/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.pl /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern gopher(g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToDTProblemTransformerProof [SOUND, 56 ms] (2) TRIPLES (3) TriplesToPiDPProof [SOUND, 0 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) TRUE ---------------------------------------- (0) Obligation: Clauses: gopher(nil, nil). gopher(X, cons(nil, T)) :- ','(no(empty(X)), ','(head(X, nil), tail(X, T))). gopher(X, Y) :- ','(no(empty(X)), ','(head(X, H), ','(no(empty(H)), ','(head(H, U), ','(tail(H, V), ','(tail(X, W), gopher(cons(U, cons(V, W)), Y))))))). head([], X1). head(.(X, X2), X). tail([], []). tail(.(X3, X), X). empty([]). no(X) :- ','(X, ','(!, failure(a))). no(X4). failure(b). Query: gopher(g,a) ---------------------------------------- (1) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 3, "program": { "directives": [], "clauses": [ [ "(gopher (nil) (nil))", null ], [ "(gopher X (cons (nil) T))", "(',' (no (empty X)) (',' (head X (nil)) (tail X T)))" ], [ "(gopher X Y)", "(',' (no (empty X)) (',' (head X H) (',' (no (empty H)) (',' (head H U) (',' (tail H V) (',' (tail X W) (gopher (cons U (cons V W)) Y)))))))" ], [ "(head ([]) X1)", null ], [ "(head (. X X2) X)", null ], [ "(tail ([]) ([]))", null ], [ "(tail (. X3 X) X)", null ], [ "(empty ([]))", null ], [ "(no X)", "(',' X (',' (!) (failure (a))))" ], [ "(no X4)", null ], [ "(failure (b))", null ] ] }, "graph": { "nodes": { "191": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (',' (call (empty T51)) (',' (!_24) (failure (a)))) (',' (head T51 X112) (',' (no (empty X112)) (',' (head X112 X113) (',' (tail X112 X114) (',' (tail T51 X115) (gopher (cons X113 (cons X114 X115)) T48)))))))" }, { "clause": 9, "scope": 24, "term": "(',' (no (empty T51)) (',' (head T51 X112) (',' (no (empty X112)) (',' (head X112 X113) (',' (tail X112 X114) (',' (tail T51 X115) (gopher (cons X113 (cons X114 X115)) T48)))))))" } ], "kb": { "nonunifying": [ [ "(gopher T51 T2)", "(gopher (nil) (nil))" ], [ "(empty T51)", "(empty ([]))" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T51"], "free": [ "X112", "X113", "X114", "X115" ], "exprvars": [] } }, "192": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (empty T51) (',' (',' (!_24) (failure (a))) (',' (head T51 X112) (',' (no (empty X112)) (',' (head X112 X113) (',' (tail X112 X114) (',' (tail T51 X115) (gopher (cons X113 (cons X114 X115)) T48))))))))" }, { "clause": -1, "scope": 25, "term": null }, { "clause": 9, "scope": 24, "term": "(',' (no (empty T51)) (',' (head T51 X112) (',' (no (empty X112)) (',' (head X112 X113) (',' (tail X112 X114) (',' (tail T51 X115) (gopher (cons X113 (cons X114 X115)) T48)))))))" } ], "kb": { "nonunifying": [ [ "(gopher T51 T2)", "(gopher (nil) (nil))" ], [ "(empty T51)", "(empty ([]))" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T51"], "free": [ "X112", "X113", "X114", "X115" ], "exprvars": [] } }, "193": { "goal": [ { "clause": 7, "scope": 26, "term": "(',' (empty T51) (',' (',' (!