YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Left Termination of the query pattern rem(g,g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1)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, 32 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, 0 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) QDPOrderProof [EQUIVALENT, 31 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": { "390": { "goal": [{ "clause": 5, "scope": 6, "term": "(geq T56 (s T62))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T56", "T62" ], "free": [], "exprvars": [] } }, "391": { "goal": [{ "clause": 6, "scope": 6, "term": "(geq T56 (s T62))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T56", "T62" ], "free": [], "exprvars": [] } }, "type": "Nodes", "370": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (notZero T57) (geq T56 T57))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T56", "T57" ], "free": [], "exprvars": [] } }, "392": { "goal": [{ "clause": -1, "scope": -1, "term": "(geq T73 T74)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T73", "T74" ], "free": [], "exprvars": [] } }, "371": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "393": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "152": { "goal": [{ "clause": 2, "scope": 3, "term": "(sub T7 (s T15) X7)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T7", "T15" ], "free": ["X7"], "exprvars": [] } }, "372": { "goal": [{ "clause": 4, "scope": 5, "term": "(',' (notZero T57) (geq T56 T57))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T56", "T57" ], "free": [], "exprvars": [] } }, "394": { "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": [] } }, "373": { "goal": [{ "clause": -1, "scope": -1, "term": "(geq T56 (s T62))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T56", "T62" ], "free": [], "exprvars": [] } }, "395": { "goal": [{ "clause": 5, "scope": 7, "term": "(geq T73 T74)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T73", "T74" ], "free": [], "exprvars": [] } }, "154": { "goal": [{ "clause": 3, "scope": 3, "term": "(sub T7 (s T15) X7)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T7", "T15" ], "free": ["X7"], "exprvars": [] } }, "374": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "396": { "goal": [{ "clause": 6, "scope": 7, "term": "(geq T73 T74)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T73", "T74" ], "free": [], "exprvars": [] } }, "375": { "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": [] } }, "397": { "goal": [{ "clause": -1, "scope": -1, "term": "(geq T85 T86)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T85", "T86" ], "free": [], "exprvars": [] } }, "310": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "398": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "399": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "312": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "50": { "goal": [{ "clause": -1, "scope": -1, "term": "(sub T7 (s T15) X7)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T7", "T15" ], "free": ["X7"], "exprvars": [] } }, "54": { "goal": [{ "clause": -1, "scope": -1, "term": "(rem T18 (s T15) T10)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T15", "T18" ], "free": [], "exprvars": [] } }, "16": { "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": [] } }, "17": { "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": [] } }, "18": { "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": [] } }, "19": { "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": [] } }, "181": { "goal": [{ "clause": -1, "scope": -1, "term": "(sub T41 T42 X64)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T41", "T42" ], "free": ["X64"], "exprvars": [] } }, "182": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "161": { "goal": [{ "clause": -1, "scope": -1, "term": "(sub T29 T30 X40)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T29", "T30" ], "free": ["X40"], "exprvars": [] } }, "162": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "163": { "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": [] } }, "164": { "goal": [{ "clause": 2, "scope": 4, "term": "(sub T29 T30 X40)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T29", "T30" ], "free": ["X40"], "exprvars": [] } }, "165": { "goal": [{ "clause": 3, "scope": 4, "term": "(sub T29 T30 X40)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T29", "T30" ], "free": ["X40"], "exprvars": [] } }, "2": { "goal": [{ "clause": -1, "scope": -1, "term": "(rem T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2" ], "free": [], "exprvars": [] } }, "147": { "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": [] } }, "367": { "goal": [{ "clause": 1, "scope": 1, "term": "(rem T7 T8 