28.68/9.25 MAYBE 28.96/9.26 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 28.96/9.26 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 28.96/9.26 28.96/9.26 28.96/9.26 Left Termination of the query pattern 28.96/9.26 28.96/9.26 insert(g,a,a) 28.96/9.26 28.96/9.26 w.r.t. the given Prolog program could not be shown: 28.96/9.26 28.96/9.26 (0) Prolog 28.96/9.26 (1) PrologToPiTRSProof [SOUND, 0 ms] 28.96/9.26 (2) PiTRS 28.96/9.26 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 28.96/9.26 (4) PiDP 28.96/9.26 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 28.96/9.26 (6) AND 28.96/9.26 (7) PiDP 28.96/9.26 (8) UsableRulesProof [EQUIVALENT, 0 ms] 28.96/9.26 (9) PiDP 28.96/9.26 (10) PiDPToQDPProof [SOUND, 0 ms] 28.96/9.26 (11) QDP 28.96/9.26 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 28.96/9.26 (13) YES 28.96/9.26 (14) PiDP 28.96/9.26 (15) UsableRulesProof [EQUIVALENT, 0 ms] 28.96/9.26 (16) PiDP 28.96/9.26 (17) PiDPToQDPProof [SOUND, 0 ms] 28.96/9.26 (18) QDP 28.96/9.26 (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] 28.96/9.26 (20) YES 28.96/9.26 (21) PiDP 28.96/9.26 (22) UsableRulesProof [EQUIVALENT, 0 ms] 28.96/9.26 (23) PiDP 28.96/9.26 (24) PiDPToQDPProof [SOUND, 4 ms] 28.96/9.26 (25) QDP 28.96/9.26 (26) TransformationProof [SOUND, 0 ms] 28.96/9.26 (27) QDP 28.96/9.26 (28) TransformationProof [SOUND, 0 ms] 28.96/9.26 (29) QDP 28.96/9.26 (30) TransformationProof [EQUIVALENT, 0 ms] 28.96/9.26 (31) QDP 28.96/9.26 (32) DependencyGraphProof [EQUIVALENT, 0 ms] 28.96/9.26 (33) AND 28.96/9.26 (34) QDP 28.96/9.26 (35) UsableRulesProof [EQUIVALENT, 0 ms] 28.96/9.26 (36) QDP 28.96/9.26 (37) QReductionProof [EQUIVALENT, 0 ms] 28.96/9.26 (38) QDP 28.96/9.26 (39) QDP 28.96/9.26 (40) TransformationProof [EQUIVALENT, 0 ms] 28.96/9.26 (41) QDP 28.96/9.26 (42) PrologToPiTRSProof [SOUND, 0 ms] 28.96/9.26 (43) PiTRS 28.96/9.26 (44) DependencyPairsProof [EQUIVALENT, 0 ms] 28.96/9.26 (45) PiDP 28.96/9.26 (46) DependencyGraphProof [EQUIVALENT, 0 ms] 28.96/9.26 (47) AND 28.96/9.26 (48) PiDP 28.96/9.26 (49) UsableRulesProof [EQUIVALENT, 0 ms] 28.96/9.26 (50) PiDP 28.96/9.26 (51) PiDPToQDPProof [SOUND, 0 ms] 28.96/9.26 (52) QDP 28.96/9.26 (53) QDPSizeChangeProof [EQUIVALENT, 0 ms] 28.96/9.26 (54) YES 28.96/9.26 (55) PiDP 28.96/9.26 (56) UsableRulesProof [EQUIVALENT, 0 ms] 28.96/9.26 (57) PiDP 28.96/9.26 (58) PiDPToQDPProof [SOUND, 0 ms] 28.96/9.26 (59) QDP 28.96/9.26 (60) QDPSizeChangeProof [EQUIVALENT, 0 ms] 28.96/9.26 (61) YES 28.96/9.26 (62) PiDP 28.96/9.26 (63) UsableRulesProof [EQUIVALENT, 0 ms] 28.96/9.26 (64) PiDP 28.96/9.26 (65) PiDPToQDPProof [SOUND, 0 ms] 28.96/9.26 (66) QDP 28.96/9.26 (67) TransformationProof [SOUND, 0 ms] 28.96/9.26 (68) QDP 28.96/9.26 (69) TransformationProof [SOUND, 0 ms] 28.96/9.26 (70) QDP 28.96/9.26 (71) TransformationProof [EQUIVALENT, 0 ms] 28.96/9.26 (72) QDP 28.96/9.26 (73) DependencyGraphProof [EQUIVALENT, 0 ms] 28.96/9.26 (74) AND 28.96/9.26 (75) QDP 28.96/9.26 (76) UsableRulesProof [EQUIVALENT, 0 ms] 28.96/9.26 (77) QDP 28.96/9.26 (78) QDP 28.96/9.26 (79) PrologToTRSTransformerProof [SOUND, 19 ms] 28.96/9.26 (80) QTRS 28.96/9.26 (81) DependencyPairsProof [EQUIVALENT, 0 ms] 28.96/9.26 (82) QDP 28.96/9.26 (83) DependencyGraphProof [EQUIVALENT, 0 ms] 28.96/9.26 (84) AND 28.96/9.26 (85) QDP 28.96/9.26 (86) UsableRulesProof [EQUIVALENT, 0 ms] 28.96/9.26 (87) QDP 28.96/9.26 (88) QDPSizeChangeProof [EQUIVALENT, 0 ms] 28.96/9.26 (89) YES 28.96/9.26 (90) QDP 28.96/9.26 (91) UsableRulesProof [EQUIVALENT, 0 ms] 28.96/9.26 (92) QDP 28.96/9.26 (93) QDPSizeChangeProof [EQUIVALENT, 0 ms] 28.96/9.26 (94) YES 28.96/9.26 (95) QDP 28.96/9.26 (96) NonTerminationLoopProof [COMPLETE, 0 ms] 28.96/9.26 (97) NO 28.96/9.26 (98) PrologToIRSwTTransformerProof [SOUND, 49 ms] 28.96/9.26 (99) AND 28.96/9.26 (100) IRSwT 28.96/9.26 (101) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] 28.96/9.26 (102) IRSwT 28.96/9.26 (103) IntTRSCompressionProof [EQUIVALENT, 21 ms] 28.96/9.26 (104) IRSwT 28.96/9.26 (105) IRSFormatTransformerProof [EQUIVALENT, 0 ms] 28.96/9.26 (106) IRSwT 28.96/9.26 (107) IRSwTTerminationDigraphProof [EQUIVALENT, 6 ms] 28.96/9.26 (108) IRSwT 28.96/9.26 (109) TempFilterProof [SOUND, 1 ms] 28.96/9.26 (110) IRSwT 28.96/9.26 (111) IRSwTToQDPProof [SOUND, 0 ms] 28.96/9.26 (112) QDP 28.96/9.26 (113) QDPSizeChangeProof [EQUIVALENT, 0 ms] 28.96/9.26 (114) YES 28.96/9.26 (115) IRSwT 28.96/9.26 (116) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] 28.96/9.26 (117) IRSwT 28.96/9.26 (118) IntTRSCompressionProof [EQUIVALENT, 1 ms] 28.96/9.26 (119) IRSwT 28.96/9.26 (120) IRSFormatTransformerProof [EQUIVALENT, 0 ms] 28.96/9.26 (121) IRSwT 28.96/9.26 (122) IRSwTTerminationDigraphProof [EQUIVALENT, 1 ms] 28.96/9.26 (123) IRSwT 28.96/9.26 (124) TempFilterProof [SOUND, 2 ms] 28.96/9.26 (125) IRSwT 28.96/9.26 (126) IRSwTToQDPProof [SOUND, 0 ms] 28.96/9.26 (127) QDP 28.96/9.26 (128) QDPSizeChangeProof [EQUIVALENT, 0 ms] 28.96/9.26 (129) YES 28.96/9.26 (130) IRSwT 28.96/9.26 (131) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] 28.96/9.26 (132) IRSwT 28.96/9.26 (133) PrologToDTProblemTransformerProof [SOUND, 38 ms] 28.96/9.26 (134) TRIPLES 28.96/9.26 (135) TriplesToPiDPProof [SOUND, 49 ms] 28.96/9.26 (136) PiDP 28.96/9.26 (137) DependencyGraphProof [EQUIVALENT, 0 ms] 28.96/9.26 (138) AND 28.96/9.26 (139) PiDP 28.96/9.26 (140) UsableRulesProof [EQUIVALENT, 0 ms] 28.96/9.26 (141) PiDP 28.96/9.26 (142) PiDPToQDPProof [SOUND, 0 ms] 28.96/9.26 (143) QDP 28.96/9.26 (144) QDPSizeChangeProof [EQUIVALENT, 0 ms] 28.96/9.26 (145) YES 28.96/9.26 (146) PiDP 28.96/9.26 (147) UsableRulesProof [EQUIVALENT, 0 ms] 28.96/9.26 (148) PiDP 28.96/9.26 (149) PiDPToQDPProof [SOUND, 0 ms] 28.96/9.26 (150) QDP 28.96/9.26 (151) QDPSizeChangeProof [EQUIVALENT, 0 ms] 28.96/9.26 (152) YES 28.96/9.26 (153) PiDP 28.96/9.26 (154) PiDPToQDPProof [SOUND, 0 ms] 28.96/9.26 (155) QDP 28.96/9.26 28.96/9.26 28.96/9.26 ---------------------------------------- 28.96/9.26 28.96/9.26 (0) 28.96/9.26 Obligation: 28.96/9.26 Clauses: 28.96/9.26 28.96/9.26 insert(X, void, tree(X, void, void)). 28.96/9.26 insert(X, tree(X, Left, Right), tree(X, Left, Right)). 28.96/9.26 insert(X, tree(Y, Left, Right), tree(Y, Left1, Right)) :- ','(less(X, Y), insert(X, Left, Left1)). 28.96/9.26 insert(X, tree(Y, Left, Right), tree(Y, Left, Right1)) :- ','(less(Y, X), insert(X, Right, Right1)). 28.96/9.26 less(0, s(X1)). 28.96/9.26 less(s(X), s(Y)) :- less(X, Y). 28.96/9.26 28.96/9.26 28.96/9.26 Query: insert(g,a,a) 28.96/9.26 ---------------------------------------- 28.96/9.26 28.96/9.26 (1) PrologToPiTRSProof (SOUND) 28.96/9.26 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 28.96/9.26 28.96/9.26 insert_in_3: (b,f,f) 28.96/9.26 28.96/9.26 less_in_2: (b,f) (f,b) 28.96/9.26 28.96/9.26 Transforming Prolog into the following Term Rewriting System: 28.96/9.26 28.96/9.26 Pi-finite rewrite system: 28.96/9.26 The TRS R consists of the following rules: 28.96/9.26 28.96/9.26 insert_in_gaa(X, void, tree(X, void, void)) -> insert_out_gaa(X, void, tree(X, void, void)) 28.96/9.26 insert_in_gaa(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_gaa(X, tree(X, Left, Right), tree(X, Left, Right)) 28.96/9.26 insert_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 28.96/9.26 less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) 28.96/9.26 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 28.96/9.26 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 28.96/9.26 U1_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> U2_gaa(X, Y, Left, Right, Left1, insert_in_gaa(X, Left, Left1)) 28.96/9.26 insert_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 28.96/9.26 less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1)) 28.96/9.26 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 28.96/9.26 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 28.96/9.26 U3_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> U4_gaa(X, Y, Left, Right, Right1, insert_in_gaa(X, Right, Right1)) 28.96/9.26 U4_gaa(X, Y, Left, Right, Right1, insert_out_gaa(X, Right, Right1)) -> insert_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) 28.96/9.26 U2_gaa(X, Y, Left, Right, Left1, insert_out_gaa(X, Left, Left1)) -> insert_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) 28.96/9.26 28.96/9.26 The argument filtering Pi contains the following mapping: 28.96/9.26 insert_in_gaa(x1, x2, x3) = insert_in_gaa(x1) 28.96/9.26 28.96/9.26 insert_out_gaa(x1, x2, x3) = insert_out_gaa 28.96/9.26 28.96/9.26 U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x1, x6) 28.96/9.26 28.96/9.26 less_in_ga(x1, x2) = less_in_ga(x1) 28.96/9.26 28.96/9.26 0 = 0 28.96/9.26 28.96/9.26 less_out_ga(x1, x2) = less_out_ga 28.96/9.26 28.96/9.26 s(x1) = s(x1) 28.96/9.26 28.96/9.26 U5_ga(x1, x2, x3) = U5_ga(x3) 28.96/9.26 28.96/9.26 U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x6) 28.96/9.26 28.96/9.26 U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x1, x6) 28.96/9.26 28.96/9.26 less_in_ag(x1, x2) = less_in_ag(x2) 28.96/9.26 28.96/9.26 less_out_ag(x1, x2) = less_out_ag(x1) 28.96/9.26 28.96/9.26 U5_ag(x1, x2, x3) = U5_ag(x3) 28.96/9.26 28.96/9.26 U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x6) 28.96/9.26 28.96/9.26 28.96/9.26 28.96/9.26 28.96/9.26 28.96/9.26 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 28.96/9.26 28.96/9.26 28.96/9.26 28.96/9.26 ---------------------------------------- 28.96/9.26 28.96/9.26 (2) 28.96/9.26 Obligation: 28.96/9.26 Pi-finite rewrite system: 28.96/9.26 The TRS R consists of the following rules: 28.96/9.26 28.96/9.26 insert_in_gaa(X, void, tree(X, void, void)) -> insert_out_gaa(X, void, tree(X, void, void)) 28.96/9.26 insert_in_gaa(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_gaa(X, tree(X, Left, Right), tree(X, Left, Right)) 28.96/9.26 insert_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 28.96/9.26 less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) 28.96/9.26 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 28.96/9.26 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 28.96/9.26 U1_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> U2_gaa(X, Y, Left, Right, Left1, insert_in_gaa(X, Left, Left1)) 28.96/9.26 insert_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 28.96/9.26 less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1)) 28.96/9.26 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 28.96/9.26 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 28.96/9.26 U3_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> U4_gaa(X, Y, Left, Right, Right1, insert_in_gaa(X, Right, Right1)) 28.96/9.26 U4_gaa(X, Y, Left, Right, Right1, insert_out_gaa(X, Right, Right1)) -> insert_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) 28.96/9.26 U2_gaa(X, Y, Left, Right, Left1, insert_out_gaa(X, Left, Left1)) -> insert_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) 28.96/9.26 28.96/9.26 The argument filtering Pi contains the following mapping: 28.96/9.26 insert_in_gaa(x1, x2, x3) = insert_in_gaa(x1) 28.96/9.26 28.96/9.26 insert_out_gaa(x1, x2, x3) = insert_out_gaa 28.96/9.26 28.96/9.26 U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x1, x6) 28.96/9.26 28.96/9.26 less_in_ga(x1, x2) = less_in_ga(x1) 28.96/9.26 28.96/9.26 0 = 0 28.96/9.26 28.96/9.26 less_out_ga(x1, x2) = less_out_ga 28.96/9.26 28.96/9.26 s(x1) = s(x1) 28.96/9.26 28.96/9.26 U5_ga(x1, x2, x3) = U5_ga(x3) 28.96/9.26 28.96/9.26 U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x6) 28.96/9.26 28.96/9.26 U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x1, x6) 28.96/9.26 28.96/9.26 less_in_ag(x1, x2) = less_in_ag(x2) 28.96/9.26 28.96/9.26 less_out_ag(x1, x2) = less_out_ag(x1) 28.96/9.26 28.96/9.26 U5_ag(x1, x2, x3) = U5_ag(x3) 28.96/9.26 28.96/9.26 U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x6) 28.96/9.26 28.96/9.26 28.96/9.26 28.96/9.26 ---------------------------------------- 28.96/9.26 28.96/9.26 (3) DependencyPairsProof (EQUIVALENT) 28.96/9.26 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 28.96/9.26 Pi DP problem: 28.96/9.26 The TRS P consists of the following rules: 28.96/9.26 28.96/9.26 INSERT_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_GAA(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 28.96/9.26 INSERT_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> LESS_IN_GA(X, Y) 28.96/9.26 LESS_IN_GA(s(X), s(Y)) -> U5_GA(X, Y, less_in_ga(X, Y)) 28.96/9.26 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 28.96/9.26 U1_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> U2_GAA(X, Y, Left, Right, Left1, insert_in_gaa(X, Left, Left1)) 28.96/9.26 U1_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> INSERT_IN_GAA(X, Left, Left1) 28.96/9.26 INSERT_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_GAA(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 28.96/9.26 INSERT_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> LESS_IN_AG(Y, X) 28.96/9.26 LESS_IN_AG(s(X), s(Y)) -> U5_AG(X, Y, less_in_ag(X, Y)) 28.96/9.26 LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) 28.96/9.26 U3_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> U4_GAA(X, Y, Left, Right, Right1, insert_in_gaa(X, Right, Right1)) 28.96/9.26 U3_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> INSERT_IN_GAA(X, Right, Right1) 28.96/9.26 28.96/9.26 The TRS R consists of the following rules: 28.96/9.26 28.96/9.26 insert_in_gaa(X, void, tree(X, void, void)) -> insert_out_gaa(X, void, tree(X, void, void)) 28.96/9.26 insert_in_gaa(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_gaa(X, tree(X, Left, Right), tree(X, Left, Right)) 28.96/9.26 insert_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 28.96/9.26 less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) 28.96/9.26 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 28.96/9.26 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 28.96/9.26 U1_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> U2_gaa(X, Y, Left, Right, Left1, insert_in_gaa(X, Left, Left1)) 28.96/9.26 insert_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 28.96/9.26 less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1)) 28.96/9.26 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 28.96/9.26 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 28.96/9.26 U3_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> U4_gaa(X, Y, Left, Right, Right1, insert_in_gaa(X, Right, Right1)) 28.96/9.26 U4_gaa(X, Y, Left, Right, Right1, insert_out_gaa(X, Right, Right1)) -> insert_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) 28.96/9.26 U2_gaa(X, Y, Left, Right, Left1, insert_out_gaa(X, Left, Left1)) -> insert_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) 28.96/9.26 28.96/9.26 The argument filtering Pi contains the following mapping: 28.96/9.26 insert_in_gaa(x1, x2, x3) = insert_in_gaa(x1) 28.96/9.26 28.96/9.26 insert_out_gaa(x1, x2, x3) = insert_out_gaa 28.96/9.26 28.96/9.26 U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x1, x6) 28.96/9.26 28.96/9.26 less_in_ga(x1, x2) = less_in_ga(x1) 28.96/9.26 28.96/9.26 0 = 0 28.96/9.26 28.96/9.26 less_out_ga(x1, x2) = less_out_ga 28.96/9.26 28.96/9.26 s(x1) = s(x1) 28.96/9.26 28.96/9.26 U5_ga(x1, x2, x3) = U5_ga(x3) 28.96/9.26 28.96/9.26 U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x6) 28.96/9.26 28.96/9.26 U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x1, x6) 28.96/9.26 28.96/9.26 less_in_ag(x1, x2) = less_in_ag(x2) 28.96/9.26 28.96/9.26 less_out_ag(x1, x2) = less_out_ag(x1) 28.96/9.26 28.96/9.26 U5_ag(x1, x2, x3) = U5_ag(x3) 28.96/9.26 28.96/9.26 U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x6) 28.96/9.26 28.96/9.26 INSERT_IN_GAA(x1, x2, x3) = INSERT_IN_GAA(x1) 28.96/9.26 28.96/9.26 U1_GAA(x1, x2, x3, x4, x5, x6) = U1_GAA(x1, x6) 28.96/9.26 28.96/9.26 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 28.96/9.26 28.96/9.26 U5_GA(x1, x2, x3) = U5_GA(x3) 28.96/9.26 28.96/9.26 U2_GAA(x1, x2, x3, x4, x5, x6) = U2_GAA(x6) 28.96/9.26 28.96/9.26 U3_GAA(x1, x2, x3, x4, x5, x6) = U3_GAA(x1, x6) 28.96/9.26 28.96/9.26 LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) 28.96/9.26 28.96/9.26 U5_AG(x1, x2, x3) = U5_AG(x3) 28.96/9.26 28.96/9.26 U4_GAA(x1, x2, x3, x4, x5, x6) = U4_GAA(x6) 28.96/9.26 28.96/9.26 28.96/9.26 We have to consider all (P,R,Pi)-chains 28.96/9.26 ---------------------------------------- 28.96/9.26 28.96/9.26 (4) 28.96/9.26 Obligation: 28.96/9.26 Pi DP problem: 28.96/9.26 The TRS P consists of the following rules: 28.96/9.26 28.96/9.26 INSERT_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_GAA(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 28.96/9.26 INSERT_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> LESS_IN_GA(X, Y) 28.96/9.26 LESS_IN_GA(s(X), s(Y)) -> U5_GA(X, Y, less_in_ga(X, Y)) 28.96/9.26 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 28.96/9.26 U1_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> U2_GAA(X, Y, Left, Right, Left1, insert_in_gaa(X, Left, Left1)) 28.96/9.26 U1_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> INSERT_IN_GAA(X, Left, Left1) 28.96/9.26 INSERT_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_GAA(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 28.96/9.26 INSERT_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> LESS_IN_AG(Y, X) 28.96/9.26 LESS_IN_AG(s(X), s(Y)) -> U5_AG(X, Y, less_in_ag(X, Y)) 28.96/9.26 LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) 28.96/9.26 U3_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> U4_GAA(X, Y, Left, Right, Right1, insert_in_gaa(X, Right, Right1)) 28.96/9.26 U3_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> INSERT_IN_GAA(X, Right, Right1) 28.96/9.26 28.96/9.26 The TRS R consists of the following rules: 28.96/9.26 28.96/9.26 insert_in_gaa(X, void, tree(X, void, void)) -> insert_out_gaa(X, void, tree(X, void, void)) 28.96/9.26 insert_in_gaa(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_gaa(X, tree(X, Left, Right), tree(X, Left, Right)) 28.96/9.26 insert_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 28.96/9.26 less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) 28.96/9.26 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 28.96/9.26 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 28.96/9.26 U1_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> U2_gaa(X, Y, Left, Right, Left1, insert_in_gaa(X, Left, Left1)) 28.96/9.26 insert_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 28.96/9.26 less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1)) 28.96/9.26 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 28.96/9.26 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 28.96/9.26 U3_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> U4_gaa(X, Y, Left, Right, Right1, insert_in_gaa(X, Right, Right1)) 28.96/9.26 U4_gaa(X, Y, Left, Right, Right1, insert_out_gaa(X, Right, Right1)) -> insert_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) 28.96/9.26 U2_gaa(X, Y, Left, Right, Left1, insert_out_gaa(X, Left, Left1)) -> insert_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) 28.96/9.26 28.96/9.26 The argument filtering Pi contains the following mapping: 28.96/9.26 insert_in_gaa(x1, x2, x3) = insert_in_gaa(x1) 28.96/9.26 28.96/9.26 insert_out_gaa(x1, x2, x3) = insert_out_gaa 28.96/9.26 28.96/9.26 U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x1, x6) 28.96/9.26 28.96/9.26 less_in_ga(x1, x2) = less_in_ga(x1) 28.96/9.26 28.96/9.26 0 = 0 28.96/9.26 28.96/9.26 less_out_ga(x1, x2) = less_out_ga 28.96/9.26 28.96/9.26 s(x1) = s(x1) 28.96/9.26 28.96/9.26 U5_ga(x1, x2, x3) = U5_ga(x3) 28.96/9.26 28.96/9.26 U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x6) 28.96/9.26 28.96/9.26 U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x1, x6) 28.96/9.26 28.96/9.26 less_in_ag(x1, x2) = less_in_ag(x2) 28.96/9.26 28.96/9.26 less_out_ag(x1, x2) = less_out_ag(x1) 28.96/9.26 28.96/9.26 U5_ag(x1, x2, x3) = U5_ag(x3) 28.96/9.26 28.96/9.26 U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x6) 28.96/9.26 28.96/9.26 INSERT_IN_GAA(x1, x2, x3) = INSERT_IN_GAA(x1) 28.96/9.26 28.96/9.26 U1_GAA(x1, x2, x3, x4, x5, x6) = U1_GAA(x1, x6) 28.96/9.26 28.96/9.26 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 28.96/9.26 28.96/9.26 U5_GA(x1, x2, x3) = U5_GA(x3) 28.96/9.26 28.96/9.26 U2_GAA(x1, x2, x3, x4, x5, x6) = U2_GAA(x6) 28.96/9.26 28.96/9.26 U3_GAA(x1, x2, x3, x4, x5, x6) = U3_GAA(x1, x6) 28.96/9.26 28.96/9.26 LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) 28.96/9.26 28.96/9.26 U5_AG(x1, x2, x3) = U5_AG(x3) 28.96/9.26 28.96/9.26 U4_GAA(x1, x2, x3, x4, x5, x6) = U4_GAA(x6) 28.96/9.26 28.96/9.26 28.96/9.26 We have to consider all (P,R,Pi)-chains 28.96/9.26 ---------------------------------------- 28.96/9.26 28.96/9.26 (5) DependencyGraphProof (EQUIVALENT) 28.96/9.26 The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 6 less nodes. 28.96/9.26 ---------------------------------------- 28.96/9.26 28.96/9.26 (6) 28.96/9.26 Complex Obligation (AND) 28.96/9.26 28.96/9.26 ---------------------------------------- 28.96/9.26 28.96/9.26 (7) 28.96/9.26 Obligation: 28.96/9.26 Pi DP problem: 28.96/9.26 The TRS P consists of the following rules: 28.96/9.26 28.96/9.26 LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) 28.96/9.26 28.96/9.26 The TRS R consists of the following rules: 28.96/9.26 28.96/9.26 insert_in_gaa(X, void, tree(X, void, void)) -> insert_out_gaa(X, void, tree(X, void, void)) 28.96/9.26 insert_in_gaa(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_gaa(X, tree(X, Left, Right), tree(X, Left, Right)) 28.96/9.26 insert_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 28.96/9.26 less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) 28.96/9.26 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 28.96/9.26 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 28.96/9.26 U1_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> U2_gaa(X, Y, Left, Right, Left1, insert_in_gaa(X, Left, Left1)) 28.96/9.26 insert_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 28.96/9.26 less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1)) 28.96/9.26 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 28.96/9.26 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 28.96/9.26 U3_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> U4_gaa(X, Y, Left, Right, Right1, insert_in_gaa(X, Right, Right1)) 28.96/9.26 U4_gaa(X, Y, Left, Right, Right1, insert_out_gaa(X, Right, Right1)) -> insert_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) 28.96/9.26 U2_gaa(X, Y, Left, Right, Left1, insert_out_gaa(X, Left, Left1)) -> insert_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) 28.96/9.26 28.96/9.26 The argument filtering Pi contains the following mapping: 28.96/9.26 insert_in_gaa(x1, x2, x3) = insert_in_gaa(x1) 28.96/9.26 28.96/9.26 insert_out_gaa(x1, x2, x3) = insert_out_gaa 28.96/9.26 28.96/9.26 U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x1, x6) 28.96/9.26 28.96/9.26 less_in_ga(x1, x2) = less_in_ga(x1) 28.96/9.26 28.96/9.26 0 = 0 28.96/9.26 28.96/9.26 less_out_ga(x1, x2) = less_out_ga 28.96/9.26 28.96/9.26 s(x1) = s(x1) 28.96/9.26 28.96/9.26 U5_ga(x1, x2, x3) = U5_ga(x3) 28.96/9.26 28.96/9.26 U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x6) 28.96/9.26 28.96/9.26 U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x1, x6) 28.96/9.26 28.96/9.26 less_in_ag(x1, x2) = less_in_ag(x2) 28.96/9.26 28.96/9.26 less_out_ag(x1, x2) = less_out_ag(x1) 28.96/9.26 28.96/9.26 U5_ag(x1, x2, x3) = U5_ag(x3) 28.96/9.26 28.96/9.26 U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x6) 28.96/9.26 28.96/9.26 LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) 28.96/9.26 28.96/9.26 28.96/9.26 We have to consider all (P,R,Pi)-chains 28.96/9.26 ---------------------------------------- 28.96/9.26 28.96/9.26 (8) UsableRulesProof (EQUIVALENT) 28.96/9.26 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 28.96/9.26 ---------------------------------------- 28.96/9.26 28.96/9.26 (9) 28.96/9.26 Obligation: 28.96/9.26 Pi DP problem: 28.96/9.26 The TRS P consists of the following rules: 28.96/9.26 28.96/9.26 LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) 28.96/9.26 28.96/9.26 R is empty. 28.96/9.26 The argument filtering Pi contains the following mapping: 28.96/9.26 s(x1) = s(x1) 28.96/9.26 28.96/9.26 LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) 28.96/9.26 28.96/9.26 28.96/9.26 We have to consider all (P,R,Pi)-chains 28.96/9.26 ---------------------------------------- 28.96/9.26 28.96/9.26 (10) PiDPToQDPProof (SOUND) 28.96/9.26 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 28.96/9.26 ---------------------------------------- 28.96/9.26 28.96/9.26 (11) 28.96/9.26 Obligation: 28.96/9.26 Q DP problem: 28.96/9.26 The TRS P consists of the following rules: 28.96/9.26 28.96/9.26 LESS_IN_AG(s(Y)) -> LESS_IN_AG(Y) 28.96/9.26 28.96/9.26 R is empty. 28.96/9.26 Q is empty. 28.96/9.26 We have to consider all (P,Q,R)-chains. 28.96/9.26 ---------------------------------------- 28.96/9.26 28.96/9.26 (12) QDPSizeChangeProof (EQUIVALENT) 28.96/9.26 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. 