/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.pl /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- MAYBE proof of /export/starexec/sandbox/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern delete(g,a,a) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2) PiTRS (3) DependencyPairsProof [EQUIVALENT, 18 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, 1 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) PrologToPiTRSProof [SOUND, 0 ms] (34) PiTRS (35) DependencyPairsProof [EQUIVALENT, 2 ms] (36) PiDP (37) DependencyGraphProof [EQUIVALENT, 0 ms] (38) AND (39) PiDP (40) UsableRulesProof [EQUIVALENT, 0 ms] (41) PiDP (42) PiDPToQDPProof [SOUND, 0 ms] (43) QDP (44) QDPSizeChangeProof [EQUIVALENT, 0 ms] (45) YES (46) PiDP (47) UsableRulesProof [EQUIVALENT, 0 ms] (48) PiDP (49) PiDPToQDPProof [SOUND, 0 ms] (50) QDP (51) QDPSizeChangeProof [EQUIVALENT, 0 ms] (52) YES (53) PiDP (54) UsableRulesProof [EQUIVALENT, 0 ms] (55) PiDP (56) PiDPToQDPProof [SOUND, 0 ms] (57) QDP (58) NonTerminationLoopProof [COMPLETE, 7 ms] (59) NO (60) PiDP (61) UsableRulesProof [EQUIVALENT, 0 ms] (62) PiDP (63) PiDPToQDPProof [SOUND, 0 ms] (64) QDP (65) TransformationProof [SOUND, 0 ms] (66) QDP (67) TransformationProof [SOUND, 0 ms] (68) QDP (69) TransformationProof [EQUIVALENT, 0 ms] (70) QDP (71) DependencyGraphProof [EQUIVALENT, 0 ms] (72) AND (73) QDP (74) UsableRulesProof [EQUIVALENT, 0 ms] (75) QDP (76) QReductionProof [EQUIVALENT, 0 ms] (77) QDP (78) NonTerminationLoopProof [COMPLETE, 0 ms] (79) NO (80) QDP (81) TransformationProof [EQUIVALENT, 0 ms] (82) QDP (83) PrologToTRSTransformerProof [SOUND, 30 ms] (84) QTRS (85) DependencyPairsProof [EQUIVALENT, 0 ms] (86) QDP (87) DependencyGraphProof [EQUIVALENT, 0 ms] (88) AND (89) QDP (90) UsableRulesProof [EQUIVALENT, 0 ms] (91) QDP (92) QDPSizeChangeProof [EQUIVALENT, 0 ms] (93) YES (94) QDP (95) UsableRulesProof [EQUIVALENT, 0 ms] (96) QDP (97) QDPSizeChangeProof [EQUIVALENT, 0 ms] (98) YES (99) QDP (100) UsableRulesProof [EQUIVALENT, 0 ms] (101) QDP (102) NonTerminationLoopProof [COMPLETE, 0 ms] (103) NO (104) QDP (105) NonTerminationLoopProof [COMPLETE, 0 ms] (106) NO (107) PrologToIRSwTTransformerProof [SOUND, 34 ms] (108) AND (109) IRSwT (110) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (111) IRSwT (112) IntTRSCompressionProof [EQUIVALENT, 45 ms] (113) IRSwT (114) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (115) IRSwT (116) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms] (117) IRSwT (118) TempFilterProof [SOUND, 4 ms] (119) IRSwT (120) IRSwTToQDPProof [SOUND, 0 ms] (121) QDP (122) QDPSizeChangeProof [EQUIVALENT, 0 ms] (123) YES (124) IRSwT (125) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (126) IRSwT (127) IntTRSCompressionProof [EQUIVALENT, 2 ms] (128) IRSwT (129) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (130) IRSwT (131) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms] (132) IRSwT (133) TempFilterProof [SOUND, 1 ms] (134) IRSwT (135) IRSwTToQDPProof [SOUND, 0 ms] (136) QDP (137) QDPSizeChangeProof [EQUIVALENT, 0 ms] (138) YES (139) IRSwT (140) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (141) IRSwT (142) IntTRSCompressionProof [EQUIVALENT, 2 ms] (143) IRSwT (144) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (145) IRSwT (146) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms] (147) IRSwT (148) FilterProof [EQUIVALENT, 0 ms] (149) IntTRS (150) IntTRSPeriodicNontermProof [COMPLETE, 4 ms] (151) NO (152) IRSwT (153) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (154) IRSwT (155) IntTRSCompressionProof [EQUIVALENT, 20 ms] (156) IRSwT (157) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (158) IRSwT (159) IRSwTTerminationDigraphProof [EQUIVALENT, 37 ms] (160) IRSwT (161) PrologToDTProblemTransformerProof [SOUND, 151 ms] (162) TRIPLES (163) TriplesToPiDPProof [SOUND, 40 ms] (164) PiDP (165) DependencyGraphProof [EQUIVALENT, 0 ms] (166) AND (167) PiDP (168) UsableRulesProof [EQUIVALENT, 0 ms] (169) PiDP (170) PiDPToQDPProof [SOUND, 0 ms] (171) QDP (172) QDPSizeChangeProof [EQUIVALENT, 0 ms] (173) YES (174) PiDP (175) UsableRulesProof [EQUIVALENT, 0 ms] (176) PiDP (177) PiDPToQDPProof [SOUND, 0 ms] (178) QDP (179) QDPSizeChangeProof [EQUIVALENT, 0 ms] (180) YES (181) PiDP (182) UsableRulesProof [EQUIVALENT, 0 ms] (183) PiDP (184) PiDPToQDPProof [SOUND, 0 ms] (185) QDP (186) NonTerminationLoopProof [COMPLETE, 0 ms] (187) NO (188) PiDP (189) PiDPToQDPProof [SOUND, 0 ms] (190) QDP ---------------------------------------- (0) Obligation: Clauses: delete(X, tree(X, void, Right), Right). delete(X, tree(X, Left, void), Left). delete(X, tree(X, Left, Right), tree(Y, Left, Right1)) :- delmin(Right, Y, Right1). delete(X, tree(Y, Left, Right), tree(Y, Left1, Right)) :- ','(less(X, Y), delete(X, Left, Left1)). delete(X, tree(Y, Left, Right), tree(Y, Left, Right1)) :- ','(less(Y, X), delete(X, Right, Right1)). delmin(tree(Y, void, Right), Y, Right). delmin(tree(X, Left, X1), Y, tree(X, Left1, X2)) :- delmin(Left, Y, Left1). less(0, s(X3)). less(s(X), s(Y)) :- less(X, Y). Query: delete(g,a,a) ---------------------------------------- (1) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: delete_in_3: (b,f,f) delmin_in_3: (f,f,f) less_in_2: (b,f) (f,b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: delete_in_gaa(X, tree(X, void, Right), Right) -> delete_out_gaa(X, tree(X, void, Right), Right) delete_in_gaa(X, tree(X, Left, void), Left) -> delete_out_gaa(X, tree(X, Left, void), Left) delete_in_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gaa(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1)) delmin_in_aaa(tree(Y, void, Right), Y, Right) -> delmin_out_aaa(tree(Y, void, Right), Y, Right) delmin_in_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_aaa(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1)) U6_aaa(X, Left, X1, Y, Left1, X2, delmin_out_aaa(Left, Y, Left1)) -> delmin_out_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) U1_gaa(X, Left, Right, Y, Right1, delmin_out_aaa(Right, Y, Right1)) -> delete_out_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3)) less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y)) U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U2_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> U3_gaa(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1)) delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3)) less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y)) U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) U4_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> U5_gaa(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1)) U5_gaa(X, Y, Left, Right, Right1, delete_out_gaa(X, Right, Right1)) -> delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) U3_gaa(X, Y, Left, Right, Left1, delete_out_gaa(X, Left, Left1)) -> delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) The argument filtering Pi contains the following mapping: delete_in_gaa(x1, x2, x3) = delete_in_gaa(x1) delete_out_gaa(x1, x2, x3) = delete_out_gaa(x1) U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x1, x6) delmin_in_aaa(x1, x2, x3) = delmin_in_aaa delmin_out_aaa(x1, x2, x3) = delmin_out_aaa U6_aaa(x1, x2, x3, x4, x5, x6, x7) = U6_aaa(x7) U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U7_ga(x1, x2, x3) = U7_ga(x1, x3) U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x1, x6) U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6) less_in_ag(x1, x2) = less_in_ag(x2) less_out_ag(x1, x2) = less_out_ag(x1, x2) U7_ag(x1, x2, x3) = U7_ag(x2, x3) U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x1, 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: delete_in_gaa(X, tree(X, void, Right), Right) -> delete_out_gaa(X, tree(X, void, Right), Right) delete_in_gaa(X, tree(X, Left, void), Left) -> delete_out_gaa(X, tree(X, Left, void), Left) delete_in_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gaa(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1)) delmin_in_aaa(tree(Y, void, Right), Y, Right) -> delmin_out_aaa(tree(Y, void, Right), Y, Right) delmin_in_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_aaa(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1)) U6_aaa(X, Left, X1, Y, Left1, X2, delmin_out_aaa(Left, Y, Left1)) -> delmin_out_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) U1_gaa(X, Left, Right, Y, Right1, delmin_out_aaa(Right, Y, Right1)) -> delete_out_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3)) less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y)) U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U2_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> U3_gaa(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1)) delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3)) less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y)) U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) U4_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> U5_gaa(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1)) U5_gaa(X, Y, Left, Right, Right1, delete_out_gaa(X, Right, Right1)) -> delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) U3_gaa(X, Y, Left, Right, Left1, delete_out_gaa(X, Left, Left1)) -> delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) The argument filtering Pi contains the following mapping: delete_in_gaa(x1, x2, x3) = delete_in_gaa(x1) delete_out_gaa(x1, x2, x3) = delete_out_gaa(x1) U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x1, x6) delmin_in_aaa(x1, x2, x3) = delmin_in_aaa delmin_out_aaa(x1, x2, x3) = delmin_out_aaa U6_aaa(x1, x2, x3, x4, x5, x6, x7) = U6_aaa(x7) U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U7_ga(x1, x2, x3) = U7_ga(x1, x3) U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x1, x6) U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6) less_in_ag(x1, x2) = less_in_ag(x2) less_out_ag(x1, x2) = less_out_ag(x1, x2) U7_ag(x1, x2, x3) = U7_ag(x2, x3) U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x1, 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: DELETE_IN_GAA(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_GAA(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1)) DELETE_IN_GAA(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> DELMIN_IN_AAA(Right, Y, Right1) DELMIN_IN_AAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_AAA(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1)) DELMIN_IN_AAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> DELMIN_IN_AAA(Left, Y, Left1) DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_GAA(X, Y, Left, Right, Left1, less_in_ga(X, Y)) DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> LESS_IN_GA(X, Y) LESS_IN_GA(s(X), s(Y)) -> U7_GA(X, Y, less_in_ga(X, Y)) LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) U2_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> U3_GAA(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1)) U2_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> DELETE_IN_GAA(X, Left, Left1) DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_GAA(X, Y, Left, Right, Right1, less_in_ag(Y, X)) DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> LESS_IN_AG(Y, X) LESS_IN_AG(s(X), s(Y)) -> U7_AG(X, Y, less_in_ag(X, Y)) LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) U4_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> U5_GAA(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1)) U4_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> DELETE_IN_GAA(X, Right, Right1) The TRS R consists of the following rules: delete_in_gaa(X, tree(X, void, Right), Right) -> delete_out_gaa(X, tree(X, void, Right), Right) delete_in_gaa(X, tree(X, Left, void), Left) -> delete_out_gaa(X, tree(X, Left, void), Left) delete_in_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gaa(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1)) delmin_in_aaa(tree(Y, void, Right), Y, Right) -> delmin_out_aaa(tree(Y, void, Right), Y, Right) delmin_in_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_aaa(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1)) U6_aaa(X, Left, X1, Y, Left1, X2, delmin_out_aaa(Left, Y, Left1)) -> delmin_out_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) U1_gaa(X, Left, Right, Y, Right1, delmin_out_aaa(Right, Y, Right1)) -> delete_out_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3)) less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y)) U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U2_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> U3_gaa(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1)) delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3)) less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y)) U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) U4_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> U5_gaa(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1)) U5_gaa(X, Y, Left, Right, Right1, delete_out_gaa(X, Right, Right1)) -> delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) U3_gaa(X, Y, Left, Right, Left1, delete_out_gaa(X, Left, Left1)) -> delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) The argument filtering Pi contains the following mapping: delete_in_gaa(x1, x2, x3) = delete_in_gaa(x1) delete_out_gaa(x1, x2, x3) = delete_out_gaa(x1) U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x1, x6) delmin_in_aaa(x1, x2, x3) = delmin_in_aaa delmin_out_aaa(x1, x2, x3) = delmin_out_aaa U6_aaa(x1, x2, x3, x4, x5, x6, x7) = U6_aaa(x7) U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U7_ga(x1, x2, x3) = U7_ga(x1, x3) U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x1, x6) U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6) less_in_ag(x1, x2) = less_in_ag(x2) less_out_ag(x1, x2) = less_out_ag(x1, x2) U7_ag(x1, x2, x3) = U7_ag(x2, x3) U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x1, x6) DELETE_IN_GAA(x1, x2, x3) = DELETE_IN_GAA(x1) U1_GAA(x1, x2, x3, x4, x5, x6) = U1_GAA(x1, x6) DELMIN_IN_AAA(x1, x2, x3) = DELMIN_IN_AAA U6_AAA(x1, x2, x3, x4, x5, x6, x7) = U6_AAA(x7) U2_GAA(x1, x2, x3, x4, x5, x6) = U2_GAA(x1, x6) LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) U7_GA(x1, x2, x3) = U7_GA(x1, x3) U3_GAA(x1, x2, x3, x4, x5, x6) = U3_GAA(x1, x6) U4_GAA(x1, x2, x3, x4, x5, x6) = U4_GAA(x1, x6) LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) U7_AG(x1, x2, x3) = U7_AG(x2, x3) U5_GAA(x1, x2, x3, x4, x5, x6) = U5_GAA(x1, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: DELETE_IN_GAA(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_GAA(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1)) DELETE_IN_GAA(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> DELMIN_IN_AAA(Right, Y, Right1) DELMIN_IN_AAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_AAA(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1)) DELMIN_IN_AAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> DELMIN_IN_AAA(Left, Y, Left1) DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_GAA(X, Y, Left, Right, Left1, less_in_ga(X, Y)) DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> LESS_IN_GA(X, Y) LESS_IN_GA(s(X), s(Y)) -> U7_GA(X, Y, less_in_ga(X, Y)) LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) U2_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> U3_GAA(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1)) U2_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> DELETE_IN_GAA(X, Left, Left1) DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_GAA(X, Y, Left, Right, Right1, less_in_ag(Y, X)) DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> LESS_IN_AG(Y, X) LESS_IN_AG(s(X), s(Y)) -> U7_AG(X, Y, less_in_ag(X, Y)) LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) U4_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> U5_GAA(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1)) U4_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> DELETE_IN_GAA(X, Right, Right1) The TRS R consists of the following rules: delete_in_gaa(X, tree(X, void, Right), Right) -> delete_out_gaa(X, tree(X, void, Right), Right) delete_in_gaa(X, tree(X, Left, void), Left) -> delete_out_gaa(X, tree(X, Left, void), Left) delete_in_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gaa(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1)) delmin_in_aaa(tree(Y, void, Right), Y, Right) -> delmin_out_aaa(tree(Y, void, Right), Y, Right) delmin_in_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_aaa(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1)) U6_aaa(X, Left, X1, Y, Left1, X2, delmin_out_aaa(Left, Y, Left1)) -> delmin_out_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) U1_gaa(X, Left, Right, Y, Right1, delmin_out_aaa(Right, Y, Right1)) -> delete_out_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3)) less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y)) U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U2_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> U3_gaa(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1)) delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3)) less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y)) U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) U4_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> U5_gaa(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1)) U5_gaa(X, Y, Left, Right, Right1, delete_out_gaa(X, Right, Right1)) -> delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) U3_gaa(X, Y, Left, Right, Left1, delete_out_gaa(X, Left, Left1)) -> delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) The argument filtering Pi contains the following mapping: delete_in_gaa(x1, x2, x3) = delete_in_gaa(x1) delete_out_gaa(x1, x2, x3) = delete_out_gaa(x1) U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x1, x6) delmin_in_aaa(x1, x2, x3) = delmin_in_aaa delmin_out_aaa(x1, x2, x3) = delmin_out_aaa U6_aaa(x1, x2, x3, x4, x5, x6, x7) = U6_aaa(x7) U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U7_ga(x1, x2, x3) = U7_ga(x1, x3) U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x1, x6) U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6) less_in_ag(x1, x2) = less_in_ag(x2) less_out_ag(x1, x2) = less_out_ag(x1, x2) U7_ag(x1, x2, x3) = U7_ag(x2, x3) U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x1, x6) DELETE_IN_GAA(x1, x2, x3) = DELETE_IN_GAA(x1) U1_GAA(x1, x2, x3, x4, x5, x6) = U1_GAA(x1, x6) DELMIN_IN_AAA(x1, x2, x3) = DELMIN_IN_AAA U6_AAA(x1, x2, x3, x4, x5, x6, x7) = U6_AAA(x7) U2_GAA(x1, x2, x3, x4, x5, x6) = U2_GAA(x1, x6) LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) U7_GA(x1, x2, x3) = U7_GA(x1, x3) U3_GAA(x1, x2, x3, x4, x5, x6) = U3_GAA(x1, x6) U4_GAA(x1, x2, x3, x4, x5, x6) = U4_GAA(x1, x6) LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) U7_AG(x1, x2, x3) = U7_AG(x2, x3) U5_GAA(x1, x2, x3, x4, x5, x6) = U5_GAA(x1, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 9 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) The TRS R consists of the following rules: delete_in_gaa(X, tree(X, void, Right), Right) -> delete_out_gaa(X, tree(X, void, Right), Right) delete_in_gaa(X, tree(X, Left, void), Left) -> delete_out_gaa(X, tree(X, Left, void), Left) delete_in_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gaa(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1)) delmin_in_aaa(tree(Y, void, Right), Y, Right) -> delmin_out_aaa(tree(Y, void, Right), Y, Right) delmin_in_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_aaa(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1)) U6_aaa(X, Left, X1, Y, Left1, X2, delmin_out_aaa(Left, Y, Left1)) -> delmin_out_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) U1_gaa(X, Left, Right, Y, Right1, delmin_out_aaa(Right, Y, Right1)) -> delete_out_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3)) less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y)) U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U2_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> U3_gaa(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1)) delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3)) less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y)) U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) U4_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> U5_gaa(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1)) U5_gaa(X, Y, Left, Right, Right1, delete_out_gaa(X, Right, Right1)) -> delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) U3_gaa(X, Y, Left, Right, Left1, delete_out_gaa(X, Left, Left1)) -> delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) The argument filtering Pi contains the following mapping: delete_in_gaa(x1, x2, x3) = delete_in_gaa(x1) delete_out_gaa(x1, x2, x3) = delete_out_gaa(x1) U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x1, x6) delmin_in_aaa(x1, x2, x3) = delmin_in_aaa delmin_out_aaa(x1, x2, x3) = delmin_out_aaa U6_aaa(x1, x2, x3, x4, x5, x6, x7) = U6_aaa(x7) U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U7_ga(x1, x2, x3) = U7_ga(x1, x3) U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x1, x6) U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6) less_in_ag(x1, x2) = less_in_ag(x2) less_out_ag(x1, x2) = less_out_ag(x1, x2) U7_ag(x1, x2, x3) = U7_ag(x2, x3) U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x1, x6) LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) 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_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) 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_AG(s(Y)) -> LESS_IN_AG(Y) 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_AG(s(Y)) -> LESS_IN_AG(Y) The graph contains the following edges 1 > 1 ---------------------------------------- (13) YES ---------------------------------------- (14) 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: delete_in_gaa(X, tree(X, void, Right), Right) -> delete_out_gaa(X, tree(X, void, Right), Right) delete_in_gaa(X, tree(X, Left, void), Left) -> delete_out_gaa(X, tree(X, Left, void), Left) delete_in_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gaa(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1)) delmin_in_aaa(tree(Y, void, Right), Y, Right) -> delmin_out_aaa(tree(Y, void, Right), Y, Right) delmin_in_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_aaa(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1)) U6_aaa(X, Left, X1, Y, Left1, X2, delmin_out_aaa(Left, Y, Left1)) -> delmin_out_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) U1_gaa(X, Left, Right, Y, Right1, delmin_out_aaa(Right, Y, Right1)) -> delete_out_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3)) less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y)) U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U2_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> U3_gaa(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1)) delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3)) less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y)) U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) U4_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> U5_gaa(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1)) U5_gaa(X, Y, Left, Right, Right1, delete_out_gaa(X, Right, Right1)) -> delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) U3_gaa(X, Y, Left, Right, Left1, delete_out_gaa(X, Left, Left1)) -> delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) The argument filtering Pi contains the following mapping: delete_in_gaa(x1, x2, x3) = delete_in_gaa(x1) delete_out_gaa(x1, x2, x3) = delete_out_gaa(x1) U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x1, x6) delmin_in_aaa(x1, x2, x3) = delmin_in_aaa delmin_out_aaa(x1, x2, x3) = delmin_out_aaa U6_aaa(x1, x2, x3, x4, x5, x6, x7) = U6_aaa(x7) U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U7_ga(x1, x2, x3) = U7_ga(x1, x3) U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x1, x6) U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6) less_in_ag(x1, x2) = less_in_ag(x2) less_out_ag(x1, x2) = less_out_ag(x1, x2) U7_ag(x1, x2, x3) = U7_ag(x2, x3) U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x1, x6) LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 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: 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 ---------------------------------------- (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: LESS_IN_GA(s(X)) -> LESS_IN_GA(X) R is empty. Q is empty. 