30.19/8.70 MAYBE 30.19/8.71 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 30.19/8.71 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 30.19/8.71 30.19/8.71 30.19/8.71 Left Termination of the query pattern 30.19/8.71 30.19/8.71 delete(g,a,a) 30.19/8.71 30.19/8.71 w.r.t. the given Prolog program could not be shown: 30.19/8.71 30.19/8.71 (0) Prolog 30.19/8.71 (1) PrologToPiTRSProof [SOUND, 0 ms] 30.19/8.71 (2) PiTRS 30.19/8.71 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 30.19/8.71 (4) PiDP 30.19/8.71 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 30.19/8.71 (6) AND 30.19/8.71 (7) PiDP 30.19/8.71 (8) UsableRulesProof [EQUIVALENT, 0 ms] 30.19/8.71 (9) PiDP 30.19/8.71 (10) PiDPToQDPProof [SOUND, 9 ms] 30.19/8.71 (11) QDP 30.19/8.71 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 30.19/8.71 (13) YES 30.19/8.71 (14) PiDP 30.19/8.71 (15) UsableRulesProof [EQUIVALENT, 0 ms] 30.19/8.71 (16) PiDP 30.19/8.71 (17) PiDPToQDPProof [SOUND, 0 ms] 30.19/8.71 (18) QDP 30.19/8.71 (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] 30.19/8.71 (20) YES 30.19/8.71 (21) PiDP 30.19/8.71 (22) UsableRulesProof [EQUIVALENT, 0 ms] 30.19/8.71 (23) PiDP 30.19/8.71 (24) PiDPToQDPProof [SOUND, 0 ms] 30.19/8.71 (25) QDP 30.19/8.71 (26) NonTerminationLoopProof [COMPLETE, 0 ms] 30.19/8.71 (27) NO 30.19/8.71 (28) PiDP 30.19/8.71 (29) UsableRulesProof [EQUIVALENT, 0 ms] 30.19/8.71 (30) PiDP 30.19/8.71 (31) PiDPToQDPProof [SOUND, 0 ms] 30.19/8.71 (32) QDP 30.19/8.71 (33) TransformationProof [SOUND, 0 ms] 30.19/8.71 (34) QDP 30.19/8.71 (35) TransformationProof [SOUND, 0 ms] 30.19/8.71 (36) QDP 30.19/8.71 (37) TransformationProof [EQUIVALENT, 0 ms] 30.19/8.71 (38) QDP 30.19/8.71 (39) DependencyGraphProof [EQUIVALENT, 0 ms] 30.19/8.71 (40) AND 30.19/8.71 (41) QDP 30.19/8.71 (42) UsableRulesProof [EQUIVALENT, 0 ms] 30.19/8.71 (43) QDP 30.19/8.71 (44) QReductionProof [EQUIVALENT, 0 ms] 30.19/8.71 (45) QDP 30.19/8.71 (46) NonTerminationLoopProof [COMPLETE, 0 ms] 30.19/8.71 (47) NO 30.19/8.71 (48) QDP 30.19/8.71 (49) TransformationProof [EQUIVALENT, 0 ms] 30.19/8.71 (50) QDP 30.19/8.71 (51) PrologToPiTRSProof [SOUND, 0 ms] 30.19/8.71 (52) PiTRS 30.19/8.71 (53) DependencyPairsProof [EQUIVALENT, 0 ms] 30.19/8.71 (54) PiDP 30.19/8.71 (55) DependencyGraphProof [EQUIVALENT, 0 ms] 30.19/8.71 (56) AND 30.19/8.71 (57) PiDP 30.19/8.71 (58) UsableRulesProof [EQUIVALENT, 0 ms] 30.19/8.71 (59) PiDP 30.19/8.71 (60) PiDPToQDPProof [SOUND, 1 ms] 30.19/8.71 (61) QDP 30.19/8.71 (62) QDPSizeChangeProof [EQUIVALENT, 0 ms] 30.19/8.71 (63) YES 30.19/8.71 (64) PiDP 30.19/8.71 (65) UsableRulesProof [EQUIVALENT, 0 ms] 30.19/8.71 (66) PiDP 30.19/8.71 (67) PiDPToQDPProof [SOUND, 0 ms] 30.19/8.71 (68) QDP 30.19/8.71 (69) QDPSizeChangeProof [EQUIVALENT, 0 ms] 30.19/8.71 (70) YES 30.19/8.71 (71) PiDP 30.19/8.71 (72) UsableRulesProof [EQUIVALENT, 0 ms] 30.19/8.71 (73) PiDP 30.19/8.71 (74) PiDPToQDPProof [SOUND, 0 ms] 30.19/8.71 (75) QDP 30.19/8.71 (76) NonTerminationLoopProof [COMPLETE, 0 ms] 30.19/8.71 (77) NO 30.19/8.71 (78) PiDP 30.19/8.71 (79) UsableRulesProof [EQUIVALENT, 0 ms] 30.19/8.71 (80) PiDP 30.19/8.71 (81) PiDPToQDPProof [SOUND, 0 ms] 30.19/8.71 (82) QDP 30.19/8.71 (83) PrologToTRSTransformerProof [SOUND, 26 ms] 30.19/8.71 (84) QTRS 30.19/8.71 (85) DependencyPairsProof [EQUIVALENT, 0 ms] 30.19/8.71 (86) QDP 30.19/8.71 (87) DependencyGraphProof [EQUIVALENT, 0 ms] 30.19/8.71 (88) AND 30.19/8.71 (89) QDP 30.19/8.71 (90) UsableRulesProof [EQUIVALENT, 0 ms] 30.19/8.71 (91) QDP 30.19/8.71 (92) QDPSizeChangeProof [EQUIVALENT, 0 ms] 30.19/8.71 (93) YES 30.19/8.71 (94) QDP 30.19/8.71 (95) UsableRulesProof [EQUIVALENT, 0 ms] 30.19/8.71 (96) QDP 30.19/8.71 (97) QDPSizeChangeProof [EQUIVALENT, 0 ms] 30.19/8.71 (98) YES 30.19/8.71 (99) QDP 30.19/8.71 (100) UsableRulesProof [EQUIVALENT, 0 ms] 30.19/8.71 (101) QDP 30.19/8.71 (102) NonTerminationLoopProof [COMPLETE, 0 ms] 30.19/8.71 (103) NO 30.19/8.71 (104) QDP 30.19/8.71 (105) NonTerminationLoopProof [COMPLETE, 4 ms] 30.19/8.71 (106) NO 30.19/8.71 (107) PrologToIRSwTTransformerProof [SOUND, 45 ms] 30.19/8.71 (108) AND 30.19/8.71 (109) IRSwT 30.19/8.71 (110) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] 30.19/8.71 (111) IRSwT 30.19/8.71 (112) IntTRSCompressionProof [EQUIVALENT, 20 ms] 30.19/8.71 (113) IRSwT 30.19/8.71 (114) IRSFormatTransformerProof [EQUIVALENT, 0 ms] 30.19/8.71 (115) IRSwT 30.19/8.71 (116) IRSwTTerminationDigraphProof [EQUIVALENT, 2 ms] 30.19/8.71 (117) IRSwT 30.19/8.71 (118) TempFilterProof [SOUND, 3 ms] 30.19/8.71 (119) IRSwT 30.19/8.71 (120) IRSwTToQDPProof [SOUND, 0 ms] 30.19/8.71 (121) QDP 30.19/8.71 (122) QDPSizeChangeProof [EQUIVALENT, 0 ms] 30.19/8.71 (123) YES 30.19/8.71 (124) IRSwT 30.19/8.71 (125) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] 30.19/8.71 (126) IRSwT 30.19/8.71 (127) IntTRSCompressionProof [EQUIVALENT, 3 ms] 30.19/8.71 (128) IRSwT 30.19/8.71 (129) IRSFormatTransformerProof [EQUIVALENT, 0 ms] 30.19/8.71 (130) IRSwT 30.19/8.71 (131) IRSwTTerminationDigraphProof [EQUIVALENT, 2 ms] 30.19/8.71 (132) IRSwT 30.19/8.71 (133) TempFilterProof [SOUND, 1 ms] 30.19/8.71 (134) IRSwT 30.19/8.71 (135) IRSwTToQDPProof [SOUND, 0 ms] 30.19/8.71 (136) QDP 30.19/8.71 (137) QDPSizeChangeProof [EQUIVALENT, 0 ms] 30.19/8.71 (138) YES 30.19/8.71 (139) IRSwT 30.19/8.71 (140) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] 30.19/8.71 (141) IRSwT 30.19/8.71 (142) IntTRSCompressionProof [EQUIVALENT, 0 ms] 30.19/8.71 (143) IRSwT 30.19/8.71 (144) IRSFormatTransformerProof [EQUIVALENT, 0 ms] 30.19/8.71 (145) IRSwT 30.19/8.71 (146) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms] 30.19/8.71 (147) IRSwT 30.19/8.71 (148) FilterProof [EQUIVALENT, 0 ms] 30.19/8.71 (149) IntTRS 30.19/8.71 (150) IntTRSPeriodicNontermProof [COMPLETE, 7 ms] 30.19/8.71 (151) NO 30.19/8.71 (152) IRSwT 30.19/8.71 (153) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] 30.19/8.71 (154) IRSwT 30.19/8.71 (155) IntTRSCompressionProof [EQUIVALENT, 20 ms] 30.19/8.71 (156) IRSwT 30.19/8.71 (157) IRSFormatTransformerProof [EQUIVALENT, 0 ms] 30.19/8.71 (158) IRSwT 30.19/8.71 (159) IRSwTTerminationDigraphProof [EQUIVALENT, 38 ms] 30.19/8.71 (160) IRSwT 30.19/8.71 (161) PrologToDTProblemTransformerProof [SOUND, 166 ms] 30.19/8.71 (162) TRIPLES 30.19/8.71 (163) TriplesToPiDPProof [SOUND, 51 ms] 30.19/8.71 (164) PiDP 30.19/8.71 (165) DependencyGraphProof [EQUIVALENT, 0 ms] 30.19/8.71 (166) AND 30.19/8.71 (167) PiDP 30.19/8.71 (168) UsableRulesProof [EQUIVALENT, 0 ms] 30.19/8.71 (169) PiDP 30.19/8.71 (170) PiDPToQDPProof [SOUND, 0 ms] 30.19/8.71 (171) QDP 30.19/8.71 (172) QDPSizeChangeProof [EQUIVALENT, 0 ms] 30.19/8.71 (173) YES 30.19/8.71 (174) PiDP 30.19/8.71 (175) UsableRulesProof [EQUIVALENT, 0 ms] 30.19/8.71 (176) PiDP 30.19/8.71 (177) PiDPToQDPProof [SOUND, 0 ms] 30.19/8.71 (178) QDP 30.19/8.71 (179) QDPSizeChangeProof [EQUIVALENT, 0 ms] 30.19/8.71 (180) YES 30.19/8.71 (181) PiDP 30.19/8.71 (182) UsableRulesProof [EQUIVALENT, 0 ms] 30.19/8.71 (183) PiDP 30.19/8.71 (184) PiDPToQDPProof [SOUND, 0 ms] 30.19/8.71 (185) QDP 30.19/8.71 (186) NonTerminationLoopProof [COMPLETE, 0 ms] 30.19/8.71 (187) NO 30.19/8.71 (188) PiDP 30.19/8.71 (189) PiDPToQDPProof [SOUND, 0 ms] 30.19/8.71 (190) QDP 30.19/8.71 30.19/8.71 30.19/8.71 ---------------------------------------- 30.19/8.71 30.19/8.71 (0) 30.19/8.71 Obligation: 30.19/8.71 Clauses: 30.19/8.71 30.19/8.71 delete(X, tree(X, void, Right), Right). 30.19/8.71 delete(X, tree(X, Left, void), Left). 30.19/8.71 delete(X, tree(X, Left, Right), tree(Y, Left, Right1)) :- delmin(Right, Y, Right1). 30.19/8.71 delete(X, tree(Y, Left, Right), tree(Y, Left1, Right)) :- ','(less(X, Y), delete(X, Left, Left1)). 30.19/8.71 delete(X, tree(Y, Left, Right), tree(Y, Left, Right1)) :- ','(less(Y, X), delete(X, Right, Right1)). 30.19/8.72 delmin(tree(Y, void, Right), Y, Right). 30.19/8.72 delmin(tree(X, Left, X1), Y, tree(X, Left1, X2)) :- delmin(Left, Y, Left1). 30.19/8.72 less(0, s(X3)). 30.19/8.72 less(s(X), s(Y)) :- less(X, Y). 30.19/8.72 30.19/8.72 30.19/8.72 Query: delete(g,a,a) 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (1) PrologToPiTRSProof (SOUND) 30.19/8.72 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 30.19/8.72 30.19/8.72 delete_in_3: (b,f,f) 30.19/8.72 30.19/8.72 delmin_in_3: (f,f,f) 30.19/8.72 30.19/8.72 less_in_2: (b,f) (f,b) 30.19/8.72 30.19/8.72 Transforming Prolog into the following Term Rewriting System: 30.19/8.72 30.19/8.72 Pi-finite rewrite system: 30.19/8.72 The TRS R consists of the following rules: 30.19/8.72 30.19/8.72 delete_in_gaa(X, tree(X, void, Right), Right) -> delete_out_gaa(X, tree(X, void, Right), Right) 30.19/8.72 delete_in_gaa(X, tree(X, Left, void), Left) -> delete_out_gaa(X, tree(X, Left, void), Left) 30.19/8.72 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)) 30.19/8.72 delmin_in_aaa(tree(Y, void, Right), Y, Right) -> delmin_out_aaa(tree(Y, void, Right), Y, Right) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 30.19/8.72 less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3)) 30.19/8.72 less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y)) 30.19/8.72 U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 30.19/8.72 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)) 30.19/8.72 delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 30.19/8.72 less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3)) 30.19/8.72 less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y)) 30.19/8.72 U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 30.19/8.72 The argument filtering Pi contains the following mapping: 30.19/8.72 delete_in_gaa(x1, x2, x3) = delete_in_gaa(x1) 30.19/8.72 30.19/8.72 delete_out_gaa(x1, x2, x3) = delete_out_gaa 30.19/8.72 30.19/8.72 U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x6) 30.19/8.72 30.19/8.72 delmin_in_aaa(x1, x2, x3) = delmin_in_aaa 30.19/8.72 30.19/8.72 delmin_out_aaa(x1, x2, x3) = delmin_out_aaa 30.19/8.72 30.19/8.72 U6_aaa(x1, x2, x3, x4, x5, x6, x7) = U6_aaa(x7) 30.19/8.72 30.19/8.72 U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6) 30.19/8.72 30.19/8.72 less_in_ga(x1, x2) = less_in_ga(x1) 30.19/8.72 30.19/8.72 0 = 0 30.19/8.72 30.19/8.72 less_out_ga(x1, x2) = less_out_ga 30.19/8.72 30.19/8.72 s(x1) = s(x1) 30.19/8.72 30.19/8.72 U7_ga(x1, x2, x3) = U7_ga(x3) 30.19/8.72 30.19/8.72 U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x6) 30.19/8.72 30.19/8.72 U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6) 30.19/8.72 30.19/8.72 less_in_ag(x1, x2) = less_in_ag(x2) 30.19/8.72 30.19/8.72 less_out_ag(x1, x2) = less_out_ag(x1) 30.19/8.72 30.19/8.72 U7_ag(x1, x2, x3) = U7_ag(x3) 30.19/8.72 30.19/8.72 U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x6) 30.19/8.72 30.19/8.72 30.19/8.72 30.19/8.72 30.19/8.72 30.19/8.72 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 30.19/8.72 30.19/8.72 30.19/8.72 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (2) 30.19/8.72 Obligation: 30.19/8.72 Pi-finite rewrite system: 30.19/8.72 The TRS R consists of the following rules: 30.19/8.72 30.19/8.72 delete_in_gaa(X, tree(X, void, Right), Right) -> delete_out_gaa(X, tree(X, void, Right), Right) 30.19/8.72 delete_in_gaa(X, tree(X, Left, void), Left) -> delete_out_gaa(X, tree(X, Left, void), Left) 30.19/8.72 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)) 30.19/8.72 delmin_in_aaa(tree(Y, void, Right), Y, Right) -> delmin_out_aaa(tree(Y, void, Right), Y, Right) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 30.19/8.72 less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3)) 30.19/8.72 less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y)) 30.19/8.72 U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 30.19/8.72 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)) 30.19/8.72 delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 30.19/8.72 less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3)) 30.19/8.72 less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y)) 30.19/8.72 U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 30.19/8.72 The argument filtering Pi contains the following mapping: 30.19/8.72 delete_in_gaa(x1, x2, x3) = delete_in_gaa(x1) 30.19/8.72 30.19/8.72 delete_out_gaa(x1, x2, x3) = delete_out_gaa 30.19/8.72 30.19/8.72 U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x6) 30.19/8.72 30.19/8.72 delmin_in_aaa(x1, x2, x3) = delmin_in_aaa 30.19/8.72 30.19/8.72 delmin_out_aaa(x1, x2, x3) = delmin_out_aaa 30.19/8.72 30.19/8.72 U6_aaa(x1, x2, x3, x4, x5, x6, x7) = U6_aaa(x7) 30.19/8.72 30.19/8.72 U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6) 30.19/8.72 30.19/8.72 less_in_ga(x1, x2) = less_in_ga(x1) 30.19/8.72 30.19/8.72 0 = 0 30.19/8.72 30.19/8.72 less_out_ga(x1, x2) = less_out_ga 30.19/8.72 30.19/8.72 s(x1) = s(x1) 30.19/8.72 30.19/8.72 U7_ga(x1, x2, x3) = U7_ga(x3) 30.19/8.72 30.19/8.72 U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x6) 30.19/8.72 30.19/8.72 U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6) 30.19/8.72 30.19/8.72 less_in_ag(x1, x2) = less_in_ag(x2) 30.19/8.72 30.19/8.72 less_out_ag(x1, x2) = less_out_ag(x1) 30.19/8.72 30.19/8.72 U7_ag(x1, x2, x3) = U7_ag(x3) 30.19/8.72 30.19/8.72 U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x6) 30.19/8.72 30.19/8.72 30.19/8.72 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (3) DependencyPairsProof (EQUIVALENT) 30.19/8.72 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 30.19/8.72 Pi DP problem: 30.19/8.72 The TRS P consists of the following rules: 30.19/8.72 30.19/8.72 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)) 30.19/8.72 DELETE_IN_GAA(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> DELMIN_IN_AAA(Right, Y, Right1) 30.19/8.72 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)) 30.19/8.72 DELMIN_IN_AAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> DELMIN_IN_AAA(Left, Y, Left1) 30.19/8.72 DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_GAA(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 30.19/8.72 DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> LESS_IN_GA(X, Y) 30.19/8.72 LESS_IN_GA(s(X), s(Y)) -> U7_GA(X, Y, less_in_ga(X, Y)) 30.19/8.72 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 30.19/8.72 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)) 30.19/8.72 U2_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> DELETE_IN_GAA(X, Left, Left1) 30.19/8.72 DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_GAA(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 30.19/8.72 DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> LESS_IN_AG(Y, X) 30.19/8.72 LESS_IN_AG(s(X), s(Y)) -> U7_AG(X, Y, less_in_ag(X, Y)) 30.19/8.72 LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) 30.19/8.72 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)) 30.19/8.72 U4_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> DELETE_IN_GAA(X, Right, Right1) 30.19/8.72 30.19/8.72 The TRS R consists of the following rules: 30.19/8.72 30.19/8.72 delete_in_gaa(X, tree(X, void, Right), Right) -> delete_out_gaa(X, tree(X, void, Right), Right) 30.19/8.72 delete_in_gaa(X, tree(X, Left, void), Left) -> delete_out_gaa(X, tree(X, Left, void), Left) 30.19/8.72 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)) 30.19/8.72 delmin_in_aaa(tree(Y, void, Right), Y, Right) -> delmin_out_aaa(tree(Y, void, Right), Y, Right) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 30.19/8.72 less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3)) 30.19/8.72 less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y)) 30.19/8.72 U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 30.19/8.72 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)) 30.19/8.72 delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 30.19/8.72 less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3)) 30.19/8.72 less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y)) 30.19/8.72 U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 30.19/8.72 The argument filtering Pi contains the following mapping: 30.19/8.72 delete_in_gaa(x1, x2, x3) = delete_in_gaa(x1) 30.19/8.72 30.19/8.72 delete_out_gaa(x1, x2, x3) = delete_out_gaa 30.19/8.72 30.19/8.72 U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x6) 30.19/8.72 30.19/8.72 delmin_in_aaa(x1, x2, x3) = delmin_in_aaa 30.19/8.72 30.19/8.72 delmin_out_aaa(x1, x2, x3) = delmin_out_aaa 30.19/8.72 30.19/8.72 U6_aaa(x1, x2, x3, x4, x5, x6, x7) = U6_aaa(x7) 30.19/8.72 30.19/8.72 U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6) 30.19/8.72 30.19/8.72 less_in_ga(x1, x2) = less_in_ga(x1) 30.19/8.72 30.19/8.72 0 = 0 30.19/8.72 30.19/8.72 less_out_ga(x1, x2) = less_out_ga 30.19/8.72 30.19/8.72 s(x1) = s(x1) 30.19/8.72 30.19/8.72 U7_ga(x1, x2, x3) = U7_ga(x3) 30.19/8.72 30.19/8.72 U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x6) 30.19/8.72 30.19/8.72 U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6) 30.19/8.72 30.19/8.72 less_in_ag(x1, x2) = less_in_ag(x2) 30.19/8.72 30.19/8.72 less_out_ag(x1, x2) = less_out_ag(x1) 30.19/8.72 30.19/8.72 U7_ag(x1, x2, x3) = U7_ag(x3) 30.19/8.72 30.19/8.72 U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x6) 30.19/8.72 30.19/8.72 DELETE_IN_GAA(x1, x2, x3) = DELETE_IN_GAA(x1) 30.19/8.72 30.19/8.72 U1_GAA(x1, x2, x3, x4, x5, x6) = U1_GAA(x6) 30.19/8.72 30.19/8.72 DELMIN_IN_AAA(x1, x2, x3) = DELMIN_IN_AAA 30.19/8.72 30.19/8.72 U6_AAA(x1, x2, x3, x4, x5, x6, x7) = U6_AAA(x7) 30.19/8.72 30.19/8.72 U2_GAA(x1, x2, x3, x4, x5, x6) = U2_GAA(x1, x6) 30.19/8.72 30.19/8.72 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 30.19/8.72 30.19/8.72 U7_GA(x1, x2, x3) = U7_GA(x3) 30.19/8.72 30.19/8.72 U3_GAA(x1, x2, x3, x4, x5, x6) = U3_GAA(x6) 30.19/8.72 30.19/8.72 U4_GAA(x1, x2, x3, x4, x5, x6) = U4_GAA(x1, x6) 30.19/8.72 30.19/8.72 LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) 30.19/8.72 30.19/8.72 U7_AG(x1, x2, x3) = U7_AG(x3) 30.19/8.72 30.19/8.72 U5_GAA(x1, x2, x3, x4, x5, x6) = U5_GAA(x6) 30.19/8.72 30.19/8.72 30.19/8.72 We have to consider all (P,R,Pi)-chains 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (4) 30.19/8.72 Obligation: 30.19/8.72 Pi DP problem: 30.19/8.72 The TRS P consists of the following rules: 30.19/8.72 30.19/8.72 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)) 30.19/8.72 DELETE_IN_GAA(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> DELMIN_IN_AAA(Right, Y, Right1) 30.19/8.72 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)) 30.19/8.72 DELMIN_IN_AAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> DELMIN_IN_AAA(Left, Y, Left1) 30.19/8.72 DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_GAA(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 30.19/8.72 DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> LESS_IN_GA(X, Y) 30.19/8.72 LESS_IN_GA(s(X), s(Y)) -> U7_GA(X, Y, less_in_ga(X, Y)) 30.19/8.72 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 30.19/8.72 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)) 30.19/8.72 U2_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> DELETE_IN_GAA(X, Left, Left1) 30.19/8.72 DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_GAA(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 30.19/8.72 DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> LESS_IN_AG(Y, X) 30.19/8.72 LESS_IN_AG(s(X), s(Y)) -> U7_AG(X, Y, less_in_ag(X, Y)) 30.19/8.72 LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) 30.19/8.72 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)) 30.19/8.72 U4_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> DELETE_IN_GAA(X, Right, Right1) 30.19/8.72 30.19/8.72 The TRS R consists of the following rules: 30.19/8.72 30.19/8.72 delete_in_gaa(X, tree(X, void, Right), Right) -> delete_out_gaa(X, tree(X, void, Right), Right) 30.19/8.72 delete_in_gaa(X, tree(X, Left, void), Left) -> delete_out_gaa(X, tree(X, Left, void), Left) 30.19/8.72 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)) 30.19/8.72 delmin_in_aaa(tree(Y, void, Right), Y, Right) -> delmin_out_aaa(tree(Y, void, Right), Y, Right) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 30.19/8.72 less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3)) 30.19/8.72 less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y)) 30.19/8.72 U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 30.19/8.72 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)) 30.19/8.72 delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 30.19/8.72 less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3)) 30.19/8.72 less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y)) 30.19/8.72 U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 30.19/8.72 The argument filtering Pi contains the following mapping: 30.19/8.72 delete_in_gaa(x1, x2, x3) = delete_in_gaa(x1) 30.19/8.72 30.19/8.72 delete_out_gaa(x1, x2, x3) = delete_out_gaa 30.19/8.72 30.19/8.72 U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x6) 30.19/8.72 30.19/8.72 delmin_in_aaa(x1, x2, x3) = delmin_in_aaa 30.19/8.72 30.19/8.72 delmin_out_aaa(x1, x2, x3) = delmin_out_aaa 30.19/8.72 30.19/8.72 U6_aaa(x1, x2, x3, x4, x5, x6, x7) = U6_aaa(x7) 30.19/8.72 30.19/8.72 U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6) 30.19/8.72 30.19/8.72 less_in_ga(x1, x2) = less_in_ga(x1) 30.19/8.72 30.19/8.72 0 = 0 30.19/8.72 30.19/8.72 less_out_ga(x1, x2) = less_out_ga 30.19/8.72 30.19/8.72 s(x1) = s(x1) 30.19/8.72 30.19/8.72 U7_ga(x1, x2, x3) = U7_ga(x3) 30.19/8.72 30.19/8.72 U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x6) 30.19/8.72 30.19/8.72 U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6) 30.19/8.72 30.19/8.72 less_in_ag(x1, x2) = less_in_ag(x2) 30.19/8.72 30.19/8.72 less_out_ag(x1, x2) = less_out_ag(x1) 30.19/8.72 30.19/8.72 U7_ag(x1, x2, x3) = U7_ag(x3) 30.19/8.72 30.19/8.72 U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x6) 30.19/8.72 30.19/8.72 DELETE_IN_GAA(x1, x2, x3) = DELETE_IN_GAA'(x1) 30.19/8.72 30.19/8.72 U1_GAA(x1, x2, x3, x4, x5, x6) = U1_GAA(x6) 30.19/8.72 30.19/8.72 DELMIN_IN_AAA(x1, x2, x3) = DELMIN_IN_AAA 30.19/8.72 30.19/8.72 U6_AAA(x1, x2, x3, x4, x5, x6, x7) = U6_AAA(x7) 30.19/8.72 30.19/8.72 U2_GAA(x1, x2, x3, x4, x5, x6) = U2_GAA(x1, x6) 30.19/8.72 30.19/8.72 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 30.19/8.72 30.19/8.72 U7_GA(x1, x2, x3) = U7_GA(x3) 30.19/8.72 30.19/8.72 U3_GAA(x1, x2, x3, x4, x5, x6) = U3_GAA(x6) 30.19/8.72 30.19/8.72 U4_GAA(x1, x2, x3, x4, x5, x6) = U4_GAA(x1, x6) 30.19/8.72 30.19/8.72 LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) 30.19/8.72 30.19/8.72 U7_AG(x1, x2, x3) = U7_AG(x3) 30.19/8.72 30.19/8.72 U5_GAA(x1, x2, x3, x4, x5, x6) = U5_GAA(x6) 30.19/8.72 30.19/8.72 30.19/8.72 We have to consider all (P,R,Pi)-chains 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (5) DependencyGraphProof (EQUIVALENT) 30.19/8.72 The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 9 less nodes. 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (6) 30.19/8.72 Complex Obligation (AND) 30.19/8.72 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (7) 30.19/8.72 Obligation: 30.19/8.72 Pi DP problem: 30.19/8.72 The TRS P consists of the following rules: 30.19/8.72 30.19/8.72 LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) 30.19/8.72 30.19/8.72 The TRS R consists of the following rules: 30.19/8.72 30.19/8.72 delete_in_gaa(X, tree(X, void, Right), Right) -> delete_out_gaa(X, tree(X, void, Right), Right) 30.19/8.72 delete_in_gaa(X, tree(X, Left, void), Left) -> delete_out_gaa(X, tree(X, Left, void), Left) 30.19/8.72 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)) 30.19/8.72 delmin_in_aaa(tree(Y, void, Right), Y, Right) -> delmin_out_aaa(tree(Y, void, Right), Y, Right) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 30.19/8.72 less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3)) 30.19/8.72 less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y)) 30.19/8.72 U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 30.19/8.72 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)) 30.19/8.72 delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 30.19/8.72 less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3)) 30.19/8.72 less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y)) 30.19/8.72 U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 30.19/8.72 The argument filtering Pi contains the following mapping: 30.19/8.72 delete_in_gaa(x1, x2, x3) = delete_in_gaa(x1) 30.19/8.72 30.19/8.72 delete_out_gaa(x1, x2, x3) = delete_out_gaa 30.19/8.72 30.19/8.72 U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x6) 30.19/8.72 30.19/8.72 delmin_in_aaa(x1, x2, x3) = delmin_in_aaa 30.19/8.72 30.19/8.72 delmin_out_aaa(x1, x2, x3) = delmin_out_aaa 30.19/8.72 30.19/8.72 U6_aaa(x1, x2, x3, x4, x5, x6, x7) = U6_aaa(x7) 30.19/8.72 30.19/8.72 U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6) 30.19/8.72 30.19/8.72 less_in_ga(x1, x2) = less_in_ga(x1) 30.19/8.72 30.19/8.72 0 = 0 30.19/8.72 30.19/8.72 less_out_ga(x1, x2) = less_out_ga 30.19/8.72 30.19/8.72 s(x1) = s(x1) 30.19/8.72 30.19/8.72 U7_ga(x1, x2, x3) = U7_ga(x3) 30.19/8.72 30.19/8.72 U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x6) 30.19/8.72 30.19/8.72 U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6) 30.19/8.72 30.19/8.72 less_in_ag(x1, x2) = less_in_ag(x2) 30.19/8.72 30.19/8.72 less_out_ag(x1, x2) = less_out_ag(x1) 30.19/8.72 30.19/8.72 U7_ag(x1, x2, x3) = U7_ag(x3) 30.19/8.72 30.19/8.72 U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x6) 30.19/8.72 30.19/8.72 LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) 30.19/8.72 30.19/8.72 30.19/8.72 We have to consider all (P,R,Pi)-chains 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (8) UsableRulesProof (EQUIVALENT) 30.19/8.72 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (9) 30.19/8.72 Obligation: 30.19/8.72 Pi DP problem: 30.19/8.72 The TRS P consists of the following rules: 30.19/8.72 30.19/8.72 LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) 30.19/8.72 30.19/8.72 R is empty. 30.19/8.72 The argument filtering Pi contains the following mapping: 30.19/8.72 s(x1) = s(x1) 30.19/8.72 30.19/8.72 LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) 30.19/8.72 30.19/8.72 30.19/8.72 We have to consider all (P,R,Pi)-chains 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (10) PiDPToQDPProof (SOUND) 30.19/8.72 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (11) 30.19/8.72 Obligation: 30.19/8.72 Q DP problem: 30.19/8.72 The TRS P consists of the following rules: 30.19/8.72 30.19/8.72 LESS_IN_AG(s(Y)) -> LESS_IN_AG(Y) 30.19/8.72 30.19/8.72 R is empty. 30.19/8.72 Q is empty. 30.19/8.72 We have to consider all (P,Q,R)-chains. 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (12) QDPSizeChangeProof (EQUIVALENT) 30.19/8.72 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. 30.19/8.72 30.19/8.72 From the DPs we obtained the following set of size-change graphs: 30.19/8.72 *LESS_IN_AG(s(Y)) -> LESS_IN_AG(Y) 30.19/8.72 The graph contains the following edges 1 > 1 30.19/8.72 30.19/8.72 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (13) 30.19/8.72 YES 30.19/8.72 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (14) 30.19/8.72 Obligation: 30.19/8.72 Pi DP problem: 30.19/8.72 The TRS P consists of the following rules: 30.19/8.72 30.19/8.72 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 30.19/8.72 30.19/8.72 The TRS R consists of the following rules: 30.19/8.72 30.19/8.72 delete_in_gaa(X, tree(X, void, Right), Right) -> delete_out_gaa(X, tree(X, void, Right), Right) 30.19/8.72 delete_in_gaa(X, tree(X, Left, void), Left) -> delete_out_gaa(X, tree(X, Left, void), Left) 30.19/8.72 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)) 30.19/8.72 delmin_in_aaa(tree(Y, void, Right), Y, Right) -> delmin_out_aaa(tree(Y, void, Right), Y, Right) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 30.19/8.72 less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3)) 30.19/8.72 less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y)) 30.19/8.72 U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 30.19/8.72 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)) 30.19/8.72 delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 30.19/8.72 less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3)) 30.19/8.72 less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y)) 30.19/8.72 U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 30.19/8.72 The argument filtering Pi contains the following mapping: 30.19/8.72 delete_in_gaa(x1, x2, x3) = delete_in_gaa(x1) 30.19/8.72 30.19/8.72 delete_out_gaa(x1, x2, x3) = delete_out_gaa 30.19/8.72 30.19/8.72 U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x6) 30.19/8.72 30.19/8.72 delmin_in_aaa(x1, x2, x3) = delmin_in_aaa 30.19/8.72 30.19/8.72 delmin_out_aaa(x1, x2, x3) = delmin_out_aaa 30.19/8.72 30.19/8.72 U6_aaa(x1, x2, x3, x4, x5, x6, x7) = U6_aaa(x7) 30.19/8.72 30.19/8.72 U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6) 30.19/8.72 30.19/8.72 less_in_ga(x1, x2) = less_in_ga(x1) 30.19/8.72 30.19/8.72 0 = 0 30.19/8.72 30.19/8.72 less_out_ga(x1, x2) = less_out_ga 30.19/8.72 30.19/8.72 s(x1) = s(x1) 30.19/8.72 30.19/8.72 U7_ga(x1, x2, x3) = U7_ga(x3) 30.19/8.72 30.19/8.72 U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x6) 30.19/8.72 30.19/8.72 U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6) 30.19/8.72 30.19/8.72 less_in_ag(x1, x2) = less_in_ag(x2) 30.19/8.72 30.19/8.72 less_out_ag(x1, x2) = less_out_ag(x1) 30.19/8.72 30.19/8.72 U7_ag(x1, x2, x3) = U7_ag(x3) 30.19/8.72 30.19/8.72 U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x6) 30.19/8.72 30.19/8.72 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 30.19/8.72 30.19/8.72 30.19/8.72 We have to consider all (P,R,Pi)-chains 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (15) UsableRulesProof (EQUIVALENT) 30.19/8.72 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (16) 30.19/8.72 Obligation: 30.19/8.72 Pi DP problem: 30.19/8.72 The TRS P consists of the following rules: 30.19/8.72 30.19/8.72 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 30.19/8.72 30.19/8.72 R is empty. 