/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.pl /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/benchmark/theBenchmark.pl # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Left Termination of the query pattern member(a,g) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToDTProblemTransformerProof [SOUND, 0 ms] (2) TRIPLES (3) TriplesToPiDPProof [SOUND, 0 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) PiDP (7) PiDPToQDPProof [SOUND, 0 ms] (8) QDP (9) QDPSizeChangeProof [EQUIVALENT, 0 ms] (10) YES ---------------------------------------- (0) Obligation: Clauses: member(X, .(Y, Xs)) :- member(X, Xs). member(X, .(X, Xs)). Query: member(a,g) ---------------------------------------- (1) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 3, "program": { "directives": [], "clauses": [ [ "(member X (. Y Xs))", "(member X Xs)" ], [ "(member X (. X Xs))", null ] ] }, "graph": { "nodes": { "77": { "goal": [{ "clause": 0, "scope": 2, "term": "(member T9 T8)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T8"], "free": [], "exprvars": [] } }, "88": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "78": { "goal": [ { "clause": 1, "scope": 2, "term": "(member T9 T8)" }, { "clause": -1, "scope": 2, "term": null }, { "clause": 1, "scope": 1, "term": "(member T1 (. T7 T8))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T7", "T8" ], "free": [], "exprvars": [] } }, "89": { "goal": [{ "clause": 1, "scope": 1, "term": "(member T1 (. T7 T8))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T7", "T8" ], "free": [], "exprvars": [] } }, "79": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T25 T24)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T24"], "free": [], "exprvars": [] } }, "type": "Nodes", "3": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "90": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "80": { "goal": [], "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": 0, "scope": 1, "term": "(member T1 T2)" }, { "clause": 1, "scope": 1, "term": "(member T1 T2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "70": { "goal": [ { "clause": -1, "scope": -1, "term": "(member T9 T8)" }, { "clause": 1, "scope": 1, "term": "(member T1 (. T7 T8))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T7", "T8" ], "free": [], "exprvars": [] } }, "92": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "71": { "goal": [{ "clause": 1, "scope": 1, "term": "(member T1 T2)" }], "kb": { "nonunifying": [[ "(member T1 T2)", "(member X4 (. X5 X6))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [ "X4", "X5", "X6" ], "exprvars": [] } }, "93": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "84": { "goal": [{ "clause": 1, "scope": 2, "term": "(member T9 T8)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T8"], "free": [], "exprvars": [] } }, "85": { "goal": [ { "clause": -1, "scope": 2, "term": null }, { "clause": 1, "scope": 1, "term": "(member T1 (. T7 T8))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T7", "T8" ], "free": [], "exprvars": [] } }, "86": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "76": { "goal": [ { "clause": 0, "scope": 2, "term": "(member T9 T8)" }, { "clause": 1, "scope": 2, "term": "(member T9 T8)" }, { "clause": -1, "scope": 2, "term": null }, { "clause": 1, "scope": 1, "term": "(member T1 (. T7 T8))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T7", "T8" ], "free": [], "exprvars": [] } }, "87": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 3, "to": 9, "label": "CASE" }, { "from": 9, "to": 70, "label": "EVAL with clause\nmember(X4, .