/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.pl /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern div(g,g,a,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToDTProblemTransformerProof [SOUND, 42 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, 26 ms] (18) QDP (19) DependencyGraphProof [EQUIVALENT, 0 ms] (20) TRUE ---------------------------------------- (0) Obligation: Clauses: div(X, 0, Z, R) :- ','(!, fail). div(0, Y, Z, R) :- ','(!, ','(=(Z, 0), =(R, 0))). div(X, Y, s(Z), R) :- ','(minus(X, Y, U), ','(!, div(U, Y, Z, R))). div(X, Y, 0, X). minus(X, 0, X). minus(s(X), s(Y), Z) :- minus(X, Y, Z). Query: div(g,g,a,a) ---------------------------------------- (1) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 1, "program": { "directives": [], "clauses": [ [ "(div X (0) Z R)", "(',' (!) (fail))" ], [ "(div (0) Y Z R)", "(',' (!) (',' (= Z (0)) (= R (0))))" ], [ "(div X Y (s Z) R)", "(',' (minus X Y U) (',' (!) (div U Y Z R)))" ], [ "(div X Y (0) X)", null ], [ "(minus X (0) X)", null ], [ "(minus (s X) (s Y) Z)", "(minus X Y Z)" ] ] }, "graph": { "nodes": { "44": { "goal": [{ "clause": -1, "scope": -1, "term": "(fail)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "370": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (!_1) (div T43 (s T40) T28 T29))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T40", "T43" ], "free": [], "exprvars": [] } }, "330": { "goal": [{ "clause": -1, "scope": -1, "term": "(= T19 (0))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "331": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "332": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "333": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "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": [] } }, "411": { "goal": [{ "clause": -1, "scope": -1, "term": "(minus T55 T56 X63)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T55", "T56" ], "free": ["X63"], "exprvars": [] } }, "335": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (minus T24 T25 X21) (',' (!_1) (div X21 T25 T28 T29)))" }, { "clause": 3, "scope": 1, "term": "(div T24 T25 T3 T4)" } ], "kb": { "nonunifying": [ [ "(div T24 T25 T3 T4)", "(div X4 (0) X5 X6)" ], [ "(div T24 T25 T3 T4)", "(div (0) X10 X11 X12)" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T24", "T25" ], "free": [ "X4", "X5", "X6", "X10", "X11", "X12", "X21" ], "exprvars": [] } }, "412": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "413": { "goal": [{ "clause": -1, "scope": -1, "term": "(div T43 (s T40) T28 T29)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T40", "T43" ], "free": [], "exprvars": [] } }, "52": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "360": { "goal": [{ "clause": 3, "scope": 1, "term": "(div T1 T2 T3 T4)" }], "kb": { "nonunifying": [ [ "(div T1 T2 T3 T4)", "(div X4 (0) X5 X6)" ], [ "(div T1 T2 T3 T4)", "(div (0) X10 X11 X12)" ], [ "(div T1 T2 T3 T4)", "(div X17 X18 (s X19) X20)" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2" ], "free": [ "X4", "X5", "X6", "X10", "X11", "X12", "X17", "X18", "X19", "X20" ], "exprvars": [] } }, "361": { "goal": [ { "clause": 4, "scope": 2, "term": "(',' (minus T24 T25 X21) (',' (!_1) (div X21 T25 T28 T29)))" }, { "clause": 5, "scope": 2, "term": "(',' (minus T24 T25 X21) (',' (!_1) (div X21 T25 T28 T29)))" }, { "clause": -1, "scope": 2, "term": null }, { "clause": 3, "scope": 1, "term": "(div T24 T25 T3 T4)" } ], "kb": { "nonunifying": [ [ "(div T24 T25 T3 T4)", "(div X4 (0) X5 X6)" ], [ "(div T24 T25 T3 T4)", "(div (0) X10 X11 X12)" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T24", "T25" ], "free": [ "X4", "X5", "X6", "X10", "X11", "X12", "X21" ], "exprvars": [] } }, "362": { "goal": [ { "clause": 5, "scope": 2, "term": "(',' (minus T24 T25 X21) (',' (!