/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern rev(g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1)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, 195 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: rev(L, R) :- rev(L, [], R). rev([], Y, Y). rev(L, S, R) :- ','(no(empty(L)), ','(head(L, X), ','(tail(L, T), rev(T, .(X, S), R)))). head([], X1). head(.(X, X2), X). tail([], []). tail(.(X3, Xs), Xs). empty([]). no(X) :- ','(X, ','(!, failure(a))). no(X4). failure(b). 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T515 ([]))))))) T517)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T511", "T512", "T513", "T514", "T515", "T606", "T607" ], "free": [], "exprvars": [] } }, "587": { "goal": [{ "clause": 9, "scope": 52, "term": "(',' (no (empty T904)) (',' (head T904 X477) (',' (tail T904 X478) (rev X478 (. X477 (. T877 (. T878 (. T879 (. T880 (. T881 (. T882 (. T883 ([]))))))))) T885))))" }], "kb": { "nonunifying": [[ "(empty T904)", "(empty ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T877", "T878", "T879", "T880", "T881", "T882", "T883", "T904" ], "free": [ "X477", "X478" ], "exprvars": [] } }, "349": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (no (empty T363)) (',' (head T363 X285) (',' (tail T363 X286) (rev X286 (. X285 (. T364 (. T365 (. T366 (. T367 ([])))))) T369))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T363", "T364", "T365", "T366", "T367" ], "free": [ "X285", "X286" ], "exprvars": [] } } }, "edges": [ { "from": 2, "to": 6, "label": "CASE" }, { "from": 6, "to": 7, "label": "ONLY EVAL with clause\nrev(X7, X8) :- rev(X7, [], X8).\nand substitutionT1 -> T5,\nX7 -> T5,\nT2 -> T7,\nX8 -> T7,\nT6 -> T7" }, { "from": 7, "to": 8, "label": "CASE" }, { "from": 8, "to": 10, "label": "PARALLEL" }, { "from": 8, "to": 11, "label": "PARALLEL" }, { "from": 10, "to": 12, "label": "EVAL with clause\nrev([], X13, X13).\nand substitutionT5 -> [],\nX13 -> [],\nT7 -> []" }, { "from": 10, "to": 13, "label": "EVAL-BACKTRACK" }, { "from": 11, "to": 20, "label": "ONLY EVAL with clause\nrev(X26, X27, X28) :- ','(no(empty(X26)), ','(head(X26, X29), ','(tail(X26, X30), rev(X30, .(X29, X27), X28)))).\nand substitutionT5 -> T15,\nX26 -> T15,\nX27 -> [],\nT7 -> T17,\nX28 -> T17,\nT16 -> T17" }, { "from": 12, "to": 14, "label": "SUCCESS" }, { "from": 20, "to": 22, "label": "CASE" }, { "from": 22, "to": 23, "label": "ONLY EVAL with clause\nno(X37) :- ','(call(X37), ','(!_3, failure(a))).\nand substitutionT15 -> T22,\nX37 -> empty(T22)" }, { "from": 23, "to": 25, "label": "CALL" }, { "from": 25, "to": 27, "label": "CASE" }, { "from": 27, "to": 28, "label": "EVAL with clause\nempty([]).\nand substitutionT22 -> []" }, { "from": 27, "to": 29, "label": "EVAL-BACKTRACK" }, { "from": 28, "to": 31, "label": "CUT" }, { "from": 29, "to": 34, "label": "FAILURE" }, { "from": 31, "to": 32, "label": "CASE" }, { "from": 32, "to": 33, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 34, "to": 35, "label": "FAILURE" }, { "from": 35, "to": 43, "label": "ONLY EVAL with clause\nno(X48).\nand substitutionT22 -> T29,\nX48 -> empty(T29)" }, { "from": 43, "to": 115, "label": "CASE" }, { "from": 115, "to": 116, "label": "BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T29), empty([]))" }, { "from": 116, "to": 118, "label": "EVAL with clause\nhead(.(X59, X60), X59).