/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.pl /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern div(g,g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) BuiltinConflictTransformerProof [EQUIVALENT, 0 ms] (2) Prolog (3) PrologToDTProblemTransformerProof [SOUND, 0 ms] (4) TRIPLES (5) TriplesToPiDPProof [SOUND, 0 ms] (6) PiDP (7) DependencyGraphProof [EQUIVALENT, 0 ms] (8) AND (9) PiDP (10) UsableRulesProof [EQUIVALENT, 0 ms] (11) PiDP (12) PiDPToQDPProof [SOUND, 15 ms] (13) QDP (14) QDPSizeChangeProof [EQUIVALENT, 0 ms] (15) YES (16) PiDP (17) PiDPToQDPProof [SOUND, 0 ms] (18) QDP (19) QDPQMonotonicMRRProof [EQUIVALENT, 27 ms] (20) QDP (21) DependencyGraphProof [EQUIVALENT, 0 ms] (22) TRUE ---------------------------------------- (0) Obligation: Clauses: div(X, 0, Z) :- ','(!, fail). div(0, Y, Z) :- ','(!, =(Z, 0)). div(X, Y, s(Z)) :- ','(minus(X, Y, U), div(U, Y, Z)). minus(0, Y, 0). minus(X, 0, X). minus(s(X), s(Y), Z) :- minus(X, Y, Z). =(X, X). Query: div(g,g,a) ---------------------------------------- (1) BuiltinConflictTransformerProof (EQUIVALENT) Renamed defined predicates conflicting with built-in predicates [PROLOG]. ---------------------------------------- (2) Obligation: Clauses: div(X, 0, Z) :- ','(!, fail). div(0, Y, Z) :- ','(!, user_defined_=(Z, 0)). div(X, Y, s(Z)) :- ','(minus(X, Y, U), div(U, Y, Z)). minus(0, Y, 0). minus(X, 0, X). minus(s(X), s(Y), Z) :- minus(X, Y, Z). user_defined_=(X, X). Query: div(g,g,a) ---------------------------------------- (3) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 1, "program": { "directives": [], "clauses": [ [ "(div X (0) Z)", "(',' (!) (fail))" ], [ "(div (0) Y Z)", "(',' (!) (user_defined_= Z (0)))" ], [ "(div X Y (s Z))", "(',' (minus X Y U) (div U Y Z))" ], [ "(minus (0) Y (0))", null ], [ "(minus X (0) X)", null ], [ "(minus (s X) (s Y) Z)", "(minus X Y Z)" ], [ "(user_defined_= X X)", null ] ] }, "graph": { "nodes": { "44": { "goal": [{ "clause": 2, "scope": 1, "term": "(div T1 T2 T3)" }], "kb": { "nonunifying": [ [ "(div T1 T2 T3)", "(div X3 (0) X4)" ], [ "(div T1 T2 T3)", "(div (0) X7 X8)" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2" ], "free": [ "X3", "X4", "X7", "X8" ], "exprvars": [] } }, "23": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (!_1) (fail))" }, { "clause": 1, "scope": 1, "term": "(div T6 (0) T3)" }, { "clause": 2, "scope": 1, "term": "(div T6 (0) T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T6"], "free": [], "exprvars": [] } }, "45": { "goal": [{ "clause": -1, "scope": -1, "term": "(user_defined_= T12 (0))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "24": { "goal": [ { "clause": 1, "scope": 1, "term": "(div T1 T2 T3)" }, { "clause": 2, "scope": 1, "term": "(div T1 T2 T3)" } ], "kb": { "nonunifying": [[ "(div T1 T2 T3)", "(div X3 (0) X4)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2" ], "free": [ "X3", "X4" ], "exprvars": [] } }, "46": { "goal": [{ "clause": 6, "scope": 2, "term": "(user_defined_= T12 (0))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "25": { "goal": [{ "clause": -1, "scope": -1, "term": "(fail)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "47": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "48": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "49": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "330": { "goal": [{ "clause": -1, "scope": -1, "term": "(minus T29 T30 X32)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T29", "T30" ], "free": ["X32"], "exprvars": [] } }, "331": { "goal": [{ "clause": -1, "scope": -1, "term": "(div T33 (s T30) T22)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T30", "T33" ], "free": [], "exprvars": [] } }, "332": { "goal": [ { "clause": 3, "scope": 4, "term": "(minus T29 T30 X32)" }, { "clause": 4, "scope": 4, "term": "(minus T29 T30 X32)" }, { "clause": 5, "scope": 4, "term": "(minus T29 T30 X32)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T29", "T30" ], "free": ["X32"], "exprvars": [] } }, "333": { "goal": [{ "clause": 3, "scope": 4, "term": "(minus T29 T30 X32)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T29", "T30" ], "free": ["X32"], "exprvars": [] } }, "334": { "goal": [ { "clause": 4, "scope": 4, "term": "(minus T29 T30 X32)" }, { "clause": 5, "scope": 4, "term": "(minus T29 T30 X32)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T29", "T30" ], "free": ["X32"], "exprvars": [] } }, "335": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "336": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "337": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "338": { "goal": [{ "clause": 4, "scope": 4, "term": "(minus T29 T30 X32)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T29", "T30" ], "free": ["X32"], "exprvars": [] } }, "339": { "goal": [{ "clause": 5, "scope": 4, "term": "(minus T29 T30 X32)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T29", "T30" ], "free": ["X32"], "exprvars": [] } }, "50": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (minus T19 T20 X18) (div X18 T20 T22))" }], "kb": { "nonunifying": [ [ "(div T19 T20 T3)", "(div X3 (0) X4)" ], [ "(div T19 T20 T3)", "(div (0) X7 X8)" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T19", "T20" ], "free": [ "X3", "X4", "X7", "X8", "X18" ], "exprvars": [] } }, "51": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "54": { "goal": [ { "clause": 3, "scope": 3, "term": "(',' (minus T19 T20 X18) (div X18 T20 T22))" }, { "clause": 4, "scope": 3, "term": "(',' (minus T19 T20 X18) (div X18 T20 T22))" }, { "clause": 5, "scope": 3, "term": "(',' (minus T19 T20 X18) (div X18 T20 T22))" } ], "kb": { "nonunifying": [ [ "(div T19 T20 T3)", "(div X3 (0) X4)" ], [ "(div T19 T20 T3)", "(div (0) X7 X8)" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T19", "T20" ], "free": [ "X3", "X4", "X7", "X8", "X18" ], "exprvars": [] } }, "340": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "341": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1": { "goal": [{ "clause": -1, "scope": -1, "term": "(div T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2" ], "free": [], "exprvars": [] } }, "342": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "343": { "goal": [{ "clause": -1, "scope": -1, "term": "(minus T50 T51 X58)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T50", "T51" ], "free": ["X58"], "exprvars": [] } }, "3": { "goal": [ { "clause": 0, "scope": 1, "term": "(div T1 T2 T3)" }, { "clause": 1, "scope": 1, "term": "(div T1 T2 