30.19/8.72 The argument filtering Pi contains the following mapping: 30.19/8.72 s(x1) = s(x1) 30.19/8.72 30.19/8.72 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 30.19/8.72 30.19/8.72 30.19/8.72 We have to consider all (P,R,Pi)-chains 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (17) PiDPToQDPProof (SOUND) 30.19/8.72 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (18) 30.19/8.72 Obligation: 30.19/8.72 Q DP problem: 30.19/8.72 The TRS P consists of the following rules: 30.19/8.72 30.19/8.72 LESS_IN_GA(s(X)) -> LESS_IN_GA(X) 30.19/8.72 30.19/8.72 R is empty. 30.19/8.72 Q is empty. 30.19/8.72 We have to consider all (P,Q,R)-chains. 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (19) QDPSizeChangeProof (EQUIVALENT) 30.19/8.72 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. 30.19/8.72 30.19/8.72 From the DPs we obtained the following set of size-change graphs: 30.19/8.72 *LESS_IN_GA(s(X)) -> LESS_IN_GA(X) 30.19/8.72 The graph contains the following edges 1 > 1 30.19/8.72 30.19/8.72 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (20) 30.19/8.72 YES 30.19/8.72 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (21) 30.19/8.72 Obligation: 30.19/8.72 Pi DP problem: 30.19/8.72 The TRS P consists of the following rules: 30.19/8.72 30.19/8.72 DELMIN_IN_AAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> DELMIN_IN_AAA(Left, Y, Left1) 30.19/8.72 30.19/8.72 The TRS R consists of the following rules: 30.19/8.72 30.19/8.72 delete_in_gaa(X, tree(X, void, Right), Right) -> delete_out_gaa(X, tree(X, void, Right), Right) 30.19/8.72 delete_in_gaa(X, tree(X, Left, void), Left) -> delete_out_gaa(X, tree(X, Left, void), Left) 30.19/8.72 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)) 30.19/8.72 delmin_in_aaa(tree(Y, void, Right), Y, Right) -> delmin_out_aaa(tree(Y, void, Right), Y, Right) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 30.19/8.72 less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3)) 30.19/8.72 less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y)) 30.19/8.72 U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 30.19/8.72 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)) 30.19/8.72 delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 30.19/8.72 less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3)) 30.19/8.72 less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y)) 30.19/8.72 U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 30.19/8.72 The argument filtering Pi contains the following mapping: 30.19/8.72 delete_in_gaa(x1, x2, x3) = delete_in_gaa(x1) 30.19/8.72 30.19/8.72 delete_out_gaa(x1, x2, x3) = delete_out_gaa 30.19/8.72 30.19/8.72 U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x6) 30.19/8.72 30.19/8.72 delmin_in_aaa(x1, x2, x3) = delmin_in_aaa 30.19/8.72 30.19/8.72 delmin_out_aaa(x1, x2, x3) = delmin_out_aaa 30.19/8.72 30.19/8.72 U6_aaa(x1, x2, x3, x4, x5, x6, x7) = U6_aaa(x7) 30.19/8.72 30.19/8.72 U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6) 30.19/8.72 30.19/8.72 less_in_ga(x1, x2) = less_in_ga(x1) 30.19/8.72 30.19/8.72 0 = 0 30.19/8.72 30.19/8.72 less_out_ga(x1, x2) = less_out_ga 30.19/8.72 30.19/8.72 s(x1) = s(x1) 30.19/8.72 30.19/8.72 U7_ga(x1, x2, x3) = U7_ga(x3) 30.19/8.72 30.19/8.72 U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x6) 30.19/8.72 30.19/8.72 U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6) 30.19/8.72 30.19/8.72 less_in_ag(x1, x2) = less_in_ag(x2) 30.19/8.72 30.19/8.72 less_out_ag(x1, x2) = less_out_ag(x1) 30.19/8.72 30.19/8.72 U7_ag(x1, x2, x3) = U7_ag(x3) 30.19/8.72 30.19/8.72 U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x6) 30.19/8.72 30.19/8.72 DELMIN_IN_AAA(x1, x2, x3) = DELMIN_IN_AAA 30.19/8.72 30.19/8.72 30.19/8.72 We have to consider all (P,R,Pi)-chains 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (22) UsableRulesProof (EQUIVALENT) 30.19/8.72 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (23) 30.19/8.72 Obligation: 30.19/8.72 Pi DP problem: 30.19/8.72 The TRS P consists of the following rules: 30.19/8.72 30.19/8.72 DELMIN_IN_AAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> DELMIN_IN_AAA(Left, Y, Left1) 30.19/8.72 30.19/8.72 R is empty. 30.19/8.72 The argument filtering Pi contains the following mapping: 30.19/8.72 DELMIN_IN_AAA(x1, x2, x3) = DELMIN_IN_AAA 30.19/8.72 30.19/8.72 30.19/8.72 We have to consider all (P,R,Pi)-chains 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (24) PiDPToQDPProof (SOUND) 30.19/8.72 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (25) 30.19/8.72 Obligation: 30.19/8.72 Q DP problem: 30.19/8.72 The TRS P consists of the following rules: 30.19/8.72 30.19/8.72 DELMIN_IN_AAA -> DELMIN_IN_AAA 30.19/8.72 30.19/8.72 R is empty. 30.19/8.72 Q is empty. 30.19/8.72 We have to consider all (P,Q,R)-chains. 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (26) NonTerminationLoopProof (COMPLETE) 30.19/8.72 We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. 30.19/8.72 Found a loop by semiunifying a rule from P directly. 30.19/8.72 30.19/8.72 s = DELMIN_IN_AAA evaluates to t =DELMIN_IN_AAA 30.19/8.72 30.19/8.72 Thus s starts an infinite chain as s semiunifies with t with the following substitutions: 30.19/8.72 * Matcher: [ ] 30.19/8.72 * Semiunifier: [ ] 30.19/8.72 30.19/8.72 -------------------------------------------------------------------------------- 30.19/8.72 Rewriting sequence 30.19/8.72 30.19/8.72 The DP semiunifies directly so there is only one rewrite step from DELMIN_IN_AAA to DELMIN_IN_AAA. 30.19/8.72 30.19/8.72 30.19/8.72 30.19/8.72 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (27) 30.19/8.72 NO 30.19/8.72 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (28) 30.19/8.72 Obligation: 30.19/8.72 Pi DP problem: 30.19/8.72 The TRS P consists of the following rules: 30.19/8.72 30.19/8.72 DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_GAA(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 30.19/8.72 U2_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> DELETE_IN_GAA(X, Left, Left1) 30.19/8.72 DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_GAA(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 30.19/8.72 U4_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> DELETE_IN_GAA(X, Right, Right1) 30.19/8.72 30.19/8.72 The TRS R consists of the following rules: 30.19/8.72 30.19/8.72 delete_in_gaa(X, tree(X, void, Right), Right) -> delete_out_gaa(X, tree(X, void, Right), Right) 30.19/8.72 delete_in_gaa(X, tree(X, Left, void), Left) -> delete_out_gaa(X, tree(X, Left, void), Left) 30.19/8.72 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)) 30.19/8.72 delmin_in_aaa(tree(Y, void, Right), Y, Right) -> delmin_out_aaa(tree(Y, void, Right), Y, Right) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 30.19/8.72 less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3)) 30.19/8.72 less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y)) 30.19/8.72 U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 30.19/8.72 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)) 30.19/8.72 delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 30.19/8.72 less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3)) 30.19/8.72 less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y)) 30.19/8.72 U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 30.19/8.72 The argument filtering Pi contains the following mapping: 30.19/8.72 delete_in_gaa(x1, x2, x3) = delete_in_gaa(x1) 30.19/8.72 30.19/8.72 delete_out_gaa(x1, x2, x3) = delete_out_gaa 30.19/8.72 30.19/8.72 U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x6) 30.19/8.72 30.19/8.72 delmin_in_aaa(x1, x2, x3) = delmin_in_aaa 30.19/8.72 30.19/8.72 delmin_out_aaa(x1, x2, x3) = delmin_out_aaa 30.19/8.72 30.19/8.72 U6_aaa(x1, x2, x3, x4, x5, x6, x7) = U6_aaa(x7) 30.19/8.72 30.19/8.72 U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6) 30.19/8.72 30.19/8.72 less_in_ga(x1, x2) = less_in_ga(x1) 30.19/8.72 30.19/8.72 0 = 0 30.19/8.72 30.19/8.72 less_out_ga(x1, x2) = less_out_ga 30.19/8.72 30.19/8.72 s(x1) = s(x1) 30.19/8.72 30.19/8.72 U7_ga(x1, x2, x3) = U7_ga(x3) 30.19/8.72 30.19/8.72 U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x6) 30.19/8.72 30.19/8.72 U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6) 30.19/8.72 30.19/8.72 less_in_ag(x1, x2) = less_in_ag(x2) 30.19/8.72 30.19/8.72 less_out_ag(x1, x2) = less_out_ag(x1) 30.19/8.72 30.19/8.72 U7_ag(x1, x2, x3) = U7_ag(x3) 30.19/8.72 30.19/8.72 U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x6) 30.19/8.72 30.19/8.72 DELETE_IN_GAA(x1, x2, x3) = DELETE_IN_GAA(x1) 30.19/8.72 30.19/8.72 U2_GAA(x1, x2, x3, x4, x5, x6) = U2_GAA(x1, x6) 30.19/8.72 30.19/8.72 U4_GAA(x1, x2, x3, x4, x5, x6) = U4_GAA(x1, x6) 30.19/8.72 30.19/8.72 30.19/8.72 We have to consider all (P,R,Pi)-chains 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (29) UsableRulesProof (EQUIVALENT) 30.19/8.72 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (30) 30.19/8.72 Obligation: 30.19/8.72 Pi DP problem: 30.19/8.72 The TRS P consists of the following rules: 30.19/8.72 30.19/8.72 DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_GAA(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 30.19/8.72 U2_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> DELETE_IN_GAA(X, Left, Left1) 30.19/8.72 DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_GAA(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 30.19/8.72 U4_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> DELETE_IN_GAA(X, Right, Right1) 30.19/8.72 30.19/8.72 The TRS R consists of the following rules: 30.19/8.72 30.19/8.72 less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3)) 30.19/8.72 less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y)) 30.19/8.72 less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3)) 30.19/8.72 less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y)) 30.19/8.72 U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 30.19/8.72 U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 30.19/8.72 30.19/8.72 The argument filtering Pi contains the following mapping: 30.19/8.72 less_in_ga(x1, x2) = less_in_ga(x1) 30.19/8.72 30.19/8.72 0 = 0 30.19/8.72 30.19/8.72 less_out_ga(x1, x2) = less_out_ga 30.19/8.72 30.19/8.72 s(x1) = s(x1) 30.19/8.72 30.19/8.72 U7_ga(x1, x2, x3) = U7_ga(x3) 30.19/8.72 30.19/8.72 less_in_ag(x1, x2) = less_in_ag(x2) 30.19/8.72 30.19/8.72 less_out_ag(x1, x2) = less_out_ag(x1) 30.19/8.72 30.19/8.72 U7_ag(x1, x2, x3) = U7_ag(x3) 30.19/8.72 30.19/8.72 DELETE_IN_GAA(x1, x2, x3) = DELETE_IN_GAA(x1) 30.19/8.72 30.19/8.72 U2_GAA(x1, x2, x3, x4, x5, x6) = U2_GAA(x1, x6) 30.19/8.72 30.19/8.72 U4_GAA(x1, x2, x3, x4, x5, x6) = U4_GAA(x1, x6) 30.19/8.72 30.19/8.72 30.19/8.72 We have to consider all (P,R,Pi)-chains 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (31) PiDPToQDPProof (SOUND) 30.19/8.72 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (32) 30.19/8.72 Obligation: 30.19/8.72 Q DP problem: 30.19/8.72 The TRS P consists of the following rules: 30.19/8.72 30.19/8.72 DELETE_IN_GAA(X) -> U2_GAA(X, less_in_ga(X)) 30.19/8.72 U2_GAA(X, less_out_ga) -> DELETE_IN_GAA(X) 30.19/8.72 DELETE_IN_GAA(X) -> U4_GAA(X, less_in_ag(X)) 30.19/8.72 U4_GAA(X, less_out_ag(Y)) -> DELETE_IN_GAA(X) 30.19/8.72 30.19/8.72 The TRS R consists of the following rules: 30.19/8.72 30.19/8.72 less_in_ga(0) -> less_out_ga 30.19/8.72 less_in_ga(s(X)) -> U7_ga(less_in_ga(X)) 30.19/8.72 less_in_ag(s(X3)) -> less_out_ag(0) 30.19/8.72 less_in_ag(s(Y)) -> U7_ag(less_in_ag(Y)) 30.19/8.72 U7_ga(less_out_ga) -> less_out_ga 30.19/8.72 U7_ag(less_out_ag(X)) -> less_out_ag(s(X)) 30.19/8.72 30.19/8.72 The set Q consists of the following terms: 30.19/8.72 30.19/8.72 less_in_ga(x0) 30.19/8.72 less_in_ag(x0) 30.19/8.72 U7_ga(x0) 30.19/8.72 U7_ag(x0) 30.19/8.72 30.19/8.72 We have to consider all (P,Q,R)-chains. 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (33) TransformationProof (SOUND) 30.19/8.72 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]: 30.19/8.72 30.19/8.72 (DELETE_IN_GAA(0) -> U2_GAA(0, less_out_ga),DELETE_IN_GAA(0) -> U2_GAA(0, less_out_ga)) 30.19/8.72 (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)))) 30.19/8.72 30.19/8.72 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (34) 30.19/8.72 Obligation: 30.19/8.72 Q DP problem: 30.19/8.72 The TRS P consists of the following rules: 30.19/8.72 30.19/8.72 U2_GAA(X, less_out_ga) -> DELETE_IN_GAA(X) 30.19/8.72 DELETE_IN_GAA(X) -> U4_GAA(X, less_in_ag(X)) 30.19/8.72 U4_GAA(X, less_out_ag(Y)) -> DELETE_IN_GAA(X) 30.19/8.72 DELETE_IN_GAA(0) -> U2_GAA(0, less_out_ga) 30.19/8.72 DELETE_IN_GAA(s(x0)) -> U2_GAA(s(x0), U7_ga(less_in_ga(x0))) 30.19/8.72 30.19/8.72 The TRS R consists of the following rules: 30.19/8.72 30.19/8.72 less_in_ga(0) -> less_out_ga 30.19/8.72 less_in_ga(s(X)) -> U7_ga(less_in_ga(X)) 30.19/8.72 less_in_ag(s(X3)) -> less_out_ag(0) 30.19/8.72 less_in_ag(s(Y)) -> U7_ag(less_in_ag(Y)) 30.19/8.72 U7_ga(less_out_ga) -> less_out_ga 30.19/8.72 U7_ag(less_out_ag(X)) -> less_out_ag(s(X)) 30.19/8.72 30.19/8.72 The set Q consists of the following terms: 30.19/8.72 30.19/8.72 less_in_ga(x0) 30.19/8.72 less_in_ag(x0) 30.19/8.72 U7_ga(x0) 30.19/8.72 U7_ag(x0) 30.19/8.72 30.19/8.72 We have to consider all (P,Q,R)-chains. 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (35) TransformationProof (SOUND) 30.19/8.72 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]: 30.19/8.72 30.19/8.72 (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))) 30.19/8.72 (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)))) 30.19/8.72 30.19/8.72 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (36) 30.19/8.72 Obligation: 30.19/8.72 Q DP problem: 30.19/8.72 The TRS P consists of the following rules: 30.19/8.72 30.19/8.72 U2_GAA(X, less_out_ga) -> DELETE_IN_GAA(X) 30.19/8.72 U4_GAA(X, less_out_ag(Y)) -> DELETE_IN_GAA(X) 30.19/8.72 DELETE_IN_GAA(0) -> U2_GAA(0, less_out_ga) 30.19/8.72 DELETE_IN_GAA(s(x0)) -> U2_GAA(s(x0), U7_ga(less_in_ga(x0))) 30.19/8.72 DELETE_IN_GAA(s(x0)) -> U4_GAA(s(x0), less_out_ag(0)) 30.19/8.72 DELETE_IN_GAA(s(x0)) -> U4_GAA(s(x0), U7_ag(less_in_ag(x0))) 30.19/8.72 30.19/8.72 The TRS R consists of the following rules: 30.19/8.72 30.19/8.72 less_in_ga(0) -> less_out_ga 30.19/8.72 less_in_ga(s(X)) -> U7_ga(less_in_ga(X)) 30.19/8.72 less_in_ag(s(X3)) -> less_out_ag(0) 30.19/8.72 less_in_ag(s(Y)) -> U7_ag(less_in_ag(Y)) 30.19/8.72 U7_ga(less_out_ga) -> less_out_ga 30.19/8.72 U7_ag(less_out_ag(X)) -> less_out_ag(s(X)) 30.19/8.72 30.19/8.72 The set Q consists of the following terms: 30.19/8.72 30.19/8.72 less_in_ga(x0) 30.19/8.72 less_in_ag(x0) 30.19/8.72 U7_ga(x0) 30.19/8.72 U7_ag(x0) 30.19/8.72 30.19/8.72 We have to consider all (P,Q,R)-chains. 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (37) TransformationProof (EQUIVALENT) 30.19/8.72 By instantiating [LPAR04] the rule U2_GAA(X, less_out_ga) -> DELETE_IN_GAA(X) we obtained the following new rules [LPAR04]: 30.19/8.72 30.19/8.72 (U2_GAA(0, less_out_ga) -> DELETE_IN_GAA(0),U2_GAA(0, less_out_ga) -> DELETE_IN_GAA(0)) 30.19/8.72 (U2_GAA(s(z0), less_out_ga) -> DELETE_IN_GAA(s(z0)),U2_GAA(s(z0), less_out_ga) -> DELETE_IN_GAA(s(z0))) 30.19/8.72 30.19/8.72 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (38) 30.19/8.72 Obligation: 30.19/8.72 Q DP problem: 30.19/8.72 The TRS P consists of the following rules: 30.19/8.72 30.19/8.72 U4_GAA(X, less_out_ag(Y)) -> DELETE_IN_GAA(X) 30.19/8.72 DELETE_IN_GAA(0) -> U2_GAA(0, less_out_ga) 30.19/8.72 DELETE_IN_GAA(s(x0)) -> U2_GAA(s(x0), U7_ga(less_in_ga(x0))) 30.19/8.72 DELETE_IN_GAA(s(x0)) -> U4_GAA(s(x0), less_out_ag(0)) 30.19/8.72 DELETE_IN_GAA(s(x0)) -> U4_GAA(s(x0), U7_ag(less_in_ag(x0))) 30.19/8.72 U2_GAA(0, less_out_ga) -> DELETE_IN_GAA(0) 30.19/8.72 U2_GAA(s(z0), less_out_ga) -> DELETE_IN_GAA(s(z0)) 30.19/8.72 30.19/8.72 The TRS R consists of the following rules: 30.19/8.72 30.19/8.72 less_in_ga(0) -> less_out_ga 30.19/8.72 less_in_ga(s(X)) -> U7_ga(less_in_ga(X)) 30.19/8.72 less_in_ag(s(X3)) -> less_out_ag(0) 30.19/8.72 less_in_ag(s(Y)) -> U7_ag(less_in_ag(Y)) 30.19/8.72 U7_ga(less_out_ga) -> less_out_ga 30.19/8.72 U7_ag(less_out_ag(X)) -> less_out_ag(s(X)) 30.19/8.72 30.19/8.72 The set Q consists of the following terms: 30.19/8.72 30.19/8.72 less_in_ga(x0) 30.19/8.72 less_in_ag(x0) 30.19/8.72 U7_ga(x0) 30.19/8.72 U7_ag(x0) 30.19/8.72 30.19/8.72 We have to consider all (P,Q,R)-chains. 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (39) DependencyGraphProof (EQUIVALENT) 30.19/8.72 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs. 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (40) 30.19/8.72 Complex Obligation (AND) 30.19/8.72 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (41) 30.19/8.72 Obligation: 30.19/8.72 Q DP problem: 30.19/8.72 The TRS P consists of the following rules: 30.19/8.72 30.19/8.72 U2_GAA(0, less_out_ga) -> DELETE_IN_GAA(0) 30.19/8.72 DELETE_IN_GAA(0) -> U2_GAA(0, less_out_ga) 30.19/8.72 30.19/8.72 The TRS R consists of the following rules: 30.19/8.72 30.19/8.72 less_in_ga(0) -> less_out_ga 30.19/8.72 less_in_ga(s(X)) -> U7_ga(less_in_ga(X)) 30.19/8.72 less_in_ag(s(X3)) -> less_out_ag(0) 30.19/8.72 less_in_ag(s(Y)) -> U7_ag(less_in_ag(Y)) 30.19/8.72 U7_ga(less_out_ga) -> less_out_ga 30.19/8.72 U7_ag(less_out_ag(X)) -> less_out_ag(s(X)) 30.19/8.72 30.19/8.72 The set Q consists of the following terms: 30.19/8.72 30.19/8.72 less_in_ga(x0) 30.19/8.72 less_in_ag(x0) 30.19/8.72 U7_ga(x0) 30.19/8.72 U7_ag(x0) 30.19/8.72 30.19/8.72 We have to consider all (P,Q,R)-chains. 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (42) UsableRulesProof (EQUIVALENT) 30.19/8.72 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. 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (43) 30.19/8.72 Obligation: 30.19/8.72 Q DP problem: 30.19/8.72 The TRS P consists of the following rules: 30.19/8.72 30.19/8.72 U2_GAA(0, less_out_ga) -> DELETE_IN_GAA(0) 30.19/8.72 DELETE_IN_GAA(0) -> U2_GAA(0, less_out_ga) 30.19/8.72 30.19/8.72 R is empty. 30.19/8.72 The set Q consists of the following terms: 30.19/8.72 30.19/8.72 less_in_ga(x0) 30.19/8.72 less_in_ag(x0) 30.19/8.72 U7_ga(x0) 30.19/8.72 U7_ag(x0) 30.19/8.72 30.19/8.72 We have to consider all (P,Q,R)-chains. 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (44) QReductionProof (EQUIVALENT) 30.19/8.72 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 30.19/8.72 30.19/8.72 less_in_ga(x0) 30.19/8.72 less_in_ag(x0) 30.19/8.72 U7_ga(x0) 30.19/8.72 U7_ag(x0) 30.19/8.72 30.19/8.72 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (45) 30.19/8.72 Obligation: 30.19/8.72 Q DP problem: 30.19/8.72 The TRS P consists of the following rules: 30.19/8.72 30.19/8.72 U2_GAA(0, less_out_ga) -> DELETE_IN_GAA(0) 30.19/8.72 DELETE_IN_GAA(0) -> U2_GAA(0, less_out_ga) 30.19/8.72 30.19/8.72 R is empty. 30.19/8.72 Q is empty. 30.19/8.72 We have to consider all (P,Q,R)-chains. 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (46) NonTerminationLoopProof (COMPLETE) 30.19/8.72 We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. 30.19/8.72 Found a loop by narrowing to the left: 30.19/8.72 30.19/8.72 s = DELETE_IN_GAA(0) evaluates to t =DELETE_IN_GAA(0) 30.19/8.72 30.19/8.72 Thus s starts an infinite chain as s semiunifies with t with the following substitutions: 30.19/8.72 * Matcher: [ ] 30.19/8.72 * Semiunifier: [ ] 30.19/8.72 30.19/8.72 -------------------------------------------------------------------------------- 30.19/8.72 Rewriting sequence 30.19/8.72 30.19/8.72 DELETE_IN_GAA(0) -> U2_GAA(0, less_out_ga) 30.19/8.72 with rule DELETE_IN_GAA(0) -> U2_GAA(0, less_out_ga) at position [] and matcher [ ] 30.19/8.72 30.19/8.72 U2_GAA(0, less_out_ga) -> DELETE_IN_GAA(0) 30.19/8.72 with rule U2_GAA(0, less_out_ga) -> DELETE_IN_GAA(0) 30.19/8.72 30.19/8.72 Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence 30.19/8.72 30.19/8.72 30.19/8.72 All these steps are and every following step will be a correct step w.r.t to Q. 30.19/8.72 30.19/8.72 30.19/8.72 30.19/8.72 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (47) 30.19/8.72 NO 30.19/8.72 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (48) 30.19/8.72 Obligation: 30.19/8.72 Q DP problem: 30.19/8.72 The TRS P consists of the following rules: 30.19/8.72 30.19/8.72 DELETE_IN_GAA(s(x0)) -> U2_GAA(s(x0), U7_ga(less_in_ga(x0))) 30.19/8.72 U2_GAA(s(z0), less_out_ga) -> DELETE_IN_GAA(s(z0)) 30.19/8.72 DELETE_IN_GAA(s(x0)) -> U4_GAA(s(x0), less_out_ag(0)) 30.19/8.72 U4_GAA(X, less_out_ag(Y)) -> DELETE_IN_GAA(X) 30.19/8.72 DELETE_IN_GAA(s(x0)) -> U4_GAA(s(x0), U7_ag(less_in_ag(x0))) 30.19/8.72 30.19/8.72 The TRS R consists of the following rules: 30.19/8.72 30.19/8.72 less_in_ga(0) -> less_out_ga 30.19/8.72 less_in_ga(s(X)) -> U7_ga(less_in_ga(X)) 30.19/8.72 less_in_ag(s(X3)) -> less_out_ag(0) 30.19/8.72 less_in_ag(s(Y)) -> U7_ag(less_in_ag(Y)) 30.19/8.72 U7_ga(less_out_ga) -> less_out_ga 30.19/8.72 U7_ag(less_out_ag(X)) -> less_out_ag(s(X)) 30.19/8.72 30.19/8.72 The set Q consists of the following terms: 30.19/8.72 30.19/8.72 less_in_ga(x0) 30.19/8.72 less_in_ag(x0) 30.19/8.72 U7_ga(x0) 30.19/8.72 U7_ag(x0) 30.19/8.72 30.19/8.72 We have to consider all (P,Q,R)-chains. 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (49) TransformationProof (EQUIVALENT) 30.19/8.72 By instantiating [LPAR04] the rule U4_GAA(X, less_out_ag(Y)) -> DELETE_IN_GAA(X) we obtained the following new rules [LPAR04]: 30.19/8.72 30.19/8.72 (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))) 30.19/8.72 (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))) 30.19/8.72 30.19/8.72 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (50) 30.19/8.72 Obligation: 30.19/8.72 Q DP problem: 30.19/8.72 The TRS P consists of the following rules: 30.19/8.72 30.19/8.72 DELETE_IN_GAA(s(x0)) -> U2_GAA(s(x0), U7_ga(less_in_ga(x0))) 30.19/8.72 U2_GAA(s(z0), less_out_ga) -> DELETE_IN_GAA(s(z0)) 30.19/8.72 DELETE_IN_GAA(s(x0)) -> U4_GAA(s(x0), less_out_ag(0)) 30.19/8.72 DELETE_IN_GAA(s(x0)) -> U4_GAA(s(x0), U7_ag(less_in_ag(x0))) 30.19/8.72 U4_GAA(s(z0), less_out_ag(0)) -> DELETE_IN_GAA(s(z0)) 30.19/8.72 U4_GAA(s(z0), less_out_ag(x1)) -> DELETE_IN_GAA(s(z0)) 30.19/8.72 30.19/8.72 The TRS R consists of the following rules: 30.19/8.72 30.19/8.72 less_in_ga(0) -> less_out_ga 30.19/8.72 less_in_ga(s(X)) -> U7_ga(less_in_ga(X)) 30.19/8.72 less_in_ag(s(X3)) -> less_out_ag(0) 30.19/8.72 less_in_ag(s(Y)) -> U7_ag(less_in_ag(Y)) 30.19/8.72 U7_ga(less_out_ga) -> less_out_ga 30.19/8.72 U7_ag(less_out_ag(X)) -> less_out_ag(s(X)) 30.19/8.72 30.19/8.72 The set Q consists of the following terms: 30.19/8.72 30.19/8.72 less_in_ga(x0) 30.19/8.72 less_in_ag(x0) 30.19/8.72 U7_ga(x0) 30.19/8.72 U7_ag(x0) 30.19/8.72 30.19/8.72 We have to consider all (P,Q,R)-chains. 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (51) PrologToPiTRSProof (SOUND) 30.19/8.72 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 30.19/8.72 30.19/8.72 delete_in_3: (b,f,f) 30.19/8.72 30.19/8.72 delmin_in_3: (f,f,f) 30.19/8.72 30.19/8.72 less_in_2: (b,f) (f,b) 30.19/8.72 30.19/8.72 Transforming Prolog into the following Term Rewriting System: 30.19/8.72 30.19/8.72 Pi-finite rewrite system: 30.19/8.72 The TRS R consists of the following rules: 30.19/8.72 30.19/8.72 delete_in_gaa(X, tree(X, void, Right), Right) -> delete_out_gaa(X, tree(X, void, Right), Right) 30.19/8.72 delete_in_gaa(X, tree(X, Left, void), Left) -> delete_out_gaa(X, tree(X, Left, void), Left) 30.19/8.72 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)) 30.19/8.72 delmin_in_aaa(tree(Y, void, Right), Y, Right) -> delmin_out_aaa(tree(Y, void, Right), Y, Right) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 30.19/8.72 less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3)) 30.19/8.72 less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y)) 30.19/8.72 U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 30.19/8.72 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)) 30.19/8.72 delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 30.19/8.72 less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3)) 30.19/8.72 less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y)) 30.19/8.72 U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 30.19/8.72 The argument filtering Pi contains the following mapping: 30.19/8.72 delete_in_gaa(x1, x2, x3) = delete_in_gaa(x1) 30.19/8.72 30.19/8.72 delete_out_gaa(x1, x2, x3) = delete_out_gaa(x1) 30.19/8.72 30.19/8.72 U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x1, x6) 30.19/8.72 30.19/8.72 delmin_in_aaa(x1, x2, x3) = delmin_in_aaa 30.19/8.72 30.19/8.72 delmin_out_aaa(x1, x2, x3) = delmin_out_aaa 30.19/8.72 30.19/8.72 U6_aaa(x1, x2, x3, x4, x5, x6, x7) = U6_aaa(x7) 30.19/8.72 30.19/8.72 U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6) 30.19/8.72 30.19/8.72 less_in_ga(x1, x2) = less_in_ga(x1) 30.19/8.72 30.19/8.72 0 = 0 30.19/8.72 30.19/8.72 less_out_ga(x1, x2) = less_out_ga(x1) 30.19/8.72 30.19/8.72 s(x1) = s(x1) 30.19/8.72 30.19/8.72 U7_ga(x1, x2, x3) = U7_ga(x1, x3) 30.19/8.72 30.19/8.72 U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x1, x6) 30.19/8.72 30.19/8.72 U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6) 30.19/8.72 30.19/8.72 less_in_ag(x1, x2) = less_in_ag(x2) 30.19/8.72 30.19/8.72 less_out_ag(x1, x2) = less_out_ag(x1, x2) 30.19/8.72 30.19/8.72 U7_ag(x1, x2, x3) = U7_ag(x2, x3) 30.19/8.72 30.19/8.72 U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x1, x6) 30.19/8.72 30.19/8.72 30.19/8.72 30.19/8.72 30.19/8.72 30.19/8.72 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 30.19/8.72 30.19/8.72 30.19/8.72 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (52) 30.19/8.72 Obligation: 30.19/8.72 Pi-finite rewrite system: 30.19/8.72 The TRS R consists of the following rules: 30.19/8.72 30.19/8.72 delete_in_gaa(X, tree(X, void, Right), Right) -> delete_out_gaa(X, tree(X, void, Right), Right) 30.19/8.72 delete_in_gaa(X, tree(X, Left, void), Left) -> delete_out_gaa(X, tree(X, Left, void), Left) 30.19/8.72 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)) 30.19/8.72 delmin_in_aaa(tree(Y, void, Right), Y, Right) -> delmin_out_aaa(tree(Y, void, Right), Y, Right) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 30.19/8.72 less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3)) 30.19/8.72 less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y)) 30.19/8.72 U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 30.19/8.72 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)) 30.19/8.72 delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 30.19/8.72 less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3)) 30.19/8.72 less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y)) 30.19/8.72 U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 30.19/8.72 The argument filtering Pi contains the following mapping: 30.19/8.72 delete_in_gaa(x1, x2, x3) = delete_in_gaa(x1) 30.19/8.72 30.19/8.72 delete_out_gaa(x1, x2, x3) = delete_out_gaa(x1) 30.19/8.72 30.19/8.72 U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x1, x6) 30.19/8.72 30.19/8.72 delmin_in_aaa(x1, x2, x3) = delmin_in_aaa 30.19/8.72 30.19/8.72 delmin_out_aaa(x1, x2, x3) = delmin_out_aaa 30.19/8.72 30.19/8.72 U6_aaa(x1, x2, x3, x4, x5, x6, x7) = U6_aaa(x7) 30.19/8.72 30.19/8.72 U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6) 30.19/8.72 30.19/8.72 less_in_ga(x1, x2) = less_in_ga(x1) 30.19/8.72 30.19/8.72 0 = 0 30.19/8.72 30.19/8.72 less_out_ga(x1, x2) = less_out_ga(x1) 30.19/8.72 30.19/8.72 s(x1) = s(x1) 30.19/8.72 30.19/8.72 U7_ga(x1, x2, x3) = U7_ga(x1, x3) 30.19/8.72 30.19/8.72 U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x1, x6) 30.19/8.72 30.19/8.72 U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6) 30.19/8.72 30.19/8.72 less_in_ag(x1, x2) = less_in_ag(x2) 30.19/8.72 30.19/8.72 less_out_ag(x1, x2) = less_out_ag(x1, x2) 30.19/8.72 30.19/8.72 U7_ag(x1, x2, x3) = U7_ag(x2, x3) 30.19/8.72 30.19/8.72 U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x1, x6) 30.19/8.72 30.19/8.72 30.19/8.72 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (53) DependencyPairsProof (EQUIVALENT) 30.19/8.72 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 30.19/8.72 Pi DP problem: 30.19/8.72 The TRS P consists of the following rules: 30.19/8.72 30.19/8.72 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)) 30.19/8.72 DELETE_IN_GAA(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> DELMIN_IN_AAA(Right, Y, Right1) 30.19/8.72 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)) 30.19/8.72 DELMIN_IN_AAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> DELMIN_IN_AAA(Left, Y, Left1) 30.19/8.72 DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_GAA(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 30.19/8.72 DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> LESS_IN_GA(X, Y) 30.19/8.72 LESS_IN_GA(s(X), s(Y)) -> U7_GA(X, Y, less_in_ga(X, Y)) 30.19/8.72 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 30.19/8.72 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)) 30.19/8.72 U2_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> DELETE_IN_GAA(X, Left, Left1) 30.19/8.72 DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_GAA(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 30.19/8.72 DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> LESS_IN_AG(Y, X) 30.19/8.72 LESS_IN_AG(s(X), s(Y)) -> U7_AG(X, Y, less_in_ag(X, Y)) 30.19/8.72 LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) 30.19/8.72 