28.96/9.26 28.96/9.26 From the DPs we obtained the following set of size-change graphs: 28.96/9.26 *LESS_IN_AG(s(Y)) -> LESS_IN_AG(Y) 28.96/9.26 The graph contains the following edges 1 > 1 28.96/9.26 28.96/9.26 28.96/9.26 ---------------------------------------- 28.96/9.26 28.96/9.26 (13) 28.96/9.26 YES 28.96/9.26 28.96/9.26 ---------------------------------------- 28.96/9.26 28.96/9.26 (14) 28.96/9.26 Obligation: 28.96/9.26 Pi DP problem: 28.96/9.26 The TRS P consists of the following rules: 28.96/9.26 28.96/9.26 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 28.96/9.26 28.96/9.26 The TRS R consists of the following rules: 28.96/9.26 28.96/9.26 insert_in_gaa(X, void, tree(X, void, void)) -> insert_out_gaa(X, void, tree(X, void, void)) 28.96/9.26 insert_in_gaa(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_gaa(X, tree(X, Left, Right), tree(X, Left, Right)) 28.96/9.26 insert_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 28.96/9.26 less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) 28.96/9.26 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 28.96/9.26 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 28.96/9.26 U1_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> U2_gaa(X, Y, Left, Right, Left1, insert_in_gaa(X, Left, Left1)) 28.96/9.26 insert_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 28.96/9.26 less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1)) 28.96/9.26 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 28.96/9.26 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 28.96/9.27 U3_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> U4_gaa(X, Y, Left, Right, Right1, insert_in_gaa(X, Right, Right1)) 28.96/9.27 U4_gaa(X, Y, Left, Right, Right1, insert_out_gaa(X, Right, Right1)) -> insert_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) 28.96/9.27 U2_gaa(X, Y, Left, Right, Left1, insert_out_gaa(X, Left, Left1)) -> insert_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) 28.96/9.27 28.96/9.27 The argument filtering Pi contains the following mapping: 28.96/9.27 insert_in_gaa(x1, x2, x3) = insert_in_gaa(x1) 28.96/9.27 28.96/9.27 insert_out_gaa(x1, x2, x3) = insert_out_gaa 28.96/9.27 28.96/9.27 U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x1, x6) 28.96/9.27 28.96/9.27 less_in_ga(x1, x2) = less_in_ga(x1) 28.96/9.27 28.96/9.27 0 = 0 28.96/9.27 28.96/9.27 less_out_ga(x1, x2) = less_out_ga 28.96/9.27 28.96/9.27 s(x1) = s(x1) 28.96/9.27 28.96/9.27 U5_ga(x1, x2, x3) = U5_ga(x3) 28.96/9.27 28.96/9.27 U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x6) 28.96/9.27 28.96/9.27 U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x1, x6) 28.96/9.27 28.96/9.27 less_in_ag(x1, x2) = less_in_ag(x2) 28.96/9.27 28.96/9.27 less_out_ag(x1, x2) = less_out_ag(x1) 28.96/9.27 28.96/9.27 U5_ag(x1, x2, x3) = U5_ag(x3) 28.96/9.27 28.96/9.27 U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x6) 28.96/9.27 28.96/9.27 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 28.96/9.27 28.96/9.27 28.96/9.27 We have to consider all (P,R,Pi)-chains 28.96/9.27 ---------------------------------------- 28.96/9.27 28.96/9.27 (15) UsableRulesProof (EQUIVALENT) 28.96/9.27 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 28.96/9.27 ---------------------------------------- 28.96/9.27 28.96/9.27 (16) 28.96/9.27 Obligation: 28.96/9.27 Pi DP problem: 28.96/9.27 The TRS P consists of the following rules: 28.96/9.27 28.96/9.27 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 28.96/9.27 28.96/9.27 R is empty. 28.96/9.27 The argument filtering Pi contains the following mapping: 28.96/9.27 s(x1) = s(x1) 28.96/9.27 28.96/9.27 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 28.96/9.27 28.96/9.27 28.96/9.27 We have to consider all (P,R,Pi)-chains 28.96/9.27 ---------------------------------------- 28.96/9.27 28.96/9.27 (17) PiDPToQDPProof (SOUND) 28.96/9.27 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 28.96/9.27 ---------------------------------------- 28.96/9.27 28.96/9.27 (18) 28.96/9.27 Obligation: 28.96/9.27 Q DP problem: 28.96/9.27 The TRS P consists of the following rules: 28.96/9.27 28.96/9.27 LESS_IN_GA(s(X)) -> LESS_IN_GA(X) 28.96/9.27 28.96/9.27 R is empty. 28.96/9.27 Q is empty. 28.96/9.27 We have to consider all (P,Q,R)-chains. 28.96/9.27 ---------------------------------------- 28.96/9.27 28.96/9.27 (19) QDPSizeChangeProof (EQUIVALENT) 28.96/9.27 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. 28.96/9.27 28.96/9.27 From the DPs we obtained the following set of size-change graphs: 28.96/9.27 *LESS_IN_GA(s(X)) -> LESS_IN_GA(X) 28.96/9.27 The graph contains the following edges 1 > 1 28.96/9.27 28.96/9.27 28.96/9.27 ---------------------------------------- 28.96/9.27 28.96/9.27 (20) 28.96/9.27 YES 28.96/9.27 28.96/9.27 ---------------------------------------- 28.96/9.27 28.96/9.27 (21) 28.96/9.27 Obligation: 28.96/9.27 Pi DP problem: 28.96/9.27 The TRS P consists of the following rules: 28.96/9.27 28.96/9.27 U1_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> INSERT_IN_GAA(X, Left, Left1) 28.96/9.27 INSERT_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_GAA(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 28.96/9.27 INSERT_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_GAA(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 28.96/9.27 U3_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> INSERT_IN_GAA(X, Right, Right1) 28.96/9.27 28.96/9.27 The TRS R consists of the following rules: 28.96/9.27 28.96/9.27 insert_in_gaa(X, void, tree(X, void, void)) -> insert_out_gaa(X, void, tree(X, void, void)) 28.96/9.27 insert_in_gaa(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_gaa(X, tree(X, Left, Right), tree(X, Left, Right)) 28.96/9.27 insert_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 28.96/9.27 less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) 28.96/9.27 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 28.96/9.27 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 28.96/9.27 U1_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> U2_gaa(X, Y, Left, Right, Left1, insert_in_gaa(X, Left, Left1)) 28.96/9.27 insert_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 28.96/9.27 less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1)) 28.96/9.27 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 28.96/9.27 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 28.96/9.27 U3_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> U4_gaa(X, Y, Left, Right, Right1, insert_in_gaa(X, Right, Right1)) 28.96/9.27 U4_gaa(X, Y, Left, Right, Right1, insert_out_gaa(X, Right, Right1)) -> insert_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) 28.96/9.27 U2_gaa(X, Y, Left, Right, Left1, insert_out_gaa(X, Left, Left1)) -> insert_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) 28.96/9.27 28.96/9.27 The argument filtering Pi contains the following mapping: 28.96/9.27 insert_in_gaa(x1, x2, x3) = insert_in_gaa(x1) 28.96/9.27 28.96/9.27 insert_out_gaa(x1, x2, x3) = insert_out_gaa 28.96/9.27 28.96/9.27 U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x1, x6) 28.96/9.27 28.96/9.27 less_in_ga(x1, x2) = less_in_ga(x1) 28.96/9.27 28.96/9.27 0 = 0 28.96/9.27 28.96/9.27 less_out_ga(x1, x2) = less_out_ga 28.96/9.27 28.96/9.27 s(x1) = s(x1) 28.96/9.27 28.96/9.27 U5_ga(x1, x2, x3) = U5_ga(x3) 28.96/9.27 28.96/9.27 U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x6) 28.96/9.27 28.96/9.27 U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x1, x6) 28.96/9.27 28.96/9.27 less_in_ag(x1, x2) = less_in_ag(x2) 28.96/9.27 28.96/9.27 less_out_ag(x1, x2) = less_out_ag(x1) 28.96/9.27 28.96/9.27 U5_ag(x1, x2, x3) = U5_ag(x3) 28.96/9.27 28.96/9.27 U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x6) 28.96/9.27 28.96/9.27 INSERT_IN_GAA(x1, x2, x3) = INSERT_IN_GAA(x1) 28.96/9.27 28.96/9.27 U1_GAA(x1, x2, x3, x4, x5, x6) = U1_GAA(x1, x6) 28.96/9.27 28.96/9.27 U3_GAA(x1, x2, x3, x4, x5, x6) = U3_GAA(x1, x6) 28.96/9.27 28.96/9.27 28.96/9.27 We have to consider all (P,R,Pi)-chains 28.96/9.27 ---------------------------------------- 28.96/9.27 28.96/9.27 (22) UsableRulesProof (EQUIVALENT) 28.96/9.27 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 28.96/9.27 ---------------------------------------- 28.96/9.27 28.96/9.27 (23) 28.96/9.27 Obligation: 28.96/9.27 Pi DP problem: 28.96/9.27 The TRS P consists of the following rules: 28.96/9.27 28.96/9.27 U1_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> INSERT_IN_GAA(X, Left, Left1) 28.96/9.27 INSERT_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_GAA(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 28.96/9.27 INSERT_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_GAA(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 28.96/9.27 U3_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> INSERT_IN_GAA(X, Right, Right1) 28.96/9.27 28.96/9.27 The TRS R consists of the following rules: 28.96/9.27 28.96/9.27 less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) 28.96/9.27 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 28.96/9.27 less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1)) 28.96/9.27 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 28.96/9.27 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 28.96/9.27 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 28.96/9.27 28.96/9.27 The argument filtering Pi contains the following mapping: 28.96/9.27 less_in_ga(x1, x2) = less_in_ga(x1) 28.96/9.27 28.96/9.27 0 = 0 28.96/9.27 28.96/9.27 less_out_ga(x1, x2) = less_out_ga 28.96/9.27 28.96/9.27 s(x1) = s(x1) 28.96/9.27 28.96/9.27 U5_ga(x1, x2, x3) = U5_ga(x3) 28.96/9.27 28.96/9.27 less_in_ag(x1, x2) = less_in_ag(x2) 28.96/9.27 28.96/9.27 less_out_ag(x1, x2) = less_out_ag(x1) 28.96/9.27 28.96/9.27 U5_ag(x1, x2, x3) = U5_ag(x3) 28.96/9.27 28.96/9.27 INSERT_IN_GAA(x1, x2, x3) = INSERT_IN_GAA(x1) 28.96/9.27 28.96/9.27 U1_GAA(x1, x2, x3, x4, x5, x6) = U1_GAA(x1, x6) 28.96/9.27 28.96/9.27 U3_GAA(x1, x2, x3, x4, x5, x6) = U3_GAA(x1, x6) 28.96/9.27 28.96/9.27 28.96/9.27 We have to consider all (P,R,Pi)-chains 28.96/9.27 ---------------------------------------- 28.96/9.27 28.96/9.27 (24) PiDPToQDPProof (SOUND) 28.96/9.27 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 28.96/9.27 ---------------------------------------- 28.96/9.27 28.96/9.27 (25) 28.96/9.27 Obligation: 28.96/9.27 Q DP problem: 28.96/9.27 The TRS P consists of the following rules: 28.96/9.27 28.96/9.27 U1_GAA(X, less_out_ga) -> INSERT_IN_GAA(X) 28.96/9.27 INSERT_IN_GAA(X) -> U1_GAA(X, less_in_ga(X)) 28.96/9.27 INSERT_IN_GAA(X) -> U3_GAA(X, less_in_ag(X)) 28.96/9.27 U3_GAA(X, less_out_ag(Y)) -> INSERT_IN_GAA(X) 28.96/9.27 28.96/9.27 The TRS R consists of the following rules: 28.96/9.27 28.96/9.27 less_in_ga(0) -> less_out_ga 28.96/9.27 less_in_ga(s(X)) -> U5_ga(less_in_ga(X)) 28.96/9.27 less_in_ag(s(X1)) -> less_out_ag(0) 28.96/9.27 less_in_ag(s(Y)) -> U5_ag(less_in_ag(Y)) 28.96/9.27 U5_ga(less_out_ga) -> less_out_ga 28.96/9.27 U5_ag(less_out_ag(X)) -> less_out_ag(s(X)) 28.96/9.27 28.96/9.27 The set Q consists of the following terms: 28.96/9.27 28.96/9.27 less_in_ga(x0) 28.96/9.27 less_in_ag(x0) 28.96/9.27 U5_ga(x0) 28.96/9.27 U5_ag(x0) 28.96/9.27 28.96/9.27 We have to consider all (P,Q,R)-chains. 28.96/9.27 ---------------------------------------- 28.96/9.27 28.96/9.27 (26) TransformationProof (SOUND) 28.96/9.27 By narrowing [LPAR04] the rule INSERT_IN_GAA(X) -> U1_GAA(X, less_in_ga(X)) at position [1] we obtained the following new rules [LPAR04]: 28.96/9.27 28.96/9.27 (INSERT_IN_GAA(0) -> U1_GAA(0, less_out_ga),INSERT_IN_GAA(0) -> U1_GAA(0, less_out_ga)) 28.96/9.27 (INSERT_IN_GAA(s(x0)) -> U1_GAA(s(x0), U5_ga(less_in_ga(x0))),INSERT_IN_GAA(s(x0)) -> U1_GAA(s(x0), U5_ga(less_in_ga(x0)))) 28.96/9.27 28.96/9.27 28.96/9.27 ---------------------------------------- 28.96/9.27 28.96/9.27 (27) 28.96/9.27 Obligation: 28.96/9.27 Q DP problem: 28.96/9.27 The TRS P consists of the following rules: 28.96/9.27 28.96/9.27 U1_GAA(X, less_out_ga) -> INSERT_IN_GAA(X) 28.96/9.27 INSERT_IN_GAA(X) -> U3_GAA(X, less_in_ag(X)) 28.96/9.27 U3_GAA(X, less_out_ag(Y)) -> INSERT_IN_GAA(X) 28.96/9.27 INSERT_IN_GAA(0) -> U1_GAA(0, less_out_ga) 28.96/9.27 INSERT_IN_GAA(s(x0)) -> U1_GAA(s(x0), U5_ga(less_in_ga(x0))) 28.96/9.27 28.96/9.27 The TRS R consists of the following rules: 28.96/9.27 28.96/9.27 less_in_ga(0) -> less_out_ga 28.96/9.27 less_in_ga(s(X)) -> U5_ga(less_in_ga(X)) 28.96/9.27 less_in_ag(s(X1)) -> less_out_ag(0) 28.96/9.27 less_in_ag(s(Y)) -> U5_ag(less_in_ag(Y)) 28.96/9.27 U5_ga(less_out_ga) -> less_out_ga 28.96/9.27 U5_ag(less_out_ag(X)) -> less_out_ag(s(X)) 28.96/9.27 28.96/9.27 The set Q consists of the following terms: 28.96/9.27 28.96/9.27 less_in_ga(x0) 28.96/9.27 less_in_ag(x0) 28.96/9.27 U5_ga(x0) 28.96/9.27 U5_ag(x0) 28.96/9.27 28.96/9.27 We have to consider all (P,Q,R)-chains. 28.96/9.27 ---------------------------------------- 28.96/9.27 28.96/9.27 (28) TransformationProof (SOUND) 28.96/9.27 By narrowing [LPAR04] the rule INSERT_IN_GAA(X) -> U3_GAA(X, less_in_ag(X)) at position [1] we obtained the following new rules [LPAR04]: 28.96/9.27 28.96/9.27 (INSERT_IN_GAA(s(x0)) -> U3_GAA(s(x0), less_out_ag(0)),INSERT_IN_GAA(s(x0)) -> U3_GAA(s(x0), less_out_ag(0))) 28.96/9.27 (INSERT_IN_GAA(s(x0)) -> U3_GAA(s(x0), U5_ag(less_in_ag(x0))),INSERT_IN_GAA(s(x0)) -> U3_GAA(s(x0), U5_ag(less_in_ag(x0)))) 28.96/9.27 28.96/9.27 28.96/9.27 ---------------------------------------- 28.96/9.27 28.96/9.27 (29) 28.96/9.27 Obligation: 28.96/9.27 Q DP problem: 28.96/9.27 The TRS P consists of the following rules: 28.96/9.27 28.96/9.27 U1_GAA(X, less_out_ga) -> INSERT_IN_GAA(X) 28.96/9.27 U3_GAA(X, less_out_ag(Y)) -> INSERT_IN_GAA(X) 28.96/9.27 INSERT_IN_GAA(0) -> U1_GAA(0, less_out_ga) 28.96/9.27 INSERT_IN_GAA(s(x0)) -> U1_GAA(s(x0), U5_ga(less_in_ga(x0))) 28.96/9.27 INSERT_IN_GAA(s(x0)) -> U3_GAA(s(x0), less_out_ag(0)) 28.96/9.27 INSERT_IN_GAA(s(x0)) -> U3_GAA(s(x0), U5_ag(less_in_ag(x0))) 28.96/9.27 28.96/9.27 The TRS R consists of the following rules: 28.96/9.27 28.96/9.27 less_in_ga(0) -> less_out_ga 28.96/9.27 less_in_ga(s(X)) -> U5_ga(less_in_ga(X)) 28.96/9.27 less_in_ag(s(X1)) -> less_out_ag(0) 28.96/9.27 less_in_ag(s(Y)) -> U5_ag(less_in_ag(Y)) 28.96/9.27 U5_ga(less_out_ga) -> less_out_ga 28.96/9.27 U5_ag(less_out_ag(X)) -> less_out_ag(s(X)) 28.96/9.27 28.96/9.27 The set Q consists of the following terms: 28.96/9.27 28.96/9.27 less_in_ga(x0) 28.96/9.27 less_in_ag(x0) 28.96/9.27 U5_ga(x0) 28.96/9.27 U5_ag(x0) 29.00/9.27 29.00/9.27 We have to consider all (P,Q,R)-chains. 29.00/9.27 ---------------------------------------- 29.00/9.27 29.00/9.27 (30) TransformationProof (EQUIVALENT) 29.00/9.27 By instantiating [LPAR04] the rule U1_GAA(X, less_out_ga) -> INSERT_IN_GAA(X) we obtained the following new rules [LPAR04]: 29.00/9.27 29.00/9.27 (U1_GAA(0, less_out_ga) -> INSERT_IN_GAA(0),U1_GAA(0, less_out_ga) -> INSERT_IN_GAA(0)) 29.00/9.27 (U1_GAA(s(z0), less_out_ga) -> INSERT_IN_GAA(s(z0)),U1_GAA(s(z0), less_out_ga) -> INSERT_IN_GAA(s(z0))) 29.00/9.27 29.00/9.27 29.00/9.27 ---------------------------------------- 29.00/9.27 29.00/9.27 (31) 29.00/9.27 Obligation: 29.00/9.27 Q DP problem: 29.00/9.27 The TRS P consists of the following rules: 29.00/9.27 29.00/9.27 U3_GAA(X, less_out_ag(Y)) -> INSERT_IN_GAA(X) 29.00/9.27 INSERT_IN_GAA(0) -> U1_GAA(0, less_out_ga) 29.00/9.27 INSERT_IN_GAA(s(x0)) -> U1_GAA(s(x0), U5_ga(less_in_ga(x0))) 29.00/9.27 INSERT_IN_GAA(s(x0)) -> U3_GAA(s(x0), less_out_ag(0)) 29.00/9.27 INSERT_IN_GAA(s(x0)) -> U3_GAA(s(x0), U5_ag(less_in_ag(x0))) 29.00/9.27 U1_GAA(0, less_out_ga) -> INSERT_IN_GAA(0) 29.00/9.27 U1_GAA(s(z0), less_out_ga) -> INSERT_IN_GAA(s(z0)) 29.00/9.27 29.00/9.27 The TRS R consists of the following rules: 29.00/9.27 29.00/9.27 less_in_ga(0) -> less_out_ga 29.00/9.27 less_in_ga(s(X)) -> U5_ga(less_in_ga(X)) 29.00/9.27 less_in_ag(s(X1)) -> less_out_ag(0) 29.00/9.27 less_in_ag(s(Y)) -> U5_ag(less_in_ag(Y)) 29.00/9.27 U5_ga(less_out_ga) -> less_out_ga 29.00/9.27 U5_ag(less_out_ag(X)) -> less_out_ag(s(X)) 29.00/9.27 29.00/9.27 The set Q consists of the following terms: 29.00/9.27 29.00/9.27 less_in_ga(x0) 29.00/9.27 less_in_ag(x0) 29.00/9.27 U5_ga(x0) 29.00/9.27 U5_ag(x0) 29.00/9.27 29.00/9.27 We have to consider all (P,Q,R)-chains. 29.00/9.27 ---------------------------------------- 29.00/9.27 29.00/9.27 (32) DependencyGraphProof (EQUIVALENT) 29.00/9.27 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs. 29.00/9.27 ---------------------------------------- 29.00/9.27 29.00/9.27 (33) 29.00/9.27 Complex Obligation (AND) 29.00/9.27 29.00/9.27 ---------------------------------------- 29.00/9.27 29.00/9.27 (34) 29.00/9.27 Obligation: 29.00/9.27 Q DP problem: 29.00/9.27 The TRS P consists of the following rules: 29.00/9.27 29.00/9.27 U1_GAA(0, less_out_ga) -> INSERT_IN_GAA(0) 29.00/9.27 INSERT_IN_GAA(0) -> U1_GAA(0, less_out_ga) 29.00/9.27 29.00/9.27 The TRS R consists of the following rules: 29.00/9.27 29.00/9.27 less_in_ga(0) -> less_out_ga 29.00/9.27 less_in_ga(s(X)) -> U5_ga(less_in_ga(X)) 29.00/9.27 less_in_ag(s(X1)) -> less_out_ag(0) 29.00/9.27 less_in_ag(s(Y)) -> U5_ag(less_in_ag(Y)) 29.00/9.27 U5_ga(less_out_ga) -> less_out_ga 29.00/9.27 U5_ag(less_out_ag(X)) -> less_out_ag(s(X)) 29.00/9.27 29.00/9.27 The set Q consists of the following terms: 29.00/9.27 29.00/9.27 less_in_ga(x0) 29.00/9.27 less_in_ag(x0) 29.00/9.27 U5_ga(x0) 29.00/9.27 U5_ag(x0) 29.00/9.27 29.00/9.27 We have to consider all (P,Q,R)-chains. 29.00/9.27 ---------------------------------------- 29.00/9.27 29.00/9.27 (35) UsableRulesProof (EQUIVALENT) 29.00/9.27 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. 29.00/9.27 ---------------------------------------- 29.00/9.27 29.00/9.27 (36) 29.00/9.27 Obligation: 29.00/9.27 Q DP problem: 29.00/9.27 The TRS P consists of the following rules: 29.00/9.27 29.00/9.27 U1_GAA(0, less_out_ga) -> INSERT_IN_GAA(0) 29.00/9.27 INSERT_IN_GAA(0) -> U1_GAA(0, less_out_ga) 29.00/9.27 29.00/9.27 R is empty. 29.00/9.27 The set Q consists of the following terms: 29.00/9.27 29.00/9.27 less_in_ga(x0) 29.00/9.27 less_in_ag(x0) 29.00/9.27 U5_ga(x0) 29.00/9.27 U5_ag(x0) 29.00/9.27 29.00/9.27 We have to consider all (P,Q,R)-chains. 29.00/9.27 ---------------------------------------- 29.00/9.27 29.00/9.27 (37) QReductionProof (EQUIVALENT) 29.00/9.27 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 29.00/9.27 29.00/9.27 less_in_ga(x0) 29.00/9.27 less_in_ag(x0) 29.00/9.27 U5_ga(x0) 29.00/9.27 U5_ag(x0) 29.00/9.27 29.00/9.27 29.00/9.27 ---------------------------------------- 29.00/9.27 29.00/9.27 (38) 29.00/9.27 Obligation: 29.00/9.27 Q DP problem: 29.00/9.27 The TRS P consists of the following rules: 29.00/9.27 29.00/9.27 U1_GAA(0, less_out_ga) -> INSERT_IN_GAA(0) 29.00/9.27 INSERT_IN_GAA(0) -> U1_GAA(0, less_out_ga) 29.00/9.27 29.00/9.27 R is empty. 29.00/9.27 Q is empty. 29.00/9.27 We have to consider all (P,Q,R)-chains. 29.00/9.27 ---------------------------------------- 29.00/9.27 29.00/9.27 (39) 29.00/9.27 Obligation: 29.00/9.27 Q DP problem: 29.00/9.27 The TRS P consists of the following rules: 29.00/9.27 29.00/9.27 INSERT_IN_GAA(s(x0)) -> U1_GAA(s(x0), U5_ga(less_in_ga(x0))) 29.00/9.27 U1_GAA(s(z0), less_out_ga) -> INSERT_IN_GAA(s(z0)) 29.00/9.27 INSERT_IN_GAA(s(x0)) -> U3_GAA(s(x0), less_out_ag(0)) 29.00/9.27 U3_GAA(X, less_out_ag(Y)) -> INSERT_IN_GAA(X) 29.00/9.27 INSERT_IN_GAA(s(x0)) -> U3_GAA(s(x0), U5_ag(less_in_ag(x0))) 29.00/9.27 29.00/9.27 The TRS R consists of the following rules: 29.00/9.27 29.00/9.27 less_in_ga(0) -> less_out_ga 29.00/9.27 less_in_ga(s(X)) -> U5_ga(less_in_ga(X)) 29.00/9.27 less_in_ag(s(X1)) -> less_out_ag(0) 29.00/9.27 less_in_ag(s(Y)) -> U5_ag(less_in_ag(Y)) 29.00/9.27 U5_ga(less_out_ga) -> less_out_ga 29.00/9.27 U5_ag(less_out_ag(X)) -> less_out_ag(s(X)) 29.00/9.27 29.00/9.27 The set Q consists of the following terms: 29.00/9.27 29.00/9.27 less_in_ga(x0) 29.00/9.27 less_in_ag(x0) 29.00/9.27 U5_ga(x0) 29.00/9.27 U5_ag(x0) 29.00/9.27 29.00/9.27 We have to consider all (P,Q,R)-chains. 29.00/9.27 ---------------------------------------- 29.00/9.27 29.00/9.27 (40) TransformationProof (EQUIVALENT) 29.00/9.27 By instantiating [LPAR04] the rule U3_GAA(X, less_out_ag(Y)) -> INSERT_IN_GAA(X) we obtained the following new rules [LPAR04]: 29.00/9.27 29.00/9.27 (U3_GAA(s(z0), less_out_ag(0)) -> INSERT_IN_GAA(s(z0)),U3_GAA(s(z0), less_out_ag(0)) -> INSERT_IN_GAA(s(z0))) 29.00/9.27 (U3_GAA(s(z0), less_out_ag(x1)) -> INSERT_IN_GAA(s(z0)),U3_GAA(s(z0), less_out_ag(x1)) -> INSERT_IN_GAA(s(z0))) 29.00/9.27 29.00/9.27 29.00/9.27 ---------------------------------------- 29.00/9.27 29.00/9.27 (41) 29.00/9.27 Obligation: 29.00/9.27 Q DP problem: 29.00/9.27 The TRS P consists of the following rules: 29.00/9.27 29.00/9.27 INSERT_IN_GAA(s(x0)) -> U1_GAA(s(x0), U5_ga(less_in_ga(x0))) 29.00/9.27 U1_GAA(s(z0), less_out_ga) -> INSERT_IN_GAA(s(z0)) 29.00/9.27 INSERT_IN_GAA(s(x0)) -> U3_GAA(s(x0), less_out_ag(0)) 29.00/9.27 INSERT_IN_GAA(s(x0)) -> U3_GAA(s(x0), U5_ag(less_in_ag(x0))) 29.00/9.27 U3_GAA(s(z0), less_out_ag(0)) -> INSERT_IN_GAA(s(z0)) 29.00/9.27 U3_GAA(s(z0), less_out_ag(x1)) -> INSERT_IN_GAA(s(z0)) 29.00/9.27 29.00/9.27 The TRS R consists of the following rules: 29.00/9.27 29.00/9.27 less_in_ga(0) -> less_out_ga 29.00/9.27 less_in_ga(s(X)) -> U5_ga(less_in_ga(X)) 29.00/9.27 less_in_ag(s(X1)) -> less_out_ag(0) 29.00/9.27 less_in_ag(s(Y)) -> U5_ag(less_in_ag(Y)) 29.00/9.27 U5_ga(less_out_ga) -> less_out_ga 29.00/9.27 U5_ag(less_out_ag(X)) -> less_out_ag(s(X)) 29.00/9.27 29.00/9.27 The set Q consists of the following terms: 29.00/9.27 29.00/9.27 less_in_ga(x0) 29.00/9.27 less_in_ag(x0) 29.00/9.27 U5_ga(x0) 29.00/9.27 U5_ag(x0) 29.00/9.27 29.00/9.27 We have to consider all (P,Q,R)-chains. 29.00/9.27 ---------------------------------------- 29.00/9.27 29.00/9.27 (42) PrologToPiTRSProof (SOUND) 29.00/9.27 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 29.00/9.27 29.00/9.27 insert_in_3: (b,f,f) 29.00/9.27 29.00/9.27 less_in_2: (b,f) (f,b) 29.00/9.27 29.00/9.27 Transforming Prolog into the following Term Rewriting System: 29.00/9.27 29.00/9.27 Pi-finite rewrite system: 29.00/9.27 The TRS R consists of the following rules: 29.00/9.27 29.00/9.27 insert_in_gaa(X, void, tree(X, void, void)) -> insert_out_gaa(X, void, tree(X, void, void)) 29.00/9.27 insert_in_gaa(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_gaa(X, tree(X, Left, Right), tree(X, Left, Right)) 29.00/9.27 insert_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 29.00/9.27 less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) 29.00/9.27 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 29.00/9.27 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 29.00/9.27 U1_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> U2_gaa(X, Y, Left, Right, Left1, insert_in_gaa(X, Left, Left1)) 29.00/9.27 insert_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 29.00/9.27 less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1)) 29.00/9.27 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 29.00/9.27 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 29.00/9.27 U3_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> U4_gaa(X, Y, Left, Right, Right1, insert_in_gaa(X, Right, Right1)) 29.00/9.27 U4_gaa(X, Y, Left, Right, Right1, insert_out_gaa(X, Right, Right1)) -> insert_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) 29.00/9.27 U2_gaa(X, Y, Left, Right, Left1, insert_out_gaa(X, Left, Left1)) -> insert_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) 29.00/9.27 29.00/9.27 The argument filtering Pi contains the following mapping: 29.00/9.27 insert_in_gaa(x1, x2, x3) = insert_in_gaa(x1) 29.00/9.27 29.00/9.27 insert_out_gaa(x1, x2, x3) = insert_out_gaa(x1) 29.00/9.27 29.00/9.27 U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x1, x6) 29.00/9.27 29.00/9.27 less_in_ga(x1, x2) = less_in_ga(x1) 29.00/9.27 29.00/9.27 0 = 0 29.00/9.27 29.00/9.27 less_out_ga(x1, x2) = less_out_ga(x1) 29.00/9.27 29.00/9.27 s(x1) = s(x1) 29.00/9.27 29.00/9.27 U5_ga(x1, x2, x3) = U5_ga(x1, x3) 29.00/9.27 29.00/9.27 U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6) 29.00/9.27 29.00/9.27 U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x1, x6) 29.00/9.27 29.00/9.27 less_in_ag(x1, x2) = less_in_ag(x2) 29.00/9.27 29.00/9.27 less_out_ag(x1, x2) = less_out_ag(x1, x2) 29.00/9.27 29.00/9.27 U5_ag(x1, x2, x3) = U5_ag(x2, x3) 29.00/9.27 29.00/9.27 U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6) 29.00/9.27 29.00/9.27 29.00/9.27 29.00/9.27 29.00/9.27 29.00/9.27 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 29.00/9.27 29.00/9.27 29.00/9.27 29.00/9.27 ---------------------------------------- 29.00/9.27 29.00/9.27 (43) 29.00/9.27 Obligation: 29.00/9.27 Pi-finite rewrite system: 29.00/9.27 The TRS R consists of the following rules: 29.00/9.27 29.00/9.27 insert_in_gaa(X, void, tree(X, void, void)) -> insert_out_gaa(X, void, tree(X, void, void)) 29.00/9.29 insert_in_gaa(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_gaa(X, tree(X, Left, Right), tree(X, Left, Right)) 29.00/9.29 insert_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 29.00/9.29 less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) 29.00/9.29 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 29.00/9.29 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 29.00/9.29 U1_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> U2_gaa(X, Y, Left, Right, Left1, insert_in_gaa(X, Left, Left1)) 29.00/9.29 insert_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 29.00/9.29 less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1)) 29.00/9.29 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 29.00/9.29 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 29.00/9.29 U3_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> U4_gaa(X, Y, Left, Right, Right1, insert_in_gaa(X, Right, Right1)) 29.00/9.29 U4_gaa(X, Y, Left, Right, Right1, insert_out_gaa(X, Right, Right1)) -> insert_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) 29.00/9.29 U2_gaa(X, Y, Left, Right, Left1, insert_out_gaa(X, Left, Left1)) -> insert_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) 29.00/9.29 29.00/9.29 The argument filtering Pi contains the following mapping: 29.00/9.29 insert_in_gaa(x1, x2, x3) = insert_in_gaa(x1) 29.00/9.29 29.00/9.29 insert_out_gaa(x1, x2, x3) = insert_out_gaa(x1) 29.00/9.29 29.00/9.29 U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x1, x6) 29.00/9.29 29.00/9.29 less_in_ga(x1, x2) = less_in_ga(x1) 29.00/9.29 29.00/9.29 0 = 0 29.00/9.29 29.00/9.29 less_out_ga(x1, x2) = less_out_ga(x1) 29.00/9.29 29.00/9.29 s(x1) = s(x1) 29.00/9.29 29.00/9.29 U5_ga(x1, x2, x3) = U5_ga(x1, x3) 29.00/9.29 29.00/9.29 U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6) 29.00/9.29 29.00/9.29 U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x1, x6) 29.00/9.29 29.00/9.29 less_in_ag(x1, x2) = less_in_ag(x2) 29.00/9.29 29.00/9.29 less_out_ag(x1, x2) = less_out_ag(x1, x2) 29.00/9.29 29.00/9.29 U5_ag(x1, x2, x3) = U5_ag(x2, x3) 29.00/9.29 29.00/9.29 U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6) 29.00/9.29 29.00/9.29 29.00/9.29 29.00/9.29 ---------------------------------------- 29.00/9.29 29.00/9.29 (44) DependencyPairsProof (EQUIVALENT) 29.00/9.29 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 29.00/9.29 Pi DP problem: 29.00/9.29 The TRS P consists of the following rules: 29.00/9.29 29.00/9.29 INSERT_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_GAA(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 29.00/9.29 INSERT_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> LESS_IN_GA(X, Y) 29.00/9.29 LESS_IN_GA(s(X), s(Y)) -> U5_GA(X, Y, less_in_ga(X, Y)) 29.00/9.29 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 