/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern num(g) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1)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: num(0) :- !. num(X) :- ','(p(X, Y), num(Y)). p(0, 0). p(s(X), X). Query: num(g) ---------------------------------------- (1) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 2, "program": { "directives": [], "clauses": [ [ "(num (0))", "(!)" ], [ "(num X)", "(',' (p X Y) (num Y))" ], [ "(p (0) (0))", null ], [ "(p (s X) X)", null ] ] }, "graph": { "nodes": { "88": { "goal": [ { "clause": 2, "scope": 2, "term": "(',' (p T3 X3) (num X3))" }, { "clause": 3, "scope": 2, "term": "(',' (p T3 X3) (num X3))" } ], "kb": { "nonunifying": [[ "(num T3)", "(num (0))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": ["X3"], "exprvars": [] } }, "89": { "goal": [{ "clause": 3, "scope": 2, "term": "(',' (p T3 X3) (num X3))" }], "kb": { "nonunifying": [[ "(num T3)", "(num (0))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": ["X3"], "exprvars": [] } }, "2": { "goal": [{ "clause": -1, "scope": -1, "term": "(num T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "79": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "14": { "goal": [ { "clause": 0, "scope": 1, "term": "(num T1)" }, { "clause": 1, "scope": 1, "term": "(num T1)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "90": { "goal": [{ "clause": -1, "scope": -1, "term": "(num T6)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T6"], "free": [], "exprvars": [] } }, "91": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "71": { "goal": [ { "clause": -1, "scope": -1, "term": "(!_1)" }, { "clause": 1, "scope": 1, "term": "(num (0))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "83": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "73": { "goal": [{ "clause": 1, "scope": 1, "term": "(num T1)" }], "kb": { "nonunifying": [[ "(num T1)", "(num (0))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "87": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (p T3 X3) (num X3))" }], "kb": { "nonunifying": [[ "(num T3)", "(num (0))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": ["X3"], "exprvars": [] } } }, "edges": [ { "from": 2, "to": 14, "label": "CASE" }, { "from": 14, "to": 71, "label": "EVAL with clause\nnum(0) :- !_1.\nand substitutionT1 -> 0" }, { "from": 14, "to": 73, "label": "EVAL-BACKTRACK" }, { "from": 71, "to": 79, "label": "CUT" }, { "from": 73, "to": 87, "label": "ONLY EVAL with clause\nnum(X2) :- ','(p(X2, X3), num(X3)).\nand substitutionT1 -> T3,\nX2 -> T3" }, { "from": 79, "to": 83, "label": "SUCCESS" }, { "from": 87, "to": 88, "label": "CASE" }, { "from": 88, "to": 89, "label": "BACKTRACK\nfor clause: p(0, 0)\nwith clash: (num(T3), num(0))" }, { "from": 89, "to": 90, "label": "EVAL with clause\np(s(X6), X6).\nand substitutionX6 -> T6,\nT3 -> s(T6),\nX3 -> T6" }, { "from": 89, "to": 91, "label": "EVAL-BACKTRACK" }, { "from": 90, "to": 2, "label": "INSTANCE with matching:\nT1 -> T6" } ], "type": "Graph" } } ---------------------------------------- (2) Obligation: Triples: numA(s(X1)) :- numA(X1). Clauses: numcA(0). numcA(s(X1)) :- numcA(X1). Afs: numA(x1) = numA(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: numA_in_1: (b) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: NUMA_IN_G(s(X1)) -> U1_G(X1, numA_in_g(X1)) NUMA_IN_G(s(X1)) -> NUMA_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: NUMA_IN_G(s(X1)) -> U1_G(X1, numA_in_g(X1)) NUMA_IN_G(s(X1)) -> NUMA_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: NUMA_IN_G(s(X1)) -> NUMA_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: NUMA_IN_G(s(X1)) -> NUMA_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: *NUMA_IN_G(s(X1)) -> NUMA_IN_G(X1) The graph contains the following edges 1 > 1 ---------------------------------------- (10) YES