_24) (failure (a))) (',' (head T51 X112) (',' (no (empty X112)) (',' (head X112 X113) (',' (tail X112 X114) (',' (tail T51 X115) (gopher (cons X113 (cons X114 X115)) T48))))))))" }, { "clause": -1, "scope": 26, "term": null }, { "clause": -1, "scope": 25, "term": null }, { "clause": 9, "scope": 24, "term": "(',' (no (empty T51)) (',' (head T51 X112) (',' (no (empty X112)) (',' (head X112 X113) (',' (tail X112 X114) (',' (tail T51 X115) (gopher (cons X113 (cons X114 X115)) T48)))))))" } ], "kb": { "nonunifying": [ [ "(gopher T51 T2)", "(gopher (nil) (nil))" ], [ "(empty T51)", "(empty ([]))" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T51"], "free": [ "X112", "X113", "X114", "X115" ], "exprvars": [] } }, "type": "Nodes", "194": { "goal": [ { "clause": -1, "scope": 26, "term": null }, { "clause": -1, "scope": 25, "term": null }, { "clause": 9, "scope": 24, "term": "(',' (no (empty T51)) (',' (head T51 X112) (',' (no (empty X112)) (',' (head X112 X113) (',' (tail X112 X114) (',' (tail T51 X115) (gopher (cons X113 (cons X114 X115)) T48)))))))" } ], "kb": { "nonunifying": [ [ "(gopher T51 T2)", "(gopher (nil) (nil))" ], [ "(empty T51)", "(empty ([]))" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T51"], "free": [ "X112", "X113", "X114", "X115" ], "exprvars": [] } }, "195": { "goal": [ { "clause": -1, "scope": 25, "term": null }, { "clause": 9, "scope": 24, "term": "(',' (no (empty T51)) (',' (head T51 X112) (',' (no (empty X112)) (',' (head X112 X113) (',' (tail X112 X114) (',' (tail T51 X115) (gopher (cons X113 (cons X114 X115)) T48)))))))" } ], "kb": { "nonunifying": [ [ "(gopher T51 T2)", "(gopher (nil) (nil))" ], [ "(empty T51)", "(empty ([]))" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T51"], "free": [ "X112", "X113", "X114", "X115" ], "exprvars": [] } }, "196": { "goal": [{ "clause": 9, "scope": 24, "term": "(',' (no (empty T51)) (',' (head T51 X112) (',' (no (empty X112)) (',' (head X112 X113) (',' (tail X112 X114) (',' (tail T51 X115) (gopher (cons X113 (cons X114 X115)) T48)))))))" }], "kb": { "nonunifying": [ [ "(gopher T51 T2)", "(gopher (nil) (nil))" ], [ "(empty T51)", "(empty ([]))" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T51"], "free": [ "X112", "X113", "X114", "X115" ], "exprvars": [] } }, "197": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (head T54 X112) (',' (no (empty X112)) (',' (head X112 X113) (',' (tail X112 X114) (',' (tail T54 X115) (gopher (cons X113 (cons X114 X115)) T48))))))" }], "kb": { "nonunifying": [ [ "(gopher T54 T2)", "(gopher (nil) (nil))" ], [ "(empty T54)", "(empty ([]))" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T54"], "free": [ "X112", "X113", "X114", "X115" ], "exprvars": [] } }, "110": { "goal": [{ "clause": 2, "scope": 1, "term": "(gopher ([]) T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "198": { "goal": [ { "clause": 3, "scope": 27, "term": "(',' (head T54 X112) (',' (no (empty X112)) (',' (head X112 X113) (',' (tail X112 X114) (',' (tail T54 X115) (gopher (cons X113 (cons X114 X115)) T48))))))" }, { "clause": 4, "scope": 27, "term": "(',' (head T54 X112) (',' (no (empty X112)) (',' (head X112 X113) (',' (tail X112 X114) (',' (tail T54 X115) (gopher (cons X113 (cons X114 X115)) T48))))))" } ], "kb": { "nonunifying": [ [ "(gopher T54 T2)", "(gopher (nil) (nil))" ], [ "(empty T54)", "(empty ([]))" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T54"], "free": [ "X112", "X113", "X114", "X115" ], "exprvars": [] } }, "231": { "goal": [ { "clause": 8, "scope": 