\nand substitutionX59 -> T36,\nX60 -> T37,\nT29 -> .(T36, T37),\nX29 -> T36" }, { "from": 116, "to": 119, "label": "EVAL-BACKTRACK" }, { "from": 118, "to": 121, "label": "CASE" }, { "from": 121, "to": 122, "label": "BACKTRACK\nfor clause: tail([], [])because of non-unification" }, { "from": 122, "to": 125, "label": "ONLY EVAL with clause\ntail(.(X69, X70), X70).\nand substitutionT36 -> T46,\nX69 -> T46,\nT37 -> T47,\nX70 -> T47,\nX30 -> T47" }, { "from": 125, "to": 129, "label": "CASE" }, { "from": 129, "to": 130, "label": "PARALLEL" }, { "from": 129, "to": 131, "label": "PARALLEL" }, { "from": 130, "to": 132, "label": "EVAL with clause\nrev([], X77, X77).\nand substitutionT47 -> [],\nT46 -> T54,\nX77 -> .(T54, []),\nT17 -> .(T54, [])" }, { "from": 130, "to": 133, "label": "EVAL-BACKTRACK" }, { "from": 131, "to": 171, "label": "ONLY EVAL with clause\nrev(X90, X91, X92) :- ','(no(empty(X90)), ','(head(X90, X93), ','(tail(X90, X94), rev(X94, .(X93, X91), X92)))).\nand substitutionT47 -> T66,\nX90 -> T66,\nT46 -> T67,\nX91 -> .(T67, []),\nT17 -> T69,\nX92 -> T69,\nT68 -> T69" }, { "from": 132, "to": 134, "label": "SUCCESS" }, { "from": 171, "to": 173, "label": "CASE" }, { "from": 173, "to": 176, "label": "ONLY EVAL with clause\nno(X101) :- ','(call(X101), ','(!_10, failure(a))).\nand substitutionT66 -> T76,\nX101 -> empty(T76)" }, { "from": 176, "to": 177, "label": "CALL" }, { "from": 177, "to": 180, "label": "CASE" }, { "from": 180, "to": 182, "label": "EVAL with clause\nempty([]).\nand substitutionT76 -> []" }, { "from": 180, "to": 183, "label": "EVAL-BACKTRACK" }, { "from": 182, "to": 184, "label": "CUT" }, { "from": 183, "to": 187, "label": "FAILURE" }, { "from": 184, "to": 185, "label": "CASE" }, { "from": 185, "to": 186, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 187, "to": 188, "label": "FAILURE" }, { "from": 188, "to": 189, "label": "ONLY EVAL with clause\nno(X112).\nand substitutionT76 -> T87,\nX112 -> empty(T87)" }, { "from": 189, "to": 191, "label": "CASE" }, { "from": 191, "to": 192, "label": "BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T87), empty([]))" }, { "from": 192, "to": 193, "label": "EVAL with clause\nhead(.(X123, X124), X123).\nand substitutionX123 -> T96,\nX124 -> T97,\nT87 -> .(T96, T97),\nX93 -> T96" }, { "from": 192, "to": 194, "label": "EVAL-BACKTRACK" }, { "from": 193, "to": 195, "label": "CASE" }, { "from": 195, "to": 196, "label": "BACKTRACK\nfor clause: tail([], [])because of non-unification" }, { "from": 196, "to": 197, "label": "ONLY EVAL with clause\ntail(.(X133, X134), X134).\nand substitutionT96 -> T110,\nX133 -> T110,\nT97 -> T111,\nX134 -> T111,\nX94 -> T111" }, { "from": 197, "to": 205, "label": "CASE" }, { "from": 205, "to": 206, "label": "PARALLEL" }, { "from": 205, "to": 207, "label": "PARALLEL" }, { "from": 206, "to": 208, "label": "EVAL with clause\nrev([], X141, X141).\nand substitutionT111 -> [],\nT110 -> T124,\nT67 -> T125,\nX141 -> .(T124, .(T125, [])),\nT69 -> .(T124, .(T125, []))" }, { "from": 206, "to": 209, "label": "EVAL-BACKTRACK" }, { "from": 207, "to": 244, "label": "ONLY EVAL with clause\nrev(X154, X155, X156) :- ','(no(empty(X154)), ','(head(X154, X157), ','(tail(X154, X158), rev(X158, .(X157, X155), X156)))).\nand substitutionT111 -> T141,\nX154 -> T141,\nT110 -> T142,\nT67 -> T143,\nX155 -> .(T142, .