/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern goal(g) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1)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, 53 ms] (2) TRIPLES (3) UndefinedPredicateInTriplesTransformerProof [SOUND, 0 ms] (4) TRIPLES (5) TriplesToPiDPProof [SOUND, 0 ms] (6) PiDP (7) DependencyGraphProof [EQUIVALENT, 0 ms] (8) AND (9) PiDP (10) UsableRulesProof [EQUIVALENT, 0 ms] (11) PiDP (12) PiDPToQDPProof [SOUND, 0 ms] (13) QDP (14) QDPSizeChangeProof [EQUIVALENT, 0 ms] (15) YES (16) PiDP (17) UsableRulesProof [EQUIVALENT, 0 ms] (18) PiDP (19) PiDPToQDPProof [SOUND, 0 ms] (20) QDP (21) QDPSizeChangeProof [EQUIVALENT, 0 ms] (22) YES ---------------------------------------- (0) Obligation: Clauses: goal(X) :- ','(s2l(X, Xs), list(Xs)). list([]) :- !. list(X) :- ','(tail(X, T), list(T)). s2l(0, L) :- ','(!, eq(L, [])). s2l(X, .(X1, Xs)) :- ','(p(X, P), s2l(P, Xs)). tail([], []). tail(.(X2, Xs), Xs). p(0, 0). p(s(X), X). eq(X, X). Query: goal(g) ---------------------------------------- (1) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 1, "program": { "directives": [], "clauses": [ [ "(goal X)", "(',' (s2l X Xs) (list Xs))" ], [ "(list ([]))", "(!)" ], [ "(list X)", "(',' (tail X T) (list T))" ], [ "(s2l (0) L)", "(',' (!) (eq L ([])))" ], [ "(s2l X (. X1 Xs))", "(',' (p X P) (s2l P Xs))" ], [ "(tail ([]) ([]))", null ], [ "(tail (. X2 Xs) Xs)", null ], [ "(p (0) (0))", null ], [ "(p (s X) X)", null ], [ "(eq X X)", null ] ] }, "graph": { "nodes": { "390": { "goal": [{ "clause": 6, "scope": 10, "term": "(',' (tail (. X91 T21) X90) (list X90))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X91", "X90" ], "exprvars": [] } }, "391": { "goal": [{ "clause": -1, "scope": -1, "term": "(list T26)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "392": { "goal": [ { "clause": 1, "scope": 11, "term": "(list T26)" }, { "clause": 2, "scope": 11, "term": "(list T26)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "393": { "goal": [ { "clause": -1, "scope": -1, "term": "(!_11)" }, { "clause": 2, "scope": 11, "term": "(list T26)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "394": { "goal": [{ "clause": 2, "scope": 11, "term": "(list T26)" }], "kb": { "nonunifying": [[ "(list T26)", "(list ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "395": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "396": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "232": { "goal": [ { "clause": 7, "scope": 5, "term": "(',' (',' (p T6 X34) (s2l X34 X36)) (list (. X35 X36)))" }, { "clause": 8, "scope": 5, "term": "(',' (',' (p T6 X34) (s2l X34 X36)) (list (. X35 X36)))" } ], "kb": { "nonunifying": [[ "(s2l T6 X5)", "(s2l (0) X10)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T6"], "free": [ "X5", "X10", "X35", "X36", "X34" ], "exprvars": [] } }, "276": { "goal": [ { "clause": 3, "scope": 6, "term": "(s2l T9 X36)" }, { "clause": 4, "scope": 6, "term": "(s2l T9 X36)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T9"], "free": ["X36"], "exprvars": [] } }, "397": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (tail T31 X105) (list X105))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X105"], "exprvars": [] } }, "277": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (!