/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern h(g) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToDTProblemTransformerProof [SOUND, 99 ms] (2) TRIPLES (3) TriplesToPiDPProof [SOUND, 17 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [EQUIVALENT, 0 ms] (11) QDP (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] (13) YES (14) PiDP (15) UsableRulesProof [EQUIVALENT, 0 ms] (16) PiDP (17) PiDPToQDPProof [EQUIVALENT, 0 ms] (18) QDP (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] (20) YES ---------------------------------------- (0) Obligation: Clauses: h(X) :- ','(f(X), g(X)). f(c(0, X1)). f(c(X, Y)) :- ','(no(zero(X)), ','(p(X, P), f(c(P, s(Y))))). g(c(X2, 0)). g(c(X, Y)) :- ','(no(zero(Y)), ','(p(Y, P), g(c(s(X), P)))). p(0, 0). p(s(X), X). zero(0). no(X) :- ','(X, ','(!, failure(a))). no(X3). failure(b). Query: h(g) ---------------------------------------- (1) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 2, "program": { "directives": [], "clauses": [ [ "(h X)", "(',' (f X) (g X))" ], [ "(f (c (0) X1))", null ], [ "(f (c X Y))", "(',' (no (zero X)) (',' (p X P) (f (c P (s Y)))))" ], [ "(g (c X2 (0)))", null ], [ "(g (c X Y))", "(',' (no (zero Y)) (',' (p Y P) (g (c (s X) P))))" ], [ "(p (0) (0))", null ], [ "(p (s X) X)", null ], [ "(zero (0))", null ], [ "(no X)", "(',' X (',' (!) (failure (a))))" ], [ "(no X3)", null ], [ "(failure (b))", null ] ] }, "graph": { "nodes": { "350": { "goal": [ { "clause": -1, "scope": 59, "term": null }, { "clause": -1, "scope": 58, "term": null }, { "clause": 9, "scope": 57, "term": "(',' (',' (no (zero T146)) (',' (p T146 X283) (f (c X283 (s T140))))) (g (c T146 T140)))" } ], "kb": { "nonunifying": [[ "(zero T146)", "(zero (0))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T140", "T146" ], "free": ["X283"], "exprvars": [] } }, "230": { "goal": [ { "clause": 3, "scope": 33, "term": "(g (c (s (s (s (s (s (0)))))) T68))" }, { "clause": 4, "scope": 33, "term": "(g (c (s (s (s (s (s (0)))))) T68))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T68"], "free": [], "exprvars": [] } }, "351": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (failure (a)) (',' (',' (p (0) X283) (f (c X283 (s T140)))) (g (c (0) T140))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T140"], "free": ["X283"], "exprvars": [] } }, "110": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (no (zero T23)) (',' (p T23 X55) (g (c (s (s (0))) X55))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T23"], "free": ["X55"], "exprvars": [] } }, "231": { "goal": [{ "clause": 3, "scope": 33, "term": "(g (c (s (s (s (s (s (0)))))) T68))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T68"], "free": [], "exprvars": [] } }, "352": { "goal": [{ "clause": 10, "scope": 60, "term": "(',' (failure (a)) (',' (',' (p (0) X283) (f (c X283 (s T140)))) (g (c (0) T140))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T140"], "free": ["X283"], "exprvars": [] } }, "232": { "goal": [{ "clause": 4, "scope": 33, "term": "(g (c (s (s (s (s (s (0)))))) T68))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T68"], "free": [], "exprvars": [] } }, "353": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "112": { "goal": [ { "clause": 8, "scope": 10, "term": "(',' (no (zero T23)) (',' (p T23 X55) (g (c (s (s (0))) X55))))" }, { "clause": 9, "scope": 10, "term": "(',' (no (zero T23)) (',' (p T23 X55) (g (c (s (s (0))) X55))))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T23"], "free": ["X55"], "exprvars": [] } }, "233": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { 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["T68"], "free": [], "exprvars": [] } }, "349": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (',' (!