/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern rev(g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToDTProblemTransformerProof [SOUND, 101 ms] (2) TRIPLES (3) TriplesToPiDPProof [SOUND, 3 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: rev(L, R) :- rev(L, [], R). rev([], Y, Z) :- ','(!, eq(Y, Z)). rev(L, S, R) :- ','(head(L, X), ','(tail(L, T), rev(T, .(X, S), R))). head([], X1). head(.(X, X2), X). tail([], []). tail(.(X3, Xs), Xs). eq(X, X). Query: rev(g,a) ---------------------------------------- (1) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 16, "program": { "directives": [], "clauses": [ [ "(rev L R)", "(rev L ([]) R)" ], [ "(rev ([]) Y Z)", "(',' (!) (eq Y Z))" ], [ "(rev L S R)", "(',' (head L X) (',' (tail L T) (rev T (. X S) R)))" ], [ "(head ([]) X1)", null ], [ "(head (. X X2) X)", null ], [ "(tail ([]) ([]))", null ], [ "(tail (. X3 Xs) Xs)", null ], [ "(eq X X)", null ] ] }, "graph": { "nodes": { "type": "Nodes", "110": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "111": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "112": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "478": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (head T183 X175) (',' (tail T183 X176) (rev X176 (. X175 (. T184 (. T185 (. T186 ([]))))) T188)))" }], "kb": { "nonunifying": [[ "(rev T183 (. T184 (. T185 (. 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T368 ([]))))))) T370)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T364", "T365", "T366", "T367", "T368", "T401", "T402" ], "free": [], "exprvars": [] } }, "460": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "581": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (!_26) (eq (. T424 (. T425 (. T426 (. T427 (. T428 (. T429 ([]))))))) T431))" }, { "clause": 2, "scope": 26, "term": "(rev ([]) (. T424 (. T425 (. T426 (. T427 (. T428 (. T429 ([]))))))) T370)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T424", "T425", "T426", "T427", "T428", "T429" ], "free": [], "exprvars": [] } }, "582": { "goal": [{ "clause": 2, "scope": 26, "term": "(rev T402 (. T401 (. T364 (. T365 (. T366 (. T367 (. T368 ([]))))))) T370)" }], "kb": { "nonunifying": [[ "(rev T402 (. T401 (. T364 (. T365 (. T366 (. T367 (. T368 ([]))))))) T370)", "(rev ([]) X300 X301)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T364", "T365", "T366", "T367", "T368", "T401", "T402" ], "free": [ "X300", "X301" ], "exprvars": [] } }, "583": { "goal": [{ "clause": -1, "scope": -1, "term": "(eq (. T424 (. T425 (. T426 (. T427 (. T428 (. T429 ([]))))))) T431)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T424", "T425", "T426", "T427", "T428", "T429" ], "free": [], "exprvars": [] } }, "100": { "goal": [ { "clause": 1, "scope": 6, "term": "(rev T37 (. T36 ([])) T20)" }, { "clause": 2, "scope": 6, "term": "(rev T37 (. T36 ([])) T20)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T36", "T37" ], "free": [], "exprvars": [] } }, "584": { "goal": [{ "clause": 7, "scope": 27, "term": "(eq (. T424 (. T425 (. T426 (. T427 (. T428 (. T429 ([]))))))) T431)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T424", "T425", "T426", "T427", "T428", "T429" ], "free": [], "exprvars": [] } }, "102": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (!_6) (eq (. T44 ([])) T46))" }, { "clause": 2, "scope": 6, "term": "(rev ([]) (. T44 ([])) T20)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T44"], "free": [], "exprvars": [] } }, "104": { "goal": [{ "clause": 2, "scope": 6, "term": "(rev T37 (. T36 ([])) T20)" }], "kb": { "nonunifying": [[ "(rev T37 (. T36 ([])) T20)", "(rev ([]) X60 X61)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T36", "T37" ], "free": [ "X60", "X61" ], "exprvars": [] } }, "106": { "goal": [{ "clause": -1, "scope": -1, "term": "(eq (. T44 ([])) T46)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T44"], "free": [], "exprvars": [] } }, "107": { "goal": [{ "clause": 7, "scope": 7, "term": "(eq (. T44 ([])) T46)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T44"], "free": [], "exprvars": [] } }, "503": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "624": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (head T474 X319) (',' (tail T474 X320) (rev X320 (. X319 (. T475 (. T476 (. T477 (. T478 (. T479 (. T480 ([])))))))) T482)))" }], "kb": { "nonunifying": [[ "(rev T474 (. T475 (. T476 (. T477 (. T478 (. T479 (. T480 ([]))))))) T370)", "(rev ([]) X300 X301)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T474", "T475", "T476", "T477", "T478", "T479", "T480" ], "free": [ "X300", "X301", "X319", "X320" ], "exprvars": [] } }, "504": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "505": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "83": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (head T18 X31) (',' (tail T18 X32) (rev X32 (. X31 ([])) T20)))" }], "kb": { "nonunifying": [[ "(rev T18 ([]) T7)", "(rev ([]) X12 X13)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T18"], "free": [ "X12", "X13", "X31", "X32" ], "exprvars": [] } }, "84": { "goal": [ { "clause": 3, "scope": 4, "term": "(',' (head T18 X31) (',' (tail T18 X32) (rev X32 (. X31 ([])) T20)))" }, { "clause": 4, "scope": 4, "term": "(',' (head T18 X31) (',' (tail T18 X32) (rev X32 (. X31 ([])) T20)))" } ], "kb": { "nonunifying": [[ "(rev T18 ([]) T7)", "(rev ([]) X12 X13)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T18"], "free": [ "X12", "X13", "X31", "X32" ], "exprvars": [] } }, "628": { "goal": [ { "clause": 3, "scope": 28, "term": "(',' (head T474 X319) (',' (tail T474 X320) (rev X320 (. X319 (. T475 (. T476 (. T477 (. T478 (. T479 (. T480 ([])))))))) T482)))" }, { "clause": 4, "scope": 28, "term": "(',' (head T474 X319) (',' (tail T474 X320) (rev X320 (. X319 (. T475 (. T476 (. T477 (. T478 (. T479 (. T480 ([])))))))) T482)))" } ], "kb": { "nonunifying": [[ "(rev T474 (. T475 (. T476 (. T477 (. T478 (. T479 (. T480 ([]))))))) T370)", "(rev ([]) X300 X301)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T474", "T475", "T476", "T477", "T478", "T479", "T480" ], "free": [ "X300", "X301", "X319", "X320" ], "exprvars": [] } }, "85": { "goal": [{ "clause": 4, "scope": 4, "term": "(',' (head T18 X31) (',' (tail T18 X32) (rev X32 (. X31 ([])) T20)))" }], "kb": { "nonunifying": [[ "(rev T18 ([]) T7)", "(rev ([]) X12 X13)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T18"], "free": [ "X12", "X13", "X31", "X32" ], "exprvars": [] } }, "629": { "goal": [{ "clause": 4, "scope": 28, "term": "(',' (head T474 X319) (',' (tail T474 X320) (rev X320 (. X319 (. T475 (. T476 (. T477 (. T478 (. T479 (. T480 ([])))))))) T482)))" }], "kb": { "nonunifying": [[ "(rev T474 (. T475 (. T476 (. T477 (. T478 (. T479 (. T480 ([]))))))) T370)", "(rev ([]) X300 X301)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T474", "T475", "T476", "T477", "T478", "T479", "T480" ], "free": [ "X300", "X301", "X319", "X320" ], "exprvars": [] } } }, "edges": [ { "from": 16, "to": 20, "label": "CASE" }, { "from": 20, "to": 21, "label": "ONLY EVAL with clause\nrev(X6, X7) :- rev(X6, [], X7).