/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern div(g,g,a,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToDTProblemTransformerProof [SOUND, 33 ms] (2) TRIPLES (3) TriplesToPiDPProof [SOUND, 0 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [SOUND, 0 ms] (11) QDP (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] (13) YES (14) PiDP (15) PiDPToQDPProof [SOUND, 0 ms] (16) QDP (17) QDPQMonotonicMRRProof [EQUIVALENT, 23 ms] (18) QDP (19) DependencyGraphProof [EQUIVALENT, 0 ms] (20) TRUE ---------------------------------------- (0) Obligation: Clauses: div(X1, 0, X2, X3) :- ','(!, failure(a)). div(0, X4, Z, R) :- ','(!, ','(eq(Z, 0), eq(R, 0))). div(X, Y, s(Z), R) :- ','(minus(X, Y, U), ','(!, div(U, Y, Z, R))). div(X, X5, X6, X). minus(X, 0, X). minus(s(X), s(Y), Z) :- minus(X, Y, Z). failure(b). eq(X, X). Query: div(g,g,a,a) ---------------------------------------- (1) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 18, "program": { "directives": [], "clauses": [ [ "(div X1 (0) X2 X3)", "(',' (!) (failure (a)))" ], [ "(div (0) X4 Z R)", "(',' (!) (',' (eq Z (0)) (eq R (0))))" ], [ "(div X Y (s Z) R)", "(',' (minus X Y U) (',' (!) (div U Y Z R)))" ], [ "(div X X5 X6 X)", null ], [ "(minus X (0) X)", null ], [ "(minus (s X) (s Y) Z)", "(minus X Y Z)" ], [ "(failure (b))", null ], [ "(eq X X)", null ] ] }, "graph": { "nodes": { "type": "Nodes", "174": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (!_1) (failure (a)))" }, { "clause": 1, "scope": 1, "term": "(div T8 (0) T3 T4)" }, { "clause": 2, "scope": 1, "term": "(div T8 (0) T3 T4)" }, { "clause": 3, "scope": 1, "term": "(div T8 (0) T3 T4)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T8"], "free": [], "exprvars": [] } }, "175": { "goal": [ { "clause": 1, "scope": 1, "term": "(div T1 T2 T3 T4)" }, { "clause": 2, "scope": 1, "term": "(div T1 T2 T3 T4)" }, { "clause": 3, "scope": 1, "term": "(div T1 T2 T3 T4)" } ], "kb": { "nonunifying": [[ "(div T1 T2 T3 T4)", "(div X10 (0) X11 X12)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2" ], "free": [ "X10", "X11", "X12" ], "exprvars": [] } }, "197": { "goal": [{ "clause": 6, "scope": 2, "term": "(failure (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "330": { "goal": [{ "clause": -1, "scope": -1, "term": "(eq T22 (0))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "352": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (minus T45 T46 X54) (',' (!_1) (div X54 (s T46) T34 T35)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T45", "T46" ], "free": ["X54"], "exprvars": [] } }, "177": { "goal": [{ "clause": -1, "scope": -1, "term": "(failure (a))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "199": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "331": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "353": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "332": { "goal": [{ "clause": 7, "scope": 4, "term": "(eq T22 (0))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "333": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "355": { "goal": [{ "clause": -1, "scope": -1, "term": "(minus T45 T46 X54)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T45", "T46" ], "free": ["X54"], "exprvars": [] } }, "410": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "334": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "335": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "357": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (!