/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.pl /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- MAYBE proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern ms(a,g) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) PrologToPiTRSProof [SOUND, 40 ms] (2) PiTRS (3) DependencyPairsProof [EQUIVALENT, 37 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, 21 ms] (40) PiTRS (41) DependencyPairsProof [EQUIVALENT, 15 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) PiDP (72) UsableRulesProof [EQUIVALENT, 0 ms] (73) PiDP (74) PiDP (75) UsableRulesProof [EQUIVALENT, 1 ms] (76) PiDP (77) PrologToTRSTransformerProof [SOUND, 29 ms] (78) QTRS (79) DependencyPairsProof [EQUIVALENT, 0 ms] (80) QDP (81) DependencyGraphProof [EQUIVALENT, 2 ms] (82) AND (83) QDP (84) UsableRulesProof [EQUIVALENT, 1 ms] (85) QDP (86) QDPSizeChangeProof [EQUIVALENT, 0 ms] (87) YES (88) QDP (89) QDPSizeChangeProof [EQUIVALENT, 0 ms] (90) YES (91) QDP (92) UsableRulesProof [EQUIVALENT, 0 ms] (93) QDP (94) NonTerminationLoopProof [COMPLETE, 0 ms] (95) NO (96) QDP (97) UsableRulesProof [EQUIVALENT, 1 ms] (98) QDP (99) QDP (100) UsableRulesProof [EQUIVALENT, 1 ms] (101) QDP (102) NonTerminationLoopProof [COMPLETE, 0 ms] (103) NO (104) QDP (105) UsableRulesProof [EQUIVALENT, 2 ms] (106) QDP (107) PrologToIRSwTTransformerProof [SOUND, 89 ms] (108) AND (109) IRSwT (110) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (111) TRUE (112) IRSwT (113) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (114) TRUE (115) IRSwT (116) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (117) TRUE (118) IRSwT (119) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (120) TRUE (121) IRSwT (122) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (123) IRSwT (124) IntTRSCompressionProof [EQUIVALENT, 19 ms] (125) IRSwT (126) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (127) IRSwT (128) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms] (129) IRSwT (130) FilterProof [EQUIVALENT, 1 ms] (131) IntTRS (132) IntTRSPeriodicNontermProof [COMPLETE, 0 ms] (133) NO (134) IRSwT (135) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (136) IRSwT (137) IntTRSCompressionProof [EQUIVALENT, 11 ms] (138) IRSwT (139) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (140) IRSwT (141) IRSwTTerminationDigraphProof [EQUIVALENT, 10 ms] (142) IRSwT (143) FilterProof [EQUIVALENT, 0 ms] (144) IntTRS (145) IntTRSPeriodicNontermProof [COMPLETE, 9 ms] (146) NO (147) PrologToDTProblemTransformerProof [SOUND, 106 ms] (148) TRIPLES (149) UndefinedPredicateInTriplesTransformerProof [SOUND, 0 ms] (150) TRIPLES (151) TriplesToPiDPProof [SOUND, 114 ms] (152) PiDP (153) DependencyGraphProof [EQUIVALENT, 0 ms] (154) AND (155) PiDP (156) UsableRulesProof [EQUIVALENT, 0 ms] (157) PiDP (158) PiDPToQDPProof [SOUND, 0 ms] (159) QDP (160) QDPSizeChangeProof [EQUIVALENT, 0 ms] (161) YES (162) PiDP (163) UsableRulesProof [EQUIVALENT, 0 ms] (164) PiDP (165) PiDPToQDPProof [SOUND, 0 ms] (166) QDP (167) QDPSizeChangeProof [EQUIVALENT, 0 ms] (168) YES (169) PiDP (170) UsableRulesProof [EQUIVALENT, 0 ms] (171) PiDP (172) PiDPToQDPProof [SOUND, 0 ms] (173) QDP (174) NonTerminationLoopProof [COMPLETE, 0 ms] (175) NO (176) PiDP (177) UsableRulesProof [EQUIVALENT, 0 ms] (178) PiDP (179) PiDPToQDPProof [SOUND, 0 ms] (180) QDP (181) PiDP (182) UsableRulesProof [EQUIVALENT, 0 ms] (183) PiDP (184) PiDP (185) UsableRulesProof [EQUIVALENT, 0 ms] (186) 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 .(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 ---------------------------------------- (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 .(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) ---------------------------------------- (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 .(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 ---------------------------------------- (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 .(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 ---------------------------------------- (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 .(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 ---------------------------------------- (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 .(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 ---------------------------------------- (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 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 ---------------------------------------- (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) -> 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. ---------------------------------------- (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) -> 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 ---------------------------------------- (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 .(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 ---------------------------------------- (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 .(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 ---------------------------------------- (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 .(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 ---------------------------------------- (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 .(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 ---------------------------------------- (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(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 ---------------------------------------- (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(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) ---------------------------------------- (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(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 ---------------------------------------- (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(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 ---------------------------------------- (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(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 ---------------------------------------- (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(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 ---------------------------------------- (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(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 ---------------------------------------- (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(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. ---------------------------------------- (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(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 ---------------------------------------- (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(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 ---------------------------------------- (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(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 ---------------------------------------- (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) 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 ---------------------------------------- (72) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (73) Obligation: Pi DP problem: The TRS P consists of the following rules: 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 ---------------------------------------- (74) 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 ---------------------------------------- (75) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (76) Obligation: Pi DP problem: The TRS P consists of the following rules: 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 ---------------------------------------- (77) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "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": { "907": { "goal": [{ "clause": 10, "scope": 10, "term": "(less T212 (s T217))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T212"], "free": [], "exprvars": [] } }, "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": [] } }, "type": "Nodes", "599": { "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": [] } }, "875": { "goal": [{ "clause": 7, "scope": 9, "term": "(merge T179 T178 T19)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T19"], "free": [], "exprvars": [] } }, "359": { "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": [] } }, "876": { "goal": [{ "clause": 8, "scope": 9, "term": "(merge T179 T178 T19)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T19"], "free": [], "exprvars": [] } }, "910": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "916": { "goal": [{ "clause": -1, "scope": -1, "term": "(less T236 T238)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T236"], "free": [], "exprvars": [] } }, "917": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "12": { "goal": [{ "clause": 0, "scope": 1, "term": "(ms T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "13": { "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": [] } }, "15": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "17": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "360": { "goal": [{ "clause": 4, "scope": 3, "term": "(split (. T36 T37) X44 X43)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X43", "X44" ], "exprvars": [] } }, "361": { "goal": [{ "clause": -1, "scope": -1, "term": "(split T45 X62 X61)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X61", "X62" ], "exprvars": [] } }, "363": { "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": [] } }, "2": { "goal": [{ "clause": -1, "scope": -1, "term": "(ms T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "244": { "goal": [ { "clause": 3, "scope": 2, "term": "(split (. 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T173 T174) T175 X200)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X200"], "exprvars": [] } }, "849": { "goal": [{ "clause": -1, "scope": -1, "term": "(ms T54 X26)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X26"], "exprvars": [] } }, "572": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "573": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "574": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "850": { "goal": [{ "clause": -1, "scope": -1, "term": "(merge T179 T178 T19)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T19"], "free": [], "exprvars": [] } }, "851": { "goal": [ { "clause": 5, "scope": 9, "term": "(merge T179 T178 T19)" }, { "clause": 6, "scope": 9, "term": "(merge T179 T178 T19)" }, { "clause": 7, "scope": 9, "term": "(merge T179 T178 T19)" }, { "clause": 8, "scope": 9, "term": "(merge T179 T178 T19)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T19"], "free": [], "exprvars": [] } }, "577": { "goal": [{ "clause": 1, "scope": 5, "term": "(ms T23 X25)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X25"], "exprvars": [] } }, "852": { "goal": [{ "clause": 5, "scope": 9, "term": "(merge T179 T178 T19)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T19"], "free": [], "exprvars": [] } }, "215": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (ms T23 X25) (',' (ms T24 X26) (merge X25 X26 T19)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T19"], "free": [ "X25", "X26" ], "exprvars": [] } }, "578": { "goal": [{ "clause": 2, "scope": 5, "term": "(ms T23 X25)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X25"], "exprvars": [] } }, "853": { "goal": [ { "clause": 6, "scope": 9, "term": "(merge T179 T178 T19)" }, { "clause": 7, "scope": 9, "term": "(merge T179 T178 T19)" }, { "clause": 8, "scope": 9, "term": "(merge T179 T178 T19)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T19"], "free": [], "exprvars": [] } }, "612": { "goal": [{ "clause": -1, "scope": -1, "term": "(ms T72 X106)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X106"], "exprvars": [] } }, "854": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "613": { "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": [] } }, "855": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "856": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "857": { "goal": [{ "clause": 6, "scope": 9, "term": "(merge T179 T178 T19)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T19"], "free": [], "exprvars": [] } }, "858": { "goal": [ { "clause": 7, "scope": 9, "term": "(merge T179 T178 T19)" }, { "clause": 8, "scope": 9, "term": "(merge T179 T178 T19)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T19"], "free": [], "exprvars": [] } }, "185": { "goal": [{ "clause": -1, "scope": -1, "term": "(split (. T20 (. T21 T22)) X23 X24)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X23", "X24" ], "exprvars": [] } }, "585": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "862": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "588": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "863": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "347": { "goal": [{ "clause": -1, "scope": -1, "term": "(split (. T36 T37) X44 X43)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X43", "X44" ], "exprvars": [] } }, "589": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "864": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "624": { "goal": [{ "clause": -1, "scope": -1, "term": "(ms T75 X107)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X107"], "exprvars": [] } }, "625": { "goal": [{ "clause": -1, "scope": -1, "term": "(merge T77 T76 X108)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X108"], "exprvars": [] } }, "905": { "goal": [ { "clause": 9, "scope": 10, "term": "(less T212 (s T217))" }, { "clause": 10, "scope": 10, "term": "(less T212 (s T217))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T212"], "free": [], "exprvars": [] } }, "906": { "goal": [{ "clause": 9, "scope": 10, "term": "(less T212 (s T217))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T212"], "free": [], "exprvars": [] } } }, "edges": [ { "from": 2, "to": 7, "label": "CASE" }, { "from": 7, "to": 12, "label": "PARALLEL" }, { "from": 7, "to": 13, "label": "PARALLEL" }, { "from": 12, "to": 15, "label": "EVAL with clause\nms([], []).\nand substitutionT1 -> [],\nT2 -> []" }, { "from": 12, "to": 17, "label": "EVAL-BACKTRACK" }, { "from": 13, "to": 32, "label": "PARALLEL" }, { "from": 13, "to": 33, "label": "PARALLEL" }, { "from": 15, "to": 28, "label": "SUCCESS" }, { "from": 32, "to": 34, "label": "EVAL with clause\nms(.(X6, []), .