\nand substitutionT1 -> T5,\nX6 -> T5,\nT2 -> T7,\nX7 -> T7,\nT6 -> T7" }, { "from": 21, "to": 22, "label": "CASE" }, { "from": 22, "to": 39, "label": "EVAL with clause\nrev([], X12, X13) :- ','(!_2, eq(X12, X13)).\nand substitutionT5 -> [],\nX12 -> [],\nT7 -> T11,\nX13 -> T11,\nT10 -> T11" }, { "from": 22, "to": 40, "label": "EVAL-BACKTRACK" }, { "from": 39, "to": 47, "label": "CUT" }, { "from": 40, "to": 83, "label": "ONLY EVAL with clause\nrev(X28, X29, X30) :- ','(head(X28, X31), ','(tail(X28, X32), rev(X32, .(X31, X29), X30))).\nand substitutionT5 -> T18,\nX28 -> T18,\nX29 -> [],\nT7 -> T20,\nX30 -> T20,\nT19 -> T20" }, { "from": 47, "to": 53, "label": "CASE" }, { "from": 53, "to": 55, "label": "EVAL with clause\neq(X16, X16).\nand substitutionX16 -> [],\nT11 -> []" }, { "from": 53, "to": 57, "label": "EVAL-BACKTRACK" }, { "from": 55, "to": 58, "label": "SUCCESS" }, { "from": 83, "to": 84, "label": "CASE" }, { "from": 84, "to": 85, "label": "BACKTRACK\nfor clause: head([], X1)\nwith clash: (rev(T18, [], T7), rev([], X12, X13))" }, { "from": 85, "to": 90, "label": "EVAL with clause\nhead(.(X42, X43), X42).\nand substitutionX42 -> T26,\nX43 -> T27,\nT18 -> .(T26, T27),\nX31 -> T26" }, { "from": 85, "to": 91, "label": "EVAL-BACKTRACK" }, { "from": 90, "to": 95, "label": "CASE" }, { "from": 95, "to": 96, "label": "BACKTRACK\nfor clause: tail([], [])because of non-unification" }, { "from": 96, "to": 97, "label": "ONLY EVAL with clause\ntail(.(X52, X53), X53).\nand substitutionT26 -> T36,\nX52 -> T36,\nT27 -> T37,\nX53 -> T37,\nX32 -> T37" }, { "from": 97, "to": 100, "label": "CASE" }, { "from": 100, "to": 102, "label": "EVAL with clause\nrev([], X60, X61) :- ','(!_6, eq(X60, X61)).\nand substitutionT37 -> [],\nT36 -> T44,\nX60 -> .(T44, []),\nT20 -> T46,\nX61 -> T46,\nT45 -> T46" }, { "from": 100, "to": 104, "label": "EVAL-BACKTRACK" }, { "from": 102, "to": 106, "label": "CUT" }, { "from": 104, "to": 206, "label": "ONLY EVAL with clause\nrev(X76, X77, X78) :- ','(head(X76, X79), ','(tail(X76, X80), rev(X80, .(X79, X77), X78))).\nand substitutionT37 -> T59,\nX76 -> T59,\nT36 -> T60,\nX77 -> .(T60, []),\nT20 -> T62,\nX78 -> T62,\nT61 -> T62" }, { "from": 106, "to": 107, "label": "CASE" }, { "from": 107, "to": 110, "label": "EVAL with clause\neq(X64, X64).\nand substitutionT44 -> T49,\nX64 -> .(T49, []),\nT46 -> .(T49, [])" }, { "from": 107, "to": 111, "label": "EVAL-BACKTRACK" }, { "from": 110, "to": 112, "label": "SUCCESS" }, { "from": 206, "to": 292, "label": "CASE" }, { "from": 292, "to": 295, "label": "BACKTRACK\nfor clause: head([], X1)\nwith clash: (rev(T59, .(T60, []), T20), rev([], X60, X61))" }, { "from": 295, "to": 296, "label": "EVAL with clause\nhead(.(X90, X91), X90).\nand substitutionX90 -> T69,\nX91 -> T70,\nT59 -> .(T69, T70),\nX79 -> T69" }, { "from": 295, "to": 297, "label": "EVAL-BACKTRACK" }, { "from": 296, "to": 298, "label": "CASE" }, { "from": 298, "to": 299, "label": "BACKTRACK\nfor clause: tail([], [])because of non-unification" }, { "from": 299, "to": 300, "label": "ONLY EVAL with clause\ntail(.(X100, X101), X101).