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: *LESS_IN_GA(s(X)) -> LESS_IN_GA(X) The graph contains the following edges 1 > 1 ---------------------------------------- (20) YES ---------------------------------------- (21) Obligation: Pi DP problem: The TRS P consists of the following rules: DELMIN_IN_AAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> DELMIN_IN_AAA(Left, Y, Left1) The TRS R consists of the following rules: delete_in_gaa(X, tree(X, void, Right), Right) -> delete_out_gaa(X, tree(X, void, Right), Right) delete_in_gaa(X, tree(X, Left, void), Left) -> delete_out_gaa(X, tree(X, Left, void), Left) delete_in_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gaa(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1)) delmin_in_aaa(tree(Y, void, Right), Y, Right) -> delmin_out_aaa(tree(Y, void, Right), Y, Right) delmin_in_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_aaa(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1)) U6_aaa(X, Left, X1, Y, Left1, X2, delmin_out_aaa(Left, Y, Left1)) -> delmin_out_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) U1_gaa(X, Left, Right, Y, Right1, delmin_out_aaa(Right, Y, Right1)) -> delete_out_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3)) less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y)) U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U2_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> U3_gaa(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1)) delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3)) less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y)) U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) U4_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> U5_gaa(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1)) U5_gaa(X, Y, Left, Right, Right1, delete_out_gaa(X, Right, Right1)) -> delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) U3_gaa(X, Y, Left, Right, Left1, delete_out_gaa(X, Left, Left1)) -> delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) The argument filtering Pi contains the following mapping: delete_in_gaa(x1, x2, x3) = delete_in_gaa(x1) delete_out_gaa(x1, x2, x3) = delete_out_gaa(x1) U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x1, x6) delmin_in_aaa(x1, x2, x3) = delmin_in_aaa delmin_out_aaa(x1, x2, x3) = delmin_out_aaa U6_aaa(x1, x2, x3, x4, x5, x6, x7) = U6_aaa(x7) U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U7_ga(x1, x2, x3) = U7_ga(x1, x3) U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x1, x6) U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6) less_in_ag(x1, x2) = less_in_ag(x2) less_out_ag(x1, x2) = less_out_ag(x1, x2) U7_ag(x1, x2, x3) = U7_ag(x2, x3) U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x1, x6) DELMIN_IN_AAA(x1, x2, x3) = DELMIN_IN_AAA 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: DELMIN_IN_AAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> DELMIN_IN_AAA(Left, Y, Left1) R is empty. The argument filtering Pi contains the following mapping: DELMIN_IN_AAA(x1, x2, x3) = DELMIN_IN_AAA 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: DELMIN_IN_AAA -> DELMIN_IN_AAA 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 = DELMIN_IN_AAA evaluates to t =DELMIN_IN_AAA Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence The DP semiunifies directly so there is only one rewrite step from DELMIN_IN_AAA to DELMIN_IN_AAA. ---------------------------------------- (27) NO ---------------------------------------- (28) Obligation: Pi DP problem: The TRS P consists of the following rules: DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_GAA(X, Y, Left, Right, Left1, less_in_ga(X, Y)) U2_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> DELETE_IN_GAA(X, Left, Left1) DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_GAA(X, Y, Left, Right, Right1, less_in_ag(Y, X)) U4_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> DELETE_IN_GAA(X, Right, Right1) The TRS R consists of the following rules: delete_in_gaa(X, tree(X, void, Right), Right) -> delete_out_gaa(X, tree(X, void, Right), Right) delete_in_gaa(X, tree(X, Left, void), Left) -> delete_out_gaa(X, tree(X, Left, void), Left) delete_in_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gaa(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1)) delmin_in_aaa(tree(Y, void, Right), Y, Right) -> delmin_out_aaa(tree(Y, void, Right), Y, Right) delmin_in_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_aaa(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1)) U6_aaa(X, Left, X1, Y, Left1, X2, delmin_out_aaa(Left, Y, Left1)) -> delmin_out_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) U1_gaa(X, Left, Right, Y, Right1, delmin_out_aaa(Right, Y, Right1)) -> delete_out_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3)) less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y)) U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U2_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> U3_gaa(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1)) delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3)) less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y)) U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) U4_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> U5_gaa(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1)) U5_gaa(X, Y, Left, Right, Right1, delete_out_gaa(X, Right, Right1)) -> delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) U3_gaa(X, Y, Left, Right, Left1, delete_out_gaa(X, Left, Left1)) -> delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) The argument filtering Pi contains the following mapping: delete_in_gaa(x1, x2, x3) = delete_in_gaa(x1) delete_out_gaa(x1, x2, x3) = delete_out_gaa(x1) U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x1, x6) delmin_in_aaa(x1, x2, x3) = delmin_in_aaa delmin_out_aaa(x1, x2, x3) = delmin_out_aaa U6_aaa(x1, x2, x3, x4, x5, x6, x7) = U6_aaa(x7) U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U7_ga(x1, x2, x3) = U7_ga(x1, x3) U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x1, x6) U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6) less_in_ag(x1, x2) = less_in_ag(x2) less_out_ag(x1, x2) = less_out_ag(x1, x2) U7_ag(x1, x2, x3) = U7_ag(x2, x3) U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x1, x6) DELETE_IN_GAA(x1, x2, x3) = DELETE_IN_GAA(x1) U2_GAA(x1, x2, x3, x4, x5, x6) = U2_GAA(x1, x6) U4_GAA(x1, x2, x3, x4, x5, x6) = U4_GAA(x1, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (29) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (30) Obligation: Pi DP problem: The TRS P consists of the following rules: DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_GAA(X, Y, Left, Right, Left1, less_in_ga(X, Y)) U2_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> DELETE_IN_GAA(X, Left, Left1) DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_GAA(X, Y, Left, Right, Right1, less_in_ag(Y, X)) U4_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> DELETE_IN_GAA(X, Right, Right1) The TRS R consists of the following rules: less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3)) less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y)) less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3)) less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y)) U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) The argument filtering Pi contains the following mapping: less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga(x1) s(x1) = s(x1) U7_ga(x1, x2, x3) = U7_ga(x1, x3) less_in_ag(x1, x2) = less_in_ag(x2) less_out_ag(x1, x2) = less_out_ag(x1, x2) U7_ag(x1, x2, x3) = U7_ag(x2, x3) DELETE_IN_GAA(x1, x2, x3) = DELETE_IN_GAA(x1) U2_GAA(x1, x2, x3, x4, x5, x6) = U2_GAA(x1, x6) U4_GAA(x1, x2, x3, x4, x5, x6) = U4_GAA(x1, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (31) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: DELETE_IN_GAA(X) -> U2_GAA(X, less_in_ga(X)) U2_GAA(X, less_out_ga(X)) -> DELETE_IN_GAA(X) DELETE_IN_GAA(X) -> U4_GAA(X, less_in_ag(X)) U4_GAA(X, less_out_ag(Y, X)) -> DELETE_IN_GAA(X) The TRS R consists of the following rules: less_in_ga(0) -> less_out_ga(0) less_in_ga(s(X)) -> U7_ga(X, less_in_ga(X)) less_in_ag(s(X3)) -> less_out_ag(0, s(X3)) less_in_ag(s(Y)) -> U7_ag(Y, less_in_ag(Y)) U7_ga(X, less_out_ga(X)) -> less_out_ga(s(X)) U7_ag(Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) The set Q consists of the following terms: less_in_ga(x0) less_in_ag(x0) U7_ga(x0, x1) U7_ag(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (33) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: delete_in_3: (b,f,f) delmin_in_3: (f,f,f) less_in_2: (b,f) (f,b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: delete_in_gaa(X, tree(X, void, Right), Right) -> delete_out_gaa(X, tree(X, void, Right), Right) delete_in_gaa(X, tree(X, Left, void), Left) -> delete_out_gaa(X, tree(X, Left, void), Left) delete_in_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gaa(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1)) delmin_in_aaa(tree(Y, void, Right), Y, Right) -> delmin_out_aaa(tree(Y, void, Right), Y, Right) delmin_in_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_aaa(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1)) U6_aaa(X, Left, X1, Y, Left1, X2, delmin_out_aaa(Left, Y, Left1)) -> delmin_out_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) U1_gaa(X, Left, Right, Y, Right1, delmin_out_aaa(Right, Y, Right1)) -> delete_out_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3)) less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y)) U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U2_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> U3_gaa(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1)) delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3)) less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y)) U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) U4_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> U5_gaa(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1)) U5_gaa(X, Y, Left, Right, Right1, delete_out_gaa(X, Right, Right1)) -> delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) U3_gaa(X, Y, Left, Right, Left1, delete_out_gaa(X, Left, Left1)) -> delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) The argument filtering Pi contains the following mapping: delete_in_gaa(x1, x2, x3) = delete_in_gaa(x1) delete_out_gaa(x1, x2, x3) = delete_out_gaa U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x6) delmin_in_aaa(x1, x2, x3) = delmin_in_aaa delmin_out_aaa(x1, x2, x3) = delmin_out_aaa U6_aaa(x1, x2, x3, x4, x5, x6, x7) = U6_aaa(x7) U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga s(x1) = s(x1) U7_ga(x1, x2, x3) = U7_ga(x3) U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x6) U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6) less_in_ag(x1, x2) = less_in_ag(x2) less_out_ag(x1, x2) = less_out_ag(x1) U7_ag(x1, x2, x3) = U7_ag(x3) U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x6) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (34) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: delete_in_gaa(X, tree(X, void, Right), Right) -> delete_out_gaa(X, tree(X, void, Right), Right) delete_in_gaa(X, tree(X, Left, void), Left) -> delete_out_gaa(X, tree(X, Left, void), Left) delete_in_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gaa(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1)) delmin_in_aaa(tree(Y, void, Right), Y, Right) -> delmin_out_aaa(tree(Y, void, Right), Y, Right) delmin_in_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_aaa(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1)) U6_aaa(X, Left, X1, Y, Left1, X2, delmin_out_aaa(Left, Y, Left1)) -> delmin_out_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) U1_gaa(X, Left, Right, Y, Right1, delmin_out_aaa(Right, Y, Right1)) -> delete_out_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3)) less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y)) U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U2_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> U3_gaa(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1)) delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3)) less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y)) U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) U4_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> U5_gaa(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1)) U5_gaa(X, Y, Left, Right, Right1, delete_out_gaa(X, Right, Right1)) -> delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) U3_gaa(X, Y, Left, Right, Left1, delete_out_gaa(X, Left, Left1)) -> delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) The argument filtering Pi contains the following mapping: delete_in_gaa(x1, x2, x3) = delete_in_gaa(x1) delete_out_gaa(x1, x2, x3) = delete_out_gaa U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x6) delmin_in_aaa(x1, x2, x3) = delmin_in_aaa delmin_out_aaa(x1, x2, x3) = delmin_out_aaa U6_aaa(x1, x2, x3, x4, x5, x6, x7) = U6_aaa(x7) U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga s(x1) = s(x1) U7_ga(x1, x2, x3) = U7_ga(x3) U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x6) U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6) less_in_ag(x1, x2) = less_in_ag(x2) less_out_ag(x1, x2) = less_out_ag(x1) U7_ag(x1, x2, x3) = U7_ag(x3) U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x6) ---------------------------------------- (35) 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: DELETE_IN_GAA(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_GAA(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1)) DELETE_IN_GAA(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> DELMIN_IN_AAA(Right, Y, Right1) DELMIN_IN_AAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_AAA(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1)) DELMIN_IN_AAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> DELMIN_IN_AAA(Left, Y, Left1) DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_GAA(X, Y, Left, Right, Left1, less_in_ga(X, Y)) DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> LESS_IN_GA(X, Y) LESS_IN_GA(s(X), s(Y)) -> U7_GA(X, Y, less_in_ga(X, Y)) LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) U2_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> U3_GAA(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1)) U2_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> DELETE_IN_GAA(X, Left, Left1) DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_GAA(X, Y, Left, Right, Right1, less_in_ag(Y, X)) DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> LESS_IN_AG(Y, X) LESS_IN_AG(s(X), s(Y)) -> U7_AG(X, Y, less_in_ag(X, Y)) LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) U4_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> U5_GAA(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1)) U4_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> DELETE_IN_GAA(X, Right, Right1) The TRS R consists of the following rules: delete_in_gaa(X, tree(X, void, Right), Right) -> delete_out_gaa(X, tree(X, void, Right), Right) delete_in_gaa(X, tree(X, Left, void), Left) -> delete_out_gaa(X, tree(X, Left, void), Left) delete_in_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gaa(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1)) delmin_in_aaa(tree(Y, void, Right), Y, Right) -> delmin_out_aaa(tree(Y, void, Right), Y, Right) delmin_in_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_aaa(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1)) U6_aaa(X, Left, X1, Y, Left1, X2, delmin_out_aaa(Left, Y, Left1)) -> delmin_out_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) U1_gaa(X, Left, Right, Y, Right1, delmin_out_aaa(Right, Y, Right1)) -> delete_out_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3)) less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y)) U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U2_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> U3_gaa(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1)) delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3)) less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y)) U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) U4_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> U5_gaa(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1)) U5_gaa(X, Y, Left, Right, Right1, delete_out_gaa(X, Right, Right1)) -> delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) U3_gaa(X, Y, Left, Right, Left1, delete_out_gaa(X, Left, Left1)) -> delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) The argument filtering Pi contains the following mapping: delete_in_gaa(x1, x2, x3) = delete_in_gaa(x1) delete_out_gaa(x1, x2, x3) = delete_out_gaa U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x6) delmin_in_aaa(x1, x2, x3) = delmin_in_aaa delmin_out_aaa(x1, x2, x3) = delmin_out_aaa U6_aaa(x1, x2, x3, x4, x5, x6, x7) = U6_aaa(x7) U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga s(x1) = s(x1) U7_ga(x1, x2, x3) = U7_ga(x3) U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x6) U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6) less_in_ag(x1, x2) = less_in_ag(x2) less_out_ag(x1, x2) = less_out_ag(x1) U7_ag(x1, x2, x3) = U7_ag(x3) U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x6) DELETE_IN_GAA(x1, x2, x3) = DELETE_IN_GAA(x1) U1_GAA(x1, x2, x3, x4, x5, x6) = U1_GAA(x6) DELMIN_IN_AAA(x1, x2, x3) = DELMIN_IN_AAA U6_AAA(x1, x2, x3, x4, x5, x6, x7) = U6_AAA(x7) U2_GAA(x1, x2, x3, x4, x5, x6) = U2_GAA(x1, x6) LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) U7_GA(x1, x2, x3) = U7_GA(x3) U3_GAA(x1, x2, x3, x4, x5, x6) = U3_GAA(x6) U4_GAA(x1, x2, x3, x4, x5, x6) = U4_GAA(x1, x6) LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) U7_AG(x1, x2, x3) = U7_AG(x3) U5_GAA(x1, x2, x3, x4, x5, x6) = U5_GAA(x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (36) Obligation: Pi DP problem: The TRS P consists of the following rules: DELETE_IN_GAA(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_GAA(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1)) DELETE_IN_GAA(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> DELMIN_IN_AAA(Right, Y, Right1) DELMIN_IN_AAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_AAA(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1)) DELMIN_IN_AAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> DELMIN_IN_AAA(Left, Y, Left1) DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_GAA(X, Y, Left, Right, Left1, less_in_ga(X, Y)) DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> LESS_IN_GA(X, Y) LESS_IN_GA(s(X), s(Y)) -> U7_GA(X, Y, less_in_ga(X, Y)) LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) U2_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> U3_GAA(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1)) U2_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> DELETE_IN_GAA(X, Left, Left1) DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_GAA(X, Y, Left, Right, Right1, less_in_ag(Y, X)) DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> LESS_IN_AG(Y, X) LESS_IN_AG(s(X), s(Y)) -> U7_AG(X, Y, less_in_ag(X, Y)) LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) U4_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> U5_GAA(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1)) U4_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> DELETE_IN_GAA(X, Right, Right1) The TRS R consists of the following rules: delete_in_gaa(X, tree(X, void, Right), Right) -> delete_out_gaa(X, tree(X, void, Right), Right) delete_in_gaa(X, tree(X, Left, void), Left) -> delete_out_gaa(X, tree(X, Left, void), Left) delete_in_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gaa(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1)) delmin_in_aaa(tree(Y, void, Right), Y, Right) -> delmin_out_aaa(tree(Y, void, Right), Y, Right) delmin_in_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_aaa(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1)) U6_aaa(X, Left, X1, Y, Left1, X2, delmin_out_aaa(Left, Y, Left1)) -> delmin_out_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) U1_gaa(X, Left, Right, Y, Right1, delmin_out_aaa(Right, Y, Right1)) -> delete_out_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3)) less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y)) U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U2_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> U3_gaa(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1)) delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3)) less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y)) U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) U4_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> U5_gaa(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1)) U5_gaa(X, Y, Left, Right, Right1, delete_out_gaa(X, Right, Right1)) -> delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) U3_gaa(X, Y, Left, Right, Left1, delete_out_gaa(X, Left, Left1)) -> delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) The