29.00/9.29 U1_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> U2_GAA(X, Y, Left, Right, Left1, insert_in_gaa(X, Left, Left1)) 29.00/9.29 U1_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> INSERT_IN_GAA(X, Left, Left1) 29.00/9.29 INSERT_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_GAA(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 29.00/9.29 INSERT_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> LESS_IN_AG(Y, X) 29.00/9.29 LESS_IN_AG(s(X), s(Y)) -> U5_AG(X, Y, less_in_ag(X, Y)) 29.00/9.29 LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) 29.00/9.29 U3_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> U4_GAA(X, Y, Left, Right, Right1, insert_in_gaa(X, Right, Right1)) 29.00/9.29 U3_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> INSERT_IN_GAA(X, Right, Right1) 29.00/9.29 29.00/9.29 The TRS R consists of the following rules: 29.00/9.29 29.00/9.29 insert_in_gaa(X, void, tree(X, void, void)) -> insert_out_gaa(X, void, tree(X, void, void)) 29.00/9.29 insert_in_gaa(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_gaa(X, tree(X, Left, Right), tree(X, Left, Right)) 29.00/9.29 insert_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 29.00/9.29 less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) 29.00/9.29 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 29.00/9.29 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 29.00/9.29 U1_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> U2_gaa(X, Y, Left, Right, Left1, insert_in_gaa(X, Left, Left1)) 29.00/9.29 insert_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 29.00/9.29 less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1)) 29.00/9.29 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 29.00/9.29 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 29.00/9.29 U3_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> U4_gaa(X, Y, Left, Right, Right1, insert_in_gaa(X, Right, Right1)) 29.00/9.29 U4_gaa(X, Y, Left, Right, Right1, insert_out_gaa(X, Right, Right1)) -> insert_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) 29.00/9.29 U2_gaa(X, Y, Left, Right, Left1, insert_out_gaa(X, Left, Left1)) -> insert_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) 29.00/9.29 29.00/9.29 The argument filtering Pi contains the following mapping: 29.00/9.29 insert_in_gaa(x1, x2, x3) = insert_in_gaa(x1) 29.00/9.29 29.00/9.29 insert_out_gaa(x1, x2, x3) = insert_out_gaa(x1) 29.00/9.29 29.00/9.29 U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x1, x6) 29.00/9.29 29.00/9.29 less_in_ga(x1, x2) = less_in_ga(x1) 29.00/9.29 29.00/9.29 0 = 0 29.00/9.29 29.00/9.29 less_out_ga(x1, x2) = less_out_ga(x1) 29.00/9.29 29.00/9.29 s(x1) = s(x1) 29.00/9.29 29.00/9.29 U5_ga(x1, x2, x3) = U5_ga(x1, x3) 29.00/9.29 29.00/9.29 U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6) 29.00/9.29 29.00/9.29 U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x1, x6) 29.00/9.29 29.00/9.29 less_in_ag(x1, x2) = less_in_ag(x2) 29.00/9.29 29.00/9.29 less_out_ag(x1, x2) = less_out_ag(x1, x2) 29.00/9.29 29.00/9.29 U5_ag(x1, x2, x3) = U5_ag(x2, x3) 29.00/9.29 29.00/9.29 U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6) 29.00/9.29 29.00/9.29 INSERT_IN_GAA(x1, x2, x3) = INSERT_IN_GAA(x1) 29.00/9.29 29.00/9.29 U1_GAA(x1, x2, x3, x4, x5, x6) = U1_GAA(x1, x6) 29.00/9.29 29.00/9.29 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 29.00/9.29 29.00/9.29 U5_GA(x1, x2, x3) = U5_GA(x1, x3) 29.00/9.29 29.00/9.29 U2_GAA(x1, x2, x3, x4, x5, x6) = U2_GAA(x1, x6) 29.00/9.29 29.00/9.29 U3_GAA(x1, x2, x3, x4, x5, x6) = U3_GAA(x1, x6) 29.00/9.29 29.00/9.29 LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) 29.00/9.29 29.00/9.29 U5_AG(x1, x2, x3) = U5_AG(x2, x3) 29.00/9.29 29.00/9.29 U4_GAA(x1, x2, x3, x4, x5, x6) = U4_GAA(x1, x6) 29.00/9.29 29.00/9.29 29.00/9.29 We have to consider all (P,R,Pi)-chains 29.00/9.29 ---------------------------------------- 29.00/9.29 29.00/9.29 (45) 29.00/9.29 Obligation: 29.00/9.29 Pi DP problem: 29.00/9.29 The TRS P consists of the following rules: 29.00/9.29 29.00/9.29 INSERT_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_GAA(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 29.00/9.29 INSERT_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> LESS_IN_GA(X, Y) 29.00/9.29 LESS_IN_GA(s(X), s(Y)) -> U5_GA(X, Y, less_in_ga(X, Y)) 29.00/9.29 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 29.00/9.29 U1_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> U2_GAA(X, Y, Left, Right, Left1, insert_in_gaa(X, Left, Left1)) 29.00/9.29 U1_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> INSERT_IN_GAA(X, Left, Left1) 29.00/9.29 INSERT_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_GAA(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 29.00/9.29 INSERT_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> LESS_IN_AG(Y, X) 29.00/9.29 LESS_IN_AG(s(X), s(Y)) -> U5_AG(X, Y, less_in_ag(X, Y)) 29.00/9.29 LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) 29.00/9.29 U3_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> U4_GAA(X, Y, Left, Right, Right1, insert_in_gaa(X, Right, Right1)) 29.00/9.29 U3_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> INSERT_IN_GAA(X, Right, Right1) 29.00/9.29 29.00/9.29 The TRS R consists of the following rules: 29.00/9.29 29.00/9.29 insert_in_gaa(X, void, tree(X, void, void)) -> insert_out_gaa(X, void, tree(X, void, void)) 29.00/9.29 insert_in_gaa(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_gaa(X, tree(X, Left, Right), tree(X, Left, Right)) 29.00/9.29 insert_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 29.00/9.29 less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) 29.00/9.29 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 29.00/9.29 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 29.00/9.29 U1_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> U2_gaa(X, Y, Left, Right, Left1, insert_in_gaa(X, Left, Left1)) 29.00/9.29 insert_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 29.00/9.29 less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1)) 29.00/9.29 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 29.00/9.29 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 29.00/9.29 U3_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> U4_gaa(X, Y, Left, Right, Right1, insert_in_gaa(X, Right, Right1)) 29.00/9.29 U4_gaa(X, Y, Left, Right, Right1, insert_out_gaa(X, Right, Right1)) -> insert_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) 29.00/9.29 U2_gaa(X, Y, Left, Right, Left1, insert_out_gaa(X, Left, Left1)) -> insert_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) 29.00/9.29 29.00/9.29 The argument filtering Pi contains the following mapping: 29.00/9.29 insert_in_gaa(x1, x2, x3) = insert_in_gaa(x1) 29.00/9.29 29.00/9.29 insert_out_gaa(x1, x2, x3) = insert_out_gaa(x1) 29.00/9.29 29.00/9.29 U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x1, x6) 29.00/9.29 29.00/9.29 less_in_ga(x1, x2) = less_in_ga(x1) 29.00/9.29 29.00/9.29 0 = 0 29.00/9.29 29.00/9.29 less_out_ga(x1, x2) = less_out_ga(x1) 29.00/9.29 29.00/9.29 s(x1) = s(x1) 29.00/9.29 29.00/9.29 U5_ga(x1, x2, x3) = U5_ga(x1, x3) 29.00/9.29 29.00/9.29 U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6) 29.00/9.29 29.00/9.29 U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x1, x6) 29.00/9.29 29.00/9.29 less_in_ag(x1, x2) = less_in_ag(x2) 29.00/9.29 29.00/9.29 less_out_ag(x1, x2) = less_out_ag(x1, x2) 29.00/9.29 29.00/9.29 U5_ag(x1, x2, x3) = U5_ag(x2, x3) 29.00/9.29 29.00/9.29 U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6) 29.00/9.29 29.00/9.29 INSERT_IN_GAA(x1, x2, x3) = INSERT_IN_GAA(x1) 29.00/9.29 29.00/9.29 U1_GAA(x1, x2, x3, x4, x5, x6) = U1_GAA(x1, x6) 29.00/9.29 29.00/9.29 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 29.00/9.29 29.00/9.29 U5_GA(x1, x2, x3) = U5_GA(x1, x3) 29.00/9.29 29.00/9.29 U2_GAA(x1, x2, x3, x4, x5, x6) = U2_GAA(x1, x6) 29.00/9.29 29.00/9.29 U3_GAA(x1, x2, x3, x4, x5, x6) = U3_GAA(x1, x6) 29.00/9.29 29.00/9.29 LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) 29.00/9.29 29.00/9.29 U5_AG(x1, x2, x3) = U5_AG(x2, x3) 29.00/9.29 29.00/9.29 U4_GAA(x1, x2, x3, x4, x5, x6) = U4_GAA(x1, x6) 29.00/9.29 29.00/9.29 29.00/9.29 We have to consider all (P,R,Pi)-chains 29.00/9.29 ---------------------------------------- 29.00/9.29 29.00/9.29 (46) DependencyGraphProof (EQUIVALENT) 29.00/9.29 The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 6 less nodes. 29.00/9.29 ---------------------------------------- 29.00/9.29 29.00/9.29 (47) 29.00/9.29 Complex Obligation (AND) 29.00/9.29 29.00/9.29 ---------------------------------------- 29.00/9.29 29.00/9.29 (48) 29.00/9.29 Obligation: 29.00/9.29 Pi DP problem: 29.00/9.29 The TRS P consists of the following rules: 29.00/9.29 29.00/9.29 LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) 29.00/9.29 29.00/9.29 The TRS R consists of the following rules: 29.00/9.29 29.00/9.29 insert_in_gaa(X, void, tree(X, void, void)) -> insert_out_gaa(X, void, tree(X, void, void)) 29.00/9.29 insert_in_gaa(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_gaa(X, tree(X, Left, Right), tree(X, Left, Right)) 29.00/9.29 insert_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 29.00/9.29 less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) 29.00/9.29 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 29.00/9.29 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 29.00/9.29 U1_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> U2_gaa(X, Y, Left, Right, Left1, insert_in_gaa(X, Left, Left1)) 29.00/9.29 insert_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 29.00/9.29 less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1)) 29.00/9.29 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 29.00/9.29 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 29.00/9.29 U3_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> U4_gaa(X, Y, Left, Right, Right1, insert_in_gaa(X, Right, Right1)) 29.00/9.29 U4_gaa(X, Y, Left, Right, Right1, insert_out_gaa(X, Right, Right1)) -> insert_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) 29.00/9.29 U2_gaa(X, Y, Left, Right, Left1, insert_out_gaa(X, Left, Left1)) -> insert_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) 29.00/9.29 29.00/9.29 The argument filtering Pi contains the following mapping: 29.00/9.29 insert_in_gaa(x1, x2, x3) = insert_in_gaa(x1) 29.00/9.29 29.00/9.29 insert_out_gaa(x1, x2, x3) = insert_out_gaa(x1) 29.00/9.29 29.00/9.29 U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x1, x6) 29.00/9.29 29.00/9.29 less_in_ga(x1, x2) = less_in_ga(x1) 29.00/9.29 29.00/9.29 0 = 0 29.00/9.29 29.00/9.29 less_out_ga(x1, x2) = less_out_ga(x1) 29.00/9.29 29.00/9.29 s(x1) = s(x1) 29.00/9.29 29.00/9.29 U5_ga(x1, x2, x3) = U5_ga(x1, x3) 29.00/9.29 29.00/9.29 U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6) 29.00/9.29 29.00/9.29 U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x1, x6) 29.00/9.29 29.00/9.29 less_in_ag(x1, x2) = less_in_ag(x2) 29.00/9.29 29.00/9.29 less_out_ag(x1, x2) = less_out_ag(x1, x2) 29.00/9.29 29.00/9.29 U5_ag(x1, x2, x3) = U5_ag(x2, x3) 29.00/9.29 29.00/9.29 U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6) 29.00/9.29 29.00/9.29 LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) 29.00/9.29 29.00/9.29 29.00/9.29 We have to consider all (P,R,Pi)-chains 29.00/9.29 ---------------------------------------- 29.00/9.29 29.00/9.29 (49) UsableRulesProof (EQUIVALENT) 29.00/9.29 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 29.00/9.29 ---------------------------------------- 29.00/9.29 29.00/9.29 (50) 29.00/9.29 Obligation: 29.00/9.29 Pi DP problem: 29.00/9.29 The TRS P consists of the following rules: 29.00/9.29 29.00/9.29 LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) 29.00/9.29 29.00/9.29 R is empty. 29.00/9.29 The argument filtering Pi contains the following mapping: 29.00/9.29 s(x1) = s(x1) 29.00/9.29 29.00/9.29 LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) 29.00/9.29 29.00/9.29 29.00/9.29 We have to consider all (P,R,Pi)-chains 29.00/9.29 ---------------------------------------- 29.00/9.29 29.00/9.29 (51) PiDPToQDPProof (SOUND) 29.00/9.29 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 29.00/9.29 ---------------------------------------- 29.00/9.29 29.00/9.29 (52) 29.00/9.29 Obligation: 29.00/9.29 Q DP problem: 29.00/9.29 The TRS P consists of the following rules: 29.00/9.29 29.00/9.29 LESS_IN_AG(s(Y)) -> LESS_IN_AG(Y) 29.00/9.29 29.00/9.29 R is empty. 29.00/9.29 Q is empty. 29.00/9.29 We have to consider all (P,Q,R)-chains. 29.00/9.29 ---------------------------------------- 29.00/9.29 29.00/9.29 (53) QDPSizeChangeProof (EQUIVALENT) 29.00/9.29 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. 29.00/9.29 29.00/9.29 From the DPs we obtained the following set of size-change graphs: 29.00/9.29 *LESS_IN_AG(s(Y)) -> LESS_IN_AG(Y) 29.00/9.29 The graph contains the following edges 1 > 1 29.00/9.29 29.00/9.29 29.00/9.29 ---------------------------------------- 29.00/9.29 29.00/9.29 (54) 29.00/9.29 YES 29.00/9.29 29.00/9.29 ---------------------------------------- 29.00/9.29 29.00/9.29 (55) 29.00/9.29 Obligation: 29.00/9.29 Pi DP problem: 29.00/9.29 The TRS P consists of the following rules: 29.00/9.29 29.00/9.29 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 29.00/9.29 29.00/9.29 The TRS R consists of the following rules: 29.00/9.29 29.00/9.29 insert_in_gaa(X, void, tree(X, void, void)) -> insert_out_gaa(X, void, tree(X, void, void)) 29.00/9.29 insert_in_gaa(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_gaa(X, tree(X, Left, Right), tree(X, Left, Right)) 29.00/9.29 insert_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 29.00/9.29 less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) 29.00/9.29 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 29.00/9.29 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 29.00/9.29 U1_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> U2_gaa(X, Y, Left, Right, Left1, insert_in_gaa(X, Left, Left1)) 29.00/9.29 insert_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 29.00/9.29 less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1)) 29.00/9.29 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 29.00/9.29 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 29.00/9.29 U3_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> U4_gaa(X, Y, Left, Right, Right1, insert_in_gaa(X, Right, Right1)) 29.00/9.29 U4_gaa(X, Y, Left, Right, Right1, insert_out_gaa(X, Right, Right1)) -> insert_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) 29.00/9.29 U2_gaa(X, Y, Left, Right, Left1, insert_out_gaa(X, Left, Left1)) -> insert_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) 29.00/9.29 29.00/9.29 The argument filtering Pi contains the following mapping: 29.00/9.29 insert_in_gaa(x1, x2, x3) = insert_in_gaa(x1) 29.00/9.29 29.00/9.29 insert_out_gaa(x1, x2, x3) = insert_out_gaa(x1) 29.00/9.29 29.00/9.29 U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x1, x6) 29.00/9.29 29.00/9.29 less_in_ga(x1, x2) = less_in_ga(x1) 29.00/9.29 29.00/9.29 0 = 0 29.00/9.29 29.00/9.29 less_out_ga(x1, x2) = less_out_ga(x1) 29.00/9.29 29.00/9.29 s(x1) = s(x1) 29.00/9.29 29.00/9.29 U5_ga(x1, x2, x3) = U5_ga(x1, x3) 29.00/9.29 29.00/9.29 U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6) 29.00/9.29 29.00/9.29 U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x1, x6) 29.00/9.29 29.00/9.29 less_in_ag(x1, x2) = less_in_ag(x2) 29.00/9.29 29.00/9.29 less_out_ag(x1, x2) = less_out_ag(x1, x2) 29.00/9.29 29.00/9.29 U5_ag(x1, x2, x3) = U5_ag(x2, x3) 29.00/9.29 29.00/9.29 U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6) 29.00/9.29 29.00/9.29 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 29.00/9.29 29.00/9.29 29.00/9.29 We have to consider all (P,R,Pi)-chains 29.00/9.29 ---------------------------------------- 29.00/9.29 29.00/9.29 (56) UsableRulesProof (EQUIVALENT) 29.00/9.29 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 29.00/9.29 ---------------------------------------- 29.00/9.29 29.00/9.29 (57) 29.00/9.29 Obligation: 29.00/9.29 Pi DP problem: 29.00/9.29 The TRS P consists of the following rules: 29.00/9.29 29.00/9.29 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 29.00/9.29 29.00/9.29 R is empty. 29.00/9.29 The argument filtering Pi contains the following mapping: 29.00/9.29 s(x1) = s(x1) 29.00/9.29 29.00/9.29 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 29.00/9.29 29.00/9.29 29.00/9.29 We have to consider all (P,R,Pi)-chains 29.00/9.29 ---------------------------------------- 29.00/9.29 29.00/9.29 (58) PiDPToQDPProof (SOUND) 29.00/9.29 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 29.00/9.29 ---------------------------------------- 29.00/9.29 29.00/9.29 (59) 29.00/9.29 Obligation: 29.00/9.29 Q DP problem: 29.00/9.29 The TRS P consists of the following rules: 29.00/9.29 29.00/9.29 LESS_IN_GA(s(X)) -> LESS_IN_GA(X) 29.00/9.29 29.00/9.29 R is empty. 29.00/9.29 Q is empty. 29.00/9.29 We have to consider all (P,Q,R)-chains. 29.00/9.29 ---------------------------------------- 29.00/9.29 29.00/9.29 (60) QDPSizeChangeProof (EQUIVALENT) 29.00/9.29 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. 29.00/9.29 29.00/9.29 From the DPs we obtained the following set of size-change graphs: 29.00/9.29 *LESS_IN_GA(s(X)) -> LESS_IN_GA(X) 29.00/9.29 The graph contains the following edges 1 > 1 29.00/9.29 29.00/9.29 29.00/9.29 ---------------------------------------- 29.00/9.29 29.00/9.29 (61) 29.00/9.29 YES 29.00/9.29 29.00/9.29 ---------------------------------------- 29.00/9.29 29.00/9.29 (62) 29.00/9.29 Obligation: 29.00/9.29 Pi DP problem: 29.00/9.29 The TRS P consists of the following rules: 29.00/9.29 29.00/9.29 U1_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> INSERT_IN_GAA(X, Left, Left1) 29.00/9.29 INSERT_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_GAA(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 29.00/9.29 INSERT_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_GAA(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 29.00/9.29 U3_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> INSERT_IN_GAA(X, Right, Right1) 29.00/9.29 29.00/9.29 The TRS R consists of the following rules: 29.00/9.29 29.00/9.29 insert_in_gaa(X, void, tree(X, void, void)) -> insert_out_gaa(X, void, tree(X, void, void)) 29.00/9.29 insert_in_gaa(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_gaa(X, tree(X, Left, Right), tree(X, Left, Right)) 29.00/9.29 insert_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 29.00/9.29 less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) 29.00/9.29 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 29.00/9.29 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 29.00/9.29 U1_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> U2_gaa(X, Y, Left, Right, Left1, insert_in_gaa(X, Left, Left1)) 29.00/9.29 insert_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 29.00/9.29 less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1)) 29.00/9.29 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 29.00/9.29 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 29.00/9.29 U3_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> U4_gaa(X, Y, Left, Right, Right1, insert_in_gaa(X, Right, Right1)) 29.00/9.29 U4_gaa(X, Y, Left, Right, Right1, insert_out_gaa(X, Right, Right1)) -> insert_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) 29.00/9.29 U2_gaa(X, Y, Left, Right, Left1, insert_out_gaa(X, Left, Left1)) -> insert_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) 29.00/9.29 29.00/9.29 The argument filtering Pi contains the following mapping: 29.00/9.29 insert_in_gaa(x1, x2, x3) = insert_in_gaa(x1) 29.00/9.29 29.00/9.29 insert_out_gaa(x1, x2, x3) = insert_out_gaa(x1) 29.00/9.29 29.00/9.29 U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x1, x6) 29.00/9.29 29.00/9.29 less_in_ga(x1, x2) = less_in_ga(x1) 29.00/9.29 29.00/9.29 0 = 0 29.00/9.29 29.00/9.29 less_out_ga(x1, x2) = less_out_ga(x1) 29.00/9.29 29.00/9.29 s(x1) = s(x1) 29.00/9.29 29.00/9.29 U5_ga(x1, x2, x3) = U5_ga(x1, x3) 29.00/9.29 29.00/9.29 U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6) 29.00/9.29 29.00/9.29 U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x1, x6) 29.00/9.29 29.00/9.29 less_in_ag(x1, x2) = less_in_ag(x2) 29.00/9.29 29.00/9.29 less_out_ag(x1, x2) = less_out_ag(x1, x2) 29.00/9.29 29.00/9.29 U5_ag(x1, x2, x3) = U5_ag(x2, x3) 29.00/9.29 29.00/9.29 U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6) 29.00/9.29 29.00/9.29 INSERT_IN_GAA(x1, x2, x3) = INSERT_IN_GAA(x1) 29.00/9.29 29.00/9.29 U1_GAA(x1, x2, x3, x4, x5, x6) = U1_GAA(x1, x6) 29.00/9.29 29.00/9.29 U3_GAA(x1, x2, x3, x4, x5, x6) = U3_GAA(x1, x6) 29.00/9.29 29.00/9.29 29.00/9.29 We have to consider all (P,R,Pi)-chains 29.00/9.29 ---------------------------------------- 29.00/9.29 29.00/9.29 (63) UsableRulesProof (EQUIVALENT) 29.00/9.29 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 29.00/9.29 ---------------------------------------- 29.00/9.29 29.00/9.29 (64) 29.00/9.29 Obligation: 29.00/9.29 Pi DP problem: 29.00/9.29 The TRS P consists of the following rules: 29.00/9.29 29.00/9.29 U1_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) -> INSERT_IN_GAA(X, Left, Left1) 29.00/9.29 INSERT_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_GAA(X, Y, Left, Right, Left1, less_in_ga(X, Y)) 29.00/9.29 INSERT_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_GAA(X, Y, Left, Right, Right1, less_in_ag(Y, X)) 29.00/9.29 U3_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) -> INSERT_IN_GAA(X, Right, Right1) 29.00/9.29 29.00/9.29 The TRS R consists of the following rules: 29.00/9.29 29.00/9.29 less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) 29.00/9.29 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 29.00/9.29 less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1)) 29.00/9.29 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 29.00/9.29 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 29.00/9.29 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 29.00/9.29 29.00/9.29 The argument filtering Pi contains the following mapping: 29.00/9.29 less_in_ga(x1, x2) = less_in_ga(x1) 29.00/9.29 29.00/9.29 0 = 0 29.00/9.29 29.00/9.29 less_out_ga(x1, x2) = less_out_ga(x1) 29.00/9.29 29.00/9.29 s(x1) = s(x1) 29.00/9.29 29.00/9.29 U5_ga(x1, x2, x3) = U5_ga(x1, x3) 29.00/9.29 29.00/9.29 less_in_ag(x1, x2) = less_in_ag(x2) 29.00/9.29 29.00/9.29 less_out_ag(x1, x2) = less_out_ag(x1, x2) 29.00/9.29 29.00/9.29 U5_ag(x1, x2, x3) = U5_ag(x2, x3) 29.00/9.29 29.00/9.29 INSERT_IN_GAA(x1, x2, x3) = INSERT_IN_GAA(x1) 29.00/9.29 29.00/9.29 U1_GAA(x1, x2, x3, x4, x5, x6) = U1_GAA(x1, x6) 29.00/9.29 29.00/9.29 U3_GAA(x1, x2, x3, x4, x5, x6) = U3_GAA(x1, x6) 29.00/9.29 29.00/9.29 29.00/9.29 We have to consider all (P,R,Pi)-chains 29.00/9.29 ---------------------------------------- 29.00/9.29 29.00/9.29 (65) PiDPToQDPProof (SOUND) 29.00/9.29 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 29.00/9.29 ---------------------------------------- 29.00/9.29 29.00/9.29 (66) 29.00/9.29 Obligation: 29.00/9.29 Q DP problem: 29.00/9.29 The TRS P consists of the following rules: 29.00/9.29 29.00/9.29 U1_GAA(X, less_out_ga(X)) -> INSERT_IN_GAA(X) 29.00/9.29 INSERT_IN_GAA(X) -> U1_GAA(X, less_in_ga(X)) 29.00/9.29 INSERT_IN_GAA(X) -> U3_GAA(X, less_in_ag(X)) 29.00/9.29 U3_GAA(X, less_out_ag(Y, X)) -> INSERT_IN_GAA(X) 29.00/9.29 29.00/9.29 The TRS R consists of the following rules: 29.00/9.29 29.00/9.29 less_in_ga(0) -> less_out_ga(0) 29.00/9.29 less_in_ga(s(X)) -> U5_ga(X, less_in_ga(X)) 29.00/9.29 less_in_ag(s(X1)) -> less_out_ag(0, s(X1)) 29.00/9.29 less_in_ag(s(Y)) -> U5_ag(Y, less_in_ag(Y)) 29.00/9.29 U5_ga(X, less_out_ga(X)) -> less_out_ga(s(X)) 29.00/9.29 U5_ag(Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 29.00/9.29 29.00/9.29 The set Q consists of the following terms: 29.00/9.29 29.00/9.29 less_in_ga(x0) 29.00/9.29 less_in_ag(x0) 29.00/9.29 U5_ga(x0, x1) 29.00/9.29 U5_ag(x0, x1) 29.00/9.29 29.00/9.29 We have to consider all (P,Q,R)-chains. 29.00/9.29 ---------------------------------------- 29.00/9.29 29.00/9.29 (67) TransformationProof (SOUND) 29.00/9.29 By narrowing [LPAR04] the rule INSERT_IN_GAA(X) -> U1_GAA(X, less_in_ga(X)) at position [1] we obtained the following new rules [LPAR04]: 29.00/9.29 29.00/9.29 (INSERT_IN_GAA(0) -> U1_GAA(0, less_out_ga(0)),INSERT_IN_GAA(0) -> U1_GAA(0, less_out_ga(0))) 29.00/9.29 (INSERT_IN_GAA(s(x0)) -> U1_GAA(s(x0), U5_ga(x0, less_in_ga(x0))),INSERT_IN_GAA(s(x0)) -> U1_GAA(s(x0), U5_ga(x0, less_in_ga(x0)))) 29.00/9.29 29.00/9.29 29.00/9.29 ---------------------------------------- 29.00/9.29 29.00/9.29 (68) 29.00/9.29 Obligation: 29.00/9.29 Q DP problem: 29.00/9.29 The TRS P consists of the following rules: 29.00/9.29 29.00/9.29 U1_GAA(X, less_out_ga(X)) -> INSERT_IN_GAA(X) 29.00/9.29 INSERT_IN_GAA(X) -> U3_GAA(X, less_in_ag(X)) 29.00/9.29 U3_GAA(X, less_out_ag(Y, X)) -> INSERT_IN_GAA(X) 29.00/9.29 INSERT_IN_GAA(0) -> U1_GAA(0, less_out_ga(0)) 29.00/9.29 INSERT_IN_GAA(s(x0)) -> U1_GAA(s(x0), U5_ga(x0, less_in_ga(x0))) 29.00/9.29 29.00/9.29 The TRS R consists of the following rules: 29.00/9.29 29.00/9.29 less_in_ga(0) -> less_out_ga(0) 29.00/9.29 less_in_ga(s(X)) -> U5_ga(X, less_in_ga(X)) 29.00/9.29 less_in_ag(s(X1)) -> less_out_ag(0, s(X1)) 29.00/9.29 less_in_ag(s(Y)) -> U5_ag(Y, less_in_ag(Y)) 29.00/9.29 U5_ga(X, less_out_ga(X)) -> less_out_ga(s(X)) 29.00/9.29 U5_ag(Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 29.00/9.29 29.00/9.29 The set Q consists of the following terms: 29.00/9.29 29.00/9.29 less_in_ga(x0) 29.00/9.29 less_in_ag(x0) 29.00/9.29 U5_ga(x0, x1) 29.00/9.29 U5_ag(x0, x1) 29.00/9.29 29.00/9.29 We have to consider all (P,Q,R)-chains. 