/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.pl /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- MAYBE proof of /export/starexec/sandbox/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern p(g) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2) PiTRS (3) DependencyPairsProof [EQUIVALENT, 0 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) PiDP (7) UsableRulesProof [EQUIVALENT, 0 ms] (8) PiDP (9) PiDPToQDPProof [SOUND, 0 ms] (10) QDP (11) PrologToPiTRSProof [SOUND, 0 ms] (12) PiTRS (13) DependencyPairsProof [EQUIVALENT, 0 ms] (14) PiDP (15) DependencyGraphProof [EQUIVALENT, 0 ms] (16) PiDP (17) UsableRulesProof [EQUIVALENT, 0 ms] (18) PiDP (19) PiDPToQDPProof [SOUND, 0 ms] (20) QDP (21) PrologToTRSTransformerProof [SOUND, 0 ms] (22) QTRS (23) QTRSRRRProof [EQUIVALENT, 54 ms] (24) QTRS (25) QTRSRRRProof [EQUIVALENT, 0 ms] (26) QTRS (27) QTRSRRRProof [EQUIVALENT, 5 ms] (28) QTRS (29) QTRSRRRProof [EQUIVALENT, 5 ms] (30) QTRS (31) Overlay + Local Confluence [EQUIVALENT, 0 ms] (32) QTRS (33) DependencyPairsProof [EQUIVALENT, 0 ms] (34) QDP (35) UsableRulesProof [EQUIVALENT, 0 ms] (36) QDP (37) QReductionProof [EQUIVALENT, 0 ms] (38) QDP (39) PrologToIRSwTTransformerProof [SOUND, 0 ms] (40) IRSwT (41) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (42) IRSwT (43) IntTRSCompressionProof [EQUIVALENT, 5 ms] (44) IRSwT (45) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (46) IRSwT (47) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms] (48) IRSwT (49) FilterProof [EQUIVALENT, 0 ms] (50) IntTRS (51) IntTRSPeriodicNontermProof [COMPLETE, 0 ms] (52) NO (53) PrologToDTProblemTransformerProof [SOUND, 0 ms] (54) TRIPLES (55) TriplesToPiDPProof [SOUND, 7 ms] (56) PiDP (57) DependencyGraphProof [EQUIVALENT, 0 ms] (58) PiDP (59) PiDPToQDPProof [SOUND, 0 ms] (60) QDP ---------------------------------------- (0) Obligation: Clauses: p(b). p(a) :- p1(X). p1(b). p1(a) :- p1(X). Query: p(g) ---------------------------------------- (1) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: p_in_1: (b) p1_in_1: (f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: p_in_g(b) -> p_out_g(b) p_in_g(a) -> U1_g(p1_in_a(X)) p1_in_a(b) -> p1_out_a(b) p1_in_a(a) -> U2_a(p1_in_a(X)) U2_a(p1_out_a(X)) -> p1_out_a(a) U1_g(p1_out_a(X)) -> p_out_g(a) The argument filtering Pi contains the following mapping: p_in_g(x1) = p_in_g(x1) b = b p_out_g(x1) = p_out_g a = a U1_g(x1) = U1_g(x1) p1_in_a(x1) = p1_in_a p1_out_a(x1) = p1_out_a(x1) U2_a(x1) = U2_a(x1) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: p_in_g(b) -> p_out_g(b) p_in_g(a) -> U1_g(p1_in_a(X)) p1_in_a(b) -> p1_out_a(b) p1_in_a(a) -> U2_a(p1_in_a(X)) U2_a(p1_out_a(X)) -> p1_out_a(a) U1_g(p1_out_a(X)) -> p_out_g(a) The argument filtering Pi contains the following mapping: p_in_g(x1) = p_in_g(x1) b = b p_out_g(x1) = p_out_g a = a U1_g(x1) = U1_g(x1) p1_in_a(x1) = p1_in_a p1_out_a(x1) = p1_out_a(x1) U2_a(x1) = U2_a(x1) ---------------------------------------- (3) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: Pi DP problem: The TRS P consists of the following rules: P_IN_G(a) -> U1_G(p1_in_a(X)) P_IN_G(a) -> P1_IN_A(X) P1_IN_A(a) -> U2_A(p1_in_a(X)) P1_IN_A(a) -> P1_IN_A(X) The TRS R consists of the following rules: p_in_g(b) -> p_out_g(b) p_in_g(a) -> U1_g(p1_in_a(X)) p1_in_a(b) -> p1_out_a(b) p1_in_a(a) -> U2_a(p1_in_a(X)) U2_a(p1_out_a(X)) -> p1_out_a(a) U1_g(p1_out_a(X)) -> p_out_g(a) The argument filtering Pi contains the following mapping: p_in_g(x1) = p_in_g(x1) b = b p_out_g(x1) = p_out_g a = a