/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern mergesort(g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToDTProblemTransformerProof [SOUND, 47 ms] (2) TRIPLES (3) TriplesToPiDPProof [SOUND, 0 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [SOUND, 0 ms] (11) QDP (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] (13) YES (14) PiDP (15) UsableRulesProof [EQUIVALENT, 0 ms] (16) PiDP (17) PiDPToQDPProof [SOUND, 0 ms] (18) QDP (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] (20) YES (21) PiDP (22) PiDPToQDPProof [SOUND, 0 ms] (23) QDP (24) TransformationProof [EQUIVALENT, 0 ms] (25) QDP (26) QDPQMonotonicMRRProof [EQUIVALENT, 60 ms] (27) QDP (28) DependencyGraphProof [EQUIVALENT, 1 ms] (29) QDP (30) UsableRulesProof [EQUIVALENT, 0 ms] (31) QDP (32) QReductionProof [EQUIVALENT, 0 ms] (33) QDP (34) QDPQMonotonicMRRProof [EQUIVALENT, 7 ms] (35) QDP (36) DependencyGraphProof [EQUIVALENT, 0 ms] (37) TRUE ---------------------------------------- (0) Obligation: Clauses: mergesort([], []). mergesort(.(X, []), .(X, [])). mergesort(.(X, .(Y, Xs)), Ys) :- ','(split(.(X, .(Y, Xs)), X1s, X2s), ','(mergesort(X1s, Y1s), ','(mergesort(X2s, Y2s), merge(Y1s, Y2s, Ys)))). split([], [], []). split(.(X, Xs), .(X, Ys), Zs) :- split(Xs, Zs, Ys). merge([], Xs, Xs). merge(Xs, [], Xs). merge(.(X, Xs), .(Y, Ys), .(X, Zs)) :- ','(=(X, Y), merge(.(X, Xs), Ys, Zs)). Query: mergesort(g,a) ---------------------------------------- (1) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 1, "program": { "directives": [], "clauses": [ [ "(mergesort ([]) ([]))", null ], [ "(mergesort (. X ([])) (. X ([])))", null ], [ "(mergesort (. X (. Y Xs)) Ys)", "(',' (split (. X (. Y Xs)) X1s X2s) (',' (mergesort X1s Y1s) (',' (mergesort X2s Y2s) (merge Y1s Y2s Ys))))" ], [ "(split ([]) ([]) ([]))", null ], [ "(split (. X Xs) (. X Ys) Zs)", "(split Xs Zs Ys)" ], [ "(merge ([]) Xs Xs)", null ], [ "(merge Xs ([]) Xs)", null ], [ "(merge (. X Xs) (. Y Ys) (. X Zs))", "(',' (= X Y) (merge (. X Xs) Ys Zs))" ] ] }, "graph": { "nodes": { "type": "Nodes", "154": { "goal": [ { "clause": -1, "scope": -1, "term": "(true)" }, { "clause": 2, "scope": 1, "term": "(mergesort (. T4 ([])) T2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "156": { "goal": [{ "clause": 2, "scope": 1, "term": "(mergesort T1 T2)" }], "kb": { "nonunifying": [ [ "(mergesort T1 T2)", "(mergesort ([]) ([]))" ], [ "(mergesort T1 T2)", "(mergesort (. X7 ([])) (. X7 ([])))" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": ["X7"], "exprvars": [] } }, "113": { "goal": [{ "clause": 2, "scope": 1, "term": "(mergesort ([]) T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "158": { "goal": [{ "clause": 2, "scope": 1, "term": "(mergesort (. T4 ([])) T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "434": { "goal": [{ "clause": -1, "scope": -1, "term": "(mergesort (. T21 T25) X22)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T21", "T25" ], "free": ["X22"], "exprvars": [] } }, "435": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (mergesort T24 X23) (merge T38 X23 T14))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T24", "T38" ], "free": ["X23"], "exprvars": [] } }, "92": { "goal": [ { "clause": -1, "scope": -1, "term": "(true)" }, { "clause": 1, "scope": 1, "term": "(mergesort ([]) T2)" }, { "clause": 2, "scope": 1, "term": "(mergesort ([]) T2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "439": { "goal": [{ "clause": -1, "scope": -1, "term": "(mergesort T24 X23)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T24"], "free": ["X23"], "exprvars": [] } }, "160": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "440": { "goal": [{ "clause": -1, "scope": -1, "term": "(merge T38 T39 T14)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T38", "T39" ], "free": [], "exprvars": [] } }, "1": { "goal": [{ "clause": -1, "scope": -1, "term": "(mergesort T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "166": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (split (. T10 (. T11 T12)) X20 X21) (',' (mergesort X20 X22) (',' (mergesort X21 X23) (merge X22 X23 T14))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T10", "T11", "T12" ], "free": [ "X20", "X21", "X22", "X23" ], "exprvars": [] } }, "244": { "goal": [{ "clause": -1, "scope": -1, "term": "(split (. T22 T23) X41 X40)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T22", "T23" ], "free": [ "X40", "X41" ], "exprvars": [] } }, "168": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "169": { "goal": [ { "clause": 3, "scope": 2, "term": "(',' (split (. T10 (. T11 T12)) X20 X21) (',' (mergesort X20 X22) (',' (mergesort X21 X23) (merge X22 X23 T14))))" }, { "clause": 4, "scope": 2, "term": "(',' (split (. T10 (. T11 T12)) X20 X21) (',' (mergesort X20 X22) (',' (mergesort X21 X23) (merge X22 X23 T14))))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T10", "T11", "T12" ], "free": [ "X20", "X21", "X22", "X23" ], "exprvars": [] } }, "246": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (mergesort (. T21 T25) X22) (',' (mergesort T24 X23) (merge X22 X23 T14)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T21", "T24", "T25" ], "free": [ "X22", "X23" ], "exprvars": [] } }, "248": { "goal": [ { "clause": 3, "scope": 3, "term": "(split (. T22 T23) X41 X40)" }, { "clause": 4, "scope": 3, "term": "(split (. T22 T23) X41 X40)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T22", "T23" ], "free": [ "X40", "X41" ], "exprvars": [] } }, "446": { "goal": [ { "clause": 5, "scope": 5, "term": "(merge T38 T39 T14)" }, { "clause": 6, "scope": 5, "term": "(merge T38 T39 T14)" }, { "clause": 7, "scope": 5, "term": "(merge T38 T39 T14)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T38", "T39" ], "free": [], "exprvars": [] } }, "447": { "goal": [{ "clause": 5, "scope": 5, "term": "(merge T38 T39 T14)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T38", "T39" ], "free": [], "exprvars": [] } }, "449": { "goal": [ { "clause": 6, "scope": 5, "term": "(merge T38 T39 T14)" }, { "clause": 7, "scope": 5, "term": "(merge T38 T39 