/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern goal(g) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToDTProblemTransformerProof [SOUND, 129 ms] (2) TRIPLES (3) UndefinedPredicateInTriplesTransformerProof [SOUND, 0 ms] (4) TRIPLES (5) TriplesToPiDPProof [SOUND, 0 ms] (6) PiDP (7) DependencyGraphProof [EQUIVALENT, 0 ms] (8) PiDP (9) PiDPToQDPProof [SOUND, 2 ms] (10) QDP (11) QDPSizeChangeProof [EQUIVALENT, 0 ms] (12) YES ---------------------------------------- (0) Obligation: Clauses: goal(X) :- ','(s2t(X, T), tree(T)). tree(nil). tree(X) :- ','(no(empty(X)), ','(left(T, L), ','(right(T, R), ','(tree(L), tree(R))))). s2t(0, nil). s2t(X, node(T, X1, T)) :- ','(no(zero(X)), ','(p(X, P), s2t(P, T))). s2t(X, node(nil, X2, T)) :- ','(no(zero(X)), ','(p(X, P), s2t(P, T))). s2t(X, node(T, X3, nil)) :- ','(no(zero(X)), ','(p(X, P), s2t(P, T))). s2t(X, node(nil, X4, nil)) :- no(zero(X)). left(nil, nil). left(node(L, X5, X6), L). right(nil, nil). right(node(X7, X8, R), R). p(0, 0). p(s(X), X). empty(nil). zero(0). no(X) :- ','(X, ','(!, failure(a))). no(X9). failure(b). Query: goal(g) ---------------------------------------- (1) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 7, "program": { "directives": [], "clauses": [ [ "(goal X)", "(',' (s2t X T) (tree T))" ], [ "(tree (nil))", null ], [ "(tree X)", "(',' (no (empty X)) (',' (left T L) (',' (right T R) (',' (tree L) (tree R)))))" ], [ "(s2t (0) (nil))", null ], [ "(s2t X (node T X1 T))", "(',' (no (zero X)) (',' (p X P) (s2t P T)))" ], [ "(s2t X (node (nil) X2 T))", "(',' (no (zero X)) (',' (p X P) (s2t P T)))" ], [ "(s2t X (node T X3 (nil)))", "(',' (no (zero X)) (',' (p X P) (s2t P T)))" ], [ "(s2t X (node (nil) X4 (nil)))", "(no (zero X))" ], [ "(left (nil) (nil))", null ], [ "(left (node L X5 X6) L)", null ], [ "(right (nil) (nil))", null ], [ "(right (node X7 X8 R) R)", null ], [ "(p (0) (0))", null ], [ "(p (s X) X)", null ], [ "(empty (nil))", null ], [ "(zero (0))", null ], [ "(no X)", "(',' X (',' (!) (failure (a))))" ], [ "(no X9)", null ], [ "(failure (b))", null ] ] }, "graph": { "nodes": { "470": { "goal": [{ "clause": 11, "scope": 56, "term": "(',' (right (node X348 X349 X350) X316) (',' (tree X348) (tree X316)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X316", "X348", "X349", "X350" ], "exprvars": [] } }, "471": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (tree X366) (tree X368))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X366", "X368" ], "exprvars": [] } }, "472": { "goal": [{ "clause": 6, "scope": 2, "term": "(',' (s2t T3 X12) (tree X12))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": ["X12"], "exprvars": [] } }, "473": { "goal": [{ "clause": 7, "scope": 2, "term": "(',' (s2t T3 X12) (tree X12))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": ["X12"], "exprvars": [] } }, "474": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (',' (no (zero T116)) (',' (p T116 X394) (s2t X394 X395))) (tree (node X395 X396 (nil))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T116"], "free": [ "X395", "X396", "X394" ], "exprvars": [] } }, "475": { "goal": [ { "clause": 16, "scope": 57, "term": "(',' (',' (no (zero T116)) (',' (p T116 X394) (s2t X394 X395))) (tree (node X395 X396 (nil))))" }, { "clause": 17, "scope": 57, "term": "(',' (',' (no (zero T116)) (',' (p T116 X394) (s2t X394 X395))) (tree (node X395 X396 (nil))))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T116"], "free": [ "X395", "X396", "X394" ], "exprvars": [] } }, "476": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (',' (call (zero T119)) (',' (!_57) (failure (a)))) (',' (',' (p T119 X394) (s2t X394 X395)) (tree (node X395 X396 (nil)))))" }, { "clause": 17, "scope": 57, "term": "(',' (',' (no (zero T119)) (',' (p T119 X394) (s2t X394 X395))) (tree (node X395 X396 (nil))))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T119"], "free": [ "X395", "X396", "X394" ], "exprvars": [] } }, "477": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (zero T119) (',' (',' (!