/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern q(a) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) CutEliminatorProof [SOUND, 0 ms] (2) Prolog (3) PrologToPiTRSProof [SOUND, 0 ms] (4) PiTRS (5) DependencyPairsProof [EQUIVALENT, 0 ms] (6) PiDP (7) DependencyGraphProof [EQUIVALENT, 0 ms] (8) PiDP (9) UsableRulesProof [EQUIVALENT, 0 ms] (10) PiDP (11) PiDPToQDPProof [SOUND, 0 ms] (12) QDP (13) PrologToPiTRSProof [SOUND, 0 ms] (14) PiTRS (15) DependencyPairsProof [EQUIVALENT, 0 ms] (16) PiDP (17) DependencyGraphProof [EQUIVALENT, 0 ms] (18) PiDP (19) UsableRulesProof [EQUIVALENT, 0 ms] (20) PiDP (21) PiDPToQDPProof [SOUND, 0 ms] (22) QDP (23) PrologToTRSTransformerProof [SOUND, 0 ms] (24) QTRS (25) QTRSRRRProof [EQUIVALENT, 61 ms] (26) QTRS (27) QTRSRRRProof [EQUIVALENT, 13 ms] (28) QTRS (29) QTRSRRRProof [EQUIVALENT, 0 ms] (30) QTRS (31) QTRSRRRProof [EQUIVALENT, 0 ms] (32) QTRS (33) QTRSRRRProof [EQUIVALENT, 5 ms] (34) QTRS (35) QTRSRRRProof [EQUIVALENT, 5 ms] (36) QTRS (37) QTRSRRRProof [EQUIVALENT, 0 ms] (38) QTRS (39) PrologToIRSwTTransformerProof [SOUND, 0 ms] (40) IRSwT (41) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (42) IRSwT (43) IntTRSCompressionProof [EQUIVALENT, 9 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, 1 ms] (52) NO (53) PrologToDTProblemTransformerProof [SOUND, 0 ms] (54) TRIPLES (55) TriplesToPiDPProof [SOUND, 6 ms] (56) PiDP (57) DependencyGraphProof [EQUIVALENT, 0 ms] (58) PiDP (59) PiDPToQDPProof [SOUND, 0 ms] (60) QDP ---------------------------------------- (0) Obligation: Clauses: q(X) :- ','(p(X), !). p(0). p(s(X)) :- p(X). Query: q(a) ---------------------------------------- (1) CutEliminatorProof (SOUND) Eliminated all cuts by simply ignoring them[PROLOG]. ---------------------------------------- (2) Obligation: Clauses: q(X) :- p(X). p(0). p(s(X)) :- p(X). Query: q(a) ---------------------------------------- (3) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: q_in_1: (f) p_in_1: (f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: q_in_a(X) -> U1_a(X, p_in_a(X)) p_in_a(0) -> p_out_a(0) p_in_a(s(X)) -> U2_a(X, p_in_a(X)) U2_a(X, p_out_a(X)) -> p_out_a(s(X)) U1_a(X, p_out_a(X)) -> q_out_a(X) The argument filtering Pi contains the following mapping: q_in_a(x1) = q_in_a U1_a(x1, x2) = U1_a(x2) p_in_a(x1) = p_in_a p_out_a(x1) = p_out_a(x1) U2_a(x1, x2) = U2_a(x2) q_out_a(x1) = q_out_a(x1) s(x1) = s(x1) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (4) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: q_in_a(X) -> U1_a(X, p_in_a(X)) p_in_a(0) -> p_out_a(0) p_in_a(s(X)) -> U2_a(X, p_in_a(X)) U2_a(X, p_out_a(X)) -> p_out_a(s(X)) U1_a(X, p_out_a(X)) -> q_out_a(X) The argument filtering Pi contains the following mapping: q_in_a(x1) = q_in_a U1_a(x1, x2) = U1_a(x2) p_in_a(x1) = p_in_a p_out_a(x1) = p_out_a(x1) U2_a(x1, x2) = U2_a(x2) q_out_a(x1) = q_out_a(x1) s(x1) = s(x1) ---------------------------------------- (5) 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: Q_IN_A(X) -> U1_A(X, p_in_a(X)) Q_IN_A(X) -> P_IN_A(X) P_IN_A(s(X)) -> U2_A(X, p_in_a(X)) P_IN_A(s(X)) -> P_IN_A(X) The TRS R consists of the following rules: q_in_a(X) -> U1_a(X, p_in_a(X)) p_in_a(0) -> p_out_a(0) p_in_a(s(X)) -> U2_a(X, p_in_a(X)) U2_a(X, p_out_a(X)) -> p_out_a(s(X)) U1_a(X, p_out_a(X)) -> q_out_a(X) The argument filtering Pi contains the following mapping: q_in_a(x1) = q_in_a U1_a(x1, x2) = U1_a(x2) p_in_a(x1) = p_in_a