argument filtering Pi contains the following mapping: delete_in_gaa(x1, x2, x3) = delete_in_gaa(x1) delete_out_gaa(x1, x2, x3) = delete_out_gaa U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x6) delmin_in_aaa(x1, x2, x3) = delmin_in_aaa delmin_out_aaa(x1, x2, x3) = delmin_out_aaa U6_aaa(x1, x2, x3, x4, x5, x6, x7) = U6_aaa(x7) U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga s(x1) = s(x1) U7_ga(x1, x2, x3) = U7_ga(x3) U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x6) U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6) less_in_ag(x1, x2) = less_in_ag(x2) less_out_ag(x1, x2) = less_out_ag(x1) U7_ag(x1, x2, x3) = U7_ag(x3) U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x6) DELETE_IN_GAA(x1, x2, x3) = DELETE_IN_GAA(x1) U1_GAA(x1, x2, x3, x4, x5, x6) = U1_GAA(x6) DELMIN_IN_AAA(x1, x2, x3) = DELMIN_IN_AAA U6_AAA(x1, x2, x3, x4, x5, x6, x7) = U6_AAA(x7) U2_GAA(x1, x2, x3, x4, x5, x6) = U2_GAA(x1, x6) LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) U7_GA(x1, x2, x3) = U7_GA(x3) U3_GAA(x1, x2, x3, x4, x5, x6) = U3_GAA(x6) U4_GAA(x1, x2, x3, x4, x5, x6) = U4_GAA(x1, x6) LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) U7_AG(x1, x2, x3) = U7_AG(x3) U5_GAA(x1, x2, x3, x4, x5, x6) = U5_GAA(x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (37) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 9 less nodes. ---------------------------------------- (38) Complex Obligation (AND) ---------------------------------------- (39) Obligation: Pi DP problem: The TRS P consists of the following rules: LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) The TRS R consists of the following rules: delete_in_gaa(X, tree(X, void, Right), Right) -> delete_out_gaa(X, tree(X, void, Right), Right) delete_in_gaa(X, tree(X, Left, void), Left) -> delete_out_gaa(X, tree(X, Left, void), Left) delete_in_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gaa(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1)) delmin_in_aaa(tree(Y, void, Right), Y, Right) -> delmin_out_aaa(tree(Y, void, Right), Y, Right) delmin_in_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_aaa(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1)) U6_aaa(X, Left, X1, Y, Left1, X2, delmin_out_aaa(Left, Y, Left1)) -> delmin_out_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) U1_gaa(X, Left, Right, Y, Right1, delmin_out_aaa(Right, Y, Right1)) -> delete_out_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3)) less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y)) U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U2_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> U3_gaa(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1)) delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3)) less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y)) U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) U4_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> U5_gaa(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1)) U5_gaa(X, Y, Left, Right, Right1, delete_out_gaa(X, Right, Right1)) -> delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) U3_gaa(X, Y, Left, Right, Left1, delete_out_gaa(X, Left, Left1)) -> delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) The argument filtering Pi contains the following mapping: delete_in_gaa(x1, x2, x3) = delete_in_gaa(x1) delete_out_gaa(x1, x2, x3) = delete_out_gaa U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x6) delmin_in_aaa(x1, x2, x3) = delmin_in_aaa delmin_out_aaa(x1, x2, x3) = delmin_out_aaa U6_aaa(x1, x2, x3, x4, x5, x6, x7) = U6_aaa(x7) U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga s(x1) = s(x1) U7_ga(x1, x2, x3) = U7_ga(x3) U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x6) U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6) less_in_ag(x1, x2) = less_in_ag(x2) less_out_ag(x1, x2) = less_out_ag(x1) U7_ag(x1, x2, x3) = U7_ag(x3) U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x6) LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (40) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (41) Obligation: Pi DP problem: The TRS P consists of the following rules: LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (42) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (43) Obligation: Q DP problem: The TRS P consists of the following rules: LESS_IN_AG(s(Y)) -> LESS_IN_AG(Y) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (44) 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_AG(s(Y)) -> LESS_IN_AG(Y) The graph contains the following edges 1 > 1 ---------------------------------------- (45) YES ---------------------------------------- (46) 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: delete_in_gaa(X, tree(X, void, Right), Right) -> delete_out_gaa(X, tree(X, void, Right), Right) delete_in_gaa(X, tree(X, Left, void), Left) -> delete_out_gaa(X, tree(X, Left, void), Left) delete_in_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gaa(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1)) delmin_in_aaa(tree(Y, void, Right), Y, Right) -> delmin_out_aaa(tree(Y, void, Right), Y, Right) delmin_in_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_aaa(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1)) U6_aaa(X, Left, X1, Y, Left1, X2, delmin_out_aaa(Left, Y, Left1)) -> delmin_out_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) U1_gaa(X, Left, Right, Y, Right1, delmin_out_aaa(Right, Y, Right1)) -> delete_out_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3)) less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y)) U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U2_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> U3_gaa(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1)) delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3)) less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y)) U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) U4_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> U5_gaa(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1)) U5_gaa(X, Y, Left, Right, Right1, delete_out_gaa(X, Right, Right1)) -> delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) U3_gaa(X, Y, Left, Right, Left1, delete_out_gaa(X, Left, Left1)) -> delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) The argument filtering Pi contains the following mapping: delete_in_gaa(x1, x2, x3) = delete_in_gaa(x1) delete_out_gaa(x1, x2, x3) = delete_out_gaa U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x6) delmin_in_aaa(x1, x2, x3) = delmin_in_aaa delmin_out_aaa(x1, x2, x3) = delmin_out_aaa U6_aaa(x1, x2, x3, x4, x5, x6, x7) = U6_aaa(x7) U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga s(x1) = s(x1) U7_ga(x1, x2, x3) = U7_ga(x3) U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x6) U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6) less_in_ag(x1, x2) = less_in_ag(x2) less_out_ag(x1, x2) = less_out_ag(x1) U7_ag(x1, x2, x3) = U7_ag(x3) U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x6) LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (47) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (48) Obligation: Pi DP problem: The TRS P consists of the following rules: 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 ---------------------------------------- (49) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (50) Obligation: Q DP problem: The TRS P consists of the following rules: LESS_IN_GA(s(X)) -> LESS_IN_GA(X) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (51) 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 ---------------------------------------- (52) YES ---------------------------------------- (53) Obligation: Pi DP problem: The TRS P consists of the following rules: DELMIN_IN_AAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> DELMIN_IN_AAA(Left, Y, Left1) The TRS R consists of the following rules: delete_in_gaa(X, tree(X, void, Right), Right) -> delete_out_gaa(X, tree(X, void, Right), Right) delete_in_gaa(X, tree(X, Left, void), Left) -> delete_out_gaa(X, tree(X, Left, void), Left) delete_in_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gaa(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1)) delmin_in_aaa(tree(Y, void, Right), Y, Right) -> delmin_out_aaa(tree(Y, void, Right), Y, Right) delmin_in_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_aaa(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1)) U6_aaa(X, Left, X1, Y, Left1, X2, delmin_out_aaa(Left, Y, Left1)) -> delmin_out_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) U1_gaa(X, Left, Right, Y, Right1, delmin_out_aaa(Right, Y, Right1)) -> delete_out_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3)) less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y)) U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U2_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> U3_gaa(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1)) delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3)) less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y)) U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) U4_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> U5_gaa(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1)) U5_gaa(X, Y, Left, Right, Right1, delete_out_gaa(X, Right, Right1)) -> delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) U3_gaa(X, Y, Left, Right, Left1, delete_out_gaa(X, Left, Left1)) -> delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) The argument filtering Pi contains the following mapping: delete_in_gaa(x1, x2, x3) = delete_in_gaa(x1) delete_out_gaa(x1, x2, x3) = delete_out_gaa U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x6) delmin_in_aaa(x1, x2, x3) = delmin_in_aaa delmin_out_aaa(x1, x2, x3) = delmin_out_aaa U6_aaa(x1, x2, x3, x4, x5, x6, x7) = U6_aaa(x7) U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga s(x1) = s(x1) U7_ga(x1, x2, x3) = U7_ga(x3) U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x6) U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6) less_in_ag(x1, x2) = less_in_ag(x2) less_out_ag(x1, x2) = less_out_ag(x1) U7_ag(x1, x2, x3) = U7_ag(x3) U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x6) DELMIN_IN_AAA(x1, x2, x3) = DELMIN_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (54) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (55) Obligation: Pi DP problem: The TRS P consists of the following rules: DELMIN_IN_AAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> DELMIN_IN_AAA(Left, Y, Left1) R is empty. The argument filtering Pi contains the following mapping: DELMIN_IN_AAA(x1, x2, x3) = DELMIN_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (56) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (57) Obligation: Q DP problem: The TRS P consists of the following rules: DELMIN_IN_AAA -> DELMIN_IN_AAA R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (58) 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 = DELMIN_IN_AAA evaluates to t =DELMIN_IN_AAA Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence The DP semiunifies directly so there is only one rewrite step from DELMIN_IN_AAA to DELMIN_IN_AAA. ---------------------------------------- (59) NO ---------------------------------------- (60) Obligation: Pi DP problem: The TRS P consists of the following rules: DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_GAA(X, Y, Left, Right, Left1, less_in_ga(X, Y)) U2_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> DELETE_IN_GAA(X, Left, Left1) DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_GAA(X, Y, Left, Right, Right1, less_in_ag(Y, X)) U4_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> DELETE_IN_GAA(X, Right, Right1) The TRS R consists of the following rules: delete_in_gaa(X, tree(X, void, Right), Right) -> delete_out_gaa(X, tree(X, void, Right), Right) delete_in_gaa(X, tree(X, Left, void), Left) -> delete_out_gaa(X, tree(X, Left, void), Left) delete_in_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gaa(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1)) delmin_in_aaa(tree(Y, void, Right), Y, Right) -> delmin_out_aaa(tree(Y, void, Right), Y, Right) delmin_in_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_aaa(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1)) U6_aaa(X, Left, X1, Y, Left1, X2, delmin_out_aaa(Left, Y, Left1)) -> delmin_out_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) U1_gaa(X, Left, Right, Y, Right1, delmin_out_aaa(Right, Y, Right1)) -> delete_out_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3)) less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y)) U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U2_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> U3_gaa(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1)) delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3)) less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y)) U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) U4_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> U5_gaa(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1)) U5_gaa(X, Y, Left, Right, Right1, delete_out_gaa(X, Right, Right1)) -> delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) U3_gaa(X, Y, Left, Right, Left1, delete_out_gaa(X, Left, Left1)) -> delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) The argument filtering Pi contains the following mapping: delete_in_gaa(x1, x2, x3) = delete_in_gaa(x1) delete_out_gaa(x1, x2, x3) = delete_out_gaa U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x6) delmin_in_aaa(x1, x2, x3) = delmin_in_aaa delmin_out_aaa(x1, x2, x3) = delmin_out_aaa U6_aaa(x1, x2, x3, x4, x5, x6, x7) = U6_aaa(x7) U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6) less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga s(x1) = s(x1) U7_ga(x1, x2, x3) = U7_ga(x3) U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x6) U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6) less_in_ag(x1, x2) = less_in_ag(x2) less_out_ag(x1, x2) = less_out_ag(x1) U7_ag(x1, x2, x3) = U7_ag(x3) U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x6) DELETE_IN_GAA(x1, x2, x3) = DELETE_IN_GAA(x1) U2_GAA(x1, x2, x3, x4, x5, x6) = U2_GAA(x1, x6) U4_GAA(x1, x2, x3, x4, x5, x6) = U4_GAA(x1, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (61) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (62) Obligation: Pi DP problem: The TRS P consists of the following rules: DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_GAA(X, Y, Left, Right, Left1, less_in_ga(X, Y)) U2_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> DELETE_IN_GAA(X, Left, Left1) DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_GAA(X, Y, Left, Right, Right1, less_in_ag(Y, X)) U4_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> DELETE_IN_GAA(X, Right, Right1) The TRS R consists of the following rules: less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3)) less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y)) less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3)) less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y)) U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) The argument filtering Pi contains the following mapping: less_in_ga(x1, x2) = less_in_ga(x1) 0 = 0 less_out_ga(x1, x2) = less_out_ga s(x1) = s(x1) U7_ga(x1, x2, x3) = U7_ga(x3) less_in_ag(x1, x2) = less_in_ag(x2) less_out_ag(x1, x2) = less_out_ag(x1) U7_ag(x1, x2, x3) = U7_ag(x3) DELETE_IN_GAA(x1, x2, x3) = DELETE_IN_GAA(x1) U2_GAA(x1, x2, x3, x4, x5, x6) = U2_GAA(x1, x6) U4_GAA(x1, x2, x3, x4, x5, x6) = U4_GAA(x1, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (63) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (64) Obligation: Q DP problem: The TRS P consists of the following rules: DELETE_IN_GAA(X) -> U2_GAA(X, less_in_ga(X)) U2_GAA(X, less_out_ga) -> DELETE_IN_GAA(X) DELETE_IN_GAA(X) -> U4_GAA(X, less_in_ag(X)) U4_GAA(X, less_out_ag(Y)) -> DELETE_IN_GAA(X) The TRS R consists of the following rules: less_in_ga(0) -> less_out_ga less_in_ga(s(X)) -> U7_ga(less_in_ga(X)) less_in_ag(s(X3)) -> less_out_ag(0) less_in_ag(s(Y)) -> U7_ag(less_in_ag(Y)) U7_ga(less_out_ga) -> less_out_ga U7_ag(less_out_ag(X)) -> less_out_ag(s(X)) The set Q consists of the following terms: less_in_ga(x0) less_in_ag(x0) U7_ga(x0) U7_ag(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (65) TransformationProof (SOUND) By narrowing [LPAR04] the rule DELETE_IN_GAA(X) -> U2_GAA(X, less_in_ga(X)) at position [1] we obtained the following new rules [LPAR04]: (DELETE_IN_GAA(0) -> U2_GAA(0, less_out_ga),DELETE_IN_GAA(0) -> U2_GAA(0, less_out_ga)) (DELETE_IN_GAA(s(x0)) -> U2_GAA(s(x0), U7_ga(less_in_ga(x0))),DELETE_IN_GAA(s(x0)) -> U2_GAA(s(x0), U7_ga(less_in_ga(x0)))) ---------------------------------------- (66) Obligation: Q DP problem: The TRS P consists of the following rules: U2_GAA(X, less_out_ga) -> DELETE_IN_GAA(X) DELETE_IN_GAA(X) -> U4_GAA(X, less_in_ag(X)) U4_GAA(X, less_out_ag(Y)) -> DELETE_IN_GAA(X) DELETE_IN_GAA(0) -> U2_GAA(0, less_out_ga) DELETE_IN_GAA(s(x0)) -> U2_GAA(s(x0), U7_ga(less_in_ga(x0))) The TRS R consists of the following rules: less_in_ga(0) -> less_out_ga less_in_ga(s(X)) -> U7_ga(less_in_ga(X)) less_in_ag(s(X3)) -> less_out_ag(0) less_in_ag(s(Y)) -> U7_ag(less_in_ag(Y)) U7_ga(less_out_ga) -> less_out_ga U7_ag(less_out_ag(X)) -> less_out_ag(s(X)) The set Q consists of the following terms: less_in_ga(x0) less_in_ag(x0) U7_ga(x0) U7_ag(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (67) TransformationProof (SOUND) By narrowing [LPAR04] the rule DELETE_IN_GAA(X) -> U4_GAA(X, less_in_ag(X)) at position [1] we obtained the following new rules [LPAR04]: (DELETE_IN_GAA(s(x0)) -> U4_GAA(s(x0), less_out_ag(0)),DELETE_IN_GAA(s(x0)) -> U4_GAA(s(x0), less_out_ag(0))) (DELETE_IN_GAA(s(x0)) -> U4_GAA(s(x0), U7_ag(less_in_ag(x0))),DELETE_IN_GAA(s(x0)) -> U4_GAA(s(x0), U7_ag(less_in_ag(x0)))) ---------------------------------------- (68) Obligation: Q DP problem: The TRS P consists of the following rules: U2_GAA(X, less_out_ga) -> DELETE_IN_GAA(X) U4_GAA(X, less_out_ag(Y)) -> DELETE_IN_GAA(X) DELETE_IN_GAA(0) -> U2_GAA(0, less_out_ga) DELETE_IN_GAA(s(x0)) -> U2_GAA(s(x0), U7_ga(less_in_ga(x0))) DELETE_IN_GAA(s(x0)) -> U4_GAA(s(x0), less_out_ag(0)) DELETE_IN_GAA(s(x0)) -> U4_GAA(s(x0), U7_ag(less_in_ag(x0))) The TRS R consists of the following rules: less_in_ga(0) -> less_out_ga less_in_ga(s(X)) -> U7_ga(less_in_ga(X)) less_in_ag(s(X3)) -> less_out_ag(0) less_in_ag(s(Y)) -> U7_ag(less_in_ag(Y)) U7_ga(less_out_ga) -> less_out_ga U7_ag(less_out_ag(X)) -> less_out_ag(s(X)) The set Q consists of the following terms: less_in_ga(x0) less_in_ag(x0) U7_ga(x0) U7_ag(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (69) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U2_GAA(X, less_out_ga) -> DELETE_IN_GAA(X) we obtained the following new rules [LPAR04]: (U2_GAA(0, less_out_ga) -> DELETE_IN_GAA(0),U2_GAA(0, less_out_ga) -> DELETE_IN_GAA(0)) (U2_GAA(s(z0), less_out_ga) -> DELETE_IN_GAA(s(z0)),U2_GAA(s(z0), less_out_ga) -> DELETE_IN_GAA(s(z0))) ---------------------------------------- (70) Obligation: Q DP problem: The TRS P consists of the following rules: U4_GAA(X, less_out_ag(Y)) -> DELETE_IN_GAA(X) DELETE_IN_GAA(0) -> U2_GAA(0, less_out_ga) DELETE_IN_GAA(s(x0)) -> U2_GAA(s(x0), U7_ga(less_in_ga(x0))) DELETE_IN_GAA(s(x0)) -> U4_GAA(s(x0), less_out_ag(0)) DELETE_IN_GAA(s(x0)) -> U4_GAA(s(x0), U7_ag(less_in_ag(x0))) U2_GAA(0, less_out_ga) -> DELETE_IN_GAA(0) U2_GAA(s(z0), less_out_ga) -> DELETE_IN_GAA(s(z0)) The TRS R consists of the following rules: less_in_ga(0) -> less_out_ga less_in_ga(s(X)) -> U7_ga(less_in_ga(X)) less_in_ag(s(X3)) -> less_out_ag(0) less_in_ag(s(Y)) -> U7_ag(less_in_ag(Y)) U7_ga(less_out_ga) -> less_out_ga U7_ag(less_out_ag(X)) -> less_out_ag(s(X)) The set Q consists of the following terms: less_in_ga(x0) less_in_ag(x0) U7_ga(x0) U7_ag(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (71) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs. ---------------------------------------- (72) Complex Obligation (AND) ---------------------------------------- (73) Obligation: Q DP problem: The TRS P consists of the following rules: U2_GAA(0, less_out_ga) -> DELETE_IN_GAA(0) DELETE_IN_GAA(0) -> U2_GAA(0, less_out_ga) The TRS R consists of the following rules: less_in_ga(0) -> less_out_ga less_in_ga(s(X)) -> U7_ga(less_in_ga(X)) less_in_ag(s(X3)) -> less_out_ag(0) less_in_ag(s(Y)) -> U7_ag(less_in_ag(Y)) U7_ga(less_out_ga) -> less_out_ga U7_ag(less_out_ag(X)) -> less_out_ag(s(X)) The set Q consists of the following terms: less_in_ga(x0) less_in_ag(x0) U7_ga(x0) U7_ag(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (74) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (75) Obligation: Q DP problem: The TRS P consists of the following rules: U2_GAA(0, less_out_ga) -> DELETE_IN_GAA(0) DELETE_IN_GAA(0) -> U2_GAA(0, less_out_ga) R is empty. The set Q consists of the following terms: less_in_ga(x0) less_in_ag(x0) U7_ga(x0) U7_ag(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (76) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. less_in_ga(x0) less_in_ag(x0) U7_ga(x0) U7_ag(x0) ---------------------------------------- (77) Obligation: Q DP problem: The TRS P consists of the following rules: U2_GAA(0, less_out_ga) -> DELETE_IN_GAA(0) DELETE_IN_GAA(0) -> U2_GAA(0, less_out_ga) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (78) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by narrowing to the left: s = DELETE_IN_GAA(0) evaluates to t =DELETE_IN_GAA(0) Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence DELETE_IN_GAA(0) -> U2_GAA(0, less_out_ga) with rule DELETE_IN_GAA(0) -> U2_GAA(0, less_out_ga) at position [] and matcher [ ] U2_GAA(0, less_out_ga) -> DELETE_IN_GAA(0) with rule U2_GAA(0, less_out_ga) -> DELETE_IN_GAA(0) Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence All these steps are and every following step will be a correct step w.r.t to Q. ---------------------------------------- (79) NO ---------------------------------------- (80) Obligation: Q DP problem: The TRS P consists of the following rules: DELETE_IN_GAA(s(x0)) -> U2_GAA(s(x0), U7_ga(less_in_ga(x0))) U2_GAA(s(z0), less_out_ga) -> DELETE_IN_GAA(s(z0)) DELETE_IN_GAA(s(x0)) -> U4_GAA(s(x0), less_out_ag(0)) U4_GAA(X, less_out_ag(Y)) -> DELETE_IN_GAA(X) DELETE_IN_GAA(s(x0)) -> U4_GAA(s(x0), U7_ag(less_in_ag(x0))) The TRS R consists of the following rules: less_in_ga(0) -> less_out_ga less_in_ga(s(X)) -> U7_ga(less_in_ga(X)) less_in_ag(s(X3)) -> less_out_ag(0) less_in_ag(s(Y)) -> U7_ag(less_in_ag(Y)) U7_ga(less_out_ga) -> less_out_ga