/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern evenSpaced2(g) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToDTProblemTransformerProof [SOUND, 57 ms] (2) TRIPLES (3) TriplesToPiDPProof [SOUND, 0 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) PiDP (7) PiDPToQDPProof [SOUND, 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)). evenSpaced2(.(X1, .(X2, []))). evenSpaced2(.(X, .(Y, Ys))) :- ','(is(Diff, -(Y, X)), evenSpacedWithDiff(.(Y, Ys), Diff)). Query: evenSpaced2(g) ---------------------------------------- (1) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 2, "program": { "directives": [], "clauses": [ [ "(evenSpacedWithDiff (. X (. Y ([]))) Diff)", "(=:= Diff (- Y X))" ], [ "(evenSpacedWithDiff (. X (. Y Tail)) Diff)", "(',' (=:= Diff (- Y X)) (evenSpacedWithDiff (. Y Tail) Diff))" ], [ "(evenSpaced2 (. X1 (. X2 ([]))))", null ], [ "(evenSpaced2 (. X (. Y Ys)))", "(',' (is Diff (- Y X)) (evenSpacedWithDiff (. Y Ys) Diff))" ] ] }, "graph": { "nodes": { "45": { "goal": [{ "clause": -1, "scope": -1, "term": "(=:= T34 (- T33 T32))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [ { "lhs": { "name": "T19", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T17", "type": "PlainIntegerVariable" }, { "name": "T16", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }, { "lhs": { "arguments": [ { "name": "T32", "type": "PlainIntegerVariable" }, { "name": "T16", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T17", "type": "PlainIntegerVariable" }, { "name": "T16", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" } ] }, "ground": [ "T32", "T33", "T34" ], "free": [], "exprvars": [ "T19", "T34", "T17", "T16", "T32" ] } }, "46": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [{ "lhs": { "name": "T19", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T17", "type": "PlainIntegerVariable" }, { "name": "T16", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }] }, "ground": [], "free": [], "exprvars": [ "T19", "T17", "T16" ] } }, "47": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "28": { "goal": [{ "clause": -1, "scope": -1, "term": "(evenSpacedWithDiff (. 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T17 T18) T19)" }], "kb": { "nonunifying": [[ "(evenSpaced2 (. T16 (. T17 T18)))", "(evenSpaced2 (. X5 (. X6 ([]))))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [{ "lhs": { "name": "T19", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T17", "type": "PlainIntegerVariable" }, { "name": "T16", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }] }, "ground": [ "T19", "T18", "T17", "T16" ], "free": [ "X5", "X6" ], "exprvars": [ "T19", "T17", "T16" ] } }, "21": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "43": { "goal": [{ "clause": 1, "scope": 3, "term": "(evenSpacedWithDiff (. T17 T18) T19)" }], "kb": { "nonunifying": [[ "(evenSpaced2 (. T16 (. T17 T18)))", "(evenSpaced2 (. X5 (. X6 ([]))))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [{ "lhs": { "name": "T19", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T17", "type": "PlainIntegerVariable" }, { "name": "T16", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "=" }] }, "ground": [ "T19", "T18", "T17", "T16" ], "free": [ "X5", "X6" ], "exprvars": [ "T19", "T17", "T16" ] } } }, "edges": [ { "from": 2, "to": 6, "label": "CASE" }, { "from": 6, "to": 12, "label": "EVAL with clause\nevenSpaced2(.