/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.pl /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern p(g,g) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1)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 [EQUIVALENT, 1 ms] (8) QDP (9) QDPSizeChangeProof [EQUIVALENT, 0 ms] (10) YES ---------------------------------------- (0) Obligation: Clauses: p(X, Y) :- ','(q(X, Y), r(X)). q(a, 0). q(X, s(Y)) :- q(X, Y). r(b) :- r(b). Query: p(g,g) ---------------------------------------- (1) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 1, "program": { "directives": [], "clauses": [ [ "(p X Y)", "(',' (q X Y) (r X))" ], [ "(q (a) (0))", null ], [ "(q X (s Y))", "(q X Y)" ], [ "(r (b))", "(r (b))" ] ] }, "graph": { "nodes": { "66": { "goal": [{ "clause": 1, "scope": 2, "term": "(',' (q T5 T6) (r T5))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T5", "T6" ], "free": [], "exprvars": [] } }, "67": { "goal": [{ "clause": 2, "scope": 2, "term": "(',' (q T5 T6) (r T5))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T5", "T6" ], "free": [], "exprvars": [] } }, "68": { "goal": [{ "clause": -1, "scope": -1, "term": "(r (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "47": { "goal": [ { "clause": 1, "scope": 2, "term": "(',' (q T5 T6) (r T5))" }, { "clause": 2, "scope": 2, "term": "(',' (q T5 T6) (r T5))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T5", "T6" ], "free": [], "exprvars": [] } }, "69": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "131": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (q T11 T12) (r T11))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T11", "T12" ], "free": [], "exprvars": [] } }, "1": { "goal": [{ "clause": -1, "scope": -1, "term": "(p T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2" ], "free": [], "exprvars": [] } }, "133": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "7": { "goal": [{ "clause": 0, "scope": 1, "term": "(p T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2" ], "free": [], "exprvars": [] } }, "31": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (q T5 T6) (r T5))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T5", "T6" ], "free": [], "exprvars": [] } }, "86": { "goal": [{ "clause": 3, "scope": 3, "term": "(r (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "87": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 7, "label": "CASE" }, { "from": 7, "to": 31, "label": "ONLY EVAL with clause\np(X3, X4) :- ','(q(X3, X4), r(X3)).\nand substitutionT1 -> T5,\nX3 -> T5,\nT2 -> T6,\nX4 -> T6" }, { "from": 31, "to": 47, "label": "CASE" }, { "from": 47, "to": 66, "label": "PARALLEL" }, { "from": 47, "to": 67, "label": "PARALLEL" }, { "from": 66, "to": 68, "label": "EVAL with clause\nq(a, 0).\nand substitutionT5 -> a,\nT6 -> 0" }, { "from": 66, "to": 69, "label": "EVAL-BACKTRACK" }, { "from": 67, "to": 131, "label": "EVAL with clause\nq(X9, s(X10)) :- q(X9, X10).\nand substitutionT5 -> T11,\nX9 -> T11,\nX10 -> T12,\nT6 -> s(T12)" }, { "from": 67, "to": 133, "label": "EVAL-BACKTRACK" }, { "from": 68, "to": 86, "label": "CASE" }, { "from": 86, "to": 87, "label": "BACKTRACK\nfor clause: r(b) :- r(b)because of non-unification" }, { "from": 131, "to": 31, "label": "INSTANCE with matching:\nT5 -> T11\nT6 -> T12" } ], "type": "Graph" } } ---------------------------------------- (2) Obligation: Triples: pA(X1, s(X2)) :- pA(X1, X2). pB(X1, X2) :- pA(X1, X2). Clauses: qcA(X1, s(X2)) :- qcA(X1, X2). Afs: pB(x1, x2) = pB(x1, 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: pB_in_2: (b,b) pA_in_2: (b,b) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: PB_IN_GG(X1, X2) -> U2_GG(X1, X2, pA_in_gg(X1, X2)) PB_IN_GG(X1, X2) -> PA_IN_GG(X1, X2) PA_IN_GG(X1, s(X2)) -> U1_GG(X1, X2, pA_in_gg(X1, X2)) PA_IN_GG(X1, s(X2)) -> PA_IN_GG(X1, X2) 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 ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: PB_IN_GG(X1, X2) -> U2_GG(X1, X2, pA_in_gg(X1, X2)) PB_IN_GG(X1, X2) -> PA_IN_GG(X1, X2) PA_IN_GG(X1, s(X2)) -> U1_GG(X1, X2, pA_in_gg(X1, X2)) PA_IN_GG(X1, s(X2)) -> PA_IN_GG(X1, X2) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes. ---------------------------------------- (6) Obligation: Pi DP problem: The TRS P consists of the following rules: PA_IN_GG(X1, s(X2)) -> PA_IN_GG(X1, X2) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (7) PiDPToQDPProof (EQUIVALENT) 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: PA_IN_GG(X1, s(X2)) -> PA_IN_GG(X1, X2) 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: *PA_IN_GG(X1, s(X2)) -> PA_IN_GG(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (10) YES