35, "term": "(',' (no (empty T90)) (',' (head T90 X160) (',' (no (empty X160)) (',' (head X160 X161) (',' (tail X160 X162) (',' (tail T90 X163) (gopher (cons X161 (cons X162 X163)) T92)))))))" }, { "clause": 9, "scope": 35, "term": "(',' (no (empty T90)) (',' (head T90 X160) (',' (no (empty X160)) (',' (head X160 X161) (',' (tail X160 X162) (',' (tail T90 X163) (gopher (cons X161 (cons X162 X163)) T92)))))))" } ], "kb": { "nonunifying": [ [ "(gopher T90 T2)", "(gopher (nil) (nil))" ], [ "(gopher T90 T2)", "(gopher X62 (cons (nil) X63))" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T90"], "free": [ "X62", "X63", "X160", "X161", "X162", "X163" ], "exprvars": [] } }, "111": { "goal": [{ "clause": 10, "scope": 17, "term": "(',' (failure (a)) (',' (head ([]) (nil)) (tail ([]) T18)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "199": { "goal": [{ "clause": 4, "scope": 27, "term": "(',' (head T54 X112) (',' (no (empty X112)) (',' (head X112 X113) (',' (tail X112 X114) (',' (tail T54 X115) (gopher (cons X113 (cons X114 X115)) T48))))))" }], "kb": { "nonunifying": [ [ "(gopher T54 T2)", "(gopher (nil) (nil))" ], [ "(empty T54)", "(empty ([]))" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T54"], "free": [ "X112", "X113", "X114", "X115" ], "exprvars": [] } }, "112": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "115": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (no (empty ([]))) (',' (head ([]) X77) (',' (no (empty X77)) (',' (head X77 X78) (',' (tail X77 X79) (',' (tail ([]) X80) (gopher (cons X78 (cons X79 X80)) T26)))))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X77", "X78", "X79", "X80" ], "exprvars": [] } }, "90": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (no (empty T16)) (',' (head T16 (nil)) (tail T16 T18)))" }, { "clause": 2, "scope": 1, "term": "(gopher T16 T2)" } ], "kb": { "nonunifying": [[ "(gopher T16 T2)", "(gopher (nil) (nil))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T16"], "free": [], "exprvars": [] } }, "117": { "goal": [ { "clause": 8, "scope": 18, "term": "(',' (no (empty ([]))) (',' (head ([]) X77) (',' (no (empty X77)) (',' (head X77 X78) (',' (tail X77 X79) (',' (tail ([]) X80) (gopher (cons X78 (cons X79 X80)) T26)))))))" }, { "clause": 9, "scope": 18, "term": "(',' (no (empty ([]))) (',' (head ([]) X77) (',' (no (empty X77)) (',' (head X77 X78) (',' (tail X77 X79) (',' (tail ([]) X80) (gopher (cons X78 (cons X79 X80)) T26)))))))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X77", "X78", "X79", "X80" ], "exprvars": [] } }, "94": { "goal": [{ "clause": 2, "scope": 1, "term": "(gopher T1 T2)" }], "kb": { "nonunifying": [ [ "(gopher T1 T2)", "(gopher (nil) (nil))" ], [ "(gopher T1 T2)", "(gopher X62 (cons (nil) X63))" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [ "X62", "X63" ], "exprvars": [] } }, "95": { "goal": [ { "clause": 8, "scope": 14, "term": "(',' (no (empty T16)) (',' (head T16 (nil)) (tail T16 T18)))" }, { "clause": 9, "scope": 14, "term": "(',' (no (empty T16)) (',' (head T16 (nil)) (tail T16 T18)))" }, { "clause": -1, "scope": 14, "term": null }, { "clause": 2, "scope": 1, "term": "(gopher T16 T2)" } ], "kb": { "nonunifying": [[ "(gopher T16 T2)", "(gopher (nil) (nil))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T16"], "free": [], "exprvars": [] } }, "17": { "goal": [ { "clause": -1, "scope": -1, "term": "(true)" }, { "clause": 1, "scope": 1, "term": "(gopher (nil) T2)" }, { "clause": 2, "scope": 1, "term": "(gopher (nil) T2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "18": { "goal": [ { "clause": 1, "scope": 1, "term": "(gopher T1 T2)" }, { "clause": 2, "scope": 1, "term": "(gopher T1 T2)" } ], "kb": { "nonunifying": [[ "(gopher T1 T2)", "(gopher (nil) (nil))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "19": { "goal": [ { "clause": 1, "scope": 1, "term": "(gopher (nil) T2)" }, { "clause": 2, "scope": 1, "term": "(gopher (nil) T2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "120": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (',' (call (empty ([]))) (',' (!_18) (failure (a)))) (',' (head ([]) X77) (',' (no (empty X77)) (',' (head X77 X78) (',' (tail X77 X79) (',' (tail ([]) X80) (gopher (cons X78 (cons X79 X80)) T26)))))))" }, { 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(nil) X47) (',' (no (empty X47)) (',' (head X47 X48) (',' (tail X47 X49) (',' (tail (nil) X50) (gopher (cons X48 (cons X49 X50)) T13))))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X47", "X48", "X49", "X50" ], "exprvars": [] } }, "86": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 3, "to": 5, "label": "CASE" }, { "from": 5, "to": 17, "label": "EVAL with clause\ngopher(nil, nil).\nand substitutionT1 -> nil,\nT2 -> nil" }, { "from": 5, "to": 18, "label": "EVAL-BACKTRACK" }, { "from": 17, "to": 19, "label": "SUCCESS" }, { "from": 18, "to": 90, "label": "EVAL with clause\ngopher(X62, cons(nil, X63)) :- ','(no(empty(X62)), ','(head(X62, nil), tail(X62, X63))).\nand substitutionT1 -> T16,\nX62 -> T16,\nX63 -> T18,\nT2 -> cons(nil, T18),\nT17 -> T18" }, { "from": 18, "to": 94, "label": "EVAL-BACKTRACK" }, { "from": 19, "to": 22, "label": "EVAL with clause\ngopher(X7, cons(nil, X8)) :- ','(no(empty(X7)), ','(head(X7, nil), tail(X7, X8))).\nand substitutionX7 -> nil,\nX8 -> T5,\nT2 -> cons(nil, T5),\nT4 -> T5" }, { "from": 19, "to": 23, "label": "EVAL-BACKTRACK" }, { "from": 22, "to": 24, "label": "CASE" }, { "from": 23, "to": 70, "label": "ONLY EVAL with clause\ngopher(X45, X46) :- ','(no(empty(X45)), ','(head(X45, X47), ','(no(empty(X47)), ','(head(X47, X48), ','(tail(X47, X49), ','(tail(X45, X50), gopher(cons(X48, cons(X49, X50)), X46))))))).\nand substitutionX45 -> nil,\nT2 -> T13,\nX46 -> T13,\nT12 -> T13" }, { "from": 24, "to": 28, "label": "ONLY EVAL with clause\nno(X11) :- ','(call(X11), ','(!_2, failure(a))).\nand substitutionX11 -> empty(nil)" }, { "from": 28, "to": 30, "label": "CALL" }, { "from": 30, "to": 32, "label": "CASE" }, { "from": 32, "to": 33, "label": "BACKTRACK\nfor clause: empty([])because of non-unification" }, { "from": 33, "to": 34, "label": "FAILURE" }, { "from": 34, "to": 35, "label": "FAILURE" }, { "from": 35, "to": 36, "label": "PARALLEL" }, { "from": 35, "to": 37, "label": "PARALLEL" }, { "from": 36, "to": 38, "label": "ONLY EVAL with clause\nno(X16).\nand substitutionX16 -> empty(nil)" }, { "from": 37, "to": 42, "label": "FAILURE" }, { "from": 38, "to": 39, "label": "CASE" }, { "from": 39, "to": 40, "label": "BACKTRACK\nfor clause: head([], X1)because of non-unification" }, { "from": 40, "to": 41, "label": "BACKTRACK\nfor clause: head(.