 PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_GAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) 53.13/15.20 PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_GAAA(X, Xs, Ls, Bs) 53.13/15.20 PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_AAAA(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) 53.13/15.20 PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_AAAA(X, Xs, Ls, Bs) 53.13/15.20 U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AG(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.20 U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> QS_IN_AA(Littles, Ls) 53.13/15.20 QS_IN_AA(.(X, Xs), Ys) -> U1_AA(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.20 QS_IN_AA(.(X, Xs), Ys) -> PART_IN_AAAA(X, Xs, Littles, Bigs) 53.13/15.20 U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AA(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.20 U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> QS_IN_AA(Littles, Ls) 53.13/15.20 U2_AA(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_AA(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.20 U2_AA(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> QS_IN_AA(Bigs, Bs) 53.13/15.20 U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_AA(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) 53.13/15.20 U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> APP_IN_AAA(Ls, .(X, Bs), Ys) 53.13/15.20 APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> U8_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) 53.13/15.20 APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAA(Xs, Ys, Zs) 53.13/15.20 U2_AG(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_AG(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.20 U2_AG(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> QS_IN_AA(Bigs, Bs) 53.13/15.20 U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_AG(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) 53.13/15.20 U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> APP_IN_AAG(Ls, .(X, Bs), Ys) 53.13/15.20 APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> U8_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 53.13/15.20 APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs) 53.13/15.20 53.13/15.20 The TRS R consists of the following rules: 53.13/15.20 53.13/15.20 qs_in_ag([], []) -> qs_out_ag([], []) 53.13/15.20 qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.20 part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) 53.13/15.20 less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) 53.13/15.20 less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) 53.13/15.20 U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) 53.13/15.20 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)) 53.13/15.20 part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) 53.13/15.20 less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) 53.13/15.20 less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) 53.13/15.20 U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 53.13/15.20 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)) 53.13/15.20 part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) 53.13/15.20 part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) 53.13/15.20 U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.13/15.20 U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.13/15.20 U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.13/15.20 part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) 53.13/15.20 part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) 53.13/15.20 U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.13/15.20 U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.20 qs_in_aa([], []) -> qs_out_aa([], []) 53.13/15.20 qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.20 U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.20 U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.20 U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) 53.13/15.20 app_in_aaa([], X, X) -> app_out_aaa([], X, X) 53.13/15.20 app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) 53.13/15.20 U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) 53.13/15.20 U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) 53.13/15.20 U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.20 U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) 53.13/15.20 app_in_aag([], X, X) -> app_out_aag([], X, X) 53.13/15.20 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 53.13/15.20 U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 53.13/15.20 U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) 53.13/15.20 53.13/15.20 The argument filtering Pi contains the following mapping: 53.13/15.20 qs_in_ag(x1, x2) = qs_in_ag(x2) 53.13/15.20 53.13/15.20 [] = [] 53.13/15.20 53.13/15.20 qs_out_ag(x1, x2) = qs_out_ag(x2) 53.13/15.20 53.13/15.20 U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) 53.13/15.20 53.13/15.20 part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa 53.13/15.20 53.13/15.20 U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) 53.13/15.20 53.13/15.20 less_in_aa(x1, x2) = less_in_aa 53.13/15.20 53.13/15.20 less_out_aa(x1, x2) = less_out_aa(x1) 53.13/15.20 53.13/15.20 U9_aa(x1, x2, x3) = U9_aa(x3) 53.13/15.20 53.13/15.20 U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) 53.13/15.20 53.13/15.20 part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) 53.13/15.20 53.13/15.20 U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) 53.13/15.20 53.13/15.20 less_in_ga(x1, x2) = less_in_ga(x1) 53.13/15.20 53.13/15.20 0 = 0 53.13/15.20 53.13/15.20 less_out_ga(x1, x2) = less_out_ga(x1) 53.13/15.20 53.13/15.20 s(x1) = s(x1) 53.13/15.20 53.13/15.20 U9_ga(x1, x2, x3) = U9_ga(x1, x3) 53.13/15.20 53.13/15.20 U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x1, x6) 53.13/15.20 53.13/15.20 U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x1, x6) 53.13/15.20 53.13/15.20 part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa(x1) 53.13/15.20 53.13/15.20 part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa 53.13/15.20 53.13/15.20 U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) 53.13/15.20 53.13/15.20 U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) 53.13/15.20 53.13/15.20 qs_in_aa(x1, x2) = qs_in_aa 53.13/15.20 53.13/15.20 qs_out_aa(x1, x2) = qs_out_aa 53.13/15.20 53.13/15.20 U1_aa(x1, x2, x3, x4) = U1_aa(x4) 53.13/15.20 53.13/15.20 U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) 53.13/15.20 53.13/15.20 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) 53.13/15.20 53.13/15.20 U4_aa(x1, x2, x3, x4) = U4_aa(x4) 53.13/15.20 53.13/15.20 .(x1, x2) = .(x1, x2) 53.13/15.20 53.13/15.20 app_in_aaa(x1, x2, x3) = app_in_aaa 53.13/15.20 53.13/15.20 app_out_aaa(x1, x2, x3) = app_out_aaa 53.13/15.20 53.13/15.20 U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) 53.13/15.20 53.13/15.20 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) 53.13/15.20 53.13/15.20 U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4) 53.13/15.20 53.13/15.20 app_in_aag(x1, x2, x3) = app_in_aag(x3) 53.13/15.20 53.13/15.20 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2, x3) 53.13/15.20 53.13/15.20 U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x4, x5) 53.13/15.20 53.13/15.20 QS_IN_AG(x1, x2) = QS_IN_AG(x2) 53.13/15.20 53.13/15.20 U1_AG(x1, x2, x3, x4) = U1_AG(x3, x4) 53.13/15.20 53.13/15.20 PART_IN_AAAA(x1, x2, x3, x4) = PART_IN_AAAA 53.13/15.20 53.13/15.20 U5_AAAA(x1, x2, x3, x4, x5, x6) = U5_AAAA(x6) 53.13/15.20 53.13/15.20 LESS_IN_AA(x1, x2) = LESS_IN_AA 53.13/15.20 53.13/15.20 U9_AA(x1, x2, x3) = U9_AA(x3) 53.13/15.20 53.13/15.20 U6_AAAA(x1, x2, x3, x4, x5, x6) = U6_AAAA(x6) 53.13/15.20 53.13/15.20 PART_IN_GAAA(x1, x2, x3, x4) = PART_IN_GAAA(x1) 53.13/15.20 53.13/15.20 U5_GAAA(x1, x2, x3, x4, x5, x6) = U5_GAAA(x1, x6) 53.13/15.20 53.13/15.20 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 53.13/15.20 53.13/15.20 U9_GA(x1, x2, x3) = U9_GA(x1, x3) 53.13/15.20 53.13/15.20 U6_GAAA(x1, x2, x3, x4, x5, x6) = U6_GAAA(x1, x6) 53.13/15.20 53.13/15.20 U7_GAAA(x1, x2, x3, x4, x5, x6) = U7_GAAA(x1, x6) 53.13/15.20 53.13/15.20 U7_AAAA(x1, x2, x3, x4, x5, x6) = U7_AAAA(x6) 53.13/15.20 53.13/15.20 U2_AG(x1, x2, x3, x4, x5) = U2_AG(x3, x5) 53.13/15.20 53.13/15.20 QS_IN_AA(x1, x2) = QS_IN_AA 53.13/15.20 53.13/15.20 U1_AA(x1, x2, x3, x4) = U1_AA(x4) 53.13/15.20 53.13/15.20 U2_AA(x1, x2, x3, x4, x5) = U2_AA(x5) 53.13/15.20 53.13/15.20 U3_AA(x1, x2, x3, x4, x5) = U3_AA(x5) 53.13/15.20 53.13/15.20 U4_AA(x1, x2, x3, x4) = U4_AA(x4) 53.13/15.20 53.13/15.20 APP_IN_AAA(x1, x2, x3) = APP_IN_AAA 53.13/15.20 53.13/15.20 U8_AAA(x1, x2, x3, x4, x5) = U8_AAA(x5) 53.13/15.20 53.13/15.20 U3_AG(x1, x2, x3, x4, x5) = U3_AG(x3, x5) 53.13/15.20 53.13/15.20 U4_AG(x1, x2, x3, x4) = U4_AG(x3, x4) 53.13/15.20 53.13/15.20 APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3) 53.13/15.20 53.13/15.20 U8_AAG(x1, x2, x3, x4, x5) = U8_AAG(x1, x4, x5) 53.13/15.20 53.13/15.20 53.13/15.20 We have to consider all (P,R,Pi)-chains 53.13/15.20 ---------------------------------------- 53.13/15.20 53.13/15.20 (53) 53.13/15.20 Obligation: 53.13/15.20 Pi DP problem: 53.13/15.20 The TRS P consists of the following rules: 53.13/15.20 53.13/15.20 QS_IN_AG(.(X, Xs), Ys) -> U1_AG(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.20 QS_IN_AG(.(X, Xs), Ys) -> PART_IN_AAAA(X, Xs, Littles, Bigs) 53.13/15.20 PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_AAAA(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) 53.13/15.20 PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> LESS_IN_AA(X, Y) 53.13/15.20 LESS_IN_AA(s(X), s(Y)) -> U9_AA(X, Y, less_in_aa(X, Y)) 53.13/15.20 LESS_IN_AA(s(X), s(Y)) -> LESS_IN_AA(X, Y) 53.13/15.20 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)) 53.13/15.20 U5_AAAA(X, Y, Xs, Ls, Bs, less_out_aa(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs) 53.13/15.20 PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GAAA(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) 53.13/15.20 PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> LESS_IN_GA(X, Y) 53.13/15.20 LESS_IN_GA(s(X), s(Y)) -> U9_GA(X, Y, less_in_ga(X, Y)) 53.13/15.20 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 53.13/15.20 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)) 53.13/15.20 U5_GAAA(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs) 53.13/15.20 PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_GAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) 53.13/15.20 PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_GAAA(X, Xs, Ls, Bs) 53.13/15.20 PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_AAAA(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) 53.13/15.20 PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_AAAA(X, Xs, Ls, Bs) 53.13/15.20 U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AG(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.20 U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> QS_IN_AA(Littles, Ls) 53.13/15.20 QS_IN_AA(.(X, Xs), Ys) -> U1_AA(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.20 QS_IN_AA(.(X, Xs), Ys) -> PART_IN_AAAA(X, Xs, Littles, Bigs) 53.13/15.20 U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AA(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.20 U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> QS_IN_AA(Littles, Ls) 53.13/15.20 U2_AA(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_AA(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.20 U2_AA(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> QS_IN_AA(Bigs, Bs) 53.13/15.20 U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_AA(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) 53.13/15.20 U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> APP_IN_AAA(Ls, .(X, Bs), Ys) 53.13/15.20 APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> U8_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) 53.13/15.20 APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAA(Xs, Ys, Zs) 53.13/15.20 U2_AG(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_AG(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.20 U2_AG(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> QS_IN_AA(Bigs, Bs) 53.13/15.20 U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_AG(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) 53.13/15.20 U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> APP_IN_AAG(Ls, .(X, Bs), Ys) 53.13/15.20 APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> U8_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 53.13/15.20 APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs) 53.13/15.20 53.13/15.20 The TRS R consists of the following rules: 53.13/15.20 53.13/15.20 qs_in_ag([], []) -> qs_out_ag([], []) 53.13/15.20 qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.20 part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) 53.13/15.20 less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) 53.13/15.20 less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) 53.13/15.20 U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) 53.13/15.20 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)) 53.13/15.20 part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) 53.13/15.20 less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) 53.13/15.20 less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) 53.13/15.20 U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 53.13/15.20 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)) 53.13/15.20 part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) 53.13/15.20 part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) 53.13/15.20 U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.13/15.20 U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.13/15.20 U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.13/15.20 part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) 53.13/15.20 part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) 53.13/15.20 U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.13/15.20 U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.20 qs_in_aa([], []) -> qs_out_aa([], []) 53.13/15.20 qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.20 U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.20 U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.20 U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) 53.13/15.20 app_in_aaa([], X, X) -> app_out_aaa([], X, X) 53.13/15.20 app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) 53.13/15.20 U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) 53.13/15.20 U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) 53.13/15.20 U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.20 U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) 53.13/15.20 app_in_aag([], X, X) -> app_out_aag([], X, X) 53.13/15.20 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 53.13/15.20 U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 53.13/15.20 U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) 53.13/15.20 53.13/15.20 The argument filtering Pi contains the following mapping: 53.13/15.20 qs_in_ag(x1, x2) = qs_in_ag(x2) 53.13/15.20 53.13/15.20 [] = [] 53.13/15.20 53.13/15.20 qs_out_ag(x1, x2) = qs_out_ag(x2) 53.13/15.20 53.13/15.20 U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) 53.13/15.20 53.13/15.20 part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa 53.13/15.20 53.13/15.20 U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) 53.13/15.20 53.13/15.20 less_in_aa(x1, x2) = less_in_aa 53.13/15.20 53.13/15.20 less_out_aa(x1, x2) = less_out_aa(x1) 53.13/15.20 53.13/15.20 U9_aa(x1, x2, x3) = U9_aa(x3) 53.13/15.20 53.13/15.20 U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) 53.13/15.20 53.13/15.20 part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) 53.13/15.20 53.13/15.20 U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) 53.13/15.20 53.13/15.20 less_in_ga(x1, x2) = less_in_ga(x1) 53.13/15.20 53.13/15.20 0 = 0 53.13/15.20 53.13/15.20 less_out_ga(x1, x2) = less_out_ga(x1) 53.13/15.20 53.13/15.20 s(x1) = s(x1) 53.13/15.20 53.13/15.20 U9_ga(x1, x2, x3) = U9_ga(x1, x3) 53.13/15.20 53.13/15.20 U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x1, x6) 53.13/15.20 53.13/15.20 U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x1, x6) 53.13/15.20 53.13/15.20 part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa(x1) 53.13/15.20 53.13/15.20 part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa 53.13/15.20 53.13/15.20 U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) 53.13/15.20 53.13/15.20 U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) 53.13/15.20 53.13/15.20 qs_in_aa(x1, x2) = qs_in_aa 53.13/15.20 53.13/15.20 qs_out_aa(x1, x2) = qs_out_aa 53.13/15.20 53.13/15.20 U1_aa(x1, x2, x3, x4) = U1_aa(x4) 53.13/15.20 53.13/15.20 U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) 53.13/15.20 53.13/15.20 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) 53.13/15.20 53.13/15.20 U4_aa(x1, x2, x3, x4) = U4_aa(x4) 53.13/15.20 53.13/15.20 .(x1, x2) = .(x1, x2) 53.13/15.20 53.13/15.20 app_in_aaa(x1, x2, x3) = app_in_aaa 53.13/15.20 53.13/15.20 app_out_aaa(x1, x2, x3) = app_out_aaa 53.13/15.20 53.13/15.20 U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) 53.13/15.20 53.13/15.20 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) 53.13/15.20 53.13/15.20 U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4) 53.13/15.20 53.13/15.20 app_in_aag(x1, x2, x3) = app_in_aag(x3) 53.13/15.20 53.13/15.20 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2, x3) 53.13/15.20 53.13/15.20 U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x4, x5) 53.13/15.20 53.13/15.20 QS_IN_AG(x1, x2) = QS_IN_AG(x2) 53.13/15.20 53.13/15.20 U1_AG(x1, x2, x3, x4) = U1_AG(x3, x4) 53.13/15.20 53.13/15.20 PART_IN_AAAA(x1, x2, x3, x4) = PART_IN_AAAA 53.13/15.20 53.13/15.20 U5_AAAA(x1, x2, x3, x4, x5, x6) = U5_AAAA(x6) 53.13/15.20 53.13/15.20 LESS_IN_AA(x1, x2) = LESS_IN_AA 53.13/15.20 53.13/15.20 U9_AA(x1, x2, x3) = U9_AA(x3) 53.13/15.20 53.13/15.20 U6_AAAA(x1, x2, x3, x4, x5, x6) = U6_AAAA(x6) 53.13/15.20 53.13/15.20 PART_IN_GAAA(x1, x2, x3, x4) = PART_IN_GAAA(x1) 53.13/15.20 53.13/15.20 U5_GAAA(x1, x2, x3, x4, x5, x6) = U5_GAAA(x1, x6) 53.13/15.20 53.13/15.20 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 53.13/15.20 53.13/15.20 U9_GA(x1, x2, x3) = U9_GA(x1, x3) 53.13/15.20 53.13/15.20 U6_GAAA(x1, x2, x3, x4, x5, x6) = U6_GAAA(x1, x6) 53.13/15.20 53.13/15.20 U7_GAAA(x1, x2, x3, x4, x5, x6) = U7_GAAA(x1, x6) 53.13/15.20 53.13/15.20 U7_AAAA(x1, x2, x3, x4, x5, x6) = U7_AAAA(x6) 53.13/15.20 53.13/15.20 U2_AG(x1, x2, x3, x4, x5) = U2_AG(x3, x5) 53.13/15.20 53.13/15.20 QS_IN_AA(x1, x2) = QS_IN_AA 53.13/15.20 53.13/15.20 U1_AA(x1, x2, x3, x4) = U1_AA(x4) 53.13/15.20 53.13/15.20 U2_AA(x1, x2, x3, x4, x5) = U2_AA(x5) 53.13/15.20 53.13/15.20 U3_AA(x1, x2, x3, x4, x5) = U3_AA(x5) 53.13/15.20 53.13/15.20 U4_AA(x1, x2, x3, x4) = U4_AA(x4) 53.13/15.20 53.13/15.20 APP_IN_AAA(x1, x2, x3) = APP_IN_AAA 53.13/15.20 53.13/15.20 U8_AAA(x1, x2, x3, x4, x5) = U8_AAA(x5) 53.13/15.20 53.13/15.20 U3_AG(x1, x2, x3, x4, x5) = U3_AG(x3, x5) 53.13/15.20 53.13/15.20 U4_AG(x1, x2, x3, x4) = U4_AG(x3, x4) 53.13/15.20 53.13/15.20 APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3) 53.13/15.20 53.13/15.20 U8_AAG(x1, x2, x3, x4, x5) = U8_AAG(x1, x4, x5) 53.13/15.20 53.13/15.20 53.13/15.20 We have to consider all (P,R,Pi)-chains 53.13/15.20 ---------------------------------------- 53.13/15.20 53.13/15.20 (54) DependencyGraphProof (EQUIVALENT) 53.13/15.20 The approximation of the Dependency Graph [LOPSTR] contains 7 SCCs with 24 less nodes. 53.13/15.20 ---------------------------------------- 53.13/15.20 53.13/15.20 (55) 53.13/15.20 Complex Obligation (AND) 53.13/15.20 53.13/15.20 ---------------------------------------- 53.13/15.20 53.13/15.20 (56) 53.13/15.20 Obligation: 53.13/15.20 Pi DP problem: 53.13/15.20 The TRS P consists of the following rules: 53.13/15.20 53.13/15.20 APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs) 53.13/15.20 53.13/15.20 The TRS R consists of the following rules: 53.13/15.20 53.13/15.20 qs_in_ag([], []) -> qs_out_ag([], []) 53.13/15.20 qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.21 part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) 53.13/15.21 less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) 53.13/15.21 less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) 53.13/15.21 U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) 53.13/15.21 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)) 53.13/15.21 part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) 53.13/15.21 less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) 53.13/15.21 less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) 53.13/15.21 U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 53.13/15.21 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)) 53.13/15.21 part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) 53.13/15.21 part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) 53.13/15.21 U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.13/15.21 U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.13/15.21 U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.13/15.21 part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) 53.13/15.21 part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) 53.13/15.21 U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.13/15.21 U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.21 qs_in_aa([], []) -> qs_out_aa([], []) 53.13/15.21 qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.21 U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.21 U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.21 U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) 53.13/15.21 app_in_aaa([], X, X) -> app_out_aaa([], X, X) 53.13/15.21 app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) 53.13/15.21 U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) 53.13/15.21 U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) 53.13/15.21 U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.21 U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) 53.13/15.21 app_in_aag([], X, X) -> app_out_aag([], X, X) 53.13/15.21 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 53.13/15.21 U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 53.13/15.21 U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) 53.13/15.21 53.13/15.21 The argument filtering Pi contains the following mapping: 53.13/15.21 qs_in_ag(x1, x2) = qs_in_ag(x2) 53.13/15.21 53.13/15.21 [] = [] 53.13/15.21 53.13/15.21 qs_out_ag(x1, x2) = qs_out_ag(x2) 53.13/15.21 53.13/15.21 U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) 53.13/15.21 53.13/15.21 part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa 53.13/15.21 53.13/15.21 U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) 53.13/15.21 53.13/15.21 less_in_aa(x1, x2) = less_in_aa 53.13/15.21 53.13/15.21 less_out_aa(x1, x2) = less_out_aa(x1) 53.13/15.21 53.13/15.21 U9_aa(x1, x2, x3) = U9_aa(x3) 53.13/15.21 53.13/15.21 U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) 53.13/15.21 53.13/15.21 part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) 53.13/15.21 53.13/15.21 U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) 53.13/15.21 53.13/15.21 less_in_ga(x1, x2) = less_in_ga(x1) 53.13/15.21 53.13/15.21 0 = 0 53.13/15.21 53.13/15.21 less_out_ga(x1, x2) = less_out_ga(x1) 53.13/15.21 53.13/15.21 s(x1) = s(x1) 53.13/15.21 53.13/15.21 U9_ga(x1, x2, x3) = U9_ga(x1, x3) 53.13/15.21 53.13/15.21 U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x1, x6) 53.13/15.21 53.13/15.21 U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x1, x6) 53.13/15.21 53.13/15.21 part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa(x1) 53.13/15.21 53.13/15.21 part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa 53.13/15.21 53.13/15.21 U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) 53.13/15.21 53.13/15.21 U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) 53.13/15.21 53.13/15.21 qs_in_aa(x1, x2) = qs_in_aa 53.13/15.21 53.13/15.21 qs_out_aa(x1, x2) = qs_out_aa 53.13/15.21 53.13/15.21 U1_aa(x1, x2, x3, x4) = U1_aa(x4) 53.13/15.21 53.13/15.21 U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) 53.13/15.21 53.13/15.21 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) 53.13/15.21 53.13/15.21 U4_aa(x1, x2, x3, x4) = U4_aa(x4) 53.13/15.21 53.13/15.21 .(x1, x2) = .(x1, x2) 53.13/15.21 53.13/15.21 app_in_aaa(x1, x2, x3) = app_in_aaa 53.13/15.21 53.13/15.21 app_out_aaa(x1, x2, x3) = app_out_aaa 53.13/15.21 53.13/15.21 U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) 53.13/15.21 53.13/15.21 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) 53.13/15.21 53.13/15.21 U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4) 53.13/15.21 53.13/15.21 app_in_aag(x1, x2, x3) = app_in_aag(x3) 53.13/15.21 53.13/15.21 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2, x3) 53.13/15.21 53.13/15.21 U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x4, x5) 53.13/15.21 53.13/15.21 APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3) 53.13/15.21 53.13/15.21 53.13/15.21 We have to consider all (P,R,Pi)-chains 53.13/15.21 ---------------------------------------- 53.13/15.21 53.13/15.21 (57) UsableRulesProof (EQUIVALENT) 53.13/15.21 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 53.13/15.21 ---------------------------------------- 53.13/15.21 53.13/15.21 (58) 53.13/15.21 Obligation: 53.13/15.21 Pi DP problem: 53.13/15.21 The TRS P consists of the following rules: 53.13/15.21 53.13/15.21 APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs) 53.13/15.21 53.13/15.21 R is empty. 53.13/15.21 The argument filtering Pi contains the following mapping: 53.13/15.21 .(x1, x2) = .(x1, x2) 53.13/15.21 53.13/15.21 APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3) 53.13/15.21 53.13/15.21 53.13/15.21 We have to consider all (P,R,Pi)-chains 53.13/15.21 ---------------------------------------- 53.13/15.21 53.13/15.21 (59) PiDPToQDPProof (SOUND) 53.13/15.21 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 53.13/15.21 ---------------------------------------- 53.13/15.21 53.13/15.21 (60) 53.13/15.21 Obligation: 53.13/15.21 Q DP problem: 53.13/15.21 The TRS P consists of the following rules: 53.13/15.21 53.13/15.21 APP_IN_AAG(.(X, Zs)) -> APP_IN_AAG(Zs) 53.13/15.21 53.13/15.21 R is empty. 53.13/15.21 Q is empty. 53.13/15.21 We have to consider all (P,Q,R)-chains. 53.13/15.21 ---------------------------------------- 53.13/15.21 53.13/15.21 (61) QDPSizeChangeProof (EQUIVALENT) 53.13/15.21 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. 53.13/15.21 53.13/15.21 From the DPs we obtained the following set of size-change graphs: 53.13/15.21 *APP_IN_AAG(.(X, Zs)) -> APP_IN_AAG(Zs) 53.13/15.21 The graph contains the following edges 1 > 1 53.13/15.21 53.13/15.21 53.13/15.21 ---------------------------------------- 53.13/15.21 53.13/15.21 (62) 53.13/15.21 YES 53.13/15.21 53.13/15.21 ---------------------------------------- 53.13/15.21 53.13/15.21 (63) 53.13/15.21 Obligation: 53.13/15.21 Pi DP problem: 53.13/15.21 The TRS P consists of the following rules: 53.13/15.21 53.13/15.21 APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAA(Xs, Ys, Zs) 53.13/15.21 53.13/15.21 The TRS R consists of the following rules: 53.13/15.21 53.13/15.21 qs_in_ag([], []) -> qs_out_ag([], []) 53.13/15.21 qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.21 part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) 53.13/15.21 less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) 53.13/15.21 less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) 53.13/15.21 U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) 53.13/15.21 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)) 53.13/15.21 part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) 53.13/15.21 less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) 53.13/15.21 less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) 53.13/15.21 U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 53.13/15.21 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)) 53.13/15.21 part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) 53.13/15.21 part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) 53.13/15.21 U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.13/15.21 U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.13/15.21 U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.13/15.21 part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) 53.13/15.21 part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) 53.13/15.21 U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.13/15.21 U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.21 qs_in_aa([], []) -> qs_out_aa([], []) 53.13/15.21 qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.13/15.21 U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.13/15.21 U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.21 U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) 53.13/15.21 app_in_aaa([], X, X) -> app_out_aaa([], X, X) 53.13/15.21 app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) 53.13/15.21 U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) 53.13/15.21 U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) 53.13/15.21 U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.13/15.21 U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) 53.13/15.21 app_in_aag([], X, X) -> app_out_aag([], X, X) 53.13/15.21 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 53.13/15.21 U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 53.13/15.21 U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) 53.13/15.21 53.13/15.21 The argument filtering Pi contains the following mapping: 53.13/15.21 qs_in_ag(x1, x2) = qs_in_ag(x2) 53.13/15.21 53.13/15.21 [] = [] 53.13/15.21 53.13/15.21 qs_out_ag(x1, x2) = qs_out_ag(x2) 53.13/15.21 53.13/15.21 U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) 53.54/15.23 53.54/15.23 part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa 53.54/15.23 53.54/15.23 U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) 53.54/15.23 53.54/15.23 less_in_aa(x1, x2) = less_in_aa 53.54/15.23 53.54/15.23 less_out_aa(x1, x2) = less_out_aa(x1) 53.54/15.23 53.54/15.23 U9_aa(x1, x2, x3) = U9_aa(x3) 53.54/15.23 53.54/15.23 U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) 53.54/15.23 53.54/15.23 part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) 53.54/15.23 53.54/15.23 U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) 53.54/15.23 53.54/15.23 less_in_ga(x1, x2) = less_in_ga(x1) 53.54/15.23 53.54/15.23 0 = 0 53.54/15.23 53.54/15.23 less_out_ga(x1, x2) = less_out_ga(x1) 53.54/15.23 53.54/15.23 s(x1) = s(x1) 53.54/15.23 53.54/15.23 U9_ga(x1, x2, x3) = U9_ga(x1, x3) 53.54/15.23 53.54/15.23 U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x1, x6) 53.54/15.23 53.54/15.23 U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x1, x6) 53.54/15.23 53.54/15.23 part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa(x1) 53.54/15.23 53.54/15.23 part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa 53.54/15.23 53.54/15.23 U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) 53.54/15.23 53.54/15.23 U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) 53.54/15.23 53.54/15.23 qs_in_aa(x1, x2) = qs_in_aa 53.54/15.23 53.54/15.23 qs_out_aa(x1, x2) = qs_out_aa 53.54/15.23 53.54/15.23 U1_aa(x1, x2, x3, x4) = U1_aa(x4) 53.54/15.23 53.54/15.23 U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) 53.54/15.23 53.54/15.23 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) 53.54/15.23 53.54/15.23 U4_aa(x1, x2, x3, x4) = U4_aa(x4) 53.54/15.23 53.54/15.23 .(x1, x2) = .(x1, x2) 53.54/15.23 53.54/15.23 app_in_aaa(x1, x2, x3) = app_in_aaa 53.54/15.23 53.54/15.23 app_out_aaa(x1, x2, x3) = app_out_aaa 53.54/15.23 53.54/15.23 U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) 53.54/15.23 53.54/15.23 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) 53.54/15.23 53.54/15.23 U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4) 53.54/15.23 53.54/15.23 app_in_aag(x1, x2, x3) = app_in_aag(x3) 53.54/15.23 53.54/15.23 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2, x3) 53.54/15.23 53.54/15.23 U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x4, x5) 53.54/15.23 53.54/15.23 APP_IN_AAA(x1, x2, x3) = APP_IN_AAA 53.54/15.23 53.54/15.23 53.54/15.23 We have to consider all (P,R,Pi)-chains 53.54/15.23 ---------------------------------------- 53.54/15.23 53.54/15.23 (64) UsableRulesProof (EQUIVALENT) 53.54/15.23 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 53.54/15.23 ---------------------------------------- 53.54/15.23 53.54/15.23 (65) 53.54/15.23 Obligation: 53.54/15.23 Pi DP problem: 53.54/15.23 The TRS P consists of the following rules: 53.54/15.23 53.54/15.23 APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAA(Xs, Ys, Zs) 53.54/15.23 53.54/15.23 R is empty. 53.54/15.23 The argument filtering Pi contains the following mapping: 53.54/15.23 .(x1, x2) = .(x1, x2) 53.54/15.23 53.54/15.23 APP_IN_AAA(x1, x2, x3) = APP_IN_AAA 53.54/15.23 53.54/15.23 53.54/15.23 We have to consider all (P,R,Pi)-chains 53.54/15.23 ---------------------------------------- 53.54/15.23 53.54/15.23 (66) PiDPToQDPProof (SOUND) 53.54/15.23 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 53.54/15.23 ---------------------------------------- 53.54/15.23 53.54/15.23 (67) 53.54/15.23 Obligation: 53.54/15.23 Q DP problem: 53.54/15.23 The TRS P consists of the following rules: 53.54/15.23 53.54/15.23 APP_IN_AAA -> APP_IN_AAA 53.54/15.23 53.54/15.23 R is empty. 53.54/15.23 Q is empty. 53.54/15.23 We have to consider all (P,Q,R)-chains. 53.54/15.23 ---------------------------------------- 53.54/15.23 53.54/15.23 (68) NonTerminationLoopProof (COMPLETE) 53.54/15.23 We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. 53.54/15.23 Found a loop by semiunifying a rule from P directly. 53.54/15.23 53.54/15.23 s = APP_IN_AAA evaluates to t =APP_IN_AAA 53.54/15.23 53.54/15.23 Thus s starts an infinite chain as s semiunifies with t with the following substitutions: 53.54/15.23 * Matcher: [ ] 53.54/15.23 * Semiunifier: [ ] 53.54/15.23 53.54/15.23 -------------------------------------------------------------------------------- 53.54/15.23 Rewriting sequence 53.54/15.23 53.54/15.23 The DP semiunifies directly so there is only one rewrite step from APP_IN_AAA to APP_IN_AAA. 53.54/15.23 53.54/15.23 53.54/15.23 53.54/15.23 53.54/15.23 ---------------------------------------- 53.54/15.23 53.54/15.23 (69) 53.54/15.23 NO 53.54/15.23 53.54/15.23 ---------------------------------------- 53.54/15.23 53.54/15.23 (70) 53.54/15.23 Obligation: 53.54/15.23 Pi DP problem: 53.54/15.23 The TRS P consists of the following rules: 53.54/15.23 53.54/15.23 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 53.54/15.23 53.54/15.23 The TRS R consists of the following rules: 53.54/15.23 53.54/15.23 qs_in_ag([], []) -> qs_out_ag([], []) 53.54/15.23 qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.54/15.23 part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) 53.54/15.23 less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) 53.54/15.23 less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) 53.54/15.23 U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) 53.54/15.23 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)) 53.54/15.23 part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) 53.54/15.23 less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) 53.54/15.23 less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) 53.54/15.23 U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 53.54/15.23 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)) 53.54/15.23 part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) 53.54/15.23 part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) 53.54/15.23 U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.54/15.23 U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.54/15.23 U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.54/15.23 part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) 53.54/15.23 part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) 53.54/15.23 U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.54/15.23 U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.54/15.23 qs_in_aa([], []) -> qs_out_aa([], []) 53.54/15.23 qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.54/15.23 U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.54/15.23 U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.54/15.23 U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) 53.54/15.23 app_in_aaa([], X, X) -> app_out_aaa([], X, X) 53.54/15.23 app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) 53.54/15.23 U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) 53.54/15.23 U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) 53.54/15.23 U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.54/15.23 U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) 53.54/15.23 app_in_aag([], X, X) -> app_out_aag([], X, X) 53.54/15.23 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 53.54/15.23 U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 53.54/15.23 U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) 53.54/15.23 53.54/15.23 The argument filtering Pi contains the following mapping: 53.54/15.23 qs_in_ag(x1, x2) = qs_in_ag(x2) 53.54/15.23 53.54/15.23 [] = [] 53.54/15.23 53.54/15.23 qs_out_ag(x1, x2) = qs_out_ag(x2) 53.54/15.23 53.54/15.23 U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) 53.54/15.23 53.54/15.23 part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa 53.54/15.23 53.54/15.23 U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) 53.54/15.23 53.54/15.23 less_in_aa(x1, x2) = less_in_aa 53.54/15.23 53.54/15.23 less_out_aa(x1, x2) = less_out_aa(x1) 53.54/15.23 53.54/15.23 U9_aa(x1, x2, x3) = U9_aa(x3) 53.54/15.23 53.54/15.23 U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) 53.54/15.23 53.54/15.23 part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) 53.54/15.23 53.54/15.23 U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) 53.54/15.23 53.54/15.23 less_in_ga(x1, x2) = less_in_ga(x1) 53.54/15.23 53.54/15.23 0 = 0 53.54/15.23 53.54/15.23 less_out_ga(x1, x2) = less_out_ga(x1) 53.54/15.23 53.54/15.23 s(x1) = s(x1) 53.54/15.23 53.54/15.23 U9_ga(x1, x2, x3) = U9_ga(x1, x3) 53.54/15.23 53.54/15.23 U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x1, x6) 53.54/15.23 53.54/15.23 U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x1, x6) 53.54/15.23 53.54/15.23 part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa(x1) 53.54/15.23 53.54/15.23 part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa 53.54/15.23 53.54/15.23 U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) 53.54/15.23 53.54/15.23 U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) 53.54/15.23 53.54/15.23 qs_in_aa(x1, x2) = qs_in_aa 53.54/15.23 53.54/15.23 qs_out_aa(x1, x2) = qs_out_aa 53.54/15.23 53.54/15.23 U1_aa(x1, x2, x3, x4) = U1_aa(x4) 53.54/15.23 53.54/15.23 U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) 53.54/15.23 53.54/15.23 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) 53.54/15.23 53.54/15.23 U4_aa(x1, x2, x3, x4) = U4_aa(x4) 53.54/15.23 53.54/15.23 .(x1, x2) = .(x1, x2) 53.54/15.23 53.54/15.23 app_in_aaa(x1, x2, x3) = app_in_aaa 53.54/15.23 53.54/15.23 app_out_aaa(x1, x2, x3) = app_out_aaa 53.54/15.23 53.54/15.23 U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) 53.54/15.23 53.54/15.23 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) 53.54/15.23 53.54/15.23 U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4) 53.54/15.23 53.54/15.23 app_in_aag(x1, x2, x3) = app_in_aag(x3) 53.54/15.23 53.54/15.23 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2, x3) 53.54/15.23 53.54/15.23 U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x4, x5) 53.54/15.23 53.54/15.23 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 53.54/15.23 53.54/15.23 53.54/15.23 We have to consider all (P,R,Pi)-chains 53.54/15.23 ---------------------------------------- 53.54/15.23 53.54/15.23 (71) UsableRulesProof (EQUIVALENT) 53.54/15.23 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 53.54/15.23 ---------------------------------------- 53.54/15.23 53.54/15.23 (72) 53.54/15.23 Obligation: 53.54/15.23 Pi DP problem: 53.54/15.23 The TRS P consists of the following rules: 53.54/15.23 53.54/15.23 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 53.54/15.23 53.54/15.23 R is empty. 53.54/15.23 The argument filtering Pi contains the following mapping: 53.54/15.23 s(x1) = s(x1) 53.54/15.23 53.54/15.23 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 53.54/15.23 53.54/15.23 53.54/15.23 We have to consider all (P,R,Pi)-chains 53.54/15.23 ---------------------------------------- 53.54/15.23 53.54/15.23 (73) PiDPToQDPProof (SOUND) 53.54/15.23 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 53.54/15.23 ---------------------------------------- 53.54/15.23 53.54/15.23 (74) 53.54/15.23 Obligation: 53.54/15.23 Q DP problem: 53.54/15.23 The TRS P consists of the following rules: 53.54/15.23 53.54/15.23 LESS_IN_GA(s(X)) -> LESS_IN_GA(X) 53.54/15.23 53.54/15.23 R is empty. 53.54/15.23 Q is empty. 53.54/15.23 We have to consider all (P,Q,R)-chains. 53.54/15.23 ---------------------------------------- 53.54/15.23 53.54/15.23 (75) QDPSizeChangeProof (EQUIVALENT) 53.54/15.23 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. 53.54/15.23 53.54/15.23 From the DPs we obtained the following set of size-change graphs: 53.54/15.23 *LESS_IN_GA(s(X)) -> LESS_IN_GA(X) 53.54/15.23 The graph contains the following edges 1 > 1 53.54/15.23 53.54/15.23 53.54/15.23 ---------------------------------------- 53.54/15.23 53.54/15.23 (76) 53.54/15.23 YES 53.54/15.23 53.54/15.23 ---------------------------------------- 53.54/15.23 53.54/15.23 (77) 53.54/15.23 Obligation: 53.54/15.23 Pi DP problem: 53.54/15.23 The TRS P consists of the following rules: 53.54/15.23 53.54/15.23 U5_GAAA(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs) 53.54/15.23 PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GAAA(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) 53.54/15.23 PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_GAAA(X, Xs, Ls, Bs) 53.54/15.23 53.54/15.23 The TRS R consists of the following rules: 53.54/15.23 53.54/15.23 qs_in_ag([], []) -> qs_out_ag([], []) 53.54/15.23 qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.54/15.23 part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) 53.54/15.23 less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) 53.54/15.23 less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) 53.54/15.23 U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) 53.54/15.23 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)) 53.54/15.23 part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) 53.54/15.23 less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) 53.54/15.23 less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) 53.54/15.23 U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 53.54/15.23 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)) 53.54/15.23 part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) 53.54/15.23 part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) 53.54/15.23 U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.54/15.23 U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.54/15.23 U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.54/15.23 part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) 53.54/15.23 part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) 53.54/15.23 U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.54/15.23 U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.54/15.23 qs_in_aa([], []) -> qs_out_aa([], []) 53.54/15.23 qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.54/15.23 U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.54/15.23 U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.54/15.23 U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) 53.54/15.23 app_in_aaa([], X, X) -> app_out_aaa([], X, X) 53.54/15.23 app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) 53.54/15.23 U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) 53.54/15.23 U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) 53.54/15.23 U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.54/15.23 U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) 53.54/15.23 app_in_aag([], X, X) -> app_out_aag([], X, X) 53.54/15.23 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 53.54/15.23 U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 53.54/15.23 U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) 53.54/15.23 53.54/15.23 The argument filtering Pi contains the following mapping: 53.54/15.23 qs_in_ag(x1, x2) = qs_in_ag(x2) 53.54/15.23 53.54/15.23 [] = [] 53.54/15.23 53.54/15.23 qs_out_ag(x1, x2) = qs_out_ag(x2) 53.54/15.23 53.54/15.23 U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) 53.54/15.23 53.54/15.23 part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa 53.54/15.23 53.54/15.23 U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) 53.54/15.23 53.54/15.23 less_in_aa(x1, x2) = less_in_aa 53.54/15.23 53.54/15.23 less_out_aa(x1, x2) = less_out_aa(x1) 53.54/15.23 53.54/15.23 U9_aa(x1, x2, x3) = U9_aa(x3) 53.54/15.23 53.54/15.23 U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) 53.54/15.23 53.54/15.23 part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) 53.54/15.23 53.54/15.23 U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) 53.54/15.23 53.54/15.23 less_in_ga(x1, x2) = less_in_ga(x1) 53.54/15.23 53.54/15.23 0 = 0 53.54/15.23 53.54/15.23 less_out_ga(x1, x2) = less_out_ga(x1) 53.54/15.23 53.54/15.23 s(x1) = s(x1) 53.54/15.23 53.54/15.23 U9_ga(x1, x2, x3) = U9_ga(x1, x3) 53.54/15.23 53.54/15.23 U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x1, x6) 53.54/15.23 53.54/15.23 U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x1, x6) 53.54/15.23 53.54/15.23 part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa(x1) 53.54/15.23 53.54/15.23 part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa 53.54/15.23 53.54/15.23 U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) 53.54/15.23 53.54/15.23 U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) 53.54/15.23 53.54/15.23 qs_in_aa(x1, x2) = qs_in_aa 53.54/15.23 53.54/15.23 qs_out_aa(x1, x2) = qs_out_aa 53.54/15.23 53.54/15.23 U1_aa(x1, x2, x3, x4) = U1_aa(x4) 53.54/15.23 53.54/15.23 U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) 53.54/15.23 53.54/15.23 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) 53.54/15.23 53.54/15.23 U4_aa(x1, x2, x3, x4) = U4_aa(x4) 53.54/15.23 53.54/15.23 .(x1, x2) = .(x1, x2) 53.54/15.23 53.54/15.23 app_in_aaa(x1, x2, x3) = app_in_aaa 53.54/15.23 53.54/15.23 app_out_aaa(x1, x2, x3) = app_out_aaa 53.54/15.23 53.54/15.23 U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) 53.54/15.23 53.54/15.23 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) 53.54/15.23 53.54/15.23 U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4) 53.54/15.23 53.54/15.23 app_in_aag(x1, x2, x3) = app_in_aag(x3) 53.54/15.23 53.54/15.23 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2, x3) 53.54/15.23 53.54/15.23 U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x4, x5) 53.54/15.23 53.54/15.23 PART_IN_GAAA(x1, x2, x3, x4) = PART_IN_GAAA(x1) 53.54/15.23 53.54/15.23 U5_GAAA(x1, x2, x3, x4, x5, x6) = U5_GAAA(x1, x6) 53.54/15.23 53.54/15.23 53.54/15.23 We have to consider all (P,R,Pi)-chains 53.54/15.23 ---------------------------------------- 53.54/15.23 53.54/15.23 (78) UsableRulesProof (EQUIVALENT) 53.54/15.23 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 53.54/15.23 ---------------------------------------- 53.54/15.23 53.54/15.23 (79) 53.54/15.23 Obligation: 53.54/15.23 Pi DP problem: 53.54/15.23 The TRS P consists of the following rules: 53.54/15.23 53.54/15.23 U5_GAAA(X, Y, Xs, Ls, Bs, less_out_ga(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs) 53.54/15.23 PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GAAA(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) 53.54/15.23 PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_GAAA(X, Xs, Ls, Bs) 53.54/15.23 53.54/15.23 The TRS R consists of the following rules: 53.54/15.23 53.54/15.23 less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) 53.54/15.23 less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) 53.54/15.23 U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 53.54/15.23 53.54/15.23 The argument filtering Pi contains the following mapping: 53.54/15.23 less_in_ga(x1, x2) = less_in_ga(x1) 53.54/15.23 53.54/15.23 0 = 0 53.54/15.23 53.54/15.23 less_out_ga(x1, x2) = less_out_ga(x1) 53.54/15.23 53.54/15.23 s(x1) = s(x1) 53.54/15.23 53.54/15.23 U9_ga(x1, x2, x3) = U9_ga(x1, x3) 53.54/15.23 53.54/15.23 .(x1, x2) = .(x1, x2) 53.54/15.23 53.54/15.23 PART_IN_GAAA(x1, x2, x3, x4) = PART_IN_GAAA(x1) 53.54/15.23 53.54/15.23 U5_GAAA(x1, x2, x3, x4, x5, x6) = U5_GAAA(x1, x6) 53.54/15.23 53.54/15.23 53.54/15.23 We have to consider all (P,R,Pi)-chains 53.54/15.23 ---------------------------------------- 53.54/15.23 53.54/15.23 (80) PiDPToQDPProof (SOUND) 53.54/15.23 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 53.54/15.23 ---------------------------------------- 53.54/15.23 53.54/15.23 (81) 53.54/15.23 Obligation: 53.54/15.23 Q DP problem: 53.54/15.23 The TRS P consists of the following rules: 53.54/15.23 53.54/15.23 U5_GAAA(X, less_out_ga(X)) -> PART_IN_GAAA(X) 53.54/15.23 PART_IN_GAAA(X) -> U5_GAAA(X, less_in_ga(X)) 53.54/15.23 PART_IN_GAAA(X) -> PART_IN_GAAA(X) 53.54/15.23 53.54/15.23 The TRS R consists of the following rules: 53.54/15.23 53.54/15.23 less_in_ga(0) -> less_out_ga(0) 53.54/15.23 less_in_ga(s(X)) -> U9_ga(X, less_in_ga(X)) 53.54/15.23 U9_ga(X, less_out_ga(X)) -> less_out_ga(s(X)) 53.54/15.23 53.54/15.23 The set Q consists of the following terms: 53.54/15.23 53.54/15.23 less_in_ga(x0) 53.54/15.23 U9_ga(x0, x1) 53.54/15.23 53.54/15.23 We have to consider all (P,Q,R)-chains. 53.54/15.23 ---------------------------------------- 53.54/15.23 53.54/15.23 (82) TransformationProof (SOUND) 53.54/15.23 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]: 53.54/15.23 53.54/15.23 (PART_IN_GAAA(0) -> U5_GAAA(0, less_out_ga(0)),PART_IN_GAAA(0) -> U5_GAAA(0, less_out_ga(0))) 53.54/15.23 (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)))) 53.54/15.23 53.54/15.23 53.54/15.23 ---------------------------------------- 53.54/15.23 53.54/15.23 (83) 53.54/15.23 Obligation: 53.54/15.23 Q DP problem: 53.54/15.23 The TRS P consists of the following rules: 53.54/15.23 53.54/15.23 U5_GAAA(X, less_out_ga(X)) -> PART_IN_GAAA(X) 53.54/15.23 PART_IN_GAAA(X) -> PART_IN_GAAA(X) 53.54/15.23 PART_IN_GAAA(0) -> U5_GAAA(0, less_out_ga(0)) 53.54/15.23 PART_IN_GAAA(s(x0)) -> U5_GAAA(s(x0), U9_ga(x0, less_in_ga(x0))) 53.54/15.23 53.54/15.23 The TRS R consists of the following rules: 53.54/15.23 53.54/15.23 less_in_ga(0) -> less_out_ga(0) 53.54/15.23 less_in_ga(s(X)) -> U9_ga(X, less_in_ga(X)) 53.54/15.23 U9_ga(X, less_out_ga(X)) -> less_out_ga(s(X)) 53.54/15.23 53.54/15.23 The set Q consists of the following terms: 53.54/15.23 53.54/15.23 less_in_ga(x0) 53.54/15.23 U9_ga(x0, x1) 53.54/15.23 53.54/15.23 We have to consider all (P,Q,R)-chains. 53.54/15.23 ---------------------------------------- 53.54/15.23 53.54/15.23 (84) TransformationProof (EQUIVALENT) 53.54/15.23 By instantiating [LPAR04] the rule U5_GAAA(X, less_out_ga(X)) -> PART_IN_GAAA(X) we obtained the following new rules [LPAR04]: 53.54/15.23 53.54/15.23 (U5_GAAA(0, less_out_ga(0)) -> PART_IN_GAAA(0),U5_GAAA(0, less_out_ga(0)) -> PART_IN_GAAA(0)) 53.54/15.23 (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))) 53.54/15.23 53.54/15.23 53.54/15.23 ---------------------------------------- 53.54/15.23 53.54/15.23 (85) 53.54/15.23 Obligation: 53.54/15.23 Q DP problem: 53.54/15.23 The TRS P consists of the following rules: 53.54/15.23 53.54/15.23 PART_IN_GAAA(X) -> PART_IN_GAAA(X) 53.54/15.23 PART_IN_GAAA(0) -> U5_GAAA(0, less_out_ga(0)) 53.54/15.23 PART_IN_GAAA(s(x0)) -> U5_GAAA(s(x0), U9_ga(x0, less_in_ga(x0))) 53.54/15.23 U5_GAAA(0, less_out_ga(0)) -> PART_IN_GAAA(0) 53.54/15.23 U5_GAAA(s(z0), less_out_ga(s(z0))) -> PART_IN_GAAA(s(z0)) 53.54/15.23 53.54/15.23 The TRS R consists of the following rules: 53.54/15.23 53.54/15.23 less_in_ga(0) -> less_out_ga(0) 53.54/15.23 less_in_ga(s(X)) -> U9_ga(X, less_in_ga(X)) 53.54/15.23 U9_ga(X, less_out_ga(X)) -> less_out_ga(s(X)) 53.54/15.23 53.54/15.23 The set Q consists of the following terms: 53.54/15.23 53.54/15.23 less_in_ga(x0) 53.54/15.23 U9_ga(x0, x1) 53.54/15.23 53.54/15.23 We have to consider all (P,Q,R)-chains. 53.54/15.23 ---------------------------------------- 53.54/15.23 53.54/15.23 (86) 53.54/15.23 Obligation: 53.54/15.23 Pi DP problem: 53.54/15.23 The TRS P consists of the following rules: 53.54/15.23 53.54/15.23 LESS_IN_AA(s(X), s(Y)) -> LESS_IN_AA(X, Y) 53.54/15.23 53.54/15.23 The TRS R consists of the following rules: 53.54/15.23 53.54/15.23 qs_in_ag([], []) -> qs_out_ag([], []) 53.54/15.23 qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.54/15.23 part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) 53.54/15.23 less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) 53.54/15.23 less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) 53.54/15.23 U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) 53.54/15.23 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)) 53.54/15.23 part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) 53.54/15.23 less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) 53.54/15.23 less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) 53.54/15.23 U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 53.54/15.23 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)) 53.54/15.23 part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) 53.54/15.23 part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) 53.54/15.23 U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.54/15.23 U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.54/15.23 U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.54/15.23 part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) 53.54/15.23 part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) 53.54/15.23 U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.54/15.23 U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.54/15.23 qs_in_aa([], []) -> qs_out_aa([], []) 53.54/15.23 qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.54/15.23 U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.54/15.23 U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.54/15.23 U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) 53.54/15.23 app_in_aaa([], X, X) -> app_out_aaa([], X, X) 53.54/15.23 app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) 53.54/15.23 U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) 53.54/15.23 U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) 53.54/15.23 U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.54/15.23 U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) 53.54/15.23 app_in_aag([], X, X) -> app_out_aag([], X, X) 53.54/15.23 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 53.54/15.23 U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 53.54/15.23 U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) 53.54/15.23 53.54/15.23 The argument filtering Pi contains the following mapping: 53.54/15.23 qs_in_ag(x1, x2) = qs_in_ag(x2) 53.54/15.23 53.54/15.23 [] = [] 53.54/15.23 53.54/15.23 qs_out_ag(x1, x2) = qs_out_ag(x2) 53.54/15.23 53.54/15.23 U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) 53.54/15.23 53.54/15.23 part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa 53.54/15.23 53.54/15.23 U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) 53.54/15.23 53.54/15.23 less_in_aa(x1, x2) = less_in_aa 53.54/15.23 53.54/15.23 less_out_aa(x1, x2) = less_out_aa(x1) 53.54/15.23 53.54/15.23 U9_aa(x1, x2, x3) = U9_aa(x3) 53.54/15.23 53.54/15.23 U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) 53.54/15.23 53.54/15.23 part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) 53.54/15.23 53.54/15.23 U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) 53.54/15.23 53.54/15.23 less_in_ga(x1, x2) = less_in_ga(x1) 53.54/15.23 53.54/15.23 0 = 0 53.54/15.23 53.54/15.23 less_out_ga(x1, x2) = less_out_ga(x1) 53.54/15.23 53.54/15.23 s(x1) = s(x1) 53.54/15.23 53.54/15.23 U9_ga(x1, x2, x3) = U9_ga(x1, x3) 53.54/15.23 53.54/15.23 U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x1, x6) 53.54/15.23 53.54/15.23 U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x1, x6) 53.54/15.23 53.54/15.23 part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa(x1) 53.54/15.23 53.54/15.23 part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa 53.54/15.23 53.54/15.23 U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) 53.54/15.23 53.54/15.23 U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) 53.54/15.23 53.54/15.23 qs_in_aa(x1, x2) = qs_in_aa 53.54/15.23 53.54/15.23 qs_out_aa(x1, x2) = qs_out_aa 53.54/15.23 53.54/15.23 U1_aa(x1, x2, x3, x4) = U1_aa(x4) 53.54/15.23 53.54/15.23 U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) 53.54/15.23 53.54/15.23 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) 53.54/15.23 53.54/15.23 U4_aa(x1, x2, x3, x4) = U4_aa(x4) 53.54/15.23 53.54/15.23 .(x1, x2) = .(x1, x2) 53.54/15.23 53.54/15.23 app_in_aaa(x1, x2, x3) = app_in_aaa 53.54/15.23 53.54/15.23 app_out_aaa(x1, x2, x3) = app_out_aaa 53.54/15.23 53.54/15.23 U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) 53.54/15.23 53.54/15.23 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) 53.54/15.23 53.54/15.23 U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4) 53.54/15.23 53.54/15.23 app_in_aag(x1, x2, x3) = app_in_aag(x3) 53.54/15.23 53.54/15.23 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2, x3) 53.54/15.23 53.54/15.23 U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x4, x5) 53.54/15.23 53.54/15.23 LESS_IN_AA(x1, x2) = LESS_IN_AA 53.54/15.23 53.54/15.23 53.54/15.23 We have to consider all (P,R,Pi)-chains 53.54/15.23 ---------------------------------------- 53.54/15.23 53.54/15.23 (87) UsableRulesProof (EQUIVALENT) 53.54/15.23 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 53.54/15.23 ---------------------------------------- 53.54/15.23 53.54/15.23 (88) 53.54/15.23 Obligation: 53.54/15.23 Pi DP problem: 53.54/15.23 The TRS P consists of the following rules: 53.54/15.23 53.54/15.23 LESS_IN_AA(s(X), s(Y)) -> LESS_IN_AA(X, Y) 53.54/15.23 53.54/15.23 R is empty. 53.54/15.23 The argument filtering Pi contains the following mapping: 53.54/15.23 s(x1) = s(x1) 53.54/15.23 53.54/15.23 LESS_IN_AA(x1, x2) = LESS_IN_AA 53.54/15.23 53.54/15.23 53.54/15.23 We have to consider all (P,R,Pi)-chains 53.54/15.23 ---------------------------------------- 53.54/15.23 53.54/15.23 (89) 53.54/15.23 Obligation: 53.54/15.23 Pi DP problem: 53.54/15.23 The TRS P consists of the following rules: 53.54/15.23 53.54/15.23 PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_AAAA(X, Xs, Ls, Bs) 53.54/15.23 53.54/15.23 The TRS R consists of the following rules: 53.54/15.23 53.54/15.23 qs_in_ag([], []) -> qs_out_ag([], []) 53.54/15.23 qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.54/15.23 part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) 53.54/15.23 less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) 53.54/15.23 less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) 53.54/15.23 U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) 53.54/15.23 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)) 53.54/15.23 part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) 53.54/15.23 less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) 53.54/15.23 less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) 53.54/15.23 U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 53.54/15.25 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)) 53.54/15.25 part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) 53.54/15.25 part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) 53.54/15.25 U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.54/15.25 U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.54/15.25 U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.54/15.25 part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) 53.54/15.25 part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) 53.54/15.25 U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.54/15.25 U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.54/15.25 qs_in_aa([], []) -> qs_out_aa([], []) 53.54/15.25 qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.54/15.25 U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.54/15.25 U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.54/15.25 U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) 53.54/15.25 app_in_aaa([], X, X) -> app_out_aaa([], X, X) 53.54/15.25 app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) 53.54/15.25 U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) 53.54/15.25 U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) 53.54/15.25 U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.54/15.25 U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) 53.54/15.25 app_in_aag([], X, X) -> app_out_aag([], X, X) 53.54/15.25 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 53.54/15.25 U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 53.54/15.25 U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) 53.54/15.25 53.54/15.25 The argument filtering Pi contains the following mapping: 53.54/15.25 qs_in_ag(x1, x2) = qs_in_ag(x2) 53.54/15.25 53.54/15.25 [] = [] 53.54/15.25 53.54/15.25 qs_out_ag(x1, x2) = qs_out_ag(x2) 53.54/15.25 53.54/15.25 U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) 53.54/15.25 53.54/15.25 part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa 53.54/15.25 53.54/15.25 U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) 53.54/15.25 53.54/15.25 less_in_aa(x1, x2) = less_in_aa 53.54/15.25 53.54/15.25 less_out_aa(x1, x2) = less_out_aa(x1) 53.54/15.25 53.54/15.25 U9_aa(x1, x2, x3) = U9_aa(x3) 53.54/15.25 53.54/15.25 U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) 53.54/15.25 53.54/15.25 part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) 53.54/15.25 53.54/15.25 U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) 53.54/15.25 53.54/15.25 less_in_ga(x1, x2) = less_in_ga(x1) 53.54/15.25 53.54/15.25 0 = 0 53.54/15.25 53.54/15.25 less_out_ga(x1, x2) = less_out_ga(x1) 53.54/15.25 53.54/15.25 s(x1) = s(x1) 53.54/15.25 53.54/15.25 U9_ga(x1, x2, x3) = U9_ga(x1, x3) 53.54/15.25 53.54/15.25 U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x1, x6) 53.54/15.25 53.54/15.25 U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x1, x6) 53.54/15.25 53.54/15.25 part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa(x1) 53.54/15.25 53.54/15.25 part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa 53.54/15.25 53.54/15.25 U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) 53.54/15.25 53.54/15.25 U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) 53.54/15.25 53.54/15.25 qs_in_aa(x1, x2) = qs_in_aa 53.54/15.25 53.54/15.25 qs_out_aa(x1, x2) = qs_out_aa 53.54/15.25 53.54/15.25 U1_aa(x1, x2, x3, x4) = U1_aa(x4) 53.54/15.25 53.54/15.25 U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) 53.54/15.25 53.54/15.25 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) 53.54/15.25 53.54/15.25 U4_aa(x1, x2, x3, x4) = U4_aa(x4) 53.54/15.25 53.54/15.25 .