U7_ag(less_out_ag(X)) -> less_out_ag(s(X)) The set Q consists of the following terms: less_in_ga(x0) less_in_ag(x0) U7_ga(x0) U7_ag(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (81) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U4_GAA(X, less_out_ag(Y)) -> DELETE_IN_GAA(X) we obtained the following new rules [LPAR04]: (U4_GAA(s(z0), less_out_ag(0)) -> DELETE_IN_GAA(s(z0)),U4_GAA(s(z0), less_out_ag(0)) -> DELETE_IN_GAA(s(z0))) (U4_GAA(s(z0), less_out_ag(x1)) -> DELETE_IN_GAA(s(z0)),U4_GAA(s(z0), less_out_ag(x1)) -> DELETE_IN_GAA(s(z0))) ---------------------------------------- (82) Obligation: Q DP problem: The TRS P consists of the following rules: DELETE_IN_GAA(s(x0)) -> U2_GAA(s(x0), U7_ga(less_in_ga(x0))) U2_GAA(s(z0), less_out_ga) -> DELETE_IN_GAA(s(z0)) DELETE_IN_GAA(s(x0)) -> U4_GAA(s(x0), less_out_ag(0)) DELETE_IN_GAA(s(x0)) -> U4_GAA(s(x0), U7_ag(less_in_ag(x0))) U4_GAA(s(z0), less_out_ag(0)) -> DELETE_IN_GAA(s(z0)) U4_GAA(s(z0), less_out_ag(x1)) -> DELETE_IN_GAA(s(z0)) The TRS R consists of the following rules: less_in_ga(0) -> less_out_ga less_in_ga(s(X)) -> U7_ga(less_in_ga(X)) less_in_ag(s(X3)) -> less_out_ag(0) less_in_ag(s(Y)) -> U7_ag(less_in_ag(Y)) U7_ga(less_out_ga) -> less_out_ga U7_ag(less_out_ag(X)) -> less_out_ag(s(X)) The set Q consists of the following terms: less_in_ga(x0) less_in_ag(x0) U7_ga(x0) U7_ag(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (83) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 2, "program": { "directives": [], "clauses": [ [ "(delete X (tree X (void) Right) Right)", null ], [ "(delete X (tree X Left (void)) Left)", null ], [ "(delete X (tree X Left Right) (tree Y Left Right1))", "(delmin Right Y Right1)" ], [ "(delete X (tree Y Left Right) (tree Y Left1 Right))", "(',' (less X Y) (delete X Left Left1))" ], [ "(delete X (tree Y Left Right) (tree Y Left Right1))", "(',' (less Y X) (delete X Right Right1))" ], [ "(delmin (tree Y (void) Right) Y Right)", null ], [ "(delmin (tree X Left X1) Y (tree X Left1 X2))", "(delmin Left Y Left1)" ], [ "(less (0) (s X3))", null ], [ "(less (s X) (s Y))", "(less X Y)" ] ] }, "graph": { "nodes": { "190": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "191": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "192": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "193": { "goal": [{ "clause": 1, "scope": 1, "term": "(delete T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "type": "Nodes", "194": { "goal": [ { "clause": 2, "scope": 1, "term": "(delete T1 T2 T3)" }, { "clause": 3, "scope": 1, "term": "(delete T1 T2 T3)" }, { "clause": 4, "scope": 1, "term": "(delete T1 T2 T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "271": { "goal": [ { "clause": 3, "scope": 1, "term": "(delete T1 T2 T3)" }, { "clause": 4, "scope": 1, "term": "(delete T1 T2 T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "353": { "goal": [{ "clause": 5, "scope": 2, "term": "(delmin T49 T50 T51)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "355": { "goal": [{ "clause": 6, "scope": 2, "term": "(delmin T49 T50 T51)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "434": { "goal": [{ "clause": 3, "scope": 1, "term": "(delete T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "435": { "goal": [{ "clause": 4, "scope": 1, "term": "(delete T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "436": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (less T111 T116) (delete T111 T117 T118))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T111"], "free": [], "exprvars": [] } }, "437": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "2": { "goal": [{ "clause": -1, "scope": -1, "term": "(delete T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "200": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "4": { "goal": [ { "clause": 0, "scope": 1, "term": "(delete T1 T2 T3)" }, { "clause": 1, "scope": 1, "term": "(delete T1 T2 T3)" }, { "clause": 2, "scope": 1, "term": "(delete T1 T2 T3)" }, { "clause": 3, "scope": 1, "term": "(delete T1 T2 T3)" }, { "clause": 4, "scope": 1, "term": "(delete T1 T2 T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "523": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (less T158 T153) (delete T153 T159 T160))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T153"], "free": [], "exprvars": [] } }, "524": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "404": { "goal": [{ "clause": -1, "scope": -1, "term": "(delmin T84 T85 T86)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "525": { "goal": [{ "clause": -1, "scope": -1, "term": "(less T158 T153)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T153"], "free": [], "exprvars": [] } }, "526": { "goal": [{ "clause": -1, "scope": -1, "term": "(delete T153 T163 T164)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T153"], "free": [], "exprvars": [] } }, "527": { "goal": [ { "clause": 7, "scope": 4, "term": "(less T158 T153)" }, { "clause": 8, "scope": 4, "term": "(less T158 T153)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T153"], "free": [], "exprvars": [] } }, "407": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "528": { "goal": [{ "clause": 7, "scope": 4, "term": "(less T158 T153)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T153"], "free": [], "exprvars": [] } }, "529": { "goal": [{ "clause": 8, "scope": 4, "term": "(less T158 T153)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T153"], "free": [], "exprvars": [] } }, "210": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "375": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "376": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "530": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "212": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "333": { "goal": [{ "clause": -1, "scope": -1, "term": "(delmin T49 T50 T51)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "454": { "goal": [{ "clause": -1, "scope": -1, "term": "(less T111 T116)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T111"], "free": [], "exprvars": [] } }, "531": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "378": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "455": { "goal": [{ "clause": -1, "scope": -1, "term": "(delete T111 T121 T122)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T111"], "free": [], "exprvars": [] } }, "532": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "335": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "456": { "goal": [ { "clause": 7, "scope": 3, "term": "(less T111 T116)" }, { "clause": 8, "scope": 3, "term": "(less T111 T116)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T111"], "free": [], "exprvars": [] } }, "533": { "goal": [{ "clause": -1, "scope": -1, "term": "(less T178 T177)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T177"], "free": [], "exprvars": [] } }, "534": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "458": { "goal": [{ "clause": 7, "scope": 3, "term": "(less T111 T116)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T111"], "free": [], "exprvars": [] } }, "459": { "goal": [{ "clause": 8, "scope": 3, "term": "(less T111 T116)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T111"], "free": [], "exprvars": [] } }, "460": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "461": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "462": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "463": { "goal": [{ "clause": -1, "scope": -1, "term": "(less T134 T136)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T134"], "free": [], "exprvars": [] } }, "266": { "goal": [{ "clause": 2, "scope": 1, "term": "(delete T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "464": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "347": { "goal": [ { "clause": 5, "scope": 2, "term": "(delmin T49 T50 T51)" }, { "clause": 6, "scope": 2, "term": "(delmin T49 T50 T51)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "42": { "goal": [{ "clause": 0, "scope": 1, "term": "(delete T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "43": { "goal": [ { "clause": 1, "scope": 1, "term": "(delete T1 T2 T3)" }, { "clause": 2, "scope": 1, "term": "(delete T1 T2 T3)" }, { "clause": 3, "scope": 1, "term": "(delete T1 T2 T3)" }, { "clause": 4, "scope": 1, "term": "(delete T1 T2 T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } } }, "edges": [ { "from": 2, "to": 4, "label": "CASE" }, { "from": 4, "to": 42, "label": "PARALLEL" }, { "from": 4, "to": 43, "label": "PARALLEL" }, { "from": 42, "to": 190, "label": "EVAL with clause\ndelete(X12, tree(X12, void, X13), X13).\nand substitutionT1 -> T12,\nX12 -> T12,\nX13 -> T13,\nT2 -> tree(T12, void, T13),\nT3 -> T13" }, { "from": 42, "to": 191, "label": "EVAL-BACKTRACK" }, { "from": 43, "to": 193, "label": "PARALLEL" }, { "from": 43, "to": 194, "label": "PARALLEL" }, { "from": 190, "to": 192, "label": "SUCCESS" }, { "from": 193, "to": 200, "label": "EVAL with clause\ndelete(X22, tree(X22, X23, void), X23).\nand substitutionT1 -> T22,\nX22 -> T22,\nX23 -> T23,\nT2 -> tree(T22, T23, void),\nT3 -> T23" }, { "from": 193, "to": 210, "label": "EVAL-BACKTRACK" }, { "from": 194, "to": 266, "label": "PARALLEL" }, { "from": 194, "to": 271, "label": "PARALLEL" }, { "from": 200, "to": 212, "label": "SUCCESS" }, { "from": 266, "to": 333, "label": "EVAL with clause\ndelete(X44, tree(X44, X45, X46), tree(X47, X45, X48)) :- delmin(X46, X47, X48).\nand substitutionT1 -> T44,\nX44 -> T44,\nX45 -> T45,\nX46 -> T49,\nT2 -> tree(T44, T45, T49),\nX47 -> T50,\nX48 -> T51,\nT3 -> tree(T50, T45, T51),\nT46 -> T49,\nT47 -> T50,\nT48 -> T51" }, { "from": 266, "to": 335, "label": "EVAL-BACKTRACK" }, { "from": 271, "to": 434, "label": "PARALLEL" }, { "from": 271, "to": 435, "label": "PARALLEL" }, { "from": 333, "to": 347, "label": "CASE" }, { "from": 347, "to": 353, "label": "PARALLEL" }, { "from": 347, "to": 355, "label": "PARALLEL" }, { "from": 353, "to": 375, "label": "EVAL with clause\ndelmin(tree(X61, void, X62), X61, X62).\nand substitutionX61 -> T64,\nX62 -> T65,\nT49 -> tree(T64, void, T65),\nT50 -> T64,\nT51 -> T65" }, { "from": 353, "to": 376, "label": "EVAL-BACKTRACK" }, { "from": 355, "to": 404, "label": "EVAL with clause\ndelmin(tree(X75, X76, X77), X78, tree(X75, X79, X80)) :- delmin(X76, X78, X79).\nand substitutionX75 -> T78,\nX76 -> T84,\nX77 -> T80,\nT49 -> tree(T78, T84, T80),\nT50 -> T85,\nX78 -> T85,\nX79 -> T86,\nX80 -> T83,\nT51 -> tree(T78, T86, T83),\nT79 -> T84,\nT81 -> T85,\nT82 -> T86" }, { "from": 355, "to": 407, "label": "EVAL-BACKTRACK" }, { "from": 375, "to": 378, "label": "SUCCESS" }, { "from": 404, "to": 333, "label": "INSTANCE with matching:\nT49 -> T84\nT50 -> T85\nT51 -> T86" }, { "from": 434, "to": 436, "label": "EVAL with clause\ndelete(X105, tree(X106, X107, X108), tree(X106, X109, X108)) :- ','(less(X105, X106), delete(X105, X107, X109)).\nand substitutionT1 -> T111,\nX105 -> T111,\nX106 -> T116,\nX107 -> T117,\nX108 -> T114,\nT2 -> tree(T116, T117, T114),\nX109 -> T118,\nT3 -> tree(T116, T118, T114),\nT112 -> T116,\nT113 -> T117,\nT115 -> T118" }, { "from": 434, "to": 437, "label": "EVAL-BACKTRACK" }, { "from": 435, "to": 523, "label": "EVAL with clause\ndelete(X141, tree(X142, X143, X144), tree(X142, X143, X145)) :- ','(less(X142, X141), delete(X141, X144, X145)).\nand substitutionT1 -> T153,\nX141 -> T153,\nX142 -> T158,\nX143 -> T155,\nX144 -> T159,\nT2 -> tree(T158, T155, T159),\nX145 -> T160,\nT3 -> tree(T158, T155, T160),\nT154 -> T158,\nT156 -> T159,\nT157 -> T160" }, { "from": 435, "to": 524, "label": "EVAL-BACKTRACK" }, { "from": 436, "to": 454, "label": "SPLIT 1" }, { "from": 436, "to": 455, "label": "SPLIT 2\nnew knowledge:\nT111 is ground\nreplacements:T117 -> T121,\nT118 -> T122" }, { "from": 454, "to": 456, "label": "CASE" }, { "from": 455, "to": 2, "label": "INSTANCE with matching:\nT1 -> T111\nT2 -> T121\nT3 -> T122" }, { "from": 456, "to": 458, "label": "PARALLEL" }, { "from": 456, "to": 459, "label": "PARALLEL" }, { "from": 458, "to": 460, "label": "EVAL with clause\nless(0, s(X118)).\nand substitutionT111 -> 0,\nX118 -> T129,\nT116 -> s(T129)" }, { "from": 458, "to": 461, "label": "EVAL-BACKTRACK" }, { "from": 459, "to": 463, "label": "EVAL with clause\nless(s(X123), s(X124)) :- less(X123, X124).\nand substitutionX123 -> T134,\nT111 -> s(T134),\nX124 -> T136,\nT116 -> s(T136),\nT135 -> T136" }, { "from": 459, "to": 464, "label": "EVAL-BACKTRACK" }, { "from": 460, "to": 462, "label": "SUCCESS" }, { "from": 463, "to": 454, "label": "INSTANCE with matching:\nT111 -> T134\nT116 -> T136" }, { "from": 523, "to": 525, "label": "SPLIT 1" }, { "from": 523, "to": 526, "label": "SPLIT 2\nnew knowledge:\nT158 is ground\nT153 is ground\nreplacements:T159 -> T163,\nT160 -> T164" }, { "from": 525, "to": 527, "label": "CASE" }, { "from": 526, "to": 2, "label": "INSTANCE with matching:\nT1 -> T153\nT2 -> T163\nT3 -> T164" }, { "from": 527, "to": 528, "label": "PARALLEL" }, { "from": 527, "to": 529, "label": "PARALLEL" }, { "from": 528, "to": 530, "label": "EVAL with clause\nless(0, s(X154)).\nand substitutionT158 -> 0,\nX154 -> T171,\nT153 -> s(T171)" }, { "from": 528, "to": 531, "label": "EVAL-BACKTRACK" }, { "from": 529, "to": 533, "label": "EVAL with clause\nless(s(X159), s(X160)) :- less(X159, X160).\nand substitutionX159 -> T178,\nT158 -> s(T178),\nX160 -> T177,\nT153 -> s(T177),\nT176 -> T178" }, { "from": 529, "to": 534, "label": "EVAL-BACKTRACK" }, { "from": 530, "to": 532, "label": "SUCCESS" }, { "from": 533, "to": 525, "label": "INSTANCE with matching:\nT158 -> T178\nT153 -> T177" } ], "type": "Graph" } } ---------------------------------------- (84) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f2_in(T12) -> f2_out1 f2_in(T44) -> U1(f333_in, T44) U1(f333_out1, T44) -> f2_out1 f2_in(T111) -> U2(f436_in(T111), T111) U2(f436_out1, T111) -> f2_out1 f2_in(T153) -> U3(f523_in(T153), T153) U3(f523_out1(T158), T153) -> f2_out1 f333_in -> f333_out1 f333_in -> U4(f333_in) U4(f333_out1) -> f333_out1 f454_in(0) -> f454_out1 f454_in(s(T134)) -> U5(f454_in(T134), s(T134)) U5(f454_out1, s(T134)) -> f454_out1 f525_in(s(T171)) -> f525_out1(0) f525_in(s(T177)) -> U6(f525_in(T177), s(T177)) U6(f525_out1(T178), s(T177)) -> f525_out1(s(T178)) f436_in(T111) -> U7(f454_in(T111), T111) U7(f454_out1, T111) -> U8(f2_in(T111), T111) U8(f2_out1, T111) -> f436_out1 f523_in(T153) -> U9(f525_in(T153), T153) U9(f525_out1(T158), T153) -> U10(f2_in(T153), T153, T158) U10(f2_out1, T153, T158) -> f523_out1(T158) Q is empty. ---------------------------------------- (85) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (86) Obligation: Q DP problem: The TRS P consists of the following rules: F2_IN(T44) -> U1^1(f333_in, T44) F2_IN(T44) -> F333_IN F2_IN(T111) -> U2^1(f436_in(T111), T111) F2_IN(T111) -> F436_IN(T111) F2_IN(T153) -> U3^1(f523_in(T153), T153) F2_IN(T153) -> F523_IN(T153) F333_IN -> U4^1(f333_in) F333_IN -> F333_IN F454_IN(s(T134)) -> U5^1(f454_in(T134), s(T134)) F454_IN(s(T134)) -> F454_IN(T134) F525_IN(s(T177)) -> U6^1(f525_in(T177), s(T177)) F525_IN(s(T177)) -> F525_IN(T177) F436_IN(T111) -> U7^1(f454_in(T111), T111) F436_IN(T111) -> F454_IN(T111) U7^1(f454_out1, T111) -> U8^1(f2_in(T111), T111) U7^1(f454_out1, T111) -> F2_IN(T111) F523_IN(T153) -> U9^1(f525_in(T153), T153) F523_IN(T153) -> F525_IN(T153) U9^1(f525_out1(T158), T153) -> U10^1(f2_in(T153), T153, T158) U9^1(f525_out1(T158), T153) -> F2_IN(T153) The TRS R consists of the following rules: f2_in(T12) -> f2_out1 f2_in(T44) -> U1(f333_in, T44) U1(f333_out1, T44) -> f2_out1 f2_in(T111) -> U2(f436_in(T111), T111) U2(f436_out1, T111) -> f2_out1 f2_in(T153) -> U3(f523_in(T153), T153) U3(f523_out1(T158), T153) -> f2_out1 f333_in -> f333_out1 f333_in -> U4(f333_in) U4(f333_out1) -> f333_out1 f454_in(0) -> f454_out1 f454_in(s(T134)) -> U5(f454_in(T134), s(T134)) U5(f454_out1, s(T134)) -> f454_out1 f525_in(s(T171)) -> f525_out1(0) f525_in(s(T177)) -> U6(f525_in(T177), s(T177)) U6(f525_out1(T178), s(T177)) -> f525_out1(s(T178)) f436_in(T111) -> U7(f454_in(T111), T111) U7(f454_out1, T111) -> U8(f2_in(T111), T111) U8(f2_out1, T111) -> f436_out1 f523_in(T153) -> U9(f525_in(T153), T153) U9(f525_out1(T158), T153) -> U10(f2_in(T153), T153, T158) U10(f2_out1, T153, T158) -> f523_out1(T158) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (87) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 11 less nodes. ---------------------------------------- (88) Complex Obligation (AND) ---------------------------------------- (89) Obligation: Q DP problem: The TRS P consists of the following rules: F525_IN(s(T177)) -> F525_IN(T177) The TRS R consists of the following rules: f2_in(T12) -> f2_out1 f2_in(T44) -> U1(f333_in, T44) U1(f333_out1, T44) -> f2_out1 f2_in(T111) -> U2(f436_in(T111), T111) U2(f436_out1, T111) -> f2_out1 f2_in(T153) -> U3(f523_in(T153), T153) U3(f523_out1(T158), T153) -> f2_out1 f333_in -> f333_out1 f333_in -> U4(f333_in) U4(f333_out1) -> f333_out1 f454_in(0) -> f454_out1 f454_in(s(T134)) -> U5(f454_in(T134), s(T134)) U5(f454_out1, s(T134)) -> f454_out1 f525_in(s(T171)) -> f525_out1(0) f525_in(s(T177)) -> U6(f525_in(T177), s(T177)) U6(f525_out1(T178), s(T177)) -> f525_out1(s(T178)) f436_in(T111) -> U7(f454_in(T111), T111) U7(f454_out1, T111) -> U8(f2_in(T111), T111) U8(f2_out1, T111) -> f436_out1 f523_in(T153) -> U9(f525_in(T153), T153) U9(f525_out1(T158), T153) -> U10(f2_in(T153), T153, T158) U10(f2_out1, T153, T158) -> f523_out1(T158) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (90) 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. ---------------------------------------- (91) Obligation: Q DP problem: The TRS P consists of the following rules: F525_IN(s(T177)) -> F525_IN(T177) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (92) 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: *F525_IN(s(T177)) -> F525_IN(T177) The graph contains the following edges 1 > 1 ---------------------------------------- (93) YES ---------------------------------------- (94) Obligation: Q DP problem: The TRS P consists of the following rules: F454_IN(s(T134)) -> F454_IN(T134) The TRS R consists of the following rules: f2_in(T12) -> f2_out1 f2_in(T44) -> U1(f333_in, T44) U1(f333_out1, T44) -> f2_out1 f2_in(T111) -> U2(f436_in(T111), T111) U2(f436_out1, T111) -> f2_out1 f2_in(T153) -> U3(f523_in(T153), T153) U3(f523_out1(T158), T153) -> f2_out1 f333_in -> f333_out1 f333_in -> U4(f333_in) U4(f333_out1) -> f333_out1 f454_in(0) -> f454_out1 f454_in(s(T134)) -> U5(f454_in(T134), s(T134)) U5(f454_out1, s(T134)) -> f454_out1 f525_in(s(T171)) -> f525_out1(0) f525_in(s(T177)) -> U6(f525_in(T177), s(T177)) U6(f525_out1(T178), s(T177)) -> f525_out1(s(T178)) f436_in(T111) -> U7(f454_in(T111), T111) U7(f454_out1, T111) -> U8(f2_in(T111), T111) U8(f2_out1, T111) -> f436_out1 f523_in(T153) -> U9(f525_in(T153), T153) U9(f525_out1(T158), T153) -> U10(f2_in(T153), T153, T158) U10(f2_out1, T153, T158) -> f523_out1(T158) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (95) 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. ---------------------------------------- (96) Obligation: Q DP problem: The TRS P consists of the following rules: F454_IN(s(T134)) -> F454_IN(T134) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (97) 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: *F454_IN(s(T134)) -> F454_IN(T134) The graph contains the following edges 1 > 1 ---------------------------------------- (98) YES ---------------------------------------- (99) Obligation: Q DP problem: The TRS P consists of the following rules: F333_IN -> F333_IN The TRS R consists of the following rules: f2_in(T12) -> f2_out1 f2_in(T44) -> U1(f333_in, T44) U1(f333_out1, T44) -> f2_out1 f2_in(T111) -> U2(f436_in(T111), T111) U2(f436_out1, T111) -> f2_out1 f2_in(T153) -> U3(f523_in(T153), T153) U3(f523_out1(T158), T153) -> f2_out1 f333_in -> f333_out1 f333_in -> U4(f333_in) U4(f333_out1) -> f333_out1 f454_in(0) -> f454_out1 f454_in(s(T134)) -> U5(f454_in(T134), s(T134)) U5(f454_out1, s(T134)) -> f454_out1 f525_in(s(T171)) -> f525_out1(0) f525_in(s(T177)) -> U6(f525_in(T177), s(T177)) U6(f525_out1(T178), s(T177)) -> f525_out1(s(T178)) f436_in(T111) -> U7(f454_in(T111), T111) U7(f454_out1, T111) -> U8(f2_in(T111), T111) U8(f2_out1, T111) -> f436_out1 f523_in(T153) -> U9(f525_in(T153), T153) U9(f525_out1(T158), T153) -> U10(f2_in(T153), T153, T158) U10(f2_out1, T153, T158) -> f523_out1(T158) 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: F333_IN -> F333_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 = F333_IN evaluates to t =F333_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 F333_IN to F333_IN. ---------------------------------------- (103) NO ---------------------------------------- (104) Obligation: Q DP problem: The TRS P consists of the following rules: F2_IN(T111) -> F436_IN(T111) F436_IN(T111) -> U7^1(f454_in(T111), T111) U7^1(f454_out1, T111) -> F2_IN(T111) F2_IN(T153) -> F523_IN(T153) F523_IN(T153) -> U9^1(f525_in(T153), T153) U9^1(f525_out1(T158), T153) -> F2_IN(T153) The TRS R consists of the following rules: f2_in(T12) -> f2_out1 f2_in(T44) -> U1(f333_in, T44) U1(f333_out1, T44) -> f2_out1 f2_in(T111) -> U2(f436_in(T111), T111) U2(f436_out1, T111) -> f2_out1 f2_in(T153) -> U3(f523_in(T153), T153) U3(f523_out1(T158), T153) -> f2_out1 f333_in -> f333_out1 f333_in -> U4(f333_in) U4(f333_out1) -> f333_out1 f454_in(0) -> f454_out1 f454_in(s(T134)) -> U5(f454_in(T134), s(T134)) U5(f454_out1, s(T134)) -> f454_out1 f525_in(s(T171)) -> f525_out1(0) f525_in(s(T177)) -> U6(f525_in(T177), s(T177)) U6(f525_out1(T178), s(T177)) -> f525_out1(s(T178)) f436_in(T111) -> U7(f454_in(T111), T111) U7(f454_out1, T111) -> U8(f2_in(T111), T111) U8(f2_out1, T111) -> f436_out1 f523_in(T153) -> U9(f525_in(T153), T153) U9(f525_out1(T158), T153) -> U10(f2_in(T153), T153, T158) U10(f2_out1, T153, T158) -> f523_out1(T158) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (105) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by narrowing to the left: s = F436_IN(0) evaluates to t =F436_IN(0) Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence F436_IN(0) -> U7^1(f454_in(0), 0) with rule F436_IN(T111) -> U7^1(f454_in(T111), T111) at position [] and matcher [T111 / 0] U7^1(f454_in(0), 0) -> U7^1(f454_out1, 0) with rule f454_in(0) -> f454_out1 at position [0] and matcher [ ] U7^1(f454_out1, 0) -> F2_IN(0) with rule U7^1(f454_out1, T111') -> F2_IN(T111') at position [] and matcher [T111' / 0] F2_IN(0) -> F436_IN(0) with rule F2_IN(T111) -> F436_IN(T111) Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence All these steps are and every following step will be a correct step w.r.t to