T3)" }, { "clause": 2, "scope": 1, "term": "(div T1 T2 T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2" ], "free": [], "exprvars": [] } }, "344": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "125": { "goal": [{ "clause": 5, "scope": 3, "term": "(',' (minus T19 T20 X18) (div X18 T20 T22))" }], "kb": { "nonunifying": [ [ "(div T19 T20 T3)", "(div X3 (0) X4)" ], [ "(div T19 T20 T3)", "(div (0) X7 X8)" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T19", "T20" ], "free": [ "X3", "X4", "X7", "X8", "X18" ], "exprvars": [] } }, "328": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (minus T29 T30 X32) (div X32 (s T30) T22))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T29", "T30" ], "free": ["X32"], "exprvars": [] } }, "82": { "goal": [ { "clause": 4, "scope": 3, "term": "(',' (minus T19 T20 X18) (div X18 T20 T22))" }, { "clause": 5, "scope": 3, "term": "(',' (minus T19 T20 X18) (div X18 T20 T22))" } ], "kb": { "nonunifying": [ [ "(div T19 T20 T3)", "(div X3 (0) X4)" ], [ "(div T19 T20 T3)", "(div (0) X7 X8)" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T19", "T20" ], "free": [ "X3", "X4", "X7", "X8", "X18" ], "exprvars": [] } }, "329": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "42": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "43": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (!_1) (user_defined_= T12 (0)))" }, { "clause": 2, "scope": 1, "term": "(div (0) T10 T3)" } ], "kb": { "nonunifying": [[ "(div (0) T10 T3)", "(div X3 (0) X4)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": [ "X3", "X4" ], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 3, "label": "CASE" }, { "from": 3, "to": 23, "label": "EVAL with clause\ndiv(X3, 0, X4) :- ','(!_1, fail).\nand substitutionT1 -> T6,\nX3 -> T6,\nT2 -> 0,\nT3 -> T7,\nX4 -> T7" }, { "from": 3, "to": 24, "label": "EVAL-BACKTRACK" }, { "from": 23, "to": 25, "label": "CUT" }, { "from": 24, "to": 43, "label": "EVAL with clause\ndiv(0, X7, X8) :- ','(!_1, user_defined_=(X8, 0)).\nand substitutionT1 -> 0,\nT2 -> T10,\nX7 -> T10,\nT3 -> T12,\nX8 -> T12,\nT11 -> T12" }, { "from": 24, "to": 44, "label": "EVAL-BACKTRACK" }, { "from": 25, "to": 42, "label": "FAILURE" }, { "from": 43, "to": 45, "label": "CUT" }, { "from": 44, "to": 50, "label": "EVAL with clause\ndiv(X15, X16, s(X17)) :- ','(minus(X15, X16, X18), div(X18, X16, X17)).\nand substitutionT1 -> T19,\nX15 -> T19,\nT2 -> T20,\nX16 -> T20,\nX17 -> T22,\nT3 -> s(T22),\nT21 -> T22" }, { "from": 44, "to": 51, "label": "EVAL-BACKTRACK" }, { "from": 45, "to": 46, "label": "CASE" }, { "from": 46, "to": 47, "label": "EVAL with clause\nuser_defined_=(X11, X11).\nand substitutionT12 -> 0,\nX11 -> 0,\nT15 -> 0" }, { "from": 46, "to": 48, "label": "EVAL-BACKTRACK" }, { "from": 47, "to": 49, "label": "SUCCESS" }, { "from": 50, "to": 54, "label": "CASE" }, { "from": 54, "to": 82, "label": "BACKTRACK\nfor clause: minus(0, Y, 0)\nwith clash: (div(T19, T20, T3), div(0, X7, X8))" }, { "from": 82, "to": 125, "label": "BACKTRACK\nfor clause: minus(X, 0, X)\nwith clash: (div(T19, T20, T3), div(X3, 0, X4))" }, { "from": 125, "to": 328, "label": "EVAL with clause\nminus(s(X29), s(X30), X31) :- minus(X29, X30, X31).