(x1, x2) = .(x1, x2) 53.54/15.25 53.54/15.25 app_in_aaa(x1, x2, x3) = app_in_aaa 53.54/15.25 53.54/15.25 app_out_aaa(x1, x2, x3) = app_out_aaa 53.54/15.25 53.54/15.25 U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) 53.54/15.25 53.54/15.25 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) 53.54/15.25 53.54/15.25 U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4) 53.54/15.25 53.54/15.25 app_in_aag(x1, x2, x3) = app_in_aag(x3) 53.54/15.25 53.54/15.25 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2, x3) 53.54/15.25 53.54/15.25 U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x4, x5) 53.54/15.25 53.54/15.25 PART_IN_AAAA(x1, x2, x3, x4) = PART_IN_AAAA 53.54/15.25 53.54/15.25 53.54/15.25 We have to consider all (P,R,Pi)-chains 53.54/15.25 ---------------------------------------- 53.54/15.25 53.54/15.25 (90) UsableRulesProof (EQUIVALENT) 53.54/15.25 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 53.54/15.25 ---------------------------------------- 53.54/15.25 53.54/15.25 (91) 53.54/15.25 Obligation: 53.54/15.25 Pi DP problem: 53.54/15.25 The TRS P consists of the following rules: 53.54/15.25 53.54/15.25 PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> PART_IN_AAAA(X, Xs, Ls, Bs) 53.54/15.25 53.54/15.25 R is empty. 53.54/15.25 The argument filtering Pi contains the following mapping: 53.54/15.25 .(x1, x2) = .(x1, x2) 53.54/15.25 53.54/15.25 PART_IN_AAAA(x1, x2, x3, x4) = PART_IN_AAAA 53.54/15.25 53.54/15.25 53.54/15.25 We have to consider all (P,R,Pi)-chains 53.54/15.25 ---------------------------------------- 53.54/15.25 53.54/15.25 (92) 53.54/15.25 Obligation: 53.54/15.25 Pi DP problem: 53.54/15.25 The TRS P consists of the following rules: 53.54/15.25 53.54/15.25 QS_IN_AA(.(X, Xs), Ys) -> U1_AA(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.54/15.25 U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AA(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.54/15.25 U2_AA(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> QS_IN_AA(Bigs, Bs) 53.54/15.25 U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> QS_IN_AA(Littles, Ls) 53.54/15.25 53.54/15.25 The TRS R consists of the following rules: 53.54/15.25 53.54/15.25 qs_in_ag([], []) -> qs_out_ag([], []) 53.54/15.25 qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.54/15.25 part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) 53.54/15.25 less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) 53.54/15.25 less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) 53.54/15.25 U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) 53.54/15.25 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)) 53.54/15.25 part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) 53.54/15.25 less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) 53.54/15.25 less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) 53.54/15.25 U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 53.54/15.25 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)) 53.54/15.25 part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) 53.54/15.25 part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) 53.54/15.25 U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.54/15.25 U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.54/15.25 U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.54/15.25 part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) 53.54/15.25 part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) 53.54/15.25 U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.54/15.25 U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.54/15.25 qs_in_aa([], []) -> qs_out_aa([], []) 53.54/15.25 qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.54/15.25 U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.54/15.25 U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.54/15.25 U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) 53.54/15.25 app_in_aaa([], X, X) -> app_out_aaa([], X, X) 53.54/15.25 app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) 53.54/15.25 U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) 53.54/15.25 U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) 53.54/15.25 U2_ag(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.54/15.25 U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_aag(Ls, .(X, Bs), Ys)) 53.54/15.25 app_in_aag([], X, X) -> app_out_aag([], X, X) 53.54/15.25 app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U8_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs)) 53.54/15.25 U8_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs)) 53.54/15.25 U4_ag(X, Xs, Ys, app_out_aag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys) 53.54/15.25 53.54/15.25 The argument filtering Pi contains the following mapping: 53.54/15.25 qs_in_ag(x1, x2) = qs_in_ag(x2) 53.54/15.25 53.54/15.25 [] = [] 53.54/15.25 53.54/15.25 qs_out_ag(x1, x2) = qs_out_ag(x2) 53.54/15.25 53.54/15.25 U1_ag(x1, x2, x3, x4) = U1_ag(x3, x4) 53.54/15.25 53.54/15.25 part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa 53.54/15.25 53.54/15.25 U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) 53.54/15.25 53.54/15.25 less_in_aa(x1, x2) = less_in_aa 53.54/15.25 53.54/15.25 less_out_aa(x1, x2) = less_out_aa(x1) 53.54/15.25 53.54/15.25 U9_aa(x1, x2, x3) = U9_aa(x3) 53.54/15.25 53.54/15.25 U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) 53.54/15.25 53.54/15.25 part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) 53.54/15.25 53.54/15.25 U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) 53.54/15.25 53.54/15.25 less_in_ga(x1, x2) = less_in_ga(x1) 53.54/15.25 53.54/15.25 0 = 0 53.54/15.25 53.54/15.25 less_out_ga(x1, x2) = less_out_ga(x1) 53.54/15.25 53.54/15.25 s(x1) = s(x1) 53.54/15.25 53.54/15.25 U9_ga(x1, x2, x3) = U9_ga(x1, x3) 53.54/15.25 53.54/15.25 U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x1, x6) 53.54/15.25 53.54/15.25 U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x1, x6) 53.54/15.25 53.54/15.25 part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa(x1) 53.54/15.25 53.54/15.25 part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa 53.54/15.25 53.54/15.25 U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) 53.54/15.25 53.54/15.25 U2_ag(x1, x2, x3, x4, x5) = U2_ag(x3, x5) 53.54/15.25 53.54/15.25 qs_in_aa(x1, x2) = qs_in_aa 53.54/15.25 53.54/15.25 qs_out_aa(x1, x2) = qs_out_aa 53.54/15.25 53.54/15.25 U1_aa(x1, x2, x3, x4) = U1_aa(x4) 53.54/15.25 53.54/15.25 U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) 53.54/15.25 53.54/15.25 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) 53.54/15.25 53.54/15.25 U4_aa(x1, x2, x3, x4) = U4_aa(x4) 53.54/15.25 53.54/15.25 .(x1, x2) = .(x1, x2) 53.54/15.25 53.54/15.25 app_in_aaa(x1, x2, x3) = app_in_aaa 53.54/15.25 53.54/15.25 app_out_aaa(x1, x2, x3) = app_out_aaa 53.54/15.25 53.54/15.25 U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) 53.54/15.25 53.54/15.25 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x3, x5) 53.54/15.25 53.54/15.25 U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4) 53.54/15.25 53.54/15.25 app_in_aag(x1, x2, x3) = app_in_aag(x3) 53.54/15.25 53.54/15.25 app_out_aag(x1, x2, x3) = app_out_aag(x1, x2, x3) 53.54/15.25 53.54/15.25 U8_aag(x1, x2, x3, x4, x5) = U8_aag(x1, x4, x5) 53.54/15.25 53.54/15.25 QS_IN_AA(x1, x2) = QS_IN_AA 53.54/15.25 53.54/15.25 U1_AA(x1, x2, x3, x4) = U1_AA(x4) 53.54/15.25 53.54/15.25 U2_AA(x1, x2, x3, x4, x5) = U2_AA(x5) 53.54/15.25 53.54/15.25 53.54/15.25 We have to consider all (P,R,Pi)-chains 53.54/15.25 ---------------------------------------- 53.54/15.25 53.54/15.25 (93) UsableRulesProof (EQUIVALENT) 53.54/15.25 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 53.54/15.25 ---------------------------------------- 53.54/15.25 53.54/15.25 (94) 53.54/15.25 Obligation: 53.54/15.25 Pi DP problem: 53.54/15.25 The TRS P consists of the following rules: 53.54/15.25 53.54/15.25 QS_IN_AA(.(X, Xs), Ys) -> U1_AA(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.54/15.25 U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AA(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.54/15.25 U2_AA(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> QS_IN_AA(Bigs, Bs) 53.54/15.25 U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> QS_IN_AA(Littles, Ls) 53.54/15.25 53.54/15.25 The TRS R consists of the following rules: 53.54/15.25 53.54/15.25 part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, less_in_aa(X, Y)) 53.54/15.25 part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, part_in_aaaa(X, Xs, Ls, Bs)) 53.54/15.25 part_in_aaaa(X1, [], [], []) -> part_out_aaaa(X1, [], [], []) 53.54/15.25 qs_in_aa([], []) -> qs_out_aa([], []) 53.54/15.25 qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs)) 53.54/15.25 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)) 53.54/15.25 U7_aaaa(X, Y, Xs, Ls, Bs, part_out_aaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.54/15.25 U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_aa(Littles, Ls)) 53.54/15.25 less_in_aa(0, s(X2)) -> less_out_aa(0, s(X2)) 53.54/15.25 less_in_aa(s(X), s(Y)) -> U9_aa(X, Y, less_in_aa(X, Y)) 53.54/15.25 U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.54/15.25 U2_aa(X, Xs, Ys, Bigs, qs_out_aa(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs)) 53.54/15.25 U9_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) 53.54/15.25 part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, less_in_ga(X, Y)) 53.54/15.25 part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs)) 53.54/15.25 part_in_gaaa(X1, [], [], []) -> part_out_gaaa(X1, [], [], []) 53.54/15.25 U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_aaa(Ls, .(X, Bs), Ys)) 53.54/15.25 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)) 53.54/15.25 U7_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) 53.54/15.25 U4_aa(X, Xs, Ys, app_out_aaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys) 53.54/15.25 less_in_ga(0, s(X2)) -> less_out_ga(0, s(X2)) 53.54/15.25 less_in_ga(s(X), s(Y)) -> U9_ga(X, Y, less_in_ga(X, Y)) 53.54/15.25 U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) 53.54/15.25 app_in_aaa([], X, X) -> app_out_aaa([], X, X) 53.54/15.25 app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U8_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs)) 53.54/15.25 U9_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 53.54/15.25 U8_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs)) 53.54/15.25 53.54/15.25 The argument filtering Pi contains the following mapping: 53.54/15.25 [] = [] 53.54/15.25 53.54/15.25 part_in_aaaa(x1, x2, x3, x4) = part_in_aaaa 53.54/15.25 53.54/15.25 U5_aaaa(x1, x2, x3, x4, x5, x6) = U5_aaaa(x6) 53.54/15.25 53.54/15.25 less_in_aa(x1, x2) = less_in_aa 53.54/15.25 53.54/15.25 less_out_aa(x1, x2) = less_out_aa(x1) 53.54/15.25 53.54/15.25 U9_aa(x1, x2, x3) = U9_aa(x3) 53.54/15.25 53.54/15.25 U6_aaaa(x1, x2, x3, x4, x5, x6) = U6_aaaa(x6) 53.54/15.25 53.54/15.25 part_in_gaaa(x1, x2, x3, x4) = part_in_gaaa(x1) 53.54/15.25 53.54/15.25 U5_gaaa(x1, x2, x3, x4, x5, x6) = U5_gaaa(x1, x6) 53.54/15.25 53.54/15.25 less_in_ga(x1, x2) = less_in_ga(x1) 53.54/15.25 53.54/15.25 0 = 0 53.54/15.25 53.54/15.25 less_out_ga(x1, x2) = less_out_ga(x1) 53.54/15.25 53.54/15.25 s(x1) = s(x1) 53.54/15.25 53.54/15.25 U9_ga(x1, x2, x3) = U9_ga(x1, x3) 53.54/15.25 53.54/15.25 U6_gaaa(x1, x2, x3, x4, x5, x6) = U6_gaaa(x1, x6) 53.54/15.25 53.54/15.25 U7_gaaa(x1, x2, x3, x4, x5, x6) = U7_gaaa(x1, x6) 53.54/15.25 53.54/15.25 part_out_gaaa(x1, x2, x3, x4) = part_out_gaaa(x1) 53.54/15.25 53.54/15.25 part_out_aaaa(x1, x2, x3, x4) = part_out_aaaa 53.54/15.25 53.54/15.25 U7_aaaa(x1, x2, x3, x4, x5, x6) = U7_aaaa(x6) 53.54/15.25 53.54/15.25 qs_in_aa(x1, x2) = qs_in_aa 53.54/15.25 53.54/15.25 qs_out_aa(x1, x2) = qs_out_aa 53.54/15.26 53.54/15.26 U1_aa(x1, x2, x3, x4) = U1_aa(x4) 53.54/15.26 53.54/15.26 U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5) 53.54/15.26 53.54/15.26 U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) 53.54/15.26 53.54/15.26 U4_aa(x1, x2, x3, x4) = U4_aa(x4) 53.54/15.26 53.54/15.26 .(x1, x2) = .(x1, x2) 53.54/15.26 53.54/15.26 app_in_aaa(x1, x2, x3) = app_in_aaa 53.54/15.26 53.54/15.26 app_out_aaa(x1, x2, x3) = app_out_aaa 53.54/15.26 53.54/15.26 U8_aaa(x1, x2, x3, x4, x5) = U8_aaa(x5) 53.54/15.26 53.54/15.26 QS_IN_AA(x1, x2) = QS_IN_AA 53.54/15.26 53.54/15.26 U1_AA(x1, x2, x3, x4) = U1_AA(x4) 53.54/15.26 53.54/15.26 U2_AA(x1, x2, x3, x4, x5) = U2_AA(x5) 53.54/15.26 53.54/15.26 53.54/15.26 We have to consider all (P,R,Pi)-chains 53.54/15.26 ---------------------------------------- 53.54/15.26 53.54/15.26 (95) PrologToTRSTransformerProof (SOUND) 53.54/15.26 Transformed Prolog program to TRS. 53.54/15.26 53.54/15.26 { 53.54/15.26 "root": 83, 53.54/15.26 "program": { 53.54/15.26 "directives": [], 53.54/15.26 "clauses": [ 53.54/15.26 [ 53.54/15.26 "(qs ([]) ([]))", 53.54/15.26 null 53.54/15.26 ], 53.54/15.26 [ 53.54/15.26 "(qs (. X Xs) Ys)", 53.54/15.26 "(',' (part X Xs Littles Bigs) (',' (qs Littles Ls) (',' (qs Bigs Bs) (app Ls (. X Bs) Ys))))" 53.54/15.26 ], 53.54/15.26 [ 53.54/15.26 "(part X (. Y Xs) (. Y Ls) Bs)", 53.54/15.26 "(',' (less X Y) (part X Xs Ls Bs))" 53.54/15.26 ], 53.54/15.26 [ 53.54/15.26 "(part X (. Y Xs) Ls (. Y Bs))", 53.54/15.26 "(part X Xs Ls Bs)" 53.54/15.26 ], 53.54/15.26 [ 53.54/15.26 "(part X1 ([]) ([]) ([]))", 53.54/15.26 null 53.54/15.26 ], 53.54/15.26 [ 53.54/15.26 "(app ([]) X X)", 53.54/15.26 null 53.54/15.26 ], 53.54/15.26 [ 53.54/15.26 "(app (. X Xs) Ys (. X Zs))", 53.54/15.26 "(app Xs Ys Zs)" 53.54/15.26 ], 53.54/15.26 [ 53.54/15.26 "(less (0) (s X2))", 53.54/15.26 null 53.54/15.26 ], 53.54/15.26 [ 53.54/15.26 "(less (s X) (s Y))", 53.54/15.26 "(less X Y)" 53.54/15.26 ] 53.54/15.26 ] 53.54/15.26 }, 53.54/15.26 "graph": { 53.54/15.26 "nodes": { 53.54/15.26 "1022": { 53.54/15.26 "goal": [{ 53.54/15.26 "clause": 5, 53.54/15.26 "scope": 8, 53.54/15.26 "term": "(app T209 (. T210 T208) T11)" 53.54/15.26 }], 53.54/15.26 "kb": { 53.54/15.26 "nonunifying": [], 53.54/15.26 "intvars": {}, 53.54/15.26 "arithmetic": { 53.54/15.26 "type": "PlainIntegerRelationState", 53.54/15.26 "relations": [] 53.54/15.26 }, 53.54/15.26 "ground": ["T11"], 53.54/15.26 "free": [], 53.54/15.26 "exprvars": [] 53.54/15.26 } 53.54/15.26 }, 53.54/15.26 "1021": { 53.54/15.26 "goal": [ 53.54/15.26 { 53.54/15.26 "clause": 5, 53.54/15.26 "scope": 8, 53.54/15.26 "term": "(app T209 (. T210 T208) T11)" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "clause": 6, 53.54/15.26 "scope": 8, 53.54/15.26 "term": "(app T209 (. T210 T208) T11)" 53.54/15.26 } 53.54/15.26 ], 53.54/15.26 "kb": { 53.54/15.26 "nonunifying": [], 53.54/15.26 "intvars": {}, 53.54/15.26 "arithmetic": { 53.54/15.26 "type": "PlainIntegerRelationState", 53.54/15.26 "relations": [] 53.54/15.26 }, 53.54/15.26 "ground": ["T11"], 53.54/15.26 "free": [], 53.54/15.26 "exprvars": [] 53.54/15.26 } 53.54/15.26 }, 53.54/15.26 "type": "Nodes", 53.54/15.26 "197": { 53.54/15.26 "goal": [{ 53.54/15.26 "clause": -1, 53.54/15.26 "scope": -1, 53.54/15.26 "term": "(true)" 53.54/15.26 }], 53.54/15.26 "kb": { 53.54/15.26 "nonunifying": [], 53.54/15.26 "intvars": {}, 53.54/15.26 "arithmetic": { 53.54/15.26 "type": "PlainIntegerRelationState", 53.54/15.26 "relations": [] 53.54/15.26 }, 53.54/15.26 "ground": [], 53.54/15.26 "free": [], 53.54/15.26 "exprvars": [] 53.54/15.26 } 53.54/15.26 }, 53.54/15.26 "199": { 53.54/15.26 "goal": [], 53.54/15.26 "kb": { 53.54/15.26 "nonunifying": [], 53.54/15.26 "intvars": {}, 53.54/15.26 "arithmetic": { 53.54/15.26 "type": "PlainIntegerRelationState", 53.54/15.26 "relations": [] 53.54/15.26 }, 53.54/15.26 "ground": [], 53.54/15.26 "free": [], 53.54/15.26 "exprvars": [] 53.54/15.26 } 53.54/15.26 }, 53.54/15.26 "397": { 53.54/15.26 "goal": [{ 53.54/15.26 "clause": -1, 53.54/15.26 "scope": -1, 53.54/15.26 "term": "(',' (less T38 T39) (part T38 T40 X67 X68))" 53.54/15.26 }], 53.54/15.26 "kb": { 53.54/15.26 "nonunifying": [], 53.54/15.26 "intvars": {}, 53.54/15.26 "arithmetic": { 53.54/15.26 "type": "PlainIntegerRelationState", 53.54/15.26 "relations": [] 53.54/15.26 }, 53.54/15.26 "ground": [], 53.54/15.26 "free": [ 53.54/15.26 "X67", 53.54/15.26 "X68" 53.54/15.26 ], 53.54/15.26 "exprvars": [] 53.54/15.26 } 53.54/15.26 }, 53.54/15.26 "430": { 53.54/15.26 "goal": [{ 53.54/15.26 "clause": -1, 53.54/15.26 "scope": -1, 53.54/15.26 "term": "(less T58 T59)" 53.54/15.26 }], 53.54/15.26 "kb": { 53.54/15.26 "nonunifying": [], 53.54/15.26 "intvars": {}, 53.54/15.26 "arithmetic": { 53.54/15.26 "type": "PlainIntegerRelationState", 53.54/15.26 "relations": [] 53.54/15.26 }, 53.54/15.26 "ground": [], 53.54/15.26 "free": [], 53.54/15.26 "exprvars": [] 53.54/15.26 } 53.54/15.26 }, 53.54/15.26 "991": { 53.54/15.26 "goal": [ 53.54/15.26 { 53.54/15.26 "clause": 5, 53.54/15.26 "scope": 7, 53.54/15.26 "term": "(app T173 (. T174 T172) X261)" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "clause": 6, 53.54/15.26 "scope": 7, 53.54/15.26 "term": "(app T173 (. 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"arithmetic": { 53.54/15.26 "type": "PlainIntegerRelationState", 53.54/15.26 "relations": [] 53.54/15.26 }, 53.54/15.26 "ground": [], 53.54/15.26 "free": [], 53.54/15.26 "exprvars": [] 53.54/15.26 } 53.54/15.26 }, 53.54/15.26 "422": { 53.54/15.26 "goal": [{ 53.54/15.26 "clause": -1, 53.54/15.26 "scope": -1, 53.54/15.26 "term": "(true)" 53.54/15.26 }], 53.54/15.26 "kb": { 53.54/15.26 "nonunifying": [], 53.54/15.26 "intvars": {}, 53.54/15.26 "arithmetic": { 53.54/15.26 "type": "PlainIntegerRelationState", 53.54/15.26 "relations": [] 53.54/15.26 }, 53.54/15.26 "ground": [], 53.54/15.26 "free": [], 53.54/15.26 "exprvars": [] 53.54/15.26 } 53.54/15.26 }, 53.54/15.26 "543": { 53.54/15.26 "goal": [], 53.54/15.26 "kb": { 53.54/15.26 "nonunifying": [], 53.54/15.26 "intvars": {}, 53.54/15.26 "arithmetic": { 53.54/15.26 "type": "PlainIntegerRelationState", 53.54/15.26 "relations": [] 53.54/15.26 }, 53.54/15.26 "ground": [], 53.54/15.26 "free": [], 53.54/15.26 "exprvars": [] 53.54/15.26 } 53.54/15.26 }, 53.54/15.26 "862": { 53.54/15.26 "goal": [{ 53.54/15.26 "clause": 4, 53.54/15.26 "scope": 4, 53.54/15.26 "term": "(part T43 T44 X67 X68)" 53.54/15.26 }], 53.54/15.26 "kb": { 53.54/15.26 "nonunifying": [], 53.54/15.26 "intvars": {}, 53.54/15.26 "arithmetic": { 53.54/15.26 "type": "PlainIntegerRelationState", 53.54/15.26 "relations": [] 53.54/15.26 }, 53.54/15.26 "ground": ["T43"], 53.54/15.26 "free": [ 53.54/15.26 "X67", 53.54/15.26 "X68" 53.54/15.26 ], 53.54/15.26 "exprvars": [] 53.54/15.26 } 53.54/15.26 }, 53.54/15.26 "423": { 53.54/15.26 "goal": [], 53.54/15.26 "kb": { 53.54/15.26 "nonunifying": [], 53.54/15.26 "intvars": {}, 53.54/15.26 "arithmetic": { 53.54/15.26 "type": "PlainIntegerRelationState", 53.54/15.26 "relations": [] 53.54/15.26 }, 53.54/15.26 "ground": [], 53.54/15.26 "free": [], 53.54/15.26 "exprvars": [] 53.54/15.26 } 53.54/15.26 }, 53.54/15.26 "467": { 53.54/15.26 "goal": [{ 53.54/15.26 "clause": 2, 53.54/15.26 "scope": 4, 53.54/15.26 "term": "(part T43 T44 X67 X68)" 53.54/15.26 }], 53.54/15.26 "kb": { 53.54/15.26 "nonunifying": [], 53.54/15.26 "intvars": {}, 53.54/15.26 "arithmetic": { 53.54/15.26 "type": "PlainIntegerRelationState", 53.54/15.26 "relations": [] 53.54/15.26 }, 53.54/15.26 "ground": ["T43"], 53.54/15.26 "free": [ 53.54/15.26 "X67", 53.54/15.26 "X68" 53.54/15.26 ], 53.54/15.26 "exprvars": [] 53.54/15.26 } 53.54/15.26 }, 53.54/15.26 "940": { 53.54/15.26 "goal": [{ 53.54/15.26 "clause": -1, 53.54/15.26 "scope": -1, 53.54/15.26 "term": "(true)" 53.54/15.26 }], 53.54/15.26 "kb": { 53.54/15.26 "nonunifying": [], 53.54/15.26 "intvars": {}, 53.54/15.26 "arithmetic": { 53.54/15.26 "type": "PlainIntegerRelationState", 53.54/15.26 "relations": [] 53.54/15.26 }, 53.54/15.26 "ground": [], 53.54/15.26 "free": [], 53.54/15.26 "exprvars": [] 53.54/15.26 } 53.54/15.26 }, 53.54/15.26 "984": { 53.54/15.26 "goal": [{ 53.54/15.26 "clause": -1, 53.54/15.26 "scope": -1, 53.54/15.26 "term": "(qs T170 X260)" 53.54/15.26 }], 53.54/15.26 "kb": { 53.54/15.26 "nonunifying": [], 53.54/15.26 "intvars": {}, 53.54/15.26 "arithmetic": { 53.54/15.26 "type": "PlainIntegerRelationState", 53.54/15.26 "relations": [] 53.54/15.26 }, 53.54/15.26 "ground": [], 53.54/15.26 "free": ["X260"], 53.54/15.26 "exprvars": [] 53.54/15.26 } 53.54/15.26 }, 53.54/15.26 "424": { 53.54/15.26 "goal": [], 53.54/15.26 "kb": { 53.54/15.26 "nonunifying": [], 53.54/15.26 "intvars": {}, 53.54/15.26 "arithmetic": { 53.54/15.26 "type": "PlainIntegerRelationState", 53.54/15.26 "relations": [] 53.54/15.26 }, 53.54/15.26 "ground": [], 53.54/15.26 "free": [], 53.54/15.26 "exprvars": [] 53.54/15.26 } 53.54/15.26 }, 53.54/15.26 "545": { 53.54/15.26 "goal": [], 53.54/15.26 "kb": { 53.54/15.26 "nonunifying": [], 53.54/15.26 "intvars": {}, 53.54/15.26 "arithmetic": { 53.54/15.26 "type": "PlainIntegerRelationState", 53.54/15.26 "relations": [] 53.54/15.26 }, 53.54/15.26 "ground": [], 53.54/15.26 "free": [], 53.54/15.26 "exprvars": [] 53.54/15.26 } 53.54/15.26 }, 53.54/15.26 "941": { 53.54/15.26 "goal": [], 53.54/15.26 "kb": { 53.54/15.26 "nonunifying": [], 53.54/15.26 "intvars": {}, 53.54/15.26 "arithmetic": { 53.54/15.26 "type": "PlainIntegerRelationState", 53.54/15.26 "relations": [] 53.54/15.26 }, 53.54/15.26 "ground": [], 53.54/15.26 "free": [], 53.54/15.26 "exprvars": [] 53.54/15.26 } 53.54/15.26 }, 53.54/15.26 "985": { 53.54/15.26 "goal": [{ 53.54/15.26 "clause": -1, 53.54/15.26 "scope": -1, 53.54/15.26 "term": "(app T173 (. T174 T172) X261)" 53.54/15.26 }], 53.54/15.26 "kb": { 53.54/15.26 "nonunifying": [], 53.54/15.26 "intvars": {}, 53.54/15.26 "arithmetic": { 53.54/15.26 "type": "PlainIntegerRelationState", 53.54/15.26 "relations": [] 53.54/15.26 }, 53.54/15.26 "ground": [], 53.54/15.26 "free": ["X261"], 53.54/15.26 "exprvars": [] 53.54/15.26 } 53.54/15.26 }, 53.54/15.26 "469": { 53.54/15.26 "goal": [ 53.54/15.26 { 53.54/15.26 "clause": 3, 53.54/15.26 "scope": 4, 53.54/15.26 "term": "(part T43 T44 X67 X68)" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "clause": 4, 53.54/15.26 "scope": 4, 53.54/15.26 "term": "(part T43 T44 X67 X68)" 53.54/15.26 } 53.54/15.26 ], 53.54/15.26 "kb": { 53.54/15.26 "nonunifying": [], 53.54/15.26 "intvars": {}, 53.54/15.26 "arithmetic": { 53.54/15.26 "type": "PlainIntegerRelationState", 53.54/15.26 "relations": [] 53.54/15.26 }, 53.54/15.26 "ground": ["T43"], 53.54/15.26 "free": [ 53.54/15.26 "X67", 53.54/15.26 "X68" 53.54/15.26 ], 53.54/15.26 "exprvars": [] 53.54/15.26 } 53.54/15.26 }, 53.54/15.26 "942": { 53.54/15.26 "goal": [], 53.54/15.26 "kb": { 53.54/15.26 "nonunifying": [], 53.54/15.26 "intvars": {}, 53.54/15.26 "arithmetic": { 53.54/15.26 "type": "PlainIntegerRelationState", 53.54/15.26 "relations": [] 53.54/15.26 }, 53.54/15.26 "ground": [], 53.54/15.26 "free": [], 53.54/15.26 "exprvars": [] 53.54/15.26 } 53.54/15.26 }, 53.54/15.26 "83": { 53.54/15.26 "goal": [{ 53.54/15.26 "clause": -1, 53.54/15.26 "scope": -1, 53.54/15.26 "term": "(qs T1 T2)" 53.54/15.26 }], 53.54/15.26 "kb": { 53.54/15.26 "nonunifying": [], 53.54/15.26 "intvars": {}, 53.54/15.26 "arithmetic": { 53.54/15.26 "type": "PlainIntegerRelationState", 53.54/15.26 "relations": [] 53.54/15.26 }, 53.54/15.26 "ground": ["T2"], 53.54/15.26 "free": [], 53.54/15.26 "exprvars": [] 53.54/15.26 } 53.54/15.26 } 53.54/15.26 }, 53.54/15.26 "edges": [ 53.54/15.26 { 53.54/15.26 "from": 83, 53.54/15.26 "to": 100, 53.54/15.26 "label": "CASE" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 100, 53.54/15.26 "to": 136, 53.54/15.26 "label": "PARALLEL" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 100, 53.54/15.26 "to": 140, 53.54/15.26 "label": "PARALLEL" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 136, 53.54/15.26 "to": 197, 53.54/15.26 "label": "EVAL with clause\nqs([], []).\nand substitutionT1 -> [],\nT2 -> []" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 136, 53.54/15.26 "to": 199, 53.54/15.26 "label": "EVAL-BACKTRACK" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 140, 53.54/15.26 "to": 207, 53.54/15.26 "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" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 140, 53.54/15.26 "to": 210, 53.54/15.26 "label": "EVAL-BACKTRACK" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 197, 53.54/15.26 "to": 203, 53.54/15.26 "label": "SUCCESS" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 207, 53.54/15.26 "to": 217, 53.54/15.26 "label": "SPLIT 1" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 207, 53.54/15.26 "to": 218, 53.54/15.26 "label": "SPLIT 2\nreplacements:X16 -> T17,\nX17 -> T18,\nT12 -> T19" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 217, 53.54/15.26 "to": 261, 53.54/15.26 "label": "CASE" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 218, 53.54/15.26 "to": 935, 53.54/15.26 "label": "SPLIT 1" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 218, 53.54/15.26 "to": 936, 53.54/15.26 "label": "SPLIT 2\nreplacements:X18 -> T149,\nT18 -> T150,\nT19 -> T151" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 261, 53.54/15.26 "to": 335, 53.54/15.26 "label": "PARALLEL" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 261, 53.54/15.26 "to": 336, 53.54/15.26 "label": "PARALLEL" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 335, 53.54/15.26 "to": 397, 53.54/15.26 "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" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 335, 53.54/15.26 "to": 399, 53.54/15.26 "label": "EVAL-BACKTRACK" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 336, 53.54/15.26 "to": 920, 53.54/15.26 "label": "PARALLEL" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 336, 53.54/15.26 "to": 921, 53.54/15.26 "label": "PARALLEL" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 397, 53.54/15.26 "to": 408, 53.54/15.26 "label": "SPLIT 1" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 397, 53.54/15.26 "to": 410, 53.54/15.26 "label": "SPLIT 2\nnew knowledge:\nT43 is ground\nreplacements:T38 -> T43,\nT40 -> T44" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 408, 53.54/15.26 "to": 416, 53.54/15.26 "label": "CASE" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 410, 53.54/15.26 "to": 452, 53.54/15.26 "label": "CASE" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 416, 53.54/15.26 "to": 420, 53.54/15.26 "label": "PARALLEL" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 416, 53.54/15.26 "to": 421, 53.54/15.26 "label": "PARALLEL" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 420, 53.54/15.26 "to": 422, 53.54/15.26 "label": "EVAL with clause\nless(0, s(X77)).\nand substitutionT38 -> 0,\nX77 -> T51,\nT39 -> s(T51)" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 420, 53.54/15.26 "to": 423, 53.54/15.26 "label": "EVAL-BACKTRACK" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 421, 53.54/15.26 "to": 430, 53.54/15.26 "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" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 421, 53.54/15.26 "to": 432, 53.54/15.26 "label": "EVAL-BACKTRACK" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 422, 53.54/15.26 "to": 424, 53.54/15.26 "label": "SUCCESS" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 430, 53.54/15.26 "to": 408, 53.54/15.26 "label": "INSTANCE with matching:\nT38 -> T58\nT39 -> T59" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 452, 53.54/15.26 "to": 467, 53.54/15.26 "label": "PARALLEL" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 452, 53.54/15.26 "to": 469, 53.54/15.26 "label": "PARALLEL" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 467, 53.54/15.26 "to": 492, 53.54/15.26 "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" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 467, 53.54/15.26 "to": 496, 53.54/15.26 "label": "EVAL-BACKTRACK" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 469, 53.54/15.26 "to": 860, 53.54/15.26 "label": "PARALLEL" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 469, 53.54/15.26 "to": 862, 53.54/15.26 "label": "PARALLEL" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 492, 53.54/15.26 "to": 512, 53.54/15.26 "label": "SPLIT 1" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 492, 53.54/15.26 "to": 514, 53.54/15.26 "label": "SPLIT 2\nnew knowledge:\nT77 is ground\nreplacements:T81 -> T84" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 512, 53.54/15.26 "to": 527, 53.54/15.26 "label": "CASE" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 514, 53.54/15.26 "to": 410, 53.54/15.26 "label": "INSTANCE with matching:\nT43 -> T77\nT44 -> T84\nX67 -> X126\nX68 -> X127" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 527, 53.54/15.26 "to": 531, 53.54/15.26 "label": "PARALLEL" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 527, 53.54/15.26 "to": 533, 53.54/15.26 "label": "PARALLEL" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 531, 53.54/15.26 "to": 541, 53.54/15.26 "label": "EVAL with clause\nless(0, s(X136)).\nand substitutionT77 -> 0,\nX136 -> T91,\nT80 -> s(T91)" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 531, 53.54/15.26 "to": 543, 53.54/15.26 "label": "EVAL-BACKTRACK" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 533, 53.54/15.26 "to": 561, 53.54/15.26 "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" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 533, 53.54/15.26 "to": 563, 53.54/15.26 "label": "EVAL-BACKTRACK" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 541, 53.54/15.26 "to": 545, 53.54/15.26 "label": "SUCCESS" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 561, 53.54/15.26 "to": 512, 53.54/15.26 "label": "INSTANCE with matching:\nT77 -> T96\nT80 -> T98" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 860, 53.54/15.26 "to": 911, 53.54/15.26 "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" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 860, 53.54/15.26 "to": 912, 53.54/15.26 "label": "EVAL-BACKTRACK" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 862, 53.54/15.26 "to": 916, 53.54/15.26 "label": "EVAL with clause\npart(X196, [], [], []).\nand substitutionT43 -> T125,\nX196 -> T125,\nT44 -> [],\nX67 -> [],\nX68 -> []" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 862, 53.54/15.26 "to": 918, 53.54/15.26 "label": "EVAL-BACKTRACK" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 911, 53.54/15.26 "to": 410, 53.54/15.26 "label": "INSTANCE with matching:\nT43 -> T116\nT44 -> T119\nX67 -> X185\nX68 -> X186" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 916, 53.54/15.26 "to": 919, 53.54/15.26 "label": "SUCCESS" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 920, 53.54/15.26 "to": 927, 53.54/15.26 "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" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 920, 53.54/15.26 "to": 928, 53.54/15.26 "label": "EVAL-BACKTRACK" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 921, 53.54/15.26 "to": 931, 53.54/15.26 "label": "EVAL with clause\npart(X241, [], [], []).\nand substitutionT12 -> T148,\nX241 -> T148,\nT13 -> [],\nX16 -> [],\nX17 -> []" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 921, 53.54/15.26 "to": 932, 53.54/15.26 "label": "EVAL-BACKTRACK" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 927, 53.54/15.26 "to": 217, 53.54/15.26 "label": "INSTANCE with matching:\nT12 -> T141\nT13 -> T142\nX16 -> X230\nX17 -> X231" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 931, 53.54/15.26 "to": 933, 53.54/15.26 "label": "SUCCESS" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 935, 53.54/15.26 "to": 937, 53.54/15.26 "label": "CASE" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 936, 53.54/15.26 "to": 1014, 53.54/15.26 "label": "SPLIT 1" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 936, 53.54/15.26 "to": 1015, 53.54/15.26 "label": "SPLIT 2\nreplacements:X19 -> T208,\nT149 -> T209,\nT151 -> T210" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 937, 53.54/15.26 "to": 938, 53.54/15.26 "label": "PARALLEL" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 937, 53.54/15.26 "to": 939, 53.54/15.26 "label": "PARALLEL" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 938, 53.54/15.26 "to": 940, 53.54/15.26 "label": "EVAL with clause\nqs([], []).\nand substitutionT17 -> [],\nX18 -> []" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 938, 53.54/15.26 "to": 941, 53.54/15.26 "label": "EVAL-BACKTRACK" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 939, 53.54/15.26 "to": 973, 53.54/15.26 "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" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 939, 53.54/15.26 "to": 974, 53.54/15.26 "label": "EVAL-BACKTRACK" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 940, 53.54/15.26 "to": 942, 53.54/15.26 "label": "SUCCESS" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 973, 53.54/15.26 "to": 977, 53.54/15.26 "label": "SPLIT 1" 53.54/15.26 }, 53.54/15.26 { 53.54/15.26 "from": 973, 53.54/15.26 "to": 978, 53.54/15.27 "label": "SPLIT 2\nreplacements:X257 -> T163,\nX258 -> T164,\nT158 -> T165" 53.54/15.27 }, 53.54/15.27 { 53.54/15.27 "from": 977, 53.54/15.27 "to": 217, 53.54/15.27 "label": "INSTANCE with matching:\nT12 -> T158\nT13 -> T159\nX16 -> X257\nX17 -> X258" 53.54/15.27 }, 53.54/15.27 { 53.54/15.27 "from": 978, 53.54/15.27 "to": 979, 53.54/15.27 "label": "SPLIT 1" 53.54/15.27 }, 53.54/15.27 { 53.54/15.27 "from": 978, 53.54/15.27 "to": 980, 53.54/15.27 "label": "SPLIT 2\nreplacements:X259 -> T169,\nT164 -> T170,\nT165 -> T171" 53.54/15.27 }, 53.54/15.27 { 53.54/15.27 "from": 979, 53.54/15.27 "to": 935, 53.54/15.27 "label": "INSTANCE with matching:\nT17 -> T163\nX18 -> X259" 53.54/15.27 }, 53.54/15.27 { 53.54/15.27 "from": 980, 53.54/15.27 "to": 984, 53.54/15.27 "label": "SPLIT 1" 53.54/15.27 }, 53.54/15.27 { 53.54/15.27 "from": 980, 53.54/15.27 "to": 985, 53.54/15.27 "label": "SPLIT 2\nreplacements:X260 -> T172,\nT169 -> T173,\nT171 -> T174" 53.54/15.27 }, 53.54/15.27 { 53.54/15.27 "from": 984, 53.54/15.27 "to": 935, 53.54/15.27 "label": "INSTANCE with matching:\nT17 -> T170\nX18 -> X260" 53.54/15.27 }, 53.54/15.27 { 53.54/15.27 "from": 985, 53.54/15.27 "to": 991, 53.54/15.27 "label": "CASE" 53.54/15.27 }, 53.54/15.27 { 53.54/15.27 "from": 991, 53.54/15.27 "to": 995, 53.54/15.27 "label": "PARALLEL" 53.54/15.27 }, 53.54/15.27 { 53.54/15.27 "from": 991, 53.54/15.27 "to": 996, 53.54/15.27 "label": "PARALLEL" 53.54/15.27 }, 53.54/15.27 { 53.54/15.27 "from": 995, 53.54/15.27 "to": 997, 53.54/15.27 "label": "EVAL with clause\napp([], X282, X282).\nand substitutionT173 -> [],\nT174 -> T187,\nT172 -> T188,\nX282 -> .(T187, T188),\nX261 -> .(T187, T188)" 53.54/15.27 }, 53.54/15.27 { 53.54/15.27 "from": 995, 53.54/15.27 "to": 998, 53.54/15.27 "label": "EVAL-BACKTRACK" 53.54/15.27 }, 53.54/15.27 { 53.54/15.27 "from": 996, 53.54/15.27 "to": 1007, 53.54/15.27 "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" 53.54/15.27 }, 53.54/15.27 { 53.54/15.27 "from": 996, 53.54/15.27 "to": 1008, 53.54/15.27 "label": "EVAL-BACKTRACK" 53.54/15.27 }, 53.54/15.27 { 53.54/15.27 "from": 997, 53.54/15.27 "to": 999, 53.54/15.27 "label": "SUCCESS" 53.54/15.27 }, 53.54/15.27 { 53.54/15.27 "from": 1007, 53.54/15.27 "to": 985, 53.54/15.27 "label": "INSTANCE with matching:\nT173 -> T201\nT174 -> T202\nT172 -> T203\nX261 -> X297" 53.54/15.27 }, 53.54/15.27 { 53.54/15.27 "from": 1014, 53.54/15.27 "to": 935, 53.54/15.27 "label": "INSTANCE with matching:\nT17 -> T150\nX18 -> X19" 53.54/15.27 }, 53.54/15.27 { 53.54/15.27 "from": 1015, 53.54/15.27 "to": 1021, 53.54/15.27 "label": "CASE" 53.54/15.27 }, 53.54/15.27 { 53.54/15.27 "from": 1021, 53.54/15.27 "to": 1022, 53.54/15.27 "label": "PARALLEL" 53.54/15.27 }, 53.54/15.27 { 53.54/15.27 "from": 1021, 53.54/15.27 "to": 1023, 53.54/15.27 "label": "PARALLEL" 53.54/15.27 }, 53.54/15.27 { 53.54/15.27 "from": 1022, 53.54/15.27 "to": 1029, 53.54/15.27 "label": "EVAL with clause\napp([], X306, X306).\nand substitutionT209 -> [],\nT210 -> T223,\nT208 -> T224,\nX306 -> .(T223, T224),\nT11 -> .(T223, T224)" 53.54/15.27 }, 53.54/15.27 { 53.54/15.27 "from": 1022, 53.54/15.27 "to": 1030, 53.54/15.27 "label": "EVAL-BACKTRACK" 53.54/15.27 }, 53.54/15.27 { 53.54/15.27 "from": 1023, 53.54/15.27 "to": 1034, 53.54/15.27 "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" 53.54/15.27 }, 53.54/15.27 { 53.54/15.27 "from": 1023, 53.54/15.27 "to": 1035, 53.54/15.27 "label": "EVAL-BACKTRACK" 53.54/15.27 }, 53.54/15.27 { 53.54/15.27 "from": 1029, 53.54/15.27 "to": 1031, 53.54/15.27 "label": "SUCCESS" 53.54/15.27 }, 53.54/15.27 { 53.54/15.27 "from": 1034, 53.54/15.27 "to": 1015, 53.54/15.27 "label": "INSTANCE with matching:\nT209 -> T240\nT210 -> T241\nT208 -> T242\nT11 -> T239" 53.54/15.27 } 53.54/15.27 ], 53.54/15.27 "type": "Graph" 53.54/15.27 } 53.54/15.27 } 53.54/15.27 53.54/15.27 ---------------------------------------- 53.54/15.27 53.54/15.27 (96) 53.54/15.27 Obligation: 53.54/15.27 Q restricted rewrite system: 53.54/15.27 The TRS R consists of the following rules: 53.54/15.27 53.54/15.27 f83_in([]) -> f83_out1 53.54/15.27 f83_in(T11) -> U1(f207_in(T11), T11) 53.54/15.27 U1(f207_out1(X18, T12, X19), T11) -> f83_out1 53.54/15.27 f408_in -> f408_out1(0) 53.54/15.27 f408_in -> U2(f408_in) 53.54/15.27 U2(f408_out1(T58)) -> f408_out1(s(T58)) 53.54/15.27 f410_in(T77) -> U3(f492_in(T77), T77) 53.54/15.27 U3(f492_out1, T77) -> f410_out1 53.54/15.27 f410_in(T116) -> U4(f410_in(T116), T116) 53.54/15.27 U4(f410_out1, T116) -> f410_out1 53.54/15.27 f410_in(T125) -> f410_out1 53.54/15.27 f512_in(0) -> f512_out1 53.54/15.27 f512_in(s(T96)) -> U5(f512_in(T96), s(T96)) 53.54/15.27 U5(f512_out1, s(T96)) -> f512_out1 53.54/15.27 f217_in -> U6(f397_in) 53.54/15.27 U6(f397_out1(T38)) -> f217_out1 53.54/15.27 f217_in -> U7(f217_in) 53.54/15.27 U7(f217_out1) -> f217_out1 53.54/15.27 f217_in -> f217_out1 53.54/15.27 f935_in -> f935_out1 53.54/15.27 f935_in -> U8(f973_in) 53.54/15.27 U8(f973_out1) -> f935_out1 53.54/15.27 f985_in -> f985_out1 53.54/15.27 f985_in -> U9(f985_in) 53.54/15.27 U9(f985_out1) -> f985_out1 53.54/15.27 f1015_in(.(T223, T224)) -> f1015_out1([], T223, T224) 53.54/15.27 f1015_in(.(T235, T239)) -> U10(f1015_in(T239), .(T235, T239)) 53.54/15.27 U10(f1015_out1(T240, T241, T242), .(T235, T239)) -> f1015_out1(.(T235, T240), T241, T242) 53.54/15.27 f207_in(T11) -> U11(f217_in, T11) 53.54/15.27 U11(f217_out1, T11) -> U12(f218_in(T11), T11) 53.54/15.27 U12(f218_out1(X18, T19, X19), T11) -> f207_out1(X18, T19, X19) 53.54/15.27 f218_in(T11) -> U13(f935_in, T11) 53.54/15.27 U13(f935_out1, T11) -> U14(f936_in(T11), T11) 53.54/15.27 U14(f936_out1(T149, T151, X19), T11) -> f218_out1(T149, T151, X19) 53.54/15.27 f397_in -> U15(f408_in) 53.54/15.27 U15(f408_out1(T43)) -> U16(f410_in(T43), T43) 53.54/15.27 U16(f410_out1, T43) -> f397_out1(T43) 53.54/15.27 f492_in(T77) -> U17(f512_in(T77), T77) 53.54/15.27 U17(f512_out1, T77) -> U18(f410_in(T77), T77) 53.54/15.27 U18(f410_out1, T77) -> f492_out1 53.54/15.27 f936_in(T11) -> U19(f935_in, T11) 53.54/15.27 U19(f935_out1, T11) -> U20(f1015_in(T11), T11) 53.54/15.27 U20(f1015_out1(T209, T210, T208), T11) -> f936_out1(T209, T210, T208) 53.54/15.27 f973_in -> U21(f217_in) 53.54/15.27 U21(f217_out1) -> U22(f978_in) 53.54/15.27 U22(f978_out1) -> f973_out1 53.54/15.27 f978_in -> U23(f935_in) 53.54/15.27 U23(f935_out1) -> U24(f980_in) 53.54/15.27 U24(f980_out1) -> f978_out1 53.54/15.27 f980_in -> U25(f935_in) 53.54/15.27 U25(f935_out1) -> U26(f985_in) 53.54/15.27 U26(f985_out1) -> f980_out1 53.54/15.27 53.54/15.27 Q is empty. 53.54/15.27 53.54/15.27 ---------------------------------------- 53.54/15.27 53.54/15.27 (97) DependencyPairsProof (EQUIVALENT) 53.54/15.27 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 53.54/15.27 ---------------------------------------- 53.54/15.27 53.54/15.27 (98) 53.54/15.27 Obligation: 53.54/15.27 Q DP problem: 53.54/15.27 The TRS P consists of the following rules: 53.54/15.27 53.54/15.27 F83_IN(T11) -> U1^1(f207_in(T11), T11) 53.54/15.27 F83_IN(T11) -> F207_IN(T11) 53.54/15.27 F408_IN -> U2^1(f408_in) 53.54/15.27 F408_IN -> F408_IN 53.54/15.27 F410_IN(T77) -> U3^1(f492_in(T77), T77) 53.54/15.27 F410_IN(T77) -> F492_IN(T77) 53.54/15.27 F410_IN(T116) -> U4^1(f410_in(T116), T116) 53.54/15.27 F410_IN(T116) -> F410_IN(T116) 53.54/15.27 F512_IN(s(T96)) -> U5^1(f512_in(T96), s(T96)) 53.54/15.27 F512_IN(s(T96)) -> F512_IN(T96) 53.54/15.27 F217_IN -> U6^1(f397_in) 53.54/15.27 F217_IN -> F397_IN 53.54/15.27 F217_IN -> U7^1(f217_in) 53.54/15.27 F217_IN -> F217_IN 53.54/15.27 F935_IN -> U8^1(f973_in) 53.54/15.27 F935_IN -> F973_IN 53.54/15.27 F985_IN -> U9^1(f985_in) 53.54/15.27 F985_IN -> F985_IN 53.54/15.27 F1015_IN(.(T235, T239)) -> U10^1(f1015_in(T239), .(T235, T239)) 53.54/15.27 F1015_IN(.(T235, T239)) -> F1015_IN(T239) 53.54/15.27 F207_IN(T11) -> U11^1(f217_in, T11) 53.54/15.27 F207_IN(T11) -> F217_IN 53.54/15.27 U11^1(f217_out1, T11) -> U12^1(f218_in(T11), T11) 53.54/15.27 U11^1(f217_out1, T11) -> F218_IN(T11) 53.54/15.27 F218_IN(T11) -> U13^1(f935_in, T11) 53.54/15.27 F218_IN(T11) -> F935_IN 53.54/15.27 U13^1(f935_out1, T11) -> U14^1(f936_in(T11), T11) 53.54/15.27 U13^1(f935_out1, T11) -> F936_IN(T11) 53.54/15.27 F397_IN -> U15^1(f408_in) 53.54/15.27 F397_IN -> F408_IN 53.54/15.27 U15^1(f408_out1(T43)) -> U16^1(f410_in(T43), T43) 53.54/15.27 U15^1(f408_out1(T43)) -> F410_IN(T43) 53.54/15.27 F492_IN(T77) -> U17^1(f512_in(T77), T77) 53.54/15.27 F492_IN(T77) -> F512_IN(T77) 53.54/15.27 U17^1(f512_out1, T77) -> U18^1(f410_in(T77), T77) 53.54/15.27 U17^1(f512_out1, T77) -> F410_IN(T77) 53.54/15.27 F936_IN(T11) -> U19^1(f935_in, T11) 53.54/15.27 F936_IN(T11) -> F935_IN 53.54/15.27 U19^1(f935_out1, T11) -> U20^1(f1015_in(T11), T11) 53.54/15.27 U19^1(f935_out1, T11) -> F1015_IN(T11) 53.54/15.27 F973_IN -> U21^1(f217_in) 53.54/15.27 F973_IN -> F217_IN 53.54/15.27 U21^1(f217_out1) -> U22^1(f978_in) 53.54/15.27 U21^1(f217_out1) -> F978_IN 53.54/15.27 F978_IN -> U23^1(f935_in) 53.54/15.27 F978_IN -> F935_IN 53.54/15.27 U23^1(f935_out1) -> U24^1(f980_in) 53.54/15.27 U23^1(f935_out1) -> F980_IN 53.54/15.27 F980_IN -> U25^1(f935_in) 53.54/15.27 F980_IN -> F935_IN 53.54/15.27 U25^1(f935_out1) -> U26^1(f985_in) 53.54/15.27 U25^1(f935_out1) -> F985_IN 53.54/15.27 53.54/15.27 The TRS R consists of the following rules: 53.54/15.27 53.54/15.27 f83_in([]) -> f83_out1 53.54/15.27 f83_in(T11) -> U1(f207_in(T11), T11) 53.54/15.27 U1(f207_out1(X18, T12, X19), T11) -> f83_out1 53.54/15.27 f408_in -> f408_out1(0) 53.54/15.27 f408_in -> U2(f408_in) 53.54/15.27 U2(f408_out1(T58)) -> f408_out1(s(T58)) 53.54/15.27 f410_in(T77) -> U3(f492_in(T77), T77) 53.54/15.27 U3(f492_out1, T77) -> f410_out1 53.54/15.27 f410_in(T116) -> U4(f410_in(T116), T116) 53.54/15.27 U4(f410_out1, T116) -> f410_out1 53.54/15.27 f410_in(T125) -> f410_out1 53.54/15.27 f512_in(0) -> f512_out1 53.54/15.27 f512_in(s(T96)) -> U5(f512_in(T96), s(T96)) 53.54/15.27 U5(f512_out1, s(T96)) -> f512_out1 53.54/15.27 f217_in -> U6(f397_in) 53.54/15.27 U6(f397_out1(T38)) -> f217_out1 53.54/15.27 f217_in -> U7(f217_in) 53.54/15.27 U7(f217_out1) -> f217_out1 53.54/15.27 f217_in -> f217_out1 53.54/15.27 f935_in -> f935_out1 53.54/15.27 f935_in -> U8(f973_in) 53.54/15.27 U8(f973_out1) -> f935_out1 53.54/15.27 f985_in -> f985_out1 53.54/15.27 f985_in -> U9(f985_in) 53.54/15.27 U9(f985_out1) -> f985_out1 53.54/15.27 f1015_in(.(T223, T224)) -> f1015_out1([], T223, T224) 53.54/15.27 f1015_in(.(T235, T239)) -> U10(f1015_in(T239), .(T235, T239)) 53.54/15.27 U10(f1015_out1(T240, T241, T242), .(T235, T239)) -> f1015_out1(.(T235, T240), T241, T242) 53.54/15.27 f207_in(T11) -> U11(f217_in, T11) 53.54/15.27 U11(f217_out1, T11) -> U12(f218_in(T11), T11) 53.54/15.27 U12(f218_out1(X18, T19, X19), T11) -> f207_out1(X18, T19, X19) 53.54/15.27 f218_in(T11) -> U13(f935_in, T11) 53.54/15.27 U13(f935_out1, T11) -> U14(f936_in(T11), T11) 53.54/15.27 U14(f936_out1(T149, T151, X19), T11) -> f218_out1(T149, T151, X19) 53.54/15.27 f397_in -> U15(f408_in) 53.54/15.27 U15(f408_out1(T43)) -> U16(f410_in(T43), T43) 53.54/15.27 U16(f410_out1, T43) -> f397_out1(T43) 53.54/15.27 f492_in(T77) -> U17(f512_in(T77), T77) 53.54/15.27 U17(f512_out1, T77) -> U18(f410_in(T77), T77) 53.54/15.27 U18(f410_out1, T77) -> f492_out1 53.54/15.27 f936_in(T11) -> U19(f935_in, T11) 53.54/15.27 U19(f935_out1, T11) -> U20(f1015_in(T11), T11) 53.54/15.27 U20(f1015_out1(T209, T210, T208), T11) -> f936_out1(T209, T210, T208) 53.54/15.27 f973_in -> U21(f217_in) 53.54/15.27 U21(f217_out1) -> U22(f978_in) 53.54/15.27 U22(f978_out1) -> f973_out1 53.54/15.27 f978_in -> U23(f935_in) 53.54/15.27 U23(f935_out1) -> U24(f980_in) 53.54/15.27 U24(f980_out1) -> f978_out1 53.54/15.27 f980_in -> U25(f935_in) 53.54/15.27 U25(f935_out1) -> U26(f985_in) 53.54/15.27 U26(f985_out1) -> f980_out1 53.54/15.27 53.54/15.27 Q is empty. 53.54/15.27 We have to consider all minimal (P,Q,R)-chains. 53.54/15.27 ---------------------------------------- 53.54/15.27 53.54/15.27 (99) DependencyGraphProof (EQUIVALENT) 53.54/15.27 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 7 SCCs with 36 less nodes. 53.54/15.27 ---------------------------------------- 53.54/15.27 53.54/15.27 (100) 53.54/15.27 Complex Obligation (AND) 53.54/15.27 53.54/15.27 ---------------------------------------- 53.54/15.27 53.54/15.27 (101) 53.54/15.27 Obligation: 53.54/15.27 Q DP problem: 53.54/15.27 The TRS P consists of the following rules: 53.54/15.27 53.54/15.27 F1015_IN(.(T235, T239)) -> F1015_IN(T239) 53.54/15.27 53.54/15.27 The TRS R consists of the following rules: 53.54/15.27 53.54/15.27 f83_in([]) -> f83_out1 53.54/15.27 f83_in(T11) -> U1(f207_in(T11), T11) 53.54/15.27 U1(f207_out1(X18, T12, X19), T11) -> f83_out1 53.54/15.27 f408_in -> f408_out1(0) 53.54/15.27 f408_in -> U2(f408_in) 53.54/15.27 U2(f408_out1(T58)) -> f408_out1(s(T58)) 53.54/15.27 f410_in(T77) -> U3(f492_in(T77), T77) 53.54/15.27 U3(f492_out1, T77) -> f410_out1 53.54/15.27 f410_in(T116) -> U4(f410_in(T116), T116) 53.54/15.27 U4(f410_out1, T116) -> f410_out1 53.54/15.27 f410_in(T125) -> f410_out1 53.54/15.27 f512_in(0) -> f512_out1 53.54/15.27 f512_in(s(T96)) -> U5(f512_in(T96), s(T96)) 53.54/15.27 U5(f512_out1, s(T96)) -> f512_out1 53.54/15.27 f217_in -> U6(f397_in) 53.54/15.27 U6(f397_out1(T38)) -> f217_out1 53.54/15.27 f217_in -> U7(f217_in) 53.54/15.27 U7(f217_out1) -> f217_out1 53.54/15.27 f217_in -> f217_out1 53.54/15.27 f935_in -> f935_out1 53.54/15.27 f935_in -> U8(f973_in) 53.54/15.27 U8(f973_out1) -> f935_out1 53.54/15.27 f985_in -> f985_out1 53.54/15.27 f985_in -> U9(f985_in) 53.54/15.27 U9(f985_out1) -> f985_out1 53.54/15.27 f1015_in(.(T223, T224)) -> f1015_out1([], T223, T224) 53.54/15.27 f1015_in(.(T235, T239)) -> U10(f1015_in(T239), .(T235, T239)) 53.54/15.27 U10(f1015_out1(T240, T241, T242), .(T235, T239)) -> f1015_out1(.(T235, T240), T241, T242) 53.54/15.27 f207_in(T11) -> U11(f217_in, T11) 53.54/15.27 U11(f217_out1, T11) -> U12(f218_in(T11), T11) 53.54/15.27 U12(f218_out1(X18, T19, X19), T11) -> f207_out1(X18, T19, X19) 53.54/15.27 f218_in(T11) -> U13(f935_in, T11) 53.54/15.27 U13(f935_out1, T11) -> U14(f936_in(T11), T11) 53.54/15.27 U14(f936_out1(T149, T151, X19), T11) -> f218_out1(T149, T151, X19) 53.54/15.27 f397_in -> U15(f408_in) 53.54/15.27 U15(f408_out1(T43)) -> U16(f410_in(T43), T43) 53.54/15.27 U16(f410_out1, T43) -> f397_out1(T43) 53.54/15.27 f492_in(T77) -> U17(f512_in(T77), T77) 53.54/15.27 U17(f512_out1, T77) -> U18(f410_in(T77), T77) 53.54/15.27 U18(f410_out1, T77) -> f492_out1 53.54/15.27 f936_in(T11) -> U19(f935_in, T11) 53.54/15.27 U19(f935_out1, T11) -> U20(f1015_in(T11), T11) 53.54/15.27 U20(f1015_out1(T209, T210, T208), T11) -> f936_out1(T209, T210, T208) 53.54/15.27 f973_in -> U21(f217_in) 53.54/15.27 U21(f217_out1) -> U22(f978_in) 53.54/15.27 U22(f978_out1) -> f973_out1 53.54/15.27 f978_in -> U23(f935_in) 53.54/15.27 U23(f935_out1) -> U24(f980_in) 53.54/15.27 U24(f980_out1) -> f978_out1 53.54/15.27 f980_in -> U25(f935_in) 53.54/15.27 U25(f935_out1) -> U26(f985_in) 53.54/15.27 U26(f985_out1) -> f980_out1 53.54/15.27 53.54/15.27 Q is empty. 53.54/15.27 We have to consider all minimal (P,Q,R)-chains. 53.54/15.27 ---------------------------------------- 53.54/15.27 53.54/15.27 (102) UsableRulesProof (EQUIVALENT) 53.54/15.27 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. 53.54/15.27 ---------------------------------------- 53.54/15.27 53.54/15.27 (103) 53.54/15.27 Obligation: 53.54/15.27 Q DP problem: 53.54/15.27 The TRS P consists of the following rules: 53.54/15.27 53.54/15.27 F1015_IN(.(T235, T239)) -> F1015_IN(T239) 53.54/15.27 53.54/15.27 R is empty. 53.54/15.27 Q is empty. 53.54/15.27 We have to consider all minimal (P,Q,R)-chains. 53.54/15.27 ---------------------------------------- 53.54/15.27 53.54/15.27 (104) QDPSizeChangeProof (EQUIVALENT) 53.54/15.27 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. 53.54/15.27 53.54/15.27 From the DPs we obtained the following set of size-change graphs: 53.54/15.27 *F1015_IN(.(T235, T239)) -> F1015_IN(T239) 53.54/15.27 The graph contains the following edges 1 > 1 53.54/15.27 53.54/15.27 53.54/15.27 ---------------------------------------- 53.54/15.27 53.54/15.27 (105) 53.54/15.27 YES 53.54/15.27 53.54/15.27 ---------------------------------------- 53.54/15.27 53.54/15.27 (106) 53.54/15.27 Obligation: 53.54/15.27 Q DP problem: 53.54/15.27 The TRS P consists of the following rules: 53.54/15.27 53.54/15.27 F985_IN -> F985_IN 53.54/15.27 53.54/15.27 The TRS R consists of the following rules: 53.54/15.27 53.54/15.27 f83_in([]) -> f83_out1 53.54/15.27 f83_in(T11) -> U1(f207_in(T11), T11) 53.54/15.27 U1(f207_out1(X18, T12, X19), T11) -> f83_out1 53.54/15.27 f408_in -> f408_out1(0) 53.54/15.27 f408_in -> U2(f408_in) 53.54/15.27 U2(f408_out1(T58)) -> f408_out1(s(T58)) 53.54/15.27 f410_in(T77) -> U3(f492_in(T77), T77) 53.54/15.27 U3(f492_out1, T77) -> f410_out1 53.54/15.27 f410_in(T116) -> U4(f410_in(T116), T116) 53.54/15.27 U4(f410_out1, T116) -> f410_out1 53.54/15.27 f410_in(T125) -> f410_out1 53.54/15.27 f512_in(0) -> f512_out1 53.54/15.27 f512_in(s(T96)) -> U5(f512_in(T96), s(T96)) 53.54/15.27 U5(f512_out1, s(T96)) -> f512_out1 53.54/15.27 f217_in -> U6(f397_in) 53.54/15.27 U6(f397_out1(T38)) -> f217_out1 53.54/15.27 f217_in -> U7(f217_in) 53.54/15.27 U7(f217_out1) -> f217_out1 53.54/15.27 f217_in -> f217_out1 53.54/15.27 f935_in -> f935_out1 53.54/15.27 f935_in -> U8(f973_in) 53.54/15.27 U8(f973_out1) -> f935_out1 53.54/15.27 f985_in -> f985_out1 53.54/15.27 f985_in -> U9(f985_in) 53.54/15.27 U9(f985_out1) -> f985_out1 53.54/15.27 f1015_in(.(T223, T224)) -> f1015_out1([], T223, T224) 53.54/15.27 f1015_in(.(T235, T239)) -> U10(f1015_in(T239), .(T235, T239)) 53.54/15.27 U10(f1015_out1(T240, T241, T242), .(T235, T239)) -> f1015_out1(.(T235, T240), T241, T242) 53.54/15.27 f207_in(T11) -> U11(f217_in, T11) 53.54/15.27 U11(f217_out1, T11) -> U12(f218_in(T11), T11) 53.54/15.27 U12(f218_out1(X18, T19, X19), T11) -> f207_out1(X18, T19, X19) 53.54/15.27 f218_in(T11) -> U13(f935_in, T11) 53.54/15.27 U13(f935_out1, T11) -> U14(f936_in(T11), T11) 53.54/15.27 U14(f936_out1(T149, T151, X19), T11) -> f218_out1(T149, T151, X19) 53.54/15.27 f397_in -> U15(f408_in) 53.54/15.27 U15(f408_out1(T43)) -> U16(f410_in(T43), T43) 53.54/15.27 U16(f410_out1, T43) -> f397_out1(T43) 53.54/15.27 f492_in(T77) -> U17(f512_in(T77), T77) 53.54/15.27 U17(f512_out1, T77) -> U18(f410_in(T77), T77) 53.54/15.27 U18(f410_out1, T77) -> f492_out1 53.54/15.27 f936_in(T11) -> U19(f935_in, T11) 53.54/15.27 U19(f935_out1, T11) -> U20(f1015_in(T11), T11) 53.54/15.27 U20(f1015_out1(T209, T210, T208), T11) -> f936_out1(T209, T210, T208) 53.54/15.27 f973_in -> U21(f217_in) 53.54/15.27 U21(f217_out1) -> U22(f978_in) 53.54/15.27 U22(f978_out1) -> f973_out1 53.54/15.27 f978_in -> U23(f935_in) 53.54/15.27 U23(f935_out1) -> U24(f980_in) 53.54/15.27 U24(f980_out1) -> f978_out1 53.54/15.27 f980_in -> U25(f935_in) 53.54/15.27 U25(f935_out1) -> U26(f985_in) 53.54/15.27 U26(f985_out1) -> f980_out1 53.54/15.27 53.54/15.27 Q is empty. 53.54/15.27 We have to consider all minimal (P,Q,R)-chains. 53.54/15.27 ---------------------------------------- 53.54/15.27 53.54/15.27 (107) UsableRulesProof (EQUIVALENT) 53.54/15.27 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. 53.54/15.27 ---------------------------------------- 53.54/15.27 53.54/15.27 (108) 53.54/15.27 Obligation: 53.54/15.27 Q DP problem: 53.54/15.27 The TRS P consists of the following rules: 53.54/15.27 53.54/15.27 F985_IN -> F985_IN 53.54/15.27 53.54/15.27 R is empty. 53.54/15.27 Q is empty. 53.54/15.27 We have to consider all minimal (P,Q,R)-chains. 53.54/15.27 ---------------------------------------- 53.54/15.27 53.54/15.27 (109) NonTerminationLoopProof (COMPLETE) 53.54/15.27 We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. 53.54/15.27 Found a loop by semiunifying a rule from P directly. 53.54/15.27 53.54/15.27 s = F985_IN evaluates to t =F985_IN 53.54/15.27 53.54/15.27 Thus s starts an infinite chain as s semiunifies with t with the following substitutions: 53.54/15.27 * Matcher: [ ] 53.54/15.27 * Semiunifier: [ ] 53.54/15.27 53.54/15.27 -------------------------------------------------------------------------------- 53.54/15.27 Rewriting sequence 53.54/15.27 53.54/15.27 The DP semiunifies directly so there is only one rewrite step from F985_IN to F985_IN. 53.54/15.27 53.54/15.27 53.54/15.27 53.54/15.27 53.54/15.27 ---------------------------------------- 53.54/15.27 53.54/15.27 (110) 53.54/15.27 NO 53.54/15.27 53.54/15.27 ---------------------------------------- 53.54/15.27 53.54/15.27 (111) 53.54/15.27 Obligation: 53.54/15.27 Q DP problem: 53.54/15.27 The TRS P consists of the following rules: 53.54/15.27 53.54/15.27 F512_IN(s(T96)) -> F512_IN(T96) 53.54/15.27 53.54/15.27 The TRS R consists of the following rules: 53.54/15.27 53.54/15.27 f83_in([]) -> f83_out1 53.54/15.27 f83_in(T11) -> U1(f207_in(T11), T11) 53.54/15.27 U1(f207_out1(X18, T12, X19), T11) -> f83_out1 53.54/15.27 f408_in -> f408_out1(0) 53.54/15.27 f408_in -> U2(f408_in) 53.54/15.27 U2(f408_out1(T58)) -> f408_out1(s(T58)) 53.54/15.27 f410_in(T77) -> U3(f492_in(T77), T77) 53.54/15.27 U3(f492_out1, T77) -> f410_out1 53.54/15.27 f410_in(T116) -> U4(f410_in(T116), T116) 53.54/15.27 U4(f410_out1, T116) -> f410_out1 53.54/15.27 f410_in(T125) -> f410_out1 53.54/15.27 f512_in(0) -> f512_out1 53.54/15.27 f512_in(s(T96)) -> U5(f512_in(T96), s(T96)) 53.54/15.27 U5(f512_out1, s(T96)) -> f512_out1 53.54/15.27 f217_in -> U6(f397_in) 53.54/15.27 U6(f397_out1(T38)) -> f217_out1 53.54/15.27 f217_in -> U7(f217_in) 53.54/15.27 U7(f217_out1) -> f217_out1 53.54/15.27 f217_in -> f217_out1 53.54/15.27 f935_in -> f935_out1 53.54/15.27 f935_in -> U8(f973_in) 53.54/15.27 U8(f973_out1) -> f935_out1 53.54/15.27 f985_in -> f985_out1 53.54/15.27 f985_in -> U9(f985_in) 53.54/15.27 U9(f985_out1) -> f985_out1 53.54/15.27 f1015_in(.(T223, T224)) -> f1015_out1([], T223, T224) 53.54/15.27 f1015_in(.(T235, T239)) -> U10(f1015_in(T239), .(T235, T239)) 53.54/15.27 U10(f1015_out1(T240, T241, T242), .(T235, T239)) -> f1015_out1(.(T235, T240), T241, T242) 53.54/15.27 f207_in(T11) -> U11(f217_in, T11) 53.54/15.27 U11(f217_out1, T11) -> U12(f218_in(T11), T11) 53.54/15.27 U12(f218_out1(X18, T19, X19), T11) -> f207_out1(X18, T19, X19) 53.54/15.27 f218_in(T11) -> U13(f935_in, T11) 53.54/15.27 U13(f935_out1, T11) -> U14(f936_in(T11), T11) 53.54/15.27 U14(f936_out1(T149, T151, X19), T11) -> f218_out1(T149, T151, X19) 53.54/15.27 f397_in -> U15(f408_in) 53.54/15.27 U15(f408_out1(T43)) -> U16(f410_in(T43), T43) 53.54/15.27 U16(f410_out1, T43) -> f397_out1(T43) 53.54/15.27 f492_in(T77) -> U17(f512_in(T77), T77) 53.54/15.27 U17(f512_out1, T77) -> U18(f410_in(T77), T77) 53.54/15.27 U18(f410_out1, T77) -> f492_out1 53.54/15.27 f936_in(T11) -> U19(f935_in, T11) 53.54/15.27 U19(f935_out1, T11) -> U20(f1015_in(T11), T11) 53.54/15.27 U20(f1015_out1(T209, T210, T208), T11) -> f936_out1(T209, T210, T208) 53.54/15.27 f973_in -> U21(f217_in) 53.54/15.27 U21(f217_out1) -> U22(f978_in) 53.54/15.27 U22(f978_out1) -> f973_out1 53.54/15.27 f978_in -> U23(f935_in) 53.54/15.27 U23(f935_out1) -> U24(f980_in) 53.54/15.27 U24(f980_out1) -> f978_out1 53.54/15.27 f980_in -> U25(f935_in) 53.54/15.27 U25(f935_out1) -> U26(f985_in) 53.54/15.27 U26(f985_out1) -> f980_out1 53.54/15.27 53.54/15.27 Q is empty. 53.54/15.27 We have to consider all minimal (P,Q,R)-chains. 53.54/15.27 ---------------------------------------- 53.54/15.27 53.54/15.27 (112) UsableRulesProof (EQUIVALENT) 53.54/15.27 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. 53.54/15.27 ---------------------------------------- 53.54/15.27 53.54/15.27 (113) 53.54/15.27 Obligation: 53.54/15.27 Q DP problem: 53.54/15.27 The TRS P consists of the following rules: 53.54/15.27 53.54/15.27 F512_IN(s(T96)) -> F512_IN(T96) 53.54/15.27 53.54/15.27 R is empty. 53.54/15.27 Q is empty. 53.54/15.27 We have to consider all minimal (P,Q,R)-chains. 53.54/15.27 ---------------------------------------- 53.54/15.27 53.54/15.27 (114) QDPSizeChangeProof (EQUIVALENT) 53.54/15.27 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. 53.54/15.27 53.54/15.27 From the DPs we obtained the following set of size-change graphs: 53.54/15.27 *F512_IN(s(T96)) -> F512_IN(T96) 53.54/15.27 The graph contains the following edges 1 > 1 53.54/15.27 53.54/15.27 53.54/15.27 ---------------------------------------- 53.54/15.27 53.54/15.27 (115) 53.54/15.27 YES 53.54/15.27 53.54/15.27 ---------------------------------------- 53.54/15.27 53.54/15.27 (116) 53.54/15.27 Obligation: 53.54/15.27 Q DP problem: 53.54/15.27 The TRS P consists of the following rules: 53.54/15.27 53.54/15.27 F410_IN(T77) -> F492_IN(T77) 53.54/15.27 F492_IN(T77) -> U17^1(f512_in(T77), T77) 53.54/15.27 U17^1(f512_out1, T77) -> F410_IN(T77) 53.54/15.27 F410_IN(T116) -> F410_IN(T116) 53.54/15.27 53.54/15.27 The TRS R consists of the following rules: 53.54/15.27 53.54/15.27 f83_in([]) -> f83_out1 53.54/15.27 f83_in(T11) -> U1(f207_in(T11), T11) 53.54/15.27 U1(f207_out1(X18, T12, X19), T11) -> f83_out1 53.54/15.27 f408_in -> f408_out1(0) 53.54/15.27 f408_in -> U2(f408_in) 53.54/15.27 U2(f408_out1(T58)) -> f408_out1(s(T58)) 53.54/15.27 f410_in(T77) -> U3(f492_in(T77), T77) 53.54/15.27 U3(f492_out1, T77) -> f410_out1 53.54/15.27 f410_in(T116) -> U4(f410_in(T116), T116) 53.54/15.27 U4(f410_out1, T116) -> f410_out1 53.54/15.27 f410_in(T125) -> f410_out1 53.54/15.27 f512_in(0) -> f512_out1 53.54/15.27 f512_in(s(T96)) -> U5(f512_in(T96), s(T96)) 53.54/15.27 U5(f512_out1, s(T96)) -> f512_out1 53.54/15.27 f217_in -> U6(f397_in) 53.54/15.27 U6(f397_out1(T38)) -> f217_out1 53.54/15.27 f217_in -> U7(f217_in) 53.54/15.27 U7(f217_out1) -> f217_out1 53.54/15.27 f217_in -> f217_out1 53.54/15.27 f935_in -> f935_out1 53.54/15.27 f935_in -> U8(f973_in) 53.54/15.27 U8(f973_out1) -> f935_out1 53.54/15.27 f985_in -> f985_out1 53.54/15.27 f985_in -> U9(f985_in) 53.54/15.27 U9(f985_out1) -> f985_out1 53.54/15.27 f1015_in(.(T223, T224)) -> f1015_out1([], T223, T224) 53.54/15.27 f1015_in(.(T235, T239)) -> U10(f1015_in(T239), .(T235, T239)) 53.54/15.27 U10(f1015_out1(T240, T241, T242), .(T235, T239)) -> f1015_out1(.