(T143, [])),\nT69 -> T145,\nX156 -> T145,\nT144 -> T145" }, { "from": 208, "to": 210, "label": "SUCCESS" }, { "from": 244, "to": 247, "label": "CASE" }, { "from": 247, "to": 248, "label": "ONLY EVAL with clause\nno(X165) :- ','(call(X165), ','(!_17, failure(a))).\nand substitutionT141 -> T154,\nX165 -> empty(T154)" }, { "from": 248, "to": 250, "label": "CALL" }, { "from": 250, "to": 252, "label": "CASE" }, { "from": 252, "to": 253, "label": "EVAL with clause\nempty([]).\nand substitutionT154 -> []" }, { "from": 252, "to": 254, "label": "EVAL-BACKTRACK" }, { "from": 253, "to": 255, "label": "CUT" }, { "from": 254, "to": 258, "label": "FAILURE" }, { "from": 255, "to": 256, "label": "CASE" }, { "from": 256, "to": 257, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 258, "to": 259, "label": "FAILURE" }, { "from": 259, "to": 261, "label": "ONLY EVAL with clause\nno(X176).\nand substitutionT154 -> T169,\nX176 -> empty(T169)" }, { "from": 261, "to": 263, "label": "CASE" }, { "from": 263, "to": 264, "label": "BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T169), empty([]))" }, { "from": 264, "to": 267, "label": "EVAL with clause\nhead(.(X187, X188), X187).\nand substitutionX187 -> T180,\nX188 -> T181,\nT169 -> .(T180, T181),\nX157 -> T180" }, { "from": 264, "to": 268, "label": "EVAL-BACKTRACK" }, { "from": 267, "to": 269, "label": "CASE" }, { "from": 269, "to": 270, "label": "BACKTRACK\nfor clause: tail([], [])because of non-unification" }, { "from": 270, "to": 274, "label": "ONLY EVAL with clause\ntail(.(X197, X198), X198).\nand substitutionT180 -> T198,\nX197 -> T198,\nT181 -> T199,\nX198 -> T199,\nX158 -> T199" }, { "from": 274, "to": 275, "label": "CASE" }, { "from": 275, "to": 276, "label": "PARALLEL" }, { "from": 275, "to": 277, "label": "PARALLEL" }, { "from": 276, "to": 278, "label": "EVAL with clause\nrev([], X205, X205).\nand substitutionT199 -> [],\nT198 -> T218,\nT142 -> T219,\nT143 -> T220,\nX205 -> .(T218, .(T219, .(T220, []))),\nT145 -> .(T218, .(T219, .(T220, [])))" }, { "from": 276, "to": 279, "label": "EVAL-BACKTRACK" }, { "from": 277, "to": 287, "label": "ONLY EVAL with clause\nrev(X218, X219, X220) :- ','(no(empty(X218)), ','(head(X218, X221), ','(tail(X218, X222), rev(X222, .(X221, X219), X220)))).\nand substitutionT199 -> T240,\nX218 -> T240,\nT198 -> T241,\nT142 -> T242,\nT143 -> T243,\nX219 -> .(T241, .(T242, .(T243, []))),\nT145 -> T245,\nX220 -> T245,\nT244 -> T245" }, { "from": 278, "to": 280, "label": "SUCCESS" }, { "from": 287, "to": 297, "label": "CASE" }, { "from": 297, "to": 298, "label": "ONLY EVAL with clause\nno(X229) :- ','(call(X229), ','(!_24, failure(a))).\nand substitutionT240 -> T256,\nX229 -> empty(T256)" }, { "from": 298, "to": 299, "label": "CALL" }, { "from": 299, "to": 300, "label": "CASE" }, { "from": 300, "to": 301, "label": "EVAL with clause\nempty([]).\nand substitutionT256 -> []" }, { "from": 300, "to": 302, "label": "EVAL-BACKTRACK" }, { "from": 301, "to": 303, "label": "CUT" }, { "from": 302, "to": 310, "label": "FAILURE" }, { "from": 303, "to": 308, "label": "CASE" }, { "from": 308, "to": 309, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 310, "to": 311, "label": "FAILURE" }, { "from": 311, "to": 314, "label": "ONLY EVAL with clause\nno(X240).