_57) (failure (a))) (',' (',' (p T119 X394) (s2t X394 X395)) (tree (node X395 X396 (nil))))))" }, { "clause": -1, "scope": 58, "term": null }, { "clause": 17, "scope": 57, "term": "(',' (',' (no (zero T119)) (',' (p T119 X394) (s2t X394 X395))) (tree (node X395 X396 (nil))))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T119"], "free": [ "X395", "X396", "X394" ], "exprvars": [] } }, "478": { "goal": [ { "clause": 15, "scope": 59, "term": "(',' (zero T119) (',' (',' (!_57) (failure (a))) (',' (',' (p T119 X394) (s2t X394 X395)) (tree (node X395 X396 (nil))))))" }, { "clause": -1, "scope": 59, "term": null }, { "clause": -1, "scope": 58, "term": null }, { "clause": 17, "scope": 57, "term": "(',' (',' (no (zero T119)) (',' (p T119 X394) (s2t X394 X395))) (tree (node X395 X396 (nil))))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T119"], "free": [ "X395", "X396", "X394" ], "exprvars": [] } }, "479": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (',' (!_57) (failure (a))) (',' (',' (p (0) X394) (s2t X394 X395)) (tree (node X395 X396 (nil)))))" }, { "clause": -1, "scope": 59, "term": null }, { "clause": -1, "scope": 58, "term": null }, { "clause": 17, 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}], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T125"], "free": [ "X395", "X396" ], "exprvars": [] } }, "127": { "goal": [ { "clause": 3, "scope": 13, "term": "(s2t T17 X51)" }, { "clause": 4, "scope": 13, "term": "(s2t T17 X51)" }, { "clause": 5, "scope": 13, "term": "(s2t T17 X51)" }, { "clause": 6, "scope": 13, "term": "(s2t T17 X51)" }, { "clause": 7, "scope": 13, "term": "(s2t T17 X51)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T17"], "free": ["X51"], "exprvars": [] } }, "7": { "goal": [{ "clause": -1, "scope": -1, "term": "(goal T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "128": { "goal": [{ "clause": 3, "scope": 13, "term": "(s2t T17 X51)" }], "kb": { "nonunifying": [], "intvars": {}, 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[], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T17"], "free": [ "X51", "X52" ], "exprvars": [] } }, "109": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 7, "to": 8, "label": "CASE" }, { "from": 8, "to": 9, "label": "ONLY EVAL with clause\ngoal(X11) :- ','(s2t(X11, X12), tree(X12)).\nand substitutionT1 -> T3,\nX11 -> T3" }, { "from": 9, "to": 10, "label": "CASE" }, { "from": 10, "to": 14, "label": "PARALLEL" }, { "from": 10, "to": 15, "label": "PARALLEL" }, { "from": 14, "to": 16, "label": "EVAL with clause\ns2t(0, nil).\nand substitutionT3 -> 0,\nX12 -> nil" }, { "from": 14, "to": 17, "label": "EVAL-BACKTRACK" }, { "from": 15, "to": 47, "label": "PARALLEL" }, { "from": 15, "to": 49, "label": "PARALLEL" }, { "from": 16, "to": 18, "label": "CASE" }, { "from": 18, "to": 19, "label": "PARALLEL" }, { "from": 18, "to": 20, "label": "PARALLEL" }, { "from": 19, "to": 23, "label": "ONLY EVAL with clause\ntree(nil).\nand substitution" }, { "from": 20, "to": 30, "label": "ONLY EVAL with clause\ntree(X18) :- ','(no(empty(X18)), ','(left(X19, X20), ','(right(X19, X21), ','(tree(X20), tree(X21))))).\nand substitutionX18 -> nil" }, { "from": 23, "to": 24, "label": "SUCCESS" }, { "from": 30, "to": 31, "label": "CASE" }, { "from": 31, "to": 32, "label": "ONLY EVAL with clause\nno(X24) :- ','(call(X24), ','(!_4, failure(a))).\nand substitutionX24 -> empty(nil)" }, { "from": 32, "to": 33, "label": "CALL" }, { "from": 33, "to": 34, "label": "CASE" }, { "from": 34, "to": 35, "label": "ONLY EVAL with clause\nempty(nil).\nand substitution" }, { "from": 35, "to": 37, "label": "CUT" }, { "from": 37, "to": 39, "label": "CASE" }, { "from": 39, "to": 40, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 47, "to": 53, "label": "ONLY EVAL with clause\ns2t(X47, node(X48, X49, X48)) :- ','(no(zero(X47)), ','(p(X47, X50), s2t(X50, X48))).\nand substitutionT3 -> T8,\nX47 -> T8,\nX48 -> X51,\nX49 -> X52,\nX12 -> node(X51, X52, X51)" }, { "from": 49, "to": 427, "label": "PARALLEL" }, { "from": 49, "to": 428, "label": "PARALLEL" }, { "from": 53, "to": 56, "label": "CASE" }, { "from": 56, "to": 58, "label": "ONLY EVAL with clause\nno(X55) :- ','(call(X55), ','(!_8, failure(a))).\nand substitutionT8 -> T11,\nX55 -> zero(T11)" }, { "from": 58, "to": 94, "label": "CALL" }, { "from": 94, "to": 95, "label": "CASE" }, { "from": 95, "to": 96, "label": "EVAL with clause\nzero(0).