/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern list(g) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1)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: list([]). list(.(X1, Ts)) :- list(Ts). Query: list(g) ---------------------------------------- (1) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 1, "program": { "directives": [], "clauses": [ [ "(list ([]))", null ], [ "(list (. X1 Ts))", "(list Ts)" ] ] }, "graph": { "nodes": { "77": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "67": { "goal": [ { "clause": 0, "scope": 1, "term": "(list T1)" }, { "clause": 1, "scope": 1, "term": "(list T1)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "78": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "68": { "goal": [ { "clause": -1, "scope": -1, "term": "(true)" }, { "clause": 1, "scope": 1, "term": "(list ([]))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "79": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "69": { "goal": [{ "clause": 1, "scope": 1, "term": "(list T1)" }], "kb": { "nonunifying": [[ "(list T1)", "(list ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "type": "Nodes", "1": { "goal": [{ "clause": -1, "scope": -1, "term": "(list T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "80": { "goal": [{ "clause": -1, "scope": -1, "term": "(list T11)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T11"], "free": [], "exprvars": [] } }, "70": { "goal": [{ "clause": 1, "scope": 1, "term": "(list ([]))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "81": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "71": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "72": { "goal": [{ "clause": -1, "scope": -1, "term": "(list T5)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T5"], "free": [], "exprvars": [] } }, "73": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "74": { "goal": [ { "clause": 0, "scope": 2, "term": "(list T5)" }, { "clause": 1, "scope": 2, "term": "(list T5)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T5"], "free": [], "exprvars": [] } }, "75": { "goal": [{ "clause": 0, "scope": 2, "term": "(list T5)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T5"], "free": [], "exprvars": [] } }, "76": { "goal": [{ "clause": 1, "scope": 2, "term": "(list T5)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T5"], "free": [], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 67, "label": "CASE" }, { "from": 67, "to": 68, "label": "EVAL with clause\nlist([]).\nand substitutionT1 -> []" }, { "from": 67, "to": 69, "label": "EVAL-BACKTRACK" }, { "from": 68, "to": 70, "label": "SUCCESS" }, { "from": 69, "to": 72, "label": "EVAL with clause\nlist(.(X6, X7)) :- list(X7).\nand substitutionX6 -> T4,\nX7 -> T5,\nT1 -> .(T4, T5)" }, { "from": 69, "to": 73, "label": "EVAL-BACKTRACK" }, { "from": 70, "to": 71, "label": "BACKTRACK\nfor clause: list(.(X1, Ts)) :- list(Ts)because of non-unification" }, { "from": 72, "to": 74, "label": "CASE" }, { "from": 74, "to": 75, "label": "PARALLEL" }, { "from": 74, "to": 76, "label": "PARALLEL" }, { "from": 75, "to": 77, "label": "EVAL with clause\nlist([]).\nand substitutionT5 -> []" }, { "from": 75, "to": 78, "label": "EVAL-BACKTRACK" }, { "from": 76, "to": 80, "label": "EVAL with clause\nlist(.(X12, X13)) :- list(X13).\nand substitutionX12 -> T10,\nX13 -> T11,\nT5 -> .(T10, T11)" }, { "from": 76, "to": 81, "label": "EVAL-BACKTRACK" }, { "from": 77, "to": 79, "label": "SUCCESS" }, { "from": 80, "to": 1, "label": "INSTANCE with matching:\nT1 -> T11" } ], "type": "Graph" } } ---------------------------------------- (2) Obligation: Triples: listA(.(X1, .(X2, X3))) :- listA(X3). Clauses: listcA([]). listcA(.(X1, [])). listcA(.(X1, .(X2, X3))) :- listcA(X3). Afs: listA(x1) = listA(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: listA_in_1: (b) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: LISTA_IN_G(.(X1, .(X2, X3))) -> U1_G(X1, X2, X3, listA_in_g(X3)) LISTA_IN_G(.(X1, .(X2, X3))) -> LISTA_IN_G(X3) 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: LISTA_IN_G(.(X1, .(X2, X3))) -> U1_G(X1, X2, X3, listA_in_g(X3)) LISTA_IN_G(.(X1, .(X2, X3))) -> LISTA_IN_G(X3) 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: LISTA_IN_G(.(X1, .(X2, X3))) -> LISTA_IN_G(X3) 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: LISTA_IN_G(.(X1, .(X2, X3))) -> LISTA_IN_G(X3) 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: *LISTA_IN_G(.(X1, .(X2, X3))) -> LISTA_IN_G(X3) The graph contains the following edges 1 > 1 ---------------------------------------- (10) YES