p_out_a(x1) = p_out_a(x1) U2_a(x1, x2) = U2_a(x2) q_out_a(x1) = q_out_a(x1) s(x1) = s(x1) Q_IN_A(x1) = Q_IN_A U1_A(x1, x2) = U1_A(x2) P_IN_A(x1) = P_IN_A U2_A(x1, x2) = U2_A(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (6) Obligation: Pi DP problem: The TRS P consists of the following rules: Q_IN_A(X) -> U1_A(X, p_in_a(X)) Q_IN_A(X) -> P_IN_A(X) P_IN_A(s(X)) -> U2_A(X, p_in_a(X)) P_IN_A(s(X)) -> P_IN_A(X) The TRS R consists of the following rules: q_in_a(X) -> U1_a(X, p_in_a(X)) p_in_a(0) -> p_out_a(0) p_in_a(s(X)) -> U2_a(X, p_in_a(X)) U2_a(X, p_out_a(X)) -> p_out_a(s(X)) U1_a(X, p_out_a(X)) -> q_out_a(X) The argument filtering Pi contains the following mapping: q_in_a(x1) = q_in_a U1_a(x1, x2) = U1_a(x2) p_in_a(x1) = p_in_a p_out_a(x1) = p_out_a(x1) U2_a(x1, x2) = U2_a(x2) q_out_a(x1) = q_out_a(x1) s(x1) = s(x1) Q_IN_A(x1) = Q_IN_A U1_A(x1, x2) = U1_A(x2) P_IN_A(x1) = P_IN_A U2_A(x1, x2) = U2_A(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (7) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes. ---------------------------------------- (8) Obligation: Pi DP problem: The TRS P consists of the following rules: P_IN_A(s(X)) -> P_IN_A(X) The TRS R consists of the following rules: q_in_a(X) -> U1_a(X, p_in_a(X)) p_in_a(0) -> p_out_a(0) p_in_a(s(X)) -> U2_a(X, p_in_a(X)) U2_a(X, p_out_a(X)) -> p_out_a(s(X)) U1_a(X, p_out_a(X)) -> q_out_a(X) The argument filtering Pi contains the following mapping: q_in_a(x1) = q_in_a U1_a(x1, x2) = U1_a(x2) p_in_a(x1) = p_in_a p_out_a(x1) = p_out_a(x1) U2_a(x1, x2) = U2_a(x2) q_out_a(x1) = q_out_a(x1) s(x1) = s(x1) P_IN_A(x1) = P_IN_A We have to consider all (P,R,Pi)-chains ---------------------------------------- (9) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (10) Obligation: Pi DP problem: The TRS P consists of the following rules: P_IN_A(s(X)) -> P_IN_A(X) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) P_IN_A(x1) = P_IN_A We have to consider all (P,R,Pi)-chains ---------------------------------------- (11) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: P_IN_A -> P_IN_A R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (13) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: q_in_1: (f) p_in_1: (f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: q_in_a(X) -> U1_a(X, p_in_a(X)) p_in_a(0) -> p_out_a(0) p_in_a(s(X)) -> U2_a(X, p_in_a(X)) U2_a(X, p_out_a(X)) -> p_out_a(s(X)) U1_a(X, p_out_a(X)) -> q_out_a(X) The argument filtering Pi contains the following mapping: q_in_a(x1) = q_in_a U1_a(x1, x2) = U1_a(x2) p_in_a(x1) = p_in_a p_out_a(x1) = p_out_a(x1) U2_a(x1, x2) = U2_a(x2) q_out_a(x1) = q_out_a(x1) s(x1) = s(x1) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (14) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: q_in_a(X) -> U1_a(X, p_in_a(X)) p_in_a(0) -> p_out_a(0) p_in_a(s(X)) -> U2_a(X, p_in_a(X)) U2_a(X, p_out_a(X)) -> p_out_a(s(X)) U1_a(X, p_out_a(X)) -> q_out_a(X) The argument filtering Pi contains the following mapping: q_in_a(x1) = q_in_a U1_a(x1, x2) = U1_a(x2) p_in_a(x1) = p_in_a p_out_a(x1) = p_out_a(x1) U2_a(x1, x2) = U2_a(x2) q_out_a(x1) = q_out_a(x1) s(x1) = s(x1) ---------------------------------------- (15) 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: Q_IN_A(X) -> U1_A(X, p_in_a(X)) Q_IN_A(X) -> P_IN_A(X) P_IN_A(s(X)) -> U2_A(X, p_in_a(X)) P_IN_A(s(X)) -> P_IN_A(X) The TRS R consists of the following rules: q_in_a(X) -> U1_a(X, p_in_a(X)) p_in_a(0) -> p_out_a(0) p_in_a(s(X)) -> U2_a(X, p_in_a(X)) U2_a(X, p_out_a(X)) -> p_out_a(s(X)) U1_a(X, p_out_a(X)) -> q_out_a(X) The argument filtering Pi contains the following mapping: q_in_a(x1) = q_in_a U1_a(x1, x2) = U1_a(x2) p_in_a(x1) = p_in_a p_out_a(x1) = p_out_a(x1) U2_a(x1, x2) = U2_a(x2) q_out_a(x1) = q_out_a(x1) s(x1) = s(x1) Q_IN_A(x1) = Q_IN_A U1_A(x1, x2) = U1_A(x2) P_IN_A(x1) = P_IN_A U2_A(x1, x2) = U2_A(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: Q_IN_A(X) -> U1_A(X, p_in_a(X)) Q_IN_A(X) -> P_IN_A(X) P_IN_A(s(X)) -> U2_A(X, p_in_a(X)) P_IN_A(s(X)) -> P_IN_A(X) The TRS R consists of the following rules: q_in_a(X) -> U1_a(X, p_in_a(X)) p_in_a(0) -> p_out_a(0) p_in_a(s(X)) -> U2_a(X, p_in_a(X)) U2_a(X, p_out_a(X)) -> p_out_a(s(X)) U1_a(X, p_out_a(X)) -> q_out_a(X) The argument filtering Pi contains the following mapping: q_in_a(x1) = q_in_a U1_a(x1, x2) = U1_a(x2) p_in_a(x1) = p_in_a p_out_a(x1) = p_out_a(x1) U2_a(x1, x2) = U2_a(x2) q_out_a(x1) = q_out_a(x1) s(x1) = s(x1) Q_IN_A(x1) = Q_IN_A U1_A(x1, x2) = U1_A(x2) P_IN_A(x1) = P_IN_A U2_A(x1, x2) = U2_A(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes. ---------------------------------------- (18) Obligation: Pi DP problem: The TRS P consists of the following rules: P_IN_A(s(X)) -> P_IN_A(X) The TRS R consists of the following rules: q_in_a(X) -> U1_a(X, p_in_a(X)) p_in_a(0) -> p_out_a(0) p_in_a(s(X)) -> U2_a(X, p_in_a(X)) U2_a(X, p_out_a(X)) -> p_out_a(s(X)) U1_a(X, p_out_a(X)) -> q_out_a(X) The argument filtering Pi contains the following mapping: q_in_a(x1) = q_in_a U1_a(x1, x2) = U1_a(x2) p_in_a(x1) = p_in_a p_out_a(x1) = p_out_a(x1) U2_a(x1, x2) = U2_a(x2) q_out_a(x1) = q_out_a(x1) s(x1) = s(x1) P_IN_A(x1) = P_IN_A We have to consider all (P,R,Pi)-chains ---------------------------------------- (19) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (20) Obligation: Pi DP problem: The TRS P consists of the following rules: P_IN_A(s(X)) -> P_IN_A(X) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) P_IN_A(x1) = P_IN_A We have to consider all (P,R,Pi)-chains ---------------------------------------- (21) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: P_IN_A -> P_IN_A R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (23) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 11, "program": { "directives": [], "clauses": [ [ "(q X)", "(',' (p X) (!))" ], [ "(p (0))", null ], [ "(p (s X))", "(p X)" ] ] }, "graph": { "nodes": { "11": { "goal": [{ "clause": -1, "scope": -1, "term": "(q T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "77": { "goal": [ { "clause": 1, "scope": 2, "term": "(p T6)" }, { "clause": 2, "scope": 2, "term": "(p T6)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "12": { "goal": [{ "clause": 0, "scope": 1, "term": "(q T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "78": { "goal": [{ "clause": 1, "scope": 2, "term": "(p T6)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "79": { "goal": [{ "clause": 2, "scope": 2, "term": "(p T6)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "80": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "81": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "82": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "72": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (p T6) (!