MAYBE proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Left Termination of the query pattern ms(a,g) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) PrologToPiTRSProof [SOUND, 20 ms] (2) PiTRS (3) DependencyPairsProof [EQUIVALENT, 33 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [SOUND, 0 ms] (11) QDP (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] (13) YES (14) PiDP (15) UsableRulesProof [EQUIVALENT, 0 ms] (16) PiDP (17) PiDPToQDPProof [SOUND, 0 ms] (18) QDP (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] (20) YES (21) PiDP (22) UsableRulesProof [EQUIVALENT, 0 ms] (23) PiDP (24) PiDPToQDPProof [SOUND, 0 ms] (25) QDP (26) NonTerminationLoopProof [COMPLETE, 0 ms] (27) NO (28) PiDP (29) UsableRulesProof [EQUIVALENT, 0 ms] (30) PiDP (31) PiDPToQDPProof [SOUND, 0 ms] (32) QDP (33) PiDP (34) UsableRulesProof [EQUIVALENT, 0 ms] (35) PiDP (36) PiDP (37) UsableRulesProof [EQUIVALENT, 0 ms] (38) PiDP (39) PrologToPiTRSProof [SOUND, 18 ms] (40) PiTRS (41) DependencyPairsProof [EQUIVALENT, 20 ms] (42) PiDP (43) DependencyGraphProof [EQUIVALENT, 0 ms] (44) AND (45) PiDP (46) UsableRulesProof [EQUIVALENT, 0 ms] (47) PiDP (48) PiDPToQDPProof [SOUND, 0 ms] (49) QDP (50) QDPSizeChangeProof [EQUIVALENT, 0 ms] (51) YES (52) PiDP (53) UsableRulesProof [EQUIVALENT, 0 ms] (54) PiDP (55) PiDPToQDPProof [SOUND, 0 ms] (56) QDP (57) QDPSizeChangeProof [EQUIVALENT, 0 ms] (58) YES (59) PiDP (60) UsableRulesProof [EQUIVALENT, 0 ms] (61) PiDP (62) PiDPToQDPProof [SOUND, 0 ms] (63) QDP (64) NonTerminationLoopProof [COMPLETE, 0 ms] (65) NO (66) PiDP (67) UsableRulesProof [EQUIVALENT, 0 ms] (68) PiDP (69) PiDPToQDPProof [SOUND, 0 ms] (70) QDP (71) TransformationProof [SOUND, 0 ms] (72) QDP (73) TransformationProof [SOUND, 0 ms] (74) QDP (75) PiDP (76) UsableRulesProof [EQUIVALENT, 0 ms] (77) PiDP (78) PiDP (79) UsableRulesProof [EQUIVALENT, 0 ms] (80) PiDP (81) PrologToTRSTransformerProof [SOUND, 66 ms] (82) QTRS (83) DependencyPairsProof [EQUIVALENT, 9 ms] (84) QDP (85) DependencyGraphProof [EQUIVALENT, 0 ms] (86) AND (87) QDP (88) UsableRulesProof [EQUIVALENT, 0 ms] (89) QDP (90) QDPSizeChangeProof [EQUIVALENT, 0 ms] (91) YES (92) QDP (93) QDPSizeChangeProof [EQUIVALENT, 0 ms] (94) YES (95) QDP (96) UsableRulesProof [EQUIVALENT, 0 ms] (97) QDP (98) NonTerminationLoopProof [COMPLETE, 0 ms] (99) NO (100) QDP (101) UsableRulesProof [EQUIVALENT, 0 ms] (102) QDP (103) TransformationProof [EQUIVALENT, 0 ms] (104) QDP (105) UsableRulesProof [EQUIVALENT, 0 ms] (106) QDP (107) TransformationProof [EQUIVALENT, 0 ms] (108) QDP (109) QDP (110) UsableRulesProof [EQUIVALENT, 0 ms] (111) QDP (112) NonTerminationLoopProof [COMPLETE, 0 ms] (113) NO (114) QDP (115) UsableRulesProof [EQUIVALENT, 0 ms] (116) QDP (117) TransformationProof [EQUIVALENT, 0 ms] (118) QDP (119) TransformationProof [EQUIVALENT, 0 ms] (120) QDP (121) PrologToIRSwTTransformerProof [SOUND, 116 ms] (122) AND (123) IRSwT (124) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (125) TRUE (126) IRSwT (127) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (128) TRUE (129) IRSwT (130) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (131) TRUE (132) IRSwT (133) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (134) TRUE (135) IRSwT (136) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (137) IRSwT (138) IntTRSCompressionProof [EQUIVALENT, 8 ms] (139) IRSwT (140) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (141) IRSwT (142) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms] (143) IRSwT (144) FilterProof [EQUIVALENT, 0 ms] (145) IntTRS (146) IntTRSNonPeriodicNontermProof [COMPLETE, 6 ms] (147) NO (148) IRSwT (149) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (150) IRSwT (151) IntTRSCompressionProof [EQUIVALENT, 9 ms] (152) IRSwT (153) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (154) IRSwT (155) IRSwTTerminationDigraphProof [EQUIVALENT, 9 ms] (156) IRSwT (157) FilterProof [EQUIVALENT, 0 ms] (158) IntTRS (159) IntTRSPeriodicNontermProof [COMPLETE, 0 ms] (160) NO (161) PrologToDTProblemTransformerProof [SOUND, 202 ms] (162) TRIPLES (163) UndefinedPredicateInTriplesTransformerProof [SOUND, 0 ms] (164) TRIPLES (165) TriplesToPiDPProof [SOUND, 59 ms] (166) PiDP (167) DependencyGraphProof [EQUIVALENT, 0 ms] (168) AND (169) PiDP (170) UsableRulesProof [EQUIVALENT, 0 ms] (171) PiDP (172) PiDPToQDPProof [SOUND, 0 ms] (173) QDP (174) QDPSizeChangeProof [EQUIVALENT, 0 ms] (175) YES (176) PiDP (177) UsableRulesProof [EQUIVALENT, 0 ms] (178) PiDP (179) PiDPToQDPProof [SOUND, 0 ms] (180) QDP (181) QDPSizeChangeProof [EQUIVALENT, 0 ms] (182) YES (183) PiDP (184) UsableRulesProof [EQUIVALENT, 0 ms] (185) PiDP (186) PiDPToQDPProof [SOUND, 0 ms] (187) QDP (188) NonTerminationLoopProof [COMPLETE, 0 ms] (189) NO (190) PiDP (191) UsableRulesProof [EQUIVALENT, 0 ms] (192) PiDP (193) PiDPToQDPProof [SOUND, 0 ms] (194) QDP (195) PiDP (196) UsableRulesProof [EQUIVALENT, 0 ms] (197) PiDP (198) PiDP (199) UsableRulesProof [EQUIVALENT, 0 ms] (200) PiDP ---------------------------------------- (0) Obligation: Clauses: ms([], []). ms(.(X, []), .(X, [])). ms(.(X, .(Y, Xs)), Ys) :- ','(split(.(X, .(Y, Xs)), X1s, X2s), ','(ms(X1s, Y1s), ','(ms(X2s, Y2s), merge(Y1s, Y2s, Ys)))). split([], [], []). split(.(X, Xs), .(X, Ys), Zs) :- split(Xs, Zs, Ys). merge([], Xs, Xs). merge(Xs, [], Xs). merge(.(X, Xs), .(Y, Ys), .(X, Zs)) :- ','(less(X, s(Y)), merge(Xs, .(Y, Ys), Zs)). merge(.(X, Xs), .(Y, Ys), .(Y, Zs)) :- ','(less(Y, X), merge(.(X, Xs), Ys, Zs)). less(0, s(X1)). less(s(X), s(Y)) :- less(X, Y). Query: ms(a,g) ---------------------------------------- (1) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: ms_in_2: (f,b) (f,f) split_in_3: (f,f,f) merge_in_3: (f,f,f) (f,f,b) less_in_2: (f,f) (b,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: ms_in_ag([], []) -> ms_out_ag([], []) ms_in_ag(.(X, []), .(X, [])) -> ms_out_ag(.(X, []), .(X, [])) ms_in_ag(.(X, .(Y, Xs)), Ys) -> U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) split_in_aaa([], [], []) -> split_out_aaa([], [], []) split_in_aaa(.(X, Xs), .(X, Ys), Zs) -> U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys)) U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) -> split_out_aaa(.(X, Xs), .(X, Ys), Zs) U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) ms_in_aa([], []) -> ms_out_aa([], []) ms_in_aa(.(X, []), .(X, [])) -> ms_out_aa(.(X, []), .(X, [])) ms_in_aa(.(X, .(Y, Xs)), Ys) -> U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys)) merge_in_aaa([], Xs, Xs) -> merge_out_aaa([], Xs, Xs) merge_in_aaa(Xs, [], Xs) -> merge_out_aaa(Xs, [], Xs) merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y))) less_in_aa(0, s(X1)) -> less_out_aa(0, s(X1)) less_in_aa(s(X), s(Y)) -> U10_aa(X, Y, less_in_aa(X, Y)) U10_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) -> U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs)) merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X)) U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) -> U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs)) U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) -> ms_out_aa(.(X, .(Y, Xs)), Ys) U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys)) merge_in_aag([], Xs, Xs) -> merge_out_aag([], Xs, Xs) merge_in_aag(Xs, [], Xs) -> merge_out_aag(Xs, [], Xs) merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y))) less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) less_in_ga(s(X), s(Y)) -> U10_ga(X, Y, less_in_ga(X, Y)) U10_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) -> U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs)) merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X)) U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) -> U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs)) U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) -> ms_out_ag(.(X, .(Y, Xs)), Ys) The argument filtering Pi contains the following mapping: ms_in_ag(x1, x2) = ms_in_ag(x2) [] = [] ms_out_ag(x1, x2) = ms_out_ag(x2) .(x1, x2) = .(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) split_in_aaa(x1, x2, x3) = split_in_aaa split_out_aaa(x1, x2, x3) = split_out_aaa U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5) U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x4, x6) ms_in_aa(x1, x2) = ms_in_aa ms_out_aa(x1, x2) = ms_out_aa U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x6) U3_aa(x1, x2, x3, x4, x5, x6) = U3_aa(x6) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x5) merge_in_aaa(x1, x2, x3) = merge_in_aaa merge_out_aaa(x1, x2, x3) = merge_out_aaa U6_aaa(x1, x2, x3, x4, x5, x6) = U6_aaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U10_aa(x1, x2, x3) = U10_aa(x3) U7_aaa(x1, x2, x3, x4, x5, x6) = U7_aaa(x6) U8_aaa(x1, x2, x3, x4, x5, x6) = U8_aaa(x6) U9_aaa(x1, x2, x3, x4, x5, x6) = U9_aaa(x6) U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x4, x6) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x4, x5) merge_in_aag(x1, x2, x3) = merge_in_aag(x3) merge_out_aag(x1, x2, x3) = merge_out_aag(x1, x2, x3) U6_aag(x1, x2, x3, x4, x5, x6) = U6_aag(x1, x5, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U10_ga(x1, x2, x3) = U10_ga(x1, x3) U7_aag(x1, x2, x3, x4, x5, x6) = U7_aag(x1, x5, x6) U8_aag(x1, x2, x3, x4, x5, x6) = U8_aag(x3, x5, x6) U9_aag(x1, x2, x3, x4, x5, x6) = U9_aag(x3, x5, x6) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: ms_in_ag([], []) -> ms_out_ag([], []) ms_in_ag(.(X, []), .(X, [])) -> ms_out_ag(.(X, []), .(X, [])) ms_in_ag(.(X, .(Y, Xs)), Ys) -> U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) split_in_aaa([], [], []) -> split_out_aaa([], [], []) split_in_aaa(.(X, Xs), .(X, Ys), Zs) -> U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys)) U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) -> split_out_aaa(.(X, Xs), .(X, Ys), Zs) U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) ms_in_aa([], []) -> ms_out_aa([], []) ms_in_aa(.(X, []), .(X, [])) -> ms_out_aa(.(X, []), .(X, [])) ms_in_aa(.(X, .(Y, Xs)), Ys) -> U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys)) merge_in_aaa([], Xs, Xs) -> merge_out_aaa([], Xs, Xs) merge_in_aaa(Xs, [], Xs) -> merge_out_aaa(Xs, [], Xs) merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y))) less_in_aa(0, s(X1)) -> less_out_aa(0, s(X1)) less_in_aa(s(X), s(Y)) -> U10_aa(X, Y, less_in_aa(X, Y)) U10_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) -> U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs)) merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X)) U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) -> U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs)) U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) -> ms_out_aa(.(X, .(Y, Xs)), Ys) U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys)) merge_in_aag([], Xs, Xs) -> merge_out_aag([], Xs, Xs) merge_in_aag(Xs, [], Xs) -> merge_out_aag(Xs, [], Xs) merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y))) less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) less_in_ga(s(X), s(Y)) -> U10_ga(X, Y, less_in_ga(X, Y)) U10_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) -> U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs)) merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X)) U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) -> U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs)) U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) -> ms_out_ag(.(X, .(Y, Xs)), Ys) The argument filtering Pi contains the following mapping: ms_in_ag(x1, x2) = ms_in_ag(x2) [] = [] ms_out_ag(x1, x2) = ms_out_ag(x2) .(x1, x2) = .(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) split_in_aaa(x1, x2, x3) = split_in_aaa split_out_aaa(x1, x2, x3) = split_out_aaa U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5) U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x4, x6) ms_in_aa(x1, x2) = ms_in_aa ms_out_aa(x1, x2) = ms_out_aa U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x6) U3_aa(x1, x2, x3, x4, x5, x6) = U3_aa(x6) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x5) merge_in_aaa(x1, x2, x3) = merge_in_aaa merge_out_aaa(x1, x2, x3) = merge_out_aaa U6_aaa(x1, x2, x3, x4, x5, x6) = U6_aaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U10_aa(x1, x2, x3) = U10_aa(x3) U7_aaa(x1, x2, x3, x4, x5, x6) = U7_aaa(x6) U8_aaa(x1, x2, x3, x4, x5, x6) = U8_aaa(x6) U9_aaa(x1, x2, x3, x4, x5, x6) = U9_aaa(x6) U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x4, x6) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x4, x5) merge_in_aag(x1, x2, x3) = merge_in_aag(x3) merge_out_aag(x1, x2, x3) = merge_out_aag(x1, x2, x3) U6_aag(x1, x2, x3, x4, x5, x6) = U6_aag(x1, x5, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U10_ga(x1, x2, x3) = U10_ga(x1, x3) U7_aag(x1, x2, x3, x4, x5, x6) = U7_aag(x1, x5, x6) U8_aag(x1, x2, x3, x4, x5, x6) = U8_aag(x3, x5, x6) U9_aag(x1, x2, x3, x4, x5, x6) = U9_aag(x3, x5, x6) ---------------------------------------- (3) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: Pi DP problem: The TRS P consists of the following rules: MS_IN_AG(.(X, .(Y, Xs)), Ys) -> U1_AG(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) MS_IN_AG(.(X, .(Y, Xs)), Ys) -> SPLIT_IN_AAA(.(X, .(Y, Xs)), X1s, X2s) SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) -> U5_AAA(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys)) SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) -> SPLIT_IN_AAA(Xs, Zs, Ys) U1_AG(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_AG(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) U1_AG(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> MS_IN_AA(X1s, Y1s) MS_IN_AA(.(X, .(Y, Xs)), Ys) -> U1_AA(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) MS_IN_AA(.(X, .(Y, Xs)), Ys) -> SPLIT_IN_AAA(.(X, .(Y, Xs)), X1s, X2s) U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_AA(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> MS_IN_AA(X1s, Y1s) U2_AA(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_AA(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U2_AA(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> MS_IN_AA(X2s, Y2s) U3_AA(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_AA(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys)) U3_AA(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> MERGE_IN_AAA(Y1s, Y2s, Ys) MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_AAA(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y))) MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(X, Zs)) -> LESS_IN_AA(X, s(Y)) LESS_IN_AA(s(X), s(Y)) -> U10_AA(X, Y, less_in_aa(X, Y)) LESS_IN_AA(s(X), s(Y)) -> LESS_IN_AA(X, Y) U6_AAA(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) -> U7_AAA(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs)) U6_AAA(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) -> MERGE_IN_AAA(Xs, .(Y, Ys), Zs) MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_AAA(X, Xs, Y, Ys, Zs, less_in_aa(Y, X)) MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> LESS_IN_AA(Y, X) U8_AAA(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) -> U9_AAA(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs)) U8_AAA(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) -> MERGE_IN_AAA(.(X, Xs), Ys, Zs) U2_AG(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_AG(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U2_AG(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> MS_IN_AA(X2s, Y2s) U3_AG(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_AG(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys)) U3_AG(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> MERGE_IN_AAG(Y1s, Y2s, Ys) MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_AAG(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y))) MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(X, Zs)) -> LESS_IN_GA(X, s(Y)) LESS_IN_GA(s(X), s(Y)) -> U10_GA(X, Y, less_in_ga(X, Y)) LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) U6_AAG(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) -> U7_AAG(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs)) U6_AAG(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) -> MERGE_IN_AAG(Xs, .(Y, Ys), Zs) MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_AAG(X, Xs, Y, Ys, Zs, less_in_ga(Y, X)) MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> LESS_IN_GA(Y, X) U8_AAG(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) -> U9_AAG(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs)) U8_AAG(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) -> MERGE_IN_AAG(.(X, Xs), Ys, Zs) The TRS R consists of the following rules: ms_in_ag([], []) -> ms_out_ag([], []) ms_in_ag(.(X, []), .(X, [])) -> ms_out_ag(.(X, []), .(X, [])) ms_in_ag(.(X, .(Y, Xs)), Ys) -> U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) split_in_aaa([], [], []) -> split_out_aaa([], [], []) split_in_aaa(.(X, Xs), .(X, Ys), Zs) -> U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys)) U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) -> split_out_aaa(.(X, Xs), .(X, Ys), Zs) U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) ms_in_aa([], []) -> ms_out_aa([], []) ms_in_aa(.(X, []), .(X, [])) -> ms_out_aa(.(X, []), .(X, [])) ms_in_aa(.(X, .(Y, Xs)), Ys) -> U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys)) merge_in_aaa([], Xs, Xs) -> merge_out_aaa([], Xs, Xs) merge_in_aaa(Xs, [], Xs) -> merge_out_aaa(Xs, [], Xs) merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y))) less_in_aa(0, s(X1)) -> less_out_aa(0, s(X1)) less_in_aa(s(X), s(Y)) -> U10_aa(X, Y, less_in_aa(X, Y)) U10_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) -> U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs)) merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X)) U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) -> U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs)) U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) -> ms_out_aa(.(X, .(Y, Xs)), Ys) U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys)) merge_in_aag([], Xs, Xs) -> merge_out_aag([], Xs, Xs) merge_in_aag(Xs, [], Xs) -> merge_out_aag(Xs, [], Xs) merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y))) less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) less_in_ga(s(X), s(Y)) -> U10_ga(X, Y, less_in_ga(X, Y)) U10_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) -> U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs)) merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X)) U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) -> U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs)) U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) -> ms_out_ag(.(X, .(Y, Xs)), Ys) The argument filtering Pi contains the following mapping: ms_in_ag(x1, x2) = ms_in_ag(x2) [] = [] ms_out_ag(x1, x2) = ms_out_ag(x2) .(x1, x2) = .(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) split_in_aaa(x1, x2, x3) = split_in_aaa split_out_aaa(x1, x2, x3) = split_out_aaa U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5) U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x4, x6) ms_in_aa(x1, x2) = ms_in_aa ms_out_aa(x1, x2) = ms_out_aa U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x6) U3_aa(x1, x2, x3, x4, x5, x6) = U3_aa(x6) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x5) merge_in_aaa(x1, x2, x3) = merge_in_aaa merge_out_aaa(x1, x2, x3) = merge_out_aaa U6_aaa(x1, x2, x3, x4, x5, x6) = U6_aaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U10_aa(x1, x2, x3) = U10_aa(x3) U7_aaa(x1, x2, x3, x4, x5, x6) = U7_aaa(x6) U8_aaa(x1, x2, x3, x4, x5, x6) = U8_aaa(x6) U9_aaa(x1, x2, x3, x4, x5, x6) = U9_aaa(x6) U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x4, x6) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x4, x5) merge_in_aag(x1, x2, x3) = merge_in_aag(x3) merge_out_aag(x1, x2, x3) = merge_out_aag(x1, x2, x3) U6_aag(x1, x2, x3, x4, x5, x6) = U6_aag(x1, x5, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U10_ga(x1, x2, x3) = U10_ga(x1, x3) U7_aag(x1, x2, x3, x4, x5, x6) = U7_aag(x1, x5, x6) U8_aag(x1, x2, x3, x4, x5, x6) = U8_aag(x3, x5, x6) U9_aag(x1, x2, x3, x4, x5, x6) = U9_aag(x3, x5, x6) MS_IN_AG(x1, x2) = MS_IN_AG(x2) U1_AG(x1, x2, x3, x4, x5) = U1_AG(x4, x5) SPLIT_IN_AAA(x1, x2, x3) = SPLIT_IN_AAA U5_AAA(x1, x2, x3, x4, x5) = U5_AAA(x5) U2_AG(x1, x2, x3, x4, x5, x6) = U2_AG(x4, x6) MS_IN_AA(x1, x2) = MS_IN_AA U1_AA(x1, x2, x3, x4, x5) = U1_AA(x5) U2_AA(x1, x2, x3, x4, x5, x6) = U2_AA(x6) U3_AA(x1, x2, x3, x4, x5, x6) = U3_AA(x6) U4_AA(x1, x2, x3, x4, x5) = U4_AA(x5) MERGE_IN_AAA(x1, x2, x3) = MERGE_IN_AAA U6_AAA(x1, x2, x3, x4, x5, x6) = U6_AAA(x6) LESS_IN_AA(x1, x2) = LESS_IN_AA U10_AA(x1, x2, x3) = U10_AA(x3) U7_AAA(x1, x2, x3, x4, x5, x6) = U7_AAA(x6) U8_AAA(x1, x2, x3, x4, x5, x6) = U8_AAA(x6) U9_AAA(x1, x2, x3, x4, x5, x6) = U9_AAA(x6) U3_AG(x1, x2, x3, x4, x5, x6) = U3_AG(x4, x6) U4_AG(x1, x2, x3, x4, x5) = U4_AG(x4, x5) MERGE_IN_AAG(x1, x2, x3) = MERGE_IN_AAG(x3) U6_AAG(x1, x2, x3, x4, x5, x6) = U6_AAG(x1, x5, x6) LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) U10_GA(x1, x2, x3) = U10_GA(x1, x3) U7_AAG(x1, x2, x3, x4, x5, x6) = U7_AAG(x1, x5, x6) U8_AAG(x1, x2, x3, x4, x5, x6) = U8_AAG(x3, x5, x6) U9_AAG(x1, x2, x3, x4, x5, x6) = U9_AAG(x3, x5, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: MS_IN_AG(.(X, .(Y, Xs)), Ys) -> U1_AG(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) MS_IN_AG(.(X, .(Y, Xs)), Ys) -> SPLIT_IN_AAA(.(X, .(Y, Xs)), X1s, X2s) SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) -> U5_AAA(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys)) SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) -> SPLIT_IN_AAA(Xs, Zs, Ys) U1_AG(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_AG(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) U1_AG(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> MS_IN_AA(X1s, Y1s) MS_IN_AA(.(X, .(Y, Xs)), Ys) -> U1_AA(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) MS_IN_AA(.(X, .(Y, Xs)), Ys) -> SPLIT_IN_AAA(.(X, .(Y, Xs)), X1s, X2s) U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_AA(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> MS_IN_AA(X1s, Y1s) U2_AA(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_AA(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U2_AA(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> MS_IN_AA(X2s, Y2s) U3_AA(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_AA(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys)) U3_AA(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> MERGE_IN_AAA(Y1s, Y2s, Ys) MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_AAA(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y))) MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(X, Zs)) -> LESS_IN_AA(X, s(Y)) LESS_IN_AA(s(X), s(Y)) -> U10_AA(X, Y, less_in_aa(X, Y)) LESS_IN_AA(s(X), s(Y)) -> LESS_IN_AA(X, Y) U6_AAA(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) -> U7_AAA(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs)) U6_AAA(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) -> MERGE_IN_AAA(Xs, .(Y, Ys), Zs) MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_AAA(X, Xs, Y, Ys, Zs, less_in_aa(Y, X)) MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> LESS_IN_AA(Y, X) U8_AAA(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) -> U9_AAA(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs)) U8_AAA(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) -> MERGE_IN_AAA(.(X, Xs), Ys, Zs) U2_AG(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_AG(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U2_AG(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> MS_IN_AA(X2s, Y2s) U3_AG(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_AG(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys)) U3_AG(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> MERGE_IN_AAG(Y1s, Y2s, Ys) MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_AAG(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y))) MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(X, Zs)) -> LESS_IN_GA(X, s(Y)) LESS_IN_GA(s(X), s(Y)) -> U10_GA(X, Y, less_in_ga(X, Y)) LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) U6_AAG(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) -> U7_AAG(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs)) U6_AAG(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) -> MERGE_IN_AAG(Xs, .(Y, Ys), Zs) MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_AAG(X, Xs, Y, Ys, Zs, less_in_ga(Y, X)) MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> LESS_IN_GA(Y, X) U8_AAG(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) -> U9_AAG(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs)) U8_AAG(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) -> MERGE_IN_AAG(.(X, Xs), Ys, Zs) The TRS R consists of the following rules: ms_in_ag([], []) -> ms_out_ag([], []) ms_in_ag(.(X, []), .(X, [])) -> ms_out_ag(.(X, []), .(X, [])) ms_in_ag(.(X, .(Y, Xs)), Ys) -> U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) split_in_aaa([], [], []) -> split_out_aaa([], [], []) split_in_aaa(.(X, Xs), .(X, Ys), Zs) -> U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys)) U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) -> split_out_aaa(.(X, Xs), .(X, Ys), Zs) U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) ms_in_aa([], []) -> ms_out_aa([], []) ms_in_aa(.(X, []), .(X, [])) -> ms_out_aa(.(X, []), .(X, [])) ms_in_aa(.(X, .(Y, Xs)), Ys) -> U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys)) merge_in_aaa([], Xs, Xs) -> merge_out_aaa([], Xs, Xs) merge_in_aaa(Xs, [], Xs) -> merge_out_aaa(Xs, [], Xs) merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y))) less_in_aa(0, s(X1)) -> less_out_aa(0, s(X1)) less_in_aa(s(X), s(Y)) -> U10_aa(X, Y, less_in_aa(X, Y)) U10_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) -> U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs)) merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X)) U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) -> U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs)) U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) -> ms_out_aa(.(X, .(Y, Xs)), Ys) U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys)) merge_in_aag([], Xs, Xs) -> merge_out_aag([], Xs, Xs) merge_in_aag(Xs, [], Xs) -> merge_out_aag(Xs, [], Xs) merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y))) less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) less_in_ga(s(X), s(Y)) -> U10_ga(X, Y, less_in_ga(X, Y)) U10_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) -> U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs)) merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X)) U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) -> U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs)) U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) -> ms_out_ag(.(X, .(Y, Xs)), Ys) The argument filtering Pi contains the following mapping: ms_in_ag(x1, x2) = ms_in_ag(x2) [] = [] ms_out_ag(x1, x2) = ms_out_ag(x2) .(x1, x2) = .(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) split_in_aaa(x1, x2, x3) = split_in_aaa split_out_aaa(x1, x2, x3) = split_out_aaa U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5) U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x4, x6) ms_in_aa(x1, x2) = ms_in_aa ms_out_aa(x1, x2) = ms_out_aa U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x6) U3_aa(x1, x2, x3, x4, x5, x6) = U3_aa(x6) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x5) merge_in_aaa(x1, x2, x3) = merge_in_aaa merge_out_aaa(x1, x2, x3) = merge_out_aaa U6_aaa(x1, x2, x3, x4, x5, x6) = U6_aaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U10_aa(x1, x2, x3) = U10_aa(x3) U7_aaa(x1, x2, x3, x4, x5, x6) = U7_aaa(x6) U8_aaa(x1, x2, x3, x4, x5, x6) = U8_aaa(x6) U9_aaa(x1, x2, x3, x4, x5, x6) = U9_aaa(x6) U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x4, x6) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x4, x5) merge_in_aag(x1, x2, x3) = merge_in_aag(x3) merge_out_aag(x1, x2, x3) = merge_out_aag(x1, x2, x3) U6_aag(x1, x2, x3, x4, x5, x6) = U6_aag(x1, x5, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U10_ga(x1, x2, x3) = U10_ga(x1, x3) U7_aag(x1, x2, x3, x4, x5, x6) = U7_aag(x1, x5, x6) U8_aag(x1, x2, x3, x4, x5, x6) = U8_aag(x3, x5, x6) U9_aag(x1, x2, x3, x4, x5, x6) = U9_aag(x3, x5, x6) MS_IN_AG(x1, x2) = MS_IN_AG(x2) U1_AG(x1, x2, x3, x4, x5) = U1_AG(x4, x5) SPLIT_IN_AAA(x1, x2, x3) = SPLIT_IN_AAA U5_AAA(x1, x2, x3, x4, x5) = U5_AAA(x5) U2_AG(x1, x2, x3, x4, x5, x6) = U2_AG(x4, x6) MS_IN_AA(x1, x2) = MS_IN_AA U1_AA(x1, x2, x3, x4, x5) = U1_AA(x5) U2_AA(x1, x2, x3, x4, x5, x6) = U2_AA(x6) U3_AA(x1, x2, x3, x4, x5, x6) = U3_AA(x6) U4_AA(x1, x2, x3, x4, x5) = U4_AA(x5) MERGE_IN_AAA(x1, x2, x3) = MERGE_IN_AAA U6_AAA(x1, x2, x3, x4, x5, x6) = U6_AAA(x6) LESS_IN_AA(x1, x2) = LESS_IN_AA U10_AA(x1, x2, x3) = U10_AA(x3) U7_AAA(x1, x2, x3, x4, x5, x6) = U7_AAA(x6) U8_AAA(x1, x2, x3, x4, x5, x6) = U8_AAA(x6) U9_AAA(x1, x2, x3, x4, x5, x6) = U9_AAA(x6) U3_AG(x1, x2, x3, x4, x5, x6) = U3_AG(x4, x6) U4_AG(x1, x2, x3, x4, x5) = U4_AG(x4, x5) MERGE_IN_AAG(x1, x2, x3) = MERGE_IN_AAG(x3) U6_AAG(x1, x2, x3, x4, x5, x6) = U6_AAG(x1, x5, x6) LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) U10_GA(x1, x2, x3) = U10_GA(x1, x3) U7_AAG(x1, x2, x3, x4, x5, x6) = U7_AAG(x1, x5, x6) U8_AAG(x1, x2, x3, x4, x5, x6) = U8_AAG(x3, x5, x6) U9_AAG(x1, x2, x3, x4, x5, x6) = U9_AAG(x3, x5, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 23 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) The TRS R consists of the following rules: ms_in_ag([], []) -> ms_out_ag([], []) ms_in_ag(.(X, []), .(X, [])) -> ms_out_ag(.(X, []), .(X, [])) ms_in_ag(.(X, .(Y, Xs)), Ys) -> U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) split_in_aaa([], [], []) -> split_out_aaa([], [], []) split_in_aaa(.(X, Xs), .(X, Ys), Zs) -> U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys)) U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) -> split_out_aaa(.(X, Xs), .(X, Ys), Zs) U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) ms_in_aa([], []) -> ms_out_aa([], []) ms_in_aa(.(X, []), .(X, [])) -> ms_out_aa(.(X, []), .(X, [])) ms_in_aa(.(X, .(Y, Xs)), Ys) -> U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys)) merge_in_aaa([], Xs, Xs) -> merge_out_aaa([], Xs, Xs) merge_in_aaa(Xs, [], Xs) -> merge_out_aaa(Xs, [], Xs) merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y))) less_in_aa(0, s(X1)) -> less_out_aa(0, s(X1)) less_in_aa(s(X), s(Y)) -> U10_aa(X, Y, less_in_aa(X, Y)) U10_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) -> U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs)) merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X)) U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) -> U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs)) U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) -> ms_out_aa(.(X, .(Y, Xs)), Ys) U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys)) merge_in_aag([], Xs, Xs) -> merge_out_aag([], Xs, Xs) merge_in_aag(Xs, [], Xs) -> merge_out_aag(Xs, [], Xs) merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y))) less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) less_in_ga(s(X), s(Y)) -> U10_ga(X, Y, less_in_ga(X, Y)) U10_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) -> U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs)) merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X)) U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) -> U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs)) U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) -> ms_out_ag(.(X, .(Y, Xs)), Ys) The argument filtering Pi contains the following mapping: ms_in_ag(x1, x2) = ms_in_ag(x2) [] = [] ms_out_ag(x1, x2) = ms_out_ag(x2) .(x1, x2) = .(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) split_in_aaa(x1, x2, x3) = split_in_aaa split_out_aaa(x1, x2, x3) = split_out_aaa U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5) U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x4, x6) ms_in_aa(x1, x2) = ms_in_aa ms_out_aa(x1, x2) = ms_out_aa U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x6) U3_aa(x1, x2, x3, x4, x5, x6) = U3_aa(x6) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x5) merge_in_aaa(x1, x2, x3) = merge_in_aaa merge_out_aaa(x1, x2, x3) = merge_out_aaa U6_aaa(x1, x2, x3, x4, x5, x6) = U6_aaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U10_aa(x1, x2, x3) = U10_aa(x3) U7_aaa(x1, x2, x3, x4, x5, x6) = U7_aaa(x6) U8_aaa(x1, x2, x3, x4, x5, x6) = U8_aaa(x6) U9_aaa(x1, x2, x3, x4, x5, x6) = U9_aaa(x6) U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x4, x6) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x4, x5) merge_in_aag(x1, x2, x3) = merge_in_aag(x3) merge_out_aag(x1, x2, x3) = merge_out_aag(x1, x2, x3) U6_aag(x1, x2, x3, x4, x5, x6) = U6_aag(x1, x5, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U10_ga(x1, x2, x3) = U10_ga(x1, x3) U7_aag(x1, x2, x3, x4, x5, x6) = U7_aag(x1, x5, x6) U8_aag(x1, x2, x3, x4, x5, x6) = U8_aag(x3, x5, x6) U9_aag(x1, x2, x3, x4, x5, x6) = U9_aag(x3, x5, x6) LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (8) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (9) Obligation: Pi DP problem: The TRS P consists of the following rules: LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (10) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: LESS_IN_GA(s(X)) -> LESS_IN_GA(X) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (12) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *LESS_IN_GA(s(X)) -> LESS_IN_GA(X) The graph contains the following edges 1 > 1 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: U6_AAG(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) -> MERGE_IN_AAG(Xs, .(Y, Ys), Zs) MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_AAG(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y))) MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_AAG(X, Xs, Y, Ys, Zs, less_in_ga(Y, X)) U8_AAG(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) -> MERGE_IN_AAG(.(X, Xs), Ys, Zs) The TRS R consists of the following rules: ms_in_ag([], []) -> ms_out_ag([], []) ms_in_ag(.(X, []), .(X, [])) -> ms_out_ag(.(X, []), .(X, [])) ms_in_ag(.(X, .(Y, Xs)), Ys) -> U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) split_in_aaa([], [], []) -> split_out_aaa([], [], []) split_in_aaa(.(X, Xs), .(X, Ys), Zs) -> U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys)) U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) -> split_out_aaa(.(X, Xs), .(X, Ys), Zs) U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) ms_in_aa([], []) -> ms_out_aa([], []) ms_in_aa(.(X, []), .(X, [])) -> ms_out_aa(.(X, []), .(X, [])) ms_in_aa(.(X, .(Y, Xs)), Ys) -> U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys)) merge_in_aaa([], Xs, Xs) -> merge_out_aaa([], Xs, Xs) merge_in_aaa(Xs, [], Xs) -> merge_out_aaa(Xs, [], Xs) merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y))) less_in_aa(0, s(X1)) -> less_out_aa(0, s(X1)) less_in_aa(s(X), s(Y)) -> U10_aa(X, Y, less_in_aa(X, Y)) U10_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) -> U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs)) merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X)) U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) -> U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs)) U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) -> ms_out_aa(.(X, .(Y, Xs)), Ys) U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys)) merge_in_aag([], Xs, Xs) -> merge_out_aag([], Xs, Xs) merge_in_aag(Xs, [], Xs) -> merge_out_aag(Xs, [], Xs) merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y))) less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) less_in_ga(s(X), s(Y)) -> U10_ga(X, Y, less_in_ga(X, Y)) U10_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) -> U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs)) merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X)) U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) -> U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs)) U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) -> ms_out_ag(.(X, .(Y, Xs)), Ys) The argument filtering Pi contains the following mapping: ms_in_ag(x1, x2) = ms_in_ag(x2) [] = [] ms_out_ag(x1, x2) = ms_out_ag(x2) .(x1, x2) = .(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) split_in_aaa(x1, x2, x3) = split_in_aaa split_out_aaa(x1, x2, x3) = split_out_aaa U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5) U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x4, x6) ms_in_aa(x1, x2) = ms_in_aa ms_out_aa(x1, x2) = ms_out_aa U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x6) U3_aa(x1, x2, x3, x4, x5, x6) = U3_aa(x6) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x5) merge_in_aaa(x1, x2, x3) = merge_in_aaa merge_out_aaa(x1, x2, x3) = merge_out_aaa U6_aaa(x1, x2, x3, x4, x5, x6) = U6_aaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U10_aa(x1, x2, x3) = U10_aa(x3) U7_aaa(x1, x2, x3, x4, x5, x6) = U7_aaa(x6) U8_aaa(x1, x2, x3, x4, x5, x6) = U8_aaa(x6) U9_aaa(x1, x2, x3, x4, x5, x6) = U9_aaa(x6) U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x4, x6) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x4, x5) merge_in_aag(x1, x2, x3) = merge_in_aag(x3) merge_out_aag(x1, x2, x3) = merge_out_aag(x1, x2, x3) U6_aag(x1, x2, x3, x4, x5, x6) = U6_aag(x1, x5, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U10_ga(x1, x2, x3) = U10_ga(x1, x3) U7_aag(x1, x2, x3, x4, x5, x6) = U7_aag(x1, x5, x6) U8_aag(x1, x2, x3, x4, x5, x6) = U8_aag(x3, x5, x6) U9_aag(x1, x2, x3, x4, x5, x6) = U9_aag(x3, x5, x6) MERGE_IN_AAG(x1, x2, x3) = MERGE_IN_AAG(x3) U6_AAG(x1, x2, x3, x4, x5, x6) = U6_AAG(x1, x5, x6) U8_AAG(x1, x2, x3, x4, x5, x6) = U8_AAG(x3, x5, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (15) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: U6_AAG(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) -> MERGE_IN_AAG(Xs, .(Y, Ys), Zs) MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_AAG(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y))) MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_AAG(X, Xs, Y, Ys, Zs, less_in_ga(Y, X)) U8_AAG(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) -> MERGE_IN_AAG(.(X, Xs), Ys, Zs) The TRS R consists of the following rules: less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) less_in_ga(s(X), s(Y)) -> U10_ga(X, Y, less_in_ga(X, Y)) U10_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U10_ga(x1, x2, x3) = U10_ga(x1, x3) MERGE_IN_AAG(x1, x2, x3) = MERGE_IN_AAG(x3) U6_AAG(x1, x2, x3, x4, x5, x6) = U6_AAG(x1, x5, x6) U8_AAG(x1, x2, x3, x4, x5, x6) = U8_AAG(x3, x5, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: U6_AAG(X, Zs, less_out_ga(X)) -> MERGE_IN_AAG(Zs) MERGE_IN_AAG(.(X, Zs)) -> U6_AAG(X, Zs, less_in_ga(X)) MERGE_IN_AAG(.(Y, Zs)) -> U8_AAG(Y, Zs, less_in_ga(Y)) U8_AAG(Y, Zs, less_out_ga(Y)) -> MERGE_IN_AAG(Zs) The TRS R consists of the following rules: less_in_ga(0) -> less_out_ga(0) less_in_ga(s(X)) -> U10_ga(X, less_in_ga(X)) U10_ga(X, less_out_ga(X)) -> less_out_ga(s(X)) The set Q consists of the following terms: less_in_ga(x0) U10_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (19) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *MERGE_IN_AAG(.(X, Zs)) -> U6_AAG(X, Zs, less_in_ga(X)) The graph contains the following edges 1 > 1, 1 > 2 *MERGE_IN_AAG(.(Y, Zs)) -> U8_AAG(Y, Zs, less_in_ga(Y)) The graph contains the following edges 1 > 1, 1 > 2 *U6_AAG(X, Zs, less_out_ga(X)) -> MERGE_IN_AAG(Zs) The graph contains the following edges 2 >= 1 *U8_AAG(Y, Zs, less_out_ga(Y)) -> MERGE_IN_AAG(Zs) The graph contains the following edges 2 >= 1 ---------------------------------------- (20) YES ---------------------------------------- (21) Obligation: Pi DP problem: The TRS P consists of the following rules: LESS_IN_AA(s(X), s(Y)) -> LESS_IN_AA(X, Y) The TRS R consists of the following rules: ms_in_ag([], []) -> ms_out_ag([], []) ms_in_ag(.(X, []), .(X, [])) -> ms_out_ag(.(X, []), .(X, [])) ms_in_ag(.(X, .(Y, Xs)), Ys) -> U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) split_in_aaa([], [], []) -> split_out_aaa([], [], []) split_in_aaa(.(X, Xs), .(X, Ys), Zs) -> U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys)) U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) -> split_out_aaa(.(X, Xs), .(X, Ys), Zs) U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) ms_in_aa([], []) -> ms_out_aa([], []) ms_in_aa(.(X, []), .(X, [])) -> ms_out_aa(.(X, []), .(X, [])) ms_in_aa(.(X, .(Y, Xs)), Ys) -> U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys)) merge_in_aaa([], Xs, Xs) -> merge_out_aaa([], Xs, Xs) merge_in_aaa(Xs, [], Xs) -> merge_out_aaa(Xs, [], Xs) merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y))) less_in_aa(0, s(X1)) -> less_out_aa(0, s(X1)) less_in_aa(s(X), s(Y)) -> U10_aa(X, Y, less_in_aa(X, Y)) U10_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) -> U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs)) merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X)) U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) -> U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs)) U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) -> ms_out_aa(.(X, .(Y, Xs)), Ys) U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys)) merge_in_aag([], Xs, Xs) -> merge_out_aag([], Xs, Xs) merge_in_aag(Xs, [], Xs) -> merge_out_aag(Xs, [], Xs) merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y))) less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) less_in_ga(s(X), s(Y)) -> U10_ga(X, Y, less_in_ga(X, Y)) U10_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) -> U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs)) merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X)) U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) -> U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs)) U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) -> ms_out_ag(.(X, .(Y, Xs)), Ys) The argument filtering Pi contains the following mapping: ms_in_ag(x1, x2) = ms_in_ag(x2) [] = [] ms_out_ag(x1, x2) = ms_out_ag(x2) .(x1, x2) = .(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) split_in_aaa(x1, x2, x3) = split_in_aaa split_out_aaa(x1, x2, x3) = split_out_aaa U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5) U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x4, x6) ms_in_aa(x1, x2) = ms_in_aa ms_out_aa(x1, x2) = ms_out_aa U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x6) U3_aa(x1, x2, x3, x4, x5, x6) = U3_aa(x6) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x5) merge_in_aaa(x1, x2, x3) = merge_in_aaa merge_out_aaa(x1, x2, x3) = merge_out_aaa U6_aaa(x1, x2, x3, x4, x5, x6) = U6_aaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U10_aa(x1, x2, x3) = U10_aa(x3) U7_aaa(x1, x2, x3, x4, x5, x6) = U7_aaa(x6) U8_aaa(x1, x2, x3, x4, x5, x6) = U8_aaa(x6) U9_aaa(x1, x2, x3, x4, x5, x6) = U9_aaa(x6) U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x4, x6) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x4, x5) merge_in_aag(x1, x2, x3) = merge_in_aag(x3) merge_out_aag(x1, x2, x3) = merge_out_aag(x1, x2, x3) U6_aag(x1, x2, x3, x4, x5, x6) = U6_aag(x1, x5, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U10_ga(x1, x2, x3) = U10_ga(x1, x3) U7_aag(x1, x2, x3, x4, x5, x6) = U7_aag(x1, x5, x6) U8_aag(x1, x2, x3, x4, x5, x6) = U8_aag(x3, x5, x6) U9_aag(x1, x2, x3, x4, x5, x6) = U9_aag(x3, x5, x6) LESS_IN_AA(x1, x2) = LESS_IN_AA We have to consider all (P,R,Pi)-chains ---------------------------------------- (22) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (23) Obligation: Pi DP problem: The TRS P consists of the following rules: LESS_IN_AA(s(X), s(Y)) -> LESS_IN_AA(X, Y) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) LESS_IN_AA(x1, x2) = LESS_IN_AA We have to consider all (P,R,Pi)-chains ---------------------------------------- (24) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (25) Obligation: Q DP problem: The TRS P consists of the following rules: LESS_IN_AA -> LESS_IN_AA R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (26) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by semiunifying a rule from P directly. s = LESS_IN_AA evaluates to t =LESS_IN_AA Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence The DP semiunifies directly so there is only one rewrite step from LESS_IN_AA to LESS_IN_AA. ---------------------------------------- (27) NO ---------------------------------------- (28) Obligation: Pi DP problem: The TRS P consists of the following rules: U6_AAA(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) -> MERGE_IN_AAA(Xs, .