\nand substitutionT11 -> 0" }, { "from": 95, "to": 97, "label": "EVAL-BACKTRACK" }, { "from": 96, "to": 98, "label": "CUT" }, { "from": 97, "to": 101, "label": "FAILURE" }, { "from": 98, "to": 99, "label": "CASE" }, { "from": 99, "to": 100, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 101, "to": 102, "label": "FAILURE" }, { "from": 102, "to": 105, "label": "ONLY EVAL with clause\nno(X58).\nand substitutionT11 -> T14,\nX58 -> zero(T14)" }, { "from": 105, "to": 106, "label": "CASE" }, { "from": 106, "to": 107, "label": "BACKTRACK\nfor clause: p(0, 0)\nwith clash: (zero(T14), zero(0))" }, { "from": 107, "to": 108, "label": "EVAL with clause\np(s(X61), X61).\nand substitutionX61 -> T17,\nT14 -> s(T17),\nX50 -> T17" }, { "from": 107, "to": 109, "label": "EVAL-BACKTRACK" }, { "from": 108, "to": 125, "label": "SPLIT 1" }, { "from": 108, "to": 126, "label": "SPLIT 2\nnew knowledge:\nT17 is ground\nreplacements:X51 -> T18" }, { "from": 125, "to": 127, "label": "CASE" }, { "from": 126, "to": 361, "label": "CASE" }, { "from": 127, "to": 128, "label": "PARALLEL" }, { "from": 127, "to": 129, "label": "PARALLEL" }, { "from": 128, "to": 130, "label": "EVAL with clause\ns2t(0, nil).\nand substitutionT17 -> 0,\nX51 -> nil" }, { "from": 128, "to": 131, "label": "EVAL-BACKTRACK" }, { "from": 129, "to": 133, "label": "PARALLEL" }, { "from": 129, "to": 134, "label": "PARALLEL" }, { "from": 130, "to": 132, "label": "SUCCESS" }, { "from": 133, "to": 138, "label": "ONLY EVAL with clause\ns2t(X84, node(X85, X86, X85)) :- ','(no(zero(X84)), ','(p(X84, X87), s2t(X87, X85))).\nand substitutionT17 -> T23,\nX84 -> T23,\nX85 -> X88,\nX86 -> X89,\nX51 -> node(X88, X89, X88)" }, { "from": 134, "to": 163, "label": "PARALLEL" }, { "from": 134, "to": 164, "label": "PARALLEL" }, { "from": 138, "to": 142, "label": "CASE" }, { "from": 142, "to": 147, "label": "ONLY EVAL with clause\nno(X92) :- ','(call(X92), ','(!_14, failure(a))).\nand substitutionT23 -> T26,\nX92 -> zero(T26)" }, { "from": 147, "to": 148, "label": "CALL" }, { "from": 148, "to": 149, "label": "CASE" }, { "from": 149, "to": 151, "label": "EVAL with clause\nzero(0).\nand substitutionT26 -> 0" }, { "from": 149, "to": 152, "label": "EVAL-BACKTRACK" }, { "from": 151, "to": 153, "label": "CUT" }, { "from": 152, "to": 156, "label": "FAILURE" }, { "from": 153, "to": 154, "label": "CASE" }, { "from": 154, "to": 155, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 156, "to": 157, "label": "FAILURE" }, { "from": 157, "to": 158, "label": "ONLY EVAL with clause\nno(X95).\nand substitutionT26 -> T29,\nX95 -> zero(T29)" }, { "from": 158, "to": 159, "label": "CASE" }, { "from": 159, "to": 160, "label": "BACKTRACK\nfor clause: p(0, 0)\nwith clash: (zero(T29), zero(0))" }, { "from": 160, "to": 161, "label": "EVAL with clause\np(s(X98), X98).\nand substitutionX98 -> T32,\nT29 -> s(T32),\nX87 -> T32" }, { "from": 160, "to": 162, "label": "EVAL-BACKTRACK" }, { "from": 161, "to": 125, "label": "INSTANCE with matching:\nT17 -> T32\nX51 -> X88" }, { "from": 163, "to": 169, "label": "ONLY EVAL with clause\ns2t(X121, node(nil, X122, X123)) :- ','(no(zero(X121)), ','(p(X121, X124), s2t(X124, X123))).\nand substitutionT17 -> T37,\nX121 -> T37,\nX122 -> X125,\nX123 -> X126,\nX51 -> node(nil, X125, X126)" }, { "from": 164, "to": 276, "label": "PARALLEL" }, { "from": 164, "to": 277, "label": "PARALLEL" }, { "from": 169, "to": 170, "label": "CASE" }, { "from": 170, "to": 171, "label": "ONLY EVAL with clause\nno(X129) :- ','(call(X129), ','(!_19, failure(a))).\nand substitutionT37 -> T40,\nX129 -> zero(T40)" }, { "from": 171, "to": 172, "label": "CALL" }, { "from": 172, "to": 173, "label": "CASE" }, { "from": 173, "to": 174, "label": "EVAL with clause\nzero(0).\nand substitutionT40 -> 0" }, { "from": 173, "to": 177, "label": "EVAL-BACKTRACK" }, { "from": 174, "to": 178, "label": "CUT" }, { "from": 177, "to": 181, "label": "FAILURE" }, { "from": 178, "to": 179, "label": "CASE" }, { "from": 179, "to": 180, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 181, "to": 182, "label": "FAILURE" }, { "from": 182, "to": 256, "label": "ONLY EVAL with clause\nno(X132).