_57) (failure (a))) (',' (',' (p (0) X283) (f (c X283 (s T140)))) (g (c (0) T140))))" }, { "clause": -1, "scope": 59, "term": null }, { "clause": -1, "scope": 58, "term": null }, { "clause": 9, "scope": 57, "term": "(',' (',' (no (zero (0))) (',' (p (0) X283) (f (c X283 (s T140))))) (g (c (0) T140)))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T140"], "free": ["X283"], "exprvars": [] } }, "229": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 2, "to": 4, "label": "CASE" }, { "from": 4, "to": 7, "label": "ONLY EVAL with clause\nh(X5) :- ','(f(X5), g(X5)).\nand substitutionT1 -> T3,\nX5 -> T3" }, { "from": 7, "to": 8, "label": "CASE" }, { "from": 8, "to": 10, "label": "PARALLEL" }, { "from": 8, "to": 11, "label": "PARALLEL" }, { "from": 10, "to": 12, "label": "EVAL with clause\nf(c(0, X10)).\nand substitutionX10 -> T8,\nT3 -> c(0, T8)" }, { "from": 10, "to": 13, "label": "EVAL-BACKTRACK" }, { "from": 11, "to": 343, "label": "EVAL with clause\nf(c(X281, X282)) :- ','(no(zero(X281)), ','(p(X281, X283), f(c(X283, s(X282))))).\nand substitutionX281 -> T139,\nX282 -> T140,\nT3 -> c(T139, T140)" }, { "from": 11, "to": 344, "label": "EVAL-BACKTRACK" }, { "from": 12, "to": 19, "label": "CASE" }, { "from": 19, "to": 20, "label": "PARALLEL" }, { "from": 19, "to": 21, "label": "PARALLEL" }, { "from": 20, "to": 25, "label": "EVAL with clause\ng(c(X17, 0)).\nand substitutionX17 -> 0,\nT8 -> 0" }, { "from": 20, "to": 27, "label": "EVAL-BACKTRACK" }, { "from": 21, "to": 65, "label": "ONLY EVAL with clause\ng(c(X24, X25)) :- ','(no(zero(X25)), ','(p(X25, X26), g(c(s(X24), X26)))).\nand substitutionX24 -> 0,\nT8 -> T11,\nX25 -> T11" }, { "from": 25, "to": 30, "label": "SUCCESS" }, { "from": 65, "to": 69, "label": "CASE" }, { "from": 69, "to": 70, "label": "ONLY EVAL with clause\nno(X30) :- ','(call(X30), ','(!_4, failure(a))).\nand substitutionT11 -> T14,\nX30 -> zero(T14)" }, { "from": 70, "to": 72, "label": "CALL" }, { "from": 72, "to": 73, "label": "CASE" }, { "from": 73, "to": 76, "label": "EVAL with clause\nzero(0).\nand substitutionT14 -> 0" }, { "from": 73, "to": 77, "label": "EVAL-BACKTRACK" }, { "from": 76, "to": 78, "label": "CUT" }, { "from": 77, "to": 83, "label": "FAILURE" }, { "from": 78, "to": 81, "label": "CASE" }, { "from": 81, "to": 82, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 83, "to": 84, "label": "FAILURE" }, { "from": 84, "to": 85, "label": "ONLY EVAL with clause\nno(X35).\nand substitutionT14 -> T17,\nX35 -> zero(T17)" }, { "from": 85, "to": 87, "label": "CASE" }, { "from": 87, "to": 90, "label": "BACKTRACK\nfor clause: p(0, 0)\nwith clash: (zero(T17), zero(0))" }, { "from": 90, "to": 94, "label": "EVAL with clause\np(s(X39), X39).\nand substitutionX39 -> T20,\nT17 -> s(T20),\nX26 -> T20" }, { "from": 90, "to": 95, "label": "EVAL-BACKTRACK" }, { "from": 94, "to": 97, "label": "CASE" }, { "from": 97, "to": 99, "label": "PARALLEL" }, { "from": 97, "to": 100, "label": "PARALLEL" }, { "from": 99, "to": 102, "label": "EVAL with clause\ng(c(X46, 0)).\nand substitutionX46 -> s(0),\nT20 -> 0" }, { "from": 99, "to": 103, "label": "EVAL-BACKTRACK" }, { "from": 100, "to": 110, "label": "ONLY EVAL with clause\ng(c(X53, X54)) :- ','(no(zero(X54)), ','(p(X54, X55), g(c(s(X53), X55)))).