U1_g(x1) = U1_g(x1) p1_in_a(x1) = p1_in_a p1_out_a(x1) = p1_out_a(x1) U2_a(x1) = U2_a(x1) P_IN_G(x1) = P_IN_G(x1) U1_G(x1) = U1_G(x1) P1_IN_A(x1) = P1_IN_A U2_A(x1) = U2_A(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: P_IN_G(a) -> U1_G(p1_in_a(X)) P_IN_G(a) -> P1_IN_A(X) P1_IN_A(a) -> U2_A(p1_in_a(X)) P1_IN_A(a) -> P1_IN_A(X) The TRS R consists of the following rules: p_in_g(b) -> p_out_g(b) p_in_g(a) -> U1_g(p1_in_a(X)) p1_in_a(b) -> p1_out_a(b) p1_in_a(a) -> U2_a(p1_in_a(X)) U2_a(p1_out_a(X)) -> p1_out_a(a) U1_g(p1_out_a(X)) -> p_out_g(a) The argument filtering Pi contains the following mapping: p_in_g(x1) = p_in_g(x1) b = b p_out_g(x1) = p_out_g a = a U1_g(x1) = U1_g(x1) p1_in_a(x1) = p1_in_a p1_out_a(x1) = p1_out_a(x1) U2_a(x1) = U2_a(x1) P_IN_G(x1) = P_IN_G(x1) U1_G(x1) = U1_G(x1) P1_IN_A(x1) = P1_IN_A U2_A(x1) = U2_A(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes. ---------------------------------------- (6) Obligation: Pi DP problem: The TRS P consists of the following rules: P1_IN_A(a) -> P1_IN_A(X) The TRS R consists of the following rules: p_in_g(b) -> p_out_g(b) p_in_g(a) -> U1_g(p1_in_a(X)) p1_in_a(b) -> p1_out_a(b) p1_in_a(a) -> U2_a(p1_in_a(X)) U2_a(p1_out_a(X)) -> p1_out_a(a) U1_g(p1_out_a(X)) -> p_out_g(a) The argument filtering Pi contains the following mapping: p_in_g(x1) = p_in_g(x1) b = b p_out_g(x1) = p_out_g a = a U1_g(x1) = U1_g(x1) p1_in_a(x1) = p1_in_a p1_out_a(x1) = p1_out_a(x1) U2_a(x1) = U2_a(x1) P1_IN_A(x1) = P1_IN_A We have to consider all (P,R,Pi)-chains ---------------------------------------- (7) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (8) Obligation: Pi DP problem: The TRS P consists of the following rules: P1_IN_A(a) -> P1_IN_A(X) R is empty. The argument filtering Pi contains the following mapping: a = a P1_IN_A(x1) = P1_IN_A We have to consider all (P,R,Pi)-chains ---------------------------------------- (9) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: P1_IN_A -> P1_IN_A R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (11) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: p_in_1: (b) p1_in_1: (f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: p_in_g(b) -> p_out_g(b) p_in_g(a) -> U1_g(p1_in_a(X)) p1_in_a(b) -> p1_out_a(b) p1_in_a(a) -> U2_a(p1_in_a(X)) U2_a(p1_out_a(X)) -> p1_out_a(a) U1_g(p1_out_a(X)) -> p_out_g(a) The argument filtering Pi contains the following mapping: p_in_g(x1) = p_in_g(x1) b = b p_out_g(x1) = p_out_g(x1) a = a U1_g(x1) = U1_g(x1) p1_in_a(x1) = p1_in_a p1_out_a(x1) = p1_out_a(x1) U2_a(x1) = U2_a(x1) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (12) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: p_in_g(b) -> p_out_g(b) p_in_g(a) -> U1_g(p1_in_a(X)) p1_in_a(b) -> p1_out_a(b) p1_in_a(a) -> U2_a(p1_in_a(X)) U2_a(p1_out_a(X)) -> p1_out_a(a) U1_g(p1_out_a(X)) -> p_out_g(a) The argument filtering Pi contains the following mapping: p_in_g(x1) = p_in_g(x1) b = b p_out_g(x1) = p_out_g(x1) a = a U1_g(x1) = U1_g(x1) p1_in_a(x1) = p1_in_a p1_out_a(x1) = p1_out_a(x1) U2_a(x1) = U2_a(x1) ---------------------------------------- (13) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: Pi DP problem: The TRS P consists of the following rules: P_IN_G(a) -> U1_G(p1_in_a(X)) P_IN_G(a) -> P1_IN_A(X) P1_IN_A(a) -> U2_A(p1_in_a(X)) P1_IN_A(a) -> P1_IN_A(X) The TRS R consists of the following rules: p_in_g(b) -> p_out_g(b) p_in_g(a) -> U1_g(p1_in_a(X)) p1_in_a(b) -> p1_out_a(b) p1_in_a(a) -> U2_a(p1_in_a(X)) U2_a(p1_out_a(X)) -> p1_out_a(a) U1_g(p1_out_a(X)) -> p_out_g(a) The argument filtering Pi