(T235, T240), T241, T242) 53.54/15.27 f207_in(T11) -> U11(f217_in, T11) 53.54/15.27 U11(f217_out1, T11) -> U12(f218_in(T11), T11) 53.54/15.27 U12(f218_out1(X18, T19, X19), T11) -> f207_out1(X18, T19, X19) 53.54/15.27 f218_in(T11) -> U13(f935_in, T11) 53.54/15.27 U13(f935_out1, T11) -> U14(f936_in(T11), T11) 53.54/15.27 U14(f936_out1(T149, T151, X19), T11) -> f218_out1(T149, T151, X19) 53.54/15.27 f397_in -> U15(f408_in) 53.54/15.27 U15(f408_out1(T43)) -> U16(f410_in(T43), T43) 53.54/15.27 U16(f410_out1, T43) -> f397_out1(T43) 53.54/15.27 f492_in(T77) -> U17(f512_in(T77), T77) 53.54/15.27 U17(f512_out1, T77) -> U18(f410_in(T77), T77) 53.54/15.27 U18(f410_out1, T77) -> f492_out1 53.54/15.27 f936_in(T11) -> U19(f935_in, T11) 53.54/15.27 U19(f935_out1, T11) -> U20(f1015_in(T11), T11) 53.54/15.27 U20(f1015_out1(T209, T210, T208), T11) -> f936_out1(T209, T210, T208) 53.54/15.27 f973_in -> U21(f217_in) 53.54/15.27 U21(f217_out1) -> U22(f978_in) 53.54/15.27 U22(f978_out1) -> f973_out1 53.54/15.27 f978_in -> U23(f935_in) 53.54/15.27 U23(f935_out1) -> U24(f980_in) 53.54/15.27 U24(f980_out1) -> f978_out1 53.54/15.27 f980_in -> U25(f935_in) 53.54/15.27 U25(f935_out1) -> U26(f985_in) 53.54/15.27 U26(f985_out1) -> f980_out1 53.54/15.27 53.54/15.27 Q is empty. 53.54/15.27 We have to consider all minimal (P,Q,R)-chains. 53.54/15.27 ---------------------------------------- 53.54/15.27 53.54/15.27 (117) NonTerminationLoopProof (COMPLETE) 53.54/15.27 We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. 53.54/15.27 Found a loop by semiunifying a rule from P directly. 53.54/15.27 53.54/15.27 s = F410_IN(T116) evaluates to t =F410_IN(T116) 53.54/15.27 53.54/15.27 Thus s starts an infinite chain as s semiunifies with t with the following substitutions: 53.54/15.27 * Matcher: [ ] 53.54/15.27 * Semiunifier: [ ] 53.54/15.27 53.54/15.27 -------------------------------------------------------------------------------- 53.54/15.27 Rewriting sequence 53.54/15.27 53.54/15.27 The DP semiunifies directly so there is only one rewrite step from F410_IN(T116) to F410_IN(T116). 53.54/15.27 53.54/15.27 53.54/15.27 53.54/15.27 53.54/15.27 ---------------------------------------- 53.54/15.27 53.54/15.27 (118) 53.54/15.27 NO 53.54/15.27 53.54/15.27 ---------------------------------------- 53.54/15.27 53.54/15.27 (119) 53.54/15.27 Obligation: 53.54/15.27 Q DP problem: 53.54/15.27 The TRS P consists of the following rules: 53.54/15.27 53.54/15.27 F408_IN -> F408_IN 53.54/15.27 53.54/15.27 The TRS R consists of the following rules: 53.54/15.27 53.54/15.27 f83_in([]) -> f83_out1 53.54/15.27 f83_in(T11) -> U1(f207_in(T11), T11) 53.54/15.27 U1(f207_out1(X18, T12, X19), T11) -> f83_out1 53.54/15.27 f408_in -> f408_out1(0) 53.54/15.27 f408_in -> U2(f408_in) 53.54/15.27 U2(f408_out1(T58)) -> f408_out1(s(T58)) 53.54/15.27 f410_in(T77) -> U3(f492_in(T77), T77) 53.54/15.27 U3(f492_out1, T77) -> f410_out1 53.54/15.27 f410_in(T116) -> U4(f410_in(T116), T116) 53.54/15.27 U4(f410_out1, T116) -> f410_out1 53.54/15.27 f410_in(T125) -> f410_out1 53.54/15.27 f512_in(0) -> f512_out1 53.54/15.27 f512_in(s(T96)) -> U5(f512_in(T96), s(T96)) 53.54/15.27 U5(f512_out1, s(T96)) -> f512_out1 53.54/15.27 f217_in -> U6(f397_in) 53.54/15.27 U6(f397_out1(T38)) -> f217_out1 53.54/15.27 f217_in -> U7(f217_in) 53.54/15.27 U7(f217_out1) -> f217_out1 53.54/15.27 f217_in -> f217_out1 53.54/15.27 f935_in -> f935_out1 53.54/15.27 f935_in -> U8(f973_in) 53.54/15.27 U8(f973_out1) -> f935_out1 53.54/15.27 f985_in -> f985_out1 53.54/15.27 f985_in -> U9(f985_in) 53.54/15.27 U9(f985_out1) -> f985_out1 53.54/15.27 f1015_in(.(T223, T224)) -> f1015_out1([], T223, T224) 53.54/15.27 f1015_in(.(T235, T239)) -> U10(f1015_in(T239), .(T235, T239)) 53.54/15.27 U10(f1015_out1(T240, T241, T242), .(T235, T239)) -> f1015_out1(.(T235, T240), T241, T242) 53.54/15.27 f207_in(T11) -> U11(f217_in, T11) 53.54/15.27 U11(f217_out1, T11) -> U12(f218_in(T11), T11) 53.54/15.27 U12(f218_out1(X18, T19, X19), T11) -> f207_out1(X18, T19, X19) 53.54/15.27 f218_in(T11) -> U13(f935_in, T11) 53.54/15.27 U13(f935_out1, T11) -> U14(f936_in(T11), T11) 53.54/15.27 U14(f936_out1(T149, T151, X19), T11) -> f218_out1(T149, T151, X19) 53.54/15.27 f397_in -> U15(f408_in) 53.54/15.27 U15(f408_out1(T43)) -> U16(f410_in(T43), T43) 53.54/15.27 U16(f410_out1, T43) -> f397_out1(T43) 53.54/15.27 f492_in(T77) -> U17(f512_in(T77), T77) 53.54/15.27 U17(f512_out1, T77) -> U18(f410_in(T77), T77) 53.54/15.27 U18(f410_out1, T77) -> f492_out1 53.54/15.27 f936_in(T11) -> U19(f935_in, T11) 53.54/15.27 U19(f935_out1, T11) -> U20(f1015_in(T11), T11) 53.54/15.27 U20(f1015_out1(T209, T210, T208), T11) -> f936_out1(T209, T210, T208) 53.54/15.27 f973_in -> U21(f217_in) 53.54/15.27 U21(f217_out1) -> U22(f978_in) 53.54/15.27 U22(f978_out1) -> f973_out1 53.54/15.27 f978_in -> U23(f935_in) 53.54/15.27 U23(f935_out1) -> U24(f980_in) 53.54/15.27 U24(f980_out1) -> f978_out1 53.54/15.27 f980_in -> U25(f935_in) 53.54/15.27 U25(f935_out1) -> U26(f985_in) 53.54/15.27 U26(f985_out1) -> f980_out1 53.54/15.27 53.54/15.27 Q is empty. 53.54/15.27 We have to consider all minimal (P,Q,R)-chains. 53.54/15.27 ---------------------------------------- 53.54/15.27 53.54/15.27 (120) UsableRulesProof (EQUIVALENT) 53.54/15.27 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. 53.54/15.27 ---------------------------------------- 53.54/15.27 53.54/15.27 (121) 53.54/15.27 Obligation: 53.54/15.27 Q DP problem: 53.54/15.27 The TRS P consists of the following rules: 53.54/15.27 53.54/15.27 F408_IN -> F408_IN 53.54/15.27 53.54/15.27 R is empty. 53.54/15.27 Q is empty. 53.54/15.27 We have to consider all minimal (P,Q,R)-chains. 53.54/15.27 ---------------------------------------- 53.54/15.27 53.54/15.27 (122) NonTerminationLoopProof (COMPLETE) 53.54/15.27 We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. 53.54/15.27 Found a loop by semiunifying a rule from P directly. 53.54/15.27 53.54/15.27 s = F408_IN evaluates to t =F408_IN 53.54/15.27 53.54/15.27 Thus s starts an infinite chain as s semiunifies with t with the following substitutions: 53.54/15.27 * Matcher: [ ] 53.54/15.27 * Semiunifier: [ ] 53.54/15.27 53.54/15.27 -------------------------------------------------------------------------------- 53.54/15.27 Rewriting sequence 53.54/15.27 53.54/15.27 The DP semiunifies directly so there is only one rewrite step from F408_IN to F408_IN. 53.54/15.27 53.54/15.27 53.54/15.27 53.54/15.27 53.54/15.27 ---------------------------------------- 53.54/15.27 53.54/15.27 (123) 53.54/15.27 NO 53.54/15.27 53.54/15.27 ---------------------------------------- 53.54/15.27 53.54/15.27 (124) 53.54/15.27 Obligation: 53.54/15.27 Q DP problem: 53.54/15.27 The TRS P consists of the following rules: 53.54/15.27 53.54/15.27 F217_IN -> F217_IN 53.54/15.27 53.54/15.27 The TRS R consists of the following rules: 53.54/15.27 53.54/15.27 f83_in([]) -> f83_out1 53.54/15.27 f83_in(T11) -> U1(f207_in(T11), T11) 53.54/15.27 U1(f207_out1(X18, T12, X19), T11) -> f83_out1 53.54/15.27 f408_in -> f408_out1(0) 53.54/15.27 f408_in -> U2(f408_in) 53.54/15.27 U2(f408_out1(T58)) -> f408_out1(s(T58)) 53.54/15.27 f410_in(T77) -> U3(f492_in(T77), T77) 53.54/15.27 U3(f492_out1, T77) -> f410_out1 53.54/15.27 f410_in(T116) -> U4(f410_in(T116), T116) 53.54/15.27 U4(f410_out1, T116) -> f410_out1 53.54/15.27 f410_in(T125) -> f410_out1 53.54/15.27 f512_in(0) -> f512_out1 53.54/15.27 f512_in(s(T96)) -> U5(f512_in(T96), s(T96)) 53.54/15.27 U5(f512_out1, s(T96)) -> f512_out1 53.54/15.27 f217_in -> U6(f397_in) 53.54/15.27 U6(f397_out1(T38)) -> f217_out1 53.54/15.27 f217_in -> U7(f217_in) 53.54/15.27 U7(f217_out1) -> f217_out1 53.54/15.27 f217_in -> f217_out1 53.54/15.27 f935_in -> f935_out1 53.54/15.27 f935_in -> U8(f973_in) 53.54/15.27 U8(f973_out1) -> f935_out1 53.54/15.27 f985_in -> f985_out1 53.54/15.27 f985_in -> U9(f985_in) 53.54/15.27 U9(f985_out1) -> f985_out1 53.54/15.27 f1015_in(.(T223, T224)) -> f1015_out1([], T223, T224) 53.54/15.27 f1015_in(.(T235, T239)) -> U10(f1015_in(T239), .(T235, T239)) 53.54/15.27 U10(f1015_out1(T240, T241, T242), .(T235, T239)) -> f1015_out1(.(T235, T240), T241, T242) 53.54/15.27 f207_in(T11) -> U11(f217_in, T11) 53.54/15.27 U11(f217_out1, T11) -> U12(f218_in(T11), T11) 53.54/15.27 U12(f218_out1(X18, T19, X19), T11) -> f207_out1(X18, T19, X19) 53.54/15.27 f218_in(T11) -> U13(f935_in, T11) 53.54/15.27 U13(f935_out1, T11) -> U14(f936_in(T11), T11) 53.54/15.27 U14(f936_out1(T149, T151, X19), T11) -> f218_out1(T149, T151, X19) 53.54/15.27 f397_in -> U15(f408_in) 53.54/15.27 U15(f408_out1(T43)) -> U16(f410_in(T43), T43) 53.54/15.27 U16(f410_out1, T43) -> f397_out1(T43) 53.54/15.27 f492_in(T77) -> U17(f512_in(T77), T77) 53.54/15.27 U17(f512_out1, T77) -> U18(f410_in(T77), T77) 53.54/15.27 U18(f410_out1, T77) -> f492_out1 53.54/15.27 f936_in(T11) -> U19(f935_in, T11) 53.54/15.27 U19(f935_out1, T11) -> U20(f1015_in(T11), T11) 53.54/15.27 U20(f1015_out1(T209, T210, T208), T11) -> f936_out1(T209, T210, T208) 53.54/15.27 f973_in -> U21(f217_in) 53.54/15.27 U21(f217_out1) -> U22(f978_in) 53.54/15.27 U22(f978_out1) -> f973_out1 53.54/15.27 f978_in -> U23(f935_in) 53.54/15.27 U23(f935_out1) -> U24(f980_in) 53.54/15.27 U24(f980_out1) -> f978_out1 53.54/15.27 f980_in -> U25(f935_in) 53.54/15.27 U25(f935_out1) -> U26(f985_in) 53.54/15.27 U26(f985_out1) -> f980_out1 53.54/15.27 53.54/15.27 Q is empty. 53.54/15.27 We have to consider all minimal (P,Q,R)-chains. 53.54/15.27 ---------------------------------------- 53.54/15.27 53.54/15.27 (125) UsableRulesProof (EQUIVALENT) 53.54/15.27 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. 53.54/15.27 ---------------------------------------- 53.54/15.27 53.54/15.27 (126) 53.54/15.27 Obligation: 53.54/15.27 Q DP problem: 53.54/15.27 The TRS P consists of the following rules: 53.54/15.27 53.54/15.27 F217_IN -> F217_IN 53.54/15.27 53.54/15.27 R is empty. 53.54/15.27 Q is empty. 53.54/15.27 We have to consider all minimal (P,Q,R)-chains. 53.54/15.27 ---------------------------------------- 53.54/15.27 53.54/15.27 (127) NonTerminationLoopProof (COMPLETE) 53.54/15.27 We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. 53.54/15.27 Found a loop by semiunifying a rule from P directly. 53.54/15.27 53.54/15.27 s = F217_IN evaluates to t =F217_IN 53.54/15.27 53.54/15.27 Thus s starts an infinite chain as s semiunifies with t with the following substitutions: 53.54/15.27 * Matcher: [ ] 53.54/15.27 * Semiunifier: [ ] 53.54/15.27 53.54/15.27 -------------------------------------------------------------------------------- 53.54/15.27 Rewriting sequence 53.54/15.27 53.54/15.27 The DP semiunifies directly so there is only one rewrite step from F217_IN to F217_IN. 53.54/15.27 53.54/15.27 53.54/15.27 53.54/15.27 53.54/15.27 ---------------------------------------- 53.54/15.27 53.54/15.27 (128) 53.54/15.27 NO 53.54/15.27 53.54/15.27 ---------------------------------------- 53.54/15.27 53.54/15.27 (129) 53.54/15.27 Obligation: 53.54/15.27 Q DP problem: 53.54/15.27 The TRS P consists of the following rules: 53.54/15.27 53.54/15.27 F973_IN -> U21^1(f217_in) 53.54/15.27 U21^1(f217_out1) -> F978_IN 53.54/15.27 F978_IN -> U23^1(f935_in) 53.54/15.27 U23^1(f935_out1) -> F980_IN 53.54/15.27 F980_IN -> F935_IN 53.54/15.27 F935_IN -> F973_IN 53.54/15.27 F978_IN -> F935_IN 53.54/15.27 53.54/15.27 The TRS R consists of the following rules: 53.54/15.27 53.54/15.27 f83_in([]) -> f83_out1 53.54/15.27 f83_in(T11) -> U1(f207_in(T11), T11) 53.54/15.27 U1(f207_out1(X18, T12, X19), T11) -> f83_out1 53.54/15.27 f408_in -> f408_out1(0) 53.54/15.27 f408_in -> U2(f408_in) 53.54/15.27 U2(f408_out1(T58)) -> f408_out1(s(T58)) 53.54/15.27 f410_in(T77) -> U3(f492_in(T77), T77) 53.54/15.27 U3(f492_out1, T77) -> f410_out1 53.54/15.27 f410_in(T116) -> U4(f410_in(T116), T116) 53.54/15.27 U4(f410_out1, T116) -> f410_out1 53.54/15.27 f410_in(T125) -> f410_out1 53.54/15.27 f512_in(0) -> f512_out1 53.54/15.27 f512_in(s(T96)) -> U5(f512_in(T96), s(T96)) 53.54/15.27 U5(f512_out1, s(T96)) -> f512_out1 53.54/15.27 f217_in -> U6(f397_in) 53.54/15.27 U6(f397_out1(T38)) -> f217_out1 53.54/15.27 f217_in -> U7(f217_in) 53.54/15.27 U7(f217_out1) -> f217_out1 53.54/15.27 f217_in -> f217_out1 53.54/15.27 f935_in -> f935_out1 53.54/15.27 f935_in -> U8(f973_in) 53.54/15.27 U8(f973_out1) -> f935_out1 53.54/15.27 f985_in -> f985_out1 53.54/15.27 f985_in -> U9(f985_in) 53.54/15.27 U9(f985_out1) -> f985_out1 53.54/15.27 f1015_in(.(T223, T224)) -> f1015_out1([], T223, T224) 53.54/15.27 f1015_in(.(T235, T239)) -> U10(f1015_in(T239), .(T235, T239)) 53.54/15.27 U10(f1015_out1(T240, T241, T242), .(T235, T239)) -> f1015_out1(.(T235, T240), T241, T242) 53.54/15.27 f207_in(T11) -> U11(f217_in, T11) 53.54/15.27 U11(f217_out1, T11) -> U12(f218_in(T11), T11) 53.54/15.27 U12(f218_out1(X18, T19, X19), T11) -> f207_out1(X18, T19, X19) 53.54/15.27 f218_in(T11) -> U13(f935_in, T11) 53.54/15.27 U13(f935_out1, T11) -> U14(f936_in(T11), T11) 53.54/15.27 U14(f936_out1(T149, T151, X19), T11) -> f218_out1(T149, T151, X19) 53.54/15.27 f397_in -> U15(f408_in) 53.54/15.27 U15(f408_out1(T43)) -> U16(f410_in(T43), T43) 53.54/15.27 U16(f410_out1, T43) -> f397_out1(T43) 53.54/15.27 f492_in(T77) -> U17(f512_in(T77), T77) 53.54/15.27 U17(f512_out1, T77) -> U18(f410_in(T77), T77) 53.54/15.27 U18(f410_out1, T77) -> f492_out1 53.54/15.27 f936_in(T11) -> U19(f935_in, T11) 53.54/15.27 U19(f935_out1, T11) -> U20(f1015_in(T11), T11) 53.54/15.27 U20(f1015_out1(T209, T210, T208), T11) -> f936_out1(T209, T210, T208) 53.54/15.27 f973_in -> U21(f217_in) 53.54/15.27 U21(f217_out1) -> U22(f978_in) 53.54/15.27 U22(f978_out1) -> f973_out1 53.54/15.27 f978_in -> U23(f935_in) 53.54/15.27 U23(f935_out1) -> U24(f980_in) 53.54/15.27 U24(f980_out1) -> f978_out1 53.54/15.27 f980_in -> U25(f935_in) 53.54/15.27 U25(f935_out1) -> U26(f985_in) 53.54/15.27 U26(f985_out1) -> f980_out1 53.54/15.27 53.54/15.27 Q is empty. 53.54/15.27 We have to consider all minimal (P,Q,R)-chains. 53.54/15.27 ---------------------------------------- 53.54/15.27 53.54/15.27 (130) NonLoopProof (COMPLETE) 53.54/15.27 By Theorem 8 [NONLOOP] we deduce infiniteness of the QDP. 53.54/15.27 We apply the theorem with m = 1, b = 0, 53.54/15.27 σ' = [ ], and μ' = [ ] on the rule 53.54/15.27 F978_IN[ ]^n[ ] -> F978_IN[ ]^n[ ] 53.54/15.27 This rule is correct for the QDP as the following derivation shows: 53.54/15.27 53.54/15.27 F978_IN[ ]^n[ ] -> F978_IN[ ]^n[ ] 53.54/15.27 by Narrowing at position: [] 53.54/15.27 F978_IN[ ]^n[ ] -> F935_IN[ ]^n[ ] 53.54/15.27 by Rule from TRS P 53.54/15.27 53.54/15.27 F935_IN[ ]^n[ ] -> F978_IN[ ]^n[ ] 53.54/15.27 by Narrowing at position: [] 53.54/15.27 F935_IN[ ]^n[ ] -> F973_IN[ ]^n[ ] 53.54/15.27 by Rule from TRS P 53.54/15.27 53.54/15.27 F973_IN[ ]^n[ ] -> F978_IN[ ]^n[ ] 53.54/15.27 by Narrowing at position: [] 53.54/15.27 F973_IN[ ]^n[ ] -> U21^1(f217_out1)[ ]^n[ ] 53.54/15.27 by Narrowing at position: [0] 53.54/15.27 F973_IN[ ]^n[ ] -> U21^1(f217_in)[ ]^n[ ] 53.54/15.27 by Rule from TRS P 53.54/15.27 53.54/15.27 f217_in[ ]^n[ ] -> f217_out1[ ]^n[ ] 53.54/15.27 by Rule from TRS R 53.54/15.27 53.54/15.27 U21^1(f217_out1)[ ]^n[ ] -> F978_IN[ ]^n[ ] 53.54/15.27 by Rule from TRS P 53.54/15.27 ---------------------------------------- 53.54/15.27 53.54/15.27 (131) 53.54/15.27 NO 53.54/15.27 53.54/15.27 ---------------------------------------- 53.54/15.27 53.54/15.27 (132) PrologToDTProblemTransformerProof (SOUND) 53.54/15.27 Built DT problem from termination graph DT10. 53.54/15.27 53.54/15.27 { 53.54/15.27 "root": 39, 53.54/15.27 "program": { 53.54/15.27 "directives": [], 53.54/15.27 "clauses": [ 53.54/15.27 [ 53.54/15.27 "(qs ([]) ([]))", 53.54/15.27 null 53.54/15.27 ], 53.54/15.27 [ 53.54/15.27 "(qs (. 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T433 X566) X438) (app X437 (. T431 X438) T305))))" 53.54/15.28 }], 53.54/15.28 "kb": { 53.54/15.28 "nonunifying": [[ 53.54/15.28 "(qs T1 T305)", 53.54/15.28 "(qs ([]) ([]))" 53.54/15.28 ]], 53.54/15.28 "intvars": {}, 53.54/15.28 "arithmetic": { 53.54/15.28 "type": "PlainIntegerRelationState", 53.54/15.28 "relations": [] 53.54/15.28 }, 53.54/15.28 "ground": ["T305"], 53.54/15.28 "free": [ 53.54/15.28 "X437", 53.54/15.28 "X438", 53.54/15.28 "X565", 53.54/15.28 "X566" 53.54/15.28 ], 53.54/15.28 "exprvars": [] 53.54/15.28 } 53.54/15.28 }, 53.54/15.28 "1172": { 53.54/15.28 "goal": [{ 53.54/15.28 "clause": 4, 53.54/15.28 "scope": 11, 53.54/15.28 "term": "(',' (part T306 T307 X435 X436) (',' (qs X435 X437) (',' (qs X436 X438) (app X437 (. T306 X438) T305))))" 53.54/15.28 }], 53.54/15.28 "kb": { 53.54/15.28 "nonunifying": [[ 53.54/15.28 "(qs T1 T305)", 53.54/15.28 "(qs ([]) ([]))" 53.54/15.28 ]], 53.54/15.28 "intvars": {}, 53.54/15.28 "arithmetic": { 53.54/15.28 "type": "PlainIntegerRelationState", 53.54/15.28 "relations": [] 53.54/15.28 }, 53.54/15.28 "ground": ["T305"], 53.54/15.28 "free": [ 53.54/15.28 "X435", 53.54/15.28 "X436", 53.54/15.28 "X437", 53.54/15.28 "X438" 53.54/15.28 ], 53.54/15.28 "exprvars": [] 53.54/15.28 } 53.54/15.28 }, 53.54/15.28 "1050": { 53.54/15.28 "goal": [], 53.54/15.28 "kb": { 53.54/15.28 "nonunifying": [], 53.54/15.28 "intvars": {}, 53.54/15.28 "arithmetic": { 53.54/15.28 "type": "PlainIntegerRelationState", 53.54/15.28 "relations": [] 53.54/15.28 }, 53.54/15.28 "ground": [], 53.54/15.28 "free": [], 53.54/15.28 "exprvars": [] 53.54/15.28 } 53.54/15.28 }, 53.54/15.28 "1171": { 53.54/15.28 "goal": [{ 53.54/15.28 "clause": 3, 53.54/15.28 "scope": 11, 53.54/15.28 "term": "(',' (part T306 T307 X435 X436) (',' (qs X435 X437) (',' (qs X436 X438) (app X437 (. T306 X438) T305))))" 53.54/15.28 }], 53.54/15.28 "kb": { 53.54/15.28 "nonunifying": [[ 53.54/15.28 "(qs T1 T305)", 53.54/15.28 "(qs ([]) ([]))" 53.54/15.28 ]], 53.54/15.28 "intvars": {}, 53.54/15.28 "arithmetic": { 53.54/15.28 "type": "PlainIntegerRelationState", 53.54/15.28 "relations": [] 53.54/15.28 }, 53.54/15.28 "ground": ["T305"], 53.54/15.28 "free": [ 53.54/15.28 "X435", 53.54/15.28 "X436", 53.54/15.28 "X437", 53.54/15.28 "X438" 53.54/15.28 ], 53.54/15.28 "exprvars": [] 53.54/15.28 } 53.54/15.28 }, 53.54/15.28 "1170": { 53.54/15.28 "goal": [], 53.54/15.28 "kb": { 53.54/15.28 "nonunifying": [], 53.54/15.28 "intvars": {}, 53.54/15.28 "arithmetic": { 53.54/15.28 "type": "PlainIntegerRelationState", 53.54/15.28 "relations": [] 53.54/15.28 }, 53.54/15.28 "ground": [], 53.54/15.28 "free": [], 53.54/15.28 "exprvars": [] 53.54/15.28 } 53.54/15.28 }, 53.54/15.28 "220": { 53.54/15.28 "goal": [], 53.54/15.28 "kb": { 53.54/15.28 "nonunifying": [], 53.54/15.28 "intvars": {}, 53.54/15.28 "arithmetic": { 53.54/15.28 "type": "PlainIntegerRelationState", 53.54/15.28 "relations": [] 53.54/15.28 }, 53.54/15.28 "ground": [], 53.54/15.28 "free": [], 53.54/15.28 "exprvars": [] 53.54/15.28 } 53.54/15.28 }, 53.54/15.28 "981": { 53.54/15.28 "goal": [{ 53.54/15.28 "clause": -1, 53.54/15.28 "scope": -1, 53.54/15.28 "term": "(true)" 53.54/15.28 }], 53.54/15.28 "kb": { 53.54/15.28 "nonunifying": [], 53.54/15.28 "intvars": {}, 53.54/15.28 "arithmetic": { 53.54/15.28 "type": "PlainIntegerRelationState", 53.54/15.28 "relations": [] 53.54/15.28 }, 53.54/15.28 "ground": [], 53.54/15.28 "free": [], 53.54/15.28 "exprvars": [] 53.54/15.28 } 53.54/15.28 }, 53.54/15.28 "982": { 53.54/15.28 "goal": [], 53.54/15.28 "kb": { 53.54/15.28 "nonunifying": [], 53.54/15.28 "intvars": {}, 53.54/15.28 "arithmetic": { 53.54/15.28 "type": "PlainIntegerRelationState", 53.54/15.28 "relations": [] 53.54/15.28 }, 53.54/15.28 "ground": [], 53.54/15.28 "free": [], 53.54/15.28 "exprvars": [] 53.54/15.28 } 53.54/15.28 }, 53.54/15.28 "1049": { 53.54/15.28 "goal": [{ 53.54/15.28 "clause": -1, 53.54/15.28 "scope": -1, 53.54/15.28 "term": "(',' (part T256 T257 X393 X394) (',' (qs X393 X11) (',' (qs (. T258 X394) X12) (app X11 (. T256 X12) ([])))))" 53.54/15.28 }], 53.54/15.28 "kb": { 53.54/15.28 "nonunifying": [], 53.54/15.28 "intvars": {}, 53.54/15.28 "arithmetic": { 53.54/15.28 "type": "PlainIntegerRelationState", 53.54/15.28 "relations": [] 53.54/15.28 }, 53.54/15.28 "ground": [], 53.54/15.28 "free": [ 53.54/15.28 "X11", 53.54/15.28 "X12", 53.54/15.28 "X393", 53.54/15.28 "X394" 53.54/15.28 ], 53.54/15.28 "exprvars": [] 53.54/15.28 } 53.54/15.28 }, 53.54/15.28 "983": { 53.54/15.28 "goal": [], 53.54/15.28 "kb": { 53.54/15.28 "nonunifying": [], 53.54/15.28 "intvars": {}, 53.54/15.28 "arithmetic": { 53.54/15.28 "type": "PlainIntegerRelationState", 53.54/15.28 "relations": [] 53.54/15.28 }, 53.54/15.28 "ground": [], 53.54/15.28 "free": [], 53.54/15.28 "exprvars": [] 53.54/15.28 } 53.54/15.28 }, 53.54/15.28 "1169": { 53.54/15.28 "goal": [{ 53.54/15.28 "clause": -1, 53.54/15.28 "scope": -1, 53.54/15.28 "term": "(app T410 (. T407 T411) T409)" 53.54/15.28 }], 53.54/15.28 "kb": { 53.54/15.28 "nonunifying": [], 53.54/15.28 "intvars": {}, 53.54/15.28 "arithmetic": { 53.54/15.28 "type": "PlainIntegerRelationState", 53.54/15.28 "relations": [] 53.54/15.28 }, 53.54/15.28 "ground": [ 53.54/15.28 "T407", 53.54/15.28 "T409" 53.54/15.28 ], 53.54/15.28 "free": [], 53.54/15.28 "exprvars": [] 53.54/15.28 } 53.54/15.28 }, 53.54/15.28 "1202": { 53.54/15.28 "goal": [{ 53.54/15.28 "clause": -1, 53.54/15.28 "scope": -1, 53.54/15.28 "term": "(app T530 (. T525 T535) T305)" 53.54/15.28 }], 53.54/15.28 "kb": { 53.54/15.28 "nonunifying": [[ 53.54/15.28 "(qs T1 T305)", 53.54/15.28 "(qs ([]) ([]))" 53.54/15.28 ]], 53.54/15.28 "intvars": {}, 53.54/15.28 "arithmetic": { 53.54/15.28 "type": "PlainIntegerRelationState", 53.54/15.28 "relations": [] 53.54/15.28 }, 53.54/15.28 "ground": [ 53.54/15.28 "T305", 53.54/15.28 "T530", 53.54/15.28 "T535" 53.54/15.28 ], 53.54/15.28 "free": [], 53.54/15.28 "exprvars": [] 53.54/15.28 } 53.54/15.28 }, 53.54/15.28 "1168": { 53.54/15.28 "goal": [], 53.54/15.28 "kb": { 53.54/15.28 "nonunifying": [], 53.54/15.28 "intvars": {}, 53.54/15.28 "arithmetic": { 53.54/15.28 "type": "PlainIntegerRelationState", 53.54/15.28 "relations": [] 53.54/15.29 }, 53.54/15.29 "ground": [], 53.54/15.29 "free": [], 53.54/15.29 "exprvars": [] 53.54/15.29 } 53.54/15.29 }, 53.54/15.29 "1201": { 53.54/15.29 "goal": [{ 53.54/15.29 "clause": -1, 53.54/15.29 "scope": -1, 53.54/15.29 "term": "(qs ([]) X438)" 53.54/15.29 }], 53.54/15.29 "kb": { 53.54/15.29 "nonunifying": [], 53.54/15.29 "intvars": {}, 53.54/15.29 "arithmetic": { 53.54/15.29 "type": "PlainIntegerRelationState", 53.54/15.29 "relations": [] 53.54/15.29 }, 53.54/15.29 "ground": [], 53.54/15.29 "free": ["X438"], 53.54/15.29 "exprvars": [] 53.54/15.29 } 53.54/15.29 }, 53.54/15.29 "1167": { 53.54/15.29 "goal": [], 53.54/15.29 "kb": { 53.54/15.29 "nonunifying": [], 53.54/15.29 "intvars": {}, 53.54/15.29 "arithmetic": { 53.54/15.29 "type": "PlainIntegerRelationState", 53.54/15.29 "relations": [] 53.54/15.29 }, 53.54/15.29 "ground": [], 53.54/15.29 "free": [], 53.54/15.29 "exprvars": [] 53.54/15.29 } 53.54/15.29 }, 53.54/15.29 "1200": { 53.54/15.29 "goal": [{ 53.54/15.29 "clause": -1, 53.54/15.29 "scope": -1, 53.54/15.29 "term": "(',' (qs ([]) X438) (app T530 (. T525 X438) T305))" 53.54/15.29 }], 53.54/15.29 "kb": { 53.54/15.29 "nonunifying": [[ 53.54/15.29 "(qs T1 T305)", 53.54/15.29 "(qs ([]) ([]))" 53.54/15.29 ]], 53.54/15.29 "intvars": {}, 53.54/15.29 "arithmetic": { 53.54/15.29 "type": "PlainIntegerRelationState", 53.54/15.29 "relations": [] 53.54/15.29 }, 53.54/15.29 "ground": [ 53.54/15.29 "T305", 53.54/15.29 "T530" 53.54/15.29 ], 53.54/15.29 "free": ["X438"], 53.54/15.29 "exprvars": [] 53.54/15.29 } 53.54/15.29 }, 53.54/15.29 "986": { 53.54/15.29 "goal": [{ 53.54/15.29 "clause": -1, 53.54/15.29 "scope": -1, 53.54/15.29 "term": "(qs T123 X194)" 53.54/15.29 }], 53.54/15.29 "kb": { 53.54/15.29 "nonunifying": [], 53.54/15.29 "intvars": {}, 53.54/15.29 "arithmetic": { 53.54/15.29 "type": "PlainIntegerRelationState", 53.54/15.29 "relations": [] 53.54/15.29 }, 53.54/15.29 "ground": [], 53.54/15.29 "free": ["X194"], 53.54/15.29 "exprvars": [] 53.54/15.29 } 53.54/15.29 }, 53.54/15.29 "1166": { 53.54/15.29 "goal": [{ 53.54/15.29 "clause": -1, 53.54/15.29 "scope": -1, 53.54/15.29 "term": "(true)" 53.54/15.29 }], 53.54/15.29 "kb": { 53.54/15.29 "nonunifying": [], 53.54/15.29 "intvars": {}, 53.54/15.29 "arithmetic": { 53.54/15.29 "type": "PlainIntegerRelationState", 53.54/15.29 "relations": [] 53.54/15.29 }, 53.54/15.29 "ground": [], 53.54/15.29 "free": [], 53.54/15.29 "exprvars": [] 53.54/15.29 } 53.54/15.29 }, 53.54/15.29 "987": { 53.54/15.29 "goal": [{ 53.54/15.29 "clause": -1, 53.54/15.29 "scope": -1, 53.54/15.29 "term": "(',' (qs T180 X195) (app T179 (. T181 X195) X196))" 53.54/15.29 }], 53.54/15.29 "kb": { 53.54/15.29 "nonunifying": [], 53.54/15.29 "intvars": {}, 53.54/15.29 "arithmetic": { 53.54/15.29 "type": "PlainIntegerRelationState", 53.54/15.29 "relations": [] 53.54/15.29 }, 53.54/15.29 "ground": [], 53.54/15.29 "free": [ 53.54/15.29 "X196", 53.54/15.29 "X195" 53.54/15.29 ], 53.54/15.29 "exprvars": [] 53.54/15.29 } 53.54/15.29 }, 53.54/15.29 "988": { 53.54/15.29 "goal": [ 53.54/15.29 { 53.54/15.29 "clause": 0, 53.54/15.29 "scope": 7, 53.54/15.29 "term": "(qs T123 X194)" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "clause": 1, 53.54/15.29 "scope": 7, 53.54/15.29 "term": "(qs T123 X194)" 53.54/15.29 } 53.54/15.29 ], 53.54/15.29 "kb": { 53.54/15.29 "nonunifying": [], 53.54/15.29 "intvars": {}, 53.54/15.29 "arithmetic": { 53.54/15.29 "type": "PlainIntegerRelationState", 53.54/15.29 "relations": [] 53.54/15.29 }, 53.54/15.29 "ground": [], 53.54/15.29 "free": ["X194"], 53.54/15.29 "exprvars": [] 53.54/15.29 } 53.54/15.29 }, 53.54/15.29 "989": { 53.54/15.29 "goal": [{ 53.54/15.29 "clause": 0, 53.54/15.29 "scope": 7, 53.54/15.29 "term": "(qs T123 X194)" 53.54/15.29 }], 53.54/15.29 "kb": { 53.54/15.29 "nonunifying": [], 53.54/15.29 "intvars": {}, 53.54/15.29 "arithmetic": { 53.54/15.29 "type": "PlainIntegerRelationState", 53.54/15.29 "relations": [] 53.54/15.29 }, 53.54/15.29 "ground": [], 53.54/15.29 "free": ["X194"], 53.54/15.29 "exprvars": [] 53.54/15.29 } 53.54/15.29 } 53.54/15.29 }, 53.54/15.29 "edges": [ 53.54/15.29 { 53.54/15.29 "from": 39, 53.54/15.29 "to": 41, 53.54/15.29 "label": "CASE" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 41, 53.54/15.29 "to": 198, 53.54/15.29 "label": "EVAL with clause\nqs([], []).\nand substitutionT1 -> [],\nT2 -> []" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 41, 53.54/15.29 "to": 200, 53.54/15.29 "label": "EVAL-BACKTRACK" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 198, 53.54/15.29 "to": 211, 53.54/15.29 "label": "SUCCESS" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 200, 53.54/15.29 "to": 1104, 53.54/15.29 "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" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 200, 53.54/15.29 "to": 1105, 53.54/15.29 "label": "EVAL-BACKTRACK" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 211, 53.54/15.29 "to": 212, 53.54/15.29 "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" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 211, 53.54/15.29 "to": 213, 53.54/15.29 "label": "EVAL-BACKTRACK" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 212, 53.54/15.29 "to": 214, 53.54/15.29 "label": "CASE" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 214, 53.54/15.29 "to": 215, 53.54/15.29 "label": "PARALLEL" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 214, 53.54/15.29 "to": 216, 53.54/15.29 "label": "PARALLEL" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 215, 53.54/15.29 "to": 219, 53.54/15.29 "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" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 215, 53.54/15.29 "to": 220, 53.54/15.29 "label": "EVAL-BACKTRACK" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 216, 53.54/15.29 "to": 1041, 53.54/15.29 "label": "PARALLEL" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 216, 53.54/15.29 "to": 1042, 53.54/15.29 "label": "PARALLEL" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 219, 53.54/15.29 "to": 247, 53.54/15.29 "label": "SPLIT 1" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 219, 53.54/15.29 "to": 257, 53.54/15.29 "label": "SPLIT 2\nnew knowledge:\nT29 is ground\nreplacements:T24 -> T29,\nT26 -> T30,\nT25 -> T31" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 247, 53.54/15.29 "to": 299, 53.54/15.29 "label": "CASE" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 257, 53.54/15.29 "to": 387, 53.54/15.29 "label": "SPLIT 1" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 257, 53.54/15.29 "to": 388, 53.54/15.29 "label": "SPLIT 2\nnew knowledge:\nT29 is ground\nreplacements:X46 -> T52,\nX47 -> T53,\nT31 -> T54" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 299, 53.54/15.29 "to": 322, 53.54/15.29 "label": "PARALLEL" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 299, 53.54/15.29 "to": 326, 53.54/15.29 "label": "PARALLEL" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 322, 53.54/15.29 "to": 378, 53.54/15.29 "label": "EVAL with clause\nless(0, s(X56)).\nand substitutionT24 -> 0,\nX56 -> T38,\nT25 -> s(T38)" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 322, 53.54/15.29 "to": 380, 53.54/15.29 "label": "EVAL-BACKTRACK" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 326, 53.54/15.29 "to": 382, 53.54/15.29 "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" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 326, 53.54/15.29 "to": 383, 53.54/15.29 "label": "EVAL-BACKTRACK" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 378, 53.54/15.29 "to": 381, 53.54/15.29 "label": "SUCCESS" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 382, 53.54/15.29 "to": 247, 53.54/15.29 "label": "INSTANCE with matching:\nT24 -> T45\nT25 -> T46" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 387, 53.54/15.29 "to": 393, 53.54/15.29 "label": "CASE" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 388, 53.54/15.29 "to": 925, 53.54/15.29 "label": "SPLIT 1" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 388, 53.54/15.29 "to": 926, 53.54/15.29 "label": "SPLIT 2\nreplacements:X11 -> T105,\nT53 -> T106" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 393, 53.54/15.29 "to": 394, 53.54/15.29 "label": "PARALLEL" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 393, 53.54/15.29 "to": 395, 53.54/15.29 "label": "PARALLEL" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 394, 53.54/15.29 "to": 413, 53.54/15.29 "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" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 394, 53.54/15.29 "to": 417, 53.54/15.29 "label": "EVAL-BACKTRACK" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 395, 53.54/15.29 "to": 913, 53.54/15.29 "label": "PARALLEL" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 395, 53.54/15.29 "to": 914, 53.54/15.29 "label": "PARALLEL" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 413, 53.54/15.29 "to": 425, 53.54/15.29 "label": "SPLIT 1" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 413, 53.54/15.29 "to": 426, 53.54/15.29 "label": "SPLIT 2\nnew knowledge:\nT70 is ground\nreplacements:T74 -> T77" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 425, 53.54/15.29 "to": 247, 53.54/15.29 "label": "INSTANCE with matching:\nT24 -> T70\nT25 -> T73" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 426, 53.54/15.29 "to": 387, 53.54/15.29 "label": "INSTANCE with matching:\nT29 -> T70\nT30 -> T77\nX46 -> X112\nX47 -> X113" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 913, 53.54/15.29 "to": 915, 53.54/15.29 "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" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 913, 53.54/15.29 "to": 917, 53.54/15.29 "label": "EVAL-BACKTRACK" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 914, 53.54/15.29 "to": 922, 53.54/15.29 "label": "EVAL with clause\npart(X169, [], [], []).\nand substitutionT29 -> T104,\nX169 -> T104,\nT30 -> [],\nX46 -> [],\nX47 -> []" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 914, 53.54/15.29 "to": 923, 53.54/15.29 "label": "EVAL-BACKTRACK" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 915, 53.54/15.29 "to": 387, 53.54/15.29 "label": "INSTANCE with matching:\nT29 -> T95\nT30 -> T98\nX46 -> X158\nX47 -> X159" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 922, 53.54/15.29 "to": 924, 53.54/15.29 "label": "SUCCESS" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 925, 53.54/15.29 "to": 929, 53.54/15.29 "label": "CASE" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 926, 53.54/15.29 "to": 1036, 53.54/15.29 "label": "SPLIT 1" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 926, 53.54/15.29 "to": 1037, 53.54/15.29 "label": "SPLIT 2\nreplacements:X12 -> T229,\nT105 -> T230" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 929, 53.54/15.29 "to": 930, 53.54/15.29 "label": "BACKTRACK\nfor clause: qs([], [])because of non-unification" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 930, 53.54/15.29 "to": 934, 53.54/15.29 "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" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 934, 53.54/15.29 "to": 943, 53.54/15.29 "label": "SPLIT 1" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 934, 53.54/15.29 "to": 944, 53.54/15.29 "label": "SPLIT 2\nreplacements:X192 -> T123,\nX193 -> T124,\nT118 -> T125" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 943, 53.54/15.29 "to": 945, 53.54/15.29 "label": "CASE" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 944, 53.54/15.29 "to": 986, 53.54/15.29 "label": "SPLIT 1" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 944, 53.54/15.29 "to": 987, 53.54/15.29 "label": "SPLIT 2\nreplacements:X194 -> T179,\nT124 -> T180,\nT125 -> T181" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 945, 53.54/15.29 "to": 951, 53.54/15.29 "label": "PARALLEL" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 945, 53.54/15.29 "to": 952, 53.54/15.29 "label": "PARALLEL" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 951, 53.54/15.29 "to": 961, 53.54/15.29 "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" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 951, 53.54/15.29 "to": 962, 53.54/15.29 "label": "EVAL-BACKTRACK" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 952, 53.54/15.29 "to": 969, 53.54/15.29 "label": "PARALLEL" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 952, 53.54/15.29 "to": 970, 53.54/15.29 "label": "PARALLEL" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 961, 53.54/15.29 "to": 963, 53.54/15.29 "label": "SPLIT 1" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 961, 53.54/15.29 "to": 964, 53.54/15.29 "label": "SPLIT 2\nnew knowledge:\nT149 is ground\nreplacements:T144 -> T149,\nT146 -> T150" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 963, 53.54/15.29 "to": 247, 53.54/15.29 "label": "INSTANCE with matching:\nT24 -> T144\nT25 -> T145" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 964, 53.54/15.29 "to": 387, 53.54/15.29 "label": "INSTANCE with matching:\nT29 -> T149\nT30 -> T150\nX46 -> X244\nX47 -> X245" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 969, 53.54/15.29 "to": 975, 53.54/15.29 "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" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 969, 53.54/15.29 "to": 976, 53.54/15.29 "label": "EVAL-BACKTRACK" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 970, 53.54/15.29 "to": 981, 53.54/15.29 "label": "EVAL with clause\npart(X301, [], [], []).\nand substitutionT118 -> T178,\nX301 -> T178,\nT119 -> [],\nX192 -> [],\nX193 -> []" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 970, 53.54/15.29 "to": 982, 53.54/15.29 "label": "EVAL-BACKTRACK" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 975, 53.54/15.29 "to": 943, 53.54/15.29 "label": "INSTANCE with matching:\nT118 -> T171\nT119 -> T172\nX192 -> X290\nX193 -> X291" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 981, 53.54/15.29 "to": 983, 53.54/15.29 "label": "SUCCESS" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 986, 53.54/15.29 "to": 988, 53.54/15.29 "label": "CASE" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 987, 53.54/15.29 "to": 1012, 53.54/15.29 "label": "SPLIT 1" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 987, 53.54/15.29 "to": 1013, 53.54/15.29 "label": "SPLIT 2\nreplacements:X195 -> T193,\nT179 -> T194,\nT181 -> T195" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 988, 53.54/15.29 "to": 989, 53.54/15.29 "label": "PARALLEL" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 988, 53.54/15.29 "to": 990, 53.54/15.29 "label": "PARALLEL" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 989, 53.54/15.29 "to": 992, 53.54/15.29 "label": "EVAL with clause\nqs([], []).\nand substitutionT123 -> [],\nX194 -> []" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 989, 53.54/15.29 "to": 993, 53.54/15.29 "label": "EVAL-BACKTRACK" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 990, 53.54/15.29 "to": 1002, 53.54/15.29 "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" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 990, 53.54/15.29 "to": 1003, 53.54/15.29 "label": "EVAL-BACKTRACK" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 992, 53.54/15.29 "to": 994, 53.54/15.29 "label": "SUCCESS" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1002, 53.54/15.29 "to": 934, 53.54/15.29 "label": "INSTANCE with matching:\nT118 -> T188\nT119 -> T189\nX192 -> X317\nX193 -> X318\nX194 -> X319\nX195 -> X320\nX196 -> X321" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1012, 53.54/15.29 "to": 986, 53.54/15.29 "label": "INSTANCE with matching:\nT123 -> T180\nX194 -> X195" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1013, 53.54/15.29 "to": 1016, 53.54/15.29 "label": "CASE" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1016, 53.54/15.29 "to": 1019, 53.54/15.29 "label": "PARALLEL" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1016, 53.54/15.29 "to": 1020, 53.54/15.29 "label": "PARALLEL" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1019, 53.54/15.29 "to": 1024, 53.54/15.29 "label": "EVAL with clause\napp([], X335, X335).\nand substitutionT194 -> [],\nT195 -> T208,\nT193 -> T209,\nX335 -> .(T208, T209),\nX196 -> .(T208, T209)" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1019, 53.54/15.29 "to": 1025, 53.54/15.29 "label": "EVAL-BACKTRACK" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1020, 53.54/15.29 "to": 1032, 53.54/15.29 "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" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1020, 53.54/15.29 "to": 1033, 53.54/15.29 "label": "EVAL-BACKTRACK" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1024, 53.54/15.29 "to": 1026, 53.54/15.29 "label": "SUCCESS" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1032, 53.54/15.29 "to": 1013, 53.54/15.29 "label": "INSTANCE with matching:\nT194 -> T222\nT195 -> T223\nT193 -> T224\nX196 -> X350" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1036, 53.54/15.29 "to": 986, 53.54/15.29 "label": "INSTANCE with matching:\nT123 -> T106\nX194 -> X12" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1037, 53.54/15.29 "to": 1038, 53.54/15.29 "label": "CASE" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1038, 53.54/15.29 "to": 1039, 53.54/15.29 "label": "BACKTRACK\nfor clause: app([], X, X)because of non-unification" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1039, 53.54/15.29 "to": 1040, 53.54/15.29 "label": "BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1041, 53.54/15.29 "to": 1049, 53.54/15.29 "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" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1041, 53.54/15.29 "to": 1050, 53.54/15.29 "label": "EVAL-BACKTRACK" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1042, 53.54/15.29 "to": 1062, 53.54/15.29 "label": "EVAL with clause\npart(X418, [], [], []).\nand substitutionT7 -> T289,\nX418 -> T289,\nT8 -> [],\nX9 -> [],\nX10 -> [],\nT288 -> T289" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1042, 53.54/15.29 "to": 1063, 53.54/15.29 "label": "EVAL-BACKTRACK" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1049, 53.54/15.29 "to": 1053, 53.54/15.29 "label": "SPLIT 1" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1049, 53.54/15.29 "to": 1054, 53.54/15.29 "label": "SPLIT 2\nreplacements:X393 -> T262,\nX394 -> T263,\nT258 -> T264,\nT256 -> T265" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1053, 53.54/15.29 "to": 943, 53.54/15.29 "label": "INSTANCE with matching:\nT118 -> T256\nT119 -> T257\nX192 -> X393\nX193 -> X394" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1054, 53.54/15.29 "to": 1055, 53.54/15.29 "label": "SPLIT 1" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1054, 53.54/15.29 "to": 1056, 53.54/15.29 "label": "SPLIT 2\nreplacements:X11 -> T269,\nT264 -> T270,\nT263 -> T271,\nT265 -> T272" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1055, 53.54/15.29 "to": 986, 53.54/15.29 "label": "INSTANCE with matching:\nT123 -> T262\nX194 -> X11" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1056, 53.54/15.29 "to": 1057, 53.54/15.29 "label": "SPLIT 1" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1056, 53.54/15.29 "to": 1058, 53.54/15.29 "label": "SPLIT 2\nreplacements:X12 -> T273,\nT269 -> T274,\nT272 -> T275" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1057, 53.54/15.29 "to": 925, 53.54/15.29 "label": "INSTANCE with matching:\nT54 -> T270\nT52 -> T271\nX11 -> X12" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1058, 53.54/15.29 "to": 1059, 53.54/15.29 "label": "CASE" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1059, 53.54/15.29 "to": 1060, 53.54/15.29 "label": "BACKTRACK\nfor clause: app([], X, X)because of non-unification" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1060, 53.54/15.29 "to": 1061, 53.54/15.29 "label": "BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1062, 53.54/15.29 "to": 1064, 53.54/15.29 "label": "SPLIT 1" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1062, 53.54/15.29 "to": 1065, 53.54/15.29 "label": "SPLIT 2\nnew knowledge:\nT292 is ground\nreplacements:X11 -> T292" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1064, 53.54/15.29 "to": 986, 53.54/15.29 "label": "INSTANCE with matching:\nT123 -> []\nX194 -> X11" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1065, 53.54/15.29 "to": 1102, 53.54/15.29 "label": "SPLIT 1" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1065, 53.54/15.29 "to": 1103, 53.54/15.29 "label": "SPLIT 2\nnew knowledge:\nT295 is ground\nreplacements:X12 -> T295" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1102, 53.54/15.29 "to": 986, 53.54/15.29 "label": "INSTANCE with matching:\nT123 -> []\nX194 -> X12" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1103, 53.54/15.29 "to": 1058, 53.54/15.29 "label": "INSTANCE with matching:\nT274 -> T292\nT275 -> T289\nT273 -> T295" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1104, 53.54/15.29 "to": 1106, 53.54/15.29 "label": "CASE" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1106, 53.54/15.29 "to": 1143, 53.54/15.29 "label": "PARALLEL" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1106, 53.54/15.29 "to": 1144, 53.54/15.29 "label": "PARALLEL" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1143, 53.54/15.29 "to": 1145, 53.54/15.29 "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" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1143, 53.54/15.29 "to": 1146, 53.54/15.29 "label": "EVAL-BACKTRACK" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1144, 53.54/15.29 "to": 1171, 53.54/15.29 "label": "PARALLEL" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1144, 53.54/15.29 "to": 1172, 53.54/15.29 "label": "PARALLEL" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1145, 53.54/15.29 "to": 1147, 53.54/15.29 "label": "SPLIT 1" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1145, 53.54/15.29 "to": 1148, 53.54/15.29 "label": "SPLIT 2\nnew knowledge:\nT328 is ground\nreplacements:T323 -> T328,\nT325 -> T329,\nT324 -> T330,\nT1 -> T331" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1147, 53.54/15.29 "to": 247, 53.54/15.29 "label": "INSTANCE with matching:\nT24 -> T323\nT25 -> T324" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1148, 53.54/15.29 "to": 1149, 53.54/15.29 "label": "SPLIT 1" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1148, 53.54/15.29 "to": 1150, 53.54/15.29 "label": "SPLIT 2\nnew knowledge:\nT328 is ground\nreplacements:X472 -> T337,\nX473 -> T338,\nT330 -> T339,\nT331 -> T340" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1149, 53.54/15.29 "to": 387, 53.54/15.29 "label": "INSTANCE with matching:\nT29 -> T328\nT30 -> T329\nX46 -> X472\nX47 -> X473" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1150, 53.54/15.29 "to": 1151, 53.54/15.29 "label": "SPLIT 1" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1150, 53.54/15.29 "to": 1152, 53.54/15.29 "label": "SPLIT 2\nreplacements:X437 -> T344,\nT338 -> T345,\nT340 -> T346" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1151, 53.54/15.29 "to": 925, 53.54/15.29 "label": "INSTANCE with matching:\nT54 -> T339\nT52 -> T337\nX11 -> X437" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1152, 53.54/15.29 "to": 1153, 53.54/15.29 "label": "SPLIT 1" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1152, 53.54/15.29 "to": 1154, 53.54/15.29 "label": "SPLIT 2\nreplacements:X438 -> T347,\nT344 -> T348,\nT346 -> T349" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1153, 53.54/15.29 "to": 986, 53.54/15.29 "label": "INSTANCE with matching:\nT123 -> T345\nX194 -> X438" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1154, 53.54/15.29 "to": 1155, 53.54/15.29 "label": "CASE" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1155, 53.54/15.29 "to": 1156, 53.54/15.29 "label": "PARALLEL" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1155, 53.54/15.29 "to": 1157, 53.54/15.29 "label": "PARALLEL" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1156, 53.54/15.29 "to": 1158, 53.54/15.29 "label": "EVAL with clause\napp([], X498, X498).\nand substitutionT348 -> [],\nT328 -> T362,\nT347 -> T363,\nX498 -> .(T362, T363),\nT305 -> .(T362, T363)" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1156, 53.54/15.29 "to": 1159, 53.54/15.29 "label": "EVAL-BACKTRACK" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1157, 53.54/15.29 "to": 1161, 53.54/15.29 "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" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1157, 53.54/15.29 "to": 1162, 53.54/15.29 "label": "EVAL-BACKTRACK" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1158, 53.54/15.29 "to": 1160, 53.54/15.29 "label": "SUCCESS" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1161, 53.54/15.29 "to": 1163, 53.54/15.29 "label": "CASE" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1163, 53.54/15.29 "to": 1164, 53.54/15.29 "label": "PARALLEL" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1163, 53.54/15.29 "to": 1165, 53.54/15.29 "label": "PARALLEL" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1164, 53.54/15.29 "to": 1166, 53.54/15.29 "label": "EVAL with clause\napp([], X517, X517).\nand substitutionT379 -> [],\nT376 -> T393,\nT380 -> T394,\nX517 -> .(T393, T394),\nT378 -> .(T393, T394)" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1164, 53.54/15.29 "to": 1167, 53.54/15.29 "label": "EVAL-BACKTRACK" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1165, 53.54/15.29 "to": 1169, 53.54/15.29 "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" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1165, 53.54/15.29 "to": 1170, 53.54/15.29 "label": "EVAL-BACKTRACK" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1166, 53.54/15.29 "to": 1168, 53.54/15.29 "label": "SUCCESS" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1169, 53.54/15.29 "to": 1161, 53.54/15.29 "label": "INSTANCE with matching:\nT379 -> T410\nT376 -> T407\nT380 -> T411\nT378 -> T409" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1171, 53.54/15.29 "to": 1173, 53.54/15.29 "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" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1171, 53.54/15.29 "to": 1174, 53.54/15.29 "label": "EVAL-BACKTRACK" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1172, 53.54/15.29 "to": 1197, 53.54/15.29 "label": "EVAL with clause\npart(X623, [], [], []).\nand substitutionT306 -> T525,\nX623 -> T525,\nT307 -> [],\nX435 -> [],\nX436 -> [],\nT524 -> T525" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1172, 53.54/15.29 "to": 1198, 53.54/15.29 "label": "EVAL-BACKTRACK" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1173, 53.54/15.29 "to": 1175, 53.54/15.29 "label": "SPLIT 1" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1173, 53.54/15.29 "to": 1176, 53.54/15.29 "label": "SPLIT 2\nreplacements:X565 -> T437,\nX566 -> T438,\nT433 -> T439,\nT431 -> T440,\nT1 -> T441" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1175, 53.54/15.29 "to": 943, 53.54/15.29 "label": "INSTANCE with matching:\nT118 -> T431\nT119 -> T432\nX192 -> X565\nX193 -> X566" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1176, 53.54/15.29 "to": 1177, 53.54/15.29 "label": "SPLIT 1" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1176, 53.54/15.29 "to": 1178, 53.54/15.29 "label": "SPLIT 2\nreplacements:X437 -> T445,\nT439 -> T446,\nT438 -> T447,\nT440 -> T448,\nT441 -> T449" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1177, 53.54/15.29 "to": 986, 53.54/15.29 "label": "INSTANCE with matching:\nT123 -> T437\nX194 -> X437" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1178, 53.54/15.29 "to": 1179, 53.54/15.29 "label": "SPLIT 1" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1178, 53.54/15.29 "to": 1180, 53.54/15.29 "label": "SPLIT 2\nreplacements:X438 -> T450,\nT445 -> T451,\nT448 -> T452,\nT449 -> T453" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1179, 53.54/15.29 "to": 925, 53.54/15.29 "label": "INSTANCE with matching:\nT54 -> T446\nT52 -> T447\nX11 -> X438" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1180, 53.54/15.29 "to": 1181, 53.54/15.29 "label": "CASE" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1181, 53.54/15.29 "to": 1182, 53.54/15.29 "label": "PARALLEL" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1181, 53.54/15.29 "to": 1183, 53.54/15.29 "label": "PARALLEL" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1182, 53.54/15.29 "to": 1184, 53.54/15.29 "label": "EVAL with clause\napp([], X587, X587).\nand substitutionT451 -> [],\nT452 -> T466,\nT450 -> T467,\nX587 -> .(T466, T467),\nT305 -> .(T466, T467)" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1182, 53.54/15.29 "to": 1185, 53.54/15.29 "label": "EVAL-BACKTRACK" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1183, 53.54/15.29 "to": 1187, 53.54/15.29 "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" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1183, 53.54/15.29 "to": 1188, 53.54/15.29 "label": "EVAL-BACKTRACK" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1184, 53.54/15.29 "to": 1186, 53.54/15.29 "label": "SUCCESS" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1187, 53.54/15.29 "to": 1189, 53.54/15.29 "label": "CASE" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1189, 53.54/15.29 "to": 1190, 53.54/15.29 "label": "PARALLEL" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1189, 53.54/15.29 "to": 1191, 53.54/15.29 "label": "PARALLEL" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1190, 53.54/15.29 "to": 1192, 53.54/15.29 "label": "EVAL with clause\napp([], X606, X606).\nand substitutionT483 -> [],\nT484 -> T498,\nT485 -> T499,\nX606 -> .(T498, T499),\nT482 -> .(T498, T499)" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1190, 53.54/15.29 "to": 1193, 53.54/15.29 "label": "EVAL-BACKTRACK" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1191, 53.54/15.29 "to": 1195, 53.54/15.29 "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" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1191, 53.54/15.29 "to": 1196, 53.54/15.29 "label": "EVAL-BACKTRACK" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1192, 53.54/15.29 "to": 1194, 53.54/15.29 "label": "SUCCESS" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1195, 53.54/15.29 "to": 1187, 53.54/15.29 "label": "INSTANCE with matching:\nT483 -> T515\nT484 -> T516\nT485 -> T517\nT482 -> T514" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1197, 53.54/15.29 "to": 1199, 53.54/15.29 "label": "SPLIT 1" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1197, 53.54/15.29 "to": 1200, 53.54/15.29 "label": "SPLIT 2\nnew knowledge:\nT530 is ground\nreplacements:X437 -> T530" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1199, 53.54/15.29 "to": 986, 53.54/15.29 "label": "INSTANCE with matching:\nT123 -> []\nX194 -> X437" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1200, 53.54/15.29 "to": 1201, 53.54/15.29 "label": "SPLIT 1" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1200, 53.54/15.29 "to": 1202, 53.54/15.29 "label": "SPLIT 2\nnew knowledge:\nT535 is ground\nreplacements:X438 -> T535" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1201, 53.54/15.29 "to": 986, 53.54/15.29 "label": "INSTANCE with matching:\nT123 -> []\nX194 -> X438" 53.54/15.29 }, 53.54/15.29 { 53.54/15.29 "from": 1202, 53.54/15.29 "to": 1180, 53.54/15.29 "label": "INSTANCE with matching:\nT451 -> T530\nT452 -> T525\nT450 -> T535\nT453 -> T1" 53.54/15.29 } 53.54/15.29 ], 53.54/15.29 "type": "Graph" 53.54/15.29 } 53.54/15.29 } 53.54/15.29 53.54/15.29 ---------------------------------------- 53.54/15.29 53.54/15.29 (133) 53.54/15.29 Obligation: 53.54/15.29 Triples: 53.54/15.29 53.54/15.29 lessA(s(X1), s(X2)) :- lessA(X1, X2). 53.54/15.29 partB(X1, .(X2, X3), .(X2, X4), X5) :- lessA(X1, X2). 53.54/15.29 partB(X1, .(X2, X3), .(X2, X4), X5) :- ','(lesscA(X1, X2), partB(X1, X3, X4, X5)). 53.54/15.29 partB(X1, .(X2, X3), X4, .(X2, X5)) :- partB(X1, X3, X4, X5). 53.54/15.29 partC(X1, .(X2, X3), .(X2, X4), X5) :- lessA(X1, X2). 53.54/15.29 partC(X1, .(X2, X3), .(X2, X4), X5) :- ','(lesscA(X1, X2), partB(X1, X3, X4, X5)). 53.54/15.29 partC(X1, .(X2, X3), X4, .(X2, X5)) :- partC(X1, X3, X4, X5). 53.54/15.29 pD(X1, X2, X3, X4, X5, X6, X7) :- partC(X1, X2, X3, X4). 53.54/15.29 pD(X1, X2, X3, X4, X5, X6, X7) :- ','(partcC(X1, X2, X3, X4), qsF(X3, X5)). 53.54/15.29 pD(X1, X2, X3, X4, X5, X6, X7) :- ','(partcC(X1, X2, X3, X4), ','(qscF(X3, X5), qsF(X4, X6))). 53.54/15.29 pD(X1, X2, X3, X4, X5, X6, X7) :- ','(partcC(X1, X2, X3, X4), ','(qscF(X3, X5), ','(qscF(X4, X6), appE(X5, X1, X6, X7)))). 53.54/15.29 qsF(.(X1, X2), X3) :- pD(X1, X2, X4, X5, X6, X7, X3). 53.54/15.29 appE(.(X1, X2), X3, X4, .(X1, X5)) :- appE(X2, X3, X4, X5). 53.54/15.29 qsG(X1, X2, X3) :- pD(X1, X2, X4, X5, X6, X7, X3). 53.54/15.29 appH(.(X1, X2), X3, X4, .(X1, X5)) :- appH(X2, X3, X4, X5). 53.54/15.29 appI(.(X1, X2), X3, X4, .(X1, X5)) :- appI(X2, X3, X4, X5). 53.54/15.29 appJ(.(X1, X2), X3, X4, .(X1, X5)) :- appI(X2, X3, X4, X5). 53.54/15.29 qsK(.(X1, .(X2, X3)), []) :- lessA(X1, X2). 53.54/15.29 qsK(.(X1, .(X2, X3)), []) :- ','(lesscA(X1, X2), partB(X1, X3, X4, X5)). 53.54/15.29 qsK(.(X1, .(X2, X3)), []) :- ','(lesscA(X1, X2), ','(partcB(X1, X3, X4, X5), qsG(X2, X4, X6))). 53.54/15.29 qsK(.(X1, .(X2, X3)), []) :- ','(lesscA(X1, X2), ','(partcB(X1, X3, X4, X5), ','(qscG(X2, X4, X6), qsF(X5, X7)))). 53.54/15.29 qsK(.(X1, .(X2, X3)), []) :- partC(X1, X3, X4, X5). 53.54/15.29 qsK(.(X1, .(X2, X3)), []) :- ','(partcC(X1, X3, X4, X5), qsF(X4, X6)). 53.54/15.29 qsK(.(X1, .(X2, X3)), []) :- ','(partcC(X1, X3, X4, X5), ','(qscF(X4, X6), qsG(X2, X5, X7))). 53.54/15.29 qsK(.(X1, .(X2, X3)), []) :- ','(partcC(X1, X3, X4, X5), ','(qscF(X4, X6), ','(qscG(X2, X5, X7), appL(X6, X1, X7)))). 53.54/15.29 qsK(.(X1, []), []) :- qsF([], X2). 53.54/15.29 qsK(.(X1, []), []) :- ','(qscF([], X2), qsF([], X3)). 53.54/15.29 qsK(.(X1, []), []) :- ','(qscF([], X2), ','(qscF([], X3), appL(X2, X1, X3))). 53.54/15.29 qsK(.(X1, .(X2, X3)), X4) :- lessA(X1, X2). 53.54/15.29 qsK(.(X1, .(X2, X3)), X4) :- ','(lesscA(X1, X2), partB(X1, X3, X5, X6)). 53.54/15.29 qsK(.(X1, .(X2, X3)), X4) :- ','(lesscA(X1, X2), ','(partcB(X1, X3, X5, X6), qsG(X2, X5, X7))). 53.54/15.29 qsK(.(X1, .(X2, X3)), X4) :- ','(lesscA(X1, X2), ','(partcB(X1, X3, X5, X6), ','(qscG(X2, X5, X7), qsF(X6, X8)))). 53.54/15.29 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))))). 53.54/15.29 qsK(.(X1, .(X2, X3)), X4) :- partC(X1, X3, X5, X6). 53.54/15.29 qsK(.(X1, .(X2, X3)), X4) :- ','(partcC(X1, X3, X5, X6), qsF(X5, X7)). 53.54/15.29 qsK(.(X1, .(X2, X3)), X4) :- ','(partcC(X1, X3, X5, X6), ','(qscF(X5, X7), qsG(X2, X6, X8))). 53.54/15.29 qsK(.(X1, .(X2, X3)), X4) :- ','(partcC(X1, X3, X5, X6), ','(qscF(X5, X7), ','(qscG(X2, X6, X8), appJ(X7, X1, X8, X4)))). 53.54/15.29 qsK(.(X1, []), X2) :- qsF([], X3). 53.54/15.29 qsK(.(X1, []), X2) :- ','(qscF([], X3), qsF([], X4)). 53.54/15.29 qsK(.(X1, []), X2) :- ','(qscF([], X3), ','(qscF([], X4), appJ(X3, X1, X4, X2))). 53.54/15.29 53.54/15.29 Clauses: 53.54/15.29 53.54/15.29 lesscA(0, s(X1)). 53.54/15.29 lesscA(s(X1), s(X2)) :- lesscA(X1, X2). 53.54/15.29 partcB(X1, .(X2, X3), .(X2, X4), X5) :- ','(lesscA(X1, X2), partcB(X1, X3, X4, X5)). 53.54/15.29 partcB(X1, .(X2, X3), X4, .(X2, X5)) :- partcB(X1, X3, X4, X5). 53.54/15.29 partcB(X1, [], [], []). 53.54/15.29 partcC(X1, .(X2, X3), .(X2, X4), X5) :- ','(lesscA(X1, X2), partcB(X1, X3, X4, X5)). 53.54/15.29 partcC(X1, .(X2, X3), X4, .(X2, X5)) :- partcC(X1, X3, X4, X5). 53.54/15.29 partcC(X1, [], [], []). 53.54/15.29 qcD(X1, X2, X3, X4, X5, X6, X7) :- ','(partcC(X1, X2, X3, X4), ','(qscF(X3, X5), ','(qscF(X4, X6), appcE(X5, X1, X6, X7)))). 53.54/15.29 qscF([], []). 53.54/15.29 qscF(.(X1, X2), X3) :- qcD(X1, X2, X4, X5, X6, X7, X3). 53.54/15.29 appcE([], X1, X2, .(X1, X2)). 53.54/15.29 appcE(.(X1, X2), X3, X4, .(X1, X5)) :- appcE(X2, X3, X4, X5). 53.54/15.29 qscG(X1, X2, X3) :- qcD(X1, X2, X4, X5, X6, X7, X3). 53.54/15.29 appcH([], X1, X2, .(X1, X2)). 53.54/15.29 appcH(.(X1, X2), X3, X4, .(X1, X5)) :- appcH(X2, X3, X4, X5). 53.54/15.29 appcI([], X1, X2, .(X1, X2)). 53.54/15.29 appcI(.(X1, X2), X3, X4, .(X1, X5)) :- appcI(X2, X3, X4, X5). 53.54/15.29 appcJ([], X1, X2, .(X1, X2)). 53.54/15.29 appcJ(.(X1, X2), X3, X4, .(X1, X5)) :- appcI(X2, X3, X4, X5). 53.54/15.29 53.54/15.29 Afs: 53.54/15.29 53.54/15.29 qsK(x1, x2) = qsK(x2) 53.54/15.29 53.54/15.29 53.54/15.29 ---------------------------------------- 53.54/15.29 53.54/15.29 (134) UndefinedPredicateInTriplesTransformerProof (SOUND) 53.54/15.29 Deleted triples and predicates having undefined goals [DT09]. 53.54/15.29 ---------------------------------------- 53.54/15.29 53.54/15.29 (135) 53.54/15.29 Obligation: 53.54/15.29 Triples: 53.54/15.29 53.54/15.29 lessA(s(X1), s(X2)) :- lessA(X1, X2). 53.54/15.29 partB(X1, .(X2, X3), .(X2, X4), X5) :- lessA(X1, X2). 53.54/15.29 partB(X1, .(X2, X3), .(X2, X4), X5) :- ','(lesscA(X1, X2), partB(X1, X3, X4, X5)). 53.54/15.29 partB(X1, .(X2, X3), X4, .(X2, X5)) :- partB(X1, X3, X4, X5). 53.54/15.29 partC(X1, .(X2, X3), .(X2, X4), X5) :- lessA(X1, X2). 53.54/15.29 partC(X1, .(X2, X3), .(X2, X4), X5) :- ','(lesscA(X1, X2), partB(X1, X3, X4, X5)). 53.54/15.29 partC(X1, .(X2, X3), X4, .(X2, X5)) :- partC(X1, X3, X4, X5). 53.54/15.29 pD(X1, X2, X3, X4, X5, X6, X7) :- partC(X1, X2, X3, X4). 53.54/15.29 pD(X1, X2, X3, X4, X5, X6, X7) :- ','(partcC(X1, X2, X3, X4), qsF(X3, X5)). 53.54/15.29 pD(X1, X2, X3, X4, X5, X6, X7) :- ','(partcC(X1, X2, X3, X4), ','(qscF(X3, X5), qsF(X4, X6))). 53.54/15.29 pD(X1, X2, X3, X4, X5, X6, X7) :- ','(partcC(X1, X2, X3, X4), ','(qscF(X3, X5), ','(qscF(X4, X6), appE(X5, X1, X6, X7)))). 53.54/15.29 qsF(.(X1, X2), X3) :- pD(X1, X2, X4, X5, X6, X7, X3). 53.54/15.29 appE(.(X1, X2), X3, X4, .(X1, X5)) :- appE(X2, X3, X4, X5). 53.54/15.29 qsG(X1, X2, X3) :- pD(X1, X2, X4, X5, X6, X7, X3). 53.54/15.29 appH(.(X1, X2), X3, X4, .(X1, X5)) :- appH(X2, X3, X4, X5). 53.54/15.29 appI(.(X1, X2), X3, X4, .(X1, X5)) :- appI(X2, X3, X4, X5). 53.54/15.29 appJ(.(X1, X2), X3, X4, .(X1, X5)) :- appI(X2, X3, X4, X5). 53.54/15.29 qsK(.(X1, .(X2, X3)), []) :- lessA(X1, X2). 53.54/15.29 qsK(.(X1, .(X2, X3)), []) :- ','(lesscA(X1, X2), partB(X1, X3, X4, X5)). 53.54/15.29 qsK(.(X1, .(X2, X3)), []) :- ','(lesscA(X1, X2), ','(partcB(X1, X3, X4, X5), qsG(X2, X4, X6))). 53.54/15.29 qsK(.(X1, .(X2, X3)), []) :- ','(lesscA(X1, X2), ','(partcB(X1, X3, X4, X5), ','(qscG(X2, X4, X6), qsF(X5, X7)))). 53.54/15.29 qsK(.(X1, .(X2, X3)), []) :- partC(X1, X3, X4, X5). 53.54/15.29 qsK(.(X1, .(X2, X3)), []) :- ','(partcC(X1, X3, X4, X5), qsF(X4, X6)). 53.54/15.29 qsK(.(X1, .(X2, X3)), []) :- ','(partcC(X1, X3, X4, X5), ','(qscF(X4, X6), qsG(X2, X5, X7))). 53.54/15.29 qsK(.(X1, []), []) :- qsF([], X2). 53.54/15.29 qsK(.(X1, []), []) :- ','(qscF([], X2), qsF([], X3)). 53.54/15.29 qsK(.(X1, .(X2, X3)), X4) :- lessA(X1, X2). 53.54/15.29 qsK(.(X1, .(X2, X3)), X4) :- ','(lesscA(X1, X2), partB(X1, X3, X5, X6)). 53.54/15.29 qsK(.(X1, .(X2, X3)), X4) :- ','(lesscA(X1, X2), ','(partcB(X1, X3, X5, X6), qsG(X2, X5, X7))). 53.54/15.29 qsK(.(X1, .(X2, X3)), X4) :- ','(lesscA(X1, X2), ','(partcB(X1, X3, X5, X6), ','(qscG(X2, X5, X7), qsF(X6, X8)))). 53.54/15.29 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))))). 53.54/15.29 qsK(.(X1, .(X2, X3)), X4) :- partC(X1, X3, X5, X6). 53.54/15.29 qsK(.(X1, .(X2, X3)), X4) :- ','(partcC(X1, X3, X5, X6), qsF(X5, X7)). 53.54/15.29 qsK(.(X1, .(X2, X3)), X4) :- ','(partcC(X1, X3, X5, X6), ','(qscF(X5, X7), qsG(X2, X6, X8))). 53.54/15.29 qsK(.(X1, .(X2, X3)), X4) :- ','(partcC(X1, X3, X5, X6), ','(qscF(X5, X7), ','(qscG(X2, X6, X8), appJ(X7, X1, X8, X4)))). 53.54/15.29 qsK(.(X1, []), X2) :- qsF([], X3). 53.54/15.29 qsK(.(X1, []), X2) :- ','(qscF([], X3), qsF([], X4)). 53.54/15.29 qsK(.(X1, []), X2) :- ','(qscF([], X3), ','(qscF([], X4), appJ(X3, X1, X4, X2))). 53.54/15.29 53.54/15.29 Clauses: 53.54/15.29 53.54/15.29 lesscA(0, s(X1)). 53.54/15.29 lesscA(s(X1), s(X2)) :- lesscA(X1, X2). 53.54/15.29 partcB(X1, .(X2, X3), .(X2, X4), X5) :- ','(lesscA(X1, X2), partcB(X1, X3, X4, X5)). 53.54/15.29 partcB(X1, .(X2, X3), X4, .(X2, X5)) :- partcB(X1, X3, X4, X5). 53.54/15.29 partcB(X1, [], [], []). 53.54/15.29 partcC(X1, .(X2, X3), .(X2, X4), X5) :- ','(lesscA(X1, X2), partcB(X1, X3, X4, X5)). 53.54/15.29 partcC(X1, .(X2, X3), X4, .(X2, X5)) :- partcC(X1, X3, X4, X5). 53.54/15.29 partcC(X1, [], [], []). 53.54/15.29 qcD(X1, X2, X3, X4, X5, X6, X7) :- ','(partcC(X1, X2, X3, X4), ','(qscF(X3, X5), ','(qscF(X4, X6), appcE(X5, X1, X6, X7)))). 53.54/15.29 qscF([], []). 53.54/15.29 qscF(.(X1, X2), X3) :- qcD(X1, X2, X4, X5, X6, X7, X3). 