Q. ---------------------------------------- (106) NO ---------------------------------------- (107) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 1, "program": { "directives": [], "clauses": [ [ "(delete X (tree X (void) Right) Right)", null ], [ "(delete X (tree X Left (void)) Left)", null ], [ "(delete X (tree X Left Right) (tree Y Left Right1))", "(delmin Right Y Right1)" ], [ "(delete X (tree Y Left Right) (tree Y Left1 Right))", "(',' (less X Y) (delete X Left Left1))" ], [ "(delete X (tree Y Left Right) (tree Y Left Right1))", "(',' (less Y X) (delete X Right Right1))" ], [ "(delmin (tree Y (void) Right) Y Right)", null ], [ "(delmin (tree X Left X1) Y (tree X Left1 X2))", "(delmin Left Y Left1)" ], [ "(less (0) (s X3))", null ], [ "(less (s X) (s Y))", "(less X Y)" ] ] }, "graph": { "nodes": { "44": { "goal": [ { "clause": 1, "scope": 1, "term": "(delete T1 T2 T3)" }, { "clause": 2, "scope": 1, "term": "(delete T1 T2 T3)" }, { "clause": 3, "scope": 1, "term": "(delete T1 T2 T3)" }, { "clause": 4, "scope": 1, "term": "(delete T1 T2 T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "type": "Nodes", "471": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (less T111 T116) (delete T111 T117 T118))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T111"], "free": [], "exprvars": [] } }, "472": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "475": { "goal": [{ "clause": -1, "scope": -1, "term": "(less T111 T116)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T111"], "free": [], "exprvars": [] } }, "476": { "goal": [{ "clause": -1, "scope": -1, "term": "(delete T111 T121 T122)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": 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"goal": [{ "clause": 7, "scope": 4, "term": "(less T158 T153)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T153"], "free": [], "exprvars": [] } }, "517": { "goal": [{ "clause": 8, "scope": 4, "term": "(less T158 T153)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T153"], "free": [], "exprvars": [] } }, "518": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "519": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "480": { "goal": [ { "clause": 7, "scope": 3, "term": "(less T111 T116)" }, { "clause": 8, "scope": 3, "term": "(less T111 T116)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T111"], "free": [], "exprvars": [] } }, "360": { "goal": [{ "clause": 1, "scope": 1, "term": "(delete T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "481": { "goal": [{ "clause": 7, "scope": 3, "term": "(less T111 T116)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T111"], "free": [], "exprvars": [] } }, "361": { "goal": [ { "clause": 2, "scope": 1, "term": "(delete T1 T2 T3)" }, { "clause": 3, "scope": 1, "term": "(delete T1 T2 T3)" }, { "clause": 4, "scope": 1, "term": "(delete T1 T2 T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "482": { "goal": [{ "clause": 8, "scope": 3, "term": "(less T111 T116)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T111"], "free": [], "exprvars": [] } }, "362": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "363": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1": { "goal": [{ "clause": -1, "scope": -1, "term": "(delete T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "364": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": 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"PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "522": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "369": { "goal": [ { "clause": 5, "scope": 2, "term": "(delmin T49 T50 T51)" }, { "clause": 6, "scope": 2, "term": "(delmin T49 T50 T51)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "370": { "goal": [{ "clause": 5, "scope": 2, "term": "(delmin T49 T50 T51)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "491": { "goal": [{ "clause": -1, "scope": -1, "term": "(less T134 T136)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T134"], "free": [], "exprvars": [] } }, "371": { "goal": [{ "clause": 6, "scope": 2, "term": "(delmin T49 T50 T51)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "492": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "372": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "373": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "494": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (less T158 T153) (delete T153 T159 T160))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T153"], "free": [], "exprvars": [] } }, "374": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "496": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "497": { "goal": [{ "clause": -1, "scope": -1, "term": "(less T158 T153)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T153"], "free": [], "exprvars": [] } }, "498": { "goal": [{ "clause": -1, "scope": -1, "term": "(delete T153 T163 T164)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T153"], "free": [], "exprvars": [] } }, "467": { "goal": [{ "clause": 3, "scope": 1, "term": "(delete T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "468": { "goal": [{ "clause": 4, "scope": 1, "term": "(delete T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "427": { "goal": [{ "clause": -1, "scope": -1, "term": "(delmin T84 T85 T86)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "428": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "40": { "goal": [ { "clause": 0, "scope": 1, "term": "(delete T1 T2 T3)" }, { "clause": 1, "scope": 1, "term": "(delete T1 T2 T3)" }, { "clause": 2, "scope": 1, "term": "(delete T1 T2 T3)" }, { "clause": 3, "scope": 1, "term": "(delete T1 T2 T3)" }, { "clause": 4, "scope": 1, "term": "(delete T1 T2 T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "41": { "goal": [{ "clause": 0, "scope": 1, "term": "(delete T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 40, "label": "CASE" }, { "from": 40, "to": 41, "label": "PARALLEL" }, { "from": 40, "to": 44, "label": "PARALLEL" }, { "from": 41, "to": 357, "label": "EVAL with clause\ndelete(X12, tree(X12, void, X13), X13).\nand substitutionT1 -> T12,\nX12 -> T12,\nX13 -> T13,\nT2 -> tree(T12, void, T13),\nT3 -> T13" }, { "from": 41, "to": 358, "label": "EVAL-BACKTRACK" }, { "from": 44, "to": 360, "label": "PARALLEL" }, { "from": 44, "to": 361, "label": "PARALLEL" }, { "from": 357, "to": 359, "label": "SUCCESS" }, { "from": 360, "to": 362, "label": "EVAL with clause\ndelete(X22, tree(X22, X23, void), X23).\nand substitutionT1 -> T22,\nX22 -> T22,\nX23 -> T23,\nT2 -> tree(T22, T23, void),\nT3 -> T23" }, { "from": 360, "to": 363, "label": "EVAL-BACKTRACK" }, { "from": 361, "to": 365, "label": "PARALLEL" }, { "from": 361, "to": 366, "label": "PARALLEL" }, { "from": 362, "to": 364, "label": "SUCCESS" }, { "from": 365, "to": 367, "label": "EVAL with clause\ndelete(X44, tree(X44, X45, X46), tree(X47, X45, X48)) :- delmin(X46, X47, X48).\nand substitutionT1 -> T44,\nX44 -> T44,\nX45 -> T45,\nX46 -> T49,\nT2 -> tree(T44, T45, T49),\nX47 -> T50,\nX48 -> T51,\nT3 -> tree(T50, T45, T51),\nT46 -> T49,\nT47 -> T50,\nT48 -> T51" }, { "from": 365, "to": 368, "label": "EVAL-BACKTRACK" }, { "from": 366, "to": 467, "label": "PARALLEL" }, { "from": 366, "to": 468, "label": "PARALLEL" }, { "from": 367, "to": 369, "label": "CASE" }, { "from": 369, "to": 370, "label": "PARALLEL" }, { "from": 369, "to": 371, "label": "PARALLEL" }, { "from": 370, "to": 372, "label": "EVAL with clause\ndelmin(tree(X61, void, X62), X61, X62).\nand substitutionX61 -> T64,\nX62 -> T65,\nT49 -> tree(T64, void, T65),\nT50 -> T64,\nT51 -> T65" }, { "from": 370, "to": 373, "label": "EVAL-BACKTRACK" }, { "from": 371, "to": 427, "label": "EVAL with clause\ndelmin(tree(X75, X76, X77), X78, tree(X75, X79, X80)) :- delmin(X76, X78, X79).\nand substitutionX75 -> T78,\nX76 -> T84,\nX77 -> T80,\nT49 -> tree(T78, T84, T80),\nT50 -> T85,\nX78 -> T85,\nX79 -> T86,\nX80 -> T83,\nT51 -> tree(T78, T86, T83),\nT79 -> T84,\nT81 -> T85,\nT82 -> T86" }, { "from": 371, "to": 428, "label": "EVAL-BACKTRACK" }, { "from": 372, "to": 374, "label": "SUCCESS" }, { "from": 427, "to": 367, "label": "INSTANCE with matching:\nT49 -> T84\nT50 -> T85\nT51 -> T86" }, { "from": 467, "to": 471, "label": "EVAL with clause\ndelete(X105, tree(X106, X107, X108), tree(X106, X109, X108)) :- ','(less(X105, X106), delete(X105, X107, X109)).\nand substitutionT1 -> T111,\nX105 -> T111,\nX106 -> T116,\nX107 -> T117,\nX108 -> T114,\nT2 -> tree(T116, T117, T114),\nX109 -> T118,\nT3 -> tree(T116, T118, T114),\nT112 -> T116,\nT113 -> T117,\nT115 -> T118" }, { "from": 467, "to": 472, "label": "EVAL-BACKTRACK" }, { "from": 468, "to": 494, "label": "EVAL with clause\ndelete(X141, tree(X142, X143, X144), tree(X142, X143, X145)) :- ','(less(X142, X141), delete(X141, X144, X145)).\nand substitutionT1 -> T153,\nX141 -> T153,\nX142 -> T158,\nX143 -> T155,\nX144 -> T159,\nT2 -> tree(T158, T155, T159),\nX145 -> T160,\nT3 -> tree(T158, T155, T160),\nT154 -> T158,\nT156 -> T159,\nT157 -> T160" }, { "from": 468, "to": 496, "label": "EVAL-BACKTRACK" }, { "from": 471, "to": 475, "label": "SPLIT 1" }, { "from": 471, "to": 476, "label": "SPLIT 2\nnew knowledge:\nT111 is ground\nreplacements:T117 -> T121,\nT118 -> T122" }, { "from": 475, "to": 480, "label": "CASE" }, { "from": 476, "to": 1, "label": "INSTANCE with matching:\nT1 -> T111\nT2 -> T121\nT3 -> T122" }, { "from": 480, "to": 481, "label": "PARALLEL" }, { "from": 480, "to": 482, "label": "PARALLEL" }, { "from": 481, "to": 486, "label": "EVAL with clause\nless(0, s(X118)).\nand substitutionT111 -> 0,\nX118 -> T129,\nT116 -> s(T129)" }, { "from": 481, "to": 487, "label": "EVAL-BACKTRACK" }, { "from": 482, "to": 491, "label": "EVAL with clause\nless(s(X123), s(X124)) :- less(X123, X124).\nand substitutionX123 -> T134,\nT111 -> s(T134),\nX124 -> T136,\nT116 -> s(T136),\nT135 -> T136" }, { "from": 482, "to": 492, "label": "EVAL-BACKTRACK" }, { "from": 486, "to": 488, "label": "SUCCESS" }, { "from": 491, "to": 475, "label": "INSTANCE with matching:\nT111 -> T134\nT116 -> T136" }, { "from": 494, "to": 497, "label": "SPLIT 1" }, { "from": 494, "to": 498, "label": "SPLIT 2\nnew knowledge:\nT158 is ground\nT153 is ground\nreplacements:T159 -> T163,\nT160 -> T164" }, { "from": 497, "to": 515, "label": "CASE" }, { "from": 498, "to": 1, "label": "INSTANCE with matching:\nT1 -> T153\nT2 -> T163\nT3 -> T164" }, { "from": 515, "to": 516, "label": "PARALLEL" }, { "from": 515, "to": 517, "label": "PARALLEL" }, { "from": 516, "to": 518, "label": "EVAL with clause\nless(0, s(X154)).\nand substitutionT158 -> 0,\nX154 -> T171,\nT153 -> s(T171)" }, { "from": 516, "to": 519, "label": "EVAL-BACKTRACK" }, { "from": 517, "to": 521, "label": "EVAL with clause\nless(s(X159), s(X160)) :- less(X159, X160).\nand substitutionX159 -> T178,\nT158 -> s(T178),\nX160 -> T177,\nT153 -> s(T177),\nT176 -> T178" }, { "from": 517, "to": 522, "label": "EVAL-BACKTRACK" }, { "from": 518, "to": 520, "label": "SUCCESS" }, { "from": 521, "to": 497, "label": "INSTANCE with matching:\nT158 -> T178\nT153 -> T177" } ], "type": "Graph" } } ---------------------------------------- (108) Complex Obligation (AND) ---------------------------------------- (109) Obligation: Rules: f521_in(T177) -> f497_in(T177) :|: TRUE f497_out(x) -> f521_out(x) :|: TRUE f516_out(T153) -> f515_out(T153) :|: TRUE f517_out(x1) -> f515_out(x1) :|: TRUE f515_in(x2) -> f516_in(x2) :|: TRUE f515_in(x3) -> f517_in(x3) :|: TRUE f497_in(x4) -> f515_in(x4) :|: TRUE f515_out(x5) -> f497_out(x5) :|: TRUE f521_out(x6) -> f517_out(s(x6)) :|: TRUE f517_in(x7) -> f522_in :|: TRUE f517_in(s(x8)) -> f521_in(x8) :|: TRUE f522_out -> f517_out(x9) :|: TRUE f1_in(T1) -> f40_in(T1) :|: TRUE f40_out(x10) -> f1_out(x10) :|: TRUE f41_out(x11) -> f40_out(x11) :|: TRUE f44_out(x12) -> f40_out(x12) :|: TRUE f40_in(x13) -> f41_in(x13) :|: TRUE f40_in(x14) -> f44_in(x14) :|: TRUE f361_out(x15) -> f44_out(x15) :|: TRUE f44_in(x16) -> f360_in(x16) :|: TRUE f360_out(x17) -> f44_out(x17) :|: TRUE f44_in(x18) -> f361_in(x18) :|: TRUE f365_out(x19) -> f361_out(x19) :|: TRUE f361_in(x20) -> f366_in(x20) :|: TRUE f366_out(x21) -> f361_out(x21) :|: TRUE f361_in(x22) -> f365_in(x22) :|: TRUE f366_in(x23) -> f468_in(x23) :|: TRUE f468_out(x24) -> f366_out(x24) :|: TRUE f467_out(x25) -> f366_out(x25) :|: TRUE f366_in(x26) -> f467_in(x26) :|: TRUE f468_in(x27) -> f496_in :|: TRUE f496_out -> f468_out(x28) :|: TRUE f494_out(x29) -> f468_out(x29) :|: TRUE f468_in(x30) -> f494_in(x30) :|: TRUE f497_out(x31) -> f498_in(x31) :|: TRUE f498_out(x32) -> f494_out(x32) :|: TRUE f494_in(x33) -> f497_in(x33) :|: TRUE Start term: f1_in(T1) ---------------------------------------- (110) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f521_in(T177) -> f497_in(T177) :|: TRUE f515_in(x3) -> f517_in(x3) :|: TRUE f497_in(x4) -> f515_in(x4) :|: TRUE f517_in(s(x8)) -> f521_in(x8) :|: TRUE ---------------------------------------- (111) Obligation: Rules: f521_in(T177) -> f497_in(T177) :|: TRUE f515_in(x3) -> f517_in(x3) :|: TRUE f497_in(x4) -> f515_in(x4) :|: TRUE f517_in(s(x8)) -> f521_in(x8) :|: TRUE ---------------------------------------- (112) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (113) Obligation: Rules: f521_in(s(x8:0)) -> f521_in(x8:0) :|: TRUE ---------------------------------------- (114) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (115) Obligation: Rules: f521_in(s(x8:0)) -> f521_in(x8:0) :|: TRUE ---------------------------------------- (116) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f521_in(s(x8:0)) -> f521_in(x8:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (117) Obligation: Termination digraph: Nodes: (1) f521_in(s(x8:0)) -> f521_in(x8:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (118) TempFilterProof (SOUND) Used the following sort dictionary for filtering: f521_in(VARIABLE) s(VARIABLE) Removed predefined arithmetic. ---------------------------------------- (119) Obligation: Rules: f521_in(s(x8:0)) -> f521_in(x8:0) ---------------------------------------- (120) IRSwTToQDPProof (SOUND) Removed the integers and created a QDP-Problem. ---------------------------------------- (121) Obligation: Q DP problem: The TRS P consists of the following rules: f521_in(s(x8:0)) -> f521_in(x8:0) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (122) 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: *f521_in(s(x8:0)) -> f521_in(x8:0) The graph contains the following edges 1 > 1 ---------------------------------------- (123) YES ---------------------------------------- (124) Obligation: Rules: f480_in(T111) -> f481_in(T111) :|: TRUE f480_in(x) -> f482_in(x) :|: TRUE f481_out(x1) -> f480_out(x1) :|: TRUE f482_out(x2) -> f480_out(x2) :|: TRUE f475_out(T134) -> f491_out(T134) :|: TRUE f491_in(x3) -> f475_in(x3) :|: TRUE f475_in(x4) -> f480_in(x4) :|: TRUE f480_out(x5) -> f475_out(x5) :|: TRUE f492_out -> f482_out(x6) :|: TRUE f482_in(x7) -> f492_in :|: TRUE f491_out(x8) -> f482_out(s(x8)) :|: TRUE f482_in(s(x9)) -> f491_in(x9) :|: TRUE f1_in(T1) -> f40_in(T1) :|: TRUE f40_out(x10) -> f1_out(x10) :|: TRUE f41_out(x11) -> f40_out(x11) :|: TRUE f44_out(x12) -> f40_out(x12) :|: TRUE f40_in(x13) -> f41_in(x13) :|: TRUE f40_in(x14) -> f44_in(x14) :|: TRUE f361_out(x15) -> f44_out(x15) :|: TRUE f44_in(x16) -> f360_in(x16) :|: TRUE f360_out(x17) -> f44_out(x17) :|: TRUE f44_in(x18) -> f361_in(x18) :|: TRUE f365_out(x19) -> f361_out(x19) :|: TRUE f361_in(x20) -> f366_in(x20) :|: TRUE f366_out(x21) -> f361_out(x21) :|: TRUE f361_in(x22) -> f365_in(x22) :|: TRUE f366_in(x23) -> f468_in(x23) :|: TRUE f468_out(x24) -> f366_out(x24) :|: TRUE f467_out(x25) -> f366_out(x25) :|: TRUE f366_in(x26) -> f467_in(x26) :|: TRUE f467_in(x27) -> f472_in :|: TRUE f467_in(x28) -> f471_in(x28) :|: TRUE f471_out(x29) -> f467_out(x29) :|: TRUE f472_out -> f467_out(x30) :|: TRUE f475_out(x31) -> f476_in(x31) :|: TRUE f471_in(x32) -> f475_in(x32) :|: TRUE f476_out(x33) -> f471_out(x33) :|: TRUE Start term: f1_in(T1) ---------------------------------------- (125) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f480_in(x) -> f482_in(x) :|: TRUE f491_in(x3) -> f475_in(x3) :|: TRUE f475_in(x4) -> f480_in(x4) :|: TRUE f482_in(s(x9)) -> f491_in(x9) :|: TRUE ---------------------------------------- (126) Obligation: Rules: f480_in(x) -> f482_in(x) :|: TRUE f491_in(x3) -> f475_in(x3) :|: TRUE f475_in(x4) -> f480_in(x4) :|: TRUE f482_in(s(x9)) -> f491_in(x9) :|: TRUE ---------------------------------------- (127) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (128) Obligation: Rules: f491_in(s(x9:0)) -> f491_in(x9:0) :|: TRUE ---------------------------------------- (129) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (130) Obligation: Rules: f491_in(s(x9:0)) -> f491_in(x9:0) :|: TRUE ---------------------------------------- (131) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f491_in(s(x9:0)) -> f491_in(x9:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (132) Obligation: Termination digraph: Nodes: (1) f491_in(s(x9:0)) -> f491_in(x9:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (133) TempFilterProof (SOUND) Used the following sort dictionary for filtering: f491_in(VARIABLE) s(VARIABLE) Removed predefined arithmetic. ---------------------------------------- (134) Obligation: Rules: f491_in(s(x9:0)) -> f491_in(x9:0) ---------------------------------------- (135) IRSwTToQDPProof (SOUND) Removed the integers and created a QDP-Problem. ---------------------------------------- (136) Obligation: Q DP problem: The TRS P consists of the following rules: f491_in(s(x9:0)) -> f491_in(x9:0) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (137) 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: *f491_in(s(x9:0)) -> f491_in(x9:0) The graph contains the following edges 1 > 1 ---------------------------------------- (138) YES ---------------------------------------- (139) Obligation: Rules: f371_in -> f427_in :|: TRUE f427_out -> f371_out :|: TRUE f371_in -> f428_in :|: TRUE f428_out -> f371_out :|: TRUE f427_in -> f367_in :|: TRUE f367_out -> f427_out :|: TRUE f367_in -> f369_in :|: TRUE f369_out -> f367_out :|: TRUE f369_in -> f371_in :|: TRUE f370_out -> f369_out :|: TRUE f371_out -> f369_out :|: TRUE f369_in -> f370_in :|: TRUE f1_in(T1) -> f40_in(T1) :|: TRUE f40_out(x) -> f1_out(x) :|: TRUE f41_out(x1) -> f40_out(x1) :|: TRUE f44_out(x2) -> f40_out(x2) :|: TRUE f40_in(x3) -> f41_in(x3) :|: TRUE f40_in(x4) -> f44_in(x4) :|: TRUE f361_out(x5) -> f44_out(x5) :|: TRUE f44_in(x6) -> f360_in(x6) :|: TRUE f360_out(x7) -> f44_out(x7) :|: TRUE f44_in(x8) -> f361_in(x8) :|: TRUE f365_out(x9) -> f361_out(x9) :|: TRUE f361_in(x10) -> f366_in(x10) :|: TRUE f366_out(x11) -> f361_out(x11) :|: TRUE f361_in(x12) -> f365_in(x12) :|: TRUE f367_out -> f365_out(T44) :|: TRUE f365_in(x13) -> f368_in :|: TRUE f365_in(x14) -> f367_in :|: TRUE f368_out -> f365_out(x15) :|: TRUE Start term: f1_in(T1) ---------------------------------------- (140) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f371_in -> f427_in :|: TRUE f427_in -> f367_in :|: TRUE f367_in -> f369_in :|: TRUE f369_in -> f371_in :|: TRUE ---------------------------------------- (141) Obligation: Rules: f371_in -> f427_in :|: TRUE f427_in -> f367_in :|: TRUE f367_in -> f369_in :|: TRUE f369_in -> f371_in :|: TRUE ---------------------------------------- (142) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (143) Obligation: Rules: f371_in -> f371_in :|: TRUE ---------------------------------------- (144) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (145) Obligation: Rules: f371_in -> f371_in :|: TRUE ---------------------------------------- (146) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f371_in -> f371_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (147) Obligation: Termination digraph: Nodes: (1) f371_in -> f371_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (148) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f371_in() Replaced non-predefined constructor symbols by 0. ---------------------------------------- (149) Obligation: Rules: f371_in -> f371_in :|: TRUE ---------------------------------------- (150) IntTRSPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc) -> f(1) :|: pc = 1 && TRUE Witness term starting non-terminating reduction: f(1) ---------------------------------------- (151) NO ---------------------------------------- (152) Obligation: Rules: f518_in -> f518_out :|: TRUE f486_in -> f486_out :|: TRUE f361_out(T1) -> f44_out(T1) :|: TRUE f44_in(x) -> f360_in(x) :|: TRUE f360_out(x1) -> f44_out(x1) :|: TRUE f44_in(x2) -> f361_in(x2) :|: TRUE f521_in(T177) -> f497_in(T177) :|: TRUE f497_out(x3) -> f521_out(x3) :|: TRUE f366_in(x4) -> f468_in(x4) :|: TRUE f468_out(x5) -> f366_out(x5) :|: TRUE f467_out(x6) -> f366_out(x6) :|: TRUE f366_in(x7) -> f467_in(x7) :|: TRUE f480_in(T111) -> f481_in(T111) :|: TRUE f480_in(x8) -> f482_in(x8) :|: TRUE f481_out(x9) -> f480_out(x9) :|: TRUE f482_out(x10) -> f480_out(x10) :|: TRUE f1_out(T153) -> f498_out(T153) :|: TRUE f498_in(x11) -> f1_in(x11) :|: TRUE f516_in(x12) -> f519_in :|: TRUE f518_out -> f516_out(s(T171)) :|: TRUE f519_out -> f516_out(x13) :|: TRUE f516_in(s(x14)) -> f518_in :|: TRUE f475_in(x15) -> f480_in(x15) :|: TRUE f480_out(x16) -> f475_out(x16) :|: TRUE f475_out(x17) -> f476_in(x17) :|: TRUE f471_in(x18) -> f475_in(x18) :|: TRUE f476_out(x19) -> f471_out(x19) :|: TRUE f475_out(T134) -> f491_out(T134) :|: TRUE f491_in(x20) -> f475_in(x20) :|: TRUE f497_in(x21) -> f515_in(x21) :|: TRUE f515_out(x22) -> f497_out(x22) :|: TRUE f492_out -> f482_out(x23) :|: TRUE f482_in(x24) -> f492_in :|: TRUE f491_out(x25) -> f482_out(s(x25)) :|: TRUE f482_in(s(x26)) -> f491_in(x26) :|: TRUE f365_out(x27) -> f361_out(x27) :|: TRUE f361_in(x28) -> f366_in(x28) :|: TRUE f366_out(x29) -> f361_out(x29) :|: TRUE f361_in(x30) -> f365_in(x30) :|: TRUE f1_in(x31) -> f40_in(x31) :|: TRUE f40_out(x32) -> f1_out(x32) :|: TRUE f521_out(x33) -> f517_out(s(x33)) :|: TRUE f517_in(x34) -> f522_in :|: TRUE f517_in(s(x35)) -> f521_in(x35) :|: TRUE f522_out -> f517_out(x36) :|: TRUE f41_out(x37) -> f40_out(x37) :|: TRUE f44_out(x38) -> f40_out(x38) :|: TRUE f40_in(x39) -> f41_in(x39) :|: TRUE f40_in(x40) -> f44_in(x40) :|: TRUE f468_in(x41) -> f496_in :|: TRUE f496_out -> f468_out(x42) :|: TRUE f494_out(x43) -> f468_out(x43) :|: TRUE f468_in(x44) -> f494_in(x44) :|: TRUE f516_out(x45) -> f515_out(x45) :|: TRUE f517_out(x46) -> f515_out(x46) :|: TRUE f515_in(x47) -> f516_in(x47) :|: TRUE f515_in(x48) -> f517_in(x48) :|: TRUE f467_in(x49) -> f472_in :|: TRUE f467_in(x50) -> f471_in(x50) :|: TRUE f471_out(x51) -> f467_out(x51) :|: TRUE f472_out -> f467_out(x52) :|: TRUE f476_in(x53) -> f1_in(x53) :|: TRUE f1_out(x54) -> f476_out(x54) :|: TRUE