/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern even(g) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1)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, 0 ms] (8) QDP (9) QDPSizeChangeProof [EQUIVALENT, 0 ms] (10) YES ---------------------------------------- (0) Obligation: Clauses: even(0). even(s(s(0))). even(s(s(s(X)))) :- odd(X). odd(s(0)). odd(s(X)) :- even(s(s(X))). Query: even(g) ---------------------------------------- (1) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 1, "program": { "directives": [], "clauses": [ [ "(even (0))", null ], [ "(even (s (s (0))))", null ], [ "(even (s (s (s X))))", "(odd X)" ], [ "(odd (s (0)))", null ], [ "(odd (s X))", "(even (s (s X)))" ] ] }, "graph": { "nodes": { "45": { "goal": [ { "clause": 0, "scope": 1, "term": "(even T1)" }, { "clause": 1, "scope": 1, "term": "(even T1)" }, { "clause": 2, "scope": 1, "term": "(even T1)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "type": "Nodes", "140": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "130": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "131": { "goal": [{ "clause": -1, "scope": -1, "term": "(odd T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "132": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1": { "goal": [{ "clause": -1, "scope": -1, "term": "(even T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "122": { "goal": [ { "clause": -1, "scope": -1, "term": "(true)" }, { "clause": 1, "scope": 1, "term": "(even (0))" }, { "clause": 2, "scope": 1, "term": "(even (0))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "133": { "goal": [ { "clause": 3, "scope": 2, "term": "(odd T3)" }, { "clause": 4, "scope": 2, "term": "(odd T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "123": { "goal": [ { "clause": 1, "scope": 1, "term": "(even T1)" }, { "clause": 2, "scope": 1, "term": "(even T1)" } ], "kb": { "nonunifying": [[ "(even T1)", "(even (0))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "134": { "goal": [{ "clause": 3, "scope": 2, "term": "(odd T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "124": { "goal": [ { "clause": 1, "scope": 1, "term": "(even (0))" }, { "clause": 2, "scope": 1, "term": "(even (0))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "135": { "goal": [{ "clause": 4, "scope": 2, "term": "(odd T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "125": { "goal": [{ "clause": 2, "scope": 1, "term": "(even (0))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "136": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "126": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "137": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "127": { "goal": [ { "clause": -1, "scope": -1, "term": "(true)" }, { "clause": 2, "scope": 1, "term": "(even (s (s (0))))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "138": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "128": { "goal": [{ "clause": 2, "scope": 1, "term": "(even T1)" }], "kb": { "nonunifying": [ [ "(even T1)", "(even (0))" ], [ "(even T1)", "(even (s (s (0))))" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "139": { "goal": [{ "clause": -1, "scope": -1, "term": "(even (s (s T6)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T6"], "free": [], "exprvars": [] } }, "129": { "goal": [{ "clause": 2, "scope": 1, "term": "(even (s (s (0))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 45, "label": "CASE" }, { "from": 45, "to": 122, "label": "EVAL with clause\neven(0).\nand substitutionT1 -> 0" }, { "from": 45, "to": 123, "label": "EVAL-BACKTRACK" }, { "from": 122, "to": 124, "label": "SUCCESS" }, { "from": 123, "to": 127, "label": "EVAL with clause\neven(s(s(0))).\nand substitutionT1 -> s(s(0))" }, { "from": 123, "to": 128, "label": "EVAL-BACKTRACK" }, { "from": 124, "to": 125, "label": "BACKTRACK\nfor clause: even(s(s(0)))because of non-unification" }, { "from": 125, "to": 126, "label": "BACKTRACK\nfor clause: even(s(s(s(X)))) :- odd(X)because of non-unification" }, { "from": 127, "to": 129, "label": "SUCCESS" }, { "from": 128, "to": 131, "label": "EVAL with clause\neven(s(s(s(X4)))) :- odd(X4).\nand substitutionX4 -> T3,\nT1 -> s(s(s(T3)))" }, { "from": 128, "to": 132, "label": "EVAL-BACKTRACK" }, { "from": 129, "to": 130, "label": "BACKTRACK\nfor clause: even(s(s(s(X)))) :- odd(X)because of non-unification" }, { "from": 131, "to": 133, "label": "CASE" }, { "from": 133, "to": 134, "label": "PARALLEL" }, { "from": 133, "to": 135, "label": "PARALLEL" }, { "from": 134, "to": 136, "label": "EVAL with clause\nodd(s(0)).\nand substitutionT3 -> s(0)" }, { "from": 134, "to": 137, "label": "EVAL-BACKTRACK" }, { "from": 135, "to": 139, "label": "EVAL with clause\nodd(s(X7)) :- even(s(s(X7))).\nand substitutionX7 -> T6,\nT3 -> s(T6)" }, { "from": 135, "to": 140, "label": "EVAL-BACKTRACK" }, { "from": 136, "to": 138, "label": "SUCCESS" }, { "from": 139, "to": 1, "label": "INSTANCE with matching:\nT1 -> s(s(T6))" } ], "type": "Graph" } } ---------------------------------------- (2) Obligation: Triples: evenA(s(s(s(s(X1))))) :- evenA(s(s(X1))). Clauses: evencA(0). evencA(s(s(0))). evencA(s(s(s(s(0))))). evencA(s(s(s(s(X1))))) :- evencA(s(s(X1))). Afs: evenA(x1) = evenA(x1) ---------------------------------------- (3) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: evenA_in_1: (b) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: EVENA_IN_G(s(s(s(s(X1))))) -> U1_G(X1, evenA_in_g(s(s(X1)))) EVENA_IN_G(s(s(s(s(X1))))) -> EVENA_IN_G(s(s(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 ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: EVENA_IN_G(s(s(s(s(X1))))) -> U1_G(X1, evenA_in_g(s(s(X1)))) EVENA_IN_G(s(s(s(s(X1))))) -> EVENA_IN_G(s(s(X1))) 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 1 less node. ---------------------------------------- (6) Obligation: Pi DP problem: The TRS P consists of the following rules: EVENA_IN_G(s(s(s(s(X1))))) -> EVENA_IN_G(s(s(X1))) 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: EVENA_IN_G(s(s(s(s(X1))))) -> EVENA_IN_G(s(s(X1))) 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: *EVENA_IN_G(s(s(s(s(X1))))) -> EVENA_IN_G(s(s(X1))) The graph contains the following edges 1 > 1 ---------------------------------------- (10) YES