contains the following mapping: p_in_g(x1) = p_in_g(x1) b = b p_out_g(x1) = p_out_g(x1) a = a U1_g(x1) = U1_g(x1) p1_in_a(x1) = p1_in_a p1_out_a(x1) = p1_out_a(x1) U2_a(x1) = U2_a(x1) P_IN_G(x1) = P_IN_G(x1) U1_G(x1) = U1_G(x1) P1_IN_A(x1) = P1_IN_A U2_A(x1) = U2_A(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: P_IN_G(a) -> U1_G(p1_in_a(X)) P_IN_G(a) -> P1_IN_A(X) P1_IN_A(a) -> U2_A(p1_in_a(X)) P1_IN_A(a) -> P1_IN_A(X) The TRS R consists of the following rules: p_in_g(b) -> p_out_g(b) p_in_g(a) -> U1_g(p1_in_a(X)) p1_in_a(b) -> p1_out_a(b) p1_in_a(a) -> U2_a(p1_in_a(X)) U2_a(p1_out_a(X)) -> p1_out_a(a) U1_g(p1_out_a(X)) -> p_out_g(a) The argument filtering Pi contains the following mapping: p_in_g(x1) = p_in_g(x1) b = b p_out_g(x1) = p_out_g(x1) a = a U1_g(x1) = U1_g(x1) p1_in_a(x1) = p1_in_a p1_out_a(x1) = p1_out_a(x1) U2_a(x1) = U2_a(x1) P_IN_G(x1) = P_IN_G(x1) U1_G(x1) = U1_G(x1) P1_IN_A(x1) = P1_IN_A U2_A(x1) = U2_A(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (15) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes. ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: P1_IN_A(a) -> P1_IN_A(X) The TRS R consists of the following rules: p_in_g(b) -> p_out_g(b) p_in_g(a) -> U1_g(p1_in_a(X)) p1_in_a(b) -> p1_out_a(b) p1_in_a(a) -> U2_a(p1_in_a(X)) U2_a(p1_out_a(X)) -> p1_out_a(a) U1_g(p1_out_a(X)) -> p_out_g(a) The argument filtering Pi contains the following mapping: p_in_g(x1) = p_in_g(x1) b = b p_out_g(x1) = p_out_g(x1) a = a U1_g(x1) = U1_g(x1) p1_in_a(x1) = p1_in_a p1_out_a(x1) = p1_out_a(x1) U2_a(x1) = U2_a(x1) P1_IN_A(x1) = P1_IN_A We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (18) Obligation: Pi DP problem: The TRS P consists of the following rules: P1_IN_A(a) -> P1_IN_A(X) R is empty. The argument filtering Pi contains the following mapping: a = a P1_IN_A(x1) = P1_IN_A We have to consider all (P,R,Pi)-chains ---------------------------------------- (19) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: P1_IN_A -> P1_IN_A R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (21) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 3, "program": { "directives": [], "clauses": [ [ "(p (b))", null ], [ "(p (a))", "(p1 X)" ], [ "(p1 (b))", null ], [ "(p1 (a))", "(p1 X)" ] ] }, "graph": { "nodes": { "99": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "36": { "goal": [ { "clause": 0, "scope": 1, "term": "(p T1)" }, { "clause": 1, "scope": 1, "term": "(p T1)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "37": { "goal": [{ "clause": 0, "scope": 1, "term": "(p T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "38": { "goal": [{ "clause": 1, "scope": 1, "term": "(p T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "type": "Nodes", "101": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "3": { "goal": [{ "clause": -1, "scope": -1, "term": "(p T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "103": { "goal": [{ "clause": -1, "scope": -1, "term": "(p1 X4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X4"], "exprvars": [] } }, "81": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "82": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "93": { "goal": [ { "clause": 2, "scope": 2, "term": "(p1 X2)" }, { "clause": 3, "scope": 2, "term": "(p1 X2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X2"], "exprvars": [] } }, "83": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "84": { "goal": [{ "clause": -1, "scope": -1, "term": "(p1 X2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X2"], "exprvars": [] } }, "95": { "goal": [{ "clause": 2, "scope": 2, "term": "(p1 X2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X2"], "exprvars": [] } }, "85": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "96": { "goal": [{ "clause": 3, "scope": 2, "term": "(p1 X2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X2"], "exprvars": [] } } }, "edges": [ { "from": 3, "to": 36, "label": "CASE" }, { "from": 36, "to": 37, "label": "PARALLEL" }, { "from": 36, "to": 38, "label": "PARALLEL" }, { "from": 37, "to": 81, "label": "EVAL with clause\np(b).