53.54/15.29 appcE([], X1, X2, .(X1, X2)). 53.54/15.29 appcE(.(X1, X2), X3, X4, .(X1, X5)) :- appcE(X2, X3, X4, X5). 53.54/15.29 qscG(X1, X2, X3) :- qcD(X1, X2, X4, X5, X6, X7, X3). 53.54/15.29 appcH([], X1, X2, .(X1, X2)). 53.54/15.29 appcH(.(X1, X2), X3, X4, .(X1, X5)) :- appcH(X2, X3, X4, X5). 53.54/15.29 appcI([], X1, X2, .(X1, X2)). 53.54/15.29 appcI(.(X1, X2), X3, X4, .(X1, X5)) :- appcI(X2, X3, X4, X5). 53.54/15.29 appcJ([], X1, X2, .(X1, X2)). 53.54/15.29 appcJ(.(X1, X2), X3, X4, .(X1, X5)) :- appcI(X2, X3, X4, X5). 53.54/15.29 53.54/15.29 Afs: 53.54/15.29 53.54/15.29 qsK(x1, x2) = qsK(x2) 53.54/15.29 53.54/15.29 53.54/15.29 ---------------------------------------- 53.54/15.29 53.54/15.29 (136) TriplesToPiDPProof (SOUND) 53.54/15.29 We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: 53.54/15.29 53.54/15.29 qsK_in_2: (f,b) 53.54/15.29 53.54/15.29 lessA_in_2: (f,f) (b,f) (b,b) 53.54/15.29 53.54/15.29 lesscA_in_2: (f,f) (b,f) (b,b) 53.54/15.29 53.54/15.29 partB_in_4: (b,f,f,f) (b,b,f,f) 53.54/15.29 53.54/15.29 partcB_in_4: (b,f,f,f) (b,b,f,f) 53.54/15.29 53.54/15.29 qsG_in_3: (f,f,f) 53.54/15.29 53.54/15.29 pD_in_7: (f,f,f,f,f,f,f) (b,b,f,f,f,f,f) 53.54/15.29 53.54/15.29 partC_in_4: (f,f,f,f) (b,b,f,f) 53.54/15.29 53.54/15.29 partcC_in_4: (f,f,f,f) (b,b,f,f) 53.54/15.29 53.54/15.29 qsF_in_2: (f,f) (b,f) 53.54/15.29 53.54/15.29 qscF_in_2: (f,f) (b,f) 53.54/15.29 53.54/15.29 qcD_in_7: (f,f,f,f,f,f,f) (b,b,f,f,f,f,f) 53.54/15.29 53.54/15.29 appcE_in_4: (f,f,f,f) (b,b,b,f) 53.54/15.29 53.54/15.29 appE_in_4: (f,f,f,f) (b,b,b,f) 53.54/15.29 53.54/15.29 qscG_in_3: (f,f,f) 53.54/15.29 53.54/15.29 appH_in_4: (f,b,f,b) 53.54/15.29 53.54/15.29 appJ_in_4: (f,f,f,b) (b,f,b,b) 53.54/15.29 53.54/15.29 appI_in_4: (f,f,f,b) (b,f,b,b) 53.54/15.29 53.54/15.29 Transforming TRIPLES into the following Term Rewriting System: 53.54/15.29 53.54/15.29 Pi DP problem: 53.54/15.29 The TRS P consists of the following rules: 53.54/15.29 53.54/15.29 QSK_IN_AG(.(X1, .(X2, X3)), []) -> U23_AG(X1, X2, X3, lessA_in_aa(X1, X2)) 53.54/15.29 QSK_IN_AG(.(X1, .(X2, X3)), []) -> LESSA_IN_AA(X1, X2) 53.54/15.29 LESSA_IN_AA(s(X1), s(X2)) -> U1_AA(X1, X2, lessA_in_aa(X1, X2)) 53.54/15.29 LESSA_IN_AA(s(X1), s(X2)) -> LESSA_IN_AA(X1, X2) 53.54/15.29 QSK_IN_AG(.(X1, .(X2, X3)), []) -> U24_AG(X1, X2, X3, lesscA_in_aa(X1, X2)) 53.54/15.29 U24_AG(X1, X2, X3, lesscA_out_aa(X1, X2)) -> U25_AG(X1, X2, X3, partB_in_gaaa(X1, X3, X4, X5)) 53.54/15.29 U24_AG(X1, X2, X3, lesscA_out_aa(X1, X2)) -> PARTB_IN_GAAA(X1, X3, X4, X5) 53.54/15.29 PARTB_IN_GAAA(X1, .(X2, X3), .(X2, X4), X5) -> U2_GAAA(X1, X2, X3, X4, X5, lessA_in_ga(X1, X2)) 53.54/15.29 PARTB_IN_GAAA(X1, .(X2, X3), .(X2, X4), X5) -> LESSA_IN_GA(X1, X2) 53.54/15.29 LESSA_IN_GA(s(X1), s(X2)) -> U1_GA(X1, X2, lessA_in_ga(X1, X2)) 53.54/15.29 LESSA_IN_GA(s(X1), s(X2)) -> LESSA_IN_GA(X1, X2) 53.54/15.29 PARTB_IN_GAAA(X1, .(X2, X3), .(X2, X4), X5) -> U3_GAAA(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) 53.54/15.29 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)) 53.54/15.29 U3_GAAA(X1, X2, X3, X4, X5, lesscA_out_ga(X1, X2)) -> PARTB_IN_GAAA(X1, X3, X4, X5) 53.54/15.29 PARTB_IN_GAAA(X1, .(X2, X3), X4, .(X2, X5)) -> U5_GAAA(X1, X2, X3, X4, X5, partB_in_gaaa(X1, X3, X4, X5)) 53.54/15.29 PARTB_IN_GAAA(X1, .(X2, X3), X4, .(X2, X5)) -> PARTB_IN_GAAA(X1, X3, X4, X5) 53.54/15.29 U24_AG(X1, X2, X3, lesscA_out_aa(X1, X2)) -> U26_AG(X1, X2, X3, partcB_in_gaaa(X1, X3, X4, X5)) 53.54/15.29 U26_AG(X1, X2, X3, partcB_out_gaaa(X1, X3, X4, X5)) -> U27_AG(X1, X2, X3, qsG_in_aaa(X2, X4, X6)) 53.54/15.29 U26_AG(X1, X2, X3, partcB_out_gaaa(X1, X3, X4, X5)) -> QSG_IN_AAA(X2, X4, X6) 53.54/15.29 QSG_IN_AAA(X1, X2, X3) -> U19_AAA(X1, X2, X3, pD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.54/15.29 QSG_IN_AAA(X1, X2, X3) -> PD_IN_AAAAAAA(X1, X2, X4, X5, X6, X7, X3) 53.54/15.29 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)) 53.54/15.29 PD_IN_AAAAAAA(X1, X2, X3, X4, X5, X6, X7) -> PARTC_IN_AAAA(X1, X2, X3, X4) 53.54/15.29 PARTC_IN_AAAA(X1, .(X2, X3), .(X2, X4), X5) -> U6_AAAA(X1, X2, X3, X4, X5, lessA_in_aa(X1, X2)) 53.54/15.29 PARTC_IN_AAAA(X1, .(X2, X3), .(X2, X4), X5) -> LESSA_IN_AA(X1, X2) 53.54/15.29 PARTC_IN_AAAA(X1, .(X2, X3), .(X2, X4), X5) -> U7_AAAA(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) 53.54/15.29 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)) 53.54/15.29 U7_AAAA(X1, X2, X3, X4, X5, lesscA_out_aa(X1, X2)) -> PARTB_IN_GAAA(X1, X3, X4, X5) 53.54/15.29 PARTC_IN_AAAA(X1, .(X2, X3), X4, .(X2, X5)) -> U9_AAAA(X1, X2, X3, X4, X5, partC_in_aaaa(X1, X3, X4, X5)) 53.54/15.29 PARTC_IN_AAAA(X1, .(X2, X3), X4, .(X2, X5)) -> PARTC_IN_AAAA(X1, X3, X4, X5) 53.54/15.29 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)) 53.54/15.29 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)) 53.54/15.29 U11_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, partcC_out_aaaa(X1, X2, X3, X4)) -> QSF_IN_AA(X3, X5) 53.54/15.29 QSF_IN_AA(.(X1, X2), X3) -> U17_AA(X1, X2, X3, pD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.54/15.29 QSF_IN_AA(.(X1, X2), X3) -> PD_IN_AAAAAAA(X1, X2, X4, X5, X6, X7, X3) 53.54/15.29 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)) 53.54/15.29 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)) 53.54/15.29 U13_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X3, X5)) -> QSF_IN_AA(X4, X6) 53.54/15.29 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)) 53.54/15.29 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)) 53.54/15.29 U15_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X4, X6)) -> APPE_IN_AAAA(X5, X1, X6, X7) 53.54/15.29 APPE_IN_AAAA(.(X1, X2), X3, X4, .(X1, X5)) -> U18_AAAA(X1, X2, X3, X4, X5, appE_in_aaaa(X2, X3, X4, X5)) 53.54/15.29 APPE_IN_AAAA(.(X1, X2), X3, X4, .(X1, X5)) -> APPE_IN_AAAA(X2, X3, X4, X5) 53.54/15.29 U26_AG(X1, X2, X3, partcB_out_gaaa(X1, X3, X4, X5)) -> U28_AG(X1, X2, X3, X5, qscG_in_aaa(X2, X4, X6)) 53.54/15.29 U28_AG(X1, X2, X3, X5, qscG_out_aaa(X2, X4, X6)) -> U29_AG(X1, X2, X3, qsF_in_aa(X5, X7)) 53.54/15.29 U28_AG(X1, X2, X3, X5, qscG_out_aaa(X2, X4, X6)) -> QSF_IN_AA(X5, X7) 53.54/15.29 QSK_IN_AG(.(X1, .(X2, X3)), []) -> U30_AG(X1, X2, X3, partC_in_aaaa(X1, X3, X4, X5)) 53.54/15.29 QSK_IN_AG(.(X1, .(X2, X3)), []) -> PARTC_IN_AAAA(X1, X3, X4, X5) 53.54/15.29 QSK_IN_AG(.(X1, .(X2, X3)), []) -> U31_AG(X1, X2, X3, partcC_in_aaaa(X1, X3, X4, X5)) 53.54/15.29 U31_AG(X1, X2, X3, partcC_out_aaaa(X1, X3, X4, X5)) -> U32_AG(X1, X2, X3, qsF_in_aa(X4, X6)) 53.54/15.29 U31_AG(X1, X2, X3, partcC_out_aaaa(X1, X3, X4, X5)) -> QSF_IN_AA(X4, X6) 53.54/15.29 U31_AG(X1, X2, X3, partcC_out_aaaa(X1, X3, X4, X5)) -> U33_AG(X1, X2, X3, X5, qscF_in_aa(X4, X6)) 53.54/15.29 U33_AG(X1, X2, X3, X5, qscF_out_aa(X4, X6)) -> U34_AG(X1, X2, X3, qsG_in_aaa(X2, X5, X7)) 53.54/15.29 U33_AG(X1, X2, X3, X5, qscF_out_aa(X4, X6)) -> QSG_IN_AAA(X2, X5, X7) 53.54/15.29 QSK_IN_AG(.(X1, []), []) -> U35_AG(X1, qsF_in_ga([], X2)) 53.54/15.29 QSK_IN_AG(.(X1, []), []) -> QSF_IN_GA([], X2) 53.54/15.29 QSF_IN_GA(.(X1, X2), X3) -> U17_GA(X1, X2, X3, pD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.54/15.29 QSF_IN_GA(.(X1, X2), X3) -> PD_IN_GGAAAAA(X1, X2, X4, X5, X6, X7, X3) 53.54/15.29 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)) 53.54/15.29 PD_IN_GGAAAAA(X1, X2, X3, X4, X5, X6, X7) -> PARTC_IN_GGAA(X1, X2, X3, X4) 53.54/15.29 PARTC_IN_GGAA(X1, .(X2, X3), .(X2, X4), X5) -> U6_GGAA(X1, X2, X3, X4, X5, lessA_in_gg(X1, X2)) 53.54/15.29 PARTC_IN_GGAA(X1, .(X2, X3), .(X2, X4), X5) -> LESSA_IN_GG(X1, X2) 53.54/15.29 LESSA_IN_GG(s(X1), s(X2)) -> U1_GG(X1, X2, lessA_in_gg(X1, X2)) 53.54/15.29 LESSA_IN_GG(s(X1), s(X2)) -> LESSA_IN_GG(X1, X2) 53.54/15.29 PARTC_IN_GGAA(X1, .(X2, X3), .(X2, X4), X5) -> U7_GGAA(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.54/15.29 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)) 53.54/15.29 U7_GGAA(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> PARTB_IN_GGAA(X1, X3, X4, X5) 53.54/15.29 PARTB_IN_GGAA(X1, .(X2, X3), .(X2, X4), X5) -> U2_GGAA(X1, X2, X3, X4, X5, lessA_in_gg(X1, X2)) 53.54/15.29 PARTB_IN_GGAA(X1, .(X2, X3), .(X2, X4), X5) -> LESSA_IN_GG(X1, X2) 53.54/15.29 PARTB_IN_GGAA(X1, .(X2, X3), .(X2, X4), X5) -> U3_GGAA(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.54/15.29 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)) 53.54/15.29 U3_GGAA(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> PARTB_IN_GGAA(X1, X3, X4, X5) 53.54/15.29 PARTB_IN_GGAA(X1, .(X2, X3), X4, .(X2, X5)) -> U5_GGAA(X1, X2, X3, X4, X5, partB_in_ggaa(X1, X3, X4, X5)) 53.54/15.29 PARTB_IN_GGAA(X1, .(X2, X3), X4, .(X2, X5)) -> PARTB_IN_GGAA(X1, X3, X4, X5) 53.54/15.29 PARTC_IN_GGAA(X1, .(X2, X3), X4, .(X2, X5)) -> U9_GGAA(X1, X2, X3, X4, X5, partC_in_ggaa(X1, X3, X4, X5)) 53.54/15.29 PARTC_IN_GGAA(X1, .(X2, X3), X4, .(X2, X5)) -> PARTC_IN_GGAA(X1, X3, X4, X5) 53.54/15.29 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)) 53.54/15.29 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)) 53.54/15.29 U11_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, partcC_out_ggaa(X1, X2, X3, X4)) -> QSF_IN_GA(X3, X5) 53.54/15.29 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)) 53.54/15.29 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)) 53.54/15.29 U13_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> QSF_IN_GA(X4, X6) 53.54/15.29 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)) 53.54/15.29 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)) 53.54/15.29 U15_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> APPE_IN_GGGA(X5, X1, X6, X7) 53.54/15.29 APPE_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) -> U18_GGGA(X1, X2, X3, X4, X5, appE_in_ggga(X2, X3, X4, X5)) 53.54/15.29 APPE_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) -> APPE_IN_GGGA(X2, X3, X4, X5) 53.54/15.29 QSK_IN_AG(.(X1, []), []) -> U36_AG(X1, qscF_in_ga([], X2)) 53.54/15.29 U36_AG(X1, qscF_out_ga([], X2)) -> U37_AG(X1, qsF_in_ga([], X3)) 53.54/15.29 U36_AG(X1, qscF_out_ga([], X2)) -> QSF_IN_GA([], X3) 53.54/15.29 QSK_IN_AG(.(X1, .(X2, X3)), X4) -> U38_AG(X1, X2, X3, X4, lessA_in_aa(X1, X2)) 53.54/15.29 QSK_IN_AG(.(X1, .(X2, X3)), X4) -> LESSA_IN_AA(X1, X2) 53.54/15.29 QSK_IN_AG(.(X1, .(X2, X3)), X4) -> U39_AG(X1, X2, X3, X4, lesscA_in_aa(X1, X2)) 53.54/15.29 U39_AG(X1, X2, X3, X4, lesscA_out_aa(X1, X2)) -> U40_AG(X1, X2, X3, X4, partB_in_gaaa(X1, X3, X5, X6)) 53.54/15.29 U39_AG(X1, X2, X3, X4, lesscA_out_aa(X1, X2)) -> PARTB_IN_GAAA(X1, X3, X5, X6) 53.54/15.29 U39_AG(X1, X2, X3, X4, lesscA_out_aa(X1, X2)) -> U41_AG(X1, X2, X3, X4, partcB_in_gaaa(X1, X3, X5, X6)) 53.54/15.29 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)) 53.54/15.29 U41_AG(X1, X2, X3, X4, partcB_out_gaaa(X1, X3, X5, X6)) -> QSG_IN_AAA(X2, X5, X7) 53.54/15.29 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)) 53.54/15.29 U43_AG(X1, X2, X3, X4, X6, qscG_out_aaa(X2, X5, X7)) -> U44_AG(X1, X2, X3, X4, qsF_in_aa(X6, X8)) 53.54/15.29 U43_AG(X1, X2, X3, X4, X6, qscG_out_aaa(X2, X5, X7)) -> QSF_IN_AA(X6, X8) 53.54/15.29 QSK_IN_AG(.(X1, .(X2, X3)), .(X4, X5)) -> U45_AG(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) 53.54/15.29 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)) 53.54/15.29 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))) 53.54/15.29 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)) 53.54/15.29 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)) 53.54/15.29 U48_AG(X1, X2, X3, X4, X5, X8, qscF_out_aa(X7, X9)) -> APPH_IN_AGAG(X8, X1, X9, X5) 53.54/15.29 APPH_IN_AGAG(.(X1, X2), X3, X4, .(X1, X5)) -> U20_AGAG(X1, X2, X3, X4, X5, appH_in_agag(X2, X3, X4, X5)) 53.54/15.29 APPH_IN_AGAG(.(X1, X2), X3, X4, .(X1, X5)) -> APPH_IN_AGAG(X2, X3, X4, X5) 53.54/15.29 QSK_IN_AG(.(X1, .(X2, X3)), X4) -> U50_AG(X1, X2, X3, X4, partC_in_aaaa(X1, X3, X5, X6)) 53.54/15.29 QSK_IN_AG(.(X1, .(X2, X3)), X4) -> PARTC_IN_AAAA(X1, X3, X5, X6) 53.54/15.29 QSK_IN_AG(.(X1, .(X2, X3)), X4) -> U51_AG(X1, X2, X3, X4, partcC_in_aaaa(X1, X3, X5, X6)) 53.54/15.29 U51_AG(X1, X2, X3, X4, partcC_out_aaaa(X1, X3, X5, X6)) -> U52_AG(X1, X2, X3, X4, qsF_in_aa(X5, X7)) 53.54/15.29 U51_AG(X1, X2, X3, X4, partcC_out_aaaa(X1, X3, X5, X6)) -> QSF_IN_AA(X5, X7) 53.54/15.29 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)) 53.54/15.29 U53_AG(X1, X2, X3, X4, X6, qscF_out_aa(X5, X7)) -> U54_AG(X1, X2, X3, X4, qsG_in_aaa(X2, X6, X8)) 53.54/15.29 U53_AG(X1, X2, X3, X4, X6, qscF_out_aa(X5, X7)) -> QSG_IN_AAA(X2, X6, X8) 53.54/15.29 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)) 53.54/15.29 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)) 53.54/15.29 U55_AG(X1, X2, X3, X4, X7, qscG_out_aaa(X2, X6, X8)) -> APPJ_IN_AAAG(X7, X1, X8, X4) 53.54/15.29 APPJ_IN_AAAG(.(X1, X2), X3, X4, .(X1, X5)) -> U22_AAAG(X1, X2, X3, X4, X5, appI_in_aaag(X2, X3, X4, X5)) 53.54/15.29 APPJ_IN_AAAG(.(X1, X2), X3, X4, .(X1, X5)) -> APPI_IN_AAAG(X2, X3, X4, X5) 53.54/15.29 APPI_IN_AAAG(.(X1, X2), X3, X4, .(X1, X5)) -> U21_AAAG(X1, X2, X3, X4, X5, appI_in_aaag(X2, X3, X4, X5)) 53.54/15.29 APPI_IN_AAAG(.(X1, X2), X3, X4, .(X1, X5)) -> APPI_IN_AAAG(X2, X3, X4, X5) 53.54/15.29 QSK_IN_AG(.(X1, []), X2) -> U57_AG(X1, X2, qsF_in_ga([], X3)) 53.54/15.29 QSK_IN_AG(.(X1, []), X2) -> QSF_IN_GA([], X3) 53.54/15.29 QSK_IN_AG(.(X1, []), X2) -> U58_AG(X1, X2, qscF_in_ga([], X3)) 53.54/15.29 U58_AG(X1, X2, qscF_out_ga([], X3)) -> U59_AG(X1, X2, qsF_in_ga([], X4)) 53.54/15.29 U58_AG(X1, X2, qscF_out_ga([], X3)) -> QSF_IN_GA([], X4) 53.54/15.29 U58_AG(X1, X2, qscF_out_ga([], X3)) -> U60_AG(X1, X2, X3, qscF_in_ga([], X4)) 53.54/15.29 U60_AG(X1, X2, X3, qscF_out_ga([], X4)) -> U61_AG(X1, X2, appJ_in_gagg(X3, X1, X4, X2)) 53.54/15.29 U60_AG(X1, X2, X3, qscF_out_ga([], X4)) -> APPJ_IN_GAGG(X3, X1, X4, X2) 53.54/15.29 APPJ_IN_GAGG(.(X1, X2), X3, X4, .(X1, X5)) -> U22_GAGG(X1, X2, X3, X4, X5, appI_in_gagg(X2, X3, X4, X5)) 53.54/15.29 APPJ_IN_GAGG(.(X1, X2), X3, X4, .(X1, X5)) -> APPI_IN_GAGG(X2, X3, X4, X5) 53.54/15.29 APPI_IN_GAGG(.(X1, X2), X3, X4, .(X1, X5)) -> U21_GAGG(X1, X2, X3, X4, X5, appI_in_gagg(X2, X3, X4, X5)) 53.54/15.29 APPI_IN_GAGG(.(X1, X2), X3, X4, .(X1, X5)) -> APPI_IN_GAGG(X2, X3, X4, X5) 53.54/15.29 53.54/15.29 The TRS R consists of the following rules: 53.54/15.29 53.54/15.29 lesscA_in_aa(0, s(X1)) -> lesscA_out_aa(0, s(X1)) 53.54/15.29 lesscA_in_aa(s(X1), s(X2)) -> U63_aa(X1, X2, lesscA_in_aa(X1, X2)) 53.54/15.29 U63_aa(X1, X2, lesscA_out_aa(X1, X2)) -> lesscA_out_aa(s(X1), s(X2)) 53.54/15.29 lesscA_in_ga(0, s(X1)) -> lesscA_out_ga(0, s(X1)) 53.54/15.29 lesscA_in_ga(s(X1), s(X2)) -> U63_ga(X1, X2, lesscA_in_ga(X1, X2)) 53.54/15.29 U63_ga(X1, X2, lesscA_out_ga(X1, X2)) -> lesscA_out_ga(s(X1), s(X2)) 53.54/15.29 partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_gaaa(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) 53.54/15.29 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)) 53.54/15.29 partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) 53.54/15.29 partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], []) 53.54/15.29 U66_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5)) 53.54/15.29 U65_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5) 53.54/15.29 partcC_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_aaaa(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) 53.54/15.29 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)) 53.54/15.29 U68_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), .(X2, X4), X5) 53.54/15.29 partcC_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_aaaa(X1, X2, X3, X4, X5, partcC_in_aaaa(X1, X3, X4, X5)) 53.54/15.29 partcC_in_aaaa(X1, [], [], []) -> partcC_out_aaaa(X1, [], [], []) 53.54/15.29 U69_aaaa(X1, X2, X3, X4, X5, partcC_out_aaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), X4, .(X2, X5)) 53.54/15.29 qscF_in_aa([], []) -> qscF_out_aa([], []) 53.54/15.29 qscF_in_aa(.(X1, X2), X3) -> U74_aa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.54/15.29 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)) 53.54/15.29 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)) 53.54/15.29 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)) 53.54/15.29 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)) 53.54/15.29 appcE_in_aaaa([], X1, X2, .(X1, X2)) -> appcE_out_aaaa([], X1, X2, .(X1, X2)) 53.54/15.29 appcE_in_aaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U75_aaaa(X1, X2, X3, X4, X5, appcE_in_aaaa(X2, X3, X4, X5)) 53.54/15.29 U75_aaaa(X1, X2, X3, X4, X5, appcE_out_aaaa(X2, X3, X4, X5)) -> appcE_out_aaaa(.(X1, X2), X3, X4, .(X1, X5)) 53.54/15.29 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) 53.54/15.29 U74_aa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_aa(.(X1, X2), X3) 53.54/15.29 qscG_in_aaa(X1, X2, X3) -> U76_aaa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.54/15.29 U76_aaa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscG_out_aaa(X1, X2, X3) 53.54/15.29 lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) 53.54/15.29 lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) 53.54/15.29 U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) 53.54/15.29 partcC_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.54/15.29 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)) 53.54/15.29 partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.54/15.29 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)) 53.54/15.29 partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) 53.54/15.29 partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], []) 53.54/15.29 U66_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.54/15.29 U65_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.54/15.29 U68_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.54/15.29 partcC_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_ggaa(X1, X2, X3, X4, X5, partcC_in_ggaa(X1, X3, X4, X5)) 53.54/15.29 partcC_in_ggaa(X1, [], [], []) -> partcC_out_ggaa(X1, [], [], []) 53.54/15.29 U69_ggaa(X1, X2, X3, X4, X5, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.54/15.29 qscF_in_ga([], []) -> qscF_out_ga([], []) 53.54/15.29 qscF_in_ga(.(X1, X2), X3) -> U74_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.54/15.29 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)) 53.54/15.29 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)) 53.54/15.29 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)) 53.54/15.29 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)) 53.54/15.29 appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2)) 53.54/15.29 appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U75_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5)) 53.54/15.29 U75_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5)) 53.54/15.29 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) 53.54/15.29 U74_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3) 53.54/15.29 53.54/15.29 The argument filtering Pi contains the following mapping: 53.54/15.29 [] = [] 53.54/15.29 53.54/15.29 lessA_in_aa(x1, x2) = lessA_in_aa 53.54/15.29 53.54/15.29 lesscA_in_aa(x1, x2) = lesscA_in_aa 53.54/15.29 53.54/15.29 lesscA_out_aa(x1, x2) = lesscA_out_aa(x1) 53.54/15.29 53.54/15.29 U63_aa(x1, x2, x3) = U63_aa(x3) 53.54/15.29 53.54/15.29 partB_in_gaaa(x1, x2, x3, x4) = partB_in_gaaa(x1) 53.54/15.29 53.54/15.29 lessA_in_ga(x1, x2) = lessA_in_ga(x1) 53.54/15.29 53.54/15.29 s(x1) = s(x1) 53.54/15.29 53.54/15.29 lesscA_in_ga(x1, x2) = lesscA_in_ga(x1) 53.54/15.29 53.54/15.29 0 = 0 53.54/15.29 53.54/15.29 lesscA_out_ga(x1, x2) = lesscA_out_ga(x1) 53.54/15.29 53.54/15.29 U63_ga(x1, x2, x3) = U63_ga(x1, x3) 53.54/15.29 53.54/15.29 partcB_in_gaaa(x1, x2, x3, x4) = partcB_in_gaaa(x1) 53.54/15.29 53.54/15.29 U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6) 53.54/15.29 53.54/15.29 U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x6) 53.54/15.29 53.54/15.29 U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6) 53.54/15.29 53.54/15.29 partcB_out_gaaa(x1, x2, x3, x4) = partcB_out_gaaa(x1) 53.54/15.29 53.54/15.29 qsG_in_aaa(x1, x2, x3) = qsG_in_aaa 53.54/15.29 53.54/15.29 pD_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = pD_in_aaaaaaa 53.54/15.29 53.54/15.29 partC_in_aaaa(x1, x2, x3, x4) = partC_in_aaaa 53.54/15.29 53.54/15.29 partcC_in_aaaa(x1, x2, x3, x4) = partcC_in_aaaa 53.54/15.29 53.54/15.29 U67_aaaa(x1, x2, x3, x4, x5, x6) = U67_aaaa(x6) 53.54/15.29 53.54/15.29 U68_aaaa(x1, x2, x3, x4, x5, x6) = U68_aaaa(x6) 53.54/15.29 53.54/15.29 partcC_out_aaaa(x1, x2, x3, x4) = partcC_out_aaaa 53.54/15.29 53.54/15.29 U69_aaaa(x1, x2, x3, x4, x5, x6) = U69_aaaa(x6) 53.54/15.29 53.54/15.29 qsF_in_aa(x1, x2) = qsF_in_aa 53.54/15.29 53.54/15.29 qscF_in_aa(x1, x2) = qscF_in_aa 53.54/15.29 53.54/15.29 qscF_out_aa(x1, x2) = qscF_out_aa 53.54/15.29 53.54/15.29 U74_aa(x1, x2, x3, x4) = U74_aa(x4) 53.54/15.29 53.54/15.29 qcD_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_aaaaaaa 53.54/15.29 53.54/15.29 U70_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_aaaaaaa(x8) 53.54/15.29 53.54/15.29 U71_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_aaaaaaa(x8) 53.54/15.29 53.54/15.29 U72_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_aaaaaaa(x8) 53.54/15.29 53.54/15.29 U73_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_aaaaaaa(x8) 53.54/15.29 53.54/15.29 .(x1, x2) = .(x1, x2) 53.54/15.29 53.54/15.29 qcD_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_aaaaaaa 53.54/15.29 53.54/15.29 appcE_in_aaaa(x1, x2, x3, x4) = appcE_in_aaaa 53.54/15.29 53.54/15.29 appcE_out_aaaa(x1, x2, x3, x4) = appcE_out_aaaa 53.54/15.29 53.54/15.29 U75_aaaa(x1, x2, x3, x4, x5, x6) = U75_aaaa(x6) 53.54/15.29 53.54/15.29 appE_in_aaaa(x1, x2, x3, x4) = appE_in_aaaa 53.54/15.29 53.54/15.29 qscG_in_aaa(x1, x2, x3) = qscG_in_aaa 53.54/15.29 53.54/15.29 U76_aaa(x1, x2, x3, x4) = U76_aaa(x4) 53.54/15.29 53.54/15.29 qscG_out_aaa(x1, x2, x3) = qscG_out_aaa 53.54/15.29 53.54/15.29 qsF_in_ga(x1, x2) = qsF_in_ga(x1) 53.54/15.29 53.54/15.29 pD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = pD_in_ggaaaaa(x1, x2) 53.54/15.29 53.54/15.29 partC_in_ggaa(x1, x2, x3, x4) = partC_in_ggaa(x1, x2) 53.54/15.29 53.54/15.29 lessA_in_gg(x1, x2) = lessA_in_gg(x1, x2) 53.54/15.29 53.54/15.29 lesscA_in_gg(x1, x2) = lesscA_in_gg(x1, x2) 53.54/15.29 53.54/15.29 lesscA_out_gg(x1, x2) = lesscA_out_gg(x1, x2) 53.54/15.29 53.54/15.29 U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3) 53.54/15.29 53.54/15.29 partB_in_ggaa(x1, x2, x3, x4) = partB_in_ggaa(x1, x2) 53.54/15.29 53.54/15.29 partcC_in_ggaa(x1, x2, x3, x4) = partcC_in_ggaa(x1, x2) 53.54/15.29 53.54/15.29 U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6) 53.54/15.29 53.54/15.29 U68_ggaa(x1, x2, x3, x4, x5, x6) = U68_ggaa(x1, x2, x3, x6) 53.54/15.29 53.54/15.29 partcB_in_ggaa(x1, x2, x3, x4) = partcB_in_ggaa(x1, x2) 53.54/15.29 53.54/15.29 U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6) 53.54/15.29 53.54/15.29 U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6) 53.54/15.29 53.54/15.29 U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6) 53.54/15.29 53.54/15.29 partcB_out_ggaa(x1, x2, x3, x4) = partcB_out_ggaa(x1, x2, x3, x4) 53.54/15.29 53.54/15.29 partcC_out_ggaa(x1, x2, x3, x4) = partcC_out_ggaa(x1, x2, x3, x4) 53.54/15.29 53.54/15.29 U69_ggaa(x1, x2, x3, x4, x5, x6) = U69_ggaa(x1, x2, x3, x6) 53.54/15.29 53.54/15.29 qscF_in_ga(x1, x2) = qscF_in_ga(x1) 53.54/15.29 53.54/15.29 qscF_out_ga(x1, x2) = qscF_out_ga(x1, x2) 53.54/15.29 53.54/15.29 U74_ga(x1, x2, x3, x4) = U74_ga(x1, x2, x4) 53.54/15.29 53.54/15.29 qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_ggaaaaa(x1, x2) 53.54/15.29 53.54/15.29 U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x8) 53.54/15.29 53.54/15.29 U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x8) 53.54/15.29 53.54/15.29 U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x8) 53.54/15.29 53.54/15.29 U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x8) 53.54/15.29 53.54/15.29 appcE_in_ggga(x1, x2, x3, x4) = appcE_in_ggga(x1, x2, x3) 53.54/15.29 53.54/15.29 appcE_out_ggga(x1, x2, x3, x4) = appcE_out_ggga(x1, x2, x3, x4) 53.54/15.29 53.54/15.29 U75_ggga(x1, x2, x3, x4, x5, x6) = U75_ggga(x1, x2, x3, x4, x6) 53.54/15.29 53.54/15.29 qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) 53.54/15.29 53.54/15.29 appE_in_ggga(x1, x2, x3, x4) = appE_in_ggga(x1, x2, x3) 53.54/15.29 53.54/15.29 appH_in_agag(x1, x2, x3, x4) = appH_in_agag(x2, x4) 53.54/15.29 53.54/15.29 appJ_in_aaag(x1, x2, x3, x4) = appJ_in_aaag(x4) 53.54/15.29 53.54/15.29 appI_in_aaag(x1, x2, x3, x4) = appI_in_aaag(x4) 53.54/15.29 53.54/15.29 appJ_in_gagg(x1, x2, x3, x4) = appJ_in_gagg(x1, x3, x4) 53.54/15.29 53.54/15.29 appI_in_gagg(x1, x2, x3, x4) = appI_in_gagg(x1, x3, x4) 53.54/15.29 53.54/15.29 QSK_IN_AG(x1, x2) = QSK_IN_AG(x2) 53.54/15.29 53.54/15.29 U23_AG(x1, x2, x3, x4) = U23_AG(x4) 53.54/15.29 53.54/15.29 LESSA_IN_AA(x1, x2) = LESSA_IN_AA 53.54/15.29 53.54/15.29 U1_AA(x1, x2, x3) = U1_AA(x3) 53.54/15.29 53.54/15.29 U24_AG(x1, x2, x3, x4) = U24_AG(x4) 53.54/15.29 53.54/15.29 U25_AG(x1, x2, x3, x4) = U25_AG(x4) 53.54/15.29 53.54/15.29 PARTB_IN_GAAA(x1, x2, x3, x4) = PARTB_IN_GAAA(x1) 53.54/15.29 53.54/15.29 U2_GAAA(x1, x2, x3, x4, x5, x6) = U2_GAAA(x1, x6) 53.54/15.29 53.54/15.29 LESSA_IN_GA(x1, x2) = LESSA_IN_GA(x1) 53.54/15.29 53.54/15.29 U1_GA(x1, x2, x3) = U1_GA(x1, x3) 53.54/15.29 53.54/15.29 U3_GAAA(x1, x2, x3, x4, x5, x6) = U3_GAAA(x1, x6) 53.54/15.29 53.54/15.29 U4_GAAA(x1, x2, x3, x4, x5, x6) = U4_GAAA(x1, x6) 53.54/15.29 53.54/15.29 U5_GAAA(x1, x2, x3, x4, x5, x6) = U5_GAAA(x1, x6) 53.54/15.29 53.54/15.29 U26_AG(x1, x2, x3, x4) = U26_AG(x4) 53.54/15.29 53.54/15.29 U27_AG(x1, x2, x3, x4) = U27_AG(x4) 53.54/15.29 53.54/15.29 QSG_IN_AAA(x1, x2, x3) = QSG_IN_AAA 53.54/15.29 53.54/15.29 U19_AAA(x1, x2, x3, x4) = U19_AAA(x4) 53.54/15.29 53.54/15.29 PD_IN_AAAAAAA(x1, x2, x3, x4, x5, x6, x7) = PD_IN_AAAAAAA 53.54/15.29 53.54/15.29 U10_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U10_AAAAAAA(x8) 53.54/15.29 53.54/15.29 PARTC_IN_AAAA(x1, x2, x3, x4) = PARTC_IN_AAAA 53.54/15.29 53.54/15.29 U6_AAAA(x1, x2, x3, x4, x5, x6) = U6_AAAA(x6) 53.54/15.29 53.54/15.29 U7_AAAA(x1, x2, x3, x4, x5, x6) = U7_AAAA(x6) 53.54/15.29 53.54/15.29 U8_AAAA(x1, x2, x3, x4, x5, x6) = U8_AAAA(x1, x6) 53.54/15.29 53.54/15.29 U9_AAAA(x1, x2, x3, x4, x5, x6) = U9_AAAA(x6) 53.54/15.29 53.54/15.29 U11_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U11_AAAAAAA(x8) 53.54/15.29 53.54/15.29 U12_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U12_AAAAAAA(x8) 53.54/15.29 53.54/15.29 QSF_IN_AA(x1, x2) = QSF_IN_AA 53.54/15.29 53.54/15.29 U17_AA(x1, x2, x3, x4) = U17_AA(x4) 53.54/15.29 53.54/15.29 U13_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U13_AAAAAAA(x8) 53.54/15.29 53.54/15.29 U14_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U14_AAAAAAA(x8) 53.54/15.29 53.54/15.29 U15_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U15_AAAAAAA(x8) 53.54/15.29 53.54/15.29 U16_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U16_AAAAAAA(x8) 53.54/15.29 53.54/15.29 APPE_IN_AAAA(x1, x2, x3, x4) = APPE_IN_AAAA 53.54/15.29 53.54/15.29 U18_AAAA(x1, x2, x3, x4, x5, x6) = U18_AAAA(x6) 53.54/15.29 53.54/15.29 U28_AG(x1, x2, x3, x4, x5) = U28_AG(x5) 53.54/15.29 53.54/15.29 U29_AG(x1, x2, x3, x4) = U29_AG(x4) 53.54/15.29 53.54/15.29 U30_AG(x1, x2, x3, x4) = U30_AG(x4) 53.54/15.29 53.54/15.29 U31_AG(x1, x2, x3, x4) = U31_AG(x4) 53.54/15.29 53.54/15.29 U32_AG(x1, x2, x3, x4) = U32_AG(x4) 53.54/15.29 53.54/15.29 U33_AG(x1, x2, x3, x4, x5) = U33_AG(x5) 53.54/15.29 53.54/15.29 U34_AG(x1, x2, x3, x4) = U34_AG(x4) 53.54/15.29 53.54/15.29 U35_AG(x1, x2) = U35_AG(x2) 53.54/15.29 53.54/15.29 QSF_IN_GA(x1, x2) = QSF_IN_GA(x1) 53.54/15.29 53.54/15.29 U17_GA(x1, x2, x3, x4) = U17_GA(x1, x2, x4) 53.54/15.29 53.54/15.29 PD_IN_GGAAAAA(x1, x2, x3, x4, x5, x6, x7) = PD_IN_GGAAAAA(x1, x2) 53.54/15.29 53.54/15.29 U10_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U10_GGAAAAA(x1, x2, x8) 53.54/15.29 53.54/15.29 PARTC_IN_GGAA(x1, x2, x3, x4) = PARTC_IN_GGAA(x1, x2) 53.54/15.29 53.54/15.29 U6_GGAA(x1, x2, x3, x4, x5, x6) = U6_GGAA(x1, x2, x3, x6) 53.54/15.29 53.54/15.29 LESSA_IN_GG(x1, x2) = LESSA_IN_GG(x1, x2) 53.54/15.29 53.54/15.29 U1_GG(x1, x2, x3) = U1_GG(x1, x2, x3) 53.54/15.29 53.54/15.29 U7_GGAA(x1, x2, x3, x4, x5, x6) = U7_GGAA(x1, x2, x3, x6) 53.54/15.29 53.54/15.29 U8_GGAA(x1, x2, x3, x4, x5, x6) = U8_GGAA(x1, x2, x3, x6) 53.54/15.29 53.54/15.29 PARTB_IN_GGAA(x1, x2, x3, x4) = PARTB_IN_GGAA(x1, x2) 53.54/15.29 53.54/15.29 U2_GGAA(x1, x2, x3, x4, x5, x6) = U2_GGAA(x1, x2, x3, x6) 53.54/15.29 53.54/15.29 U3_GGAA(x1, x2, x3, x4, x5, x6) = U3_GGAA(x1, x2, x3, x6) 53.54/15.29 53.54/15.29 U4_GGAA(x1, x2, x3, x4, x5, x6) = U4_GGAA(x1, x2, x3, x6) 53.54/15.29 53.54/15.29 U5_GGAA(x1, x2, x3, x4, x5, x6) = U5_GGAA(x1, x2, x3, x6) 53.54/15.29 53.54/15.29 U9_GGAA(x1, x2, x3, x4, x5, x6) = U9_GGAA(x1, x2, x3, x6) 53.54/15.29 53.54/15.29 U11_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U11_GGAAAAA(x1, x2, x8) 53.54/15.29 53.54/15.29 U12_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U12_GGAAAAA(x1, x2, x8) 53.54/15.29 53.54/15.29 U13_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U13_GGAAAAA(x1, x2, x4, x8) 53.54/15.29 53.54/15.29 U14_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U14_GGAAAAA(x1, x2, x8) 53.54/15.29 53.54/15.29 U15_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U15_GGAAAAA(x1, x2, x5, x8) 53.54/15.29 53.54/15.29 U16_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U16_GGAAAAA(x1, x2, x8) 53.54/15.29 53.54/15.29 APPE_IN_GGGA(x1, x2, x3, x4) = APPE_IN_GGGA(x1, x2, x3) 53.54/15.29 53.54/15.29 U18_GGGA(x1, x2, x3, x4, x5, x6) = U18_GGGA(x1, x2, x3, x4, x6) 53.54/15.29 53.54/15.29 U36_AG(x1, x2) = U36_AG(x2) 53.54/15.29 53.54/15.29 U37_AG(x1, x2) = U37_AG(x2) 53.54/15.29 53.54/15.29 U38_AG(x1, x2, x3, x4, x5) = U38_AG(x4, x5) 53.54/15.29 53.54/15.29 U39_AG(x1, x2, x3, x4, x5) = U39_AG(x4, x5) 53.54/15.29 53.54/15.29 U40_AG(x1, x2, x3, x4, x5) = U40_AG(x4, x5) 53.54/15.29 53.54/15.29 U41_AG(x1, x2, x3, x4, x5) = U41_AG(x4, x5) 53.54/15.29 53.54/15.29 U42_AG(x1, x2, x3, x4, x5) = U42_AG(x4, x5) 53.54/15.29 53.54/15.29 U43_AG(x1, x2, x3, x4, x5, x6) = U43_AG(x4, x6) 53.54/15.29 53.54/15.29 U44_AG(x1, x2, x3, x4, x5) = U44_AG(x4, x5) 53.54/15.29 53.54/15.29 U45_AG(x1, x2, x3, x4, x5, x6) = U45_AG(x4, x5, x6) 53.54/15.29 53.54/15.29 U46_AG(x1, x2, x3, x4, x5, x6) = U46_AG(x1, x4, x5, x6) 53.54/15.29 53.54/15.29 U47_AG(x1, x2, x3, x4, x5, x6, x7) = U47_AG(x1, x4, x5, x7) 53.54/15.29 53.54/15.29 U48_AG(x1, x2, x3, x4, x5, x6, x7) = U48_AG(x1, x4, x5, x7) 53.54/15.29 53.54/15.29 U49_AG(x1, x2, x3, x4, x5, x6) = U49_AG(x4, x5, x6) 53.54/15.29 53.54/15.29 APPH_IN_AGAG(x1, x2, x3, x4) = APPH_IN_AGAG(x2, x4) 53.54/15.29 53.54/15.29 U20_AGAG(x1, x2, x3, x4, x5, x6) = U20_AGAG(x1, x3, x5, x6) 53.54/15.29 53.54/15.29 U50_AG(x1, x2, x3, x4, x5) = U50_AG(x4, x5) 53.54/15.29 53.54/15.29 U51_AG(x1, x2, x3, x4, x5) = U51_AG(x4, x5) 53.54/15.29 53.54/15.29 U52_AG(x1, x2, x3, x4, x5) = U52_AG(x4, x5) 53.54/15.29 53.54/15.29 U53_AG(x1, x2, x3, x4, x5, x6) = U53_AG(x4, x6) 53.54/15.29 53.54/15.29 U54_AG(x1, x2, x3, x4, x5) = U54_AG(x4, x5) 53.54/15.29 53.54/15.29 U55_AG(x1, x2, x3, x4, x5, x6) = U55_AG(x4, x6) 53.54/15.29 53.54/15.29 U56_AG(x1, x2, x3, x4, x5) = U56_AG(x4, x5) 53.54/15.29 53.54/15.29 APPJ_IN_AAAG(x1, x2, x3, x4) = APPJ_IN_AAAG(x4) 53.54/15.29 53.54/15.29 U22_AAAG(x1, x2, x3, x4, x5, x6) = U22_AAAG(x1, x5, x6) 53.54/15.29 53.54/15.29 APPI_IN_AAAG(x1, x2, x3, x4) = APPI_IN_AAAG(x4) 53.54/15.29 53.54/15.29 U21_AAAG(x1, x2, x3, x4, x5, x6) = U21_AAAG(x1, x5, x6) 53.54/15.29 53.54/15.29 U57_AG(x1, x2, x3) = U57_AG(x2, x3) 53.54/15.29 53.54/15.29 U58_AG(x1, x2, x3) = U58_AG(x2, x3) 53.54/15.29 53.54/15.29 U59_AG(x1, x2, x3) = U59_AG(x2, x3) 53.54/15.29 53.54/15.29 U60_AG(x1, x2, x3, x4) = U60_AG(x2, x3, x4) 53.54/15.29 53.54/15.29 U61_AG(x1, x2, x3) = U61_AG(x2, x3) 53.54/15.29 53.54/15.29 APPJ_IN_GAGG(x1, x2, x3, x4) = APPJ_IN_GAGG(x1, x3, x4) 53.54/15.29 53.54/15.29 U22_GAGG(x1, x2, x3, x4, x5, x6) = U22_GAGG(x1, x2, x4, x5, x6) 53.54/15.29 53.54/15.29 APPI_IN_GAGG(x1, x2, x3, x4) = APPI_IN_GAGG(x1, x3, x4) 53.54/15.29 53.54/15.29 U21_GAGG(x1, x2, x3, x4, x5, x6) = U21_GAGG(x1, x2, x4, x5, x6) 53.54/15.29 53.54/15.29 53.54/15.29 We have to consider all (P,R,Pi)-chains 53.54/15.29 53.54/15.29 53.54/15.29 Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES 53.54/15.29 53.54/15.29 53.54/15.29 53.54/15.29 ---------------------------------------- 53.54/15.29 53.54/15.29 (137) 53.54/15.29 Obligation: 53.54/15.29 Pi DP problem: 53.54/15.29 The TRS P consists of the following rules: 53.54/15.29 53.54/15.29 QSK_IN_AG(.(X1, .(X2, X3)), []) -> U23_AG(X1, X2, X3, lessA_in_aa(X1, X2)) 53.54/15.29 QSK_IN_AG(.(X1, .(X2, X3)), []) -> LESSA_IN_AA(X1, X2) 53.54/15.29 LESSA_IN_AA(s(X1), s(X2)) -> U1_AA(X1, X2, lessA_in_aa(X1, X2)) 53.54/15.29 LESSA_IN_AA(s(X1), s(X2)) -> LESSA_IN_AA(X1, X2) 53.54/15.29 QSK_IN_AG(.(X1, .(X2, X3)), []) -> U24_AG(X1, X2, X3, lesscA_in_aa(X1, X2)) 53.54/15.29 U24_AG(X1, X2, X3, lesscA_out_aa(X1, X2)) -> U25_AG(X1, X2, X3, partB_in_gaaa(X1, X3, X4, X5)) 53.54/15.29 U24_AG(X1, X2, X3, lesscA_out_aa(X1, X2)) -> PARTB_IN_GAAA(X1, X3, X4, X5) 53.54/15.29 PARTB_IN_GAAA(X1, .(X2, X3), .(X2, X4), X5) -> U2_GAAA(X1, X2, X3, X4, X5, lessA_in_ga(X1, X2)) 53.54/15.29 PARTB_IN_GAAA(X1, .(X2, X3), .(X2, X4), X5) -> LESSA_IN_GA(X1, X2) 53.54/15.29 LESSA_IN_GA(s(X1), s(X2)) -> U1_GA(X1, X2, lessA_in_ga(X1, X2)) 53.54/15.29 LESSA_IN_GA(s(X1), s(X2)) -> LESSA_IN_GA(X1, X2) 53.54/15.29 PARTB_IN_GAAA(X1, .(X2, X3), .(X2, X4), X5) -> U3_GAAA(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) 53.54/15.29 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)) 53.54/15.29 U3_GAAA(X1, X2, X3, X4, X5, lesscA_out_ga(X1, X2)) -> PARTB_IN_GAAA(X1, X3, X4, X5) 53.54/15.29 PARTB_IN_GAAA(X1, .(X2, X3), X4, .(X2, X5)) -> U5_GAAA(X1, X2, X3, X4, X5, partB_in_gaaa(X1, X3, X4, X5)) 53.54/15.29 PARTB_IN_GAAA(X1, .(X2, X3), X4, .(X2, X5)) -> PARTB_IN_GAAA(X1, X3, X4, X5) 53.54/15.29 U24_AG(X1, X2, X3, lesscA_out_aa(X1, X2)) -> U26_AG(X1, X2, X3, partcB_in_gaaa(X1, X3, X4, X5)) 53.54/15.29 U26_AG(X1, X2, X3, partcB_out_gaaa(X1, X3, X4, X5)) -> U27_AG(X1, X2, X3, qsG_in_aaa(X2, X4, X6)) 53.54/15.29 U26_AG(X1, X2, X3, partcB_out_gaaa(X1, X3, X4, X5)) -> QSG_IN_AAA(X2, X4, X6) 53.54/15.29 QSG_IN_AAA(X1, X2, X3) -> U19_AAA(X1, X2, X3, pD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.54/15.29 QSG_IN_AAA(X1, X2, X3) -> PD_IN_AAAAAAA(X1, X2, X4, X5, X6, X7, X3) 53.54/15.29 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)) 53.54/15.29 PD_IN_AAAAAAA(X1, X2, X3, X4, X5, X6, X7) -> PARTC_IN_AAAA(X1, X2, X3, X4) 53.54/15.29 PARTC_IN_AAAA(X1, .(X2, X3), .(X2, X4), X5) -> U6_AAAA(X1, X2, X3, X4, X5, lessA_in_aa(X1, X2)) 53.54/15.29 PARTC_IN_AAAA(X1, .(X2, X3), .(X2, X4), X5) -> LESSA_IN_AA(X1, X2) 53.54/15.29 PARTC_IN_AAAA(X1, .(X2, X3), .(X2, X4), X5) -> U7_AAAA(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) 53.54/15.29 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)) 53.54/15.29 U7_AAAA(X1, X2, X3, X4, X5, lesscA_out_aa(X1, X2)) -> PARTB_IN_GAAA(X1, X3, X4, X5) 53.54/15.29 PARTC_IN_AAAA(X1, .(X2, X3), X4, .(X2, X5)) -> U9_AAAA(X1, X2, X3, X4, X5, partC_in_aaaa(X1, X3, X4, X5)) 53.54/15.29 PARTC_IN_AAAA(X1, .(X2, X3), X4, .(X2, X5)) -> PARTC_IN_AAAA(X1, X3, X4, X5) 53.54/15.29 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)) 53.54/15.29 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)) 53.54/15.29 U11_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, partcC_out_aaaa(X1, X2, X3, X4)) -> QSF_IN_AA(X3, X5) 53.54/15.29 QSF_IN_AA(.(X1, X2), X3) -> U17_AA(X1, X2, X3, pD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.54/15.29 QSF_IN_AA(.(X1, X2), X3) -> PD_IN_AAAAAAA(X1, X2, X4, X5, X6, X7, X3) 53.54/15.29 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)) 53.54/15.29 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)) 53.54/15.29 U13_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X3, X5)) -> QSF_IN_AA(X4, X6) 53.54/15.29 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)) 53.54/15.29 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)) 53.54/15.29 U15_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X4, X6)) -> APPE_IN_AAAA(X5, X1, X6, X7) 53.54/15.29 APPE_IN_AAAA(.(X1, X2), X3, X4, .(X1, X5)) -> U18_AAAA(X1, X2, X3, X4, X5, appE_in_aaaa(X2, X3, X4, X5)) 53.54/15.29 APPE_IN_AAAA(.(X1, X2), X3, X4, .(X1, X5)) -> APPE_IN_AAAA(X2, X3, X4, X5) 53.54/15.29 U26_AG(X1, X2, X3, partcB_out_gaaa(X1, X3, X4, X5)) -> U28_AG(X1, X2, X3, X5, qscG_in_aaa(X2, X4, X6)) 53.54/15.29 U28_AG(X1, X2, X3, X5, qscG_out_aaa(X2, X4, X6)) -> U29_AG(X1, X2, X3, qsF_in_aa(X5, X7)) 53.54/15.29 U28_AG(X1, X2, X3, X5, qscG_out_aaa(X2, X4, X6)) -> QSF_IN_AA(X5, X7) 53.54/15.29 QSK_IN_AG(.(X1, .(X2, X3)), []) -> U30_AG(X1, X2, X3, partC_in_aaaa(X1, X3, X4, X5)) 53.54/15.29 QSK_IN_AG(.(X1, .(X2, X3)), []) -> PARTC_IN_AAAA(X1, X3, X4, X5) 53.54/15.29 QSK_IN_AG(.(X1, .(X2, X3)), []) -> U31_AG(X1, X2, X3, partcC_in_aaaa(X1, X3, X4, X5)) 53.54/15.29 U31_AG(X1, X2, X3, partcC_out_aaaa(X1, X3, X4, X5)) -> U32_AG(X1, X2, X3, qsF_in_aa(X4, X6)) 53.54/15.29 U31_AG(X1, X2, X3, partcC_out_aaaa(X1, X3, X4, X5)) -> QSF_IN_AA(X4, X6) 53.54/15.29 U31_AG(X1, X2, X3, partcC_out_aaaa(X1, X3, X4, X5)) -> U33_AG(X1, X2, X3, X5, qscF_in_aa(X4, X6)) 53.54/15.29 U33_AG(X1, X2, X3, X5, qscF_out_aa(X4, X6)) -> U34_AG(X1, X2, X3, qsG_in_aaa(X2, X5, X7)) 53.54/15.29 U33_AG(X1, X2, X3, X5, qscF_out_aa(X4, X6)) -> QSG_IN_AAA(X2, X5, X7) 53.54/15.29 QSK_IN_AG(.(X1, []), []) -> U35_AG(X1, qsF_in_ga([], X2)) 53.54/15.29 QSK_IN_AG(.(X1, []), []) -> QSF_IN_GA([], X2) 53.54/15.29 QSF_IN_GA(.(X1, X2), X3) -> U17_GA(X1, X2, X3, pD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.54/15.29 QSF_IN_GA(.(X1, X2), X3) -> PD_IN_GGAAAAA(X1, X2, X4, X5, X6, X7, X3) 53.54/15.29 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)) 53.54/15.29 PD_IN_GGAAAAA(X1, X2, X3, X4, X5, X6, X7) -> PARTC_IN_GGAA(X1, X2, X3, X4) 53.54/15.29 PARTC_IN_GGAA(X1, .(X2, X3), .(X2, X4), X5) -> U6_GGAA(X1, X2, X3, X4, X5, lessA_in_gg(X1, X2)) 53.54/15.29 PARTC_IN_GGAA(X1, .(X2, X3), .(X2, X4), X5) -> LESSA_IN_GG(X1, X2) 53.54/15.29 LESSA_IN_GG(s(X1), s(X2)) -> U1_GG(X1, X2, lessA_in_gg(X1, X2)) 53.54/15.29 LESSA_IN_GG(s(X1), s(X2)) -> LESSA_IN_GG(X1, X2) 53.54/15.29 PARTC_IN_GGAA(X1, .(X2, X3), .(X2, X4), X5) -> U7_GGAA(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.54/15.29 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)) 53.54/15.29 U7_GGAA(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> PARTB_IN_GGAA(X1, X3, X4, X5) 53.54/15.29 PARTB_IN_GGAA(X1, .(X2, X3), .(X2, X4), X5) -> U2_GGAA(X1, X2, X3, X4, X5, lessA_in_gg(X1, X2)) 53.54/15.29 PARTB_IN_GGAA(X1, .(X2, X3), .(X2, X4), X5) -> LESSA_IN_GG(X1, X2) 53.54/15.29 PARTB_IN_GGAA(X1, .(X2, X3), .(X2, X4), X5) -> U3_GGAA(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.54/15.29 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)) 53.54/15.29 U3_GGAA(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> PARTB_IN_GGAA(X1, X3, X4, X5) 53.54/15.29 PARTB_IN_GGAA(X1, .(X2, X3), X4, .(X2, X5)) -> U5_GGAA(X1, X2, X3, X4, X5, partB_in_ggaa(X1, X3, X4, X5)) 53.54/15.29 PARTB_IN_GGAA(X1, .(X2, X3), X4, .(X2, X5)) -> PARTB_IN_GGAA(X1, X3, X4, X5) 53.54/15.29 PARTC_IN_GGAA(X1, .(X2, X3), X4, .(X2, X5)) -> U9_GGAA(X1, X2, X3, X4, X5, partC_in_ggaa(X1, X3, X4, X5)) 53.54/15.29 PARTC_IN_GGAA(X1, .(X2, X3), X4, .(X2, X5)) -> PARTC_IN_GGAA(X1, X3, X4, X5) 53.54/15.29 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)) 53.54/15.29 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)) 53.54/15.29 U11_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, partcC_out_ggaa(X1, X2, X3, X4)) -> QSF_IN_GA(X3, X5) 53.54/15.29 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)) 53.54/15.29 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)) 53.54/15.29 U13_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> QSF_IN_GA(X4, X6) 53.54/15.29 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)) 53.54/15.29 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)) 53.54/15.29 U15_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> APPE_IN_GGGA(X5, X1, X6, X7) 53.54/15.29 APPE_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) -> U18_GGGA(X1, X2, X3, X4, X5, appE_in_ggga(X2, X3, X4, X5)) 53.54/15.29 APPE_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) -> APPE_IN_GGGA(X2, X3, X4, X5) 53.54/15.29 QSK_IN_AG(.(X1, []), []) -> U36_AG(X1, qscF_in_ga([], X2)) 53.54/15.29 U36_AG(X1, qscF_out_ga([], X2)) -> U37_AG(X1, qsF_in_ga([], X3)) 53.54/15.29 U36_AG(X1, qscF_out_ga([], X2)) -> QSF_IN_GA([], X3) 53.54/15.29 QSK_IN_AG(.(X1, .(X2, X3)), X4) -> U38_AG(X1, X2, X3, X4, lessA_in_aa(X1, X2)) 53.54/15.29 QSK_IN_AG(.(X1, .(X2, X3)), X4) -> LESSA_IN_AA(X1, X2) 53.54/15.29 QSK_IN_AG(.(X1, .(X2, X3)), X4) -> U39_AG(X1, X2, X3, X4, lesscA_in_aa(X1, X2)) 53.54/15.29 U39_AG(X1, X2, X3, X4, lesscA_out_aa(X1, X2)) -> U40_AG(X1, X2, X3, X4, partB_in_gaaa(X1, X3, X5, X6)) 53.54/15.29 U39_AG(X1, X2, X3, X4, lesscA_out_aa(X1, X2)) -> PARTB_IN_GAAA(X1, X3, X5, X6) 53.54/15.29 U39_AG(X1, X2, X3, X4, lesscA_out_aa(X1, X2)) -> U41_AG(X1, X2, X3, X4, partcB_in_gaaa(X1, X3, X5, X6)) 53.54/15.29 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)) 53.54/15.29 U41_AG(X1, X2, X3, X4, partcB_out_gaaa(X1, X3, X5, X6)) -> QSG_IN_AAA(X2, X5, X7) 53.54/15.29 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)) 53.54/15.29 U43_AG(X1, X2, X3, X4, X6, qscG_out_aaa(X2, X5, X7)) -> U44_AG(X1, X2, X3, X4, qsF_in_aa(X6, X8)) 53.54/15.29 U43_AG(X1, X2, X3, X4, X6, qscG_out_aaa(X2, X5, X7)) -> QSF_IN_AA(X6, X8) 53.54/15.29 QSK_IN_AG(.(X1, .(X2, X3)), .(X4, X5)) -> U45_AG(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) 53.54/15.29 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)) 53.54/15.29 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))) 53.54/15.29 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)) 53.54/15.29 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)) 53.54/15.29 U48_AG(X1, X2, X3, X4, X5, X8, qscF_out_aa(X7, X9)) -> APPH_IN_AGAG(X8, X1, X9, X5) 53.54/15.29 APPH_IN_AGAG(.(X1, X2), X3, X4, .(X1, X5)) -> U20_AGAG(X1, X2, X3, X4, X5, appH_in_agag(X2, X3, X4, X5)) 53.54/15.29 APPH_IN_AGAG(.(X1, X2), X3, X4, .(X1, X5)) -> APPH_IN_AGAG(X2, X3, X4, X5) 53.54/15.29 QSK_IN_AG(.(X1, .(X2, X3)), X4) -> U50_AG(X1, X2, X3, X4, partC_in_aaaa(X1, X3, X5, X6)) 53.54/15.29 QSK_IN_AG(.(X1, .(X2, X3)), X4) -> PARTC_IN_AAAA(X1, X3, X5, X6) 53.54/15.29 QSK_IN_AG(.(X1, .(X2, X3)), X4) -> U51_AG(X1, X2, X3, X4, partcC_in_aaaa(X1, X3, X5, X6)) 53.54/15.29 U51_AG(X1, X2, X3, X4, partcC_out_aaaa(X1, X3, X5, X6)) -> U52_AG(X1, X2, X3, X4, qsF_in_aa(X5, X7)) 53.54/15.29 U51_AG(X1, X2, X3, X4, partcC_out_aaaa(X1, X3, X5, X6)) -> QSF_IN_AA(X5, X7) 53.54/15.29 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)) 53.54/15.29 U53_AG(X1, X2, X3, X4, X6, qscF_out_aa(X5, X7)) -> U54_AG(X1, X2, X3, X4, qsG_in_aaa(X2, X6, X8)) 53.54/15.29 U53_AG(X1, X2, X3, X4, X6, qscF_out_aa(X5, X7)) -> QSG_IN_AAA(X2, X6, X8) 53.54/15.29 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)) 53.54/15.29 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)) 53.54/15.29 U55_AG(X1, X2, X3, X4, X7, qscG_out_aaa(X2, X6, X8)) -> APPJ_IN_AAAG(X7, X1, X8, X4) 53.54/15.29 APPJ_IN_AAAG(.(X1, X2), X3, X4, .(X1, X5)) -> U22_AAAG(X1, X2, X3, X4, X5, appI_in_aaag(X2, X3, X4, X5)) 53.54/15.29 APPJ_IN_AAAG(.(X1, X2), X3, X4, .(X1, X5)) -> APPI_IN_AAAG(X2, X3, X4, X5) 53.54/15.29 APPI_IN_AAAG(.(X1, X2), X3, X4, .(X1, X5)) -> U21_AAAG(X1, X2, X3, X4, X5, appI_in_aaag(X2, X3, X4, X5)) 53.54/15.29 APPI_IN_AAAG(.(X1, X2), X3, X4, .(X1, X5)) -> APPI_IN_AAAG(X2, X3, X4, X5) 53.54/15.29 QSK_IN_AG(.(X1, []), X2) -> U57_AG(X1, X2, qsF_in_ga([], X3)) 53.54/15.29 QSK_IN_AG(.(X1, []), X2) -> QSF_IN_GA([], X3) 53.54/15.29 QSK_IN_AG(.(X1, []), X2) -> U58_AG(X1, X2, qscF_in_ga([], X3)) 53.54/15.29 U58_AG(X1, X2, qscF_out_ga([], X3)) -> U59_AG(X1, X2, qsF_in_ga([], X4)) 53.54/15.29 U58_AG(X1, X2, qscF_out_ga([], X3)) -> QSF_IN_GA([], X4) 53.54/15.29 U58_AG(X1, X2, qscF_out_ga([], X3)) -> U60_AG(X1, X2, X3, qscF_in_ga([], X4)) 53.54/15.29 U60_AG(X1, X2, X3, qscF_out_ga([], X4)) -> U61_AG(X1, X2, appJ_in_gagg(X3, X1, X4, X2)) 53.54/15.29 U60_AG(X1, X2, X3, qscF_out_ga([], X4)) -> APPJ_IN_GAGG(X3, X1, X4, X2) 53.54/15.29 APPJ_IN_GAGG(.(X1, X2), X3, X4, .(X1, X5)) -> U22_GAGG(X1, X2, X3, X4, X5, appI_in_gagg(X2, X3, X4, X5)) 53.54/15.29 APPJ_IN_GAGG(.(X1, X2), X3, X4, .(X1, X5)) -> APPI_IN_GAGG(X2, X3, X4, X5) 53.54/15.29 APPI_IN_GAGG(.(X1, X2), X3, X4, .(X1, X5)) -> U21_GAGG(X1, X2, X3, X4, X5, appI_in_gagg(X2, X3, X4, X5)) 53.54/15.29 APPI_IN_GAGG(.(X1, X2), X3, X4, .(X1, X5)) -> APPI_IN_GAGG(X2, X3, X4, X5) 53.54/15.29 53.54/15.29 The TRS R consists of the following rules: 53.54/15.29 53.54/15.29 lesscA_in_aa(0, s(X1)) -> lesscA_out_aa(0, s(X1)) 53.54/15.29 lesscA_in_aa(s(X1), s(X2)) -> U63_aa(X1, X2, lesscA_in_aa(X1, X2)) 53.54/15.29 U63_aa(X1, X2, lesscA_out_aa(X1, X2)) -> lesscA_out_aa(s(X1), s(X2)) 53.54/15.29 lesscA_in_ga(0, s(X1)) -> lesscA_out_ga(0, s(X1)) 53.54/15.29 lesscA_in_ga(s(X1), s(X2)) -> U63_ga(X1, X2, lesscA_in_ga(X1, X2)) 53.54/15.29 U63_ga(X1, X2, lesscA_out_ga(X1, X2)) -> lesscA_out_ga(s(X1), s(X2)) 53.54/15.29 partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_gaaa(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) 53.54/15.29 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)) 53.54/15.29 partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) 53.54/15.29 partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], []) 53.54/15.29 U66_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5)) 53.54/15.29 U65_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5) 53.54/15.29 partcC_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_aaaa(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) 53.54/15.29 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)) 53.54/15.29 U68_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), .(X2, X4), X5) 53.54/15.29 partcC_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_aaaa(X1, X2, X3, X4, X5, partcC_in_aaaa(X1, X3, X4, X5)) 53.54/15.29 partcC_in_aaaa(X1, [], [], []) -> partcC_out_aaaa(X1, [], [], []) 53.54/15.29 U69_aaaa(X1, X2, X3, X4, X5, partcC_out_aaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), X4, .(X2, X5)) 53.54/15.29 qscF_in_aa([], []) -> qscF_out_aa([], []) 53.54/15.29 qscF_in_aa(.(X1, X2), X3) -> U74_aa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.54/15.30 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)) 53.54/15.30 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)) 53.54/15.30 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)) 53.54/15.30 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)) 53.54/15.30 appcE_in_aaaa([], X1, X2, .(X1, X2)) -> appcE_out_aaaa([], X1, X2, .(X1, X2)) 53.54/15.30 appcE_in_aaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U75_aaaa(X1, X2, X3, X4, X5, appcE_in_aaaa(X2, X3, X4, X5)) 53.54/15.30 U75_aaaa(X1, X2, X3, X4, X5, appcE_out_aaaa(X2, X3, X4, X5)) -> appcE_out_aaaa(.(X1, X2), X3, X4, .(X1, X5)) 53.54/15.30 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) 53.54/15.30 U74_aa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_aa(.(X1, X2), X3) 53.54/15.30 qscG_in_aaa(X1, X2, X3) -> U76_aaa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.54/15.30 U76_aaa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscG_out_aaa(X1, X2, X3) 53.54/15.30 lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) 53.54/15.30 lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) 53.54/15.30 U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) 53.54/15.30 partcC_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.54/15.30 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)) 53.54/15.30 partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.54/15.30 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)) 53.54/15.30 partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) 53.54/15.30 partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], []) 53.54/15.30 U66_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.54/15.30 U65_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.54/15.30 U68_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.54/15.30 partcC_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_ggaa(X1, X2, X3, X4, X5, partcC_in_ggaa(X1, X3, X4, X5)) 53.54/15.30 partcC_in_ggaa(X1, [], [], []) -> partcC_out_ggaa(X1, [], [], []) 53.54/15.30 U69_ggaa(X1, X2, X3, X4, X5, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.54/15.30 qscF_in_ga([], []) -> qscF_out_ga([], []) 53.54/15.30 qscF_in_ga(.(X1, X2), X3) -> U74_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.54/15.30 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)) 53.54/15.30 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)) 53.54/15.30 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)) 53.54/15.30 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)) 53.54/15.30 appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2)) 53.54/15.30 appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U75_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5)) 53.54/15.30 U75_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5)) 53.54/15.30 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) 53.54/15.30 U74_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3) 53.54/15.30 53.54/15.30 The argument filtering Pi contains the following mapping: 53.54/15.30 [] = [] 53.54/15.30 53.54/15.30 lessA_in_aa(x1, x2) = lessA_in_aa 53.54/15.30 53.54/15.30 lesscA_in_aa(x1, x2) = lesscA_in_aa 53.54/15.30 53.54/15.30 lesscA_out_aa(x1, x2) = lesscA_out_aa(x1) 53.54/15.30 53.54/15.30 U63_aa(x1, x2, x3) = U63_aa(x3) 53.54/15.30 53.54/15.30 partB_in_gaaa(x1, x2, x3, x4) = partB_in_gaaa(x1) 53.54/15.30 53.54/15.30 lessA_in_ga(x1, x2) = lessA_in_ga(x1) 53.54/15.30 53.54/15.30 s(x1) = s(x1) 53.54/15.30 53.54/15.30 lesscA_in_ga(x1, x2) = lesscA_in_ga(x1) 53.54/15.30 53.54/15.30 0 = 0 53.54/15.30 53.54/15.30 lesscA_out_ga(x1, x2) = lesscA_out_ga(x1) 53.54/15.30 53.54/15.30 U63_ga(x1, x2, x3) = U63_ga(x1, x3) 53.54/15.30 53.54/15.30 partcB_in_gaaa(x1, x2, x3, x4) = partcB_in_gaaa(x1) 53.54/15.30 53.54/15.30 U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6) 53.54/15.30 53.54/15.30 U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x6) 53.54/15.30 53.54/15.30 U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6) 53.54/15.30 53.54/15.30 partcB_out_gaaa(x1, x2, x3, x4) = partcB_out_gaaa(x1) 53.54/15.30 53.54/15.30 qsG_in_aaa(x1, x2, x3) = qsG_in_aaa 53.54/15.30 53.54/15.30 pD_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = pD_in_aaaaaaa 53.54/15.30 53.54/15.30 partC_in_aaaa(x1, x2, x3, x4) = partC_in_aaaa 53.54/15.30 53.54/15.30 partcC_in_aaaa(x1, x2, x3, x4) = partcC_in_aaaa 53.54/15.30 53.54/15.30 U67_aaaa(x1, x2, x3, x4, x5, x6) = U67_aaaa(x6) 53.54/15.30 53.54/15.30 U68_aaaa(x1, x2, x3, x4, x5, x6) = U68_aaaa(x6) 53.54/15.30 53.54/15.30 partcC_out_aaaa(x1, x2, x3, x4) = partcC_out_aaaa 53.54/15.30 53.54/15.30 U69_aaaa(x1, x2, x3, x4, x5, x6) = U69_aaaa(x6) 53.54/15.30 53.54/15.30 qsF_in_aa(x1, x2) = qsF_in_aa 53.54/15.30 53.54/15.30 qscF_in_aa(x1, x2) = qscF_in_aa 53.54/15.30 53.54/15.30 qscF_out_aa(x1, x2) = qscF_out_aa 53.54/15.30 53.54/15.30 U74_aa(x1, x2, x3, x4) = U74_aa(x4) 53.54/15.30 53.54/15.30 qcD_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_aaaaaaa 53.54/15.30 53.54/15.30 U70_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_aaaaaaa(x8) 53.54/15.30 53.54/15.30 U71_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_aaaaaaa(x8) 53.54/15.30 53.54/15.30 U72_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_aaaaaaa(x8) 53.54/15.30 53.54/15.30 U73_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_aaaaaaa(x8) 53.54/15.30 53.54/15.30 .(x1, x2) = .(x1, x2) 53.54/15.30 53.54/15.30 qcD_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_aaaaaaa 53.54/15.30 53.54/15.30 appcE_in_aaaa(x1, x2, x3, x4) = appcE_in_aaaa 53.54/15.30 53.54/15.30 appcE_out_aaaa(x1, x2, x3, x4) = appcE_out_aaaa 53.54/15.30 53.54/15.30 U75_aaaa(x1, x2, x3, x4, x5, x6) = U75_aaaa(x6) 53.54/15.30 53.54/15.30 appE_in_aaaa(x1, x2, x3, x4) = appE_in_aaaa 53.54/15.30 53.54/15.30 qscG_in_aaa(x1, x2, x3) = qscG_in_aaa 53.54/15.30 53.54/15.30 U76_aaa(x1, x2, x3, x4) = U76_aaa(x4) 53.54/15.30 53.54/15.30 qscG_out_aaa(x1, x2, x3) = qscG_out_aaa 53.54/15.30 53.54/15.30 qsF_in_ga(x1, x2) = qsF_in_ga(x1) 53.54/15.30 53.54/15.30 pD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = pD_in_ggaaaaa(x1, x2) 53.54/15.30 53.54/15.30 partC_in_ggaa(x1, x2, x3, x4) = partC_in_ggaa(x1, x2) 53.54/15.30 53.54/15.30 lessA_in_gg(x1, x2) = lessA_in_gg(x1, x2) 53.54/15.30 53.54/15.30 lesscA_in_gg(x1, x2) = lesscA_in_gg(x1, x2) 53.54/15.30 53.54/15.30 lesscA_out_gg(x1, x2) = lesscA_out_gg(x1, x2) 53.54/15.30 53.54/15.30 U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3) 53.54/15.30 53.54/15.30 partB_in_ggaa(x1, x2, x3, x4) = partB_in_ggaa(x1, x2) 53.54/15.30 53.54/15.30 partcC_in_ggaa(x1, x2, x3, x4) = partcC_in_ggaa(x1, x2) 53.54/15.30 53.54/15.30 U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6) 53.54/15.30 53.54/15.30 U68_ggaa(x1, x2, x3, x4, x5, x6) = U68_ggaa(x1, x2, x3, x6) 53.54/15.30 53.54/15.30 partcB_in_ggaa(x1, x2, x3, x4) = partcB_in_ggaa(x1, x2) 53.54/15.30 53.54/15.30 U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6) 53.54/15.30 53.54/15.30 U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6) 53.54/15.30 53.54/15.30 U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6) 53.54/15.30 53.54/15.30 partcB_out_ggaa(x1, x2, x3, x4) = partcB_out_ggaa(x1, x2, x3, x4) 53.54/15.30 53.54/15.30 partcC_out_ggaa(x1, x2, x3, x4) = partcC_out_ggaa(x1, x2, x3, x4) 53.54/15.30 53.54/15.30 U69_ggaa(x1, x2, x3, x4, x5, x6) = U69_ggaa(x1, x2, x3, x6) 53.54/15.30 53.54/15.30 qscF_in_ga(x1, x2) = qscF_in_ga(x1) 53.54/15.30 53.54/15.30 qscF_out_ga(x1, x2) = qscF_out_ga(x1, x2) 53.54/15.30 53.54/15.30 U74_ga(x1, x2, x3, x4) = U74_ga(x1, x2, x4) 53.54/15.30 53.54/15.30 qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_ggaaaaa(x1, x2) 53.54/15.30 53.54/15.30 U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x8) 53.54/15.30 53.54/15.30 U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x8) 53.54/15.30 53.54/15.30 U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x8) 53.54/15.30 53.54/15.30 U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x8) 53.54/15.30 53.54/15.30 appcE_in_ggga(x1, x2, x3, x4) = appcE_in_ggga(x1, x2, x3) 53.54/15.30 53.54/15.30 appcE_out_ggga(x1, x2, x3, x4) = appcE_out_ggga(x1, x2, x3, x4) 53.54/15.30 53.54/15.30 U75_ggga(x1, x2, x3, x4, x5, x6) = U75_ggga(x1, x2, x3, x4, x6) 53.54/15.30 53.54/15.30 qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) 53.54/15.30 53.54/15.30 appE_in_ggga(x1, x2, x3, x4) = appE_in_ggga(x1, x2, x3) 53.54/15.30 53.54/15.30 appH_in_agag(x1, x2, x3, x4) = appH_in_agag(x2, x4) 53.54/15.30 53.54/15.30 appJ_in_aaag(x1, x2, x3, x4) = appJ_in_aaag(x4) 53.54/15.30 53.54/15.30 appI_in_aaag(x1, x2, x3, x4) = appI_in_aaag(x4) 53.54/15.30 53.54/15.30 appJ_in_gagg(x1, x2, x3, x4) = appJ_in_gagg(x1, x3, x4) 53.54/15.30 53.54/15.30 appI_in_gagg(x1, x2, x3, x4) = appI_in_gagg(x1, x3, x4) 53.54/15.30 53.54/15.30 QSK_IN_AG(x1, x2) = QSK_IN_AG(x2) 53.54/15.30 53.54/15.30 U23_AG(x1, x2, x3, x4) = U23_AG(x4) 53.54/15.30 53.54/15.30 LESSA_IN_AA(x1, x2) = LESSA_IN_AA 53.54/15.30 53.54/15.30 U1_AA(x1, x2, x3) = U1_AA(x3) 53.54/15.30 53.54/15.30 U24_AG(x1, x2, x3, x4) = U24_AG(x4) 53.54/15.30 53.54/15.30 U25_AG(x1, x2, x3, x4) = U25_AG(x4) 53.54/15.30 53.54/15.30 PARTB_IN_GAAA(x1, x2, x3, x4) = PARTB_IN_GAAA(x1) 53.54/15.30 53.54/15.30 U2_GAAA(x1, x2, x3, x4, x5, x6) = U2_GAAA(x1, x6) 53.54/15.30 53.54/15.30 LESSA_IN_GA(x1, x2) = LESSA_IN_GA(x1) 53.54/15.30 53.54/15.30 U1_GA(x1, x2, x3) = U1_GA(x1, x3) 53.54/15.30 53.54/15.30 U3_GAAA(x1, x2, x3, x4, x5, x6) = U3_GAAA(x1, x6) 53.54/15.30 53.54/15.30 U4_GAAA(x1, x2, x3, x4, x5, x6) = U4_GAAA(x1, x6) 53.54/15.30 53.54/15.30 U5_GAAA(x1, x2, x3, x4, x5, x6) = U5_GAAA(x1, x6) 53.54/15.30 53.54/15.30 U26_AG(x1, x2, x3, x4) = U26_AG(x4) 53.54/15.30 53.54/15.30 U27_AG(x1, x2, x3, x4) = U27_AG(x4) 53.54/15.30 53.54/15.30 QSG_IN_AAA(x1, x2, x3) = QSG_IN_AAA 53.54/15.30 53.54/15.30 U19_AAA(x1, x2, x3, x4) = U19_AAA(x4) 53.54/15.30 53.54/15.30 PD_IN_AAAAAAA(x1, x2, x3, x4, x5, x6, x7) = PD_IN_AAAAAAA 53.54/15.30 53.54/15.30 U10_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U10_AAAAAAA(x8) 53.54/15.30 53.54/15.30 PARTC_IN_AAAA(x1, x2, x3, x4) = PARTC_IN_AAAA 53.54/15.30 53.54/15.30 U6_AAAA(x1, x2, x3, x4, x5, x6) = U6_AAAA(x6) 53.54/15.30 53.54/15.30 U7_AAAA(x1, x2, x3, x4, x5, x6) = U7_AAAA(x6) 53.54/15.30 53.54/15.30 U8_AAAA(x1, x2, x3, x4, x5, x6) = U8_AAAA(x1, x6) 53.54/15.30 53.54/15.30 U9_AAAA(x1, x2, x3, x4, x5, x6) = U9_AAAA(x6) 53.54/15.30 53.54/15.30 U11_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U11_AAAAAAA(x8) 53.54/15.30 53.54/15.30 U12_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U12_AAAAAAA(x8) 53.54/15.30 53.54/15.30 QSF_IN_AA(x1, x2) = QSF_IN_AA 53.54/15.30 53.54/15.30 U17_AA(x1, x2, x3, x4) = U17_AA(x4) 53.54/15.30 53.54/15.30 U13_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U13_AAAAAAA(x8) 53.54/15.30 53.54/15.30 U14_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U14_AAAAAAA(x8) 53.54/15.30 53.54/15.30 U15_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U15_AAAAAAA(x8) 53.54/15.30 53.54/15.30 U16_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U16_AAAAAAA(x8) 53.54/15.30 53.54/15.30 APPE_IN_AAAA(x1, x2, x3, x4) = APPE_IN_AAAA 53.54/15.30 53.54/15.30 U18_AAAA(x1, x2, x3, x4, x5, x6) = U18_AAAA(x6) 53.54/15.30 53.54/15.30 U28_AG(x1, x2, x3, x4, x5) = U28_AG(x5) 53.54/15.30 53.54/15.30 U29_AG(x1, x2, x3, x4) = U29_AG(x4) 53.54/15.30 53.54/15.30 U30_AG(x1, x2, x3, x4) = U30_AG(x4) 53.54/15.30 53.54/15.30 U31_AG(x1, x2, x3, x4) = U31_AG(x4) 53.54/15.30 53.54/15.30 U32_AG(x1, x2, x3, x4) = U32_AG(x4) 53.54/15.30 53.54/15.30 U33_AG(x1, x2, x3, x4, x5) = U33_AG(x5) 53.54/15.30 53.54/15.30 U34_AG(x1, x2, x3, x4) = U34_AG(x4) 53.54/15.30 53.54/15.30 U35_AG(x1, x2) = U35_AG(x2) 53.54/15.30 53.54/15.30 QSF_IN_GA(x1, x2) = QSF_IN_GA(x1) 53.54/15.30 53.54/15.30 U17_GA(x1, x2, x3, x4) = U17_GA(x1, x2, x4) 53.54/15.30 53.54/15.30 PD_IN_GGAAAAA(x1, x2, x3, x4, x5, x6, x7) = PD_IN_GGAAAAA(x1, x2) 53.54/15.30 53.54/15.30 U10_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U10_GGAAAAA(x1, x2, x8) 53.54/15.30 53.54/15.30 PARTC_IN_GGAA(x1, x2, x3, x4) = PARTC_IN_GGAA(x1, x2) 53.54/15.30 53.54/15.30 U6_GGAA(x1, x2, x3, x4, x5, x6) = U6_GGAA(x1, x2, x3, x6) 53.54/15.30 53.54/15.30 LESSA_IN_GG(x1, x2) = LESSA_IN_GG(x1, x2) 53.54/15.30 53.54/15.30 U1_GG(x1, x2, x3) = U1_GG(x1, x2, x3) 53.54/15.30 53.54/15.30 U7_GGAA(x1, x2, x3, x4, x5, x6) = U7_GGAA(x1, x2, x3, x6) 53.54/15.30 53.54/15.30 U8_GGAA(x1, x2, x3, x4, x5, x6) = U8_GGAA(x1, x2, x3, x6) 53.54/15.30 53.54/15.30 PARTB_IN_GGAA(x1, x2, x3, x4) = PARTB_IN_GGAA(x1, x2) 53.54/15.30 53.54/15.30 U2_GGAA(x1, x2, x3, x4, x5, x6) = U2_GGAA(x1, x2, x3, x6) 53.54/15.30 53.54/15.30 U3_GGAA(x1, x2, x3, x4, x5, x6) = U3_GGAA(x1, x2, x3, x6) 53.54/15.30 53.54/15.30 U4_GGAA(x1, x2, x3, x4, x5, x6) = U4_GGAA(x1, x2, x3, x6) 53.54/15.30 53.54/15.30 U5_GGAA(x1, x2, x3, x4, x5, x6) = U5_GGAA(x1, x2, x3, x6) 53.54/15.30 53.54/15.30 U9_GGAA(x1, x2, x3, x4, x5, x6) = U9_GGAA(x1, x2, x3, x6) 53.54/15.30 53.54/15.30 U11_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U11_GGAAAAA(x1, x2, x8) 53.54/15.30 53.54/15.30 U12_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U12_GGAAAAA(x1, x2, x8) 53.54/15.30 53.54/15.30 U13_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U13_GGAAAAA(x1, x2, x4, x8) 53.54/15.30 53.54/15.30 U14_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U14_GGAAAAA(x1, x2, x8) 53.54/15.30 53.54/15.30 U15_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U15_GGAAAAA(x1, x2, x5, x8) 53.54/15.30 53.54/15.30 U16_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U16_GGAAAAA(x1, x2, x8) 53.54/15.30 53.54/15.30 APPE_IN_GGGA(x1, x2, x3, x4) = APPE_IN_GGGA(x1, x2, x3) 53.54/15.30 53.54/15.30 U18_GGGA(x1, x2, x3, x4, x5, x6) = U18_GGGA(x1, x2, x3, x4, x6) 53.54/15.30 53.54/15.30 U36_AG(x1, x2) = U36_AG(x2) 53.54/15.30 53.54/15.30 U37_AG(x1, x2) = U37_AG(x2) 53.54/15.30 53.54/15.30 U38_AG(x1, x2, x3, x4, x5) = U38_AG(x4, x5) 53.54/15.30 53.54/15.30 U39_AG(x1, x2, x3, x4, x5) = U39_AG(x4, x5) 53.54/15.30 53.54/15.30 U40_AG(x1, x2, x3, x4, x5) = U40_AG(x4, x5) 53.54/15.30 53.54/15.30 U41_AG(x1, x2, x3, x4, x5) = U41_AG(x4, x5) 53.54/15.30 53.54/15.30 U42_AG(x1, x2, x3, x4, x5) = U42_AG(x4, x5) 53.54/15.30 53.54/15.30 U43_AG(x1, x2, x3, x4, x5, x6) = U43_AG(x4, x6) 53.54/15.30 53.54/15.30 U44_AG(x1, x2, x3, x4, x5) = U44_AG(x4, x5) 53.54/15.30 53.54/15.30 U45_AG(x1, x2, x3, x4, x5, x6) = U45_AG(x4, x5, x6) 53.54/15.30 53.54/15.30 U46_AG(x1, x2, x3, x4, x5, x6) = U46_AG(x1, x4, x5, x6) 53.54/15.30 53.54/15.30 U47_AG(x1, x2, x3, x4, x5, x6, x7) = U47_AG(x1, x4, x5, x7) 53.54/15.30 53.54/15.30 U48_AG(x1, x2, x3, x4, x5, x6, x7) = U48_AG(x1, x4, x5, x7) 53.54/15.30 53.54/15.30 U49_AG(x1, x2, x3, x4, x5, x6) = U49_AG(x4, x5, x6) 53.54/15.30 53.54/15.30 APPH_IN_AGAG(x1, x2, x3, x4) = APPH_IN_AGAG(x2, x4) 53.54/15.30 53.54/15.30 U20_AGAG(x1, x2, x3, x4, x5, x6) = U20_AGAG(x1, x3, x5, x6) 53.54/15.30 53.54/15.30 U50_AG(x1, x2, x3, x4, x5) = U50_AG(x4, x5) 53.54/15.30 53.54/15.30 U51_AG(x1, x2, x3, x4, x5) = U51_AG(x4, x5) 53.54/15.30 53.54/15.30 U52_AG(x1, x2, x3, x4, x5) = U52_AG(x4, x5) 53.54/15.30 53.54/15.30 U53_AG(x1, x2, x3, x4, x5, x6) = U53_AG(x4, x6) 53.54/15.30 53.54/15.30 U54_AG(x1, x2, x3, x4, x5) = U54_AG(x4, x5) 53.54/15.30 53.54/15.30 U55_AG(x1, x2, x3, x4, x5, x6) = U55_AG(x4, x6) 53.54/15.30 53.54/15.30 U56_AG(x1, x2, x3, x4, x5) = U56_AG(x4, x5) 53.54/15.30 53.54/15.30 APPJ_IN_AAAG(x1, x2, x3, x4) = APPJ_IN_AAAG(x4) 53.54/15.30 53.54/15.30 U22_AAAG(x1, x2, x3, x4, x5, x6) = U22_AAAG(x1, x5, x6) 53.54/15.30 53.54/15.30 APPI_IN_AAAG(x1, x2, x3, x4) = APPI_IN_AAAG(x4) 53.54/15.30 53.54/15.30 U21_AAAG(x1, x2, x3, x4, x5, x6) = U21_AAAG(x1, x5, x6) 53.54/15.30 53.54/15.30 U57_AG(x1, x2, x3) = U57_AG(x2, x3) 53.54/15.30 53.54/15.30 U58_AG(x1, x2, x3) = U58_AG(x2, x3) 53.54/15.30 53.54/15.30 U59_AG(x1, x2, x3) = U59_AG(x2, x3) 53.54/15.30 53.54/15.30 U60_AG(x1, x2, x3, x4) = U60_AG(x2, x3, x4) 53.54/15.30 53.54/15.30 U61_AG(x1, x2, x3) = U61_AG(x2, x3) 53.54/15.30 53.54/15.30 APPJ_IN_GAGG(x1, x2, x3, x4) = APPJ_IN_GAGG(x1, x3, x4) 53.54/15.30 53.54/15.30 U22_GAGG(x1, x2, x3, x4, x5, x6) = U22_GAGG(x1, x2, x4, x5, x6) 53.54/15.30 53.54/15.30 APPI_IN_GAGG(x1, x2, x3, x4) = APPI_IN_GAGG(x1, x3, x4) 53.54/15.30 53.54/15.30 U21_GAGG(x1, x2, x3, x4, x5, x6) = U21_GAGG(x1, x2, x4, x5, x6) 53.54/15.30 53.54/15.30 53.54/15.30 We have to consider all (P,R,Pi)-chains 53.54/15.30 ---------------------------------------- 53.54/15.30 53.54/15.30 (138) DependencyGraphProof (EQUIVALENT) 53.54/15.30 The approximation of the Dependency Graph [LOPSTR] contains 14 SCCs with 110 less nodes. 