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)) 30.19/8.72 U4_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> DELETE_IN_GAA(X, Right, Right1) 30.19/8.72 30.19/8.72 The TRS R consists of the following rules: 30.19/8.72 30.19/8.72 delete_in_gaa(X, tree(X, void, Right), Right) -> delete_out_gaa(X, tree(X, void, Right), Right) 30.19/8.72 delete_in_gaa(X, tree(X, Left, void), Left) -> delete_out_gaa(X, tree(X, Left, void), Left) 30.19/8.72 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)) 30.19/8.72 delmin_in_aaa(tree(Y, void, Right), Y, Right) -> delmin_out_aaa(tree(Y, void, Right), Y, Right) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 30.19/8.72 less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3)) 30.19/8.72 less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y)) 30.19/8.72 U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 30.19/8.72 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)) 30.19/8.72 delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 30.19/8.72 less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3)) 30.19/8.72 less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y)) 30.19/8.72 U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 30.19/8.72 The argument filtering Pi contains the following mapping: 30.19/8.72 delete_in_gaa(x1, x2, x3) = delete_in_gaa(x1) 30.19/8.72 30.19/8.72 delete_out_gaa(x1, x2, x3) = delete_out_gaa(x1) 30.19/8.72 30.19/8.72 U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x1, x6) 30.19/8.72 30.19/8.72 delmin_in_aaa(x1, x2, x3) = delmin_in_aaa 30.19/8.72 30.19/8.72 delmin_out_aaa(x1, x2, x3) = delmin_out_aaa 30.19/8.72 30.19/8.72 U6_aaa(x1, x2, x3, x4, x5, x6, x7) = U6_aaa(x7) 30.19/8.72 30.19/8.72 U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6) 30.19/8.72 30.19/8.72 less_in_ga(x1, x2) = less_in_ga(x1) 30.19/8.72 30.19/8.72 0 = 0 30.19/8.72 30.19/8.72 less_out_ga(x1, x2) = less_out_ga(x1) 30.19/8.72 30.19/8.72 s(x1) = s(x1) 30.19/8.72 30.19/8.72 U7_ga(x1, x2, x3) = U7_ga(x1, x3) 30.19/8.72 30.19/8.72 U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x1, x6) 30.19/8.72 30.19/8.72 U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6) 30.19/8.72 30.19/8.72 less_in_ag(x1, x2) = less_in_ag(x2) 30.19/8.72 30.19/8.72 less_out_ag(x1, x2) = less_out_ag(x1, x2) 30.19/8.72 30.19/8.72 U7_ag(x1, x2, x3) = U7_ag(x2, x3) 30.19/8.72 30.19/8.72 U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x1, x6) 30.19/8.72 30.19/8.72 DELETE_IN_GAA(x1, x2, x3) = DELETE_IN_GAA(x1) 30.19/8.72 30.19/8.72 U1_GAA(x1, x2, x3, x4, x5, x6) = U1_GAA(x1, x6) 30.19/8.72 30.19/8.72 DELMIN_IN_AAA(x1, x2, x3) = DELMIN_IN_AAA 30.19/8.72 30.19/8.72 U6_AAA(x1, x2, x3, x4, x5, x6, x7) = U6_AAA(x7) 30.19/8.72 30.19/8.72 U2_GAA(x1, x2, x3, x4, x5, x6) = U2_GAA(x1, x6) 30.19/8.72 30.19/8.72 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 30.19/8.72 30.19/8.72 U7_GA(x1, x2, x3) = U7_GA(x1, x3) 30.19/8.72 30.19/8.72 U3_GAA(x1, x2, x3, x4, x5, x6) = U3_GAA(x1, x6) 30.19/8.72 30.19/8.72 U4_GAA(x1, x2, x3, x4, x5, x6) = U4_GAA(x1, x6) 30.19/8.72 30.19/8.72 LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) 30.19/8.72 30.19/8.72 U7_AG(x1, x2, x3) = U7_AG(x2, x3) 30.19/8.72 30.19/8.72 U5_GAA(x1, x2, x3, x4, x5, x6) = U5_GAA(x1, x6) 30.19/8.72 30.19/8.72 30.19/8.72 We have to consider all (P,R,Pi)-chains 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (54) 30.19/8.72 Obligation: 30.19/8.72 Pi DP problem: 30.19/8.72 The TRS P consists of the following rules: 30.19/8.72 30.19/8.72 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)) 30.19/8.72 DELETE_IN_GAA(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> DELMIN_IN_AAA(Right, Y, Right1) 30.19/8.72 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)) 30.19/8.72 DELMIN_IN_AAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> DELMIN_IN_AAA(Left, Y, Left1) 30.19/8.72 DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_GAA(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 30.19/8.72 DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> LESS_IN_GA(X, Y) 30.19/8.72 LESS_IN_GA(s(X), s(Y)) -> U7_GA(X, Y, less_in_ga(X, Y)) 30.19/8.72 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 30.19/8.72 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)) 30.19/8.72 U2_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> DELETE_IN_GAA(X, Left, Left1) 30.19/8.72 DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_GAA(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 30.19/8.72 DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> LESS_IN_AG(Y, X) 30.19/8.72 LESS_IN_AG(s(X), s(Y)) -> U7_AG(X, Y, less_in_ag(X, Y)) 30.19/8.72 LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) 30.19/8.72 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)) 30.19/8.72 U4_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> DELETE_IN_GAA(X, Right, Right1) 30.19/8.72 30.19/8.72 The TRS R consists of the following rules: 30.19/8.72 30.19/8.72 delete_in_gaa(X, tree(X, void, Right), Right) -> delete_out_gaa(X, tree(X, void, Right), Right) 30.19/8.72 delete_in_gaa(X, tree(X, Left, void), Left) -> delete_out_gaa(X, tree(X, Left, void), Left) 30.19/8.72 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)) 30.19/8.72 delmin_in_aaa(tree(Y, void, Right), Y, Right) -> delmin_out_aaa(tree(Y, void, Right), Y, Right) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 30.19/8.72 less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3)) 30.19/8.72 less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y)) 30.19/8.72 U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 30.19/8.72 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)) 30.19/8.72 delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 30.19/8.72 less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3)) 30.19/8.72 less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y)) 30.19/8.72 U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 30.19/8.72 The argument filtering Pi contains the following mapping: 30.19/8.72 delete_in_gaa(x1, x2, x3) = delete_in_gaa(x1) 30.19/8.72 30.19/8.72 delete_out_gaa(x1, x2, x3) = delete_out_gaa(x1) 30.19/8.72 30.19/8.72 U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x1, x6) 30.19/8.72 30.19/8.72 delmin_in_aaa(x1, x2, x3) = delmin_in_aaa 30.19/8.72 30.19/8.72 delmin_out_aaa(x1, x2, x3) = delmin_out_aaa 30.19/8.72 30.19/8.72 U6_aaa(x1, x2, x3, x4, x5, x6, x7) = U6_aaa(x7) 30.19/8.72 30.19/8.72 U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6) 30.19/8.72 30.19/8.72 less_in_ga(x1, x2) = less_in_ga(x1) 30.19/8.72 30.19/8.72 0 = 0 30.19/8.72 30.19/8.72 less_out_ga(x1, x2) = less_out_ga(x1) 30.19/8.72 30.19/8.72 s(x1) = s(x1) 30.19/8.72 30.19/8.72 U7_ga(x1, x2, x3) = U7_ga(x1, x3) 30.19/8.72 30.19/8.72 U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x1, x6) 30.19/8.72 30.19/8.72 U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6) 30.19/8.72 30.19/8.72 less_in_ag(x1, x2) = less_in_ag(x2) 30.19/8.72 30.19/8.72 less_out_ag(x1, x2) = less_out_ag(x1, x2) 30.19/8.72 30.19/8.72 U7_ag(x1, x2, x3) = U7_ag(x2, x3) 30.19/8.72 30.19/8.72 U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x1, x6) 30.19/8.72 30.19/8.72 DELETE_IN_GAA(x1, x2, x3) = DELETE_IN_GAA(x1) 30.19/8.72 30.19/8.72 U1_GAA(x1, x2, x3, x4, x5, x6) = U1_GAA(x1, x6) 30.19/8.72 30.19/8.72 DELMIN_IN_AAA(x1, x2, x3) = DELMIN_IN_AAA 30.19/8.72 30.19/8.72 U6_AAA(x1, x2, x3, x4, x5, x6, x7) = U6_AAA(x7) 30.19/8.72 30.19/8.72 U2_GAA(x1, x2, x3, x4, x5, x6) = U2_GAA(x1, x6) 30.19/8.72 30.19/8.72 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 30.19/8.72 30.19/8.72 U7_GA(x1, x2, x3) = U7_GA(x1, x3) 30.19/8.72 30.19/8.72 U3_GAA(x1, x2, x3, x4, x5, x6) = U3_GAA(x1, x6) 30.19/8.72 30.19/8.72 U4_GAA(x1, x2, x3, x4, x5, x6) = U4_GAA(x1, x6) 30.19/8.72 30.19/8.72 LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) 30.19/8.72 30.19/8.72 U7_AG(x1, x2, x3) = U7_AG(x2, x3) 30.19/8.72 30.19/8.72 U5_GAA(x1, x2, x3, x4, x5, x6) = U5_GAA(x1, x6) 30.19/8.72 30.19/8.72 30.19/8.72 We have to consider all (P,R,Pi)-chains 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (55) DependencyGraphProof (EQUIVALENT) 30.19/8.72 The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 9 less nodes. 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (56) 30.19/8.72 Complex Obligation (AND) 30.19/8.72 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (57) 30.19/8.72 Obligation: 30.19/8.72 Pi DP problem: 30.19/8.72 The TRS P consists of the following rules: 30.19/8.72 30.19/8.72 LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) 30.19/8.72 30.19/8.72 The TRS R consists of the following rules: 30.19/8.72 30.19/8.72 delete_in_gaa(X, tree(X, void, Right), Right) -> delete_out_gaa(X, tree(X, void, Right), Right) 30.19/8.72 delete_in_gaa(X, tree(X, Left, void), Left) -> delete_out_gaa(X, tree(X, Left, void), Left) 30.19/8.72 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)) 30.19/8.72 delmin_in_aaa(tree(Y, void, Right), Y, Right) -> delmin_out_aaa(tree(Y, void, Right), Y, Right) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 30.19/8.72 less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3)) 30.19/8.72 less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y)) 30.19/8.72 U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 30.19/8.72 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)) 30.19/8.72 delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 30.19/8.72 less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3)) 30.19/8.72 less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y)) 30.19/8.72 U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 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)) 30.19/8.72 30.19/8.72 The argument filtering Pi contains the following mapping: 30.19/8.72 delete_in_gaa(x1, x2, x3) = delete_in_gaa(x1) 30.19/8.72 30.19/8.72 delete_out_gaa(x1, x2, x3) = delete_out_gaa(x1) 30.19/8.72 30.19/8.72 U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x1, x6) 30.19/8.72 30.19/8.72 delmin_in_aaa(x1, x2, x3) = delmin_in_aaa 30.19/8.72 30.19/8.72 delmin_out_aaa(x1, x2, x3) = delmin_out_aaa 30.19/8.72 30.19/8.72 U6_aaa(x1, x2, x3, x4, x5, x6, x7) = U6_aaa(x7) 30.19/8.72 30.19/8.72 U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6) 30.19/8.72 30.19/8.72 less_in_ga(x1, x2) = less_in_ga(x1) 30.19/8.72 30.19/8.72 0 = 0 30.19/8.72 30.19/8.72 less_out_ga(x1, x2) = less_out_ga(x1) 30.19/8.72 30.19/8.72 s(x1) = s(x1) 30.19/8.72 30.19/8.72 U7_ga(x1, x2, x3) = U7_ga(x1, x3) 30.19/8.72 30.19/8.72 U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x1, x6) 30.19/8.72 30.19/8.72 U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6) 30.19/8.72 30.19/8.72 less_in_ag(x1, x2) = less_in_ag(x2) 30.19/8.72 30.19/8.72 less_out_ag(x1, x2) = less_out_ag(x1, x2) 30.19/8.72 30.19/8.72 U7_ag(x1, x2, x3) = U7_ag(x2, x3) 30.19/8.72 30.19/8.72 U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x1, x6) 30.19/8.72 30.19/8.72 LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) 30.19/8.72 30.19/8.72 30.19/8.72 We have to consider all (P,R,Pi)-chains 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (58) UsableRulesProof (EQUIVALENT) 30.19/8.72 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (59) 30.19/8.72 Obligation: 30.19/8.72 Pi DP problem: 30.19/8.72 The TRS P consists of the following rules: 30.19/8.72 30.19/8.72 LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) 30.19/8.72 30.19/8.72 R is empty. 30.19/8.72 The argument filtering Pi contains the following mapping: 30.19/8.72 s(x1) = s(x1) 30.19/8.72 30.19/8.72 LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) 30.19/8.72 30.19/8.72 30.19/8.72 We have to consider all (P,R,Pi)-chains 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (60) PiDPToQDPProof (SOUND) 30.19/8.72 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (61) 30.19/8.72 Obligation: 30.19/8.72 Q DP problem: 30.19/8.72 The TRS P consists of the following rules: 30.19/8.72 30.19/8.72 LESS_IN_AG(s(Y)) -> LESS_IN_AG(Y) 30.19/8.72 30.19/8.72 R is empty. 30.19/8.72 Q is empty. 30.19/8.72 We have to consider all (P,Q,R)-chains. 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (62) QDPSizeChangeProof (EQUIVALENT) 30.19/8.72 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. 30.19/8.72 30.19/8.72 From the DPs we obtained the following set of size-change graphs: 30.19/8.72 *LESS_IN_AG(s(Y)) -> LESS_IN_AG(Y) 30.19/8.72 The graph contains the following edges 1 > 1 30.19/8.72 30.19/8.72 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (63) 30.19/8.72 YES 30.19/8.72 30.19/8.72 ---------------------------------------- 30.19/8.72 30.19/8.72 (64) 30.19/8.72 Obligation: 30.19/8.72 Pi DP problem: 30.19/8.72 The TRS P consists of the following rules: 30.19/8.72 30.19/8.72 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 30.19/8.72 30.19/8.72 The TRS R consists of the following rules: 30.19/8.72 30.19/8.72 delete_in_gaa(X, tree(X, void, Right), Right) -> delete_out_gaa(X, tree(X, void, Right), Right) 30.19/8.72 delete_in_gaa(X, tree(X, Left, void), Left) -> delete_out_gaa(X, tree(X, Left, void), Left) 30.19/8.72 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)) 30.19/8.72 delmin_in_aaa(tree(Y, void, Right), Y, Right) -> delmin_out_aaa(tree(Y, void, Right), Y, Right) 30.19/8.72 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)) 30.43/8.75 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)) 30.43/8.75 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)) 30.43/8.75 delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 30.43/8.75 less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3)) 30.43/8.75 less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y)) 30.43/8.75 U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 30.43/8.75 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)) 30.43/8.75 delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 30.43/8.75 less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3)) 30.43/8.75 less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y)) 30.43/8.75 U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 30.43/8.75 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)) 30.43/8.75 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)) 30.43/8.75 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)) 30.43/8.75 30.43/8.75 The argument filtering Pi contains the following mapping: 30.43/8.75 delete_in_gaa(x1, x2, x3) = delete_in_gaa(x1) 30.43/8.75 30.43/8.75 delete_out_gaa(x1, x2, x3) = delete_out_gaa(x1) 30.43/8.75 30.43/8.75 U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x1, x6) 30.43/8.75 30.43/8.75 delmin_in_aaa(x1, x2, x3) = delmin_in_aaa 30.43/8.75 30.43/8.75 delmin_out_aaa(x1, x2, x3) = delmin_out_aaa 30.43/8.75 30.43/8.75 U6_aaa(x1, x2, x3, x4, x5, x6, x7) = U6_aaa(x7) 30.43/8.75 30.43/8.75 U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6) 30.43/8.75 30.43/8.75 less_in_ga(x1, x2) = less_in_ga(x1) 30.43/8.75 30.43/8.75 0 = 0 30.43/8.75 30.43/8.75 less_out_ga(x1, x2) = less_out_ga(x1) 30.43/8.75 30.43/8.75 s(x1) = s(x1) 30.43/8.75 30.43/8.75 U7_ga(x1, x2, x3) = U7_ga(x1, x3) 30.43/8.75 30.43/8.75 U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x1, x6) 30.43/8.75 30.43/8.75 U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6) 30.43/8.75 30.43/8.75 less_in_ag(x1, x2) = less_in_ag(x2) 30.43/8.75 30.43/8.75 less_out_ag(x1, x2) = less_out_ag(x1, x2) 30.43/8.75 30.43/8.75 U7_ag(x1, x2, x3) = U7_ag(x2, x3) 30.43/8.75 30.43/8.75 U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x1, x6) 30.43/8.75 30.43/8.75 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 30.43/8.75 30.43/8.75 30.43/8.75 We have to consider all (P,R,Pi)-chains 30.43/8.75 ---------------------------------------- 30.43/8.75 30.43/8.75 (65) UsableRulesProof (EQUIVALENT) 30.43/8.75 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 30.43/8.75 ---------------------------------------- 30.43/8.75 30.43/8.75 (66) 30.43/8.75 Obligation: 30.43/8.75 Pi DP problem: 30.43/8.75 The TRS P consists of the following rules: 30.43/8.75 30.43/8.75 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 30.43/8.75 30.43/8.75 R is empty. 30.43/8.75 The argument filtering Pi contains the following mapping: 30.43/8.75 s(x1) = s(x1) 30.43/8.75 30.43/8.75 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 30.43/8.75 30.43/8.75 30.43/8.75 We have to consider all (P,R,Pi)-chains 30.43/8.75 ---------------------------------------- 30.43/8.75 30.43/8.75 (67) PiDPToQDPProof (SOUND) 30.43/8.75 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 30.43/8.75 ---------------------------------------- 30.43/8.75 30.43/8.75 (68) 30.43/8.75 Obligation: 30.43/8.75 Q DP problem: 30.43/8.75 The TRS P consists of the following rules: 30.43/8.75 30.43/8.75 LESS_IN_GA(s(X)) -> LESS_IN_GA(X) 30.43/8.75 30.43/8.75 R is empty. 30.43/8.75 Q is empty. 30.43/8.75 We have to consider all (P,Q,R)-chains. 30.43/8.75 ---------------------------------------- 30.43/8.75 30.43/8.75 (69) QDPSizeChangeProof (EQUIVALENT) 30.43/8.75 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. 30.43/8.75 30.43/8.75 From the DPs we obtained the following set of size-change graphs: 30.43/8.75 *LESS_IN_GA(s(X)) -> LESS_IN_GA(X) 30.43/8.75 The graph contains the following edges 1 > 1 30.43/8.75 30.43/8.75 30.43/8.75 ---------------------------------------- 30.43/8.75 30.43/8.75 (70) 30.43/8.75 YES 30.43/8.75 30.43/8.75 ---------------------------------------- 30.43/8.75 30.43/8.75 (71) 30.43/8.75 Obligation: 30.43/8.75 Pi DP problem: 30.43/8.75 The TRS P consists of the following rules: 30.43/8.75 30.43/8.75 DELMIN_IN_AAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> DELMIN_IN_AAA(Left, Y, Left1) 30.43/8.75 30.43/8.75 The TRS R consists of the following rules: 30.43/8.75 30.43/8.75 delete_in_gaa(X, tree(X, void, Right), Right) -> delete_out_gaa(X, tree(X, void, Right), Right) 30.43/8.75 delete_in_gaa(X, tree(X, Left, void), Left) -> delete_out_gaa(X, tree(X, Left, void), Left) 30.43/8.75 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)) 30.43/8.75 delmin_in_aaa(tree(Y, void, Right), Y, Right) -> delmin_out_aaa(tree(Y, void, Right), Y, Right) 30.43/8.75 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)) 30.43/8.75 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)) 30.43/8.75 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)) 30.43/8.75 delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 30.43/8.75 less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3)) 30.43/8.75 less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y)) 30.43/8.75 U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 30.43/8.75 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)) 30.43/8.75 delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 30.43/8.75 less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3)) 30.43/8.75 less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y)) 30.43/8.75 U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 30.43/8.75 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)) 30.43/8.75 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)) 30.43/8.75 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)) 30.43/8.75 30.43/8.75 The argument filtering Pi contains the following mapping: 30.43/8.75 delete_in_gaa(x1, x2, x3) = delete_in_gaa(x1) 30.43/8.75 30.43/8.75 delete_out_gaa(x1, x2, x3) = delete_out_gaa(x1) 30.43/8.75 30.43/8.75 U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x1, x6) 30.43/8.75 30.43/8.75 delmin_in_aaa(x1, x2, x3) = delmin_in_aaa 30.43/8.75 30.43/8.75 delmin_out_aaa(x1, x2, x3) = delmin_out_aaa 30.43/8.75 30.43/8.75 U6_aaa(x1, x2, x3, x4, x5, x6, x7) = U6_aaa(x7) 30.43/8.75 30.43/8.75 U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6) 30.43/8.75 30.43/8.75 less_in_ga(x1, x2) = less_in_ga(x1) 30.43/8.75 30.43/8.75 0 = 0 30.43/8.75 30.43/8.75 less_out_ga(x1, x2) = less_out_ga(x1) 30.43/8.75 30.43/8.75 s(x1) = s(x1) 30.43/8.75 30.43/8.75 U7_ga(x1, x2, x3) = U7_ga(x1, x3) 30.43/8.75 30.43/8.75 U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x1, x6) 30.43/8.75 30.43/8.75 U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6) 30.43/8.75 30.43/8.75 less_in_ag(x1, x2) = less_in_ag(x2) 30.43/8.75 30.43/8.75 less_out_ag(x1, x2) = less_out_ag(x1, x2) 30.43/8.75 30.43/8.75 U7_ag(x1, x2, x3) = U7_ag(x2, x3) 30.43/8.75 30.43/8.75 U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x1, x6) 30.43/8.75 30.43/8.75 DELMIN_IN_AAA(x1, x2, x3) = DELMIN_IN_AAA 30.43/8.75 30.43/8.75 30.43/8.75 We have to consider all (P,R,Pi)-chains 30.43/8.75 ---------------------------------------- 30.43/8.75 30.43/8.75 (72) UsableRulesProof (EQUIVALENT) 30.43/8.75 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 30.43/8.75 ---------------------------------------- 30.43/8.75 30.43/8.75 (73) 30.43/8.75 Obligation: 30.43/8.75 Pi DP problem: 30.43/8.75 The TRS P consists of the following rules: 30.43/8.75 30.43/8.75 DELMIN_IN_AAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> DELMIN_IN_AAA(Left, Y, Left1) 30.43/8.75 30.43/8.75 R is empty. 30.43/8.75 The argument filtering Pi contains the following mapping: 30.43/8.75 DELMIN_IN_AAA(x1, x2, x3) = DELMIN_IN_AAA 30.43/8.75 30.43/8.75 30.43/8.75 We have to consider all (P,R,Pi)-chains 30.43/8.75 ---------------------------------------- 30.43/8.75 30.43/8.75 (74) PiDPToQDPProof (SOUND) 30.43/8.75 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 30.43/8.75 ---------------------------------------- 30.43/8.75 30.43/8.75 (75) 30.43/8.75 Obligation: 30.43/8.75 Q DP problem: 30.43/8.75 The TRS P consists of the following rules: 30.43/8.75 30.43/8.75 DELMIN_IN_AAA -> DELMIN_IN_AAA 30.43/8.75 30.43/8.75 R is empty. 30.43/8.75 Q is empty. 30.43/8.75 We have to consider all (P,Q,R)-chains. 30.43/8.75 ---------------------------------------- 30.43/8.75 30.43/8.75 (76) NonTerminationLoopProof (COMPLETE) 30.43/8.75 We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. 30.43/8.75 Found a loop by semiunifying a rule from P directly. 30.43/8.75 30.43/8.75 s = DELMIN_IN_AAA evaluates to t =DELMIN_IN_AAA 30.43/8.75 30.43/8.75 Thus s starts an infinite chain as s semiunifies with t with the following substitutions: 30.43/8.75 * Matcher: [ ] 30.43/8.75 * Semiunifier: [ ] 30.43/8.75 30.43/8.75 -------------------------------------------------------------------------------- 30.43/8.75 Rewriting sequence 30.43/8.75 30.43/8.75 The DP semiunifies directly so there is only one rewrite step from DELMIN_IN_AAA to DELMIN_IN_AAA. 30.43/8.75 30.43/8.75 30.43/8.75 30.43/8.75 30.43/8.75 ---------------------------------------- 30.43/8.75 30.43/8.75 (77) 30.43/8.75 NO 30.43/8.75 30.43/8.75 ---------------------------------------- 30.43/8.75 30.43/8.75 (78) 30.43/8.75 Obligation: 30.43/8.75 Pi DP problem: 30.43/8.75 The TRS P consists of the following rules: 30.43/8.75 30.43/8.75 DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_GAA(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 30.43/8.75 U2_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> DELETE_IN_GAA(X, Left, Left1) 30.43/8.75 DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_GAA(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 30.43/8.75 U4_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> DELETE_IN_GAA(X, Right, Right1) 30.43/8.75 30.43/8.75 The TRS R consists of the following rules: 30.43/8.75 30.43/8.75 delete_in_gaa(X, tree(X, void, Right), Right) -> delete_out_gaa(X, tree(X, void, Right), Right) 30.43/8.75 delete_in_gaa(X, tree(X, Left, void), Left) -> delete_out_gaa(X, tree(X, Left, void), Left) 30.43/8.75 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)) 30.43/8.75 delmin_in_aaa(tree(Y, void, Right), Y, Right) -> delmin_out_aaa(tree(Y, void, Right), Y, Right) 30.43/8.75 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)) 30.43/8.75 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)) 30.43/8.75 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)) 30.43/8.75 delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 30.43/8.75 less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3)) 30.43/8.75 less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y)) 30.43/8.75 U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 30.43/8.75 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)) 30.43/8.75 delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 30.43/8.75 less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3)) 30.43/8.75 less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y)) 30.43/8.75 U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 30.43/8.75 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)) 30.43/8.75 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)) 30.43/8.75 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)) 30.43/8.75 30.43/8.75 The argument filtering Pi contains the following mapping: 30.43/8.75 delete_in_gaa(x1, x2, x3) = delete_in_gaa(x1) 30.43/8.75 30.43/8.75 delete_out_gaa(x1, x2, x3) = delete_out_gaa(x1) 30.43/8.75 30.43/8.75 U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x1, x6) 30.43/8.75 30.43/8.75 delmin_in_aaa(x1, x2, x3) = delmin_in_aaa 30.43/8.75 30.43/8.75 delmin_out_aaa(x1, x2, x3) = delmin_out_aaa 30.43/8.75 30.43/8.75 U6_aaa(x1, x2, x3, x4, x5, x6, x7) = U6_aaa(x7) 30.43/8.75 30.43/8.75 U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6) 30.43/8.75 30.43/8.75 less_in_ga(x1, x2) = less_in_ga(x1) 30.43/8.75 30.43/8.75 0 = 0 30.43/8.75 30.43/8.75 less_out_ga(x1, x2) = less_out_ga(x1) 30.43/8.75 30.43/8.75 s(x1) = s(x1) 30.43/8.75 30.43/8.75 U7_ga(x1, x2, x3) = U7_ga(x1, x3) 30.43/8.75 30.43/8.75 U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x1, x6) 30.43/8.75 30.43/8.75 U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6) 30.43/8.75 30.43/8.75 less_in_ag(x1, x2) = less_in_ag(x2) 30.43/8.75 30.43/8.75 less_out_ag(x1, x2) = less_out_ag(x1, x2) 30.43/8.75 30.43/8.75 U7_ag(x1, x2, x3) = U7_ag(x2, x3) 30.43/8.75 30.43/8.75 U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x1, x6) 30.43/8.75 30.43/8.75 DELETE_IN_GAA(x1, x2, x3) = DELETE_IN_GAA(x1) 30.43/8.75 30.43/8.75 U2_GAA(x1, x2, x3, x4, x5, x6) = U2_GAA(x1, x6) 30.43/8.75 30.43/8.75 U4_GAA(x1, x2, x3, x4, x5, x6) = U4_GAA(x1, x6) 30.43/8.75 30.43/8.75 30.43/8.75 We have to consider all (P,R,Pi)-chains 30.43/8.75 ---------------------------------------- 30.43/8.75 30.43/8.75 (79) UsableRulesProof (EQUIVALENT) 30.43/8.75 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 30.43/8.75 ---------------------------------------- 30.43/8.75 30.43/8.75 (80) 30.43/8.75 Obligation: 30.43/8.75 Pi DP problem: 30.43/8.75 The TRS P consists of the following rules: 30.43/8.75 30.43/8.75 DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_GAA(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 30.43/8.75 U2_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> DELETE_IN_GAA(X, Left, Left1) 30.43/8.75 DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_GAA(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 30.43/8.75 U4_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> DELETE_IN_GAA(X, Right, Right1) 30.43/8.75 30.43/8.75 The TRS R consists of the following rules: 30.43/8.75 30.43/8.75 less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3)) 30.43/8.75 less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y)) 30.43/8.75 less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3)) 30.43/8.75 less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y)) 30.43/8.75 U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 30.43/8.75 U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 30.43/8.75 30.43/8.75 The argument filtering Pi contains the following mapping: 30.43/8.75 less_in_ga(x1, x2) = less_in_ga(x1) 30.43/8.75 30.43/8.75 0 = 0 30.43/8.75 30.43/8.75 less_out_ga(x1, x2) = less_out_ga(x1) 30.43/8.75 30.43/8.75 s(x1) = s(x1) 30.43/8.75 30.43/8.75 U7_ga(x1, x2, x3) = U7_ga(x1, x3) 30.43/8.75 30.43/8.75 less_in_ag(x1, x2) = less_in_ag(x2) 30.43/8.75 30.43/8.75 less_out_ag(x1, x2) = less_out_ag(x1, x2) 30.43/8.75 30.43/8.75 U7_ag(x1, x2, x3) = U7_ag(x2, x3) 30.43/8.75 30.43/8.75 DELETE_IN_GAA(x1, x2, x3) = DELETE_IN_GAA(x1) 30.43/8.75 30.43/8.75 U2_GAA(x1, x2, x3, x4, x5, x6) = U2_GAA(x1, x6) 30.43/8.75 30.43/8.75 U4_GAA(x1, x2, x3, x4, x5, x6) = U4_GAA(x1, x6) 30.43/8.75 30.43/8.75 30.43/8.75 We have to consider all (P,R,Pi)-chains 30.43/8.75 ---------------------------------------- 30.43/8.75 30.43/8.75 (81) PiDPToQDPProof (SOUND) 30.43/8.75 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 30.43/8.75 ---------------------------------------- 30.43/8.75 30.43/8.75 (82) 30.43/8.75 Obligation: 30.43/8.75 Q DP problem: 30.43/8.75 The TRS P consists of the following rules: 30.43/8.75 30.43/8.75 DELETE_IN_GAA(X) -> U2_GAA(X, less_in_ga(X)) 30.43/8.75 U2_GAA(X, less_out_ga(X)) -> DELETE_IN_GAA(X) 30.43/8.75 DELETE_IN_GAA(X) -> U4_GAA(X, less_in_ag(X)) 30.43/8.75 U4_GAA(X, less_out_ag(Y, X)) -> DELETE_IN_GAA(X) 30.43/8.75 30.43/8.75 The TRS R consists of the following rules: 30.43/8.75 30.43/8.75 less_in_ga(0) -> less_out_ga(0) 30.43/8.75 less_in_ga(s(X)) -> U7_ga(X, less_in_ga(X)) 30.43/8.75 less_in_ag(s(X3)) -> less_out_ag(0, s(X3)) 30.43/8.75 less_in_ag(s(Y)) -> U7_ag(Y, less_in_ag(Y)) 30.43/8.75 U7_ga(X, less_out_ga(X)) -> less_out_ga(s(X)) 30.43/8.75 U7_ag(Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 30.43/8.75 30.43/8.75 The set Q consists of the following terms: 30.43/8.75 30.43/8.75 less_in_ga(x0) 30.43/8.75 less_in_ag(x0) 30.43/8.75 U7_ga(x0, x1) 30.43/8.75 U7_ag(x0, x1) 30.43/8.75 30.43/8.75 We have to consider all (P,Q,R)-chains. 