/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern thief(g) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) CutEliminatorProof [SOUND, 0 ms] (2)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 [EQUIVALENT, 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 [EQUIVALENT, 0 ms] (22) QDP (23) PrologToDTProblemTransformerProof [SOUND, 0 ms] (24) TRIPLES (25) TriplesToPiDPProof [SOUND, 2 ms] (26) PiDP (27) DependencyGraphProof [EQUIVALENT, 0 ms] (28) PiDP (29) PiDPToQDPProof [EQUIVALENT, 0 ms] (30) QDP (31) PrologToTRSTransformerProof [SOUND, 0 ms] (32) QTRS (33) DependencyPairsProof [EQUIVALENT, 0 ms] (34) QDP (35) DependencyGraphProof [EQUIVALENT, 0 ms] (36) QDP (37) UsableRulesProof [EQUIVALENT, 0 ms] (38) QDP (39) PrologToIRSwTTransformerProof [SOUND, 0 ms] (40) IRSwT (41) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (42) IRSwT (43) IntTRSCompressionProof [EQUIVALENT, 36 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 ---------------------------------------- (0) Obligation: Clauses: thief(john). thief(X) :- ','(thief(X), !). Query: thief(g) ---------------------------------------- (1) CutEliminatorProof (SOUND) Eliminated all cuts by simply ignoring them[PROLOG]. ---------------------------------------- (2) Obligation: Clauses: thief(john). thief(X) :- thief(X). Query: thief(g) ---------------------------------------- (3) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: thief_in_1: (b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: thief_in_g(john) -> thief_out_g(john) thief_in_g(X) -> U1_g(X, thief_in_g(X)) U1_g(X, thief_out_g(X)) -> thief_out_g(X) The argument filtering Pi contains the following mapping: thief_in_g(x1) = thief_in_g(x1) john = john thief_out_g(x1) = thief_out_g U1_g(x1, x2) = U1_g(x2) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (4) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: thief_in_g(john) -> thief_out_g(john) thief_in_g(X) -> U1_g(X, thief_in_g(X)) U1_g(X, thief_out_g(X)) -> thief_out_g(X) The argument filtering Pi contains the following mapping: thief_in_g(x1) = thief_in_g(x1) john = john thief_out_g(x1) = thief_out_g U1_g(x1, x2) = U1_g(x2) ---------------------------------------- (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: THIEF_IN_G(X) -> U1_G(X, thief_in_g(X)) THIEF_IN_G(X) -> THIEF_IN_G(X) The TRS R consists of the following rules: thief_in_g(john) -> thief_out_g(john) thief_in_g(X) -> U1_g(X, thief_in_g(X)) U1_g(X, thief_out_g(X)) -> thief_out_g(X) The argument filtering Pi contains the following mapping: thief_in_g(x1) = thief_in_g(x1) john = john thief_out_g(x1) = thief_out_g U1_g(x1, x2) = U1_g(x2) THIEF_IN_G(x1) = THIEF_IN_G(x1) U1_G(x1, x2) = U1_G(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (6) Obligation: Pi DP problem: The TRS P consists of the following rules: THIEF_IN_G(X) -> U1_G(X, thief_in_g(X)) THIEF_IN_G(X) -> THIEF_IN_G(X) The TRS R consists of the following rules: thief_in_g(john) -> thief_out_g(john) thief_in_g(X) -> U1_g(X, thief_in_g(X)) U1_g(X, thief_out_g(X)) -> thief_out_g(X) The argument filtering Pi contains the following mapping: thief_in_g(x1) = thief_in_g(x1) john = john thief_out_g(x1) = thief_out_g U1_g(x1, x2) = U1_g(x2) THIEF_IN_G(x1) = THIEF_IN_G(x1) U1_G(x1, x2) = U1_G(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (7) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node. ---------------------------------------- (8) Obligation: Pi DP problem: The TRS P consists of the following rules: THIEF_IN_G(X) -> THIEF_IN_G(X) The TRS R consists of the following rules: thief_in_g(john) -> thief_out_g(john) thief_in_g(X) -> U1_g(X, thief_in_g(X)) U1_g(X, thief_out_g(X)) -> thief_out_g(X) The argument filtering Pi contains the following mapping: thief_in_g(x1) = thief_in_g(x1) john = john thief_out_g(x1) = thief_out_g U1_g(x1, x2) = U1_g(x2) THIEF_IN_G(x1) = THIEF_IN_G(x1) 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: THIEF_IN_G(X) -> THIEF_IN_G(X) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (11) PiDPToQDPProof (EQUIVALENT) 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: THIEF_IN_G(X) -> THIEF_IN_G(X) 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: thief_in_1: (b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: thief_in_g(john) -> thief_out_g(john) thief_in_g(X) -> U1_g(X, thief_in_g(X)) U1_g(X, thief_out_g(X)) -> thief_out_g(X) Pi is empty. Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (14) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: thief_in_g(john) -> thief_out_g(john) thief_in_g(X) -> U1_g(X, thief_in_g(X)) U1_g(X, thief_out_g(X)) -> thief_out_g(X) Pi is empty. ---------------------------------------- (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: THIEF_IN_G(X) -> U1_G(X, thief_in_g(X)) THIEF_IN_G(X) -> THIEF_IN_G(X) The TRS R consists of the following rules: thief_in_g(john) -> thief_out_g(john) thief_in_g(X) -> U1_g(X, thief_in_g(X)) U1_g(X, thief_out_g(X)) -> thief_out_g(X) Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: THIEF_IN_G(X) -> U1_G(X, thief_in_g(X)) THIEF_IN_G(X) -> THIEF_IN_G(X) The TRS R consists of the following rules: thief_in_g(john) -> thief_out_g(john) thief_in_g(X) -> U1_g(X, thief_in_g(X)) U1_g(X, thief_out_g(X)) -> thief_out_g(X) Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node. ---------------------------------------- (18) Obligation: Pi DP problem: The TRS P consists of the following rules: THIEF_IN_G(X) -> THIEF_IN_G(X) The TRS R consists of the following rules: thief_in_g(john) -> thief_out_g(john) thief_in_g(X) -> U1_g(X, thief_in_g(X)) U1_g(X, thief_out_g(X)) -> thief_out_g(X) Pi is empty. 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: THIEF_IN_G(X) -> THIEF_IN_G(X) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (21) PiDPToQDPProof (EQUIVALENT) 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: THIEF_IN_G(X) -> THIEF_IN_G(X) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (23) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 3, "program": { "directives": [], "clauses": [ [ "(thief (john))", null ], [ "(thief X)", "(',' (thief X) (!))" ] ] }, "graph": { "nodes": { "11": { "goal": [{ "clause": 1, "scope": 1, "term": "(thief T1)" }], "kb": { "nonunifying": [[ "(thief T1)", "(thief (john))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "22": { "goal": [ { "clause": -1, "scope": -1, "term": "(!_1)" }, { "clause": 1, "scope": 2, "term": "(',' (thief (john)) (!_1))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "23": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "34": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (',' (thief T7) (!_3)) (!_1))" }], "kb": { "nonunifying": [[ "(thief T7)", "(thief (john))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T7"], "free": [], "exprvars": [] } }, "13": { "goal": [{ "clause": 1, "scope": 1, "term": "(thief (john))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "25": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "69": { "goal": [{ "clause": -1, "scope": -1, "term": "(!_1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "27": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (thief T3) (!_1))" }], "kb": { "nonunifying": [[ "(thief T3)", "(thief (john))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "49": { "goal": [{ "clause": -1, "scope": -1, "term": "(thief T7)" }], "kb": { "nonunifying": [[ "(thief T7)", "(thief (john))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T7"], "free": [], "exprvars": [] } }, "29": { "goal": [ { "clause": 0, "scope": 3, "term": "(',' (thief T3) (!_1))" }, { "clause": 1, "scope": 3, "term": "(',' (thief T3) (!_1))" } ], "kb": { "nonunifying": [[ "(thief T3)", "(thief (john))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "type": "Nodes", "3": { "goal": [{ "clause": -1, "scope": -1, "term": "(thief T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "4": { "goal": [ { "clause": 0, "scope": 1, "term": "(thief T1)" }, { "clause": 1, "scope": 1, "term": "(thief T1)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "70": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "71": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "50": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (!_3) (!_1))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "30": { "goal": [{ "clause": 1, "scope": 3, "term": "(',' (thief T3) (!_1))" }], "kb": { "nonunifying": [[ "(thief T3)", "(thief (john))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "20": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (thief (john)) (!