27.18/8.82 MAYBE 27.18/8.84 proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl 27.18/8.84 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 27.18/8.84 27.18/8.84 27.18/8.84 Left Termination of the query pattern 27.18/8.84 27.18/8.84 in(g,a) 27.18/8.84 27.18/8.84 w.r.t. the given Prolog program could not be shown: 27.18/8.84 27.18/8.84 (0) Prolog 27.18/8.84 (1) PrologToPiTRSProof [SOUND, 0 ms] 27.18/8.84 (2) PiTRS 27.18/8.84 (3) DependencyPairsProof [EQUIVALENT, 5 ms] 27.18/8.84 (4) PiDP 27.18/8.84 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 27.18/8.84 (6) AND 27.18/8.84 (7) PiDP 27.18/8.84 (8) UsableRulesProof [EQUIVALENT, 0 ms] 27.18/8.84 (9) PiDP 27.18/8.84 (10) PiDPToQDPProof [SOUND, 0 ms] 27.18/8.84 (11) QDP 27.18/8.84 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 27.18/8.84 (13) YES 27.18/8.84 (14) PiDP 27.18/8.84 (15) UsableRulesProof [EQUIVALENT, 0 ms] 27.18/8.84 (16) PiDP 27.18/8.84 (17) PiDPToQDPProof [SOUND, 0 ms] 27.18/8.84 (18) QDP 27.18/8.84 (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] 27.18/8.84 (20) YES 27.18/8.84 (21) PiDP 27.18/8.84 (22) UsableRulesProof [EQUIVALENT, 0 ms] 27.18/8.84 (23) PiDP 27.18/8.84 (24) PiDPToQDPProof [SOUND, 0 ms] 27.18/8.84 (25) QDP 27.18/8.84 (26) TransformationProof [SOUND, 0 ms] 27.18/8.84 (27) QDP 27.18/8.84 (28) TransformationProof [SOUND, 0 ms] 27.18/8.84 (29) QDP 27.18/8.84 (30) PrologToPiTRSProof [SOUND, 0 ms] 27.18/8.84 (31) PiTRS 27.18/8.84 (32) DependencyPairsProof [EQUIVALENT, 5 ms] 27.18/8.84 (33) PiDP 27.18/8.84 (34) DependencyGraphProof [EQUIVALENT, 0 ms] 27.18/8.84 (35) AND 27.18/8.84 (36) PiDP 27.18/8.84 (37) UsableRulesProof [EQUIVALENT, 0 ms] 27.18/8.84 (38) PiDP 27.18/8.84 (39) PiDPToQDPProof [SOUND, 0 ms] 27.18/8.84 (40) QDP 27.18/8.84 (41) QDPSizeChangeProof [EQUIVALENT, 0 ms] 27.18/8.84 (42) YES 27.18/8.84 (43) PiDP 27.18/8.84 (44) UsableRulesProof [EQUIVALENT, 0 ms] 27.18/8.84 (45) PiDP 27.18/8.84 (46) PiDPToQDPProof [SOUND, 0 ms] 27.18/8.84 (47) QDP 27.18/8.84 (48) QDPSizeChangeProof [EQUIVALENT, 0 ms] 27.18/8.84 (49) YES 27.18/8.84 (50) PiDP 27.18/8.84 (51) UsableRulesProof [EQUIVALENT, 0 ms] 27.18/8.84 (52) PiDP 27.18/8.84 (53) PiDPToQDPProof [SOUND, 0 ms] 27.18/8.84 (54) QDP 27.18/8.84 (55) TransformationProof [SOUND, 0 ms] 27.18/8.84 (56) QDP 27.18/8.84 (57) TransformationProof [SOUND, 0 ms] 27.18/8.84 (58) QDP 27.18/8.84 (59) TransformationProof [EQUIVALENT, 0 ms] 27.18/8.84 (60) QDP 27.18/8.84 (61) DependencyGraphProof [EQUIVALENT, 0 ms] 27.18/8.84 (62) AND 27.18/8.84 (63) QDP 27.18/8.84 (64) UsableRulesProof [EQUIVALENT, 0 ms] 27.18/8.84 (65) QDP 27.18/8.84 (66) QReductionProof [EQUIVALENT, 0 ms] 27.18/8.84 (67) QDP 27.18/8.84 (68) QDP 27.18/8.84 (69) TransformationProof [EQUIVALENT, 0 ms] 27.18/8.84 (70) QDP 27.18/8.84 (71) PrologToTRSTransformerProof [SOUND, 0 ms] 27.18/8.84 (72) QTRS 27.18/8.84 (73) DependencyPairsProof [EQUIVALENT, 0 ms] 27.18/8.84 (74) QDP 27.18/8.84 (75) DependencyGraphProof [EQUIVALENT, 0 ms] 27.18/8.84 (76) AND 27.18/8.84 (77) QDP 27.18/8.84 (78) UsableRulesProof [EQUIVALENT, 0 ms] 27.18/8.84 (79) QDP 27.18/8.84 (80) QDPSizeChangeProof [EQUIVALENT, 0 ms] 27.18/8.84 (81) YES 27.18/8.84 (82) QDP 27.18/8.84 (83) UsableRulesProof [EQUIVALENT, 0 ms] 27.18/8.84 (84) QDP 27.18/8.84 (85) QDPSizeChangeProof [EQUIVALENT, 0 ms] 27.18/8.84 (86) YES 27.18/8.84 (87) QDP 27.18/8.84 (88) NonTerminationLoopProof [COMPLETE, 0 ms] 27.18/8.84 (89) NO 27.18/8.84 (90) PrologToIRSwTTransformerProof [SOUND, 59 ms] 27.18/8.84 (91) AND 27.18/8.84 (92) IRSwT 27.18/8.84 (93) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] 27.18/8.84 (94) IRSwT 27.18/8.84 (95) IntTRSCompressionProof [EQUIVALENT, 35 ms] 27.18/8.84 (96) IRSwT 27.18/8.84 (97) IRSFormatTransformerProof [EQUIVALENT, 0 ms] 27.18/8.84 (98) IRSwT 27.18/8.84 (99) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms] 27.18/8.84 (100) IRSwT 27.18/8.84 (101) TempFilterProof [SOUND, 4 ms] 27.18/8.84 (102) IRSwT 27.18/8.84 (103) IRSwTToQDPProof [SOUND, 0 ms] 27.18/8.84 (104) QDP 27.18/8.84 (105) QDPSizeChangeProof [EQUIVALENT, 0 ms] 27.18/8.84 (106) YES 27.18/8.84 (107) IRSwT 27.18/8.84 (108) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] 27.18/8.84 (109) IRSwT 27.18/8.84 (110) IntTRSCompressionProof [EQUIVALENT, 1 ms] 27.18/8.84 (111) IRSwT 27.18/8.84 (112) IRSFormatTransformerProof [EQUIVALENT, 0 ms] 27.18/8.84 (113) IRSwT 27.18/8.84 (114) IRSwTTerminationDigraphProof [EQUIVALENT, 1 ms] 27.18/8.84 (115) IRSwT 27.18/8.84 (116) TempFilterProof [SOUND, 2 ms] 27.18/8.84 (117) IRSwT 27.18/8.84 (118) IRSwTToQDPProof [SOUND, 0 ms] 27.18/8.84 (119) QDP 27.18/8.84 (120) QDPSizeChangeProof [EQUIVALENT, 0 ms] 27.18/8.84 (121) YES 27.18/8.84 (122) IRSwT 27.18/8.84 (123) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] 27.18/8.84 (124) IRSwT 27.18/8.84 (125) IntTRSCompressionProof [EQUIVALENT, 21 ms] 27.18/8.84 (126) IRSwT 27.18/8.84 (127) PrologToDTProblemTransformerProof [SOUND, 44 ms] 27.18/8.84 (128) TRIPLES 27.18/8.84 (129) TriplesToPiDPProof [SOUND, 0 ms] 27.18/8.84 (130) PiDP 27.18/8.84 (131) DependencyGraphProof [EQUIVALENT, 0 ms] 27.18/8.84 (132) AND 27.18/8.84 (133) PiDP 27.18/8.84 (134) UsableRulesProof [EQUIVALENT, 0 ms] 27.18/8.84 (135) PiDP 27.18/8.84 (136) PiDPToQDPProof [SOUND, 0 ms] 27.18/8.84 (137) QDP 27.18/8.84 (138) QDPSizeChangeProof [EQUIVALENT, 0 ms] 27.18/8.84 (139) YES 27.18/8.84 (140) PiDP 27.18/8.84 (141) UsableRulesProof [EQUIVALENT, 0 ms] 27.18/8.84 (142) PiDP 27.18/8.84 (143) PiDPToQDPProof [SOUND, 0 ms] 27.18/8.84 (144) QDP 27.18/8.84 (145) QDPSizeChangeProof [EQUIVALENT, 0 ms] 27.18/8.84 (146) YES 27.18/8.84 (147) PiDP 27.18/8.84 (148) PiDPToQDPProof [SOUND, 1 ms] 27.18/8.84 (149) QDP 27.18/8.84 (150) TransformationProof [SOUND, 0 ms] 27.18/8.84 (151) QDP 27.18/8.84 (152) TransformationProof [SOUND, 0 ms] 27.18/8.84 (153) QDP 27.18/8.84 (154) TransformationProof [SOUND, 0 ms] 27.18/8.84 (155) QDP 27.18/8.84 (156) TransformationProof [EQUIVALENT, 0 ms] 27.18/8.84 (157) QDP 27.18/8.84 (158) DependencyGraphProof [EQUIVALENT, 0 ms] 27.18/8.84 (159) AND 27.18/8.84 (160) QDP 27.18/8.84 (161) QDP 27.18/8.84 (162) UsableRulesProof [EQUIVALENT, 0 ms] 27.18/8.84 (163) QDP 27.18/8.84 (164) QReductionProof [EQUIVALENT, 0 ms] 27.18/8.84 (165) QDP 27.18/8.84 27.18/8.84 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (0) 27.18/8.84 Obligation: 27.18/8.84 Clauses: 27.18/8.84 27.18/8.84 in(X, tree(X, X1, X2)). 27.18/8.84 in(X, tree(Y, Left, X3)) :- ','(less(X, Y), in(X, Left)). 27.18/8.84 in(X, tree(Y, X4, Right)) :- ','(less(Y, X), in(X, Right)). 27.18/8.84 less(0, s(X5)). 27.18/8.84 less(s(X), s(Y)) :- less(X, Y). 27.18/8.84 27.18/8.84 27.18/8.84 Query: in(g,a) 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (1) PrologToPiTRSProof (SOUND) 27.18/8.84 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 27.18/8.84 27.18/8.84 in_in_2: (b,f) 27.18/8.84 27.18/8.84 less_in_2: (b,f) (f,b) 27.18/8.84 27.18/8.84 Transforming Prolog into the following Term Rewriting System: 27.18/8.84 27.18/8.84 Pi-finite rewrite system: 27.18/8.84 The TRS R consists of the following rules: 27.18/8.84 27.18/8.84 in_in_ga(X, tree(X, X1, X2)) -> in_out_ga(X, tree(X, X1, X2)) 27.18/8.84 in_in_ga(X, tree(Y, Left, X3)) -> U1_ga(X, Y, Left, X3, less_in_ga(X, Y)) 27.18/8.84 less_in_ga(0, s(X5)) -> less_out_ga(0, s(X5)) 27.18/8.84 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 27.18/8.84 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 27.18/8.84 U1_ga(X, Y, Left, X3, less_out_ga(X, Y)) -> U2_ga(X, Y, Left, X3, in_in_ga(X, Left)) 27.18/8.84 in_in_ga(X, tree(Y, X4, Right)) -> U3_ga(X, Y, X4, Right, less_in_ag(Y, X)) 27.18/8.84 less_in_ag(0, s(X5)) -> less_out_ag(0, s(X5)) 27.18/8.84 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 27.18/8.84 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 27.18/8.84 U3_ga(X, Y, X4, Right, less_out_ag(Y, X)) -> U4_ga(X, Y, X4, Right, in_in_ga(X, Right)) 27.18/8.84 U4_ga(X, Y, X4, Right, in_out_ga(X, Right)) -> in_out_ga(X, tree(Y, X4, Right)) 27.18/8.84 U2_ga(X, Y, Left, X3, in_out_ga(X, Left)) -> in_out_ga(X, tree(Y, Left, X3)) 27.18/8.84 27.18/8.84 The argument filtering Pi contains the following mapping: 27.18/8.84 in_in_ga(x1, x2) = in_in_ga(x1) 27.18/8.84 27.18/8.84 in_out_ga(x1, x2) = in_out_ga(x1) 27.18/8.84 27.18/8.84 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x5) 27.18/8.84 27.18/8.84 less_in_ga(x1, x2) = less_in_ga(x1) 27.18/8.84 27.18/8.84 0 = 0 27.18/8.84 27.18/8.84 less_out_ga(x1, x2) = less_out_ga(x1) 27.18/8.84 27.18/8.84 s(x1) = s(x1) 27.18/8.84 27.18/8.84 U5_ga(x1, x2, x3) = U5_ga(x1, x3) 27.18/8.84 27.18/8.84 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x5) 27.18/8.84 27.18/8.84 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x5) 27.18/8.84 27.18/8.84 less_in_ag(x1, x2) = less_in_ag(x2) 27.18/8.84 27.18/8.84 less_out_ag(x1, x2) = less_out_ag(x1, x2) 27.18/8.84 27.18/8.84 U5_ag(x1, x2, x3) = U5_ag(x2, x3) 27.18/8.84 27.18/8.84 U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x5) 27.18/8.84 27.18/8.84 27.18/8.84 27.18/8.84 27.18/8.84 27.18/8.84 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 27.18/8.84 27.18/8.84 27.18/8.84 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (2) 27.18/8.84 Obligation: 27.18/8.84 Pi-finite rewrite system: 27.18/8.84 The TRS R consists of the following rules: 27.18/8.84 27.18/8.84 in_in_ga(X, tree(X, X1, X2)) -> in_out_ga(X, tree(X, X1, X2)) 27.18/8.84 in_in_ga(X, tree(Y, Left, X3)) -> U1_ga(X, Y, Left, X3, less_in_ga(X, Y)) 27.18/8.84 less_in_ga(0, s(X5)) -> less_out_ga(0, s(X5)) 27.18/8.84 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 27.18/8.84 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 27.18/8.84 U1_ga(X, Y, Left, X3, less_out_ga(X, Y)) -> U2_ga(X, Y, Left, X3, in_in_ga(X, Left)) 27.18/8.84 in_in_ga(X, tree(Y, X4, Right)) -> U3_ga(X, Y, X4, Right, less_in_ag(Y, X)) 27.18/8.84 less_in_ag(0, s(X5)) -> less_out_ag(0, s(X5)) 27.18/8.84 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 27.18/8.84 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 27.18/8.84 U3_ga(X, Y, X4, Right, less_out_ag(Y, X)) -> U4_ga(X, Y, X4, Right, in_in_ga(X, Right)) 27.18/8.84 U4_ga(X, Y, X4, Right, in_out_ga(X, Right)) -> in_out_ga(X, tree(Y, X4, Right)) 27.18/8.84 U2_ga(X, Y, Left, X3, in_out_ga(X, Left)) -> in_out_ga(X, tree(Y, Left, X3)) 27.18/8.84 27.18/8.84 The argument filtering Pi contains the following mapping: 27.18/8.84 in_in_ga(x1, x2) = in_in_ga(x1) 27.18/8.84 27.18/8.84 in_out_ga(x1, x2) = in_out_ga(x1) 27.18/8.84 27.18/8.84 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x5) 27.18/8.84 27.18/8.84 less_in_ga(x1, x2) = less_in_ga(x1) 27.18/8.84 27.18/8.84 0 = 0 27.18/8.84 27.18/8.84 less_out_ga(x1, x2) = less_out_ga(x1) 27.18/8.84 27.18/8.84 s(x1) = s(x1) 27.18/8.84 27.18/8.84 U5_ga(x1, x2, x3) = U5_ga(x1, x3) 27.18/8.84 27.18/8.84 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x5) 27.18/8.84 27.18/8.84 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x5) 27.18/8.84 27.18/8.84 less_in_ag(x1, x2) = less_in_ag(x2) 27.18/8.84 27.18/8.84 less_out_ag(x1, x2) = less_out_ag(x1, x2) 27.18/8.84 27.18/8.84 U5_ag(x1, x2, x3) = U5_ag(x2, x3) 27.18/8.84 27.18/8.84 U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x5) 27.18/8.84 27.18/8.84 27.18/8.84 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (3) DependencyPairsProof (EQUIVALENT) 27.18/8.84 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 27.18/8.84 Pi DP problem: 27.18/8.84 The TRS P consists of the following rules: 27.18/8.84 27.18/8.84 IN_IN_GA(X, tree(Y, Left, X3)) -> U1_GA(X, Y, Left, X3, less_in_ga(X, Y)) 27.18/8.84 IN_IN_GA(X, tree(Y, Left, X3)) -> LESS_IN_GA(X, Y) 27.18/8.84 LESS_IN_GA(s(X), s(Y)) -> U5_GA(X, Y, less_in_ga(X, Y)) 27.18/8.84 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 27.18/8.84 U1_GA(X, Y, Left, X3, less_out_ga(X, Y)) -> U2_GA(X, Y, Left, X3, in_in_ga(X, Left)) 27.18/8.84 U1_GA(X, Y, Left, X3, less_out_ga(X, Y)) -> IN_IN_GA(X, Left) 27.18/8.84 IN_IN_GA(X, tree(Y, X4, Right)) -> U3_GA(X, Y, X4, Right, less_in_ag(Y, X)) 27.18/8.84 IN_IN_GA(X, tree(Y, X4, Right)) -> LESS_IN_AG(Y, X) 27.18/8.84 LESS_IN_AG(s(X), s(Y)) -> U5_AG(X, Y, less_in_ag(X, Y)) 27.18/8.84 LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) 27.18/8.84 U3_GA(X, Y, X4, Right, less_out_ag(Y, X)) -> U4_GA(X, Y, X4, Right, in_in_ga(X, Right)) 27.18/8.84 U3_GA(X, Y, X4, Right, less_out_ag(Y, X)) -> IN_IN_GA(X, Right) 27.18/8.84 27.18/8.84 The TRS R consists of the following rules: 27.18/8.84 27.18/8.84 in_in_ga(X, tree(X, X1, X2)) -> in_out_ga(X, tree(X, X1, X2)) 27.18/8.84 in_in_ga(X, tree(Y, Left, X3)) -> U1_ga(X, Y, Left, X3, less_in_ga(X, Y)) 27.18/8.84 less_in_ga(0, s(X5)) -> less_out_ga(0, s(X5)) 27.18/8.84 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 27.18/8.84 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 27.18/8.84 U1_ga(X, Y, Left, X3, less_out_ga(X, Y)) -> U2_ga(X, Y, Left, X3, in_in_ga(X, Left)) 27.18/8.84 in_in_ga(X, tree(Y, X4, Right)) -> U3_ga(X, Y, X4, Right, less_in_ag(Y, X)) 27.18/8.84 less_in_ag(0, s(X5)) -> less_out_ag(0, s(X5)) 27.18/8.84 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 27.18/8.84 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 27.18/8.84 U3_ga(X, Y, X4, Right, less_out_ag(Y, X)) -> U4_ga(X, Y, X4, Right, in_in_ga(X, Right)) 27.18/8.84 U4_ga(X, Y, X4, Right, in_out_ga(X, Right)) -> in_out_ga(X, tree(Y, X4, Right)) 27.18/8.84 U2_ga(X, Y, Left, X3, in_out_ga(X, Left)) -> in_out_ga(X, tree(Y, Left, X3)) 27.18/8.84 27.18/8.84 The argument filtering Pi contains the following mapping: 27.18/8.84 in_in_ga(x1, x2) = in_in_ga(x1) 27.18/8.84 27.18/8.84 in_out_ga(x1, x2) = in_out_ga(x1) 27.18/8.84 27.18/8.84 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x5) 27.18/8.84 27.18/8.84 less_in_ga(x1, x2) = less_in_ga(x1) 27.18/8.84 27.18/8.84 0 = 0 27.18/8.84 27.18/8.84 less_out_ga(x1, x2) = less_out_ga(x1) 27.18/8.84 27.18/8.84 s(x1) = s(x1) 27.18/8.84 27.18/8.84 U5_ga(x1, x2, x3) = U5_ga(x1, x3) 27.18/8.84 27.18/8.84 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x5) 27.18/8.84 27.18/8.84 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x5) 27.18/8.84 27.18/8.84 less_in_ag(x1, x2) = less_in_ag(x2) 27.18/8.84 27.18/8.84 less_out_ag(x1, x2) = less_out_ag(x1, x2) 27.18/8.84 27.18/8.84 U5_ag(x1, x2, x3) = U5_ag(x2, x3) 27.18/8.84 27.18/8.84 U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x5) 27.18/8.84 27.18/8.84 IN_IN_GA(x1, x2) = IN_IN_GA(x1) 27.18/8.84 27.18/8.84 U1_GA(x1, x2, x3, x4, x5) = U1_GA(x1, x5) 27.18/8.84 27.18/8.84 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 27.18/8.84 27.18/8.84 U5_GA(x1, x2, x3) = U5_GA(x1, x3) 27.18/8.84 27.18/8.84 U2_GA(x1, x2, x3, x4, x5) = U2_GA(x1, x5) 27.18/8.84 27.18/8.84 U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x5) 27.18/8.84 27.18/8.84 LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) 27.18/8.84 27.18/8.84 U5_AG(x1, x2, x3) = U5_AG(x2, x3) 27.18/8.84 27.18/8.84 U4_GA(x1, x2, x3, x4, x5) = U4_GA(x1, x5) 27.18/8.84 27.18/8.84 27.18/8.84 We have to consider all (P,R,Pi)-chains 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (4) 27.18/8.84 Obligation: 27.18/8.84 Pi DP problem: 27.18/8.84 The TRS P consists of the following rules: 27.18/8.84 27.18/8.84 IN_IN_GA(X, tree(Y, Left, X3)) -> U1_GA(X, Y, Left, X3, less_in_ga(X, Y)) 27.18/8.84 IN_IN_GA(X, tree(Y, Left, X3)) -> LESS_IN_GA(X, Y) 27.18/8.84 LESS_IN_GA(s(X), s(Y)) -> U5_GA(X, Y, less_in_ga(X, Y)) 27.18/8.84 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 27.18/8.84 U1_GA(X, Y, Left, X3, less_out_ga(X, Y)) -> U2_GA(X, Y, Left, X3, in_in_ga(X, Left)) 27.18/8.84 U1_GA(X, Y, Left, X3, less_out_ga(X, Y)) -> IN_IN_GA(X, Left) 27.18/8.84 IN_IN_GA(X, tree(Y, X4, Right)) -> U3_GA(X, Y, X4, Right, less_in_ag(Y, X)) 27.18/8.84 IN_IN_GA(X, tree(Y, X4, Right)) -> LESS_IN_AG(Y, X) 27.18/8.84 LESS_IN_AG(s(X), s(Y)) -> U5_AG(X, Y, less_in_ag(X, Y)) 27.18/8.84 LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) 27.18/8.84 U3_GA(X, Y, X4, Right, less_out_ag(Y, X)) -> U4_GA(X, Y, X4, Right, in_in_ga(X, Right)) 27.18/8.84 U3_GA(X, Y, X4, Right, less_out_ag(Y, X)) -> IN_IN_GA(X, Right) 27.18/8.84 27.18/8.84 The TRS R consists of the following rules: 27.18/8.84 27.18/8.84 in_in_ga(X, tree(X, X1, X2)) -> in_out_ga(X, tree(X, X1, X2)) 27.18/8.84 in_in_ga(X, tree(Y, Left, X3)) -> U1_ga(X, Y, Left, X3, less_in_ga(X, Y)) 27.18/8.84 less_in_ga(0, s(X5)) -> less_out_ga(0, s(X5)) 27.18/8.84 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 27.18/8.84 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 27.18/8.84 U1_ga(X, Y, Left, X3, less_out_ga(X, Y)) -> U2_ga(X, Y, Left, X3, in_in_ga(X, Left)) 27.18/8.84 in_in_ga(X, tree(Y, X4, Right)) -> U3_ga(X, Y, X4, Right, less_in_ag(Y, X)) 27.18/8.84 less_in_ag(0, s(X5)) -> less_out_ag(0, s(X5)) 27.18/8.84 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 27.18/8.84 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 27.18/8.84 U3_ga(X, Y, X4, Right, less_out_ag(Y, X)) -> U4_ga(X, Y, X4, Right, in_in_ga(X, Right)) 27.18/8.84 U4_ga(X, Y, X4, Right, in_out_ga(X, Right)) -> in_out_ga(X, tree(Y, X4, Right)) 27.18/8.84 U2_ga(X, Y, Left, X3, in_out_ga(X, Left)) -> in_out_ga(X, tree(Y, Left, X3)) 27.18/8.84 27.18/8.84 The argument filtering Pi contains the following mapping: 27.18/8.84 in_in_ga(x1, x2) = in_in_ga(x1) 27.18/8.84 27.18/8.84 in_out_ga(x1, x2) = in_out_ga(x1) 27.18/8.84 27.18/8.84 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x5) 27.18/8.84 27.18/8.84 less_in_ga(x1, x2) = less_in_ga(x1) 27.18/8.84 27.18/8.84 0 = 0 27.18/8.84 27.18/8.84 less_out_ga(x1, x2) = less_out_ga(x1) 27.18/8.84 27.18/8.84 s(x1) = s(x1) 27.18/8.84 27.18/8.84 U5_ga(x1, x2, x3) = U5_ga(x1, x3) 27.18/8.84 27.18/8.84 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x5) 27.18/8.84 27.18/8.84 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x5) 27.18/8.84 27.18/8.84 less_in_ag(x1, x2) = less_in_ag(x2) 27.18/8.84 27.18/8.84 less_out_ag(x1, x2) = less_out_ag(x1, x2) 27.18/8.84 27.18/8.84 U5_ag(x1, x2, x3) = U5_ag(x2, x3) 27.18/8.84 27.18/8.84 U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x5) 27.18/8.84 27.18/8.84 IN_IN_GA(x1, x2) = IN_IN_GA(x1) 27.18/8.84 27.18/8.84 U1_GA(x1, x2, x3, x4, x5) = U1_GA(x1, x5) 27.18/8.84 27.18/8.84 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 27.18/8.84 27.18/8.84 U5_GA(x1, x2, x3) = U5_GA(x1, x3) 27.18/8.84 27.18/8.84 U2_GA(x1, x2, x3, x4, x5) = U2_GA(x1, x5) 27.18/8.84 27.18/8.84 U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x5) 27.18/8.84 27.18/8.84 LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) 27.18/8.84 27.18/8.84 U5_AG(x1, x2, x3) = U5_AG(x2, x3) 27.18/8.84 27.18/8.84 U4_GA(x1, x2, x3, x4, x5) = U4_GA(x1, x5) 27.18/8.84 27.18/8.84 27.18/8.84 We have to consider all (P,R,Pi)-chains 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (5) DependencyGraphProof (EQUIVALENT) 27.18/8.84 The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 6 less nodes. 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (6) 27.18/8.84 Complex Obligation (AND) 27.18/8.84 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (7) 27.18/8.84 Obligation: 27.18/8.84 Pi DP problem: 27.18/8.84 The TRS P consists of the following rules: 27.18/8.84 27.18/8.84 LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) 27.18/8.84 27.18/8.84 The TRS R consists of the following rules: 27.18/8.84 27.18/8.84 in_in_ga(X, tree(X, X1, X2)) -> in_out_ga(X, tree(X, X1, X2)) 27.18/8.84 in_in_ga(X, tree(Y, Left, X3)) -> U1_ga(X, Y, Left, X3, less_in_ga(X, Y)) 27.18/8.84 less_in_ga(0, s(X5)) -> less_out_ga(0, s(X5)) 27.18/8.84 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 27.18/8.84 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 27.18/8.84 U1_ga(X, Y, Left, X3, less_out_ga(X, Y)) -> U2_ga(X, Y, Left, X3, in_in_ga(X, Left)) 27.18/8.84 in_in_ga(X, tree(Y, X4, Right)) -> U3_ga(X, Y, X4, Right, less_in_ag(Y, X)) 27.18/8.84 less_in_ag(0, s(X5)) -> less_out_ag(0, s(X5)) 27.18/8.84 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 27.18/8.84 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 27.18/8.84 U3_ga(X, Y, X4, Right, less_out_ag(Y, X)) -> U4_ga(X, Y, X4, Right, in_in_ga(X, Right)) 27.18/8.84 U4_ga(X, Y, X4, Right, in_out_ga(X, Right)) -> in_out_ga(X, tree(Y, X4, Right)) 27.18/8.84 U2_ga(X, Y, Left, X3, in_out_ga(X, Left)) -> in_out_ga(X, tree(Y, Left, X3)) 27.18/8.84 27.18/8.84 The argument filtering Pi contains the following mapping: 27.18/8.84 in_in_ga(x1, x2) = in_in_ga(x1) 27.18/8.84 27.18/8.84 in_out_ga(x1, x2) = in_out_ga(x1) 27.18/8.84 27.18/8.84 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x5) 27.18/8.84 27.18/8.84 less_in_ga(x1, x2) = less_in_ga(x1) 27.18/8.84 27.18/8.84 0 = 0 27.18/8.84 27.18/8.84 less_out_ga(x1, x2) = less_out_ga(x1) 27.18/8.84 27.18/8.84 s(x1) = s(x1) 27.18/8.84 27.18/8.84 U5_ga(x1, x2, x3) = U5_ga(x1, x3) 27.18/8.84 27.18/8.84 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x5) 27.18/8.84 27.18/8.84 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x5) 27.18/8.84 27.18/8.84 less_in_ag(x1, x2) = less_in_ag(x2) 27.18/8.84 27.18/8.84 less_out_ag(x1, x2) = less_out_ag(x1, x2) 27.18/8.84 27.18/8.84 U5_ag(x1, x2, x3) = U5_ag(x2, x3) 27.18/8.84 27.18/8.84 U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x5) 27.18/8.84 27.18/8.84 LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) 27.18/8.84 27.18/8.84 27.18/8.84 We have to consider all (P,R,Pi)-chains 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (8) UsableRulesProof (EQUIVALENT) 27.18/8.84 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (9) 27.18/8.84 Obligation: 27.18/8.84 Pi DP problem: 27.18/8.84 The TRS P consists of the following rules: 27.18/8.84 27.18/8.84 LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) 27.18/8.84 27.18/8.84 R is empty. 27.18/8.84 The argument filtering Pi contains the following mapping: 27.18/8.84 s(x1) = s(x1) 27.18/8.84 27.18/8.84 LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) 27.18/8.84 27.18/8.84 27.18/8.84 We have to consider all (P,R,Pi)-chains 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (10) PiDPToQDPProof (SOUND) 27.18/8.84 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (11) 27.18/8.84 Obligation: 27.18/8.84 Q DP problem: 27.18/8.84 The TRS P consists of the following rules: 27.18/8.84 27.18/8.84 LESS_IN_AG(s(Y)) -> LESS_IN_AG(Y) 27.18/8.84 27.18/8.84 R is empty. 27.18/8.84 Q is empty. 27.18/8.84 We have to consider all (P,Q,R)-chains. 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (12) QDPSizeChangeProof (EQUIVALENT) 27.18/8.84 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. 