53.54/15.30 ---------------------------------------- 53.54/15.30 53.54/15.30 (139) 53.54/15.30 Complex Obligation (AND) 53.54/15.30 53.54/15.30 ---------------------------------------- 53.54/15.30 53.54/15.30 (140) 53.54/15.30 Obligation: 53.54/15.30 Pi DP problem: 53.54/15.30 The TRS P consists of the following rules: 53.54/15.30 53.54/15.30 APPI_IN_GAGG(.(X1, X2), X3, X4, .(X1, X5)) -> APPI_IN_GAGG(X2, X3, X4, X5) 53.54/15.30 53.54/15.30 The TRS R consists of the following rules: 53.54/15.30 53.54/15.30 lesscA_in_aa(0, s(X1)) -> lesscA_out_aa(0, s(X1)) 53.54/15.30 lesscA_in_aa(s(X1), s(X2)) -> U63_aa(X1, X2, lesscA_in_aa(X1, X2)) 53.54/15.30 U63_aa(X1, X2, lesscA_out_aa(X1, X2)) -> lesscA_out_aa(s(X1), s(X2)) 53.54/15.30 lesscA_in_ga(0, s(X1)) -> lesscA_out_ga(0, s(X1)) 53.54/15.30 lesscA_in_ga(s(X1), s(X2)) -> U63_ga(X1, X2, lesscA_in_ga(X1, X2)) 53.54/15.30 U63_ga(X1, X2, lesscA_out_ga(X1, X2)) -> lesscA_out_ga(s(X1), s(X2)) 53.54/15.30 partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_gaaa(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) 53.54/15.30 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)) 53.54/15.30 partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) 53.54/15.30 partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], []) 53.54/15.30 U66_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5)) 53.54/15.30 U65_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5) 53.54/15.30 partcC_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_aaaa(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) 53.54/15.30 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)) 53.54/15.30 U68_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), .(X2, X4), X5) 53.54/15.30 partcC_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_aaaa(X1, X2, X3, X4, X5, partcC_in_aaaa(X1, X3, X4, X5)) 53.54/15.30 partcC_in_aaaa(X1, [], [], []) -> partcC_out_aaaa(X1, [], [], []) 53.54/15.30 U69_aaaa(X1, X2, X3, X4, X5, partcC_out_aaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), X4, .(X2, X5)) 53.54/15.30 qscF_in_aa([], []) -> qscF_out_aa([], []) 53.54/15.30 qscF_in_aa(.(X1, X2), X3) -> U74_aa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.54/15.30 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)) 53.54/15.30 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)) 53.54/15.30 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)) 53.54/15.30 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)) 53.54/15.30 appcE_in_aaaa([], X1, X2, .(X1, X2)) -> appcE_out_aaaa([], X1, X2, .(X1, X2)) 53.54/15.30 appcE_in_aaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U75_aaaa(X1, X2, X3, X4, X5, appcE_in_aaaa(X2, X3, X4, X5)) 53.54/15.30 U75_aaaa(X1, X2, X3, X4, X5, appcE_out_aaaa(X2, X3, X4, X5)) -> appcE_out_aaaa(.(X1, X2), X3, X4, .(X1, X5)) 53.54/15.30 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) 53.54/15.30 U74_aa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_aa(.(X1, X2), X3) 53.54/15.30 qscG_in_aaa(X1, X2, X3) -> U76_aaa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.54/15.30 U76_aaa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscG_out_aaa(X1, X2, X3) 53.54/15.30 lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) 53.54/15.30 lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) 53.54/15.30 U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) 53.54/15.30 partcC_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.54/15.30 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)) 53.54/15.30 partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.54/15.30 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)) 53.54/15.30 partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) 53.54/15.30 partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], []) 53.54/15.30 U66_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.54/15.30 U65_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.54/15.30 U68_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.54/15.30 partcC_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_ggaa(X1, X2, X3, X4, X5, partcC_in_ggaa(X1, X3, X4, X5)) 53.54/15.30 partcC_in_ggaa(X1, [], [], []) -> partcC_out_ggaa(X1, [], [], []) 53.54/15.30 U69_ggaa(X1, X2, X3, X4, X5, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.54/15.30 qscF_in_ga([], []) -> qscF_out_ga([], []) 53.54/15.30 qscF_in_ga(.(X1, X2), X3) -> U74_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.54/15.30 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)) 53.54/15.30 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)) 53.54/15.30 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)) 53.54/15.30 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)) 53.54/15.30 appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2)) 53.54/15.30 appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U75_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5)) 53.54/15.30 U75_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5)) 53.54/15.30 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) 53.54/15.30 U74_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3) 53.54/15.30 53.54/15.30 The argument filtering Pi contains the following mapping: 53.54/15.30 [] = [] 53.54/15.30 53.54/15.30 lesscA_in_aa(x1, x2) = lesscA_in_aa 53.54/15.30 53.54/15.30 lesscA_out_aa(x1, x2) = lesscA_out_aa(x1) 53.54/15.30 53.54/15.30 U63_aa(x1, x2, x3) = U63_aa(x3) 53.54/15.30 53.54/15.30 s(x1) = s(x1) 53.54/15.30 53.54/15.30 lesscA_in_ga(x1, x2) = lesscA_in_ga(x1) 53.54/15.30 53.54/15.30 0 = 0 53.54/15.30 53.54/15.30 lesscA_out_ga(x1, x2) = lesscA_out_ga(x1) 53.54/15.30 53.54/15.30 U63_ga(x1, x2, x3) = U63_ga(x1, x3) 53.54/15.30 53.54/15.30 partcB_in_gaaa(x1, x2, x3, x4) = partcB_in_gaaa(x1) 53.54/15.30 53.54/15.30 U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6) 53.54/15.30 53.54/15.30 U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x6) 53.54/15.30 53.54/15.30 U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6) 53.54/15.30 53.54/15.30 partcB_out_gaaa(x1, x2, x3, x4) = partcB_out_gaaa(x1) 53.54/15.30 53.54/15.30 partcC_in_aaaa(x1, x2, x3, x4) = partcC_in_aaaa 53.54/15.30 53.54/15.30 U67_aaaa(x1, x2, x3, x4, x5, x6) = U67_aaaa(x6) 53.54/15.30 53.54/15.30 U68_aaaa(x1, x2, x3, x4, x5, x6) = U68_aaaa(x6) 53.54/15.30 53.54/15.30 partcC_out_aaaa(x1, x2, x3, x4) = partcC_out_aaaa 53.54/15.30 53.54/15.30 U69_aaaa(x1, x2, x3, x4, x5, x6) = U69_aaaa(x6) 53.54/15.30 53.54/15.30 qscF_in_aa(x1, x2) = qscF_in_aa 53.54/15.30 53.54/15.30 qscF_out_aa(x1, x2) = qscF_out_aa 53.54/15.30 53.54/15.30 U74_aa(x1, x2, x3, x4) = U74_aa(x4) 53.54/15.30 53.54/15.30 qcD_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_aaaaaaa 53.54/15.30 53.54/15.30 U70_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_aaaaaaa(x8) 53.54/15.30 53.54/15.30 U71_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_aaaaaaa(x8) 53.54/15.30 53.54/15.30 U72_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_aaaaaaa(x8) 53.54/15.30 53.54/15.30 U73_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_aaaaaaa(x8) 53.54/15.30 53.54/15.30 .(x1, x2) = .(x1, x2) 53.54/15.30 53.54/15.30 qcD_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_aaaaaaa 53.54/15.30 53.54/15.30 appcE_in_aaaa(x1, x2, x3, x4) = appcE_in_aaaa 53.54/15.30 53.54/15.30 appcE_out_aaaa(x1, x2, x3, x4) = appcE_out_aaaa 53.54/15.30 53.54/15.30 U75_aaaa(x1, x2, x3, x4, x5, x6) = U75_aaaa(x6) 53.54/15.30 53.54/15.30 qscG_in_aaa(x1, x2, x3) = qscG_in_aaa 53.54/15.30 53.54/15.30 U76_aaa(x1, x2, x3, x4) = U76_aaa(x4) 53.54/15.30 53.54/15.30 qscG_out_aaa(x1, x2, x3) = qscG_out_aaa 53.54/15.30 53.54/15.30 lesscA_in_gg(x1, x2) = lesscA_in_gg(x1, x2) 53.54/15.30 53.54/15.30 lesscA_out_gg(x1, x2) = lesscA_out_gg(x1, x2) 53.54/15.30 53.54/15.30 U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3) 53.54/15.30 53.54/15.30 partcC_in_ggaa(x1, x2, x3, x4) = partcC_in_ggaa(x1, x2) 53.54/15.30 53.54/15.30 U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6) 53.54/15.30 53.54/15.30 U68_ggaa(x1, x2, x3, x4, x5, x6) = U68_ggaa(x1, x2, x3, x6) 53.54/15.30 53.54/15.30 partcB_in_ggaa(x1, x2, x3, x4) = partcB_in_ggaa(x1, x2) 53.54/15.30 53.54/15.30 U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6) 53.54/15.30 53.54/15.30 U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6) 53.54/15.30 53.54/15.30 U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6) 53.54/15.30 53.54/15.30 partcB_out_ggaa(x1, x2, x3, x4) = partcB_out_ggaa(x1, x2, x3, x4) 53.54/15.30 53.54/15.30 partcC_out_ggaa(x1, x2, x3, x4) = partcC_out_ggaa(x1, x2, x3, x4) 53.54/15.30 53.54/15.30 U69_ggaa(x1, x2, x3, x4, x5, x6) = U69_ggaa(x1, x2, x3, x6) 53.54/15.30 53.54/15.30 qscF_in_ga(x1, x2) = qscF_in_ga(x1) 53.54/15.30 53.54/15.30 qscF_out_ga(x1, x2) = qscF_out_ga(x1, x2) 53.54/15.30 53.54/15.30 U74_ga(x1, x2, x3, x4) = U74_ga(x1, x2, x4) 53.54/15.30 53.54/15.30 qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_ggaaaaa(x1, x2) 53.54/15.30 53.54/15.30 U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x8) 53.54/15.30 53.54/15.30 U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x8) 53.54/15.30 53.54/15.30 U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x8) 53.54/15.30 53.54/15.30 U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x8) 53.54/15.30 53.54/15.30 appcE_in_ggga(x1, x2, x3, x4) = appcE_in_ggga(x1, x2, x3) 53.54/15.30 53.54/15.30 appcE_out_ggga(x1, x2, x3, x4) = appcE_out_ggga(x1, x2, x3, x4) 53.54/15.30 53.54/15.30 U75_ggga(x1, x2, x3, x4, x5, x6) = U75_ggga(x1, x2, x3, x4, x6) 53.54/15.30 53.54/15.30 qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) 53.54/15.30 53.54/15.30 APPI_IN_GAGG(x1, x2, x3, x4) = APPI_IN_GAGG(x1, x3, x4) 53.54/15.30 53.54/15.30 53.54/15.30 We have to consider all (P,R,Pi)-chains 53.54/15.30 ---------------------------------------- 53.54/15.30 53.54/15.30 (141) UsableRulesProof (EQUIVALENT) 53.54/15.30 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 53.54/15.30 ---------------------------------------- 53.54/15.30 53.54/15.30 (142) 53.54/15.30 Obligation: 53.54/15.30 Pi DP problem: 53.54/15.30 The TRS P consists of the following rules: 53.54/15.30 53.54/15.30 APPI_IN_GAGG(.(X1, X2), X3, X4, .(X1, X5)) -> APPI_IN_GAGG(X2, X3, X4, X5) 53.54/15.30 53.54/15.30 R is empty. 53.54/15.30 The argument filtering Pi contains the following mapping: 53.54/15.30 .(x1, x2) = .(x1, x2) 53.54/15.30 53.54/15.30 APPI_IN_GAGG(x1, x2, x3, x4) = APPI_IN_GAGG(x1, x3, x4) 53.54/15.30 53.54/15.30 53.54/15.30 We have to consider all (P,R,Pi)-chains 53.54/15.30 ---------------------------------------- 53.83/15.30 53.83/15.30 (143) PiDPToQDPProof (SOUND) 53.83/15.30 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 53.83/15.30 ---------------------------------------- 53.83/15.30 53.83/15.30 (144) 53.83/15.30 Obligation: 53.83/15.30 Q DP problem: 53.83/15.30 The TRS P consists of the following rules: 53.83/15.30 53.83/15.30 APPI_IN_GAGG(.(X1, X2), X4, .(X1, X5)) -> APPI_IN_GAGG(X2, X4, X5) 53.83/15.30 53.83/15.30 R is empty. 53.83/15.30 Q is empty. 53.83/15.30 We have to consider all (P,Q,R)-chains. 53.83/15.30 ---------------------------------------- 53.83/15.30 53.83/15.30 (145) QDPSizeChangeProof (EQUIVALENT) 53.83/15.30 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. 53.83/15.30 53.83/15.30 From the DPs we obtained the following set of size-change graphs: 53.83/15.30 *APPI_IN_GAGG(.(X1, X2), X4, .(X1, X5)) -> APPI_IN_GAGG(X2, X4, X5) 53.83/15.30 The graph contains the following edges 1 > 1, 2 >= 2, 3 > 3 53.83/15.30 53.83/15.30 53.83/15.30 ---------------------------------------- 53.83/15.30 53.83/15.30 (146) 53.83/15.30 YES 53.83/15.30 53.83/15.30 ---------------------------------------- 53.83/15.30 53.83/15.30 (147) 53.83/15.30 Obligation: 53.83/15.30 Pi DP problem: 53.83/15.30 The TRS P consists of the following rules: 53.83/15.30 53.83/15.30 APPI_IN_AAAG(.(X1, X2), X3, X4, .(X1, X5)) -> APPI_IN_AAAG(X2, X3, X4, X5) 53.83/15.30 53.83/15.30 The TRS R consists of the following rules: 53.83/15.30 53.83/15.30 lesscA_in_aa(0, s(X1)) -> lesscA_out_aa(0, s(X1)) 53.83/15.30 lesscA_in_aa(s(X1), s(X2)) -> U63_aa(X1, X2, lesscA_in_aa(X1, X2)) 53.83/15.30 U63_aa(X1, X2, lesscA_out_aa(X1, X2)) -> lesscA_out_aa(s(X1), s(X2)) 53.83/15.30 lesscA_in_ga(0, s(X1)) -> lesscA_out_ga(0, s(X1)) 53.83/15.30 lesscA_in_ga(s(X1), s(X2)) -> U63_ga(X1, X2, lesscA_in_ga(X1, X2)) 53.83/15.30 U63_ga(X1, X2, lesscA_out_ga(X1, X2)) -> lesscA_out_ga(s(X1), s(X2)) 53.83/15.30 partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_gaaa(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) 53.83/15.30 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)) 53.83/15.30 partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) 53.83/15.30 partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], []) 53.83/15.30 U66_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.30 U65_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.30 partcC_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_aaaa(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) 53.83/15.30 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)) 53.83/15.30 U68_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.30 partcC_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_aaaa(X1, X2, X3, X4, X5, partcC_in_aaaa(X1, X3, X4, X5)) 53.83/15.30 partcC_in_aaaa(X1, [], [], []) -> partcC_out_aaaa(X1, [], [], []) 53.83/15.30 U69_aaaa(X1, X2, X3, X4, X5, partcC_out_aaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.30 qscF_in_aa([], []) -> qscF_out_aa([], []) 53.83/15.30 qscF_in_aa(.(X1, X2), X3) -> U74_aa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.30 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)) 53.83/15.30 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)) 53.83/15.30 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)) 53.83/15.30 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)) 53.83/15.30 appcE_in_aaaa([], X1, X2, .(X1, X2)) -> appcE_out_aaaa([], X1, X2, .(X1, X2)) 53.83/15.30 appcE_in_aaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U75_aaaa(X1, X2, X3, X4, X5, appcE_in_aaaa(X2, X3, X4, X5)) 53.83/15.30 U75_aaaa(X1, X2, X3, X4, X5, appcE_out_aaaa(X2, X3, X4, X5)) -> appcE_out_aaaa(.(X1, X2), X3, X4, .(X1, X5)) 53.83/15.30 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) 53.83/15.30 U74_aa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_aa(.(X1, X2), X3) 53.83/15.30 qscG_in_aaa(X1, X2, X3) -> U76_aaa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.30 U76_aaa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscG_out_aaa(X1, X2, X3) 53.83/15.30 lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) 53.83/15.30 lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) 53.83/15.30 U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) 53.83/15.30 partcC_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.83/15.30 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)) 53.83/15.30 partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.83/15.30 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)) 53.83/15.30 partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) 53.83/15.30 partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], []) 53.83/15.30 U66_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.30 U65_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.30 U68_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.30 partcC_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_ggaa(X1, X2, X3, X4, X5, partcC_in_ggaa(X1, X3, X4, X5)) 53.83/15.30 partcC_in_ggaa(X1, [], [], []) -> partcC_out_ggaa(X1, [], [], []) 53.83/15.30 U69_ggaa(X1, X2, X3, X4, X5, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.30 qscF_in_ga([], []) -> qscF_out_ga([], []) 53.83/15.30 qscF_in_ga(.(X1, X2), X3) -> U74_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.30 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)) 53.83/15.30 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)) 53.83/15.30 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)) 53.83/15.30 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)) 53.83/15.30 appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2)) 53.83/15.30 appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U75_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5)) 53.83/15.30 U75_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5)) 53.83/15.30 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) 53.83/15.30 U74_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3) 53.83/15.30 53.83/15.30 The argument filtering Pi contains the following mapping: 53.83/15.30 [] = [] 53.83/15.30 53.83/15.30 lesscA_in_aa(x1, x2) = lesscA_in_aa 53.83/15.30 53.83/15.30 lesscA_out_aa(x1, x2) = lesscA_out_aa(x1) 53.83/15.30 53.83/15.30 U63_aa(x1, x2, x3) = U63_aa(x3) 53.83/15.30 53.83/15.30 s(x1) = s(x1) 53.83/15.30 53.83/15.30 lesscA_in_ga(x1, x2) = lesscA_in_ga(x1) 53.83/15.30 53.83/15.30 0 = 0 53.83/15.30 53.83/15.30 lesscA_out_ga(x1, x2) = lesscA_out_ga(x1) 53.83/15.30 53.83/15.30 U63_ga(x1, x2, x3) = U63_ga(x1, x3) 53.83/15.30 53.83/15.30 partcB_in_gaaa(x1, x2, x3, x4) = partcB_in_gaaa(x1) 53.83/15.30 53.83/15.30 U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6) 53.83/15.30 53.83/15.30 U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x6) 53.83/15.30 53.83/15.30 U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6) 53.83/15.30 53.83/15.30 partcB_out_gaaa(x1, x2, x3, x4) = partcB_out_gaaa(x1) 53.83/15.30 53.83/15.30 partcC_in_aaaa(x1, x2, x3, x4) = partcC_in_aaaa 53.83/15.30 53.83/15.30 U67_aaaa(x1, x2, x3, x4, x5, x6) = U67_aaaa(x6) 53.83/15.30 53.83/15.30 U68_aaaa(x1, x2, x3, x4, x5, x6) = U68_aaaa(x6) 53.83/15.30 53.83/15.30 partcC_out_aaaa(x1, x2, x3, x4) = partcC_out_aaaa 53.83/15.30 53.83/15.30 U69_aaaa(x1, x2, x3, x4, x5, x6) = U69_aaaa(x6) 53.83/15.30 53.83/15.30 qscF_in_aa(x1, x2) = qscF_in_aa 53.83/15.30 53.83/15.30 qscF_out_aa(x1, x2) = qscF_out_aa 53.83/15.30 53.83/15.30 U74_aa(x1, x2, x3, x4) = U74_aa(x4) 53.83/15.30 53.83/15.30 qcD_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_aaaaaaa 53.83/15.30 53.83/15.30 U70_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_aaaaaaa(x8) 53.83/15.30 53.83/15.30 U71_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_aaaaaaa(x8) 53.83/15.30 53.83/15.30 U72_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_aaaaaaa(x8) 53.83/15.30 53.83/15.30 U73_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_aaaaaaa(x8) 53.83/15.30 53.83/15.30 .(x1, x2) = .(x1, x2) 53.83/15.30 53.83/15.30 qcD_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_aaaaaaa 53.83/15.30 53.83/15.30 appcE_in_aaaa(x1, x2, x3, x4) = appcE_in_aaaa 53.83/15.30 53.83/15.30 appcE_out_aaaa(x1, x2, x3, x4) = appcE_out_aaaa 53.83/15.30 53.83/15.30 U75_aaaa(x1, x2, x3, x4, x5, x6) = U75_aaaa(x6) 53.83/15.30 53.83/15.30 qscG_in_aaa(x1, x2, x3) = qscG_in_aaa 53.83/15.30 53.83/15.30 U76_aaa(x1, x2, x3, x4) = U76_aaa(x4) 53.83/15.30 53.83/15.30 qscG_out_aaa(x1, x2, x3) = qscG_out_aaa 53.83/15.30 53.83/15.30 lesscA_in_gg(x1, x2) = lesscA_in_gg(x1, x2) 53.83/15.30 53.83/15.30 lesscA_out_gg(x1, x2) = lesscA_out_gg(x1, x2) 53.83/15.30 53.83/15.30 U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3) 53.83/15.30 53.83/15.30 partcC_in_ggaa(x1, x2, x3, x4) = partcC_in_ggaa(x1, x2) 53.83/15.30 53.83/15.30 U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6) 53.83/15.30 53.83/15.30 U68_ggaa(x1, x2, x3, x4, x5, x6) = U68_ggaa(x1, x2, x3, x6) 53.83/15.30 53.83/15.30 partcB_in_ggaa(x1, x2, x3, x4) = partcB_in_ggaa(x1, x2) 53.83/15.30 53.83/15.30 U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6) 53.83/15.30 53.83/15.30 U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6) 53.83/15.30 53.83/15.30 U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6) 53.83/15.30 53.83/15.30 partcB_out_ggaa(x1, x2, x3, x4) = partcB_out_ggaa(x1, x2, x3, x4) 53.83/15.30 53.83/15.30 partcC_out_ggaa(x1, x2, x3, x4) = partcC_out_ggaa(x1, x2, x3, x4) 53.83/15.30 53.83/15.30 U69_ggaa(x1, x2, x3, x4, x5, x6) = U69_ggaa(x1, x2, x3, x6) 53.83/15.30 53.83/15.30 qscF_in_ga(x1, x2) = qscF_in_ga(x1) 53.83/15.30 53.83/15.30 qscF_out_ga(x1, x2) = qscF_out_ga(x1, x2) 53.83/15.30 53.83/15.30 U74_ga(x1, x2, x3, x4) = U74_ga(x1, x2, x4) 53.83/15.30 53.83/15.30 qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_ggaaaaa(x1, x2) 53.83/15.30 53.83/15.30 U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x8) 53.83/15.30 53.83/15.30 U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x8) 53.83/15.30 53.83/15.30 U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x8) 53.83/15.30 53.83/15.30 U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x8) 53.83/15.30 53.83/15.30 appcE_in_ggga(x1, x2, x3, x4) = appcE_in_ggga(x1, x2, x3) 53.83/15.30 53.83/15.30 appcE_out_ggga(x1, x2, x3, x4) = appcE_out_ggga(x1, x2, x3, x4) 53.83/15.30 53.83/15.30 U75_ggga(x1, x2, x3, x4, x5, x6) = U75_ggga(x1, x2, x3, x4, x6) 53.83/15.30 53.83/15.30 qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) 53.83/15.30 53.83/15.30 APPI_IN_AAAG(x1, x2, x3, x4) = APPI_IN_AAAG(x4) 53.83/15.30 53.83/15.30 53.83/15.30 We have to consider all (P,R,Pi)-chains 53.83/15.30 ---------------------------------------- 53.83/15.30 53.83/15.30 (148) UsableRulesProof (EQUIVALENT) 53.83/15.30 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 53.83/15.30 ---------------------------------------- 53.83/15.30 53.83/15.30 (149) 53.83/15.30 Obligation: 53.83/15.30 Pi DP problem: 53.83/15.30 The TRS P consists of the following rules: 53.83/15.30 53.83/15.30 APPI_IN_AAAG(.(X1, X2), X3, X4, .(X1, X5)) -> APPI_IN_AAAG(X2, X3, X4, X5) 53.83/15.30 53.83/15.30 R is empty. 53.83/15.30 The argument filtering Pi contains the following mapping: 53.83/15.30 .(x1, x2) = .(x1, x2) 53.83/15.30 53.83/15.30 APPI_IN_AAAG(x1, x2, x3, x4) = APPI_IN_AAAG(x4) 53.83/15.30 53.83/15.30 53.83/15.30 We have to consider all (P,R,Pi)-chains 53.83/15.30 ---------------------------------------- 53.83/15.30 53.83/15.30 (150) PiDPToQDPProof (SOUND) 53.83/15.30 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 53.83/15.30 ---------------------------------------- 53.83/15.30 53.83/15.30 (151) 53.83/15.30 Obligation: 53.83/15.30 Q DP problem: 53.83/15.30 The TRS P consists of the following rules: 53.83/15.30 53.83/15.30 APPI_IN_AAAG(.(X1, X5)) -> APPI_IN_AAAG(X5) 53.83/15.30 53.83/15.30 R is empty. 53.83/15.30 Q is empty. 53.83/15.30 We have to consider all (P,Q,R)-chains. 53.83/15.30 ---------------------------------------- 53.83/15.30 53.83/15.30 (152) QDPSizeChangeProof (EQUIVALENT) 53.83/15.30 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. 53.83/15.30 53.83/15.30 From the DPs we obtained the following set of size-change graphs: 53.83/15.30 *APPI_IN_AAAG(.(X1, X5)) -> APPI_IN_AAAG(X5) 53.83/15.30 The graph contains the following edges 1 > 1 53.83/15.30 53.83/15.30 53.83/15.30 ---------------------------------------- 53.83/15.30 53.83/15.30 (153) 53.83/15.30 YES 53.83/15.30 53.83/15.30 ---------------------------------------- 53.83/15.30 53.83/15.30 (154) 53.83/15.30 Obligation: 53.83/15.30 Pi DP problem: 53.83/15.30 The TRS P consists of the following rules: 53.83/15.30 53.83/15.30 APPH_IN_AGAG(.(X1, X2), X3, X4, .(X1, X5)) -> APPH_IN_AGAG(X2, X3, X4, X5) 53.83/15.30 53.83/15.30 The TRS R consists of the following rules: 53.83/15.30 53.83/15.30 lesscA_in_aa(0, s(X1)) -> lesscA_out_aa(0, s(X1)) 53.83/15.30 lesscA_in_aa(s(X1), s(X2)) -> U63_aa(X1, X2, lesscA_in_aa(X1, X2)) 53.83/15.30 U63_aa(X1, X2, lesscA_out_aa(X1, X2)) -> lesscA_out_aa(s(X1), s(X2)) 53.83/15.30 lesscA_in_ga(0, s(X1)) -> lesscA_out_ga(0, s(X1)) 53.83/15.30 lesscA_in_ga(s(X1), s(X2)) -> U63_ga(X1, X2, lesscA_in_ga(X1, X2)) 53.83/15.30 U63_ga(X1, X2, lesscA_out_ga(X1, X2)) -> lesscA_out_ga(s(X1), s(X2)) 53.83/15.30 partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_gaaa(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) 53.83/15.30 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)) 53.83/15.30 partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) 53.83/15.30 partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], []) 53.83/15.30 U66_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.30 U65_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.30 partcC_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_aaaa(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) 53.83/15.30 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)) 53.83/15.30 U68_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.30 partcC_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_aaaa(X1, X2, X3, X4, X5, partcC_in_aaaa(X1, X3, X4, X5)) 53.83/15.30 partcC_in_aaaa(X1, [], [], []) -> partcC_out_aaaa(X1, [], [], []) 53.83/15.30 U69_aaaa(X1, X2, X3, X4, X5, partcC_out_aaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.30 qscF_in_aa([], []) -> qscF_out_aa([], []) 53.83/15.30 qscF_in_aa(.(X1, X2), X3) -> U74_aa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.30 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)) 53.83/15.30 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)) 53.83/15.30 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)) 53.83/15.30 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)) 53.83/15.30 appcE_in_aaaa([], X1, X2, .(X1, X2)) -> appcE_out_aaaa([], X1, X2, .(X1, X2)) 53.83/15.30 appcE_in_aaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U75_aaaa(X1, X2, X3, X4, X5, appcE_in_aaaa(X2, X3, X4, X5)) 53.83/15.30 U75_aaaa(X1, X2, X3, X4, X5, appcE_out_aaaa(X2, X3, X4, X5)) -> appcE_out_aaaa(.(X1, X2), X3, X4, .(X1, X5)) 53.83/15.30 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) 53.83/15.30 U74_aa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_aa(.(X1, X2), X3) 53.83/15.30 qscG_in_aaa(X1, X2, X3) -> U76_aaa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.30 U76_aaa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscG_out_aaa(X1, X2, X3) 53.83/15.30 lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) 53.83/15.30 lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) 53.83/15.30 U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) 53.83/15.30 partcC_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.83/15.30 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)) 53.83/15.30 partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.83/15.30 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)) 53.83/15.30 partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) 53.83/15.30 partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], []) 53.83/15.30 U66_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.30 U65_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.30 U68_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.30 partcC_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_ggaa(X1, X2, X3, X4, X5, partcC_in_ggaa(X1, X3, X4, X5)) 53.83/15.30 partcC_in_ggaa(X1, [], [], []) -> partcC_out_ggaa(X1, [], [], []) 53.83/15.30 U69_ggaa(X1, X2, X3, X4, X5, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.30 qscF_in_ga([], []) -> qscF_out_ga([], []) 53.83/15.30 qscF_in_ga(.(X1, X2), X3) -> U74_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.30 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)) 53.83/15.30 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)) 53.83/15.30 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)) 53.83/15.30 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)) 53.83/15.30 appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2)) 53.83/15.30 appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U75_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5)) 53.83/15.30 U75_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5)) 53.83/15.30 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) 53.83/15.30 U74_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3) 53.83/15.30 53.83/15.30 The argument filtering Pi contains the following mapping: 53.83/15.30 [] = [] 53.83/15.30 53.83/15.30 lesscA_in_aa(x1, x2) = lesscA_in_aa 53.83/15.30 53.83/15.30 lesscA_out_aa(x1, x2) = lesscA_out_aa(x1) 53.83/15.30 53.83/15.30 U63_aa(x1, x2, x3) = U63_aa(x3) 53.83/15.30 53.83/15.30 s(x1) = s(x1) 53.83/15.30 53.83/15.30 lesscA_in_ga(x1, x2) = lesscA_in_ga(x1) 53.83/15.30 53.83/15.30 0 = 0 53.83/15.30 53.83/15.30 lesscA_out_ga(x1, x2) = lesscA_out_ga(x1) 53.83/15.30 53.83/15.30 U63_ga(x1, x2, x3) = U63_ga(x1, x3) 53.83/15.30 53.83/15.30 partcB_in_gaaa(x1, x2, x3, x4) = partcB_in_gaaa(x1) 53.83/15.30 53.83/15.30 U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6) 53.83/15.30 53.83/15.30 U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x6) 53.83/15.30 53.83/15.30 U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6) 53.83/15.30 53.83/15.30 partcB_out_gaaa(x1, x2, x3, x4) = partcB_out_gaaa(x1) 53.83/15.30 53.83/15.30 partcC_in_aaaa(x1, x2, x3, x4) = partcC_in_aaaa 53.83/15.30 53.83/15.30 U67_aaaa(x1, x2, x3, x4, x5, x6) = U67_aaaa(x6) 53.83/15.30 53.83/15.30 U68_aaaa(x1, x2, x3, x4, x5, x6) = U68_aaaa(x6) 53.83/15.30 53.83/15.30 partcC_out_aaaa(x1, x2, x3, x4) = partcC_out_aaaa 53.83/15.30 53.83/15.30 U69_aaaa(x1, x2, x3, x4, x5, x6) = U69_aaaa(x6) 53.83/15.30 53.83/15.30 qscF_in_aa(x1, x2) = qscF_in_aa 53.83/15.30 53.83/15.30 qscF_out_aa(x1, x2) = qscF_out_aa 53.83/15.30 53.83/15.30 U74_aa(x1, x2, x3, x4) = U74_aa(x4) 53.83/15.30 53.83/15.30 qcD_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_aaaaaaa 53.83/15.30 53.83/15.30 U70_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_aaaaaaa(x8) 53.83/15.30 53.83/15.30 U71_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_aaaaaaa(x8) 53.83/15.30 53.83/15.30 U72_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_aaaaaaa(x8) 53.83/15.30 53.83/15.30 U73_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_aaaaaaa(x8) 53.83/15.30 53.83/15.30 .(x1, x2) = .(x1, x2) 53.83/15.30 53.83/15.30 qcD_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_aaaaaaa 53.83/15.30 53.83/15.30 appcE_in_aaaa(x1, x2, x3, x4) = appcE_in_aaaa 53.83/15.30 53.83/15.30 appcE_out_aaaa(x1, x2, x3, x4) = appcE_out_aaaa 53.83/15.30 53.83/15.30 U75_aaaa(x1, x2, x3, x4, x5, x6) = U75_aaaa(x6) 53.83/15.30 53.83/15.30 qscG_in_aaa(x1, x2, x3) = qscG_in_aaa 53.83/15.30 53.83/15.30 U76_aaa(x1, x2, x3, x4) = U76_aaa(x4) 53.83/15.30 53.83/15.30 qscG_out_aaa(x1, x2, x3) = qscG_out_aaa 53.83/15.30 53.83/15.30 lesscA_in_gg(x1, x2) = lesscA_in_gg(x1, x2) 53.83/15.30 53.83/15.30 lesscA_out_gg(x1, x2) = lesscA_out_gg(x1, x2) 53.83/15.30 53.83/15.30 U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3) 53.83/15.30 53.83/15.30 partcC_in_ggaa(x1, x2, x3, x4) = partcC_in_ggaa(x1, x2) 53.83/15.30 53.83/15.30 U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6) 53.83/15.30 53.83/15.30 U68_ggaa(x1, x2, x3, x4, x5, x6) = U68_ggaa(x1, x2, x3, x6) 53.83/15.30 53.83/15.30 partcB_in_ggaa(x1, x2, x3, x4) = partcB_in_ggaa(x1, x2) 53.83/15.30 53.83/15.30 U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6) 53.83/15.30 53.83/15.30 U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6) 53.83/15.30 53.83/15.30 U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6) 53.83/15.30 53.83/15.30 partcB_out_ggaa(x1, x2, x3, x4) = partcB_out_ggaa(x1, x2, x3, x4) 53.83/15.30 53.83/15.30 partcC_out_ggaa(x1, x2, x3, x4) = partcC_out_ggaa(x1, x2, x3, x4) 53.83/15.30 53.83/15.30 U69_ggaa(x1, x2, x3, x4, x5, x6) = U69_ggaa(x1, x2, x3, x6) 53.83/15.30 53.83/15.30 qscF_in_ga(x1, x2) = qscF_in_ga(x1) 53.83/15.30 53.83/15.30 qscF_out_ga(x1, x2) = qscF_out_ga(x1, x2) 53.83/15.30 53.83/15.30 U74_ga(x1, x2, x3, x4) = U74_ga(x1, x2, x4) 53.83/15.30 53.83/15.30 qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_ggaaaaa(x1, x2) 53.83/15.30 53.83/15.30 U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x8) 53.83/15.30 53.83/15.30 U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x8) 53.83/15.30 53.83/15.30 U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x8) 53.83/15.30 53.83/15.30 U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x8) 53.83/15.30 53.83/15.30 appcE_in_ggga(x1, x2, x3, x4) = appcE_in_ggga(x1, x2, x3) 53.83/15.30 53.83/15.30 appcE_out_ggga(x1, x2, x3, x4) = appcE_out_ggga(x1, x2, x3, x4) 53.83/15.30 53.83/15.30 U75_ggga(x1, x2, x3, x4, x5, x6) = U75_ggga(x1, x2, x3, x4, x6) 53.83/15.30 53.83/15.30 qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) 53.83/15.30 53.83/15.30 APPH_IN_AGAG(x1, x2, x3, x4) = APPH_IN_AGAG(x2, x4) 53.83/15.30 53.83/15.30 53.83/15.30 We have to consider all (P,R,Pi)-chains 53.83/15.30 ---------------------------------------- 53.83/15.30 53.83/15.30 (155) UsableRulesProof (EQUIVALENT) 53.83/15.30 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 53.83/15.30 ---------------------------------------- 53.83/15.30 53.83/15.30 (156) 53.83/15.30 Obligation: 53.83/15.30 Pi DP problem: 53.83/15.30 The TRS P consists of the following rules: 53.83/15.30 53.83/15.30 APPH_IN_AGAG(.(X1, X2), X3, X4, .(X1, X5)) -> APPH_IN_AGAG(X2, X3, X4, X5) 53.83/15.30 53.83/15.30 R is empty. 53.83/15.30 The argument filtering Pi contains the following mapping: 53.83/15.30 .(x1, x2) = .(x1, x2) 53.83/15.30 53.83/15.30 APPH_IN_AGAG(x1, x2, x3, x4) = APPH_IN_AGAG(x2, x4) 53.83/15.30 53.83/15.30 53.83/15.30 We have to consider all (P,R,Pi)-chains 53.83/15.30 ---------------------------------------- 53.83/15.30 53.83/15.30 (157) PiDPToQDPProof (SOUND) 53.83/15.30 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 53.83/15.30 ---------------------------------------- 53.83/15.30 53.83/15.30 (158) 53.83/15.30 Obligation: 53.83/15.30 Q DP problem: 53.83/15.30 The TRS P consists of the following rules: 53.83/15.30 53.83/15.30 APPH_IN_AGAG(X3, .(X1, X5)) -> APPH_IN_AGAG(X3, X5) 53.83/15.30 53.83/15.30 R is empty. 53.83/15.30 Q is empty. 53.83/15.30 We have to consider all (P,Q,R)-chains. 53.83/15.30 ---------------------------------------- 53.83/15.30 53.83/15.30 (159) QDPSizeChangeProof (EQUIVALENT) 53.83/15.30 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. 53.83/15.30 53.83/15.30 From the DPs we obtained the following set of size-change graphs: 53.83/15.30 *APPH_IN_AGAG(X3, .(X1, X5)) -> APPH_IN_AGAG(X3, X5) 53.83/15.30 The graph contains the following edges 1 >= 1, 2 > 2 53.83/15.30 53.83/15.30 53.83/15.30 ---------------------------------------- 53.83/15.30 53.83/15.30 (160) 53.83/15.30 YES 53.83/15.30 53.83/15.30 ---------------------------------------- 53.83/15.30 53.83/15.30 (161) 53.83/15.30 Obligation: 53.83/15.30 Pi DP problem: 53.83/15.30 The TRS P consists of the following rules: 53.83/15.30 53.83/15.30 APPE_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) -> APPE_IN_GGGA(X2, X3, X4, X5) 53.83/15.30 53.83/15.30 The TRS R consists of the following rules: 53.83/15.30 53.83/15.30 lesscA_in_aa(0, s(X1)) -> lesscA_out_aa(0, s(X1)) 53.83/15.30 lesscA_in_aa(s(X1), s(X2)) -> U63_aa(X1, X2, lesscA_in_aa(X1, X2)) 53.83/15.30 U63_aa(X1, X2, lesscA_out_aa(X1, X2)) -> lesscA_out_aa(s(X1), s(X2)) 53.83/15.30 lesscA_in_ga(0, s(X1)) -> lesscA_out_ga(0, s(X1)) 53.83/15.30 lesscA_in_ga(s(X1), s(X2)) -> U63_ga(X1, X2, lesscA_in_ga(X1, X2)) 53.83/15.30 U63_ga(X1, X2, lesscA_out_ga(X1, X2)) -> lesscA_out_ga(s(X1), s(X2)) 53.83/15.30 partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_gaaa(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) 53.83/15.30 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)) 53.83/15.30 partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) 53.83/15.30 partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], []) 53.83/15.30 U66_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.30 U65_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.30 partcC_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_aaaa(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) 53.83/15.30 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)) 53.83/15.30 U68_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.30 partcC_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_aaaa(X1, X2, X3, X4, X5, partcC_in_aaaa(X1, X3, X4, X5)) 53.83/15.30 partcC_in_aaaa(X1, [], [], []) -> partcC_out_aaaa(X1, [], [], []) 53.83/15.30 U69_aaaa(X1, X2, X3, X4, X5, partcC_out_aaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.30 qscF_in_aa([], []) -> qscF_out_aa([], []) 53.83/15.30 qscF_in_aa(.(X1, X2), X3) -> U74_aa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.30 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)) 53.83/15.30 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)) 53.83/15.30 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)) 53.83/15.30 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)) 53.83/15.30 appcE_in_aaaa([], X1, X2, .(X1, X2)) -> appcE_out_aaaa([], X1, X2, .(X1, X2)) 53.83/15.30 appcE_in_aaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U75_aaaa(X1, X2, X3, X4, X5, appcE_in_aaaa(X2, X3, X4, X5)) 53.83/15.30 U75_aaaa(X1, X2, X3, X4, X5, appcE_out_aaaa(X2, X3, X4, X5)) -> appcE_out_aaaa(.(X1, X2), X3, X4, .(X1, X5)) 53.83/15.30 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) 53.83/15.30 U74_aa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_aa(.(X1, X2), X3) 53.83/15.30 qscG_in_aaa(X1, X2, X3) -> U76_aaa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.30 U76_aaa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscG_out_aaa(X1, X2, X3) 53.83/15.30 lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) 53.83/15.30 lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) 53.83/15.30 U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) 53.83/15.30 partcC_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.83/15.30 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)) 53.83/15.30 partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.83/15.30 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)) 53.83/15.30 partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) 53.83/15.30 partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], []) 53.83/15.30 U66_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.30 U65_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.30 U68_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.30 partcC_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_ggaa(X1, X2, X3, X4, X5, partcC_in_ggaa(X1, X3, X4, X5)) 53.83/15.30 partcC_in_ggaa(X1, [], [], []) -> partcC_out_ggaa(X1, [], [], []) 53.83/15.30 U69_ggaa(X1, X2, X3, X4, X5, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.30 qscF_in_ga([], []) -> qscF_out_ga([], []) 53.83/15.30 qscF_in_ga(.(X1, X2), X3) -> U74_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.30 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)) 53.83/15.30 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)) 53.83/15.30 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)) 53.83/15.30 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)) 53.83/15.30 appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2)) 53.83/15.30 appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U75_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5)) 53.83/15.30 U75_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5)) 53.83/15.30 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) 53.83/15.30 U74_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3) 53.83/15.30 53.83/15.30 The argument filtering Pi contains the following mapping: 53.83/15.30 [] = [] 53.83/15.30 53.83/15.30 lesscA_in_aa(x1, x2) = lesscA_in_aa 53.83/15.30 53.83/15.30 lesscA_out_aa(x1, x2) = lesscA_out_aa(x1) 53.83/15.30 53.83/15.30 U63_aa(x1, x2, x3) = U63_aa(x3) 53.83/15.30 53.83/15.30 s(x1) = s(x1) 53.83/15.30 53.83/15.30 lesscA_in_ga(x1, x2) = lesscA_in_ga(x1) 53.83/15.30 53.83/15.30 0 = 0 53.83/15.30 53.83/15.30 lesscA_out_ga(x1, x2) = lesscA_out_ga(x1) 53.83/15.30 53.83/15.30 U63_ga(x1, x2, x3) = U63_ga(x1, x3) 53.83/15.30 53.83/15.30 partcB_in_gaaa(x1, x2, x3, x4) = partcB_in_gaaa(x1) 53.83/15.30 53.83/15.30 U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6) 53.83/15.30 53.83/15.30 U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x6) 53.83/15.30 53.83/15.30 U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6) 53.83/15.30 53.83/15.30 partcB_out_gaaa(x1, x2, x3, x4) = partcB_out_gaaa(x1) 53.83/15.30 53.83/15.30 partcC_in_aaaa(x1, x2, x3, x4) = partcC_in_aaaa 53.83/15.30 53.83/15.30 U67_aaaa(x1, x2, x3, x4, x5, x6) = U67_aaaa(x6) 53.83/15.30 53.83/15.30 U68_aaaa(x1, x2, x3, x4, x5, x6) = U68_aaaa(x6) 53.83/15.30 53.83/15.30 partcC_out_aaaa(x1, x2, x3, x4) = partcC_out_aaaa 53.83/15.30 53.83/15.30 U69_aaaa(x1, x2, x3, x4, x5, x6) = U69_aaaa(x6) 53.83/15.30 53.83/15.30 qscF_in_aa(x1, x2) = qscF_in_aa 53.83/15.30 53.83/15.30 qscF_out_aa(x1, x2) = qscF_out_aa 53.83/15.30 53.83/15.30 U74_aa(x1, x2, x3, x4) = U74_aa(x4) 53.83/15.30 53.83/15.30 qcD_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_aaaaaaa 53.83/15.30 53.83/15.30 U70_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_aaaaaaa(x8) 53.83/15.30 53.83/15.30 U71_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_aaaaaaa(x8) 53.83/15.30 53.83/15.30 U72_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_aaaaaaa(x8) 53.83/15.30 53.83/15.30 U73_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_aaaaaaa(x8) 53.83/15.30 53.83/15.30 .(x1, x2) = .(x1, x2) 53.83/15.30 53.83/15.30 qcD_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_aaaaaaa 53.83/15.30 53.83/15.30 appcE_in_aaaa(x1, x2, x3, x4) = appcE_in_aaaa 53.83/15.30 53.83/15.30 appcE_out_aaaa(x1, x2, x3, x4) = appcE_out_aaaa 53.83/15.30 53.83/15.30 U75_aaaa(x1, x2, x3, x4, x5, x6) = U75_aaaa(x6) 53.83/15.30 53.83/15.30 qscG_in_aaa(x1, x2, x3) = qscG_in_aaa 53.83/15.30 53.83/15.30 U76_aaa(x1, x2, x3, x4) = U76_aaa(x4) 53.83/15.30 53.83/15.30 qscG_out_aaa(x1, x2, x3) = qscG_out_aaa 53.83/15.30 53.83/15.30 lesscA_in_gg(x1, x2) = lesscA_in_gg(x1, x2) 53.83/15.30 53.83/15.30 lesscA_out_gg(x1, x2) = lesscA_out_gg(x1, x2) 53.83/15.30 53.83/15.30 U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3) 53.83/15.30 53.83/15.30 partcC_in_ggaa(x1, x2, x3, x4) = partcC_in_ggaa(x1, x2) 53.83/15.30 53.83/15.30 U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6) 53.83/15.30 53.83/15.30 U68_ggaa(x1, x2, x3, x4, x5, x6) = U68_ggaa(x1, x2, x3, x6) 53.83/15.30 53.83/15.30 partcB_in_ggaa(x1, x2, x3, x4) = partcB_in_ggaa(x1, x2) 53.83/15.30 53.83/15.30 U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6) 53.83/15.30 53.83/15.30 U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6) 53.83/15.30 53.83/15.30 U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6) 53.83/15.30 53.83/15.30 partcB_out_ggaa(x1, x2, x3, x4) = partcB_out_ggaa(x1, x2, x3, x4) 53.83/15.30 53.83/15.30 partcC_out_ggaa(x1, x2, x3, x4) = partcC_out_ggaa(x1, x2, x3, x4) 53.83/15.30 53.83/15.30 U69_ggaa(x1, x2, x3, x4, x5, x6) = U69_ggaa(x1, x2, x3, x6) 53.83/15.30 53.83/15.30 qscF_in_ga(x1, x2) = qscF_in_ga(x1) 53.83/15.30 53.83/15.30 qscF_out_ga(x1, x2) = qscF_out_ga(x1, x2) 53.83/15.30 53.83/15.30 U74_ga(x1, x2, x3, x4) = U74_ga(x1, x2, x4) 53.83/15.30 53.83/15.30 qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_ggaaaaa(x1, x2) 53.83/15.30 53.83/15.30 U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x8) 53.83/15.30 53.83/15.30 U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x8) 53.83/15.30 53.83/15.30 U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x8) 53.83/15.30 53.83/15.30 U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x8) 53.83/15.30 53.83/15.30 appcE_in_ggga(x1, x2, x3, x4) = appcE_in_ggga(x1, x2, x3) 53.83/15.30 53.83/15.30 appcE_out_ggga(x1, x2, x3, x4) = appcE_out_ggga(x1, x2, x3, x4) 53.83/15.30 53.83/15.30 U75_ggga(x1, x2, x3, x4, x5, x6) = U75_ggga(x1, x2, x3, x4, x6) 53.83/15.30 53.83/15.30 qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) 53.83/15.30 53.83/15.30 APPE_IN_GGGA(x1, x2, x3, x4) = APPE_IN_GGGA(x1, x2, x3) 53.83/15.30 53.83/15.30 53.83/15.30 We have to consider all (P,R,Pi)-chains 53.83/15.30 ---------------------------------------- 53.83/15.30 53.83/15.30 (162) UsableRulesProof (EQUIVALENT) 53.83/15.30 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 53.83/15.30 ---------------------------------------- 53.83/15.30 53.83/15.30 (163) 53.83/15.30 Obligation: 53.83/15.30 Pi DP problem: 53.83/15.30 The TRS P consists of the following rules: 53.83/15.30 53.83/15.30 APPE_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) -> APPE_IN_GGGA(X2, X3, X4, X5) 53.83/15.30 53.83/15.30 R is empty. 53.83/15.30 The argument filtering Pi contains the following mapping: 53.83/15.30 .(x1, x2) = .(x1, x2) 53.83/15.30 53.83/15.30 APPE_IN_GGGA(x1, x2, x3, x4) = APPE_IN_GGGA(x1, x2, x3) 53.83/15.30 53.83/15.30 53.83/15.30 We have to consider all (P,R,Pi)-chains 53.83/15.30 ---------------------------------------- 53.83/15.30 53.83/15.30 (164) PiDPToQDPProof (SOUND) 53.83/15.30 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 53.83/15.30 ---------------------------------------- 53.83/15.30 53.83/15.30 (165) 53.83/15.30 Obligation: 53.83/15.30 Q DP problem: 53.83/15.30 The TRS P consists of the following rules: 53.83/15.30 53.83/15.30 APPE_IN_GGGA(.(X1, X2), X3, X4) -> APPE_IN_GGGA(X2, X3, X4) 53.83/15.30 53.83/15.30 R is empty. 53.83/15.30 Q is empty. 53.83/15.30 We have to consider all (P,Q,R)-chains. 53.83/15.30 ---------------------------------------- 53.83/15.30 53.83/15.30 (166) QDPSizeChangeProof (EQUIVALENT) 53.83/15.30 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. 53.83/15.30 53.83/15.30 From the DPs we obtained the following set of size-change graphs: 53.83/15.30 *APPE_IN_GGGA(.(X1, X2), X3, X4) -> APPE_IN_GGGA(X2, X3, X4) 53.83/15.30 The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 53.83/15.30 53.83/15.30 53.83/15.30 ---------------------------------------- 53.83/15.30 53.83/15.30 (167) 53.83/15.30 YES 53.83/15.30 53.83/15.30 ---------------------------------------- 53.83/15.30 53.83/15.30 (168) 53.83/15.30 Obligation: 53.83/15.30 Pi DP problem: 53.83/15.30 The TRS P consists of the following rules: 53.83/15.30 53.83/15.30 LESSA_IN_GG(s(X1), s(X2)) -> LESSA_IN_GG(X1, X2) 53.83/15.30 53.83/15.30 The TRS R consists of the following rules: 53.83/15.30 53.83/15.30 lesscA_in_aa(0, s(X1)) -> lesscA_out_aa(0, s(X1)) 53.83/15.30 lesscA_in_aa(s(X1), s(X2)) -> U63_aa(X1, X2, lesscA_in_aa(X1, X2)) 53.83/15.30 U63_aa(X1, X2, lesscA_out_aa(X1, X2)) -> lesscA_out_aa(s(X1), s(X2)) 53.83/15.30 lesscA_in_ga(0, s(X1)) -> lesscA_out_ga(0, s(X1)) 53.83/15.30 lesscA_in_ga(s(X1), s(X2)) -> U63_ga(X1, X2, lesscA_in_ga(X1, X2)) 53.83/15.30 U63_ga(X1, X2, lesscA_out_ga(X1, X2)) -> lesscA_out_ga(s(X1), s(X2)) 53.83/15.30 partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_gaaa(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) 53.83/15.30 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)) 53.83/15.30 partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) 53.83/15.30 partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], []) 53.83/15.30 U66_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.30 U65_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.30 partcC_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_aaaa(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) 53.83/15.30 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)) 53.83/15.30 U68_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.30 partcC_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_aaaa(X1, X2, X3, X4, X5, partcC_in_aaaa(X1, X3, X4, X5)) 53.83/15.30 partcC_in_aaaa(X1, [], [], []) -> partcC_out_aaaa(X1, [], [], []) 53.83/15.30 U69_aaaa(X1, X2, X3, X4, X5, partcC_out_aaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.30 qscF_in_aa([], []) -> qscF_out_aa([], []) 53.83/15.30 qscF_in_aa(.(X1, X2), X3) -> U74_aa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.30 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)) 53.83/15.30 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)) 53.83/15.30 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)) 53.83/15.30 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)) 53.83/15.30 appcE_in_aaaa([], X1, X2, .(X1, X2)) -> appcE_out_aaaa([], X1, X2, .(X1, X2)) 53.83/15.30 appcE_in_aaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U75_aaaa(X1, X2, X3, X4, X5, appcE_in_aaaa(X2, X3, X4, X5)) 53.83/15.30 U75_aaaa(X1, X2, X3, X4, X5, appcE_out_aaaa(X2, X3, X4, X5)) -> appcE_out_aaaa(.(X1, X2), X3, X4, .(X1, X5)) 53.83/15.30 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) 53.83/15.30 U74_aa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_aa(.(X1, X2), X3) 53.83/15.30 qscG_in_aaa(X1, X2, X3) -> U76_aaa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.30 U76_aaa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscG_out_aaa(X1, X2, X3) 53.83/15.30 lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) 53.83/15.30 lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) 53.83/15.30 U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) 53.83/15.30 partcC_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.83/15.30 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)) 53.83/15.30 partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.83/15.30 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)) 53.83/15.30 partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) 53.83/15.30 partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], []) 53.83/15.30 U66_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.30 U65_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.30 U68_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.30 partcC_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_ggaa(X1, X2, X3, X4, X5, partcC_in_ggaa(X1, X3, X4, X5)) 53.83/15.30 partcC_in_ggaa(X1, [], [], []) -> partcC_out_ggaa(X1, [], [], []) 53.83/15.30 U69_ggaa(X1, X2, X3, X4, X5, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.30 qscF_in_ga([], []) -> qscF_out_ga([], []) 53.83/15.30 qscF_in_ga(.(X1, X2), X3) -> U74_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.30 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)) 53.83/15.30 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)) 53.83/15.30 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)) 53.83/15.30 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)) 53.83/15.30 appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2)) 53.83/15.30 appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U75_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5)) 53.83/15.30 U75_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5)) 53.83/15.30 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) 53.83/15.30 U74_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3) 53.83/15.30 53.83/15.30 The argument filtering Pi contains the following mapping: 53.83/15.30 [] = [] 53.83/15.30 53.83/15.30 lesscA_in_aa(x1, x2) = lesscA_in_aa 53.83/15.30 53.83/15.30 lesscA_out_aa(x1, x2) = lesscA_out_aa(x1) 53.83/15.30 53.83/15.30 U63_aa(x1, x2, x3) = U63_aa(x3) 53.83/15.30 53.83/15.30 s(x1) = s(x1) 53.83/15.30 53.83/15.30 lesscA_in_ga(x1, x2) = lesscA_in_ga(x1) 53.83/15.30 53.83/15.30 0 = 0 53.83/15.30 53.83/15.30 lesscA_out_ga(x1, x2) = lesscA_out_ga(x1) 53.83/15.30 53.83/15.30 U63_ga(x1, x2, x3) = U63_ga(x1, x3) 53.83/15.30 53.83/15.30 partcB_in_gaaa(x1, x2, x3, x4) = partcB_in_gaaa(x1) 53.83/15.30 53.83/15.30 U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6) 53.83/15.30 53.83/15.30 U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x6) 53.83/15.30 53.83/15.30 U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6) 53.83/15.30 53.83/15.30 partcB_out_gaaa(x1, x2, x3, x4) = partcB_out_gaaa(x1) 53.83/15.30 53.83/15.30 partcC_in_aaaa(x1, x2, x3, x4) = partcC_in_aaaa 53.83/15.30 53.83/15.30 U67_aaaa(x1, x2, x3, x4, x5, x6) = U67_aaaa(x6) 53.83/15.30 53.83/15.30 U68_aaaa(x1, x2, x3, x4, x5, x6) = U68_aaaa(x6) 53.83/15.30 53.83/15.30 partcC_out_aaaa(x1, x2, x3, x4) = partcC_out_aaaa 53.83/15.30 53.83/15.30 U69_aaaa(x1, x2, x3, x4, x5, x6) = U69_aaaa(x6) 53.83/15.30 53.83/15.30 qscF_in_aa(x1, x2) = qscF_in_aa 53.83/15.30 53.83/15.30 qscF_out_aa(x1, x2) = qscF_out_aa 53.83/15.30 53.83/15.30 U74_aa(x1, x2, x3, x4) = U74_aa(x4) 53.83/15.30 53.83/15.30 qcD_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_aaaaaaa 53.83/15.30 53.83/15.30 U70_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_aaaaaaa(x8) 53.83/15.30 53.83/15.30 U71_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_aaaaaaa(x8) 53.83/15.30 53.83/15.30 U72_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_aaaaaaa(x8) 53.83/15.30 53.83/15.30 U73_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_aaaaaaa(x8) 53.83/15.30 53.83/15.30 .(x1, x2) = .(x1, x2) 53.83/15.30 53.83/15.30 qcD_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_aaaaaaa 53.83/15.30 53.83/15.30 appcE_in_aaaa(x1, x2, x3, x4) = appcE_in_aaaa 53.83/15.30 53.83/15.30 appcE_out_aaaa(x1, x2, x3, x4) = appcE_out_aaaa 53.83/15.30 53.83/15.30 U75_aaaa(x1, x2, x3, x4, x5, x6) = U75_aaaa(x6) 53.83/15.30 53.83/15.30 qscG_in_aaa(x1, x2, x3) = qscG_in_aaa 53.83/15.30 53.83/15.30 U76_aaa(x1, x2, x3, x4) = U76_aaa(x4) 53.83/15.30 53.83/15.30 qscG_out_aaa(x1, x2, x3) = qscG_out_aaa 53.83/15.30 53.83/15.30 lesscA_in_gg(x1, x2) = lesscA_in_gg(x1, x2) 53.83/15.30 53.83/15.30 lesscA_out_gg(x1, x2) = lesscA_out_gg(x1, x2) 53.83/15.30 53.83/15.30 U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3) 53.83/15.30 53.83/15.30 partcC_in_ggaa(x1, x2, x3, x4) = partcC_in_ggaa(x1, x2) 53.83/15.30 53.83/15.30 U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6) 53.83/15.30 53.83/15.30 U68_ggaa(x1, x2, x3, x4, x5, x6) = U68_ggaa(x1, x2, x3, x6) 53.83/15.30 53.83/15.30 partcB_in_ggaa(x1, x2, x3, x4) = partcB_in_ggaa(x1, x2) 53.83/15.30 53.83/15.30 U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6) 53.83/15.30 53.83/15.30 U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6) 53.83/15.30 53.83/15.30 U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6) 53.83/15.30 53.83/15.30 partcB_out_ggaa(x1, x2, x3, x4) = partcB_out_ggaa(x1, x2, x3, x4) 53.83/15.30 53.83/15.30 partcC_out_ggaa(x1, x2, x3, x4) = partcC_out_ggaa(x1, x2, x3, x4) 53.83/15.30 53.83/15.30 U69_ggaa(x1, x2, x3, x4, x5, x6) = U69_ggaa(x1, x2, x3, x6) 53.83/15.30 53.83/15.30 qscF_in_ga(x1, x2) = qscF_in_ga(x1) 53.83/15.30 53.83/15.30 qscF_out_ga(x1, x2) = qscF_out_ga(x1, x2) 53.83/15.30 53.83/15.30 U74_ga(x1, x2, x3, x4) = U74_ga(x1, x2, x4) 53.83/15.30 53.83/15.30 qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_ggaaaaa(x1, x2) 53.83/15.30 53.83/15.30 U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x8) 53.83/15.30 53.83/15.30 U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x8) 53.83/15.30 53.83/15.30 U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x8) 53.83/15.30 53.83/15.30 U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x8) 53.83/15.30 53.83/15.30 appcE_in_ggga(x1, x2, x3, x4) = appcE_in_ggga(x1, x2, x3) 53.83/15.30 53.83/15.30 appcE_out_ggga(x1, x2, x3, x4) = appcE_out_ggga(x1, x2, x3, x4) 53.83/15.30 53.83/15.30 U75_ggga(x1, x2, x3, x4, x5, x6) = U75_ggga(x1, x2, x3, x4, x6) 53.83/15.30 53.83/15.30 qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) 53.83/15.30 53.83/15.30 LESSA_IN_GG(x1, x2) = LESSA_IN_GG(x1, x2) 53.83/15.30 53.83/15.30 53.83/15.30 We have to consider all (P,R,Pi)-chains 53.83/15.30 ---------------------------------------- 53.83/15.30 53.83/15.30 (169) UsableRulesProof (EQUIVALENT) 53.83/15.30 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 53.83/15.30 ---------------------------------------- 53.83/15.30 53.83/15.30 (170) 53.83/15.30 Obligation: 53.83/15.30 Pi DP problem: 53.83/15.30 The TRS P consists of the following rules: 53.83/15.30 53.83/15.30 LESSA_IN_GG(s(X1), s(X2)) -> LESSA_IN_GG(X1, X2) 53.83/15.30 53.83/15.30 R is empty. 53.83/15.30 Pi is empty. 53.83/15.30 We have to consider all (P,R,Pi)-chains 53.83/15.30 ---------------------------------------- 53.83/15.30 53.83/15.30 (171) PiDPToQDPProof (EQUIVALENT) 53.83/15.30 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 53.83/15.30 ---------------------------------------- 53.83/15.30 53.83/15.30 (172) 53.83/15.30 Obligation: 53.83/15.30 Q DP problem: 53.83/15.30 The TRS P consists of the following rules: 53.83/15.30 53.83/15.30 LESSA_IN_GG(s(X1), s(X2)) -> LESSA_IN_GG(X1, X2) 53.83/15.30 53.83/15.30 R is empty. 53.83/15.30 Q is empty. 53.83/15.30 We have to consider all (P,Q,R)-chains. 53.83/15.30 ---------------------------------------- 53.83/15.30 53.83/15.30 (173) QDPSizeChangeProof (EQUIVALENT) 53.83/15.30 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. 53.83/15.30 53.83/15.30 From the DPs we obtained the following set of size-change graphs: 53.83/15.30 *LESSA_IN_GG(s(X1), s(X2)) -> LESSA_IN_GG(X1, X2) 53.83/15.30 The graph contains the following edges 1 > 1, 2 > 2 53.83/15.30 53.83/15.30 53.83/15.30 ---------------------------------------- 53.83/15.30 53.83/15.30 (174) 53.83/15.30 YES 53.83/15.30 53.83/15.30 ---------------------------------------- 53.83/15.30 53.83/15.30 (175) 53.83/15.30 Obligation: 53.83/15.30 Pi DP problem: 53.83/15.30 The TRS P consists of the following rules: 53.83/15.30 53.83/15.30 PARTB_IN_GGAA(X1, .(X2, X3), .(X2, X4), X5) -> U3_GGAA(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.83/15.30 U3_GGAA(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> PARTB_IN_GGAA(X1, X3, X4, X5) 53.83/15.30 PARTB_IN_GGAA(X1, .(X2, X3), X4, .(X2, X5)) -> PARTB_IN_GGAA(X1, X3, X4, X5) 53.83/15.30 53.83/15.30 The TRS R consists of the following rules: 53.83/15.30 53.83/15.30 lesscA_in_aa(0, s(X1)) -> lesscA_out_aa(0, s(X1)) 53.83/15.30 lesscA_in_aa(s(X1), s(X2)) -> U63_aa(X1, X2, lesscA_in_aa(X1, X2)) 53.83/15.30 U63_aa(X1, X2, lesscA_out_aa(X1, X2)) -> lesscA_out_aa(s(X1), s(X2)) 53.83/15.30 lesscA_in_ga(0, s(X1)) -> lesscA_out_ga(0, s(X1)) 53.83/15.30 lesscA_in_ga(s(X1), s(X2)) -> U63_ga(X1, X2, lesscA_in_ga(X1, X2)) 53.83/15.30 U63_ga(X1, X2, lesscA_out_ga(X1, X2)) -> lesscA_out_ga(s(X1), s(X2)) 53.83/15.30 partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_gaaa(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) 53.83/15.30 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)) 53.83/15.30 partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) 53.83/15.30 partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], []) 53.83/15.30 U66_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.30 U65_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.30 partcC_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_aaaa(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) 53.83/15.30 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)) 53.83/15.30 U68_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.30 partcC_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_aaaa(X1, X2, X3, X4, X5, partcC_in_aaaa(X1, X3, X4, X5)) 53.83/15.30 partcC_in_aaaa(X1, [], [], []) -> partcC_out_aaaa(X1, [], [], []) 53.83/15.30 U69_aaaa(X1, X2, X3, X4, X5, partcC_out_aaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.30 qscF_in_aa([], []) -> qscF_out_aa([], []) 53.83/15.30 qscF_in_aa(.(X1, X2), X3) -> U74_aa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.30 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)) 53.83/15.30 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)) 53.83/15.30 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)) 53.83/15.30 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)) 53.83/15.30 appcE_in_aaaa([], X1, X2, .(X1, X2)) -> appcE_out_aaaa([], X1, X2, .(X1, X2)) 53.83/15.30 appcE_in_aaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U75_aaaa(X1, X2, X3, X4, X5, appcE_in_aaaa(X2, X3, X4, X5)) 53.83/15.30 U75_aaaa(X1, X2, X3, X4, X5, appcE_out_aaaa(X2, X3, X4, X5)) -> appcE_out_aaaa(.(X1, X2), X3, X4, .(X1, X5)) 53.83/15.30 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) 53.83/15.30 U74_aa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_aa(.(X1, X2), X3) 53.83/15.30 qscG_in_aaa(X1, X2, X3) -> U76_aaa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.30 U76_aaa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscG_out_aaa(X1, X2, X3) 53.83/15.30 lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) 53.83/15.30 lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) 53.83/15.30 U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) 53.83/15.30 partcC_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.83/15.30 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)) 53.83/15.30 partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.83/15.30 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)) 53.83/15.30 partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) 53.83/15.30 partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], []) 53.83/15.30 U66_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.30 U65_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.30 U68_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.30 partcC_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_ggaa(X1, X2, X3, X4, X5, partcC_in_ggaa(X1, X3, X4, X5)) 53.83/15.30 partcC_in_ggaa(X1, [], [], []) -> partcC_out_ggaa(X1, [], [], []) 53.83/15.30 U69_ggaa(X1, X2, X3, X4, X5, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.30 qscF_in_ga([], []) -> qscF_out_ga([], []) 53.83/15.30 qscF_in_ga(.