_1) (div T49 (s T46) T34 T35))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T46", "T49" ], "free": [], "exprvars": [] } }, "336": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (minus T30 T31 X33) (',' (!_1) (div X33 T31 T34 T35)))" }, { "clause": 3, "scope": 1, "term": "(div T30 T31 T3 T4)" } ], "kb": { "nonunifying": [ [ "(div T30 T31 T3 T4)", "(div X10 (0) X11 X12)" ], [ "(div T30 T31 T3 T4)", "(div (0) X16 X17 X18)" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T30", "T31" ], "free": [ "X10", "X11", "X12", "X16", "X17", "X18", "X33" ], "exprvars": [] } }, "358": { "goal": [ { "clause": 4, "scope": 6, "term": "(minus T45 T46 X54)" }, { "clause": 5, "scope": 6, "term": "(minus T45 T46 X54)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T45", "T46" ], "free": ["X54"], "exprvars": [] } }, "359": { "goal": [{ "clause": 4, "scope": 6, "term": "(minus T45 T46 X54)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T45", "T46" ], "free": ["X54"], "exprvars": [] } }, "339": { "goal": [{ "clause": 3, "scope": 1, "term": "(div T1 T2 T3 T4)" }], "kb": { "nonunifying": [ [ "(div T1 T2 T3 T4)", "(div X10 (0) X11 X12)" ], [ "(div T1 T2 T3 T4)", "(div (0) X16 X17 X18)" ], [ "(div T1 T2 T3 T4)", "(div X29 X30 (s X31) X32)" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2" ], "free": [ "X10", "X11", "X12", "X16", "X17", "X18", "X29", "X30", "X31", "X32" ], "exprvars": [] } }, "18": { "goal": [{ "clause": -1, "scope": -1, "term": "(div T1 T2 T3 T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2" ], "free": [], "exprvars": [] } }, "19": { "goal": [ { "clause": 0, "scope": 1, "term": "(div T1 T2 T3 T4)" }, { "clause": 1, "scope": 1, "term": "(div T1 T2 T3 T4)" }, { "clause": 2, "scope": 1, "term": "(div T1 T2 T3 T4)" }, { "clause": 3, "scope": 1, "term": "(div T1 T2 T3 T4)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2" ], "free": [], "exprvars": [] } }, "360": { "goal": [{ "clause": 5, "scope": 6, "term": "(minus T45 T46 X54)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T45", "T46" ], "free": ["X54"], "exprvars": [] } }, "361": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "340": { "goal": [ { "clause": 4, "scope": 5, "term": "(',' (minus T30 T31 X33) (',' (!_1) (div X33 T31 T34 T35)))" }, { "clause": 5, "scope": 5, "term": "(',' (minus T30 T31 X33) (',' (!_1) (div X33 T31 T34 T35)))" }, { "clause": -1, "scope": 5, "term": null }, { "clause": 3, "scope": 1, "term": "(div T30 T31 T3 T4)" } ], "kb": { "nonunifying": [ [ "(div T30 T31 T3 T4)", "(div X10 (0) X11 X12)" ], [ "(div T30 T31 T3 T4)", "(div (0) X16 X17 X18)" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T30", "T31" ], "free": [ "X10", "X11", "X12", "X16", "X17", "X18", "X33" ], "exprvars": [] } }, "362": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "341": { "goal": [ { "clause": 5, "scope": 5, "term": "(',' (minus T30 T31 X33) (',' (!