(X6, [])).\nand substitutionX6 -> T7,\nT1 -> .(T7, []),\nT2 -> .(T7, [])" }, { "from": 32, "to": 35, "label": "EVAL-BACKTRACK" }, { "from": 33, "to": 43, "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": 33, "to": 44, "label": "EVAL-BACKTRACK" }, { "from": 34, "to": 36, "label": "SUCCESS" }, { "from": 43, "to": 185, "label": "SPLIT 1" }, { "from": 43, "to": 215, "label": "SPLIT 2\nreplacements:X23 -> T23,\nX24 -> T24" }, { "from": 185, "to": 244, "label": "CASE" }, { "from": 215, "to": 563, "label": "SPLIT 1" }, { "from": 215, "to": 564, "label": "SPLIT 2\nreplacements:X25 -> T53,\nT24 -> T54" }, { "from": 244, "to": 250, "label": "BACKTRACK\nfor clause: split([], [], [])because of non-unification" }, { "from": 250, "to": 347, "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": 347, "to": 359, "label": "CASE" }, { "from": 359, "to": 360, "label": "BACKTRACK\nfor clause: split([], [], [])because of non-unification" }, { "from": 360, "to": 361, "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": 361, "to": 363, "label": "CASE" }, { "from": 363, "to": 365, "label": "PARALLEL" }, { "from": 363, "to": 367, "label": "PARALLEL" }, { "from": 365, "to": 368, "label": "EVAL with clause\nsplit([], [], []).\nand substitutionT45 -> [],\nX62 -> [],\nX61 -> []" }, { "from": 365, "to": 369, "label": "EVAL-BACKTRACK" }, { "from": 367, "to": 371, "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": 367, "to": 372, "label": "EVAL-BACKTRACK" }, { "from": 368, "to": 370, "label": "SUCCESS" }, { "from": 371, "to": 361, "label": "INSTANCE with matching:\nT45 -> T52\nX62 -> X80\nX61 -> X79" }, { "from": 563, "to": 567, "label": "CASE" }, { "from": 564, "to": 849, "label": "SPLIT 1" }, { "from": 564, "to": 850, "label": "SPLIT 2\nreplacements:X26 -> T178,\nT53 -> T179" }, { "from": 567, "to": 568, "label": "PARALLEL" }, { "from": 567, "to": 569, "label": "PARALLEL" }, { "from": 568, "to": 572, "label": "EVAL with clause\nms([], []).\nand substitutionT23 -> [],\nX25 -> []" }, { "from": 568, "to": 573, "label": "EVAL-BACKTRACK" }, { "from": 569, "to": 577, "label": "PARALLEL" }, { "from": 569, "to": 578, "label": "PARALLEL" }, { "from": 572, "to": 574, "label": "SUCCESS" }, { "from": 577, "to": 585, "label": "EVAL with clause\nms(.(X85, []), .(X85, [])).\nand substitutionX85 -> T59,\nT23 -> .(T59, []),\nX25 -> .(T59, [])" }, { "from": 577, "to": 588, "label": "EVAL-BACKTRACK" }, { "from": 578, "to": 599, "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": 578, "to": 601, "label": "EVAL-BACKTRACK" }, { "from": 585, "to": 589, "label": "SUCCESS" }, { "from": 599, "to": 604, "label": "SPLIT 1" }, { "from": 599, "to": 605, "label": "SPLIT 2\nreplacements:X104 -> T72,\nX105 -> T73" }, { "from": 604, "to": 185, "label": "INSTANCE with matching:\nT20 -> T69\nT21 -> T70\nT22 -> T71\nX23 -> X104\nX24 -> X105" }, { "from": 605, "to": 612, "label": "SPLIT 1" }, { "from": 605, "to": 613, "label": "SPLIT 2\nreplacements:X106 -> T74,\nT73 -> T75" }, { "from": 612, "to": 563, "label": "INSTANCE with matching:\nT23 -> T72\nX25 -> X106" }, { "from": 613, "to": 624, "label": "SPLIT 1" }, { "from": 613, "to": 625, "label": "SPLIT 2\nreplacements:X107 -> T76,\nT74 -> T77" }, { "from": 624, "to": 563, "label": "INSTANCE with matching:\nT23 -> T75\nX25 -> X107" }, { "from": 625, "to": 799, "label": "CASE" }, { "from": 799, "to": 800, "label": "PARALLEL" }, { "from": 799, "to": 801, "label": "PARALLEL" }, { "from": 800, "to": 802, "label": "EVAL with clause\nmerge([], X115, X115).\nand substitutionT77 -> [],\nT76 -> T84,\nX115 -> T84,\nX108 -> T84" }, { "from": 800, "to": 803, "label": "EVAL-BACKTRACK" }, { "from": 801, "to": 805, "label": "PARALLEL" }, { "from": 801, "to": 806, "label": "PARALLEL" }, { "from": 802, "to": 804, "label": "SUCCESS" }, { "from": 805, "to": 807, "label": "EVAL with clause\nmerge(X120, [], X120).\nand substitutionT77 -> T89,\nX120 -> T89,\nT76 -> [],\nX108 -> T89" }, { "from": 805, "to": 808, "label": "EVAL-BACKTRACK" }, { "from": 806, "to": 810, "label": "PARALLEL" }, { "from": 806, "to": 811, "label": "PARALLEL" }, { "from": 807, "to": 809, "label": "SUCCESS" }, { "from": 810, "to": 812, "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": 810, "to": 813, "label": "EVAL-BACKTRACK" }, { "from": 811, "to": 845, "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": 811, "to": 846, "label": "EVAL-BACKTRACK" }, { "from": 812, "to": 814, "label": "SPLIT 1" }, { "from": 812, "to": 815, "label": "SPLIT 2\nnew knowledge:\nT110 is ground\nreplacements:T112 -> T116,\nT111 -> T117,\nT113 -> T118" }, { "from": 814, "to": 818, "label": "CASE" }, { "from": 815, "to": 625, "label": "INSTANCE with matching:\nT77 -> T116\nT76 -> .(T117, T118)\nX108 -> X150" }, { "from": 818, "to": 819, "label": "PARALLEL" }, { "from": 818, "to": 820, "label": "PARALLEL" }, { "from": 819, "to": 824, "label": "EVAL with clause\nless(0, s(X159)).\nand substitutionT110 -> 0,\nT111 -> T125,\nX159 -> T125" }, { "from": 819, "to": 825, "label": "EVAL-BACKTRACK" }, { "from": 820, "to": 835, "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": 820, "to": 836, "label": "EVAL-BACKTRACK" }, { "from": 824, "to": 826, "label": "SUCCESS" }, { "from": 835, "to": 837, "label": "CASE" }, { "from": 837, "to": 838, "label": "PARALLEL" }, { "from": 837, "to": 839, "label": "PARALLEL" }, { "from": 838, "to": 840, "label": "EVAL with clause\nless(0, s(X172)).\nand substitutionT132 -> 0,\nX172 -> T140,\nT133 -> s(T140)" }, { "from": 838, "to": 841, "label": "EVAL-BACKTRACK" }, { "from": 839, "to": 843, "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": 839, "to": 844, "label": "EVAL-BACKTRACK" }, { "from": 840, "to": 842, "label": "SUCCESS" }, { "from": 843, "to": 835, "label": "INSTANCE with matching:\nT132 -> T147\nT133 -> T148" }, { "from": 845, "to": 847, "label": "SPLIT 1" }, { "from": 845, "to": 848, "label": "SPLIT 2\nnew knowledge:\nT167 is ground\nreplacements:T168 -> T173,\nT170 -> T174,\nT169 -> T175" }, { "from": 847, "to": 835, "label": "INSTANCE with matching:\nT132 -> T167\nT133 -> T168" }, { "from": 848, "to": 625, "label": "INSTANCE with matching:\nT77 -> .(T173, T174)\nT76 -> T175\nX108 -> X200" }, { "from": 849, "to": 563, "label": "INSTANCE with matching:\nT23 -> T54\nX25 -> X26" }, { "from": 850, "to": 851, "label": "CASE" }, { "from": 851, "to": 852, "label": "PARALLEL" }, { "from": 851, "to": 853, "label": "PARALLEL" }, { "from": 852, "to": 854, "label": "EVAL with clause\nmerge([], X213, X213).\nand substitutionT179 -> [],\nT178 -> T186,\nX213 -> T186,\nT19 -> T186" }, { "from": 852, "to": 855, "label": "EVAL-BACKTRACK" }, { "from": 853, "to": 857, "label": "PARALLEL" }, { "from": 853, "to": 858, "label": "PARALLEL" }, { "from": 854, "to": 856, "label": "SUCCESS" }, { "from": 857, "to": 862, "label": "EVAL with clause\nmerge(X218, [], X218).\nand substitutionT179 -> T191,\nX218 -> T191,\nT178 -> [],\nT19 -> T191" }, { "from": 857, "to": 863, "label": "EVAL-BACKTRACK" }, { "from": 858, "to": 875, "label": "PARALLEL" }, { "from": 858, "to": 876, "label": "PARALLEL" }, { "from": 862, "to": 864, "label": "SUCCESS" }, { "from": 875, "to": 887, "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": 875, "to": 889, "label": "EVAL-BACKTRACK" }, { "from": 876, "to": 946, "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": 876, "to": 947, "label": "EVAL-BACKTRACK" }, { "from": 887, "to": 897, "label": "SPLIT 1" }, { "from": 887, "to": 898, "label": "SPLIT 2\nnew knowledge:\nT212 is ground\nreplacements:T218 -> T222,\nT217 -> T223,\nT219 -> T224" }, { "from": 897, "to": 905, "label": "CASE" }, { "from": 898, "to": 850, "label": "INSTANCE with matching:\nT179 -> T222\nT178 -> .(T223, T224)\nT19 -> T216" }, { "from": 905, "to": 906, "label": "PARALLEL" }, { "from": 905, "to": 907, "label": "PARALLEL" }, { "from": 906, "to": 908, "label": "EVAL with clause\nless(0, s(X252)).\nand substitutionT212 -> 0,\nT217 -> T231,\nX252 -> T231" }, { "from": 906, "to": 909, "label": "EVAL-BACKTRACK" }, { "from": 907, "to": 916, "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": 907, "to": 917, "label": "EVAL-BACKTRACK" }, { "from": 908, "to": 910, "label": "SUCCESS" }, { "from": 916, "to": 921, "label": "CASE" }, { "from": 921, "to": 922, "label": "PARALLEL" }, { "from": 921, "to": 923, "label": "PARALLEL" }, { "from": 922, "to": 928, "label": "EVAL with clause\nless(0, s(X265)).\nand substitutionT236 -> 0,\nX265 -> T245,\nT238 -> s(T245)" }, { "from": 922, "to": 930, "label": "EVAL-BACKTRACK" }, { "from": 923, "to": 934, "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": 923, "to": 935, "label": "EVAL-BACKTRACK" }, { "from": 928, "to": 931, "label": "SUCCESS" }, { "from": 934, "to": 916, "label": "INSTANCE with matching:\nT236 -> T250\nT238 -> T252" }, { "from": 946, "to": 948, "label": "SPLIT 1" }, { "from": 946, "to": 950, "label": "SPLIT 2\nnew knowledge:\nT271 is ground\nreplacements:T274 -> T279,\nT276 -> T280,\nT275 -> T281" }, { "from": 948, "to": 916, "label": "INSTANCE with matching:\nT236 -> T271\nT238 -> T274" }, { "from": 950, "to": 850, "label": "INSTANCE with matching:\nT179 -> .(T279, T280)\nT178 -> T281\nT19 -> T273" } ], "type": "Graph" } } ---------------------------------------- (78) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f2_in([]) -> f2_out1 f2_in(.(T7, [])) -> f2_out1 f2_in(T19) -> U1(f43_in(T19), T19) U1(f43_out1(X25, X26), T19) -> f2_out1 f361_in -> f361_out1 f361_in -> U2(f361_in) U2(f361_out1) -> f361_out1 f185_in -> U3(f361_in) U3(f361_out1) -> f185_out1 f563_in -> f563_out1 f563_in -> U4(f599_in) U4(f599_out1) -> f563_out1 f625_in -> f625_out1 f625_in -> U5(f812_in) U5(f812_out1(T110)) -> f625_out1 f625_in -> U6(f845_in) U6(f845_out1(T167)) -> f625_out1 f835_in -> f835_out1(0) f835_in -> U7(f835_in) U7(f835_out1(T147)) -> f835_out1(s(T147)) f850_in(T186) -> f850_out1([], T186) f850_in(T191) -> f850_out1(T191, []) f850_in(.(T212, T216)) -> U8(f887_in(T212, T216), .(T212, T216)) U8(f887_out1(T218, T217, T219), .(T212, T216)) -> f850_out1(.(T212, T218), .(T217, T219)) f850_in(.(T271, T273)) -> U9(f946_in(T271, T273), .(T271, T273)) U9(f946_out1(T274, T276, T275), .(T271, T273)) -> f850_out1(.(T274, T276), .(T271, T275)) f916_in(0) -> f916_out1 f916_in(s(T250)) -> U10(f916_in(T250), s(T250)) U10(f916_out1, s(T250)) -> f916_out1 f814_in -> f814_out1(0) f814_in -> U11(f835_in) U11(f835_out1(T132)) -> f814_out1(s(T132)) f897_in(0) -> f897_out1 f897_in(s(T236)) -> U12(f916_in(T236), s(T236)) U12(f916_out1, s(T236)) -> f897_out1 f43_in(T19) -> U13(f185_in, T19) U13(f185_out1, T19) -> U14(f215_in(T19), T19) U14(f215_out1(X25, X26), T19) -> f43_out1(X25, X26) f215_in(T19) -> U15(f563_in, T19) U15(f563_out1, T19) -> U16(f564_in(T19), T19) U16(f564_out1(T53, X26), T19) -> f215_out1(T53, X26) f564_in(T19) -> U17(f563_in, T19) U17(f563_out1, T19) -> U18(f850_in(T19), T19) U18(f850_out1(T179, T178), T19) -> f564_out1(T179, T178) f599_in -> U19(f185_in) U19(f185_out1) -> U20(f605_in) U20(f605_out1) -> f599_out1 f605_in -> U21(f563_in) U21(f563_out1) -> U22(f613_in) U22(f613_out1) -> f605_out1 f613_in -> U23(f563_in) U23(f563_out1) -> U24(f625_in) U24(f625_out1) -> f613_out1 f812_in -> U25(f814_in) U25(f814_out1(T110)) -> U26(f625_in, T110) U26(f625_out1, T110) -> f812_out1(T110) f845_in -> U27(f835_in) U27(f835_out1(T167)) -> U28(f625_in, T167) U28(f625_out1, T167) -> f845_out1(T167) f887_in(T212, T216) -> U29(f897_in(T212), T212, T216) U29(f897_out1, T212, T216) -> U30(f850_in(T216), T212, T216) U30(f850_out1(T222, .(T223, T224)), T212, T216) -> f887_out1(T222, T223, T224) f946_in(T271, T273) -> U31(f916_in(T271), T271, T273) U31(f916_out1, T271, T273) -> U32(f850_in(T273), T271, T273) U32(f850_out1(.(T279, T280), T281), T271, T273) -> f946_out1(T279, T280, T281) Q is empty. ---------------------------------------- (79) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (80) Obligation: Q DP problem: The TRS P consists of the following rules: F2_IN(T19) -> U1^1(f43_in(T19), T19) F2_IN(T19) -> F43_IN(T19) F361_IN -> U2^1(f361_in) F361_IN -> F361_IN F185_IN -> U3^1(f361_in) F185_IN -> F361_IN F563_IN -> U4^1(f599_in) F563_IN -> F599_IN F625_IN -> U5^1(f812_in) F625_IN -> F812_IN F625_IN -> U6^1(f845_in) F625_IN -> F845_IN F835_IN -> U7^1(f835_in) F835_IN -> F835_IN F850_IN(.(T212, T216)) -> U8^1(f887_in(T212, T216), .(T212, T216)) F850_IN(.(T212, T216)) -> F887_IN(T212, T216) F850_IN(.(T271, T273)) -> U9^1(f946_in(T271, T273), .(T271, T273)) F850_IN(.(T271, T273)) -> F946_IN(T271, T273) F916_IN(s(T250)) -> U10^1(f916_in(T250), s(T250)) F916_IN(s(T250)) -> F916_IN(T250) F814_IN -> U11^1(f835_in) F814_IN -> F835_IN F897_IN(s(T236)) -> U12^1(f916_in(T236), s(T236)) F897_IN(s(T236)) -> F916_IN(T236) F43_IN(T19) -> U13^1(f185_in, T19) F43_IN(T19) -> F185_IN U13^1(f185_out1, T19) -> U14^1(f215_in(T19), T19) U13^1(f185_out1, T19) -> F215_IN(T19) F215_IN(T19) -> U15^1(f563_in, T19) F215_IN(T19) -> F563_IN U15^1(f563_out1, T19) -> U16^1(f564_in(T19), T19) U15^1(f563_out1, T19) -> F564_IN(T19) F564_IN(T19) -> U17^1(f563_in, T19) F564_IN(T19) -> F563_IN U17^1(f563_out1, T19) -> U18^1(f850_in(T19), T19) U17^1(f563_out1, T19) -> F850_IN(T19) F599_IN -> U19^1(f185_in) F599_IN -> F185_IN U19^1(f185_out1) -> U20^1(f605_in) U19^1(f185_out1) -> F605_IN F605_IN -> U21^1(f563_in) F605_IN -> F563_IN U21^1(f563_out1) -> U22^1(f613_in) U21^1(f563_out1) -> F613_IN F613_IN -> U23^1(f563_in) F613_IN -> F563_IN U23^1(f563_out1) -> U24^1(f625_in) U23^1(f563_out1) -> F625_IN F812_IN -> U25^1(f814_in) F812_IN -> F814_IN U25^1(f814_out1(T110)) -> U26^1(f625_in, T110) U25^1(f814_out1(T110)) -> F625_IN F845_IN -> U27^1(f835_in) F845_IN -> F835_IN U27^1(f835_out1(T167)) -> U28^1(f625_in, T167) U27^1(f835_out1(T167)) -> F625_IN F887_IN(T212, T216) -> U29^1(f897_in(T212), T212, T216) F887_IN(T212, T216) -> F897_IN(T212) U29^1(f897_out1, T212, T216) -> U30^1(f850_in(T216), T212, T216) U29^1(f897_out1, T212, T216) -> F850_IN(T216) F946_IN(T271, T273) -> U31^1(f916_in(T271), T271, T273) F946_IN(T271, T273) -> F916_IN(T271) U31^1(f916_out1, T271, T273) -> U32^1(f850_in(T273), T271, T273) U31^1(f916_out1, T271, T273) -> F850_IN(T273) The TRS R consists of the following rules: f2_in([]) -> f2_out1 f2_in(.