\nand substitutionT40 -> T43,\nX132 -> zero(T43)" }, { "from": 256, "to": 257, "label": "CASE" }, { "from": 257, "to": 258, "label": "BACKTRACK\nfor clause: p(0, 0)\nwith clash: (zero(T43), zero(0))" }, { "from": 258, "to": 262, "label": "EVAL with clause\np(s(X135), X135).\nand substitutionX135 -> T46,\nT43 -> s(T46),\nX124 -> T46" }, { "from": 258, "to": 265, "label": "EVAL-BACKTRACK" }, { "from": 262, "to": 125, "label": "INSTANCE with matching:\nT17 -> T46\nX51 -> X126" }, { "from": 276, "to": 278, "label": "ONLY EVAL with clause\ns2t(X158, node(X159, X160, nil)) :- ','(no(zero(X158)), ','(p(X158, X161), s2t(X161, X159))).\nand substitutionT17 -> T51,\nX158 -> T51,\nX159 -> X162,\nX160 -> X163,\nX51 -> node(X162, X163, nil)" }, { "from": 277, "to": 297, "label": "ONLY EVAL with clause\ns2t(X179, node(nil, X180, nil)) :- no(zero(X179)).\nand substitutionT17 -> T63,\nX179 -> T63,\nX180 -> X181,\nX51 -> node(nil, X181, nil)" }, { "from": 278, "to": 279, "label": "CASE" }, { "from": 279, "to": 280, "label": "ONLY EVAL with clause\nno(X166) :- ','(call(X166), ','(!_24, failure(a))).\nand substitutionT51 -> T54,\nX166 -> zero(T54)" }, { "from": 280, "to": 281, "label": "CALL" }, { "from": 281, "to": 282, "label": "CASE" }, { "from": 282, "to": 283, "label": "EVAL with clause\nzero(0).\nand substitutionT54 -> 0" }, { "from": 282, "to": 284, "label": "EVAL-BACKTRACK" }, { "from": 283, "to": 285, "label": "CUT" }, { "from": 284, "to": 288, "label": "FAILURE" }, { "from": 285, "to": 286, "label": "CASE" }, { "from": 286, "to": 287, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 288, "to": 289, "label": "FAILURE" }, { "from": 289, "to": 292, "label": "ONLY EVAL with clause\nno(X169).\nand substitutionT54 -> T57,\nX169 -> zero(T57)" }, { "from": 292, "to": 293, "label": "CASE" }, { "from": 293, "to": 294, "label": "BACKTRACK\nfor clause: p(0, 0)\nwith clash: (zero(T57), zero(0))" }, { "from": 294, "to": 295, "label": "EVAL with clause\np(s(X172), X172).\nand substitutionX172 -> T60,\nT57 -> s(T60),\nX161 -> T60" }, { "from": 294, "to": 296, "label": "EVAL-BACKTRACK" }, { "from": 295, "to": 125, "label": "INSTANCE with matching:\nT17 -> T60\nX51 -> X162" }, { "from": 297, "to": 298, "label": "CASE" }, { "from": 298, "to": 299, "label": "ONLY EVAL with clause\nno(X184) :- ','(call(X184), ','(!_29, failure(a))).\nand substitutionT63 -> T66,\nX184 -> zero(T66)" }, { "from": 299, "to": 300, "label": "CALL" }, { "from": 300, "to": 301, "label": "CASE" }, { "from": 301, "to": 302, "label": "EVAL with clause\nzero(0).\nand substitutionT66 -> 0" }, { "from": 301, "to": 303, "label": "EVAL-BACKTRACK" }, { "from": 302, "to": 304, "label": "CUT" }, { "from": 303, "to": 307, "label": "FAILURE" }, { "from": 304, "to": 305, "label": "CASE" }, { "from": 305, "to": 306, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 307, "to": 310, "label": "FAILURE" }, { "from": 310, "to": 314, "label": "ONLY EVAL with clause\nno(X187).\nand substitutionT66 -> T69,\nX187 -> zero(T69)" }, { "from": 314, "to": 316, "label": "SUCCESS" }, { "from": 361, "to": 364, "label": "BACKTRACK\nfor clause: tree(nil)because of non-unification" }, { "from": 364, "to": 374, "label": "ONLY EVAL with clause\ntree(X195) :- ','(no(empty(X195)), ','(left(X196, X197), ','(right(X196, X198), ','(tree(X197), tree(X198))))).\nand substitutionT18 -> T74,\nX52 -> X199,\nX195 -> node(T74, X199, T74),\nT73 -> T74" }, { "from": 374, "to": 377, "label": "CASE" }, { "from": 377, "to": 380, "label": "ONLY EVAL with clause\nno(X204) :- ','(call(X204), ','(!_34, failure(a))).\nand substitutionT74 -> T79,\nX199 -> X205,\nX204 -> empty(node(T79, X205, T79)),\nT78 -> T79" }, { "from": 380, "to": 381, "label": "CALL" }, { "from": 381, "to": 382, "label": "CASE" }, { "from": 382, "to": 383, "label": "BACKTRACK\nfor clause: empty(nil)because of non-unification" }, { "from": 383, "to": 385, "label": "FAILURE" }, { "from": 385, "to": 386, "label": "FAILURE" }, { "from": 386, "to": 387, "label": "ONLY EVAL with clause\nno(X210).\nand substitutionT74 -> T82,\nX199 -> X211,\nX210 -> empty(node(T82, X211, T82))" }, { "from": 387, "to": 389, "label": "CASE" }, { "from": 389, "to": 390, "label": "PARALLEL" }, { "from": 389, "to": 391, "label": "PARALLEL" }, { "from": 390, "to": 392, "label": "ONLY EVAL with clause\nleft(nil, nil).