27.18/8.84 27.18/8.84 From the DPs we obtained the following set of size-change graphs: 27.18/8.84 *LESS_IN_AG(s(Y)) -> LESS_IN_AG(Y) 27.18/8.84 The graph contains the following edges 1 > 1 27.18/8.84 27.18/8.84 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (13) 27.18/8.84 YES 27.18/8.84 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (14) 27.18/8.84 Obligation: 27.18/8.84 Pi DP problem: 27.18/8.84 The TRS P consists of the following rules: 27.18/8.84 27.18/8.84 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 27.18/8.84 27.18/8.84 The TRS R consists of the following rules: 27.18/8.84 27.18/8.84 in_in_ga(X, tree(X, X1, X2)) -> in_out_ga(X, tree(X, X1, X2)) 27.18/8.84 in_in_ga(X, tree(Y, Left, X3)) -> U1_ga(X, Y, Left, X3, less_in_ga(X, Y)) 27.18/8.84 less_in_ga(0, s(X5)) -> less_out_ga(0, s(X5)) 27.18/8.84 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 27.18/8.84 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 27.18/8.84 U1_ga(X, Y, Left, X3, less_out_ga(X, Y)) -> U2_ga(X, Y, Left, X3, in_in_ga(X, Left)) 27.18/8.84 in_in_ga(X, tree(Y, X4, Right)) -> U3_ga(X, Y, X4, Right, less_in_ag(Y, X)) 27.18/8.84 less_in_ag(0, s(X5)) -> less_out_ag(0, s(X5)) 27.18/8.84 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 27.18/8.84 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 27.18/8.84 U3_ga(X, Y, X4, Right, less_out_ag(Y, X)) -> U4_ga(X, Y, X4, Right, in_in_ga(X, Right)) 27.18/8.84 U4_ga(X, Y, X4, Right, in_out_ga(X, Right)) -> in_out_ga(X, tree(Y, X4, Right)) 27.18/8.84 U2_ga(X, Y, Left, X3, in_out_ga(X, Left)) -> in_out_ga(X, tree(Y, Left, X3)) 27.18/8.84 27.18/8.84 The argument filtering Pi contains the following mapping: 27.18/8.84 in_in_ga(x1, x2) = in_in_ga(x1) 27.18/8.84 27.18/8.84 in_out_ga(x1, x2) = in_out_ga(x1) 27.18/8.84 27.18/8.84 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x5) 27.18/8.84 27.18/8.84 less_in_ga(x1, x2) = less_in_ga(x1) 27.18/8.84 27.18/8.84 0 = 0 27.18/8.84 27.18/8.84 less_out_ga(x1, x2) = less_out_ga(x1) 27.18/8.84 27.18/8.84 s(x1) = s(x1) 27.18/8.84 27.18/8.84 U5_ga(x1, x2, x3) = U5_ga(x1, x3) 27.18/8.84 27.18/8.84 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x5) 27.18/8.84 27.18/8.84 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x5) 27.18/8.84 27.18/8.84 less_in_ag(x1, x2) = less_in_ag(x2) 27.18/8.84 27.18/8.84 less_out_ag(x1, x2) = less_out_ag(x1, x2) 27.18/8.84 27.18/8.84 U5_ag(x1, x2, x3) = U5_ag(x2, x3) 27.18/8.84 27.18/8.84 U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x5) 27.18/8.84 27.18/8.84 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 27.18/8.84 27.18/8.84 27.18/8.84 We have to consider all (P,R,Pi)-chains 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (15) UsableRulesProof (EQUIVALENT) 27.18/8.84 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (16) 27.18/8.84 Obligation: 27.18/8.84 Pi DP problem: 27.18/8.84 The TRS P consists of the following rules: 27.18/8.84 27.18/8.84 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 27.18/8.84 27.18/8.84 R is empty. 27.18/8.84 The argument filtering Pi contains the following mapping: 27.18/8.84 s(x1) = s(x1) 27.18/8.84 27.18/8.84 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 27.18/8.84 27.18/8.84 27.18/8.84 We have to consider all (P,R,Pi)-chains 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (17) PiDPToQDPProof (SOUND) 27.18/8.84 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (18) 27.18/8.84 Obligation: 27.18/8.84 Q DP problem: 27.18/8.84 The TRS P consists of the following rules: 27.18/8.84 27.18/8.84 LESS_IN_GA(s(X)) -> LESS_IN_GA(X) 27.18/8.84 27.18/8.84 R is empty. 27.18/8.84 Q is empty. 27.18/8.84 We have to consider all (P,Q,R)-chains. 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (19) QDPSizeChangeProof (EQUIVALENT) 27.18/8.84 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. 27.18/8.84 27.18/8.84 From the DPs we obtained the following set of size-change graphs: 27.18/8.84 *LESS_IN_GA(s(X)) -> LESS_IN_GA(X) 27.18/8.84 The graph contains the following edges 1 > 1 27.18/8.84 27.18/8.84 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (20) 27.18/8.84 YES 27.18/8.84 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (21) 27.18/8.84 Obligation: 27.18/8.84 Pi DP problem: 27.18/8.84 The TRS P consists of the following rules: 27.18/8.84 27.18/8.84 U1_GA(X, Y, Left, X3, less_out_ga(X, Y)) -> IN_IN_GA(X, Left) 27.18/8.84 IN_IN_GA(X, tree(Y, Left, X3)) -> U1_GA(X, Y, Left, X3, less_in_ga(X, Y)) 27.18/8.84 IN_IN_GA(X, tree(Y, X4, Right)) -> U3_GA(X, Y, X4, Right, less_in_ag(Y, X)) 27.18/8.84 U3_GA(X, Y, X4, Right, less_out_ag(Y, X)) -> IN_IN_GA(X, Right) 27.18/8.84 27.18/8.84 The TRS R consists of the following rules: 27.18/8.84 27.18/8.84 in_in_ga(X, tree(X, X1, X2)) -> in_out_ga(X, tree(X, X1, X2)) 27.18/8.84 in_in_ga(X, tree(Y, Left, X3)) -> U1_ga(X, Y, Left, X3, less_in_ga(X, Y)) 27.18/8.84 less_in_ga(0, s(X5)) -> less_out_ga(0, s(X5)) 27.18/8.84 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 27.18/8.84 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 27.18/8.84 U1_ga(X, Y, Left, X3, less_out_ga(X, Y)) -> U2_ga(X, Y, Left, X3, in_in_ga(X, Left)) 27.18/8.84 in_in_ga(X, tree(Y, X4, Right)) -> U3_ga(X, Y, X4, Right, less_in_ag(Y, X)) 27.18/8.84 less_in_ag(0, s(X5)) -> less_out_ag(0, s(X5)) 27.18/8.84 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 27.18/8.84 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 27.18/8.84 U3_ga(X, Y, X4, Right, less_out_ag(Y, X)) -> U4_ga(X, Y, X4, Right, in_in_ga(X, Right)) 27.18/8.84 U4_ga(X, Y, X4, Right, in_out_ga(X, Right)) -> in_out_ga(X, tree(Y, X4, Right)) 27.18/8.84 U2_ga(X, Y, Left, X3, in_out_ga(X, Left)) -> in_out_ga(X, tree(Y, Left, X3)) 27.18/8.84 27.18/8.84 The argument filtering Pi contains the following mapping: 27.18/8.84 in_in_ga(x1, x2) = in_in_ga(x1) 27.18/8.84 27.18/8.84 in_out_ga(x1, x2) = in_out_ga(x1) 27.18/8.84 27.18/8.84 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x5) 27.18/8.84 27.18/8.84 less_in_ga(x1, x2) = less_in_ga(x1) 27.18/8.84 27.18/8.84 0 = 0 27.18/8.84 27.18/8.84 less_out_ga(x1, x2) = less_out_ga(x1) 27.18/8.84 27.18/8.84 s(x1) = s(x1) 27.18/8.84 27.18/8.84 U5_ga(x1, x2, x3) = U5_ga(x1, x3) 27.18/8.84 27.18/8.84 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x5) 27.18/8.84 27.18/8.84 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x5) 27.18/8.84 27.18/8.84 less_in_ag(x1, x2) = less_in_ag(x2) 27.18/8.84 27.18/8.84 less_out_ag(x1, x2) = less_out_ag(x1, x2) 27.18/8.84 27.18/8.84 U5_ag(x1, x2, x3) = U5_ag(x2, x3) 27.18/8.84 27.18/8.84 U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x5) 27.18/8.84 27.18/8.84 IN_IN_GA(x1, x2) = IN_IN_GA(x1) 27.18/8.84 27.18/8.84 U1_GA(x1, x2, x3, x4, x5) = U1_GA(x1, x5) 27.18/8.84 27.18/8.84 U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x5) 27.18/8.84 27.18/8.84 27.18/8.84 We have to consider all (P,R,Pi)-chains 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (22) UsableRulesProof (EQUIVALENT) 27.18/8.84 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (23) 27.18/8.84 Obligation: 27.18/8.84 Pi DP problem: 27.18/8.84 The TRS P consists of the following rules: 27.18/8.84 27.18/8.84 U1_GA(X, Y, Left, X3, less_out_ga(X, Y)) -> IN_IN_GA(X, Left) 27.18/8.84 IN_IN_GA(X, tree(Y, Left, X3)) -> U1_GA(X, Y, Left, X3, less_in_ga(X, Y)) 27.18/8.84 IN_IN_GA(X, tree(Y, X4, Right)) -> U3_GA(X, Y, X4, Right, less_in_ag(Y, X)) 27.18/8.84 U3_GA(X, Y, X4, Right, less_out_ag(Y, X)) -> IN_IN_GA(X, Right) 27.18/8.84 27.18/8.84 The TRS R consists of the following rules: 27.18/8.84 27.18/8.84 less_in_ga(0, s(X5)) -> less_out_ga(0, s(X5)) 27.18/8.84 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 27.18/8.84 less_in_ag(0, s(X5)) -> less_out_ag(0, s(X5)) 27.18/8.84 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 27.18/8.84 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 27.18/8.84 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 27.18/8.84 27.18/8.84 The argument filtering Pi contains the following mapping: 27.18/8.84 less_in_ga(x1, x2) = less_in_ga(x1) 27.18/8.84 27.18/8.84 0 = 0 27.18/8.84 27.18/8.84 less_out_ga(x1, x2) = less_out_ga(x1) 27.18/8.84 27.18/8.84 s(x1) = s(x1) 27.18/8.84 27.18/8.84 U5_ga(x1, x2, x3) = U5_ga(x1, x3) 27.18/8.84 27.18/8.84 less_in_ag(x1, x2) = less_in_ag(x2) 27.18/8.84 27.18/8.84 less_out_ag(x1, x2) = less_out_ag(x1, x2) 27.18/8.84 27.18/8.84 U5_ag(x1, x2, x3) = U5_ag(x2, x3) 27.18/8.84 27.18/8.84 IN_IN_GA(x1, x2) = IN_IN_GA(x1) 27.18/8.84 27.18/8.84 U1_GA(x1, x2, x3, x4, x5) = U1_GA(x1, x5) 27.18/8.84 27.18/8.84 U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x5) 27.18/8.84 27.18/8.84 27.18/8.84 We have to consider all (P,R,Pi)-chains 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (24) PiDPToQDPProof (SOUND) 27.18/8.84 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (25) 27.18/8.84 Obligation: 27.18/8.84 Q DP problem: 27.18/8.84 The TRS P consists of the following rules: 27.18/8.84 27.18/8.84 U1_GA(X, less_out_ga(X)) -> IN_IN_GA(X) 27.18/8.84 IN_IN_GA(X) -> U1_GA(X, less_in_ga(X)) 27.18/8.84 IN_IN_GA(X) -> U3_GA(X, less_in_ag(X)) 27.18/8.84 U3_GA(X, less_out_ag(Y, X)) -> IN_IN_GA(X) 27.18/8.84 27.18/8.84 The TRS R consists of the following rules: 27.18/8.84 27.18/8.84 less_in_ga(0) -> less_out_ga(0) 27.18/8.84 less_in_ga(s(X)) -> U5_ga(X, less_in_ga(X)) 27.18/8.84 less_in_ag(s(X5)) -> less_out_ag(0, s(X5)) 27.18/8.84 less_in_ag(s(Y)) -> U5_ag(Y, less_in_ag(Y)) 27.18/8.84 U5_ga(X, less_out_ga(X)) -> less_out_ga(s(X)) 27.18/8.84 U5_ag(Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 27.18/8.84 27.18/8.84 The set Q consists of the following terms: 27.18/8.84 27.18/8.84 less_in_ga(x0) 27.18/8.84 less_in_ag(x0) 27.18/8.84 U5_ga(x0, x1) 27.18/8.84 U5_ag(x0, x1) 27.18/8.84 27.18/8.84 We have to consider all (P,Q,R)-chains. 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (26) TransformationProof (SOUND) 27.18/8.84 By narrowing [LPAR04] the rule IN_IN_GA(X) -> U1_GA(X, less_in_ga(X)) at position [1] we obtained the following new rules [LPAR04]: 27.18/8.84 27.18/8.84 (IN_IN_GA(0) -> U1_GA(0, less_out_ga(0)),IN_IN_GA(0) -> U1_GA(0, less_out_ga(0))) 27.18/8.84 (IN_IN_GA(s(x0)) -> U1_GA(s(x0), U5_ga(x0, less_in_ga(x0))),IN_IN_GA(s(x0)) -> U1_GA(s(x0), U5_ga(x0, less_in_ga(x0)))) 27.18/8.84 27.18/8.84 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (27) 27.18/8.84 Obligation: 27.18/8.84 Q DP problem: 27.18/8.84 The TRS P consists of the following rules: 27.18/8.84 27.18/8.84 U1_GA(X, less_out_ga(X)) -> IN_IN_GA(X) 27.18/8.84 IN_IN_GA(X) -> U3_GA(X, less_in_ag(X)) 27.18/8.84 U3_GA(X, less_out_ag(Y, X)) -> IN_IN_GA(X) 27.18/8.84 IN_IN_GA(0) -> U1_GA(0, less_out_ga(0)) 27.18/8.84 IN_IN_GA(s(x0)) -> U1_GA(s(x0), U5_ga(x0, less_in_ga(x0))) 27.18/8.84 27.18/8.84 The TRS R consists of the following rules: 27.18/8.84 27.18/8.84 less_in_ga(0) -> less_out_ga(0) 27.18/8.84 less_in_ga(s(X)) -> U5_ga(X, less_in_ga(X)) 27.18/8.84 less_in_ag(s(X5)) -> less_out_ag(0, s(X5)) 27.18/8.84 less_in_ag(s(Y)) -> U5_ag(Y, less_in_ag(Y)) 27.18/8.84 U5_ga(X, less_out_ga(X)) -> less_out_ga(s(X)) 27.18/8.84 U5_ag(Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 27.18/8.84 27.18/8.84 The set Q consists of the following terms: 27.18/8.84 27.18/8.84 less_in_ga(x0) 27.18/8.84 less_in_ag(x0) 27.18/8.84 U5_ga(x0, x1) 27.18/8.84 U5_ag(x0, x1) 27.18/8.84 27.18/8.84 We have to consider all (P,Q,R)-chains. 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (28) TransformationProof (SOUND) 27.18/8.84 By narrowing [LPAR04] the rule IN_IN_GA(X) -> U3_GA(X, less_in_ag(X)) at position [1] we obtained the following new rules [LPAR04]: 27.18/8.84 27.18/8.84 (IN_IN_GA(s(x0)) -> U3_GA(s(x0), less_out_ag(0, s(x0))),IN_IN_GA(s(x0)) -> U3_GA(s(x0), less_out_ag(0, s(x0)))) 27.18/8.84 (IN_IN_GA(s(x0)) -> U3_GA(s(x0), U5_ag(x0, less_in_ag(x0))),IN_IN_GA(s(x0)) -> U3_GA(s(x0), U5_ag(x0, less_in_ag(x0)))) 27.18/8.84 27.18/8.84 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (29) 27.18/8.84 Obligation: 27.18/8.84 Q DP problem: 27.18/8.84 The TRS P consists of the following rules: 27.18/8.84 27.18/8.84 U1_GA(X, less_out_ga(X)) -> IN_IN_GA(X) 27.18/8.84 U3_GA(X, less_out_ag(Y, X)) -> IN_IN_GA(X) 27.18/8.84 IN_IN_GA(0) -> U1_GA(0, less_out_ga(0)) 27.18/8.84 IN_IN_GA(s(x0)) -> U1_GA(s(x0), U5_ga(x0, less_in_ga(x0))) 27.18/8.84 IN_IN_GA(s(x0)) -> U3_GA(s(x0), less_out_ag(0, s(x0))) 27.18/8.84 IN_IN_GA(s(x0)) -> U3_GA(s(x0), U5_ag(x0, less_in_ag(x0))) 27.18/8.84 27.18/8.84 The TRS R consists of the following rules: 27.18/8.84 27.18/8.84 less_in_ga(0) -> less_out_ga(0) 27.18/8.84 less_in_ga(s(X)) -> U5_ga(X, less_in_ga(X)) 27.18/8.84 less_in_ag(s(X5)) -> less_out_ag(0, s(X5)) 27.18/8.84 less_in_ag(s(Y)) -> U5_ag(Y, less_in_ag(Y)) 27.18/8.84 U5_ga(X, less_out_ga(X)) -> less_out_ga(s(X)) 27.18/8.84 U5_ag(Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 27.18/8.84 27.18/8.84 The set Q consists of the following terms: 27.18/8.84 27.18/8.84 less_in_ga(x0) 27.18/8.84 less_in_ag(x0) 27.18/8.84 U5_ga(x0, x1) 27.18/8.84 U5_ag(x0, x1) 27.18/8.84 27.18/8.84 We have to consider all (P,Q,R)-chains. 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (30) PrologToPiTRSProof (SOUND) 27.18/8.84 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 27.18/8.84 27.18/8.84 in_in_2: (b,f) 27.18/8.84 27.18/8.84 less_in_2: (b,f) (f,b) 27.18/8.84 27.18/8.84 Transforming Prolog into the following Term Rewriting System: 27.18/8.84 27.18/8.84 Pi-finite rewrite system: 27.18/8.84 The TRS R consists of the following rules: 27.18/8.84 27.18/8.84 in_in_ga(X, tree(X, X1, X2)) -> in_out_ga(X, tree(X, X1, X2)) 27.18/8.84 in_in_ga(X, tree(Y, Left, X3)) -> U1_ga(X, Y, Left, X3, less_in_ga(X, Y)) 27.18/8.84 less_in_ga(0, s(X5)) -> less_out_ga(0, s(X5)) 27.18/8.84 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 27.18/8.84 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 27.18/8.84 U1_ga(X, Y, Left, X3, less_out_ga(X, Y)) -> U2_ga(X, Y, Left, X3, in_in_ga(X, Left)) 27.18/8.84 in_in_ga(X, tree(Y, X4, Right)) -> U3_ga(X, Y, X4, Right, less_in_ag(Y, X)) 27.18/8.84 less_in_ag(0, s(X5)) -> less_out_ag(0, s(X5)) 27.18/8.84 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 27.18/8.84 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 27.18/8.84 U3_ga(X, Y, X4, Right, less_out_ag(Y, X)) -> U4_ga(X, Y, X4, Right, in_in_ga(X, Right)) 27.18/8.84 U4_ga(X, Y, X4, Right, in_out_ga(X, Right)) -> in_out_ga(X, tree(Y, X4, Right)) 27.18/8.84 U2_ga(X, Y, Left, X3, in_out_ga(X, Left)) -> in_out_ga(X, tree(Y, Left, X3)) 27.18/8.84 27.18/8.84 The argument filtering Pi contains the following mapping: 27.18/8.84 in_in_ga(x1, x2) = in_in_ga(x1) 27.18/8.84 27.18/8.84 in_out_ga(x1, x2) = in_out_ga 27.18/8.84 27.18/8.84 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x5) 27.18/8.84 27.18/8.84 less_in_ga(x1, x2) = less_in_ga(x1) 27.18/8.84 27.18/8.84 0 = 0 27.18/8.84 27.18/8.84 less_out_ga(x1, x2) = less_out_ga 27.18/8.84 27.18/8.84 s(x1) = s(x1) 27.18/8.84 27.18/8.84 U5_ga(x1, x2, x3) = U5_ga(x3) 27.18/8.84 27.18/8.84 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5) 27.18/8.84 27.18/8.84 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x5) 27.18/8.84 27.18/8.84 less_in_ag(x1, x2) = less_in_ag(x2) 27.18/8.84 27.18/8.84 less_out_ag(x1, x2) = less_out_ag(x1) 27.18/8.84 27.18/8.84 U5_ag(x1, x2, x3) = U5_ag(x3) 27.18/8.84 27.18/8.84 U4_ga(x1, x2, x3, x4, x5) = U4_ga(x5) 27.18/8.84 27.18/8.84 27.18/8.84 27.18/8.84 27.18/8.84 27.18/8.84 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 27.18/8.84 27.18/8.84 27.18/8.84 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (31) 27.18/8.84 Obligation: 27.18/8.84 Pi-finite rewrite system: 27.18/8.84 The TRS R consists of the following rules: 27.18/8.84 27.18/8.84 in_in_ga(X, tree(X, X1, X2)) -> in_out_ga(X, tree(X, X1, X2)) 27.18/8.84 in_in_ga(X, tree(Y, Left, X3)) -> U1_ga(X, Y, Left, X3, less_in_ga(X, Y)) 27.18/8.84 less_in_ga(0, s(X5)) -> less_out_ga(0, s(X5)) 27.18/8.84 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 27.18/8.84 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 27.18/8.84 U1_ga(X, Y, Left, X3, less_out_ga(X, Y)) -> U2_ga(X, Y, Left, X3, in_in_ga(X, Left)) 27.18/8.84 in_in_ga(X, tree(Y, X4, Right)) -> U3_ga(X, Y, X4, Right, less_in_ag(Y, X)) 27.18/8.84 less_in_ag(0, s(X5)) -> less_out_ag(0, s(X5)) 27.18/8.84 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 27.18/8.84 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 27.18/8.84 U3_ga(X, Y, X4, Right, less_out_ag(Y, X)) -> U4_ga(X, Y, X4, Right, in_in_ga(X, Right)) 27.18/8.84 U4_ga(X, Y, X4, Right, in_out_ga(X, Right)) -> in_out_ga(X, tree(Y, X4, Right)) 27.18/8.84 U2_ga(X, Y, Left, X3, in_out_ga(X, Left)) -> in_out_ga(X, tree(Y, Left, X3)) 27.18/8.84 27.18/8.84 The argument filtering Pi contains the following mapping: 27.18/8.84 in_in_ga(x1, x2) = in_in_ga(x1) 27.18/8.84 27.18/8.84 in_out_ga(x1, x2) = in_out_ga 27.18/8.84 27.18/8.84 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x5) 27.18/8.84 27.18/8.84 less_in_ga(x1, x2) = less_in_ga(x1) 27.18/8.84 27.18/8.84 0 = 0 27.18/8.84 27.18/8.84 less_out_ga(x1, x2) = less_out_ga 27.18/8.84 27.18/8.84 s(x1) = s(x1) 27.18/8.84 27.18/8.84 U5_ga(x1, x2, x3) = U5_ga(x3) 27.18/8.84 27.18/8.84 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5) 27.18/8.84 27.18/8.84 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x5) 27.18/8.84 27.18/8.84 less_in_ag(x1, x2) = less_in_ag(x2) 27.18/8.84 27.18/8.84 less_out_ag(x1, x2) = less_out_ag(x1) 27.18/8.84 27.18/8.84 U5_ag(x1, x2, x3) = U5_ag(x3) 27.18/8.84 27.18/8.84 U4_ga(x1, x2, x3, x4, x5) = U4_ga(x5) 27.18/8.84 27.18/8.84 27.18/8.84 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (32) DependencyPairsProof (EQUIVALENT) 27.18/8.84 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 27.18/8.84 Pi DP problem: 27.18/8.84 The TRS P consists of the following rules: 27.18/8.84 27.18/8.84 IN_IN_GA(X, tree(Y, Left, X3)) -> U1_GA(X, Y, Left, X3, less_in_ga(X, Y)) 27.18/8.84 IN_IN_GA(X, tree(Y, Left, X3)) -> LESS_IN_GA(X, Y) 27.18/8.84 LESS_IN_GA(s(X), s(Y)) -> U5_GA(X, Y, less_in_ga(X, Y)) 27.18/8.84 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 27.18/8.84 U1_GA(X, Y, Left, X3, less_out_ga(X, Y)) -> U2_GA(X, Y, Left, X3, in_in_ga(X, Left)) 27.18/8.84 U1_GA(X, Y, Left, X3, less_out_ga(X, Y)) -> IN_IN_GA(X, Left) 27.18/8.84 IN_IN_GA(X, tree(Y, X4, Right)) -> U3_GA(X, Y, X4, Right, less_in_ag(Y, X)) 27.18/8.84 IN_IN_GA(X, tree(Y, X4, Right)) -> LESS_IN_AG(Y, X) 27.18/8.84 LESS_IN_AG(s(X), s(Y)) -> U5_AG(X, Y, less_in_ag(X, Y)) 27.18/8.84 LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) 27.18/8.84 U3_GA(X, Y, X4, Right, less_out_ag(Y, X)) -> U4_GA(X, Y, X4, Right, in_in_ga(X, Right)) 27.18/8.84 U3_GA(X, Y, X4, Right, less_out_ag(Y, X)) -> IN_IN_GA(X, Right) 27.18/8.84 27.18/8.84 The TRS R consists of the following rules: 27.18/8.84 27.18/8.84 in_in_ga(X, tree(X, X1, X2)) -> in_out_ga(X, tree(X, X1, X2)) 27.18/8.84 in_in_ga(X, tree(Y, Left, X3)) -> U1_ga(X, Y, Left, X3, less_in_ga(X, Y)) 27.18/8.84 less_in_ga(0, s(X5)) -> less_out_ga(0, s(X5)) 27.18/8.84 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 27.18/8.84 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 27.18/8.84 U1_ga(X, Y, Left, X3, less_out_ga(X, Y)) -> U2_ga(X, Y, Left, X3, in_in_ga(X, Left)) 27.18/8.84 in_in_ga(X, tree(Y, X4, Right)) -> U3_ga(X, Y, X4, Right, less_in_ag(Y, X)) 27.18/8.84 less_in_ag(0, s(X5)) -> less_out_ag(0, s(X5)) 27.18/8.84 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 27.18/8.84 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 27.18/8.84 U3_ga(X, Y, X4, Right, less_out_ag(Y, X)) -> U4_ga(X, Y, X4, Right, in_in_ga(X, Right)) 27.18/8.84 U4_ga(X, Y, X4, Right, in_out_ga(X, Right)) -> in_out_ga(X, tree(Y, X4, Right)) 27.18/8.84 U2_ga(X, Y, Left, X3, in_out_ga(X, Left)) -> in_out_ga(X, tree(Y, Left, X3)) 27.18/8.84 27.18/8.84 The argument filtering Pi contains the following mapping: 27.18/8.84 in_in_ga(x1, x2) = in_in_ga(x1) 27.18/8.84 27.18/8.84 in_out_ga(x1, x2) = in_out_ga 27.18/8.84 27.18/8.84 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x5) 27.18/8.84 27.18/8.84 less_in_ga(x1, x2) = less_in_ga(x1) 27.18/8.84 27.18/8.84 0 = 0 27.18/8.84 27.18/8.84 less_out_ga(x1, x2) = less_out_ga 27.18/8.84 27.18/8.84 s(x1) = s(x1) 27.18/8.84 27.18/8.84 U5_ga(x1, x2, x3) = U5_ga(x3) 27.18/8.84 27.18/8.84 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5) 27.18/8.84 27.18/8.84 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x5) 27.18/8.84 27.18/8.84 less_in_ag(x1, x2) = less_in_ag(x2) 27.18/8.84 27.18/8.84 less_out_ag(x1, x2) = less_out_ag(x1) 27.18/8.84 27.18/8.84 U5_ag(x1, x2, x3) = U5_ag(x3) 27.18/8.84 27.18/8.84 U4_ga(x1, x2, x3, x4, x5) = U4_ga(x5) 27.18/8.84 27.18/8.84 IN_IN_GA(x1, x2) = IN_IN_GA(x1) 27.18/8.84 27.18/8.84 U1_GA(x1, x2, x3, x4, x5) = U1_GA(x1, x5) 27.18/8.84 27.18/8.84 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 27.18/8.84 27.18/8.84 U5_GA(x1, x2, x3) = U5_GA(x3) 27.18/8.84 27.18/8.84 U2_GA(x1, x2, x3, x4, x5) = U2_GA(x5) 27.18/8.84 27.18/8.84 U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x5) 27.18/8.84 27.18/8.84 LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) 27.18/8.84 27.18/8.84 U5_AG(x1, x2, x3) = U5_AG(x3) 27.18/8.84 27.18/8.84 U4_GA(x1, x2, x3, x4, x5) = U4_GA(x5) 27.18/8.84 27.18/8.84 27.18/8.84 We have to consider all (P,R,Pi)-chains 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (33) 27.18/8.84 Obligation: 27.18/8.84 Pi DP problem: 27.18/8.84 The TRS P consists of the following rules: 27.18/8.84 27.18/8.84 IN_IN_GA(X, tree(Y, Left, X3)) -> U1_GA(X, Y, Left, X3, less_in_ga(X, Y)) 27.18/8.84 IN_IN_GA(X, tree(Y, Left, X3)) -> LESS_IN_GA(X, Y) 27.18/8.84 LESS_IN_GA(s(X), s(Y)) -> U5_GA(X, Y, less_in_ga(X, Y)) 27.18/8.84 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 27.18/8.84 U1_GA(X, Y, Left, X3, less_out_ga(X, Y)) -> U2_GA(X, Y, Left, X3, in_in_ga(X, Left)) 27.18/8.84 U1_GA(X, Y, Left, X3, less_out_ga(X, Y)) -> IN_IN_GA(X, Left) 27.18/8.84 IN_IN_GA(X, tree(Y, X4, Right)) -> U3_GA(X, Y, X4, Right, less_in_ag(Y, X)) 27.18/8.84 IN_IN_GA(X, tree(Y, X4, Right)) -> LESS_IN_AG(Y, X) 27.18/8.84 LESS_IN_AG(s(X), s(Y)) -> U5_AG(X, Y, less_in_ag(X, Y)) 27.18/8.84 LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) 27.18/8.84 U3_GA(X, Y, X4, Right, less_out_ag(Y, X)) -> U4_GA(X, Y, X4, Right, in_in_ga(X, Right)) 27.18/8.84 U3_GA(X, Y, X4, Right, less_out_ag(Y, X)) -> IN_IN_GA(X, Right) 27.18/8.84 27.18/8.84 The TRS R consists of the following rules: 27.18/8.84 27.18/8.84 in_in_ga(X, tree(X, X1, X2)) -> in_out_ga(X, tree(X, X1, X2)) 27.18/8.84 in_in_ga(X, tree(Y, Left, X3)) -> U1_ga(X, Y, Left, X3, less_in_ga(X, Y)) 27.18/8.84 less_in_ga(0, s(X5)) -> less_out_ga(0, s(X5)) 27.18/8.84 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 27.18/8.84 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 27.18/8.84 U1_ga(X, Y, Left, X3, less_out_ga(X, Y)) -> U2_ga(X, Y, Left, X3, in_in_ga(X, Left)) 27.18/8.84 in_in_ga(X, tree(Y, X4, Right)) -> U3_ga(X, Y, X4, Right, less_in_ag(Y, X)) 27.18/8.84 less_in_ag(0, s(X5)) -> less_out_ag(0, s(X5)) 27.18/8.84 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 27.18/8.84 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 27.18/8.84 U3_ga(X, Y, X4, Right, less_out_ag(Y, X)) -> U4_ga(X, Y, X4, Right, in_in_ga(X, Right)) 27.18/8.84 U4_ga(X, Y, X4, Right, in_out_ga(X, Right)) -> in_out_ga(X, tree(Y, X4, Right)) 27.18/8.84 U2_ga(X, Y, Left, X3, in_out_ga(X, Left)) -> in_out_ga(X, tree(Y, Left, X3)) 27.18/8.84 27.18/8.84 The argument filtering Pi contains the following mapping: 27.18/8.84 in_in_ga(x1, x2) = in_in_ga(x1) 27.18/8.84 27.18/8.84 in_out_ga(x1, x2) = in_out_ga 27.18/8.84 27.18/8.84 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x5) 27.18/8.84 27.18/8.84 less_in_ga(x1, x2) = less_in_ga(x1) 27.18/8.84 27.18/8.84 0 = 0 27.18/8.84 27.18/8.84 less_out_ga(x1, x2) = less_out_ga 27.18/8.84 27.18/8.84 s(x1) = s(x1) 27.18/8.84 27.18/8.84 U5_ga(x1, x2, x3) = U5_ga(x3) 27.18/8.84 27.18/8.84 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5) 27.18/8.84 27.18/8.84 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x5) 27.18/8.84 27.18/8.84 less_in_ag(x1, x2) = less_in_ag(x2) 27.18/8.84 27.18/8.84 less_out_ag(x1, x2) = less_out_ag(x1) 27.18/8.84 27.18/8.84 U5_ag(x1, x2, x3) = U5_ag(x3) 27.18/8.84 27.18/8.84 U4_ga(x1, x2, x3, x4, x5) = U4_ga(x5) 27.18/8.84 27.18/8.84 IN_IN_GA(x1, x2) = IN_IN_GA(x1) 27.18/8.84 27.18/8.84 U1_GA(x1, x2, x3, x4, x5) = U1_GA(x1, x5) 27.18/8.84 27.18/8.84 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 27.18/8.84 27.18/8.84 U5_GA(x1, x2, x3) = U5_GA(x3) 27.18/8.84 27.18/8.84 U2_GA(x1, x2, x3, x4, x5) = U2_GA(x5) 27.18/8.84 27.18/8.84 U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x5) 27.18/8.84 27.18/8.84 LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) 27.18/8.84 27.18/8.84 U5_AG(x1, x2, x3) = U5_AG(x3) 27.18/8.84 27.18/8.84 U4_GA(x1, x2, x3, x4, x5) = U4_GA(x5) 27.18/8.84 27.18/8.84 27.18/8.84 We have to consider all (P,R,Pi)-chains 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (34) DependencyGraphProof (EQUIVALENT) 27.18/8.84 The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 6 less nodes. 