/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern rem(g,g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToDTProblemTransformerProof [SOUND, 37 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 [EQUIVALENT, 14 ms] (11) QDP (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] (13) YES (14) PiDP (15) UsableRulesProof [EQUIVALENT, 0 ms] (16) PiDP (17) PiDPToQDPProof [SOUND, 0 ms] (18) QDP (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] (20) YES (21) PiDP (22) PiDPToQDPProof [SOUND, 0 ms] (23) QDP (24) QDPQMonotonicMRRProof [EQUIVALENT, 23 ms] (25) QDP (26) PisEmptyProof [EQUIVALENT, 0 ms] (27) YES ---------------------------------------- (0) Obligation: Clauses: rem(X, Y, R) :- ','(notZero(Y), ','(sub(X, Y, Z), rem(Z, Y, R))). rem(X, Y, X) :- ','(notZero(Y), geq(X, Y)). sub(s(X), s(Y), Z) :- sub(X, Y, Z). sub(X, 0, X). notZero(s(X)). geq(s(X), s(Y)) :- geq(X, Y). geq(X, 0). Query: rem(g,g,a) ---------------------------------------- (1) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 2, "program": { "directives": [], "clauses": [ [ "(rem X Y R)", "(',' (notZero Y) (',' (sub X Y Z) (rem Z Y R)))" ], [ "(rem X Y X)", "(',' (notZero Y) (geq X Y))" ], [ "(sub (s X) (s Y) Z)", "(sub X Y Z)" ], [ "(sub X (0) X)", null ], [ "(notZero (s X))", null ], [ "(geq (s X) (s Y))", "(geq X Y)" ], [ "(geq X (0))", null ] ] }, "graph": { "nodes": { "47": { "goal": [ { "clause": 0, "scope": 1, "term": "(rem T1 T2 T3)" }, { "clause": 1, "scope": 1, "term": "(rem T1 T2 T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2" ], "free": [], "exprvars": [] } }, "390": { "goal": [{ "clause": 4, "scope": 5, "term": "(',' (notZero T57) (geq T56 T57))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T56", "T57" ], "free": [], "exprvars": [] } }, "391": { "goal": [{ "clause": -1, "scope": -1, "term": "(geq T56 (s T62))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T56", "T62" ], "free": [], "exprvars": [] } }, "type": "Nodes", "392": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "393": { "goal": [ { "clause": 5, "scope": 6, "term": "(geq T56 (s T62))" }, { "clause": 6, "scope": 6, "term": "(geq T56 (s T62))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T56", "T62" ], "free": [], "exprvars": [] } }, "174": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (sub T7 (s T15) X7) (rem X7 (s T15) T10))" }], "kb": { "nonunifying": 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"kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T56", "T62" ], "free": [], "exprvars": [] } }, "297": { "goal": [ { "clause": 2, "scope": 3, "term": "(sub T7 (s T15) X7)" }, { "clause": 3, "scope": 3, "term": "(sub T7 (s T15) X7)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T7", "T15" ], "free": ["X7"], "exprvars": [] } }, "396": { "goal": [{ "clause": -1, "scope": -1, "term": "(geq T73 T74)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T73", "T74" ], "free": [], "exprvars": [] } }, "397": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "398": { "goal": [ { "clause": 5, "scope": 7, "term": "(geq T73 T74)" }, { "clause": 6, "scope": 7, "term": "(geq T73 