_1))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "10": { "goal": [ { "clause": -1, "scope": -1, "term": "(true)" }, { "clause": 1, "scope": 1, "term": "(thief (john))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "21": { "goal": [ { "clause": 0, "scope": 2, "term": "(',' (thief (john)) (!_1))" }, { "clause": 1, "scope": 2, "term": "(',' (thief (john)) (!_1))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 3, "to": 4, "label": "CASE" }, { "from": 4, "to": 10, "label": "EVAL with clause\nthief(john).\nand substitutionT1 -> john" }, { "from": 4, "to": 11, "label": "EVAL-BACKTRACK" }, { "from": 10, "to": 13, "label": "SUCCESS" }, { "from": 11, "to": 27, "label": "ONLY EVAL with clause\nthief(X4) :- ','(thief(X4), !_1).\nand substitutionT1 -> T3,\nX4 -> T3" }, { "from": 13, "to": 20, "label": "ONLY EVAL with clause\nthief(X2) :- ','(thief(X2), !_1).\nand substitutionX2 -> john" }, { "from": 20, "to": 21, "label": "CASE" }, { "from": 21, "to": 22, "label": "ONLY EVAL with clause\nthief(john).\nand substitution" }, { "from": 22, "to": 23, "label": "CUT" }, { "from": 23, "to": 25, "label": "SUCCESS" }, { "from": 27, "to": 29, "label": "CASE" }, { "from": 29, "to": 30, "label": "BACKTRACK\nfor clause: thief(john)\nwith clash: (thief(T3), thief(john))" }, { "from": 30, "to": 34, "label": "ONLY EVAL with clause\nthief(X8) :- ','(thief(X8), !_3).\nand substitutionT3 -> T7,\nX8 -> T7" }, { "from": 34, "to": 49, "label": "SPLIT 1" }, { "from": 34, "to": 50, "label": "SPLIT 2\nnew knowledge:\nT7 is ground" }, { "from": 49, "to": 3, "label": "INSTANCE with matching:\nT1 -> T7" }, { "from": 50, "to": 69, "label": "CUT" }, { "from": 69, "to": 70, "label": "CUT" }, { "from": 70, "to": 71, "label": "SUCCESS" } ], "type": "Graph" } } ---------------------------------------- (24) Obligation: Triples: thiefA(X1) :- thiefA(X1). Clauses: thiefcA(john). thiefcA(john). thiefcA(X1) :- thiefcA(X1). Afs: thiefA(x1) = thiefA(x1) ---------------------------------------- (25) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: thiefA_in_1: (b) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: THIEFA_IN_G(X1) -> U1_G(X1, thiefA_in_g(X1)) THIEFA_IN_G(X1) -> THIEFA_IN_G(X1) 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 ---------------------------------------- (26) Obligation: Pi DP problem: The TRS P consists of the following rules: THIEFA_IN_G(X1) -> U1_G(X1, thiefA_in_g(X1)) THIEFA_IN_G(X1) -> THIEFA_IN_G(X1) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (27) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node. ---------------------------------------- (28) Obligation: Pi DP problem: The TRS P consists of the following rules: THIEFA_IN_G(X1) -> THIEFA_IN_G(X1) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (29) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (30) Obligation: Q DP problem: The TRS P consists of the following rules: THIEFA_IN_G(X1) -> THIEFA_IN_G(X1) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (31) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 2, "program": { "directives": [], "clauses": [ [ "(thief (john))", null ], [ "(thief X)", "(',' (thief X) (!))" ] ] }, "graph": { "nodes": { "33": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "44": { "goal": [{ "clause": -1, "scope": -1, "term": "(thief T5)" }], "kb": { "nonunifying": [[ "(thief T5)", "(thief (john))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T5"], "free": [], "exprvars": [] } }, "12": { "goal": [{ "clause": 1, "scope": 1, "term": "(thief T1)" }], "kb": { "nonunifying": [[ "(thief T1)", "(thief (john))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "45": { "goal": [{ "clause": -1, "scope": -1, "term": "(!_1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "24": { "goal": [{ "clause": 1, "scope": 1, "term": "(thief (john))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "35": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (thief T5) (!_1))" }], "kb": { "nonunifying": [[ "(thief T5)", "(thief (john))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T5"], "free": [], "exprvars": [] } }, "26": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (thief (john)) (!_1))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "28": { "goal": [ { "clause": 0, "scope": 2, "term": "(',' (thief (john)) (!_1))" }, { "clause": 1, "scope": 2, "term": "(',' (thief (john)) (!