f497_out(x55) -> f498_in(x55) :|: TRUE f498_out(x56) -> f494_out(x56) :|: TRUE f494_in(x57) -> f497_in(x57) :|: TRUE f481_in(0) -> f486_in :|: TRUE f486_out -> f481_out(0) :|: TRUE f481_in(x58) -> f487_in :|: TRUE f487_out -> f481_out(x59) :|: TRUE Start term: f1_in(T1) ---------------------------------------- (153) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f518_in -> f518_out :|: TRUE f486_in -> f486_out :|: TRUE f44_in(x2) -> f361_in(x2) :|: TRUE f521_in(T177) -> f497_in(T177) :|: TRUE f497_out(x3) -> f521_out(x3) :|: TRUE f366_in(x4) -> f468_in(x4) :|: TRUE f366_in(x7) -> f467_in(x7) :|: TRUE f480_in(T111) -> f481_in(T111) :|: TRUE f480_in(x8) -> f482_in(x8) :|: TRUE f481_out(x9) -> f480_out(x9) :|: TRUE f482_out(x10) -> f480_out(x10) :|: TRUE f498_in(x11) -> f1_in(x11) :|: TRUE f518_out -> f516_out(s(T171)) :|: TRUE f516_in(s(x14)) -> f518_in :|: TRUE f475_in(x15) -> f480_in(x15) :|: TRUE f480_out(x16) -> f475_out(x16) :|: TRUE f475_out(x17) -> f476_in(x17) :|: TRUE f471_in(x18) -> f475_in(x18) :|: TRUE f475_out(T134) -> f491_out(T134) :|: TRUE f491_in(x20) -> f475_in(x20) :|: TRUE f497_in(x21) -> f515_in(x21) :|: TRUE f515_out(x22) -> f497_out(x22) :|: TRUE f491_out(x25) -> f482_out(s(x25)) :|: TRUE f482_in(s(x26)) -> f491_in(x26) :|: TRUE f361_in(x28) -> f366_in(x28) :|: TRUE f1_in(x31) -> f40_in(x31) :|: TRUE f521_out(x33) -> f517_out(s(x33)) :|: TRUE f517_in(s(x35)) -> f521_in(x35) :|: TRUE f40_in(x40) -> f44_in(x40) :|: TRUE f468_in(x44) -> f494_in(x44) :|: TRUE f516_out(x45) -> f515_out(x45) :|: TRUE f517_out(x46) -> f515_out(x46) :|: TRUE f515_in(x47) -> f516_in(x47) :|: TRUE f515_in(x48) -> f517_in(x48) :|: TRUE f467_in(x50) -> f471_in(x50) :|: TRUE f476_in(x53) -> f1_in(x53) :|: TRUE f497_out(x55) -> f498_in(x55) :|: TRUE f494_in(x57) -> f497_in(x57) :|: TRUE f481_in(0) -> f486_in :|: TRUE f486_out -> f481_out(0) :|: TRUE ---------------------------------------- (154) Obligation: Rules: f518_in -> f518_out :|: TRUE f486_in -> f486_out :|: TRUE f44_in(x2) -> f361_in(x2) :|: TRUE f521_in(T177) -> f497_in(T177) :|: TRUE f497_out(x3) -> f521_out(x3) :|: TRUE f366_in(x4) -> f468_in(x4) :|: TRUE f366_in(x7) -> f467_in(x7) :|: TRUE f480_in(T111) -> f481_in(T111) :|: TRUE f480_in(x8) -> f482_in(x8) :|: TRUE f481_out(x9) -> f480_out(x9) :|: TRUE f482_out(x10) -> f480_out(x10) :|: TRUE f498_in(x11) -> f1_in(x11) :|: TRUE f518_out -> f516_out(s(T171)) :|: TRUE f516_in(s(x14)) -> f518_in :|: TRUE f475_in(x15) -> f480_in(x15) :|: TRUE f480_out(x16) -> f475_out(x16) :|: TRUE f475_out(x17) -> f476_in(x17) :|: TRUE f471_in(x18) -> f475_in(x18) :|: TRUE f475_out(T134) -> f491_out(T134) :|: TRUE f491_in(x20) -> f475_in(x20) :|: TRUE f497_in(x21) -> f515_in(x21) :|: TRUE f515_out(x22) -> f497_out(x22) :|: TRUE f491_out(x25) -> f482_out(s(x25)) :|: TRUE f482_in(s(x26)) -> f491_in(x26) :|: TRUE f361_in(x28) -> f366_in(x28) :|: TRUE f1_in(x31) -> f40_in(x31) :|: TRUE f521_out(x33) -> f517_out(s(x33)) :|: TRUE f517_in(s(x35)) -> f521_in(x35) :|: TRUE f40_in(x40) -> f44_in(x40) :|: TRUE f468_in(x44) -> f494_in(x44) :|: TRUE f516_out(x45) -> f515_out(x45) :|: TRUE f517_out(x46) -> f515_out(x46) :|: TRUE f515_in(x47) -> f516_in(x47) :|: TRUE f515_in(x48) -> f517_in(x48) :|: TRUE f467_in(x50) -> f471_in(x50) :|: TRUE f476_in(x53) -> f1_in(x53) :|: TRUE f497_out(x55) -> f498_in(x55) :|: TRUE f494_in(x57) -> f497_in(x57) :|: TRUE f481_in(0) -> f486_in :|: TRUE f486_out -> f481_out(0) :|: TRUE ---------------------------------------- (155) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (156) Obligation: Rules: f44_in(x2:0) -> f515_in(x2:0) :|: TRUE f515_in(s(x35:0)) -> f515_in(x35:0) :|: TRUE f44_in(x) -> f475_in(x) :|: TRUE f515_out(x22:0) -> f44_in(x22:0) :|: TRUE f475_in(s(x26:0)) -> f475_in(x26:0) :|: TRUE f475_in(cons_0) -> f475_out(0) :|: TRUE && cons_0 = 0 f475_out(x17:0) -> f44_in(x17:0) :|: TRUE f515_in(s(x14:0)) -> f515_out(s(T171:0)) :|: TRUE f515_out(x1) -> f515_out(s(x1)) :|: TRUE f475_out(T134:0) -> f475_out(s(T134:0)) :|: TRUE ---------------------------------------- (157) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (158) Obligation: Rules: f44_in(x2:0) -> f515_in(x2:0) :|: TRUE f515_in(s(x35:0)) -> f515_in(x35:0) :|: TRUE f44_in(x) -> f475_in(x) :|: TRUE f515_out(x22:0) -> f44_in(x22:0) :|: TRUE f475_in(s(x26:0)) -> f475_in(x26:0) :|: TRUE f475_in(cons_0) -> f475_out(0) :|: TRUE && cons_0 = 0 f475_out(x17:0) -> f44_in(x17:0) :|: TRUE f515_in(s(x14:0)) -> f515_out(s(T171:0)) :|: TRUE f515_out(x1) -> f515_out(s(x1)) :|: TRUE f475_out(T134:0) -> f475_out(s(T134:0)) :|: TRUE ---------------------------------------- (159) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f44_in(x2:0) -> f515_in(x2:0) :|: TRUE (2) f515_in(s(x35:0)) -> f515_in(x35:0) :|: TRUE (3) f44_in(x) -> f475_in(x) :|: TRUE (4) f515_out(x22:0) -> f44_in(x22:0) :|: TRUE (5) f475_in(s(x26:0)) -> f475_in(x26:0) :|: TRUE (6) f475_in(cons_0) -> f475_out(0) :|: TRUE && cons_0 = 0 (7) f475_out(x17:0) -> f44_in(x17:0) :|: TRUE (8) f515_in(s(x14:0)) -> f515_out(s(T171:0)) :|: TRUE (9) f515_out(x1) -> f515_out(s(x1)) :|: TRUE (10) f475_out(T134:0) -> f475_out(s(T134:0)) :|: TRUE Arcs: (1) -> (2), (8) (2) -> (2), (8) (3) -> (5), (6) (4) -> (1), (3) (5) -> (5), (6) (6) -> (7), (10) (7) -> (1), (3) (8) -> (4), (9) (9) -> (4), (9) (10) -> (7), (10) This digraph is fully evaluated! ---------------------------------------- (160) Obligation: Termination digraph: Nodes: (1) f44_in(x2:0) -> f515_in(x2:0) :|: TRUE (2) f475_out(x17:0) -> f44_in(x17:0) :|: TRUE (3) f475_out(T134:0) -> f475_out(s(T134:0)) :|: TRUE (4) f475_in(cons_0) -> f475_out(0) :|: TRUE && cons_0 = 0 (5) f475_in(s(x26:0)) -> f475_in(x26:0) :|: TRUE (6) f44_in(x) -> f475_in(x) :|: TRUE (7) f515_out(x22:0) -> f44_in(x22:0) :|: TRUE (8) f515_out(x1) -> f515_out(s(x1)) :|: TRUE (9) f515_in(s(x14:0)) -> f515_out(s(T171:0)) :|: TRUE (10) f515_in(s(x35:0)) -> f515_in(x35:0) :|: TRUE Arcs: (1) -> (9), (10) (2) -> (1), (6) (3) -> (2), (3) (4) -> (2), (3) (5) -> (4), (5) (6) -> (4), (5) (7) -> (1), (6) (8) -> (7), (8) (9) -> (7), (8) (10) -> (9), (10) This digraph is fully evaluated! ---------------------------------------- (161) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 3, "program": { "directives": [], "clauses": [ [ "(delete X (tree X (void) Right) Right)", null ], [ "(delete X (tree X Left (void)) Left)", null ], [ "(delete X (tree X Left Right) (tree Y Left Right1))", "(delmin Right Y Right1)" ], [ "(delete X (tree Y Left Right) (tree Y Left1 Right))", "(',' (less X Y) (delete X Left Left1))" ], [ "(delete X (tree Y Left Right) (tree Y Left Right1))", "(',' (less Y X) (delete X Right Right1))" ], [ "(delmin (tree Y (void) Right) Y Right)", null ], [ "(delmin (tree X Left X1) Y (tree X Left1 X2))", "(delmin Left Y Left1)" ], [ "(less (0) (s X3))", null ], [ "(less (s X) (s Y))", "(less X Y)" ] ] }, "graph": { "nodes": { "470": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "473": { "goal": [{ "clause": -1, "scope": -1, "term": "(less T127 T132)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T127"], "free": [], "exprvars": [] } }, "474": { "goal": [{ "clause": -1, "scope": -1, "term": "(delete T127 T137 T138)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T127"], "free": [], "exprvars": [] } }, "597": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (less T298 T297) (delete (s T297) T299 T300))" }], "kb": { "nonunifying": [ [ "(delete (s T297) T2 T3)", "(delete X17 (tree X17 X18 X19) (tree X20 X18 X21))" ], [ "(delete (s T297) T2 T3)", "(delete X185 (tree X186 X187 X188) (tree X186 X189 X188))" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T297"], "free": [ "X17", "X18", "X19", "X20", "X21", "X185", "X186", "X187", "X188", "X189" ], "exprvars": [] } }, "477": { "goal": [ { "clause": 7, "scope": 4, "term": "(less T127 T132)" }, { "clause": 8, "scope": 4, "term": "(less T127 T132)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T127"], "free": [], "exprvars": [] } }, "598": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "478": { "goal": [{ "clause": 7, "scope": 4, "term": "(less T127 T132)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T127"], "free": [], "exprvars": [] } }, "599": { "goal": [{ "clause": -1, "scope": -1, "term": "(less T298 T297)" }], "kb": { "nonunifying": [ [ "(delete (s T297) T2 T3)", "(delete X17 (tree X17 X18 X19) (tree X20 X18 X21))" ], [ "(delete (s T297) T2 T3)", "(delete X185 (tree X186 X187 X188) (tree X186 X189 X188))" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T297"], "free": [ "X17", "X18", "X19", "X20", "X21", "X185", "X186", "X187", "X188", "X189" ], "exprvars": [] } }, "479": { "goal": [{ "clause": 8, "scope": 4, "term": "(less T127 T132)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T127"], "free": [], "exprvars": [] } }, "483": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "121": { "goal": [ { "clause": 1, "scope": 1, "term": "(delete T6 T2 T3)" }, { "clause": 2, "scope": 1, "term": "(delete T6 T2 T3)" }, { "clause": 3, "scope": 1, "term": "(delete T6 T2 T3)" }, { "clause": 4, "scope": 1, "term": "(delete T6 T2 T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T6"], "free": [], "exprvars": [] } }, "484": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "485": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "3": { "goal": [{ "clause": -1, "scope": -1, "term": "(delete T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "489": { "goal": [{ "clause": -1, "scope": -1, "term": "(less T150 T152)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T150"], "free": [], "exprvars": [] } }, "8": { "goal": [ { "clause": 0, "scope": 1, "term": "(delete T1 T2 T3)" }, { "clause": 1, "scope": 1, "term": "(delete T1 T2 T3)" }, { "clause": 2, "scope": 1, "term": "(delete T1 T2 T3)" }, { "clause": 3, "scope": 1, "term": "(delete T1 T2 T3)" }, { "clause": 4, "scope": 1, "term": "(delete T1 T2 T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "490": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "493": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (less T174 T169) (delete T169 T175 T176))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T169"], "free": [], "exprvars": [] } }, "495": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "377": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "379": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "380": { "goal": [], "kb": { 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"699": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "465": { "goal": [{ "clause": 3, "scope": 1, "term": "(delete T17 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T17"], "free": [], "exprvars": [] } }, "466": { "goal": [{ "clause": 4, "scope": 1, "term": "(delete T17 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T17"], "free": [], "exprvars": [] } }, "469": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (less T127 T132) (delete T127 T133 T134))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T127"], "free": [], "exprvars": [] } } }, "edges": [ { "from": 3, "to": 8, "label": "CASE" }, { "from": 8, "to": 70, "label": "EVAL with clause\ndelete(X6, tree(X6, void, X7), X7).\nand substitutionT1 -> T6,\nX6 -> T6,\nX7 -> T7,\nT2 -> tree(T6, void, T7),\nT3 -> T7" }, { "from": 8, "to": 92, "label": "EVAL-BACKTRACK" }, { "from": 70, "to": 121, "label": "SUCCESS" }, { "from": 92, "to": 643, "label": "EVAL with clause\ndelete(X442, tree(X442, X443, void), X443).\nand substitutionT1 -> T517,\nX442 -> T517,\nX443 -> T518,\nT2 -> tree(T517, T518, void),\nT3 -> T518" }, { "from": 92, "to": 644, "label": "EVAL-BACKTRACK" }, { "from": 121, "to": 158, "label": "EVAL with clause\ndelete(X10, tree(X10, X11, void), X11).\nand substitutionT6 -> T10,\nX10 -> T10,\nX11 -> T11,\nT2 -> tree(T10, T11, void),\nT3 -> T11" }, { "from": 121, "to": 171, "label": "EVAL-BACKTRACK" }, { "from": 158, "to": 178, "label": "SUCCESS" }, { "from": 171, "to": 601, "label": "EVAL with clause\ndelete(X272, tree(X272, X273, X274), tree(X275, X273, X276)) :- delmin(X274, X275, X276).\nand substitutionT6 -> T318,\nX272 -> T318,\nX273 -> T319,\nX274 -> T323,\nT2 -> tree(T318, T319, T323),\nX275 -> T324,\nX276 -> T325,\nT3 -> tree(T324, T319, T325),\nT320 -> T323,\nT321 -> T324,\nT322 -> T325" }, { "from": 171, "to": 602, "label": "EVAL-BACKTRACK" }, { "from": 178, "to": 184, "label": "EVAL with clause\ndelete(X17, tree(X17, X18, X19), tree(X20, X18, X21)) :- delmin(X19, X20, X21).\nand substitutionT10 -> T17,\nX17 -> T17,\nX18 -> T18,\nX19 -> T22,\nT2 -> tree(T17, T18, T22),\nX20 -> T23,\nX21 -> T24,\nT3 -> tree(T23, T18, T24),\nT19 -> T22,\nT20 -> T23,\nT21 -> T24" }, { "from": 178, "to": 185, "label": "EVAL-BACKTRACK" }, { "from": 184, "to": 186, "label": "CASE" }, { "from": 185, "to": 545, "label": "EVAL with clause\ndelete(X185, tree(X186, X187, X188), tree(X186, X189, X188)) :- ','(less(X185, X186), delete(X185, X187, X189)).\nand substitutionT10 -> T206,\nX185 -> T206,\nX186 -> T211,\nX187 -> T212,\nX188 -> T209,\nT2 -> tree(T211, T212, T209),\nX189 -> T213,\nT3 -> tree(T211, T213, T209),\nT207 -> T211,\nT208 -> T212,\nT210 -> T213" }, { "from": 185, "to": 546, "label": "EVAL-BACKTRACK" }, { "from": 186, "to": 188, "label": "PARALLEL" }, { "from": 186, "to": 189, "label": "PARALLEL" }, { "from": 188, "to": 377, "label": "EVAL with clause\ndelmin(tree(X30, void, X31), X30, X31).\nand substitutionX30 -> T33,\nX31 -> T34,\nT22 -> tree(T33, void, T34),\nT23 -> T33,\nT24 -> T34" }, { "from": 188, "to": 379, "label": "EVAL-BACKTRACK" }, { "from": 189, "to": 398, "label": "PARALLEL" }, { "from": 189, "to": 399, "label": "PARALLEL" }, { "from": 377, "to": 380, "label": "SUCCESS" }, { "from": 398, "to": 420, "label": "EVAL with clause\ndelmin(tree(X56, X57, X58), X59, tree(X56, X60, X61)) :- delmin(X57, X59, X60).\nand substitutionX56 -> T59,\nX57 -> T65,\nX58 -> T61,\nT22 -> tree(T59, T65, T61),\nT23 -> T66,\nX59 -> T66,\nX60 -> T67,\nX61 -> T64,\nT24 -> tree(T59, T67, T64),\nT60 -> T65,\nT62 -> T66,\nT63 -> T67" }, { "from": 398, "to": 423, "label": "EVAL-BACKTRACK" }, { "from": 399, "to": 457, "label": "FAILURE" }, { "from": 420, "to": 424, "label": "CASE" }, { "from": 424, "to": 425, "label": "PARALLEL" }, { "from": 424, "to": 426, "label": "PARALLEL" }, { "from": 425, "to": 429, "label": "EVAL with clause\ndelmin(tree(X74, void, X75), X74, X75).\nand substitutionX74 -> T80,\nX75 -> T81,\nT65 -> tree(T80, void, T81),\nT66 -> T80,\nT67 -> T81" }, { "from": 425, "to": 430, "label": "EVAL-BACKTRACK" }, { "from": 426, "to": 432, "label": "EVAL with clause\ndelmin(tree(X88, X89, X90), X91, tree(X88, X92, X93)) :- delmin(X89, X91, X92).\nand substitutionX88 -> T94,\nX89 -> T100,\nX90 -> T96,\nT65 -> tree(T94, T100, T96),\nT66 -> T101,\nX91 -> T101,\nX92 -> T102,\nX93 -> T99,\nT67 -> tree(T94, T102, T99),\nT95 -> T100,\nT97 -> T101,\nT98 -> T102" }, { "from": 426, "to": 433, "label": "EVAL-BACKTRACK" }, { "from": 429, "to": 431, "label": "SUCCESS" }, { "from": 432, "to": 420, "label": "INSTANCE with matching:\nT65 -> T100\nT66 -> T101\nT67 -> T102" }, { "from": 457, "to": 465, "label": "PARALLEL" }, { "from": 457, "to": 466, "label": "PARALLEL" }, { "from": 465, "to": 469, "label": "EVAL with clause\ndelete(X118, tree(X119, X120, X121), tree(X119, X122, X121)) :- ','(less(X118, X119), delete(X118, X120, X122)).\nand substitutionT17 -> T127,\nX118 -> T127,\nX119 -> T132,\nX120 -> T133,\nX121 -> T130,\nT2 -> tree(T132, T133, T130),\nX122 -> T134,\nT3 -> tree(T132, T134, T130),\nT128 -> T132,\nT129 -> T133,\nT131 -> T134" }, { "from": 465, "to": 470, "label": "EVAL-BACKTRACK" }, { "from": 466, "to": 493, "label": "EVAL with clause\ndelete(X154, tree(X155, X156, X157), tree(X155, X156, X158)) :- ','(less(X155, X154), delete(X154, X157, X158)).\nand substitutionT17 -> T169,\nX154 -> T169,\nX155 -> T174,\nX156 -> T171,\nX157 -> T175,\nT2 -> tree(T174, T171, T175),\nX158 -> T176,\nT3 -> tree(T174, T171, T176),\nT170 -> T174,\nT172 -> T175,\nT173 -> T176" }, { "from": 466, "to": 495, "label": "EVAL-BACKTRACK" }, { "from": 469, "to": 473, "label": "SPLIT 1" }, { "from": 469, "to": 474, "label": "SPLIT 2\nnew knowledge:\nT127 is ground\nreplacements:T133 -> T137,\nT134 -> T138" }, { "from": 473, "to": 477, "label": "CASE" }, { "from": 474, "to": 3, "label": "INSTANCE with matching:\nT1 -> T127\nT2 -> T137\nT3 -> T138" }, { "from": 477, "to": 478, "label": "PARALLEL" }, { "from": 477, "to": 479, "label": "PARALLEL" }, { "from": 478, "to": 483, "label": "EVAL with clause\nless(0, s(X131)).\nand substitutionT127 -> 0,\nX131 -> T145,\nT132 -> s(T145)" }, { "from": 478, "to": 484, "label": "EVAL-BACKTRACK" }, { "from": 479, "to": 489, "label": "EVAL with clause\nless(s(X136), s(X137)) :- less(X136, X137).\nand substitutionX136 -> T150,\nT127 -> s(T150),\nX137 -> T152,\nT132 -> s(T152),\nT151 -> T152" }, { "from": 479, "to": 490, "label": "EVAL-BACKTRACK" }, { "from": 483, "to": 485, "label": "SUCCESS" }, { "from": 489, "to": 473, "label": "INSTANCE with matching:\nT127 -> T150\nT132 -> T152" }, { "from": 493, "to": 535, "label": "SPLIT 1" }, { "from": 493, "to": 536, "label": "SPLIT 2\nnew knowledge:\nT174 is ground\nT169 is ground\nreplacements:T175 -> T179,\nT176 -> T180" }, { "from": 535, "to": 537, "label": "CASE" }, { "from": 536, "to": 3, "label": "INSTANCE with matching:\nT1 -> T169\nT2 -> T179\nT3 -> T180" }, { "from": 537, "to": 538, "label": "PARALLEL" }, { "from": 537, "to": 539, "label": "PARALLEL" }, { "from": 538, "to": 540, "label": "EVAL with clause\nless(0, s(X167)).\nand substitutionT174 -> 0,\nX167 -> T187,\nT169 -> s(T187)" }, { "from": 538, "to": 541, "label": "EVAL-BACKTRACK" }, { "from": 539, "to": 543, "label": "EVAL with clause\nless(s(X172), s(X173)) :- less(X172, X173).\nand substitutionX172 -> T194,\nT174 -> s(T194),\nX173 -> T193,\nT169 -> s(T193),\nT192 -> T194" }, { "from": 539, "to": 544, "label": "EVAL-BACKTRACK" }, { "from": 540, "to": 542, "label": "SUCCESS" }, { "from": 543, "to": 535, "label": "INSTANCE with matching:\nT174 -> T194\nT169 -> T193" }, { "from": 545, "to": 547, "label": "CASE" }, { "from": 546, "to": 561, "label": "EVAL with clause\ndelete(X239, tree(X240, X241, X242), tree(X240, X241, X243)) :- ','(less(X240, X239), delete(X239, X242, X243)).\nand substitutionT10 -> T273,\nX239 -> T273,\nX240 -> T278,\nX241 -> T275,\nX242 -> T279,\nT2 -> tree(T278, T275, T279),\nX243 -> T280,\nT3 -> tree(T278, T275, T280),\nT274 -> T278,\nT276 -> T279,\nT277 -> T280" }, { "from": 546, "to": 562, "label": "EVAL-BACKTRACK" }, { "from": 547, "to": 548, "label": "PARALLEL" }, { "from": 547, "to": 549, "label": "PARALLEL" }, { "from": 548, "to": 550, "label": "EVAL with clause\nless(0, s(X194)).\nand substitutionT206 -> 0,\nX194 -> T218,\nT211 -> s(T218),\nT212 -> T219,\nT213 -> T220" }, { "from": 548, "to": 551, "label": "EVAL-BACKTRACK" }, { "from": 549, "to": 552, "label": "PARALLEL" }, { "from": 549, "to": 553, "label": "PARALLEL" }, { "from": 550, "to": 3, "label": "INSTANCE with matching:\nT1 -> 0\nT2 -> T219\nT3 -> T220" }, { "from": 552, "to": 554, "label": "EVAL with clause\nless(s(X207), s(X208)) :- less(X207, X208).\nand substitutionX207 -> T231,\nT206 -> s(T231),\nX208 -> T233,\nT211 -> s(T233),\nT232 -> T233,\nT212 -> T234,\nT213 -> T235" }, { "from": 552, "to": 555, "label": "EVAL-BACKTRACK" }, { "from": 553, "to": 558, "label": "FAILURE" }, { "from": 554, "to": 556, "label": "SPLIT 1" }, { "from": 554, "to": 557, "label": "SPLIT 2\nnew knowledge:\nT231 is ground\nreplacements:T234 -> T238,\nT235 -> T239,\nT2 -> T240,\nT3 -> T241" }, { "from": 556, "to": 473, "label": "INSTANCE with matching:\nT127 -> T231\nT132 -> T233" }, { "from": 557, "to": 3, "label": "INSTANCE with matching:\nT1 -> s(T231)\nT2 -> T238\nT3 -> T239" }, { "from": 558, "to": 559, "label": "EVAL with clause\ndelete(X227, tree(X228, X229, X230), tree(X228, X229, X231)) :- ','(less(X228, X227), delete(X227, X230, X231)).\nand substitutionT206 -> T258,\nX227 -> T258,\nX228 -> T263,\nX229 -> T260,\nX230 -> T264,\nT2 -> tree(T263, T260, T264),\nX231 -> T265,\nT3 -> tree(T263, T260, T265),\nT259 -> T263,\nT261 -> T264,\nT262 -> T265" }, { "from": 558, "to": 560, "label": "EVAL-BACKTRACK" }, { "from": 559, "to": 493, "label": "INSTANCE with matching:\nT174 -> T263\nT169 -> T258\nT175 -> T264\nT176 -> T265" }, { "from": 561, "to": 563, "label": "CASE" }, { "from": 563, "to": 564, "label": "PARALLEL" }, { "from": 563, "to": 565, "label": "PARALLEL" }, { "from": 564, "to": 566, "label": "EVAL with clause\nless(0, s(X248)).\nand substitutionT278 -> 0,\nX248 -> T285,\nT273 -> s(T285),\nT279 -> T286,\nT280 -> T287" }, { "from": 564, "to": 567, "label": "EVAL-BACKTRACK" }, { "from": 565, "to": 597, "label": "EVAL with clause\nless(s(X257), s(X258)) :- less(X257, X258).\nand substitutionX257 -> T298,\nT278 -> s(T298),\nX258 -> T297,\nT273 -> s(T297),\nT296 -> T298,\nT279 -> T299,\nT280 -> T300" }, { "from": 565, "to": 598, "label": "EVAL-BACKTRACK" }, { "from": 566, "to": 3, "label": "INSTANCE with matching:\nT1 -> s(T285)\nT2 -> T286\nT3 -> T287" }, { "from": 597, "to": 599, "label": "SPLIT 1" }, { "from": 597, "to": 600, "label": "SPLIT 2\nnew knowledge:\nT298 is ground\nT297 is ground\nreplacements:T299 -> T303,\nT300 -> T304,\nT2 -> T305,\nT3 -> T306" }, { "from": 599, "to": 535, "label": "INSTANCE with matching:\nT174 -> T298\nT169 -> T297" }, { "from": 600, "to": 3, "label": "INSTANCE with matching:\nT1 -> s(T297)\nT2 -> T303\nT3 -> T304" }, { "from": 601, "to": 603, "label": "CASE" }, { "from": 602, "to": 620, "label": "EVAL with clause\ndelete(X370, tree(X371, X372, X373), tree(X371, X374, X373)) :- ','(less(X370, X371), delete(X370, X372, X374)).\nand substitutionT6 -> T428,\nX370 -> T428,\nX371 -> T433,\nX372 -> T434,\nX373 -> T431,\nT2 -> tree(T433, T434, T431),\nX374 -> T435,\nT3 -> tree(T433, T435, T431),\nT429 -> T433,\nT430 -> T434,\nT432 -> T435" }, { "from": 602, "to": 621, "label": "EVAL-BACKTRACK" }, { "from": 603, "to": 604, "label": "PARALLEL" }, { "from": 603, "to": 605, "label": "PARALLEL" }, { "from": 604, "to": 606, "label": "EVAL with clause\ndelmin(tree(X285, void, X286), X285, X286).\nand substitutionX285 -> T334,\nX286 -> T335,\nT323 -> tree(T334, void, T335),\nT324 -> T334,\nT325 -> T335" }, { "from": 604, "to": 607, "label": "EVAL-BACKTRACK" }, { "from": 605, "to": 609, "label": "PARALLEL" }, { "from": 605, "to": 610, "label": "PARALLEL" }, { "from": 606, "to": 608, "label": "SUCCESS" }, { "from": 609, "to": 611, "label": "EVAL with clause\ndelmin(tree(X311, X312, X313), X314, tree(X311, X315, X316)) :- delmin(X312, X314, X315).