T74)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T73", "T74" ], "free": [], "exprvars": [] } }, "179": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "399": { "goal": [{ "clause": 5, "scope": 7, "term": "(geq T73 T74)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T73", "T74" ], "free": [], "exprvars": [] } }, "313": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "314": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "315": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "316": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "317": { "goal": [{ "clause": 1, "scope": 1, "term": "(rem T7 T8 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T7", "T8" ], "free": [], "exprvars": [] } }, "318": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (notZero T57) (geq T56 T57))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T56", "T57" ], "free": [], "exprvars": [] } }, "319": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "160": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (notZero T8) (',' (sub T7 T8 X7) (rem X7 T8 T10)))" }, { "clause": 1, "scope": 1, "term": "(rem T7 T8 T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T7", "T8" ], "free": ["X7"], "exprvars": [] } }, "164": { "goal": [ { "clause": 4, "scope": 2, "term": "(',' (notZero T8) (',' (sub T7 T8 X7) (rem X7 T8 T10)))" }, { "clause": -1, "scope": 2, "term": null }, { "clause": 1, "scope": 1, "term": "(rem T7 T8 T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T7", "T8" ], "free": ["X7"], "exprvars": [] } }, "2": { "goal": [{ "clause": -1, "scope": -1, "term": "(rem T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2" ], "free": [], "exprvars": [] } }, "167": { "goal": [{ "clause": 4, "scope": 2, "term": "(',' (notZero T8) (',' (sub T7 T8 X7) (rem X7 T8 T10)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T7", "T8" ], "free": ["X7"], "exprvars": [] } }, "168": { "goal": [ { "clause": -1, "scope": 2, "term": null }, { "clause": 1, "scope": 1, "term": "(rem T7 T8 T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T7", "T8" ], "free": [], "exprvars": [] } }, "300": { "goal": [{ "clause": 2, "scope": 3, "term": "(sub T7 (s T15) X7)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T7", "T15" ], "free": ["X7"], "exprvars": [] } }, "301": { "goal": [{ "clause": 3, "scope": 3, "term": "(sub T7 (s T15) X7)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T7", "T15" ], "free": ["X7"], "exprvars": [] } }, "400": { "goal": [{ "clause": 6, "scope": 7, "term": "(geq T73 T74)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T73", "T74" ], "free": [], "exprvars": [] } }, "302": { "goal": [{ "clause": -1, "scope": -1, "term": "(sub T29 T30 X40)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T29", "T30" ], "free": ["X40"], "exprvars": [] } }, "401": { "goal": [{ "clause": -1, "scope": -1, "term": "(geq T85 T86)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T85", "T86" ], "free": [], "exprvars": [] } }, "303": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "402": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "304": { "goal": [ { "clause": 2, "scope": 4, "term": "(sub T29 T30 X40)" }, { "clause": 3, "scope": 4, "term": "(sub