(X1, X2), X3) -> U74_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.30 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)) 53.83/15.30 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)) 53.83/15.30 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)) 53.83/15.30 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)) 53.83/15.30 appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2)) 53.83/15.30 appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U75_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5)) 53.83/15.30 U75_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5)) 53.83/15.30 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) 53.83/15.30 U74_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3) 53.83/15.30 53.83/15.30 The argument filtering Pi contains the following mapping: 53.83/15.30 [] = [] 53.83/15.30 53.83/15.30 lesscA_in_aa(x1, x2) = lesscA_in_aa 53.83/15.30 53.83/15.30 lesscA_out_aa(x1, x2) = lesscA_out_aa(x1) 53.83/15.30 53.83/15.30 U63_aa(x1, x2, x3) = U63_aa(x3) 53.83/15.30 53.83/15.30 s(x1) = s(x1) 53.83/15.30 53.83/15.30 lesscA_in_ga(x1, x2) = lesscA_in_ga(x1) 53.83/15.30 53.83/15.30 0 = 0 53.83/15.30 53.83/15.30 lesscA_out_ga(x1, x2) = lesscA_out_ga(x1) 53.83/15.30 53.83/15.30 U63_ga(x1, x2, x3) = U63_ga(x1, x3) 53.83/15.30 53.83/15.30 partcB_in_gaaa(x1, x2, x3, x4) = partcB_in_gaaa(x1) 53.83/15.30 53.83/15.30 U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6) 53.83/15.30 53.83/15.30 U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x6) 53.83/15.30 53.83/15.30 U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6) 53.83/15.30 53.83/15.30 partcB_out_gaaa(x1, x2, x3, x4) = partcB_out_gaaa(x1) 53.83/15.30 53.83/15.30 partcC_in_aaaa(x1, x2, x3, x4) = partcC_in_aaaa 53.83/15.30 53.83/15.30 U67_aaaa(x1, x2, x3, x4, x5, x6) = U67_aaaa(x6) 53.83/15.30 53.83/15.30 U68_aaaa(x1, x2, x3, x4, x5, x6) = U68_aaaa(x6) 53.83/15.30 53.83/15.30 partcC_out_aaaa(x1, x2, x3, x4) = partcC_out_aaaa 53.83/15.30 53.83/15.30 U69_aaaa(x1, x2, x3, x4, x5, x6) = U69_aaaa(x6) 53.83/15.30 53.83/15.30 qscF_in_aa(x1, x2) = qscF_in_aa 53.83/15.30 53.83/15.30 qscF_out_aa(x1, x2) = qscF_out_aa 53.83/15.30 53.83/15.30 U74_aa(x1, x2, x3, x4) = U74_aa(x4) 53.83/15.30 53.83/15.30 qcD_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_aaaaaaa 53.83/15.30 53.83/15.30 U70_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_aaaaaaa(x8) 53.83/15.30 53.83/15.30 U71_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_aaaaaaa(x8) 53.83/15.30 53.83/15.30 U72_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_aaaaaaa(x8) 53.83/15.30 53.83/15.30 U73_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_aaaaaaa(x8) 53.83/15.30 53.83/15.30 .(x1, x2) = .(x1, x2) 53.83/15.30 53.83/15.30 qcD_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_aaaaaaa 53.83/15.30 53.83/15.30 appcE_in_aaaa(x1, x2, x3, x4) = appcE_in_aaaa 53.83/15.30 53.83/15.30 appcE_out_aaaa(x1, x2, x3, x4) = appcE_out_aaaa 53.83/15.30 53.83/15.30 U75_aaaa(x1, x2, x3, x4, x5, x6) = U75_aaaa(x6) 53.83/15.30 53.83/15.30 qscG_in_aaa(x1, x2, x3) = qscG_in_aaa 53.83/15.30 53.83/15.30 U76_aaa(x1, x2, x3, x4) = U76_aaa(x4) 53.83/15.30 53.83/15.30 qscG_out_aaa(x1, x2, x3) = qscG_out_aaa 53.83/15.30 53.83/15.30 lesscA_in_gg(x1, x2) = lesscA_in_gg(x1, x2) 53.83/15.30 53.83/15.30 lesscA_out_gg(x1, x2) = lesscA_out_gg(x1, x2) 53.83/15.30 53.83/15.30 U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3) 53.83/15.30 53.83/15.30 partcC_in_ggaa(x1, x2, x3, x4) = partcC_in_ggaa(x1, x2) 53.83/15.30 53.83/15.30 U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6) 53.83/15.30 53.83/15.30 U68_ggaa(x1, x2, x3, x4, x5, x6) = U68_ggaa(x1, x2, x3, x6) 53.83/15.30 53.83/15.30 partcB_in_ggaa(x1, x2, x3, x4) = partcB_in_ggaa(x1, x2) 53.83/15.30 53.83/15.30 U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6) 53.83/15.30 53.83/15.30 U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6) 53.83/15.30 53.83/15.30 U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6) 53.83/15.30 53.83/15.30 partcB_out_ggaa(x1, x2, x3, x4) = partcB_out_ggaa(x1, x2, x3, x4) 53.83/15.30 53.83/15.30 partcC_out_ggaa(x1, x2, x3, x4) = partcC_out_ggaa(x1, x2, x3, x4) 53.83/15.30 53.83/15.30 U69_ggaa(x1, x2, x3, x4, x5, x6) = U69_ggaa(x1, x2, x3, x6) 53.83/15.30 53.83/15.30 qscF_in_ga(x1, x2) = qscF_in_ga(x1) 53.83/15.30 53.83/15.30 qscF_out_ga(x1, x2) = qscF_out_ga(x1, x2) 53.83/15.30 53.83/15.30 U74_ga(x1, x2, x3, x4) = U74_ga(x1, x2, x4) 53.83/15.30 53.83/15.30 qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_ggaaaaa(x1, x2) 53.83/15.30 53.83/15.30 U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x8) 53.83/15.30 53.83/15.30 U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x8) 53.83/15.30 53.83/15.30 U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x8) 53.83/15.30 53.83/15.30 U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x8) 53.83/15.30 53.83/15.30 appcE_in_ggga(x1, x2, x3, x4) = appcE_in_ggga(x1, x2, x3) 53.83/15.30 53.83/15.30 appcE_out_ggga(x1, x2, x3, x4) = appcE_out_ggga(x1, x2, x3, x4) 53.83/15.30 53.83/15.30 U75_ggga(x1, x2, x3, x4, x5, x6) = U75_ggga(x1, x2, x3, x4, x6) 53.83/15.30 53.83/15.30 qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) 53.83/15.30 53.83/15.30 PARTB_IN_GGAA(x1, x2, x3, x4) = PARTB_IN_GGAA(x1, x2) 53.83/15.30 53.83/15.30 U3_GGAA(x1, x2, x3, x4, x5, x6) = U3_GGAA(x1, x2, x3, x6) 53.83/15.30 53.83/15.30 53.83/15.30 We have to consider all (P,R,Pi)-chains 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (176) UsableRulesProof (EQUIVALENT) 53.83/15.31 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (177) 53.83/15.31 Obligation: 53.83/15.31 Pi DP problem: 53.83/15.31 The TRS P consists of the following rules: 53.83/15.31 53.83/15.31 PARTB_IN_GGAA(X1, .(X2, X3), .(X2, X4), X5) -> U3_GGAA(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.83/15.31 U3_GGAA(X1, X2, X3, X4, X5, lesscA_out_gg(X1, X2)) -> PARTB_IN_GGAA(X1, X3, X4, X5) 53.83/15.31 PARTB_IN_GGAA(X1, .(X2, X3), X4, .(X2, X5)) -> PARTB_IN_GGAA(X1, X3, X4, X5) 53.83/15.31 53.83/15.31 The TRS R consists of the following rules: 53.83/15.31 53.83/15.31 lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) 53.83/15.31 lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) 53.83/15.31 U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) 53.83/15.31 53.83/15.31 The argument filtering Pi contains the following mapping: 53.83/15.31 s(x1) = s(x1) 53.83/15.31 53.83/15.31 0 = 0 53.83/15.31 53.83/15.31 .(x1, x2) = .(x1, x2) 53.83/15.31 53.83/15.31 lesscA_in_gg(x1, x2) = lesscA_in_gg(x1, x2) 53.83/15.31 53.83/15.31 lesscA_out_gg(x1, x2) = lesscA_out_gg(x1, x2) 53.83/15.31 53.83/15.31 U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3) 53.83/15.31 53.83/15.31 PARTB_IN_GGAA(x1, x2, x3, x4) = PARTB_IN_GGAA(x1, x2) 53.83/15.31 53.83/15.31 U3_GGAA(x1, x2, x3, x4, x5, x6) = U3_GGAA(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 53.83/15.31 We have to consider all (P,R,Pi)-chains 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (178) PiDPToQDPProof (SOUND) 53.83/15.31 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (179) 53.83/15.31 Obligation: 53.83/15.31 Q DP problem: 53.83/15.31 The TRS P consists of the following rules: 53.83/15.31 53.83/15.31 PARTB_IN_GGAA(X1, .(X2, X3)) -> U3_GGAA(X1, X2, X3, lesscA_in_gg(X1, X2)) 53.83/15.31 U3_GGAA(X1, X2, X3, lesscA_out_gg(X1, X2)) -> PARTB_IN_GGAA(X1, X3) 53.83/15.31 PARTB_IN_GGAA(X1, .(X2, X3)) -> PARTB_IN_GGAA(X1, X3) 53.83/15.31 53.83/15.31 The TRS R consists of the following rules: 53.83/15.31 53.83/15.31 lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) 53.83/15.31 lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) 53.83/15.31 U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) 53.83/15.31 53.83/15.31 The set Q consists of the following terms: 53.83/15.31 53.83/15.31 lesscA_in_gg(x0, x1) 53.83/15.31 U63_gg(x0, x1, x2) 53.83/15.31 53.83/15.31 We have to consider all (P,Q,R)-chains. 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (180) QDPSizeChangeProof (EQUIVALENT) 53.83/15.31 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. 53.83/15.31 53.83/15.31 From the DPs we obtained the following set of size-change graphs: 53.83/15.31 *U3_GGAA(X1, X2, X3, lesscA_out_gg(X1, X2)) -> PARTB_IN_GGAA(X1, X3) 53.83/15.31 The graph contains the following edges 1 >= 1, 4 > 1, 3 >= 2 53.83/15.31 53.83/15.31 53.83/15.31 *PARTB_IN_GGAA(X1, .(X2, X3)) -> PARTB_IN_GGAA(X1, X3) 53.83/15.31 The graph contains the following edges 1 >= 1, 2 > 2 53.83/15.31 53.83/15.31 53.83/15.31 *PARTB_IN_GGAA(X1, .(X2, X3)) -> U3_GGAA(X1, X2, X3, lesscA_in_gg(X1, X2)) 53.83/15.31 The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3 53.83/15.31 53.83/15.31 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (181) 53.83/15.31 YES 53.83/15.31 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (182) 53.83/15.31 Obligation: 53.83/15.31 Pi DP problem: 53.83/15.31 The TRS P consists of the following rules: 53.83/15.31 53.83/15.31 PARTC_IN_GGAA(X1, .(X2, X3), X4, .(X2, X5)) -> PARTC_IN_GGAA(X1, X3, X4, X5) 53.83/15.31 53.83/15.31 The TRS R consists of the following rules: 53.83/15.31 53.83/15.31 lesscA_in_aa(0, s(X1)) -> lesscA_out_aa(0, s(X1)) 53.83/15.31 lesscA_in_aa(s(X1), s(X2)) -> U63_aa(X1, X2, lesscA_in_aa(X1, X2)) 53.83/15.31 U63_aa(X1, X2, lesscA_out_aa(X1, X2)) -> lesscA_out_aa(s(X1), s(X2)) 53.83/15.31 lesscA_in_ga(0, s(X1)) -> lesscA_out_ga(0, s(X1)) 53.83/15.31 lesscA_in_ga(s(X1), s(X2)) -> U63_ga(X1, X2, lesscA_in_ga(X1, X2)) 53.83/15.31 U63_ga(X1, X2, lesscA_out_ga(X1, X2)) -> lesscA_out_ga(s(X1), s(X2)) 53.83/15.31 partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_gaaa(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) 53.83/15.31 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)) 53.83/15.31 partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) 53.83/15.31 partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], []) 53.83/15.31 U66_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.31 U65_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.31 partcC_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_aaaa(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) 53.83/15.31 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)) 53.83/15.31 U68_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.31 partcC_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_aaaa(X1, X2, X3, X4, X5, partcC_in_aaaa(X1, X3, X4, X5)) 53.83/15.31 partcC_in_aaaa(X1, [], [], []) -> partcC_out_aaaa(X1, [], [], []) 53.83/15.31 U69_aaaa(X1, X2, X3, X4, X5, partcC_out_aaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.31 qscF_in_aa([], []) -> qscF_out_aa([], []) 53.83/15.31 qscF_in_aa(.(X1, X2), X3) -> U74_aa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 appcE_in_aaaa([], X1, X2, .(X1, X2)) -> appcE_out_aaaa([], X1, X2, .(X1, X2)) 53.83/15.31 appcE_in_aaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U75_aaaa(X1, X2, X3, X4, X5, appcE_in_aaaa(X2, X3, X4, X5)) 53.83/15.31 U75_aaaa(X1, X2, X3, X4, X5, appcE_out_aaaa(X2, X3, X4, X5)) -> appcE_out_aaaa(.(X1, X2), X3, X4, .(X1, X5)) 53.83/15.31 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) 53.83/15.31 U74_aa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_aa(.(X1, X2), X3) 53.83/15.31 qscG_in_aaa(X1, X2, X3) -> U76_aaa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.31 U76_aaa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscG_out_aaa(X1, X2, X3) 53.83/15.31 lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) 53.83/15.31 lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) 53.83/15.31 U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) 53.83/15.31 partcC_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.83/15.31 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)) 53.83/15.31 partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.83/15.31 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)) 53.83/15.31 partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) 53.83/15.31 partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], []) 53.83/15.31 U66_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.31 U65_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.31 U68_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.31 partcC_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_ggaa(X1, X2, X3, X4, X5, partcC_in_ggaa(X1, X3, X4, X5)) 53.83/15.31 partcC_in_ggaa(X1, [], [], []) -> partcC_out_ggaa(X1, [], [], []) 53.83/15.31 U69_ggaa(X1, X2, X3, X4, X5, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.31 qscF_in_ga([], []) -> qscF_out_ga([], []) 53.83/15.31 qscF_in_ga(.(X1, X2), X3) -> U74_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2)) 53.83/15.31 appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U75_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5)) 53.83/15.31 U75_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5)) 53.83/15.31 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) 53.83/15.31 U74_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3) 53.83/15.31 53.83/15.31 The argument filtering Pi contains the following mapping: 53.83/15.31 [] = [] 53.83/15.31 53.83/15.31 lesscA_in_aa(x1, x2) = lesscA_in_aa 53.83/15.31 53.83/15.31 lesscA_out_aa(x1, x2) = lesscA_out_aa(x1) 53.83/15.31 53.83/15.31 U63_aa(x1, x2, x3) = U63_aa(x3) 53.83/15.31 53.83/15.31 s(x1) = s(x1) 53.83/15.31 53.83/15.31 lesscA_in_ga(x1, x2) = lesscA_in_ga(x1) 53.83/15.31 53.83/15.31 0 = 0 53.83/15.31 53.83/15.31 lesscA_out_ga(x1, x2) = lesscA_out_ga(x1) 53.83/15.31 53.83/15.31 U63_ga(x1, x2, x3) = U63_ga(x1, x3) 53.83/15.31 53.83/15.31 partcB_in_gaaa(x1, x2, x3, x4) = partcB_in_gaaa(x1) 53.83/15.31 53.83/15.31 U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6) 53.83/15.31 53.83/15.31 U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x6) 53.83/15.31 53.83/15.31 U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6) 53.83/15.31 53.83/15.31 partcB_out_gaaa(x1, x2, x3, x4) = partcB_out_gaaa(x1) 53.83/15.31 53.83/15.31 partcC_in_aaaa(x1, x2, x3, x4) = partcC_in_aaaa 53.83/15.31 53.83/15.31 U67_aaaa(x1, x2, x3, x4, x5, x6) = U67_aaaa(x6) 53.83/15.31 53.83/15.31 U68_aaaa(x1, x2, x3, x4, x5, x6) = U68_aaaa(x6) 53.83/15.31 53.83/15.31 partcC_out_aaaa(x1, x2, x3, x4) = partcC_out_aaaa 53.83/15.31 53.83/15.31 U69_aaaa(x1, x2, x3, x4, x5, x6) = U69_aaaa(x6) 53.83/15.31 53.83/15.31 qscF_in_aa(x1, x2) = qscF_in_aa 53.83/15.31 53.83/15.31 qscF_out_aa(x1, x2) = qscF_out_aa 53.83/15.31 53.83/15.31 U74_aa(x1, x2, x3, x4) = U74_aa(x4) 53.83/15.31 53.83/15.31 qcD_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_aaaaaaa 53.83/15.31 53.83/15.31 U70_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_aaaaaaa(x8) 53.83/15.31 53.83/15.31 U71_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_aaaaaaa(x8) 53.83/15.31 53.83/15.31 U72_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_aaaaaaa(x8) 53.83/15.31 53.83/15.31 U73_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_aaaaaaa(x8) 53.83/15.31 53.83/15.31 .(x1, x2) = .(x1, x2) 53.83/15.31 53.83/15.31 qcD_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_aaaaaaa 53.83/15.31 53.83/15.31 appcE_in_aaaa(x1, x2, x3, x4) = appcE_in_aaaa 53.83/15.31 53.83/15.31 appcE_out_aaaa(x1, x2, x3, x4) = appcE_out_aaaa 53.83/15.31 53.83/15.31 U75_aaaa(x1, x2, x3, x4, x5, x6) = U75_aaaa(x6) 53.83/15.31 53.83/15.31 qscG_in_aaa(x1, x2, x3) = qscG_in_aaa 53.83/15.31 53.83/15.31 U76_aaa(x1, x2, x3, x4) = U76_aaa(x4) 53.83/15.31 53.83/15.31 qscG_out_aaa(x1, x2, x3) = qscG_out_aaa 53.83/15.31 53.83/15.31 lesscA_in_gg(x1, x2) = lesscA_in_gg(x1, x2) 53.83/15.31 53.83/15.31 lesscA_out_gg(x1, x2) = lesscA_out_gg(x1, x2) 53.83/15.31 53.83/15.31 U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3) 53.83/15.31 53.83/15.31 partcC_in_ggaa(x1, x2, x3, x4) = partcC_in_ggaa(x1, x2) 53.83/15.31 53.83/15.31 U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 U68_ggaa(x1, x2, x3, x4, x5, x6) = U68_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 partcB_in_ggaa(x1, x2, x3, x4) = partcB_in_ggaa(x1, x2) 53.83/15.31 53.83/15.31 U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 partcB_out_ggaa(x1, x2, x3, x4) = partcB_out_ggaa(x1, x2, x3, x4) 53.83/15.31 53.83/15.31 partcC_out_ggaa(x1, x2, x3, x4) = partcC_out_ggaa(x1, x2, x3, x4) 53.83/15.31 53.83/15.31 U69_ggaa(x1, x2, x3, x4, x5, x6) = U69_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 qscF_in_ga(x1, x2) = qscF_in_ga(x1) 53.83/15.31 53.83/15.31 qscF_out_ga(x1, x2) = qscF_out_ga(x1, x2) 53.83/15.31 53.83/15.31 U74_ga(x1, x2, x3, x4) = U74_ga(x1, x2, x4) 53.83/15.31 53.83/15.31 qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_ggaaaaa(x1, x2) 53.83/15.31 53.83/15.31 U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x8) 53.83/15.31 53.83/15.31 U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x8) 53.83/15.31 53.83/15.31 U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x8) 53.83/15.31 53.83/15.31 U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x8) 53.83/15.31 53.83/15.31 appcE_in_ggga(x1, x2, x3, x4) = appcE_in_ggga(x1, x2, x3) 53.83/15.31 53.83/15.31 appcE_out_ggga(x1, x2, x3, x4) = appcE_out_ggga(x1, x2, x3, x4) 53.83/15.31 53.83/15.31 U75_ggga(x1, x2, x3, x4, x5, x6) = U75_ggga(x1, x2, x3, x4, x6) 53.83/15.31 53.83/15.31 qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) 53.83/15.31 53.83/15.31 PARTC_IN_GGAA(x1, x2, x3, x4) = PARTC_IN_GGAA(x1, x2) 53.83/15.31 53.83/15.31 53.83/15.31 We have to consider all (P,R,Pi)-chains 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (183) UsableRulesProof (EQUIVALENT) 53.83/15.31 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (184) 53.83/15.31 Obligation: 53.83/15.31 Pi DP problem: 53.83/15.31 The TRS P consists of the following rules: 53.83/15.31 53.83/15.31 PARTC_IN_GGAA(X1, .(X2, X3), X4, .(X2, X5)) -> PARTC_IN_GGAA(X1, X3, X4, X5) 53.83/15.31 53.83/15.31 R is empty. 53.83/15.31 The argument filtering Pi contains the following mapping: 53.83/15.31 .(x1, x2) = .(x1, x2) 53.83/15.31 53.83/15.31 PARTC_IN_GGAA(x1, x2, x3, x4) = PARTC_IN_GGAA(x1, x2) 53.83/15.31 53.83/15.31 53.83/15.31 We have to consider all (P,R,Pi)-chains 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (185) PiDPToQDPProof (SOUND) 53.83/15.31 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (186) 53.83/15.31 Obligation: 53.83/15.31 Q DP problem: 53.83/15.31 The TRS P consists of the following rules: 53.83/15.31 53.83/15.31 PARTC_IN_GGAA(X1, .(X2, X3)) -> PARTC_IN_GGAA(X1, X3) 53.83/15.31 53.83/15.31 R is empty. 53.83/15.31 Q is empty. 53.83/15.31 We have to consider all (P,Q,R)-chains. 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (187) QDPSizeChangeProof (EQUIVALENT) 53.83/15.31 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. 53.83/15.31 53.83/15.31 From the DPs we obtained the following set of size-change graphs: 53.83/15.31 *PARTC_IN_GGAA(X1, .(X2, X3)) -> PARTC_IN_GGAA(X1, X3) 53.83/15.31 The graph contains the following edges 1 >= 1, 2 > 2 53.83/15.31 53.83/15.31 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (188) 53.83/15.31 YES 53.83/15.31 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (189) 53.83/15.31 Obligation: 53.83/15.31 Pi DP problem: 53.83/15.31 The TRS P consists of the following rules: 53.83/15.31 53.83/15.31 QSF_IN_GA(.(X1, X2), X3) -> PD_IN_GGAAAAA(X1, X2, X4, X5, X6, X7, X3) 53.83/15.31 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)) 53.83/15.31 U11_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, partcC_out_ggaa(X1, X2, X3, X4)) -> QSF_IN_GA(X3, X5) 53.83/15.31 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)) 53.83/15.31 U13_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> QSF_IN_GA(X4, X6) 53.83/15.31 53.83/15.31 The TRS R consists of the following rules: 53.83/15.31 53.83/15.31 lesscA_in_aa(0, s(X1)) -> lesscA_out_aa(0, s(X1)) 53.83/15.31 lesscA_in_aa(s(X1), s(X2)) -> U63_aa(X1, X2, lesscA_in_aa(X1, X2)) 53.83/15.31 U63_aa(X1, X2, lesscA_out_aa(X1, X2)) -> lesscA_out_aa(s(X1), s(X2)) 53.83/15.31 lesscA_in_ga(0, s(X1)) -> lesscA_out_ga(0, s(X1)) 53.83/15.31 lesscA_in_ga(s(X1), s(X2)) -> U63_ga(X1, X2, lesscA_in_ga(X1, X2)) 53.83/15.31 U63_ga(X1, X2, lesscA_out_ga(X1, X2)) -> lesscA_out_ga(s(X1), s(X2)) 53.83/15.31 partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_gaaa(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) 53.83/15.31 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)) 53.83/15.31 partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) 53.83/15.31 partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], []) 53.83/15.31 U66_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.31 U65_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.31 partcC_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_aaaa(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) 53.83/15.31 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)) 53.83/15.31 U68_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.31 partcC_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_aaaa(X1, X2, X3, X4, X5, partcC_in_aaaa(X1, X3, X4, X5)) 53.83/15.31 partcC_in_aaaa(X1, [], [], []) -> partcC_out_aaaa(X1, [], [], []) 53.83/15.31 U69_aaaa(X1, X2, X3, X4, X5, partcC_out_aaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.31 qscF_in_aa([], []) -> qscF_out_aa([], []) 53.83/15.31 qscF_in_aa(.(X1, X2), X3) -> U74_aa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 appcE_in_aaaa([], X1, X2, .(X1, X2)) -> appcE_out_aaaa([], X1, X2, .(X1, X2)) 53.83/15.31 appcE_in_aaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U75_aaaa(X1, X2, X3, X4, X5, appcE_in_aaaa(X2, X3, X4, X5)) 53.83/15.31 U75_aaaa(X1, X2, X3, X4, X5, appcE_out_aaaa(X2, X3, X4, X5)) -> appcE_out_aaaa(.(X1, X2), X3, X4, .(X1, X5)) 53.83/15.31 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) 53.83/15.31 U74_aa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_aa(.(X1, X2), X3) 53.83/15.31 qscG_in_aaa(X1, X2, X3) -> U76_aaa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.31 U76_aaa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscG_out_aaa(X1, X2, X3) 53.83/15.31 lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) 53.83/15.31 lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) 53.83/15.31 U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) 53.83/15.31 partcC_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.83/15.31 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)) 53.83/15.31 partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.83/15.31 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)) 53.83/15.31 partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) 53.83/15.31 partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], []) 53.83/15.31 U66_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.31 U65_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.31 U68_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.31 partcC_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_ggaa(X1, X2, X3, X4, X5, partcC_in_ggaa(X1, X3, X4, X5)) 53.83/15.31 partcC_in_ggaa(X1, [], [], []) -> partcC_out_ggaa(X1, [], [], []) 53.83/15.31 U69_ggaa(X1, X2, X3, X4, X5, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.31 qscF_in_ga([], []) -> qscF_out_ga([], []) 53.83/15.31 qscF_in_ga(.(X1, X2), X3) -> U74_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2)) 53.83/15.31 appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U75_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5)) 53.83/15.31 U75_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5)) 53.83/15.31 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) 53.83/15.31 U74_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3) 53.83/15.31 53.83/15.31 The argument filtering Pi contains the following mapping: 53.83/15.31 [] = [] 53.83/15.31 53.83/15.31 lesscA_in_aa(x1, x2) = lesscA_in_aa 53.83/15.31 53.83/15.31 lesscA_out_aa(x1, x2) = lesscA_out_aa(x1) 53.83/15.31 53.83/15.31 U63_aa(x1, x2, x3) = U63_aa(x3) 53.83/15.31 53.83/15.31 s(x1) = s(x1) 53.83/15.31 53.83/15.31 lesscA_in_ga(x1, x2) = lesscA_in_ga(x1) 53.83/15.31 53.83/15.31 0 = 0 53.83/15.31 53.83/15.31 lesscA_out_ga(x1, x2) = lesscA_out_ga(x1) 53.83/15.31 53.83/15.31 U63_ga(x1, x2, x3) = U63_ga(x1, x3) 53.83/15.31 53.83/15.31 partcB_in_gaaa(x1, x2, x3, x4) = partcB_in_gaaa(x1) 53.83/15.31 53.83/15.31 U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6) 53.83/15.31 53.83/15.31 U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x6) 53.83/15.31 53.83/15.31 U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6) 53.83/15.31 53.83/15.31 partcB_out_gaaa(x1, x2, x3, x4) = partcB_out_gaaa(x1) 53.83/15.31 53.83/15.31 partcC_in_aaaa(x1, x2, x3, x4) = partcC_in_aaaa 53.83/15.31 53.83/15.31 U67_aaaa(x1, x2, x3, x4, x5, x6) = U67_aaaa(x6) 53.83/15.31 53.83/15.31 U68_aaaa(x1, x2, x3, x4, x5, x6) = U68_aaaa(x6) 53.83/15.31 53.83/15.31 partcC_out_aaaa(x1, x2, x3, x4) = partcC_out_aaaa 53.83/15.31 53.83/15.31 U69_aaaa(x1, x2, x3, x4, x5, x6) = U69_aaaa(x6) 53.83/15.31 53.83/15.31 qscF_in_aa(x1, x2) = qscF_in_aa 53.83/15.31 53.83/15.31 qscF_out_aa(x1, x2) = qscF_out_aa 53.83/15.31 53.83/15.31 U74_aa(x1, x2, x3, x4) = U74_aa(x4) 53.83/15.31 53.83/15.31 qcD_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_aaaaaaa 53.83/15.31 53.83/15.31 U70_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_aaaaaaa(x8) 53.83/15.31 53.83/15.31 U71_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_aaaaaaa(x8) 53.83/15.31 53.83/15.31 U72_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_aaaaaaa(x8) 53.83/15.31 53.83/15.31 U73_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_aaaaaaa(x8) 53.83/15.31 53.83/15.31 .(x1, x2) = .(x1, x2) 53.83/15.31 53.83/15.31 qcD_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_aaaaaaa 53.83/15.31 53.83/15.31 appcE_in_aaaa(x1, x2, x3, x4) = appcE_in_aaaa 53.83/15.31 53.83/15.31 appcE_out_aaaa(x1, x2, x3, x4) = appcE_out_aaaa 53.83/15.31 53.83/15.31 U75_aaaa(x1, x2, x3, x4, x5, x6) = U75_aaaa(x6) 53.83/15.31 53.83/15.31 qscG_in_aaa(x1, x2, x3) = qscG_in_aaa 53.83/15.31 53.83/15.31 U76_aaa(x1, x2, x3, x4) = U76_aaa(x4) 53.83/15.31 53.83/15.31 qscG_out_aaa(x1, x2, x3) = qscG_out_aaa 53.83/15.31 53.83/15.31 lesscA_in_gg(x1, x2) = lesscA_in_gg(x1, x2) 53.83/15.31 53.83/15.31 lesscA_out_gg(x1, x2) = lesscA_out_gg(x1, x2) 53.83/15.31 53.83/15.31 U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3) 53.83/15.31 53.83/15.31 partcC_in_ggaa(x1, x2, x3, x4) = partcC_in_ggaa(x1, x2) 53.83/15.31 53.83/15.31 U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 U68_ggaa(x1, x2, x3, x4, x5, x6) = U68_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 partcB_in_ggaa(x1, x2, x3, x4) = partcB_in_ggaa(x1, x2) 53.83/15.31 53.83/15.31 U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 partcB_out_ggaa(x1, x2, x3, x4) = partcB_out_ggaa(x1, x2, x3, x4) 53.83/15.31 53.83/15.31 partcC_out_ggaa(x1, x2, x3, x4) = partcC_out_ggaa(x1, x2, x3, x4) 53.83/15.31 53.83/15.31 U69_ggaa(x1, x2, x3, x4, x5, x6) = U69_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 qscF_in_ga(x1, x2) = qscF_in_ga(x1) 53.83/15.31 53.83/15.31 qscF_out_ga(x1, x2) = qscF_out_ga(x1, x2) 53.83/15.31 53.83/15.31 U74_ga(x1, x2, x3, x4) = U74_ga(x1, x2, x4) 53.83/15.31 53.83/15.31 qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_ggaaaaa(x1, x2) 53.83/15.31 53.83/15.31 U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x8) 53.83/15.31 53.83/15.31 U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x8) 53.83/15.31 53.83/15.31 U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x8) 53.83/15.31 53.83/15.31 U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x8) 53.83/15.31 53.83/15.31 appcE_in_ggga(x1, x2, x3, x4) = appcE_in_ggga(x1, x2, x3) 53.83/15.31 53.83/15.31 appcE_out_ggga(x1, x2, x3, x4) = appcE_out_ggga(x1, x2, x3, x4) 53.83/15.31 53.83/15.31 U75_ggga(x1, x2, x3, x4, x5, x6) = U75_ggga(x1, x2, x3, x4, x6) 53.83/15.31 53.83/15.31 qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) 53.83/15.31 53.83/15.31 QSF_IN_GA(x1, x2) = QSF_IN_GA(x1) 53.83/15.31 53.83/15.31 PD_IN_GGAAAAA(x1, x2, x3, x4, x5, x6, x7) = PD_IN_GGAAAAA(x1, x2) 53.83/15.31 53.83/15.31 U11_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U11_GGAAAAA(x1, x2, x8) 53.83/15.31 53.83/15.31 U13_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U13_GGAAAAA(x1, x2, x4, x8) 53.83/15.31 53.83/15.31 53.83/15.31 We have to consider all (P,R,Pi)-chains 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (190) UsableRulesProof (EQUIVALENT) 53.83/15.31 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (191) 53.83/15.31 Obligation: 53.83/15.31 Pi DP problem: 53.83/15.31 The TRS P consists of the following rules: 53.83/15.31 53.83/15.31 QSF_IN_GA(.(X1, X2), X3) -> PD_IN_GGAAAAA(X1, X2, X4, X5, X6, X7, X3) 53.83/15.31 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)) 53.83/15.31 U11_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, partcC_out_ggaa(X1, X2, X3, X4)) -> QSF_IN_GA(X3, X5) 53.83/15.31 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)) 53.83/15.31 U13_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> QSF_IN_GA(X4, X6) 53.83/15.31 53.83/15.31 The TRS R consists of the following rules: 53.83/15.31 53.83/15.31 partcC_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.83/15.31 partcC_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_ggaa(X1, X2, X3, X4, X5, partcC_in_ggaa(X1, X3, X4, X5)) 53.83/15.31 partcC_in_ggaa(X1, [], [], []) -> partcC_out_ggaa(X1, [], [], []) 53.83/15.31 qscF_in_ga([], []) -> qscF_out_ga([], []) 53.83/15.31 qscF_in_ga(.(X1, X2), X3) -> U74_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.31 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)) 53.83/15.31 U69_ggaa(X1, X2, X3, X4, X5, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.31 U74_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3) 53.83/15.31 lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) 53.83/15.31 lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) 53.83/15.31 U68_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.31 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)) 53.83/15.31 U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) 53.83/15.31 partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.83/15.31 partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) 53.83/15.31 partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], []) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 U66_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.31 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)) 53.83/15.31 U65_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.31 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)) 53.83/15.31 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) 53.83/15.31 appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2)) 53.83/15.31 appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U75_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5)) 53.83/15.31 U75_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5)) 53.83/15.31 53.83/15.31 The argument filtering Pi contains the following mapping: 53.83/15.31 [] = [] 53.83/15.31 53.83/15.31 s(x1) = s(x1) 53.83/15.31 53.83/15.31 0 = 0 53.83/15.31 53.83/15.31 .(x1, x2) = .(x1, x2) 53.83/15.31 53.83/15.31 lesscA_in_gg(x1, x2) = lesscA_in_gg(x1, x2) 53.83/15.31 53.83/15.31 lesscA_out_gg(x1, x2) = lesscA_out_gg(x1, x2) 53.83/15.31 53.83/15.31 U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3) 53.83/15.31 53.83/15.31 partcC_in_ggaa(x1, x2, x3, x4) = partcC_in_ggaa(x1, x2) 53.83/15.31 53.83/15.31 U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 U68_ggaa(x1, x2, x3, x4, x5, x6) = U68_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 partcB_in_ggaa(x1, x2, x3, x4) = partcB_in_ggaa(x1, x2) 53.83/15.31 53.83/15.31 U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 partcB_out_ggaa(x1, x2, x3, x4) = partcB_out_ggaa(x1, x2, x3, x4) 53.83/15.31 53.83/15.31 partcC_out_ggaa(x1, x2, x3, x4) = partcC_out_ggaa(x1, x2, x3, x4) 53.83/15.31 53.83/15.31 U69_ggaa(x1, x2, x3, x4, x5, x6) = U69_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 qscF_in_ga(x1, x2) = qscF_in_ga(x1) 53.83/15.31 53.83/15.31 qscF_out_ga(x1, x2) = qscF_out_ga(x1, x2) 53.83/15.31 53.83/15.31 U74_ga(x1, x2, x3, x4) = U74_ga(x1, x2, x4) 53.83/15.31 53.83/15.31 qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_ggaaaaa(x1, x2) 53.83/15.31 53.83/15.31 U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x8) 53.83/15.31 53.83/15.31 U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x8) 53.83/15.31 53.83/15.31 U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x8) 53.83/15.31 53.83/15.31 U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x8) 53.83/15.31 53.83/15.31 appcE_in_ggga(x1, x2, x3, x4) = appcE_in_ggga(x1, x2, x3) 53.83/15.31 53.83/15.31 appcE_out_ggga(x1, x2, x3, x4) = appcE_out_ggga(x1, x2, x3, x4) 53.83/15.31 53.83/15.31 U75_ggga(x1, x2, x3, x4, x5, x6) = U75_ggga(x1, x2, x3, x4, x6) 53.83/15.31 53.83/15.31 qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) 53.83/15.31 53.83/15.31 QSF_IN_GA(x1, x2) = QSF_IN_GA(x1) 53.83/15.31 53.83/15.31 PD_IN_GGAAAAA(x1, x2, x3, x4, x5, x6, x7) = PD_IN_GGAAAAA(x1, x2) 53.83/15.31 53.83/15.31 U11_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U11_GGAAAAA(x1, x2, x8) 53.83/15.31 53.83/15.31 U13_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U13_GGAAAAA(x1, x2, x4, x8) 53.83/15.31 53.83/15.31 53.83/15.31 We have to consider all (P,R,Pi)-chains 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (192) PiDPToQDPProof (SOUND) 53.83/15.31 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (193) 53.83/15.31 Obligation: 53.83/15.31 Q DP problem: 53.83/15.31 The TRS P consists of the following rules: 53.83/15.31 53.83/15.31 QSF_IN_GA(.(X1, X2)) -> PD_IN_GGAAAAA(X1, X2) 53.83/15.31 PD_IN_GGAAAAA(X1, X2) -> U11_GGAAAAA(X1, X2, partcC_in_ggaa(X1, X2)) 53.83/15.31 U11_GGAAAAA(X1, X2, partcC_out_ggaa(X1, X2, X3, X4)) -> QSF_IN_GA(X3) 53.83/15.31 U11_GGAAAAA(X1, X2, partcC_out_ggaa(X1, X2, X3, X4)) -> U13_GGAAAAA(X1, X2, X4, qscF_in_ga(X3)) 53.83/15.31 U13_GGAAAAA(X1, X2, X4, qscF_out_ga(X3, X5)) -> QSF_IN_GA(X4) 53.83/15.31 53.83/15.31 The TRS R consists of the following rules: 53.83/15.31 53.83/15.31 partcC_in_ggaa(X1, .(X2, X3)) -> U67_ggaa(X1, X2, X3, lesscA_in_gg(X1, X2)) 53.83/15.31 partcC_in_ggaa(X1, .(X2, X3)) -> U69_ggaa(X1, X2, X3, partcC_in_ggaa(X1, X3)) 53.83/15.31 partcC_in_ggaa(X1, []) -> partcC_out_ggaa(X1, [], [], []) 53.83/15.31 qscF_in_ga([]) -> qscF_out_ga([], []) 53.83/15.31 qscF_in_ga(.(X1, X2)) -> U74_ga(X1, X2, qcD_in_ggaaaaa(X1, X2)) 53.83/15.31 U67_ggaa(X1, X2, X3, lesscA_out_gg(X1, X2)) -> U68_ggaa(X1, X2, X3, partcB_in_ggaa(X1, X3)) 53.83/15.31 U69_ggaa(X1, X2, X3, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.31 U74_ga(X1, X2, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3) 53.83/15.31 lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) 53.83/15.31 lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) 53.83/15.31 U68_ggaa(X1, X2, X3, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.31 qcD_in_ggaaaaa(X1, X2) -> U70_ggaaaaa(X1, X2, partcC_in_ggaa(X1, X2)) 53.83/15.31 U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) 53.83/15.31 partcB_in_ggaa(X1, .(X2, X3)) -> U64_ggaa(X1, X2, X3, lesscA_in_gg(X1, X2)) 53.83/15.31 partcB_in_ggaa(X1, .(X2, X3)) -> U66_ggaa(X1, X2, X3, partcB_in_ggaa(X1, X3)) 53.83/15.31 partcB_in_ggaa(X1, []) -> partcB_out_ggaa(X1, [], [], []) 53.83/15.31 U70_ggaaaaa(X1, X2, partcC_out_ggaa(X1, X2, X3, X4)) -> U71_ggaaaaa(X1, X2, X3, X4, qscF_in_ga(X3)) 53.83/15.31 U64_ggaa(X1, X2, X3, lesscA_out_gg(X1, X2)) -> U65_ggaa(X1, X2, X3, partcB_in_ggaa(X1, X3)) 53.83/15.31 U66_ggaa(X1, X2, X3, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.31 U71_ggaaaaa(X1, X2, X3, X4, qscF_out_ga(X3, X5)) -> U72_ggaaaaa(X1, X2, X3, X4, X5, qscF_in_ga(X4)) 53.83/15.31 U65_ggaa(X1, X2, X3, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.31 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)) 53.83/15.31 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) 53.83/15.31 appcE_in_ggga([], X1, X2) -> appcE_out_ggga([], X1, X2, .(X1, X2)) 53.83/15.31 appcE_in_ggga(.(X1, X2), X3, X4) -> U75_ggga(X1, X2, X3, X4, appcE_in_ggga(X2, X3, X4)) 53.83/15.31 U75_ggga(X1, X2, X3, X4, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5)) 53.83/15.31 53.83/15.31 The set Q consists of the following terms: 53.83/15.31 53.83/15.31 partcC_in_ggaa(x0, x1) 53.83/15.31 qscF_in_ga(x0) 53.83/15.31 U67_ggaa(x0, x1, x2, x3) 53.83/15.31 U69_ggaa(x0, x1, x2, x3) 53.83/15.31 U74_ga(x0, x1, x2) 53.83/15.31 lesscA_in_gg(x0, x1) 53.83/15.31 U68_ggaa(x0, x1, x2, x3) 53.83/15.31 qcD_in_ggaaaaa(x0, x1) 53.83/15.31 U63_gg(x0, x1, x2) 53.83/15.31 partcB_in_ggaa(x0, x1) 53.83/15.31 U70_ggaaaaa(x0, x1, x2) 53.83/15.31 U64_ggaa(x0, x1, x2, x3) 53.83/15.31 U66_ggaa(x0, x1, x2, x3) 53.83/15.31 U71_ggaaaaa(x0, x1, x2, x3, x4) 53.83/15.31 U65_ggaa(x0, x1, x2, x3) 53.83/15.31 U72_ggaaaaa(x0, x1, x2, x3, x4, x5) 53.83/15.31 U73_ggaaaaa(x0, x1, x2, x3, x4, x5, x6) 53.83/15.31 appcE_in_ggga(x0, x1, x2) 53.83/15.31 U75_ggga(x0, x1, x2, x3, x4) 53.83/15.31 53.83/15.31 We have to consider all (P,Q,R)-chains. 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (194) QDPOrderProof (EQUIVALENT) 53.83/15.31 We use the reduction pair processor [LPAR04,JAR06]. 53.83/15.31 53.83/15.31 53.83/15.31 The following pairs can be oriented strictly and are deleted. 53.83/15.31 53.83/15.31 U11_GGAAAAA(X1, X2, partcC_out_ggaa(X1, X2, X3, X4)) -> QSF_IN_GA(X3) 53.83/15.31 U11_GGAAAAA(X1, X2, partcC_out_ggaa(X1, X2, X3, X4)) -> U13_GGAAAAA(X1, X2, X4, qscF_in_ga(X3)) 53.83/15.31 The remaining pairs can at least be oriented weakly. 53.83/15.31 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 53.83/15.31 53.83/15.31 POL( U11_GGAAAAA_3(x_1, ..., x_3) ) = x_3 + 2 53.83/15.31 POL( U13_GGAAAAA_4(x_1, ..., x_4) ) = 2x_3 53.83/15.31 POL( U70_ggaaaaa_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2} 53.83/15.31 POL( partcC_in_ggaa_2(x_1, x_2) ) = x_1 + 2x_2 53.83/15.31 POL( ._2(x_1, x_2) ) = x_1 + x_2 + 1 53.83/15.31 POL( U67_ggaa_4(x_1, ..., x_4) ) = 2x_3 + x_4 + 2 53.83/15.31 POL( lesscA_in_gg_2(x_1, x_2) ) = x_1 + x_2 53.83/15.31 POL( U69_ggaa_4(x_1, ..., x_4) ) = 2x_2 + x_4 + 2 53.83/15.31 POL( [] ) = 0 53.83/15.31 POL( partcC_out_ggaa_4(x_1, ..., x_4) ) = max{0, 2x_3 + 2x_4 - 1} 53.83/15.31 POL( qscF_in_ga_1(x_1) ) = 2 53.83/15.31 POL( qscF_out_ga_2(x_1, x_2) ) = max{0, x_2 - 2} 53.83/15.31 POL( U74_ga_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 - 2} 53.83/15.31 POL( qcD_in_ggaaaaa_2(x_1, x_2) ) = 2x_1 + 2 53.83/15.31 POL( U71_ggaaaaa_5(x_1, ..., x_5) ) = 2x_1 + x_2 + 2x_3 + 2x_4 + 2x_5 + 2 53.83/15.31 POL( qcD_out_ggaaaaa_7(x_1, ..., x_7) ) = x_1 + 2x_4 + 2x_6 + 2x_7 + 2 53.83/15.31 POL( U72_ggaaaaa_6(x_1, ..., x_6) ) = max{0, x_3 - 2} 53.83/15.31 POL( U73_ggaaaaa_7(x_1, ..., x_7) ) = max{0, 2x_4 + 2x_5 + 2x_6 - 2} 53.83/15.31 POL( appcE_in_ggga_3(x_1, ..., x_3) ) = x_2 53.83/15.31 POL( U64_ggaa_4(x_1, ..., x_4) ) = x_2 + x_3 + 1 53.83/15.31 POL( U68_ggaa_4(x_1, ..., x_4) ) = 2x_2 + 2x_4 + 1 53.83/15.31 POL( 0 ) = 2 53.83/15.31 POL( s_1(x_1) ) = 1 53.83/15.31 POL( lesscA_out_gg_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 1} 53.83/15.31 POL( U63_gg_3(x_1, ..., x_3) ) = 2 53.83/15.31 POL( partcB_in_ggaa_2(x_1, x_2) ) = x_2 53.83/15.31 POL( U66_ggaa_4(x_1, ..., x_4) ) = x_2 + x_4 + 1 53.83/15.31 POL( partcB_out_ggaa_4(x_1, ..., x_4) ) = x_3 + x_4 53.83/15.31 POL( U65_ggaa_4(x_1, ..., x_4) ) = x_2 + x_4 + 1 53.83/15.31 POL( appcE_out_ggga_4(x_1, ..., x_4) ) = max{0, 2x_3 - 2} 53.83/15.31 POL( U75_ggga_5(x_1, ..., x_5) ) = 2x_1 + 2x_2 + x_3 + 2 53.83/15.31 POL( QSF_IN_GA_1(x_1) ) = 2x_1 53.83/15.31 POL( PD_IN_GGAAAAA_2(x_1, x_2) ) = x_1 + 2x_2 + 2 53.83/15.31 53.83/15.31 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 53.83/15.31 53.83/15.31 partcC_in_ggaa(X1, .(X2, X3)) -> U67_ggaa(X1, X2, X3, lesscA_in_gg(X1, X2)) 53.83/15.31 partcC_in_ggaa(X1, .(X2, X3)) -> U69_ggaa(X1, X2, X3, partcC_in_ggaa(X1, X3)) 53.83/15.31 partcC_in_ggaa(X1, []) -> partcC_out_ggaa(X1, [], [], []) 53.83/15.31 U69_ggaa(X1, X2, X3, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.31 lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) 53.83/15.31 lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) 53.83/15.31 U67_ggaa(X1, X2, X3, lesscA_out_gg(X1, X2)) -> U68_ggaa(X1, X2, X3, partcB_in_ggaa(X1, X3)) 53.83/15.31 partcB_in_ggaa(X1, .(X2, X3)) -> U64_ggaa(X1, X2, X3, lesscA_in_gg(X1, X2)) 53.83/15.31 partcB_in_ggaa(X1, .(X2, X3)) -> U66_ggaa(X1, X2, X3, partcB_in_ggaa(X1, X3)) 53.83/15.31 partcB_in_ggaa(X1, []) -> partcB_out_ggaa(X1, [], [], []) 53.83/15.31 U68_ggaa(X1, X2, X3, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.31 U64_ggaa(X1, X2, X3, lesscA_out_gg(X1, X2)) -> U65_ggaa(X1, X2, X3, partcB_in_ggaa(X1, X3)) 53.83/15.31 U65_ggaa(X1, X2, X3, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.31 U66_ggaa(X1, X2, X3, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.31 U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) 53.83/15.31 53.83/15.31 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (195) 53.83/15.31 Obligation: 53.83/15.31 Q DP problem: 53.83/15.31 The TRS P consists of the following rules: 53.83/15.31 53.83/15.31 QSF_IN_GA(.(X1, X2)) -> PD_IN_GGAAAAA(X1, X2) 53.83/15.31 PD_IN_GGAAAAA(X1, X2) -> U11_GGAAAAA(X1, X2, partcC_in_ggaa(X1, X2)) 53.83/15.31 U13_GGAAAAA(X1, X2, X4, qscF_out_ga(X3, X5)) -> QSF_IN_GA(X4) 53.83/15.31 53.83/15.31 The TRS R consists of the following rules: 53.83/15.31 53.83/15.31 partcC_in_ggaa(X1, .(X2, X3)) -> U67_ggaa(X1, X2, X3, lesscA_in_gg(X1, X2)) 53.83/15.31 partcC_in_ggaa(X1, .(X2, X3)) -> U69_ggaa(X1, X2, X3, partcC_in_ggaa(X1, X3)) 53.83/15.31 partcC_in_ggaa(X1, []) -> partcC_out_ggaa(X1, [], [], []) 53.83/15.31 qscF_in_ga([]) -> qscF_out_ga([], []) 53.83/15.31 qscF_in_ga(.(X1, X2)) -> U74_ga(X1, X2, qcD_in_ggaaaaa(X1, X2)) 53.83/15.31 U67_ggaa(X1, X2, X3, lesscA_out_gg(X1, X2)) -> U68_ggaa(X1, X2, X3, partcB_in_ggaa(X1, X3)) 53.83/15.31 U69_ggaa(X1, X2, X3, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.31 U74_ga(X1, X2, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3) 53.83/15.31 lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) 53.83/15.31 lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) 53.83/15.31 U68_ggaa(X1, X2, X3, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.31 qcD_in_ggaaaaa(X1, X2) -> U70_ggaaaaa(X1, X2, partcC_in_ggaa(X1, X2)) 53.83/15.31 U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) 53.83/15.31 partcB_in_ggaa(X1, .(X2, X3)) -> U64_ggaa(X1, X2, X3, lesscA_in_gg(X1, X2)) 53.83/15.31 partcB_in_ggaa(X1, .(X2, X3)) -> U66_ggaa(X1, X2, X3, partcB_in_ggaa(X1, X3)) 53.83/15.31 partcB_in_ggaa(X1, []) -> partcB_out_ggaa(X1, [], [], []) 53.83/15.31 U70_ggaaaaa(X1, X2, partcC_out_ggaa(X1, X2, X3, X4)) -> U71_ggaaaaa(X1, X2, X3, X4, qscF_in_ga(X3)) 53.83/15.31 U64_ggaa(X1, X2, X3, lesscA_out_gg(X1, X2)) -> U65_ggaa(X1, X2, X3, partcB_in_ggaa(X1, X3)) 53.83/15.31 U66_ggaa(X1, X2, X3, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.31 U71_ggaaaaa(X1, X2, X3, X4, qscF_out_ga(X3, X5)) -> U72_ggaaaaa(X1, X2, X3, X4, X5, qscF_in_ga(X4)) 53.83/15.31 U65_ggaa(X1, X2, X3, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.31 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)) 53.83/15.31 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) 53.83/15.31 appcE_in_ggga([], X1, X2) -> appcE_out_ggga([], X1, X2, .(X1, X2)) 53.83/15.31 appcE_in_ggga(.(X1, X2), X3, X4) -> U75_ggga(X1, X2, X3, X4, appcE_in_ggga(X2, X3, X4)) 53.83/15.31 U75_ggga(X1, X2, X3, X4, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5)) 53.83/15.31 53.83/15.31 The set Q consists of the following terms: 53.83/15.31 53.83/15.31 partcC_in_ggaa(x0, x1) 53.83/15.31 qscF_in_ga(x0) 53.83/15.31 U67_ggaa(x0, x1, x2, x3) 53.83/15.31 U69_ggaa(x0, x1, x2, x3) 53.83/15.31 U74_ga(x0, x1, x2) 53.83/15.31 lesscA_in_gg(x0, x1) 53.83/15.31 U68_ggaa(x0, x1, x2, x3) 53.83/15.31 qcD_in_ggaaaaa(x0, x1) 53.83/15.31 U63_gg(x0, x1, x2) 53.83/15.31 partcB_in_ggaa(x0, x1) 53.83/15.31 U70_ggaaaaa(x0, x1, x2) 53.83/15.31 U64_ggaa(x0, x1, x2, x3) 53.83/15.31 U66_ggaa(x0, x1, x2, x3) 53.83/15.31 U71_ggaaaaa(x0, x1, x2, x3, x4) 53.83/15.31 U65_ggaa(x0, x1, x2, x3) 53.83/15.31 U72_ggaaaaa(x0, x1, x2, x3, x4, x5) 53.83/15.31 U73_ggaaaaa(x0, x1, x2, x3, x4, x5, x6) 53.83/15.31 appcE_in_ggga(x0, x1, x2) 53.83/15.31 U75_ggga(x0, x1, x2, x3, x4) 53.83/15.31 53.83/15.31 We have to consider all (P,Q,R)-chains. 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (196) DependencyGraphProof (EQUIVALENT) 53.83/15.31 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes. 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (197) 53.83/15.31 TRUE 53.83/15.31 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (198) 53.83/15.31 Obligation: 53.83/15.31 Pi DP problem: 53.83/15.31 The TRS P consists of the following rules: 53.83/15.31 53.83/15.31 APPE_IN_AAAA(.(X1, X2), X3, X4, .(X1, X5)) -> APPE_IN_AAAA(X2, X3, X4, X5) 53.83/15.31 53.83/15.31 The TRS R consists of the following rules: 53.83/15.31 53.83/15.31 lesscA_in_aa(0, s(X1)) -> lesscA_out_aa(0, s(X1)) 53.83/15.31 lesscA_in_aa(s(X1), s(X2)) -> U63_aa(X1, X2, lesscA_in_aa(X1, X2)) 53.83/15.31 U63_aa(X1, X2, lesscA_out_aa(X1, X2)) -> lesscA_out_aa(s(X1), s(X2)) 53.83/15.31 lesscA_in_ga(0, s(X1)) -> lesscA_out_ga(0, s(X1)) 53.83/15.31 lesscA_in_ga(s(X1), s(X2)) -> U63_ga(X1, X2, lesscA_in_ga(X1, X2)) 53.83/15.31 U63_ga(X1, X2, lesscA_out_ga(X1, X2)) -> lesscA_out_ga(s(X1), s(X2)) 53.83/15.31 partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_gaaa(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) 53.83/15.31 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)) 53.83/15.31 partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) 53.83/15.31 partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], []) 53.83/15.31 U66_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.31 U65_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.31 partcC_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_aaaa(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) 53.83/15.31 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)) 53.83/15.31 U68_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.31 partcC_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_aaaa(X1, X2, X3, X4, X5, partcC_in_aaaa(X1, X3, X4, X5)) 53.83/15.31 partcC_in_aaaa(X1, [], [], []) -> partcC_out_aaaa(X1, [], [], []) 53.83/15.31 U69_aaaa(X1, X2, X3, X4, X5, partcC_out_aaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.31 qscF_in_aa([], []) -> qscF_out_aa([], []) 53.83/15.31 qscF_in_aa(.(X1, X2), X3) -> U74_aa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 appcE_in_aaaa([], X1, X2, .(X1, X2)) -> appcE_out_aaaa([], X1, X2, .(X1, X2)) 53.83/15.31 appcE_in_aaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U75_aaaa(X1, X2, X3, X4, X5, appcE_in_aaaa(X2, X3, X4, X5)) 53.83/15.31 U75_aaaa(X1, X2, X3, X4, X5, appcE_out_aaaa(X2, X3, X4, X5)) -> appcE_out_aaaa(.(X1, X2), X3, X4, .(X1, X5)) 53.83/15.31 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) 53.83/15.31 U74_aa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_aa(.(X1, X2), X3) 53.83/15.31 qscG_in_aaa(X1, X2, X3) -> U76_aaa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.31 U76_aaa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscG_out_aaa(X1, X2, X3) 53.83/15.31 lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) 53.83/15.31 lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) 53.83/15.31 U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) 53.83/15.31 partcC_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.83/15.31 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)) 53.83/15.31 partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.83/15.31 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)) 53.83/15.31 partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) 53.83/15.31 partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], []) 53.83/15.31 U66_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.31 U65_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.31 U68_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.31 partcC_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_ggaa(X1, X2, X3, X4, X5, partcC_in_ggaa(X1, X3, X4, X5)) 53.83/15.31 partcC_in_ggaa(X1, [], [], []) -> partcC_out_ggaa(X1, [], [], []) 53.83/15.31 U69_ggaa(X1, X2, X3, X4, X5, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.31 qscF_in_ga([], []) -> qscF_out_ga([], []) 53.83/15.31 qscF_in_ga(.(X1, X2), X3) -> U74_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2)) 53.83/15.31 appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U75_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5)) 53.83/15.31 U75_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5)) 53.83/15.31 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) 53.83/15.31 U74_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3) 53.83/15.31 53.83/15.31 The argument filtering Pi contains the following mapping: 53.83/15.31 [] = [] 53.83/15.31 53.83/15.31 lesscA_in_aa(x1, x2) = lesscA_in_aa 53.83/15.31 53.83/15.31 lesscA_out_aa(x1, x2) = lesscA_out_aa(x1) 53.83/15.31 53.83/15.31 U63_aa(x1, x2, x3) = U63_aa(x3) 53.83/15.31 53.83/15.31 s(x1) = s(x1) 53.83/15.31 53.83/15.31 lesscA_in_ga(x1, x2) = lesscA_in_ga(x1) 53.83/15.31 53.83/15.31 0 = 0 53.83/15.31 53.83/15.31 lesscA_out_ga(x1, x2) = lesscA_out_ga(x1) 53.83/15.31 53.83/15.31 U63_ga(x1, x2, x3) = U63_ga(x1, x3) 53.83/15.31 53.83/15.31 partcB_in_gaaa(x1, x2, x3, x4) = partcB_in_gaaa(x1) 53.83/15.31 53.83/15.31 U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6) 53.83/15.31 53.83/15.31 U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x6) 53.83/15.31 53.83/15.31 U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6) 53.83/15.31 53.83/15.31 partcB_out_gaaa(x1, x2, x3, x4) = partcB_out_gaaa(x1) 53.83/15.31 53.83/15.31 partcC_in_aaaa(x1, x2, x3, x4) = partcC_in_aaaa 53.83/15.31 53.83/15.31 U67_aaaa(x1, x2, x3, x4, x5, x6) = U67_aaaa(x6) 53.83/15.31 53.83/15.31 U68_aaaa(x1, x2, x3, x4, x5, x6) = U68_aaaa(x6) 53.83/15.31 53.83/15.31 partcC_out_aaaa(x1, x2, x3, x4) = partcC_out_aaaa 53.83/15.31 53.83/15.31 U69_aaaa(x1, x2, x3, x4, x5, x6) = U69_aaaa(x6) 53.83/15.31 53.83/15.31 qscF_in_aa(x1, x2) = qscF_in_aa 53.83/15.31 53.83/15.31 qscF_out_aa(x1, x2) = qscF_out_aa 53.83/15.31 53.83/15.31 U74_aa(x1, x2, x3, x4) = U74_aa(x4) 53.83/15.31 53.83/15.31 qcD_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_aaaaaaa 53.83/15.31 53.83/15.31 U70_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_aaaaaaa(x8) 53.83/15.31 53.83/15.31 U71_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_aaaaaaa(x8) 53.83/15.31 53.83/15.31 U72_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_aaaaaaa(x8) 53.83/15.31 53.83/15.31 U73_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_aaaaaaa(x8) 53.83/15.31 53.83/15.31 .(x1, x2) = .(x1, x2) 53.83/15.31 53.83/15.31 qcD_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_aaaaaaa 53.83/15.31 53.83/15.31 appcE_in_aaaa(x1, x2, x3, x4) = appcE_in_aaaa 53.83/15.31 53.83/15.31 appcE_out_aaaa(x1, x2, x3, x4) = appcE_out_aaaa 53.83/15.31 53.83/15.31 U75_aaaa(x1, x2, x3, x4, x5, x6) = U75_aaaa(x6) 53.83/15.31 53.83/15.31 qscG_in_aaa(x1, x2, x3) = qscG_in_aaa 53.83/15.31 53.83/15.31 U76_aaa(x1, x2, x3, x4) = U76_aaa(x4) 53.83/15.31 53.83/15.31 qscG_out_aaa(x1, x2, x3) = qscG_out_aaa 53.83/15.31 53.83/15.31 lesscA_in_gg(x1, x2) = lesscA_in_gg(x1, x2) 53.83/15.31 53.83/15.31 lesscA_out_gg(x1, x2) = lesscA_out_gg(x1, x2) 53.83/15.31 53.83/15.31 U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3) 53.83/15.31 53.83/15.31 partcC_in_ggaa(x1, x2, x3, x4) = partcC_in_ggaa(x1, x2) 53.83/15.31 53.83/15.31 U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 U68_ggaa(x1, x2, x3, x4, x5, x6) = U68_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 partcB_in_ggaa(x1, x2, x3, x4) = partcB_in_ggaa(x1, x2) 53.83/15.31 53.83/15.31 U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 partcB_out_ggaa(x1, x2, x3, x4) = partcB_out_ggaa(x1, x2, x3, x4) 53.83/15.31 53.83/15.31 partcC_out_ggaa(x1, x2, x3, x4) = partcC_out_ggaa(x1, x2, x3, x4) 53.83/15.31 53.83/15.31 U69_ggaa(x1, x2, x3, x4, x5, x6) = U69_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 qscF_in_ga(x1, x2) = qscF_in_ga(x1) 53.83/15.31 53.83/15.31 qscF_out_ga(x1, x2) = qscF_out_ga(x1, x2) 53.83/15.31 53.83/15.31 U74_ga(x1, x2, x3, x4) = U74_ga(x1, x2, x4) 53.83/15.31 53.83/15.31 qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_ggaaaaa(x1, x2) 53.83/15.31 53.83/15.31 U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x8) 53.83/15.31 53.83/15.31 U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x8) 53.83/15.31 53.83/15.31 U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x8) 53.83/15.31 53.83/15.31 U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x8) 53.83/15.31 53.83/15.31 appcE_in_ggga(x1, x2, x3, x4) = appcE_in_ggga(x1, x2, x3) 53.83/15.31 53.83/15.31 appcE_out_ggga(x1, x2, x3, x4) = appcE_out_ggga(x1, x2, x3, x4) 53.83/15.31 53.83/15.31 U75_ggga(x1, x2, x3, x4, x5, x6) = U75_ggga(x1, x2, x3, x4, x6) 53.83/15.31 53.83/15.31 qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) 53.83/15.31 53.83/15.31 APPE_IN_AAAA(x1, x2, x3, x4) = APPE_IN_AAAA 53.83/15.31 53.83/15.31 53.83/15.31 We have to consider all (P,R,Pi)-chains 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (199) UsableRulesProof (EQUIVALENT) 53.83/15.31 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (200) 53.83/15.31 Obligation: 53.83/15.31 Pi DP problem: 53.83/15.31 The TRS P consists of the following rules: 53.83/15.31 53.83/15.31 APPE_IN_AAAA(.(X1, X2), X3, X4, .(X1, X5)) -> APPE_IN_AAAA(X2, X3, X4, X5) 53.83/15.31 53.83/15.31 R is empty. 53.83/15.31 The argument filtering Pi contains the following mapping: 53.83/15.31 .(x1, x2) = .(x1, x2) 53.83/15.31 53.83/15.31 APPE_IN_AAAA(x1, x2, x3, x4) = APPE_IN_AAAA 53.83/15.31 53.83/15.31 53.83/15.31 We have to consider all (P,R,Pi)-chains 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (201) PiDPToQDPProof (SOUND) 53.83/15.31 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (202) 53.83/15.31 Obligation: 53.83/15.31 Q DP problem: 53.83/15.31 The TRS P consists of the following rules: 53.83/15.31 53.83/15.31 APPE_IN_AAAA -> APPE_IN_AAAA 53.83/15.31 53.83/15.31 R is empty. 53.83/15.31 Q is empty. 53.83/15.31 We have to consider all (P,Q,R)-chains. 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (203) NonTerminationLoopProof (COMPLETE) 53.83/15.31 We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. 53.83/15.31 Found a loop by semiunifying a rule from P directly. 53.83/15.31 53.83/15.31 s = APPE_IN_AAAA evaluates to t =APPE_IN_AAAA 53.83/15.31 53.83/15.31 Thus s starts an infinite chain as s semiunifies with t with the following substitutions: 53.83/15.31 * Matcher: [ ] 53.83/15.31 * Semiunifier: [ ] 53.83/15.31 53.83/15.31 -------------------------------------------------------------------------------- 53.83/15.31 Rewriting sequence 53.83/15.31 53.83/15.31 The DP semiunifies directly so there is only one rewrite step from APPE_IN_AAAA to APPE_IN_AAAA. 53.83/15.31 53.83/15.31 53.83/15.31 53.83/15.31 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (204) 53.83/15.31 NO 53.83/15.31 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (205) 53.83/15.31 Obligation: 53.83/15.31 Pi DP problem: 53.83/15.31 The TRS P consists of the following rules: 53.83/15.31 53.83/15.31 LESSA_IN_GA(s(X1), s(X2)) -> LESSA_IN_GA(X1, X2) 53.83/15.31 53.83/15.31 The TRS R consists of the following rules: 53.83/15.31 53.83/15.31 lesscA_in_aa(0, s(X1)) -> lesscA_out_aa(0, s(X1)) 53.83/15.31 lesscA_in_aa(s(X1), s(X2)) -> U63_aa(X1, X2, lesscA_in_aa(X1, X2)) 53.83/15.31 U63_aa(X1, X2, lesscA_out_aa(X1, X2)) -> lesscA_out_aa(s(X1), s(X2)) 53.83/15.31 lesscA_in_ga(0, s(X1)) -> lesscA_out_ga(0, s(X1)) 53.83/15.31 lesscA_in_ga(s(X1), s(X2)) -> U63_ga(X1, X2, lesscA_in_ga(X1, X2)) 53.83/15.31 U63_ga(X1, X2, lesscA_out_ga(X1, X2)) -> lesscA_out_ga(s(X1), s(X2)) 53.83/15.31 partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_gaaa(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) 53.83/15.31 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)) 53.83/15.31 partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) 53.83/15.31 partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], []) 53.83/15.31 U66_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.31 U65_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.31 partcC_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_aaaa(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) 53.83/15.31 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)) 53.83/15.31 U68_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.31 partcC_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_aaaa(X1, X2, X3, X4, X5, partcC_in_aaaa(X1, X3, X4, X5)) 53.83/15.31 partcC_in_aaaa(X1, [], [], []) -> partcC_out_aaaa(X1, [], [], []) 53.83/15.31 U69_aaaa(X1, X2, X3, X4, X5, partcC_out_aaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.31 qscF_in_aa([], []) -> qscF_out_aa([], []) 53.83/15.31 qscF_in_aa(.(X1, X2), X3) -> U74_aa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 appcE_in_aaaa([], X1, X2, .(X1, X2)) -> appcE_out_aaaa([], X1, X2, .(X1, X2)) 53.83/15.31 appcE_in_aaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U75_aaaa(X1, X2, X3, X4, X5, appcE_in_aaaa(X2, X3, X4, X5)) 53.83/15.31 U75_aaaa(X1, X2, X3, X4, X5, appcE_out_aaaa(X2, X3, X4, X5)) -> appcE_out_aaaa(.(X1, X2), X3, X4, .(X1, X5)) 53.83/15.31 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) 53.83/15.31 U74_aa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_aa(.(X1, X2), X3) 53.83/15.31 qscG_in_aaa(X1, X2, X3) -> U76_aaa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.31 U76_aaa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscG_out_aaa(X1, X2, X3) 53.83/15.31 lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) 53.83/15.31 lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) 53.83/15.31 U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) 53.83/15.31 partcC_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.83/15.31 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)) 53.83/15.31 partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.83/15.31 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)) 53.83/15.31 partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) 53.83/15.31 partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], []) 53.83/15.31 U66_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.31 U65_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.31 U68_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.31 partcC_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_ggaa(X1, X2, X3, X4, X5, partcC_in_ggaa(X1, X3, X4, X5)) 53.83/15.31 partcC_in_ggaa(X1, [], [], []) -> partcC_out_ggaa(X1, [], [], []) 53.83/15.31 U69_ggaa(X1, X2, X3, X4, X5, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.31 qscF_in_ga([], []) -> qscF_out_ga([], []) 53.83/15.31 qscF_in_ga(.(X1, X2), X3) -> U74_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2)) 53.83/15.31 appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U75_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5)) 53.83/15.31 U75_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5)) 53.83/15.31 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) 53.83/15.31 U74_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3) 53.83/15.31 53.83/15.31 The argument filtering Pi contains the following mapping: 53.83/15.31 [] = [] 53.83/15.31 53.83/15.31 lesscA_in_aa(x1, x2) = lesscA_in_aa 53.83/15.31 53.83/15.31 lesscA_out_aa(x1, x2) = lesscA_out_aa(x1) 53.83/15.31 53.83/15.31 U63_aa(x1, x2, x3) = U63_aa(x3) 53.83/15.31 53.83/15.31 s(x1) = s(x1) 53.83/15.31 53.83/15.31 lesscA_in_ga(x1, x2) = lesscA_in_ga(x1) 53.83/15.31 53.83/15.31 0 = 0 53.83/15.31 53.83/15.31 lesscA_out_ga(x1, x2) = lesscA_out_ga(x1) 53.83/15.31 53.83/15.31 U63_ga(x1, x2, x3) = U63_ga(x1, x3) 53.83/15.31 53.83/15.31 partcB_in_gaaa(x1, x2, x3, x4) = partcB_in_gaaa(x1) 53.83/15.31 53.83/15.31 U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6) 53.83/15.31 53.83/15.31 U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x6) 53.83/15.31 53.83/15.31 U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6) 53.83/15.31 53.83/15.31 partcB_out_gaaa(x1, x2, x3, x4) = partcB_out_gaaa(x1) 53.83/15.31 53.83/15.31 partcC_in_aaaa(x1, x2, x3, x4) = partcC_in_aaaa 53.83/15.31 53.83/15.31 U67_aaaa(x1, x2, x3, x4, x5, x6) = U67_aaaa(x6) 53.83/15.31 53.83/15.31 U68_aaaa(x1, x2, x3, x4, x5, x6) = U68_aaaa(x6) 53.83/15.31 53.83/15.31 partcC_out_aaaa(x1, x2, x3, x4) = partcC_out_aaaa 53.83/15.31 53.83/15.31 U69_aaaa(x1, x2, x3, x4, x5, x6) = U69_aaaa(x6) 53.83/15.31 53.83/15.31 qscF_in_aa(x1, x2) = qscF_in_aa 53.83/15.31 53.83/15.31 qscF_out_aa(x1, x2) = qscF_out_aa 53.83/15.31 53.83/15.31 U74_aa(x1, x2, x3, x4) = U74_aa(x4) 53.83/15.31 53.83/15.31 qcD_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_aaaaaaa 53.83/15.31 53.83/15.31 U70_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_aaaaaaa(x8) 53.83/15.31 53.83/15.31 U71_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_aaaaaaa(x8) 53.83/15.31 53.83/15.31 U72_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_aaaaaaa(x8) 53.83/15.31 53.83/15.31 U73_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_aaaaaaa(x8) 53.83/15.31 53.83/15.31 .