/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern m(g,g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1)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, 34 ms] (4) TRIPLES (5) TriplesToPiDPProof [SOUND, 7 ms] (6) PiDP (7) DependencyGraphProof [EQUIVALENT, 0 ms] (8) AND (9) PiDP (10) UsableRulesProof [EQUIVALENT, 0 ms] (11) PiDP (12) PiDPToQDPProof [SOUND, 0 ms] (13) QDP (14) QDPSizeChangeProof [EQUIVALENT, 0 ms] (15) YES (16) PiDP (17) UsableRulesProof [EQUIVALENT, 0 ms] (18) PiDP (19) PiDPToQDPProof [SOUND, 3 ms] (20) QDP (21) QDPQMonotonicMRRProof [EQUIVALENT, 21 ms] (22) QDP (23) DependencyGraphProof [EQUIVALENT, 0 ms] (24) TRUE ---------------------------------------- (0) Obligation: Clauses: m(X, 0, Z) :- ','(!, =(Z, X)). m(0, Y, Z) :- ','(!, =(Z, 0)). m(X, Y, Z) :- ','(p(X, A), ','(p(Y, B), m(A, B, Z))). p(0, 0). p(s(0), 0). p(s(s(X)), s(Y)) :- p(s(X), Y). =(X, X). Query: m(g,g,a) ---------------------------------------- (1) BuiltinConflictTransformerProof (EQUIVALENT) Renamed defined predicates conflicting with built-in predicates [PROLOG]. ---------------------------------------- (2) Obligation: Clauses: m(X, 0, Z) :- ','(!, user_defined_=(Z, X)). m(0, Y, Z) :- ','(!, user_defined_=(Z, 0)). m(X, Y, Z) :- ','(p(X, A), ','(p(Y, B), m(A, B, Z))). p(0, 0). p(s(0), 0). p(s(s(X)), s(Y)) :- p(s(X), Y). user_defined_=(X, X). Query: m(g,g,a) ---------------------------------------- (3) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 2, "program": { "directives": [], "clauses": [ [ "(m X (0) Z)", "(',' (!) (user_defined_= Z X))" ], [ "(m (0) Y Z)", "(',' (!) 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"relations": [] }, "ground": [ "T23", "T24" ], "free": [ "X3", "X4", "X10", "X11", "X21", "X22" ], "exprvars": [] } }, "508": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (s T42) X60)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T42"], "free": ["X60"], "exprvars": [] } }, "509": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 2, "to": 4, "label": "CASE" }, { "from": 4, "to": 21, "label": "EVAL with clause\nm(X3, 0, X4) :- ','(!_1, user_defined_=(X4, X3)).\nand substitutionT1 -> T6,\nX3 -> T6,\nT2 -> 0,\nT3 -> T8,\nX4 -> T8,\nT7 -> T8" }, { "from": 4, "to": 22, "label": "EVAL-BACKTRACK" }, { "from": 21, "to": 23, "label": "CUT" }, { "from": 22, "to": 28, "label": "EVAL with clause\nm(0, X10, X11) :- ','(!_1, user_defined_=(X11, 0)).