(Y, Ys), Zs) MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_AAA(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y))) MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_AAA(X, Xs, Y, Ys, Zs, less_in_aa(Y, X)) U8_AAA(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) -> MERGE_IN_AAA(.(X, Xs), Ys, Zs) The TRS R consists of the following rules: ms_in_ag([], []) -> ms_out_ag([], []) ms_in_ag(.(X, []), .(X, [])) -> ms_out_ag(.(X, []), .(X, [])) ms_in_ag(.(X, .(Y, Xs)), Ys) -> U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) split_in_aaa([], [], []) -> split_out_aaa([], [], []) split_in_aaa(.(X, Xs), .(X, Ys), Zs) -> U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys)) U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) -> split_out_aaa(.(X, Xs), .(X, Ys), Zs) U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) ms_in_aa([], []) -> ms_out_aa([], []) ms_in_aa(.(X, []), .(X, [])) -> ms_out_aa(.(X, []), .(X, [])) ms_in_aa(.(X, .(Y, Xs)), Ys) -> U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys)) merge_in_aaa([], Xs, Xs) -> merge_out_aaa([], Xs, Xs) merge_in_aaa(Xs, [], Xs) -> merge_out_aaa(Xs, [], Xs) merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y))) less_in_aa(0, s(X1)) -> less_out_aa(0, s(X1)) less_in_aa(s(X), s(Y)) -> U10_aa(X, Y, less_in_aa(X, Y)) U10_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) -> U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs)) merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X)) U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) -> U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs)) U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) -> ms_out_aa(.(X, .(Y, Xs)), Ys) U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys)) merge_in_aag([], Xs, Xs) -> merge_out_aag([], Xs, Xs) merge_in_aag(Xs, [], Xs) -> merge_out_aag(Xs, [], Xs) merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y))) less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) less_in_ga(s(X), s(Y)) -> U10_ga(X, Y, less_in_ga(X, Y)) U10_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) -> U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs)) merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X)) U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) -> U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs)) U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) -> ms_out_ag(.(X, .(Y, Xs)), Ys) The argument filtering Pi contains the following mapping: ms_in_ag(x1, x2) = ms_in_ag(x2) [] = [] ms_out_ag(x1, x2) = ms_out_ag(x2) .(x1, x2) = .(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) split_in_aaa(x1, x2, x3) = split_in_aaa split_out_aaa(x1, x2, x3) = split_out_aaa U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5) U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x4, x6) ms_in_aa(x1, x2) = ms_in_aa ms_out_aa(x1, x2) = ms_out_aa U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x6) U3_aa(x1, x2, x3, x4, x5, x6) = U3_aa(x6) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x5) merge_in_aaa(x1, x2, x3) = merge_in_aaa merge_out_aaa(x1, x2, x3) = merge_out_aaa U6_aaa(x1, x2, x3, x4, x5, x6) = U6_aaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U10_aa(x1, x2, x3) = U10_aa(x3) U7_aaa(x1, x2, x3, x4, x5, x6) = U7_aaa(x6) U8_aaa(x1, x2, x3, x4, x5, x6) = U8_aaa(x6) U9_aaa(x1, x2, x3, x4, x5, x6) = U9_aaa(x6) U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x4, x6) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x4, x5) merge_in_aag(x1, x2, x3) = merge_in_aag(x3) merge_out_aag(x1, x2, x3) = merge_out_aag(x1, x2, x3) U6_aag(x1, x2, x3, x4, x5, x6) = U6_aag(x1, x5, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U10_ga(x1, x2, x3) = U10_ga(x1, x3) U7_aag(x1, x2, x3, x4, x5, x6) = U7_aag(x1, x5, x6) U8_aag(x1, x2, x3, x4, x5, x6) = U8_aag(x3, x5, x6) U9_aag(x1, x2, x3, x4, x5, x6) = U9_aag(x3, x5, x6) MERGE_IN_AAA(x1, x2, x3) = MERGE_IN_AAA U6_AAA(x1, x2, x3, x4, x5, x6) = U6_AAA(x6) U8_AAA(x1, x2, x3, x4, x5, x6) = U8_AAA(x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (29) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (30) Obligation: Pi DP problem: The TRS P consists of the following rules: U6_AAA(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) -> MERGE_IN_AAA(Xs, .(Y, Ys), Zs) MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_AAA(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y))) MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_AAA(X, Xs, Y, Ys, Zs, less_in_aa(Y, X)) U8_AAA(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) -> MERGE_IN_AAA(.(X, Xs), Ys, Zs) The TRS R consists of the following rules: less_in_aa(0, s(X1)) -> less_out_aa(0, s(X1)) less_in_aa(s(X), s(Y)) -> U10_aa(X, Y, less_in_aa(X, Y)) U10_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U10_aa(x1, x2, x3) = U10_aa(x3) 0 = 0 s(x1) = s(x1) MERGE_IN_AAA(x1, x2, x3) = MERGE_IN_AAA U6_AAA(x1, x2, x3, x4, x5, x6) = U6_AAA(x6) U8_AAA(x1, x2, x3, x4, x5, x6) = U8_AAA(x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (31) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: U6_AAA(less_out_aa(X)) -> MERGE_IN_AAA MERGE_IN_AAA -> U6_AAA(less_in_aa) MERGE_IN_AAA -> U8_AAA(less_in_aa) U8_AAA(less_out_aa(Y)) -> MERGE_IN_AAA The TRS R consists of the following rules: less_in_aa -> less_out_aa(0) less_in_aa -> U10_aa(less_in_aa) U10_aa(less_out_aa(X)) -> less_out_aa(s(X)) The set Q consists of the following terms: less_in_aa U10_aa(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (33) Obligation: Pi DP problem: The TRS P consists of the following rules: SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) -> SPLIT_IN_AAA(Xs, Zs, Ys) The TRS R consists of the following rules: ms_in_ag([], []) -> ms_out_ag([], []) ms_in_ag(.(X, []), .(X, [])) -> ms_out_ag(.(X, []), .(X, [])) ms_in_ag(.(X, .(Y, Xs)), Ys) -> U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) split_in_aaa([], [], []) -> split_out_aaa([], [], []) split_in_aaa(.(X, Xs), .(X, Ys), Zs) -> U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys)) U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) -> split_out_aaa(.(X, Xs), .(X, Ys), Zs) U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) ms_in_aa([], []) -> ms_out_aa([], []) ms_in_aa(.(X, []), .(X, [])) -> ms_out_aa(.(X, []), .(X, [])) ms_in_aa(.(X, .(Y, Xs)), Ys) -> U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys)) merge_in_aaa([], Xs, Xs) -> merge_out_aaa([], Xs, Xs) merge_in_aaa(Xs, [], Xs) -> merge_out_aaa(Xs, [], Xs) merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y))) less_in_aa(0, s(X1)) -> less_out_aa(0, s(X1)) less_in_aa(s(X), s(Y)) -> U10_aa(X, Y, less_in_aa(X, Y)) U10_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) -> U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs)) merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X)) U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) -> U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs)) U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) -> ms_out_aa(.(X, .(Y, Xs)), Ys) U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys)) merge_in_aag([], Xs, Xs) -> merge_out_aag([], Xs, Xs) merge_in_aag(Xs, [], Xs) -> merge_out_aag(Xs, [], Xs) merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y))) less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) less_in_ga(s(X), s(Y)) -> U10_ga(X, Y, less_in_ga(X, Y)) U10_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) -> U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs)) merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X)) U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) -> U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs)) U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) -> ms_out_ag(.(X, .(Y, Xs)), Ys) The argument filtering Pi contains the following mapping: ms_in_ag(x1, x2) = ms_in_ag(x2) [] = [] ms_out_ag(x1, x2) = ms_out_ag(x2) .(x1, x2) = .(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) split_in_aaa(x1, x2, x3) = split_in_aaa split_out_aaa(x1, x2, x3) = split_out_aaa U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5) U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x4, x6) ms_in_aa(x1, x2) = ms_in_aa ms_out_aa(x1, x2) = ms_out_aa U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x6) U3_aa(x1, x2, x3, x4, x5, x6) = U3_aa(x6) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x5) merge_in_aaa(x1, x2, x3) = merge_in_aaa merge_out_aaa(x1, x2, x3) = merge_out_aaa U6_aaa(x1, x2, x3, x4, x5, x6) = U6_aaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U10_aa(x1, x2, x3) = U10_aa(x3) U7_aaa(x1, x2, x3, x4, x5, x6) = U7_aaa(x6) U8_aaa(x1, x2, x3, x4, x5, x6) = U8_aaa(x6) U9_aaa(x1, x2, x3, x4, x5, x6) = U9_aaa(x6) U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x4, x6) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x4, x5) merge_in_aag(x1, x2, x3) = merge_in_aag(x3) merge_out_aag(x1, x2, x3) = merge_out_aag(x1, x2, x3) U6_aag(x1, x2, x3, x4, x5, x6) = U6_aag(x1, x5, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U10_ga(x1, x2, x3) = U10_ga(x1, x3) U7_aag(x1, x2, x3, x4, x5, x6) = U7_aag(x1, x5, x6) U8_aag(x1, x2, x3, x4, x5, x6) = U8_aag(x3, x5, x6) U9_aag(x1, x2, x3, x4, x5, x6) = U9_aag(x3, x5, x6) SPLIT_IN_AAA(x1, x2, x3) = SPLIT_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (34) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (35) Obligation: Pi DP problem: The TRS P consists of the following rules: SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) -> SPLIT_IN_AAA(Xs, Zs, Ys) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) SPLIT_IN_AAA(x1, x2, x3) = SPLIT_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (36) Obligation: Pi DP problem: The TRS P consists of the following rules: MS_IN_AA(.(X, .(Y, Xs)), Ys) -> U1_AA(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_AA(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) U2_AA(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> MS_IN_AA(X2s, Y2s) U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> MS_IN_AA(X1s, Y1s) The TRS R consists of the following rules: ms_in_ag([], []) -> ms_out_ag([], []) ms_in_ag(.(X, []), .(X, [])) -> ms_out_ag(.(X, []), .(X, [])) ms_in_ag(.(X, .(Y, Xs)), Ys) -> U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) split_in_aaa([], [], []) -> split_out_aaa([], [], []) split_in_aaa(.(X, Xs), .(X, Ys), Zs) -> U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys)) U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) -> split_out_aaa(.(X, Xs), .(X, Ys), Zs) U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) ms_in_aa([], []) -> ms_out_aa([], []) ms_in_aa(.(X, []), .(X, [])) -> ms_out_aa(.(X, []), .(X, [])) ms_in_aa(.(X, .(Y, Xs)), Ys) -> U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys)) merge_in_aaa([], Xs, Xs) -> merge_out_aaa([], Xs, Xs) merge_in_aaa(Xs, [], Xs) -> merge_out_aaa(Xs, [], Xs) merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y))) less_in_aa(0, s(X1)) -> less_out_aa(0, s(X1)) less_in_aa(s(X), s(Y)) -> U10_aa(X, Y, less_in_aa(X, Y)) U10_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) -> U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs)) merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X)) U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) -> U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs)) U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) -> ms_out_aa(.(X, .(Y, Xs)), Ys) U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys)) merge_in_aag([], Xs, Xs) -> merge_out_aag([], Xs, Xs) merge_in_aag(Xs, [], Xs) -> merge_out_aag(Xs, [], Xs) merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y))) less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) less_in_ga(s(X), s(Y)) -> U10_ga(X, Y, less_in_ga(X, Y)) U10_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) -> U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs)) merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X)) U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) -> U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs)) U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) -> ms_out_ag(.(X, .(Y, Xs)), Ys) The argument filtering Pi contains the following mapping: ms_in_ag(x1, x2) = ms_in_ag(x2) [] = [] ms_out_ag(x1, x2) = ms_out_ag(x2) .(x1, x2) = .(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) split_in_aaa(x1, x2, x3) = split_in_aaa split_out_aaa(x1, x2, x3) = split_out_aaa U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5) U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x4, x6) ms_in_aa(x1, x2) = ms_in_aa ms_out_aa(x1, x2) = ms_out_aa U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x6) U3_aa(x1, x2, x3, x4, x5, x6) = U3_aa(x6) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x5) merge_in_aaa(x1, x2, x3) = merge_in_aaa merge_out_aaa(x1, x2, x3) = merge_out_aaa U6_aaa(x1, x2, x3, x4, x5, x6) = U6_aaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U10_aa(x1, x2, x3) = U10_aa(x3) U7_aaa(x1, x2, x3, x4, x5, x6) = U7_aaa(x6) U8_aaa(x1, x2, x3, x4, x5, x6) = U8_aaa(x6) U9_aaa(x1, x2, x3, x4, x5, x6) = U9_aaa(x6) U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x4, x6) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x4, x5) merge_in_aag(x1, x2, x3) = merge_in_aag(x3) merge_out_aag(x1, x2, x3) = merge_out_aag(x1, x2, x3) U6_aag(x1, x2, x3, x4, x5, x6) = U6_aag(x1, x5, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U10_ga(x1, x2, x3) = U10_ga(x1, x3) U7_aag(x1, x2, x3, x4, x5, x6) = U7_aag(x1, x5, x6) U8_aag(x1, x2, x3, x4, x5, x6) = U8_aag(x3, x5, x6) U9_aag(x1, x2, x3, x4, x5, x6) = U9_aag(x3, x5, x6) MS_IN_AA(x1, x2) = MS_IN_AA U1_AA(x1, x2, x3, x4, x5) = U1_AA(x5) U2_AA(x1, x2, x3, x4, x5, x6) = U2_AA(x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (37) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (38) Obligation: Pi DP problem: The TRS P consists of the following rules: MS_IN_AA(.(X, .(Y, Xs)), Ys) -> U1_AA(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_AA(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) U2_AA(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> MS_IN_AA(X2s, Y2s) U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> MS_IN_AA(X1s, Y1s) The TRS R consists of the following rules: split_in_aaa(.(X, Xs), .(X, Ys), Zs) -> U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys)) ms_in_aa([], []) -> ms_out_aa([], []) ms_in_aa(.(X, []), .(X, [])) -> ms_out_aa(.(X, []), .(X, [])) ms_in_aa(.(X, .(Y, Xs)), Ys) -> U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) -> split_out_aaa(.(X, Xs), .(X, Ys), Zs) U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) split_in_aaa([], [], []) -> split_out_aaa([], [], []) U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys)) U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) -> ms_out_aa(.(X, .(Y, Xs)), Ys) merge_in_aaa([], Xs, Xs) -> merge_out_aaa([], Xs, Xs) merge_in_aaa(Xs, [], Xs) -> merge_out_aaa(Xs, [], Xs) merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y))) merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X)) U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) -> U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs)) U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) -> U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs)) less_in_aa(0, s(X1)) -> less_out_aa(0, s(X1)) less_in_aa(s(X), s(Y)) -> U10_aa(X, Y, less_in_aa(X, Y)) U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) U10_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) The argument filtering Pi contains the following mapping: [] = [] .(x1, x2) = .(x1, x2) split_in_aaa(x1, x2, x3) = split_in_aaa split_out_aaa(x1, x2, x3) = split_out_aaa U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5) ms_in_aa(x1, x2) = ms_in_aa ms_out_aa(x1, x2) = ms_out_aa U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x6) U3_aa(x1, x2, x3, x4, x5, x6) = U3_aa(x6) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x5) merge_in_aaa(x1, x2, x3) = merge_in_aaa merge_out_aaa(x1, x2, x3) = merge_out_aaa U6_aaa(x1, x2, x3, x4, x5, x6) = U6_aaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U10_aa(x1, x2, x3) = U10_aa(x3) U7_aaa(x1, x2, x3, x4, x5, x6) = U7_aaa(x6) U8_aaa(x1, x2, x3, x4, x5, x6) = U8_aaa(x6) U9_aaa(x1, x2, x3, x4, x5, x6) = U9_aaa(x6) 0 = 0 s(x1) = s(x1) MS_IN_AA(x1, x2) = MS_IN_AA U1_AA(x1, x2, x3, x4, x5) = U1_AA(x5) U2_AA(x1, x2, x3, x4, x5, x6) = U2_AA(x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (39) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: ms_in_2: (f,b) (f,f) split_in_3: (f,f,f) merge_in_3: (f,f,f) (f,f,b) less_in_2: (f,f) (b,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: ms_in_ag([], []) -> ms_out_ag([], []) ms_in_ag(.(X, []), .(X, [])) -> ms_out_ag(.(X, []), .(X, [])) ms_in_ag(.(X, .(Y, Xs)), Ys) -> U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) split_in_aaa([], [], []) -> split_out_aaa([], [], []) split_in_aaa(.(X, Xs), .(X, Ys), Zs) -> U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys)) U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) -> split_out_aaa(.(X, Xs), .(X, Ys), Zs) U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) ms_in_aa([], []) -> ms_out_aa([], []) ms_in_aa(.(X, []), .(X, [])) -> ms_out_aa(.(X, []), .(X, [])) ms_in_aa(.(X, .(Y, Xs)), Ys) -> U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys)) merge_in_aaa([], Xs, Xs) -> merge_out_aaa([], Xs, Xs) merge_in_aaa(Xs, [], Xs) -> merge_out_aaa(Xs, [], Xs) merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y))) less_in_aa(0, s(X1)) -> less_out_aa(0, s(X1)) less_in_aa(s(X), s(Y)) -> U10_aa(X, Y, less_in_aa(X, Y)) U10_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) -> U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs)) merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X)) U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) -> U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs)) U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) -> ms_out_aa(.(X, .(Y, Xs)), Ys) U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys)) merge_in_aag([], Xs, Xs) -> merge_out_aag([], Xs, Xs) merge_in_aag(Xs, [], Xs) -> merge_out_aag(Xs, [], Xs) merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y))) less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) less_in_ga(s(X), s(Y)) -> U10_ga(X, Y, less_in_ga(X, Y)) U10_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) -> U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs)) merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X)) U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) -> U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs)) U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) -> ms_out_ag(.(X, .(Y, Xs)), Ys) The argument filtering Pi contains the following mapping: ms_in_ag(x1, x2) = ms_in_ag(x2) [] = [] ms_out_ag(x1, x2) = ms_out_ag .(x1, x2) = .(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) split_in_aaa(x1, x2, x3) = split_in_aaa split_out_aaa(x1, x2, x3) = split_out_aaa U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5) U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x4, x6) ms_in_aa(x1, x2) = ms_in_aa ms_out_aa(x1, x2) = ms_out_aa U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x6) U3_aa(x1, x2, x3, x4, x5, x6) = U3_aa(x6) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x5) merge_in_aaa(x1, x2, x3) = merge_in_aaa merge_out_aaa(x1, x2, x3) = merge_out_aaa U6_aaa(x1, x2, x3, x4, x5, x6) = U6_aaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U10_aa(x1, x2, x3) = U10_aa(x3) U7_aaa(x1, x2, x3, x4, x5, x6) = U7_aaa(x6) U8_aaa(x1, x2, x3, x4, x5, x6) = U8_aaa(x6) U9_aaa(x1, x2, x3, x4, x5, x6) = U9_aaa(x6) U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x4, x6) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x5) merge_in_aag(x1, x2, x3) = merge_in_aag(x3) merge_out_aag(x1, x2, x3) = merge_out_aag(x1, x2) U6_aag(x1, x2, x3, x4, x5, x6) = U6_aag(x1, x5, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga s(x1) = s(x1) U10_ga(x1, x2, x3) = U10_ga(x3) U7_aag(x1, x2, x3, x4, x5, x6) = U7_aag(x1, x6) U8_aag(x1, x2, x3, x4, x5, x6) = U8_aag(x3, x5, x6) U9_aag(x1, x2, x3, x4, x5, x6) = U9_aag(x3, x6) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (40) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: ms_in_ag([], []) -> ms_out_ag([], []) ms_in_ag(.(X, []), .(X, [])) -> ms_out_ag(.(X, []), .(X, [])) ms_in_ag(.(X, .(Y, Xs)), Ys) -> U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) split_in_aaa([], [], []) -> split_out_aaa([], [], []) split_in_aaa(.(X, Xs), .(X, Ys), Zs) -> U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys)) U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) -> split_out_aaa(.(X, Xs), .(X, Ys), Zs) U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) ms_in_aa([], []) -> ms_out_aa([], []) ms_in_aa(.(X, []), .(X, [])) -> ms_out_aa(.(X, []), .(X, [])) ms_in_aa(.(X, .(Y, Xs)), Ys) -> U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys)) merge_in_aaa([], Xs, Xs) -> merge_out_aaa([], Xs, Xs) merge_in_aaa(Xs, [], Xs) -> merge_out_aaa(Xs, [], Xs) merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y))) less_in_aa(0, s(X1)) -> less_out_aa(0, s(X1)) less_in_aa(s(X), s(Y)) -> U10_aa(X, Y, less_in_aa(X, Y)) U10_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) -> U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs)) merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X)) U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) -> U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs)) U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) -> ms_out_aa(.(X, .(Y, Xs)), Ys) U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys)) merge_in_aag([], Xs, Xs) -> merge_out_aag([], Xs, Xs) merge_in_aag(Xs, [], Xs) -> merge_out_aag(Xs, [], Xs) merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y))) less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) less_in_ga(s(X), s(Y)) -> U10_ga(X, Y, less_in_ga(X, Y)) U10_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) -> U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs)) merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X)) U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) -> U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs)) U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) -> ms_out_ag(.(X, .(Y, Xs)), Ys) The argument filtering Pi contains the following mapping: ms_in_ag(x1, x2) = ms_in_ag(x2) [] = [] ms_out_ag(x1, x2) = ms_out_ag .(x1, x2) = .(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) split_in_aaa(x1, x2, x3) = split_in_aaa split_out_aaa(x1, x2, x3) = split_out_aaa U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5) U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x4, x6) ms_in_aa(x1, x2) = ms_in_aa ms_out_aa(x1, x2) = ms_out_aa U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x6) U3_aa(x1, x2, x3, x4, x5, x6) = U3_aa(x6) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x5) merge_in_aaa(x1, x2, x3) = merge_in_aaa merge_out_aaa(x1, x2, x3) = merge_out_aaa U6_aaa(x1, x2, x3, x4, x5, x6) = U6_aaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U10_aa(x1, x2, x3) = U10_aa(x3) U7_aaa(x1, x2, x3, x4, x5, x6) = U7_aaa(x6) U8_aaa(x1, x2, x3, x4, x5, x6) = U8_aaa(x6) U9_aaa(x1, x2, x3, x4, x5, x6) = U9_aaa(x6) U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x4, x6) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x5) merge_in_aag(x1, x2, x3) = merge_in_aag(x3) merge_out_aag(x1, x2, x3) = merge_out_aag(x1, x2) U6_aag(x1, x2, x3, x4, x5, x6) = U6_aag(x1, x5, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga s(x1) = s(x1) U10_ga(x1, x2, x3) = U10_ga(x3) U7_aag(x1, x2, x3, x4, x5, x6) = U7_aag(x1, x6) U8_aag(x1, x2, x3, x4, x5, x6) = U8_aag(x3, x5, x6) U9_aag(x1, x2, x3, x4, x5, x6) = U9_aag(x3, x6) ---------------------------------------- (41) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: Pi DP problem: The TRS P consists of the following rules: MS_IN_AG(.(X, .(Y, Xs)), Ys) -> U1_AG(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) MS_IN_AG(.(X, .(Y, Xs)), Ys) -> SPLIT_IN_AAA(.(X, .(Y, Xs)), X1s, X2s) SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) -> U5_AAA(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys)) SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) -> SPLIT_IN_AAA(Xs, Zs, Ys) U1_AG(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_AG(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) U1_AG(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> MS_IN_AA(X1s, Y1s) MS_IN_AA(.(X, .(Y, Xs)), Ys) -> U1_AA(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) MS_IN_AA(.(X, .(Y, Xs)), Ys) -> SPLIT_IN_AAA(.(X, .(Y, Xs)), X1s, X2s) U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_AA(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> MS_IN_AA(X1s, Y1s) U2_AA(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_AA(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U2_AA(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> MS_IN_AA(X2s, Y2s) U3_AA(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_AA(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys)) U3_AA(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> MERGE_IN_AAA(Y1s, Y2s, Ys) MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_AAA(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y))) MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(X, Zs)) -> LESS_IN_AA(X, s(Y)) LESS_IN_AA(s(X), s(Y)) -> U10_AA(X, Y, less_in_aa(X, Y)) LESS_IN_AA(s(X), s(Y)) -> LESS_IN_AA(X, Y) U6_AAA(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) -> U7_AAA(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs)) U6_AAA(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) -> MERGE_IN_AAA(Xs, .(Y, Ys), Zs) MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_AAA(X, Xs, Y, Ys, Zs, less_in_aa(Y, X)) MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> LESS_IN_AA(Y, X) U8_AAA(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) -> U9_AAA(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs)) U8_AAA(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) -> MERGE_IN_AAA(.(X, Xs), Ys, Zs) U2_AG(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_AG(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U2_AG(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> MS_IN_AA(X2s, Y2s) U3_AG(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_AG(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys)) U3_AG(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> MERGE_IN_AAG(Y1s, Y2s, Ys) MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_AAG(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y))) MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(X, Zs)) -> LESS_IN_GA(X, s(Y)) LESS_IN_GA(s(X), s(Y)) -> U10_GA(X, Y, less_in_ga(X, Y)) LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) U6_AAG(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) -> U7_AAG(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs)) U6_AAG(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) -> MERGE_IN_AAG(Xs, .(Y, Ys), Zs) MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_AAG(X, Xs, Y, Ys, Zs, less_in_ga(Y, X)) MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> LESS_IN_GA(Y, X) U8_AAG(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) -> U9_AAG(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs)) U8_AAG(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) -> MERGE_IN_AAG(.(X, Xs), Ys, Zs) The TRS R consists of the following rules: ms_in_ag([], []) -> ms_out_ag([], []) ms_in_ag(.(X, []), .(X, [])) -> ms_out_ag(.(X, []), .(X, [])) ms_in_ag(.(X, .(Y, Xs)), Ys) -> U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) split_in_aaa([], [], []) -> split_out_aaa([], [], []) split_in_aaa(.(X, Xs), .(X, Ys), Zs) -> U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys)) U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) -> split_out_aaa(.(X, Xs), .(X, Ys), Zs) U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) ms_in_aa([], []) -> ms_out_aa([], []) ms_in_aa(.(X, []), .(X, [])) -> ms_out_aa(.(X, []), .(X, [])) ms_in_aa(.(X, .(Y, Xs)), Ys) -> U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys)) merge_in_aaa([], Xs, Xs) -> merge_out_aaa([], Xs, Xs) merge_in_aaa(Xs, [], Xs) -> merge_out_aaa(Xs, [], Xs) merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y))) less_in_aa(0, s(X1)) -> less_out_aa(0, s(X1)) less_in_aa(s(X), s(Y)) -> U10_aa(X, Y, less_in_aa(X, Y)) U10_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) -> U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs)) merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X)) U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) -> U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs)) U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) -> ms_out_aa(.(X, .(Y, Xs)), Ys) U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys)) merge_in_aag([], Xs, Xs) -> merge_out_aag([], Xs, Xs) merge_in_aag(Xs, [], Xs) -> merge_out_aag(Xs, [], Xs) merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y))) less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) less_in_ga(s(X), s(Y)) -> U10_ga(X, Y, less_in_ga(X, Y)) U10_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) -> U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs)) merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X)) U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) -> U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs)) U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) -> ms_out_ag(.(X, .(Y, Xs)), Ys) The argument filtering Pi contains the following mapping: ms_in_ag(x1, x2) = ms_in_ag(x2) [] = [] ms_out_ag(x1, x2) = ms_out_ag .(x1, x2) = .(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) split_in_aaa(x1, x2, x3) = split_in_aaa split_out_aaa(x1, x2, x3) = split_out_aaa U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5) U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x4, x6) ms_in_aa(x1, x2) = ms_in_aa ms_out_aa(x1, x2) = ms_out_aa U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x6) U3_aa(x1, x2, x3, x4, x5, x6) = U3_aa(x6) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x5) merge_in_aaa(x1, x2, x3) = merge_in_aaa merge_out_aaa(x1, x2, x3) = merge_out_aaa U6_aaa(x1, x2, x3, x4, x5, x6) = U6_aaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U10_aa(x1, x2, x3) = U10_aa(x3) U7_aaa(x1, x2, x3, x4, x5, x6) = U7_aaa(x6) U8_aaa(x1, x2, x3, x4, x5, x6) = U8_aaa(x6) U9_aaa(x1, x2, x3, x4, x5, x6) = U9_aaa(x6) U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x4, x6) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x5) merge_in_aag(x1, x2, x3) = merge_in_aag(x3) merge_out_aag(x1, x2, x3) = merge_out_aag(x1, x2) U6_aag(x1, x2, x3, x4, x5, x6) = U6_aag(x1, x5, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga s(x1) = s(x1) U10_ga(x1, x2, x3) = U10_ga(x3) U7_aag(x1, x2, x3, x4, x5, x6) = U7_aag(x1, x6) U8_aag(x1, x2, x3, x4, x5, x6) = U8_aag(x3, x5, x6) U9_aag(x1, x2, x3, x4, x5, x6) = U9_aag(x3, x6) MS_IN_AG(x1, x2) = MS_IN_AG(x2) U1_AG(x1, x2, x3, x4, x5) = U1_AG(x4, x5) SPLIT_IN_AAA(x1, x2, x3) = SPLIT_IN_AAA U5_AAA(x1, x2, x3, x4, x5) = U5_AAA(x5) U2_AG(x1, x2, x3, x4, x5, x6) = U2_AG(x4, x6) MS_IN_AA(x1, x2) = MS_IN_AA U1_AA(x1, x2, x3, x4, x5) = U1_AA(x5) U2_AA(x1, x2, x3, x4, x5, x6) = U2_AA(x6) U3_AA(x1, x2, x3, x4, x5, x6) = U3_AA(x6) U4_AA(x1, x2, x3, x4, x5) = U4_AA(x5) MERGE_IN_AAA(x1, x2, x3) = MERGE_IN_AAA U6_AAA(x1, x2, x3, x4, x5, x6) = U6_AAA(x6) LESS_IN_AA(x1, x2) = LESS_IN_AA U10_AA(x1, x2, x3) = U10_AA(x3) U7_AAA(x1, x2, x3, x4, x5, x6) = U7_AAA(x6) U8_AAA(x1, x2, x3, x4, x5, x6) = U8_AAA(x6) U9_AAA(x1, x2, x3, x4, x5, x6) = U9_AAA(x6) U3_AG(x1, x2, x3, x4, x5, x6) = U3_AG(x4, x6) U4_AG(x1, x2, x3, x4, x5) = U4_AG(x5) MERGE_IN_AAG(x1, x2, x3) = MERGE_IN_AAG(x3) U6_AAG(x1, x2, x3, x4, x5, x6) = U6_AAG(x1, x5, x6) LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) U10_GA(x1, x2, x3) = U10_GA(x3) U7_AAG(x1, x2, x3, x4, x5, x6) = U7_AAG(x1, x6) U8_AAG(x1, x2, x3, x4, x5, x6) = U8_AAG(x3, x5, x6) U9_AAG(x1, x2, x3, x4, x5, x6) = U9_AAG(x3, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (42) Obligation: Pi DP problem: The TRS P consists of the following rules: MS_IN_AG(.(X, .(Y, Xs)), Ys) -> U1_AG(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) MS_IN_AG(.(X, .(Y, Xs)), Ys) -> SPLIT_IN_AAA(.(X, .(Y, Xs)), X1s, X2s) SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) -> U5_AAA(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys)) SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) -> SPLIT_IN_AAA(Xs, Zs, Ys) U1_AG(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_AG(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) U1_AG(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> MS_IN_AA(X1s, Y1s) MS_IN_AA(.(X, .(Y, Xs)), Ys) -> U1_AA(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) MS_IN_AA(.(X, .(Y, Xs)), Ys) -> SPLIT_IN_AAA(.(X, .(Y, Xs)), X1s, X2s) U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_AA(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> MS_IN_AA(X1s, Y1s) U2_AA(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_AA(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U2_AA(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> MS_IN_AA(X2s, Y2s) U3_AA(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_AA(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys)) U3_AA(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> MERGE_IN_AAA(Y1s, Y2s, Ys) MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_AAA(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y))) MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(X, Zs)) -> LESS_IN_AA(X, s(Y)) LESS_IN_AA(s(X), s(Y)) -> U10_AA(X, Y, less_in_aa(X, Y)) LESS_IN_AA(s(X), s(Y)) -> LESS_IN_AA(X, Y) U6_AAA(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) -> U7_AAA(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs)) U6_AAA(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) -> MERGE_IN_AAA(Xs, .(Y, Ys), Zs) MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_AAA(X, Xs, Y, Ys, Zs, less_in_aa(Y, X)) MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> LESS_IN_AA(Y, X) U8_AAA(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) -> U9_AAA(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs)) U8_AAA(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) -> MERGE_IN_AAA(.(X, Xs), Ys, Zs) U2_AG(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_AG(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U2_AG(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> MS_IN_AA(X2s, Y2s) U3_AG(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_AG(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys)) U3_AG(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> MERGE_IN_AAG(Y1s, Y2s, Ys) MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_AAG(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y))) MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(X, Zs)) -> LESS_IN_GA(X, s(Y)) LESS_IN_GA(s(X), s(Y)) -> U10_GA(X, Y, less_in_ga(X, Y)) LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) U6_AAG(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) -> U7_AAG(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs)) U6_AAG(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) -> MERGE_IN_AAG(Xs, .(Y, Ys), Zs) MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_AAG(X, Xs, Y, Ys, Zs, less_in_ga(Y, X)) MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> LESS_IN_GA(Y, X) U8_AAG(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) -> U9_AAG(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs)) U8_AAG(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) -> MERGE_IN_AAG(.(X, Xs), Ys, Zs) The TRS R consists of the following rules: ms_in_ag([], []) -> ms_out_ag([], []) ms_in_ag(.(X, []), .(X, [])) -> ms_out_ag(.(X, []), .(X, [])) ms_in_ag(.(X, .(Y, Xs)), Ys) -> U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) split_in_aaa([], [], []) -> split_out_aaa([], [], []) split_in_aaa(.(X, Xs), .(X, Ys), Zs) -> U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys)) U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) -> split_out_aaa(.(X, Xs), .(X, Ys), Zs) U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) ms_in_aa([], []) -> ms_out_aa([], []) ms_in_aa(.(X, []), .(X, [])) -> ms_out_aa(.(X, []), .(X, [])) ms_in_aa(.(X, .(Y, Xs)), Ys) -> U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys)) merge_in_aaa([], Xs, Xs) -> merge_out_aaa([], Xs, Xs) merge_in_aaa(Xs, [], Xs) -> merge_out_aaa(Xs, [], Xs) merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y))) less_in_aa(0, s(X1)) -> less_out_aa(0, s(X1)) less_in_aa(s(X), s(Y)) -> U10_aa(X, Y, less_in_aa(X, Y)) U10_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) -> U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs)) merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X)) U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) -> U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs)) U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) -> ms_out_aa(.(X, .(Y, Xs)), Ys) U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys)) merge_in_aag([], Xs, Xs) -> merge_out_aag([], Xs, Xs) merge_in_aag(Xs, [], Xs) -> merge_out_aag(Xs, [], Xs) merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y))) less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) less_in_ga(s(X), s(Y)) -> U10_ga(X, Y, less_in_ga(X, Y)) U10_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) -> U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs)) merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X)) U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) -> U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs)) U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) -> ms_out_ag(.(X, .(Y, Xs)), Ys) The argument filtering Pi contains the following mapping: ms_in_ag(x1, x2) = ms_in_ag(x2) [] = [] ms_out_ag(x1, x2) = ms_out_ag .(x1, x2) = .(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) split_in_aaa(x1, x2, x3) = split_in_aaa split_out_aaa(x1, x2, x3) = split_out_aaa U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5) U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x4, x6) ms_in_aa(x1, x2) = ms_in_aa ms_out_aa(x1, x2) = ms_out_aa U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x6) U3_aa(x1, x2, x3, x4, x5, x6) = U3_aa(x6) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x5) merge_in_aaa(x1, x2, x3) = merge_in_aaa merge_out_aaa(x1, x2, x3) = merge_out_aaa U6_aaa(x1, x2, x3, x4, x5, x6) = U6_aaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U10_aa(x1, x2, x3) = U10_aa(x3) U7_aaa(x1, x2, x3, x4, x5, x6) = U7_aaa(x6) U8_aaa(x1, x2, x3, x4, x5, x6) = U8_aaa(x6) U9_aaa(x1, x2, x3, x4, x5, x6) = U9_aaa(x6) U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x4, x6) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x5) merge_in_aag(x1, x2, x3) = merge_in_aag(x3) merge_out_aag(x1, x2, x3) = merge_out_aag(x1, x2) U6_aag(x1, x2, x3, x4, x5, x6) = U6_aag(x1, x5, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga s(x1) = s(x1) U10_ga(x1, x2, x3) = U10_ga(x3) U7_aag(x1, x2, x3, x4, x5, x6) = U7_aag(x1, x6) U8_aag(x1, x2, x3, x4, x5, x6) = U8_aag(x3, x5, x6) U9_aag(x1, x2, x3, x4, x5, x6) = U9_aag(x3, x6) MS_IN_AG(x1, x2) = MS_IN_AG(x2) U1_AG(x1, x2, x3, x4, x5) = U1_AG(x4, x5) SPLIT_IN_AAA(x1, x2, x3) = SPLIT_IN_AAA U5_AAA(x1, x2, x3, x4, x5) = U5_AAA(x5) U2_AG(x1, x2, x3, x4, x5, x6) = U2_AG(x4, x6) MS_IN_AA(x1, x2) = MS_IN_AA U1_AA(x1, x2, x3, x4, x5) = U1_AA(x5) U2_AA(x1, x2, x3, x4, x5, x6) = U2_AA(x6) U3_AA(x1, x2, x3, x4, x5, x6) = U3_AA(x6) U4_AA(x1, x2, x3, x4, x5) = U4_AA(x5) MERGE_IN_AAA(x1, x2, x3) = MERGE_IN_AAA U6_AAA(x1, x2, x3, x4, x5, x6) = U6_AAA(x6) LESS_IN_AA(x1, x2) = LESS_IN_AA U10_AA(x1, x2, x3) = U10_AA(x3) U7_AAA(x1, x2, x3, x4, x5, x6) = U7_AAA(x6) U8_AAA(x1, x2, x3, x4, x5, x6) = U8_AAA(x6) U9_AAA(x1, x2, x3, x4, x5, x6) = U9_AAA(x6) U3_AG(x1, x2, x3, x4, x5, x6) = U3_AG(x4, x6) U4_AG(x1, x2, x3, x4, x5) = U4_AG(x5) MERGE_IN_AAG(x1, x2, x3) = MERGE_IN_AAG(x3) U6_AAG(x1, x2, x3, x4, x5, x6) = U6_AAG(x1, x5, x6) LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) U10_GA(x1, x2, x3) = U10_GA(x3) U7_AAG(x1, x2, x3, x4, x5, x6) = U7_AAG(x1, x6) U8_AAG(x1, x2, x3, x4, x5, x6) = U8_AAG(x3, x5, x6) U9_AAG(x1, x2, x3, x4, x5, x6) = U9_AAG(x3, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (43) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 23 less nodes. ---------------------------------------- (44) Complex Obligation (AND) ---------------------------------------- (45) Obligation: Pi DP problem: The TRS P consists of the following rules: LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) The TRS R consists of the following rules: ms_in_ag([], []) -> ms_out_ag([], []) ms_in_ag(.(X, []), .(X, [])) -> ms_out_ag(.(X, []), .(X, [])) ms_in_ag(.(X, .(Y, Xs)), Ys) -> U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) split_in_aaa([], [], []) -> split_out_aaa([], [], []) split_in_aaa(.(X, Xs), .(X, Ys), Zs) -> U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys)) U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) -> split_out_aaa(.(X, Xs), .(X, Ys), Zs) U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) ms_in_aa([], []) -> ms_out_aa([], []) ms_in_aa(.(X, []), .(X, [])) -> ms_out_aa(.(X, []), .(X, [])) ms_in_aa(.(X, .(Y, Xs)), Ys) -> U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys)) merge_in_aaa([], Xs, Xs) -> merge_out_aaa([], Xs, Xs) merge_in_aaa(Xs, [], Xs) -> merge_out_aaa(Xs, [], Xs) merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y))) less_in_aa(0, s(X1)) -> less_out_aa(0, s(X1)) less_in_aa(s(X), s(Y)) -> U10_aa(X, Y, less_in_aa(X, Y)) U10_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) -> U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs)) merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X)) U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) -> U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs)) U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) -> ms_out_aa(.(X, .(Y, Xs)), Ys) U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys)) merge_in_aag([], Xs, Xs) -> merge_out_aag([], Xs, Xs) merge_in_aag(Xs, [], Xs) -> merge_out_aag(Xs, [], Xs) merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y))) less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) less_in_ga(s(X), s(Y)) -> U10_ga(X, Y, less_in_ga(X, Y)) U10_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) -> U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs)) merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X)) U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) -> U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs)) U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) -> ms_out_ag(.(X, .(Y, Xs)), Ys) The argument filtering Pi contains the following mapping: ms_in_ag(x1, x2) = ms_in_ag(x2) [] = [] ms_out_ag(x1, x2) = ms_out_ag .(x1, x2) = .