\nand substitutionT69 -> T81,\nX100 -> T81,\nT70 -> T82,\nX101 -> T82,\nX80 -> T82" }, { "from": 300, "to": 301, "label": "CASE" }, { "from": 301, "to": 314, "label": "EVAL with clause\nrev([], X108, X109) :- ','(!_10, eq(X108, X109)).\nand substitutionT82 -> [],\nT81 -> T92,\nT60 -> T93,\nX108 -> .(T92, .(T93, [])),\nT62 -> T95,\nX109 -> T95,\nT94 -> T95" }, { "from": 301, "to": 318, "label": "EVAL-BACKTRACK" }, { "from": 314, "to": 319, "label": "CUT" }, { "from": 318, "to": 375, "label": "ONLY EVAL with clause\nrev(X124, X125, X126) :- ','(head(X124, X127), ','(tail(X124, X128), rev(X128, .(X127, X125), X126))).\nand substitutionT82 -> T114,\nX124 -> T114,\nT81 -> T115,\nT60 -> T116,\nX125 -> .(T115, .(T116, [])),\nT62 -> T118,\nX126 -> T118,\nT117 -> T118" }, { "from": 319, "to": 320, "label": "CASE" }, { "from": 320, "to": 323, "label": "EVAL with clause\neq(X112, X112).\nand substitutionT92 -> T100,\nT93 -> T101,\nX112 -> .(T100, .(T101, [])),\nT95 -> .(T100, .(T101, []))" }, { "from": 320, "to": 325, "label": "EVAL-BACKTRACK" }, { "from": 323, "to": 327, "label": "SUCCESS" }, { "from": 375, "to": 376, "label": "CASE" }, { "from": 376, "to": 377, "label": "BACKTRACK\nfor clause: head([], X1)\nwith clash: (rev(T114, .(T115, .(T116, [])), T62), rev([], X108, X109))" }, { "from": 377, "to": 380, "label": "EVAL with clause\nhead(.(X138, X139), X138).\nand substitutionX138 -> T126,\nX139 -> T127,\nT114 -> .(T126, T127),\nX127 -> T126" }, { "from": 377, "to": 381, "label": "EVAL-BACKTRACK" }, { "from": 380, "to": 382, "label": "CASE" }, { "from": 382, "to": 383, "label": "BACKTRACK\nfor clause: tail([], [])because of non-unification" }, { "from": 383, "to": 449, "label": "ONLY EVAL with clause\ntail(.(X148, X149), X149).\nand substitutionT126 -> T140,\nX148 -> T140,\nT127 -> T141,\nX149 -> T141,\nX128 -> T141" }, { "from": 449, "to": 451, "label": "CASE" }, { "from": 451, "to": 453, "label": "EVAL with clause\nrev([], X156, X157) :- ','(!_14, eq(X156, X157)).\nand substitutionT141 -> [],\nT140 -> T154,\nT115 -> T155,\nT116 -> T156,\nX156 -> .(T154, .(T155, .(T156, []))),\nT118 -> T158,\nX157 -> T158,\nT157 -> T158" }, { "from": 451, "to": 454, "label": "EVAL-BACKTRACK" }, { "from": 453, "to": 455, "label": "CUT" }, { "from": 454, "to": 478, "label": "ONLY EVAL with clause\nrev(X172, X173, X174) :- ','(head(X172, X175), ','(tail(X172, X176), rev(X176, .(X175, X173), X174))).\nand substitutionT141 -> T183,\nX172 -> T183,\nT140 -> T184,\nT115 -> T185,\nT116 -> T186,\nX173 -> .(T184, .(T185, .(T186, []))),\nT118 -> T188,\nX174 -> T188,\nT187 -> T188" }, { "from": 455, "to": 456, "label": "CASE" }, { "from": 456, "to": 458, "label": "EVAL with clause\neq(X160, X160).\nand substitutionT154 -> T165,\nT155 -> T166,\nT156 -> T167,\nX160 -> .(T165, .(T166, .(T167, []))),\nT158 -> .(T165, .(T166, .(T167, [])))" }, { "from": 456, "to": 459, "label": "EVAL-BACKTRACK" }, { "from": 458, "to": 460, "label": "SUCCESS" }, { "from": 478, "to": 479, "label": "CASE" }, { "from": 479, "to": 481, "label": "BACKTRACK\nfor clause: head([], X1)\nwith clash: (rev(T183, .(T184, .(T185, .(T186, []))), T118), rev([], X156, X157))" }, { "from": 481, "to": 482, "label": "EVAL with clause\nhead(.(X186, X187), X186).\nand substitutionX186 -> T197,\nX187 -> T198,\nT183 -> .(T197, T198),\nX175 -> T197" }, { "from": 481, "to": 483, "label": "EVAL-BACKTRACK" }, { "from": 482, "to": 486, "label": "CASE" }, { "from": 486, "to": 487, "label": "BACKTRACK\nfor clause: tail([], [])because of non-unification" }, { "from": 487, "to": 490, "label": "ONLY EVAL with clause\ntail(.