/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern evenSpacedWithDiff(g,g) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToDTProblemTransformerProof [SOUND, 61 ms] (2) TRIPLES (3) TriplesToPiDPProof [SOUND, 3 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: evenSpacedWithDiff(.(X, .(Y, [])), Diff) :- =:=(Diff, -(Y, X)). evenSpacedWithDiff(.(X, .(Y, Tail)), Diff) :- ','(=:=(Diff, -(Y, X)), evenSpacedWithDiff(.(Y, Tail), Diff)). Query: evenSpacedWithDiff(g,g) ---------------------------------------- (1) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 1, "program": { "directives": [], "clauses": [ [ "(evenSpacedWithDiff (. X (. Y ([]))) Diff)", "(=:= Diff (- Y X))" ], [ "(evenSpacedWithDiff (. X (. Y Tail)) Diff)", "(',' (=:= Diff (- Y X)) (evenSpacedWithDiff (. Y Tail) Diff))" ] ] }, "graph": { "nodes": { "1462": { "goal": [{ "clause": 1, "scope": 1, "term": "(evenSpacedWithDiff (. T6 (. 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"intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [ { "lhs": { "name": "T32", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T30", "type": "PlainIntegerVariable" }, { "name": "T29", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "arguments": [ { "name": "T45", "type": "PlainIntegerVariable" }, { "name": "T29", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T30", "type": "PlainIntegerVariable" }, { "name": "T29", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "name": "T47", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T46", "type": "PlainIntegerVariable" }, { "name": "T45", "type": 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"from": 1460, "to": 1462, "label": "SUCCESS" }, { "from": 1461, "to": 1488, "label": "ONLY EVAL with clause\nevenSpacedWithDiff(.(X26, .(X27, X28)), X29) :- ','(=:=(X29, -(X27, X26)), evenSpacedWithDiff(.(X27, X28), X29)).\nand substitutionT6 -> T20,\nX26 -> T20,\nT7 -> T21,\nX27 -> T21,\nX28 -> [],\nT8 -> T22,\nX29 -> T22" }, { "from": 1462, "to": 1463, "label": "ONLY EVAL with clause\nevenSpacedWithDiff(.(X11, .(X12, X13)), X14) :- ','(=:=(X14, -(X12, X11)), evenSpacedWithDiff(.(X12, X13), X14)).\nand substitutionT6 -> T12,\nX11 -> T12,\nT7 -> T13,\nX12 -> T13,\nX13 -> [],\nT8 -> T14,\nX14 -> T14" }, { "from": 1463, "to": 1481, "label": "ARITHCOMP SUCCESS" }, { "from": 1463, "to": 1482, "label": "ARITHCOMP FAIL" }, { "from": 1481, "to": 1483, "label": "CASE" }, { "from": 1483, "to": 1486, "label": "BACKTRACK\nfor clause: evenSpacedWithDiff(.(X, .(Y, [])), Diff) :- =:=(Diff, -(Y, X))because of non-unification" }, { "from": 1486, "to": 1487, "label": "BACKTRACK\nfor clause: evenSpacedWithDiff(.(X, .(Y, Tail)), Diff) :- ','(=:=(Diff, -(Y, X)), evenSpacedWithDiff(.(Y, Tail), Diff))because of non-unification" }, { "from": 1488, "to": 1490, "label": "ARITHCOMP SUCCESS" }, { "from": 1488, "to": 1491, "label": "ARITHCOMP FAIL" }, { "from": 1490, "to": 1492, "label": "CASE" }, { "from": 1492, "to": 1493, "label": "BACKTRACK\nfor clause: evenSpacedWithDiff(.(X, .(Y, [])), Diff) :- =:=(Diff, -(Y, X))because of non-unification" }, { "from": 1493, "to": 1494, "label": "BACKTRACK\nfor clause: evenSpacedWithDiff(.