(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) split_in_aaa(x1, x2, x3) = split_in_aaa split_out_aaa(x1, x2, x3) = split_out_aaa U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5) U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x4, x6) ms_in_aa(x1, x2) = ms_in_aa ms_out_aa(x1, x2) = ms_out_aa U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x6) U3_aa(x1, x2, x3, x4, x5, x6) = U3_aa(x6) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x5) merge_in_aaa(x1, x2, x3) = merge_in_aaa merge_out_aaa(x1, x2, x3) = merge_out_aaa U6_aaa(x1, x2, x3, x4, x5, x6) = U6_aaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U10_aa(x1, x2, x3) = U10_aa(x3) U7_aaa(x1, x2, x3, x4, x5, x6) = U7_aaa(x6) U8_aaa(x1, x2, x3, x4, x5, x6) = U8_aaa(x6) U9_aaa(x1, x2, x3, x4, x5, x6) = U9_aaa(x6) U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x4, x6) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x5) merge_in_aag(x1, x2, x3) = merge_in_aag(x3) merge_out_aag(x1, x2, x3) = merge_out_aag(x1, x2) U6_aag(x1, x2, x3, x4, x5, x6) = U6_aag(x1, x5, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga s(x1) = s(x1) U10_ga(x1, x2, x3) = U10_ga(x3) U7_aag(x1, x2, x3, x4, x5, x6) = U7_aag(x1, x6) U8_aag(x1, x2, x3, x4, x5, x6) = U8_aag(x3, x5, x6) U9_aag(x1, x2, x3, x4, x5, x6) = U9_aag(x3, x6) LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (46) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (47) Obligation: Pi DP problem: The TRS P consists of the following rules: LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (48) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (49) Obligation: Q DP problem: The TRS P consists of the following rules: LESS_IN_GA(s(X)) -> LESS_IN_GA(X) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (50) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *LESS_IN_GA(s(X)) -> LESS_IN_GA(X) The graph contains the following edges 1 > 1 ---------------------------------------- (51) YES ---------------------------------------- (52) Obligation: Pi DP problem: The TRS P consists of the following rules: U6_AAG(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) -> MERGE_IN_AAG(Xs, .(Y, Ys), Zs) MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_AAG(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y))) MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_AAG(X, Xs, Y, Ys, Zs, less_in_ga(Y, X)) U8_AAG(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) -> MERGE_IN_AAG(.(X, Xs), Ys, Zs) The TRS R consists of the following rules: ms_in_ag([], []) -> ms_out_ag([], []) ms_in_ag(.(X, []), .(X, [])) -> ms_out_ag(.(X, []), .(X, [])) ms_in_ag(.(X, .(Y, Xs)), Ys) -> U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) split_in_aaa([], [], []) -> split_out_aaa([], [], []) split_in_aaa(.(X, Xs), .(X, Ys), Zs) -> U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys)) U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) -> split_out_aaa(.(X, Xs), .(X, Ys), Zs) U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) ms_in_aa([], []) -> ms_out_aa([], []) ms_in_aa(.(X, []), .(X, [])) -> ms_out_aa(.(X, []), .(X, [])) ms_in_aa(.(X, .(Y, Xs)), Ys) -> U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys)) merge_in_aaa([], Xs, Xs) -> merge_out_aaa([], Xs, Xs) merge_in_aaa(Xs, [], Xs) -> merge_out_aaa(Xs, [], Xs) merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y))) less_in_aa(0, s(X1)) -> less_out_aa(0, s(X1)) less_in_aa(s(X), s(Y)) -> U10_aa(X, Y, less_in_aa(X, Y)) U10_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) -> U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs)) merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X)) U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) -> U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs)) U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) -> ms_out_aa(.(X, .(Y, Xs)), Ys) U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys)) merge_in_aag([], Xs, Xs) -> merge_out_aag([], Xs, Xs) merge_in_aag(Xs, [], Xs) -> merge_out_aag(Xs, [], Xs) merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y))) less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) less_in_ga(s(X), s(Y)) -> U10_ga(X, Y, less_in_ga(X, Y)) U10_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) -> U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs)) merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X)) U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) -> U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs)) U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) -> ms_out_ag(.(X, .(Y, Xs)), Ys) The argument filtering Pi contains the following mapping: ms_in_ag(x1, x2) = ms_in_ag(x2) [] = [] ms_out_ag(x1, x2) = ms_out_ag .(x1, x2) = .(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) split_in_aaa(x1, x2, x3) = split_in_aaa split_out_aaa(x1, x2, x3) = split_out_aaa U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5) U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x4, x6) ms_in_aa(x1, x2) = ms_in_aa ms_out_aa(x1, x2) = ms_out_aa U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x6) U3_aa(x1, x2, x3, x4, x5, x6) = U3_aa(x6) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x5) merge_in_aaa(x1, x2, x3) = merge_in_aaa merge_out_aaa(x1, x2, x3) = merge_out_aaa U6_aaa(x1, x2, x3, x4, x5, x6) = U6_aaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U10_aa(x1, x2, x3) = U10_aa(x3) U7_aaa(x1, x2, x3, x4, x5, x6) = U7_aaa(x6) U8_aaa(x1, x2, x3, x4, x5, x6) = U8_aaa(x6) U9_aaa(x1, x2, x3, x4, x5, x6) = U9_aaa(x6) U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x4, x6) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x5) merge_in_aag(x1, x2, x3) = merge_in_aag(x3) merge_out_aag(x1, x2, x3) = merge_out_aag(x1, x2) U6_aag(x1, x2, x3, x4, x5, x6) = U6_aag(x1, x5, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga s(x1) = s(x1) U10_ga(x1, x2, x3) = U10_ga(x3) U7_aag(x1, x2, x3, x4, x5, x6) = U7_aag(x1, x6) U8_aag(x1, x2, x3, x4, x5, x6) = U8_aag(x3, x5, x6) U9_aag(x1, x2, x3, x4, x5, x6) = U9_aag(x3, x6) MERGE_IN_AAG(x1, x2, x3) = MERGE_IN_AAG(x3) U6_AAG(x1, x2, x3, x4, x5, x6) = U6_AAG(x1, x5, x6) U8_AAG(x1, x2, x3, x4, x5, x6) = U8_AAG(x3, x5, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (53) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (54) Obligation: Pi DP problem: The TRS P consists of the following rules: U6_AAG(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) -> MERGE_IN_AAG(Xs, .(Y, Ys), Zs) MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_AAG(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y))) MERGE_IN_AAG(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_AAG(X, Xs, Y, Ys, Zs, less_in_ga(Y, X)) U8_AAG(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) -> MERGE_IN_AAG(.(X, Xs), Ys, Zs) The TRS R consists of the following rules: less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) less_in_ga(s(X), s(Y)) -> U10_ga(X, Y, less_in_ga(X, Y)) U10_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga s(x1) = s(x1) U10_ga(x1, x2, x3) = U10_ga(x3) MERGE_IN_AAG(x1, x2, x3) = MERGE_IN_AAG(x3) U6_AAG(x1, x2, x3, x4, x5, x6) = U6_AAG(x1, x5, x6) U8_AAG(x1, x2, x3, x4, x5, x6) = U8_AAG(x3, x5, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (55) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (56) Obligation: Q DP problem: The TRS P consists of the following rules: U6_AAG(X, Zs, less_out_ga) -> MERGE_IN_AAG(Zs) MERGE_IN_AAG(.(X, Zs)) -> U6_AAG(X, Zs, less_in_ga(X)) MERGE_IN_AAG(.(Y, Zs)) -> U8_AAG(Y, Zs, less_in_ga(Y)) U8_AAG(Y, Zs, less_out_ga) -> MERGE_IN_AAG(Zs) The TRS R consists of the following rules: less_in_ga(0) -> less_out_ga less_in_ga(s(X)) -> U10_ga(less_in_ga(X)) U10_ga(less_out_ga) -> less_out_ga The set Q consists of the following terms: less_in_ga(x0) U10_ga(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (57) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *MERGE_IN_AAG(.(X, Zs)) -> U6_AAG(X, Zs, less_in_ga(X)) The graph contains the following edges 1 > 1, 1 > 2 *MERGE_IN_AAG(.(Y, Zs)) -> U8_AAG(Y, Zs, less_in_ga(Y)) The graph contains the following edges 1 > 1, 1 > 2 *U6_AAG(X, Zs, less_out_ga) -> MERGE_IN_AAG(Zs) The graph contains the following edges 2 >= 1 *U8_AAG(Y, Zs, less_out_ga) -> MERGE_IN_AAG(Zs) The graph contains the following edges 2 >= 1 ---------------------------------------- (58) YES ---------------------------------------- (59) Obligation: Pi DP problem: The TRS P consists of the following rules: LESS_IN_AA(s(X), s(Y)) -> LESS_IN_AA(X, Y) The TRS R consists of the following rules: ms_in_ag([], []) -> ms_out_ag([], []) ms_in_ag(.(X, []), .(X, [])) -> ms_out_ag(.(X, []), .(X, [])) ms_in_ag(.(X, .(Y, Xs)), Ys) -> U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) split_in_aaa([], [], []) -> split_out_aaa([], [], []) split_in_aaa(.(X, Xs), .(X, Ys), Zs) -> U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys)) U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) -> split_out_aaa(.(X, Xs), .(X, Ys), Zs) U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) ms_in_aa([], []) -> ms_out_aa([], []) ms_in_aa(.(X, []), .(X, [])) -> ms_out_aa(.(X, []), .(X, [])) ms_in_aa(.(X, .(Y, Xs)), Ys) -> U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys)) merge_in_aaa([], Xs, Xs) -> merge_out_aaa([], Xs, Xs) merge_in_aaa(Xs, [], Xs) -> merge_out_aaa(Xs, [], Xs) merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y))) less_in_aa(0, s(X1)) -> less_out_aa(0, s(X1)) less_in_aa(s(X), s(Y)) -> U10_aa(X, Y, less_in_aa(X, Y)) U10_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) -> U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs)) merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X)) U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) -> U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs)) U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) -> ms_out_aa(.(X, .(Y, Xs)), Ys) U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys)) merge_in_aag([], Xs, Xs) -> merge_out_aag([], Xs, Xs) merge_in_aag(Xs, [], Xs) -> merge_out_aag(Xs, [], Xs) merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y))) less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) less_in_ga(s(X), s(Y)) -> U10_ga(X, Y, less_in_ga(X, Y)) U10_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) -> U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs)) merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X)) U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) -> U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs)) U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) -> ms_out_ag(.(X, .(Y, Xs)), Ys) The argument filtering Pi contains the following mapping: ms_in_ag(x1, x2) = ms_in_ag(x2) [] = [] ms_out_ag(x1, x2) = ms_out_ag .(x1, x2) = .(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) split_in_aaa(x1, x2, x3) = split_in_aaa split_out_aaa(x1, x2, x3) = split_out_aaa U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5) U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x4, x6) ms_in_aa(x1, x2) = ms_in_aa ms_out_aa(x1, x2) = ms_out_aa U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x6) U3_aa(x1, x2, x3, x4, x5, x6) = U3_aa(x6) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x5) merge_in_aaa(x1, x2, x3) = merge_in_aaa merge_out_aaa(x1, x2, x3) = merge_out_aaa U6_aaa(x1, x2, x3, x4, x5, x6) = U6_aaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U10_aa(x1, x2, x3) = U10_aa(x3) U7_aaa(x1, x2, x3, x4, x5, x6) = U7_aaa(x6) U8_aaa(x1, x2, x3, x4, x5, x6) = U8_aaa(x6) U9_aaa(x1, x2, x3, x4, x5, x6) = U9_aaa(x6) U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x4, x6) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x5) merge_in_aag(x1, x2, x3) = merge_in_aag(x3) merge_out_aag(x1, x2, x3) = merge_out_aag(x1, x2) U6_aag(x1, x2, x3, x4, x5, x6) = U6_aag(x1, x5, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga s(x1) = s(x1) U10_ga(x1, x2, x3) = U10_ga(x3) U7_aag(x1, x2, x3, x4, x5, x6) = U7_aag(x1, x6) U8_aag(x1, x2, x3, x4, x5, x6) = U8_aag(x3, x5, x6) U9_aag(x1, x2, x3, x4, x5, x6) = U9_aag(x3, x6) LESS_IN_AA(x1, x2) = LESS_IN_AA We have to consider all (P,R,Pi)-chains ---------------------------------------- (60) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (61) Obligation: Pi DP problem: The TRS P consists of the following rules: LESS_IN_AA(s(X), s(Y)) -> LESS_IN_AA(X, Y) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) LESS_IN_AA(x1, x2) = LESS_IN_AA We have to consider all (P,R,Pi)-chains ---------------------------------------- (62) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (63) Obligation: Q DP problem: The TRS P consists of the following rules: LESS_IN_AA -> LESS_IN_AA R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (64) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by semiunifying a rule from P directly. s = LESS_IN_AA evaluates to t =LESS_IN_AA Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence The DP semiunifies directly so there is only one rewrite step from LESS_IN_AA to LESS_IN_AA. ---------------------------------------- (65) NO ---------------------------------------- (66) Obligation: Pi DP problem: The TRS P consists of the following rules: U6_AAA(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) -> MERGE_IN_AAA(Xs, .(Y, Ys), Zs) MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_AAA(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y))) MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_AAA(X, Xs, Y, Ys, Zs, less_in_aa(Y, X)) U8_AAA(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) -> MERGE_IN_AAA(.(X, Xs), Ys, Zs) The TRS R consists of the following rules: ms_in_ag([], []) -> ms_out_ag([], []) ms_in_ag(.(X, []), .(X, [])) -> ms_out_ag(.(X, []), .(X, [])) ms_in_ag(.(X, .(Y, Xs)), Ys) -> U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) split_in_aaa([], [], []) -> split_out_aaa([], [], []) split_in_aaa(.(X, Xs), .(X, Ys), Zs) -> U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys)) U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) -> split_out_aaa(.(X, Xs), .(X, Ys), Zs) U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) ms_in_aa([], []) -> ms_out_aa([], []) ms_in_aa(.(X, []), .(X, [])) -> ms_out_aa(.(X, []), .(X, [])) ms_in_aa(.(X, .(Y, Xs)), Ys) -> U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys)) merge_in_aaa([], Xs, Xs) -> merge_out_aaa([], Xs, Xs) merge_in_aaa(Xs, [], Xs) -> merge_out_aaa(Xs, [], Xs) merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y))) less_in_aa(0, s(X1)) -> less_out_aa(0, s(X1)) less_in_aa(s(X), s(Y)) -> U10_aa(X, Y, less_in_aa(X, Y)) U10_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) -> U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs)) merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X)) U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) -> U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs)) U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) -> ms_out_aa(.(X, .(Y, Xs)), Ys) U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys)) merge_in_aag([], Xs, Xs) -> merge_out_aag([], Xs, Xs) merge_in_aag(Xs, [], Xs) -> merge_out_aag(Xs, [], Xs) merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y))) less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) less_in_ga(s(X), s(Y)) -> U10_ga(X, Y, less_in_ga(X, Y)) U10_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) -> U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs)) merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X)) U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) -> U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs)) U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) -> ms_out_ag(.(X, .(Y, Xs)), Ys) The argument filtering Pi contains the following mapping: ms_in_ag(x1, x2) = ms_in_ag(x2) [] = [] ms_out_ag(x1, x2) = ms_out_ag .(x1, x2) = .(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) split_in_aaa(x1, x2, x3) = split_in_aaa split_out_aaa(x1, x2, x3) = split_out_aaa U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5) U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x4, x6) ms_in_aa(x1, x2) = ms_in_aa ms_out_aa(x1, x2) = ms_out_aa U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x6) U3_aa(x1, x2, x3, x4, x5, x6) = U3_aa(x6) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x5) merge_in_aaa(x1, x2, x3) = merge_in_aaa merge_out_aaa(x1, x2, x3) = merge_out_aaa U6_aaa(x1, x2, x3, x4, x5, x6) = U6_aaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U10_aa(x1, x2, x3) = U10_aa(x3) U7_aaa(x1, x2, x3, x4, x5, x6) = U7_aaa(x6) U8_aaa(x1, x2, x3, x4, x5, x6) = U8_aaa(x6) U9_aaa(x1, x2, x3, x4, x5, x6) = U9_aaa(x6) U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x4, x6) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x5) merge_in_aag(x1, x2, x3) = merge_in_aag(x3) merge_out_aag(x1, x2, x3) = merge_out_aag(x1, x2) U6_aag(x1, x2, x3, x4, x5, x6) = U6_aag(x1, x5, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga s(x1) = s(x1) U10_ga(x1, x2, x3) = U10_ga(x3) U7_aag(x1, x2, x3, x4, x5, x6) = U7_aag(x1, x6) U8_aag(x1, x2, x3, x4, x5, x6) = U8_aag(x3, x5, x6) U9_aag(x1, x2, x3, x4, x5, x6) = U9_aag(x3, x6) MERGE_IN_AAA(x1, x2, x3) = MERGE_IN_AAA U6_AAA(x1, x2, x3, x4, x5, x6) = U6_AAA(x6) U8_AAA(x1, x2, x3, x4, x5, x6) = U8_AAA(x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (67) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (68) Obligation: Pi DP problem: The TRS P consists of the following rules: U6_AAA(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) -> MERGE_IN_AAA(Xs, .(Y, Ys), Zs) MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_AAA(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y))) MERGE_IN_AAA(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_AAA(X, Xs, Y, Ys, Zs, less_in_aa(Y, X)) U8_AAA(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) -> MERGE_IN_AAA(.(X, Xs), Ys, Zs) The TRS R consists of the following rules: less_in_aa(0, s(X1)) -> less_out_aa(0, s(X1)) less_in_aa(s(X), s(Y)) -> U10_aa(X, Y, less_in_aa(X, Y)) U10_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U10_aa(x1, x2, x3) = U10_aa(x3) 0 = 0 s(x1) = s(x1) MERGE_IN_AAA(x1, x2, x3) = MERGE_IN_AAA U6_AAA(x1, x2, x3, x4, x5, x6) = U6_AAA(x6) U8_AAA(x1, x2, x3, x4, x5, x6) = U8_AAA(x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (69) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (70) Obligation: Q DP problem: The TRS P consists of the following rules: U6_AAA(less_out_aa(X)) -> MERGE_IN_AAA MERGE_IN_AAA -> U6_AAA(less_in_aa) MERGE_IN_AAA -> U8_AAA(less_in_aa) U8_AAA(less_out_aa(Y)) -> MERGE_IN_AAA The TRS R consists of the following rules: less_in_aa -> less_out_aa(0) less_in_aa -> U10_aa(less_in_aa) U10_aa(less_out_aa(X)) -> less_out_aa(s(X)) The set Q consists of the following terms: less_in_aa U10_aa(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (71) TransformationProof (SOUND) By narrowing [LPAR04] the rule MERGE_IN_AAA -> U6_AAA(less_in_aa) at position [0] we obtained the following new rules [LPAR04]: (MERGE_IN_AAA -> U6_AAA(less_out_aa(0)),MERGE_IN_AAA -> U6_AAA(less_out_aa(0))) (MERGE_IN_AAA -> U6_AAA(U10_aa(less_in_aa)),MERGE_IN_AAA -> U6_AAA(U10_aa(less_in_aa))) ---------------------------------------- (72) Obligation: Q DP problem: The TRS P consists of the following rules: U6_AAA(less_out_aa(X)) -> MERGE_IN_AAA MERGE_IN_AAA -> U8_AAA(less_in_aa) U8_AAA(less_out_aa(Y)) -> MERGE_IN_AAA MERGE_IN_AAA -> U6_AAA(less_out_aa(0)) MERGE_IN_AAA -> U6_AAA(U10_aa(less_in_aa)) The TRS R consists of the following rules: less_in_aa -> less_out_aa(0) less_in_aa -> U10_aa(less_in_aa) U10_aa(less_out_aa(X)) -> less_out_aa(s(X)) The set Q consists of the following terms: less_in_aa U10_aa(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (73) TransformationProof (SOUND) By narrowing [LPAR04] the rule MERGE_IN_AAA -> U8_AAA(less_in_aa) at position [0] we obtained the following new rules [LPAR04]: (MERGE_IN_AAA -> U8_AAA(less_out_aa(0)),MERGE_IN_AAA -> U8_AAA(less_out_aa(0))) (MERGE_IN_AAA -> U8_AAA(U10_aa(less_in_aa)),MERGE_IN_AAA -> U8_AAA(U10_aa(less_in_aa))) ---------------------------------------- (74) Obligation: Q DP problem: The TRS P consists of the following rules: U6_AAA(less_out_aa(X)) -> MERGE_IN_AAA U8_AAA(less_out_aa(Y)) -> MERGE_IN_AAA MERGE_IN_AAA -> U6_AAA(less_out_aa(0)) MERGE_IN_AAA -> U6_AAA(U10_aa(less_in_aa)) MERGE_IN_AAA -> U8_AAA(less_out_aa(0)) MERGE_IN_AAA -> U8_AAA(U10_aa(less_in_aa)) The TRS R consists of the following rules: less_in_aa -> less_out_aa(0) less_in_aa -> U10_aa(less_in_aa) U10_aa(less_out_aa(X)) -> less_out_aa(s(X)) The set Q consists of the following terms: less_in_aa U10_aa(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (75) Obligation: Pi DP problem: The TRS P consists of the following rules: SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) -> SPLIT_IN_AAA(Xs, Zs, Ys) The TRS R consists of the following rules: ms_in_ag([], []) -> ms_out_ag([], []) ms_in_ag(.(X, []), .(X, [])) -> ms_out_ag(.(X, []), .(X, [])) ms_in_ag(.(X, .(Y, Xs)), Ys) -> U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) split_in_aaa([], [], []) -> split_out_aaa([], [], []) split_in_aaa(.(X, Xs), .(X, Ys), Zs) -> U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys)) U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) -> split_out_aaa(.(X, Xs), .(X, Ys), Zs) U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) ms_in_aa([], []) -> ms_out_aa([], []) ms_in_aa(.(X, []), .(X, [])) -> ms_out_aa(.(X, []), .(X, [])) ms_in_aa(.(X, .(Y, Xs)), Ys) -> U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys)) merge_in_aaa([], Xs, Xs) -> merge_out_aaa([], Xs, Xs) merge_in_aaa(Xs, [], Xs) -> merge_out_aaa(Xs, [], Xs) merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y))) less_in_aa(0, s(X1)) -> less_out_aa(0, s(X1)) less_in_aa(s(X), s(Y)) -> U10_aa(X, Y, less_in_aa(X, Y)) U10_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) -> U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs)) merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X)) U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) -> U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs)) U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) -> ms_out_aa(.(X, .(Y, Xs)), Ys) U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys)) merge_in_aag([], Xs, Xs) -> merge_out_aag([], Xs, Xs) merge_in_aag(Xs, [], Xs) -> merge_out_aag(Xs, [], Xs) merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y))) less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) less_in_ga(s(X), s(Y)) -> U10_ga(X, Y, less_in_ga(X, Y)) U10_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) -> U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs)) merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X)) U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) -> U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs)) U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) -> ms_out_ag(.(X, .(Y, Xs)), Ys) The argument filtering Pi contains the following mapping: ms_in_ag(x1, x2) = ms_in_ag(x2) [] = [] ms_out_ag(x1, x2) = ms_out_ag .(x1, x2) = .(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) split_in_aaa(x1, x2, x3) = split_in_aaa split_out_aaa(x1, x2, x3) = split_out_aaa U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5) U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x4, x6) ms_in_aa(x1, x2) = ms_in_aa ms_out_aa(x1, x2) = ms_out_aa U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x6) U3_aa(x1, x2, x3, x4, x5, x6) = U3_aa(x6) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x5) merge_in_aaa(x1, x2, x3) = merge_in_aaa merge_out_aaa(x1, x2, x3) = merge_out_aaa U6_aaa(x1, x2, x3, x4, x5, x6) = U6_aaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U10_aa(x1, x2, x3) = U10_aa(x3) U7_aaa(x1, x2, x3, x4, x5, x6) = U7_aaa(x6) U8_aaa(x1, x2, x3, x4, x5, x6) = U8_aaa(x6) U9_aaa(x1, x2, x3, x4, x5, x6) = U9_aaa(x6) U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x4, x6) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x5) merge_in_aag(x1, x2, x3) = merge_in_aag(x3) merge_out_aag(x1, x2, x3) = merge_out_aag(x1, x2) U6_aag(x1, x2, x3, x4, x5, x6) = U6_aag(x1, x5, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga s(x1) = s(x1) U10_ga(x1, x2, x3) = U10_ga(x3) U7_aag(x1, x2, x3, x4, x5, x6) = U7_aag(x1, x6) U8_aag(x1, x2, x3, x4, x5, x6) = U8_aag(x3, x5, x6) U9_aag(x1, x2, x3, x4, x5, x6) = U9_aag(x3, x6) SPLIT_IN_AAA(x1, x2, x3) = SPLIT_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (76) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (77) Obligation: Pi DP problem: The TRS P consists of the following rules: SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) -> SPLIT_IN_AAA(Xs, Zs, Ys) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) SPLIT_IN_AAA(x1, x2, x3) = SPLIT_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (78) Obligation: Pi DP problem: The TRS P consists of the following rules: MS_IN_AA(.(X, .(Y, Xs)), Ys) -> U1_AA(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_AA(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) U2_AA(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> MS_IN_AA(X2s, Y2s) U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> MS_IN_AA(X1s, Y1s) The TRS R consists of the following rules: ms_in_ag([], []) -> ms_out_ag([], []) ms_in_ag(.(X, []), .(X, [])) -> ms_out_ag(.(X, []), .(X, [])) ms_in_ag(.(X, .(Y, Xs)), Ys) -> U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) split_in_aaa([], [], []) -> split_out_aaa([], [], []) split_in_aaa(.(X, Xs), .(X, Ys), Zs) -> U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys)) U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) -> split_out_aaa(.(X, Xs), .(X, Ys), Zs) U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_ag(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) ms_in_aa([], []) -> ms_out_aa([], []) ms_in_aa(.(X, []), .(X, [])) -> ms_out_aa(.(X, []), .(X, [])) ms_in_aa(.(X, .(Y, Xs)), Ys) -> U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys)) merge_in_aaa([], Xs, Xs) -> merge_out_aaa([], Xs, Xs) merge_in_aaa(Xs, [], Xs) -> merge_out_aaa(Xs, [], Xs) merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y))) less_in_aa(0, s(X1)) -> less_out_aa(0, s(X1)) less_in_aa(s(X), s(Y)) -> U10_aa(X, Y, less_in_aa(X, Y)) U10_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) -> U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs)) merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X)) U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) -> U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs)) U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) -> ms_out_aa(.(X, .(Y, Xs)), Ys) U2_ag(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_ag(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_ag(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_ag(X, Y, Xs, Ys, merge_in_aag(Y1s, Y2s, Ys)) merge_in_aag([], Xs, Xs) -> merge_out_aag([], Xs, Xs) merge_in_aag(Xs, [], Xs) -> merge_out_aag(Xs, [], Xs) merge_in_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aag(X, Xs, Y, Ys, Zs, less_in_ga(X, s(Y))) less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) less_in_ga(s(X), s(Y)) -> U10_ga(X, Y, less_in_ga(X, Y)) U10_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U6_aag(X, Xs, Y, Ys, Zs, less_out_ga(X, s(Y))) -> U7_aag(X, Xs, Y, Ys, Zs, merge_in_aag(Xs, .(Y, Ys), Zs)) merge_in_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aag(X, Xs, Y, Ys, Zs, less_in_ga(Y, X)) U8_aag(X, Xs, Y, Ys, Zs, less_out_ga(Y, X)) -> U9_aag(X, Xs, Y, Ys, Zs, merge_in_aag(.(X, Xs), Ys, Zs)) U9_aag(X, Xs, Y, Ys, Zs, merge_out_aag(.(X, Xs), Ys, Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(Y, Zs)) U7_aag(X, Xs, Y, Ys, Zs, merge_out_aag(Xs, .(Y, Ys), Zs)) -> merge_out_aag(.(X, Xs), .(Y, Ys), .(X, Zs)) U4_ag(X, Y, Xs, Ys, merge_out_aag(Y1s, Y2s, Ys)) -> ms_out_ag(.(X, .(Y, Xs)), Ys) The argument filtering Pi contains the following mapping: ms_in_ag(x1, x2) = ms_in_ag(x2) [] = [] ms_out_ag(x1, x2) = ms_out_ag .(x1, x2) = .(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) split_in_aaa(x1, x2, x3) = split_in_aaa split_out_aaa(x1, x2, x3) = split_out_aaa U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5) U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x4, x6) ms_in_aa(x1, x2) = ms_in_aa ms_out_aa(x1, x2) = ms_out_aa U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x6) U3_aa(x1, x2, x3, x4, x5, x6) = U3_aa(x6) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x5) merge_in_aaa(x1, x2, x3) = merge_in_aaa merge_out_aaa(x1, x2, x3) = merge_out_aaa U6_aaa(x1, x2, x3, x4, x5, x6) = U6_aaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U10_aa(x1, x2, x3) = U10_aa(x3) U7_aaa(x1, x2, x3, x4, x5, x6) = U7_aaa(x6) U8_aaa(x1, x2, x3, x4, x5, x6) = U8_aaa(x6) U9_aaa(x1, x2, x3, x4, x5, x6) = U9_aaa(x6) U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x4, x6) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x5) merge_in_aag(x1, x2, x3) = merge_in_aag(x3) merge_out_aag(x1, x2, x3) = merge_out_aag(x1, x2) U6_aag(x1, x2, x3, x4, x5, x6) = U6_aag(x1, x5, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga s(x1) = s(x1) U10_ga(x1, x2, x3) = U10_ga(x3) U7_aag(x1, x2, x3, x4, x5, x6) = U7_aag(x1, x6) U8_aag(x1, x2, x3, x4, x5, x6) = U8_aag(x3, x5, x6) U9_aag(x1, x2, x3, x4, x5, x6) = U9_aag(x3, x6) MS_IN_AA(x1, x2) = MS_IN_AA U1_AA(x1, x2, x3, x4, x5) = U1_AA(x5) U2_AA(x1, x2, x3, x4, x5, x6) = U2_AA(x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (79) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (80) Obligation: Pi DP problem: The TRS P consists of the following rules: MS_IN_AA(.(X, .(Y, Xs)), Ys) -> U1_AA(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_AA(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) U2_AA(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> MS_IN_AA(X2s, Y2s) U1_AA(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> MS_IN_AA(X1s, Y1s) The TRS R consists of the following rules: split_in_aaa(.(X, Xs), .(X, Ys), Zs) -> U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys)) ms_in_aa([], []) -> ms_out_aa([], []) ms_in_aa(.(X, []), .(X, [])) -> ms_out_aa(.(X, []), .(X, [])) ms_in_aa(.(X, .(Y, Xs)), Ys) -> U1_aa(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s)) U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) -> split_out_aaa(.(X, Xs), .(X, Ys), Zs) U1_aa(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) -> U2_aa(X, Y, Xs, Ys, X2s, ms_in_aa(X1s, Y1s)) split_in_aaa([], [], []) -> split_out_aaa([], [], []) U2_aa(X, Y, Xs, Ys, X2s, ms_out_aa(X1s, Y1s)) -> U3_aa(X, Y, Xs, Ys, Y1s, ms_in_aa(X2s, Y2s)) U3_aa(X, Y, Xs, Ys, Y1s, ms_out_aa(X2s, Y2s)) -> U4_aa(X, Y, Xs, Ys, merge_in_aaa(Y1s, Y2s, Ys)) U4_aa(X, Y, Xs, Ys, merge_out_aaa(Y1s, Y2s, Ys)) -> ms_out_aa(.(X, .(Y, Xs)), Ys) merge_in_aaa([], Xs, Xs) -> merge_out_aaa([], Xs, Xs) merge_in_aaa(Xs, [], Xs) -> merge_out_aaa(Xs, [], Xs) merge_in_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) -> U6_aaa(X, Xs, Y, Ys, Zs, less_in_aa(X, s(Y))) merge_in_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) -> U8_aaa(X, Xs, Y, Ys, Zs, less_in_aa(Y, X)) U6_aaa(X, Xs, Y, Ys, Zs, less_out_aa(X, s(Y))) -> U7_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(Xs, .(Y, Ys), Zs)) U8_aaa(X, Xs, Y, Ys, Zs, less_out_aa(Y, X)) -> U9_aaa(X, Xs, Y, Ys, Zs, merge_in_aaa(.(X, Xs), Ys, Zs)) less_in_aa(0, s(X1)) -> less_out_aa(0, s(X1)) less_in_aa(s(X), s(Y)) -> U10_aa(X, Y, less_in_aa(X, Y)) U7_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(Xs, .(Y, Ys), Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(X, Zs)) U9_aaa(X, Xs, Y, Ys, Zs, merge_out_aaa(.(X, Xs), Ys, Zs)) -> merge_out_aaa(.(X, Xs), .(Y, Ys), .(Y, Zs)) U10_aa(X, Y, less_out_aa(X, Y)) -> less_out_aa(s(X), s(Y)) The argument filtering Pi contains the following mapping: [] = [] .(x1, x2) = .(x1, x2) split_in_aaa(x1, x2, x3) = split_in_aaa split_out_aaa(x1, x2, x3) = split_out_aaa U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5) ms_in_aa(x1, x2) = ms_in_aa ms_out_aa(x1, x2) = ms_out_aa U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U2_aa(x1, x2, x3, x4, x5, x6) = U2_aa(x6) U3_aa(x1, x2, x3, x4, x5, x6) = U3_aa(x6) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x5) merge_in_aaa(x1, x2, x3) = merge_in_aaa merge_out_aaa(x1, x2, x3) = merge_out_aaa U6_aaa(x1, x2, x3, x4, x5, x6) = U6_aaa(x6) less_in_aa(x1, x2) = less_in_aa less_out_aa(x1, x2) = less_out_aa(x1) U10_aa(x1, x2, x3) = U10_aa(x3) U7_aaa(x1, x2, x3, x4, x5, x6) = U7_aaa(x6) U8_aaa(x1, x2, x3, x4, x5, x6) = U8_aaa(x6) U9_aaa(x1, x2, x3, x4, x5, x6) = U9_aaa(x6) 0 = 0 s(x1) = s(x1) MS_IN_AA(x1, x2) = MS_IN_AA U1_AA(x1, x2, x3, x4, x5) = U1_AA(x5) U2_AA(x1, x2, x3, x4, x5, x6) = U2_AA(x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (81) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 1, "program": { "directives": [], "clauses": [ [ "(ms ([]) ([]))", null ], [ "(ms (. 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"exprvars": [] } }, "705": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (ms T54 X26) (merge T53 X26 T19))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T19"], "free": ["X26"], "exprvars": [] } }, "707": { "goal": [ { "clause": 0, "scope": 5, "term": "(ms T23 X25)" }, { "clause": 1, "scope": 5, "term": "(ms T23 X25)" }, { "clause": 2, "scope": 5, "term": "(ms T23 X25)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X25"], "exprvars": [] } }, "708": { "goal": [{ "clause": 0, "scope": 5, "term": "(ms T23 X25)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X25"], "exprvars": [] } }, "709": { "goal": [ { "clause": 1, "scope": 5, "term": "(ms T23 X25)" }, { "clause": 2, "scope": 5, "term": "(ms T23 X25)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X25"], "exprvars": [] } }, "554": { "goal": [{ "clause": -1, "scope": -1, "term": "(split T45 X62 X61)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X61", "X62" ], "exprvars": [] } }, "555": { "goal": [ { "clause": 3, "scope": 4, "term": "(split T45 X62 X61)" }, { "clause": 4, "scope": 4, "term": "(split T45 X62 X61)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X61", "X62" ], "exprvars": [] } }, "556": { "goal": [{ "clause": 3, "scope": 4, "term": "(split T45 X62 X61)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X61", "X62" ], "exprvars": [] } }, "710": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "557": { "goal": [{ "clause": 4, "scope": 4, "term": "(split T45 X62 X61)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X61", "X62" ], "exprvars": [] } }, "711": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "558": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "712": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "559": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "713": { "goal": [{ "clause": 1, "scope": 5, "term": "(ms T23 X25)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X25"], "exprvars": [] } }, "714": { "goal": [{ "clause": 2, "scope": 5, "term": "(ms T23 X25)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X25"], "exprvars": [] } }, "715": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "716": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "717": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "560": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "561": { "goal": [{ "clause": -1, "scope": -1, "term": "(split T52 X80 X79)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X79", "X80" ], "exprvars": [] } }, "569": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1055": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1054": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (less T167 T168) (merge (. T168 T170) T169 X200))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X200"], "exprvars": [] } }, "100": { "goal": [{ "clause": 4, "scope": 2, "term": "(split (. T20 (. T21 T22)) X23 X24)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X23", "X24" ], "exprvars": [] } }, "103": { "goal": [{ "clause": -1, "scope": -1, "term": "(split (. T36 T37) X44 X43)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X43", "X44" ], "exprvars": [] } }, "862": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (split (. T69 (. T70 T71)) X104 X105) (',' (ms X104 X106) (',' (ms X105 X107) (merge X106 X107 X108))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X108", "X104", "X105", "X106", "X107" ], "exprvars": [] } }, "104": { "goal": [ { "clause": 3, "scope": 3, "term": "(split (. T36 T37) X44 X43)" }, { "clause": 4, "scope": 3, "term": "(split (. T36 T37) X44 X43)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X43", "X44" ], "exprvars": [] } }, "863": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "105": { "goal": [{ "clause": 4, "scope": 3, "term": "(split (. T36 T37) X44 X43)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X43", "X44" ], "exprvars": [] } }, "864": { "goal": [{ "clause": -1, "scope": -1, "term": "(split (. T69 (. T70 T71)) X104 X105)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X104", "X105" ], "exprvars": [] } }, "865": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (ms T72 X106) (',' (ms T73 X107) (merge X106 X107 X108)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X108", "X106", "X107" ], "exprvars": [] } }, "900": { "goal": [{ "clause": 9, "scope": 7, "term": "(less T110 (s T111))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "868": { "goal": [{ "clause": -1, "scope": -1, "term": "(ms T72 X106)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X106"], "exprvars": [] } }, "901": { "goal": [{ "clause": 10, "scope": 7, "term": "(less T110 (s T111))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "869": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (ms T75 X107) (merge T74 X107 X108))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X108", "X107" ], "exprvars": [] } }, "903": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "904": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "905": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 5, "label": "CASE" }, { "from": 5, "to": 7, "label": "PARALLEL" }, { "from": 5, "to": 8, "label": "PARALLEL" }, { "from": 7, "to": 11, "label": "EVAL with clause\nms([], []).\nand substitutionT1 -> [],\nT2 -> []" }, { "from": 7, "to": 14, "label": "EVAL-BACKTRACK" }, { "from": 8, "to": 26, "label": "PARALLEL" }, { "from": 8, "to": 27, "label": "PARALLEL" }, { "from": 11, "to": 15, "label": "SUCCESS" }, { "from": 26, "to": 29, "label": "EVAL with clause\nms(.(X6, []), .(X6, [])).\nand substitutionX6 -> T7,\nT1 -> .(T7, []),\nT2 -> .(T7, [])" }, { "from": 26, "to": 31, "label": "EVAL-BACKTRACK" }, { "from": 27, "to": 35, "label": "EVAL with clause\nms(.(X19, .(X20, X21)), X22) :- ','(split(.(X19, .(X20, X21)), X23, X24), ','(ms(X23, X25), ','(ms(X24, X26), merge(X25, X26, X22)))).\nand substitutionX19 -> T20,\nX20 -> T21,\nX21 -> T22,\nT1 -> .(T20, .(T21, T22)),\nT2 -> T19,\nX22 -> T19,\nT16 -> T20,\nT17 -> T21,\nT18 -> T22" }, { "from": 27, "to": 36, "label": "EVAL-BACKTRACK" }, { "from": 29, "to": 32, "label": "SUCCESS" }, { "from": 35, "to": 97, "label": "SPLIT 1" }, { "from": 35, "to": 98, "label": "SPLIT 2\nreplacements:X23 -> T23,\nX24 -> T24" }, { "from": 97, "to": 99, "label": "CASE" }, { "from": 98, "to": 704, "label": "SPLIT 1" }, { "from": 98, "to": 705, "label": "SPLIT 2\nreplacements:X25 -> T53,\nT24 -> T54" }, { "from": 99, "to": 100, "label": "BACKTRACK\nfor clause: split([], [], [])because of non-unification" }, { "from": 100, "to": 103, "label": "ONLY EVAL with clause\nsplit(.(X39, X40), .(X39, X41), X42) :- split(X40, X42, X41).\nand substitutionT20 -> T33,\nX39 -> T33,\nT21 -> T36,\nT22 -> T37,\nX40 -> .(T36, T37),\nX41 -> X43,\nX23 -> .(T33, X43),\nX24 -> X44,\nX42 -> X44,\nT34 -> T36,\nT35 -> T37" }, { "from": 103, "to": 104, "label": "CASE" }, { "from": 104, "to": 105, "label": "BACKTRACK\nfor clause: split([], [], [])because of non-unification" }, { "from": 105, "to": 554, "label": "ONLY EVAL with clause\nsplit(.(X57, X58), .(X57, X59), X60) :- split(X58, X60, X59).\nand substitutionT36 -> T43,\nX57 -> T43,\nT37 -> T45,\nX58 -> T45,\nX59 -> X61,\nX44 -> .(T43, X61),\nX43 -> X62,\nX60 -> X62,\nT44 -> T45" }, { "from": 554, "to": 555, "label": "CASE" }, { "from": 555, "to": 556, "label": "PARALLEL" }, { "from": 555, "to": 557, "label": "PARALLEL" }, { "from": 556, "to": 558, "label": "EVAL with clause\nsplit([], [], []).\nand substitutionT45 -> [],\nX62 -> [],\nX61 -> []" }, { "from": 556, "to": 559, "label": "EVAL-BACKTRACK" }, { "from": 557, "to": 561, "label": "EVAL with clause\nsplit(.(X75, X76), .(X75, X77), X78) :- split(X76, X78, X77).\nand substitutionX75 -> T50,\nX76 -> T52,\nT45 -> .(T50, T52),\nX77 -> X79,\nX62 -> .(T50, X79),\nX61 -> X80,\nX78 -> X80,\nT51 -> T52" }, { "from": 557, "to": 569, "label": "EVAL-BACKTRACK" }, { "from": 558, "to": 560, "label": "SUCCESS" }, { "from": 561, "to": 554, "label": "INSTANCE with matching:\nT45 -> T52\nX62 -> X80\nX61 -> X79" }, { "from": 704, "to": 707, "label": "CASE" }, { "from": 705, "to": 1058, "label": "SPLIT 1" }, { "from": 705, "to": 1059, "label": "SPLIT 2\nreplacements:X26 -> T178,\nT53 -> T179" }, { "from": 707, "to": 708, "label": "PARALLEL" }, { "from": 707, "to": 709, "label": "PARALLEL" }, { "from": 708, "to": 710, "label": "EVAL with clause\nms([], []).\nand substitutionT23 -> [],\nX25 -> []" }, { "from": 708, "to": 711, "label": "EVAL-BACKTRACK" }, { "from": 709, "to": 713, "label": "PARALLEL" }, { "from": 709, "to": 714, "label": "PARALLEL" }, { "from": 710, "to": 712, "label": "SUCCESS" }, { "from": 713, "to": 715, "label": "EVAL with clause\nms(.(X85, []), .(X85, [])).\nand substitutionX85 -> T59,\nT23 -> .(T59, []),\nX25 -> .(T59, [])" }, { "from": 713, "to": 716, "label": "EVAL-BACKTRACK" }, { "from": 714, "to": 862, "label": "EVAL with clause\nms(.(X100, .(X101, X102)), X103) :- ','(split(.(X100, .(X101, X102)), X104, X105), ','(ms(X104, X106), ','(ms(X105, X107), merge(X106, X107, X103)))).\nand substitutionX100 -> T69,\nX101 -> T70,\nX102 -> T71,\nT23 -> .(T69, .(T70, T71)),\nX25 -> X108,\nX103 -> X108,\nT66 -> T69,\nT67 -> T70,\nT68 -> T71" }, { "from": 714, "to": 863, "label": "EVAL-BACKTRACK" }, { "from": 715, "to": 717, "label": "SUCCESS" }, { "from": 862, "to": 864, "label": "SPLIT 1" }, { "from": 862, "to": 865, "label": "SPLIT 2\nreplacements:X104 -> T72,\nX105 -> T73" }, { "from": 864, "to": 97, "label": "INSTANCE with matching:\nT20 -> T69\nT21 -> T70\nT22 -> T71\nX23 -> X104\nX24 -> X105" }, { "from": 865, "to": 868, "label": "SPLIT 1" }, { "from": 865, "to": 869, "label": "SPLIT 2\nreplacements:X106 -> T74,\nT73 -> T75" }, { "from": 868, "to": 704, "label": "INSTANCE with matching:\nT23 -> T72\nX25 -> X106" }, { "from": 869, "to": 872, "label": "SPLIT 1" }, { "from": 869, "to": 873, "label": "SPLIT 2\nreplacements:X107 -> T76,\nT74 -> T77" }, { "from": 872, "to": 704, "label": "INSTANCE with matching:\nT23 -> T75\nX25 -> X107" }, { "from": 873, "to": 874, "label": "CASE" }, { "from": 874, "to": 875, "label": "PARALLEL" }, { "from": 874, "to": 876, "label": "PARALLEL" }, { "from": 875, "to": 877, "label": "EVAL with clause\nmerge([], X115, X115).\nand substitutionT77 -> [],\nT76 -> T84,\nX115 -> T84,\nX108 -> T84" }, { "from": 875, "to": 878, "label": "EVAL-BACKTRACK" }, { "from": 876, "to": 880, "label": "PARALLEL" }, { "from": 876, "to": 881, "label": "PARALLEL" }, { "from": 877, "to": 879, "label": "SUCCESS" }, { "from": 880, "to": 882, "label": "EVAL with clause\nmerge(X120, [], X120).\nand substitutionT77 -> T89,\nX120 -> T89,\nT76 -> [],\nX108 -> T89" }, { "from": 880, "to": 883, "label": "EVAL-BACKTRACK" }, { "from": 881, "to": 885, "label": "PARALLEL" }, { "from": 881, "to": 886, "label": "PARALLEL" }, { "from": 882, "to": 884, "label": "SUCCESS" }, { "from": 885, "to": 889, "label": "EVAL with clause\nmerge(.(X145, X146), .(X147, X148), .(X145, X149)) :- ','(less(X145, s(X147)), merge(X146, .(X147, X148), X149)).\nand substitutionX145 -> T110,\nX146 -> T112,\nT77 -> .(T110, T112),\nX147 -> T111,\nX148 -> T113,\nT76 -> .(T111, T113),\nX149 -> X150,\nX108 -> .(T110, X150),\nT106 -> T110,\nT108 -> T111,\nT107 -> T112,\nT109 -> T113" }, { "from": 885, "to": 895, "label": "EVAL-BACKTRACK" }, { "from": 886, "to": 1054, "label": "EVAL with clause\nmerge(.(X195, X196), .(X197, X198), .(X197, X199)) :- ','(less(X197, X195), merge(.(X195, X196), X198, X199)).\nand substitutionX195 -> T168,\nX196 -> T170,\nT77 -> .(T168, T170),\nX197 -> T167,\nX198 -> T169,\nT76 -> .(T167, T169),\nX199 -> X200,\nX108 -> .(T167, X200),\nT165 -> T167,\nT163 -> T168,\nT166 -> T169,\nT164 -> T170" }, { "from": 886, "to": 1055, "label": "EVAL-BACKTRACK" }, { "from": 889, "to": 896, "label": "SPLIT 1" }, { "from": 889, "to": 897, "label": "SPLIT 2\nnew knowledge:\nT110 is ground\nreplacements:T112 -> T116,\nT111 -> T117,\nT113 -> T118" }, { "from": 896, "to": 898, "label": "CASE" }, { "from": 897, "to": 873, "label": "INSTANCE with matching:\nT77 -> T116\nT76 -> .(T117, T118)\nX108 -> X150" }, { "from": 898, "to": 900, "label": "PARALLEL" }, { "from": 898, "to": 901, "label": "PARALLEL" }, { "from": 900, "to": 903, "label": "EVAL with clause\nless(0, s(X159)).\nand substitutionT110 -> 0,\nT111 -> T125,\nX159 -> T125" }, { "from": 900, "to": 904, "label": "EVAL-BACKTRACK" }, { "from": 901, "to": 918, "label": "EVAL with clause\nless(s(X164), s(X165)) :- less(X164, X165).