_1) (div X33 T31 T34 T35)))" }, { "clause": -1, "scope": 5, "term": null }, { "clause": 3, "scope": 1, "term": "(div T30 T31 T3 T4)" } ], "kb": { "nonunifying": [ [ "(div T30 T31 T3 T4)", "(div X10 (0) X11 X12)" ], [ "(div T30 T31 T3 T4)", "(div (0) X16 X17 X18)" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T30", "T31" ], "free": [ "X10", "X11", "X12", "X16", "X17", "X18", "X33" ], "exprvars": [] } }, "363": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "364": { "goal": [{ "clause": -1, "scope": -1, "term": "(minus T61 T62 X75)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T61", "T62" ], "free": ["X75"], "exprvars": [] } }, "365": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "344": { "goal": [{ "clause": 5, "scope": 5, "term": "(',' (minus T30 T31 X33) (',' (!_1) (div X33 T31 T34 T35)))" }], "kb": { "nonunifying": [ [ "(div T30 T31 T3 T4)", "(div X10 (0) X11 X12)" ], [ "(div T30 T31 T3 T4)", "(div (0) X16 X17 X18)" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T30", "T31" ], "free": [ "X10", "X11", "X12", "X16", "X17", "X18", "X33" ], "exprvars": [] } }, "346": { "goal": [ { "clause": -1, "scope": 5, "term": null }, { "clause": 3, "scope": 1, "term": "(div T30 T31 T3 T4)" } ], "kb": { "nonunifying": [ [ "(div T30 T31 T3 T4)", "(div X10 (0) X11 X12)" ], [ "(div T30 T31 T3 T4)", "(div (0) X16 X17 X18)" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T30", "T31" ], "free": [ "X10", "X11", "X12", "X16", "X17", "X18" ], "exprvars": [] } }, "205": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (!_1) (',' (eq T17 (0)) (eq T18 (0))))" }, { "clause": 2, "scope": 1, "term": "(div (0) T14 T3 T4)" }, { "clause": 3, "scope": 1, "term": "(div (0) T14 T3 T4)" } ], "kb": { "nonunifying": [[ "(div (0) T14 T3 T4)", "(div X10 (0) X11 X12)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T14"], "free": [ "X10", "X11", "X12" ], "exprvars": [] } }, "403": { "goal": [{ "clause": -1, "scope": -1, "term": "(div T49 (s T46) T34 T35)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T46", "T49" ], "free": [], "exprvars": [] } }, "327": { "goal": [ { "clause": 2, "scope": 1, "term": "(div T1 T2 T3 T4)" }, { "clause": 3, "scope": 1, "term": "(div T1 T2 T3 T4)" } ], "kb": { "nonunifying": [ [ "(div T1 T2 T3 T4)", "(div X10 (0) X11 X12)" ], [ "(div T1 T2 T3 T4)", "(div (0) X16 X17 X18)" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2" ], "free": [ "X10", "X11", "X12", "X16", "X17", "X18" ], "exprvars": [] } }, "404": { "goal": [{ "clause": 3, "scope": 1, "term": "(div T30 T31 T3 T4)" }], "kb": { "nonunifying": [ [ "(div T30 T31 T3 T4)", "(div X10 (0) X11 X12)" ], [ "(div T30 T31 T3 T4)", "(div (0) X16 X17 X18)" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T30", "T31" ], "free": [ "X10", "X11", "X12", "X16", "X17", "X18" ], "exprvars": [] } }, "328": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (eq T17 (0)) (eq T18 (0)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "405": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "329": { "goal": [{ "clause": 7, "scope": 3, "term": "(',' (eq T17 (0)) (eq T18 (0)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "406": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "407": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "408": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "409": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 18, "to": 19, "label": "CASE" }, { "from": 19, "to": 174, "label": "EVAL with clause\ndiv(X10, 0, X11, X12) :- ','(!_1, failure(a)).\nand substitutionT1 -> T8,\nX10 -> T8,\nT2 -> 0,\nT3 -> T9,\nX11 -> T9,\nT4 -> T10,\nX12 -> T10" }, { "from": 19, "to": 175, "label": "EVAL-BACKTRACK" }, { "from": 174, "to": 177, "label": "CUT" }, { "from": 175, "to": 205, "label": "EVAL with clause\ndiv(0, X16, X17, X18) :- ','(!_1, ','(eq(X17, 0), eq(X18, 0))).\nand substitutionT1 -> 0,\nT2 -> T14,\nX16 -> T14,\nT3 -> T17,\nX17 -> T17,\nT4 -> T18,\nX18 -> T18,\nT15 -> T17,\nT16 -> T18" }, { "from": 175, "to": 327, "label": "EVAL-BACKTRACK" }, { "from": 177, "to": 197, "label": "CASE" }, { "from": 197, "to": 199, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 205, "to": 328, "label": "CUT" }, { "from": 327, "to": 336, "label": "EVAL with clause\ndiv(X29, X30, s(X31), X32) :- ','(minus(X29, X30, X33), ','(!_1, div(X33, X30, X31, X32))).\nand substitutionT1 -> T30,\nX29 -> T30,\nT2 -> T31,\nX30 -> T31,\nX31 -> T34,\nT3 -> s(T34),\nT4 -> T35,\nX32 -> T35,\nT32 -> T34,\nT33 -> T35" }, { "from": 327, "to": 339, "label": "EVAL-BACKTRACK" }, { "from": 328, "to": 329, "label": "CASE" }, { "from": 329, "to": 330, "label": "EVAL with clause\neq(X21, X21).\nand substitutionT17 -> 0,\nX21 -> 0,\nT21 -> 0,\nT18 -> T22" }, { "from": 329, "to": 331, "label": "EVAL-BACKTRACK" }, { "from": 330, "to": 332, "label": "CASE" }, { "from": 332, "to": 333, "label": "EVAL with clause\neq(X24, X24).\nand substitutionT22 -> 0,\nX24 -> 0,\nT25 -> 0" }, { "from": 332, "to": 334, "label": "EVAL-BACKTRACK" }, { "from": 333, "to": 335, "label": "SUCCESS" }, { "from": 336, "to": 340, "label": "CASE" }, { "from": 339, "to": 408, "label": "EVAL with clause\ndiv(X93, X94, X95, X93).\nand substitutionT1 -> T78,\nX93 -> T78,\nT2 -> T79,\nX94 -> T79,\nT3 -> T80,\nX95 -> T80,\nT4 -> T78" }, { "from": 339, "to": 409, "label": "EVAL-BACKTRACK" }, { "from": 340, "to": 341, "label": "BACKTRACK\nfor clause: minus(X, 0, X)\nwith clash: (div(T30, T31, T3, T4), div(X10, 0, X11, X12))" }, { "from": 341, "to": 344, "label": "PARALLEL" }, { "from": 341, "to": 346, "label": "PARALLEL" }, { "from": 344, "to": 352, "label": "EVAL with clause\nminus(s(X51), s(X52), X53) :- minus(X51, X52, X53).