/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern fold(g,g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToDTProblemTransformerProof [SOUND, 0 ms] (2) TRIPLES (3) TriplesToPiDPProof [SOUND, 4 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) TRUE ---------------------------------------- (0) Obligation: Clauses: fold(X, [], X). fold(X, Y, Z) :- ','(no(empty(Y)), ','(head(Y, H), ','(tail(Y, T), ','(myop(X, H, V), fold(V, T, Z))))). myop(a, b, c). head([], X1). head(.(H, X2), H). tail([], []). tail(.(X3, T), T). empty([]). no(X) :- ','(X, ','(!, failure(a))). no(X4). failure(b). Query: fold(g,g,a) ---------------------------------------- (1) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 3, "program": { "directives": [], "clauses": [ [ "(fold X ([]) X)", null ], [ "(fold X Y Z)", "(',' (no (empty Y)) (',' (head Y H) (',' (tail Y T) (',' (myop X H V) (fold V T Z)))))" ], [ "(myop (a) (b) (c))", null ], [ "(head ([]) X1)", null ], [ "(head (. H X2) H)", null ], [ "(tail ([]) ([]))", null ], [ "(tail (. X3 T) T)", null ], [ "(empty ([]))", null ], [ "(no X)", "(',' X (',' (!) (failure (a))))" ], [ "(no X4)", null ], [ "(failure (b))", null ] ] }, "graph": { "nodes": { "88": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (tail (. T32 T33) X28) (',' (myop T15 T32 X29) (fold X29 X28 T18)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T15", "T32", "T33" ], "free": [ "X28", "X29" ], "exprvars": [] } }, "23": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (',' (call (empty ([]))) (',' (!_2) (failure (a)))) (',' (head ([]) X13) (',' (tail ([]) X14) (',' (myop T8 X13 X15) (fold X15 X14 T10)))))" }, { "clause": 9, "scope": 2, "term": "(',' (no (empty ([]))) (',' (head ([]) X13) (',' (tail ([]) X14) (',' (myop T8 X13 X15) (fold X15 X14 T10)))))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T8"], "free": [ "X13", "X14", "X15" ], "exprvars": [] } }, "89": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "25": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (empty ([])) (',' (',' (!_2) (failure (a))) (',' (head ([]) X13) (',' (tail ([]) X14) (',' (myop T8 X13 X15) (fold X15 X14 T10))))))" }, { "clause": -1, "scope": 3, "term": null }, { "clause": 9, "scope": 2, "term": "(',' (no (empty ([]))) (',' (head ([]) X13) (',' (tail ([]) X14) (',' (myop T8 X13 X15) (fold X15 X14 T10)))))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T8"], "free": [ "X13", "X14", "X15" ], "exprvars": [] } }, "47": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (empty T21) (',' (',' (!_6) (failure (a))) (',' (head T21 X27) (',' (tail T21 X28) (',' (myop T15 X27 X29) (fold X29 X28 T18))))))" }, { "clause": -1, "scope": 7, "term": null }, { "clause": 9, "scope": 6, "term": "(',' (no (empty T21)) (',' (head T21 X27) (',' (tail T21 X28) (',' (myop T15 X27 X29) (fold X29 X28 T18)))))" } ], "kb": { "nonunifying": [[ "(fold T15 T21 T3)", "(fold X6 ([]) X6)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T15", "T21" ], "free": [ "X6", "X27", "X28", "X29" ], "exprvars": [] } }, "48": { "goal": [ { "clause": 7, "scope": 8, "term": "(',' (empty T21) (',' (',' (!_6) (failure (a))) (',' (head T21 X27) (',' (tail T21 X28) (',' (myop T15 X27 X29) (fold X29 X28 T18))))))" }, { "clause": -1, "scope": 8, "term": null }, { "clause": -1, "scope": 7, "term": null }, { "clause": 9, "scope": 6, "term": "(',' (no (empty T21)) (',' (head T21 X27) (',' (tail T21 X28) (',' (myop T15 X27 X29) (fold X29 X28 T18)))))" } ], "kb": { "nonunifying": [[ "(fold T15 T21 T3)", "(fold X6 ([]) X6)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T15", "T21" ], "free": [ "X6", "X27", "X28", "X29" ], "exprvars": [] } }, "28": { "goal": [ { "clause": 7, "scope": 4, "term": "(',' (empty ([])) (',' (',' (!