30.43/8.75 ---------------------------------------- 30.43/8.75 30.43/8.75 (83) PrologToTRSTransformerProof (SOUND) 30.43/8.75 Transformed Prolog program to TRS. 30.43/8.75 30.43/8.75 { 30.43/8.75 "root": 23, 30.43/8.75 "program": { 30.43/8.75 "directives": [], 30.43/8.75 "clauses": [ 30.43/8.75 [ 30.43/8.75 "(delete X (tree X (void) Right) Right)", 30.43/8.75 null 30.43/8.75 ], 30.43/8.75 [ 30.43/8.75 "(delete X (tree X Left (void)) Left)", 30.43/8.75 null 30.43/8.75 ], 30.43/8.75 [ 30.43/8.75 "(delete X (tree X Left Right) (tree Y Left Right1))", 30.43/8.75 "(delmin Right Y Right1)" 30.43/8.75 ], 30.43/8.75 [ 30.43/8.75 "(delete X (tree Y Left Right) (tree Y Left1 Right))", 30.43/8.75 "(',' (less X Y) (delete X Left Left1))" 30.43/8.75 ], 30.43/8.75 [ 30.43/8.75 "(delete X (tree Y Left Right) (tree Y Left Right1))", 30.43/8.75 "(',' (less Y X) (delete X Right Right1))" 30.43/8.75 ], 30.43/8.75 [ 30.43/8.75 "(delmin (tree Y (void) Right) Y Right)", 30.43/8.75 null 30.43/8.75 ], 30.43/8.75 [ 30.43/8.75 "(delmin (tree X Left X1) Y (tree X Left1 X2))", 30.43/8.75 "(delmin Left Y Left1)" 30.43/8.75 ], 30.43/8.75 [ 30.43/8.75 "(less (0) (s X3))", 30.43/8.75 null 30.43/8.75 ], 30.43/8.75 [ 30.43/8.75 "(less (s X) (s Y))", 30.43/8.75 "(less X Y)" 30.43/8.75 ] 30.43/8.75 ] 30.43/8.75 }, 30.43/8.75 "graph": { 30.43/8.75 "nodes": { 30.43/8.75 "type": "Nodes", 30.43/8.75 "510": { 30.43/8.75 "goal": [{ 30.43/8.75 "clause": 7, 30.43/8.75 "scope": 4, 30.43/8.75 "term": "(less T158 T153)" 30.43/8.75 }], 30.43/8.75 "kb": { 30.43/8.75 "nonunifying": [], 30.43/8.75 "intvars": {}, 30.43/8.75 "arithmetic": { 30.43/8.75 "type": "PlainIntegerRelationState", 30.43/8.75 "relations": [] 30.43/8.75 }, 30.43/8.75 "ground": ["T153"], 30.43/8.75 "free": [], 30.43/8.75 "exprvars": [] 30.48/8.75 } 30.48/8.75 }, 30.48/8.75 "511": { 30.48/8.75 "goal": [{ 30.48/8.75 "clause": 8, 30.48/8.75 "scope": 4, 30.48/8.75 "term": "(less T158 T153)" 30.48/8.75 }], 30.48/8.75 "kb": { 30.48/8.75 "nonunifying": [], 30.48/8.75 "intvars": {}, 30.48/8.75 "arithmetic": { 30.48/8.75 "type": "PlainIntegerRelationState", 30.48/8.75 "relations": [] 30.48/8.75 }, 30.48/8.75 "ground": ["T153"], 30.48/8.75 "free": [], 30.48/8.75 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[], 30.48/8.75 "exprvars": [] 30.48/8.75 } 30.48/8.75 }, 30.48/8.75 "497": { 30.48/8.75 "goal": [], 30.48/8.75 "kb": { 30.48/8.75 "nonunifying": [], 30.48/8.75 "intvars": {}, 30.48/8.75 "arithmetic": { 30.48/8.75 "type": "PlainIntegerRelationState", 30.48/8.75 "relations": [] 30.48/8.75 }, 30.48/8.75 "ground": [], 30.48/8.75 "free": [], 30.48/8.75 "exprvars": [] 30.48/8.75 } 30.48/8.75 }, 30.48/8.75 "454": { 30.48/8.75 "goal": [{ 30.48/8.75 "clause": -1, 30.48/8.75 "scope": -1, 30.48/8.75 "term": "(true)" 30.48/8.75 }], 30.48/8.75 "kb": { 30.48/8.75 "nonunifying": [], 30.48/8.75 "intvars": {}, 30.48/8.75 "arithmetic": { 30.48/8.75 "type": "PlainIntegerRelationState", 30.48/8.75 "relations": [] 30.48/8.75 }, 30.48/8.75 "ground": [], 30.48/8.75 "free": [], 30.48/8.75 "exprvars": [] 30.48/8.75 } 30.48/8.75 }, 30.48/8.75 "498": { 30.48/8.75 "goal": [], 30.48/8.75 "kb": { 30.48/8.75 "nonunifying": [], 30.48/8.75 "intvars": {}, 30.48/8.75 "arithmetic": { 30.48/8.75 "type": "PlainIntegerRelationState", 30.48/8.75 "relations": [] 30.48/8.75 }, 30.48/8.75 "ground": [], 30.48/8.75 "free": [], 30.48/8.75 "exprvars": [] 30.48/8.75 } 30.48/8.75 }, 30.48/8.75 "456": { 30.48/8.75 "goal": [], 30.48/8.75 "kb": { 30.48/8.75 "nonunifying": [], 30.48/8.75 "intvars": {}, 30.48/8.75 "arithmetic": { 30.48/8.75 "type": "PlainIntegerRelationState", 30.48/8.75 "relations": [] 30.48/8.75 }, 30.48/8.75 "ground": [], 30.48/8.75 "free": [], 30.48/8.75 "exprvars": [] 30.48/8.75 } 30.48/8.75 }, 30.48/8.75 "457": { 30.48/8.75 "goal": [], 30.48/8.75 "kb": { 30.48/8.75 "nonunifying": [], 30.48/8.75 "intvars": {}, 30.48/8.75 "arithmetic": { 30.48/8.75 "type": "PlainIntegerRelationState", 30.48/8.75 "relations": [] 30.48/8.75 }, 30.48/8.75 "ground": [], 30.48/8.75 "free": [], 30.48/8.75 "exprvars": [] 30.48/8.75 } 30.48/8.75 }, 30.48/8.75 "180": { 30.48/8.75 "goal": [{ 30.48/8.75 "clause": -1, 30.48/8.75 "scope": -1, 30.48/8.75 "term": "(delmin T49 T50 T51)" 30.48/8.75 }], 30.48/8.75 "kb": { 30.48/8.75 "nonunifying": [], 30.48/8.75 "intvars": {}, 30.48/8.75 "arithmetic": { 30.48/8.75 "type": "PlainIntegerRelationState", 30.48/8.75 "relations": [] 30.48/8.75 }, 30.48/8.75 "ground": [], 30.48/8.75 "free": [], 30.48/8.75 "exprvars": [] 30.48/8.75 } 30.48/8.75 }, 30.48/8.75 "463": { 30.48/8.75 "goal": [{ 30.48/8.75 "clause": -1, 30.48/8.75 "scope": -1, 30.48/8.75 "term": "(delmin T84 T85 T86)" 30.48/8.75 }], 30.48/8.75 "kb": { 30.48/8.75 "nonunifying": [], 30.48/8.75 "intvars": {}, 30.48/8.75 "arithmetic": { 30.48/8.75 "type": "PlainIntegerRelationState", 30.48/8.75 "relations": [] 30.48/8.75 }, 30.48/8.75 "ground": [], 30.48/8.75 "free": [], 30.48/8.75 "exprvars": [] 30.48/8.75 } 30.48/8.75 }, 30.48/8.75 "145": { 30.48/8.75 "goal": [{ 30.48/8.75 "clause": -1, 30.48/8.75 "scope": -1, 30.48/8.75 "term": "(true)" 30.48/8.75 }], 30.48/8.75 "kb": { 30.48/8.75 "nonunifying": [], 30.48/8.75 "intvars": {}, 30.48/8.75 "arithmetic": { 30.48/8.75 "type": "PlainIntegerRelationState", 30.48/8.75 "relations": [] 30.48/8.75 }, 30.48/8.75 "ground": [], 30.48/8.75 "free": [], 30.48/8.75 "exprvars": [] 30.48/8.75 } 30.48/8.75 }, 30.48/8.75 "464": { 30.48/8.75 "goal": [], 30.48/8.75 "kb": { 30.48/8.75 "nonunifying": [], 30.48/8.75 "intvars": {}, 30.48/8.75 "arithmetic": { 30.48/8.75 "type": "PlainIntegerRelationState", 30.48/8.75 "relations": [] 30.48/8.75 }, 30.48/8.75 "ground": [], 30.48/8.75 "free": [], 30.48/8.75 "exprvars": [] 30.48/8.75 } 30.48/8.75 }, 30.48/8.75 "146": { 30.48/8.75 "goal": [], 30.48/8.75 "kb": { 30.48/8.75 "nonunifying": [], 30.48/8.75 "intvars": {}, 30.48/8.75 "arithmetic": { 30.48/8.75 "type": "PlainIntegerRelationState", 30.48/8.75 "relations": [] 30.48/8.75 }, 30.48/8.75 "ground": [], 30.48/8.75 "free": [], 30.48/8.75 "exprvars": [] 30.48/8.75 } 30.48/8.75 }, 30.48/8.75 "147": { 30.48/8.75 "goal": [], 30.48/8.75 "kb": { 30.48/8.75 "nonunifying": [], 30.48/8.75 "intvars": {}, 30.48/8.75 "arithmetic": { 30.48/8.75 "type": "PlainIntegerRelationState", 30.48/8.75 "relations": [] 30.48/8.75 }, 30.48/8.75 "ground": [], 30.48/8.75 "free": [], 30.48/8.75 "exprvars": [] 30.48/8.75 } 30.48/8.75 }, 30.48/8.75 "500": { 30.48/8.75 "goal": [{ 30.48/8.75 "clause": -1, 30.48/8.75 "scope": -1, 30.48/8.75 "term": "(less T134 T136)" 30.48/8.75 }], 30.48/8.75 "kb": { 30.48/8.75 "nonunifying": [], 30.48/8.75 "intvars": {}, 30.48/8.75 "arithmetic": { 30.48/8.75 "type": "PlainIntegerRelationState", 30.48/8.75 "relations": [] 30.48/8.75 }, 30.48/8.75 "ground": ["T134"], 30.48/8.75 "free": [], 30.48/8.75 "exprvars": [] 30.48/8.75 } 30.48/8.75 }, 30.48/8.75 "501": { 30.48/8.75 "goal": [], 30.48/8.75 "kb": { 30.48/8.75 "nonunifying": [], 30.48/8.75 "intvars": {}, 30.48/8.75 "arithmetic": { 30.48/8.75 "type": "PlainIntegerRelationState", 30.48/8.75 "relations": [] 30.48/8.75 }, 30.48/8.75 "ground": [], 30.48/8.75 "free": [], 30.48/8.75 "exprvars": [] 30.48/8.75 } 30.48/8.75 }, 30.48/8.75 "505": { 30.48/8.75 "goal": [{ 30.48/8.75 "clause": -1, 30.48/8.75 "scope": -1, 30.48/8.75 "term": "(',' (less T158 T153) (delete T153 T159 T160))" 30.48/8.75 }], 30.48/8.75 "kb": { 30.48/8.75 "nonunifying": [], 30.48/8.75 "intvars": {}, 30.48/8.75 "arithmetic": { 30.48/8.75 "type": "PlainIntegerRelationState", 30.48/8.75 "relations": [] 30.48/8.75 }, 30.48/8.75 "ground": ["T153"], 30.48/8.75 "free": [], 30.48/8.75 "exprvars": [] 30.48/8.75 } 30.48/8.75 }, 30.48/8.75 "506": { 30.48/8.75 "goal": [], 30.48/8.75 "kb": { 30.48/8.75 "nonunifying": [], 30.48/8.75 "intvars": {}, 30.48/8.75 "arithmetic": { 30.48/8.75 "type": "PlainIntegerRelationState", 30.48/8.75 "relations": [] 30.48/8.75 }, 30.48/8.75 "ground": [], 30.48/8.75 "free": [], 30.48/8.75 "exprvars": [] 30.48/8.75 } 30.48/8.75 }, 30.48/8.75 "507": { 30.48/8.75 "goal": [{ 30.48/8.75 "clause": -1, 30.48/8.75 "scope": -1, 30.48/8.75 "term": "(less T158 T153)" 30.48/8.75 }], 30.48/8.75 "kb": { 30.48/8.75 "nonunifying": [], 30.48/8.75 "intvars": {}, 30.48/8.75 "arithmetic": { 30.48/8.75 "type": "PlainIntegerRelationState", 30.48/8.75 "relations": [] 30.48/8.75 }, 30.48/8.75 "ground": ["T153"], 30.48/8.75 "free": [], 30.48/8.75 "exprvars": [] 30.48/8.75 } 30.48/8.75 }, 30.48/8.75 "508": { 30.48/8.75 "goal": [{ 30.48/8.75 "clause": -1, 30.48/8.75 "scope": -1, 30.48/8.75 "term": "(delete T153 T163 T164)" 30.48/8.75 }], 30.48/8.75 "kb": { 30.48/8.75 "nonunifying": [], 30.48/8.75 "intvars": {}, 30.48/8.75 "arithmetic": { 30.48/8.75 "type": "PlainIntegerRelationState", 30.48/8.75 "relations": [] 30.48/8.75 }, 30.48/8.75 "ground": ["T153"], 30.48/8.75 "free": [], 30.48/8.75 "exprvars": [] 30.48/8.75 } 30.48/8.75 }, 30.48/8.75 "509": { 30.48/8.75 "goal": [ 30.48/8.75 { 30.48/8.75 "clause": 7, 30.48/8.75 "scope": 4, 30.48/8.75 "term": "(less T158 T153)" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "clause": 8, 30.48/8.75 "scope": 4, 30.48/8.75 "term": "(less T158 T153)" 30.48/8.75 } 30.48/8.75 ], 30.48/8.75 "kb": { 30.48/8.75 "nonunifying": [], 30.48/8.75 "intvars": {}, 30.48/8.75 "arithmetic": { 30.48/8.75 "type": "PlainIntegerRelationState", 30.48/8.75 "relations": [] 30.48/8.75 }, 30.48/8.75 "ground": ["T153"], 30.48/8.75 "free": [], 30.48/8.75 "exprvars": [] 30.48/8.75 } 30.48/8.75 } 30.48/8.75 }, 30.48/8.75 "edges": [ 30.48/8.75 { 30.48/8.75 "from": 23, 30.48/8.75 "to": 24, 30.48/8.75 "label": "CASE" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 24, 30.48/8.75 "to": 25, 30.48/8.75 "label": "PARALLEL" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 24, 30.48/8.75 "to": 26, 30.48/8.75 "label": "PARALLEL" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 25, 30.48/8.75 "to": 145, 30.48/8.75 "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" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 25, 30.48/8.75 "to": 146, 30.48/8.75 "label": "EVAL-BACKTRACK" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 26, 30.48/8.75 "to": 167, 30.48/8.75 "label": "PARALLEL" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 26, 30.48/8.75 "to": 168, 30.48/8.75 "label": "PARALLEL" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 145, 30.48/8.75 "to": 147, 30.48/8.75 "label": "SUCCESS" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 167, 30.48/8.75 "to": 172, 30.48/8.75 "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" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 167, 30.48/8.75 "to": 173, 30.48/8.75 "label": "EVAL-BACKTRACK" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 168, 30.48/8.75 "to": 177, 30.48/8.75 "label": "PARALLEL" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 168, 30.48/8.75 "to": 178, 30.48/8.75 "label": "PARALLEL" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 172, 30.48/8.75 "to": 174, 30.48/8.75 "label": "SUCCESS" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 177, 30.48/8.75 "to": 180, 30.48/8.75 "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" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 177, 30.48/8.75 "to": 447, 30.48/8.75 "label": "EVAL-BACKTRACK" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 178, 30.48/8.75 "to": 485, 30.48/8.75 "label": "PARALLEL" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 178, 30.48/8.75 "to": 486, 30.48/8.75 "label": "PARALLEL" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 180, 30.48/8.75 "to": 449, 30.48/8.75 "label": "CASE" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 449, 30.48/8.75 "to": 450, 30.48/8.75 "label": "PARALLEL" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 449, 30.48/8.75 "to": 452, 30.48/8.75 "label": "PARALLEL" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 450, 30.48/8.75 "to": 454, 30.48/8.75 "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" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 450, 30.48/8.75 "to": 456, 30.48/8.75 "label": "EVAL-BACKTRACK" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 452, 30.48/8.75 "to": 463, 30.48/8.75 "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" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 452, 30.48/8.75 "to": 464, 30.48/8.75 "label": "EVAL-BACKTRACK" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 454, 30.48/8.75 "to": 457, 30.48/8.75 "label": "SUCCESS" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 463, 30.48/8.75 "to": 180, 30.48/8.75 "label": "INSTANCE with matching:\nT49 -> T84\nT50 -> T85\nT51 -> T86" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 485, 30.48/8.75 "to": 489, 30.48/8.75 "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" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 485, 30.48/8.75 "to": 490, 30.48/8.75 "label": "EVAL-BACKTRACK" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 486, 30.48/8.75 "to": 505, 30.48/8.75 "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" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 486, 30.48/8.75 "to": 506, 30.48/8.75 "label": "EVAL-BACKTRACK" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 489, 30.48/8.75 "to": 491, 30.48/8.75 "label": "SPLIT 1" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 489, 30.48/8.75 "to": 492, 30.48/8.75 "label": "SPLIT 2\nnew knowledge:\nT111 is ground\nreplacements:T117 -> T121,\nT118 -> T122" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 491, 30.48/8.75 "to": 493, 30.48/8.75 "label": "CASE" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 492, 30.48/8.75 "to": 23, 30.48/8.75 "label": "INSTANCE with matching:\nT1 -> T111\nT2 -> T121\nT3 -> T122" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 493, 30.48/8.75 "to": 494, 30.48/8.75 "label": "PARALLEL" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 493, 30.48/8.75 "to": 495, 30.48/8.75 "label": "PARALLEL" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 494, 30.48/8.75 "to": 496, 30.48/8.75 "label": "EVAL with clause\nless(0, s(X118)).\nand substitutionT111 -> 0,\nX118 -> T129,\nT116 -> s(T129)" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 494, 30.48/8.75 "to": 497, 30.48/8.75 "label": "EVAL-BACKTRACK" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 495, 30.48/8.75 "to": 500, 30.48/8.75 "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" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 495, 30.48/8.75 "to": 501, 30.48/8.75 "label": "EVAL-BACKTRACK" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 496, 30.48/8.75 "to": 498, 30.48/8.75 "label": "SUCCESS" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 500, 30.48/8.75 "to": 491, 30.48/8.75 "label": "INSTANCE with matching:\nT111 -> T134\nT116 -> T136" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 505, 30.48/8.75 "to": 507, 30.48/8.75 "label": "SPLIT 1" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 505, 30.48/8.75 "to": 508, 30.48/8.75 "label": "SPLIT 2\nnew knowledge:\nT158 is ground\nT153 is ground\nreplacements:T159 -> T163,\nT160 -> T164" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 507, 30.48/8.75 "to": 509, 30.48/8.75 "label": "CASE" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 508, 30.48/8.75 "to": 23, 30.48/8.75 "label": "INSTANCE with matching:\nT1 -> T153\nT2 -> T163\nT3 -> T164" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 509, 30.48/8.75 "to": 510, 30.48/8.75 "label": "PARALLEL" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 509, 30.48/8.75 "to": 511, 30.48/8.75 "label": "PARALLEL" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 510, 30.48/8.75 "to": 512, 30.48/8.75 "label": "EVAL with clause\nless(0, s(X154)).\nand substitutionT158 -> 0,\nX154 -> T171,\nT153 -> s(T171)" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 510, 30.48/8.75 "to": 513, 30.48/8.75 "label": "EVAL-BACKTRACK" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 511, 30.48/8.75 "to": 516, 30.48/8.75 "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" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 511, 30.48/8.75 "to": 517, 30.48/8.75 "label": "EVAL-BACKTRACK" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 512, 30.48/8.75 "to": 514, 30.48/8.75 "label": "SUCCESS" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "from": 516, 30.48/8.75 "to": 507, 30.48/8.75 "label": "INSTANCE with matching:\nT158 -> T178\nT153 -> T177" 30.48/8.75 } 30.48/8.75 ], 30.48/8.75 "type": "Graph" 30.48/8.75 } 30.48/8.75 } 30.48/8.75 30.48/8.75 ---------------------------------------- 30.48/8.75 30.48/8.75 (84) 30.48/8.75 Obligation: 30.48/8.75 Q restricted rewrite system: 30.48/8.75 The TRS R consists of the following rules: 30.48/8.75 30.48/8.75 f23_in(T12) -> f23_out1 30.48/8.75 f23_in(T44) -> U1(f180_in, T44) 30.48/8.75 U1(f180_out1, T44) -> f23_out1 30.48/8.75 f23_in(T111) -> U2(f489_in(T111), T111) 30.48/8.75 U2(f489_out1, T111) -> f23_out1 30.48/8.75 f23_in(T153) -> U3(f505_in(T153), T153) 30.48/8.75 U3(f505_out1(T158), T153) -> f23_out1 30.48/8.75 f180_in -> f180_out1 30.48/8.75 f180_in -> U4(f180_in) 30.48/8.75 U4(f180_out1) -> f180_out1 30.48/8.75 f491_in(0) -> f491_out1 30.48/8.75 f491_in(s(T134)) -> U5(f491_in(T134), s(T134)) 30.48/8.75 U5(f491_out1, s(T134)) -> f491_out1 30.48/8.75 f507_in(s(T171)) -> f507_out1(0) 30.48/8.75 f507_in(s(T177)) -> U6(f507_in(T177), s(T177)) 30.48/8.75 U6(f507_out1(T178), s(T177)) -> f507_out1(s(T178)) 30.48/8.75 f489_in(T111) -> U7(f491_in(T111), T111) 30.48/8.75 U7(f491_out1, T111) -> U8(f23_in(T111), T111) 30.48/8.75 U8(f23_out1, T111) -> f489_out1 30.48/8.75 f505_in(T153) -> U9(f507_in(T153), T153) 30.48/8.75 U9(f507_out1(T158), T153) -> U10(f23_in(T153), T153, T158) 30.48/8.75 U10(f23_out1, T153, T158) -> f505_out1(T158) 30.48/8.75 30.48/8.75 Q is empty. 30.48/8.75 30.48/8.75 ---------------------------------------- 30.48/8.75 30.48/8.75 (85) DependencyPairsProof (EQUIVALENT) 30.48/8.75 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 30.48/8.75 ---------------------------------------- 30.48/8.75 30.48/8.75 (86) 30.48/8.75 Obligation: 30.48/8.75 Q DP problem: 30.48/8.75 The TRS P consists of the following rules: 30.48/8.75 30.48/8.75 F23_IN(T44) -> U1^1(f180_in, T44) 30.48/8.75 F23_IN(T44) -> F180_IN 30.48/8.75 F23_IN(T111) -> U2^1(f489_in(T111), T111) 30.48/8.75 F23_IN(T111) -> F489_IN(T111) 30.48/8.75 F23_IN(T153) -> U3^1(f505_in(T153), T153) 30.48/8.75 F23_IN(T153) -> F505_IN(T153) 30.48/8.75 F180_IN -> U4^1(f180_in) 30.48/8.75 F180_IN -> F180_IN 30.48/8.75 F491_IN(s(T134)) -> U5^1(f491_in(T134), s(T134)) 30.48/8.75 F491_IN(s(T134)) -> F491_IN(T134) 30.48/8.75 F507_IN(s(T177)) -> U6^1(f507_in(T177), s(T177)) 30.48/8.75 F507_IN(s(T177)) -> F507_IN(T177) 30.48/8.75 F489_IN(T111) -> U7^1(f491_in(T111), T111) 30.48/8.75 F489_IN(T111) -> F491_IN(T111) 30.48/8.75 U7^1(f491_out1, T111) -> U8^1(f23_in(T111), T111) 30.48/8.75 U7^1(f491_out1, T111) -> F23_IN(T111) 30.48/8.75 F505_IN(T153) -> U9^1(f507_in(T153), T153) 30.48/8.75 F505_IN(T153) -> F507_IN(T153) 30.48/8.75 U9^1(f507_out1(T158), T153) -> U10^1(f23_in(T153), T153, T158) 30.48/8.75 U9^1(f507_out1(T158), T153) -> F23_IN(T153) 30.48/8.75 30.48/8.75 The TRS R consists of the following rules: 30.48/8.75 30.48/8.75 f23_in(T12) -> f23_out1 30.48/8.75 f23_in(T44) -> U1(f180_in, T44) 30.48/8.75 U1(f180_out1, T44) -> f23_out1 30.48/8.75 f23_in(T111) -> U2(f489_in(T111), T111) 30.48/8.75 U2(f489_out1, T111) -> f23_out1 30.48/8.75 f23_in(T153) -> U3(f505_in(T153), T153) 30.48/8.75 U3(f505_out1(T158), T153) -> f23_out1 30.48/8.75 f180_in -> f180_out1 30.48/8.75 f180_in -> U4(f180_in) 30.48/8.75 U4(f180_out1) -> f180_out1 30.48/8.75 f491_in(0) -> f491_out1 30.48/8.75 f491_in(s(T134)) -> U5(f491_in(T134), s(T134)) 30.48/8.75 U5(f491_out1, s(T134)) -> f491_out1 30.48/8.75 f507_in(s(T171)) -> f507_out1(0) 30.48/8.75 f507_in(s(T177)) -> U6(f507_in(T177), s(T177)) 30.48/8.75 U6(f507_out1(T178), s(T177)) -> f507_out1(s(T178)) 30.48/8.75 f489_in(T111) -> U7(f491_in(T111), T111) 30.48/8.75 U7(f491_out1, T111) -> U8(f23_in(T111), T111) 30.48/8.75 U8(f23_out1, T111) -> f489_out1 30.48/8.75 f505_in(T153) -> U9(f507_in(T153), T153) 30.48/8.75 U9(f507_out1(T158), T153) -> U10(f23_in(T153), T153, T158) 30.48/8.75 U10(f23_out1, T153, T158) -> f505_out1(T158) 30.48/8.75 30.48/8.75 Q is empty. 30.48/8.75 We have to consider all minimal (P,Q,R)-chains. 30.48/8.75 ---------------------------------------- 30.48/8.75 30.48/8.75 (87) DependencyGraphProof (EQUIVALENT) 30.48/8.75 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 11 less nodes. 30.48/8.75 ---------------------------------------- 30.48/8.75 30.48/8.75 (88) 30.48/8.75 Complex Obligation (AND) 30.48/8.75 30.48/8.75 ---------------------------------------- 30.48/8.75 30.48/8.75 (89) 30.48/8.75 Obligation: 30.48/8.75 Q DP problem: 30.48/8.75 The TRS P consists of the following rules: 30.48/8.75 30.48/8.75 F507_IN(s(T177)) -> F507_IN(T177) 30.48/8.75 30.48/8.75 The TRS R consists of the following rules: 30.48/8.75 30.48/8.75 f23_in(T12) -> f23_out1 30.48/8.75 f23_in(T44) -> U1(f180_in, T44) 30.48/8.75 U1(f180_out1, T44) -> f23_out1 30.48/8.75 f23_in(T111) -> U2(f489_in(T111), T111) 30.48/8.75 U2(f489_out1, T111) -> f23_out1 30.48/8.75 f23_in(T153) -> U3(f505_in(T153), T153) 30.48/8.75 U3(f505_out1(T158), T153) -> f23_out1 30.48/8.75 f180_in -> f180_out1 30.48/8.75 f180_in -> U4(f180_in) 30.48/8.75 U4(f180_out1) -> f180_out1 30.48/8.75 f491_in(0) -> f491_out1 30.48/8.75 f491_in(s(T134)) -> U5(f491_in(T134), s(T134)) 30.48/8.75 U5(f491_out1, s(T134)) -> f491_out1 30.48/8.75 f507_in(s(T171)) -> f507_out1(0) 30.48/8.75 f507_in(s(T177)) -> U6(f507_in(T177), s(T177)) 30.48/8.75 U6(f507_out1(T178), s(T177)) -> f507_out1(s(T178)) 30.48/8.75 f489_in(T111) -> U7(f491_in(T111), T111) 30.48/8.75 U7(f491_out1, T111) -> U8(f23_in(T111), T111) 30.48/8.75 U8(f23_out1, T111) -> f489_out1 30.48/8.75 f505_in(T153) -> U9(f507_in(T153), T153) 30.48/8.75 U9(f507_out1(T158), T153) -> U10(f23_in(T153), T153, T158) 30.48/8.75 U10(f23_out1, T153, T158) -> f505_out1(T158) 30.48/8.75 30.48/8.75 Q is empty. 30.48/8.75 We have to consider all minimal (P,Q,R)-chains. 30.48/8.75 ---------------------------------------- 30.48/8.75 30.48/8.75 (90) UsableRulesProof (EQUIVALENT) 30.48/8.75 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. 30.48/8.75 ---------------------------------------- 30.48/8.75 30.48/8.75 (91) 30.48/8.75 Obligation: 30.48/8.75 Q DP problem: 30.48/8.75 The TRS P consists of the following rules: 30.48/8.75 30.48/8.75 F507_IN(s(T177)) -> F507_IN(T177) 30.48/8.75 30.48/8.75 R is empty. 30.48/8.75 Q is empty. 30.48/8.75 We have to consider all minimal (P,Q,R)-chains. 30.48/8.75 ---------------------------------------- 30.48/8.75 30.48/8.75 (92) QDPSizeChangeProof (EQUIVALENT) 30.48/8.75 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. 30.48/8.75 30.48/8.75 From the DPs we obtained the following set of size-change graphs: 30.48/8.75 *F507_IN(s(T177)) -> F507_IN(T177) 30.48/8.75 The graph contains the following edges 1 > 1 30.48/8.75 30.48/8.75 30.48/8.75 ---------------------------------------- 30.48/8.75 30.48/8.75 (93) 30.48/8.75 YES 30.48/8.75 30.48/8.75 ---------------------------------------- 30.48/8.75 30.48/8.75 (94) 30.48/8.75 Obligation: 30.48/8.75 Q DP problem: 30.48/8.75 The TRS P consists of the following rules: 30.48/8.75 30.48/8.75 F491_IN(s(T134)) -> F491_IN(T134) 30.48/8.75 30.48/8.75 The TRS R consists of the following rules: 30.48/8.75 30.48/8.75 f23_in(T12) -> f23_out1 30.48/8.75 f23_in(T44) -> U1(f180_in, T44) 30.48/8.75 U1(f180_out1, T44) -> f23_out1 30.48/8.75 f23_in(T111) -> U2(f489_in(T111), T111) 30.48/8.75 U2(f489_out1, T111) -> f23_out1 30.48/8.75 f23_in(T153) -> U3(f505_in(T153), T153) 30.48/8.75 U3(f505_out1(T158), T153) -> f23_out1 30.48/8.75 f180_in -> f180_out1 30.48/8.75 f180_in -> U4(f180_in) 30.48/8.75 U4(f180_out1) -> f180_out1 30.48/8.75 f491_in(0) -> f491_out1 30.48/8.75 f491_in(s(T134)) -> U5(f491_in(T134), s(T134)) 30.48/8.75 U5(f491_out1, s(T134)) -> f491_out1 30.48/8.75 f507_in(s(T171)) -> f507_out1(0) 30.48/8.75 f507_in(s(T177)) -> U6(f507_in(T177), s(T177)) 30.48/8.75 U6(f507_out1(T178), s(T177)) -> f507_out1(s(T178)) 30.48/8.75 f489_in(T111) -> U7(f491_in(T111), T111) 30.48/8.75 U7(f491_out1, T111) -> U8(f23_in(T111), T111) 30.48/8.75 U8(f23_out1, T111) -> f489_out1 30.48/8.75 f505_in(T153) -> U9(f507_in(T153), T153) 30.48/8.75 U9(f507_out1(T158), T153) -> U10(f23_in(T153), T153, T158) 30.48/8.75 U10(f23_out1, T153, T158) -> f505_out1(T158) 30.48/8.75 30.48/8.75 Q is empty. 30.48/8.75 We have to consider all minimal (P,Q,R)-chains. 30.48/8.75 ---------------------------------------- 30.48/8.75 30.48/8.75 (95) UsableRulesProof (EQUIVALENT) 30.48/8.75 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. 30.48/8.75 ---------------------------------------- 30.48/8.75 30.48/8.75 (96) 30.48/8.75 Obligation: 30.48/8.75 Q DP problem: 30.48/8.75 The TRS P consists of the following rules: 30.48/8.75 30.48/8.75 F491_IN(s(T134)) -> F491_IN(T134) 30.48/8.75 30.48/8.75 R is empty. 30.48/8.75 Q is empty. 30.48/8.75 We have to consider all minimal (P,Q,R)-chains. 30.48/8.75 ---------------------------------------- 30.48/8.75 30.48/8.75 (97) QDPSizeChangeProof (EQUIVALENT) 30.48/8.75 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. 30.48/8.75 30.48/8.75 From the DPs we obtained the following set of size-change graphs: 30.48/8.75 *F491_IN(s(T134)) -> F491_IN(T134) 30.48/8.75 The graph contains the following edges 1 > 1 30.48/8.75 30.48/8.75 30.48/8.75 ---------------------------------------- 30.48/8.75 30.48/8.75 (98) 30.48/8.75 YES 30.48/8.75 30.48/8.75 ---------------------------------------- 30.48/8.75 30.48/8.75 (99) 30.48/8.75 Obligation: 30.48/8.75 Q DP problem: 30.48/8.75 The TRS P consists of the following rules: 30.48/8.75 30.48/8.75 F180_IN -> F180_IN 30.48/8.75 30.48/8.75 The TRS R consists of the following rules: 30.48/8.75 30.48/8.75 f23_in(T12) -> f23_out1 30.48/8.75 f23_in(T44) -> U1(f180_in, T44) 30.48/8.75 U1(f180_out1, T44) -> f23_out1 30.48/8.75 f23_in(T111) -> U2(f489_in(T111), T111) 30.48/8.75 U2(f489_out1, T111) -> f23_out1 30.48/8.75 f23_in(T153) -> U3(f505_in(T153), T153) 30.48/8.75 U3(f505_out1(T158), T153) -> f23_out1 30.48/8.75 f180_in -> f180_out1 30.48/8.75 f180_in -> U4(f180_in) 30.48/8.75 U4(f180_out1) -> f180_out1 30.48/8.75 f491_in(0) -> f491_out1 30.48/8.75 f491_in(s(T134)) -> U5(f491_in(T134), s(T134)) 30.48/8.75 U5(f491_out1, s(T134)) -> f491_out1 30.48/8.75 f507_in(s(T171)) -> f507_out1(0) 30.48/8.75 f507_in(s(T177)) -> U6(f507_in(T177), s(T177)) 30.48/8.75 U6(f507_out1(T178), s(T177)) -> f507_out1(s(T178)) 30.48/8.75 f489_in(T111) -> U7(f491_in(T111), T111) 30.48/8.75 U7(f491_out1, T111) -> U8(f23_in(T111), T111) 30.48/8.75 U8(f23_out1, T111) -> f489_out1 30.48/8.75 f505_in(T153) -> U9(f507_in(T153), T153) 30.48/8.75 U9(f507_out1(T158), T153) -> U10(f23_in(T153), T153, T158) 30.48/8.75 U10(f23_out1, T153, T158) -> f505_out1(T158) 30.48/8.75 30.48/8.75 Q is empty. 30.48/8.75 We have to consider all minimal (P,Q,R)-chains. 30.48/8.75 ---------------------------------------- 30.48/8.75 30.48/8.75 (100) UsableRulesProof (EQUIVALENT) 30.48/8.75 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. 30.48/8.75 ---------------------------------------- 30.48/8.75 30.48/8.75 (101) 30.48/8.75 Obligation: 30.48/8.75 Q DP problem: 30.48/8.75 The TRS P consists of the following rules: 30.48/8.75 30.48/8.75 F180_IN -> F180_IN 30.48/8.75 30.48/8.75 R is empty. 30.48/8.75 Q is empty. 30.48/8.75 We have to consider all minimal (P,Q,R)-chains. 30.48/8.75 ---------------------------------------- 30.48/8.75 30.48/8.75 (102) NonTerminationLoopProof (COMPLETE) 30.48/8.75 We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. 30.48/8.75 Found a loop by semiunifying a rule from P directly. 30.48/8.75 30.48/8.75 s = F180_IN evaluates to t =F180_IN 30.48/8.75 30.48/8.75 Thus s starts an infinite chain as s semiunifies with t with the following substitutions: 30.48/8.75 * Matcher: [ ] 30.48/8.75 * Semiunifier: [ ] 30.48/8.75 30.48/8.75 -------------------------------------------------------------------------------- 30.48/8.75 Rewriting sequence 30.48/8.75 30.48/8.75 The DP semiunifies directly so there is only one rewrite step from F180_IN to F180_IN. 30.48/8.75 30.48/8.75 30.48/8.75 30.48/8.75 30.48/8.75 ---------------------------------------- 30.48/8.75 30.48/8.75 (103) 30.48/8.75 NO 30.48/8.75 30.48/8.75 ---------------------------------------- 30.48/8.75 30.48/8.75 (104) 30.48/8.75 Obligation: 30.48/8.75 Q DP problem: 30.48/8.75 The TRS P consists of the following rules: 30.48/8.75 30.48/8.75 F23_IN(T111) -> F489_IN(T111) 30.48/8.75 F489_IN(T111) -> U7^1(f491_in(T111), T111) 30.48/8.75 U7^1(f491_out1, T111) -> F23_IN(T111) 30.48/8.75 F23_IN(T153) -> F505_IN(T153) 30.48/8.75 F505_IN(T153) -> U9^1(f507_in(T153), T153) 30.48/8.75 U9^1(f507_out1(T158), T153) -> F23_IN(T153) 30.48/8.75 30.48/8.75 The TRS R consists of the following rules: 30.48/8.75 30.48/8.75 f23_in(T12) -> f23_out1 30.48/8.75 f23_in(T44) -> U1(f180_in, T44) 30.48/8.75 U1(f180_out1, T44) -> f23_out1 30.48/8.75 f23_in(T111) -> U2(f489_in(T111), T111) 30.48/8.75 U2(f489_out1, T111) -> f23_out1 30.48/8.75 f23_in(T153) -> U3(f505_in(T153), T153) 30.48/8.75 U3(f505_out1(T158), T153) -> f23_out1 30.48/8.75 f180_in -> f180_out1 30.48/8.75 f180_in -> U4(f180_in) 30.48/8.75 U4(f180_out1) -> f180_out1 30.48/8.75 f491_in(0) -> f491_out1 30.48/8.75 f491_in(s(T134)) -> U5(f491_in(T134), s(T134)) 30.48/8.75 U5(f491_out1, s(T134)) -> f491_out1 30.48/8.75 f507_in(s(T171)) -> f507_out1(0) 30.48/8.75 f507_in(s(T177)) -> U6(f507_in(T177), s(T177)) 30.48/8.75 U6(f507_out1(T178), s(T177)) -> f507_out1(s(T178)) 30.48/8.75 f489_in(T111) -> U7(f491_in(T111), T111) 30.48/8.75 U7(f491_out1, T111) -> U8(f23_in(T111), T111) 30.48/8.75 U8(f23_out1, T111) -> f489_out1 30.48/8.75 f505_in(T153) -> U9(f507_in(T153), T153) 30.48/8.75 U9(f507_out1(T158), T153) -> U10(f23_in(T153), T153, T158) 30.48/8.75 U10(f23_out1, T153, T158) -> f505_out1(T158) 30.48/8.75 30.48/8.75 Q is empty. 