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (35) 27.18/8.84 Complex Obligation (AND) 27.18/8.84 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (36) 27.18/8.84 Obligation: 27.18/8.84 Pi DP problem: 27.18/8.84 The TRS P consists of the following rules: 27.18/8.84 27.18/8.84 LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) 27.18/8.84 27.18/8.84 The TRS R consists of the following rules: 27.18/8.84 27.18/8.84 in_in_ga(X, tree(X, X1, X2)) -> in_out_ga(X, tree(X, X1, X2)) 27.18/8.84 in_in_ga(X, tree(Y, Left, X3)) -> U1_ga(X, Y, Left, X3, less_in_ga(X, Y)) 27.18/8.84 less_in_ga(0, s(X5)) -> less_out_ga(0, s(X5)) 27.18/8.84 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 27.18/8.84 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 27.18/8.84 U1_ga(X, Y, Left, X3, less_out_ga(X, Y)) -> U2_ga(X, Y, Left, X3, in_in_ga(X, Left)) 27.18/8.84 in_in_ga(X, tree(Y, X4, Right)) -> U3_ga(X, Y, X4, Right, less_in_ag(Y, X)) 27.18/8.84 less_in_ag(0, s(X5)) -> less_out_ag(0, s(X5)) 27.18/8.84 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 27.18/8.84 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 27.18/8.84 U3_ga(X, Y, X4, Right, less_out_ag(Y, X)) -> U4_ga(X, Y, X4, Right, in_in_ga(X, Right)) 27.18/8.84 U4_ga(X, Y, X4, Right, in_out_ga(X, Right)) -> in_out_ga(X, tree(Y, X4, Right)) 27.18/8.84 U2_ga(X, Y, Left, X3, in_out_ga(X, Left)) -> in_out_ga(X, tree(Y, Left, X3)) 27.18/8.84 27.18/8.84 The argument filtering Pi contains the following mapping: 27.18/8.84 in_in_ga(x1, x2) = in_in_ga(x1) 27.18/8.84 27.18/8.84 in_out_ga(x1, x2) = in_out_ga 27.18/8.84 27.18/8.84 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x5) 27.18/8.84 27.18/8.84 less_in_ga(x1, x2) = less_in_ga(x1) 27.18/8.84 27.18/8.84 0 = 0 27.18/8.84 27.18/8.84 less_out_ga(x1, x2) = less_out_ga 27.18/8.84 27.18/8.84 s(x1) = s(x1) 27.18/8.84 27.18/8.84 U5_ga(x1, x2, x3) = U5_ga(x3) 27.18/8.84 27.18/8.84 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5) 27.18/8.84 27.18/8.84 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x5) 27.18/8.84 27.18/8.84 less_in_ag(x1, x2) = less_in_ag(x2) 27.18/8.84 27.18/8.84 less_out_ag(x1, x2) = less_out_ag(x1) 27.18/8.84 27.18/8.84 U5_ag(x1, x2, x3) = U5_ag(x3) 27.18/8.84 27.18/8.84 U4_ga(x1, x2, x3, x4, x5) = U4_ga(x5) 27.18/8.84 27.18/8.84 LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) 27.18/8.84 27.18/8.84 27.18/8.84 We have to consider all (P,R,Pi)-chains 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (37) UsableRulesProof (EQUIVALENT) 27.18/8.84 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (38) 27.18/8.84 Obligation: 27.18/8.84 Pi DP problem: 27.18/8.84 The TRS P consists of the following rules: 27.18/8.84 27.18/8.84 LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) 27.18/8.84 27.18/8.84 R is empty. 27.18/8.84 The argument filtering Pi contains the following mapping: 27.18/8.84 s(x1) = s(x1) 27.18/8.84 27.18/8.84 LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) 27.18/8.84 27.18/8.84 27.18/8.84 We have to consider all (P,R,Pi)-chains 27.18/8.84 ---------------------------------------- 27.18/8.84 27.18/8.84 (39) PiDPToQDPProof (SOUND) 27.18/8.84 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 27.18/8.87 ---------------------------------------- 27.18/8.87 27.18/8.87 (40) 27.18/8.87 Obligation: 27.18/8.87 Q DP problem: 27.18/8.87 The TRS P consists of the following rules: 27.18/8.87 27.18/8.87 LESS_IN_AG(s(Y)) -> LESS_IN_AG(Y) 27.18/8.87 27.18/8.87 R is empty. 27.18/8.87 Q is empty. 27.18/8.87 We have to consider all (P,Q,R)-chains. 27.18/8.87 ---------------------------------------- 27.18/8.87 27.18/8.87 (41) QDPSizeChangeProof (EQUIVALENT) 27.18/8.87 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. 27.18/8.87 27.18/8.87 From the DPs we obtained the following set of size-change graphs: 27.18/8.87 *LESS_IN_AG(s(Y)) -> LESS_IN_AG(Y) 27.18/8.87 The graph contains the following edges 1 > 1 27.18/8.87 27.18/8.87 27.18/8.87 ---------------------------------------- 27.18/8.87 27.18/8.87 (42) 27.18/8.87 YES 27.18/8.87 27.18/8.87 ---------------------------------------- 27.18/8.87 27.18/8.87 (43) 27.18/8.87 Obligation: 27.18/8.87 Pi DP problem: 27.18/8.87 The TRS P consists of the following rules: 27.18/8.87 27.18/8.87 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 27.18/8.87 27.18/8.87 The TRS R consists of the following rules: 27.18/8.87 27.18/8.87 in_in_ga(X, tree(X, X1, X2)) -> in_out_ga(X, tree(X, X1, X2)) 27.18/8.87 in_in_ga(X, tree(Y, Left, X3)) -> U1_ga(X, Y, Left, X3, less_in_ga(X, Y)) 27.18/8.87 less_in_ga(0, s(X5)) -> less_out_ga(0, s(X5)) 27.18/8.87 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 27.18/8.87 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 27.18/8.87 U1_ga(X, Y, Left, X3, less_out_ga(X, Y)) -> U2_ga(X, Y, Left, X3, in_in_ga(X, Left)) 27.18/8.87 in_in_ga(X, tree(Y, X4, Right)) -> U3_ga(X, Y, X4, Right, less_in_ag(Y, X)) 27.18/8.87 less_in_ag(0, s(X5)) -> less_out_ag(0, s(X5)) 27.18/8.87 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 27.18/8.87 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 27.18/8.87 U3_ga(X, Y, X4, Right, less_out_ag(Y, X)) -> U4_ga(X, Y, X4, Right, in_in_ga(X, Right)) 27.18/8.87 U4_ga(X, Y, X4, Right, in_out_ga(X, Right)) -> in_out_ga(X, tree(Y, X4, Right)) 27.18/8.87 U2_ga(X, Y, Left, X3, in_out_ga(X, Left)) -> in_out_ga(X, tree(Y, Left, X3)) 27.18/8.87 27.18/8.87 The argument filtering Pi contains the following mapping: 27.18/8.87 in_in_ga(x1, x2) = in_in_ga(x1) 27.18/8.87 27.18/8.87 in_out_ga(x1, x2) = in_out_ga 27.18/8.87 27.18/8.87 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x5) 27.18/8.87 27.18/8.87 less_in_ga(x1, x2) = less_in_ga(x1) 27.18/8.87 27.18/8.87 0 = 0 27.18/8.87 27.18/8.87 less_out_ga(x1, x2) = less_out_ga 27.18/8.87 27.18/8.87 s(x1) = s(x1) 27.18/8.87 27.18/8.87 U5_ga(x1, x2, x3) = U5_ga(x3) 27.18/8.87 27.18/8.87 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5) 27.18/8.87 27.18/8.87 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x5) 27.18/8.87 27.18/8.87 less_in_ag(x1, x2) = less_in_ag(x2) 27.18/8.87 27.18/8.87 less_out_ag(x1, x2) = less_out_ag(x1) 27.18/8.87 27.18/8.87 U5_ag(x1, x2, x3) = U5_ag(x3) 27.18/8.87 27.18/8.87 U4_ga(x1, x2, x3, x4, x5) = U4_ga(x5) 27.18/8.87 27.18/8.87 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 27.18/8.87 27.18/8.87 27.18/8.87 We have to consider all (P,R,Pi)-chains 27.18/8.87 ---------------------------------------- 27.18/8.87 27.18/8.87 (44) UsableRulesProof (EQUIVALENT) 27.18/8.87 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 27.18/8.87 ---------------------------------------- 27.18/8.87 27.18/8.87 (45) 27.18/8.87 Obligation: 27.18/8.87 Pi DP problem: 27.18/8.87 The TRS P consists of the following rules: 27.18/8.87 27.18/8.87 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 27.18/8.87 27.18/8.87 R is empty. 27.18/8.87 The argument filtering Pi contains the following mapping: 27.18/8.87 s(x1) = s(x1) 27.18/8.87 27.18/8.87 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 27.18/8.87 27.18/8.87 27.18/8.87 We have to consider all (P,R,Pi)-chains 27.18/8.87 ---------------------------------------- 27.18/8.87 27.18/8.87 (46) PiDPToQDPProof (SOUND) 27.18/8.87 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 27.18/8.87 ---------------------------------------- 27.18/8.87 27.18/8.87 (47) 27.18/8.87 Obligation: 27.18/8.87 Q DP problem: 27.18/8.87 The TRS P consists of the following rules: 27.18/8.87 27.18/8.87 LESS_IN_GA(s(X)) -> LESS_IN_GA(X) 27.18/8.87 27.18/8.87 R is empty. 27.18/8.87 Q is empty. 27.18/8.87 We have to consider all (P,Q,R)-chains. 27.18/8.87 ---------------------------------------- 27.18/8.87 27.18/8.87 (48) QDPSizeChangeProof (EQUIVALENT) 27.18/8.87 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. 27.18/8.87 27.18/8.87 From the DPs we obtained the following set of size-change graphs: 27.18/8.87 *LESS_IN_GA(s(X)) -> LESS_IN_GA(X) 27.18/8.87 The graph contains the following edges 1 > 1 27.18/8.87 27.18/8.87 27.18/8.87 ---------------------------------------- 27.18/8.87 27.18/8.87 (49) 27.18/8.87 YES 27.18/8.87 27.18/8.87 ---------------------------------------- 27.18/8.87 27.18/8.87 (50) 27.18/8.87 Obligation: 27.18/8.87 Pi DP problem: 27.18/8.87 The TRS P consists of the following rules: 27.18/8.87 27.18/8.87 U1_GA(X, Y, Left, X3, less_out_ga(X, Y)) -> IN_IN_GA(X, Left) 27.18/8.87 IN_IN_GA(X, tree(Y, Left, X3)) -> U1_GA(X, Y, Left, X3, less_in_ga(X, Y)) 27.18/8.87 IN_IN_GA(X, tree(Y, X4, Right)) -> U3_GA(X, Y, X4, Right, less_in_ag(Y, X)) 27.18/8.87 U3_GA(X, Y, X4, Right, less_out_ag(Y, X)) -> IN_IN_GA(X, Right) 27.18/8.87 27.18/8.87 The TRS R consists of the following rules: 27.18/8.87 27.18/8.87 in_in_ga(X, tree(X, X1, X2)) -> in_out_ga(X, tree(X, X1, X2)) 27.18/8.87 in_in_ga(X, tree(Y, Left, X3)) -> U1_ga(X, Y, Left, X3, less_in_ga(X, Y)) 27.18/8.87 less_in_ga(0, s(X5)) -> less_out_ga(0, s(X5)) 27.18/8.87 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 27.18/8.87 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 27.18/8.87 U1_ga(X, Y, Left, X3, less_out_ga(X, Y)) -> U2_ga(X, Y, Left, X3, in_in_ga(X, Left)) 27.18/8.87 in_in_ga(X, tree(Y, X4, Right)) -> U3_ga(X, Y, X4, Right, less_in_ag(Y, X)) 27.18/8.87 less_in_ag(0, s(X5)) -> less_out_ag(0, s(X5)) 27.18/8.87 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 27.18/8.87 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 27.18/8.87 U3_ga(X, Y, X4, Right, less_out_ag(Y, X)) -> U4_ga(X, Y, X4, Right, in_in_ga(X, Right)) 27.18/8.87 U4_ga(X, Y, X4, Right, in_out_ga(X, Right)) -> in_out_ga(X, tree(Y, X4, Right)) 27.18/8.87 U2_ga(X, Y, Left, X3, in_out_ga(X, Left)) -> in_out_ga(X, tree(Y, Left, X3)) 27.18/8.87 27.18/8.87 The argument filtering Pi contains the following mapping: 27.18/8.87 in_in_ga(x1, x2) = in_in_ga(x1) 27.18/8.87 27.18/8.87 in_out_ga(x1, x2) = in_out_ga 27.18/8.87 27.18/8.87 U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x5) 27.18/8.87 27.18/8.87 less_in_ga(x1, x2) = less_in_ga(x1) 27.18/8.87 27.18/8.87 0 = 0 27.18/8.87 27.18/8.87 less_out_ga(x1, x2) = less_out_ga 27.18/8.87 27.18/8.87 s(x1) = s(x1) 27.18/8.87 27.18/8.87 U5_ga(x1, x2, x3) = U5_ga(x3) 27.18/8.87 27.18/8.87 U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5) 27.18/8.87 27.18/8.87 U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x5) 27.18/8.87 27.18/8.87 less_in_ag(x1, x2) = less_in_ag(x2) 27.18/8.87 27.18/8.87 less_out_ag(x1, x2) = less_out_ag(x1) 27.18/8.87 27.18/8.87 U5_ag(x1, x2, x3) = U5_ag(x3) 27.18/8.87 27.18/8.87 U4_ga(x1, x2, x3, x4, x5) = U4_ga(x5) 27.18/8.87 27.18/8.87 IN_IN_GA(x1, x2) = IN_IN_GA(x1) 27.18/8.87 27.18/8.87 U1_GA(x1, x2, x3, x4, x5) = U1_GA(x1, x5) 27.18/8.87 27.18/8.87 U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x5) 27.18/8.87 27.18/8.87 27.18/8.87 We have to consider all (P,R,Pi)-chains 27.18/8.87 ---------------------------------------- 27.18/8.87 27.18/8.87 (51) UsableRulesProof (EQUIVALENT) 27.18/8.87 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 27.18/8.87 ---------------------------------------- 27.18/8.87 27.18/8.87 (52) 27.18/8.87 Obligation: 27.18/8.87 Pi DP problem: 27.18/8.87 The TRS P consists of the following rules: 27.18/8.87 27.18/8.87 U1_GA(X, Y, Left, X3, less_out_ga(X, Y)) -> IN_IN_GA(X, Left) 27.18/8.87 IN_IN_GA(X, tree(Y, Left, X3)) -> U1_GA(X, Y, Left, X3, less_in_ga(X, Y)) 27.18/8.87 IN_IN_GA(X, tree(Y, X4, Right)) -> U3_GA(X, Y, X4, Right, less_in_ag(Y, X)) 27.18/8.87 U3_GA(X, Y, X4, Right, less_out_ag(Y, X)) -> IN_IN_GA(X, Right) 27.18/8.87 27.18/8.87 The TRS R consists of the following rules: 27.18/8.87 27.18/8.87 less_in_ga(0, s(X5)) -> less_out_ga(0, s(X5)) 27.18/8.87 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 27.18/8.87 less_in_ag(0, s(X5)) -> less_out_ag(0, s(X5)) 27.18/8.87 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 27.18/8.87 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 27.18/8.87 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 27.18/8.87 27.18/8.87 The argument filtering Pi contains the following mapping: 27.18/8.87 less_in_ga(x1, x2) = less_in_ga(x1) 27.18/8.87 27.18/8.87 0 = 0 27.18/8.87 27.18/8.87 less_out_ga(x1, x2) = less_out_ga 27.18/8.87 27.18/8.87 s(x1) = s(x1) 27.18/8.87 27.18/8.87 U5_ga(x1, x2, x3) = U5_ga(x3) 27.18/8.87 27.18/8.87 less_in_ag(x1, x2) = less_in_ag(x2) 27.18/8.87 27.18/8.87 less_out_ag(x1, x2) = less_out_ag(x1) 27.18/8.87 27.18/8.87 U5_ag(x1, x2, x3) = U5_ag(x3) 27.18/8.87 27.18/8.87 IN_IN_GA(x1, x2) = IN_IN_GA(x1) 27.18/8.87 27.18/8.87 U1_GA(x1, x2, x3, x4, x5) = U1_GA(x1, x5) 27.18/8.87 27.18/8.87 U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x5) 27.18/8.87 27.18/8.87 27.18/8.87 We have to consider all (P,R,Pi)-chains 27.18/8.87 ---------------------------------------- 27.18/8.87 27.18/8.87 (53) PiDPToQDPProof (SOUND) 27.18/8.87 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 27.18/8.87 ---------------------------------------- 27.18/8.87 27.18/8.87 (54) 27.18/8.87 Obligation: 27.18/8.87 Q DP problem: 27.18/8.87 The TRS P consists of the following rules: 27.18/8.87 27.18/8.87 U1_GA(X, less_out_ga) -> IN_IN_GA(X) 27.18/8.87 IN_IN_GA(X) -> U1_GA(X, less_in_ga(X)) 27.18/8.87 IN_IN_GA(X) -> U3_GA(X, less_in_ag(X)) 27.18/8.87 U3_GA(X, less_out_ag(Y)) -> IN_IN_GA(X) 27.18/8.87 27.18/8.87 The TRS R consists of the following rules: 27.18/8.87 27.18/8.87 less_in_ga(0) -> less_out_ga 27.18/8.87 less_in_ga(s(X)) -> U5_ga(less_in_ga(X)) 27.18/8.87 less_in_ag(s(X5)) -> less_out_ag(0) 27.18/8.87 less_in_ag(s(Y)) -> U5_ag(less_in_ag(Y)) 27.18/8.87 U5_ga(less_out_ga) -> less_out_ga 27.18/8.87 U5_ag(less_out_ag(X)) -> less_out_ag(s(X)) 27.18/8.87 27.18/8.87 The set Q consists of the following terms: 27.18/8.87 27.18/8.87 less_in_ga(x0) 27.18/8.87 less_in_ag(x0) 27.18/8.87 U5_ga(x0) 27.18/8.87 U5_ag(x0) 27.18/8.87 27.18/8.87 We have to consider all (P,Q,R)-chains. 27.18/8.87 ---------------------------------------- 27.18/8.87 27.18/8.87 (55) TransformationProof (SOUND) 27.18/8.87 By narrowing [LPAR04] the rule IN_IN_GA(X) -> U1_GA(X, less_in_ga(X)) at position [1] we obtained the following new rules [LPAR04]: 27.18/8.87 27.18/8.87 (IN_IN_GA(0) -> U1_GA(0, less_out_ga),IN_IN_GA(0) -> U1_GA(0, less_out_ga)) 27.18/8.87 (IN_IN_GA(s(x0)) -> U1_GA(s(x0), U5_ga(less_in_ga(x0))),IN_IN_GA(s(x0)) -> U1_GA(s(x0), U5_ga(less_in_ga(x0)))) 27.18/8.87 27.18/8.87 27.18/8.87 ---------------------------------------- 27.18/8.87 27.18/8.87 (56) 27.18/8.87 Obligation: 27.18/8.87 Q DP problem: 27.18/8.87 The TRS P consists of the following rules: 27.18/8.87 27.18/8.87 U1_GA(X, less_out_ga) -> IN_IN_GA(X) 27.18/8.87 IN_IN_GA(X) -> U3_GA(X, less_in_ag(X)) 27.18/8.87 U3_GA(X, less_out_ag(Y)) -> IN_IN_GA(X) 27.18/8.87 IN_IN_GA(0) -> U1_GA(0, less_out_ga) 27.18/8.87 IN_IN_GA(s(x0)) -> U1_GA(s(x0), U5_ga(less_in_ga(x0))) 27.18/8.87 27.18/8.87 The TRS R consists of the following rules: 27.18/8.87 27.18/8.87 less_in_ga(0) -> less_out_ga 27.18/8.87 less_in_ga(s(X)) -> U5_ga(less_in_ga(X)) 27.18/8.87 less_in_ag(s(X5)) -> less_out_ag(0) 27.18/8.87 less_in_ag(s(Y)) -> U5_ag(less_in_ag(Y)) 27.18/8.87 U5_ga(less_out_ga) -> less_out_ga 27.18/8.87 U5_ag(less_out_ag(X)) -> less_out_ag(s(X)) 27.18/8.87 27.18/8.87 The set Q consists of the following terms: 27.18/8.87 27.18/8.87 less_in_ga(x0) 27.18/8.87 less_in_ag(x0) 27.18/8.87 U5_ga(x0) 27.18/8.87 U5_ag(x0) 27.18/8.87 27.18/8.87 We have to consider all (P,Q,R)-chains. 27.18/8.87 ---------------------------------------- 27.18/8.87 27.18/8.87 (57) TransformationProof (SOUND) 27.18/8.87 By narrowing [LPAR04] the rule IN_IN_GA(X) -> U3_GA(X, less_in_ag(X)) at position [1] we obtained the following new rules [LPAR04]: 27.18/8.87 27.18/8.87 (IN_IN_GA(s(x0)) -> U3_GA(s(x0), less_out_ag(0)),IN_IN_GA(s(x0)) -> U3_GA(s(x0), less_out_ag(0))) 27.18/8.87 (IN_IN_GA(s(x0)) -> U3_GA(s(x0), U5_ag(less_in_ag(x0))),IN_IN_GA(s(x0)) -> U3_GA(s(x0), U5_ag(less_in_ag(x0)))) 27.18/8.87 27.18/8.87 27.18/8.87 ---------------------------------------- 27.18/8.87 27.18/8.87 (58) 27.18/8.87 Obligation: 27.18/8.87 Q DP problem: 27.18/8.87 The TRS P consists of the following rules: 27.18/8.87 27.18/8.87 U1_GA(X, less_out_ga) -> IN_IN_GA(X) 27.18/8.87 U3_GA(X, less_out_ag(Y)) -> IN_IN_GA(X) 27.18/8.87 IN_IN_GA(0) -> U1_GA(0, less_out_ga) 27.18/8.87 IN_IN_GA(s(x0)) -> U1_GA(s(x0), U5_ga(less_in_ga(x0))) 27.18/8.87 IN_IN_GA(s(x0)) -> U3_GA(s(x0), less_out_ag(0)) 27.18/8.87 IN_IN_GA(s(x0)) -> U3_GA(s(x0), U5_ag(less_in_ag(x0))) 27.18/8.87 27.18/8.87 The TRS R consists of the following rules: 27.18/8.87 27.18/8.87 less_in_ga(0) -> less_out_ga 27.18/8.87 less_in_ga(s(X)) -> U5_ga(less_in_ga(X)) 27.18/8.87 less_in_ag(s(X5)) -> less_out_ag(0) 27.18/8.87 less_in_ag(s(Y)) -> U5_ag(less_in_ag(Y)) 27.18/8.87 U5_ga(less_out_ga) -> less_out_ga 27.18/8.87 U5_ag(less_out_ag(X)) -> less_out_ag(s(X)) 27.18/8.87 27.18/8.87 The set Q consists of the following terms: 27.18/8.87 27.18/8.87 less_in_ga(x0) 27.18/8.87 less_in_ag(x0) 27.18/8.87 U5_ga(x0) 27.18/8.87 U5_ag(x0) 27.18/8.87 27.18/8.87 We have to consider all (P,Q,R)-chains. 27.18/8.87 ---------------------------------------- 27.18/8.87 27.18/8.87 (59) TransformationProof (EQUIVALENT) 27.18/8.87 By instantiating [LPAR04] the rule U1_GA(X, less_out_ga) -> IN_IN_GA(X) we obtained the following new rules [LPAR04]: 27.18/8.87 27.18/8.87 (U1_GA(0, less_out_ga) -> IN_IN_GA(0),U1_GA(0, less_out_ga) -> IN_IN_GA(0)) 27.18/8.87 (U1_GA(s(z0), less_out_ga) -> IN_IN_GA(s(z0)),U1_GA(s(z0), less_out_ga) -> IN_IN_GA(s(z0))) 27.18/8.87 27.18/8.87 27.18/8.87 ---------------------------------------- 27.18/8.87 27.18/8.87 (60) 27.18/8.87 Obligation: 27.18/8.87 Q DP problem: 27.18/8.87 The TRS P consists of the following rules: 27.18/8.87 27.18/8.87 U3_GA(X, less_out_ag(Y)) -> IN_IN_GA(X) 27.18/8.87 IN_IN_GA(0) -> U1_GA(0, less_out_ga) 27.18/8.87 IN_IN_GA(s(x0)) -> U1_GA(s(x0), U5_ga(less_in_ga(x0))) 27.18/8.87 IN_IN_GA(s(x0)) -> U3_GA(s(x0), less_out_ag(0)) 27.18/8.87 IN_IN_GA(s(x0)) -> U3_GA(s(x0), U5_ag(less_in_ag(x0))) 27.18/8.87 U1_GA(0, less_out_ga) -> IN_IN_GA(0) 27.18/8.87 U1_GA(s(z0), less_out_ga) -> IN_IN_GA(s(z0)) 27.18/8.87 27.18/8.87 The TRS R consists of the following rules: 27.18/8.87 27.18/8.87 less_in_ga(0) -> less_out_ga 27.18/8.87 less_in_ga(s(X)) -> U5_ga(less_in_ga(X)) 27.18/8.87 less_in_ag(s(X5)) -> less_out_ag(0) 27.18/8.87 less_in_ag(s(Y)) -> U5_ag(less_in_ag(Y)) 27.18/8.87 U5_ga(less_out_ga) -> less_out_ga 27.18/8.87 U5_ag(less_out_ag(X)) -> less_out_ag(s(X)) 27.18/8.87 27.18/8.87 The set Q consists of the following terms: 27.18/8.87 27.18/8.87 less_in_ga(x0) 27.18/8.87 less_in_ag(x0) 27.18/8.87 U5_ga(x0) 27.18/8.87 U5_ag(x0) 27.18/8.87 27.18/8.87 We have to consider all (P,Q,R)-chains. 27.18/8.87 ---------------------------------------- 27.18/8.87 27.18/8.87 (61) DependencyGraphProof (EQUIVALENT) 27.18/8.87 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs. 27.18/8.87 ---------------------------------------- 27.18/8.87 27.18/8.87 (62) 27.18/8.87 Complex Obligation (AND) 27.18/8.87 27.18/8.87 ---------------------------------------- 27.18/8.87 27.18/8.87 (63) 27.18/8.87 Obligation: 27.18/8.87 Q DP problem: 27.18/8.87 The TRS P consists of the following rules: 27.18/8.87 27.18/8.87 U1_GA(0, less_out_ga) -> IN_IN_GA(0) 27.18/8.87 IN_IN_GA(0) -> U1_GA(0, less_out_ga) 27.18/8.87 27.18/8.87 The TRS R consists of the following rules: 27.18/8.87 27.18/8.87 less_in_ga(0) -> less_out_ga 27.18/8.87 less_in_ga(s(X)) -> U5_ga(less_in_ga(X)) 27.18/8.87 less_in_ag(s(X5)) -> less_out_ag(0) 27.18/8.87 less_in_ag(s(Y)) -> U5_ag(less_in_ag(Y)) 27.18/8.87 U5_ga(less_out_ga) -> less_out_ga 27.18/8.87 U5_ag(less_out_ag(X)) -> less_out_ag(s(X)) 27.18/8.87 27.18/8.87 The set Q consists of the following terms: 27.18/8.87 27.18/8.87 less_in_ga(x0) 27.18/8.87 less_in_ag(x0) 27.18/8.87 U5_ga(x0) 27.18/8.87 U5_ag(x0) 27.18/8.87 27.18/8.87 We have to consider all (P,Q,R)-chains. 27.18/8.87 ---------------------------------------- 27.18/8.87 27.18/8.87 (64) UsableRulesProof (EQUIVALENT) 27.18/8.87 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. 27.18/8.87 ---------------------------------------- 27.18/8.87 27.18/8.87 (65) 27.18/8.87 Obligation: 27.18/8.87 Q DP problem: 27.18/8.87 The TRS P consists of the following rules: 27.18/8.87 27.18/8.87 U1_GA(0, less_out_ga) -> IN_IN_GA(0) 27.18/8.87 IN_IN_GA(0) -> U1_GA(0, less_out_ga) 27.18/8.87 27.18/8.87 R is empty. 27.18/8.87 The set Q consists of the following terms: 27.18/8.87 27.18/8.87 less_in_ga(x0) 27.18/8.87 less_in_ag(x0) 27.18/8.87 U5_ga(x0) 27.18/8.87 U5_ag(x0) 27.18/8.87 27.18/8.87 We have to consider all (P,Q,R)-chains. 27.18/8.87 ---------------------------------------- 27.18/8.87 27.18/8.87 (66) QReductionProof (EQUIVALENT) 27.18/8.87 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 27.18/8.87 27.18/8.87 less_in_ga(x0) 27.18/8.87 less_in_ag(x0) 27.18/8.87 U5_ga(x0) 27.18/8.87 U5_ag(x0) 27.18/8.87 27.18/8.87 27.18/8.87 ---------------------------------------- 27.18/8.87 27.18/8.87 (67) 27.18/8.87 Obligation: 27.18/8.87 Q DP problem: 27.18/8.87 The TRS P consists of the following rules: 27.18/8.87 27.18/8.87 U1_GA(0, less_out_ga) -> IN_IN_GA(0) 27.18/8.87 IN_IN_GA(0) -> U1_GA(0, less_out_ga) 27.18/8.87 27.18/8.87 R is empty. 27.18/8.87 Q is empty. 27.18/8.87 We have to consider all (P,Q,R)-chains. 27.18/8.87 ---------------------------------------- 27.18/8.87 27.18/8.87 (68) 27.18/8.87 Obligation: 27.18/8.87 Q DP problem: 27.18/8.87 The TRS P consists of the following rules: 27.18/8.87 27.18/8.87 IN_IN_GA(s(x0)) -> U1_GA(s(x0), U5_ga(less_in_ga(x0))) 27.18/8.87 U1_GA(s(z0), less_out_ga) -> IN_IN_GA(s(z0)) 27.18/8.87 IN_IN_GA(s(x0)) -> U3_GA(s(x0), less_out_ag(0)) 27.18/8.87 U3_GA(X, less_out_ag(Y)) -> IN_IN_GA(X) 27.18/8.87 IN_IN_GA(s(x0)) -> U3_GA(s(x0), U5_ag(less_in_ag(x0))) 27.18/8.87 27.18/8.87 The TRS R consists of the following rules: 27.18/8.87 27.18/8.87 less_in_ga(0) -> less_out_ga 27.18/8.87 less_in_ga(s(X)) -> U5_ga(less_in_ga(X)) 27.18/8.87 less_in_ag(s(X5)) -> less_out_ag(0) 27.18/8.87 less_in_ag(s(Y)) -> U5_ag(less_in_ag(Y)) 27.18/8.87 U5_ga(less_out_ga) -> less_out_ga 27.18/8.87 U5_ag(less_out_ag(X)) -> less_out_ag(s(X)) 27.18/8.87 27.18/8.87 The set Q consists of the following terms: 27.18/8.87 27.18/8.87 less_in_ga(x0) 27.18/8.87 less_in_ag(x0) 27.18/8.87 U5_ga(x0) 27.18/8.87 U5_ag(x0) 27.18/8.87 27.18/8.87 We have to consider all (P,Q,R)-chains. 27.18/8.87 ---------------------------------------- 27.18/8.87 27.18/8.87 (69) TransformationProof (EQUIVALENT) 27.18/8.87 By instantiating [LPAR04] the rule U3_GA(X, less_out_ag(Y)) -> IN_IN_GA(X) we obtained the following new rules [LPAR04]: 27.18/8.87 27.18/8.87 (U3_GA(s(z0), less_out_ag(0)) -> IN_IN_GA(s(z0)),U3_GA(s(z0), less_out_ag(0)) -> IN_IN_GA(s(z0))) 27.18/8.87 (U3_GA(s(z0), less_out_ag(x1)) -> IN_IN_GA(s(z0)),U3_GA(s(z0), less_out_ag(x1)) -> IN_IN_GA(s(z0))) 27.18/8.87 27.18/8.87 27.18/8.87 ---------------------------------------- 27.18/8.87 27.18/8.87 (70) 27.18/8.87 Obligation: 27.18/8.87 Q DP problem: 27.18/8.87 The TRS P consists of the following rules: 27.18/8.87 27.18/8.87 IN_IN_GA(s(x0)) -> U1_GA(s(x0), U5_ga(less_in_ga(x0))) 27.18/8.87 U1_GA(s(z0), less_out_ga) -> IN_IN_GA(s(z0)) 27.18/8.87 IN_IN_GA(s(x0)) -> U3_GA(s(x0), less_out_ag(0)) 27.18/8.87 IN_IN_GA(s(x0)) -> U3_GA(s(x0), U5_ag(less_in_ag(x0))) 27.18/8.87 U3_GA(s(z0), less_out_ag(0)) -> IN_IN_GA(s(z0)) 27.18/8.87 U3_GA(s(z0), less_out_ag(x1)) -> IN_IN_GA(s(z0)) 27.18/8.87 27.18/8.87 The TRS R consists of the following rules: 27.18/8.87 27.18/8.87 less_in_ga(0) -> less_out_ga 27.18/8.87 less_in_ga(s(X)) -> U5_ga(less_in_ga(X)) 27.18/8.87 less_in_ag(s(X5)) -> less_out_ag(0) 27.18/8.87 less_in_ag(s(Y)) -> U5_ag(less_in_ag(Y)) 27.18/8.87 U5_ga(less_out_ga) -> less_out_ga 27.18/8.87 U5_ag(less_out_ag(X)) -> less_out_ag(s(X)) 27.18/8.87 27.18/8.87 The set Q consists of the following terms: 27.18/8.87 27.18/8.87 less_in_ga(x0) 27.18/8.87 less_in_ag(x0) 27.18/8.87 U5_ga(x0) 27.18/8.87 U5_ag(x0) 27.18/8.87 27.18/8.87 We have to consider all (P,Q,R)-chains. 