\nand substitutionX53 -> s(0),\nT20 -> T23,\nX54 -> T23" }, { "from": 102, "to": 104, "label": "SUCCESS" }, { "from": 110, "to": 112, "label": "CASE" }, { "from": 112, "to": 117, "label": "ONLY EVAL with clause\nno(X59) :- ','(call(X59), ','(!_10, failure(a))).\nand substitutionT23 -> T26,\nX59 -> zero(T26)" }, { "from": 117, "to": 118, "label": "CALL" }, { "from": 118, "to": 119, "label": "CASE" }, { "from": 119, "to": 121, "label": "EVAL with clause\nzero(0).\nand substitutionT26 -> 0" }, { "from": 119, "to": 122, "label": "EVAL-BACKTRACK" }, { "from": 121, "to": 123, "label": "CUT" }, { "from": 122, "to": 154, "label": "FAILURE" }, { "from": 123, "to": 126, "label": "CASE" }, { "from": 126, "to": 127, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 154, "to": 155, "label": "FAILURE" }, { "from": 155, "to": 156, "label": "ONLY EVAL with clause\nno(X64).\nand substitutionT26 -> T29,\nX64 -> zero(T29)" }, { "from": 156, "to": 157, "label": "CASE" }, { "from": 157, "to": 158, "label": "BACKTRACK\nfor clause: p(0, 0)\nwith clash: (zero(T29), zero(0))" }, { "from": 158, "to": 159, "label": "EVAL with clause\np(s(X68), X68).\nand substitutionX68 -> T32,\nT29 -> s(T32),\nX55 -> T32" }, { "from": 158, "to": 160, "label": "EVAL-BACKTRACK" }, { "from": 159, "to": 161, "label": "CASE" }, { "from": 161, "to": 162, "label": "PARALLEL" }, { "from": 161, "to": 163, "label": "PARALLEL" }, { "from": 162, "to": 164, "label": "EVAL with clause\ng(c(X75, 0)).\nand substitutionX75 -> s(s(0)),\nT32 -> 0" }, { "from": 162, "to": 165, "label": "EVAL-BACKTRACK" }, { "from": 163, "to": 167, "label": "ONLY EVAL with clause\ng(c(X82, X83)) :- ','(no(zero(X83)), ','(p(X83, X84), g(c(s(X82), X84)))).\nand substitutionX82 -> s(s(0)),\nT32 -> T35,\nX83 -> T35" }, { "from": 164, "to": 166, "label": "SUCCESS" }, { "from": 167, "to": 168, "label": "CASE" }, { "from": 168, "to": 169, "label": "ONLY EVAL with clause\nno(X88) :- ','(call(X88), ','(!_16, failure(a))).\nand substitutionT35 -> T38,\nX88 -> zero(T38)" }, { "from": 169, "to": 170, "label": "CALL" }, { "from": 170, "to": 171, "label": "CASE" }, { "from": 171, "to": 172, "label": "EVAL with clause\nzero(0).\nand substitutionT38 -> 0" }, { "from": 171, "to": 173, "label": "EVAL-BACKTRACK" }, { "from": 172, "to": 174, "label": "CUT" }, { "from": 173, "to": 177, "label": "FAILURE" }, { "from": 174, "to": 175, "label": "CASE" }, { "from": 175, "to": 176, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 177, "to": 178, "label": "FAILURE" }, { "from": 178, "to": 179, "label": "ONLY EVAL with clause\nno(X93).\nand substitutionT38 -> T41,\nX93 -> zero(T41)" }, { "from": 179, "to": 180, "label": "CASE" }, { "from": 180, "to": 181, "label": "BACKTRACK\nfor clause: p(0, 0)\nwith clash: (zero(T41), zero(0))" }, { "from": 181, "to": 182, "label": "EVAL with clause\np(s(X97), X97).\nand substitutionX97 -> T44,\nT41 -> s(T44),\nX84 -> T44" }, { "from": 181, "to": 183, "label": "EVAL-BACKTRACK" }, { "from": 182, "to": 184, "label": "CASE" }, { "from": 184, "to": 185, "label": "PARALLEL" }, { "from": 184, "to": 186, "label": "PARALLEL" }, { "from": 185, "to": 187, "label": "EVAL with clause\ng(c(X104, 0)).\nand substitutionX104 -> s(s(s(0))),\nT44 -> 0" }, { "from": 185, "to": 188, "label": "EVAL-BACKTRACK" }, { "from": 186, "to": 190, "label": "ONLY EVAL with clause\ng(c(X111, X112)) :- ','(no(zero(X112)), ','(p(X112, X113), g(c(s(X111), X113)))).