_1))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "2": { "goal": [{ "clause": -1, "scope": -1, "term": "(thief T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "5": { "goal": [ { "clause": 0, "scope": 1, "term": "(thief T1)" }, { "clause": 1, "scope": 1, "term": "(thief T1)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "9": { "goal": [ { "clause": -1, "scope": -1, "term": "(true)" }, { "clause": 1, "scope": 1, "term": "(thief (john))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "83": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "84": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "31": { "goal": [ { "clause": -1, "scope": -1, "term": "(!_1)" }, { "clause": 1, "scope": 2, "term": "(',' (thief (john)) (!_1))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "32": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 2, "to": 5, "label": "CASE" }, { "from": 5, "to": 9, "label": "EVAL with clause\nthief(john).\nand substitutionT1 -> john" }, { "from": 5, "to": 12, "label": "EVAL-BACKTRACK" }, { "from": 9, "to": 24, "label": "SUCCESS" }, { "from": 12, "to": 35, "label": "ONLY EVAL with clause\nthief(X7) :- ','(thief(X7), !_1).\nand substitutionT1 -> T5,\nX7 -> T5" }, { "from": 24, "to": 26, "label": "ONLY EVAL with clause\nthief(X3) :- ','(thief(X3), !_1).\nand substitutionX3 -> john" }, { "from": 26, "to": 28, "label": "CASE" }, { "from": 28, "to": 31, "label": "ONLY EVAL with clause\nthief(john).\nand substitution" }, { "from": 31, "to": 32, "label": "CUT" }, { "from": 32, "to": 33, "label": "SUCCESS" }, { "from": 35, "to": 44, "label": "SPLIT 1" }, { "from": 35, "to": 45, "label": "SPLIT 2\nnew knowledge:\nT5 is ground" }, { "from": 44, "to": 2, "label": "INSTANCE with matching:\nT1 -> T5" }, { "from": 45, "to": 83, "label": "CUT" }, { "from": 83, "to": 84, "label": "SUCCESS" } ], "type": "Graph" } } ---------------------------------------- (32) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f2_in(john) -> f2_out1 f2_in(T5) -> U1(f35_in(T5), T5) U1(f35_out1, T5) -> f2_out1 f45_in -> f45_out1 f35_in(T5) -> U2(f2_in(T5), T5) U2(f2_out1, T5) -> U3(f45_in, T5) U3(f45_out1, T5) -> f35_out1 Q is empty. ---------------------------------------- (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: F2_IN(T5) -> U1^1(f35_in(T5), T5) F2_IN(T5) -> F35_IN(T5) F35_IN(T5) -> U2^1(f2_in(T5), T5) F35_IN(T5) -> F2_IN(T5) U2^1(f2_out1, T5) -> U3^1(f45_in, T5) U2^1(f2_out1, T5) -> F45_IN The TRS R consists of the following rules: f2_in(john) -> f2_out1 f2_in(T5) -> U1(f35_in(T5), T5) U1(f35_out1, T5) -> f2_out1 f45_in -> f45_out1 f35_in(T5) -> U2(f2_in(T5), T5) U2(f2_out1, T5) -> U3(f45_in, T5) U3(f45_out1, T5) -> f35_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (35) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes. ---------------------------------------- (36) Obligation: Q DP problem: The TRS P consists of the following rules: F2_IN(T5) -> F35_IN(T5) F35_IN(T5) -> F2_IN(T5) The TRS R consists of the following rules: f2_in(john) -> f2_out1 f2_in(T5) -> U1(f35_in(T5), T5) U1(f35_out1, T5) -> f2_out1 f45_in -> f45_out1 f35_in(T5) -> U2(f2_in(T5), T5) U2(f2_out1, T5) -> U3(f45_in, T5) U3(f45_out1, T5) -> f35_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (37) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (38) Obligation: Q DP problem: The TRS P consists of the following rules: F2_IN(T5) -> F35_IN(T5) F35_IN(T5) -> F2_IN(T5) 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": 1, "program": { "directives": [], "clauses": [ [ "(thief (john))", null ], [ "(thief X)", "(',' (thief X) (!))" ] ] }, "graph": { "nodes": { "46": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (thief T5) (!_1))" }], "kb": { "nonunifying": [[ "(thief T5)", "(thief (john))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T5"], "free": [], "exprvars": [] } }, "36": { "goal": [ { "clause": -1, "scope": -1, "term": "(true)" }, { "clause": 1, "scope": 1, "term": "(thief (john))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "47": { "goal": [{ "clause": -1, "scope": -1, "term": "(thief T5)" }], "kb": { "nonunifying": [[ "(thief T5)", "(thief (john))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T5"], "free": [], "exprvars": [] } }, "37": { "goal": [{ "clause": 1, "scope": 1, "term": "(thief T1)" }], "kb": { "nonunifying": [[ "(thief T1)", "(thief (john))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "48": { "goal": [{ "clause": -1, "scope": -1, "term": "(!