T29 T30 X40)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T29", "T30" ], "free": ["X40"], "exprvars": [] } }, "403": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "305": { "goal": [{ "clause": 2, "scope": 4, "term": "(sub T29 T30 X40)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T29", "T30" ], "free": ["X40"], "exprvars": [] } }, "404": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "306": { "goal": [{ "clause": 3, "scope": 4, "term": "(sub T29 T30 X40)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T29", "T30" ], "free": ["X40"], "exprvars": [] } }, "405": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "307": { "goal": [{ "clause": -1, "scope": -1, "term": "(sub T41 T42 X64)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T41", "T42" ], "free": ["X64"], "exprvars": [] } }, "406": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "308": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 2, "to": 47, "label": "CASE" }, { "from": 47, "to": 160, "label": "ONLY EVAL with clause\nrem(X4, X5, X6) :- ','(notZero(X5), ','(sub(X4, X5, X7), rem(X7, X5, X6))).\nand substitutionT1 -> T7,\nX4 -> T7,\nT2 -> T8,\nX5 -> T8,\nT3 -> T10,\nX6 -> T10,\nT9 -> T10" }, { "from": 160, "to": 164, "label": "CASE" }, { "from": 164, "to": 167, "label": "PARALLEL" }, { "from": 164, "to": 168, "label": "PARALLEL" }, { "from": 167, "to": 174, "label": "EVAL with clause\nnotZero(s(X12)).\nand substitutionX12 -> T15,\nT8 -> s(T15)" }, { "from": 167, "to": 179, "label": "EVAL-BACKTRACK" }, { "from": 168, "to": 317, "label": "FAILURE" }, { "from": 174, "to": 295, "label": "SPLIT 1" }, { "from": 174, "to": 296, "label": "SPLIT 2\nnew knowledge:\nT7 is ground\nT15 is ground\nT18 is ground\nreplacements:X7 -> T18" }, { "from": 295, "to": 297, "label": "CASE" }, { "from": 296, "to": 2, "label": "INSTANCE with matching:\nT1 -> T18\nT2 -> s(T15)\nT3 -> T10" }, { "from": 297, "to": 300, "label": "PARALLEL" }, { "from": 297, "to": 301, "label": "PARALLEL" }, { "from": 300, "to": 302, "label": "EVAL with clause\nsub(s(X37), s(X38), X39) :- sub(X37, X38, X39).\nand substitutionX37 -> T29,\nT7 -> s(T29),\nT15 -> T30,\nX38 -> T30,\nX7 -> X40,\nX39 -> X40" }, { "from": 300, "to": 303, "label": "EVAL-BACKTRACK" }, { "from": 301, "to": 316, "label": "BACKTRACK\nfor clause: sub(X, 0, X)because of non-unification" }, { "from": 302, "to": 304, "label": "CASE" }, { "from": 304, "to": 305, "label": "PARALLEL" }, { "from": 304, "to": 306, "label": "PARALLEL" }, { "from": 305, "to": 307, "label": "EVAL with clause\nsub(s(X61), s(X62), X63) :- sub(X61, X62, X63).\nand substitutionX61 -> T41,\nT29 -> s(T41),\nX62 -> T42,\nT30 -> s(T42),\nX40 -> X64,\nX63 -> X64" }, { "from": 305, "to": 308, "label": "EVAL-BACKTRACK" }, { "from": 306, "to": 313, "label": "EVAL with clause\nsub(X71, 0, X71).\nand substitutionT29 -> T47,\nX71 -> T47,\nT30 -> 0,\nX40 -> T47" }, { "from": 306, "to": 314, "label": "EVAL-BACKTRACK" }, { "from": 307, "to": 302, "label": "INSTANCE with matching:\nT29 -> T41\nT30 -> T42\nX40 -> X64" }, { "from": 313, "to": 315, "label": "SUCCESS" }, { "from": 317, "to": 318, "label": "EVAL with clause\nrem(X81, X82, X81) :- ','(notZero(X82), geq(X81, X82)).