\nand substitutionX51 -> T45,\nT30 -> s(T45),\nX52 -> T46,\nT31 -> s(T46),\nX33 -> X54,\nX53 -> X54" }, { "from": 344, "to": 353, "label": "EVAL-BACKTRACK" }, { "from": 346, "to": 404, "label": "FAILURE" }, { "from": 352, "to": 355, "label": "SPLIT 1" }, { "from": 352, "to": 357, "label": "SPLIT 2\nnew knowledge:\nT45 is ground\nT46 is ground\nT49 is ground\nreplacements:X54 -> T49" }, { "from": 355, "to": 358, "label": "CASE" }, { "from": 357, "to": 403, "label": "CUT" }, { "from": 358, "to": 359, "label": "PARALLEL" }, { "from": 358, "to": 360, "label": "PARALLEL" }, { "from": 359, "to": 361, "label": "EVAL with clause\nminus(X63, 0, X63).\nand substitutionT45 -> T56,\nX63 -> T56,\nT46 -> 0,\nX54 -> T56" }, { "from": 359, "to": 362, "label": "EVAL-BACKTRACK" }, { "from": 360, "to": 364, "label": "EVAL with clause\nminus(s(X72), s(X73), X74) :- minus(X72, X73, X74).\nand substitutionX72 -> T61,\nT45 -> s(T61),\nX73 -> T62,\nT46 -> s(T62),\nX54 -> X75,\nX74 -> X75" }, { "from": 360, "to": 365, "label": "EVAL-BACKTRACK" }, { "from": 361, "to": 363, "label": "SUCCESS" }, { "from": 364, "to": 355, "label": "INSTANCE with matching:\nT45 -> T61\nT46 -> T62\nX54 -> X75" }, { "from": 403, "to": 18, "label": "INSTANCE with matching:\nT1 -> T49\nT2 -> s(T46)\nT3 -> T34\nT4 -> T35" }, { "from": 404, "to": 405, "label": "EVAL with clause\ndiv(X87, X88, X89, X87).\nand substitutionT30 -> T72,\nX87 -> T72,\nT31 -> T73,\nX88 -> T73,\nT3 -> T74,\nX89 -> T74,\nT4 -> T72" }, { "from": 404, "to": 406, "label": "EVAL-BACKTRACK" }, { "from": 405, "to": 407, "label": "SUCCESS" }, { "from": 408, "to": 410, "label": "SUCCESS" } ], "type": "Graph" } } ---------------------------------------- (2) Obligation: Triples: minusA(s(X1), s(X2), X3) :- minusA(X1, X2, X3). divB(s(X1), s(X2), s(X3), X4) :- minusA(X1, X2, X5). divB(s(X1), s(X2), s(X3), X4) :- ','(minuscA(X1, X2, X5), divB(X5, s(X2), X3, X4)). Clauses: minuscA(X1, 0, X1). minuscA(s(X1), s(X2), X3) :- minuscA(X1, X2, X3). divcB(0, X1, 0, 0). divcB(s(X1), s(X2), s(X3), X4) :- ','(minuscA(X1, X2, X5), divcB(X5, s(X2), X3, X4)). divcB(X1, X2, X3, X1). divcB(X1, X2, X3, X1). Afs: divB(x1, x2, x3, x4) = divB(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: divB_in_4: (b,b,f,f) minusA_in_3: (b,b,f) minuscA_in_3: (b,b,f) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: DIVB_IN_GGAA(s(X1), s(X2), s(X3), X4) -> U2_GGAA(X1, X2, X3, X4, minusA_in_gga(X1, X2, X5)) DIVB_IN_GGAA(s(X1), s(X2), s(X3), X4) -> MINUSA_IN_GGA(X1, X2, X5) MINUSA_IN_GGA(s(X1), s(X2), X3) -> U1_GGA(X1, X2, X3, minusA_in_gga(X1, X2, X3)) MINUSA_IN_GGA(s(X1), s(X2), X3) -> MINUSA_IN_GGA(X1, X2, X3) DIVB_IN_GGAA(s(X1), s(X2), s(X3), X4) -> U3_GGAA(X1, X2, X3, X4, minuscA_in_gga(X1, X2, X5)) U3_GGAA(X1, X2, X3, X4, minuscA_out_gga(X1, X2, X5)) -> U4_GGAA(X1, X2, X3, X4, divB_in_ggaa(X5, s(X2), X3, X4)) U3_GGAA(X1, X2, X3, X4, minuscA_out_gga(X1, X2, X5)) -> DIVB_IN_GGAA(X5, s(X2), X3, X4) The TRS R consists of the following rules: minuscA_in_gga(X1, 0, X1) -> minuscA_out_gga(X1, 0, X1) minuscA_in_gga(s(X1), s(X2), X3) -> U6_gga(X1, X2, X3, minuscA_in_gga(X1, X2, X3)) U6_gga(X1, X2, X3, minuscA_out_gga(X1, X2, X3)) -> minuscA_out_gga(s(X1), s(X2), X3) The argument filtering Pi contains the