\nand substitutionX311 -> T360,\nX312 -> T366,\nX313 -> T362,\nT323 -> tree(T360, T366, T362),\nT324 -> T367,\nX314 -> T367,\nX315 -> T368,\nX316 -> T365,\nT325 -> tree(T360, T368, T365),\nT361 -> T366,\nT363 -> T367,\nT364 -> T368" }, { "from": 609, "to": 612, "label": "EVAL-BACKTRACK" }, { "from": 610, "to": 613, "label": "FAILURE" }, { "from": 611, "to": 420, "label": "INSTANCE with matching:\nT65 -> T366\nT66 -> T367\nT67 -> T368" }, { "from": 613, "to": 614, "label": "PARALLEL" }, { "from": 613, "to": 615, "label": "PARALLEL" }, { "from": 614, "to": 616, "label": "EVAL with clause\ndelete(X341, tree(X342, X343, X344), tree(X342, X345, X344)) :- ','(less(X341, X342), delete(X341, X343, X345)).\nand substitutionT318 -> T393,\nX341 -> T393,\nX342 -> T398,\nX343 -> T399,\nX344 -> T396,\nT2 -> tree(T398, T399, T396),\nX345 -> T400,\nT3 -> tree(T398, T400, T396),\nT394 -> T398,\nT395 -> T399,\nT397 -> T400" }, { "from": 614, "to": 617, "label": "EVAL-BACKTRACK" }, { "from": 615, "to": 618, "label": "EVAL with clause\ndelete(X358, tree(X359, X360, X361), tree(X359, X360, X362)) :- ','(less(X359, X358), delete(X358, X361, X362)).\nand substitutionT318 -> T413,\nX358 -> T413,\nX359 -> T418,\nX360 -> T415,\nX361 -> T419,\nT2 -> tree(T418, T415, T419),\nX362 -> T420,\nT3 -> tree(T418, T415, T420),\nT414 -> T418,\nT416 -> T419,\nT417 -> T420" }, { "from": 615, "to": 619, "label": "EVAL-BACKTRACK" }, { "from": 616, "to": 469, "label": "INSTANCE with matching:\nT127 -> T393\nT132 -> T398\nT133 -> T399\nT134 -> T400" }, { "from": 618, "to": 493, "label": "INSTANCE with matching:\nT174 -> T418\nT169 -> T413\nT175 -> T419\nT176 -> T420" }, { "from": 620, "to": 622, "label": "CASE" }, { "from": 621, "to": 634, "label": "EVAL with clause\ndelete(X418, tree(X419, X420, X421), tree(X419, X420, X422)) :- ','(less(X419, X418), delete(X418, X421, X422)).\nand substitutionT6 -> T485,\nX418 -> T485,\nX419 -> T490,\nX420 -> T487,\nX421 -> T491,\nT2 -> tree(T490, T487, T491),\nX422 -> T492,\nT3 -> tree(T490, T487, T492),\nT486 -> T490,\nT488 -> T491,\nT489 -> T492" }, { "from": 621, "to": 635, "label": "EVAL-BACKTRACK" }, { "from": 622, "to": 623, "label": "PARALLEL" }, { "from": 622, "to": 624, "label": "PARALLEL" }, { "from": 623, "to": 625, "label": "EVAL with clause\nless(0, s(X379)).\nand substitutionT428 -> 0,\nX379 -> T440,\nT433 -> s(T440),\nT434 -> T441,\nT435 -> T442" }, { "from": 623, "to": 626, "label": "EVAL-BACKTRACK" }, { "from": 624, "to": 627, "label": "PARALLEL" }, { "from": 624, "to": 628, "label": "PARALLEL" }, { "from": 625, "to": 3, "label": "INSTANCE with matching:\nT1 -> 0\nT2 -> T441\nT3 -> T442" }, { "from": 627, "to": 629, "label": "EVAL with clause\nless(s(X392), s(X393)) :- less(X392, X393).\nand substitutionX392 -> T453,\nT428 -> s(T453),\nX393 -> T455,\nT433 -> s(T455),\nT454 -> T455,\nT434 -> T456,\nT435 -> T457" }, { "from": 627, "to": 630, "label": "EVAL-BACKTRACK" }, { "from": 628, "to": 631, "label": "FAILURE" }, { "from": 629, "to": 554, "label": "INSTANCE with matching:\nT231 -> T453\nT233 -> T455\nT234 -> T456\nT235 -> T457\nX17 -> X272\nX18 -> X273\nX19 -> X274\nX20 -> X275\nX21 -> X276" }, { "from": 631, "to": 632, "label": "EVAL with clause\ndelete(X406, tree(X407, X408, X409), tree(X407, X408, X410)) :- ','(less(X407, X406), delete(X406, X409, X410)).\nand substitutionT428 -> T470,\nX406 -> T470,\nX407 -> T475,\nX408 -> T472,\nX409 -> T476,\nT2 -> tree(T475, T472, T476),\nX410 -> T477,\nT3 -> tree(T475, T472, T477),\nT471 -> T475,\nT473 -> T476,\nT474 -> T477" }, { "from": 631, "to": 633, "label": "EVAL-BACKTRACK" }, { "from": 632, "to": 493, "label": "INSTANCE with matching:\nT174 -> T475\nT169 -> T470\nT175 -> T476\nT176 -> T477" }, { "from": 634, "to": 636, "label": "CASE" }, { "from": 636, "to": 637, "label": "PARALLEL" }, { "from": 636, "to": 638, "label": "PARALLEL" }, { "from": 637, "to": 639, "label": "EVAL with clause\nless(0, s(X427)).\nand substitutionT490 -> 0,\nX427 -> T497,\nT485 -> s(T497),\nT491 -> T498,\nT492 -> T499" }, { "from": 637, "to": 640, "label": "EVAL-BACKTRACK" }, { "from": 638, "to": 641, "label": "EVAL with clause\nless(s(X436), s(X437)) :- less(X436, X437).\nand substitutionX436 -> T510,\nT490 -> s(T510),\nX437 -> T509,\nT485 -> s(T509),\nT508 -> T510,\nT491 -> T511,\nT492 -> T512" }, { "from": 638, "to": 642, "label": "EVAL-BACKTRACK" }, { "from": 639, "to": 3, "label": "INSTANCE with matching:\nT1 -> s(T497)\nT2 -> T498\nT3 -> T499" }, { "from": 641, "to": 597, "label": "INSTANCE with matching:\nT298 -> T510\nT297 -> T509\nT299 -> T511\nT300 -> T512\nX17 -> X272\nX18 -> X273\nX19 -> X274\nX20 -> X275\nX21 -> X276\nX185 -> X370\nX186 -> X371\nX187 -> X372\nX188 -> X373\nX189 -> X374" }, { "from": 643, "to": 645, "label": "SUCCESS" }, { "from": 644, "to": 688, "label": "EVAL with clause\ndelete(X622, tree(X622, X623, X624), tree(X625, X623, X626)) :- delmin(X624, X625, X626).\nand substitutionT1 -> T726,\nX622 -> T726,\nX623 -> T727,\nX624 -> T731,\nT2 -> tree(T726, T727, T731),\nX625 -> T732,\nX626 -> T733,\nT3 -> tree(T732, T727, T733),\nT728 -> T731,\nT729 -> T732,\nT730 -> T733" }, { "from": 644, "to": 689, "label": "EVAL-BACKTRACK" }, { "from": 645, "to": 646, "label": "EVAL with clause\ndelete(X449, tree(X449, X450, X451), tree(X452, X450, X453)) :- delmin(X451, X452, X453).\nand substitutionT517 -> T524,\nX449 -> T524,\nX450 -> T525,\nX451 -> T529,\nT2 -> tree(T524, T525, T529),\nX452 -> T530,\nX453 -> T531,\nT3 -> tree(T530, T525, T531),\nT526 -> T529,\nT527 -> T530,\nT528 -> T531" }, { "from": 645, "to": 647, "label": "EVAL-BACKTRACK" }, { "from": 646, "to": 648, "label": "CASE" }, { "from": 647, "to": 665, "label": "EVAL with clause\ndelete(X547, tree(X548, X549, X550), tree(X548, X551, X550)) :- ','(less(X547, X548), delete(X547, X549, X551)).\nand substitutionT517 -> T634,\nX547 -> T634,\nX548 -> T639,\nX549 -> T640,\nX550 -> T637,\nT2 -> tree(T639, T640, T637),\nX551 -> T641,\nT3 -> tree(T639, T641, T637),\nT635 -> T639,\nT636 -> T640,\nT638 -> T641" }, { "from": 647, "to": 666, "label": "EVAL-BACKTRACK" }, { "from": 648, "to": 649, "label": "PARALLEL" }, { "from": 648, "to": 650, "label": "PARALLEL" }, { "from": 649, "to": 651, "label": "EVAL with clause\ndelmin(tree(X462, void, X463), X462, X463).\nand substitutionX462 -> T540,\nX463 -> T541,\nT529 -> tree(T540, void, T541),\nT530 -> T540,\nT531 -> T541" }, { "from": 649, "to": 652, "label": "EVAL-BACKTRACK" }, { "from": 650, "to": 654, "label": "PARALLEL" }, { "from": 650, "to": 655, "label": "PARALLEL" }, { "from": 651, "to": 653, "label": "SUCCESS" }, { "from": 654, "to": 656, "label": "EVAL with clause\ndelmin(tree(X488, X489, X490), X491, tree(X488, X492, X493)) :- delmin(X489, X491, X492).\nand substitutionX488 -> T566,\nX489 -> T572,\nX490 -> T568,\nT529 -> tree(T566, T572, T568),\nT530 -> T573,\nX491 -> T573,\nX492 -> T574,\nX493 -> T571,\nT531 -> tree(T566, T574, T571),\nT567 -> T572,\nT569 -> T573,\nT570 -> T574" }, { "from": 654, "to": 657, "label": "EVAL-BACKTRACK" }, { "from": 655, "to": 658, "label": "FAILURE" }, { "from": 656, "to": 420, "label": "INSTANCE with matching:\nT65 -> T572\nT66 -> T573\nT67 -> T574" }, { "from": 658, "to": 659, "label": "PARALLEL" }, { "from": 658, "to": 660, "label": "PARALLEL" }, { "from": 659, "to": 661, "label": "EVAL with clause\ndelete(X518, tree(X519, X520, X521), tree(X519, X522, X521)) :- ','(less(X518, X519), delete(X518, X520, X522)).\nand substitutionT524 -> T599,\nX518 -> T599,\nX519 -> T604,\nX520 -> T605,\nX521 -> T602,\nT2 -> tree(T604, T605, T602),\nX522 -> T606,\nT3 -> tree(T604, T606, T602),\nT600 -> T604,\nT601 -> T605,\nT603 -> T606" }, { "from": 659, "to": 662, "label": "EVAL-BACKTRACK" }, { "from": 660, "to": 663, "label": "EVAL with clause\ndelete(X535, tree(X536, X537, X538), tree(X536, X537, X539)) :- ','(less(X536, X535), delete(X535, X538, X539)).\nand substitutionT524 -> T619,\nX535 -> T619,\nX536 -> T624,\nX537 -> T621,\nX538 -> T625,\nT2 -> tree(T624, T621, T625),\nX539 -> T626,\nT3 -> tree(T624, T621, T626),\nT620 -> T624,\nT622 -> T625,\nT623 -> T626" }, { "from": 660, "to": 664, "label": "EVAL-BACKTRACK" }, { "from": 661, "to": 469, "label": "INSTANCE with matching:\nT127 -> T599\nT132 -> T604\nT133 -> T605\nT134 -> T606" }, { "from": 663, "to": 493, "label": "INSTANCE with matching:\nT174 -> T624\nT169 -> T619\nT175 -> T625\nT176 -> T626" }, { "from": 665, "to": 667, "label": "CASE" }, { "from": 666, "to": 679, "label": "EVAL with clause\ndelete(X595, tree(X596, X597, X598), tree(X596, X597, X599)) :- ','(less(X596, X595), delete(X595, X598, X599)).\nand substitutionT517 -> T691,\nX595 -> T691,\nX596 -> T696,\nX597 -> T693,\nX598 -> T697,\nT2 -> tree(T696, T693, T697),\nX599 -> T698,\nT3 -> tree(T696, T693, T698),\nT692 -> T696,\nT694 -> T697,\nT695 -> T698" }, { "from": 666, "to": 680, "label": "EVAL-BACKTRACK" }, { "from": 667, "to": 668, "label": "PARALLEL" }, { "from": 667, "to": 669, "label": "PARALLEL" }, { "from": 668, "to": 670, "label": "EVAL with clause\nless(0, s(X556)).\nand substitutionT634 -> 0,\nX556 -> T646,\nT639 -> s(T646),\nT640 -> T647,\nT641 -> T648" }, { "from": 668, "to": 671, "label": "EVAL-BACKTRACK" }, { "from": 669, "to": 672, "label": "PARALLEL" }, { "from": 669, "to": 673, "label": "PARALLEL" }, { "from": 670, "to": 3, "label": "INSTANCE with matching:\nT1 -> 0\nT2 -> T647\nT3 -> T648" }, { "from": 672, "to": 674, "label": "EVAL with clause\nless(s(X569), s(X570)) :- less(X569, X570).\nand substitutionX569 -> T659,\nT634 -> s(T659),\nX570 -> T661,\nT639 -> s(T661),\nT660 -> T661,\nT640 -> T662,\nT641 -> T663" }, { "from": 672, "to": 675, "label": "EVAL-BACKTRACK" }, { "from": 673, "to": 676, "label": "FAILURE" }, { "from": 674, "to": 554, "label": "INSTANCE with matching:\nT231 -> T659\nT233 -> T661\nT234 -> T662\nT235 -> T663\nX17 -> X449\nX18 -> X450\nX19 -> X451\nX20 -> X452\nX21 -> X453" }, { "from": 676, "to": 677, "label": "EVAL with clause\ndelete(X583, tree(X584, X585, X586), tree(X584, X585, X587)) :- ','(less(X584, X583), delete(X583, X586, X587)).\nand substitutionT634 -> T676,\nX583 -> T676,\nX584 -> T681,\nX585 -> T678,\nX586 -> T682,\nT2 -> tree(T681, T678, T682),\nX587 -> T683,\nT3 -> tree(T681, T678, T683),\nT677 -> T681,\nT679 -> T682,\nT680 -> T683" }, { "from": 676, "to": 678, "label": "EVAL-BACKTRACK" }, { "from": 677, "to": 493, "label": "INSTANCE with matching:\nT174 -> T681\nT169 -> T676\nT175 -> T682\nT176 -> T683" }, { "from": 679, "to": 681, "label": "CASE" }, { "from": 681, "to": 682, "label": "PARALLEL" }, { "from": 681, "to": 683, "label": "PARALLEL" }, { "from": 682, "to": 684, "label": "EVAL with clause\nless(0, s(X604)).\nand substitutionT696 -> 0,\nX604 -> T703,\nT691 -> s(T703),\nT697 -> T704,\nT698 -> T705" }, { "from": 682, "to": 685, "label": "EVAL-BACKTRACK" }, { "from": 683, "to": 686, "label": "EVAL with clause\nless(s(X613), s(X614)) :- less(X613, X614).\nand substitutionX613 -> T716,\nT696 -> s(T716),\nX614 -> T715,\nT691 -> s(T715),\nT714 -> T716,\nT697 -> T717,\nT698 -> T718" }, { "from": 683, "to": 687, "label": "EVAL-BACKTRACK" }, { "from": 684, "to": 3, "label": "INSTANCE with matching:\nT1 -> s(T703)\nT2 -> T704\nT3 -> T705" }, { "from": 686, "to": 597, "label": "INSTANCE with matching:\nT298 -> T716\nT297 -> T715\nT299 -> T717\nT300 -> T718\nX17 -> X449\nX18 -> X450\nX19 -> X451\nX20 -> X452\nX21 -> X453\nX185 -> X547\nX186 -> X548\nX187 -> X549\nX188 -> X550\nX189 -> X551" }, { "from": 688, "to": 690, "label": "CASE" }, { "from": 689, "to": 710, "label": "EVAL with clause\ndelete(X720, tree(X721, X722, X723), tree(X721, X724, X723)) :- ','(less(X720, X721), delete(X720, X722, X724)).\nand substitutionT1 -> T836,\nX720 -> T836,\nX721 -> T841,\nX722 -> T842,\nX723 -> T839,\nT2 -> tree(T841, T842, T839),\nX724 -> T843,\nT3 -> tree(T841, T843, T839),\nT837 -> T841,\nT838 -> T842,\nT840 -> T843" }, { "from": 689, "to": 711, "label": "EVAL-BACKTRACK" }, { "from": 690, "to": 691, "label": "PARALLEL" }, { "from": 690, "to": 692, "label": "PARALLEL" }, { "from": 691, "to": 693, "label": "EVAL with clause\ndelmin(tree(X635, void, X636), X635, X636).\nand substitutionX635 -> T742,\nX636 -> T743,\nT731 -> tree(T742, void, T743),\nT732 -> T742,\nT733 -> T743" }, { "from": 691, "to": 694, "label": "EVAL-BACKTRACK" }, { "from": 692, "to": 696, "label": "PARALLEL" }, { "from": 692, "to": 697, "label": "PARALLEL" }, { "from": 693, "to": 695, "label": "SUCCESS" }, { "from": 696, "to": 698, "label": "EVAL with clause\ndelmin(tree(X661, X662, X663), X664, tree(X661, X665, X666)) :- delmin(X662, X664, X665).\nand substitutionX661 -> T768,\nX662 -> T774,\nX663 -> T770,\nT731 -> tree(T768, T774, T770),\nT732 -> T775,\nX664 -> T775,\nX665 -> T776,\nX666 -> T773,\nT733 -> tree(T768, T776, T773),\nT769 -> T774,\nT771 -> T775,\nT772 -> T776" }, { "from": 696, "to": 699, "label": "EVAL-BACKTRACK" }, { "from": 697, "to": 700, "label": "FAILURE" }, { "from": 698, "to": 420, "label": "INSTANCE with matching:\nT65 -> T774\nT66 -> T775\nT67 -> T776" }, { "from": 700, "to": 701, "label": "PARALLEL" }, { "from": 700, "to": 702, "label": "PARALLEL" }, { "from": 701, "to": 703, "label": "EVAL with clause\ndelete(X691, tree(X692, X693, X694), tree(X692, X695, X694)) :- ','(less(X691, X692), delete(X691, X693, X695)).\nand substitutionT726 -> T801,\nX691 -> T801,\nX692 -> T806,\nX693 -> T807,\nX694 -> T804,\nT2 -> tree(T806, T807, T804),\nX695 -> T808,\nT3 -> tree(T806, T808, T804),\nT802 -> T806,\nT803 -> T807,\nT805 -> T808" }, { "from": 701, "to": 704, "label": "EVAL-BACKTRACK" }, { "from": 702, "to": 708, "label": "EVAL with clause\ndelete(X708, tree(X709, X710, X711), tree(X709, X710, X712)) :- ','(less(X709, X708), delete(X708, X711, X712)).\nand substitutionT726 -> T821,\nX708 -> T821,\nX709 -> T826,\nX710 -> T823,\nX711 -> T827,\nT2 -> tree(T826, T823, T827),\nX712 -> T828,\nT3 -> tree(T826, T823, T828),\nT822 -> T826,\nT824 -> T827,\nT825 -> T828" }, { "from": 702, "to": 709, "label": "EVAL-BACKTRACK" }, { "from": 703, "to": 469, "label": "INSTANCE with matching:\nT127 -> T801\nT132 -> T806\nT133 -> T807\nT134 -> T808" }, { "from": 708, "to": 493, "label": "INSTANCE with matching:\nT174 -> T826\nT169 -> T821\nT175 -> T827\nT176 -> T828" }, { "from": 710, "to": 712, "label": "CASE" }, { "from": 711, "to": 725, "label": "EVAL with clause\ndelete(X768, tree(X769, X770, X771), tree(X769, X770, X772)) :- ','(less(X769, X768), delete(X768, X771, X772)).\nand substitutionT1 -> T893,\nX768 -> T893,\nX769 -> T898,\nX770 -> T895,\nX771 -> T899,\nT2 -> tree(T898, T895, T899),\nX772 -> T900,\nT3 -> tree(T898, T895, T900),\nT894 -> T898,\nT896 -> T899,\nT897 -> T900" }, { "from": 711, "to": 726, "label": "EVAL-BACKTRACK" }, { "from": 712, "to": 713, "label": "PARALLEL" }, { "from": 712, "to": 714, "label": "PARALLEL" }, { "from": 713, "to": 715, "label": "EVAL with clause\nless(0, s(X729)).\nand substitutionT836 -> 0,\nX729 -> T848,\nT841 -> s(T848),\nT842 -> T849,\nT843 -> T850" }, { "from": 713, "to": 716, "label": "EVAL-BACKTRACK" }, { "from": 714, "to": 718, "label": "PARALLEL" }, { "from": 714, "to": 719, "label": "PARALLEL" }, { "from": 715, "to": 3, "label": "INSTANCE with matching:\nT1 -> 0\nT2 -> T849\nT3 -> T850" }, { "from": 718, "to": 720, "label": "EVAL with clause\nless(s(X742), s(X743)) :- less(X742, X743).\nand substitutionX742 -> T861,\nT836 -> s(T861),\nX743 -> T863,\nT841 -> s(T863),\nT862 -> T863,\nT842 -> T864,\nT843 -> T865" }, { "from": 718, "to": 721, "label": "EVAL-BACKTRACK" }, { "from": 719, "to": 722, "label": "FAILURE" }, { "from": 720, "to": 554, "label": "INSTANCE with matching:\nT231 -> T861\nT233 -> T863\nT234 -> T864\nT235 -> T865\nX17 -> X622\nX18 -> X623\nX19 -> X624\nX20 -> X625\nX21 -> X626" }, { "from": 722, "to": 723, "label": "EVAL with clause\ndelete(X756, tree(X757, X758, X759), tree(X757, X758, X760)) :- ','(less(X757, X756), delete(X756, X759, X760)).\nand substitutionT836 -> T878,\nX756 -> T878,\nX757 -> T883,\nX758 -> T880,\nX759 -> T884,\nT2 -> tree(T883, T880, T884),\nX760 -> T885,\nT3 -> tree(T883, T880, T885),\nT879 -> T883,\nT881 -> T884,\nT882 -> T885" }, { "from": 722, "to": 724, "label": "EVAL-BACKTRACK" }, { "from": 723, "to": 493, "label": "INSTANCE with matching:\nT174 -> T883\nT169 -> T878\nT175 -> T884\nT176 -> T885" }, { "from": 725, "to": 727, "label": "CASE" }, { "from": 727, "to": 728, "label": "PARALLEL" }, { "from": 727, "to": 729, "label": "PARALLEL" }, { "from": 728, "to": 733, "label": "EVAL with clause\nless(0, s(X777)).\nand substitutionT898 -> 0,\nX777 -> T905,\nT893 -> s(T905),\nT899 -> T906,\nT900 -> T907" }, { "from": 728, "to": 734, "label": "EVAL-BACKTRACK" }, { "from": 729, "to": 767, "label": "EVAL with clause\nless(s(X786), s(X787)) :- less(X786, X787).\nand substitutionX786 -> T918,\nT898 -> s(T918),\nX787 -> T917,\nT893 -> s(T917),\nT916 -> T918,\nT899 -> T919,\nT900 -> T920" }, { "from": 729, "to": 768, "label": "EVAL-BACKTRACK" }, { "from": 733, "to": 3, "label": "INSTANCE with matching:\nT1 -> s(T905)\nT2 -> T906\nT3 -> T907" }, { "from": 767, "to": 597, "label": "INSTANCE with matching:\nT298 -> T918\nT297 -> T917\nT299 -> T919\nT300 -> T920\nX17 -> X622\nX18 -> X623\nX19 -> X624\nX20 -> X625\nX21 -> X626\nX185 -> X720\nX186 -> X721\nX187 -> X722\nX188 -> X723\nX189 -> X724" } ], "type": "Graph" } } ---------------------------------------- (162) Obligation: Triples: delminA(tree(X1, X2, X3), X4, tree(X1, X5, X6)) :- delminA(X2, X4, X5). lessG(s(X1), s(X2)) :- lessG(X1, X2). lessH(s(X1), s(X2)) :- lessH(X1, X2). pD(X1, X2, X3, X4) :- lessH(X1, X2). pD(X1, X2, X3, X4) :- ','(lesscH(X1, X2), deleteB(X2, X3, X4)). pC(X1, X2, X3, X4) :- lessG(X1, X2). pC(X1, X2, X3, X4) :- ','(lesscG(X1, X2), deleteB(X1, X3, X4)). pE(X1, X2, X3, X4) :- lessG(X1, X2). pE(X1, X2, X3, X4) :- ','(lesscG(X1, X2), deleteB(s(X1), X3, X4)). pF(X1, X2, X3, X4) :- lessH(X1, X2). pF(X1, X2, X3, X4) :- ','(lesscH(X1, X2), deleteB(s(X2), X3, X4)). deleteB(X1, tree(X1, X2, tree(X3, X4, X5)), tree(X6, X2, tree(X3, X7, X8))) :- delminA(X4, X6, X7). deleteB(X1, tree(X2, X3, X4), tree(X2, X5, X4)) :- pC(X1, X2, X3, X5). deleteB(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- pD(X2, X1, X4, X5). deleteB(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) :- deleteB(0, X2, X4). deleteB(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) :- pE(X1, X2, X3, X5). deleteB(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- pD(X2, X1, X4, X5). deleteB(s(X1), tree(0, X2, X3), tree(0, X2, X4)) :- deleteB(s(X1), X3, X4). deleteB(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) :- pF(X2, X1, X4, X5). deleteB(X1, tree(X1, X2, tree(X3, X4, X5)), tree(X6, X2, tree(X3, X7, X8))) :- delminA(X4, X6, X7). deleteB(X1, tree(X2, X3, X4), tree(X2, X5, X4)) :- pC(X1, X2, X3, X5). deleteB(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- pD(X2, X1, X4, X5). deleteB(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) :- deleteB(0, X2, X4). deleteB(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) :- pE(X1, X2, X3, X5). deleteB(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- pD(X2, X1, X4, X5). deleteB(s(X1), tree(0, X2, X3), tree(0, X2, X4)) :- deleteB(s(X1), X3, X4). deleteB(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) :- pF(X2, X1, X4, X5). deleteB(X1, tree(X1, X2, tree(X3, X4, X5)), tree(X6, X2, tree(X3, X7, X8))) :- delminA(X4, X6, X7). deleteB(X1, tree(X2, X3, X4), tree(X2, X5, X4)) :- pC(X1, X2, X3, X5). deleteB(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- pD(X2, X1, X4, X5). deleteB(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) :- deleteB(0, X2, X4). deleteB(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) :- pE(X1, X2, X3, X5). deleteB(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- pD(X2, X1, X4, X5). deleteB(s(X1), tree(0, X2, X3), tree(0, X2, X4)) :- deleteB(s(X1), X3, X4). deleteB(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) :- pF(X2, X1, X4, X5). deleteB(X1, tree(X1, X2, tree(X3, X4, X5)), tree(X6, X2, tree(X3, X7, X8))) :- delminA(X4, X6, X7). deleteB(X1, tree(X2, X3, X4), tree(X2, X5, X4)) :- pC(X1, X2, X3, X5). deleteB(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- pD(X2, X1, X4, X5). deleteB(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) :- deleteB(0, X2, X4). deleteB(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) :- pE(X1, X2, X3, X5). deleteB(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- pD(X2, X1, X4, X5). deleteB(s(X1), tree(0, X2, X3), tree(0, X2, X4)) :- deleteB(s(X1), X3, X4). deleteB(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) :- pF(X2, X1, X4, X5). Clauses: delmincA(tree(X1, void, X2), X1, X2). delmincA(tree(X1, X2, X3), X4, tree(X1, X5, X6)) :- delmincA(X2, X4, X5). deletecB(X1, tree(X1, void, X2), X2). deletecB(X1, tree(X1, X2, void), X2). deletecB(X1, tree(X1, X2, tree(X3, void, X4)), tree(X3, X2, X4)). deletecB(X1, tree(X1, X2, tree(X3, X4, X5)), tree(X6, X2, tree(X3, X7, X8))) :- delmincA(X4, X6, X7). deletecB(X1, tree(X2, X3, X4), tree(X2, X5, X4)) :- qcC(X1, X2, X3, X5). deletecB(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- qcD(X2, X1, X4, X5). deletecB(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) :- deletecB(0, X2, X4). deletecB(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) :- qcE(X1, X2, X3, X5). deletecB(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- qcD(X2, X1, X4, X5). deletecB(s(X1), tree(0, X2, X3), tree(0, X2, X4)) :- deletecB(s(X1), X3, X4). deletecB(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) :- qcF(X2, X1, X4, X5). deletecB(X1, tree(X1, X2, tree(X3, void, X4)), tree(X3, X2, X4)). deletecB(X1, tree(X1, X2, tree(X3, X4, X5)), tree(X6, X2, tree(X3, X7, X8))) :- delmincA(X4, X6, X7). deletecB(X1, tree(X2, X3, X4), tree(X2, X5, X4)) :- qcC(X1, X2, X3, X5). deletecB(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- qcD(X2, X1, X4, X5). deletecB(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) :- deletecB(0, X2, X4). deletecB(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) :- qcE(X1, X2, X3, X5). deletecB(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- qcD(X2, X1, X4, X5). deletecB(s(X1), tree(0, X2, X3), tree(0, X2, X4)) :- deletecB(s(X1), X3, X4). deletecB(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) :- qcF(X2, X1, X4, X5). deletecB(X1, tree(X1, X2, void), X2). deletecB(X1, tree(X1, X2, tree(X3, void, X4)), tree(X3, X2, X4)). deletecB(X1, tree(X1, X2, tree(X3, X4, X5)), tree(X6, X2, tree(X3, X7, X8))) :- delmincA(X4, X6, X7). deletecB(X1, tree(X2, X3, X4), tree(X2, X5, X4)) :- qcC(X1, X2, X3, X5). deletecB(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- qcD(X2, X1, X4, X5). deletecB(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) :- deletecB(0, X2, X4). deletecB(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) :- qcE(X1, X2, X3, X5). deletecB(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- qcD(X2, X1, X4, X5). deletecB(s(X1), tree(0, X2, X3), tree(0, X2, X4)) :- deletecB(s(X1), X3, X4). deletecB(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) :- qcF(X2, X1, X4, X5). deletecB(X1, tree(X1, X2, tree(X3, void, X4)), tree(X3, X2, X4)). deletecB(X1, tree(X1, X2, tree(X3, X4, X5)), tree(X6, X2, tree(X3, X7, X8))) :- delmincA(X4, X6, X7). deletecB(X1, tree(X2, X3, X4), tree(X2, X5, X4)) :- qcC(X1, X2, X3, X5). deletecB(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- qcD(X2, X1, X4, X5). deletecB(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) :- deletecB(0, X2, X4). deletecB(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) :- qcE(X1, X2, X3, X5). deletecB(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- qcD(X2, X1, X4, X5). deletecB(s(X1), tree(0, X2, X3), tree(0, X2, X4)) :- deletecB(s(X1), X3, X4). deletecB(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) :- qcF(X2, X1, X4, X5). lesscG(0, s(X1)). lesscG(s(X1), s(X2)) :- lesscG(X1, X2). lesscH(0, s(X1)). lesscH(s(X1), s(X2)) :- lesscH(X1, X2). qcD(X1, X2, X3, X4) :- ','(lesscH(X1, X2), deletecB(X2, X3, X4)). qcC(X1, X2, X3, X4) :- ','(lesscG(X1, X2), deletecB(X1, X3, X4)). qcE(X1, X2, X3, X4) :- ','(lesscG(X1, X2), deletecB(s(X1), X3, X4)). qcF(X1, X2, X3, X4) :- ','(lesscH(X1, X2), deletecB(s(X2), X3, X4)). Afs: deleteB(x1, x2, x3) = deleteB(x1) ---------------------------------------- (163) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: deleteB_in_3: (b,f,f) delminA_in_3: (f,f,f) pC_in_4: (b,f,f,f) lessG_in_2: (b,f) lesscG_in_2: (b,f) pD_in_4: (f,b,f,f) lessH_in_2: (f,b) lesscH_in_2: (f,b) pE_in_4: (b,f,f,f) pF_in_4: (f,b,f,f) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: DELETEB_IN_GAA(X1, tree(X1, X2, tree(X3, X4, X5)), tree(X6, X2, tree(X3, X7, X8))) -> U16_GAA(X1, X2, X3, X4, X5, X6, X7, X8, delminA_in_aaa(X4, X6, X7)) DELETEB_IN_GAA(X1, tree(X1, X2, tree(X3, X4, X5)), tree(X6, X2, tree(X3, X7, X8))) -> DELMINA_IN_AAA(X4, X6, X7) DELMINA_IN_AAA(tree(X1, X2, X3), X4, tree(X1, X5, X6)) -> U1_AAA(X1, X2, X3, X4, X5, X6, delminA_in_aaa(X2, X4, X5)) DELMINA_IN_AAA(tree(X1, X2, X3), X4, tree(X1, X5, X6)) -> DELMINA_IN_AAA(X2, X4, X5) DELETEB_IN_GAA(X1, tree(X2, X3, X4), tree(X2, X5, X4)) -> U17_GAA(X1, X2, X3, X4, X5, pC_in_gaaa(X1, X2, X3, X5)) DELETEB_IN_GAA(X1, tree(X2, X3, X4), tree(X2, X5, X4)) -> PC_IN_GAAA(X1, X2, X3, X5) PC_IN_GAAA(X1, X2, X3, X4) -> U7_GAAA(X1, X2, X3, X4, lessG_in_ga(X1, X2)) PC_IN_GAAA(X1, X2, X3, X4) -> LESSG_IN_GA(X1, X2) LESSG_IN_GA(s(X1), s(X2)) -> U2_GA(X1, X2, lessG_in_ga(X1, X2)) LESSG_IN_GA(s(X1), s(X2)) -> LESSG_IN_GA(X1, X2) PC_IN_GAAA(X1, X2, X3, X4) -> U8_GAAA(X1, X2, X3, X4, lesscG_in_ga(X1, X2)) U8_GAAA(X1, X2, X3, X4, lesscG_out_ga(X1, X2)) -> U9_GAAA(X1, X2, X3, X4, deleteB_in_gaa(X1, X3, X4)) U8_GAAA(X1, X2, X3, X4, lesscG_out_ga(X1, X2)) -> DELETEB_IN_GAA(X1, X3, X4) DELETEB_IN_GAA(X1, tree(X2, X3, X4), tree(X2, X3, X5)) -> U18_GAA(X1, X2, X3, X4, X5, pD_in_agaa(X2, X1, X4, X5)) DELETEB_IN_GAA(X1, tree(X2, X3, X4), tree(X2, X3, X5)) -> PD_IN_AGAA(X2, X1, X4, X5) PD_IN_AGAA(X1, X2, X3, X4) -> U4_AGAA(X1, X2, X3, X4, lessH_in_ag(X1, X2)) PD_IN_AGAA(X1, X2, X3, X4) -> LESSH_IN_AG(X1, X2) LESSH_IN_AG(s(X1), s(X2)) -> U3_AG(X1, X2, lessH_in_ag(X1, X2)) LESSH_IN_AG(s(X1), s(X2)) -> LESSH_IN_AG(X1, X2) PD_IN_AGAA(X1, X2, X3, X4) -> U5_AGAA(X1, X2, X3, X4, lesscH_in_ag(X1, X2)) U5_AGAA(X1, X2, X3, X4, lesscH_out_ag(X1, X2)) -> U6_AGAA(X1, X2, X3, X4, deleteB_in_gaa(X2, X3, X4)) U5_AGAA(X1, X2, X3, X4, lesscH_out_ag(X1, X2)) -> DELETEB_IN_GAA(X2, X3, X4) DELETEB_IN_GAA(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) -> U19_GAA(X1, X2, X3, X4, deleteB_in_gaa(0, X2, X4)) DELETEB_IN_GAA(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) -> DELETEB_IN_GAA(0, X2, X4) DELETEB_IN_GAA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) -> U20_GAA(X1, X2, X3, X4, X5, pE_in_gaaa(X1, X2, X3, X5)) DELETEB_IN_GAA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) -> PE_IN_GAAA(X1, X2, X3, X5) PE_IN_GAAA(X1, X2, X3, X4) -> U10_GAAA(X1, X2, X3, X4, lessG_in_ga(X1, X2)) PE_IN_GAAA(X1, X2, X3, X4) -> LESSG_IN_GA(X1, X2) PE_IN_GAAA(X1, X2, X3, X4) -> U11_GAAA(X1, X2, X3, X4, lesscG_in_ga(X1, X2)) U11_GAAA(X1, X2, X3, X4, lesscG_out_ga(X1, X2)) -> U12_GAAA(X1, X2, X3, X4, deleteB_in_gaa(s(X1), X3, X4)) U11_GAAA(X1, X2, X3, X4, lesscG_out_ga(X1, X2)) -> DELETEB_IN_GAA(s(X1), X3, X4) DELETEB_IN_GAA(s(X1), tree(0, X2, X3), tree(0, X2, X4)) -> U21_GAA(X1, X2, X3, X4, deleteB_in_gaa(s(X1), X3, X4)) DELETEB_IN_GAA(s(X1), tree(0, X2, X3), tree(0, X2, X4)) -> DELETEB_IN_GAA(s(X1), X3, X4) DELETEB_IN_GAA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) -> U22_GAA(X1, X2, X3, X4, X5, pF_in_agaa(X2, X1, X4, X5)) DELETEB_IN_GAA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) -> PF_IN_AGAA(X2, X1, X4, X5) PF_IN_AGAA(X1, X2, X3, X4) -> U13_AGAA(X1, X2, X3, X4, lessH_in_ag(X1, X2)) PF_IN_AGAA(X1, X2, X3, X4) -> LESSH_IN_AG(X1, X2) PF_IN_AGAA(X1, X2, X3, X4) -> U14_AGAA(X1, X2, X3, X4, lesscH_in_ag(X1, X2)) U14_AGAA(X1, X2, X3, X4, lesscH_out_ag(X1, X2)) -> U15_AGAA(X1, X2, X3, X4, deleteB_in_gaa(s(X2), X3, X4)) U14_AGAA(X1, X2, X3, X4, lesscH_out_ag(X1, X2)) -> DELETEB_IN_GAA(s(X2), X3, X4) The TRS R consists of the following rules: lesscG_in_ga(0, s(X1)) -> lesscG_out_ga(0, s(X1)) lesscG_in_ga(s(X1), s(X2)) -> U32_ga(X1, X2, lesscG_in_ga(X1, X2)) U32_ga(X1, X2, lesscG_out_ga(X1, X2)) -> lesscG_out_ga(s(X1), s(X2)) lesscH_in_ag(0, s(X1)) -> lesscH_out_ag(0, s(X1)) lesscH_in_ag(s(X1), s(X2)) -> U33_ag(X1, X2, lesscH_in_ag(X1, X2)) U33_ag(X1, X2, lesscH_out_ag(X1, X2)) -> lesscH_out_ag(s(X1), s(X2)) The argument filtering Pi contains the following mapping: deleteB_in_gaa(x1, x2, x3) = deleteB_in_gaa(x1) delminA_in_aaa(x1, x2, x3) = delminA_in_aaa pC_in_gaaa(x1, x2, x3, x4) = pC_in_gaaa(x1) lessG_in_ga(x1, x2) = lessG_in_ga(x1) s(x1) = s(x1) lesscG_in_ga(x1, x2) = lesscG_in_ga(x1) 0 = 0 lesscG_out_ga(x1, x2) = lesscG_out_ga(x1) U32_ga(x1, x2, x3) = U32_ga(x1, x3) pD_in_agaa(x1, x2, x3, x4) = pD_in_agaa(x2) lessH_in_ag(x1, x2) = lessH_in_ag(x2) lesscH_in_ag(x1, x2) = lesscH_in_ag(x2) lesscH_out_ag(x1, x2) = lesscH_out_ag(x1, x2) U33_ag(x1, x2, x3) = U33_ag(x2, x3) pE_in_gaaa(x1, x2, x3, x4) = pE_in_gaaa(x1) pF_in_agaa(x1, x2, x3, x4) = pF_in_agaa(x2) DELETEB_IN_GAA(x1, x2, x3) = DELETEB_IN_GAA(x1) U16_GAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U16_GAA(x1, x9) DELMINA_IN_AAA(x1, x2, x3) = DELMINA_IN_AAA U1_AAA(x1, x2, x3, x4, x5, x6, x7) = U1_AAA(x7) U17_GAA(x1, x2, x3, x4, x5, x6) = U17_GAA(x1, x6) PC_IN_GAAA(x1, x2, x3, x4) = PC_IN_GAAA(x1) U7_GAAA(x1, x2, x3, x4, x5) = U7_GAAA(x1, x5) LESSG_IN_GA(x1, x2) = LESSG_IN_GA(x1) U2_GA(x1, x2, x3) = U2_GA(x1, x3) U8_GAAA(x1, x2, x3, x4, x5) = U8_GAAA(x1, x5) U9_GAAA(x1, x2, x3, x4, x5) = U9_GAAA(x1, x5) U18_GAA(x1, x2, x3, x4, x5, x6) = U18_GAA(x1, x6) PD_IN_AGAA(x1, x2, x3, x4) = PD_IN_AGAA(x2) U4_AGAA(x1, x2, x3, x4, x5) = U4_AGAA(x2, x5) LESSH_IN_AG(x1, x2) = LESSH_IN_AG(x2) U3_AG(x1, x2, x3) = U3_AG(x2, x3) U5_AGAA(x1, x2, x3, x4, x5) = U5_AGAA(x2, x5) U6_AGAA(x1, x2, x3, x4, x5) = U6_AGAA(x1, x2, x5) U19_GAA(x1, x2, x3, x4, x5) = U19_GAA(x5) U20_GAA(x1, x2, x3, x4, x5, x6) = U20_GAA(x1, x6) PE_IN_GAAA(x1, x2, x3, x4) = PE_IN_GAAA(x1) U10_GAAA(x1, x2, x3, x4, x5) = U10_GAAA(x1, x5) U11_GAAA(x1, x2, x3, x4, x5) = U11_GAAA(x1, x5) U12_GAAA(x1, x2, x3, x4, x5) = U12_GAAA(x1, x5) U21_GAA(x1, x2, x3, x4, x5) = U21_GAA(x1, x5) U22_GAA(x1, x2, x3, x4, x5, x6) = U22_GAA(x1, x6) PF_IN_AGAA(x1, x2, x3, x4) = PF_IN_AGAA(x2) U13_AGAA(x1, x2, x3, x4, x5) = U13_AGAA(x2, x5) U14_AGAA(x1, x2, x3, x4, x5) = U14_AGAA(x2, x5) U15_AGAA(x1, x2, x3, x4, x5) = U15_AGAA(x1, x2, x5) We have to consider all (P,R,Pi)-chains Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES ---------------------------------------- (164) Obligation: Pi DP problem: The TRS P consists of the following rules: DELETEB_IN_GAA(X1, tree(X1, X2, tree(X3, X4, X5)), tree(X6, X2, tree(X3, X7, X8))) -> U16_GAA(X1, X2, X3, X4, X5, X6, X7, X8, delminA_in_aaa(X4, X6, X7)) DELETEB_IN_GAA(X1, tree(X1, X2, tree(X3, X4, X5)), tree(X6, X2, tree(X3, X7, X8))) -> DELMINA_IN_AAA(X4, X6, X7) DELMINA_IN_AAA(tree(X1, X2, X3), X4, tree(X1, X5, X6)) -> U1_AAA(X1, X2, X3, X4, X5, X6, delminA_in_aaa(X2, X4, X5)) DELMINA_IN_AAA(tree(X1, X2, X3), X4, tree(X1, X5, X6)) -> DELMINA_IN_AAA(X2, X4, X5) DELETEB_IN_GAA(X1, tree(X2, X3, X4), tree(X2, X5, X4)) -> U17_GAA(X1, X2, X3, X4, X5, pC_in_gaaa(X1, X2, X3, X5)) DELETEB_IN_GAA(X1, tree(X2, X3, X4), tree(X2, X5, X4)) -> PC_IN_GAAA(X1, X2, X3, X5) PC_IN_GAAA(X1, X2, X3, X4) -> U7_GAAA(X1, X2, X3, X4, lessG_in_ga(X1, X2)) PC_IN_GAAA(X1, X2, X3, X4) -> LESSG_IN_GA(X1, X2) LESSG_IN_GA(s(X1), s(X2)) -> U2_GA(X1, X2, lessG_in_ga(X1, X2)) LESSG_IN_GA(s(X1), s(X2)) -> LESSG_IN_GA(X1, X2) PC_IN_GAAA(X1, X2, X3, X4) -> U8_GAAA(X1, X2, X3, X4, lesscG_in_ga(X1, X2)) U8_GAAA(X1, X2, X3, X4, lesscG_out_ga(X1, X2)) -> U9_GAAA(X1, X2, X3, X4, deleteB_in_gaa(X1, X3, X4)) U8_GAAA(X1, X2, X3, X4, lesscG_out_ga(X1, X2)) -> DELETEB_IN_GAA(X1, X3, X4) DELETEB_IN_GAA(X1, tree(X2, X3, X4), tree(X2, X3, X5)) -> U18_GAA(X1, X2, X3, X4, X5, pD_in_agaa(X2, X1, X4, X5)) DELETEB_IN_GAA(X1, tree(X2, X3, X4), tree(X2, X3, X5)) -> PD_IN_AGAA(X2, X1, X4, X5) PD_IN_AGAA(X1, X2, X3, X4) -> U4_AGAA(X1, X2, X3, X4, lessH_in_ag(X1, X2)) PD_IN_AGAA(X1, X2, X3, X4) -> LESSH_IN_AG(X1, X2) LESSH_IN_AG(s(X1), s(X2)) -> U3_AG(X1, X2, lessH_in_ag(X1, X2)) LESSH_IN_AG(s(X1), s(X2)) -> LESSH_IN_AG(X1, X2) PD_IN_AGAA(X1, X2, X3, X4) -> U5_AGAA(X1, X2, X3, X4, lesscH_in_ag(X1, X2)) U5_AGAA(X1, X2, X3, X4, lesscH_out_ag(X1, X2)) -> U6_AGAA(X1, X2, X3, X4, deleteB_in_gaa(X2, X3, X4)) U5_AGAA(X1, X2, X3, X4, lesscH_out_ag(X1, X2)) -> DELETEB_IN_GAA(X2, X3, X4) DELETEB_IN_GAA(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) -> U19_GAA(X1, X2, X3, X4, deleteB_in_gaa(0, X2, X4)) DELETEB_IN_GAA(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) -> DELETEB_IN_GAA(0, X2, X4) DELETEB_IN_GAA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) -> U20_GAA(X1, X2, X3, X4, X5, pE_in_gaaa(X1, X2, X3, X5)) DELETEB_IN_GAA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) -> PE_IN_GAAA(X1, X2, X3, X5) PE_IN_GAAA(X1, X2, X3, X4) -> U10_GAAA(X1, X2, X3, X4, lessG_in_ga(X1, X2)) PE_IN_GAAA(X1, X2, X3, X4) -> LESSG_IN_GA(X1, X2) PE_IN_GAAA(X1, X2, X3, X4) -> U11_GAAA(X1, X2, X3, X4, lesscG_in_ga(X1, X2)) U11_GAAA(X1, X2, X3, X4, lesscG_out_ga(X1, X2)) -> U12_GAAA(X1, X2, X3, X4, deleteB_in_gaa(s(X1), X3, X4)) U11_GAAA(X1, X2, X3, X4, lesscG_out_ga(X1, X2)) -> DELETEB_IN_GAA(s(X1), X3, X4) DELETEB_IN_GAA(s(X1), tree(0, X2, X3), tree(0, X2, X4)) -> U21_GAA(X1, X2, X3, X4, deleteB_in_gaa(s(X1), X3, X4)) DELETEB_IN_GAA(s(X1), tree(0, X2, X3), tree(0, X2, X4)) -> DELETEB_IN_GAA(s(X1), X3, X4) DELETEB_IN_GAA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) -> U22_GAA(X1, X2, X3, X4, X5, pF_in_agaa(X2, X1, X4, X5)) DELETEB_IN_GAA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) -> PF_IN_AGAA(X2, X1, X4, X5) PF_IN_AGAA(X1, X2, X3, X4) -> U13_AGAA(X1, X2, X3, X4, lessH_in_ag(X1, X2)) PF_IN_AGAA(X1, X2, X3, X4) -> LESSH_IN_AG(X1, X2) PF_IN_AGAA(X1, X2, X3, X4) -> U14_AGAA(X1, X2, X3, X4, lesscH_in_ag(X1, X2)) U14_AGAA(X1, X2, X3, X4, lesscH_out_ag(X1, X2)) -> U15_AGAA(X1, X2, X3, X4, deleteB_in_gaa(s(X2), X3, X4)) U14_AGAA(X1, X2, X3, X4, lesscH_out_ag(X1, X2)) -> DELETEB_IN_GAA(s(X2), X3, X4) The TRS R consists of the following rules: lesscG_in_ga(0, s(X1)) -> lesscG_out_ga(0, s(X1)) lesscG_in_ga(s(X1), s(X2)) -> U32_ga(X1, X2, lesscG_in_ga(X1, X2)) U32_ga(X1, X2, lesscG_out_ga(X1, X2)) -> lesscG_out_ga(s(X1), s(X2)) lesscH_in_ag(0, s(X1)) -> lesscH_out_ag(0, s(X1)) lesscH_in_ag(s(X1), s(X2)) -> U33_ag(X1, X2, lesscH_in_ag(X1, X2)) U33_ag(X1, X2, lesscH_out_ag(X1, X2)) -> lesscH_out_ag(s(X1), s(X2)) The argument filtering Pi contains the following mapping: deleteB_in_gaa(x1, x2, x3) = deleteB_in_gaa(x1) delminA_in_aaa(x1, x2, x3) = delminA_in_aaa pC_in_gaaa(x1, x2, x3, x4) = pC_in_gaaa(x1) lessG_in_ga(x1, x2) = lessG_in_ga(x1) s(x1) = s(x1) lesscG_in_ga(x1, x2) = lesscG_in_ga(x1) 0 = 0 lesscG_out_ga(x1, x2) = lesscG_out_ga(x1) U32_ga(x1, x2, x3) = U32_ga(x1, x3) pD_in_agaa(x1, x2, x3, x4) = pD_in_agaa(x2) lessH_in_ag(x1, x2) = lessH_in_ag(x2) lesscH_in_ag(x1, x2) = lesscH_in_ag(x2) lesscH_out_ag(x1, x2) = lesscH_out_ag(x1, x2) U33_ag(x1, x2, x3) = U33_ag(x2, x3) pE_in_gaaa(x1, x2, x3, x4) = pE_in_gaaa(x1) pF_in_agaa(x1, x2, x3, x4) = pF_in_agaa(x2) DELETEB_IN_GAA(x1, x2, x3) = DELETEB_IN_GAA(x1) U16_GAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U16_GAA(x1, x9) DELMINA_IN_AAA(x1, x2, x3) = DELMINA_IN_AAA U1_AAA(x1, x2, x3, x4, x5, x6, x7) = U1_AAA(x7) U17_GAA(x1, x2, x3, x4, x5, x6) = U17_GAA(x1, x6) PC_IN_GAAA(x1, x2, x3, x4) = PC_IN_GAAA(x1) U7_GAAA(x1, x2, x3, x4, x5) = U7_GAAA(x1, x5) LESSG_IN_GA(x1, x2) = LESSG_IN_GA(x1) U2_GA(x1, x2, x3) = U2_GA(x1, x3) U8_GAAA(x1, x2, x3, x4, x5) = U8_GAAA(x1, x5) U9_GAAA(x1, x2, x3, x4, x5) = U9_GAAA(x1, x5) U18_GAA(x1, x2, x3, x4, x5, x6) = U18_GAA(x1, x6) PD_IN_AGAA(x1, x2, x3, x4) = PD_IN_AGAA(x2) U4_AGAA(x1, x2, x3, x4, x5) = U4_AGAA(x2, x5) LESSH_IN_AG(x1, x2) = LESSH_IN_AG(x2) U3_AG(x1, x2, x3) = U3_AG(x2, x3) U5_AGAA(x1, x2, x3, x4, x5) = U5_AGAA(x2, x5) U6_AGAA(x1, x2, x3, x4, x5) = U6_AGAA(x1, x2, x5) U19_GAA(x1, x2, x3, x4, x5) = U19_GAA(x5) U20_GAA(x1, x2, x3, x4, x5, x6) = U20_GAA(x1, x6) PE_IN_GAAA(x1, x2, x3, x4) = PE_IN_GAAA(x1) U10_GAAA(x1, x2, x3, x4, x5) = U10_GAAA(x1, x5) U11_GAAA(x1, x2, x3, x4, x5) = U11_GAAA(x1, x5) U12_GAAA(x1, x2, x3, x4, x5) = U12_GAAA(x1, x5) U21_GAA(x1, x2, x3, x4, x5) = U21_GAA(x1, x5) U22_GAA(x1, x2, x3, x4, x5, x6) = U22_GAA(x1, x6) PF_IN_AGAA(x1, x2, x3, x4) = PF_IN_AGAA(x2) U13_AGAA(x1, x2, x3, x4, x5) = U13_AGAA(x2, x5) U14_AGAA(x1, x2, x3, x4, x5) = U14_AGAA(x2, x5) U15_AGAA(x1, x2, x3, x4, x5) = U15_AGAA(x1, x2, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (165) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 23 less nodes. ---------------------------------------- (166) Complex Obligation (AND) ---------------------------------------- (167) Obligation: Pi DP problem: The TRS P consists of the following rules: LESSH_IN_AG(s(X1), s(X2)) -> LESSH_IN_AG(X1, X2) The TRS R consists of the following rules: lesscG_in_ga(0, s(X1)) -> lesscG_out_ga(0, s(X1)) lesscG_in_ga(s(X1), s(X2)) -> U32_ga(X1, X2, lesscG_in_ga(X1, X2)) U32_ga(X1, X2, lesscG_out_ga(X1, X2)) -> lesscG_out_ga(s(X1), s(X2)) lesscH_in_ag(0, s(X1)) -> lesscH_out_ag(0, s(X1)) lesscH_in_ag(s(X1), s(X2)) -> U33_ag(X1, X2, lesscH_in_ag(X1, X2)) U33_ag(X1, X2, lesscH_out_ag(X1, X2)) -> lesscH_out_ag(s(X1), s(X2)) The argument filtering Pi contains the following mapping: s(x1) = s(x1) lesscG_in_ga(x1, x2) = lesscG_in_ga(x1) 0 = 0 lesscG_out_ga(x1, x2) = lesscG_out_ga(x1) U32_ga(x1, x2, x3) = U32_ga(x1, x3) lesscH_in_ag(x1, x2) = lesscH_in_ag(x2) lesscH_out_ag(x1, x2) = lesscH_out_ag(x1, x2) U33_ag(x1, x2, x3) = U33_ag(x2, x3) LESSH_IN_AG(x1, x2) = LESSH_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (168) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (169) Obligation: Pi DP problem: The TRS P consists of the following rules: LESSH_IN_AG(s(X1), s(X2)) -> LESSH_IN_AG(X1, X2) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) LESSH_IN_AG(x1, x2) = LESSH_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (170) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (171) Obligation: Q DP problem: The TRS P consists of the following rules: LESSH_IN_AG(s(X2)) -> LESSH_IN_AG(X2) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (172) 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: *LESSH_IN_AG(s(X2)) -> LESSH_IN_AG(X2) The graph contains the following edges 1 > 1 ---------------------------------------- (173) YES ---------------------------------------- (174) 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: lesscG_in_ga(0, s(X1)) -> lesscG_out_ga(0, s(X1)) lesscG_in_ga(s(X1), s(X2)) -> U32_ga(X1, X2, lesscG_in_ga(X1, X2)) U32_ga(X1, X2, lesscG_out_ga(X1, X2)) -> lesscG_out_ga(s(X1), s(X2)) lesscH_in_ag(0, s(X1)) -> lesscH_out_ag(0, s(X1)) lesscH_in_ag(s(X1), s(X2)) -> U33_ag(X1, X2, lesscH_in_ag(X1, X2)) U33_ag(X1, X2, lesscH_out_ag(X1, X2)) -> lesscH_out_ag(s(X1), s(X2)) The argument filtering Pi contains the following mapping: s(x1) = s(x1) lesscG_in_ga(x1, x2) = lesscG_in_ga(x1) 0 = 0 lesscG_out_ga(x1, x2) = lesscG_out_ga(x1) U32_ga(x1, x2, x3) = U32_ga(x1, x3) lesscH_in_ag(x1, x2) = lesscH_in_ag(x2) lesscH_out_ag(x1, x2) = lesscH_out_ag(x1, x2) U33_ag(x1, x2, x3) = U33_ag(x2, x3) LESSG_IN_GA(x1, x2) = LESSG_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (175) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (176) 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 ---------------------------------------- (177) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (178) 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. ---------------------------------------- (179) 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 ---------------------------------------- (180) YES ---------------------------------------- (181) Obligation: Pi DP problem: The TRS P consists of the following rules: DELMINA_IN_AAA(tree(X1, X2, X3), X4, tree(X1, X5, X6)) -> DELMINA_IN_AAA(X2, X4, X5) The TRS R consists of the following rules: lesscG_in_ga(0, s(X1)) -> lesscG_out_ga(0, s(X1)) lesscG_in_ga(s(X1), s(X2)) -> U32_ga(X1, X2, lesscG_in_ga(X1, X2)) U32_ga(X1, X2, lesscG_out_ga(X1, X2)) -> lesscG_out_ga(s(X1), s(X2)) lesscH_in_ag(0, s(X1)) -> lesscH_out_ag(0, s(X1)) lesscH_in_ag(s(X1), s(X2)) -> U33_ag(X1, X2, lesscH_in_ag(X1, X2)) U33_ag(X1, X2, lesscH_out_ag(X1, X2)) -> lesscH_out_ag(s(X1), s(X2)) The argument filtering Pi contains the following mapping: s(x1) = s(x1) lesscG_in_ga(x1, x2) = lesscG_in_ga(x1) 0 = 0 lesscG_out_ga(x1, x2) = lesscG_out_ga(x1) U32_ga(x1, x2, x3) = U32_ga(x1, x3) lesscH_in_ag(x1, x2) = lesscH_in_ag(x2) lesscH_out_ag(x1, x2) = lesscH_out_ag(x1, x2) U33_ag(x1, x2, x3) = U33_ag(x2, x3) DELMINA_IN_AAA(x1, x2, x3) = DELMINA_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: DELMINA_IN_AAA(tree(X1, X2, X3), X4, tree(X1, X5, X6)) -> DELMINA_IN_AAA(X2, X4, X5) R is empty. The argument filtering Pi contains the following mapping: DELMINA_IN_AAA(x1, x2, x3) = DELMINA_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (184) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (185) Obligation: Q DP problem: The TRS P consists of the following rules: DELMINA_IN_AAA -> DELMINA_IN_AAA R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (186) 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 = DELMINA_IN_AAA evaluates to t =DELMINA_IN_AAA Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence The DP semiunifies directly so there is only one rewrite step from DELMINA_IN_AAA to DELMINA_IN_AAA. ---------------------------------------- (187) NO ---------------------------------------- (188) Obligation: Pi DP problem: The TRS P consists of the following rules: DELETEB_IN_GAA(X1, tree(X2, X3, X4), tree(X2, X5, X4)) -> PC_IN_GAAA(X1, X2, X3, X5) PC_IN_GAAA(X1, X2, X3, X4) -> U8_GAAA(X1, X2, X3, X4, lesscG_in_ga(X1, X2)) U8_GAAA(X1, X2, X3, X4, lesscG_out_ga(X1, X2)) -> DELETEB_IN_GAA(X1, X3, X4) DELETEB_IN_GAA(X1, tree(X2, X3, X4), tree(X2, X3, X5)) -> PD_IN_AGAA(X2, X1, X4, X5) PD_IN_AGAA(X1, X2, X3, X4) -> U5_AGAA(X1, X2, X3, X4, lesscH_in_ag(X1, X2)) U5_AGAA(X1, X2, X3, X4, lesscH_out_ag(X1, X2)) -> DELETEB_IN_GAA(X2, X3, X4) DELETEB_IN_GAA(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) -> DELETEB_IN_GAA(0, X2, X4) DELETEB_IN_GAA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) -> PE_IN_GAAA(X1, X2, X3, X5) PE_IN_GAAA(X1, X2, X3, X4) -> U11_GAAA(X1, X2, X3, X4, lesscG_in_ga(X1, X2)) U11_GAAA(X1, X2, X3, X4, lesscG_out_ga(X1, X2)) -> DELETEB_IN_GAA(s(X1), X3, X4) DELETEB_IN_GAA(s(X1), tree(0, X2, X3), tree(0, X2, X4)) -> DELETEB_IN_GAA(s(X1), X3, X4) DELETEB_IN_GAA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) -> PF_IN_AGAA(X2, X1, X4, X5) PF_IN_AGAA(X1, X2, X3, X4) -> U14_AGAA(X1, X2, X3, X4, lesscH_in_ag(X1, X2)) U14_AGAA(X1, X2, X3, X4, lesscH_out_ag(X1, X2)) -> DELETEB_IN_GAA(s(X2), X3, X4) The TRS R consists of the following rules: lesscG_in_ga(0, s(X1)) -> lesscG_out_ga(0, s(X1)) lesscG_in_ga(s(X1), s(X2)) -> U32_ga(X1, X2, lesscG_in_ga(X1, X2)) U32_ga(X1, X2, lesscG_out_ga(X1, X2)) -> lesscG_out_ga(s(X1), s(X2)) lesscH_in_ag(0, s(X1)) -> lesscH_out_ag(0, s(X1)) lesscH_in_ag(s(X1), s(X2)) -> U33_ag(X1, X2, lesscH_in_ag(X1, X2)) U33_ag(X1, X2, lesscH_out_ag(X1, X2)) -> lesscH_out_ag(s(X1), s(X2)) The argument filtering Pi contains the following mapping: s(x1) = s(x1) lesscG_in_ga(x1, x2) = lesscG_in_ga(x1) 0 = 0 lesscG_out_ga(x1, x2) = lesscG_out_ga(x1) U32_ga(x1, x2, x3) = U32_ga(x1, x3) lesscH_in_ag(x1, x2) = lesscH_in_ag(x2) lesscH_out_ag(x1, x2) = lesscH_out_ag(x1, x2) U33_ag(x1, x2, x3) = U33_ag(x2, x3) DELETEB_IN_GAA(x1, x2, x3) = DELETEB_IN_GAA(x1) PC_IN_GAAA(x1, x2, x3, x4) = PC_IN_GAAA(x1) U8_GAAA(x1, x2, x3, x4, x5) = U8_GAAA(x1, x5) PD_IN_AGAA(x1, x2, x3, x4) = PD_IN_AGAA(x2) U5_AGAA(x1, x2, x3, x4, x5) = U5_AGAA(x2, x5) PE_IN_GAAA(x1, x2, x3, x4) = PE_IN_GAAA(x1) U11_GAAA(x1, x2, x3, x4, x5) = U11_GAAA(x1, x5) PF_IN_AGAA(x1, x2, x3, x4) = PF_IN_AGAA(x2) U14_AGAA(x1, x2, x3, x4, x5) = U14_AGAA(x2, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (189) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (190) Obligation: Q DP problem: The TRS P consists of the following rules: DELETEB_IN_GAA(X1) -> PC_IN_GAAA(X1) PC_IN_GAAA(X1) -> U8_GAAA(X1, lesscG_in_ga(X1)) U8_GAAA(X1, lesscG_out_ga(X1)) -> DELETEB_IN_GAA(X1) DELETEB_IN_GAA(X1) -> PD_IN_AGAA(X1) PD_IN_AGAA(X2) -> U5_AGAA(X2, lesscH_in_ag(X2)) U5_AGAA(X2, lesscH_out_ag(X1, X2)) -> DELETEB_IN_GAA(X2) DELETEB_IN_GAA(0) -> DELETEB_IN_GAA(0) DELETEB_IN_GAA(s(X1)) -> PE_IN_GAAA(X1) PE_IN_GAAA(X1) -> U11_GAAA(X1, lesscG_in_ga(X1)) U11_GAAA(X1, lesscG_out_ga(X1)) -> DELETEB_IN_GAA(s(X1)) DELETEB_IN_GAA(s(X1)) -> DELETEB_IN_GAA(s(X1)) DELETEB_IN_GAA(s(X1)) -> PF_IN_AGAA(X1) PF_IN_AGAA(X2) -> U14_AGAA(X2, lesscH_in_ag(X2)) U14_AGAA(X2, lesscH_out_ag(X1, X2)) -> DELETEB_IN_GAA(s(X2)) The TRS R consists of the following rules: lesscG_in_ga(0) -> lesscG_out_ga(0) lesscG_in_ga(s(X1)) -> U32_ga(X1, lesscG_in_ga(X1)) U32_ga(X1, lesscG_out_ga(X1)) -> lesscG_out_ga(s(X1)) lesscH_in_ag(s(X1)) -> lesscH_out_ag(0, s(X1)) lesscH_in_ag(s(X2)) -> U33_ag(X2, lesscH_in_ag(X2)) U33_ag(X2, lesscH_out_ag(X1, X2)) -> lesscH_out_ag(s(X1), s(X2)) The set Q consists of the following terms: lesscG_in_ga(x0) U32_ga(x0, x1) lesscH_in_ag(x0) U33_ag(x0, x1) We have to consider all (P,Q,R)-chains.