30.48/8.75 We have to consider all minimal (P,Q,R)-chains. 30.48/8.75 ---------------------------------------- 30.48/8.75 30.48/8.75 (105) NonTerminationLoopProof (COMPLETE) 30.48/8.75 We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. 30.48/8.75 Found a loop by narrowing to the left: 30.48/8.75 30.48/8.75 s = F489_IN(0) evaluates to t =F489_IN(0) 30.48/8.75 30.48/8.75 Thus s starts an infinite chain as s semiunifies with t with the following substitutions: 30.48/8.75 * Matcher: [ ] 30.48/8.75 * Semiunifier: [ ] 30.48/8.75 30.48/8.75 -------------------------------------------------------------------------------- 30.48/8.75 Rewriting sequence 30.48/8.75 30.48/8.75 F489_IN(0) -> U7^1(f491_in(0), 0) 30.48/8.75 with rule F489_IN(T111) -> U7^1(f491_in(T111), T111) at position [] and matcher [T111 / 0] 30.48/8.75 30.48/8.75 U7^1(f491_in(0), 0) -> U7^1(f491_out1, 0) 30.48/8.75 with rule f491_in(0) -> f491_out1 at position [0] and matcher [ ] 30.48/8.75 30.48/8.75 U7^1(f491_out1, 0) -> F23_IN(0) 30.48/8.75 with rule U7^1(f491_out1, T111') -> F23_IN(T111') at position [] and matcher [T111' / 0] 30.48/8.75 30.48/8.75 F23_IN(0) -> F489_IN(0) 30.48/8.75 with rule F23_IN(T111) -> F489_IN(T111) 30.48/8.75 30.48/8.75 Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence 30.48/8.75 30.48/8.75 30.48/8.75 All these steps are and every following step will be a correct step w.r.t to Q. 30.48/8.75 30.48/8.75 30.48/8.75 30.48/8.75 30.48/8.75 ---------------------------------------- 30.48/8.75 30.48/8.75 (106) 30.48/8.75 NO 30.48/8.75 30.48/8.75 ---------------------------------------- 30.48/8.75 30.48/8.75 (107) PrologToIRSwTTransformerProof (SOUND) 30.48/8.75 Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert 30.48/8.75 30.48/8.75 { 30.48/8.75 "root": 2, 30.48/8.75 "program": { 30.48/8.75 "directives": [], 30.48/8.75 "clauses": [ 30.48/8.75 [ 30.48/8.75 "(delete X (tree X (void) Right) Right)", 30.48/8.75 null 30.48/8.75 ], 30.48/8.75 [ 30.48/8.75 "(delete X (tree X Left (void)) Left)", 30.48/8.75 null 30.48/8.75 ], 30.48/8.75 [ 30.48/8.75 "(delete X (tree X Left Right) (tree Y Left Right1))", 30.48/8.75 "(delmin Right Y Right1)" 30.48/8.75 ], 30.48/8.75 [ 30.48/8.75 "(delete X (tree Y Left Right) (tree Y Left1 Right))", 30.48/8.75 "(',' (less X Y) (delete X Left Left1))" 30.48/8.75 ], 30.48/8.75 [ 30.48/8.75 "(delete X (tree Y Left Right) (tree Y Left Right1))", 30.48/8.75 "(',' (less Y X) (delete X Right Right1))" 30.48/8.75 ], 30.48/8.75 [ 30.48/8.75 "(delmin (tree Y (void) Right) Y Right)", 30.48/8.75 null 30.48/8.75 ], 30.48/8.75 [ 30.48/8.75 "(delmin (tree X Left X1) Y (tree X Left1 X2))", 30.48/8.75 "(delmin Left Y Left1)" 30.48/8.75 ], 30.48/8.75 [ 30.48/8.75 "(less (0) (s X3))", 30.48/8.75 null 30.48/8.75 ], 30.48/8.75 [ 30.48/8.75 "(less (s X) (s Y))", 30.48/8.75 "(less X Y)" 30.48/8.75 ] 30.48/8.75 ] 30.48/8.75 }, 30.48/8.75 "graph": { 30.48/8.75 "nodes": { 30.48/8.75 "190": { 30.48/8.75 "goal": [], 30.48/8.75 "kb": { 30.48/8.75 "nonunifying": [], 30.48/8.75 "intvars": {}, 30.48/8.75 "arithmetic": { 30.48/8.75 "type": "PlainIntegerRelationState", 30.48/8.75 "relations": [] 30.48/8.75 }, 30.48/8.75 "ground": [], 30.48/8.75 "free": [], 30.48/8.75 "exprvars": [] 30.48/8.75 } 30.48/8.75 }, 30.48/8.75 "type": "Nodes", 30.48/8.75 "151": { 30.48/8.75 "goal": [{ 30.48/8.75 "clause": -1, 30.48/8.75 "scope": -1, 30.48/8.75 "term": "(true)" 30.48/8.75 }], 30.48/8.75 "kb": { 30.48/8.75 "nonunifying": [], 30.48/8.75 "intvars": {}, 30.48/8.75 "arithmetic": { 30.48/8.75 "type": "PlainIntegerRelationState", 30.48/8.75 "relations": [] 30.48/8.75 }, 30.48/8.75 "ground": [], 30.48/8.75 "free": [], 30.48/8.75 "exprvars": [] 30.48/8.75 } 30.48/8.75 }, 30.48/8.75 "152": { 30.48/8.75 "goal": [], 30.48/8.75 "kb": { 30.48/8.75 "nonunifying": [], 30.48/8.75 "intvars": {}, 30.48/8.75 "arithmetic": { 30.48/8.75 "type": "PlainIntegerRelationState", 30.48/8.75 "relations": [] 30.48/8.75 }, 30.48/8.75 "ground": [], 30.48/8.75 "free": [], 30.48/8.75 "exprvars": [] 30.48/8.75 } 30.48/8.75 }, 30.48/8.75 "153": { 30.48/8.75 "goal": [], 30.48/8.75 "kb": { 30.48/8.75 "nonunifying": [], 30.48/8.75 "intvars": {}, 30.48/8.75 "arithmetic": { 30.48/8.75 "type": "PlainIntegerRelationState", 30.48/8.75 "relations": [] 30.48/8.75 }, 30.48/8.75 "ground": [], 30.48/8.75 "free": [], 30.48/8.75 "exprvars": [] 30.48/8.75 } 30.48/8.75 }, 30.48/8.75 "156": { 30.48/8.75 "goal": [{ 30.48/8.75 "clause": 2, 30.48/8.75 "scope": 1, 30.48/8.75 "term": "(delete T1 T2 T3)" 30.48/8.75 }], 30.48/8.75 "kb": { 30.48/8.75 "nonunifying": [], 30.48/8.75 "intvars": {}, 30.48/8.75 "arithmetic": { 30.48/8.75 "type": "PlainIntegerRelationState", 30.48/8.75 "relations": [] 30.48/8.75 }, 30.48/8.75 "ground": ["T1"], 30.48/8.75 "free": [], 30.48/8.75 "exprvars": [] 30.48/8.75 } 30.48/8.75 }, 30.48/8.75 "157": { 30.48/8.75 "goal": [ 30.48/8.75 { 30.48/8.75 "clause": 3, 30.48/8.75 "scope": 1, 30.48/8.75 "term": "(delete T1 T2 T3)" 30.48/8.75 }, 30.48/8.75 { 30.48/8.75 "clause": 4, 30.48/8.75 "scope": 1, 30.48/8.75 "term": "(delete T1 T2 T3)" 30.48/8.75 } 30.48/8.75 ], 30.48/8.75 "kb": { 30.48/8.75 "nonunifying": [], 30.48/8.75 "intvars": {}, 30.48/8.75 "arithmetic": { 30.48/8.75 "type": "PlainIntegerRelationState", 30.48/8.75 "relations": [] 30.48/8.75 }, 30.48/8.75 "ground": ["T1"], 30.48/8.75 "free": [], 30.48/8.75 "exprvars": [] 30.48/8.75 } 30.48/8.75 }, 30.48/8.75 "13": { 30.48/8.75 "goal": [{ 30.48/8.75 "clause": 0, 30.48/8.75 "scope": 1, 30.48/8.75 "term": "(delete T1 T2 T3)" 30.48/8.75 }], 30.48/8.75 "kb": { 30.48/8.75 "nonunifying": [], 30.48/8.75 "intvars": {}, 30.48/8.75 "arithmetic": { 30.48/8.75 "type": "PlainIntegerRelationState", 30.48/8.75 "relations": [] 30.48/8.75 }, 30.48/8.75 "ground": ["T1"], 30.48/8.75 "free": [], 30.48/8.75 "exprvars": [] 30.48/8.75 } 30.48/8.75 }, 30.48/8.75 "163": { 30.48/8.75 "goal": [{ 30.48/8.75 "clause": -1, 30.48/8.75 "scope": -1, 30.48/8.75 "term": "(delmin T49 T50 T51)" 30.48/8.75 }], 30.48/8.75 "kb": { 30.48/8.75 "nonunifying": [], 30.48/8.75 "intvars": {}, 30.48/8.75 "arithmetic": { 30.48/8.75 "type": "PlainIntegerRelationState", 30.48/8.75 "relations": [] 30.48/8.75 }, 30.48/8.75 "ground": [], 30.48/8.75 "free": [], 30.48/8.75 "exprvars": [] 30.48/8.75 } 30.48/8.75 }, 30.48/8.75 "164": { 30.48/8.75 "goal": [], 30.48/8.75 "kb": { 30.48/8.75 "nonunifying": [], 30.48/8.75 "intvars": {}, 30.48/8.75 "arithmetic": { 30.48/8.75 "type": "PlainIntegerRelationState", 30.48/8.75 "relations": [] 30.48/8.75 }, 30.48/8.75 "ground": [], 30.48/8.75 "free": [], 30.48/8.75 "exprvars": [] 30.48/8.75 } 30.48/8.75 }, 30.48/8.75 "364": { 30.48/8.75 "goal": [{ 30.48/8.75 "clause": 7, 30.48/8.75 "scope": 3, 30.48/8.75 "term": "(less 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[], 30.48/8.76 "free": [], 30.48/8.76 "exprvars": [] 30.48/8.76 } 30.48/8.76 }, 30.48/8.76 "461": { 30.48/8.76 "goal": [], 30.48/8.76 "kb": { 30.48/8.76 "nonunifying": [], 30.48/8.76 "intvars": {}, 30.48/8.76 "arithmetic": { 30.48/8.76 "type": "PlainIntegerRelationState", 30.48/8.76 "relations": [] 30.48/8.76 }, 30.48/8.76 "ground": [], 30.48/8.76 "free": [], 30.48/8.76 "exprvars": [] 30.48/8.76 } 30.48/8.76 }, 30.48/8.76 "187": { 30.48/8.76 "goal": [], 30.48/8.76 "kb": { 30.48/8.76 "nonunifying": [], 30.48/8.76 "intvars": {}, 30.48/8.76 "arithmetic": { 30.48/8.76 "type": "PlainIntegerRelationState", 30.48/8.76 "relations": [] 30.48/8.76 }, 30.48/8.76 "ground": [], 30.48/8.76 "free": [], 30.48/8.76 "exprvars": [] 30.48/8.76 } 30.48/8.76 }, 30.48/8.76 "341": { 30.48/8.76 "goal": [ 30.48/8.76 { 30.48/8.76 "clause": 7, 30.48/8.76 "scope": 3, 30.48/8.76 "term": "(less T111 T116)" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "clause": 8, 30.48/8.76 "scope": 3, 30.48/8.76 "term": "(less T111 T116)" 30.48/8.76 } 30.48/8.76 ], 30.48/8.76 "kb": { 30.48/8.76 "nonunifying": [], 30.48/8.76 "intvars": {}, 30.48/8.76 "arithmetic": { 30.48/8.76 "type": "PlainIntegerRelationState", 30.48/8.76 "relations": [] 30.48/8.76 }, 30.48/8.76 "ground": ["T111"], 30.48/8.76 "free": [], 30.48/8.76 "exprvars": [] 30.48/8.76 } 30.48/8.76 }, 30.48/8.76 "462": { 30.48/8.76 "goal": [], 30.48/8.76 "kb": { 30.48/8.76 "nonunifying": [], 30.48/8.76 "intvars": {}, 30.48/8.76 "arithmetic": { 30.48/8.76 "type": "PlainIntegerRelationState", 30.48/8.76 "relations": [] 30.48/8.76 }, 30.48/8.76 "ground": [], 30.48/8.76 "free": [], 30.48/8.76 "exprvars": [] 30.48/8.76 } 30.48/8.76 }, 30.48/8.76 "188": { 30.48/8.76 "goal": [], 30.48/8.76 "kb": { 30.48/8.76 "nonunifying": [], 30.48/8.76 "intvars": {}, 30.48/8.76 "arithmetic": { 30.48/8.76 "type": "PlainIntegerRelationState", 30.48/8.76 "relations": [] 30.48/8.76 }, 30.48/8.76 "ground": [], 30.48/8.76 "free": [], 30.48/8.76 "exprvars": [] 30.48/8.76 } 30.48/8.76 }, 30.48/8.76 "189": { 30.48/8.76 "goal": [{ 30.48/8.76 "clause": -1, 30.48/8.76 "scope": -1, 30.48/8.76 "term": "(delmin T84 T85 T86)" 30.48/8.76 }], 30.48/8.76 "kb": { 30.48/8.76 "nonunifying": [], 30.48/8.76 "intvars": {}, 30.48/8.76 "arithmetic": { 30.48/8.76 "type": "PlainIntegerRelationState", 30.48/8.76 "relations": [] 30.48/8.76 }, 30.48/8.76 "ground": [], 30.48/8.76 "free": [], 30.48/8.76 "exprvars": [] 30.48/8.76 } 30.48/8.76 }, 30.48/8.76 "465": { 30.48/8.76 "goal": [{ 30.48/8.76 "clause": -1, 30.48/8.76 "scope": -1, 30.48/8.76 "term": "(less T178 T177)" 30.48/8.76 }], 30.48/8.76 "kb": { 30.48/8.76 "nonunifying": [], 30.48/8.76 "intvars": {}, 30.48/8.76 "arithmetic": { 30.48/8.76 "type": "PlainIntegerRelationState", 30.48/8.76 "relations": [] 30.48/8.76 }, 30.48/8.76 "ground": ["T177"], 30.48/8.76 "free": [], 30.48/8.76 "exprvars": [] 30.48/8.76 } 30.48/8.76 }, 30.48/8.76 "466": { 30.48/8.76 "goal": [], 30.48/8.76 "kb": { 30.48/8.76 "nonunifying": [], 30.48/8.76 "intvars": {}, 30.48/8.76 "arithmetic": { 30.48/8.76 "type": "PlainIntegerRelationState", 30.48/8.76 "relations": [] 30.48/8.76 }, 30.48/8.76 "ground": [], 30.48/8.76 "free": [], 30.48/8.76 "exprvars": [] 30.48/8.76 } 30.48/8.76 } 30.48/8.76 }, 30.48/8.76 "edges": [ 30.48/8.76 { 30.48/8.76 "from": 2, 30.48/8.76 "to": 3, 30.48/8.76 "label": "CASE" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 3, 30.48/8.76 "to": 13, 30.48/8.76 "label": "PARALLEL" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 3, 30.48/8.76 "to": 20, 30.48/8.76 "label": "PARALLEL" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 13, 30.48/8.76 "to": 27, 30.48/8.76 "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" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 13, 30.48/8.76 "to": 28, 30.48/8.76 "label": "EVAL-BACKTRACK" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 20, 30.48/8.76 "to": 63, 30.48/8.76 "label": "PARALLEL" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 20, 30.48/8.76 "to": 64, 30.48/8.76 "label": "PARALLEL" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 27, 30.48/8.76 "to": 29, 30.48/8.76 "label": "SUCCESS" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 63, 30.48/8.76 "to": 151, 30.48/8.76 "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" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 63, 30.48/8.76 "to": 152, 30.48/8.76 "label": "EVAL-BACKTRACK" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 64, 30.48/8.76 "to": 156, 30.48/8.76 "label": "PARALLEL" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 64, 30.48/8.76 "to": 157, 30.48/8.76 "label": "PARALLEL" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 151, 30.48/8.76 "to": 153, 30.48/8.76 "label": "SUCCESS" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 156, 30.48/8.76 "to": 163, 30.48/8.76 "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" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 156, 30.48/8.76 "to": 164, 30.48/8.76 "label": "EVAL-BACKTRACK" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 157, 30.48/8.76 "to": 245, 30.48/8.76 "label": "PARALLEL" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 157, 30.48/8.76 "to": 246, 30.48/8.76 "label": "PARALLEL" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 163, 30.48/8.76 "to": 181, 30.48/8.76 "label": "CASE" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 181, 30.48/8.76 "to": 182, 30.48/8.76 "label": "PARALLEL" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 181, 30.48/8.76 "to": 183, 30.48/8.76 "label": "PARALLEL" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 182, 30.48/8.76 "to": 186, 30.48/8.76 "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" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 182, 30.48/8.76 "to": 187, 30.48/8.76 "label": "EVAL-BACKTRACK" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 183, 30.48/8.76 "to": 189, 30.48/8.76 "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" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 183, 30.48/8.76 "to": 190, 30.48/8.76 "label": "EVAL-BACKTRACK" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 186, 30.48/8.76 "to": 188, 30.48/8.76 "label": "SUCCESS" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 189, 30.48/8.76 "to": 163, 30.48/8.76 "label": "INSTANCE with matching:\nT49 -> T84\nT50 -> T85\nT51 -> T86" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 245, 30.48/8.76 "to": 259, 30.48/8.76 "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" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 245, 30.48/8.76 "to": 260, 30.48/8.76 "label": "EVAL-BACKTRACK" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 246, 30.48/8.76 "to": 446, 30.48/8.76 "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" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 246, 30.48/8.76 "to": 448, 30.48/8.76 "label": "EVAL-BACKTRACK" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 259, 30.48/8.76 "to": 261, 30.48/8.76 "label": "SPLIT 1" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 259, 30.48/8.76 "to": 262, 30.48/8.76 "label": "SPLIT 2\nnew knowledge:\nT111 is ground\nreplacements:T117 -> T121,\nT118 -> T122" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 261, 30.48/8.76 "to": 341, 30.48/8.76 "label": "CASE" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 262, 30.48/8.76 "to": 2, 30.48/8.76 "label": "INSTANCE with matching:\nT1 -> T111\nT2 -> T121\nT3 -> T122" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 341, 30.48/8.76 "to": 364, 30.48/8.76 "label": "PARALLEL" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 341, 30.48/8.76 "to": 366, 30.48/8.76 "label": "PARALLEL" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 364, 30.48/8.76 "to": 404, 30.48/8.76 "label": "EVAL with clause\nless(0, s(X118)).\nand substitutionT111 -> 0,\nX118 -> T129,\nT116 -> s(T129)" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 364, 30.48/8.76 "to": 405, 30.48/8.76 "label": "EVAL-BACKTRACK" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 366, 30.48/8.76 "to": 412, 30.48/8.76 "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" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 366, 30.48/8.76 "to": 415, 30.48/8.76 "label": "EVAL-BACKTRACK" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 404, 30.48/8.76 "to": 406, 30.48/8.76 "label": "SUCCESS" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 412, 30.48/8.76 "to": 261, 30.48/8.76 "label": "INSTANCE with matching:\nT111 -> T134\nT116 -> T136" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 446, 30.48/8.76 "to": 451, 30.48/8.76 "label": "SPLIT 1" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 446, 30.48/8.76 "to": 453, 30.48/8.76 "label": "SPLIT 2\nnew knowledge:\nT158 is ground\nT153 is ground\nreplacements:T159 -> T163,\nT160 -> T164" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 451, 30.48/8.76 "to": 455, 30.48/8.76 "label": "CASE" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 453, 30.48/8.76 "to": 2, 30.48/8.76 "label": "INSTANCE with matching:\nT1 -> T153\nT2 -> T163\nT3 -> T164" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 455, 30.48/8.76 "to": 458, 30.48/8.76 "label": "PARALLEL" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 455, 30.48/8.76 "to": 459, 30.48/8.76 "label": "PARALLEL" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 458, 30.48/8.76 "to": 460, 30.48/8.76 "label": "EVAL with clause\nless(0, s(X154)).\nand substitutionT158 -> 0,\nX154 -> T171,\nT153 -> s(T171)" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 458, 30.48/8.76 "to": 461, 30.48/8.76 "label": "EVAL-BACKTRACK" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 459, 30.48/8.76 "to": 465, 30.48/8.76 "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" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 459, 30.48/8.76 "to": 466, 30.48/8.76 "label": "EVAL-BACKTRACK" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 460, 30.48/8.76 "to": 462, 30.48/8.76 "label": "SUCCESS" 30.48/8.76 }, 30.48/8.76 { 30.48/8.76 "from": 465, 30.48/8.76 "to": 451, 30.48/8.76 "label": "INSTANCE with matching:\nT158 -> T178\nT153 -> T177" 30.48/8.76 } 30.48/8.76 ], 30.48/8.76 "type": "Graph" 30.48/8.76 } 30.48/8.76 } 30.48/8.76 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (108) 30.48/8.76 Complex Obligation (AND) 30.48/8.76 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (109) 30.48/8.76 Obligation: 30.48/8.76 Rules: 30.48/8.76 f455_in(T153) -> f458_in(T153) :|: TRUE 30.48/8.76 f455_in(x) -> f459_in(x) :|: TRUE 30.48/8.76 f458_out(x1) -> f455_out(x1) :|: TRUE 30.48/8.76 f459_out(x2) -> f455_out(x2) :|: TRUE 30.48/8.76 f455_out(x3) -> f451_out(x3) :|: TRUE 30.48/8.76 f451_in(x4) -> f455_in(x4) :|: TRUE 30.48/8.76 f459_in(s(T177)) -> f465_in(T177) :|: TRUE 30.48/8.76 f465_out(x5) -> f459_out(s(x5)) :|: TRUE 30.48/8.76 f466_out -> f459_out(x6) :|: TRUE 30.48/8.76 f459_in(x7) -> f466_in :|: TRUE 30.48/8.76 f465_in(x8) -> f451_in(x8) :|: TRUE 30.48/8.76 f451_out(x9) -> f465_out(x9) :|: TRUE 30.48/8.76 f2_in(T1) -> f3_in(T1) :|: TRUE 30.48/8.76 f3_out(x10) -> f2_out(x10) :|: TRUE 30.48/8.76 f3_in(x11) -> f20_in(x11) :|: TRUE 30.48/8.76 f3_in(x12) -> f13_in(x12) :|: TRUE 30.48/8.76 f20_out(x13) -> f3_out(x13) :|: TRUE 30.48/8.76 f13_out(x14) -> f3_out(x14) :|: TRUE 30.48/8.76 f64_out(x15) -> f20_out(x15) :|: TRUE 30.48/8.76 f20_in(x16) -> f64_in(x16) :|: TRUE 30.48/8.76 f20_in(x17) -> f63_in(x17) :|: TRUE 30.48/8.76 f63_out(x18) -> f20_out(x18) :|: TRUE 30.48/8.76 f64_in(x19) -> f157_in(x19) :|: TRUE 30.48/8.76 f157_out(x20) -> f64_out(x20) :|: TRUE 30.48/8.76 f64_in(x21) -> f156_in(x21) :|: TRUE 30.48/8.76 f156_out(x22) -> f64_out(x22) :|: TRUE 30.48/8.76 f157_in(x23) -> f245_in(x23) :|: TRUE 30.48/8.76 f157_in(x24) -> f246_in(x24) :|: TRUE 30.48/8.76 f246_out(x25) -> f157_out(x25) :|: TRUE 30.48/8.76 f245_out(x26) -> f157_out(x26) :|: TRUE 30.48/8.76 f246_in(x27) -> f446_in(x27) :|: TRUE 30.48/8.76 f448_out -> f246_out(x28) :|: TRUE 30.48/8.76 f446_out(x29) -> f246_out(x29) :|: TRUE 30.48/8.76 f246_in(x30) -> f448_in :|: TRUE 30.48/8.76 f453_out(x31) -> f446_out(x31) :|: TRUE 30.48/8.76 f446_in(x32) -> f451_in(x32) :|: TRUE 30.48/8.76 f451_out(x33) -> f453_in(x33) :|: TRUE 30.48/8.76 Start term: f2_in(T1) 30.48/8.76 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (110) IRSwTSimpleDependencyGraphProof (EQUIVALENT) 30.48/8.76 Constructed simple dependency graph. 30.48/8.76 30.48/8.76 Simplified to the following IRSwTs: 30.48/8.76 30.48/8.76 intTRSProblem: 30.48/8.76 f455_in(x) -> f459_in(x) :|: TRUE 30.48/8.76 f451_in(x4) -> f455_in(x4) :|: TRUE 30.48/8.76 f459_in(s(T177)) -> f465_in(T177) :|: TRUE 30.48/8.76 f465_in(x8) -> f451_in(x8) :|: TRUE 30.48/8.76 30.48/8.76 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (111) 30.48/8.76 Obligation: 30.48/8.76 Rules: 30.48/8.76 f455_in(x) -> f459_in(x) :|: TRUE 30.48/8.76 f451_in(x4) -> f455_in(x4) :|: TRUE 30.48/8.76 f459_in(s(T177)) -> f465_in(T177) :|: TRUE 30.48/8.76 f465_in(x8) -> f451_in(x8) :|: TRUE 30.48/8.76 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (112) IntTRSCompressionProof (EQUIVALENT) 30.48/8.76 Compressed rules. 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (113) 30.48/8.76 Obligation: 30.48/8.76 Rules: 30.48/8.76 f451_in(s(T177:0)) -> f451_in(T177:0) :|: TRUE 30.48/8.76 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (114) IRSFormatTransformerProof (EQUIVALENT) 30.48/8.76 Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (115) 30.48/8.76 Obligation: 30.48/8.76 Rules: 30.48/8.76 f451_in(s(T177:0)) -> f451_in(T177:0) :|: TRUE 30.48/8.76 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (116) IRSwTTerminationDigraphProof (EQUIVALENT) 30.48/8.76 Constructed termination digraph! 30.48/8.76 Nodes: 30.48/8.76 (1) f451_in(s(T177:0)) -> f451_in(T177:0) :|: TRUE 30.48/8.76 30.48/8.76 Arcs: 30.48/8.76 (1) -> (1) 30.48/8.76 30.48/8.76 This digraph is fully evaluated! 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (117) 30.48/8.76 Obligation: 30.48/8.76 30.48/8.76 Termination digraph: 30.48/8.76 Nodes: 30.48/8.76 (1) f451_in(s(T177:0)) -> f451_in(T177:0) :|: TRUE 30.48/8.76 30.48/8.76 Arcs: 30.48/8.76 (1) -> (1) 30.48/8.76 30.48/8.76 This digraph is fully evaluated! 30.48/8.76 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (118) TempFilterProof (SOUND) 30.48/8.76 Used the following sort dictionary for filtering: 30.48/8.76 f451_in(VARIABLE) 30.48/8.76 s(VARIABLE) 30.48/8.76 Removed predefined arithmetic. 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (119) 30.48/8.76 Obligation: 30.48/8.76 Rules: 30.48/8.76 f451_in(s(T177:0)) -> f451_in(T177:0) 30.48/8.76 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (120) IRSwTToQDPProof (SOUND) 30.48/8.76 Removed the integers and created a QDP-Problem. 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (121) 30.48/8.76 Obligation: 30.48/8.76 Q DP problem: 30.48/8.76 The TRS P consists of the following rules: 30.48/8.76 30.48/8.76 f451_in(s(T177:0)) -> f451_in(T177:0) 30.48/8.76 30.48/8.76 R is empty. 30.48/8.76 Q is empty. 30.48/8.76 We have to consider all (P,Q,R)-chains. 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (122) QDPSizeChangeProof (EQUIVALENT) 30.48/8.76 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. 30.48/8.76 30.48/8.76 From the DPs we obtained the following set of size-change graphs: 30.48/8.76 *f451_in(s(T177:0)) -> f451_in(T177:0) 30.48/8.76 The graph contains the following edges 1 > 1 30.48/8.76 30.48/8.76 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (123) 30.48/8.76 YES 30.48/8.76 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (124) 30.48/8.76 Obligation: 30.48/8.76 Rules: 30.48/8.76 f261_in(T111) -> f341_in(T111) :|: TRUE 30.48/8.76 f341_out(x) -> f261_out(x) :|: TRUE 30.48/8.76 f415_out -> f366_out(x1) :|: TRUE 30.48/8.76 f366_in(x2) -> f415_in :|: TRUE 30.48/8.76 f366_in(s(T134)) -> f412_in(T134) :|: TRUE 30.48/8.76 f412_out(x3) -> f366_out(s(x3)) :|: TRUE 30.48/8.76 f261_out(x4) -> f412_out(x4) :|: TRUE 30.48/8.76 f412_in(x5) -> f261_in(x5) :|: TRUE 30.48/8.76 f341_in(x6) -> f364_in(x6) :|: TRUE 30.48/8.76 f366_out(x7) -> f341_out(x7) :|: TRUE 30.48/8.76 f364_out(x8) -> f341_out(x8) :|: TRUE 30.48/8.76 f341_in(x9) -> f366_in(x9) :|: TRUE 30.48/8.76 f2_in(T1) -> f3_in(T1) :|: TRUE 30.48/8.76 f3_out(x10) -> f2_out(x10) :|: TRUE 30.48/8.76 f3_in(x11) -> f20_in(x11) :|: TRUE 30.48/8.76 f3_in(x12) -> f13_in(x12) :|: TRUE 30.48/8.76 f20_out(x13) -> f3_out(x13) :|: TRUE 30.48/8.76 f13_out(x14) -> f3_out(x14) :|: TRUE 30.48/8.76 f64_out(x15) -> f20_out(x15) :|: TRUE 30.48/8.76 f20_in(x16) -> f64_in(x16) :|: TRUE 30.48/8.76 f20_in(x17) -> f63_in(x17) :|: TRUE 30.48/8.76 f63_out(x18) -> f20_out(x18) :|: TRUE 30.48/8.76 f64_in(x19) -> f157_in(x19) :|: TRUE 30.48/8.76 f157_out(x20) -> f64_out(x20) :|: TRUE 30.48/8.76 f64_in(x21) -> f156_in(x21) :|: TRUE 30.48/8.76 f156_out(x22) -> f64_out(x22) :|: TRUE 30.48/8.76 f157_in(x23) -> f245_in(x23) :|: TRUE 30.48/8.76 f157_in(x24) -> f246_in(x24) :|: TRUE 30.48/8.76 f246_out(x25) -> f157_out(x25) :|: TRUE 30.48/8.76 f245_out(x26) -> f157_out(x26) :|: TRUE 30.48/8.76 f245_in(x27) -> f259_in(x27) :|: TRUE 30.48/8.76 f260_out -> f245_out(x28) :|: TRUE 30.48/8.76 f259_out(x29) -> f245_out(x29) :|: TRUE 30.48/8.76 f245_in(x30) -> f260_in :|: TRUE 30.48/8.76 f262_out(x31) -> f259_out(x31) :|: TRUE 30.48/8.76 f259_in(x32) -> f261_in(x32) :|: TRUE 30.48/8.76 f261_out(x33) -> f262_in(x33) :|: TRUE 30.48/8.76 Start term: f2_in(T1) 30.48/8.76 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (125) IRSwTSimpleDependencyGraphProof (EQUIVALENT) 30.48/8.76 Constructed simple dependency graph. 30.48/8.76 30.48/8.76 Simplified to the following IRSwTs: 30.48/8.76 30.48/8.76 intTRSProblem: 30.48/8.76 f261_in(T111) -> f341_in(T111) :|: TRUE 30.48/8.76 f366_in(s(T134)) -> f412_in(T134) :|: TRUE 30.48/8.76 f412_in(x5) -> f261_in(x5) :|: TRUE 30.48/8.76 f341_in(x9) -> f366_in(x9) :|: TRUE 30.48/8.76 30.48/8.76 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (126) 30.48/8.76 Obligation: 30.48/8.76 Rules: 30.48/8.76 f261_in(T111) -> f341_in(T111) :|: TRUE 30.48/8.76 f366_in(s(T134)) -> f412_in(T134) :|: TRUE 30.48/8.76 f412_in(x5) -> f261_in(x5) :|: TRUE 30.48/8.76 f341_in(x9) -> f366_in(x9) :|: TRUE 30.48/8.76 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (127) IntTRSCompressionProof (EQUIVALENT) 30.48/8.76 Compressed rules. 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (128) 30.48/8.76 Obligation: 30.48/8.76 Rules: 30.48/8.76 f366_in(s(T134:0)) -> f366_in(T134:0) :|: TRUE 30.48/8.76 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (129) IRSFormatTransformerProof (EQUIVALENT) 30.48/8.76 Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (130) 30.48/8.76 Obligation: 30.48/8.76 Rules: 30.48/8.76 f366_in(s(T134:0)) -> f366_in(T134:0) :|: TRUE 30.48/8.76 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (131) IRSwTTerminationDigraphProof (EQUIVALENT) 30.48/8.76 Constructed termination digraph! 30.48/8.76 Nodes: 30.48/8.76 (1) f366_in(s(T134:0)) -> f366_in(T134:0) :|: TRUE 30.48/8.76 30.48/8.76 Arcs: 30.48/8.76 (1) -> (1) 30.48/8.76 30.48/8.76 This digraph is fully evaluated! 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (132) 30.48/8.76 Obligation: 30.48/8.76 30.48/8.76 Termination digraph: 30.48/8.76 Nodes: 30.48/8.76 (1) f366_in(s(T134:0)) -> f366_in(T134:0) :|: TRUE 30.48/8.76 30.48/8.76 Arcs: 30.48/8.76 (1) -> (1) 30.48/8.76 30.48/8.76 This digraph is fully evaluated! 30.48/8.76 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (133) TempFilterProof (SOUND) 30.48/8.76 Used the following sort dictionary for filtering: 30.48/8.76 f366_in(VARIABLE) 30.48/8.76 s(VARIABLE) 30.48/8.76 Removed predefined arithmetic. 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (134) 30.48/8.76 Obligation: 30.48/8.76 Rules: 30.48/8.76 f366_in(s(T134:0)) -> f366_in(T134:0) 30.48/8.76 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (135) IRSwTToQDPProof (SOUND) 30.48/8.76 Removed the integers and created a QDP-Problem. 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (136) 30.48/8.76 Obligation: 30.48/8.76 Q DP problem: 30.48/8.76 The TRS P consists of the following rules: 30.48/8.76 30.48/8.76 f366_in(s(T134:0)) -> f366_in(T134:0) 30.48/8.76 30.48/8.76 R is empty. 30.48/8.76 Q is empty. 30.48/8.76 We have to consider all (P,Q,R)-chains. 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (137) QDPSizeChangeProof (EQUIVALENT) 30.48/8.76 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. 30.48/8.76 30.48/8.76 From the DPs we obtained the following set of size-change graphs: 30.48/8.76 *f366_in(s(T134:0)) -> f366_in(T134:0) 30.48/8.76 The graph contains the following edges 1 > 1 30.48/8.76 30.48/8.76 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (138) 30.48/8.76 YES 30.48/8.76 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (139) 30.48/8.76 Obligation: 30.48/8.76 Rules: 30.48/8.76 f189_in -> f163_in :|: TRUE 30.48/8.76 f163_out -> f189_out :|: TRUE 30.48/8.76 f183_out -> f181_out :|: TRUE 30.48/8.76 f182_out -> f181_out :|: TRUE 30.48/8.76 f181_in -> f182_in :|: TRUE 30.48/8.76 f181_in -> f183_in :|: TRUE 30.48/8.76 f181_out -> f163_out :|: TRUE 30.48/8.76 f163_in -> f181_in :|: TRUE 30.48/8.76 f190_out -> f183_out :|: TRUE 30.48/8.76 f183_in -> f190_in :|: TRUE 30.48/8.76 f189_out -> f183_out :|: TRUE 30.48/8.76 f183_in -> f189_in :|: TRUE 30.48/8.76 f2_in(T1) -> f3_in(T1) :|: TRUE 30.48/8.76 f3_out(x) -> f2_out(x) :|: TRUE 30.48/8.76 f3_in(x1) -> f20_in(x1) :|: TRUE 30.48/8.76 f3_in(x2) -> f13_in(x2) :|: TRUE 30.48/8.76 f20_out(x3) -> f3_out(x3) :|: TRUE 30.48/8.76 f13_out(x4) -> f3_out(x4) :|: TRUE 30.48/8.76 f64_out(x5) -> f20_out(x5) :|: TRUE 30.48/8.76 f20_in(x6) -> f64_in(x6) :|: TRUE 30.48/8.76 f20_in(x7) -> f63_in(x7) :|: TRUE 30.48/8.76 f63_out(x8) -> f20_out(x8) :|: TRUE 30.48/8.76 f64_in(x9) -> f157_in(x9) :|: TRUE 30.48/8.76 f157_out(x10) -> f64_out(x10) :|: TRUE 30.48/8.76 f64_in(x11) -> f156_in(x11) :|: TRUE 30.48/8.76 f156_out(x12) -> f64_out(x12) :|: TRUE 30.48/8.76 f156_in(x13) -> f164_in :|: TRUE 30.48/8.76 f163_out -> f156_out(T44) :|: TRUE 30.48/8.76 f164_out -> f156_out(x14) :|: TRUE 30.48/8.76 f156_in(x15) -> f163_in :|: TRUE 30.48/8.76 Start term: f2_in(T1) 30.48/8.76 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (140) IRSwTSimpleDependencyGraphProof (EQUIVALENT) 30.48/8.76 Constructed simple dependency graph. 30.48/8.76 30.48/8.76 Simplified to the following IRSwTs: 30.48/8.76 30.48/8.76 intTRSProblem: 30.48/8.76 f189_in -> f163_in :|: TRUE 30.48/8.76 f181_in -> f183_in :|: TRUE 30.48/8.76 f163_in -> f181_in :|: TRUE 30.48/8.76 f183_in -> f189_in :|: TRUE 30.48/8.76 30.48/8.76 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (141) 30.48/8.76 Obligation: 30.48/8.76 Rules: 30.48/8.76 f189_in -> f163_in :|: TRUE 30.48/8.76 f181_in -> f183_in :|: TRUE 30.48/8.76 f163_in -> f181_in :|: TRUE 30.48/8.76 f183_in -> f189_in :|: TRUE 30.48/8.76 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (142) IntTRSCompressionProof (EQUIVALENT) 30.48/8.76 Compressed rules. 