_6) (eq X51 ([])))" }, { "clause": 4, "scope": 6, "term": "(s2l (0) X36)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X36", "X51" ], "exprvars": [] } }, "398": { "goal": [ { "clause": 5, "scope": 12, "term": "(',' (tail T31 X105) (list X105))" }, { "clause": 6, "scope": 12, "term": "(',' (tail T31 X105) (list X105))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X105"], "exprvars": [] } }, "278": { "goal": [{ "clause": 4, "scope": 6, "term": "(s2l T9 X36)" }], "kb": { "nonunifying": [[ "(s2l T9 X36)", "(s2l (0) X50)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T9"], "free": [ "X36", "X50" ], "exprvars": [] } }, "399": { "goal": [{ "clause": 5, "scope": 12, "term": "(',' (tail T31 X105) (list X105))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X105"], "exprvars": [] } }, "279": { "goal": [{ "clause": -1, "scope": -1, "term": "(eq X51 ([]))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X51"], "exprvars": [] } }, "239": { "goal": [{ "clause": 8, "scope": 5, "term": "(',' (',' (p T6 X34) (s2l X34 X36)) (list (. X35 X36)))" }], "kb": { "nonunifying": [[ "(s2l T6 X5)", "(s2l (0) X10)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T6"], "free": [ "X5", "X10", "X35", "X36", "X34" ], "exprvars": [] } }, "10": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (s2l T3 X5) (list X5))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": ["X5"], "exprvars": [] } }, "17": { "goal": [ { "clause": 3, "scope": 2, "term": "(',' (s2l T3 X5) (list X5))" }, { "clause": 4, "scope": 2, "term": "(',' (s2l T3 X5) (list X5))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": ["X5"], "exprvars": [] } }, "280": { "goal": [{ "clause": 9, "scope": 7, "term": "(eq X51 ([]))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X51"], "exprvars": [] } }, "281": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "282": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1": { "goal": [{ "clause": -1, "scope": -1, "term": "(goal T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "4": { "goal": [{ "clause": 0, "scope": 1, "term": "(goal T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "367": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (p T13 X74) (s2l X74 X76))" }], "kb": { "nonunifying": [[ "(s2l T13 X36)", "(s2l (0) X50)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T13"], "free": [ "X36", "X50", "X76", "X74" ], "exprvars": [] } }, "400": { "goal": [{ "clause": 6, "scope": 12, "term": "(',' (tail T31 X105) (list X105))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X105"], "exprvars": [] } }, "126": { "goal": [ { "clause": 1, "scope": 4, "term": "(list ([]))" }, { "clause": 2, "scope": 4, "term": "(list ([]))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "247": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (s2l T9 X36) (list (. X35 X36)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T9"], "free": [ "X35", "X36" ], "exprvars": [] } }, "368": { "goal": [ { "clause": 7, "scope": 8, "term": "(',' (p T13 X74) (s2l X74 X76))" }, { "clause": 8, "scope": 8, "term": "(',' (p T13 X74) (s2l X74 X76))" } ], "kb": { "nonunifying": [[ "(s2l T13 X36)", "(s2l (0) X50)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T13"], "free": [ "X36", "X50", "X76", "X74" ], "exprvars": [] } }, "401": { "goal": [{ "clause": -1, "scope": -1, "term": "(list ([]))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "248": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "369": { "goal": [{ "clause": 8, "scope": 8, "term": "(',' (p T13 X74) (s2l X74 X76))" }], "kb": { "nonunifying": [[ "(s2l T13 X36)", "(s2l (0) X50)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T13"], "free": [ "X36", "X50", "X76", "X74" ], "exprvars": [] } }, "402": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "403": { "goal": [{ "clause": -1, "scope": -1, "term": "(list T38)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "404": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "207": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (',' (p T6 X34) (s2l X34 X36)) (list (. X35 X36)))" }], "kb": { "nonunifying": [[ "(s2l T6 X5)", "(s2l (0) X10)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T6"], "free": [ "X5", "X10", "X35", "X36", "X34" ], "exprvars": [] } }, "28": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (',' (!_2) (eq X11 ([]))) (list X11))" }, { "clause": 4, "scope": 2, "term": "(',' (s2l (0) X5) (list X5))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X5", "X11" ], "exprvars": [] } }, "29": { "goal": [{ "clause": 4, "scope": 2, "term": "(',' (s2l T3 X5) (list X5))" }], "kb": { "nonunifying": [[ "(s2l T3 X5)", "(s2l (0) X10)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [ "X5", "X10" ], "exprvars": [] } }, "370": { "goal": [{ "clause": -1, "scope": -1, "term": "(s2l T16 X76)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T16"], "free": ["X76"], "exprvars": [] } }, "371": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "130": { "goal": [ { "clause": -1, "scope": -1, "term": "(!_4)" }, { "clause": 2, "scope": 4, "term": "(list ([]))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "372": { "goal": [ { "clause": 1, "scope": 9, "term": "(list (. X35 T10))" }, { "clause": 2, "scope": 9, "term": "(list (. X35 T10))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X35"], "exprvars": [] } }, "373": { "goal": [{ "clause": 2, "scope": 9, "term": "(list (. X35 T10))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X35"], "exprvars": [] } }, "134": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "139": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "30": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (eq X11 ([])) (list X11))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X11"], "exprvars": [] } }, "31": { "goal": [{ "clause": 9, "scope": 3, "term": "(',' (eq X11 ([])) (list X11))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X11"], "exprvars": [] } }, "32": { "goal": [{ "clause": -1, "scope": -1, "term": "(list ([]))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "265": { "goal": [{ "clause": -1, "scope": -1, "term": "(s2l T9 X36)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T9"], "free": ["X36"], "exprvars": [] } }, "267": { "goal": [{ "clause": -1, "scope": -1, "term": "(list (. X35 T10))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X35"], "exprvars": [] } }, "388": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (tail (. X91 T21) X90) (list X90))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X91", "X90" ], "exprvars": [] } }, "389": { "goal": [ { "clause": 5, "scope": 10, "term": "(',' (tail (. X91 T21) X90) (list X90))" }, { "clause": 6, "scope": 10, "term": "(',' (tail (. X91 T21) X90) (list X90))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X91", "X90" ], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 4, "label": "CASE" }, { "from": 4, "to": 10, "label": "ONLY EVAL with clause\ngoal(X4) :- ','(s2l(X4, X5), list(X5)).\nand substitutionT1 -> T3,\nX4 -> T3" }, { "from": 10, "to": 17, "label": "CASE" }, { "from": 17, "to": 28, "label": "EVAL with clause\ns2l(0, X10) :- ','(!_2, eq(X10, [])).\nand substitutionT3 -> 0,\nX5 -> X11,\nX10 -> X11" }, { "from": 17, "to": 29, "label": "EVAL-BACKTRACK" }, { "from": 28, "to": 30, "label": "CUT" }, { "from": 29, "to": 207, "label": "ONLY EVAL with clause\ns2l(X31, .