\nand substitutionT1 -> 0,\nT2 -> T14,\nX10 -> T14,\nT3 -> T16,\nX11 -> T16,\nT15 -> T16" }, { "from": 22, "to": 29, "label": "EVAL-BACKTRACK" }, { "from": 23, "to": 24, "label": "CASE" }, { "from": 24, "to": 25, "label": "EVAL with clause\nuser_defined_=(X7, X7).\nand substitutionT8 -> T11,\nX7 -> T11,\nT6 -> T11" }, { "from": 24, "to": 26, "label": "EVAL-BACKTRACK" }, { "from": 25, "to": 27, "label": "SUCCESS" }, { "from": 28, "to": 30, "label": "CUT" }, { "from": 29, "to": 35, "label": "ONLY EVAL with clause\nm(X18, X19, X20) :- ','(p(X18, X21), ','(p(X19, X22), m(X21, X22, X20))).\nand substitutionT1 -> T23,\nX18 -> T23,\nT2 -> T24,\nX19 -> T24,\nT3 -> T26,\nX20 -> T26,\nT25 -> T26" }, { "from": 30, "to": 31, "label": "CASE" }, { "from": 31, "to": 32, "label": "EVAL with clause\nuser_defined_=(X14, X14).\nand substitutionT16 -> 0,\nX14 -> 0,\nT19 -> 0" }, { "from": 31, "to": 33, "label": "EVAL-BACKTRACK" }, { "from": 32, "to": 34, "label": "SUCCESS" }, { "from": 35, "to": 37, "label": "CASE" }, { "from": 37, "to": 41, "label": "BACKTRACK\nfor clause: p(0, 0)\nwith clash: (m(T23, T24, T3), m(0, X10, X11))" }, { "from": 41, "to": 361, "label": "PARALLEL" }, { "from": 41, "to": 363, "label": "PARALLEL" }, { "from": 361, "to": 366, "label": "EVAL with clause\np(s(0), 0).\nand substitutionT23 -> s(0),\nX21 -> 0" }, { "from": 361, "to": 367, "label": "EVAL-BACKTRACK" }, { "from": 363, "to": 466, "label": "EVAL with clause\np(s(s(X49)), s(X50)) :- p(s(X49), X50).\nand substitutionX49 -> T37,\nT23 -> s(s(T37)),\nX50 -> X51,\nX21 -> s(X51)" }, { "from": 363, "to": 467, "label": "EVAL-BACKTRACK" }, { "from": 366, "to": 370, "label": "SPLIT 1" }, { "from": 366, "to": 372, "label": "SPLIT 2\nnew knowledge:\nT24 is ground\nT27 is ground\nreplacements:X22 -> T27" }, { "from": 370, "to": 422, "label": "CASE" }, { "from": 372, "to": 2, "label": "INSTANCE with matching:\nT1 -> 0\nT2 -> T27\nT3 -> T26" }, { "from": 422, "to": 423, "label": "BACKTRACK\nfor clause: p(0, 0)\nwith clash: (m(s(0), T24, T3), m(X3, 0, X4))" }, { "from": 423, "to": 424, "label": "PARALLEL" }, { "from": 423, "to": 425, "label": "PARALLEL" }, { "from": 424, "to": 427, "label": "EVAL with clause\np(s(0), 0).\nand substitutionT24 -> s(0),\nX22 -> 0" }, { "from": 424, "to": 429, "label": "EVAL-BACKTRACK" }, { "from": 425, "to": 436, "label": "EVAL with clause\np(s(s(X29)), s(X30)) :- p(s(X29), X30).