(T7, [])) -> f2_out1 f2_in(T19) -> U1(f43_in(T19), T19) U1(f43_out1(X25, X26), T19) -> f2_out1 f361_in -> f361_out1 f361_in -> U2(f361_in) U2(f361_out1) -> f361_out1 f185_in -> U3(f361_in) U3(f361_out1) -> f185_out1 f563_in -> f563_out1 f563_in -> U4(f599_in) U4(f599_out1) -> f563_out1 f625_in -> f625_out1 f625_in -> U5(f812_in) U5(f812_out1(T110)) -> f625_out1 f625_in -> U6(f845_in) U6(f845_out1(T167)) -> f625_out1 f835_in -> f835_out1(0) f835_in -> U7(f835_in) U7(f835_out1(T147)) -> f835_out1(s(T147)) f850_in(T186) -> f850_out1([], T186) f850_in(T191) -> f850_out1(T191, []) f850_in(.(T212, T216)) -> U8(f887_in(T212, T216), .(T212, T216)) U8(f887_out1(T218, T217, T219), .(T212, T216)) -> f850_out1(.(T212, T218), .(T217, T219)) f850_in(.(T271, T273)) -> U9(f946_in(T271, T273), .(T271, T273)) U9(f946_out1(T274, T276, T275), .(T271, T273)) -> f850_out1(.(T274, T276), .(T271, T275)) f916_in(0) -> f916_out1 f916_in(s(T250)) -> U10(f916_in(T250), s(T250)) U10(f916_out1, s(T250)) -> f916_out1 f814_in -> f814_out1(0) f814_in -> U11(f835_in) U11(f835_out1(T132)) -> f814_out1(s(T132)) f897_in(0) -> f897_out1 f897_in(s(T236)) -> U12(f916_in(T236), s(T236)) U12(f916_out1, s(T236)) -> f897_out1 f43_in(T19) -> U13(f185_in, T19) U13(f185_out1, T19) -> U14(f215_in(T19), T19) U14(f215_out1(X25, X26), T19) -> f43_out1(X25, X26) f215_in(T19) -> U15(f563_in, T19) U15(f563_out1, T19) -> U16(f564_in(T19), T19) U16(f564_out1(T53, X26), T19) -> f215_out1(T53, X26) f564_in(T19) -> U17(f563_in, T19) U17(f563_out1, T19) -> U18(f850_in(T19), T19) U18(f850_out1(T179, T178), T19) -> f564_out1(T179, T178) f599_in -> U19(f185_in) U19(f185_out1) -> U20(f605_in) U20(f605_out1) -> f599_out1 f605_in -> U21(f563_in) U21(f563_out1) -> U22(f613_in) U22(f613_out1) -> f605_out1 f613_in -> U23(f563_in) U23(f563_out1) -> U24(f625_in) U24(f625_out1) -> f613_out1 f812_in -> U25(f814_in) U25(f814_out1(T110)) -> U26(f625_in, T110) U26(f625_out1, T110) -> f812_out1(T110) f845_in -> U27(f835_in) U27(f835_out1(T167)) -> U28(f625_in, T167) U28(f625_out1, T167) -> f845_out1(T167) f887_in(T212, T216) -> U29(f897_in(T212), T212, T216) U29(f897_out1, T212, T216) -> U30(f850_in(T216), T212, T216) U30(f850_out1(T222, .(T223, T224)), T212, T216) -> f887_out1(T222, T223, T224) f946_in(T271, T273) -> U31(f916_in(T271), T271, T273) U31(f916_out1, T271, T273) -> U32(f850_in(T273), T271, T273) U32(f850_out1(.(T279, T280), T281), T271, T273) -> f946_out1(T279, T280, T281) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (81) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 6 SCCs with 42 less nodes. ---------------------------------------- (82) Complex Obligation (AND) ---------------------------------------- (83) Obligation: Q DP problem: The TRS P consists of the following rules: F916_IN(s(T250)) -> F916_IN(T250) The TRS R consists of the following rules: f2_in([]) -> f2_out1 f2_in(.(T7, [])) -> f2_out1 f2_in(T19) -> U1(f43_in(T19), T19) U1(f43_out1(X25, X26), T19) -> f2_out1 f361_in -> f361_out1 f361_in -> U2(f361_in) U2(f361_out1) -> f361_out1 f185_in -> U3(f361_in) U3(f361_out1) -> f185_out1 f563_in -> f563_out1 f563_in -> U4(f599_in) U4(f599_out1) -> f563_out1 f625_in -> f625_out1 f625_in -> U5(f812_in) U5(f812_out1(T110)) -> f625_out1 f625_in -> U6(f845_in) U6(f845_out1(T167)) -> f625_out1 f835_in -> f835_out1(0) f835_in -> U7(f835_in) U7(f835_out1(T147)) -> f835_out1(s(T147)) f850_in(T186) -> f850_out1([], T186) f850_in(T191) -> f850_out1(T191, []) f850_in(.(T212, T216)) -> U8(f887_in(T212, T216), .(T212, T216)) U8(f887_out1(T218, T217, T219), .(T212, T216)) -> f850_out1(.(T212, T218), .(T217, T219)) f850_in(.(T271, T273)) -> U9(f946_in(T271, T273), .(T271, T273)) U9(f946_out1(T274, T276, T275), .(T271, T273)) -> f850_out1(.(T274, T276), .(T271, T275)) f916_in(0) -> f916_out1 f916_in(s(T250)) -> U10(f916_in(T250), s(T250)) U10(f916_out1, s(T250)) -> f916_out1 f814_in -> f814_out1(0) f814_in -> U11(f835_in) U11(f835_out1(T132)) -> f814_out1(s(T132)) f897_in(0) -> f897_out1 f897_in(s(T236)) -> U12(f916_in(T236), s(T236)) U12(f916_out1, s(T236)) -> f897_out1 f43_in(T19) -> U13(f185_in, T19) U13(f185_out1, T19) -> U14(f215_in(T19), T19) U14(f215_out1(X25, X26), T19) -> f43_out1(X25, X26) f215_in(T19) -> U15(f563_in, T19) U15(f563_out1, T19) -> U16(f564_in(T19), T19) U16(f564_out1(T53, X26), T19) -> f215_out1(T53, X26) f564_in(T19) -> U17(f563_in, T19) U17(f563_out1, T19) -> U18(f850_in(T19), T19) U18(f850_out1(T179, T178), T19) -> f564_out1(T179, T178) f599_in -> U19(f185_in) U19(f185_out1) -> U20(f605_in) U20(f605_out1) -> f599_out1 f605_in -> U21(f563_in) U21(f563_out1) -> U22(f613_in) U22(f613_out1) -> f605_out1 f613_in -> U23(f563_in) U23(f563_out1) -> U24(f625_in) U24(f625_out1) -> f613_out1 f812_in -> U25(f814_in) U25(f814_out1(T110)) -> U26(f625_in, T110) U26(f625_out1, T110) -> f812_out1(T110) f845_in -> U27(f835_in) U27(f835_out1(T167)) -> U28(f625_in, T167) U28(f625_out1, T167) -> f845_out1(T167) f887_in(T212, T216) -> U29(f897_in(T212), T212, T216) U29(f897_out1, T212, T216) -> U30(f850_in(T216), T212, T216) U30(f850_out1(T222, .(T223, T224)), T212, T216) -> f887_out1(T222, T223, T224) f946_in(T271, T273) -> U31(f916_in(T271), T271, T273) U31(f916_out1, T271, T273) -> U32(f850_in(T273), T271, T273) U32(f850_out1(.(T279, T280), T281), T271, T273) -> f946_out1(T279, T280, T281) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (84) 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. ---------------------------------------- (85) Obligation: Q DP problem: The TRS P consists of the following rules: F916_IN(s(T250)) -> F916_IN(T250) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (86) 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: *F916_IN(s(T250)) -> F916_IN(T250) The graph contains the following edges 1 > 1 ---------------------------------------- (87) YES ---------------------------------------- (88) Obligation: Q DP problem: The TRS P consists of the following rules: F850_IN(.(T212, T216)) -> F887_IN(T212, T216) F887_IN(T212, T216) -> U29^1(f897_in(T212), T212, T216) U29^1(f897_out1, T212, T216) -> F850_IN(T216) F850_IN(.(T271, T273)) -> F946_IN(T271, T273) F946_IN(T271, T273) -> U31^1(f916_in(T271), T271, T273) U31^1(f916_out1, T271, T273) -> F850_IN(T273) The TRS R consists of the following rules: f2_in([]) -> f2_out1 f2_in(.(T7, [])) -> f2_out1 f2_in(T19) -> U1(f43_in(T19), T19) U1(f43_out1(X25, X26), T19) -> f2_out1 f361_in -> f361_out1 f361_in -> U2(f361_in) U2(f361_out1) -> f361_out1 f185_in -> U3(f361_in) U3(f361_out1) -> f185_out1 f563_in -> f563_out1 f563_in -> U4(f599_in) U4(f599_out1) -> f563_out1 f625_in -> f625_out1 f625_in -> U5(f812_in) U5(f812_out1(T110)) -> f625_out1 f625_in -> U6(f845_in) U6(f845_out1(T167)) -> f625_out1 f835_in -> f835_out1(0) f835_in -> U7(f835_in) U7(f835_out1(T147)) -> f835_out1(s(T147)) f850_in(T186) -> f850_out1([], T186) f850_in(T191) -> f850_out1(T191, []) f850_in(.(T212, T216)) -> U8(f887_in(T212, T216), .(T212, T216)) U8(f887_out1(T218, T217, T219), .(T212, T216)) -> f850_out1(.(T212, T218), .(T217, T219)) f850_in(.(T271, T273)) -> U9(f946_in(T271, T273), .(T271, T273)) U9(f946_out1(T274, T276, T275), .(T271, T273)) -> f850_out1(.(T274, T276), .(T271, T275)) f916_in(0) -> f916_out1 f916_in(s(T250)) -> U10(f916_in(T250), s(T250)) U10(f916_out1, s(T250)) -> f916_out1 f814_in -> f814_out1(0) f814_in -> U11(f835_in) U11(f835_out1(T132)) -> f814_out1(s(T132)) f897_in(0) -> f897_out1 f897_in(s(T236)) -> U12(f916_in(T236), s(T236)) U12(f916_out1, s(T236)) -> f897_out1 f43_in(T19) -> U13(f185_in, T19) U13(f185_out1, T19) -> U14(f215_in(T19), T19) U14(f215_out1(X25, X26), T19) -> f43_out1(X25, X26) f215_in(T19) -> U15(f563_in, T19) U15(f563_out1, T19) -> U16(f564_in(T19), T19) U16(f564_out1(T53, X26), T19) -> f215_out1(T53, X26) f564_in(T19) -> U17(f563_in, T19) U17(f563_out1, T19) -> U18(f850_in(T19), T19) U18(f850_out1(T179, T178), T19) -> f564_out1(T179, T178) f599_in -> U19(f185_in) U19(f185_out1) -> U20(f605_in) U20(f605_out1) -> f599_out1 f605_in -> U21(f563_in) U21(f563_out1) -> U22(f613_in) U22(f613_out1) -> f605_out1 f613_in -> U23(f563_in) U23(f563_out1) -> U24(f625_in) U24(f625_out1) -> f613_out1 f812_in -> U25(f814_in) U25(f814_out1(T110)) -> U26(f625_in, T110) U26(f625_out1, T110) -> f812_out1(T110) f845_in -> U27(f835_in) U27(f835_out1(T167)) -> U28(f625_in, T167) U28(f625_out1, T167) -> f845_out1(T167) f887_in(T212, T216) -> U29(f897_in(T212), T212, T216) U29(f897_out1, T212, T216) -> U30(f850_in(T216), T212, T216) U30(f850_out1(T222, .(T223, T224)), T212, T216) -> f887_out1(T222, T223, T224) f946_in(T271, T273) -> U31(f916_in(T271), T271, T273) U31(f916_out1, T271, T273) -> U32(f850_in(T273), T271, T273) U32(f850_out1(.(T279, T280), T281), T271, T273) -> f946_out1(T279, T280, T281) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (89) 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: *F887_IN(T212, T216) -> U29^1(f897_in(T212), T212, T216) The graph contains the following edges 1 >= 2, 2 >= 3 *U29^1(f897_out1, T212, T216) -> F850_IN(T216) The graph contains the following edges 3 >= 1 *U31^1(f916_out1, T271, T273) -> F850_IN(T273) The graph contains the following edges 3 >= 1 *F850_IN(.(T212, T216)) -> F887_IN(T212, T216) The graph contains the following edges 1 > 1, 1 > 2 *F850_IN(.(T271, T273)) -> F946_IN(T271, T273) The graph contains the following edges 1 > 1, 1 > 2 *F946_IN(T271, T273) -> U31^1(f916_in(T271), T271, T273) The graph contains the following edges 1 >= 2, 2 >= 3 ---------------------------------------- (90) YES ---------------------------------------- (91) Obligation: Q DP problem: The TRS P consists of the following rules: F835_IN -> F835_IN The TRS R consists of the following rules: f2_in([]) -> f2_out1 f2_in(.(T7, [])) -> f2_out1 f2_in(T19) -> U1(f43_in(T19), T19) U1(f43_out1(X25, X26), T19) -> f2_out1 f361_in -> f361_out1 f361_in -> U2(f361_in) U2(f361_out1) -> f361_out1 f185_in -> U3(f361_in) U3(f361_out1) -> f185_out1 f563_in -> f563_out1 f563_in -> U4(f599_in) U4(f599_out1) -> f563_out1 f625_in -> f625_out1 f625_in -> U5(f812_in) U5(f812_out1(T110)) -> f625_out1 f625_in -> U6(f845_in) U6(f845_out1(T167)) -> f625_out1 f835_in -> f835_out1(0) f835_in -> U7(f835_in) U7(f835_out1(T147)) -> f835_out1(s(T147)) f850_in(T186) -> f850_out1([], T186) f850_in(T191) -> f850_out1(T191, []) f850_in(.(T212, T216)) -> U8(f887_in(T212, T216), .(T212, T216)) U8(f887_out1(T218, T217, T219), .(T212, T216)) -> f850_out1(.(T212, T218), .(T217, T219)) f850_in(.(T271, T273)) -> U9(f946_in(T271, T273), .(T271, T273)) U9(f946_out1(T274, T276, T275), .(T271, T273)) -> f850_out1(.(T274, T276), .(T271, T275)) f916_in(0) -> f916_out1 f916_in(s(T250)) -> U10(f916_in(T250), s(T250)) U10(f916_out1, s(T250)) -> f916_out1 f814_in -> f814_out1(0) f814_in -> U11(f835_in) U11(f835_out1(T132)) -> f814_out1(s(T132)) f897_in(0) -> f897_out1 f897_in(s(T236)) -> U12(f916_in(T236), s(T236)) U12(f916_out1, s(T236)) -> f897_out1 f43_in(T19) -> U13(f185_in, T19) U13(f185_out1, T19) -> U14(f215_in(T19), T19) U14(f215_out1(X25, X26), T19) -> f43_out1(X25, X26) f215_in(T19) -> U15(f563_in, T19) U15(f563_out1, T19) -> U16(f564_in(T19), T19) U16(f564_out1(T53, X26), T19) -> f215_out1(T53, X26) f564_in(T19) -> U17(f563_in, T19) U17(f563_out1, T19) -> U18(f850_in(T19), T19) U18(f850_out1(T179, T178), T19) -> f564_out1(T179, T178) f599_in -> U19(f185_in) U19(f185_out1) -> U20(f605_in) U20(f605_out1) -> f599_out1 f605_in -> U21(f563_in) U21(f563_out1) -> U22(f613_in) U22(f613_out1) -> f605_out1 f613_in -> U23(f563_in) U23(f563_out1) -> U24(f625_in) U24(f625_out1) -> f613_out1 f812_in -> U25(f814_in) U25(f814_out1(T110)) -> U26(f625_in, T110) U26(f625_out1, T110) -> f812_out1(T110) f845_in -> U27(f835_in) U27(f835_out1(T167)) -> U28(f625_in, T167) U28(f625_out1, T167) -> f845_out1(T167) f887_in(T212, T216) -> U29(f897_in(T212), T212, T216) U29(f897_out1, T212, T216) -> U30(f850_in(T216), T212, T216) U30(f850_out1(T222, .(T223, T224)), T212, T216) -> f887_out1(T222, T223, T224) f946_in(T271, T273) -> U31(f916_in(T271), T271, T273) U31(f916_out1, T271, T273) -> U32(f850_in(T273), T271, T273) U32(f850_out1(.(T279, T280), T281), T271, T273) -> f946_out1(T279, T280, T281) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (92) 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. ---------------------------------------- (93) Obligation: Q DP problem: The TRS P consists of the following rules: F835_IN -> F835_IN R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (94) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by semiunifying a rule from P directly. s = F835_IN evaluates to t =F835_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 F835_IN to F835_IN. ---------------------------------------- (95) NO ---------------------------------------- (96) Obligation: Q DP problem: The TRS P consists of the following rules: F625_IN -> F812_IN F812_IN -> U25^1(f814_in) U25^1(f814_out1(T110)) -> F625_IN F625_IN -> F845_IN F845_IN -> U27^1(f835_in) U27^1(f835_out1(T167)) -> F625_IN The TRS R consists of the following rules: f2_in([]) -> f2_out1 f2_in(.(T7, [])) -> f2_out1 f2_in(T19) -> U1(f43_in(T19), T19) U1(f43_out1(X25, X26), T19) -> f2_out1 f361_in -> f361_out1 f361_in -> U2(f361_in) U2(f361_out1) -> f361_out1 f185_in -> U3(f361_in) U3(f361_out1) -> f185_out1 f563_in -> f563_out1 f563_in -> U4(f599_in) U4(f599_out1) -> f563_out1 f625_in -> f625_out1 f625_in -> U5(f812_in) U5(f812_out1(T110)) -> f625_out1 f625_in -> U6(f845_in) U6(f845_out1(T167)) -> f625_out1 f835_in -> f835_out1(0) f835_in -> U7(f835_in) U7(f835_out1(T147)) -> f835_out1(s(T147)) f850_in(T186) -> f850_out1([], T186) f850_in(T191) -> f850_out1(T191, []) f850_in(.(T212, T216)) -> U8(f887_in(T212, T216), .(T212, T216)) U8(f887_out1(T218, T217, T219), .(T212, T216)) -> f850_out1(.(T212, T218), .(T217, T219)) f850_in(.(T271, T273)) -> U9(f946_in(T271, T273), .(T271, T273)) U9(f946_out1(T274, T276, T275), .(T271, T273)) -> f850_out1(.(T274, T276), .(T271, T275)) f916_in(0) -> f916_out1 f916_in(s(T250)) -> U10(f916_in(T250), s(T250)) U10(f916_out1, s(T250)) -> f916_out1 f814_in -> f814_out1(0) f814_in -> U11(f835_in) U11(f835_out1(T132)) -> f814_out1(s(T132)) f897_in(0) -> f897_out1 f897_in(s(T236)) -> U12(f916_in(T236), s(T236)) U12(f916_out1, s(T236)) -> f897_out1 f43_in(T19) -> U13(f185_in, T19) U13(f185_out1, T19) -> U14(f215_in(T19), T19) U14(f215_out1(X25, X26), T19) -> f43_out1(X25, X26) f215_in(T19) -> U15(f563_in, T19) U15(f563_out1, T19) -> U16(f564_in(T19), T19) U16(f564_out1(T53, X26), T19) -> f215_out1(T53, X26) f564_in(T19) -> U17(f563_in, T19) U17(f563_out1, T19) -> U18(f850_in(T19), T19) U18(f850_out1(T179, T178), T19) -> f564_out1(T179, T178) f599_in -> U19(f185_in) U19(f185_out1) -> U20(f605_in) U20(f605_out1) -> f599_out1 f605_in -> U21(f563_in) U21(f563_out1) -> U22(f613_in) U22(f613_out1) -> f605_out1 f613_in -> U23(f563_in) U23(f563_out1) -> U24(f625_in) U24(f625_out1) -> f613_out1 f812_in -> U25(f814_in) U25(f814_out1(T110)) -> U26(f625_in, T110) U26(f625_out1, T110) -> f812_out1(T110) f845_in -> U27(f835_in) U27(f835_out1(T167)) -> U28(f625_in, T167) U28(f625_out1, T167) -> f845_out1(T167) f887_in(T212, T216) -> U29(f897_in(T212), T212, T216) U29(f897_out1, T212, T216) -> U30(f850_in(T216), T212, T216) U30(f850_out1(T222, .