_2) (failure (a))) (',' (head ([]) X13) (',' (tail ([]) X14) (',' (myop T8 X13 X15) (fold X15 X14 T10))))))" }, { "clause": -1, "scope": 4, "term": null }, { "clause": -1, "scope": 3, "term": null }, { "clause": 9, "scope": 2, "term": "(',' (no (empty ([]))) (',' (head ([]) X13) (',' (tail ([]) X14) (',' (myop T8 X13 X15) (fold X15 X14 T10)))))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T8"], "free": [ "X13", "X14", "X15" ], "exprvars": [] } }, "29": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (',' (!_2) (failure (a))) (',' (head ([]) X13) (',' (tail ([]) X14) (',' (myop T8 X13 X15) (fold X15 X14 T10)))))" }, { "clause": -1, "scope": 4, "term": null }, { "clause": -1, "scope": 3, "term": null }, { "clause": 9, "scope": 2, "term": "(',' (no (empty ([]))) (',' (head ([]) X13) (',' (tail ([]) X14) (',' (myop T8 X13 X15) (fold X15 X14 T10)))))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T8"], "free": [ "X13", "X14", "X15" ], "exprvars": [] } }, "type": "Nodes", "90": { "goal": [ { "clause": 5, "scope": 11, "term": "(',' (tail (. T32 T33) X28) (',' (myop T15 T32 X29) (fold X29 X28 T18)))" }, { "clause": 6, "scope": 11, "term": "(',' (tail (. T32 T33) X28) (',' (myop T15 T32 X29) (fold X29 X28 T18)))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T15", "T32", "T33" ], "free": [ "X28", "X29" ], "exprvars": [] } }, "91": { "goal": [{ "clause": 6, "scope": 11, "term": "(',' (tail (. T32 T33) X28) (',' (myop T15 T32 X29) (fold X29 X28 T18)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T15", "T32", "T33" ], "free": [ "X28", "X29" ], "exprvars": [] } }, "92": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (myop T15 T41 X29) (fold X29 T42 T18))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T15", "T41", "T42" ], "free": ["X29"], "exprvars": [] } }, "93": { "goal": [{ "clause": 2, "scope": 12, "term": "(',' (myop T15 T41 X29) (fold X29 T42 T18))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T15", "T41", "T42" ], "free": ["X29"], "exprvars": [] } }, "50": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (',' (!_6) (failure (a))) (',' (head ([]) X27) (',' (tail ([]) X28) (',' (myop T15 X27 X29) (fold X29 X28 T18)))))" }, { "clause": -1, "scope": 8, "term": null }, { "clause": -1, "scope": 7, "term": null }, { "clause": 9, "scope": 6, "term": "(',' (no (empty ([]))) (',' (head ([]) X27) (',' (tail ([]) X28) (',' (myop T15 X27 X29) (fold X29 X28 T18)))))" } ], "kb": { "nonunifying": [[ "(fold T15 ([]) T3)", "(fold X6 ([]) X6)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T15"], "free": [ "X6", "X27", "X28", "X29" ], "exprvars": [] } }, "94": { "goal": [{ "clause": -1, "scope": -1, "term": "(fold (c) T42 T18)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T42"], "free": [], "exprvars": [] } }, "51": { "goal": [ { "clause": -1, "scope": 8, "term": null }, { "clause": -1, "scope": 7, "term": null }, { "clause": 9, "scope": 6, "term": "(',' (no (empty T21)) (',' (head T21 X27) (',' (tail T21 X28) (',' (myop T15 X27 X29) (fold X29 X28 T18)))))" } ], "kb": { "nonunifying": [ [ "(fold T15 T21 T3)", "(fold X6 ([]) X6)" ], [ "(empty T21)", "(empty ([]))" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T15", "T21" ], "free": [ "X6", "X27", "X28", "X29" ], "exprvars": [] } }, "95": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "30": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (failure (a)) (',' (head ([]) X13) (',' (tail ([]) X14) (',' (myop T8 X13 X15) (fold X15 X14 