29.00/9.29 ---------------------------------------- 29.00/9.29 29.00/9.29 (69) TransformationProof (SOUND) 29.00/9.29 By narrowing [LPAR04] the rule INSERT_IN_GAA(X) -> U3_GAA(X, less_in_ag(X)) at position [1] we obtained the following new rules [LPAR04]: 29.00/9.29 29.00/9.29 (INSERT_IN_GAA(s(x0)) -> U3_GAA(s(x0), less_out_ag(0, s(x0))),INSERT_IN_GAA(s(x0)) -> U3_GAA(s(x0), less_out_ag(0, s(x0)))) 29.00/9.29 (INSERT_IN_GAA(s(x0)) -> U3_GAA(s(x0), U5_ag(x0, less_in_ag(x0))),INSERT_IN_GAA(s(x0)) -> U3_GAA(s(x0), U5_ag(x0, less_in_ag(x0)))) 29.00/9.29 29.00/9.29 29.00/9.29 ---------------------------------------- 29.00/9.29 29.00/9.29 (70) 29.00/9.29 Obligation: 29.00/9.29 Q DP problem: 29.00/9.29 The TRS P consists of the following rules: 29.00/9.29 29.00/9.29 U1_GAA(X, less_out_ga(X)) -> INSERT_IN_GAA(X) 29.00/9.29 U3_GAA(X, less_out_ag(Y, X)) -> INSERT_IN_GAA(X) 29.00/9.29 INSERT_IN_GAA(0) -> U1_GAA(0, less_out_ga(0)) 29.00/9.29 INSERT_IN_GAA(s(x0)) -> U1_GAA(s(x0), U5_ga(x0, less_in_ga(x0))) 29.00/9.29 INSERT_IN_GAA(s(x0)) -> U3_GAA(s(x0), less_out_ag(0, s(x0))) 29.00/9.29 INSERT_IN_GAA(s(x0)) -> U3_GAA(s(x0), U5_ag(x0, less_in_ag(x0))) 29.00/9.29 29.00/9.29 The TRS R consists of the following rules: 29.00/9.29 29.00/9.29 less_in_ga(0) -> less_out_ga(0) 29.00/9.29 less_in_ga(s(X)) -> U5_ga(X, less_in_ga(X)) 29.00/9.29 less_in_ag(s(X1)) -> less_out_ag(0, s(X1)) 29.00/9.29 less_in_ag(s(Y)) -> U5_ag(Y, less_in_ag(Y)) 29.00/9.29 U5_ga(X, less_out_ga(X)) -> less_out_ga(s(X)) 29.00/9.29 U5_ag(Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 29.00/9.29 29.00/9.29 The set Q consists of the following terms: 29.00/9.29 29.00/9.29 less_in_ga(x0) 29.00/9.29 less_in_ag(x0) 29.00/9.29 U5_ga(x0, x1) 29.00/9.29 U5_ag(x0, x1) 29.00/9.29 29.00/9.29 We have to consider all (P,Q,R)-chains. 29.00/9.29 ---------------------------------------- 29.00/9.29 29.00/9.29 (71) TransformationProof (EQUIVALENT) 29.00/9.29 By instantiating [LPAR04] the rule U1_GAA(X, less_out_ga(X)) -> INSERT_IN_GAA(X) we obtained the following new rules [LPAR04]: 29.00/9.29 29.00/9.29 (U1_GAA(0, less_out_ga(0)) -> INSERT_IN_GAA(0),U1_GAA(0, less_out_ga(0)) -> INSERT_IN_GAA(0)) 29.00/9.29 (U1_GAA(s(z0), less_out_ga(s(z0))) -> INSERT_IN_GAA(s(z0)),U1_GAA(s(z0), less_out_ga(s(z0))) -> INSERT_IN_GAA(s(z0))) 29.00/9.29 29.00/9.29 29.00/9.29 ---------------------------------------- 29.00/9.29 29.00/9.29 (72) 29.00/9.29 Obligation: 29.00/9.29 Q DP problem: 29.00/9.29 The TRS P consists of the following rules: 29.00/9.29 29.00/9.29 U3_GAA(X, less_out_ag(Y, X)) -> INSERT_IN_GAA(X) 29.00/9.29 INSERT_IN_GAA(0) -> U1_GAA(0, less_out_ga(0)) 29.00/9.29 INSERT_IN_GAA(s(x0)) -> U1_GAA(s(x0), U5_ga(x0, less_in_ga(x0))) 29.00/9.29 INSERT_IN_GAA(s(x0)) -> U3_GAA(s(x0), less_out_ag(0, s(x0))) 29.00/9.29 INSERT_IN_GAA(s(x0)) -> U3_GAA(s(x0), U5_ag(x0, less_in_ag(x0))) 29.00/9.29 U1_GAA(0, less_out_ga(0)) -> INSERT_IN_GAA(0) 29.00/9.29 U1_GAA(s(z0), less_out_ga(s(z0))) -> INSERT_IN_GAA(s(z0)) 29.00/9.29 29.00/9.29 The TRS R consists of the following rules: 29.00/9.29 29.00/9.29 less_in_ga(0) -> less_out_ga(0) 29.00/9.29 less_in_ga(s(X)) -> U5_ga(X, less_in_ga(X)) 29.00/9.29 less_in_ag(s(X1)) -> less_out_ag(0, s(X1)) 29.00/9.29 less_in_ag(s(Y)) -> U5_ag(Y, less_in_ag(Y)) 29.00/9.29 U5_ga(X, less_out_ga(X)) -> less_out_ga(s(X)) 29.00/9.29 U5_ag(Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 29.00/9.29 29.00/9.29 The set Q consists of the following terms: 29.00/9.29 29.00/9.29 less_in_ga(x0) 29.00/9.29 less_in_ag(x0) 29.00/9.29 U5_ga(x0, x1) 29.00/9.29 U5_ag(x0, x1) 29.00/9.29 29.00/9.29 We have to consider all (P,Q,R)-chains. 29.00/9.29 ---------------------------------------- 29.00/9.29 29.00/9.29 (73) DependencyGraphProof (EQUIVALENT) 29.00/9.29 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs. 29.00/9.29 ---------------------------------------- 29.00/9.29 29.00/9.29 (74) 29.00/9.29 Complex Obligation (AND) 29.00/9.29 29.00/9.29 ---------------------------------------- 29.00/9.29 29.00/9.29 (75) 29.00/9.29 Obligation: 29.00/9.29 Q DP problem: 29.00/9.29 The TRS P consists of the following rules: 29.00/9.29 29.00/9.29 U1_GAA(0, less_out_ga(0)) -> INSERT_IN_GAA(0) 29.00/9.29 INSERT_IN_GAA(0) -> U1_GAA(0, less_out_ga(0)) 29.00/9.29 29.00/9.29 The TRS R consists of the following rules: 29.00/9.29 29.00/9.29 less_in_ga(0) -> less_out_ga(0) 29.00/9.29 less_in_ga(s(X)) -> U5_ga(X, less_in_ga(X)) 29.00/9.29 less_in_ag(s(X1)) -> less_out_ag(0, s(X1)) 29.00/9.29 less_in_ag(s(Y)) -> U5_ag(Y, less_in_ag(Y)) 29.00/9.29 U5_ga(X, less_out_ga(X)) -> less_out_ga(s(X)) 29.00/9.29 U5_ag(Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 29.00/9.29 29.00/9.29 The set Q consists of the following terms: 29.00/9.29 29.00/9.29 less_in_ga(x0) 29.00/9.29 less_in_ag(x0) 29.00/9.29 U5_ga(x0, x1) 29.00/9.29 U5_ag(x0, x1) 29.00/9.29 29.00/9.29 We have to consider all (P,Q,R)-chains. 29.00/9.29 ---------------------------------------- 29.00/9.29 29.00/9.29 (76) UsableRulesProof (EQUIVALENT) 29.00/9.29 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. 29.00/9.29 ---------------------------------------- 29.00/9.29 29.00/9.29 (77) 29.00/9.29 Obligation: 29.00/9.29 Q DP problem: 29.00/9.29 The TRS P consists of the following rules: 29.00/9.29 29.00/9.29 U1_GAA(0, less_out_ga(0)) -> INSERT_IN_GAA(0) 29.00/9.29 INSERT_IN_GAA(0) -> U1_GAA(0, less_out_ga(0)) 29.00/9.29 29.00/9.29 R is empty. 29.00/9.29 The set Q consists of the following terms: 29.00/9.29 29.00/9.29 less_in_ga(x0) 29.00/9.29 less_in_ag(x0) 29.00/9.29 U5_ga(x0, x1) 29.00/9.29 U5_ag(x0, x1) 29.00/9.29 29.00/9.29 We have to consider all (P,Q,R)-chains. 29.00/9.29 ---------------------------------------- 29.00/9.29 29.00/9.29 (78) 29.00/9.29 Obligation: 29.00/9.29 Q DP problem: 29.00/9.29 The TRS P consists of the following rules: 29.00/9.29 29.00/9.29 INSERT_IN_GAA(s(x0)) -> U1_GAA(s(x0), U5_ga(x0, less_in_ga(x0))) 29.00/9.29 U1_GAA(s(z0), less_out_ga(s(z0))) -> INSERT_IN_GAA(s(z0)) 29.00/9.29 INSERT_IN_GAA(s(x0)) -> U3_GAA(s(x0), less_out_ag(0, s(x0))) 29.00/9.29 U3_GAA(X, less_out_ag(Y, X)) -> INSERT_IN_GAA(X) 29.00/9.29 INSERT_IN_GAA(s(x0)) -> U3_GAA(s(x0), U5_ag(x0, less_in_ag(x0))) 29.00/9.29 29.00/9.29 The TRS R consists of the following rules: 29.00/9.29 29.00/9.29 less_in_ga(0) -> less_out_ga(0) 29.00/9.29 less_in_ga(s(X)) -> U5_ga(X, less_in_ga(X)) 29.00/9.29 less_in_ag(s(X1)) -> less_out_ag(0, s(X1)) 29.00/9.29 less_in_ag(s(Y)) -> U5_ag(Y, less_in_ag(Y)) 29.00/9.29 U5_ga(X, less_out_ga(X)) -> less_out_ga(s(X)) 29.00/9.29 U5_ag(Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 29.00/9.29 29.00/9.29 The set Q consists of the following terms: 29.00/9.29 29.00/9.29 less_in_ga(x0) 29.00/9.29 less_in_ag(x0) 29.00/9.29 U5_ga(x0, x1) 29.00/9.29 U5_ag(x0, x1) 29.00/9.29 29.00/9.29 We have to consider all (P,Q,R)-chains. 29.00/9.29 ---------------------------------------- 29.00/9.29 29.00/9.29 (79) PrologToTRSTransformerProof (SOUND) 29.00/9.29 Transformed Prolog program to TRS. 29.00/9.29 29.00/9.29 { 29.00/9.29 "root": 2, 29.00/9.29 "program": { 29.00/9.29 "directives": [], 29.00/9.29 "clauses": [ 29.00/9.29 [ 29.00/9.29 "(insert X (void) (tree X (void) (void)))", 29.00/9.29 null 29.00/9.29 ], 29.00/9.29 [ 29.00/9.29 "(insert X (tree X Left Right) (tree X Left Right))", 29.00/9.29 null 29.00/9.29 ], 29.00/9.29 [ 29.00/9.29 "(insert X (tree Y Left Right) (tree Y Left1 Right))", 29.00/9.29 "(',' (less X Y) (insert X Left Left1))" 29.00/9.29 ], 29.00/9.29 [ 29.00/9.29 "(insert X (tree Y Left Right) (tree Y Left Right1))", 29.00/9.29 "(',' (less Y X) (insert X Right Right1))" 29.00/9.29 ], 29.00/9.29 [ 29.00/9.29 "(less (0) (s X1))", 29.00/9.29 null 29.00/9.29 ], 29.00/9.29 [ 29.00/9.29 "(less (s X) (s Y))", 29.00/9.29 "(less X Y)" 29.00/9.29 ] 29.00/9.29 ] 29.00/9.29 }, 29.00/9.29 "graph": { 29.00/9.29 "nodes": { 29.00/9.29 "27": { 29.00/9.29 "goal": [{ 29.00/9.29 "clause": -1, 29.00/9.29 "scope": -1, 29.00/9.29 "term": "(true)" 29.00/9.29 }], 29.00/9.29 "kb": { 29.00/9.29 "nonunifying": [], 29.00/9.29 "intvars": {}, 29.00/9.29 "arithmetic": { 29.00/9.29 "type": "PlainIntegerRelationState", 29.00/9.29 "relations": [] 29.00/9.29 }, 29.00/9.29 "ground": [], 29.00/9.29 "free": [], 29.00/9.29 "exprvars": [] 29.00/9.29 } 29.00/9.29 }, 29.00/9.29 "28": { 29.00/9.29 "goal": [], 29.00/9.29 "kb": { 29.00/9.29 "nonunifying": [], 29.00/9.29 "intvars": {}, 29.00/9.29 "arithmetic": { 29.00/9.29 "type": "PlainIntegerRelationState", 29.00/9.29 "relations": [] 29.00/9.29 }, 29.00/9.29 "ground": [], 29.00/9.29 "free": [], 29.00/9.29 "exprvars": [] 29.00/9.29 } 29.00/9.29 }, 29.00/9.29 "290": { 29.00/9.29 "goal": [{ 29.00/9.29 "clause": 5, 29.00/9.29 "scope": 2, 29.00/9.29 "term": "(less T44 T49)" 29.00/9.29 }], 29.00/9.29 "kb": { 29.00/9.29 "nonunifying": [], 29.00/9.29 "intvars": {}, 29.00/9.29 "arithmetic": { 29.00/9.29 "type": "PlainIntegerRelationState", 29.00/9.29 "relations": [] 29.00/9.29 }, 29.00/9.29 "ground": ["T44"], 29.00/9.29 "free": [], 29.00/9.29 "exprvars": [] 29.00/9.29 } 29.00/9.29 }, 29.00/9.29 "type": "Nodes", 29.00/9.29 "250": { 29.00/9.29 "goal": [], 29.00/9.29 "kb": { 29.00/9.29 "nonunifying": [], 29.00/9.29 "intvars": {}, 29.00/9.29 "arithmetic": { 29.00/9.29 "type": "PlainIntegerRelationState", 29.00/9.29 "relations": [] 29.00/9.29 }, 29.00/9.29 "ground": [], 29.00/9.29 "free": [], 29.00/9.29 "exprvars": [] 29.00/9.29 } 29.00/9.29 }, 29.00/9.29 "251": { 29.00/9.29 "goal": [{ 29.00/9.29 "clause": 1, 29.00/9.29 "scope": 1, 29.00/9.29 "term": "(insert T1 T2 T3)" 29.00/9.29 }], 29.00/9.29 "kb": { 29.00/9.29 "nonunifying": [], 29.00/9.29 "intvars": {}, 29.00/9.29 "arithmetic": { 29.00/9.29 "type": "PlainIntegerRelationState", 29.00/9.29 "relations": [] 29.00/9.29 }, 29.00/9.29 "ground": ["T1"], 29.00/9.29 "free": [], 29.00/9.29 "exprvars": [] 29.00/9.29 } 29.00/9.29 }, 29.00/9.29 "295": { 29.00/9.29 "goal": [{ 29.00/9.29 "clause": -1, 29.00/9.29 "scope": -1, 29.00/9.29 "term": "(true)" 29.00/9.29 }], 29.00/9.29 "kb": { 29.00/9.29 "nonunifying": [], 29.00/9.29 "intvars": {}, 29.00/9.29 "arithmetic": { 29.00/9.29 "type": "PlainIntegerRelationState", 29.00/9.29 "relations": [] 29.00/9.29 }, 29.00/9.29 "ground": [], 29.00/9.29 "free": [], 29.00/9.29 "exprvars": [] 29.00/9.29 } 29.00/9.29 }, 29.00/9.29 "252": { 29.00/9.29 "goal": [ 29.00/9.29 { 29.00/9.29 "clause": 2, 29.00/9.29 "scope": 1, 29.00/9.29 "term": "(insert T1 T2 T3)" 29.00/9.29 }, 29.00/9.29 { 29.00/9.29 "clause": 3, 29.00/9.29 "scope": 1, 29.00/9.29 "term": "(insert T1 T2 T3)" 29.00/9.29 } 29.00/9.29 ], 29.00/9.29 "kb": { 29.00/9.29 "nonunifying": [], 29.00/9.29 "intvars": {}, 29.00/9.29 "arithmetic": { 29.00/9.29 "type": "PlainIntegerRelationState", 29.00/9.29 "relations": [] 29.00/9.29 }, 29.00/9.29 "ground": ["T1"], 29.00/9.29 "free": [], 29.00/9.29 "exprvars": [] 29.00/9.29 } 29.00/9.29 }, 29.00/9.29 "296": { 29.00/9.29 "goal": [], 29.00/9.29 "kb": { 29.00/9.29 "nonunifying": [], 29.00/9.29 "intvars": {}, 29.00/9.29 "arithmetic": { 29.00/9.29 "type": "PlainIntegerRelationState", 29.00/9.29 "relations": [] 29.00/9.29 }, 29.00/9.29 "ground": [], 29.00/9.29 "free": [], 29.00/9.29 "exprvars": [] 29.00/9.29 } 29.00/9.29 }, 29.00/9.29 "297": { 29.00/9.29 "goal": [], 29.00/9.29 "kb": { 29.00/9.29 "nonunifying": [], 29.00/9.29 "intvars": {}, 29.00/9.29 "arithmetic": { 29.00/9.29 "type": "PlainIntegerRelationState", 29.00/9.29 "relations": [] 29.00/9.29 }, 29.00/9.29 "ground": [], 29.00/9.29 "free": [], 29.00/9.29 "exprvars": [] 29.00/9.29 } 29.00/9.29 }, 29.00/9.29 "254": { 29.00/9.29 "goal": [{ 29.00/9.29 "clause": -1, 29.00/9.29 "scope": -1, 29.00/9.29 "term": "(true)" 29.00/9.29 }], 29.00/9.29 "kb": { 29.00/9.29 "nonunifying": [], 29.00/9.29 "intvars": {}, 29.00/9.29 "arithmetic": { 29.00/9.29 "type": "PlainIntegerRelationState", 29.00/9.29 "relations": [] 29.00/9.29 }, 29.00/9.29 "ground": [], 29.00/9.29 "free": [], 29.00/9.29 "exprvars": [] 29.00/9.29 } 29.00/9.29 }, 29.00/9.29 "276": { 29.00/9.29 "goal": [{ 29.00/9.29 "clause": -1, 29.00/9.29 "scope": -1, 29.00/9.29 "term": "(less T44 T49)" 29.00/9.29 }], 29.00/9.29 "kb": { 29.00/9.29 "nonunifying": [], 29.00/9.29 "intvars": {}, 29.00/9.29 "arithmetic": { 29.00/9.29 "type": "PlainIntegerRelationState", 29.00/9.29 "relations": [] 29.00/9.29 }, 29.00/9.29 "ground": ["T44"], 29.00/9.29 "free": [], 29.00/9.29 "exprvars": [] 29.00/9.29 } 29.00/9.29 }, 29.00/9.29 "298": { 29.00/9.29 "goal": [{ 29.00/9.29 "clause": -1, 29.00/9.29 "scope": -1, 29.00/9.29 "term": "(less T67 T69)" 29.00/9.29 }], 29.00/9.29 "kb": { 29.00/9.29 "nonunifying": [], 29.00/9.29 "intvars": {}, 29.00/9.29 "arithmetic": { 29.00/9.29 "type": "PlainIntegerRelationState", 29.00/9.29 "relations": [] 29.00/9.29 }, 29.00/9.29 "ground": ["T67"], 29.00/9.29 "free": [], 29.00/9.29 "exprvars": [] 29.00/9.29 } 29.00/9.29 }, 29.00/9.29 "277": { 29.00/9.29 "goal": [{ 29.00/9.29 "clause": -1, 29.00/9.29 "scope": -1, 29.00/9.29 "term": "(insert T44 T54 T55)" 29.00/9.29 }], 29.00/9.29 "kb": { 29.00/9.29 "nonunifying": [], 29.00/9.29 "intvars": {}, 29.00/9.29 "arithmetic": { 29.00/9.29 "type": "PlainIntegerRelationState", 29.00/9.29 "relations": [] 29.00/9.29 }, 29.00/9.29 "ground": ["T44"], 29.00/9.29 "free": [], 29.00/9.29 "exprvars": [] 29.00/9.29 } 29.00/9.29 }, 29.00/9.29 "299": { 29.00/9.29 "goal": [], 29.00/9.29 "kb": { 29.00/9.29 "nonunifying": [], 29.00/9.29 "intvars": {}, 29.00/9.29 "arithmetic": { 29.00/9.29 "type": "PlainIntegerRelationState", 29.00/9.29 "relations": [] 29.00/9.29 }, 29.00/9.29 "ground": [], 29.00/9.29 "free": [], 29.00/9.29 "exprvars": [] 29.00/9.29 } 29.00/9.29 }, 29.00/9.29 "256": { 29.00/9.29 "goal": [], 29.00/9.29 "kb": { 29.00/9.29 "nonunifying": [], 29.00/9.29 "intvars": {}, 29.00/9.29 "arithmetic": { 29.00/9.29 "type": "PlainIntegerRelationState", 29.00/9.29 "relations": [] 29.00/9.29 }, 29.00/9.29 "ground": [], 29.00/9.29 "free": [], 29.00/9.29 "exprvars": [] 29.00/9.29 } 29.00/9.29 }, 29.00/9.29 "311": { 29.00/9.29 "goal": [ 29.00/9.29 { 29.00/9.29 "clause": 4, 29.00/9.29 "scope": 3, 29.00/9.29 "term": "(less T89 T84)" 29.00/9.29 }, 29.00/9.29 { 29.00/9.29 "clause": 5, 29.00/9.29 "scope": 3, 29.00/9.29 "term": "(less T89 T84)" 29.00/9.29 } 29.00/9.29 ], 29.00/9.29 "kb": { 29.00/9.29 "nonunifying": [], 29.00/9.29 "intvars": {}, 29.00/9.29 "arithmetic": { 29.00/9.29 "type": "PlainIntegerRelationState", 29.00/9.29 "relations": [] 29.00/9.29 }, 29.00/9.29 "ground": ["T84"], 29.00/9.29 "free": [], 29.00/9.29 "exprvars": [] 29.00/9.29 } 29.00/9.29 }, 29.00/9.29 "257": { 29.00/9.29 "goal": [], 29.00/9.29 "kb": { 29.00/9.29 "nonunifying": [], 29.00/9.29 "intvars": {}, 29.00/9.29 "arithmetic": { 29.00/9.29 "type": "PlainIntegerRelationState", 29.00/9.29 "relations": [] 29.00/9.29 }, 29.00/9.29 "ground": [], 29.00/9.29 "free": [], 29.00/9.29 "exprvars": [] 29.00/9.29 } 29.00/9.29 }, 29.00/9.29 "258": { 29.00/9.29 "goal": [{ 29.00/9.29 "clause": 2, 29.00/9.29 "scope": 1, 29.00/9.29 "term": "(insert T1 T2 T3)" 29.00/9.29 }], 29.00/9.29 "kb": { 29.00/9.29 "nonunifying": [], 29.00/9.29 "intvars": {}, 29.00/9.29 "arithmetic": { 29.00/9.29 "type": "PlainIntegerRelationState", 29.00/9.29 "relations": [] 29.00/9.29 }, 29.00/9.29 "ground": ["T1"], 29.00/9.29 "free": [], 29.00/9.29 "exprvars": [] 29.00/9.29 } 29.00/9.29 }, 29.00/9.29 "259": { 29.00/9.29 "goal": [{ 29.00/9.29 "clause": 3, 29.00/9.29 "scope": 1, 29.00/9.29 "term": "(insert T1 T2 T3)" 29.00/9.29 }], 29.00/9.29 "kb": { 29.00/9.29 "nonunifying": [], 29.00/9.29 "intvars": {}, 29.00/9.29 "arithmetic": { 29.00/9.29 "type": "PlainIntegerRelationState", 29.00/9.29 "relations": [] 29.00/9.29 }, 29.00/9.29 "ground": ["T1"], 29.00/9.29 "free": [], 29.00/9.29 "exprvars": [] 29.00/9.29 } 29.00/9.29 }, 29.00/9.29 "316": { 29.00/9.29 "goal": [{ 29.00/9.29 "clause": 4, 29.00/9.29 "scope": 3, 29.00/9.29 "term": "(less T89 T84)" 29.00/9.29 }], 29.00/9.29 "kb": { 29.00/9.29 "nonunifying": [], 29.00/9.29 "intvars": {}, 29.00/9.29 "arithmetic": { 29.00/9.29 "type": "PlainIntegerRelationState", 29.00/9.29 "relations": [] 29.00/9.29 }, 29.00/9.29 "ground": ["T84"], 29.00/9.29 "free": [], 29.00/9.29 "exprvars": [] 29.00/9.29 } 29.00/9.29 }, 29.00/9.29 "318": { 29.00/9.29 "goal": [{ 29.00/9.29 "clause": 5, 29.00/9.29 "scope": 3, 29.00/9.29 "term": "(less T89 T84)" 29.00/9.29 }], 29.00/9.29 "kb": { 29.00/9.29 "nonunifying": [], 29.00/9.29 "intvars": {}, 29.00/9.29 "arithmetic": { 29.00/9.29 "type": "PlainIntegerRelationState", 29.00/9.29 "relations": [] 29.00/9.29 }, 29.00/9.29 "ground": ["T84"], 29.00/9.29 "free": [], 29.00/9.29 "exprvars": [] 29.00/9.29 } 29.00/9.29 }, 29.00/9.29 "10": { 29.00/9.29 "goal": [{ 29.00/9.29 "clause": 0, 29.00/9.29 "scope": 1, 29.00/9.29 "term": "(insert T1 T2 T3)" 29.00/9.29 }], 29.00/9.29 "kb": { 29.00/9.29 "nonunifying": [], 29.00/9.29 "intvars": {}, 29.00/9.29 "arithmetic": { 29.00/9.29 "type": "PlainIntegerRelationState", 29.00/9.29 "relations": [] 29.00/9.29 }, 29.00/9.29 "ground": ["T1"], 29.00/9.29 "free": [], 29.00/9.29 "exprvars": [] 29.00/9.29 } 29.00/9.29 }, 29.00/9.29 "11": { 29.00/9.29 "goal": [ 29.00/9.29 { 29.00/9.29 "clause": 1, 29.00/9.29 "scope": 1, 29.00/9.29 "term": "(insert T1 T2 T3)" 29.00/9.29 }, 29.00/9.29 { 29.00/9.29 "clause": 2, 29.00/9.29 "scope": 1, 29.00/9.29 "term": "(insert T1 T2 T3)" 29.00/9.29 }, 29.00/9.29 { 29.00/9.29 "clause": 3, 29.00/9.29 "scope": 1, 29.00/9.29 "term": "(insert T1 T2 T3)" 29.00/9.29 } 29.00/9.29 ], 29.00/9.29 "kb": { 29.00/9.29 "nonunifying": [], 29.00/9.29 "intvars": {}, 29.00/9.29 "arithmetic": { 29.00/9.29 "type": "PlainIntegerRelationState", 29.00/9.29 "relations": [] 29.00/9.29 }, 29.00/9.29 "ground": ["T1"], 29.00/9.29 "free": [], 29.00/9.29 "exprvars": [] 29.00/9.29 } 29.00/9.29 }, 29.00/9.29 "284": { 29.00/9.29 "goal": [ 29.00/9.29 { 29.00/9.29 "clause": 4, 29.00/9.29 "scope": 2, 29.00/9.29 "term": "(less T44 T49)" 29.00/9.29 }, 29.00/9.29 { 29.00/9.29 "clause": 5, 29.00/9.29 "scope": 2, 29.00/9.29 "term": "(less T44 T49)" 29.00/9.29 } 29.00/9.29 ], 29.00/9.29 "kb": { 29.00/9.29 "nonunifying": [], 29.00/9.29 "intvars": {}, 29.00/9.29 "arithmetic": { 29.00/9.29 "type": "PlainIntegerRelationState", 29.00/9.29 "relations": [] 29.00/9.29 }, 29.00/9.29 "ground": ["T44"], 29.00/9.29 "free": [], 29.00/9.29 "exprvars": [] 29.00/9.29 } 29.00/9.29 }, 29.00/9.29 "2": { 29.00/9.29 "goal": [{ 29.00/9.29 "clause": -1, 29.00/9.29 "scope": -1, 29.00/9.29 "term": "(insert T1 T2 T3)" 29.00/9.29 }], 29.00/9.29 "kb": { 29.00/9.29 "nonunifying": [], 29.00/9.29 "intvars": {}, 29.00/9.29 "arithmetic": { 29.00/9.29 "type": "PlainIntegerRelationState", 29.00/9.29 "relations": [] 29.00/9.29 }, 29.00/9.29 "ground": ["T1"], 29.00/9.29 "free": [], 29.00/9.29 "exprvars": [] 29.00/9.29 } 29.00/9.29 }, 29.00/9.29 "288": { 29.00/9.29 "goal": [{ 29.00/9.29 "clause": 4, 29.00/9.29 "scope": 2, 29.00/9.29 "term": "(less T44 T49)" 29.00/9.29 }], 29.00/9.29 "kb": { 29.00/9.29 "nonunifying": [], 29.00/9.29 "intvars": {}, 29.00/9.29 "arithmetic": { 29.00/9.29 "type": "PlainIntegerRelationState", 29.00/9.29 "relations": [] 29.00/9.29 }, 29.00/9.29 "ground": ["T44"], 29.00/9.29 "free": [], 29.00/9.29 "exprvars": [] 29.00/9.29 } 29.00/9.29 }, 29.00/9.29 "267": { 29.00/9.29 "goal": [{ 29.00/9.29 "clause": -1, 29.00/9.29 "scope": -1, 29.00/9.29 "term": "(',' (less T44 T49) (insert T44 T50 T51))" 29.00/9.29 }], 29.00/9.29 "kb": { 29.00/9.29 "nonunifying": [], 29.00/9.29 "intvars": {}, 29.00/9.29 "arithmetic": { 29.00/9.29 "type": "PlainIntegerRelationState", 29.00/9.29 "relations": [] 29.00/9.29 }, 29.00/9.29 "ground": ["T44"], 29.00/9.29 "free": [], 29.00/9.29 "exprvars": [] 29.00/9.29 } 29.00/9.29 }, 29.00/9.29 "300": { 29.00/9.29 "goal": [{ 29.00/9.29 "clause": -1, 29.00/9.29 "scope": -1, 29.00/9.29 "term": "(',' (less T89 T84) (insert T84 T90 T91))" 29.00/9.29 }], 29.00/9.29 "kb": { 29.00/9.29 "nonunifying": [], 29.00/9.29 "intvars": {}, 29.00/9.29 "arithmetic": { 29.00/9.29 "type": "PlainIntegerRelationState", 29.00/9.29 "relations": [] 29.00/9.29 }, 29.00/9.29 "ground": ["T84"], 29.00/9.29 "free": [], 29.00/9.29 "exprvars": [] 29.00/9.29 } 29.00/9.29 }, 29.00/9.29 "268": { 29.00/9.29 "goal": [], 29.00/9.29 "kb": { 29.00/9.29 "nonunifying": [], 29.00/9.29 "intvars": {}, 29.00/9.29 "arithmetic": { 29.00/9.29 "type": "PlainIntegerRelationState", 29.00/9.29 "relations": [] 29.00/9.29 }, 29.00/9.29 "ground": [], 29.00/9.29 "free": [], 29.00/9.29 "exprvars": [] 29.00/9.29 } 29.00/9.29 }, 29.00/9.29 "301": { 29.00/9.29 "goal": [], 29.00/9.29 "kb": { 29.00/9.29 "nonunifying": [], 29.00/9.29 "intvars": {}, 29.00/9.29 "arithmetic": { 29.00/9.29 "type": "PlainIntegerRelationState", 29.00/9.29 "relations": [] 29.00/9.29 }, 29.00/9.29 "ground": [], 29.00/9.29 "free": [], 29.00/9.29 "exprvars": [] 29.00/9.29 } 29.00/9.29 }, 29.00/9.29 "323": { 29.00/9.29 "goal": [{ 29.00/9.29 "clause": -1, 29.00/9.29 "scope": -1, 29.00/9.29 "term": "(true)" 29.00/9.29 }], 29.00/9.29 "kb": { 29.00/9.29 "nonunifying": [], 29.00/9.29 "intvars": {}, 29.00/9.29 "arithmetic": { 29.00/9.30 "type": "PlainIntegerRelationState", 29.00/9.30 "relations": [] 29.00/9.30 }, 29.00/9.30 "ground": [], 29.00/9.30 "free": [], 29.00/9.30 "exprvars": [] 29.00/9.30 } 29.00/9.30 }, 29.00/9.30 "5": { 29.00/9.30 "goal": [ 29.00/9.30 { 29.00/9.30 "clause": 0, 29.00/9.30 "scope": 1, 29.00/9.30 "term": "(insert T1 T2 T3)" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "clause": 1, 29.00/9.30 "scope": 1, 29.00/9.30 "term": "(insert T1 T2 T3)" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "clause": 2, 29.00/9.30 "scope": 1, 29.00/9.30 "term": "(insert T1 T2 T3)" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "clause": 3, 29.00/9.30 "scope": 1, 29.00/9.30 "term": "(insert T1 T2 T3)" 29.00/9.30 } 29.00/9.30 ], 29.00/9.30 "kb": { 29.00/9.30 "nonunifying": [], 29.00/9.30 "intvars": {}, 29.00/9.30 "arithmetic": { 29.00/9.30 "type": "PlainIntegerRelationState", 29.00/9.30 "relations": [] 29.00/9.30 }, 29.00/9.30 "ground": ["T1"], 29.00/9.30 "free": [], 29.00/9.30 "exprvars": [] 29.00/9.30 } 29.00/9.30 }, 29.00/9.30 "324": { 29.00/9.30 "goal": [], 29.00/9.30 "kb": { 29.00/9.30 "nonunifying": [], 29.00/9.30 "intvars": {}, 29.00/9.30 "arithmetic": { 29.00/9.30 "type": "PlainIntegerRelationState", 29.00/9.30 "relations": [] 29.00/9.30 }, 29.00/9.30 "ground": [], 29.00/9.30 "free": [], 29.00/9.30 "exprvars": [] 29.00/9.30 } 29.00/9.30 }, 29.00/9.30 "325": { 29.00/9.30 "goal": [], 29.00/9.30 "kb": { 29.00/9.30 "nonunifying": [], 29.00/9.30 "intvars": {}, 29.00/9.30 "arithmetic": { 29.00/9.30 "type": "PlainIntegerRelationState", 29.00/9.30 "relations": [] 29.00/9.30 }, 29.00/9.30 "ground": [], 29.00/9.30 "free": [], 29.00/9.30 "exprvars": [] 29.00/9.30 } 29.00/9.30 }, 29.00/9.30 "304": { 29.00/9.30 "goal": [{ 29.00/9.30 "clause": -1, 29.00/9.30 "scope": -1, 29.00/9.30 "term": "(less T89 T84)" 29.00/9.30 }], 29.00/9.30 "kb": { 29.00/9.30 "nonunifying": [], 29.00/9.30 "intvars": {}, 29.00/9.30 "arithmetic": { 29.00/9.30 "type": "PlainIntegerRelationState", 29.00/9.30 "relations": [] 29.00/9.30 }, 29.00/9.30 "ground": ["T84"], 29.00/9.30 "free": [], 29.00/9.30 "exprvars": [] 29.00/9.30 } 29.00/9.30 }, 29.00/9.30 "348": { 29.00/9.30 "goal": [{ 29.00/9.30 "clause": -1, 29.00/9.30 "scope": -1, 29.00/9.30 "term": "(less T109 T108)" 29.00/9.30 }], 29.00/9.30 "kb": { 29.00/9.30 "nonunifying": [], 29.00/9.30 "intvars": {}, 29.00/9.30 "arithmetic": { 29.00/9.30 "type": "PlainIntegerRelationState", 29.00/9.30 "relations": [] 29.00/9.30 }, 29.00/9.30 "ground": ["T108"], 29.00/9.30 "free": [], 29.00/9.30 "exprvars": [] 29.00/9.30 } 29.00/9.30 }, 29.00/9.30 "305": { 29.00/9.30 "goal": [{ 29.00/9.30 "clause": -1, 29.00/9.30 "scope": -1, 29.00/9.30 "term": "(insert T84 T94 T95)" 29.00/9.30 }], 29.00/9.30 "kb": { 29.00/9.30 "nonunifying": [], 29.00/9.30 "intvars": {}, 29.00/9.30 "arithmetic": { 29.00/9.30 "type": "PlainIntegerRelationState", 29.00/9.30 "relations": [] 29.00/9.30 }, 29.00/9.30 "ground": ["T84"], 29.00/9.30 "free": [], 29.00/9.30 "exprvars": [] 29.00/9.30 } 29.00/9.30 }, 29.00/9.30 "349": { 29.00/9.30 "goal": [], 29.00/9.30 "kb": { 29.00/9.30 "nonunifying": [], 29.00/9.30 "intvars": {}, 29.00/9.30 "arithmetic": { 29.00/9.30 "type": "PlainIntegerRelationState", 29.00/9.30 "relations": [] 29.00/9.30 }, 29.00/9.30 "ground": [], 29.00/9.30 "free": [], 29.00/9.30 "exprvars": [] 29.00/9.30 } 29.00/9.30 } 29.00/9.30 }, 29.00/9.30 "edges": [ 29.00/9.30 { 29.00/9.30 "from": 2, 29.00/9.30 "to": 5, 29.00/9.30 "label": "CASE" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 5, 29.00/9.30 "to": 10, 29.00/9.30 "label": "PARALLEL" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 5, 29.00/9.30 "to": 11, 29.00/9.30 "label": "PARALLEL" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 10, 29.00/9.30 "to": 27, 29.00/9.30 "label": "EVAL with clause\ninsert(X6, void, tree(X6, void, void)).\nand substitutionT1 -> T8,\nX6 -> T8,\nT2 -> void,\nT3 -> tree(T8, void, void)" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 10, 29.00/9.30 "to": 28, 29.00/9.30 "label": "EVAL-BACKTRACK" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 11, 29.00/9.30 "to": 251, 29.00/9.30 "label": "PARALLEL" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 11, 29.00/9.30 "to": 252, 29.00/9.30 "label": "PARALLEL" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 27, 29.00/9.30 "to": 250, 29.00/9.30 "label": "SUCCESS" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 251, 29.00/9.30 "to": 254, 29.00/9.30 "label": "EVAL with clause\ninsert(X19, tree(X19, X20, X21), tree(X19, X20, X21)).\nand substitutionT1 -> T21,\nX19 -> T21,\nX20 -> T22,\nX21 -> T23,\nT2 -> tree(T21, T22, T23),\nT3 -> tree(T21, T22, T23)" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 251, 29.00/9.30 "to": 256, 29.00/9.30 "label": "EVAL-BACKTRACK" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 252, 29.00/9.30 "to": 258, 29.00/9.30 "label": "PARALLEL" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 252, 29.00/9.30 "to": 259, 29.00/9.30 "label": "PARALLEL" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 254, 29.00/9.30 "to": 257, 29.00/9.30 "label": "SUCCESS" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 258, 29.00/9.30 "to": 267, 29.00/9.30 "label": "EVAL with clause\ninsert(X42, tree(X43, X44, X45), tree(X43, X46, X45)) :- ','(less(X42, X43), insert(X42, X44, X46)).\nand substitutionT1 -> T44,\nX42 -> T44,\nX43 -> T49,\nX44 -> T50,\nX45 -> T47,\nT2 -> tree(T49, T50, T47),\nX46 -> T51,\nT3 -> tree(T49, T51, T47),\nT45 -> T49,\nT46 -> T50,\nT48 -> T51" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 258, 29.00/9.30 "to": 268, 29.00/9.30 "label": "EVAL-BACKTRACK" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 259, 29.00/9.30 "to": 300, 29.00/9.30 "label": "EVAL with clause\ninsert(X76, tree(X77, X78, X79), tree(X77, X78, X80)) :- ','(less(X77, X76), insert(X76, X79, X80)).