(X32, X33)) :- ','(p(X31, X34), s2l(X34, X33)).\nand substitutionT3 -> T6,\nX31 -> T6,\nX32 -> X35,\nX33 -> X36,\nX5 -> .(X35, X36)" }, { "from": 30, "to": 31, "label": "CASE" }, { "from": 31, "to": 32, "label": "ONLY EVAL with clause\neq(X16, X16).\nand substitutionX11 -> [],\nX16 -> [],\nX17 -> []" }, { "from": 32, "to": 126, "label": "CASE" }, { "from": 126, "to": 130, "label": "ONLY EVAL with clause\nlist([]) :- !_4.\nand substitution" }, { "from": 130, "to": 134, "label": "CUT" }, { "from": 134, "to": 139, "label": "SUCCESS" }, { "from": 207, "to": 232, "label": "CASE" }, { "from": 232, "to": 239, "label": "BACKTRACK\nfor clause: p(0, 0)\nwith clash: (s2l(T6, X5), s2l(0, X10))" }, { "from": 239, "to": 247, "label": "EVAL with clause\np(s(X41), X41).\nand substitutionX41 -> T9,\nT6 -> s(T9),\nX34 -> T9" }, { "from": 239, "to": 248, "label": "EVAL-BACKTRACK" }, { "from": 247, "to": 265, "label": "SPLIT 1" }, { "from": 247, "to": 267, "label": "SPLIT 2\nnew knowledge:\nT9 is ground\nreplacements:X36 -> T10" }, { "from": 265, "to": 276, "label": "CASE" }, { "from": 267, "to": 372, "label": "CASE" }, { "from": 276, "to": 277, "label": "EVAL with clause\ns2l(0, X50) :- ','(!_6, eq(X50, [])).\nand substitutionT9 -> 0,\nX36 -> X51,\nX50 -> X51" }, { "from": 276, "to": 278, "label": "EVAL-BACKTRACK" }, { "from": 277, "to": 279, "label": "CUT" }, { "from": 278, "to": 367, "label": "ONLY EVAL with clause\ns2l(X71, .(X72, X73)) :- ','(p(X71, X74), s2l(X74, X73)).\nand substitutionT9 -> T13,\nX71 -> T13,\nX72 -> X75,\nX73 -> X76,\nX36 -> .(X75, X76)" }, { "from": 279, "to": 280, "label": "CASE" }, { "from": 280, "to": 281, "label": "ONLY EVAL with clause\neq(X56, X56).\nand substitutionX51 -> [],\nX56 -> [],\nX57 -> []" }, { "from": 281, "to": 282, "label": "SUCCESS" }, { "from": 367, "to": 368, "label": "CASE" }, { "from": 368, "to": 369, "label": "BACKTRACK\nfor clause: p(0, 0)\nwith clash: (s2l(T13, X36), s2l(0, X50))" }, { "from": 369, "to": 370, "label": "EVAL with clause\np(s(X81), X81).\nand substitutionX81 -> T16,\nT13 -> s(T16),\nX74 -> T16" }, { "from": 369, "to": 371, "label": "EVAL-BACKTRACK" }, { "from": 370, "to": 265, "label": "INSTANCE with matching:\nT9 -> T16\nX36 -> X76" }, { "from": 372, "to": 373, "label": "BACKTRACK\nfor clause: list([]) :- !because of non-unification" }, { "from": 373, "to": 388, "label": "ONLY EVAL with clause\nlist(X89) :- ','(tail(X89, X90), list(X90)).\nand substitutionX35 -> X91,\nT10 -> T21,\nX89 -> .(X91, T21),\nT20 -> T21" }, { "from": 388, "to": 389, "label": "CASE" }, { "from": 389, "to": 390, "label": "BACKTRACK\nfor clause: tail([], [])because of non-unification" }, { "from": 390, "to": 391, "label": "ONLY EVAL with clause\ntail(.(X98, X99), X99).\nand substitutionX91 -> X100,\nX98 -> X100,\nT21 -> T26,\nX99 -> T26,\nX90 -> T26,\nT25 -> T26" }, { "from": 391, "to": 392, "label": "CASE" }, { "from": 392, "to": 393, "label": "EVAL with clause\nlist([]) :- !_11.\nand substitutionT26 -> []" }, { "from": 392, "to": 394, "label": "EVAL-BACKTRACK" }, { "from": 393, "to": 395, "label": "CUT" }, { "from": 394, "to": 397, "label": "ONLY EVAL with clause\nlist(X104) :- ','(tail(X104, X105), list(X105)).\nand substitutionT26 -> T31,\nX104 -> T31,\nT30 -> T31" }, { "from": 395, "to": 396, "label": "SUCCESS" }, { "from": 397, "to": 398, "label": "CASE" }, { "from": 398, "to": 399, "label": "PARALLEL" }, { "from": 398, "to": 400, "label": "PARALLEL" }, { "from": 399, "to": 401, "label": "EVAL with clause\ntail([], []).