\nand substitutionT256 -> T275,\nX240 -> empty(T275)" }, { "from": 314, "to": 316, "label": "CASE" }, { "from": 316, "to": 318, "label": "BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T275), empty([]))" }, { "from": 318, "to": 326, "label": "EVAL with clause\nhead(.(X251, X252), X251).\nand substitutionX251 -> T288,\nX252 -> T289,\nT275 -> .(T288, T289),\nX221 -> T288" }, { "from": 318, "to": 327, "label": "EVAL-BACKTRACK" }, { "from": 326, "to": 334, "label": "CASE" }, { "from": 334, "to": 335, "label": "BACKTRACK\nfor clause: tail([], [])because of non-unification" }, { "from": 335, "to": 338, "label": "ONLY EVAL with clause\ntail(.(X261, X262), X262).\nand substitutionT288 -> T310,\nX261 -> T310,\nT289 -> T311,\nX262 -> T311,\nX222 -> T311" }, { "from": 338, "to": 339, "label": "CASE" }, { "from": 339, "to": 340, "label": "PARALLEL" }, { "from": 339, "to": 341, "label": "PARALLEL" }, { "from": 340, "to": 343, "label": "EVAL with clause\nrev([], X269, X269).\nand substitutionT311 -> [],\nT310 -> T336,\nT241 -> T337,\nT242 -> T338,\nT243 -> T339,\nX269 -> .(T336, .(T337, .(T338, .(T339, [])))),\nT245 -> .(T336, .(T337, .(T338, .(T339, []))))" }, { "from": 340, "to": 344, "label": "EVAL-BACKTRACK" }, { "from": 341, "to": 349, "label": "ONLY EVAL with clause\nrev(X282, X283, X284) :- ','(no(empty(X282)), ','(head(X282, X285), ','(tail(X282, X286), rev(X286, .(X285, X283), X284)))).\nand substitutionT311 -> T363,\nX282 -> T363,\nT310 -> T364,\nT241 -> T365,\nT242 -> T366,\nT243 -> T367,\nX283 -> .(T364, .(T365, .(T366, .(T367, [])))),\nT245 -> T369,\nX284 -> T369,\nT368 -> T369" }, { "from": 343, "to": 345, "label": "SUCCESS" }, { "from": 349, "to": 353, "label": "CASE" }, { "from": 353, "to": 354, "label": "ONLY EVAL with clause\nno(X293) :- ','(call(X293), ','(!_31, failure(a))).\nand substitutionT363 -> T382,\nX293 -> empty(T382)" }, { "from": 354, "to": 355, "label": "CALL" }, { "from": 355, "to": 356, "label": "CASE" }, { "from": 356, "to": 357, "label": "EVAL with clause\nempty([]).\nand substitutionT382 -> []" }, { "from": 356, "to": 358, "label": "EVAL-BACKTRACK" }, { "from": 357, "to": 359, "label": "CUT" }, { "from": 358, "to": 362, "label": "FAILURE" }, { "from": 359, "to": 360, "label": "CASE" }, { "from": 360, "to": 361, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 362, "to": 363, "label": "FAILURE" }, { "from": 363, "to": 365, "label": "ONLY EVAL with clause\nno(X304).\nand substitutionT382 -> T405,\nX304 -> empty(T405)" }, { "from": 365, "to": 367, "label": "CASE" }, { "from": 367, "to": 368, "label": "BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T405), empty([]))" }, { "from": 368, "to": 370, "label": "EVAL with clause\nhead(.(X315, X316), X315).\nand substitutionX315 -> T420,\nX316 -> T421,\nT405 -> .(T420, T421),\nX285 -> T420" }, { "from": 368, "to": 371, "label": "EVAL-BACKTRACK" }, { "from": 370, "to": 399, "label": "CASE" }, { "from": 399, "to": 400, "label": "BACKTRACK\nfor clause: tail([], [])because of non-unification" }, { "from": 400, "to": 404, "label": "ONLY EVAL with clause\ntail(.(X325, X326), X326).\nand substitutionT420 -> T446,\nX325 -> T446,\nT421 -> T447,\nX326 -> T447,\nX286 -> T447" }, { "from": 404, "to": 405, "label": "CASE" }, { "from": 405, "to": 407, "label": "PARALLEL" }, { "from": 405, "to": 408, "label": "PARALLEL" }, { "from": 407, "to": 413, "label": "EVAL with clause\nrev([], X333, X333).\nand substitutionT447 -> [],\nT446 -> T478,\nT364 -> T479,\nT365 -> T480,\nT366 -> T481,\nT367 -> T482,\nX333 -> .