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T7", "T8" ], "free": [], "exprvars": [] } }, "400": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "401": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "6": { "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": [] } }, "402": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "307": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "309": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "20": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (sub T7 (s T15) X7) (rem X7 (s T15) T10))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T7", "T15" ], "free": ["X7"], "exprvars": [] } }, "21": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 2, "to": 6, "label": "CASE" }, { "from": 6, "to": 16, "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": 16, "to": 17, "label": "CASE" }, { "from": 17, "to": 18, "label": "PARALLEL" }, { "from": 17, "to": 19, "label": "PARALLEL" }, { "from": 18, "to": 20, "label": "EVAL with clause\nnotZero(s(X12)).\nand substitutionX12 -> T15,\nT8 -> s(T15)" }, { "from": 18, "to": 21, "label": "EVAL-BACKTRACK" }, { "from": 19, "to": 367, "label": "FAILURE" }, { "from": 20, "to": 50, "label": "SPLIT 1" }, { "from": 20, "to": 54, "label": "SPLIT 2\nnew knowledge:\nT7 is ground\nT15 is ground\nT18 is ground\nreplacements:X7 -> T18" }, { "from": 50, "to": 147, "label": "CASE" }, { "from": 54, "to": 2, "label": "INSTANCE with matching:\nT1 -> T18\nT2 -> s(T15)\nT3 -> T10" }, { "from": 147, "to": 152, "label": "PARALLEL" }, { "from": 147, "to": 154, "label": "PARALLEL" }, { "from": 152, "to": 161, "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": 152, "to": 162, "label": "EVAL-BACKTRACK" }, { "from": 154, "to": 312, "label": "BACKTRACK\nfor clause: sub(X, 0, X)because of non-unification" }, { "from": 161, "to": 163, "label": "CASE" }, { "from": 163, "to": 164, "label": "PARALLEL" }, { "from": 163, "to": 165, "label": "PARALLEL" }, { "from": 164, "to": 181, "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": 164, "to": 182, "label": "EVAL-BACKTRACK" }, { "from": 165, "to": 307, "label": "EVAL with clause\nsub(X71, 0, X71).\nand substitutionT29 -> T47,\nX71 -> T47,\nT30 -> 0,\nX40 -> T47" }, { "from": 165, "to": 309, "label": "EVAL-BACKTRACK" }, { "from": 181, "to": 161, "label": "INSTANCE with matching:\nT29 -> T41\nT30 -> T42\nX40 -> X64" }, { "from": 307, "to": 310, "label": "SUCCESS" }, { "from": 367, "to": 370, "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": 367, "to": 371, "label": "EVAL-BACKTRACK" }, { "from": 370, "to": 372, "label": "CASE" }, { "from": 372, "to": 373, "label": "EVAL with clause\nnotZero(s(X87)).\nand substitutionX87 -> T62,\nT57 -> s(T62)" }, { "from": 372, "to": 374, "label": "EVAL-BACKTRACK" }, { "from": 373, "to": 375, "label": "CASE" }, { "from": 375, "to": 390, "label": "PARALLEL" }, { "from": 375, "to": 391, "label": "PARALLEL" }, { "from": 390, "to": 392, "label": "EVAL with clause\ngeq(s(X98), s(X99)) :- geq(X98, X99).\nand substitutionX98 -> T73,\nT56 -> s(T73),\nT62 -> T74,\nX99 -> T74" }, { "from": 390, "to": 393, "label": "EVAL-BACKTRACK" }, { "from": 391, "to": 402, "label": "BACKTRACK\nfor clause: geq(X, 0)because of non-unification" }, { "from": 392, "to": 394, "label": "CASE" }, { "from": 394, "to": 395, "label": "PARALLEL" }, { "from": 394, "to": 396, "label": "PARALLEL" }, { "from": 395, "to": 397, "label": "EVAL with clause\ngeq(s(X110), s(X111)) :- geq(X110, X111).\nand substitutionX110 -> T85,\nT73 -> s(T85),\nX111 -> T86,\nT74 -> s(T86)" }, { "from": 395, "to": 398, "label": "EVAL-BACKTRACK" }, { "from": 396, "to": 399, "label": "EVAL with clause\ngeq(X116, 0).\nand substitutionT73 -> T91,\nX116 -> T91,\nT74 -> 0" }, { "from": 396, "to": 400, "label": "EVAL-BACKTRACK" }, { "from": 397, "to": 392, "label": "INSTANCE with matching:\nT73 -> T85\nT74 -> T86" }, { "from": 399, "to": 401, "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) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 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 remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( U4_GGA_3(x_1, ..., x_3) ) = x_3 POL( subcB_in_gga_2(x_1, x_2) ) = 2x_1 POL( s_1(x_1) ) = 2x_1 + 1 POL( U13_gga_3(x_1, ..., x_3) ) = x_3 + 2 POL( subcD_in_gga_2(x_1, x_2) ) = 2x_1 POL( U11_gga_3(x_1, ..., x_3) ) = x_1 + x_3 + 1 POL( 0 ) = 0 POL( subcD_out_gga_3(x_1, ..., x_3) ) = 2x_3 POL( subcB_out_gga_3(x_1, ..., x_3) ) = 2x_3 + 2 POL( REMA_IN_GGA_2(x_1, x_2) ) = 2x_1 + 1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 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) U13_gga(X1, X2, subcD_out_gga(X1, X2, X3)) -> subcB_out_gga(s(X1), X2, X3) U11_gga(X1, X2, subcD_out_gga(X1, X2, X3)) -> subcD_out_gga(s(X1), s(X2), X3) ---------------------------------------- (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(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. ---------------------------------------- (26) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (27) YES