(X5, X6)) :- member(X4, X6).\nand substitutionT1 -> T9,\nX4 -> T9,\nX5 -> T7,\nX6 -> T8,\nT2 -> .(T7, T8),\nT6 -> T9" }, { "from": 9, "to": 71, "label": "EVAL-BACKTRACK" }, { "from": 70, "to": 76, "label": "CASE" }, { "from": 71, "to": 93, "label": "BACKTRACK\nfor clause: member(X, .(X, Xs))\nwith clash: (member(T1, T2), member(X4, .(X5, X6)))" }, { "from": 76, "to": 77, "label": "PARALLEL" }, { "from": 76, "to": 78, "label": "PARALLEL" }, { "from": 77, "to": 79, "label": "EVAL with clause\nmember(X19, .(X20, X21)) :- member(X19, X21).\nand substitutionT9 -> T25,\nX19 -> T25,\nX20 -> T23,\nX21 -> T24,\nT8 -> .(T23, T24),\nT22 -> T25" }, { "from": 77, "to": 80, "label": "EVAL-BACKTRACK" }, { "from": 78, "to": 84, "label": "PARALLEL" }, { "from": 78, "to": 85, "label": "PARALLEL" }, { "from": 79, "to": 3, "label": "INSTANCE with matching:\nT1 -> T25\nT2 -> T24" }, { "from": 84, "to": 86, "label": "EVAL with clause\nmember(X33, .(X33, X34)).\nand substitutionT9 -> T37,\nX33 -> T37,\nX34 -> T38,\nT8 -> .(T37, T38)" }, { "from": 84, "to": 87, "label": "EVAL-BACKTRACK" }, { "from": 85, "to": 89, "label": "FAILURE" }, { "from": 86, "to": 88, "label": "SUCCESS" }, { "from": 89, "to": 90, "label": "EVAL with clause\nmember(X39, .(X39, X40)).\nand substitutionT1 -> T43,\nX39 -> T43,\nT7 -> T43,\nT8 -> T44,\nX40 -> T44" }, { "from": 89, "to": 91, "label": "EVAL-BACKTRACK" }, { "from": 90, "to": 92, "label": "SUCCESS" } ], "type": "Graph" } } ---------------------------------------- (2) Obligation: Triples: memberA(X1, .(X2, .(X3, X4))) :- memberA(X1, X4). Clauses: membercA(X1, .(X2, .(X3, X4))) :- membercA(X1, X4). membercA(X1, .(X2, .(X1, X3))). membercA(X1, .(X1, X2)). Afs: memberA(x1, x2) = memberA(x2) ---------------------------------------- (3) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: memberA_in_2: (f,b) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: MEMBERA_IN_AG(X1, .(X2, .(X3, X4))) -> U1_AG(X1, X2, X3, X4, memberA_in_ag(X1, X4)) MEMBERA_IN_AG(X1, .(X2, .(X3, X4))) -> MEMBERA_IN_AG(X1, X4) R is empty. The argument filtering Pi contains the following mapping: memberA_in_ag(x1, x2) = memberA_in_ag(x2) .(x1, x2) = .(x1, x2) MEMBERA_IN_AG(x1, x2) = MEMBERA_IN_AG(x2) U1_AG(x1, x2, x3, x4, x5) = U1_AG(x2, x3, x4, x5) We have to consider all (P,R,Pi)-chains Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBERA_IN_AG(X1, .(X2, .(X3, X4))) -> U1_AG(X1, X2, X3, X4, memberA_in_ag(X1, X4)) MEMBERA_IN_AG(X1, .(X2, .(X3, X4))) -> MEMBERA_IN_AG(X1, X4) R is empty. The argument filtering Pi contains the following mapping: memberA_in_ag(x1, x2) = memberA_in_ag(x2) .(x1, x2) = .(x1, x2) MEMBERA_IN_AG(x1, x2) = MEMBERA_IN_AG(x2) U1_AG(x1, x2, x3, x4, x5) = U1_AG(x2, x3, x4, x5) 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: MEMBERA_IN_AG(X1, .(X2, .(X3, X4))) -> MEMBERA_IN_AG(X1, X4) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) MEMBERA_IN_AG(x1, x2) = MEMBERA_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (7) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (8) Obligation: Q DP problem: The TRS P consists of the following rules: MEMBERA_IN_AG(.(X2, .(X3, X4))) -> MEMBERA_IN_AG(X4) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (9) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *MEMBERA_IN_AG(.(X2, .(X3, X4))) -> MEMBERA_IN_AG(X4) The graph contains the following edges 1 > 1 ---------------------------------------- (10) YES