\nand substitutionX29 -> T30,\nT24 -> s(s(T30)),\nX30 -> X31,\nX22 -> s(X31)" }, { "from": 425, "to": 437, "label": "EVAL-BACKTRACK" }, { "from": 427, "to": 432, "label": "SUCCESS" }, { "from": 436, "to": 438, "label": "CASE" }, { "from": 438, "to": 439, "label": "BACKTRACK\nfor clause: p(0, 0)because of non-unification" }, { "from": 439, "to": 440, "label": "PARALLEL" }, { "from": 439, "to": 441, "label": "PARALLEL" }, { "from": 440, "to": 442, "label": "EVAL with clause\np(s(0), 0).\nand substitutionT30 -> 0,\nX31 -> 0" }, { "from": 440, "to": 444, "label": "EVAL-BACKTRACK" }, { "from": 441, "to": 447, "label": "EVAL with clause\np(s(s(X38)), s(X39)) :- p(s(X38), X39).\nand substitutionX38 -> T33,\nT30 -> s(T33),\nX39 -> X40,\nX31 -> s(X40)" }, { "from": 441, "to": 448, "label": "EVAL-BACKTRACK" }, { "from": 442, "to": 445, "label": "SUCCESS" }, { "from": 447, "to": 436, "label": "INSTANCE with matching:\nT30 -> T33\nX31 -> X40" }, { "from": 466, "to": 468, "label": "SPLIT 1" }, { "from": 466, "to": 469, "label": "SPLIT 2\nnew knowledge:\nT37 is ground\nT38 is ground\nreplacements:X51 -> T38" }, { "from": 468, "to": 436, "label": "INSTANCE with matching:\nT30 -> T37\nX31 -> X51" }, { "from": 469, "to": 472, "label": "SPLIT 1" }, { "from": 469, "to": 473, "label": "SPLIT 2\nnew knowledge:\nT24 is ground\nT39 is ground\nreplacements:X22 -> T39" }, { "from": 472, "to": 474, "label": "CASE" }, { "from": 473, "to": 2, "label": "INSTANCE with matching:\nT1 -> s(T38)\nT2 -> T39\nT3 -> T26" }, { "from": 474, "to": 475, "label": "BACKTRACK\nfor clause: p(0, 0)\nwith clash: (m(s(s(T37)), T24, T3), m(X3, 0, X4))" }, { "from": 475, "to": 476, "label": "PARALLEL" }, { "from": 475, "to": 477, "label": "PARALLEL" }, { "from": 476, "to": 480, "label": "EVAL with clause\np(s(0), 0).\nand substitutionT24 -> s(0),\nX22 -> 0" }, { "from": 476, "to": 482, "label": "EVAL-BACKTRACK" }, { "from": 477, "to": 508, "label": "EVAL with clause\np(s(s(X58)), s(X59)) :- p(s(X58), X59).\nand substitutionX58 -> T42,\nT24 -> s(s(T42)),\nX59 -> X60,\nX22 -> s(X60)" }, { "from": 477, "to": 509, "label": "EVAL-BACKTRACK" }, { "from": 480, "to": 483, "label": "SUCCESS" }, { "from": 508, "to": 436, "label": "INSTANCE with matching:\nT30 -> T42\nX31 -> X60" } ], "type": "Graph" } } ---------------------------------------- (4) Obligation: Triples: pC(s(X1), s(X2)) :- pC(X1, X2). mA(s(0), s(s(X1)), X2) :- pC(X1, X3). mA(s(0), X1, X2) :- ','(pcB(X1, X3), mA(0, X3, X2)). mA(s(s(X1)), X2, X3) :- pC(X1, X4). mA(s(s(X1)), s(s(X2)), X3) :- ','(pcC(X1, X4), pC(X2, X5)). mA(s(s(X1)), X2, X3) :- ','(pcC(X1, X4), ','(pcD(X2, X5), mA(s(X4), X5, X3))). Clauses: mcA(X1, 0, X1). mcA(0, X1, 0). mcA(s(0), X1, X2) :- ','(pcB(X1, X3), mcA(0, X3, X2)). mcA(s(s(X1)), X2, X3) :- ','(pcC(X1, X4), ','(pcD(X2, X5), mcA(s(X4), X5, X3))). pcC(0, 0). pcC(s(X1), s(X2)) :- pcC(X1, X2). pcB(s(0), 0). pcB(s(s(X1)), s(X2)) :- pcC(X1, X2). pcD(s(0), 0). pcD(s(s(X1)), s(X2)) :- pcC(X1, X2). Afs: mA(x1, x2, x3) = mA(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: mA_in_3: (b,b,f) pC_in_2: (b,f) pcB_in_2: (b,f) pcC_in_2: (b,f) pcD_in_2: (b,f) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: MA_IN_GGA(s(0), s(s(X1)), X2) -> U2_GGA(X1, X2, pC_in_ga(X1, X3)) MA_IN_GGA(s(0), s(s(X1)), X2) -> PC_IN_GA(X1, X3) PC_IN_GA(s(X1), s(X2)) -> U1_GA(X1, X2, pC_in_ga(X1, X2)) PC_IN_GA(s(X1), s(X2)) -> PC_IN_GA(X1, X2) MA_IN_GGA(s(0), X1, X2) -> U3_GGA(X1, X2, pcB_in_ga(X1, X3)) U3_GGA(X1, X2, pcB_out_ga(X1, X3)) -> U4_GGA(X1, X2, mA_in_gga(0, X3, X2)) U3_GGA(X1, X2, pcB_out_ga(X1, X3)) -> MA_IN_GGA(0, X3, X2) MA_IN_GGA(s(s(X1)), X2, X3) -> U5_GGA(X1, X2, X3, pC_in_ga(X1, X4)) MA_IN_GGA(s(s(X1)), X2, X3) -> PC_IN_GA(X1, X4) MA_IN_GGA(s(s(X1)), s(s(X2)), X3) -> U6_GGA(X1, X2, X3, pcC_in_ga(X1, X4)) U6_GGA(X1, X2, X3, pcC_out_ga(X1, X4)) -> U7_GGA(X1, X2, X3, pC_in_ga(X2, X5)) U6_GGA(X1, X2, X3, pcC_out_ga(X1, X4)) -> PC_IN_GA(X2, X5) MA_IN_GGA(s(s(X1)), X2, X3) -> U8_GGA(X1, X2, X3, pcC_in_ga(X1, X4)) U8_GGA(X1, X2, X3, pcC_out_ga(X1, X4)) -> U9_GGA(X1, X2, X3, X4, pcD_in_ga(X2, X5)) U9_GGA(X1, X2, X3, X4, pcD_out_ga(X2, X5)) -> U10_GGA(X1, X2, X3, mA_in_gga(s(X4), X5, X3)) U9_GGA(X1, X2, X3, X4, pcD_out_ga(X2, X5)) -> MA_IN_GGA(s(X4), X5, X3) The TRS R consists of the following rules: pcB_in_ga(s(0), 0) -> pcB_out_ga(s(0), 0) pcB_in_ga(s(s(X1)), s(X2)) -> U18_ga(X1, X2, pcC_in_ga(X1, X2)) pcC_in_ga(0, 0) -> pcC_out_ga(0, 0) pcC_in_ga(s(X1), s(X2)) -> U17_ga(X1, X2, pcC_in_ga(X1, X2)) U17_ga(X1, X2, pcC_out_ga(X1, X2)) -> pcC_out_ga(s(X1), s(X2)) U18_ga(X1, X2, pcC_out_ga(X1, X2)) -> pcB_out_ga(s(s(X1)), s(X2)) pcD_in_ga(s(0), 0) -> pcD_out_ga(s(0), 0) pcD_in_ga(s(s(X1)), s(X2)) -> U19_ga(X1, X2, pcC_in_ga(X1, X2)) U19_ga(X1, X2, pcC_out_ga(X1, X2)) -> pcD_out_ga(s(s(X1)), s(X2)) The argument filtering Pi contains the following mapping: mA_in_gga(x1, x2, x3) = mA_in_gga(x1, x2) s(x1) = s(x1) 