27.18/8.87 ---------------------------------------- 27.18/8.87 27.18/8.87 (71) PrologToTRSTransformerProof (SOUND) 27.18/8.87 Transformed Prolog program to TRS. 27.18/8.87 27.18/8.87 { 27.18/8.87 "root": 11, 27.18/8.87 "program": { 27.18/8.87 "directives": [], 27.18/8.87 "clauses": [ 27.18/8.87 [ 27.18/8.87 "(in X (tree X X1 X2))", 27.18/8.87 null 27.18/8.87 ], 27.18/8.87 [ 27.18/8.87 "(in X (tree Y Left X3))", 27.18/8.87 "(',' (less X Y) (in X Left))" 27.18/8.87 ], 27.18/8.87 [ 27.18/8.87 "(in X (tree Y X4 Right))", 27.18/8.87 "(',' (less Y X) (in X Right))" 27.18/8.87 ], 27.18/8.87 [ 27.18/8.87 "(less (0) (s X5))", 27.18/8.87 null 27.18/8.87 ], 27.18/8.87 [ 27.18/8.87 "(less (s X) (s Y))", 27.18/8.87 "(less X Y)" 27.18/8.87 ] 27.18/8.87 ] 27.18/8.87 }, 27.18/8.87 "graph": { 27.18/8.87 "nodes": { 27.18/8.87 "23": { 27.18/8.87 "goal": [ 27.18/8.87 { 27.18/8.87 "clause": 1, 27.18/8.87 "scope": 1, 27.18/8.87 "term": "(in T1 T2)" 27.18/8.87 }, 27.18/8.87 { 27.18/8.87 "clause": 2, 27.18/8.87 "scope": 1, 27.18/8.87 "term": "(in T1 T2)" 27.18/8.87 } 27.18/8.87 ], 27.18/8.87 "kb": { 27.18/8.87 "nonunifying": [], 27.18/8.87 "intvars": {}, 27.18/8.87 "arithmetic": { 27.18/8.87 "type": "PlainIntegerRelationState", 27.18/8.87 "relations": [] 27.18/8.87 }, 27.18/8.87 "ground": ["T1"], 27.18/8.87 "free": [], 27.18/8.87 "exprvars": [] 27.18/8.87 } 27.18/8.87 }, 27.18/8.87 "270": { 27.18/8.87 "goal": [], 27.18/8.87 "kb": { 27.18/8.87 "nonunifying": [], 27.18/8.87 "intvars": {}, 27.18/8.87 "arithmetic": { 27.18/8.87 "type": "PlainIntegerRelationState", 27.18/8.87 "relations": [] 27.18/8.87 }, 27.18/8.87 "ground": [], 27.18/8.87 "free": [], 27.18/8.87 "exprvars": [] 27.18/8.87 } 27.18/8.87 }, 27.18/8.87 "type": "Nodes", 27.18/8.87 "271": { 27.18/8.87 "goal": [], 27.18/8.87 "kb": { 27.18/8.87 "nonunifying": [], 27.18/8.87 "intvars": {}, 27.18/8.87 "arithmetic": { 27.18/8.87 "type": "PlainIntegerRelationState", 27.18/8.87 "relations": [] 27.18/8.87 }, 27.18/8.87 "ground": [], 27.18/8.87 "free": [], 27.18/8.87 "exprvars": [] 27.18/8.87 } 27.18/8.87 }, 27.18/8.87 "272": { 27.18/8.87 "goal": [{ 27.18/8.87 "clause": -1, 27.18/8.87 "scope": -1, 27.18/8.87 "term": "(less T54 T56)" 27.18/8.87 }], 27.18/8.87 "kb": { 27.18/8.87 "nonunifying": [], 27.18/8.87 "intvars": {}, 27.18/8.87 "arithmetic": { 27.18/8.87 "type": "PlainIntegerRelationState", 27.18/8.87 "relations": [] 27.18/8.87 }, 27.18/8.87 "ground": ["T54"], 27.18/8.87 "free": [], 27.18/8.87 "exprvars": [] 27.18/8.87 } 27.18/8.87 }, 27.18/8.87 "273": { 27.18/8.87 "goal": [], 27.18/8.87 "kb": { 27.18/8.87 "nonunifying": [], 27.18/8.87 "intvars": {}, 27.18/8.87 "arithmetic": { 27.18/8.87 "type": "PlainIntegerRelationState", 27.18/8.87 "relations": [] 27.18/8.87 }, 27.18/8.87 "ground": [], 27.18/8.87 "free": [], 27.18/8.87 "exprvars": [] 27.18/8.87 } 27.18/8.87 }, 27.18/8.87 "274": { 27.18/8.87 "goal": [{ 27.18/8.87 "clause": -1, 27.18/8.87 "scope": -1, 27.18/8.87 "term": "(',' (less T77 T73) (in T73 T78))" 27.18/8.87 }], 27.18/8.87 "kb": { 27.18/8.87 "nonunifying": [], 27.18/8.87 "intvars": {}, 27.18/8.87 "arithmetic": { 27.18/8.87 "type": "PlainIntegerRelationState", 27.18/8.87 "relations": [] 27.18/8.87 }, 27.18/8.87 "ground": ["T73"], 27.18/8.87 "free": [], 27.18/8.87 "exprvars": [] 27.18/8.87 } 27.18/8.87 }, 27.18/8.87 "132": { 27.18/8.87 "goal": [{ 27.18/8.87 "clause": 1, 27.18/8.87 "scope": 1, 27.18/8.87 "term": "(in T1 T2)" 27.18/8.87 }], 27.18/8.87 "kb": { 27.18/8.87 "nonunifying": [], 27.18/8.87 "intvars": {}, 27.18/8.87 "arithmetic": { 27.18/8.87 "type": "PlainIntegerRelationState", 27.18/8.87 "relations": [] 27.18/8.87 }, 27.18/8.87 "ground": ["T1"], 27.18/8.87 "free": [], 27.18/8.87 "exprvars": [] 27.18/8.87 } 27.18/8.87 }, 27.18/8.87 "275": { 27.18/8.87 "goal": [], 27.18/8.87 "kb": { 27.18/8.87 "nonunifying": [], 27.18/8.87 "intvars": {}, 27.18/8.87 "arithmetic": { 27.18/8.87 "type": "PlainIntegerRelationState", 27.18/8.87 "relations": [] 27.18/8.87 }, 27.18/8.87 "ground": [], 27.18/8.87 "free": [], 27.18/8.87 "exprvars": [] 27.18/8.87 } 27.18/8.87 }, 27.18/8.87 "133": { 27.18/8.87 "goal": [{ 27.18/8.87 "clause": 2, 27.18/8.87 "scope": 1, 27.18/8.87 "term": "(in T1 T2)" 27.18/8.87 }], 27.18/8.87 "kb": { 27.18/8.87 "nonunifying": [], 27.18/8.87 "intvars": {}, 27.18/8.87 "arithmetic": { 27.18/8.87 "type": "PlainIntegerRelationState", 27.18/8.87 "relations": [] 27.18/8.87 }, 27.18/8.87 "ground": ["T1"], 27.18/8.87 "free": [], 27.18/8.87 "exprvars": [] 27.18/8.87 } 27.18/8.87 }, 27.18/8.87 "276": { 27.18/8.87 "goal": [{ 27.18/8.87 "clause": -1, 27.18/8.87 "scope": -1, 27.18/8.87 "term": "(less T77 T73)" 27.18/8.87 }], 27.18/8.87 "kb": { 27.18/8.87 "nonunifying": [], 27.18/8.87 "intvars": {}, 27.18/8.87 "arithmetic": { 27.18/8.87 "type": "PlainIntegerRelationState", 27.18/8.87 "relations": [] 27.18/8.87 }, 27.18/8.87 "ground": ["T73"], 27.18/8.87 "free": [], 27.18/8.87 "exprvars": [] 27.18/8.87 } 27.18/8.87 }, 27.18/8.87 "277": { 27.18/8.87 "goal": [{ 27.18/8.87 "clause": -1, 27.18/8.87 "scope": -1, 27.18/8.87 "term": "(in T73 T81)" 27.18/8.87 }], 27.18/8.87 "kb": { 27.18/8.87 "nonunifying": [], 27.18/8.87 "intvars": {}, 27.18/8.87 "arithmetic": { 27.18/8.87 "type": "PlainIntegerRelationState", 27.18/8.87 "relations": [] 27.18/8.87 }, 27.18/8.87 "ground": ["T73"], 27.18/8.87 "free": [], 27.18/8.87 "exprvars": [] 27.18/8.87 } 27.18/8.87 }, 27.18/8.87 "11": { 27.18/8.87 "goal": [{ 27.18/8.87 "clause": -1, 27.18/8.87 "scope": -1, 27.18/8.87 "term": "(in T1 T2)" 27.18/8.87 }], 27.18/8.87 "kb": { 27.18/8.87 "nonunifying": [], 27.18/8.87 "intvars": {}, 27.18/8.87 "arithmetic": { 27.18/8.87 "type": "PlainIntegerRelationState", 27.18/8.87 "relations": [] 27.18/8.87 }, 27.18/8.87 "ground": ["T1"], 27.18/8.87 "free": [], 27.18/8.87 "exprvars": [] 27.18/8.87 } 27.18/8.87 }, 27.18/8.87 "12": { 27.18/8.87 "goal": [ 27.18/8.87 { 27.18/8.87 "clause": 0, 27.18/8.87 "scope": 1, 27.18/8.87 "term": "(in T1 T2)" 27.18/8.87 }, 27.18/8.87 { 27.18/8.87 "clause": 1, 27.18/8.87 "scope": 1, 27.18/8.87 "term": "(in T1 T2)" 27.18/8.87 }, 27.18/8.87 { 27.18/8.87 "clause": 2, 27.18/8.87 "scope": 1, 27.18/8.87 "term": "(in T1 T2)" 27.18/8.87 } 27.18/8.87 ], 27.18/8.87 "kb": { 27.18/8.87 "nonunifying": [], 27.18/8.87 "intvars": {}, 27.18/8.87 "arithmetic": { 27.18/8.87 "type": "PlainIntegerRelationState", 27.18/8.87 "relations": [] 27.18/8.87 }, 27.18/8.87 "ground": ["T1"], 27.18/8.87 "free": [], 27.18/8.87 "exprvars": [] 27.18/8.87 } 27.18/8.87 }, 27.18/8.87 "280": { 27.18/8.87 "goal": [ 27.18/8.87 { 27.18/8.87 "clause": 3, 27.18/8.87 "scope": 3, 27.18/8.87 "term": "(less T77 T73)" 27.18/8.87 }, 27.18/8.87 { 27.18/8.87 "clause": 4, 27.18/8.87 "scope": 3, 27.18/8.87 "term": "(less T77 T73)" 27.18/8.87 } 27.18/8.87 ], 27.18/8.87 "kb": { 27.18/8.87 "nonunifying": [], 27.18/8.87 "intvars": {}, 27.18/8.87 "arithmetic": { 27.18/8.87 "type": "PlainIntegerRelationState", 27.18/8.87 "relations": [] 27.18/8.87 }, 27.18/8.87 "ground": ["T73"], 27.18/8.87 "free": [], 27.18/8.87 "exprvars": [] 27.18/8.87 } 27.18/8.87 }, 27.18/8.87 "281": { 27.18/8.87 "goal": [{ 27.18/8.87 "clause": 3, 27.18/8.87 "scope": 3, 27.18/8.87 "term": "(less T77 T73)" 27.18/8.87 }], 27.18/8.87 "kb": { 27.18/8.87 "nonunifying": [], 27.18/8.87 "intvars": {}, 27.18/8.87 "arithmetic": { 27.18/8.87 "type": "PlainIntegerRelationState", 27.18/8.87 "relations": [] 27.18/8.87 }, 27.18/8.87 "ground": ["T73"], 27.18/8.87 "free": [], 27.18/8.87 "exprvars": [] 27.18/8.87 } 27.18/8.87 }, 27.18/8.87 "282": { 27.18/8.87 "goal": [{ 27.18/8.87 "clause": 4, 27.18/8.87 "scope": 3, 27.18/8.87 "term": "(less T77 T73)" 27.18/8.87 }], 27.18/8.87 "kb": { 27.18/8.87 "nonunifying": [], 27.18/8.87 "intvars": {}, 27.18/8.87 "arithmetic": { 27.18/8.87 "type": "PlainIntegerRelationState", 27.18/8.87 "relations": [] 27.18/8.87 }, 27.18/8.87 "ground": ["T73"], 27.18/8.87 "free": [], 27.18/8.87 "exprvars": [] 27.18/8.87 } 27.18/8.87 }, 27.18/8.87 "120": { 27.18/8.87 "goal": [{ 27.18/8.87 "clause": -1, 27.18/8.87 "scope": -1, 27.18/8.87 "term": "(true)" 27.18/8.87 }], 27.18/8.87 "kb": { 27.18/8.87 "nonunifying": [], 27.18/8.87 "intvars": {}, 27.18/8.87 "arithmetic": { 27.18/8.87 "type": "PlainIntegerRelationState", 27.18/8.87 "relations": [] 27.18/8.87 }, 27.18/8.87 "ground": [], 27.18/8.87 "free": [], 27.18/8.87 "exprvars": [] 27.18/8.87 } 27.18/8.87 }, 27.18/8.87 "285": { 27.18/8.87 "goal": [{ 27.18/8.87 "clause": -1, 27.18/8.87 "scope": -1, 27.18/8.87 "term": "(true)" 27.18/8.87 }], 27.18/8.87 "kb": { 27.18/8.87 "nonunifying": [], 27.18/8.87 "intvars": {}, 27.18/8.87 "arithmetic": { 27.18/8.87 "type": "PlainIntegerRelationState", 27.18/8.87 "relations": [] 27.18/8.87 }, 27.18/8.87 "ground": [], 27.18/8.87 "free": [], 27.18/8.87 "exprvars": [] 27.18/8.87 } 27.18/8.87 }, 27.18/8.87 "264": { 27.18/8.87 "goal": [{ 27.18/8.87 "clause": -1, 27.18/8.87 "scope": -1, 27.18/8.87 "term": "(less T34 T38)" 27.18/8.87 }], 27.18/8.87 "kb": { 27.18/8.87 "nonunifying": [], 27.18/8.87 "intvars": {}, 27.18/8.87 "arithmetic": { 27.18/8.87 "type": "PlainIntegerRelationState", 27.18/8.87 "relations": [] 27.18/8.87 }, 27.18/8.87 "ground": ["T34"], 27.18/8.87 "free": [], 27.18/8.87 "exprvars": [] 27.18/8.87 } 27.18/8.87 }, 27.18/8.87 "286": { 27.18/8.87 "goal": [], 27.18/8.87 "kb": { 27.18/8.87 "nonunifying": [], 27.18/8.87 "intvars": {}, 27.18/8.87 "arithmetic": { 27.18/8.87 "type": "PlainIntegerRelationState", 27.18/8.87 "relations": [] 27.18/8.87 }, 27.18/8.87 "ground": [], 27.18/8.87 "free": [], 27.18/8.87 "exprvars": [] 27.18/8.87 } 27.18/8.87 }, 27.18/8.87 "122": { 27.18/8.87 "goal": [], 27.18/8.87 "kb": { 27.18/8.87 "nonunifying": [], 27.18/8.87 "intvars": {}, 27.18/8.87 "arithmetic": { 27.18/8.87 "type": "PlainIntegerRelationState", 27.18/8.87 "relations": [] 27.18/8.87 }, 27.18/8.87 "ground": [], 27.18/8.87 "free": [], 27.18/8.87 "exprvars": [] 27.18/8.87 } 27.18/8.87 }, 27.18/8.87 "265": { 27.18/8.87 "goal": [{ 27.18/8.87 "clause": -1, 27.18/8.87 "scope": -1, 27.18/8.87 "term": "(in T34 T42)" 27.18/8.87 }], 27.18/8.87 "kb": { 27.18/8.87 "nonunifying": [], 27.18/8.87 "intvars": {}, 27.18/8.87 "arithmetic": { 27.18/8.87 "type": "PlainIntegerRelationState", 27.18/8.87 "relations": [] 27.18/8.87 }, 27.18/8.87 "ground": ["T34"], 27.18/8.87 "free": [], 27.18/8.87 "exprvars": [] 27.18/8.87 } 27.18/8.87 }, 27.18/8.87 "287": { 27.18/8.87 "goal": [], 27.18/8.87 "kb": { 27.18/8.87 "nonunifying": [], 27.18/8.87 "intvars": {}, 27.18/8.87 "arithmetic": { 27.18/8.87 "type": "PlainIntegerRelationState", 27.38/8.87 "relations": [] 27.38/8.87 }, 27.38/8.87 "ground": [], 27.38/8.87 "free": [], 27.38/8.87 "exprvars": [] 27.38/8.87 } 27.38/8.87 }, 27.38/8.87 "123": { 27.38/8.87 "goal": [], 27.38/8.87 "kb": { 27.38/8.87 "nonunifying": [], 27.38/8.87 "intvars": {}, 27.38/8.87 "arithmetic": { 27.38/8.87 "type": "PlainIntegerRelationState", 27.38/8.87 "relations": [] 27.38/8.87 }, 27.38/8.87 "ground": [], 27.38/8.87 "free": [], 27.38/8.87 "exprvars": [] 27.38/8.87 } 27.38/8.87 }, 27.38/8.87 "266": { 27.38/8.87 "goal": [ 27.38/8.87 { 27.38/8.87 "clause": 3, 27.38/8.87 "scope": 2, 27.38/8.87 "term": "(less T34 T38)" 27.38/8.87 }, 27.38/8.87 { 27.38/8.87 "clause": 4, 27.38/8.87 "scope": 2, 27.38/8.87 "term": "(less T34 T38)" 27.38/8.87 } 27.38/8.87 ], 27.38/8.87 "kb": { 27.38/8.87 "nonunifying": [], 27.38/8.87 "intvars": {}, 27.38/8.87 "arithmetic": { 27.38/8.87 "type": "PlainIntegerRelationState", 27.38/8.87 "relations": [] 27.38/8.87 }, 27.38/8.87 "ground": ["T34"], 27.38/8.87 "free": [], 27.38/8.87 "exprvars": [] 27.38/8.87 } 27.38/8.87 }, 27.38/8.87 "288": { 27.38/8.87 "goal": [{ 27.38/8.87 "clause": -1, 27.38/8.87 "scope": -1, 27.38/8.87 "term": "(less T95 T94)" 27.38/8.87 }], 27.38/8.87 "kb": { 27.38/8.87 "nonunifying": [], 27.38/8.87 "intvars": {}, 27.38/8.87 "arithmetic": { 27.38/8.87 "type": "PlainIntegerRelationState", 27.38/8.87 "relations": [] 27.38/8.87 }, 27.38/8.87 "ground": ["T94"], 27.38/8.87 "free": [], 27.38/8.87 "exprvars": [] 27.38/8.87 } 27.38/8.87 }, 27.38/8.87 "267": { 27.38/8.87 "goal": [{ 27.38/8.87 "clause": 3, 27.38/8.87 "scope": 2, 27.38/8.87 "term": "(less T34 T38)" 27.38/8.87 }], 27.38/8.87 "kb": { 27.38/8.87 "nonunifying": [], 27.38/8.87 "intvars": {}, 27.38/8.87 "arithmetic": { 27.38/8.87 "type": "PlainIntegerRelationState", 27.38/8.87 "relations": [] 27.38/8.87 }, 27.38/8.87 "ground": ["T34"], 27.38/8.87 "free": [], 27.38/8.87 "exprvars": [] 27.38/8.87 } 27.38/8.87 }, 27.38/8.87 "289": { 27.38/8.87 "goal": [], 27.38/8.87 "kb": { 27.38/8.87 "nonunifying": [], 27.38/8.87 "intvars": {}, 27.38/8.87 "arithmetic": { 27.38/8.87 "type": "PlainIntegerRelationState", 27.38/8.87 "relations": [] 27.38/8.87 }, 27.38/8.87 "ground": [], 27.38/8.87 "free": [], 27.38/8.87 "exprvars": [] 27.38/8.87 } 27.38/8.87 }, 27.38/8.87 "147": { 27.38/8.87 "goal": [{ 27.38/8.87 "clause": -1, 27.38/8.87 "scope": -1, 27.38/8.87 "term": "(',' (less T34 T38) (in T34 T39))" 27.38/8.87 }], 27.38/8.87 "kb": { 27.38/8.87 "nonunifying": [], 27.38/8.87 "intvars": {}, 27.38/8.87 "arithmetic": { 27.38/8.87 "type": "PlainIntegerRelationState", 27.38/8.87 "relations": [] 27.38/8.87 }, 27.38/8.87 "ground": ["T34"], 27.38/8.87 "free": [], 27.38/8.87 "exprvars": [] 27.38/8.87 } 27.38/8.87 }, 27.38/8.87 "268": { 27.38/8.87 "goal": [{ 27.38/8.87 "clause": 4, 27.38/8.87 "scope": 2, 27.38/8.87 "term": "(less T34 T38)" 27.38/8.87 }], 27.38/8.87 "kb": { 27.38/8.87 "nonunifying": [], 27.38/8.87 "intvars": {}, 27.38/8.87 "arithmetic": { 27.38/8.87 "type": "PlainIntegerRelationState", 27.38/8.87 "relations": [] 27.38/8.87 }, 27.38/8.87 "ground": ["T34"], 27.38/8.87 "free": [], 27.38/8.87 "exprvars": [] 27.38/8.87 } 27.38/8.87 }, 27.38/8.87 "269": { 27.38/8.87 "goal": [{ 27.38/8.87 "clause": -1, 27.38/8.87 "scope": -1, 27.38/8.87 "term": "(true)" 27.38/8.87 }], 27.38/8.87 "kb": { 27.38/8.87 "nonunifying": [], 27.38/8.87 "intvars": {}, 27.38/8.87 "arithmetic": { 27.38/8.87 "type": "PlainIntegerRelationState", 27.38/8.87 "relations": [] 27.38/8.87 }, 27.38/8.87 "ground": [], 27.38/8.87 "free": [], 27.38/8.87 "exprvars": [] 27.38/8.87 } 27.38/8.87 }, 27.38/8.87 "149": { 27.38/8.87 "goal": [], 27.38/8.87 "kb": { 27.38/8.87 "nonunifying": [], 27.38/8.87 "intvars": {}, 27.38/8.87 "arithmetic": { 27.38/8.87 "type": "PlainIntegerRelationState", 27.38/8.87 "relations": [] 27.38/8.87 }, 27.38/8.87 "ground": [], 27.38/8.87 "free": [], 27.38/8.87 "exprvars": [] 27.38/8.87 } 27.38/8.87 }, 27.38/8.87 "21": { 27.38/8.87 "goal": [{ 27.38/8.87 "clause": 0, 27.38/8.87 "scope": 1, 27.38/8.87 "term": "(in T1 T2)" 27.38/8.87 }], 27.38/8.87 "kb": { 27.38/8.87 "nonunifying": [], 27.38/8.87 "intvars": {}, 27.38/8.87 "arithmetic": { 27.38/8.87 "type": "PlainIntegerRelationState", 27.38/8.87 "relations": [] 27.38/8.87 }, 27.38/8.87 "ground": ["T1"], 27.38/8.87 "free": [], 27.38/8.87 "exprvars": [] 27.38/8.87 } 27.38/8.87 } 27.38/8.87 }, 27.38/8.87 "edges": [ 27.38/8.87 { 27.38/8.87 "from": 11, 27.38/8.87 "to": 12, 27.38/8.87 "label": "CASE" 27.38/8.87 }, 27.38/8.87 { 27.38/8.87 "from": 12, 27.38/8.87 "to": 21, 27.38/8.87 "label": "PARALLEL" 27.38/8.87 }, 27.38/8.87 { 27.38/8.87 "from": 12, 27.38/8.87 "to": 23, 27.38/8.87 "label": "PARALLEL" 27.38/8.87 }, 27.38/8.87 { 27.38/8.87 "from": 21, 27.38/8.87 "to": 120, 27.38/8.87 "label": "EVAL with clause\nin(X18, tree(X18, X19, X20)).\nand substitutionT1 -> T15,\nX18 -> T15,\nX19 -> T16,\nX20 -> T17,\nT2 -> tree(T15, T16, T17)" 27.38/8.87 }, 27.38/8.87 { 27.38/8.87 "from": 21, 27.38/8.87 "to": 122, 27.38/8.87 "label": "EVAL-BACKTRACK" 27.38/8.87 }, 27.38/8.87 { 27.38/8.87 "from": 23, 27.38/8.87 "to": 132, 27.38/8.87 "label": "PARALLEL" 27.38/8.87 }, 27.38/8.87 { 27.38/8.87 "from": 23, 27.38/8.87 "to": 133, 27.38/8.87 "label": "PARALLEL" 27.38/8.87 }, 27.38/8.87 { 27.38/8.87 "from": 120, 27.38/8.87 "to": 123, 27.38/8.87 "label": "SUCCESS" 27.38/8.87 }, 27.38/8.87 { 27.38/8.87 "from": 132, 27.38/8.87 "to": 147, 27.38/8.87 "label": "EVAL with clause\nin(X37, tree(X38, X39, X40)) :- ','(less(X37, X38), in(X37, X39)).\nand substitutionT1 -> T34,\nX37 -> T34,\nX38 -> T38,\nX39 -> T39,\nX40 -> T37,\nT2 -> tree(T38, T39, T37),\nT35 -> T38,\nT36 -> T39" 27.38/8.87 }, 27.38/8.87 { 27.38/8.87 "from": 132, 27.38/8.87 "to": 149, 27.38/8.87 "label": "EVAL-BACKTRACK" 27.38/8.87 }, 27.38/8.87 { 27.38/8.87 "from": 133, 27.38/8.87 "to": 274, 27.38/8.87 "label": "EVAL with clause\nin(X72, tree(X73, X74, X75)) :- ','(less(X73, X72), in(X72, X75)).\nand substitutionT1 -> T73,\nX72 -> T73,\nX73 -> T77,\nX74 -> T75,\nX75 -> T78,\nT2 -> tree(T77, T75, T78),\nT74 -> T77,\nT76 -> T78" 27.38/8.87 }, 27.38/8.87 { 27.38/8.87 "from": 133, 27.38/8.87 "to": 275, 27.38/8.87 "label": "EVAL-BACKTRACK" 27.38/8.87 }, 27.38/8.87 { 27.38/8.87 "from": 147, 27.38/8.87 "to": 264, 27.38/8.87 "label": "SPLIT 1" 27.38/8.87 }, 27.38/8.87 { 27.38/8.87 "from": 147, 27.38/8.87 "to": 265, 27.38/8.87 "label": "SPLIT 2\nnew knowledge:\nT34 is ground\nreplacements:T39 -> T42" 27.38/8.87 }, 27.38/8.87 { 27.38/8.87 "from": 264, 27.38/8.87 "to": 266, 27.38/8.87 "label": "CASE" 27.38/8.87 }, 27.38/8.87 { 27.38/8.87 "from": 265, 27.38/8.87 "to": 11, 27.38/8.87 "label": "INSTANCE with matching:\nT1 -> T34\nT2 -> T42" 27.38/8.87 }, 27.38/8.87 { 27.38/8.87 "from": 266, 27.38/8.87 "to": 267, 27.38/8.87 "label": "PARALLEL" 27.38/8.87 }, 27.38/8.87 { 27.38/8.87 "from": 266, 27.38/8.87 "to": 268, 27.38/8.87 "label": "PARALLEL" 27.38/8.87 }, 27.38/8.87 { 27.38/8.87 "from": 267, 27.38/8.87 "to": 269, 27.38/8.87 "label": "EVAL with clause\nless(0, s(X49)).\nand substitutionT34 -> 0,\nX49 -> T49,\nT38 -> s(T49)" 27.38/8.87 }, 27.38/8.87 { 27.38/8.87 "from": 267, 27.38/8.87 "to": 270, 27.38/8.87 "label": "EVAL-BACKTRACK" 27.38/8.87 }, 27.38/8.87 { 27.38/8.87 "from": 268, 27.38/8.87 "to": 272, 27.38/8.87 "label": "EVAL with clause\nless(s(X54), s(X55)) :- less(X54, X55).\nand substitutionX54 -> T54,\nT34 -> s(T54),\nX55 -> T56,\nT38 -> s(T56),\nT55 -> T56" 27.38/8.87 }, 27.38/8.87 { 27.38/8.87 "from": 268, 27.38/8.87 "to": 273, 27.38/8.87 "label": "EVAL-BACKTRACK" 27.38/8.87 }, 27.38/8.87 { 27.38/8.87 "from": 269, 27.38/8.87 "to": 271, 27.38/8.87 "label": "SUCCESS" 27.38/8.87 }, 27.38/8.87 { 27.38/8.87 "from": 272, 27.38/8.87 "to": 264, 27.38/8.87 "label": "INSTANCE with matching:\nT34 -> T54\nT38 -> T56" 27.38/8.87 }, 27.38/8.87 { 27.38/8.87 "from": 274, 27.38/8.87 "to": 276, 27.38/8.87 "label": "SPLIT 1" 27.38/8.87 }, 27.38/8.87 { 27.38/8.87 "from": 274, 27.38/8.87 "to": 277, 27.38/8.87 "label": "SPLIT 2\nnew knowledge:\nT77 is ground\nT73 is ground\nreplacements:T78 -> T81" 27.38/8.87 }, 27.38/8.87 { 27.38/8.87 "from": 276, 27.38/8.87 "to": 280, 27.38/8.87 "label": "CASE" 27.38/8.87 }, 27.38/8.87 { 27.38/8.87 "from": 277, 27.38/8.87 "to": 11, 27.38/8.87 "label": "INSTANCE with matching:\nT1 -> T73\nT2 -> T81" 27.38/8.87 }, 27.38/8.87 { 27.38/8.87 "from": 280, 27.38/8.87 "to": 281, 27.38/8.87 "label": "PARALLEL" 27.38/8.87 }, 27.38/8.87 { 27.38/8.87 "from": 280, 27.38/8.87 "to": 282, 27.38/8.87 "label": "PARALLEL" 27.38/8.87 }, 27.38/8.87 { 27.38/8.87 "from": 281, 27.38/8.87 "to": 285, 27.38/8.87 "label": "EVAL with clause\nless(0, s(X84)).\nand substitutionT77 -> 0,\nX84 -> T88,\nT73 -> s(T88)" 27.38/8.87 }, 27.38/8.87 { 27.38/8.87 "from": 281, 27.38/8.87 "to": 286, 27.38/8.87 "label": "EVAL-BACKTRACK" 27.38/8.87 }, 27.38/8.87 { 27.38/8.87 "from": 282, 27.38/8.87 "to": 288, 27.38/8.87 "label": "EVAL with clause\nless(s(X89), s(X90)) :- less(X89, X90).\nand substitutionX89 -> T95,\nT77 -> s(T95),\nX90 -> T94,\nT73 -> s(T94),\nT93 -> T95" 27.38/8.87 }, 27.38/8.87 { 27.38/8.87 "from": 282, 27.38/8.87 "to": 289, 27.38/8.87 "label": "EVAL-BACKTRACK" 27.38/8.87 }, 27.38/8.87 { 27.38/8.87 "from": 285, 27.38/8.87 "to": 287, 27.38/8.87 "label": "SUCCESS" 27.38/8.87 }, 27.38/8.87 { 27.38/8.87 "from": 288, 27.38/8.87 "to": 276, 27.38/8.87 "label": "INSTANCE with matching:\nT77 -> T95\nT73 -> T94" 27.38/8.87 } 27.38/8.87 ], 27.38/8.87 "type": "Graph" 27.38/8.87 } 27.38/8.87 } 27.38/8.87 27.38/8.87 ---------------------------------------- 27.38/8.87 27.38/8.87 (72) 27.38/8.87 Obligation: 27.38/8.87 Q restricted rewrite system: 27.38/8.87 The TRS R consists of the following rules: 27.38/8.87 27.38/8.87 f11_in(T15) -> f11_out1 27.38/8.87 f11_in(T34) -> U1(f147_in(T34), T34) 27.38/8.87 U1(f147_out1, T34) -> f11_out1 27.38/8.87 f11_in(T73) -> U2(f274_in(T73), T73) 27.38/8.87 U2(f274_out1(T77), T73) -> f11_out1 27.38/8.87 f264_in(0) -> f264_out1 27.38/8.87 f264_in(s(T54)) -> U3(f264_in(T54), s(T54)) 27.38/8.87 U3(f264_out1, s(T54)) -> f264_out1 27.38/8.87 f276_in(s(T88)) -> f276_out1(0) 27.38/8.87 f276_in(s(T94)) -> U4(f276_in(T94), s(T94)) 27.38/8.87 U4(f276_out1(T95), s(T94)) -> f276_out1(s(T95)) 27.38/8.87 f147_in(T34) -> U5(f264_in(T34), T34) 27.38/8.87 U5(f264_out1, T34) -> U6(f11_in(T34), T34) 27.38/8.87 U6(f11_out1, T34) -> f147_out1 27.38/8.87 f274_in(T73) -> U7(f276_in(T73), T73) 27.38/8.87 U7(f276_out1(T77), T73) -> U8(f11_in(T73), T73, T77) 27.38/8.87 U8(f11_out1, T73, T77) -> f274_out1(T77) 27.38/8.87 27.38/8.87 Q is empty. 27.38/8.87 27.38/8.87 ---------------------------------------- 27.38/8.87 27.38/8.87 (73) DependencyPairsProof (EQUIVALENT) 27.38/8.87 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 27.38/8.87 ---------------------------------------- 27.38/8.87 27.38/8.87 (74) 27.38/8.87 Obligation: 27.38/8.87 Q DP problem: 27.38/8.87 The TRS P consists of the following rules: 27.38/8.87 27.38/8.87 F11_IN(T34) -> U1^1(f147_in(T34), T34) 27.38/8.87 F11_IN(T34) -> F147_IN(T34) 27.38/8.87 F11_IN(T73) -> U2^1(f274_in(T73), T73) 27.38/8.87 F11_IN(T73) -> F274_IN(T73) 27.38/8.87 F264_IN(s(T54)) -> U3^1(f264_in(T54), s(T54)) 27.38/8.87 F264_IN(s(T54)) -> F264_IN(T54) 27.38/8.87 F276_IN(s(T94)) -> U4^1(f276_in(T94), s(T94)) 27.38/8.87 F276_IN(s(T94)) -> F276_IN(T94) 27.38/8.87 F147_IN(T34) -> U5^1(f264_in(T34), T34) 27.38/8.87 F147_IN(T34) -> F264_IN(T34) 27.38/8.87 U5^1(f264_out1, T34) -> U6^1(f11_in(T34), T34) 27.38/8.87 U5^1(f264_out1, T34) -> F11_IN(T34) 27.38/8.87 F274_IN(T73) -> U7^1(f276_in(T73), T73) 27.38/8.87 F274_IN(T73) -> F276_IN(T73) 27.38/8.87 U7^1(f276_out1(T77), T73) -> U8^1(f11_in(T73), T73, T77) 27.38/8.87 U7^1(f276_out1(T77), T73) -> F11_IN(T73) 27.38/8.87 27.38/8.87 The TRS R consists of the following rules: 27.38/8.87 27.38/8.87 f11_in(T15) -> f11_out1 27.38/8.87 f11_in(T34) -> U1(f147_in(T34), T34) 27.38/8.87 U1(f147_out1, T34) -> f11_out1 27.38/8.87 f11_in(T73) -> U2(f274_in(T73), T73) 27.38/8.87 U2(f274_out1(T77), T73) -> f11_out1 27.38/8.87 f264_in(0) -> f264_out1 27.38/8.87 f264_in(s(T54)) -> U3(f264_in(T54), s(T54)) 27.38/8.87 U3(f264_out1, s(T54)) -> f264_out1 27.38/8.87 f276_in(s(T88)) -> f276_out1(0) 27.38/8.87 f276_in(s(T94)) -> U4(f276_in(T94), s(T94)) 27.38/8.87 U4(f276_out1(T95), s(T94)) -> f276_out1(s(T95)) 27.38/8.87 f147_in(T34) -> U5(f264_in(T34), T34) 27.38/8.87 U5(f264_out1, T34) -> U6(f11_in(T34), T34) 27.38/8.87 U6(f11_out1, T34) -> f147_out1 27.38/8.87 f274_in(T73) -> U7(f276_in(T73), T73) 27.38/8.87 U7(f276_out1(T77), T73) -> U8(f11_in(T73), T73, T77) 27.38/8.87 U8(f11_out1, T73, T77) -> f274_out1(T77) 27.38/8.87 27.38/8.87 Q is empty. 27.38/8.87 We have to consider all minimal (P,Q,R)-chains. 27.38/8.87 ---------------------------------------- 27.38/8.87 27.38/8.87 (75) DependencyGraphProof (EQUIVALENT) 27.38/8.87 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 8 less nodes. 