\nand substitutionX111 -> s(s(s(0))),\nT44 -> T47,\nX112 -> T47" }, { "from": 187, "to": 189, "label": "SUCCESS" }, { "from": 190, "to": 191, "label": "CASE" }, { "from": 191, "to": 192, "label": "ONLY EVAL with clause\nno(X117) :- ','(call(X117), ','(!_22, failure(a))).\nand substitutionT47 -> T50,\nX117 -> zero(T50)" }, { "from": 192, "to": 193, "label": "CALL" }, { "from": 193, "to": 194, "label": "CASE" }, { "from": 194, "to": 195, "label": "EVAL with clause\nzero(0).\nand substitutionT50 -> 0" }, { "from": 194, "to": 196, "label": "EVAL-BACKTRACK" }, { "from": 195, "to": 197, "label": "CUT" }, { "from": 196, "to": 200, "label": "FAILURE" }, { "from": 197, "to": 198, "label": "CASE" }, { "from": 198, "to": 199, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 200, "to": 201, "label": "FAILURE" }, { "from": 201, "to": 202, "label": "ONLY EVAL with clause\nno(X122).\nand substitutionT50 -> T53,\nX122 -> zero(T53)" }, { "from": 202, "to": 203, "label": "CASE" }, { "from": 203, "to": 204, "label": "BACKTRACK\nfor clause: p(0, 0)\nwith clash: (zero(T53), zero(0))" }, { "from": 204, "to": 205, "label": "EVAL with clause\np(s(X126), X126).\nand substitutionX126 -> T56,\nT53 -> s(T56),\nX113 -> T56" }, { "from": 204, "to": 206, "label": "EVAL-BACKTRACK" }, { "from": 205, "to": 207, "label": "CASE" }, { "from": 207, "to": 208, "label": "PARALLEL" }, { "from": 207, "to": 209, "label": "PARALLEL" }, { "from": 208, "to": 210, "label": "EVAL with clause\ng(c(X133, 0)).\nand substitutionX133 -> s(s(s(s(0)))),\nT56 -> 0" }, { "from": 208, "to": 211, "label": "EVAL-BACKTRACK" }, { "from": 209, "to": 213, "label": "ONLY EVAL with clause\ng(c(X140, X141)) :- ','(no(zero(X141)), ','(p(X141, X142), g(c(s(X140), X142)))).\nand substitutionX140 -> s(s(s(s(0)))),\nT56 -> T59,\nX141 -> T59" }, { "from": 210, "to": 212, "label": "SUCCESS" }, { "from": 213, "to": 214, "label": "CASE" }, { "from": 214, "to": 215, "label": "ONLY EVAL with clause\nno(X146) :- ','(call(X146), ','(!_28, failure(a))).\nand substitutionT59 -> T62,\nX146 -> zero(T62)" }, { "from": 215, "to": 216, "label": "CALL" }, { "from": 216, "to": 217, "label": "CASE" }, { "from": 217, "to": 218, "label": "EVAL with clause\nzero(0).\nand substitutionT62 -> 0" }, { "from": 217, "to": 219, "label": "EVAL-BACKTRACK" }, { "from": 218, "to": 220, "label": "CUT" }, { "from": 219, "to": 223, "label": "FAILURE" }, { "from": 220, "to": 221, "label": "CASE" }, { "from": 221, "to": 222, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 223, "to": 224, "label": "FAILURE" }, { "from": 224, "to": 225, "label": "ONLY EVAL with clause\nno(X151).\nand substitutionT62 -> T65,\nX151 -> zero(T65)" }, { "from": 225, "to": 226, "label": "CASE" }, { "from": 226, "to": 227, "label": "BACKTRACK\nfor clause: p(0, 0)\nwith clash: (zero(T65), zero(0))" }, { "from": 227, "to": 228, "label": "EVAL with clause\np(s(X155), X155).\nand substitutionX155 -> T68,\nT65 -> s(T68),\nX142 -> T68" }, { "from": 227, "to": 229, "label": "EVAL-BACKTRACK" }, { "from": 228, "to": 230, "label": "CASE" }, { "from": 230, "to": 231, "label": "PARALLEL" }, { "from": 230, "to": 232, "label": "PARALLEL" }, { "from": 231, "to": 233, "label": "EVAL with clause\ng(c(X162, 0)).\nand substitutionX162 -> s(s(s(s(s(0))))),\nT68 -> 0" }, { "from": 231, "to": 234, "label": "EVAL-BACKTRACK" }, { "from": 232, "to": 236, "label": "ONLY EVAL with clause\ng(c(X169, X170)) :- ','(no(zero(X170)), ','(p(X170, X171), g(c(s(X169), X171)))).