_1))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "83": { "goal": [{ "clause": -1, "scope": -1, "term": "(p T10)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "84": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "74": { "goal": [{ "clause": -1, "scope": -1, "term": "(p T6)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "85": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "75": { "goal": [{ "clause": -1, "scope": -1, "term": "(!_1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "86": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 11, "to": 12, "label": "CASE" }, { "from": 12, "to": 72, "label": "ONLY EVAL with clause\nq(X3) :- ','(p(X3), !_1).\nand substitutionT1 -> T6,\nX3 -> T6,\nT5 -> T6" }, { "from": 72, "to": 74, "label": "SPLIT 1" }, { "from": 72, "to": 75, "label": "SPLIT 2\nnew knowledge:\nT6 is ground" }, { "from": 74, "to": 77, "label": "CASE" }, { "from": 75, "to": 85, "label": "CUT" }, { "from": 77, "to": 78, "label": "PARALLEL" }, { "from": 77, "to": 79, "label": "PARALLEL" }, { "from": 78, "to": 80, "label": "EVAL with clause\np(0).\nand substitutionT6 -> 0" }, { "from": 78, "to": 81, "label": "EVAL-BACKTRACK" }, { "from": 79, "to": 83, "label": "EVAL with clause\np(s(X6)) :- p(X6).\nand substitutionX6 -> T10,\nT6 -> s(T10),\nT9 -> T10" }, { "from": 79, "to": 84, "label": "EVAL-BACKTRACK" }, { "from": 80, "to": 82, "label": "SUCCESS" }, { "from": 83, "to": 74, "label": "INSTANCE with matching:\nT6 -> T10" }, { "from": 85, "to": 86, "label": "SUCCESS" } ], "type": "Graph" } } ---------------------------------------- (24) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f11_in -> U1(f72_in) U1(f72_out1(T6)) -> f11_out1(T6) f74_in -> f74_out1(0) f74_in -> U2(f74_in) U2(f74_out1(T10)) -> f74_out1(s(T10)) f75_in -> f75_out1 f72_in -> U3(f74_in) U3(f74_out1(T6)) -> U4(f75_in, T6) U4(f75_out1, T6) -> f72_out1(T6) Q is empty. ---------------------------------------- (25) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(U1(x_1)) = 2*x_1 POL(U2(x_1)) = 2*x_1 POL(U3(x_1)) = 2*x_1 POL(U4(x_1, x_2)) = x_1 + 2*x_2 POL(f11_in) = 1 POL(f11_out1(x_1)) = 2*x_1 POL(f72_in) = 0 POL(f72_out1(x_1)) = x_1 POL(f74_in) = 0 POL(f74_out1(x_1)) = x_1 POL(f75_in) = 0 POL(f75_out1) = 0 POL(s(x_1)) = x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: f11_in -> U1(f72_in) ---------------------------------------- (26) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: U1(f72_out1(T6)) -> f11_out1(T6) f74_in -> f74_out1(0) f74_in -> U2(f74_in) U2(f74_out1(T10)) -> f74_out1(s(T10)) f75_in -> f75_out1 f72_in -> U3(f74_in) U3(f74_out1(T6)) -> U4(f75_in, T6) U4(f75_out1, T6) -> f72_out1(T6) Q is empty. ---------------------------------------- (27) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(U1(x_1)) = x_1 POL(U2(x_1)) = 2*x_1 POL(U3(x_1)) = x_1 POL(U4(x_1, x_2)) = x_1 + 2*x_2 POL(f11_out1(x_1)) = 2*x_1 POL(f72_in) = 2 POL(f72_out1(x_1)) = 2*x_1 POL(f74_in) = 0 POL(f74_out1(x_1)) = 2*x_1 POL(f75_in) = 0 POL(f75_out1) = 0 POL(s(x_1)) = x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: f72_in -> U3(f74_in) ---------------------------------------- (28) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: U1(f72_out1(T6)) -> f11_out1(T6) f74_in -> f74_out1(0) f74_in -> U2(f74_in) U2(f74_out1(T10)) -> f74_out1(s(T10)) f75_in -> f75_out1 U3(f74_out1(T6)) -> U4(f75_in, T6) U4(f75_out1, T6) -> f72_out1(T6) Q is empty. ---------------------------------------- (29) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(U1(x_1)) = 