/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern duplicate(g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1)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, 16 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) PiDP (7) PiDPToQDPProof [SOUND, 0 ms] (8) QDP (9) QDPSizeChangeProof [EQUIVALENT, 0 ms] (10) YES ---------------------------------------- (0) Obligation: Clauses: duplicate([], []). duplicate(X, .(H, .(H, Z))) :- ','(no(empty(X)), ','(head(X, H), ','(tail(X, T), duplicate(T, Z)))). head([], X1). head(.(H, X2), H). tail([], []). tail(.(X3, T), T). empty([]). no(X) :- ','(X, ','(!, failure(a))). no(X4). failure(b). Query: duplicate(g,a) ---------------------------------------- (1) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 1, "program": { "directives": [], "clauses": [ [ "(duplicate ([]) ([]))", null ], [ "(duplicate X (. H (. H Z)))", "(',' (no (empty X)) (',' (head X H) (',' (tail X T) (duplicate T Z))))" ], [ "(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": { "44": { "goal": [ { "clause": 6, "scope": 8, "term": "(',' (empty T19) (',' (',' (!_6) (failure (a))) (',' (head T19 T15) (',' (tail T19 X21) (duplicate X21 T16)))))" }, { "clause": -1, "scope": 8, "term": null }, { "clause": -1, "scope": 7, "term": null }, { "clause": 8, "scope": 6, "term": "(',' (no (empty T19)) (',' (head T19 T15) (',' (tail T19 X21) (duplicate X21 T16))))" } ], "kb": { "nonunifying": [[ "(duplicate T19 T2)", "(duplicate ([]) ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T19"], "free": ["X21"], "exprvars": [] } }, "23": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (',' (call (empty ([]))) (',' (!_2) (failure (a)))) (',' (head ([]) T7) (',' (tail ([]) X11) (duplicate X11 T8))))" }, { "clause": 8, "scope": 2, "term": "(',' (no (empty ([]))) (',' (head ([]) T7) (',' (tail ([]) X11) (duplicate X11 T8))))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X11"], "exprvars": [] } }, "24": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (empty ([])) (',' (',' (!_2) (failure (a))) (',' (head ([]) T7) (',' (tail ([]) X11) (duplicate X11 T8)))))" }, { "clause": -1, "scope": 3, "term": null }, { "clause": 8, "scope": 2, "term": "(',' (no (empty ([]))) (',' (head ([]) T7) (',' (tail ([]) X11) (duplicate X11 T8))))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X11"], "exprvars": [] } }, "46": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (',' (!_6) (failure (a))) (',' (head ([]) T15) (',' (tail ([]) X21) (duplicate X21 T16))))" }, { "clause": -1, "scope": 8, "term": null }, { "clause": -1, "scope": 7, "term": null }, { "clause": 8, "scope": 6, "term": "(',' (no (empty ([]))) (',' (head ([]) T15) (',' (tail ([]) X21) (duplicate X21 T16))))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X21"], "exprvars": [] } }, "48": { "goal": [ { "clause": -1, "scope": 8, "term": null }, { "clause": -1, "scope": 7, "term": null }, { "clause": 8, "scope": 6, "term": "(',' (no (empty T19)) (',' (head T19 T15) (',' (tail T19 X21) (duplicate X21 T16))))" } ], "kb": { "nonunifying": [ [ "(duplicate T19 T2)", "(duplicate ([]) ([]))" ], [ "(empty T19)", "(empty ([]))" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T19"], "free": ["X21"], "exprvars": [] } }, "27": { "goal": [ { "clause": 6, "scope": 4, "term": "(',' (empty ([])) (',' (',' (!_2) (failure (a))) (',' (head ([]) T7) (',' (tail ([]) X11) (duplicate X11 T8)))))" }, { "clause": -1, "scope": 4, "term": null }, { "clause": -1, "scope": 3, "term": null }, { "clause": 8, "scope": 2, "term": "(',' (no (empty ([]))) (',' (head ([]) T7) (',' (tail ([]) X11) (duplicate X11 T8))))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X11"], "exprvars": [] } }, "49": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (failure (a)) (',' (head ([]) T15) (',' (tail ([]) X21) (duplicate X21 T16))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X21"], "exprvars": [] } }, "29": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (',' (!