\nand substitutionX169 -> s(s(s(s(s(0))))),\nT68 -> T71,\nX170 -> T71" }, { "from": 233, "to": 235, "label": "SUCCESS" }, { "from": 236, "to": 237, "label": "CASE" }, { "from": 237, "to": 238, "label": "ONLY EVAL with clause\nno(X175) :- ','(call(X175), ','(!_34, failure(a))).\nand substitutionT71 -> T74,\nX175 -> zero(T74)" }, { "from": 238, "to": 239, "label": "CALL" }, { "from": 239, "to": 240, "label": "CASE" }, { "from": 240, "to": 241, "label": "EVAL with clause\nzero(0).\nand substitutionT74 -> 0" }, { "from": 240, "to": 242, "label": "EVAL-BACKTRACK" }, { "from": 241, "to": 243, "label": "CUT" }, { "from": 242, "to": 246, "label": "FAILURE" }, { "from": 243, "to": 244, "label": "CASE" }, { "from": 244, "to": 245, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 246, "to": 247, "label": "FAILURE" }, { "from": 247, "to": 248, "label": "ONLY EVAL with clause\nno(X180).\nand substitutionT74 -> T77,\nX180 -> zero(T77)" }, { "from": 248, "to": 249, "label": "CASE" }, { "from": 249, "to": 250, "label": "BACKTRACK\nfor clause: p(0, 0)\nwith clash: (zero(T77), zero(0))" }, { "from": 250, "to": 251, "label": "EVAL with clause\np(s(X184), X184).\nand substitutionX184 -> T80,\nT77 -> s(T80),\nX171 -> T80" }, { "from": 250, "to": 252, "label": "EVAL-BACKTRACK" }, { "from": 251, "to": 253, "label": "CASE" }, { "from": 253, "to": 254, "label": "PARALLEL" }, { "from": 253, "to": 255, "label": "PARALLEL" }, { "from": 254, "to": 256, "label": "EVAL with clause\ng(c(X191, 0)).\nand substitutionX191 -> s(s(s(s(s(s(0)))))),\nT80 -> 0" }, { "from": 254, "to": 257, "label": "EVAL-BACKTRACK" }, { "from": 255, "to": 259, "label": "ONLY EVAL with clause\ng(c(X198, X199)) :- ','(no(zero(X199)), ','(p(X199, X200), g(c(s(X198), X200)))).\nand substitutionX198 -> s(s(s(s(s(s(0)))))),\nT80 -> T83,\nX199 -> T83" }, { "from": 256, "to": 258, "label": "SUCCESS" }, { "from": 259, "to": 260, "label": "CASE" }, { "from": 260, "to": 261, "label": "ONLY EVAL with clause\nno(X204) :- ','(call(X204), ','(!_40, failure(a))).\nand substitutionT83 -> T86,\nX204 -> zero(T86)" }, { "from": 261, "to": 262, "label": "CALL" }, { "from": 262, "to": 263, "label": "CASE" }, { "from": 263, "to": 264, "label": "EVAL with clause\nzero(0).\nand substitutionT86 -> 0" }, { "from": 263, "to": 265, "label": "EVAL-BACKTRACK" }, { "from": 264, "to": 266, "label": "CUT" }, { "from": 265, "to": 269, "label": "FAILURE" }, { "from": 266, "to": 267, "label": "CASE" }, { "from": 267, "to": 268, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 269, "to": 270, "label": "FAILURE" }, { "from": 270, "to": 271, "label": "ONLY EVAL with clause\nno(X209).\nand substitutionT86 -> T89,\nX209 -> zero(T89)" }, { "from": 271, "to": 272, "label": "CASE" }, { "from": 272, "to": 273, "label": "BACKTRACK\nfor clause: p(0, 0)\nwith clash: (zero(T89), zero(0))" }, { "from": 273, "to": 274, "label": "EVAL with clause\np(s(X213), X213).\nand substitutionX213 -> T92,\nT89 -> s(T92),\nX200 -> T92" }, { "from": 273, "to": 275, "label": "EVAL-BACKTRACK" }, { "from": 274, "to": 276, "label": "CASE" }, { "from": 276, "to": 277, "label": "PARALLEL" }, { "from": 276, "to": 278, "label": "PARALLEL" }, { "from": 277, "to": 279, "label": "EVAL with clause\ng(c(X220, 0)).\nand substitutionX220 -> s(s(s(s(s(s(s(0))))))),\nT92 -> 0" }, { "from": 277, "to": 280, "label": "EVAL-BACKTRACK" }, { "from": 278, "to": 282, "label": "ONLY EVAL with clause\ng(c(X227, X228)) :- ','(no(zero(X228)), ','(p(X228, X229), g(c(s(X227), X229)))).