27.38/8.87 ---------------------------------------- 27.38/8.87 27.38/8.87 (76) 27.38/8.87 Complex Obligation (AND) 27.38/8.87 27.38/8.87 ---------------------------------------- 27.38/8.88 27.38/8.88 (77) 27.38/8.88 Obligation: 27.38/8.88 Q DP problem: 27.38/8.88 The TRS P consists of the following rules: 27.38/8.88 27.38/8.88 F276_IN(s(T94)) -> F276_IN(T94) 27.38/8.88 27.38/8.88 The TRS R consists of the following rules: 27.38/8.88 27.38/8.88 f11_in(T15) -> f11_out1 27.38/8.88 f11_in(T34) -> U1(f147_in(T34), T34) 27.38/8.88 U1(f147_out1, T34) -> f11_out1 27.38/8.88 f11_in(T73) -> U2(f274_in(T73), T73) 27.38/8.88 U2(f274_out1(T77), T73) -> f11_out1 27.38/8.88 f264_in(0) -> f264_out1 27.38/8.88 f264_in(s(T54)) -> U3(f264_in(T54), s(T54)) 27.38/8.88 U3(f264_out1, s(T54)) -> f264_out1 27.38/8.88 f276_in(s(T88)) -> f276_out1(0) 27.38/8.88 f276_in(s(T94)) -> U4(f276_in(T94), s(T94)) 27.38/8.88 U4(f276_out1(T95), s(T94)) -> f276_out1(s(T95)) 27.38/8.88 f147_in(T34) -> U5(f264_in(T34), T34) 27.38/8.88 U5(f264_out1, T34) -> U6(f11_in(T34), T34) 27.38/8.88 U6(f11_out1, T34) -> f147_out1 27.38/8.88 f274_in(T73) -> U7(f276_in(T73), T73) 27.38/8.88 U7(f276_out1(T77), T73) -> U8(f11_in(T73), T73, T77) 27.38/8.88 U8(f11_out1, T73, T77) -> f274_out1(T77) 27.38/8.88 27.38/8.88 Q is empty. 27.38/8.88 We have to consider all minimal (P,Q,R)-chains. 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (78) UsableRulesProof (EQUIVALENT) 27.38/8.88 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. 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (79) 27.38/8.88 Obligation: 27.38/8.88 Q DP problem: 27.38/8.88 The TRS P consists of the following rules: 27.38/8.88 27.38/8.88 F276_IN(s(T94)) -> F276_IN(T94) 27.38/8.88 27.38/8.88 R is empty. 27.38/8.88 Q is empty. 27.38/8.88 We have to consider all minimal (P,Q,R)-chains. 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (80) QDPSizeChangeProof (EQUIVALENT) 27.38/8.88 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. 27.38/8.88 27.38/8.88 From the DPs we obtained the following set of size-change graphs: 27.38/8.88 *F276_IN(s(T94)) -> F276_IN(T94) 27.38/8.88 The graph contains the following edges 1 > 1 27.38/8.88 27.38/8.88 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (81) 27.38/8.88 YES 27.38/8.88 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (82) 27.38/8.88 Obligation: 27.38/8.88 Q DP problem: 27.38/8.88 The TRS P consists of the following rules: 27.38/8.88 27.38/8.88 F264_IN(s(T54)) -> F264_IN(T54) 27.38/8.88 27.38/8.88 The TRS R consists of the following rules: 27.38/8.88 27.38/8.88 f11_in(T15) -> f11_out1 27.38/8.88 f11_in(T34) -> U1(f147_in(T34), T34) 27.38/8.88 U1(f147_out1, T34) -> f11_out1 27.38/8.88 f11_in(T73) -> U2(f274_in(T73), T73) 27.38/8.88 U2(f274_out1(T77), T73) -> f11_out1 27.38/8.88 f264_in(0) -> f264_out1 27.38/8.88 f264_in(s(T54)) -> U3(f264_in(T54), s(T54)) 27.38/8.88 U3(f264_out1, s(T54)) -> f264_out1 27.38/8.88 f276_in(s(T88)) -> f276_out1(0) 27.38/8.88 f276_in(s(T94)) -> U4(f276_in(T94), s(T94)) 27.38/8.88 U4(f276_out1(T95), s(T94)) -> f276_out1(s(T95)) 27.38/8.88 f147_in(T34) -> U5(f264_in(T34), T34) 27.38/8.88 U5(f264_out1, T34) -> U6(f11_in(T34), T34) 27.38/8.88 U6(f11_out1, T34) -> f147_out1 27.38/8.88 f274_in(T73) -> U7(f276_in(T73), T73) 27.38/8.88 U7(f276_out1(T77), T73) -> U8(f11_in(T73), T73, T77) 27.38/8.88 U8(f11_out1, T73, T77) -> f274_out1(T77) 27.38/8.88 27.38/8.88 Q is empty. 27.38/8.88 We have to consider all minimal (P,Q,R)-chains. 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (83) UsableRulesProof (EQUIVALENT) 27.38/8.88 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. 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (84) 27.38/8.88 Obligation: 27.38/8.88 Q DP problem: 27.38/8.88 The TRS P consists of the following rules: 27.38/8.88 27.38/8.88 F264_IN(s(T54)) -> F264_IN(T54) 27.38/8.88 27.38/8.88 R is empty. 27.38/8.88 Q is empty. 27.38/8.88 We have to consider all minimal (P,Q,R)-chains. 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (85) QDPSizeChangeProof (EQUIVALENT) 27.38/8.88 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. 27.38/8.88 27.38/8.88 From the DPs we obtained the following set of size-change graphs: 27.38/8.88 *F264_IN(s(T54)) -> F264_IN(T54) 27.38/8.88 The graph contains the following edges 1 > 1 27.38/8.88 27.38/8.88 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (86) 27.38/8.88 YES 27.38/8.88 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (87) 27.38/8.88 Obligation: 27.38/8.88 Q DP problem: 27.38/8.88 The TRS P consists of the following rules: 27.38/8.88 27.38/8.88 F11_IN(T34) -> F147_IN(T34) 27.38/8.88 F147_IN(T34) -> U5^1(f264_in(T34), T34) 27.38/8.88 U5^1(f264_out1, T34) -> F11_IN(T34) 27.38/8.88 F11_IN(T73) -> F274_IN(T73) 27.38/8.88 F274_IN(T73) -> U7^1(f276_in(T73), T73) 27.38/8.88 U7^1(f276_out1(T77), T73) -> F11_IN(T73) 27.38/8.88 27.38/8.88 The TRS R consists of the following rules: 27.38/8.88 27.38/8.88 f11_in(T15) -> f11_out1 27.38/8.88 f11_in(T34) -> U1(f147_in(T34), T34) 27.38/8.88 U1(f147_out1, T34) -> f11_out1 27.38/8.88 f11_in(T73) -> U2(f274_in(T73), T73) 27.38/8.88 U2(f274_out1(T77), T73) -> f11_out1 27.38/8.88 f264_in(0) -> f264_out1 27.38/8.88 f264_in(s(T54)) -> U3(f264_in(T54), s(T54)) 27.38/8.88 U3(f264_out1, s(T54)) -> f264_out1 27.38/8.88 f276_in(s(T88)) -> f276_out1(0) 27.38/8.88 f276_in(s(T94)) -> U4(f276_in(T94), s(T94)) 27.38/8.88 U4(f276_out1(T95), s(T94)) -> f276_out1(s(T95)) 27.38/8.88 f147_in(T34) -> U5(f264_in(T34), T34) 27.38/8.88 U5(f264_out1, T34) -> U6(f11_in(T34), T34) 27.38/8.88 U6(f11_out1, T34) -> f147_out1 27.38/8.88 f274_in(T73) -> U7(f276_in(T73), T73) 27.38/8.88 U7(f276_out1(T77), T73) -> U8(f11_in(T73), T73, T77) 27.38/8.88 U8(f11_out1, T73, T77) -> f274_out1(T77) 27.38/8.88 27.38/8.88 Q is empty. 27.38/8.88 We have to consider all minimal (P,Q,R)-chains. 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (88) NonTerminationLoopProof (COMPLETE) 27.38/8.88 We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. 27.38/8.88 Found a loop by narrowing to the left: 27.38/8.88 27.38/8.88 s = F147_IN(0) evaluates to t =F147_IN(0) 27.38/8.88 27.38/8.88 Thus s starts an infinite chain as s semiunifies with t with the following substitutions: 27.38/8.88 * Matcher: [ ] 27.38/8.88 * Semiunifier: [ ] 27.38/8.88 27.38/8.88 -------------------------------------------------------------------------------- 27.38/8.88 Rewriting sequence 27.38/8.88 27.38/8.88 F147_IN(0) -> U5^1(f264_in(0), 0) 27.38/8.88 with rule F147_IN(T34) -> U5^1(f264_in(T34), T34) at position [] and matcher [T34 / 0] 27.38/8.88 27.38/8.88 U5^1(f264_in(0), 0) -> U5^1(f264_out1, 0) 27.38/8.88 with rule f264_in(0) -> f264_out1 at position [0] and matcher [ ] 27.38/8.88 27.38/8.88 U5^1(f264_out1, 0) -> F11_IN(0) 27.38/8.88 with rule U5^1(f264_out1, T34') -> F11_IN(T34') at position [] and matcher [T34' / 0] 27.38/8.88 27.38/8.88 F11_IN(0) -> F147_IN(0) 27.38/8.88 with rule F11_IN(T34) -> F147_IN(T34) 27.38/8.88 27.38/8.88 Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence 27.38/8.88 27.38/8.88 27.38/8.88 All these steps are and every following step will be a correct step w.r.t to Q. 27.38/8.88 27.38/8.88 27.38/8.88 27.38/8.88 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (89) 27.38/8.88 NO 27.38/8.88 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (90) PrologToIRSwTTransformerProof (SOUND) 27.38/8.88 Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert 27.38/8.88 27.38/8.88 { 27.38/8.88 "root": 4, 27.38/8.88 "program": { 27.38/8.88 "directives": [], 27.38/8.88 "clauses": [ 27.38/8.88 [ 27.38/8.88 "(in X (tree X X1 X2))", 27.38/8.88 null 27.38/8.88 ], 27.38/8.88 [ 27.38/8.88 "(in X (tree Y Left X3))", 27.38/8.88 "(',' (less X Y) (in X Left))" 27.38/8.88 ], 27.38/8.88 [ 27.38/8.88 "(in X (tree Y X4 Right))", 27.38/8.88 "(',' (less Y X) (in X Right))" 27.38/8.88 ], 27.38/8.88 [ 27.38/8.88 "(less (0) (s X5))", 27.38/8.88 null 27.38/8.88 ], 27.38/8.88 [ 27.38/8.88 "(less (s X) (s Y))", 27.38/8.88 "(less X Y)" 27.38/8.88 ] 27.38/8.88 ] 27.38/8.88 }, 27.38/8.88 "graph": { 27.38/8.88 "nodes": { 27.38/8.88 "type": "Nodes", 27.38/8.88 "250": { 27.38/8.88 "goal": [{ 27.38/8.88 "clause": 4, 27.38/8.88 "scope": 3, 27.38/8.88 "term": "(less T77 T73)" 27.38/8.88 }], 27.38/8.88 "kb": { 27.38/8.88 "nonunifying": [], 27.38/8.88 "intvars": {}, 27.38/8.88 "arithmetic": { 27.38/8.88 "type": "PlainIntegerRelationState", 27.38/8.88 "relations": [] 27.38/8.88 }, 27.38/8.88 "ground": ["T73"], 27.38/8.88 "free": [], 27.38/8.88 "exprvars": [] 27.38/8.88 } 27.38/8.88 }, 27.38/8.88 "130": { 27.38/8.88 "goal": [], 27.38/8.88 "kb": { 27.38/8.88 "nonunifying": [], 27.38/8.88 "intvars": {}, 27.38/8.88 "arithmetic": { 27.38/8.88 "type": "PlainIntegerRelationState", 27.38/8.88 "relations": [] 27.38/8.88 }, 27.38/8.88 "ground": [], 27.38/8.88 "free": [], 27.38/8.88 "exprvars": [] 27.38/8.88 } 27.38/8.88 }, 27.38/8.88 "251": { 27.38/8.88 "goal": [{ 27.38/8.88 "clause": -1, 27.38/8.88 "scope": -1, 27.38/8.88 "term": "(true)" 27.38/8.88 }], 27.38/8.88 "kb": { 27.38/8.88 "nonunifying": [], 27.38/8.88 "intvars": {}, 27.38/8.88 "arithmetic": { 27.38/8.88 "type": "PlainIntegerRelationState", 27.38/8.88 "relations": [] 27.38/8.88 }, 27.38/8.88 "ground": [], 27.38/8.88 "free": [], 27.38/8.88 "exprvars": [] 27.38/8.88 } 27.38/8.88 }, 27.38/8.88 "252": { 27.38/8.88 "goal": [], 27.38/8.88 "kb": { 27.38/8.88 "nonunifying": [], 27.38/8.88 "intvars": {}, 27.38/8.88 "arithmetic": { 27.38/8.88 "type": "PlainIntegerRelationState", 27.38/8.88 "relations": [] 27.38/8.88 }, 27.38/8.88 "ground": [], 27.38/8.88 "free": [], 27.38/8.88 "exprvars": [] 27.38/8.88 } 27.38/8.88 }, 27.38/8.88 "253": { 27.38/8.88 "goal": [], 27.38/8.88 "kb": { 27.38/8.88 "nonunifying": [], 27.38/8.88 "intvars": {}, 27.38/8.88 "arithmetic": { 27.38/8.88 "type": "PlainIntegerRelationState", 27.38/8.88 "relations": [] 27.38/8.88 }, 27.38/8.88 "ground": [], 27.38/8.88 "free": [], 27.38/8.88 "exprvars": [] 27.38/8.88 } 27.38/8.88 }, 27.38/8.88 "254": { 27.38/8.88 "goal": [{ 27.38/8.88 "clause": -1, 27.38/8.88 "scope": -1, 27.38/8.88 "term": "(less T95 T94)" 27.38/8.88 }], 27.38/8.88 "kb": { 27.38/8.88 "nonunifying": [], 27.38/8.88 "intvars": {}, 27.38/8.88 "arithmetic": { 27.38/8.88 "type": "PlainIntegerRelationState", 27.38/8.88 "relations": [] 27.38/8.88 }, 27.38/8.88 "ground": ["T94"], 27.38/8.88 "free": [], 27.38/8.88 "exprvars": [] 27.38/8.88 } 27.38/8.88 }, 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"ground": ["T73"], 27.38/8.88 "free": [], 27.38/8.88 "exprvars": [] 27.38/8.88 } 27.38/8.88 }, 27.38/8.88 "126": { 27.38/8.88 "goal": [{ 27.38/8.88 "clause": -1, 27.38/8.88 "scope": -1, 27.38/8.88 "term": "(true)" 27.38/8.88 }], 27.38/8.88 "kb": { 27.38/8.88 "nonunifying": [], 27.38/8.88 "intvars": {}, 27.38/8.88 "arithmetic": { 27.38/8.88 "type": "PlainIntegerRelationState", 27.38/8.88 "relations": [] 27.38/8.88 }, 27.38/8.88 "ground": [], 27.38/8.88 "free": [], 27.38/8.88 "exprvars": [] 27.38/8.88 } 27.38/8.88 }, 27.38/8.88 "247": { 27.38/8.88 "goal": [{ 27.38/8.88 "clause": -1, 27.38/8.88 "scope": -1, 27.38/8.88 "term": "(in T73 T81)" 27.38/8.88 }], 27.38/8.88 "kb": { 27.38/8.88 "nonunifying": [], 27.38/8.88 "intvars": {}, 27.38/8.88 "arithmetic": { 27.38/8.88 "type": "PlainIntegerRelationState", 27.38/8.88 "relations": [] 27.38/8.88 }, 27.38/8.88 "ground": ["T73"], 27.38/8.88 "free": [], 27.38/8.88 "exprvars": [] 27.38/8.88 } 27.38/8.88 }, 27.38/8.88 "6": { 27.38/8.88 "goal": [ 27.38/8.88 { 27.38/8.88 "clause": 0, 27.38/8.88 "scope": 1, 27.38/8.88 "term": "(in T1 T2)" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "clause": 1, 27.38/8.88 "scope": 1, 27.38/8.88 "term": "(in T1 T2)" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "clause": 2, 27.38/8.88 "scope": 1, 27.38/8.88 "term": "(in T1 T2)" 27.38/8.88 } 27.38/8.88 ], 27.38/8.88 "kb": { 27.38/8.88 "nonunifying": [], 27.38/8.88 "intvars": {}, 27.38/8.88 "arithmetic": { 27.38/8.88 "type": "PlainIntegerRelationState", 27.38/8.88 "relations": [] 27.38/8.88 }, 27.38/8.88 "ground": ["T1"], 27.38/8.88 "free": [], 27.38/8.88 "exprvars": [] 27.38/8.88 } 27.38/8.88 }, 27.38/8.88 "248": { 27.38/8.88 "goal": [ 27.38/8.88 { 27.38/8.88 "clause": 3, 27.38/8.88 "scope": 3, 27.38/8.88 "term": "(less T77 T73)" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "clause": 4, 27.38/8.88 "scope": 3, 27.38/8.88 "term": "(less T77 T73)" 27.38/8.88 } 27.38/8.88 ], 27.38/8.88 "kb": { 27.38/8.88 "nonunifying": [], 27.38/8.88 "intvars": {}, 27.38/8.88 "arithmetic": { 27.38/8.88 "type": "PlainIntegerRelationState", 27.38/8.88 "relations": [] 27.38/8.88 }, 27.38/8.88 "ground": ["T73"], 27.38/8.88 "free": [], 27.38/8.88 "exprvars": [] 27.38/8.88 } 27.38/8.88 }, 27.38/8.88 "128": { 27.38/8.88 "goal": [], 27.38/8.88 "kb": { 27.38/8.88 "nonunifying": [], 27.38/8.88 "intvars": {}, 27.38/8.88 "arithmetic": { 27.38/8.88 "type": "PlainIntegerRelationState", 27.38/8.88 "relations": [] 27.38/8.88 }, 27.38/8.88 "ground": [], 27.38/8.88 "free": [], 27.38/8.88 "exprvars": [] 27.38/8.88 } 27.38/8.88 }, 27.38/8.88 "249": { 27.38/8.88 "goal": [{ 27.38/8.88 "clause": 3, 27.38/8.88 "scope": 3, 27.38/8.88 "term": "(less T77 T73)" 27.38/8.88 }], 27.38/8.88 "kb": { 27.38/8.88 "nonunifying": [], 27.38/8.88 "intvars": {}, 27.38/8.88 "arithmetic": { 27.38/8.88 "type": "PlainIntegerRelationState", 27.38/8.88 "relations": [] 27.38/8.88 }, 27.38/8.88 "ground": ["T73"], 27.38/8.88 "free": [], 27.38/8.88 "exprvars": [] 27.38/8.88 } 27.38/8.88 }, 27.38/8.88 "9": { 27.38/8.88 "goal": [{ 27.38/8.88 "clause": 0, 27.38/8.88 "scope": 1, 27.38/8.88 "term": "(in T1 T2)" 27.38/8.88 }], 27.38/8.88 "kb": { 27.38/8.88 "nonunifying": [], 27.38/8.88 "intvars": {}, 27.38/8.88 "arithmetic": { 27.38/8.88 "type": "PlainIntegerRelationState", 27.38/8.88 "relations": [] 27.38/8.88 }, 27.38/8.88 "ground": ["T1"], 27.38/8.88 "free": [], 27.38/8.88 "exprvars": [] 27.38/8.88 } 27.38/8.88 }, 27.38/8.88 "41": { 27.38/8.88 "goal": [], 27.38/8.88 "kb": { 27.38/8.88 "nonunifying": [], 27.38/8.88 "intvars": {}, 27.38/8.88 "arithmetic": { 27.38/8.88 "type": "PlainIntegerRelationState", 27.38/8.88 "relations": [] 27.38/8.88 }, 27.38/8.88 "ground": [], 27.38/8.88 "free": [], 27.38/8.88 "exprvars": [] 27.38/8.88 } 27.38/8.88 } 27.38/8.88 }, 27.38/8.88 "edges": [ 27.38/8.88 { 27.38/8.88 "from": 4, 27.38/8.88 "to": 6, 27.38/8.88 "label": "CASE" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "from": 6, 27.38/8.88 "to": 9, 27.38/8.88 "label": "PARALLEL" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "from": 6, 27.38/8.88 "to": 10, 27.38/8.88 "label": "PARALLEL" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "from": 9, 27.38/8.88 "to": 33, 27.38/8.88 "label": "EVAL with clause\nin(X18, tree(X18, X19, X20)).\nand substitutionT1 -> T15,\nX18 -> T15,\nX19 -> T16,\nX20 -> T17,\nT2 -> tree(T15, T16, T17)" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "from": 9, 27.38/8.88 "to": 34, 27.38/8.88 "label": "EVAL-BACKTRACK" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "from": 10, 27.38/8.88 "to": 36, 27.38/8.88 "label": "PARALLEL" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "from": 10, 27.38/8.88 "to": 37, 27.38/8.88 "label": "PARALLEL" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "from": 33, 27.38/8.88 "to": 35, 27.38/8.88 "label": "SUCCESS" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "from": 36, 27.38/8.88 "to": 38, 27.38/8.88 "label": "EVAL with clause\nin(X37, tree(X38, X39, X40)) :- ','(less(X37, X38), in(X37, X39)).\nand substitutionT1 -> T34,\nX37 -> T34,\nX38 -> T38,\nX39 -> T39,\nX40 -> T37,\nT2 -> tree(T38, T39, T37),\nT35 -> T38,\nT36 -> T39" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "from": 36, 27.38/8.88 "to": 41, 27.38/8.88 "label": "EVAL-BACKTRACK" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "from": 37, 27.38/8.88 "to": 244, 27.38/8.88 "label": "EVAL with clause\nin(X72, tree(X73, X74, X75)) :- ','(less(X73, X72), in(X72, X75)).\nand substitutionT1 -> T73,\nX72 -> T73,\nX73 -> T77,\nX74 -> T75,\nX75 -> T78,\nT2 -> tree(T77, T75, T78),\nT74 -> T77,\nT76 -> T78" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "from": 37, 27.38/8.88 "to": 245, 27.38/8.88 "label": "EVAL-BACKTRACK" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "from": 38, 27.38/8.88 "to": 114, 27.38/8.88 "label": "SPLIT 1" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "from": 38, 27.38/8.88 "to": 115, 27.38/8.88 "label": "SPLIT 2\nnew knowledge:\nT34 is ground\nreplacements:T39 -> T42" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "from": 114, 27.38/8.88 "to": 116, 27.38/8.88 "label": "CASE" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "from": 115, 27.38/8.88 "to": 4, 27.38/8.88 "label": "INSTANCE with matching:\nT1 -> T34\nT2 -> T42" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "from": 116, 27.38/8.88 "to": 118, 27.38/8.88 "label": "PARALLEL" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "from": 116, 27.38/8.88 "to": 119, 27.38/8.88 "label": "PARALLEL" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "from": 118, 27.38/8.88 "to": 126, 27.38/8.88 "label": "EVAL with clause\nless(0, s(X49)).\nand substitutionT34 -> 0,\nX49 -> T49,\nT38 -> s(T49)" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "from": 118, 27.38/8.88 "to": 128, 27.38/8.88 "label": "EVAL-BACKTRACK" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "from": 119, 27.38/8.88 "to": 141, 27.38/8.88 "label": "EVAL with clause\nless(s(X54), s(X55)) :- less(X54, X55).\nand substitutionX54 -> T54,\nT34 -> s(T54),\nX55 -> T56,\nT38 -> s(T56),\nT55 -> T56" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "from": 119, 27.38/8.88 "to": 142, 27.38/8.88 "label": "EVAL-BACKTRACK" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "from": 126, 27.38/8.88 "to": 130, 27.38/8.88 "label": "SUCCESS" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "from": 141, 27.38/8.88 "to": 114, 27.38/8.88 "label": "INSTANCE with matching:\nT34 -> T54\nT38 -> T56" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "from": 244, 27.38/8.88 "to": 246, 27.38/8.88 "label": "SPLIT 1" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "from": 244, 27.38/8.88 "to": 247, 27.38/8.88 "label": "SPLIT 2\nnew knowledge:\nT77 is ground\nT73 is ground\nreplacements:T78 -> T81" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "from": 246, 27.38/8.88 "to": 248, 27.38/8.88 "label": "CASE" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "from": 247, 27.38/8.88 "to": 4, 27.38/8.88 "label": "INSTANCE with matching:\nT1 -> T73\nT2 -> T81" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "from": 248, 27.38/8.88 "to": 249, 27.38/8.88 "label": "PARALLEL" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "from": 248, 27.38/8.88 "to": 250, 27.38/8.88 "label": "PARALLEL" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "from": 249, 27.38/8.88 "to": 251, 27.38/8.88 "label": "EVAL with clause\nless(0, s(X84)).\nand substitutionT77 -> 0,\nX84 -> T88,\nT73 -> s(T88)" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "from": 249, 27.38/8.88 "to": 252, 27.38/8.88 "label": "EVAL-BACKTRACK" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "from": 250, 27.38/8.88 "to": 254, 27.38/8.88 "label": "EVAL with clause\nless(s(X89), s(X90)) :- less(X89, X90).\nand substitutionX89 -> T95,\nT77 -> s(T95),\nX90 -> T94,\nT73 -> s(T94),\nT93 -> T95" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "from": 250, 27.38/8.88 "to": 255, 27.38/8.88 "label": "EVAL-BACKTRACK" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "from": 251, 27.38/8.88 "to": 253, 27.38/8.88 "label": "SUCCESS" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "from": 254, 27.38/8.88 "to": 246, 27.38/8.88 "label": "INSTANCE with matching:\nT77 -> T95\nT73 -> T94" 27.38/8.88 } 27.38/8.88 ], 27.38/8.88 "type": "Graph" 27.38/8.88 } 27.38/8.88 } 27.38/8.88 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (91) 27.38/8.88 Complex Obligation (AND) 27.38/8.88 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (92) 27.38/8.88 Obligation: 27.38/8.88 Rules: 27.38/8.88 f246_out(T94) -> f254_out(T94) :|: TRUE 27.38/8.88 f254_in(x) -> f246_in(x) :|: TRUE 27.38/8.88 f254_out(x1) -> f250_out(s(x1)) :|: TRUE 27.38/8.88 f250_in(s(x2)) -> f254_in(x2) :|: TRUE 27.38/8.88 f255_out -> f250_out(T73) :|: TRUE 27.38/8.88 f250_in(x3) -> f255_in :|: TRUE 27.38/8.88 f246_in(x4) -> f248_in(x4) :|: TRUE 27.38/8.88 f248_out(x5) -> f246_out(x5) :|: TRUE 27.38/8.88 f248_in(x6) -> f249_in(x6) :|: TRUE 27.38/8.88 f249_out(x7) -> f248_out(x7) :|: TRUE 27.38/8.88 f250_out(x8) -> f248_out(x8) :|: TRUE 27.38/8.88 f248_in(x9) -> f250_in(x9) :|: TRUE 27.38/8.88 f4_in(T1) -> f6_in(T1) :|: TRUE 27.38/8.88 f6_out(x10) -> f4_out(x10) :|: TRUE 27.38/8.88 f6_in(x11) -> f10_in(x11) :|: TRUE 27.38/8.88 f6_in(x12) -> f9_in(x12) :|: TRUE 27.38/8.88 f9_out(x13) -> f6_out(x13) :|: TRUE 27.38/8.88 f10_out(x14) -> f6_out(x14) :|: TRUE 27.38/8.88 f10_in(x15) -> f37_in(x15) :|: TRUE 27.38/8.88 f10_in(x16) -> f36_in(x16) :|: TRUE 27.38/8.88 f36_out(x17) -> f10_out(x17) :|: TRUE 27.38/8.88 f37_out(x18) -> f10_out(x18) :|: TRUE 27.38/8.88 f244_out(x19) -> f37_out(x19) :|: TRUE 27.38/8.88 f37_in(x20) -> f245_in :|: TRUE 27.38/8.88 f245_out -> f37_out(x21) :|: TRUE 27.38/8.88 f37_in(x22) -> f244_in(x22) :|: TRUE 27.38/8.88 f244_in(x23) -> f246_in(x23) :|: TRUE 27.38/8.88 f246_out(x24) -> f247_in(x24) :|: TRUE 27.38/8.88 f247_out(x25) -> f244_out(x25) :|: TRUE 27.38/8.88 Start term: f4_in(T1) 27.38/8.88 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (93) IRSwTSimpleDependencyGraphProof (EQUIVALENT) 27.38/8.88 Constructed simple dependency graph. 27.38/8.88 27.38/8.88 Simplified to the following IRSwTs: 27.38/8.88 27.38/8.88 intTRSProblem: 27.38/8.88 f254_in(x) -> f246_in(x) :|: TRUE 27.38/8.88 f250_in(s(x2)) -> f254_in(x2) :|: TRUE 27.38/8.88 f246_in(x4) -> f248_in(x4) :|: TRUE 27.38/8.88 f248_in(x9) -> f250_in(x9) :|: TRUE 27.38/8.88 27.38/8.88 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (94) 27.38/8.88 Obligation: 27.38/8.88 Rules: 27.38/8.88 f254_in(x) -> f246_in(x) :|: TRUE 27.38/8.88 f250_in(s(x2)) -> f254_in(x2) :|: TRUE 27.38/8.88 f246_in(x4) -> f248_in(x4) :|: TRUE 27.38/8.88 f248_in(x9) -> f250_in(x9) :|: TRUE 27.38/8.88 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (95) IntTRSCompressionProof (EQUIVALENT) 27.38/8.88 Compressed rules. 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (96) 27.38/8.88 Obligation: 27.38/8.88 Rules: 27.38/8.88 f250_in(s(x2:0)) -> f250_in(x2:0) :|: TRUE 27.38/8.88 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (97) IRSFormatTransformerProof (EQUIVALENT) 27.38/8.88 Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (98) 27.38/8.88 Obligation: 27.38/8.88 Rules: 27.38/8.88 f250_in(s(x2:0)) -> f250_in(x2:0) :|: TRUE 27.38/8.88 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (99) IRSwTTerminationDigraphProof (EQUIVALENT) 27.38/8.88 Constructed termination digraph! 27.38/8.88 Nodes: 27.38/8.88 (1) f250_in(s(x2:0)) -> f250_in(x2:0) :|: TRUE 27.38/8.88 27.38/8.88 Arcs: 27.38/8.88 (1) -> (1) 27.38/8.88 27.38/8.88 This digraph is fully evaluated! 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (100) 27.38/8.88 Obligation: 27.38/8.88 27.38/8.88 Termination digraph: 27.38/8.88 Nodes: 27.38/8.88 (1) f250_in(s(x2:0)) -> f250_in(x2:0) :|: TRUE 27.38/8.88 27.38/8.88 Arcs: 27.38/8.88 (1) -> (1) 27.38/8.88 27.38/8.88 This digraph is fully evaluated! 27.38/8.88 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (101) TempFilterProof (SOUND) 27.38/8.88 Used the following sort dictionary for filtering: 27.38/8.88 f250_in(VARIABLE) 27.38/8.88 s(VARIABLE) 27.38/8.88 Removed predefined arithmetic. 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (102) 27.38/8.88 Obligation: 27.38/8.88 Rules: 27.38/8.88 f250_in(s(x2:0)) -> f250_in(x2:0) 27.38/8.88 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (103) IRSwTToQDPProof (SOUND) 27.38/8.88 Removed the integers and created a QDP-Problem. 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (104) 27.38/8.88 Obligation: 27.38/8.88 Q DP problem: 27.38/8.88 The TRS P consists of the following rules: 27.38/8.88 27.38/8.88 f250_in(s(x2:0)) -> f250_in(x2:0) 27.38/8.88 27.38/8.88 R is empty. 27.38/8.88 Q is empty. 27.38/8.88 We have to consider all (P,Q,R)-chains. 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (105) QDPSizeChangeProof (EQUIVALENT) 27.38/8.88 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. 