/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern rem(g,a,g) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToDTProblemTransformerProof [SOUND, 46 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) UsableRulesProof [EQUIVALENT, 0 ms] (16) PiDP (17) PiDPToQDPProof [EQUIVALENT, 0 ms] (18) QDP (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] (20) YES (21) PiDP (22) UsableRulesProof [EQUIVALENT, 0 ms] (23) PiDP (24) PiDPToQDPProof [SOUND, 0 ms] (25) QDP (26) QDPSizeChangeProof [EQUIVALENT, 0 ms] (27) YES (28) PiDP (29) UsableRulesProof [EQUIVALENT, 0 ms] (30) PiDP (31) PiDPToQDPProof [SOUND, 0 ms] (32) QDP (33) QDPQMonotonicMRRProof [EQUIVALENT, 29 ms] (34) QDP (35) PisEmptyProof [EQUIVALENT, 0 ms] (36) YES (37) PiDP (38) UsableRulesProof [EQUIVALENT, 0 ms] (39) PiDP (40) PiDPToQDPProof [SOUND, 0 ms] (41) QDP (42) QDPSizeChangeProof [EQUIVALENT, 0 ms] (43) 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,a,g) ---------------------------------------- (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": { "193": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "294": { "goal": [{ "clause": -1, "scope": -1, "term": "(geq T60 (s T68))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T60"], "free": [], "exprvars": [] } }, "251": { "goal": [{ "clause": -1, "scope": -1, "term": "(sub T44 T46 X64)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T44"], "free": ["X64"], "exprvars": [] } }, "295": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "231": { "goal": [{ "clause": -1, "scope": -1, "term": "(rem T19 (s T20) T9)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T9", "T20", "T19" ], "free": [], "exprvars": [] } }, "330": { "goal": [{ "clause": 5, "scope": 7, "term": "(geq T79 T81)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T79"], "free": [], "exprvars": [] } }, "254": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "331": { "goal": [{ "clause": 6, "scope": 7, "term": "(geq T79 T81)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T79"], "free": [], "exprvars": [] } }, "156": { "goal": [{ "clause": 4, "scope": 2, "term": "(',' (notZero T10) (',' (sub T7 T10 X7) (rem X7 T10 T9)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T7", "T9" ], "free": ["X7"], "exprvars": [] } }, "310": { "goal": [{ "clause": 6, "scope": 6, "term": "(geq T60 (s T68))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T60"], "free": [], "exprvars": [] } }, "135": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (notZero T10) (',' (sub T7 T10 X7) (rem X7 T10 T9)))" }, { "clause": 1, "scope": 1, "term": "(rem T7 T2 T9)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T7", "T9" ], "free": ["X7"], "exprvars": [] } }, "136": { "goal": [ { "clause": 4, "scope": 2, "term": "(',' (notZero T10) (',' (sub T7 T10 X7) (rem X7 T10 T9)))" }, { "clause": -1, "scope": 2, "term": null }, { "clause": 1, "scope": 1, "term": "(rem T7 T2 T9)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T7", "T9" ], "free": ["X7"], "exprvars": [] } }, "235": { "goal": [ { "clause": 2, "scope": 3, "term": "(sub T7 (s T16) X7)" }, { "clause": 3, "scope": 3, "term": "(sub T7 (s T16) X7)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T7"], "free": ["X7"], "exprvars": [] } }, "238": { "goal": [{ "clause": 2, "scope": 3, "term": "(sub T7 (s