MAYBE proof of /export/starexec/sandbox/benchmark/theBenchmark.pl # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Left Termination of the query pattern p(g) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2) PiTRS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 [EQUIVALENT, 0 ms] (10) QDP (11) TransformationProof [EQUIVALENT, 0 ms] (12) QDP (13) NonTerminationLoopProof [COMPLETE, 0 ms] (14) NO (15) PrologToPiTRSProof [SOUND, 0 ms] (16) PiTRS (17) DependencyPairsProof [EQUIVALENT, 0 ms] (18) PiDP (19) DependencyGraphProof [EQUIVALENT, 0 ms] (20) PiDP (21) UsableRulesProof [EQUIVALENT, 0 ms] (22) PiDP (23) PiDPToQDPProof [EQUIVALENT, 0 ms] (24) QDP (25) TransformationProof [EQUIVALENT, 0 ms] (26) QDP (27) NonTerminationLoopProof [COMPLETE, 0 ms] (28) NO (29) PrologToDTProblemTransformerProof [SOUND, 0 ms] (30) TRIPLES (31) TriplesToPiDPProof [SOUND, 0 ms] (32) PiDP (33) DependencyGraphProof [EQUIVALENT, 0 ms] (34) PiDP (35) PiDPToQDPProof [EQUIVALENT, 0 ms] (36) QDP (37) TransformationProof [EQUIVALENT, 0 ms] (38) QDP (39) NonTerminationLoopProof [COMPLETE, 0 ms] (40) NO (41) PrologToTRSTransformerProof [SOUND, 0 ms] (42) QTRS (43) QTRSRRRProof [EQUIVALENT, 43 ms] (44) QTRS (45) QTRSRRRProof [EQUIVALENT, 4 ms] (46) QTRS (47) Overlay + Local Confluence [EQUIVALENT, 0 ms] (48) QTRS (49) DependencyPairsProof [EQUIVALENT, 0 ms] (50) QDP (51) UsableRulesProof [EQUIVALENT, 0 ms] (52) QDP (53) QReductionProof [EQUIVALENT, 0 ms] (54) QDP (55) TransformationProof [EQUIVALENT, 0 ms] (56) QDP (57) NonTerminationLoopProof [COMPLETE, 0 ms] (58) NO (59) PrologToIRSwTTransformerProof [SOUND, 0 ms] (60) IRSwT (61) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (62) IRSwT (63) IntTRSCompressionProof [EQUIVALENT, 55 ms] (64) IRSwT (65) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (66) IRSwT (67) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms] (68) IRSwT (69) IRSwTToIntTRSProof [SOUND, 10 ms] (70) IRSwT ---------------------------------------- (0) Obligation: Clauses: p(a). p(X) :- p(a). q(b). 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) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: p_in_g(a) -> p_out_g(a) p_in_g(X) -> U1_g(X, p_in_g(a)) U1_g(X, p_out_g(a)) -> p_out_g(X) The argument filtering Pi contains the following mapping: p_in_g(x1) = p_in_g(x1) a = a p_out_g(x1) = p_out_g U1_g(x1, x2) = U1_g(x2) 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(a) -> p_out_g(a) p_in_g(X) -> U1_g(X, p_in_g(a)) U1_g(X, p_out_g(a)) -> p_out_g(X) The argument filtering Pi contains the following mapping: p_in_g(x1) = p_in_g(x1) a = a p_out_g(x1) = p_out_g U1_g(x1, x2) = U1_g(x2) ---------------------------------------- (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(X) -> U1_G(X, p_in_g(a)) P_IN_G(X) -> P_IN_G(a) The TRS R consists of the following rules: p_in_g(a) -> p_out_g(a) p_in_g(X) -> U1_g(X, p_in_g(a)) U1_g(X, p_out_g(a)) -> p_out_g(X) The argument filtering Pi contains the following mapping: p_in_g(x1) = p_in_g(x1) a = a p_out_g(x1) = p_out_g U1_g(x1, x2) = U1_g(x2) P_IN_G(x1) = P_IN_G(x1) U1_G(x1, x2) = U1_G(x2) 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(X) -> U1_G(X, p_in_g(a)) P_IN_G(X) -> P_IN_G(a) The TRS R consists of the following rules: p_in_g(a) -> p_out_g(a) p_in_g(X) -> U1_g(X, p_in_g(a)) U1_g(X, p_out_g(a)) -> p_out_g(X) The argument filtering Pi contains the following mapping: p_in_g(x1) = p_in_g(x1) a = a p_out_g(x1) = p_out_g U1_g(x1, x2) = U1_g(x2) P_IN_G(x1) = P_IN_G(x1) U1_G(x1, x2) = U1_G(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node. ---------------------------------------- (6) Obligation: Pi DP problem: The TRS P consists of the following rules: P_IN_G(X) -> P_IN_G(a) The TRS R consists of the following rules: p_in_g(a) -> p_out_g(a) p_in_g(X) -> U1_g(X, p_in_g(a)) U1_g(X, p_out_g(a)) -> p_out_g(X) The argument filtering Pi contains the following mapping: p_in_g(x1) = p_in_g(x1) a = a p_out_g(x1) = p_out_g U1_g(x1, x2) = U1_g(x2) P_IN_G(x1) = P_IN_G(x1) 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: P_IN_G(X) -> P_IN_G(a) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (9) PiDPToQDPProof (EQUIVALENT) 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: P_IN_G(X) -> P_IN_G(a) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (11) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule P_IN_G(X) -> P_IN_G(a) we obtained the following new rules [LPAR04]: (P_IN_G(a) -> P_IN_G(a),P_IN_G(a) -> P_IN_G(a)) ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: P_IN_G(a) -> P_IN_G(a) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (13) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by semiunifying a rule from P directly. s = P_IN_G(a) evaluates to t =P_IN_G(a) Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence The DP semiunifies directly so there is only one rewrite step from P_IN_G(a) to P_IN_G(a). ---------------------------------------- (14) NO ---------------------------------------- (15) 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) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: p_in_g(a) -> p_out_g(a) p_in_g(X) -> U1_g(X, p_in_g(a)) U1_g(X, p_out_g(a)) -> p_out_g(X) Pi is empty. Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (16) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: p_in_g(a) -> p_out_g(a) p_in_g(X) -> U1_g(X, p_in_g(a)) U1_g(X, p_out_g(a)) -> p_out_g(X) Pi is empty. ---------------------------------------- (17) 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(X) -> U1_G(X, p_in_g(a)) P_IN_G(X) -> P_IN_G(a) The TRS R consists of the following rules: p_in_g(a) -> p_out_g(a) p_in_g(X) -> U1_g(X, p_in_g(a)) U1_g(X, p_out_g(a)) -> p_out_g(X) Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (18) Obligation: Pi DP problem: The TRS P consists of the following rules: P_IN_G(X) -> U1_G(X, p_in_g(a)) P_IN_G(X) -> P_IN_G(a) The TRS R consists of the following rules: p_in_g(a) -> p_out_g(a) p_in_g(X) -> U1_g(X, p_in_g(a)) U1_g(X, p_out_g(a)) -> p_out_g(X) Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (19) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node. ---------------------------------------- (20) Obligation: Pi DP problem: The TRS P consists of the following rules: P_IN_G(X) -> P_IN_G(a) The TRS R consists of the following rules: p_in_g(a) -> p_out_g(a) p_in_g(X) -> U1_g(X, p_in_g(a)) U1_g(X, p_out_g(a)) -> p_out_g(X) Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (21) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (22) Obligation: Pi DP problem: The TRS P consists of the following rules: P_IN_G(X) -> P_IN_G(a) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (23) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (24) Obligation: Q DP problem: The TRS P consists of the following rules: P_IN_G(X) -> P_IN_G(a) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (25) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule P_IN_G(X) -> P_IN_G(a) we obtained the following new rules [LPAR04]: (P_IN_G(a) -> P_IN_G(a),P_IN_G(a) -> P_IN_G(a)) ---------------------------------------- (26) Obligation: Q DP problem: The TRS P consists of the following rules: P_IN_G(a) -> P_IN_G(a) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (27) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by semiunifying a rule from P directly. s = P_IN_G(a) evaluates to t =P_IN_G(a) Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence The DP semiunifies directly so there is only one rewrite step from P_IN_G(a) to P_IN_G(a). ---------------------------------------- (28) NO ---------------------------------------- (29) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 1, "program": { "directives": [], "clauses": [ [ "(p (a))", null ], [ "(p X)", "(p (a))" ], [ "(q (b))", null ] ] }, "graph": { "nodes": { "22": { "goal": [{ "clause": 1, "scope": 1, "term": "(p T1)" }], "kb": { "nonunifying": [[ "(p T1)", "(p (a))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "33": { "goal": [ { "clause": 0, "scope": 2, "term": "(p (a))" }, { "clause": 1, "scope": 2, "term": "(p (a))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "44": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "34": { "goal": [{ "clause": 0, "scope": 2, "term": "(p (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "45": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "35": { "goal": [{ "clause": 1, "scope": 2, "term": "(p (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "36": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "37": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "38": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "39": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "19": { "goal": [ { "clause": -1, "scope": -1, "term": "(true)" }, { "clause": 1, "scope": 1, "term": "(p (a))" } ], "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": [] } }, "4": { "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": [] } }, "40": { "goal": [ { "clause": 0, "scope": 3, "term": "(p (a))" }, { "clause": 1, "scope": 3, "term": "(p (a))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "41": { "goal": [{ "clause": 0, "scope": 3, "term": "(p (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "31": { "goal": [{ "clause": 1, "scope": 1, "term": "(p (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "42": { "goal": [{ "clause": 1, "scope": 3, "term": "(p (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "32": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "43": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 4, "label": "CASE" }, { "from": 4, "to": 19, "label": "EVAL with clause\np(a).\nand substitutionT1 -> a" }, { "from": 4, "to": 22, "label": "EVAL-BACKTRACK" }, { "from": 19, "to": 31, "label": "SUCCESS" }, { "from": 22, "to": 39, "label": "ONLY EVAL with clause\np(X7) :- p(a).\nand substitutionT1 -> T3,\nX7 -> T3" }, { "from": 31, "to": 32, "label": "ONLY EVAL with clause\np(X2) :- p(a).\nand substitutionX2 -> a" }, { "from": 32, "to": 33, "label": "CASE" }, { "from": 33, "to": 34, "label": "PARALLEL" }, { "from": 33, "to": 35, "label": "PARALLEL" }, { "from": 34, "to": 36, "label": "ONLY EVAL with clause\np(a).\nand substitution" }, { "from": 35, "to": 38, "label": "ONLY EVAL with clause\np(X5) :- p(a).\nand substitutionX5 -> a" }, { "from": 36, "to": 37, "label": "SUCCESS" }, { "from": 38, "to": 1, "label": "INSTANCE with matching:\nT1 -> a" }, { "from": 39, "to": 40, "label": "CASE" }, { "from": 40, "to": 41, "label": "PARALLEL" }, { "from": 40, "to": 42, "label": "PARALLEL" }, { "from": 41, "to": 43, "label": "ONLY EVAL with clause\np(a).\nand substitution" }, { "from": 42, "to": 45, "label": "ONLY EVAL with clause\np(X10) :- p(a).\nand substitutionX10 -> a" }, { "from": 43, "to": 44, "label": "SUCCESS" }, { "from": 45, "to": 1, "label": "INSTANCE with matching:\nT1 -> a" } ], "type": "Graph" } } ---------------------------------------- (30) Obligation: Triples: pA(a) :- pA(a). pA(X1) :- pA(a). Clauses: pcA(a). pcA(a). pcA(a) :- pcA(a). pcA(X1). pcA(X1) :- pcA(a). Afs: pA(x1) = pA(x1) ---------------------------------------- (31) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: pA_in_1: (b) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: PA_IN_G(a) -> U1_G(pA_in_g(a)) PA_IN_G(a) -> PA_IN_G(a) PA_IN_G(X1) -> U2_G(X1, pA_in_g(a)) PA_IN_G(X1) -> PA_IN_G(a) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES ---------------------------------------- (32) Obligation: Pi DP problem: The TRS P consists of the following rules: PA_IN_G(a) -> U1_G(pA_in_g(a)) PA_IN_G(a) -> PA_IN_G(a) PA_IN_G(X1) -> U2_G(X1, pA_in_g(a)) PA_IN_G(X1) -> PA_IN_G(a) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (33) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes. ---------------------------------------- (34) Obligation: Pi DP problem: The TRS P consists of the following rules: PA_IN_G(X1) -> PA_IN_G(a) PA_IN_G(a) -> PA_IN_G(a) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (35) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (36) Obligation: Q DP problem: The TRS P consists of the following rules: PA_IN_G(X1) -> PA_IN_G(a) PA_IN_G(a) -> PA_IN_G(a) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (37) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule PA_IN_G(X1) -> PA_IN_G(a) we obtained the following new rules [LPAR04]: (PA_IN_G(a) -> PA_IN_G(a),PA_IN_G(a) -> PA_IN_G(a)) ---------------------------------------- (38) Obligation: Q DP problem: The TRS P consists of the following rules: PA_IN_G(a) -> PA_IN_G(a) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (39) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by semiunifying a rule from P directly. s = PA_IN_G(a) evaluates to t =PA_IN_G(a) Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence The DP semiunifies directly so there is only one rewrite step from PA_IN_G(a) to PA_IN_G(a). ---------------------------------------- (40) NO ---------------------------------------- (41) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 15, "program": { "directives": [], "clauses": [ [ "(p (a))", null ], [ "(p X)", "(p (a))" ], [ "(q (b))", null ] ] }, "graph": { "nodes": { "23": { "goal": [{ "clause": 0, "scope": 1, "term": "(p T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "24": { "goal": [{ "clause": 1, "scope": 1, "term": "(p T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "25": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "15": { "goal": [{ "clause": -1, "scope": -1, "term": "(p T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "16": { "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": [] } }, "27": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "28": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "29": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes" }, "edges": [ { "from": 15, "to": 16, "label": "CASE" }, { "from": 16, "to": 23, "label": "PARALLEL" }, { "from": 16, "to": 24, "label": "PARALLEL" }, { "from": 23, "to": 25, "label": "EVAL with clause\np(a).\nand substitutionT1 -> a" }, { "from": 23, "to": 27, "label": "EVAL-BACKTRACK" }, { "from": 24, "to": 29, "label": "ONLY EVAL with clause\np(X3) :- p(a).\nand substitutionT1 -> T4,\nX3 -> T4" }, { "from": 25, "to": 28, "label": "SUCCESS" }, { "from": 29, "to": 15, "label": "INSTANCE with matching:\nT1 -> a" } ], "type": "Graph" } } ---------------------------------------- (42) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f15_in(a) -> f15_out1 f15_in(T4) -> U1(f15_in(a), T4) U1(f15_out1, T4) -> f15_out1 Q is empty. ---------------------------------------- (43) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U1(x_1, x_2)) = x_1 + 2*x_2 POL(a) = 0 POL(f15_in(x_1)) = 1 + 2*x_1 POL(f15_out1) = 0 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: f15_in(a) -> f15_out1 ---------------------------------------- (44) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f15_in(T4) -> U1(f15_in(a), T4) U1(f15_out1, T4) -> f15_out1 Q is empty. ---------------------------------------- (45) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U1(x_1, x_2)) = 2*x_1 + 2*x_2 POL(a) = 0 POL(f15_in(x_1)) = 2*x_1 POL(f15_out1) = 1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U1(f15_out1, T4) -> f15_out1 ---------------------------------------- (46) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f15_in(T4) -> U1(f15_in(a), T4) Q is empty. ---------------------------------------- (47) Overlay + Local Confluence (EQUIVALENT) The TRS is overlay and locally confluent. By [NOC] we can switch to innermost. ---------------------------------------- (48) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f15_in(T4) -> U1(f15_in(a), T4) The set Q consists of the following terms: f15_in(x0) ---------------------------------------- (49) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (50) Obligation: Q DP problem: The TRS P consists of the following rules: F15_IN(T4) -> F15_IN(a) The TRS R consists of the following rules: f15_in(T4) -> U1(f15_in(a), T4) The set Q consists of the following terms: f15_in(x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (51) 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. ---------------------------------------- (52) Obligation: Q DP problem: The TRS P consists of the following rules: F15_IN(T4) -> F15_IN(a) R