\nand substitutionX29 -> T29,\nT19 -> s(T29),\nX30 -> T30,\nT20 -> s(T30),\nX18 -> X32,\nX31 -> X32" }, { "from": 125, "to": 329, "label": "EVAL-BACKTRACK" }, { "from": 328, "to": 330, "label": "SPLIT 1" }, { "from": 328, "to": 331, "label": "SPLIT 2\nnew knowledge:\nT29 is ground\nT30 is ground\nT33 is ground\nreplacements:X32 -> T33" }, { "from": 330, "to": 332, "label": "CASE" }, { "from": 331, "to": 1, "label": "INSTANCE with matching:\nT1 -> T33\nT2 -> s(T30)\nT3 -> T22" }, { "from": 332, "to": 333, "label": "PARALLEL" }, { "from": 332, "to": 334, "label": "PARALLEL" }, { "from": 333, "to": 335, "label": "EVAL with clause\nminus(0, X41, 0).\nand substitutionT29 -> 0,\nT30 -> T40,\nX41 -> T40,\nX32 -> 0" }, { "from": 333, "to": 336, "label": "EVAL-BACKTRACK" }, { "from": 334, "to": 338, "label": "PARALLEL" }, { "from": 334, "to": 339, "label": "PARALLEL" }, { "from": 335, "to": 337, "label": "SUCCESS" }, { "from": 338, "to": 340, "label": "EVAL with clause\nminus(X46, 0, X46).\nand substitutionT29 -> T45,\nX46 -> T45,\nT30 -> 0,\nX32 -> T45" }, { "from": 338, "to": 341, "label": "EVAL-BACKTRACK" }, { "from": 339, "to": 343, "label": "EVAL with clause\nminus(s(X55), s(X56), X57) :- minus(X55, X56, X57).\nand substitutionX55 -> T50,\nT29 -> s(T50),\nX56 -> T51,\nT30 -> s(T51),\nX32 -> X58,\nX57 -> X58" }, { "from": 339, "to": 344, "label": "EVAL-BACKTRACK" }, { "from": 340, "to": 342, "label": "SUCCESS" }, { "from": 343, "to": 330, "label": "INSTANCE with matching:\nT29 -> T50\nT30 -> T51\nX32 -> X58" } ], "type": "Graph" } } ---------------------------------------- (4) Obligation: Triples: minusB(s(X1), s(X2), X3) :- minusB(X1, X2, X3). divA(s(X1), s(X2), s(X3)) :- minusB(X1, X2, X4). divA(s(X1), s(X2), s(X3)) :- ','(minuscB(X1, X2, X4), divA(X4, s(X2), X3)). Clauses: divcA(0, X1, 0). divcA(s(X1), s(X2), s(X3)) :- ','(minuscB(X1, X2, X4), divcA(X4, s(X2), X3)). minuscB(0, X1, 0). minuscB(X1, 0, X1). minuscB(s(X1), s(X2), X3) :- minuscB(X1, X2, X3). Afs: divA(x1, x2, x3) = divA(x1, x2) ---------------------------------------- (5) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: divA_in_3: (b,b,f) minusB_in_3: (b,b,f) minuscB_in_3: (b,b,f) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: DIVA_IN_GGA(s(X1), s(X2), s(X3)) -> U2_GGA(X1, X2, X3, minusB_in_gga(X1, X2, X4)) DIVA_IN_GGA(s(X1), s(X2), s(X3)) -> MINUSB_IN_GGA(X1, X2, X4) MINUSB_IN_GGA(s(X1), s(X2), X3) -> U1_GGA(X1, X2, X3, minusB_in_gga(X1, X2, X3)) MINUSB_IN_GGA(s(X1), s(X2), X3) -> MINUSB_IN_GGA(X1, X2, X3) DIVA_IN_GGA(s(X1), s(X2), s(X3)) -> U3_GGA(X1, X2, X3, minuscB_in_gga(X1, X2, X4)) U3_GGA(X1, X2, X3, minuscB_out_gga(X1, X2, X4)) -> U4_GGA(X1, X2, X3, divA_in_gga(X4, s(X2), X3)) U3_GGA(X1, X2, X3, minuscB_out_gga(X1, X2, X4)) -> DIVA_IN_GGA(X4, s(X2), X3) The TRS R consists of the following rules: minuscB_in_gga(0, X1, 0) -> minuscB_out_gga(0, X1, 0) minuscB_in_gga(X1, 0, X1) -> minuscB_out_gga(X1, 0, X1) minuscB_in_gga(s(X1), s(X2), X3) -> U8_gga(X1, X2, X3, minuscB_in_gga(X1, X2, X3)) U8_gga(X1, X2, X3, minuscB_out_gga(X1, X2, X3)) -> minuscB_out_gga(s(X1), s(X2), X3) The argument filtering Pi contains the following mapping: divA_in_gga(x1, x2, x3) = divA_in_gga(x1, x2) s(x1) = s(x1) minusB_in_gga(x1, x2, x3) = minusB_in_gga(x1, x2) minuscB_in_gga(x1, x2, x3) = minuscB_in_gga(x1, x2) 