/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern fl(g,a,g) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1)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) PiDP (7) PiDPToQDPProof [SOUND, 0 ms] (8) QDP (9) QDPSizeChangeProof [EQUIVALENT, 0 ms] (10) YES ---------------------------------------- (0) Obligation: Clauses: fl([], [], 0). fl(.(E, X), R, s(Z)) :- ','(append(E, Y, R), fl(X, Y, Z)). append([], X, X). append(.(X, Xs), Ys, .(X, Zs)) :- append(Xs, Ys, Zs). Query: fl(g,a,g) ---------------------------------------- (1) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 3, "program": { "directives": [], "clauses": [ [ "(fl ([]) ([]) (0))", null ], [ "(fl (. E X) R (s Z))", "(',' (append E Y R) (fl X Y Z))" ], [ "(append ([]) X X)", null ], [ "(append (. X Xs) Ys (. X Zs))", "(append Xs Ys Zs)" ] ] }, "graph": { "nodes": { "190": { "goal": [{ "clause": 3, "scope": 2, "term": "(',' (append T8 X13 T12) (fl T9 X13 T11))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T8", "T9", "T11" ], "free": ["X13"], "exprvars": [] } }, "191": { "goal": [{ "clause": -1, "scope": -1, "term": "(fl T9 T18 T11)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T9", "T11" ], "free": [], "exprvars": [] } }, "192": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "197": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (append T26 X38 T28) (fl T9 X38 T11))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T9", "T11", "T26" ], "free": ["X38"], "exprvars": [] } }, "198": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "188": { "goal": [ { "clause": 2, "scope": 2, "term": "(',' (append T8 X13 T12) (fl T9 X13 T11))" }, { "clause": 3, "scope": 2, "term": "(',' (append T8 X13 T12) (fl T9 X13 T11))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T8", "T9", "T11" ], "free": ["X13"], "exprvars": [] } }, "101": { "goal": [ { "clause": -1, "scope": -1, "term": "(true)" }, { "clause": 1, "scope": 1, "term": "(fl ([]) T2 (0))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "189": { "goal": [{ "clause": 2, "scope": 2, "term": "(',' (append T8 X13 T12) (fl T9 X13 T11))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T8", "T9", "T11" ], "free": ["X13"], "exprvars": [] } }, "3": { "goal": [{ "clause": -1, "scope": -1, "term": "(fl T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T3" ], "free": [], "exprvars": [] } }, "102": { "goal": [{ "clause": 1, "scope": 1, "term": "(fl T1 T2 T3)" }], "kb": { "nonunifying": [[ "(fl T1 T2 T3)", "(fl ([]) ([]) (0))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T3" ], "free": [], "exprvars": [] } }, "103": { "goal": [{ "clause": 1, "scope": 1, "term": "(fl ([]) T2 (0))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "104": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "107": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (append T8 X13 T12) (fl T9 X13 