\nand substitutionT1 -> b" }, { "from": 37, "to": 82, "label": "EVAL-BACKTRACK" }, { "from": 38, "to": 84, "label": "EVAL with clause\np(a) :- p1(X2).\nand substitutionT1 -> a" }, { "from": 38, "to": 85, "label": "EVAL-BACKTRACK" }, { "from": 81, "to": 83, "label": "SUCCESS" }, { "from": 84, "to": 93, "label": "CASE" }, { "from": 93, "to": 95, "label": "PARALLEL" }, { "from": 93, "to": 96, "label": "PARALLEL" }, { "from": 95, "to": 99, "label": "ONLY EVAL with clause\np1(b).\nand substitutionX2 -> b" }, { "from": 96, "to": 103, "label": "ONLY EVAL with clause\np1(a) :- p1(X4).\nand substitutionX2 -> a" }, { "from": 99, "to": 101, "label": "SUCCESS" }, { "from": 103, "to": 84, "label": "INSTANCE with matching:\nX2 -> X4" } ], "type": "Graph" } } ---------------------------------------- (22) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f3_in(b) -> f3_out1 f3_in(a) -> U1(f84_in, a) U1(f84_out1(X2), a) -> f3_out1 f84_in -> f84_out1(b) f84_in -> U2(f84_in) U2(f84_out1(X4)) -> f84_out1(a) Q is empty. ---------------------------------------- (23) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U1(x_1, x_2)) = x_1 + 2*x_2 POL(U2(x_1)) = 2*x_1 POL(a) = 0 POL(b) = 0 POL(f3_in(x_1)) = 1 + 2*x_1 POL(f3_out1) = 0 POL(f84_in) = 0 POL(f84_out1(x_1)) = x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: f3_in(b) -> f3_out1 f3_in(a) -> U1(f84_in, a) ---------------------------------------- (24) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: U1(f84_out1(X2), a) -> f3_out1 f84_in -> f84_out1(b) f84_in -> U2(f84_in) U2(f84_out1(X4)) -> f84_out1(a) Q is empty. ---------------------------------------- (25) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U1(x_1, x_2)) = 2 + x_1 + 2*x_2 POL(U2(x_1)) = 2*x_1 POL(a) = 0 POL(b) = 0 POL(f3_out1) = 0 POL(f84_in) = 0 POL(f84_out1(x_1)) = 2*x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U1(f84_out1(X2), a) -> f3_out1 ---------------------------------------- (26) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f84_in -> f84_out1(b) f84_in -> U2(f84_in) U2(f84_out1(X4)) -> f84_out1(a) Q is empty. ---------------------------------------- (27) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U2(x_1)) = x_1 POL(a) = 0 POL(b) = 0 POL(f84_in) = 1 POL(f84_out1(x_1)) = 2*x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: f84_in -> f84_out1(b) ---------------------------------------- (28) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f84_in -> U2(f84_in) U2(f84_out1(X4)) -> f84_out1(a) Q is empty. ---------------------------------------- (29) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U2(x_1)) = 2*x_1 POL(a) = 0 POL(f84_in) = 0 POL(f84_out1(x_1)) = 1 + 2*x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U2(f84_out1(X4)) -> f84_out1(a) ---------------------------------------- (30) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f84_in -> U2(f84_in) Q is empty. ---------------------------------------- (31) Overlay + Local Confluence (EQUIVALENT) The TRS is overlay and locally confluent. By [NOC] we