(T223, T224)), T212, T216) -> f887_out1(T222, T223, T224) f946_in(T271, T273) -> U31(f916_in(T271), T271, T273) U31(f916_out1, T271, T273) -> U32(f850_in(T273), T271, T273) U32(f850_out1(.(T279, T280), T281), T271, T273) -> f946_out1(T279, T280, T281) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (97) 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. ---------------------------------------- (98) Obligation: Q DP problem: The TRS P consists of the following rules: F625_IN -> F812_IN F812_IN -> U25^1(f814_in) U25^1(f814_out1(T110)) -> F625_IN F625_IN -> F845_IN F845_IN -> U27^1(f835_in) U27^1(f835_out1(T167)) -> F625_IN The TRS R consists of the following rules: f835_in -> f835_out1(0) f835_in -> U7(f835_in) U7(f835_out1(T147)) -> f835_out1(s(T147)) f814_in -> f814_out1(0) f814_in -> U11(f835_in) U11(f835_out1(T132)) -> f814_out1(s(T132)) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (99) Obligation: Q DP problem: The TRS P consists of the following rules: F361_IN -> F361_IN The TRS R consists of the following rules: f2_in([]) -> f2_out1 f2_in(.(T7, [])) -> f2_out1 f2_in(T19) -> U1(f43_in(T19), T19) U1(f43_out1(X25, X26), T19) -> f2_out1 f361_in -> f361_out1 f361_in -> U2(f361_in) U2(f361_out1) -> f361_out1 f185_in -> U3(f361_in) U3(f361_out1) -> f185_out1 f563_in -> f563_out1 f563_in -> U4(f599_in) U4(f599_out1) -> f563_out1 f625_in -> f625_out1 f625_in -> U5(f812_in) U5(f812_out1(T110)) -> f625_out1 f625_in -> U6(f845_in) U6(f845_out1(T167)) -> f625_out1 f835_in -> f835_out1(0) f835_in -> U7(f835_in) U7(f835_out1(T147)) -> f835_out1(s(T147)) f850_in(T186) -> f850_out1([], T186) f850_in(T191) -> f850_out1(T191, []) f850_in(.(T212, T216)) -> U8(f887_in(T212, T216), .(T212, T216)) U8(f887_out1(T218, T217, T219), .(T212, T216)) -> f850_out1(.(T212, T218), .(T217, T219)) f850_in(.(T271, T273)) -> U9(f946_in(T271, T273), .(T271, T273)) U9(f946_out1(T274, T276, T275), .(T271, T273)) -> f850_out1(.(T274, T276), .(T271, T275)) f916_in(0) -> f916_out1 f916_in(s(T250)) -> U10(f916_in(T250), s(T250)) U10(f916_out1, s(T250)) -> f916_out1 f814_in -> f814_out1(0) f814_in -> U11(f835_in) U11(f835_out1(T132)) -> f814_out1(s(T132)) f897_in(0) -> f897_out1 f897_in(s(T236)) -> U12(f916_in(T236), s(T236)) U12(f916_out1, s(T236)) -> f897_out1 f43_in(T19) -> U13(f185_in, T19) U13(f185_out1, T19) -> U14(f215_in(T19), T19) U14(f215_out1(X25, X26), T19) -> f43_out1(X25, X26) f215_in(T19) -> U15(f563_in, T19) U15(f563_out1, T19) -> U16(f564_in(T19), T19) U16(f564_out1(T53, X26), T19) -> f215_out1(T53, X26) f564_in(T19) -> U17(f563_in, T19) U17(f563_out1, T19) -> U18(f850_in(T19), T19) U18(f850_out1(T179, T178), T19) -> f564_out1(T179, T178) f599_in -> U19(f185_in) U19(f185_out1) -> U20(f605_in) U20(f605_out1) -> f599_out1 f605_in -> U21(f563_in) U21(f563_out1) -> U22(f613_in) U22(f613_out1) -> f605_out1 f613_in -> U23(f563_in) U23(f563_out1) -> U24(f625_in) U24(f625_out1) -> f613_out1 f812_in -> U25(f814_in) U25(f814_out1(T110)) -> U26(f625_in, T110) U26(f625_out1, T110) -> f812_out1(T110) f845_in -> U27(f835_in) U27(f835_out1(T167)) -> U28(f625_in, T167) U28(f625_out1, T167) -> f845_out1(T167) f887_in(T212, T216) -> U29(f897_in(T212), T212, T216) U29(f897_out1, T212, T216) -> U30(f850_in(T216), T212, T216) U30(f850_out1(T222, .(T223, T224)), T212, T216) -> f887_out1(T222, T223, T224) f946_in(T271, T273) -> U31(f916_in(T271), T271, T273) U31(f916_out1, T271, T273) -> U32(f850_in(T273), T271, T273) U32(f850_out1(.(T279, T280), T281), T271, T273) -> f946_out1(T279, T280, T281) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (100) 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. ---------------------------------------- (101) Obligation: Q DP problem: The TRS P consists of the following rules: F361_IN -> F361_IN R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (102) 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 = F361_IN evaluates to t =F361_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 F361_IN to F361_IN. ---------------------------------------- (103) NO ---------------------------------------- (104) Obligation: Q DP problem: The TRS P consists of the following rules: F599_IN -> U19^1(f185_in) U19^1(f185_out1) -> F605_IN F605_IN -> U21^1(f563_in) U21^1(f563_out1) -> F613_IN F613_IN -> F563_IN F563_IN -> F599_IN F605_IN -> F563_IN The TRS R consists of the following rules: f2_in([]) -> f2_out1 f2_in(.(T7, [])) -> f2_out1 f2_in(T19) -> U1(f43_in(T19), T19) U1(f43_out1(X25, X26), T19) -> f2_out1 f361_in -> f361_out1 f361_in -> U2(f361_in) U2(f361_out1) -> f361_out1 f185_in -> U3(f361_in) U3(f361_out1) -> f185_out1 f563_in -> f563_out1 f563_in -> U4(f599_in) U4(f599_out1) -> f563_out1 f625_in -> f625_out1 f625_in -> U5(f812_in) U5(f812_out1(T110)) -> f625_out1 f625_in -> U6(f845_in) U6(f845_out1(T167)) -> f625_out1 f835_in -> f835_out1(0) f835_in -> U7(f835_in) U7(f835_out1(T147)) -> f835_out1(s(T147)) f850_in(T186) -> f850_out1([], T186) f850_in(T191) -> f850_out1(T191, []) f850_in(.(T212, T216)) -> U8(f887_in(T212, T216), .(T212, T216)) U8(f887_out1(T218, T217, T219), .(T212, T216)) -> f850_out1(.(T212, T218), .(T217, T219)) f850_in(.(T271, T273)) -> U9(f946_in(T271, T273), .(T271, T273)) U9(f946_out1(T274, T276, T275), .(T271, T273)) -> f850_out1(.(T274, T276), .(T271, T275)) f916_in(0) -> f916_out1 f916_in(s(T250)) -> U10(f916_in(T250), s(T250)) U10(f916_out1, s(T250)) -> f916_out1 f814_in -> f814_out1(0) f814_in -> U11(f835_in) U11(f835_out1(T132)) -> f814_out1(s(T132)) f897_in(0) -> f897_out1 f897_in(s(T236)) -> U12(f916_in(T236), s(T236)) U12(f916_out1, s(T236)) -> f897_out1 f43_in(T19) -> U13(f185_in, T19) U13(f185_out1, T19) -> U14(f215_in(T19), T19) U14(f215_out1(X25, X26), T19) -> f43_out1(X25, X26) f215_in(T19) -> U15(f563_in, T19) U15(f563_out1, T19) -> U16(f564_in(T19), T19) U16(f564_out1(T53, X26), T19) -> f215_out1(T53, X26) f564_in(T19) -> U17(f563_in, T19) U17(f563_out1, T19) -> U18(f850_in(T19), T19) U18(f850_out1(T179, T178), T19) -> f564_out1(T179, T178) f599_in -> U19(f185_in) U19(f185_out1) -> U20(f605_in) U20(f605_out1) -> f599_out1 f605_in -> U21(f563_in) U21(f563_out1) -> U22(f613_in) U22(f613_out1) -> f605_out1 f613_in -> U23(f563_in) U23(f563_out1) -> U24(f625_in) U24(f625_out1) -> f613_out1 f812_in -> U25(f814_in) U25(f814_out1(T110)) -> U26(f625_in, T110) U26(f625_out1, T110) -> f812_out1(T110) f845_in -> U27(f835_in) U27(f835_out1(T167)) -> U28(f625_in, T167) U28(f625_out1, T167) -> f845_out1(T167) f887_in(T212, T216) -> U29(f897_in(T212), T212, T216) U29(f897_out1, T212, T216) -> U30(f850_in(T216), T212, T216) U30(f850_out1(T222, .(T223, T224)), T212, T216) -> f887_out1(T222, T223, T224) f946_in(T271, T273) -> U31(f916_in(T271), T271, T273) U31(f916_out1, T271, T273) -> U32(f850_in(T273), T271, T273) U32(f850_out1(.(T279, T280), T281), T271, T273) -> f946_out1(T279, T280, T281) 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: F599_IN -> U19^1(f185_in) U19^1(f185_out1) -> F605_IN F605_IN -> U21^1(f563_in) U21^1(f563_out1) -> F613_IN F613_IN -> F563_IN F563_IN -> F599_IN F605_IN -> F563_IN The TRS R consists of the following rules: f563_in -> f563_out1 f563_in -> U4(f599_in) f599_in -> U19(f185_in) U4(f599_out1) -> f563_out1 f185_in -> U3(f361_in) U19(f185_out1) -> U20(f605_in) f605_in -> U21(f563_in) U20(f605_out1) -> f599_out1 U21(f563_out1) -> U22(f613_in) f613_in -> U23(f563_in) U22(f613_out1) -> f605_out1 U23(f563_out1) -> U24(f625_in) f625_in -> f625_out1 f625_in -> U5(f812_in) f625_in -> U6(f845_in) U24(f625_out1) -> f613_out1 f845_in -> U27(f835_in) U6(f845_out1(T167)) -> f625_out1 f835_in -> f835_out1(0) f835_in -> U7(f835_in) U27(f835_out1(T167)) -> U28(f625_in, T167) U28(f625_out1, T167) -> f845_out1(T167) U7(f835_out1(T147)) -> f835_out1(s(T147)) f812_in -> U25(f814_in) U5(f812_out1(T110)) -> f625_out1 f814_in -> f814_out1(0) f814_in -> U11(f835_in) U25(f814_out1(T110)) -> U26(f625_in, T110) U26(f625_out1, T110) -> f812_out1(T110) U11(f835_out1(T132)) -> f814_out1(s(T132)) f361_in -> f361_out1 f361_in -> U2(f361_in) U3(f361_out1) -> f185_out1 U2(f361_out1) -> f361_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (107) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 8, "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": { "1066": { "goal": [{ "clause": -1, "scope": -1, "term": "(less T271 T274)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T271"], "free": [], "exprvars": [] } }, "type": "Nodes", "590": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "591": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "592": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "593": { "goal": [{ "clause": 1, "scope": 5, "term": "(ms T23 X25)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X25"], "exprvars": [] } }, "594": { "goal": [{ "clause": 2, "scope": 5, "term": "(ms T23 X25)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X25"], "exprvars": [] } }, "990": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "595": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "870": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "475": { "goal": [{ "clause": -1, "scope": -1, "term": "(split T52 X80 X79)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X79", "X80" ], "exprvars": [] } }, "596": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "871": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "597": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "872": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": 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T223 T224) T216)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T216"], "free": [], "exprvars": [] } }, "859": { "goal": [ { "clause": 5, "scope": 6, "term": "(merge T77 T76 X108)" }, { "clause": 6, "scope": 6, "term": "(merge T77 T76 X108)" }, { "clause": 7, "scope": 6, "term": "(merge T77 T76 X108)" }, { "clause": 8, "scope": 6, "term": "(merge T77 T76 X108)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X108"], "exprvars": [] } }, "580": { "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": [] } }, "460": { "goal": [{ "clause": 3, "scope": 4, "term": "(split T45 X62 X61)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X61", "X62" ], "exprvars": [] } }, "461": { "goal": [{ "clause": 4, "scope": 4, "term": "(split T45 X62 X61)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X61", "X62" ], "exprvars": [] } }, "462": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "463": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "584": { "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": [] } }, "980": { "goal": [ { "clause": 9, "scope": 10, "term": "(less T212 (s T217))" }, { "clause": 10, "scope": 10, "term": "(less T212 (s T217))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T212"], "free": [], "exprvars": [] } }, "464": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "586": { "goal": [{ "clause": 0, "scope": 5, "term": "(ms T23 X25)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X25"], "exprvars": [] } }, "982": { "goal": [{ "clause": 9, "scope": 10, "term": "(less T212 (s T217))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T212"], "free": [], "exprvars": [] } }, "587": { "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": [] } }, "984": { "goal": [{ "clause": 10, "scope": 10, "term": "(less T212 (s T217))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T212"], "free": [], "exprvars": [] } }, "622": { "goal": [{ "clause": -1, "scope": -1, "term": "(ms T72 X106)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X106"], "exprvars": [] } }, "623": { "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": [] } }, "865": { "goal": [{ "clause": 5, "scope": 6, "term": "(merge T77 T76 X108)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X108"], "exprvars": [] } }, "866": { "goal": [ { "clause": 6, "scope": 6, "term": "(merge T77 T76 X108)" }, { "clause": 7, "scope": 6, "term": "(merge T77 T76 X108)" }, { "clause": 8, "scope": 6, "term": "(merge T77 T76 X108)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X108"], "exprvars": [] } }, "988": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "901": { "goal": [ { "clause": 9, "scope": 7, "term": "(less T110 (s T111))" }, { "clause": 10, "scope": 7, "term": "(less T110 (s T111))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "989": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "903": { "goal": [{ "clause": 9, "scope": 7, "term": "(less T110 (s T111))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "904": { "goal": [{ "clause": 10, "scope": 7, "term": "(less T110 (s T111))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 8, "to": 9, "label": "CASE" }, { "from": 9, "to": 10, "label": "PARALLEL" }, { "from": 9, "to": 11, "label": "PARALLEL" }, { "from": 10, "to": 14, "label": "EVAL with clause\nms([], []).\nand substitutionT1 -> [],\nT2 -> []" }, { "from": 10, "to": 16, "label": "EVAL-BACKTRACK" }, { "from": 11, "to": 19, "label": "PARALLEL" }, { "from": 11, "to": 21, "label": "PARALLEL" }, { "from": 14, "to": 18, "label": "SUCCESS" }, { "from": 19, "to": 29, "label": "EVAL with clause\nms(.(X6, []), .(X6, [])).\nand substitutionX6 -> T7,\nT1 -> .(T7, []),\nT2 -> .(T7, [])" }, { "from": 19, "to": 30, "label": "EVAL-BACKTRACK" }, { "from": 21, "to": 41, "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": 42, "label": "EVAL-BACKTRACK" }, { "from": 29, "to": 31, "label": "SUCCESS" }, { "from": 41, "to": 376, "label": "SPLIT 1" }, { "from": 41, "to": 377, "label": "SPLIT 2\nreplacements:X23 -> T23,\nX24 -> T24" }, { "from": 376, "to": 378, "label": "CASE" }, { "from": 377, "to": 579, "label": "SPLIT 1" }, { "from": 377, "to": 580, "label": "SPLIT 2\nreplacements:X25 -> T53,\nT24 -> T54" }, { "from": 378, "to": 379, "label": "BACKTRACK\nfor clause: split([], [], [])because of non-unification" }, { "from": 379, "to": 380, "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": 380, "to": 382, "label": "CASE" }, { "from": 382, "to": 386, "label": "BACKTRACK\nfor clause: split([], [], [])because of non-unification" }, { "from": 386, "to": 458, "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": 458, "to": 459, "label": "CASE" }, { "from": 459, "to": 460, "label": "PARALLEL" }, { "from": 459, "to": 461, "label": "PARALLEL" }, { "from": 460, "to": 462, "label": "EVAL with clause\nsplit([], [], []).