T11))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T8", "T9", "T11" ], "free": ["X13"], "exprvars": [] } }, "108": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "10": { "goal": [ { "clause": 0, "scope": 1, "term": "(fl T1 T2 T3)" }, { "clause": 1, "scope": 1, "term": "(fl T1 T2 T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T3" ], "free": [], "exprvars": [] } } }, "edges": [ { "from": 3, "to": 10, "label": "CASE" }, { "from": 10, "to": 101, "label": "EVAL with clause\nfl([], [], 0).\nand substitutionT1 -> [],\nT2 -> [],\nT3 -> 0" }, { "from": 10, "to": 102, "label": "EVAL-BACKTRACK" }, { "from": 101, "to": 103, "label": "SUCCESS" }, { "from": 102, "to": 107, "label": "EVAL with clause\nfl(.(X9, X10), X11, s(X12)) :- ','(append(X9, X13, X11), fl(X10, X13, X12)).\nand substitutionX9 -> T8,\nX10 -> T9,\nT1 -> .(T8, T9),\nT2 -> T12,\nX11 -> T12,\nX12 -> T11,\nT3 -> s(T11),\nT10 -> T12" }, { "from": 102, "to": 108, "label": "EVAL-BACKTRACK" }, { "from": 103, "to": 104, "label": "BACKTRACK\nfor clause: fl(.(E, X), R, s(Z)) :- ','(append(E, Y, R), fl(X, Y, Z))because of non-unification" }, { "from": 107, "to": 188, "label": "CASE" }, { "from": 188, "to": 189, "label": "PARALLEL" }, { "from": 188, "to": 190, "label": "PARALLEL" }, { "from": 189, "to": 191, "label": "EVAL with clause\nappend([], X22, X22).\nand substitutionT8 -> [],\nX13 -> T18,\nX22 -> T18,\nT12 -> T18,\nX23 -> T18,\nT17 -> T18" }, { "from": 189, "to": 192, "label": "EVAL-BACKTRACK" }, { "from": 190, "to": 197, "label": "EVAL with clause\nappend(.(X34, X35), X36, .(X34, X37)) :- append(X35, X36, X37).\nand substitutionX34 -> T25,\nX35 -> T26,\nT8 -> .(T25, T26),\nX13 -> X38,\nX36 -> X38,\nX37 -> T28,\nT12 -> .(T25, T28),\nT27 -> T28" }, { "from": 190, "to": 198, "label": "EVAL-BACKTRACK" }, { "from": 191, "to": 3, "label": "INSTANCE with matching:\nT1 -> T9\nT2 -> T18\nT3 -> T11" }, { "from": 197, "to": 107, "label": "INSTANCE with matching:\nT8 -> T26\nX13 -> X38\nT12 -> T28" } ], "type": "Graph" } } ---------------------------------------- (2) Obligation: Triples: pB([], X1, X1, X2, X3) :- flA(X2, X1, X3). pB(.(X1, X2), X3, .(X1, X4), X5, X6) :- pB(X2, X3, X4, X5, X6). flA(.(X1, X2), X3, s(X4)) :- pB(X1, X5, X3, X2, X4). Clauses: flcA([], [], 0). flcA(.(X1, X2), X3, s(X4)) :- qcB(X1, X5, X3, X2, X4). qcB([], X1, X1, X2, X3) :- flcA(X2, X1, X3). qcB(.(X1, X2), X3, .(X1, X4), X5, X6) :- qcB(X2, X3, X4, X5, X6). Afs: flA(x1, x2, x3) = flA(x1, x3) ---------------------------------------- (3) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: flA_in_3: (b,f,b) pB_in_5: (b,f,f,b,b) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: FLA_IN_GAG(.(X1, X2), X3, s(X4)) -> U3_GAG(X1, X2, X3, X4, pB_in_gaagg(X1, X5, X3, X2, X4)) FLA_IN_GAG(.(X1, X2), X3, s(X4)) -> PB_IN_GAAGG(X1, X5, X3, X2, X4) PB_IN_GAAGG([], X1, X1, X2, X3) -> U1_GAAGG(X1, X2, X3, flA_in_gag(X2, X1, X3)) PB_IN_GAAGG([], X1, X1, X2, X3) -> FLA_IN_GAG(X2, X1, X3) PB_IN_GAAGG(.