T16) X7)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T7"], "free": ["X7"], "exprvars": [] } }, "239": { "goal": [{ "clause": 3, "scope": 3, "term": "(sub T7 (s T16) X7)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T7"], "free": ["X7"], "exprvars": [] } }, "161": { "goal": [ { "clause": -1, "scope": 2, "term": null }, { "clause": 1, "scope": 1, "term": "(rem T7 T2 T9)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T7", "T9" ], "free": [], "exprvars": [] } }, "260": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "282": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (notZero T62) (geq T60 T62))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T60"], "free": [], "exprvars": [] } }, "261": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "262": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "241": { "goal": [{ "clause": -1, "scope": -1, "term": "(sub T31 T33 X40)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T31"], "free": ["X40"], "exprvars": [] } }, "263": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "285": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "461": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "187": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (sub T7 (s T16) X7) (rem X7 (s T16) T9))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T7", "T9" ], "free": ["X7"], "exprvars": [] } }, "462": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "2": { "goal": [{ "clause": -1, "scope": -1, "term": "(rem T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T3" ], "free": [], "exprvars": [] } }, "244": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "464": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "245": { "goal": [ { "clause": 2, "scope": 4, "term": "(sub T31 T33 X40)" }, { "clause": 3, "scope": 4, "term": "(sub T31 T33 X40)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T31"], "free": ["X40"], "exprvars": [] } }, "289": { "goal": [{ "clause": 4, "scope": 5, "term": "(',' (notZero T62) (geq T60 T62))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T60"], "free": [], "exprvars": [] } }, "465": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "4": { "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", "T3" ], "free": [], "exprvars": [] } }, "269": { "goal": [{ "clause": 1, "scope": 1, "term": "(rem T7 T2 T9)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T7", "T9" ], "free": [], "exprvars": [] } }, "248": { "goal": [{ "clause": 2, "scope": 4, "term": "(sub T31 T33 X40)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T31"], "free": ["X40"], "exprvars": [] } }, "303": { "goal": [ { "clause": 5, "scope": 6, "term": "(geq T60 (s T68))" }, { "clause": 6, "scope": 6, "term": "(geq T60 (s T68))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T60"], "free": [], "exprvars": [] } }, "325": { "goal": [{ "clause": -1, "scope": -1, "term": "(geq T79 T81)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T79"], "free": [], "exprvars": [] } }, "424": { "goal": [{ "clause": -1, "scope": -1, "term": "(geq T92 T94)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T92"], "free": [], "exprvars": [] } }, "249": { "goal": [{ "clause": 3, "scope": 4, "term": "(sub T31 T33 X40)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T31"], "free": ["X40"], "exprvars": [] } }, "425": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "228": { "goal": [{ "clause": -1, "scope": -1, "term": "(sub T7 (s T16) X7)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T7"], "free": ["X7"], "exprvars": [] } }, "328": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "329": { "goal": [ { "clause": 5, "scope": 7, "term": "(geq T79 T81)" }, { "clause": 6, "scope": 7, "term": "(geq T79 T81)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T79"], "free": [], "exprvars": [] } }, "308": { "goal": [{ "clause": 5, "scope": 6, "term": "(geq T60 (s T68))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T60"], "free": [], "exprvars": [] } } }, "edges": [ { "from": 2, "to": 4, "label": "CASE" }, { "from": 4, "to": 135, "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 -> T10,\nX5 -> T10,\nT3 -> T9,\nX6 -> T9,\nT8 -> T10" }, { "from": 135, "to": 136, "label": "CASE" }, { "from": 136, "to": 156, "label": "PARALLEL" }, { "from": 136, "to": 161, "label": "PARALLEL" }, { "from": 156, "to": 187, "label": "EVAL with clause\nnotZero(s(X12)).\nand substitutionX12 -> T16,\nT10 -> s(T16),\nT15 -> T16" }, { "from": 156, "to": 193, "label": "EVAL-BACKTRACK" }, { "from": 161, "to": 269, "label": "FAILURE" }, { "from": 187, "to": 228, "label": "SPLIT 1" }, { "from": 187, "to": 231, "label": "SPLIT 2\nnew knowledge:\nT7 is ground\nT20 is ground\nT19 is ground\nreplacements:X7 -> T19,\nT16 -> T20" }, { "from": 228, "to": 235, "label": "CASE" }, { "from": 231, "to": 2, "label": "INSTANCE with matching:\nT1 -> T19\nT2 -> s(T20)\nT3 -> T9" }, { "from": 235, "to": 238, "label": "PARALLEL" }, { "from": 235, "to": 239, "label": "PARALLEL" }, { "from": 238, "to": 241, "label": "EVAL with clause\nsub(s(X37), s(X38), X39) :- sub(X37, X38, X39).\nand substitutionX37 -> T31,\nT7 -> s(T31),\nT16 -> T33,\nX38 -> T33,\nX7 -> X40,\nX39 -> X40,\nT32 -> T33" }, { "from": 238, "to": 244, "label": "EVAL-BACKTRACK" }, { "from": 239, "to": 263, "label": "BACKTRACK\nfor clause: sub(X, 0, X)because of non-unification" }, { "from": 241, "to": 245, "label": "CASE" }, { "from": 245, "to": 248, "label": "PARALLEL" }, { "from": 245, "to": 249, "label": "PARALLEL" }, { "from": 248, "to": 251, "label": "EVAL with clause\nsub(s(X61), s(X62), X63) :- sub(X61, X62, X63).\nand substitutionX61 -> T44,\nT31 -> s(T44),\nX62 -> T46,\nT33 -> s(T46),\nX40 -> X64,\nX63 -> X64,\nT45 -> T46" }, { "from": 248, "to": 254, "label": "EVAL-BACKTRACK" }, { "from": 249, "to": 260, "label": "EVAL with clause\nsub(X71, 0, X71).\nand substitutionT31 -> T51,\nX71 -> T51,\nT33 -> 0,\nX40 -> T51" }, { "from": 249, "to": 261, "label": "EVAL-BACKTRACK" }, { "from": 251, "to": 241, "label": "INSTANCE with matching:\nT31 -> T44\nT33 -> T46\nX40 -> X64" }, { "from": 260, "to": 262, "label": "SUCCESS" }, { "from": 269, "to": 282, "label": "EVAL with clause\nrem(X81, X82, X81) :- ','(notZero(X82), geq(X81, X82)).\nand substitutionT7 -> T60,\nX81 -> T60,\nT2 -> T62,\nX82 -> T62,\nT9 -> T60,\nT61 -> T62" }, { "from": 269, "to": 285, "label": "EVAL-BACKTRACK" }, { "from": 282, "to": 289, "label": "CASE" }, { "from": 289, "to": 294, "label": "EVAL with clause\nnotZero(s(X87)).\nand substitutionX87 -> T68,\nT62 -> s(T68),\nT67 -> T68" }, { "from": 289, "to": 295, "label": "EVAL-BACKTRACK" }, { "from": 294, "to": 303, "label": "CASE" }, { "from": 303, "to": 308, "label": "PARALLEL" }, { "from": 303, "to": 310, "label": "PARALLEL" }, { "from": 308, "to": 325, "label": "EVAL with clause\ngeq(s(X98), s(X99)) :- geq(X98, X99).