_2) (failure (a))) (',' (head ([]) T7) (',' (tail ([]) X11) (duplicate X11 T8))))" }, { "clause": -1, "scope": 4, "term": null }, { "clause": -1, "scope": 3, "term": null }, { "clause": 8, "scope": 2, "term": "(',' (no (empty ([]))) (',' (head ([]) T7) (',' (tail ([]) X11) (duplicate X11 T8))))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X11"], "exprvars": [] } }, "type": "Nodes", "72": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (head T22 T15) (',' (tail T22 X21) (duplicate X21 T16)))" }], "kb": { "nonunifying": [ [ "(duplicate T22 T2)", "(duplicate ([]) ([]))" ], [ "(empty T22)", "(empty ([]))" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T22"], "free": ["X21"], "exprvars": [] } }, "51": { "goal": [{ "clause": 9, "scope": 9, "term": "(',' (failure (a)) (',' (head ([]) T15) (',' (tail ([]) X21) (duplicate X21 T16))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X21"], "exprvars": [] } }, "73": { "goal": [ { "clause": 2, "scope": 10, "term": "(',' (head T22 T15) (',' (tail T22 X21) (duplicate X21 T16)))" }, { "clause": 3, "scope": 10, "term": "(',' (head T22 T15) (',' (tail T22 X21) (duplicate X21 T16)))" } ], "kb": { "nonunifying": [ [ "(duplicate T22 T2)", "(duplicate ([]) ([]))" ], [ "(empty T22)", "(empty ([]))" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T22"], "free": ["X21"], "exprvars": [] } }, "52": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "31": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (failure (a)) (',' (head ([]) T7) (',' (tail ([]) X11) (duplicate X11 T8))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X11"], "exprvars": [] } }, "53": { "goal": [ { "clause": -1, "scope": 7, "term": null }, { "clause": 8, "scope": 6, "term": "(',' (no (empty T19)) (',' (head T19 T15) (',' (tail T19 X21) (duplicate X21 T16))))" } ], "kb": { "nonunifying": [ [ 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T28 T29) X21) (duplicate X21 T30))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T28", "T29" ], "free": ["X21"], "exprvars": [] } }, "14": { "goal": [{ "clause": 1, "scope": 1, "term": "(duplicate T1 T2)" }], "kb": { "nonunifying": [[ "(duplicate T1 T2)", "(duplicate ([]) ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "36": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (no (empty T12)) (',' (head T12 T15) (',' (tail T12 X21) (duplicate X21 T16))))" }], "kb": { "nonunifying": [[ "(duplicate T12 T2)", "(duplicate ([]) ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T12"], "free": ["X21"], "exprvars": [] } }, "37": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "16": { "goal": [{ "clause": 1, "scope": 1, "term": "(duplicate ([]) T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "38": { "goal": [ { "clause": 7, "scope": 6, "term": "(',' (no (empty T12)) (',' (head T12 T15) (',' (tail T12 X21) (duplicate X21 T16))))" }, { "clause": 8, "scope": 6, "term": "(',' (no (empty T12)) (',' (head T12 T15) (',' (tail T12 X21) (duplicate X21 T16))))" } ], "kb": { "nonunifying": [[ "(duplicate T12 T2)", "(duplicate ([]) ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T12"], "free": ["X21"], "exprvars": [] } }, "17": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (no (empty ([]))) (',' (head ([]) T7) (',' (tail ([]) X11) (duplicate X11 T8))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X11"], "exprvars": [] } }, "18": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1": { "goal": [{ "clause": -1, "scope": -1, "term": "(duplicate T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "2": { "goal": [ { "clause": 0, "scope": 1, "term": "(duplicate T1 T2)" }, { "clause": 1, "scope": 1, "term": "(duplicate T1 T2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "80": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "82": { "goal": [ { "clause": 4, "scope": 11, "term": "(',' (tail (. T28 T29) X21) (duplicate X21 T30))" }, { "clause": 5, "scope": 11, "term": "(',' (tail (. T28 T29) X21) (duplicate X21 T30))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T28", "T29" ], "free": ["X21"], "exprvars": [] } }, "84": { "goal": [{ "clause": 5, "scope": 11, "term": "(',' (tail (. T28 T29) X21) (duplicate X21 T30))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T28", "T29" ], "free": ["X21"], "exprvars": [] } }, "20": { "goal": [ { "clause": 7, "scope": 2, "term": "(',' (no (empty ([]))) (',' (head ([]) T7) (',' (tail ([]) X11) (duplicate X11 T8))))" }, { "clause": 8, "scope": 2, "term": "(',' (no (empty ([]))) (',' (head ([]) T7) (',' (tail ([]) X11) (duplicate X11 T8))))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X11"], "exprvars": [] } }, "42": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (',' (call (empty T19)) (',' (!_6) (failure (a)))) (',' (head T19 T15) (',' (tail T19 X21) (duplicate X21 T16))))" }, { "clause": 8, "scope": 6, "term": "(',' (no (empty T19)) (',' (head T19 T15) (',' (tail T19 X21) (duplicate X21 T16))))" } ], "kb": { "nonunifying": [[ "(duplicate T19 T2)", "(duplicate ([]) ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T19"], "free": ["X21"], "exprvars": [] } }, "43": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (empty T19) (',' (',' (!_6) (failure (a))) (',' (head T19 T15) (',' (tail T19 X21) (duplicate X21 T16)))))" }, { "clause": -1, "scope": 7, "term": null }, { "clause": 8, "scope": 6, "term": "(',' (no (empty T19)) (',' (head T19 T15) (',' (tail T19 X21) (duplicate X21 T16))))" } ], "kb": { "nonunifying": [[ "(duplicate T19 T2)", "(duplicate ([]) ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T19"], "free": ["X21"], "exprvars": [] } }, "87": { "goal": [{ "clause": -1, "scope": -1, "term": "(duplicate T36 T30)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T36"], "free": [], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 2, "label": "CASE" }, { "from": 2, "to": 11, "label": "EVAL with clause\nduplicate([], []).\nand substitutionT1 -> [],\nT2 -> []" }, { "from": 2, "to": 14, "label": "EVAL-BACKTRACK" }, { "from": 11, "to": 16, "label": "SUCCESS" }, { "from": 14, "to": 36, "label": "EVAL with clause\nduplicate(X18, .(X19, .(X19, X20))) :- ','(no(empty(X18)), ','(head(X18, X19), ','(tail(X18, X21), duplicate(X21, X20)))).\nand substitutionT1 -> T12,\nX18 -> T12,\nX19 -> T15,\nX20 -> T16,\nT2 -> .(T15, .(T15, T16)),\nT13 -> T15,\nT14 -> T16" }, { "from": 14, "to": 37, "label": "EVAL-BACKTRACK" }, { "from": 16, "to": 17, "label": "EVAL with clause\nduplicate(X8, .(X9, .(X9, X10))) :- ','(no(empty(X8)), ','(head(X8, X9), ','(tail(X8, X11), duplicate(X11, X10)))).\nand substitutionX8 -> [],\nX9 -> T7,\nX10 -> T8,\nT2 -> .(T7, .(T7, T8)),\nT5 -> T7,\nT6 -> T8" }, { "from": 16, "to": 18, "label": "EVAL-BACKTRACK" }, { "from": 17, "to": 20, "label": "CASE" }, { "from": 20, "to": 23, "label": "ONLY EVAL with clause\nno(X14) :- ','(call(X14), ','(!_2, failure(a))).\nand substitutionX14 -> empty([])" }, { "from": 23, "to": 24, "label": "CALL" }, { "from": 24, "to": 27, "label": "CASE" }, { "from": 27, "to": 29, "label": "ONLY EVAL with clause\nempty([]).