\nand substitutionX164 -> T132,\nT110 -> s(T132),\nT111 -> T133,\nX165 -> T133,\nT130 -> T132,\nT131 -> T133" }, { "from": 901, "to": 920, "label": "EVAL-BACKTRACK" }, { "from": 903, "to": 905, "label": "SUCCESS" }, { "from": 918, "to": 923, "label": "CASE" }, { "from": 923, "to": 924, "label": "PARALLEL" }, { "from": 923, "to": 925, "label": "PARALLEL" }, { "from": 924, "to": 926, "label": "EVAL with clause\nless(0, s(X172)).\nand substitutionT132 -> 0,\nX172 -> T140,\nT133 -> s(T140)" }, { "from": 924, "to": 927, "label": "EVAL-BACKTRACK" }, { "from": 925, "to": 933, "label": "EVAL with clause\nless(s(X177), s(X178)) :- less(X177, X178).\nand substitutionX177 -> T147,\nT132 -> s(T147),\nX178 -> T148,\nT133 -> s(T148),\nT145 -> T147,\nT146 -> T148" }, { "from": 925, "to": 934, "label": "EVAL-BACKTRACK" }, { "from": 926, "to": 928, "label": "SUCCESS" }, { "from": 933, "to": 918, "label": "INSTANCE with matching:\nT132 -> T147\nT133 -> T148" }, { "from": 1054, "to": 1056, "label": "SPLIT 1" }, { "from": 1054, "to": 1057, "label": "SPLIT 2\nnew knowledge:\nT167 is ground\nreplacements:T168 -> T173,\nT170 -> T174,\nT169 -> T175" }, { "from": 1056, "to": 918, "label": "INSTANCE with matching:\nT132 -> T167\nT133 -> T168" }, { "from": 1057, "to": 873, "label": "INSTANCE with matching:\nT77 -> .(T173, T174)\nT76 -> T175\nX108 -> X200" }, { "from": 1058, "to": 704, "label": "INSTANCE with matching:\nT23 -> T54\nX25 -> X26" }, { "from": 1059, "to": 1060, "label": "CASE" }, { "from": 1060, "to": 1061, "label": "PARALLEL" }, { "from": 1060, "to": 1062, "label": "PARALLEL" }, { "from": 1061, "to": 1063, "label": "EVAL with clause\nmerge([], X213, X213).\nand substitutionT179 -> [],\nT178 -> T186,\nX213 -> T186,\nT19 -> T186" }, { "from": 1061, "to": 1064, "label": "EVAL-BACKTRACK" }, { "from": 1062, "to": 1066, "label": "PARALLEL" }, { "from": 1062, "to": 1067, "label": "PARALLEL" }, { "from": 1063, "to": 1065, "label": "SUCCESS" }, { "from": 1066, "to": 1068, "label": "EVAL with clause\nmerge(X218, [], X218).\nand substitutionT179 -> T191,\nX218 -> T191,\nT178 -> [],\nT19 -> T191" }, { "from": 1066, "to": 1069, "label": "EVAL-BACKTRACK" }, { "from": 1067, "to": 1071, "label": "PARALLEL" }, { "from": 1067, "to": 1072, "label": "PARALLEL" }, { "from": 1068, "to": 1070, "label": "SUCCESS" }, { "from": 1071, "to": 1083, "label": "EVAL with clause\nmerge(.(X239, X240), .(X241, X242), .(X239, X243)) :- ','(less(X239, s(X241)), merge(X240, .(X241, X242), X243)).\nand substitutionX239 -> T212,\nX240 -> T218,\nT179 -> .(T212, T218),\nX241 -> T217,\nX242 -> T219,\nT178 -> .(T217, T219),\nX243 -> T216,\nT19 -> .(T212, T216),\nT214 -> T217,\nT213 -> T218,\nT215 -> T219" }, { "from": 1071, "to": 1084, "label": "EVAL-BACKTRACK" }, { "from": 1072, "to": 1111, "label": "EVAL with clause\nmerge(.(X286, X287), .(X288, X289), .(X288, X290)) :- ','(less(X288, X286), merge(.(X286, X287), X289, X290)).\nand substitutionX286 -> T274,\nX287 -> T276,\nT179 -> .(T274, T276),\nX288 -> T271,\nX289 -> T275,\nT178 -> .(T271, T275),\nX290 -> T273,\nT19 -> .(T271, T273),\nT269 -> T274,\nT272 -> T275,\nT270 -> T276" }, { "from": 1072, "to": 1112, "label": "EVAL-BACKTRACK" }, { "from": 1083, "to": 1085, "label": "SPLIT 1" }, { "from": 1083, "to": 1086, "label": "SPLIT 2\nnew knowledge:\nT212 is ground\nreplacements:T218 -> T222,\nT217 -> T223,\nT219 -> T224" }, { "from": 1085, "to": 1088, "label": "CASE" }, { "from": 1086, "to": 1059, "label": "INSTANCE with matching:\nT179 -> T222\nT178 -> .(T223, T224)\nT19 -> T216" }, { "from": 1088, "to": 1090, "label": "PARALLEL" }, { "from": 1088, "to": 1091, "label": "PARALLEL" }, { "from": 1090, "to": 1092, "label": "EVAL with clause\nless(0, s(X252)).\nand substitutionT212 -> 0,\nT217 -> T231,\nX252 -> T231" }, { "from": 1090, "to": 1093, "label": "EVAL-BACKTRACK" }, { "from": 1091, "to": 1095, "label": "EVAL with clause\nless(s(X257), s(X258)) :- less(X257, X258).\nand substitutionX257 -> T236,\nT212 -> s(T236),\nT217 -> T238,\nX258 -> T238,\nT237 -> T238" }, { "from": 1091, "to": 1096, "label": "EVAL-BACKTRACK" }, { "from": 1092, "to": 1094, "label": "SUCCESS" }, { "from": 1095, "to": 1099, "label": "CASE" }, { "from": 1099, "to": 1100, "label": "PARALLEL" }, { "from": 1099, "to": 1101, "label": "PARALLEL" }, { "from": 1100, "to": 1102, "label": "EVAL with clause\nless(0, s(X265)).\nand substitutionT236 -> 0,\nX265 -> T245,\nT238 -> s(T245)" }, { "from": 1100, "to": 1103, "label": "EVAL-BACKTRACK" }, { "from": 1101, "to": 1107, "label": "EVAL with clause\nless(s(X270), s(X271)) :- less(X270, X271).\nand substitutionX270 -> T250,\nT236 -> s(T250),\nX271 -> T252,\nT238 -> s(T252),\nT251 -> T252" }, { "from": 1101, "to": 1108, "label": "EVAL-BACKTRACK" }, { "from": 1102, "to": 1106, "label": "SUCCESS" }, { "from": 1107, "to": 1095, "label": "INSTANCE with matching:\nT236 -> T250\nT238 -> T252" }, { "from": 1111, "to": 1115, "label": "SPLIT 1" }, { "from": 1111, "to": 1116, "label": "SPLIT 2\nnew knowledge:\nT271 is ground\nreplacements:T274 -> T279,\nT276 -> T280,\nT275 -> T281" }, { "from": 1115, "to": 1095, "label": "INSTANCE with matching:\nT236 -> T271\nT238 -> T274" }, { "from": 1116, "to": 1059, "label": "INSTANCE with matching:\nT179 -> .(T279, T280)\nT178 -> T281\nT19 -> T273" } ], "type": "Graph" } } ---------------------------------------- (82) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f1_in([]) -> f1_out1 f1_in(.(T7, [])) -> f1_out1 f1_in(T19) -> U1(f35_in(T19), T19) U1(f35_out1(X25, X26), T19) -> f1_out1 f554_in -> f554_out1 f554_in -> U2(f554_in) U2(f554_out1) -> f554_out1 f97_in -> U3(f554_in) U3(f554_out1) -> f97_out1 f704_in -> f704_out1 f704_in -> U4(f862_in) U4(f862_out1) -> f704_out1 f873_in -> f873_out1 f873_in -> U5(f889_in) U5(f889_out1(T110)) -> f873_out1 f873_in -> U6(f1054_in) U6(f1054_out1(T167)) -> f873_out1 f918_in -> f918_out1(0) f918_in -> U7(f918_in) U7(f918_out1(T147)) -> f918_out1(s(T147)) f1059_in(T186) -> f1059_out1([], T186) f1059_in(T191) -> f1059_out1(T191, []) f1059_in(.(T212, T216)) -> U8(f1083_in(T212, T216), .(T212, T216)) U8(f1083_out1(T218, T217, T219), .(T212, T216)) -> f1059_out1(.(T212, T218), .(T217, T219)) f1059_in(.(T271, T273)) -> U9(f1111_in(T271, T273), .(T271, T273)) U9(f1111_out1(T274, T276, T275), .(T271, T273)) -> f1059_out1(.(T274, T276), .(T271, T275)) f1095_in(0) -> f1095_out1 f1095_in(s(T250)) -> U10(f1095_in(T250), s(T250)) U10(f1095_out1, s(T250)) -> f1095_out1 f896_in -> f896_out1(0) f896_in -> U11(f918_in) U11(f918_out1(T132)) -> f896_out1(s(T132)) f1085_in(0) -> f1085_out1 f1085_in(s(T236)) -> U12(f1095_in(T236), s(T236)) U12(f1095_out1, s(T236)) -> f1085_out1 f35_in(T19) -> U13(f97_in, T19) U13(f97_out1, T19) -> U14(f98_in(T19), T19) U14(f98_out1(X25, X26), T19) -> f35_out1(X25, X26) f98_in(T19) -> U15(f704_in, T19) U15(f704_out1, T19) -> U16(f705_in(T19), T19) U16(f705_out1(T53, X26), T19) -> f98_out1(T53, X26) f705_in(T19) -> U17(f704_in, T19) U17(f704_out1, T19) -> U18(f1059_in(T19), T19) U18(f1059_out1(T179, T178), T19) -> f705_out1(T179, T178) f862_in -> U19(f97_in) U19(f97_out1) -> U20(f865_in) U20(f865_out1) -> f862_out1 f865_in -> U21(f704_in) U21(f704_out1) -> U22(f869_in) U22(f869_out1) -> f865_out1 f869_in -> U23(f704_in) U23(f704_out1) -> U24(f873_in) U24(f873_out1) -> f869_out1 f889_in -> U25(f896_in) U25(f896_out1(T110)) -> U26(f873_in, T110) U26(f873_out1, T110) -> f889_out1(T110) f1054_in -> U27(f918_in) U27(f918_out1(T167)) -> U28(f873_in, T167) U28(f873_out1, T167) -> f1054_out1(T167) f1083_in(T212, T216) -> U29(f1085_in(T212), T212, T216) U29(f1085_out1, T212, T216) -> U30(f1059_in(T216), T212, T216) U30(f1059_out1(T222, .(T223, T224)), T212, T216) -> f1083_out1(T222, T223, T224) f1111_in(T271, T273) -> U31(f1095_in(T271), T271, T273) U31(f1095_out1, T271, T273) -> U32(f1059_in(T273), T271, T273) U32(f1059_out1(.(T279, T280), T281), T271, T273) -> f1111_out1(T279, T280, T281) Q is empty. ---------------------------------------- (83) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (84) Obligation: Q DP problem: The TRS P consists of the following rules: F1_IN(T19) -> U1^1(f35_in(T19), T19) F1_IN(T19) -> F35_IN(T19) F554_IN -> U2^1(f554_in) F554_IN -> F554_IN F97_IN -> U3^1(f554_in) F97_IN -> F554_IN F704_IN -> U4^1(f862_in) F704_IN -> F862_IN F873_IN -> U5^1(f889_in) F873_IN -> F889_IN F873_IN -> U6^1(f1054_in) F873_IN -> F1054_IN F918_IN -> U7^1(f918_in) F918_IN -> F918_IN F1059_IN(.(T212, T216)) -> U8^1(f1083_in(T212, T216), .(T212, T216)) F1059_IN(.(T212, T216)) -> F1083_IN(T212, T216) F1059_IN(.(T271, T273)) -> U9^1(f1111_in(T271, T273), .(T271, T273)) F1059_IN(.(T271, T273)) -> F1111_IN(T271, T273) F1095_IN(s(T250)) -> U10^1(f1095_in(T250), s(T250)) F1095_IN(s(T250)) -> F1095_IN(T250) F896_IN -> U11^1(f918_in) F896_IN -> F918_IN F1085_IN(s(T236)) -> U12^1(f1095_in(T236), s(T236)) F1085_IN(s(T236)) -> F1095_IN(T236) F35_IN(T19) -> U13^1(f97_in, T19) F35_IN(T19) -> F97_IN U13^1(f97_out1, T19) -> U14^1(f98_in(T19), T19) U13^1(f97_out1, T19) -> F98_IN(T19) F98_IN(T19) -> U15^1(f704_in, T19) F98_IN(T19) -> F704_IN U15^1(f704_out1, T19) -> U16^1(f705_in(T19), T19) U15^1(f704_out1, T19) -> F705_IN(T19) F705_IN(T19) -> U17^1(f704_in, T19) F705_IN(T19) -> F704_IN U17^1(f704_out1, T19) -> U18^1(f1059_in(T19), T19) U17^1(f704_out1, T19) -> F1059_IN(T19) F862_IN -> U19^1(f97_in) F862_IN -> F97_IN U19^1(f97_out1) -> U20^1(f865_in) U19^1(f97_out1) -> F865_IN F865_IN -> U21^1(f704_in) F865_IN -> F704_IN U21^1(f704_out1) -> U22^1(f869_in) U21^1(f704_out1) -> F869_IN F869_IN -> U23^1(f704_in) F869_IN -> F704_IN U23^1(f704_out1) -> U24^1(f873_in) U23^1(f704_out1) -> F873_IN F889_IN -> U25^1(f896_in) F889_IN -> F896_IN U25^1(f896_out1(T110)) -> U26^1(f873_in, T110) U25^1(f896_out1(T110)) -> F873_IN F1054_IN -> U27^1(f918_in) F1054_IN -> F918_IN U27^1(f918_out1(T167)) -> U28^1(f873_in, T167) U27^1(f918_out1(T167)) -> F873_IN F1083_IN(T212, T216) -> U29^1(f1085_in(T212), T212, T216) F1083_IN(T212, T216) -> F1085_IN(T212) U29^1(f1085_out1, T212, T216) -> U30^1(f1059_in(T216), T212, T216) U29^1(f1085_out1, T212, T216) -> F1059_IN(T216) F1111_IN(T271, T273) -> U31^1(f1095_in(T271), T271, T273) F1111_IN(T271, T273) -> F1095_IN(T271) U31^1(f1095_out1, T271, T273) -> U32^1(f1059_in(T273), T271, T273) U31^1(f1095_out1, T271, T273) -> F1059_IN(T273) The TRS R consists of the following rules: f1_in([]) -> f1_out1 f1_in(.(T7, [])) -> f1_out1 f1_in(T19) -> U1(f35_in(T19), T19) U1(f35_out1(X25, X26), T19) -> f1_out1 f554_in -> f554_out1 f554_in -> U2(f554_in) U2(f554_out1) -> f554_out1 f97_in -> U3(f554_in) U3(f554_out1) -> f97_out1 f704_in -> f704_out1 f704_in -> U4(f862_in) U4(f862_out1) -> f704_out1 f873_in -> f873_out1 f873_in -> U5(f889_in) U5(f889_out1(T110)) -> f873_out1 f873_in -> U6(f1054_in) U6(f1054_out1(T167)) -> f873_out1 f918_in -> f918_out1(0) f918_in -> U7(f918_in) U7(f918_out1(T147)) -> f918_out1(s(T147)) f1059_in(T186) -> f1059_out1([], T186) f1059_in(T191) -> f1059_out1(T191, []) f1059_in(.(T212, T216)) -> U8(f1083_in(T212, T216), .(T212, T216)) U8(f1083_out1(T218, T217, T219), .(T212, T216)) -> f1059_out1(.(T212, T218), .(T217, T219)) f1059_in(.(T271, T273)) -> U9(f1111_in(T271, T273), .(T271, T273)) U9(f1111_out1(T274, T276, T275), .(T271, T273)) -> f1059_out1(.(T274, T276), .(T271, T275)) f1095_in(0) -> f1095_out1 f1095_in(s(T250)) -> U10(f1095_in(T250), s(T250)) U10(f1095_out1, s(T250)) -> f1095_out1 f896_in -> f896_out1(0) f896_in -> U11(f918_in) U11(f918_out1(T132)) -> f896_out1(s(T132)) f1085_in(0) -> f1085_out1 f1085_in(s(T236)) -> U12(f1095_in(T236), s(T236)) U12(f1095_out1, s(T236)) -> f1085_out1 f35_in(T19) -> U13(f97_in, T19) U13(f97_out1, T19) -> U14(f98_in(T19), T19) U14(f98_out1(X25, X26), T19) -> f35_out1(X25, X26) f98_in(T19) -> U15(f704_in, T19) U15(f704_out1, T19) -> U16(f705_in(T19), T19) U16(f705_out1(T53, X26), T19) -> f98_out1(T53, X26) f705_in(T19) -> U17(f704_in, T19) U17(f704_out1, T19) -> U18(f1059_in(T19), T19) U18(f1059_out1(T179, T178), T19) -> f705_out1(T179, T178) f862_in -> U19(f97_in) U19(f97_out1) -> U20(f865_in) U20(f865_out1) -> f862_out1 f865_in -> U21(f704_in) U21(f704_out1) -> U22(f869_in) U22(f869_out1) -> f865_out1 f869_in -> U23(f704_in) U23(f704_out1) -> U24(f873_in) U24(f873_out1) -> f869_out1 f889_in -> U25(f896_in) U25(f896_out1(T110)) -> U26(f873_in, T110) U26(f873_out1, T110) -> f889_out1(T110) f1054_in -> U27(f918_in) U27(f918_out1(T167)) -> U28(f873_in, T167) U28(f873_out1, T167) -> f1054_out1(T167) f1083_in(T212, T216) -> U29(f1085_in(T212), T212, T216) U29(f1085_out1, T212, T216) -> U30(f1059_in(T216), T212, T216) U30(f1059_out1(T222, .(T223, T224)), T212, T216) -> f1083_out1(T222, T223, T224) f1111_in(T271, T273) -> U31(f1095_in(T271), T271, T273) U31(f1095_out1, T271, T273) -> U32(f1059_in(T273), T271, T273) U32(f1059_out1(.(T279, T280), T281), T271, T273) -> f1111_out1(T279, T280, T281) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (85) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 6 SCCs with 42 less nodes. ---------------------------------------- (86) Complex Obligation (AND) ---------------------------------------- (87) Obligation: Q DP problem: The TRS P consists of the following rules: F1095_IN(s(T250)) -> F1095_IN(T250) The TRS R consists of the following rules: f1_in([]) -> f1_out1 f1_in(.(T7, [])) -> f1_out1 f1_in(T19) -> U1(f35_in(T19), T19) U1(f35_out1(X25, X26), T19) -> f1_out1 f554_in -> f554_out1 f554_in -> U2(f554_in) U2(f554_out1) -> f554_out1 f97_in -> U3(f554_in) U3(f554_out1) -> f97_out1 f704_in -> f704_out1 f704_in -> U4(f862_in) U4(f862_out1) -> f704_out1 f873_in -> f873_out1 f873_in -> U5(f889_in) U5(f889_out1(T110)) -> f873_out1 f873_in -> U6(f1054_in) U6(f1054_out1(T167)) -> f873_out1 f918_in -> f918_out1(0) f918_in -> U7(f918_in) U7(f918_out1(T147)) -> f918_out1(s(T147)) f1059_in(T186) -> f1059_out1([], T186) f1059_in(T191) -> f1059_out1(T191, []) f1059_in(.(T212, T216)) -> U8(f1083_in(T212, T216), .(T212, T216)) U8(f1083_out1(T218, T217, T219), .(T212, T216)) -> f1059_out1(.(T212, T218), .(T217, T219)) f1059_in(.(T271, T273)) -> U9(f1111_in(T271, T273), .(T271, T273)) U9(f1111_out1(T274, T276, T275), .(T271, T273)) -> f1059_out1(.(T274, T276), .(T271, T275)) f1095_in(0) -> f1095_out1 f1095_in(s(T250)) -> U10(f1095_in(T250), s(T250)) U10(f1095_out1, s(T250)) -> f1095_out1 f896_in -> f896_out1(0) f896_in -> U11(f918_in) U11(f918_out1(T132)) -> f896_out1(s(T132)) f1085_in(0) -> f1085_out1 f1085_in(s(T236)) -> U12(f1095_in(T236), s(T236)) U12(f1095_out1, s(T236)) -> f1085_out1 f35_in(T19) -> U13(f97_in, T19) U13(f97_out1, T19) -> U14(f98_in(T19), T19) U14(f98_out1(X25, X26), T19) -> f35_out1(X25, X26) f98_in(T19) -> U15(f704_in, T19) U15(f704_out1, T19) -> U16(f705_in(T19), T19) U16(f705_out1(T53, X26), T19) -> f98_out1(T53, X26) f705_in(T19) -> U17(f704_in, T19) U17(f704_out1, T19) -> U18(f1059_in(T19), T19) U18(f1059_out1(T179, T178), T19) -> f705_out1(T179, T178) f862_in -> U19(f97_in) U19(f97_out1) -> U20(f865_in) U20(f865_out1) -> f862_out1 f865_in -> U21(f704_in) U21(f704_out1) -> U22(f869_in) U22(f869_out1) -> f865_out1 f869_in -> U23(f704_in) U23(f704_out1) -> U24(f873_in) U24(f873_out1) -> f869_out1 f889_in -> U25(f896_in) U25(f896_out1(T110)) -> U26(f873_in, T110) U26(f873_out1, T110) -> f889_out1(T110) f1054_in -> U27(f918_in) U27(f918_out1(T167)) -> U28(f873_in, T167) U28(f873_out1, T167) -> f1054_out1(T167) f1083_in(T212, T216) -> U29(f1085_in(T212), T212, T216) U29(f1085_out1, T212, T216) -> U30(f1059_in(T216), T212, T216) U30(f1059_out1(T222, .(T223, T224)), T212, T216) -> f1083_out1(T222, T223, T224) f1111_in(T271, T273) -> U31(f1095_in(T271), T271, T273) U31(f1095_out1, T271, T273) -> U32(f1059_in(T273), T271, T273) U32(f1059_out1(.(T279, T280), T281), T271, T273) -> f1111_out1(T279, T280, T281) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (88) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (89) Obligation: Q DP problem: The TRS P consists of the following rules: F1095_IN(s(T250)) -> F1095_IN(T250) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (90) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *F1095_IN(s(T250)) -> F1095_IN(T250) The graph contains the following edges 1 > 1 ---------------------------------------- (91) YES ---------------------------------------- (92) Obligation: Q DP problem: The TRS P consists of the following rules: F1059_IN(.(T212, T216)) -> F1083_IN(T212, T216) F1083_IN(T212, T216) -> U29^1(f1085_in(T212), T212, T216) U29^1(f1085_out1, T212, T216) -> F1059_IN(T216) F1059_IN(.(T271, T273)) -> F1111_IN(T271, T273) F1111_IN(T271, T273) -> U31^1(f1095_in(T271), T271, T273) U31^1(f1095_out1, T271, T273) -> F1059_IN(T273) The TRS R consists of the following rules: f1_in([]) -> f1_out1 f1_in(.(T7, [])) -> f1_out1 f1_in(T19) -> U1(f35_in(T19), T19) U1(f35_out1(X25, X26), T19) -> f1_out1 f554_in -> f554_out1 f554_in -> U2(f554_in) U2(f554_out1) -> f554_out1 f97_in -> U3(f554_in) U3(f554_out1) -> f97_out1 f704_in -> f704_out1 f704_in -> U4(f862_in) U4(f862_out1) -> f704_out1 f873_in -> f873_out1 f873_in -> U5(f889_in) U5(f889_out1(T110)) -> f873_out1 f873_in -> U6(f1054_in) U6(f1054_out1(T167)) -> f873_out1 f918_in -> f918_out1(0) f918_in -> U7(f918_in) U7(f918_out1(T147)) -> f918_out1(s(T147)) f1059_in(T186) -> f1059_out1([], T186) f1059_in(T191) -> f1059_out1(T191, []) f1059_in(.(T212, T216)) -> U8(f1083_in(T212, T216), .(T212, T216)) U8(f1083_out1(T218, T217, T219), .(T212, T216)) -> f1059_out1(.(T212, T218), .(T217, T219)) f1059_in(.(T271, T273)) -> U9(f1111_in(T271, T273), .(T271, T273)) U9(f1111_out1(T274, T276, T275), .(T271, T273)) -> f1059_out1(.(T274, T276), .(T271, T275)) f1095_in(0) -> f1095_out1 f1095_in(s(T250)) -> U10(f1095_in(T250), s(T250)) U10(f1095_out1, s(T250)) -> f1095_out1 f896_in -> f896_out1(0) f896_in -> U11(f918_in) U11(f918_out1(T132)) -> f896_out1(s(T132)) f1085_in(0) -> f1085_out1 f1085_in(s(T236)) -> U12(f1095_in(T236), s(T236)) U12(f1095_out1, s(T236)) -> f1085_out1 f35_in(T19) -> U13(f97_in, T19) U13(f97_out1, T19) -> U14(f98_in(T19), T19) U14(f98_out1(X25, X26), T19) -> f35_out1(X25, X26) f98_in(T19) -> U15(f704_in, T19) U15(f704_out1, T19) -> U16(f705_in(T19), T19) U16(f705_out1(T53, X26), T19) -> f98_out1(T53, X26) f705_in(T19) -> U17(f704_in, T19) U17(f704_out1, T19) -> U18(f1059_in(T19), T19) U18(f1059_out1(T179, T178), T19) -> f705_out1(T179, T178) f862_in -> U19(f97_in) U19(f97_out1) -> U20(f865_in) U20(f865_out1) -> f862_out1 f865_in -> U21(f704_in) U21(f704_out1) -> U22(f869_in) U22(f869_out1) -> f865_out1 f869_in -> U23(f704_in) U23(f704_out1) -> U24(f873_in) U24(f873_out1) -> f869_out1 f889_in -> U25(f896_in) U25(f896_out1(T110)) -> U26(f873_in, T110) U26(f873_out1, T110) -> f889_out1(T110) f1054_in -> U27(f918_in) U27(f918_out1(T167)) -> U28(f873_in, T167) U28(f873_out1, T167) -> f1054_out1(T167) f1083_in(T212, T216) -> U29(f1085_in(T212), T212, T216) U29(f1085_out1, T212, T216) -> U30(f1059_in(T216), T212, T216) U30(f1059_out1(T222, .(T223, T224)), T212, T216) -> f1083_out1(T222, T223, T224) f1111_in(T271, T273) -> U31(f1095_in(T271), T271, T273) U31(f1095_out1, T271, T273) -> U32(f1059_in(T273), T271, T273) U32(f1059_out1(.(T279, T280), T281), T271, T273) -> f1111_out1(T279, T280, T281) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (93) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *F1083_IN(T212, T216) -> U29^1(f1085_in(T212), T212, T216) The graph contains the following edges 1 >= 2, 2 >= 3 *U29^1(f1085_out1, T212, T216) -> F1059_IN(T216) The graph contains the following edges 3 >= 1 *U31^1(f1095_out1, T271, T273) -> F1059_IN(T273) The graph contains the following edges 3 >= 1 *F1059_IN(.(T212, T216)) -> F1083_IN(T212, T216) The graph contains the following edges 1 > 1, 1 > 2 *F1059_IN(.(T271, T273)) -> F1111_IN(T271, T273) The graph contains the following edges 1 > 1, 1 > 2 *F1111_IN(T271, T273) -> U31^1(f1095_in(T271), T271, T273) The graph contains the following edges 1 >= 2, 2 >= 3 ---------------------------------------- (94) YES ---------------------------------------- (95) Obligation: Q DP problem: The TRS P consists of the following rules: F918_IN -> F918_IN The TRS R consists of the following rules: f1_in([]) -> f1_out1 f1_in(.(T7, [])) -> f1_out1 f1_in(T19) -> U1(f35_in(T19), T19) U1(f35_out1(X25, X26), T19) -> f1_out1 f554_in -> f554_out1 f554_in -> U2(f554_in) U2(f554_out1) -> f554_out1 f97_in -> U3(f554_in) U3(f554_out1) -> f97_out1 f704_in -> f704_out1 f704_in -> U4(f862_in) U4(f862_out1) -> f704_out1 f873_in -> f873_out1 f873_in -> U5(f889_in) U5(f889_out1(T110)) -> f873_out1 f873_in -> U6(f1054_in) U6(f1054_out1(T167)) -> f873_out1 f918_in -> f918_out1(0) f918_in -> U7(f918_in) U7(f918_out1(T147)) -> f918_out1(s(T147)) f1059_in(T186) -> f1059_out1([], T186) f1059_in(T191) -> f1059_out1(T191, []) f1059_in(.(T212, T216)) -> U8(f1083_in(T212, T216), .(T212, T216)) U8(f1083_out1(T218, T217, T219), .(T212, T216)) -> f1059_out1(.(T212, T218), .(T217, T219)) f1059_in(.(T271, T273)) -> U9(f1111_in(T271, T273), .(T271, T273)) U9(f1111_out1(T274, T276, T275), .(T271, T273)) -> f1059_out1(.(T274, T276), .(T271, T275)) f1095_in(0) -> f1095_out1 f1095_in(s(T250)) -> U10(f1095_in(T250), s(T250)) U10(f1095_out1, s(T250)) -> f1095_out1 f896_in -> f896_out1(0) f896_in -> U11(f918_in) U11(f918_out1(T132)) -> f896_out1(s(T132)) f1085_in(0) -> f1085_out1 f1085_in(s(T236)) -> U12(f1095_in(T236), s(T236)) U12(f1095_out1, s(T236)) -> f1085_out1 f35_in(T19) -> U13(f97_in, T19) U13(f97_out1, T19) -> U14(f98_in(T19), T19) U14(f98_out1(X25, X26), T19) -> f35_out1(X25, X26) f98_in(T19) -> U15(f704_in, T19) U15(f704_out1, T19) -> U16(f705_in(T19), T19) U16(f705_out1(T53, X26), T19) -> f98_out1(T53, X26) f705_in(T19) -> U17(f704_in, T19) U17(f704_out1, T19) -> U18(f1059_in(T19), T19) U18(f1059_out1(T179, T178), T19) -> f705_out1(T179, T178) f862_in -> U19(f97_in) U19(f97_out1) -> U20(f865_in) U20(f865_out1) -> f862_out1 f865_in -> U21(f704_in) U21(f704_out1) -> U22(f869_in) U22(f869_out1) -> f865_out1 f869_in -> U23(f704_in) U23(f704_out1) -> U24(f873_in) U24(f873_out1) -> f869_out1 f889_in -> U25(f896_in) U25(f896_out1(T110)) -> U26(f873_in, T110) U26(f873_out1, T110) -> f889_out1(T110) f1054_in -> U27(f918_in) U27(f918_out1(T167)) -> U28(f873_in, T167) U28(f873_out1, T167) -> f1054_out1(T167) f1083_in(T212, T216) -> U29(f1085_in(T212), T212, T216) U29(f1085_out1, T212, T216) -> U30(f1059_in(T216), T212, T216) U30(f1059_out1(T222, .(T223, T224)), T212, T216) -> f1083_out1(T222, T223, T224) f1111_in(T271, T273) -> U31(f1095_in(T271), T271, T273) U31(f1095_out1, T271, T273) -> U32(f1059_in(T273), T271, T273) U32(f1059_out1(.(T279, T280), T281), T271, T273) -> f1111_out1(T279, T280, T281) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (96) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (97) Obligation: Q DP problem: The TRS P consists of the following rules: F918_IN -> F918_IN R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (98) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by semiunifying a rule from P directly. s = F918_IN evaluates to t =F918_IN Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence The DP semiunifies directly so there is only one rewrite step from F918_IN to F918_IN. ---------------------------------------- (99) NO ---------------------------------------- (100) Obligation: Q DP problem: The TRS P consists of the following rules: F873_IN -> F889_IN F889_IN -> U25^1(f896_in) U25^1(f896_out1(T110)) -> F873_IN F873_IN -> F1054_IN F1054_IN -> U27^1(f918_in) U27^1(f918_out1(T167)) -> F873_IN The TRS R consists of the following rules: f1_in([]) -> f1_out1 f1_in(.(T7, [])) -> f1_out1 f1_in(T19) -> U1(f35_in(T19), T19) U1(f35_out1(X25, X26), T19) -> f1_out1 f554_in -> f554_out1 f554_in -> U2(f554_in) U2(f554_out1) -> f554_out1 f97_in -> U3(f554_in) U3(f554_out1) -> f97_out1 f704_in -> f704_out1 f704_in -> U4(f862_in) U4(f862_out1) -> f704_out1 f873_in -> f873_out1 f873_in -> U5(f889_in) U5(f889_out1(T110)) -> f873_out1 f873_in -> U6(f1054_in) U6(f1054_out1(T167)) -> f873_out1 f918_in -> f918_out1(0) f918_in -> U7(f918_in) U7(f918_out1(T147)) -> f918_out1(s(T147)) f1059_in(T186) -> f1059_out1([], T186) f1059_in(T191) -> f1059_out1(T191, []) f1059_in(.(T212, T216)) -> U8(f1083_in(T212, T216), .(T212, T216)) U8(f1083_out1(T218, T217, T219), .(T212, T216)) -> f1059_out1(.(T212, T218), .(T217, T219)) f1059_in(.(T271, T273)) -> U9(f1111_in(T271, T273), .(T271, T273)) U9(f1111_out1(T274, T276, T275), .(T271, T273)) -> f1059_out1(.(T274, T276), .(T271, T275)) f1095_in(0) -> f1095_out1 f1095_in(s(T250)) -> U10(f1095_in(T250), s(T250)) U10(f1095_out1, s(T250)) -> f1095_out1 f896_in -> f896_out1(0) f896_in -> U11(f918_in) U11(f918_out1(T132)) -> f896_out1(s(T132)) f1085_in(0) -> f1085_out1 f1085_in(s(T236)) -> U12(f1095_in(T236), s(T236)) U12(f1095_out1, s(T236)) -> f1085_out1 f35_in(T19) -> U13(f97_in, T19) U13(f97_out1, T19) -> U14(f98_in(T19), T19) U14(f98_out1(X25, X26), T19) -> f35_out1(X25, X26) f98_in(T19) -> U15(f704_in, T19) U15(f704_out1, T19) -> U16(f705_in(T19), T19) U16(f705_out1(T53, X26), T19) -> f98_out1(T53, X26) f705_in(T19) -> U17(f704_in, T19) U17(f704_out1, T19) -> U18(f1059_in(T19), T19) U18(f1059_out1(T179, T178), T19) -> f705_out1(T179, T178) f862_in -> U19(f97_in) U19(f97_out1) -> U20(f865_in) U20(f865_out1) -> f862_out1 f865_in -> U21(f704_in) U21(f704_out1) -> U22(f869_in) U22(f869_out1) -> f865_out1 f869_in -> U23(f704_in) U23(f704_out1) -> U24(f873_in) U24(f873_out1) -> f869_out1 f889_in -> U25(f896_in) U25(f896_out1(T110)) -> U26(f873_in, T110) U26(f873_out1, T110) -> f889_out1(T110) f1054_in -> U27(f918_in) U27(f918_out1(T167)) -> U28(f873_in, T167) U28(f873_out1, T167) -> f1054_out1(T167) f1083_in(T212, T216) -> U29(f1085_in(T212), T212, T216) U29(f1085_out1, T212, T216) -> U30(f1059_in(T216), T212, T216) U30(f1059_out1(T222, .(T223, T224)), T212, T216) -> f1083_out1(T222, T223, T224) f1111_in(T271, T273) -> U31(f1095_in(T271), T271, T273) U31(f1095_out1, T271, T273) -> U32(f1059_in(T273), T271, T273) U32(f1059_out1(.(T279, T280), T281), T271, T273) -> f1111_out1(T279, T280, T281) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (101) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (102) Obligation: Q DP problem: The TRS P consists of the following rules: F873_IN -> F889_IN F889_IN -> U25^1(f896_in) U25^1(f896_out1(T110)) -> F873_IN F873_IN -> F1054_IN F1054_IN -> U27^1(f918_in) U27^1(f918_out1(T167)) -> F873_IN The TRS R consists of the following rules: f918_in -> f918_out1(0) f918_in -> U7(f918_in) U7(f918_out1(T147)) -> f918_out1(s(T147)) f896_in -> f896_out1(0) f896_in -> U11(f918_in) U11(f918_out1(T132)) -> f896_out1(s(T132)) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (103) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule F889_IN -> U25^1(f896_in) at position [0] we obtained the following new rules [LPAR04]: (F889_IN -> U25^1(f896_out1(0)),F889_IN -> U25^1(f896_out1(0))) (F889_IN -> U25^1(U11(f918_in)),F889_IN -> U25^1(U11(f918_in))) ---------------------------------------- (104) Obligation: Q DP problem: The TRS P consists of the following rules: F873_IN -> F889_IN U25^1(f896_out1(T110)) -> F873_IN F873_IN -> F1054_IN F1054_IN -> U27^1(f918_in) U27^1(f918_out1(T167)) -> F873_IN F889_IN -> U25^1(f896_out1(0)) F889_IN -> U25^1(U11(f918_in)) The TRS R consists of the following rules: f918_in -> f918_out1(0) f918_in -> U7(f918_in) U7(f918_out1(T147)) -> f918_out1(s(T147)) f896_in -> f896_out1(0) f896_in -> U11(f918_in) U11(f918_out1(T132)) -> f896_out1(s(T132)) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (105) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (106) Obligation: Q DP problem: The TRS P consists of the following rules: F873_IN -> F889_IN U25^1(f896_out1(T110)) -> F873_IN F873_IN -> F1054_IN F1054_IN -> U27^1(f918_in) U27^1(f918_out1(T167)) -> F873_IN F889_IN -> U25^1(f896_out1(0)) F889_IN -> U25^1(U11(f918_in)) The TRS R consists of the following rules: f918_in -> f918_out1(0) f918_in -> U7(f918_in) U11(f918_out1(T132)) -> f896_out1(s(T132)) U7(f918_out1(T147)) -> f918_out1(s(T147)) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (107) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule F1054_IN -> U27^1(f918_in) at position [0] we obtained the following new rules [LPAR04]: (F1054_IN -> U27^1(f918_out1(0)),F1054_IN -> U27^1(f918_out1(0))) (F1054_IN -> U27^1(U7(f918_in)),F1054_IN -> U27^1(U7(f918_in))) ---------------------------------------- (108) Obligation: Q DP problem: The TRS P consists of the following rules: F873_IN -> F889_IN U25^1(f896_out1(T110)) -> F873_IN F873_IN -> F1054_IN U27^1(f918_out1(T167)) -> F873_IN F889_IN -> U25^1(f896_out1(0)) F889_IN -> U25^1(U11(f918_in)) F1054_IN -> U27^1(f918_out1(0)) F1054_IN -> U27^1(U7(f918_in)) The TRS R consists of the following rules: f918_in -> f918_out1(0) f918_in -> U7(f918_in) U11(f918_out1(T132)) -> f896_out1(s(T132)) U7(f918_out1(T147)) -> f918_out1(s(T147)) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (109) Obligation: Q DP problem: The TRS P consists of the following rules: F554_IN -> F554_IN The TRS R consists of the following rules: f1_in([]) -> f1_out1 f1_in(.(T7, [])) -> f1_out1 f1_in(T19) -> U1(f35_in(T19), T19) U1(f35_out1(X25, X26), T19) -> f1_out1 f554_in -> f554_out1 f554_in -> U2(f554_in) U2(f554_out1) -> f554_out1 f97_in -> U3(f554_in) U3(f554_out1) -> f97_out1 f704_in -> f704_out1 f704_in -> U4(f862_in) U4(f862_out1) -> f704_out1 f873_in -> f873_out1 f873_in -> U5(f889_in) U5(f889_out1(T110)) -> f873_out1 f873_in -> U6(f1054_in) U6(f1054_out1(T167)) -> f873_out1 f918_in -> f918_out1(0) f918_in -> U7(f918_in) U7(f918_out1(T147)) -> f918_out1(s(T147)) f1059_in(T186) -> f1059_out1([], T186) f1059_in(T191) -> f1059_out1(T191, []) f1059_in(.(T212, T216)) -> U8(f1083_in(T212, T216), .(T212, T216)) U8(f1083_out1(T218, T217, T219), .(T212, T216)) -> f1059_out1(.(T212, T218), .(T217, T219)) f1059_in(.(T271, T273)) -> U9(f1111_in(T271, T273), .(T271, T273)) U9(f1111_out1(T274, T276, T275), .(T271, T273)) -> f1059_out1(.(T274, T276), .(T271, T275)) f1095_in(0) -> f1095_out1 f1095_in(s(T250)) -> U10(f1095_in(T250), s(T250)) U10(f1095_out1, s(T250)) -> f1095_out1 f896_in -> f896_out1(0) f896_in -> U11(f918_in) U11(f918_out1(T132)) -> f896_out1(s(T132)) f1085_in(0) -> f1085_out1 f1085_in(s(T236)) -> U12(f1095_in(T236), s(T236)) U12(f1095_out1, s(T236)) -> f1085_out1 f35_in(T19) -> U13(f97_in, T19) U13(f97_out1, T19) -> U14(f98_in(T19), T19) U14(f98_out1(X25, X26), T19) -> f35_out1(X25, X26) f98_in(T19) -> U15(f704_in, T19) U15(f704_out1, T19) -> U16(f705_in(T19), T19) U16(f705_out1(T53, X26), T19) -> f98_out1(T53, X26) f705_in(T19) -> U17(f704_in, T19) U17(f704_out1, T19) -> U18(f1059_in(T19), T19) U18(f1059_out1(T179, T178), T19) -> f705_out1(T179, T178) f862_in -> U19(f97_in) U19(f97_out1) -> U20(f865_in) U20(f865_out1) -> f862_out1 f865_in -> U21(f704_in) U21(f704_out1) -> U22(f869_in) U22(f869_out1) -> f865_out1 f869_in -> U23(f704_in) U23(f704_out1) -> U24(f873_in) U24(f873_out1) -> f869_out1 f889_in -> U25(f896_in) U25(f896_out1(T110)) -> U26(f873_in, T110) U26(f873_out1, T110) -> f889_out1(T110) f1054_in -> U27(f918_in) U27(f918_out1(T167)) -> U28(f873_in, T167) U28(f873_out1, T167) -> f1054_out1(T167) f1083_in(T212, T216) -> U29(f1085_in(T212), T212, T216) U29(f1085_out1, T212, T216) -> U30(f1059_in(T216), T212, T216) U30(f1059_out1(T222, .(T223, T224)), T212, T216) -> f1083_out1(T222, T223, T224) f1111_in(T271, T273) -> U31(f1095_in(T271), T271, T273) U31(f1095_out1, T271, T273) -> U32(f1059_in(T273), T271, T273) U32(f1059_out1(.(T279, T280), T281), T271, T273) -> f1111_out1(T279, T280, T281) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (110) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (111) Obligation: Q DP problem: The TRS P consists of the following rules: F554_IN -> F554_IN R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (112) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by semiunifying a rule from P directly. s = F554_IN evaluates to t =F554_IN Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence The DP semiunifies directly so there is only one rewrite step from F554_IN to F554_IN. ---------------------------------------- (113) NO ---------------------------------------- (114) Obligation: Q DP problem: The TRS P consists of the following rules: F862_IN -> U19^1(f97_in) U19^1(f97_out1) -> F865_IN F865_IN -> U21^1(f704_in) U21^1(f704_out1) -> F869_IN F869_IN -> F704_IN F704_IN -> F862_IN F865_IN -> F704_IN The TRS R consists of the following rules: f1_in([]) -> f1_out1 f1_in(.(T7, [])) -> f1_out1 f1_in(T19) -> U1(f35_in(T19), T19) U1(f35_out1(X25, X26), T19) -> f1_out1 f554_in -> f554_out1 f554_in -> U2(f554_in) U2(f554_out1) -> f554_out1 f97_in -> U3(f554_in) U3(f554_out1) -> f97_out1 f704_in -> f704_out1 f704_in -> U4(f862_in) U4(f862_out1) -> f704_out1 f873_in -> f873_out1 f873_in -> U5(f889_in) U5(f889_out1(T110)) -> f873_out1 f873_in -> U6(f1054_in) U6(f1054_out1(T167)) -> f873_out1 f918_in -> f918_out1(0) f918_in -> U7(f918_in) U7(f918_out1(T147)) -> f918_out1(s(T147)) f1059_in(T186) -> f1059_out1([], T186) f1059_in(T191) -> f1059_out1(T191, []) f1059_in(.(T212, T216)) -> U8(f1083_in(T212, T216), .(T212, T216)) U8(f1083_out1(T218, T217, T219), .(T212, T216)) -> f1059_out1(.(T212, T218), .(T217, T219)) f1059_in(.(T271, T273)) -> U9(f1111_in(T271, T273), .(T271, T273)) U9(f1111_out1(T274, T276, T275), .(T271, T273)) -> f1059_out1(.(T274, T276), .(T271, T275)) f1095_in(0) -> f1095_out1 f1095_in(s(T250)) -> U10(f1095_in(T250), s(T250)) U10(f1095_out1, s(T250)) -> f1095_out1 f896_in -> f896_out1(0) f896_in -> U11(f918_in) U11(f918_out1(T132)) -> f896_out1(s(T132)) f1085_in(0) -> f1085_out1 f1085_in(s(T236)) -> U12(f1095_in(T236), s(T236)) U12(f1095_out1, s(T236)) -> f1085_out1 f35_in(T19) -> U13(f97_in, T19) U13(f97_out1, T19) -> U14(f98_in(T19), T19) U14(f98_out1(X25, X26), T19) -> f35_out1(X25, X26) f98_in(T19) -> U15(f704_in, T19) U15(f704_out1, T19) -> U16(f705_in(T19), T19) U16(f705_out1(T53, X26), T19) -> f98_out1(T53, X26) f705_in(T19) -> U17(f704_in, T19) U17(f704_out1, T19) -> U18(f1059_in(T19), T19) U18(f1059_out1(T179, T178), T19) -> f705_out1(T179, T178) f862_in -> U19(f97_in) U19(f97_out1) -> U20(f865_in) U20(f865_out1) -> f862_out1 f865_in -> U21(f704_in) U21(f704_out1) -> U22(f869_in) U22(f869_out1) -> f865_out1 f869_in -> U23(f704_in) U23(f704_out1) -> U24(f873_in) U24(f873_out1) -> f869_out1 f889_in -> U25(f896_in) U25(f896_out1(T110)) -> U26(f873_in, T110) U26(f873_out1, T110) -> f889_out1(T110) f1054_in -> U27(f918_in) U27(f918_out1(T167)) -> U28(f873_in, T167) U28(f873_out1, T167) -> f1054_out1(T167) f1083_in(T212, T216) -> U29(f1085_in(T212), T212, T216) U29(f1085_out1, T212, T216) -> U30(f1059_in(T216), T212, T216) U30(f1059_out1(T222, .(T223, T224)), T212, T216) -> f1083_out1(T222, T223, T224) f1111_in(T271, T273) -> U31(f1095_in(T271), T271, T273) U31(f1095_out1, T271, T273) -> U32(f1059_in(T273), T271, T273) U32(f1059_out1(.(T279, T280), T281), T271, T273) -> f1111_out1(T279, T280, T281) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (115) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (116) Obligation: Q DP problem: The TRS P consists of the following rules: F862_IN -> U19^1(f97_in) U19^1(f97_out1) -> F865_IN F865_IN -> U21^1(f704_in) U21^1(f704_out1) -> F869_IN F869_IN -> F704_IN F704_IN -> F862_IN F865_IN -> F704_IN The TRS R consists of the following rules: f704_in -> f704_out1 f704_in -> U4(f862_in) f862_in -> U19(f97_in) U4(f862_out1) -> f704_out1 f97_in -> U3(f554_in) U19(f97_out1) -> U20(f865_in) f865_in -> U21(f704_in) U20(f865_out1) -> f862_out1 U21(f704_out1) -> U22(f869_in) f869_in -> U23(f704_in) U22(f869_out1) -> f865_out1 U23(f704_out1) -> U24(f873_in) f873_in -> f873_out1 f873_in -> U5(f889_in) f873_in -> U6(f1054_in) U24(f873_out1) -> f869_out1 f1054_in -> U27(f918_in) U6(f1054_out1(T167)) -> f873_out1 f918_in -> f918_out1(0) f918_in -> U7(f918_in) U27(f918_out1(T167)) -> U28(f873_in, T167) U28(f873_out1, T167) -> f1054_out1(T167) U7(f918_out1(T147)) -> f918_out1(s(T147)) f889_in -> U25(f896_in) U5(f889_out1(T110)) -> f873_out1 f896_in -> f896_out1(0) f896_in -> U11(f918_in) U25(f896_out1(T110)) -> U26(f873_in, T110) U26(f873_out1, T110) -> f889_out1(T110) U11(f918_out1(T132)) -> f896_out1(s(T132)) f554_in -> f554_out1 f554_in -> U2(f554_in) U3(f554_out1) -> f97_out1 U2(f554_out1) -> f554_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (117) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule F862_IN -> U19^1(f97_in) at position [0] we obtained the following new rules [LPAR04]: (F862_IN -> U19^1(U3(f554_in)),F862_IN -> U19^1(U3(f554_in))) ---------------------------------------- (118) Obligation: Q DP problem: The TRS P consists of the following rules: U19^1(f97_out1) -> F865_IN F865_IN -> U21^1(f704_in) U21^1(f704_out1) -> F869_IN F869_IN -> F704_IN F704_IN -> F862_IN F865_IN -> F704_IN F862_IN -> U19^1(U3(f554_in)) The TRS R consists of the following rules: f704_in -> f704_out1 f704_in -> U4(f862_in) f862_in -> U19(f97_in) U4(f862_out1) -> f704_out1 f97_in -> U3(f554_in) U19(f97_out1) -> U20(f865_in) f865_in -> U21(f704_in) U20(f865_out1) -> f862_out1 U21(f704_out1) -> U22(f869_in) f869_in -> U23(f704_in) U22(f869_out1) -> f865_out1 U23(f704_out1) -> U24(f873_in) f873_in -> f873_out1 f873_in -> U5(f889_in) f873_in -> U6(f1054_in) U24(f873_out1) -> f869_out1 f1054_in -> U27(f918_in) U6(f1054_out1(T167)) -> f873_out1 f918_in -> f918_out1(0) f918_in -> U7(f918_in) U27(f918_out1(T167)) -> U28(f873_in, T167) U28(f873_out1, T167) -> f1054_out1(T167) U7(f918_out1(T147)) -> f918_out1(s(T147)) f889_in -> U25(f896_in) U5(f889_out1(T110)) -> f873_out1 f896_in -> f896_out1(0) f896_in -> U11(f918_in) U25(f896_out1(T110)) -> U26(f873_in, T110) U26(f873_out1, T110) -> f889_out1(T110) U11(f918_out1(T132)) -> f896_out1(s(T132)) f554_in -> f554_out1 f554_in -> U2(f554_in) U3(f554_out1) -> f97_out1 U2(f554_out1) -> f554_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (119) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule F865_IN -> U21^1(f704_in) at position [0] we obtained the following new rules [LPAR04]: (F865_IN -> U21^1(f704_out1),F865_IN -> U21^1(f704_out1)) (F865_IN -> U21^1(U4(f862_in)),F865_IN -> U21^1(U4(f862_in))) ---------------------------------------- (120) Obligation: Q DP problem: The TRS P consists of the following rules: U19^1(f97_out1) -> F865_IN U21^1(f704_out1) -> F869_IN F869_IN -> F704_IN F704_IN -> F862_IN F865_IN -> F704_IN F862_IN -> U19^1(U3(f554_in)) F865_IN -> U21^1(f704_out1) F865_IN -> U21^1(U4(f862_in)) The TRS R consists of the following rules: f704_in -> f704_out1 f704_in -> U4(f862_in) f862_in -> U19(f97_in) U4(f862_out1) -> f704_out1 f97_in -> U3(f554_in) U19(f97_out1) -> U20(f865_in) f865_in -> U21(f704_in) U20(f865_out1) -> f862_out1 U21(f704_out1) -> U22(f869_in) f869_in -> U23(f704_in) U22(f869_out1) -> f865_out1 U23(f704_out1) -> U24(f873_in) f873_in -> f873_out1 f873_in -> U5(f889_in) f873_in -> U6(f1054_in) U24(f873_out1) -> f869_out1 f1054_in -> U27(f918_in) U6(f1054_out1(T167)) -> f873_out1 f918_in -> f918_out1(0) f918_in -> U7(f918_in) U27(f918_out1(T167)) -> U28(f873_in, T167) U28(f873_out1, T167) -> f1054_out1(T167) U7(f918_out1(T147)) -> f918_out1(s(T147)) f889_in -> U25(f896_in) U5(f889_out1(T110)) -> f873_out1 f896_in -> f896_out1(0) f896_in -> U11(f918_in) U25(f896_out1(T110)) -> U26(f873_in, T110) U26(f873_out1, T110) -> f889_out1(T110) U11(f918_out1(T132)) -> f896_out1(s(T132)) f554_in -> f554_out1 f554_in -> U2(f554_in) U3(f554_out1) -> f97_out1 U2(f554_out1) -> f554_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (121) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 3, "program": { "directives": [], "clauses": [ [ "(ms ([]) ([]))", null ], [ "(ms (. X ([])) (. X ([])))", null ], [ "(ms (. X (. Y Xs)) Ys)", "(',' (split (. X (. Y Xs)) X1s X2s) (',' (ms X1s Y1s) (',' (ms X2s Y2s) (merge Y1s Y2s Ys))))" ], [ "(split ([]) ([]) ([]))", null ], [ "(split (. X Xs) (. X Ys) Zs)", "(split Xs Zs Ys)" ], [ "(merge ([]) Xs Xs)", null ], [ "(merge Xs ([]) Xs)", null ], [ "(merge (. X Xs) (. Y Ys) (. X Zs))", "(',' (less X (s Y)) (merge Xs (. Y Ys) Zs))" ], [ "(merge (. X Xs) (. Y Ys) (. Y Zs))", "(',' (less Y X) (merge (. X Xs) Ys Zs))" ], [ "(less (0) (s X1))", null ], [ "(less (s X) (s Y))", "(less X Y)" ] ] }, "graph": { "nodes": { "907": { "goal": [{ "clause": -1, "scope": -1, "term": "(merge (. T173 T174) T175 X200)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X200"], "exprvars": [] } }, "type": "Nodes", "759": { "goal": [{ "clause": -1, "scope": -1, "term": "(ms T72 X106)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X106"], "exprvars": [] } }, "10": { "goal": [ { "clause": 1, "scope": 1, "term": "(ms T1 T2)" }, { "clause": 2, "scope": 1, "term": "(ms T1 T2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "17": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "18": { "goal": [], "kb": { 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"ground": [], "free": [], "exprvars": [] } }, "1047": { "goal": [{ "clause": 10, "scope": 10, "term": "(less T212 (s T217))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T212"], "free": [], "exprvars": [] } }, "1046": { "goal": [{ "clause": 9, "scope": 10, "term": "(less T212 (s T217))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T212"], "free": [], "exprvars": [] } }, "902": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "906": { "goal": [{ "clause": -1, "scope": -1, "term": "(less T167 T168)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 3, "to": 6, "label": "CASE" }, { "from": 6, "to": 9, "label": "PARALLEL" }, { "from": 6, "to": 10, "label": "PARALLEL" }, { "from": 9, "to": 17, "label": "EVAL with clause\nms([], []).\nand substitutionT1 -> [],\nT2 -> []" }, { "from": 9, "to": 18, "label": "EVAL-BACKTRACK" }, { "from": 10, "to": 20, "label": "PARALLEL" }, { "from": 10, "to": 21, "label": "PARALLEL" }, { "from": 17, "to": 19, "label": "SUCCESS" }, { "from": 20, "to": 22, "label": "EVAL with clause\nms(.(X6, []), .(X6, [])).\nand substitutionX6 -> T7,\nT1 -> .(T7, []),\nT2 -> .(T7, [])" }, { "from": 20, "to": 23, "label": "EVAL-BACKTRACK" }, { "from": 21, "to": 38, "label": "EVAL with clause\nms(.(X19, .(X20, X21)), X22) :- ','(split(.(X19, .(X20, X21)), X23, X24), ','(ms(X23, X25), ','(ms(X24, X26), merge(X25, X26, X22)))).\nand substitutionX19 -> T20,\nX20 -> T21,\nX21 -> T22,\nT1 -> .(T20, .(T21, T22)),\nT2 -> T19,\nX22 -> T19,\nT16 -> T20,\nT17 -> T21,\nT18 -> T22" }, { "from": 21, "to": 39, "label": "EVAL-BACKTRACK" }, { "from": 22, "to": 24, "label": "SUCCESS" }, { "from": 38, "to": 101, "label": "SPLIT 1" }, { "from": 38, "to": 102, "label": "SPLIT 2\nreplacements:X23 -> T23,\nX24 -> T24" }, { "from": 101, "to": 208, "label": "CASE" }, { "from": 102, "to": 567, "label": "SPLIT 1" }, { "from": 102, "to": 568, "label": "SPLIT 2\nreplacements:X25 -> T53,\nT24 -> T54" }, { "from": 208, "to": 217, "label": "BACKTRACK\nfor clause: split([], [], [])because of non-unification" }, { "from": 217, "to": 277, "label": "ONLY EVAL with clause\nsplit(.(X39, X40), .(X39, X41), X42) :- split(X40, X42, X41).\nand substitutionT20 -> T33,\nX39 -> T33,\nT21 -> T36,\nT22 -> T37,\nX40 -> .(T36, T37),\nX41 -> X43,\nX23 -> .(T33, X43),\nX24 -> X44,\nX42 -> X44,\nT34 -> T36,\nT35 -> T37" }, { "from": 277, "to": 278, "label": "CASE" }, { "from": 278, "to": 279, "label": "BACKTRACK\nfor clause: split([], [], [])because of non-unification" }, { "from": 279, "to": 280, "label": "ONLY EVAL with clause\nsplit(.(X57, X58), .(X57, X59), X60) :- split(X58, X60, X59).\nand substitutionT36 -> T43,\nX57 -> T43,\nT37 -> T45,\nX58 -> T45,\nX59 -> X61,\nX44 -> .(T43, X61),\nX43 -> X62,\nX60 -> X62,\nT44 -> T45" }, { "from": 280, "to": 281, "label": "CASE" }, { "from": 281, "to": 282, "label": "PARALLEL" }, { "from": 281, "to": 283, "label": "PARALLEL" }, { "from": 282, "to": 285, "label": "EVAL with clause\nsplit([], [], []).\nand substitutionT45 -> [],\nX62 -> [],\nX61 -> []" }, { "from": 282, "to": 287, "label": "EVAL-BACKTRACK" }, { "from": 283, "to": 335, "label": "EVAL with clause\nsplit(.(X75, X76), .(X75, X77), X78) :- split(X76, X78, X77).\nand substitutionX75 -> T50,\nX76 -> T52,\nT45 -> .(T50, T52),\nX77 -> X79,\nX62 -> .(T50, X79),\nX61 -> X80,\nX78 -> X80,\nT51 -> T52" }, { "from": 283, "to": 345, "label": "EVAL-BACKTRACK" }, { "from": 285, "to": 289, "label": "SUCCESS" }, { "from": 335, "to": 280, "label": "INSTANCE with matching:\nT45 -> T52\nX62 -> X80\nX61 -> X79" }, { "from": 567, "to": 570, "label": "CASE" }, { "from": 568, "to": 936, "label": "SPLIT 1" }, { "from": 568, "to": 1003, "label": "SPLIT 2\nreplacements:X26 -> T178,\nT53 -> T179" }, { "from": 570, "to": 571, "label": "PARALLEL" }, { "from": 570, "to": 572, "label": "PARALLEL" }, { "from": 571, "to": 573, "label": "EVAL with clause\nms([], []).