27.38/8.88 27.38/8.88 From the DPs we obtained the following set of size-change graphs: 27.38/8.88 *f250_in(s(x2:0)) -> f250_in(x2:0) 27.38/8.88 The graph contains the following edges 1 > 1 27.38/8.88 27.38/8.88 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (106) 27.38/8.88 YES 27.38/8.88 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (107) 27.38/8.88 Obligation: 27.38/8.88 Rules: 27.38/8.88 f141_out(T54) -> f119_out(s(T54)) :|: TRUE 27.38/8.88 f119_in(s(x)) -> f141_in(x) :|: TRUE 27.38/8.88 f142_out -> f119_out(T34) :|: TRUE 27.38/8.88 f119_in(x1) -> f142_in :|: TRUE 27.38/8.88 f118_out(x2) -> f116_out(x2) :|: TRUE 27.38/8.88 f119_out(x3) -> f116_out(x3) :|: TRUE 27.38/8.88 f116_in(x4) -> f118_in(x4) :|: TRUE 27.38/8.88 f116_in(x5) -> f119_in(x5) :|: TRUE 27.38/8.88 f116_out(x6) -> f114_out(x6) :|: TRUE 27.38/8.88 f114_in(x7) -> f116_in(x7) :|: TRUE 27.38/8.88 f141_in(x8) -> f114_in(x8) :|: TRUE 27.38/8.88 f114_out(x9) -> f141_out(x9) :|: TRUE 27.38/8.88 f4_in(T1) -> f6_in(T1) :|: TRUE 27.38/8.88 f6_out(x10) -> f4_out(x10) :|: TRUE 27.38/8.88 f6_in(x11) -> f10_in(x11) :|: TRUE 27.38/8.88 f6_in(x12) -> f9_in(x12) :|: TRUE 27.38/8.88 f9_out(x13) -> f6_out(x13) :|: TRUE 27.38/8.88 f10_out(x14) -> f6_out(x14) :|: TRUE 27.38/8.88 f10_in(x15) -> f37_in(x15) :|: TRUE 27.38/8.88 f10_in(x16) -> f36_in(x16) :|: TRUE 27.38/8.88 f36_out(x17) -> f10_out(x17) :|: TRUE 27.38/8.88 f37_out(x18) -> f10_out(x18) :|: TRUE 27.38/8.88 f36_in(x19) -> f38_in(x19) :|: TRUE 27.38/8.88 f36_in(x20) -> f41_in :|: TRUE 27.38/8.88 f41_out -> f36_out(x21) :|: TRUE 27.38/8.88 f38_out(x22) -> f36_out(x22) :|: TRUE 27.38/8.88 f38_in(x23) -> f114_in(x23) :|: TRUE 27.38/8.88 f115_out(x24) -> f38_out(x24) :|: TRUE 27.38/8.88 f114_out(x25) -> f115_in(x25) :|: TRUE 27.38/8.88 Start term: f4_in(T1) 27.38/8.88 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (108) IRSwTSimpleDependencyGraphProof (EQUIVALENT) 27.38/8.88 Constructed simple dependency graph. 27.38/8.88 27.38/8.88 Simplified to the following IRSwTs: 27.38/8.88 27.38/8.88 intTRSProblem: 27.38/8.88 f119_in(s(x)) -> f141_in(x) :|: TRUE 27.38/8.88 f116_in(x5) -> f119_in(x5) :|: TRUE 27.38/8.88 f114_in(x7) -> f116_in(x7) :|: TRUE 27.38/8.88 f141_in(x8) -> f114_in(x8) :|: TRUE 27.38/8.88 27.38/8.88 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (109) 27.38/8.88 Obligation: 27.38/8.88 Rules: 27.38/8.88 f119_in(s(x)) -> f141_in(x) :|: TRUE 27.38/8.88 f116_in(x5) -> f119_in(x5) :|: TRUE 27.38/8.88 f114_in(x7) -> f116_in(x7) :|: TRUE 27.38/8.88 f141_in(x8) -> f114_in(x8) :|: TRUE 27.38/8.88 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (110) IntTRSCompressionProof (EQUIVALENT) 27.38/8.88 Compressed rules. 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (111) 27.38/8.88 Obligation: 27.38/8.88 Rules: 27.38/8.88 f114_in(s(x:0)) -> f114_in(x:0) :|: TRUE 27.38/8.88 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (112) IRSFormatTransformerProof (EQUIVALENT) 27.38/8.88 Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (113) 27.38/8.88 Obligation: 27.38/8.88 Rules: 27.38/8.88 f114_in(s(x:0)) -> f114_in(x:0) :|: TRUE 27.38/8.88 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (114) IRSwTTerminationDigraphProof (EQUIVALENT) 27.38/8.88 Constructed termination digraph! 27.38/8.88 Nodes: 27.38/8.88 (1) f114_in(s(x:0)) -> f114_in(x:0) :|: TRUE 27.38/8.88 27.38/8.88 Arcs: 27.38/8.88 (1) -> (1) 27.38/8.88 27.38/8.88 This digraph is fully evaluated! 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (115) 27.38/8.88 Obligation: 27.38/8.88 27.38/8.88 Termination digraph: 27.38/8.88 Nodes: 27.38/8.88 (1) f114_in(s(x:0)) -> f114_in(x:0) :|: TRUE 27.38/8.88 27.38/8.88 Arcs: 27.38/8.88 (1) -> (1) 27.38/8.88 27.38/8.88 This digraph is fully evaluated! 27.38/8.88 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (116) TempFilterProof (SOUND) 27.38/8.88 Used the following sort dictionary for filtering: 27.38/8.88 f114_in(VARIABLE) 27.38/8.88 s(VARIABLE) 27.38/8.88 Removed predefined arithmetic. 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (117) 27.38/8.88 Obligation: 27.38/8.88 Rules: 27.38/8.88 f114_in(s(x:0)) -> f114_in(x:0) 27.38/8.88 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (118) IRSwTToQDPProof (SOUND) 27.38/8.88 Removed the integers and created a QDP-Problem. 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (119) 27.38/8.88 Obligation: 27.38/8.88 Q DP problem: 27.38/8.88 The TRS P consists of the following rules: 27.38/8.88 27.38/8.88 f114_in(s(x:0)) -> f114_in(x:0) 27.38/8.88 27.38/8.88 R is empty. 27.38/8.88 Q is empty. 27.38/8.88 We have to consider all (P,Q,R)-chains. 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (120) QDPSizeChangeProof (EQUIVALENT) 27.38/8.88 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. 27.38/8.88 27.38/8.88 From the DPs we obtained the following set of size-change graphs: 27.38/8.88 *f114_in(s(x:0)) -> f114_in(x:0) 27.38/8.88 The graph contains the following edges 1 > 1 27.38/8.88 27.38/8.88 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (121) 27.38/8.88 YES 27.38/8.88 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (122) 27.38/8.88 Obligation: 27.38/8.88 Rules: 27.38/8.88 f251_out -> f249_out(s(T88)) :|: TRUE 27.38/8.88 f249_in(T73) -> f252_in :|: TRUE 27.38/8.88 f249_in(s(x)) -> f251_in :|: TRUE 27.38/8.88 f252_out -> f249_out(x1) :|: TRUE 27.38/8.88 f116_out(T34) -> f114_out(T34) :|: TRUE 27.38/8.88 f114_in(x2) -> f116_in(x2) :|: TRUE 27.38/8.88 f141_in(T54) -> f114_in(T54) :|: TRUE 27.38/8.88 f114_out(x3) -> f141_out(x3) :|: TRUE 27.38/8.88 f248_in(x4) -> f249_in(x4) :|: TRUE 27.38/8.88 f249_out(x5) -> f248_out(x5) :|: TRUE 27.38/8.88 f250_out(x6) -> f248_out(x6) :|: TRUE 27.38/8.88 f248_in(x7) -> f250_in(x7) :|: TRUE 27.38/8.88 f115_in(x8) -> f4_in(x8) :|: TRUE 27.38/8.88 f4_out(x9) -> f115_out(x9) :|: TRUE 27.38/8.88 f4_in(T1) -> f6_in(T1) :|: TRUE 27.38/8.88 f6_out(x10) -> f4_out(x10) :|: TRUE 27.38/8.88 f244_in(x11) -> f246_in(x11) :|: TRUE 27.38/8.88 f246_out(x12) -> f247_in(x12) :|: TRUE 27.38/8.88 f247_out(x13) -> f244_out(x13) :|: TRUE 27.38/8.88 f254_out(T94) -> f250_out(s(T94)) :|: TRUE 27.38/8.88 f250_in(s(x14)) -> f254_in(x14) :|: TRUE 27.38/8.88 f255_out -> f250_out(x15) :|: TRUE 27.38/8.88 f250_in(x16) -> f255_in :|: TRUE 27.38/8.88 f118_in(x17) -> f128_in :|: TRUE 27.38/8.88 f118_in(0) -> f126_in :|: TRUE 27.38/8.88 f128_out -> f118_out(x18) :|: TRUE 27.38/8.88 f126_out -> f118_out(0) :|: TRUE 27.38/8.88 f141_out(x19) -> f119_out(s(x19)) :|: TRUE 27.38/8.88 f119_in(s(x20)) -> f141_in(x20) :|: TRUE 27.38/8.88 f142_out -> f119_out(x21) :|: TRUE 27.38/8.88 f119_in(x22) -> f142_in :|: TRUE 27.38/8.88 f38_in(x23) -> f114_in(x23) :|: TRUE 27.38/8.88 f115_out(x24) -> f38_out(x24) :|: TRUE 27.38/8.88 f114_out(x25) -> f115_in(x25) :|: TRUE 27.38/8.88 f246_out(x26) -> f254_out(x26) :|: TRUE 27.38/8.88 f254_in(x27) -> f246_in(x27) :|: TRUE 27.38/8.88 f246_in(x28) -> f248_in(x28) :|: TRUE 27.38/8.88 f248_out(x29) -> f246_out(x29) :|: TRUE 27.38/8.88 f126_in -> f126_out :|: TRUE 27.38/8.88 f251_in -> f251_out :|: TRUE 27.38/8.88 f6_in(x30) -> f10_in(x30) :|: TRUE 27.38/8.88 f6_in(x31) -> f9_in(x31) :|: TRUE 27.38/8.88 f9_out(x32) -> f6_out(x32) :|: TRUE 27.38/8.88 f10_out(x33) -> f6_out(x33) :|: TRUE 27.38/8.88 f36_in(x34) -> f38_in(x34) :|: TRUE 27.38/8.88 f36_in(x35) -> f41_in :|: TRUE 27.38/8.88 f41_out -> f36_out(x36) :|: TRUE 27.38/8.88 f38_out(x37) -> f36_out(x37) :|: TRUE 27.38/8.88 f118_out(x38) -> f116_out(x38) :|: TRUE 27.38/8.88 f119_out(x39) -> f116_out(x39) :|: TRUE 27.38/8.88 f116_in(x40) -> f118_in(x40) :|: TRUE 27.38/8.88 f116_in(x41) -> f119_in(x41) :|: TRUE 27.38/8.88 f10_in(x42) -> f37_in(x42) :|: TRUE 27.38/8.88 f10_in(x43) -> f36_in(x43) :|: TRUE 27.38/8.88 f36_out(x44) -> f10_out(x44) :|: TRUE 27.38/8.88 f37_out(x45) -> f10_out(x45) :|: TRUE 27.38/8.88 f247_in(x46) -> f4_in(x46) :|: TRUE 27.38/8.88 f4_out(x47) -> f247_out(x47) :|: TRUE 27.38/8.88 f244_out(x48) -> f37_out(x48) :|: TRUE 27.38/8.88 f37_in(x49) -> f245_in :|: TRUE 27.38/8.88 f245_out -> f37_out(x50) :|: TRUE 27.38/8.88 f37_in(x51) -> f244_in(x51) :|: TRUE 27.38/8.88 Start term: f4_in(T1) 27.38/8.88 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (123) IRSwTSimpleDependencyGraphProof (EQUIVALENT) 27.38/8.88 Constructed simple dependency graph. 27.38/8.88 27.38/8.88 Simplified to the following IRSwTs: 27.38/8.88 27.38/8.88 intTRSProblem: 27.38/8.88 f251_out -> f249_out(s(T88)) :|: TRUE 27.38/8.88 f249_in(s(x)) -> f251_in :|: TRUE 27.38/8.88 f116_out(T34) -> f114_out(T34) :|: TRUE 27.38/8.88 f114_in(x2) -> f116_in(x2) :|: TRUE 27.38/8.88 f141_in(T54) -> f114_in(T54) :|: TRUE 27.38/8.88 f114_out(x3) -> f141_out(x3) :|: TRUE 27.38/8.88 f248_in(x4) -> f249_in(x4) :|: TRUE 27.38/8.88 f249_out(x5) -> f248_out(x5) :|: TRUE 27.38/8.88 f250_out(x6) -> f248_out(x6) :|: TRUE 27.38/8.88 f248_in(x7) -> f250_in(x7) :|: TRUE 27.38/8.88 f115_in(x8) -> f4_in(x8) :|: TRUE 27.38/8.88 f4_in(T1) -> f6_in(T1) :|: TRUE 27.38/8.88 f244_in(x11) -> f246_in(x11) :|: TRUE 27.38/8.88 f246_out(x12) -> f247_in(x12) :|: TRUE 27.38/8.88 f254_out(T94) -> f250_out(s(T94)) :|: TRUE 27.38/8.88 f250_in(s(x14)) -> f254_in(x14) :|: TRUE 27.38/8.88 f118_in(0) -> f126_in :|: TRUE 27.38/8.88 f126_out -> f118_out(0) :|: TRUE 27.38/8.88 f141_out(x19) -> f119_out(s(x19)) :|: TRUE 27.38/8.88 f119_in(s(x20)) -> f141_in(x20) :|: TRUE 27.38/8.88 f38_in(x23) -> f114_in(x23) :|: TRUE 27.38/8.88 f114_out(x25) -> f115_in(x25) :|: TRUE 27.38/8.88 f246_out(x26) -> f254_out(x26) :|: TRUE 27.38/8.88 f254_in(x27) -> f246_in(x27) :|: TRUE 27.38/8.88 f246_in(x28) -> f248_in(x28) :|: TRUE 27.38/8.88 f248_out(x29) -> f246_out(x29) :|: TRUE 27.38/8.88 f126_in -> f126_out :|: TRUE 27.38/8.88 f251_in -> f251_out :|: TRUE 27.38/8.88 f6_in(x30) -> f10_in(x30) :|: TRUE 27.38/8.88 f36_in(x34) -> f38_in(x34) :|: TRUE 27.38/8.88 f118_out(x38) -> f116_out(x38) :|: TRUE 27.38/8.88 f119_out(x39) -> f116_out(x39) :|: TRUE 27.38/8.88 f116_in(x40) -> f118_in(x40) :|: TRUE 27.38/8.88 f116_in(x41) -> f119_in(x41) :|: TRUE 27.38/8.88 f10_in(x42) -> f37_in(x42) :|: TRUE 27.38/8.88 f10_in(x43) -> f36_in(x43) :|: TRUE 27.38/8.88 f247_in(x46) -> f4_in(x46) :|: TRUE 27.38/8.88 f37_in(x51) -> f244_in(x51) :|: TRUE 27.38/8.88 27.38/8.88 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (124) 27.38/8.88 Obligation: 27.38/8.88 Rules: 27.38/8.88 f251_out -> f249_out(s(T88)) :|: TRUE 27.38/8.88 f249_in(s(x)) -> f251_in :|: TRUE 27.38/8.88 f116_out(T34) -> f114_out(T34) :|: TRUE 27.38/8.88 f114_in(x2) -> f116_in(x2) :|: TRUE 27.38/8.88 f141_in(T54) -> f114_in(T54) :|: TRUE 27.38/8.88 f114_out(x3) -> f141_out(x3) :|: TRUE 27.38/8.88 f248_in(x4) -> f249_in(x4) :|: TRUE 27.38/8.88 f249_out(x5) -> f248_out(x5) :|: TRUE 27.38/8.88 f250_out(x6) -> f248_out(x6) :|: TRUE 27.38/8.88 f248_in(x7) -> f250_in(x7) :|: TRUE 27.38/8.88 f115_in(x8) -> f4_in(x8) :|: TRUE 27.38/8.88 f4_in(T1) -> f6_in(T1) :|: TRUE 27.38/8.88 f244_in(x11) -> f246_in(x11) :|: TRUE 27.38/8.88 f246_out(x12) -> f247_in(x12) :|: TRUE 27.38/8.88 f254_out(T94) -> f250_out(s(T94)) :|: TRUE 27.38/8.88 f250_in(s(x14)) -> f254_in(x14) :|: TRUE 27.38/8.88 f118_in(0) -> f126_in :|: TRUE 27.38/8.88 f126_out -> f118_out(0) :|: TRUE 27.38/8.88 f141_out(x19) -> f119_out(s(x19)) :|: TRUE 27.38/8.88 f119_in(s(x20)) -> f141_in(x20) :|: TRUE 27.38/8.88 f38_in(x23) -> f114_in(x23) :|: TRUE 27.38/8.88 f114_out(x25) -> f115_in(x25) :|: TRUE 27.38/8.88 f246_out(x26) -> f254_out(x26) :|: TRUE 27.38/8.88 f254_in(x27) -> f246_in(x27) :|: TRUE 27.38/8.88 f246_in(x28) -> f248_in(x28) :|: TRUE 27.38/8.88 f248_out(x29) -> f246_out(x29) :|: TRUE 27.38/8.88 f126_in -> f126_out :|: TRUE 27.38/8.88 f251_in -> f251_out :|: TRUE 27.38/8.88 f6_in(x30) -> f10_in(x30) :|: TRUE 27.38/8.88 f36_in(x34) -> f38_in(x34) :|: TRUE 27.38/8.88 f118_out(x38) -> f116_out(x38) :|: TRUE 27.38/8.88 f119_out(x39) -> f116_out(x39) :|: TRUE 27.38/8.88 f116_in(x40) -> f118_in(x40) :|: TRUE 27.38/8.88 f116_in(x41) -> f119_in(x41) :|: TRUE 27.38/8.88 f10_in(x42) -> f37_in(x42) :|: TRUE 27.38/8.88 f10_in(x43) -> f36_in(x43) :|: TRUE 27.38/8.88 f247_in(x46) -> f4_in(x46) :|: TRUE 27.38/8.88 f37_in(x51) -> f244_in(x51) :|: TRUE 27.38/8.88 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (125) IntTRSCompressionProof (EQUIVALENT) 27.38/8.88 Compressed rules. 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (126) 27.38/8.88 Obligation: 27.38/8.88 Rules: 27.38/8.88 f116_out(T34:0) -> f10_in(T34:0) :|: TRUE 27.38/8.88 f246_out(x26:0) -> f246_out(s(x26:0)) :|: TRUE 27.38/8.88 f248_in(s(x:0)) -> f246_out(s(T88:0)) :|: TRUE 27.38/8.88 f116_out(x) -> f116_out(s(x)) :|: TRUE 27.38/8.88 f10_in(x43:0) -> f114_in(x43:0) :|: TRUE 27.38/8.88 f248_in(s(x14:0)) -> f248_in(x14:0) :|: TRUE 27.38/8.88 f246_out(x12:0) -> f10_in(x12:0) :|: TRUE 27.38/8.88 f114_in(s(x20:0)) -> f114_in(x20:0) :|: TRUE 27.38/8.88 f114_in(cons_0) -> f116_out(0) :|: TRUE && cons_0 = 0 27.38/8.88 f10_in(x42:0) -> f248_in(x42:0) :|: TRUE 27.38/8.88 27.38/8.88 ---------------------------------------- 27.38/8.88 27.38/8.88 (127) PrologToDTProblemTransformerProof (SOUND) 27.38/8.88 Built DT problem from termination graph DT10. 27.38/8.88 27.38/8.88 { 27.38/8.88 "root": 3, 27.38/8.88 "program": { 27.38/8.88 "directives": [], 27.38/8.88 "clauses": [ 27.38/8.88 [ 27.38/8.88 "(in X (tree X X1 X2))", 27.38/8.88 null 27.38/8.88 ], 27.38/8.88 [ 27.38/8.88 "(in X (tree Y Left X3))", 27.38/8.88 "(',' (less X Y) (in X Left))" 27.38/8.88 ], 27.38/8.88 [ 27.38/8.88 "(in X (tree Y X4 Right))", 27.38/8.88 "(',' (less Y X) (in X Right))" 27.38/8.88 ], 27.38/8.88 [ 27.38/8.88 "(less (0) (s X5))", 27.38/8.88 null 27.38/8.88 ], 27.38/8.88 [ 27.38/8.88 "(less (s X) (s Y))", 27.38/8.88 "(less X Y)" 27.38/8.88 ] 27.38/8.88 ] 27.38/8.88 }, 27.38/8.88 "graph": { 27.38/8.88 "nodes": { 27.38/8.88 "type": "Nodes", 27.38/8.88 "350": { 27.38/8.88 "goal": [ 27.38/8.88 { 27.38/8.88 "clause": -1, 27.38/8.88 "scope": -1, 27.38/8.88 "term": "(',' (less T151 T155) (in T151 T156))" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "clause": 2, 27.38/8.88 "scope": 1, 27.38/8.88 "term": "(in T151 T2)" 27.38/8.88 } 27.38/8.88 ], 27.38/8.88 "kb": { 27.38/8.88 "nonunifying": [[ 27.38/8.88 "(in T151 T2)", 27.38/8.88 "(in X9 (tree X9 X10 X11))" 27.38/8.88 ]], 27.38/8.88 "intvars": {}, 27.38/8.88 "arithmetic": { 27.38/8.88 "type": "PlainIntegerRelationState", 27.38/8.88 "relations": [] 27.38/8.88 }, 27.38/8.88 "ground": ["T151"], 27.38/8.88 "free": [ 27.38/8.88 "X9", 27.38/8.88 "X10", 27.38/8.88 "X11" 27.38/8.88 ], 27.38/8.88 "exprvars": [] 27.38/8.88 } 27.38/8.88 }, 27.38/8.88 "351": { 27.38/8.88 "goal": [{ 27.38/8.88 "clause": 2, 27.38/8.88 "scope": 1, 27.38/8.88 "term": "(in T1 T2)" 27.38/8.88 }], 27.38/8.88 "kb": { 27.38/8.88 "nonunifying": [[ 27.38/8.88 "(in T1 T2)", 27.38/8.88 "(in X9 (tree X9 X10 X11))" 27.38/8.88 ]], 27.38/8.88 "intvars": {}, 27.38/8.88 "arithmetic": { 27.38/8.88 "type": "PlainIntegerRelationState", 27.38/8.88 "relations": [] 27.38/8.88 }, 27.38/8.88 "ground": ["T1"], 27.38/8.88 "free": [ 27.38/8.88 "X9", 27.38/8.88 "X10", 27.38/8.88 "X11" 27.38/8.88 ], 27.38/8.88 "exprvars": [] 27.38/8.88 } 27.38/8.88 }, 27.38/8.88 "352": { 27.38/8.88 "goal": [ 27.38/8.88 { 27.38/8.88 "clause": 3, 27.38/8.88 "scope": 6, 27.38/8.88 "term": "(',' (less T151 T155) (in T151 T156))" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "clause": 4, 27.38/8.88 "scope": 6, 27.38/8.88 "term": "(',' (less T151 T155) (in T151 T156))" 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "clause": -1, 27.38/8.88 "scope": 6, 27.38/8.88 "term": null 27.38/8.88 }, 27.38/8.88 { 27.38/8.88 "clause": 2, 27.38/8.88 "scope": 1, 27.38/8.88 "term": "(in T151 T2)" 27.38/8.88 } 27.38/8.88 ], 27.38/8.88 "kb": { 27.38/8.88 "nonunifying": [[ 27.38/8.88 "(in T151 T2)", 27.38/8.88 "(in X9 (tree X9 X10 X11))" 27.38/8.88 ]], 27.38/8.88 "intvars": {}, 27.38/8.88 "arithmetic": { 27.38/8.88 "type": "PlainIntegerRelationState", 27.38/8.88 "relations": [] 27.38/8.88 }, 27.38/8.88 "ground": ["T151"], 27.38/8.88 "free": [ 27.38/8.88 "X9", 27.38/8.88 "X10", 27.38/8.88 "X11" 27.38/8.88 ], 27.38/8.88 "exprvars": [] 27.38/8.88 } 27.38/8.88 }, 27.38/8.88 "353": { 27.38/8.88 "goal": [{ 27.38/8.88 "clause": 3, 27.38/8.88 "scope": 6, 27.38/8.88 "term": "(',' (less T151 T155) (in T151 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27.38/8.89 "relations": [] 27.38/8.89 }, 27.38/8.89 "ground": ["T13"], 27.38/8.89 "free": [], 27.38/8.89 "exprvars": [] 27.38/8.89 } 27.38/8.89 }, 27.38/8.89 "346": { 27.38/8.89 "goal": [{ 27.38/8.89 "clause": -1, 27.38/8.89 "scope": -1, 27.38/8.89 "term": "(',' (less T133 T132) (in (s T132) T134))" 27.38/8.89 }], 27.38/8.89 "kb": { 27.38/8.89 "nonunifying": [[ 27.38/8.89 "(in (s T132) T2)", 27.38/8.89 "(in X16 (tree X17 X18 X19))" 27.38/8.89 ]], 27.38/8.89 "intvars": {}, 27.38/8.89 "arithmetic": { 27.38/8.89 "type": "PlainIntegerRelationState", 27.38/8.89 "relations": [] 27.38/8.89 }, 27.38/8.89 "ground": ["T132"], 27.38/8.89 "free": [ 27.38/8.89 "X16", 27.38/8.89 "X17", 27.38/8.89 "X18", 27.38/8.89 "X19" 27.38/8.89 ], 27.38/8.89 "exprvars": [] 27.38/8.89 } 27.38/8.89 }, 27.38/8.89 "303": { 27.38/8.89 "goal": [{ 27.38/8.89 "clause": -1, 27.38/8.89 "scope": -1, 27.38/8.89 "term": "(',' (less T78 T74) (in T74 T79))" 27.38/8.89 }], 27.38/8.89 "kb": { 27.38/8.89 "nonunifying": [], 27.38/8.89 "intvars": {}, 27.38/8.89 "arithmetic": { 27.38/8.89 "type": "PlainIntegerRelationState", 27.38/8.89 "relations": [] 27.38/8.89 }, 27.38/8.89 "ground": ["T74"], 27.38/8.89 "free": [], 27.38/8.89 "exprvars": [] 27.38/8.89 } 27.38/8.89 }, 27.38/8.89 "347": { 27.38/8.89 "goal": [], 27.38/8.89 "kb": { 27.38/8.89 "nonunifying": [], 27.38/8.89 "intvars": {}, 27.38/8.89 "arithmetic": { 27.38/8.89 "type": "PlainIntegerRelationState", 27.38/8.89 "relations": [] 27.38/8.89 }, 27.38/8.89 "ground": [], 27.38/8.89 "free": [], 27.38/8.89 "exprvars": [] 27.38/8.89 } 27.38/8.89 }, 27.38/8.89 "304": { 27.38/8.89 "goal": [], 27.38/8.89 "kb": { 27.38/8.89 "nonunifying": [], 27.38/8.89 "intvars": {}, 27.38/8.89 "arithmetic": { 27.38/8.89 "type": "PlainIntegerRelationState", 27.38/8.89 "relations": [] 27.38/8.89 }, 27.38/8.89 "ground": [], 27.38/8.89 "free": [], 27.38/8.89 "exprvars": [] 27.38/8.89 } 27.38/8.89 }, 27.38/8.89 "348": { 27.38/8.89 "goal": [{ 27.38/8.89 "clause": -1, 27.38/8.89 "scope": -1, 27.38/8.89 "term": "(less T133 T132)" 27.38/8.89 }], 27.38/8.89 "kb": { 27.38/8.89 "nonunifying": [[ 27.38/8.89 "(in (s T132) T2)", 27.38/8.89 "(in X16 (tree X17 X18 X19))" 27.38/8.89 ]], 27.38/8.89 "intvars": {}, 27.38/8.89 "arithmetic": { 27.38/8.89 "type": "PlainIntegerRelationState", 27.38/8.89 "relations": [] 27.38/8.89 }, 27.38/8.89 "ground": ["T132"], 27.38/8.89 "free": [ 27.38/8.89 "X16", 27.38/8.89 "X17", 27.38/8.89 "X18", 27.38/8.89 "X19" 27.38/8.89 ], 27.38/8.89 "exprvars": [] 27.38/8.89 } 27.38/8.89 }, 27.38/8.89 "305": { 27.38/8.89 "goal": [{ 27.38/8.89 "clause": -1, 27.38/8.89 "scope": -1, 27.38/8.89 "term": "(less T78 T74)" 27.38/8.89 }], 27.38/8.89 "kb": { 27.38/8.89 "nonunifying": [], 27.38/8.89 "intvars": {}, 27.38/8.89 "arithmetic": { 27.38/8.89 "type": "PlainIntegerRelationState", 27.38/8.89 "relations": [] 27.38/8.89 }, 27.38/8.89 "ground": ["T74"], 27.38/8.89 "free": [], 27.38/8.89 "exprvars": [] 27.38/8.89 } 27.38/8.89 }, 27.38/8.89 "349": { 27.38/8.89 "goal": [{ 27.38/8.89 "clause": -1, 27.38/8.89 "scope": -1, 27.38/8.89 "term": "(in (s T132) T137)" 27.38/8.89 }], 27.38/8.89 "kb": { 27.38/8.89 "nonunifying": [[ 27.38/8.89 "(in (s T132) T138)", 27.38/8.89 "(in X16 (tree X17 X18 X19))" 27.38/8.89 ]], 27.38/8.89 "intvars": {}, 27.38/8.89 "arithmetic": { 27.38/8.89 "type": "PlainIntegerRelationState", 27.38/8.89 "relations": [] 27.38/8.89 }, 27.38/8.89 "ground": ["T132"], 27.38/8.89 "free": [ 27.38/8.89 "X16", 27.38/8.89 "X17", 27.38/8.89 "X18", 27.38/8.89 "X19" 27.38/8.89 ], 27.38/8.89 "exprvars": [] 27.38/8.89 } 27.38/8.89 }, 27.38/8.89 "306": { 27.38/8.89 "goal": [{ 27.38/8.89 "clause": -1, 27.38/8.89 "scope": -1, 27.38/8.89 "term": "(in T74 T82)" 27.38/8.89 }], 27.38/8.89 "kb": { 27.38/8.89 "nonunifying": [], 27.38/8.89 "intvars": {}, 27.38/8.89 "arithmetic": { 27.38/8.89 "type": "PlainIntegerRelationState", 27.38/8.89 "relations": [] 27.38/8.89 }, 27.38/8.89 "ground": ["T74"], 27.38/8.89 "free": [], 27.38/8.89 "exprvars": [] 27.38/8.89 } 27.38/8.89 }, 27.38/8.89 "307": { 27.38/8.89 "goal": [ 27.38/8.89 { 27.38/8.89 "clause": 3, 27.38/8.89 "scope": 4, 27.38/8.89 "term": "(less T78 T74)" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "clause": 4, 27.38/8.89 "scope": 4, 27.38/8.89 "term": "(less T78 T74)" 27.38/8.89 } 27.38/8.89 ], 27.38/8.89 "kb": { 27.38/8.89 "nonunifying": [], 27.38/8.89 "intvars": {}, 27.38/8.89 "arithmetic": { 27.38/8.89 "type": "PlainIntegerRelationState", 27.38/8.89 "relations": [] 27.38/8.89 }, 27.38/8.89 "ground": ["T74"], 27.38/8.89 "free": [], 27.38/8.89 "exprvars": [] 27.38/8.89 } 27.38/8.89 }, 27.38/8.89 "308": { 27.38/8.89 "goal": [{ 27.38/8.89 "clause": 3, 27.38/8.89 "scope": 4, 27.38/8.89 "term": "(less T78 T74)" 27.38/8.89 }], 27.38/8.89 "kb": { 27.38/8.89 "nonunifying": [], 27.38/8.89 "intvars": {}, 27.38/8.89 "arithmetic": { 27.38/8.89 "type": "PlainIntegerRelationState", 27.38/8.89 "relations": [] 27.38/8.89 }, 27.38/8.89 "ground": ["T74"], 27.38/8.89 "free": [], 27.38/8.89 "exprvars": [] 27.38/8.89 } 27.38/8.89 }, 27.38/8.89 "309": { 27.38/8.89 "goal": [{ 27.38/8.89 "clause": 4, 27.38/8.89 "scope": 4, 27.38/8.89 "term": "(less T78 T74)" 27.38/8.89 }], 27.38/8.89 "kb": { 27.38/8.89 "nonunifying": [], 27.38/8.89 "intvars": {}, 27.38/8.89 "arithmetic": { 27.38/8.89 "type": "PlainIntegerRelationState", 27.38/8.89 "relations": [] 27.38/8.89 }, 27.38/8.89 "ground": ["T74"], 27.38/8.89 "free": [], 27.38/8.89 "exprvars": [] 27.38/8.89 } 27.38/8.89 } 27.38/8.89 }, 27.38/8.89 "edges": [ 27.38/8.89 { 27.38/8.89 "from": 3, 27.38/8.89 "to": 5, 27.38/8.89 "label": "CASE" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 5, 27.38/8.89 "to": 256, 27.38/8.89 "label": "EVAL with clause\nin(X9, tree(X9, X10, X11)).\nand substitutionT1 -> T6,\nX9 -> T6,\nX10 -> T7,\nX11 -> T8,\nT2 -> tree(T6, T7, T8)" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 5, 27.38/8.89 "to": 257, 27.38/8.89 "label": "EVAL-BACKTRACK" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 256, 27.38/8.89 "to": 258, 27.38/8.89 "label": "SUCCESS" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 257, 27.38/8.89 "to": 350, 27.38/8.89 "label": "EVAL with clause\nin(X138, tree(X139, X140, X141)) :- ','(less(X138, X139), in(X138, X140)).\nand substitutionT1 -> T151,\nX138 -> T151,\nX139 -> T155,\nX140 -> T156,\nX141 -> T154,\nT2 -> tree(T155, T156, T154),\nT152 -> T155,\nT153 -> T156" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 257, 27.38/8.89 "to": 351, 27.38/8.89 "label": "EVAL-BACKTRACK" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 258, 27.38/8.89 "to": 259, 27.38/8.89 "label": "EVAL with clause\nin(X16, tree(X17, X18, X19)) :- ','(less(X16, X17), in(X16, X18)).\nand substitutionT6 -> T13,\nX16 -> T13,\nX17 -> T17,\nX18 -> T18,\nX19 -> T16,\nT2 -> tree(T17, T18, T16),\nT14 -> T17,\nT15 -> T18" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 258, 27.38/8.89 "to": 260, 27.38/8.89 "label": "EVAL-BACKTRACK" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 259, 27.38/8.89 "to": 261, 27.38/8.89 "label": "CASE" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 260, 27.38/8.89 "to": 339, 27.38/8.89 "label": "EVAL with clause\nin(X103, tree(X104, X105, X106)) :- ','(less(X104, X103), in(X103, X106)).\nand substitutionT6 -> T109,\nX103 -> T109,\nX104 -> T113,\nX105 -> T111,\nX106 -> T114,\nT2 -> tree(T113, T111, T114),\nT110 -> T113,\nT112 -> T114" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 260, 27.38/8.89 "to": 340, 27.38/8.89 "label": "EVAL-BACKTRACK" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 261, 27.38/8.89 "to": 262, 27.38/8.89 "label": "PARALLEL" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 261, 27.38/8.89 "to": 263, 27.38/8.89 "label": "PARALLEL" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 262, 27.38/8.89 "to": 278, 27.38/8.89 "label": "EVAL with clause\nless(0, s(X24)).