\nand substitutionT45 -> [],\nX62 -> [],\nX61 -> []" }, { "from": 460, "to": 463, "label": "EVAL-BACKTRACK" }, { "from": 461, "to": 475, "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": 461, "to": 478, "label": "EVAL-BACKTRACK" }, { "from": 462, "to": 464, "label": "SUCCESS" }, { "from": 475, "to": 458, "label": "INSTANCE with matching:\nT45 -> T52\nX62 -> X80\nX61 -> X79" }, { "from": 579, "to": 584, "label": "CASE" }, { "from": 580, "to": 954, "label": "SPLIT 1" }, { "from": 580, "to": 955, "label": "SPLIT 2\nreplacements:X26 -> T178,\nT53 -> T179" }, { "from": 584, "to": 586, "label": "PARALLEL" }, { "from": 584, "to": 587, "label": "PARALLEL" }, { "from": 586, "to": 590, "label": "EVAL with clause\nms([], []).\nand substitutionT23 -> [],\nX25 -> []" }, { "from": 586, "to": 591, "label": "EVAL-BACKTRACK" }, { "from": 587, "to": 593, "label": "PARALLEL" }, { "from": 587, "to": 594, "label": "PARALLEL" }, { "from": 590, "to": 592, "label": "SUCCESS" }, { "from": 593, "to": 595, "label": "EVAL with clause\nms(.(X85, []), .(X85, [])).\nand substitutionX85 -> T59,\nT23 -> .(T59, []),\nX25 -> .(T59, [])" }, { "from": 593, "to": 596, "label": "EVAL-BACKTRACK" }, { "from": 594, "to": 606, "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": 594, "to": 607, "label": "EVAL-BACKTRACK" }, { "from": 595, "to": 597, "label": "SUCCESS" }, { "from": 606, "to": 610, "label": "SPLIT 1" }, { "from": 606, "to": 611, "label": "SPLIT 2\nreplacements:X104 -> T72,\nX105 -> T73" }, { "from": 610, "to": 376, "label": "INSTANCE with matching:\nT20 -> T69\nT21 -> T70\nT22 -> T71\nX23 -> X104\nX24 -> X105" }, { "from": 611, "to": 622, "label": "SPLIT 1" }, { "from": 611, "to": 623, "label": "SPLIT 2\nreplacements:X106 -> T74,\nT73 -> T75" }, { "from": 622, "to": 579, "label": "INSTANCE with matching:\nT23 -> T72\nX25 -> X106" }, { "from": 623, "to": 821, "label": "SPLIT 1" }, { "from": 623, "to": 822, "label": "SPLIT 2\nreplacements:X107 -> T76,\nT74 -> T77" }, { "from": 821, "to": 579, "label": "INSTANCE with matching:\nT23 -> T75\nX25 -> X107" }, { "from": 822, "to": 859, "label": "CASE" }, { "from": 859, "to": 865, "label": "PARALLEL" }, { "from": 859, "to": 866, "label": "PARALLEL" }, { "from": 865, "to": 870, "label": "EVAL with clause\nmerge([], X115, X115).\nand substitutionT77 -> [],\nT76 -> T84,\nX115 -> T84,\nX108 -> T84" }, { "from": 865, "to": 871, "label": "EVAL-BACKTRACK" }, { "from": 866, "to": 873, "label": "PARALLEL" }, { "from": 866, "to": 874, "label": "PARALLEL" }, { "from": 870, "to": 872, "label": "SUCCESS" }, { "from": 873, "to": 877, "label": "EVAL with clause\nmerge(X120, [], X120).\nand substitutionT77 -> T89,\nX120 -> T89,\nT76 -> [],\nX108 -> T89" }, { "from": 873, "to": 878, "label": "EVAL-BACKTRACK" }, { "from": 874, "to": 882, "label": "PARALLEL" }, { "from": 874, "to": 883, "label": "PARALLEL" }, { "from": 877, "to": 879, "label": "SUCCESS" }, { "from": 882, "to": 886, "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": 882, "to": 888, "label": "EVAL-BACKTRACK" }, { "from": 883, "to": 949, "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": 883, "to": 951, "label": "EVAL-BACKTRACK" }, { "from": 886, "to": 895, "label": "SPLIT 1" }, { "from": 886, "to": 896, "label": "SPLIT 2\nnew knowledge:\nT110 is ground\nreplacements:T112 -> T116,\nT111 -> T117,\nT113 -> T118" }, { "from": 895, "to": 901, "label": "CASE" }, { "from": 896, "to": 822, "label": "INSTANCE with matching:\nT77 -> T116\nT76 -> .(T117, T118)\nX108 -> X150" }, { "from": 901, "to": 903, "label": "PARALLEL" }, { "from": 901, "to": 904, "label": "PARALLEL" }, { "from": 903, "to": 926, "label": "EVAL with clause\nless(0, s(X159)).\nand substitutionT110 -> 0,\nT111 -> T125,\nX159 -> T125" }, { "from": 903, "to": 927, "label": "EVAL-BACKTRACK" }, { "from": 904, "to": 932, "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": 904, "to": 933, "label": "EVAL-BACKTRACK" }, { "from": 926, "to": 929, "label": "SUCCESS" }, { "from": 932, "to": 936, "label": "CASE" }, { "from": 936, "to": 937, "label": "PARALLEL" }, { "from": 936, "to": 938, "label": "PARALLEL" }, { "from": 937, "to": 939, "label": "EVAL with clause\nless(0, s(X172)).\nand substitutionT132 -> 0,\nX172 -> T140,\nT133 -> s(T140)" }, { "from": 937, "to": 940, "label": "EVAL-BACKTRACK" }, { "from": 938, "to": 943, "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": 938, "to": 945, "label": "EVAL-BACKTRACK" }, { "from": 939, "to": 941, "label": "SUCCESS" }, { "from": 943, "to": 932, "label": "INSTANCE with matching:\nT132 -> T147\nT133 -> T148" }, { "from": 949, "to": 952, "label": "SPLIT 1" }, { "from": 949, "to": 953, "label": "SPLIT 2\nnew knowledge:\nT167 is ground\nreplacements:T168 -> T173,\nT170 -> T174,\nT169 -> T175" }, { "from": 952, "to": 932, "label": "INSTANCE with matching:\nT132 -> T167\nT133 -> T168" }, { "from": 953, "to": 822, "label": "INSTANCE with matching:\nT77 -> .(T173, T174)\nT76 -> T175\nX108 -> X200" }, { "from": 954, "to": 579, "label": "INSTANCE with matching:\nT23 -> T54\nX25 -> X26" }, { "from": 955, "to": 956, "label": "CASE" }, { "from": 956, "to": 957, "label": "PARALLEL" }, { "from": 956, "to": 958, "label": "PARALLEL" }, { "from": 957, "to": 959, "label": "EVAL with clause\nmerge([], X213, X213).\nand substitutionT179 -> [],\nT178 -> T186,\nX213 -> T186,\nT19 -> T186" }, { "from": 957, "to": 960, "label": "EVAL-BACKTRACK" }, { "from": 958, "to": 962, "label": "PARALLEL" }, { "from": 958, "to": 963, "label": "PARALLEL" }, { "from": 959, "to": 961, "label": "SUCCESS" }, { "from": 962, "to": 964, "label": "EVAL with clause\nmerge(X218, [], X218).\nand substitutionT179 -> T191,\nX218 -> T191,\nT178 -> [],\nT19 -> T191" }, { "from": 962, "to": 965, "label": "EVAL-BACKTRACK" }, { "from": 963, "to": 969, "label": "PARALLEL" }, { "from": 963, "to": 970, "label": "PARALLEL" }, { "from": 964, "to": 966, "label": "SUCCESS" }, { "from": 969, "to": 971, "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": 969, "to": 972, "label": "EVAL-BACKTRACK" }, { "from": 970, "to": 1023, "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": 970, "to": 1024, "label": "EVAL-BACKTRACK" }, { "from": 971, "to": 975, "label": "SPLIT 1" }, { "from": 971, "to": 976, "label": "SPLIT 2\nnew knowledge:\nT212 is ground\nreplacements:T218 -> T222,\nT217 -> T223,\nT219 -> T224" }, { "from": 975, "to": 980, "label": "CASE" }, { "from": 976, "to": 955, "label": "INSTANCE with matching:\nT179 -> T222\nT178 -> .(T223, T224)\nT19 -> T216" }, { "from": 980, "to": 982, "label": "PARALLEL" }, { "from": 980, "to": 984, "label": "PARALLEL" }, { "from": 982, "to": 988, "label": "EVAL with clause\nless(0, s(X252)).\nand substitutionT212 -> 0,\nT217 -> T231,\nX252 -> T231" }, { "from": 982, "to": 989, "label": "EVAL-BACKTRACK" }, { "from": 984, "to": 996, "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": 984, "to": 997, "label": "EVAL-BACKTRACK" }, { "from": 988, "to": 990, "label": "SUCCESS" }, { "from": 996, "to": 1001, "label": "CASE" }, { "from": 1001, "to": 1002, "label": "PARALLEL" }, { "from": 1001, "to": 1003, "label": "PARALLEL" }, { "from": 1002, "to": 1004, "label": "EVAL with clause\nless(0, s(X265)).\nand substitutionT236 -> 0,\nX265 -> T245,\nT238 -> s(T245)" }, { "from": 1002, "to": 1005, "label": "EVAL-BACKTRACK" }, { "from": 1003, "to": 1007, "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": 1003, "to": 1008, "label": "EVAL-BACKTRACK" }, { "from": 1004, "to": 1006, "label": "SUCCESS" }, { "from": 1007, "to": 996, "label": "INSTANCE with matching:\nT236 -> T250\nT238 -> T252" }, { "from": 1023, "to": 1066, "label": "SPLIT 1" }, { "from": 1023, "to": 1067, "label": "SPLIT 2\nnew knowledge:\nT271 is ground\nreplacements:T274 -> T279,\nT276 -> T280,\nT275 -> T281" }, { "from": 1066, "to": 996, "label": "INSTANCE with matching:\nT236 -> T271\nT238 -> T274" }, { "from": 1067, "to": 955, "label": "INSTANCE with matching:\nT179 -> .(T279, T280)\nT178 -> T281\nT19 -> T273" } ], "type": "Graph" } } ---------------------------------------- (108) Complex Obligation (AND) ---------------------------------------- (109) Obligation: Rules: f1003_in(s(T250)) -> f1007_in(T250) :|: TRUE f1008_out -> f1003_out(T236) :|: TRUE f1003_in(x) -> f1008_in :|: TRUE f1007_out(x1) -> f1003_out(s(x1)) :|: TRUE f996_out(x2) -> f1007_out(x2) :|: TRUE f1007_in(x3) -> f996_in(x3) :|: TRUE f1001_out(x4) -> f996_out(x4) :|: TRUE f996_in(x5) -> f1001_in(x5) :|: TRUE f1001_in(x6) -> f1002_in(x6) :|: TRUE f1002_out(x7) -> f1001_out(x7) :|: TRUE f1003_out(x8) -> f1001_out(x8) :|: TRUE f1001_in(x9) -> f1003_in(x9) :|: TRUE f9_out(T2) -> f8_out(T2) :|: TRUE f8_in(x10) -> f9_in(x10) :|: TRUE f9_in(x11) -> f11_in(x11) :|: TRUE f11_out(x12) -> f9_out(x12) :|: TRUE f9_in(x13) -> f10_in(x13) :|: TRUE f10_out(x14) -> f9_out(x14) :|: TRUE f11_in(x15) -> f19_in(x15) :|: TRUE f19_out(x16) -> f11_out(x16) :|: TRUE f11_in(x17) -> f21_in(x17) :|: TRUE f21_out(x18) -> f11_out(x18) :|: TRUE f41_out(T19) -> f21_out(T19) :|: TRUE f21_in(x19) -> f41_in(x19) :|: TRUE f42_out -> f21_out(x20) :|: TRUE f21_in(x21) -> f42_in :|: TRUE f377_out(x22) -> f41_out(x22) :|: TRUE f41_in(x23) -> f376_in :|: TRUE f376_out -> f377_in(x24) :|: TRUE f377_in(x25) -> f579_in :|: TRUE f580_out(x26) -> f377_out(x26) :|: TRUE f579_out -> f580_in(x27) :|: TRUE f955_out(x28) -> f580_out(x28) :|: TRUE f954_out -> f955_in(x29) :|: TRUE f580_in(x30) -> f954_in :|: TRUE f956_out(x31) -> f955_out(x31) :|: TRUE f955_in(x32) -> f956_in(x32) :|: TRUE f956_in(x33) -> f957_in(x33) :|: TRUE f956_in(x34) -> f958_in(x34) :|: TRUE f957_out(x35) -> f956_out(x35) :|: TRUE f958_out(x36) -> f956_out(x36) :|: TRUE f962_out(x37) -> f958_out(x37) :|: TRUE f963_out(x38) -> f958_out(x38) :|: TRUE f958_in(x39) -> f963_in(x39) :|: TRUE f958_in(x40) -> f962_in(x40) :|: TRUE f963_in(x41) -> f970_in(x41) :|: TRUE f969_out(x42) -> f963_out(x42) :|: TRUE f970_out(x43) -> f963_out(x43) :|: TRUE f963_in(x44) -> f969_in(x44) :|: TRUE f1023_out(T271, T273) -> f970_out(.(T271, T273)) :|: TRUE f970_in(x45) -> f1024_in :|: TRUE f970_in(.(x46, x47)) -> f1023_in(x46, x47) :|: TRUE f1024_out -> f970_out(x48) :|: TRUE f1023_in(x49, x50) -> f1066_in(x49) :|: TRUE f1067_out(x51) -> f1023_out(x52, x51) :|: TRUE f1066_out(x53) -> f1067_in(x54) :|: TRUE f1066_in(x55) -> f996_in(x55) :|: TRUE f996_out(x56) -> f1066_out(x56) :|: TRUE f969_in(x57) -> f972_in :|: TRUE f971_out(T212, T216) -> f969_out(.(T212, T216)) :|: TRUE f969_in(.(x58, x59)) -> f971_in(x58, x59) :|: TRUE f972_out -> f969_out(x60) :|: TRUE f971_in(x61, x62) -> f975_in(x61) :|: TRUE f975_out(x63) -> f976_in(x64) :|: TRUE f976_out(x65) -> f971_out(x66, x65) :|: TRUE f975_in(x67) -> f980_in(x67) :|: TRUE f980_out(x68) -> f975_out(x68) :|: TRUE f980_in(x69) -> f982_in(x69) :|: TRUE f980_in(x70) -> f984_in(x70) :|: TRUE f982_out(x71) -> f980_out(x71) :|: TRUE f984_out(x72) -> f980_out(x72) :|: TRUE f984_in(x73) -> f997_in :|: TRUE f996_out(x74) -> f984_out(s(x74)) :|: TRUE f984_in(s(x75)) -> f996_in(x75) :|: TRUE f997_out -> f984_out(x76) :|: TRUE Start term: f8_in(T2) ---------------------------------------- (110) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (111) TRUE ---------------------------------------- (112) Obligation: Rules: f1023_in(T271, T273) -> f1066_in(T271) :|: TRUE f1067_out(x) -> f1023_out(x1, x) :|: TRUE f1066_out(x2) -> f1067_in(x3) :|: TRUE f962_out(T19) -> f958_out(T19) :|: TRUE f963_out(x4) -> f958_out(x4) :|: TRUE f958_in(x5) -> f963_in(x5) :|: TRUE f958_in(x6) -> f962_in(x6) :|: TRUE f988_in -> f988_out :|: TRUE f1001_in(T236) -> f1002_in(T236) :|: TRUE f1002_out(x7) -> f1001_out(x7) :|: TRUE f1003_out(x8) -> f1001_out(x8) :|: TRUE f1001_in(x9) -> f1003_in(x9) :|: TRUE f975_in(T212) -> f980_in(T212) :|: TRUE f980_out(x10) -> f975_out(x10) :|: TRUE f980_in(x11) -> f982_in(x11) :|: TRUE f980_in(x12) -> f984_in(x12) :|: TRUE f982_out(x13) -> f980_out(x13) :|: TRUE f984_out(x14) -> f980_out(x14) :|: TRUE f956_in(x15) -> f957_in(x15) :|: TRUE f956_in(x16) -> f958_in(x16) :|: TRUE f957_out(x17) -> f956_out(x17) :|: TRUE f958_out(x18) -> f956_out(x18) :|: TRUE f1003_in(s(T250)) -> f1007_in(T250) :|: TRUE f1008_out -> f1003_out(x19) :|: TRUE f1003_in(x20) -> f1008_in :|: TRUE f1007_out(x21) -> f1003_out(s(x21)) :|: TRUE f996_out(x22) -> f1007_out(x22) :|: TRUE f1007_in(x23) -> f996_in(x23) :|: TRUE f1023_out(x24, x25) -> f970_out(.(x24, x25)) :|: TRUE f970_in(x26) -> f1024_in :|: TRUE f970_in(.(x27, x28)) -> f1023_in(x27, x28) :|: TRUE f1024_out -> f970_out(x29) :|: TRUE f1066_in(x30) -> f996_in(x30) :|: TRUE f996_out(x31) -> f1066_out(x31) :|: TRUE f984_in(x32) -> f997_in :|: TRUE f996_out(x33) -> f984_out(s(x33)) :|: TRUE f984_in(s(x34)) -> f996_in(x34) :|: TRUE f997_out -> f984_out(x35) :|: TRUE f976_in(T216) -> f955_in(T216) :|: TRUE f955_out(x36) -> f976_out(x36) :|: TRUE f1067_in(x37) -> f955_in(x37) :|: TRUE f955_out(x38) -> f1067_out(x38) :|: TRUE f1001_out(x39) -> f996_out(x39) :|: TRUE f996_in(x40) -> f1001_in(x40) :|: TRUE f956_out(x41) -> f955_out(x41) :|: TRUE f955_in(x42) -> f956_in(x42) :|: TRUE f963_in(x43) -> f970_in(x43) :|: TRUE f969_out(x44) -> f963_out(x44) :|: TRUE f970_out(x45) -> f963_out(x45) :|: TRUE f963_in(x46) -> f969_in(x46) :|: TRUE f969_in(x47) -> f972_in :|: TRUE f971_out(x48, x49) -> f969_out(.(x48, x49)) :|: TRUE f969_in(.