(X196, X197), X197).\nand substitutionT197 -> T213,\nX196 -> T213,\nT198 -> T214,\nX197 -> T214,\nX176 -> T214" }, { "from": 490, "to": 492, "label": "CASE" }, { "from": 492, "to": 496, "label": "EVAL with clause\nrev([], X204, X205) :- ','(!_18, eq(X204, X205)).\nand substitutionT214 -> [],\nT213 -> T230,\nT184 -> T231,\nT185 -> T232,\nT186 -> T233,\nX204 -> .(T230, .(T231, .(T232, .(T233, [])))),\nT188 -> T235,\nX205 -> T235,\nT234 -> T235" }, { "from": 492, "to": 497, "label": "EVAL-BACKTRACK" }, { "from": 496, "to": 498, "label": "CUT" }, { "from": 497, "to": 520, "label": "ONLY EVAL with clause\nrev(X220, X221, X222) :- ','(head(X220, X223), ','(tail(X220, X224), rev(X224, .(X223, X221), X222))).\nand substitutionT214 -> T266,\nX220 -> T266,\nT213 -> T267,\nT184 -> T268,\nT185 -> T269,\nT186 -> T270,\nX221 -> .(T267, .(T268, .(T269, .(T270, [])))),\nT188 -> T272,\nX222 -> T272,\nT271 -> T272" }, { "from": 498, "to": 499, "label": "CASE" }, { "from": 499, "to": 503, "label": "EVAL with clause\neq(X208, X208).\nand substitutionT230 -> T244,\nT231 -> T245,\nT232 -> T246,\nT233 -> T247,\nX208 -> .(T244, .(T245, .(T246, .(T247, [])))),\nT235 -> .(T244, .(T245, .(T246, .(T247, []))))" }, { "from": 499, "to": 504, "label": "EVAL-BACKTRACK" }, { "from": 503, "to": 505, "label": "SUCCESS" }, { "from": 520, "to": 524, "label": "CASE" }, { "from": 524, "to": 525, "label": "BACKTRACK\nfor clause: head([], X1)\nwith clash: (rev(T266, .(T267, .(T268, .(T269, .(T270, [])))), T188), rev([], X204, X205))" }, { "from": 525, "to": 526, "label": "EVAL with clause\nhead(.(X234, X235), X234).\nand substitutionX234 -> T282,\nX235 -> T283,\nT266 -> .(T282, T283),\nX223 -> T282" }, { "from": 525, "to": 527, "label": "EVAL-BACKTRACK" }, { "from": 526, "to": 528, "label": "CASE" }, { "from": 528, "to": 529, "label": "BACKTRACK\nfor clause: tail([], [])because of non-unification" }, { "from": 529, "to": 532, "label": "ONLY EVAL with clause\ntail(.(X244, X245), X245).\nand substitutionT282 -> T300,\nX244 -> T300,\nT283 -> T301,\nX245 -> T301,\nX224 -> T301" }, { "from": 532, "to": 543, "label": "CASE" }, { "from": 543, "to": 545, "label": "EVAL with clause\nrev([], X252, X253) :- ','(!_22, eq(X252, X253)).\nand substitutionT301 -> [],\nT300 -> T320,\nT267 -> T321,\nT268 -> T322,\nT269 -> T323,\nT270 -> T324,\nX252 -> .(T320, .(T321, .(T322, .(T323, .(T324, []))))),\nT272 -> T326,\nX253 -> T326,\nT325 -> T326" }, { "from": 543, "to": 546, "label": "EVAL-BACKTRACK" }, { "from": 545, "to": 547, "label": "CUT" }, { "from": 546, "to": 560, "label": "ONLY EVAL with clause\nrev(X268, X269, X270) :- ','(head(X268, X271), ','(tail(X268, X272), rev(X272, .(X271, X269), X270))).\nand substitutionT301 -> T363,\nX268 -> T363,\nT300 -> T364,\nT267 -> T365,\nT268 -> T366,\nT269 -> T367,\nT270 -> T368,\nX269 -> .(T364, .(T365, .(T366, .(T367, .(T368, []))))),\nT272 -> T370,\nX270 -> T370,\nT369 -> T370" }, { "from": 547, "to": 548, "label": "CASE" }, { "from": 548, "to": 550, "label": "EVAL with clause\neq(X256, X256).\nand substitutionT320 -> T337,\nT321 -> T338,\nT322 -> T339,\nT323 -> T340,\nT324 -> T341,\nX256 -> .(T337, .(T338, .(T339, .(T340, .(T341, []))))),\nT326 -> .(T337, .(T338, .(T339, .(T340, .