\nand substitutionT13 -> 0,\nX24 -> T23,\nT17 -> s(T23),\nT18 -> T24" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 262, 27.38/8.89 "to": 279, 27.38/8.89 "label": "EVAL-BACKTRACK" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 263, 27.38/8.89 "to": 283, 27.38/8.89 "label": "PARALLEL" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 263, 27.38/8.89 "to": 284, 27.38/8.89 "label": "PARALLEL" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 278, 27.38/8.89 "to": 3, 27.38/8.89 "label": "INSTANCE with matching:\nT1 -> 0\nT2 -> T24" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 283, 27.38/8.89 "to": 290, 27.38/8.89 "label": "EVAL with clause\nless(s(X39), s(X40)) :- less(X39, X40).\nand substitutionX39 -> T37,\nT13 -> s(T37),\nX40 -> T39,\nT17 -> s(T39),\nT38 -> T39,\nT18 -> T40" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 283, 27.38/8.89 "to": 291, 27.38/8.89 "label": "EVAL-BACKTRACK" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 284, 27.38/8.89 "to": 302, 27.38/8.89 "label": "FAILURE" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 290, 27.38/8.89 "to": 292, 27.38/8.89 "label": "SPLIT 1" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 290, 27.38/8.89 "to": 293, 27.38/8.89 "label": "SPLIT 2\nnew knowledge:\nT37 is ground\nreplacements:T40 -> T43" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 292, 27.38/8.89 "to": 294, 27.38/8.89 "label": "CASE" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 293, 27.38/8.89 "to": 3, 27.38/8.89 "label": "INSTANCE with matching:\nT1 -> s(T37)\nT2 -> T43" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 294, 27.38/8.89 "to": 295, 27.38/8.89 "label": "PARALLEL" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 294, 27.38/8.89 "to": 296, 27.38/8.89 "label": "PARALLEL" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 295, 27.38/8.89 "to": 297, 27.38/8.89 "label": "EVAL with clause\nless(0, s(X49)).\nand substitutionT37 -> 0,\nX49 -> T50,\nT39 -> s(T50)" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 295, 27.38/8.89 "to": 298, 27.38/8.89 "label": "EVAL-BACKTRACK" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 296, 27.38/8.89 "to": 300, 27.38/8.89 "label": "EVAL with clause\nless(s(X54), s(X55)) :- less(X54, X55).\nand substitutionX54 -> T55,\nT37 -> s(T55),\nX55 -> T57,\nT39 -> s(T57),\nT56 -> T57" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 296, 27.38/8.89 "to": 301, 27.38/8.89 "label": "EVAL-BACKTRACK" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 297, 27.38/8.89 "to": 299, 27.38/8.89 "label": "SUCCESS" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 300, 27.38/8.89 "to": 292, 27.38/8.89 "label": "INSTANCE with matching:\nT37 -> T55\nT39 -> T57" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 302, 27.38/8.89 "to": 303, 27.38/8.89 "label": "EVAL with clause\nin(X72, tree(X73, X74, X75)) :- ','(less(X73, X72), in(X72, X75)).\nand substitutionT13 -> T74,\nX72 -> T74,\nX73 -> T78,\nX74 -> T76,\nX75 -> T79,\nT2 -> tree(T78, T76, T79),\nT75 -> T78,\nT77 -> T79" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 302, 27.38/8.89 "to": 304, 27.38/8.89 "label": "EVAL-BACKTRACK" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 303, 27.38/8.89 "to": 305, 27.38/8.89 "label": "SPLIT 1" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 303, 27.38/8.89 "to": 306, 27.38/8.89 "label": "SPLIT 2\nnew knowledge:\nT78 is ground\nT74 is ground\nreplacements:T79 -> T82" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 305, 27.38/8.89 "to": 307, 27.38/8.89 "label": "CASE" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 306, 27.38/8.89 "to": 3, 27.38/8.89 "label": "INSTANCE with matching:\nT1 -> T74\nT2 -> T82" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 307, 27.38/8.89 "to": 308, 27.38/8.89 "label": "PARALLEL" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 307, 27.38/8.89 "to": 309, 27.38/8.89 "label": "PARALLEL" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 308, 27.38/8.89 "to": 310, 27.38/8.89 "label": "EVAL with clause\nless(0, s(X84)).\nand substitutionT78 -> 0,\nX84 -> T89,\nT74 -> s(T89)" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 308, 27.38/8.89 "to": 311, 27.38/8.89 "label": "EVAL-BACKTRACK" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 309, 27.38/8.89 "to": 313, 27.38/8.89 "label": "EVAL with clause\nless(s(X89), s(X90)) :- less(X89, X90).\nand substitutionX89 -> T96,\nT78 -> s(T96),\nX90 -> T95,\nT74 -> s(T95),\nT94 -> T96" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 309, 27.38/8.89 "to": 314, 27.38/8.89 "label": "EVAL-BACKTRACK" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 310, 27.38/8.89 "to": 312, 27.38/8.89 "label": "SUCCESS" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 313, 27.38/8.89 "to": 305, 27.38/8.89 "label": "INSTANCE with matching:\nT78 -> T96\nT74 -> T95" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 339, 27.38/8.89 "to": 341, 27.38/8.89 "label": "CASE" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 341, 27.38/8.89 "to": 342, 27.38/8.89 "label": "PARALLEL" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 341, 27.38/8.89 "to": 343, 27.38/8.89 "label": "PARALLEL" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 342, 27.38/8.89 "to": 344, 27.38/8.89 "label": "EVAL with clause\nless(0, s(X111)).\nand substitutionT113 -> 0,\nX111 -> T119,\nT109 -> s(T119),\nT114 -> T120" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 342, 27.38/8.89 "to": 345, 27.38/8.89 "label": "EVAL-BACKTRACK" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 343, 27.38/8.89 "to": 346, 27.38/8.89 "label": "EVAL with clause\nless(s(X122), s(X123)) :- less(X122, X123).\nand substitutionX122 -> T133,\nT113 -> s(T133),\nX123 -> T132,\nT109 -> s(T132),\nT131 -> T133,\nT114 -> T134" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 343, 27.38/8.89 "to": 347, 27.38/8.89 "label": "EVAL-BACKTRACK" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 344, 27.38/8.89 "to": 3, 27.38/8.89 "label": "INSTANCE with matching:\nT1 -> s(T119)\nT2 -> T120" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 346, 27.38/8.89 "to": 348, 27.38/8.89 "label": "SPLIT 1" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 346, 27.38/8.89 "to": 349, 27.38/8.89 "label": "SPLIT 2\nnew knowledge:\nT133 is ground\nT132 is ground\nreplacements:T134 -> T137,\nT2 -> T138" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 348, 27.38/8.89 "to": 305, 27.38/8.89 "label": "INSTANCE with matching:\nT78 -> T133\nT74 -> T132" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 349, 27.38/8.89 "to": 3, 27.38/8.89 "label": "INSTANCE with matching:\nT1 -> s(T132)\nT2 -> T137" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 350, 27.38/8.89 "to": 352, 27.38/8.89 "label": "CASE" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 351, 27.38/8.89 "to": 364, 27.38/8.89 "label": "EVAL with clause\nin(X183, tree(X184, X185, X186)) :- ','(less(X184, X183), in(X183, X186)).\nand substitutionT1 -> T201,\nX183 -> T201,\nX184 -> T205,\nX185 -> T203,\nX186 -> T206,\nT2 -> tree(T205, T203, T206),\nT202 -> T205,\nT204 -> T206" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 351, 27.38/8.89 "to": 365, 27.38/8.89 "label": "EVAL-BACKTRACK" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 352, 27.38/8.89 "to": 353, 27.38/8.89 "label": "PARALLEL" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 352, 27.38/8.89 "to": 354, 27.38/8.89 "label": "PARALLEL" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 353, 27.38/8.89 "to": 355, 27.38/8.89 "label": "EVAL with clause\nless(0, s(X146)).\nand substitutionT151 -> 0,\nX146 -> T161,\nT155 -> s(T161),\nT156 -> T162" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 353, 27.38/8.89 "to": 356, 27.38/8.89 "label": "EVAL-BACKTRACK" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 354, 27.38/8.89 "to": 357, 27.38/8.89 "label": "PARALLEL" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 354, 27.38/8.89 "to": 358, 27.38/8.89 "label": "PARALLEL" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 355, 27.38/8.89 "to": 3, 27.38/8.89 "label": "INSTANCE with matching:\nT1 -> 0\nT2 -> T162" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 357, 27.38/8.89 "to": 359, 27.38/8.89 "label": "EVAL with clause\nless(s(X161), s(X162)) :- less(X161, X162).\nand substitutionX161 -> T175,\nT151 -> s(T175),\nX162 -> T177,\nT155 -> s(T177),\nT176 -> T177,\nT156 -> T178" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 357, 27.38/8.89 "to": 360, 27.38/8.89 "label": "EVAL-BACKTRACK" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 358, 27.38/8.89 "to": 361, 27.38/8.89 "label": "FAILURE" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 359, 27.38/8.89 "to": 290, 27.38/8.89 "label": "INSTANCE with matching:\nT37 -> T175\nT39 -> T177\nT40 -> T178" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 361, 27.38/8.89 "to": 362, 27.38/8.89 "label": "EVAL with clause\nin(X173, tree(X174, X175, X176)) :- ','(less(X174, X173), in(X173, X176)).\nand substitutionT151 -> T189,\nX173 -> T189,\nX174 -> T193,\nX175 -> T191,\nX176 -> T194,\nT2 -> tree(T193, T191, T194),\nT190 -> T193,\nT192 -> T194" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 361, 27.38/8.89 "to": 363, 27.38/8.89 "label": "EVAL-BACKTRACK" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 362, 27.38/8.89 "to": 303, 27.38/8.89 "label": "INSTANCE with matching:\nT78 -> T193\nT74 -> T189\nT79 -> T194" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 364, 27.38/8.89 "to": 366, 27.38/8.89 "label": "CASE" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 366, 27.38/8.89 "to": 367, 27.38/8.89 "label": "PARALLEL" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 366, 27.38/8.89 "to": 368, 27.38/8.89 "label": "PARALLEL" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 367, 27.38/8.89 "to": 369, 27.38/8.89 "label": "EVAL with clause\nless(0, s(X191)).\nand substitutionT205 -> 0,\nX191 -> T211,\nT201 -> s(T211),\nT206 -> T212" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 367, 27.38/8.89 "to": 370, 27.38/8.89 "label": "EVAL-BACKTRACK" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 368, 27.38/8.89 "to": 371, 27.38/8.89 "label": "EVAL with clause\nless(s(X202), s(X203)) :- less(X202, X203).\nand substitutionX202 -> T225,\nT205 -> s(T225),\nX203 -> T224,\nT201 -> s(T224),\nT223 -> T225,\nT206 -> T226" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 368, 27.38/8.89 "to": 372, 27.38/8.89 "label": "EVAL-BACKTRACK" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 369, 27.38/8.89 "to": 3, 27.38/8.89 "label": "INSTANCE with matching:\nT1 -> s(T211)\nT2 -> T212" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 371, 27.38/8.89 "to": 373, 27.38/8.89 "label": "SPLIT 1" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 371, 27.38/8.89 "to": 374, 27.38/8.89 "label": "SPLIT 2\nnew knowledge:\nT225 is ground\nT224 is ground\nreplacements:T226 -> T229,\nT2 -> T230" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 373, 27.38/8.89 "to": 305, 27.38/8.89 "label": "INSTANCE with matching:\nT78 -> T225\nT74 -> T224" 27.38/8.89 }, 27.38/8.89 { 27.38/8.89 "from": 374, 27.38/8.89 "to": 3, 27.38/8.89 "label": "INSTANCE with matching:\nT1 -> s(T224)\nT2 -> T229" 27.38/8.89 } 27.38/8.89 ], 27.38/8.89 "type": "Graph" 27.38/8.89 } 27.38/8.89 } 27.38/8.89 27.38/8.89 ---------------------------------------- 27.38/8.89 27.38/8.89 (128) 27.38/8.89 Obligation: 27.38/8.89 Triples: 27.38/8.89 27.38/8.89 lessE(s(X1), s(X2)) :- lessE(X1, X2). 27.38/8.89 lessD(s(X1), s(X2)) :- lessD(X1, X2). 27.38/8.89 pB(X1, X2, X3) :- lessE(X1, X2). 27.38/8.89 pB(X1, X2, X3) :- ','(lesscE(X1, X2), inA(s(X1), X3)). 27.38/8.89 pC(X1, X2, X3) :- lessD(X1, X2). 27.38/8.89 pC(X1, X2, X3) :- ','(lesscD(X1, X2), inA(X2, X3)). 27.38/8.89 inA(0, tree(s(X1), X2, X3)) :- inA(0, X2). 27.38/8.89 inA(s(X1), tree(s(X2), X3, X4)) :- pB(X1, X2, X3). 27.38/8.89 inA(X1, tree(X2, X3, X4)) :- pC(X2, X1, X4). 27.38/8.89 inA(s(X1), tree(0, X2, X3)) :- inA(s(X1), X3). 27.38/8.89 inA(s(X1), tree(s(X2), X3, X4)) :- lessD(X2, X1). 27.38/8.89 inA(s(X1), tree(s(X2), X3, X4)) :- ','(lesscD(X2, X1), inA(s(X1), X4)). 27.38/8.89 inA(0, tree(s(X1), X2, X3)) :- inA(0, X2). 27.38/8.89 inA(s(X1), tree(s(X2), X3, X4)) :- pB(X1, X2, X3). 27.38/8.89 inA(X1, tree(X2, X3, X4)) :- pC(X2, X1, X4). 27.38/8.89 inA(s(X1), tree(0, X2, X3)) :- inA(s(X1), X3). 27.38/8.89 inA(s(X1), tree(s(X2), X3, X4)) :- lessD(X2, X1). 27.38/8.89 inA(s(X1), tree(s(X2), X3, X4)) :- ','(lesscD(X2, X1), inA(s(X1), X4)). 27.38/8.89 27.38/8.89 Clauses: 27.38/8.89 27.38/8.89 incA(X1, tree(X1, X2, X3)). 27.38/8.89 incA(0, tree(s(X1), X2, X3)) :- incA(0, X2). 27.38/8.89 incA(s(X1), tree(s(X2), X3, X4)) :- qcB(X1, X2, X3). 27.38/8.89 incA(X1, tree(X2, X3, X4)) :- qcC(X2, X1, X4). 27.38/8.89 incA(s(X1), tree(0, X2, X3)) :- incA(s(X1), X3). 27.38/8.89 incA(s(X1), tree(s(X2), X3, X4)) :- ','(lesscD(X2, X1), incA(s(X1), X4)). 27.38/8.89 incA(0, tree(s(X1), X2, X3)) :- incA(0, X2). 27.38/8.89 incA(s(X1), tree(s(X2), X3, X4)) :- qcB(X1, X2, X3). 27.38/8.89 incA(X1, tree(X2, X3, X4)) :- qcC(X2, X1, X4). 27.38/8.89 incA(s(X1), tree(0, X2, X3)) :- incA(s(X1), X3). 27.38/8.89 incA(s(X1), tree(s(X2), X3, X4)) :- ','(lesscD(X2, X1), incA(s(X1), X4)). 27.38/8.89 lesscE(0, s(X1)). 27.38/8.89 lesscE(s(X1), s(X2)) :- lesscE(X1, X2). 27.38/8.89 lesscD(0, s(X1)). 27.38/8.89 lesscD(s(X1), s(X2)) :- lesscD(X1, X2). 27.38/8.89 qcB(X1, X2, X3) :- ','(lesscE(X1, X2), incA(s(X1), X3)). 27.38/8.89 qcC(X1, X2, X3) :- ','(lesscD(X1, X2), incA(X2, X3)). 27.38/8.89 27.38/8.89 Afs: 27.38/8.89 27.38/8.89 inA(x1, x2) = inA(x1) 27.38/8.89 27.38/8.89 27.38/8.89 ---------------------------------------- 27.38/8.89 27.38/8.89 (129) TriplesToPiDPProof (SOUND) 27.38/8.89 We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: 27.38/8.89 27.38/8.89 inA_in_2: (b,f) 27.38/8.89 27.38/8.89 pB_in_3: (b,f,f) 27.38/8.89 27.38/8.89 lessE_in_2: (b,f) 27.38/8.89 27.38/8.89 lesscE_in_2: (b,f) 27.38/8.89 27.38/8.89 pC_in_3: (f,b,f) 27.38/8.89 27.38/8.89 lessD_in_2: (f,b) 27.38/8.89 27.38/8.89 lesscD_in_2: (f,b) 27.38/8.89 27.38/8.89 Transforming TRIPLES into the following Term Rewriting System: 27.38/8.89 27.38/8.89 Pi DP problem: 27.38/8.89 The TRS P consists of the following rules: 27.38/8.89 27.38/8.89 INA_IN_GA(0, tree(s(X1), X2, X3)) -> U9_GA(X1, X2, X3, inA_in_ga(0, X2)) 27.38/8.89 INA_IN_GA(0, tree(s(X1), X2, X3)) -> INA_IN_GA(0, X2) 27.38/8.89 INA_IN_GA(s(X1), tree(s(X2), X3, X4)) -> U10_GA(X1, X2, X3, X4, pB_in_gaa(X1, X2, X3)) 27.38/8.89 INA_IN_GA(s(X1), tree(s(X2), X3, X4)) -> PB_IN_GAA(X1, X2, X3) 27.38/8.89 PB_IN_GAA(X1, X2, X3) -> U3_GAA(X1, X2, X3, lessE_in_ga(X1, X2)) 27.38/8.89 PB_IN_GAA(X1, X2, X3) -> LESSE_IN_GA(X1, X2) 27.38/8.89 LESSE_IN_GA(s(X1), s(X2)) -> U1_GA(X1, X2, lessE_in_ga(X1, X2)) 27.38/8.89 LESSE_IN_GA(s(X1), s(X2)) -> LESSE_IN_GA(X1, X2) 27.38/8.89 PB_IN_GAA(X1, X2, X3) -> U4_GAA(X1, X2, X3, lesscE_in_ga(X1, X2)) 27.38/8.89 U4_GAA(X1, X2, X3, lesscE_out_ga(X1, X2)) -> U5_GAA(X1, X2, X3, inA_in_ga(s(X1), X3)) 27.38/8.89 U4_GAA(X1, X2, X3, lesscE_out_ga(X1, X2)) -> INA_IN_GA(s(X1), X3) 27.38/8.89 INA_IN_GA(X1, tree(X2, X3, X4)) -> U11_GA(X1, X2, X3, X4, pC_in_aga(X2, X1, X4)) 27.38/8.89 INA_IN_GA(X1, tree(X2, X3, X4)) -> PC_IN_AGA(X2, X1, X4) 27.38/8.89 PC_IN_AGA(X1, X2, X3) -> U6_AGA(X1, X2, X3, lessD_in_ag(X1, X2)) 27.38/8.89 PC_IN_AGA(X1, X2, X3) -> LESSD_IN_AG(X1, X2) 27.38/8.89 LESSD_IN_AG(s(X1), s(X2)) -> U2_AG(X1, X2, lessD_in_ag(X1, X2)) 27.38/8.89 LESSD_IN_AG(s(X1), s(X2)) -> LESSD_IN_AG(X1, X2) 27.38/8.89 PC_IN_AGA(X1, X2, X3) -> U7_AGA(X1, X2, X3, lesscD_in_ag(X1, X2)) 27.38/8.89 U7_AGA(X1, X2, X3, lesscD_out_ag(X1, X2)) -> U8_AGA(X1, X2, X3, inA_in_ga(X2, X3)) 27.38/8.89 U7_AGA(X1, X2, X3, lesscD_out_ag(X1, X2)) -> INA_IN_GA(X2, X3) 27.38/8.89 INA_IN_GA(s(X1), tree(0, X2, X3)) -> U12_GA(X1, X2, X3, inA_in_ga(s(X1), X3)) 27.38/8.89 INA_IN_GA(s(X1), tree(0, X2, X3)) -> INA_IN_GA(s(X1), X3) 27.38/8.89 INA_IN_GA(s(X1), tree(s(X2), X3, X4)) -> U13_GA(X1, X2, X3, X4, lessD_in_ag(X2, X1)) 27.38/8.89 INA_IN_GA(s(X1), tree(s(X2), X3, X4)) -> LESSD_IN_AG(X2, X1) 27.38/8.89 INA_IN_GA(s(X1), tree(s(X2), X3, X4)) -> U14_GA(X1, X2, X3, X4, lesscD_in_ag(X2, X1)) 27.38/8.89 U14_GA(X1, X2, X3, X4, lesscD_out_ag(X2, X1)) -> U15_GA(X1, X2, X3, X4, inA_in_ga(s(X1), X4)) 27.38/8.89 U14_GA(X1, X2, X3, X4, lesscD_out_ag(X2, X1)) -> INA_IN_GA(s(X1), X4) 27.38/8.89 27.38/8.89 The TRS R consists of the following rules: 27.38/8.89 27.38/8.89 lesscE_in_ga(0, s(X1)) -> lesscE_out_ga(0, s(X1)) 27.38/8.89 lesscE_in_ga(s(X1), s(X2)) -> U23_ga(X1, X2, lesscE_in_ga(X1, X2)) 27.38/8.89 U23_ga(X1, X2, lesscE_out_ga(X1, X2)) -> lesscE_out_ga(s(X1), s(X2)) 27.38/8.89 lesscD_in_ag(0, s(X1)) -> lesscD_out_ag(0, s(X1)) 27.38/8.89 lesscD_in_ag(s(X1), s(X2)) -> U24_ag(X1, X2, lesscD_in_ag(X1, X2)) 27.38/8.89 U24_ag(X1, X2, lesscD_out_ag(X1, X2)) -> lesscD_out_ag(s(X1), s(X2)) 27.38/8.89 27.38/8.89 The argument filtering Pi contains the following mapping: 27.38/8.89 inA_in_ga(x1, x2) = inA_in_ga(x1) 27.38/8.89 27.38/8.89 0 = 0 27.38/8.89 27.38/8.89 s(x1) = s(x1) 27.38/8.89 27.38/8.89 pB_in_gaa(x1, x2, x3) = pB_in_gaa(x1) 27.38/8.89 27.38/8.89 lessE_in_ga(x1, x2) = lessE_in_ga(x1) 27.38/8.89 27.38/8.89 lesscE_in_ga(x1, x2) = lesscE_in_ga(x1) 27.38/8.89 27.38/8.89 lesscE_out_ga(x1, x2) = lesscE_out_ga(x1) 27.38/8.89 27.38/8.89 U23_ga(x1, x2, x3) = U23_ga(x1, x3) 27.38/8.89 27.38/8.89 pC_in_aga(x1, x2, x3) = pC_in_aga(x2) 27.38/8.89 27.38/8.89 lessD_in_ag(x1, x2) = lessD_in_ag(x2) 27.38/8.89 27.38/8.89 lesscD_in_ag(x1, x2) = lesscD_in_ag(x2) 27.38/8.89 27.38/8.89 lesscD_out_ag(x1, x2) = lesscD_out_ag(x1, x2) 27.38/8.89 27.38/8.89 U24_ag(x1, x2, x3) = U24_ag(x2, x3) 27.38/8.89 27.38/8.89 INA_IN_GA(x1, x2) = INA_IN_GA(x1) 27.38/8.89 27.38/8.89 U9_GA(x1, x2, x3, x4) = U9_GA(x4) 27.38/8.89 27.38/8.89 U10_GA(x1, x2, x3, x4, x5) = U10_GA(x1, x5) 27.38/8.89 27.38/8.89 PB_IN_GAA(x1, x2, x3) = PB_IN_GAA(x1) 27.38/8.89 27.38/8.89 U3_GAA(x1, x2, x3, x4) = U3_GAA(x1, x4) 27.38/8.89 27.38/8.89 LESSE_IN_GA(x1, x2) = LESSE_IN_GA(x1) 27.38/8.89 27.38/8.89 U1_GA(x1, x2, x3) = U1_GA(x1, x3) 27.38/8.89 27.38/8.89 U4_GAA(x1, x2, x3, x4) = U4_GAA(x1, x4) 27.38/8.89 27.38/8.89 U5_GAA(x1, x2, x3, x4) = U5_GAA(x1, x4) 27.38/8.89 27.38/8.89 U11_GA(x1, x2, x3, x4, x5) = U11_GA(x1, x5) 27.38/8.89 27.38/8.89 PC_IN_AGA(x1, x2, x3) = PC_IN_AGA(x2) 27.38/8.89 27.38/8.89 U6_AGA(x1, x2, x3, x4) = U6_AGA(x2, x4) 27.38/8.89 27.38/8.89 LESSD_IN_AG(x1, x2) = LESSD_IN_AG(x2) 27.38/8.89 27.38/8.89 U2_AG(x1, x2, x3) = U2_AG(x2, x3) 27.38/8.89 27.38/8.89 U7_AGA(x1, x2, x3, x4) = U7_AGA(x2, x4) 27.38/8.89 27.38/8.89 U8_AGA(x1, x2, x3, x4) = U8_AGA(x1, x2, x4) 27.38/8.89 27.38/8.89 U12_GA(x1, x2, x3, x4) = U12_GA(x1, x4) 27.38/8.89 27.38/8.89 U13_GA(x1, x2, x3, x4, x5) = U13_GA(x1, x5) 27.38/8.89 27.38/8.89 U14_GA(x1, x2, x3, x4, x5) = U14_GA(x1, x5) 27.38/8.89 27.38/8.89 U15_GA(x1, x2, x3, x4, x5) = U15_GA(x1, x5) 27.38/8.89 27.38/8.89 27.38/8.89 We have to consider all (P,R,Pi)-chains 27.38/8.89 27.38/8.89 27.38/8.89 Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES 27.38/8.89 27.38/8.89 27.38/8.89 27.38/8.89 ---------------------------------------- 27.38/8.89 27.38/8.89 (130) 27.38/8.89 Obligation: 27.38/8.89 Pi DP problem: 27.38/8.89 The TRS P consists of the following rules: 27.38/8.89 27.38/8.89 INA_IN_GA(0, tree(s(X1), X2, X3)) -> U9_GA(X1, X2, X3, inA_in_ga(0, X2)) 27.38/8.89 INA_IN_GA(0, tree(s(X1), X2, X3)) -> INA_IN_GA(0, X2) 27.38/8.89 INA_IN_GA(s(X1), tree(s(X2), X3, X4)) -> U10_GA(X1, X2, X3, X4, pB_in_gaa(X1, X2, X3)) 27.38/8.89 INA_IN_GA(s(X1), tree(s(X2), X3, X4)) -> PB_IN_GAA(X1, X2, X3) 27.38/8.89 PB_IN_GAA(X1, X2, X3) -> U3_GAA(X1, X2, X3, lessE_in_ga(X1, X2)) 27.38/8.89 PB_IN_GAA(X1, X2, X3) -> LESSE_IN_GA(X1, X2) 27.38/8.89 LESSE_IN_GA(s(X1), s(X2)) -> U1_GA(X1, X2, lessE_in_ga(X1, X2)) 27.38/8.89 LESSE_IN_GA(s(X1), s(X2)) -> LESSE_IN_GA(X1, X2) 27.38/8.89 PB_IN_GAA(X1, X2, X3) -> U4_GAA(X1, X2, X3, lesscE_in_ga(X1, X2)) 27.38/8.89 U4_GAA(X1, X2, X3, lesscE_out_ga(X1, X2)) -> U5_GAA(X1, X2, X3, inA_in_ga(s(X1), X3)) 27.38/8.89 U4_GAA(X1, X2, X3, lesscE_out_ga(X1, X2)) -> INA_IN_GA(s(X1), X3) 27.38/8.89 INA_IN_GA(X1, tree(X2, X3, X4)) -> U11_GA(X1, X2, X3, X4, pC_in_aga(X2, X1, X4)) 27.38/8.89 INA_IN_GA(X1, tree(X2, X3, X4)) -> PC_IN_AGA(X2, X1, X4) 27.38/8.89 PC_IN_AGA(X1, X2, X3) -> U6_AGA(X1, X2, X3, lessD_in_ag(X1, X2)) 27.38/8.89 PC_IN_AGA(X1, X2, X3) -> LESSD_IN_AG(X1, X2) 27.38/8.89 LESSD_IN_AG(s(X1), s(X2)) -> U2_AG(X1, X2, lessD_in_ag(X1, X2)) 27.38/8.89 LESSD_IN_AG(s(X1), s(X2)) -> LESSD_IN_AG(X1, X2) 27.38/8.89 PC_IN_AGA(X1, X2, X3) -> U7_AGA(X1, X2, X3, lesscD_in_ag(X1, X2)) 27.38/8.89 U7_AGA(X1, X2, X3, lesscD_out_ag(X1, X2)) -> U8_AGA(X1, X2, X3, inA_in_ga(X2, X3)) 27.38/8.89 U7_AGA(X1, X2, X3, lesscD_out_ag(X1, X2)) -> INA_IN_GA(X2, X3) 27.38/8.89 INA_IN_GA(s(X1), tree(0, X2, X3)) -> U12_GA(X1, X2, X3, inA_in_ga(s(X1), X3)) 27.38/8.89 INA_IN_GA(s(X1), tree(0, X2, X3)) -> INA_IN_GA(s(X1), X3) 27.38/8.89 INA_IN_GA(s(X1), tree(s(X2), X3, X4)) -> U13_GA(X1, X2, X3, X4, lessD_in_ag(X2, X1)) 27.38/8.89 INA_IN_GA(s(X1), tree(s(X2), X3, X4)) -> LESSD_IN_AG(X2, X1) 27.38/8.89 INA_IN_GA(s(X1), tree(s(X2), X3, X4)) -> U14_GA(X1, X2, X3, X4, lesscD_in_ag(X2, X1)) 27.38/8.89 U14_GA(X1, X2, X3, X4, lesscD_out_ag(X2, X1)) -> U15_GA(X1, X2, X3, X4, inA_in_ga(s(X1), X4)) 27.38/8.89 U14_GA(X1, X2, X3, X4, lesscD_out_ag(X2, X1)) -> INA_IN_GA(s(X1), X4) 27.38/8.89 27.38/8.89 The TRS R consists of the following rules: 27.38/8.89 27.38/8.89 lesscE_in_ga(0, s(X1)) -> lesscE_out_ga(0, s(X1)) 27.38/8.89 lesscE_in_ga(s(X1), s(X2)) -> U23_ga(X1, X2, lesscE_in_ga(X1, X2)) 27.38/8.89 U23_ga(X1, X2, lesscE_out_ga(X1, X2)) -> lesscE_out_ga(s(X1), s(X2)) 27.38/8.89 lesscD_in_ag(0, s(X1)) -> lesscD_out_ag(0, s(X1)) 27.38/8.89 lesscD_in_ag(s(X1), s(X2)) -> U24_ag(X1, X2, lesscD_in_ag(X1, X2)) 27.38/8.89 U24_ag(X1, X2, lesscD_out_ag(X1, X2)) -> lesscD_out_ag(s(X1), s(X2)) 27.38/8.89 27.38/8.89 The argument filtering Pi contains the following mapping: 27.38/8.89 inA_in_ga(x1, x2) = inA_in_ga(x1) 27.38/8.89 27.38/8.89 0 = 0 27.38/8.89 27.38/8.89 s(x1) = s(x1) 27.38/8.89 27.38/8.89 pB_in_gaa(x1, x2, x3) = pB_in_gaa(x1) 27.38/8.89 27.38/8.89 lessE_in_ga(x1, x2) = lessE_in_ga(x1) 27.38/8.89 27.38/8.89 lesscE_in_ga(x1, x2) = lesscE_in_ga(x1) 27.38/8.89 27.38/8.89 lesscE_out_ga(x1, x2) = lesscE_out_ga(x1) 27.38/8.89 27.38/8.89 U23_ga(x1, x2, x3) = U23_ga(x1, x3) 27.38/8.89 27.38/8.89 pC_in_aga(x1, x2, x3) = pC_in_aga(x2) 27.38/8.89 27.38/8.89 lessD_in_ag(x1, x2) = lessD_in_ag(x2) 27.38/8.89 27.38/8.89 lesscD_in_ag(x1, x2) = lesscD_in_ag(x2) 27.38/8.89 27.38/8.89 lesscD_out_ag(x1, x2) = lesscD_out_ag(x1, x2) 27.38/8.89 27.38/8.89 U24_ag(x1, x2, x3) = U24_ag(x2, x3) 27.38/8.89 27.38/8.89 INA_IN_GA(x1, x2) = INA_IN_GA(x1) 27.38/8.89 27.38/8.89 U9_GA(x1, x2, x3, x4) = U9_GA(x4) 27.38/8.89 27.38/8.89 U10_GA(x1, x2, x3, x4, x5) = U10_GA(x1, x5) 27.38/8.89 27.38/8.89 PB_IN_GAA(x1, x2, x3) = PB_IN_GAA(x1) 27.38/8.89 27.38/8.89 U3_GAA(x1, x2, x3, x4) = U3_GAA(x1, x4) 27.38/8.89 27.38/8.89 LESSE_IN_GA(x1, x2) = LESSE_IN_GA(x1) 27.38/8.89 27.38/8.89 U1_GA(x1, x2, x3) = U1_GA(x1, x3) 27.38/8.89 27.38/8.89 U4_GAA(x1, x2, x3, x4) = U4_GAA(x1, x4) 27.38/8.89 27.38/8.89 U5_GAA(x1, x2, x3, x4) = U5_GAA(x1, x4) 27.38/8.89 27.38/8.89 U11_GA(x1, x2, x3, x4, x5) = U11_GA(x1, x5) 27.38/8.89 27.38/8.89 PC_IN_AGA(x1, x2, x3) = PC_IN_AGA(x2) 27.38/8.89 27.38/8.89 U6_AGA(x1, x2, x3, x4) = U6_AGA(x2, x4) 27.38/8.89 27.38/8.89 LESSD_IN_AG(x1, x2) = LESSD_IN_AG(x2) 27.38/8.89 27.38/8.89 U2_AG(x1, x2, x3) = U2_AG(x2, x3) 27.38/8.89 27.38/8.89 U7_AGA(x1, x2, x3, x4) = U7_AGA(x2, x4) 27.38/8.89 27.38/8.89 U8_AGA(x1, x2, x3, x4) = U8_AGA(x1, x2, x4) 27.38/8.89 27.38/8.89 U12_GA(x1, x2, x3, x4) = U12_GA(x1, x4) 27.38/8.89 27.38/8.89 U13_GA(x1, x2, x3, x4, x5) = U13_GA(x1, x5) 27.38/8.89 27.38/8.89 U14_GA(x1, x2, x3, x4, x5) = U14_GA(x1, x5) 27.38/8.89 27.38/8.89 U15_GA(x1, x2, x3, x4, x5) = U15_GA(x1, x5) 27.38/8.89 27.38/8.89 27.38/8.89 We have to consider all (P,R,Pi)-chains 27.38/8.89 ---------------------------------------- 27.38/8.89 27.38/8.89 (131) DependencyGraphProof (EQUIVALENT) 27.38/8.89 The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 15 less nodes. 