\nand substitutionX196 -> nil,\nX197 -> nil" }, { "from": 391, "to": 407, "label": "ONLY EVAL with clause\nleft(node(X227, X228, X229), X227).\nand substitutionX227 -> X230,\nX228 -> X231,\nX229 -> X232,\nX196 -> node(X230, X231, X232),\nX197 -> X230" }, { "from": 392, "to": 393, "label": "CASE" }, { "from": 393, "to": 394, "label": "PARALLEL" }, { "from": 393, "to": 395, "label": "PARALLEL" }, { "from": 394, "to": 399, "label": "ONLY EVAL with clause\nright(nil, nil).\nand substitutionX198 -> nil" }, { "from": 395, "to": 406, "label": "BACKTRACK\nfor clause: right(node(X7, X8, R), R)because of non-unification" }, { "from": 399, "to": 402, "label": "SPLIT 1" }, { "from": 399, "to": 403, "label": "SPLIT 2" }, { "from": 402, "to": 16, "label": "INSTANCE" }, { "from": 403, "to": 16, "label": "INSTANCE" }, { "from": 407, "to": 408, "label": "CASE" }, { "from": 408, "to": 409, "label": "BACKTRACK\nfor clause: right(nil, nil)because of non-unification" }, { "from": 409, "to": 410, "label": "ONLY EVAL with clause\nright(node(X245, X246, X247), X247).\nand substitutionX230 -> X248,\nX245 -> X248,\nX231 -> X249,\nX246 -> X249,\nX232 -> X250,\nX247 -> X250,\nX198 -> X250" }, { "from": 410, "to": 411, "label": "SPLIT 1" }, { "from": 410, "to": 412, "label": "SPLIT 2\nreplacements:X248 -> T83" }, { "from": 411, "to": 412, "label": "INSTANCE with matching:\nX250 -> X248" }, { "from": 412, "to": 413, "label": "CASE" }, { "from": 413, "to": 414, "label": "PARALLEL" }, { "from": 413, "to": 415, "label": "PARALLEL" }, { "from": 414, "to": 416, "label": "ONLY EVAL with clause\ntree(nil).\nand substitutionX250 -> nil" }, { "from": 415, "to": 418, "label": "ONLY EVAL with clause\ntree(X258) :- ','(no(empty(X258)), ','(left(X259, X260), ','(right(X259, X261), ','(tree(X260), tree(X261))))).\nand substitutionX250 -> X262,\nX258 -> X262" }, { "from": 416, "to": 417, "label": "SUCCESS" }, { "from": 418, "to": 419, "label": "CASE" }, { "from": 419, "to": 420, "label": "ONLY EVAL with clause\nno(X267) :- ','(call(X267), ','(!_41, failure(a))).\nand substitutionX262 -> X268,\nX267 -> empty(X268)" }, { "from": 420, "to": 421, "label": "CALL" }, { "from": 421, "to": 422, "label": "CASE" }, { "from": 422, "to": 423, "label": "ONLY EVAL with clause\nempty(nil).\nand substitutionX268 -> nil" }, { "from": 423, "to": 424, "label": "CUT" }, { "from": 424, "to": 425, "label": "CASE" }, { "from": 425, "to": 426, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 427, "to": 429, "label": "ONLY EVAL with clause\ns2t(X291, node(nil, X292, X293)) :- ','(no(zero(X291)), ','(p(X291, X294), s2t(X294, X293))).\nand substitutionT3 -> T88,\nX291 -> T88,\nX292 -> X295,\nX293 -> X296,\nX12 -> node(nil, X295, X296)" }, { "from": 428, "to": 472, "label": "PARALLEL" }, { "from": 428, "to": 473, "label": "PARALLEL" }, { "from": 429, "to": 430, "label": "CASE" }, { "from": 430, "to": 431, "label": "ONLY EVAL with clause\nno(X299) :- ','(call(X299), ','(!_45, failure(a))).\nand substitutionT88 -> T91,\nX299 -> zero(T91)" }, { "from": 431, "to": 432, "label": "CALL" }, { "from": 432, "to": 433, "label": "CASE" }, { "from": 433, "to": 434, "label": "EVAL with clause\nzero(0).\nand substitutionT91 -> 0" }, { "from": 433, "to": 435, "label": "EVAL-BACKTRACK" }, { "from": 434, "to": 436, "label": "CUT" }, { "from": 435, "to": 439, "label": "FAILURE" }, { "from": 436, "to": 437, "label": "CASE" }, { "from": 437, "to": 438, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 439, "to": 440, "label": "FAILURE" }, { "from": 440, "to": 441, "label": "ONLY EVAL with clause\nno(X302).\nand substitutionT91 -> T94,\nX302 -> zero(T94)" }, { "from": 441, "to": 442, "label": "CASE" }, { "from": 442, "to": 443, "label": "BACKTRACK\nfor clause: p(0, 0)\nwith clash: (zero(T94), zero(0))" }, { "from": 443, "to": 444, "label": "EVAL with clause\np(s(X305), X305).