T14)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T38", "T39" ], "free": [], "exprvars": [] } }, "170": { "goal": [{ "clause": 4, "scope": 2, "term": "(',' (split (. T10 (. T11 T12)) X20 X21) (',' (mergesort X20 X22) (',' (mergesort X21 X23) (merge X22 X23 T14))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T10", "T11", "T12" ], "free": [ "X20", "X21", "X22", "X23" ], "exprvars": [] } }, "250": { "goal": [{ "clause": 4, "scope": 3, "term": "(split (. T22 T23) X41 X40)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T22", "T23" ], "free": [ "X40", "X41" ], "exprvars": [] } }, "252": { "goal": [{ "clause": -1, "scope": -1, "term": "(split T31 X59 X58)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T31"], "free": [ "X58", "X59" ], "exprvars": [] } }, "176": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (split (. T22 T23) X41 X40) (',' (mergesort (. T21 X40) X22) (',' (mergesort X41 X23) (merge X22 X23 T14))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T21", "T22", "T23" ], "free": [ "X22", "X23", "X40", "X41" ], "exprvars": [] } }, "452": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "255": { "goal": [ { "clause": 3, "scope": 4, "term": "(split T31 X59 X58)" }, { "clause": 4, "scope": 4, "term": "(split T31 X59 X58)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T31"], "free": [ "X58", "X59" ], "exprvars": [] } }, "453": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "256": { "goal": [{ "clause": 3, "scope": 4, "term": "(split T31 X59 X58)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T31"], "free": [ "X58", "X59" ], "exprvars": [] } }, "454": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "257": { "goal": [{ "clause": 4, "scope": 4, "term": "(split T31 X59 X58)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T31"], "free": [ "X58", "X59" ], "exprvars": [] } }, "455": { "goal": [{ "clause": 6, "scope": 5, "term": "(merge T38 T39 T14)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T38", "T39" ], "free": [], "exprvars": [] } }, "258": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "456": { "goal": [{ "clause": 7, "scope": 5, "term": "(merge T38 T39 T14)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T38", "T39" ], "free": [], "exprvars": [] } }, "259": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "457": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "458": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "459": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "74": { "goal": [ { "clause": 0, "scope": 1, "term": "(mergesort T1 T2)" }, { "clause": 1, "scope": 1, "term": "(mergesort T1 T2)" }, { "clause": 2, "scope": 1, "term": "(mergesort T1 T2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "260": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "140": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "263": { "goal": [{ "clause": -1, "scope": -1, "term": "(split T37 X77 X76)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T37"], "free": [ "X76", "X77" ], "exprvars": [] } }, "264": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "101": { "goal": [ { "clause": 1, "scope": 1, "term": "(mergesort T1 T2)" }, { "clause": 2, "scope": 1, "term": "(mergesort T1 T2)" } ], "kb": { "nonunifying": [[ "(mergesort T1 T2)", "(mergesort ([]) ([]))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "464": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (= T62 T64) (merge (. T62 T63) T65 T67))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T62", "T63", "T64", "T65" ], "free": [], "exprvars": [] } }, "465": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "466": { "goal": [{ "clause": -1, "scope": -1, "term": "(merge (. T69 T63) T65 T67)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T63", "T65", "T69" ], "free": [], "exprvars": [] } }, "467": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "106": { "goal": [ { "clause": 1, "scope": 1, "term": "(mergesort ([]) T2)" }, { "clause": 2, "scope": 1, "term": "(mergesort ([]) T2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 74, "label": "CASE" }, { "from": 74, "to": 92, "label": "EVAL with clause\nmergesort([], []).\nand substitutionT1 -> [],\nT2 -> []" }, { "from": 74, "to": 101, "label": "EVAL-BACKTRACK" }, { "from": 92, "to": 106, "label": "SUCCESS" }, { "from": 101, "to": 154, "label": "EVAL with clause\nmergesort(.(X7, []), .(X7, [])).\nand substitutionX7 -> T4,\nT1 -> .(T4, []),\nT2 -> .(T4, [])" }, { "from": 101, "to": 156, "label": "EVAL-BACKTRACK" }, { "from": 106, "to": 113, "label": "BACKTRACK\nfor clause: mergesort(.(X, []), .(X, []))because of non-unification" }, { "from": 113, "to": 140, "label": "BACKTRACK\nfor clause: mergesort(.(X, .(Y, Xs)), Ys) :- ','(split(.(X, .(Y, Xs)), X1s, X2s), ','(mergesort(X1s, Y1s), ','(mergesort(X2s, Y2s), merge(Y1s, Y2s, Ys))))because of non-unification" }, { "from": 154, "to": 158, "label": "SUCCESS" }, { "from": 156, "to": 166, "label": "EVAL with clause\nmergesort(.(X16, .(X17, X18)), X19) :- ','(split(.(X16, .(X17, X18)), X20, X21), ','(mergesort(X20, X22), ','(mergesort(X21, X23), merge(X22, X23, X19)))).\nand substitutionX16 -> T10,\nX17 -> T11,\nX18 -> T12,\nT1 -> .(T10, .(T11, T12)),\nT2 -> T14,\nX19 -> T14,\nT13 -> T14" }, { "from": 156, "to": 168, "label": "EVAL-BACKTRACK" }, { "from": 158, "to": 160, "label": "BACKTRACK\nfor clause: mergesort(.(X, .(Y, Xs)), Ys) :- ','(split(.(X, .(Y, Xs)), X1s, X2s), ','(mergesort(X1s, Y1s), ','(mergesort(X2s, Y2s), merge(Y1s, Y2s, Ys))))because of non-unification" }, { "from": 166, "to": 169, "label": "CASE" }, { "from": 169, "to": 170, "label": "BACKTRACK\nfor clause: split([], [], [])because of non-unification" }, { "from": 170, "to": 176, "label": "ONLY EVAL with clause\nsplit(.(X36, X37), .(X36, X38), X39) :- split(X37, X39, X38).\nand substitutionT10 -> T21,\nX36 -> T21,\nT11 -> T22,\nT12 -> T23,\nX37 -> .(T22, T23),\nX38 -> X40,\nX20 -> .