(x1, x2) = .(x1, x2) 53.83/15.31 53.83/15.31 qcD_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_aaaaaaa 53.83/15.31 53.83/15.31 appcE_in_aaaa(x1, x2, x3, x4) = appcE_in_aaaa 53.83/15.31 53.83/15.31 appcE_out_aaaa(x1, x2, x3, x4) = appcE_out_aaaa 53.83/15.31 53.83/15.31 U75_aaaa(x1, x2, x3, x4, x5, x6) = U75_aaaa(x6) 53.83/15.31 53.83/15.31 qscG_in_aaa(x1, x2, x3) = qscG_in_aaa 53.83/15.31 53.83/15.31 U76_aaa(x1, x2, x3, x4) = U76_aaa(x4) 53.83/15.31 53.83/15.31 qscG_out_aaa(x1, x2, x3) = qscG_out_aaa 53.83/15.31 53.83/15.31 lesscA_in_gg(x1, x2) = lesscA_in_gg(x1, x2) 53.83/15.31 53.83/15.31 lesscA_out_gg(x1, x2) = lesscA_out_gg(x1, x2) 53.83/15.31 53.83/15.31 U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3) 53.83/15.31 53.83/15.31 partcC_in_ggaa(x1, x2, x3, x4) = partcC_in_ggaa(x1, x2) 53.83/15.31 53.83/15.31 U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 U68_ggaa(x1, x2, x3, x4, x5, x6) = U68_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 partcB_in_ggaa(x1, x2, x3, x4) = partcB_in_ggaa(x1, x2) 53.83/15.31 53.83/15.31 U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 partcB_out_ggaa(x1, x2, x3, x4) = partcB_out_ggaa(x1, x2, x3, x4) 53.83/15.31 53.83/15.31 partcC_out_ggaa(x1, x2, x3, x4) = partcC_out_ggaa(x1, x2, x3, x4) 53.83/15.31 53.83/15.31 U69_ggaa(x1, x2, x3, x4, x5, x6) = U69_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 qscF_in_ga(x1, x2) = qscF_in_ga(x1) 53.83/15.31 53.83/15.31 qscF_out_ga(x1, x2) = qscF_out_ga(x1, x2) 53.83/15.31 53.83/15.31 U74_ga(x1, x2, x3, x4) = U74_ga(x1, x2, x4) 53.83/15.31 53.83/15.31 qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_ggaaaaa(x1, x2) 53.83/15.31 53.83/15.31 U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x8) 53.83/15.31 53.83/15.31 U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x8) 53.83/15.31 53.83/15.31 U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x8) 53.83/15.31 53.83/15.31 U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x8) 53.83/15.31 53.83/15.31 appcE_in_ggga(x1, x2, x3, x4) = appcE_in_ggga(x1, x2, x3) 53.83/15.31 53.83/15.31 appcE_out_ggga(x1, x2, x3, x4) = appcE_out_ggga(x1, x2, x3, x4) 53.83/15.31 53.83/15.31 U75_ggga(x1, x2, x3, x4, x5, x6) = U75_ggga(x1, x2, x3, x4, x6) 53.83/15.31 53.83/15.31 qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) 53.83/15.31 53.83/15.31 LESSA_IN_GA(x1, x2) = LESSA_IN_GA(x1) 53.83/15.31 53.83/15.31 53.83/15.31 We have to consider all (P,R,Pi)-chains 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (206) UsableRulesProof (EQUIVALENT) 53.83/15.31 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (207) 53.83/15.31 Obligation: 53.83/15.31 Pi DP problem: 53.83/15.31 The TRS P consists of the following rules: 53.83/15.31 53.83/15.31 LESSA_IN_GA(s(X1), s(X2)) -> LESSA_IN_GA(X1, X2) 53.83/15.31 53.83/15.31 R is empty. 53.83/15.31 The argument filtering Pi contains the following mapping: 53.83/15.31 s(x1) = s(x1) 53.83/15.31 53.83/15.31 LESSA_IN_GA(x1, x2) = LESSA_IN_GA(x1) 53.83/15.31 53.83/15.31 53.83/15.31 We have to consider all (P,R,Pi)-chains 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (208) PiDPToQDPProof (SOUND) 53.83/15.31 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (209) 53.83/15.31 Obligation: 53.83/15.31 Q DP problem: 53.83/15.31 The TRS P consists of the following rules: 53.83/15.31 53.83/15.31 LESSA_IN_GA(s(X1)) -> LESSA_IN_GA(X1) 53.83/15.31 53.83/15.31 R is empty. 53.83/15.31 Q is empty. 53.83/15.31 We have to consider all (P,Q,R)-chains. 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (210) QDPSizeChangeProof (EQUIVALENT) 53.83/15.31 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. 53.83/15.31 53.83/15.31 From the DPs we obtained the following set of size-change graphs: 53.83/15.31 *LESSA_IN_GA(s(X1)) -> LESSA_IN_GA(X1) 53.83/15.31 The graph contains the following edges 1 > 1 53.83/15.31 53.83/15.31 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (211) 53.83/15.31 YES 53.83/15.31 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (212) 53.83/15.31 Obligation: 53.83/15.31 Pi DP problem: 53.83/15.31 The TRS P consists of the following rules: 53.83/15.31 53.83/15.31 PARTB_IN_GAAA(X1, .(X2, X3), .(X2, X4), X5) -> U3_GAAA(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) 53.83/15.31 U3_GAAA(X1, X2, X3, X4, X5, lesscA_out_ga(X1, X2)) -> PARTB_IN_GAAA(X1, X3, X4, X5) 53.83/15.31 PARTB_IN_GAAA(X1, .(X2, X3), X4, .(X2, X5)) -> PARTB_IN_GAAA(X1, X3, X4, X5) 53.83/15.31 53.83/15.31 The TRS R consists of the following rules: 53.83/15.31 53.83/15.31 lesscA_in_aa(0, s(X1)) -> lesscA_out_aa(0, s(X1)) 53.83/15.31 lesscA_in_aa(s(X1), s(X2)) -> U63_aa(X1, X2, lesscA_in_aa(X1, X2)) 53.83/15.31 U63_aa(X1, X2, lesscA_out_aa(X1, X2)) -> lesscA_out_aa(s(X1), s(X2)) 53.83/15.31 lesscA_in_ga(0, s(X1)) -> lesscA_out_ga(0, s(X1)) 53.83/15.31 lesscA_in_ga(s(X1), s(X2)) -> U63_ga(X1, X2, lesscA_in_ga(X1, X2)) 53.83/15.31 U63_ga(X1, X2, lesscA_out_ga(X1, X2)) -> lesscA_out_ga(s(X1), s(X2)) 53.83/15.31 partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_gaaa(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) 53.83/15.31 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)) 53.83/15.31 partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) 53.83/15.31 partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], []) 53.83/15.31 U66_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.31 U65_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.31 partcC_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_aaaa(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) 53.83/15.31 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)) 53.83/15.31 U68_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.31 partcC_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_aaaa(X1, X2, X3, X4, X5, partcC_in_aaaa(X1, X3, X4, X5)) 53.83/15.31 partcC_in_aaaa(X1, [], [], []) -> partcC_out_aaaa(X1, [], [], []) 53.83/15.31 U69_aaaa(X1, X2, X3, X4, X5, partcC_out_aaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.31 qscF_in_aa([], []) -> qscF_out_aa([], []) 53.83/15.31 qscF_in_aa(.(X1, X2), X3) -> U74_aa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 appcE_in_aaaa([], X1, X2, .(X1, X2)) -> appcE_out_aaaa([], X1, X2, .(X1, X2)) 53.83/15.31 appcE_in_aaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U75_aaaa(X1, X2, X3, X4, X5, appcE_in_aaaa(X2, X3, X4, X5)) 53.83/15.31 U75_aaaa(X1, X2, X3, X4, X5, appcE_out_aaaa(X2, X3, X4, X5)) -> appcE_out_aaaa(.(X1, X2), X3, X4, .(X1, X5)) 53.83/15.31 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) 53.83/15.31 U74_aa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_aa(.(X1, X2), X3) 53.83/15.31 qscG_in_aaa(X1, X2, X3) -> U76_aaa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.31 U76_aaa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscG_out_aaa(X1, X2, X3) 53.83/15.31 lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) 53.83/15.31 lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) 53.83/15.31 U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) 53.83/15.31 partcC_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.83/15.31 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)) 53.83/15.31 partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.83/15.31 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)) 53.83/15.31 partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) 53.83/15.31 partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], []) 53.83/15.31 U66_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.31 U65_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.31 U68_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.31 partcC_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_ggaa(X1, X2, X3, X4, X5, partcC_in_ggaa(X1, X3, X4, X5)) 53.83/15.31 partcC_in_ggaa(X1, [], [], []) -> partcC_out_ggaa(X1, [], [], []) 53.83/15.31 U69_ggaa(X1, X2, X3, X4, X5, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.31 qscF_in_ga([], []) -> qscF_out_ga([], []) 53.83/15.31 qscF_in_ga(.(X1, X2), X3) -> U74_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2)) 53.83/15.31 appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U75_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5)) 53.83/15.31 U75_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5)) 53.83/15.31 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) 53.83/15.31 U74_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3) 53.83/15.31 53.83/15.31 The argument filtering Pi contains the following mapping: 53.83/15.31 [] = [] 53.83/15.31 53.83/15.31 lesscA_in_aa(x1, x2) = lesscA_in_aa 53.83/15.31 53.83/15.31 lesscA_out_aa(x1, x2) = lesscA_out_aa(x1) 53.83/15.31 53.83/15.31 U63_aa(x1, x2, x3) = U63_aa(x3) 53.83/15.31 53.83/15.31 s(x1) = s(x1) 53.83/15.31 53.83/15.31 lesscA_in_ga(x1, x2) = lesscA_in_ga(x1) 53.83/15.31 53.83/15.31 0 = 0 53.83/15.31 53.83/15.31 lesscA_out_ga(x1, x2) = lesscA_out_ga(x1) 53.83/15.31 53.83/15.31 U63_ga(x1, x2, x3) = U63_ga(x1, x3) 53.83/15.31 53.83/15.31 partcB_in_gaaa(x1, x2, x3, x4) = partcB_in_gaaa(x1) 53.83/15.31 53.83/15.31 U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6) 53.83/15.31 53.83/15.31 U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x6) 53.83/15.31 53.83/15.31 U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6) 53.83/15.31 53.83/15.31 partcB_out_gaaa(x1, x2, x3, x4) = partcB_out_gaaa(x1) 53.83/15.31 53.83/15.31 partcC_in_aaaa(x1, x2, x3, x4) = partcC_in_aaaa 53.83/15.31 53.83/15.31 U67_aaaa(x1, x2, x3, x4, x5, x6) = U67_aaaa(x6) 53.83/15.31 53.83/15.31 U68_aaaa(x1, x2, x3, x4, x5, x6) = U68_aaaa(x6) 53.83/15.31 53.83/15.31 partcC_out_aaaa(x1, x2, x3, x4) = partcC_out_aaaa 53.83/15.31 53.83/15.31 U69_aaaa(x1, x2, x3, x4, x5, x6) = U69_aaaa(x6) 53.83/15.31 53.83/15.31 qscF_in_aa(x1, x2) = qscF_in_aa 53.83/15.31 53.83/15.31 qscF_out_aa(x1, x2) = qscF_out_aa 53.83/15.31 53.83/15.31 U74_aa(x1, x2, x3, x4) = U74_aa(x4) 53.83/15.31 53.83/15.31 qcD_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_aaaaaaa 53.83/15.31 53.83/15.31 U70_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_aaaaaaa(x8) 53.83/15.31 53.83/15.31 U71_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_aaaaaaa(x8) 53.83/15.31 53.83/15.31 U72_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_aaaaaaa(x8) 53.83/15.31 53.83/15.31 U73_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_aaaaaaa(x8) 53.83/15.31 53.83/15.31 .(x1, x2) = .(x1, x2) 53.83/15.31 53.83/15.31 qcD_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_aaaaaaa 53.83/15.31 53.83/15.31 appcE_in_aaaa(x1, x2, x3, x4) = appcE_in_aaaa 53.83/15.31 53.83/15.31 appcE_out_aaaa(x1, x2, x3, x4) = appcE_out_aaaa 53.83/15.31 53.83/15.31 U75_aaaa(x1, x2, x3, x4, x5, x6) = U75_aaaa(x6) 53.83/15.31 53.83/15.31 qscG_in_aaa(x1, x2, x3) = qscG_in_aaa 53.83/15.31 53.83/15.31 U76_aaa(x1, x2, x3, x4) = U76_aaa(x4) 53.83/15.31 53.83/15.31 qscG_out_aaa(x1, x2, x3) = qscG_out_aaa 53.83/15.31 53.83/15.31 lesscA_in_gg(x1, x2) = lesscA_in_gg(x1, x2) 53.83/15.31 53.83/15.31 lesscA_out_gg(x1, x2) = lesscA_out_gg(x1, x2) 53.83/15.31 53.83/15.31 U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3) 53.83/15.31 53.83/15.31 partcC_in_ggaa(x1, x2, x3, x4) = partcC_in_ggaa(x1, x2) 53.83/15.31 53.83/15.31 U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 U68_ggaa(x1, x2, x3, x4, x5, x6) = U68_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 partcB_in_ggaa(x1, x2, x3, x4) = partcB_in_ggaa(x1, x2) 53.83/15.31 53.83/15.31 U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 partcB_out_ggaa(x1, x2, x3, x4) = partcB_out_ggaa(x1, x2, x3, x4) 53.83/15.31 53.83/15.31 partcC_out_ggaa(x1, x2, x3, x4) = partcC_out_ggaa(x1, x2, x3, x4) 53.83/15.31 53.83/15.31 U69_ggaa(x1, x2, x3, x4, x5, x6) = U69_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 qscF_in_ga(x1, x2) = qscF_in_ga(x1) 53.83/15.31 53.83/15.31 qscF_out_ga(x1, x2) = qscF_out_ga(x1, x2) 53.83/15.31 53.83/15.31 U74_ga(x1, x2, x3, x4) = U74_ga(x1, x2, x4) 53.83/15.31 53.83/15.31 qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_ggaaaaa(x1, x2) 53.83/15.31 53.83/15.31 U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x8) 53.83/15.31 53.83/15.31 U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x8) 53.83/15.31 53.83/15.31 U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x8) 53.83/15.31 53.83/15.31 U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x8) 53.83/15.31 53.83/15.31 appcE_in_ggga(x1, x2, x3, x4) = appcE_in_ggga(x1, x2, x3) 53.83/15.31 53.83/15.31 appcE_out_ggga(x1, x2, x3, x4) = appcE_out_ggga(x1, x2, x3, x4) 53.83/15.31 53.83/15.31 U75_ggga(x1, x2, x3, x4, x5, x6) = U75_ggga(x1, x2, x3, x4, x6) 53.83/15.31 53.83/15.31 qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) 53.83/15.31 53.83/15.31 PARTB_IN_GAAA(x1, x2, x3, x4) = PARTB_IN_GAAA(x1) 53.83/15.31 53.83/15.31 U3_GAAA(x1, x2, x3, x4, x5, x6) = U3_GAAA(x1, x6) 53.83/15.31 53.83/15.31 53.83/15.31 We have to consider all (P,R,Pi)-chains 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (213) UsableRulesProof (EQUIVALENT) 53.83/15.31 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (214) 53.83/15.31 Obligation: 53.83/15.31 Pi DP problem: 53.83/15.31 The TRS P consists of the following rules: 53.83/15.31 53.83/15.31 PARTB_IN_GAAA(X1, .(X2, X3), .(X2, X4), X5) -> U3_GAAA(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) 53.83/15.31 U3_GAAA(X1, X2, X3, X4, X5, lesscA_out_ga(X1, X2)) -> PARTB_IN_GAAA(X1, X3, X4, X5) 53.83/15.31 PARTB_IN_GAAA(X1, .(X2, X3), X4, .(X2, X5)) -> PARTB_IN_GAAA(X1, X3, X4, X5) 53.83/15.31 53.83/15.31 The TRS R consists of the following rules: 53.83/15.31 53.83/15.31 lesscA_in_ga(0, s(X1)) -> lesscA_out_ga(0, s(X1)) 53.83/15.31 lesscA_in_ga(s(X1), s(X2)) -> U63_ga(X1, X2, lesscA_in_ga(X1, X2)) 53.83/15.31 U63_ga(X1, X2, lesscA_out_ga(X1, X2)) -> lesscA_out_ga(s(X1), s(X2)) 53.83/15.31 53.83/15.31 The argument filtering Pi contains the following mapping: 53.83/15.31 s(x1) = s(x1) 53.83/15.31 53.83/15.31 lesscA_in_ga(x1, x2) = lesscA_in_ga(x1) 53.83/15.31 53.83/15.31 0 = 0 53.83/15.31 53.83/15.31 lesscA_out_ga(x1, x2) = lesscA_out_ga(x1) 53.83/15.31 53.83/15.31 U63_ga(x1, x2, x3) = U63_ga(x1, x3) 53.83/15.31 53.83/15.31 .(x1, x2) = .(x1, x2) 53.83/15.31 53.83/15.31 PARTB_IN_GAAA(x1, x2, x3, x4) = PARTB_IN_GAAA(x1) 53.83/15.31 53.83/15.31 U3_GAAA(x1, x2, x3, x4, x5, x6) = U3_GAAA(x1, x6) 53.83/15.31 53.83/15.31 53.83/15.31 We have to consider all (P,R,Pi)-chains 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (215) PiDPToQDPProof (SOUND) 53.83/15.31 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (216) 53.83/15.31 Obligation: 53.83/15.31 Q DP problem: 53.83/15.31 The TRS P consists of the following rules: 53.83/15.31 53.83/15.31 PARTB_IN_GAAA(X1) -> U3_GAAA(X1, lesscA_in_ga(X1)) 53.83/15.31 U3_GAAA(X1, lesscA_out_ga(X1)) -> PARTB_IN_GAAA(X1) 53.83/15.31 PARTB_IN_GAAA(X1) -> PARTB_IN_GAAA(X1) 53.83/15.31 53.83/15.31 The TRS R consists of the following rules: 53.83/15.31 53.83/15.31 lesscA_in_ga(0) -> lesscA_out_ga(0) 53.83/15.31 lesscA_in_ga(s(X1)) -> U63_ga(X1, lesscA_in_ga(X1)) 53.83/15.31 U63_ga(X1, lesscA_out_ga(X1)) -> lesscA_out_ga(s(X1)) 53.83/15.31 53.83/15.31 The set Q consists of the following terms: 53.83/15.31 53.83/15.31 lesscA_in_ga(x0) 53.83/15.31 U63_ga(x0, x1) 53.83/15.31 53.83/15.31 We have to consider all (P,Q,R)-chains. 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (217) TransformationProof (SOUND) 53.83/15.31 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]: 53.83/15.31 53.83/15.31 (PARTB_IN_GAAA(0) -> U3_GAAA(0, lesscA_out_ga(0)),PARTB_IN_GAAA(0) -> U3_GAAA(0, lesscA_out_ga(0))) 53.83/15.31 (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)))) 53.83/15.31 53.83/15.31 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (218) 53.83/15.31 Obligation: 53.83/15.31 Q DP problem: 53.83/15.31 The TRS P consists of the following rules: 53.83/15.31 53.83/15.31 U3_GAAA(X1, lesscA_out_ga(X1)) -> PARTB_IN_GAAA(X1) 53.83/15.31 PARTB_IN_GAAA(X1) -> PARTB_IN_GAAA(X1) 53.83/15.31 PARTB_IN_GAAA(0) -> U3_GAAA(0, lesscA_out_ga(0)) 53.83/15.31 PARTB_IN_GAAA(s(x0)) -> U3_GAAA(s(x0), U63_ga(x0, lesscA_in_ga(x0))) 53.83/15.31 53.83/15.31 The TRS R consists of the following rules: 53.83/15.31 53.83/15.31 lesscA_in_ga(0) -> lesscA_out_ga(0) 53.83/15.31 lesscA_in_ga(s(X1)) -> U63_ga(X1, lesscA_in_ga(X1)) 53.83/15.31 U63_ga(X1, lesscA_out_ga(X1)) -> lesscA_out_ga(s(X1)) 53.83/15.31 53.83/15.31 The set Q consists of the following terms: 53.83/15.31 53.83/15.31 lesscA_in_ga(x0) 53.83/15.31 U63_ga(x0, x1) 53.83/15.31 53.83/15.31 We have to consider all (P,Q,R)-chains. 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (219) TransformationProof (EQUIVALENT) 53.83/15.31 By instantiating [LPAR04] the rule U3_GAAA(X1, lesscA_out_ga(X1)) -> PARTB_IN_GAAA(X1) we obtained the following new rules [LPAR04]: 53.83/15.31 53.83/15.31 (U3_GAAA(0, lesscA_out_ga(0)) -> PARTB_IN_GAAA(0),U3_GAAA(0, lesscA_out_ga(0)) -> PARTB_IN_GAAA(0)) 53.83/15.31 (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))) 53.83/15.31 53.83/15.31 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (220) 53.83/15.31 Obligation: 53.83/15.31 Q DP problem: 53.83/15.31 The TRS P consists of the following rules: 53.83/15.31 53.83/15.31 PARTB_IN_GAAA(X1) -> PARTB_IN_GAAA(X1) 53.83/15.31 PARTB_IN_GAAA(0) -> U3_GAAA(0, lesscA_out_ga(0)) 53.83/15.31 PARTB_IN_GAAA(s(x0)) -> U3_GAAA(s(x0), U63_ga(x0, lesscA_in_ga(x0))) 53.83/15.31 U3_GAAA(0, lesscA_out_ga(0)) -> PARTB_IN_GAAA(0) 53.83/15.31 U3_GAAA(s(z0), lesscA_out_ga(s(z0))) -> PARTB_IN_GAAA(s(z0)) 53.83/15.31 53.83/15.31 The TRS R consists of the following rules: 53.83/15.31 53.83/15.31 lesscA_in_ga(0) -> lesscA_out_ga(0) 53.83/15.31 lesscA_in_ga(s(X1)) -> U63_ga(X1, lesscA_in_ga(X1)) 53.83/15.31 U63_ga(X1, lesscA_out_ga(X1)) -> lesscA_out_ga(s(X1)) 53.83/15.31 53.83/15.31 The set Q consists of the following terms: 53.83/15.31 53.83/15.31 lesscA_in_ga(x0) 53.83/15.31 U63_ga(x0, x1) 53.83/15.31 53.83/15.31 We have to consider all (P,Q,R)-chains. 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (221) 53.83/15.31 Obligation: 53.83/15.31 Pi DP problem: 53.83/15.31 The TRS P consists of the following rules: 53.83/15.31 53.83/15.31 LESSA_IN_AA(s(X1), s(X2)) -> LESSA_IN_AA(X1, X2) 53.83/15.31 53.83/15.31 The TRS R consists of the following rules: 53.83/15.31 53.83/15.31 lesscA_in_aa(0, s(X1)) -> lesscA_out_aa(0, s(X1)) 53.83/15.31 lesscA_in_aa(s(X1), s(X2)) -> U63_aa(X1, X2, lesscA_in_aa(X1, X2)) 53.83/15.31 U63_aa(X1, X2, lesscA_out_aa(X1, X2)) -> lesscA_out_aa(s(X1), s(X2)) 53.83/15.31 lesscA_in_ga(0, s(X1)) -> lesscA_out_ga(0, s(X1)) 53.83/15.31 lesscA_in_ga(s(X1), s(X2)) -> U63_ga(X1, X2, lesscA_in_ga(X1, X2)) 53.83/15.31 U63_ga(X1, X2, lesscA_out_ga(X1, X2)) -> lesscA_out_ga(s(X1), s(X2)) 53.83/15.31 partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_gaaa(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) 53.83/15.31 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)) 53.83/15.31 partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) 53.83/15.31 partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], []) 53.83/15.31 U66_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.31 U65_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.31 partcC_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_aaaa(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) 53.83/15.31 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)) 53.83/15.31 U68_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.31 partcC_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_aaaa(X1, X2, X3, X4, X5, partcC_in_aaaa(X1, X3, X4, X5)) 53.83/15.31 partcC_in_aaaa(X1, [], [], []) -> partcC_out_aaaa(X1, [], [], []) 53.83/15.31 U69_aaaa(X1, X2, X3, X4, X5, partcC_out_aaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.31 qscF_in_aa([], []) -> qscF_out_aa([], []) 53.83/15.31 qscF_in_aa(.(X1, X2), X3) -> U74_aa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 appcE_in_aaaa([], X1, X2, .(X1, X2)) -> appcE_out_aaaa([], X1, X2, .(X1, X2)) 53.83/15.31 appcE_in_aaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U75_aaaa(X1, X2, X3, X4, X5, appcE_in_aaaa(X2, X3, X4, X5)) 53.83/15.31 U75_aaaa(X1, X2, X3, X4, X5, appcE_out_aaaa(X2, X3, X4, X5)) -> appcE_out_aaaa(.(X1, X2), X3, X4, .(X1, X5)) 53.83/15.31 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) 53.83/15.31 U74_aa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_aa(.(X1, X2), X3) 53.83/15.31 qscG_in_aaa(X1, X2, X3) -> U76_aaa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.31 U76_aaa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscG_out_aaa(X1, X2, X3) 53.83/15.31 lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) 53.83/15.31 lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) 53.83/15.31 U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) 53.83/15.31 partcC_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.83/15.31 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)) 53.83/15.31 partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.83/15.31 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)) 53.83/15.31 partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) 53.83/15.31 partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], []) 53.83/15.31 U66_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.31 U65_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.31 U68_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.31 partcC_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_ggaa(X1, X2, X3, X4, X5, partcC_in_ggaa(X1, X3, X4, X5)) 53.83/15.31 partcC_in_ggaa(X1, [], [], []) -> partcC_out_ggaa(X1, [], [], []) 53.83/15.31 U69_ggaa(X1, X2, X3, X4, X5, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.31 qscF_in_ga([], []) -> qscF_out_ga([], []) 53.83/15.31 qscF_in_ga(.(X1, X2), X3) -> U74_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2)) 53.83/15.31 appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U75_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5)) 53.83/15.31 U75_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5)) 53.83/15.31 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) 53.83/15.31 U74_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3) 53.83/15.31 53.83/15.31 The argument filtering Pi contains the following mapping: 53.83/15.31 [] = [] 53.83/15.31 53.83/15.31 lesscA_in_aa(x1, x2) = lesscA_in_aa 53.83/15.31 53.83/15.31 lesscA_out_aa(x1, x2) = lesscA_out_aa(x1) 53.83/15.31 53.83/15.31 U63_aa(x1, x2, x3) = U63_aa(x3) 53.83/15.31 53.83/15.31 s(x1) = s(x1) 53.83/15.31 53.83/15.31 lesscA_in_ga(x1, x2) = lesscA_in_ga(x1) 53.83/15.31 53.83/15.31 0 = 0 53.83/15.31 53.83/15.31 lesscA_out_ga(x1, x2) = lesscA_out_ga(x1) 53.83/15.31 53.83/15.31 U63_ga(x1, x2, x3) = U63_ga(x1, x3) 53.83/15.31 53.83/15.31 partcB_in_gaaa(x1, x2, x3, x4) = partcB_in_gaaa(x1) 53.83/15.31 53.83/15.31 U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6) 53.83/15.31 53.83/15.31 U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x6) 53.83/15.31 53.83/15.31 U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6) 53.83/15.31 53.83/15.31 partcB_out_gaaa(x1, x2, x3, x4) = partcB_out_gaaa(x1) 53.83/15.31 53.83/15.31 partcC_in_aaaa(x1, x2, x3, x4) = partcC_in_aaaa 53.83/15.31 53.83/15.31 U67_aaaa(x1, x2, x3, x4, x5, x6) = U67_aaaa(x6) 53.83/15.31 53.83/15.31 U68_aaaa(x1, x2, x3, x4, x5, x6) = U68_aaaa(x6) 53.83/15.31 53.83/15.31 partcC_out_aaaa(x1, x2, x3, x4) = partcC_out_aaaa 53.83/15.31 53.83/15.31 U69_aaaa(x1, x2, x3, x4, x5, x6) = U69_aaaa(x6) 53.83/15.31 53.83/15.31 qscF_in_aa(x1, x2) = qscF_in_aa 53.83/15.31 53.83/15.31 qscF_out_aa(x1, x2) = qscF_out_aa 53.83/15.31 53.83/15.31 U74_aa(x1, x2, x3, x4) = U74_aa(x4) 53.83/15.31 53.83/15.31 qcD_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_aaaaaaa 53.83/15.31 53.83/15.31 U70_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_aaaaaaa(x8) 53.83/15.31 53.83/15.31 U71_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_aaaaaaa(x8) 53.83/15.31 53.83/15.31 U72_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_aaaaaaa(x8) 53.83/15.31 53.83/15.31 U73_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_aaaaaaa(x8) 53.83/15.31 53.83/15.31 .(x1, x2) = .(x1, x2) 53.83/15.31 53.83/15.31 qcD_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_aaaaaaa 53.83/15.31 53.83/15.31 appcE_in_aaaa(x1, x2, x3, x4) = appcE_in_aaaa 53.83/15.31 53.83/15.31 appcE_out_aaaa(x1, x2, x3, x4) = appcE_out_aaaa 53.83/15.31 53.83/15.31 U75_aaaa(x1, x2, x3, x4, x5, x6) = U75_aaaa(x6) 53.83/15.31 53.83/15.31 qscG_in_aaa(x1, x2, x3) = qscG_in_aaa 53.83/15.31 53.83/15.31 U76_aaa(x1, x2, x3, x4) = U76_aaa(x4) 53.83/15.31 53.83/15.31 qscG_out_aaa(x1, x2, x3) = qscG_out_aaa 53.83/15.31 53.83/15.31 lesscA_in_gg(x1, x2) = lesscA_in_gg(x1, x2) 53.83/15.31 53.83/15.31 lesscA_out_gg(x1, x2) = lesscA_out_gg(x1, x2) 53.83/15.31 53.83/15.31 U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3) 53.83/15.31 53.83/15.31 partcC_in_ggaa(x1, x2, x3, x4) = partcC_in_ggaa(x1, x2) 53.83/15.31 53.83/15.31 U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 U68_ggaa(x1, x2, x3, x4, x5, x6) = U68_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 partcB_in_ggaa(x1, x2, x3, x4) = partcB_in_ggaa(x1, x2) 53.83/15.31 53.83/15.31 U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 partcB_out_ggaa(x1, x2, x3, x4) = partcB_out_ggaa(x1, x2, x3, x4) 53.83/15.31 53.83/15.31 partcC_out_ggaa(x1, x2, x3, x4) = partcC_out_ggaa(x1, x2, x3, x4) 53.83/15.31 53.83/15.31 U69_ggaa(x1, x2, x3, x4, x5, x6) = U69_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 qscF_in_ga(x1, x2) = qscF_in_ga(x1) 53.83/15.31 53.83/15.31 qscF_out_ga(x1, x2) = qscF_out_ga(x1, x2) 53.83/15.31 53.83/15.31 U74_ga(x1, x2, x3, x4) = U74_ga(x1, x2, x4) 53.83/15.31 53.83/15.31 qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_ggaaaaa(x1, x2) 53.83/15.31 53.83/15.31 U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x8) 53.83/15.31 53.83/15.31 U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x8) 53.83/15.31 53.83/15.31 U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x8) 53.83/15.31 53.83/15.31 U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x8) 53.83/15.31 53.83/15.31 appcE_in_ggga(x1, x2, x3, x4) = appcE_in_ggga(x1, x2, x3) 53.83/15.31 53.83/15.31 appcE_out_ggga(x1, x2, x3, x4) = appcE_out_ggga(x1, x2, x3, x4) 53.83/15.31 53.83/15.31 U75_ggga(x1, x2, x3, x4, x5, x6) = U75_ggga(x1, x2, x3, x4, x6) 53.83/15.31 53.83/15.31 qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) 53.83/15.31 53.83/15.31 LESSA_IN_AA(x1, x2) = LESSA_IN_AA 53.83/15.31 53.83/15.31 53.83/15.31 We have to consider all (P,R,Pi)-chains 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (222) UsableRulesProof (EQUIVALENT) 53.83/15.31 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (223) 53.83/15.31 Obligation: 53.83/15.31 Pi DP problem: 53.83/15.31 The TRS P consists of the following rules: 53.83/15.31 53.83/15.31 LESSA_IN_AA(s(X1), s(X2)) -> LESSA_IN_AA(X1, X2) 53.83/15.31 53.83/15.31 R is empty. 53.83/15.31 The argument filtering Pi contains the following mapping: 53.83/15.31 s(x1) = s(x1) 53.83/15.31 53.83/15.31 LESSA_IN_AA(x1, x2) = LESSA_IN_AA 53.83/15.31 53.83/15.31 53.83/15.31 We have to consider all (P,R,Pi)-chains 53.83/15.31 ---------------------------------------- 53.83/15.31 53.83/15.31 (224) 53.83/15.31 Obligation: 53.83/15.31 Pi DP problem: 53.83/15.31 The TRS P consists of the following rules: 53.83/15.31 53.83/15.31 PARTC_IN_AAAA(X1, .(X2, X3), X4, .(X2, X5)) -> PARTC_IN_AAAA(X1, X3, X4, X5) 53.83/15.31 53.83/15.31 The TRS R consists of the following rules: 53.83/15.31 53.83/15.31 lesscA_in_aa(0, s(X1)) -> lesscA_out_aa(0, s(X1)) 53.83/15.31 lesscA_in_aa(s(X1), s(X2)) -> U63_aa(X1, X2, lesscA_in_aa(X1, X2)) 53.83/15.31 U63_aa(X1, X2, lesscA_out_aa(X1, X2)) -> lesscA_out_aa(s(X1), s(X2)) 53.83/15.31 lesscA_in_ga(0, s(X1)) -> lesscA_out_ga(0, s(X1)) 53.83/15.31 lesscA_in_ga(s(X1), s(X2)) -> U63_ga(X1, X2, lesscA_in_ga(X1, X2)) 53.83/15.31 U63_ga(X1, X2, lesscA_out_ga(X1, X2)) -> lesscA_out_ga(s(X1), s(X2)) 53.83/15.31 partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_gaaa(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) 53.83/15.31 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)) 53.83/15.31 partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) 53.83/15.31 partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], []) 53.83/15.31 U66_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.31 U65_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.31 partcC_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_aaaa(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) 53.83/15.31 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)) 53.83/15.31 U68_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.31 partcC_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_aaaa(X1, X2, X3, X4, X5, partcC_in_aaaa(X1, X3, X4, X5)) 53.83/15.31 partcC_in_aaaa(X1, [], [], []) -> partcC_out_aaaa(X1, [], [], []) 53.83/15.31 U69_aaaa(X1, X2, X3, X4, X5, partcC_out_aaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.31 qscF_in_aa([], []) -> qscF_out_aa([], []) 53.83/15.31 qscF_in_aa(.(X1, X2), X3) -> U74_aa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 appcE_in_aaaa([], X1, X2, .(X1, X2)) -> appcE_out_aaaa([], X1, X2, .(X1, X2)) 53.83/15.31 appcE_in_aaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U75_aaaa(X1, X2, X3, X4, X5, appcE_in_aaaa(X2, X3, X4, X5)) 53.83/15.31 U75_aaaa(X1, X2, X3, X4, X5, appcE_out_aaaa(X2, X3, X4, X5)) -> appcE_out_aaaa(.(X1, X2), X3, X4, .(X1, X5)) 53.83/15.31 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) 53.83/15.31 U74_aa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_aa(.(X1, X2), X3) 53.83/15.31 qscG_in_aaa(X1, X2, X3) -> U76_aaa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.31 U76_aaa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscG_out_aaa(X1, X2, X3) 53.83/15.31 lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) 53.83/15.31 lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) 53.83/15.31 U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) 53.83/15.31 partcC_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.83/15.31 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)) 53.83/15.31 partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.83/15.31 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)) 53.83/15.31 partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) 53.83/15.31 partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], []) 53.83/15.31 U66_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.31 U65_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.31 U68_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.31 partcC_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_ggaa(X1, X2, X3, X4, X5, partcC_in_ggaa(X1, X3, X4, X5)) 53.83/15.31 partcC_in_ggaa(X1, [], [], []) -> partcC_out_ggaa(X1, [], [], []) 53.83/15.31 U69_ggaa(X1, X2, X3, X4, X5, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.31 qscF_in_ga([], []) -> qscF_out_ga([], []) 53.83/15.31 qscF_in_ga(.(X1, X2), X3) -> U74_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 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)) 53.83/15.31 appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2)) 53.83/15.31 appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U75_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5)) 53.83/15.31 U75_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5)) 53.83/15.31 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) 53.83/15.31 U74_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3) 53.83/15.31 53.83/15.31 The argument filtering Pi contains the following mapping: 53.83/15.31 [] = [] 53.83/15.31 53.83/15.31 lesscA_in_aa(x1, x2) = lesscA_in_aa 53.83/15.31 53.83/15.31 lesscA_out_aa(x1, x2) = lesscA_out_aa(x1) 53.83/15.31 53.83/15.31 U63_aa(x1, x2, x3) = U63_aa(x3) 53.83/15.31 53.83/15.31 s(x1) = s(x1) 53.83/15.31 53.83/15.31 lesscA_in_ga(x1, x2) = lesscA_in_ga(x1) 53.83/15.31 53.83/15.31 0 = 0 53.83/15.31 53.83/15.31 lesscA_out_ga(x1, x2) = lesscA_out_ga(x1) 53.83/15.31 53.83/15.31 U63_ga(x1, x2, x3) = U63_ga(x1, x3) 53.83/15.31 53.83/15.31 partcB_in_gaaa(x1, x2, x3, x4) = partcB_in_gaaa(x1) 53.83/15.31 53.83/15.31 U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6) 53.83/15.31 53.83/15.31 U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x6) 53.83/15.31 53.83/15.31 U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6) 53.83/15.31 53.83/15.31 partcB_out_gaaa(x1, x2, x3, x4) = partcB_out_gaaa(x1) 53.83/15.31 53.83/15.31 partcC_in_aaaa(x1, x2, x3, x4) = partcC_in_aaaa 53.83/15.31 53.83/15.31 U67_aaaa(x1, x2, x3, x4, x5, x6) = U67_aaaa(x6) 53.83/15.31 53.83/15.31 U68_aaaa(x1, x2, x3, x4, x5, x6) = U68_aaaa(x6) 53.83/15.31 53.83/15.31 partcC_out_aaaa(x1, x2, x3, x4) = partcC_out_aaaa 53.83/15.31 53.83/15.31 U69_aaaa(x1, x2, x3, x4, x5, x6) = U69_aaaa(x6) 53.83/15.31 53.83/15.31 qscF_in_aa(x1, x2) = qscF_in_aa 53.83/15.31 53.83/15.31 qscF_out_aa(x1, x2) = qscF_out_aa 53.83/15.31 53.83/15.31 U74_aa(x1, x2, x3, x4) = U74_aa(x4) 53.83/15.31 53.83/15.31 qcD_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_aaaaaaa 53.83/15.31 53.83/15.31 U70_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_aaaaaaa(x8) 53.83/15.31 53.83/15.31 U71_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_aaaaaaa(x8) 53.83/15.31 53.83/15.31 U72_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_aaaaaaa(x8) 53.83/15.31 53.83/15.31 U73_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_aaaaaaa(x8) 53.83/15.31 53.83/15.31 .(x1, x2) = .(x1, x2) 53.83/15.31 53.83/15.31 qcD_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_aaaaaaa 53.83/15.31 53.83/15.31 appcE_in_aaaa(x1, x2, x3, x4) = appcE_in_aaaa 53.83/15.31 53.83/15.31 appcE_out_aaaa(x1, x2, x3, x4) = appcE_out_aaaa 53.83/15.31 53.83/15.31 U75_aaaa(x1, x2, x3, x4, x5, x6) = U75_aaaa(x6) 53.83/15.31 53.83/15.31 qscG_in_aaa(x1, x2, x3) = qscG_in_aaa 53.83/15.31 53.83/15.31 U76_aaa(x1, x2, x3, x4) = U76_aaa(x4) 53.83/15.31 53.83/15.31 qscG_out_aaa(x1, x2, x3) = qscG_out_aaa 53.83/15.31 53.83/15.31 lesscA_in_gg(x1, x2) = lesscA_in_gg(x1, x2) 53.83/15.31 53.83/15.31 lesscA_out_gg(x1, x2) = lesscA_out_gg(x1, x2) 53.83/15.31 53.83/15.31 U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3) 53.83/15.31 53.83/15.31 partcC_in_ggaa(x1, x2, x3, x4) = partcC_in_ggaa(x1, x2) 53.83/15.31 53.83/15.31 U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 U68_ggaa(x1, x2, x3, x4, x5, x6) = U68_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 partcB_in_ggaa(x1, x2, x3, x4) = partcB_in_ggaa(x1, x2) 53.83/15.31 53.83/15.31 U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 partcB_out_ggaa(x1, x2, x3, x4) = partcB_out_ggaa(x1, x2, x3, x4) 53.83/15.31 53.83/15.31 partcC_out_ggaa(x1, x2, x3, x4) = partcC_out_ggaa(x1, x2, x3, x4) 53.83/15.31 53.83/15.31 U69_ggaa(x1, x2, x3, x4, x5, x6) = U69_ggaa(x1, x2, x3, x6) 53.83/15.31 53.83/15.31 qscF_in_ga(x1, x2) = qscF_in_ga(x1) 53.83/15.31 53.83/15.31 qscF_out_ga(x1, x2) = qscF_out_ga(x1, x2) 53.83/15.31 53.83/15.31 U74_ga(x1, x2, x3, x4) = U74_ga(x1, x2, x4) 53.83/15.31 53.83/15.31 qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_ggaaaaa(x1, x2) 53.83/15.31 53.83/15.31 U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x8) 53.83/15.31 53.83/15.31 U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x8) 53.83/15.32 53.83/15.32 U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x8) 53.83/15.32 53.83/15.32 U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x8) 53.83/15.32 53.83/15.32 appcE_in_ggga(x1, x2, x3, x4) = appcE_in_ggga(x1, x2, x3) 53.83/15.32 53.83/15.32 appcE_out_ggga(x1, x2, x3, x4) = appcE_out_ggga(x1, x2, x3, x4) 53.83/15.32 53.83/15.32 U75_ggga(x1, x2, x3, x4, x5, x6) = U75_ggga(x1, x2, x3, x4, x6) 53.83/15.32 53.83/15.32 qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) 53.83/15.32 53.83/15.32 PARTC_IN_AAAA(x1, x2, x3, x4) = PARTC_IN_AAAA 53.83/15.32 53.83/15.32 53.83/15.32 We have to consider all (P,R,Pi)-chains 53.83/15.32 ---------------------------------------- 53.83/15.32 53.83/15.32 (225) UsableRulesProof (EQUIVALENT) 53.83/15.32 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 53.83/15.32 ---------------------------------------- 53.83/15.32 53.83/15.32 (226) 53.83/15.32 Obligation: 53.83/15.32 Pi DP problem: 53.83/15.32 The TRS P consists of the following rules: 53.83/15.32 53.83/15.32 PARTC_IN_AAAA(X1, .(X2, X3), X4, .(X2, X5)) -> PARTC_IN_AAAA(X1, X3, X4, X5) 53.83/15.32 53.83/15.32 R is empty. 53.83/15.32 The argument filtering Pi contains the following mapping: 53.83/15.32 .(x1, x2) = .(x1, x2) 53.83/15.32 53.83/15.32 PARTC_IN_AAAA(x1, x2, x3, x4) = PARTC_IN_AAAA 53.83/15.32 53.83/15.32 53.83/15.32 We have to consider all (P,R,Pi)-chains 53.83/15.32 ---------------------------------------- 53.83/15.32 53.83/15.32 (227) 53.83/15.32 Obligation: 53.83/15.32 Pi DP problem: 53.83/15.32 The TRS P consists of the following rules: 53.83/15.32 53.83/15.32 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)) 53.83/15.32 U11_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, partcC_out_aaaa(X1, X2, X3, X4)) -> QSF_IN_AA(X3, X5) 53.83/15.32 QSF_IN_AA(.(X1, X2), X3) -> PD_IN_AAAAAAA(X1, X2, X4, X5, X6, X7, X3) 53.83/15.32 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)) 53.83/15.32 U13_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X3, X5)) -> QSF_IN_AA(X4, X6) 53.83/15.32 53.83/15.32 The TRS R consists of the following rules: 53.83/15.32 53.83/15.32 lesscA_in_aa(0, s(X1)) -> lesscA_out_aa(0, s(X1)) 53.83/15.32 lesscA_in_aa(s(X1), s(X2)) -> U63_aa(X1, X2, lesscA_in_aa(X1, X2)) 53.83/15.32 U63_aa(X1, X2, lesscA_out_aa(X1, X2)) -> lesscA_out_aa(s(X1), s(X2)) 53.83/15.32 lesscA_in_ga(0, s(X1)) -> lesscA_out_ga(0, s(X1)) 53.83/15.32 lesscA_in_ga(s(X1), s(X2)) -> U63_ga(X1, X2, lesscA_in_ga(X1, X2)) 53.83/15.32 U63_ga(X1, X2, lesscA_out_ga(X1, X2)) -> lesscA_out_ga(s(X1), s(X2)) 53.83/15.32 partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_gaaa(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) 53.83/15.32 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)) 53.83/15.32 partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) 53.83/15.32 partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], []) 53.83/15.32 U66_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.32 U65_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.32 partcC_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_aaaa(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) 53.83/15.32 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)) 53.83/15.32 U68_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.32 partcC_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_aaaa(X1, X2, X3, X4, X5, partcC_in_aaaa(X1, X3, X4, X5)) 53.83/15.32 partcC_in_aaaa(X1, [], [], []) -> partcC_out_aaaa(X1, [], [], []) 53.83/15.32 U69_aaaa(X1, X2, X3, X4, X5, partcC_out_aaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.32 qscF_in_aa([], []) -> qscF_out_aa([], []) 53.83/15.32 qscF_in_aa(.(X1, X2), X3) -> U74_aa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.32 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)) 53.83/15.32 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)) 53.83/15.32 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)) 53.83/15.32 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)) 53.83/15.32 appcE_in_aaaa([], X1, X2, .(X1, X2)) -> appcE_out_aaaa([], X1, X2, .(X1, X2)) 53.83/15.32 appcE_in_aaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U75_aaaa(X1, X2, X3, X4, X5, appcE_in_aaaa(X2, X3, X4, X5)) 53.83/15.32 U75_aaaa(X1, X2, X3, X4, X5, appcE_out_aaaa(X2, X3, X4, X5)) -> appcE_out_aaaa(.(X1, X2), X3, X4, .(X1, X5)) 53.83/15.32 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) 53.83/15.32 U74_aa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_aa(.(X1, X2), X3) 53.83/15.32 qscG_in_aaa(X1, X2, X3) -> U76_aaa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.32 U76_aaa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscG_out_aaa(X1, X2, X3) 53.83/15.32 lesscA_in_gg(0, s(X1)) -> lesscA_out_gg(0, s(X1)) 53.83/15.32 lesscA_in_gg(s(X1), s(X2)) -> U63_gg(X1, X2, lesscA_in_gg(X1, X2)) 53.83/15.32 U63_gg(X1, X2, lesscA_out_gg(X1, X2)) -> lesscA_out_gg(s(X1), s(X2)) 53.83/15.32 partcC_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.83/15.32 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)) 53.83/15.32 partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_ggaa(X1, X2, X3, X4, X5, lesscA_in_gg(X1, X2)) 53.83/15.32 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)) 53.83/15.32 partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5)) 53.83/15.32 partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], []) 53.83/15.32 U66_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.32 U65_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.32 U68_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.32 partcC_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_ggaa(X1, X2, X3, X4, X5, partcC_in_ggaa(X1, X3, X4, X5)) 53.83/15.32 partcC_in_ggaa(X1, [], [], []) -> partcC_out_ggaa(X1, [], [], []) 53.83/15.32 U69_ggaa(X1, X2, X3, X4, X5, partcC_out_ggaa(X1, X3, X4, X5)) -> partcC_out_ggaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.32 qscF_in_ga([], []) -> qscF_out_ga([], []) 53.83/15.32 qscF_in_ga(.(X1, X2), X3) -> U74_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.32 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)) 53.83/15.32 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)) 53.83/15.32 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)) 53.83/15.32 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)) 53.83/15.32 appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2)) 53.83/15.32 appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U75_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5)) 53.83/15.32 U75_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5)) 53.83/15.32 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) 53.83/15.32 U74_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3) 53.83/15.32 53.83/15.32 The argument filtering Pi contains the following mapping: 53.83/15.32 [] = [] 53.83/15.32 53.83/15.32 lesscA_in_aa(x1, x2) = lesscA_in_aa 53.83/15.32 53.83/15.32 lesscA_out_aa(x1, x2) = lesscA_out_aa(x1) 53.83/15.32 53.83/15.32 U63_aa(x1, x2, x3) = U63_aa(x3) 53.83/15.32 53.83/15.32 s(x1) = s(x1) 53.83/15.32 53.83/15.32 lesscA_in_ga(x1, x2) = lesscA_in_ga(x1) 53.83/15.32 53.83/15.32 0 = 0 53.83/15.32 53.83/15.32 lesscA_out_ga(x1, x2) = lesscA_out_ga(x1) 53.83/15.32 53.83/15.32 U63_ga(x1, x2, x3) = U63_ga(x1, x3) 53.83/15.32 53.83/15.32 partcB_in_gaaa(x1, x2, x3, x4) = partcB_in_gaaa(x1) 53.83/15.32 53.83/15.32 U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6) 53.83/15.32 53.83/15.32 U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x6) 53.83/15.32 53.83/15.32 U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6) 53.83/15.32 53.83/15.32 partcB_out_gaaa(x1, x2, x3, x4) = partcB_out_gaaa(x1) 53.83/15.32 53.83/15.32 partcC_in_aaaa(x1, x2, x3, x4) = partcC_in_aaaa 53.83/15.32 53.83/15.32 U67_aaaa(x1, x2, x3, x4, x5, x6) = U67_aaaa(x6) 53.83/15.32 53.83/15.32 U68_aaaa(x1, x2, x3, x4, x5, x6) = U68_aaaa(x6) 53.83/15.32 53.83/15.32 partcC_out_aaaa(x1, x2, x3, x4) = partcC_out_aaaa 53.83/15.32 53.83/15.32 U69_aaaa(x1, x2, x3, x4, x5, x6) = U69_aaaa(x6) 53.83/15.32 53.83/15.32 qscF_in_aa(x1, x2) = qscF_in_aa 53.83/15.32 53.83/15.32 qscF_out_aa(x1, x2) = qscF_out_aa 53.83/15.32 53.83/15.32 U74_aa(x1, x2, x3, x4) = U74_aa(x4) 53.83/15.32 53.83/15.32 qcD_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_aaaaaaa 53.83/15.32 53.83/15.32 U70_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_aaaaaaa(x8) 53.83/15.32 53.83/15.32 U71_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_aaaaaaa(x8) 53.83/15.32 53.83/15.32 U72_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_aaaaaaa(x8) 53.83/15.32 53.83/15.32 U73_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_aaaaaaa(x8) 53.83/15.32 53.83/15.32 .(x1, x2) = .(x1, x2) 53.83/15.32 53.83/15.32 qcD_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_aaaaaaa 53.83/15.32 53.83/15.32 appcE_in_aaaa(x1, x2, x3, x4) = appcE_in_aaaa 53.83/15.32 53.83/15.32 appcE_out_aaaa(x1, x2, x3, x4) = appcE_out_aaaa 53.83/15.32 53.83/15.32 U75_aaaa(x1, x2, x3, x4, x5, x6) = U75_aaaa(x6) 53.83/15.32 53.83/15.32 qscG_in_aaa(x1, x2, x3) = qscG_in_aaa 53.83/15.32 53.83/15.32 U76_aaa(x1, x2, x3, x4) = U76_aaa(x4) 53.83/15.32 53.83/15.32 qscG_out_aaa(x1, x2, x3) = qscG_out_aaa 53.83/15.32 53.83/15.32 lesscA_in_gg(x1, x2) = lesscA_in_gg(x1, x2) 53.83/15.32 53.83/15.32 lesscA_out_gg(x1, x2) = lesscA_out_gg(x1, x2) 53.83/15.32 53.83/15.32 U63_gg(x1, x2, x3) = U63_gg(x1, x2, x3) 53.83/15.32 53.83/15.32 partcC_in_ggaa(x1, x2, x3, x4) = partcC_in_ggaa(x1, x2) 53.83/15.32 53.83/15.32 U67_ggaa(x1, x2, x3, x4, x5, x6) = U67_ggaa(x1, x2, x3, x6) 53.83/15.32 53.83/15.32 U68_ggaa(x1, x2, x3, x4, x5, x6) = U68_ggaa(x1, x2, x3, x6) 53.83/15.32 53.83/15.32 partcB_in_ggaa(x1, x2, x3, x4) = partcB_in_ggaa(x1, x2) 53.83/15.32 53.83/15.32 U64_ggaa(x1, x2, x3, x4, x5, x6) = U64_ggaa(x1, x2, x3, x6) 53.83/15.32 53.83/15.32 U65_ggaa(x1, x2, x3, x4, x5, x6) = U65_ggaa(x1, x2, x3, x6) 53.83/15.32 53.83/15.32 U66_ggaa(x1, x2, x3, x4, x5, x6) = U66_ggaa(x1, x2, x3, x6) 53.83/15.32 53.83/15.32 partcB_out_ggaa(x1, x2, x3, x4) = partcB_out_ggaa(x1, x2, x3, x4) 53.83/15.32 53.83/15.32 partcC_out_ggaa(x1, x2, x3, x4) = partcC_out_ggaa(x1, x2, x3, x4) 53.83/15.32 53.83/15.32 U69_ggaa(x1, x2, x3, x4, x5, x6) = U69_ggaa(x1, x2, x3, x6) 53.83/15.32 53.83/15.32 qscF_in_ga(x1, x2) = qscF_in_ga(x1) 53.83/15.32 53.83/15.32 qscF_out_ga(x1, x2) = qscF_out_ga(x1, x2) 53.83/15.32 53.83/15.32 U74_ga(x1, x2, x3, x4) = U74_ga(x1, x2, x4) 53.83/15.32 53.83/15.32 qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_ggaaaaa(x1, x2) 53.83/15.32 53.83/15.32 U70_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_ggaaaaa(x1, x2, x8) 53.83/15.32 53.83/15.32 U71_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_ggaaaaa(x1, x2, x3, x4, x8) 53.83/15.32 53.83/15.32 U72_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_ggaaaaa(x1, x2, x3, x4, x5, x8) 53.83/15.32 53.83/15.32 U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_ggaaaaa(x1, x2, x3, x4, x5, x6, x8) 53.83/15.32 53.83/15.32 appcE_in_ggga(x1, x2, x3, x4) = appcE_in_ggga(x1, x2, x3) 53.83/15.32 53.83/15.32 appcE_out_ggga(x1, x2, x3, x4) = appcE_out_ggga(x1, x2, x3, x4) 53.83/15.32 53.83/15.32 U75_ggga(x1, x2, x3, x4, x5, x6) = U75_ggga(x1, x2, x3, x4, x6) 53.83/15.32 53.83/15.32 qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) 53.83/15.32 53.83/15.32 PD_IN_AAAAAAA(x1, x2, x3, x4, x5, x6, x7) = PD_IN_AAAAAAA 53.83/15.32 53.83/15.32 U11_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U11_AAAAAAA(x8) 53.83/15.32 53.83/15.32 QSF_IN_AA(x1, x2) = QSF_IN_AA 53.83/15.32 53.83/15.32 U13_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U13_AAAAAAA(x8) 53.83/15.32 53.83/15.32 53.83/15.32 We have to consider all (P,R,Pi)-chains 53.83/15.32 ---------------------------------------- 53.83/15.32 53.83/15.32 (228) UsableRulesProof (EQUIVALENT) 53.83/15.32 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 53.83/15.32 ---------------------------------------- 53.83/15.32 53.83/15.32 (229) 53.83/15.32 Obligation: 53.83/15.32 Pi DP problem: 53.83/15.32 The TRS P consists of the following rules: 53.83/15.32 53.83/15.32 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)) 53.83/15.32 U11_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, partcC_out_aaaa(X1, X2, X3, X4)) -> QSF_IN_AA(X3, X5) 53.83/15.32 QSF_IN_AA(.(X1, X2), X3) -> PD_IN_AAAAAAA(X1, X2, X4, X5, X6, X7, X3) 53.83/15.32 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)) 53.83/15.32 U13_AAAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_aa(X3, X5)) -> QSF_IN_AA(X4, X6) 53.83/15.32 53.83/15.32 The TRS R consists of the following rules: 53.83/15.32 53.83/15.32 partcC_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U67_aaaa(X1, X2, X3, X4, X5, lesscA_in_aa(X1, X2)) 53.83/15.32 partcC_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U69_aaaa(X1, X2, X3, X4, X5, partcC_in_aaaa(X1, X3, X4, X5)) 53.83/15.32 partcC_in_aaaa(X1, [], [], []) -> partcC_out_aaaa(X1, [], [], []) 53.83/15.32 qscF_in_aa([], []) -> qscF_out_aa([], []) 53.83/15.32 qscF_in_aa(.(X1, X2), X3) -> U74_aa(X1, X2, X3, qcD_in_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) 53.83/15.32 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)) 53.83/15.32 U69_aaaa(X1, X2, X3, X4, X5, partcC_out_aaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.32 U74_aa(X1, X2, X3, qcD_out_aaaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_aa(.(X1, X2), X3) 53.83/15.32 lesscA_in_aa(0, s(X1)) -> lesscA_out_aa(0, s(X1)) 53.83/15.32 lesscA_in_aa(s(X1), s(X2)) -> U63_aa(X1, X2, lesscA_in_aa(X1, X2)) 53.83/15.32 U68_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcC_out_aaaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.32 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)) 53.83/15.32 U63_aa(X1, X2, lesscA_out_aa(X1, X2)) -> lesscA_out_aa(s(X1), s(X2)) 53.83/15.32 partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U64_gaaa(X1, X2, X3, X4, X5, lesscA_in_ga(X1, X2)) 53.83/15.32 partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U66_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5)) 53.83/15.32 partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], []) 53.83/15.32 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)) 53.83/15.32 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)) 53.83/15.32 U66_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5)) 53.83/15.32 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)) 53.83/15.32 lesscA_in_ga(0, s(X1)) -> lesscA_out_ga(0, s(X1)) 53.83/15.32 lesscA_in_ga(s(X1), s(X2)) -> U63_ga(X1, X2, lesscA_in_ga(X1, X2)) 53.83/15.32 U65_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5) 53.83/15.32 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)) 53.83/15.32 U63_ga(X1, X2, lesscA_out_ga(X1, X2)) -> lesscA_out_ga(s(X1), s(X2)) 53.83/15.32 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) 53.83/15.32 appcE_in_aaaa([], X1, X2, .(X1, X2)) -> appcE_out_aaaa([], X1, X2, .(X1, X2)) 53.83/15.32 appcE_in_aaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U75_aaaa(X1, X2, X3, X4, X5, appcE_in_aaaa(X2, X3, X4, X5)) 53.83/15.32 U75_aaaa(X1, X2, X3, X4, X5, appcE_out_aaaa(X2, X3, X4, X5)) -> appcE_out_aaaa(.(X1, X2), X3, X4, .(X1, X5)) 53.83/15.32 53.83/15.32 The argument filtering Pi contains the following mapping: 53.83/15.32 [] = [] 53.83/15.32 53.83/15.32 lesscA_in_aa(x1, x2) = lesscA_in_aa 53.83/15.32 53.83/15.32 lesscA_out_aa(x1, x2) = lesscA_out_aa(x1) 53.83/15.32 53.83/15.32 U63_aa(x1, x2, x3) = U63_aa(x3) 53.83/15.32 53.83/15.32 s(x1) = s(x1) 53.83/15.32 53.83/15.32 lesscA_in_ga(x1, x2) = lesscA_in_ga(x1) 53.83/15.32 53.83/15.32 0 = 0 53.83/15.32 53.83/15.32 lesscA_out_ga(x1, x2) = lesscA_out_ga(x1) 53.83/15.32 53.83/15.32 U63_ga(x1, x2, x3) = U63_ga(x1, x3) 53.83/15.32 53.83/15.32 partcB_in_gaaa(x1, x2, x3, x4) = partcB_in_gaaa(x1) 53.83/15.32 53.83/15.32 U64_gaaa(x1, x2, x3, x4, x5, x6) = U64_gaaa(x1, x6) 53.83/15.32 53.83/15.32 U65_gaaa(x1, x2, x3, x4, x5, x6) = U65_gaaa(x1, x6) 53.83/15.32 53.83/15.32 U66_gaaa(x1, x2, x3, x4, x5, x6) = U66_gaaa(x1, x6) 53.83/15.32 53.83/15.32 partcB_out_gaaa(x1, x2, x3, x4) = partcB_out_gaaa(x1) 53.83/15.32 53.83/15.32 partcC_in_aaaa(x1, x2, x3, x4) = partcC_in_aaaa 53.83/15.32 53.83/15.32 U67_aaaa(x1, x2, x3, x4, x5, x6) = U67_aaaa(x6) 53.83/15.32 53.83/15.32 U68_aaaa(x1, x2, x3, x4, x5, x6) = U68_aaaa(x6) 53.83/15.32 53.83/15.32 partcC_out_aaaa(x1, x2, x3, x4) = partcC_out_aaaa 53.83/15.32 53.83/15.32 U69_aaaa(x1, x2, x3, x4, x5, x6) = U69_aaaa(x6) 53.83/15.32 53.83/15.32 qscF_in_aa(x1, x2) = qscF_in_aa 53.83/15.32 53.83/15.32 qscF_out_aa(x1, x2) = qscF_out_aa 53.83/15.32 53.83/15.32 U74_aa(x1, x2, x3, x4) = U74_aa(x4) 53.83/15.32 53.83/15.32 qcD_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_in_aaaaaaa 53.83/15.32 53.83/15.32 U70_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U70_aaaaaaa(x8) 53.83/15.32 53.83/15.32 U71_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U71_aaaaaaa(x8) 53.83/15.32 53.83/15.32 U72_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U72_aaaaaaa(x8) 53.83/15.32 53.83/15.32 U73_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U73_aaaaaaa(x8) 53.83/15.32 53.83/15.32 .(x1, x2) = .(x1, x2) 53.83/15.32 53.83/15.32 qcD_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7) = qcD_out_aaaaaaa 53.83/15.32 53.83/15.32 appcE_in_aaaa(x1, x2, x3, x4) = appcE_in_aaaa 53.83/15.32 53.83/15.32 appcE_out_aaaa(x1, x2, x3, x4) = appcE_out_aaaa 53.83/15.32 53.83/15.32 U75_aaaa(x1, x2, x3, x4, x5, x6) = U75_aaaa(x6) 53.83/15.32 53.83/15.32 PD_IN_AAAAAAA(x1, x2, x3, x4, x5, x6, x7) = PD_IN_AAAAAAA 53.83/15.32 53.83/15.32 U11_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U11_AAAAAAA(x8) 53.83/15.32 53.83/15.32 QSF_IN_AA(x1, x2) = QSF_IN_AA 53.83/15.32 53.83/15.32 U13_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U13_AAAAAAA(x8) 53.83/15.32 53.83/15.32 53.83/15.32 We have to consider all (P,R,Pi)-chains 53.83/15.32 ---------------------------------------- 53.83/15.32 53.83/15.32 (230) PrologToIRSwTTransformerProof (SOUND) 53.83/15.32 Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert 53.83/15.32 53.83/15.32 { 53.83/15.32 "root": 40, 53.83/15.32 "program": { 53.83/15.32 "directives": [], 53.83/15.32 "clauses": [ 53.83/15.32 [ 53.83/15.32 "(qs ([]) ([]))", 53.83/15.32 null 53.83/15.32 ], 53.83/15.32 [ 53.83/15.32 "(qs (. X Xs) Ys)", 53.83/15.32 "(',' (part X Xs Littles Bigs) (',' (qs Littles Ls) (',' (qs Bigs Bs) (app Ls (. X Bs) Ys))))" 53.83/15.32 ], 53.83/15.32 [ 53.83/15.32 "(part X (. Y Xs) (. Y Ls) Bs)", 53.83/15.32 "(',' (less X Y) (part X Xs Ls Bs))" 53.83/15.32 ], 53.83/15.32 [ 53.83/15.32 "(part X (. Y Xs) Ls (. Y Bs))", 53.83/15.32 "(part X Xs Ls Bs)" 53.83/15.32 ], 53.83/15.32 [ 53.83/15.32 "(part X1 ([]) ([]) ([]))", 53.83/15.32 null 53.83/15.32 ], 53.83/15.32 [ 53.83/15.32 "(app ([]) X X)", 53.83/15.32 null 53.83/15.32 ], 53.83/15.32 [ 53.83/15.32 "(app (. X Xs) Ys (. X Zs))", 53.83/15.32 "(app Xs Ys Zs)" 53.83/15.32 ], 53.83/15.32 [ 53.83/15.32 "(less (0) (s X2))", 53.83/15.32 null 53.83/15.32 ], 53.83/15.32 [ 53.83/15.32 "(less (s X) (s Y))", 53.83/15.32 "(less X Y)" 53.83/15.32 ] 53.83/15.32 ] 53.83/15.32 }, 53.83/15.32 "graph": { 53.83/15.32 "nodes": { 53.83/15.32 "type": "Nodes", 53.83/15.32 "471": { 53.83/15.32 "goal": [], 53.83/15.32 "kb": { 53.83/15.32 "nonunifying": [], 53.83/15.32 "intvars": {}, 53.83/15.32 "arithmetic": { 53.83/15.32 "type": "PlainIntegerRelationState", 53.83/15.32 "relations": [] 53.83/15.32 }, 53.83/15.32 "ground": [], 53.83/15.32 "free": [], 53.83/15.32 "exprvars": [] 53.83/15.32 } 53.83/15.32 }, 53.83/15.32 "791": { 53.83/15.32 "goal": [{ 53.83/15.32 "clause": -1, 53.83/15.32 "scope": -1, 53.83/15.32 "term": "(part T116 T119 X185 X186)" 53.83/15.32 }], 53.83/15.32 "kb": { 53.83/15.32 "nonunifying": [], 53.83/15.32 "intvars": {}, 53.83/15.32 "arithmetic": { 53.83/15.32 "type": "PlainIntegerRelationState", 53.83/15.32 "relations": [] 53.83/15.32 }, 53.83/15.32 "ground": ["T116"], 53.83/15.32 "free": [ 53.83/15.32 "X185", 53.83/15.32 "X186" 53.83/15.32 ], 53.83/15.32 "exprvars": [] 53.83/15.32 } 53.83/15.32 }, 53.83/15.32 "792": { 53.83/15.32 "goal": [], 53.83/15.32 "kb": { 53.83/15.32 "nonunifying": [], 53.83/15.32 "intvars": {}, 53.83/15.32 "arithmetic": { 53.83/15.32 "type": "PlainIntegerRelationState", 53.83/15.32 "relations": [] 53.83/15.32 }, 53.83/15.32 "ground": [], 53.83/15.32 "free": [], 53.83/15.32 "exprvars": [] 53.83/15.32 } 53.83/15.32 }, 53.83/15.32 "793": { 53.83/15.32 "goal": [{ 53.83/15.32 "clause": -1, 53.83/15.32 "scope": -1, 53.83/15.32 "term": "(true)" 53.83/15.32 }], 53.83/15.32 "kb": { 53.83/15.32 "nonunifying": [], 53.83/15.32 "intvars": {}, 53.83/15.32 "arithmetic": { 53.83/15.32 "type": "PlainIntegerRelationState", 53.83/15.32 "relations": [] 53.83/15.32 }, 53.83/15.32 "ground": [], 53.83/15.32 "free": [], 53.83/15.32 "exprvars": [] 53.83/15.32 } 53.83/15.32 }, 53.83/15.32 "1018": { 53.83/15.32 "goal": [], 53.83/15.32 "kb": { 53.83/15.32 "nonunifying": [], 53.83/15.32 "intvars": {}, 53.83/15.32 "arithmetic": { 53.83/15.32 "type": "PlainIntegerRelationState", 53.83/15.32 "relations": [] 53.83/15.32 }, 53.83/15.32 "ground": [], 53.83/15.32 "free": [], 53.83/15.32 "exprvars": [] 53.83/15.32 } 53.83/15.32 }, 53.83/15.32 "794": { 53.83/15.32 "goal": [], 53.83/15.32 "kb": { 53.83/15.32 "nonunifying": [], 53.83/15.32 "intvars": {}, 53.83/15.32 "arithmetic": { 53.83/15.32 "type": "PlainIntegerRelationState", 53.83/15.32 "relations": [] 53.83/15.32 }, 53.83/15.32 "ground": [], 53.83/15.32 "free": [], 53.83/15.32 "exprvars": [] 53.83/15.32 } 53.83/15.32 }, 53.83/15.32 "1017": { 53.83/15.32 "goal": [{ 53.83/15.32 "clause": -1, 53.83/15.32 "scope": -1, 53.83/15.32 "term": "(app T201 (. T202 T203) X297)" 53.83/15.32 }], 53.83/15.32 "kb": { 53.83/15.32 "nonunifying": [], 53.83/15.32 "intvars": {}, 53.83/15.32 "arithmetic": { 53.83/15.32 "type": "PlainIntegerRelationState", 53.83/15.32 "relations": [] 53.83/15.32 }, 53.83/15.32 "ground": [], 53.83/15.32 "free": ["X297"], 53.83/15.32 "exprvars": [] 53.83/15.32 } 53.83/15.32 }, 53.83/15.32 "795": { 53.83/15.32 "goal": [], 53.83/15.32 "kb": { 53.83/15.32 "nonunifying": [], 53.83/15.32 "intvars": {}, 53.83/15.32 "arithmetic": { 53.83/15.32 "type": "PlainIntegerRelationState", 53.83/15.32 "relations": [] 53.83/15.32 }, 53.83/15.32 "ground": [], 53.83/15.32 "free": [], 53.83/15.32 "exprvars": [] 53.83/15.32 } 53.83/15.32 }, 53.83/15.32 "950": { 53.83/15.32 "goal": [], 53.83/15.32 "kb": { 53.83/15.32 "nonunifying": [], 53.83/15.32 "intvars": {}, 53.83/15.32 "arithmetic": { 53.83/15.32 "type": "PlainIntegerRelationState", 53.83/15.32 "relations": [] 53.83/15.32 }, 53.83/15.32 "ground": [], 53.83/15.32 "free": [], 53.83/15.32 "exprvars": [] 53.83/15.32 } 53.83/15.32 }, 53.83/15.32 "953": { 53.83/15.32 "goal": [{ 53.83/15.32 "clause": -1, 53.83/15.32 "scope": -1, 53.83/15.32 "term": "(qs T17 X18)" 53.83/15.32 }], 53.83/15.32 "kb": { 53.83/15.32 "nonunifying": [], 53.83/15.32 "intvars": {}, 53.83/15.32 "arithmetic": { 53.83/15.32 "type": "PlainIntegerRelationState", 53.83/15.32 "relations": [] 53.83/15.32 }, 53.83/15.32 "ground": [], 53.83/15.32 "free": ["X18"], 53.83/15.32 "exprvars": [] 53.83/15.32 } 53.83/15.32 }, 53.83/15.32 "954": { 53.83/15.32 "goal": [{ 53.83/15.32 "clause": -1, 53.83/15.32 "scope": -1, 53.83/15.32 "term": "(',' (qs T150 X19) (app T149 (. T151 X19) T11))" 53.83/15.32 }], 53.83/15.32 "kb": { 53.83/15.32 "nonunifying": [], 53.83/15.32 "intvars": {}, 53.83/15.32 "arithmetic": { 53.83/15.32 "type": "PlainIntegerRelationState", 53.83/15.32 "relations": [] 53.83/15.32 }, 53.83/15.32 "ground": ["T11"], 53.83/15.32 "free": ["X19"], 53.83/15.32 "exprvars": [] 53.83/15.32 } 53.83/15.32 }, 53.83/15.32 "636": { 53.83/15.32 "goal": [ 53.83/15.32 { 53.83/15.32 "clause": 7, 53.83/15.32 "scope": 5, 53.83/15.32 "term": "(less T77 T80)" 53.83/15.32 }, 53.83/15.32 { 53.83/15.32 "clause": 8, 53.83/15.32 "scope": 5, 53.83/15.32 "term": "(less T77 T80)" 53.83/15.32 } 53.83/15.32 ], 53.83/15.32 "kb": { 53.83/15.32 "nonunifying": [], 53.83/15.32 "intvars": {}, 53.83/15.32 "arithmetic": { 53.83/15.32 "type": "PlainIntegerRelationState", 53.83/15.32 "relations": [] 53.83/15.32 }, 53.83/15.32 "ground": ["T77"], 53.83/15.32 "free": [], 53.83/15.32 "exprvars": [] 53.83/15.32 } 53.83/15.32 }, 53.83/15.32 "955": { 53.83/15.32 "goal": [ 53.83/15.32 { 53.83/15.32 "clause": 0, 53.83/15.32 "scope": 6, 53.83/15.32 "term": "(qs T17 X18)" 53.83/15.32 }, 53.83/15.32 { 53.83/15.32 "clause": 1, 53.83/15.32 "scope": 6, 53.83/15.32 "term": "(qs T17 X18)" 53.83/15.32 } 53.83/15.32 ], 53.83/15.32 "kb": { 53.83/15.32 "nonunifying": [], 53.83/15.32 "intvars": {}, 53.83/15.32 "arithmetic": { 53.83/15.32 "type": "PlainIntegerRelationState", 53.83/15.32 "relations": [] 53.83/15.32 }, 53.83/15.32 "ground": [], 53.83/15.32 "free": ["X18"], 53.83/15.32 "exprvars": [] 53.83/15.32 } 53.83/15.32 }, 53.83/15.32 "956": { 53.83/15.32 "goal": [{ 53.83/15.32 "clause": 0, 53.83/15.32 "scope": 6, 53.83/15.32 "term": "(qs T17 X18)" 53.83/15.32 }], 53.83/15.32 "kb": { 53.83/15.32 "nonunifying": [], 53.83/15.32 "intvars": {}, 53.83/15.32 "arithmetic": { 53.83/15.32 "type": "PlainIntegerRelationState", 53.83/15.32 "relations": [] 53.83/15.32 }, 53.83/15.32 "ground": [], 53.83/15.32 "free": ["X18"], 53.83/15.32 "exprvars": [] 53.83/15.32 } 53.83/15.32 }, 53.83/15.32 "957": { 53.83/15.32 "goal": [{ 53.83/15.32 "clause": 1, 53.83/15.32 "scope": 6, 53.83/15.32 "term": "(qs T17 X18)" 53.83/15.32 }], 53.83/15.32 "kb": { 53.83/15.32 "nonunifying": [], 53.83/15.32 "intvars": {}, 53.83/15.32 "arithmetic": { 53.83/15.32 "type": "PlainIntegerRelationState", 53.83/15.32 "relations": [] 53.83/15.32 }, 53.83/15.32 "ground": [], 53.83/15.32 "free": ["X18"], 53.83/15.32 "exprvars": [] 53.83/15.32 } 53.83/15.32 }, 53.83/15.32 "958": { 53.83/15.32 "goal": [{ 53.83/15.32 "clause": -1, 53.83/15.32 "scope": -1, 53.83/15.32 "term": "(true)" 53.83/15.32 }], 53.83/15.32 "kb": { 53.83/15.32 "nonunifying": [], 53.83/15.32 "intvars": {}, 53.83/15.32 "arithmetic": { 53.83/15.32 "type": "PlainIntegerRelationState", 53.83/15.32 "relations": [] 53.83/15.32 }, 53.83/15.32 "ground": [], 53.83/15.32 "free": [], 53.83/15.32 "exprvars": [] 53.83/15.32 } 53.83/15.32 }, 53.83/15.32 "959": { 53.83/15.32 "goal": [], 53.83/15.32 "kb": { 53.83/15.32 "nonunifying": [], 53.83/15.32 "intvars": {}, 53.83/15.32 "arithmetic": { 53.83/15.32 "type": "PlainIntegerRelationState", 53.83/15.32 "relations": [] 53.83/15.32 }, 53.83/15.32 "ground": [], 53.83/15.32 "free": [], 53.83/15.32 "exprvars": [] 53.83/15.32 } 53.83/15.32 }, 53.83/15.32 "480": { 53.83/15.32 "goal": [{ 53.83/15.32 "clause": -1, 53.83/15.32 "scope": -1, 53.83/15.32 "term": "(less T38 T39)" 53.83/15.32 }], 53.83/15.32 "kb": { 53.83/15.32 "nonunifying": [], 53.83/15.32 "intvars": {}, 53.83/15.32 "arithmetic": { 53.83/15.32 "type": "PlainIntegerRelationState", 53.83/15.32 "relations": [] 53.83/15.32 }, 53.83/15.32 "ground": [], 53.83/15.32 "free": [], 53.83/15.32 "exprvars": [] 53.83/15.32 } 53.83/15.32 }, 53.83/15.32 "481": { 53.83/15.32 "goal": [{ 53.83/15.32 "clause": -1, 53.83/15.32 "scope": -1, 53.83/15.32 "term": "(part T43 T44 X67 X68)" 53.83/15.32 }], 53.83/15.32 "kb": { 53.83/15.32 "nonunifying": [], 53.83/15.32 "intvars": {}, 53.83/15.32 "arithmetic": { 53.83/15.32 "type": "PlainIntegerRelationState", 53.83/15.32 "relations": [] 53.83/15.32 }, 53.83/15.32 "ground": ["T43"], 53.83/15.32 "free": [ 53.83/15.32 "X67", 53.83/15.32 "X68" 53.83/15.32 ], 53.83/15.32 "exprvars": [] 53.83/15.32 } 53.83/15.32 }, 53.83/15.32 "680": { 53.83/15.32 "goal": [{ 53.83/15.32 "clause": -1, 53.83/15.32 "scope": -1, 53.83/15.32 "term": "(less T96 T98)" 53.83/15.32 }], 53.83/15.32 "kb": { 53.83/15.32 "nonunifying": [], 53.83/15.32 "intvars": {}, 53.83/15.32 "arithmetic": { 53.83/15.32 "type": "PlainIntegerRelationState", 53.83/15.32 "relations": [] 53.83/15.32 }, 53.83/15.32 "ground": ["T96"], 53.83/15.32 "free": [], 53.83/15.32 "exprvars": [] 53.83/15.32 } 53.83/15.32 }, 53.83/15.32 "683": { 53.83/15.32 "goal": [], 53.83/15.32 "kb": { 53.83/15.32 "nonunifying": [], 53.83/15.32 "intvars": {}, 53.83/15.32 "arithmetic": { 53.83/15.32 "type": "PlainIntegerRelationState", 53.83/15.32 "relations": [] 53.83/15.32 }, 53.83/15.32 "ground": [], 53.83/15.32 "free": [], 53.83/15.32 "exprvars": [] 53.83/15.32 } 53.83/15.32 }, 53.83/15.32 "1028": { 53.83/15.32 "goal": [{ 53.83/15.32 "clause": -1, 53.83/15.32 "scope": -1, 53.83/15.32 "term": "(app T209 (. T210 T208) T11)" 53.83/15.32 }], 53.83/15.32 "kb": { 53.83/15.32 "nonunifying": [], 53.83/15.32 "intvars": {}, 53.83/15.32 "arithmetic": { 53.83/15.32 "type": "PlainIntegerRelationState", 53.83/15.32 "relations": [] 53.83/15.32 }, 53.83/15.32 "ground": ["T11"], 53.83/15.32 "free": [], 53.83/15.32 "exprvars": [] 53.83/15.32 } 53.83/15.32 }, 53.83/15.32 "201": { 53.83/15.32 "goal": [{ 53.83/15.32 "clause": 0, 53.83/15.32 "scope": 1, 53.83/15.32 "term": "(qs T1 T2)" 53.83/15.32 }], 53.83/15.32 "kb": { 53.83/15.32 "nonunifying": [], 53.83/15.32 "intvars": {}, 53.83/15.32 "arithmetic": { 53.83/15.32 "type": "PlainIntegerRelationState", 53.83/15.32 "relations": [] 53.83/15.32 }, 53.83/15.32 "ground": ["T2"], 53.83/15.32 "free": [], 53.83/15.32 "exprvars": [] 53.83/15.32 } 53.83/15.32 }, 53.83/15.32 "960": { 53.83/15.32 "goal": [], 53.83/15.32 "kb": { 53.83/15.32 "nonunifying": [], 53.83/15.32 "intvars": {}, 53.83/15.32 "arithmetic": { 53.83/15.32 "type": "PlainIntegerRelationState", 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53.83/15.32 "X231" 53.83/15.32 ], 53.83/15.32 "exprvars": [] 53.83/15.32 } 53.83/15.32 }, 53.83/15.32 "40": { 53.83/15.32 "goal": [{ 53.83/15.32 "clause": -1, 53.83/15.32 "scope": -1, 53.83/15.32 "term": "(qs T1 T2)" 53.83/15.32 }], 53.83/15.32 "kb": { 53.83/15.32 "nonunifying": [], 53.83/15.32 "intvars": {}, 53.83/15.32 "arithmetic": { 53.83/15.32 "type": "PlainIntegerRelationState", 53.83/15.32 "relations": [] 53.83/15.32 }, 53.83/15.32 "ground": ["T2"], 53.83/15.32 "free": [], 53.83/15.32 "exprvars": [] 53.83/15.32 } 53.83/15.32 }, 53.83/15.32 "507": { 53.83/15.32 "goal": [], 53.83/15.32 "kb": { 53.83/15.32 "nonunifying": [], 53.83/15.32 "intvars": {}, 53.83/15.32 "arithmetic": { 53.83/15.32 "type": "PlainIntegerRelationState", 53.83/15.32 "relations": [] 53.83/15.32 }, 53.83/15.32 "ground": [], 53.83/15.32 "free": [], 53.83/15.32 "exprvars": [] 53.83/15.32 } 53.83/15.32 }, 53.83/15.32 "947": { 53.83/15.32 "goal": [], 53.83/15.32 "kb": { 53.83/15.32 "nonunifying": [], 53.83/15.32 "intvars": {}, 53.83/15.32 "arithmetic": { 53.83/15.32 "type": "PlainIntegerRelationState", 53.83/15.32 "relations": [] 53.83/15.32 }, 53.83/15.32 "ground": [], 53.83/15.32 "free": [], 53.83/15.32 "exprvars": [] 53.83/15.32 } 53.83/15.32 }, 53.83/15.32 "948": { 53.83/15.32 "goal": [{ 53.83/15.32 "clause": -1, 53.83/15.32 "scope": -1, 53.83/15.32 "term": "(true)" 53.83/15.32 }], 53.83/15.32 "kb": { 53.83/15.32 "nonunifying": [], 53.83/15.32 "intvars": {}, 53.83/15.32 "arithmetic": { 53.83/15.32 "type": "PlainIntegerRelationState", 53.83/15.32 "relations": [] 53.83/15.32 }, 53.83/15.32 "ground": [], 53.83/15.32 "free": [], 53.83/15.32 "exprvars": [] 53.83/15.32 } 53.83/15.32 }, 53.83/15.32 "42": { 53.83/15.32 "goal": [ 53.83/15.32 { 53.83/15.32 "clause": 0, 53.83/15.32 "scope": 1, 53.83/15.32 "term": "(qs T1 T2)" 53.83/15.32 }, 53.83/15.32 { 53.83/15.32 "clause": 1, 53.83/15.32 "scope": 1, 53.83/15.32 "term": "(qs T1 T2)" 53.83/15.32 } 53.83/15.32 ], 53.83/15.32 "kb": { 53.83/15.32 "nonunifying": [], 53.83/15.32 "intvars": {}, 53.83/15.32 "arithmetic": { 53.83/15.32 "type": "PlainIntegerRelationState", 53.83/15.32 "relations": [] 53.83/15.32 }, 53.83/15.32 "ground": ["T2"], 53.83/15.32 "free": [], 53.83/15.32 "exprvars": [] 53.83/15.32 } 53.83/15.32 }, 53.83/15.32 "509": { 53.83/15.32 "goal": [], 53.83/15.32 "kb": { 53.83/15.32 "nonunifying": [], 53.83/15.32 "intvars": {}, 53.83/15.32 "arithmetic": { 53.83/15.32 "type": "PlainIntegerRelationState", 53.83/15.32 "relations": [] 53.83/15.32 }, 53.83/15.32 "ground": [], 53.83/15.32 "free": [], 53.83/15.32 "exprvars": [] 53.83/15.32 } 53.83/15.32 }, 53.83/15.32 "949": { 53.83/15.32 "goal": [], 53.83/15.32 "kb": { 53.83/15.32 "nonunifying": [], 53.83/15.32 "intvars": {}, 53.83/15.32 "arithmetic": { 53.83/15.32 "type": "PlainIntegerRelationState", 53.83/15.32 "relations": [] 53.83/15.32 }, 53.83/15.32 "ground": [], 53.83/15.32 "free": [], 53.83/15.32 "exprvars": [] 53.83/15.32 } 53.83/15.32 }, 53.83/15.32 "1009": { 53.83/15.32 "goal": [{ 53.83/15.32 "clause": -1, 53.83/15.32 "scope": -1, 53.83/15.32 "term": "(true)" 53.83/15.32 }], 53.83/15.32 "kb": { 53.83/15.32 "nonunifying": [], 53.83/15.32 "intvars": {}, 53.83/15.32 "arithmetic": { 53.83/15.32 "type": "PlainIntegerRelationState", 53.83/15.32 "relations": [] 53.83/15.32 }, 53.83/15.33 "ground": [], 53.83/15.33 "free": [], 53.83/15.33 "exprvars": [] 53.83/15.33 } 53.83/15.33 } 53.83/15.33 }, 53.83/15.33 "edges": [ 53.83/15.33 { 53.83/15.33 "from": 40, 53.83/15.33 "to": 42, 53.83/15.33 "label": "CASE" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 42, 53.83/15.33 "to": 201, 53.83/15.33 "label": "PARALLEL" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 42, 53.83/15.33 "to": 202, 53.83/15.33 "label": "PARALLEL" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 201, 53.83/15.33 "to": 204, 53.83/15.33 "label": "EVAL with clause\nqs([], []).