0 = 0 pC_in_ga(x1, x2) = pC_in_ga(x1) pcB_in_ga(x1, x2) = pcB_in_ga(x1) pcB_out_ga(x1, x2) = pcB_out_ga(x1, x2) U18_ga(x1, x2, x3) = U18_ga(x1, x3) pcC_in_ga(x1, x2) = pcC_in_ga(x1) pcC_out_ga(x1, x2) = pcC_out_ga(x1, x2) U17_ga(x1, x2, x3) = U17_ga(x1, x3) pcD_in_ga(x1, x2) = pcD_in_ga(x1) pcD_out_ga(x1, x2) = pcD_out_ga(x1, x2) U19_ga(x1, x2, x3) = U19_ga(x1, x3) MA_IN_GGA(x1, x2, x3) = MA_IN_GGA(x1, x2) U2_GGA(x1, x2, x3) = U2_GGA(x1, x3) PC_IN_GA(x1, x2) = PC_IN_GA(x1) U1_GA(x1, x2, x3) = U1_GA(x1, x3) U3_GGA(x1, x2, x3) = U3_GGA(x1, x3) U4_GGA(x1, x2, x3) = U4_GGA(x1, x3) U5_GGA(x1, x2, x3, x4) = U5_GGA(x1, x2, x4) U6_GGA(x1, x2, x3, x4) = U6_GGA(x1, x2, x4) U7_GGA(x1, x2, x3, x4) = U7_GGA(x1, x2, x4) U8_GGA(x1, x2, x3, x4) = U8_GGA(x1, x2, x4) U9_GGA(x1, x2, x3, x4, x5) = U9_GGA(x1, x2, x4, x5) U10_GGA(x1, x2, x3, x4) = U10_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: MA_IN_GGA(s(0), s(s(X1)), X2) -> U2_GGA(X1, X2, pC_in_ga(X1, X3)) MA_IN_GGA(s(0), s(s(X1)), X2) -> PC_IN_GA(X1, X3) PC_IN_GA(s(X1), s(X2)) -> U1_GA(X1, X2, pC_in_ga(X1, X2)) PC_IN_GA(s(X1), s(X2)) -> PC_IN_GA(X1, X2) MA_IN_GGA(s(0), X1, X2) -> U3_GGA(X1, X2, pcB_in_ga(X1, X3)) U3_GGA(X1, X2, pcB_out_ga(X1, X3)) -> U4_GGA(X1, X2, mA_in_gga(0, X3, X2)) U3_GGA(X1, X2, pcB_out_ga(X1, X3)) -> MA_IN_GGA(0, X3, X2) MA_IN_GGA(s(s(X1)), X2, X3) -> U5_GGA(X1, X2, X3, pC_in_ga(X1, X4)) MA_IN_GGA(s(s(X1)), X2, X3) -> PC_IN_GA(X1, X4) MA_IN_GGA(s(s(X1)), s(s(X2)), X3) -> U6_GGA(X1, X2, X3, pcC_in_ga(X1, X4)) U6_GGA(X1, X2, X3, pcC_out_ga(X1, X4)) -> U7_GGA(X1, X2, X3, pC_in_ga(X2, X5)) U6_GGA(X1, X2, X3, pcC_out_ga(X1, X4)) -> PC_IN_GA(X2, X5) MA_IN_GGA(s(s(X1)), X2, X3) -> U8_GGA(X1, X2, X3, pcC_in_ga(X1, X4)) U8_GGA(X1, X2, X3, pcC_out_ga(X1, X4)) -> U9_GGA(X1, X2, X3, X4, pcD_in_ga(X2, X5)) U9_GGA(X1, X2, X3, X4, pcD_out_ga(X2, X5)) -> U10_GGA(X1, X2, X3, mA_in_gga(s(X4), X5, X3)) U9_GGA(X1, X2, X3, X4, pcD_out_ga(X2, X5)) -> MA_IN_GGA(s(X4), X5, X3) The TRS R consists of the following rules: pcB_in_ga(s(0), 0) -> pcB_out_ga(s(0), 0) pcB_in_ga(s(s(X1)), s(X2)) -> U18_ga(X1, X2, pcC_in_ga(X1, X2)) pcC_in_ga(0, 0) -> pcC_out_ga(0, 0) pcC_in_ga(s(X1), s(X2)) -> U17_ga(X1, X2, pcC_in_ga(X1, X2)) U17_ga(X1, X2, pcC_out_ga(X1, X2)) -> pcC_out_ga(s(X1), s(X2)) U18_ga(X1, X2, pcC_out_ga(X1, X2)) -> pcB_out_ga(s(s(X1)), s(X2)) pcD_in_ga(s(0), 0) -> pcD_out_ga(s(0), 