T10)))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T8"], "free": [ "X13", "X14", "X15" ], "exprvars": [] } }, "53": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (failure (a)) (',' (head ([]) X27) (',' (tail ([]) X28) (',' (myop T15 X27 X29) (fold X29 X28 T18)))))" }], "kb": { "nonunifying": [[ "(fold T15 ([]) T3)", "(fold X6 ([]) X6)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T15"], "free": [ "X6", "X27", "X28", "X29" ], "exprvars": [] } }, "32": { "goal": [{ "clause": 10, "scope": 5, "term": "(',' (failure (a)) (',' (head ([]) X13) (',' (tail ([]) X14) (',' (myop T8 X13 X15) (fold X15 X14 T10)))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T8"], "free": [ "X13", "X14", "X15" ], "exprvars": [] } }, "33": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "55": { "goal": [{ "clause": 10, "scope": 9, "term": "(',' (failure (a)) (',' (head ([]) X27) (',' (tail ([]) X28) (',' (myop T15 X27 X29) (fold X29 X28 T18)))))" }], "kb": { "nonunifying": [[ "(fold T15 ([]) T3)", "(fold X6 ([]) X6)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T15"], "free": [ "X6", "X27", "X28", "X29" ], "exprvars": [] } }, "12": { "goal": [ { "clause": -1, "scope": -1, "term": "(true)" }, { "clause": 1, "scope": 1, "term": "(fold T5 ([]) T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T5"], "free": [], "exprvars": [] } }, "57": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "14": { "goal": [{ "clause": 1, "scope": 1, "term": "(fold T1 T2 T3)" }], "kb": { "nonunifying": [[ "(fold T1 T2 T3)", "(fold X6 ([]) X6)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2" ], "free": ["X6"], "exprvars": [] } }, "58": { "goal": [ { "clause": -1, "scope": 7, "term": null }, { "clause": 9, "scope": 6, "term": "(',' (no (empty T21)) (',' (head T21 X27) (',' (tail T21 X28) (',' (myop T15 X27 X29) (fold X29 X28 T18)))))" } ], "kb": { "nonunifying": [ [ "(fold T15 T21 T3)", "(fold X6 ([]) X6)" ], [ "(empty T21)", "(empty ([]))" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T15", "T21" ], "free": [ "X6", "X27", "X28", "X29" ], "exprvars": [] } }, "37": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (no (empty T16)) (',' (head T16 X27) (',' (tail T16 X28) (',' (myop T15 X27 X29) (fold X29 X28 T18)))))" }], "kb": { "nonunifying": [[ "(fold T15 T16 T3)", "(fold X6 ([]) X6)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T15", "T16" ], "free": [ "X6", "X27", "X28", "X29" ], "exprvars": [] } }, "59": { "goal": [{ "clause": 9, "scope": 6, "term": "(',' (no (empty T21)) (',' (head T21 X27) (',' (tail T21 X28) (',' (myop T15 X27 X29) (fold X29 X28 T18)))))" }], "kb": { "nonunifying": [ [ "(fold T15 T21 T3)", "(fold X6 ([]) X6)" ], [ "(empty T21)", "(empty ([]))" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T15", "T21" ], "free": [ "X6", "X27", "X28", "X29" ], "exprvars": [] } }, "17": { "goal": [{ "clause": 1, "scope": 1, "term": "(fold T5 ([]) T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T5"], "free": [], "exprvars": [] } }, "39": { "goal": [ { "clause": 8, "scope": 6, "term": "(',' (no (empty T16)) (',' (head T16 X27) (',' (tail T16 X28) (',' (myop T15 X27 X29) (fold X29 X28 T18)))))" }, { "clause": 9, "scope": 6, "term": "(',' (no (empty T16)) (',' (head T16 X27) (',' (tail T16 X28) (',' (myop T15 X27 X29) (fold X29 X28 T18)))))" } ], "kb": { "nonunifying": [[ "(fold T15 T16 T3)", "(fold X6 ([]) X6)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T15", "T16" ], "free": [ "X6", "X27", "X28", "X29" ], "exprvars": [] } }, "3": { "goal": [{ "clause": -1, "scope": -1, "term": "(fold T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2" ], "free": [], "exprvars": [] } }, "6": { "goal": [ { "clause": 0, "scope": 1, "term": "(fold T1 T2 T3)" }, { "clause": 1, "scope": 1, "term": "(fold T1 T2 T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2" ], "free": [], "exprvars": [] } }, "20": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (no (empty ([]))) (',' (head ([]) X13) (',' (tail ([]) X14) (',' (myop T8 X13 X15) (fold X15 X14 T10)))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T8"], "free": [ "X13", "X14", "X15" ], "exprvars": [] } }, "64": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (head T26 X27) (',' (tail T26 X28) (',' (myop T15 X27 X29) (fold X29 X28 T18))))" }], "kb": { "nonunifying": [ [ "(fold T15 T26 T3)", "(fold X6 ([]) X6)" ], [ "(empty T26)", "(empty ([]))" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T15", "T26" ], "free": [ "X6", "X27", "X28", "X29" ], "exprvars": [] } }, "86": { "goal": [ { "clause": 3, "scope": 10, "term": "(',' (head T26 X27) (',' (tail T26 X28) (',' (myop T15 X27 X29) (fold X29 X28 T18))))" }, { "clause": 4, "scope": 10, "term": "(',' (head T26 X27) (',' (tail T26 X28) (',' (myop T15 X27 X29) (fold X29 X28 T18))))" } ], "kb": { "nonunifying": [ [ "(fold T15 T26 T3)", "(fold X6 ([]) X6)" ], [ "(empty T26)", "(empty ([]))" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T15", "T26" ], "free": [ "X6", "X27", "X28", "X29" ], "exprvars": [] } }, "21": { "goal": [ { "clause": 8, "scope": 2, "term": "(',' (no (empty ([]))) (',' (head ([]) X13) (',' (tail ([]) X14) (',' (myop T8 X13 X15) (fold X15 X14 T10)))))" }, { "clause": 9, "scope": 2, "term": "(',' (no (empty ([]))) (',' (head ([]) X13) (',' (tail ([]) X14) (',' (myop T8 X13 X15) (fold X15 X14 T10)))))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T8"], "free": [ "X13", "X14", "X15" ], "exprvars": [] } }, "43": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (',' (call (empty T21)) (',' (!_6) (failure (a)))) (',' (head T21 X27) (',' (tail T21 X28) (',' (myop T15 X27 X29) (fold X29 X28 T18)))))" }, { "clause": 9, "scope": 6, "term": "(',' (no (empty T21)) (',' (head T21 X27) (',' (tail T21 X28) (',' (myop T15 X27 X29) (fold X29 X28 T18)))))" } ], "kb": { "nonunifying": [[ "(fold T15 T21 T3)", "(fold X6 ([]) X6)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T15", "T21" ], "free": [ "X6", "X27", "X28", "X29" ], "exprvars": [] } }, "87": { "goal": [{ "clause": 4, "scope": 10, "term": "(',' (head T26 X27) (',' (tail T26 X28) (',' (myop T15 X27 X29) (fold X29 X28 T18))))" }], "kb": { "nonunifying": [ [ "(fold T15 T26 T3)", "(fold X6 ([]) X6)" ], [ "(empty T26)", "(empty ([]))" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T15", "T26" ], "free": [ "X6", "X27", "X28", "X29" ], "exprvars": [] } } }, "edges": [ { "from": 3, "to": 6, "label": "CASE" }, { "from": 6, "to": 12, "label": "EVAL with clause\nfold(X6, [], X6).\nand substitutionT1 -> T5,\nX6 -> T5,\nT2 -> [],\nT3 -> T5" }, { "from": 6, "to": 14, "label": "EVAL-BACKTRACK" }, { "from": 12, "to": 17, "label": "SUCCESS" }, { "from": 14, "to": 37, "label": "ONLY EVAL with clause\nfold(X24, X25, X26) :- ','(no(empty(X25)), ','(head(X25, X27), ','(tail(X25, X28), ','(myop(X24, X27, X29), fold(X29, X28, X26))))).\nand substitutionT1 -> T15,\nX24 -> T15,\nT2 -> T16,\nX25 -> T16,\nT3 -> T18,\nX26 -> T18,\nT17 -> T18" }, { "from": 17, "to": 20, "label": "ONLY EVAL with clause\nfold(X10, X11, X12) :- ','(no(empty(X11)), ','(head(X11, X13), ','(tail(X11, X14), ','(myop(X10, X13, X15), fold(X15, X14, X12))))).\nand substitutionT5 -> T8,\nX10 -> T8,\nX11 -> [],\nT3 -> T10,\nX12 -> T10,\nT9 -> T10" }, { "from": 20, "to": 21, "label": "CASE" }, { "from": 21, "to": 23, "label": "ONLY EVAL with clause\nno(X18) :- ','(call(X18), ','(!