(X, X2), X)because of non-unification" }, { "from": 42, "to": 43, "label": "ONLY EVAL with clause\ngopher(X28, X29) :- ','(no(empty(X28)), ','(head(X28, X30), ','(no(empty(X30)), ','(head(X30, X31), ','(tail(X30, X32), ','(tail(X28, X33), gopher(cons(X31, cons(X32, X33)), X29))))))).\nand substitutionX28 -> nil,\nT2 -> T10,\nX29 -> T10,\nT9 -> T10" }, { "from": 43, "to": 44, "label": "CASE" }, { "from": 44, "to": 47, "label": "ONLY EVAL with clause\nno(X36) :- ','(call(X36), ','(!_6, failure(a))).\nand substitutionX36 -> empty(nil)" }, { "from": 47, "to": 48, "label": "CALL" }, { "from": 48, "to": 49, "label": "CASE" }, { "from": 49, "to": 50, "label": "BACKTRACK\nfor clause: empty([])because of non-unification" }, { "from": 50, "to": 52, "label": "FAILURE" }, { "from": 52, "to": 54, "label": "FAILURE" }, { "from": 54, "to": 56, "label": "ONLY EVAL with clause\nno(X39).\nand substitutionX39 -> empty(nil)" }, { "from": 56, "to": 58, "label": "CASE" }, { "from": 58, "to": 60, "label": "BACKTRACK\nfor clause: head([], X1)because of non-unification" }, { "from": 60, "to": 63, "label": "BACKTRACK\nfor clause: head(.(X, X2), X)because of non-unification" }, { "from": 70, "to": 71, "label": "CASE" }, { "from": 71, "to": 73, "label": "ONLY EVAL with clause\nno(X53) :- ','(call(X53), ','(!_10, failure(a))).\nand substitutionX53 -> empty(nil)" }, { "from": 73, "to": 74, "label": "CALL" }, { "from": 74, "to": 75, "label": "CASE" }, { "from": 75, "to": 77, "label": "BACKTRACK\nfor clause: empty([])because of non-unification" }, { "from": 77, "to": 79, "label": "FAILURE" }, { "from": 79, "to": 82, "label": "FAILURE" }, { "from": 82, "to": 83, "label": "ONLY EVAL with clause\nno(X56).\nand substitutionX56 -> empty(nil)" }, { "from": 83, "to": 84, "label": "CASE" }, { "from": 84, "to": 85, "label": "BACKTRACK\nfor clause: head([], X1)because of non-unification" }, { "from": 85, "to": 86, "label": "BACKTRACK\nfor clause: head(.(X, X2), X)because of non-unification" }, { "from": 90, "to": 95, "label": "CASE" }, { "from": 94, "to": 228, "label": "ONLY EVAL with clause\ngopher(X158, X159) :- ','(no(empty(X158)), ','(head(X158, X160), ','(no(empty(X160)), ','(head(X160, X161), ','(tail(X160, X162), ','(tail(X158, X163), gopher(cons(X161, cons(X162, X163)), X159))))))).\nand substitutionT1 -> T90,\nX158 -> T90,\nT2 -> T92,\nX159 -> T92,\nT91 -> T92" }, { "from": 95, "to": 100, "label": "ONLY EVAL with clause\nno(X66) :- ','(call(X66), ','(!_14, failure(a))).\nand substitutionT16 -> T21,\nX66 -> empty(T21)" }, { "from": 100, "to": 101, "label": "CALL" }, { "from": 101, "to": 103, "label": "CASE" }, { "from": 103, "to": 106, "label": "EVAL with clause\nempty([]).\nand substitutionT21 -> []" }, { "from": 103, "to": 107, "label": "EVAL-BACKTRACK" }, { "from": 106, "to": 108, "label": "CUT" }, { "from": 107, "to": 134, "label": "FAILURE" }, { "from": 108, "to": 109, "label": "PARALLEL" }, { "from": 108, "to": 110, "label": "PARALLEL" }, { "from": 109, "to": 111, "label": "CASE" }, { "from": 110, "to": 115, "label": "ONLY EVAL with clause\ngopher(X75, X76) :- ','(no(empty(X75)), ','(head(X75, X77), ','(no(empty(X77)), ','(head(X77, X78), ','(tail(X77, X79), ','(tail(X75, X80), gopher(cons(X78, cons(X79, X80)), X76))))))).