0) pcD_in_ga(s(s(X1)), s(X2)) -> U19_ga(X1, X2, pcC_in_ga(X1, X2)) U19_ga(X1, X2, pcC_out_ga(X1, X2)) -> pcD_out_ga(s(s(X1)), s(X2)) The argument filtering Pi contains the following mapping: mA_in_gga(x1, x2, x3) = mA_in_gga(x1, x2) s(x1) = s(x1) 0 = 0 pC_in_ga(x1, x2) = pC_in_ga(x1) pcB_in_ga(x1, x2) = pcB_in_ga(x1) pcB_out_ga(x1, x2) = pcB_out_ga(x1, x2) U18_ga(x1, x2, x3) = U18_ga(x1, x3) pcC_in_ga(x1, x2) = pcC_in_ga(x1) pcC_out_ga(x1, x2) = pcC_out_ga(x1, x2) U17_ga(x1, x2, x3) = U17_ga(x1, x3) pcD_in_ga(x1, x2) = pcD_in_ga(x1) pcD_out_ga(x1, x2) = pcD_out_ga(x1, x2) U19_ga(x1, x2, x3) = U19_ga(x1, x3) MA_IN_GGA(x1, x2, x3) = MA_IN_GGA(x1, x2) U2_GGA(x1, x2, x3) = U2_GGA(x1, x3) PC_IN_GA(x1, x2) = PC_IN_GA(x1) U1_GA(x1, x2, x3) = U1_GA(x1, x3) U3_GGA(x1, x2, x3) = U3_GGA(x1, x3) U4_GGA(x1, x2, x3) = U4_GGA(x1, x3) U5_GGA(x1, x2, x3, x4) = U5_GGA(x1, x2, x4) U6_GGA(x1, x2, x3, x4) = U6_GGA(x1, x2, x4) U7_GGA(x1, x2, x3, x4) = U7_GGA(x1, x2, x4) U8_GGA(x1, x2, x3, x4) = U8_GGA(x1, x2, x4) U9_GGA(x1, x2, x3, x4, x5) = U9_GGA(x1, x2, x4, x5) U10_GGA(x1, x2, x3, x4) = U10_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 12 less nodes. ---------------------------------------- (8) Complex Obligation (AND) ---------------------------------------- (9) Obligation: Pi DP problem: The TRS P consists of the following rules: PC_IN_GA(s(X1), s(X2)) -> PC_IN_GA(X1, X2) The TRS R consists of the following rules: pcB_in_ga(s(0), 0) -> pcB_out_ga(s(0), 0) pcB_in_ga(s(s(X1)), s(X2)) -> U18_ga(X1, X2, pcC_in_ga(X1, X2)) pcC_in_ga(0, 0) -> pcC_out_ga(0, 0) pcC_in_ga(s(X1), s(X2)) -> U17_ga(X1, X2, pcC_in_ga(X1, X2)) U17_ga(X1, X2, pcC_out_ga(X1, X2)) -> pcC_out_ga(s(X1), s(X2)) U18_ga(X1, X2, pcC_out_ga(X1, X2)) -> pcB_out_ga(s(s(X1)), s(X2)) pcD_in_ga(s(0), 0) -> pcD_out_ga(s(0), 0) pcD_in_ga(s(s(X1)), s(X2)) -> U19_ga(X1, X2, pcC_in_ga(X1, X2)) U19_ga(X1, X2, pcC_out_ga(X1, X2)) -> pcD_out_ga(s(s(X1)), s(X2)) The argument filtering Pi contains the following mapping: s(x1) = s(x1) 0 = 0 pcB_in_ga(x1, x2) = pcB_in_ga(x1) pcB_out_ga(x1, x2) = pcB_out_ga(x1, x2) U18_ga(x1, x2, x3) = U18_ga(x1, x3) pcC_in_ga(x1, x2) = pcC_in_ga(x1) pcC_out_ga(x1, x2) = pcC_out_ga(x1, x2) U17_ga(x1, x2, x3) = U17_ga(x1, x3) pcD_in_ga(x1, x2) = pcD_in_ga(x1) pcD_out_ga(x1, x2) = pcD_out_ga(x1, x2) U19_ga(x1, x2, x3) = U19_ga(x1, x3) PC_IN_GA(x1, x2) = PC_IN_GA(x1) 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: PC_IN_GA(s(X1), s(X2)) -> PC_IN_GA(X1, X2) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) PC_IN_GA(x1, x2) = PC_IN_GA(x1) 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: PC_IN_GA(s(X1)) -> PC_IN_GA(X1) 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: *PC_IN_GA(s(X1)) -> PC_IN_GA(X1) The graph contains the following edges 1 > 1 ---------------------------------------- (15) YES ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: MA_IN_GGA(s(s(X1)), X2, X3) -> U8_GGA(X1, X2, X3, pcC_in_ga(X1, X4)) U8_GGA(X1, X2, X3, pcC_out_ga(X1, X4)) -> U9_GGA(X1, X2, X3, X4, pcD_in_ga(X2, X5)) U9_GGA(X1, X2, X3, X4, pcD_out_ga(X2, X5)) -> MA_IN_GGA(s(X4), X5, X3) The TRS R consists of the following rules: pcB_in_ga(s(0), 0) -> pcB_out_ga(s(0), 0) pcB_in_ga(s(s(X1)), s(X2)) -> U18_ga(X1, X2, pcC_in_ga(X1, X2)) pcC_in_ga(0, 0) -> pcC_out_ga(0, 0) pcC_in_ga(s(X1), s(X2)) -> U17_ga(X1, X2, pcC_in_ga(X1, X2)) U17_ga(X1, X2, pcC_out_ga(X1, X2)) -> pcC_out_ga(s(X1), s(X2)) U18_ga(X1, X2, pcC_out_ga(X1, X2)) -> pcB_out_ga(s(s(X1)), s(X2)) pcD_in_ga(s(0), 0) -> pcD_out_ga(s(0), 0) pcD_in_ga(s(s(X1)), s(X2)) -> U19_ga(X1, X2, pcC_in_ga(X1, X2)) U19_ga(X1, X2, pcC_out_ga(X1, X2)) -> pcD_out_ga(s(s(X1)), s(X2)) The argument filtering Pi contains the following mapping: s(x1) = s(x1) 0 = 0 pcB_in_ga(x1, x2) = pcB_in_ga(x1) pcB_out_ga(x1, x2) = pcB_out_ga(x1, x2) U18_ga(x1, x2, x3) = U18_ga(x1, x3) pcC_in_ga(x1, x2) = pcC_in_ga(x1) pcC_out_ga(x1, x2) = pcC_out_ga(x1, x2) U17_ga(x1, x2, x3) = U17_ga(x1, x3) pcD_in_ga(x1, x2) = pcD_in_ga(x1) pcD_out_ga(x1, x2) = pcD_out_ga(x1, x2) U19_ga(x1, x2, x3) = U19_ga(x1, x3) MA_IN_GGA(x1, x2, x3) = MA_IN_GGA(x1, x2) U8_GGA(x1, x2, x3, x4) = U8_GGA(x1, x2, x4) U9_GGA(x1, x2, x3, x4, x5) = U9_GGA(x1, x2, x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (18) Obligation: Pi DP problem: The TRS P consists of the following rules: MA_IN_GGA(s(s(X1)), X2, X3) -> U8_GGA(X1, X2, X3, pcC_in_ga(X1, X4)) U8_GGA(X1, X2, X3, pcC_out_ga(X1, X4)) -> U9_GGA(X1, X2, X3, X4, pcD_in_ga(X2, X5)) U9_GGA(X1, X2, X3, X4, pcD_out_ga(X2, X5)) -> MA_IN_GGA(s(X4), X5, X3) The TRS R consists of the following rules: pcC_in_ga(0, 0) -> pcC_out_ga(0, 0) pcC_in_ga(s(X1), s(X2)) -> U17_ga(X1, X2, pcC_in_ga(X1, X2)) pcD_in_ga(s(0), 0) -> pcD_out_ga(s(0), 0) pcD_in_ga(s(s(X1)), s(X2)) -> U19_ga(X1, X2, pcC_in_ga(X1, X2)) U17_ga(X1, X2, pcC_out_ga(X1, X2)) -> pcC_out_ga(s(X1), s(X2)) U19_ga(X1, X2, pcC_out_ga(X1, X2)) -> pcD_out_ga(s(s(X1)), s(X2)) The