(T21, X40),\nX21 -> X41,\nX39 -> X41" }, { "from": 176, "to": 244, "label": "SPLIT 1" }, { "from": 176, "to": 246, "label": "SPLIT 2\nnew knowledge:\nT22 is ground\nT23 is ground\nT24 is ground\nT25 is ground\nreplacements:X41 -> T24,\nX40 -> T25" }, { "from": 244, "to": 248, "label": "CASE" }, { "from": 246, "to": 434, "label": "SPLIT 1" }, { "from": 246, "to": 435, "label": "SPLIT 2\nnew knowledge:\nT21 is ground\nT25 is ground\nT38 is ground\nreplacements:X22 -> T38" }, { "from": 248, "to": 250, "label": "BACKTRACK\nfor clause: split([], [], [])because of non-unification" }, { "from": 250, "to": 252, "label": "ONLY EVAL with clause\nsplit(.(X54, X55), .(X54, X56), X57) :- split(X55, X57, X56).\nand substitutionT22 -> T30,\nX54 -> T30,\nT23 -> T31,\nX55 -> T31,\nX56 -> X58,\nX41 -> .(T30, X58),\nX40 -> X59,\nX57 -> X59" }, { "from": 252, "to": 255, "label": "CASE" }, { "from": 255, "to": 256, "label": "PARALLEL" }, { "from": 255, "to": 257, "label": "PARALLEL" }, { "from": 256, "to": 258, "label": "EVAL with clause\nsplit([], [], []).\nand substitutionT31 -> [],\nX59 -> [],\nX58 -> []" }, { "from": 256, "to": 259, "label": "EVAL-BACKTRACK" }, { "from": 257, "to": 263, "label": "EVAL with clause\nsplit(.(X72, X73), .(X72, X74), X75) :- split(X73, X75, X74).\nand substitutionX72 -> T36,\nX73 -> T37,\nT31 -> .(T36, T37),\nX74 -> X76,\nX59 -> .(T36, X76),\nX58 -> X77,\nX75 -> X77" }, { "from": 257, "to": 264, "label": "EVAL-BACKTRACK" }, { "from": 258, "to": 260, "label": "SUCCESS" }, { "from": 263, "to": 252, "label": "INSTANCE with matching:\nT31 -> T37\nX59 -> X77\nX58 -> X76" }, { "from": 434, "to": 1, "label": "INSTANCE with matching:\nT1 -> .(T21, T25)\nT2 -> X22" }, { "from": 435, "to": 439, "label": "SPLIT 1" }, { "from": 435, "to": 440, "label": "SPLIT 2\nnew knowledge:\nT24 is ground\nT39 is ground\nreplacements:X23 -> T39" }, { "from": 439, "to": 1, "label": "INSTANCE with matching:\nT1 -> T24\nT2 -> X23" }, { "from": 440, "to": 446, "label": "CASE" }, { "from": 446, "to": 447, "label": "PARALLEL" }, { "from": 446, "to": 449, "label": "PARALLEL" }, { "from": 447, "to": 452, "label": "EVAL with clause\nmerge([], X84, X84).\nand substitutionT38 -> [],\nT39 -> T46,\nX84 -> T46,\nT14 -> T46" }, { "from": 447, "to": 453, "label": "EVAL-BACKTRACK" }, { "from": 449, "to": 455, "label": "PARALLEL" }, { "from": 449, "to": 456, "label": "PARALLEL" }, { "from": 452, "to": 454, "label": "SUCCESS" }, { "from": 455, "to": 457, "label": "EVAL with clause\nmerge(X89, [], X89).\nand substitutionT38 -> T51,\nX89 -> T51,\nT39 -> [],\nT14 -> T51" }, { "from": 455, "to": 458, "label": "EVAL-BACKTRACK" }, { "from": 456, "to": 464, "label": "EVAL with clause\nmerge(.(X100, X101), .(X102, X103), .(X100, X104)) :- ','(=(X100, X102), merge(.(X100, X101), X103, X104)).\nand substitutionX100 -> T62,\nX101 -> T63,\nT38 -> .(T62, T63),\nX102 -> T64,\nX103 -> T65,\nT39 -> .(T64, T65),\nX104 -> T67,\nT14 -> .(T62, T67),\nT66 -> T67" }, { "from": 456, "to": 465, "label": "EVAL-BACKTRACK" }, { "from": 457, "to": 459, "label": "SUCCESS" }, { "from": 464, "to": 466, "label": "UNIFY CASE with substitutionT62 -> T69,\nT64 -> T69" }, { "from": 464, "to": 467, "label": "UNIFY-BACKTRACK" }, { "from": 466, "to": 440, "label": "INSTANCE with matching:\nT38 -> .(T69, T63)\nT39 -> T65\nT14 -> T67" } ], "type": "Graph" } } ---------------------------------------- (2) Obligation: Triples: splitA(.(X1, X2), .(X1, X3), X4) :- splitA(X2, X4, X3). mergeC(.(X1, X2), .(X1, X3), .(X1, X4)) :- mergeC(.(X1, X2), X3, X4). mergesortB(.(X1, .(X2, X3)), X4) :- splitA(X3, X5, X6). mergesortB(.(X1, .(X2, X3)), X4) :- ','(splitcD(X2, X3, X5, X6), mergesortB(.(X1, X6), X7)). mergesortB(.(X1, .(X2, X3)), X4) :- ','(splitcD(X2, X3, X5, X6), ','(mergesortcB(.(X1, X6), X7), mergesortB(X5, X8))). mergesortB(.(X1, .(X2, X3)), X4) :- ','(splitcD(X2, X3, X5, X6), ','(mergesortcB(.(X1, X6), X7), ','(mergesortcB(X5, X8), mergeC(X7, X8, X4)))). Clauses: splitcA([], [], []). splitcA(.(X1, X2), .(X1, X3), X4) :- splitcA(X2, X4, X3). mergesortcB([], []). mergesortcB(.(X1, []), .(X1, [])). mergesortcB(.(X1, .(X2, X3)), X4) :- ','(splitcD(X2, X3, X5, X6), ','(mergesortcB(.(X1, X6), X7), ','(mergesortcB(X5, X8), mergecC(X7, X8, X4)))). mergecC([], X1, X1). mergecC(X1, [], X1). mergecC(.(X1, X2), .(X1, X3), .(X1, X4)) :- mergecC(.(X1, X2), X3, X4). splitcD(X1, X2, .(X1, X3), X4) :- splitcA(X2, X4, X3). Afs: mergesortB(x1, x2) = mergesortB(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: mergesortB_in_2: (b,f) splitA_in_3: (b,f,f) splitcD_in_4: (b,b,f,f) splitcA_in_3: (b,f,f) mergesortcB_in_2: (b,f) mergecC_in_3: (b,b,f) mergeC_in_3: (b,b,f) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: MERGESORTB_IN_GA(.(X1, .(X2, X3)), X4) -> U3_GA(X1, X2, X3, X4, splitA_in_gaa(X3, X5, X6)) MERGESORTB_IN_GA(.(X1, .(X2, X3)), X4) -> SPLITA_IN_GAA(X3, X5, X6) SPLITA_IN_GAA(.(X1, X2), .(X1, X3), X4) -> U1_GAA(X1, X2, X3, X4, splitA_in_gaa(X2, X4, X3)) SPLITA_IN_GAA(.(X1, X2), .(X1, X3), X4) -> SPLITA_IN_GAA(X2, X4, X3) MERGESORTB_IN_GA(.(X1, .(X2, X3)), X4) -> U4_GA(X1, X2, X3, X4, splitcD_in_ggaa(X2, X3, X5, X6)) U4_GA(X1, X2, X3, X4, splitcD_out_ggaa(X2, X3, X5, X6)) -> U5_GA(X1, X2, X3, X4, mergesortB_in_ga(.(X1, X6), X7)) U4_GA(X1, X2, X3, X4, splitcD_out_ggaa(X2, X3, X5, X6)) -> MERGESORTB_IN_GA(.(X1, X6), X7) U4_GA(X1, X2, X3, X4, splitcD_out_ggaa(X2, X3, X5, X6)) -> U6_GA(X1, X2, X3, X4, X5, mergesortcB_in_ga(.(X1, X6), X7)) U6_GA(X1, X2, X3, X4, X5, mergesortcB_out_ga(.(X1, X6), X7)) -> U7_GA(X1, X2, X3, X4, mergesortB_in_ga(X5, X8)) U6_GA(X1, X2, X3, X4, X5, mergesortcB_out_ga(.(X1, X6), X7)) -> MERGESORTB_IN_GA(X5, X8) U6_GA(X1, X2, X3, X4, X5, mergesortcB_out_ga(.(X1, X6), X7)) -> U8_GA(X1, X2, X3, X4, X7, mergesortcB_in_ga(X5, X8)) U8_GA(X1, X2, X3, X4, X7, mergesortcB_out_ga(X5, X8)) -> U9_GA(X1, X2, X3, X4, mergeC_in_gga(X7, X8, X4)) U8_GA(X1, X2, X3, X4, X7, mergesortcB_out_ga(X5, X8)) -> MERGEC_IN_GGA(X7, X8, X4) MERGEC_IN_GGA(.(X1, X2), .(X1, X3), .(X1, X4)) -> U2_GGA(X1, X2, X3, X4, mergeC_in_gga(.(X1, X2), X3, X4)) MERGEC_IN_GGA(.(X1, X2), .(X1, X3), .(X1, X4)) -> MERGEC_IN_GGA(.(X1, X2), X3, X4) The TRS R consists of the following rules: splitcD_in_ggaa(X1, X2, .(X1, X3), X4) -> U17_ggaa(X1, X2, X3, X4, splitcA_in_gaa(X2, X4, X3)) splitcA_in_gaa([], [], []) -> splitcA_out_gaa([], [], []) splitcA_in_gaa(.(X1, X2), .