\nand substitutionT23 -> [],\nX25 -> []" }, { "from": 571, "to": 574, "label": "EVAL-BACKTRACK" }, { "from": 572, "to": 670, "label": "PARALLEL" }, { "from": 572, "to": 672, "label": "PARALLEL" }, { "from": 573, "to": 575, "label": "SUCCESS" }, { "from": 670, "to": 679, "label": "EVAL with clause\nms(.(X85, []), .(X85, [])).\nand substitutionX85 -> T59,\nT23 -> .(T59, []),\nX25 -> .(T59, [])" }, { "from": 670, "to": 682, "label": "EVAL-BACKTRACK" }, { "from": 672, "to": 706, "label": "EVAL with clause\nms(.(X100, .(X101, X102)), X103) :- ','(split(.(X100, .(X101, X102)), X104, X105), ','(ms(X104, X106), ','(ms(X105, X107), merge(X106, X107, X103)))).\nand substitutionX100 -> T69,\nX101 -> T70,\nX102 -> T71,\nT23 -> .(T69, .(T70, T71)),\nX25 -> X108,\nX103 -> X108,\nT66 -> T69,\nT67 -> T70,\nT68 -> T71" }, { "from": 672, "to": 718, "label": "EVAL-BACKTRACK" }, { "from": 679, "to": 684, "label": "SUCCESS" }, { "from": 706, "to": 733, "label": "SPLIT 1" }, { "from": 706, "to": 735, "label": "SPLIT 2\nreplacements:X104 -> T72,\nX105 -> T73" }, { "from": 733, "to": 101, "label": "INSTANCE with matching:\nT20 -> T69\nT21 -> T70\nT22 -> T71\nX23 -> X104\nX24 -> X105" }, { "from": 735, "to": 759, "label": "SPLIT 1" }, { "from": 735, "to": 761, "label": "SPLIT 2\nreplacements:X106 -> T74,\nT73 -> T75" }, { "from": 759, "to": 567, "label": "INSTANCE with matching:\nT23 -> T72\nX25 -> X106" }, { "from": 761, "to": 790, "label": "SPLIT 1" }, { "from": 761, "to": 791, "label": "SPLIT 2\nreplacements:X107 -> T76,\nT74 -> T77" }, { "from": 790, "to": 567, "label": "INSTANCE with matching:\nT23 -> T75\nX25 -> X107" }, { "from": 791, "to": 808, "label": "CASE" }, { "from": 808, "to": 815, "label": "PARALLEL" }, { "from": 808, "to": 817, "label": "PARALLEL" }, { "from": 815, "to": 824, "label": "EVAL with clause\nmerge([], X115, X115).\nand substitutionT77 -> [],\nT76 -> T84,\nX115 -> T84,\nX108 -> T84" }, { "from": 815, "to": 825, "label": "EVAL-BACKTRACK" }, { "from": 817, "to": 827, "label": "PARALLEL" }, { "from": 817, "to": 828, "label": "PARALLEL" }, { "from": 824, "to": 826, "label": "SUCCESS" }, { "from": 827, "to": 829, "label": "EVAL with clause\nmerge(X120, [], X120).\nand substitutionT77 -> T89,\nX120 -> T89,\nT76 -> [],\nX108 -> T89" }, { "from": 827, "to": 830, "label": "EVAL-BACKTRACK" }, { "from": 828, "to": 832, "label": "PARALLEL" }, { "from": 828, "to": 833, "label": "PARALLEL" }, { "from": 829, "to": 831, "label": "SUCCESS" }, { "from": 832, "to": 834, "label": "EVAL with clause\nmerge(.(X145, X146), .(X147, X148), .(X145, X149)) :- ','(less(X145, s(X147)), merge(X146, .(X147, X148), X149)).\nand substitutionX145 -> T110,\nX146 -> T112,\nT77 -> .(T110, T112),\nX147 -> T111,\nX148 -> T113,\nT76 -> .(T111, T113),\nX149 -> X150,\nX108 -> .(T110, X150),\nT106 -> T110,\nT108 -> T111,\nT107 -> T112,\nT109 -> T113" }, { "from": 832, "to": 835, "label": "EVAL-BACKTRACK" }, { "from": 833, "to": 899, "label": "EVAL with clause\nmerge(.(X195, X196), .(X197, X198), .(X197, X199)) :- ','(less(X197, X195), merge(.(X195, X196), X198, X199)).\nand substitutionX195 -> T168,\nX196 -> T170,\nT77 -> .(T168, T170),\nX197 -> T167,\nX198 -> T169,\nT76 -> .(T167, T169),\nX199 -> X200,\nX108 -> .(T167, X200),\nT165 -> T167,\nT163 -> T168,\nT166 -> T169,\nT164 -> T170" }, { "from": 833, "to": 902, "label": "EVAL-BACKTRACK" }, { "from": 834, "to": 838, "label": "SPLIT 1" }, { "from": 834, "to": 839, "label": "SPLIT 2\nnew knowledge:\nT110 is ground\nreplacements:T112 -> T116,\nT111 -> T117,\nT113 -> T118" }, { "from": 838, "to": 841, "label": "CASE" }, { "from": 839, "to": 791, "label": "INSTANCE with matching:\nT77 -> T116\nT76 -> .(T117, T118)\nX108 -> X150" }, { "from": 841, "to": 844, "label": "PARALLEL" }, { "from": 841, "to": 845, "label": "PARALLEL" }, { "from": 844, "to": 847, "label": "EVAL with clause\nless(0, s(X159)).\nand substitutionT110 -> 0,\nT111 -> T125,\nX159 -> T125" }, { "from": 844, "to": 849, "label": "EVAL-BACKTRACK" }, { "from": 845, "to": 852, "label": "EVAL with clause\nless(s(X164), s(X165)) :- less(X164, X165).\nand substitutionX164 -> T132,\nT110 -> s(T132),\nT111 -> T133,\nX165 -> T133,\nT130 -> T132,\nT131 -> T133" }, { "from": 845, "to": 853, "label": "EVAL-BACKTRACK" }, { "from": 847, "to": 851, "label": "SUCCESS" }, { "from": 852, "to": 856, "label": "CASE" }, { "from": 856, "to": 857, "label": "PARALLEL" }, { "from": 856, "to": 858, "label": "PARALLEL" }, { "from": 857, "to": 859, "label": "EVAL with clause\nless(0, s(X172)).\nand substitutionT132 -> 0,\nX172 -> T140,\nT133 -> s(T140)" }, { "from": 857, "to": 860, "label": "EVAL-BACKTRACK" }, { "from": 858, "to": 887, "label": "EVAL with clause\nless(s(X177), s(X178)) :- less(X177, X178).\nand substitutionX177 -> T147,\nT132 -> s(T147),\nX178 -> T148,\nT133 -> s(T148),\nT145 -> T147,\nT146 -> T148" }, { "from": 858, "to": 888, "label": "EVAL-BACKTRACK" }, { "from": 859, "to": 861, "label": "SUCCESS" }, { "from": 887, "to": 852, "label": "INSTANCE with matching:\nT132 -> T147\nT133 -> T148" }, { "from": 899, "to": 906, "label": "SPLIT 1" }, { "from": 899, "to": 907, "label": "SPLIT 2\nnew knowledge:\nT167 is ground\nreplacements:T168 -> T173,\nT170 -> T174,\nT169 -> T175" }, { "from": 906, "to": 852, "label": "INSTANCE with matching:\nT132 -> T167\nT133 -> T168" }, { "from": 907, "to": 791, "label": "INSTANCE with matching:\nT77 -> .(T173, T174)\nT76 -> T175\nX108 -> X200" }, { "from": 936, "to": 567, "label": "INSTANCE with matching:\nT23 -> T54\nX25 -> X26" }, { "from": 1003, "to": 1014, "label": "CASE" }, { "from": 1014, "to": 1015, "label": "PARALLEL" }, { "from": 1014, "to": 1016, "label": "PARALLEL" }, { "from": 1015, "to": 1018, "label": "EVAL with clause\nmerge([], X213, X213).\nand substitutionT179 -> [],\nT178 -> T186,\nX213 -> T186,\nT19 -> T186" }, { "from": 1015, "to": 1020, "label": "EVAL-BACKTRACK" }, { "from": 1016, "to": 1022, "label": "PARALLEL" }, { "from": 1016, "to": 1023, "label": "PARALLEL" }, { "from": 1018, "to": 1021, "label": "SUCCESS" }, { "from": 1022, "to": 1024, "label": "EVAL with clause\nmerge(X218, [], X218).\nand substitutionT179 -> T191,\nX218 -> T191,\nT178 -> [],\nT19 -> T191" }, { "from": 1022, "to": 1025, "label": "EVAL-BACKTRACK" }, { "from": 1023, "to": 1031, "label": "PARALLEL" }, { "from": 1023, "to": 1032, "label": "PARALLEL" }, { "from": 1024, "to": 1026, "label": "SUCCESS" }, { "from": 1031, "to": 1036, "label": "EVAL with clause\nmerge(.(X239, X240), .(X241, X242), .(X239, X243)) :- ','(less(X239, s(X241)), merge(X240, .(X241, X242), X243)).\nand substitutionX239 -> T212,\nX240 -> T218,\nT179 -> .(T212, T218),\nX241 -> T217,\nX242 -> T219,\nT178 -> .(T217, T219),\nX243 -> T216,\nT19 -> .(T212, T216),\nT214 -> T217,\nT213 -> T218,\nT215 -> T219" }, { "from": 1031, "to": 1037, "label": "EVAL-BACKTRACK" }, { "from": 1032, "to": 1113, "label": "EVAL with clause\nmerge(.(X286, X287), .(X288, X289), .(X288, X290)) :- ','(less(X288, X286), merge(.(X286, X287), X289, X290)).\nand substitutionX286 -> T274,\nX287 -> T276,\nT179 -> .(T274, T276),\nX288 -> T271,\nX289 -> T275,\nT178 -> .(T271, T275),\nX290 -> T273,\nT19 -> .(T271, T273),\nT269 -> T274,\nT272 -> T275,\nT270 -> T276" }, { "from": 1032, "to": 1114, "label": "EVAL-BACKTRACK" }, { "from": 1036, "to": 1038, "label": "SPLIT 1" }, { "from": 1036, "to": 1039, "label": "SPLIT 2\nnew knowledge:\nT212 is ground\nreplacements:T218 -> T222,\nT217 -> T223,\nT219 -> T224" }, { "from": 1038, "to": 1044, "label": "CASE" }, { "from": 1039, "to": 1003, "label": "INSTANCE with matching:\nT179 -> T222\nT178 -> .(T223, T224)\nT19 -> T216" }, { "from": 1044, "to": 1046, "label": "PARALLEL" }, { "from": 1044, "to": 1047, "label": "PARALLEL" }, { "from": 1046, "to": 1051, "label": "EVAL with clause\nless(0, s(X252)).\nand substitutionT212 -> 0,\nT217 -> T231,\nX252 -> T231" }, { "from": 1046, "to": 1052, "label": "EVAL-BACKTRACK" }, { "from": 1047, "to": 1073, "label": "EVAL with clause\nless(s(X257), s(X258)) :- less(X257, X258).\nand substitutionX257 -> T236,\nT212 -> s(T236),\nT217 -> T238,\nX258 -> T238,\nT237 -> T238" }, { "from": 1047, "to": 1074, "label": "EVAL-BACKTRACK" }, { "from": 1051, "to": 1053, "label": "SUCCESS" }, { "from": 1073, "to": 1075, "label": "CASE" }, { "from": 1075, "to": 1076, "label": "PARALLEL" }, { "from": 1075, "to": 1077, "label": "PARALLEL" }, { "from": 1076, "to": 1078, "label": "EVAL with clause\nless(0, s(X265)).\nand substitutionT236 -> 0,\nX265 -> T245,\nT238 -> s(T245)" }, { "from": 1076, "to": 1079, "label": "EVAL-BACKTRACK" }, { "from": 1077, "to": 1081, "label": "EVAL with clause\nless(s(X270), s(X271)) :- less(X270, X271).\nand substitutionX270 -> T250,\nT236 -> s(T250),\nX271 -> T252,\nT238 -> s(T252),\nT251 -> T252" }, { "from": 1077, "to": 1082, "label": "EVAL-BACKTRACK" }, { "from": 1078, "to": 1080, "label": "SUCCESS" }, { "from": 1081, "to": 1073, "label": "INSTANCE with matching:\nT236 -> T250\nT238 -> T252" }, { "from": 1113, "to": 1123, "label": "SPLIT 1" }, { "from": 1113, "to": 1124, "label": "SPLIT 2\nnew knowledge:\nT271 is ground\nreplacements:T274 -> T279,\nT276 -> T280,\nT275 -> T281" }, { "from": 1123, "to": 1073, "label": "INSTANCE with matching:\nT236 -> T271\nT238 -> T274" }, { "from": 1124, "to": 1003, "label": "INSTANCE with matching:\nT179 -> .(T279, T280)\nT178 -> T281\nT19 -> T273" } ], "type": "Graph" } } ---------------------------------------- (122) Complex Obligation (AND) ---------------------------------------- (123) Obligation: Rules: f1073_out(T250) -> f1081_out(T250) :|: TRUE f1081_in(x) -> f1073_in(x) :|: TRUE f1075_in(T236) -> f1076_in(T236) :|: TRUE f1076_out(x1) -> f1075_out(x1) :|: TRUE f1077_out(x2) -> f1075_out(x2) :|: TRUE f1075_in(x3) -> f1077_in(x3) :|: TRUE f1075_out(x4) -> f1073_out(x4) :|: TRUE f1073_in(x5) -> f1075_in(x5) :|: TRUE f1077_in(x6) -> f1082_in :|: TRUE f1082_out -> f1077_out(x7) :|: TRUE f1077_in(s(x8)) -> f1081_in(x8) :|: TRUE f1081_out(x9) -> f1077_out(s(x9)) :|: TRUE f6_out(T2) -> f3_out(T2) :|: TRUE f3_in(x10) -> f6_in(x10) :|: TRUE f6_in(x11) -> f10_in(x11) :|: TRUE f6_in(x12) -> f9_in(x12) :|: TRUE f9_out(x13) -> f6_out(x13) :|: TRUE f10_out(x14) -> f6_out(x14) :|: TRUE f10_in(x15) -> f21_in(x15) :|: TRUE f10_in(x16) -> f20_in(x16) :|: TRUE f20_out(x17) -> f10_out(x17) :|: TRUE f21_out(x18) -> f10_out(x18) :|: TRUE f21_in(T19) -> f38_in(T19) :|: TRUE f39_out -> f21_out(x19) :|: TRUE f21_in(x20) -> f39_in :|: TRUE f38_out(x21) -> f21_out(x21) :|: TRUE f38_in(x22) -> f101_in :|: TRUE f102_out(x23) -> f38_out(x23) :|: TRUE f101_out -> f102_in(x24) :|: TRUE f567_out -> f568_in(x25) :|: TRUE f102_in(x26) -> f567_in :|: TRUE f568_out(x27) -> f102_out(x27) :|: TRUE f568_in(x28) -> f936_in :|: TRUE f936_out -> f1003_in(x29) :|: TRUE f1003_out(x30) -> f568_out(x30) :|: TRUE f1014_out(x31) -> f1003_out(x31) :|: TRUE f1003_in(x32) -> f1014_in(x32) :|: TRUE f1014_in(x33) -> f1016_in(x33) :|: TRUE f1014_in(x34) -> f1015_in(x34) :|: TRUE f1016_out(x35) -> f1014_out(x35) :|: TRUE f1015_out(x36) -> f1014_out(x36) :|: TRUE f1016_in(x37) -> f1023_in(x37) :|: TRUE f1022_out(x38) -> f1016_out(x38) :|: TRUE f1023_out(x39) -> f1016_out(x39) :|: TRUE f1016_in(x40) -> f1022_in(x40) :|: TRUE f1031_out(x41) -> f1023_out(x41) :|: TRUE f1032_out(x42) -> f1023_out(x42) :|: TRUE f1023_in(x43) -> f1032_in(x43) :|: TRUE f1023_in(x44) -> f1031_in(x44) :|: TRUE f1036_out(T212, T216) -> f1031_out(.(T212, T216)) :|: TRUE f1037_out -> f1031_out(x45) :|: TRUE f1031_in(.(x46, x47)) -> f1036_in(x46, x47) :|: TRUE f1031_in(x48) -> f1037_in :|: TRUE f1036_in(x49, x50) -> f1038_in(x49) :|: TRUE f1038_out(x51) -> f1039_in(x52) :|: TRUE f1039_out(x53) -> f1036_out(x54, x53) :|: TRUE f1044_out(x55) -> f1038_out(x55) :|: TRUE f1038_in(x56) -> f1044_in(x56) :|: TRUE f1047_out(x57) -> f1044_out(x57) :|: TRUE f1044_in(x58) -> f1047_in(x58) :|: TRUE f1044_in(x59) -> f1046_in(x59) :|: TRUE f1046_out(x60) -> f1044_out(x60) :|: TRUE f1047_in(s(x61)) -> f1073_in(x61) :|: TRUE f1074_out -> f1047_out(x62) :|: TRUE f1073_out(x63) -> f1047_out(s(x63)) :|: TRUE f1047_in(x64) -> f1074_in :|: TRUE f1032_in(x65) -> f1114_in :|: TRUE f1114_out -> f1032_out(x66) :|: TRUE f1032_in(.(T271, T273)) -> f1113_in(T271, T273) :|: TRUE f1113_out(x67, x68) -> f1032_out(.(x67, x68)) :|: TRUE f1123_out(x69) -> f1124_in(x70) :|: TRUE f1113_in(x71, x72) -> f1123_in(x71) :|: TRUE f1124_out(x73) -> f1113_out(x74, x73) :|: TRUE f1073_out(x75) -> f1123_out(x75) :|: TRUE f1123_in(x76) -> f1073_in(x76) :|: TRUE Start term: f3_in(T2) ---------------------------------------- (124) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (125) TRUE ---------------------------------------- (126) Obligation: Rules: f1036_out(T212, T216) -> f1031_out(.(T212, T216)) :|: TRUE f1037_out -> f1031_out(T19) :|: TRUE f1031_in(.(x, x1)) -> f1036_in(x, x1) :|: TRUE f1031_in(x2) -> f1037_in :|: TRUE f1003_out(x3) -> f1039_out(x3) :|: TRUE f1039_in(x4) -> f1003_in(x4) :|: TRUE f1014_in(x5) -> f1016_in(x5) :|: TRUE f1014_in(x6) -> f1015_in(x6) :|: TRUE f1016_out(x7) -> f1014_out(x7) :|: TRUE f1015_out(x8) -> f1014_out(x8) :|: TRUE f1036_in(x9, x10) -> f1038_in(x9) :|: TRUE f1038_out(x11) -> f1039_in(x12) :|: TRUE f1039_out(x13) -> f1036_out(x14, x13) :|: TRUE f1016_in(x15) -> f1023_in(x15) :|: TRUE f1022_out(x16) -> f1016_out(x16) :|: TRUE f1023_out(x17) -> f1016_out(x17) :|: TRUE f1016_in(x18) -> f1022_in(x18) :|: TRUE f1046_in(x19) -> f1052_in :|: TRUE f1051_out -> f1046_out(0) :|: TRUE f1052_out -> f1046_out(x20) :|: TRUE f1046_in(0) -> f1051_in :|: TRUE f1073_out(T271) -> f1123_out(T271) :|: TRUE f1123_in(x21) -> f1073_in(x21) :|: TRUE f1073_out(T250) -> f1081_out(T250) :|: TRUE f1081_in(x22) -> f1073_in(x22) :|: TRUE f1075_in(T236) -> f1076_in(T236) :|: TRUE f1076_out(x23) -> f1075_out(x23) :|: TRUE f1077_out(x24) -> f1075_out(x24) :|: TRUE f1075_in(x25) -> f1077_in(x25) :|: TRUE f1123_out(x26) -> f1124_in(x27) :|: TRUE f1113_in(x28, x29) -> f1123_in(x28) :|: TRUE f1124_out(x30) -> f1113_out(x31, x30) :|: TRUE f1076_in(0) -> f1078_in :|: TRUE f1078_out -> f1076_out(0) :|: TRUE f1079_out -> f1076_out(x32) :|: TRUE f1076_in(x33) -> f1079_in :|: TRUE f1077_in(x34) -> f1082_in :|: TRUE f1082_out -> f1077_out(x35) :|: TRUE f1077_in(s(x36)) -> f1081_in(x36) :|: TRUE f1081_out(x37) -> f1077_out(s(x37)) :|: TRUE f1078_in -> f1078_out :|: TRUE f1047_out(x38) -> f1044_out(x38) :|: TRUE f1044_in(x39) -> f1047_in(x39) :|: TRUE f1044_in(x40) -> f1046_in(x40) :|: TRUE f1046_out(x41) -> f1044_out(x41) :|: TRUE f1047_in(s(x42)) -> f1073_in(x42) :|: TRUE f1074_out -> f1047_out(x43) :|: TRUE f1073_out(x44) -> f1047_out(s(x44)) :|: TRUE f1047_in(x45) -> f1074_in :|: TRUE f1051_in -> f1051_out :|: TRUE f1003_out(T273) -> f1124_out(T273) :|: TRUE f1124_in(x46) -> f1003_in(x46) :|: TRUE f1044_out(x47) -> f1038_out(x47) :|: TRUE f1038_in(x48) -> f1044_in(x48) :|: TRUE f1031_out(x49) -> f1023_out(x49) :|: TRUE f1032_out(x50) -> f1023_out(x50) :|: TRUE f1023_in(x51) -> f1032_in(x51) :|: TRUE f1023_in(x52) -> f1031_in(x52) :|: TRUE f1014_out(x53) -> f1003_out(x53) :|: TRUE f1003_in(x54) -> f1014_in(x54) :|: TRUE f1032_in(x55) -> f1114_in :|: TRUE f1114_out -> f1032_out(x56) :|: TRUE f1032_in(.(x57, x58)) -> f1113_in(x57, x58) :|: TRUE f1113_out(x59, x60) -> f1032_out(.(x59, x60)) :|: TRUE f1075_out(x61) -> f1073_out(x61) :|: TRUE f1073_in(x62) -> f1075_in(x62) :|: TRUE f6_out(T2) -> f3_out(T2) :|: TRUE f3_in(x63) -> f6_in(x63) :|: TRUE f6_in(x64) -> f10_in(x64) :|: TRUE f6_in(x65) -> f9_in(x65) :|: TRUE f9_out(x66) -> f6_out(x66) :|: TRUE f10_out(x67) -> f6_out(x67) :|: TRUE f10_in(x68) -> f21_in(x68) :|: TRUE f10_in(x69) -> f20_in(x69) :|: TRUE f20_out(x70) -> f10_out(x70) :|: TRUE f21_out(x71) -> f10_out(x71) :|: TRUE f21_in(x72) -> f38_in(x72) :|: TRUE f39_out -> f21_out(x73) :|: TRUE f21_in(x74) -> f39_in :|: TRUE f38_out(x75) -> f21_out(x75) :|: TRUE f38_in(x76) -> f101_in :|: TRUE f102_out(x77) -> f38_out(x77) :|: TRUE f101_out -> f102_in(x78) :|: TRUE f567_out -> f568_in(x79) :|: TRUE f102_in(x80) -> f567_in :|: TRUE f568_out(x81) -> f102_out(x81) :|: TRUE f568_in(x82) -> f936_in :|: TRUE f936_out -> f1003_in(x83) :|: TRUE f1003_out(x84) -> f568_out(x84) :|: TRUE Start term: f3_in(T2) ---------------------------------------- (127) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (128) TRUE ---------------------------------------- (129) Obligation: Rules: f852_in -> f856_in :|: TRUE f856_out -> f852_out :|: TRUE f858_out -> f856_out :|: TRUE f856_in -> f858_in :|: TRUE f856_in -> f857_in :|: TRUE f857_out -> f856_out :|: TRUE f887_in -> f852_in :|: TRUE f852_out -> f887_out :|: TRUE f858_in -> f887_in :|: TRUE f888_out -> f858_out :|: TRUE f887_out -> f858_out :|: TRUE f858_in -> f888_in :|: TRUE f6_out(T2) -> f3_out(T2) :|: TRUE f3_in(x) -> f6_in(x) :|: TRUE f6_in(x1) -> f10_in(x1) :|: TRUE f6_in(x2) -> f9_in(x2) :|: TRUE f9_out(x3) -> f6_out(x3) :|: TRUE f10_out(x4) -> f6_out(x4) :|: TRUE f10_in(x5) -> f21_in(x5) :|: TRUE f10_in(x6) -> f20_in(x6) :|: TRUE f20_out(x7) -> f10_out(x7) :|: TRUE f21_out(x8) -> f10_out(x8) :|: TRUE f21_in(T19) -> f38_in(T19) :|: TRUE f39_out -> f21_out(x9) :|: TRUE f21_in(x10) -> f39_in :|: TRUE f38_out(x11) -> f21_out(x11) :|: TRUE f38_in(x12) -> f101_in :|: TRUE f102_out(x13) -> f38_out(x13) :|: TRUE f101_out -> f102_in(x14) :|: TRUE f567_out -> f568_in(x15) :|: TRUE f102_in(x16) -> f567_in :|: TRUE f568_out(x17) -> f102_out(x17) :|: TRUE f568_in(x18) -> f936_in :|: TRUE f936_out -> f1003_in(x19) :|: TRUE f1003_out(x20) -> f568_out(x20) :|: TRUE f936_in -> f567_in :|: TRUE f567_out -> f936_out :|: TRUE f570_out -> f567_out :|: TRUE f567_in -> f570_in :|: TRUE f571_out -> f570_out :|: TRUE f572_out -> f570_out :|: TRUE f570_in -> f571_in :|: TRUE f570_in -> f572_in :|: TRUE f572_in -> f672_in :|: TRUE f672_out -> f572_out :|: TRUE f670_out -> f572_out :|: TRUE f572_in -> f670_in :|: TRUE f718_out -> f672_out :|: TRUE f672_in -> f718_in :|: TRUE f672_in -> f706_in :|: TRUE f706_out -> f672_out :|: TRUE f733_out -> f735_in :|: TRUE f735_out -> f706_out :|: TRUE f706_in -> f733_in :|: TRUE f761_out -> f735_out :|: TRUE f759_out -> f761_in :|: TRUE f735_in -> f759_in :|: TRUE f761_in -> f790_in :|: TRUE f790_out -> f791_in :|: TRUE f791_out -> f761_out :|: TRUE f791_in -> f808_in :|: TRUE f808_out -> f791_out :|: TRUE f808_in -> f817_in :|: TRUE f817_out -> f808_out :|: TRUE f808_in -> f815_in :|: TRUE f815_out -> f808_out :|: TRUE f828_out -> f817_out :|: TRUE f817_in -> f828_in :|: TRUE f817_in -> f827_in :|: TRUE f827_out -> f817_out :|: TRUE f832_out -> f828_out :|: TRUE f833_out -> f828_out :|: TRUE f828_in -> f832_in :|: TRUE f828_in -> f833_in :|: TRUE f833_in -> f902_in :|: TRUE f902_out -> f833_out :|: TRUE f833_in -> f899_in :|: TRUE f899_out -> f833_out :|: TRUE f907_out -> f899_out :|: TRUE f906_out -> f907_in :|: TRUE f899_in -> f906_in :|: TRUE f906_in -> f852_in :|: TRUE f852_out -> f906_out :|: TRUE f832_in -> f835_in :|: TRUE f832_in -> f834_in :|: TRUE f834_out -> f832_out :|: TRUE f835_out -> f832_out :|: TRUE f834_in -> f838_in :|: TRUE f839_out -> f834_out :|: TRUE f838_out -> f839_in :|: TRUE f838_in -> f841_in :|: TRUE f841_out -> f838_out :|: TRUE f844_out -> f841_out :|: TRUE f841_in -> f845_in :|: TRUE f841_in -> f844_in :|: TRUE f845_out -> f841_out :|: TRUE f845_in -> f853_in :|: TRUE f852_out -> f845_out :|: TRUE f853_out -> f845_out :|: TRUE f845_in -> f852_in :|: TRUE Start term: f3_in(T2) ---------------------------------------- (130) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (131) TRUE ---------------------------------------- (132) Obligation: Rules: f832_out -> f828_out :|: TRUE f833_out -> f828_out :|: TRUE f828_in -> f832_in :|: TRUE f828_in -> f833_in :|: TRUE f907_out -> f899_out :|: TRUE f906_out -> f907_in :|: TRUE f899_in -> f906_in :|: TRUE f858_in -> f887_in :|: TRUE f888_out -> f858_out :|: TRUE f887_out -> f858_out :|: TRUE f858_in -> f888_in :|: TRUE f808_in -> f817_in :|: TRUE f817_out -> f808_out :|: TRUE f808_in -> f815_in :|: TRUE f815_out -> f808_out :|: TRUE f847_out -> f844_out :|: TRUE f849_out -> f844_out :|: TRUE f844_in -> f849_in :|: TRUE f844_in -> f847_in :|: TRUE f834_in -> f838_in :|: TRUE f839_out -> f834_out :|: TRUE f838_out -> f839_in :|: TRUE f828_out -> f817_out :|: TRUE f817_in -> f828_in :|: TRUE f817_in -> f827_in :|: TRUE f827_out -> f817_out :|: TRUE f857_in -> f859_in :|: TRUE f860_out -> f857_out :|: TRUE f859_out -> f857_out :|: TRUE f857_in -> f860_in :|: TRUE f845_in -> f853_in :|: TRUE f852_out -> f845_out :|: TRUE f853_out -> f845_out :|: TRUE f845_in -> f852_in :|: TRUE f852_in -> f856_in :|: TRUE f856_out -> f852_out :|: TRUE f832_in -> f835_in :|: TRUE f832_in -> f834_in :|: TRUE f834_out -> f832_out :|: TRUE f835_out -> f832_out :|: TRUE f858_out -> f856_out :|: TRUE f856_in -> f858_in :|: TRUE f856_in -> f857_in :|: TRUE f857_out -> f856_out :|: TRUE f838_in -> f841_in :|: TRUE f841_out -> f838_out :|: TRUE f887_in -> f852_in :|: TRUE f852_out -> f887_out :|: TRUE f847_in -> f847_out :|: TRUE f833_in -> f902_in :|: TRUE f902_out -> f833_out :|: TRUE f833_in -> f899_in :|: TRUE f899_out -> f833_out :|: TRUE f859_in -> f859_out :|: TRUE f906_in -> f852_in :|: TRUE f852_out -> f906_out :|: TRUE f844_out -> f841_out :|: TRUE f841_in -> f845_in :|: TRUE f841_in -> f844_in :|: TRUE f845_out -> f841_out :|: TRUE f839_in -> f791_in :|: TRUE f791_out -> f839_out :|: TRUE f791_out -> f907_out :|: TRUE f907_in -> f791_in :|: TRUE f791_in -> f808_in :|: TRUE f808_out -> f791_out :|: TRUE f6_out(T2) -> f3_out(T2) :|: TRUE f3_in(x) -> f6_in(x) :|: TRUE f6_in(x1) -> f10_in(x1) :|: TRUE f6_in(x2) -> f9_in(x2) :|: TRUE f9_out(x3) -> f6_out(x3) :|: TRUE f10_out(x4) -> f6_out(x4) :|: TRUE f10_in(x5) -> f21_in(x5) :|: TRUE f10_in(x6) -> f20_in(x6) :|: TRUE f20_out(x7) -> f10_out(x7) :|: TRUE f21_out(x8) -> f10_out(x8) :|: TRUE f21_in(T19) -> f38_in(T19) :|: TRUE f39_out -> f21_out(x9) :|: TRUE f21_in(x10) -> f39_in :|: TRUE f38_out(x11) -> f21_out(x11) :|: TRUE f38_in(x12) -> f101_in :|: TRUE f102_out(x13) -> f38_out(x13) :|: TRUE f101_out -> f102_in(x14) :|: TRUE f567_out -> f568_in(x15) :|: TRUE f102_in(x16) -> f567_in :|: TRUE f568_out(x17) -> f102_out(x17) :|: TRUE f570_out -> f567_out :|: TRUE f567_in -> f570_in :|: TRUE f571_out -> f570_out :|: TRUE f572_out -> f570_out :|: TRUE f570_in -> f571_in :|: TRUE f570_in -> f572_in :|: TRUE f572_in -> f672_in :|: TRUE f672_out -> f572_out :|: TRUE f670_out -> f572_out :|: TRUE f572_in -> f670_in :|: TRUE f718_out -> f672_out :|: TRUE f672_in -> f718_in :|: TRUE f672_in -> f706_in :|: TRUE f706_out -> f672_out :|: TRUE f733_out -> f735_in :|: TRUE f735_out -> f706_out :|: TRUE f706_in -> f733_in :|: TRUE f761_out -> f735_out :|: TRUE f759_out -> f761_in :|: TRUE f735_in -> f759_in :|: TRUE f761_in -> f790_in :|: TRUE f790_out -> f791_in :|: TRUE f791_out -> f761_out :|: TRUE f568_in(x18) -> f936_in :|: TRUE f936_out -> f1003_in(x19) :|: TRUE f1003_out(x20) -> f568_out(x20) :|: TRUE f936_in -> f567_in :|: TRUE f567_out -> f936_out :|: TRUE Start term: f3_in(T2) ---------------------------------------- (133) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (134) TRUE ---------------------------------------- (135) Obligation: Rules: f282_out -> f281_out :|: TRUE f281_in -> f283_in :|: TRUE f281_in -> f282_in :|: TRUE f283_out -> f281_out :|: TRUE f280_out -> f335_out :|: TRUE f335_in -> f280_in :|: TRUE f283_in -> f335_in :|: TRUE f345_out -> f283_out :|: TRUE f335_out -> f283_out :|: TRUE f283_in -> f345_in :|: TRUE f281_out -> f280_out :|: TRUE f280_in -> f281_in :|: TRUE f6_out(T2) -> f3_out(T2) :|: TRUE f3_in(x) -> f6_in(x) :|: TRUE f6_in(x1) -> f10_in(x1) :|: TRUE f6_in(x2) -> f9_in(x2) :|: TRUE f9_out(x3) -> f6_out(x3) :|: TRUE f10_out(x4) -> f6_out(x4) :|: TRUE f10_in(x5) -> f21_in(x5) :|: TRUE f10_in(x6) -> f20_in(x6) :|: TRUE f20_out(x7) -> f10_out(x7) :|: TRUE f21_out(x8) -> f10_out(x8) :|: TRUE f21_in(T19) -> f38_in(T19) :|: TRUE f39_out -> f21_out(x9) :|: TRUE f21_in(x10) -> f39_in :|: TRUE f38_out(x11) -> f21_out(x11) :|: TRUE f38_in(x12) -> f101_in :|: TRUE f102_out(x13) -> f38_out(x13) :|: TRUE f101_out -> f102_in(x14) :|: TRUE f101_in -> f208_in :|: TRUE f208_out -> f101_out :|: TRUE f217_out -> f208_out :|: TRUE f208_in -> f217_in :|: TRUE f217_in -> f277_in :|: TRUE f277_out -> f217_out :|: TRUE f278_out -> f277_out :|: TRUE f277_in -> f278_in :|: TRUE f279_out -> f278_out :|: TRUE f278_in -> f279_in :|: TRUE f280_out -> f279_out :|: TRUE f279_in -> f280_in :|: TRUE f567_out -> f568_in(x15) :|: TRUE f102_in(x16) -> f567_in :|: TRUE f568_out(x17) -> f102_out(x17) :|: TRUE f570_out -> f567_out :|: TRUE f567_in -> f570_in :|: TRUE f571_out -> f570_out :|: TRUE f572_out -> f570_out :|: TRUE f570_in -> f571_in :|: TRUE f570_in -> f572_in :|: TRUE f572_in -> f672_in :|: TRUE f672_out -> f572_out :|: TRUE f670_out -> f572_out :|: TRUE f572_in -> f670_in :|: TRUE f718_out -> f672_out :|: TRUE f672_in -> f718_in :|: TRUE f672_in -> f706_in :|: TRUE f706_out -> f672_out :|: TRUE f733_out -> f735_in :|: TRUE f735_out -> f706_out :|: TRUE f706_in -> f733_in :|: TRUE f733_in -> f101_in :|: TRUE f101_out -> f733_out :|: TRUE f568_in(x18) -> f936_in :|: TRUE f936_out -> f1003_in(x19) :|: TRUE f1003_out(x20) -> f568_out(x20) :|: TRUE f936_in -> f567_in :|: TRUE f567_out -> f936_out :|: TRUE Start term: f3_in(T2) ---------------------------------------- (136) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f281_in -> f283_in :|: TRUE f335_in -> f280_in :|: TRUE f283_in -> f335_in :|: TRUE f280_in -> f281_in :|: TRUE ---------------------------------------- (137) Obligation: Rules: f281_in -> f283_in :|: TRUE f335_in -> f280_in :|: TRUE f283_in -> f335_in :|: TRUE f280_in -> f281_in :|: TRUE ---------------------------------------- (138) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (139) Obligation: Rules: f281_in -> f281_in :|: TRUE ---------------------------------------- (140) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (141) Obligation: Rules: f281_in -> f281_in :|: TRUE ---------------------------------------- (142) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f281_in -> f281_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (143) Obligation: Termination digraph: Nodes: (1) f281_in -> f281_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (144) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f281_in() Replaced non-predefined constructor symbols by 0. ---------------------------------------- (145) Obligation: Rules: f281_in -> f281_in :|: TRUE ---------------------------------------- (146) IntTRSNonPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc) -> f(1) :|: pc = 1 && TRUE Proved unsatisfiability of the following formula, indicating that the system is never left after entering: ((run2_0 = ((1 * 1)) and (((run1_0 * 1)) = ((1 * 1)) and T)) and !(((run2_0 * 1)) = ((1 * 1)) and T)) Proved satisfiability of the following formula, indicating that the system is entered at least once: (run2_0 = ((1 * 1)) and (((run1_0 * 1)) = ((1 * 1)) and T)) ---------------------------------------- (147) NO ---------------------------------------- (148) Obligation: Rules: f832_out -> f828_out :|: TRUE f833_out -> f828_out :|: TRUE f828_in -> f832_in :|: TRUE f828_in -> f833_in :|: TRUE f907_out -> f899_out :|: TRUE f906_out -> f907_in :|: TRUE f899_in -> f906_in :|: TRUE f571_out -> f570_out :|: TRUE f572_out -> f570_out :|: TRUE f570_in -> f571_in :|: TRUE f570_in -> f572_in :|: TRUE f858_in -> f887_in :|: TRUE f888_out -> f858_out :|: TRUE f887_out -> f858_out :|: TRUE f858_in -> f888_in :|: TRUE f285_in -> f285_out :|: TRUE f808_in -> f817_in :|: TRUE f817_out -> f808_out :|: TRUE f808_in -> f815_in :|: TRUE f815_out -> f808_out :|: TRUE f847_out -> f844_out :|: TRUE f849_out -> f844_out :|: TRUE f844_in -> f849_in :|: TRUE f844_in -> f847_in :|: TRUE f834_in -> f838_in :|: TRUE f839_out -> f834_out :|: TRUE f838_out -> f839_in :|: TRUE f733_in -> f101_in :|: TRUE f101_out -> f733_out :|: TRUE f278_out -> f277_out :|: TRUE f277_in -> f278_in :|: TRUE f828_out -> f817_out :|: TRUE f817_in -> f828_in :|: TRUE f817_in -> f827_in :|: TRUE f827_out -> f817_out :|: TRUE f824_in -> f824_out :|: TRUE f217_in -> f277_in :|: TRUE f277_out -> f217_out :|: TRUE f852_in -> f856_in :|: TRUE f856_out -> f852_out :|: TRUE f858_out -> f856_out :|: TRUE f856_in -> f858_in :|: TRUE f856_in -> f857_in :|: TRUE f857_out -> f856_out :|: TRUE f829_in -> f829_out :|: TRUE f838_in -> f841_in :|: TRUE f841_out -> f838_out :|: TRUE f279_out -> f278_out :|: TRUE f278_in -> f279_in :|: TRUE f887_in -> f852_in :|: TRUE f852_out -> f887_out :|: TRUE f101_in -> f208_in :|: TRUE f208_out -> f101_out :|: TRUE f859_in -> f859_out :|: TRUE f280_out -> f279_out :|: TRUE f279_in -> f280_in :|: TRUE f844_out -> f841_out :|: TRUE f841_in -> f845_in :|: TRUE f841_in -> f844_in :|: TRUE f845_out -> f841_out :|: TRUE f761_out -> f735_out :|: TRUE f759_out -> f761_in :|: TRUE f735_in -> f759_in :|: TRUE f791_out -> f907_out :|: TRUE f907_in -> f791_in :|: TRUE f791_in -> f808_in :|: TRUE f808_out -> f791_out :|: TRUE f570_out -> f567_out :|: TRUE f567_in -> f570_in :|: TRUE f733_out -> f735_in :|: TRUE f735_out -> f706_out :|: TRUE f706_in -> f733_in :|: TRUE f718_out -> f672_out :|: TRUE f672_in -> f718_in :|: TRUE f672_in -> f706_in :|: TRUE f706_out -> f672_out :|: TRUE f285_out -> f282_out :|: TRUE f287_out -> f282_out :|: TRUE f282_in -> f285_in :|: TRUE f282_in -> f287_in :|: TRUE f827_in -> f829_in :|: TRUE f827_in -> f830_in :|: TRUE f830_out -> f827_out :|: TRUE f829_out -> f827_out :|: TRUE f282_out -> f281_out :|: TRUE f281_in -> f283_in :|: TRUE f281_in -> f282_in :|: TRUE f283_out -> f281_out :|: TRUE f280_out -> f335_out :|: TRUE f335_in -> f280_in :|: TRUE f217_out -> f208_out :|: TRUE f208_in -> f217_in :|: TRUE f759_in -> f567_in :|: TRUE f567_out -> f759_out :|: TRUE f761_in -> f790_in :|: TRUE f790_out -> f791_in :|: TRUE f791_out -> f761_out :|: TRUE f283_in -> f335_in :|: TRUE f345_out -> f283_out :|: TRUE f335_out -> f283_out :|: TRUE f283_in -> f345_in :|: TRUE f857_in -> f859_in :|: TRUE f860_out -> f857_out :|: TRUE f859_out -> f857_out :|: TRUE f857_in -> f860_in :|: TRUE f845_in -> f853_in :|: TRUE f852_out -> f845_out :|: TRUE f853_out -> f845_out :|: TRUE f845_in -> f852_in :|: TRUE f815_in -> f824_in :|: TRUE f824_out -> f815_out :|: TRUE f815_in -> f825_in :|: TRUE f825_out -> f815_out :|: TRUE f832_in -> f835_in :|: TRUE f832_in -> f834_in :|: TRUE f834_out -> f832_out :|: TRUE f835_out -> f832_out :|: TRUE f847_in -> f847_out :|: TRUE f833_in -> f902_in :|: TRUE f902_out -> f833_out :|: TRUE f833_in -> f899_in :|: TRUE f899_out -> f833_out :|: TRUE f281_out -> f280_out :|: TRUE f280_in -> f281_in :|: TRUE f790_in -> f567_in :|: TRUE f567_out -> f790_out :|: TRUE f906_in -> f852_in :|: TRUE f852_out -> f906_out :|: TRUE f572_in -> f672_in :|: TRUE f672_out -> f572_out :|: TRUE f670_out -> f572_out :|: TRUE f572_in -> f670_in :|: TRUE f839_in -> f791_in :|: TRUE f791_out -> f839_out :|: TRUE f6_out(T2) -> f3_out(T2) :|: TRUE f3_in(x) -> f6_in(x) :|: TRUE f6_in(x1) -> f10_in(x1) :|: TRUE f6_in(x2) -> f9_in(x2) :|: TRUE f9_out(x3) -> f6_out(x3) :|: TRUE f10_out(x4) -> f6_out(x4) :|: TRUE f10_in(x5) -> f21_in(x5) :|: TRUE f10_in(x6) -> f20_in(x6) :|: TRUE f20_out(x7) -> f10_out(x7) :|: TRUE f21_out(x8) -> f10_out(x8) :|: TRUE f21_in(T19) -> f38_in(T19) :|: TRUE f39_out -> f21_out(x9) :|: TRUE f21_in(x10) -> f39_in :|: TRUE f38_out(x11) -> f21_out(x11) :|: TRUE f38_in(x12) -> f101_in :|: TRUE f102_out(x13) -> f38_out(x13) :|: TRUE f101_out -> f102_in(x14) :|: TRUE f567_out -> f568_in(x15) :|: TRUE f102_in(x16) -> f567_in :|: TRUE f568_out(x17) -> f102_out(x17) :|: TRUE f568_in(x18) -> f936_in :|: TRUE f936_out -> f1003_in(x19) :|: TRUE f1003_out(x20) -> f568_out(x20) :|: TRUE f936_in -> f567_in :|: TRUE f567_out -> f936_out :|: TRUE Start term: f3_in(T2) ---------------------------------------- (149) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f570_in -> f572_in :|: TRUE f285_in -> f285_out :|: TRUE f733_in -> f101_in :|: TRUE f101_out -> f733_out :|: TRUE f278_out -> f277_out :|: TRUE f277_in -> f278_in :|: TRUE f217_in -> f277_in :|: TRUE f277_out -> f217_out :|: TRUE f279_out -> f278_out :|: TRUE f278_in -> f279_in :|: TRUE f101_in -> f208_in :|: TRUE f208_out -> f101_out :|: TRUE f280_out -> f279_out :|: TRUE f279_in -> f280_in :|: TRUE f735_in -> f759_in :|: TRUE f567_in -> f570_in :|: TRUE f733_out -> f735_in :|: TRUE f706_in -> f733_in :|: TRUE f672_in -> f706_in :|: TRUE f285_out -> f282_out :|: TRUE f282_in -> f285_in :|: TRUE f282_out -> f281_out :|: TRUE f281_in -> f283_in :|: TRUE f281_in -> f282_in :|: TRUE f283_out -> f281_out :|: TRUE f280_out -> f335_out :|: TRUE f335_in -> f280_in :|: TRUE f217_out -> f208_out :|: TRUE f208_in -> f217_in :|: TRUE f759_in -> f567_in :|: TRUE f283_in -> f335_in :|: TRUE f335_out -> f283_out :|: TRUE f281_out -> f280_out :|: TRUE f280_in -> f281_in :|: TRUE f572_in -> f672_in :|: TRUE f101_out -> f102_in(x14) :|: TRUE f102_in(x16) -> f567_in :|: TRUE ---------------------------------------- (150) Obligation: Rules: f570_in -> f572_in :|: TRUE f285_in -> f285_out :|: TRUE f733_in -> f101_in :|: TRUE f101_out -> f733_out :|: TRUE f278_out -> f277_out :|: TRUE f277_in -> f278_in :|: TRUE f217_in -> f277_in :|: TRUE f277_out -> f217_out :|: TRUE f279_out -> f278_out :|: TRUE f278_in -> f279_in :|: TRUE f101_in -> f208_in :|: TRUE f208_out -> f101_out :|: TRUE f280_out -> f279_out :|: TRUE f279_in -> f280_in :|: TRUE f735_in -> f759_in :|: TRUE f567_in -> f570_in :|: TRUE f733_out -> f735_in :|: TRUE f706_in -> f733_in :|: TRUE f672_in -> f706_in :|: TRUE f285_out -> f282_out :|: TRUE f282_in -> f285_in :|: TRUE f282_out -> f281_out :|: TRUE f281_in -> f283_in :|: TRUE f281_in -> f282_in :|: TRUE f283_out -> f281_out :|: TRUE f280_out -> f335_out :|: TRUE f335_in -> f280_in :|: TRUE f217_out -> f208_out :|: TRUE f208_in -> f217_in :|: TRUE f759_in -> f567_in :|: TRUE f283_in -> f335_in :|: TRUE f335_out -> f283_out :|: TRUE f281_out -> f280_out :|: TRUE f280_in -> f281_in :|: TRUE f572_in -> f672_in :|: TRUE f101_out -> f102_in(x14) :|: TRUE f102_in(x16) -> f567_in :|: TRUE ---------------------------------------- (151) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (152) Obligation: Rules: f280_out -> f281_in :|: TRUE f281_in -> f280_out :|: TRUE f281_in -> f281_in :|: TRUE f280_out -> f280_out :|: TRUE ---------------------------------------- (153) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (154) Obligation: Rules: f280_out -> f281_in :|: TRUE f281_in -> f280_out :|: TRUE f281_in -> f281_in :|: TRUE f280_out -> f280_out :|: TRUE ---------------------------------------- (155) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f280_out -> f281_in :|: TRUE (2) f281_in -> f280_out :|: TRUE (3) f281_in -> f281_in :|: TRUE (4) f280_out -> f280_out :|: TRUE Arcs: (1) -> (2), (3) (2) -> (1), (4) (3) -> (2), (3) (4) -> (1), (4) This digraph is fully evaluated! ---------------------------------------- (156) Obligation: Termination digraph: Nodes: (1) f280_out -> f281_in :|: TRUE (2) f280_out -> f280_out :|: TRUE (3) f281_in -> f280_out :|: TRUE (4) f281_in -> f281_in :|: TRUE Arcs: (1) -> (3), (4) (2) -> (1), (2) (3) -> (1), (2) (4) -> (3), (4) This digraph is fully evaluated! ---------------------------------------- (157) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f280_out() f281_in() Replaced non-predefined constructor symbols by 0. ---------------------------------------- (158) Obligation: Rules: f280_out -> f281_in :|: TRUE f280_out -> f280_out :|: TRUE f281_in -> f280_out :|: TRUE f281_in -> f281_in :|: TRUE ---------------------------------------- (159) IntTRSPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc) -> f(2) :|: pc = 1 && TRUE f(pc) -> f(1) :|: pc = 1 && TRUE f(pc) -> f(1) :|: pc = 2 && TRUE f(pc) -> f(2) :|: pc = 2 && TRUE Witness term starting non-terminating reduction: f(2) ---------------------------------------- (160) NO ---------------------------------------- (161) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 2, "program": { "directives": [], "clauses": [ [ "(ms ([]) ([]))", null ], [ "(ms (. X ([])) (. X ([])))", null ], [ "(ms (. X (. Y Xs)) Ys)", "(',' (split (. X (. Y Xs)) X1s X2s) (',' (ms X1s Y1s) (',' (ms X2s Y2s) (merge Y1s Y2s Ys))))" ], [ "(split ([]) ([]) ([]))", null ], [ "(split (. X Xs) (. X Ys) Zs)", "(split Xs Zs Ys)" ], [ "(merge ([]) Xs Xs)", null ], [ "(merge Xs ([]) Xs)", null ], [ "(merge (. X Xs) (. Y Ys) (. X Zs))", "(',' (less X (s Y)) (merge Xs (. Y Ys) Zs))" ], [ "(merge (. X Xs) (. Y Ys) (. Y Zs))", "(',' (less Y X) (merge (. X Xs) Ys Zs))" ], [ "(less (0) (s X1))", null ], [ "(less (s X) (s Y))", "(less X Y)" ] ] }, "graph": { "nodes": { "908": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "909": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "470": { "goal": [ { "clause": 0, "scope": 5, "term": "(ms (. T30 T29) X13)" }, { "clause": 1, "scope": 5, "term": "(ms (. T30 T29) X13)" }, { "clause": 2, "scope": 5, "term": "(ms (. T30 T29) X13)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X13"], "exprvars": [] } }, "592": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "472": { "goal": [ { "clause": 1, "scope": 5, "term": "(ms (. T30 T29) X13)" }, { "clause": 2, "scope": 5, "term": "(ms (. T30 T29) X13)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X13"], "exprvars": [] } }, "473": { "goal": [{ "clause": 1, "scope": 5, "term": "(ms (. T30 T29) X13)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X13"], "exprvars": [] } }, "594": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "474": { "goal": [{ "clause": 2, "scope": 5, "term": "(ms (. T30 T29) X13)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X13"], "exprvars": [] } }, "475": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "596": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "476": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "477": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "910": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "911": { "goal": [{ "clause": 6, "scope": 10, "term": "(merge T189 T188 ([]))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "912": { "goal": [ { "clause": 7, "scope": 10, "term": "(merge T189 T188 ([]))" }, { "clause": 8, "scope": 10, "term": "(merge T189 T188 ([]))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "913": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "914": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "915": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "916": { "goal": [{ "clause": 8, "scope": 10, "term": "(merge T189 T188 ([]))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "917": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "12": { "goal": [ { "clause": -1, "scope": -1, "term": "(true)" }, { "clause": 1, "scope": 1, "term": "(ms T1 ([]))" }, { "clause": 2, "scope": 1, "term": "(ms T1 ([]))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "919": { "goal": [ { "clause": -1, "scope": -1, "term": "(true)" }, { "clause": 2, "scope": 1, "term": "(ms T1 (. T211 ([])))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T211"], "free": [], "exprvars": [] } }, "13": { "goal": [ { "clause": 1, "scope": 1, "term": "(ms T1 T2)" }, { "clause": 2, "scope": 1, "term": "(ms T1 T2)" } ], "kb": { "nonunifying": [[ "(ms T1 T2)", "(ms ([]) ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "16": { "goal": [ { "clause": 1, "scope": 1, "term": "(ms T1 ([]))" }, { "clause": 2, "scope": 1, "term": "(ms T1 ([]))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1110": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "2": { "goal": [{ "clause": -1, "scope": -1, "term": "(ms T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "1105": { "goal": [{ "clause": -1, "scope": -1, "term": "(merge T294 (. T295 T296) ([]))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1104": { "goal": [{ "clause": -1, "scope": -1, "term": "(less T285 (s T289))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T285"], "free": [], "exprvars": [] } }, "4": { "goal": [ { "clause": 0, "scope": 1, "term": "(ms T1 T2)" }, { "clause": 1, "scope": 1, "term": "(ms T1 T2)" }, { "clause": 2, "scope": 1, "term": "(ms T1 T2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "921": { "goal": [{ "clause": 2, "scope": 1, "term": "(ms T1 T2)" }], "kb": { "nonunifying": [ [ "(ms T1 T2)", "(ms ([]) ([]))" ], [ "(ms T1 T2)", "(ms (. X246 ([])) (. 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T219 ([])))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T219"], "free": [], "exprvars": [] } }, "890": { "goal": [{ "clause": -1, "scope": -1, "term": "(ms T47 X14)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X14"], "exprvars": [] } }, "891": { "goal": [{ "clause": -1, "scope": -1, "term": "(merge T189 T188 ([]))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "650": { "goal": [ { "clause": 7, "scope": 7, "term": "(merge T87 T86 X96)" }, { "clause": 8, "scope": 7, "term": "(merge T87 T86 X96)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X96"], "exprvars": [] } }, "892": { "goal": [ { "clause": 5, "scope": 10, "term": "(merge T189 T188 ([]))" }, { "clause": 6, "scope": 10, "term": "(merge T189 T188 ([]))" }, { "clause": 7, "scope": 10, "term": "(merge T189 T188 ([]))" }, { "clause": 8, "scope": 10, "term": "(merge T189 T188 ([]))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "651": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "893": { "goal": [{ "clause": 5, "scope": 10, "term": "(merge T189 T188 ([]))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "652": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "894": { "goal": [ { "clause": 6, "scope": 10, "term": "(merge T189 T188 ([]))" }, { "clause": 7, "scope": 10, "term": "(merge T189 T188 ([]))" }, { "clause": 8, "scope": 10, "term": "(merge T189 T188 ([]))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "653": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "654": { "goal": [{ "clause": 7, "scope": 7, "term": "(merge T87 T86 X96)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X96"], "exprvars": [] } }, "655": { "goal": [{ "clause": 8, "scope": 7, "term": "(merge T87 T86 X96)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X96"], "exprvars": [] } }, "656": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (less T120 (s T121)) (merge T122 (. T121 T123) X166))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X166"], "exprvars": [] } }, "657": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "658": { "goal": [{ "clause": -1, "scope": -1, "term": "(less T120 (s T121))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "659": { "goal": [{ "clause": -1, "scope": -1, "term": "(merge T126 (. T127 T128) X166)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X166"], "exprvars": [] } }, "1098": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1097": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (less T285 (s T289)) (merge T290 (. T289 T291) ([])))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T285"], "free": [], "exprvars": [] } }, "660": { "goal": [ { "clause": 9, "scope": 8, "term": "(less T120 (s T121))" }, { "clause": 10, "scope": 8, "term": "(less T120 (s T121))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "661": { "goal": [{ "clause": 9, "scope": 8, "term": "(less T120 (s T121))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "541": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (split (. T62 (. T63 T64)) X92 X93) (',' (ms X92 X94) (',' (ms X93 X95) (merge X94 X95 X96))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X96", "X92", "X93", "X94", "X95" ], "exprvars": [] } }, "662": { "goal": [{ "clause": 10, "scope": 8, "term": "(less T120 (s T121))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "663": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "543": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "664": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "665": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "545": { "goal": [{ "clause": -1, "scope": -1, "term": "(split (. T62 (. T63 T64)) X92 X93)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X92", "X93" ], "exprvars": [] } }, "546": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (ms T65 X94) (',' (ms T66 X95) (merge X94 X95 X96)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X96", "X94", "X95" ], "exprvars": [] } }, "1089": { "goal": [{ "clause": 8, "scope": 12, "term": "(merge T244 T243 (. T219 ([])))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T219"], "free": [], "exprvars": [] } }, "547": { "goal": [{ "clause": -1, "scope": -1, "term": "(ms T65 X94)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X94"], "exprvars": [] } }, "548": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (ms T68 X95) (merge T67 X95 X96))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X96", "X95" ], "exprvars": [] } }, "549": { "goal": [ { "clause": 0, "scope": 6, "term": "(ms T65 X94)" }, { "clause": 1, "scope": 6, "term": "(ms T65 X94)" }, { "clause": 2, "scope": 6, "term": "(ms T65 X94)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X94"], "exprvars": [] } }, "550": { "goal": [{ "clause": 0, "scope": 6, "term": "(ms T65 X94)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X94"], "exprvars": [] } }, "551": { "goal": [ { "clause": 1, "scope": 6, "term": "(ms T65 X94)" }, { "clause": 2, "scope": 6, "term": "(ms T65 X94)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X94"], "exprvars": [] } }, "552": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "553": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "576": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "582": { "goal": [{ "clause": 1, "scope": 6, "term": "(ms T65 X94)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X94"], "exprvars": [] } }, "583": { "goal": [{ "clause": 2, "scope": 6, "term": "(ms T65 X94)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X94"], "exprvars": [] } }, "464": { "goal": [{ "clause": -1, "scope": -1, "term": "(ms (. T30 T29) X13)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X13"], "exprvars": [] } }, "466": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (ms T47 X14) (merge T46 X14 ([])))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X14"], "exprvars": [] } } }, "edges": [ { "from": 2, "to": 4, "label": "CASE" }, { "from": 4, "to": 12, "label": "EVAL with clause\nms([], []).