\nand substitutionX98 -> T79,\nT60 -> s(T79),\nT68 -> T81,\nX99 -> T81,\nT80 -> T81" }, { "from": 308, "to": 328, "label": "EVAL-BACKTRACK" }, { "from": 310, "to": 465, "label": "BACKTRACK\nfor clause: geq(X, 0)because of non-unification" }, { "from": 325, "to": 329, "label": "CASE" }, { "from": 329, "to": 330, "label": "PARALLEL" }, { "from": 329, "to": 331, "label": "PARALLEL" }, { "from": 330, "to": 424, "label": "EVAL with clause\ngeq(s(X110), s(X111)) :- geq(X110, X111).\nand substitutionX110 -> T92,\nT79 -> s(T92),\nX111 -> T94,\nT81 -> s(T94),\nT93 -> T94" }, { "from": 330, "to": 425, "label": "EVAL-BACKTRACK" }, { "from": 331, "to": 461, "label": "EVAL with clause\ngeq(X116, 0).\nand substitutionT79 -> T99,\nX116 -> T99,\nT81 -> 0" }, { "from": 331, "to": 462, "label": "EVAL-BACKTRACK" }, { "from": 424, "to": 325, "label": "INSTANCE with matching:\nT79 -> T92\nT81 -> T94" }, { "from": 461, "to": 464, "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, x3) ---------------------------------------- (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,f,b) (b,b,b) subD_in_3: (b,f,f) (b,b,f) subcB_in_3: (b,f,f) (b,b,f) subcD_in_3: (b,f,f) (b,b,f) geqC_in_2: (b,b) (b,f) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: REMA_IN_GAG(s(X1), s(X2), X3) -> U3_GAG(X1, X2, X3, subD_in_gaa(X1, X2, X4)) REMA_IN_GAG(s(X1), s(X2), X3) -> SUBD_IN_GAA(X1, X2, X4) SUBD_IN_GAA(s(X1), s(X2), X3) -> U1_GAA(X1, X2, X3, subD_in_gaa(X1, X2, X3)) SUBD_IN_GAA(s(X1), s(X2), X3) -> SUBD_IN_GAA(X1, X2, X3) REMA_IN_GAG(X1, s(X2), X3) -> U4_GAG(X1, X2, X3, subcB_in_gaa(X1, X2, X4)) U4_GAG(X1, X2, X3, subcB_out_gaa(X1, X2, X4)) -> U5_GAG(X1, X2, X3, remA_in_ggg(X4, s(X2), X3)) U4_GAG(X1, X2, X3, subcB_out_gaa(X1, X2, X4)) -> REMA_IN_GGG(X4, s(X2), X3) REMA_IN_GGG(s(X1), s(X2), X3) -> U3_GGG(X1, X2, X3, subD_in_gga(X1, X2, X4)) REMA_IN_GGG(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_GGG(X1, s(X2), X3) -> U4_GGG(X1, X2, X3, subcB_in_gga(X1, X2, X4)) U4_GGG(X1, X2, X3, subcB_out_gga(X1, X2, X4)) -> U5_GGG(X1, X2, X3, remA_in_ggg(X4, s(X2), X3)) U4_GGG(X1, X2, X3, subcB_out_gga(X1, X2, X4)) -> REMA_IN_GGG(X4, s(X2), X3) REMA_IN_GGG(s(X1), s(X2), s(X1)) -> U6_GGG(X1, X2, geqC_in_gg(X1, X2)) REMA_IN_GGG(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) REMA_IN_GAG(s(X1), s(X2), s(X1)) -> U6_GAG(X1, X2, geqC_in_ga(X1, X2)) REMA_IN_GAG(s(X1), s(X2), s(X1)) -> GEQC_IN_GA(X1, X2) GEQC_IN_GA(s(X1), s(X2)) -> U2_GA(X1, X2, geqC_in_ga(X1, X2)) GEQC_IN_GA(s(X1), s(X2)) -> GEQC_IN_GA(X1, X2) The TRS R consists of the following rules: subcB_in_gaa(s(X1), X2, X3) -> U13_gaa(X1, X2, X3, subcD_in_gaa(X1, X2, X3)) subcD_in_gaa(s(X1), s(X2), X3) -> U11_gaa(X1, X2, X3, subcD_in_gaa(X1, X2, X3)) subcD_in_gaa(X1, 0, X1) -> subcD_out_gaa(X1, 0, X1) U11_gaa(X1, X2, X3, subcD_out_gaa(X1, X2, X3)) -> subcD_out_gaa(s(X1), s(X2), X3) U13_gaa(X1, X2, X3, subcD_out_gaa(X1, X2, X3)) -> subcB_out_gaa(s(X1), X2, X3) 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) subD_in_gaa(x1, x2, x3) = subD_in_gaa(x1) subcB_in_gaa(x1, x2, x3) = subcB_in_gaa(x1) U13_gaa(x1, x2, x3, x4) = U13_gaa(x1, x4) subcD_in_gaa(x1, x2, x3) = subcD_in_gaa(x1) U11_gaa(x1, x2, x3, x4) = U11_gaa(x1, x4) subcD_out_gaa(x1, x2, x3) = subcD_out_gaa(x1, x2, x3) subcB_out_gaa(x1, x2, x3) = subcB_out_gaa(x1, x2, x3) remA_in_ggg(x1, x2, x3) = remA_in_ggg(x1, x2, x3) 