27.38/8.89 ---------------------------------------- 27.38/8.89 27.38/8.89 (132) 27.38/8.89 Complex Obligation (AND) 27.38/8.89 27.38/8.89 ---------------------------------------- 27.38/8.89 27.38/8.89 (133) 27.38/8.89 Obligation: 27.38/8.89 Pi DP problem: 27.38/8.89 The TRS P consists of the following rules: 27.38/8.89 27.38/8.89 LESSD_IN_AG(s(X1), s(X2)) -> LESSD_IN_AG(X1, X2) 27.38/8.89 27.38/8.89 The TRS R consists of the following rules: 27.38/8.89 27.38/8.89 lesscE_in_ga(0, s(X1)) -> lesscE_out_ga(0, s(X1)) 27.38/8.89 lesscE_in_ga(s(X1), s(X2)) -> U23_ga(X1, X2, lesscE_in_ga(X1, X2)) 27.38/8.89 U23_ga(X1, X2, lesscE_out_ga(X1, X2)) -> lesscE_out_ga(s(X1), s(X2)) 27.38/8.89 lesscD_in_ag(0, s(X1)) -> lesscD_out_ag(0, s(X1)) 27.38/8.89 lesscD_in_ag(s(X1), s(X2)) -> U24_ag(X1, X2, lesscD_in_ag(X1, X2)) 27.38/8.89 U24_ag(X1, X2, lesscD_out_ag(X1, X2)) -> lesscD_out_ag(s(X1), s(X2)) 27.38/8.89 27.38/8.89 The argument filtering Pi contains the following mapping: 27.38/8.89 0 = 0 27.38/8.89 27.38/8.89 s(x1) = s(x1) 27.38/8.89 27.38/8.89 lesscE_in_ga(x1, x2) = lesscE_in_ga(x1) 27.38/8.89 27.38/8.89 lesscE_out_ga(x1, x2) = lesscE_out_ga(x1) 27.38/8.89 27.38/8.89 U23_ga(x1, x2, x3) = U23_ga(x1, x3) 27.38/8.89 27.38/8.89 lesscD_in_ag(x1, x2) = lesscD_in_ag(x2) 27.38/8.89 27.38/8.89 lesscD_out_ag(x1, x2) = lesscD_out_ag(x1, x2) 27.38/8.89 27.38/8.89 U24_ag(x1, x2, x3) = U24_ag(x2, x3) 27.38/8.89 27.38/8.89 LESSD_IN_AG(x1, x2) = LESSD_IN_AG(x2) 27.38/8.89 27.38/8.89 27.38/8.89 We have to consider all (P,R,Pi)-chains 27.38/8.89 ---------------------------------------- 27.38/8.89 27.38/8.89 (134) UsableRulesProof (EQUIVALENT) 27.38/8.89 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 27.38/8.89 ---------------------------------------- 27.38/8.89 27.38/8.89 (135) 27.38/8.89 Obligation: 27.38/8.89 Pi DP problem: 27.38/8.89 The TRS P consists of the following rules: 27.38/8.89 27.38/8.89 LESSD_IN_AG(s(X1), s(X2)) -> LESSD_IN_AG(X1, X2) 27.38/8.89 27.38/8.89 R is empty. 27.38/8.89 The argument filtering Pi contains the following mapping: 27.38/8.89 s(x1) = s(x1) 27.38/8.89 27.38/8.89 LESSD_IN_AG(x1, x2) = LESSD_IN_AG(x2) 27.38/8.89 27.38/8.89 27.38/8.89 We have to consider all (P,R,Pi)-chains 27.38/8.89 ---------------------------------------- 27.38/8.89 27.38/8.89 (136) PiDPToQDPProof (SOUND) 27.38/8.89 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 27.38/8.89 ---------------------------------------- 27.38/8.89 27.38/8.89 (137) 27.38/8.89 Obligation: 27.38/8.89 Q DP problem: 27.38/8.89 The TRS P consists of the following rules: 27.38/8.89 27.38/8.89 LESSD_IN_AG(s(X2)) -> LESSD_IN_AG(X2) 27.38/8.89 27.38/8.89 R is empty. 27.38/8.89 Q is empty. 27.38/8.89 We have to consider all (P,Q,R)-chains. 27.38/8.89 ---------------------------------------- 27.38/8.89 27.38/8.89 (138) QDPSizeChangeProof (EQUIVALENT) 27.38/8.89 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. 27.38/8.89 27.38/8.89 From the DPs we obtained the following set of size-change graphs: 27.38/8.89 *LESSD_IN_AG(s(X2)) -> LESSD_IN_AG(X2) 27.38/8.89 The graph contains the following edges 1 > 1 27.38/8.89 27.38/8.89 27.38/8.89 ---------------------------------------- 27.38/8.89 27.38/8.89 (139) 27.38/8.89 YES 27.38/8.89 27.38/8.89 ---------------------------------------- 27.38/8.89 27.38/8.89 (140) 27.38/8.89 Obligation: 27.38/8.89 Pi DP problem: 27.38/8.89 The TRS P consists of the following rules: 27.38/8.89 27.38/8.89 LESSE_IN_GA(s(X1), s(X2)) -> LESSE_IN_GA(X1, X2) 27.38/8.89 27.38/8.89 The TRS R consists of the following rules: 27.38/8.89 27.38/8.89 lesscE_in_ga(0, s(X1)) -> lesscE_out_ga(0, s(X1)) 27.38/8.89 lesscE_in_ga(s(X1), s(X2)) -> U23_ga(X1, X2, lesscE_in_ga(X1, X2)) 27.38/8.89 U23_ga(X1, X2, lesscE_out_ga(X1, X2)) -> lesscE_out_ga(s(X1), s(X2)) 27.38/8.89 lesscD_in_ag(0, s(X1)) -> lesscD_out_ag(0, s(X1)) 27.38/8.89 lesscD_in_ag(s(X1), s(X2)) -> U24_ag(X1, X2, lesscD_in_ag(X1, X2)) 27.38/8.89 U24_ag(X1, X2, lesscD_out_ag(X1, X2)) -> lesscD_out_ag(s(X1), s(X2)) 27.38/8.89 27.38/8.89 The argument filtering Pi contains the following mapping: 27.38/8.89 0 = 0 27.38/8.89 27.38/8.89 s(x1) = s(x1) 27.38/8.89 27.38/8.89 lesscE_in_ga(x1, x2) = lesscE_in_ga(x1) 27.38/8.89 27.38/8.89 lesscE_out_ga(x1, x2) = lesscE_out_ga(x1) 27.38/8.89 27.38/8.89 U23_ga(x1, x2, x3) = U23_ga(x1, x3) 27.38/8.89 27.38/8.89 lesscD_in_ag(x1, x2) = lesscD_in_ag(x2) 27.38/8.89 27.38/8.89 lesscD_out_ag(x1, x2) = lesscD_out_ag(x1, x2) 27.38/8.89 27.38/8.89 U24_ag(x1, x2, x3) = U24_ag(x2, x3) 27.38/8.89 27.38/8.89 LESSE_IN_GA(x1, x2) = LESSE_IN_GA(x1) 27.38/8.89 27.38/8.89 27.38/8.89 We have to consider all (P,R,Pi)-chains 27.38/8.89 ---------------------------------------- 27.38/8.89 27.38/8.89 (141) UsableRulesProof (EQUIVALENT) 27.38/8.89 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 27.38/8.89 ---------------------------------------- 27.38/8.89 27.38/8.89 (142) 27.38/8.89 Obligation: 27.38/8.89 Pi DP problem: 27.38/8.89 The TRS P consists of the following rules: 27.38/8.89 27.38/8.89 LESSE_IN_GA(s(X1), s(X2)) -> LESSE_IN_GA(X1, X2) 27.38/8.89 27.38/8.89 R is empty. 27.38/8.89 The argument filtering Pi contains the following mapping: 27.38/8.89 s(x1) = s(x1) 27.38/8.89 27.38/8.89 LESSE_IN_GA(x1, x2) = LESSE_IN_GA(x1) 27.38/8.89 27.38/8.89 27.38/8.89 We have to consider all (P,R,Pi)-chains 27.38/8.89 ---------------------------------------- 27.38/8.89 27.38/8.89 (143) PiDPToQDPProof (SOUND) 27.38/8.89 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 27.38/8.89 ---------------------------------------- 27.38/8.89 27.38/8.89 (144) 27.38/8.89 Obligation: 27.38/8.89 Q DP problem: 27.38/8.89 The TRS P consists of the following rules: 27.38/8.89 27.38/8.89 LESSE_IN_GA(s(X1)) -> LESSE_IN_GA(X1) 27.38/8.89 27.38/8.89 R is empty. 27.38/8.89 Q is empty. 27.38/8.89 We have to consider all (P,Q,R)-chains. 27.38/8.89 ---------------------------------------- 27.38/8.89 27.38/8.89 (145) QDPSizeChangeProof (EQUIVALENT) 27.38/8.89 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. 27.38/8.89 27.38/8.89 From the DPs we obtained the following set of size-change graphs: 27.38/8.89 *LESSE_IN_GA(s(X1)) -> LESSE_IN_GA(X1) 27.38/8.89 The graph contains the following edges 1 > 1 27.38/8.89 27.38/8.89 27.38/8.89 ---------------------------------------- 27.38/8.89 27.38/8.89 (146) 27.38/8.89 YES 27.38/8.89 27.38/8.89 ---------------------------------------- 27.38/8.89 27.38/8.89 (147) 27.38/8.89 Obligation: 27.38/8.89 Pi DP problem: 27.38/8.89 The TRS P consists of the following rules: 27.38/8.89 27.38/8.89 INA_IN_GA(X1, tree(X2, X3, X4)) -> PC_IN_AGA(X2, X1, X4) 27.38/8.89 PC_IN_AGA(X1, X2, X3) -> U7_AGA(X1, X2, X3, lesscD_in_ag(X1, X2)) 27.38/8.89 U7_AGA(X1, X2, X3, lesscD_out_ag(X1, X2)) -> INA_IN_GA(X2, X3) 27.38/8.89 INA_IN_GA(0, tree(s(X1), X2, X3)) -> INA_IN_GA(0, X2) 27.38/8.89 INA_IN_GA(s(X1), tree(s(X2), X3, X4)) -> PB_IN_GAA(X1, X2, X3) 27.38/8.89 PB_IN_GAA(X1, X2, X3) -> U4_GAA(X1, X2, X3, lesscE_in_ga(X1, X2)) 27.38/8.89 U4_GAA(X1, X2, X3, lesscE_out_ga(X1, X2)) -> INA_IN_GA(s(X1), X3) 27.38/8.89 INA_IN_GA(s(X1), tree(0, X2, X3)) -> INA_IN_GA(s(X1), X3) 27.38/8.89 INA_IN_GA(s(X1), tree(s(X2), X3, X4)) -> U14_GA(X1, X2, X3, X4, lesscD_in_ag(X2, X1)) 27.38/8.89 U14_GA(X1, X2, X3, X4, lesscD_out_ag(X2, X1)) -> INA_IN_GA(s(X1), X4) 27.38/8.89 27.38/8.89 The TRS R consists of the following rules: 27.38/8.89 27.38/8.89 lesscE_in_ga(0, s(X1)) -> lesscE_out_ga(0, s(X1)) 27.38/8.89 lesscE_in_ga(s(X1), s(X2)) -> U23_ga(X1, X2, lesscE_in_ga(X1, X2)) 27.38/8.89 U23_ga(X1, X2, lesscE_out_ga(X1, X2)) -> lesscE_out_ga(s(X1), s(X2)) 27.38/8.89 lesscD_in_ag(0, s(X1)) -> lesscD_out_ag(0, s(X1)) 27.38/8.89 lesscD_in_ag(s(X1), s(X2)) -> U24_ag(X1, X2, lesscD_in_ag(X1, X2)) 27.38/8.89 U24_ag(X1, X2, lesscD_out_ag(X1, X2)) -> lesscD_out_ag(s(X1), s(X2)) 27.38/8.89 27.38/8.89 The argument filtering Pi contains the following mapping: 27.38/8.89 0 = 0 27.38/8.89 27.38/8.89 s(x1) = s(x1) 27.38/8.89 27.38/8.89 lesscE_in_ga(x1, x2) = lesscE_in_ga(x1) 27.38/8.89 27.38/8.89 lesscE_out_ga(x1, x2) = lesscE_out_ga(x1) 27.38/8.89 27.38/8.89 U23_ga(x1, x2, x3) = U23_ga(x1, x3) 27.38/8.89 27.38/8.89 lesscD_in_ag(x1, x2) = lesscD_in_ag(x2) 27.38/8.89 27.38/8.89 lesscD_out_ag(x1, x2) = lesscD_out_ag(x1, x2) 27.38/8.89 27.38/8.89 U24_ag(x1, x2, x3) = U24_ag(x2, x3) 27.38/8.89 27.38/8.89 INA_IN_GA(x1, x2) = INA_IN_GA(x1) 27.38/8.89 27.38/8.89 PB_IN_GAA(x1, x2, x3) = PB_IN_GAA(x1) 27.38/8.89 27.38/8.89 U4_GAA(x1, x2, x3, x4) = U4_GAA(x1, x4) 27.38/8.89 27.38/8.89 PC_IN_AGA(x1, x2, x3) = PC_IN_AGA(x2) 27.38/8.89 27.38/8.89 U7_AGA(x1, x2, x3, x4) = U7_AGA(x2, x4) 27.38/8.89 27.38/8.89 U14_GA(x1, x2, x3, x4, x5) = U14_GA(x1, x5) 27.38/8.89 27.38/8.89 27.38/8.89 We have to consider all (P,R,Pi)-chains 27.38/8.89 ---------------------------------------- 27.38/8.89 27.38/8.89 (148) PiDPToQDPProof (SOUND) 27.38/8.89 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 27.38/8.90 ---------------------------------------- 27.38/8.90 27.38/8.90 (149) 27.38/8.90 Obligation: 27.38/8.90 Q DP problem: 27.38/8.90 The TRS P consists of the following rules: 27.38/8.90 27.38/8.90 INA_IN_GA(X1) -> PC_IN_AGA(X1) 27.38/8.90 PC_IN_AGA(X2) -> U7_AGA(X2, lesscD_in_ag(X2)) 27.38/8.90 U7_AGA(X2, lesscD_out_ag(X1, X2)) -> INA_IN_GA(X2) 27.38/8.90 INA_IN_GA(0) -> INA_IN_GA(0) 27.38/8.90 INA_IN_GA(s(X1)) -> PB_IN_GAA(X1) 27.38/8.90 PB_IN_GAA(X1) -> U4_GAA(X1, lesscE_in_ga(X1)) 27.38/8.90 U4_GAA(X1, lesscE_out_ga(X1)) -> INA_IN_GA(s(X1)) 27.38/8.90 INA_IN_GA(s(X1)) -> INA_IN_GA(s(X1)) 27.38/8.90 INA_IN_GA(s(X1)) -> U14_GA(X1, lesscD_in_ag(X1)) 27.38/8.90 U14_GA(X1, lesscD_out_ag(X2, X1)) -> INA_IN_GA(s(X1)) 27.38/8.90 27.38/8.90 The TRS R consists of the following rules: 27.38/8.90 27.38/8.90 lesscE_in_ga(0) -> lesscE_out_ga(0) 27.38/8.90 lesscE_in_ga(s(X1)) -> U23_ga(X1, lesscE_in_ga(X1)) 27.38/8.90 U23_ga(X1, lesscE_out_ga(X1)) -> lesscE_out_ga(s(X1)) 27.38/8.90 lesscD_in_ag(s(X1)) -> lesscD_out_ag(0, s(X1)) 27.38/8.90 lesscD_in_ag(s(X2)) -> U24_ag(X2, lesscD_in_ag(X2)) 27.38/8.90 U24_ag(X2, lesscD_out_ag(X1, X2)) -> lesscD_out_ag(s(X1), s(X2)) 27.38/8.90 27.38/8.90 The set Q consists of the following terms: 27.38/8.90 27.38/8.90 lesscE_in_ga(x0) 27.38/8.90 U23_ga(x0, x1) 27.38/8.90 lesscD_in_ag(x0) 27.38/8.90 U24_ag(x0, x1) 27.38/8.90 27.38/8.90 We have to consider all (P,Q,R)-chains. 27.38/8.90 ---------------------------------------- 27.38/8.90 27.38/8.90 (150) TransformationProof (SOUND) 27.38/8.90 By narrowing [LPAR04] the rule PC_IN_AGA(X2) -> U7_AGA(X2, lesscD_in_ag(X2)) at position [1] we obtained the following new rules [LPAR04]: 27.38/8.90 27.38/8.90 (PC_IN_AGA(s(x0)) -> U7_AGA(s(x0), lesscD_out_ag(0, s(x0))),PC_IN_AGA(s(x0)) -> U7_AGA(s(x0), lesscD_out_ag(0, s(x0)))) 27.38/8.90 (PC_IN_AGA(s(x0)) -> U7_AGA(s(x0), U24_ag(x0, lesscD_in_ag(x0))),PC_IN_AGA(s(x0)) -> U7_AGA(s(x0), U24_ag(x0, lesscD_in_ag(x0)))) 27.38/8.90 27.38/8.90 27.38/8.90 ---------------------------------------- 27.38/8.90 27.38/8.90 (151) 27.38/8.90 Obligation: 27.38/8.90 Q DP problem: 27.38/8.90 The TRS P consists of the following rules: 27.38/8.90 27.38/8.90 INA_IN_GA(X1) -> PC_IN_AGA(X1) 27.38/8.90 U7_AGA(X2, lesscD_out_ag(X1, X2)) -> INA_IN_GA(X2) 27.38/8.90 INA_IN_GA(0) -> INA_IN_GA(0) 27.38/8.90 INA_IN_GA(s(X1)) -> PB_IN_GAA(X1) 27.38/8.90 PB_IN_GAA(X1) -> U4_GAA(X1, lesscE_in_ga(X1)) 27.38/8.90 U4_GAA(X1, lesscE_out_ga(X1)) -> INA_IN_GA(s(X1)) 27.38/8.90 INA_IN_GA(s(X1)) -> INA_IN_GA(s(X1)) 27.38/8.90 INA_IN_GA(s(X1)) -> U14_GA(X1, lesscD_in_ag(X1)) 27.38/8.90 U14_GA(X1, lesscD_out_ag(X2, X1)) -> INA_IN_GA(s(X1)) 27.38/8.90 PC_IN_AGA(s(x0)) -> U7_AGA(s(x0), lesscD_out_ag(0, s(x0))) 27.38/8.90 PC_IN_AGA(s(x0)) -> U7_AGA(s(x0), U24_ag(x0, lesscD_in_ag(x0))) 27.38/8.90 27.38/8.90 The TRS R consists of the following rules: 27.38/8.90 27.38/8.90 lesscE_in_ga(0) -> lesscE_out_ga(0) 27.38/8.90 lesscE_in_ga(s(X1)) -> U23_ga(X1, lesscE_in_ga(X1)) 27.38/8.90 U23_ga(X1, lesscE_out_ga(X1)) -> lesscE_out_ga(s(X1)) 27.38/8.90 lesscD_in_ag(s(X1)) -> lesscD_out_ag(0, s(X1)) 27.38/8.90 lesscD_in_ag(s(X2)) -> U24_ag(X2, lesscD_in_ag(X2)) 27.38/8.90 U24_ag(X2, lesscD_out_ag(X1, X2)) -> lesscD_out_ag(s(X1), s(X2)) 27.38/8.90 27.38/8.90 The set Q consists of the following terms: 27.38/8.90 27.38/8.90 lesscE_in_ga(x0) 27.38/8.90 U23_ga(x0, x1) 27.38/8.90 lesscD_in_ag(x0) 27.38/8.90 U24_ag(x0, x1) 27.38/8.90 27.38/8.90 We have to consider all (P,Q,R)-chains. 27.38/8.90 ---------------------------------------- 27.38/8.90 27.38/8.90 (152) TransformationProof (SOUND) 27.38/8.90 By narrowing [LPAR04] the rule PB_IN_GAA(X1) -> U4_GAA(X1, lesscE_in_ga(X1)) at position [1] we obtained the following new rules [LPAR04]: 27.38/8.90 27.38/8.90 (PB_IN_GAA(0) -> U4_GAA(0, lesscE_out_ga(0)),PB_IN_GAA(0) -> U4_GAA(0, lesscE_out_ga(0))) 27.38/8.90 (PB_IN_GAA(s(x0)) -> U4_GAA(s(x0), U23_ga(x0, lesscE_in_ga(x0))),PB_IN_GAA(s(x0)) -> U4_GAA(s(x0), U23_ga(x0, lesscE_in_ga(x0)))) 27.38/8.90 27.38/8.90 27.38/8.90 ---------------------------------------- 27.38/8.90 27.38/8.90 (153) 27.38/8.90 Obligation: 27.38/8.90 Q DP problem: 27.38/8.90 The TRS P consists of the following rules: 27.38/8.90 27.38/8.90 INA_IN_GA(X1) -> PC_IN_AGA(X1) 27.38/8.90 U7_AGA(X2, lesscD_out_ag(X1, X2)) -> INA_IN_GA(X2) 27.38/8.90 INA_IN_GA(0) -> INA_IN_GA(0) 27.38/8.90 INA_IN_GA(s(X1)) -> PB_IN_GAA(X1) 27.38/8.90 U4_GAA(X1, lesscE_out_ga(X1)) -> INA_IN_GA(s(X1)) 27.38/8.90 INA_IN_GA(s(X1)) -> INA_IN_GA(s(X1)) 27.38/8.90 INA_IN_GA(s(X1)) -> U14_GA(X1, lesscD_in_ag(X1)) 27.38/8.90 U14_GA(X1, lesscD_out_ag(X2, X1)) -> INA_IN_GA(s(X1)) 27.38/8.90 PC_IN_AGA(s(x0)) -> U7_AGA(s(x0), lesscD_out_ag(0, s(x0))) 27.38/8.90 PC_IN_AGA(s(x0)) -> U7_AGA(s(x0), U24_ag(x0, lesscD_in_ag(x0))) 27.38/8.90 PB_IN_GAA(0) -> U4_GAA(0, lesscE_out_ga(0)) 27.38/8.90 PB_IN_GAA(s(x0)) -> U4_GAA(s(x0), U23_ga(x0, lesscE_in_ga(x0))) 27.38/8.90 27.38/8.90 The TRS R consists of the following rules: 27.38/8.90 27.38/8.90 lesscE_in_ga(0) -> lesscE_out_ga(0) 27.38/8.90 lesscE_in_ga(s(X1)) -> U23_ga(X1, lesscE_in_ga(X1)) 27.38/8.90 U23_ga(X1, lesscE_out_ga(X1)) -> lesscE_out_ga(s(X1)) 27.38/8.90 lesscD_in_ag(s(X1)) -> lesscD_out_ag(0, s(X1)) 27.38/8.90 lesscD_in_ag(s(X2)) -> U24_ag(X2, lesscD_in_ag(X2)) 27.38/8.90 U24_ag(X2, lesscD_out_ag(X1, X2)) -> lesscD_out_ag(s(X1), s(X2)) 27.38/8.90 27.38/8.90 The set Q consists of the following terms: 27.38/8.90 27.38/8.90 lesscE_in_ga(x0) 27.38/8.90 U23_ga(x0, x1) 27.38/8.90 lesscD_in_ag(x0) 27.38/8.90 U24_ag(x0, x1) 27.38/8.90 27.38/8.90 We have to consider all (P,Q,R)-chains. 27.38/8.90 ---------------------------------------- 27.38/8.90 27.38/8.90 (154) TransformationProof (SOUND) 27.38/8.90 By narrowing [LPAR04] the rule INA_IN_GA(s(X1)) -> U14_GA(X1, lesscD_in_ag(X1)) at position [1] we obtained the following new rules [LPAR04]: 27.38/8.90 27.38/8.90 (INA_IN_GA(s(s(x0))) -> U14_GA(s(x0), lesscD_out_ag(0, s(x0))),INA_IN_GA(s(s(x0))) -> U14_GA(s(x0), lesscD_out_ag(0, s(x0)))) 27.38/8.90 (INA_IN_GA(s(s(x0))) -> U14_GA(s(x0), U24_ag(x0, lesscD_in_ag(x0))),INA_IN_GA(s(s(x0))) -> U14_GA(s(x0), U24_ag(x0, lesscD_in_ag(x0)))) 27.38/8.90 27.38/8.90 27.38/8.90 ---------------------------------------- 27.38/8.90 27.38/8.90 (155) 27.38/8.90 Obligation: 27.38/8.90 Q DP problem: 27.38/8.90 The TRS P consists of the following rules: 27.38/8.90 27.38/8.90 INA_IN_GA(X1) -> PC_IN_AGA(X1) 27.38/8.90 U7_AGA(X2, lesscD_out_ag(X1, X2)) -> INA_IN_GA(X2) 27.38/8.90 INA_IN_GA(0) -> INA_IN_GA(0) 27.38/8.90 INA_IN_GA(s(X1)) -> PB_IN_GAA(X1) 27.38/8.90 U4_GAA(X1, lesscE_out_ga(X1)) -> INA_IN_GA(s(X1)) 27.38/8.90 INA_IN_GA(s(X1)) -> INA_IN_GA(s(X1)) 27.38/8.90 U14_GA(X1, lesscD_out_ag(X2, X1)) -> INA_IN_GA(s(X1)) 27.38/8.90 PC_IN_AGA(s(x0)) -> U7_AGA(s(x0), lesscD_out_ag(0, s(x0))) 27.38/8.90 PC_IN_AGA(s(x0)) -> U7_AGA(s(x0), U24_ag(x0, lesscD_in_ag(x0))) 27.38/8.90 PB_IN_GAA(0) -> U4_GAA(0, lesscE_out_ga(0)) 27.38/8.90 PB_IN_GAA(s(x0)) -> U4_GAA(s(x0), U23_ga(x0, lesscE_in_ga(x0))) 27.38/8.90 INA_IN_GA(s(s(x0))) -> U14_GA(s(x0), lesscD_out_ag(0, s(x0))) 27.38/8.90 INA_IN_GA(s(s(x0))) -> U14_GA(s(x0), U24_ag(x0, lesscD_in_ag(x0))) 27.38/8.90 27.38/8.90 The TRS R consists of the following rules: 27.38/8.90 27.38/8.90 lesscE_in_ga(0) -> lesscE_out_ga(0) 27.38/8.90 lesscE_in_ga(s(X1)) -> U23_ga(X1, lesscE_in_ga(X1)) 27.38/8.90 U23_ga(X1, lesscE_out_ga(X1)) -> lesscE_out_ga(s(X1)) 27.38/8.90 lesscD_in_ag(s(X1)) -> lesscD_out_ag(0, s(X1)) 27.38/8.90 lesscD_in_ag(s(X2)) -> U24_ag(X2, lesscD_in_ag(X2)) 27.38/8.90 U24_ag(X2, lesscD_out_ag(X1, X2)) -> lesscD_out_ag(s(X1), s(X2)) 27.38/8.90 27.38/8.90 The set Q consists of the following terms: 27.38/8.90 27.38/8.90 lesscE_in_ga(x0) 27.38/8.90 U23_ga(x0, x1) 27.38/8.90 lesscD_in_ag(x0) 27.38/8.90 U24_ag(x0, x1) 27.38/8.90 27.38/8.90 We have to consider all (P,Q,R)-chains. 27.38/8.90 ---------------------------------------- 27.38/8.90 27.38/8.90 (156) TransformationProof (EQUIVALENT) 27.38/8.90 By instantiating [LPAR04] the rule U7_AGA(X2, lesscD_out_ag(X1, X2)) -> INA_IN_GA(X2) we obtained the following new rules [LPAR04]: 27.38/8.90 27.38/8.90 (U7_AGA(s(z0), lesscD_out_ag(0, s(z0))) -> INA_IN_GA(s(z0)),U7_AGA(s(z0), lesscD_out_ag(0, s(z0))) -> INA_IN_GA(s(z0))) 27.38/8.90 (U7_AGA(s(z0), lesscD_out_ag(x1, s(z0))) -> INA_IN_GA(s(z0)),U7_AGA(s(z0), lesscD_out_ag(x1, s(z0))) -> INA_IN_GA(s(z0))) 27.38/8.90 27.38/8.90 27.38/8.90 ---------------------------------------- 27.38/8.90 27.38/8.90 (157) 27.38/8.90 Obligation: 27.38/8.90 Q DP problem: 27.38/8.90 The TRS P consists of the following rules: 27.38/8.90 27.38/8.90 INA_IN_GA(X1) -> PC_IN_AGA(X1) 27.38/8.90 INA_IN_GA(0) -> INA_IN_GA(0) 27.38/8.90 INA_IN_GA(s(X1)) -> PB_IN_GAA(X1) 27.38/8.90 U4_GAA(X1, lesscE_out_ga(X1)) -> INA_IN_GA(s(X1)) 27.38/8.90 INA_IN_GA(s(X1)) -> INA_IN_GA(s(X1)) 27.38/8.90 U14_GA(X1, lesscD_out_ag(X2, X1)) -> INA_IN_GA(s(X1)) 27.38/8.90 PC_IN_AGA(s(x0)) -> U7_AGA(s(x0), lesscD_out_ag(0, s(x0))) 27.38/8.90 PC_IN_AGA(s(x0)) -> U7_AGA(s(x0), U24_ag(x0, lesscD_in_ag(x0))) 27.38/8.90 PB_IN_GAA(0) -> U4_GAA(0, lesscE_out_ga(0)) 27.38/8.90 PB_IN_GAA(s(x0)) -> U4_GAA(s(x0), U23_ga(x0, lesscE_in_ga(x0))) 27.38/8.90 INA_IN_GA(s(s(x0))) -> U14_GA(s(x0), lesscD_out_ag(0, s(x0))) 27.38/8.90 INA_IN_GA(s(s(x0))) -> U14_GA(s(x0), U24_ag(x0, lesscD_in_ag(x0))) 27.38/8.90 U7_AGA(s(z0), lesscD_out_ag(0, s(z0))) -> INA_IN_GA(s(z0)) 27.38/8.90 U7_AGA(s(z0), lesscD_out_ag(x1, s(z0))) -> INA_IN_GA(s(z0)) 27.38/8.90 27.38/8.90 The TRS R consists of the following rules: 27.38/8.90 27.38/8.90 lesscE_in_ga(0) -> lesscE_out_ga(0) 27.38/8.90 lesscE_in_ga(s(X1)) -> U23_ga(X1, lesscE_in_ga(X1)) 27.38/8.90 U23_ga(X1, lesscE_out_ga(X1)) -> lesscE_out_ga(s(X1)) 27.38/8.90 lesscD_in_ag(s(X1)) -> lesscD_out_ag(0, s(X1)) 27.38/8.90 lesscD_in_ag(s(X2)) -> U24_ag(X2, lesscD_in_ag(X2)) 27.38/8.90 U24_ag(X2, lesscD_out_ag(X1, X2)) -> lesscD_out_ag(s(X1), s(X2)) 27.38/8.90 27.38/8.90 The set Q consists of the following terms: 27.38/8.90 27.38/8.90 lesscE_in_ga(x0) 27.38/8.90 U23_ga(x0, x1) 27.38/8.90 lesscD_in_ag(x0) 27.38/8.90 U24_ag(x0, x1) 27.38/8.90 27.38/8.90 We have to consider all (P,Q,R)-chains. 27.38/8.90 ---------------------------------------- 27.38/8.90 27.38/8.90 (158) DependencyGraphProof (EQUIVALENT) 27.38/8.90 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs. 27.38/8.90 ---------------------------------------- 27.38/8.90 27.38/8.90 (159) 27.38/8.90 Complex Obligation (AND) 27.38/8.90 27.38/8.90 ---------------------------------------- 27.38/8.90 27.38/8.90 (160) 27.38/8.90 Obligation: 27.38/8.90 Q DP problem: 27.38/8.90 The TRS P consists of the following rules: 27.38/8.90 27.38/8.90 PC_IN_AGA(s(x0)) -> U7_AGA(s(x0), lesscD_out_ag(0, s(x0))) 27.38/8.90 U7_AGA(s(z0), lesscD_out_ag(0, s(z0))) -> INA_IN_GA(s(z0)) 27.38/8.90 INA_IN_GA(X1) -> PC_IN_AGA(X1) 27.38/8.90 PC_IN_AGA(s(x0)) -> U7_AGA(s(x0), U24_ag(x0, lesscD_in_ag(x0))) 27.38/8.90 U7_AGA(s(z0), lesscD_out_ag(x1, s(z0))) -> INA_IN_GA(s(z0)) 27.38/8.90 INA_IN_GA(s(X1)) -> PB_IN_GAA(X1) 27.38/8.90 PB_IN_GAA(0) -> U4_GAA(0, lesscE_out_ga(0)) 27.38/8.90 U4_GAA(X1, lesscE_out_ga(X1)) -> INA_IN_GA(s(X1)) 27.38/8.90 INA_IN_GA(s(X1)) -> INA_IN_GA(s(X1)) 27.38/8.90 INA_IN_GA(s(s(x0))) -> U14_GA(s(x0), lesscD_out_ag(0, s(x0))) 27.38/8.90 U14_GA(X1, lesscD_out_ag(X2, X1)) -> INA_IN_GA(s(X1)) 27.38/8.90 INA_IN_GA(s(s(x0))) -> U14_GA(s(x0), U24_ag(x0, lesscD_in_ag(x0))) 27.38/8.90 PB_IN_GAA(s(x0)) -> U4_GAA(s(x0), U23_ga(x0, lesscE_in_ga(x0))) 27.38/8.90 27.38/8.90 The TRS R consists of the following rules: 27.38/8.90 27.38/8.90 lesscE_in_ga(0) -> lesscE_out_ga(0) 27.38/8.90 lesscE_in_ga(s(X1)) -> U23_ga(X1, lesscE_in_ga(X1)) 27.38/8.90 U23_ga(X1, lesscE_out_ga(X1)) -> lesscE_out_ga(s(X1)) 27.38/8.90 lesscD_in_ag(s(X1)) -> lesscD_out_ag(0, s(X1)) 27.38/8.90 lesscD_in_ag(s(X2)) -> U24_ag(X2, lesscD_in_ag(X2)) 27.38/8.90 U24_ag(X2, lesscD_out_ag(X1, X2)) -> lesscD_out_ag(s(X1), s(X2)) 27.38/8.90 27.38/8.90 The set Q consists of the following terms: 27.38/8.90 27.38/8.90 lesscE_in_ga(x0) 27.38/8.90 U23_ga(x0, x1) 27.38/8.90 lesscD_in_ag(x0) 27.38/8.90 U24_ag(x0, x1) 27.38/8.90 27.38/8.90 We have to consider all (P,Q,R)-chains. 27.38/8.90 ---------------------------------------- 27.38/8.90 27.38/8.90 (161) 27.38/8.90 Obligation: 27.38/8.90 Q DP problem: 27.38/8.90 The TRS P consists of the following rules: 27.38/8.90 27.38/8.90 INA_IN_GA(0) -> INA_IN_GA(0) 27.38/8.90 27.38/8.90 The TRS R consists of the following rules: 27.38/8.90 27.38/8.90 lesscE_in_ga(0) -> lesscE_out_ga(0) 27.38/8.90 lesscE_in_ga(s(X1)) -> U23_ga(X1, lesscE_in_ga(X1)) 27.38/8.90 U23_ga(X1, lesscE_out_ga(X1)) -> lesscE_out_ga(s(X1)) 27.38/8.90 lesscD_in_ag(s(X1)) -> lesscD_out_ag(0, s(X1)) 27.38/8.90 lesscD_in_ag(s(X2)) -> U24_ag(X2, lesscD_in_ag(X2)) 27.38/8.90 U24_ag(X2, lesscD_out_ag(X1, X2)) -> lesscD_out_ag(s(X1), s(X2)) 27.38/8.90 27.38/8.90 The set Q consists of the following terms: 27.38/8.90 27.38/8.90 lesscE_in_ga(x0) 27.38/8.90 U23_ga(x0, x1) 27.38/8.90 lesscD_in_ag(x0) 27.38/8.90 U24_ag(x0, x1) 27.38/8.90 27.38/8.90 We have to consider all (P,Q,R)-chains. 27.38/8.90 ---------------------------------------- 27.38/8.90 27.38/8.90 (162) UsableRulesProof (EQUIVALENT) 27.38/8.90 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. 27.38/8.90 ---------------------------------------- 27.38/8.90 27.38/8.90 (163) 27.38/8.90 Obligation: 27.38/8.90 Q DP problem: 27.38/8.90 The TRS P consists of the following rules: 27.38/8.90 27.38/8.90 INA_IN_GA(0) -> INA_IN_GA(0) 27.38/8.90 27.38/8.90 R is empty. 27.38/8.90 The set Q consists of the following terms: 27.38/8.90 27.38/8.90 lesscE_in_ga(x0) 27.38/8.90 U23_ga(x0, x1) 27.38/8.90 lesscD_in_ag(x0) 27.38/8.90 U24_ag(x0, x1) 27.38/8.90 27.38/8.90 We have to consider all (P,Q,R)-chains. 27.38/8.90 ---------------------------------------- 27.38/8.90 27.38/8.90 (164) QReductionProof (EQUIVALENT) 27.38/8.90 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 27.38/8.90 27.38/8.90 lesscE_in_ga(x0) 27.38/8.90 U23_ga(x0, x1) 27.38/8.90 lesscD_in_ag(x0) 27.38/8.90 U24_ag(x0, x1) 27.38/8.90 27.38/8.90 27.38/8.90 ---------------------------------------- 27.38/8.90 27.38/8.90 (165) 27.38/8.90 Obligation: 27.38/8.90 Q DP problem: 27.38/8.90 The TRS P consists of the following rules: 27.38/8.90 27.38/8.90 INA_IN_GA(0) -> INA_IN_GA(0) 27.38/8.90 27.38/8.90 R is empty. 27.38/8.90 Q is empty. 27.38/8.90 We have to consider all (P,Q,R)-chains. 27.38/8.93 EOF