\nand substitutionT1 -> [],\nT2 -> []" }, { "from": 4, "to": 13, "label": "EVAL-BACKTRACK" }, { "from": 12, "to": 16, "label": "SUCCESS" }, { "from": 13, "to": 919, "label": "EVAL with clause\nms(.(X246, []), .(X246, [])).\nand substitutionX246 -> T211,\nT1 -> .(T211, []),\nT2 -> .(T211, [])" }, { "from": 13, "to": 921, "label": "EVAL-BACKTRACK" }, { "from": 16, "to": 25, "label": "BACKTRACK\nfor clause: ms(.(X, []), .(X, []))because of non-unification" }, { "from": 25, "to": 28, "label": "EVAL with clause\nms(.(X7, .(X8, X9)), X10) :- ','(split(.(X7, .(X8, X9)), X11, X12), ','(ms(X11, X13), ','(ms(X12, X14), merge(X13, X14, X10)))).\nand substitutionX7 -> T10,\nX8 -> T11,\nX9 -> T12,\nT1 -> .(T10, .(T11, T12)),\nX10 -> [],\nT7 -> T10,\nT8 -> T11,\nT9 -> T12" }, { "from": 25, "to": 30, "label": "EVAL-BACKTRACK" }, { "from": 28, "to": 33, "label": "CASE" }, { "from": 33, "to": 34, "label": "BACKTRACK\nfor clause: split([], [], [])because of non-unification" }, { "from": 34, "to": 37, "label": "ONLY EVAL with clause\nsplit(.(X27, X28), .(X27, X29), X30) :- split(X28, X30, X29).\nand substitutionT10 -> T27,\nX27 -> T27,\nT11 -> T25,\nT12 -> T26,\nX28 -> .(T25, T26),\nX29 -> X31,\nX11 -> .(T27, X31),\nX12 -> X32,\nX30 -> X32,\nT23 -> T25,\nT24 -> T26,\nT22 -> T27" }, { "from": 37, "to": 40, "label": "SPLIT 1" }, { "from": 37, "to": 41, "label": "SPLIT 2\nreplacements:X32 -> T28,\nX31 -> T29,\nT27 -> T30" }, { "from": 40, "to": 46, "label": "CASE" }, { "from": 41, "to": 464, "label": "SPLIT 1" }, { "from": 41, "to": 466, "label": "SPLIT 2\nreplacements:X13 -> T46,\nT28 -> T47" }, { "from": 46, "to": 56, "label": "BACKTRACK\nfor clause: split([], [], [])because of non-unification" }, { "from": 56, "to": 66, "label": "ONLY EVAL with clause\nsplit(.(X45, X46), .(X45, X47), X48) :- split(X46, X48, X47).\nand substitutionT25 -> T36,\nX45 -> T36,\nT26 -> T38,\nX46 -> T38,\nX47 -> X49,\nX32 -> .(T36, X49),\nX31 -> X50,\nX48 -> X50,\nT37 -> T38" }, { "from": 66, "to": 67, "label": "CASE" }, { "from": 67, "to": 68, "label": "PARALLEL" }, { "from": 67, "to": 69, "label": "PARALLEL" }, { "from": 68, "to": 70, "label": "EVAL with clause\nsplit([], [], []).\nand substitutionT38 -> [],\nX50 -> [],\nX49 -> []" }, { "from": 68, "to": 71, "label": "EVAL-BACKTRACK" }, { "from": 69, "to": 86, "label": "EVAL with clause\nsplit(.(X63, X64), .(X63, X65), X66) :- split(X64, X66, X65).\nand substitutionX63 -> T43,\nX64 -> T45,\nT38 -> .(T43, T45),\nX65 -> X67,\nX50 -> .(T43, X67),\nX49 -> X68,\nX66 -> X68,\nT44 -> T45" }, { "from": 69, "to": 96, "label": "EVAL-BACKTRACK" }, { "from": 70, "to": 72, "label": "SUCCESS" }, { "from": 86, "to": 66, "label": "INSTANCE with matching:\nT38 -> T45\nX50 -> X68\nX49 -> X67" }, { "from": 464, "to": 470, "label": "CASE" }, { "from": 466, "to": 890, "label": "SPLIT 1" }, { "from": 466, "to": 891, "label": "SPLIT 2\nreplacements:X14 -> T188,\nT46 -> T189" }, { "from": 470, "to": 472, "label": "BACKTRACK\nfor clause: ms([], [])because of non-unification" }, { "from": 472, "to": 473, "label": "PARALLEL" }, { "from": 472, "to": 474, "label": "PARALLEL" }, { "from": 473, "to": 475, "label": "EVAL with clause\nms(.(X73, []), .(X73, [])).\nand substitutionT30 -> T52,\nX73 -> T52,\nT29 -> [],\nX13 -> .(T52, [])" }, { "from": 473, "to": 476, "label": "EVAL-BACKTRACK" }, { "from": 474, "to": 541, "label": "EVAL with clause\nms(.(X88, .(X89, X90)), X91) :- ','(split(.(X88, .(X89, X90)), X92, X93), ','(ms(X92, X94), ','(ms(X93, X95), merge(X94, X95, X91)))).\nand substitutionT30 -> T62,\nX88 -> T62,\nX89 -> T63,\nX90 -> T64,\nT29 -> .(T63, T64),\nX13 -> X96,\nX91 -> X96,\nT59 -> T62,\nT60 -> T63,\nT61 -> T64" }, { "from": 474, "to": 543, "label": "EVAL-BACKTRACK" }, { "from": 475, "to": 477, "label": "SUCCESS" }, { "from": 541, "to": 545, "label": "SPLIT 1" }, { "from": 541, "to": 546, "label": "SPLIT 2\nreplacements:X92 -> T65,\nX93 -> T66" }, { "from": 545, "to": 40, "label": "INSTANCE with matching:\nT25 -> T62\nT26 -> .(T63, T64)\nX32 -> X92\nX31 -> X93" }, { "from": 546, "to": 547, "label": "SPLIT 1" }, { "from": 546, "to": 548, "label": "SPLIT 2\nreplacements:X94 -> T67,\nT66 -> T68" }, { "from": 547, "to": 549, "label": "CASE" }, { "from": 548, "to": 641, "label": "SPLIT 1" }, { "from": 548, "to": 642, "label": "SPLIT 2\nreplacements:X95 -> T86,\nT67 -> T87" }, { "from": 549, "to": 550, "label": "PARALLEL" }, { "from": 549, "to": 551, "label": "PARALLEL" }, { "from": 550, "to": 552, "label": "EVAL with clause\nms([], []).\nand substitutionT65 -> [],\nX94 -> []" }, { "from": 550, "to": 553, "label": "EVAL-BACKTRACK" }, { "from": 551, "to": 582, "label": "PARALLEL" }, { "from": 551, "to": 583, "label": "PARALLEL" }, { "from": 552, "to": 576, "label": "SUCCESS" }, { "from": 582, "to": 592, "label": "EVAL with clause\nms(.(X101, []), .(X101, [])).\nand substitutionX101 -> T73,\nT65 -> .(T73, []),\nX94 -> .(T73, [])" }, { "from": 582, "to": 594, "label": "EVAL-BACKTRACK" }, { "from": 583, "to": 626, "label": "EVAL with clause\nms(.(X116, .(X117, X118)), X119) :- ','(split(.(X116, .(X117, X118)), X120, X121), ','(ms(X120, X122), ','(ms(X121, X123), merge(X122, X123, X119)))).\nand substitutionX116 -> T83,\nX117 -> T84,\nX118 -> T85,\nT65 -> .(T83, .(T84, T85)),\nX94 -> X124,\nX119 -> X124,\nT80 -> T83,\nT81 -> T84,\nT82 -> T85" }, { "from": 583, "to": 628, "label": "EVAL-BACKTRACK" }, { "from": 592, "to": 596, "label": "SUCCESS" }, { "from": 626, "to": 541, "label": "INSTANCE with matching:\nT62 -> T83\nT63 -> T84\nT64 -> T85\nX92 -> X120\nX93 -> X121\nX94 -> X122\nX95 -> X123\nX96 -> X124" }, { "from": 641, "to": 547, "label": "INSTANCE with matching:\nT65 -> T68\nX94 -> X95" }, { "from": 642, "to": 643, "label": "CASE" }, { "from": 643, "to": 644, "label": "PARALLEL" }, { "from": 643, "to": 645, "label": "PARALLEL" }, { "from": 644, "to": 646, "label": "EVAL with clause\nmerge([], X131, X131).\nand substitutionT87 -> [],\nT86 -> T94,\nX131 -> T94,\nX96 -> T94" }, { "from": 644, "to": 647, "label": "EVAL-BACKTRACK" }, { "from": 645, "to": 649, "label": "PARALLEL" }, { "from": 645, "to": 650, "label": "PARALLEL" }, { "from": 646, "to": 648, "label": "SUCCESS" }, { "from": 649, "to": 651, "label": "EVAL with clause\nmerge(X136, [], X136).\nand substitutionT87 -> T99,\nX136 -> T99,\nT86 -> [],\nX96 -> T99" }, { "from": 649, "to": 652, "label": "EVAL-BACKTRACK" }, { "from": 650, "to": 654, "label": "PARALLEL" }, { "from": 650, "to": 655, "label": "PARALLEL" }, { "from": 651, "to": 653, "label": "SUCCESS" }, { "from": 654, "to": 656, "label": "EVAL with clause\nmerge(.(X161, X162), .(X163, X164), .(X161, X165)) :- ','(less(X161, s(X163)), merge(X162, .(X163, X164), X165)).\nand substitutionX161 -> T120,\nX162 -> T122,\nT87 -> .(T120, T122),\nX163 -> T121,\nX164 -> T123,\nT86 -> .(T121, T123),\nX165 -> X166,\nX96 -> .(T120, X166),\nT116 -> T120,\nT118 -> T121,\nT117 -> T122,\nT119 -> T123" }, { "from": 654, "to": 657, "label": "EVAL-BACKTRACK" }, { "from": 655, "to": 866, "label": "EVAL with clause\nmerge(.(X211, X212), .(X213, X214), .(X213, X215)) :- ','(less(X213, X211), merge(.(X211, X212), X214, X215)).\nand substitutionX211 -> T178,\nX212 -> T180,\nT87 -> .(T178, T180),\nX213 -> T177,\nX214 -> T179,\nT86 -> .(T177, T179),\nX215 -> X216,\nX96 -> .(T177, X216),\nT175 -> T177,\nT173 -> T178,\nT176 -> T179,\nT174 -> T180" }, { "from": 655, "to": 867, "label": "EVAL-BACKTRACK" }, { "from": 656, "to": 658, "label": "SPLIT 1" }, { "from": 656, "to": 659, "label": "SPLIT 2\nnew knowledge:\nT120 is ground\nreplacements:T122 -> T126,\nT121 -> T127,\nT123 -> T128" }, { "from": 658, "to": 660, "label": "CASE" }, { "from": 659, "to": 642, "label": "INSTANCE with matching:\nT87 -> T126\nT86 -> .(T127, T128)\nX96 -> X166" }, { "from": 660, "to": 661, "label": "PARALLEL" }, { "from": 660, "to": 662, "label": "PARALLEL" }, { "from": 661, "to": 663, "label": "EVAL with clause\nless(0, s(X175)).\nand substitutionT120 -> 0,\nT121 -> T135,\nX175 -> T135" }, { "from": 661, "to": 664, "label": "EVAL-BACKTRACK" }, { "from": 662, "to": 836, "label": "EVAL with clause\nless(s(X180), s(X181)) :- less(X180, X181).\nand substitutionX180 -> T142,\nT120 -> s(T142),\nT121 -> T143,\nX181 -> T143,\nT140 -> T142,\nT141 -> T143" }, { "from": 662, "to": 837, "label": "EVAL-BACKTRACK" }, { "from": 663, "to": 665, "label": "SUCCESS" }, { "from": 836, "to": 840, "label": "CASE" }, { "from": 840, "to": 842, "label": "PARALLEL" }, { "from": 840, "to": 843, "label": "PARALLEL" }, { "from": 842, "to": 846, "label": "EVAL with clause\nless(0, s(X188)).\nand substitutionT142 -> 0,\nX188 -> T150,\nT143 -> s(T150)" }, { "from": 842, "to": 848, "label": "EVAL-BACKTRACK" }, { "from": 843, "to": 854, "label": "EVAL with clause\nless(s(X193), s(X194)) :- less(X193, X194).\nand substitutionX193 -> T157,\nT142 -> s(T157),\nX194 -> T158,\nT143 -> s(T158),\nT155 -> T157,\nT156 -> T158" }, { "from": 843, "to": 855, "label": "EVAL-BACKTRACK" }, { "from": 846, "to": 850, "label": "SUCCESS" }, { "from": 854, "to": 836, "label": "INSTANCE with matching:\nT142 -> T157\nT143 -> T158" }, { "from": 866, "to": 870, "label": "SPLIT 1" }, { "from": 866, "to": 871, "label": "SPLIT 2\nnew knowledge:\nT177 is ground\nreplacements:T178 -> T183,\nT180 -> T184,\nT179 -> T185" }, { "from": 870, "to": 836, "label": "INSTANCE with matching:\nT142 -> T177\nT143 -> T178" }, { "from": 871, "to": 642, "label": "INSTANCE with matching:\nT87 -> .(T183, T184)\nT86 -> T185\nX96 -> X216" }, { "from": 890, "to": 547, "label": "INSTANCE with matching:\nT65 -> T47\nX94 -> X14" }, { "from": 891, "to": 892, "label": "CASE" }, { "from": 892, "to": 893, "label": "PARALLEL" }, { "from": 892, "to": 894, "label": "PARALLEL" }, { "from": 893, "to": 908, "label": "EVAL with clause\nmerge([], X229, X229).\nand substitutionT189 -> [],\nT188 -> [],\nX229 -> [],\nT196 -> []" }, { "from": 893, "to": 909, "label": "EVAL-BACKTRACK" }, { "from": 894, "to": 911, "label": "PARALLEL" }, { "from": 894, "to": 912, "label": "PARALLEL" }, { "from": 908, "to": 910, "label": "SUCCESS" }, { "from": 911, "to": 913, "label": "EVAL with clause\nmerge(X234, [], X234).\nand substitutionT189 -> [],\nX234 -> [],\nT188 -> [],\nT201 -> []" }, { "from": 911, "to": 914, "label": "EVAL-BACKTRACK" }, { "from": 912, "to": 916, "label": "BACKTRACK\nfor clause: merge(.(X, Xs), .(Y, Ys), .(X, Zs)) :- ','(less(X, s(Y)), merge(Xs, .(Y, Ys), Zs))because of non-unification" }, { "from": 913, "to": 915, "label": "SUCCESS" }, { "from": 916, "to": 917, "label": "BACKTRACK\nfor clause: merge(.(X, Xs), .(Y, Ys), .(Y, Zs)) :- ','(less(Y, X), merge(.(X, Xs), Ys, Zs))because of non-unification" }, { "from": 919, "to": 922, "label": "SUCCESS" }, { "from": 921, "to": 1119, "label": "EVAL with clause\nms(.(X345, .(X346, X347)), X348) :- ','(split(.(X345, .(X346, X347)), X349, X350), ','(ms(X349, X351), ','(ms(X350, X352), merge(X351, X352, X348)))).\nand substitutionX345 -> T333,\nX346 -> T334,\nX347 -> T335,\nT1 -> .(T333, .(T334, T335)),\nT2 -> T332,\nX348 -> T332,\nT329 -> T333,\nT330 -> T334,\nT331 -> T335" }, { "from": 921, "to": 1120, "label": "EVAL-BACKTRACK" }, { "from": 922, "to": 929, "label": "EVAL with clause\nms(.(X251, .(X252, X253)), X254) :- ','(split(.(X251, .(X252, X253)), X255, X256), ','(ms(X255, X257), ','(ms(X256, X258), merge(X257, X258, X254)))).\nand substitutionX251 -> T220,\nX252 -> T221,\nX253 -> T222,\nT1 -> .(T220, .(T221, T222)),\nT211 -> T219,\nX254 -> .(T219, []),\nT216 -> T220,\nT217 -> T221,\nT218 -> T222" }, { "from": 922, "to": 930, "label": "EVAL-BACKTRACK" }, { "from": 929, "to": 931, "label": "CASE" }, { "from": 931, "to": 932, "label": "BACKTRACK\nfor clause: split([], [], [])because of non-unification" }, { "from": 932, "to": 935, "label": "ONLY EVAL with clause\nsplit(.(X271, X272), .(X271, X273), X274) :- split(X272, X274, X273).\nand substitutionT220 -> T237,\nX271 -> T237,\nT221 -> T235,\nT222 -> T236,\nX272 -> .(T235, T236),\nX273 -> X275,\nX255 -> .(T237, X275),\nX256 -> X276,\nX274 -> X276,\nT233 -> T235,\nT234 -> T236,\nT232 -> T237" }, { "from": 935, "to": 1017, "label": "SPLIT 1" }, { "from": 935, "to": 1019, "label": "SPLIT 2\nreplacements:X276 -> T238,\nX275 -> T239,\nT237 -> T240" }, { "from": 1017, "to": 40, "label": "INSTANCE with matching:\nT25 -> T235\nT26 -> T236\nX32 -> X276\nX31 -> X275" }, { "from": 1019, "to": 1027, "label": "SPLIT 1" }, { "from": 1019, "to": 1028, "label": "SPLIT 2\nreplacements:X257 -> T241,\nT238 -> T242" }, { "from": 1027, "to": 464, "label": "INSTANCE with matching:\nT30 -> T240\nT29 -> T239\nX13 -> X257" }, { "from": 1028, "to": 1029, "label": "SPLIT 1" }, { "from": 1028, "to": 1030, "label": "SPLIT 2\nreplacements:X258 -> T243,\nT241 -> T244" }, { "from": 1029, "to": 547, "label": "INSTANCE with matching:\nT65 -> T242\nX94 -> X258" }, { "from": 1030, "to": 1033, "label": "CASE" }, { "from": 1033, "to": 1034, "label": "PARALLEL" }, { "from": 1033, "to": 1035, "label": "PARALLEL" }, { "from": 1034, "to": 1040, "label": "EVAL with clause\nmerge([], X283, X283).\nand substitutionT244 -> [],\nT243 -> .(T258, []),\nX283 -> .(T258, []),\nT219 -> T258,\nT257 -> .(T258, [])" }, { "from": 1034, "to": 1041, "label": "EVAL-BACKTRACK" }, { "from": 1035, "to": 1043, "label": "PARALLEL" }, { "from": 1035, "to": 1045, "label": "PARALLEL" }, { "from": 1040, "to": 1042, "label": "SUCCESS" }, { "from": 1043, "to": 1048, "label": "EVAL with clause\nmerge(X288, [], X288).\nand substitutionT244 -> .(T268, []),\nX288 -> .(T268, []),\nT243 -> [],\nT219 -> T268,\nT267 -> .(T268, [])" }, { "from": 1043, "to": 1049, "label": "EVAL-BACKTRACK" }, { "from": 1045, "to": 1087, "label": "PARALLEL" }, { "from": 1045, "to": 1089, "label": "PARALLEL" }, { "from": 1048, "to": 1050, "label": "SUCCESS" }, { "from": 1087, "to": 1097, "label": "EVAL with clause\nmerge(.(X309, X310), .(X311, X312), .(X309, X313)) :- ','(less(X309, s(X311)), merge(X310, .(X311, X312), X313)).\nand substitutionX309 -> T285,\nX310 -> T290,\nT244 -> .(T285, T290),\nX311 -> T289,\nX312 -> T291,\nT243 -> .(T289, T291),\nT219 -> T285,\nX313 -> [],\nT287 -> T289,\nT286 -> T290,\nT288 -> T291" }, { "from": 1087, "to": 1098, "label": "EVAL-BACKTRACK" }, { "from": 1089, "to": 1109, "label": "EVAL with clause\nmerge(.(X330, X331), .(X332, X333), .(X332, X334)) :- ','(less(X332, X330), merge(.(X330, X331), X333, X334)).\nand substitutionX330 -> T315,\nX331 -> T317,\nT244 -> .(T315, T317),\nX332 -> T313,\nX333 -> T316,\nT243 -> .(T313, T316),\nT219 -> T313,\nX334 -> [],\nT311 -> T315,\nT314 -> T316,\nT312 -> T317" }, { "from": 1089, "to": 1110, "label": "EVAL-BACKTRACK" }, { "from": 1097, "to": 1104, "label": "SPLIT 1" }, { "from": 1097, "to": 1105, "label": "SPLIT 2\nnew knowledge:\nT285 is ground\nreplacements:T290 -> T294,\nT289 -> T295,\nT291 -> T296" }, { "from": 1104, "to": 658, "label": "INSTANCE with matching:\nT120 -> T285\nT121 -> T289" }, { "from": 1105, "to": 891, "label": "INSTANCE with matching:\nT189 -> T294\nT188 -> .(T295, T296)" }, { "from": 1109, "to": 1117, "label": "SPLIT 1" }, { "from": 1109, "to": 1118, "label": "SPLIT 2\nnew knowledge:\nT313 is ground\nreplacements:T315 -> T320,\nT317 -> T321,\nT316 -> T322" }, { "from": 1117, "to": 836, "label": "INSTANCE with matching:\nT142 -> T313\nT143 -> T315" }, { "from": 1118, "to": 891, "label": "INSTANCE with matching:\nT189 -> .(T320, T321)\nT188 -> T322" }, { "from": 1119, "to": 1121, "label": "CASE" }, { "from": 1121, "to": 1122, "label": "BACKTRACK\nfor clause: split([], [], [])because of non-unification" }, { "from": 1122, "to": 1125, "label": "ONLY EVAL with clause\nsplit(.(X365, X366), .(X365, X367), X368) :- split(X366, X368, X367).\nand substitutionT333 -> T350,\nX365 -> T350,\nT334 -> T348,\nT335 -> T349,\nX366 -> .(T348, T349),\nX367 -> X369,\nX349 -> .(T350, X369),\nX350 -> X370,\nX368 -> X370,\nT346 -> T348,\nT347 -> T349,\nT345 -> T350" }, { "from": 1125, "to": 1126, "label": "SPLIT 1" }, { "from": 1125, "to": 1127, "label": "SPLIT 2\nreplacements:X370 -> T351,\nX369 -> T352,\nT350 -> T353,\nT1 -> T354" }, { "from": 1126, "to": 40, "label": "INSTANCE with matching:\nT25 -> T348\nT26 -> T349\nX32 -> X370\nX31 -> X369" }, { "from": 1127, "to": 1128, "label": "SPLIT 1" }, { "from": 1127, "to": 1129, "label": "SPLIT 2\nreplacements:X351 -> T355,\nT351 -> T356,\nT354 -> T357" }, { "from": 1128, "to": 464, "label": "INSTANCE with matching:\nT30 -> T353\nT29 -> T352\nX13 -> X351" }, { "from": 1129, "to": 1130, "label": "SPLIT 1" }, { "from": 1129, "to": 1131, "label": "SPLIT 2\nreplacements:X352 -> T358,\nT355 -> T359,\nT357 -> T360" }, { "from": 1130, "to": 547, "label": "INSTANCE with matching:\nT65 -> T356\nX94 -> X352" }, { "from": 1131, "to": 1132, "label": "CASE" }, { "from": 1132, "to": 1133, "label": "PARALLEL" }, { "from": 1132, "to": 1134, "label": "PARALLEL" }, { "from": 1133, "to": 1135, "label": "EVAL with clause\nmerge([], X377, X377).\nand substitutionT359 -> [],\nT358 -> T367,\nX377 -> T367,\nT332 -> T367" }, { "from": 1133, "to": 1136, "label": "EVAL-BACKTRACK" }, { "from": 1134, "to": 1138, "label": "PARALLEL" }, { "from": 1134, "to": 1139, "label": "PARALLEL" }, { "from": 1135, "to": 1137, "label": "SUCCESS" }, { "from": 1138, "to": 1140, "label": "EVAL with clause\nmerge(X382, [], X382).\nand substitutionT359 -> T372,\nX382 -> T372,\nT358 -> [],\nT332 -> T372" }, { "from": 1138, "to": 1141, "label": "EVAL-BACKTRACK" }, { "from": 1139, "to": 1143, "label": "PARALLEL" }, { "from": 1139, "to": 1144, "label": "PARALLEL" }, { "from": 1140, "to": 1142, "label": "SUCCESS" }, { "from": 1143, "to": 1145, "label": "EVAL with clause\nmerge(.(X403, X404), .(X405, X406), .(X403, X407)) :- ','(less(X403, s(X405)), merge(X404, .(X405, X406), X407)).\nand substitutionX403 -> T393,\nX404 -> T399,\nT359 -> .(T393, T399),\nX405 -> T398,\nX406 -> T400,\nT358 -> .(T398, T400),\nX407 -> T397,\nT332 -> .(T393, T397),\nT395 -> T398,\nT394 -> T399,\nT396 -> T400" }, { "from": 1143, "to": 1146, "label": "EVAL-BACKTRACK" }, { "from": 1144, "to": 1192, "label": "EVAL with clause\nmerge(.(X582, X583), .(X584, X585), .(X584, X586)) :- ','(less(X584, X582), merge(.(X582, X583), X585, X586)).\nand substitutionX582 -> T624,\nX583 -> T626,\nT359 -> .(T624, T626),\nX584 -> T621,\nX585 -> T625,\nT358 -> .(T621, T625),\nX586 -> T623,\nT332 -> .(T621, T623),\nT619 -> T624,\nT622 -> T625,\nT620 -> T626" }, { "from": 1144, "to": 1193, "label": "EVAL-BACKTRACK" }, { "from": 1145, "to": 1147, "label": "SPLIT 1" }, { "from": 1145, "to": 1148, "label": "SPLIT 2\nnew knowledge:\nT393 is ground\nreplacements:T399 -> T403,\nT398 -> T404,\nT400 -> T405,\nT360 -> T406" }, { "from": 1147, "to": 658, "label": "INSTANCE with matching:\nT120 -> T393\nT121 -> T398" }, { "from": 1148, "to": 1149, "label": "CASE" }, { "from": 1149, "to": 1150, "label": "PARALLEL" }, { "from": 1149, "to": 1151, "label": "PARALLEL" }, { "from": 1150, "to": 1152, "label": "EVAL with clause\nmerge([], X418, X418).\nand substitutionT403 -> [],\nT404 -> T421,\nT405 -> T422,\nX418 -> .(T421, T422),\nT397 -> .(T421, T422)" }, { "from": 1150, "to": 1153, "label": "EVAL-BACKTRACK" }, { "from": 1151, "to": 1155, "label": "BACKTRACK\nfor clause: merge(Xs, [], Xs)because of non-unification" }, { "from": 1152, "to": 1154, "label": "SUCCESS" }, { "from": 1155, "to": 1156, "label": "PARALLEL" }, { "from": 1155, "to": 1157, "label": "PARALLEL" }, { "from": 1156, "to": 1158, "label": "EVAL with clause\nmerge(.(X440, X441), .(X442, X443), .(X440, X444)) :- ','(less(X440, s(X442)), merge(X441, .(X442, X443), X444)).\nand substitutionX440 -> T444,\nX441 -> T450,\nT403 -> .(T444, T450),\nT404 -> T449,\nX442 -> T449,\nT405 -> T451,\nX443 -> T451,\nX444 -> T448,\nT397 -> .(T444, T448),\nT446 -> T449,\nT445 -> T450,\nT447 -> T451" }, { "from": 1156, "to": 1159, "label": "EVAL-BACKTRACK" }, { "from": 1157, "to": 1190, "label": "EVAL with clause\nmerge(.(X565, X566), .(X567, X568), .(X567, X569)) :- ','(less(X567, X565), merge(.(X565, X566), X568, X569)).\nand substitutionX565 -> T604,\nX566 -> T606,\nT403 -> .(T604, T606),\nT404 -> T601,\nX567 -> T601,\nT405 -> T605,\nX568 -> T605,\nX569 -> T603,\nT397 -> .(T601, T603),\nT599 -> T604,\nT602 -> T605,\nT600 -> T606" }, { "from": 1157, "to": 1191, "label": "EVAL-BACKTRACK" }, { "from": 1158, "to": 1160, "label": "SPLIT 1" }, { "from": 1158, "to": 1161, "label": "SPLIT 2\nnew knowledge:\nT444 is ground\nreplacements:T450 -> T454,\nT449 -> T455,\nT451 -> T456" }, { "from": 1160, "to": 658, "label": "INSTANCE with matching:\nT120 -> T444\nT121 -> T449" }, { "from": 1161, "to": 1162, "label": "CASE" }, { "from": 1162, "to": 1163, "label": "PARALLEL" }, { "from": 1162, "to": 1164, "label": "PARALLEL" }, { "from": 1163, "to": 1165, "label": "EVAL with clause\nmerge([], X455, X455).\nand substitutionT454 -> [],\nT455 -> T471,\nT456 -> T472,\nX455 -> .(T471, T472),\nT448 -> .(T471, T472)" }, { "from": 1163, "to": 1166, "label": "EVAL-BACKTRACK" }, { "from": 1164, "to": 1168, "label": "BACKTRACK\nfor clause: merge(Xs, [], Xs)because of non-unification" }, { "from": 1165, "to": 1167, "label": "SUCCESS" }, { "from": 1168, "to": 1169, "label": "PARALLEL" }, { "from": 1168, "to": 1170, "label": "PARALLEL" }, { "from": 1169, "to": 1171, "label": "EVAL with clause\nmerge(.(X477, X478), .(X479, X480), .(X477, X481)) :- ','(less(X477, s(X479)), merge(X478, .(X479, X480), X481)).\nand substitutionX477 -> T494,\nX478 -> T500,\nT454 -> .(T494, T500),\nT455 -> T499,\nX479 -> T499,\nT456 -> T501,\nX480 -> T501,\nX481 -> T498,\nT448 -> .(T494, T498),\nT496 -> T499,\nT495 -> T500,\nT497 -> T501" }, { "from": 1169, "to": 1172, "label": "EVAL-BACKTRACK" }, { "from": 1170, "to": 1173, "label": "EVAL with clause\nmerge(.(X494, X495), .(X496, X497), .(X496, X498)) :- ','(less(X496, X494), merge(.(X494, X495), X497, X498)).\nand substitutionX494 -> T519,\nX495 -> T521,\nT454 -> .(T519, T521),\nT455 -> T516,\nX496 -> T516,\nT456 -> T520,\nX497 -> T520,\nX498 -> T518,\nT448 -> .(T516, T518),\nT514 -> T519,\nT517 -> T520,\nT515 -> T521" }, { "from": 1170, "to": 1174, "label": "EVAL-BACKTRACK" }, { "from": 1171, "to": 1158, "label": "INSTANCE with matching:\nT444 -> T494\nT449 -> T499\nT450 -> T500\nT451 -> T501\nT448 -> T498" }, { "from": 1173, "to": 1175, "label": "SPLIT 1" }, { "from": 1173, "to": 1176, "label": "SPLIT 2\nnew knowledge:\nT516 is ground\nreplacements:T519 -> T524,\nT521 -> T525,\nT520 -> T526" }, { "from": 1175, "to": 836, "label": "INSTANCE with matching:\nT142 -> T516\nT143 -> T519" }, { "from": 1176, "to": 1177, "label": "CASE" }, { "from": 1177, "to": 1178, "label": "BACKTRACK\nfor clause: merge([], Xs, Xs)because of non-unification" }, { "from": 1178, "to": 1179, "label": "PARALLEL" }, { "from": 1178, "to": 1180, "label": "PARALLEL" }, { "from": 1179, "to": 1181, "label": "EVAL with clause\nmerge(X510, [], X510).\nand substitutionT524 -> T537,\nT525 -> T538,\nX510 -> .(T537, T538),\nT526 -> [],\nT518 -> .(T537, T538)" }, { "from": 1179, "to": 1182, "label": "EVAL-BACKTRACK" }, { "from": 1180, "to": 1184, "label": "PARALLEL" }, { "from": 1180, "to": 1185, "label": "PARALLEL" }, { "from": 1181, "to": 1183, "label": "SUCCESS" }, { "from": 1184, "to": 1186, "label": "EVAL with clause\nmerge(.(X531, X532), .(X533, X534), .(X531, X535)) :- ','(less(X531, s(X533)), merge(X532, .(X533, X534), X535)).\nand substitutionT524 -> T559,\nX531 -> T559,\nT525 -> T565,\nX532 -> T565,\nX533 -> T564,\nX534 -> T566,\nT526 -> .(T564, T566),\nX535 -> T563,\nT518 -> .(T559, T563),\nT561 -> T564,\nT560 -> T565,\nT562 -> T566" }, { "from": 1184, "to": 1187, "label": "EVAL-BACKTRACK" }, { "from": 1185, "to": 1188, "label": "EVAL with clause\nmerge(.(X548, X549), .(X550, X551), .(X550, X552)) :- ','(less(X550, X548), merge(.(X548, X549), X551, X552)).\nand substitutionT524 -> T584,\nX548 -> T584,\nT525 -> T586,\nX549 -> T586,\nX550 -> T581,\nX551 -> T585,\nT526 -> .(T581, T585),\nX552 -> T583,\nT518 -> .(T581, T583),\nT579 -> T584,\nT582 -> T585,\nT580 -> T586" }, { "from": 1185, "to": 1189, "label": "EVAL-BACKTRACK" }, { "from": 1186, "to": 1158, "label": "INSTANCE with matching:\nT444 -> T559\nT449 -> T564\nT450 -> T565\nT451 -> T566\nT448 -> T563" }, { "from": 1188, "to": 1173, "label": "INSTANCE with matching:\nT516 -> T581\nT519 -> T584\nT521 -> T586\nT520 -> T585\nT518 -> T583" }, { "from": 1190, "to": 1173, "label": "INSTANCE with matching:\nT516 -> T601\nT519 -> T604\nT521 -> T606\nT520 -> T605\nT518 -> T603" }, { "from": 1192, "to": 1173, "label": "INSTANCE with matching:\nT516 -> T621\nT519 -> T624\nT521 -> T626\nT520 -> T625\nT518 -> T623" } ], "type": "Graph" } } ---------------------------------------- (162) Obligation: Triples: splitA(.(X1, X2), .(X1, X3), X4) :- splitA(X2, X4, X3). splitB(X1, X2, .(X1, X3), X4) :- splitA(X2, X4, X3). pC(X1, X2, X3, X4, X5, X6, X7, X8) :- splitB(X1, .(X2, X3), X4, X5). pC(X1, X2, X3, X4, X5, X6, X7, X8) :- ','(splitcB(X1, .(X2, X3), X4, X5), msE(X4, X6)). pC(X1, X2, X3, X4, X5, X6, X7, X8) :- ','(splitcB(X1, .(X2, X3), X4, X5), ','(mscE(X4, X6), msE(X5, X7))). pC(X1, X2, X3, X4, X5, X6, X7, X8) :- ','(splitcB(X1, .(X2, X3), X4, X5), ','(mscE(X4, X6), ','(mscE(X5, X7), mergeD(X6, X7, X8)))). msE(.(X1, .(X2, X3)), X4) :- pC(X1, X2, X3, X5, X6, X7, X8, X4). mergeD(.(X1, X2), .(X3, X4), .(X1, X5)) :- lessF(X1, X3). mergeD(.(X1, X2), .(X3, X4), .(X1, X5)) :- ','(lesscF(X1, X3), mergeD(X2, .(X3, X4), X5)). mergeD(.(X1, X2), .(X3, X4), .(X3, X5)) :- lessG(X3, X1). mergeD(.(X1, X2), .(X3, X4), .(X3, X5)) :- ','(lesscG(X3, X1), mergeD(.(X1, X2), X4, X5)). lessG(s(X1), s(X2)) :- lessG(X1, X2). msH(X1, .(X2, X3), X4) :- pC(X1, X2, X3, X5, X6, X7, X8, X4). lessF(s(X1), X2) :- lessG(X1, X2). pJ(X1, X2, X3, X4, X5) :- lessF(X1, X2). pJ(X1, X2, .(X3, X4), X5, .(X3, X6)) :- ','(lesscF(X1, X2), pJ(X3, X2, X4, X5, X6)). pJ(X1, X2, .(X3, X4), X5, .(X2, X6)) :- ','(lesscF(X1, X2), pK(X2, X3, X4, X5, X6)). pK(X1, X2, X3, X4, X5) :- lessG(X1, X2). pK(X1, X2, X3, .(X4, X5), .(X2, X6)) :- ','(lesscG(X1, X2), pJ(X2, X4, X3, X5, X6)). pK(X1, X2, X3, .(X4, X5), .(X4, X6)) :- ','(lesscG(X1, X2), pK(X4, X2, X3, X5, X6)). msL(.(X1, .(X2, X3)), []) :- splitB(X2, X3, X4, X5). msL(.(X1, .(X2, X3)), []) :- ','(splitcB(X2, X3, X4, X5), msH(X1, X5, X6)). msL(.(X1, .(X2, X3)), []) :- ','(splitcB(X2, X3, X4, X5), ','(mscH(X1, X5, X6), msE(X4, X7))). msL(.(X1, .(X2, X3)), []) :- ','(splitcB(X2, X3, X4, X5), ','(mscH(X1, X5, X6), ','(mscE(X4, X7), mergeI(X6, X7)))). msL(.(X1, .(X2, X3)), .(X4, [])) :- splitB(X2, X3, X5, X6). msL(.(X1, .(X2, X3)), .(X4, [])) :- ','(splitcB(X2, X3, X5, X6), msH(X1, X6, X7)). msL(.(X1, .(X2, X3)), .(X4, [])) :- ','(splitcB(X2, X3, X5, X6), ','(mscH(X1, X6, X7), msE(X5, X8))). msL(.(X1, .(X2, X3)), .(X4, [])) :- ','(splitcB(X2, X3, X5, X6), ','(mscH(X1, X6, .(X4, X7)), ','(mscE(X5, .(X8, X9)), lessF(X4, X8)))). msL(.(X1, .(X2, X3)), .(X4, [])) :- ','(splitcB(X2, X3, X5, X6), ','(mscH(X1, X6, .(X4, X7)), ','(mscE(X5, .(X8, X9)), ','(lesscF(X4, X8), mergeI(X7, .(X8, X9)))))). msL(.(X1, .(X2, X3)), .(X4, [])) :- ','(splitcB(X2, X3, X5, X6), ','(mscH(X1, X6, .(X7, X8)), ','(mscE(X5, .(X4, X9)), lessG(X4, X7)))). msL(.(X1, .(X2, X3)), .(X4, [])) :- ','(splitcB(X2, X3, X5, X6), ','(mscH(X1, X6, .(X7, X8)), ','(mscE(X5, .(X4, X9)), ','(lesscG(X4, X7), mergeI(.(X7, X8), X9))))). msL(.(X1, .(X2, X3)), X4) :- splitB(X2, X3, X5, X6). msL(.(X1, .(X2, X3)), X4) :- ','(splitcB(X2, X3, X5, X6), msH(X1, X6, X7)). msL(.(X1, .(X2, X3)), X4) :- ','(splitcB(X2, X3, X5, X6), ','(mscH(X1, X6, X7), msE(X5, X8))). msL(.(X1, .(X2, X3)), .(X4, X5)) :- ','(splitcB(X2, X3, X6, X7), ','(mscH(X1, X7, .(X4, X8)), ','(mscE(X6, .(X9, X10)), lessF(X4, X9)))). msL(.(X1, .(X2, X3)), .(X4, .(X5, X6))) :- ','(splitcB(X2, X3, X7, X8), ','(mscH(X1, X8, .(X4, .(X5, X9))), ','(mscE(X7, .(X10, X11)), ','(lesscF(X4, X10), pJ(X5, X10, X9, X11, X6))))). msL(.(X1, .(X2, X3)), .(X4, .(X5, X6))) :- ','(splitcB(X2, X3, X7, X8), ','(mscH(X1, X8, .(X4, .(X9, X10))), ','(mscE(X7, .(X5, X11)), ','(lesscF(X4, X5), pK(X5, X9, X10, X11, X6))))). msL(.(X1, .(X2, X3)), .(X4, X5)) :- ','(splitcB(X2, X3, X6, X7), ','(mscH(X1, X7, .(X8, X9)), ','(mscE(X6, .(X4, X10)), pK(X4, X8, X9, X10, X5)))). Clauses: splitcA([], [], []). splitcA(.(X1, X2), .(X1, X3), X4) :- splitcA(X2, X4, X3). splitcB(X1, X2, .(X1, X3), X4) :- splitcA(X2, X4, X3). qcC(X1, X2, X3, X4, X5, X6, X7, X8) :- ','(splitcB(X1, .(X2, X3), X4, X5), ','(mscE(X4, X6), ','(mscE(X5, X7), mergecD(X6, X7, X8)))). mscE([], []). mscE(.(X1, []), .(X1, [])). mscE(.(X1, .(X2, X3)), X4) :- qcC(X1, X2, X3, X5, X6, X7, X8, X4). mergecD([], X1, X1). mergecD(X1, [], X1). mergecD(.(X1, X2), .(X3, X4), .(X1, X5)) :- ','(lesscF(X1, X3), mergecD(X2, .(X3, X4), X5)). mergecD(.(X1, X2), .(X3, X4), .(X3, X5)) :- ','(lesscG(X3, X1), mergecD(.(X1, X2), X4, X5)). lesscG(0, s(X1)). lesscG(s(X1), s(X2)) :- lesscG(X1, X2). mscH(X1, [], .(X1, [])). mscH(X1, .(X2, X3), X4) :- qcC(X1, X2, X3, X5, X6, X7, X8, X4). lesscF(0, X1). lesscF(s(X1), X2) :- lesscG(X1, X2). mergecI([], []). mergecI([], []). qcJ(X1, X2, [], X3, .(X2, X3)) :- lesscF(X1, X2). qcJ(X1, X2, .(X3, X4), X5, .(X3, X6)) :- ','(lesscF(X1, X2), qcJ(X3, X2, X4, X5, X6)). qcJ(X1, X2, .(X3, X4), X5, .(X2, X6)) :- ','(lesscF(X1, X2), qcK(X2, X3, X4, X5, X6)). qcK(X1, X2, X3, [], .(X2, X3)) :- lesscG(X1, X2). qcK(X1, X2, X3, .(X4, X5), .(X2, X6)) :- ','(lesscG(X1, X2), qcJ(X2, X4, X3, X5, X6)). qcK(X1, X2, X3, .(X4, X5), .(X4, X6)) :- ','(lesscG(X1, X2), qcK(X4, X2, X3, X5, X6)). Afs: msL(x1, x2) = msL(x2) ---------------------------------------- (163) UndefinedPredicateInTriplesTransformerProof (SOUND) Deleted triples and predicates having undefined goals [DT09]. ---------------------------------------- (164) Obligation: Triples: splitA(.(X1, X2), .(X1, X3), X4) :- splitA(X2, X4, X3). splitB(X1, X2, .(X1, X3), X4) :- splitA(X2, X4, X3). pC(X1, X2, X3, X4, X5, X6, X7, X8) :- splitB(X1, .(X2, X3), X4, X5). pC(X1, X2, X3, X4, X5, X6, X7, X8) :- ','(splitcB(X1, .(X2, X3), X4, X5), msE(X4, X6)). pC(X1, X2, X3, X4, X5, X6, X7, X8) :- ','(splitcB(X1, .(X2, X3), X4, X5), ','(mscE(X4, X6), msE(X5, X7))). pC(X1, X2, X3, X4, X5, X6, X7, X8) :- ','(splitcB(X1, .(X2, X3), X4, X5), ','(mscE(X4, X6), ','(mscE(X5, X7), mergeD(X6, X7, X8)))). msE(.(X1, .(X2, X3)), X4) :- pC(X1, X2, X3, X5, X6, X7, X8, X4). mergeD(.(X1, X2), .(X3, X4), .(X1, X5)) :- lessF(X1, X3). mergeD(.(X1, X2), .(X3, X4), .(X1, X5)) :- ','(lesscF(X1, X3), mergeD(X2, .(X3, X4), X5)). mergeD(.(X1, X2), .(X3, X4), .(X3, X5)) :- lessG(X3, X1). mergeD(.(X1, X2), .(X3, X4), .(X3, X5)) :- ','(lesscG(X3, X1), mergeD(.(X1, X2), X4, X5)). lessG(s(X1), s(X2)) :- lessG(X1, X2). msH(X1, .(X2, X3), X4) :- pC(X1, X2, X3, X5, X6, X7, X8, X4). lessF(s(X1), X2) :- lessG(X1, X2). pJ(X1, X2, X3, X4, X5) :- lessF(X1, X2). pJ(X1, X2, .(X3, X4), X5, .(X3, X6)) :- ','(lesscF(X1, X2), pJ(X3, X2, X4, X5, X6)). pJ(X1, X2, .(X3, X4), X5, .(X2, X6)) :- ','(lesscF(X1, X2), pK(X2, X3, X4, X5, X6)). pK(X1, X2, X3, X4, X5) :- lessG(X1, X2). pK(X1, X2, X3, .(X4, X5), .(X2, X6)) :- ','(lesscG(X1, X2), pJ(X2, X4, X3, X5, X6)). pK(X1, X2, X3, .(X4, X5), .(X4, X6)) :- ','(lesscG(X1, X2), pK(X4, X2, X3, X5, X6)). msL(.(X1, .(X2, X3)), []) :- splitB(X2, X3, X4, X5). msL(.(X1, .(X2, X3)), []) :- ','(splitcB(X2, X3, X4, X5), msH(X1, X5, X6)). msL(.(X1, .(X2, X3)), []) :- ','(splitcB(X2, X3, X4, X5), ','(mscH(X1, X5, X6), msE(X4, X7))). msL(.(X1, .(X2, X3)), .(X4, [])) :- splitB(X2, X3, X5, X6). msL(.(X1, .(X2, X3)), .(X4, [])) :- ','(splitcB(X2, X3, X5, X6), msH(X1, X6, X7)). msL(.(X1, .(X2, X3)), .(X4, [])) :- ','(splitcB(X2, X3, X5, X6), ','(mscH(X1, X6, X7), msE(X5, X8))). msL(.(X1, .(X2, X3)), .(X4, [])) :- ','(splitcB(X2, X3, X5, X6), ','(mscH(X1, X6, .(X4, X7)), ','(mscE(X5, .(X8, X9)), lessF(X4, X8)))). msL(.(X1, .(X2, X3)), .(X4, [])) :- ','(splitcB(X2, X3, X5, X6), ','(mscH(X1, X6, .(X7, X8)), ','(mscE(X5, .(X4, X9)), lessG(X4, X7)))). msL(.(X1, .(X2, X3)), X4) :- splitB(X2, X3, X5, X6). msL(.(X1, .(X2, X3)), X4) :- ','(splitcB(X2, X3, X5, X6), msH(X1, X6, X7)). msL(.(X1, .(X2, X3)), X4) :- ','(splitcB(X2, X3, X5, X6), ','(mscH(X1, X6, X7), msE(X5, X8))). msL(.(X1, .(X2, X3)), .(X4, X5)) :- ','(splitcB(X2, X3, X6, X7), ','(mscH(X1, X7, .(X4, X8)), ','(mscE(X6, .(X9, X10)), lessF(X4, X9)))). msL(.(X1, .(X2, X3)), .(X4, .(X5, X6))) :- ','(splitcB(X2, X3, X7, X8), ','(mscH(X1, X8, .(X4, .(X5, X9))), ','(mscE(X7, .(X10, X11)), ','(lesscF(X4, X10), pJ(X5, X10, X9, X11, X6))))). msL(.(X1, .(X2, X3)), .(X4, .(X5, X6))) :- ','(splitcB(X2, X3, X7, X8), ','(mscH(X1, X8, .(X4, .(X9, X10))), ','(mscE(X7, .(X5, X11)), ','(lesscF(X4, X5), pK(X5, X9, X10, X11, X6))))). msL(.(X1, .(X2, X3)), .(X4, X5)) :- ','(splitcB(X2, X3, X6, X7), ','(mscH(X1, X7, .(X8, X9)), ','(mscE(X6, .(X4, X10)), pK(X4, X8, X9, X10, X5)))). Clauses: splitcA([], [], []). splitcA(.(X1, X2), .(X1, X3), X4) :- splitcA(X2, X4, X3). splitcB(X1, X2, .(X1, X3), X4) :- splitcA(X2, X4, X3). qcC(X1, X2, X3, X4, X5, X6, X7, X8) :- ','(splitcB(X1, .(X2, X3), X4, X5), ','(mscE(X4, X6), ','(mscE(X5, X7), mergecD(X6, X7, X8)))). mscE([], []). mscE(.(X1, []), .(X1, [])). mscE(.(X1, .(X2, X3)), X4) :- qcC(X1, X2, X3, X5, X6, X7, X8, X4). mergecD([], X1, X1). mergecD(X1, [], X1). mergecD(.(X1, X2), .(X3, X4), .(X1, X5)) :- ','(lesscF(X1, X3), mergecD(X2, .(X3, X4), X5)). mergecD(.(X1, X2), .(X3, X4), .(X3, X5)) :- ','(lesscG(X3, X1), mergecD(.(X1, X2), X4, X5)). lesscG(0, s(X1)). lesscG(s(X1), s(X2)) :- lesscG(X1, X2). mscH(X1, [], .(X1, [])). mscH(X1, .(X2, X3), X4) :- qcC(X1, X2, X3, X5, X6, X7, X8, X4). lesscF(0, X1). lesscF(s(X1), X2) :- lesscG(X1, X2). mergecI([], []). mergecI([], []). qcJ(X1, X2, [], X3, .(X2, X3)) :- lesscF(X1, X2). qcJ(X1, X2, .(X3, X4), X5, .(X3, X6)) :- ','(lesscF(X1, X2), qcJ(X3, X2, X4, X5, X6)). qcJ(X1, X2, .(X3, X4), X5, .(X2, X6)) :- ','(lesscF(X1, X2), qcK(X2, X3, X4, X5, X6)). qcK(X1, X2, X3, [], .(X2, X3)) :- lesscG(X1, X2). qcK(X1, X2, X3, .(X4, X5), .(X2, X6)) :- ','(lesscG(X1, X2), qcJ(X2, X4, X3, X5, X6)). qcK(X1, X2, X3, .(X4, X5), .(X4, X6)) :- ','(lesscG(X1, X2), qcK(X4, X2, X3, X5, X6)). Afs: msL(x1, x2) = msL(x2) ---------------------------------------- (165) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: msL_in_2: (f,b) splitB_in_4: (f,f,f,f) splitA_in_3: (f,f,f) splitcB_in_4: (f,f,f,f) splitcA_in_3: (f,f,f) msH_in_3: (f,f,f) pC_in_8: (f,f,f,f,f,f,f,f) msE_in_2: (f,f) mscE_in_2: (f,f) qcC_in_8: (f,f,f,f,f,f,f,f) mergecD_in_3: (f,f,f) lesscF_in_2: (f,f) (b,f) (b,b) lesscG_in_2: (f,f) (b,b) (b,f) mergeD_in_3: (f,f,f) lessF_in_2: (f,f) (b,f) lessG_in_2: (f,f) (b,f) mscH_in_3: (f,f,f) pK_in_5: (b,f,f,f,b) pJ_in_5: (b,f,f,f,b) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: MSL_IN_AG(.(X1, .(X2, X3)), []) -> U30_AG(X1, X2, X3, splitB_in_aaaa(X2, X3, X4, X5)) MSL_IN_AG(.(X1, .(X2, X3)), []) -> SPLITB_IN_AAAA(X2, X3, X4, X5) SPLITB_IN_AAAA(X1, X2, .(X1, X3), X4) -> U2_AAAA(X1, X2, X3, X4, splitA_in_aaa(X2, X4, X3)) SPLITB_IN_AAAA(X1, X2, .(X1, X3), X4) -> SPLITA_IN_AAA(X2, X4, X3) SPLITA_IN_AAA(.(X1, X2), .(X1, X3), X4) -> U1_AAA(X1, X2, X3, X4, splitA_in_aaa(X2, X4, X3)) SPLITA_IN_AAA(.(X1, X2), .(X1, X3), X4) -> SPLITA_IN_AAA(X2, X4, X3) MSL_IN_AG(.(X1, .(X2, X3)), []) -> U31_AG(X1, X2, X3, splitcB_in_aaaa(X2, X3, X4, X5)) U31_AG(X1, X2, X3, splitcB_out_aaaa(X2, X3, X4, X5)) -> U32_AG(X1, X2, X3, msH_in_aaa(X1, X5, X6)) U31_AG(X1, X2, X3, splitcB_out_aaaa(X2, X3, X4, X5)) -> MSH_IN_AAA(X1, X5, X6) MSH_IN_AAA(X1, .(X2, X3), X4) -> U18_AAA(X1, X2, X3, X4, pC_in_aaaaaaaa(X1, X2, X3, X5, X6, X7, X8, X4)) MSH_IN_AAA(X1, .(X2, X3), X4) -> PC_IN_AAAAAAAA(X1, X2, X3, X5, X6, X7, X8, X4) PC_IN_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8) -> U3_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, splitB_in_aaaa(X1, .(X2, X3), X4, X5)) PC_IN_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8) -> SPLITB_IN_AAAA(X1, .(X2, X3), X4, X5) PC_IN_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8) -> U4_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, splitcB_in_aaaa(X1, .(X2, X3), X4, X5)) U4_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, splitcB_out_aaaa(X1, .(X2, X3), X4, X5)) -> U5_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, msE_in_aa(X4, X6)) U4_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, splitcB_out_aaaa(X1, .(X2, X3), X4, X5)) -> MSE_IN_AA(X4, X6) MSE_IN_AA(.(X1, .(X2, X3)), X4) -> U10_AA(X1, X2, X3, X4, pC_in_aaaaaaaa(X1, X2, X3, X5, X6, X7, X8, X4)) MSE_IN_AA(.(X1, .(X2, X3)), X4) -> PC_IN_AAAAAAAA(X1, X2, X3, X5, X6, X7, X8, X4) U4_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, splitcB_out_aaaa(X1, .(X2, X3), X4, X5)) -> U6_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, mscE_in_aa(X4, X6)) U6_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, mscE_out_aa(X4, X6)) -> U7_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, msE_in_aa(X5, X7)) U6_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, mscE_out_aa(X4, X6)) -> MSE_IN_AA(X5, X7) U6_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, mscE_out_aa(X4, X6)) -> U8_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, mscE_in_aa(X5, X7)) U8_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, mscE_out_aa(X5, X7)) -> U9_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, mergeD_in_aaa(X6, X7, X8)) U8_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, mscE_out_aa(X5, X7)) -> MERGED_IN_AAA(X6, X7, X8) MERGED_IN_AAA(.(X1, X2), .(X3, X4), .(X1, X5)) -> U11_AAA(X1, X2, X3, X4, X5, lessF_in_aa(X1, X3)) MERGED_IN_AAA(.(X1, X2), .(X3, X4), .(X1, X5)) -> LESSF_IN_AA(X1, X3) LESSF_IN_AA(s(X1), X2) -> U19_AA(X1, X2, lessG_in_aa(X1, X2)) LESSF_IN_AA(s(X1), X2) -> LESSG_IN_AA(X1, X2) LESSG_IN_AA(s(X1), s(X2)) -> U17_AA(X1, X2, lessG_in_aa(X1, X2)) LESSG_IN_AA(s(X1), s(X2)) -> LESSG_IN_AA(X1, X2) MERGED_IN_AAA(.(X1, X2), .(X3, X4), .(X1, X5)) -> U12_AAA(X1, X2, X3, X4, X5, lesscF_in_aa(X1, X3)) U12_AAA(X1, X2, X3, X4, X5, lesscF_out_aa(X1, X3)) -> U13_AAA(X1, X2, X3, X4, X5, mergeD_in_aaa(X2, .(X3, X4), X5)) U12_AAA(X1, X2, X3, X4, X5, lesscF_out_aa(X1, X3)) -> MERGED_IN_AAA(X2, .(X3, X4), X5) MERGED_IN_AAA(.(X1, X2), .(X3, X4), .(X3, X5)) -> U14_AAA(X1, X2, X3, X4, X5, lessG_in_aa(X3, X1)) MERGED_IN_AAA(.(X1, X2), .(X3, X4), .(X3, X5)) -> LESSG_IN_AA(X3, X1) MERGED_IN_AAA(.(X1, X2), .(X3, X4), .(X3, X5)) -> U15_AAA(X1, X2, X3, X4, X5, lesscG_in_aa(X3, X1)) U15_AAA(X1, X2, X3, X4, X5, lesscG_out_aa(X3, X1)) -> U16_AAA(X1, X2, X3, X4, X5, mergeD_in_aaa(.(X1, X2), X4, X5)) U15_AAA(X1, X2, X3, X4, X5, lesscG_out_aa(X3, X1)) -> MERGED_IN_AAA(.(X1, X2), X4, X5) U31_AG(X1, X2, X3, splitcB_out_aaaa(X2, X3, X4, X5)) -> U33_AG(X1, X2, X3, X4, mscH_in_aaa(X1, X5, X6)) U33_AG(X1, X2, X3, X4, mscH_out_aaa(X1, X5, X6)) -> U34_AG(X1, X2, X3, msE_in_aa(X4, X7)) U33_AG(X1, X2, X3, X4, mscH_out_aaa(X1, X5, X6)) -> MSE_IN_AA(X4, X7) MSL_IN_AG(.(X1, .(X2, X3)), .(X4, [])) -> U35_AG(X1, X2, X3, X4, splitB_in_aaaa(X2, X3, X5, X6)) MSL_IN_AG(.(X1, .(X2, X3)), .(X4, [])) -> SPLITB_IN_AAAA(X2, X3, X5, X6) MSL_IN_AG(.(X1, .(X2, X3)), .(X4, [])) -> U36_AG(X1, X2, X3, X4, splitcB_in_aaaa(X2, X3, X5, X6)) U36_AG(X1, X2, X3, X4, splitcB_out_aaaa(X2, X3, X5, X6)) -> U37_AG(X1, X2, X3, X4, msH_in_aaa(X1, X6, X7)) U36_AG(X1, X2, X3, X4, splitcB_out_aaaa(X2, X3, X5, X6)) -> MSH_IN_AAA(X1, X6, X7) U36_AG(X1, X2, X3, X4, splitcB_out_aaaa(X2, X3, X5, X6)) -> U38_AG(X1, X2, X3, X4, X5, mscH_in_aaa(X1, X6, X7)) U38_AG(X1, X2, X3, X4, X5, mscH_out_aaa(X1, X6, X7)) -> U39_AG(X1, X2, X3, X4, msE_in_aa(X5, X8)) U38_AG(X1, X2, X3, X4, X5, mscH_out_aaa(X1, X6, X7)) -> MSE_IN_AA(X5, X8) U36_AG(X1, X2, X3, X4, splitcB_out_aaaa(X2, X3, X5, X6)) -> U40_AG(X1, X2, X3, X4, X5, mscH_in_aaa(X1, X6, .(X4, X7))) U40_AG(X1, X2, X3, X4, X5, mscH_out_aaa(X1, X6, .(X4, X7))) -> U41_AG(X1, X2, X3, X4, mscE_in_aa(X5, .(X8, X9))) U41_AG(X1, X2, X3, X4, mscE_out_aa(X5, .(X8, X9))) -> U42_AG(X1, X2, X3, X4, lessF_in_ga(X4, X8)) U41_AG(X1, X2, X3, X4, mscE_out_aa(X5, .(X8, X9))) -> LESSF_IN_GA(X4, X8) LESSF_IN_GA(s(X1), X2) -> U19_GA(X1, X2, lessG_in_ga(X1, X2)) LESSF_IN_GA(s(X1), X2) -> LESSG_IN_GA(X1, X2) LESSG_IN_GA(s(X1), s(X2)) -> U17_GA(X1, X2, lessG_in_ga(X1, X2)) LESSG_IN_GA(s(X1), s(X2)) -> LESSG_IN_GA(X1, X2) U36_AG(X1, X2, X3, X4, splitcB_out_aaaa(X2, X3, X5, X6)) -> U43_AG(X1, X2, X3, X4, X5, mscH_in_aaa(X1, X6, .(X7, X8))) U43_AG(X1, X2, X3, X4, X5, mscH_out_aaa(X1, X6, .(X7, X8))) -> U44_AG(X1, X2, X3, X4, X7, mscE_in_aa(X5, .(X4, X9))) U44_AG(X1, X2, X3, X4, X7, mscE_out_aa(X5, .(X4, X9))) -> U45_AG(X1, X2, X3, X4, lessG_in_ga(X4, X7)) U44_AG(X1, X2, X3, X4, X7, mscE_out_aa(X5, .(X4, X9))) -> LESSG_IN_GA(X4, X7) MSL_IN_AG(.(X1, .(X2, X3)), X4) -> U46_AG(X1, X2, X3, X4, splitB_in_aaaa(X2, X3, X5, X6)) MSL_IN_AG(.(X1, .(X2, X3)), X4) -> SPLITB_IN_AAAA(X2, X3, X5, X6) MSL_IN_AG(.(X1, .(X2, X3)), X4) -> U47_AG(X1, X2, X3, X4, splitcB_in_aaaa(X2, X3, X5, X6)) U47_AG(X1, X2, X3, X4, splitcB_out_aaaa(X2, X3, X5, X6)) -> U48_AG(X1, X2, X3, X4, msH_in_aaa(X1, X6, X7)) U47_AG(X1, X2, X3, X4, splitcB_out_aaaa(X2, X3, X5, X6)) -> MSH_IN_AAA(X1, X6, X7) U47_AG(X1, X2, X3, X4, splitcB_out_aaaa(X2, X3, X5, X6)) -> U49_AG(X1, X2, X3, X4, X5, mscH_in_aaa(X1, X6, X7)) U49_AG(X1, X2, X3, X4, X5, mscH_out_aaa(X1, X6, X7)) -> U50_AG(X1, X2, X3, X4, msE_in_aa(X5, X8)) U49_AG(X1, X2, X3, X4, X5, mscH_out_aaa(X1, X6, X7)) -> MSE_IN_AA(X5, X8) MSL_IN_AG(.(X1, .(X2, X3)), .(X4, X5)) -> U51_AG(X1, X2, X3, X4, X5, splitcB_in_aaaa(X2, X3, X6, X7)) U51_AG(X1, X2, X3, X4, X5, splitcB_out_aaaa(X2, X3, X6, X7)) -> U52_AG(X1, X2, X3, X4, X5, X6, mscH_in_aaa(X1, X7, .(X4, X8))) U52_AG(X1, X2, X3, X4, X5, X6, mscH_out_aaa(X1, X7, .(X4, X8))) -> U53_AG(X1, X2, X3, X4, X5, mscE_in_aa(X6, .(X9, X10))) U53_AG(X1, X2, X3, X4, X5, mscE_out_aa(X6, .(X9, X10))) -> U54_AG(X1, X2, X3, X4, X5, lessF_in_ga(X4, X9)) U53_AG(X1, X2, X3, X4, X5, mscE_out_aa(X6, .(X9, X10))) -> LESSF_IN_GA(X4, X9) U51_AG(X1, X2, X3, X4, X5, splitcB_out_aaaa(X2, X3, X6, X7)) -> U55_AG(X1, X2, X3, X4, X5, X6, mscH_in_aaa(X1, X7, .(X8, X9))) U55_AG(X1, X2, X3, X4, X5, X6, mscH_out_aaa(X1, X7, .(X8, X9))) -> U56_AG(X1, X2, X3, X4, X5, X8, X9, mscE_in_aa(X6, .(X4, X10))) U56_AG(X1, X2, X3, X4, X5, X8, X9, mscE_out_aa(X6, .