(X5, .(X6, []))).\nand substitutionX5 -> T4,\nX6 -> T5,\nT1 -> .(T4, .(T5, []))" }, { "from": 6, "to": 14, "label": "EVAL-BACKTRACK" }, { "from": 12, "to": 16, "label": "SUCCESS" }, { "from": 14, "to": 35, "label": "EVAL with clause\nevenSpaced2(.(X24, .(X25, X26))) :- ','(is(X27, -(X25, X24)), evenSpacedWithDiff(.(X25, X26), X27)).\nand substitutionX24 -> T16,\nX25 -> T17,\nX26 -> T18,\nT1 -> .(T16, .(T17, T18))" }, { "from": 14, "to": 36, "label": "EVAL-BACKTRACK" }, { "from": 16, "to": 17, "label": "ONLY EVAL with clause\nevenSpaced2(.(X10, .(X11, X12))) :- ','(is(X13, -(X11, X10)), evenSpacedWithDiff(.(X11, X12), X13)).\nand substitutionT4 -> T8,\nX10 -> T8,\nT5 -> T9,\nX11 -> T9,\nX12 -> []" }, { "from": 17, "to": 21, "label": "IS ERROR" }, { "from": 17, "to": 28, "label": "\nX13 -> T10" }, { "from": 28, "to": 31, "label": "CASE" }, { "from": 31, "to": 32, "label": "BACKTRACK\nfor clause: evenSpacedWithDiff(.(X, .(Y, [])), Diff) :- =:=(Diff, -(Y, X))because of non-unification" }, { "from": 32, "to": 33, "label": "BACKTRACK\nfor clause: evenSpacedWithDiff(.(X, .(Y, Tail)), Diff) :- ','(=:=(Diff, -(Y, X)), evenSpacedWithDiff(.(Y, Tail), Diff))because of non-unification" }, { "from": 35, "to": 37, "label": "IS ERROR" }, { "from": 35, "to": 40, "label": "\nX27 -> T19" }, { "from": 40, "to": 41, "label": "CASE" }, { "from": 41, "to": 42, "label": "PARALLEL" }, { "from": 41, "to": 43, "label": "PARALLEL" }, { "from": 42, "to": 45, "label": "EVAL with clause\nevenSpacedWithDiff(.(X40, .(X41, [])), X42) :- =:=(X42, -(X41, X40)).\nand substitutionT17 -> T32,\nX40 -> T32,\nX41 -> T33,\nT18 -> .(T33, []),\nT19 -> T34,\nX42 -> T34" }, { "from": 42, "to": 46, "label": "EVAL-BACKTRACK" }, { "from": 43, "to": 1704, "label": "EVAL with clause\nevenSpacedWithDiff(.(X51, .(X52, X53)), X54) :- ','(=:=(X54, -(X52, X51)), evenSpacedWithDiff(.(X52, X53), X54)).\nand substitutionT17 -> T43,\nX51 -> T43,\nX52 -> T44,\nX53 -> T45,\nT18 -> .(T44, T45),\nT19 -> T46,\nX54 -> T46" }, { "from": 43, "to": 1705, "label": "EVAL-BACKTRACK" }, { "from": 45, "to": 47, "label": "IS ERROR" }, { "from": 45, "to": 1695, "label": "ARITHCOMP SUCCESS" }, { "from": 45, "to": 1696, "label": "ARITHCOMP FAIL" }, { "from": 1695, "to": 1697, "label": "SUCCESS" }, { "from": 1704, "to": 1706, "label": "IS ERROR" }, { "from": 1704, "to": 1712, "label": "ARITHCOMP SUCCESS" }, { "from": 1704, "to": 1713, "label": "ARITHCOMP FAIL" }, { "from": 1712, "to": 1714, "label": "CASE" }, { "from": 1714, "to": 1715, "label": "PARALLEL" }, { "from": 1714, "to": 1716, "label": "PARALLEL" }, { "from": 1715, "to": 1717, "label": "EVAL with clause\nevenSpacedWithDiff(.(X70, .(X71, [])), X72) :- =:=(X72, -(X71, X70)).\nand substitutionT44 -> T62,\nX70 -> T62,\nX71 -> T63,\nT45 -> .(T63, []),\nT46 -> T64,\nX72 -> T64" }, { "from": 1715, "to": 1718, "label": "EVAL-BACKTRACK" }, { "from": 1716, "to": 1739, "label": "EVAL with clause\nevenSpacedWithDiff(.(X81, .(X82, X83)), X84) :- ','(=:=(X84, -(X82, X81)), evenSpacedWithDiff(.(X82, X83), X84)).\nand substitutionT44 -> T73,\nX81 -> T73,\nX82 -> T74,\nX83 -> T75,\nT45 -> .