\nand substitutionT31 -> [],\nX105 -> []" }, { "from": 399, "to": 402, "label": "EVAL-BACKTRACK" }, { "from": 400, "to": 403, "label": "EVAL with clause\ntail(.(X110, X111), X111).\nand substitutionX110 -> T36,\nX111 -> T38,\nT31 -> .(T36, T38),\nX105 -> T38,\nT37 -> T38" }, { "from": 400, "to": 404, "label": "EVAL-BACKTRACK" }, { "from": 401, "to": 32, "label": "INSTANCE" }, { "from": 403, "to": 391, "label": "INSTANCE with matching:\nT26 -> T38" } ], "type": "Graph" } } ---------------------------------------- (2) Obligation: Triples: s2lA(s(X1), .(X2, X3)) :- s2lA(X1, X3). listC([]) :- listB. listC(.(X1, X2)) :- listC(X2). goalD(0) :- listB. goalD(s(X1)) :- s2lA(X1, X2). goalD(s(X1)) :- ','(s2lcA(X1, X2), listC(X2)). Clauses: s2lcA(0, []). s2lcA(s(X1), .(X2, X3)) :- s2lcA(X1, X3). listcB. listcC([]). listcC([]) :- listcB. listcC(.(X1, X2)) :- listcC(X2). Afs: goalD(x1) = goalD(x1) ---------------------------------------- (3) UndefinedPredicateInTriplesTransformerProof (SOUND) Deleted triples and predicates having undefined goals [DT09]. ---------------------------------------- (4) Obligation: Triples: s2lA(s(X1), .(X2, X3)) :- s2lA(X1, X3). listC(.(X1, X2)) :- listC(X2). goalD(s(X1)) :- s2lA(X1, X2). goalD(s(X1)) :- ','(s2lcA(X1, X2), listC(X2)). Clauses: s2lcA(0, []). s2lcA(s(X1), .(X2, X3)) :- s2lcA(X1, X3). listcB. listcC([]). listcC([]) :- listcB. listcC(.(X1, X2)) :- listcC(X2). Afs: goalD(x1) = goalD(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: goalD_in_1: (b) s2lA_in_2: (b,f) s2lcA_in_2: (b,f) listC_in_1: (b) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: GOALD_IN_G(s(X1)) -> U3_G(X1, s2lA_in_ga(X1, X2)) GOALD_IN_G(s(X1)) -> S2LA_IN_GA(X1, X2) S2LA_IN_GA(s(X1), .(X2, X3)) -> U1_GA(X1, X2, X3, s2lA_in_ga(X1, X3)) S2LA_IN_GA(s(X1), .(X2, X3)) -> S2LA_IN_GA(X1, X3) GOALD_IN_G(s(X1)) -> U4_G(X1, s2lcA_in_ga(X1, X2)) U4_G(X1, s2lcA_out_ga(X1, X2)) -> U5_G(X1, listC_in_g(X2)) U4_G(X1, s2lcA_out_ga(X1, X2)) -> LISTC_IN_G(X2) LISTC_IN_G(.(X1, X2)) -> U2_G(X1, X2, listC_in_g(X2)) LISTC_IN_G(.(X1, X2)) -> LISTC_IN_G(X2) The TRS R consists of the following rules: s2lcA_in_ga(0, []) -> s2lcA_out_ga(0, []) s2lcA_in_ga(s(X1), .(X2, X3)) -> U7_ga(X1, X2, X3, s2lcA_in_ga(X1, X3)) U7_ga(X1, X2, X3, s2lcA_out_ga(X1, X3)) -> s2lcA_out_ga(s(X1), .(X2, X3)) The argument filtering Pi contains the following mapping: s(x1) = s(x1) s2lA_in_ga(x1, x2) = s2lA_in_ga(x1) .(x1, x2) = .(x2) s2lcA_in_ga(x1, x2) = s2lcA_in_ga(x1) 0 = 0 s2lcA_out_ga(x1, x2) = s2lcA_out_ga(x1, x2) U7_ga(x1, x2, x3, x4) = U7_ga(x1, x4) listC_in_g(x1) = listC_in_g(x1) GOALD_IN_G(x1) = GOALD_IN_G(x1) U3_G(x1, x2) = U3_G(x1, x2) S2LA_IN_GA(x1, x2) = S2LA_IN_GA(x1) U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4) U4_G(x1, x2) = U4_G(x1, x2) U5_G(x1, x2) = U5_G(x1, x2) LISTC_IN_G(x1) = LISTC_IN_G(x1) U2_G(x1, x2, x3) = U2_G(x2, x3) 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: GOALD_IN_G(s(X1)) -> U3_G(X1, s2lA_in_ga(X1, X2)) GOALD_IN_G(s(X1)) -> S2LA_IN_GA(X1, X2) S2LA_IN_GA(s(X1), .(X2, X3)) -> U1_GA(X1, X2, X3, s2lA_in_ga(X1, X3)) S2LA_IN_GA(s(X1), .(X2, X3)) -> S2LA_IN_GA(X1, X3) GOALD_IN_G(s(X1)) -> U4_G(X1, s2lcA_in_ga(X1, X2)) U4_G(X1, s2lcA_out_ga(X1, X2)) -> U5_G(X1, listC_in_g(X2)) U4_G(X1, s2lcA_out_ga(X1, X2)) -> LISTC_IN_G(X2) LISTC_IN_G(.(X1, X2)) -> U2_G(X1, X2, listC_in_g(X2)) LISTC_IN_G(.(X1, X2)) -> LISTC_IN_G(X2) The TRS R consists of the following rules: s2lcA_in_ga(0, []) -> s2lcA_out_ga(0, []) s2lcA_in_ga(s(X1), .