_1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "38": { "goal": [{ "clause": 1, "scope": 1, "term": "(thief (john))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "39": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (thief (john)) (!_1))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "1": { "goal": [{ "clause": -1, "scope": -1, "term": "(thief T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "6": { "goal": [ { "clause": 0, "scope": 1, "term": "(thief T1)" }, { "clause": 1, "scope": 1, "term": "(thief T1)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "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": [] } }, "40": { "goal": [ { "clause": 0, "scope": 2, "term": "(',' (thief (john)) (!_1))" }, { "clause": 1, "scope": 2, "term": "(',' (thief (john)) (!_1))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "41": { "goal": [ { "clause": -1, "scope": -1, "term": "(!_1)" }, { "clause": 1, "scope": 2, "term": "(',' (thief (john)) (!_1))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "42": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "43": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 6, "label": "CASE" }, { "from": 6, "to": 36, "label": "EVAL with clause\nthief(john).\nand substitutionT1 -> john" }, { "from": 6, "to": 37, "label": "EVAL-BACKTRACK" }, { "from": 36, "to": 38, "label": "SUCCESS" }, { "from": 37, "to": 46, "label": "ONLY EVAL with clause\nthief(X7) :- ','(thief(X7), !_1).\nand substitutionT1 -> T5,\nX7 -> T5" }, { "from": 38, "to": 39, "label": "ONLY EVAL with clause\nthief(X3) :- ','(thief(X3), !_1).\nand substitutionX3 -> john" }, { "from": 39, "to": 40, "label": "CASE" }, { "from": 40, "to": 41, "label": "ONLY EVAL with clause\nthief(john).\nand substitution" }, { "from": 41, "to": 42, "label": "CUT" }, { "from": 42, "to": 43, "label": "SUCCESS" }, { "from": 46, "to": 47, "label": "SPLIT 1" }, { "from": 46, "to": 48, "label": "SPLIT 2\nnew knowledge:\nT5 is ground" }, { "from": 47, "to": 1, "label": "INSTANCE with matching:\nT1 -> T5" }, { "from": 48, "to": 81, "label": "CUT" }, { "from": 81, "to": 82, "label": "SUCCESS" } ], "type": "Graph" } } ---------------------------------------- (40) Obligation: Rules: f1_in(T1) -> f6_in(T1) :|: TRUE f6_out(x) -> f1_out(x) :|: TRUE f37_out(x1) -> f6_out(x1) :|: TRUE f6_in(x2) -> f37_in(x2) :|: TRUE f36_out -> f6_out(john) :|: TRUE f6_in(john) -> f36_in :|: TRUE f46_out(T5) -> f37_out(T5) :|: TRUE f37_in(x3) -> f46_in(x3) :|: TRUE f1_out(x4) -> f47_out(x4) :|: TRUE f47_in(x5) -> f1_in(x5) :|: TRUE f81_in -> f81_out :|: TRUE f81_out -> f48_out :|: TRUE f48_in -> f81_in :|: TRUE f48_out -> f46_out(x6) :|: TRUE f46_in(x7) -> f47_in(x7) :|: TRUE f47_out(x8) -> f48_in :|: TRUE Start term: f1_in(T1) ---------------------------------------- (41) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f1_in(T1) -> f6_in(T1) :|: TRUE f6_in(x2) -> f37_in(x2) :|: TRUE f37_in(x3) -> f46_in(x3) :|: TRUE f47_in(x5) -> f1_in(x5) :|: TRUE f46_in(x7) -> f47_in(x7) :|: TRUE ---------------------------------------- (42) Obligation: Rules: f1_in(T1) -> f6_in(T1) :|: TRUE f6_in(x2) -> f37_in(x2) :|: TRUE f37_in(x3) -> f46_in(x3) :|: TRUE f47_in(x5) -> f1_in(x5) :|: TRUE f46_in(x7) -> f47_in(x7) :|: TRUE ---------------------------------------- (43) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (44) Obligation: Rules: f47_in(x5:0) -> f47_in(x5:0) :|: TRUE ---------------------------------------- (45) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (46) Obligation: Rules: f47_in(x5:0) -> f47_in(x5:0) :|: TRUE ---------------------------------------- (47) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f47_in(x5:0) -> f47_in(x5:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (48) Obligation: Termination digraph: Nodes: (1) f47_in(x5:0) -> f47_in(x5:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (49) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f47_in(VARIABLE) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (50) Obligation: Rules: f47_in(x5:0) -> f47_in(x5:0) :|: TRUE ---------------------------------------- (51) IntTRSPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc, x5:0) -> f(1, x5:0) :|: pc = 1 && TRUE Witness term starting non-terminating reduction: f(1, -8) ---------------------------------------- (52) NO