(T341, [])))))" }, { "from": 548, "to": 551, "label": "EVAL-BACKTRACK" }, { "from": 550, "to": 552, "label": "SUCCESS" }, { "from": 560, "to": 561, "label": "CASE" }, { "from": 561, "to": 562, "label": "BACKTRACK\nfor clause: head([], X1)\nwith clash: (rev(T363, .(T364, .(T365, .(T366, .(T367, .(T368, []))))), T272), rev([], X252, X253))" }, { "from": 562, "to": 563, "label": "EVAL with clause\nhead(.(X282, X283), X282).\nand substitutionX282 -> T381,\nX283 -> T382,\nT363 -> .(T381, T382),\nX271 -> T381" }, { "from": 562, "to": 564, "label": "EVAL-BACKTRACK" }, { "from": 563, "to": 565, "label": "CASE" }, { "from": 565, "to": 567, "label": "BACKTRACK\nfor clause: tail([], [])because of non-unification" }, { "from": 567, "to": 570, "label": "ONLY EVAL with clause\ntail(.(X292, X293), X293).\nand substitutionT381 -> T401,\nX292 -> T401,\nT382 -> T402,\nX293 -> T402,\nX272 -> T402" }, { "from": 570, "to": 580, "label": "CASE" }, { "from": 580, "to": 581, "label": "EVAL with clause\nrev([], X300, X301) :- ','(!_26, eq(X300, X301)).\nand substitutionT402 -> [],\nT401 -> T424,\nT364 -> T425,\nT365 -> T426,\nT366 -> T427,\nT367 -> T428,\nT368 -> T429,\nX300 -> .(T424, .(T425, .(T426, .(T427, .(T428, .(T429, [])))))),\nT370 -> T431,\nX301 -> T431,\nT430 -> T431" }, { "from": 580, "to": 582, "label": "EVAL-BACKTRACK" }, { "from": 581, "to": 583, "label": "CUT" }, { "from": 582, "to": 624, "label": "ONLY EVAL with clause\nrev(X316, X317, X318) :- ','(head(X316, X319), ','(tail(X316, X320), rev(X320, .(X319, X317), X318))).\nand substitutionT402 -> T474,\nX316 -> T474,\nT401 -> T475,\nT364 -> T476,\nT365 -> T477,\nT366 -> T478,\nT367 -> T479,\nT368 -> T480,\nX317 -> .(T475, .(T476, .(T477, .(T478, .(T479, .(T480, [])))))),\nT370 -> T482,\nX318 -> T482,\nT481 -> T482" }, { "from": 583, "to": 584, "label": "CASE" }, { "from": 584, "to": 607, "label": "EVAL with clause\neq(X304, X304).\nand substitutionT424 -> T444,\nT425 -> T445,\nT426 -> T446,\nT427 -> T447,\nT428 -> T448,\nT429 -> T449,\nX304 -> .(T444, .(T445, .(T446, .(T447, .(T448, .(T449, [])))))),\nT431 -> .(T444, .(T445, .(T446, .(T447, .(T448, .(T449, []))))))" }, { "from": 584, "to": 608, "label": "EVAL-BACKTRACK" }, { "from": 607, "to": 609, "label": "SUCCESS" }, { "from": 624, "to": 628, "label": "CASE" }, { "from": 628, "to": 629, "label": "BACKTRACK\nfor clause: head([], X1)\nwith clash: (rev(T474, .(T475, .(T476, .(T477, .(T478, .(T479, .(T480, [])))))), T370), rev([], X300, X301))" }, { "from": 629, "to": 633, "label": "EVAL with clause\nhead(.(X330, X331), X330).\nand substitutionX330 -> T494,\nX331 -> T495,\nT474 -> .(T494, T495),\nX319 -> T494" }, { "from": 629, "to": 634, "label": "EVAL-BACKTRACK" }, { "from": 633, "to": 636, "label": "CASE" }, { "from": 636, "to": 637, "label": "BACKTRACK\nfor clause: tail([], [])because of non-unification" }, { "from": 637, "to": 638, "label": "ONLY EVAL with clause\ntail(.(X340, X341), X341).\nand substitutionT494 -> T516,\nX340 -> T516,\nT495 -> T517,\nX341 -> T517,\nX320 -> T517" }, { "from": 638, "to": 646, "label": "CASE" }, { "from": 646, "to": 648, "label": "EVAL with clause\nrev([], X348, X349) :- ','(!_30, eq(X348, X349)).\nand substitutionT517 -> [],\nT516 -> T542,\nT475 -> T543,\nT476 -> T544,\nT477 -> T545,\nT478 -> T546,\nT479 -> T547,\nT480 -> T548,\nX348 -> .(T542, .(T543, .(T544, .(T545, .(T546, .(T547, .