argument filtering Pi contains the following mapping: s(x1) = s(x1) 0 = 0 pcC_in_ga(x1, x2) = pcC_in_ga(x1) pcC_out_ga(x1, x2) = pcC_out_ga(x1, x2) U17_ga(x1, x2, x3) = U17_ga(x1, x3) pcD_in_ga(x1, x2) = pcD_in_ga(x1) pcD_out_ga(x1, x2) = pcD_out_ga(x1, x2) U19_ga(x1, x2, x3) = U19_ga(x1, x3) MA_IN_GGA(x1, x2, x3) = MA_IN_GGA(x1, x2) U8_GGA(x1, x2, x3, x4) = U8_GGA(x1, x2, x4) U9_GGA(x1, x2, x3, x4, x5) = U9_GGA(x1, x2, x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (19) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: MA_IN_GGA(s(s(X1)), X2) -> U8_GGA(X1, X2, pcC_in_ga(X1)) U8_GGA(X1, X2, pcC_out_ga(X1, X4)) -> U9_GGA(X1, X2, X4, pcD_in_ga(X2)) U9_GGA(X1, X2, X4, pcD_out_ga(X2, X5)) -> MA_IN_GGA(s(X4), X5) The TRS R consists of the following rules: pcC_in_ga(0) -> pcC_out_ga(0, 0) pcC_in_ga(s(X1)) -> U17_ga(X1, pcC_in_ga(X1)) pcD_in_ga(s(0)) -> pcD_out_ga(s(0), 0) pcD_in_ga(s(s(X1))) -> U19_ga(X1, pcC_in_ga(X1)) U17_ga(X1, pcC_out_ga(X1, X2)) -> pcC_out_ga(s(X1), s(X2)) U19_ga(X1, pcC_out_ga(X1, X2)) -> pcD_out_ga(s(s(X1)), s(X2)) The set Q consists of the following terms: pcC_in_ga(x0) pcD_in_ga(x0) U17_ga(x0, x1) U19_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (21) 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: MA_IN_GGA(s(s(X1)), X2) -> U8_GGA(X1, X2, pcC_in_ga(X1)) U8_GGA(X1, X2, pcC_out_ga(X1, X4)) -> U9_GGA(X1, X2, X4, pcD_in_ga(X2)) Used ordering: Polynomial interpretation [POLO]: POL(0) = 1 POL(MA_IN_GGA(x_1, x_2)) = 1 + x_1 POL(U17_ga(x_1, x_2)) = 2*x_2 POL(U19_ga(x_1, x_2)) = 0 POL(U8_GGA(x_1, x_2, x_3)) = 2*x_3 POL(U9_GGA(x_1, x_2, x_3, x_4)) = 1 + 2*x_3 POL(pcC_in_ga(x_1)) = 2*x_1 POL(pcC_out_ga(x_1, x_2)) = 1 + x_2 POL(pcD_in_ga(x_1)) = 2 POL(pcD_out_ga(x_1, x_2)) = 0 POL(s(x_1)) = 2*x_1 ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: U9_GGA(X1, X2, X4, pcD_out_ga(X2, X5)) -> MA_IN_GGA(s(X4), X5) The TRS R consists of the following rules: pcC_in_ga(0) -> pcC_out_ga(0, 0) pcC_in_ga(s(X1)) -> U17_ga(X1, pcC_in_ga(X1)) pcD_in_ga(s(0)) -> pcD_out_ga(s(0), 0) pcD_in_ga(s(s(X1))) -> U19_ga(X1, pcC_in_ga(X1)) U17_ga(X1, pcC_out_ga(X1, X2)) -> pcC_out_ga(s(X1), s(X2)) U19_ga(X1, pcC_out_ga(X1, X2)) -> pcD_out_ga(s(s(X1)), s(X2)) The set Q consists of the following terms: pcC_in_ga(x0) pcD_in_ga(x0) U17_ga(x0, x1) U19_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (23) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (24) TRUE