(T478, .(T479, .(T480, .(T481, .(T482, []))))),\nT369 -> .(T478, .(T479, .(T480, .(T481, .(T482, [])))))" }, { "from": 407, "to": 415, "label": "EVAL-BACKTRACK" }, { "from": 408, "to": 425, "label": "ONLY EVAL with clause\nrev(X346, X347, X348) :- ','(no(empty(X346)), ','(head(X346, X349), ','(tail(X346, X350), rev(X350, .(X349, X347), X348)))).\nand substitutionT447 -> T510,\nX346 -> T510,\nT446 -> T511,\nT364 -> T512,\nT365 -> T513,\nT366 -> T514,\nT367 -> T515,\nX347 -> .(T511, .(T512, .(T513, .(T514, .(T515, []))))),\nT369 -> T517,\nX348 -> T517,\nT516 -> T517" }, { "from": 413, "to": 417, "label": "SUCCESS" }, { "from": 425, "to": 432, "label": "CASE" }, { "from": 432, "to": 434, "label": "ONLY EVAL with clause\nno(X357) :- ','(call(X357), ','(!_38, failure(a))).\nand substitutionT510 -> T532,\nX357 -> empty(T532)" }, { "from": 434, "to": 435, "label": "CALL" }, { "from": 435, "to": 436, "label": "CASE" }, { "from": 436, "to": 437, "label": "EVAL with clause\nempty([]).\nand substitutionT532 -> []" }, { "from": 436, "to": 438, "label": "EVAL-BACKTRACK" }, { "from": 437, "to": 439, "label": "CUT" }, { "from": 438, "to": 445, "label": "FAILURE" }, { "from": 439, "to": 443, "label": "CASE" }, { "from": 443, "to": 444, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 445, "to": 446, "label": "FAILURE" }, { "from": 446, "to": 450, "label": "ONLY EVAL with clause\nno(X368).\nand substitutionT532 -> T559,\nX368 -> empty(T559)" }, { "from": 450, "to": 451, "label": "CASE" }, { "from": 451, "to": 452, "label": "BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T559), empty([]))" }, { "from": 452, "to": 453, "label": "EVAL with clause\nhead(.(X379, X380), X379).\nand substitutionX379 -> T576,\nX380 -> T577,\nT559 -> .(T576, T577),\nX349 -> T576" }, { "from": 452, "to": 455, "label": "EVAL-BACKTRACK" }, { "from": 453, "to": 456, "label": "CASE" }, { "from": 456, "to": 457, "label": "BACKTRACK\nfor clause: tail([], [])because of non-unification" }, { "from": 457, "to": 463, "label": "ONLY EVAL with clause\ntail(.(X389, X390), X390).\nand substitutionT576 -> T606,\nX389 -> T606,\nT577 -> T607,\nX390 -> T607,\nX350 -> T607" }, { "from": 463, "to": 466, "label": "CASE" }, { "from": 466, "to": 473, "label": "PARALLEL" }, { "from": 466, "to": 474, "label": "PARALLEL" }, { "from": 473, "to": 479, "label": "EVAL with clause\nrev([], X397, X397).\nand substitutionT607 -> [],\nT606 -> T644,\nT511 -> T645,\nT512 -> T646,\nT513 -> T647,\nT514 -> T648,\nT515 -> T649,\nX397 -> .(T644, .(T645, .(T646, .(T647, .(T648, .(T649, [])))))),\nT517 -> .(T644, .(T645, .(T646, .(T647, .(T648, .(T649, []))))))" }, { "from": 473, "to": 480, "label": "EVAL-BACKTRACK" }, { "from": 474, "to": 489, "label": "ONLY EVAL with clause\nrev(X410, X411, X412) :- ','(no(empty(X410)), ','(head(X410, X413), ','(tail(X410, X414), rev(X414, .(X413, X411), X412)))).\nand substitutionT607 -> T681,\nX410 -> T681,\nT606 -> T682,\nT511 -> T683,\nT512 -> T684,\nT513 -> T685,\nT514 -> T686,\nT515 -> T687,\nX411 -> .(T682, .(T683, .(T684, .(T685, .(T686, .(T687, [])))))),\nT517 -> T689,\nX412 -> T689,\nT688 -> T689" }, { "from": 479, "to": 481, "label": "SUCCESS" }, { "from": 489, "to": 490, "label": "CASE" }, { "from": 490, "to": 491, "label": "ONLY EVAL with clause\nno(X421) :- ','(call(X421), ','(!