\nand substitution" }, { "from": 29, "to": 31, "label": "CUT" }, { "from": 31, "to": 34, "label": "CASE" }, { "from": 34, "to": 35, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 36, "to": 38, "label": "CASE" }, { "from": 38, "to": 42, "label": "ONLY EVAL with clause\nno(X24) :- ','(call(X24), ','(!_6, failure(a))).\nand substitutionT12 -> T19,\nX24 -> empty(T19)" }, { "from": 42, "to": 43, "label": "CALL" }, { "from": 43, "to": 44, "label": "CASE" }, { "from": 44, "to": 46, "label": "EVAL with clause\nempty([]).\nand substitutionT19 -> []" }, { "from": 44, "to": 48, "label": "EVAL-BACKTRACK" }, { "from": 46, "to": 49, "label": "CUT" }, { "from": 48, "to": 53, "label": "FAILURE" }, { "from": 49, "to": 51, "label": "CASE" }, { "from": 51, "to": 52, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 53, "to": 56, "label": "FAILURE" }, { "from": 56, "to": 72, "label": "ONLY EVAL with clause\nno(X27).\nand substitutionT19 -> T22,\nX27 -> empty(T22)" }, { "from": 72, "to": 73, "label": "CASE" }, { "from": 73, "to": 76, "label": "BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T22), empty([]))" }, { "from": 76, "to": 79, "label": "EVAL with clause\nhead(.(X33, X34), X33).\nand substitutionX33 -> T28,\nX34 -> T29,\nT22 -> .(T28, T29),\nT15 -> T28,\nT16 -> T30" }, { "from": 76, "to": 80, "label": "EVAL-BACKTRACK" }, { "from": 79, "to": 82, "label": "CASE" }, { "from": 82, "to": 84, "label": "BACKTRACK\nfor clause: tail([], [])because of non-unification" }, { "from": 84, "to": 87, "label": "ONLY EVAL with clause\ntail(.(X39, X40), X40).\nand substitutionT28 -> T35,\nX39 -> T35,\nT29 -> T36,\nX40 -> T36,\nX21 -> T36" }, { "from": 87, "to": 1, "label": "INSTANCE with matching:\nT1 -> T36\nT2 -> T30" } ], "type": "Graph" } } ---------------------------------------- (2) Obligation: Triples: duplicateA(.(X1, X2), .(X1, .(X1, X3))) :- duplicateA(X2, X3). Clauses: duplicatecA([], []). duplicatecA(.(X1, X2), .(X1, .(X1, X3))) :- duplicatecA(X2, X3). Afs: duplicateA(x1, x2) = duplicateA(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: duplicateA_in_2: (b,f) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: DUPLICATEA_IN_GA(.(X1, X2), .(X1, .(X1, X3))) -> U1_GA(X1, X2, X3, duplicateA_in_ga(X2, X3)) DUPLICATEA_IN_GA(.(X1, X2), .(X1, .(X1, X3))) -> DUPLICATEA_IN_GA(X2, X3) R is empty. The argument filtering Pi contains the following mapping: duplicateA_in_ga(x1, x2) = duplicateA_in_ga(x1) .(x1, x2) = .(x1, x2) DUPLICATEA_IN_GA(x1, x2) = DUPLICATEA_IN_GA(x1) U1_GA(x1, x2, x3, x4) = U1_GA(x1, x2, x4) 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: DUPLICATEA_IN_GA(.(X1, X2), .(X1, .(X1, X3))) -> U1_GA(X1, X2, X3, duplicateA_in_ga(X2, X3)) DUPLICATEA_IN_GA(.(X1, X2), .(X1, .(X1, X3))) -> DUPLICATEA_IN_GA(X2, X3) R is empty. The argument filtering Pi contains the following mapping: duplicateA_in_ga(x1, x2) = duplicateA_in_ga(x1) .(x1, x2) = .(x1, x2) DUPLICATEA_IN_GA(x1, x2) = DUPLICATEA_IN_GA(x1) U1_GA(x1, x2, x3, x4) = U1_GA(x1, x2, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node. ---------------------------------------- (6) Obligation: Pi DP problem: The TRS P consists of the following rules: DUPLICATEA_IN_GA(.(X1, X2), .(X1, .(X1, X3))) -> DUPLICATEA_IN_GA(X2, X3) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) DUPLICATEA_IN_GA(x1, x2) = DUPLICATEA_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (7) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (8) Obligation: Q DP problem: The TRS P consists of the following rules: DUPLICATEA_IN_GA(.(X1, X2)) -> DUPLICATEA_IN_GA(X2) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (9) 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: *DUPLICATEA_IN_GA(.(X1, X2)) -> DUPLICATEA_IN_GA(X2) The graph contains the following edges 1 > 1 ---------------------------------------- (10) YES