2 + 2*x_1 POL(U2(x_1)) = 2*x_1 POL(U3(x_1)) = x_1 POL(U4(x_1, x_2)) = x_1 + x_2 POL(f11_out1(x_1)) = 2*x_1 POL(f72_out1(x_1)) = x_1 POL(f74_in) = 0 POL(f74_out1(x_1)) = x_1 POL(f75_in) = 0 POL(f75_out1) = 0 POL(s(x_1)) = x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U1(f72_out1(T6)) -> f11_out1(T6) ---------------------------------------- (30) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f74_in -> f74_out1(0) f74_in -> U2(f74_in) U2(f74_out1(T10)) -> f74_out1(s(T10)) f75_in -> f75_out1 U3(f74_out1(T6)) -> U4(f75_in, T6) U4(f75_out1, T6) -> f72_out1(T6) Q is empty. ---------------------------------------- (31) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(U2(x_1)) = x_1 POL(U3(x_1)) = 1 + 2*x_1 POL(U4(x_1, x_2)) = 2 + x_1 + 2*x_2 POL(f72_out1(x_1)) = 1 + 2*x_1 POL(f74_in) = 2 POL(f74_out1(x_1)) = 2 + 2*x_1 POL(f75_in) = 0 POL(f75_out1) = 0 POL(s(x_1)) = x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U3(f74_out1(T6)) -> U4(f75_in, T6) U4(f75_out1, T6) -> f72_out1(T6) ---------------------------------------- (32) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f74_in -> f74_out1(0) f74_in -> U2(f74_in) U2(f74_out1(T10)) -> f74_out1(s(T10)) f75_in -> f75_out1 Q is empty. ---------------------------------------- (33) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(U2(x_1)) = 2*x_1 POL(f74_in) = 0 POL(f74_out1(x_1)) = 2*x_1 POL(f75_in) = 1 POL(f75_out1) = 0 POL(s(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: f75_in -> f75_out1 ---------------------------------------- (34) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f74_in -> f74_out1(0) f74_in -> U2(f74_in) U2(f74_out1(T10)) -> f74_out1(s(T10)) Q is empty. ---------------------------------------- (35) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(U2(x_1)) = x_1 POL(f74_in) = 1 POL(f74_out1(x_1)) = 2*x_1 POL(s(x_1)) = x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: f74_in -> f74_out1(0) ---------------------------------------- (36) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f74_in -> U2(f74_in) U2(f74_out1(T10)) -> f74_out1(s(T10)) Q is empty. ---------------------------------------- (37) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U2(x_1)) = 2*x_1 POL(f74_in) = 0 POL(f74_out1(x_1)) = 2 + x_1 POL(s(x_1)) = x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U2(f74_out1(T10)) -> f74_out1(s(T10)) ---------------------------------------- (38) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f74_in -> U2(f74_in) Q is empty. ---------------------------------------- (39) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 9, "program": { "directives": [], "clauses": [ [ "(q X)", "(',' (p X) (!))" ], [ "(p (0))", null ], [ "(p (s X))", "(p X)" ] ] }, "graph": { "nodes": { "88": { "goal": [{ "clause": 1, "scope": 2, "term": "(p T6)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "89": { "goal": [{ "clause": 2, "scope": 2, "term": "(p T6)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "90": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "91": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "9": { "goal": [{ "clause": -1, "scope": -1, "term": "(q T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "92": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "71": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (p T6) (!