\nand substitutionT7 -> T56,\nX81 -> T56,\nT8 -> T57,\nX82 -> T57,\nT3 -> T56" }, { "from": 317, "to": 319, "label": "EVAL-BACKTRACK" }, { "from": 318, "to": 390, "label": "CASE" }, { "from": 390, "to": 391, "label": "EVAL with clause\nnotZero(s(X87)).\nand substitutionX87 -> T62,\nT57 -> s(T62)" }, { "from": 390, "to": 392, "label": "EVAL-BACKTRACK" }, { "from": 391, "to": 393, "label": "CASE" }, { "from": 393, "to": 394, "label": "PARALLEL" }, { "from": 393, "to": 395, "label": "PARALLEL" }, { "from": 394, "to": 396, "label": "EVAL with clause\ngeq(s(X98), s(X99)) :- geq(X98, X99).\nand substitutionX98 -> T73,\nT56 -> s(T73),\nT62 -> T74,\nX99 -> T74" }, { "from": 394, "to": 397, "label": "EVAL-BACKTRACK" }, { "from": 395, "to": 406, "label": "BACKTRACK\nfor clause: geq(X, 0)because of non-unification" }, { "from": 396, "to": 398, "label": "CASE" }, { "from": 398, "to": 399, "label": "PARALLEL" }, { "from": 398, "to": 400, "label": "PARALLEL" }, { "from": 399, "to": 401, "label": "EVAL with clause\ngeq(s(X110), s(X111)) :- geq(X110, X111).\nand substitutionX110 -> T85,\nT73 -> s(T85),\nX111 -> T86,\nT74 -> s(T86)" }, { "from": 399, "to": 402, "label": "EVAL-BACKTRACK" }, { "from": 400, "to": 403, "label": "EVAL with clause\ngeq(X116, 0).\nand substitutionT73 -> T91,\nX116 -> T91,\nT74 -> 0" }, { "from": 400, "to": 404, "label": "EVAL-BACKTRACK" }, { "from": 401, "to": 396, "label": "INSTANCE with matching:\nT73 -> T85\nT74 -> T86" }, { "from": 403, "to": 405, "label": "SUCCESS" } ], "type": "Graph" } } ---------------------------------------- (2) Obligation: Triples: subD(s(X1), s(X2), X3) :- subD(X1, X2, X3). geqC(s(X1), s(X2)) :- geqC(X1, X2). remA(s(X1), s(X2), X3) :- subD(X1, X2, X4). remA(X1, s(X2), X3) :- ','(subcB(X1, X2, X4), remA(X4, s(X2), X3)). remA(s(X1), s(X2), s(X1)) :- geqC(X1, X2). Clauses: remcA(X1, s(X2), X3) :- ','(subcB(X1, X2, X4), remcA(X4, s(X2), X3)). remcA(s(X1), s(X2), s(X1)) :- geqcC(X1, X2). subcD(s(X1), s(X2), X3) :- subcD(X1, X2, X3). subcD(X1, 0, X1). geqcC(s(X1), s(X2)) :- geqcC(X1, X2). geqcC(X1, 0). subcB(s(X1), X2, X3) :- subcD(X1, X2, X3). Afs: remA(x1, x2, x3) = remA(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: remA_in_3: (b,b,f) subD_in_3: (b,b,f) subcB_in_3: (b,b,f) subcD_in_3: (b,b,f) geqC_in_2: (b,b) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: REMA_IN_GGA(s(X1), s(X2), X3) -> U3_GGA(X1, X2, X3, subD_in_gga(X1, X2, X4)) REMA_IN_GGA(s(X1), s(X2), X3) -> SUBD_IN_GGA(X1, X2, X4) SUBD_IN_GGA(s(X1), s(X2), X3) -> U1_GGA(X1, X2, X3, subD_in_gga(X1, X2, X3)) SUBD_IN_GGA(s(X1), s(X2), X3) -> SUBD_IN_GGA(X1, X2, X3) REMA_IN_GGA(X1, s(X2), X3) -> U4_GGA(X1, X2, X3, subcB_in_gga(X1, X2, X4)) U4_GGA(X1, X2, X3, subcB_out_gga(X1, X2, X4)) -> U5_GGA(X1, X2, X3, remA_in_gga(X4, s(X2), X3)) U4_GGA(X1, X2, X3, subcB_out_gga(X1, X2, X4)) -> REMA_IN_GGA(X4, s(X2), X3) REMA_IN_GGA(s(X1), s(X2), s(X1)) -> U6_GGA(X1, X2, geqC_in_gg(X1, X2)) REMA_IN_GGA(s(X1), s(X2), s(X1)) -> GEQC_IN_GG(X1, X2) GEQC_IN_GG(s(X1), s(X2)) -> U2_GG(X1, X2, geqC_in_gg(X1, X2)) GEQC_IN_GG(s(X1), s(X2)) -> GEQC_IN_GG(X1, X2) The TRS R consists of the following rules: subcB_in_gga(s(X1), X2, X3) -> U13_gga(X1, X2, X3, subcD_in_gga(X1, X2, X3)) subcD_in_gga(s(X1), s(X2), X3) -> U11_gga(X1, X2, X3, subcD_in_gga(X1, X2, X3)) subcD_in_gga(X1, 0, X1) -> subcD_out_gga(X1, 0, X1) U11_gga(X1, X2, X3, subcD_out_gga(X1, X2, X3)) -> subcD_out_gga(s(X1), s(X2), X3) U13_gga(X1, X2, X3, subcD_out_gga(X1, X2, X3)) -> subcB_out_gga(s(X1), X2, X3) The argument filtering Pi contains the following mapping: remA_in_gga(x1, x2, x3) = remA_in_gga(x1, x2) s(x1) = s(x1) subD_in_gga(x1, x2, x3) = subD_in_gga(x1, x2) subcB_in_gga(x1, x2, x3) = subcB_in_gga(x1, x2) U13_gga(x1, x2, x3, x4) = U13_gga(x1, x2, x4) subcD_in_gga(x1, x2, x3) = subcD_in_gga(x1, x2) U11_gga(x1, x2, x3, x4) = U11_gga(x1, x2, x4) 0 = 0 subcD_out_gga(x1, x2, x3) = subcD_out_gga(x1, x2, x3) subcB_out_gga(x1, x2, x3) = subcB_out_gga(x1, x2, x3) geqC_in_gg(x1, x2) = geqC_in_gg(x1, x2) REMA_IN_GGA(x1, x2, x3) = REMA_IN_GGA(x1, x2) U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4) SUBD_IN_GGA(x1, x2, x3) = SUBD_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x2, x4) U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4) U5_GGA(x1, x2, x3, x4) = U5_GGA(x1, x2, x4) U6_GGA(x1, x2, x3) = U6_GGA(x1, x2, x3) GEQC_IN_GG(x1, x2) = GEQC_IN_GG(x1, x2) U2_GG(x1, x2, x3) = U2_GG(x1, x2, x3) 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: REMA_IN_GGA(s(X1), s(X2), X3) -> U3_GGA(X1, X2, X3, subD_in_gga(X1, X2, X4)) REMA_IN_GGA(s(X1), s(X2), X3) -> SUBD_IN_GGA(X1, X2, X4) SUBD_IN_GGA(s(X1), s(X2), X3) -> U1_GGA(X1, X2, X3, subD_in_gga(X1, X2, X3)) SUBD_IN_GGA(s(X1), s(X2), X3) -> SUBD_IN_GGA(X1, X2, X3) REMA_IN_GGA(X1, s(X2), X3) -> U4_GGA(X1, X2, X3, subcB_in_gga(X1, X2, X4)) U4_GGA(X1, X2, X3, subcB_out_gga(X1, X2, X4)) -> U5_GGA(X1, X2, X3, remA_in_gga(X4, s(X2), X3)) U4_GGA(X1, X2, X3, subcB_out_gga(X1, X2, X4)) -> REMA_IN_GGA(X4, s(X2), X3) REMA_IN_GGA(s(X1), s(X2), s(X1)) -> U6_GGA(X1, X2, geqC_in_gg(X1, X2)) REMA_IN_GGA(s(X1), s(X2), s(X1)) -> GEQC_IN_GG(X1, X2) GEQC_IN_GG(s(X1), s(X2)) -> U2_GG(X1, X2, geqC_in_gg(X1, X2)) GEQC_IN_GG(s(X1), s(X2)) -> GEQC_IN_GG(X1, X2) The TRS R consists of the following rules: subcB_in_gga(s(X1), X2, X3) -> U13_gga(X1, X2, X3, subcD_in_gga(X1, X2, X3)) subcD_in_gga(s(X1), s(X2), X3) -> U11_gga(X1, X2, X3, subcD_in_gga(X1, X2, X3)) subcD_in_gga(X1, 0, X1) -> subcD_out_gga(X1, 0, X1) U11_gga(X1, X2, X3, subcD_out_gga(X1, X2, X3)) -> subcD_out_gga(s(X1), s(X2), X3) U13_gga(X1, X2, X3, subcD_out_gga(X1, X2, X3)) -> subcB_out_gga(s(X1), X2, X3) The argument filtering Pi contains the following mapping: remA_in_gga(x1, x2, x3) = remA_in_gga(x1, x2) s(x1) = s(x1) subD_in_gga(x1, x2, x3) = subD_in_gga(x1, x2) subcB_in_gga(x1, x2, x3) = subcB_in_gga(x1, x2) U13_gga(x1, x2, x3, x4) = U13_gga(x1, x2, x4) subcD_in_gga(x1, x2, x3) = subcD_in_gga(x1, x2) U11_gga(x1, x2, x3, x4) = U11_gga(x1, x2, x4) 0 = 0 subcD_out_gga(x1, x2, x3) = subcD_out_gga(x1, x2, x3) subcB_out_gga(x1, x2, x3) = subcB_out_gga(x1, x2, x3) geqC_in_gg(x1, x2) = geqC_in_gg(x1, x2) REMA_IN_GGA(x1, x2, x3) = REMA_IN_GGA(x1, x2) U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4) SUBD_IN_GGA(x1, x2, x3) = SUBD_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x2, x4) U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4) U5_GGA(x1, x2, x3, x4) = U5_GGA(x1, x2, x4) U6_GGA(x1, x2, x3) = U6_GGA(x1, x2, x3) GEQC_IN_GG(x1, x2) = GEQC_IN_GG(x1, x2) U2_GG(x1, x2, x3) = U2_GG(x1, x2, x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 