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (143) 30.48/8.76 Obligation: 30.48/8.76 Rules: 30.48/8.76 f189_in -> f189_in :|: TRUE 30.48/8.76 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (144) IRSFormatTransformerProof (EQUIVALENT) 30.48/8.76 Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (145) 30.48/8.76 Obligation: 30.48/8.76 Rules: 30.48/8.76 f189_in -> f189_in :|: TRUE 30.48/8.76 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (146) IRSwTTerminationDigraphProof (EQUIVALENT) 30.48/8.76 Constructed termination digraph! 30.48/8.76 Nodes: 30.48/8.76 (1) f189_in -> f189_in :|: TRUE 30.48/8.76 30.48/8.76 Arcs: 30.48/8.76 (1) -> (1) 30.48/8.76 30.48/8.76 This digraph is fully evaluated! 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (147) 30.48/8.76 Obligation: 30.48/8.76 30.48/8.76 Termination digraph: 30.48/8.76 Nodes: 30.48/8.76 (1) f189_in -> f189_in :|: TRUE 30.48/8.76 30.48/8.76 Arcs: 30.48/8.76 (1) -> (1) 30.48/8.76 30.48/8.76 This digraph is fully evaluated! 30.48/8.76 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (148) FilterProof (EQUIVALENT) 30.48/8.76 Used the following sort dictionary for filtering: 30.48/8.76 f189_in() 30.48/8.76 Replaced non-predefined constructor symbols by 0. 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (149) 30.48/8.76 Obligation: 30.48/8.76 Rules: 30.48/8.76 f189_in -> f189_in :|: TRUE 30.48/8.76 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (150) IntTRSPeriodicNontermProof (COMPLETE) 30.48/8.76 Normalized system to the following form: 30.48/8.76 f(pc) -> f(1) :|: pc = 1 && TRUE 30.48/8.76 Witness term starting non-terminating reduction: f(1) 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (151) 30.48/8.76 NO 30.48/8.76 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (152) 30.48/8.76 Obligation: 30.48/8.76 Rules: 30.48/8.76 f455_in(T153) -> f458_in(T153) :|: TRUE 30.48/8.76 f455_in(x) -> f459_in(x) :|: TRUE 30.48/8.76 f458_out(x1) -> f455_out(x1) :|: TRUE 30.48/8.76 f459_out(x2) -> f455_out(x2) :|: TRUE 30.48/8.76 f460_in -> f460_out :|: TRUE 30.48/8.76 f64_in(T1) -> f157_in(T1) :|: TRUE 30.48/8.76 f157_out(x3) -> f64_out(x3) :|: TRUE 30.48/8.76 f64_in(x4) -> f156_in(x4) :|: TRUE 30.48/8.76 f156_out(x5) -> f64_out(x5) :|: TRUE 30.48/8.76 f341_in(T111) -> f364_in(T111) :|: TRUE 30.48/8.76 f366_out(x6) -> f341_out(x6) :|: TRUE 30.48/8.76 f364_out(x7) -> f341_out(x7) :|: TRUE 30.48/8.76 f341_in(x8) -> f366_in(x8) :|: TRUE 30.48/8.76 f404_in -> f404_out :|: TRUE 30.48/8.76 f261_in(x9) -> f341_in(x9) :|: TRUE 30.48/8.76 f341_out(x10) -> f261_out(x10) :|: TRUE 30.48/8.76 f459_in(s(T177)) -> f465_in(T177) :|: TRUE 30.48/8.76 f465_out(x11) -> f459_out(s(x11)) :|: TRUE 30.48/8.76 f466_out -> f459_out(x12) :|: TRUE 30.48/8.76 f459_in(x13) -> f466_in :|: TRUE 30.48/8.76 f157_in(x14) -> f245_in(x14) :|: TRUE 30.48/8.76 f157_in(x15) -> f246_in(x15) :|: TRUE 30.48/8.76 f246_out(x16) -> f157_out(x16) :|: TRUE 30.48/8.76 f245_out(x17) -> f157_out(x17) :|: TRUE 30.48/8.76 f262_in(x18) -> f2_in(x18) :|: TRUE 30.48/8.76 f2_out(x19) -> f262_out(x19) :|: TRUE 30.48/8.76 f64_out(x20) -> f20_out(x20) :|: TRUE 30.48/8.76 f20_in(x21) -> f64_in(x21) :|: TRUE 30.48/8.76 f20_in(x22) -> f63_in(x22) :|: TRUE 30.48/8.76 f63_out(x23) -> f20_out(x23) :|: TRUE 30.48/8.76 f2_in(x24) -> f3_in(x24) :|: TRUE 30.48/8.76 f3_out(x25) -> f2_out(x25) :|: TRUE 30.48/8.76 f453_out(x26) -> f446_out(x26) :|: TRUE 30.48/8.76 f446_in(x27) -> f451_in(x27) :|: TRUE 30.48/8.76 f451_out(x28) -> f453_in(x28) :|: TRUE 30.48/8.76 f364_in(0) -> f404_in :|: TRUE 30.48/8.76 f364_in(x29) -> f405_in :|: TRUE 30.48/8.76 f405_out -> f364_out(x30) :|: TRUE 30.48/8.76 f404_out -> f364_out(0) :|: TRUE 30.48/8.76 f262_out(x31) -> f259_out(x31) :|: TRUE 30.48/8.76 f259_in(x32) -> f261_in(x32) :|: TRUE 30.48/8.76 f261_out(x33) -> f262_in(x33) :|: TRUE 30.48/8.76 f3_in(x34) -> f20_in(x34) :|: TRUE 30.48/8.76 f3_in(x35) -> f13_in(x35) :|: TRUE 30.48/8.76 f20_out(x36) -> f3_out(x36) :|: TRUE 30.48/8.76 f13_out(x37) -> f3_out(x37) :|: TRUE 30.48/8.76 f245_in(x38) -> f259_in(x38) :|: TRUE 30.48/8.76 f260_out -> f245_out(x39) :|: TRUE 30.48/8.76 f259_out(x40) -> f245_out(x40) :|: TRUE 30.48/8.76 f245_in(x41) -> f260_in :|: TRUE 30.48/8.76 f415_out -> f366_out(x42) :|: TRUE 30.48/8.76 f366_in(x43) -> f415_in :|: TRUE 30.48/8.76 f366_in(s(T134)) -> f412_in(T134) :|: TRUE 30.48/8.76 f412_out(x44) -> f366_out(s(x44)) :|: TRUE 30.48/8.76 f453_in(x45) -> f2_in(x45) :|: TRUE 30.48/8.76 f2_out(x46) -> f453_out(x46) :|: TRUE 30.48/8.76 f465_in(x47) -> f451_in(x47) :|: TRUE 30.48/8.76 f451_out(x48) -> f465_out(x48) :|: TRUE 30.48/8.76 f455_out(x49) -> f451_out(x49) :|: TRUE 30.48/8.76 f451_in(x50) -> f455_in(x50) :|: TRUE 30.48/8.76 f246_in(x51) -> f446_in(x51) :|: TRUE 30.48/8.76 f448_out -> f246_out(x52) :|: TRUE 30.48/8.76 f446_out(x53) -> f246_out(x53) :|: TRUE 30.48/8.76 f246_in(x54) -> f448_in :|: TRUE 30.48/8.76 f458_in(x55) -> f461_in :|: TRUE 30.48/8.76 f460_out -> f458_out(s(T171)) :|: TRUE 30.48/8.76 f458_in(s(x56)) -> f460_in :|: TRUE 30.48/8.76 f461_out -> f458_out(x57) :|: TRUE 30.48/8.76 f261_out(x58) -> f412_out(x58) :|: TRUE 30.48/8.76 f412_in(x59) -> f261_in(x59) :|: TRUE 30.48/8.76 Start term: f2_in(T1) 30.48/8.76 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (153) IRSwTSimpleDependencyGraphProof (EQUIVALENT) 30.48/8.76 Constructed simple dependency graph. 30.48/8.76 30.48/8.76 Simplified to the following IRSwTs: 30.48/8.76 30.48/8.76 intTRSProblem: 30.48/8.76 f455_in(T153) -> f458_in(T153) :|: TRUE 30.48/8.76 f455_in(x) -> f459_in(x) :|: TRUE 30.48/8.76 f458_out(x1) -> f455_out(x1) :|: TRUE 30.48/8.76 f459_out(x2) -> f455_out(x2) :|: TRUE 30.48/8.76 f460_in -> f460_out :|: TRUE 30.48/8.76 f64_in(T1) -> f157_in(T1) :|: TRUE 30.48/8.76 f341_in(T111) -> f364_in(T111) :|: TRUE 30.48/8.76 f366_out(x6) -> f341_out(x6) :|: TRUE 30.48/8.76 f364_out(x7) -> f341_out(x7) :|: TRUE 30.48/8.76 f341_in(x8) -> f366_in(x8) :|: TRUE 30.48/8.76 f404_in -> f404_out :|: TRUE 30.48/8.76 f261_in(x9) -> f341_in(x9) :|: TRUE 30.48/8.76 f341_out(x10) -> f261_out(x10) :|: TRUE 30.48/8.76 f459_in(s(T177)) -> f465_in(T177) :|: TRUE 30.48/8.76 f465_out(x11) -> f459_out(s(x11)) :|: TRUE 30.48/8.76 f157_in(x14) -> f245_in(x14) :|: TRUE 30.48/8.76 f157_in(x15) -> f246_in(x15) :|: TRUE 30.48/8.76 f262_in(x18) -> f2_in(x18) :|: TRUE 30.48/8.76 f20_in(x21) -> f64_in(x21) :|: TRUE 30.48/8.76 f2_in(x24) -> f3_in(x24) :|: TRUE 30.48/8.76 f446_in(x27) -> f451_in(x27) :|: TRUE 30.48/8.76 f451_out(x28) -> f453_in(x28) :|: TRUE 30.48/8.76 f364_in(0) -> f404_in :|: TRUE 30.48/8.76 f404_out -> f364_out(0) :|: TRUE 30.48/8.76 f259_in(x32) -> f261_in(x32) :|: TRUE 30.48/8.76 f261_out(x33) -> f262_in(x33) :|: TRUE 30.48/8.76 f3_in(x34) -> f20_in(x34) :|: TRUE 30.48/8.76 f245_in(x38) -> f259_in(x38) :|: TRUE 30.48/8.76 f366_in(s(T134)) -> f412_in(T134) :|: TRUE 30.48/8.76 f412_out(x44) -> f366_out(s(x44)) :|: TRUE 30.48/8.76 f453_in(x45) -> f2_in(x45) :|: TRUE 30.48/8.76 f465_in(x47) -> f451_in(x47) :|: TRUE 30.48/8.76 f451_out(x48) -> f465_out(x48) :|: TRUE 30.48/8.76 f455_out(x49) -> f451_out(x49) :|: TRUE 30.48/8.76 f451_in(x50) -> f455_in(x50) :|: TRUE 30.48/8.76 f246_in(x51) -> f446_in(x51) :|: TRUE 30.48/8.76 f460_out -> f458_out(s(T171)) :|: TRUE 30.48/8.76 f458_in(s(x56)) -> f460_in :|: TRUE 30.48/8.76 f261_out(x58) -> f412_out(x58) :|: TRUE 30.48/8.76 f412_in(x59) -> f261_in(x59) :|: TRUE 30.48/8.76 30.48/8.76 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (154) 30.48/8.76 Obligation: 30.48/8.76 Rules: 30.48/8.76 f455_in(T153) -> f458_in(T153) :|: TRUE 30.48/8.76 f455_in(x) -> f459_in(x) :|: TRUE 30.48/8.76 f458_out(x1) -> f455_out(x1) :|: TRUE 30.48/8.76 f459_out(x2) -> f455_out(x2) :|: TRUE 30.48/8.76 f460_in -> f460_out :|: TRUE 30.48/8.76 f64_in(T1) -> f157_in(T1) :|: TRUE 30.48/8.76 f341_in(T111) -> f364_in(T111) :|: TRUE 30.48/8.76 f366_out(x6) -> f341_out(x6) :|: TRUE 30.48/8.76 f364_out(x7) -> f341_out(x7) :|: TRUE 30.48/8.76 f341_in(x8) -> f366_in(x8) :|: TRUE 30.48/8.76 f404_in -> f404_out :|: TRUE 30.48/8.76 f261_in(x9) -> f341_in(x9) :|: TRUE 30.48/8.76 f341_out(x10) -> f261_out(x10) :|: TRUE 30.48/8.76 f459_in(s(T177)) -> f465_in(T177) :|: TRUE 30.48/8.76 f465_out(x11) -> f459_out(s(x11)) :|: TRUE 30.48/8.76 f157_in(x14) -> f245_in(x14) :|: TRUE 30.48/8.76 f157_in(x15) -> f246_in(x15) :|: TRUE 30.48/8.76 f262_in(x18) -> f2_in(x18) :|: TRUE 30.48/8.76 f20_in(x21) -> f64_in(x21) :|: TRUE 30.48/8.76 f2_in(x24) -> f3_in(x24) :|: TRUE 30.48/8.76 f446_in(x27) -> f451_in(x27) :|: TRUE 30.48/8.76 f451_out(x28) -> f453_in(x28) :|: TRUE 30.48/8.76 f364_in(0) -> f404_in :|: TRUE 30.48/8.76 f404_out -> f364_out(0) :|: TRUE 30.48/8.76 f259_in(x32) -> f261_in(x32) :|: TRUE 30.48/8.76 f261_out(x33) -> f262_in(x33) :|: TRUE 30.48/8.76 f3_in(x34) -> f20_in(x34) :|: TRUE 30.48/8.76 f245_in(x38) -> f259_in(x38) :|: TRUE 30.48/8.76 f366_in(s(T134)) -> f412_in(T134) :|: TRUE 30.48/8.76 f412_out(x44) -> f366_out(s(x44)) :|: TRUE 30.48/8.76 f453_in(x45) -> f2_in(x45) :|: TRUE 30.48/8.76 f465_in(x47) -> f451_in(x47) :|: TRUE 30.48/8.76 f451_out(x48) -> f465_out(x48) :|: TRUE 30.48/8.76 f455_out(x49) -> f451_out(x49) :|: TRUE 30.48/8.76 f451_in(x50) -> f455_in(x50) :|: TRUE 30.48/8.76 f246_in(x51) -> f446_in(x51) :|: TRUE 30.48/8.76 f460_out -> f458_out(s(T171)) :|: TRUE 30.48/8.76 f458_in(s(x56)) -> f460_in :|: TRUE 30.48/8.76 f261_out(x58) -> f412_out(x58) :|: TRUE 30.48/8.76 f412_in(x59) -> f261_in(x59) :|: TRUE 30.48/8.76 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (155) IntTRSCompressionProof (EQUIVALENT) 30.48/8.76 Compressed rules. 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (156) 30.48/8.76 Obligation: 30.48/8.76 Rules: 30.48/8.76 f451_out(x28:0) -> f20_in(x28:0) :|: TRUE 30.48/8.76 f20_in(x21:0) -> f261_in(x21:0) :|: TRUE 30.48/8.76 f261_out(x33:0) -> f20_in(x33:0) :|: TRUE 30.48/8.76 f20_in(x) -> f455_in(x) :|: TRUE 30.48/8.76 f261_in(cons_0) -> f261_out(0) :|: TRUE && cons_0 = 0 30.48/8.76 f455_in(s(x56:0)) -> f451_out(s(T171:0)) :|: TRUE 30.48/8.76 f261_in(s(T134:0)) -> f261_in(T134:0) :|: TRUE 30.48/8.76 f455_in(s(T177:0)) -> f455_in(T177:0) :|: TRUE 30.48/8.76 f451_out(x48:0) -> f451_out(s(x48:0)) :|: TRUE 30.48/8.76 f261_out(x58:0) -> f261_out(s(x58:0)) :|: TRUE 30.48/8.76 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (157) IRSFormatTransformerProof (EQUIVALENT) 30.48/8.76 Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (158) 30.48/8.76 Obligation: 30.48/8.76 Rules: 30.48/8.76 f451_out(x28:0) -> f20_in(x28:0) :|: TRUE 30.48/8.76 f20_in(x21:0) -> f261_in(x21:0) :|: TRUE 30.48/8.76 f261_out(x33:0) -> f20_in(x33:0) :|: TRUE 30.48/8.76 f20_in(x) -> f455_in(x) :|: TRUE 30.48/8.76 f261_in(cons_0) -> f261_out(0) :|: TRUE && cons_0 = 0 30.48/8.76 f455_in(s(x56:0)) -> f451_out(s(T171:0)) :|: TRUE 30.48/8.76 f261_in(s(T134:0)) -> f261_in(T134:0) :|: TRUE 30.48/8.76 f455_in(s(T177:0)) -> f455_in(T177:0) :|: TRUE 30.48/8.76 f451_out(x48:0) -> f451_out(s(x48:0)) :|: TRUE 30.48/8.76 f261_out(x58:0) -> f261_out(s(x58:0)) :|: TRUE 30.48/8.76 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (159) IRSwTTerminationDigraphProof (EQUIVALENT) 30.48/8.76 Constructed termination digraph! 30.48/8.76 Nodes: 30.48/8.76 (1) f451_out(x28:0) -> f20_in(x28:0) :|: TRUE 30.48/8.76 (2) f20_in(x21:0) -> f261_in(x21:0) :|: TRUE 30.48/8.76 (3) f261_out(x33:0) -> f20_in(x33:0) :|: TRUE 30.48/8.76 (4) f20_in(x) -> f455_in(x) :|: TRUE 30.48/8.76 (5) f261_in(cons_0) -> f261_out(0) :|: TRUE && cons_0 = 0 30.48/8.76 (6) f455_in(s(x56:0)) -> f451_out(s(T171:0)) :|: TRUE 30.48/8.76 (7) f261_in(s(T134:0)) -> f261_in(T134:0) :|: TRUE 30.48/8.76 (8) f455_in(s(T177:0)) -> f455_in(T177:0) :|: TRUE 30.48/8.76 (9) f451_out(x48:0) -> f451_out(s(x48:0)) :|: TRUE 30.48/8.76 (10) f261_out(x58:0) -> f261_out(s(x58:0)) :|: TRUE 30.48/8.76 30.48/8.76 Arcs: 30.48/8.76 (1) -> (2), (4) 30.48/8.76 (2) -> (5), (7) 30.48/8.76 (3) -> (2), (4) 30.48/8.76 (4) -> (6), (8) 30.48/8.76 (5) -> (3), (10) 30.48/8.76 (6) -> (1), (9) 30.48/8.76 (7) -> (5), (7) 30.48/8.76 (8) -> (6), (8) 30.48/8.76 (9) -> (1), (9) 30.48/8.76 (10) -> (3), (10) 30.48/8.76 30.48/8.76 This digraph is fully evaluated! 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (160) 30.48/8.76 Obligation: 30.48/8.76 30.48/8.76 Termination digraph: 30.48/8.76 Nodes: 30.48/8.76 (1) f451_out(x28:0) -> f20_in(x28:0) :|: TRUE 30.48/8.76 (2) f451_out(x48:0) -> f451_out(s(x48:0)) :|: TRUE 30.48/8.76 (3) f455_in(s(x56:0)) -> f451_out(s(T171:0)) :|: TRUE 30.48/8.76 (4) f455_in(s(T177:0)) -> f455_in(T177:0) :|: TRUE 30.48/8.76 (5) f20_in(x) -> f455_in(x) :|: TRUE 30.48/8.76 (6) f261_out(x33:0) -> f20_in(x33:0) :|: TRUE 30.48/8.76 (7) f261_out(x58:0) -> f261_out(s(x58:0)) :|: TRUE 30.48/8.76 (8) f261_in(cons_0) -> f261_out(0) :|: TRUE && cons_0 = 0 30.48/8.76 (9) f261_in(s(T134:0)) -> f261_in(T134:0) :|: TRUE 30.48/8.76 (10) f20_in(x21:0) -> f261_in(x21:0) :|: TRUE 30.48/8.76 30.48/8.76 Arcs: 30.48/8.76 (1) -> (5), (10) 30.48/8.76 (2) -> (1), (2) 30.48/8.76 (3) -> (1), (2) 30.48/8.76 (4) -> (3), (4) 30.48/8.76 (5) -> (3), (4) 30.48/8.76 (6) -> (5), (10) 30.48/8.76 (7) -> (6), (7) 30.48/8.76 (8) -> (6), (7) 30.48/8.76 (9) -> (8), (9) 30.48/8.76 (10) -> (8), (9) 30.48/8.76 30.48/8.76 This digraph is fully evaluated! 30.48/8.76 30.48/8.76 ---------------------------------------- 30.48/8.76 30.48/8.76 (161) PrologToDTProblemTransformerProof (SOUND) 30.48/8.76 Built DT problem from termination graph DT10. 30.48/8.76 30.48/8.76 { 30.48/8.76 "root": 1, 30.48/8.76 "program": { 30.48/8.76 "directives": [], 30.48/8.76 "clauses": [ 30.48/8.76 [ 30.48/8.76 "(delete X (tree X (void) Right) Right)", 30.48/8.76 null 30.48/8.76 ], 30.48/8.76 [ 30.48/8.76 "(delete X (tree X Left (void)) Left)", 30.48/8.76 null 30.48/8.76 ], 30.48/8.76 [ 30.48/8.76 "(delete X (tree X Left Right) (tree Y Left Right1))", 30.48/8.76 "(delmin Right Y Right1)" 30.48/8.76 ], 30.48/8.76 [ 30.48/8.76 "(delete X (tree Y Left Right) (tree Y Left1 Right))", 30.48/8.76 "(',' (less X Y) (delete X Left Left1))" 30.48/8.76 ], 30.48/8.76 [ 30.48/8.76 "(delete X (tree Y Left Right) (tree Y Left Right1))", 30.48/8.76 "(',' (less Y X) (delete X Right Right1))" 30.48/8.76 ], 30.48/8.76 [ 30.48/8.76 "(delmin (tree Y (void) Right) Y Right)", 30.48/8.76 null 30.48/8.76 ], 30.48/8.76 [ 30.48/8.76 "(delmin (tree X Left X1) Y (tree X Left1 X2))", 30.48/8.76 "(delmin Left Y Left1)" 30.48/8.76 ], 30.48/8.76 [ 30.48/8.76 "(less (0) (s X3))", 30.48/8.76 null 30.48/8.76 ], 30.48/8.76 [ 30.48/8.76 "(less (s X) (s Y))", 30.48/8.76 "(less X Y)" 30.48/8.76 ] 30.48/8.76 ] 30.48/8.76 }, 30.48/8.76 "graph": { 30.48/8.76 "nodes": { 30.48/8.76 "590": { 30.48/8.76 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"(delete T10 T2 T3)" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "clause": 4, 30.48/8.78 "scope": 1, 30.48/8.78 "term": "(delete T10 T2 T3)" 30.48/8.78 } 30.48/8.78 ], 30.48/8.78 "kb": { 30.48/8.78 "nonunifying": [], 30.48/8.78 "intvars": {}, 30.48/8.78 "arithmetic": { 30.48/8.78 "type": "PlainIntegerRelationState", 30.48/8.78 "relations": [] 30.48/8.78 }, 30.48/8.78 "ground": ["T10"], 30.48/8.78 "free": [], 30.48/8.78 "exprvars": [] 30.48/8.78 } 30.48/8.78 }, 30.48/8.78 "467": { 30.48/8.78 "goal": [ 30.48/8.78 { 30.48/8.78 "clause": 7, 30.48/8.78 "scope": 4, 30.48/8.78 "term": "(less T127 T132)" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "clause": 8, 30.48/8.78 "scope": 4, 30.48/8.78 "term": "(less T127 T132)" 30.48/8.78 } 30.48/8.78 ], 30.48/8.78 "kb": { 30.48/8.78 "nonunifying": [], 30.48/8.78 "intvars": {}, 30.48/8.78 "arithmetic": { 30.48/8.78 "type": "PlainIntegerRelationState", 30.48/8.78 "relations": [] 30.48/8.78 }, 30.48/8.78 "ground": ["T127"], 30.48/8.78 "free": [], 30.48/8.78 "exprvars": [] 30.48/8.78 } 30.48/8.78 }, 30.48/8.78 "588": { 30.48/8.78 "goal": [], 30.48/8.78 "kb": { 30.48/8.78 "nonunifying": [], 30.48/8.78 "intvars": {}, 30.48/8.78 "arithmetic": { 30.48/8.78 "type": "PlainIntegerRelationState", 30.48/8.78 "relations": [] 30.48/8.78 }, 30.48/8.78 "ground": [], 30.48/8.78 "free": [], 30.48/8.78 "exprvars": [] 30.48/8.78 } 30.48/8.78 }, 30.48/8.78 "468": { 30.48/8.78 "goal": [{ 30.48/8.78 "clause": 7, 30.48/8.78 "scope": 4, 30.48/8.78 "term": "(less T127 T132)" 30.48/8.78 }], 30.48/8.78 "kb": { 30.48/8.78 "nonunifying": [], 30.48/8.78 "intvars": {}, 30.48/8.78 "arithmetic": { 30.48/8.78 "type": "PlainIntegerRelationState", 30.48/8.78 "relations": [] 30.48/8.78 }, 30.48/8.78 "ground": ["T127"], 30.48/8.78 "free": [], 30.48/8.78 "exprvars": [] 30.48/8.78 } 30.48/8.78 }, 30.48/8.78 "589": { 30.48/8.78 "goal": [{ 30.48/8.78 "clause": -1, 30.48/8.78 "scope": -1, 30.48/8.78 "term": "(',' (less T418 T413) (delete T413 T419 T420))" 30.48/8.78 }], 30.48/8.78 "kb": { 30.48/8.78 "nonunifying": [[ 30.48/8.78 "(delete T413 T2 T3)", 30.48/8.78 "(delete X10 (tree X10 X11 (void)) X11)" 30.48/8.78 ]], 30.48/8.78 "intvars": {}, 30.48/8.78 "arithmetic": { 30.48/8.78 "type": "PlainIntegerRelationState", 30.48/8.78 "relations": [] 30.48/8.78 }, 30.48/8.78 "ground": ["T413"], 30.48/8.78 "free": [ 30.48/8.78 "X10", 30.48/8.78 "X11" 30.48/8.78 ], 30.48/8.78 "exprvars": [] 30.48/8.78 } 30.48/8.78 }, 30.48/8.78 "469": { 30.48/8.78 "goal": [{ 30.48/8.78 "clause": 8, 30.48/8.78 "scope": 4, 30.48/8.78 "term": "(less T127 T132)" 30.48/8.78 }], 30.48/8.78 "kb": { 30.48/8.78 "nonunifying": [], 30.48/8.78 "intvars": {}, 30.48/8.78 "arithmetic": { 30.48/8.78 "type": "PlainIntegerRelationState", 30.48/8.78 "relations": [] 30.48/8.78 }, 30.48/8.78 "ground": ["T127"], 30.48/8.78 "free": [], 30.48/8.78 "exprvars": [] 30.48/8.78 } 30.48/8.78 } 30.48/8.78 }, 30.48/8.78 "edges": [ 30.48/8.78 { 30.48/8.78 "from": 1, 30.48/8.78 "to": 30, 30.48/8.78 "label": "CASE" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 30, 30.48/8.78 "to": 43, 30.48/8.78 "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" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 30, 30.48/8.78 "to": 44, 30.48/8.78 "label": "EVAL-BACKTRACK" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 43, 30.48/8.78 "to": 45, 30.48/8.78 "label": "SUCCESS" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 44, 30.48/8.78 "to": 643, 30.48/8.78 "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" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 44, 30.48/8.78 "to": 644, 30.48/8.78 "label": "EVAL-BACKTRACK" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 45, 30.48/8.78 "to": 81, 30.48/8.78 "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" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 45, 30.48/8.78 "to": 97, 30.48/8.78 "label": "EVAL-BACKTRACK" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 81, 30.48/8.78 "to": 104, 30.48/8.78 "label": "SUCCESS" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 97, 30.48/8.78 "to": 572, 30.48/8.78 "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" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 97, 30.48/8.78 "to": 573, 30.48/8.78 "label": "EVAL-BACKTRACK" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 104, 30.48/8.78 "to": 148, 30.48/8.78 "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" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 104, 30.48/8.78 "to": 149, 30.48/8.78 "label": "EVAL-BACKTRACK" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 148, 30.48/8.78 "to": 150, 30.48/8.78 "label": "CASE" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 149, 30.48/8.78 "to": 499, 30.48/8.78 "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" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 149, 30.48/8.78 "to": 502, 30.48/8.78 "label": "EVAL-BACKTRACK" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 150, 30.48/8.78 "to": 154, 30.48/8.78 "label": "PARALLEL" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 150, 30.48/8.78 "to": 155, 30.48/8.78 "label": "PARALLEL" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 154, 30.48/8.78 "to": 158, 30.48/8.78 "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" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 154, 30.48/8.78 "to": 159, 30.48/8.78 "label": "EVAL-BACKTRACK" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 155, 30.48/8.78 "to": 161, 30.48/8.78 "label": "PARALLEL" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 155, 30.48/8.78 "to": 162, 30.48/8.78 "label": "PARALLEL" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 158, 30.48/8.78 "to": 160, 30.48/8.78 "label": "SUCCESS" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 161, 30.48/8.78 "to": 165, 30.48/8.78 "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" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 161, 30.48/8.78 "to": 166, 30.48/8.78 "label": "EVAL-BACKTRACK" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 162, 30.48/8.78 "to": 235, 30.48/8.78 "label": "FAILURE" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 165, 30.48/8.78 "to": 169, 30.48/8.78 "label": "CASE" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 169, 30.48/8.78 "to": 170, 30.48/8.78 "label": "PARALLEL" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 169, 30.48/8.78 "to": 171, 30.48/8.78 "label": "PARALLEL" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 170, 30.48/8.78 "to": 175, 30.48/8.78 "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" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 170, 30.48/8.78 "to": 176, 30.48/8.78 "label": "EVAL-BACKTRACK" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 171, 30.48/8.78 "to": 184, 30.48/8.78 "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" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 171, 30.48/8.78 "to": 185, 30.48/8.78 "label": "EVAL-BACKTRACK" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 175, 30.48/8.78 "to": 179, 30.48/8.78 "label": "SUCCESS" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 184, 30.48/8.78 "to": 165, 30.48/8.78 "label": "INSTANCE with matching:\nT65 -> T100\nT66 -> T101\nT67 -> T102" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 235, 30.48/8.78 "to": 236, 30.48/8.78 "label": "PARALLEL" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 235, 30.48/8.78 "to": 237, 30.48/8.78 "label": "PARALLEL" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 236, 30.48/8.78 "to": 263, 30.48/8.78 "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" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 236, 30.48/8.78 "to": 264, 30.48/8.78 "label": "EVAL-BACKTRACK" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 237, 30.48/8.78 "to": 475, 30.48/8.78 "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" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 237, 30.48/8.78 "to": 476, 30.48/8.78 "label": "EVAL-BACKTRACK" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 263, 30.48/8.78 "to": 265, 30.48/8.78 "label": "SPLIT 1" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 263, 30.48/8.78 "to": 266, 30.48/8.78 "label": "SPLIT 2\nnew knowledge:\nT127 is ground\nreplacements:T133 -> T137,\nT134 -> T138" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 265, 30.48/8.78 "to": 467, 30.48/8.78 "label": "CASE" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 266, 30.48/8.78 "to": 1, 30.48/8.78 "label": "INSTANCE with matching:\nT1 -> T127\nT2 -> T137\nT3 -> T138" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 467, 30.48/8.78 "to": 468, 30.48/8.78 "label": "PARALLEL" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 467, 30.48/8.78 "to": 469, 30.48/8.78 "label": "PARALLEL" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 468, 30.48/8.78 "to": 470, 30.48/8.78 "label": "EVAL with clause\nless(0, s(X131)).\nand substitutionT127 -> 0,\nX131 -> T145,\nT132 -> s(T145)" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 468, 30.48/8.78 "to": 471, 30.48/8.78 "label": "EVAL-BACKTRACK" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 469, 30.48/8.78 "to": 473, 30.48/8.78 "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" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 469, 30.48/8.78 "to": 474, 30.48/8.78 "label": "EVAL-BACKTRACK" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 470, 30.48/8.78 "to": 472, 30.48/8.78 "label": "SUCCESS" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 473, 30.48/8.78 "to": 265, 30.48/8.78 "label": "INSTANCE with matching:\nT127 -> T150\nT132 -> T152" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 475, 30.48/8.78 "to": 477, 30.48/8.78 "label": "SPLIT 1" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 475, 30.48/8.78 "to": 478, 30.48/8.78 "label": "SPLIT 2\nnew knowledge:\nT174 is ground\nT169 is ground\nreplacements:T175 -> T179,\nT176 -> T180" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 477, 30.48/8.78 "to": 479, 30.48/8.78 "label": "CASE" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 478, 30.48/8.78 "to": 1, 30.48/8.78 "label": "INSTANCE with matching:\nT1 -> T169\nT2 -> T179\nT3 -> T180" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 479, 30.48/8.78 "to": 480, 30.48/8.78 "label": "PARALLEL" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 479, 30.48/8.78 "to": 481, 30.48/8.78 "label": "PARALLEL" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 480, 30.48/8.78 "to": 482, 30.48/8.78 "label": "EVAL with clause\nless(0, s(X167)).\nand substitutionT174 -> 0,\nX167 -> T187,\nT169 -> s(T187)" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 480, 30.48/8.78 "to": 483, 30.48/8.78 "label": "EVAL-BACKTRACK" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 481, 30.48/8.78 "to": 487, 30.48/8.78 "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" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 481, 30.48/8.78 "to": 488, 30.48/8.78 "label": "EVAL-BACKTRACK" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 482, 30.48/8.78 "to": 484, 30.48/8.78 "label": "SUCCESS" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 487, 30.48/8.78 "to": 477, 30.48/8.78 "label": "INSTANCE with matching:\nT174 -> T194\nT169 -> T193" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 499, 30.48/8.78 "to": 503, 30.48/8.78 "label": "CASE" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 502, 30.48/8.78 "to": 561, 30.48/8.78 "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" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 502, 30.48/8.78 "to": 562, 30.48/8.78 "label": "EVAL-BACKTRACK" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 503, 30.48/8.78 "to": 504, 30.48/8.78 "label": "PARALLEL" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 503, 30.48/8.78 "to": 515, 30.48/8.78 "label": "PARALLEL" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 504, 30.48/8.78 "to": 518, 30.48/8.78 "label": "EVAL with clause\nless(0, s(X194)).\nand substitutionT206 -> 0,\nX194 -> T218,\nT211 -> s(T218),\nT212 -> T219,\nT213 -> T220" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 504, 30.48/8.78 "to": 519, 30.48/8.78 "label": "EVAL-BACKTRACK" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 515, 30.48/8.78 "to": 520, 30.48/8.78 "label": "PARALLEL" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 515, 30.48/8.78 "to": 521, 30.48/8.78 "label": "PARALLEL" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 518, 30.48/8.78 "to": 1, 30.48/8.78 "label": "INSTANCE with matching:\nT1 -> 0\nT2 -> T219\nT3 -> T220" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 520, 30.48/8.78 "to": 522, 30.48/8.78 "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" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 520, 30.48/8.78 "to": 523, 30.48/8.78 "label": "EVAL-BACKTRACK" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 521, 30.48/8.78 "to": 526, 30.48/8.78 "label": "FAILURE" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 522, 30.48/8.78 "to": 524, 30.48/8.78 "label": "SPLIT 1" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 522, 30.48/8.78 "to": 525, 30.48/8.78 "label": "SPLIT 2\nnew knowledge:\nT231 is ground\nreplacements:T234 -> T238,\nT235 -> T239,\nT2 -> T240,\nT3 -> T241" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 524, 30.48/8.78 "to": 265, 30.48/8.78 "label": "INSTANCE with matching:\nT127 -> T231\nT132 -> T233" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 525, 30.48/8.78 "to": 1, 30.48/8.78 "label": "INSTANCE with matching:\nT1 -> s(T231)\nT2 -> T238\nT3 -> T239" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 526, 30.48/8.78 "to": 527, 30.48/8.78 "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" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 526, 30.48/8.78 "to": 528, 30.48/8.78 "label": "EVAL-BACKTRACK" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 527, 30.48/8.78 "to": 475, 30.48/8.78 "label": "INSTANCE with matching:\nT174 -> T263\nT169 -> T258\nT175 -> T264\nT176 -> T265" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 561, 30.48/8.78 "to": 563, 30.48/8.78 "label": "CASE" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 563, 30.48/8.78 "to": 564, 30.48/8.78 "label": "PARALLEL" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 563, 30.48/8.78 "to": 565, 30.48/8.78 "label": "PARALLEL" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 564, 30.48/8.78 "to": 566, 30.48/8.78 "label": "EVAL with clause\nless(0, s(X248)).