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) geqC_in_ga(x1, x2) = geqC_in_ga(x1) REMA_IN_GAG(x1, x2, x3) = REMA_IN_GAG(x1, x3) U3_GAG(x1, x2, x3, x4) = U3_GAG(x1, x3, x4) SUBD_IN_GAA(x1, x2, x3) = SUBD_IN_GAA(x1) U1_GAA(x1, x2, x3, x4) = U1_GAA(x1, x4) U4_GAG(x1, x2, x3, x4) = U4_GAG(x1, x3, x4) U5_GAG(x1, x2, x3, x4) = U5_GAG(x1, x2, x3, x4) REMA_IN_GGG(x1, x2, x3) = REMA_IN_GGG(x1, x2, x3) U3_GGG(x1, x2, x3, x4) = U3_GGG(x1, x2, x3, x4) SUBD_IN_GGA(x1, x2, x3) = SUBD_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x2, x4) U4_GGG(x1, x2, x3, x4) = U4_GGG(x1, x2, x3, x4) U5_GGG(x1, x2, x3, x4) = U5_GGG(x1, x2, x3, x4) U6_GGG(x1, x2, x3) = U6_GGG(x1, x2, x3) GEQC_IN_GG(x1, x2) = GEQC_IN_GG(x1, x2) U2_GG(x1, x2, x3) = U2_GG(x1, x2, x3) U6_GAG(x1, x2, x3) = U6_GAG(x1, x3) GEQC_IN_GA(x1, x2) = GEQC_IN_GA(x1) U2_GA(x1, x2, x3) = U2_GA(x1, 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_GAG(s(X1), s(X2), X3) -> U3_GAG(X1, X2, X3, subD_in_gaa(X1, X2, X4)) REMA_IN_GAG(s(X1), s(X2), X3) -> SUBD_IN_GAA(X1, X2, X4) SUBD_IN_GAA(s(X1), s(X2), X3) -> U1_GAA(X1, X2, X3, subD_in_gaa(X1, X2, X3)) SUBD_IN_GAA(s(X1), s(X2), X3) -> SUBD_IN_GAA(X1, X2, X3) REMA_IN_GAG(X1, s(X2), X3) -> U4_GAG(X1, X2, X3, subcB_in_gaa(X1, X2, X4)) U4_GAG(X1, X2, X3, subcB_out_gaa(X1, X2, X4)) -> U5_GAG(X1, X2, X3, remA_in_ggg(X4, s(X2), X3)) U4_GAG(X1, X2, X3, subcB_out_gaa(X1, X2, X4)) -> REMA_IN_GGG(X4, s(X2), X3) REMA_IN_GGG(s(X1), s(X2), X3) -> U3_GGG(X1, X2, X3, subD_in_gga(X1, X2, X4)) REMA_IN_GGG(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_GGG(X1, s(X2), X3) -> U4_GGG(X1, X2, X3, subcB_in_gga(X1, X2, X4)) U4_GGG(X1, X2, X3, subcB_out_gga(X1, X2, X4)) -> U5_GGG(X1, X2, X3, remA_in_ggg(X4, s(X2), X3)) U4_GGG(X1, X2, X3, subcB_out_gga(X1, X2, X4)) -> REMA_IN_GGG(X4, s(X2), X3) REMA_IN_GGG(s(X1), s(X2), s(X1)) -> U6_GGG(X1, X2, geqC_in_gg(X1, X2)) REMA_IN_GGG(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) REMA_IN_GAG(s(X1), s(X2), s(X1)) -> U6_GAG(X1, X2, geqC_in_ga(X1, X2)) REMA_IN_GAG(s(X1), s(X2), s(X1)) -> GEQC_IN_GA(X1, X2) GEQC_IN_GA(s(X1), s(X2)) -> U2_GA(X1, X2, geqC_in_ga(X1, X2)) GEQC_IN_GA(s(X1), s(X2)) -> GEQC_IN_GA(X1, X2) The TRS R consists of the following rules: subcB_in_gaa(s(X1), X2, X3) -> U13_gaa(X1, X2, X3, subcD_in_gaa(X1, X2, X3)) subcD_in_gaa(s(X1), s(X2), X3) -> U11_gaa(X1, X2, X3, subcD_in_gaa(X1, X2, X3)) subcD_in_gaa(X1, 0, X1) -> subcD_out_gaa(X1, 0, X1) U11_gaa(X1, X2, X3, subcD_out_gaa(X1, X2, X3)) -> subcD_out_gaa(s(X1), s(X2), X3) U13_gaa(X1, X2, X3, subcD_out_gaa(X1, X2, X3)) -> subcB_out_gaa(s(X1), X2, X3) 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) subD_in_gaa(x1, x2, x3) = subD_in_gaa(x1) subcB_in_gaa(x1, x2, x3) = subcB_in_gaa(x1) U13_gaa(x1, x2, x3, x4) = U13_gaa(x1, x4) subcD_in_gaa(x1, x2, x3) = subcD_in_gaa(x1) U11_gaa(x1, x2, x3, x4) = U11_gaa(x1, x4) subcD_out_gaa(x1, x2, x3) = subcD_out_gaa(x1, x2, x3) subcB_out_gaa(x1, x2, x3) = subcB_out_gaa(x1, x2, x3) remA_in_ggg(x1, x2, x3) = remA_in_ggg(x1, x2, x3) 