\nand substitutionX227 -> s(s(s(s(s(s(s(0))))))),\nT92 -> T95,\nX228 -> T95" }, { "from": 279, "to": 281, "label": "SUCCESS" }, { "from": 282, "to": 283, "label": "CASE" }, { "from": 283, "to": 284, "label": "ONLY EVAL with clause\nno(X233) :- ','(call(X233), ','(!_46, failure(a))).\nand substitutionT95 -> T98,\nX233 -> zero(T98)" }, { "from": 284, "to": 285, "label": "CALL" }, { "from": 285, "to": 286, "label": "CASE" }, { "from": 286, "to": 287, "label": "EVAL with clause\nzero(0).\nand substitutionT98 -> 0" }, { "from": 286, "to": 288, "label": "EVAL-BACKTRACK" }, { "from": 287, "to": 289, "label": "CUT" }, { "from": 288, "to": 292, "label": "FAILURE" }, { "from": 289, "to": 290, "label": "CASE" }, { "from": 290, "to": 291, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 292, "to": 293, "label": "FAILURE" }, { "from": 293, "to": 294, "label": "ONLY EVAL with clause\nno(X238).\nand substitutionT98 -> T101,\nX238 -> zero(T101)" }, { "from": 294, "to": 295, "label": "CASE" }, { "from": 295, "to": 296, "label": "BACKTRACK\nfor clause: p(0, 0)\nwith clash: (zero(T101), zero(0))" }, { "from": 296, "to": 297, "label": "EVAL with clause\np(s(X242), X242).\nand substitutionX242 -> T104,\nT101 -> s(T104),\nX229 -> T104" }, { "from": 296, "to": 298, "label": "EVAL-BACKTRACK" }, { "from": 297, "to": 299, "label": "GENERALIZATION\nT105 <-- s(s(s(s(s(s(s(0)))))))\n\nNew Knowledge:\nT105 is ground" }, { "from": 299, "to": 300, "label": "CASE" }, { "from": 300, "to": 301, "label": "PARALLEL" }, { "from": 300, "to": 302, "label": "PARALLEL" }, { "from": 301, "to": 303, "label": "EVAL with clause\ng(c(X251, 0)).\nand substitutionT105 -> T112,\nX251 -> s(T112),\nT104 -> 0" }, { "from": 301, "to": 304, "label": "EVAL-BACKTRACK" }, { "from": 302, "to": 306, "label": "ONLY EVAL with clause\ng(c(X258, X259)) :- ','(no(zero(X259)), ','(p(X259, X260), g(c(s(X258), X260)))).\nand substitutionT105 -> T118,\nX258 -> s(T118),\nT104 -> T119,\nX259 -> T119" }, { "from": 303, "to": 305, "label": "SUCCESS" }, { "from": 306, "to": 307, "label": "CASE" }, { "from": 307, "to": 311, "label": "ONLY EVAL with clause\nno(X264) :- ','(call(X264), ','(!_52, failure(a))).\nand substitutionT119 -> T123,\nX264 -> zero(T123)" }, { "from": 311, "to": 312, "label": "CALL" }, { "from": 312, "to": 313, "label": "CASE" }, { "from": 313, "to": 314, "label": "EVAL with clause\nzero(0).\nand substitutionT123 -> 0" }, { "from": 313, "to": 315, "label": "EVAL-BACKTRACK" }, { "from": 314, "to": 316, "label": "CUT" }, { "from": 315, "to": 336, "label": "FAILURE" }, { "from": 316, "to": 317, "label": "CASE" }, { "from": 317, "to": 318, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 336, "to": 337, "label": "FAILURE" }, { "from": 337, "to": 338, "label": "ONLY EVAL with clause\nno(X269).\nand substitutionT123 -> T128,\nX269 -> zero(T128)" }, { "from": 338, "to": 339, "label": "CASE" }, { "from": 339, "to": 340, "label": "BACKTRACK\nfor clause: p(0, 0)\nwith clash: (zero(T128), zero(0))" }, { "from": 340, "to": 341, "label": "EVAL with clause\np(s(X273), X273).\nand substitutionX273 -> T132,\nT128 -> s(T132),\nX260 -> T132" }, { "from": 340, "to": 342, "label": "EVAL-BACKTRACK" }, { "from": 341, "to": 299, "label": "INSTANCE with matching:\nT105 -> s(T118)\nT104 -> T132" }, { "from": 343, "to": 345, "label": "CASE" }, { "from": 345, "to": 346, "label": "ONLY EVAL with clause\nno(X289) :- ','(call(X289), ','(!_57, failure(a))).