0 = 0 minuscB_out_gga(x1, x2, x3) = minuscB_out_gga(x1, x2, x3) U8_gga(x1, x2, x3, x4) = U8_gga(x1, x2, x4) DIVA_IN_GGA(x1, x2, x3) = DIVA_IN_GGA(x1, x2) U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x2, x4) MINUSB_IN_GGA(x1, x2, x3) = MINUSB_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x2, x4) U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4) U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4) We have to consider all (P,R,Pi)-chains Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES ---------------------------------------- (6) Obligation: Pi DP problem: The TRS P consists of the following rules: DIVA_IN_GGA(s(X1), s(X2), s(X3)) -> U2_GGA(X1, X2, X3, minusB_in_gga(X1, X2, X4)) DIVA_IN_GGA(s(X1), s(X2), s(X3)) -> MINUSB_IN_GGA(X1, X2, X4) MINUSB_IN_GGA(s(X1), s(X2), X3) -> U1_GGA(X1, X2, X3, minusB_in_gga(X1, X2, X3)) MINUSB_IN_GGA(s(X1), s(X2), X3) -> MINUSB_IN_GGA(X1, X2, X3) DIVA_IN_GGA(s(X1), s(X2), s(X3)) -> U3_GGA(X1, X2, X3, minuscB_in_gga(X1, X2, X4)) U3_GGA(X1, X2, X3, minuscB_out_gga(X1, X2, X4)) -> U4_GGA(X1, X2, X3, divA_in_gga(X4, s(X2), X3)) U3_GGA(X1, X2, X3, minuscB_out_gga(X1, X2, X4)) -> DIVA_IN_GGA(X4, s(X2), X3) The TRS R consists of the following rules: minuscB_in_gga(0, X1, 0) -> minuscB_out_gga(0, X1, 0) minuscB_in_gga(X1, 0, X1) -> minuscB_out_gga(X1, 0, X1) minuscB_in_gga(s(X1), s(X2), X3) -> U8_gga(X1, X2, X3, minuscB_in_gga(X1, X2, X3)) U8_gga(X1, X2, X3, minuscB_out_gga(X1, X2, X3)) -> minuscB_out_gga(s(X1), s(X2), X3) The argument filtering Pi contains the following mapping: divA_in_gga(x1, x2, x3) = divA_in_gga(x1, x2) s(x1) = s(x1) minusB_in_gga(x1, x2, x3) = minusB_in_gga(x1, x2) minuscB_in_gga(x1, x2, x3) = minuscB_in_gga(x1, x2) 0 = 0 minuscB_out_gga(x1, x2, x3) = minuscB_out_gga(x1, x2, x3) U8_gga(x1, x2, x3, x4) = U8_gga(x1, x2, x4) DIVA_IN_GGA(x1, x2, x3) = DIVA_IN_GGA(x1, x2) U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x2, x4) MINUSB_IN_GGA(x1, x2, x3) = MINUSB_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x2, x4) U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4) U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (7) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 4 less nodes. ---------------------------------------- (8) Complex Obligation (AND) ---------------------------------------- (9) Obligation: Pi DP problem: The TRS P consists of the following rules: MINUSB_IN_GGA(s(X1), s(X2), X3) -> MINUSB_IN_GGA(X1, X2, X3) The TRS R consists of the following rules: minuscB_in_gga(0, X1, 0) -> minuscB_out_gga(0, X1, 0) minuscB_in_gga(X1, 0, X1) -> minuscB_out_gga(X1, 0, X1) minuscB_in_gga(s(X1), s(X2), X3) -> U8_gga(X1, X2, X3, minuscB_in_gga(X1, X2, X3)) U8_gga(X1, X2, X3, minuscB_out_gga(X1, X2, X3)) -> minuscB_out_gga(s(X1), s(X2), X3) The argument filtering Pi contains the following mapping: s(x1) = s(x1) minuscB_in_gga(x1, x2, x3) = minuscB_in_gga(x1, x2) 0 = 0 minuscB_out_gga(x1, x2, x3) = minuscB_out_gga(x1, x2, x3) U8_gga(x1, x2, x3, x4) = U8_gga(x1, x2, x4) MINUSB_IN_GGA(x1, x2, x3) = MINUSB_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (10) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (11) Obligation: Pi DP problem: The TRS P consists