(X1, X3), X4) -> U11_gaa(X1, X2, X3, X4, splitcA_in_gaa(X2, X4, X3)) U11_gaa(X1, X2, X3, X4, splitcA_out_gaa(X2, X4, X3)) -> splitcA_out_gaa(.(X1, X2), .(X1, X3), X4) U17_ggaa(X1, X2, X3, X4, splitcA_out_gaa(X2, X4, X3)) -> splitcD_out_ggaa(X1, X2, .(X1, X3), X4) mergesortcB_in_ga([], []) -> mergesortcB_out_ga([], []) mergesortcB_in_ga(.(X1, []), .(X1, [])) -> mergesortcB_out_ga(.(X1, []), .(X1, [])) mergesortcB_in_ga(.(X1, .(X2, X3)), X4) -> U12_ga(X1, X2, X3, X4, splitcD_in_ggaa(X2, X3, X5, X6)) U12_ga(X1, X2, X3, X4, splitcD_out_ggaa(X2, X3, X5, X6)) -> U13_ga(X1, X2, X3, X4, X5, X6, mergesortcB_in_ga(.(X1, X6), X7)) U13_ga(X1, X2, X3, X4, X5, X6, mergesortcB_out_ga(.(X1, X6), X7)) -> U14_ga(X1, X2, X3, X4, X5, X6, X7, mergesortcB_in_ga(X5, X8)) U14_ga(X1, X2, X3, X4, X5, X6, X7, mergesortcB_out_ga(X5, X8)) -> U15_ga(X1, X2, X3, X4, mergecC_in_gga(X7, X8, X4)) mergecC_in_gga([], X1, X1) -> mergecC_out_gga([], X1, X1) mergecC_in_gga(X1, [], X1) -> mergecC_out_gga(X1, [], X1) mergecC_in_gga(.(X1, X2), .(X1, X3), .(X1, X4)) -> U16_gga(X1, X2, X3, X4, mergecC_in_gga(.(X1, X2), X3, X4)) U16_gga(X1, X2, X3, X4, mergecC_out_gga(.(X1, X2), X3, X4)) -> mergecC_out_gga(.(X1, X2), .(X1, X3), .(X1, X4)) U15_ga(X1, X2, X3, X4, mergecC_out_gga(X7, X8, X4)) -> mergesortcB_out_ga(.(X1, .(X2, X3)), X4) The argument filtering Pi contains the following mapping: mergesortB_in_ga(x1, x2) = mergesortB_in_ga(x1) .(x1, x2) = .(x1, x2) splitA_in_gaa(x1, x2, x3) = splitA_in_gaa(x1) splitcD_in_ggaa(x1, x2, x3, x4) = splitcD_in_ggaa(x1, x2) U17_ggaa(x1, x2, x3, x4, x5) = U17_ggaa(x1, x2, x5) splitcA_in_gaa(x1, x2, x3) = splitcA_in_gaa(x1) [] = [] splitcA_out_gaa(x1, x2, x3) = splitcA_out_gaa(x1, x2, x3) U11_gaa(x1, x2, x3, x4, x5) = U11_gaa(x1, x2, x5) splitcD_out_ggaa(x1, x2, x3, x4) = splitcD_out_ggaa(x1, x2, x3, x4) mergesortcB_in_ga(x1, x2) = mergesortcB_in_ga(x1) mergesortcB_out_ga(x1, x2) = mergesortcB_out_ga(x1, x2) U12_ga(x1, x2, x3, x4, x5) = U12_ga(x1, x2, x3, x5) U13_ga(x1, x2, x3, x4, x5, x6, x7) = U13_ga(x1, x2, x3, x5, x7) U14_ga(x1, x2, x3, x4, x5, x6, x7, x8) = U14_ga(x1, x2, x3, x7, x8) U15_ga(x1, x2, x3, x4, x5) = U15_ga(x1, x2, x3, x5) mergecC_in_gga(x1, x2, x3) = mergecC_in_gga(x1, x2) mergecC_out_gga(x1, x2, x3) = mergecC_out_gga(x1, x2, x3) U16_gga(x1, x2, x3, x4, x5) = U16_gga(x1, x2, x3, x5) mergeC_in_gga(x1, x2, x3) = mergeC_in_gga(x1, x2) MERGESORTB_IN_GA(x1, x2) = MERGESORTB_IN_GA(x1) U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x2, x3, x5) SPLITA_IN_GAA(x1, x2, x3) = SPLITA_IN_GAA(x1) U1_GAA(x1, x2, x3, x4, x5) = U1_GAA(x1, x2, x5) U4_GA(x1, x2, x3, x4, x5) = U4_GA(x1, x2, x3, x5) U5_GA(x1, x2, x3, x4, x5) = U5_GA(x1, x2, x3, x5) U6_GA(x1, x2, x3, x4, x5, x6) = U6_GA(x1, x2, x3, x5, x6) U7_GA(x1, x2, x3, x4, x5) = U7_GA(x1, x2, x3, x5) U8_GA(x1, x2, x3, x4, x5, x6) = U8_GA(x1, x2, x3, x5, x6) U9_GA(x1, x2, x3, x4, x5) = U9_GA(x1, x2, x3, x5) MERGEC_IN_GGA(x1, x2, x3) = MERGEC_IN_GGA(x1, x2) U2_GGA(x1, x2, x3, x4, x5) = U2_GGA(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: MERGESORTB_IN_GA(.(X1, .(X2, X3)), X4) -> U3_GA(X1, X2, X3, X4, splitA_in_gaa(X3, X5, X6)) MERGESORTB_IN_GA(.(X1, .(X2, X3)), X4) -> SPLITA_IN_GAA(X3, X5, X6) SPLITA_IN_GAA(.(X1, X2), .(X1, X3), X4) -> U1_GAA(X1, X2, X3, X4, splitA_in_gaa(X2, X4, X3)) SPLITA_IN_GAA(.(X1, X2), .(X1, X3), X4) -> SPLITA_IN_GAA(X2, X4, X3) MERGESORTB_IN_GA(.(X1, .(X2, X3)), X4) -> U4_GA(X1, X2, X3, X4, splitcD_in_ggaa(X2, X3, X5, X6)) U4_GA(X1, X2, X3, X4, splitcD_out_ggaa(X2, X3, X5, X6)) -> U5_GA(X1, X2, X3, X4, mergesortB_in_ga(.(X1, X6), X7)) U4_GA(X1, X2, X3, X4, splitcD_out_ggaa(X2, X3, X5, X6)) -> MERGESORTB_IN_GA(.(X1, X6), X7) U4_GA(X1, X2, X3, X4, splitcD_out_ggaa(X2, X3, X5, X6)) -> U6_GA(X1, X2, X3, X4, X5, mergesortcB_in_ga(.(X1, X6), X7)) U6_GA(X1, X2, X3, X4, X5, mergesortcB_out_ga(.(X1, X6), X7)) -> U7_GA(X1, X2, X3, X4, mergesortB_in_ga(X5, X8)) U6_GA(X1, X2, X3, X4, X5, mergesortcB_out_ga(.(X1, X6), X7)) -> MERGESORTB_IN_GA(X5, X8) U6_GA(X1, X2, X3, X4, X5, mergesortcB_out_ga(.(X1, X6), X7)) -> U8_GA(X1, X2, X3, X4, X7, mergesortcB_in_ga(X5, X8)) U8_GA(X1, X2, X3, X4, X7, mergesortcB_out_ga(X5, X8)) -> U9_GA(X1, X2, X3, X4, mergeC_in_gga(X7, X8, X4)) U8_GA(X1, X2, X3, X4, X7, mergesortcB_out_ga(X5, X8)) -> MERGEC_IN_GGA(X7, X8, X4) MERGEC_IN_GGA(.(X1, X2), .(X1, X3), .(X1, X4)) -> U2_GGA(X1, X2, X3, X4, mergeC_in_gga(.(X1, X2), X3, X4)) MERGEC_IN_GGA(.(X1, X2), .(X1, X3), .(X1, X4)) -> MERGEC_IN_GGA(.(X1, X2), X3, X4) The TRS R consists of the following rules: splitcD_in_ggaa(X1, X2, .(X1, X3), X4) -> U17_ggaa(X1, X2, X3, X4, splitcA_in_gaa(X2, X4, X3)) splitcA_in_gaa([], [], []) -> splitcA_out_gaa([], [], []) splitcA_in_gaa(.(X1, X2), .(X1, X3), X4) -> U11_gaa(X1, X2, X3, X4, splitcA_in_gaa(X2, X4, X3)) U11_gaa(X1, X2, X3, X4, splitcA_out_gaa(X2, X4, X3)) -> splitcA_out_gaa(.(X1, X2), .(X1, X3), X4) U17_ggaa(X1, X2, X3, X4, splitcA_out_gaa(X2, X4, X3)) -> splitcD_out_ggaa(X1, X2, .(X1, X3), X4) mergesortcB_in_ga([], []) -> mergesortcB_out_ga([], []) mergesortcB_in_ga(.(X1, []), .(X1, [])) -> mergesortcB_out_ga(.(X1, []), .(X1, [])) mergesortcB_in_ga(.(X1, .(X2, X3)), X4) -> U12_ga(X1, X2, X3, X4, splitcD_in_ggaa(X2, X3, X5, X6)) U12_ga(X1, X2, X3, X4, splitcD_out_ggaa(X2, X3, X5, X6)) -> U13_ga(X1, X2, X3, X4, X5, X6, mergesortcB_in_ga(.(X1, X6), X7)) U13_ga(X1, X2, X3, X4, X5, X6, mergesortcB_out_ga(.(X1, X6), X7)) -> U14_ga(X1, X2, X3, X4, X5, X6, X7, mergesortcB_in_ga(X5, X8)) U14_ga(X1, X2, X3, X4, X5, X6, X7, mergesortcB_out_ga(X5, X8)) -> U15_ga(X1, X2, X3, X4, mergecC_in_gga(X7, X8, X4)) mergecC_in_gga([], X1, X1) -> mergecC_out_gga([], X1, X1) mergecC_in_gga(X1, [], X1) -> mergecC_out_gga(X1, [], X1) mergecC_in_gga(.(X1, X2), .(X1, X3), .(X1, X4)) -> U16_gga(X1, X2, X3, X4, mergecC_in_gga(.(X1, X2), X3, X4)) U16_gga(X1, X2, X3, X4, mergecC_out_gga(.(X1, X2), X3, X4)) -> mergecC_out_gga(.(X1, X2), .(X1, X3), .(X1, X4)) U15_ga(X1, X2, X3, X4, mergecC_out_gga(X7, X8, X4)) -> mergesortcB_out_ga(.(X1, .(X2, X3)), X4) The argument filtering Pi contains the following mapping: mergesortB_in_ga(x1, x2) = mergesortB_in_ga(x1) .(x1, x2) = .