\nand substitutionX305 -> T97,\nT94 -> s(T97),\nX294 -> T97" }, { "from": 443, "to": 445, "label": "EVAL-BACKTRACK" }, { "from": 444, "to": 446, "label": "SPLIT 1" }, { "from": 444, "to": 447, "label": "SPLIT 2\nnew knowledge:\nT97 is ground\nreplacements:X296 -> T98" }, { "from": 446, "to": 125, "label": "INSTANCE with matching:\nT17 -> T97\nX51 -> X296" }, { "from": 447, "to": 448, "label": "CASE" }, { "from": 448, "to": 449, "label": "BACKTRACK\nfor clause: tree(nil)because of non-unification" }, { "from": 449, "to": 450, "label": "ONLY EVAL with clause\ntree(X313) :- ','(no(empty(X313)), ','(left(X314, X315), ','(right(X314, X316), ','(tree(X315), tree(X316))))).\nand substitutionX295 -> X317,\nT98 -> T103,\nX313 -> node(nil, X317, T103),\nT102 -> T103" }, { "from": 450, "to": 451, "label": "CASE" }, { "from": 451, "to": 452, "label": "ONLY EVAL with clause\nno(X322) :- ','(call(X322), ','(!_51, failure(a))).\nand substitutionX317 -> X323,\nT103 -> T108,\nX322 -> empty(node(nil, X323, T108)),\nT107 -> T108" }, { "from": 452, "to": 453, "label": "CALL" }, { "from": 453, "to": 454, "label": "CASE" }, { "from": 454, "to": 455, "label": "BACKTRACK\nfor clause: empty(nil)because of non-unification" }, { "from": 455, "to": 456, "label": "FAILURE" }, { "from": 456, "to": 457, "label": "FAILURE" }, { "from": 457, "to": 458, "label": "ONLY EVAL with clause\nno(X328).\nand substitutionX317 -> X329,\nT103 -> T111,\nX328 -> empty(node(nil, X329, T111))" }, { "from": 458, "to": 459, "label": "CASE" }, { "from": 459, "to": 460, "label": "PARALLEL" }, { "from": 459, "to": 461, "label": "PARALLEL" }, { "from": 460, "to": 462, "label": "ONLY EVAL with clause\nleft(nil, nil).\nand substitutionX314 -> nil,\nX315 -> nil" }, { "from": 461, "to": 468, "label": "ONLY EVAL with clause\nleft(node(X345, X346, X347), X345).\nand substitutionX345 -> X348,\nX346 -> X349,\nX347 -> X350,\nX314 -> node(X348, X349, X350),\nX315 -> X348" }, { "from": 462, "to": 463, "label": "CASE" }, { "from": 463, "to": 464, "label": "PARALLEL" }, { "from": 463, "to": 465, "label": "PARALLEL" }, { "from": 464, "to": 466, "label": "ONLY EVAL with clause\nright(nil, nil).\nand substitutionX316 -> nil" }, { "from": 465, "to": 467, "label": "BACKTRACK\nfor clause: right(node(X7, X8, R), R)because of non-unification" }, { "from": 466, "to": 399, "label": "INSTANCE" }, { "from": 468, "to": 469, "label": "CASE" }, { "from": 469, "to": 470, "label": "BACKTRACK\nfor clause: right(nil, nil)because of non-unification" }, { "from": 470, "to": 471, "label": "ONLY EVAL with clause\nright(node(X363, X364, X365), X365).\nand substitutionX348 -> X366,\nX363 -> X366,\nX349 -> X367,\nX364 -> X367,\nX350 -> X368,\nX365 -> X368,\nX316 -> X368" }, { "from": 471, "to": 410, "label": "INSTANCE with matching:\nX248 -> X366\nX250 -> X368" }, { "from": 472, "to": 474, "label": "ONLY EVAL with clause\ns2t(X391, node(X392, X393, nil)) :- ','(no(zero(X391)), ','(p(X391, X394), s2t(X394, X392))).\nand substitutionT3 -> T116,\nX391 -> T116,\nX392 -> X395,\nX393 -> X396,\nX12 -> node(X395, X396, nil)" }, { "from": 473, "to": 517, "label": "ONLY EVAL with clause\ns2t(X475, node(nil, X476, nil)) :- no(zero(X475)).\nand substitutionT3 -> T142,\nX475 -> T142,\nX476 -> X477,\nX12 -> node(nil, X477, nil)" }, { "from": 474, "to": 475, "label": "CASE" }, { "from": 475, "to": 476, "label": "ONLY EVAL with clause\nno(X399) :- ','(call(X399), ','(!_57, failure(a))).\nand substitutionT116 -> T119,\nX399 -> zero(T119)" }, { "from": 476, "to": 477, "label": "CALL" }, { "from": 477, "to": 478, "label": "CASE" }, { "from": 478, "to": 479, "label": "EVAL with clause\nzero(0).\nand substitutionT119 -> 0" }, { "from": 478, "to": 480, "label": "EVAL-BACKTRACK" }, { "from": 479, "to": 481, "label": "CUT" }, { "from": 480, "to": 484, "label": "FAILURE" }, { "from": 481, "to": 482, "label": "CASE" }, { "from": 482, "to": 483, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 484, "to": 485, "label": "FAILURE" }, { "from": 485, "to": 486, "label": "ONLY EVAL with clause\nno(X402).\nand substitutionT119 -> T122,\nX402 -> zero(T122)" }, { "from": 486, "to": 487, "label": "CASE" }, { "from": 487, "to": 488, "label": "BACKTRACK\nfor clause: p(0, 0)\nwith clash: (zero(T122), zero(0))" }, { "from": 488, "to": 489, "label": "EVAL with clause\np(s(X405), X405).