(X4, X10))) -> U57_AG(X1, X2, X3, X4, X5, pK_in_gaaag(X4, X8, X9, X10, X5)) U56_AG(X1, X2, X3, X4, X5, X8, X9, mscE_out_aa(X6, .(X4, X10))) -> PK_IN_GAAAG(X4, X8, X9, X10, X5) PK_IN_GAAAG(X1, X2, X3, X4, X5) -> U25_GAAAG(X1, X2, X3, X4, X5, lessG_in_ga(X1, X2)) PK_IN_GAAAG(X1, X2, X3, X4, X5) -> LESSG_IN_GA(X1, X2) PK_IN_GAAAG(X1, X2, X3, .(X4, X5), .(X2, X6)) -> U26_GAAAG(X1, X2, X3, X4, X5, X6, lesscG_in_gg(X1, X2)) U26_GAAAG(X1, X2, X3, X4, X5, X6, lesscG_out_gg(X1, X2)) -> U27_GAAAG(X1, X2, X3, X4, X5, X6, pJ_in_gaaag(X2, X4, X3, X5, X6)) U26_GAAAG(X1, X2, X3, X4, X5, X6, lesscG_out_gg(X1, X2)) -> PJ_IN_GAAAG(X2, X4, X3, X5, X6) PJ_IN_GAAAG(X1, X2, X3, X4, X5) -> U20_GAAAG(X1, X2, X3, X4, X5, lessF_in_ga(X1, X2)) PJ_IN_GAAAG(X1, X2, X3, X4, X5) -> LESSF_IN_GA(X1, X2) PJ_IN_GAAAG(X1, X2, .(X3, X4), X5, .(X3, X6)) -> U21_GAAAG(X1, X2, X3, X4, X5, X6, lesscF_in_ga(X1, X2)) U21_GAAAG(X1, X2, X3, X4, X5, X6, lesscF_out_ga(X1, X2)) -> U22_GAAAG(X1, X2, X3, X4, X5, X6, pJ_in_gaaag(X3, X2, X4, X5, X6)) U21_GAAAG(X1, X2, X3, X4, X5, X6, lesscF_out_ga(X1, X2)) -> PJ_IN_GAAAG(X3, X2, X4, X5, X6) PJ_IN_GAAAG(X1, X2, .(X3, X4), X5, .(X2, X6)) -> U23_GAAAG(X1, X2, X3, X4, X5, X6, lesscF_in_gg(X1, X2)) U23_GAAAG(X1, X2, X3, X4, X5, X6, lesscF_out_gg(X1, X2)) -> U24_GAAAG(X1, X2, X3, X4, X5, X6, pK_in_gaaag(X2, X3, X4, X5, X6)) U23_GAAAG(X1, X2, X3, X4, X5, X6, lesscF_out_gg(X1, X2)) -> PK_IN_GAAAG(X2, X3, X4, X5, X6) PK_IN_GAAAG(X1, X2, X3, .(X4, X5), .(X4, X6)) -> U28_GAAAG(X1, X2, X3, X4, X5, X6, lesscG_in_ga(X1, X2)) U28_GAAAG(X1, X2, X3, X4, X5, X6, lesscG_out_ga(X1, X2)) -> U29_GAAAG(X1, X2, X3, X4, X5, X6, pK_in_gaaag(X4, X2, X3, X5, X6)) U28_GAAAG(X1, X2, X3, X4, X5, X6, lesscG_out_ga(X1, X2)) -> PK_IN_GAAAG(X4, X2, X3, X5, X6) MSL_IN_AG(.(X1, .(X2, X3)), .(X4, .(X5, X6))) -> U58_AG(X1, X2, X3, X4, X5, X6, splitcB_in_aaaa(X2, X3, X7, X8)) U58_AG(X1, X2, X3, X4, X5, X6, splitcB_out_aaaa(X2, X3, X7, X8)) -> U59_AG(X1, X2, X3, X4, X5, X6, X7, mscH_in_aaa(X1, X8, .(X4, .(X5, X9)))) U59_AG(X1, X2, X3, X4, X5, X6, X7, mscH_out_aaa(X1, X8, .(X4, .(X5, X9)))) -> U60_AG(X1, X2, X3, X4, X5, X6, X9, mscE_in_aa(X7, .(X10, X11))) U60_AG(X1, X2, X3, X4, X5, X6, X9, mscE_out_aa(X7, .(X10, X11))) -> U61_AG(X1, X2, X3, X4, X5, X6, X9, X10, X11, lesscF_in_ga(X4, X10)) U61_AG(X1, X2, X3, X4, X5, X6, X9, X10, X11, lesscF_out_ga(X4, X10)) -> U62_AG(X1, X2, X3, X4, X5, X6, pJ_in_gaaag(X5, X10, X9, X11, X6)) U61_AG(X1, X2, X3, X4, X5, X6, X9, X10, X11, lesscF_out_ga(X4, X10)) -> PJ_IN_GAAAG(X5, X10, X9, X11, X6) U58_AG(X1, X2, X3, X4, X5, X6, splitcB_out_aaaa(X2, X3, X7, X8)) -> U63_AG(X1, X2, X3, X4, X5, X6, X7, mscH_in_aaa(X1, X8, .(X4, .(X9, X10)))) U63_AG(X1, X2, X3, X4, X5, X6, X7, mscH_out_aaa(X1, X8, .(X4, .(X9, X10)))) -> U64_AG(X1, X2, X3, X4, X5, X6, X9, X10, mscE_in_aa(X7, .(X5, X11))) U64_AG(X1, X2, X3, X4, X5, X6, X9, X10, mscE_out_aa(X7, .(X5, X11))) -> U65_AG(X1, X2, X3, X4, X5, X6, X9, X10, X11, lesscF_in_gg(X4, X5)) U65_AG(X1, X2, X3, X4, X5, X6, X9, X10, X11, lesscF_out_gg(X4, X5)) -> U66_AG(X1, X2, X3, X4, X5, X6, pK_in_gaaag(X5, X9, X10, X11, X6)) U65_AG(X1, X2, X3, X4, X5, X6, X9, X10, X11, lesscF_out_gg(X4, X5)) -> PK_IN_GAAAG(X5, X9, X10, X11, X6) The TRS R consists of the following rules: splitcB_in_aaaa(X1, X2, .(X1, X3), X4) -> U69_aaaa(X1, X2, X3, X4, splitcA_in_aaa(X2, X4, X3)) splitcA_in_aaa([], [], []) -> splitcA_out_aaa([], [], []) splitcA_in_aaa(.(X1, X2), .(X1, X3), X4) -> U68_aaa(X1, X2, X3, X4, splitcA_in_aaa(X2, X4, X3)) U68_aaa(X1, X2, X3, X4, splitcA_out_aaa(X2, X4, X3)) -> splitcA_out_aaa(.(X1, X2), .(X1, X3), X4) U69_aaaa(X1, X2, X3, X4, splitcA_out_aaa(X2, X4, X3)) -> splitcB_out_aaaa(X1, X2, .(X1, X3), X4) mscE_in_aa([], []) -> mscE_out_aa([], []) mscE_in_aa(.(X1, []), .(X1, [])) -> mscE_out_aa(.(X1, []), .(X1, [])) mscE_in_aa(.(X1, .(X2, X3)), X4) -> U74_aa(X1, X2, X3, X4, qcC_in_aaaaaaaa(X1, X2, X3, X5, X6, X7, X8, X4)) qcC_in_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8) -> U70_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, splitcB_in_aaaa(X1, .(X2, X3), X4, X5)) U70_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, splitcB_out_aaaa(X1, .(X2, X3), X4, X5)) -> U71_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mscE_in_aa(X4, X6)) U71_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mscE_out_aa(X4, X6)) -> U72_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mscE_in_aa(X5, X7)) U72_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mscE_out_aa(X5, X7)) -> U73_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mergecD_in_aaa(X6, X7, X8)) mergecD_in_aaa([], X1, X1) -> mergecD_out_aaa([], X1, X1) mergecD_in_aaa(X1, [], X1) -> mergecD_out_aaa(X1, [], X1) mergecD_in_aaa(.(X1, X2), .(X3, X4), .(X1, X5)) -> U75_aaa(X1, X2, X3, X4, X5, lesscF_in_aa(X1, X3)) lesscF_in_aa(0, X1) -> lesscF_out_aa(0, X1) lesscF_in_aa(s(X1), X2) -> U81_aa(X1, X2, lesscG_in_aa(X1, X2)) lesscG_in_aa(0, s(X1)) -> lesscG_out_aa(0, s(X1)) lesscG_in_aa(s(X1), s(X2)) -> U79_aa(X1, X2, lesscG_in_aa(X1, X2)) U79_aa(X1, X2, lesscG_out_aa(X1, X2)) -> lesscG_out_aa(s(X1), s(X2)) U81_aa(X1, X2, lesscG_out_aa(X1, X2)) -> lesscF_out_aa(s(X1), X2) U75_aaa(X1, X2, X3, X4, X5, lesscF_out_aa(X1, X3)) -> U76_aaa(X1, X2, X3, X4, X5, mergecD_in_aaa(X2, .(X3, X4), X5)) mergecD_in_aaa(.(X1, X2), .(X3, X4), .(X3, X5)) -> U77_aaa(X1, X2, X3, X4, X5, lesscG_in_aa(X3, X1)) U77_aaa(X1, X2, X3, X4, X5, lesscG_out_aa(X3, X1)) -> U78_aaa(X1, X2, X3, X4, X5, mergecD_in_aaa(.(X1, X2), X4, X5)) U78_aaa(X1, X2, X3, X4, X5, mergecD_out_aaa(.(X1, X2), X4, X5)) -> mergecD_out_aaa(.(X1, X2), .(X3, X4), .(X3, X5)) U76_aaa(X1, X2, X3, X4, X5, mergecD_out_aaa(X2, .(X3, X4), X5)) -> mergecD_out_aaa(.(X1, X2), .(X3, X4), .(X1, X5)) U73_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mergecD_out_aaa(X6, X7, X8)) -> qcC_out_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8) U74_aa(X1, X2, X3, X4, qcC_out_aaaaaaaa(X1, X2, X3, X5, X6, X7, X8, X4)) -> mscE_out_aa(.(X1, .(X2, X3)), X4) mscH_in_aaa(X1, [], .(X1, [])) -> mscH_out_aaa(X1, [], .(X1, [])) mscH_in_aaa(X1, .(X2, X3), X4) -> U80_aaa(X1, X2, X3, X4, qcC_in_aaaaaaaa(X1, X2, X3, X5, X6, X7, X8, X4)) U80_aaa(X1, X2, X3, X4, qcC_out_aaaaaaaa(X1, X2, X3, X5, X6, X7, X8, X4)) -> mscH_out_aaa(X1, .(X2, X3), X4) lesscG_in_gg(0, s(X1)) -> lesscG_out_gg(0, s(X1)) lesscG_in_gg(s(X1), s(X2)) -> U79_gg(X1, X2, lesscG_in_gg(X1, X2)) U79_gg(X1, X2, lesscG_out_gg(X1, X2)) -> lesscG_out_gg(s(X1), s(X2)) lesscF_in_ga(0, X1) -> lesscF_out_ga(0, X1) lesscF_in_ga(s(X1), X2) -> U81_ga(X1, X2, lesscG_in_ga(X1, X2)) lesscG_in_ga(0, s(X1)) -> lesscG_out_ga(0, s(X1)) lesscG_in_ga(s(X1), s(X2)) -> U79_ga(X1, X2, lesscG_in_ga(X1, X2)) U79_ga(X1, X2, lesscG_out_ga(X1, X2)) -> lesscG_out_ga(s(X1), s(X2)) U81_ga(X1, X2, lesscG_out_ga(X1, X2)) -> lesscF_out_ga(s(X1), X2) lesscF_in_gg(0, X1) -> lesscF_out_gg(0, X1) lesscF_in_gg(s(X1), X2) -> U81_gg(X1, X2, lesscG_in_gg(X1, X2)) U81_gg(X1, X2, lesscG_out_gg(X1, X2)) -> lesscF_out_gg(s(X1), X2) The argument filtering Pi contains the following mapping: [] = [] splitB_in_aaaa(x1, x2, x3, x4) = splitB_in_aaaa splitA_in_aaa(x1, x2, x3) = splitA_in_aaa splitcB_in_aaaa(x1, x2, x3, x4) = splitcB_in_aaaa U69_aaaa(x1, x2, x3, x4, x5) = U69_aaaa(x5) splitcA_in_aaa(x1, x2, x3) = splitcA_in_aaa splitcA_out_aaa(x1, x2, x3) = splitcA_out_aaa U68_aaa(x1, x2, x3, x4, x5) = U68_aaa(x5) splitcB_out_aaaa(x1, x2, x3, x4) = splitcB_out_aaaa msH_in_aaa(x1, x2, x3) = msH_in_aaa pC_in_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = pC_in_aaaaaaaa msE_in_aa(x1, x2) = msE_in_aa mscE_in_aa(x1, x2) = mscE_in_aa mscE_out_aa(x1, x2) = mscE_out_aa U74_aa(x1, x2, x3, x4, x5) = U74_aa(x5) qcC_in_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = qcC_in_aaaaaaaa U70_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U70_aaaaaaaa(x9) U71_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U71_aaaaaaaa(x9) U72_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U72_aaaaaaaa(x9) U73_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U73_aaaaaaaa(x9) mergecD_in_aaa(x1, x2, x3) = mergecD_in_aaa mergecD_out_aaa(x1, x2, x3) = mergecD_out_aaa U75_aaa(x1, x2, x3, x4, x5, x6) = U75_aaa(x6) lesscF_in_aa(x1, x2) = lesscF_in_aa lesscF_out_aa(x1, x2) = lesscF_out_aa(x1) U81_aa(x1, x2, x3) = U81_aa(x3) lesscG_in_aa(x1, x2) = lesscG_in_aa lesscG_out_aa(x1, x2) = lesscG_out_aa(x1) U79_aa(x1, x2, x3) = U79_aa(x3) U76_aaa(x1, x2, x3, x4, x5, x6) = U76_aaa(x6) U77_aaa(x1, x2, x3, x4, x5, x6) = U77_aaa(x6) U78_aaa(x1, x2, x3, x4, x5, x6) = U78_aaa(x6) qcC_out_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = qcC_out_aaaaaaaa mergeD_in_aaa(x1, x2, x3) = mergeD_in_aaa lessF_in_aa(x1, x2) = lessF_in_aa lessG_in_aa(x1, x2) = lessG_in_aa mscH_in_aaa(x1, x2, x3) = mscH_in_aaa mscH_out_aaa(x1, x2, x3) = mscH_out_aaa U80_aaa(x1, x2, x3, x4, x5) = U80_aaa(x5) .(x1, x2) = .(x1, x2) lessF_in_ga(x1, x2) = lessF_in_ga(x1) s(x1) = s(x1) lessG_in_ga(x1, x2) = lessG_in_ga(x1) pK_in_gaaag(x1, x2, x3, x4, x5) = pK_in_gaaag(x1, x5) lesscG_in_gg(x1, x2) = lesscG_in_gg(x1, x2) 0 = 0 lesscG_out_gg(x1, x2) = lesscG_out_gg(x1, x2) U79_gg(x1, x2, x3) = U79_gg(x1, x2, x3) pJ_in_gaaag(x1, x2, x3, x4, x5) = pJ_in_gaaag(x1, x5) lesscF_in_ga(x1, x2) = lesscF_in_ga(x1) lesscF_out_ga(x1, x2) = lesscF_out_ga(x1) U81_ga(x1, x2, x3) = U81_ga(x1, x3) lesscG_in_ga(x1, x2) = lesscG_in_ga(x1) lesscG_out_ga(x1, x2) = lesscG_out_ga(x1) U79_ga(x1, x2, x3) = U79_ga(x1, x3) lesscF_in_gg(x1, x2) = lesscF_in_gg(x1, x2) lesscF_out_gg(x1, x2) = lesscF_out_gg(x1, x2) U81_gg(x1, x2, x3) = U81_gg(x1, x2, x3) MSL_IN_AG(x1, x2) = MSL_IN_AG(x2) U30_AG(x1, x2, x3, x4) = U30_AG(x4) SPLITB_IN_AAAA(x1, x2, x3, x4) = SPLITB_IN_AAAA U2_AAAA(x1, x2, x3, x4, x5) = U2_AAAA(x5) SPLITA_IN_AAA(x1, x2, x3) = SPLITA_IN_AAA U1_AAA(x1, x2, x3, x4, x5) = U1_AAA(x5) U31_AG(x1, x2, x3, x4) = U31_AG(x4) U32_AG(x1, x2, x3, x4) = U32_AG(x4) MSH_IN_AAA(x1, x2, x3) = MSH_IN_AAA U18_AAA(x1, x2, x3, x4, x5) = U18_AAA(x5) PC_IN_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = PC_IN_AAAAAAAA U3_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U3_AAAAAAAA(x9) U4_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U4_AAAAAAAA(x9) U5_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_AAAAAAAA(x9) MSE_IN_AA(x1, x2) = MSE_IN_AA U10_AA(x1, x2, x3, x4, x5) = U10_AA(x5) U6_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_AAAAAAAA(x9) U7_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_AAAAAAAA(x9) U8_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U8_AAAAAAAA(x9) U9_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_AAAAAAAA(x9) MERGED_IN_AAA(x1, x2, x3) = MERGED_IN_AAA U11_AAA(x1, x2, x3, x4, x5, x6) = U11_AAA(x6) LESSF_IN_AA(x1, x2) = LESSF_IN_AA U19_AA(x1, x2, x3) = U19_AA(x3) LESSG_IN_AA(x1, x2) = LESSG_IN_AA U17_AA(x1, x2, x3) = U17_AA(x3) U12_AAA(x1, x2, x3, x4, x5, x6) = U12_AAA(x6) U13_AAA(x1, x2, x3, x4, x5, x6) = U13_AAA(x6) U14_AAA(x1, x2, x3, x4, x5, x6) = U14_AAA(x6) U15_AAA(x1, x2, x3, x4, x5, x6) = U15_AAA(x6) U16_AAA(x1, x2, x3, x4, x5, x6) = U16_AAA(x6) U33_AG(x1, x2, x3, x4, x5) = U33_AG(x5) U34_AG(x1, x2, x3, x4) = U34_AG(x4) U35_AG(x1, x2, x3, x4, x5) = U35_AG(x4, x5) U36_AG(x1, x2, x3, x4, x5) = U36_AG(x4, x5) U37_AG(x1, x2, x3, x4, x5) = U37_AG(x4, x5) U38_AG(x1, x2, x3, x4, x5, x6) = U38_AG(x4, x6) U39_AG(x1, x2, x3, x4, x5) = U39_AG(x4, x5) U40_AG(x1, x2, x3, x4, x5, x6) = U40_AG(x4, x6) U41_AG(x1, x2, x3, x4, x5) = U41_AG(x4, x5) U42_AG(x1, x2, x3, x4, x5) = U42_AG(x4, x5) LESSF_IN_GA(x1, x2) = LESSF_IN_GA(x1) U19_GA(x1, x2, x3) = U19_GA(x1, x3) LESSG_IN_GA(x1, x2) = LESSG_IN_GA(x1) U17_GA(x1, x2, x3) = U17_GA(x1, x3) U43_AG(x1, x2, x3, x4, x5, x6) = U43_AG(x4, x6) U44_AG(x1, x2, x3, x4, x5, x6) = U44_AG(x4, x6) U45_AG(x1, x2, x3, x4, x5) = U45_AG(x4, x5) U46_AG(x1, x2, x3, x4, x5) = U46_AG(x4, x5) U47_AG(x1, x2, x3, x4, x5) = U47_AG(x4, x5) U48_AG(x1, x2, x3, x4, x5) = U48_AG(x4, x5) U49_AG(x1, x2, x3, x4, x5, x6) = U49_AG(x4, x6) U50_AG(x1, x2, x3, x4, x5) = U50_AG(x4, x5) U51_AG(x1, x2, x3, x4, x5, x6) = U51_AG(x4, x5, x6) U52_AG(x1, x2, x3, x4, x5, x6, x7) = U52_AG(x4, x5, x7) U53_AG(x1, x2, x3, x4, x5, x6) = U53_AG(x4, x5, x6) U54_AG(x1, x2, x3, x4, x5, x6) = U54_AG(x4, x5, x6) U55_AG(x1, x2, x3, x4, x5, x6, x7) = U55_AG(x4, x5, x7) U56_AG(x1, x2, x3, x4, x5, x6, x7, x8) = U56_AG(x4, x5, x8) U57_AG(x1, x2, x3, x4, x5, x6) = U57_AG(x4, x5, x6) PK_IN_GAAAG(x1, x2, x3, x4, x5) = PK_IN_GAAAG(x1, x5) U25_GAAAG(x1, x2, x3, x4, x5, x6) = U25_GAAAG(x1, x5, x6) U26_GAAAG(x1, x2, x3, x4, x5, x6, x7) = U26_GAAAG(x1, x2, x6, x7) U27_GAAAG(x1, x2, x3, x4, x5, x6, x7) = U27_GAAAG(x1, x2, x6, x7) PJ_IN_GAAAG(x1, x2, x3, x4, x5) = PJ_IN_GAAAG(x1, x5) U20_GAAAG(x1, x2, x3, x4, x5, x6) = U20_GAAAG(x1, x5, x6) U21_GAAAG(x1, x2, x3, x4, x5, x6, x7) = U21_GAAAG(x1, x3, x6, x7) U22_GAAAG(x1, x2, x3, x4, x5, x6, x7) = U22_GAAAG(x1, x3, x6, x7) U23_GAAAG(x1, x2, x3, x4, x5, x6, x7) = U23_GAAAG(x1, x2, x6, x7) U24_GAAAG(x1, x2, x3, x4, x5, x6, x7) = U24_GAAAG(x1, x2, x6, x7) U28_GAAAG(x1, x2, x3, x4, x5, x6, x7) = U28_GAAAG(x1, x4, x6, x7) U29_GAAAG(x1, x2, x3, x4, x5, x6, x7) = U29_GAAAG(x1, x4, x6, x7) U58_AG(x1, x2, x3, x4, x5, x6, x7) = U58_AG(x4, x5, x6, x7) U59_AG(x1, x2, x3, x4, x5, x6, x7, x8) = U59_AG(x4, x5, x6, x8) U60_AG(x1, x2, x3, x4, x5, x6, x7, x8) = U60_AG(x4, x5, x6, x8) U61_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U61_AG(x4, x5, x6, x10) U62_AG(x1, x2, x3, x4, x5, x6, x7) = U62_AG(x4, x5, x6, x7) U63_AG(x1, x2, x3, x4, x5, x6, x7, x8) = U63_AG(x4, x5, x6, x8) U64_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U64_AG(x4, x5, x6, x9) U65_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U65_AG(x4, x5, x6, x10) U66_AG(x1, x2, x3, x4, x5, x6, x7) = U66_AG(x4, x5, x6, x7) We have to consider all (P,R,Pi)-chains Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES ---------------------------------------- (166) Obligation: Pi DP problem: The TRS P consists of the following rules: MSL_IN_AG(.(X1, .(X2, X3)), []) -> U30_AG(X1, X2, X3, splitB_in_aaaa(X2, X3, X4, X5)) MSL_IN_AG(.(X1, .(X2, X3)), []) -> SPLITB_IN_AAAA(X2, X3, X4, X5) SPLITB_IN_AAAA(X1, X2, .(X1, X3), X4) -> U2_AAAA(X1, X2, X3, X4, splitA_in_aaa(X2, X4, X3)) SPLITB_IN_AAAA(X1, X2, .(X1, X3), X4) -> SPLITA_IN_AAA(X2, X4, X3) SPLITA_IN_AAA(.(X1, X2), .(X1, X3), X4) -> U1_AAA(X1, X2, X3, X4, splitA_in_aaa(X2, X4, X3)) SPLITA_IN_AAA(.(X1, X2), .(X1, X3), X4) -> SPLITA_IN_AAA(X2, X4, X3) MSL_IN_AG(.(X1, .(X2, X3)), []) -> U31_AG(X1, X2, X3, splitcB_in_aaaa(X2, X3, X4, X5)) U31_AG(X1, X2, X3, splitcB_out_aaaa(X2, X3, X4, X5)) -> U32_AG(X1, X2, X3, msH_in_aaa(X1, X5, X6)) U31_AG(X1, X2, X3, splitcB_out_aaaa(X2, X3, X4, X5)) -> MSH_IN_AAA(X1, X5, X6) MSH_IN_AAA(X1, .(X2, X3), X4) -> U18_AAA(X1, X2, X3, X4, pC_in_aaaaaaaa(X1, X2, X3, X5, X6, X7, X8, X4)) MSH_IN_AAA(X1, .(X2, X3), X4) -> PC_IN_AAAAAAAA(X1, X2, X3, X5, X6, X7, X8, X4) PC_IN_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8) -> U3_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, splitB_in_aaaa(X1, .(X2, X3), X4, X5)) PC_IN_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8) -> SPLITB_IN_AAAA(X1, .(X2, X3), X4, X5) PC_IN_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8) -> U4_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, splitcB_in_aaaa(X1, .(X2, X3), X4, X5)) U4_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, splitcB_out_aaaa(X1, .(X2, X3), X4, X5)) -> U5_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, msE_in_aa(X4, X6)) U4_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, splitcB_out_aaaa(X1, .(X2, X3), X4, X5)) -> MSE_IN_AA(X4, X6) MSE_IN_AA(.(X1, .(X2, X3)), X4) -> U10_AA(X1, X2, X3, X4, pC_in_aaaaaaaa(X1, X2, X3, X5, X6, X7, X8, X4)) MSE_IN_AA(.(X1, .(X2, X3)), X4) -> PC_IN_AAAAAAAA(X1, X2, X3, X5, X6, X7, X8, X4) U4_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, splitcB_out_aaaa(X1, .(X2, X3), X4, X5)) -> U6_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, mscE_in_aa(X4, X6)) U6_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, mscE_out_aa(X4, X6)) -> U7_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, msE_in_aa(X5, X7)) U6_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, mscE_out_aa(X4, X6)) -> MSE_IN_AA(X5, X7) U6_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, mscE_out_aa(X4, X6)) -> U8_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, mscE_in_aa(X5, X7)) U8_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, mscE_out_aa(X5, X7)) -> U9_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, mergeD_in_aaa(X6, X7, X8)) U8_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, mscE_out_aa(X5, X7)) -> MERGED_IN_AAA(X6, X7, X8) MERGED_IN_AAA(.(X1, X2), .(X3, X4), .(X1, X5)) -> U11_AAA(X1, X2, X3, X4, X5, lessF_in_aa(X1, X3)) MERGED_IN_AAA(.(X1, X2), .(X3, X4), .(X1, X5)) -> LESSF_IN_AA(X1, X3) LESSF_IN_AA(s(X1), X2) -> U19_AA(X1, X2, lessG_in_aa(X1, X2)) LESSF_IN_AA(s(X1), X2) -> LESSG_IN_AA(X1, X2) LESSG_IN_AA(s(X1), s(X2)) -> U17_AA(X1, X2, lessG_in_aa(X1, X2)) LESSG_IN_AA(s(X1), s(X2)) -> LESSG_IN_AA(X1, X2) MERGED_IN_AAA(.(X1, X2), .(X3, X4), .(X1, X5)) -> U12_AAA(X1, X2, X3, X4, X5, lesscF_in_aa(X1, X3)) U12_AAA(X1, X2, X3, X4, X5, lesscF_out_aa(X1, X3)) -> U13_AAA(X1, X2, X3, X4, X5, mergeD_in_aaa(X2, .(X3, X4), X5)) U12_AAA(X1, X2, X3, X4, X5, lesscF_out_aa(X1, X3)) -> MERGED_IN_AAA(X2, .(X3, X4), X5) MERGED_IN_AAA(.(X1, X2), .(X3, X4), .(X3, X5)) -> U14_AAA(X1, X2, X3, X4, X5, lessG_in_aa(X3, X1)) MERGED_IN_AAA(.(X1, X2), .(X3, X4), .(X3, X5)) -> LESSG_IN_AA(X3, X1) MERGED_IN_AAA(.(X1, X2), .(X3, X4), .(X3, X5)) -> U15_AAA(X1, X2, X3, X4, X5, lesscG_in_aa(X3, X1)) U15_AAA(X1, X2, X3, X4, X5, lesscG_out_aa(X3, X1)) -> U16_AAA(X1, X2, X3, X4, X5, mergeD_in_aaa(.(X1, X2), X4, X5)) U15_AAA(X1, X2, X3, X4, X5, lesscG_out_aa(X3, X1)) -> MERGED_IN_AAA(.(X1, X2), X4, X5) U31_AG(X1, X2, X3, splitcB_out_aaaa(X2, X3, X4, X5)) -> U33_AG(X1, X2, X3, X4, mscH_in_aaa(X1, X5, X6)) U33_AG(X1, X2, X3, X4, mscH_out_aaa(X1, X5, X6)) -> U34_AG(X1, X2, X3, msE_in_aa(X4, X7)) U33_AG(X1, X2, X3, X4, mscH_out_aaa(X1, X5, X6)) -> MSE_IN_AA(X4, X7) MSL_IN_AG(.(X1, .(X2, X3)), .(X4, [])) -> U35_AG(X1, X2, X3, X4, splitB_in_aaaa(X2, X3, X5, X6)) MSL_IN_AG(.(X1, .(X2, X3)), .(X4, [])) -> SPLITB_IN_AAAA(X2, X3, X5, X6) MSL_IN_AG(.(X1, .(X2, X3)), .(X4, [])) -> U36_AG(X1, X2, X3, X4, splitcB_in_aaaa(X2, X3, X5, X6)) U36_AG(X1, X2, X3, X4, splitcB_out_aaaa(X2, X3, X5, X6)) -> U37_AG(X1, X2, X3, X4, msH_in_aaa(X1, X6, X7)) U36_AG(X1, X2, X3, X4, splitcB_out_aaaa(X2, X3, X5, X6)) -> MSH_IN_AAA(X1, X6, X7) U36_AG(X1, X2, X3, X4, splitcB_out_aaaa(X2, X3, X5, X6)) -> U38_AG(X1, X2, X3, X4, X5, mscH_in_aaa(X1, X6, X7)) U38_AG(X1, X2, X3, X4, X5, mscH_out_aaa(X1, X6, X7)) -> U39_AG(X1, X2, X3, X4, msE_in_aa(X5, X8)) U38_AG(X1, X2, X3, X4, X5, mscH_out_aaa(X1, X6, X7)) -> MSE_IN_AA(X5, X8) U36_AG(X1, X2, X3, X4, splitcB_out_aaaa(X2, X3, X5, X6)) -> U40_AG(X1, X2, X3, X4, X5, mscH_in_aaa(X1, X6, .(X4, X7))) U40_AG(X1, X2, X3, X4, X5, mscH_out_aaa(X1, X6, .(X4, X7))) -> U41_AG(X1, X2, X3, X4, mscE_in_aa(X5, .(X8, X9))) U41_AG(X1, X2, X3, X4, mscE_out_aa(X5, .(X8, X9))) -> U42_AG(X1, X2, X3, X4, lessF_in_ga(X4, X8)) U41_AG(X1, X2, X3, X4, mscE_out_aa(X5, .(X8, X9))) -> LESSF_IN_GA(X4, X8) LESSF_IN_GA(s(X1), X2) -> U19_GA(X1, X2, lessG_in_ga(X1, X2)) LESSF_IN_GA(s(X1), X2) -> LESSG_IN_GA(X1, X2) LESSG_IN_GA(s(X1), s(X2)) -> U17_GA(X1, X2, lessG_in_ga(X1, X2)) LESSG_IN_GA(s(X1), s(X2)) -> LESSG_IN_GA(X1, X2) U36_AG(X1, X2, X3, X4, splitcB_out_aaaa(X2, X3, X5, X6)) -> U43_AG(X1, X2, X3, X4, X5, mscH_in_aaa(X1, X6, .(X7, X8))) U43_AG(X1, X2, X3, X4, X5, mscH_out_aaa(X1, X6, .(X7, X8))) -> U44_AG(X1, X2, X3, X4, X7, mscE_in_aa(X5, .(X4, X9))) U44_AG(X1, X2, X3, X4, X7, mscE_out_aa(X5, .(X4, X9))) -> U45_AG(X1, X2, X3, X4, lessG_in_ga(X4, X7)) U44_AG(X1, X2, X3, X4, X7, mscE_out_aa(X5, .(X4, X9))) -> LESSG_IN_GA(X4, X7) MSL_IN_AG(.(X1, .(X2, X3)), X4) -> U46_AG(X1, X2, X3, X4, splitB_in_aaaa(X2, X3, X5, X6)) MSL_IN_AG(.(X1, .(X2, X3)), X4) -> SPLITB_IN_AAAA(X2, X3, X5, X6) MSL_IN_AG(.(X1, .(X2, X3)), X4) -> U47_AG(X1, X2, X3, X4, splitcB_in_aaaa(X2, X3, X5, X6)) U47_AG(X1, X2, X3, X4, splitcB_out_aaaa(X2, X3, X5, X6)) -> U48_AG(X1, X2, X3, X4, msH_in_aaa(X1, X6, X7)) U47_AG(X1, X2, X3, X4, splitcB_out_aaaa(X2, X3, X5, X6)) -> MSH_IN_AAA(X1, X6, X7) U47_AG(X1, X2, X3, X4, splitcB_out_aaaa(X2, X3, X5, X6)) -> U49_AG(X1, X2, X3, X4, X5, mscH_in_aaa(X1, X6, X7)) U49_AG(X1, X2, X3, X4, X5, mscH_out_aaa(X1, X6, X7)) -> U50_AG(X1, X2, X3, X4, msE_in_aa(X5, X8)) U49_AG(X1, X2, X3, X4, X5, mscH_out_aaa(X1, X6, X7)) -> MSE_IN_AA(X5, X8) MSL_IN_AG(.(X1, .(X2, X3)), .(X4, X5)) -> U51_AG(X1, X2, X3, X4, X5, splitcB_in_aaaa(X2, X3, X6, X7)) U51_AG(X1, X2, X3, X4, X5, splitcB_out_aaaa(X2, X3, X6, X7)) -> U52_AG(X1, X2, X3, X4, X5, X6, mscH_in_aaa(X1, X7, .(X4, X8))) U52_AG(X1, X2, X3, X4, X5, X6, mscH_out_aaa(X1, X7, .(X4, X8))) -> U53_AG(X1, X2, X3, X4, X5, mscE_in_aa(X6, .(X9, X10))) U53_AG(X1, X2, X3, X4, X5, mscE_out_aa(X6, .(X9, X10))) -> U54_AG(X1, X2, X3, X4, X5, lessF_in_ga(X4, X9)) U53_AG(X1, X2, X3, X4, X5, mscE_out_aa(X6, .(X9, X10))) -> LESSF_IN_GA(X4, X9) U51_AG(X1, X2, X3, X4, X5, splitcB_out_aaaa(X2, X3, X6, X7)) -> U55_AG(X1, X2, X3, X4, X5, X6, mscH_in_aaa(X1, X7, .(X8, X9))) U55_AG(X1, X2, X3, X4, X5, X6, mscH_out_aaa(X1, X7, .(X8, X9))) -> U56_AG(X1, X2, X3, X4, X5, X8, X9, mscE_in_aa(X6, .(X4, X10))) U56_AG(X1, X2, X3, X4, X5, X8, X9, mscE_out_aa(X6, .(X4, X10))) -> U57_AG(X1, X2, X3, X4, X5, pK_in_gaaag(X4, X8, X9, X10, X5)) U56_AG(X1, X2, X3, X4, X5, X8, X9, mscE_out_aa(X6, .(X4, X10))) -> PK_IN_GAAAG(X4, X8, X9, X10, X5) PK_IN_GAAAG(X1, X2, X3, X4, X5) -> U25_GAAAG(X1, X2, X3, X4, X5, lessG_in_ga(X1, X2)) PK_IN_GAAAG(X1, X2, X3, X4, X5) -> LESSG_IN_GA(X1, X2) PK_IN_GAAAG(X1, X2, X3, .(X4, X5), .(X2, X6)) -> U26_GAAAG(X1, X2, X3, X4, X5, X6, lesscG_in_gg(X1, X2)) U26_GAAAG(X1, X2, X3, X4, X5, X6, lesscG_out_gg(X1, X2)) -> U27_GAAAG(X1, X2, X3, X4, X5, X6, pJ_in_gaaag(X2, X4, X3, X5, X6)) U26_GAAAG(X1, X2, X3, X4, X5, X6, lesscG_out_gg(X1, X2)) -> PJ_IN_GAAAG(X2, X4, X3, X5, X6) PJ_IN_GAAAG(X1, X2, X3, X4, X5) -> U20_GAAAG(X1, X2, X3, X4, X5, lessF_in_ga(X1, X2)) PJ_IN_GAAAG(X1, X2, X3, X4, X5) -> LESSF_IN_GA(X1, X2) PJ_IN_GAAAG(X1, X2, .(X3, X4), X5, .(X3, X6)) -> U21_GAAAG(X1, X2, X3, X4, X5, X6, lesscF_in_ga(X1, X2)) U21_GAAAG(X1, X2, X3, X4, X5, X6, lesscF_out_ga(X1, X2)) -> U22_GAAAG(X1, X2, X3, X4, X5, X6, pJ_in_gaaag(X3, X2, X4, X5, X6)) U21_GAAAG(X1, X2, X3, X4, X5, X6, lesscF_out_ga(X1, X2)) -> PJ_IN_GAAAG(X3, X2, X4, X5, X6) PJ_IN_GAAAG(X1, X2, .(X3, X4), X5, .(X2, X6)) -> U23_GAAAG(X1, X2, X3, X4, X5, X6, lesscF_in_gg(X1, X2)) U23_GAAAG(X1, X2, X3, X4, X5, X6, lesscF_out_gg(X1, X2)) -> U24_GAAAG(X1, X2, X3, X4, X5, X6, pK_in_gaaag(X2, X3, X4, X5, X6)) U23_GAAAG(X1, X2, X3, X4, X5, X6, lesscF_out_gg(X1, X2)) -> PK_IN_GAAAG(X2, X3, X4, X5, X6) PK_IN_GAAAG(X1, X2, X3, .(X4, X5), .(X4, X6)) -> U28_GAAAG(X1, X2, X3, X4, X5, X6, lesscG_in_ga(X1, X2)) U28_GAAAG(X1, X2, X3, X4, X5, X6, lesscG_out_ga(X1, X2)) -> U29_GAAAG(X1, X2, X3, X4, X5, X6, pK_in_gaaag(X4, X2, X3, X5, X6)) U28_GAAAG(X1, X2, X3, X4, X5, X6, lesscG_out_ga(X1, X2)) -> PK_IN_GAAAG(X4, X2, X3, X5, X6) MSL_IN_AG(.(X1, .(X2, X3)), .(X4, .(X5, X6))) -> U58_AG(X1, X2, X3, X4, X5, X6, splitcB_in_aaaa(X2, X3, X7, X8)) U58_AG(X1, X2, X3, X4, X5, X6, splitcB_out_aaaa(X2, X3, X7, X8)) -> U59_AG(X1, X2, X3, X4, X5, X6, X7, mscH_in_aaa(X1, X8, .(X4, .(X5, X9)))) U59_AG(X1, X2, X3, X4, X5, X6, X7, mscH_out_aaa(X1, X8, .(X4, .(X5, X9)))) -> U60_AG(X1, X2, X3, X4, X5, X6, X9, mscE_in_aa(X7, .(X10, X11))) U60_AG(X1, X2, X3, X4, X5, X6, X9, mscE_out_aa(X7, .(X10, X11))) -> U61_AG(X1, X2, X3, X4, X5, X6, X9, X10, X11, lesscF_in_ga(X4, X10)) U61_AG(X1, X2, X3, X4, X5, X6, X9, X10, X11, lesscF_out_ga(X4, X10)) -> U62_AG(X1, X2, X3, X4, X5, X6, pJ_in_gaaag(X5, X10, X9, X11, X6)) U61_AG(X1, X2, X3, X4, X5, X6, X9, X10, X11, lesscF_out_ga(X4, X10)) -> PJ_IN_GAAAG(X5, X10, X9, X11, X6) U58_AG(X1, X2, X3, X4, X5, X6, splitcB_out_aaaa(X2, X3, X7, X8)) -> U63_AG(X1, X2, X3, X4, X5, X6, X7, mscH_in_aaa(X1, X8, .(X4, .(X9, X10)))) U63_AG(X1, X2, X3, X4, X5, X6, X7, mscH_out_aaa(X1, X8, .(X4, .(X9, X10)))) -> U64_AG(X1, X2, X3, X4, X5, X6, X9, X10, mscE_in_aa(X7, .(X5, X11))) U64_AG(X1, X2, X3, X4, X5, X6, X9, X10, mscE_out_aa(X7, .(X5, X11))) -> U65_AG(X1, X2, X3, X4, X5, X6, X9, X10, X11, lesscF_in_gg(X4, X5)) U65_AG(X1, X2, X3, X4, X5, X6, X9, X10, X11, lesscF_out_gg(X4, X5)) -> U66_AG(X1, X2, X3, X4, X5, X6, pK_in_gaaag(X5, X9, X10, X11, X6)) U65_AG(X1, X2, X3, X4, X5, X6, X9, X10, X11, lesscF_out_gg(X4, X5)) -> PK_IN_GAAAG(X5, X9, X10, X11, X6) The TRS R consists of the following rules: splitcB_in_aaaa(X1, X2, .(X1, X3), X4) -> U69_aaaa(X1, X2, X3, X4, splitcA_in_aaa(X2, X4, X3)) splitcA_in_aaa([], [], []) -> splitcA_out_aaa([], [], []) splitcA_in_aaa(.(X1, X2), .(X1, X3), X4) -> U68_aaa(X1, X2, X3, X4, splitcA_in_aaa(X2, X4, X3)) U68_aaa(X1, X2, X3, X4, splitcA_out_aaa(X2, X4, X3)) -> splitcA_out_aaa(.(X1, X2), .(X1, X3), X4) U69_aaaa(X1, X2, X3, X4, splitcA_out_aaa(X2, X4, X3)) -> splitcB_out_aaaa(X1, X2, .(X1, X3), X4) mscE_in_aa([], []) -> mscE_out_aa([], []) mscE_in_aa(.(X1, []), .(X1, [])) -> mscE_out_aa(.(X1, []), .(X1, [])) mscE_in_aa(.(X1, .(X2, X3)), X4) -> U74_aa(X1, X2, X3, X4, qcC_in_aaaaaaaa(X1, X2, X3, X5, X6, X7, X8, X4)) qcC_in_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8) -> U70_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, splitcB_in_aaaa(X1, .(X2, X3), X4, X5)) U70_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, splitcB_out_aaaa(X1, .(X2, X3), X4, X5)) -> U71_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mscE_in_aa(X4, X6)) U71_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mscE_out_aa(X4, X6)) -> U72_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mscE_in_aa(X5, X7)) U72_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mscE_out_aa(X5, X7)) -> U73_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mergecD_in_aaa(X6, X7, X8)) mergecD_in_aaa([], X1, X1) -> mergecD_out_aaa([], X1, X1) mergecD_in_aaa(X1, [], X1) -> mergecD_out_aaa(X1, [], X1) mergecD_in_aaa(.(X1, X2), .(X3, X4), .(X1, X5)) -> U75_aaa(X1, X2, X3, X4, X5, lesscF_in_aa(X1, X3)) lesscF_in_aa(0, X1) -> lesscF_out_aa(0, X1) lesscF_in_aa(s(X1), X2) -> U81_aa(X1, X2, lesscG_in_aa(X1, X2)) lesscG_in_aa(0, s(X1)) -> lesscG_out_aa(0, s(X1)) lesscG_in_aa(s(X1), s(X2)) -> U79_aa(X1, X2, lesscG_in_aa(X1, X2)) U79_aa(X1, X2, lesscG_out_aa(X1, X2)) -> lesscG_out_aa(s(X1), s(X2)) U81_aa(X1, X2, lesscG_out_aa(X1, X2)) -> lesscF_out_aa(s(X1), X2) U75_aaa(X1, X2, X3, X4, X5, lesscF_out_aa(X1, X3)) -> U76_aaa(X1, X2, X3, X4, X5, mergecD_in_aaa(X2, .(X3, X4), X5)) mergecD_in_aaa(.(X1, X2), .(X3, X4), .(X3, X5)) -> U77_aaa(X1, X2, X3, X4, X5, lesscG_in_aa(X3, X1)) U77_aaa(X1, X2, X3, X4, X5, lesscG_out_aa(X3, X1)) -> U78_aaa(X1, X2, X3, X4, X5, mergecD_in_aaa(.(X1, X2), X4, X5)) U78_aaa(X1, X2, X3, X4, X5, mergecD_out_aaa(.(X1, X2), X4, X5)) -> mergecD_out_aaa(.(X1, X2), .(X3, X4), .(X3, X5)) U76_aaa(X1, X2, X3, X4, X5, mergecD_out_aaa(X2, .(X3, X4), X5)) -> mergecD_out_aaa(.(X1, X2), .(X3, X4), .(X1, X5)) U73_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mergecD_out_aaa(X6, X7, X8)) -> qcC_out_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8) U74_aa(X1, X2, X3, X4, qcC_out_aaaaaaaa(X1, X2, X3, X5, X6, X7, X8, X4)) -> mscE_out_aa(.(X1, .(X2, X3)), X4) mscH_in_aaa(X1, [], .(X1, [])) -> mscH_out_aaa(X1, [], .(X1, [])) mscH_in_aaa(X1, .(X2, X3), X4) -> U80_aaa(X1, X2, X3, X4, qcC_in_aaaaaaaa(X1, X2, X3, X5, X6, X7, X8, X4)) U80_aaa(X1, X2, X3, X4, qcC_out_aaaaaaaa(X1, X2, X3, X5, X6, X7, X8, X4)) -> mscH_out_aaa(X1, .(X2, X3), X4) lesscG_in_gg(0, s(X1)) -> lesscG_out_gg(0, s(X1)) lesscG_in_gg(s(X1), s(X2)) -> U79_gg(X1, X2, lesscG_in_gg(X1, X2)) U79_gg(X1, X2, lesscG_out_gg(X1, X2)) -> lesscG_out_gg(s(X1), s(X2)) lesscF_in_ga(0, X1) -> lesscF_out_ga(0, X1) lesscF_in_ga(s(X1), X2) -> U81_ga(X1, X2, lesscG_in_ga(X1, X2)) lesscG_in_ga(0, s(X1)) -> lesscG_out_ga(0, s(X1)) lesscG_in_ga(s(X1), s(X2)) -> U79_ga(X1, X2, lesscG_in_ga(X1, X2)) U79_ga(X1, X2, lesscG_out_ga(X1, X2)) -> lesscG_out_ga(s(X1), s(X2)) U81_ga(X1, X2, lesscG_out_ga(X1, X2)) -> lesscF_out_ga(s(X1), X2) lesscF_in_gg(0, X1) -> lesscF_out_gg(0, X1) lesscF_in_gg(s(X1), X2) -> U81_gg(X1, X2, lesscG_in_gg(X1, X2)) U81_gg(X1, X2, lesscG_out_gg(X1, X2)) -> lesscF_out_gg(s(X1), X2) The argument filtering Pi contains the following mapping: [] = [] splitB_in_aaaa(x1, x2, x3, x4) = splitB_in_aaaa splitA_in_aaa(x1, x2, x3) = splitA_in_aaa splitcB_in_aaaa(x1, x2, x3, x4) = splitcB_in_aaaa U69_aaaa(x1, x2, x3, x4, x5) = U69_aaaa(x5) splitcA_in_aaa(x1, x2, x3) = splitcA_in_aaa splitcA_out_aaa(x1, x2, x3) = splitcA_out_aaa U68_aaa(x1, x2, x3, x4, x5) = U68_aaa(x5) splitcB_out_aaaa(x1, x2, x3, x4) = splitcB_out_aaaa msH_in_aaa(x1, x2, x3) = msH_in_aaa pC_in_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = pC_in_aaaaaaaa msE_in_aa(x1, x2) = msE_in_aa mscE_in_aa(x1, x2) = mscE_in_aa mscE_out_aa(x1, x2) = mscE_out_aa U74_aa(x1, x2, x3, x4, x5) = U74_aa(x5) qcC_in_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = qcC_in_aaaaaaaa U70_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U70_aaaaaaaa(x9) U71_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U71_aaaaaaaa(x9) U72_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U72_aaaaaaaa(x9) U73_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U73_aaaaaaaa(x9) mergecD_in_aaa(x1, x2, x3) = mergecD_in_aaa mergecD_out_aaa(x1, x2, x3) = mergecD_out_aaa U75_aaa(x1, x2, x3, x4, x5, x6) = U75_aaa(x6) lesscF_in_aa(x1, x2) = lesscF_in_aa lesscF_out_aa(x1, x2) = lesscF_out_aa(x1) U81_aa(x1, x2, x3) = U81_aa(x3) lesscG_in_aa(x1, x2) = lesscG_in_aa lesscG_out_aa(x1, x2) = lesscG_out_aa(x1) U79_aa(x1, x2, x3) = U79_aa(x3) U76_aaa(x1, x2, x3, x4, x5, x6) = U76_aaa(x6) U77_aaa(x1, x2, x3, x4, x5, x6) = U77_aaa(x6) U78_aaa(x1, x2, x3, x4, x5, x6) = U78_aaa(x6) qcC_out_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = qcC_out_aaaaaaaa mergeD_in_aaa(x1, x2, x3) = mergeD_in_aaa lessF_in_aa(x1, x2) = lessF_in_aa lessG_in_aa(x1, x2) = lessG_in_aa mscH_in_aaa(x1, x2, x3) = mscH_in_aaa mscH_out_aaa(x1, x2, x3) = mscH_out_aaa U80_aaa(x1, x2, x3, x4, x5) = U80_aaa(x5) .(x1, x2) = .(x1, x2) lessF_in_ga(x1, x2) = lessF_in_ga(x1) s(x1) = s(x1) lessG_in_ga(x1, x2) = lessG_in_ga(x1) pK_in_gaaag(x1, x2, x3, x4, x5) = pK_in_gaaag(x1, x5) lesscG_in_gg(x1, x2) = lesscG_in_gg(x1, x2) 0 = 0 lesscG_out_gg(x1, x2) = lesscG_out_gg(x1, x2) U79_gg(x1, x2, x3) = U79_gg(x1, x2, x3) pJ_in_gaaag(x1, x2, x3, x4, x5) = pJ_in_gaaag(x1, x5) lesscF_in_ga(x1, x2) = lesscF_in_ga(x1) lesscF_out_ga(x1, x2) = lesscF_out_ga(x1) U81_ga(x1, x2, x3) = U81_ga(x1, x3) lesscG_in_ga(x1, x2) = lesscG_in_ga(x1) lesscG_out_ga(x1, x2) = lesscG_out_ga(x1) U79_ga(x1, x2, x3) = U79_ga(x1, x3) lesscF_in_gg(x1, x2) = lesscF_in_gg(x1, x2) lesscF_out_gg(x1, x2) = lesscF_out_gg(x1, x2) U81_gg(x1, x2, x3) = U81_gg(x1, x2, x3) MSL_IN_AG(x1, x2) = MSL_IN_AG(x2) U30_AG(x1, x2, x3, x4) = U30_AG(x4) SPLITB_IN_AAAA(x1, x2, x3, x4) = SPLITB_IN_AAAA U2_AAAA(x1, x2, x3, x4, x5) = U2_AAAA(x5) SPLITA_IN_AAA(x1, x2, x3) = SPLITA_IN_AAA U1_AAA(x1, x2, x3, x4, x5) = U1_AAA(x5) U31_AG(x1, x2, x3, x4) = U31_AG(x4) U32_AG(x1, x2, x3, x4) = U32_AG(x4) MSH_IN_AAA(x1, x2, x3) = MSH_IN_AAA U18_AAA(x1, x2, x3, x4, x5) = U18_AAA(x5) PC_IN_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = PC_IN_AAAAAAAA U3_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U3_AAAAAAAA(x9) U4_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U4_AAAAAAAA(x9) U5_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U5_AAAAAAAA(x9) MSE_IN_AA(x1, x2) = MSE_IN_AA U10_AA(x1, x2, x3, x4, x5) = U10_AA(x5) U6_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_AAAAAAAA(x9) U7_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U7_AAAAAAAA(x9) U8_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U8_AAAAAAAA(x9) U9_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U9_AAAAAAAA(x9) MERGED_IN_AAA(x1, x2, x3) = MERGED_IN_AAA U11_AAA(x1, x2, x3, x4, x5, x6) = U11_AAA(x6) LESSF_IN_AA(x1, x2) = LESSF_IN_AA U19_AA(x1, x2, x3) = U19_AA(x3) LESSG_IN_AA(x1, x2) = LESSG_IN_AA U17_AA(x1, x2, x3) = U17_AA(x3) U12_AAA(x1, x2, x3, x4, x5, x6) = U12_AAA(x6) U13_AAA(x1, x2, x3, x4, x5, x6) = U13_AAA(x6) U14_AAA(x1, x2, x3, x4, x5, x6) = U14_AAA(x6) U15_AAA(x1, x2, x3, x4, x5, x6) = U15_AAA(x6) U16_AAA(x1, x2, x3, x4, x5, x6) = U16_AAA(x6) U33_AG(x1, x2, x3, x4, x5) = U33_AG(x5) U34_AG(x1, x2, x3, x4) = U34_AG(x4) U35_AG(x1, x2, x3, x4, x5) = U35_AG(x4, x5) U36_AG(x1, x2, x3, x4, x5) = U36_AG(x4, x5) U37_AG(x1, x2, x3, x4, x5) = U37_AG(x4, x5) U38_AG(x1, x2, x3, x4, x5, x6) = U38_AG(x4, x6) U39_AG(x1, x2, x3, x4, x5) = U39_AG(x4, x5) U40_AG(x1, x2, x3, x4, x5, x6) = U40_AG(x4, x6) U41_AG(x1, x2, x3, x4, x5) = U41_AG(x4, x5) U42_AG(x1, x2, x3, x4, x5) = U42_AG(x4, x5) LESSF_IN_GA(x1, x2) = LESSF_IN_GA(x1) U19_GA(x1, x2, x3) = U19_GA(x1, x3) LESSG_IN_GA(x1, x2) = LESSG_IN_GA(x1) U17_GA(x1, x2, x3) = U17_GA(x1, x3) U43_AG(x1, x2, x3, x4, x5, x6) = U43_AG(x4, x6) U44_AG(x1, x2, x3, x4, x5, x6) = U44_AG(x4, x6) U45_AG(x1, x2, x3, x4, x5) = U45_AG(x4, x5) U46_AG(x1, x2, x3, x4, x5) = U46_AG(x4, x5) U47_AG(x1, x2, x3, x4, x5) = U47_AG(x4, x5) U48_AG(x1, x2, x3, x4, x5) = U48_AG(x4, x5) U49_AG(x1, x2, x3, x4, x5, x6) = U49_AG(x4, x6) U50_AG(x1, x2, x3, x4, x5) = U50_AG(x4, x5) U51_AG(x1, x2, x3, x4, x5, x6) = U51_AG(x4, x5, x6) U52_AG(x1, x2, x3, x4, x5, x6, x7) = U52_AG(x4, x5, x7) U53_AG(x1, x2, x3, x4, x5, x6) = U53_AG(x4, x5, x6) U54_AG(x1, x2, x3, x4, x5, x6) = U54_AG(x4, x5, x6) U55_AG(x1, x2, x3, x4, x5, x6, x7) = U55_AG(x4, x5, x7) U56_AG(x1, x2, x3, x4, x5, x6, x7, x8) = U56_AG(x4, x5, x8) U57_AG(x1, x2, x3, x4, x5, x6) = U57_AG(x4, x5, x6) PK_IN_GAAAG(x1, x2, x3, x4, x5) = PK_IN_GAAAG(x1, x5) U25_GAAAG(x1, x2, x3, x4, x5, x6) = U25_GAAAG(x1, x5, x6) U26_GAAAG(x1, x2, x3, x4, x5, x6, x7) = U26_GAAAG(x1, x2, x6, x7) U27_GAAAG(x1, x2, x3, x4, x5, x6, x7) = U27_GAAAG(x1, x2, x6, x7) PJ_IN_GAAAG(x1, x2, x3, x4, x5) = PJ_IN_GAAAG(x1, x5) U20_GAAAG(x1, x2, x3, x4, x5, x6) = U20_GAAAG(x1, x5, x6) U21_GAAAG(x1, x2, x3, x4, x5, x6, x7) = U21_GAAAG(x1, x3, x6, x7) U22_GAAAG(x1, x2, x3, x4, x5, x6, x7) = U22_GAAAG(x1, x3, x6, x7) U23_GAAAG(x1, x2, x3, x4, x5, x6, x7) = U23_GAAAG(x1, x2, x6, x7) U24_GAAAG(x1, x2, x3, x4, x5, x6, x7) = U24_GAAAG(x1, x2, x6, x7) U28_GAAAG(x1, x2, x3, x4, x5, x6, x7) = U28_GAAAG(x1, x4, x6, x7) U29_GAAAG(x1, x2, x3, x4, x5, x6, x7) = U29_GAAAG(x1, x4, x6, x7) U58_AG(x1, x2, x3, x4, x5, x6, x7) = U58_AG(x4, x5, x6, x7) U59_AG(x1, x2, x3, x4, x5, x6, x7, x8) = U59_AG(x4, x5, x6, x8) U60_AG(x1, x2, x3, x4, x5, x6, x7, x8) = U60_AG(x4, x5, x6, x8) U61_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U61_AG(x4, x5, x6, x10) U62_AG(x1, x2, x3, x4, x5, x6, x7) = U62_AG(x4, x5, x6, x7) U63_AG(x1, x2, x3, x4, x5, x6, x7, x8) = U63_AG(x4, x5, x6, x8) U64_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U64_AG(x4, x5, x6, x9) U65_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U65_AG(x4, x5, x6, x10) U66_AG(x1, x2, x3, x4, x5, x6, x7) = U66_AG(x4, x5, x6, x7) We have to consider all (P,R,Pi)-chains ---------------------------------------- (167) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 85 less nodes. ---------------------------------------- (168) Complex Obligation (AND) ---------------------------------------- (169) Obligation: Pi DP problem: The TRS P consists of the following rules: LESSG_IN_GA(s(X1), s(X2)) -> LESSG_IN_GA(X1, X2) The TRS R consists of the following rules: splitcB_in_aaaa(X1, X2, .(X1, X3), X4) -> U69_aaaa(X1, X2, X3, X4, splitcA_in_aaa(X2, X4, X3)) splitcA_in_aaa([], [], []) -> splitcA_out_aaa([], [], []) splitcA_in_aaa(.(X1, X2), .(X1, X3), X4) -> U68_aaa(X1, X2, X3, X4, splitcA_in_aaa(X2, X4, X3)) U68_aaa(X1, X2, X3, X4, splitcA_out_aaa(X2, X4, X3)) -> splitcA_out_aaa(.(X1, X2), .(X1, X3), X4) U69_aaaa(X1, X2, X3, X4, splitcA_out_aaa(X2, X4, X3)) -> splitcB_out_aaaa(X1, X2, .(X1, X3), X4) mscE_in_aa([], []) -> mscE_out_aa([], []) mscE_in_aa(.(X1, []), .(X1, [])) -> mscE_out_aa(.(X1, []), .(X1, [])) mscE_in_aa(.(X1, .(X2, X3)), X4) -> U74_aa(X1, X2, X3, X4, qcC_in_aaaaaaaa(X1, X2, X3, X5, X6, X7, X8, X4)) qcC_in_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8) -> U70_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, splitcB_in_aaaa(X1, .(X2, X3), X4, X5)) U70_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, splitcB_out_aaaa(X1, .(X2, X3), X4, X5)) -> U71_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mscE_in_aa(X4, X6)) U71_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mscE_out_aa(X4, X6)) -> U72_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mscE_in_aa(X5, X7)) U72_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mscE_out_aa(X5, X7)) -> U73_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mergecD_in_aaa(X6, X7, X8)) mergecD_in_aaa([], X1, X1) -> mergecD_out_aaa([], X1, X1) mergecD_in_aaa(X1, [], X1) -> mergecD_out_aaa(X1, [], X1) mergecD_in_aaa(.(X1, X2), .(X3, X4), .(X1, X5)) -> U75_aaa(X1, X2, X3, X4, X5, lesscF_in_aa(X1, X3)) lesscF_in_aa(0, X1) -> lesscF_out_aa(0, X1) lesscF_in_aa(s(X1), X2) -> U81_aa(X1, X2, lesscG_in_aa(X1, X2)) lesscG_in_aa(0, s(X1)) -> lesscG_out_aa(0, s(X1)) lesscG_in_aa(s(X1), s(X2)) -> U79_aa(X1, X2, lesscG_in_aa(X1, X2)) U79_aa(X1, X2, lesscG_out_aa(X1, X2)) -> lesscG_out_aa(s(X1), s(X2)) U81_aa(X1, X2, lesscG_out_aa(X1, X2)) -> lesscF_out_aa(s(X1), X2) U75_aaa(X1, X2, X3, X4, X5, lesscF_out_aa(X1, X3)) -> U76_aaa(X1, X2, X3, X4, X5, mergecD_in_aaa(X2, .(X3, X4), X5)) mergecD_in_aaa(.(X1, X2), .(X3, X4), .(X3, X5)) -> U77_aaa(X1, X2, X3, X4, X5, lesscG_in_aa(X3, X1)) U77_aaa(X1, X2, X3, X4, X5, lesscG_out_aa(X3, X1)) -> U78_aaa(X1, X2, X3, X4, X5, mergecD_in_aaa(.(X1, X2), X4, X5)) U78_aaa(X1, X2, X3, X4, X5, mergecD_out_aaa(.(X1, X2), X4, X5)) -> mergecD_out_aaa(.(X1, X2), .(X3, X4), .(X3, X5)) U76_aaa(X1, X2, X3, X4, X5, mergecD_out_aaa(X2, .(X3, X4), X5)) -> mergecD_out_aaa(.(X1, X2), .(X3, X4), .(X1, X5)) U73_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mergecD_out_aaa(X6, X7, X8)) -> qcC_out_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8) U74_aa(X1, X2, X3, X4, qcC_out_aaaaaaaa(X1, X2, X3, X5, X6, X7, X8, X4)) -> mscE_out_aa(.(X1, .(X2, X3)), X4) mscH_in_aaa(X1, [], .(X1, [])) -> mscH_out_aaa(X1, [], .(X1, [])) mscH_in_aaa(X1, .(X2, X3), X4) -> U80_aaa(X1, X2, X3, X4, qcC_in_aaaaaaaa(X1, X2, X3, X5, X6, X7, X8, X4)) U80_aaa(X1, X2, X3, X4, qcC_out_aaaaaaaa(X1, X2, X3, X5, X6, X7, X8, X4)) -> mscH_out_aaa(X1, .(X2, X3), X4) lesscG_in_gg(0, s(X1)) -> lesscG_out_gg(0, s(X1)) lesscG_in_gg(s(X1), s(X2)) -> U79_gg(X1, X2, lesscG_in_gg(X1, X2)) U79_gg(X1, X2, lesscG_out_gg(X1, X2)) -> lesscG_out_gg(s(X1), s(X2)) lesscF_in_ga(0, X1) -> lesscF_out_ga(0, X1) lesscF_in_ga(s(X1), X2) -> U81_ga(X1, X2, lesscG_in_ga(X1, X2)) lesscG_in_ga(0, s(X1)) -> lesscG_out_ga(0, s(X1)) lesscG_in_ga(s(X1), s(X2)) -> U79_ga(X1, X2, lesscG_in_ga(X1, X2)) U79_ga(X1, X2, lesscG_out_ga(X1, X2)) -> lesscG_out_ga(s(X1), s(X2)) U81_ga(X1, X2, lesscG_out_ga(X1, X2)) -> lesscF_out_ga(s(X1), X2) lesscF_in_gg(0, X1) -> lesscF_out_gg(0, X1) lesscF_in_gg(s(X1), X2) -> U81_gg(X1, X2, lesscG_in_gg(X1, X2)) U81_gg(X1, X2, lesscG_out_gg(X1, X2)) -> lesscF_out_gg(s(X1), X2) The argument filtering Pi contains the following mapping: [] = [] splitcB_in_aaaa(x1, x2, x3, x4) = splitcB_in_aaaa U69_aaaa(x1, x2, x3, x4, x5) = U69_aaaa(x5) splitcA_in_aaa(x1, x2, x3) = splitcA_in_aaa splitcA_out_aaa(x1, x2, x3) = splitcA_out_aaa U68_aaa(x1, x2, x3, x4, x5) = U68_aaa(x5) splitcB_out_aaaa(x1, x2, x3, x4) = splitcB_out_aaaa mscE_in_aa(x1, x2) = mscE_in_aa mscE_out_aa(x1, x2) = mscE_out_aa U74_aa(x1, x2, x3, x4, x5) = U74_aa(x5) qcC_in_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = qcC_in_aaaaaaaa U70_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U70_aaaaaaaa(x9) U71_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U71_aaaaaaaa(x9) U72_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U72_aaaaaaaa(x9) U73_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U73_aaaaaaaa(x9) mergecD_in_aaa(x1, x2, x3) = mergecD_in_aaa mergecD_out_aaa(x1, x2, x3) = mergecD_out_aaa U75_aaa(x1, x2, x3, x4, x5, x6) = U75_aaa(x6) lesscF_in_aa(x1, x2) = lesscF_in_aa lesscF_out_aa(x1, x2) = lesscF_out_aa(x1) U81_aa(x1, x2, x3) = U81_aa(x3) lesscG_in_aa(x1, x2) = lesscG_in_aa lesscG_out_aa(x1, x2) = lesscG_out_aa(x1) U79_aa(x1, x2, x3) = U79_aa(x3) U76_aaa(x1, x2, x3, x4, x5, x6) = U76_aaa(x6) U77_aaa(x1, x2, x3, x4, x5, x6) = U77_aaa(x6) U78_aaa(x1, x2, x3, x4, x5, x6) = U78_aaa(x6) qcC_out_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = qcC_out_aaaaaaaa mscH_in_aaa(x1, x2, x3) = mscH_in_aaa mscH_out_aaa(x1, x2, x3) = mscH_out_aaa U80_aaa(x1, x2, x3, x4, x5) = U80_aaa(x5) .(x1, x2) = .(x1, x2) s(x1) = s(x1) lesscG_in_gg(x1, x2) = lesscG_in_gg(x1, x2) 0 = 0 lesscG_out_gg(x1, x2) = lesscG_out_gg(x1, x2) U79_gg(x1, x2, x3) = U79_gg(x1, x2, x3) lesscF_in_ga(x1, x2) = lesscF_in_ga(x1) lesscF_out_ga(x1, x2) = lesscF_out_ga(x1) U81_ga(x1, x2, x3) = U81_ga(x1, x3) lesscG_in_ga(x1, x2) = lesscG_in_ga(x1) lesscG_out_ga(x1, x2) = lesscG_out_ga(x1) U79_ga(x1, x2, x3) = U79_ga(x1, x3) lesscF_in_gg(x1, x2) = lesscF_in_gg(x1, x2) lesscF_out_gg(x1, x2) = lesscF_out_gg(x1, x2) U81_gg(x1, x2, x3) = U81_gg(x1, x2, x3) LESSG_IN_GA(x1, x2) = LESSG_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (170) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (171) Obligation: Pi DP problem: The TRS P consists of the following rules: LESSG_IN_GA(s(X1), s(X2)) -> LESSG_IN_GA(X1, X2) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) LESSG_IN_GA(x1, x2) = LESSG_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (172) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (173) Obligation: Q DP problem: The TRS P consists of the following rules: LESSG_IN_GA(s(X1)) -> LESSG_IN_GA(X1) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (174) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *LESSG_IN_GA(s(X1)) -> LESSG_IN_GA(X1) The graph contains the following edges 1 > 1 ---------------------------------------- (175) YES ---------------------------------------- (176) Obligation: Pi DP problem: The TRS P consists of the following rules: PJ_IN_GAAAG(X1, X2, .(X3, X4), X5, .(X3, X6)) -> U21_GAAAG(X1, X2, X3, X4, X5, X6, lesscF_in_ga(X1, X2)) U21_GAAAG(X1, X2, X3, X4, X5, X6, lesscF_out_ga(X1, X2)) -> PJ_IN_GAAAG(X3, X2, X4, X5, X6) PJ_IN_GAAAG(X1, X2, .(X3, X4), X5, .(X2, X6)) -> U23_GAAAG(X1, X2, X3, X4, X5, X6, lesscF_in_gg(X1, X2)) U23_GAAAG(X1, X2, X3, X4, X5, X6, lesscF_out_gg(X1, X2)) -> PK_IN_GAAAG(X2, X3, X4, X5, X6) PK_IN_GAAAG(X1, X2, X3, .(X4, X5), .(X2, X6)) -> U26_GAAAG(X1, X2, X3, X4, X5, X6, lesscG_in_gg(X1, X2)) U26_GAAAG(X1, X2, X3, X4, X5, X6, lesscG_out_gg(X1, X2)) -> PJ_IN_GAAAG(X2, X4, X3, X5, X6) PK_IN_GAAAG(X1, X2, X3, .(X4, X5), .(X4, X6)) -> U28_GAAAG(X1, X2, X3, X4, X5, X6, lesscG_in_ga(X1, X2)) U28_GAAAG(X1, X2, X3, X4, X5, X6, lesscG_out_ga(X1, X2)) -> PK_IN_GAAAG(X4, X2, X3, X5, X6) The TRS R consists of the following rules: splitcB_in_aaaa(X1, X2, .(X1, X3), X4) -> U69_aaaa(X1, X2, X3, X4, splitcA_in_aaa(X2, X4, X3)) splitcA_in_aaa([], [], []) -> splitcA_out_aaa([], [], []) splitcA_in_aaa(.(X1, X2), .(X1, X3), X4) -> U68_aaa(X1, X2, X3, X4, splitcA_in_aaa(X2, X4, X3)) U68_aaa(X1, X2, X3, X4, splitcA_out_aaa(X2, X4, X3)) -> splitcA_out_aaa(.(X1, X2), .(X1, X3), X4) U69_aaaa(X1, X2, X3, X4, splitcA_out_aaa(X2, X4, X3)) -> splitcB_out_aaaa(X1, X2, .(X1, X3), X4) mscE_in_aa([], []) -> mscE_out_aa([], []) mscE_in_aa(.(X1, []), .(X1, [])) -> mscE_out_aa(.(X1, []), .(X1, [])) mscE_in_aa(.(X1, .(X2, X3)), X4) -> U74_aa(X1, X2, X3, X4, qcC_in_aaaaaaaa(X1, X2, X3, X5, X6, X7, X8, X4)) qcC_in_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8) -> U70_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, splitcB_in_aaaa(X1, .(X2, X3), X4, X5)) U70_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, splitcB_out_aaaa(X1, .(X2, X3), X4, X5)) -> U71_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mscE_in_aa(X4, X6)) U71_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mscE_out_aa(X4, X6)) -> U72_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mscE_in_aa(X5, X7)) U72_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mscE_out_aa(X5, X7)) -> U73_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mergecD_in_aaa(X6, X7, X8)) mergecD_in_aaa([], X1, X1) -> mergecD_out_aaa([], X1, X1) mergecD_in_aaa(X1, [], X1) -> mergecD_out_aaa(X1, [], X1) mergecD_in_aaa(.