(x1, x2) splitA_in_gaa(x1, x2, x3) = splitA_in_gaa(x1) splitcD_in_ggaa(x1, x2, x3, x4) = splitcD_in_ggaa(x1, x2) U17_ggaa(x1, x2, x3, x4, x5) = U17_ggaa(x1, x2, x5) splitcA_in_gaa(x1, x2, x3) = splitcA_in_gaa(x1) [] = [] splitcA_out_gaa(x1, x2, x3) = splitcA_out_gaa(x1, x2, x3) U11_gaa(x1, x2, x3, x4, x5) = U11_gaa(x1, x2, x5) splitcD_out_ggaa(x1, x2, x3, x4) = splitcD_out_ggaa(x1, x2, x3, x4) mergesortcB_in_ga(x1, x2) = mergesortcB_in_ga(x1) mergesortcB_out_ga(x1, x2) = mergesortcB_out_ga(x1, x2) U12_ga(x1, x2, x3, x4, x5) = U12_ga(x1, x2, x3, x5) U13_ga(x1, x2, x3, x4, x5, x6, x7) = U13_ga(x1, x2, x3, x5, x7) U14_ga(x1, x2, x3, x4, x5, x6, x7, x8) = U14_ga(x1, x2, x3, x7, x8) U15_ga(x1, x2, x3, x4, x5) = U15_ga(x1, x2, x3, x5) mergecC_in_gga(x1, x2, x3) = mergecC_in_gga(x1, x2) mergecC_out_gga(x1, x2, x3) = mergecC_out_gga(x1, x2, x3) U16_gga(x1, x2, x3, x4, x5) = U16_gga(x1, x2, x3, x5) mergeC_in_gga(x1, x2, x3) = mergeC_in_gga(x1, x2) MERGESORTB_IN_GA(x1, x2) = MERGESORTB_IN_GA(x1) U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x2, x3, x5) SPLITA_IN_GAA(x1, x2, x3) = SPLITA_IN_GAA(x1) U1_GAA(x1, x2, x3, x4, x5) = U1_GAA(x1, x2, x5) U4_GA(x1, x2, x3, x4, x5) = U4_GA(x1, x2, x3, x5) U5_GA(x1, x2, x3, x4, x5) = U5_GA(x1, x2, x3, x5) U6_GA(x1, x2, x3, x4, x5, x6) = U6_GA(x1, x2, x3, x5, x6) U7_GA(x1, x2, x3, x4, x5) = U7_GA(x1, x2, x3, x5) U8_GA(x1, x2, x3, x4, x5, x6) = U8_GA(x1, x2, x3, x5, x6) U9_GA(x1, x2, x3, x4, x5) = U9_GA(x1, x2, x3, x5) MERGEC_IN_GGA(x1, x2, x3) = MERGEC_IN_GGA(x1, x2) U2_GGA(x1, x2, x3, x4, x5) = U2_GGA(x1, x2, x3, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 9 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: MERGEC_IN_GGA(.(X1, X2), .(X1, X3), .(X1, X4)) -> MERGEC_IN_GGA(.(X1, X2), X3, X4) The TRS R consists of the following rules: splitcD_in_ggaa(X1, X2, .(X1, X3), X4) -> U17_ggaa(X1, X2, X3, X4, splitcA_in_gaa(X2, X4, X3)) splitcA_in_gaa([], [], []) -> splitcA_out_gaa([], [], []) splitcA_in_gaa(.(X1, X2), .(X1, X3), X4) -> U11_gaa(X1, X2, X3, X4, splitcA_in_gaa(X2, X4, X3)) U11_gaa(X1, X2, X3, X4, splitcA_out_gaa(X2, X4, X3)) -> splitcA_out_gaa(.(X1, X2), .(X1, X3), X4) U17_ggaa(X1, X2, X3, X4, splitcA_out_gaa(X2, X4, X3)) -> splitcD_out_ggaa(X1, X2, .(X1, X3), X4) mergesortcB_in_ga([], []) -> mergesortcB_out_ga([], []) mergesortcB_in_ga(.(X1, []), .(X1, [])) -> mergesortcB_out_ga(.(X1, []), .(X1, [])) mergesortcB_in_ga(.(X1, .(X2, X3)), X4) -> U12_ga(X1, X2, X3, X4, splitcD_in_ggaa(X2, X3, X5, X6)) U12_ga(X1, X2, X3, X4, splitcD_out_ggaa(X2, X3, X5, X6)) -> U13_ga(X1, X2, X3, X4, X5, X6, mergesortcB_in_ga(.(X1, X6), X7)) U13_ga(X1, X2, X3, X4, X5, X6, mergesortcB_out_ga(.(X1, X6), X7)) -> U14_ga(X1, X2, X3, X4, X5, X6, X7, mergesortcB_in_ga(X5, X8)) U14_ga(X1, X2, X3, X4, X5, X6, X7, mergesortcB_out_ga(X5, X8)) -> U15_ga(X1, X2, X3, X4, mergecC_in_gga(X7, X8, X4)) mergecC_in_gga([], X1, X1) -> mergecC_out_gga([], X1, X1) mergecC_in_gga(X1, [], X1) -> mergecC_out_gga(X1, [], X1) mergecC_in_gga(.(X1, X2), .(X1, X3), .(X1, X4)) -> U16_gga(X1, X2, X3, X4, mergecC_in_gga(.(X1, X2), X3, X4)) U16_gga(X1, X2, X3, X4, mergecC_out_gga(.(X1, X2), X3, X4)) -> mergecC_out_gga(.(X1, X2), .(X1, X3), .(X1, X4)) U15_ga(X1, X2, X3, X4, mergecC_out_gga(X7, X8, X4)) -> mergesortcB_out_ga(.(X1, .(X2, X3)), X4) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) splitcD_in_ggaa(x1, x2, x3, x4) = splitcD_in_ggaa(x1, x2) U17_ggaa(x1, x2, x3, x4, x5) = U17_ggaa(x1, x2, x5) splitcA_in_gaa(x1, x2, x3) = splitcA_in_gaa(x1) [] = [] splitcA_out_gaa(x1, x2, x3) = splitcA_out_gaa(x1, x2, x3) U11_gaa(x1, x2, x3, x4, x5) = U11_gaa(x1, x2, x5) splitcD_out_ggaa(x1, x2, x3, x4) = splitcD_out_ggaa(x1, x2, x3, x4) mergesortcB_in_ga(x1, x2) = mergesortcB_in_ga(x1) mergesortcB_out_ga(x1, x2) = mergesortcB_out_ga(x1, x2) U12_ga(x1, x2, x3, x4, x5) = U12_ga(x1, x2, x3, x5) U13_ga(x1, x2, x3, x4, x5, x6, x7) = U13_ga(x1, x2, x3, x5, x7) U14_ga(x1, x2, x3, x4, x5, x6, x7, x8) = U14_ga(x1, x2, x3, x7, x8) U15_ga(x1, x2, x3, x4, x5) = U15_ga(x1, x2, x3, x5) mergecC_in_gga(x1, x2, x3) = mergecC_in_gga(x1, x2) mergecC_out_gga(x1, x2, x3) = mergecC_out_gga(x1, x2, x3) U16_gga(x1, x2, x3, x4, x5) = U16_gga(x1, x2, x3, x5) MERGEC_IN_GGA(x1, x2, x3) = MERGEC_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (8) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (9) Obligation: Pi DP problem: The TRS P consists of the following rules: MERGEC_IN_GGA(.(X1, X2), .(X1, X3), .(X1, X4)) -> MERGEC_IN_GGA(.(X1, X2), X3, X4) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) MERGEC_IN_GGA(x1, x2, x3) = MERGEC_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (10) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: MERGEC_IN_GGA(.(X1, X2), .(X1, X3)) -> MERGEC_IN_GGA(.(X1, X2), X3) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (12) 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: *MERGEC_IN_GGA(.(X1, X2), .(X1, X3)) -> MERGEC_IN_GGA(.(X1, X2), X3) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: SPLITA_IN_GAA(.(X1, X2), .(X1, X3), X4) -> SPLITA_IN_GAA(X2, X4, X3) The TRS R consists of the following rules: splitcD_in_ggaa(X1, X2, .(X1, X3), X4) -> U17_ggaa(X1, X2, X3, X4, splitcA_in_gaa(X2, X4, X3)) splitcA_in_gaa([], [], []) -> splitcA_out_gaa([], [], []) splitcA_in_gaa(.(X1, X2), .(X1, X3), X4) -> U11_gaa(X1, X2, X3, X4, splitcA_in_gaa(X2, X4, X3)) U11_gaa(X1, X2, X3, X4, splitcA_out_gaa(X2, X4, X3)) -> splitcA_out_gaa(.(X1, X2), .(X1, X3), X4) U17_ggaa(X1, X2, X3, X4, splitcA_out_gaa(X2, X4, X3)) -> splitcD_out_ggaa(X1, X2, .(X1, X3), X4) mergesortcB_in_ga([], []) -> mergesortcB_out_ga([], []) mergesortcB_in_ga(.(X1, []), .(X1, [])) -> mergesortcB_out_ga(.(X1, []), .(X1, [])) mergesortcB_in_ga(.(X1, .(X2, X3)), X4) -> U12_ga(X1, X2, X3, X4, splitcD_in_ggaa(X2, X3, X5, X6)) U12_ga(X1, X2, X3, X4, splitcD_out_ggaa(X2, X3, X5, X6)) -> U13_ga(X1, X2, X3, X4, X5, X6, mergesortcB_in_ga(.(X1, X6), X7)) U13_ga(X1, X2, X3, X4, X5, X6, mergesortcB_out_ga(.