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) geqC_in_ga(x1, x2) = geqC_in_ga(x1) REMA_IN_GAG(x1, x2, x3) = REMA_IN_GAG(x1, x3) U3_GAG(x1, x2, x3, x4) = U3_GAG(x1, x3, x4) SUBD_IN_GAA(x1, x2, x3) = SUBD_IN_GAA(x1) U1_GAA(x1, x2, x3, x4) = U1_GAA(x1, x4) U4_GAG(x1, x2, x3, x4) = U4_GAG(x1, x3, x4) U5_GAG(x1, x2, x3, x4) = U5_GAG(x1, x2, x3, x4) REMA_IN_GGG(x1, x2, x3) = REMA_IN_GGG(x1, x2, x3) U3_GGG(x1, x2, x3, x4) = U3_GGG(x1, x2, x3, x4) SUBD_IN_GGA(x1, x2, x3) = SUBD_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x2, x4) U4_GGG(x1, x2, x3, x4) = U4_GGG(x1, x2, x3, x4) U5_GGG(x1, x2, x3, x4) = U5_GGG(x1, x2, x3, x4) U6_GGG(x1, x2, x3) = U6_GGG(x1, x2, x3) GEQC_IN_GG(x1, x2) = GEQC_IN_GG(x1, x2) U2_GG(x1, x2, x3) = U2_GG(x1, x2, x3) U6_GAG(x1, x2, x3) = U6_GAG(x1, x3) GEQC_IN_GA(x1, x2) = GEQC_IN_GA(x1) U2_GA(x1, x2, x3) = U2_GA(x1, x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 16 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: GEQC_IN_GA(s(X1), s(X2)) -> GEQC_IN_GA(X1, X2) The TRS R consists of the following rules: subcB_in_gaa(s(X1), X2, X3) -> U13_gaa(X1, X2, X3, subcD_in_gaa(X1, X2, X3)) subcD_in_gaa(s(X1), s(X2), X3) -> U11_gaa(X1, X2, X3, subcD_in_gaa(X1, X2, X3)) subcD_in_gaa(X1, 0, X1) -> subcD_out_gaa(X1, 0, X1) U11_gaa(X1, X2, X3, subcD_out_gaa(X1, X2, X3)) -> subcD_out_gaa(s(X1), s(X2), X3) U13_gaa(X1, X2, X3, subcD_out_gaa(X1, X2, X3)) -> subcB_out_gaa(s(X1), X2, X3) 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_gaa(x1, x2, x3) = subcB_in_gaa(x1) U13_gaa(x1, x2, x3, x4) = U13_gaa(x1, x4) subcD_in_gaa(x1, x2, x3) = subcD_in_gaa(x1) U11_gaa(x1, x2, x3, x4) = U11_gaa(x1, x4) subcD_out_gaa(x1, x2, x3) = subcD_out_gaa(x1, x2, x3) subcB_out_gaa(x1, x2, x3) = subcB_out_gaa(x1, x2, x3) 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_GA(x1, x2) = GEQC_IN_GA(x1) 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_GA(s(X1), s(X2)) -> GEQC_IN_GA(X1, X2) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) GEQC_IN_GA(x1, x2) = GEQC_IN_GA(x1) 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: GEQC_IN_GA(s(X1)) -> GEQC_IN_GA(X1) 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_GA(s(X1)) -> GEQC_IN_GA(X1) The graph contains the following edges 1 > 1 ---------------------------------------- (13) YES ---------------------------------------- (14) 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_gaa(s(X1), X2, X3) -> U13_gaa(X1, X2, X3, subcD_in_gaa(X1, X2, X3)) subcD_in_gaa(s(X1), s(X2), X3) -> U11_gaa(X1, X2, X3, subcD_in_gaa(X1, X2, X3)) subcD_in_gaa(X1, 0, X1) -> subcD_out_gaa(X1, 0, X1) U11_gaa(X1, X2, X3, subcD_out_gaa(X1, X2, X3)) -> subcD_out_gaa(s(X1), s(X2), X3) U13_gaa(X1, X2, X3, subcD_out_gaa(X1, X2, X3)) -> subcB_out_gaa(s(X1), X2, X3) 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_gaa(x1, x2, x3) = subcB_in_gaa(x1) U13_gaa(x1, x2, x3, x4) = U13_gaa(x1, x4) subcD_in_gaa(x1, x2, x3) = subcD_in_gaa(x1) U11_gaa(x1, x2, x3, x4) = U11_gaa(x1, x4) subcD_out_gaa(x1, x2, x3) = subcD_out_gaa(x1, x2, x3) subcB_out_gaa(x1, x2, x3) = subcB_out_gaa(x1, x2, x3) 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 ---------------------------------------- (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: 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 ---------------------------------------- (17) PiDPToQDPProof (EQUIVALENT) 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: 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. ---------------------------------------- (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: *GEQC_IN_GG(s(X1), s(X2)) -> GEQC_IN_GG(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: SUBD_IN_GGA(s(X1), s(X2), X3) -> SUBD_IN_GGA(X1, X2, X3) The TRS R consists of the following rules: subcB_in_gaa(s(X1), X2, X3) -> U13_gaa(X1, X2, X3, subcD_in_gaa(X1, X2, X3)) subcD_in_gaa(s(X1), s(X2), X3) -> U11_gaa(X1, X2, X3, subcD_in_gaa(X1, X2, X3)) subcD_in_gaa(X1, 0, X1) -> subcD_out_gaa(X1, 0, X1) U11_gaa(X1, X2, X3, subcD_out_gaa(X1, X2, X3)) -> subcD_out_gaa(s(X1), s(X2), X3) U13_gaa(X1, X2, X3, subcD_out_gaa(X1, X2, X3)) -> subcB_out_gaa(s(X1), X2, X3) 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_gaa(x1, x2, x3) = subcB_in_gaa(x1) U13_gaa(x1, x2, x3, x4) = U13_gaa(x1, x4) subcD_in_gaa(x1, x2, x3) = subcD_in_gaa(x1) U11_gaa(x1, x2, x3, x4) = U11_gaa(x1, x4) subcD_out_gaa(x1, x2, x3) = subcD_out_gaa(x1, x2, x3) subcB_out_gaa(x1, x2, x3) = subcB_out_gaa(x1, x2, x3) 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 ---------------------------------------- (22) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (23) 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 ---------------------------------------- (24) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (25) 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. ---------------------------------------- (26) 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 ---------------------------------------- (27) YES ---------------------------------------- (28) Obligation: Pi DP problem: The TRS P consists of the following rules: REMA_IN_GGG(X1, s(X2), X3) -> U4_GGG(X1, X2, X3, subcB_in_gga(X1, X2, X4)) U4_GGG(X1, X2, X3, subcB_out_gga(X1, X2, X4)) -> REMA_IN_GGG(X4, s(X2), X3) The TRS R consists of the following rules: subcB_in_gaa(s(X1), X2, X3) -> U13_gaa(X1, X2, X3, subcD_in_gaa(X1, X2, X3)) subcD_in_gaa(s(X1), s(X2), X3) -> U11_gaa(X1, X2, X3, subcD_in_gaa(X1, X2, X3)) subcD_in_gaa(X1, 0, X1) -> subcD_out_gaa(X1, 0, X1) U11_gaa(X1, X2, X3, subcD_out_gaa(X1, X2, X3)) -> subcD_out_gaa(s(X1), s(X2), X3) U13_gaa(X1, X2, X3, subcD_out_gaa(X1, X2, X3)) -> subcB_out_gaa(s(X1), X2, X3) 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_gaa(x1, x2, x3) = subcB_in_gaa(x1) U13_gaa(x1, x2, x3, x4) = U13_gaa(x1, x4) subcD_in_gaa(x1, x2, x3) = subcD_in_gaa(x1) U11_gaa(x1, x2, x3, x4) = U11_gaa(x1, x4) subcD_out_gaa(x1, x2, x3) = subcD_out_gaa(x1, x2, x3) subcB_out_gaa(x1, x2, x3) = subcB_out_gaa(x1, x2, x3) 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_GGG(x1, x2, x3) = REMA_IN_GGG(x1, x2, x3) U4_GGG(x1, x2, x3, x4) = U4_GGG(x1, x2, x3, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (29) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (30) Obligation: Pi DP problem: The TRS P consists of the following rules: REMA_IN_GGG(X1, s(X2), X3) -> U4_GGG(X1, X2, X3, subcB_in_gga(X1, X2, X4)) U4_GGG(X1, X2, X3, subcB_out_gga(X1, X2, X4)) -> REMA_IN_GGG(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)) U13_gga(X1, X2, X3, subcD_out_gga(X1, X2, X3)) -> subcB_out_gga(s(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) 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_GGG(x1, x2, x3) = REMA_IN_GGG(x1, x2, x3) U4_GGG(x1, x2, x3, x4) = U4_GGG(x1, x2, x3, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (31) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: REMA_IN_GGG(X1, s(X2), X3) -> U4_GGG(X1, X2, X3, subcB_in_gga(X1, X2)) U4_GGG(X1, X2, X3, subcB_out_gga(X1, X2, X4)) -> REMA_IN_GGG(X4, s(X2), X3) The