/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Left Termination of the query pattern f(g) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToDTProblemTransformerProof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, 9 ms] (8) QDP (9) QDPSizeChangeProof [EQUIVALENT, 0 ms] (10) YES ---------------------------------------- (0) Obligation: Clauses: f(X) :- g(s(s(s(X)))). f(s(X)) :- f(X). g(s(s(s(s(X))))) :- f(X). Query: f(g) ---------------------------------------- (1) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 3, "program": { "directives": [], "clauses": [ [ "(f X)", "(g (s (s (s X))))" ], [ "(f (s X))", "(f X)" ], [ "(g (s (s (s (s X)))))", "(f X)" ] ] }, "graph": { "nodes": { "34": { "goal": [ { "clause": 2, "scope": 2, "term": "(g (s (s (s T3))))" }, { "clause": -1, "scope": 2, "term": null }, { "clause": 1, "scope": 1, "term": "(f T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "35": { "goal": [{ "clause": 2, "scope": 2, "term": "(g (s (s (s T3))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "101": { "goal": [{ "clause": 1, "scope": 1, "term": "(f T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "3": { "goal": [{ "clause": -1, "scope": -1, "term": "(f T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "36": { "goal": [ { "clause": -1, "scope": 2, "term": null }, { "clause": 1, "scope": 1, "term": "(f T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "102": { "goal": [{ "clause": -1, "scope": -1, "term": "(f T12)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T12"], "free": [], "exprvars": [] } }, "103": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "5": { "goal": [ { "clause": 0, "scope": 1, "term": "(f T1)" }, { "clause": 1, "scope": 1, "term": "(f T1)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "38": { "goal": [{ "clause": -1, "scope": -1, "term": "(f T8)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T8"], "free": [], "exprvars": [] } }, "39": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "31": { "goal": [ { "clause": -1, "scope": -1, "term": "(g (s (s (s T3))))" }, { "clause": 1, "scope": 1, "term": "(f T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } } }, "edges": [ { "from": 3, "to": 5, "label": "CASE" }, { "from": 5, "to": 31, "label": "ONLY EVAL with clause\nf(X2) :- g(s(s(s(X2)))).\nand substitutionT1 -> T3,\nX2 -> T3" }, { "from": 31, "to": 34, "label": "CASE" }, { "from": 34, "to": 35, "label": "PARALLEL" }, { "from": 34, "to": 36, "label": "PARALLEL" }, { "from": 35, "to": 38, "label": "EVAL with clause\ng(s(s(s(s(X7))))) :- f(X7).\nand substitutionX7 -> T8,\nT3 -> s(T8)" }, { "from": 35, "to": 39, "label": "EVAL-BACKTRACK" }, { "from": 36, "to": 101, "label": "FAILURE" }, { "from": 38, "to": 3, "label": "INSTANCE with matching:\nT1 -> T8" }, { "from": 101, "to": 102, "label": "EVAL with clause\nf(s(X11)) :- f(X11).\nand substitutionX11 -> T12,\nT3 -> s(T12)" }, { "from": 101, "to": 103, "label": "EVAL-BACKTRACK" }, { "from": 102, "to": 3, "label": "INSTANCE with matching:\nT1 -> T12" } ], "type": "Graph" } } ---------------------------------------- (2) Obligation: Triples: fA(s(X1)) :- fA(X1). fA(s(X1)) :- fA(X1). Clauses: fcA(s(X1)) :- fcA(X1). fcA(s(X1)) :- fcA(X1). Afs: fA(x1) = fA(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: fA_in_1: (b) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: FA_IN_G(s(X1)) -> U1_G(X1, fA_in_g(X1)) FA_IN_G(s(X1)) -> FA_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 ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: FA_IN_G(s(X1)) -> U1_G(X1, fA_in_g(X1)) FA_IN_G(s(X1)) -> FA_IN_G(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: FA_IN_G(s(X1)) -> FA_IN_G(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: FA_IN_G(s(X1)) -> FA_IN_G(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: *FA_IN_G(s(X1)) -> FA_IN_G(X1) The graph contains the following edges 1 > 1 ---------------------------------------- (10) YES