(T548, []))))))),\nT482 -> T550,\nX349 -> T550,\nT549 -> T550" }, { "from": 646, "to": 649, "label": "EVAL-BACKTRACK" }, { "from": 648, "to": 650, "label": "CUT" }, { "from": 649, "to": 660, "label": "ONLY EVAL with clause\nrev(X364, X365, X366) :- ','(head(X364, X367), ','(tail(X364, X368), rev(X368, .(X367, X365), X366))).\nand substitutionT517 -> T599,\nX364 -> T599,\nT516 -> T600,\nT475 -> T601,\nT476 -> T602,\nT477 -> T603,\nT478 -> T604,\nT479 -> T605,\nT480 -> T606,\nX365 -> .(T600, .(T601, .(T602, .(T603, .(T604, .(T605, .(T606, []))))))),\nT482 -> T608,\nX366 -> T608,\nT607 -> T608" }, { "from": 650, "to": 651, "label": "CASE" }, { "from": 651, "to": 653, "label": "EVAL with clause\neq(X352, X352).\nand substitutionT542 -> T565,\nT543 -> T566,\nT544 -> T567,\nT545 -> T568,\nT546 -> T569,\nT547 -> T570,\nT548 -> T571,\nX352 -> .(T565, .(T566, .(T567, .(T568, .(T569, .(T570, .(T571, []))))))),\nT550 -> .(T565, .(T566, .(T567, .(T568, .(T569, .(T570, .(T571, [])))))))" }, { "from": 651, "to": 654, "label": "EVAL-BACKTRACK" }, { "from": 653, "to": 655, "label": "SUCCESS" }, { "from": 660, "to": 666, "label": "CASE" }, { "from": 666, "to": 667, "label": "BACKTRACK\nfor clause: head([], X1)\nwith clash: (rev(T599, .(T600, .(T601, .(T602, .(T603, .(T604, .(T605, .(T606, []))))))), T482), rev([], X348, X349))" }, { "from": 667, "to": 671, "label": "EVAL with clause\nhead(.(X378, X379), X378).\nand substitutionX378 -> T621,\nX379 -> T622,\nT599 -> .(T621, T622),\nX367 -> T621" }, { "from": 667, "to": 672, "label": "EVAL-BACKTRACK" }, { "from": 671, "to": 676, "label": "CASE" }, { "from": 676, "to": 677, "label": "BACKTRACK\nfor clause: tail([], [])because of non-unification" }, { "from": 677, "to": 679, "label": "ONLY EVAL with clause\ntail(.(X388, X389), X389).\nand substitutionT621 -> T645,\nX388 -> T645,\nT622 -> T646,\nX389 -> T646,\nX368 -> T646" }, { "from": 679, "to": 685, "label": "GENERALIZATION\nT656 <-- .(T600, .(T601, .(T602, .(T603, .(T604, .(T605, .(T606, [])))))))\n\nNew Knowledge:\nT656 is ground" }, { "from": 685, "to": 687, "label": "CASE" }, { "from": 687, "to": 689, "label": "EVAL with clause\nrev([], X398, X399) :- ','(!_34, eq(X398, X399)).\nand substitutionT646 -> [],\nT645 -> T666,\nT656 -> T667,\nX398 -> .(T666, T667),\nT608 -> T669,\nX399 -> T669,\nT668 -> T669" }, { "from": 687, "to": 690, "label": "EVAL-BACKTRACK" }, { "from": 689, "to": 691, "label": "CUT" }, { "from": 690, "to": 702, "label": "ONLY EVAL with clause\nrev(X414, X415, X416) :- ','(head(X414, X417), ','(tail(X414, X418), rev(X418, .(X417, X415), X416))).\nand substitutionT646 -> T688,\nX414 -> T688,\nT645 -> T689,\nT656 -> T690,\nX415 -> .(T689, T690),\nT608 -> T692,\nX416 -> T692,\nT691 -> T692" }, { "from": 691, "to": 692, "label": "CASE" }, { "from": 692, "to": 694, "label": "EVAL with clause\neq(X402, X402).\nand substitutionT666 -> T674,\nT667 -> T675,\nX402 -> .(T674, T675),\nT669 -> .(T674, T675)" }, { "from": 692, "to": 695, "label": "EVAL-BACKTRACK" }, { "from": 694, "to": 696, "label": "SUCCESS" }, { "from": 702, "to": 705, "label": "CASE" }, { "from": 705, "to": 706, "label": "BACKTRACK\nfor clause: head([], X1)\nwith clash: (rev(T688, .(T689, T690), T608), rev([], X398, X399))" }, { "from": 706, "to": 707, "label": "EVAL with clause\nhead(.(X428, X429), X428).\nand substitutionX428 -> T700,\nX429 -> T701,\nT688 -> .(T700, T701),\nX417 -> T700" }, { "from": 706, "to": 708, "label": "EVAL-BACKTRACK" }, { "from": 707, "to": 710, "label": "CASE" }, { "from": 710, "to": 731, "label": "BACKTRACK\nfor clause: tail([], [])because of non-unification" }, { "from": 731, "to": 733, "label": "ONLY EVAL with clause\ntail(.