TRS R consists of the following rules: subcB_in_gga(s(X1), X2) -> U13_gga(X1, X2, subcD_in_gga(X1, X2)) U13_gga(X1, X2, subcD_out_gga(X1, X2, X3)) -> subcB_out_gga(s(X1), X2, X3) 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) The set Q consists of the following terms: subcB_in_gga(x0, x1) U13_gga(x0, x1, x2) subcD_in_gga(x0, x1) U11_gga(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (33) 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_GGG(X1, s(X2), X3) -> U4_GGG(X1, X2, X3, subcB_in_gga(X1, X2)) U4_GGG(X1, X2, X3, subcB_out_gga(X1, X2, X4)) -> REMA_IN_GGG(X4, s(X2), X3) Strictly oriented rules of the TRS R: subcD_in_gga(s(X1), s(X2)) -> U11_gga(X1, X2, subcD_in_gga(X1, X2)) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(REMA_IN_GGG(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_3 POL(U11_gga(x_1, x_2, x_3)) = 2*x_1 + x_3 POL(U13_gga(x_1, x_2, x_3)) = 2 + x_3 POL(U4_GGG(x_1, x_2, x_3, x_4)) = 2*x_3 + 2*x_4 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)) = 2*x_1 POL(subcD_out_gga(x_1, x_2, x_3)) = 2*x_3 ---------------------------------------- (34) 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)) U13_gga(X1, X2, subcD_out_gga(X1, X2, X3)) -> subcB_out_gga(s(X1), X2, X3) 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) The set Q consists of the following terms: subcB_in_gga(x0, x1) U13_gga(x0, x1, x2) subcD_in_gga(x0, x1) U11_gga(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (35) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (36) YES ---------------------------------------- (37) Obligation: Pi DP problem: The TRS P consists of the following rules: SUBD_IN_GAA(s(X1), s(X2), X3) -> SUBD_IN_GAA(X1, X2, X3) The TRS R consists of the following rules: subcB_in_gaa(s(X1), X2, X3) -> U13_gaa(X1, X2, X3, subcD_in_gaa(X1, X2, X3)) subcD_in_gaa(s(X1), s(X2), X3) -> U11_gaa(X1, X2, X3, subcD_in_gaa(X1, X2, X3)) subcD_in_gaa(X1, 0, X1) -> subcD_out_gaa(X1, 0, X1) U11_gaa(X1, X2, X3, subcD_out_gaa(X1, X2, X3)) -> subcD_out_gaa(s(X1), s(X2), X3) U13_gaa(X1, X2, X3, subcD_out_gaa(X1, X2, X3)) -> subcB_out_gaa(s(X1), X2, X3) 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_gaa(x1, x2, x3) = subcB_in_gaa(x1) U13_gaa(x1, x2, x3, x4) = U13_gaa(x1, x4) subcD_in_gaa(x1, x2, x3) = subcD_in_gaa(x1) U11_gaa(x1, x2, x3, x4) = U11_gaa(x1, x4) subcD_out_gaa(x1, x2, x3) = subcD_out_gaa(x1, x2, x3) subcB_out_gaa(x1, x2, x3) = subcB_out_gaa(x1, x2, x3) 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_GAA(x1, x2, x3) = SUBD_IN_GAA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (38) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (39) Obligation: Pi DP problem: The TRS P consists of the following rules: SUBD_IN_GAA(s(X1), s(X2), X3) -> SUBD_IN_GAA(X1, X2, X3) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) SUBD_IN_GAA(x1, x2, x3) = SUBD_IN_GAA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (40) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (41) Obligation: Q DP problem: The TRS P consists of the following rules: SUBD_IN_GAA(s(X1)) -> SUBD_IN_GAA(X1) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (42) 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_GAA(s(X1)) -> SUBD_IN_GAA(X1) The graph contains the following edges 1 > 1 ---------------------------------------- (43) YES