\nand substitutionT278 -> 0,\nX248 -> T285,\nT273 -> s(T285),\nT279 -> T286,\nT280 -> T287" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 564, 30.48/8.78 "to": 567, 30.48/8.78 "label": "EVAL-BACKTRACK" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 565, 30.48/8.78 "to": 568, 30.48/8.78 "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" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 565, 30.48/8.78 "to": 569, 30.48/8.78 "label": "EVAL-BACKTRACK" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 566, 30.48/8.78 "to": 1, 30.48/8.78 "label": "INSTANCE with matching:\nT1 -> s(T285)\nT2 -> T286\nT3 -> T287" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 568, 30.48/8.78 "to": 570, 30.48/8.78 "label": "SPLIT 1" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 568, 30.48/8.78 "to": 571, 30.48/8.78 "label": "SPLIT 2\nnew knowledge:\nT298 is ground\nT297 is ground\nreplacements:T299 -> T303,\nT300 -> T304,\nT2 -> T305,\nT3 -> T306" 30.48/8.78 }, 30.48/8.78 { 30.48/8.78 "from": 570, 30.48/8.78 "to": 477, 30.48/8.79 "label": "INSTANCE with matching:\nT174 -> T298\nT169 -> T297" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 571, 30.48/8.79 "to": 1, 30.48/8.79 "label": "INSTANCE with matching:\nT1 -> s(T297)\nT2 -> T303\nT3 -> T304" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 572, 30.48/8.79 "to": 574, 30.48/8.79 "label": "CASE" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 573, 30.48/8.79 "to": 620, 30.48/8.79 "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" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 573, 30.48/8.79 "to": 621, 30.48/8.79 "label": "EVAL-BACKTRACK" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 574, 30.48/8.79 "to": 575, 30.48/8.79 "label": "PARALLEL" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 574, 30.48/8.79 "to": 576, 30.48/8.79 "label": "PARALLEL" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 575, 30.48/8.79 "to": 577, 30.48/8.79 "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" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 575, 30.48/8.79 "to": 578, 30.48/8.79 "label": "EVAL-BACKTRACK" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 576, 30.48/8.79 "to": 580, 30.48/8.79 "label": "PARALLEL" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 576, 30.48/8.79 "to": 581, 30.48/8.79 "label": "PARALLEL" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 577, 30.48/8.79 "to": 579, 30.48/8.79 "label": "SUCCESS" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 580, 30.48/8.79 "to": 582, 30.48/8.79 "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" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 580, 30.48/8.79 "to": 583, 30.48/8.79 "label": "EVAL-BACKTRACK" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 581, 30.48/8.79 "to": 584, 30.48/8.79 "label": "FAILURE" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 582, 30.48/8.79 "to": 165, 30.48/8.79 "label": "INSTANCE with matching:\nT65 -> T366\nT66 -> T367\nT67 -> T368" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 584, 30.48/8.79 "to": 585, 30.48/8.79 "label": "PARALLEL" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 584, 30.48/8.79 "to": 586, 30.48/8.79 "label": "PARALLEL" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 585, 30.48/8.79 "to": 587, 30.48/8.79 "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" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 585, 30.48/8.79 "to": 588, 30.48/8.79 "label": "EVAL-BACKTRACK" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 586, 30.48/8.79 "to": 589, 30.48/8.79 "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" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 586, 30.48/8.79 "to": 590, 30.48/8.79 "label": "EVAL-BACKTRACK" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 587, 30.48/8.79 "to": 263, 30.48/8.79 "label": "INSTANCE with matching:\nT127 -> T393\nT132 -> T398\nT133 -> T399\nT134 -> T400" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 589, 30.48/8.79 "to": 475, 30.48/8.79 "label": "INSTANCE with matching:\nT174 -> T418\nT169 -> T413\nT175 -> T419\nT176 -> T420" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 620, 30.48/8.79 "to": 622, 30.48/8.79 "label": "CASE" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 621, 30.48/8.79 "to": 634, 30.48/8.79 "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" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 621, 30.48/8.79 "to": 635, 30.48/8.79 "label": "EVAL-BACKTRACK" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 622, 30.48/8.79 "to": 623, 30.48/8.79 "label": "PARALLEL" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 622, 30.48/8.79 "to": 624, 30.48/8.79 "label": "PARALLEL" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 623, 30.48/8.79 "to": 625, 30.48/8.79 "label": "EVAL with clause\nless(0, s(X379)).\nand substitutionT428 -> 0,\nX379 -> T440,\nT433 -> s(T440),\nT434 -> T441,\nT435 -> T442" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 623, 30.48/8.79 "to": 626, 30.48/8.79 "label": "EVAL-BACKTRACK" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 624, 30.48/8.79 "to": 627, 30.48/8.79 "label": "PARALLEL" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 624, 30.48/8.79 "to": 628, 30.48/8.79 "label": "PARALLEL" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 625, 30.48/8.79 "to": 1, 30.48/8.79 "label": "INSTANCE with matching:\nT1 -> 0\nT2 -> T441\nT3 -> T442" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 627, 30.48/8.79 "to": 629, 30.48/8.79 "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" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 627, 30.48/8.79 "to": 630, 30.48/8.79 "label": "EVAL-BACKTRACK" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 628, 30.48/8.79 "to": 631, 30.48/8.79 "label": "FAILURE" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 629, 30.48/8.79 "to": 522, 30.48/8.79 "label": "INSTANCE with matching:\nT231 -> T453\nT233 -> T455\nT234 -> T456\nT235 -> T457\nX17 -> X272\nX18 -> X273\nX19 -> X274\nX20 -> X275\nX21 -> X276" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 631, 30.48/8.79 "to": 632, 30.48/8.79 "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" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 631, 30.48/8.79 "to": 633, 30.48/8.79 "label": "EVAL-BACKTRACK" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 632, 30.48/8.79 "to": 475, 30.48/8.79 "label": "INSTANCE with matching:\nT174 -> T475\nT169 -> T470\nT175 -> T476\nT176 -> T477" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 634, 30.48/8.79 "to": 636, 30.48/8.79 "label": "CASE" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 636, 30.48/8.79 "to": 637, 30.48/8.79 "label": "PARALLEL" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 636, 30.48/8.79 "to": 638, 30.48/8.79 "label": "PARALLEL" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 637, 30.48/8.79 "to": 639, 30.48/8.79 "label": "EVAL with clause\nless(0, s(X427)).\nand substitutionT490 -> 0,\nX427 -> T497,\nT485 -> s(T497),\nT491 -> T498,\nT492 -> T499" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 637, 30.48/8.79 "to": 640, 30.48/8.79 "label": "EVAL-BACKTRACK" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 638, 30.48/8.79 "to": 641, 30.48/8.79 "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" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 638, 30.48/8.79 "to": 642, 30.48/8.79 "label": "EVAL-BACKTRACK" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 639, 30.48/8.79 "to": 1, 30.48/8.79 "label": "INSTANCE with matching:\nT1 -> s(T497)\nT2 -> T498\nT3 -> T499" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 641, 30.48/8.79 "to": 568, 30.48/8.79 "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" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 643, 30.48/8.79 "to": 645, 30.48/8.79 "label": "SUCCESS" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 644, 30.48/8.79 "to": 692, 30.48/8.79 "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" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 644, 30.48/8.79 "to": 693, 30.48/8.79 "label": "EVAL-BACKTRACK" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 645, 30.48/8.79 "to": 646, 30.48/8.79 "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" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 645, 30.48/8.79 "to": 647, 30.48/8.79 "label": "EVAL-BACKTRACK" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 646, 30.48/8.79 "to": 648, 30.48/8.79 "label": "CASE" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 647, 30.48/8.79 "to": 665, 30.48/8.79 "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" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 647, 30.48/8.79 "to": 666, 30.48/8.79 "label": "EVAL-BACKTRACK" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 648, 30.48/8.79 "to": 649, 30.48/8.79 "label": "PARALLEL" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 648, 30.48/8.79 "to": 650, 30.48/8.79 "label": "PARALLEL" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 649, 30.48/8.79 "to": 651, 30.48/8.79 "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" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 649, 30.48/8.79 "to": 652, 30.48/8.79 "label": "EVAL-BACKTRACK" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 650, 30.48/8.79 "to": 654, 30.48/8.79 "label": "PARALLEL" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 650, 30.48/8.79 "to": 655, 30.48/8.79 "label": "PARALLEL" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 651, 30.48/8.79 "to": 653, 30.48/8.79 "label": "SUCCESS" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 654, 30.48/8.79 "to": 656, 30.48/8.79 "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" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 654, 30.48/8.79 "to": 657, 30.48/8.79 "label": "EVAL-BACKTRACK" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 655, 30.48/8.79 "to": 658, 30.48/8.79 "label": "FAILURE" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 656, 30.48/8.79 "to": 165, 30.48/8.79 "label": "INSTANCE with matching:\nT65 -> T572\nT66 -> T573\nT67 -> T574" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 658, 30.48/8.79 "to": 659, 30.48/8.79 "label": "PARALLEL" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 658, 30.48/8.79 "to": 660, 30.48/8.79 "label": "PARALLEL" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 659, 30.48/8.79 "to": 661, 30.48/8.79 "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" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 659, 30.48/8.79 "to": 662, 30.48/8.79 "label": "EVAL-BACKTRACK" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 660, 30.48/8.79 "to": 663, 30.48/8.79 "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" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 660, 30.48/8.79 "to": 664, 30.48/8.79 "label": "EVAL-BACKTRACK" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 661, 30.48/8.79 "to": 263, 30.48/8.79 "label": "INSTANCE with matching:\nT127 -> T599\nT132 -> T604\nT133 -> T605\nT134 -> T606" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 663, 30.48/8.79 "to": 475, 30.48/8.79 "label": "INSTANCE with matching:\nT174 -> T624\nT169 -> T619\nT175 -> T625\nT176 -> T626" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 665, 30.48/8.79 "to": 667, 30.48/8.79 "label": "CASE" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 666, 30.48/8.79 "to": 683, 30.48/8.79 "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" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 666, 30.48/8.79 "to": 684, 30.48/8.79 "label": "EVAL-BACKTRACK" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 667, 30.48/8.79 "to": 668, 30.48/8.79 "label": "PARALLEL" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 667, 30.48/8.79 "to": 669, 30.48/8.79 "label": "PARALLEL" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 668, 30.48/8.79 "to": 670, 30.48/8.79 "label": "EVAL with clause\nless(0, s(X556)).\nand substitutionT634 -> 0,\nX556 -> T646,\nT639 -> s(T646),\nT640 -> T647,\nT641 -> T648" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 668, 30.48/8.79 "to": 671, 30.48/8.79 "label": "EVAL-BACKTRACK" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 669, 30.48/8.79 "to": 672, 30.48/8.79 "label": "PARALLEL" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 669, 30.48/8.79 "to": 673, 30.48/8.79 "label": "PARALLEL" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 670, 30.48/8.79 "to": 1, 30.48/8.79 "label": "INSTANCE with matching:\nT1 -> 0\nT2 -> T647\nT3 -> T648" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 672, 30.48/8.79 "to": 674, 30.48/8.79 "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" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 672, 30.48/8.79 "to": 675, 30.48/8.79 "label": "EVAL-BACKTRACK" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 673, 30.48/8.79 "to": 676, 30.48/8.79 "label": "FAILURE" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 674, 30.48/8.79 "to": 522, 30.48/8.79 "label": "INSTANCE with matching:\nT231 -> T659\nT233 -> T661\nT234 -> T662\nT235 -> T663\nX17 -> X449\nX18 -> X450\nX19 -> X451\nX20 -> X452\nX21 -> X453" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 676, 30.48/8.79 "to": 677, 30.48/8.79 "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" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 676, 30.48/8.79 "to": 678, 30.48/8.79 "label": "EVAL-BACKTRACK" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 677, 30.48/8.79 "to": 475, 30.48/8.79 "label": "INSTANCE with matching:\nT174 -> T681\nT169 -> T676\nT175 -> T682\nT176 -> T683" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 683, 30.48/8.79 "to": 685, 30.48/8.79 "label": "CASE" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 685, 30.48/8.79 "to": 686, 30.48/8.79 "label": "PARALLEL" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 685, 30.48/8.79 "to": 687, 30.48/8.79 "label": "PARALLEL" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 686, 30.48/8.79 "to": 688, 30.48/8.79 "label": "EVAL with clause\nless(0, s(X604)).\nand substitutionT696 -> 0,\nX604 -> T703,\nT691 -> s(T703),\nT697 -> T704,\nT698 -> T705" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 686, 30.48/8.79 "to": 689, 30.48/8.79 "label": "EVAL-BACKTRACK" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 687, 30.48/8.79 "to": 690, 30.48/8.79 "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" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 687, 30.48/8.79 "to": 691, 30.48/8.79 "label": "EVAL-BACKTRACK" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 688, 30.48/8.79 "to": 1, 30.48/8.79 "label": "INSTANCE with matching:\nT1 -> s(T703)\nT2 -> T704\nT3 -> T705" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 690, 30.48/8.79 "to": 568, 30.48/8.79 "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" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 692, 30.48/8.79 "to": 694, 30.48/8.79 "label": "CASE" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 693, 30.48/8.79 "to": 746, 30.48/8.79 "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" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 693, 30.48/8.79 "to": 747, 30.48/8.79 "label": "EVAL-BACKTRACK" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 694, 30.48/8.79 "to": 695, 30.48/8.79 "label": "PARALLEL" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 694, 30.48/8.79 "to": 696, 30.48/8.79 "label": "PARALLEL" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 695, 30.48/8.79 "to": 697, 30.48/8.79 "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" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 695, 30.48/8.79 "to": 698, 30.48/8.79 "label": "EVAL-BACKTRACK" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 696, 30.48/8.79 "to": 700, 30.48/8.79 "label": "PARALLEL" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 696, 30.48/8.79 "to": 701, 30.48/8.79 "label": "PARALLEL" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 697, 30.48/8.79 "to": 699, 30.48/8.79 "label": "SUCCESS" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 700, 30.48/8.79 "to": 705, 30.48/8.79 "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" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 700, 30.48/8.79 "to": 706, 30.48/8.79 "label": "EVAL-BACKTRACK" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 701, 30.48/8.79 "to": 707, 30.48/8.79 "label": "FAILURE" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 705, 30.48/8.79 "to": 165, 30.48/8.79 "label": "INSTANCE with matching:\nT65 -> T774\nT66 -> T775\nT67 -> T776" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 707, 30.48/8.79 "to": 740, 30.48/8.79 "label": "PARALLEL" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 707, 30.48/8.79 "to": 741, 30.48/8.79 "label": "PARALLEL" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 740, 30.48/8.79 "to": 742, 30.48/8.79 "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" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 740, 30.48/8.79 "to": 743, 30.48/8.79 "label": "EVAL-BACKTRACK" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 741, 30.48/8.79 "to": 744, 30.48/8.79 "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" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 741, 30.48/8.79 "to": 745, 30.48/8.79 "label": "EVAL-BACKTRACK" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 742, 30.48/8.79 "to": 263, 30.48/8.79 "label": "INSTANCE with matching:\nT127 -> T801\nT132 -> T806\nT133 -> T807\nT134 -> T808" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 744, 30.48/8.79 "to": 475, 30.48/8.79 "label": "INSTANCE with matching:\nT174 -> T826\nT169 -> T821\nT175 -> T827\nT176 -> T828" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 746, 30.48/8.79 "to": 748, 30.48/8.79 "label": "CASE" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 747, 30.48/8.79 "to": 760, 30.48/8.79 "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" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 747, 30.48/8.79 "to": 761, 30.48/8.79 "label": "EVAL-BACKTRACK" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 748, 30.48/8.79 "to": 749, 30.48/8.79 "label": "PARALLEL" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 748, 30.48/8.79 "to": 750, 30.48/8.79 "label": "PARALLEL" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 749, 30.48/8.79 "to": 751, 30.48/8.79 "label": "EVAL with clause\nless(0, s(X729)).\nand substitutionT836 -> 0,\nX729 -> T848,\nT841 -> s(T848),\nT842 -> T849,\nT843 -> T850" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 749, 30.48/8.79 "to": 752, 30.48/8.79 "label": "EVAL-BACKTRACK" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 750, 30.48/8.79 "to": 753, 30.48/8.79 "label": "PARALLEL" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 750, 30.48/8.79 "to": 754, 30.48/8.79 "label": "PARALLEL" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 751, 30.48/8.79 "to": 1, 30.48/8.79 "label": "INSTANCE with matching:\nT1 -> 0\nT2 -> T849\nT3 -> T850" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 753, 30.48/8.79 "to": 755, 30.48/8.79 "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" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 753, 30.48/8.79 "to": 756, 30.48/8.79 "label": "EVAL-BACKTRACK" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 754, 30.48/8.79 "to": 757, 30.48/8.79 "label": "FAILURE" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 755, 30.48/8.79 "to": 522, 30.48/8.79 "label": "INSTANCE with matching:\nT231 -> T861\nT233 -> T863\nT234 -> T864\nT235 -> T865\nX17 -> X622\nX18 -> X623\nX19 -> X624\nX20 -> X625\nX21 -> X626" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 757, 30.48/8.79 "to": 758, 30.48/8.79 "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" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 757, 30.48/8.79 "to": 759, 30.48/8.79 "label": "EVAL-BACKTRACK" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 758, 30.48/8.79 "to": 475, 30.48/8.79 "label": "INSTANCE with matching:\nT174 -> T883\nT169 -> T878\nT175 -> T884\nT176 -> T885" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 760, 30.48/8.79 "to": 762, 30.48/8.79 "label": "CASE" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 762, 30.48/8.79 "to": 763, 30.48/8.79 "label": "PARALLEL" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 762, 30.48/8.79 "to": 764, 30.48/8.79 "label": "PARALLEL" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 763, 30.48/8.79 "to": 765, 30.48/8.79 "label": "EVAL with clause\nless(0, s(X777)).\nand substitutionT898 -> 0,\nX777 -> T905,\nT893 -> s(T905),\nT899 -> T906,\nT900 -> T907" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 763, 30.48/8.79 "to": 766, 30.48/8.79 "label": "EVAL-BACKTRACK" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 764, 30.48/8.79 "to": 767, 30.48/8.79 "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" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 764, 30.48/8.79 "to": 768, 30.48/8.79 "label": "EVAL-BACKTRACK" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 765, 30.48/8.79 "to": 1, 30.48/8.79 "label": "INSTANCE with matching:\nT1 -> s(T905)\nT2 -> T906\nT3 -> T907" 30.48/8.79 }, 30.48/8.79 { 30.48/8.79 "from": 767, 30.48/8.79 "to": 568, 30.48/8.79 "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" 30.48/8.79 } 30.48/8.79 ], 30.48/8.79 "type": "Graph" 30.48/8.79 } 30.48/8.79 } 30.48/8.79 30.48/8.79 ---------------------------------------- 30.48/8.79 30.48/8.79 (162) 30.48/8.79 Obligation: 30.48/8.79 Triples: 30.48/8.79 30.48/8.79 delminA(tree(X1, X2, X3), X4, tree(X1, X5, X6)) :- delminA(X2, X4, X5). 30.48/8.79 lessG(s(X1), s(X2)) :- lessG(X1, X2). 30.48/8.79 lessH(s(X1), s(X2)) :- lessH(X1, X2). 30.48/8.79 pD(X1, X2, X3, X4) :- lessH(X1, X2). 30.48/8.79 pD(X1, X2, X3, X4) :- ','(lesscH(X1, X2), deleteB(X2, X3, X4)). 30.48/8.79 pC(X1, X2, X3, X4) :- lessG(X1, X2). 30.48/8.79 pC(X1, X2, X3, X4) :- ','(lesscG(X1, X2), deleteB(X1, X3, X4)). 30.48/8.79 pE(X1, X2, X3, X4) :- lessG(X1, X2). 30.48/8.79 pE(X1, X2, X3, X4) :- ','(lesscG(X1, X2), deleteB(s(X1), X3, X4)). 30.48/8.79 pF(X1, X2, X3, X4) :- lessH(X1, X2). 30.48/8.79 pF(X1, X2, X3, X4) :- ','(lesscH(X1, X2), deleteB(s(X2), X3, X4)). 30.48/8.79 deleteB(X1, tree(X1, X2, tree(X3, X4, X5)), tree(X6, X2, tree(X3, X7, X8))) :- delminA(X4, X6, X7). 30.48/8.79 deleteB(X1, tree(X2, X3, X4), tree(X2, X5, X4)) :- pC(X1, X2, X3, X5). 30.48/8.79 deleteB(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- pD(X2, X1, X4, X5). 30.48/8.79 deleteB(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) :- deleteB(0, X2, X4). 30.48/8.79 deleteB(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) :- pE(X1, X2, X3, X5). 30.48/8.79 deleteB(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- pD(X2, X1, X4, X5). 30.48/8.79 deleteB(s(X1), tree(0, X2, X3), tree(0, X2, X4)) :- deleteB(s(X1), X3, X4). 30.48/8.79 deleteB(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) :- pF(X2, X1, X4, X5). 30.48/8.79 deleteB(X1, tree(X1, X2, tree(X3, X4, X5)), tree(X6, X2, tree(X3, X7, X8))) :- delminA(X4, X6, X7). 30.48/8.79 deleteB(X1, tree(X2, X3, X4), tree(X2, X5, X4)) :- pC(X1, X2, X3, X5). 30.48/8.79 deleteB(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- pD(X2, X1, X4, X5). 30.48/8.79 deleteB(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) :- deleteB(0, X2, X4). 30.48/8.79 deleteB(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) :- pE(X1, X2, X3, X5). 30.48/8.79 deleteB(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- pD(X2, X1, X4, X5). 30.48/8.79 deleteB(s(X1), tree(0, X2, X3), tree(0, X2, X4)) :- deleteB(s(X1), X3, X4). 30.48/8.79 deleteB(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) :- pF(X2, X1, X4, X5). 30.48/8.79 deleteB(X1, tree(X1, X2, tree(X3, X4, X5)), tree(X6, X2, tree(X3, X7, X8))) :- delminA(X4, X6, X7). 30.48/8.79 deleteB(X1, tree(X2, X3, X4), tree(X2, X5, X4)) :- pC(X1, X2, X3, X5). 30.48/8.79 deleteB(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- pD(X2, X1, X4, X5). 30.48/8.79 deleteB(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) :- deleteB(0, X2, X4). 30.48/8.79 deleteB(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) :- pE(X1, X2, X3, X5). 30.48/8.79 deleteB(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- pD(X2, X1, X4, X5). 30.48/8.79 deleteB(s(X1), tree(0, X2, X3), tree(0, X2, X4)) :- deleteB(s(X1), X3, X4). 30.48/8.79 deleteB(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) :- pF(X2, X1, X4, X5). 30.48/8.79 deleteB(X1, tree(X1, X2, tree(X3, X4, X5)), tree(X6, X2, tree(X3, X7, X8))) :- delminA(X4, X6, X7). 30.48/8.79 deleteB(X1, tree(X2, X3, X4), tree(X2, X5, X4)) :- pC(X1, X2, X3, X5). 30.48/8.79 deleteB(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- pD(X2, X1, X4, X5). 30.48/8.79 deleteB(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) :- deleteB(0, X2, X4). 30.48/8.79 deleteB(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) :- pE(X1, X2, X3, X5). 30.48/8.79 deleteB(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- pD(X2, X1, X4, X5). 30.48/8.79 deleteB(s(X1), tree(0, X2, X3), tree(0, X2, X4)) :- deleteB(s(X1), X3, X4). 30.48/8.79 deleteB(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) :- pF(X2, X1, X4, X5). 30.48/8.79 30.48/8.79 Clauses: 30.48/8.79 30.48/8.79 delmincA(tree(X1, void, X2), X1, X2). 30.48/8.79 delmincA(tree(X1, X2, X3), X4, tree(X1, X5, X6)) :- delmincA(X2, X4, X5). 30.48/8.79 deletecB(X1, tree(X1, void, X2), X2). 30.48/8.79 deletecB(X1, tree(X1, X2, void), X2). 30.48/8.79 deletecB(X1, tree(X1, X2, tree(X3, void, X4)), tree(X3, X2, X4)). 30.48/8.79 deletecB(X1, tree(X1, X2, tree(X3, X4, X5)), tree(X6, X2, tree(X3, X7, X8))) :- delmincA(X4, X6, X7). 30.48/8.79 deletecB(X1, tree(X2, X3, X4), tree(X2, X5, X4)) :- qcC(X1, X2, X3, X5). 30.48/8.79 deletecB(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- qcD(X2, X1, X4, X5). 30.48/8.79 deletecB(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) :- deletecB(0, X2, X4). 30.48/8.79 deletecB(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) :- qcE(X1, X2, X3, X5). 30.48/8.79 deletecB(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- qcD(X2, X1, X4, X5). 30.48/8.79 deletecB(s(X1), tree(0, X2, X3), tree(0, X2, X4)) :- deletecB(s(X1), X3, X4). 30.48/8.79 deletecB(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) :- qcF(X2, X1, X4, X5). 30.48/8.79 deletecB(X1, tree(X1, X2, tree(X3, void, X4)), tree(X3, X2, X4)). 30.48/8.79 deletecB(X1, tree(X1, X2, tree(X3, X4, X5)), tree(X6, X2, tree(X3, X7, X8))) :- delmincA(X4, X6, X7). 30.48/8.79 deletecB(X1, tree(X2, X3, X4), tree(X2, X5, X4)) :- qcC(X1, X2, X3, X5). 30.48/8.79 deletecB(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- qcD(X2, X1, X4, X5). 30.48/8.79 deletecB(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) :- deletecB(0, X2, X4). 30.48/8.79 deletecB(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) :- qcE(X1, X2, X3, X5). 30.48/8.79 deletecB(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- qcD(X2, X1, X4, X5). 30.48/8.79 deletecB(s(X1), tree(0, X2, X3), tree(0, X2, X4)) :- deletecB(s(X1), X3, X4). 30.48/8.79 deletecB(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) :- qcF(X2, X1, X4, X5). 30.48/8.79 deletecB(X1, tree(X1, X2, void), X2). 30.48/8.79 deletecB(X1, tree(X1, X2, tree(X3, void, X4)), tree(X3, X2, X4)). 30.48/8.79 deletecB(X1, tree(X1, X2, tree(X3, X4, X5)), tree(X6, X2, tree(X3, X7, X8))) :- delmincA(X4, X6, X7). 30.48/8.79 deletecB(X1, tree(X2, X3, X4), tree(X2, X5, X4)) :- qcC(X1, X2, X3, X5). 30.48/8.79 deletecB(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- qcD(X2, X1, X4, X5). 30.48/8.79 deletecB(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) :- deletecB(0, X2, X4). 30.48/8.79 deletecB(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) :- qcE(X1, X2, X3, X5). 30.48/8.79 deletecB(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- qcD(X2, X1, X4, X5). 30.48/8.79 deletecB(s(X1), tree(0, X2, X3), tree(0, X2, X4)) :- deletecB(s(X1), X3, X4). 30.48/8.79 deletecB(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) :- qcF(X2, X1, X4, X5). 30.48/8.79 deletecB(X1, tree(X1, X2, tree(X3, void, X4)), tree(X3, X2, X4)). 30.48/8.79 deletecB(X1, tree(X1, X2, tree(X3, X4, X5)), tree(X6, X2, tree(X3, X7, X8))) :- delmincA(X4, X6, X7). 30.48/8.79 deletecB(X1, tree(X2, X3, X4), tree(X2, X5, X4)) :- qcC(X1, X2, X3, X5). 30.48/8.79 deletecB(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- qcD(X2, X1, X4, X5). 30.48/8.79 deletecB(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) :- deletecB(0, X2, X4). 30.48/8.79 deletecB(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) :- qcE(X1, X2, X3, X5). 30.48/8.79 deletecB(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- qcD(X2, X1, X4, X5). 30.48/8.79 deletecB(s(X1), tree(0, X2, X3), tree(0, X2, X4)) :- deletecB(s(X1), X3, X4). 30.48/8.79 deletecB(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) :- qcF(X2, X1, X4, X5). 30.48/8.79 lesscG(0, s(X1)). 30.48/8.79 lesscG(s(X1), s(X2)) :- lesscG(X1, X2). 30.48/8.79 lesscH(0, s(X1)). 30.48/8.79 lesscH(s(X1), s(X2)) :- lesscH(X1, X2). 30.48/8.79 qcD(X1, X2, X3, X4) :- ','(lesscH(X1, X2), deletecB(X2, X3, X4)). 30.48/8.79 qcC(X1, X2, X3, X4) :- ','(lesscG(X1, X2), deletecB(X1, X3, X4)). 30.48/8.79 qcE(X1, X2, X3, X4) :- ','(lesscG(X1, X2), deletecB(s(X1), X3, X4)). 30.48/8.79 qcF(X1, X2, X3, X4) :- ','(lesscH(X1, X2), deletecB(s(X2), X3, X4)). 