_1))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "93": { "goal": [{ "clause": -1, "scope": -1, "term": "(p T10)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "94": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "73": { "goal": [{ "clause": -1, "scope": -1, "term": "(p T6)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "95": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "96": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "10": { "goal": [{ "clause": 0, "scope": 1, "term": "(q T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "76": { "goal": [{ "clause": -1, "scope": -1, "term": "(!_1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "87": { "goal": [ { "clause": 1, "scope": 2, "term": "(p T6)" }, { "clause": 2, "scope": 2, "term": "(p T6)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 9, "to": 10, "label": "CASE" }, { "from": 10, "to": 71, "label": "ONLY EVAL with clause\nq(X3) :- ','(p(X3), !_1).\nand substitutionT1 -> T6,\nX3 -> T6,\nT5 -> T6" }, { "from": 71, "to": 73, "label": "SPLIT 1" }, { "from": 71, "to": 76, "label": "SPLIT 2\nnew knowledge:\nT6 is ground" }, { "from": 73, "to": 87, "label": "CASE" }, { "from": 76, "to": 95, "label": "CUT" }, { "from": 87, "to": 88, "label": "PARALLEL" }, { "from": 87, "to": 89, "label": "PARALLEL" }, { "from": 88, "to": 90, "label": "EVAL with clause\np(0).\nand substitutionT6 -> 0" }, { "from": 88, "to": 91, "label": "EVAL-BACKTRACK" }, { "from": 89, "to": 93, "label": "EVAL with clause\np(s(X6)) :- p(X6).\nand substitutionX6 -> T10,\nT6 -> s(T10),\nT9 -> T10" }, { "from": 89, "to": 94, "label": "EVAL-BACKTRACK" }, { "from": 90, "to": 92, "label": "SUCCESS" }, { "from": 93, "to": 73, "label": "INSTANCE with matching:\nT6 -> T10" }, { "from": 95, "to": 96, "label": "SUCCESS" } ], "type": "Graph" } } ---------------------------------------- (40) Obligation: Rules: f87_in -> f88_in :|: TRUE f89_out -> f87_out :|: TRUE f88_out -> f87_out :|: TRUE f87_in -> f89_in :|: TRUE f73_out -> f93_out :|: TRUE f93_in -> f73_in :|: TRUE f87_out -> f73_out :|: TRUE f73_in -> f87_in :|: TRUE f89_in -> f93_in :|: TRUE f89_in -> f94_in :|: TRUE f94_out -> f89_out :|: TRUE f93_out -> f89_out :|: TRUE f10_out -> f9_out :|: TRUE f9_in -> f10_in :|: TRUE f71_out -> f10_out :|: TRUE f10_in -> f71_in :|: TRUE f71_in -> f73_in :|: TRUE f76_out -> f71_out :|: TRUE f73_out -> f76_in :|: TRUE Start term: f9_in ---------------------------------------- (41) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f87_in -> f89_in :|: TRUE f93_in -> f73_in :|: TRUE f73_in -> f87_in :|: TRUE f89_in -> f93_in :|: TRUE ---------------------------------------- (42) Obligation: Rules: f87_in -> f89_in :|: TRUE f93_in -> f73_in :|: TRUE f73_in -> f87_in :|: TRUE f89_in -> f93_in :|: TRUE ---------------------------------------- (43) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (44) Obligation: Rules: f93_in -> f93_in :|: TRUE ---------------------------------------- (45) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (46) Obligation: Rules: f93_in -> f93_in :|: TRUE ---------------------------------------- (47) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f93_in -> f93_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (48) Obligation: Termination digraph: Nodes: (1) f93_in -> f93_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (49) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f93_in() Replaced non-predefined constructor symbols by 0. ---------------------------------------- (50) Obligation: Rules: f93_in -> f93_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": [ [ "(q X)", "(',' (p X) (!))" ], [ "(p (0))", null ], [ "(p (s X))", "(p X)" ] ] }, "graph": { "nodes": { "1": { "goal": [{ "clause": -1, "scope": -1, "term": "(q T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "2": { "goal": [{ "clause": 0, "scope": 1, "term": "(q T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "57": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "25": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (p T4) (!_1))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "58": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "60": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (p T8) (!_1))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "61": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "40": { "goal": [ { "clause": 1, "scope": 2, "term": "(',' (p T4) (!_1))" }, { "clause": 2, "scope": 2, "term": "(',' (p T4) (!_1))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "41": { "goal": [ { "clause": -1, "scope": -1, "term": "(!_1)" }, { "clause": 2, "scope": 2, "term": "(',' (p T4) (!_1))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "42": { "goal": [{ "clause": 2, "scope": 2, "term": "(',' (p T4) (!_1))" }], "kb": { "nonunifying": [[ "(p T4)", "(p (0))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 2, "label": "CASE" }, { "from": 2, "to": 25, "label": "ONLY EVAL with clause\nq(X2) :- ','(p(X2), !_1).\nand substitutionT1 -> T4,\nX2 -> T4,\nT3 -> T4" }, { "from": 25, "to": 40, "label": "CASE" }, { "from": 40, "to": 41, "label": "EVAL with clause\np(0).\nand substitutionT4 -> 0" }, { "from": 40, "to": 42, "label": "EVAL-BACKTRACK" }, { "from": 41, "to": 57, "label": "CUT" }, { "from": 42, "to": 60, "label": "EVAL with clause\np(s(X5)) :- p(X5).\nand substitutionX5 -> T8,\nT4 -> s(T8),\nT7 -> T8" }, { "from": 42, "to": 61, "label": "EVAL-BACKTRACK" }, { "from": 57, "to": 58, "label": "SUCCESS" }, { "from": 60, "to": 25, "label": "INSTANCE with matching:\nT4 -> T8" } ], "type": "Graph" } } ---------------------------------------- (54) Obligation: Triples: pA(s(X1)) :- pA(X1). qB(X1) :- pA(X1). Clauses: qcA(0). qcA(s(X1)) :- qcA(X1). Afs: qB(x1) = qB ---------------------------------------- (55) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: qB_in_1: (f) pA_in_1: (f) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: QB_IN_A(X1) -> U2_A(X1, pA_in_a(X1)) QB_IN_A(X1) -> PA_IN_A(X1) PA_IN_A(s(X1)) -> U1_A(X1, pA_in_a(X1)) PA_IN_A(s(X1)) -> PA_IN_A(X1) R is empty. The argument filtering Pi contains the following mapping: pA_in_a(x1) = pA_in_a s(x1) = s(x1) QB_IN_A(x1) = QB_IN_A U2_A(x1, x2) = U2_A(x2) PA_IN_A(x1) = PA_IN_A U1_A(x1, x2) = U1_A(x2) 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: QB_IN_A(X1) -> U2_A(X1, pA_in_a(X1)) QB_IN_A(X1) -> PA_IN_A(X1) PA_IN_A(s(X1)) -> U1_A(X1, pA_in_a(X1)) PA_IN_A(s(X1)) -> PA_IN_A(X1) R is empty. The argument filtering Pi contains the following mapping: pA_in_a(x1) = pA_in_a s(x1) = s(x1) QB_IN_A(x1) = QB_IN_A U2_A(x1, x2) = U2_A(x2) PA_IN_A(x1) = PA_IN_A U1_A(x1, x2) = U1_A(x2) 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: PA_IN_A(s(X1)) -> PA_IN_A(X1) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) PA_IN_A(x1) = PA_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: PA_IN_A -> PA_IN_A R is empty. Q is empty. We have to consider all (P,Q,R)-chains.