/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern app(g,a,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToDTProblemTransformerProof [SOUND, 0 ms] (2) TRIPLES (3) TriplesToPiDPProof [SOUND, 9 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: app([], Y, Y). app(X, Y, .(H, Z)) :- ','(no(empty(X)), ','(head(X, H), ','(tail(X, T), app(T, Y, Z)))). head([], X1). head(.(X, X2), X). tail([], []). tail(.(X3, Xs), Xs). empty([]). no(X) :- ','(X, ','(!, failure(a))). no(X4). failure(b). Query: app(g,a,a) ---------------------------------------- (1) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 3, "program": { "directives": [], "clauses": [ [ "(app ([]) Y Y)", null ], [ "(app X Y (. H Z))", "(',' (no (empty X)) (',' (head X H) (',' (tail X T) (app T Y Z))))" ], [ "(head ([]) X1)", null ], [ "(head (. X X2) X)", null ], [ "(tail ([]) ([]))", null ], [ "(tail (. X3 Xs) Xs)", null ], [ "(empty ([]))", null ], [ "(no X)", "(',' X (',' (!) (failure (a))))" ], [ "(no X4)", null ], [ "(failure (b))", null ] ] }, "graph": { "nodes": { "45": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "67": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (head T37 T25) (',' (tail T37 X29) (app X29 T26 T27)))" }], "kb": { "nonunifying": [ [ "(app T37 T2 T3)", "(app ([]) X6 X6)" ], [ "(empty T37)", "(empty ([]))" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T37"], "free": [ "X6", "X29" ], "exprvars": [] } }, "24": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (',' (call (empty ([]))) (',' (!_2) (failure (a)))) (',' (head ([]) T12) (',' (tail ([]) X15) (app X15 T13 T14))))" }, { "clause": 8, "scope": 2, "term": "(',' (no (empty ([]))) (',' (head ([]) T12) (',' (tail ([]) X15) (app X15 T13 T14))))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X15"], "exprvars": [] } }, "48": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (no (empty T21)) (',' (head T21 T25) (',' (tail T21 X29) (app X29 T26 T27))))" }], "kb": { "nonunifying": [[ "(app T21 T2 T3)", "(app ([]) X6 X6)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T21"], "free": [ "X6", "X29" ], "exprvars": [] } }, "49": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "28": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (empty ([])) (',' (',' (!_2) (failure (a))) (',' (head ([]) T12) (',' (tail ([]) X15) (app X15 T13 T14)))))" }, { "clause": -1, "scope": 3, "term": null }, { "clause": 8, "scope": 2, "term": "(',' (no (empty ([]))) (',' (head ([]) T12) (',' (tail ([]) X15) (app X15 T13 T14))))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X15"], "exprvars": [] } }, "type": "Nodes", "91": { "goal": [{ "clause": -1, "scope": -1, "term": "(app T58 T47 T48)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T58"], "free": [], "exprvars": [] } }, "71": { "goal": [ { "clause": 2, "scope": 10, "term": "(',' (head T37 T25) (',' (tail T37 X29) (app X29 T26 T27)))" }, { "clause": 3, "scope": 10, "term": "(',' (head T37 T25) (',' (tail T37 X29) (app X29 T26 T27)))" } ], "kb": { "nonunifying": [ [ "(app T37 T2 T3)", "(app ([]) X6 X6)" ], [ "(empty T37)", "(empty ([]))" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T37"], "free": [ "X6", "X29" ], "exprvars": [] } }, "73": { "goal": [{ "clause": 3, "scope": 10, "term": "(',' (head T37 T25) (',' 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T45 T46) X29) (app X29 T47 T48))" }, { "clause": 5, "scope": 11, "term": "(',' (tail (. T45 T46) X29) (app X29 T47 T48))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T45", "T46" ], "free": ["X29"], "exprvars": [] } }, "61": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "83": { "goal": [{ "clause": 5, "scope": 11, "term": "(',' (tail (. T45 T46) X29) (app X29 T47 T48))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T45", "T46" ], "free": ["X29"], "exprvars": [] } }, "62": { "goal": [ { "clause": -1, "scope": 7, "term": null }, { "clause": 8, "scope": 6, "term": "(',' (no (empty T30)) (',' (head T30 T25) (',' (tail T30 X29) (app X29 T26 T27))))" } ], "kb": { "nonunifying": [ [ "(app T30 T2 T3)", "(app ([]) X6 X6)" ], [ "(empty T30)", "(empty ([]))" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T30"], "free": [ "X6", "X29" ], "exprvars": [] } }, "41": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (',' (!_2) (failure (a))) (',' (head ([]) T12) (',' (tail ([]) X15) (app X15 T13 T14))))" }, { "clause": -1, "scope": 4, "term": null }, { "clause": -1, "scope": 3, "term": null }, { "clause": 8, "scope": 2, "term": "(',' (no (empty ([]))) (',' (head ([]) T12) (',' (tail ([]) X15) (app X15 T13 T14))))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X15"], "exprvars": [] } }, "63": { "goal": [{ "clause": 8, "scope": 6, "term": "(',' (no (empty T30)) (',' (head T30 T25) (',' (tail T30 X29) (app X29 T26 T27))))" }], "kb": { "nonunifying": [ [ "(app T30 T2 T3)", "(app ([]) X6 X6)" ], [ "(empty T30)", "(empty ([]))" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T30"], "free": [ "X6", "X29" ], "exprvars": [] } }, "20": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "42": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (failure (a)) (',' (head ([]) T12) (',' (tail ([]) X15) (app X15 T13 T14))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X15"], "exprvars": [] } }, "21": { "goal": [ { "clause": 7, "scope": 2, "term": "(',' (no (empty ([]))) (',' (head ([]) T12) (',' (tail ([]) X15) (app X15 T13 T14))))" }, { "clause": 8, "scope": 2, "term": "(',' (no (empty ([]))) (',' (head ([]) T12) (',' (tail ([]) X15) (app X15 T13 T14))))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X15"], "exprvars": [] } }, "43": { "goal": [{ "clause": 9, "scope": 5, "term": "(',' (failure (a)) (',' (head ([]) T12) (',' (tail ([]) X15) (app X15 T13 T14))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X15"], "exprvars": [] } } }, "edges": [ { "from": 3, "to": 4, "label": "CASE" }, { "from": 4, "to": 16, "label": "EVAL with clause\napp([], X6, X6).\nand substitutionT1 -> [],\nT2 -> T5,\nX6 -> T5,\nT3 -> T5" }, { "from": 4, "to": 17, "label": "EVAL-BACKTRACK" }, { "from": 16, "to": 18, "label": "SUCCESS" }, { "from": 17, "to": 48, "label": "EVAL with clause\napp(X25, X26, .(X27, X28)) :- ','(no(empty(X25)), ','(head(X25, X27), ','(tail(X25, X29), app(X29, X26, X28)))).\nand substitutionT1 -> T21,\nX25 -> T21,\nT2 -> T26,\nX26 -> T26,\nX27 -> T25,\nX28 -> T27,\nT3 -> .(T25, T27),\nT23 -> T25,\nT22 -> T26,\nT24 -> T27" }, { "from": 17, "to": 49, "label": "EVAL-BACKTRACK" }, { "from": 18, "to": 19, "label": "EVAL with clause\napp(X11, X12, .(X13, X14)) :- ','(no(empty(X11)), ','(head(X11, X13), ','(tail(X11, X15), app(X15, X12, X14)))).\nand substitutionX11 -> [],\nT2 -> T13,\nX12 -> T13,\nX13 -> T12,\nX14 -> T14,\nT3 -> .(T12, T14),\nT10 -> T12,\nT9 -> T13,\nT11 -> T14" }, { "from": 18, "to": 20, "label": "EVAL-BACKTRACK" }, { "from": 19, "to": 21, "label": "CASE" }, { "from": 21, "to": 24, "label": "ONLY EVAL with clause\nno(X18) :- ','(call(X18), ','(!_2, failure(a))).\nand substitutionX18 -> empty([])" }, { "from": 24, "to": 28, "label": "CALL" }, { "from": 28, "to": 39, "label": "CASE" }, { "from": 39, "to": 41, "label": "ONLY EVAL with clause\nempty([]).