\nand substitutionT1 -> T84,\nX76 -> T84,\nX77 -> T89,\nX78 -> T86,\nX79 -> T90,\nT2 -> tree(T89, T86, T90),\nX80 -> T91,\nT3 -> tree(T89, T86, T91),\nT85 -> T89,\nT87 -> T90,\nT88 -> T91" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 259, 29.00/9.30 "to": 301, 29.00/9.30 "label": "EVAL-BACKTRACK" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 267, 29.00/9.30 "to": 276, 29.00/9.30 "label": "SPLIT 1" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 267, 29.00/9.30 "to": 277, 29.00/9.30 "label": "SPLIT 2\nnew knowledge:\nT44 is ground\nreplacements:T50 -> T54,\nT51 -> T55" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 276, 29.00/9.30 "to": 284, 29.00/9.30 "label": "CASE" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 277, 29.00/9.30 "to": 2, 29.00/9.30 "label": "INSTANCE with matching:\nT1 -> T44\nT2 -> T54\nT3 -> T55" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 284, 29.00/9.30 "to": 288, 29.00/9.30 "label": "PARALLEL" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 284, 29.00/9.30 "to": 290, 29.00/9.30 "label": "PARALLEL" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 288, 29.00/9.30 "to": 295, 29.00/9.30 "label": "EVAL with clause\nless(0, s(X55)).\nand substitutionT44 -> 0,\nX55 -> T62,\nT49 -> s(T62)" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 288, 29.00/9.30 "to": 296, 29.00/9.30 "label": "EVAL-BACKTRACK" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 290, 29.00/9.30 "to": 298, 29.00/9.30 "label": "EVAL with clause\nless(s(X60), s(X61)) :- less(X60, X61).\nand substitutionX60 -> T67,\nT44 -> s(T67),\nX61 -> T69,\nT49 -> s(T69),\nT68 -> T69" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 290, 29.00/9.30 "to": 299, 29.00/9.30 "label": "EVAL-BACKTRACK" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 295, 29.00/9.30 "to": 297, 29.00/9.30 "label": "SUCCESS" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 298, 29.00/9.30 "to": 276, 29.00/9.30 "label": "INSTANCE with matching:\nT44 -> T67\nT49 -> T69" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 300, 29.00/9.30 "to": 304, 29.00/9.30 "label": "SPLIT 1" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 300, 29.00/9.30 "to": 305, 29.00/9.30 "label": "SPLIT 2\nnew knowledge:\nT89 is ground\nT84 is ground\nreplacements:T90 -> T94,\nT91 -> T95" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 304, 29.00/9.30 "to": 311, 29.00/9.30 "label": "CASE" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 305, 29.00/9.30 "to": 2, 29.00/9.30 "label": "INSTANCE with matching:\nT1 -> T84\nT2 -> T94\nT3 -> T95" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 311, 29.00/9.30 "to": 316, 29.00/9.30 "label": "PARALLEL" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 311, 29.00/9.30 "to": 318, 29.00/9.30 "label": "PARALLEL" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 316, 29.00/9.30 "to": 323, 29.00/9.30 "label": "EVAL with clause\nless(0, s(X89)).\nand substitutionT89 -> 0,\nX89 -> T102,\nT84 -> s(T102)" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 316, 29.00/9.30 "to": 324, 29.00/9.30 "label": "EVAL-BACKTRACK" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 318, 29.00/9.30 "to": 348, 29.00/9.30 "label": "EVAL with clause\nless(s(X94), s(X95)) :- less(X94, X95).\nand substitutionX94 -> T109,\nT89 -> s(T109),\nX95 -> T108,\nT84 -> s(T108),\nT107 -> T109" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 318, 29.00/9.30 "to": 349, 29.00/9.30 "label": "EVAL-BACKTRACK" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 323, 29.00/9.30 "to": 325, 29.00/9.30 "label": "SUCCESS" 29.00/9.30 }, 29.00/9.30 { 29.00/9.30 "from": 348, 29.00/9.30 "to": 304, 29.00/9.30 "label": "INSTANCE with matching:\nT89 -> T109\nT84 -> T108" 29.00/9.30 } 29.00/9.30 ], 29.00/9.30 "type": "Graph" 29.00/9.30 } 29.00/9.30 } 29.00/9.30 29.00/9.30 ---------------------------------------- 29.00/9.30 29.00/9.30 (80) 29.00/9.30 Obligation: 29.00/9.30 Q restricted rewrite system: 29.00/9.30 The TRS R consists of the following rules: 29.00/9.30 29.00/9.30 f2_in(T8) -> f2_out1 29.00/9.30 f2_in(T44) -> U1(f267_in(T44), T44) 29.00/9.30 U1(f267_out1, T44) -> f2_out1 29.00/9.30 f2_in(T84) -> U2(f300_in(T84), T84) 29.00/9.30 U2(f300_out1(T89), T84) -> f2_out1 29.00/9.30 f276_in(0) -> f276_out1 29.00/9.30 f276_in(s(T67)) -> U3(f276_in(T67), s(T67)) 29.00/9.30 U3(f276_out1, s(T67)) -> f276_out1 29.00/9.30 f304_in(s(T102)) -> f304_out1(0) 29.00/9.30 f304_in(s(T108)) -> U4(f304_in(T108), s(T108)) 29.00/9.30 U4(f304_out1(T109), s(T108)) -> f304_out1(s(T109)) 29.00/9.30 f267_in(T44) -> U5(f276_in(T44), T44) 29.00/9.30 U5(f276_out1, T44) -> U6(f2_in(T44), T44) 29.00/9.30 U6(f2_out1, T44) -> f267_out1 29.00/9.30 f300_in(T84) -> U7(f304_in(T84), T84) 29.00/9.30 U7(f304_out1(T89), T84) -> U8(f2_in(T84), T84, T89) 29.00/9.30 U8(f2_out1, T84, T89) -> f300_out1(T89) 29.00/9.30 29.00/9.30 Q is empty. 29.00/9.30 29.00/9.30 ---------------------------------------- 29.00/9.30 29.00/9.30 (81) DependencyPairsProof (EQUIVALENT) 29.00/9.30 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 29.00/9.30 ---------------------------------------- 29.00/9.30 29.00/9.30 (82) 29.00/9.30 Obligation: 29.00/9.30 Q DP problem: 29.00/9.30 The TRS P consists of the following rules: 29.00/9.30 29.00/9.30 F2_IN(T44) -> U1^1(f267_in(T44), T44) 29.00/9.30 F2_IN(T44) -> F267_IN(T44) 29.00/9.30 F2_IN(T84) -> U2^1(f300_in(T84), T84) 29.00/9.30 F2_IN(T84) -> F300_IN(T84) 29.00/9.30 F276_IN(s(T67)) -> U3^1(f276_in(T67), s(T67)) 29.00/9.30 F276_IN(s(T67)) -> F276_IN(T67) 29.00/9.30 F304_IN(s(T108)) -> U4^1(f304_in(T108), s(T108)) 29.00/9.30 F304_IN(s(T108)) -> F304_IN(T108) 29.00/9.30 F267_IN(T44) -> U5^1(f276_in(T44), T44) 29.00/9.30 F267_IN(T44) -> F276_IN(T44) 29.00/9.30 U5^1(f276_out1, T44) -> U6^1(f2_in(T44), T44) 29.00/9.30 U5^1(f276_out1, T44) -> F2_IN(T44) 29.00/9.30 F300_IN(T84) -> U7^1(f304_in(T84), T84) 29.00/9.30 F300_IN(T84) -> F304_IN(T84) 29.00/9.30 U7^1(f304_out1(T89), T84) -> U8^1(f2_in(T84), T84, T89) 29.00/9.30 U7^1(f304_out1(T89), T84) -> F2_IN(T84) 29.00/9.30 29.00/9.30 The TRS R consists of the following rules: 29.00/9.30 29.00/9.30 f2_in(T8) -> f2_out1 29.00/9.30 f2_in(T44) -> U1(f267_in(T44), T44) 29.00/9.30 U1(f267_out1, T44) -> f2_out1 29.00/9.30 f2_in(T84) -> U2(f300_in(T84), T84) 29.00/9.30 U2(f300_out1(T89), T84) -> f2_out1 29.00/9.30 f276_in(0) -> f276_out1 29.00/9.30 f276_in(s(T67)) -> U3(f276_in(T67), s(T67)) 29.00/9.30 U3(f276_out1, s(T67)) -> f276_out1 29.00/9.30 f304_in(s(T102)) -> f304_out1(0) 29.00/9.30 f304_in(s(T108)) -> U4(f304_in(T108), s(T108)) 29.00/9.30 U4(f304_out1(T109), s(T108)) -> f304_out1(s(T109)) 29.00/9.30 f267_in(T44) -> U5(f276_in(T44), T44) 29.00/9.30 U5(f276_out1, T44) -> U6(f2_in(T44), T44) 29.00/9.30 U6(f2_out1, T44) -> f267_out1 29.00/9.30 f300_in(T84) -> U7(f304_in(T84), T84) 29.00/9.30 U7(f304_out1(T89), T84) -> U8(f2_in(T84), T84, T89) 29.00/9.30 U8(f2_out1, T84, T89) -> f300_out1(T89) 29.00/9.30 29.00/9.30 Q is empty. 29.00/9.30 We have to consider all minimal (P,Q,R)-chains. 29.00/9.30 ---------------------------------------- 29.00/9.30 29.00/9.30 (83) DependencyGraphProof (EQUIVALENT) 29.00/9.30 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 8 less nodes. 29.00/9.30 ---------------------------------------- 29.00/9.30 29.00/9.30 (84) 29.00/9.30 Complex Obligation (AND) 29.00/9.30 29.00/9.30 ---------------------------------------- 29.00/9.30 29.00/9.30 (85) 29.00/9.30 Obligation: 29.00/9.30 Q DP problem: 29.00/9.30 The TRS P consists of the following rules: 29.00/9.30 29.00/9.30 F304_IN(s(T108)) -> F304_IN(T108) 29.00/9.30 29.00/9.30 The TRS R consists of the following rules: 29.00/9.30 29.00/9.30 f2_in(T8) -> f2_out1 29.00/9.30 f2_in(T44) -> U1(f267_in(T44), T44) 29.00/9.30 U1(f267_out1, T44) -> f2_out1 29.00/9.30 f2_in(T84) -> U2(f300_in(T84), T84) 29.00/9.30 U2(f300_out1(T89), T84) -> f2_out1 29.00/9.30 f276_in(0) -> f276_out1 29.00/9.30 f276_in(s(T67)) -> U3(f276_in(T67), s(T67)) 29.00/9.30 U3(f276_out1, s(T67)) -> f276_out1 29.00/9.30 f304_in(s(T102)) -> f304_out1(0) 29.00/9.30 f304_in(s(T108)) -> U4(f304_in(T108), s(T108)) 29.00/9.30 U4(f304_out1(T109), s(T108)) -> f304_out1(s(T109)) 29.00/9.30 f267_in(T44) -> U5(f276_in(T44), T44) 29.00/9.30 U5(f276_out1, T44) -> U6(f2_in(T44), T44) 29.00/9.30 U6(f2_out1, T44) -> f267_out1 29.00/9.30 f300_in(T84) -> U7(f304_in(T84), T84) 29.00/9.30 U7(f304_out1(T89), T84) -> U8(f2_in(T84), T84, T89) 29.00/9.30 U8(f2_out1, T84, T89) -> f300_out1(T89) 29.00/9.30 29.00/9.30 Q is empty. 29.00/9.30 We have to consider all minimal (P,Q,R)-chains. 29.00/9.30 ---------------------------------------- 29.00/9.30 29.00/9.30 (86) UsableRulesProof (EQUIVALENT) 29.00/9.30 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. 29.00/9.30 ---------------------------------------- 29.00/9.30 29.00/9.30 (87) 29.00/9.30 Obligation: 29.00/9.30 Q DP problem: 29.00/9.30 The TRS P consists of the following rules: 29.00/9.30 29.00/9.30 F304_IN(s(T108)) -> F304_IN(T108) 29.00/9.30 29.00/9.30 R is empty. 29.00/9.30 Q is empty. 29.00/9.30 We have to consider all minimal (P,Q,R)-chains. 29.00/9.30 ---------------------------------------- 29.00/9.30 29.00/9.30 (88) QDPSizeChangeProof (EQUIVALENT) 29.00/9.30 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. 29.00/9.30 29.00/9.30 From the DPs we obtained the following set of size-change graphs: 29.00/9.30 *F304_IN(s(T108)) -> F304_IN(T108) 29.00/9.30 The graph contains the following edges 1 > 1 29.00/9.30 29.00/9.30 29.00/9.30 ---------------------------------------- 29.00/9.30 29.00/9.30 (89) 29.00/9.30 YES 29.00/9.30 29.00/9.30 ---------------------------------------- 29.00/9.30 29.00/9.30 (90) 29.00/9.30 Obligation: 29.00/9.30 Q DP problem: 29.00/9.30 The TRS P consists of the following rules: 29.00/9.30 29.00/9.30 F276_IN(s(T67)) -> F276_IN(T67) 29.00/9.30 29.00/9.30 The TRS R consists of the following rules: 29.00/9.30 29.00/9.30 f2_in(T8) -> f2_out1 29.00/9.30 f2_in(T44) -> U1(f267_in(T44), T44) 29.00/9.30 U1(f267_out1, T44) -> f2_out1 29.00/9.30 f2_in(T84) -> U2(f300_in(T84), T84) 29.00/9.30 U2(f300_out1(T89), T84) -> f2_out1 29.00/9.30 f276_in(0) -> f276_out1 29.00/9.30 f276_in(s(T67)) -> U3(f276_in(T67), s(T67)) 29.00/9.30 U3(f276_out1, s(T67)) -> f276_out1 29.00/9.30 f304_in(s(T102)) -> f304_out1(0) 29.00/9.30 f304_in(s(T108)) -> U4(f304_in(T108), s(T108)) 29.00/9.30 U4(f304_out1(T109), s(T108)) -> f304_out1(s(T109)) 29.00/9.30 f267_in(T44) -> U5(f276_in(T44), T44) 29.00/9.30 U5(f276_out1, T44) -> U6(f2_in(T44), T44) 29.00/9.30 U6(f2_out1, T44) -> f267_out1 29.00/9.30 f300_in(T84) -> U7(f304_in(T84), T84) 29.00/9.30 U7(f304_out1(T89), T84) -> U8(f2_in(T84), T84, T89) 29.00/9.30 U8(f2_out1, T84, T89) -> f300_out1(T89) 29.00/9.30 29.00/9.30 Q is empty. 29.00/9.30 We have to consider all minimal (P,Q,R)-chains. 29.00/9.30 ---------------------------------------- 29.00/9.30 29.00/9.30 (91) UsableRulesProof (EQUIVALENT) 29.00/9.30 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. 29.00/9.30 ---------------------------------------- 29.00/9.30 29.00/9.30 (92) 29.00/9.30 Obligation: 29.00/9.30 Q DP problem: 29.00/9.30 The TRS P consists of the following rules: 29.00/9.30 29.00/9.30 F276_IN(s(T67)) -> F276_IN(T67) 29.00/9.30 29.00/9.30 R is empty. 29.00/9.30 Q is empty. 29.00/9.30 We have to consider all minimal (P,Q,R)-chains. 29.00/9.30 ---------------------------------------- 29.00/9.30 29.00/9.30 (93) QDPSizeChangeProof (EQUIVALENT) 29.00/9.30 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. 29.00/9.30 29.00/9.30 From the DPs we obtained the following set of size-change graphs: 29.00/9.30 *F276_IN(s(T67)) -> F276_IN(T67) 29.00/9.30 The graph contains the following edges 1 > 1 29.00/9.30 29.00/9.30 29.00/9.30 ---------------------------------------- 29.00/9.30 29.00/9.30 (94) 29.00/9.30 YES 29.00/9.30 29.00/9.30 ---------------------------------------- 29.00/9.30 29.00/9.30 (95) 29.00/9.30 Obligation: 29.00/9.30 Q DP problem: 29.00/9.30 The TRS P consists of the following rules: 29.00/9.30 29.00/9.30 F2_IN(T44) -> F267_IN(T44) 29.00/9.30 F267_IN(T44) -> U5^1(f276_in(T44), T44) 29.00/9.30 U5^1(f276_out1, T44) -> F2_IN(T44) 29.00/9.30 F2_IN(T84) -> F300_IN(T84) 29.00/9.30 F300_IN(T84) -> U7^1(f304_in(T84), T84) 29.00/9.30 U7^1(f304_out1(T89), T84) -> F2_IN(T84) 29.00/9.30 29.00/9.30 The TRS R consists of the following rules: 29.00/9.30 29.00/9.30 f2_in(T8) -> f2_out1 29.00/9.30 f2_in(T44) -> U1(f267_in(T44), T44) 29.00/9.30 U1(f267_out1, T44) -> f2_out1 29.00/9.30 f2_in(T84) -> U2(f300_in(T84), T84) 29.00/9.30 U2(f300_out1(T89), T84) -> f2_out1 29.00/9.30 f276_in(0) -> f276_out1 29.00/9.30 f276_in(s(T67)) -> U3(f276_in(T67), s(T67)) 29.00/9.30 U3(f276_out1, s(T67)) -> f276_out1 29.00/9.30 f304_in(s(T102)) -> f304_out1(0) 29.00/9.30 f304_in(s(T108)) -> U4(f304_in(T108), s(T108)) 29.00/9.30 U4(f304_out1(T109), s(T108)) -> f304_out1(s(T109)) 29.00/9.30 f267_in(T44) -> U5(f276_in(T44), T44) 29.00/9.30 U5(f276_out1, T44) -> U6(f2_in(T44), T44) 29.00/9.30 U6(f2_out1, T44) -> f267_out1 29.00/9.30 f300_in(T84) -> U7(f304_in(T84), T84) 29.00/9.30 U7(f304_out1(T89), T84) -> U8(f2_in(T84), T84, T89) 29.00/9.30 U8(f2_out1, T84, T89) -> f300_out1(T89) 29.00/9.30 29.00/9.30 Q is empty. 29.00/9.30 We have to consider all minimal (P,Q,R)-chains. 29.00/9.30 ---------------------------------------- 29.00/9.30 29.00/9.30 (96) NonTerminationLoopProof (COMPLETE) 29.00/9.30 We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. 29.00/9.30 Found a loop by narrowing to the left: 29.00/9.30 29.00/9.30 s = F267_IN(0) evaluates to t =F267_IN(0) 29.00/9.30 29.00/9.30 Thus s starts an infinite chain as s semiunifies with t with the following substitutions: 29.00/9.30 * Matcher: [ ] 29.00/9.30 * Semiunifier: [ ] 29.00/9.30 29.00/9.30 -------------------------------------------------------------------------------- 29.00/9.30 Rewriting sequence 29.00/9.30 29.00/9.30 F267_IN(0) -> U5^1(f276_in(0), 0) 29.00/9.30 with rule F267_IN(T44) -> U5^1(f276_in(T44), T44) at position [] and matcher [T44 / 0] 29.00/9.30 29.00/9.30 U5^1(f276_in(0), 0) -> U5^1(f276_out1, 0) 29.00/9.30 with rule f276_in(0) -> f276_out1 at position [0] and matcher [ ] 29.00/9.30 29.00/9.30 U5^1(f276_out1, 0) -> F2_IN(0) 29.00/9.30 with rule U5^1(f276_out1, T44') -> F2_IN(T44') at position [] and matcher [T44' / 0] 29.00/9.30 29.00/9.30 F2_IN(0) -> F267_IN(0) 29.00/9.30 with rule F2_IN(T44) -> F267_IN(T44) 29.00/9.30 29.00/9.30 Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence 29.00/9.30 29.00/9.30 29.00/9.30 All these steps are and every following step will be a correct step w.r.t to Q. 29.00/9.30 29.00/9.30 29.00/9.30 29.00/9.30 29.00/9.30 ---------------------------------------- 29.00/9.30 29.00/9.30 (97) 29.00/9.30 NO 29.00/9.30 29.00/9.30 ---------------------------------------- 29.00/9.30 29.00/9.30 (98) PrologToIRSwTTransformerProof (SOUND) 29.00/9.30 Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert 29.00/9.30 29.00/9.30 { 29.00/9.30 "root": 1, 29.00/9.30 "program": { 29.00/9.30 "directives": [], 29.00/9.30 "clauses": [ 29.00/9.30 [ 29.00/9.30 "(insert X (void) (tree X (void) (void)))", 29.00/9.30 null 29.00/9.30 ], 29.00/9.30 [ 29.00/9.30 "(insert X (tree X Left Right) (tree X Left Right))", 29.00/9.30 null 29.00/9.30 ], 29.00/9.30 [ 29.00/9.30 "(insert X (tree Y Left Right) (tree Y Left1 Right))", 29.00/9.30 "(',' (less X Y) (insert X Left Left1))" 29.00/9.30 ], 29.00/9.30 [ 29.00/9.30 "(insert X (tree Y Left Right) (tree Y Left Right1))", 29.00/9.30 "(',' (less Y X) (insert X Right Right1))" 29.00/9.30 ], 29.00/9.30 [ 29.00/9.30 "(less (0) (s X1))", 29.00/9.30 null 29.00/9.30 ], 29.00/9.30 [ 29.00/9.30 "(less (s X) (s Y))", 29.00/9.30 "(less X Y)" 29.00/9.30 ] 29.00/9.30 ] 29.00/9.30 }, 29.00/9.30 "graph": { 29.00/9.30 "nodes": { 29.00/9.30 "22": { 29.00/9.30 "goal": [{ 29.00/9.30 "clause": 1, 29.00/9.30 "scope": 1, 29.00/9.30 "term": "(insert T1 T2 T3)" 29.00/9.30 }], 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29.00/9.31 "exprvars": [] 29.00/9.31 } 29.00/9.31 }, 29.00/9.31 "21": { 29.00/9.31 "goal": [], 29.00/9.31 "kb": { 29.00/9.31 "nonunifying": [], 29.00/9.31 "intvars": {}, 29.00/9.31 "arithmetic": { 29.00/9.31 "type": "PlainIntegerRelationState", 29.00/9.31 "relations": [] 29.00/9.31 }, 29.00/9.31 "ground": [], 29.00/9.31 "free": [], 29.00/9.31 "exprvars": [] 29.00/9.31 } 29.00/9.31 } 29.00/9.31 }, 29.00/9.31 "edges": [ 29.00/9.31 { 29.00/9.31 "from": 1, 29.00/9.31 "to": 4, 29.00/9.31 "label": "CASE" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 4, 29.00/9.31 "to": 15, 29.00/9.31 "label": "PARALLEL" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 4, 29.00/9.31 "to": 16, 29.00/9.31 "label": "PARALLEL" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 15, 29.00/9.31 "to": 17, 29.00/9.31 "label": "EVAL with clause\ninsert(X6, void, tree(X6, void, void)).\nand substitutionT1 -> T8,\nX6 -> T8,\nT2 -> void,\nT3 -> tree(T8, void, void)" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 15, 29.00/9.31 "to": 19, 29.00/9.31 "label": "EVAL-BACKTRACK" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 16, 29.00/9.31 "to": 22, 29.00/9.31 "label": "PARALLEL" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 16, 29.00/9.31 "to": 23, 29.00/9.31 "label": "PARALLEL" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 17, 29.00/9.31 "to": 21, 29.00/9.31 "label": "SUCCESS" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 22, 29.00/9.31 "to": 24, 29.00/9.31 "label": "EVAL with clause\ninsert(X19, tree(X19, X20, X21), tree(X19, X20, X21)).\nand substitutionT1 -> T21,\nX19 -> T21,\nX20 -> T22,\nX21 -> T23,\nT2 -> tree(T21, T22, T23),\nT3 -> tree(T21, T22, T23)" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 22, 29.00/9.31 "to": 25, 29.00/9.31 "label": "EVAL-BACKTRACK" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 23, 29.00/9.31 "to": 58, 29.00/9.31 "label": "PARALLEL" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 23, 29.00/9.31 "to": 59, 29.00/9.31 "label": "PARALLEL" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 24, 29.00/9.31 "to": 26, 29.00/9.31 "label": "SUCCESS" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 58, 29.00/9.31 "to": 60, 29.00/9.31 "label": "EVAL with clause\ninsert(X42, tree(X43, X44, X45), tree(X43, X46, X45)) :- ','(less(X42, X43), insert(X42, X44, X46)).\nand substitutionT1 -> T44,\nX42 -> T44,\nX43 -> T49,\nX44 -> T50,\nX45 -> T47,\nT2 -> tree(T49, T50, T47),\nX46 -> T51,\nT3 -> tree(T49, T51, T47),\nT45 -> T49,\nT46 -> T50,\nT48 -> T51" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 58, 29.00/9.31 "to": 61, 29.00/9.31 "label": "EVAL-BACKTRACK" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 59, 29.00/9.31 "to": 331, 29.00/9.31 "label": "EVAL with clause\ninsert(X76, tree(X77, X78, X79), tree(X77, X78, X80)) :- ','(less(X77, X76), insert(X76, X79, X80)).\nand substitutionT1 -> T84,\nX76 -> T84,\nX77 -> T89,\nX78 -> T86,\nX79 -> T90,\nT2 -> tree(T89, T86, T90),\nX80 -> T91,\nT3 -> tree(T89, T86, T91),\nT85 -> T89,\nT87 -> T90,\nT88 -> T91" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 59, 29.00/9.31 "to": 332, 29.00/9.31 "label": "EVAL-BACKTRACK" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 60, 29.00/9.31 "to": 245, 29.00/9.31 "label": "SPLIT 1" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 60, 29.00/9.31 "to": 246, 29.00/9.31 "label": "SPLIT 2\nnew knowledge:\nT44 is ground\nreplacements:T50 -> T54,\nT51 -> T55" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 245, 29.00/9.31 "to": 247, 29.00/9.31 "label": "CASE" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 246, 29.00/9.31 "to": 1, 29.00/9.31 "label": "INSTANCE with matching:\nT1 -> T44\nT2 -> T54\nT3 -> T55" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 247, 29.00/9.31 "to": 248, 29.00/9.31 "label": "PARALLEL" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 247, 29.00/9.31 "to": 249, 29.00/9.31 "label": "PARALLEL" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 248, 29.00/9.31 "to": 326, 29.00/9.31 "label": "EVAL with clause\nless(0, s(X55)).\nand substitutionT44 -> 0,\nX55 -> T62,\nT49 -> s(T62)" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 248, 29.00/9.31 "to": 327, 29.00/9.31 "label": "EVAL-BACKTRACK" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 249, 29.00/9.31 "to": 329, 29.00/9.31 "label": "EVAL with clause\nless(s(X60), s(X61)) :- less(X60, X61).\nand substitutionX60 -> T67,\nT44 -> s(T67),\nX61 -> T69,\nT49 -> s(T69),\nT68 -> T69" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 249, 29.00/9.31 "to": 330, 29.00/9.31 "label": "EVAL-BACKTRACK" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 326, 29.00/9.31 "to": 328, 29.00/9.31 "label": "SUCCESS" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 329, 29.00/9.31 "to": 245, 29.00/9.31 "label": "INSTANCE with matching:\nT44 -> T67\nT49 -> T69" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 331, 29.00/9.31 "to": 333, 29.00/9.31 "label": "SPLIT 1" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 331, 29.00/9.31 "to": 334, 29.00/9.31 "label": "SPLIT 2\nnew knowledge:\nT89 is ground\nT84 is ground\nreplacements:T90 -> T94,\nT91 -> T95" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 333, 29.00/9.31 "to": 335, 29.00/9.31 "label": "CASE" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 334, 29.00/9.31 "to": 1, 29.00/9.31 "label": "INSTANCE with matching:\nT1 -> T84\nT2 -> T94\nT3 -> T95" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 335, 29.00/9.31 "to": 336, 29.00/9.31 "label": "PARALLEL" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 335, 29.00/9.31 "to": 337, 29.00/9.31 "label": "PARALLEL" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 336, 29.00/9.31 "to": 338, 29.00/9.31 "label": "EVAL with clause\nless(0, s(X89)).\nand substitutionT89 -> 0,\nX89 -> T102,\nT84 -> s(T102)" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 336, 29.00/9.31 "to": 339, 29.00/9.31 "label": "EVAL-BACKTRACK" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 337, 29.00/9.31 "to": 341, 29.00/9.31 "label": "EVAL with clause\nless(s(X94), s(X95)) :- less(X94, X95).\nand substitutionX94 -> T109,\nT89 -> s(T109),\nX95 -> T108,\nT84 -> s(T108),\nT107 -> T109" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 337, 29.00/9.31 "to": 342, 29.00/9.31 "label": "EVAL-BACKTRACK" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 338, 29.00/9.31 "to": 340, 29.00/9.31 "label": "SUCCESS" 29.00/9.31 }, 29.00/9.31 { 29.00/9.31 "from": 341, 29.00/9.31 "to": 333, 29.00/9.31 "label": "INSTANCE with matching:\nT89 -> T109\nT84 -> T108" 29.00/9.31 } 29.00/9.31 ], 29.00/9.31 "type": "Graph" 29.00/9.31 } 29.00/9.31 } 29.00/9.31 29.00/9.31 ---------------------------------------- 29.00/9.31 29.00/9.31 (99) 29.00/9.31 Complex Obligation (AND) 29.00/9.31 29.00/9.31 ---------------------------------------- 29.00/9.31 29.00/9.31 (100) 29.00/9.31 Obligation: 29.00/9.31 Rules: 29.00/9.31 f341_in(T108) -> f333_in(T108) :|: TRUE 29.00/9.31 f333_out(x) -> f341_out(x) :|: TRUE 29.00/9.31 f335_in(T84) -> f336_in(T84) :|: TRUE 29.00/9.31 f336_out(x1) -> f335_out(x1) :|: TRUE 29.00/9.31 f337_out(x2) -> f335_out(x2) :|: TRUE 29.00/9.31 f335_in(x3) -> f337_in(x3) :|: TRUE 29.00/9.31 f342_out -> f337_out(x4) :|: TRUE 29.00/9.31 f341_out(x5) -> f337_out(s(x5)) :|: TRUE 29.00/9.31 f337_in(s(x6)) -> f341_in(x6) :|: TRUE 29.00/9.31 f337_in(x7) -> f342_in :|: TRUE 29.00/9.31 f333_in(x8) -> f335_in(x8) :|: TRUE 29.00/9.31 f335_out(x9) -> f333_out(x9) :|: TRUE 29.00/9.31 f4_out(T1) -> f1_out(T1) :|: TRUE 29.00/9.31 f1_in(x10) -> f4_in(x10) :|: TRUE 29.00/9.31 f15_out(x11) -> f4_out(x11) :|: TRUE 29.00/9.31 f4_in(x12) -> f15_in(x12) :|: TRUE 29.00/9.31 f4_in(x13) -> f16_in(x13) :|: TRUE 29.00/9.31 f16_out(x14) -> f4_out(x14) :|: TRUE 29.00/9.31 f16_in(x15) -> f23_in(x15) :|: TRUE 29.00/9.31 f22_out(x16) -> f16_out(x16) :|: TRUE 29.00/9.31 f16_in(x17) -> f22_in(x17) :|: TRUE 29.00/9.31 f23_out(x18) -> f16_out(x18) :|: TRUE 29.00/9.31 f23_in(x19) -> f59_in(x19) :|: TRUE 29.00/9.31 f58_out(x20) -> f23_out(x20) :|: TRUE 29.00/9.31 f59_out(x21) -> f23_out(x21) :|: TRUE 29.00/9.31 f23_in(x22) -> f58_in(x22) :|: TRUE 29.00/9.31 f59_in(x23) -> f332_in :|: TRUE 29.00/9.31 f332_out -> f59_out(x24) :|: TRUE 29.00/9.31 f331_out(x25) -> f59_out(x25) :|: TRUE 29.00/9.31 f59_in(x26) -> f331_in(x26) :|: TRUE 29.00/9.31 f334_out(x27) -> f331_out(x27) :|: TRUE 29.00/9.31 f331_in(x28) -> f333_in(x28) :|: TRUE 29.00/9.31 f333_out(x29) -> f334_in(x29) :|: TRUE 29.00/9.31 Start term: f1_in(T1) 29.00/9.31 29.00/9.31 ---------------------------------------- 29.00/9.31 29.00/9.31 (101) IRSwTSimpleDependencyGraphProof (EQUIVALENT) 29.00/9.31 Constructed simple dependency graph. 29.00/9.31 29.00/9.31 Simplified to the following IRSwTs: 29.00/9.31 29.00/9.31 intTRSProblem: 29.00/9.31 f341_in(T108) -> f333_in(T108) :|: TRUE 29.00/9.31 f335_in(x3) -> f337_in(x3) :|: TRUE 29.00/9.31 f337_in(s(x6)) -> f341_in(x6) :|: TRUE 29.00/9.31 f333_in(x8) -> f335_in(x8) :|: TRUE 29.00/9.31 29.00/9.31 29.00/9.31 ---------------------------------------- 29.00/9.31 29.00/9.31 (102) 29.00/9.31 Obligation: 29.00/9.31 Rules: 29.00/9.31 f341_in(T108) -> f333_in(T108) :|: TRUE 29.00/9.31 f335_in(x3) -> f337_in(x3) :|: TRUE 29.00/9.31 f337_in(s(x6)) -> f341_in(x6) :|: TRUE 29.00/9.31 f333_in(x8) -> f335_in(x8) :|: TRUE 29.00/9.31 29.00/9.31 ---------------------------------------- 29.00/9.31 29.00/9.31 (103) IntTRSCompressionProof (EQUIVALENT) 29.00/9.31 Compressed rules. 29.00/9.31 ---------------------------------------- 29.00/9.31 29.00/9.31 (104) 29.00/9.31 Obligation: 29.00/9.31 Rules: 29.00/9.31 f335_in(s(x6:0)) -> f335_in(x6:0) :|: TRUE 29.00/9.31 29.00/9.31 ---------------------------------------- 29.00/9.31 29.00/9.31 (105) IRSFormatTransformerProof (EQUIVALENT) 29.00/9.31 Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). 29.00/9.31 ---------------------------------------- 29.00/9.31 29.00/9.31 (106) 29.00/9.31 Obligation: 29.00/9.31 Rules: 29.00/9.31 f335_in(s(x6:0)) -> f335_in(x6:0) :|: TRUE 29.00/9.31 29.00/9.31 ---------------------------------------- 29.00/9.31 29.00/9.31 (107) IRSwTTerminationDigraphProof (EQUIVALENT) 29.00/9.31 Constructed termination digraph! 29.00/9.31 Nodes: 29.00/9.31 (1) f335_in(s(x6:0)) -> f335_in(x6:0) :|: TRUE 29.00/9.31 29.00/9.31 Arcs: 29.00/9.31 (1) -> (1) 29.00/9.31 29.00/9.31 This digraph is fully evaluated! 29.00/9.31 ---------------------------------------- 29.00/9.31 29.00/9.31 (108) 29.00/9.31 Obligation: 29.00/9.31 29.00/9.31 Termination digraph: 29.00/9.31 Nodes: 29.00/9.31 (1) f335_in(s(x6:0)) -> f335_in(x6:0) :|: TRUE 29.00/9.31 29.00/9.31 Arcs: 29.00/9.31 (1) -> (1) 29.00/9.31 29.00/9.31 This digraph is fully evaluated! 