(x50, x51)) -> f971_in(x50, x51) :|: TRUE f972_out -> f969_out(x52) :|: TRUE f1002_in(x53) -> f1005_in :|: TRUE f1004_out -> f1002_out(0) :|: TRUE f1002_in(0) -> f1004_in :|: TRUE f1005_out -> f1002_out(x54) :|: TRUE f1004_in -> f1004_out :|: TRUE f982_in(0) -> f988_in :|: TRUE f989_out -> f982_out(x55) :|: TRUE f988_out -> f982_out(0) :|: TRUE f982_in(x56) -> f989_in :|: TRUE f971_in(x57, x58) -> f975_in(x57) :|: TRUE f975_out(x59) -> f976_in(x60) :|: TRUE f976_out(x61) -> f971_out(x62, x61) :|: TRUE f9_out(T2) -> f8_out(T2) :|: TRUE f8_in(x63) -> f9_in(x63) :|: TRUE f9_in(x64) -> f11_in(x64) :|: TRUE f11_out(x65) -> f9_out(x65) :|: TRUE f9_in(x66) -> f10_in(x66) :|: TRUE f10_out(x67) -> f9_out(x67) :|: TRUE f11_in(x68) -> f19_in(x68) :|: TRUE f19_out(x69) -> f11_out(x69) :|: TRUE f11_in(x70) -> f21_in(x70) :|: TRUE f21_out(x71) -> f11_out(x71) :|: TRUE f41_out(x72) -> f21_out(x72) :|: TRUE f21_in(x73) -> f41_in(x73) :|: TRUE f42_out -> f21_out(x74) :|: TRUE f21_in(x75) -> f42_in :|: TRUE f377_out(x76) -> f41_out(x76) :|: TRUE f41_in(x77) -> f376_in :|: TRUE f376_out -> f377_in(x78) :|: TRUE f377_in(x79) -> f579_in :|: TRUE f580_out(x80) -> f377_out(x80) :|: TRUE f579_out -> f580_in(x81) :|: TRUE f955_out(x82) -> f580_out(x82) :|: TRUE f954_out -> f955_in(x83) :|: TRUE f580_in(x84) -> f954_in :|: TRUE Start term: f8_in(T2) ---------------------------------------- (113) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (114) TRUE ---------------------------------------- (115) Obligation: Rules: f938_out -> f936_out :|: TRUE f937_out -> f936_out :|: TRUE f936_in -> f938_in :|: TRUE f936_in -> f937_in :|: TRUE f945_out -> f938_out :|: TRUE f943_out -> f938_out :|: TRUE f938_in -> f945_in :|: TRUE f938_in -> f943_in :|: TRUE f943_in -> f932_in :|: TRUE f932_out -> f943_out :|: TRUE f932_in -> f936_in :|: TRUE f936_out -> f932_out :|: TRUE f9_out(T2) -> f8_out(T2) :|: TRUE f8_in(x) -> f9_in(x) :|: TRUE f9_in(x1) -> f11_in(x1) :|: TRUE f11_out(x2) -> f9_out(x2) :|: TRUE f9_in(x3) -> f10_in(x3) :|: TRUE f10_out(x4) -> f9_out(x4) :|: TRUE f11_in(x5) -> f19_in(x5) :|: TRUE f19_out(x6) -> f11_out(x6) :|: TRUE f11_in(x7) -> f21_in(x7) :|: TRUE f21_out(x8) -> f11_out(x8) :|: TRUE f41_out(T19) -> f21_out(T19) :|: TRUE f21_in(x9) -> f41_in(x9) :|: TRUE f42_out -> f21_out(x10) :|: TRUE f21_in(x11) -> f42_in :|: TRUE f377_out(x12) -> f41_out(x12) :|: TRUE f41_in(x13) -> f376_in :|: TRUE f376_out -> f377_in(x14) :|: TRUE f377_in(x15) -> f579_in :|: TRUE f580_out(x16) -> f377_out(x16) :|: TRUE f579_out -> f580_in(x17) :|: TRUE f579_in -> f584_in :|: TRUE f584_out -> f579_out :|: TRUE f584_in -> f586_in :|: TRUE f584_in -> f587_in :|: TRUE f587_out -> f584_out :|: TRUE f586_out -> f584_out :|: TRUE f593_out -> f587_out :|: TRUE f587_in -> f594_in :|: TRUE f594_out -> f587_out :|: TRUE f587_in -> f593_in :|: TRUE f594_in -> f607_in :|: TRUE f607_out -> f594_out :|: TRUE f594_in -> f606_in :|: TRUE f606_out -> f594_out :|: TRUE f610_out -> f611_in :|: TRUE f611_out -> f606_out :|: TRUE f606_in -> f610_in :|: TRUE f622_out -> f623_in :|: TRUE f623_out -> f611_out :|: TRUE f611_in -> f622_in :|: TRUE f822_out -> f623_out :|: TRUE f623_in -> f821_in :|: TRUE f821_out -> f822_in :|: TRUE f859_out -> f822_out :|: TRUE f822_in -> f859_in :|: TRUE f866_out -> f859_out :|: TRUE f859_in -> f866_in :|: TRUE f859_in -> f865_in :|: TRUE f865_out -> f859_out :|: TRUE f866_in -> f874_in :|: TRUE f874_out -> f866_out :|: TRUE f866_in -> f873_in :|: TRUE f873_out -> f866_out :|: TRUE f883_out -> f874_out :|: TRUE f882_out -> f874_out :|: TRUE f874_in -> f883_in :|: TRUE f874_in -> f882_in :|: TRUE f883_in -> f951_in :|: TRUE f883_in -> f949_in :|: TRUE f951_out -> f883_out :|: TRUE f949_out -> f883_out :|: TRUE f953_out -> f949_out :|: TRUE f949_in -> f952_in :|: TRUE f952_out -> f953_in :|: TRUE f952_in -> f932_in :|: TRUE f932_out -> f952_out :|: TRUE f955_out(x18) -> f580_out(x18) :|: TRUE f954_out -> f955_in(x19) :|: TRUE f580_in(x20) -> f954_in :|: TRUE f954_in -> f579_in :|: TRUE f579_out -> f954_out :|: TRUE f882_in -> f886_in :|: TRUE f888_out -> f882_out :|: TRUE f886_out -> f882_out :|: TRUE f882_in -> f888_in :|: TRUE f896_out -> f886_out :|: TRUE f895_out -> f896_in :|: TRUE f886_in -> f895_in :|: TRUE f901_out -> f895_out :|: TRUE f895_in -> f901_in :|: TRUE f901_in -> f903_in :|: TRUE f901_in -> f904_in :|: TRUE f903_out -> f901_out :|: TRUE f904_out -> f901_out :|: TRUE f933_out -> f904_out :|: TRUE f904_in -> f933_in :|: TRUE f904_in -> f932_in :|: TRUE f932_out -> f904_out :|: TRUE Start term: f8_in(T2) ---------------------------------------- (116) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (117) TRUE ---------------------------------------- (118) Obligation: Rules: f938_out -> f936_out :|: TRUE f937_out -> f936_out :|: TRUE f936_in -> f938_in :|: TRUE f936_in -> f937_in :|: TRUE f939_in -> f939_out :|: TRUE f952_in -> f932_in :|: TRUE f932_out -> f952_out :|: TRUE f896_out -> f886_out :|: TRUE f895_out -> f896_in :|: TRUE f886_in -> f895_in :|: TRUE f926_in -> f926_out :|: TRUE f822_out -> f953_out :|: TRUE f953_in -> f822_in :|: TRUE f932_in -> f936_in :|: TRUE f936_out -> f932_out :|: TRUE f883_out -> f874_out :|: TRUE f882_out -> f874_out :|: TRUE f874_in -> f883_in :|: TRUE f874_in -> f882_in :|: TRUE f901_out -> f895_out :|: TRUE f895_in -> f901_in :|: TRUE f945_out -> f938_out :|: TRUE f943_out -> f938_out :|: TRUE f938_in -> f945_in :|: TRUE f938_in -> f943_in :|: TRUE f866_out -> f859_out :|: TRUE f859_in -> f866_in :|: TRUE f859_in -> f865_in :|: TRUE f865_out -> f859_out :|: TRUE f933_out -> f904_out :|: TRUE f904_in -> f933_in :|: TRUE f904_in -> f932_in :|: TRUE f932_out -> f904_out :|: TRUE f882_in -> f886_in :|: TRUE f888_out -> f882_out :|: TRUE f886_out -> f882_out :|: TRUE f882_in -> f888_in :|: TRUE f901_in -> f903_in :|: TRUE f901_in -> f904_in :|: TRUE f903_out -> f901_out :|: TRUE f904_out -> f901_out :|: TRUE f927_out -> f903_out :|: TRUE f926_out -> f903_out :|: TRUE f903_in -> f926_in :|: TRUE f903_in -> f927_in :|: TRUE f859_out -> f822_out :|: TRUE f822_in -> f859_in :|: TRUE f883_in -> f951_in :|: TRUE f883_in -> f949_in :|: TRUE f951_out -> f883_out :|: TRUE f949_out -> f883_out :|: TRUE f943_in -> f932_in :|: TRUE f932_out -> f943_out :|: TRUE f953_out -> f949_out :|: TRUE f949_in -> f952_in :|: TRUE f952_out -> f953_in :|: TRUE f939_out -> f937_out :|: TRUE f937_in -> f940_in :|: TRUE f937_in -> f939_in :|: TRUE f940_out -> f937_out :|: TRUE f866_in -> f874_in :|: TRUE f874_out -> f866_out :|: TRUE f866_in -> f873_in :|: TRUE f873_out -> f866_out :|: TRUE f822_out -> f896_out :|: TRUE f896_in -> f822_in :|: TRUE f9_out(T2) -> f8_out(T2) :|: TRUE f8_in(x) -> f9_in(x) :|: TRUE f9_in(x1) -> f11_in(x1) :|: TRUE f11_out(x2) -> f9_out(x2) :|: TRUE f9_in(x3) -> f10_in(x3) :|: TRUE f10_out(x4) -> f9_out(x4) :|: TRUE f11_in(x5) -> f19_in(x5) :|: TRUE f19_out(x6) -> f11_out(x6) :|: TRUE f11_in(x7) -> f21_in(x7) :|: TRUE f21_out(x8) -> f11_out(x8) :|: TRUE f41_out(T19) -> f21_out(T19) :|: TRUE f21_in(x9) -> f41_in(x9) :|: TRUE f42_out -> f21_out(x10) :|: TRUE f21_in(x11) -> f42_in :|: TRUE f377_out(x12) -> f41_out(x12) :|: TRUE f41_in(x13) -> f376_in :|: TRUE f376_out -> f377_in(x14) :|: TRUE f377_in(x15) -> f579_in :|: TRUE f580_out(x16) -> f377_out(x16) :|: TRUE f579_out -> f580_in(x17) :|: TRUE f579_in -> f584_in :|: TRUE f584_out -> f579_out :|: TRUE f584_in -> f586_in :|: TRUE f584_in -> f587_in :|: TRUE f587_out -> f584_out :|: TRUE f586_out -> f584_out :|: TRUE f593_out -> f587_out :|: TRUE f587_in -> f594_in :|: TRUE f594_out -> f587_out :|: TRUE f587_in -> f593_in :|: TRUE f594_in -> f607_in :|: TRUE f607_out -> f594_out :|: TRUE f594_in -> f606_in :|: TRUE f606_out -> f594_out :|: TRUE f610_out -> f611_in :|: TRUE f611_out -> f606_out :|: TRUE f606_in -> f610_in :|: TRUE f622_out -> f623_in :|: TRUE f623_out -> f611_out :|: TRUE f611_in -> f622_in :|: TRUE f822_out -> f623_out :|: TRUE f623_in -> f821_in :|: TRUE f821_out -> f822_in :|: TRUE f955_out(x18) -> f580_out(x18) :|: TRUE f954_out -> f955_in(x19) :|: TRUE f580_in(x20) -> f954_in :|: TRUE f954_in -> f579_in :|: TRUE f579_out -> f954_out :|: TRUE Start term: f8_in(T2) ---------------------------------------- (119) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (120) TRUE ---------------------------------------- (121) Obligation: Rules: f478_out -> f461_out :|: TRUE f461_in -> f478_in :|: TRUE f461_in -> f475_in :|: TRUE f475_out -> f461_out :|: TRUE f458_out -> f475_out :|: TRUE f475_in -> f458_in :|: TRUE f461_out -> f459_out :|: TRUE f459_in -> f461_in :|: TRUE f459_in -> f460_in :|: TRUE f460_out -> f459_out :|: TRUE f459_out -> f458_out :|: TRUE f458_in -> f459_in :|: TRUE f9_out(T2) -> f8_out(T2) :|: TRUE f8_in(x) -> f9_in(x) :|: TRUE f9_in(x1) -> f11_in(x1) :|: TRUE f11_out(x2) -> f9_out(x2) :|: TRUE f9_in(x3) -> f10_in(x3) :|: TRUE f10_out(x4) -> f9_out(x4) :|: TRUE f11_in(x5) -> f19_in(x5) :|: TRUE f19_out(x6) -> f11_out(x6) :|: TRUE f11_in(x7) -> f21_in(x7) :|: TRUE f21_out(x8) -> f11_out(x8) :|: TRUE f41_out(T19) -> f21_out(T19) :|: TRUE f21_in(x9) -> f41_in(x9) :|: TRUE f42_out -> f21_out(x10) :|: TRUE f21_in(x11) -> f42_in :|: TRUE f377_out(x12) -> f41_out(x12) :|: TRUE f41_in(x13) -> f376_in :|: TRUE f376_out -> f377_in(x14) :|: TRUE f376_in -> f378_in :|: TRUE f378_out -> f376_out :|: TRUE f379_out -> f378_out :|: TRUE f378_in -> f379_in :|: TRUE f379_in -> f380_in :|: TRUE f380_out -> f379_out :|: TRUE f380_in -> f382_in :|: TRUE f382_out -> f380_out :|: TRUE f382_in -> f386_in :|: TRUE f386_out -> f382_out :|: TRUE f458_out -> f386_out :|: TRUE f386_in -> f458_in :|: TRUE f377_in(x15) -> f579_in :|: TRUE f580_out(x16) -> f377_out(x16) :|: TRUE f579_out -> f580_in(x17) :|: TRUE f579_in -> f584_in :|: TRUE f584_out -> f579_out :|: TRUE f584_in -> f586_in :|: TRUE f584_in -> f587_in :|: TRUE f587_out -> f584_out :|: TRUE f586_out -> f584_out :|: TRUE f593_out -> f587_out :|: TRUE f587_in -> f594_in :|: TRUE f594_out -> f587_out :|: TRUE f587_in -> f593_in :|: TRUE f594_in -> f607_in :|: TRUE f607_out -> f594_out :|: TRUE f594_in -> f606_in :|: TRUE f606_out -> f594_out :|: TRUE f610_out -> f611_in :|: TRUE f611_out -> f606_out :|: TRUE f606_in -> f610_in :|: TRUE f376_out -> f610_out :|: TRUE f610_in -> f376_in :|: TRUE f955_out(x18) -> f580_out(x18) :|: TRUE f954_out -> f955_in(x19) :|: TRUE f580_in(x20) -> f954_in :|: TRUE f954_in -> f579_in :|: TRUE f579_out -> f954_out :|: TRUE Start term: f8_in(T2) ---------------------------------------- (122) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f461_in -> f475_in :|: TRUE f475_in -> f458_in :|: TRUE f459_in -> f461_in :|: TRUE f458_in -> f459_in :|: TRUE ---------------------------------------- (123) Obligation: Rules: f461_in -> f475_in :|: TRUE f475_in -> f458_in :|: TRUE f459_in -> f461_in :|: TRUE f458_in -> f459_in :|: TRUE ---------------------------------------- (124) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (125) Obligation: Rules: f459_in -> f459_in :|: TRUE ---------------------------------------- (126) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (127) Obligation: Rules: f459_in -> f459_in :|: TRUE ---------------------------------------- (128) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f459_in -> f459_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (129) Obligation: Termination digraph: Nodes: (1) f459_in -> f459_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (130) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f459_in() Replaced non-predefined constructor symbols by 0. ---------------------------------------- (131) Obligation: Rules: f459_in -> f459_in :|: TRUE ---------------------------------------- (132) IntTRSPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc) -> f(1) :|: pc = 1 && TRUE Witness term starting non-terminating reduction: f(1) ---------------------------------------- (133) NO ---------------------------------------- (134) Obligation: Rules: f952_in -> f932_in :|: TRUE f932_out -> f952_out :|: TRUE f939_in -> f939_out :|: TRUE f877_in -> f877_out :|: TRUE f380_in -> f382_in :|: TRUE f382_out -> f380_out :|: TRUE f896_out -> f886_out :|: TRUE f895_out -> f896_in :|: TRUE f886_in -> f895_in :|: TRUE f926_in -> f926_out :|: TRUE f870_in -> f870_out :|: TRUE f822_out -> f953_out :|: TRUE f953_in -> f822_in :|: TRUE f932_in -> f936_in :|: TRUE f936_out -> f932_out :|: TRUE f622_out -> f623_in :|: TRUE f623_out -> f611_out :|: TRUE f611_in -> f622_in :|: TRUE f376_out -> f610_out :|: TRUE f610_in -> f376_in :|: TRUE f822_out -> f623_out :|: TRUE f623_in -> f821_in :|: TRUE f821_out -> f822_in :|: TRUE f382_in -> f386_in :|: TRUE f386_out -> f382_out :|: TRUE f866_out -> f859_out :|: TRUE f859_in -> f866_in :|: TRUE f859_in -> f865_in :|: TRUE f865_out -> f859_out :|: TRUE f933_out -> f904_out :|: TRUE f904_in -> f933_in :|: TRUE f904_in -> f932_in :|: TRUE f932_out -> f904_out :|: TRUE f593_out -> f587_out :|: TRUE f587_in -> f594_in :|: TRUE f594_out -> f587_out :|: TRUE f587_in -> f593_in :|: TRUE f859_out -> f822_out :|: TRUE f822_in -> f859_in :|: TRUE f943_in -> f932_in :|: TRUE f932_out -> f943_out :|: TRUE f939_out -> f937_out :|: TRUE f937_in -> f940_in :|: TRUE f937_in -> f939_in :|: TRUE f940_out -> f937_out :|: TRUE f822_out -> f896_out :|: TRUE f896_in -> f822_in :|: TRUE f938_out -> f936_out :|: TRUE f937_out -> f936_out :|: TRUE f936_in -> f938_in :|: TRUE f936_in -> f937_in :|: TRUE f579_out -> f622_out :|: TRUE f622_in -> f579_in :|: TRUE f462_in -> f462_out :|: TRUE f871_out -> f865_out :|: TRUE f865_in -> f870_in :|: TRUE f870_out -> f865_out :|: TRUE f865_in -> f871_in :|: TRUE f458_out -> f386_out :|: TRUE f386_in -> f458_in :|: TRUE f610_out -> f611_in :|: TRUE f611_out -> f606_out :|: TRUE f606_in -> f610_in :|: TRUE f877_out -> f873_out :|: TRUE f878_out -> f873_out :|: TRUE f873_in -> f878_in :|: TRUE f873_in -> f877_in :|: TRUE f883_out -> f874_out :|: TRUE f882_out -> f874_out :|: TRUE f874_in -> f883_in :|: TRUE f874_in -> f882_in :|: TRUE f594_in -> f607_in :|: TRUE f607_out -> f594_out :|: TRUE f594_in -> f606_in :|: TRUE f606_out -> f594_out :|: TRUE f376_in -> f378_in :|: TRUE f378_out -> f376_out :|: TRUE f584_in -> f586_in :|: TRUE f584_in -> f587_in :|: TRUE f587_out -> f584_out :|: TRUE f586_out -> f584_out :|: TRUE f901_out -> f895_out :|: TRUE f895_in -> f901_in :|: TRUE f459_out -> f458_out :|: TRUE f458_in -> f459_in :|: TRUE f945_out -> f938_out :|: TRUE f943_out -> f938_out :|: TRUE f938_in -> f945_in :|: TRUE f938_in -> f943_in :|: TRUE f379_in -> f380_in :|: TRUE f380_out -> f379_out :|: TRUE f463_out -> f460_out :|: TRUE f462_out -> f460_out :|: TRUE f460_in -> f463_in :|: TRUE f460_in -> f462_in :|: TRUE f579_out -> f821_out :|: TRUE f821_in -> f579_in :|: TRUE f478_out -> f461_out :|: TRUE f461_in -> f478_in :|: TRUE f461_in -> f475_in :|: TRUE f475_out -> f461_out :|: TRUE f379_out -> f378_out :|: TRUE f378_in -> f379_in :|: TRUE f882_in -> f886_in :|: TRUE f888_out -> f882_out :|: TRUE f886_out -> f882_out :|: TRUE f882_in -> f888_in :|: TRUE f901_in -> f903_in :|: TRUE f901_in -> f904_in :|: TRUE f903_out -> f901_out :|: TRUE f904_out -> f901_out :|: TRUE f927_out -> f903_out :|: TRUE f926_out -> f903_out :|: TRUE f903_in -> f926_in :|: TRUE f903_in -> f927_in :|: TRUE f579_in -> f584_in :|: TRUE f584_out -> f579_out :|: TRUE f883_in -> f951_in :|: TRUE f883_in -> f949_in :|: TRUE f951_out -> f883_out :|: TRUE f949_out -> f883_out :|: TRUE f953_out -> f949_out :|: TRUE f949_in -> f952_in :|: TRUE f952_out -> f953_in :|: TRUE f458_out -> f475_out :|: TRUE f475_in -> f458_in :|: TRUE f461_out -> f459_out :|: TRUE f459_in -> f461_in :|: TRUE f459_in -> f460_in :|: TRUE f460_out -> f459_out :|: TRUE f866_in -> f874_in :|: TRUE f874_out -> f866_out :|: TRUE f866_in -> f873_in :|: TRUE f873_out -> f866_out :|: TRUE f9_out(T2) -> f8_out(T2) :|: TRUE f8_in(x) -> f9_in(x) :|: TRUE f9_in(x1) -> f11_in(x1) :|: TRUE f11_out(x2) -> f9_out(x2) :|: TRUE f9_in(x3) -> f10_in(x3) :|: TRUE f10_out(x4) -> f9_out(x4) :|: TRUE f11_in(x5) -> f19_in(x5) :|: TRUE f19_out(x6) -> f11_out(x6) :|: TRUE f11_in(x7) -> f21_in(x7) :|: TRUE f21_out(x8) -> f11_out(x8) :|: TRUE f41_out(T19) -> f21_out(T19) :|: TRUE f21_in(x9) -> f41_in(x9) :|: TRUE f42_out -> f21_out(x10) :|: TRUE f21_in(x11) -> f42_in :|: TRUE f377_out(x12) -> f41_out(x12) :|: TRUE f41_in(x13) -> f376_in :|: TRUE f376_out -> f377_in(x14) :|: TRUE f377_in(x15) -> f579_in :|: TRUE f580_out(x16) -> f377_out(x16) :|: TRUE f579_out -> f580_in(x17) :|: TRUE f955_out(x18) -> f580_out(x18) :|: TRUE f954_out -> f955_in(x19) :|: TRUE f580_in(x20) -> f954_in :|: TRUE f954_in -> f579_in :|: TRUE f579_out -> f954_out :|: TRUE Start term: f8_in(T2) ---------------------------------------- (135) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f380_in -> f382_in :|: TRUE f382_out -> f380_out :|: TRUE f611_in -> f622_in :|: TRUE f376_out -> f610_out :|: TRUE f610_in -> f376_in :|: TRUE f382_in -> f386_in :|: TRUE f386_out -> f382_out :|: TRUE f587_in -> f594_in :|: TRUE f622_in -> f579_in :|: TRUE f462_in -> f462_out :|: TRUE f458_out -> f386_out :|: TRUE f386_in -> f458_in :|: TRUE f610_out -> f611_in :|: TRUE f606_in -> f610_in :|: TRUE f594_in -> f606_in :|: TRUE f376_in -> f378_in :|: TRUE f378_out -> f376_out :|: TRUE f584_in -> f587_in :|: TRUE f459_out -> f458_out :|: TRUE f458_in -> f459_in :|: TRUE f379_in -> f380_in :|: TRUE f380_out -> f379_out :|: TRUE f462_out -> f460_out :|: TRUE f460_in -> f462_in :|: TRUE f461_in -> f475_in :|: TRUE f475_out -> f461_out :|: TRUE f379_out -> f378_out :|: TRUE f378_in -> f379_in :|: TRUE f579_in -> f584_in :|: TRUE f458_out -> f475_out :|: TRUE f475_in -> f458_in :|: TRUE f461_out -> f459_out :|: TRUE f459_in -> f461_in :|: TRUE f459_in -> f460_in :|: TRUE f460_out -> f459_out :|: TRUE f376_out -> f377_in(x14) :|: TRUE f377_in(x15) -> f579_in :|: TRUE ---------------------------------------- (136) Obligation: Rules: f380_in -> f382_in :|: TRUE f382_out -> f380_out :|: TRUE f611_in -> f622_in :|: TRUE f376_out -> f610_out :|: TRUE f610_in -> f376_in :|: TRUE f382_in -> f386_in :|: TRUE f386_out -> f382_out :|: TRUE f587_in -> f594_in :|: TRUE f622_in -> f579_in :|: TRUE f462_in -> f462_out :|: TRUE f458_out -> f386_out :|: TRUE f386_in -> f458_in :|: TRUE f610_out -> f611_in :|: TRUE f606_in -> f610_in :|: TRUE f594_in -> f606_in :|: TRUE f376_in -> f378_in :|: TRUE f378_out -> f376_out :|: TRUE f584_in -> f587_in :|: TRUE f459_out -> f458_out :|: TRUE f458_in -> f459_in :|: TRUE f379_in -> f380_in :|: TRUE f380_out -> f379_out :|: TRUE f462_out -> f460_out :|: TRUE f460_in -> f462_in :|: TRUE f461_in -> f475_in :|: TRUE f475_out -> f461_out :|: TRUE f379_out -> f378_out :|: TRUE f378_in -> f379_in :|: TRUE f579_in -> f584_in :|: TRUE f458_out -> f475_out :|: TRUE f475_in -> f458_in :|: TRUE f461_out -> f459_out :|: TRUE f459_in -> f461_in :|: TRUE f459_in -> f460_in :|: TRUE f460_out -> f459_out :|: TRUE f376_out -> f377_in(x14) :|: TRUE f377_in(x15) -> f579_in :|: TRUE ---------------------------------------- (137) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (138) Obligation: Rules: f459_out -> f459_in :|: TRUE f459_in -> f459_out :|: TRUE f459_out -> f459_out :|: TRUE f459_in -> f459_in :|: TRUE ---------------------------------------- (139) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (140) Obligation: Rules: f459_out -> f459_in :|: TRUE f459_in -> f459_out :|: TRUE f459_out -> f459_out :|: TRUE f459_in -> f459_in :|: TRUE ---------------------------------------- (141) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f459_out -> f459_in :|: TRUE (2) f459_in -> f459_out :|: TRUE (3) f459_out -> f459_out :|: TRUE (4) f459_in -> f459_in :|: TRUE Arcs: (1) -> (2), (4) (2) -> (1), (3) (3) -> (1), (3) (4) -> (2), (4) This digraph is fully evaluated! ---------------------------------------- (142) Obligation: Termination digraph: Nodes: (1) f459_out -> f459_in :|: TRUE (2) f459_out -> f459_out :|: TRUE (3) f459_in -> f459_out :|: TRUE (4) f459_in -> f459_in :|: TRUE Arcs: (1) -> (3), (4) (2) -> (1), (2) (3) -> (1), (2) (4) -> (3), (4) This digraph is fully evaluated! ---------------------------------------- (143) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f459_out() f459_in() Replaced non-predefined constructor symbols by 0. ---------------------------------------- (144) Obligation: Rules: f459_out -> f459_in :|: TRUE f459_out -> f459_out :|: TRUE f459_in -> f459_out :|: TRUE f459_in -> f459_in :|: TRUE ---------------------------------------- (145) 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) ---------------------------------------- (146) NO ---------------------------------------- (147) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 1, "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": { "598": { "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": [] } }, "911": { "goal": [{ "clause": -1, "scope": -1, "term": "(less T142 T143)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "912": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "913": { "goal": [ { "clause": 9, "scope": 9, "term": "(less T142 T143)" }, { "clause": 10, "scope": 9, "term": "(less T142 T143)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "914": { "goal": [{ "clause": 9, "scope": 9, "term": "(less T142 T143)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "915": { "goal": [{ "clause": 10, "scope": 9, "term": "(less T142 T143)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "918": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "919": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1110": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1107": { "goal": [ { "clause": 7, "scope": 12, "term": "(merge T244 T243 (. T219 ([])))" }, { "clause": 8, "scope": 12, "term": "(merge T244 T243 (. T219 ([])))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T219"], "free": [], "exprvars": [] } }, "1": { "goal": [{ "clause": -1, "scope": -1, "term": "(ms T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "1106": { "goal": [{ "clause": 6, "scope": 12, "term": "(merge T244 T243 (. T219 ([])))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T219"], "free": [], "exprvars": [] } }, "3": { "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": [] } }, "920": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "924": { "goal": [{ "clause": -1, "scope": -1, "term": "(less T157 T158)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "925": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "20": { "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": [] } }, "1109": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1108": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "22": { "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": [] } }, "23": { "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": [] } }, "27": { "goal": [{ "clause": 2, "scope": 1, "term": "(ms T1 ([]))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1000": { "goal": [{ "clause": 2, "scope": 1, "term": "(ms T1 (. T211 ([])))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T211"], "free": [], "exprvars": [] } }, "1121": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (split (. T333 (. T334 T335)) X349 X350) (',' (ms X349 X351) (',' (ms X350 X352) (merge X351 X352 T332))))" }], "kb": { "nonunifying": [ [ "(ms T1 T332)", "(ms ([]) ([]))" ], [ "(ms T1 T332)", "(ms (. X246 ([])) (. X246 ([])))" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T332"], "free": [ "X246", "X349", "X350", "X351", "X352" ], "exprvars": [] } }, "1120": { "goal": [{ "clause": -1, "scope": -1, "term": "(merge (. T320 T321) T322 ([]))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1118": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1117": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (less T313 T315) (merge (. T315 T317) T316 ([])))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T313"], "free": [], "exprvars": [] } }, "1116": { "goal": [{ "clause": -1, "scope": -1, "term": "(merge T294 (. T295 T296) ([]))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1115": { "goal": [{ "clause": -1, "scope": -1, "term": "(less T285 (s T289))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T285"], "free": [], "exprvars": [] } }, "1114": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1113": { "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": [] } }, "1112": { "goal": [{ "clause": 8, "scope": 12, "term": "(merge T244 T243 (. T219 ([])))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T219"], "free": [], "exprvars": [] } }, "1111": { "goal": [{ "clause": 7, "scope": 12, "term": "(merge T244 T243 (. T219 ([])))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T219"], "free": [], "exprvars": [] } }, "816": { "goal": [{ "clause": -1, "scope": -1, "term": "(ms T68 X95)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X95"], "exprvars": [] } }, "817": { "goal": [{ "clause": -1, "scope": -1, "term": "(merge T87 T86 X96)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X96"], "exprvars": [] } }, "1119": { "goal": [{ "clause": -1, "scope": -1, "term": "(less T313 T315)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T313"], "free": [], "exprvars": [] } }, "37": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (split (. T10 (. 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T127 T128) X166)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X166"], "exprvars": [] } }, "892": { "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": [] } }, "893": { "goal": [{ "clause": 9, "scope": 8, "term": "(less T120 (s T121))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "894": { "goal": [{ "clause": 10, "scope": 8, "term": "(less T120 (s T121))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "532": { "goal": [{ "clause": -1, "scope": -1, "term": "(split T38 X50 X49)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X49", "X50" ], "exprvars": [] } }, "899": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "537": { "goal": [ { "clause": 3, "scope": 4, "term": "(split T38 X50 X49)" }, { "clause": 4, "scope": 4, "term": "(split T38 X50 X49)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X49", "X50" ], "exprvars": [] } }, "540": { "goal": [{ "clause": 3, "scope": 4, "term": "(split T38 X50 X49)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X49", "X50" ], "exprvars": [] } }, "541": { "goal": [{ "clause": 4, "scope": 4, "term": "(split T38 X50 X49)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X49", "X50" ], "exprvars": [] } }, "546": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "549": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "550": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "794": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "795": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "796": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "797": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (split (. T83 (. T84 T85)) X120 X121) (',' (ms X120 X122) (',' (ms X121 X123) (merge X122 X123 X124))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X124", "X120", "X121", "X122", "X123" ], "exprvars": [] } }, "798": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "561": { "goal": [{ "clause": -1, "scope": -1, "term": "(split T45 X68 X67)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X67", "X68" ], "exprvars": [] } }, "562": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "565": { "goal": [{ "clause": -1, "scope": -1, "term": "(ms (. T30 T29) X13)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X13"], "exprvars": [] } }, "566": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (ms T47 X14) (merge T46 X14 ([])))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X14"], "exprvars": [] } }, "570": { "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": [] } }, "571": { "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": [] } }, "575": { "goal": [{ "clause": 1, "scope": 5, "term": "(ms (. T30 T29) X13)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X13"], "exprvars": [] } }, "576": { "goal": [{ "clause": 2, "scope": 5, "term": "(ms (. T30 T29) X13)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X13"], "exprvars": [] } }, "581": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "582": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "583": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "900": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "902": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 3, "label": "CASE" }, { "from": 3, "to": 20, "label": "EVAL with clause\nms([], []).\nand substitutionT1 -> [],\nT2 -> []" }, { "from": 3, "to": 22, "label": "EVAL-BACKTRACK" }, { "from": 20, "to": 23, "label": "SUCCESS" }, { "from": 22, "to": 998, "label": "EVAL with clause\nms(.(X246, []), .(X246, [])).\nand substitutionX246 -> T211,\nT1 -> .(T211, []),\nT2 -> .(T211, [])" }, { "from": 22, "to": 999, "label": "EVAL-BACKTRACK" }, { "from": 23, "to": 27, "label": "BACKTRACK\nfor clause: ms(.(X, []), .(X, []))because of non-unification" }, { "from": 27, "to": 37, "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": 27, "to": 38, "label": "EVAL-BACKTRACK" }, { "from": 37, "to": 39, "label": "CASE" }, { "from": 39, "to": 40, "label": "BACKTRACK\nfor clause: split([], [], [])because of non-unification" }, { "from": 40, "to": 45, "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": 45, "to": 65, "label": "SPLIT 1" }, { "from": 45, "to": 66, "label": "SPLIT 2\nreplacements:X32 -> T28,\nX31 -> T29,\nT27 -> T30" }, { "from": 65, "to": 67, "label": "CASE" }, { "from": 66, "to": 565, "label": "SPLIT 1" }, { "from": 66, "to": 566, "label": "SPLIT 2\nreplacements:X13 -> T46,\nT28 -> T47" }, { "from": 67, "to": 68, "label": "BACKTRACK\nfor clause: split([], [], [])because of non-unification" }, { "from": 68, "to": 532, "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": 532, "to": 537, "label": "CASE" }, { "from": 537, "to": 540, "label": "PARALLEL" }, { "from": 537, "to": 541, "label": "PARALLEL" }, { "from": 540, "to": 546, "label": "EVAL with clause\nsplit([], [], []).