\nand substitutionX405 -> T125,\nT122 -> s(T125),\nX394 -> T125" }, { "from": 488, "to": 490, "label": "EVAL-BACKTRACK" }, { "from": 489, "to": 491, "label": "SPLIT 1" }, { "from": 489, "to": 492, "label": "SPLIT 2\nnew knowledge:\nT125 is ground\nreplacements:X395 -> T126" }, { "from": 491, "to": 125, "label": "INSTANCE with matching:\nT17 -> T125\nX51 -> X395" }, { "from": 492, "to": 493, "label": "CASE" }, { "from": 493, "to": 494, "label": "BACKTRACK\nfor clause: tree(nil)because of non-unification" }, { "from": 494, "to": 495, "label": "ONLY EVAL with clause\ntree(X413) :- ','(no(empty(X413)), ','(left(X414, X415), ','(right(X414, X416), ','(tree(X415), tree(X416))))).\nand substitutionT126 -> T131,\nX396 -> X417,\nX413 -> node(T131, X417, nil),\nT130 -> T131" }, { "from": 495, "to": 496, "label": "CASE" }, { "from": 496, "to": 497, "label": "ONLY EVAL with clause\nno(X422) :- ','(call(X422), ','(!_63, failure(a))).\nand substitutionT131 -> T136,\nX417 -> X423,\nX422 -> empty(node(T136, X423, nil)),\nT135 -> T136" }, { "from": 497, "to": 498, "label": "CALL" }, { "from": 498, "to": 499, "label": "CASE" }, { "from": 499, "to": 500, "label": "BACKTRACK\nfor clause: empty(nil)because of non-unification" }, { "from": 500, "to": 501, "label": "FAILURE" }, { "from": 501, "to": 502, "label": "FAILURE" }, { "from": 502, "to": 503, "label": "ONLY EVAL with clause\nno(X428).\nand substitutionT131 -> T139,\nX417 -> X429,\nX428 -> empty(node(T139, X429, nil))" }, { "from": 503, "to": 504, "label": "CASE" }, { "from": 504, "to": 505, "label": "PARALLEL" }, { "from": 504, "to": 506, "label": "PARALLEL" }, { "from": 505, "to": 507, "label": "ONLY EVAL with clause\nleft(nil, nil).\nand substitutionX414 -> nil,\nX415 -> nil" }, { "from": 506, "to": 513, "label": "ONLY EVAL with clause\nleft(node(X445, X446, X447), X445).\nand substitutionX445 -> X448,\nX446 -> X449,\nX447 -> X450,\nX414 -> node(X448, X449, X450),\nX415 -> X448" }, { "from": 507, "to": 508, "label": "CASE" }, { "from": 508, "to": 509, "label": "PARALLEL" }, { "from": 508, "to": 510, "label": "PARALLEL" }, { "from": 509, "to": 511, "label": "ONLY EVAL with clause\nright(nil, nil).\nand substitutionX416 -> nil" }, { "from": 510, "to": 512, "label": "BACKTRACK\nfor clause: right(node(X7, X8, R), R)because of non-unification" }, { "from": 511, "to": 399, "label": "INSTANCE" }, { "from": 513, "to": 514, "label": "CASE" }, { "from": 514, "to": 515, "label": "BACKTRACK\nfor clause: right(nil, nil)because of non-unification" }, { "from": 515, "to": 516, "label": "ONLY EVAL with clause\nright(node(X463, X464, X465), X465).\nand substitutionX448 -> X466,\nX463 -> X466,\nX449 -> X467,\nX464 -> X467,\nX450 -> X468,\nX465 -> X468,\nX416 -> X468" }, { "from": 516, "to": 410, "label": "INSTANCE with matching:\nX248 -> X466\nX250 -> X468" }, { "from": 517, "to": 518, "label": "CASE" }, { "from": 518, "to": 519, "label": "ONLY EVAL with clause\nno(X480) :- ','(call(X480), ','(!_69, failure(a))).\nand substitutionT142 -> T145,\nX480 -> zero(T145)" }, { "from": 519, "to": 520, "label": "CALL" }, { "from": 520, "to": 521, "label": "CASE" }, { "from": 521, "to": 522, "label": "EVAL with clause\nzero(0).\nand substitutionT145 -> 0" }, { "from": 521, "to": 523, "label": "EVAL-BACKTRACK" }, { "from": 522, "to": 524, "label": "CUT" }, { "from": 523, "to": 527, "label": "FAILURE" }, { "from": 524, "to": 525, "label": "CASE" }, { "from": 525, "to": 526, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 527, "to": 528, "label": "FAILURE" }, { "from": 528, "to": 529, "label": "ONLY EVAL with clause\nno(X483).\nand substitutionT145 -> T148,\nX483 -> zero(T148)" }, { "from": 529, "to": 126, "label": "INSTANCE with matching:\nT18 -> nil\nX52 -> X477" } ], "type": "Graph" } } ---------------------------------------- (2) Obligation: Triples: s2tA(s(X1), node(X2, X3, X2)) :- s2tA(X1, X2). s2tA(s(X1), node(nil, X2, X3)) :- s2tA(X1, X3). s2tA(s(X1), node(X2, X3, nil)) :- s2tA(X1, X2). pD :- treeB. pD :- ','(treecB, treeB). pE(X1, X2) :- treeC(X1). pE(X1, X2) :- ','(treecC(X1), treeC(X2)). treeF(X1, X2) :- pD. treeF(X1, X2) :- pE(X3, X4). goalG(0) :- treeB. goalG(s(X1)) :- s2tA(X1, X2). goalG(s(X1)) :- ','(s2tcA(X1, X2), treeF(X2, X3)). goalG(s(X1)) :- s2tA(X1, X2). goalG(s(X1)) :- ','(s2tcA(X1, X2), pD). goalG(s(X1)) :- ','(s2tcA(X1, X2), pE(X3, X4)). goalG(s(X1)) :- s2tA(X1, X2). goalG(s(X1)) :- ','(s2tcA(X1, X2), pD). goalG(s(X1)) :- ','(s2tcA(X1, X2), pE(X3, X4)). goalG(X1) :- treeF(nil, X2). Clauses: s2tcA(0, nil). s2tcA(s(X1), node(X2, X3, X2)) :- s2tcA(X1, X2). s2tcA(s(X1), node(nil, X2, X3)) :- s2tcA(X1, X3). s2tcA(s(X1), node(X2, X3, nil)) :- s2tcA(X1, X2). s2tcA(X1, node(nil, X2, nil)). treecB. treecC(nil). qcD :- ','(treecB, treecB). qcE(X1, X2) :- ','(treecC(X1), treecC(X2)). treecF(X1, X2) :- qcD. treecF(X1, X2) :- qcE(X3, X4). Afs: goalG(x1) = goalG(x1) ---------------------------------------- (3) UndefinedPredicateInTriplesTransformerProof (SOUND) Deleted triples and predicates having undefined goals [DT09]. ---------------------------------------- (4) Obligation: Triples: s2tA(s(X1), node(X2, X3, X2)) :- s2tA(X1, X2). s2tA(s(X1), node(nil, X2, X3)) :- s2tA(X1, X3). s2tA(s(X1), node(X2, X3, nil)) :- s2tA(X1, X2). goalG(s(X1)) :- s2tA(X1, X2). goalG(s(X1)) :- s2tA(X1, X2). goalG(s(X1)) :- s2tA(X1, X2). Clauses: s2tcA(0, nil). s2tcA(s(X1), node(X2, X3, X2)) :- s2tcA(X1, X2). s2tcA(s(X1), node(nil, X2, X3)) :- s2tcA(X1, X3). s2tcA(s(X1), node(X2, X3, nil)) :- s2tcA(X1, X2). s2tcA(X1, node(nil, X2, nil)). treecB. treecC(nil). qcD :- ','(treecB, treecB). qcE(X1, X2) :- ','(treecC(X1), treecC(X2)). treecF(X1, X2) :- qcD. treecF(X1, X2) :- qcE(X3, X4). Afs: goalG(x1) = goalG(x1) ---------------------------------------- (5) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: goalG_in_1: (b) s2tA_in_2: (b,f) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: GOALG_IN_G(s(X1)) -> U4_G(X1, s2tA_in_ga(X1, X2)) GOALG_IN_G(s(X1)) -> S2TA_IN_GA(X1, X2) S2TA_IN_GA(s(X1), node(X2, X3, X2)) -> U1_GA(X1, X2, X3, s2tA_in_ga(X1, X2)) S2TA_IN_GA(s(X1), node(X2, X3, X2)) -> S2TA_IN_GA(X1, X2) S2TA_IN_GA(s(X1), node(nil, X2, X3)) -> U2_GA(X1, X2, X3, s2tA_in_ga(X1, X3)) S2TA_IN_GA(s(X1), node(nil, X2, X3)) -> S2TA_IN_GA(X1, X3) S2TA_IN_GA(s(X1), node(X2, X3, nil)) -> U3_GA(X1, X2, X3, s2tA_in_ga(X1, X2)) S2TA_IN_GA(s(X1), node(X2, X3, nil)) -> S2TA_IN_GA(X1, X2) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) s2tA_in_ga(x1, x2) = s2tA_in_ga(x1) node(x1, x2, x3) = node(x1, x3) nil = nil GOALG_IN_G(x1) = GOALG_IN_G(x1) U4_G(x1, x2) = U4_G(x1, x2) S2TA_IN_GA(x1, x2) = S2TA_IN_GA(x1) U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4) U2_GA(x1, x2, x3, x4) = U2_GA(x1, x4) U3_GA(x1, x2, x3, x4) = U3_GA(x1, 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: GOALG_IN_G(s(X1)) -> U4_G(X1, s2tA_in_ga(X1, X2)) GOALG_IN_G(s(X1)) -> S2TA_IN_GA(X1, X2) S2TA_IN_GA(s(X1), node(X2, X3, X2)) -> U1_GA(X1, X2, X3, s2tA_in_ga(X1, X2)) S2TA_IN_GA(s(X1), node(X2, X3, X2)) -> S2TA_IN_GA(X1, X2) S2TA_IN_GA(s(X1), node(nil, X2, X3)) -> U2_GA(X1, X2, X3, s2tA_in_ga(X1, X3)) S2TA_IN_GA(s(X1), node(nil, X2, X3)) -> S2TA_IN_GA(X1, X3) S2TA_IN_GA(s(X1), node(X2, X3, nil)) -> U3_GA(X1, X2, X3, s2tA_in_ga(X1, X2)) S2TA_IN_GA(s(X1), node(X2, X3, nil)) -> S2TA_IN_GA(X1, X2) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) s2tA_in_ga(x1, x2) = s2tA_in_ga(x1) node(x1, x2, x3) = node(x1, x3) nil = nil GOALG_IN_G(x1) = GOALG_IN_G(x1) U4_G(x1, x2) = U4_G(x1, x2) S2TA_IN_GA(x1, x2) = S2TA_IN_GA(x1) U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4) U2_GA(x1, x2, x3, x4) = U2_GA(x1, x4) U3_GA(x1, x2, x3, x4) = U3_GA(x1, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (7) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 5 less nodes. ---------------------------------------- (8) Obligation: Pi DP problem: The TRS P consists of the following rules: S2TA_IN_GA(s(X1), node(nil, X2, X3)) -> S2TA_IN_GA(X1, X3) S2TA_IN_GA(s(X1), node(X2, X3, X2)) -> S2TA_IN_GA(X1, X2) S2TA_IN_GA(s(X1), node(X2, X3, nil)) -> S2TA_IN_GA(X1, X2) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) node(x1, x2, x3) = node(x1, x3) nil = nil S2TA_IN_GA(x1, x2) = S2TA_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (9) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: S2TA_IN_GA(s(X1)) -> S2TA_IN_GA(X1) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (11) 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: *S2TA_IN_GA(s(X1)) -> S2TA_IN_GA(X1) The graph contains the following edges 1 > 1 ---------------------------------------- (12) YES