\nand substitutionT139 -> T146,\nX289 -> zero(T146)" }, { "from": 346, "to": 347, "label": "CALL" }, { "from": 347, "to": 348, "label": "CASE" }, { "from": 348, "to": 349, "label": "EVAL with clause\nzero(0).\nand substitutionT146 -> 0" }, { "from": 348, "to": 350, "label": "EVAL-BACKTRACK" }, { "from": 349, "to": 351, "label": "CUT" }, { "from": 350, "to": 354, "label": "FAILURE" }, { "from": 351, "to": 352, "label": "CASE" }, { "from": 352, "to": 353, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 354, "to": 355, "label": "FAILURE" }, { "from": 355, "to": 356, "label": "ONLY EVAL with clause\nno(X298).\nand substitutionT146 -> T153,\nX298 -> zero(T153)" }, { "from": 356, "to": 357, "label": "CASE" }, { "from": 357, "to": 358, "label": "BACKTRACK\nfor clause: p(0, 0)\nwith clash: (zero(T153), zero(0))" }, { "from": 358, "to": 359, "label": "EVAL with clause\np(s(X304), X304).\nand substitutionX304 -> T159,\nT153 -> s(T159),\nX283 -> T159" }, { "from": 358, "to": 360, "label": "EVAL-BACKTRACK" }, { "from": 359, "to": 361, "label": "SPLIT 1" }, { "from": 359, "to": 362, "label": "SPLIT 2\nnew knowledge:\nT159 is ground\nT140 is ground" }, { "from": 361, "to": 363, "label": "CASE" }, { "from": 362, "to": 299, "label": "INSTANCE with matching:\nT105 -> T159\nT104 -> T140" }, { "from": 363, "to": 364, "label": "PARALLEL" }, { "from": 363, "to": 365, "label": "PARALLEL" }, { "from": 364, "to": 366, "label": "EVAL with clause\nf(c(0, X313)).\nand substitutionT159 -> 0,\nT140 -> T168,\nX313 -> s(T168)" }, { "from": 364, "to": 367, "label": "EVAL-BACKTRACK" }, { "from": 365, "to": 369, "label": "ONLY EVAL with clause\nf(c(X320, X321)) :- ','(no(zero(X320)), ','(p(X320, X322), f(c(X322, s(X321))))).\nand substitutionT159 -> T174,\nX320 -> T174,\nT140 -> T175,\nX321 -> s(T175)" }, { "from": 366, "to": 368, "label": "SUCCESS" }, { "from": 369, "to": 370, "label": "CASE" }, { "from": 370, "to": 371, "label": "ONLY EVAL with clause\nno(X326) :- ','(call(X326), ','(!_63, failure(a))).\nand substitutionT174 -> T179,\nX326 -> zero(T179)" }, { "from": 371, "to": 372, "label": "CALL" }, { "from": 372, "to": 373, "label": "CASE" }, { "from": 373, "to": 374, "label": "EVAL with clause\nzero(0).\nand substitutionT179 -> 0" }, { "from": 373, "to": 375, "label": "EVAL-BACKTRACK" }, { "from": 374, "to": 376, "label": "CUT" }, { "from": 375, "to": 379, "label": "FAILURE" }, { "from": 376, "to": 377, "label": "CASE" }, { "from": 377, "to": 378, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 379, "to": 380, "label": "FAILURE" }, { "from": 380, "to": 381, "label": "ONLY EVAL with clause\nno(X331).\nand substitutionT179 -> T184,\nX331 -> zero(T184)" }, { "from": 381, "to": 382, "label": "CASE" }, { "from": 382, "to": 383, "label": "BACKTRACK\nfor clause: p(0, 0)\nwith clash: (zero(T184), zero(0))" }, { "from": 383, "to": 384, "label": "EVAL with clause\np(s(X335), X335).\nand substitutionX335 -> T188,\nT184 -> s(T188),\nX322 -> T188" }, { "from": 383, "to": 385, "label": "EVAL-BACKTRACK" }, { "from": 384, "to": 361, "label": "INSTANCE with matching:\nT159 -> T188\nT140 -> s(T175)" } ], "type": "Graph" } } ---------------------------------------- (2) Obligation: Triples: gA(X1, s(X2)) :- gA(s(X1), X2). fB(s(X1), X2) :- fB(X1, s(X2)). hC(c(0, s(s(s(s(s(s(s(s(X1)))))))))) :- gA(s(s(s(s(s(s(s(0))))))), X1). hC(c(s(X1), X2)) :- fB(X1, X2). hC(c(s(X1), X2)) :- ','(fcB(X1, X2), gA(X1, X2)). Clauses: gcA(X1, 0). gcA(X1, s(X2)) :- gcA(s(X1), X2). fcB(0, X1). fcB(s(X1), X2) :- fcB(X1, s(X2)). Afs: hC(x1) = hC(x1) ---------------------------------------- (3) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: hC_in_1: (b) gA_in_2: (b,b) fB_in_2: (b,b) fcB_in_2: (b,b) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: HC_IN_G(c(0, s(s(s(s(s(s(s(s(X1)))))))))) -> U3_G(X1, gA_in_gg(s(s(s(s(s(s(s(0))))))), X1)) HC_IN_G(c(0, s(s(s(s(s(s(s(s(X1)))))))))) -> GA_IN_GG(s(s(s(s(s(s(s(0))))))), X1) GA_IN_GG(X1, s(X2)) -> U1_GG(X1, X2, gA_in_gg(s(X1), X2)) GA_IN_GG(X1, s(X2)) -> GA_IN_GG(s(X1), X2) HC_IN_G(c(s(X1), X2)) -> U4_G(X1, X2, fB_in_gg(X1, X2)) HC_IN_G(c(s(X1), X2)) -> FB_IN_GG(X1, X2) FB_IN_GG(s(X1), X2) -> U2_GG(X1, X2, fB_in_gg(X1, s(X2))) FB_IN_GG(s(X1), X2) -> FB_IN_GG(X1, s(X2)) HC_IN_G(c(s(X1), X2)) -> U5_G(X1, X2, fcB_in_gg(X1, X2)) U5_G(X1, X2, fcB_out_gg(X1, X2)) -> U6_G(X1, X2, gA_in_gg(X1, X2)) U5_G(X1, X2, fcB_out_gg(X1, X2)) -> GA_IN_GG(X1, X2) The TRS R consists of the following rules: fcB_in_gg(0, X1) -> fcB_out_gg(0, X1) fcB_in_gg(s(X1), X2) -> U9_gg(X1, X2, fcB_in_gg(X1, s(X2))) U9_gg(X1, X2, fcB_out_gg(X1, s(X2))) -> fcB_out_gg(s(X1), X2) Pi is empty. 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: HC_IN_G(c(0, s(s(s(s(s(s(s(s(X1)))))))))) -> U3_G(X1, gA_in_gg(s(s(s(s(s(s(s(0))))))), X1)) HC_IN_G(c(0, s(s(s(s(s(s(s(s(X1)))))))))) -> GA_IN_GG(s(s(s(s(s(s(s(0))))))), X1) GA_IN_GG(X1, s(X2)) -> U1_GG(X1, X2, gA_in_gg(s(X1), X2)) GA_IN_GG(X1, s(X2)) -> GA_IN_GG(s(X1), X2) HC_IN_G(c(s(X1), X2)) -> U4_G(X1, X2, fB_in_gg(X1, X2)) HC_IN_G(c(s(X1), X2)) -> FB_IN_GG(X1, X2) FB_IN_GG(s(X1), X2) -> U2_GG(X1, X2, fB_in_gg(X1, s(X2))) FB_IN_GG(s(X1), X2) -> FB_IN_GG(X1, s(X2)) HC_IN_G(c(s(X1), X2)) -> U5_G(X1, X2, fcB_in_gg(X1, X2)) U5_G(X1, X2, fcB_out_gg(X1, X2)) -> U6_G(X1, X2, gA_in_gg(X1, X2)) U5_G(X1, X2, fcB_out_gg(X1, X2)) -> GA_IN_GG(X1, X2) The TRS R consists of the following rules: fcB_in_gg(0, X1) -> fcB_out_gg(0, X1) fcB_in_gg(s(X1), X2) -> U9_gg(X1, X2, fcB_in_gg(X1, s(X2))) U9_gg(X1, X2, fcB_out_gg(X1, s(X2))) -> fcB_out_gg(s(X1), X2) Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 9 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: FB_IN_GG(s(X1), X2) -> FB_IN_GG(X1, s(X2)) The TRS R consists of the following rules: fcB_in_gg(0, X1) -> fcB_out_gg(0, X1) fcB_in_gg(s(X1), X2) -> U9_gg(X1, X2, fcB_in_gg(X1, s(X2))) U9_gg(X1, X2, fcB_out_gg(X1, s(X2))) -> fcB_out_gg(s(X1), X2) Pi is empty. 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: FB_IN_GG(s(X1), X2) -> FB_IN_GG(X1, s(X2)) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (10) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: FB_IN_GG(s(X1), X2) -> FB_IN_GG(X1, s(X2)) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (12) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *FB_IN_GG(s(X1), X2) -> FB_IN_GG(X1, s(X2)) The graph contains the following edges 1 > 1 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: GA_IN_GG(X1, s(X2)) -> GA_IN_GG(s(X1), X2) The TRS R consists of the following rules: fcB_in_gg(0, X1) -> fcB_out_gg(0, X1) fcB_in_gg(s(X1), X2) -> U9_gg(X1, X2, fcB_in_gg(X1, s(X2))) U9_gg(X1, X2, fcB_out_gg(X1, s(X2))) -> fcB_out_gg(s(X1), X2) Pi is empty. 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: GA_IN_GG(X1, s(X2)) -> GA_IN_GG(s(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: GA_IN_GG(X1, s(X2)) -> GA_IN_GG(s(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: *GA_IN_GG(X1, s(X2)) -> GA_IN_GG(s(X1), X2) The graph contains the following edges 2 > 2 ---------------------------------------- (20) YES