29.00/9.31 29.00/9.31 ---------------------------------------- 29.00/9.31 29.00/9.31 (109) TempFilterProof (SOUND) 29.00/9.31 Used the following sort dictionary for filtering: 29.00/9.31 f335_in(VARIABLE) 29.00/9.31 s(VARIABLE) 29.00/9.31 Removed predefined arithmetic. 29.00/9.31 ---------------------------------------- 29.00/9.31 29.00/9.31 (110) 29.00/9.31 Obligation: 29.00/9.31 Rules: 29.00/9.31 f335_in(s(x6:0)) -> f335_in(x6:0) 29.00/9.31 29.00/9.31 ---------------------------------------- 29.00/9.31 29.00/9.31 (111) IRSwTToQDPProof (SOUND) 29.00/9.31 Removed the integers and created a QDP-Problem. 29.00/9.31 ---------------------------------------- 29.00/9.31 29.00/9.31 (112) 29.00/9.31 Obligation: 29.00/9.31 Q DP problem: 29.00/9.31 The TRS P consists of the following rules: 29.00/9.31 29.00/9.31 f335_in(s(x6:0)) -> f335_in(x6:0) 29.00/9.31 29.00/9.31 R is empty. 29.00/9.31 Q is empty. 29.00/9.31 We have to consider all (P,Q,R)-chains. 29.00/9.31 ---------------------------------------- 29.00/9.31 29.00/9.31 (113) QDPSizeChangeProof (EQUIVALENT) 29.00/9.31 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. 29.00/9.31 29.00/9.31 From the DPs we obtained the following set of size-change graphs: 29.00/9.31 *f335_in(s(x6:0)) -> f335_in(x6:0) 29.00/9.31 The graph contains the following edges 1 > 1 29.00/9.31 29.00/9.31 29.00/9.31 ---------------------------------------- 29.00/9.31 29.00/9.31 (114) 29.00/9.31 YES 29.00/9.31 29.00/9.31 ---------------------------------------- 29.00/9.31 29.00/9.31 (115) 29.00/9.31 Obligation: 29.00/9.31 Rules: 29.00/9.31 f245_out(T67) -> f329_out(T67) :|: TRUE 29.00/9.31 f329_in(x) -> f245_in(x) :|: TRUE 29.00/9.31 f330_out -> f249_out(T44) :|: TRUE 29.00/9.31 f329_out(x1) -> f249_out(s(x1)) :|: TRUE 29.00/9.31 f249_in(x2) -> f330_in :|: TRUE 29.00/9.31 f249_in(s(x3)) -> f329_in(x3) :|: TRUE 29.00/9.31 f247_in(x4) -> f248_in(x4) :|: TRUE 29.00/9.31 f248_out(x5) -> f247_out(x5) :|: TRUE 29.00/9.31 f247_in(x6) -> f249_in(x6) :|: TRUE 29.00/9.31 f249_out(x7) -> f247_out(x7) :|: TRUE 29.00/9.31 f245_in(x8) -> f247_in(x8) :|: TRUE 29.00/9.31 f247_out(x9) -> f245_out(x9) :|: TRUE 29.00/9.31 f4_out(T1) -> f1_out(T1) :|: TRUE 29.00/9.31 f1_in(x10) -> f4_in(x10) :|: TRUE 29.00/9.31 f15_out(x11) -> f4_out(x11) :|: TRUE 29.00/9.31 f4_in(x12) -> f15_in(x12) :|: TRUE 29.00/9.31 f4_in(x13) -> f16_in(x13) :|: TRUE 29.00/9.31 f16_out(x14) -> f4_out(x14) :|: TRUE 29.00/9.31 f16_in(x15) -> f23_in(x15) :|: TRUE 29.00/9.31 f22_out(x16) -> f16_out(x16) :|: TRUE 29.00/9.31 f16_in(x17) -> f22_in(x17) :|: TRUE 29.00/9.31 f23_out(x18) -> f16_out(x18) :|: TRUE 29.00/9.31 f23_in(x19) -> f59_in(x19) :|: TRUE 29.00/9.31 f58_out(x20) -> f23_out(x20) :|: TRUE 29.00/9.31 f59_out(x21) -> f23_out(x21) :|: TRUE 29.00/9.31 f23_in(x22) -> f58_in(x22) :|: TRUE 29.00/9.31 f58_in(x23) -> f60_in(x23) :|: TRUE 29.00/9.31 f60_out(x24) -> f58_out(x24) :|: TRUE 29.00/9.31 f61_out -> f58_out(x25) :|: TRUE 29.00/9.31 f58_in(x26) -> f61_in :|: TRUE 29.00/9.31 f246_out(x27) -> f60_out(x27) :|: TRUE 29.00/9.31 f245_out(x28) -> f246_in(x28) :|: TRUE 29.00/9.31 f60_in(x29) -> f245_in(x29) :|: TRUE 29.00/9.31 Start term: f1_in(T1) 29.00/9.31 29.00/9.31 ---------------------------------------- 29.00/9.31 29.00/9.31 (116) IRSwTSimpleDependencyGraphProof (EQUIVALENT) 29.00/9.31 Constructed simple dependency graph. 29.00/9.31 29.00/9.31 Simplified to the following IRSwTs: 29.00/9.31 29.00/9.31 intTRSProblem: 29.00/9.31 f329_in(x) -> f245_in(x) :|: TRUE 29.00/9.31 f249_in(s(x3)) -> f329_in(x3) :|: TRUE 29.00/9.31 f247_in(x6) -> f249_in(x6) :|: TRUE 29.00/9.31 f245_in(x8) -> f247_in(x8) :|: TRUE 29.00/9.31 29.00/9.31 29.00/9.31 ---------------------------------------- 29.00/9.31 29.00/9.31 (117) 29.00/9.31 Obligation: 29.00/9.31 Rules: 29.00/9.31 f329_in(x) -> f245_in(x) :|: TRUE 29.00/9.31 f249_in(s(x3)) -> f329_in(x3) :|: TRUE 29.00/9.31 f247_in(x6) -> f249_in(x6) :|: TRUE 29.00/9.31 f245_in(x8) -> f247_in(x8) :|: TRUE 29.00/9.31 29.00/9.31 ---------------------------------------- 29.00/9.31 29.00/9.31 (118) IntTRSCompressionProof (EQUIVALENT) 29.00/9.31 Compressed rules. 29.00/9.31 ---------------------------------------- 29.00/9.31 29.00/9.31 (119) 29.00/9.31 Obligation: 29.00/9.31 Rules: 29.00/9.31 f247_in(s(x3:0)) -> f247_in(x3:0) :|: TRUE 29.00/9.31 29.00/9.31 ---------------------------------------- 29.00/9.31 29.00/9.31 (120) IRSFormatTransformerProof (EQUIVALENT) 29.00/9.31 Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). 29.00/9.31 ---------------------------------------- 29.00/9.31 29.00/9.31 (121) 29.00/9.31 Obligation: 29.00/9.31 Rules: 29.00/9.31 f247_in(s(x3:0)) -> f247_in(x3:0) :|: TRUE 29.00/9.31 29.00/9.31 ---------------------------------------- 29.00/9.31 29.00/9.31 (122) IRSwTTerminationDigraphProof (EQUIVALENT) 29.00/9.31 Constructed termination digraph! 29.00/9.31 Nodes: 29.00/9.31 (1) f247_in(s(x3:0)) -> f247_in(x3:0) :|: TRUE 29.00/9.31 29.00/9.31 Arcs: 29.00/9.31 (1) -> (1) 29.00/9.31 29.00/9.31 This digraph is fully evaluated! 29.00/9.31 ---------------------------------------- 29.00/9.31 29.00/9.31 (123) 29.00/9.31 Obligation: 29.00/9.31 29.00/9.31 Termination digraph: 29.00/9.31 Nodes: 29.00/9.31 (1) f247_in(s(x3:0)) -> f247_in(x3:0) :|: TRUE 29.00/9.31 29.00/9.31 Arcs: 29.00/9.31 (1) -> (1) 29.00/9.31 29.00/9.31 This digraph is fully evaluated! 29.00/9.31 29.00/9.31 ---------------------------------------- 29.00/9.31 29.00/9.31 (124) TempFilterProof (SOUND) 29.00/9.31 Used the following sort dictionary for filtering: 29.00/9.31 f247_in(VARIABLE) 29.00/9.31 s(VARIABLE) 29.00/9.31 Removed predefined arithmetic. 29.00/9.31 ---------------------------------------- 29.00/9.31 29.00/9.31 (125) 29.00/9.31 Obligation: 29.00/9.31 Rules: 29.00/9.31 f247_in(s(x3:0)) -> f247_in(x3:0) 29.00/9.31 29.00/9.31 ---------------------------------------- 29.00/9.31 29.00/9.31 (126) IRSwTToQDPProof (SOUND) 29.00/9.31 Removed the integers and created a QDP-Problem. 29.00/9.31 ---------------------------------------- 29.00/9.31 29.00/9.31 (127) 29.00/9.31 Obligation: 29.00/9.31 Q DP problem: 29.00/9.31 The TRS P consists of the following rules: 29.00/9.31 29.00/9.31 f247_in(s(x3:0)) -> f247_in(x3:0) 29.00/9.31 29.00/9.31 R is empty. 29.00/9.31 Q is empty. 29.00/9.31 We have to consider all (P,Q,R)-chains. 29.00/9.31 ---------------------------------------- 29.00/9.31 29.00/9.31 (128) QDPSizeChangeProof (EQUIVALENT) 29.00/9.31 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. 29.00/9.31 29.00/9.31 From the DPs we obtained the following set of size-change graphs: 29.00/9.31 *f247_in(s(x3:0)) -> f247_in(x3:0) 29.00/9.31 The graph contains the following edges 1 > 1 29.00/9.31 29.00/9.31 29.00/9.31 ---------------------------------------- 29.00/9.31 29.00/9.31 (129) 29.00/9.31 YES 29.00/9.31 29.00/9.31 ---------------------------------------- 29.00/9.31 29.00/9.31 (130) 29.00/9.31 Obligation: 29.00/9.31 Rules: 29.00/9.31 f341_in(T108) -> f333_in(T108) :|: TRUE 29.00/9.31 f333_out(x) -> f341_out(x) :|: TRUE 29.00/9.31 f339_out -> f336_out(T84) :|: TRUE 29.00/9.31 f336_in(x1) -> f339_in :|: TRUE 29.00/9.31 f336_in(s(T102)) -> f338_in :|: TRUE 29.00/9.31 f338_out -> f336_out(s(x2)) :|: TRUE 29.00/9.31 f326_in -> f326_out :|: TRUE 29.00/9.31 f333_in(x3) -> f335_in(x3) :|: TRUE 29.00/9.31 f335_out(x4) -> f333_out(x4) :|: TRUE 29.00/9.31 f246_out(T44) -> f60_out(T44) :|: TRUE 29.00/9.31 f245_out(x5) -> f246_in(x5) :|: TRUE 29.00/9.31 f60_in(x6) -> f245_in(x6) :|: TRUE 29.00/9.31 f245_in(x7) -> f247_in(x7) :|: TRUE 29.00/9.31 f247_out(x8) -> f245_out(x8) :|: TRUE 29.00/9.31 f338_in -> f338_out :|: TRUE 29.00/9.31 f59_in(T1) -> f332_in :|: TRUE 29.00/9.31 f332_out -> f59_out(x9) :|: TRUE 29.00/9.31 f331_out(x10) -> f59_out(x10) :|: TRUE 29.00/9.31 f59_in(x11) -> f331_in(x11) :|: TRUE 29.00/9.31 f245_out(T67) -> f329_out(T67) :|: TRUE 29.00/9.31 f329_in(x12) -> f245_in(x12) :|: TRUE 29.00/9.31 f16_in(x13) -> f23_in(x13) :|: TRUE 29.00/9.31 f22_out(x14) -> f16_out(x14) :|: TRUE 29.00/9.31 f16_in(x15) -> f22_in(x15) :|: TRUE 29.00/9.31 f23_out(x16) -> f16_out(x16) :|: TRUE 29.00/9.31 f330_out -> f249_out(x17) :|: TRUE 29.00/9.31 f329_out(x18) -> f249_out(s(x18)) :|: TRUE 29.00/9.31 f249_in(x19) -> f330_in :|: TRUE 29.00/9.31 f249_in(s(x20)) -> f329_in(x20) :|: TRUE 29.00/9.31 f342_out -> f337_out(x21) :|: TRUE 29.00/9.31 f341_out(x22) -> f337_out(s(x22)) :|: TRUE 29.00/9.31 f337_in(s(x23)) -> f341_in(x23) :|: TRUE 29.00/9.31 f337_in(x24) -> f342_in :|: TRUE 29.00/9.31 f23_in(x25) -> f59_in(x25) :|: TRUE 29.00/9.31 f58_out(x26) -> f23_out(x26) :|: TRUE 29.00/9.31 f59_out(x27) -> f23_out(x27) :|: TRUE 29.00/9.31 f23_in(x28) -> f58_in(x28) :|: TRUE 29.00/9.31 f246_in(x29) -> f1_in(x29) :|: TRUE 29.00/9.31 f1_out(x30) -> f246_out(x30) :|: TRUE 29.00/9.31 f4_out(x31) -> f1_out(x31) :|: TRUE 29.00/9.31 f1_in(x32) -> f4_in(x32) :|: TRUE 29.00/9.31 f326_out -> f248_out(0) :|: TRUE 29.00/9.31 f248_in(x33) -> f327_in :|: TRUE 29.00/9.31 f327_out -> f248_out(x34) :|: TRUE 29.00/9.31 f248_in(0) -> f326_in :|: TRUE 29.00/9.31 f335_in(x35) -> f336_in(x35) :|: TRUE 29.00/9.31 f336_out(x36) -> f335_out(x36) :|: TRUE 29.00/9.31 f337_out(x37) -> f335_out(x37) :|: TRUE 29.00/9.31 f335_in(x38) -> f337_in(x38) :|: TRUE 29.00/9.31 f15_out(x39) -> f4_out(x39) :|: TRUE 29.00/9.31 f4_in(x40) -> f15_in(x40) :|: TRUE 29.00/9.31 f4_in(x41) -> f16_in(x41) :|: TRUE 29.00/9.31 f16_out(x42) -> f4_out(x42) :|: TRUE 29.00/9.31 f334_in(x43) -> f1_in(x43) :|: TRUE 29.00/9.31 f1_out(x44) -> f334_out(x44) :|: TRUE 29.00/9.31 f58_in(x45) -> f60_in(x45) :|: TRUE 29.00/9.31 f60_out(x46) -> f58_out(x46) :|: TRUE 29.00/9.31 f61_out -> f58_out(x47) :|: TRUE 29.00/9.31 f58_in(x48) -> f61_in :|: TRUE 29.00/9.31 f247_in(x49) -> f248_in(x49) :|: TRUE 29.00/9.31 f248_out(x50) -> f247_out(x50) :|: TRUE 29.00/9.31 f247_in(x51) -> f249_in(x51) :|: TRUE 29.00/9.31 f249_out(x52) -> f247_out(x52) :|: TRUE 29.00/9.31 f334_out(x53) -> f331_out(x53) :|: TRUE 29.00/9.31 f331_in(x54) -> f333_in(x54) :|: TRUE 29.00/9.31 f333_out(x55) -> f334_in(x55) :|: TRUE 29.00/9.31 Start term: f1_in(T1) 29.00/9.31 29.00/9.31 ---------------------------------------- 29.00/9.31 29.00/9.31 (131) IRSwTSimpleDependencyGraphProof (EQUIVALENT) 29.00/9.31 Constructed simple dependency graph. 29.00/9.31 29.00/9.31 Simplified to the following IRSwTs: 29.00/9.31 29.00/9.31 intTRSProblem: 29.00/9.31 f341_in(T108) -> f333_in(T108) :|: TRUE 29.00/9.31 f333_out(x) -> f341_out(x) :|: TRUE 29.00/9.31 f336_in(s(T102)) -> f338_in :|: TRUE 29.00/9.31 f338_out -> f336_out(s(x2)) :|: TRUE 29.00/9.31 f326_in -> f326_out :|: TRUE 29.00/9.31 f333_in(x3) -> f335_in(x3) :|: TRUE 29.00/9.31 f335_out(x4) -> f333_out(x4) :|: TRUE 29.00/9.31 f245_out(x5) -> f246_in(x5) :|: TRUE 29.00/9.31 f60_in(x6) -> f245_in(x6) :|: TRUE 29.00/9.31 f245_in(x7) -> f247_in(x7) :|: TRUE 29.00/9.31 f247_out(x8) -> f245_out(x8) :|: TRUE 29.00/9.31 f338_in -> f338_out :|: TRUE 29.00/9.31 f59_in(x11) -> f331_in(x11) :|: TRUE 29.00/9.31 f245_out(T67) -> f329_out(T67) :|: TRUE 29.00/9.31 f329_in(x12) -> f245_in(x12) :|: TRUE 29.00/9.31 f16_in(x13) -> f23_in(x13) :|: TRUE 29.00/9.31 f329_out(x18) -> f249_out(s(x18)) :|: TRUE 29.00/9.31 f249_in(s(x20)) -> f329_in(x20) :|: TRUE 29.00/9.31 f341_out(x22) -> f337_out(s(x22)) :|: TRUE 29.00/9.31 f337_in(s(x23)) -> f341_in(x23) :|: TRUE 29.00/9.31 f23_in(x25) -> f59_in(x25) :|: TRUE 29.00/9.31 f23_in(x28) -> f58_in(x28) :|: TRUE 29.00/9.31 f246_in(x29) -> f1_in(x29) :|: TRUE 29.00/9.31 f1_in(x32) -> f4_in(x32) :|: TRUE 29.00/9.31 f326_out -> f248_out(0) :|: TRUE 29.00/9.31 f248_in(0) -> f326_in :|: TRUE 29.00/9.31 f335_in(x35) -> f336_in(x35) :|: TRUE 29.00/9.31 f336_out(x36) -> f335_out(x36) :|: TRUE 29.00/9.31 f337_out(x37) -> f335_out(x37) :|: TRUE 29.00/9.31 f335_in(x38) -> f337_in(x38) :|: TRUE 29.00/9.31 f4_in(x41) -> f16_in(x41) :|: TRUE 29.00/9.31 f334_in(x43) -> f1_in(x43) :|: TRUE 29.00/9.31 f58_in(x45) -> f60_in(x45) :|: TRUE 29.00/9.31 f247_in(x49) -> f248_in(x49) :|: TRUE 29.00/9.31 f248_out(x50) -> f247_out(x50) :|: TRUE 29.00/9.31 f247_in(x51) -> f249_in(x51) :|: TRUE 29.00/9.31 f249_out(x52) -> f247_out(x52) :|: TRUE 29.00/9.31 f331_in(x54) -> f333_in(x54) :|: TRUE 29.00/9.31 f333_out(x55) -> f334_in(x55) :|: TRUE 29.00/9.31 29.00/9.31 29.00/9.31 ---------------------------------------- 29.00/9.31 29.00/9.31 (132) 29.00/9.31 Obligation: 29.00/9.31 Rules: 29.00/9.31 f341_in(T108) -> f333_in(T108) :|: TRUE 29.00/9.31 f333_out(x) -> f341_out(x) :|: TRUE 29.00/9.31 f336_in(s(T102)) -> f338_in :|: TRUE 29.00/9.31 f338_out -> f336_out(s(x2)) :|: TRUE 29.00/9.31 f326_in -> f326_out :|: TRUE 29.00/9.31 f333_in(x3) -> f335_in(x3) :|: TRUE 29.00/9.31 f335_out(x4) -> f333_out(x4) :|: TRUE 29.00/9.31 f245_out(x5) -> f246_in(x5) :|: TRUE 29.00/9.31 f60_in(x6) -> f245_in(x6) :|: TRUE 29.00/9.31 f245_in(x7) -> f247_in(x7) :|: TRUE 29.00/9.31 f247_out(x8) -> f245_out(x8) :|: TRUE 29.00/9.31 f338_in -> f338_out :|: TRUE 29.00/9.31 f59_in(x11) -> f331_in(x11) :|: TRUE 29.00/9.31 f245_out(T67) -> f329_out(T67) :|: TRUE 29.00/9.31 f329_in(x12) -> f245_in(x12) :|: TRUE 29.00/9.31 f16_in(x13) -> f23_in(x13) :|: TRUE 29.00/9.31 f329_out(x18) -> f249_out(s(x18)) :|: TRUE 29.00/9.31 f249_in(s(x20)) -> f329_in(x20) :|: TRUE 29.00/9.31 f341_out(x22) -> f337_out(s(x22)) :|: TRUE 29.00/9.31 f337_in(s(x23)) -> f341_in(x23) :|: TRUE 29.00/9.31 f23_in(x25) -> f59_in(x25) :|: TRUE 29.00/9.31 f23_in(x28) -> f58_in(x28) :|: TRUE 29.00/9.31 f246_in(x29) -> f1_in(x29) :|: TRUE 29.00/9.31 f1_in(x32) -> f4_in(x32) :|: TRUE 29.00/9.31 f326_out -> f248_out(0) :|: TRUE 29.00/9.31 f248_in(0) -> f326_in :|: TRUE 29.00/9.31 f335_in(x35) -> f336_in(x35) :|: TRUE 29.00/9.31 f336_out(x36) -> f335_out(x36) :|: TRUE 29.00/9.31 f337_out(x37) -> f335_out(x37) :|: TRUE 29.00/9.31 f335_in(x38) -> f337_in(x38) :|: TRUE 29.00/9.31 f4_in(x41) -> f16_in(x41) :|: TRUE 29.00/9.31 f334_in(x43) -> f1_in(x43) :|: TRUE 29.00/9.31 f58_in(x45) -> f60_in(x45) :|: TRUE 29.00/9.31 f247_in(x49) -> f248_in(x49) :|: TRUE 29.00/9.31 f248_out(x50) -> f247_out(x50) :|: TRUE 29.00/9.31 f247_in(x51) -> f249_in(x51) :|: TRUE 29.00/9.31 f249_out(x52) -> f247_out(x52) :|: TRUE 29.00/9.31 f331_in(x54) -> f333_in(x54) :|: TRUE 29.00/9.31 f333_out(x55) -> f334_in(x55) :|: TRUE 29.00/9.31 29.00/9.31 ---------------------------------------- 29.00/9.31 29.00/9.31 (133) PrologToDTProblemTransformerProof (SOUND) 29.00/9.31 Built DT problem from termination graph DT10. 29.00/9.31 29.00/9.31 { 29.00/9.31 "root": 3, 29.00/9.31 "program": { 29.00/9.31 "directives": [], 29.00/9.31 "clauses": [ 29.00/9.31 [ 29.00/9.31 "(insert X (void) (tree X (void) (void)))", 29.00/9.31 null 29.00/9.31 ], 29.00/9.31 [ 29.00/9.31 "(insert X (tree X Left Right) (tree X Left Right))", 29.00/9.31 null 29.00/9.31 ], 29.00/9.31 [ 29.00/9.31 "(insert X (tree Y Left Right) (tree Y Left1 Right))", 29.00/9.31 "(',' (less X Y) (insert X Left Left1))" 29.00/9.31 ], 29.00/9.31 [ 29.00/9.31 "(insert X (tree Y Left Right) (tree Y Left Right1))", 29.00/9.31 "(',' (less Y X) (insert X Right Right1))" 29.00/9.31 ], 29.00/9.31 [ 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"464": { 29.13/9.32 "goal": [], 29.13/9.32 "kb": { 29.13/9.32 "nonunifying": [], 29.13/9.32 "intvars": {}, 29.13/9.32 "arithmetic": { 29.13/9.32 "type": "PlainIntegerRelationState", 29.13/9.32 "relations": [] 29.13/9.32 }, 29.13/9.32 "ground": [], 29.13/9.32 "free": [], 29.13/9.32 "exprvars": [] 29.13/9.32 } 29.13/9.32 }, 29.13/9.32 "344": { 29.13/9.32 "goal": [{ 29.13/9.32 "clause": 5, 29.13/9.32 "scope": 3, 29.13/9.32 "term": "(less T40 T42)" 29.13/9.32 }], 29.13/9.32 "kb": { 29.13/9.32 "nonunifying": [], 29.13/9.32 "intvars": {}, 29.13/9.32 "arithmetic": { 29.13/9.32 "type": "PlainIntegerRelationState", 29.13/9.32 "relations": [] 29.13/9.32 }, 29.13/9.32 "ground": ["T40"], 29.13/9.32 "free": [], 29.13/9.32 "exprvars": [] 29.13/9.32 } 29.13/9.32 }, 29.13/9.32 "465": { 29.13/9.32 "goal": [{ 29.13/9.32 "clause": 3, 29.13/9.32 "scope": 1, 29.13/9.32 "term": "(insert T255 T2 T3)" 29.13/9.32 }], 29.13/9.32 "kb": { 29.13/9.32 "nonunifying": [[ 29.13/9.32 "(insert T255 T2 T3)", 29.13/9.32 "(insert X3 (void) (tree X3 (void) (void)))" 29.13/9.32 ]], 29.13/9.32 "intvars": {}, 29.13/9.32 "arithmetic": { 29.13/9.32 "type": "PlainIntegerRelationState", 29.13/9.32 "relations": [] 29.13/9.32 }, 29.13/9.32 "ground": ["T255"], 29.13/9.32 "free": ["X3"], 29.13/9.32 "exprvars": [] 29.13/9.32 } 29.13/9.32 }, 29.13/9.32 "345": { 29.13/9.32 "goal": [{ 29.13/9.32 "clause": -1, 29.13/9.32 "scope": -1, 29.13/9.32 "term": "(true)" 29.13/9.32 }], 29.13/9.32 "kb": { 29.13/9.32 "nonunifying": [], 29.13/9.32 "intvars": {}, 29.13/9.32 "arithmetic": { 29.13/9.32 "type": "PlainIntegerRelationState", 29.13/9.32 "relations": [] 29.13/9.32 }, 29.13/9.32 "ground": [], 29.13/9.32 "free": [], 29.13/9.32 "exprvars": [] 29.13/9.32 } 29.13/9.32 }, 29.13/9.32 "466": { 29.13/9.32 "goal": [{ 29.13/9.32 "clause": -1, 29.13/9.32 "scope": -1, 29.13/9.32 "term": "(',' (less T300 T295) (insert T295 T301 T302))" 29.13/9.32 }], 29.13/9.32 "kb": { 29.13/9.32 "nonunifying": [[ 29.13/9.32 "(insert T295 T2 T3)", 29.13/9.32 "(insert X3 (void) (tree X3 (void) (void)))" 29.13/9.32 ]], 29.13/9.32 "intvars": {}, 29.13/9.32 "arithmetic": { 29.13/9.32 "type": "PlainIntegerRelationState", 29.13/9.32 "relations": [] 29.13/9.32 }, 29.13/9.32 "ground": ["T295"], 29.13/9.32 "free": ["X3"], 29.13/9.32 "exprvars": [] 29.13/9.32 } 29.13/9.32 }, 29.13/9.32 "346": { 29.13/9.32 "goal": [], 29.13/9.32 "kb": { 29.13/9.32 "nonunifying": [], 29.13/9.32 "intvars": {}, 29.13/9.32 "arithmetic": { 29.13/9.32 "type": "PlainIntegerRelationState", 29.13/9.32 "relations": [] 29.13/9.32 }, 29.13/9.32 "ground": [], 29.13/9.32 "free": [], 29.13/9.32 "exprvars": [] 29.13/9.32 } 29.13/9.32 }, 29.13/9.32 "467": { 29.13/9.32 "goal": [], 29.13/9.32 "kb": { 29.13/9.32 "nonunifying": [], 29.13/9.32 "intvars": {}, 29.13/9.32 "arithmetic": { 29.13/9.32 "type": "PlainIntegerRelationState", 29.13/9.32 "relations": [] 29.13/9.32 }, 29.13/9.32 "ground": [], 29.13/9.32 "free": [], 29.13/9.32 "exprvars": [] 29.13/9.32 } 29.13/9.32 }, 29.13/9.32 "347": { 29.13/9.32 "goal": [], 29.13/9.32 "kb": { 29.13/9.32 "nonunifying": [], 29.13/9.32 "intvars": {}, 29.13/9.32 "arithmetic": { 29.13/9.32 "type": "PlainIntegerRelationState", 29.13/9.32 "relations": [] 29.13/9.32 }, 29.13/9.32 "ground": [], 29.13/9.32 "free": [], 29.13/9.32 "exprvars": [] 29.13/9.32 } 29.13/9.32 }, 29.13/9.32 "468": { 29.13/9.32 "goal": [{ 29.13/9.32 "clause": -1, 29.13/9.32 "scope": -1, 29.13/9.32 "term": "(',' (less T315 T310) (insert T310 T316 T317))" 29.13/9.32 }], 29.13/9.32 "kb": { 29.13/9.32 "nonunifying": [ 29.13/9.32 [ 29.13/9.32 "(insert T310 T2 T3)", 29.13/9.32 "(insert X3 (void) (tree X3 (void) (void)))" 29.13/9.32 ], 29.13/9.32 [ 29.13/9.32 "(insert T310 T2 T3)", 29.13/9.32 "(insert X205 (tree X206 X207 X208) (tree X206 X209 X208))" 29.13/9.32 ] 29.13/9.32 ], 29.13/9.32 "intvars": {}, 29.13/9.32 "arithmetic": { 29.13/9.32 "type": "PlainIntegerRelationState", 29.13/9.32 "relations": [] 29.13/9.32 }, 29.13/9.32 "ground": ["T310"], 29.13/9.32 "free": [ 29.13/9.32 "X3", 29.13/9.32 "X205", 29.13/9.32 "X206", 29.13/9.32 "X207", 29.13/9.32 "X208", 29.13/9.32 "X209" 29.13/9.32 ], 29.13/9.32 "exprvars": [] 29.13/9.32 } 29.13/9.32 }, 29.13/9.32 "469": { 29.13/9.32 "goal": [], 29.13/9.32 "kb": { 29.13/9.32 "nonunifying": [], 29.13/9.32 "intvars": {}, 29.13/9.32 "arithmetic": { 29.13/9.32 "type": "PlainIntegerRelationState", 29.13/9.32 "relations": [] 29.13/9.32 }, 29.13/9.32 "ground": [], 29.13/9.32 "free": [], 29.13/9.32 "exprvars": [] 29.13/9.32 } 29.13/9.32 } 29.13/9.32 }, 29.13/9.32 "edges": [ 29.13/9.32 { 29.13/9.32 "from": 3, 29.13/9.32 "to": 9, 29.13/9.32 "label": "CASE" 29.13/9.32 }, 29.13/9.32 { 29.13/9.32 "from": 9, 29.13/9.32 "to": 18, 29.13/9.32 "label": "EVAL with clause\ninsert(X3, void, tree(X3, void, void)).\nand substitutionT1 -> T5,\nX3 -> T5,\nT2 -> void,\nT3 -> tree(T5, void, void)" 29.13/9.32 }, 29.13/9.32 { 29.13/9.32 "from": 9, 29.13/9.32 "to": 20, 29.13/9.32 "label": "EVAL-BACKTRACK" 29.13/9.32 }, 29.13/9.32 { 29.13/9.32 "from": 18, 29.13/9.32 "to": 29, 29.13/9.32 "label": "SUCCESS" 29.13/9.32 }, 29.13/9.32 { 29.13/9.32 "from": 20, 29.13/9.32 "to": 425, 29.13/9.32 "label": "EVAL with clause\ninsert(X197, tree(X197, X198, X199), tree(X197, X198, X199)).\nand substitutionT1 -> T247,\nX197 -> T247,\nX198 -> T248,\nX199 -> T249,\nT2 -> tree(T247, T248, T249),\nT3 -> tree(T247, T248, T249)" 29.13/9.32 }, 29.13/9.32 { 29.13/9.32 "from": 20, 29.13/9.32 "to": 426, 29.13/9.32 "label": "EVAL-BACKTRACK" 29.13/9.32 }, 29.13/9.32 { 29.13/9.32 "from": 29, 29.13/9.32 "to": 30, 29.13/9.32 "label": "EVAL with clause\ninsert(X7, tree(X7, X8, X9), tree(X7, X8, X9)).\nand substitutionT5 -> T9,\nX7 -> T9,\nX8 -> T10,\nX9 -> T11,\nT2 -> tree(T9, T10, T11),\nT3 -> tree(T9, T10, T11)" 29.13/9.32 }, 29.13/9.32 { 29.13/9.32 "from": 29, 29.13/9.32 "to": 31, 29.13/9.32 "label": "EVAL-BACKTRACK" 29.13/9.32 }, 29.13/9.32 { 29.13/9.32 "from": 30, 29.13/9.32 "to": 32, 29.13/9.32 "label": "SUCCESS" 29.13/9.32 }, 29.13/9.32 { 29.13/9.32 "from": 31, 29.13/9.32 "to": 400, 29.13/9.32 "label": "EVAL with clause\ninsert(X124, tree(X125, X126, X127), tree(X125, X128, X127)) :- ','(less(X124, X125), insert(X124, X126, X128)).\nand substitutionT5 -> T153,\nX124 -> T153,\nX125 -> T158,\nX126 -> T159,\nX127 -> T156,\nT2 -> tree(T158, T159, T156),\nX128 -> T160,\nT3 -> tree(T158, T160, T156),\nT154 -> T158,\nT155 -> T159,\nT157 -> T160" 29.13/9.32 }, 29.13/9.32 { 29.13/9.32 "from": 31, 29.13/9.32 "to": 401, 29.13/9.32 "label": "EVAL-BACKTRACK" 29.13/9.32 }, 29.13/9.32 { 29.13/9.32 "from": 32, 29.13/9.32 "to": 33, 29.13/9.32 "label": "EVAL with clause\ninsert(X15, tree(X16, X17, X18), tree(X16, X19, X18)) :- ','(less(X15, X16), insert(X15, X17, X19)).\nand substitutionT9 -> T17,\nX15 -> T17,\nX16 -> T22,\nX17 -> T23,\nX18 -> T20,\nT2 -> tree(T22, T23, T20),\nX19 -> T24,\nT3 -> tree(T22, T24, T20),\nT18 -> T22,\nT19 -> T23,\nT21 -> T24" 29.13/9.32 }, 29.13/9.32 { 29.13/9.32 "from": 32, 29.13/9.32 "to": 34, 29.13/9.32 "label": "EVAL-BACKTRACK" 29.13/9.32 }, 29.13/9.32 { 29.13/9.32 "from": 33, 29.13/9.32 "to": 35, 29.13/9.32 "label": "CASE" 29.13/9.32 }, 29.13/9.32 { 29.13/9.32 "from": 34, 29.13/9.32 "to": 365, 29.13/9.32 "label": "EVAL with clause\ninsert(X95, tree(X96, X97, X98), tree(X96, X97, X99)) :- ','(less(X96, X95), insert(X95, X98, X99)).\nand substitutionT9 -> T112,\nX95 -> T112,\nX96 -> T117,\nX97 -> T114,\nX98 -> T118,\nT2 -> tree(T117, T114, T118),\nX99 -> T119,\nT3 -> tree(T117, T114, T119),\nT113 -> T117,\nT115 -> T118,\nT116 -> T119" 29.13/9.32 }, 29.13/9.32 { 29.13/9.32 "from": 34, 29.13/9.32 "to": 366, 29.13/9.32 "label": "EVAL-BACKTRACK" 29.13/9.32 }, 29.13/9.32 { 29.13/9.32 "from": 35, 29.13/9.32 "to": 45, 29.13/9.32 "label": "PARALLEL" 29.13/9.32 }, 29.13/9.32 { 29.13/9.32 "from": 35, 29.13/9.33 "to": 46, 29.13/9.33 "label": "PARALLEL" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 45, 29.13/9.33 "to": 56, 29.13/9.33 "label": "EVAL with clause\nless(0, s(X24)).\nand substitutionT17 -> 0,\nX24 -> T29,\nT22 -> s(T29),\nT23 -> T30,\nT24 -> T31" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 45, 29.13/9.33 "to": 57, 29.13/9.33 "label": "EVAL-BACKTRACK" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 46, 29.13/9.33 "to": 150, 29.13/9.33 "label": "PARALLEL" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 46, 29.13/9.33 "to": 151, 29.13/9.33 "label": "PARALLEL" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 56, 29.13/9.33 "to": 3, 29.13/9.33 "label": "INSTANCE with matching:\nT1 -> 0\nT2 -> T30\nT3 -> T31" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 150, 29.13/9.33 "to": 152, 29.13/9.33 "label": "EVAL with clause\nless(s(X35), s(X36)) :- less(X35, X36).\nand substitutionX35 -> T40,\nT17 -> s(T40),\nX36 -> T42,\nT22 -> s(T42),\nT41 -> T42,\nT23 -> T43,\nT24 -> T44" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 150, 29.13/9.33 "to": 153, 29.13/9.33 "label": "EVAL-BACKTRACK" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 151, 29.13/9.33 "to": 352, 29.13/9.33 "label": "FAILURE" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 152, 29.13/9.33 "to": 203, 29.13/9.33 "label": "SPLIT 1" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 152, 29.13/9.33 "to": 205, 29.13/9.33 "label": "SPLIT 2\nnew knowledge:\nT40 is ground\nreplacements:T43 -> T47,\nT44 -> T48" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 203, 29.13/9.33 "to": 244, 29.13/9.33 "label": "CASE" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 205, 29.13/9.33 "to": 3, 29.13/9.33 "label": "INSTANCE with matching:\nT1 -> s(T40)\nT2 -> T47\nT3 -> T48" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 244, 29.13/9.33 "to": 343, 29.13/9.33 "label": "PARALLEL" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 244, 29.13/9.33 "to": 344, 29.13/9.33 "label": "PARALLEL" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 343, 29.13/9.33 "to": 345, 29.13/9.33 "label": "EVAL with clause\nless(0, s(X45)).\nand substitutionT40 -> 0,\nX45 -> T55,\nT42 -> s(T55)" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 343, 29.13/9.33 "to": 346, 29.13/9.33 "label": "EVAL-BACKTRACK" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 344, 29.13/9.33 "to": 350, 29.13/9.33 "label": "EVAL with clause\nless(s(X50), s(X51)) :- less(X50, X51).\nand substitutionX50 -> T60,\nT40 -> s(T60),\nX51 -> T62,\nT42 -> s(T62),\nT61 -> T62" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 344, 29.13/9.33 "to": 351, 29.13/9.33 "label": "EVAL-BACKTRACK" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 345, 29.13/9.33 "to": 347, 29.13/9.33 "label": "SUCCESS" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 350, 29.13/9.33 "to": 203, 29.13/9.33 "label": "INSTANCE with matching:\nT40 -> T60\nT42 -> T62" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 352, 29.13/9.33 "to": 353, 29.13/9.33 "label": "EVAL with clause\ninsert(X66, tree(X67, X68, X69), tree(X67, X68, X70)) :- ','(less(X67, X66), insert(X66, X69, X70)).