_2, failure(a))).\nand substitutionX18 -> empty([])" }, { "from": 23, "to": 25, "label": "CALL" }, { "from": 25, "to": 28, "label": "CASE" }, { "from": 28, "to": 29, "label": "ONLY EVAL with clause\nempty([]).\nand substitution" }, { "from": 29, "to": 30, "label": "CUT" }, { "from": 30, "to": 32, "label": "CASE" }, { "from": 32, "to": 33, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 37, "to": 39, "label": "CASE" }, { "from": 39, "to": 43, "label": "ONLY EVAL with clause\nno(X32) :- ','(call(X32), ','(!_6, failure(a))).\nand substitutionT16 -> T21,\nX32 -> empty(T21)" }, { "from": 43, "to": 47, "label": "CALL" }, { "from": 47, "to": 48, "label": "CASE" }, { "from": 48, "to": 50, "label": "EVAL with clause\nempty([]).\nand substitutionT21 -> []" }, { "from": 48, "to": 51, "label": "EVAL-BACKTRACK" }, { "from": 50, "to": 53, "label": "CUT" }, { "from": 51, "to": 58, "label": "FAILURE" }, { "from": 53, "to": 55, "label": "CASE" }, { "from": 55, "to": 57, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 58, "to": 59, "label": "FAILURE" }, { "from": 59, "to": 64, "label": "ONLY EVAL with clause\nno(X39).\nand substitutionT21 -> T26,\nX39 -> empty(T26)" }, { "from": 64, "to": 86, "label": "CASE" }, { "from": 86, "to": 87, "label": "BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T26), empty([]))" }, { "from": 87, "to": 88, "label": "EVAL with clause\nhead(.(X48, X49), X48).\nand substitutionX48 -> T32,\nX49 -> T33,\nT26 -> .(T32, T33),\nX27 -> T32" }, { "from": 87, "to": 89, "label": "EVAL-BACKTRACK" }, { "from": 88, "to": 90, "label": "CASE" }, { "from": 90, "to": 91, "label": "BACKTRACK\nfor clause: tail([], [])because of non-unification" }, { "from": 91, "to": 92, "label": "ONLY EVAL with clause\ntail(.(X60, X61), X61).\nand substitutionT32 -> T41,\nX60 -> T41,\nT33 -> T42,\nX61 -> T42,\nX28 -> T42" }, { "from": 92, "to": 93, "label": "CASE" }, { "from": 93, "to": 94, "label": "EVAL with clause\nmyop(a, b, c).\nand substitutionT15 -> a,\nT41 -> b,\nX29 -> c" }, { "from": 93, "to": 95, "label": "EVAL-BACKTRACK" }, { "from": 94, "to": 3, "label": "INSTANCE with matching:\nT1 -> c\nT2 -> T42\nT3 -> T18" } ], "type": "Graph" } } ---------------------------------------- (2) Obligation: Triples: foldA(a, .(b, X1), X2) :- foldA(c, X1, X2). Clauses: foldcA(X1, [], X1). foldcA(a, .(b, X1), X2) :- foldcA(c, X1, X2). Afs: foldA(x1, x2, x3) = foldA(x1, x2) ---------------------------------------- (3) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: foldA_in_3: (b,b,f) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: FOLDA_IN_GGA(a, .(b, X1), X2) -> U1_GGA(X1, X2, foldA_in_gga(c, X1, X2)) FOLDA_IN_GGA(a, .(b, X1), X2) -> FOLDA_IN_GGA(c, X1, X2) R is empty. The argument filtering Pi contains the following mapping: foldA_in_gga(x1, x2, x3) = foldA_in_gga(x1, x2) a = a .(x1, x2) = .(x1, x2) b = b c = c FOLDA_IN_GGA(x1, x2, x3) = FOLDA_IN_GGA(x1, x2) U1_GGA(x1, x2, x3) = U1_GGA(x1, x3) We have to consider all (P,R,Pi)-chains Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: FOLDA_IN_GGA(a, .(b, X1), X2) -> U1_GGA(X1, X2, foldA_in_gga(c, X1, X2)) FOLDA_IN_GGA(a, .(b, X1), X2) -> FOLDA_IN_GGA(c, X1, X2) R is empty. The argument filtering Pi contains the following mapping: foldA_in_gga(x1, x2, x3) = foldA_in_gga(x1, x2) a = a .(x1, x2) = .(x1, x2) b = b c = c FOLDA_IN_GGA(x1, x2, x3) = FOLDA_IN_GGA(x1, x2) U1_GGA(x1, x2, x3) = U1_GGA(x1, x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 0 SCCs with 2 less nodes. ---------------------------------------- (6) TRUE