\nand substitutionX75 -> [],\nT2 -> T26,\nX76 -> T26,\nT25 -> T26" }, { "from": 111, "to": 112, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 115, "to": 117, "label": "CASE" }, { "from": 117, "to": 120, "label": "ONLY EVAL with clause\nno(X83) :- ','(call(X83), ','(!_18, failure(a))).\nand substitutionX83 -> empty([])" }, { "from": 120, "to": 122, "label": "CALL" }, { "from": 122, "to": 125, "label": "CASE" }, { "from": 125, "to": 128, "label": "ONLY EVAL with clause\nempty([]).\nand substitution" }, { "from": 128, "to": 130, "label": "CUT" }, { "from": 130, "to": 132, "label": "CASE" }, { "from": 132, "to": 133, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 134, "to": 135, "label": "FAILURE" }, { "from": 135, "to": 138, "label": "PARALLEL" }, { "from": 135, "to": 139, "label": "PARALLEL" }, { "from": 138, "to": 141, "label": "ONLY EVAL with clause\nno(X88).\nand substitutionT21 -> T31,\nX88 -> empty(T31)" }, { "from": 139, "to": 165, "label": "FAILURE" }, { "from": 141, "to": 143, "label": "CASE" }, { "from": 143, "to": 145, "label": "BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T31), empty([]))" }, { "from": 145, "to": 147, "label": "EVAL with clause\nhead(.(X94, X95), X94).\nand substitutionX94 -> nil,\nX95 -> T37,\nT31 -> .(nil, T37),\nT36 -> nil" }, { "from": 145, "to": 149, "label": "EVAL-BACKTRACK" }, { "from": 147, "to": 153, "label": "CASE" }, { "from": 153, "to": 154, "label": "BACKTRACK\nfor clause: tail([], [])because of non-unification" }, { "from": 154, "to": 162, "label": "EVAL with clause\ntail(.(X100, X101), X101).\nand substitutionX100 -> nil,\nT37 -> T40,\nX101 -> T40,\nT18 -> T40" }, { "from": 154, "to": 163, "label": "EVAL-BACKTRACK" }, { "from": 162, "to": 164, "label": "SUCCESS" }, { "from": 165, "to": 173, "label": "ONLY EVAL with clause\ngopher(X110, X111) :- ','(no(empty(X110)), ','(head(X110, X112), ','(no(empty(X112)), ','(head(X112, X113), ','(tail(X112, X114), ','(tail(X110, X115), gopher(cons(X113, cons(X114, X115)), X111))))))).\nand substitutionT21 -> T46,\nX110 -> T46,\nT2 -> T48,\nX111 -> T48,\nT47 -> T48" }, { "from": 173, "to": 183, "label": "CASE" }, { "from": 183, "to": 191, "label": "ONLY EVAL with clause\nno(X118) :- ','(call(X118), ','(!_24, failure(a))).\nand substitutionT46 -> T51,\nX118 -> empty(T51)" }, { "from": 191, "to": 192, "label": "CALL" }, { "from": 192, "to": 193, "label": "CASE" }, { "from": 193, "to": 194, "label": "BACKTRACK\nfor clause: empty([])\nwith clash: (empty(T51), empty([]))" }, { "from": 194, "to": 195, "label": "FAILURE" }, { "from": 195, "to": 196, "label": "FAILURE" }, { "from": 196, "to": 197, "label": "ONLY EVAL with clause\nno(X121).\nand substitutionT51 -> T54,\nX121 -> empty(T54)" }, { "from": 197, "to": 198, "label": "CASE" }, { "from": 198, "to": 199, "label": "BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T54), empty([]))" }, { "from": 199, "to": 200, "label": "EVAL with clause\nhead(.(X128, X129), X128).\nand substitutionX128 -> T59,\nX129 -> T60,\nT54 -> .(T59, T60),\nX112 -> T59" }, { "from": 199, "to": 201, "label": "EVAL-BACKTRACK" }, { "from": 200, "to": 202, "label": "CASE" }, { "from": 202, "to": 203, "label": "ONLY EVAL with clause\nno(X132) :- ','(call(X132), ','(!_28, failure(a))).