of the following rules: MINUSB_IN_GGA(s(X1), s(X2), X3) -> MINUSB_IN_GGA(X1, X2, X3) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) MINUSB_IN_GGA(x1, x2, x3) = MINUSB_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (12) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (13) Obligation: Q DP problem: The TRS P consists of the following rules: MINUSB_IN_GGA(s(X1), s(X2)) -> MINUSB_IN_GGA(X1, X2) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (14) 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: *MINUSB_IN_GGA(s(X1), s(X2)) -> MINUSB_IN_GGA(X1, X2) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (15) YES ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: DIVA_IN_GGA(s(X1), s(X2), s(X3)) -> U3_GGA(X1, X2, X3, minuscB_in_gga(X1, X2, X4)) U3_GGA(X1, X2, X3, minuscB_out_gga(X1, X2, X4)) -> DIVA_IN_GGA(X4, s(X2), X3) The TRS R consists of the following rules: minuscB_in_gga(0, X1, 0) -> minuscB_out_gga(0, X1, 0) minuscB_in_gga(X1, 0, X1) -> minuscB_out_gga(X1, 0, X1) minuscB_in_gga(s(X1), s(X2), X3) -> U8_gga(X1, X2, X3, minuscB_in_gga(X1, X2, X3)) U8_gga(X1, X2, X3, minuscB_out_gga(X1, X2, X3)) -> minuscB_out_gga(s(X1), s(X2), X3) The argument filtering Pi contains the following mapping: s(x1) = s(x1) minuscB_in_gga(x1, x2, x3) = minuscB_in_gga(x1, x2) 0 = 0 minuscB_out_gga(x1, x2, x3) = minuscB_out_gga(x1, x2, x3) U8_gga(x1, x2, x3, x4) = U8_gga(x1, x2, x4) DIVA_IN_GGA(x1, x2, x3) = DIVA_IN_GGA(x1, x2) U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4) 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: DIVA_IN_GGA(s(X1), s(X2)) -> U3_GGA(X1, X2, minuscB_in_gga(X1, X2)) U3_GGA(X1, X2, minuscB_out_gga(X1, X2, X4)) -> DIVA_IN_GGA(X4, s(X2)) The TRS R consists of the following rules: minuscB_in_gga(0, X1) -> minuscB_out_gga(0, X1, 0) minuscB_in_gga(X1, 0) -> minuscB_out_gga(X1, 0, X1) minuscB_in_gga(s(X1), s(X2)) -> U8_gga(X1, X2, minuscB_in_gga(X1, X2)) U8_gga(X1, X2, minuscB_out_gga(X1, X2, X3)) -> minuscB_out_gga(s(X1), s(X2), X3) The set Q consists of the following terms: minuscB_in_gga(x0, x1) U8_gga(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (19) QDPQMonotonicMRRProof (EQUIVALENT) By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain. Strictly oriented dependency pairs: U3_GGA(X1, X2, minuscB_out_gga(X1, X2, X4)) -> DIVA_IN_GGA(X4, s(X2)) Strictly oriented rules of the TRS R: U8_gga(X1, X2, minuscB_out_gga(X1, X2, X3)) -> minuscB_out_gga(s(X1), s(X2), X3) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(DIVA_IN_GGA(x_1, x_2)) = x_1 POL(U3_GGA(x_1, x_2, x_3)) = 2 + x_3 POL(U8_gga(x_1, x_2, x_3)) = 2 + x_3 POL(minuscB_in_gga(x_1, x_2)) = x_1 POL(minuscB_out_gga(x_1, x_2, x_3)) = x_3 POL(s(x_1)) = 2 + 2*x_1 ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: DIVA_IN_GGA(s(X1), s(X2)) -> U3_GGA(X1, X2, minuscB_in_gga(X1, X2)) The TRS R consists of the following rules: minuscB_in_gga(0, X1) -> minuscB_out_gga(0, X1, 0) minuscB_in_gga(X1, 0) -> minuscB_out_gga(X1, 0, X1) minuscB_in_gga(s(X1), s(X2)) -> U8_gga(X1, X2, minuscB_in_gga(X1, X2)) The set Q consists of the following terms: minuscB_in_gga(x0, x1) U8_gga(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (21) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (22) TRUE