\nand substitutionT38 -> [],\nX50 -> [],\nX49 -> []" }, { "from": 540, "to": 549, "label": "EVAL-BACKTRACK" }, { "from": 541, "to": 561, "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": 541, "to": 562, "label": "EVAL-BACKTRACK" }, { "from": 546, "to": 550, "label": "SUCCESS" }, { "from": 561, "to": 532, "label": "INSTANCE with matching:\nT38 -> T45\nX50 -> X68\nX49 -> X67" }, { "from": 565, "to": 570, "label": "CASE" }, { "from": 566, "to": 973, "label": "SPLIT 1" }, { "from": 566, "to": 974, "label": "SPLIT 2\nreplacements:X14 -> T188,\nT46 -> T189" }, { "from": 570, "to": 571, "label": "BACKTRACK\nfor clause: ms([], [])because of non-unification" }, { "from": 571, "to": 575, "label": "PARALLEL" }, { "from": 571, "to": 576, "label": "PARALLEL" }, { "from": 575, "to": 581, "label": "EVAL with clause\nms(.(X73, []), .(X73, [])).\nand substitutionT30 -> T52,\nX73 -> T52,\nT29 -> [],\nX13 -> .(T52, [])" }, { "from": 575, "to": 582, "label": "EVAL-BACKTRACK" }, { "from": 576, "to": 598, "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": 576, "to": 600, "label": "EVAL-BACKTRACK" }, { "from": 581, "to": 583, "label": "SUCCESS" }, { "from": 598, "to": 602, "label": "SPLIT 1" }, { "from": 598, "to": 603, "label": "SPLIT 2\nreplacements:X92 -> T65,\nX93 -> T66" }, { "from": 602, "to": 65, "label": "INSTANCE with matching:\nT25 -> T62\nT26 -> .(T63, T64)\nX32 -> X92\nX31 -> X93" }, { "from": 603, "to": 608, "label": "SPLIT 1" }, { "from": 603, "to": 609, "label": "SPLIT 2\nreplacements:X94 -> T67,\nT66 -> T68" }, { "from": 608, "to": 614, "label": "CASE" }, { "from": 609, "to": 816, "label": "SPLIT 1" }, { "from": 609, "to": 817, "label": "SPLIT 2\nreplacements:X95 -> T86,\nT67 -> T87" }, { "from": 614, "to": 615, "label": "PARALLEL" }, { "from": 614, "to": 616, "label": "PARALLEL" }, { "from": 615, "to": 617, "label": "EVAL with clause\nms([], []).\nand substitutionT65 -> [],\nX94 -> []" }, { "from": 615, "to": 618, "label": "EVAL-BACKTRACK" }, { "from": 616, "to": 620, "label": "PARALLEL" }, { "from": 616, "to": 621, "label": "PARALLEL" }, { "from": 617, "to": 619, "label": "SUCCESS" }, { "from": 620, "to": 794, "label": "EVAL with clause\nms(.(X101, []), .(X101, [])).\nand substitutionX101 -> T73,\nT65 -> .(T73, []),\nX94 -> .(T73, [])" }, { "from": 620, "to": 795, "label": "EVAL-BACKTRACK" }, { "from": 621, "to": 797, "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": 621, "to": 798, "label": "EVAL-BACKTRACK" }, { "from": 794, "to": 796, "label": "SUCCESS" }, { "from": 797, "to": 598, "label": "INSTANCE with matching:\nT62 -> T83\nT63 -> T84\nT64 -> T85\nX92 -> X120\nX93 -> X121\nX94 -> X122\nX95 -> X123\nX96 -> X124" }, { "from": 816, "to": 608, "label": "INSTANCE with matching:\nT65 -> T68\nX94 -> X95" }, { "from": 817, "to": 823, "label": "CASE" }, { "from": 823, "to": 827, "label": "PARALLEL" }, { "from": 823, "to": 828, "label": "PARALLEL" }, { "from": 827, "to": 829, "label": "EVAL with clause\nmerge([], X131, X131).\nand substitutionT87 -> [],\nT86 -> T94,\nX131 -> T94,\nX96 -> T94" }, { "from": 827, "to": 830, "label": "EVAL-BACKTRACK" }, { "from": 828, "to": 860, "label": "PARALLEL" }, { "from": 828, "to": 861, "label": "PARALLEL" }, { "from": 829, "to": 831, "label": "SUCCESS" }, { "from": 860, "to": 867, "label": "EVAL with clause\nmerge(X136, [], X136).\nand substitutionT87 -> T99,\nX136 -> T99,\nT86 -> [],\nX96 -> T99" }, { "from": 860, "to": 868, "label": "EVAL-BACKTRACK" }, { "from": 861, "to": 880, "label": "PARALLEL" }, { "from": 861, "to": 881, "label": "PARALLEL" }, { "from": 867, "to": 869, "label": "SUCCESS" }, { "from": 880, "to": 884, "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": 880, "to": 885, "label": "EVAL-BACKTRACK" }, { "from": 881, "to": 942, "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": 881, "to": 944, "label": "EVAL-BACKTRACK" }, { "from": 884, "to": 890, "label": "SPLIT 1" }, { "from": 884, "to": 891, "label": "SPLIT 2\nnew knowledge:\nT120 is ground\nreplacements:T122 -> T126,\nT121 -> T127,\nT123 -> T128" }, { "from": 890, "to": 892, "label": "CASE" }, { "from": 891, "to": 817, "label": "INSTANCE with matching:\nT87 -> T126\nT86 -> .(T127, T128)\nX96 -> X166" }, { "from": 892, "to": 893, "label": "PARALLEL" }, { "from": 892, "to": 894, "label": "PARALLEL" }, { "from": 893, "to": 899, "label": "EVAL with clause\nless(0, s(X175)).\nand substitutionT120 -> 0,\nT121 -> T135,\nX175 -> T135" }, { "from": 893, "to": 900, "label": "EVAL-BACKTRACK" }, { "from": 894, "to": 911, "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": 894, "to": 912, "label": "EVAL-BACKTRACK" }, { "from": 899, "to": 902, "label": "SUCCESS" }, { "from": 911, "to": 913, "label": "CASE" }, { "from": 913, "to": 914, "label": "PARALLEL" }, { "from": 913, "to": 915, "label": "PARALLEL" }, { "from": 914, "to": 918, "label": "EVAL with clause\nless(0, s(X188)).\nand substitutionT142 -> 0,\nX188 -> T150,\nT143 -> s(T150)" }, { "from": 914, "to": 919, "label": "EVAL-BACKTRACK" }, { "from": 915, "to": 924, "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": 915, "to": 925, "label": "EVAL-BACKTRACK" }, { "from": 918, "to": 920, "label": "SUCCESS" }, { "from": 924, "to": 911, "label": "INSTANCE with matching:\nT142 -> T157\nT143 -> T158" }, { "from": 942, "to": 967, "label": "SPLIT 1" }, { "from": 942, "to": 968, "label": "SPLIT 2\nnew knowledge:\nT177 is ground\nreplacements:T178 -> T183,\nT180 -> T184,\nT179 -> T185" }, { "from": 967, "to": 911, "label": "INSTANCE with matching:\nT142 -> T177\nT143 -> T178" }, { "from": 968, "to": 817, "label": "INSTANCE with matching:\nT87 -> .(T183, T184)\nT86 -> T185\nX96 -> X216" }, { "from": 973, "to": 608, "label": "INSTANCE with matching:\nT65 -> T47\nX94 -> X14" }, { "from": 974, "to": 977, "label": "CASE" }, { "from": 977, "to": 978, "label": "PARALLEL" }, { "from": 977, "to": 979, "label": "PARALLEL" }, { "from": 978, "to": 981, "label": "EVAL with clause\nmerge([], X229, X229).\nand substitutionT189 -> [],\nT188 -> [],\nX229 -> [],\nT196 -> []" }, { "from": 978, "to": 983, "label": "EVAL-BACKTRACK" }, { "from": 979, "to": 986, "label": "PARALLEL" }, { "from": 979, "to": 987, "label": "PARALLEL" }, { "from": 981, "to": 985, "label": "SUCCESS" }, { "from": 986, "to": 991, "label": "EVAL with clause\nmerge(X234, [], X234).\nand substitutionT189 -> [],\nX234 -> [],\nT188 -> [],\nT201 -> []" }, { "from": 986, "to": 992, "label": "EVAL-BACKTRACK" }, { "from": 987, "to": 994, "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": 991, "to": 993, "label": "SUCCESS" }, { "from": 994, "to": 995, "label": "BACKTRACK\nfor clause: merge(.(X, Xs), .(Y, Ys), .(Y, Zs)) :- ','(less(Y, X), merge(.(X, Xs), Ys, Zs))because of non-unification" }, { "from": 998, "to": 1000, "label": "SUCCESS" }, { "from": 999, "to": 1121, "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": 999, "to": 1122, "label": "EVAL-BACKTRACK" }, { "from": 1000, "to": 1009, "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": 1000, "to": 1010, "label": "EVAL-BACKTRACK" }, { "from": 1009, "to": 1011, "label": "CASE" }, { "from": 1011, "to": 1012, "label": "BACKTRACK\nfor clause: split([], [], [])because of non-unification" }, { "from": 1012, "to": 1013, "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": 1013, "to": 1014, "label": "SPLIT 1" }, { "from": 1013, "to": 1015, "label": "SPLIT 2\nreplacements:X276 -> T238,\nX275 -> T239,\nT237 -> T240" }, { "from": 1014, "to": 65, "label": "INSTANCE with matching:\nT25 -> T235\nT26 -> T236\nX32 -> X276\nX31 -> X275" }, { "from": 1015, "to": 1016, "label": "SPLIT 1" }, { "from": 1015, "to": 1017, "label": "SPLIT 2\nreplacements:X257 -> T241,\nT238 -> T242" }, { "from": 1016, "to": 565, "label": "INSTANCE with matching:\nT30 -> T240\nT29 -> T239\nX13 -> X257" }, { "from": 1017, "to": 1018, "label": "SPLIT 1" }, { "from": 1017, "to": 1019, "label": "SPLIT 2\nreplacements:X258 -> T243,\nT241 -> T244" }, { "from": 1018, "to": 608, "label": "INSTANCE with matching:\nT65 -> T242\nX94 -> X258" }, { "from": 1019, "to": 1020, "label": "CASE" }, { "from": 1020, "to": 1021, "label": "PARALLEL" }, { "from": 1020, "to": 1022, "label": "PARALLEL" }, { "from": 1021, "to": 1025, "label": "EVAL with clause\nmerge([], X283, X283).\nand substitutionT244 -> [],\nT243 -> .(T258, []),\nX283 -> .(T258, []),\nT219 -> T258,\nT257 -> .(T258, [])" }, { "from": 1021, "to": 1026, "label": "EVAL-BACKTRACK" }, { "from": 1022, "to": 1106, "label": "PARALLEL" }, { "from": 1022, "to": 1107, "label": "PARALLEL" }, { "from": 1025, "to": 1027, "label": "SUCCESS" }, { "from": 1106, "to": 1108, "label": "EVAL with clause\nmerge(X288, [], X288).\nand substitutionT244 -> .(T268, []),\nX288 -> .(T268, []),\nT243 -> [],\nT219 -> T268,\nT267 -> .(T268, [])" }, { "from": 1106, "to": 1109, "label": "EVAL-BACKTRACK" }, { "from": 1107, "to": 1111, "label": "PARALLEL" }, { "from": 1107, "to": 1112, "label": "PARALLEL" }, { "from": 1108, "to": 1110, "label": "SUCCESS" }, { "from": 1111, "to": 1113, "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": 1111, "to": 1114, "label": "EVAL-BACKTRACK" }, { "from": 1112, "to": 1117, "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": 1112, "to": 1118, "label": "EVAL-BACKTRACK" }, { "from": 1113, "to": 1115, "label": "SPLIT 1" }, { "from": 1113, "to": 1116, "label": "SPLIT 2\nnew knowledge:\nT285 is ground\nreplacements:T290 -> T294,\nT289 -> T295,\nT291 -> T296" }, { "from": 1115, "to": 890, "label": "INSTANCE with matching:\nT120 -> T285\nT121 -> T289" }, { "from": 1116, "to": 974, "label": "INSTANCE with matching:\nT189 -> T294\nT188 -> .(T295, T296)" }, { "from": 1117, "to": 1119, "label": "SPLIT 1" }, { "from": 1117, "to": 1120, "label": "SPLIT 2\nnew knowledge:\nT313 is ground\nreplacements:T315 -> T320,\nT317 -> T321,\nT316 -> T322" }, { "from": 1119, "to": 911, "label": "INSTANCE with matching:\nT142 -> T313\nT143 -> T315" }, { "from": 1120, "to": 974, "label": "INSTANCE with matching:\nT189 -> .(T320, T321)\nT188 -> T322" }, { "from": 1121, "to": 1123, "label": "CASE" }, { "from": 1123, "to": 1124, "label": "BACKTRACK\nfor clause: split([], [], [])because of non-unification" }, { "from": 1124, "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": 65, "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": 565, "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": 608, "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": 890, "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": 890, "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": 911, "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" } } ---------------------------------------- (148) 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) ---------------------------------------- (149) UndefinedPredicateInTriplesTransformerProof (SOUND) Deleted triples and predicates having undefined goals [DT09]. ---------------------------------------- (150) 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) ---------------------------------------- (151) 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 ---------------------------------------- (152) 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 ---------------------------------------- (153) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 85 less nodes. ---------------------------------------- (154) Complex Obligation (AND) ---------------------------------------- (155) 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 ---------------------------------------- (156) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (157) 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 ---------------------------------------- (158) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (159) 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. ---------------------------------------- (160) 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 ---------------------------------------- (161) YES ---------------------------------------- (162) 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 ---------------------------------------- (163) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (164) 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 ---------------------------------------- (165) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (166) 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. ---------------------------------------- (167) 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 ---------------------------------------- (168) YES ---------------------------------------- (169) 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 ---------------------------------------- (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_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 ---------------------------------------- (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_AA -> LESSG_IN_AA R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (174) 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. ---------------------------------------- (175) NO ---------------------------------------- (176) 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 ---------------------------------------- (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: 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 ---------------------------------------- (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: 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. ---------------------------------------- (181) 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 ---------------------------------------- (182) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (183) 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 ---------------------------------------- (184) 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 ---------------------------------------- (185) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (186) 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