\nand substitutionT59 -> T63,\nX132 -> empty(T63)" }, { "from": 203, "to": 204, "label": "CALL" }, { "from": 204, "to": 205, "label": "CASE" }, { "from": 205, "to": 206, "label": "EVAL with clause\nempty([]).\nand substitutionT63 -> []" }, { "from": 205, "to": 207, "label": "EVAL-BACKTRACK" }, { "from": 206, "to": 208, "label": "CUT" }, { "from": 207, "to": 211, "label": "FAILURE" }, { "from": 208, "to": 209, "label": "CASE" }, { "from": 209, "to": 210, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 211, "to": 212, "label": "FAILURE" }, { "from": 212, "to": 213, "label": "ONLY EVAL with clause\nno(X135).\nand substitutionT63 -> T66,\nX135 -> empty(T66)" }, { "from": 213, "to": 214, "label": "CASE" }, { "from": 214, "to": 215, "label": "BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T66), empty([]))" }, { "from": 215, "to": 216, "label": "EVAL with clause\nhead(.(X142, X143), X142).\nand substitutionX142 -> T71,\nX143 -> T72,\nT66 -> .(T71, T72),\nX113 -> T71" }, { "from": 215, "to": 217, "label": "EVAL-BACKTRACK" }, { "from": 216, "to": 218, "label": "CASE" }, { "from": 218, "to": 219, "label": "BACKTRACK\nfor clause: tail([], [])because of non-unification" }, { "from": 219, "to": 220, "label": "ONLY EVAL with clause\ntail(.(X148, X149), X149).\nand substitutionT71 -> T77,\nX148 -> T77,\nT72 -> T78,\nX149 -> T78,\nX114 -> T78" }, { "from": 220, "to": 221, "label": "CASE" }, { "from": 221, "to": 222, "label": "BACKTRACK\nfor clause: tail([], [])because of non-unification" }, { "from": 222, "to": 223, "label": "ONLY EVAL with clause\ntail(.(X154, X155), X155).\nand substitutionT77 -> T85,\nT78 -> T86,\nX154 -> .(T85, T86),\nT60 -> T87,\nX155 -> T87,\nX115 -> T87" }, { "from": 223, "to": 3, "label": "INSTANCE with matching:\nT1 -> cons(T85, cons(T86, T87))\nT2 -> T48" }, { "from": 228, "to": 231, "label": "CASE" }, { "from": 231, "to": 262, "label": "ONLY EVAL with clause\nno(X166) :- ','(call(X166), ','(!_35, failure(a))).\nand substitutionT90 -> T95,\nX166 -> empty(T95)" }, { "from": 262, "to": 263, "label": "CALL" }, { "from": 263, "to": 264, "label": "CASE" }, { "from": 264, "to": 265, "label": "EVAL with clause\nempty([]).\nand substitutionT95 -> []" }, { "from": 264, "to": 266, "label": "EVAL-BACKTRACK" }, { "from": 265, "to": 267, "label": "CUT" }, { "from": 266, "to": 270, "label": "FAILURE" }, { "from": 267, "to": 268, "label": "CASE" }, { "from": 268, "to": 269, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 270, "to": 271, "label": "FAILURE" }, { "from": 271, "to": 272, "label": "ONLY EVAL with clause\nno(X169).\nand substitutionT95 -> T98,\nX169 -> empty(T98)" }, { "from": 272, "to": 273, "label": "CASE" }, { "from": 273, "to": 274, "label": "BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T98), empty([]))" }, { "from": 274, "to": 275, "label": "EVAL with clause\nhead(.(X176, X177), X176).\nand substitutionX176 -> T103,\nX177 -> T104,\nT98 -> .(T103, T104),\nX160 -> T103" }, { "from": 274, "to": 276, "label": "EVAL-BACKTRACK" }, { "from": 275, "to": 277, "label": "CASE" }, { "from": 277, "to": 278, "label": "ONLY EVAL with clause\nno(X180) :- ','(call(X180), ','(!_40, failure(a))).\nand substitutionT103 -> T107,\nX180 -> empty(T107)" }, { "from": 278, "to": 279, "label": "CALL" }, { "from": 279, "to": 280, "label": "CASE" }, { "from": 280, "to": 281, "label": "EVAL with clause\nempty([]).