(X1, X2), .(X3, X4), .(X1, X5)) -> U75_aaa(X1, X2, X3, X4, X5, lesscF_in_aa(X1, X3)) lesscF_in_aa(0, X1) -> lesscF_out_aa(0, X1) lesscF_in_aa(s(X1), X2) -> U81_aa(X1, X2, lesscG_in_aa(X1, X2)) lesscG_in_aa(0, s(X1)) -> lesscG_out_aa(0, s(X1)) lesscG_in_aa(s(X1), s(X2)) -> U79_aa(X1, X2, lesscG_in_aa(X1, X2)) U79_aa(X1, X2, lesscG_out_aa(X1, X2)) -> lesscG_out_aa(s(X1), s(X2)) U81_aa(X1, X2, lesscG_out_aa(X1, X2)) -> lesscF_out_aa(s(X1), X2) U75_aaa(X1, X2, X3, X4, X5, lesscF_out_aa(X1, X3)) -> U76_aaa(X1, X2, X3, X4, X5, mergecD_in_aaa(X2, .(X3, X4), X5)) mergecD_in_aaa(.(X1, X2), .(X3, X4), .(X3, X5)) -> U77_aaa(X1, X2, X3, X4, X5, lesscG_in_aa(X3, X1)) U77_aaa(X1, X2, X3, X4, X5, lesscG_out_aa(X3, X1)) -> U78_aaa(X1, X2, X3, X4, X5, mergecD_in_aaa(.(X1, X2), X4, X5)) U78_aaa(X1, X2, X3, X4, X5, mergecD_out_aaa(.(X1, X2), X4, X5)) -> mergecD_out_aaa(.(X1, X2), .(X3, X4), .(X3, X5)) U76_aaa(X1, X2, X3, X4, X5, mergecD_out_aaa(X2, .(X3, X4), X5)) -> mergecD_out_aaa(.(X1, X2), .(X3, X4), .(X1, X5)) U73_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mergecD_out_aaa(X6, X7, X8)) -> qcC_out_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8) U74_aa(X1, X2, X3, X4, qcC_out_aaaaaaaa(X1, X2, X3, X5, X6, X7, X8, X4)) -> mscE_out_aa(.(X1, .(X2, X3)), X4) mscH_in_aaa(X1, [], .(X1, [])) -> mscH_out_aaa(X1, [], .(X1, [])) mscH_in_aaa(X1, .(X2, X3), X4) -> U80_aaa(X1, X2, X3, X4, qcC_in_aaaaaaaa(X1, X2, X3, X5, X6, X7, X8, X4)) U80_aaa(X1, X2, X3, X4, qcC_out_aaaaaaaa(X1, X2, X3, X5, X6, X7, X8, X4)) -> mscH_out_aaa(X1, .(X2, X3), X4) lesscG_in_gg(0, s(X1)) -> lesscG_out_gg(0, s(X1)) lesscG_in_gg(s(X1), s(X2)) -> U79_gg(X1, X2, lesscG_in_gg(X1, X2)) U79_gg(X1, X2, lesscG_out_gg(X1, X2)) -> lesscG_out_gg(s(X1), s(X2)) lesscF_in_ga(0, X1) -> lesscF_out_ga(0, X1) lesscF_in_ga(s(X1), X2) -> U81_ga(X1, X2, lesscG_in_ga(X1, X2)) lesscG_in_ga(0, s(X1)) -> lesscG_out_ga(0, s(X1)) lesscG_in_ga(s(X1), s(X2)) -> U79_ga(X1, X2, lesscG_in_ga(X1, X2)) U79_ga(X1, X2, lesscG_out_ga(X1, X2)) -> lesscG_out_ga(s(X1), s(X2)) U81_ga(X1, X2, lesscG_out_ga(X1, X2)) -> lesscF_out_ga(s(X1), X2) lesscF_in_gg(0, X1) -> lesscF_out_gg(0, X1) lesscF_in_gg(s(X1), X2) -> U81_gg(X1, X2, lesscG_in_gg(X1, X2)) U81_gg(X1, X2, lesscG_out_gg(X1, X2)) -> lesscF_out_gg(s(X1), X2) The argument filtering Pi contains the following mapping: [] = [] splitcB_in_aaaa(x1, x2, x3, x4) = splitcB_in_aaaa U69_aaaa(x1, x2, x3, x4, x5) = U69_aaaa(x5) splitcA_in_aaa(x1, x2, x3) = splitcA_in_aaa splitcA_out_aaa(x1, x2, x3) = splitcA_out_aaa U68_aaa(x1, x2, x3, x4, x5) = U68_aaa(x5) splitcB_out_aaaa(x1, x2, x3, x4) = splitcB_out_aaaa mscE_in_aa(x1, x2) = mscE_in_aa mscE_out_aa(x1, x2) = mscE_out_aa U74_aa(x1, x2, x3, x4, x5) = U74_aa(x5) qcC_in_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = qcC_in_aaaaaaaa U70_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U70_aaaaaaaa(x9) U71_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U71_aaaaaaaa(x9) U72_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U72_aaaaaaaa(x9) U73_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U73_aaaaaaaa(x9) mergecD_in_aaa(x1, x2, x3) = mergecD_in_aaa mergecD_out_aaa(x1, x2, x3) = mergecD_out_aaa U75_aaa(x1, x2, x3, x4, x5, x6) = U75_aaa(x6) lesscF_in_aa(x1, x2) = lesscF_in_aa lesscF_out_aa(x1, x2) = lesscF_out_aa(x1) U81_aa(x1, x2, x3) = U81_aa(x3) lesscG_in_aa(x1, x2) = lesscG_in_aa lesscG_out_aa(x1, x2) = lesscG_out_aa(x1) U79_aa(x1, x2, x3) = U79_aa(x3) U76_aaa(x1, x2, x3, x4, x5, x6) = U76_aaa(x6) U77_aaa(x1, x2, x3, x4, x5, x6) = U77_aaa(x6) U78_aaa(x1, x2, x3, x4, x5, x6) = U78_aaa(x6) qcC_out_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = qcC_out_aaaaaaaa mscH_in_aaa(x1, x2, x3) = mscH_in_aaa mscH_out_aaa(x1, x2, x3) = mscH_out_aaa U80_aaa(x1, x2, x3, x4, x5) = U80_aaa(x5) .(x1, x2) = .(x1, x2) s(x1) = s(x1) lesscG_in_gg(x1, x2) = lesscG_in_gg(x1, x2) 0 = 0 lesscG_out_gg(x1, x2) = lesscG_out_gg(x1, x2) U79_gg(x1, x2, x3) = U79_gg(x1, x2, x3) lesscF_in_ga(x1, x2) = lesscF_in_ga(x1) lesscF_out_ga(x1, x2) = lesscF_out_ga(x1) U81_ga(x1, x2, x3) = U81_ga(x1, x3) lesscG_in_ga(x1, x2) = lesscG_in_ga(x1) lesscG_out_ga(x1, x2) = lesscG_out_ga(x1) U79_ga(x1, x2, x3) = U79_ga(x1, x3) lesscF_in_gg(x1, x2) = lesscF_in_gg(x1, x2) lesscF_out_gg(x1, x2) = lesscF_out_gg(x1, x2) U81_gg(x1, x2, x3) = U81_gg(x1, x2, x3) PK_IN_GAAAG(x1, x2, x3, x4, x5) = PK_IN_GAAAG(x1, x5) U26_GAAAG(x1, x2, x3, x4, x5, x6, x7) = U26_GAAAG(x1, x2, x6, x7) PJ_IN_GAAAG(x1, x2, x3, x4, x5) = PJ_IN_GAAAG(x1, x5) U21_GAAAG(x1, x2, x3, x4, x5, x6, x7) = U21_GAAAG(x1, x3, x6, x7) U23_GAAAG(x1, x2, x3, x4, x5, x6, x7) = U23_GAAAG(x1, x2, x6, x7) U28_GAAAG(x1, x2, x3, x4, x5, x6, x7) = U28_GAAAG(x1, x4, x6, x7) We have to consider all (P,R,Pi)-chains ---------------------------------------- (177) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (178) Obligation: Pi DP problem: The TRS P consists of the following rules: PJ_IN_GAAAG(X1, X2, .(X3, X4), X5, .(X3, X6)) -> U21_GAAAG(X1, X2, X3, X4, X5, X6, lesscF_in_ga(X1, X2)) U21_GAAAG(X1, X2, X3, X4, X5, X6, lesscF_out_ga(X1, X2)) -> PJ_IN_GAAAG(X3, X2, X4, X5, X6) PJ_IN_GAAAG(X1, X2, .(X3, X4), X5, .(X2, X6)) -> U23_GAAAG(X1, X2, X3, X4, X5, X6, lesscF_in_gg(X1, X2)) U23_GAAAG(X1, X2, X3, X4, X5, X6, lesscF_out_gg(X1, X2)) -> PK_IN_GAAAG(X2, X3, X4, X5, X6) PK_IN_GAAAG(X1, X2, X3, .(X4, X5), .(X2, X6)) -> U26_GAAAG(X1, X2, X3, X4, X5, X6, lesscG_in_gg(X1, X2)) U26_GAAAG(X1, X2, X3, X4, X5, X6, lesscG_out_gg(X1, X2)) -> PJ_IN_GAAAG(X2, X4, X3, X5, X6) PK_IN_GAAAG(X1, X2, X3, .(X4, X5), .(X4, X6)) -> U28_GAAAG(X1, X2, X3, X4, X5, X6, lesscG_in_ga(X1, X2)) U28_GAAAG(X1, X2, X3, X4, X5, X6, lesscG_out_ga(X1, X2)) -> PK_IN_GAAAG(X4, X2, X3, X5, X6) The TRS R consists of the following rules: lesscF_in_ga(0, X1) -> lesscF_out_ga(0, X1) lesscF_in_ga(s(X1), X2) -> U81_ga(X1, X2, lesscG_in_ga(X1, X2)) lesscF_in_gg(0, X1) -> lesscF_out_gg(0, X1) lesscF_in_gg(s(X1), X2) -> U81_gg(X1, X2, lesscG_in_gg(X1, X2)) lesscG_in_gg(0, s(X1)) -> lesscG_out_gg(0, s(X1)) lesscG_in_gg(s(X1), s(X2)) -> U79_gg(X1, X2, lesscG_in_gg(X1, X2)) lesscG_in_ga(0, s(X1)) -> lesscG_out_ga(0, s(X1)) lesscG_in_ga(s(X1), s(X2)) -> U79_ga(X1, X2, lesscG_in_ga(X1, X2)) U81_ga(X1, X2, lesscG_out_ga(X1, X2)) -> lesscF_out_ga(s(X1), X2) U81_gg(X1, X2, lesscG_out_gg(X1, X2)) -> lesscF_out_gg(s(X1), X2) U79_gg(X1, X2, lesscG_out_gg(X1, X2)) -> lesscG_out_gg(s(X1), s(X2)) U79_ga(X1, X2, lesscG_out_ga(X1, X2)) -> lesscG_out_ga(s(X1), s(X2)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) s(x1) = s(x1) lesscG_in_gg(x1, x2) = lesscG_in_gg(x1, x2) 0 = 0 lesscG_out_gg(x1, x2) = lesscG_out_gg(x1, x2) U79_gg(x1, x2, x3) = U79_gg(x1, x2, x3) lesscF_in_ga(x1, x2) = lesscF_in_ga(x1) lesscF_out_ga(x1, x2) = lesscF_out_ga(x1) U81_ga(x1, x2, x3) = U81_ga(x1, x3) lesscG_in_ga(x1, x2) = lesscG_in_ga(x1) lesscG_out_ga(x1, x2) = lesscG_out_ga(x1) U79_ga(x1, x2, x3) = U79_ga(x1, x3) lesscF_in_gg(x1, x2) = lesscF_in_gg(x1, x2) lesscF_out_gg(x1, x2) = lesscF_out_gg(x1, x2) U81_gg(x1, x2, x3) = U81_gg(x1, x2, x3) PK_IN_GAAAG(x1, x2, x3, x4, x5) = PK_IN_GAAAG(x1, x5) U26_GAAAG(x1, x2, x3, x4, x5, x6, x7) = U26_GAAAG(x1, x2, x6, x7) PJ_IN_GAAAG(x1, x2, x3, x4, x5) = PJ_IN_GAAAG(x1, x5) U21_GAAAG(x1, x2, x3, x4, x5, x6, x7) = U21_GAAAG(x1, x3, x6, x7) U23_GAAAG(x1, x2, x3, x4, x5, x6, x7) = U23_GAAAG(x1, x2, x6, x7) U28_GAAAG(x1, x2, x3, x4, x5, x6, x7) = U28_GAAAG(x1, x4, x6, x7) We have to consider all (P,R,Pi)-chains ---------------------------------------- (179) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (180) Obligation: Q DP problem: The TRS P consists of the following rules: PJ_IN_GAAAG(X1, .(X3, X6)) -> U21_GAAAG(X1, X3, X6, lesscF_in_ga(X1)) U21_GAAAG(X1, X3, X6, lesscF_out_ga(X1)) -> PJ_IN_GAAAG(X3, X6) PJ_IN_GAAAG(X1, .(X2, X6)) -> U23_GAAAG(X1, X2, X6, lesscF_in_gg(X1, X2)) U23_GAAAG(X1, X2, X6, lesscF_out_gg(X1, X2)) -> PK_IN_GAAAG(X2, X6) PK_IN_GAAAG(X1, .(X2, X6)) -> U26_GAAAG(X1, X2, X6, lesscG_in_gg(X1, X2)) U26_GAAAG(X1, X2, X6, lesscG_out_gg(X1, X2)) -> PJ_IN_GAAAG(X2, X6) PK_IN_GAAAG(X1, .(X4, X6)) -> U28_GAAAG(X1, X4, X6, lesscG_in_ga(X1)) U28_GAAAG(X1, X4, X6, lesscG_out_ga(X1)) -> PK_IN_GAAAG(X4, X6) The TRS R consists of the following rules: lesscF_in_ga(0) -> lesscF_out_ga(0) lesscF_in_ga(s(X1)) -> U81_ga(X1, lesscG_in_ga(X1)) lesscF_in_gg(0, X1) -> lesscF_out_gg(0, X1) lesscF_in_gg(s(X1), X2) -> U81_gg(X1, X2, lesscG_in_gg(X1, X2)) lesscG_in_gg(0, s(X1)) -> lesscG_out_gg(0, s(X1)) lesscG_in_gg(s(X1), s(X2)) -> U79_gg(X1, X2, lesscG_in_gg(X1, X2)) lesscG_in_ga(0) -> lesscG_out_ga(0) lesscG_in_ga(s(X1)) -> U79_ga(X1, lesscG_in_ga(X1)) U81_ga(X1, lesscG_out_ga(X1)) -> lesscF_out_ga(s(X1)) U81_gg(X1, X2, lesscG_out_gg(X1, X2)) -> lesscF_out_gg(s(X1), X2) U79_gg(X1, X2, lesscG_out_gg(X1, X2)) -> lesscG_out_gg(s(X1), s(X2)) U79_ga(X1, lesscG_out_ga(X1)) -> lesscG_out_ga(s(X1)) The set Q consists of the following terms: lesscF_in_ga(x0) lesscF_in_gg(x0, x1) lesscG_in_gg(x0, x1) lesscG_in_ga(x0) U81_ga(x0, x1) U81_gg(x0, x1, x2) U79_gg(x0, x1, x2) U79_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (181) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *U21_GAAAG(X1, X3, X6, lesscF_out_ga(X1)) -> PJ_IN_GAAAG(X3, X6) The graph contains the following edges 2 >= 1, 3 >= 2 *U26_GAAAG(X1, X2, X6, lesscG_out_gg(X1, X2)) -> PJ_IN_GAAAG(X2, X6) The graph contains the following edges 2 >= 1, 4 > 1, 3 >= 2 *PJ_IN_GAAAG(X1, .(X3, X6)) -> U21_GAAAG(X1, X3, X6, lesscF_in_ga(X1)) The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3 *PJ_IN_GAAAG(X1, .(X2, X6)) -> U23_GAAAG(X1, X2, X6, lesscF_in_gg(X1, X2)) The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3 *U23_GAAAG(X1, X2, X6, lesscF_out_gg(X1, X2)) -> PK_IN_GAAAG(X2, X6) The graph contains the following edges 2 >= 1, 4 > 1, 3 >= 2 *U28_GAAAG(X1, X4, X6, lesscG_out_ga(X1)) -> PK_IN_GAAAG(X4, X6) The graph contains the following edges 2 >= 1, 3 >= 2 *PK_IN_GAAAG(X1, .(X2, X6)) -> U26_GAAAG(X1, X2, X6, lesscG_in_gg(X1, X2)) The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3 *PK_IN_GAAAG(X1, .(X4, X6)) -> U28_GAAAG(X1, X4, X6, lesscG_in_ga(X1)) The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3 ---------------------------------------- (182) YES ---------------------------------------- (183) Obligation: Pi DP problem: The TRS P consists of the following rules: LESSG_IN_AA(s(X1), s(X2)) -> LESSG_IN_AA(X1, X2) The TRS R consists of the following rules: splitcB_in_aaaa(X1, X2, .(X1, X3), X4) -> U69_aaaa(X1, X2, X3, X4, splitcA_in_aaa(X2, X4, X3)) splitcA_in_aaa([], [], []) -> splitcA_out_aaa([], [], []) splitcA_in_aaa(.(X1, X2), .(X1, X3), X4) -> U68_aaa(X1, X2, X3, X4, splitcA_in_aaa(X2, X4, X3)) U68_aaa(X1, X2, X3, X4, splitcA_out_aaa(X2, X4, X3)) -> splitcA_out_aaa(.(X1, X2), .(X1, X3), X4) U69_aaaa(X1, X2, X3, X4, splitcA_out_aaa(X2, X4, X3)) -> splitcB_out_aaaa(X1, X2, .(X1, X3), X4) mscE_in_aa([], []) -> mscE_out_aa([], []) mscE_in_aa(.(X1, []), .(X1, [])) -> mscE_out_aa(.(X1, []), .(X1, [])) mscE_in_aa(.(X1, .(X2, X3)), X4) -> U74_aa(X1, X2, X3, X4, qcC_in_aaaaaaaa(X1, X2, X3, X5, X6, X7, X8, X4)) qcC_in_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8) -> U70_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, splitcB_in_aaaa(X1, .(X2, X3), X4, X5)) U70_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, splitcB_out_aaaa(X1, .(X2, X3), X4, X5)) -> U71_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mscE_in_aa(X4, X6)) U71_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mscE_out_aa(X4, X6)) -> U72_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mscE_in_aa(X5, X7)) U72_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mscE_out_aa(X5, X7)) -> U73_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mergecD_in_aaa(X6, X7, X8)) mergecD_in_aaa([], X1, X1) -> mergecD_out_aaa([], X1, X1) mergecD_in_aaa(X1, [], X1) -> mergecD_out_aaa(X1, [], X1) mergecD_in_aaa(.(X1, X2), .(X3, X4), .(X1, X5)) -> U75_aaa(X1, X2, X3, X4, X5, lesscF_in_aa(X1, X3)) lesscF_in_aa(0, X1) -> lesscF_out_aa(0, X1) lesscF_in_aa(s(X1), X2) -> U81_aa(X1, X2, lesscG_in_aa(X1, X2)) lesscG_in_aa(0, s(X1)) -> lesscG_out_aa(0, s(X1)) lesscG_in_aa(s(X1), s(X2)) -> U79_aa(X1, X2, lesscG_in_aa(X1, X2)) U79_aa(X1, X2, lesscG_out_aa(X1, X2)) -> lesscG_out_aa(s(X1), s(X2)) U81_aa(X1, X2, lesscG_out_aa(X1, X2)) -> lesscF_out_aa(s(X1), X2) U75_aaa(X1, X2, X3, X4, X5, lesscF_out_aa(X1, X3)) -> U76_aaa(X1, X2, X3, X4, X5, mergecD_in_aaa(X2, .(X3, X4), X5)) mergecD_in_aaa(.(X1, X2), .(X3, X4), .(X3, X5)) -> U77_aaa(X1, X2, X3, X4, X5, lesscG_in_aa(X3, X1)) U77_aaa(X1, X2, X3, X4, X5, lesscG_out_aa(X3, X1)) -> U78_aaa(X1, X2, X3, X4, X5, mergecD_in_aaa(.(X1, X2), X4, X5)) U78_aaa(X1, X2, X3, X4, X5, mergecD_out_aaa(.(X1, X2), X4, X5)) -> mergecD_out_aaa(.(X1, X2), .(X3, X4), .(X3, X5)) U76_aaa(X1, X2, X3, X4, X5, mergecD_out_aaa(X2, .(X3, X4), X5)) -> mergecD_out_aaa(.(X1, X2), .(X3, X4), .(X1, X5)) U73_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mergecD_out_aaa(X6, X7, X8)) -> qcC_out_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8) U74_aa(X1, X2, X3, X4, qcC_out_aaaaaaaa(X1, X2, X3, X5, X6, X7, X8, X4)) -> mscE_out_aa(.(X1, .(X2, X3)), X4) mscH_in_aaa(X1, [], .(X1, [])) -> mscH_out_aaa(X1, [], .(X1, [])) mscH_in_aaa(X1, .(X2, X3), X4) -> U80_aaa(X1, X2, X3, X4, qcC_in_aaaaaaaa(X1, X2, X3, X5, X6, X7, X8, X4)) U80_aaa(X1, X2, X3, X4, qcC_out_aaaaaaaa(X1, X2, X3, X5, X6, X7, X8, X4)) -> mscH_out_aaa(X1, .(X2, X3), X4) lesscG_in_gg(0, s(X1)) -> lesscG_out_gg(0, s(X1)) lesscG_in_gg(s(X1), s(X2)) -> U79_gg(X1, X2, lesscG_in_gg(X1, X2)) U79_gg(X1, X2, lesscG_out_gg(X1, X2)) -> lesscG_out_gg(s(X1), s(X2)) lesscF_in_ga(0, X1) -> lesscF_out_ga(0, X1) lesscF_in_ga(s(X1), X2) -> U81_ga(X1, X2, lesscG_in_ga(X1, X2)) lesscG_in_ga(0, s(X1)) -> lesscG_out_ga(0, s(X1)) lesscG_in_ga(s(X1), s(X2)) -> U79_ga(X1, X2, lesscG_in_ga(X1, X2)) U79_ga(X1, X2, lesscG_out_ga(X1, X2)) -> lesscG_out_ga(s(X1), s(X2)) U81_ga(X1, X2, lesscG_out_ga(X1, X2)) -> lesscF_out_ga(s(X1), X2) lesscF_in_gg(0, X1) -> lesscF_out_gg(0, X1) lesscF_in_gg(s(X1), X2) -> U81_gg(X1, X2, lesscG_in_gg(X1, X2)) U81_gg(X1, X2, lesscG_out_gg(X1, X2)) -> lesscF_out_gg(s(X1), X2) The argument filtering Pi contains the following mapping: [] = [] splitcB_in_aaaa(x1, x2, x3, x4) = splitcB_in_aaaa U69_aaaa(x1, x2, x3, x4, x5) = U69_aaaa(x5) splitcA_in_aaa(x1, x2, x3) = splitcA_in_aaa splitcA_out_aaa(x1, x2, x3) = splitcA_out_aaa U68_aaa(x1, x2, x3, x4, x5) = U68_aaa(x5) splitcB_out_aaaa(x1, x2, x3, x4) = splitcB_out_aaaa mscE_in_aa(x1, x2) = mscE_in_aa mscE_out_aa(x1, x2) = mscE_out_aa U74_aa(x1, x2, x3, x4, x5) = U74_aa(x5) qcC_in_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = qcC_in_aaaaaaaa U70_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U70_aaaaaaaa(x9) U71_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U71_aaaaaaaa(x9) U72_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U72_aaaaaaaa(x9) U73_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U73_aaaaaaaa(x9) mergecD_in_aaa(x1, x2, x3) = mergecD_in_aaa mergecD_out_aaa(x1, x2, x3) = mergecD_out_aaa U75_aaa(x1, x2, x3, x4, x5, x6) = U75_aaa(x6) lesscF_in_aa(x1, x2) = lesscF_in_aa lesscF_out_aa(x1, x2) = lesscF_out_aa(x1) U81_aa(x1, x2, x3) = U81_aa(x3) lesscG_in_aa(x1, x2) = lesscG_in_aa lesscG_out_aa(x1, x2) = lesscG_out_aa(x1) U79_aa(x1, x2, x3) = U79_aa(x3) U76_aaa(x1, x2, x3, x4, x5, x6) = U76_aaa(x6) U77_aaa(x1, x2, x3, x4, x5, x6) = U77_aaa(x6) U78_aaa(x1, x2, x3, x4, x5, x6) = U78_aaa(x6) qcC_out_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = qcC_out_aaaaaaaa mscH_in_aaa(x1, x2, x3) = mscH_in_aaa mscH_out_aaa(x1, x2, x3) = mscH_out_aaa U80_aaa(x1, x2, x3, x4, x5) = U80_aaa(x5) .(x1, x2) = .(x1, x2) s(x1) = s(x1) lesscG_in_gg(x1, x2) = lesscG_in_gg(x1, x2) 0 = 0 lesscG_out_gg(x1, x2) = lesscG_out_gg(x1, x2) U79_gg(x1, x2, x3) = U79_gg(x1, x2, x3) lesscF_in_ga(x1, x2) = lesscF_in_ga(x1) lesscF_out_ga(x1, x2) = lesscF_out_ga(x1) U81_ga(x1, x2, x3) = U81_ga(x1, x3) lesscG_in_ga(x1, x2) = lesscG_in_ga(x1) lesscG_out_ga(x1, x2) = lesscG_out_ga(x1) U79_ga(x1, x2, x3) = U79_ga(x1, x3) lesscF_in_gg(x1, x2) = lesscF_in_gg(x1, x2) lesscF_out_gg(x1, x2) = lesscF_out_gg(x1, x2) U81_gg(x1, x2, x3) = U81_gg(x1, x2, x3) LESSG_IN_AA(x1, x2) = LESSG_IN_AA We have to consider all (P,R,Pi)-chains ---------------------------------------- (184) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (185) Obligation: Pi DP problem: The TRS P consists of the following rules: LESSG_IN_AA(s(X1), s(X2)) -> LESSG_IN_AA(X1, X2) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) LESSG_IN_AA(x1, x2) = LESSG_IN_AA We have to consider all (P,R,Pi)-chains ---------------------------------------- (186) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (187) Obligation: Q DP problem: The TRS P consists of the following rules: LESSG_IN_AA -> LESSG_IN_AA R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (188) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by semiunifying a rule from P directly. s = LESSG_IN_AA evaluates to t =LESSG_IN_AA Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence The DP semiunifies directly so there is only one rewrite step from LESSG_IN_AA to LESSG_IN_AA. ---------------------------------------- (189) NO ---------------------------------------- (190) Obligation: Pi DP problem: The TRS P consists of the following rules: MERGED_IN_AAA(.(X1, X2), .(X3, X4), .(X1, X5)) -> U12_AAA(X1, X2, X3, X4, X5, lesscF_in_aa(X1, X3)) U12_AAA(X1, X2, X3, X4, X5, lesscF_out_aa(X1, X3)) -> MERGED_IN_AAA(X2, .(X3, X4), X5) MERGED_IN_AAA(.(X1, X2), .(X3, X4), .(X3, X5)) -> U15_AAA(X1, X2, X3, X4, X5, lesscG_in_aa(X3, X1)) U15_AAA(X1, X2, X3, X4, X5, lesscG_out_aa(X3, X1)) -> MERGED_IN_AAA(.(X1, X2), X4, X5) The TRS R consists of the following rules: splitcB_in_aaaa(X1, X2, .(X1, X3), X4) -> U69_aaaa(X1, X2, X3, X4, splitcA_in_aaa(X2, X4, X3)) splitcA_in_aaa([], [], []) -> splitcA_out_aaa([], [], []) splitcA_in_aaa(.(X1, X2), .(X1, X3), X4) -> U68_aaa(X1, X2, X3, X4, splitcA_in_aaa(X2, X4, X3)) U68_aaa(X1, X2, X3, X4, splitcA_out_aaa(X2, X4, X3)) -> splitcA_out_aaa(.(X1, X2), .(X1, X3), X4) U69_aaaa(X1, X2, X3, X4, splitcA_out_aaa(X2, X4, X3)) -> splitcB_out_aaaa(X1, X2, .(X1, X3), X4) mscE_in_aa([], []) -> mscE_out_aa([], []) mscE_in_aa(.(X1, []), .(X1, [])) -> mscE_out_aa(.(X1, []), .(X1, [])) mscE_in_aa(.(X1, .(X2, X3)), X4) -> U74_aa(X1, X2, X3, X4, qcC_in_aaaaaaaa(X1, X2, X3, X5, X6, X7, X8, X4)) qcC_in_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8) -> U70_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, splitcB_in_aaaa(X1, .(X2, X3), X4, X5)) U70_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, splitcB_out_aaaa(X1, .(X2, X3), X4, X5)) -> U71_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mscE_in_aa(X4, X6)) U71_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mscE_out_aa(X4, X6)) -> U72_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mscE_in_aa(X5, X7)) U72_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mscE_out_aa(X5, X7)) -> U73_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mergecD_in_aaa(X6, X7, X8)) mergecD_in_aaa([], X1, X1) -> mergecD_out_aaa([], X1, X1) mergecD_in_aaa(X1, [], X1) -> mergecD_out_aaa(X1, [], X1) mergecD_in_aaa(.(X1, X2), .(X3, X4), .(X1, X5)) -> U75_aaa(X1, X2, X3, X4, X5, lesscF_in_aa(X1, X3)) lesscF_in_aa(0, X1) -> lesscF_out_aa(0, X1) lesscF_in_aa(s(X1), X2) -> U81_aa(X1, X2, lesscG_in_aa(X1, X2)) lesscG_in_aa(0, s(X1)) -> lesscG_out_aa(0, s(X1)) lesscG_in_aa(s(X1), s(X2)) -> U79_aa(X1, X2, lesscG_in_aa(X1, X2)) U79_aa(X1, X2, lesscG_out_aa(X1, X2)) -> lesscG_out_aa(s(X1), s(X2)) U81_aa(X1, X2, lesscG_out_aa(X1, X2)) -> lesscF_out_aa(s(X1), X2) U75_aaa(X1, X2, X3, X4, X5, lesscF_out_aa(X1, X3)) -> U76_aaa(X1, X2, X3, X4, X5, mergecD_in_aaa(X2, .(X3, X4), X5)) mergecD_in_aaa(.(X1, X2), .(X3, X4), .(X3, X5)) -> U77_aaa(X1, X2, X3, X4, X5, lesscG_in_aa(X3, X1)) U77_aaa(X1, X2, X3, X4, X5, lesscG_out_aa(X3, X1)) -> U78_aaa(X1, X2, X3, X4, X5, mergecD_in_aaa(.(X1, X2), X4, X5)) U78_aaa(X1, X2, X3, X4, X5, mergecD_out_aaa(.(X1, X2), X4, X5)) -> mergecD_out_aaa(.(X1, X2), .(X3, X4), .(X3, X5)) U76_aaa(X1, X2, X3, X4, X5, mergecD_out_aaa(X2, .(X3, X4), X5)) -> mergecD_out_aaa(.(X1, X2), .(X3, X4), .(X1, X5)) U73_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mergecD_out_aaa(X6, X7, X8)) -> qcC_out_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8) U74_aa(X1, X2, X3, X4, qcC_out_aaaaaaaa(X1, X2, X3, X5, X6, X7, X8, X4)) -> mscE_out_aa(.(X1, .(X2, X3)), X4) mscH_in_aaa(X1, [], .(X1, [])) -> mscH_out_aaa(X1, [], .(X1, [])) mscH_in_aaa(X1, .(X2, X3), X4) -> U80_aaa(X1, X2, X3, X4, qcC_in_aaaaaaaa(X1, X2, X3, X5, X6, X7, X8, X4)) U80_aaa(X1, X2, X3, X4, qcC_out_aaaaaaaa(X1, X2, X3, X5, X6, X7, X8, X4)) -> mscH_out_aaa(X1, .(X2, X3), X4) lesscG_in_gg(0, s(X1)) -> lesscG_out_gg(0, s(X1)) lesscG_in_gg(s(X1), s(X2)) -> U79_gg(X1, X2, lesscG_in_gg(X1, X2)) U79_gg(X1, X2, lesscG_out_gg(X1, X2)) -> lesscG_out_gg(s(X1), s(X2)) lesscF_in_ga(0, X1) -> lesscF_out_ga(0, X1) lesscF_in_ga(s(X1), X2) -> U81_ga(X1, X2, lesscG_in_ga(X1, X2)) lesscG_in_ga(0, s(X1)) -> lesscG_out_ga(0, s(X1)) lesscG_in_ga(s(X1), s(X2)) -> U79_ga(X1, X2, lesscG_in_ga(X1, X2)) U79_ga(X1, X2, lesscG_out_ga(X1, X2)) -> lesscG_out_ga(s(X1), s(X2)) U81_ga(X1, X2, lesscG_out_ga(X1, X2)) -> lesscF_out_ga(s(X1), X2) lesscF_in_gg(0, X1) -> lesscF_out_gg(0, X1) lesscF_in_gg(s(X1), X2) -> U81_gg(X1, X2, lesscG_in_gg(X1, X2)) U81_gg(X1, X2, lesscG_out_gg(X1, X2)) -> lesscF_out_gg(s(X1), X2) The argument filtering Pi contains the following mapping: [] = [] splitcB_in_aaaa(x1, x2, x3, x4) = splitcB_in_aaaa U69_aaaa(x1, x2, x3, x4, x5) = U69_aaaa(x5) splitcA_in_aaa(x1, x2, x3) = splitcA_in_aaa splitcA_out_aaa(x1, x2, x3) = splitcA_out_aaa U68_aaa(x1, x2, x3, x4, x5) = U68_aaa(x5) splitcB_out_aaaa(x1, x2, x3, x4) = splitcB_out_aaaa mscE_in_aa(x1, x2) = mscE_in_aa mscE_out_aa(x1, x2) = mscE_out_aa U74_aa(x1, x2, x3, x4, x5) = U74_aa(x5) qcC_in_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = qcC_in_aaaaaaaa U70_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U70_aaaaaaaa(x9) U71_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U71_aaaaaaaa(x9) U72_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U72_aaaaaaaa(x9) U73_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U73_aaaaaaaa(x9) mergecD_in_aaa(x1, x2, x3) = mergecD_in_aaa mergecD_out_aaa(x1, x2, x3) = mergecD_out_aaa U75_aaa(x1, x2, x3, x4, x5, x6) = U75_aaa(x6) lesscF_in_aa(x1, x2) = lesscF_in_aa lesscF_out_aa(x1, x2) = lesscF_out_aa(x1) U81_aa(x1, x2, x3) = U81_aa(x3) lesscG_in_aa(x1, x2) = lesscG_in_aa lesscG_out_aa(x1, x2) = lesscG_out_aa(x1) U79_aa(x1, x2, x3) = U79_aa(x3) U76_aaa(x1, x2, x3, x4, x5, x6) = U76_aaa(x6) U77_aaa(x1, x2, x3, x4, x5, x6) = U77_aaa(x6) U78_aaa(x1, x2, x3, x4, x5, x6) = U78_aaa(x6) qcC_out_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = qcC_out_aaaaaaaa mscH_in_aaa(x1, x2, x3) = mscH_in_aaa mscH_out_aaa(x1, x2, x3) = mscH_out_aaa U80_aaa(x1, x2, x3, x4, x5) = U80_aaa(x5) .(x1, x2) = .(x1, x2) s(x1) = s(x1) lesscG_in_gg(x1, x2) = lesscG_in_gg(x1, x2) 0 = 0 lesscG_out_gg(x1, x2) = lesscG_out_gg(x1, x2) U79_gg(x1, x2, x3) = U79_gg(x1, x2, x3) lesscF_in_ga(x1, x2) = lesscF_in_ga(x1) lesscF_out_ga(x1, x2) = lesscF_out_ga(x1) U81_ga(x1, x2, x3) = U81_ga(x1, x3) lesscG_in_ga(x1, x2) = lesscG_in_ga(x1) lesscG_out_ga(x1, x2) = lesscG_out_ga(x1) U79_ga(x1, x2, x3) = U79_ga(x1, x3) lesscF_in_gg(x1, x2) = lesscF_in_gg(x1, x2) lesscF_out_gg(x1, x2) = lesscF_out_gg(x1, x2) U81_gg(x1, x2, x3) = U81_gg(x1, x2, x3) MERGED_IN_AAA(x1, x2, x3) = MERGED_IN_AAA U12_AAA(x1, x2, x3, x4, x5, x6) = U12_AAA(x6) U15_AAA(x1, x2, x3, x4, x5, x6) = U15_AAA(x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (191) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (192) Obligation: Pi DP problem: The TRS P consists of the following rules: MERGED_IN_AAA(.(X1, X2), .(X3, X4), .(X1, X5)) -> U12_AAA(X1, X2, X3, X4, X5, lesscF_in_aa(X1, X3)) U12_AAA(X1, X2, X3, X4, X5, lesscF_out_aa(X1, X3)) -> MERGED_IN_AAA(X2, .(X3, X4), X5) MERGED_IN_AAA(.(X1, X2), .(X3, X4), .(X3, X5)) -> U15_AAA(X1, X2, X3, X4, X5, lesscG_in_aa(X3, X1)) U15_AAA(X1, X2, X3, X4, X5, lesscG_out_aa(X3, X1)) -> MERGED_IN_AAA(.(X1, X2), X4, X5) The TRS R consists of the following rules: lesscF_in_aa(0, X1) -> lesscF_out_aa(0, X1) lesscF_in_aa(s(X1), X2) -> U81_aa(X1, X2, lesscG_in_aa(X1, X2)) lesscG_in_aa(0, s(X1)) -> lesscG_out_aa(0, s(X1)) lesscG_in_aa(s(X1), s(X2)) -> U79_aa(X1, X2, lesscG_in_aa(X1, X2)) U81_aa(X1, X2, lesscG_out_aa(X1, X2)) -> lesscF_out_aa(s(X1), X2) U79_aa(X1, X2, lesscG_out_aa(X1, X2)) -> lesscG_out_aa(s(X1), s(X2)) The argument filtering Pi contains the following mapping: lesscF_in_aa(x1, x2) = lesscF_in_aa lesscF_out_aa(x1, x2) = lesscF_out_aa(x1) U81_aa(x1, x2, x3) = U81_aa(x3) lesscG_in_aa(x1, x2) = lesscG_in_aa lesscG_out_aa(x1, x2) = lesscG_out_aa(x1) U79_aa(x1, x2, x3) = U79_aa(x3) .(x1, x2) = .(x1, x2) s(x1) = s(x1) 0 = 0 MERGED_IN_AAA(x1, x2, x3) = MERGED_IN_AAA U12_AAA(x1, x2, x3, x4, x5, x6) = U12_AAA(x6) U15_AAA(x1, x2, x3, x4, x5, x6) = U15_AAA(x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (193) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (194) Obligation: Q DP problem: The TRS P consists of the following rules: MERGED_IN_AAA -> U12_AAA(lesscF_in_aa) U12_AAA(lesscF_out_aa(X1)) -> MERGED_IN_AAA MERGED_IN_AAA -> U15_AAA(lesscG_in_aa) U15_AAA(lesscG_out_aa(X3)) -> MERGED_IN_AAA The TRS R consists of the following rules: lesscF_in_aa -> lesscF_out_aa(0) lesscF_in_aa -> U81_aa(lesscG_in_aa) lesscG_in_aa -> lesscG_out_aa(0) lesscG_in_aa -> U79_aa(lesscG_in_aa) U81_aa(lesscG_out_aa(X1)) -> lesscF_out_aa(s(X1)) U79_aa(lesscG_out_aa(X1)) -> lesscG_out_aa(s(X1)) The set Q consists of the following terms: lesscF_in_aa lesscG_in_aa U81_aa(x0) U79_aa(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (195) Obligation: Pi DP problem: The TRS P consists of the following rules: SPLITA_IN_AAA(.(X1, X2), .(X1, X3), X4) -> SPLITA_IN_AAA(X2, X4, X3) The TRS R consists of the following rules: splitcB_in_aaaa(X1, X2, .(X1, X3), X4) -> U69_aaaa(X1, X2, X3, X4, splitcA_in_aaa(X2, X4, X3)) splitcA_in_aaa([], [], []) -> splitcA_out_aaa([], [], []) splitcA_in_aaa(.(X1, X2), .(X1, X3), X4) -> U68_aaa(X1, X2, X3, X4, splitcA_in_aaa(X2, X4, X3)) U68_aaa(X1, X2, X3, X4, splitcA_out_aaa(X2, X4, X3)) -> splitcA_out_aaa(.(X1, X2), .(X1, X3), X4) U69_aaaa(X1, X2, X3, X4, splitcA_out_aaa(X2, X4, X3)) -> splitcB_out_aaaa(X1, X2, .(X1, X3), X4) mscE_in_aa([], []) -> mscE_out_aa([], []) mscE_in_aa(.(X1, []), .(X1, [])) -> mscE_out_aa(.(X1, []), .(X1, [])) mscE_in_aa(.(X1, .(X2, X3)), X4) -> U74_aa(X1, X2, X3, X4, qcC_in_aaaaaaaa(X1, X2, X3, X5, X6, X7, X8, X4)) qcC_in_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8) -> U70_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, splitcB_in_aaaa(X1, .(X2, X3), X4, X5)) U70_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, splitcB_out_aaaa(X1, .(X2, X3), X4, X5)) -> U71_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mscE_in_aa(X4, X6)) U71_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mscE_out_aa(X4, X6)) -> U72_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mscE_in_aa(X5, X7)) U72_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mscE_out_aa(X5, X7)) -> U73_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mergecD_in_aaa(X6, X7, X8)) mergecD_in_aaa([], X1, X1) -> mergecD_out_aaa([], X1, X1) mergecD_in_aaa(X1, [], X1) -> mergecD_out_aaa(X1, [], X1) mergecD_in_aaa(.(X1, X2), .(X3, X4), .(X1, X5)) -> U75_aaa(X1, X2, X3, X4, X5, lesscF_in_aa(X1, X3)) lesscF_in_aa(0, X1) -> lesscF_out_aa(0, X1) lesscF_in_aa(s(X1), X2) -> U81_aa(X1, X2, lesscG_in_aa(X1, X2)) lesscG_in_aa(0, s(X1)) -> lesscG_out_aa(0, s(X1)) lesscG_in_aa(s(X1), s(X2)) -> U79_aa(X1, X2, lesscG_in_aa(X1, X2)) U79_aa(X1, X2, lesscG_out_aa(X1, X2)) -> lesscG_out_aa(s(X1), s(X2)) U81_aa(X1, X2, lesscG_out_aa(X1, X2)) -> lesscF_out_aa(s(X1), X2) U75_aaa(X1, X2, X3, X4, X5, lesscF_out_aa(X1, X3)) -> U76_aaa(X1, X2, X3, X4, X5, mergecD_in_aaa(X2, .(X3, X4), X5)) mergecD_in_aaa(.(X1, X2), .(X3, X4), .(X3, X5)) -> U77_aaa(X1, X2, X3, X4, X5, lesscG_in_aa(X3, X1)) U77_aaa(X1, X2, X3, X4, X5, lesscG_out_aa(X3, X1)) -> U78_aaa(X1, X2, X3, X4, X5, mergecD_in_aaa(.(X1, X2), X4, X5)) U78_aaa(X1, X2, X3, X4, X5, mergecD_out_aaa(.(X1, X2), X4, X5)) -> mergecD_out_aaa(.(X1, X2), .(X3, X4), .(X3, X5)) U76_aaa(X1, X2, X3, X4, X5, mergecD_out_aaa(X2, .(X3, X4), X5)) -> mergecD_out_aaa(.(X1, X2), .(X3, X4), .(X1, X5)) U73_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mergecD_out_aaa(X6, X7, X8)) -> qcC_out_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8) U74_aa(X1, X2, X3, X4, qcC_out_aaaaaaaa(X1, X2, X3, X5, X6, X7, X8, X4)) -> mscE_out_aa(.(X1, .(X2, X3)), X4) mscH_in_aaa(X1, [], .(X1, [])) -> mscH_out_aaa(X1, [], .(X1, [])) mscH_in_aaa(X1, .(X2, X3), X4) -> U80_aaa(X1, X2, X3, X4, qcC_in_aaaaaaaa(X1, X2, X3, X5, X6, X7, X8, X4)) U80_aaa(X1, X2, X3, X4, qcC_out_aaaaaaaa(X1, X2, X3, X5, X6, X7, X8, X4)) -> mscH_out_aaa(X1, .(X2, X3), X4) lesscG_in_gg(0, s(X1)) -> lesscG_out_gg(0, s(X1)) lesscG_in_gg(s(X1), s(X2)) -> U79_gg(X1, X2, lesscG_in_gg(X1, X2)) U79_gg(X1, X2, lesscG_out_gg(X1, X2)) -> lesscG_out_gg(s(X1), s(X2)) lesscF_in_ga(0, X1) -> lesscF_out_ga(0, X1) lesscF_in_ga(s(X1), X2) -> U81_ga(X1, X2, lesscG_in_ga(X1, X2)) lesscG_in_ga(0, s(X1)) -> lesscG_out_ga(0, s(X1)) lesscG_in_ga(s(X1), s(X2)) -> U79_ga(X1, X2, lesscG_in_ga(X1, X2)) U79_ga(X1, X2, lesscG_out_ga(X1, X2)) -> lesscG_out_ga(s(X1), s(X2)) U81_ga(X1, X2, lesscG_out_ga(X1, X2)) -> lesscF_out_ga(s(X1), X2) lesscF_in_gg(0, X1) -> lesscF_out_gg(0, X1) lesscF_in_gg(s(X1), X2) -> U81_gg(X1, X2, lesscG_in_gg(X1, X2)) U81_gg(X1, X2, lesscG_out_gg(X1, X2)) -> lesscF_out_gg(s(X1), X2) The argument filtering Pi contains the following mapping: [] = [] splitcB_in_aaaa(x1, x2, x3, x4) = splitcB_in_aaaa U69_aaaa(x1, x2, x3, x4, x5) = U69_aaaa(x5) splitcA_in_aaa(x1, x2, x3) = splitcA_in_aaa splitcA_out_aaa(x1, x2, x3) = splitcA_out_aaa U68_aaa(x1, x2, x3, x4, x5) = U68_aaa(x5) splitcB_out_aaaa(x1, x2, x3, x4) = splitcB_out_aaaa mscE_in_aa(x1, x2) = mscE_in_aa mscE_out_aa(x1, x2) = mscE_out_aa U74_aa(x1, x2, x3, x4, x5) = U74_aa(x5) qcC_in_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = qcC_in_aaaaaaaa U70_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U70_aaaaaaaa(x9) U71_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U71_aaaaaaaa(x9) U72_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U72_aaaaaaaa(x9) U73_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U73_aaaaaaaa(x9) mergecD_in_aaa(x1, x2, x3) = mergecD_in_aaa mergecD_out_aaa(x1, x2, x3) = mergecD_out_aaa U75_aaa(x1, x2, x3, x4, x5, x6) = U75_aaa(x6) lesscF_in_aa(x1, x2) = lesscF_in_aa lesscF_out_aa(x1, x2) = lesscF_out_aa(x1) U81_aa(x1, x2, x3) = U81_aa(x3) lesscG_in_aa(x1, x2) = lesscG_in_aa lesscG_out_aa(x1, x2) = lesscG_out_aa(x1) U79_aa(x1, x2, x3) = U79_aa(x3) U76_aaa(x1, x2, x3, x4, x5, x6) = U76_aaa(x6) U77_aaa(x1, x2, x3, x4, x5, x6) = U77_aaa(x6) U78_aaa(x1, x2, x3, x4, x5, x6) = U78_aaa(x6) qcC_out_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = qcC_out_aaaaaaaa mscH_in_aaa(x1, x2, x3) = mscH_in_aaa mscH_out_aaa(x1, x2, x3) = mscH_out_aaa U80_aaa(x1, x2, x3, x4, x5) = U80_aaa(x5) .(x1, x2) = .(x1, x2) s(x1) = s(x1) lesscG_in_gg(x1, x2) = lesscG_in_gg(x1, x2) 0 = 0 lesscG_out_gg(x1, x2) = lesscG_out_gg(x1, x2) U79_gg(x1, x2, x3) = U79_gg(x1, x2, x3) lesscF_in_ga(x1, x2) = lesscF_in_ga(x1) lesscF_out_ga(x1, x2) = lesscF_out_ga(x1) U81_ga(x1, x2, x3) = U81_ga(x1, x3) lesscG_in_ga(x1, x2) = lesscG_in_ga(x1) lesscG_out_ga(x1, x2) = lesscG_out_ga(x1) U79_ga(x1, x2, x3) = U79_ga(x1, x3) lesscF_in_gg(x1, x2) = lesscF_in_gg(x1, x2) lesscF_out_gg(x1, x2) = lesscF_out_gg(x1, x2) U81_gg(x1, x2, x3) = U81_gg(x1, x2, x3) SPLITA_IN_AAA(x1, x2, x3) = SPLITA_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (196) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (197) Obligation: Pi DP problem: The TRS P consists of the following rules: SPLITA_IN_AAA(.(X1, X2), .(X1, X3), X4) -> SPLITA_IN_AAA(X2, X4, X3) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) SPLITA_IN_AAA(x1, x2, x3) = SPLITA_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (198) Obligation: Pi DP problem: The TRS P consists of the following rules: PC_IN_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8) -> U4_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, splitcB_in_aaaa(X1, .(X2, X3), X4, X5)) U4_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, splitcB_out_aaaa(X1, .(X2, X3), X4, X5)) -> MSE_IN_AA(X4, X6) MSE_IN_AA(.(X1, .(X2, X3)), X4) -> PC_IN_AAAAAAAA(X1, X2, X3, X5, X6, X7, X8, X4) U4_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, splitcB_out_aaaa(X1, .(X2, X3), X4, X5)) -> U6_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, mscE_in_aa(X4, X6)) U6_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, mscE_out_aa(X4, X6)) -> MSE_IN_AA(X5, X7) The TRS R consists of the following rules: splitcB_in_aaaa(X1, X2, .(X1, X3), X4) -> U69_aaaa(X1, X2, X3, X4, splitcA_in_aaa(X2, X4, X3)) splitcA_in_aaa([], [], []) -> splitcA_out_aaa([], [], []) splitcA_in_aaa(.(X1, X2), .(X1, X3), X4) -> U68_aaa(X1, X2, X3, X4, splitcA_in_aaa(X2, X4, X3)) U68_aaa(X1, X2, X3, X4, splitcA_out_aaa(X2, X4, X3)) -> splitcA_out_aaa(.(X1, X2), .(X1, X3), X4) U69_aaaa(X1, X2, X3, X4, splitcA_out_aaa(X2, X4, X3)) -> splitcB_out_aaaa(X1, X2, .(X1, X3), X4) mscE_in_aa([], []) -> mscE_out_aa([], []) mscE_in_aa(.(X1, []), .(X1, [])) -> mscE_out_aa(.(X1, []), .(X1, [])) mscE_in_aa(.(X1, .(X2, X3)), X4) -> U74_aa(X1, X2, X3, X4, qcC_in_aaaaaaaa(X1, X2, X3, X5, X6, X7, X8, X4)) qcC_in_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8) -> U70_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, splitcB_in_aaaa(X1, .(X2, X3), X4, X5)) U70_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, splitcB_out_aaaa(X1, .(X2, X3), X4, X5)) -> U71_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mscE_in_aa(X4, X6)) U71_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mscE_out_aa(X4, X6)) -> U72_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mscE_in_aa(X5, X7)) U72_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mscE_out_aa(X5, X7)) -> U73_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mergecD_in_aaa(X6, X7, X8)) mergecD_in_aaa([], X1, X1) -> mergecD_out_aaa([], X1, X1) mergecD_in_aaa(X1, [], X1) -> mergecD_out_aaa(X1, [], X1) mergecD_in_aaa(.(X1, X2), .(X3, X4), .(X1, X5)) -> U75_aaa(X1, X2, X3, X4, X5, lesscF_in_aa(X1, X3)) lesscF_in_aa(0, X1) -> lesscF_out_aa(0, X1) lesscF_in_aa(s(X1), X2) -> U81_aa(X1, X2, lesscG_in_aa(X1, X2)) lesscG_in_aa(0, s(X1)) -> lesscG_out_aa(0, s(X1)) lesscG_in_aa(s(X1), s(X2)) -> U79_aa(X1, X2, lesscG_in_aa(X1, X2)) U79_aa(X1, X2, lesscG_out_aa(X1, X2)) -> lesscG_out_aa(s(X1), s(X2)) U81_aa(X1, X2, lesscG_out_aa(X1, X2)) -> lesscF_out_aa(s(X1), X2) U75_aaa(X1, X2, X3, X4, X5, lesscF_out_aa(X1, X3)) -> U76_aaa(X1, X2, X3, X4, X5, mergecD_in_aaa(X2, .(X3, X4), X5)) mergecD_in_aaa(.(X1, X2), .(X3, X4), .(X3, X5)) -> U77_aaa(X1, X2, X3, X4, X5, lesscG_in_aa(X3, X1)) U77_aaa(X1, X2, X3, X4, X5, lesscG_out_aa(X3, X1)) -> U78_aaa(X1, X2, X3, X4, X5, mergecD_in_aaa(.(X1, X2), X4, X5)) U78_aaa(X1, X2, X3, X4, X5, mergecD_out_aaa(.(X1, X2), X4, X5)) -> mergecD_out_aaa(.(X1, X2), .(X3, X4), .(X3, X5)) U76_aaa(X1, X2, X3, X4, X5, mergecD_out_aaa(X2, .(X3, X4), X5)) -> mergecD_out_aaa(.(X1, X2), .(X3, X4), .(X1, X5)) U73_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mergecD_out_aaa(X6, X7, X8)) -> qcC_out_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8) U74_aa(X1, X2, X3, X4, qcC_out_aaaaaaaa(X1, X2, X3, X5, X6, X7, X8, X4)) -> mscE_out_aa(.(X1, .(X2, X3)), X4) mscH_in_aaa(X1, [], .(X1, [])) -> mscH_out_aaa(X1, [], .(X1, [])) mscH_in_aaa(X1, .(X2, X3), X4) -> U80_aaa(X1, X2, X3, X4, qcC_in_aaaaaaaa(X1, X2, X3, X5, X6, X7, X8, X4)) U80_aaa(X1, X2, X3, X4, qcC_out_aaaaaaaa(X1, X2, X3, X5, X6, X7, X8, X4)) -> mscH_out_aaa(X1, .(X2, X3), X4) lesscG_in_gg(0, s(X1)) -> lesscG_out_gg(0, s(X1)) lesscG_in_gg(s(X1), s(X2)) -> U79_gg(X1, X2, lesscG_in_gg(X1, X2)) U79_gg(X1, X2, lesscG_out_gg(X1, X2)) -> lesscG_out_gg(s(X1), s(X2)) lesscF_in_ga(0, X1) -> lesscF_out_ga(0, X1) lesscF_in_ga(s(X1), X2) -> U81_ga(X1, X2, lesscG_in_ga(X1, X2)) lesscG_in_ga(0, s(X1)) -> lesscG_out_ga(0, s(X1)) lesscG_in_ga(s(X1), s(X2)) -> U79_ga(X1, X2, lesscG_in_ga(X1, X2)) U79_ga(X1, X2, lesscG_out_ga(X1, X2)) -> lesscG_out_ga(s(X1), s(X2)) U81_ga(X1, X2, lesscG_out_ga(X1, X2)) -> lesscF_out_ga(s(X1), X2) lesscF_in_gg(0, X1) -> lesscF_out_gg(0, X1) lesscF_in_gg(s(X1), X2) -> U81_gg(X1, X2, lesscG_in_gg(X1, X2)) U81_gg(X1, X2, lesscG_out_gg(X1, X2)) -> lesscF_out_gg(s(X1), X2) The argument filtering Pi contains the following mapping: [] = [] splitcB_in_aaaa(x1, x2, x3, x4) = splitcB_in_aaaa U69_aaaa(x1, x2, x3, x4, x5) = U69_aaaa(x5) splitcA_in_aaa(x1, x2, x3) = splitcA_in_aaa splitcA_out_aaa(x1, x2, x3) = splitcA_out_aaa U68_aaa(x1, x2, x3, x4, x5) = U68_aaa(x5) splitcB_out_aaaa(x1, x2, x3, x4) = splitcB_out_aaaa mscE_in_aa(x1, x2) = mscE_in_aa mscE_out_aa(x1, x2) = mscE_out_aa U74_aa(x1, x2, x3, x4, x5) = U74_aa(x5) qcC_in_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = qcC_in_aaaaaaaa U70_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U70_aaaaaaaa(x9) U71_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U71_aaaaaaaa(x9) U72_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U72_aaaaaaaa(x9) U73_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U73_aaaaaaaa(x9) mergecD_in_aaa(x1, x2, x3) = mergecD_in_aaa mergecD_out_aaa(x1, x2, x3) = mergecD_out_aaa U75_aaa(x1, x2, x3, x4, x5, x6) = U75_aaa(x6) lesscF_in_aa(x1, x2) = lesscF_in_aa lesscF_out_aa(x1, x2) = lesscF_out_aa(x1) U81_aa(x1, x2, x3) = U81_aa(x3) lesscG_in_aa(x1, x2) = lesscG_in_aa lesscG_out_aa(x1, x2) = lesscG_out_aa(x1) U79_aa(x1, x2, x3) = U79_aa(x3) U76_aaa(x1, x2, x3, x4, x5, x6) = U76_aaa(x6) U77_aaa(x1, x2, x3, x4, x5, x6) = U77_aaa(x6) U78_aaa(x1, x2, x3, x4, x5, x6) = U78_aaa(x6) qcC_out_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = qcC_out_aaaaaaaa mscH_in_aaa(x1, x2, x3) = mscH_in_aaa mscH_out_aaa(x1, x2, x3) = mscH_out_aaa U80_aaa(x1, x2, x3, x4, x5) = U80_aaa(x5) .(x1, x2) = .(x1, x2) s(x1) = s(x1) lesscG_in_gg(x1, x2) = lesscG_in_gg(x1, x2) 0 = 0 lesscG_out_gg(x1, x2) = lesscG_out_gg(x1, x2) U79_gg(x1, x2, x3) = U79_gg(x1, x2, x3) lesscF_in_ga(x1, x2) = lesscF_in_ga(x1) lesscF_out_ga(x1, x2) = lesscF_out_ga(x1) U81_ga(x1, x2, x3) = U81_ga(x1, x3) lesscG_in_ga(x1, x2) = lesscG_in_ga(x1) lesscG_out_ga(x1, x2) = lesscG_out_ga(x1) U79_ga(x1, x2, x3) = U79_ga(x1, x3) lesscF_in_gg(x1, x2) = lesscF_in_gg(x1, x2) lesscF_out_gg(x1, x2) = lesscF_out_gg(x1, x2) U81_gg(x1, x2, x3) = U81_gg(x1, x2, x3) PC_IN_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = PC_IN_AAAAAAAA U4_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U4_AAAAAAAA(x9) MSE_IN_AA(x1, x2) = MSE_IN_AA U6_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_AAAAAAAA(x9) We have to consider all (P,R,Pi)-chains ---------------------------------------- (199) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (200) Obligation: Pi DP problem: The TRS P consists of the following rules: PC_IN_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8) -> U4_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, splitcB_in_aaaa(X1, .(X2, X3), X4, X5)) U4_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, splitcB_out_aaaa(X1, .(X2, X3), X4, X5)) -> MSE_IN_AA(X4, X6) MSE_IN_AA(.(X1, .(X2, X3)), X4) -> PC_IN_AAAAAAAA(X1, X2, X3, X5, X6, X7, X8, X4) U4_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, splitcB_out_aaaa(X1, .(X2, X3), X4, X5)) -> U6_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, mscE_in_aa(X4, X6)) U6_AAAAAAAA(X1, X2, X3, X4, X5, X6, X7, X8, mscE_out_aa(X4, X6)) -> MSE_IN_AA(X5, X7) The TRS R consists of the following rules: splitcB_in_aaaa(X1, X2, .(X1, X3), X4) -> U69_aaaa(X1, X2, X3, X4, splitcA_in_aaa(X2, X4, X3)) mscE_in_aa([], []) -> mscE_out_aa([], []) mscE_in_aa(.(X1, []), .(X1, [])) -> mscE_out_aa(.(X1, []), .(X1, [])) mscE_in_aa(.(X1, .(X2, X3)), X4) -> U74_aa(X1, X2, X3, X4, qcC_in_aaaaaaaa(X1, X2, X3, X5, X6, X7, X8, X4)) U69_aaaa(X1, X2, X3, X4, splitcA_out_aaa(X2, X4, X3)) -> splitcB_out_aaaa(X1, X2, .(X1, X3), X4) U74_aa(X1, X2, X3, X4, qcC_out_aaaaaaaa(X1, X2, X3, X5, X6, X7, X8, X4)) -> mscE_out_aa(.(X1, .(X2, X3)), X4) splitcA_in_aaa([], [], []) -> splitcA_out_aaa([], [], []) splitcA_in_aaa(.(X1, X2), .(X1, X3), X4) -> U68_aaa(X1, X2, X3, X4, splitcA_in_aaa(X2, X4, X3)) qcC_in_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8) -> U70_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, splitcB_in_aaaa(X1, .(X2, X3), X4, X5)) U68_aaa(X1, X2, X3, X4, splitcA_out_aaa(X2, X4, X3)) -> splitcA_out_aaa(.(X1, X2), .(X1, X3), X4) U70_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, splitcB_out_aaaa(X1, .(X2, X3), X4, X5)) -> U71_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mscE_in_aa(X4, X6)) U71_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mscE_out_aa(X4, X6)) -> U72_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mscE_in_aa(X5, X7)) U72_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mscE_out_aa(X5, X7)) -> U73_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mergecD_in_aaa(X6, X7, X8)) U73_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8, mergecD_out_aaa(X6, X7, X8)) -> qcC_out_aaaaaaaa(X1, X2, X3, X4, X5, X6, X7, X8) mergecD_in_aaa([], X1, X1) -> mergecD_out_aaa([], X1, X1) mergecD_in_aaa(X1, [], X1) -> mergecD_out_aaa(X1, [], X1) mergecD_in_aaa(.(X1, X2), .(X3, X4), .(X1, X5)) -> U75_aaa(X1, X2, X3, X4, X5, lesscF_in_aa(X1, X3)) mergecD_in_aaa(.(X1, X2), .(X3, X4), .(X3, X5)) -> U77_aaa(X1, X2, X3, X4, X5, lesscG_in_aa(X3, X1)) U75_aaa(X1, X2, X3, X4, X5, lesscF_out_aa(X1, X3)) -> U76_aaa(X1, X2, X3, X4, X5, mergecD_in_aaa(X2, .(X3, X4), X5)) U77_aaa(X1, X2, X3, X4, X5, lesscG_out_aa(X3, X1)) -> U78_aaa(X1, X2, X3, X4, X5, mergecD_in_aaa(.(X1, X2), X4, X5)) lesscF_in_aa(0, X1) -> lesscF_out_aa(0, X1) lesscF_in_aa(s(X1), X2) -> U81_aa(X1, X2, lesscG_in_aa(X1, X2)) U76_aaa(X1, X2, X3, X4, X5, mergecD_out_aaa(X2, .(X3, X4), X5)) -> mergecD_out_aaa(.(X1, X2), .(X3, X4), .(X1, X5)) lesscG_in_aa(0, s(X1)) -> lesscG_out_aa(0, s(X1)) lesscG_in_aa(s(X1), s(X2)) -> U79_aa(X1, X2, lesscG_in_aa(X1, X2)) U78_aaa(X1, X2, X3, X4, X5, mergecD_out_aaa(.(X1, X2), X4, X5)) -> mergecD_out_aaa(.(X1, X2), .(X3, X4), .(X3, X5)) U81_aa(X1, X2, lesscG_out_aa(X1, X2)) -> lesscF_out_aa(s(X1), X2) U79_aa(X1, X2, lesscG_out_aa(X1, X2)) -> lesscG_out_aa(s(X1), s(X2)) The argument filtering Pi contains the following mapping: [] = [] splitcB_in_aaaa(x1, x2, x3, x4) = splitcB_in_aaaa U69_aaaa(x1, x2, x3, x4, x5) = U69_aaaa(x5) splitcA_in_aaa(x1, x2, x3) = splitcA_in_aaa splitcA_out_aaa(x1, x2, x3) = splitcA_out_aaa U68_aaa(x1, x2, x3, x4, x5) = U68_aaa(x5) splitcB_out_aaaa(x1, x2, x3, x4) = splitcB_out_aaaa mscE_in_aa(x1, x2) = mscE_in_aa mscE_out_aa(x1, x2) = mscE_out_aa U74_aa(x1, x2, x3, x4, x5) = U74_aa(x5) qcC_in_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = qcC_in_aaaaaaaa U70_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U70_aaaaaaaa(x9) U71_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U71_aaaaaaaa(x9) U72_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U72_aaaaaaaa(x9) U73_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U73_aaaaaaaa(x9) mergecD_in_aaa(x1, x2, x3) = mergecD_in_aaa mergecD_out_aaa(x1, x2, x3) = mergecD_out_aaa U75_aaa(x1, x2, x3, x4, x5, x6) = U75_aaa(x6) lesscF_in_aa(x1, x2) = lesscF_in_aa lesscF_out_aa(x1, x2) = lesscF_out_aa(x1) U81_aa(x1, x2, x3) = U81_aa(x3) lesscG_in_aa(x1, x2) = lesscG_in_aa lesscG_out_aa(x1, x2) = lesscG_out_aa(x1) U79_aa(x1, x2, x3) = U79_aa(x3) U76_aaa(x1, x2, x3, x4, x5, x6) = U76_aaa(x6) U77_aaa(x1, x2, x3, x4, x5, x6) = U77_aaa(x6) U78_aaa(x1, x2, x3, x4, x5, x6) = U78_aaa(x6) qcC_out_aaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = qcC_out_aaaaaaaa .(x1, x2) = .(x1, x2) s(x1) = s(x1) 0 = 0 PC_IN_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = PC_IN_AAAAAAAA U4_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U4_AAAAAAAA(x9) MSE_IN_AA(x1, x2) = MSE_IN_AA U6_AAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U6_AAAAAAAA(x9) We have to consider all (P,R,Pi)-chains