is empty. The set Q consists of the following terms: f15_in(x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (53) 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]. f15_in(x0) ---------------------------------------- (54) Obligation: Q DP problem: The TRS P consists of the following rules: F15_IN(T4) -> F15_IN(a) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (55) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule F15_IN(T4) -> F15_IN(a) we obtained the following new rules [LPAR04]: (F15_IN(a) -> F15_IN(a),F15_IN(a) -> F15_IN(a)) ---------------------------------------- (56) Obligation: Q DP problem: The TRS P consists of the following rules: F15_IN(a) -> F15_IN(a) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (57) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by semiunifying a rule from P directly. s = F15_IN(a) evaluates to t =F15_IN(a) Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence The DP semiunifies directly so there is only one rewrite step from F15_IN(a) to F15_IN(a). ---------------------------------------- (58) NO ---------------------------------------- (59) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 2, "program": { "directives": [], "clauses": [ [ "(p (a))", null ], [ "(p X)", "(p (a))" ], [ "(q (b))", null ] ] }, "graph": { "nodes": { "2": { "goal": [{ "clause": -1, "scope": -1, "term": "(p T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "26": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "17": { "goal": [{ "clause": 0, "scope": 1, "term": "(p T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "18": { "goal": [{ "clause": 1, "scope": 1, "term": "(p T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "9": { "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", "30": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "20": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "21": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 2, "to": 9, "label": "CASE" }, { "from": 9, "to": 17, "label": "PARALLEL" }, { "from": 9, "to": 18, "label": "PARALLEL" }, { "from": 17, "to": 20, "label": "EVAL with clause\np(a).\nand substitutionT1 -> a" }, { "from": 17, "to": 21, "label": "EVAL-BACKTRACK" }, { "from": 18, "to": 30, "label": "ONLY EVAL with clause\np(X3) :- p(a).\nand substitutionT1 -> T4,\nX3 -> T4" }, { "from": 20, "to": 26, "label": "SUCCESS" }, { "from": 30, "to": 2, "label": "INSTANCE with matching:\nT1 -> a" } ], "type": "Graph" } } ---------------------------------------- (60) Obligation: Rules: f30_in -> f2_in(a) :|: TRUE f2_out(a) -> f30_out :|: TRUE f9_in(T1) -> f17_in(T1) :|: TRUE f9_in(x) -> f18_in(x) :|: TRUE f17_out(x1) -> f9_out(x1) :|: TRUE f18_out(x2) -> f9_out(x2) :|: TRUE f18_in(T4) -> f30_in :|: TRUE f30_out -> f18_out(x3) :|: TRUE f9_out(x4) -> f2_out(x4) :|: TRUE f2_in(x5) -> f9_in(x5) :|: TRUE Start term: f2_in(T1) ---------------------------------------- (61) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f30_in -> f2_in(a) :|: TRUE f9_in(x) -> f18_in(x) :|: TRUE f18_in(T4) -> f30_in :|: TRUE f2_in(x5) -> f9_in(x5) :|: TRUE ---------------------------------------- (62) Obligation: Rules: f30_in -> f2_in(a) :|: TRUE f9_in(x) -> f18_in(x) :|: TRUE f18_in(T4) -> f30_in :|: TRUE f2_in(x5) -> f9_in(x5) :|: TRUE ---------------------------------------- (63) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (64) Obligation: Rules: f9_in(x:0) -> f9_in(a) :|: TRUE ---------------------------------------- (65) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (66) Obligation: Rules: f9_in(x:0) -> f9_in(a) :|: TRUE ---------------------------------------- (67) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f9_in(x:0) -> f9_in(a) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (68) Obligation: Termination digraph: Nodes: (1) f9_in(x:0) -> f9_in(a) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (69) IRSwTToIntTRSProof (SOUND) Applied path-length measure to transform intTRS with terms to intTRS. ---------------------------------------- (70) Obligation: Rules: f9_in(x) -> f9_in(a) :|: TRUE