(X2, X3)) -> U7_ga(X1, X2, X3, s2lcA_in_ga(X1, X3)) U7_ga(X1, X2, X3, s2lcA_out_ga(X1, X3)) -> s2lcA_out_ga(s(X1), .(X2, X3)) The argument filtering Pi contains the following mapping: s(x1) = s(x1) s2lA_in_ga(x1, x2) = s2lA_in_ga(x1) .(x1, x2) = .(x2) s2lcA_in_ga(x1, x2) = s2lcA_in_ga(x1) 0 = 0 s2lcA_out_ga(x1, x2) = s2lcA_out_ga(x1, x2) U7_ga(x1, x2, x3, x4) = U7_ga(x1, x4) listC_in_g(x1) = listC_in_g(x1) GOALD_IN_G(x1) = GOALD_IN_G(x1) U3_G(x1, x2) = U3_G(x1, x2) S2LA_IN_GA(x1, x2) = S2LA_IN_GA(x1) U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4) U4_G(x1, x2) = U4_G(x1, x2) U5_G(x1, x2) = U5_G(x1, x2) LISTC_IN_G(x1) = LISTC_IN_G(x1) U2_G(x1, x2, x3) = U2_G(x2, x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (7) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 7 less nodes. ---------------------------------------- (8) Complex Obligation (AND) ---------------------------------------- (9) Obligation: Pi DP problem: The TRS P consists of the following rules: LISTC_IN_G(.(X1, X2)) -> LISTC_IN_G(X2) The TRS R consists of the following rules: s2lcA_in_ga(0, []) -> s2lcA_out_ga(0, []) s2lcA_in_ga(s(X1), .(X2, X3)) -> U7_ga(X1, X2, X3, s2lcA_in_ga(X1, X3)) U7_ga(X1, X2, X3, s2lcA_out_ga(X1, X3)) -> s2lcA_out_ga(s(X1), .(X2, X3)) The argument filtering Pi contains the following mapping: s(x1) = s(x1) .(x1, x2) = .(x2) s2lcA_in_ga(x1, x2) = s2lcA_in_ga(x1) 0 = 0 s2lcA_out_ga(x1, x2) = s2lcA_out_ga(x1, x2) U7_ga(x1, x2, x3, x4) = U7_ga(x1, x4) LISTC_IN_G(x1) = LISTC_IN_G(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (10) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (11) Obligation: Pi DP problem: The TRS P consists of the following rules: LISTC_IN_G(.(X1, X2)) -> LISTC_IN_G(X2) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) LISTC_IN_G(x1) = LISTC_IN_G(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (12) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (13) Obligation: Q DP problem: The TRS P consists of the following rules: LISTC_IN_G(.(X2)) -> LISTC_IN_G(X2) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (14) 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: *LISTC_IN_G(.(X2)) -> LISTC_IN_G(X2) The graph contains the following edges 1 > 1 ---------------------------------------- (15) YES ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: S2LA_IN_GA(s(X1), .(X2, X3)) -> S2LA_IN_GA(X1, X3) The TRS R consists of the following rules: s2lcA_in_ga(0, []) -> s2lcA_out_ga(0, []) s2lcA_in_ga(s(X1), .(X2, X3)) -> U7_ga(X1, X2, X3, s2lcA_in_ga(X1, X3)) U7_ga(X1, X2, X3, s2lcA_out_ga(X1, X3)) -> s2lcA_out_ga(s(X1), .(X2, X3)) The argument filtering Pi contains the following mapping: s(x1) = s(x1) .(x1, x2) = .(x2) s2lcA_in_ga(x1, x2) = s2lcA_in_ga(x1) 0 = 0 s2lcA_out_ga(x1, x2) = s2lcA_out_ga(x1, x2) U7_ga(x1, x2, x3, x4) = U7_ga(x1, x4) S2LA_IN_GA(x1, x2) = S2LA_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (18) Obligation: Pi DP problem: The TRS P consists of the following rules: S2LA_IN_GA(s(X1), .(X2, X3)) -> S2LA_IN_GA(X1, X3) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) .(x1, x2) = .(x2) S2LA_IN_GA(x1, x2) = S2LA_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (19) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: S2LA_IN_GA(s(X1)) -> S2LA_IN_GA(X1) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (21) 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: *S2LA_IN_GA(s(X1)) -> S2LA_IN_GA(X1) The graph contains the following edges 1 > 1 ---------------------------------------- (22) YES