7 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: GEQC_IN_GG(s(X1), s(X2)) -> GEQC_IN_GG(X1, X2) The TRS R consists of the following rules: subcB_in_gga(s(X1), X2, X3) -> U13_gga(X1, X2, X3, subcD_in_gga(X1, X2, X3)) subcD_in_gga(s(X1), s(X2), X3) -> U11_gga(X1, X2, X3, subcD_in_gga(X1, X2, X3)) subcD_in_gga(X1, 0, X1) -> subcD_out_gga(X1, 0, X1) U11_gga(X1, X2, X3, subcD_out_gga(X1, X2, X3)) -> subcD_out_gga(s(X1), s(X2), X3) U13_gga(X1, X2, X3, subcD_out_gga(X1, X2, X3)) -> subcB_out_gga(s(X1), X2, X3) The argument filtering Pi contains the following mapping: s(x1) = s(x1) subcB_in_gga(x1, x2, x3) = subcB_in_gga(x1, x2) U13_gga(x1, x2, x3, x4) = U13_gga(x1, x2, x4) subcD_in_gga(x1, x2, x3) = subcD_in_gga(x1, x2) U11_gga(x1, x2, x3, x4) = U11_gga(x1, x2, x4) 0 = 0 subcD_out_gga(x1, x2, x3) = subcD_out_gga(x1, x2, x3) subcB_out_gga(x1, x2, x3) = subcB_out_gga(x1, x2, x3) GEQC_IN_GG(x1, x2) = GEQC_IN_GG(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: GEQC_IN_GG(s(X1), s(X2)) -> GEQC_IN_GG(X1, X2) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (10) PiDPToQDPProof (EQUIVALENT) 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: GEQC_IN_GG(s(X1), s(X2)) -> GEQC_IN_GG(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: *GEQC_IN_GG(s(X1), s(X2)) -> GEQC_IN_GG(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: SUBD_IN_GGA(s(X1), s(X2), X3) -> SUBD_IN_GGA(X1, X2, X3) The TRS R consists of the following rules: subcB_in_gga(s(X1), X2, X3) -> U13_gga(X1, X2, X3, subcD_in_gga(X1, X2, X3)) subcD_in_gga(s(X1), s(X2), X3) -> U11_gga(X1, X2, X3, subcD_in_gga(X1, X2, X3)) subcD_in_gga(X1, 0, X1) -> subcD_out_gga(X1, 0, X1) U11_gga(X1, X2, X3, subcD_out_gga(X1, X2, X3)) -> subcD_out_gga(s(X1), s(X2), X3) U13_gga(X1, X2, X3, subcD_out_gga(X1, X2, X3)) -> subcB_out_gga(s(X1), X2, X3) The argument filtering Pi contains the following mapping: s(x1) = s(x1) subcB_in_gga(x1, x2, x3) = subcB_in_gga(x1, x2) U13_gga(x1, x2, x3, x4) = U13_gga(x1, x2, x4) subcD_in_gga(x1, x2, x3) = subcD_in_gga(x1, x2) U11_gga(x1, x2, x3, x4) = U11_gga(x1, x2, x4) 0 = 0 subcD_out_gga(x1, x2, x3) = subcD_out_gga(x1, x2, x3) subcB_out_gga(x1, x2, x3) = subcB_out_gga(x1, x2, x3) SUBD_IN_GGA(x1, x2, x3) = SUBD_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (15) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: SUBD_IN_GGA(s(X1), s(X2), X3) -> SUBD_IN_GGA(X1, X2, X3) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) SUBD_IN_GGA(x1, x2, x3) = SUBD_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: SUBD_IN_GGA(s(X1), s(X2)) -> SUBD_IN_GGA(X1, X2) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (19) 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: *SUBD_IN_GGA(s(X1), s(X2)) -> SUBD_IN_GGA(X1, X2) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (20) YES ---------------------------------------- (21) Obligation: Pi