(X1, X6), X7)) -> U14_ga(X1, X2, X3, X4, X5, X6, X7, mergesortcB_in_ga(X5, X8)) U14_ga(X1, X2, X3, X4, X5, X6, X7, mergesortcB_out_ga(X5, X8)) -> U15_ga(X1, X2, X3, X4, mergecC_in_gga(X7, X8, X4)) mergecC_in_gga([], X1, X1) -> mergecC_out_gga([], X1, X1) mergecC_in_gga(X1, [], X1) -> mergecC_out_gga(X1, [], X1) mergecC_in_gga(.(X1, X2), .(X1, X3), .(X1, X4)) -> U16_gga(X1, X2, X3, X4, mergecC_in_gga(.(X1, X2), X3, X4)) U16_gga(X1, X2, X3, X4, mergecC_out_gga(.(X1, X2), X3, X4)) -> mergecC_out_gga(.(X1, X2), .(X1, X3), .(X1, X4)) U15_ga(X1, X2, X3, X4, mergecC_out_gga(X7, X8, X4)) -> mergesortcB_out_ga(.(X1, .(X2, X3)), X4) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) splitcD_in_ggaa(x1, x2, x3, x4) = splitcD_in_ggaa(x1, x2) U17_ggaa(x1, x2, x3, x4, x5) = U17_ggaa(x1, x2, x5) splitcA_in_gaa(x1, x2, x3) = splitcA_in_gaa(x1) [] = [] splitcA_out_gaa(x1, x2, x3) = splitcA_out_gaa(x1, x2, x3) U11_gaa(x1, x2, x3, x4, x5) = U11_gaa(x1, x2, x5) splitcD_out_ggaa(x1, x2, x3, x4) = splitcD_out_ggaa(x1, x2, x3, x4) mergesortcB_in_ga(x1, x2) = mergesortcB_in_ga(x1) mergesortcB_out_ga(x1, x2) = mergesortcB_out_ga(x1, x2) U12_ga(x1, x2, x3, x4, x5) = U12_ga(x1, x2, x3, x5) U13_ga(x1, x2, x3, x4, x5, x6, x7) = U13_ga(x1, x2, x3, x5, x7) U14_ga(x1, x2, x3, x4, x5, x6, x7, x8) = U14_ga(x1, x2, x3, x7, x8) U15_ga(x1, x2, x3, x4, x5) = U15_ga(x1, x2, x3, x5) mergecC_in_gga(x1, x2, x3) = mergecC_in_gga(x1, x2) mergecC_out_gga(x1, x2, x3) = mergecC_out_gga(x1, x2, x3) U16_gga(x1, x2, x3, x4, x5) = U16_gga(x1, x2, x3, x5) SPLITA_IN_GAA(x1, x2, x3) = SPLITA_IN_GAA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (15) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: SPLITA_IN_GAA(.(X1, X2), .(X1, X3), X4) -> SPLITA_IN_GAA(X2, X4, X3) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) SPLITA_IN_GAA(x1, x2, x3) = SPLITA_IN_GAA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: SPLITA_IN_GAA(.(X1, X2)) -> SPLITA_IN_GAA(X2) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (19) 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: *SPLITA_IN_GAA(.(X1, X2)) -> SPLITA_IN_GAA(X2) The graph contains the following edges 1 > 1 ---------------------------------------- (20) YES ---------------------------------------- (21) Obligation: Pi DP problem: The TRS P consists of the following rules: MERGESORTB_IN_GA(.(X1, .(X2, X3)), X4) -> U4_GA(X1, X2, X3, X4, splitcD_in_ggaa(X2, X3, X5, X6)) U4_GA(X1, X2, X3, X4, splitcD_out_ggaa(X2, X3, X5, X6)) -> MERGESORTB_IN_GA(.(X1, X6), X7) U4_GA(X1, X2, X3, X4, splitcD_out_ggaa(X2, X3, X5, X6)) -> U6_GA(X1, X2, X3, X4, X5, mergesortcB_in_ga(.(X1, X6), X7)) U6_GA(X1, X2, X3, X4, X5, mergesortcB_out_ga(.(X1, X6), X7)) -> MERGESORTB_IN_GA(X5, X8) The TRS R consists of the following rules: splitcD_in_ggaa(X1, X2, .(X1, X3), X4) -> U17_ggaa(X1, X2, X3, X4, splitcA_in_gaa(X2, X4, X3)) splitcA_in_gaa([], [], []) -> splitcA_out_gaa([], [], []) splitcA_in_gaa(.(X1, X2), .(X1, X3), X4) -> U11_gaa(X1, X2, X3, X4, splitcA_in_gaa(X2, X4, X3)) U11_gaa(X1, X2, X3, X4, splitcA_out_gaa(X2, X4, X3)) -> splitcA_out_gaa(.(X1, X2), .(X1, X3), X4) U17_ggaa(X1, X2, X3, X4, splitcA_out_gaa(X2, X4, X3)) -> splitcD_out_ggaa(X1, X2, .(X1, X3), X4) mergesortcB_in_ga([], []) -> mergesortcB_out_ga([], []) mergesortcB_in_ga(.(X1, []), .(X1, [])) -> mergesortcB_out_ga(.(X1, []), .(X1, [])) mergesortcB_in_ga(.(X1, .(X2, X3)), X4) -> U12_ga(X1, X2, X3, X4, splitcD_in_ggaa(X2, X3, X5, X6)) U12_ga(X1, X2, X3, X4, splitcD_out_ggaa(X2, X3, X5, X6)) -> U13_ga(X1, X2, X3, X4, X5, X6, mergesortcB_in_ga(.(X1, X6), X7)) U13_ga(X1, X2, X3, X4, X5, X6, mergesortcB_out_ga(.(X1, X6), X7)) -> U14_ga(X1, X2, X3, X4, X5, X6, X7, mergesortcB_in_ga(X5, X8)) U14_ga(X1, X2, X3, X4, X5, X6, X7, mergesortcB_out_ga(X5, X8)) -> U15_ga(X1, X2, X3, X4, mergecC_in_gga(X7, X8, X4)) mergecC_in_gga([], X1, X1) -> mergecC_out_gga([], X1, X1) mergecC_in_gga(X1, [], X1) -> mergecC_out_gga(X1, [], X1) mergecC_in_gga(.(X1, X2), .(X1, X3), .(X1, X4)) -> U16_gga(X1, X2, X3, X4, mergecC_in_gga(.(X1, X2), X3, X4)) U16_gga(X1, X2, X3, X4, mergecC_out_gga(.(X1, X2), X3, X4)) -> mergecC_out_gga(.(X1, X2), .(X1, X3), .(X1, X4)) U15_ga(X1, X2, X3, X4, mergecC_out_gga(X7, X8, X4)) -> mergesortcB_out_ga(.(X1, .(X2, X3)), X4) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) splitcD_in_ggaa(x1, x2, x3, x4) = splitcD_in_ggaa(x1, x2) U17_ggaa(x1, x2, x3, x4, x5) = U17_ggaa(x1, x2, x5) splitcA_in_gaa(x1, x2, x3) = splitcA_in_gaa(x1) [] = [] splitcA_out_gaa(x1, x2, x3) = splitcA_out_gaa(x1, x2, x3) U11_gaa(x1, x2, x3, x4, x5) = U11_gaa(x1, x2, x5) splitcD_out_ggaa(x1, x2, x3, x4) = splitcD_out_ggaa(x1, x2, x3, x4) mergesortcB_in_ga(x1, x2) = mergesortcB_in_ga(x1) mergesortcB_out_ga(x1, x2) = mergesortcB_out_ga(x1, x2) U12_ga(x1, x2, x3, x4, x5) = U12_ga(x1, x2, x3, x5) U13_ga(x1, x2, x3, x4, x5, x6, x7) = U13_ga(x1, x2, x3, x5, x7) U14_ga(x1, x2, x3, x4, x5, x6, x7, x8) = U14_ga(x1, x2, x3, x7, x8) U15_ga(x1, x2, x3, x4, x5) = U15_ga(x1, x2, x3, x5) mergecC_in_gga(x1, x2, x3) = mergecC_in_gga(x1, x2) mergecC_out_gga(x1, x2, x3) = mergecC_out_gga(x1, x2, x3) U16_gga(x1, x2, x3, x4, x5) = U16_gga(x1, x2, x3, x5) MERGESORTB_IN_GA(x1, x2) = MERGESORTB_IN_GA(x1) U4_GA(x1, x2, x3, x4, x5) = U4_GA(x1, x2, x3, x5) U6_GA(x1, x2, x3, x4, x5, x6) = U6_GA(x1, x2, x3, x5, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (22) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (23) Obligation: Q DP problem: The TRS P consists of the following rules: MERGESORTB_IN_GA(.(X1, .(X2, X3))) -> U4_GA(X1, X2, X3, splitcD_in_ggaa(X2, X3)) U4_GA(X1, X2, X3, splitcD_out_ggaa(X2, X3, X5, X6)) -> MERGESORTB_IN_GA(.(X1, X6)) U4_GA(X1, X2, X3, splitcD_out_ggaa(X2, X3, X5, X6)) -> U6_GA(X1, X2, X3, X5, mergesortcB_in_ga(.(X1, X6))) U6_GA(X1, X2, X3, X5, mergesortcB_out_ga(.(X1, X6), X7)) -> MERGESORTB_IN_GA(X5) The TRS R consists of the following rules: splitcD_in_ggaa(X1, X2) -> U17_ggaa(X1, X2, splitcA_in_gaa(X2)) splitcA_in_gaa([]) -> splitcA_out_gaa([], [], []) splitcA_in_gaa(.(X1, X2)) -> U11_gaa(X1, X2, splitcA_in_gaa(X2)) U11_gaa(X1, X2, splitcA_out_gaa(X2, X4, X3)) -> splitcA_out_gaa(.(X1, X2), .(X1, X3), X4) U17_ggaa(X1, X2, splitcA_out_gaa(X2, X4, X3)) -> splitcD_out_ggaa(X1, X2, .