_45, failure(a))).\nand substitutionT681 -> T706,\nX421 -> empty(T706)" }, { "from": 491, "to": 492, "label": "CALL" }, { "from": 492, "to": 493, "label": "CASE" }, { "from": 493, "to": 494, "label": "EVAL with clause\nempty([]).\nand substitutionT706 -> []" }, { "from": 493, "to": 495, "label": "EVAL-BACKTRACK" }, { "from": 494, "to": 496, "label": "CUT" }, { "from": 495, "to": 501, "label": "FAILURE" }, { "from": 496, "to": 499, "label": "CASE" }, { "from": 499, "to": 500, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 501, "to": 502, "label": "FAILURE" }, { "from": 502, "to": 507, "label": "ONLY EVAL with clause\nno(X432).\nand substitutionT706 -> T737,\nX432 -> empty(T737)" }, { "from": 507, "to": 511, "label": "CASE" }, { "from": 511, "to": 515, "label": "BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T737), empty([]))" }, { "from": 515, "to": 517, "label": "EVAL with clause\nhead(.(X443, X444), X443).\nand substitutionX443 -> T756,\nX444 -> T757,\nT737 -> .(T756, T757),\nX413 -> T756" }, { "from": 515, "to": 518, "label": "EVAL-BACKTRACK" }, { "from": 517, "to": 520, "label": "CASE" }, { "from": 520, "to": 521, "label": "BACKTRACK\nfor clause: tail([], [])because of non-unification" }, { "from": 521, "to": 531, "label": "ONLY EVAL with clause\ntail(.(X453, X454), X454).\nand substitutionT756 -> T790,\nX453 -> T790,\nT757 -> T791,\nX454 -> T791,\nX414 -> T791" }, { "from": 531, "to": 535, "label": "CASE" }, { "from": 535, "to": 537, "label": "PARALLEL" }, { "from": 535, "to": 538, "label": "PARALLEL" }, { "from": 537, "to": 541, "label": "EVAL with clause\nrev([], X461, X461).\nand substitutionT791 -> [],\nT790 -> T834,\nT682 -> T835,\nT683 -> T836,\nT684 -> T837,\nT685 -> T838,\nT686 -> T839,\nT687 -> T840,\nX461 -> .(T834, .(T835, .(T836, .(T837, .(T838, .(T839, .(T840, []))))))),\nT689 -> .(T834, .(T835, .(T836, .(T837, .(T838, .(T839, .(T840, [])))))))" }, { "from": 537, "to": 542, "label": "EVAL-BACKTRACK" }, { "from": 538, "to": 563, "label": "ONLY EVAL with clause\nrev(X474, X475, X476) :- ','(no(empty(X474)), ','(head(X474, X477), ','(tail(X474, X478), rev(X478, .(X477, X475), X476)))).\nand substitutionT791 -> T876,\nX474 -> T876,\nT790 -> T877,\nT682 -> T878,\nT683 -> T879,\nT684 -> T880,\nT685 -> T881,\nT686 -> T882,\nT687 -> T883,\nX475 -> .(T877, .(T878, .(T879, .(T880, .(T881, .(T882, .(T883, []))))))),\nT689 -> T885,\nX476 -> T885,\nT884 -> T885" }, { "from": 541, "to": 543, "label": "SUCCESS" }, { "from": 563, "to": 569, "label": "CASE" }, { "from": 569, "to": 570, "label": "ONLY EVAL with clause\nno(X485) :- ','(call(X485), ','(!_52, failure(a))).\nand substitutionT876 -> T904,\nX485 -> empty(T904)" }, { "from": 570, "to": 571, "label": "CALL" }, { "from": 571, "to": 572, "label": "CASE" }, { "from": 572, "to": 573, "label": "EVAL with clause\nempty([]).\nand substitutionT904 -> []" }, { "from": 572, "to": 575, "label": "EVAL-BACKTRACK" }, { "from": 573, "to": 576, "label": "CUT" }, { "from": 575, "to": 586, "label": "FAILURE" }, { "from": 576, "to": 584, "label": "CASE" }, { "from": 584, "to": 585, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 586, "to": 587, "label": "FAILURE" }, { "from": 587, "to": 595, "label": "ONLY EVAL with clause\nno(X496).