(X1, X2), X3, .(X1, X4), X5, X6) -> U2_GAAGG(X1, X2, X3, X4, X5, X6, pB_in_gaagg(X2, X3, X4, X5, X6)) PB_IN_GAAGG(.(X1, X2), X3, .(X1, X4), X5, X6) -> PB_IN_GAAGG(X2, X3, X4, X5, X6) R is empty. The argument filtering Pi contains the following mapping: flA_in_gag(x1, x2, x3) = flA_in_gag(x1, x3) .(x1, x2) = .(x1, x2) s(x1) = s(x1) pB_in_gaagg(x1, x2, x3, x4, x5) = pB_in_gaagg(x1, x4, x5) [] = [] FLA_IN_GAG(x1, x2, x3) = FLA_IN_GAG(x1, x3) U3_GAG(x1, x2, x3, x4, x5) = U3_GAG(x1, x2, x4, x5) PB_IN_GAAGG(x1, x2, x3, x4, x5) = PB_IN_GAAGG(x1, x4, x5) U1_GAAGG(x1, x2, x3, x4) = U1_GAAGG(x2, x3, x4) U2_GAAGG(x1, x2, x3, x4, x5, x6, x7) = U2_GAAGG(x1, x2, x5, x6, x7) 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: FLA_IN_GAG(.(X1, X2), X3, s(X4)) -> U3_GAG(X1, X2, X3, X4, pB_in_gaagg(X1, X5, X3, X2, X4)) FLA_IN_GAG(.(X1, X2), X3, s(X4)) -> PB_IN_GAAGG(X1, X5, X3, X2, X4) PB_IN_GAAGG([], X1, X1, X2, X3) -> U1_GAAGG(X1, X2, X3, flA_in_gag(X2, X1, X3)) PB_IN_GAAGG([], X1, X1, X2, X3) -> FLA_IN_GAG(X2, X1, X3) PB_IN_GAAGG(.(X1, X2), X3, .(X1, X4), X5, X6) -> U2_GAAGG(X1, X2, X3, X4, X5, X6, pB_in_gaagg(X2, X3, X4, X5, X6)) PB_IN_GAAGG(.(X1, X2), X3, .(X1, X4), X5, X6) -> PB_IN_GAAGG(X2, X3, X4, X5, X6) R is empty. The argument filtering Pi contains the following mapping: flA_in_gag(x1, x2, x3) = flA_in_gag(x1, x3) .(x1, x2) = .(x1, x2) s(x1) = s(x1) pB_in_gaagg(x1, x2, x3, x4, x5) = pB_in_gaagg(x1, x4, x5) [] = [] FLA_IN_GAG(x1, x2, x3) = FLA_IN_GAG(x1, x3) U3_GAG(x1, x2, x3, x4, x5) = U3_GAG(x1, x2, x4, x5) PB_IN_GAAGG(x1, x2, x3, x4, x5) = PB_IN_GAAGG(x1, x4, x5) U1_GAAGG(x1, x2, x3, x4) = U1_GAAGG(x2, x3, x4) U2_GAAGG(x1, x2, x3, x4, x5, x6, x7) = U2_GAAGG(x1, x2, x5, x6, x7) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes. ---------------------------------------- (6) Obligation: Pi DP problem: The TRS P consists of the following rules: FLA_IN_GAG(.(X1, X2), X3, s(X4)) -> PB_IN_GAAGG(X1, X5, X3, X2, X4) PB_IN_GAAGG([], X1, X1, X2, X3) -> FLA_IN_GAG(X2, X1, X3) PB_IN_GAAGG(.(X1, X2), X3, .(X1, X4), X5, X6) -> PB_IN_GAAGG(X2, X3, X4, X5, X6) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) s(x1) = s(x1) [] = [] FLA_IN_GAG(x1, x2, x3) = FLA_IN_GAG(x1, x3) PB_IN_GAAGG(x1, x2, x3, x4, x5) = PB_IN_GAAGG(x1, x4, x5) 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: FLA_IN_GAG(.(X1, X2), s(X4)) -> PB_IN_GAAGG(X1, X2, X4) PB_IN_GAAGG([], X2, X3) -> FLA_IN_GAG(X2, X3) PB_IN_GAAGG(.(X1, X2), X5, X6) -> PB_IN_GAAGG(X2, X5, X6) 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: *PB_IN_GAAGG([], X2, X3) -> FLA_IN_GAG(X2, X3) The graph contains the following edges 2 >= 1, 3 >= 2 *PB_IN_GAAGG(.(X1, X2), X5, X6) -> PB_IN_GAAGG(X2, X5, X6) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *FLA_IN_GAG(.(X1, X2), s(X4)) -> PB_IN_GAAGG(X1, X2, X4) The graph contains the following edges 1 > 1, 1 > 2, 2 > 3 ---------------------------------------- (10) YES