\nand substitution" }, { "from": 41, "to": 42, "label": "CUT" }, { "from": 42, "to": 43, "label": "CASE" }, { "from": 43, "to": 45, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 48, "to": 52, "label": "CASE" }, { "from": 52, "to": 53, "label": "ONLY EVAL with clause\nno(X32) :- ','(call(X32), ','(!_6, failure(a))).\nand substitutionT21 -> T30,\nX32 -> empty(T30)" }, { "from": 53, "to": 55, "label": "CALL" }, { "from": 55, "to": 56, "label": "CASE" }, { "from": 56, "to": 57, "label": "EVAL with clause\nempty([]).\nand substitutionT30 -> []" }, { "from": 56, "to": 58, "label": "EVAL-BACKTRACK" }, { "from": 57, "to": 59, "label": "CUT" }, { "from": 58, "to": 62, "label": "FAILURE" }, { "from": 59, "to": 60, "label": "CASE" }, { "from": 60, "to": 61, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 62, "to": 63, "label": "FAILURE" }, { "from": 63, "to": 67, "label": "ONLY EVAL with clause\nno(X39).\nand substitutionT30 -> T37,\nX39 -> empty(T37)" }, { "from": 67, "to": 71, "label": "CASE" }, { "from": 71, "to": 73, "label": "BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T37), empty([]))" }, { "from": 73, "to": 79, "label": "EVAL with clause\nhead(.(X47, X48), X47).\nand substitutionX47 -> T45,\nX48 -> T46,\nT37 -> .(T45, T46),\nT25 -> T45,\nT26 -> T47,\nT27 -> T48" }, { "from": 73, "to": 81, "label": "EVAL-BACKTRACK" }, { "from": 79, "to": 82, "label": "CASE" }, { "from": 82, "to": 83, "label": "BACKTRACK\nfor clause: tail([], [])because of non-unification" }, { "from": 83, "to": 91, "label": "ONLY EVAL with clause\ntail(.(X57, X58), X58).\nand substitutionT45 -> T57,\nX57 -> T57,\nT46 -> T58,\nX58 -> T58,\nX29 -> T58" }, { "from": 91, "to": 3, "label": "INSTANCE with matching:\nT1 -> T58\nT2 -> T47\nT3 -> T48" } ], "type": "Graph" } } ---------------------------------------- (2) Obligation: Triples: appA(.(X1, X2), X3, .(X1, X4)) :- appA(X2, X3, X4). Clauses: appcA([], X1, X1). appcA(.(X1, X2), X3, .(X1, X4)) :- appcA(X2, X3, X4). Afs: appA(x1, x2, x3) = appA(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: appA_in_3: (b,f,f) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: APPA_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> U1_GAA(X1, X2, X3, X4, appA_in_gaa(X2, X3, X4)) APPA_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> APPA_IN_GAA(X2, X3, X4) R is empty. The argument filtering Pi contains the following mapping: appA_in_gaa(x1, x2, x3) = appA_in_gaa(x1) .(x1, x2) = .(x1, x2) APPA_IN_GAA(x1, x2, x3) = APPA_IN_GAA(x1) U1_GAA(x1, x2, x3, x4, x5) = U1_GAA(x1, x2, x5) 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: APPA_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> U1_GAA(X1, X2, X3, X4, appA_in_gaa(X2, X3, X4)) APPA_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> APPA_IN_GAA(X2, X3, X4) R is empty. The argument filtering Pi contains the following mapping: appA_in_gaa(x1, x2, x3) = appA_in_gaa(x1) .(x1, x2) = .(x1, x2) APPA_IN_GAA(x1, x2, x3) = APPA_IN_GAA(x1) U1_GAA(x1, x2, x3, x4, x5) = U1_GAA(x1, x2, x5) 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: APPA_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> APPA_IN_GAA(X2, X3, X4) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) APPA_IN_GAA(x1, x2, x3) = APPA_IN_GAA(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: APPA_IN_GAA(.(X1, X2)) -> APPA_IN_GAA(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: *APPA_IN_GAA(.(X1, X2)) -> APPA_IN_GAA(X2) The graph contains the following edges 1 > 1 ---------------------------------------- (10) YES