can switch to innermost. ---------------------------------------- (32) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f84_in -> U2(f84_in) The set Q consists of the following terms: f84_in ---------------------------------------- (33) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (34) Obligation: Q DP problem: The TRS P consists of the following rules: F84_IN -> F84_IN The TRS R consists of the following rules: f84_in -> U2(f84_in) The set Q consists of the following terms: f84_in We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (35) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (36) Obligation: Q DP problem: The TRS P consists of the following rules: F84_IN -> F84_IN R is empty. The set Q consists of the following terms: f84_in We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (37) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. f84_in ---------------------------------------- (38) Obligation: Q DP problem: The TRS P consists of the following rules: F84_IN -> F84_IN R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (39) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 2, "program": { "directives": [], "clauses": [ [ "(p (b))", null ], [ "(p (a))", "(p1 X)" ], [ "(p1 (b))", null ], [ "(p1 (a))", "(p1 X)" ] ] }, "graph": { "nodes": { "88": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "89": { "goal": [{ "clause": -1, "scope": -1, "term": "(p1 X2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X2"], "exprvars": [] } }, "26": { "goal": [{ "clause": 0, "scope": 1, "term": "(p T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "27": { "goal": [{ "clause": 1, "scope": 1, "term": "(p T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "19": { "goal": [ { "clause": 0, "scope": 1, "term": "(p T1)" }, { "clause": 1, "scope": 1, "term": "(p T1)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "type": "Nodes", "100": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "2": { "goal": [{ "clause": -1, "scope": -1, "term": "(p T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "102": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "104": { "goal": [{ "clause": -1, "scope": -1, "term": "(p1 X4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X4"], "exprvars": [] } }, "90": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "94": { "goal": [ { "clause": 2, "scope": 2, "term": "(p1 X2)" }, { "clause": 3, "scope": 2, "term": "(p1 X2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X2"], "exprvars": [] } }, "86": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "97": { "goal": [{ "clause": 2, "scope": 2, "term": "(p1 X2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X2"], "exprvars": [] } }, "87": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "98": { "goal": [{ "clause": 3, "scope": 2, "term": "(p1 X2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X2"], "exprvars": [] } } }, "edges": [ { "from": 2, "to": 19, "label": "CASE" }, { "from": 19, "to": 26, "label": "PARALLEL" }, { "from": 19, "to": 27, "label": "PARALLEL" }, { "from": 26, "to": 86, "label": "EVAL with clause\np(b).\nand substitutionT1 -> b" }, { "from": 26, "to": 87, "label": "EVAL-BACKTRACK" }, { "from": 27, "to": 89, "label": "EVAL with clause\np(a) :- p1(X2).\nand substitutionT1 -> a" }, { "from": 27, "to": 90, "label": "EVAL-BACKTRACK" }, { "from": 86, "to": 88, "label": "SUCCESS" }, { "from": 89, "to": 94, "label": "CASE" }, { "from": 94, "to": 97, "label": "PARALLEL" }, { "from": 94, "to": 98, "label": "PARALLEL" }, { "from": 97, "to": 100, "label": "ONLY EVAL with clause\np1(b).