\nand substitutionT904 -> T939,\nX496 -> empty(T939)" }, { "from": 595, "to": 597, "label": "CASE" }, { "from": 597, "to": 600, "label": "BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T939), empty([]))" }, { "from": 600, "to": 603, "label": "EVAL with clause\nhead(.(X507, X508), X507).\nand substitutionX507 -> T960,\nX508 -> T961,\nT939 -> .(T960, T961),\nX477 -> T960" }, { "from": 600, "to": 605, "label": "EVAL-BACKTRACK" }, { "from": 603, "to": 608, "label": "CASE" }, { "from": 608, "to": 609, "label": "BACKTRACK\nfor clause: tail([], [])because of non-unification" }, { "from": 609, "to": 613, "label": "ONLY EVAL with clause\ntail(.(X517, X518), X518).\nand substitutionT960 -> T998,\nX517 -> T998,\nT961 -> T999,\nX518 -> T999,\nX478 -> T999" }, { "from": 613, "to": 622, "label": "GENERALIZATION\nT1016 <-- .(T877, .(T878, .(T879, .(T880, .(T881, .(T882, .(T883, [])))))))\n\nNew Knowledge:\nT1016 is ground" }, { "from": 622, "to": 628, "label": "CASE" }, { "from": 628, "to": 630, "label": "PARALLEL" }, { "from": 628, "to": 631, "label": "PARALLEL" }, { "from": 630, "to": 634, "label": "EVAL with clause\nrev([], X527, X527).\nand substitutionT999 -> [],\nT998 -> T1029,\nT1016 -> T1030,\nX527 -> .(T1029, T1030),\nT885 -> .(T1029, T1030)" }, { "from": 630, "to": 635, "label": "EVAL-BACKTRACK" }, { "from": 631, "to": 645, "label": "ONLY EVAL with clause\nrev(X540, X541, X542) :- ','(no(empty(X540)), ','(head(X540, X543), ','(tail(X540, X544), rev(X544, .(X543, X541), X542)))).\nand substitutionT999 -> T1046,\nX540 -> T1046,\nT998 -> T1047,\nT1016 -> T1048,\nX541 -> .(T1047, T1048),\nT885 -> T1050,\nX542 -> T1050,\nT1049 -> T1050" }, { "from": 634, "to": 637, "label": "SUCCESS" }, { "from": 645, "to": 647, "label": "CASE" }, { "from": 647, "to": 650, "label": "ONLY EVAL with clause\nno(X551) :- ','(call(X551), ','(!_59, failure(a))).\nand substitutionT1046 -> T1059,\nX551 -> empty(T1059)" }, { "from": 650, "to": 653, "label": "CALL" }, { "from": 653, "to": 655, "label": "CASE" }, { "from": 655, "to": 657, "label": "EVAL with clause\nempty([]).\nand substitutionT1059 -> []" }, { "from": 655, "to": 658, "label": "EVAL-BACKTRACK" }, { "from": 657, "to": 660, "label": "CUT" }, { "from": 658, "to": 666, "label": "FAILURE" }, { "from": 660, "to": 663, "label": "CASE" }, { "from": 663, "to": 664, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 666, "to": 668, "label": "FAILURE" }, { "from": 668, "to": 671, "label": "ONLY EVAL with clause\nno(X562).\nand substitutionT1059 -> T1074,\nX562 -> empty(T1074)" }, { "from": 671, "to": 673, "label": "CASE" }, { "from": 673, "to": 675, "label": "BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T1074), empty([]))" }, { "from": 675, "to": 677, "label": "EVAL with clause\nhead(.(X573, X574), X573).\nand substitutionX573 -> T1085,\nX574 -> T1086,\nT1074 -> .(T1085, T1086),\nX543 -> T1085" }, { "from": 675, "to": 678, "label": "EVAL-BACKTRACK" }, { "from": 677, "to": 681, "label": "CASE" }, { "from": 681, "to": 682, "label": "BACKTRACK\nfor clause: tail([], [])because of non-unification" }, { "from": 682, "to": 685, "label": "ONLY EVAL with clause\ntail(.(X583, X584), X584).\nand substitutionT1085 -> T1103,\nX583 -> T1103,\nT1086 -> T1104,\nX584 -> T1104,\nX544 -> T1104" }, { "from": 685, "to": 622, "label": "INSTANCE with matching:\nT999 -> T1104\nT998 -> T1103\nT1016 -> .(T1047, T1048)\nT885 -> T1050" } ], "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