\nand substitutionT17 -> T77,\nX66 -> T77,\nX67 -> T82,\nX68 -> T79,\nX69 -> T83,\nT2 -> tree(T82, T79, T83),\nX70 -> T84,\nT3 -> tree(T82, T79, T84),\nT78 -> T82,\nT80 -> T83,\nT81 -> T84" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 352, 29.13/9.33 "to": 354, 29.13/9.33 "label": "EVAL-BACKTRACK" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 353, 29.13/9.33 "to": 355, 29.13/9.33 "label": "SPLIT 1" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 353, 29.13/9.33 "to": 356, 29.13/9.33 "label": "SPLIT 2\nnew knowledge:\nT82 is ground\nT77 is ground\nreplacements:T83 -> T87,\nT84 -> T88" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 355, 29.13/9.33 "to": 357, 29.13/9.33 "label": "CASE" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 356, 29.13/9.33 "to": 3, 29.13/9.33 "label": "INSTANCE with matching:\nT1 -> T77\nT2 -> T87\nT3 -> T88" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 357, 29.13/9.33 "to": 358, 29.13/9.33 "label": "PARALLEL" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 357, 29.13/9.33 "to": 359, 29.13/9.33 "label": "PARALLEL" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 358, 29.13/9.33 "to": 360, 29.13/9.33 "label": "EVAL with clause\nless(0, s(X79)).\nand substitutionT82 -> 0,\nX79 -> T95,\nT77 -> s(T95)" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 358, 29.13/9.33 "to": 361, 29.13/9.33 "label": "EVAL-BACKTRACK" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 359, 29.13/9.33 "to": 363, 29.13/9.33 "label": "EVAL with clause\nless(s(X84), s(X85)) :- less(X84, X85).\nand substitutionX84 -> T102,\nT82 -> s(T102),\nX85 -> T101,\nT77 -> s(T101),\nT100 -> T102" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 359, 29.13/9.33 "to": 364, 29.13/9.33 "label": "EVAL-BACKTRACK" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 360, 29.13/9.33 "to": 362, 29.13/9.33 "label": "SUCCESS" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 363, 29.13/9.33 "to": 355, 29.13/9.33 "label": "INSTANCE with matching:\nT82 -> T102\nT77 -> T101" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 365, 29.13/9.33 "to": 367, 29.13/9.33 "label": "CASE" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 367, 29.13/9.33 "to": 368, 29.13/9.33 "label": "PARALLEL" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 367, 29.13/9.33 "to": 369, 29.13/9.33 "label": "PARALLEL" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 368, 29.13/9.33 "to": 370, 29.13/9.33 "label": "EVAL with clause\nless(0, s(X104)).\nand substitutionT117 -> 0,\nX104 -> T124,\nT112 -> s(T124),\nT118 -> T125,\nT119 -> T126" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 368, 29.13/9.33 "to": 371, 29.13/9.33 "label": "EVAL-BACKTRACK" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 369, 29.13/9.33 "to": 372, 29.13/9.33 "label": "EVAL with clause\nless(s(X111), s(X112)) :- less(X111, X112).\nand substitutionX111 -> T135,\nT117 -> s(T135),\nX112 -> T134,\nT112 -> s(T134),\nT133 -> T135,\nT118 -> T136,\nT119 -> T137" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 369, 29.13/9.33 "to": 373, 29.13/9.33 "label": "EVAL-BACKTRACK" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 370, 29.13/9.33 "to": 3, 29.13/9.33 "label": "INSTANCE with matching:\nT1 -> s(T124)\nT2 -> T125\nT3 -> T126" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 372, 29.13/9.33 "to": 374, 29.13/9.33 "label": "SPLIT 1" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 372, 29.13/9.33 "to": 375, 29.13/9.33 "label": "SPLIT 2\nnew knowledge:\nT135 is ground\nT134 is ground\nreplacements:T136 -> T140,\nT137 -> T141,\nT2 -> T142,\nT3 -> T143" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 374, 29.13/9.33 "to": 355, 29.13/9.33 "label": "INSTANCE with matching:\nT82 -> T135\nT77 -> T134" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 375, 29.13/9.33 "to": 3, 29.13/9.33 "label": "INSTANCE with matching:\nT1 -> s(T134)\nT2 -> T140\nT3 -> T141" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 400, 29.13/9.33 "to": 402, 29.13/9.33 "label": "CASE" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 401, 29.13/9.33 "to": 414, 29.13/9.33 "label": "EVAL with clause\ninsert(X170, tree(X171, X172, X173), tree(X171, X172, X174)) :- ','(less(X171, X170), insert(X170, X173, X174)).\nand substitutionT5 -> T208,\nX170 -> T208,\nX171 -> T213,\nX172 -> T210,\nX173 -> T214,\nT2 -> tree(T213, T210, T214),\nX174 -> T215,\nT3 -> tree(T213, T210, T215),\nT209 -> T213,\nT211 -> T214,\nT212 -> T215" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 401, 29.13/9.33 "to": 415, 29.13/9.33 "label": "EVAL-BACKTRACK" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 402, 29.13/9.33 "to": 403, 29.13/9.33 "label": "PARALLEL" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 402, 29.13/9.33 "to": 404, 29.13/9.33 "label": "PARALLEL" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 403, 29.13/9.33 "to": 405, 29.13/9.33 "label": "EVAL with clause\nless(0, s(X133)).\nand substitutionT153 -> 0,\nX133 -> T165,\nT158 -> s(T165),\nT159 -> T166,\nT160 -> T167" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 403, 29.13/9.33 "to": 406, 29.13/9.33 "label": "EVAL-BACKTRACK" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 404, 29.13/9.33 "to": 407, 29.13/9.33 "label": "PARALLEL" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 404, 29.13/9.33 "to": 408, 29.13/9.33 "label": "PARALLEL" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 405, 29.13/9.33 "to": 3, 29.13/9.33 "label": "INSTANCE with matching:\nT1 -> 0\nT2 -> T166\nT3 -> T167" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 407, 29.13/9.33 "to": 409, 29.13/9.33 "label": "EVAL with clause\nless(s(X144), s(X145)) :- less(X144, X145).\nand substitutionX144 -> T176,\nT153 -> s(T176),\nX145 -> T178,\nT158 -> s(T178),\nT177 -> T178,\nT159 -> T179,\nT160 -> T180" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 407, 29.13/9.33 "to": 410, 29.13/9.33 "label": "EVAL-BACKTRACK" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 408, 29.13/9.33 "to": 411, 29.13/9.33 "label": "FAILURE" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 409, 29.13/9.33 "to": 152, 29.13/9.33 "label": "INSTANCE with matching:\nT40 -> T176\nT42 -> T178\nT43 -> T179\nT44 -> T180" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 411, 29.13/9.33 "to": 412, 29.13/9.33 "label": "EVAL with clause\ninsert(X158, tree(X159, X160, X161), tree(X159, X160, X162)) :- ','(less(X159, X158), insert(X158, X161, X162)).\nand substitutionT153 -> T193,\nX158 -> T193,\nX159 -> T198,\nX160 -> T195,\nX161 -> T199,\nT2 -> tree(T198, T195, T199),\nX162 -> T200,\nT3 -> tree(T198, T195, T200),\nT194 -> T198,\nT196 -> T199,\nT197 -> T200" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 411, 29.13/9.33 "to": 413, 29.13/9.33 "label": "EVAL-BACKTRACK" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 412, 29.13/9.33 "to": 353, 29.13/9.33 "label": "INSTANCE with matching:\nT82 -> T198\nT77 -> T193\nT83 -> T199\nT84 -> T200" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 414, 29.13/9.33 "to": 416, 29.13/9.33 "label": "CASE" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 416, 29.13/9.33 "to": 417, 29.13/9.33 "label": "PARALLEL" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 416, 29.13/9.33 "to": 418, 29.13/9.33 "label": "PARALLEL" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 417, 29.13/9.33 "to": 419, 29.13/9.33 "label": "EVAL with clause\nless(0, s(X179)).\nand substitutionT213 -> 0,\nX179 -> T220,\nT208 -> s(T220),\nT214 -> T221,\nT215 -> T222" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 417, 29.13/9.33 "to": 420, 29.13/9.33 "label": "EVAL-BACKTRACK" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 418, 29.13/9.33 "to": 421, 29.13/9.33 "label": "EVAL with clause\nless(s(X186), s(X187)) :- less(X186, X187).\nand substitutionX186 -> T231,\nT213 -> s(T231),\nX187 -> T230,\nT208 -> s(T230),\nT229 -> T231,\nT214 -> T232,\nT215 -> T233" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 418, 29.13/9.33 "to": 422, 29.13/9.33 "label": "EVAL-BACKTRACK" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 419, 29.13/9.33 "to": 3, 29.13/9.33 "label": "INSTANCE with matching:\nT1 -> s(T220)\nT2 -> T221\nT3 -> T222" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 421, 29.13/9.33 "to": 423, 29.13/9.33 "label": "SPLIT 1" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 421, 29.13/9.33 "to": 424, 29.13/9.33 "label": "SPLIT 2\nnew knowledge:\nT231 is ground\nT230 is ground\nreplacements:T232 -> T236,\nT233 -> T237,\nT2 -> T238,\nT3 -> T239" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 423, 29.13/9.33 "to": 355, 29.13/9.33 "label": "INSTANCE with matching:\nT82 -> T231\nT77 -> T230" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 424, 29.13/9.33 "to": 3, 29.13/9.33 "label": "INSTANCE with matching:\nT1 -> s(T230)\nT2 -> T236\nT3 -> T237" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 425, 29.13/9.33 "to": 427, 29.13/9.33 "label": "SUCCESS" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 426, 29.13/9.33 "to": 477, 29.13/9.33 "label": "EVAL with clause\ninsert(X276, tree(X277, X278, X279), tree(X277, X280, X279)) :- ','(less(X276, X277), insert(X276, X278, X280)).\nand substitutionT1 -> T343,\nX276 -> T343,\nX277 -> T348,\nX278 -> T349,\nX279 -> T346,\nT2 -> tree(T348, T349, T346),\nX280 -> T350,\nT3 -> tree(T348, T350, T346),\nT344 -> T348,\nT345 -> T349,\nT347 -> T350" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 426, 29.13/9.33 "to": 478, 29.13/9.33 "label": "EVAL-BACKTRACK" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 427, 29.13/9.33 "to": 454, 29.13/9.33 "label": "EVAL with clause\ninsert(X205, tree(X206, X207, X208), tree(X206, X209, X208)) :- ','(less(X205, X206), insert(X205, X207, X209)).\nand substitutionT247 -> T255,\nX205 -> T255,\nX206 -> T260,\nX207 -> T261,\nX208 -> T258,\nT2 -> tree(T260, T261, T258),\nX209 -> T262,\nT3 -> tree(T260, T262, T258),\nT256 -> T260,\nT257 -> T261,\nT259 -> T262" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 427, 29.13/9.33 "to": 455, 29.13/9.33 "label": "EVAL-BACKTRACK" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 454, 29.13/9.33 "to": 456, 29.13/9.33 "label": "CASE" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 455, 29.13/9.33 "to": 468, 29.13/9.33 "label": "EVAL with clause\ninsert(X251, tree(X252, X253, X254), tree(X252, X253, X255)) :- ','(less(X252, X251), insert(X251, X254, X255)).\nand substitutionT247 -> T310,\nX251 -> T310,\nX252 -> T315,\nX253 -> T312,\nX254 -> T316,\nT2 -> tree(T315, T312, T316),\nX255 -> T317,\nT3 -> tree(T315, T312, T317),\nT311 -> T315,\nT313 -> T316,\nT314 -> T317" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 455, 29.13/9.33 "to": 469, 29.13/9.33 "label": "EVAL-BACKTRACK" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 456, 29.13/9.33 "to": 457, 29.13/9.33 "label": "PARALLEL" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 456, 29.13/9.33 "to": 458, 29.13/9.33 "label": "PARALLEL" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 457, 29.13/9.33 "to": 459, 29.13/9.33 "label": "EVAL with clause\nless(0, s(X214)).\nand substitutionT255 -> 0,\nX214 -> T267,\nT260 -> s(T267),\nT261 -> T268,\nT262 -> T269" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 457, 29.13/9.33 "to": 460, 29.13/9.33 "label": "EVAL-BACKTRACK" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 458, 29.13/9.33 "to": 461, 29.13/9.33 "label": "PARALLEL" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 458, 29.13/9.33 "to": 462, 29.13/9.33 "label": "PARALLEL" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 459, 29.13/9.33 "to": 3, 29.13/9.33 "label": "INSTANCE with matching:\nT1 -> 0\nT2 -> T268\nT3 -> T269" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 461, 29.13/9.33 "to": 463, 29.13/9.33 "label": "EVAL with clause\nless(s(X225), s(X226)) :- less(X225, X226).\nand substitutionX225 -> T278,\nT255 -> s(T278),\nX226 -> T280,\nT260 -> s(T280),\nT279 -> T280,\nT261 -> T281,\nT262 -> T282" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 461, 29.13/9.33 "to": 464, 29.13/9.33 "label": "EVAL-BACKTRACK" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 462, 29.13/9.33 "to": 465, 29.13/9.33 "label": "FAILURE" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 463, 29.13/9.33 "to": 152, 29.13/9.33 "label": "INSTANCE with matching:\nT40 -> T278\nT42 -> T280\nT43 -> T281\nT44 -> T282" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 465, 29.13/9.33 "to": 466, 29.13/9.33 "label": "EVAL with clause\ninsert(X239, tree(X240, X241, X242), tree(X240, X241, X243)) :- ','(less(X240, X239), insert(X239, X242, X243)).\nand substitutionT255 -> T295,\nX239 -> T295,\nX240 -> T300,\nX241 -> T297,\nX242 -> T301,\nT2 -> tree(T300, T297, T301),\nX243 -> T302,\nT3 -> tree(T300, T297, T302),\nT296 -> T300,\nT298 -> T301,\nT299 -> T302" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 465, 29.13/9.33 "to": 467, 29.13/9.33 "label": "EVAL-BACKTRACK" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 466, 29.13/9.33 "to": 353, 29.13/9.33 "label": "INSTANCE with matching:\nT82 -> T300\nT77 -> T295\nT83 -> T301\nT84 -> T302" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 468, 29.13/9.33 "to": 470, 29.13/9.33 "label": "CASE" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 470, 29.13/9.33 "to": 471, 29.13/9.33 "label": "PARALLEL" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 470, 29.13/9.33 "to": 472, 29.13/9.33 "label": "PARALLEL" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 471, 29.13/9.33 "to": 473, 29.13/9.33 "label": "EVAL with clause\nless(0, s(X260)).\nand substitutionT315 -> 0,\nX260 -> T322,\nT310 -> s(T322),\nT316 -> T323,\nT317 -> T324" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 471, 29.13/9.33 "to": 474, 29.13/9.33 "label": "EVAL-BACKTRACK" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 472, 29.13/9.33 "to": 475, 29.13/9.33 "label": "EVAL with clause\nless(s(X267), s(X268)) :- less(X267, X268).\nand substitutionX267 -> T333,\nT315 -> s(T333),\nX268 -> T332,\nT310 -> s(T332),\nT331 -> T333,\nT316 -> T334,\nT317 -> T335" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 472, 29.13/9.33 "to": 476, 29.13/9.33 "label": "EVAL-BACKTRACK" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 473, 29.13/9.33 "to": 3, 29.13/9.33 "label": "INSTANCE with matching:\nT1 -> s(T322)\nT2 -> T323\nT3 -> T324" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 475, 29.13/9.33 "to": 372, 29.13/9.33 "label": "INSTANCE with matching:\nT135 -> T333\nT134 -> T332\nT136 -> T334\nT137 -> T335\nX15 -> X205\nX16 -> X206\nX17 -> X207\nX18 -> X208\nX19 -> X209" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 477, 29.13/9.33 "to": 479, 29.13/9.33 "label": "CASE" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 478, 29.13/9.33 "to": 491, 29.13/9.33 "label": "EVAL with clause\ninsert(X322, tree(X323, X324, X325), tree(X323, X324, X326)) :- ','(less(X323, X322), insert(X322, X325, X326)).\nand substitutionT1 -> T398,\nX322 -> T398,\nX323 -> T403,\nX324 -> T400,\nX325 -> T404,\nT2 -> tree(T403, T400, T404),\nX326 -> T405,\nT3 -> tree(T403, T400, T405),\nT399 -> T403,\nT401 -> T404,\nT402 -> T405" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 478, 29.13/9.33 "to": 492, 29.13/9.33 "label": "EVAL-BACKTRACK" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 479, 29.13/9.33 "to": 480, 29.13/9.33 "label": "PARALLEL" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 479, 29.13/9.33 "to": 481, 29.13/9.33 "label": "PARALLEL" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 480, 29.13/9.33 "to": 482, 29.13/9.33 "label": "EVAL with clause\nless(0, s(X285)).\nand substitutionT343 -> 0,\nX285 -> T355,\nT348 -> s(T355),\nT349 -> T356,\nT350 -> T357" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 480, 29.13/9.33 "to": 483, 29.13/9.33 "label": "EVAL-BACKTRACK" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 481, 29.13/9.33 "to": 484, 29.13/9.33 "label": "PARALLEL" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 481, 29.13/9.33 "to": 485, 29.13/9.33 "label": "PARALLEL" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 482, 29.13/9.33 "to": 3, 29.13/9.33 "label": "INSTANCE with matching:\nT1 -> 0\nT2 -> T356\nT3 -> T357" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 484, 29.13/9.33 "to": 486, 29.13/9.33 "label": "EVAL with clause\nless(s(X296), s(X297)) :- less(X296, X297).\nand substitutionX296 -> T366,\nT343 -> s(T366),\nX297 -> T368,\nT348 -> s(T368),\nT367 -> T368,\nT349 -> T369,\nT350 -> T370" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 484, 29.13/9.33 "to": 487, 29.13/9.33 "label": "EVAL-BACKTRACK" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 485, 29.13/9.33 "to": 488, 29.13/9.33 "label": "FAILURE" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 486, 29.13/9.33 "to": 152, 29.13/9.33 "label": "INSTANCE with matching:\nT40 -> T366\nT42 -> T368\nT43 -> T369\nT44 -> T370" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 488, 29.13/9.33 "to": 489, 29.13/9.33 "label": "EVAL with clause\ninsert(X310, tree(X311, X312, X313), tree(X311, X312, X314)) :- ','(less(X311, X310), insert(X310, X313, X314)).\nand substitutionT343 -> T383,\nX310 -> T383,\nX311 -> T388,\nX312 -> T385,\nX313 -> T389,\nT2 -> tree(T388, T385, T389),\nX314 -> T390,\nT3 -> tree(T388, T385, T390),\nT384 -> T388,\nT386 -> T389,\nT387 -> T390" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 488, 29.13/9.33 "to": 490, 29.13/9.33 "label": "EVAL-BACKTRACK" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 489, 29.13/9.33 "to": 414, 29.13/9.33 "label": "INSTANCE with matching:\nT213 -> T388\nT208 -> T383\nT214 -> T389\nT215 -> T390\nX7 -> X197\nX8 -> X198\nX9 -> X199" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 491, 29.13/9.33 "to": 493, 29.13/9.33 "label": "CASE" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 493, 29.13/9.33 "to": 494, 29.13/9.33 "label": "PARALLEL" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 493, 29.13/9.33 "to": 495, 29.13/9.33 "label": "PARALLEL" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 494, 29.13/9.33 "to": 496, 29.13/9.33 "label": "EVAL with clause\nless(0, s(X331)).\nand substitutionT403 -> 0,\nX331 -> T410,\nT398 -> s(T410),\nT404 -> T411,\nT405 -> T412" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 494, 29.13/9.33 "to": 497, 29.13/9.33 "label": "EVAL-BACKTRACK" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 495, 29.13/9.33 "to": 498, 29.13/9.33 "label": "EVAL with clause\nless(s(X338), s(X339)) :- less(X338, X339).\nand substitutionX338 -> T421,\nT403 -> s(T421),\nX339 -> T420,\nT398 -> s(T420),\nT419 -> T421,\nT404 -> T422,\nT405 -> T423" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 495, 29.13/9.33 "to": 499, 29.13/9.33 "label": "EVAL-BACKTRACK" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 496, 29.13/9.33 "to": 3, 29.13/9.33 "label": "INSTANCE with matching:\nT1 -> s(T410)\nT2 -> T411\nT3 -> T412" 29.13/9.33 }, 29.13/9.33 { 29.13/9.33 "from": 498, 29.13/9.33 "to": 421, 29.13/9.33 "label": "INSTANCE with matching:\nT231 -> T421\nT230 -> T420\nT232 -> T422\nT233 -> T423\nX7 -> X197\nX8 -> X198\nX9 -> X199" 29.13/9.33 } 29.13/9.33 ], 29.13/9.33 "type": "Graph" 29.13/9.33 } 29.13/9.33 } 29.13/9.33 29.13/9.33 ---------------------------------------- 29.13/9.33 29.13/9.33 (134) 29.13/9.33 Obligation: 29.13/9.33 Triples: 29.13/9.33 29.13/9.33 lessG(s(X1), s(X2)) :- lessG(X1, X2). 29.13/9.33 lessH(s(X1), s(X2)) :- lessH(X1, X2). 29.13/9.33 pB(X1, X2, X3, X4) :- lessG(X1, X2). 29.13/9.33 pB(X1, X2, X3, X4) :- ','(lesscG(X1, X2), insertA(s(X1), X3, X4)). 29.13/9.33 pC(X1, X2, X3, X4) :- lessH(X1, X2). 29.13/9.33 pC(X1, X2, X3, X4) :- ','(lesscH(X1, X2), insertA(X2, X3, X4)). 29.13/9.33 pD(X1, X2, X3, X4) :- lessH(X1, X2). 29.13/9.33 pD(X1, X2, X3, X4) :- ','(lesscH(X1, X2), insertA(s(X2), X3, X4)). 29.13/9.33 pE(0, s(X1), X2, X3) :- insertA(s(X1), X2, X3). 29.13/9.33 pE(s(X1), s(X2), X3, X4) :- pF(X1, X2, X3, X4). 29.13/9.33 pF(X1, X2, X3, X4) :- lessH(X1, X2). 29.13/9.33 pF(X1, X2, X3, X4) :- ','(lesscH(X1, X2), insertA(s(X2), X3, X4)). 29.13/9.33 insertA(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) :- insertA(0, X2, X4). 29.13/9.33 insertA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) :- pB(X1, X2, X3, X5). 29.13/9.33 insertA(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- pC(X2, X1, X4, X5). 29.13/9.33 insertA(s(X1), tree(0, X2, X3), tree(0, X2, X4)) :- insertA(s(X1), X3, X4). 29.13/9.33 insertA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) :- pD(X2, X1, X4, X5). 29.13/9.33 insertA(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) :- insertA(0, X2, X4). 29.13/9.33 insertA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) :- pB(X1, X2, X3, X5). 29.13/9.33 insertA(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- pC(X2, X1, X4, X5). 29.13/9.33 insertA(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- pE(X2, X1, X4, X5). 29.13/9.33 insertA(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) :- insertA(0, X2, X4). 29.13/9.33 insertA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) :- pB(X1, X2, X3, X5). 29.13/9.33 insertA(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- pC(X2, X1, X4, X5). 29.13/9.33 insertA(s(X1), tree(0, X2, X3), tree(0, X2, X4)) :- insertA(s(X1), X3, X4). 29.13/9.33 insertA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) :- pD(X2, X1, X4, X5). 29.13/9.33 insertA(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) :- insertA(0, X2, X4). 29.13/9.33 insertA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) :- pB(X1, X2, X3, X5). 29.13/9.33 insertA(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- pE(X2, X1, X4, X5). 29.13/9.33 insertA(s(X1), tree(0, X2, X3), tree(0, X2, X4)) :- insertA(s(X1), X3, X4). 29.13/9.33 insertA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) :- pF(X2, X1, X4, X5). 29.13/9.33 29.13/9.33 Clauses: 29.13/9.33 29.13/9.33 insertcA(X1, void, tree(X1, void, void)). 29.13/9.33 insertcA(X1, tree(X1, X2, X3), tree(X1, X2, X3)). 29.13/9.33 insertcA(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) :- insertcA(0, X2, X4). 29.13/9.33 insertcA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) :- qcB(X1, X2, X3, X5). 29.13/9.33 insertcA(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- qcC(X2, X1, X4, X5). 29.13/9.33 insertcA(s(X1), tree(0, X2, X3), tree(0, X2, X4)) :- insertcA(s(X1), X3, X4). 29.13/9.33 insertcA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) :- qcD(X2, X1, X4, X5). 29.13/9.33 insertcA(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) :- insertcA(0, X2, X4). 29.13/9.33 insertcA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) :- qcB(X1, X2, X3, X5). 29.13/9.33 insertcA(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- qcC(X2, X1, X4, X5). 29.13/9.33 insertcA(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- qcE(X2, X1, X4, X5). 29.13/9.33 insertcA(X1, tree(X1, X2, X3), tree(X1, X2, X3)). 29.13/9.33 insertcA(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) :- insertcA(0, X2, X4). 29.13/9.33 insertcA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) :- qcB(X1, X2, X3, X5). 29.13/9.33 insertcA(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- qcC(X2, X1, X4, X5). 29.13/9.33 insertcA(s(X1), tree(0, X2, X3), tree(0, X2, X4)) :- insertcA(s(X1), X3, X4). 29.13/9.33 insertcA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) :- qcD(X2, X1, X4, X5). 29.13/9.33 insertcA(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) :- insertcA(0, X2, X4). 29.13/9.33 insertcA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) :- qcB(X1, X2, X3, X5). 29.13/9.33 insertcA(X1, tree(X2, X3, X4), tree(X2, X3, X5)) :- qcE(X2, X1, X4, X5). 29.13/9.33 insertcA(s(X1), tree(0, X2, X3), tree(0, X2, X4)) :- insertcA(s(X1), X3, X4). 29.13/9.33 insertcA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) :- qcF(X2, X1, X4, X5). 29.13/9.33 lesscG(0, s(X1)). 29.13/9.33 lesscG(s(X1), s(X2)) :- lesscG(X1, X2). 29.13/9.33 lesscH(0, s(X1)). 29.13/9.33 lesscH(s(X1), s(X2)) :- lesscH(X1, X2). 29.13/9.33 qcB(X1, X2, X3, X4) :- ','(lesscG(X1, X2), insertcA(s(X1), X3, X4)). 29.13/9.33 qcC(X1, X2, X3, X4) :- ','(lesscH(X1, X2), insertcA(X2, X3, X4)). 29.13/9.33 qcD(X1, X2, X3, X4) :- ','(lesscH(X1, X2), insertcA(s(X2), X3, X4)). 29.13/9.33 qcE(0, s(X1), X2, X3) :- insertcA(s(X1), X2, X3). 29.13/9.33 qcE(s(X1), s(X2), X3, X4) :- qcF(X1, X2, X3, X4). 29.13/9.33 qcF(X1, X2, X3, X4) :- ','(lesscH(X1, X2), insertcA(s(X2), X3, X4)). 