\nand substitutionX2 -> b" }, { "from": 98, "to": 104, "label": "ONLY EVAL with clause\np1(a) :- p1(X4).\nand substitutionX2 -> a" }, { "from": 100, "to": 102, "label": "SUCCESS" }, { "from": 104, "to": 89, "label": "INSTANCE with matching:\nX2 -> X4" } ], "type": "Graph" } } ---------------------------------------- (40) Obligation: Rules: f89_in -> f94_in :|: TRUE f94_out -> f89_out :|: TRUE f94_in -> f98_in :|: TRUE f94_in -> f97_in :|: TRUE f98_out -> f94_out :|: TRUE f97_out -> f94_out :|: TRUE f89_out -> f104_out :|: TRUE f104_in -> f89_in :|: TRUE f98_in -> f104_in :|: TRUE f104_out -> f98_out :|: TRUE f19_out(T1) -> f2_out(T1) :|: TRUE f2_in(x) -> f19_in(x) :|: TRUE f27_out(x1) -> f19_out(x1) :|: TRUE f26_out(x2) -> f19_out(x2) :|: TRUE f19_in(x3) -> f26_in(x3) :|: TRUE f19_in(x4) -> f27_in(x4) :|: TRUE f27_in(a) -> f89_in :|: TRUE f27_in(x5) -> f90_in :|: TRUE f90_out -> f27_out(x6) :|: TRUE f89_out -> f27_out(a) :|: TRUE Start term: f2_in(T1) ---------------------------------------- (41) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f89_in -> f94_in :|: TRUE f94_in -> f98_in :|: TRUE f104_in -> f89_in :|: TRUE f98_in -> f104_in :|: TRUE ---------------------------------------- (42) Obligation: Rules: f89_in -> f94_in :|: TRUE f94_in -> f98_in :|: TRUE f104_in -> f89_in :|: TRUE f98_in -> f104_in :|: TRUE ---------------------------------------- (43) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (44) Obligation: Rules: f104_in -> f104_in :|: TRUE ---------------------------------------- (45) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (46) Obligation: Rules: f104_in -> f104_in :|: TRUE ---------------------------------------- (47) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f104_in -> f104_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (48) Obligation: Termination digraph: Nodes: (1) f104_in -> f104_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (49) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f104_in() Replaced non-predefined constructor symbols by 0. ---------------------------------------- (50) Obligation: Rules: f104_in -> f104_in :|: TRUE ---------------------------------------- (51) IntTRSPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc) -> f(1) :|: pc = 1 && TRUE Witness term starting non-terminating reduction: f(1) ---------------------------------------- (52) NO ---------------------------------------- (53) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 1, "program": { "directives": [], "clauses": [ [ "(p (b))", null ], [ "(p (a))", "(p1 X)" ], [ "(p1 (b))", null ], [ "(p1 (a))", "(p1 X)" ] ] }, "graph": { "nodes": { "44": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "49": { "goal": [{ "clause": -1, "scope": -1, "term": "(p1 X1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X1"], "exprvars": [] } }, "39": { "goal": [ { "clause": -1, "scope": -1, "term": "(true)" }, { "clause": 1, "scope": 1, "term": "(p (b))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "1": { "goal": [{ "clause": -1, "scope": -1, "term": "(p T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "6": { "goal": [ { "clause": 0, "scope": 1, "term": "(p T1)" }, { "clause": 1, "scope": 1, "term": "(p T1)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "91": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "92": { "goal": [{ "clause": -1, "scope": -1, "term": "(p1 X3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X3"], "exprvars": [] } }, "50": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "40": { "goal": [{ "clause": 1, "scope": 1, "term": "(p T1)" }], "kb": { "nonunifying": [[ "(p T1)", "(p (b))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "51": { "goal": [ { "clause": 2, "scope": 2, "term": "(p1 X1)" }, { "clause": 3, "scope": 2, "term": "(p1 X1)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X1"], "exprvars": [] } }, "52": { "goal": [{ "clause": 2, "scope": 2, "term": "(p1 X1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X1"], "exprvars": [] } }, "42": { "goal": [{ "clause": 1, "scope": 1, "term": "(p (b))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "53": { "goal": [{ "clause": 3, "scope": 2, "term": "(p1 X1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X1"], "exprvars": [] } }, "54": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 6, "label": "CASE" }, { "from": 6, "to": 39, "label": "EVAL with clause\np(b).\nand substitutionT1 -> b" }, { "from": 6, "to": 40, "label": "EVAL-BACKTRACK" }, { "from": 39, "to": 42, "label": "SUCCESS" }, { "from": 40, "to": 49, "label": "EVAL with clause\np(a) :- p1(X1).\nand substitutionT1 -> a" }, { "from": 40, "to": 50, "label": "EVAL-BACKTRACK" }, { "from": 42, "to": 44, "label": "BACKTRACK\nfor clause: p(a) :- p1(X)because of non-unification" }, { "from": 49, "to": 51, "label": "CASE" }, { "from": 51, "to": 52, "label": "PARALLEL" }, { "from": 51, "to": 53, "label": "PARALLEL" }, { "from": 52, "to": 54, "label": "ONLY EVAL with clause\np1(b).\nand substitutionX1 -> b" }, { "from": 53, "to": 92, "label": "ONLY EVAL with clause\np1(a) :- p1(X3).\nand substitutionX1 -> a" }, { "from": 54, "to": 91, "label": "SUCCESS" }, { "from": 92, "to": 49, "label": "INSTANCE with matching:\nX1 -> X3" } ], "type": "Graph" } } ---------------------------------------- (54) Obligation: Triples: p1A(a) :- p1A(X1). pB(a) :- p1A(X1). Clauses: p1cA(b). p1cA(a) :- p1cA(X1). Afs: pB(x1) = pB(x1) ---------------------------------------- (55) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: pB_in_1: (b) p1A_in_1: (f) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: PB_IN_G(a) -> U2_G(p1A_in_a(X1)) PB_IN_G(a) -> P1A_IN_A(X1) P1A_IN_A(a) -> U1_A(p1A_in_a(X1)) P1A_IN_A(a) -> P1A_IN_A(X1) R is empty. The argument filtering Pi contains the following mapping: a = a p1A_in_a(x1) = p1A_in_a PB_IN_G(x1) = PB_IN_G(x1) U2_G(x1) = U2_G(x1) P1A_IN_A(x1) = P1A_IN_A U1_A(x1) = U1_A(x1) We have to consider all (P,R,Pi)-chains Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES ---------------------------------------- (56) Obligation: Pi DP problem: The TRS P consists of the following rules: PB_IN_G(a) -> U2_G(p1A_in_a(X1)) PB_IN_G(a) -> P1A_IN_A(X1) P1A_IN_A(a) -> U1_A(p1A_in_a(X1)) P1A_IN_A(a) -> P1A_IN_A(X1) R is empty. The argument filtering Pi contains the following mapping: a = a p1A_in_a(x1) = p1A_in_a PB_IN_G(x1) = PB_IN_G(x1) U2_G(x1) = U2_G(x1) P1A_IN_A(x1) = P1A_IN_A U1_A(x1) = U1_A(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (57) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes. ---------------------------------------- (58) Obligation: Pi DP problem: The TRS P consists of the following rules: P1A_IN_A(a) -> P1A_IN_A(X1) R is empty. The argument filtering Pi contains the following mapping: a = a P1A_IN_A(x1) = P1A_IN_A We have to consider all (P,R,Pi)-chains ---------------------------------------- (59) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (60) Obligation: Q DP problem: The TRS P consists of the following rules: P1A_IN_A -> P1A_IN_A R is empty. Q is empty. We have to consider all (P,Q,R)-chains.