(X1, X3), X4) mergesortcB_in_ga([]) -> mergesortcB_out_ga([], []) mergesortcB_in_ga(.(X1, [])) -> mergesortcB_out_ga(.(X1, []), .(X1, [])) mergesortcB_in_ga(.(X1, .(X2, X3))) -> U12_ga(X1, X2, X3, splitcD_in_ggaa(X2, X3)) U12_ga(X1, X2, X3, splitcD_out_ggaa(X2, X3, X5, X6)) -> U13_ga(X1, X2, X3, X5, mergesortcB_in_ga(.(X1, X6))) U13_ga(X1, X2, X3, X5, mergesortcB_out_ga(.(X1, X6), X7)) -> U14_ga(X1, X2, X3, X7, mergesortcB_in_ga(X5)) U14_ga(X1, X2, X3, X7, mergesortcB_out_ga(X5, X8)) -> U15_ga(X1, X2, X3, mergecC_in_gga(X7, X8)) mergecC_in_gga([], X1) -> mergecC_out_gga([], X1, X1) mergecC_in_gga(X1, []) -> mergecC_out_gga(X1, [], X1) mergecC_in_gga(.(X1, X2), .(X1, X3)) -> U16_gga(X1, X2, X3, mergecC_in_gga(.(X1, X2), X3)) U16_gga(X1, X2, X3, mergecC_out_gga(.(X1, X2), X3, X4)) -> mergecC_out_gga(.(X1, X2), .(X1, X3), .(X1, X4)) U15_ga(X1, X2, X3, mergecC_out_gga(X7, X8, X4)) -> mergesortcB_out_ga(.(X1, .(X2, X3)), X4) The set Q consists of the following terms: splitcD_in_ggaa(x0, x1) splitcA_in_gaa(x0) U11_gaa(x0, x1, x2) U17_ggaa(x0, x1, x2) mergesortcB_in_ga(x0) U12_ga(x0, x1, x2, x3) U13_ga(x0, x1, x2, x3, x4) U14_ga(x0, x1, x2, x3, x4) mergecC_in_gga(x0, x1) U16_gga(x0, x1, x2, x3) U15_ga(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (24) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule MERGESORTB_IN_GA(.(X1, .(X2, X3))) -> U4_GA(X1, X2, X3, splitcD_in_ggaa(X2, X3)) at position [3] we obtained the following new rules [LPAR04]: (MERGESORTB_IN_GA(.(X1, .(X2, X3))) -> U4_GA(X1, X2, X3, U17_ggaa(X2, X3, splitcA_in_gaa(X3))),MERGESORTB_IN_GA(.(X1, .(X2, X3))) -> U4_GA(X1, X2, X3, U17_ggaa(X2, X3, splitcA_in_gaa(X3)))) ---------------------------------------- (25) Obligation: Q DP problem: The TRS P consists of the following rules: U4_GA(X1, X2, X3, splitcD_out_ggaa(X2, X3, X5, X6)) -> MERGESORTB_IN_GA(.(X1, X6)) U4_GA(X1, X2, X3, splitcD_out_ggaa(X2, X3, X5, X6)) -> U6_GA(X1, X2, X3, X5, mergesortcB_in_ga(.(X1, X6))) U6_GA(X1, X2, X3, X5, mergesortcB_out_ga(.(X1, X6), X7)) -> MERGESORTB_IN_GA(X5) MERGESORTB_IN_GA(.(X1, .(X2, X3))) -> U4_GA(X1, X2, X3, U17_ggaa(X2, X3, splitcA_in_gaa(X3))) The TRS R consists of the following rules: splitcD_in_ggaa(X1, X2) -> U17_ggaa(X1, X2, splitcA_in_gaa(X2)) splitcA_in_gaa([]) -> splitcA_out_gaa([], [], []) splitcA_in_gaa(.(X1, X2)) -> U11_gaa(X1, X2, splitcA_in_gaa(X2)) U11_gaa(X1, X2, splitcA_out_gaa(X2, X4, X3)) -> splitcA_out_gaa(.(X1, X2), .(X1, X3), X4) U17_ggaa(X1, X2, splitcA_out_gaa(X2, X4, X3)) -> splitcD_out_ggaa(X1, X2, .(X1, X3), X4) mergesortcB_in_ga([]) -> mergesortcB_out_ga([], []) mergesortcB_in_ga(.(X1, [])) -> mergesortcB_out_ga(.(X1, []), .(X1, [])) mergesortcB_in_ga(.(X1, .(X2, X3))) -> U12_ga(X1, X2, X3, splitcD_in_ggaa(X2, X3)) U12_ga(X1, X2, X3, splitcD_out_ggaa(X2, X3, X5, X6)) -> U13_ga(X1, X2, X3, X5, mergesortcB_in_ga(.(X1, X6))) U13_ga(X1, X2, X3, X5, mergesortcB_out_ga(.(X1, X6), X7)) -> U14_ga(X1, X2, X3, X7, mergesortcB_in_ga(X5)) U14_ga(X1, X2, X3, X7, mergesortcB_out_ga(X5, X8)) -> U15_ga(X1, X2, X3, mergecC_in_gga(X7, X8)) mergecC_in_gga([], X1) -> mergecC_out_gga([], X1, X1) mergecC_in_gga(X1, []) -> mergecC_out_gga(X1, [], X1) mergecC_in_gga(.(X1, X2), .(X1, X3)) -> U16_gga(X1, X2, X3, mergecC_in_gga(.(X1, X2), X3)) U16_gga(X1, X2, X3, mergecC_out_gga(.(X1, X2), X3, X4)) -> mergecC_out_gga(.(X1, X2), .(X1, X3), .(X1, X4)) U15_ga(X1, X2, X3, mergecC_out_gga(X7, X8, X4)) -> mergesortcB_out_ga(.(X1, .(X2, X3)), X4) The set Q consists of the following terms: splitcD_in_ggaa(x0, x1) splitcA_in_gaa(x0) U11_gaa(x0, x1, x2) U17_ggaa(x0, x1, x2) mergesortcB_in_ga(x0) U12_ga(x0, x1, x2, x3) U13_ga(x0, x1, x2, x3, x4) U14_ga(x0, x1, x2, x3, x4) mergecC_in_gga(x0, x1) U16_gga(x0, x1, x2, x3) U15_ga(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (26) QDPQMonotonicMRRProof (EQUIVALENT) By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain. Strictly oriented dependency pairs: U6_GA(X1, X2, X3, X5, mergesortcB_out_ga(.(X1, X6), X7)) -> MERGESORTB_IN_GA(X5) Used ordering: Polynomial interpretation [POLO]: POL(.(x_1, x_2)) = 2 + 2*x_2 POL(MERGESORTB_IN_GA(x_1)) = x_1 POL(U11_gaa(x_1, x_2, x_3)) = 1 + 2*x_3 POL(U12_ga(x_1, x_2, x_3, x_4)) = 0 POL(U13_ga(x_1, x_2, x_3, x_4, x_5)) = 0 POL(U14_ga(x_1, x_2, x_3, x_4, x_5)) = 0 POL(U15_ga(x_1, x_2, x_3, x_4)) = 0 POL(U16_gga(x_1, x_2, x_3, x_4)) = 0 POL(U17_ggaa(x_1, x_2, x_3)) = 2*x_3 POL(U4_GA(x_1, x_2, x_3, x_4)) = 2 + 2*x_4 POL(U6_GA(x_1, x_2, x_3, x_4, x_5)) = 2 + 2*x_4 POL([]) = 0 POL(mergecC_in_gga(x_1, x_2)) = 0 POL(mergecC_out_gga(x_1, x_2, x_3)) = 0 POL(mergesortcB_in_ga(x_1)) = 0 POL(mergesortcB_out_ga(x_1, x_2)) = 0 POL(splitcA_in_gaa(x_1)) = 1 + x_1 POL(splitcA_out_gaa(x_1, x_2, x_3)) = 1 + x_2 + 2*x_3 POL(splitcD_in_ggaa(x_1, x_2)) = 2 + 2*x_2 POL(splitcD_out_ggaa(x_1, x_2, x_3, x_4)) = x_3 + x_4 ---------------------------------------- (27) Obligation: Q DP problem: The TRS P consists of the following rules: U4_GA(X1, X2, X3, splitcD_out_ggaa(X2, X3, X5, X6)) -> MERGESORTB_IN_GA(.(X1, X6)) U4_GA(X1, X2, X3, splitcD_out_ggaa(X2, X3, X5, X6)) -> U6_GA(X1, X2, X3, X5, mergesortcB_in_ga(.(X1, X6))) MERGESORTB_IN_GA(.(X1, .(X2, X3))) -> U4_GA(X1, X2, X3, U17_ggaa(X2, X3, splitcA_in_gaa(X3))) The TRS R consists of the following rules: splitcD_in_ggaa(X1, X2) -> U17_ggaa(X1, X2, splitcA_in_gaa(X2)) splitcA_in_gaa([]) -> splitcA_out_gaa([], [], []) splitcA_in_gaa(.(X1, X2)) -> U11_gaa(X1, X2, splitcA_in_gaa(X2)) U11_gaa(X1, X2, splitcA_out_gaa(X2, X4, X3)) -> splitcA_out_gaa(.(X1, X2), .(X1, X3), X4) U17_ggaa(X1, X2, splitcA_out_gaa(X2, X4, X3)) -> splitcD_out_ggaa(X1, X2, .(X1, X3), X4) mergesortcB_in_ga([]) -> mergesortcB_out_ga([], []) mergesortcB_in_ga(.(X1, [])) -> mergesortcB_out_ga(.(X1, []), .(X1, [])) mergesortcB_in_ga(.(X1, .(X2, X3))) -> U12_ga(X1, X2, X3, splitcD_in_ggaa(X2, X3)) U12_ga(X1, X2, X3, splitcD_out_ggaa(X2, X3, X5, X6)) -> U13_ga(X1, X2, X3, X5, mergesortcB_in_ga(.(X1, X6))) U13_ga(X1, X2, X3, X5, mergesortcB_out_ga(.(X1, X6), X7)) -> U14_ga(X1, X2, X3, X7, mergesortcB_in_ga(X5)) U14_ga(X1, X2, X3, X7, mergesortcB_out_ga(X5, X8)) -> U15_ga(X1, X2, X3, mergecC_in_gga(X7, X8)) mergecC_in_gga([], X1) -> mergecC_out_gga([], X1, X1) mergecC_in_gga(X1, []) -> mergecC_out_gga(X1, [], X1) mergecC_in_gga(.