(X438, X439), X439).\nand substitutionT700 -> T714,\nX438 -> T714,\nT701 -> T715,\nX439 -> T715,\nX418 -> T715" }, { "from": 733, "to": 685, "label": "INSTANCE with matching:\nT646 -> T715\nT645 -> T714\nT656 -> .(T689, T690)\nT608 -> T692" } ], "type": "Graph" } } ---------------------------------------- (2) Obligation: Triples: revA(.(X1, X2), X3, X4, X5) :- revA(X2, X1, .(X3, X4), X5). revB(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) :- revA(X9, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))), X10). Clauses: revcA([], X1, X2, .(X1, X2)). revcA(.(X1, X2), X3, X4, X5) :- revcA(X2, X1, .(X3, X4), X5). Afs: revB(x1, x2) = revB(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: revB_in_2: (b,f) revA_in_4: (b,b,b,f) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: REVB_IN_GA(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) -> U2_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revA_in_ggga(X9, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))), X10)) REVB_IN_GA(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) -> REVA_IN_GGGA(X9, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))), X10) REVA_IN_GGGA(.(X1, X2), X3, X4, X5) -> U1_GGGA(X1, X2, X3, X4, X5, revA_in_ggga(X2, X1, .(X3, X4), X5)) REVA_IN_GGGA(.(X1, X2), X3, X4, X5) -> REVA_IN_GGGA(X2, X1, .(X3, X4), X5) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) revA_in_ggga(x1, x2, x3, x4) = revA_in_ggga(x1, x2, x3) [] = [] REVB_IN_GA(x1, x2) = REVB_IN_GA(x1) U2_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U2_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x11) REVA_IN_GGGA(x1, x2, x3, x4) = REVA_IN_GGGA(x1, x2, x3) U1_GGGA(x1, x2, x3, x4, x5, x6) = U1_GGGA(x1, x2, x3, x4, x6) 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: REVB_IN_GA(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) -> U2_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revA_in_ggga(X9, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))), X10)) REVB_IN_GA(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) -> REVA_IN_GGGA(X9, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))), X10) REVA_IN_GGGA(.(X1, X2), X3, X4, X5) -> U1_GGGA(X1, X2, X3, X4, X5, revA_in_ggga(X2, X1, .(X3, X4), X5)) REVA_IN_GGGA(.(X1, X2), X3, X4, X5) -> REVA_IN_GGGA(X2, X1, .(X3, X4), X5) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) revA_in_ggga(x1, x2, x3, x4) = revA_in_ggga(x1, x2, x3) [] = [] REVB_IN_GA(x1, x2) = REVB_IN_GA(x1) U2_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U2_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x11) REVA_IN_GGGA(x1, x2, x3, x4) = REVA_IN_GGGA(x1, x2, x3) U1_GGGA(x1, x2, x3, x4, x5, x6) = U1_GGGA(x1, x2, x3, x4, x6) 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: REVA_IN_GGGA(.(X1, X2), X3, X4, X5) -> REVA_IN_GGGA(X2, X1, .(X3, X4), X5) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) REVA_IN_GGGA(x1, x2, x3, x4) = REVA_IN_GGGA(x1, x2, x3) 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: REVA_IN_GGGA(.(X1, X2), X3, X4) -> REVA_IN_GGGA(X2, X1, .(X3, X4)) 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: *REVA_IN_GGGA(.(X1, X2), X3, X4) -> REVA_IN_GGGA(X2, X1, .(X3, X4)) The graph contains the following edges 1 > 1, 1 > 2 ---------------------------------------- (10) YES