_1) (div X21 T25 T28 T29)))" }, { "clause": -1, "scope": 2, "term": null }, { "clause": 3, "scope": 1, "term": "(div T24 T25 T3 T4)" } ], "kb": { "nonunifying": [ [ "(div T24 T25 T3 T4)", "(div X4 (0) X5 X6)" ], [ "(div T24 T25 T3 T4)", "(div (0) X10 X11 X12)" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T24", "T25" ], "free": [ "X4", "X5", "X6", "X10", "X11", "X12", "X21" ], "exprvars": [] } }, "363": { "goal": [{ "clause": 5, "scope": 2, "term": "(',' (minus T24 T25 X21) (',' (!_1) (div X21 T25 T28 T29)))" }], "kb": { "nonunifying": [ [ "(div T24 T25 T3 T4)", "(div X4 (0) X5 X6)" ], [ "(div T24 T25 T3 T4)", "(div (0) X10 X11 X12)" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T24", "T25" ], "free": [ "X4", "X5", "X6", "X10", "X11", "X12", "X21" ], "exprvars": [] } }, "1": { "goal": [{ "clause": -1, "scope": -1, "term": "(div T1 T2 T3 T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2" ], "free": [], "exprvars": [] } }, "364": { "goal": [ { "clause": -1, "scope": 2, "term": null }, { "clause": 3, "scope": 1, "term": "(div T24 T25 T3 T4)" } ], "kb": { "nonunifying": [ [ "(div T24 T25 T3 T4)", "(div X4 (0) X5 X6)" ], [ "(div T24 T25 T3 T4)", "(div (0) X10 X11 X12)" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T24", "T25" ], "free": [ "X4", "X5", "X6", "X10", "X11", "X12" ], "exprvars": [] } }, "365": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (minus T39 T40 X42) (',' (!_1) (div X42 (s T40) T28 T29)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T39", "T40" ], "free": ["X42"], "exprvars": [] } }, "420": { "goal": [{ "clause": 3, "scope": 1, "term": "(div T24 T25 T3 T4)" }], "kb": { "nonunifying": [ [ "(div T24 T25 T3 T4)", "(div X4 (0) X5 X6)" ], [ "(div T24 T25 T3 T4)", "(div (0) X10 X11 X12)" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T24", "T25" ], "free": [ "X4", "X5", "X6", "X10", "X11", "X12" ], "exprvars": [] } }, "3": { "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": [] } }, "366": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "421": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "422": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "423": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "369": { "goal": [{ "clause": -1, "scope": -1, "term": "(minus T39 T40 X42)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T39", "T40" ], "free": ["X42"], "exprvars": [] } }, "424": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "425": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "129": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (!_1) (',' (= T17 (0)) (= 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 X4 (0) X5 X6)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T14"], "free": [ "X4", "X5", "X6" ], "exprvars": [] } }, "426": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "328": { "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 X4 (0) X5 X6)" ], [ "(div T1 T2 T3 T4)", "(div (0) X10 X11 X12)" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2" ], "free": [ "X4", "X5", "X6", "X10", "X11", "X12" ], "exprvars": [] } }, "405": { "goal": [ { "clause": 4, "scope": 3, "term": "(minus T39 T40 X42)" }, { "clause": 5, "scope": 3, "term": "(minus T39 T40 X42)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T39", "T40" ], "free": ["X42"], "exprvars": [] } }, "329": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (= T17 (0)) (= T18 (0)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "406": { "goal": [{ "clause": 4, "scope": 3, "term": "(minus T39 T40 X42)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T39", "T40" ], "free": ["X42"], "exprvars": [] } }, "407": { "goal": [{ "clause": 5, "scope": 3, "term": "(minus T39 T40 X42)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T39", "T40" ], "free": ["X42"], "exprvars": [] } }, "40": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (!_1) (fail))" }, { "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": [] } }, "408": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "41": { "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 X4 (0) X5 X6)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2" ], "free": [ "X4", "X5", "X6" ], "exprvars": [] } }, "409": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 3, "label": "CASE" }, { "from": 3, "to": 40, "label": "EVAL with clause\ndiv(X4, 0, X5, X6) :- ','(!_1, fail).\nand substitutionT1 -> T8,\nX4 -> T8,\nT2 -> 0,\nT3 -> T9,\nX5 -> T9,\nT4 -> T10,\nX6 -> T10" }, { "from": 3, "to": 41, "label": "EVAL-BACKTRACK" }, { "from": 40, "to": 44, "label": "CUT" }, { "from": 41, "to": 129, "label": "EVAL with clause\ndiv(0, X10, X11, X12) :- ','(!_1, ','(=(X11, 0), =(X12, 0))).\nand substitutionT1 -> 0,\nT2 -> T14,\nX10 -> T14,\nT3 -> T17,\nX11 -> T17,\nT4 -> T18,\nX12 -> T18,\nT15 -> T17,\nT16 -> T18" }, { "from": 41, "to": 328, "label": "EVAL-BACKTRACK" }, { "from": 44, "to": 52, "label": "FAILURE" }, { "from": 129, "to": 329, "label": "CUT" }, { "from": 328, "to": 335, "label": "EVAL with clause\ndiv(X17, X18, s(X19), X20) :- ','(minus(X17, X18, X21), ','(!_1, div(X21, X18, X19, X20))).\nand substitutionT1 -> T24,\nX17 -> T24,\nT2 -> T25,\nX18 -> T25,\nX19 -> T28,\nT3 -> s(T28),\nT4 -> T29,\nX20 -> T29,\nT26 -> T28,\nT27 -> T29" }, { "from": 328, "to": 360, "label": "EVAL-BACKTRACK" }, { "from": 329, "to": 330, "label": "UNIFY CASE with substitutionT17 -> 0,\nT18 -> T19" }, { "from": 329, "to": 331, "label": "UNIFY-BACKTRACK" }, { "from": 330, "to": 332, "label": "UNIFY CASE with substitutionT19 -> 0" }, { "from": 330, "to": 333, "label": "UNIFY-BACKTRACK" }, { "from": 332, "to": 334, "label": "SUCCESS" }, { "from": 335, "to": 361, "label": "CASE" }, { "from": 360, "to": 424, "label": "EVAL with clause\ndiv(X77, X78, 0, X77).\nand substitutionT1 -> T68,\nX77 -> T68,\nT2 -> T69,\nX78 -> T69,\nT3 -> 0,\nT4 -> T68" }, { "from": 360, "to": 425, "label": "EVAL-BACKTRACK" }, { "from": 361, "to": 362, "label": "BACKTRACK\nfor clause: minus(X, 0, X)\nwith clash: (div(T24, T25, T3, T4), div(X4, 0, X5, X6))" }, { "from": 362, "to": 363, "label": "PARALLEL" }, { "from": 362, "to": 364, "label": "PARALLEL" }, { "from": 363, "to": 365, "label": "EVAL with clause\nminus(s(X39), s(X40), X41) :- minus(X39, X40, X41).\nand substitutionX39 -> T39,\nT24 -> s(T39),\nX40 -> T40,\nT25 -> s(T40),\nX21 -> X42,\nX41 -> X42" }, { "from": 363, "to": 366, "label": "EVAL-BACKTRACK" }, { "from": 364, "to": 420, "label": "FAILURE" }, { "from": 365, "to": 369, "label": "SPLIT 1" }, { "from": 365, "to": 370, "label": "SPLIT 2\nnew knowledge:\nT39 is ground\nT40 is ground\nT43 is ground\nreplacements:X42 -> T43" }, { "from": 369, "to": 405, "label": "CASE" }, { "from": 370, "to": 413, "label": "CUT" }, { "from": 405, "to": 406, "label": "PARALLEL" }, { "from": 405, "to": 407, "label": "PARALLEL" }, { "from": 406, "to": 408, "label": "EVAL with clause\nminus(X51, 0, X51).\nand substitutionT39 -> T50,\nX51 -> T50,\nT40 -> 0,\nX42 -> T50" }, { "from": 406, "to": 409, "label": "EVAL-BACKTRACK" }, { "from": 407, "to": 411, "label": "EVAL with clause\nminus(s(X60), s(X61), X62) :- minus(X60, X61, X62).\nand substitutionX60 -> T55,\nT39 -> s(T55),\nX61 -> T56,\nT40 -> s(T56),\nX42 -> X63,\nX62 -> X63" }, { "from": 407, "to": 412, "label": "EVAL-BACKTRACK" }, { "from": 408, "to": 410, "label": "SUCCESS" }, { "from": 411, "to": 369, "label": "INSTANCE with matching:\nT39 -> T55\nT40 -> T56\nX42 -> X63" }, { "from": 413, "to": 1, "label": "INSTANCE with matching:\nT1 -> T43\nT2 -> s(T40)\nT3 -> T28\nT4 -> T29" }, { "from": 420, "to": 421, "label": "EVAL with clause\ndiv(X73, X74, 0, X73).\nand substitutionT24 -> T64,\nX73 -> T64,\nT25 -> T65,\nX74 -> T65,\nT3 -> 0,\nT4 -> T64" }, { "from": 420, "to": 422, "label": "EVAL-BACKTRACK" }, { "from": 421, "to": 423, "label": "SUCCESS" }, { "from": 424, "to": 426, "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, 0, X1). divcB(X1, X2, 0, 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