(X1, X2), .(X1, X3)) -> U16_gga(X1, X2, X3, mergecC_in_gga(.(X1, X2), X3)) U16_gga(X1, X2, X3, mergecC_out_gga(.(X1, X2), X3, X4)) -> mergecC_out_gga(.(X1, X2), .(X1, X3), .(X1, X4)) U15_ga(X1, X2, X3, mergecC_out_gga(X7, X8, X4)) -> mergesortcB_out_ga(.(X1, .(X2, X3)), X4) The set Q consists of the following terms: splitcD_in_ggaa(x0, x1) splitcA_in_gaa(x0) U11_gaa(x0, x1, x2) U17_ggaa(x0, x1, x2) mergesortcB_in_ga(x0) U12_ga(x0, x1, x2, x3) U13_ga(x0, x1, x2, x3, x4) U14_ga(x0, x1, x2, x3, x4) mergecC_in_gga(x0, x1) U16_gga(x0, x1, x2, x3) U15_ga(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (28) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (29) Obligation: Q DP problem: The TRS P consists of the following rules: MERGESORTB_IN_GA(.(X1, .(X2, X3))) -> U4_GA(X1, X2, X3, U17_ggaa(X2, X3, splitcA_in_gaa(X3))) U4_GA(X1, X2, X3, splitcD_out_ggaa(X2, X3, X5, X6)) -> MERGESORTB_IN_GA(.(X1, X6)) The TRS R consists of the following rules: splitcD_in_ggaa(X1, X2) -> U17_ggaa(X1, X2, splitcA_in_gaa(X2)) splitcA_in_gaa([]) -> splitcA_out_gaa([], [], []) splitcA_in_gaa(.(X1, X2)) -> U11_gaa(X1, X2, splitcA_in_gaa(X2)) U11_gaa(X1, X2, splitcA_out_gaa(X2, X4, X3)) -> splitcA_out_gaa(.(X1, X2), .(X1, X3), X4) U17_ggaa(X1, X2, splitcA_out_gaa(X2, X4, X3)) -> splitcD_out_ggaa(X1, X2, .(X1, X3), X4) mergesortcB_in_ga([]) -> mergesortcB_out_ga([], []) mergesortcB_in_ga(.(X1, [])) -> mergesortcB_out_ga(.(X1, []), .(X1, [])) mergesortcB_in_ga(.(X1, .(X2, X3))) -> U12_ga(X1, X2, X3, splitcD_in_ggaa(X2, X3)) U12_ga(X1, X2, X3, splitcD_out_ggaa(X2, X3, X5, X6)) -> U13_ga(X1, X2, X3, X5, mergesortcB_in_ga(.(X1, X6))) U13_ga(X1, X2, X3, X5, mergesortcB_out_ga(.(X1, X6), X7)) -> U14_ga(X1, X2, X3, X7, mergesortcB_in_ga(X5)) U14_ga(X1, X2, X3, X7, mergesortcB_out_ga(X5, X8)) -> U15_ga(X1, X2, X3, mergecC_in_gga(X7, X8)) mergecC_in_gga([], X1) -> mergecC_out_gga([], X1, X1) mergecC_in_gga(X1, []) -> mergecC_out_gga(X1, [], X1) mergecC_in_gga(.(X1, X2), .(X1, X3)) -> U16_gga(X1, X2, X3, mergecC_in_gga(.(X1, X2), X3)) U16_gga(X1, X2, X3, mergecC_out_gga(.(X1, X2), X3, X4)) -> mergecC_out_gga(.(X1, X2), .(X1, X3), .(X1, X4)) U15_ga(X1, X2, X3, mergecC_out_gga(X7, X8, X4)) -> mergesortcB_out_ga(.(X1, .(X2, X3)), X4) The set Q consists of the following terms: splitcD_in_ggaa(x0, x1) splitcA_in_gaa(x0) U11_gaa(x0, x1, x2) U17_ggaa(x0, x1, x2) mergesortcB_in_ga(x0) U12_ga(x0, x1, x2, x3) U13_ga(x0, x1, x2, x3, x4) U14_ga(x0, x1, x2, x3, x4) mergecC_in_gga(x0, x1) U16_gga(x0, x1, x2, x3) U15_ga(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (30) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (31) Obligation: Q DP problem: The TRS P consists of the following rules: MERGESORTB_IN_GA(.(X1, .(X2, X3))) -> U4_GA(X1, X2, X3, U17_ggaa(X2, X3, splitcA_in_gaa(X3))) U4_GA(X1, X2, X3, splitcD_out_ggaa(X2, X3, X5, X6)) -> MERGESORTB_IN_GA(.(X1, X6)) The TRS R consists of the following rules: splitcA_in_gaa([]) -> splitcA_out_gaa([], [], []) splitcA_in_gaa(.(X1, X2)) -> U11_gaa(X1, X2, splitcA_in_gaa(X2)) U17_ggaa(X1, X2, splitcA_out_gaa(X2, X4, X3)) -> splitcD_out_ggaa(X1, X2, .(X1, X3), X4) U11_gaa(X1, X2, splitcA_out_gaa(X2, X4, X3)) -> splitcA_out_gaa(.(X1, X2), .(X1, X3), X4) The set Q consists of the following terms: splitcD_in_ggaa(x0, x1) splitcA_in_gaa(x0) U11_gaa(x0, x1, x2) U17_ggaa(x0, x1, x2) mergesortcB_in_ga(x0) U12_ga(x0, x1, x2, x3) U13_ga(x0, x1, x2, x3, x4) U14_ga(x0, x1, x2, x3, x4) mergecC_in_gga(x0, x1) U16_gga(x0, x1, x2, x3) U15_ga(x0, x1, x2, x3) We have to consider all (P,Q,R)-chains. ---------------------------------------- (32) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. splitcD_in_ggaa(x0, x1) mergesortcB_in_ga(x0) U12_ga(x0, x1, x2, x3) U13_ga(x0, x1, x2, x3, x4) U14_ga(x0, x1, x2, x3, x4) mergecC_in_gga(x0, x1) U16_gga(x0, x1, x2, x3) U15_ga(x0, x1, x2, x3) ---------------------------------------- (33) Obligation: Q DP problem: The TRS P consists of the following rules: MERGESORTB_IN_GA(.(X1, .(X2, X3))) -> U4_GA(X1, X2, X3, U17_ggaa(X2, X3, splitcA_in_gaa(X3))) U4_GA(X1, X2, X3, splitcD_out_ggaa(X2, X3, X5, X6)) -> MERGESORTB_IN_GA(.(X1, X6)) The TRS R consists of the following rules: splitcA_in_gaa([]) -> splitcA_out_gaa([], [], []) splitcA_in_gaa(.(X1, X2)) -> U11_gaa(X1, X2, splitcA_in_gaa(X2)) U17_ggaa(X1, X2, splitcA_out_gaa(X2, X4, X3)) -> splitcD_out_ggaa(X1, X2, .(X1, X3), X4) U11_gaa(X1, X2, splitcA_out_gaa(X2, X4, X3)) -> splitcA_out_gaa(.(X1, X2), .(X1, X3), X4) The set Q consists of the following terms: splitcA_in_gaa(x0) U11_gaa(x0, x1, x2) U17_ggaa(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (34) QDPQMonotonicMRRProof (EQUIVALENT) By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain. Strictly oriented dependency pairs: MERGESORTB_IN_GA(.(X1, .(X2, X3))) -> U4_GA(X1, X2, X3, U17_ggaa(X2, X3, splitcA_in_gaa(X3))) Strictly oriented rules of the TRS R: splitcA_in_gaa([]) -> splitcA_out_gaa([], [], []) Used ordering: Polynomial interpretation [POLO]: POL(.(x_1, x_2)) = 1 + x_2 POL(MERGESORTB_IN_GA(x_1)) = 1 + 2*x_1 POL(U11_gaa(x_1, x_2, x_3)) = 2 + x_3 POL(U17_ggaa(x_1, x_2, x_3)) = 1 + x_3 POL(U4_GA(x_1, x_2, x_3, x_4)) = 1 + x_4 POL([]) = 0 POL(splitcA_in_gaa(x_1)) = 2 + 2*x_1 POL(splitcA_out_gaa(x_1, x_2, x_3)) = 1 + 2*x_2 + 2*x_3 POL(splitcD_out_ggaa(x_1, x_2, x_3, x_4)) = 2 + 2*x_4 ---------------------------------------- (35) Obligation: Q DP problem: The TRS P consists of the following rules: U4_GA(X1, X2, X3, splitcD_out_ggaa(X2, X3, X5, X6)) -> MERGESORTB_IN_GA(.(X1, X6)) The TRS R consists of the following rules: splitcA_in_gaa(.(X1, X2)) -> U11_gaa(X1, X2, splitcA_in_gaa(X2)) U17_ggaa(X1, X2, splitcA_out_gaa(X2, X4, X3)) -> splitcD_out_ggaa(X1, X2, .(X1, X3), X4) U11_gaa(X1, X2, splitcA_out_gaa(X2, X4, X3)) -> splitcA_out_gaa(.(X1, X2), .(X1, X3), X4) The set Q consists of the following terms: splitcA_in_gaa(x0) U11_gaa(x0, x1, x2) U17_ggaa(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (36) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (37) TRUE