\nand substitutionT1 -> [],\nT2 -> []" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 201, 53.83/15.33 "to": 205, 53.83/15.33 "label": "EVAL-BACKTRACK" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 202, 53.83/15.33 "to": 208, 53.83/15.33 "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" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 202, 53.83/15.33 "to": 209, 53.83/15.33 "label": "EVAL-BACKTRACK" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 204, 53.83/15.33 "to": 206, 53.83/15.33 "label": "SUCCESS" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 208, 53.83/15.33 "to": 384, 53.83/15.33 "label": "SPLIT 1" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 208, 53.83/15.33 "to": 385, 53.83/15.33 "label": "SPLIT 2\nreplacements:X16 -> T17,\nX17 -> T18,\nT12 -> T19" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 384, 53.83/15.33 "to": 386, 53.83/15.33 "label": "CASE" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 385, 53.83/15.33 "to": 953, 53.83/15.33 "label": "SPLIT 1" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 385, 53.83/15.33 "to": 954, 53.83/15.33 "label": "SPLIT 2\nreplacements:X18 -> T149,\nT18 -> T150,\nT19 -> T151" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 386, 53.83/15.33 "to": 445, 53.83/15.33 "label": "PARALLEL" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 386, 53.83/15.33 "to": 446, 53.83/15.33 "label": "PARALLEL" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 445, 53.83/15.33 "to": 466, 53.83/15.33 "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" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 445, 53.83/15.33 "to": 471, 53.83/15.33 "label": "EVAL-BACKTRACK" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 446, 53.83/15.33 "to": 809, 53.83/15.33 "label": "PARALLEL" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 446, 53.83/15.33 "to": 812, 53.83/15.33 "label": "PARALLEL" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 466, 53.83/15.33 "to": 480, 53.83/15.33 "label": "SPLIT 1" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 466, 53.83/15.33 "to": 481, 53.83/15.33 "label": "SPLIT 2\nnew knowledge:\nT43 is ground\nreplacements:T38 -> T43,\nT40 -> T44" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 480, 53.83/15.33 "to": 494, 53.83/15.33 "label": "CASE" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 481, 53.83/15.33 "to": 548, 53.83/15.33 "label": "CASE" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 494, 53.83/15.33 "to": 498, 53.83/15.33 "label": "PARALLEL" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 494, 53.83/15.33 "to": 500, 53.83/15.33 "label": "PARALLEL" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 498, 53.83/15.33 "to": 506, 53.83/15.33 "label": "EVAL with clause\nless(0, s(X77)).\nand substitutionT38 -> 0,\nX77 -> T51,\nT39 -> s(T51)" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 498, 53.83/15.33 "to": 507, 53.83/15.33 "label": "EVAL-BACKTRACK" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 500, 53.83/15.33 "to": 521, 53.83/15.33 "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" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 500, 53.83/15.33 "to": 523, 53.83/15.33 "label": "EVAL-BACKTRACK" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 506, 53.83/15.33 "to": 509, 53.83/15.33 "label": "SUCCESS" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 521, 53.83/15.33 "to": 480, 53.83/15.33 "label": "INSTANCE with matching:\nT38 -> T58\nT39 -> T59" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 548, 53.83/15.33 "to": 569, 53.83/15.33 "label": "PARALLEL" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 548, 53.83/15.33 "to": 570, 53.83/15.33 "label": "PARALLEL" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 569, 53.83/15.33 "to": 603, 53.83/15.33 "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" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 569, 53.83/15.33 "to": 606, 53.83/15.33 "label": "EVAL-BACKTRACK" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 570, 53.83/15.33 "to": 767, 53.83/15.33 "label": "PARALLEL" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 570, 53.83/15.33 "to": 769, 53.83/15.33 "label": "PARALLEL" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 603, 53.83/15.33 "to": 621, 53.83/15.33 "label": "SPLIT 1" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 603, 53.83/15.33 "to": 623, 53.83/15.33 "label": "SPLIT 2\nnew knowledge:\nT77 is ground\nreplacements:T81 -> T84" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 621, 53.83/15.33 "to": 636, 53.83/15.33 "label": "CASE" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 623, 53.83/15.33 "to": 481, 53.83/15.33 "label": "INSTANCE with matching:\nT43 -> T77\nT44 -> T84\nX67 -> X126\nX68 -> X127" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 636, 53.83/15.33 "to": 643, 53.83/15.33 "label": "PARALLEL" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 636, 53.83/15.33 "to": 645, 53.83/15.33 "label": "PARALLEL" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 643, 53.83/15.33 "to": 657, 53.83/15.33 "label": "EVAL with clause\nless(0, s(X136)).\nand substitutionT77 -> 0,\nX136 -> T91,\nT80 -> s(T91)" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 643, 53.83/15.33 "to": 660, 53.83/15.33 "label": "EVAL-BACKTRACK" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 645, 53.83/15.33 "to": 680, 53.83/15.33 "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" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 645, 53.83/15.33 "to": 683, 53.83/15.33 "label": "EVAL-BACKTRACK" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 657, 53.83/15.33 "to": 662, 53.83/15.33 "label": "SUCCESS" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 680, 53.83/15.33 "to": 621, 53.83/15.33 "label": "INSTANCE with matching:\nT77 -> T96\nT80 -> T98" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 767, 53.83/15.33 "to": 791, 53.83/15.33 "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" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 767, 53.83/15.33 "to": 792, 53.83/15.33 "label": "EVAL-BACKTRACK" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 769, 53.83/15.33 "to": 793, 53.83/15.33 "label": "EVAL with clause\npart(X196, [], [], []).\nand substitutionT43 -> T125,\nX196 -> T125,\nT44 -> [],\nX67 -> [],\nX68 -> []" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 769, 53.83/15.33 "to": 794, 53.83/15.33 "label": "EVAL-BACKTRACK" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 791, 53.83/15.33 "to": 481, 53.83/15.33 "label": "INSTANCE with matching:\nT43 -> T116\nT44 -> T119\nX67 -> X185\nX68 -> X186" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 793, 53.83/15.33 "to": 795, 53.83/15.33 "label": "SUCCESS" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 809, 53.83/15.33 "to": 946, 53.83/15.33 "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" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 809, 53.83/15.33 "to": 947, 53.83/15.33 "label": "EVAL-BACKTRACK" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 812, 53.83/15.33 "to": 948, 53.83/15.33 "label": "EVAL with clause\npart(X241, [], [], []).\nand substitutionT12 -> T148,\nX241 -> T148,\nT13 -> [],\nX16 -> [],\nX17 -> []" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 812, 53.83/15.33 "to": 949, 53.83/15.33 "label": "EVAL-BACKTRACK" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 946, 53.83/15.33 "to": 384, 53.83/15.33 "label": "INSTANCE with matching:\nT12 -> T141\nT13 -> T142\nX16 -> X230\nX17 -> X231" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 948, 53.83/15.33 "to": 950, 53.83/15.33 "label": "SUCCESS" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 953, 53.83/15.33 "to": 955, 53.83/15.33 "label": "CASE" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 954, 53.83/15.33 "to": 1027, 53.83/15.33 "label": "SPLIT 1" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 954, 53.83/15.33 "to": 1028, 53.83/15.33 "label": "SPLIT 2\nreplacements:X19 -> T208,\nT149 -> T209,\nT151 -> T210" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 955, 53.83/15.33 "to": 956, 53.83/15.33 "label": "PARALLEL" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 955, 53.83/15.33 "to": 957, 53.83/15.33 "label": "PARALLEL" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 956, 53.83/15.33 "to": 958, 53.83/15.33 "label": "EVAL with clause\nqs([], []).\nand substitutionT17 -> [],\nX18 -> []" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 956, 53.83/15.33 "to": 959, 53.83/15.33 "label": "EVAL-BACKTRACK" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 957, 53.83/15.33 "to": 965, 53.83/15.33 "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" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 957, 53.83/15.33 "to": 966, 53.83/15.33 "label": "EVAL-BACKTRACK" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 958, 53.83/15.33 "to": 960, 53.83/15.33 "label": "SUCCESS" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 965, 53.83/15.33 "to": 967, 53.83/15.33 "label": "SPLIT 1" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 965, 53.83/15.33 "to": 968, 53.83/15.33 "label": "SPLIT 2\nreplacements:X257 -> T163,\nX258 -> T164,\nT158 -> T165" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 967, 53.83/15.33 "to": 384, 53.83/15.33 "label": "INSTANCE with matching:\nT12 -> T158\nT13 -> T159\nX16 -> X257\nX17 -> X258" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 968, 53.83/15.33 "to": 971, 53.83/15.33 "label": "SPLIT 1" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 968, 53.83/15.33 "to": 972, 53.83/15.33 "label": "SPLIT 2\nreplacements:X259 -> T169,\nT164 -> T170,\nT165 -> T171" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 971, 53.83/15.33 "to": 953, 53.83/15.33 "label": "INSTANCE with matching:\nT17 -> T163\nX18 -> X259" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 972, 53.83/15.33 "to": 1000, 53.83/15.33 "label": "SPLIT 1" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 972, 53.83/15.33 "to": 1001, 53.83/15.33 "label": "SPLIT 2\nreplacements:X260 -> T172,\nT169 -> T173,\nT171 -> T174" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 1000, 53.83/15.33 "to": 953, 53.83/15.33 "label": "INSTANCE with matching:\nT17 -> T170\nX18 -> X260" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 1001, 53.83/15.33 "to": 1004, 53.83/15.33 "label": "CASE" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 1004, 53.83/15.33 "to": 1005, 53.83/15.33 "label": "PARALLEL" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 1004, 53.83/15.33 "to": 1006, 53.83/15.33 "label": "PARALLEL" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 1005, 53.83/15.33 "to": 1009, 53.83/15.33 "label": "EVAL with clause\napp([], X282, X282).\nand substitutionT173 -> [],\nT174 -> T187,\nT172 -> T188,\nX282 -> .(T187, T188),\nX261 -> .(T187, T188)" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 1005, 53.83/15.33 "to": 1010, 53.83/15.33 "label": "EVAL-BACKTRACK" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 1006, 53.83/15.33 "to": 1017, 53.83/15.33 "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" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 1006, 53.83/15.33 "to": 1018, 53.83/15.33 "label": "EVAL-BACKTRACK" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 1009, 53.83/15.33 "to": 1011, 53.83/15.33 "label": "SUCCESS" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 1017, 53.83/15.33 "to": 1001, 53.83/15.33 "label": "INSTANCE with matching:\nT173 -> T201\nT174 -> T202\nT172 -> T203\nX261 -> X297" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 1027, 53.83/15.33 "to": 953, 53.83/15.33 "label": "INSTANCE with matching:\nT17 -> T150\nX18 -> X19" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 1028, 53.83/15.33 "to": 1043, 53.83/15.33 "label": "CASE" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 1043, 53.83/15.33 "to": 1044, 53.83/15.33 "label": "PARALLEL" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 1043, 53.83/15.33 "to": 1045, 53.83/15.33 "label": "PARALLEL" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 1044, 53.83/15.33 "to": 1046, 53.83/15.33 "label": "EVAL with clause\napp([], X306, X306).\nand substitutionT209 -> [],\nT210 -> T223,\nT208 -> T224,\nX306 -> .(T223, T224),\nT11 -> .(T223, T224)" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 1044, 53.83/15.33 "to": 1047, 53.83/15.33 "label": "EVAL-BACKTRACK" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 1045, 53.83/15.33 "to": 1051, 53.83/15.33 "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" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 1045, 53.83/15.33 "to": 1052, 53.83/15.33 "label": "EVAL-BACKTRACK" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 1046, 53.83/15.33 "to": 1048, 53.83/15.33 "label": "SUCCESS" 53.83/15.33 }, 53.83/15.33 { 53.83/15.33 "from": 1051, 53.83/15.33 "to": 1028, 53.83/15.33 "label": "INSTANCE with matching:\nT209 -> T240\nT210 -> T241\nT208 -> T242\nT11 -> T239" 53.83/15.33 } 53.83/15.33 ], 53.83/15.33 "type": "Graph" 53.83/15.33 } 53.83/15.33 } 53.83/15.33 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (231) 53.83/15.33 Complex Obligation (AND) 53.83/15.33 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (232) 53.83/15.33 Obligation: 53.83/15.33 Rules: 53.83/15.33 f1045_in(T11) -> f1052_in :|: TRUE 53.83/15.33 f1052_out -> f1045_out(x) :|: TRUE 53.83/15.33 f1045_in(.(T235, T239)) -> f1051_in(T239) :|: TRUE 53.83/15.33 f1051_out(x1) -> f1045_out(.(x2, x1)) :|: TRUE 53.83/15.33 f1051_in(x3) -> f1028_in(x3) :|: TRUE 53.83/15.33 f1028_out(x4) -> f1051_out(x4) :|: TRUE 53.83/15.33 f1043_out(x5) -> f1028_out(x5) :|: TRUE 53.83/15.33 f1028_in(x6) -> f1043_in(x6) :|: TRUE 53.83/15.33 f1044_out(x7) -> f1043_out(x7) :|: TRUE 53.83/15.33 f1045_out(x8) -> f1043_out(x8) :|: TRUE 53.83/15.33 f1043_in(x9) -> f1045_in(x9) :|: TRUE 53.83/15.33 f1043_in(x10) -> f1044_in(x10) :|: TRUE 53.83/15.33 f42_out(T2) -> f40_out(T2) :|: TRUE 53.83/15.33 f40_in(x11) -> f42_in(x11) :|: TRUE 53.83/15.33 f202_out(x12) -> f42_out(x12) :|: TRUE 53.83/15.33 f42_in(x13) -> f201_in(x13) :|: TRUE 53.83/15.33 f42_in(x14) -> f202_in(x14) :|: TRUE 53.83/15.33 f201_out(x15) -> f42_out(x15) :|: TRUE 53.83/15.33 f202_in(x16) -> f209_in :|: TRUE 53.83/15.33 f208_out(x17) -> f202_out(x17) :|: TRUE 53.83/15.33 f209_out -> f202_out(x18) :|: TRUE 53.83/15.33 f202_in(x19) -> f208_in(x19) :|: TRUE 53.83/15.33 f384_out -> f385_in(x20) :|: TRUE 53.83/15.33 f208_in(x21) -> f384_in :|: TRUE 53.83/15.33 f385_out(x22) -> f208_out(x22) :|: TRUE 53.83/15.33 f954_out(x23) -> f385_out(x23) :|: TRUE 53.83/15.33 f385_in(x24) -> f953_in :|: TRUE 53.83/15.33 f953_out -> f954_in(x25) :|: TRUE 53.83/15.33 f954_in(x26) -> f1027_in :|: TRUE 53.83/15.33 f1028_out(x27) -> f954_out(x27) :|: TRUE 53.83/15.33 f1027_out -> f1028_in(x28) :|: TRUE 53.83/15.33 Start term: f40_in(T2) 53.83/15.33 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (233) IRSwTSimpleDependencyGraphProof (EQUIVALENT) 53.83/15.33 Constructed simple dependency graph. 53.83/15.33 53.83/15.33 Simplified to the following IRSwTs: 53.83/15.33 53.83/15.33 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (234) 53.83/15.33 TRUE 53.83/15.33 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (235) 53.83/15.33 Obligation: 53.83/15.33 Rules: 53.83/15.33 f1017_in -> f1001_in :|: TRUE 53.83/15.33 f1001_out -> f1017_out :|: TRUE 53.83/15.33 f1006_out -> f1004_out :|: TRUE 53.83/15.33 f1004_in -> f1006_in :|: TRUE 53.83/15.33 f1005_out -> f1004_out :|: TRUE 53.83/15.33 f1004_in -> f1005_in :|: TRUE 53.83/15.33 f1017_out -> f1006_out :|: TRUE 53.83/15.33 f1006_in -> f1018_in :|: TRUE 53.83/15.33 f1018_out -> f1006_out :|: TRUE 53.83/15.33 f1006_in -> f1017_in :|: TRUE 53.83/15.33 f1004_out -> f1001_out :|: TRUE 53.83/15.33 f1001_in -> f1004_in :|: TRUE 53.83/15.33 f42_out(T2) -> f40_out(T2) :|: TRUE 53.83/15.33 f40_in(x) -> f42_in(x) :|: TRUE 53.83/15.33 f202_out(x1) -> f42_out(x1) :|: TRUE 53.83/15.33 f42_in(x2) -> f201_in(x2) :|: TRUE 53.83/15.33 f42_in(x3) -> f202_in(x3) :|: TRUE 53.83/15.33 f201_out(x4) -> f42_out(x4) :|: TRUE 53.83/15.33 f202_in(x5) -> f209_in :|: TRUE 53.83/15.33 f208_out(T11) -> f202_out(T11) :|: TRUE 53.83/15.33 f209_out -> f202_out(x6) :|: TRUE 53.83/15.33 f202_in(x7) -> f208_in(x7) :|: TRUE 53.83/15.33 f384_out -> f385_in(x8) :|: TRUE 53.83/15.33 f208_in(x9) -> f384_in :|: TRUE 53.83/15.33 f385_out(x10) -> f208_out(x10) :|: TRUE 53.83/15.33 f954_out(x11) -> f385_out(x11) :|: TRUE 53.83/15.33 f385_in(x12) -> f953_in :|: TRUE 53.83/15.33 f953_out -> f954_in(x13) :|: TRUE 53.83/15.33 f953_in -> f955_in :|: TRUE 53.83/15.33 f955_out -> f953_out :|: TRUE 53.83/15.33 f955_in -> f957_in :|: TRUE 53.83/15.33 f956_out -> f955_out :|: TRUE 53.83/15.33 f957_out -> f955_out :|: TRUE 53.83/15.33 f955_in -> f956_in :|: TRUE 53.83/15.33 f957_in -> f965_in :|: TRUE 53.83/15.33 f965_out -> f957_out :|: TRUE 53.83/15.33 f966_out -> f957_out :|: TRUE 53.83/15.33 f957_in -> f966_in :|: TRUE 53.83/15.33 f968_out -> f965_out :|: TRUE 53.83/15.33 f965_in -> f967_in :|: TRUE 53.83/15.33 f967_out -> f968_in :|: TRUE 53.83/15.33 f968_in -> f971_in :|: TRUE 53.83/15.33 f971_out -> f972_in :|: TRUE 53.83/15.33 f972_out -> f968_out :|: TRUE 53.83/15.33 f1001_out -> f972_out :|: TRUE 53.83/15.33 f972_in -> f1000_in :|: TRUE 53.83/15.33 f1000_out -> f1001_in :|: TRUE 53.83/15.33 f954_in(x14) -> f1027_in :|: TRUE 53.83/15.33 f1028_out(x15) -> f954_out(x15) :|: TRUE 53.83/15.33 f1027_out -> f1028_in(x16) :|: TRUE 53.83/15.33 f953_out -> f1027_out :|: TRUE 53.83/15.33 f1027_in -> f953_in :|: TRUE 53.83/15.33 Start term: f40_in(T2) 53.83/15.33 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (236) IRSwTSimpleDependencyGraphProof (EQUIVALENT) 53.83/15.33 Constructed simple dependency graph. 53.83/15.33 53.83/15.33 Simplified to the following IRSwTs: 53.83/15.33 53.83/15.33 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (237) 53.83/15.33 TRUE 53.83/15.33 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (238) 53.83/15.33 Obligation: 53.83/15.33 Rules: 53.83/15.33 f680_in(T96) -> f621_in(T96) :|: TRUE 53.83/15.33 f621_out(x) -> f680_out(x) :|: TRUE 53.83/15.33 f636_in(T77) -> f645_in(T77) :|: TRUE 53.83/15.33 f643_out(x1) -> f636_out(x1) :|: TRUE 53.83/15.33 f645_out(x2) -> f636_out(x2) :|: TRUE 53.83/15.33 f636_in(x3) -> f643_in(x3) :|: TRUE 53.83/15.33 f645_in(x4) -> f683_in :|: TRUE 53.83/15.33 f683_out -> f645_out(x5) :|: TRUE 53.83/15.33 f645_in(s(x6)) -> f680_in(x6) :|: TRUE 53.83/15.33 f680_out(x7) -> f645_out(s(x7)) :|: TRUE 53.83/15.33 f636_out(x8) -> f621_out(x8) :|: TRUE 53.83/15.33 f621_in(x9) -> f636_in(x9) :|: TRUE 53.83/15.33 f42_out(T2) -> f40_out(T2) :|: TRUE 53.83/15.33 f40_in(x10) -> f42_in(x10) :|: TRUE 53.83/15.33 f202_out(x11) -> f42_out(x11) :|: TRUE 53.83/15.33 f42_in(x12) -> f201_in(x12) :|: TRUE 53.83/15.33 f42_in(x13) -> f202_in(x13) :|: TRUE 53.83/15.33 f201_out(x14) -> f42_out(x14) :|: TRUE 53.83/15.33 f202_in(x15) -> f209_in :|: TRUE 53.83/15.33 f208_out(T11) -> f202_out(T11) :|: TRUE 53.83/15.33 f209_out -> f202_out(x16) :|: TRUE 53.83/15.33 f202_in(x17) -> f208_in(x17) :|: TRUE 53.83/15.33 f384_out -> f385_in(x18) :|: TRUE 53.83/15.33 f208_in(x19) -> f384_in :|: TRUE 53.83/15.33 f385_out(x20) -> f208_out(x20) :|: TRUE 53.83/15.33 f384_in -> f386_in :|: TRUE 53.83/15.33 f386_out -> f384_out :|: TRUE 53.83/15.33 f445_out -> f386_out :|: TRUE 53.83/15.33 f446_out -> f386_out :|: TRUE 53.83/15.33 f386_in -> f445_in :|: TRUE 53.83/15.33 f386_in -> f446_in :|: TRUE 53.83/15.33 f445_in -> f471_in :|: TRUE 53.83/15.33 f471_out -> f445_out :|: TRUE 53.83/15.33 f466_out -> f445_out :|: TRUE 53.83/15.33 f445_in -> f466_in :|: TRUE 53.83/15.33 f481_out(T43) -> f466_out :|: TRUE 53.83/15.33 f480_out -> f481_in(x21) :|: TRUE 53.83/15.33 f466_in -> f480_in :|: TRUE 53.83/15.33 f481_in(x22) -> f548_in(x22) :|: TRUE 53.83/15.33 f548_out(x23) -> f481_out(x23) :|: TRUE 53.83/15.33 f570_out(x24) -> f548_out(x24) :|: TRUE 53.83/15.33 f548_in(x25) -> f570_in(x25) :|: TRUE 53.83/15.33 f569_out(x26) -> f548_out(x26) :|: TRUE 53.83/15.33 f548_in(x27) -> f569_in(x27) :|: TRUE 53.83/15.33 f569_in(x28) -> f603_in(x28) :|: TRUE 53.83/15.33 f606_out -> f569_out(x29) :|: TRUE 53.83/15.33 f603_out(x30) -> f569_out(x30) :|: TRUE 53.83/15.33 f569_in(x31) -> f606_in :|: TRUE 53.83/15.33 f623_out(x32) -> f603_out(x32) :|: TRUE 53.83/15.33 f621_out(x33) -> f623_in(x33) :|: TRUE 53.83/15.33 f603_in(x34) -> f621_in(x34) :|: TRUE 53.83/15.33 f954_out(x35) -> f385_out(x35) :|: TRUE 53.83/15.33 f385_in(x36) -> f953_in :|: TRUE 53.83/15.33 f953_out -> f954_in(x37) :|: TRUE 53.83/15.33 f953_in -> f955_in :|: TRUE 53.83/15.33 f955_out -> f953_out :|: TRUE 53.83/15.33 f955_in -> f957_in :|: TRUE 53.83/15.33 f956_out -> f955_out :|: TRUE 53.83/15.33 f957_out -> f955_out :|: TRUE 53.83/15.33 f955_in -> f956_in :|: TRUE 53.83/15.33 f957_in -> f965_in :|: TRUE 53.83/15.33 f965_out -> f957_out :|: TRUE 53.83/15.33 f966_out -> f957_out :|: TRUE 53.83/15.33 f957_in -> f966_in :|: TRUE 53.83/15.33 f968_out -> f965_out :|: TRUE 53.83/15.33 f965_in -> f967_in :|: TRUE 53.83/15.33 f967_out -> f968_in :|: TRUE 53.83/15.33 f967_in -> f384_in :|: TRUE 53.83/15.33 f384_out -> f967_out :|: TRUE 53.83/15.33 f954_in(x38) -> f1027_in :|: TRUE 53.83/15.33 f1028_out(x39) -> f954_out(x39) :|: TRUE 53.83/15.33 f1027_out -> f1028_in(x40) :|: TRUE 53.83/15.33 f953_out -> f1027_out :|: TRUE 53.83/15.33 f1027_in -> f953_in :|: TRUE 53.83/15.33 Start term: f40_in(T2) 53.83/15.33 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (239) IRSwTSimpleDependencyGraphProof (EQUIVALENT) 53.83/15.33 Constructed simple dependency graph. 53.83/15.33 53.83/15.33 Simplified to the following IRSwTs: 53.83/15.33 53.83/15.33 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (240) 53.83/15.33 TRUE 53.83/15.33 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (241) 53.83/15.33 Obligation: 53.83/15.33 Rules: 53.83/15.33 f623_out(T77) -> f603_out(T77) :|: TRUE 53.83/15.33 f621_out(x) -> f623_in(x) :|: TRUE 53.83/15.33 f603_in(x1) -> f621_in(x1) :|: TRUE 53.83/15.33 f636_in(x2) -> f645_in(x2) :|: TRUE 53.83/15.33 f643_out(x3) -> f636_out(x3) :|: TRUE 53.83/15.33 f645_out(x4) -> f636_out(x4) :|: TRUE 53.83/15.33 f636_in(x5) -> f643_in(x5) :|: TRUE 53.83/15.33 f636_out(x6) -> f621_out(x6) :|: TRUE 53.83/15.33 f621_in(x7) -> f636_in(x7) :|: TRUE 53.83/15.33 f791_out(T116) -> f767_out(T116) :|: TRUE 53.83/15.33 f767_in(T43) -> f792_in :|: TRUE 53.83/15.33 f767_in(x8) -> f791_in(x8) :|: TRUE 53.83/15.33 f792_out -> f767_out(x9) :|: TRUE 53.83/15.33 f569_in(x10) -> f603_in(x10) :|: TRUE 53.83/15.33 f606_out -> f569_out(x11) :|: TRUE 53.83/15.33 f603_out(x12) -> f569_out(x12) :|: TRUE 53.83/15.33 f569_in(x13) -> f606_in :|: TRUE 53.83/15.33 f767_out(x14) -> f570_out(x14) :|: TRUE 53.83/15.33 f769_out(x15) -> f570_out(x15) :|: TRUE 53.83/15.33 f570_in(x16) -> f767_in(x16) :|: TRUE 53.83/15.33 f570_in(x17) -> f769_in(x17) :|: TRUE 53.83/15.33 f657_out -> f643_out(0) :|: TRUE 53.83/15.33 f643_in(x18) -> f660_in :|: TRUE 53.83/15.33 f643_in(0) -> f657_in :|: TRUE 53.83/15.33 f660_out -> f643_out(x19) :|: TRUE 53.83/15.33 f657_in -> f657_out :|: TRUE 53.83/15.33 f680_in(T96) -> f621_in(T96) :|: TRUE 53.83/15.33 f621_out(x20) -> f680_out(x20) :|: TRUE 53.83/15.33 f481_in(x21) -> f548_in(x21) :|: TRUE 53.83/15.33 f548_out(x22) -> f481_out(x22) :|: TRUE 53.83/15.33 f645_in(x23) -> f683_in :|: TRUE 53.83/15.33 f683_out -> f645_out(x24) :|: TRUE 53.83/15.33 f645_in(s(x25)) -> f680_in(x25) :|: TRUE 53.83/15.33 f680_out(x26) -> f645_out(s(x26)) :|: TRUE 53.83/15.33 f570_out(x27) -> f548_out(x27) :|: TRUE 53.83/15.33 f548_in(x28) -> f570_in(x28) :|: TRUE 53.83/15.33 f569_out(x29) -> f548_out(x29) :|: TRUE 53.83/15.33 f548_in(x30) -> f569_in(x30) :|: TRUE 53.83/15.33 f481_out(x31) -> f623_out(x31) :|: TRUE 53.83/15.33 f623_in(x32) -> f481_in(x32) :|: TRUE 53.83/15.33 f481_out(x33) -> f791_out(x33) :|: TRUE 53.83/15.33 f791_in(x34) -> f481_in(x34) :|: TRUE 53.83/15.33 f42_out(T2) -> f40_out(T2) :|: TRUE 53.83/15.33 f40_in(x35) -> f42_in(x35) :|: TRUE 53.83/15.33 f202_out(x36) -> f42_out(x36) :|: TRUE 53.83/15.33 f42_in(x37) -> f201_in(x37) :|: TRUE 53.83/15.33 f42_in(x38) -> f202_in(x38) :|: TRUE 53.83/15.33 f201_out(x39) -> f42_out(x39) :|: TRUE 53.83/15.33 f202_in(x40) -> f209_in :|: TRUE 53.83/15.33 f208_out(T11) -> f202_out(T11) :|: TRUE 53.83/15.33 f209_out -> f202_out(x41) :|: TRUE 53.83/15.33 f202_in(x42) -> f208_in(x42) :|: TRUE 53.83/15.33 f384_out -> f385_in(x43) :|: TRUE 53.83/15.33 f208_in(x44) -> f384_in :|: TRUE 53.83/15.33 f385_out(x45) -> f208_out(x45) :|: TRUE 53.83/15.33 f384_in -> f386_in :|: TRUE 53.83/15.33 f386_out -> f384_out :|: TRUE 53.83/15.33 f445_out -> f386_out :|: TRUE 53.83/15.33 f446_out -> f386_out :|: TRUE 53.83/15.33 f386_in -> f445_in :|: TRUE 53.83/15.33 f386_in -> f446_in :|: TRUE 53.83/15.33 f445_in -> f471_in :|: TRUE 53.83/15.33 f471_out -> f445_out :|: TRUE 53.83/15.33 f466_out -> f445_out :|: TRUE 53.83/15.33 f445_in -> f466_in :|: TRUE 53.83/15.33 f481_out(x46) -> f466_out :|: TRUE 53.83/15.33 f480_out -> f481_in(x47) :|: TRUE 53.83/15.33 f466_in -> f480_in :|: TRUE 53.83/15.33 f954_out(x48) -> f385_out(x48) :|: TRUE 53.83/15.33 f385_in(x49) -> f953_in :|: TRUE 53.83/15.33 f953_out -> f954_in(x50) :|: TRUE 53.83/15.33 f954_in(x51) -> f1027_in :|: TRUE 53.83/15.33 f1028_out(x52) -> f954_out(x52) :|: TRUE 53.83/15.33 f1027_out -> f1028_in(x53) :|: TRUE 53.83/15.33 f953_out -> f1027_out :|: TRUE 53.83/15.33 f1027_in -> f953_in :|: TRUE 53.83/15.33 f953_in -> f955_in :|: TRUE 53.83/15.33 f955_out -> f953_out :|: TRUE 53.83/15.33 f955_in -> f957_in :|: TRUE 53.83/15.33 f956_out -> f955_out :|: TRUE 53.83/15.33 f957_out -> f955_out :|: TRUE 53.83/15.33 f955_in -> f956_in :|: TRUE 53.83/15.33 f957_in -> f965_in :|: TRUE 53.83/15.33 f965_out -> f957_out :|: TRUE 53.83/15.33 f966_out -> f957_out :|: TRUE 53.83/15.33 f957_in -> f966_in :|: TRUE 53.83/15.33 f968_out -> f965_out :|: TRUE 53.83/15.33 f965_in -> f967_in :|: TRUE 53.83/15.33 f967_out -> f968_in :|: TRUE 53.83/15.33 f967_in -> f384_in :|: TRUE 53.83/15.33 f384_out -> f967_out :|: TRUE 53.83/15.33 Start term: f40_in(T2) 53.83/15.33 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (242) IRSwTSimpleDependencyGraphProof (EQUIVALENT) 53.83/15.33 Constructed simple dependency graph. 53.83/15.33 53.83/15.33 Simplified to the following IRSwTs: 53.83/15.33 53.83/15.33 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (243) 53.83/15.33 TRUE 53.83/15.33 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (244) 53.83/15.33 Obligation: 53.83/15.33 Rules: 53.83/15.33 f480_out -> f521_out :|: TRUE 53.83/15.33 f521_in -> f480_in :|: TRUE 53.83/15.33 f494_out -> f480_out :|: TRUE 53.83/15.33 f480_in -> f494_in :|: TRUE 53.83/15.33 f523_out -> f500_out :|: TRUE 53.83/15.33 f500_in -> f523_in :|: TRUE 53.83/15.33 f521_out -> f500_out :|: TRUE 53.83/15.33 f500_in -> f521_in :|: TRUE 53.83/15.33 f494_in -> f500_in :|: TRUE 53.83/15.33 f498_out -> f494_out :|: TRUE 53.83/15.33 f500_out -> f494_out :|: TRUE 53.83/15.33 f494_in -> f498_in :|: TRUE 53.83/15.33 f42_out(T2) -> f40_out(T2) :|: TRUE 53.83/15.33 f40_in(x) -> f42_in(x) :|: TRUE 53.83/15.33 f202_out(x1) -> f42_out(x1) :|: TRUE 53.83/15.33 f42_in(x2) -> f201_in(x2) :|: TRUE 53.83/15.33 f42_in(x3) -> f202_in(x3) :|: TRUE 53.83/15.33 f201_out(x4) -> f42_out(x4) :|: TRUE 53.83/15.33 f202_in(x5) -> f209_in :|: TRUE 53.83/15.33 f208_out(T11) -> f202_out(T11) :|: TRUE 53.83/15.33 f209_out -> f202_out(x6) :|: TRUE 53.83/15.33 f202_in(x7) -> f208_in(x7) :|: TRUE 53.83/15.33 f384_out -> f385_in(x8) :|: TRUE 53.83/15.33 f208_in(x9) -> f384_in :|: TRUE 53.83/15.33 f385_out(x10) -> f208_out(x10) :|: TRUE 53.83/15.33 f954_out(x11) -> f385_out(x11) :|: TRUE 53.83/15.33 f385_in(x12) -> f953_in :|: TRUE 53.83/15.33 f953_out -> f954_in(x13) :|: TRUE 53.83/15.33 f953_in -> f955_in :|: TRUE 53.83/15.33 f955_out -> f953_out :|: TRUE 53.83/15.33 f955_in -> f957_in :|: TRUE 53.83/15.33 f956_out -> f955_out :|: TRUE 53.83/15.33 f957_out -> f955_out :|: TRUE 53.83/15.33 f955_in -> f956_in :|: TRUE 53.83/15.33 f957_in -> f965_in :|: TRUE 53.83/15.33 f965_out -> f957_out :|: TRUE 53.83/15.33 f966_out -> f957_out :|: TRUE 53.83/15.33 f957_in -> f966_in :|: TRUE 53.83/15.33 f968_out -> f965_out :|: TRUE 53.83/15.33 f965_in -> f967_in :|: TRUE 53.83/15.33 f967_out -> f968_in :|: TRUE 53.83/15.33 f967_in -> f384_in :|: TRUE 53.83/15.33 f384_out -> f967_out :|: TRUE 53.83/15.33 f384_in -> f386_in :|: TRUE 53.83/15.33 f386_out -> f384_out :|: TRUE 53.83/15.33 f445_out -> f386_out :|: TRUE 53.83/15.33 f446_out -> f386_out :|: TRUE 53.83/15.33 f386_in -> f445_in :|: TRUE 53.83/15.33 f386_in -> f446_in :|: TRUE 53.83/15.33 f445_in -> f471_in :|: TRUE 53.83/15.33 f471_out -> f445_out :|: TRUE 53.83/15.33 f466_out -> f445_out :|: TRUE 53.83/15.33 f445_in -> f466_in :|: TRUE 53.83/15.33 f481_out(T43) -> f466_out :|: TRUE 53.83/15.33 f480_out -> f481_in(x14) :|: TRUE 53.83/15.33 f466_in -> f480_in :|: TRUE 53.83/15.33 f954_in(x15) -> f1027_in :|: TRUE 53.83/15.33 f1028_out(x16) -> f954_out(x16) :|: TRUE 53.83/15.33 f1027_out -> f1028_in(x17) :|: TRUE 53.83/15.33 f953_out -> f1027_out :|: TRUE 53.83/15.33 f1027_in -> f953_in :|: TRUE 53.83/15.33 Start term: f40_in(T2) 53.83/15.33 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (245) IRSwTSimpleDependencyGraphProof (EQUIVALENT) 53.83/15.33 Constructed simple dependency graph. 53.83/15.33 53.83/15.33 Simplified to the following IRSwTs: 53.83/15.33 53.83/15.33 intTRSProblem: 53.83/15.33 f521_in -> f480_in :|: TRUE 53.83/15.33 f480_in -> f494_in :|: TRUE 53.83/15.33 f500_in -> f521_in :|: TRUE 53.83/15.33 f494_in -> f500_in :|: TRUE 53.83/15.33 53.83/15.33 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (246) 53.83/15.33 Obligation: 53.83/15.33 Rules: 53.83/15.33 f521_in -> f480_in :|: TRUE 53.83/15.33 f480_in -> f494_in :|: TRUE 53.83/15.33 f500_in -> f521_in :|: TRUE 53.83/15.33 f494_in -> f500_in :|: TRUE 53.83/15.33 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (247) IntTRSCompressionProof (EQUIVALENT) 53.83/15.33 Compressed rules. 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (248) 53.83/15.33 Obligation: 53.83/15.33 Rules: 53.83/15.33 f500_in -> f500_in :|: TRUE 53.83/15.33 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (249) IRSFormatTransformerProof (EQUIVALENT) 53.83/15.33 Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (250) 53.83/15.33 Obligation: 53.83/15.33 Rules: 53.83/15.33 f500_in -> f500_in :|: TRUE 53.83/15.33 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (251) IRSwTTerminationDigraphProof (EQUIVALENT) 53.83/15.33 Constructed termination digraph! 53.83/15.33 Nodes: 53.83/15.33 (1) f500_in -> f500_in :|: TRUE 53.83/15.33 53.83/15.33 Arcs: 53.83/15.33 (1) -> (1) 53.83/15.33 53.83/15.33 This digraph is fully evaluated! 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (252) 53.83/15.33 Obligation: 53.83/15.33 53.83/15.33 Termination digraph: 53.83/15.33 Nodes: 53.83/15.33 (1) f500_in -> f500_in :|: TRUE 53.83/15.33 53.83/15.33 Arcs: 53.83/15.33 (1) -> (1) 53.83/15.33 53.83/15.33 This digraph is fully evaluated! 53.83/15.33 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (253) FilterProof (EQUIVALENT) 53.83/15.33 Used the following sort dictionary for filtering: 53.83/15.33 f500_in() 53.83/15.33 Replaced non-predefined constructor symbols by 0. 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (254) 53.83/15.33 Obligation: 53.83/15.33 Rules: 53.83/15.33 f500_in -> f500_in :|: TRUE 53.83/15.33 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (255) IntTRSNonPeriodicNontermProof (COMPLETE) 53.83/15.33 Normalized system to the following form: 53.83/15.33 f(pc) -> f(1) :|: pc = 1 && TRUE 53.83/15.33 Proved unsatisfiability of the following formula, indicating that the system is never left after entering: 53.83/15.33 ((run2_0 = ((1 * 1)) and (((run1_0 * 1)) = ((1 * 1)) and T)) and !(((run2_0 * 1)) = ((1 * 1)) and T)) 53.83/15.33 Proved satisfiability of the following formula, indicating that the system is entered at least once: 53.83/15.33 (run2_0 = ((1 * 1)) and (((run1_0 * 1)) = ((1 * 1)) and T)) 53.83/15.33 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (256) 53.83/15.33 NO 53.83/15.33 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (257) 53.83/15.33 Obligation: 53.83/15.33 Rules: 53.83/15.33 f445_out -> f386_out :|: TRUE 53.83/15.33 f446_out -> f386_out :|: TRUE 53.83/15.33 f386_in -> f445_in :|: TRUE 53.83/15.33 f386_in -> f446_in :|: TRUE 53.83/15.33 f809_out -> f446_out :|: TRUE 53.83/15.33 f446_in -> f809_in :|: TRUE 53.83/15.33 f446_in -> f812_in :|: TRUE 53.83/15.33 f812_out -> f446_out :|: TRUE 53.83/15.33 f946_out -> f809_out :|: TRUE 53.83/15.33 f809_in -> f947_in :|: TRUE 53.83/15.33 f809_in -> f946_in :|: TRUE 53.83/15.33 f947_out -> f809_out :|: TRUE 53.83/15.33 f384_in -> f386_in :|: TRUE 53.83/15.33 f386_out -> f384_out :|: TRUE 53.83/15.33 f384_out -> f946_out :|: TRUE 53.83/15.33 f946_in -> f384_in :|: TRUE 53.83/15.33 f42_out(T2) -> f40_out(T2) :|: TRUE 53.83/15.33 f40_in(x) -> f42_in(x) :|: TRUE 53.83/15.33 f202_out(x1) -> f42_out(x1) :|: TRUE 53.83/15.33 f42_in(x2) -> f201_in(x2) :|: TRUE 53.83/15.33 f42_in(x3) -> f202_in(x3) :|: TRUE 53.83/15.33 f201_out(x4) -> f42_out(x4) :|: TRUE 53.83/15.33 f202_in(x5) -> f209_in :|: TRUE 53.83/15.33 f208_out(T11) -> f202_out(T11) :|: TRUE 53.83/15.33 f209_out -> f202_out(x6) :|: TRUE 53.83/15.33 f202_in(x7) -> f208_in(x7) :|: TRUE 53.83/15.33 f384_out -> f385_in(x8) :|: TRUE 53.83/15.33 f208_in(x9) -> f384_in :|: TRUE 53.83/15.33 f385_out(x10) -> f208_out(x10) :|: TRUE 53.83/15.33 f954_out(x11) -> f385_out(x11) :|: TRUE 53.83/15.33 f385_in(x12) -> f953_in :|: TRUE 53.83/15.33 f953_out -> f954_in(x13) :|: TRUE 53.83/15.33 f953_in -> f955_in :|: TRUE 53.83/15.33 f955_out -> f953_out :|: TRUE 53.83/15.33 f955_in -> f957_in :|: TRUE 53.83/15.33 f956_out -> f955_out :|: TRUE 53.83/15.33 f957_out -> f955_out :|: TRUE 53.83/15.33 f955_in -> f956_in :|: TRUE 53.83/15.33 f957_in -> f965_in :|: TRUE 53.83/15.33 f965_out -> f957_out :|: TRUE 53.83/15.33 f966_out -> f957_out :|: TRUE 53.83/15.33 f957_in -> f966_in :|: TRUE 53.83/15.33 f968_out -> f965_out :|: TRUE 53.83/15.33 f965_in -> f967_in :|: TRUE 53.83/15.33 f967_out -> f968_in :|: TRUE 53.83/15.33 f967_in -> f384_in :|: TRUE 53.83/15.33 f384_out -> f967_out :|: TRUE 53.83/15.33 f954_in(x14) -> f1027_in :|: TRUE 53.83/15.33 f1028_out(x15) -> f954_out(x15) :|: TRUE 53.83/15.33 f1027_out -> f1028_in(x16) :|: TRUE 53.83/15.33 f953_out -> f1027_out :|: TRUE 53.83/15.33 f1027_in -> f953_in :|: TRUE 53.83/15.33 Start term: f40_in(T2) 53.83/15.33 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (258) IRSwTSimpleDependencyGraphProof (EQUIVALENT) 53.83/15.33 Constructed simple dependency graph. 53.83/15.33 53.83/15.33 Simplified to the following IRSwTs: 53.83/15.33 53.83/15.33 intTRSProblem: 53.83/15.33 f386_in -> f446_in :|: TRUE 53.83/15.33 f446_in -> f809_in :|: TRUE 53.83/15.33 f809_in -> f946_in :|: TRUE 53.83/15.33 f384_in -> f386_in :|: TRUE 53.83/15.33 f946_in -> f384_in :|: TRUE 53.83/15.33 53.83/15.33 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (259) 53.83/15.33 Obligation: 53.83/15.33 Rules: 53.83/15.33 f386_in -> f446_in :|: TRUE 53.83/15.33 f446_in -> f809_in :|: TRUE 53.83/15.33 f809_in -> f946_in :|: TRUE 53.83/15.33 f384_in -> f386_in :|: TRUE 53.83/15.33 f946_in -> f384_in :|: TRUE 53.83/15.33 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (260) IntTRSCompressionProof (EQUIVALENT) 53.83/15.33 Compressed rules. 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (261) 53.83/15.33 Obligation: 53.83/15.33 Rules: 53.83/15.33 f384_in -> f384_in :|: TRUE 53.83/15.33 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (262) IRSFormatTransformerProof (EQUIVALENT) 53.83/15.33 Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (263) 53.83/15.33 Obligation: 53.83/15.33 Rules: 53.83/15.33 f384_in -> f384_in :|: TRUE 53.83/15.33 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (264) IRSwTTerminationDigraphProof (EQUIVALENT) 53.83/15.33 Constructed termination digraph! 53.83/15.33 Nodes: 53.83/15.33 (1) f384_in -> f384_in :|: TRUE 53.83/15.33 53.83/15.33 Arcs: 53.83/15.33 (1) -> (1) 53.83/15.33 53.83/15.33 This digraph is fully evaluated! 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (265) 53.83/15.33 Obligation: 53.83/15.33 53.83/15.33 Termination digraph: 53.83/15.33 Nodes: 53.83/15.33 (1) f384_in -> f384_in :|: TRUE 53.83/15.33 53.83/15.33 Arcs: 53.83/15.33 (1) -> (1) 53.83/15.33 53.83/15.33 This digraph is fully evaluated! 53.83/15.33 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (266) FilterProof (EQUIVALENT) 53.83/15.33 Used the following sort dictionary for filtering: 53.83/15.33 f384_in() 53.83/15.33 Replaced non-predefined constructor symbols by 0. 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (267) 53.83/15.33 Obligation: 53.83/15.33 Rules: 53.83/15.33 f384_in -> f384_in :|: TRUE 53.83/15.33 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (268) IntTRSPeriodicNontermProof (COMPLETE) 53.83/15.33 Normalized system to the following form: 53.83/15.33 f(pc) -> f(1) :|: pc = 1 && TRUE 53.83/15.33 Witness term starting non-terminating reduction: f(1) 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (269) 53.83/15.33 NO 53.83/15.33 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (270) 53.83/15.33 Obligation: 53.83/15.33 Rules: 53.83/15.33 f506_in -> f506_out :|: TRUE 53.83/15.33 f968_in -> f971_in :|: TRUE 53.83/15.33 f971_out -> f972_in :|: TRUE 53.83/15.33 f972_out -> f968_out :|: TRUE 53.83/15.33 f623_out(T77) -> f603_out(T77) :|: TRUE 53.83/15.33 f621_out(x) -> f623_in(x) :|: TRUE 53.83/15.33 f603_in(x1) -> f621_in(x1) :|: TRUE 53.83/15.33 f971_in -> f953_in :|: TRUE 53.83/15.33 f953_out -> f971_out :|: TRUE 53.83/15.33 f957_in -> f965_in :|: TRUE 53.83/15.33 f965_out -> f957_out :|: TRUE 53.83/15.33 f966_out -> f957_out :|: TRUE 53.83/15.33 f957_in -> f966_in :|: TRUE 53.83/15.33 f636_in(x2) -> f645_in(x2) :|: TRUE 53.83/15.33 f643_out(x3) -> f636_out(x3) :|: TRUE 53.83/15.33 f645_out(x4) -> f636_out(x4) :|: TRUE 53.83/15.33 f636_in(x5) -> f643_in(x5) :|: TRUE 53.83/15.33 f1006_out -> f1004_out :|: TRUE 53.83/15.33 f1004_in -> f1006_in :|: TRUE 53.83/15.33 f1005_out -> f1004_out :|: TRUE 53.83/15.33 f1004_in -> f1005_in :|: TRUE 53.83/15.33 f946_out -> f809_out :|: TRUE 53.83/15.33 f809_in -> f947_in :|: TRUE 53.83/15.33 f809_in -> f946_in :|: TRUE 53.83/15.33 f947_out -> f809_out :|: TRUE 53.83/15.33 f384_out -> f946_out :|: TRUE 53.83/15.33 f946_in -> f384_in :|: TRUE 53.83/15.33 f968_out -> f965_out :|: TRUE 53.83/15.33 f965_in -> f967_in :|: TRUE 53.83/15.33 f967_out -> f968_in :|: TRUE 53.83/15.33 f645_in(x6) -> f683_in :|: TRUE 53.83/15.33 f683_out -> f645_out(x7) :|: TRUE 53.83/15.33 f645_in(s(T96)) -> f680_in(T96) :|: TRUE 53.83/15.33 f680_out(x8) -> f645_out(s(x8)) :|: TRUE 53.83/15.33 f1000_in -> f953_in :|: TRUE 53.83/15.33 f953_out -> f1000_out :|: TRUE 53.83/15.33 f953_in -> f955_in :|: TRUE 53.83/15.33 f955_out -> f953_out :|: TRUE 53.83/15.33 f1005_in -> f1009_in :|: TRUE 53.83/15.33 f1009_out -> f1005_out :|: TRUE 53.83/15.33 f1005_in -> f1010_in :|: TRUE 53.83/15.33 f1010_out -> f1005_out :|: TRUE 53.83/15.33 f948_in -> f948_out :|: TRUE 53.83/15.33 f657_in -> f657_out :|: TRUE 53.83/15.33 f1001_out -> f972_out :|: TRUE 53.83/15.33 f972_in -> f1000_in :|: TRUE 53.83/15.33 f1000_out -> f1001_in :|: TRUE 53.83/15.33 f445_out -> f386_out :|: TRUE 53.83/15.33 f446_out -> f386_out :|: TRUE 53.83/15.33 f386_in -> f445_in :|: TRUE 53.83/15.33 f386_in -> f446_in :|: TRUE 53.83/15.33 f1009_in -> f1009_out :|: TRUE 53.83/15.33 f494_out -> f480_out :|: TRUE 53.83/15.33 f480_in -> f494_in :|: TRUE 53.83/15.33 f570_out(T43) -> f548_out(T43) :|: TRUE 53.83/15.33 f548_in(x9) -> f570_in(x9) :|: TRUE 53.83/15.33 f569_out(x10) -> f548_out(x10) :|: TRUE 53.83/15.33 f548_in(x11) -> f569_in(x11) :|: TRUE 53.83/15.33 f494_in -> f500_in :|: TRUE 53.83/15.33 f498_out -> f494_out :|: TRUE 53.83/15.33 f500_out -> f494_out :|: TRUE 53.83/15.33 f494_in -> f498_in :|: TRUE 53.83/15.33 f948_out -> f812_out :|: TRUE 53.83/15.33 f949_out -> f812_out :|: TRUE 53.83/15.33 f812_in -> f949_in :|: TRUE 53.83/15.33 f812_in -> f948_in :|: TRUE 53.83/15.33 f955_in -> f957_in :|: TRUE 53.83/15.33 f956_out -> f955_out :|: TRUE 53.83/15.33 f957_out -> f955_out :|: TRUE 53.83/15.33 f955_in -> f956_in :|: TRUE 53.83/15.33 f809_out -> f446_out :|: TRUE 53.83/15.33 f446_in -> f809_in :|: TRUE 53.83/15.33 f446_in -> f812_in :|: TRUE 53.83/15.33 f812_out -> f446_out :|: TRUE 53.83/15.33 f523_out -> f500_out :|: TRUE 53.83/15.33 f500_in -> f523_in :|: TRUE 53.83/15.33 f521_out -> f500_out :|: TRUE 53.83/15.33 f500_in -> f521_in :|: TRUE 53.83/15.33 f657_out -> f643_out(0) :|: TRUE 53.83/15.33 f643_in(x12) -> f660_in :|: TRUE 53.83/15.33 f643_in(0) -> f657_in :|: TRUE 53.83/15.33 f660_out -> f643_out(x13) :|: TRUE 53.83/15.33 f1017_out -> f1006_out :|: TRUE 53.83/15.33 f1006_in -> f1018_in :|: TRUE 53.83/15.33 f1018_out -> f1006_out :|: TRUE 53.83/15.33 f1006_in -> f1017_in :|: TRUE 53.83/15.33 f445_in -> f471_in :|: TRUE 53.83/15.33 f471_out -> f445_out :|: TRUE 53.83/15.33 f466_out -> f445_out :|: TRUE 53.83/15.33 f445_in -> f466_in :|: TRUE 53.83/15.33 f680_in(x14) -> f621_in(x14) :|: TRUE 53.83/15.33 f621_out(x15) -> f680_out(x15) :|: TRUE 53.83/15.33 f793_in -> f793_out :|: TRUE 53.83/15.33 f481_out(x16) -> f466_out :|: TRUE 53.83/15.33 f480_out -> f481_in(x17) :|: TRUE 53.83/15.33 f466_in -> f480_in :|: TRUE 53.83/15.33 f481_out(x18) -> f623_out(x18) :|: TRUE 53.83/15.33 f623_in(x19) -> f481_in(x19) :|: TRUE 53.83/15.33 f480_out -> f521_out :|: TRUE 53.83/15.33 f521_in -> f480_in :|: TRUE 53.83/15.33 f967_in -> f384_in :|: TRUE 53.83/15.33 f384_out -> f967_out :|: TRUE 53.83/15.33 f1017_in -> f1001_in :|: TRUE 53.83/15.33 f1001_out -> f1017_out :|: TRUE 53.83/15.33 f636_out(x20) -> f621_out(x20) :|: TRUE 53.83/15.33 f621_in(x21) -> f636_in(x21) :|: TRUE 53.83/15.33 f791_out(T116) -> f767_out(T116) :|: TRUE 53.83/15.33 f767_in(x22) -> f792_in :|: TRUE 53.83/15.33 f767_in(x23) -> f791_in(x23) :|: TRUE 53.83/15.33 f792_out -> f767_out(x24) :|: TRUE 53.83/15.33 f569_in(x25) -> f603_in(x25) :|: TRUE 53.83/15.33 f606_out -> f569_out(x26) :|: TRUE 53.83/15.33 f603_out(x27) -> f569_out(x27) :|: TRUE 53.83/15.33 f569_in(x28) -> f606_in :|: TRUE 53.83/15.33 f767_out(x29) -> f570_out(x29) :|: TRUE 53.83/15.33 f769_out(x30) -> f570_out(x30) :|: TRUE 53.83/15.33 f570_in(x31) -> f767_in(x31) :|: TRUE 53.83/15.33 f570_in(x32) -> f769_in(x32) :|: TRUE 53.83/15.33 f384_in -> f386_in :|: TRUE 53.83/15.33 f386_out -> f384_out :|: TRUE 53.83/15.33 f498_in -> f507_in :|: TRUE 53.83/15.33 f498_in -> f506_in :|: TRUE 53.83/15.33 f506_out -> f498_out :|: TRUE 53.83/15.33 f507_out -> f498_out :|: TRUE 53.83/15.33 f1004_out -> f1001_out :|: TRUE 53.83/15.33 f1001_in -> f1004_in :|: TRUE 53.83/15.33 f481_in(x33) -> f548_in(x33) :|: TRUE 53.83/15.33 f548_out(x34) -> f481_out(x34) :|: TRUE 53.83/15.33 f794_out -> f769_out(x35) :|: TRUE 53.83/15.33 f769_in(x36) -> f794_in :|: TRUE 53.83/15.33 f769_in(T125) -> f793_in :|: TRUE 53.83/15.33 f793_out -> f769_out(x37) :|: TRUE 53.83/15.33 f481_out(x38) -> f791_out(x38) :|: TRUE 53.83/15.33 f791_in(x39) -> f481_in(x39) :|: TRUE 53.83/15.33 f42_out(T2) -> f40_out(T2) :|: TRUE 53.83/15.33 f40_in(x40) -> f42_in(x40) :|: TRUE 53.83/15.33 f202_out(x41) -> f42_out(x41) :|: TRUE 53.83/15.33 f42_in(x42) -> f201_in(x42) :|: TRUE 53.83/15.33 f42_in(x43) -> f202_in(x43) :|: TRUE 53.83/15.33 f201_out(x44) -> f42_out(x44) :|: TRUE 53.83/15.33 f202_in(x45) -> f209_in :|: TRUE 53.83/15.33 f208_out(T11) -> f202_out(T11) :|: TRUE 53.83/15.33 f209_out -> f202_out(x46) :|: TRUE 53.83/15.33 f202_in(x47) -> f208_in(x47) :|: TRUE 53.83/15.33 f384_out -> f385_in(x48) :|: TRUE 53.83/15.33 f208_in(x49) -> f384_in :|: TRUE 53.83/15.33 f385_out(x50) -> f208_out(x50) :|: TRUE 53.83/15.33 f954_out(x51) -> f385_out(x51) :|: TRUE 53.83/15.33 f385_in(x52) -> f953_in :|: TRUE 53.83/15.33 f953_out -> f954_in(x53) :|: TRUE 53.83/15.33 f954_in(x54) -> f1027_in :|: TRUE 53.83/15.33 f1028_out(x55) -> f954_out(x55) :|: TRUE 53.83/15.33 f1027_out -> f1028_in(x56) :|: TRUE 53.83/15.33 f953_out -> f1027_out :|: TRUE 53.83/15.33 f1027_in -> f953_in :|: TRUE 53.83/15.33 Start term: f40_in(T2) 53.83/15.33 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (271) IRSwTSimpleDependencyGraphProof (EQUIVALENT) 53.83/15.33 Constructed simple dependency graph. 53.83/15.33 53.83/15.33 Simplified to the following IRSwTs: 53.83/15.33 53.83/15.33 intTRSProblem: 53.83/15.33 f506_in -> f506_out :|: TRUE 53.83/15.33 f968_in -> f971_in :|: TRUE 53.83/15.33 f623_out(T77) -> f603_out(T77) :|: TRUE 53.83/15.33 f621_out(x) -> f623_in(x) :|: TRUE 53.83/15.33 f603_in(x1) -> f621_in(x1) :|: TRUE 53.83/15.33 f971_in -> f953_in :|: TRUE 53.83/15.33 f957_in -> f965_in :|: TRUE 53.83/15.33 f636_in(x2) -> f645_in(x2) :|: TRUE 53.83/15.33 f643_out(x3) -> f636_out(x3) :|: TRUE 53.83/15.33 f645_out(x4) -> f636_out(x4) :|: TRUE 53.83/15.33 f636_in(x5) -> f643_in(x5) :|: TRUE 53.83/15.33 f946_out -> f809_out :|: TRUE 53.83/15.33 f809_in -> f946_in :|: TRUE 53.83/15.33 f384_out -> f946_out :|: TRUE 53.83/15.33 f946_in -> f384_in :|: TRUE 53.83/15.33 f965_in -> f967_in :|: TRUE 53.83/15.33 f967_out -> f968_in :|: TRUE 53.83/15.33 f645_in(s(T96)) -> f680_in(T96) :|: TRUE 53.83/15.33 f680_out(x8) -> f645_out(s(x8)) :|: TRUE 53.83/15.33 f953_in -> f955_in :|: TRUE 53.83/15.33 f948_in -> f948_out :|: TRUE 53.83/15.33 f657_in -> f657_out :|: TRUE 53.83/15.33 f445_out -> f386_out :|: TRUE 53.83/15.33 f446_out -> f386_out :|: TRUE 53.83/15.33 f386_in -> f445_in :|: TRUE 53.83/15.33 f386_in -> f446_in :|: TRUE 53.83/15.33 f494_out -> f480_out :|: TRUE 53.83/15.33 f480_in -> f494_in :|: TRUE 53.83/15.33 f570_out(T43) -> f548_out(T43) :|: TRUE 53.83/15.33 f548_in(x9) -> f570_in(x9) :|: TRUE 53.83/15.33 f569_out(x10) -> f548_out(x10) :|: TRUE 53.83/15.33 f548_in(x11) -> f569_in(x11) :|: TRUE 53.83/15.33 f494_in -> f500_in :|: TRUE 53.83/15.33 f498_out -> f494_out :|: TRUE 53.83/15.33 f500_out -> f494_out :|: TRUE 53.83/15.33 f494_in -> f498_in :|: TRUE 53.83/15.33 f948_out -> f812_out :|: TRUE 53.83/15.33 f812_in -> f948_in :|: TRUE 53.83/15.33 f955_in -> f957_in :|: TRUE 53.83/15.33 f809_out -> f446_out :|: TRUE 53.83/15.33 f446_in -> f809_in :|: TRUE 53.83/15.33 f446_in -> f812_in :|: TRUE 53.83/15.33 f812_out -> f446_out :|: TRUE 53.83/15.33 f521_out -> f500_out :|: TRUE 53.83/15.33 f500_in -> f521_in :|: TRUE 53.83/15.33 f657_out -> f643_out(0) :|: TRUE 53.83/15.33 f643_in(0) -> f657_in :|: TRUE 53.83/15.33 f466_out -> f445_out :|: TRUE 53.83/15.33 f445_in -> f466_in :|: TRUE 53.83/15.33 f680_in(x14) -> f621_in(x14) :|: TRUE 53.83/15.33 f621_out(x15) -> f680_out(x15) :|: TRUE 53.83/15.33 f793_in -> f793_out :|: TRUE 53.83/15.33 f481_out(x16) -> f466_out :|: TRUE 53.83/15.33 f480_out -> f481_in(x17) :|: TRUE 53.83/15.33 f466_in -> f480_in :|: TRUE 53.83/15.33 f481_out(x18) -> f623_out(x18) :|: TRUE 53.83/15.33 f623_in(x19) -> f481_in(x19) :|: TRUE 53.83/15.33 f480_out -> f521_out :|: TRUE 53.83/15.33 f521_in -> f480_in :|: TRUE 53.83/15.33 f967_in -> f384_in :|: TRUE 53.83/15.33 f384_out -> f967_out :|: TRUE 53.83/15.33 f636_out(x20) -> f621_out(x20) :|: TRUE 53.83/15.33 f621_in(x21) -> f636_in(x21) :|: TRUE 53.83/15.33 f791_out(T116) -> f767_out(T116) :|: TRUE 53.83/15.33 f767_in(x23) -> f791_in(x23) :|: TRUE 53.83/15.33 f569_in(x25) -> f603_in(x25) :|: TRUE 53.83/15.33 f603_out(x27) -> f569_out(x27) :|: TRUE 53.83/15.33 f767_out(x29) -> f570_out(x29) :|: TRUE 53.83/15.33 f769_out(x30) -> f570_out(x30) :|: TRUE 53.83/15.33 f570_in(x31) -> f767_in(x31) :|: TRUE 53.83/15.33 f570_in(x32) -> f769_in(x32) :|: TRUE 53.83/15.33 f384_in -> f386_in :|: TRUE 53.83/15.33 f386_out -> f384_out :|: TRUE 53.83/15.33 f498_in -> f506_in :|: TRUE 53.83/15.33 f506_out -> f498_out :|: TRUE 53.83/15.33 f481_in(x33) -> f548_in(x33) :|: TRUE 53.83/15.33 f548_out(x34) -> f481_out(x34) :|: TRUE 53.83/15.33 f769_in(T125) -> f793_in :|: TRUE 53.83/15.33 f793_out -> f769_out(x37) :|: TRUE 53.83/15.33 f481_out(x38) -> f791_out(x38) :|: TRUE 53.83/15.33 f791_in(x39) -> f481_in(x39) :|: TRUE 53.83/15.33 f384_out -> f385_in(x48) :|: TRUE 53.83/15.33 f385_in(x52) -> f953_in :|: TRUE 53.83/15.33 53.83/15.33 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (272) 53.83/15.33 Obligation: 53.83/15.33 Rules: 53.83/15.33 f506_in -> f506_out :|: TRUE 53.83/15.33 f968_in -> f971_in :|: TRUE 53.83/15.33 f623_out(T77) -> f603_out(T77) :|: TRUE 53.83/15.33 f621_out(x) -> f623_in(x) :|: TRUE 53.83/15.33 f603_in(x1) -> f621_in(x1) :|: TRUE 53.83/15.33 f971_in -> f953_in :|: TRUE 53.83/15.33 f957_in -> f965_in :|: TRUE 53.83/15.33 f636_in(x2) -> f645_in(x2) :|: TRUE 53.83/15.33 f643_out(x3) -> f636_out(x3) :|: TRUE 53.83/15.33 f645_out(x4) -> f636_out(x4) :|: TRUE 53.83/15.33 f636_in(x5) -> f643_in(x5) :|: TRUE 53.83/15.33 f946_out -> f809_out :|: TRUE 53.83/15.33 f809_in -> f946_in :|: TRUE 53.83/15.33 f384_out -> f946_out :|: TRUE 53.83/15.33 f946_in -> f384_in :|: TRUE 53.83/15.33 f965_in -> f967_in :|: TRUE 53.83/15.33 f967_out -> f968_in :|: TRUE 53.83/15.33 f645_in(s(T96)) -> f680_in(T96) :|: TRUE 53.83/15.33 f680_out(x8) -> f645_out(s(x8)) :|: TRUE 53.83/15.33 f953_in -> f955_in :|: TRUE 53.83/15.33 f948_in -> f948_out :|: TRUE 53.83/15.33 f657_in -> f657_out :|: TRUE 53.83/15.33 f445_out -> f386_out :|: TRUE 53.83/15.33 f446_out -> f386_out :|: TRUE 53.83/15.33 f386_in -> f445_in :|: TRUE 53.83/15.33 f386_in -> f446_in :|: TRUE 53.83/15.33 f494_out -> f480_out :|: TRUE 53.83/15.33 f480_in -> f494_in :|: TRUE 53.83/15.33 f570_out(T43) -> f548_out(T43) :|: TRUE 53.83/15.33 f548_in(x9) -> f570_in(x9) :|: TRUE 53.83/15.33 f569_out(x10) -> f548_out(x10) :|: TRUE 53.83/15.33 f548_in(x11) -> f569_in(x11) :|: TRUE 53.83/15.33 f494_in -> f500_in :|: TRUE 53.83/15.33 f498_out -> f494_out :|: TRUE 53.83/15.33 f500_out -> f494_out :|: TRUE 53.83/15.33 f494_in -> f498_in :|: TRUE 53.83/15.33 f948_out -> f812_out :|: TRUE 53.83/15.33 f812_in -> f948_in :|: TRUE 53.83/15.33 f955_in -> f957_in :|: TRUE 53.83/15.33 f809_out -> f446_out :|: TRUE 53.83/15.33 f446_in -> f809_in :|: TRUE 53.83/15.33 f446_in -> f812_in :|: TRUE 53.83/15.33 f812_out -> f446_out :|: TRUE 53.83/15.33 f521_out -> f500_out :|: TRUE 53.83/15.33 f500_in -> f521_in :|: TRUE 53.83/15.33 f657_out -> f643_out(0) :|: TRUE 53.83/15.33 f643_in(0) -> f657_in :|: TRUE 53.83/15.33 f466_out -> f445_out :|: TRUE 53.83/15.33 f445_in -> f466_in :|: TRUE 53.83/15.33 f680_in(x14) -> f621_in(x14) :|: TRUE 53.83/15.33 f621_out(x15) -> f680_out(x15) :|: TRUE 53.83/15.33 f793_in -> f793_out :|: TRUE 53.83/15.33 f481_out(x16) -> f466_out :|: TRUE 53.83/15.33 f480_out -> f481_in(x17) :|: TRUE 53.83/15.33 f466_in -> f480_in :|: TRUE 53.83/15.33 f481_out(x18) -> f623_out(x18) :|: TRUE 53.83/15.33 f623_in(x19) -> f481_in(x19) :|: TRUE 53.83/15.33 f480_out -> f521_out :|: TRUE 53.83/15.33 f521_in -> f480_in :|: TRUE 53.83/15.33 f967_in -> f384_in :|: TRUE 53.83/15.33 f384_out -> f967_out :|: TRUE 53.83/15.33 f636_out(x20) -> f621_out(x20) :|: TRUE 53.83/15.33 f621_in(x21) -> f636_in(x21) :|: TRUE 53.83/15.33 f791_out(T116) -> f767_out(T116) :|: TRUE 53.83/15.33 f767_in(x23) -> f791_in(x23) :|: TRUE 53.83/15.33 f569_in(x25) -> f603_in(x25) :|: TRUE 53.83/15.33 f603_out(x27) -> f569_out(x27) :|: TRUE 53.83/15.33 f767_out(x29) -> f570_out(x29) :|: TRUE 53.83/15.33 f769_out(x30) -> f570_out(x30) :|: TRUE 53.83/15.33 f570_in(x31) -> f767_in(x31) :|: TRUE 53.83/15.33 f570_in(x32) -> f769_in(x32) :|: TRUE 53.83/15.33 f384_in -> f386_in :|: TRUE 53.83/15.33 f386_out -> f384_out :|: TRUE 53.83/15.33 f498_in -> f506_in :|: TRUE 53.83/15.33 f506_out -> f498_out :|: TRUE 53.83/15.33 f481_in(x33) -> f548_in(x33) :|: TRUE 53.83/15.33 f548_out(x34) -> f481_out(x34) :|: TRUE 53.83/15.33 f769_in(T125) -> f793_in :|: TRUE 53.83/15.33 f793_out -> f769_out(x37) :|: TRUE 53.83/15.33 f481_out(x38) -> f791_out(x38) :|: TRUE 53.83/15.33 f791_in(x39) -> f481_in(x39) :|: TRUE 53.83/15.33 f384_out -> f385_in(x48) :|: TRUE 53.83/15.33 f385_in(x52) -> f953_in :|: TRUE 53.83/15.33 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (273) IntTRSCompressionProof (EQUIVALENT) 53.83/15.33 Compressed rules. 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (274) 53.83/15.33 Obligation: 53.83/15.33 Rules: 53.83/15.33 f548_in(x9:0) -> f548_in(x9:0) :|: TRUE 53.83/15.33 f481_out(x38:0) -> f481_out(x38:0) :|: TRUE 53.83/15.33 f636_in(cons_0) -> f621_out(0) :|: TRUE && cons_0 = 0 53.83/15.33 f480_in -> f494_out :|: TRUE 53.83/15.33 f384_out -> f386_in :|: TRUE 53.83/15.33 f621_out(x:0) -> f548_in(x:0) :|: TRUE 53.83/15.33 f494_out -> f494_out :|: TRUE 53.83/15.33 f386_in -> f384_out :|: TRUE 53.83/15.33 f386_in -> f386_in :|: TRUE 53.83/15.33 f494_out -> f548_in(x17:0) :|: TRUE 53.83/15.33 f384_out -> f384_out :|: TRUE 53.83/15.33 f621_out(x15:0) -> f621_out(s(x15:0)) :|: TRUE 53.83/15.33 f480_in -> f480_in :|: TRUE 53.83/15.33 f548_in(x) -> f481_out(x1) :|: TRUE 53.83/15.33 f548_in(x11:0) -> f636_in(x11:0) :|: TRUE 53.83/15.33 f481_out(x16:0) -> f384_out :|: TRUE 53.83/15.33 f636_in(s(T96:0)) -> f636_in(T96:0) :|: TRUE 53.83/15.33 f386_in -> f480_in :|: TRUE 53.83/15.33 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (275) IRSFormatTransformerProof (EQUIVALENT) 53.83/15.33 Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). 53.83/15.33 ---------------------------------------- 53.83/15.33 53.83/15.33 (276) 53.83/15.33 Obligation: 53.83/15.33 Rules: 53.83/15.33 f548_in(x9:0) -> f548_in(x9:0) :|: TRUE 53.83/15.33 f481_out(x38:0) -> f481_out(x38:0) :|: TRUE 53.83/15.33 f636_in(cons_0) -> f621_out(0) :|: TRUE && cons_0 = 0 53.83/15.33 f480_in -> f494_out :|: TRUE 53.83/15.33 f384_out -> f386_in :|: TRUE 53.83/15.33 f621_out(x:0) -> f548_in(x:0) :|: TRUE 53.83/15.33 f494_out -> f494_out :|: TRUE 53.83/15.33 f386_in -> f384_out :|: TRUE 53.83/15.33 f386_in -> f386_in :|: TRUE 53.83/15.33 f494_out -> f548_in(x17:0) :|: TRUE 53.83/15.33 f384_out -> f384_out :|: TRUE 53.83/15.33 f621_out(x15:0) -> f621_out(s(x15:0)) :|: TRUE 53.83/15.33 f480_in -> f480_in :|: TRUE 53.83/15.33 f548_in(x) -> f481_out(x1) :|: TRUE 53.83/15.33 f548_in(x11:0) -> f636_in(x11:0) :|: TRUE 53.83/15.33 f481_out(x16:0) -> f384_out :|: TRUE 53.83/15.33 f636_in(s(T96:0)) -> f636_in(T96:0) :|: TRUE 53.83/15.33 f386_in -> f480_in :|: TRUE 53.96/15.41 EOF