29.13/9.33 29.13/9.33 Afs: 29.13/9.33 29.13/9.33 insertA(x1, x2, x3) = insertA(x1) 29.13/9.33 29.13/9.33 29.13/9.33 ---------------------------------------- 29.13/9.33 29.13/9.33 (135) TriplesToPiDPProof (SOUND) 29.13/9.33 We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: 29.13/9.33 29.13/9.33 insertA_in_3: (b,f,f) 29.13/9.33 29.13/9.33 pB_in_4: (b,f,f,f) 29.13/9.33 29.13/9.33 lessG_in_2: (b,f) 29.13/9.33 29.13/9.33 lesscG_in_2: (b,f) 29.13/9.33 29.13/9.33 pC_in_4: (f,b,f,f) 29.13/9.33 29.13/9.33 lessH_in_2: (f,b) 29.13/9.33 29.13/9.33 lesscH_in_2: (f,b) 29.13/9.33 29.13/9.33 pD_in_4: (f,b,f,f) 29.13/9.33 29.13/9.33 pE_in_4: (f,b,f,f) 29.13/9.33 29.13/9.33 pF_in_4: (f,b,f,f) 29.13/9.33 29.13/9.33 Transforming TRIPLES into the following Term Rewriting System: 29.13/9.33 29.13/9.33 Pi DP problem: 29.13/9.33 The TRS P consists of the following rules: 29.13/9.33 29.13/9.33 INSERTA_IN_GAA(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) -> U17_GAA(X1, X2, X3, X4, insertA_in_gaa(0, X2, X4)) 29.13/9.33 INSERTA_IN_GAA(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) -> INSERTA_IN_GAA(0, X2, X4) 29.13/9.33 INSERTA_IN_GAA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) -> U18_GAA(X1, X2, X3, X4, X5, pB_in_gaaa(X1, X2, X3, X5)) 29.13/9.33 INSERTA_IN_GAA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) -> PB_IN_GAAA(X1, X2, X3, X5) 29.13/9.33 PB_IN_GAAA(X1, X2, X3, X4) -> U3_GAAA(X1, X2, X3, X4, lessG_in_ga(X1, X2)) 29.13/9.33 PB_IN_GAAA(X1, X2, X3, X4) -> LESSG_IN_GA(X1, X2) 29.13/9.33 LESSG_IN_GA(s(X1), s(X2)) -> U1_GA(X1, X2, lessG_in_ga(X1, X2)) 29.13/9.33 LESSG_IN_GA(s(X1), s(X2)) -> LESSG_IN_GA(X1, X2) 29.13/9.33 PB_IN_GAAA(X1, X2, X3, X4) -> U4_GAAA(X1, X2, X3, X4, lesscG_in_ga(X1, X2)) 29.13/9.33 U4_GAAA(X1, X2, X3, X4, lesscG_out_ga(X1, X2)) -> U5_GAAA(X1, X2, X3, X4, insertA_in_gaa(s(X1), X3, X4)) 29.13/9.33 U4_GAAA(X1, X2, X3, X4, lesscG_out_ga(X1, X2)) -> INSERTA_IN_GAA(s(X1), X3, X4) 29.13/9.33 INSERTA_IN_GAA(X1, tree(X2, X3, X4), tree(X2, X3, X5)) -> U19_GAA(X1, X2, X3, X4, X5, pC_in_agaa(X2, X1, X4, X5)) 29.13/9.33 INSERTA_IN_GAA(X1, tree(X2, X3, X4), tree(X2, X3, X5)) -> PC_IN_AGAA(X2, X1, X4, X5) 29.13/9.33 PC_IN_AGAA(X1, X2, X3, X4) -> U6_AGAA(X1, X2, X3, X4, lessH_in_ag(X1, X2)) 29.13/9.33 PC_IN_AGAA(X1, X2, X3, X4) -> LESSH_IN_AG(X1, X2) 29.13/9.33 LESSH_IN_AG(s(X1), s(X2)) -> U2_AG(X1, X2, lessH_in_ag(X1, X2)) 29.13/9.33 LESSH_IN_AG(s(X1), s(X2)) -> LESSH_IN_AG(X1, X2) 29.13/9.33 PC_IN_AGAA(X1, X2, X3, X4) -> U7_AGAA(X1, X2, X3, X4, lesscH_in_ag(X1, X2)) 29.13/9.33 U7_AGAA(X1, X2, X3, X4, lesscH_out_ag(X1, X2)) -> U8_AGAA(X1, X2, X3, X4, insertA_in_gaa(X2, X3, X4)) 29.13/9.33 U7_AGAA(X1, X2, X3, X4, lesscH_out_ag(X1, X2)) -> INSERTA_IN_GAA(X2, X3, X4) 29.13/9.33 INSERTA_IN_GAA(s(X1), tree(0, X2, X3), tree(0, X2, X4)) -> U20_GAA(X1, X2, X3, X4, insertA_in_gaa(s(X1), X3, X4)) 29.13/9.33 INSERTA_IN_GAA(s(X1), tree(0, X2, X3), tree(0, X2, X4)) -> INSERTA_IN_GAA(s(X1), X3, X4) 29.13/9.33 INSERTA_IN_GAA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) -> U21_GAA(X1, X2, X3, X4, X5, pD_in_agaa(X2, X1, X4, X5)) 29.13/9.33 INSERTA_IN_GAA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) -> PD_IN_AGAA(X2, X1, X4, X5) 29.13/9.33 PD_IN_AGAA(X1, X2, X3, X4) -> U9_AGAA(X1, X2, X3, X4, lessH_in_ag(X1, X2)) 29.13/9.33 PD_IN_AGAA(X1, X2, X3, X4) -> LESSH_IN_AG(X1, X2) 29.13/9.33 PD_IN_AGAA(X1, X2, X3, X4) -> U10_AGAA(X1, X2, X3, X4, lesscH_in_ag(X1, X2)) 29.13/9.33 U10_AGAA(X1, X2, X3, X4, lesscH_out_ag(X1, X2)) -> U11_AGAA(X1, X2, X3, X4, insertA_in_gaa(s(X2), X3, X4)) 29.13/9.33 U10_AGAA(X1, X2, X3, X4, lesscH_out_ag(X1, X2)) -> INSERTA_IN_GAA(s(X2), X3, X4) 29.13/9.33 INSERTA_IN_GAA(X1, tree(X2, X3, X4), tree(X2, X3, X5)) -> U22_GAA(X1, X2, X3, X4, X5, pE_in_agaa(X2, X1, X4, X5)) 29.13/9.33 INSERTA_IN_GAA(X1, tree(X2, X3, X4), tree(X2, X3, X5)) -> PE_IN_AGAA(X2, X1, X4, X5) 29.13/9.33 PE_IN_AGAA(0, s(X1), X2, X3) -> U12_AGAA(X1, X2, X3, insertA_in_gaa(s(X1), X2, X3)) 29.13/9.33 PE_IN_AGAA(0, s(X1), X2, X3) -> INSERTA_IN_GAA(s(X1), X2, X3) 29.13/9.33 INSERTA_IN_GAA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) -> U23_GAA(X1, X2, X3, X4, X5, pF_in_agaa(X2, X1, X4, X5)) 29.13/9.33 INSERTA_IN_GAA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) -> PF_IN_AGAA(X2, X1, X4, X5) 29.13/9.33 PF_IN_AGAA(X1, X2, X3, X4) -> U14_AGAA(X1, X2, X3, X4, lessH_in_ag(X1, X2)) 29.13/9.33 PF_IN_AGAA(X1, X2, X3, X4) -> LESSH_IN_AG(X1, X2) 29.13/9.33 PF_IN_AGAA(X1, X2, X3, X4) -> U15_AGAA(X1, X2, X3, X4, lesscH_in_ag(X1, X2)) 29.13/9.33 U15_AGAA(X1, X2, X3, X4, lesscH_out_ag(X1, X2)) -> U16_AGAA(X1, X2, X3, X4, insertA_in_gaa(s(X2), X3, X4)) 29.13/9.33 U15_AGAA(X1, X2, X3, X4, lesscH_out_ag(X1, X2)) -> INSERTA_IN_GAA(s(X2), X3, X4) 29.13/9.33 PE_IN_AGAA(s(X1), s(X2), X3, X4) -> U13_AGAA(X1, X2, X3, X4, pF_in_agaa(X1, X2, X3, X4)) 29.13/9.33 PE_IN_AGAA(s(X1), s(X2), X3, X4) -> PF_IN_AGAA(X1, X2, X3, X4) 29.13/9.33 29.13/9.33 The TRS R consists of the following rules: 29.13/9.33 29.13/9.33 lesscG_in_ga(0, s(X1)) -> lesscG_out_ga(0, s(X1)) 29.13/9.33 lesscG_in_ga(s(X1), s(X2)) -> U32_ga(X1, X2, lesscG_in_ga(X1, X2)) 29.13/9.33 U32_ga(X1, X2, lesscG_out_ga(X1, X2)) -> lesscG_out_ga(s(X1), s(X2)) 29.13/9.33 lesscH_in_ag(0, s(X1)) -> lesscH_out_ag(0, s(X1)) 29.13/9.33 lesscH_in_ag(s(X1), s(X2)) -> U33_ag(X1, X2, lesscH_in_ag(X1, X2)) 29.13/9.33 U33_ag(X1, X2, lesscH_out_ag(X1, X2)) -> lesscH_out_ag(s(X1), s(X2)) 29.13/9.33 29.13/9.33 The argument filtering Pi contains the following mapping: 29.13/9.33 insertA_in_gaa(x1, x2, x3) = insertA_in_gaa(x1) 29.13/9.33 29.13/9.33 0 = 0 29.13/9.33 29.13/9.33 s(x1) = s(x1) 29.13/9.33 29.13/9.33 pB_in_gaaa(x1, x2, x3, x4) = pB_in_gaaa(x1) 29.13/9.33 29.13/9.33 lessG_in_ga(x1, x2) = lessG_in_ga(x1) 29.13/9.33 29.13/9.33 lesscG_in_ga(x1, x2) = lesscG_in_ga(x1) 29.13/9.33 29.13/9.33 lesscG_out_ga(x1, x2) = lesscG_out_ga(x1) 29.13/9.33 29.13/9.33 U32_ga(x1, x2, x3) = U32_ga(x1, x3) 29.13/9.33 29.13/9.33 pC_in_agaa(x1, x2, x3, x4) = pC_in_agaa(x2) 29.13/9.33 29.13/9.33 lessH_in_ag(x1, x2) = lessH_in_ag(x2) 29.13/9.33 29.13/9.33 lesscH_in_ag(x1, x2) = lesscH_in_ag(x2) 29.13/9.33 29.13/9.33 lesscH_out_ag(x1, x2) = lesscH_out_ag(x1, x2) 29.13/9.33 29.13/9.33 U33_ag(x1, x2, x3) = U33_ag(x2, x3) 29.13/9.33 29.13/9.33 pD_in_agaa(x1, x2, x3, x4) = pD_in_agaa(x2) 29.13/9.33 29.13/9.33 pE_in_agaa(x1, x2, x3, x4) = pE_in_agaa(x2) 29.13/9.33 29.13/9.33 pF_in_agaa(x1, x2, x3, x4) = pF_in_agaa(x2) 29.13/9.33 29.13/9.33 INSERTA_IN_GAA(x1, x2, x3) = INSERTA_IN_GAA(x1) 29.13/9.33 29.13/9.33 U17_GAA(x1, x2, x3, x4, x5) = U17_GAA(x5) 29.13/9.33 29.13/9.33 U18_GAA(x1, x2, x3, x4, x5, x6) = U18_GAA(x1, x6) 29.13/9.33 29.13/9.33 PB_IN_GAAA(x1, x2, x3, x4) = PB_IN_GAAA(x1) 29.13/9.33 29.13/9.33 U3_GAAA(x1, x2, x3, x4, x5) = U3_GAAA(x1, x5) 29.13/9.33 29.13/9.33 LESSG_IN_GA(x1, x2) = LESSG_IN_GA(x1) 29.13/9.33 29.13/9.33 U1_GA(x1, x2, x3) = U1_GA(x1, x3) 29.13/9.33 29.13/9.33 U4_GAAA(x1, x2, x3, x4, x5) = U4_GAAA(x1, x5) 29.13/9.33 29.13/9.33 U5_GAAA(x1, x2, x3, x4, x5) = U5_GAAA(x1, x5) 29.13/9.33 29.13/9.33 U19_GAA(x1, x2, x3, x4, x5, x6) = U19_GAA(x1, x6) 29.13/9.33 29.13/9.33 PC_IN_AGAA(x1, x2, x3, x4) = PC_IN_AGAA(x2) 29.13/9.33 29.13/9.33 U6_AGAA(x1, x2, x3, x4, x5) = U6_AGAA(x2, x5) 29.13/9.33 29.13/9.33 LESSH_IN_AG(x1, x2) = LESSH_IN_AG(x2) 29.13/9.33 29.13/9.33 U2_AG(x1, x2, x3) = U2_AG(x2, x3) 29.13/9.33 29.13/9.33 U7_AGAA(x1, x2, x3, x4, x5) = U7_AGAA(x2, x5) 29.13/9.33 29.13/9.33 U8_AGAA(x1, x2, x3, x4, x5) = U8_AGAA(x1, x2, x5) 29.13/9.33 29.13/9.33 U20_GAA(x1, x2, x3, x4, x5) = U20_GAA(x1, x5) 29.13/9.33 29.13/9.33 U21_GAA(x1, x2, x3, x4, x5, x6) = U21_GAA(x1, x6) 29.13/9.33 29.13/9.33 PD_IN_AGAA(x1, x2, x3, x4) = PD_IN_AGAA(x2) 29.13/9.33 29.13/9.33 U9_AGAA(x1, x2, x3, x4, x5) = U9_AGAA(x2, x5) 29.13/9.33 29.13/9.33 U10_AGAA(x1, x2, x3, x4, x5) = U10_AGAA(x2, x5) 29.13/9.33 29.13/9.33 U11_AGAA(x1, x2, x3, x4, x5) = U11_AGAA(x1, x2, x5) 29.13/9.33 29.13/9.33 U22_GAA(x1, x2, x3, x4, x5, x6) = U22_GAA(x1, x6) 29.13/9.33 29.13/9.33 PE_IN_AGAA(x1, x2, x3, x4) = PE_IN_AGAA(x2) 29.13/9.33 29.13/9.33 U12_AGAA(x1, x2, x3, x4) = U12_AGAA(x1, x4) 29.13/9.33 29.13/9.33 U23_GAA(x1, x2, x3, x4, x5, x6) = U23_GAA(x1, x6) 29.13/9.33 29.13/9.33 PF_IN_AGAA(x1, x2, x3, x4) = PF_IN_AGAA(x2) 29.13/9.33 29.13/9.33 U14_AGAA(x1, x2, x3, x4, x5) = U14_AGAA(x2, x5) 29.13/9.33 29.13/9.33 U15_AGAA(x1, x2, x3, x4, x5) = U15_AGAA(x2, x5) 29.13/9.33 29.13/9.33 U16_AGAA(x1, x2, x3, x4, x5) = U16_AGAA(x1, x2, x5) 29.13/9.33 29.13/9.33 U13_AGAA(x1, x2, x3, x4, x5) = U13_AGAA(x2, x5) 29.13/9.33 29.13/9.33 29.13/9.33 We have to consider all (P,R,Pi)-chains 29.13/9.33 29.13/9.33 29.13/9.33 Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES 29.13/9.33 29.13/9.33 29.13/9.33 29.13/9.33 ---------------------------------------- 29.13/9.33 29.13/9.33 (136) 29.13/9.33 Obligation: 29.13/9.33 Pi DP problem: 29.13/9.33 The TRS P consists of the following rules: 29.13/9.33 29.13/9.33 INSERTA_IN_GAA(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) -> U17_GAA(X1, X2, X3, X4, insertA_in_gaa(0, X2, X4)) 29.13/9.33 INSERTA_IN_GAA(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) -> INSERTA_IN_GAA(0, X2, X4) 29.13/9.33 INSERTA_IN_GAA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) -> U18_GAA(X1, X2, X3, X4, X5, pB_in_gaaa(X1, X2, X3, X5)) 29.13/9.33 INSERTA_IN_GAA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) -> PB_IN_GAAA(X1, X2, X3, X5) 29.13/9.33 PB_IN_GAAA(X1, X2, X3, X4) -> U3_GAAA(X1, X2, X3, X4, lessG_in_ga(X1, X2)) 29.13/9.33 PB_IN_GAAA(X1, X2, X3, X4) -> LESSG_IN_GA(X1, X2) 29.13/9.33 LESSG_IN_GA(s(X1), s(X2)) -> U1_GA(X1, X2, lessG_in_ga(X1, X2)) 29.13/9.33 LESSG_IN_GA(s(X1), s(X2)) -> LESSG_IN_GA(X1, X2) 29.13/9.33 PB_IN_GAAA(X1, X2, X3, X4) -> U4_GAAA(X1, X2, X3, X4, lesscG_in_ga(X1, X2)) 29.13/9.33 U4_GAAA(X1, X2, X3, X4, lesscG_out_ga(X1, X2)) -> U5_GAAA(X1, X2, X3, X4, insertA_in_gaa(s(X1), X3, X4)) 29.13/9.33 U4_GAAA(X1, X2, X3, X4, lesscG_out_ga(X1, X2)) -> INSERTA_IN_GAA(s(X1), X3, X4) 29.13/9.33 INSERTA_IN_GAA(X1, tree(X2, X3, X4), tree(X2, X3, X5)) -> U19_GAA(X1, X2, X3, X4, X5, pC_in_agaa(X2, X1, X4, X5)) 29.13/9.33 INSERTA_IN_GAA(X1, tree(X2, X3, X4), tree(X2, X3, X5)) -> PC_IN_AGAA(X2, X1, X4, X5) 29.13/9.33 PC_IN_AGAA(X1, X2, X3, X4) -> U6_AGAA(X1, X2, X3, X4, lessH_in_ag(X1, X2)) 29.13/9.33 PC_IN_AGAA(X1, X2, X3, X4) -> LESSH_IN_AG(X1, X2) 29.13/9.33 LESSH_IN_AG(s(X1), s(X2)) -> U2_AG(X1, X2, lessH_in_ag(X1, X2)) 29.13/9.33 LESSH_IN_AG(s(X1), s(X2)) -> LESSH_IN_AG(X1, X2) 29.13/9.33 PC_IN_AGAA(X1, X2, X3, X4) -> U7_AGAA(X1, X2, X3, X4, lesscH_in_ag(X1, X2)) 29.13/9.33 U7_AGAA(X1, X2, X3, X4, lesscH_out_ag(X1, X2)) -> U8_AGAA(X1, X2, X3, X4, insertA_in_gaa(X2, X3, X4)) 29.13/9.33 U7_AGAA(X1, X2, X3, X4, lesscH_out_ag(X1, X2)) -> INSERTA_IN_GAA(X2, X3, X4) 29.13/9.33 INSERTA_IN_GAA(s(X1), tree(0, X2, X3), tree(0, X2, X4)) -> U20_GAA(X1, X2, X3, X4, insertA_in_gaa(s(X1), X3, X4)) 29.13/9.33 INSERTA_IN_GAA(s(X1), tree(0, X2, X3), tree(0, X2, X4)) -> INSERTA_IN_GAA(s(X1), X3, X4) 29.13/9.33 INSERTA_IN_GAA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) -> U21_GAA(X1, X2, X3, X4, X5, pD_in_agaa(X2, X1, X4, X5)) 29.13/9.33 INSERTA_IN_GAA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) -> PD_IN_AGAA(X2, X1, X4, X5) 29.13/9.33 PD_IN_AGAA(X1, X2, X3, X4) -> U9_AGAA(X1, X2, X3, X4, lessH_in_ag(X1, X2)) 29.13/9.33 PD_IN_AGAA(X1, X2, X3, X4) -> LESSH_IN_AG(X1, X2) 29.13/9.33 PD_IN_AGAA(X1, X2, X3, X4) -> U10_AGAA(X1, X2, X3, X4, lesscH_in_ag(X1, X2)) 29.13/9.33 U10_AGAA(X1, X2, X3, X4, lesscH_out_ag(X1, X2)) -> U11_AGAA(X1, X2, X3, X4, insertA_in_gaa(s(X2), X3, X4)) 29.13/9.33 U10_AGAA(X1, X2, X3, X4, lesscH_out_ag(X1, X2)) -> INSERTA_IN_GAA(s(X2), X3, X4) 29.13/9.33 INSERTA_IN_GAA(X1, tree(X2, X3, X4), tree(X2, X3, X5)) -> U22_GAA(X1, X2, X3, X4, X5, pE_in_agaa(X2, X1, X4, X5)) 29.13/9.33 INSERTA_IN_GAA(X1, tree(X2, X3, X4), tree(X2, X3, X5)) -> PE_IN_AGAA(X2, X1, X4, X5) 29.13/9.33 PE_IN_AGAA(0, s(X1), X2, X3) -> U12_AGAA(X1, X2, X3, insertA_in_gaa(s(X1), X2, X3)) 29.13/9.33 PE_IN_AGAA(0, s(X1), X2, X3) -> INSERTA_IN_GAA(s(X1), X2, X3) 29.13/9.33 INSERTA_IN_GAA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) -> U23_GAA(X1, X2, X3, X4, X5, pF_in_agaa(X2, X1, X4, X5)) 29.13/9.33 INSERTA_IN_GAA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) -> PF_IN_AGAA(X2, X1, X4, X5) 29.13/9.33 PF_IN_AGAA(X1, X2, X3, X4) -> U14_AGAA(X1, X2, X3, X4, lessH_in_ag(X1, X2)) 29.13/9.33 PF_IN_AGAA(X1, X2, X3, X4) -> LESSH_IN_AG(X1, X2) 29.13/9.33 PF_IN_AGAA(X1, X2, X3, X4) -> U15_AGAA(X1, X2, X3, X4, lesscH_in_ag(X1, X2)) 29.13/9.33 U15_AGAA(X1, X2, X3, X4, lesscH_out_ag(X1, X2)) -> U16_AGAA(X1, X2, X3, X4, insertA_in_gaa(s(X2), X3, X4)) 29.13/9.33 U15_AGAA(X1, X2, X3, X4, lesscH_out_ag(X1, X2)) -> INSERTA_IN_GAA(s(X2), X3, X4) 29.13/9.33 PE_IN_AGAA(s(X1), s(X2), X3, X4) -> U13_AGAA(X1, X2, X3, X4, pF_in_agaa(X1, X2, X3, X4)) 29.13/9.33 PE_IN_AGAA(s(X1), s(X2), X3, X4) -> PF_IN_AGAA(X1, X2, X3, X4) 29.13/9.33 29.13/9.33 The TRS R consists of the following rules: 29.13/9.33 29.13/9.33 lesscG_in_ga(0, s(X1)) -> lesscG_out_ga(0, s(X1)) 29.13/9.33 lesscG_in_ga(s(X1), s(X2)) -> U32_ga(X1, X2, lesscG_in_ga(X1, X2)) 29.13/9.33 U32_ga(X1, X2, lesscG_out_ga(X1, X2)) -> lesscG_out_ga(s(X1), s(X2)) 29.13/9.33 lesscH_in_ag(0, s(X1)) -> lesscH_out_ag(0, s(X1)) 29.13/9.33 lesscH_in_ag(s(X1), s(X2)) -> U33_ag(X1, X2, lesscH_in_ag(X1, X2)) 29.13/9.33 U33_ag(X1, X2, lesscH_out_ag(X1, X2)) -> lesscH_out_ag(s(X1), s(X2)) 29.13/9.33 29.13/9.33 The argument filtering Pi contains the following mapping: 29.13/9.33 insertA_in_gaa(x1, x2, x3) = insertA_in_gaa(x1) 29.13/9.33 29.13/9.33 0 = 0 29.13/9.33 29.13/9.33 s(x1) = s(x1) 29.13/9.33 29.13/9.33 pB_in_gaaa(x1, x2, x3, x4) = pB_in_gaaa(x1) 29.13/9.33 29.13/9.33 lessG_in_ga(x1, x2) = lessG_in_ga(x1) 29.13/9.33 29.13/9.33 lesscG_in_ga(x1, x2) = lesscG_in_ga(x1) 29.13/9.33 29.13/9.33 lesscG_out_ga(x1, x2) = lesscG_out_ga(x1) 29.13/9.33 29.13/9.33 U32_ga(x1, x2, x3) = U32_ga(x1, x3) 29.13/9.33 29.13/9.33 pC_in_agaa(x1, x2, x3, x4) = pC_in_agaa(x2) 29.13/9.33 29.13/9.33 lessH_in_ag(x1, x2) = lessH_in_ag(x2) 29.13/9.33 29.13/9.33 lesscH_in_ag(x1, x2) = lesscH_in_ag(x2) 29.13/9.33 29.13/9.33 lesscH_out_ag(x1, x2) = lesscH_out_ag(x1, x2) 29.13/9.33 29.13/9.33 U33_ag(x1, x2, x3) = U33_ag(x2, x3) 29.13/9.33 29.13/9.33 pD_in_agaa(x1, x2, x3, x4) = pD_in_agaa(x2) 29.13/9.33 29.13/9.33 pE_in_agaa(x1, x2, x3, x4) = pE_in_agaa(x2) 29.13/9.33 29.13/9.33 pF_in_agaa(x1, x2, x3, x4) = pF_in_agaa(x2) 29.13/9.33 29.13/9.33 INSERTA_IN_GAA(x1, x2, x3) = INSERTA_IN_GAA(x1) 29.13/9.33 29.13/9.33 U17_GAA(x1, x2, x3, x4, x5) = U17_GAA(x5) 29.13/9.33 29.13/9.33 U18_GAA(x1, x2, x3, x4, x5, x6) = U18_GAA(x1, x6) 29.13/9.33 29.13/9.33 PB_IN_GAAA(x1, x2, x3, x4) = PB_IN_GAAA(x1) 29.13/9.33 29.13/9.33 U3_GAAA(x1, x2, x3, x4, x5) = U3_GAAA(x1, x5) 29.13/9.33 29.13/9.33 LESSG_IN_GA(x1, x2) = LESSG_IN_GA(x1) 29.13/9.33 29.13/9.33 U1_GA(x1, x2, x3) = U1_GA(x1, x3) 29.13/9.33 29.13/9.33 U4_GAAA(x1, x2, x3, x4, x5) = U4_GAAA(x1, x5) 29.13/9.33 29.13/9.33 U5_GAAA(x1, x2, x3, x4, x5) = U5_GAAA(x1, x5) 29.13/9.33 29.13/9.33 U19_GAA(x1, x2, x3, x4, x5, x6) = U19_GAA(x1, x6) 29.13/9.33 29.13/9.33 PC_IN_AGAA(x1, x2, x3, x4) = PC_IN_AGAA(x2) 29.13/9.33 29.13/9.33 U6_AGAA(x1, x2, x3, x4, x5) = U6_AGAA(x2, x5) 29.13/9.33 29.13/9.33 LESSH_IN_AG(x1, x2) = LESSH_IN_AG(x2) 29.13/9.33 29.13/9.33 U2_AG(x1, x2, x3) = U2_AG(x2, x3) 29.13/9.33 29.13/9.33 U7_AGAA(x1, x2, x3, x4, x5) = U7_AGAA(x2, x5) 29.13/9.33 29.13/9.33 U8_AGAA(x1, x2, x3, x4, x5) = U8_AGAA(x1, x2, x5) 29.13/9.33 29.13/9.33 U20_GAA(x1, x2, x3, x4, x5) = U20_GAA(x1, x5) 29.13/9.33 29.13/9.33 U21_GAA(x1, x2, x3, x4, x5, x6) = U21_GAA(x1, x6) 29.13/9.33 29.13/9.33 PD_IN_AGAA(x1, x2, x3, x4) = PD_IN_AGAA(x2) 29.13/9.33 29.13/9.33 U9_AGAA(x1, x2, x3, x4, x5) = U9_AGAA(x2, x5) 29.13/9.33 29.13/9.33 U10_AGAA(x1, x2, x3, x4, x5) = U10_AGAA(x2, x5) 29.13/9.33 29.13/9.33 U11_AGAA(x1, x2, x3, x4, x5) = U11_AGAA(x1, x2, x5) 29.13/9.33 29.13/9.33 U22_GAA(x1, x2, x3, x4, x5, x6) = U22_GAA(x1, x6) 29.13/9.33 29.13/9.33 PE_IN_AGAA(x1, x2, x3, x4) = PE_IN_AGAA(x2) 29.13/9.33 29.13/9.33 U12_AGAA(x1, x2, x3, x4) = U12_AGAA(x1, x4) 29.13/9.33 29.13/9.33 U23_GAA(x1, x2, x3, x4, x5, x6) = U23_GAA(x1, x6) 29.13/9.33 29.13/9.33 PF_IN_AGAA(x1, x2, x3, x4) = PF_IN_AGAA(x2) 29.13/9.33 29.13/9.33 U14_AGAA(x1, x2, x3, x4, x5) = U14_AGAA(x2, x5) 29.13/9.33 29.13/9.33 U15_AGAA(x1, x2, x3, x4, x5) = U15_AGAA(x2, x5) 29.13/9.33 29.13/9.33 U16_AGAA(x1, x2, x3, x4, x5) = U16_AGAA(x1, x2, x5) 29.13/9.33 29.13/9.33 U13_AGAA(x1, x2, x3, x4, x5) = U13_AGAA(x2, x5) 29.13/9.33 29.13/9.33 29.13/9.33 We have to consider all (P,R,Pi)-chains 29.13/9.33 ---------------------------------------- 29.13/9.33 29.13/9.33 (137) DependencyGraphProof (EQUIVALENT) 29.13/9.33 The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 23 less nodes. 29.13/9.33 ---------------------------------------- 29.13/9.33 29.13/9.33 (138) 29.13/9.33 Complex Obligation (AND) 29.13/9.33 29.13/9.33 ---------------------------------------- 29.13/9.33 29.13/9.33 (139) 29.13/9.33 Obligation: 29.13/9.33 Pi DP problem: 29.13/9.33 The TRS P consists of the following rules: 29.13/9.33 29.13/9.33 LESSH_IN_AG(s(X1), s(X2)) -> LESSH_IN_AG(X1, X2) 29.13/9.33 29.13/9.33 The TRS R consists of the following rules: 29.13/9.33 29.13/9.33 lesscG_in_ga(0, s(X1)) -> lesscG_out_ga(0, s(X1)) 29.13/9.33 lesscG_in_ga(s(X1), s(X2)) -> U32_ga(X1, X2, lesscG_in_ga(X1, X2)) 29.13/9.33 U32_ga(X1, X2, lesscG_out_ga(X1, X2)) -> lesscG_out_ga(s(X1), s(X2)) 29.13/9.33 lesscH_in_ag(0, s(X1)) -> lesscH_out_ag(0, s(X1)) 29.13/9.33 lesscH_in_ag(s(X1), s(X2)) -> U33_ag(X1, X2, lesscH_in_ag(X1, X2)) 29.13/9.33 U33_ag(X1, X2, lesscH_out_ag(X1, X2)) -> lesscH_out_ag(s(X1), s(X2)) 29.13/9.33 29.13/9.33 The argument filtering Pi contains the following mapping: 29.13/9.33 0 = 0 29.13/9.33 29.13/9.33 s(x1) = s(x1) 29.13/9.33 29.13/9.33 lesscG_in_ga(x1, x2) = lesscG_in_ga(x1) 29.13/9.33 29.13/9.33 lesscG_out_ga(x1, x2) = lesscG_out_ga(x1) 29.13/9.33 29.13/9.33 U32_ga(x1, x2, x3) = U32_ga(x1, x3) 29.13/9.33 29.13/9.33 lesscH_in_ag(x1, x2) = lesscH_in_ag(x2) 29.13/9.33 29.13/9.33 lesscH_out_ag(x1, x2) = lesscH_out_ag(x1, x2) 29.13/9.33 29.13/9.33 U33_ag(x1, x2, x3) = U33_ag(x2, x3) 29.13/9.33 29.13/9.33 LESSH_IN_AG(x1, x2) = LESSH_IN_AG(x2) 29.13/9.33 29.13/9.33 29.13/9.33 We have to consider all (P,R,Pi)-chains 29.13/9.33 ---------------------------------------- 29.13/9.33 29.13/9.33 (140) UsableRulesProof (EQUIVALENT) 29.13/9.33 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 29.13/9.33 ---------------------------------------- 29.13/9.33 29.13/9.33 (141) 29.13/9.33 Obligation: 29.13/9.33 Pi DP problem: 29.13/9.33 The TRS P consists of the following rules: 29.13/9.33 29.13/9.33 LESSH_IN_AG(s(X1), s(X2)) -> LESSH_IN_AG(X1, X2) 29.13/9.33 29.13/9.33 R is empty. 29.13/9.33 The argument filtering Pi contains the following mapping: 29.13/9.33 s(x1) = s(x1) 29.13/9.33 29.13/9.33 LESSH_IN_AG(x1, x2) = LESSH_IN_AG(x2) 29.13/9.33 29.13/9.33 29.13/9.33 We have to consider all (P,R,Pi)-chains 29.13/9.33 ---------------------------------------- 29.13/9.33 29.13/9.33 (142) PiDPToQDPProof (SOUND) 29.13/9.33 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 29.13/9.33 ---------------------------------------- 29.13/9.33 29.13/9.33 (143) 29.13/9.33 Obligation: 29.13/9.33 Q DP problem: 29.13/9.33 The TRS P consists of the following rules: 29.13/9.33 29.13/9.33 LESSH_IN_AG(s(X2)) -> LESSH_IN_AG(X2) 29.13/9.33 29.13/9.33 R is empty. 29.13/9.33 Q is empty. 29.13/9.33 We have to consider all (P,Q,R)-chains. 29.13/9.33 ---------------------------------------- 29.13/9.33 29.13/9.33 (144) QDPSizeChangeProof (EQUIVALENT) 29.13/9.33 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. 29.13/9.33 29.13/9.33 From the DPs we obtained the following set of size-change graphs: 29.13/9.33 *LESSH_IN_AG(s(X2)) -> LESSH_IN_AG(X2) 29.13/9.33 The graph contains the following edges 1 > 1 29.13/9.33 29.13/9.33 29.13/9.33 ---------------------------------------- 29.13/9.33 29.13/9.33 (145) 29.13/9.33 YES 29.13/9.33 29.13/9.33 ---------------------------------------- 29.13/9.33 29.13/9.33 (146) 29.13/9.33 Obligation: 29.13/9.33 Pi DP problem: 29.13/9.33 The TRS P consists of the following rules: 29.13/9.33 29.13/9.33 LESSG_IN_GA(s(X1), s(X2)) -> LESSG_IN_GA(X1, X2) 29.13/9.33 29.13/9.33 The TRS R consists of the following rules: 29.13/9.33 29.13/9.33 lesscG_in_ga(0, s(X1)) -> lesscG_out_ga(0, s(X1)) 29.13/9.33 lesscG_in_ga(s(X1), s(X2)) -> U32_ga(X1, X2, lesscG_in_ga(X1, X2)) 29.13/9.33 U32_ga(X1, X2, lesscG_out_ga(X1, X2)) -> lesscG_out_ga(s(X1), s(X2)) 29.13/9.33 lesscH_in_ag(0, s(X1)) -> lesscH_out_ag(0, s(X1)) 29.13/9.33 lesscH_in_ag(s(X1), s(X2)) -> U33_ag(X1, X2, lesscH_in_ag(X1, X2)) 29.13/9.33 U33_ag(X1, X2, lesscH_out_ag(X1, X2)) -> lesscH_out_ag(s(X1), s(X2)) 29.13/9.33 29.13/9.33 The argument filtering Pi contains the following mapping: 29.13/9.33 0 = 0 29.13/9.33 29.13/9.33 s(x1) = s(x1) 29.13/9.33 29.13/9.33 lesscG_in_ga(x1, x2) = lesscG_in_ga(x1) 29.13/9.33 29.13/9.33 lesscG_out_ga(x1, x2) = lesscG_out_ga(x1) 29.13/9.33 29.13/9.33 U32_ga(x1, x2, x3) = U32_ga(x1, x3) 29.13/9.33 29.13/9.33 lesscH_in_ag(x1, x2) = lesscH_in_ag(x2) 29.13/9.33 29.13/9.33 lesscH_out_ag(x1, x2) = lesscH_out_ag(x1, x2) 29.13/9.33 29.13/9.33 U33_ag(x1, x2, x3) = U33_ag(x2, x3) 29.13/9.33 29.13/9.33 LESSG_IN_GA(x1, x2) = LESSG_IN_GA(x1) 29.13/9.33 29.13/9.33 29.13/9.33 We have to consider all (P,R,Pi)-chains 29.13/9.33 ---------------------------------------- 29.13/9.33 29.13/9.33 (147) UsableRulesProof (EQUIVALENT) 29.13/9.33 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 29.13/9.33 ---------------------------------------- 29.13/9.33 29.13/9.33 (148) 29.13/9.33 Obligation: 29.13/9.33 Pi DP problem: 29.13/9.33 The TRS P consists of the following rules: 29.13/9.33 29.13/9.33 LESSG_IN_GA(s(X1), s(X2)) -> LESSG_IN_GA(X1, X2) 29.13/9.33 29.13/9.33 R is empty. 29.13/9.33 The argument filtering Pi contains the following mapping: 29.13/9.33 s(x1) = s(x1) 29.13/9.33 29.13/9.33 LESSG_IN_GA(x1, x2) = LESSG_IN_GA(x1) 29.13/9.33 29.13/9.33 29.13/9.33 We have to consider all (P,R,Pi)-chains 29.13/9.33 ---------------------------------------- 29.13/9.33 29.13/9.33 (149) PiDPToQDPProof (SOUND) 29.13/9.33 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 29.13/9.33 ---------------------------------------- 29.13/9.33 29.13/9.33 (150) 29.13/9.33 Obligation: 29.13/9.33 Q DP problem: 29.13/9.33 The TRS P consists of the following rules: 29.13/9.33 29.13/9.33 LESSG_IN_GA(s(X1)) -> LESSG_IN_GA(X1) 29.13/9.33 29.13/9.33 R is empty. 29.13/9.33 Q is empty. 29.13/9.33 We have to consider all (P,Q,R)-chains. 29.13/9.33 ---------------------------------------- 29.13/9.33 29.13/9.33 (151) QDPSizeChangeProof (EQUIVALENT) 29.13/9.33 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. 29.13/9.33 29.13/9.33 From the DPs we obtained the following set of size-change graphs: 29.13/9.33 *LESSG_IN_GA(s(X1)) -> LESSG_IN_GA(X1) 29.13/9.33 The graph contains the following edges 1 > 1 29.13/9.33 29.13/9.33 29.13/9.33 ---------------------------------------- 29.13/9.33 29.13/9.33 (152) 29.13/9.33 YES 29.13/9.33 29.13/9.33 ---------------------------------------- 29.13/9.33 29.13/9.33 (153) 29.13/9.33 Obligation: 29.13/9.33 Pi DP problem: 29.13/9.33 The TRS P consists of the following rules: 29.13/9.33 29.13/9.33 INSERTA_IN_GAA(X1, tree(X2, X3, X4), tree(X2, X3, X5)) -> PC_IN_AGAA(X2, X1, X4, X5) 29.13/9.33 PC_IN_AGAA(X1, X2, X3, X4) -> U7_AGAA(X1, X2, X3, X4, lesscH_in_ag(X1, X2)) 29.13/9.33 U7_AGAA(X1, X2, X3, X4, lesscH_out_ag(X1, X2)) -> INSERTA_IN_GAA(X2, X3, X4) 29.13/9.33 INSERTA_IN_GAA(0, tree(s(X1), X2, X3), tree(s(X1), X4, X3)) -> INSERTA_IN_GAA(0, X2, X4) 29.13/9.33 INSERTA_IN_GAA(X1, tree(X2, X3, X4), tree(X2, X3, X5)) -> PE_IN_AGAA(X2, X1, X4, X5) 29.13/9.33 PE_IN_AGAA(0, s(X1), X2, X3) -> INSERTA_IN_GAA(s(X1), X2, X3) 29.13/9.33 INSERTA_IN_GAA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X5, X4)) -> PB_IN_GAAA(X1, X2, X3, X5) 29.13/9.33 PB_IN_GAAA(X1, X2, X3, X4) -> U4_GAAA(X1, X2, X3, X4, lesscG_in_ga(X1, X2)) 29.13/9.33 U4_GAAA(X1, X2, X3, X4, lesscG_out_ga(X1, X2)) -> INSERTA_IN_GAA(s(X1), X3, X4) 29.13/9.33 INSERTA_IN_GAA(s(X1), tree(0, X2, X3), tree(0, X2, X4)) -> INSERTA_IN_GAA(s(X1), X3, X4) 29.13/9.33 INSERTA_IN_GAA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) -> PD_IN_AGAA(X2, X1, X4, X5) 29.13/9.33 PD_IN_AGAA(X1, X2, X3, X4) -> U10_AGAA(X1, X2, X3, X4, lesscH_in_ag(X1, X2)) 29.13/9.33 U10_AGAA(X1, X2, X3, X4, lesscH_out_ag(X1, X2)) -> INSERTA_IN_GAA(s(X2), X3, X4) 29.13/9.33 INSERTA_IN_GAA(s(X1), tree(s(X2), X3, X4), tree(s(X2), X3, X5)) -> PF_IN_AGAA(X2, X1, X4, X5) 29.13/9.33 PF_IN_AGAA(X1, X2, X3, X4) -> U15_AGAA(X1, X2, X3, X4, lesscH_in_ag(X1, X2)) 29.13/9.33 U15_AGAA(X1, X2, X3, X4, lesscH_out_ag(X1, X2)) -> INSERTA_IN_GAA(s(X2), X3, X4) 29.13/9.33 PE_IN_AGAA(s(X1), s(X2), X3, X4) -> PF_IN_AGAA(X1, X2, X3, X4) 29.13/9.33 29.13/9.33 The TRS R consists of the following rules: 29.13/9.33 29.13/9.33 lesscG_in_ga(0, s(X1)) -> lesscG_out_ga(0, s(X1)) 29.13/9.33 lesscG_in_ga(s(X1), s(X2)) -> U32_ga(X1, X2, lesscG_in_ga(X1, X2)) 29.13/9.33 U32_ga(X1, X2, lesscG_out_ga(X1, X2)) -> lesscG_out_ga(s(X1), s(X2)) 29.13/9.33 lesscH_in_ag(0, s(X1)) -> lesscH_out_ag(0, s(X1)) 29.13/9.33 lesscH_in_ag(s(X1), s(X2)) -> U33_ag(X1, X2, lesscH_in_ag(X1, X2)) 29.13/9.33 U33_ag(X1, X2, lesscH_out_ag(X1, X2)) -> lesscH_out_ag(s(X1), s(X2)) 29.13/9.33 29.13/9.33 The argument filtering Pi contains the following mapping: 29.13/9.33 0 = 0 29.13/9.33 29.13/9.33 s(x1) = s(x1) 29.13/9.33 29.13/9.33 lesscG_in_ga(x1, x2) = lesscG_in_ga(x1) 29.13/9.33 29.13/9.33 lesscG_out_ga(x1, x2) = lesscG_out_ga(x1) 29.13/9.33 29.13/9.33 U32_ga(x1, x2, x3) = U32_ga(x1, x3) 29.13/9.33 29.13/9.33 lesscH_in_ag(x1, x2) = lesscH_in_ag(x2) 29.13/9.33 29.13/9.33 lesscH_out_ag(x1, x2) = lesscH_out_ag(x1, x2) 29.13/9.33 29.13/9.33 U33_ag(x1, x2, x3) = U33_ag(x2, x3) 29.13/9.33 29.13/9.33 INSERTA_IN_GAA(x1, x2, x3) = INSERTA_IN_GAA(x1) 29.13/9.33 29.13/9.33 PB_IN_GAAA(x1, x2, x3, x4) = PB_IN_GAAA(x1) 29.13/9.33 29.13/9.33 U4_GAAA(x1, x2, x3, x4, x5) = U4_GAAA(x1, x5) 29.13/9.33 29.13/9.33 PC_IN_AGAA(x1, x2, x3, x4) = PC_IN_AGAA(x2) 29.13/9.33 29.13/9.33 U7_AGAA(x1, x2, x3, x4, x5) = U7_AGAA(x2, x5) 29.13/9.33 29.13/9.33 PD_IN_AGAA(x1, x2, x3, x4) = PD_IN_AGAA(x2) 29.13/9.33 29.13/9.33 U10_AGAA(x1, x2, x3, x4, x5) = U10_AGAA(x2, x5) 29.13/9.33 29.13/9.33 PE_IN_AGAA(x1, x2, x3, x4) = PE_IN_AGAA(x2) 29.13/9.33 29.13/9.33 PF_IN_AGAA(x1, x2, x3, x4) = PF_IN_AGAA(x2) 29.13/9.33 29.13/9.33 U15_AGAA(x1, x2, x3, x4, x5) = U15_AGAA(x2, x5) 29.13/9.33 29.13/9.33 29.13/9.33 We have to consider all (P,R,Pi)-chains 29.13/9.33 ---------------------------------------- 29.13/9.33 29.13/9.33 (154) PiDPToQDPProof (SOUND) 29.13/9.33 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 29.13/9.33 ---------------------------------------- 29.13/9.33 29.13/9.33 (155) 29.13/9.33 Obligation: 29.13/9.33 Q DP problem: 29.13/9.33 The TRS P consists of the following rules: 29.13/9.33 29.13/9.33 INSERTA_IN_GAA(X1) -> PC_IN_AGAA(X1) 29.13/9.33 PC_IN_AGAA(X2) -> U7_AGAA(X2, lesscH_in_ag(X2)) 29.13/9.33 U7_AGAA(X2, lesscH_out_ag(X1, X2)) -> INSERTA_IN_GAA(X2) 29.13/9.33 INSERTA_IN_GAA(0) -> INSERTA_IN_GAA(0) 29.13/9.33 INSERTA_IN_GAA(X1) -> PE_IN_AGAA(X1) 29.13/9.33 PE_IN_AGAA(s(X1)) -> INSERTA_IN_GAA(s(X1)) 29.13/9.33 INSERTA_IN_GAA(s(X1)) -> PB_IN_GAAA(X1) 29.13/9.33 PB_IN_GAAA(X1) -> U4_GAAA(X1, lesscG_in_ga(X1)) 29.13/9.33 U4_GAAA(X1, lesscG_out_ga(X1)) -> INSERTA_IN_GAA(s(X1)) 29.13/9.33 INSERTA_IN_GAA(s(X1)) -> INSERTA_IN_GAA(s(X1)) 29.13/9.33 INSERTA_IN_GAA(s(X1)) -> PD_IN_AGAA(X1) 29.13/9.33 PD_IN_AGAA(X2) -> U10_AGAA(X2, lesscH_in_ag(X2)) 29.13/9.33 U10_AGAA(X2, lesscH_out_ag(X1, X2)) -> INSERTA_IN_GAA(s(X2)) 29.13/9.33 INSERTA_IN_GAA(s(X1)) -> PF_IN_AGAA(X1) 29.13/9.33 PF_IN_AGAA(X2) -> U15_AGAA(X2, lesscH_in_ag(X2)) 29.13/9.33 U15_AGAA(X2, lesscH_out_ag(X1, X2)) -> INSERTA_IN_GAA(s(X2)) 29.13/9.33 PE_IN_AGAA(s(X2)) -> PF_IN_AGAA(X2) 29.13/9.33 29.13/9.33 The TRS R consists of the following rules: 29.13/9.33 29.13/9.33 lesscG_in_ga(0) -> lesscG_out_ga(0) 29.13/9.33 lesscG_in_ga(s(X1)) -> U32_ga(X1, lesscG_in_ga(X1)) 29.13/9.33 U32_ga(X1, lesscG_out_ga(X1)) -> lesscG_out_ga(s(X1)) 29.13/9.33 lesscH_in_ag(s(X1)) -> lesscH_out_ag(0, s(X1)) 29.13/9.33 lesscH_in_ag(s(X2)) -> U33_ag(X2, lesscH_in_ag(X2)) 29.13/9.33 U33_ag(X2, lesscH_out_ag(X1, X2)) -> lesscH_out_ag(s(X1), s(X2)) 29.13/9.33 29.13/9.33 The set Q consists of the following terms: 29.13/9.33 29.13/9.33 lesscG_in_ga(x0) 29.13/9.33 U32_ga(x0, x1) 29.13/9.33 lesscH_in_ag(x0) 29.13/9.33 U33_ag(x0, x1) 29.13/9.33 29.13/9.33 We have to consider all (P,Q,R)-chains. 29.17/9.39 EOF