following mapping: divB_in_ggaa(x1, x2, x3, x4) = divB_in_ggaa(x1, x2) s(x1) = s(x1) minusA_in_gga(x1, x2, x3) = minusA_in_gga(x1, x2) minuscA_in_gga(x1, x2, x3) = minuscA_in_gga(x1, x2) 0 = 0 minuscA_out_gga(x1, x2, x3) = minuscA_out_gga(x1, x2, x3) U6_gga(x1, x2, x3, x4) = U6_gga(x1, x2, x4) DIVB_IN_GGAA(x1, x2, x3, x4) = DIVB_IN_GGAA(x1, x2) U2_GGAA(x1, x2, x3, x4, x5) = U2_GGAA(x1, x2, x5) MINUSA_IN_GGA(x1, x2, x3) = MINUSA_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x2, x4) U3_GGAA(x1, x2, x3, x4, x5) = U3_GGAA(x1, x2, x5) U4_GGAA(x1, x2, x3, x4, x5) = U4_GGAA(x1, x2, 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: DIVB_IN_GGAA(s(X1), s(X2), s(X3), X4) -> U2_GGAA(X1, X2, X3, X4, minusA_in_gga(X1, X2, X5)) DIVB_IN_GGAA(s(X1), s(X2), s(X3), X4) -> MINUSA_IN_GGA(X1, X2, X5) MINUSA_IN_GGA(s(X1), s(X2), X3) -> U1_GGA(X1, X2, X3, minusA_in_gga(X1, X2, X3)) MINUSA_IN_GGA(s(X1), s(X2), X3) -> MINUSA_IN_GGA(X1, X2, X3) DIVB_IN_GGAA(s(X1), s(X2), s(X3), X4) -> U3_GGAA(X1, X2, X3, X4, minuscA_in_gga(X1, X2, X5)) U3_GGAA(X1, X2, X3, X4, minuscA_out_gga(X1, X2, X5)) -> U4_GGAA(X1, X2, X3, X4, divB_in_ggaa(X5, s(X2), X3, X4)) U3_GGAA(X1, X2, X3, X4, minuscA_out_gga(X1, X2, X5)) -> DIVB_IN_GGAA(X5, s(X2), X3, X4) The TRS R consists of the following rules: minuscA_in_gga(X1, 0, X1) -> minuscA_out_gga(X1, 0, X1) minuscA_in_gga(s(X1), s(X2), X3) -> U6_gga(X1, X2, X3, minuscA_in_gga(X1, X2, X3)) U6_gga(X1, X2, X3, minuscA_out_gga(X1, X2, X3)) -> minuscA_out_gga(s(X1), s(X2), X3) The argument filtering Pi contains the following mapping: divB_in_ggaa(x1, x2, x3, x4) = divB_in_ggaa(x1, x2) s(x1) = s(x1) minusA_in_gga(x1, x2, x3) = minusA_in_gga(x1, x2) minuscA_in_gga(x1, x2, x3) = minuscA_in_gga(x1, x2) 0 = 0 minuscA_out_gga(x1, x2, x3) = minuscA_out_gga(x1, x2, x3) U6_gga(x1, x2, x3, x4) = U6_gga(x1, x2, x4) DIVB_IN_GGAA(x1, x2, x3, x4) = DIVB_IN_GGAA(x1, x2) U2_GGAA(x1, x2, x3, x4, x5) = U2_GGAA(x1, x2, x5) MINUSA_IN_GGA(x1, x2, x3) = MINUSA_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x2, x4) U3_GGAA(x1, x2, x3, x4, x5) = U3_GGAA(x1, x2, x5) U4_GGAA(x1, x2, x3, x4, x5) = U4_GGAA(x1, x2, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 4 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: MINUSA_IN_GGA(s(X1), s(X2), X3) -> MINUSA_IN_GGA(X1, X2, X3) The TRS R consists of the following rules: minuscA_in_gga(X1, 0, X1) -> minuscA_out_gga(X1, 0, X1) minuscA_in_gga(s(X1), s(X2), X3) -> U6_gga(X1, X2, X3, minuscA_in_gga(X1, X2, X3)) U6_gga(X1, X2, X3, minuscA_out_gga(X1, X2, X3)) -> minuscA_out_gga(s(X1), s(X2), X3) The argument filtering Pi contains the following mapping: s(x1) = s(x1) minuscA_in_gga(x1, x2, x3) = minuscA_in_gga(x1, x2) 0 = 0 minuscA_out_gga(x1, x2, x3) = minuscA_out_gga(x1, x2, x3) U6_gga(x1, x2, x3, x4) = U6_gga(x1, x2, x4) MINUSA_IN_GGA(x1, x2, x3) = MINUSA_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (8) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (9) Obligation: Pi DP problem: The TRS P consists of the following rules: MINUSA_IN_GGA(s(X1), s(X2), X3) -> MINUSA_IN_GGA(X1, X2, X3) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) MINUSA_IN_GGA(x1, x2, x3) = MINUSA_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (10) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: MINUSA_IN_GGA(s(X1), s(X2)) -> MINUSA_IN_GGA(X1, X2) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (12) 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: *MINUSA_IN_GGA(s(X1), s(X2)) -> MINUSA_IN_GGA(X1, X2) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: DIVB_IN_GGAA(s(X1), s(X2), s(X3), X4) -> U3_GGAA(X1, X2, X3, X4, minuscA_in_gga(X1, X2, X5)) U3_GGAA(X1, X2, X3, X4, minuscA_out_gga(X1, X2, X5)) -> DIVB_IN_GGAA(X5, s(X2), X3, X4) The TRS R consists of the following rules: minuscA_in_gga(X1, 0, X1) -> minuscA_out_gga(X1, 0, X1) minuscA_in_gga(s(X1), s(X2), X3) -> U6_gga(X1, X2, X3, minuscA_in_gga(X1, X2, X3)) U6_gga(X1, X2, X3, minuscA_out_gga(X1, X2, X3)) -> minuscA_out_gga(s(X1), s(X2), X3) The argument filtering Pi contains the following mapping: s(x1) = s(x1) minuscA_in_gga(x1, x2, x3) = minuscA_in_gga(x1, x2) 0 = 0 minuscA_out_gga(x1, x2, x3) = minuscA_out_gga(x1, x2, x3) U6_gga(x1, x2, x3, x4) = U6_gga(x1, x2, x4) DIVB_IN_GGAA(x1, x2, x3, x4) = DIVB_IN_GGAA(x1, x2) U3_GGAA(x1, x2, x3, x4, x5) = U3_GGAA(x1, x2, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (15) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (16) Obligation: Q DP problem: The TRS P consists of the following rules: DIVB_IN_GGAA(s(X1), s(X2)) -> U3_GGAA(X1, X2, minuscA_in_gga(X1, X2)) U3_GGAA(X1, X2, minuscA_out_gga(X1, X2, X5)) -> DIVB_IN_GGAA(X5, s(X2)) The TRS R consists of the following rules: minuscA_in_gga(X1, 0) -> minuscA_out_gga(X1, 0, X1) minuscA_in_gga(s(X1), s(X2)) -> U6_gga(X1, X2, minuscA_in_gga(X1, X2)) U6_gga(X1, X2, minuscA_out_gga(X1, X2, X3)) -> minuscA_out_gga(s(X1), s(X2), X3) The set Q consists of the following terms: minuscA_in_gga(x0, x1) U6_gga(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (17) QDPQMonotonicMRRProof (EQUIVALENT) By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain. Strictly oriented dependency pairs: U3_GGAA(X1, X2, minuscA_out_gga(X1, X2, X5)) -> DIVB_IN_GGAA(X5, s(X2)) Strictly oriented rules of the TRS R: U6_gga(X1, X2, minuscA_out_gga(X1, X2, X3)) -> minuscA_out_gga(s(X1), s(X2), X3) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(DIVB_IN_GGAA(x_1, x_2)) = x_1 POL(U3_GGAA(x_1, x_2, x_3)) = 2 + x_3 POL(U6_gga(x_1, x_2, x_3)) = 2 + x_3 POL(minuscA_in_gga(x_1, x_2)) = x_1 POL(minuscA_out_gga(x_1, x_2, x_3)) = x_3 POL(s(x_1)) = 2 + 2*x_1 ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: DIVB_IN_GGAA(s(X1), s(X2)) -> U3_GGAA(X1, X2, minuscA_in_gga(X1, X2)) The TRS R consists of the following rules: minuscA_in_gga(X1, 0) -> minuscA_out_gga(X1, 0, X1) minuscA_in_gga(s(X1), s(X2)) -> U6_gga(X1, X2, minuscA_in_gga(X1, X2)) The set Q consists of the following terms: minuscA_in_gga(x0, x1) U6_gga(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (19) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (20) TRUE