(X, .(Y, Tail)), Diff) :- ','(=:=(Diff, -(Y, X)), evenSpacedWithDiff(.(Y, Tail), Diff))because of non-unification" }, { "from": 1495, "to": 1497, "label": "IS ERROR" }, { "from": 1495, "to": 1512, "label": "ARITHCOMP SUCCESS" }, { "from": 1495, "to": 1513, "label": "ARITHCOMP FAIL" }, { "from": 1512, "to": 1514, "label": "CASE" }, { "from": 1514, "to": 1515, "label": "PARALLEL" }, { "from": 1514, "to": 1516, "label": "PARALLEL" }, { "from": 1515, "to": 1517, "label": "EVAL with clause\nevenSpacedWithDiff(.(X57, .(X58, [])), X59) :- =:=(X59, -(X58, X57)).\nand substitutionT30 -> T45,\nX57 -> T45,\nX58 -> T46,\nT31 -> .(T46, []),\nT32 -> T47,\nX59 -> T47" }, { "from": 1515, "to": 1518, "label": "EVAL-BACKTRACK" }, { "from": 1516, "to": 1527, "label": "EVAL with clause\nevenSpacedWithDiff(.(X68, .(X69, X70)), X71) :- ','(=:=(X71, -(X69, X68)), evenSpacedWithDiff(.(X69, X70), X71)).\nand substitutionT30 -> T56,\nX68 -> T56,\nX69 -> T57,\nX70 -> T58,\nT31 -> .(T57, T58),\nT32 -> T59,\nX71 -> T59" }, { "from": 1516, "to": 1528, "label": "EVAL-BACKTRACK" }, { "from": 1517, "to": 1519, "label": "IS ERROR" }, { "from": 1517, "to": 1524, "label": "ARITHCOMP SUCCESS" }, { "from": 1517, "to": 1525, "label": "ARITHCOMP FAIL" }, { "from": 1524, "to": 1526, "label": "SUCCESS" }, { "from": 1527, "to": 1529, "label": "IS ERROR" }, { "from": 1527, "to": 1567, "label": "ARITHCOMP SUCCESS" }, { "from": 1527, "to": 1568, "label": "ARITHCOMP FAIL" }, { "from": 1567, "to": 1, "label": "INSTANCE with matching:\nT1 -> .(T57, T58)\nT2 -> T59" } ], "type": "Graph" } } ---------------------------------------- (2) Obligation: Triples: evenSpacedWithDiffA(.(X1, .(X2, .(X3, X4))), X5) :- evenSpacedWithDiffA(.(X3, X4), X5). Clauses: evenSpacedWithDiffcA(.(X1, .(X2, [])), X3). evenSpacedWithDiffcA(.(X1, .(X2, .(X3, []))), X4). evenSpacedWithDiffcA(.(X1, .(X2, .(X3, X4))), X5) :- evenSpacedWithDiffcA(.(X3, X4), X5). Afs: evenSpacedWithDiffA(x1, x2) = evenSpacedWithDiffA(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: evenSpacedWithDiffA_in_2: (b,b) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: EVENSPACEDWITHDIFFA_IN_GG(.(X1, .(X2, .(X3, X4))), X5) -> U1_GG(X1, X2, X3, X4, X5, evenSpacedWithDiffA_in_gg(.(X3, X4), X5)) EVENSPACEDWITHDIFFA_IN_GG(.(X1, .(X2, .(X3, X4))), X5) -> EVENSPACEDWITHDIFFA_IN_GG(.(X3, X4), X5) 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: EVENSPACEDWITHDIFFA_IN_GG(.(X1, .(X2, .(X3, X4))), X5) -> U1_GG(X1, X2, X3, X4, X5, evenSpacedWithDiffA_in_gg(.(X3, X4), X5)) EVENSPACEDWITHDIFFA_IN_GG(.(X1, .(X2, .(X3, X4))), X5) -> EVENSPACEDWITHDIFFA_IN_GG(.(X3, X4), X5) 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: EVENSPACEDWITHDIFFA_IN_GG(.(X1, .(X2, .(X3, X4))), X5) -> EVENSPACEDWITHDIFFA_IN_GG(.(X3, X4), X5) 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: EVENSPACEDWITHDIFFA_IN_GG(.(X1, .(X2, .(X3, X4))), X5) -> EVENSPACEDWITHDIFFA_IN_GG(.(X3, X4), X5) 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: *EVENSPACEDWITHDIFFA_IN_GG(.(X1, .(X2, .(X3, X4))), X5) -> EVENSPACEDWITHDIFFA_IN_GG(.(X3, X4), X5) The graph contains the following edges 1 > 1, 2 >= 2 ---------------------------------------- (10) YES