DP problem: The TRS P consists of the following rules: REMA_IN_GGA(X1, s(X2), X3) -> U4_GGA(X1, X2, X3, subcB_in_gga(X1, X2, X4)) U4_GGA(X1, X2, X3, subcB_out_gga(X1, X2, X4)) -> REMA_IN_GGA(X4, s(X2), X3) The TRS R consists of the following rules: subcB_in_gga(s(X1), X2, X3) -> U13_gga(X1, X2, X3, subcD_in_gga(X1, X2, X3)) subcD_in_gga(s(X1), s(X2), X3) -> U11_gga(X1, X2, X3, subcD_in_gga(X1, X2, X3)) subcD_in_gga(X1, 0, X1) -> subcD_out_gga(X1, 0, X1) U11_gga(X1, X2, X3, subcD_out_gga(X1, X2, X3)) -> subcD_out_gga(s(X1), s(X2), X3) U13_gga(X1, X2, X3, subcD_out_gga(X1, X2, X3)) -> subcB_out_gga(s(X1), X2, X3) The argument filtering Pi contains the following mapping: s(x1) = s(x1) subcB_in_gga(x1, x2, x3) = subcB_in_gga(x1, x2) U13_gga(x1, x2, x3, x4) = U13_gga(x1, x2, x4) subcD_in_gga(x1, x2, x3) = subcD_in_gga(x1, x2) U11_gga(x1, x2, x3, x4) = U11_gga(x1, x2, x4) 0 = 0 subcD_out_gga(x1, x2, x3) = subcD_out_gga(x1, x2, x3) subcB_out_gga(x1, x2, x3) = subcB_out_gga(x1, x2, x3) REMA_IN_GGA(x1, x2, x3) = REMA_IN_GGA(x1, x2) U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (22) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (23) Obligation: Q DP problem: The TRS P consists of the following rules: REMA_IN_GGA(X1, s(X2)) -> U4_GGA(X1, X2, subcB_in_gga(X1, X2)) U4_GGA(X1, X2, subcB_out_gga(X1, X2, X4)) -> REMA_IN_GGA(X4, s(X2)) The TRS R consists of the following rules: subcB_in_gga(s(X1), X2) -> U13_gga(X1, X2, subcD_in_gga(X1, X2)) subcD_in_gga(s(X1), s(X2)) -> U11_gga(X1, X2, subcD_in_gga(X1, X2)) subcD_in_gga(X1, 0) -> subcD_out_gga(X1, 0, X1) U11_gga(X1, X2, subcD_out_gga(X1, X2, X3)) -> subcD_out_gga(s(X1), s(X2), X3) U13_gga(X1, X2, subcD_out_gga(X1, X2, X3)) -> subcB_out_gga(s(X1), X2, X3) The set Q consists of the following terms: subcB_in_gga(x0, x1) subcD_in_gga(x0, x1) U11_gga(x0, x1, x2) U13_gga(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (24) 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: REMA_IN_GGA(X1, s(X2)) -> U4_GGA(X1, X2, subcB_in_gga(X1, X2)) U4_GGA(X1, X2, subcB_out_gga(X1, X2, X4)) -> REMA_IN_GGA(X4, s(X2)) Strictly oriented rules of the TRS R: subcD_in_gga(s(X1), s(X2)) -> U11_gga(X1, X2, subcD_in_gga(X1, X2)) U11_gga(X1, X2, subcD_out_gga(X1, X2, X3)) -> subcD_out_gga(s(X1), s(X2), X3) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(REMA_IN_GGA(x_1, x_2)) = 2 + 2*x_1 POL(U11_gga(x_1, x_2, x_3)) = 2*x_3 POL(U13_gga(x_1, x_2, x_3)) = 2*x_3 POL(U4_GGA(x_1, x_2, x_3)) = 2*x_3 POL(s(x_1)) = 2 + 2*x_1 POL(subcB_in_gga(x_1, x_2)) = x_1 POL(subcB_out_gga(x_1, x_2, x_3)) = 2 + 2*x_3 POL(subcD_in_gga(x_1, x_2)) = 1 + x_1 POL(subcD_out_gga(x_1, x_2, x_3)) = 1 + x_3 ---------------------------------------- (25) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: subcB_in_gga(s(X1), X2) -> U13_gga(X1, X2, subcD_in_gga(X1, X2)) subcD_in_gga(X1, 0) -> subcD_out_gga(X1, 0, X1) U13_gga(X1, X2, subcD_out_gga(X1, X2, X3)) -> subcB_out_gga(s(X1), X2, X3) The set Q consists of the following terms: subcB_in_gga(x0, x1) subcD_in_gga(x0, x1) U11_gga(x0, x1, x2) U13_gga(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (26) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (27) YES