30.48/8.79 30.48/8.79 Afs: 30.48/8.79 30.48/8.79 deleteB(x1, x2, x3) = deleteB(x1) 30.48/8.79 30.48/8.79 30.48/8.79 ---------------------------------------- 30.48/8.79 30.48/8.79 (163) TriplesToPiDPProof (SOUND) 30.48/8.79 We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: 30.48/8.79 30.48/8.79 deleteB_in_3: (b,f,f) 30.48/8.79 30.48/8.79 delminA_in_3: (f,f,f) 30.48/8.79 30.48/8.79 pC_in_4: (b,f,f,f) 30.48/8.79 30.48/8.79 lessG_in_2: (b,f) 30.48/8.79 30.48/8.79 lesscG_in_2: (b,f) 30.48/8.79 30.48/8.79 pD_in_4: (f,b,f,f) 30.48/8.79 30.48/8.79 lessH_in_2: (f,b) 30.48/8.79 30.48/8.79 lesscH_in_2: (f,b) 30.48/8.79 30.48/8.79 pE_in_4: (b,f,f,f) 30.48/8.79 30.48/8.79 pF_in_4: (f,b,f,f) 30.48/8.79 30.48/8.79 Transforming TRIPLES into the following Term Rewriting System: 30.48/8.79 30.48/8.79 Pi DP problem: 30.48/8.79 The TRS P consists of the following rules: 30.48/8.79 30.48/8.79 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)) 30.48/8.79 DELETEB_IN_GAA(X1, tree(X1, X2, tree(X3, X4, X5)), tree(X6, X2, tree(X3, X7, X8))) -> DELMINA_IN_AAA(X4, X6, X7) 30.48/8.79 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)) 30.48/8.79 DELMINA_IN_AAA(tree(X1, X2, X3), X4, tree(X1, X5, X6)) -> DELMINA_IN_AAA(X2, X4, X5) 30.48/8.79 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)) 30.48/8.79 DELETEB_IN_GAA(X1, tree(X2, X3, X4), tree(X2, X5, X4)) -> PC_IN_GAAA(X1, X2, X3, X5) 30.48/8.79 PC_IN_GAAA(X1, X2, X3, X4) -> U7_GAAA(X1, X2, X3, X4, lessG_in_ga(X1, X2)) 30.48/8.79 PC_IN_GAAA(X1, X2, X3, X4) -> LESSG_IN_GA(X1, X2) 30.48/8.79 LESSG_IN_GA(s(X1), s(X2)) -> U2_GA(X1, X2, lessG_in_ga(X1, X2)) 30.48/8.79 LESSG_IN_GA(s(X1), s(X2)) -> LESSG_IN_GA(X1, X2) 30.48/8.79 PC_IN_GAAA(X1, X2, X3, X4) -> U8_GAAA(X1, X2, X3, X4, lesscG_in_ga(X1, X2)) 30.48/8.79 U8_GAAA(X1, X2, X3, X4, lesscG_out_ga(X1, X2)) -> U9_GAAA(X1, X2, X3, X4, deleteB_in_gaa(X1, X3, X4)) 30.48/8.79 U8_GAAA(X1, X2, X3, X4, lesscG_out_ga(X1, X2)) -> DELETEB_IN_GAA(X1, X3, X4) 30.48/8.79 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)) 30.48/8.79 DELETEB_IN_GAA(X1, tree(X2, X3, X4), tree(X2, X3, X5)) -> PD_IN_AGAA(X2, X1, X4, X5) 30.48/8.79 PD_IN_AGAA(X1, X2, X3, X4) -> U4_AGAA(X1, X2, X3, X4, lessH_in_ag(X1, X2)) 30.48/8.79 PD_IN_AGAA(X1, X2, X3, X4) -> LESSH_IN_AG(X1, X2) 30.48/8.79 LESSH_IN_AG(s(X1), s(X2)) -> U3_AG(X1, X2, lessH_in_ag(X1, X2)) 30.48/8.79 LESSH_IN_AG(s(X1), s(X2)) -> LESSH_IN_AG(X1, X2) 30.48/8.79 PD_IN_AGAA(X1, X2, X3, X4) -> U5_AGAA(X1, X2, X3, X4, lesscH_in_ag(X1, X2)) 30.48/8.79 U5_AGAA(X1, X2, X3, X4, lesscH_out_ag(X1, X2)) -> U6_AGAA(X1, X2, X3, X4, deleteB_in_gaa(X2, X3, X4)) 30.48/8.79 U5_AGAA(X1, X2, X3, X4, lesscH_out_ag(X1, X2)) -> DELETEB_IN_GAA(X2, X3, X4) 30.48/8.79 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)) 30.48/8.79 DELETEB_IN_GAA(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) -> DELETEB_IN_GAA(0, X2, X4) 30.48/8.79 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)) 30.48/8.79 DELETEB_IN_GAA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) -> PE_IN_GAAA(X1, X2, X3, X5) 30.48/8.79 PE_IN_GAAA(X1, X2, X3, X4) -> U10_GAAA(X1, X2, X3, X4, lessG_in_ga(X1, X2)) 30.48/8.79 PE_IN_GAAA(X1, X2, X3, X4) -> LESSG_IN_GA(X1, X2) 30.48/8.79 PE_IN_GAAA(X1, X2, X3, X4) -> U11_GAAA(X1, X2, X3, X4, lesscG_in_ga(X1, X2)) 30.48/8.79 U11_GAAA(X1, X2, X3, X4, lesscG_out_ga(X1, X2)) -> U12_GAAA(X1, X2, X3, X4, deleteB_in_gaa(s(X1), X3, X4)) 30.48/8.79 U11_GAAA(X1, X2, X3, X4, lesscG_out_ga(X1, X2)) -> DELETEB_IN_GAA(s(X1), X3, X4) 30.48/8.79 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)) 30.48/8.79 DELETEB_IN_GAA(s(X1), tree(0, X2, X3), tree(0, X2, X4)) -> DELETEB_IN_GAA(s(X1), X3, X4) 30.48/8.79 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)) 30.48/8.79 DELETEB_IN_GAA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) -> PF_IN_AGAA(X2, X1, X4, X5) 30.48/8.79 PF_IN_AGAA(X1, X2, X3, X4) -> U13_AGAA(X1, X2, X3, X4, lessH_in_ag(X1, X2)) 30.48/8.79 PF_IN_AGAA(X1, X2, X3, X4) -> LESSH_IN_AG(X1, X2) 30.48/8.79 PF_IN_AGAA(X1, X2, X3, X4) -> U14_AGAA(X1, X2, X3, X4, lesscH_in_ag(X1, X2)) 30.48/8.79 U14_AGAA(X1, X2, X3, X4, lesscH_out_ag(X1, X2)) -> U15_AGAA(X1, X2, X3, X4, deleteB_in_gaa(s(X2), X3, X4)) 30.48/8.79 U14_AGAA(X1, X2, X3, X4, lesscH_out_ag(X1, X2)) -> DELETEB_IN_GAA(s(X2), X3, X4) 30.48/8.79 30.48/8.79 The TRS R consists of the following rules: 30.48/8.79 30.48/8.79 lesscG_in_ga(0, s(X1)) -> lesscG_out_ga(0, s(X1)) 30.48/8.79 lesscG_in_ga(s(X1), s(X2)) -> U32_ga(X1, X2, lesscG_in_ga(X1, X2)) 30.48/8.79 U32_ga(X1, X2, lesscG_out_ga(X1, X2)) -> lesscG_out_ga(s(X1), s(X2)) 30.48/8.79 lesscH_in_ag(0, s(X1)) -> lesscH_out_ag(0, s(X1)) 30.48/8.79 lesscH_in_ag(s(X1), s(X2)) -> U33_ag(X1, X2, lesscH_in_ag(X1, X2)) 30.48/8.79 U33_ag(X1, X2, lesscH_out_ag(X1, X2)) -> lesscH_out_ag(s(X1), s(X2)) 30.48/8.79 30.48/8.79 The argument filtering Pi contains the following mapping: 30.48/8.79 deleteB_in_gaa(x1, x2, x3) = deleteB_in_gaa(x1) 30.48/8.79 30.48/8.79 delminA_in_aaa(x1, x2, x3) = delminA_in_aaa 30.48/8.79 30.48/8.79 pC_in_gaaa(x1, x2, x3, x4) = pC_in_gaaa(x1) 30.48/8.79 30.48/8.79 lessG_in_ga(x1, x2) = lessG_in_ga(x1) 30.48/8.79 30.48/8.79 s(x1) = s(x1) 30.48/8.79 30.48/8.79 lesscG_in_ga(x1, x2) = lesscG_in_ga(x1) 30.48/8.79 30.48/8.79 0 = 0 30.48/8.79 30.48/8.79 lesscG_out_ga(x1, x2) = lesscG_out_ga(x1) 30.48/8.79 30.48/8.79 U32_ga(x1, x2, x3) = U32_ga(x1, x3) 30.48/8.79 30.48/8.79 pD_in_agaa(x1, x2, x3, x4) = pD_in_agaa(x2) 30.48/8.79 30.48/8.79 lessH_in_ag(x1, x2) = lessH_in_ag(x2) 30.48/8.79 30.48/8.79 lesscH_in_ag(x1, x2) = lesscH_in_ag(x2) 30.48/8.79 30.48/8.79 lesscH_out_ag(x1, x2) = lesscH_out_ag(x1, x2) 30.48/8.79 30.48/8.79 U33_ag(x1, x2, x3) = U33_ag(x2, x3) 30.48/8.79 30.48/8.79 pE_in_gaaa(x1, x2, x3, x4) = pE_in_gaaa(x1) 30.48/8.79 30.48/8.79 pF_in_agaa(x1, x2, x3, x4) = pF_in_agaa(x2) 30.48/8.79 30.48/8.79 DELETEB_IN_GAA(x1, x2, x3) = DELETEB_IN_GAA(x1) 30.48/8.79 30.48/8.79 U16_GAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U16_GAA(x1, x9) 30.48/8.79 30.48/8.79 DELMINA_IN_AAA(x1, x2, x3) = DELMINA_IN_AAA 30.48/8.79 30.48/8.79 U1_AAA(x1, x2, x3, x4, x5, x6, x7) = U1_AAA(x7) 30.48/8.79 30.48/8.79 U17_GAA(x1, x2, x3, x4, x5, x6) = U17_GAA(x1, x6) 30.48/8.79 30.48/8.79 PC_IN_GAAA(x1, x2, x3, x4) = PC_IN_GAAA(x1) 30.48/8.79 30.48/8.79 U7_GAAA(x1, x2, x3, x4, x5) = U7_GAAA(x1, x5) 30.48/8.79 30.48/8.79 LESSG_IN_GA(x1, x2) = LESSG_IN_GA(x1) 30.48/8.79 30.48/8.79 U2_GA(x1, x2, x3) = U2_GA(x1, x3) 30.48/8.79 30.48/8.79 U8_GAAA(x1, x2, x3, x4, x5) = U8_GAAA(x1, x5) 30.48/8.79 30.48/8.79 U9_GAAA(x1, x2, x3, x4, x5) = U9_GAAA(x1, x5) 30.48/8.79 30.48/8.79 U18_GAA(x1, x2, x3, x4, x5, x6) = U18_GAA(x1, x6) 30.48/8.79 30.48/8.79 PD_IN_AGAA(x1, x2, x3, x4) = PD_IN_AGAA(x2) 30.48/8.79 30.48/8.79 U4_AGAA(x1, x2, x3, x4, x5) = U4_AGAA(x2, x5) 30.48/8.79 30.48/8.79 LESSH_IN_AG(x1, x2) = LESSH_IN_AG(x2) 30.48/8.79 30.48/8.79 U3_AG(x1, x2, x3) = U3_AG(x2, x3) 30.48/8.79 30.48/8.79 U5_AGAA(x1, x2, x3, x4, x5) = U5_AGAA(x2, x5) 30.48/8.79 30.48/8.79 U6_AGAA(x1, x2, x3, x4, x5) = U6_AGAA(x1, x2, x5) 30.48/8.79 30.48/8.79 U19_GAA(x1, x2, x3, x4, x5) = U19_GAA(x5) 30.48/8.79 30.48/8.79 U20_GAA(x1, x2, x3, x4, x5, x6) = U20_GAA(x1, x6) 30.48/8.79 30.48/8.79 PE_IN_GAAA(x1, x2, x3, x4) = PE_IN_GAAA(x1) 30.48/8.79 30.48/8.79 U10_GAAA(x1, x2, x3, x4, x5) = U10_GAAA(x1, x5) 30.48/8.79 30.48/8.79 U11_GAAA(x1, x2, x3, x4, x5) = U11_GAAA(x1, x5) 30.48/8.79 30.48/8.79 U12_GAAA(x1, x2, x3, x4, x5) = U12_GAAA(x1, x5) 30.48/8.79 30.48/8.79 U21_GAA(x1, x2, x3, x4, x5) = U21_GAA(x1, x5) 30.48/8.79 30.48/8.79 U22_GAA(x1, x2, x3, x4, x5, x6) = U22_GAA(x1, x6) 30.48/8.79 30.48/8.79 PF_IN_AGAA(x1, x2, x3, x4) = PF_IN_AGAA(x2) 30.48/8.79 30.48/8.79 U13_AGAA(x1, x2, x3, x4, x5) = U13_AGAA(x2, x5) 30.48/8.79 30.48/8.79 U14_AGAA(x1, x2, x3, x4, x5) = U14_AGAA(x2, x5) 30.48/8.79 30.48/8.79 U15_AGAA(x1, x2, x3, x4, x5) = U15_AGAA(x1, x2, x5) 30.48/8.79 30.48/8.79 30.48/8.79 We have to consider all (P,R,Pi)-chains 30.48/8.79 30.48/8.79 30.48/8.79 Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES 30.48/8.79 30.48/8.79 30.48/8.79 30.48/8.79 ---------------------------------------- 30.48/8.79 30.48/8.79 (164) 30.48/8.79 Obligation: 30.48/8.79 Pi DP problem: 30.48/8.79 The TRS P consists of the following rules: 30.48/8.79 30.48/8.79 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)) 30.48/8.79 DELETEB_IN_GAA(X1, tree(X1, X2, tree(X3, X4, X5)), tree(X6, X2, tree(X3, X7, X8))) -> DELMINA_IN_AAA(X4, X6, X7) 30.48/8.79 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)) 30.48/8.79 DELMINA_IN_AAA(tree(X1, X2, X3), X4, tree(X1, X5, X6)) -> DELMINA_IN_AAA(X2, X4, X5) 30.48/8.79 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)) 30.48/8.79 DELETEB_IN_GAA(X1, tree(X2, X3, X4), tree(X2, X5, X4)) -> PC_IN_GAAA(X1, X2, X3, X5) 30.48/8.79 PC_IN_GAAA(X1, X2, X3, X4) -> U7_GAAA(X1, X2, X3, X4, lessG_in_ga(X1, X2)) 30.48/8.79 PC_IN_GAAA(X1, X2, X3, X4) -> LESSG_IN_GA(X1, X2) 30.48/8.79 LESSG_IN_GA(s(X1), s(X2)) -> U2_GA(X1, X2, lessG_in_ga(X1, X2)) 30.48/8.79 LESSG_IN_GA(s(X1), s(X2)) -> LESSG_IN_GA(X1, X2) 30.48/8.79 PC_IN_GAAA(X1, X2, X3, X4) -> U8_GAAA(X1, X2, X3, X4, lesscG_in_ga(X1, X2)) 30.48/8.79 U8_GAAA(X1, X2, X3, X4, lesscG_out_ga(X1, X2)) -> U9_GAAA(X1, X2, X3, X4, deleteB_in_gaa(X1, X3, X4)) 30.48/8.79 U8_GAAA(X1, X2, X3, X4, lesscG_out_ga(X1, X2)) -> DELETEB_IN_GAA(X1, X3, X4) 30.48/8.79 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)) 30.48/8.79 DELETEB_IN_GAA(X1, tree(X2, X3, X4), tree(X2, X3, X5)) -> PD_IN_AGAA(X2, X1, X4, X5) 30.48/8.79 PD_IN_AGAA(X1, X2, X3, X4) -> U4_AGAA(X1, X2, X3, X4, lessH_in_ag(X1, X2)) 30.48/8.79 PD_IN_AGAA(X1, X2, X3, X4) -> LESSH_IN_AG(X1, X2) 30.48/8.79 LESSH_IN_AG(s(X1), s(X2)) -> U3_AG(X1, X2, lessH_in_ag(X1, X2)) 30.48/8.79 LESSH_IN_AG(s(X1), s(X2)) -> LESSH_IN_AG(X1, X2) 30.48/8.79 PD_IN_AGAA(X1, X2, X3, X4) -> U5_AGAA(X1, X2, X3, X4, lesscH_in_ag(X1, X2)) 30.48/8.79 U5_AGAA(X1, X2, X3, X4, lesscH_out_ag(X1, X2)) -> U6_AGAA(X1, X2, X3, X4, deleteB_in_gaa(X2, X3, X4)) 30.48/8.79 U5_AGAA(X1, X2, X3, X4, lesscH_out_ag(X1, X2)) -> DELETEB_IN_GAA(X2, X3, X4) 30.48/8.79 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)) 30.48/8.79 DELETEB_IN_GAA(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) -> DELETEB_IN_GAA(0, X2, X4) 30.48/8.79 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)) 30.48/8.79 DELETEB_IN_GAA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) -> PE_IN_GAAA(X1, X2, X3, X5) 30.48/8.79 PE_IN_GAAA(X1, X2, X3, X4) -> U10_GAAA(X1, X2, X3, X4, lessG_in_ga(X1, X2)) 30.48/8.79 PE_IN_GAAA(X1, X2, X3, X4) -> LESSG_IN_GA(X1, X2) 30.48/8.79 PE_IN_GAAA(X1, X2, X3, X4) -> U11_GAAA(X1, X2, X3, X4, lesscG_in_ga(X1, X2)) 30.48/8.79 U11_GAAA(X1, X2, X3, X4, lesscG_out_ga(X1, X2)) -> U12_GAAA(X1, X2, X3, X4, deleteB_in_gaa(s(X1), X3, X4)) 30.48/8.79 U11_GAAA(X1, X2, X3, X4, lesscG_out_ga(X1, X2)) -> DELETEB_IN_GAA(s(X1), X3, X4) 30.48/8.79 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)) 30.48/8.79 DELETEB_IN_GAA(s(X1), tree(0, X2, X3), tree(0, X2, X4)) -> DELETEB_IN_GAA(s(X1), X3, X4) 30.48/8.79 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)) 30.48/8.79 DELETEB_IN_GAA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) -> PF_IN_AGAA(X2, X1, X4, X5) 30.48/8.79 PF_IN_AGAA(X1, X2, X3, X4) -> U13_AGAA(X1, X2, X3, X4, lessH_in_ag(X1, X2)) 30.48/8.79 PF_IN_AGAA(X1, X2, X3, X4) -> LESSH_IN_AG(X1, X2) 30.48/8.79 PF_IN_AGAA(X1, X2, X3, X4) -> U14_AGAA(X1, X2, X3, X4, lesscH_in_ag(X1, X2)) 30.48/8.79 U14_AGAA(X1, X2, X3, X4, lesscH_out_ag(X1, X2)) -> U15_AGAA(X1, X2, X3, X4, deleteB_in_gaa(s(X2), X3, X4)) 30.48/8.79 U14_AGAA(X1, X2, X3, X4, lesscH_out_ag(X1, X2)) -> DELETEB_IN_GAA(s(X2), X3, X4) 30.48/8.79 30.48/8.79 The TRS R consists of the following rules: 30.48/8.79 30.48/8.79 lesscG_in_ga(0, s(X1)) -> lesscG_out_ga(0, s(X1)) 30.48/8.79 lesscG_in_ga(s(X1), s(X2)) -> U32_ga(X1, X2, lesscG_in_ga(X1, X2)) 30.48/8.79 U32_ga(X1, X2, lesscG_out_ga(X1, X2)) -> lesscG_out_ga(s(X1), s(X2)) 30.48/8.79 lesscH_in_ag(0, s(X1)) -> lesscH_out_ag(0, s(X1)) 30.48/8.79 lesscH_in_ag(s(X1), s(X2)) -> U33_ag(X1, X2, lesscH_in_ag(X1, X2)) 30.48/8.79 U33_ag(X1, X2, lesscH_out_ag(X1, X2)) -> lesscH_out_ag(s(X1), s(X2)) 30.48/8.79 30.48/8.79 The argument filtering Pi contains the following mapping: 30.48/8.79 deleteB_in_gaa(x1, x2, x3) = deleteB_in_gaa(x1) 30.48/8.79 30.48/8.79 delminA_in_aaa(x1, x2, x3) = delminA_in_aaa 30.48/8.79 30.48/8.79 pC_in_gaaa(x1, x2, x3, x4) = pC_in_gaaa(x1) 30.48/8.79 30.48/8.79 lessG_in_ga(x1, x2) = lessG_in_ga(x1) 30.48/8.79 30.48/8.79 s(x1) = s(x1) 30.48/8.79 30.48/8.79 lesscG_in_ga(x1, x2) = lesscG_in_ga(x1) 30.48/8.79 30.48/8.79 0 = 0 30.48/8.79 30.48/8.79 lesscG_out_ga(x1, x2) = lesscG_out_ga(x1) 30.48/8.79 30.48/8.79 U32_ga(x1, x2, x3) = U32_ga(x1, x3) 30.48/8.79 30.48/8.79 pD_in_agaa(x1, x2, x3, x4) = pD_in_agaa(x2) 30.48/8.79 30.48/8.79 lessH_in_ag(x1, x2) = lessH_in_ag(x2) 30.48/8.79 30.48/8.79 lesscH_in_ag(x1, x2) = lesscH_in_ag(x2) 30.48/8.79 30.48/8.79 lesscH_out_ag(x1, x2) = lesscH_out_ag(x1, x2) 30.48/8.79 30.48/8.79 U33_ag(x1, x2, x3) = U33_ag(x2, x3) 30.48/8.79 30.48/8.79 pE_in_gaaa(x1, x2, x3, x4) = pE_in_gaaa(x1) 30.48/8.79 30.48/8.79 pF_in_agaa(x1, x2, x3, x4) = pF_in_agaa(x2) 30.48/8.79 30.48/8.79 DELETEB_IN_GAA(x1, x2, x3) = DELETEB_IN_GAA(x1) 30.48/8.79 30.48/8.79 U16_GAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U16_GAA(x1, x9) 30.48/8.79 30.48/8.79 DELMINA_IN_AAA(x1, x2, x3) = DELMINA_IN_AAA 30.48/8.79 30.48/8.79 U1_AAA(x1, x2, x3, x4, x5, x6, x7) = U1_AAA(x7) 30.48/8.79 30.48/8.79 U17_GAA(x1, x2, x3, x4, x5, x6) = U17_GAA(x1, x6) 30.48/8.79 30.48/8.79 PC_IN_GAAA(x1, x2, x3, x4) = PC_IN_GAAA(x1) 30.48/8.79 30.48/8.79 U7_GAAA(x1, x2, x3, x4, x5) = U7_GAAA(x1, x5) 30.48/8.79 30.48/8.79 LESSG_IN_GA(x1, x2) = LESSG_IN_GA(x1) 30.48/8.79 30.48/8.79 U2_GA(x1, x2, x3) = U2_GA(x1, x3) 30.48/8.79 30.48/8.79 U8_GAAA(x1, x2, x3, x4, x5) = U8_GAAA(x1, x5) 30.48/8.79 30.48/8.79 U9_GAAA(x1, x2, x3, x4, x5) = U9_GAAA(x1, x5) 30.48/8.79 30.48/8.79 U18_GAA(x1, x2, x3, x4, x5, x6) = U18_GAA(x1, x6) 30.48/8.79 30.48/8.79 PD_IN_AGAA(x1, x2, x3, x4) = PD_IN_AGAA(x2) 30.48/8.79 30.48/8.79 U4_AGAA(x1, x2, x3, x4, x5) = U4_AGAA(x2, x5) 30.48/8.79 30.48/8.79 LESSH_IN_AG(x1, x2) = LESSH_IN_AG(x2) 30.48/8.79 30.48/8.79 U3_AG(x1, x2, x3) = U3_AG(x2, x3) 30.48/8.79 30.48/8.79 U5_AGAA(x1, x2, x3, x4, x5) = U5_AGAA(x2, x5) 30.48/8.79 30.48/8.79 U6_AGAA(x1, x2, x3, x4, x5) = U6_AGAA(x1, x2, x5) 30.48/8.79 30.48/8.79 U19_GAA(x1, x2, x3, x4, x5) = U19_GAA(x5) 30.48/8.79 30.48/8.79 U20_GAA(x1, x2, x3, x4, x5, x6) = U20_GAA(x1, x6) 30.48/8.79 30.48/8.79 PE_IN_GAAA(x1, x2, x3, x4) = PE_IN_GAAA(x1) 30.48/8.79 30.48/8.79 U10_GAAA(x1, x2, x3, x4, x5) = U10_GAAA(x1, x5) 30.48/8.79 30.48/8.79 U11_GAAA(x1, x2, x3, x4, x5) = U11_GAAA(x1, x5) 30.48/8.79 30.48/8.79 U12_GAAA(x1, x2, x3, x4, x5) = U12_GAAA(x1, x5) 30.48/8.79 30.48/8.79 U21_GAA(x1, x2, x3, x4, x5) = U21_GAA(x1, x5) 30.48/8.79 30.48/8.79 U22_GAA(x1, x2, x3, x4, x5, x6) = U22_GAA(x1, x6) 30.48/8.79 30.48/8.79 PF_IN_AGAA(x1, x2, x3, x4) = PF_IN_AGAA(x2) 30.48/8.79 30.48/8.79 U13_AGAA(x1, x2, x3, x4, x5) = U13_AGAA(x2, x5) 30.48/8.79 30.48/8.79 U14_AGAA(x1, x2, x3, x4, x5) = U14_AGAA(x2, x5) 30.48/8.79 30.48/8.79 U15_AGAA(x1, x2, x3, x4, x5) = U15_AGAA(x1, x2, x5) 30.48/8.79 30.48/8.79 30.48/8.79 We have to consider all (P,R,Pi)-chains 30.48/8.79 ---------------------------------------- 30.48/8.79 30.48/8.79 (165) DependencyGraphProof (EQUIVALENT) 30.48/8.79 The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 23 less nodes. 30.48/8.79 ---------------------------------------- 30.48/8.79 30.48/8.79 (166) 30.48/8.79 Complex Obligation (AND) 30.48/8.79 30.48/8.79 ---------------------------------------- 30.48/8.79 30.48/8.79 (167) 30.48/8.79 Obligation: 30.48/8.79 Pi DP problem: 30.48/8.79 The TRS P consists of the following rules: 30.48/8.79 30.48/8.79 LESSH_IN_AG(s(X1), s(X2)) -> LESSH_IN_AG(X1, X2) 30.48/8.79 30.48/8.79 The TRS R consists of the following rules: 30.48/8.79 30.48/8.79 lesscG_in_ga(0, s(X1)) -> lesscG_out_ga(0, s(X1)) 30.48/8.79 lesscG_in_ga(s(X1), s(X2)) -> U32_ga(X1, X2, lesscG_in_ga(X1, X2)) 30.48/8.79 U32_ga(X1, X2, lesscG_out_ga(X1, X2)) -> lesscG_out_ga(s(X1), s(X2)) 30.48/8.79 lesscH_in_ag(0, s(X1)) -> lesscH_out_ag(0, s(X1)) 30.48/8.79 lesscH_in_ag(s(X1), s(X2)) -> U33_ag(X1, X2, lesscH_in_ag(X1, X2)) 30.48/8.79 U33_ag(X1, X2, lesscH_out_ag(X1, X2)) -> lesscH_out_ag(s(X1), s(X2)) 30.48/8.79 30.48/8.79 The argument filtering Pi contains the following mapping: 30.48/8.79 s(x1) = s(x1) 30.48/8.79 30.48/8.79 lesscG_in_ga(x1, x2) = lesscG_in_ga(x1) 30.48/8.79 30.48/8.79 0 = 0 30.48/8.79 30.48/8.79 lesscG_out_ga(x1, x2) = lesscG_out_ga(x1) 30.48/8.79 30.48/8.79 U32_ga(x1, x2, x3) = U32_ga(x1, x3) 30.48/8.79 30.48/8.79 lesscH_in_ag(x1, x2) = lesscH_in_ag(x2) 30.48/8.79 30.48/8.79 lesscH_out_ag(x1, x2) = lesscH_out_ag(x1, x2) 30.48/8.79 30.48/8.79 U33_ag(x1, x2, x3) = U33_ag(x2, x3) 30.48/8.79 30.48/8.79 LESSH_IN_AG(x1, x2) = LESSH_IN_AG(x2) 30.48/8.79 30.48/8.79 30.48/8.79 We have to consider all (P,R,Pi)-chains 30.48/8.79 ---------------------------------------- 30.48/8.79 30.48/8.79 (168) UsableRulesProof (EQUIVALENT) 30.48/8.79 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 30.48/8.79 ---------------------------------------- 30.48/8.79 30.48/8.79 (169) 30.48/8.79 Obligation: 30.48/8.79 Pi DP problem: 30.48/8.79 The TRS P consists of the following rules: 30.48/8.79 30.48/8.79 LESSH_IN_AG(s(X1), s(X2)) -> LESSH_IN_AG(X1, X2) 30.48/8.79 30.48/8.79 R is empty. 30.48/8.79 The argument filtering Pi contains the following mapping: 30.48/8.79 s(x1) = s(x1) 30.48/8.79 30.48/8.79 LESSH_IN_AG(x1, x2) = LESSH_IN_AG(x2) 30.48/8.79 30.48/8.79 30.48/8.79 We have to consider all (P,R,Pi)-chains 30.48/8.79 ---------------------------------------- 30.48/8.79 30.48/8.79 (170) PiDPToQDPProof (SOUND) 30.48/8.79 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 30.48/8.79 ---------------------------------------- 30.48/8.79 30.48/8.79 (171) 30.48/8.79 Obligation: 30.48/8.79 Q DP problem: 30.48/8.79 The TRS P consists of the following rules: 30.48/8.79 30.48/8.79 LESSH_IN_AG(s(X2)) -> LESSH_IN_AG(X2) 30.48/8.79 30.48/8.79 R is empty. 30.48/8.79 Q is empty. 30.48/8.79 We have to consider all (P,Q,R)-chains. 30.48/8.79 ---------------------------------------- 30.48/8.79 30.48/8.79 (172) QDPSizeChangeProof (EQUIVALENT) 30.48/8.79 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. 30.48/8.79 30.48/8.79 From the DPs we obtained the following set of size-change graphs: 30.48/8.79 *LESSH_IN_AG(s(X2)) -> LESSH_IN_AG(X2) 30.48/8.79 The graph contains the following edges 1 > 1 30.48/8.79 30.48/8.79 30.48/8.79 ---------------------------------------- 30.48/8.79 30.48/8.79 (173) 30.48/8.79 YES 30.48/8.79 30.48/8.79 ---------------------------------------- 30.48/8.79 30.48/8.79 (174) 30.48/8.79 Obligation: 30.48/8.79 Pi DP problem: 30.48/8.79 The TRS P consists of the following rules: 30.48/8.79 30.48/8.79 LESSG_IN_GA(s(X1), s(X2)) -> LESSG_IN_GA(X1, X2) 30.48/8.79 30.48/8.79 The TRS R consists of the following rules: 30.48/8.79 30.48/8.79 lesscG_in_ga(0, s(X1)) -> lesscG_out_ga(0, s(X1)) 30.48/8.79 lesscG_in_ga(s(X1), s(X2)) -> U32_ga(X1, X2, lesscG_in_ga(X1, X2)) 30.48/8.79 U32_ga(X1, X2, lesscG_out_ga(X1, X2)) -> lesscG_out_ga(s(X1), s(X2)) 30.48/8.79 lesscH_in_ag(0, s(X1)) -> lesscH_out_ag(0, s(X1)) 30.48/8.79 lesscH_in_ag(s(X1), s(X2)) -> U33_ag(X1, X2, lesscH_in_ag(X1, X2)) 30.48/8.79 U33_ag(X1, X2, lesscH_out_ag(X1, X2)) -> lesscH_out_ag(s(X1), s(X2)) 30.48/8.79 30.48/8.79 The argument filtering Pi contains the following mapping: 30.48/8.79 s(x1) = s(x1) 30.48/8.79 30.48/8.79 lesscG_in_ga(x1, x2) = lesscG_in_ga(x1) 30.48/8.79 30.48/8.79 0 = 0 30.48/8.79 30.48/8.79 lesscG_out_ga(x1, x2) = lesscG_out_ga(x1) 30.48/8.79 30.48/8.79 U32_ga(x1, x2, x3) = U32_ga(x1, x3) 30.48/8.79 30.48/8.79 lesscH_in_ag(x1, x2) = lesscH_in_ag(x2) 30.48/8.79 30.48/8.79 lesscH_out_ag(x1, x2) = lesscH_out_ag(x1, x2) 30.48/8.79 30.48/8.79 U33_ag(x1, x2, x3) = U33_ag(x2, x3) 30.48/8.79 30.48/8.79 LESSG_IN_GA(x1, x2) = LESSG_IN_GA(x1) 30.48/8.79 30.48/8.79 30.48/8.79 We have to consider all (P,R,Pi)-chains 30.48/8.79 ---------------------------------------- 30.48/8.79 30.48/8.79 (175) UsableRulesProof (EQUIVALENT) 30.48/8.79 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 30.48/8.79 ---------------------------------------- 30.48/8.79 30.48/8.79 (176) 30.48/8.79 Obligation: 30.48/8.79 Pi DP problem: 30.48/8.79 The TRS P consists of the following rules: 30.48/8.79 30.48/8.79 LESSG_IN_GA(s(X1), s(X2)) -> LESSG_IN_GA(X1, X2) 30.48/8.79 30.48/8.79 R is empty. 30.48/8.79 The argument filtering Pi contains the following mapping: 30.48/8.79 s(x1) = s(x1) 30.48/8.79 30.48/8.79 LESSG_IN_GA(x1, x2) = LESSG_IN_GA(x1) 30.48/8.79 30.48/8.79 30.48/8.79 We have to consider all (P,R,Pi)-chains 30.48/8.79 ---------------------------------------- 30.48/8.79 30.48/8.79 (177) PiDPToQDPProof (SOUND) 30.48/8.79 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 30.48/8.79 ---------------------------------------- 30.48/8.79 30.48/8.79 (178) 30.48/8.79 Obligation: 30.48/8.79 Q DP problem: 30.48/8.79 The TRS P consists of the following rules: 30.48/8.79 30.48/8.79 LESSG_IN_GA(s(X1)) -> LESSG_IN_GA(X1) 30.48/8.79 30.48/8.79 R is empty. 30.48/8.79 Q is empty. 30.48/8.79 We have to consider all (P,Q,R)-chains. 30.48/8.79 ---------------------------------------- 30.48/8.79 30.48/8.79 (179) QDPSizeChangeProof (EQUIVALENT) 30.48/8.79 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. 30.48/8.79 30.48/8.79 From the DPs we obtained the following set of size-change graphs: 30.48/8.79 *LESSG_IN_GA(s(X1)) -> LESSG_IN_GA(X1) 30.48/8.79 The graph contains the following edges 1 > 1 30.48/8.79 30.48/8.79 30.48/8.79 ---------------------------------------- 30.48/8.79 30.48/8.79 (180) 30.48/8.79 YES 30.48/8.79 30.48/8.79 ---------------------------------------- 30.48/8.79 30.48/8.79 (181) 30.48/8.79 Obligation: 30.48/8.79 Pi DP problem: 30.48/8.79 The TRS P consists of the following rules: 30.48/8.79 30.48/8.79 DELMINA_IN_AAA(tree(X1, X2, X3), X4, tree(X1, X5, X6)) -> DELMINA_IN_AAA(X2, X4, X5) 30.48/8.79 30.48/8.79 The TRS R consists of the following rules: 30.48/8.79 30.48/8.79 lesscG_in_ga(0, s(X1)) -> lesscG_out_ga(0, s(X1)) 30.48/8.79 lesscG_in_ga(s(X1), s(X2)) -> U32_ga(X1, X2, lesscG_in_ga(X1, X2)) 30.48/8.79 U32_ga(X1, X2, lesscG_out_ga(X1, X2)) -> lesscG_out_ga(s(X1), s(X2)) 30.48/8.79 lesscH_in_ag(0, s(X1)) -> lesscH_out_ag(0, s(X1)) 30.48/8.79 lesscH_in_ag(s(X1), s(X2)) -> U33_ag(X1, X2, lesscH_in_ag(X1, X2)) 30.48/8.79 U33_ag(X1, X2, lesscH_out_ag(X1, X2)) -> lesscH_out_ag(s(X1), s(X2)) 30.48/8.79 30.48/8.79 The argument filtering Pi contains the following mapping: 30.48/8.79 s(x1) = s(x1) 30.48/8.79 30.48/8.79 lesscG_in_ga(x1, x2) = lesscG_in_ga(x1) 30.48/8.79 30.48/8.79 0 = 0 30.48/8.79 30.48/8.79 lesscG_out_ga(x1, x2) = lesscG_out_ga(x1) 30.48/8.79 30.48/8.79 U32_ga(x1, x2, x3) = U32_ga(x1, x3) 30.48/8.79 30.48/8.79 lesscH_in_ag(x1, x2) = lesscH_in_ag(x2) 30.48/8.79 30.48/8.79 lesscH_out_ag(x1, x2) = lesscH_out_ag(x1, x2) 30.48/8.79 30.48/8.79 U33_ag(x1, x2, x3) = U33_ag(x2, x3) 30.48/8.79 30.48/8.79 DELMINA_IN_AAA(x1, x2, x3) = DELMINA_IN_AAA 30.48/8.79 30.48/8.79 30.48/8.79 We have to consider all (P,R,Pi)-chains 30.48/8.79 ---------------------------------------- 30.48/8.79 30.48/8.79 (182) UsableRulesProof (EQUIVALENT) 30.48/8.79 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 30.48/8.79 ---------------------------------------- 30.48/8.79 30.48/8.79 (183) 30.48/8.79 Obligation: 30.48/8.79 Pi DP problem: 30.48/8.79 The TRS P consists of the following rules: 30.48/8.79 30.48/8.79 DELMINA_IN_AAA(tree(X1, X2, X3), X4, tree(X1, X5, X6)) -> DELMINA_IN_AAA(X2, X4, X5) 30.48/8.79 30.48/8.79 R is empty. 30.48/8.79 The argument filtering Pi contains the following mapping: 30.48/8.79 DELMINA_IN_AAA(x1, x2, x3) = DELMINA_IN_AAA 30.48/8.79 30.48/8.79 30.48/8.79 We have to consider all (P,R,Pi)-chains 30.48/8.79 ---------------------------------------- 30.48/8.79 30.48/8.79 (184) PiDPToQDPProof (SOUND) 30.48/8.79 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 30.48/8.79 ---------------------------------------- 30.48/8.79 30.48/8.79 (185) 30.48/8.79 Obligation: 30.48/8.79 Q DP problem: 30.48/8.79 The TRS P consists of the following rules: 30.48/8.79 30.48/8.79 DELMINA_IN_AAA -> DELMINA_IN_AAA 30.48/8.79 30.48/8.79 R is empty. 30.48/8.79 Q is empty. 30.48/8.79 We have to consider all (P,Q,R)-chains. 30.48/8.79 ---------------------------------------- 30.48/8.79 30.48/8.79 (186) NonTerminationLoopProof (COMPLETE) 30.48/8.79 We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. 30.48/8.79 Found a loop by semiunifying a rule from P directly. 30.48/8.79 30.48/8.79 s = DELMINA_IN_AAA evaluates to t =DELMINA_IN_AAA 30.48/8.79 30.48/8.79 Thus s starts an infinite chain as s semiunifies with t with the following substitutions: 30.48/8.79 * Matcher: [ ] 30.48/8.79 * Semiunifier: [ ] 30.48/8.79 30.48/8.79 -------------------------------------------------------------------------------- 30.48/8.79 Rewriting sequence 30.48/8.79 30.48/8.79 The DP semiunifies directly so there is only one rewrite step from DELMINA_IN_AAA to DELMINA_IN_AAA. 30.48/8.79 30.48/8.79 30.48/8.79 30.48/8.79 30.48/8.79 ---------------------------------------- 30.48/8.79 30.48/8.79 (187) 30.48/8.79 NO 30.48/8.79 30.48/8.79 ---------------------------------------- 30.48/8.79 30.48/8.79 (188) 30.48/8.79 Obligation: 30.48/8.79 Pi DP problem: 30.48/8.79 The TRS P consists of the following rules: 30.48/8.79 30.48/8.79 DELETEB_IN_GAA(X1, tree(X2, X3, X4), tree(X2, X5, X4)) -> PC_IN_GAAA(X1, X2, X3, X5) 30.48/8.79 PC_IN_GAAA(X1, X2, X3, X4) -> U8_GAAA(X1, X2, X3, X4, lesscG_in_ga(X1, X2)) 30.48/8.79 U8_GAAA(X1, X2, X3, X4, lesscG_out_ga(X1, X2)) -> DELETEB_IN_GAA(X1, X3, X4) 30.48/8.79 DELETEB_IN_GAA(X1, tree(X2, X3, X4), tree(X2, X3, X5)) -> PD_IN_AGAA(X2, X1, X4, X5) 30.48/8.79 PD_IN_AGAA(X1, X2, X3, X4) -> U5_AGAA(X1, X2, X3, X4, lesscH_in_ag(X1, X2)) 30.48/8.79 U5_AGAA(X1, X2, X3, X4, lesscH_out_ag(X1, X2)) -> DELETEB_IN_GAA(X2, X3, X4) 30.48/8.79 DELETEB_IN_GAA(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) -> DELETEB_IN_GAA(0, X2, X4) 30.48/8.79 DELETEB_IN_GAA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) -> PE_IN_GAAA(X1, X2, X3, X5) 30.48/8.79 PE_IN_GAAA(X1, X2, X3, X4) -> U11_GAAA(X1, X2, X3, X4, lesscG_in_ga(X1, X2)) 30.48/8.79 U11_GAAA(X1, X2, X3, X4, lesscG_out_ga(X1, X2)) -> DELETEB_IN_GAA(s(X1), X3, X4) 30.48/8.79 DELETEB_IN_GAA(s(X1), tree(0, X2, X3), tree(0, X2, X4)) -> DELETEB_IN_GAA(s(X1), X3, X4) 30.48/8.79 DELETEB_IN_GAA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) -> PF_IN_AGAA(X2, X1, X4, X5) 30.48/8.79 PF_IN_AGAA(X1, X2, X3, X4) -> U14_AGAA(X1, X2, X3, X4, lesscH_in_ag(X1, X2)) 30.48/8.79 U14_AGAA(X1, X2, X3, X4, lesscH_out_ag(X1, X2)) -> DELETEB_IN_GAA(s(X2), X3, X4) 30.48/8.79 30.48/8.79 The TRS R consists of the following rules: 30.48/8.79 30.48/8.79 lesscG_in_ga(0, s(X1)) -> lesscG_out_ga(0, s(X1)) 30.48/8.79 lesscG_in_ga(s(X1), s(X2)) -> U32_ga(X1, X2, lesscG_in_ga(X1, X2)) 30.48/8.79 U32_ga(X1, X2, lesscG_out_ga(X1, X2)) -> lesscG_out_ga(s(X1), s(X2)) 30.48/8.79 lesscH_in_ag(0, s(X1)) -> lesscH_out_ag(0, s(X1)) 30.48/8.79 lesscH_in_ag(s(X1), s(X2)) -> U33_ag(X1, X2, lesscH_in_ag(X1, X2)) 30.48/8.79 U33_ag(X1, X2, lesscH_out_ag(X1, X2)) -> lesscH_out_ag(s(X1), s(X2)) 30.48/8.79 30.48/8.79 The argument filtering Pi contains the following mapping: 30.48/8.79 s(x1) = s(x1) 30.48/8.79 30.48/8.79 lesscG_in_ga(x1, x2) = lesscG_in_ga(x1) 30.48/8.79 30.48/8.79 0 = 0 30.48/8.79 30.48/8.79 lesscG_out_ga(x1, x2) = lesscG_out_ga(x1) 30.48/8.79 30.48/8.79 U32_ga(x1, x2, x3) = U32_ga(x1, x3) 30.48/8.79 30.48/8.79 lesscH_in_ag(x1, x2) = lesscH_in_ag(x2) 30.48/8.79 30.48/8.79 lesscH_out_ag(x1, x2) = lesscH_out_ag(x1, x2) 30.48/8.79 30.48/8.79 U33_ag(x1, x2, x3) = U33_ag(x2, x3) 30.48/8.79 30.48/8.79 DELETEB_IN_GAA(x1, x2, x3) = DELETEB_IN_GAA(x1) 30.48/8.79 30.48/8.79 PC_IN_GAAA(x1, x2, x3, x4) = PC_IN_GAAA(x1) 30.48/8.79 30.48/8.79 U8_GAAA(x1, x2, x3, x4, x5) = U8_GAAA(x1, x5) 30.48/8.79 30.48/8.79 PD_IN_AGAA(x1, x2, x3, x4) = PD_IN_AGAA(x2) 30.48/8.79 30.48/8.79 U5_AGAA(x1, x2, x3, x4, x5) = U5_AGAA(x2, x5) 30.48/8.79 30.48/8.79 PE_IN_GAAA(x1, x2, x3, x4) = PE_IN_GAAA(x1) 30.48/8.79 30.48/8.79 U11_GAAA(x1, x2, x3, x4, x5) = U11_GAAA(x1, x5) 30.48/8.79 30.48/8.79 PF_IN_AGAA(x1, x2, x3, x4) = PF_IN_AGAA(x2) 30.48/8.79 30.48/8.79 U14_AGAA(x1, x2, x3, x4, x5) = U14_AGAA(x2, x5) 30.48/8.79 30.48/8.79 30.48/8.79 We have to consider all (P,R,Pi)-chains 30.48/8.79 ---------------------------------------- 30.48/8.79 30.48/8.79 (189) PiDPToQDPProof (SOUND) 30.48/8.79 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 30.48/8.79 ---------------------------------------- 30.48/8.79 30.48/8.79 (190) 30.48/8.79 Obligation: 30.48/8.79 Q DP problem: 30.48/8.79 The TRS P consists of the following rules: 30.48/8.79 30.48/8.79 DELETEB_IN_GAA(X1) -> PC_IN_GAAA(X1) 30.48/8.79 PC_IN_GAAA(X1) -> U8_GAAA(X1, lesscG_in_ga(X1)) 30.48/8.79 U8_GAAA(X1, lesscG_out_ga(X1)) -> DELETEB_IN_GAA(X1) 30.48/8.79 DELETEB_IN_GAA(X1) -> PD_IN_AGAA(X1) 30.48/8.79 PD_IN_AGAA(X2) -> U5_AGAA(X2, lesscH_in_ag(X2)) 30.48/8.79 U5_AGAA(X2, lesscH_out_ag(X1, X2)) -> DELETEB_IN_GAA(X2) 30.48/8.79 DELETEB_IN_GAA(0) -> DELETEB_IN_GAA(0) 30.48/8.79 DELETEB_IN_GAA(s(X1)) -> PE_IN_GAAA(X1) 30.48/8.79 PE_IN_GAAA(X1) -> U11_GAAA(X1, lesscG_in_ga(X1)) 30.48/8.79 U11_GAAA(X1, lesscG_out_ga(X1)) -> DELETEB_IN_GAA(s(X1)) 30.48/8.79 DELETEB_IN_GAA(s(X1)) -> DELETEB_IN_GAA(s(X1)) 30.48/8.79 DELETEB_IN_GAA(s(X1)) -> PF_IN_AGAA(X1) 30.48/8.79 PF_IN_AGAA(X2) -> U14_AGAA(X2, lesscH_in_ag(X2)) 30.48/8.79 U14_AGAA(X2, lesscH_out_ag(X1, X2)) -> DELETEB_IN_GAA(s(X2)) 30.48/8.79 30.48/8.79 The TRS R consists of the following rules: 30.48/8.79 30.48/8.79 lesscG_in_ga(0) -> lesscG_out_ga(0) 30.48/8.79 lesscG_in_ga(s(X1)) -> U32_ga(X1, lesscG_in_ga(X1)) 30.48/8.79 U32_ga(X1, lesscG_out_ga(X1)) -> lesscG_out_ga(s(X1)) 30.48/8.79 lesscH_in_ag(s(X1)) -> lesscH_out_ag(0, s(X1)) 30.48/8.79 lesscH_in_ag(s(X2)) -> U33_ag(X2, lesscH_in_ag(X2)) 30.48/8.79 U33_ag(X2, lesscH_out_ag(X1, X2)) -> lesscH_out_ag(s(X1), s(X2)) 30.48/8.79 30.48/8.79 The set Q consists of the following terms: 30.48/8.79 30.48/8.79 lesscG_in_ga(x0) 30.48/8.79 U32_ga(x0, x1) 30.48/8.79 lesscH_in_ag(x0) 30.48/8.79 U33_ag(x0, x1) 30.48/8.79 30.48/8.79 We have to consider all (P,Q,R)-chains. 30.59/8.83 EOF