\nand substitutionT107 -> []" }, { "from": 280, "to": 282, "label": "EVAL-BACKTRACK" }, { "from": 281, "to": 283, "label": "CUT" }, { "from": 282, "to": 286, "label": "FAILURE" }, { "from": 283, "to": 284, "label": "CASE" }, { "from": 284, "to": 285, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 286, "to": 287, "label": "FAILURE" }, { "from": 287, "to": 288, "label": "ONLY EVAL with clause\nno(X183).\nand substitutionT107 -> T110,\nX183 -> empty(T110)" }, { "from": 288, "to": 289, "label": "CASE" }, { "from": 289, "to": 290, "label": "BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T110), empty([]))" }, { "from": 290, "to": 291, "label": "EVAL with clause\nhead(.(X190, X191), X190).\nand substitutionX190 -> T115,\nX191 -> T116,\nT110 -> .(T115, T116),\nX161 -> T115" }, { "from": 290, "to": 292, "label": "EVAL-BACKTRACK" }, { "from": 291, "to": 293, "label": "CASE" }, { "from": 293, "to": 294, "label": "BACKTRACK\nfor clause: tail([], [])because of non-unification" }, { "from": 294, "to": 295, "label": "ONLY EVAL with clause\ntail(.(X196, X197), X197).\nand substitutionT115 -> T121,\nX196 -> T121,\nT116 -> T122,\nX197 -> T122,\nX162 -> T122" }, { "from": 295, "to": 296, "label": "CASE" }, { "from": 296, "to": 297, "label": "BACKTRACK\nfor clause: tail([], [])because of non-unification" }, { "from": 297, "to": 298, "label": "ONLY EVAL with clause\ntail(.(X202, X203), X203).\nand substitutionT121 -> T129,\nT122 -> T130,\nX202 -> .(T129, T130),\nT104 -> T131,\nX203 -> T131,\nX163 -> T131" }, { "from": 298, "to": 3, "label": "INSTANCE with matching:\nT1 -> cons(T129, cons(T130, T131))\nT2 -> T92" } ], "type": "Graph" } } ---------------------------------------- (2) Obligation: Triples: gopherA(.(.(X1, X2), X3), X4) :- gopherA(cons(X1, cons(X2, X3)), X4). gopherA(.(.(X1, X2), X3), X4) :- gopherA(cons(X1, cons(X2, X3)), X4). Clauses: gophercA(nil, nil). gophercA(.(nil, X1), cons(nil, X1)). gophercA(.(.(X1, X2), X3), X4) :- gophercA(cons(X1, cons(X2, X3)), X4). gophercA(.(.(X1, X2), X3), X4) :- gophercA(cons(X1, cons(X2, X3)), X4). Afs: gopherA(x1, x2) = gopherA(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: gopherA_in_2: (b,f) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: GOPHERA_IN_GA(.(.(X1, X2), X3), X4) -> U1_GA(X1, X2, X3, X4, gopherA_in_ga(cons(X1, cons(X2, X3)), X4)) GOPHERA_IN_GA(.(.(X1, X2), X3), X4) -> GOPHERA_IN_GA(cons(X1, cons(X2, X3)), X4) R is empty. The argument filtering Pi contains the following mapping: gopherA_in_ga(x1, x2) = gopherA_in_ga(x1) .(x1, x2) = .(x1, x2) cons(x1, x2) = cons(x1, x2) GOPHERA_IN_GA(x1, x2) = GOPHERA_IN_GA(x1) U1_GA(x1, x2, x3, x4, x5) = U1_GA(x1, x2, x3, 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: GOPHERA_IN_GA(.(.(X1, X2), X3), X4) -> U1_GA(X1, X2, X3, X4, gopherA_in_ga(cons(X1, cons(X2, X3)), X4)) GOPHERA_IN_GA(.(.(X1, X2), X3), X4) -> GOPHERA_IN_GA(cons(X1, cons(X2, X3)), X4) R is empty. The argument filtering Pi contains the following mapping: gopherA_in_ga(x1, x2) = gopherA_in_ga(x1) .(x1, x2) = .(x1, x2) cons(x1, x2) = cons(x1, x2) GOPHERA_IN_GA(x1, x2) = GOPHERA_IN_GA(x1) U1_GA(x1, x2, x3, x4, x5) = U1_GA(x1, x2, x3, x5) 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