(T74, T75),\nT46 -> T76,\nX84 -> T76" }, { "from": 1716, "to": 1740, "label": "EVAL-BACKTRACK" }, { "from": 1717, "to": 1719, "label": "IS ERROR" }, { "from": 1717, "to": 1734, "label": "ARITHCOMP SUCCESS" }, { "from": 1717, "to": 1735, "label": "ARITHCOMP FAIL" }, { "from": 1734, "to": 1736, "label": "SUCCESS" }, { "from": 1739, "to": 1741, "label": "IS ERROR" }, { "from": 1739, "to": 1757, "label": "ARITHCOMP SUCCESS" }, { "from": 1739, "to": 1758, "label": "ARITHCOMP FAIL" }, { "from": 1757, "to": 1712, "label": "INSTANCE with matching:\nT44 -> T74\nT45 -> T75\nT46 -> T76" } ], "type": "Graph" } } ---------------------------------------- (2) Obligation: Triples: evenSpacedWithDiffA(X1, .(X2, X3), X4) :- evenSpacedWithDiffA(X2, X3, X4). evenSpaced2B(.(X1, .(X2, .(X3, X4)))) :- evenSpacedWithDiffA(X3, X4, X5). Clauses: evenSpacedWithDiffcA(X1, .(X2, []), X3). evenSpacedWithDiffcA(X1, .(X2, X3), X4) :- evenSpacedWithDiffcA(X2, X3, X4). Afs: evenSpaced2B(x1) = evenSpaced2B(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: evenSpaced2B_in_1: (b) evenSpacedWithDiffA_in_3: (b,b,f) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: EVENSPACED2B_IN_G(.(X1, .(X2, .(X3, X4)))) -> U2_G(X1, X2, X3, X4, evenSpacedWithDiffA_in_gga(X3, X4, X5)) EVENSPACED2B_IN_G(.(X1, .(X2, .(X3, X4)))) -> EVENSPACEDWITHDIFFA_IN_GGA(X3, X4, X5) EVENSPACEDWITHDIFFA_IN_GGA(X1, .(X2, X3), X4) -> U1_GGA(X1, X2, X3, X4, evenSpacedWithDiffA_in_gga(X2, X3, X4)) EVENSPACEDWITHDIFFA_IN_GGA(X1, .(X2, X3), X4) -> EVENSPACEDWITHDIFFA_IN_GGA(X2, X3, X4) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) evenSpacedWithDiffA_in_gga(x1, x2, x3) = evenSpacedWithDiffA_in_gga(x1, x2) EVENSPACED2B_IN_G(x1) = EVENSPACED2B_IN_G(x1) U2_G(x1, x2, x3, x4, x5) = U2_G(x1, x2, x3, x4, x5) EVENSPACEDWITHDIFFA_IN_GGA(x1, x2, x3) = EVENSPACEDWITHDIFFA_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4, x5) = U1_GGA(x1, x2, x3, x5) 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: EVENSPACED2B_IN_G(.(X1, .(X2, .(X3, X4)))) -> U2_G(X1, X2, X3, X4, evenSpacedWithDiffA_in_gga(X3, X4, X5)) EVENSPACED2B_IN_G(.(X1, .(X2, .(X3, X4)))) -> EVENSPACEDWITHDIFFA_IN_GGA(X3, X4, X5) EVENSPACEDWITHDIFFA_IN_GGA(X1, .(X2, X3), X4) -> U1_GGA(X1, X2, X3, X4, evenSpacedWithDiffA_in_gga(X2, X3, X4)) EVENSPACEDWITHDIFFA_IN_GGA(X1, .(X2, X3), X4) -> EVENSPACEDWITHDIFFA_IN_GGA(X2, X3, X4) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) evenSpacedWithDiffA_in_gga(x1, x2, x3) = evenSpacedWithDiffA_in_gga(x1, x2) EVENSPACED2B_IN_G(x1) = EVENSPACED2B_IN_G(x1) U2_G(x1, x2, x3, x4, x5) = U2_G(x1, x2, x3, x4, x5) EVENSPACEDWITHDIFFA_IN_GGA(x1, x2, x3) = EVENSPACEDWITHDIFFA_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4, x5) = U1_GGA(x1, x2, x3, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes. ---------------------------------------- (6) Obligation: Pi DP problem: The TRS P consists of the following rules: EVENSPACEDWITHDIFFA_IN_GGA(X1, .(X2, X3), X4) -> EVENSPACEDWITHDIFFA_IN_GGA(X2, X3, X4) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) EVENSPACEDWITHDIFFA_IN_GGA(x1, x2, x3) = EVENSPACEDWITHDIFFA_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (7) PiDPToQDPProof (SOUND) 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_GGA(X1, .(X2, X3)) -> EVENSPACEDWITHDIFFA_IN_GGA(X2, 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: *EVENSPACEDWITHDIFFA_IN_GGA(X1, .(X2, X3)) -> EVENSPACEDWITHDIFFA_IN_GGA(X2, X3) The graph contains the following edges 2 > 1, 2 > 2 ---------------------------------------- (10) YES